new organization of the Base and LambdaDelta modules

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base/blt/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base/blt/defs.ma
new file mode 100644 (file)
index 0000000..3c64c05
--- /dev/null
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base/blt/defs.ma
@@ -0,0 +1,27 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/Base/blt/defs".
+
+include "ext/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.
+

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base/blt/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base/blt/props.ma
new file mode 100644 (file)
index 0000000..34b6059
--- /dev/null
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base/blt/props.ma
@@ -0,0 +1,102 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/Base/blt/props".
+
+include "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_le_S (S n0) 
+(S n) (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).
+

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base/ext/arith.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base/ext/arith.ma
new file mode 100644 (file)
index 0000000..766a34d
--- /dev/null
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base/ext/arith.ma
@@ -0,0 +1,576 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/Base/ext/arith".
+
+include "ext/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 (n0: nat).(or (eq nat (S n) n0) ((eq nat (S n) n0) \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 n) \Rightarrow n])) (S n) 
+(S n0) H2) in (let H4 \def (eq_ind_r nat n0 (\lambda (n0: nat).((eq nat n n0) 
+\to (\forall (P: Prop).P))) H1 n H3) in (let H5 \def (eq_ind_r nat n0 
+(\lambda (n0: nat).(or (eq nat (S n) n0) ((eq nat (S n) n0) \to (\forall (P: 
+Prop).P)))) 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))).(plus_reg_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_comm n 
+p)) (plus m n) H) (plus n m) (plus_comm n m)))))).
+
+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_comm 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_reverse 
+x z y)) (plus y z) (plus_comm y z)) (plus (plus x y) z) (plus_assoc_reverse 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 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 (n: nat).(\lambda (_: (le ? n)).((eq nat n 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 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 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 (n: nat).(\lambda (_: 
+(le ? n)).((eq nat n 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 m H2) \Rightarrow (\lambda (H3: 
+(eq nat (S m) O)).((let H4 \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 H3) in (False_ind ((le (S n) m) \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 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).(match y in nat return 
+(\lambda (n0: nat).(le (minus (S n) n0) (S n))) with [O \Rightarrow (le_n (S 
+n)) | (S n0) \Rightarrow (le_S (minus n n0) n (H n0))])))) 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_comm (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)).(plus_le_reg_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 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_comm 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))).(plus_lt_reg_l n m p (let H0 \def (eq_ind nat (plus n p) 
+(\lambda (n: nat).(lt n (plus m p))) H (plus p n) (plus_comm 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_comm 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_comm (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_comm 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_irrefl 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) (lt_le_S (plus (plus d2 h2) h1) (S (plus n h1)) 
+(le_lt_n_Sm (plus (plus d2 h2) h1) (plus n h1) (plus_le_compat (plus d2 h2) n 
+h1 h1 H (le_n h1))))) (plus h2 d2) (plus_comm h2 d2)) (plus h2 (plus d2 h1)) 
+(plus_assoc 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 (_: (le (S z0) (S 
+x0))).(\lambda (H0: (le (S z0) O)).(\lambda (_: (eq nat (minus (S x0) (S z0)) 
+(minus O (S z0)))).(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 (y1: nat).(\forall (y2: nat).((le O z0) \to ((le O z0) \to ((eq 
+nat (plus n y1) (plus n y2)) \to (eq nat y2 y1))))))) H_y z0 (minus_n_O z0)) 
+in (H2 y1 y2 (le_O_n z0) (le_O_n z0) (H2 (plus z0 y2) (plus z0 y1) (le_O_n 
+z0) (le_O_n z0) (f_equal nat nat (plus z0) (plus z0 y2) (plus z0 y1) 
+(sym_equal nat (plus z0 y1) (plus z0 y2) (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 (y2: nat).((le O z0) \to ((le x3 z0) \to ((eq nat (plus n (S 
+y1)) (plus (minus z0 x3) y2)) \to (eq nat y2 (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 (y2: nat).((le O z0) \to ((le x3 z0) \to ((eq nat n (plus 
+(minus z0 x3) y2)) \to (eq nat y2 (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 (y2: nat).((le O z0) \to ((le x3 z0) \to ((eq nat (S (plus 
+z0 y1)) (plus (minus z0 x3) y2)) \to (eq nat y2 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).
+

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base/ext/preamble.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base/ext/preamble.ma
new file mode 100644 (file)
index 0000000..05abd97
--- /dev/null
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base/ext/preamble.ma
@@ -0,0 +1,95 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/Base/ext/preamble".
+
+(* FG: We should include legacy/coq.ma bit it is not working *)
+(* include "legacy/coq.ma".                                  *)
+
+default "equality"
+ cic:/Coq/Init/Logic/eq.ind
+ cic:/Coq/Init/Logic/sym_eq.con
+ cic:/Coq/Init/Logic/trans_eq.con
+ cic:/Coq/Init/Logic/eq_ind.con
+ cic:/Coq/Init/Logic/eq_ind_r.con
+ cic:/Coq/Init/Logic/f_equal.con
+ cic:/Coq/Init/Logic/f_equal1.con.
+       
+default "true"
+ cic:/Coq/Init/Logic/True.ind.
+
+default "false"
+ cic:/Coq/Init/Logic/False.ind.
+
+default "absurd"
+ cic:/Coq/Init/Logic/absurd.con.
+
+interpretation "Coq's leibnitz's equality" 'eq x y = (cic:/Coq/Init/Logic/eq.ind#xpointer(1/1) _ x y).
+interpretation "Coq's not equal to (leibnitz)" 'neq x y = (cic:/Coq/Init/Logic/not.con (cic:/Coq/Init/Logic/eq.ind#xpointer(1/1) _ x y)).
+interpretation "Coq's natural plus" 'plus x y = (cic:/Coq/Init/Peano/plus.con x y).
+interpretation "Coq's natural 'less or equal to'" 'leq x y = (cic:/Coq/Init/Peano/le.ind#xpointer(1/1) x y).
+
+alias id "land" = "cic:/Coq/Init/Logic/and.ind#xpointer(1/1)".
+         
+(* FG/CSC: These aliases should disappear: we would like to write something
+ *         like: "disambiguate in cic:/Coq/*"
+ *)
+alias id "True" = "cic:/Coq/Init/Logic/True.ind#xpointer(1/1)".
+alias id "or" = "cic:/Coq/Init/Logic/or.ind#xpointer(1/1)".
+alias id "nat" = "cic:/Coq/Init/Datatypes/nat.ind#xpointer(1/1)".
+alias id "eq" = "cic:/Coq/Init/Logic/eq.ind#xpointer(1/1)".
+alias id "plus" = "cic:/Coq/Init/Peano/plus.con".
+alias id "le_trans" = "cic:/Coq/Arith/Le/le_trans.con".
+alias id "le_plus_r" = "cic:/Coq/Arith/Plus/le_plus_r.con".
+alias id "le" = "cic:/Coq/Init/Peano/le.ind#xpointer(1/1)".
+alias id "ex" = "cic:/Coq/Init/Logic/ex.ind#xpointer(1/1)".
+alias id "ex2" = "cic:/Coq/Init/Logic/ex2.ind#xpointer(1/1)".
+alias id "true" = "cic:/Coq/Init/Datatypes/bool.ind#xpointer(1/1/1)".
+alias id "false" = "cic:/Coq/Init/Datatypes/bool.ind#xpointer(1/1/2)".
+alias id "bool" = "cic:/Coq/Init/Datatypes/bool.ind#xpointer(1/1)".
+alias id "O" = "cic:/Coq/Init/Datatypes/nat.ind#xpointer(1/1/1)".
+alias id "S" = "cic:/Coq/Init/Datatypes/nat.ind#xpointer(1/1/2)".
+alias id "eq_ind" = "cic:/Coq/Init/Logic/eq_ind.con".
+alias id "or_introl" = "cic:/Coq/Init/Logic/or.ind#xpointer(1/1/1)".
+alias id "or_intror" = "cic:/Coq/Init/Logic/or.ind#xpointer(1/1/2)".
+alias id "False_ind" = "cic:/Coq/Init/Logic/False_ind.con".
+alias id "False" = "cic:/Coq/Init/Logic/False.ind#xpointer(1/1)".
+alias id "I" = "cic:/Coq/Init/Logic/True.ind#xpointer(1/1/1)".
+alias id "minus" = "cic:/Coq/Init/Peano/minus.con".
+alias id "le_n" = "cic:/Coq/Init/Peano/le.ind#xpointer(1/1/1)".
+alias id "le_antisym" = "cic:/Coq/Arith/Le/le_antisym.con".
+alias id "eq_ind_r" = "cic:/Coq/Init/Logic/eq_ind_r.con".
+
+theorem f_equal: \forall A,B:Type. \forall f:A \to B.
+                 \forall x,y:A. x = y \to f x = f y.
+ intros. elim H. reflexivity.
+qed.
+
+theorem sym_eq: \forall A:Type. \forall x,y:A. x = y \to y = x.
+ intros. rewrite > H. reflexivity.
+qed.
+
+theorem sym_not_eq: \forall A:Type. \forall x,y:A. x \neq y \to y \neq x.
+ unfold not. intros. apply H. symmetry. assumption.
+qed.
+
+theorem plus_reg_l: \forall (n,m,p:nat). n + m = n + p \to m = p.
+ intros. apply plus_reg_l; auto.
+qed.
+
+theorem plus_le_reg_l: \forall p,n,m. p + n <= p + m \to n <= m.
+ intros. apply plus_le_reg_l; auto.
+qed.
+
+definition sym_equal \def sym_eq.

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base/ext/tactics.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base/ext/tactics.ma
new file mode 100644 (file)
index 0000000..57901bf
--- /dev/null
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base/ext/tactics.ma
@@ -0,0 +1,42 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/Base/ext/tactics".
+
+include "ext/preamble.ma".
+
+theorem insert_eq:
+ \forall (S: Set).(\forall (x: S).(\forall (P: ((S \to Prop))).(\forall (G: 
+Prop).(((\forall (y: S).((P y) \to ((eq S y x) \to G)))) \to ((P x) \to G)))))
+\def
+ \lambda (S: Set).(\lambda (x: S).(\lambda (P: ((S \to Prop))).(\lambda (G: 
+Prop).(\lambda (H: ((\forall (y: S).((P y) \to ((eq S y x) \to 
+G))))).(\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))))).
+

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base/makefile b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base/makefile
new file mode 100644 (file)
index 0000000..a9ac218
--- /dev/null
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base/makefile
@@ -0,0 +1,33 @@
+H=@
+
+RT_BASEDIR=/home/fguidi/svn/software/matita/
+OPTIONS=-bench
+MMAKE=$(RT_BASEDIR)matitamake $(OPTIONS)
+CLEAN=$(RT_BASEDIR)matitaclean $(OPTIONS) 
+MMAKEO=$(RT_BASEDIR)matitamake.opt $(OPTIONS)
+CLEANO=$(RT_BASEDIR)matitaclean.opt $(OPTIONS) 
+
+devel:=$(shell basename `pwd`)
+
+all: preall
+       $(H)MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKE) build $(devel)
+clean: preall
+       $(H)MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKE) clean $(devel)
+cleanall: preall
+       $(H)MATITA_FLAGS=$(MATITA_FLAGS) $(MCLEAN) all
+
+all.opt opt: preall
+       $(H)MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKEO) build $(devel)
+clean.opt: preall
+       $(H)MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKEO) clean $(devel)
+cleanall.opt: preall
+       $(H)MATITA_FLAGS=$(MATITA_FLAGS) $(MCLEANO) all
+
+%.mo: preall
+       $(H)MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKE) $@
+%.mo.opt: preall
+       $(H)MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKEO) $@
+       
+preall:
+       $(H)MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKE) init $(devel)
+

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base/theory.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base/theory.ma
new file mode 100644 (file)
index 0000000..31ae9dd
--- /dev/null
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base/theory.ma
@@ -0,0 +1,30 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/Base/theory".
+
+include "ext/tactics.ma".
+
+include "ext/arith.ma".
+
+include "types/defs.ma".
+
+include "types/props.ma".
+
+include "blt/defs.ma".
+
+include "blt/props.ma".
+

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base/types/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base/types/defs.ma
new file mode 100644 (file)
index 0000000..888d926
--- /dev/null
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base/types/defs.ma
@@ -0,0 +1,148 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/Base/types/defs".
+
+include "ext/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 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 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 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))))))))))))).
+

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base/types/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base/types/props.ma
new file mode 100644 (file)
index 0000000..c40648b
--- /dev/null
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base/types/props.ma
@@ -0,0 +1,32 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/Base/types/props".
+
+include "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)))).
+

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta.ma
deleted file mode 100644 (file)
index 2c90a29..0000000
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta.ma
+++ /dev/null
@@ -1,52217 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta".
-
-include "LambdaDelta/theory.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))).
-
-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 (n: 
-nat).(le (f n) (g n))))).(le_n (weight_map g (TSort n))))))) (\lambda (n: 
-nat).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H: 
-((\forall (n: nat).(le (f n) (g n))))).(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)) 
-(plus_le_compat (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_S_n (S (plus (weight_map f t0) (weight_map (wadd f O) 
-t1))) (S (plus (weight_map g t0) (weight_map (wadd g O) t1))) (le_n_S (S 
-(plus (weight_map f t0) (weight_map (wadd f O) t1))) (S (plus (weight_map g 
-t0) (weight_map (wadd g O) t1))) (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)) 
-(plus_le_compat (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_S_n (S (plus (weight_map f t0) (weight_map (wadd f O) 
-t1))) (S (plus (weight_map g t0) (weight_map (wadd g O) t1))) (le_n_S (S 
-(plus (weight_map f t0) (weight_map (wadd f O) t1))) (S (plus (weight_map g 
-t0) (weight_map (wadd g O) t1))) (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)) 
-(plus_le_compat (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 (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 (f0: 
-((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H1: ((\forall (n: 
-nat).(le (f0 n) (g n))))).(lt_le_S (plus (weight_map f0 t0) (weight_map f0 
-t1)) (S (plus (weight_map g t0) (weight_map g t1))) (le_lt_n_Sm (plus 
-(weight_map f0 t0) (weight_map f0 t1)) (plus (weight_map g t0) (weight_map g 
-t1)) (plus_le_compat (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).(le_S_n (wadd (\lambda 
-(_: nat).O) O n) (wadd (\lambda (_: nat).O) (S m) n) (le_n_S (wadd (\lambda 
-(_: nat).O) O n) (wadd (\lambda (_: nat).O) (S m) n) (wadd_le (\lambda (_: 
-nat).O) (\lambda (_: nat).O) (\lambda (_: nat).(le_n O)) O (S m) (le_O_n (S 
-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_S_n (S (weight_map (\lambda (_: nat).O) u)) (S 
-(plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_: 
-nat).O) (S (weight_map (\lambda (_: nat).O) u))) t))) (le_n_S (S (weight_map 
-(\lambda (_: nat).O) u)) (S (plus (weight_map (\lambda (_: nat).O) u) 
-(weight_map (wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) 
-u))) 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_S_n (S (weight_map (\lambda 
-(_: nat).O) u)) (S (plus (weight_map (\lambda (_: nat).O) u) (weight_map 
-(wadd (\lambda (_: nat).O) O) t))) (le_n_S (S (weight_map (\lambda (_: 
-nat).O) u)) (S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd 
-(\lambda (_: nat).O) O) 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_S_n (S (weight_map (\lambda (_: nat).O) u)) (S (plus (weight_map 
-(\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) O) t))) 
-(le_n_S (S (weight_map (\lambda (_: nat).O) u)) (S (plus (weight_map (\lambda 
-(_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) O) 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_S_n (S (weight_map 
-(\lambda (_: nat).O) u)) (S (plus (weight_map (\lambda (_: nat).O) u) 
-(weight_map (\lambda (_: nat).O) t))) (le_n_S (S (weight_map (\lambda (_: 
-nat).O) u)) (S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (\lambda 
-(_: nat).O) 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_S_n (S (weight_map (\lambda (_: 
-nat).O) t)) (S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (\lambda 
-(_: nat).O) t))) (le_n_S (S (weight_map (\lambda (_: nat).O) t)) (S (plus 
-(weight_map (\lambda (_: nat).O) u) (weight_map (\lambda (_: nat).O) 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))))) (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_S_n (S (weight_map (\lambda (_: nat).O) t)) (S (plus 
-(weight_map (\lambda (_: nat).O) u) (weight_map (\lambda (_: nat).O) t))) 
-(le_n_S (S (weight_map (\lambda (_: nat).O) t)) (S (plus (weight_map (\lambda 
-(_: nat).O) u) (weight_map (\lambda (_: nat).O) 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))))) (weight_map (wadd (\lambda (_: 
-nat).O) O) t) (weight_add_O t)))) b)) (\lambda (_: F).(\lambda (u: 
-T).(\lambda (t: T).(le_S_n (S (weight_map (\lambda (_: nat).O) t)) (S (plus 
-(weight_map (\lambda (_: nat).O) u) (weight_map (\lambda (_: nat).O) t))) 
-(le_n_S (S (weight_map (\lambda (_: nat).O) t)) (S (plus (weight_map (\lambda 
-(_: nat).O) u) (weight_map (\lambda (_: nat).O) 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 (P: ((T \to 
-Prop))).(\lambda (n0: nat).(\forall (t: T).((eq nat (weight t) n0) \to (P 
-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 (t: T).(P t)) m))))).(\lambda (t0: T).(\lambda (H1: (eq nat 
-(weight t0) n0)).(let H2 \def (eq_ind_r nat n0 (\lambda (n: nat).(\forall (m: 
-nat).((lt m n) \to (\forall (t: T).((eq nat (weight t) m) \to (P t)))))) 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)))).
-
-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 (k: K).(\forall (v1: T).(\forall (v2: T).(\forall (t1: 
-T).(\forall (t2: T).(iso (THead k v1 t1) (THead k v2 t2)))))).
-
-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 H0 
-\def (match H return (\lambda (t0: T).(\lambda (t1: T).(\lambda (_: (iso t0 
-t1)).((eq T t0 (TLRef i)) \to ((eq T t1 (THead (Bind b) v t)) \to P))))) with 
-[(iso_sort n1 n2) \Rightarrow (\lambda (H0: (eq T (TSort n1) (TLRef 
-i))).(\lambda (H1: (eq T (TSort n2) (THead (Bind b) v t))).((let H2 \def 
-(eq_ind T (TSort n1) (\lambda (e: T).(match e return (\lambda (_: T).Prop) 
-with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ 
-_) \Rightarrow False])) I (TLRef i) H0) in (False_ind ((eq T (TSort n2) 
-(THead (Bind b) v t)) \to P) H2)) H1))) | (iso_lref i1 i2) \Rightarrow 
-(\lambda (H0: (eq T (TLRef i1) (TLRef i))).(\lambda (H1: (eq T (TLRef i2) 
-(THead (Bind b) v t))).((let H2 \def (f_equal T nat (\lambda (e: T).(match e 
-return (\lambda (_: T).nat) with [(TSort _) \Rightarrow i1 | (TLRef n) 
-\Rightarrow n | (THead _ _ _) \Rightarrow i1])) (TLRef i1) (TLRef i) H0) in 
-(eq_ind nat i (\lambda (_: nat).((eq T (TLRef i2) (THead (Bind b) v t)) \to 
-P)) (\lambda (H3: (eq T (TLRef i2) (THead (Bind b) v t))).(let H4 \def 
-(eq_ind T (TLRef i2) (\lambda (e: T).(match e return (\lambda (_: T).Prop) 
-with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ 
-_) \Rightarrow False])) I (THead (Bind b) v t) H3) in (False_ind P H4))) i1 
-(sym_eq nat i1 i H2))) H1))) | (iso_head k v1 v2 t1 t2) \Rightarrow (\lambda 
-(H0: (eq T (THead k v1 t1) (TLRef i))).(\lambda (H1: (eq T (THead k v2 t2) 
-(THead (Bind b) v t))).((let H2 \def (eq_ind T (THead k v1 t1) (\lambda (e: 
-T).(match e return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | 
-(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef i) 
-H0) in (False_ind ((eq T (THead k v2 t2) (THead (Bind b) v t)) \to P) H2)) 
-H1)))]) in (H0 (refl_equal T (TLRef i)) (refl_equal T (THead (Bind b) v 
-t)))))) (\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 H1 \def (match H0 return 
-(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (iso t2 t3)).((eq T t2 (THead 
-(Flat f) t0 (THeads (Flat f) t1 (TLRef i)))) \to ((eq T t3 (THead (Bind b) v 
-t)) \to P))))) with [(iso_sort n1 n2) \Rightarrow (\lambda (H1: (eq T (TSort 
-n1) (THead (Flat f) t0 (THeads (Flat f) t1 (TLRef i))))).(\lambda (H2: (eq T 
-(TSort n2) (THead (Bind b) v t))).((let H3 \def (eq_ind T (TSort n1) (\lambda 
-(e: T).(match e return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True 
-| (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead 
-(Flat f) t0 (THeads (Flat f) t1 (TLRef i))) H1) in (False_ind ((eq T (TSort 
-n2) (THead (Bind b) v t)) \to P) H3)) H2))) | (iso_lref i1 i2) \Rightarrow 
-(\lambda (H1: (eq T (TLRef i1) (THead (Flat f) t0 (THeads (Flat f) t1 (TLRef 
-i))))).(\lambda (H2: (eq T (TLRef i2) (THead (Bind b) v t))).((let H3 \def 
-(eq_ind T (TLRef i1) (\lambda (e: T).(match e return (\lambda (_: T).Prop) 
-with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ 
-_) \Rightarrow False])) I (THead (Flat f) t0 (THeads (Flat f) t1 (TLRef i))) 
-H1) in (False_ind ((eq T (TLRef i2) (THead (Bind b) v t)) \to P) H3)) H2))) | 
-(iso_head k v1 v2 t2 t3) \Rightarrow (\lambda (H1: (eq T (THead k v1 t2) 
-(THead (Flat f) t0 (THeads (Flat f) t1 (TLRef i))))).(\lambda (H2: (eq T 
-(THead k v2 t3) (THead (Bind b) v t))).((let H3 \def (f_equal T T (\lambda 
-(e: T).(match e return (\lambda (_: T).T) with [(TSort _) \Rightarrow t2 | 
-(TLRef _) \Rightarrow t2 | (THead _ _ t) \Rightarrow t])) (THead k v1 t2) 
-(THead (Flat f) t0 (THeads (Flat f) t1 (TLRef i))) H1) in ((let H4 \def 
-(f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort 
-_) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t _) \Rightarrow t])) 
-(THead k v1 t2) (THead (Flat f) t0 (THeads (Flat f) t1 (TLRef i))) H1) in 
-((let H5 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: T).K) 
-with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) 
-\Rightarrow k])) (THead k v1 t2) (THead (Flat f) t0 (THeads (Flat f) t1 
-(TLRef i))) H1) in (eq_ind K (Flat f) (\lambda (k0: K).((eq T v1 t0) \to ((eq 
-T t2 (THeads (Flat f) t1 (TLRef i))) \to ((eq T (THead k0 v2 t3) (THead (Bind 
-b) v t)) \to P)))) (\lambda (H6: (eq T v1 t0)).(eq_ind T t0 (\lambda (_: 
-T).((eq T t2 (THeads (Flat f) t1 (TLRef i))) \to ((eq T (THead (Flat f) v2 
-t3) (THead (Bind b) v t)) \to P))) (\lambda (H7: (eq T t2 (THeads (Flat f) t1 
-(TLRef i)))).(eq_ind T (THeads (Flat f) t1 (TLRef i)) (\lambda (_: T).((eq T 
-(THead (Flat f) v2 t3) (THead (Bind b) v t)) \to P)) (\lambda (H8: (eq T 
-(THead (Flat f) v2 t3) (THead (Bind b) v t))).(let H9 \def (eq_ind T (THead 
-(Flat f) v2 t3) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
-\Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow 
-False | (Flat _) \Rightarrow True])])) I (THead (Bind b) v t) H8) in 
-(False_ind P H9))) t2 (sym_eq T t2 (THeads (Flat f) t1 (TLRef i)) H7))) v1 
-(sym_eq T v1 t0 H6))) k (sym_eq K k (Flat f) H5))) H4)) H3)) H2)))]) in (H1 
-(refl_equal T (THead (Flat f) t0 (THeads (Flat f) t1 (TLRef i)))) (refl_equal 
-T (THead (Bind b) v t))))))))) 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 H0 \def 
-(match H return (\lambda (t0: T).(\lambda (t1: T).(\lambda (_: (iso t0 
-t1)).((eq T t0 (THead (Flat f2) v2 t2)) \to ((eq T t1 (THead (Bind b) v t)) 
-\to P))))) with [(iso_sort n1 n2) \Rightarrow (\lambda (H0: (eq T (TSort n1) 
-(THead (Flat f2) v2 t2))).(\lambda (H1: (eq T (TSort n2) (THead (Bind b) v 
-t))).((let H2 \def (eq_ind T (TSort n1) (\lambda (e: T).(match e return 
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) 
-\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Flat f2) v2 
-t2) H0) in (False_ind ((eq T (TSort n2) (THead (Bind b) v t)) \to P) H2)) 
-H1))) | (iso_lref i1 i2) \Rightarrow (\lambda (H0: (eq T (TLRef i1) (THead 
-(Flat f2) v2 t2))).(\lambda (H1: (eq T (TLRef i2) (THead (Bind b) v 
-t))).((let H2 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e return 
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat f2) v2 
-t2) H0) in (False_ind ((eq T (TLRef i2) (THead (Bind b) v t)) \to P) H2)) 
-H1))) | (iso_head k v1 v0 t1 t0) \Rightarrow (\lambda (H0: (eq T (THead k v1 
-t1) (THead (Flat f2) v2 t2))).(\lambda (H1: (eq T (THead k v0 t0) (THead 
-(Bind b) v t))).((let H2 \def (f_equal T T (\lambda (e: T).(match e return 
-(\lambda (_: T).T) with [(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 
-| (THead _ _ t) \Rightarrow t])) (THead k v1 t1) (THead (Flat f2) v2 t2) H0) 
-in ((let H3 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: 
-T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t 
-_) \Rightarrow t])) (THead k v1 t1) (THead (Flat f2) v2 t2) H0) in ((let H4 
-\def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: T).K) with 
-[(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) 
-\Rightarrow k])) (THead k v1 t1) (THead (Flat f2) v2 t2) H0) in (eq_ind K 
-(Flat f2) (\lambda (k0: K).((eq T v1 v2) \to ((eq T t1 t2) \to ((eq T (THead 
-k0 v0 t0) (THead (Bind b) v t)) \to P)))) (\lambda (H5: (eq T v1 v2)).(eq_ind 
-T v2 (\lambda (_: T).((eq T t1 t2) \to ((eq T (THead (Flat f2) v0 t0) (THead 
-(Bind b) v t)) \to P))) (\lambda (H6: (eq T t1 t2)).(eq_ind T t2 (\lambda (_: 
-T).((eq T (THead (Flat f2) v0 t0) (THead (Bind b) v t)) \to P)) (\lambda (H7: 
-(eq T (THead (Flat f2) v0 t0) (THead (Bind b) v t))).(let H8 \def (eq_ind T 
-(THead (Flat f2) v0 t0) (\lambda (e: T).(match e return (\lambda (_: T).Prop) 
-with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ 
-_) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind _) 
-\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) v t) H7) 
-in (False_ind P H8))) t1 (sym_eq T t1 t2 H6))) v1 (sym_eq T v1 v2 H5))) k 
-(sym_eq K k (Flat f2) H4))) H3)) H2)) H1)))]) in (H0 (refl_equal T (THead 
-(Flat f2) v2 t2)) (refl_equal T (THead (Bind b) v t)))))) (\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 H1 \def (match H0 return 
-(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (iso t3 t4)).((eq T t3 (THead 
-(Flat f1) t0 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2)))) \to ((eq T t4 
-(THead (Bind b) v t)) \to P))))) with [(iso_sort n1 n2) \Rightarrow (\lambda 
-(H1: (eq T (TSort n1) (THead (Flat f1) t0 (THeads (Flat f1) t1 (THead (Flat 
-f2) v2 t2))))).(\lambda (H2: (eq T (TSort n2) (THead (Bind b) v t))).((let H3 
-\def (eq_ind T (TSort n1) (\lambda (e: T).(match e return (\lambda (_: 
-T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | 
-(THead _ _ _) \Rightarrow False])) I (THead (Flat f1) t0 (THeads (Flat f1) t1 
-(THead (Flat f2) v2 t2))) H1) in (False_ind ((eq T (TSort n2) (THead (Bind b) 
-v t)) \to P) H3)) H2))) | (iso_lref i1 i2) \Rightarrow (\lambda (H1: (eq T 
-(TLRef i1) (THead (Flat f1) t0 (THeads (Flat f1) t1 (THead (Flat f2) v2 
-t2))))).(\lambda (H2: (eq T (TLRef i2) (THead (Bind b) v t))).((let H3 \def 
-(eq_ind T (TLRef i1) (\lambda (e: T).(match e return (\lambda (_: T).Prop) 
-with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ 
-_) \Rightarrow False])) I (THead (Flat f1) t0 (THeads (Flat f1) t1 (THead 
-(Flat f2) v2 t2))) H1) in (False_ind ((eq T (TLRef i2) (THead (Bind b) v t)) 
-\to P) H3)) H2))) | (iso_head k v1 v0 t3 t4) \Rightarrow (\lambda (H1: (eq T 
-(THead k v1 t3) (THead (Flat f1) t0 (THeads (Flat f1) t1 (THead (Flat f2) v2 
-t2))))).(\lambda (H2: (eq T (THead k v0 t4) (THead (Bind b) v t))).((let H3 
-\def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t) 
-\Rightarrow t])) (THead k v1 t3) (THead (Flat f1) t0 (THeads (Flat f1) t1 
-(THead (Flat f2) v2 t2))) H1) in ((let H4 \def (f_equal T T (\lambda (e: 
-T).(match e return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef 
-_) \Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead k v1 t3) (THead 
-(Flat f1) t0 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2))) H1) in ((let H5 
-\def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: T).K) with 
-[(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) 
-\Rightarrow k])) (THead k v1 t3) (THead (Flat f1) t0 (THeads (Flat f1) t1 
-(THead (Flat f2) v2 t2))) H1) in (eq_ind K (Flat f1) (\lambda (k0: K).((eq T 
-v1 t0) \to ((eq T t3 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2))) \to ((eq 
-T (THead k0 v0 t4) (THead (Bind b) v t)) \to P)))) (\lambda (H6: (eq T v1 
-t0)).(eq_ind T t0 (\lambda (_: T).((eq T t3 (THeads (Flat f1) t1 (THead (Flat 
-f2) v2 t2))) \to ((eq T (THead (Flat f1) v0 t4) (THead (Bind b) v t)) \to 
-P))) (\lambda (H7: (eq T t3 (THeads (Flat f1) t1 (THead (Flat f2) v2 
-t2)))).(eq_ind T (THeads (Flat f1) t1 (THead (Flat f2) v2 t2)) (\lambda (_: 
-T).((eq T (THead (Flat f1) v0 t4) (THead (Bind b) v t)) \to P)) (\lambda (H8: 
-(eq T (THead (Flat f1) v0 t4) (THead (Bind b) v t))).(let H9 \def (eq_ind T 
-(THead (Flat f1) v0 t4) (\lambda (e: T).(match e return (\lambda (_: T).Prop) 
-with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ 
-_) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind _) 
-\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) v t) H8) 
-in (False_ind P H9))) t3 (sym_eq T t3 (THeads (Flat f1) t1 (THead (Flat f2) 
-v2 t2)) H7))) v1 (sym_eq T v1 t0 H6))) k (sym_eq K k (Flat f1) H5))) H4)) 
-H3)) H2)))]) in (H1 (refl_equal T (THead (Flat f1) t0 (THeads (Flat f1) t1 
-(THead (Flat f2) v2 t2)))) (refl_equal T (THead (Bind b) v t))))))))) 
-vs)))))))).
-
-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 H1 \def (match H0 return (\lambda (t: T).(\lambda (t0: 
-T).(\lambda (_: (iso t t0)).((eq T t (TSort n2)) \to ((eq T t0 t3) \to (iso 
-(TSort n1) t3)))))) with [(iso_sort n0 n3) \Rightarrow (\lambda (H0: (eq T 
-(TSort n0) (TSort n2))).(\lambda (H1: (eq T (TSort n3) t3)).((let H2 \def 
-(f_equal T nat (\lambda (e: T).(match e return (\lambda (_: T).nat) with 
-[(TSort n) \Rightarrow n | (TLRef _) \Rightarrow n0 | (THead _ _ _) 
-\Rightarrow n0])) (TSort n0) (TSort n2) H0) in (eq_ind nat n2 (\lambda (_: 
-nat).((eq T (TSort n3) t3) \to (iso (TSort n1) t3))) (\lambda (H3: (eq T 
-(TSort n3) t3)).(eq_ind T (TSort n3) (\lambda (t: T).(iso (TSort n1) t)) 
-(iso_sort n1 n3) t3 H3)) n0 (sym_eq nat n0 n2 H2))) H1))) | (iso_lref i1 i2) 
-\Rightarrow (\lambda (H0: (eq T (TLRef i1) (TSort n2))).(\lambda (H1: (eq T 
-(TLRef i2) t3)).((let H2 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e 
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (TSort n2) H0) in 
-(False_ind ((eq T (TLRef i2) t3) \to (iso (TSort n1) t3)) H2)) H1))) | 
-(iso_head k v1 v2 t1 t2) \Rightarrow (\lambda (H0: (eq T (THead k v1 t1) 
-(TSort n2))).(\lambda (H1: (eq T (THead k v2 t2) t3)).((let H2 \def (eq_ind T 
-(THead k v1 t1) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) 
-\Rightarrow True])) I (TSort n2) H0) in (False_ind ((eq T (THead k v2 t2) t3) 
-\to (iso (TSort n1) t3)) H2)) H1)))]) in (H1 (refl_equal T (TSort n2)) 
-(refl_equal T t3))))))) (\lambda (i1: nat).(\lambda (i2: nat).(\lambda (t3: 
-T).(\lambda (H0: (iso (TLRef i2) t3)).(let H1 \def (match H0 return (\lambda 
-(t: T).(\lambda (t0: T).(\lambda (_: (iso t t0)).((eq T t (TLRef i2)) \to 
-((eq T t0 t3) \to (iso (TLRef i1) t3)))))) with [(iso_sort n1 n2) \Rightarrow 
-(\lambda (H0: (eq T (TSort n1) (TLRef i2))).(\lambda (H1: (eq T (TSort n2) 
-t3)).((let H2 \def (eq_ind T (TSort n1) (\lambda (e: T).(match e return 
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) 
-\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef i2) H0) in 
-(False_ind ((eq T (TSort n2) t3) \to (iso (TLRef i1) t3)) H2)) H1))) | 
-(iso_lref i0 i3) \Rightarrow (\lambda (H0: (eq T (TLRef i0) (TLRef 
-i2))).(\lambda (H1: (eq T (TLRef i3) t3)).((let H2 \def (f_equal T nat 
-(\lambda (e: T).(match e return (\lambda (_: T).nat) with [(TSort _) 
-\Rightarrow i0 | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i0])) 
-(TLRef i0) (TLRef i2) H0) in (eq_ind nat i2 (\lambda (_: nat).((eq T (TLRef 
-i3) t3) \to (iso (TLRef i1) t3))) (\lambda (H3: (eq T (TLRef i3) t3)).(eq_ind 
-T (TLRef i3) (\lambda (t: T).(iso (TLRef i1) t)) (iso_lref i1 i3) t3 H3)) i0 
-(sym_eq nat i0 i2 H2))) H1))) | (iso_head k v1 v2 t1 t2) \Rightarrow (\lambda 
-(H0: (eq T (THead k v1 t1) (TLRef i2))).(\lambda (H1: (eq T (THead k v2 t2) 
-t3)).((let H2 \def (eq_ind T (THead k v1 t1) (\lambda (e: T).(match e return 
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef i2) H0) in 
-(False_ind ((eq T (THead k v2 t2) t3) \to (iso (TLRef i1) t3)) H2)) H1)))]) 
-in (H1 (refl_equal T (TLRef i2)) (refl_equal T t3))))))) (\lambda (k: 
-K).(\lambda (v1: T).(\lambda (v2: T).(\lambda (t3: T).(\lambda (t4: 
-T).(\lambda (t5: T).(\lambda (H0: (iso (THead k v2 t4) t5)).(let H1 \def 
-(match H0 return (\lambda (t: T).(\lambda (t0: T).(\lambda (_: (iso t 
-t0)).((eq T t (THead k v2 t4)) \to ((eq T t0 t5) \to (iso (THead k v1 t3) 
-t5)))))) with [(iso_sort n1 n2) \Rightarrow (\lambda (H0: (eq T (TSort n1) 
-(THead k v2 t4))).(\lambda (H1: (eq T (TSort n2) t5)).((let H2 \def (eq_ind T 
-(TSort n1) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with [(TSort 
-_) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
-False])) I (THead k v2 t4) H0) in (False_ind ((eq T (TSort n2) t5) \to (iso 
-(THead k v1 t3) t5)) H2)) H1))) | (iso_lref i1 i2) \Rightarrow (\lambda (H0: 
-(eq T (TLRef i1) (THead k v2 t4))).(\lambda (H1: (eq T (TLRef i2) t5)).((let 
-H2 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e return (\lambda (_: 
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | 
-(THead _ _ _) \Rightarrow False])) I (THead k v2 t4) H0) in (False_ind ((eq T 
-(TLRef i2) t5) \to (iso (THead k v1 t3) t5)) H2)) H1))) | (iso_head k0 v0 v3 
-t0 t4) \Rightarrow (\lambda (H0: (eq T (THead k0 v0 t0) (THead k v2 
-t4))).(\lambda (H1: (eq T (THead k0 v3 t4) t5)).((let H2 \def (f_equal T T 
-(\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) 
-(THead k0 v0 t0) (THead k v2 t4) H0) in ((let H3 \def (f_equal T T (\lambda 
-(e: T).(match e return (\lambda (_: T).T) with [(TSort _) \Rightarrow v0 | 
-(TLRef _) \Rightarrow v0 | (THead _ t _) \Rightarrow t])) (THead k0 v0 t0) 
-(THead k v2 t4) H0) in ((let H4 \def (f_equal T K (\lambda (e: T).(match e 
-return (\lambda (_: T).K) with [(TSort _) \Rightarrow k0 | (TLRef _) 
-\Rightarrow k0 | (THead k _ _) \Rightarrow k])) (THead k0 v0 t0) (THead k v2 
-t4) H0) in (eq_ind K k (\lambda (k1: K).((eq T v0 v2) \to ((eq T t0 t4) \to 
-((eq T (THead k1 v3 t4) t5) \to (iso (THead k v1 t3) t5))))) (\lambda (H5: 
-(eq T v0 v2)).(eq_ind T v2 (\lambda (_: T).((eq T t0 t4) \to ((eq T (THead k 
-v3 t4) t5) \to (iso (THead k v1 t3) t5)))) (\lambda (H6: (eq T t0 
-t4)).(eq_ind T t4 (\lambda (_: T).((eq T (THead k v3 t4) t5) \to (iso (THead 
-k v1 t3) t5))) (\lambda (H7: (eq T (THead k v3 t4) t5)).(eq_ind T (THead k v3 
-t4) (\lambda (t: T).(iso (THead k v1 t3) t)) (iso_head k v1 v3 t3 t4) t5 H7)) 
-t0 (sym_eq T t0 t4 H6))) v0 (sym_eq T v0 v2 H5))) k0 (sym_eq K k0 k H4))) 
-H3)) H2)) H1)))]) in (H1 (refl_equal T (THead k v2 t4)) (refl_equal T 
-t5)))))))))) t1 t2 H))).
-
-inductive C: Set \def
-| CSort: nat \to C
-| CHead: C \to (K \to (T \to C)).
-
-definition r:
- K \to (nat \to nat)
-\def
- \lambda (k: K).(\lambda (i: nat).(match k with [(Bind _) \Rightarrow i | 
-(Flat _) \Rightarrow (S i)])).
-
-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.
-
-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).(match k return (\lambda (k0: K).(\forall (i: nat).(eq nat (s 
-k0 (r k0 i)) (S i)))) with [(Bind _) \Rightarrow (\lambda (i: 
-nat).(refl_equal nat (S i))) | (Flat _) \Rightarrow (\lambda (i: 
-nat).(refl_equal nat (S i)))]).
-
-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)))).
-
-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.
-
-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))).
-
-theorem tweight_lt:
- \forall (t: T).(lt O (tweight t))
-\def
- \lambda (t: T).(match t return (\lambda (t0: T).(lt O (tweight t0))) with 
-[(TSort _) \Rightarrow (le_n (S O)) | (TLRef _) \Rightarrow (le_n (S O)) | 
-(THead _ t0 t1) \Rightarrow (le_S_n (S O) (S (plus (tweight t0) (tweight 
-t1))) (le_n_S (S O) (S (plus (tweight t0) (tweight t1))) (le_n_S O (plus 
-(tweight t0) (tweight t1)) (le_O_n (plus (tweight t0) (tweight t1))))))]).
-
-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_le_S (plus (cweight c) (tweight t)) 
-(plus (cweight d) (tweight t)) (plus_lt_compat_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)))) (lt_le_S (plus 
-(cweight c) O) (plus (cweight c) (tweight u)) (plus_le_lt_compat (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 (P: ((C \to 
-Prop))).(\lambda (n0: nat).(\forall (c: C).((eq nat (cweight c) n0) \to (P 
-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 (c: C).(P c)) m))))).(\lambda (c0: C).(\lambda (H1: (eq nat 
-(cweight c0) n0)).(let H2 \def (eq_ind_r nat n0 (\lambda (n: nat).(\forall 
-(m: nat).((lt m n) \to (\forall (c: C).((eq nat (cweight c) m) \to (P c)))))) 
-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)))).
-
-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.
-
-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).(match c0 return (\lambda (c1: C).(((\forall (d: 
-C).((clt d c1) \to (P d)))) \to (P c1))) with [(CSort n) \Rightarrow (\lambda 
-(_: ((\forall (d: C).((clt d (CSort n)) \to (P d))))).(H n)) | (CHead c1 k t) 
-\Rightarrow (\lambda (H1: ((\forall (d: C).((clt d (CHead c1 k t)) \to (P 
-d))))).(let H_x \def (chead_ctail c1 t k) in (let H2 \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 (H3: (eq C (CHead c1 k t) (CTail x0 x2 
-x1))).(eq_ind_r C (CTail x0 x2 x1) (\lambda (c2: C).(P c2)) (let H4 \def 
-(eq_ind C (CHead c1 k t) (\lambda (c: C).(\forall (d: C).((clt d c) \to (P 
-d)))) H1 (CTail x0 x2 x1) H3) in (H0 x1 (H4 x1 (clt_thead x0 x2 x1)) x0 x2)) 
-(CHead c1 k t) H3))))) H2))))])) c)))).
-
-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))))).
-
-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).(lt_le_S 
-(plus (cweight c) (tweight u)) (plus (cweight c) (S (plus (tweight u) 
-(tweight t)))) (plus_le_lt_compat (cweight c) (cweight c) (tweight u) (S 
-(plus (tweight u) (tweight t))) (le_n (cweight c)) (le_lt_n_Sm (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).(lt_le_S 
-(plus (cweight c) (tweight t)) (plus (cweight c) (S (plus (tweight u) 
-(tweight t)))) (plus_le_lt_compat (cweight c) (cweight c) (tweight t) (S 
-(plus (tweight u) (tweight t))) (le_n (cweight c)) (le_lt_n_Sm (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 (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).(le_S_n (S 
-(plus (cweight c) (tweight t))) (plus (plus (cweight c) (tweight t)) (S O)) 
-(lt_le_S (S (plus (cweight c) (tweight t))) (S (plus (plus (cweight c) 
-(tweight t)) (S O))) (lt_n_S (plus (cweight c) (tweight t)) (plus (plus 
-(cweight c) (tweight t)) (S O)) (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_comm (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_S_n (plus (cweight c2) (S O)) (plus (plus (cweight c2) 
-(tweight t2)) (S O)) (lt_le_S (plus (cweight c2) (S O)) (S (plus (plus 
-(cweight c2) (tweight t2)) (S O))) (le_lt_n_Sm (plus (cweight c2) (S O)) 
-(plus (plus (cweight c2) (tweight t2)) (S O)) (plus_le_compat (cweight c2) 
-(plus (cweight c2) (tweight t2)) (S O) (S O) (le_plus_l (cweight c2) (tweight 
-t2)) (le_n (S O)))))))))))))).
-
-theorem flt_wf__q_ind:
- \forall (P: ((C \to (T \to Prop)))).(((\forall (n: nat).((\lambda (P: ((C 
-\to (T \to Prop)))).(\lambda (n0: nat).(\forall (c: C).(\forall (t: T).((eq 
-nat (fweight c t) n0) \to (P 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 (n: 
-nat).(\forall (m: nat).((lt m n) \to (\forall (c: C).(\forall (t: T).((eq nat 
-(fweight c t) m) \to (P c t))))))) 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))))).
-
-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.
-
-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_equal 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_equal 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 (H0: (lt n0 d)).(let H1 \def (eq_ind T (lift h d (TLRef n0)) 
-(\lambda (t: T).(eq T (TSort n) t)) H (TLRef n0) (lift_lref_lt n0 h d H0)) in 
-(let H2 \def (match H1 return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T 
-t (TLRef n0)) \to (eq T (TLRef n0) (TSort n))))) with [refl_equal \Rightarrow 
-(\lambda (H1: (eq T (TSort n) (TLRef n0))).(let H2 \def (eq_ind T (TSort n) 
-(\lambda (e: T).(match e 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)))]) 
-in (H2 (refl_equal T (TLRef n0)))))) (\lambda (H0: (le d n0)).(let H1 \def 
-(eq_ind T (lift h d (TLRef n0)) (\lambda (t: T).(eq T (TSort n) t)) H (TLRef 
-(plus n0 h)) (lift_lref_ge n0 h d H0)) in (let H2 \def (match H1 return 
-(\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t (TLRef (plus n0 h))) \to 
-(eq T (TLRef n0) (TSort n))))) with [refl_equal \Rightarrow (\lambda (H1: (eq 
-T (TSort n) (TLRef (plus n0 h)))).(let H2 \def (eq_ind T (TSort n) (\lambda 
-(e: T).(match e 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)))]) in (H2 
-(refl_equal T (TLRef (plus n0 h)))))))))) (\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 (t: T).(eq 
-T (TSort n) t)) H1 (THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k 
-t0 t1 h d)) in (let H3 \def (match H2 return (\lambda (t: T).(\lambda (_: (eq 
-? ? t)).((eq T t (THead k (lift h d t0) (lift h (s k d) t1))) \to (eq T 
-(THead k t0 t1) (TSort n))))) with [refl_equal \Rightarrow (\lambda (H2: (eq 
-T (TSort n) (THead k (lift h d t0) (lift h (s k d) t1)))).(let H3 \def 
-(eq_ind T (TSort n) (\lambda (e: T).(match e 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)))]) in (H3 (refl_equal T 
-(THead k (lift h d t0) (lift h (s k d) t1)))))))))))) 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 (t: T).(eq T (TLRef i) t)) H (TSort 
-n) (lift_sort n h d)) in (let H1 \def (eq_ind T (TLRef i) (\lambda (ee: 
-T).(match ee 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 
-(t: T).(eq T (TLRef i) t)) H (TLRef n) (lift_lref_lt n h d H0)) in (let H2 
-\def (f_equal T nat (\lambda (e: T).(match e return (\lambda (_: T).nat) with 
-[(TSort _) \Rightarrow i | (TLRef n) \Rightarrow n | (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 (t: T).(eq T (TLRef i) t)) 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 return (\lambda 
-(_: T).nat) with [(TSort _) \Rightarrow i | (TLRef n) \Rightarrow n | (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)) (plus_le_compat 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 (t: T).(eq 
-T (TLRef i) t)) 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 
-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).(T_ind (\lambda (t0: T).((eq T (TLRef n) (lift h d t0)) 
-\to (eq T t0 (TLRef n)))) (\lambda (n0: nat).(\lambda (H0: (eq T (TLRef n) 
-(lift h d (TSort n0)))).(sym_eq T (TLRef n) (TSort n0) H0))) (\lambda (n0: 
-nat).(\lambda (H0: (eq T (TLRef n) (lift h d (TLRef n0)))).(lt_le_e n0 d (eq 
-T (TLRef n0) (TLRef n)) (\lambda (H1: (lt n0 d)).(let H2 \def (eq_ind T (lift 
-h d (TLRef n0)) (\lambda (t: T).(eq T (TLRef n) t)) H0 (TLRef n0) 
-(lift_lref_lt n0 h d H1)) in (sym_eq T (TLRef n) (TLRef n0) H2))) (\lambda 
-(H1: (le d n0)).(let H2 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t: 
-T).(eq T (TLRef n) t)) H0 (TLRef (plus n0 h)) (lift_lref_ge n0 h d H1)) in 
-(let H3 \def (match H2 return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T 
-t (TLRef (plus n0 h))) \to (eq T (TLRef n0) (TLRef n))))) with [refl_equal 
-\Rightarrow (\lambda (H2: (eq T (TLRef n) (TLRef (plus n0 h)))).(let H3 \def 
-(f_equal T nat (\lambda (e: T).(match e return (\lambda (_: T).nat) with 
-[(TSort _) \Rightarrow n | (TLRef n) \Rightarrow n | (THead _ _ _) 
-\Rightarrow n])) (TLRef n) (TLRef (plus n0 h)) H2) in (eq_ind nat (plus n0 h) 
-(\lambda (n: nat).(eq T (TLRef n0) (TLRef n))) (let H0 \def (eq_ind nat n 
-(\lambda (n: nat).(lt n d)) H (plus n0 h) H3) in (le_false d n0 (eq T (TLRef 
-n0) (TLRef (plus n0 h))) H1 (lt_le_S n0 d (le_lt_trans n0 (plus n0 h) d 
-(le_plus_l n0 h) H0)))) n (sym_eq nat n (plus n0 h) H3))))]) in (H3 
-(refl_equal T (TLRef (plus n0 h)))))))))) (\lambda (k: K).(\lambda (t0: 
-T).(\lambda (_: (((eq T (TLRef n) (lift h d t0)) \to (eq T t0 (TLRef 
-n))))).(\lambda (t1: T).(\lambda (_: (((eq T (TLRef n) (lift h d t1)) \to (eq 
-T t1 (TLRef n))))).(\lambda (H2: (eq T (TLRef n) (lift h d (THead k t0 
-t1)))).(let H3 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda (t: T).(eq 
-T (TLRef n) t)) H2 (THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k 
-t0 t1 h d)) in (let H4 \def (match H3 return (\lambda (t: T).(\lambda (_: (eq 
-? ? t)).((eq T t (THead k (lift h d t0) (lift h (s k d) t1))) \to (eq T 
-(THead k t0 t1) (TLRef n))))) with [refl_equal \Rightarrow (\lambda (H3: (eq 
-T (TLRef n) (THead k (lift h d t0) (lift h (s k d) t1)))).(let H4 \def 
-(eq_ind T (TLRef n) (\lambda (e: T).(match e 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)) H3) in 
-(False_ind (eq T (THead k t0 t1) (TLRef n)) H4)))]) in (H4 (refl_equal T 
-(THead k (lift h d t0) (lift h (s k d) t1)))))))))))) t))))).
-
-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).(T_ind (\lambda (t0: 
-T).((eq T (TLRef n) (lift h d t0)) \to (\forall (P: Prop).P))) (\lambda (n0: 
-nat).(\lambda (H1: (eq T (TLRef n) (lift h d (TSort n0)))).(\lambda (P: 
-Prop).(let H2 \def (match H1 return (\lambda (t: T).(\lambda (_: (eq ? ? 
-t)).((eq T t (lift h d (TSort n0))) \to P))) with [refl_equal \Rightarrow 
-(\lambda (H2: (eq T (TLRef n) (lift h d (TSort n0)))).(let H3 \def (eq_ind T 
-(TLRef n) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with [(TSort 
-_) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow 
-False])) I (lift h d (TSort n0)) H2) in (False_ind P H3)))]) in (H2 
-(refl_equal T (lift h d (TSort n0)))))))) (\lambda (n0: nat).(\lambda (H1: 
-(eq T (TLRef n) (lift h d (TLRef n0)))).(\lambda (P: Prop).(lt_le_e n0 d P 
-(\lambda (H2: (lt n0 d)).(let H3 \def (eq_ind T (lift h d (TLRef n0)) 
-(\lambda (t: T).(eq T (TLRef n) t)) H1 (TLRef n0) (lift_lref_lt n0 h d H2)) 
-in (let H4 \def (match H3 return (\lambda (t: T).(\lambda (_: (eq ? ? 
-t)).((eq T t (TLRef n0)) \to P))) with [refl_equal \Rightarrow (\lambda (H3: 
-(eq T (TLRef n) (TLRef n0))).(let H4 \def (f_equal T nat (\lambda (e: 
-T).(match e return (\lambda (_: T).nat) with [(TSort _) \Rightarrow n | 
-(TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef n0) 
-H3) in (eq_ind nat n0 (\lambda (_: nat).P) (let H1 \def (eq_ind_r nat n0 
-(\lambda (n: nat).(lt n d)) H2 n H4) in (le_false d n P H H1)) n (sym_eq nat 
-n n0 H4))))]) in (H4 (refl_equal T (TLRef n0)))))) (\lambda (H2: (le d 
-n0)).(let H3 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t: T).(eq T 
-(TLRef n) t)) H1 (TLRef (plus n0 h)) (lift_lref_ge n0 h d H2)) in (let H4 
-\def (match H3 return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t 
-(TLRef (plus n0 h))) \to P))) with [refl_equal \Rightarrow (\lambda (H3: (eq 
-T (TLRef n) (TLRef (plus n0 h)))).(let H4 \def (f_equal T nat (\lambda (e: 
-T).(match e return (\lambda (_: T).nat) with [(TSort _) \Rightarrow n | 
-(TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef 
-(plus n0 h)) H3) in (eq_ind nat (plus n0 h) (\lambda (_: nat).P) (let H1 \def 
-(eq_ind nat n (\lambda (n: nat).(lt n (plus d h))) H0 (plus n0 h) H4) in 
-(le_false d n0 P H2 (lt_le_S n0 d (simpl_lt_plus_r h n0 d H1)))) n (sym_eq 
-nat n (plus n0 h) H4))))]) in (H4 (refl_equal T (TLRef (plus n0 h))))))))))) 
-(\lambda (k: K).(\lambda (t0: T).(\lambda (_: (((eq T (TLRef n) (lift h d 
-t0)) \to (\forall (P: Prop).P)))).(\lambda (t1: T).(\lambda (_: (((eq T 
-(TLRef n) (lift h d t1)) \to (\forall (P: Prop).P)))).(\lambda (H3: (eq T 
-(TLRef n) (lift h d (THead k t0 t1)))).(\lambda (P: Prop).(let H4 \def 
-(eq_ind T (lift h d (THead k t0 t1)) (\lambda (t: T).(eq T (TLRef n) t)) H3 
-(THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) in (let 
-H5 \def (match H4 return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t 
-(THead k (lift h d t0) (lift h (s k d) t1))) \to P))) with [refl_equal 
-\Rightarrow (\lambda (H4: (eq T (TLRef n) (THead k (lift h d t0) (lift h (s k 
-d) t1)))).(let H5 \def (eq_ind T (TLRef n) (\lambda (e: T).(match e 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)) H4) in (False_ind P H5)))]) in (H5 (refl_equal T 
-(THead k (lift h d t0) (lift h (s k d) t1))))))))))))) t)))))).
-
-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).(T_ind (\lambda (t0: T).((eq T (TLRef (plus n h)) (lift h 
-d t0)) \to (eq T t0 (TLRef n)))) (\lambda (n0: nat).(\lambda (H0: (eq T 
-(TLRef (plus n h)) (lift h d (TSort n0)))).(let H1 \def (match H0 return 
-(\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t (lift h d (TSort n0))) \to 
-(eq T (TSort n0) (TLRef n))))) with [refl_equal \Rightarrow (\lambda (H1: (eq 
-T (TLRef (plus n h)) (lift h d (TSort n0)))).(let H2 \def (eq_ind T (TLRef 
-(plus n h)) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) 
-\Rightarrow False])) I (lift h d (TSort n0)) H1) in (False_ind (eq T (TSort 
-n0) (TLRef n)) H2)))]) in (H1 (refl_equal T (lift h d (TSort n0))))))) 
-(\lambda (n0: nat).(\lambda (H0: (eq T (TLRef (plus n h)) (lift h d (TLRef 
-n0)))).(lt_le_e n0 d (eq T (TLRef n0) (TLRef n)) (\lambda (H1: (lt n0 
-d)).(let H2 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t: T).(eq T (TLRef 
-(plus n h)) t)) H0 (TLRef n0) (lift_lref_lt n0 h d H1)) in (let H3 \def 
-(match H2 return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t (TLRef 
-n0)) \to (eq T (TLRef n0) (TLRef n))))) with [refl_equal \Rightarrow (\lambda 
-(H2: (eq T (TLRef (plus n h)) (TLRef n0))).(let H3 \def (f_equal T nat 
-(\lambda (e: T).(match e return (\lambda (_: T).nat) with [(TSort _) 
-\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) n h) | (TLRef n) \Rightarrow n | (THead _ _ _) \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) n h)])) (TLRef 
-(plus n h)) (TLRef n0) H2) in (eq_ind nat (plus n h) (\lambda (n0: nat).(eq T 
-(TLRef n0) (TLRef n))) (let H0 \def (eq_ind_r nat n0 (\lambda (n: nat).(lt n 
-d)) H1 (plus n h) H3) in (le_false d n (eq T (TLRef (plus n h)) (TLRef n)) H 
-(lt_le_S n d (le_lt_trans n (plus n h) d (le_plus_l n h) H0)))) n0 H3)))]) in 
-(H3 (refl_equal T (TLRef n0)))))) (\lambda (H1: (le d n0)).(let H2 \def 
-(eq_ind T (lift h d (TLRef n0)) (\lambda (t: T).(eq T (TLRef (plus n h)) t)) 
-H0 (TLRef (plus n0 h)) (lift_lref_ge n0 h d H1)) in (let H3 \def (match H2 
-return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t (TLRef (plus n0 h))) 
-\to (eq T (TLRef n0) (TLRef n))))) with [refl_equal \Rightarrow (\lambda (H2: 
-(eq T (TLRef (plus n h)) (TLRef (plus n0 h)))).(let H3 \def (f_equal T nat 
-(\lambda (e: T).(match e return (\lambda (_: T).nat) with [(TSort _) 
-\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) n h) | (TLRef n) \Rightarrow n | (THead _ _ _) \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) n h)])) (TLRef 
-(plus n h)) (TLRef (plus n0 h)) H2) in (eq_ind nat (plus n h) (\lambda (_: 
-nat).(eq T (TLRef n0) (TLRef n))) (f_equal nat T TLRef n0 n (simpl_plus_r h 
-n0 n (sym_eq nat (plus n h) (plus n0 h) H3))) (plus n0 h) H3)))]) in (H3 
-(refl_equal T (TLRef (plus n0 h)))))))))) (\lambda (k: K).(\lambda (t0: 
-T).(\lambda (_: (((eq T (TLRef (plus n h)) (lift h d t0)) \to (eq T t0 (TLRef 
-n))))).(\lambda (t1: T).(\lambda (_: (((eq T (TLRef (plus n h)) (lift h d 
-t1)) \to (eq T t1 (TLRef n))))).(\lambda (H2: (eq T (TLRef (plus n h)) (lift 
-h d (THead k t0 t1)))).(let H3 \def (eq_ind T (lift h d (THead k t0 t1)) 
-(\lambda (t: T).(eq T (TLRef (plus n h)) t)) H2 (THead k (lift h d t0) (lift 
-h (s k d) t1)) (lift_head k t0 t1 h d)) in (let H4 \def (match H3 return 
-(\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t (THead k (lift h d t0) 
-(lift h (s k d) t1))) \to (eq T (THead k t0 t1) (TLRef n))))) with 
-[refl_equal \Rightarrow (\lambda (H3: (eq T (TLRef (plus n h)) (THead k (lift 
-h d t0) (lift h (s k d) t1)))).(let H4 \def (eq_ind T (TLRef (plus n h)) 
-(\lambda (e: T).(match e 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)) H3) in (False_ind (eq 
-T (THead k t0 t1) (TLRef n)) H4)))]) in (H4 (refl_equal T (THead k (lift h d 
-t0) (lift h (s k d) t1)))))))))))) t))))).
-
-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 (match H return (\lambda (t0: T).(\lambda 
-(_: (eq ? ? t0)).((eq T t0 (lift h d (TSort n))) \to (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)))))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead k 
-u t) (lift h d (TSort n)))).(let H1 \def (eq_ind T (THead k u t) (\lambda (e: 
-T).(match e return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | 
-(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (lift h d 
-(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)))]) 
-in (H0 (refl_equal T (lift h d (TSort n))))))))) (\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 
-(match H1 return (\lambda (t0: T).(\lambda (_: (eq ? ? t0)).((eq T t0 (TLRef 
-n)) \to (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)))))))) with [refl_equal 
-\Rightarrow (\lambda (H1: (eq T (THead k u t) (TLRef n))).(let H2 \def 
-(eq_ind T (THead k u t) (\lambda (e: T).(match e 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)))]) in (H2 (refl_equal T (TLRef n)))))) (\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 (match H1 return (\lambda (t0: T).(\lambda (_: (eq ? ? t0)).((eq T t0 
-(TLRef (plus n h))) \to (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)))))))) with 
-[refl_equal \Rightarrow (\lambda (H1: (eq T (THead k u t) (TLRef (plus n 
-h)))).(let H2 \def (eq_ind T (THead k u t) (\lambda (e: T).(match e 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)))]) in (H2 
-(refl_equal T (TLRef (plus n h)))))))))))) (\lambda (k0: K).(\lambda (t0: 
-T).(\lambda (_: ((\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 (_: ((\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 (t0: 
-T).(eq T (THead k u t) t0)) H1 (THead k0 (lift h d t0) (lift h (s k0 d) t1)) 
-(lift_head k0 t0 t1 h d)) in (let H3 \def (match H2 return (\lambda (t2: 
-T).(\lambda (_: (eq ? ? t2)).((eq T t2 (THead k0 (lift h d t0) (lift h (s k0 
-d) t1))) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k0 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)))))))) with 
-[refl_equal \Rightarrow (\lambda (H2: (eq T (THead k u t) (THead k0 (lift h d 
-t0) (lift h (s k0 d) t1)))).(let H3 \def (f_equal T T (\lambda (e: T).(match 
-e return (\lambda (_: T).T) with [(TSort _) \Rightarrow t | (TLRef _) 
-\Rightarrow t | (THead _ _ t) \Rightarrow t])) (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 return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef 
-_) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead k u t) (THead k0 
-(lift h d t0) (lift h (s k0 d) t1)) H2) in ((let H5 \def (f_equal T K 
-(\lambda (e: T).(match e return (\lambda (_: T).K) with [(TSort _) 
-\Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) 
-(THead k u t) (THead k0 (lift h d t0) (lift h (s k0 d) t1)) H2) in (eq_ind K 
-k0 (\lambda (k: K).((eq T u (lift h d t0)) \to ((eq T t (lift h (s k0 d) t1)) 
-\to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k0 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)))))))) (\lambda (H6: (eq T 
-u (lift h d t0))).(eq_ind T (lift h d t0) (\lambda (t2: T).((eq T t (lift h 
-(s k0 d) t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k0 
-t0 t1) (THead k0 y z)))) (\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h d 
-y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k0 d) z))))))) 
-(\lambda (H7: (eq T t (lift h (s k0 d) t1))).(eq_ind T (lift h (s k0 d) t1) 
-(\lambda (t: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k0 t0 
-t1) (THead k0 y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t0) 
-(lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k0 d) 
-z)))))) (ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k0 t0 
-t1) (THead k0 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 k0 d) t1) 
-(lift h (s k0 d) z)))) t0 t1 (refl_equal T (THead k0 t0 t1)) (refl_equal T 
-(lift h d t0)) (refl_equal T (lift h (s k0 d) t1))) t (sym_eq T t (lift h (s 
-k0 d) t1) H7))) u (sym_eq T u (lift h d t0) H6))) k (sym_eq K k k0 H5))) H4)) 
-H3)))]) in (H3 (refl_equal T (THead k0 (lift h d t0) (lift h (s k0 d) 
-t1)))))))))))))) 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).(T_ind 
-(\lambda (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead (Bind b) u 
-t) (lift h d t0)) \to (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))))))))) 
-(\lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (eq T 
-(THead (Bind b) u t) (lift h d (TSort n)))).(let H0 \def (match H return 
-(\lambda (t0: T).(\lambda (_: (eq ? ? t0)).((eq T t0 (lift h d (TSort n))) 
-\to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TSort n) (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)))))))) with [refl_equal 
-\Rightarrow (\lambda (H0: (eq T (THead (Bind b) u t) (lift h d (TSort 
-n)))).(let H1 \def (eq_ind T (THead (Bind b) u t) (\lambda (e: T).(match e 
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (lift h d (TSort n)) 
-H0) in (False_ind (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TSort n) 
-(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))))) H1)))]) in 
-(H0 (refl_equal T (lift h d (TSort n))))))))) (\lambda (n: nat).(\lambda (h: 
-nat).(\lambda (d: nat).(\lambda (H: (eq T (THead (Bind b) 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 (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 (H0: (lt n d)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda 
-(t0: T).(eq T (THead (Bind b) u t) t0)) H (TLRef n) (lift_lref_lt n h d H0)) 
-in (let H2 \def (match H1 return (\lambda (t0: T).(\lambda (_: (eq ? ? 
-t0)).((eq T t0 (TLRef n)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq 
-T (TLRef n) (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)))))))) with [refl_equal \Rightarrow (\lambda (H1: (eq T (THead (Bind b) u 
-t) (TLRef n))).(let H2 \def (eq_ind T (THead (Bind b) u t) (\lambda (e: 
-T).(match e 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 (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))))) H2)))]) in 
-(H2 (refl_equal T (TLRef n)))))) (\lambda (H0: (le d n)).(let H1 \def (eq_ind 
-T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (THead (Bind b) u t) t0)) H 
-(TLRef (plus n h)) (lift_lref_ge n h d H0)) in (let H2 \def (match H1 return 
-(\lambda (t0: T).(\lambda (_: (eq ? ? t0)).((eq T t0 (TLRef (plus n h))) \to 
-(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (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)))))))) with [refl_equal 
-\Rightarrow (\lambda (H1: (eq T (THead (Bind b) u t) (TLRef (plus n 
-h)))).(let H2 \def (eq_ind T (THead (Bind b) u t) (\lambda (e: T).(match e 
-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 (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))))) H2)))]) in 
-(H2 (refl_equal T (TLRef (plus n h)))))))))))) (\lambda (k: K).(\lambda (t0: 
-T).(\lambda (_: ((\forall (h: nat).(\forall (d: nat).((eq T (THead (Bind b) u 
-t) (lift h d t0)) \to (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)))))))))).(\lambda (t1: T).(\lambda (_: ((\forall (h: nat).(\forall (d: 
-nat).((eq T (THead (Bind b) u t) (lift h d t1)) \to (ex3_2 T T (\lambda (y: 
-T).(\lambda (z: T).(eq T t1 (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 (h: nat).(\lambda (d: nat).(\lambda (H1: (eq T 
-(THead (Bind b) u t) (lift h d (THead k t0 t1)))).(let H2 \def (eq_ind T 
-(lift h d (THead k t0 t1)) (\lambda (t0: T).(eq T (THead (Bind b) u t) t0)) 
-H1 (THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) in 
-(let H3 \def (match H2 return (\lambda (t2: T).(\lambda (_: (eq ? ? t2)).((eq 
-T t2 (THead k (lift h d t0) (lift h (s k d) t1))) \to (ex3_2 T T (\lambda (y: 
-T).(\lambda (z: T).(eq T (THead k t0 t1) (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)))))))) with [refl_equal \Rightarrow (\lambda (H2: 
-(eq T (THead (Bind b) u t) (THead k (lift h d t0) (lift h (s k d) t1)))).(let 
-H3 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t) 
-\Rightarrow t])) (THead (Bind b) u t) (THead k (lift h d t0) (lift h (s k d) 
-t1)) H2) in ((let H4 \def (f_equal T T (\lambda (e: T).(match e return 
-(\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | 
-(THead _ t _) \Rightarrow t])) (THead (Bind b) u t) (THead k (lift h d t0) 
-(lift h (s k d) t1)) H2) in ((let H5 \def (f_equal T K (\lambda (e: T).(match 
-e return (\lambda (_: T).K) with [(TSort _) \Rightarrow (Bind b) | (TLRef _) 
-\Rightarrow (Bind b) | (THead k _ _) \Rightarrow k])) (THead (Bind b) u t) 
-(THead k (lift h d t0) (lift h (s k d) t1)) H2) in (eq_ind K (Bind b) 
-(\lambda (k: K).((eq T u (lift h d t0)) \to ((eq T t (lift h (s k d) t1)) \to 
-(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k t0 t1) (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 (H6: (eq T u 
-(lift h d t0))).(eq_ind T (lift h d t0) (\lambda (t2: T).((eq T t (lift h (s 
-(Bind b) d) t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead 
-(Bind b) t0 t1) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T 
-t2 (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) 
-z))))))) (\lambda (H7: (eq T t (lift h (s (Bind b) d) t1))).(eq_ind T (lift h 
-(s (Bind b) d) t1) (\lambda (t: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: 
-T).(eq T (THead (Bind b) t0 t1) (THead (Bind b) y z)))) (\lambda (y: 
-T).(\lambda (_: T).(eq T (lift h d t0) (lift h d y)))) (\lambda (_: 
-T).(\lambda (z: T).(eq T t (lift h (S d) z)))))) (ex3_2_intro T T (\lambda 
-(y: T).(\lambda (z: T).(eq T (THead (Bind b) t0 t1) (THead (Bind b) 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 (Bind b) d) t1) (lift h (S d) z)))) 
-t0 t1 (refl_equal T (THead (Bind b) t0 t1)) (refl_equal T (lift h d t0)) 
-(refl_equal T (lift h (S d) t1))) t (sym_eq T t (lift h (s (Bind b) d) t1) 
-H7))) u (sym_eq T u (lift h d t0) H6))) k H5)) H4)) H3)))]) in (H3 
-(refl_equal T (THead k (lift h d t0) (lift h (s k d) t1)))))))))))))) x)))).
-
-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).(T_ind 
-(\lambda (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead (Flat f) u 
-t) (lift h d t0)) \to (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))))))))) (\lambda 
-(n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (eq T (THead (Flat 
-f) u t) (lift h d (TSort n)))).(let H0 \def (match H return (\lambda (t0: 
-T).(\lambda (_: (eq ? ? t0)).((eq T t0 (lift h d (TSort n))) \to (ex3_2 T T 
-(\lambda (y: T).(\lambda (z: T).(eq T (TSort n) (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)))))))) with [refl_equal \Rightarrow 
-(\lambda (H0: (eq T (THead (Flat f) u t) (lift h d (TSort n)))).(let H1 \def 
-(eq_ind T (THead (Flat f) u t) (\lambda (e: T).(match e return (\lambda (_: 
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
-(THead _ _ _) \Rightarrow True])) I (lift h d (TSort n)) H0) in (False_ind 
-(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TSort n) (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))))) H1)))]) in (H0 (refl_equal T 
-(lift h d (TSort n))))))))) (\lambda (n: nat).(\lambda (h: nat).(\lambda (d: 
-nat).(\lambda (H: (eq T (THead (Flat f) 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 (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 (H0: (lt n d)).(let 
-H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (THead (Flat f) 
-u t) t0)) H (TLRef n) (lift_lref_lt n h d H0)) in (let H2 \def (match H1 
-return (\lambda (t0: T).(\lambda (_: (eq ? ? t0)).((eq T t0 (TLRef n)) \to 
-(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (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)))))))) with [refl_equal \Rightarrow 
-(\lambda (H1: (eq T (THead (Flat f) u t) (TLRef n))).(let H2 \def (eq_ind T 
-(THead (Flat f) u t) (\lambda (e: T).(match e 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 (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))))) H2)))]) in (H2 (refl_equal T (TLRef n)))))) 
-(\lambda (H0: (le d n)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda 
-(t0: T).(eq T (THead (Flat f) u t) t0)) H (TLRef (plus n h)) (lift_lref_ge n 
-h d H0)) in (let H2 \def (match H1 return (\lambda (t0: T).(\lambda (_: (eq ? 
-? t0)).((eq T t0 (TLRef (plus n h))) \to (ex3_2 T T (\lambda (y: T).(\lambda 
-(z: T).(eq T (TLRef n) (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)))))))) with [refl_equal \Rightarrow (\lambda (H1: (eq T (THead (Flat f) u 
-t) (TLRef (plus n h)))).(let H2 \def (eq_ind T (THead (Flat f) u t) (\lambda 
-(e: T).(match e 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 (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))))) H2)))]) in (H2 (refl_equal T (TLRef (plus n h)))))))))))) (\lambda (k: 
-K).(\lambda (t0: T).(\lambda (_: ((\forall (h: nat).(\forall (d: nat).((eq T 
-(THead (Flat f) u t) (lift h d t0)) \to (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)))))))))).(\lambda (t1: T).(\lambda (_: ((\forall (h: nat).(\forall (d: 
-nat).((eq T (THead (Flat f) u t) (lift h d t1)) \to (ex3_2 T T (\lambda (y: 
-T).(\lambda (z: T).(eq T t1 (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 (h: nat).(\lambda (d: nat).(\lambda (H1: (eq T 
-(THead (Flat f) u t) (lift h d (THead k t0 t1)))).(let H2 \def (eq_ind T 
-(lift h d (THead k t0 t1)) (\lambda (t0: T).(eq T (THead (Flat f) u t) t0)) 
-H1 (THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) in 
-(let H3 \def (match H2 return (\lambda (t2: T).(\lambda (_: (eq ? ? t2)).((eq 
-T t2 (THead k (lift h d t0) (lift h (s k d) t1))) \to (ex3_2 T T (\lambda (y: 
-T).(\lambda (z: T).(eq T (THead k t0 t1) (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)))))))) with [refl_equal \Rightarrow (\lambda (H2: (eq 
-T (THead (Flat f) u t) (THead k (lift h d t0) (lift h (s k d) t1)))).(let H3 
-\def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t) 
-\Rightarrow t])) (THead (Flat f) u t) (THead k (lift h d t0) (lift h (s k d) 
-t1)) H2) in ((let H4 \def (f_equal T T (\lambda (e: T).(match e return 
-(\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | 
-(THead _ t _) \Rightarrow t])) (THead (Flat f) u t) (THead k (lift h d t0) 
-(lift h (s k d) t1)) H2) in ((let H5 \def (f_equal T K (\lambda (e: T).(match 
-e return (\lambda (_: T).K) with [(TSort _) \Rightarrow (Flat f) | (TLRef _) 
-\Rightarrow (Flat f) | (THead k _ _) \Rightarrow k])) (THead (Flat f) u t) 
-(THead k (lift h d t0) (lift h (s k d) t1)) H2) in (eq_ind K (Flat f) 
-(\lambda (k: K).((eq T u (lift h d t0)) \to ((eq T t (lift h (s k d) t1)) \to 
-(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k t0 t1) (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 (H6: (eq T u 
-(lift h d t0))).(eq_ind T (lift h d t0) (\lambda (t2: T).((eq T t (lift h (s 
-(Flat f) d) t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead 
-(Flat f) t0 t1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T 
-t2 (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h d z))))))) 
-(\lambda (H7: (eq T t (lift h (s (Flat f) d) t1))).(eq_ind T (lift h (s (Flat 
-f) d) t1) (\lambda (t: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T 
-(THead (Flat f) t0 t1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: 
-T).(eq T (lift h d t0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T 
-t (lift h d z)))))) (ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T 
-(THead (Flat f) t0 t1) (THead (Flat f) 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 (Flat f) d) t1) (lift h d z)))) t0 t1 (refl_equal T (THead (Flat 
-f) t0 t1)) (refl_equal T (lift h d t0)) (refl_equal T (lift h d t1))) t 
-(sym_eq T t (lift h (s (Flat f) d) t1) H7))) u (sym_eq T u (lift h d t0) 
-H6))) k H5)) H4)) H3)))]) in (H3 (refl_equal T (THead k (lift h d t0) (lift h 
-(s k d) t1)))))))))))))) x)))).
-
-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 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 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 return (\lambda (_: T).K) with [(TSort _) 
-\Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) 
-(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 return (\lambda (_: T).T) with [(TSort 
-_) \Rightarrow v | (TLRef _) \Rightarrow v | (THead _ t _) \Rightarrow t])) 
-(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 return (\lambda (_: T).T) with [(TSort 
-_) \Rightarrow (THead k0 ((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) (\lambda (x: nat).(plus x h)) d t0) 
-((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) (\lambda (x: nat).(plus x h)) (s k0 d) t1)) | (TLRef _) \Rightarrow 
-(THead k0 ((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) (\lambda (x: nat).(plus x h)) d t0) ((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) (\lambda 
-(x: nat).(plus x h)) (s k0 d) t1)) | (THead _ _ t) \Rightarrow t])) (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 (k: 
-K).(\forall (v: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k v 
-(lift h d t1)) t1) \to (\forall (P: Prop).P)))))) H0 k0 H6) in (let H8 \def 
-(eq_ind T (lift h d (THead k0 t0 t1)) (\lambda (t: T).(eq T t 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))) (sym_equal T (THead k t0 t1) 
-(THead k (lift O d t0) (lift O (s k d) t1)) (sym_equal T (THead k (lift O d 
-t0) (lift O (s k d) t1)) (THead k t0 t1) (sym_equal T (THead k t0 t1) (THead 
-k (lift O d t0) (lift O (s k d) t1)) (f_equal3 K T T T THead k k t0 (lift O d 
-t0) t1 (lift O (s k d) t1) (refl_equal K k) (sym_eq T (lift O d t0) t0 (H d)) 
-(sym_eq T (lift O (s k d) t1) t1 (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_comm (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 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 (t: 
-T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t)) \to 
-(eq T t0 t)))))) \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 (t: T).(\forall 
-(h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t)) \to (eq T t0 
-t))))))).(\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 (t: T).(eq T t (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)) 
-(sym_equal T (THead (Bind b) x0 x1) (THead (Bind b) t t0) (sym_equal T (THead 
-(Bind b) t t0) (THead (Bind b) x0 x1) (sym_equal T (THead (Bind b) x0 x1) 
-(THead (Bind b) t t0) (f_equal3 K T T T THead (Bind b) (Bind b) x0 t x1 t0 
-(refl_equal K (Bind b)) (sym_eq T t x0 (H x0 h d H4)) (sym_eq T t0 x1 (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 (t: 
-T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t)) \to 
-(eq T t0 t))))))).(\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 (t: 
-T).(eq T t (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)) (sym_equal T (THead (Flat f) x0 x1) (THead (Flat f) t t0) 
-(sym_equal T (THead (Flat f) t t0) (THead (Flat f) x0 x1) (sym_equal T (THead 
-(Flat f) x0 x1) (THead (Flat f) t t0) (f_equal3 K T T T THead (Flat f) (Flat 
-f) x0 t x1 t0 (refl_equal K (Flat f)) (sym_eq T t x0 (H x0 h d H4)) (sym_eq T 
-t0 x1 (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) (plus_lt_compat_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_S_n (plus d2 h1) (plus n h1) (lt_le_S (plus d2 h1) (S 
-(plus n h1)) (le_lt_n_Sm (plus d2 h1) (plus n h1) (plus_le_compat 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_le_S (plus n h1) (plus (plus d2 h2) h1) 
-(plus_lt_compat_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 (n: 
-nat).(eq T (TLRef n) (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_plus_r d2 h2 n 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_plus_r d2 h2 n 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 (plus_le_compat d2 (minus n h2) h2 h2 
-(le_minus d2 n h2 H4) (le_n h2))))) n (le_plus_minus_sym h2 n 
-(le_trans_plus_r d2 h2 n 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 (t: T).(eq T t (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_S_n (S d1) (S d2) (lt_le_S (S d1) (S 
-(S d2)) (lt_n_S d1 (S d2) (le_lt_n_Sm 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 (t: T).(eq T t (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 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 (plus_le_compat 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 (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 (t1: 
-T).(\lambda (H0: ((\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 t1)) (lift (plus k 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_le_S (plus n k) (plus d k) 
-(plus_lt_compat_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_S_n e (plus n h) (lt_le_S e (S (plus n h)) 
-(le_lt_n_Sm e (plus n h) (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_S_n (plus d k) (plus n k) (lt_le_S (plus d k) (S 
-(plus n k)) (le_lt_n_Sm (plus d k) (plus n k) (plus_le_compat d n k k H1 
-(le_n k))))))))) (plus k d) (plus_comm 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 (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 (t1: T).(\lambda (H0: ((\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 t1)) (lift k 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).
-
-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_S_n d (plus n h) 
-(le_n_S d (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_equal 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_equal 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_equal 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 (m: nat).(lt m 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 (m: nat).(lt m 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 (m: nat).(lt m 
-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)).(let H0 \def (match H return (\lambda (n: 
-nat).(\lambda (_: (le ? n)).((eq nat n O) \to (eq nat (wadd f w m) (f m))))) 
-with [le_n \Rightarrow (\lambda (H0: (eq nat (S m) O)).(let H1 \def (eq_ind 
-nat (S m) (\lambda (e: nat).(match e return (\lambda (_: nat).Prop) with [O 
-\Rightarrow False | (S _) \Rightarrow True])) I O H0) in (False_ind (eq nat 
-(wadd f w m) (f m)) H1))) | (le_S m0 H0) \Rightarrow (\lambda (H1: (eq nat (S 
-m0) O)).((let H2 \def (eq_ind nat (S m0) (\lambda (e: nat).(match e return 
-(\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) 
-I O H1) in (False_ind ((le (S m) m0) \to (eq nat (wadd f w m) (f m))) H2)) 
-H0))]) in (H0 (refl_equal nat O))))) (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)))))).
-
-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 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 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 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.
-
-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 (trans PNil i)))) (\lambda (h: nat).(\lambda (d: 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 h d t) (TLRef (match (blt (trans p i) d) with [true 
-\Rightarrow (trans p i) | false \Rightarrow (plus (trans p i) h)])))) 
-(refl_equal T (TLRef (match (blt (trans p i) d) with [true \Rightarrow (trans 
-p i) | false \Rightarrow (plus (trans p i) h)]))) (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) (lift1 PNil u) (lift1 (Ss PNil) t))))) (\lambda (h: 
-nat).(\lambda (d: 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 
-h d t0) (THead (Bind b) (lift h d (lift1 p u)) (lift h (S d) (lift1 (Ss p) 
-t))))) (eq_ind_r T (THead (Bind b) (lift h d (lift1 p u)) (lift h (S d) 
-(lift1 (Ss p) t))) (\lambda (t0: T).(eq T t0 (THead (Bind b) (lift h d (lift1 
-p u)) (lift h (S d) (lift1 (Ss p) t))))) (refl_equal T (THead (Bind b) (lift 
-h d (lift1 p u)) (lift h (S d) (lift1 (Ss p) t)))) (lift h d (THead (Bind b) 
-(lift1 p u) (lift1 (Ss p) t))) (lift_bind b (lift1 p u) (lift1 (Ss p) t) h 
-d)) (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) (lift1 PNil u) (lift1 PNil t))))) (\lambda (h: nat).(\lambda 
-(d: 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 h d t0) (THead (Flat f) (lift h 
-d (lift1 p u)) (lift h d (lift1 p t))))) (eq_ind_r T (THead (Flat f) (lift h 
-d (lift1 p u)) (lift h d (lift1 p t))) (\lambda (t0: T).(eq T t0 (THead (Flat 
-f) (lift h d (lift1 p u)) (lift h d (lift1 p t))))) (refl_equal T (THead 
-(Flat f) (lift h d (lift1 p u)) (lift h d (lift1 p t)))) (lift h d (THead 
-(Flat f) (lift1 p u) (lift1 p t))) (lift_flat f (lift1 p u) (lift1 p t) h d)) 
-(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)))).
-
-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)).
-
-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))))).
-
-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)).
-
-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)))))))).
-
-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)) (and3 (eq C x (CSort n)) (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 (CSort n)) (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 (CSort n)) (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 (CSort n)) (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 return (\lambda (_: 
-C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow 
-True])) I (CSort n) H3) in (False_ind (and3 (eq C e (CSort n)) (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 (CSort n)) (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 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) (CSort n)) (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)) (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)) ((eq nat y 
-O) \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 O) \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 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 O) \to (eq C c e0))))).(\lambda (u: 
-T).(\lambda (H4: (eq nat (S d) O)).(\lambda (H5: (eq nat h O)).(let H6 \def 
-(f_equal nat nat (\lambda (e1: nat).e1) h O H5) in (let H7 \def (eq_ind nat h 
-(\lambda (n: nat).((eq nat (r k d) O) \to ((eq nat n O) \to (eq C c e0)))) H3 
-O H6) in (let H8 \def (eq_ind nat h (\lambda (n: nat).(drop n (r k d) c e0)) 
-H2 O H6) in (eq_ind_r nat O (\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 return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S 
-_) \Rightarrow True])) I O H4) in (False_ind (eq C (CHead c k (lift O (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)) (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)) ((eq C y (CHead c k u)) \to (drop (r k h) O 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)) ((eq nat y0 O) \to ((eq C y (CHead c k u)) \to (drop (r 
-k h) O 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) O c c1)))))))) (\lambda (c0: C).(\lambda (H3: (eq nat O (S h))).(\lambda 
-(_: (eq nat O O)).(\lambda (_: (eq C c0 (CHead c k u))).(let H6 \def (match 
-H3 return (\lambda (n: nat).(\lambda (_: (eq ? ? n)).((eq nat n (S h)) \to 
-(drop (r k h) O c c0)))) with [refl_equal \Rightarrow (\lambda (H2: (eq nat O 
-(S h))).(let H3 \def (eq_ind nat O (\lambda (e: nat).(match e return (\lambda 
-(_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S h) 
-H2) in (False_ind (drop (r k h) O c c0) H3)))]) in (H6 (refl_equal nat (S 
-h)))))))) (\lambda (k0: K).(\lambda (h0: nat).(\lambda (c0: C).(\lambda (e: 
-C).(\lambda (H3: (drop (r k0 h0) O c0 e)).(\lambda (_: (((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 (match H5 return (\lambda (n: nat).(\lambda (_: (eq ? ? n)).((eq nat n 
-(S h)) \to (drop (r k h) O c e)))) with [refl_equal \Rightarrow (\lambda (H4: 
-(eq nat (S h0) (S h))).(let H5 \def (f_equal nat nat (\lambda (e0: 
-nat).(match e0 return (\lambda (_: nat).nat) with [O \Rightarrow h0 | (S n) 
-\Rightarrow n])) (S h0) (S h) H4) in (eq_ind nat h (\lambda (_: nat).(drop (r 
-k h) O c e)) (let H6 \def (match H7 return (\lambda (c0: C).(\lambda (_: (eq 
-? ? c0)).((eq C c0 (CHead c k u)) \to (drop (r k h) O c e)))) with 
-[refl_equal \Rightarrow (\lambda (H4: (eq C (CHead c0 k0 u0) (CHead c k 
-u))).(let H6 \def (f_equal C T (\lambda (e0: C).(match e0 return (\lambda (_: 
-C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead 
-c0 k0 u0) (CHead c k u) H4) in ((let H7 \def (f_equal C K (\lambda (e0: 
-C).(match e0 return (\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | 
-(CHead _ k _) \Rightarrow k])) (CHead c0 k0 u0) (CHead c k u) H4) in ((let H8 
-\def (f_equal C C (\lambda (e0: C).(match e0 return (\lambda (_: C).C) with 
-[(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k0 u0) 
-(CHead c k u) H4) in (eq_ind C c (\lambda (_: C).((eq K k0 k) \to ((eq T u0 
-u) \to (drop (r k h) O c e)))) (\lambda (H9: (eq K k0 k)).(eq_ind K k 
-(\lambda (_: K).((eq T u0 u) \to (drop (r k h) O c e))) (\lambda (H10: (eq T 
-u0 u)).(eq_ind T u (\lambda (_: T).(drop (r k h) O c e)) (eq_ind nat h0 
-(\lambda (n: nat).(drop (r k n) O c e)) (eq_ind C c0 (\lambda (c: C).(drop (r 
-k h0) O c e)) (eq_ind K k0 (\lambda (k: K).(drop (r k h0) O c0 e)) H3 k H9) c 
-H8) h H5) u0 (sym_eq T u0 u H10))) k0 (sym_eq K k0 k H9))) c0 (sym_eq C c0 c 
-H8))) H7)) H6)))]) in (H6 (refl_equal C (CHead c k u)))) h0 (sym_eq nat h0 h 
-H5))))]) in (H8 (refl_equal nat (S h)))))))))))))) (\lambda (k0: K).(\lambda 
-(h0: nat).(\lambda (d: nat).(\lambda (c0: C).(\lambda (e: C).(\lambda (_: 
-(drop h0 (r k0 d) c0 e)).(\lambda (_: (((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) O c e)))))).(\lambda 
-(u0: T).(\lambda (_: (eq nat h0 (S h))).(\lambda (H6: (eq nat (S d) 
-O)).(\lambda (_: (eq C (CHead c0 k0 (lift h0 (r k0 d) u0)) (CHead c k 
-u))).(let H8 \def (match H6 return (\lambda (n: nat).(\lambda (_: (eq ? ? 
-n)).((eq nat n O) \to (drop (r k h) O c (CHead e k0 u0))))) with [refl_equal 
-\Rightarrow (\lambda (H4: (eq nat (S d) O)).(let H5 \def (eq_ind nat (S d) 
-(\lambda (e0: nat).(match e0 return (\lambda (_: nat).Prop) with [O 
-\Rightarrow False | (S _) \Rightarrow True])) I O H4) in (False_ind (drop (r 
-k h) O c (CHead e k0 u0)) H5)))]) in (H8 (refl_equal nat O)))))))))))))) 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))).(let H0 
-\def (match H return (\lambda (n: nat).(\lambda (n0: nat).(\lambda (c0: 
-C).(\lambda (c1: C).(\lambda (_: (drop n n0 c0 c1)).((eq nat n h) \to ((eq 
-nat n0 (S d)) \to ((eq C c0 x) \to ((eq C c1 (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)))))))))))) with [(drop_refl c0) \Rightarrow (\lambda 
-(H0: (eq nat O h)).(\lambda (H1: (eq nat O (S d))).(\lambda (H2: (eq C c0 
-x)).(\lambda (H3: (eq C c0 (CHead c k u))).(eq_ind nat O (\lambda (n: 
-nat).((eq nat O (S d)) \to ((eq C c0 x) \to ((eq C c0 (CHead c k u)) \to (ex2 
-C (\lambda (e: C).(eq C x (CHead e k (lift n (r k d) u)))) (\lambda (e: 
-C).(drop n (r k d) e c))))))) (\lambda (H4: (eq nat O (S d))).(let H5 \def 
-(eq_ind nat O (\lambda (e: nat).(match e return (\lambda (_: nat).Prop) with 
-[O \Rightarrow True | (S _) \Rightarrow False])) I (S d) H4) in (False_ind 
-((eq C c0 x) \to ((eq C c0 (CHead c k u)) \to (ex2 C (\lambda (e: C).(eq C x 
-(CHead e k (lift O (r k d) u)))) (\lambda (e: C).(drop O (r k d) e c))))) 
-H5))) h H0 H1 H2 H3))))) | (drop_drop k0 h0 c0 e H0 u0) \Rightarrow (\lambda 
-(H1: (eq nat (S h0) h)).(\lambda (H2: (eq nat O (S d))).(\lambda (H3: (eq C 
-(CHead c0 k0 u0) x)).(\lambda (H4: (eq C e (CHead c k u))).(eq_ind nat (S h0) 
-(\lambda (n: nat).((eq nat O (S d)) \to ((eq C (CHead c0 k0 u0) x) \to ((eq C 
-e (CHead c k u)) \to ((drop (r k0 h0) O c0 e) \to (ex2 C (\lambda (e0: C).(eq 
-C x (CHead e0 k (lift n (r k d) u)))) (\lambda (e0: C).(drop n (r k d) e0 
-c)))))))) (\lambda (H5: (eq nat O (S d))).(let H6 \def (eq_ind nat O (\lambda 
-(e0: nat).(match e0 return (\lambda (_: nat).Prop) with [O \Rightarrow True | 
-(S _) \Rightarrow False])) I (S d) H5) in (False_ind ((eq C (CHead c0 k0 u0) 
-x) \to ((eq C e (CHead c k u)) \to ((drop (r k0 h0) O c0 e) \to (ex2 C 
-(\lambda (e0: C).(eq C x (CHead e0 k (lift (S h0) (r k d) u)))) (\lambda (e0: 
-C).(drop (S h0) (r k d) e0 c)))))) H6))) h H1 H2 H3 H4 H0))))) | (drop_skip 
-k0 h0 d0 c0 e H0 u0) \Rightarrow (\lambda (H1: (eq nat h0 h)).(\lambda (H2: 
-(eq nat (S d0) (S d))).(\lambda (H3: (eq C (CHead c0 k0 (lift h0 (r k0 d0) 
-u0)) x)).(\lambda (H4: (eq C (CHead e k0 u0) (CHead c k u))).(eq_ind nat h 
-(\lambda (n: nat).((eq nat (S d0) (S d)) \to ((eq C (CHead c0 k0 (lift n (r 
-k0 d0) u0)) x) \to ((eq C (CHead e k0 u0) (CHead c k u)) \to ((drop n (r k0 
-d0) c0 e) \to (ex2 C (\lambda (e0: C).(eq C x (CHead e0 k (lift h (r k d) 
-u)))) (\lambda (e0: C).(drop h (r k d) e0 c)))))))) (\lambda (H5: (eq nat (S 
-d0) (S d))).(let H6 \def (f_equal nat nat (\lambda (e0: nat).(match e0 return 
-(\lambda (_: nat).nat) with [O \Rightarrow d0 | (S n) \Rightarrow n])) (S d0) 
-(S d) H5) in (eq_ind nat d (\lambda (n: nat).((eq C (CHead c0 k0 (lift h (r 
-k0 n) u0)) x) \to ((eq C (CHead e k0 u0) (CHead c k u)) \to ((drop h (r k0 n) 
-c0 e) \to (ex2 C (\lambda (e0: C).(eq C x (CHead e0 k (lift h (r k d) u)))) 
-(\lambda (e0: C).(drop h (r k d) e0 c))))))) (\lambda (H7: (eq C (CHead c0 k0 
-(lift h (r k0 d) u0)) x)).(eq_ind C (CHead c0 k0 (lift h (r k0 d) u0)) 
-(\lambda (c1: C).((eq C (CHead e k0 u0) (CHead c k u)) \to ((drop h (r k0 d) 
-c0 e) \to (ex2 C (\lambda (e0: C).(eq C c1 (CHead e0 k (lift h (r k d) u)))) 
-(\lambda (e0: C).(drop h (r k d) e0 c)))))) (\lambda (H8: (eq C (CHead e k0 
-u0) (CHead c k u))).(let H9 \def (f_equal C T (\lambda (e0: C).(match e0 
-return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) 
-\Rightarrow t])) (CHead e k0 u0) (CHead c k u) H8) in ((let H10 \def (f_equal 
-C K (\lambda (e0: C).(match e0 return (\lambda (_: C).K) with [(CSort _) 
-\Rightarrow k0 | (CHead _ k _) \Rightarrow k])) (CHead e k0 u0) (CHead c k u) 
-H8) in ((let H11 \def (f_equal C C (\lambda (e0: C).(match e0 return (\lambda 
-(_: C).C) with [(CSort _) \Rightarrow e | (CHead c _ _) \Rightarrow c])) 
-(CHead e k0 u0) (CHead c k u) H8) in (eq_ind C c (\lambda (c1: C).((eq K k0 
-k) \to ((eq T u0 u) \to ((drop h (r k0 d) c0 c1) \to (ex2 C (\lambda (e0: 
-C).(eq C (CHead c0 k0 (lift h (r k0 d) u0)) (CHead e0 k (lift h (r k d) u)))) 
-(\lambda (e0: C).(drop h (r k d) e0 c))))))) (\lambda (H12: (eq K k0 
-k)).(eq_ind K k (\lambda (k1: K).((eq T u0 u) \to ((drop h (r k1 d) c0 c) \to 
-(ex2 C (\lambda (e0: C).(eq C (CHead c0 k1 (lift h (r k1 d) u0)) (CHead e0 k 
-(lift h (r k d) u)))) (\lambda (e0: C).(drop h (r k d) e0 c)))))) (\lambda 
-(H13: (eq T u0 u)).(eq_ind T u (\lambda (t: T).((drop h (r k d) c0 c) \to 
-(ex2 C (\lambda (e0: C).(eq C (CHead c0 k (lift h (r k d) t)) (CHead e0 k 
-(lift h (r k d) u)))) (\lambda (e0: C).(drop h (r k d) e0 c))))) (\lambda 
-(H14: (drop h (r k d) c0 c)).(let H15 \def (eq_ind T u0 (\lambda (t: T).(eq C 
-(CHead c0 k0 (lift h (r k0 d) t)) x)) H7 u H13) in (let H16 \def (eq_ind K k0 
-(\lambda (k: K).(eq C (CHead c0 k (lift h (r k d) u)) x)) H15 k H12) in (let 
-H17 \def (eq_ind_r C x (\lambda (c0: C).(drop h (S d) c0 (CHead c k u))) H 
-(CHead c0 k (lift h (r k d) u)) H16) in (ex_intro2 C (\lambda (e0: C).(eq C 
-(CHead c0 k (lift h (r k d) u)) (CHead e0 k (lift h (r k d) u)))) (\lambda 
-(e0: C).(drop h (r k d) e0 c)) c0 (refl_equal C (CHead c0 k (lift h (r k d) 
-u))) H14))))) u0 (sym_eq T u0 u H13))) k0 (sym_eq K k0 k H12))) e (sym_eq C e 
-c H11))) H10)) H9))) x H7)) d0 (sym_eq nat d0 d H6)))) h0 (sym_eq nat h0 h 
-H1) H2 H3 H4 H0)))))]) in (H0 (refl_equal nat h) (refl_equal nat (S d)) 
-(refl_equal C x) (refl_equal C (CHead c k u)))))))))).
-
-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)).(let H0 
-\def (match H return (\lambda (n: nat).(\lambda (n0: nat).(\lambda (c0: 
-C).(\lambda (c1: C).(\lambda (_: (drop n n0 c0 c1)).((eq nat n h) \to ((eq 
-nat n0 (S d)) \to ((eq C c0 (CHead c k u)) \to ((eq C c1 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))))))))))))) with [(drop_refl c0) \Rightarrow (\lambda 
-(H0: (eq nat O h)).(\lambda (H1: (eq nat O (S d))).(\lambda (H2: (eq C c0 
-(CHead c k u))).(\lambda (H3: (eq C c0 x)).(eq_ind nat O (\lambda (n: 
-nat).((eq nat O (S d)) \to ((eq C c0 (CHead c k u)) \to ((eq C c0 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 n (r k d) v)))) (\lambda (e: C).(\lambda 
-(_: T).(drop n (r k d) c e)))))))) (\lambda (H4: (eq nat O (S d))).(let H5 
-\def (eq_ind nat O (\lambda (e: nat).(match e return (\lambda (_: nat).Prop) 
-with [O \Rightarrow True | (S _) \Rightarrow False])) I (S d) H4) in 
-(False_ind ((eq C c0 (CHead c k u)) \to ((eq C c0 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 O (r k d) v)))) (\lambda (e: C).(\lambda (_: T).(drop O (r k 
-d) c e)))))) H5))) h H0 H1 H2 H3))))) | (drop_drop k0 h0 c0 e H0 u0) 
-\Rightarrow (\lambda (H1: (eq nat (S h0) h)).(\lambda (H2: (eq nat O (S 
-d))).(\lambda (H3: (eq C (CHead c0 k0 u0) (CHead c k u))).(\lambda (H4: (eq C 
-e x)).(eq_ind nat (S h0) (\lambda (n: nat).((eq nat O (S d)) \to ((eq C 
-(CHead c0 k0 u0) (CHead c k u)) \to ((eq C e x) \to ((drop (r k0 h0) O c0 e) 
-\to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C x (CHead e0 k v)))) 
-(\lambda (_: C).(\lambda (v: T).(eq T u (lift n (r k d) v)))) (\lambda (e0: 
-C).(\lambda (_: T).(drop n (r k d) c e0))))))))) (\lambda (H5: (eq nat O (S 
-d))).(let H6 \def (eq_ind nat O (\lambda (e0: nat).(match e0 return (\lambda 
-(_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S d) 
-H5) in (False_ind ((eq C (CHead c0 k0 u0) (CHead c k u)) \to ((eq C e x) \to 
-((drop (r k0 h0) O c0 e) \to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq 
-C x (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))))))) 
-H6))) h H1 H2 H3 H4 H0))))) | (drop_skip k0 h0 d0 c0 e H0 u0) \Rightarrow 
-(\lambda (H1: (eq nat h0 h)).(\lambda (H2: (eq nat (S d0) (S d))).(\lambda 
-(H3: (eq C (CHead c0 k0 (lift h0 (r k0 d0) u0)) (CHead c k u))).(\lambda (H4: 
-(eq C (CHead e k0 u0) x)).(eq_ind nat h (\lambda (n: nat).((eq nat (S d0) (S 
-d)) \to ((eq C (CHead c0 k0 (lift n (r k0 d0) u0)) (CHead c k u)) \to ((eq C 
-(CHead e k0 u0) x) \to ((drop n (r k0 d0) c0 e) \to (ex3_2 C T (\lambda (e0: 
-C).(\lambda (v: T).(eq C x (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: 
-T).(eq T u (lift h (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r 
-k d) c e0))))))))) (\lambda (H5: (eq nat (S d0) (S d))).(let H6 \def (f_equal 
-nat nat (\lambda (e0: nat).(match e0 return (\lambda (_: nat).nat) with [O 
-\Rightarrow d0 | (S n) \Rightarrow n])) (S d0) (S d) H5) in (eq_ind nat d 
-(\lambda (n: nat).((eq C (CHead c0 k0 (lift h (r k0 n) u0)) (CHead c k u)) 
-\to ((eq C (CHead e k0 u0) x) \to ((drop h (r k0 n) c0 e) \to (ex3_2 C T 
-(\lambda (e0: C).(\lambda (v: T).(eq C x (CHead e0 k v)))) (\lambda (_: 
-C).(\lambda (v: T).(eq T u (lift h (r k d) v)))) (\lambda (e0: C).(\lambda 
-(_: T).(drop h (r k d) c e0)))))))) (\lambda (H7: (eq C (CHead c0 k0 (lift h 
-(r k0 d) u0)) (CHead c k u))).(let H8 \def (f_equal C T (\lambda (e0: 
-C).(match e0 return (\lambda (_: C).T) with [(CSort _) \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 u t0) 
-\Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) 
-(\lambda (x: nat).(plus x h)) (r k0 d) u0) | (CHead _ _ t) \Rightarrow t])) 
-(CHead c0 k0 (lift h (r k0 d) u0)) (CHead c k u) H7) in ((let H9 \def 
-(f_equal C K (\lambda (e0: C).(match e0 return (\lambda (_: C).K) with 
-[(CSort _) \Rightarrow k0 | (CHead _ k _) \Rightarrow k])) (CHead c0 k0 (lift 
-h (r k0 d) u0)) (CHead c k u) H7) in ((let H10 \def (f_equal C C (\lambda 
-(e0: C).(match e0 return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | 
-(CHead c _ _) \Rightarrow c])) (CHead c0 k0 (lift h (r k0 d) u0)) (CHead c k 
-u) H7) in (eq_ind C c (\lambda (c1: C).((eq K k0 k) \to ((eq T (lift h (r k0 
-d) u0) u) \to ((eq C (CHead e k0 u0) x) \to ((drop h (r k0 d) c1 e) \to 
-(ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C x (CHead e0 k v)))) 
-(\lambda (_: C).(\lambda (v: T).(eq T u (lift h (r k d) v)))) (\lambda (e0: 
-C).(\lambda (_: T).(drop h (r k d) c e0))))))))) (\lambda (H11: (eq K k0 
-k)).(eq_ind K k (\lambda (k1: K).((eq T (lift h (r k1 d) u0) u) \to ((eq C 
-(CHead e k1 u0) x) \to ((drop h (r k1 d) c e) \to (ex3_2 C T (\lambda (e0: 
-C).(\lambda (v: T).(eq C x (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: 
-T).(eq T u (lift h (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r 
-k d) c e0)))))))) (\lambda (H12: (eq T (lift h (r k d) u0) u)).(eq_ind T 
-(lift h (r k d) u0) (\lambda (t: T).((eq C (CHead e k u0) x) \to ((drop h (r 
-k d) c e) \to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C x (CHead e0 k 
-v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k d) v)))) (\lambda 
-(e0: C).(\lambda (_: T).(drop h (r k d) c e0))))))) (\lambda (H13: (eq C 
-(CHead e k u0) x)).(eq_ind C (CHead e k u0) (\lambda (c1: C).((drop h (r k d) 
-c e) \to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C c1 (CHead e0 k 
-v)))) (\lambda (_: C).(\lambda (v: T).(eq T (lift h (r k d) u0) (lift h (r k 
-d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r k d) c e0)))))) (\lambda 
-(H14: (drop h (r k d) c e)).(let H15 \def (eq_ind_r T u (\lambda (t: T).(drop 
-h (S d) (CHead c k t) x)) H (lift h (r k d) u0) H12) in (let H16 \def 
-(eq_ind_r C x (\lambda (c0: C).(drop h (S d) (CHead c k (lift h (r k d) u0)) 
-c0)) H15 (CHead e k u0) H13) in (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 h (r k d) u0) (lift h (r k d) v)))) (\lambda (e0: C).(\lambda 
-(_: T).(drop h (r k d) c e0))) e u0 (refl_equal C (CHead e k u0)) (refl_equal 
-T (lift h (r k d) u0)) H14)))) x H13)) u H12)) k0 (sym_eq K k0 k H11))) c0 
-(sym_eq C c0 c H10))) H9)) H8))) d0 (sym_eq nat d0 d H6)))) h0 (sym_eq nat h0 
-h H1) H2 H3 H4 H0)))))]) in (H0 (refl_equal nat h) (refl_equal nat (S d)) 
-(refl_equal C (CHead c k u)) (refl_equal C x))))))))).
-
-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 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 return (\lambda (_: C).C) with 
-[(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (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 return (\lambda (_: 
-C).K) with [(CSort _) \Rightarrow k | (CHead _ k _) \Rightarrow k])) (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 return 
-(\lambda (_: C).T) with [(CSort _) \Rightarrow t | (CHead _ _ t) \Rightarrow 
-t])) (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 (h: nat).((drop h n (CHead c2 k t) 
-c) \to (\forall (k0: K).(\forall (u: T).(drop h n (CTail k0 u (CHead c2 k t)) 
-(CTail k0 u 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 (t: T).(\forall (h: nat).((drop h n (CHead 
-c2 k t) (CHead x0 k x1)) \to (\forall (k0: K).(\forall (u: T).(drop h n 
-(CTail k0 u (CHead c2 k t)) (CTail k0 u (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 (n: nat).(eq nat n O)) H2 O H5) in (let H8 \def (eq_ind nat d 
-(\lambda (n: nat).(eq nat n 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 (c: C).(\forall (h: nat).((drop h n (CHead c0 k t) c) \to (\forall 
-(x2: C).((drop h n (CHead c0 k t) x2) \to (eq C c x2)))))) 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 (t: T).(\forall (h: nat).((drop h n 
-(CHead c0 k t) (CHead x4 k x5)) \to (\forall (x2: C).((drop h n (CHead c0 k 
-t) x2) \to (eq C (CHead x4 k x5) x2)))))) H9 (lift h (r k n) x5) H7) in (let 
-H11 \def (eq_ind T t (\lambda (t: T).(eq T t (lift h (r k n) x3))) H4 (lift h 
-(r k n) x5) H7) in (let H12 \def (eq_ind T x5 (\lambda (t: T).(\forall (h0: 
-nat).((drop h0 n (CHead c0 k (lift h (r k n) t)) (CHead x4 k t)) \to (\forall 
-(x2: C).((drop h0 n (CHead c0 k (lift h (r k n) t)) x2) \to (eq C (CHead x4 k 
-t) x2)))))) 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))) (sym_equal C (CHead x0 k x3) 
-(CHead x4 k x3) (sym_equal C (CHead x4 k x3) (CHead x0 k x3) (sym_equal C 
-(CHead x0 k x3) (CHead x4 k x3) (f_equal3 C K T C CHead x0 x4 k k x3 x3 (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 (c: C).(drop h (S 
-(plus O d)) c 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 
-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 
-(c: C).(drop h d c e)) H0 a (drop_gen_refl c a H)) in (let H3 \def (match H1 
-return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n O) \to (drop 
-(minus O h) O e a)))) with [le_n \Rightarrow (\lambda (H: (eq nat (plus d h) 
-O)).(let H3 \def (f_equal nat nat (\lambda (e0: nat).e0) (plus d h) O H) in 
-(eq_ind nat (plus d h) (\lambda (n: nat).(drop (minus n h) n e a)) (eq_ind_r 
-nat O (\lambda (n: nat).(drop (minus n h) n e a)) (and_ind (eq nat d O) (eq 
-nat h O) (drop O O e a) (\lambda (H0: (eq nat d O)).(\lambda (H1: (eq nat h 
-O)).(let H2 \def (eq_ind nat d (\lambda (n: nat).(drop h n a e)) H2 O H0) in 
-(let H4 \def (eq_ind nat h (\lambda (n: nat).(drop n O a e)) H2 O H1) in 
-(eq_ind C a (\lambda (c: C).(drop O O c a)) (drop_refl a) e (drop_gen_refl a 
-e H4)))))) (plus_O d h H3)) (plus d h) H3) O H3))) | (le_S m H) \Rightarrow 
-(\lambda (H2: (eq nat (S m) O)).((let H0 \def (eq_ind nat (S m) (\lambda (e0: 
-nat).(match e0 return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S 
-_) \Rightarrow True])) I O H2) in (False_ind ((le (plus d h) m) \to (drop 
-(minus O h) O e a)) H0)) H))]) in (H3 (refl_equal nat O)))))))))))) (\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 (n: nat).(le (plus n h) (S i0))) H2 O 
-H5) in (let H10 \def (eq_ind nat h (\lambda (n: nat).(le (plus O n) (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 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 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)).((match k return (\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)))))) with [(Bind b) 
-\Rightarrow (\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)))) | (Flat f) \Rightarrow 
-(\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))))]) (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)).(let H2 \def 
-(match H return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n O) \to 
-(ex2 C (\lambda (e1: C).(drop (S i0) O c1 e1)) (\lambda (e1: C).(drop h 
-(minus O (S i0)) e1 e2)))))) with [le_n \Rightarrow (\lambda (H2: (eq nat (S 
-i0) O)).(let H3 \def (eq_ind nat (S i0) (\lambda (e: nat).(match e return 
-(\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) 
-I O 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))) H3))) | (le_S m H2) \Rightarrow 
-(\lambda (H3: (eq nat (S m) O)).((let H4 \def (eq_ind nat (S m) (\lambda (e: 
-nat).(match e return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S 
-_) \Rightarrow True])) I O H3) in (False_ind ((le (S i0) m) \to (ex2 C 
-(\lambda (e1: C).(drop (S i0) O c1 e1)) (\lambda (e1: C).(drop h (minus O (S 
-i0)) e1 e2)))) H4)) H2))]) in (H2 (refl_equal nat O)))))))))) (\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 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 H2 \def (match H1 return (\lambda (n: nat).(\lambda (_: (le ? 
-n)).((eq nat n O) \to (drop (plus O h) O c1 c2)))) with [le_n \Rightarrow 
-(\lambda (H0: (eq nat d O)).(eq_ind nat O (\lambda (_: nat).(drop (plus O h) 
-O c1 c2)) (let H2 \def (eq_ind nat d (\lambda (n: nat).(le n O)) H1 O H0) in 
-(let H3 \def (eq_ind nat d (\lambda (n: nat).(drop h n c1 c2)) H O H0) in 
-H3)) d (sym_eq nat d O H0))) | (le_S m H0) \Rightarrow (\lambda (H2: (eq nat 
-(S m) O)).((let H1 \def (eq_ind nat (S m) (\lambda (e: nat).(match e return 
-(\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) 
-I O H2) in (False_ind ((le d m) \to (drop (plus O h) O c1 c2)) H1)) H0))]) in 
-(H2 (refl_equal nat O))) 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 (n: nat).(le n (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 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 (h: nat).((drop h d0 (CHead c2 k 
-t) c) \to (\forall (e2: C).((drop (S i0) O c e2) \to ((le d0 (S i0)) \to 
-(drop (S (plus i0 h)) O (CHead c2 k t) e2))))))) 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 (t: T).(\forall (h: 
-nat).((drop h d0 (CHead c2 k t) (CHead x0 k x1)) \to (\forall (e2: C).((drop 
-(S i0) O (CHead x0 k x1) e2) \to ((le d0 (S i0)) \to (drop (S (plus i0 h)) O 
-(CHead c2 k t) e2))))))) 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).
-
-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 ctrans:
- PList \to (nat \to (T \to T))
-\def
- let rec ctrans (hds: PList) on hds: (nat \to (T \to T)) \def (\lambda (i: 
-nat).(\lambda (t: T).(match hds with [PNil \Rightarrow t | (PCons h d hds0) 
-\Rightarrow (let j \def (trans hds0 i) in (let u \def (ctrans hds0 i t) in 
-(match (blt j d) with [true \Rightarrow (lift h (minus d (S j)) u) | false 
-\Rightarrow u])))]))) in ctrans.
-
-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 H0 \def (match H 
-return (\lambda (p: PList).(\lambda (c0: C).(\lambda (c1: C).(\lambda (_: 
-(drop1 p c0 c1)).((eq PList p PNil) \to ((eq C c0 c) \to ((eq C c1 e) \to 
-(drop1 PNil (CHead c (Bind b) u) (CHead e (Bind b) u))))))))) with 
-[(drop1_nil c0) \Rightarrow (\lambda (_: (eq PList PNil PNil)).(\lambda (H1: 
-(eq C c0 c)).(\lambda (H2: (eq C c0 e)).(eq_ind C c (\lambda (c1: C).((eq C 
-c1 e) \to (drop1 PNil (CHead c (Bind b) u) (CHead e (Bind b) u)))) (\lambda 
-(H3: (eq C c e)).(eq_ind C e (\lambda (c: C).(drop1 PNil (CHead c (Bind b) u) 
-(CHead e (Bind b) u))) (drop1_nil (CHead e (Bind b) u)) c (sym_eq C c e H3))) 
-c0 (sym_eq C c0 c H1) H2)))) | (drop1_cons c1 c2 h d H0 c3 hds H1) 
-\Rightarrow (\lambda (H2: (eq PList (PCons h d hds) PNil)).(\lambda (H3: (eq 
-C c1 c)).(\lambda (H4: (eq C c3 e)).((let H5 \def (eq_ind PList (PCons h d 
-hds) (\lambda (e0: PList).(match e0 return (\lambda (_: PList).Prop) with 
-[PNil \Rightarrow False | (PCons _ _ _) \Rightarrow True])) I PNil H2) in 
-(False_ind ((eq C c1 c) \to ((eq C c3 e) \to ((drop h d c1 c2) \to ((drop1 
-hds c2 c3) \to (drop1 PNil (CHead c (Bind b) u) (CHead e (Bind b) u)))))) 
-H5)) H3 H4 H0 H1))))]) in (H0 (refl_equal PList PNil) (refl_equal C c) 
-(refl_equal C e)))))) (\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 H1 \def (match H0 return (\lambda (p0: PList).(\lambda (c0: 
-C).(\lambda (c1: C).(\lambda (_: (drop1 p0 c0 c1)).((eq PList p0 (PCons n n0 
-p)) \to ((eq C c0 c) \to ((eq C c1 e) \to (drop1 (PCons n (S n0) (Ss p)) 
-(CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead e (Bind b) u))))))))) with 
-[(drop1_nil c0) \Rightarrow (\lambda (H1: (eq PList PNil (PCons n n0 
-p))).(\lambda (H2: (eq C c0 c)).(\lambda (H3: (eq C c0 e)).((let H4 \def 
-(eq_ind PList PNil (\lambda (e0: PList).(match e0 return (\lambda (_: 
-PList).Prop) with [PNil \Rightarrow True | (PCons _ _ _) \Rightarrow False])) 
-I (PCons n n0 p) H1) in (False_ind ((eq C c0 c) \to ((eq C c0 e) \to (drop1 
-(PCons n (S n0) (Ss p)) (CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead e 
-(Bind b) u)))) H4)) H2 H3)))) | (drop1_cons c1 c2 h d H1 c3 hds H2) 
-\Rightarrow (\lambda (H3: (eq PList (PCons h d hds) (PCons n n0 p))).(\lambda 
-(H4: (eq C c1 c)).(\lambda (H5: (eq C c3 e)).((let H6 \def (f_equal PList 
-PList (\lambda (e0: PList).(match e0 return (\lambda (_: PList).PList) with 
-[PNil \Rightarrow hds | (PCons _ _ p) \Rightarrow p])) (PCons h d hds) (PCons 
-n n0 p) H3) in ((let H7 \def (f_equal PList nat (\lambda (e0: PList).(match 
-e0 return (\lambda (_: PList).nat) with [PNil \Rightarrow d | (PCons _ n _) 
-\Rightarrow n])) (PCons h d hds) (PCons n n0 p) H3) in ((let H8 \def (f_equal 
-PList nat (\lambda (e0: PList).(match e0 return (\lambda (_: PList).nat) with 
-[PNil \Rightarrow h | (PCons n _ _) \Rightarrow n])) (PCons h d hds) (PCons n 
-n0 p) H3) in (eq_ind nat n (\lambda (n1: nat).((eq nat d n0) \to ((eq PList 
-hds p) \to ((eq C c1 c) \to ((eq C c3 e) \to ((drop n1 d c1 c2) \to ((drop1 
-hds c2 c3) \to (drop1 (PCons n (S n0) (Ss p)) (CHead c (Bind b) (lift n n0 
-(lift1 p u))) (CHead e (Bind b) u))))))))) (\lambda (H9: (eq nat d 
-n0)).(eq_ind nat n0 (\lambda (n1: nat).((eq PList hds p) \to ((eq C c1 c) \to 
-((eq C c3 e) \to ((drop n n1 c1 c2) \to ((drop1 hds c2 c3) \to (drop1 (PCons 
-n (S n0) (Ss p)) (CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead e (Bind b) 
-u)))))))) (\lambda (H10: (eq PList hds p)).(eq_ind PList p (\lambda (p0: 
-PList).((eq C c1 c) \to ((eq C c3 e) \to ((drop n n0 c1 c2) \to ((drop1 p0 c2 
-c3) \to (drop1 (PCons n (S n0) (Ss p)) (CHead c (Bind b) (lift n n0 (lift1 p 
-u))) (CHead e (Bind b) u))))))) (\lambda (H11: (eq C c1 c)).(eq_ind C c 
-(\lambda (c0: C).((eq C c3 e) \to ((drop n n0 c0 c2) \to ((drop1 p c2 c3) \to 
-(drop1 (PCons n (S n0) (Ss p)) (CHead c (Bind b) (lift n n0 (lift1 p u))) 
-(CHead e (Bind b) u)))))) (\lambda (H12: (eq C c3 e)).(eq_ind C e (\lambda 
-(c0: C).((drop n n0 c c2) \to ((drop1 p c2 c0) \to (drop1 (PCons n (S n0) (Ss 
-p)) (CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead e (Bind b) u))))) 
-(\lambda (H13: (drop n n0 c c2)).(\lambda (H14: (drop1 p c2 e)).(drop1_cons 
-(CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead c2 (Bind b) (lift1 p u)) n 
-(S n0) (drop_skip_bind n n0 c c2 H13 b (lift1 p u)) (CHead e (Bind b) u) (Ss 
-p) (H c2 u H14)))) c3 (sym_eq C c3 e H12))) c1 (sym_eq C c1 c H11))) hds 
-(sym_eq PList hds p H10))) d (sym_eq nat d n0 H9))) h (sym_eq nat h n H8))) 
-H7)) H6)) H4 H5 H1 H2))))]) in (H1 (refl_equal PList (PCons n n0 p)) 
-(refl_equal C c) (refl_equal C e)))))))))) 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 H1 \def (match 
-H0 return (\lambda (p: PList).(\lambda (c: C).(\lambda (c0: C).(\lambda (_: 
-(drop1 p c c0)).((eq PList p PNil) \to ((eq C c c1) \to ((eq C c0 c2) \to 
-(drop1 (PCons h d PNil) c1 c3)))))))) with [(drop1_nil c) \Rightarrow 
-(\lambda (_: (eq PList PNil PNil)).(\lambda (H2: (eq C c c1)).(\lambda (H3: 
-(eq C c c2)).(eq_ind C c1 (\lambda (c0: C).((eq C c0 c2) \to (drop1 (PCons h 
-d PNil) c1 c3))) (\lambda (H4: (eq C c1 c2)).(eq_ind C c2 (\lambda (c0: 
-C).(drop1 (PCons h d PNil) c0 c3)) (drop1_cons c2 c3 h d H c3 PNil (drop1_nil 
-c3)) c1 (sym_eq C c1 c2 H4))) c (sym_eq C c c1 H2) H3)))) | (drop1_cons c0 c4 
-h0 d0 H1 c5 hds H2) \Rightarrow (\lambda (H3: (eq PList (PCons h0 d0 hds) 
-PNil)).(\lambda (H4: (eq C c0 c1)).(\lambda (H5: (eq C c5 c2)).((let H6 \def 
-(eq_ind PList (PCons h0 d0 hds) (\lambda (e: PList).(match e return (\lambda 
-(_: PList).Prop) with [PNil \Rightarrow False | (PCons _ _ _) \Rightarrow 
-True])) I PNil H3) in (False_ind ((eq C c0 c1) \to ((eq C c5 c2) \to ((drop 
-h0 d0 c0 c4) \to ((drop1 hds c4 c5) \to (drop1 (PCons h d PNil) c1 c3))))) 
-H6)) H4 H5 H1 H2))))]) in (H1 (refl_equal PList PNil) (refl_equal C c1) 
-(refl_equal C c2))))) (\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 H2 \def (match H1 return (\lambda (p0: PList).(\lambda (c: 
-C).(\lambda (c0: C).(\lambda (_: (drop1 p0 c c0)).((eq PList p0 (PCons n n0 
-p)) \to ((eq C c c1) \to ((eq C c0 c2) \to (drop1 (PCons n n0 (PConsTail p h 
-d)) c1 c3)))))))) with [(drop1_nil c) \Rightarrow (\lambda (H2: (eq PList 
-PNil (PCons n n0 p))).(\lambda (H3: (eq C c c1)).(\lambda (H4: (eq C c 
-c2)).((let H5 \def (eq_ind PList PNil (\lambda (e: PList).(match e return 
-(\lambda (_: PList).Prop) with [PNil \Rightarrow True | (PCons _ _ _) 
-\Rightarrow False])) I (PCons n n0 p) H2) in (False_ind ((eq C c c1) \to ((eq 
-C c c2) \to (drop1 (PCons n n0 (PConsTail p h d)) c1 c3))) H5)) H3 H4)))) | 
-(drop1_cons c0 c4 h0 d0 H2 c5 hds H3) \Rightarrow (\lambda (H4: (eq PList 
-(PCons h0 d0 hds) (PCons n n0 p))).(\lambda (H5: (eq C c0 c1)).(\lambda (H6: 
-(eq C c5 c2)).((let H7 \def (f_equal PList PList (\lambda (e: PList).(match e 
-return (\lambda (_: PList).PList) with [PNil \Rightarrow hds | (PCons _ _ p) 
-\Rightarrow p])) (PCons h0 d0 hds) (PCons n n0 p) H4) in ((let H8 \def 
-(f_equal PList nat (\lambda (e: PList).(match e return (\lambda (_: 
-PList).nat) with [PNil \Rightarrow d0 | (PCons _ n _) \Rightarrow n])) (PCons 
-h0 d0 hds) (PCons n n0 p) H4) in ((let H9 \def (f_equal PList nat (\lambda 
-(e: PList).(match e return (\lambda (_: PList).nat) with [PNil \Rightarrow h0 
-| (PCons n _ _) \Rightarrow n])) (PCons h0 d0 hds) (PCons n n0 p) H4) in 
-(eq_ind nat n (\lambda (n1: nat).((eq nat d0 n0) \to ((eq PList hds p) \to 
-((eq C c0 c1) \to ((eq C c5 c2) \to ((drop n1 d0 c0 c4) \to ((drop1 hds c4 
-c5) \to (drop1 (PCons n n0 (PConsTail p h d)) c1 c3)))))))) (\lambda (H10: 
-(eq nat d0 n0)).(eq_ind nat n0 (\lambda (n1: nat).((eq PList hds p) \to ((eq 
-C c0 c1) \to ((eq C c5 c2) \to ((drop n n1 c0 c4) \to ((drop1 hds c4 c5) \to 
-(drop1 (PCons n n0 (PConsTail p h d)) c1 c3))))))) (\lambda (H11: (eq PList 
-hds p)).(eq_ind PList p (\lambda (p0: PList).((eq C c0 c1) \to ((eq C c5 c2) 
-\to ((drop n n0 c0 c4) \to ((drop1 p0 c4 c5) \to (drop1 (PCons n n0 
-(PConsTail p h d)) c1 c3)))))) (\lambda (H12: (eq C c0 c1)).(eq_ind C c1 
-(\lambda (c: C).((eq C c5 c2) \to ((drop n n0 c c4) \to ((drop1 p c4 c5) \to 
-(drop1 (PCons n n0 (PConsTail p h d)) c1 c3))))) (\lambda (H13: (eq C c5 
-c2)).(eq_ind C c2 (\lambda (c: C).((drop n n0 c1 c4) \to ((drop1 p c4 c) \to 
-(drop1 (PCons n n0 (PConsTail p h d)) c1 c3)))) (\lambda (H14: (drop n n0 c1 
-c4)).(\lambda (H15: (drop1 p c4 c2)).(drop1_cons c1 c4 n n0 H14 c3 (PConsTail 
-p h d) (H0 c4 H15)))) c5 (sym_eq C c5 c2 H13))) c0 (sym_eq C c0 c1 H12))) hds 
-(sym_eq PList hds p H11))) d0 (sym_eq nat d0 n0 H10))) h0 (sym_eq nat h0 n 
-H9))) H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal PList (PCons n n0 p)) 
-(refl_equal C c1) (refl_equal C c2))))))))) hds)))))).
-
-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 (ctrans 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 
-(ctrans 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 (ctrans hds0 i t))))))).(\lambda 
-(i: nat).(\lambda (t: T).(eq_ind_r T (lift (S (trans hds0 i)) O (ctrans 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 (match (blt (trans hds0 i) d) with [true \Rightarrow (lift h 
-(minus d (S (trans hds0 i))) (ctrans hds0 i t)) | false \Rightarrow (ctrans 
-hds0 i t)])))) (xinduction bool (blt (trans hds0 i) d) (\lambda (b: bool).(eq 
-T (lift h d (lift (S (trans hds0 i)) O (ctrans hds0 i t))) (lift (S (match b 
-with [true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 
-i) h)])) O (match b with [true \Rightarrow (lift h (minus d (S (trans hds0 
-i))) (ctrans hds0 i t)) | false \Rightarrow (ctrans 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 (ctrans hds0 i t))) (lift (S 
-(match b with [true \Rightarrow (trans hds0 i) | false \Rightarrow (plus 
-(trans hds0 i) h)])) O (match b with [true \Rightarrow (lift h (minus d (S 
-(trans hds0 i))) (ctrans hds0 i t)) | false \Rightarrow (ctrans 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 (ctrans hds0 i t))) (lift (S (trans 
-hds0 i)) O (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i t))))) 
-(eq_ind_r T (lift (S (trans hds0 i)) O (lift h (minus d (S (trans hds0 i))) 
-(ctrans hds0 i t))) (\lambda (t0: T).(eq T t0 (lift (S (trans hds0 i)) O 
-(lift h (minus d (S (trans hds0 i))) (ctrans hds0 i t))))) (refl_equal T 
-(lift (S (trans hds0 i)) O (lift h (minus d (S (trans hds0 i))) (ctrans hds0 
-i t)))) (lift h (plus (S (trans hds0 i)) (minus d (S (trans hds0 i)))) (lift 
-(S (trans hds0 i)) O (ctrans hds0 i t))) (lift_d (ctrans 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 (ctrans hds0 i t)) 
-(\lambda (t0: T).(eq T t0 (lift (S (plus (trans hds0 i) h)) O (ctrans hds0 i 
-t)))) (eq_ind nat (S (plus h (trans hds0 i))) (\lambda (n: nat).(eq T (lift n 
-O (ctrans hds0 i t)) (lift (S (plus (trans hds0 i) h)) O (ctrans hds0 i t)))) 
-(eq_ind_r nat (plus (trans hds0 i) h) (\lambda (n: nat).(eq T (lift (S n) O 
-(ctrans hds0 i t)) (lift (S (plus (trans hds0 i) h)) O (ctrans hds0 i t)))) 
-(refl_equal T (lift (S (plus (trans hds0 i) h)) O (ctrans hds0 i t))) (plus h 
-(trans hds0 i)) (plus_comm 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 (ctrans 
-hds0 i t))) (lift_free (ctrans 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_comm 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).
-
-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))))).
-
-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))).
-
-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)))))))))).
-
-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).(let H0 \def (match H return (\lambda (c: C).(\lambda (c0: 
-C).(\lambda (_: (clear c c0)).((eq C c (CSort n)) \to ((eq C c0 x) \to P))))) 
-with [(clear_bind b e u) \Rightarrow (\lambda (H0: (eq C (CHead e (Bind b) u) 
-(CSort n))).(\lambda (H1: (eq C (CHead e (Bind b) u) x)).((let H2 \def 
-(eq_ind C (CHead e (Bind b) u) (\lambda (e0: C).(match e0 return (\lambda (_: 
-C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow 
-True])) I (CSort n) H0) in (False_ind ((eq C (CHead e (Bind b) u) x) \to P) 
-H2)) H1))) | (clear_flat e c H0 f u) \Rightarrow (\lambda (H1: (eq C (CHead e 
-(Flat f) u) (CSort n))).(\lambda (H2: (eq C c x)).((let H3 \def (eq_ind C 
-(CHead e (Flat f) u) (\lambda (e0: C).(match e0 return (\lambda (_: C).Prop) 
-with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I 
-(CSort n) H1) in (False_ind ((eq C c x) \to ((clear e c) \to P)) H3)) H2 
-H0)))]) in (H0 (refl_equal C (CSort n)) (refl_equal C x)))))).
-
-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)).(let H0 \def (match H return (\lambda (c: 
-C).(\lambda (c0: C).(\lambda (_: (clear c c0)).((eq C c (CHead e (Bind b) u)) 
-\to ((eq C c0 x) \to (eq C x (CHead e (Bind b) u))))))) with [(clear_bind b0 
-e0 u0) \Rightarrow (\lambda (H0: (eq C (CHead e0 (Bind b0) u0) (CHead e (Bind 
-b) u))).(\lambda (H1: (eq C (CHead e0 (Bind b0) u0) x)).((let H2 \def 
-(f_equal C T (\lambda (e1: C).(match e1 return (\lambda (_: C).T) with 
-[(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead e0 (Bind 
-b0) u0) (CHead e (Bind b) u) H0) in ((let H3 \def (f_equal C B (\lambda (e1: 
-C).(match e1 return (\lambda (_: C).B) with [(CSort _) \Rightarrow b0 | 
-(CHead _ k _) \Rightarrow (match k return (\lambda (_: K).B) with [(Bind b) 
-\Rightarrow b | (Flat _) \Rightarrow b0])])) (CHead e0 (Bind b0) u0) (CHead e 
-(Bind b) u) H0) in ((let H4 \def (f_equal C C (\lambda (e1: C).(match e1 
-return (\lambda (_: C).C) with [(CSort _) \Rightarrow e0 | (CHead c _ _) 
-\Rightarrow c])) (CHead e0 (Bind b0) u0) (CHead e (Bind b) u) H0) in (eq_ind 
-C e (\lambda (c: C).((eq B b0 b) \to ((eq T u0 u) \to ((eq C (CHead c (Bind 
-b0) u0) x) \to (eq C x (CHead e (Bind b) u)))))) (\lambda (H5: (eq B b0 
-b)).(eq_ind B b (\lambda (b1: B).((eq T u0 u) \to ((eq C (CHead e (Bind b1) 
-u0) x) \to (eq C x (CHead e (Bind b) u))))) (\lambda (H6: (eq T u0 
-u)).(eq_ind T u (\lambda (t: T).((eq C (CHead e (Bind b) t) x) \to (eq C x 
-(CHead e (Bind b) u)))) (\lambda (H7: (eq C (CHead e (Bind b) u) x)).(eq_ind 
-C (CHead e (Bind b) u) (\lambda (c: C).(eq C c (CHead e (Bind b) u))) 
-(refl_equal C (CHead e (Bind b) u)) x H7)) u0 (sym_eq T u0 u H6))) b0 (sym_eq 
-B b0 b H5))) e0 (sym_eq C e0 e H4))) H3)) H2)) H1))) | (clear_flat e0 c H0 f 
-u0) \Rightarrow (\lambda (H1: (eq C (CHead e0 (Flat f) u0) (CHead e (Bind b) 
-u))).(\lambda (H2: (eq C c x)).((let H3 \def (eq_ind C (CHead e0 (Flat f) u0) 
-(\lambda (e1: C).(match e1 return (\lambda (_: C).Prop) with [(CSort _) 
-\Rightarrow False | (CHead _ k _) \Rightarrow (match k return (\lambda (_: 
-K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I 
-(CHead e (Bind b) u) H1) in (False_ind ((eq C c x) \to ((clear e0 c) \to (eq 
-C x (CHead e (Bind b) u)))) H3)) H2 H0)))]) in (H0 (refl_equal C (CHead e 
-(Bind b) u)) (refl_equal C x))))))).
-
-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)).(let H0 \def (match H return (\lambda (c: 
-C).(\lambda (c0: C).(\lambda (_: (clear c c0)).((eq C c (CHead e (Flat f) u)) 
-\to ((eq C c0 x) \to (clear e x)))))) with [(clear_bind b e0 u0) \Rightarrow 
-(\lambda (H0: (eq C (CHead e0 (Bind b) u0) (CHead e (Flat f) u))).(\lambda 
-(H1: (eq C (CHead e0 (Bind b) u0) x)).((let H2 \def (eq_ind C (CHead e0 (Bind 
-b) u0) (\lambda (e1: C).(match e1 return (\lambda (_: C).Prop) with [(CSort 
-_) \Rightarrow False | (CHead _ k _) \Rightarrow (match k return (\lambda (_: 
-K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I 
-(CHead e (Flat f) u) H0) in (False_ind ((eq C (CHead e0 (Bind b) u0) x) \to 
-(clear e x)) H2)) H1))) | (clear_flat e0 c H0 f0 u0) \Rightarrow (\lambda 
-(H1: (eq C (CHead e0 (Flat f0) u0) (CHead e (Flat f) u))).(\lambda (H2: (eq C 
-c x)).((let H3 \def (f_equal C T (\lambda (e1: C).(match e1 return (\lambda 
-(_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) 
-(CHead e0 (Flat f0) u0) (CHead e (Flat f) u) H1) in ((let H4 \def (f_equal C 
-F (\lambda (e1: C).(match e1 return (\lambda (_: C).F) with [(CSort _) 
-\Rightarrow f0 | (CHead _ k _) \Rightarrow (match k return (\lambda (_: K).F) 
-with [(Bind _) \Rightarrow f0 | (Flat f) \Rightarrow f])])) (CHead e0 (Flat 
-f0) u0) (CHead e (Flat f) u) H1) in ((let H5 \def (f_equal C C (\lambda (e1: 
-C).(match e1 return (\lambda (_: C).C) with [(CSort _) \Rightarrow e0 | 
-(CHead c _ _) \Rightarrow c])) (CHead e0 (Flat f0) u0) (CHead e (Flat f) u) 
-H1) in (eq_ind C e (\lambda (c0: C).((eq F f0 f) \to ((eq T u0 u) \to ((eq C 
-c x) \to ((clear c0 c) \to (clear e x)))))) (\lambda (H6: (eq F f0 
-f)).(eq_ind F f (\lambda (_: F).((eq T u0 u) \to ((eq C c x) \to ((clear e c) 
-\to (clear e x))))) (\lambda (H7: (eq T u0 u)).(eq_ind T u (\lambda (_: 
-T).((eq C c x) \to ((clear e c) \to (clear e x)))) (\lambda (H8: (eq C c 
-x)).(eq_ind C x (\lambda (c0: C).((clear e c0) \to (clear e x))) (\lambda 
-(H9: (clear e x)).H9) c (sym_eq C c x H8))) u0 (sym_eq T u0 u H7))) f0 
-(sym_eq F f0 f H6))) e0 (sym_eq C e0 e H5))) H4)) H3)) H2 H0)))]) in (H0 
-(refl_equal C (CHead e (Flat f) u)) (refl_equal C x))))))).
-
-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)) 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 return 
-(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) 
-\Rightarrow (match 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 (c: C).((eq C c (CHead e (Flat f) u)) \to P)) H2 (CHead 
-e (Flat f) u) H3) in (let H5 \def (eq_ind C c (\lambda (c: C).(clear e0 c)) 
-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 (e: C).(\lambda (u: T).(eq C c (CHead e (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 (c: C).(clear e c)) 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))).
-
-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 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)).((match k return (\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))))))) with [(Bind b) \Rightarrow (\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)) | (Flat f) \Rightarrow (\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)))]) (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)).((match k return 
-(\lambda (k0: K).((clear (CHead c0 k0 t) (CHead e1 (Bind b) u)) \to (drop (S 
-i) O (CHead c0 k0 t) e2))) with [(Bind b0) \Rightarrow (\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 return (\lambda (_: C).C) with [(CSort _) 
-\Rightarrow e1 | (CHead c _ _) \Rightarrow c])) (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 return (\lambda (_: C).B) with 
-[(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow (match k return (\lambda 
-(_: K).B) with [(Bind b) \Rightarrow b | (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 return 
-(\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow 
-t])) (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 (c: C).(drop i O c 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))) | (Flat f) 
-\Rightarrow (\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))]) 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)))) ((match k 
-return (\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)))))) with [(Bind b0) \Rightarrow (\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 return (\lambda (_: C).C) with 
-[(CSort _) \Rightarrow c2 | (CHead c _ _) \Rightarrow c])) (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 return (\lambda (_: C).B) 
-with [(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow (match k return 
-(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (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 
-return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) 
-\Rightarrow t])) (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)))) | (Flat f) \Rightarrow (\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))))]) H1 H3) x1 H2)))) (drop_gen_skip_r c x1 t h d k 
-H0)))))))))))))) x2).
-
-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)).((match k return (\lambda (k0: K).((clear (CHead c k0 t) 
-c2) \to (clear c2 c2))) with [(Bind b) \Rightarrow (\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))) | 
-(Flat f) \Rightarrow (\lambda (H1: (clear (CHead c (Flat f) t) c2)).(H c2 
-(clear_gen_flat f c c2 t H1)))]) 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)).((match k 
-return (\lambda (k0: K).((clear (CHead c0 k0 t) c1) \to ((clear (CHead c0 k0 
-t) c2) \to (eq C c1 c2)))) with [(Bind b) \Rightarrow (\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)))) | (Flat f) \Rightarrow (\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))))]) 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)).((match k return (\lambda 
-(k0: K).((clear (CHead c k0 t) c0) \to (clear (CHead c k0 t) c2))) with 
-[(Bind b) \Rightarrow (\lambda (H2: (clear (CHead c (Bind b) t) c0)).(let H3 
-\def (eq_ind C c0 (\lambda (c: C).(clear c 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)))) | (Flat f) \Rightarrow (\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))]) 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).(match k return (\lambda (k0: K).(clear (CHead (CSort n) k0 u1) (CHead 
-(CTail k0 u1 c2) (Bind b) u2))) with [(Bind b0) \Rightarrow (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))) | (Flat f) \Rightarrow (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)))]))))))) (\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).((match k return (\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)))) with [(Bind b0) \Rightarrow (\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 return (\lambda (_: C).C) with [(CSort _) 
-\Rightarrow c2 | (CHead c _ _) \Rightarrow c])) (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 return (\lambda (_: C).B) with 
-[(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow (match k return (\lambda 
-(_: K).B) with [(Bind b) \Rightarrow b | (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 return 
-(\lambda (_: C).T) with [(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow 
-t])) (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))) | (Flat f) \Rightarrow (\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))]) H0)))))))))) c1)).
-
-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)).(let H0 \def (match H return (\lambda (_: (getl ? ? ?)).(ex2 C (\lambda 
-(e: C).(drop i O c1 e)) (\lambda (e: C).(clear e c2)))) with [(getl_intro e 
-H0 H1) \Rightarrow (ex_intro2 C (\lambda (e0: C).(drop i O c1 e0)) (\lambda 
-(e0: C).(clear e0 c2)) e H0 H1)]) in H0)))).
-
-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_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 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))).((match k 
-return (\lambda (k0: K).((getl (S n) (CHead c k0 t) c3) \to ((clear c1 (CHead 
-c k0 t)) \to (getl (S n) c1 c3)))) with [(Bind b) \Rightarrow (\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)))) | (Flat f) \Rightarrow (\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))))]) 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_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)).((match h return (\lambda (n: nat).((drop n O 
-c x) \to (getl n (CHead c (Flat f) u) e))) with [O \Rightarrow (\lambda (H3: 
-(drop O O c x)).(let H4 \def (eq_ind_r C x (\lambda (c: C).(clear c 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)))) | (S 
-n) \Rightarrow (\lambda (H3: (drop (S n) O c x)).(getl_intro (S n) (CHead c 
-(Flat f) u) e x (drop_drop (Flat f) n c x H3 u) H2))]) H1)))) H0))))))).
-
-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 return (\lambda 
-(_: C).C) with [(CSort _) \Rightarrow e | (CHead c _ _) \Rightarrow c])) 
-(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 
-return (\lambda (_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k _) 
-\Rightarrow (match k return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | 
-(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 return (\lambda (_: C).T) with [(CSort _) 
-\Rightarrow u | (CHead _ _ t) \Rightarrow t])) (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_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)).((match k return (\lambda (k0: K).((clear 
-(CHead c0 k0 t) (CHead e1 (Bind b) v)) \to (getl (S n) (CHead c0 k0 t) e2))) 
-with [(Bind b0) \Rightarrow (\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 
-return (\lambda (_: C).C) with [(CSort _) \Rightarrow e1 | (CHead c _ _) 
-\Rightarrow c])) (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 return (\lambda (_: C).B) with [(CSort _) \Rightarrow b | (CHead 
-_ k _) \Rightarrow (match k return (\lambda (_: K).B) with [(Bind b) 
-\Rightarrow b | (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 return (\lambda (_: C).T) with 
-[(CSort _) \Rightarrow v | (CHead _ _ t) \Rightarrow t])) (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 (c: C).(getl n c 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))) | (Flat f) \Rightarrow (\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))]) 
-H0))))))))))) c)).
-
-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_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))).(match k return 
-(\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))))) with [(Bind b0) 
-\Rightarrow (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)) | (Flat f) \Rightarrow (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))]))) H0)))))) c))).
-
-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).(match i return (\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))))) with [O \Rightarrow (match k return (\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))))) with [(Bind b) \Rightarrow (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))) | (Flat f) \Rightarrow (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)))]) | (S n) \Rightarrow 
-(let H_x \def (H (r k 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 (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 (H1: (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 (H2: 
-(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) H2 t))))))) H1)) (\lambda (H1: ((\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 (H2: (getl (S n) (CHead c0 k 
-t) d)).(\lambda (P: Prop).(H1 d (getl_gen_S k c0 d t n H2) P)))))) 
-H0)))])))))) c).
-
-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)).((match k return (\lambda (k0: 
-K).((clear (CHead c k0 t) c2) \to (le (cweight c2) (plus (cweight c) (tweight 
-t))))) with [(Bind b) \Rightarrow (\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))) | (Flat f) \Rightarrow (\lambda (H1: (clear 
-(CHead c (Flat f) t) c2)).(le_S_n (cweight c2) (plus (cweight c) (tweight t)) 
-(le_n_S (cweight c2) (plus (cweight c) (tweight t)) (le_plus_trans (cweight 
-c2) (cweight c) (tweight t) (H c2 (clear_gen_flat f c c2 t H1))))))]) 
-H0))))))) c1).
-
-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).(match i return (\lambda (n: nat).((getl n (CHead c0 k 
-t) (CHead e (Bind b) u)) \to (flt e u (CHead c0 k t) (TLRef n)))) with [O 
-\Rightarrow (\lambda (H0: (getl O (CHead c0 k t) (CHead e (Bind b) 
-u))).((match k return (\lambda (k0: K).((clear (CHead c0 k0 t) (CHead e (Bind 
-b) u)) \to (flt e u (CHead c0 k0 t) (TLRef O)))) with [(Bind b0) \Rightarrow 
-(\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 return (\lambda (_: C).C) with 
-[(CSort _) \Rightarrow e | (CHead c _ _) \Rightarrow c])) (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 return (\lambda (_: 
-C).B) with [(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow (match k 
-return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (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 return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead 
-_ _ t) \Rightarrow t])) (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))) | (Flat f) \Rightarrow (\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))]) (getl_gen_O (CHead c0 k t) (CHead e (Bind b) u) H0))) | (S n) 
-\Rightarrow (\lambda (H0: (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 
-H0)) in (flt_arith2 e c0 u (r k n) H_y k t (S n))))])))))))) c)).
-
-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))))).
-
-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).(match i return (\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))))))) with [O \Rightarrow (\lambda (H: 
-(getl O (CHead (CSort n) k u1) (CHead c2 (Bind b) u2))).((match k return 
-(\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))))))) with [(Bind b0) \Rightarrow 
-(\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 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 return (\lambda (_: C).B) with [(CSort _) \Rightarrow b | (CHead 
-_ k _) \Rightarrow (match k return (\lambda (_: K).B) with [(Bind b) 
-\Rightarrow b | (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 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))) | (Flat f) \Rightarrow (\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)))))))]) (getl_gen_O (CHead 
-(CSort n) k u1) (CHead c2 (Bind b) u2) H))) | (S n0) \Rightarrow (\lambda (H: 
-(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 H) (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)))))))]))) (\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).(match i return (\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))))))) with [O \Rightarrow (\lambda 
-(H0: (getl O (CHead (CTail k u1 c) k0 t) (CHead c2 (Bind b) u2))).((match k0 
-return (\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))))))) 
-with [(Bind b0) \Rightarrow (\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 return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead 
-c _ _) \Rightarrow c])) (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 return (\lambda (_: C).B) 
-with [(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow (match k return 
-(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (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 return (\lambda (_: C).T) with [(CSort _) 
-\Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (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 (i: nat).((getl i (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 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 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))) | 
-(Flat f) \Rightarrow (\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)))]) (getl_gen_O (CHead (CTail 
-k u1 c) k0 t) (CHead c2 (Bind b) u2) H0))) | (S n) \Rightarrow (\lambda (H0: 
-(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 
-H0)) in (let H1 \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 (H2: (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 (H3: (eq C c2 (CTail k u1 x))).(\lambda (H4: (getl 
-(r k0 n) c (CHead x (Bind b) u2))).(let H5 \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 H0) (CTail k u1 x) H3) 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) H4 t))) c2 H3))))) H2)) (\lambda (H2: (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 (n: nat).(eq C c2 (CSort 
-n))))).(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 (H3: (eq nat (r k0 n) (clen 
-c))).(\lambda (H4: (eq K k (Bind b))).(\lambda (H5: (eq T u1 u2)).(\lambda 
-(H6: (eq C c2 (CSort x0))).(let H7 \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 H0) (CSort x0) H6) 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 H8 \def (eq_ind_r T u2 (\lambda (t: T).(getl (r k0 n) (CTail k 
-u1 c) (CHead (CSort x0) (Bind b) t))) H7 u1 H5) 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 H9 \def (eq_ind K k (\lambda (k: K).(getl (r k0 n) (CTail k u1 
-c) (CHead (CSort x0) (Bind b) u1))) H8 (Bind b) H4) 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) H3) k H4)) u2 H5)) c2 H6))))))) H2)) H1))))])))))) c1)))))).
-
-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))).((match h return (\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)))))) with [O \Rightarrow (\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))))) | (S n) \Rightarrow (\lambda (H0: (getl (S 
-n) (CHead c (Flat f) v) (CHead d1 (Bind b) w))).(ex_intro C (\lambda (d2: 
-C).(getl (S n) c (CHead d2 (Bind b) w))) d1 (getl_gen_S (Flat f) c (CHead d1 
-(Bind b) w) v n H0)))]) 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))).((match h return (\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)))))) with [O \Rightarrow (\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 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 return (\lambda (_: C).B) with [(CSort _) 
-\Rightarrow b0 | (CHead _ k _) \Rightarrow (match k return (\lambda (_: K).B) 
-with [(Bind b) \Rightarrow b | (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 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))) | (S n) \Rightarrow (\lambda (H1: (getl (S n) (CHead c1 
-(Bind b) v) (CHead d1 (Bind b0) w))).(let H_x \def (H b0 d1 w (r (Bind b) n) 
-(getl_gen_S (Bind b) c1 (CHead d1 (Bind b0) w) v n H1)) in (let H2 \def H_x 
-in (ex_ind C (\lambda (d2: C).(getl (r (Bind b) n) c2 (CHead d2 (Bind b0) 
-w))) (ex C (\lambda (d2: C).(getl (S n) (CHead c2 (Bind b) v) (CHead d2 (Bind 
-b0) w)))) (\lambda (x: C).(\lambda (H3: (getl (r (Bind b) n) c2 (CHead x 
-(Bind b0) w))).(ex_intro C (\lambda (d2: C).(getl (S n) (CHead c2 (Bind b) v) 
-(CHead d2 (Bind b0) w))) x (getl_head (Bind b) n c2 (CHead x (Bind b0) w) H3 
-v)))) H2))))]) 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 (c: C).(clear c x1)) H6 x (drop_mono c x O h H2 x0 
-H5)) in (clear_mono x x1 H9 x2 H3))))))) H4))))) H1))))))).
-
-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)).((match k return (\lambda 
-(k0: K).((getl (S n) (CHead c k0 t) c3) \to ((clear (CHead c k0 t) c2) \to 
-(getl (S n) c2 c3)))) with [(Bind b) \Rightarrow (\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)))) | (Flat f) \Rightarrow (\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))))]) H1 H2))))))))) c1)))) i).
-
-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))).((match x return (\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))))))) with [(CSort n) \Rightarrow (\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))))))) | (CHead c2 k0 t0) \Rightarrow (\lambda (H5: (drop i O 
-(CHead c0 k t) (CHead c2 k0 t0))).(\lambda (H6: (clear (CHead c2 k0 t0) 
-(CHead c1 (Bind b) u))).((match k0 return (\lambda (k1: K).((drop i O (CHead 
-c0 k t) (CHead c2 k1 t0)) \to ((clear (CHead c2 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))))))) with [(Bind b0) \Rightarrow 
-(\lambda (H7: (drop i O (CHead c0 k t) (CHead c2 (Bind b0) t0))).(\lambda 
-(H8: (clear (CHead c2 (Bind b0) t0) (CHead c1 (Bind b) u))).(let H9 \def 
-(f_equal C C (\lambda (e0: C).(match e0 return (\lambda (_: C).C) with 
-[(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 (Bind b) 
-u) (CHead c2 (Bind b0) t0) (clear_gen_bind b0 c2 (CHead c1 (Bind b) u) t0 
-H8)) in ((let H10 \def (f_equal C B (\lambda (e0: C).(match e0 return 
-(\lambda (_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow 
-(match k return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) 
-\Rightarrow b])])) (CHead c1 (Bind b) u) (CHead c2 (Bind b0) t0) 
-(clear_gen_bind b0 c2 (CHead c1 (Bind b) u) t0 H8)) in ((let H11 \def 
-(f_equal C T (\lambda (e0: C).(match e0 return (\lambda (_: C).T) with 
-[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 (Bind b) 
-u) (CHead c2 (Bind b0) t0) (clear_gen_bind b0 c2 (CHead c1 (Bind b) u) t0 
-H8)) in (\lambda (H12: (eq B b b0)).(\lambda (H13: (eq C c1 c2)).(let H14 
-\def (eq_ind_r T t0 (\lambda (t0: T).(drop i O (CHead c0 k t) (CHead c2 (Bind 
-b0) t0))) H7 u H11) in (let H15 \def (eq_ind_r B b0 (\lambda (b: B).(drop i O 
-(CHead c0 k t) (CHead c2 (Bind b) u))) H14 b H12) in (let H16 \def (eq_ind_r 
-C c2 (\lambda (c: C).(drop i O (CHead c0 k t) (CHead c (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 (x0: T).(\lambda (x1: 
-C).(\lambda (H17: (eq T u (lift h (r (Bind b) d) x0))).(\lambda (H18: (drop i 
-O e (CHead x1 (Bind b) x0))).(\lambda (H19: (drop h (r (Bind b) d) c1 
-x1)).(eq_ind_r T (lift h (r (Bind b) d) x0) (\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) x0) (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))) x0 x1 (refl_equal T (lift h d x0)) 
-(getl_intro i e (CHead x1 (Bind b) x0) (CHead x1 (Bind b) x0) H18 (clear_bind 
-b x1 x0)) H19) u H17)))))) (drop_conf_lt (Bind b) i u c1 (CHead c0 k t) H16 e 
-h d H1)))))))) H10)) H9)))) | (Flat f) \Rightarrow (\lambda (H7: (drop i O 
-(CHead c0 k t) (CHead c2 (Flat f) t0))).(\lambda (H8: (clear (CHead c2 (Flat 
-f) t0) (CHead c1 (Bind b) u))).((match i return (\lambda (n: nat).((drop h (S 
-(plus n d)) (CHead c0 k t) e) \to ((drop n O (CHead c0 k t) (CHead c2 (Flat 
-f) t0)) \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))))))) with [O \Rightarrow 
-(\lambda (H9: (drop h (S (plus O d)) (CHead c0 k t) e)).(\lambda (H10: (drop 
-O O (CHead c0 k t) (CHead c2 (Flat f) t0))).(let H11 \def (f_equal C C 
-(\lambda (e0: C).(match e0 return (\lambda (_: C).C) with [(CSort _) 
-\Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k t) (CHead c2 
-(Flat f) t0) (drop_gen_refl (CHead c0 k t) (CHead c2 (Flat f) t0) H10)) in 
-((let H12 \def (f_equal C K (\lambda (e0: C).(match e0 return (\lambda (_: 
-C).K) with [(CSort _) \Rightarrow k | (CHead _ k _) \Rightarrow k])) (CHead 
-c0 k t) (CHead c2 (Flat f) t0) (drop_gen_refl (CHead c0 k t) (CHead c2 (Flat 
-f) t0) H10)) in ((let H13 \def (f_equal C T (\lambda (e0: C).(match e0 return 
-(\lambda (_: C).T) with [(CSort _) \Rightarrow t | (CHead _ _ t) \Rightarrow 
-t])) (CHead c0 k t) (CHead c2 (Flat f) t0) (drop_gen_refl (CHead c0 k t) 
-(CHead c2 (Flat f) t0) H10)) in (\lambda (H14: (eq K k (Flat f))).(\lambda 
-(H15: (eq C c0 c2)).(let H16 \def (eq_ind_r C c2 (\lambda (c: C).(clear c 
-(CHead c1 (Bind b) u))) (clear_gen_flat f c2 (CHead c1 (Bind b) u) t0 H8) c0 
-H15) in (let H17 \def (eq_ind K k (\lambda (k: K).(drop h (S (plus O d)) 
-(CHead c0 k 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 (x0: C).(\lambda (x1: 
-T).(\lambda (H18: (eq C e (CHead x0 (Flat f) x1))).(\lambda (H19: (eq T t 
-(lift h (r (Flat f) (plus O d)) x1))).(\lambda (H20: (drop h (r (Flat f) 
-(plus O d)) c0 x0)).(let H21 \def (f_equal T T (\lambda (e0: T).e0) t (lift h 
-(r (Flat f) (plus O d)) x1) H19) in (eq_ind_r C (CHead x0 (Flat f) x1) 
-(\lambda (c3: 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 c3 (CHead e0 (Bind b) v)))) 
-(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))) (let H22 \def (H c1 u O 
-(getl_intro O c0 (CHead c1 (Bind b) u) c0 (drop_refl c0) H16) x0 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 O x0 (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 x0 (Flat f) x1) (CHead e0 (Bind b) v)))) (\lambda (_: 
-T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x2: T).(\lambda (x3: 
-C).(\lambda (H23: (eq T u (lift h d x2))).(\lambda (H24: (getl O x0 (CHead x3 
-(Bind b) x2))).(\lambda (H25: (drop h d c1 x3)).(let H26 \def (eq_ind T u 
-(\lambda (t: T).(clear c0 (CHead c1 (Bind b) t))) H16 (lift h d x2) H23) in 
-(eq_ind_r T (lift h d x2) (\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 x0 (Flat f) x1) (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 x2) (lift h d v)))) (\lambda (v: 
-T).(\lambda (e0: C).(getl O (CHead x0 (Flat f) x1) (CHead e0 (Bind b) v)))) 
-(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))) x2 x3 (refl_equal T (lift 
-h d x2)) (getl_flat x0 (CHead x3 (Bind b) x2) O H24 f x1) H25) u H23))))))) 
-H22)) e H18))))))) (drop_gen_skip_l c0 e t h (plus O d) (Flat f) H17))))))) 
-H12)) H11)))) | (S n) \Rightarrow (\lambda (H9: (drop h (S (plus (S n) d)) 
-(CHead c0 k t) e)).(\lambda (H10: (drop (S n) O (CHead c0 k t) (CHead c2 
-(Flat f) t0))).(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 n) 
-d)) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r k (plus (S n) 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 n) e (CHead e0 (Bind b) v)))) 
-(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x0: 
-C).(\lambda (x1: T).(\lambda (H11: (eq C e (CHead x0 k x1))).(\lambda (H12: 
-(eq T t (lift h (r k (plus (S n) d)) x1))).(\lambda (H13: (drop h (r k (plus 
-(S n) d)) c0 x0)).(let H14 \def (f_equal T T (\lambda (e0: T).e0) t (lift h 
-(r k (plus (S n) d)) x1) H12) in (eq_ind_r C (CHead x0 k x1) (\lambda (c3: 
-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 n) c3 (CHead e0 (Bind b) v)))) 
-(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))) (let H15 \def (eq_ind 
-nat (r k (plus (S n) d)) (\lambda (n: nat).(drop h n c0 x0)) H13 (plus (r k 
-(S n)) d) (r_plus k (S n) d)) in (let H16 \def (eq_ind nat (r k (S n)) 
-(\lambda (n: nat).(drop h (plus n d) c0 x0)) H15 (S (r k n)) (r_S k n)) in 
-(let H17 \def (H c1 u (r k n) (getl_intro (r k n) c0 (CHead c1 (Bind b) u) 
-(CHead c2 (Flat f) t0) (drop_gen_drop k c0 (CHead c2 (Flat f) t0) t n H10) 
-(clear_flat c2 (CHead c1 (Bind b) u) (clear_gen_flat f c2 (CHead c1 (Bind b) 
-u) t0 H8) f t0)) x0 h d H16) 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 n) x0 
-(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 n) (CHead x0 k x1) (CHead e0 (Bind b) v)))) 
-(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x2: 
-T).(\lambda (x3: C).(\lambda (H18: (eq T u (lift h d x2))).(\lambda (H19: 
-(getl (r k n) x0 (CHead x3 (Bind b) x2))).(\lambda (H20: (drop h d c1 
-x3)).(let H21 \def (eq_ind T u (\lambda (t: T).(clear c2 (CHead c1 (Bind b) 
-t))) (clear_gen_flat f c2 (CHead c1 (Bind b) u) t0 H8) (lift h d x2) H18) in 
-(eq_ind_r T (lift h d x2) (\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 n) (CHead x0 k x1) (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 x2) (lift h d v)))) (\lambda (v: 
-T).(\lambda (e0: C).(getl (S n) (CHead x0 k x1) (CHead e0 (Bind b) v)))) 
-(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))) x2 x3 (refl_equal T (lift 
-h d x2)) (getl_head k n x0 (CHead x3 (Bind b) x2) H19 x1) H20) u H18))))))) 
-H17)))) e H11))))))) (drop_gen_skip_l c0 e t h (plus (S n) d) k H9))))]) H1 
-H7)))]) H5 H6)))]) 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_comm i (S O)))) i (minus_Sx_SO i)) in 
-H3)))))))).
-
-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_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).
-
-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)).((match i return (\lambda (n: nat).((drop n O c2 x) \to (getl (plus n h) 
-c1 e2))) with [O \Rightarrow (\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))) | (S n) \Rightarrow (\lambda 
-(H4: (drop (S n) O c2 x)).(let H_y \def (getl_drop_trans c1 c2 h H x n H4) in 
-(getl_intro (plus (S n) h) c1 e2 x H_y H3)))]) H2)))) H1)))))))).
-
-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 (ex C (\lambda (e2: C).(getl (trans hds i) c2 
-(CHead e2 (Bind b) (ctrans 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 
-(ex C (\lambda (e2: C).(getl (trans p i) c2 (CHead e2 (Bind b) (ctrans 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 H1 \def (match H 
-return (\lambda (p: PList).(\lambda (c: C).(\lambda (c0: C).(\lambda (_: 
-(drop1 p c c0)).((eq PList p PNil) \to ((eq C c c2) \to ((eq C c0 c1) \to (ex 
-C (\lambda (e2: C).(getl i c2 (CHead e2 (Bind b) v))))))))))) with 
-[(drop1_nil c) \Rightarrow (\lambda (_: (eq PList PNil PNil)).(\lambda (H2: 
-(eq C c c2)).(\lambda (H3: (eq C c c1)).(eq_ind C c2 (\lambda (c0: C).((eq C 
-c0 c1) \to (ex C (\lambda (e2: C).(getl i c2 (CHead e2 (Bind b) v)))))) 
-(\lambda (H4: (eq C c2 c1)).(eq_ind C c1 (\lambda (c0: C).(ex C (\lambda (e2: 
-C).(getl i c0 (CHead e2 (Bind b) v))))) (ex_intro C (\lambda (e2: C).(getl i 
-c1 (CHead e2 (Bind b) v))) e1 H0) c2 (sym_eq C c2 c1 H4))) c (sym_eq C c c2 
-H2) H3)))) | (drop1_cons c0 c3 h d H1 c4 hds H2) \Rightarrow (\lambda (H3: 
-(eq PList (PCons h d hds) PNil)).(\lambda (H4: (eq C c0 c2)).(\lambda (H5: 
-(eq C c4 c1)).((let H6 \def (eq_ind PList (PCons h d hds) (\lambda (e: 
-PList).(match e return (\lambda (_: PList).Prop) with [PNil \Rightarrow False 
-| (PCons _ _ _) \Rightarrow True])) I PNil H3) in (False_ind ((eq C c0 c2) 
-\to ((eq C c4 c1) \to ((drop h d c0 c3) \to ((drop1 hds c3 c4) \to (ex C 
-(\lambda (e2: C).(getl i c2 (CHead e2 (Bind b) v)))))))) H6)) H4 H5 H1 
-H2))))]) in (H1 (refl_equal PList PNil) (refl_equal C c2) (refl_equal C 
-c1))))))))))) (\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 (ex C (\lambda (e2: C).(getl (trans hds0 i) 
-c2 (CHead e2 (Bind b) (ctrans 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 H2 \def (match H0 return (\lambda (p: 
-PList).(\lambda (c: C).(\lambda (c0: C).(\lambda (_: (drop1 p c c0)).((eq 
-PList p (PCons h d hds0)) \to ((eq C c c2) \to ((eq C c0 c1) \to (ex C 
-(\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) (match (blt (trans hds0 i) d) with [true \Rightarrow (lift h (minus 
-d (S (trans hds0 i))) (ctrans hds0 i v)) | false \Rightarrow (ctrans hds0 i 
-v)])))))))))))) with [(drop1_nil c) \Rightarrow (\lambda (H2: (eq PList PNil 
-(PCons h d hds0))).(\lambda (H3: (eq C c c2)).(\lambda (H4: (eq C c 
-c1)).((let H5 \def (eq_ind PList PNil (\lambda (e: PList).(match e return 
-(\lambda (_: PList).Prop) with [PNil \Rightarrow True | (PCons _ _ _) 
-\Rightarrow False])) I (PCons h d hds0) H2) in (False_ind ((eq C c c2) \to 
-((eq C c c1) \to (ex C (\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) (match (blt (trans hds0 i) d) with [true 
-\Rightarrow (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i v)) | false 
-\Rightarrow (ctrans hds0 i v)]))))))) H5)) H3 H4)))) | (drop1_cons c0 c3 h0 
-d0 H2 c4 hds0 H3) \Rightarrow (\lambda (H4: (eq PList (PCons h0 d0 hds0) 
-(PCons h d hds0))).(\lambda (H5: (eq C c0 c2)).(\lambda (H6: (eq C c4 
-c1)).((let H7 \def (f_equal PList PList (\lambda (e: PList).(match e return 
-(\lambda (_: PList).PList) with [PNil \Rightarrow hds0 | (PCons _ _ p) 
-\Rightarrow p])) (PCons h0 d0 hds0) (PCons h d hds0) H4) in ((let H8 \def 
-(f_equal PList nat (\lambda (e: PList).(match e return (\lambda (_: 
-PList).nat) with [PNil \Rightarrow d0 | (PCons _ n _) \Rightarrow n])) (PCons 
-h0 d0 hds0) (PCons h d hds0) H4) in ((let H9 \def (f_equal PList nat (\lambda 
-(e: PList).(match e return (\lambda (_: PList).nat) with [PNil \Rightarrow h0 
-| (PCons n _ _) \Rightarrow n])) (PCons h0 d0 hds0) (PCons h d hds0) H4) in 
-(eq_ind nat h (\lambda (n: nat).((eq nat d0 d) \to ((eq PList hds0 hds0) \to 
-((eq C c0 c2) \to ((eq C c4 c1) \to ((drop n d0 c0 c3) \to ((drop1 hds0 c3 
-c4) \to (ex C (\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) (match (blt (trans hds0 i) d) with [true \Rightarrow (lift 
-h (minus d (S (trans hds0 i))) (ctrans hds0 i v)) | false \Rightarrow (ctrans 
-hds0 i v)])))))))))))) (\lambda (H10: (eq nat d0 d)).(eq_ind nat d (\lambda 
-(n: nat).((eq PList hds0 hds0) \to ((eq C c0 c2) \to ((eq C c4 c1) \to ((drop 
-h n c0 c3) \to ((drop1 hds0 c3 c4) \to (ex C (\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) (match (blt 
-(trans hds0 i) d) with [true \Rightarrow (lift h (minus d (S (trans hds0 i))) 
-(ctrans hds0 i v)) | false \Rightarrow (ctrans hds0 i v)]))))))))))) (\lambda 
-(H11: (eq PList hds0 hds0)).(eq_ind PList hds0 (\lambda (p: PList).((eq C c0 
-c2) \to ((eq C c4 c1) \to ((drop h d c0 c3) \to ((drop1 p c3 c4) \to (ex C 
-(\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) (match (blt (trans hds0 i) d) with [true \Rightarrow (lift h (minus 
-d (S (trans hds0 i))) (ctrans hds0 i v)) | false \Rightarrow (ctrans hds0 i 
-v)])))))))))) (\lambda (H12: (eq C c0 c2)).(eq_ind C c2 (\lambda (c: C).((eq 
-C c4 c1) \to ((drop h d c c3) \to ((drop1 hds0 c3 c4) \to (ex C (\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) (match 
-(blt (trans hds0 i) d) with [true \Rightarrow (lift h (minus d (S (trans hds0 
-i))) (ctrans hds0 i v)) | false \Rightarrow (ctrans hds0 i v)]))))))))) 
-(\lambda (H13: (eq C c4 c1)).(eq_ind C c1 (\lambda (c: C).((drop h d c2 c3) 
-\to ((drop1 hds0 c3 c) \to (ex C (\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) (match (blt (trans hds0 i) d) with 
-[true \Rightarrow (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i v)) | 
-false \Rightarrow (ctrans hds0 i v)])))))))) (\lambda (H14: (drop h d c2 
-c3)).(\lambda (H15: (drop1 hds0 c3 c1)).(xinduction bool (blt (trans hds0 i) 
-d) (\lambda (b0: bool).(ex C (\lambda (e2: C).(getl (match b0 with [true 
-\Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 
-(CHead e2 (Bind b) (match b0 with [true \Rightarrow (lift h (minus d (S 
-(trans hds0 i))) (ctrans hds0 i v)) | false \Rightarrow (ctrans hds0 i 
-v)])))))) (\lambda (x_x: bool).(bool_ind (\lambda (b0: bool).((eq bool (blt 
-(trans hds0 i) d) b0) \to (ex C (\lambda (e2: C).(getl (match b0 with [true 
-\Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 
-(CHead e2 (Bind b) (match b0 with [true \Rightarrow (lift h (minus d (S 
-(trans hds0 i))) (ctrans hds0 i v)) | false \Rightarrow (ctrans hds0 i 
-v)]))))))) (\lambda (H0: (eq bool (blt (trans hds0 i) d) true)).(let H_x \def 
-(H c1 c3 H15 b e1 v i H1) in (let H16 \def H_x in (ex_ind C (\lambda (e2: 
-C).(getl (trans hds0 i) c3 (CHead e2 (Bind b) (ctrans hds0 i v)))) (ex C 
-(\lambda (e2: C).(getl (trans hds0 i) c2 (CHead e2 (Bind b) (lift h (minus d 
-(S (trans hds0 i))) (ctrans hds0 i v)))))) (\lambda (x: C).(\lambda (H17: 
-(getl (trans hds0 i) c3 (CHead x (Bind b) (ctrans hds0 i v)))).(let H_x0 \def 
-(drop_getl_trans_lt (trans hds0 i) d (le_S_n (S (trans hds0 i)) d (lt_le_S (S 
-(trans hds0 i)) (S d) (blt_lt (S d) (S (trans hds0 i)) H0))) c2 c3 h H14 b x 
-(ctrans hds0 i v) H17) in (let H \def H_x0 in (ex2_ind C (\lambda (e1: 
-C).(getl (trans hds0 i) c2 (CHead e1 (Bind b) (lift h (minus d (S (trans hds0 
-i))) (ctrans hds0 i v))))) (\lambda (e1: C).(drop h (minus d (S (trans hds0 
-i))) e1 x)) (ex C (\lambda (e2: C).(getl (trans hds0 i) c2 (CHead e2 (Bind b) 
-(lift h (minus d (S (trans hds0 i))) (ctrans hds0 i v)))))) (\lambda (x0: 
-C).(\lambda (H1: (getl (trans hds0 i) c2 (CHead x0 (Bind b) (lift h (minus d 
-(S (trans hds0 i))) (ctrans hds0 i v))))).(\lambda (_: (drop h (minus d (S 
-(trans hds0 i))) x0 x)).(ex_intro C (\lambda (e2: C).(getl (trans hds0 i) c2 
-(CHead e2 (Bind b) (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i v))))) 
-x0 H1)))) H))))) H16)))) (\lambda (H0: (eq bool (blt (trans hds0 i) d) 
-false)).(let H_x \def (H c1 c3 H15 b e1 v i H1) in (let H16 \def H_x in 
-(ex_ind C (\lambda (e2: C).(getl (trans hds0 i) c3 (CHead e2 (Bind b) (ctrans 
-hds0 i v)))) (ex C (\lambda (e2: C).(getl (plus (trans hds0 i) h) c2 (CHead 
-e2 (Bind b) (ctrans hds0 i v))))) (\lambda (x: C).(\lambda (H17: (getl (trans 
-hds0 i) c3 (CHead x (Bind b) (ctrans hds0 i v)))).(let H \def 
-(drop_getl_trans_ge (trans hds0 i) c2 c3 d h H14 (CHead x (Bind b) (ctrans 
-hds0 i v)) H17) in (ex_intro C (\lambda (e2: C).(getl (plus (trans hds0 i) h) 
-c2 (CHead e2 (Bind b) (ctrans hds0 i v)))) x (H (bge_le d (trans hds0 i) 
-H0)))))) H16)))) x_x))))) c4 (sym_eq C c4 c1 H13))) c0 (sym_eq C c0 c2 H12))) 
-hds0 (sym_eq PList hds0 hds0 H11))) d0 (sym_eq nat d0 d H10))) h0 (sym_eq nat 
-h0 h H9))) H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal PList (PCons h d 
-hds0)) (refl_equal C c2) (refl_equal C c1))))))))))))))) hds).
-
-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 (plus (S 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 (plus (S h) i) c2 (CHead x0 
-(Bind b0) w0))).(let H_y0 \def (getl_conf_le (plus (S h) i) (CHead x0 (Bind 
-b0) w0) c2 H5 (CHead x (Bind b) w) i H2) in (let H6 \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))) (H_y0 (le_plus_r (S h) i)) (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 H6)))))) H4))))))))) H2))) 
-H1)))))))))).
-
-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)))))))))).
-
-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).(let H0 \def (match H return 
-(\lambda (n0: nat).(\lambda (t: T).(\lambda (t0: T).(\lambda (t1: T).(\lambda 
-(_: (subst0 n0 t t0 t1)).((eq nat n0 i) \to ((eq T t v) \to ((eq T t0 (TSort 
-n)) \to ((eq T t1 x) \to P))))))))) with [(subst0_lref v0 i0) \Rightarrow 
-(\lambda (H0: (eq nat i0 i)).(\lambda (H1: (eq T v0 v)).(\lambda (H2: (eq T 
-(TLRef i0) (TSort n))).(\lambda (H3: (eq T (lift (S i0) O v0) x)).(eq_ind nat 
-i (\lambda (n0: nat).((eq T v0 v) \to ((eq T (TLRef n0) (TSort n)) \to ((eq T 
-(lift (S n0) O v0) x) \to P)))) (\lambda (H4: (eq T v0 v)).(eq_ind T v 
-(\lambda (t: T).((eq T (TLRef i) (TSort n)) \to ((eq T (lift (S i) O t) x) 
-\to P))) (\lambda (H5: (eq T (TLRef i) (TSort n))).(let H6 \def (eq_ind T 
-(TLRef i) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with [(TSort 
-_) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow 
-False])) I (TSort n) H5) in (False_ind ((eq T (lift (S i) O v) x) \to P) 
-H6))) v0 (sym_eq T v0 v H4))) i0 (sym_eq nat i0 i H0) H1 H2 H3))))) | 
-(subst0_fst v0 u2 u1 i0 H0 t k) \Rightarrow (\lambda (H1: (eq nat i0 
-i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3: (eq T (THead k u1 t) (TSort 
-n))).(\lambda (H4: (eq T (THead k u2 t) x)).(eq_ind nat i (\lambda (n0: 
-nat).((eq T v0 v) \to ((eq T (THead k u1 t) (TSort n)) \to ((eq T (THead k u2 
-t) x) \to ((subst0 n0 v0 u1 u2) \to P))))) (\lambda (H5: (eq T v0 v)).(eq_ind 
-T v (\lambda (t0: T).((eq T (THead k u1 t) (TSort n)) \to ((eq T (THead k u2 
-t) x) \to ((subst0 i t0 u1 u2) \to P)))) (\lambda (H6: (eq T (THead k u1 t) 
-(TSort n))).(let H7 \def (eq_ind T (THead k u1 t) (\lambda (e: T).(match e 
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H6) in 
-(False_ind ((eq T (THead k u2 t) x) \to ((subst0 i v u1 u2) \to P)) H7))) v0 
-(sym_eq T v0 v H5))) i0 (sym_eq nat i0 i H1) H2 H3 H4 H0))))) | (subst0_snd k 
-v0 t2 t1 i0 H0 u) \Rightarrow (\lambda (H1: (eq nat i0 i)).(\lambda (H2: (eq 
-T v0 v)).(\lambda (H3: (eq T (THead k u t1) (TSort n))).(\lambda (H4: (eq T 
-(THead k u t2) x)).(eq_ind nat i (\lambda (n0: nat).((eq T v0 v) \to ((eq T 
-(THead k u t1) (TSort n)) \to ((eq T (THead k u t2) x) \to ((subst0 (s k n0) 
-v0 t1 t2) \to P))))) (\lambda (H5: (eq T v0 v)).(eq_ind T v (\lambda (t: 
-T).((eq T (THead k u t1) (TSort n)) \to ((eq T (THead k u t2) x) \to ((subst0 
-(s k i) t t1 t2) \to P)))) (\lambda (H6: (eq T (THead k u t1) (TSort 
-n))).(let H7 \def (eq_ind T (THead k u t1) (\lambda (e: T).(match e return 
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H6) in 
-(False_ind ((eq T (THead k u t2) x) \to ((subst0 (s k i) v t1 t2) \to P)) 
-H7))) v0 (sym_eq T v0 v H5))) i0 (sym_eq nat i0 i H1) H2 H3 H4 H0))))) | 
-(subst0_both v0 u1 u2 i0 H0 k t1 t2 H1) \Rightarrow (\lambda (H2: (eq nat i0 
-i)).(\lambda (H3: (eq T v0 v)).(\lambda (H4: (eq T (THead k u1 t1) (TSort 
-n))).(\lambda (H5: (eq T (THead k u2 t2) x)).(eq_ind nat i (\lambda (n0: 
-nat).((eq T v0 v) \to ((eq T (THead k u1 t1) (TSort n)) \to ((eq T (THead k 
-u2 t2) x) \to ((subst0 n0 v0 u1 u2) \to ((subst0 (s k n0) v0 t1 t2) \to 
-P)))))) (\lambda (H6: (eq T v0 v)).(eq_ind T v (\lambda (t: T).((eq T (THead 
-k u1 t1) (TSort n)) \to ((eq T (THead k u2 t2) x) \to ((subst0 i t u1 u2) \to 
-((subst0 (s k i) t t1 t2) \to P))))) (\lambda (H7: (eq T (THead k u1 t1) 
-(TSort n))).(let H8 \def (eq_ind T (THead k u1 t1) (\lambda (e: T).(match e 
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H7) in 
-(False_ind ((eq T (THead k u2 t2) x) \to ((subst0 i v u1 u2) \to ((subst0 (s 
-k i) v t1 t2) \to P))) H8))) v0 (sym_eq T v0 v H6))) i0 (sym_eq nat i0 i H2) 
-H3 H4 H5 H0 H1)))))]) in (H0 (refl_equal nat i) (refl_equal T v) (refl_equal 
-T (TSort n)) (refl_equal T x)))))))).
-
-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)).(let H0 \def (match H return (\lambda (n0: 
-nat).(\lambda (t: T).(\lambda (t0: T).(\lambda (t1: T).(\lambda (_: (subst0 
-n0 t t0 t1)).((eq nat n0 i) \to ((eq T t v) \to ((eq T t0 (TLRef n)) \to ((eq 
-T t1 x) \to (land (eq nat n i) (eq T x (lift (S n) O v)))))))))))) with 
-[(subst0_lref v0 i0) \Rightarrow (\lambda (H0: (eq nat i0 i)).(\lambda (H1: 
-(eq T v0 v)).(\lambda (H2: (eq T (TLRef i0) (TLRef n))).(\lambda (H3: (eq T 
-(lift (S i0) O v0) x)).(eq_ind nat i (\lambda (n0: nat).((eq T v0 v) \to ((eq 
-T (TLRef n0) (TLRef n)) \to ((eq T (lift (S n0) O v0) x) \to (land (eq nat n 
-i) (eq T x (lift (S n) O v))))))) (\lambda (H4: (eq T v0 v)).(eq_ind T v 
-(\lambda (t: T).((eq T (TLRef i) (TLRef n)) \to ((eq T (lift (S i) O t) x) 
-\to (land (eq nat n i) (eq T x (lift (S n) O v)))))) (\lambda (H5: (eq T 
-(TLRef i) (TLRef n))).(let H6 \def (f_equal T nat (\lambda (e: T).(match e 
-return (\lambda (_: T).nat) with [(TSort _) \Rightarrow i | (TLRef n) 
-\Rightarrow n | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef n) H5) in 
-(eq_ind nat n (\lambda (n0: nat).((eq T (lift (S n0) O v) x) \to (land (eq 
-nat n n0) (eq T x (lift (S n) O v))))) (\lambda (H7: (eq T (lift (S n) O v) 
-x)).(eq_ind T (lift (S n) O v) (\lambda (t: T).(land (eq nat n n) (eq T t 
-(lift (S n) O v)))) (conj (eq nat n n) (eq T (lift (S n) O v) (lift (S n) O 
-v)) (refl_equal nat n) (refl_equal T (lift (S n) O v))) x H7)) i (sym_eq nat 
-i n H6)))) v0 (sym_eq T v0 v H4))) i0 (sym_eq nat i0 i H0) H1 H2 H3))))) | 
-(subst0_fst v0 u2 u1 i0 H0 t k) \Rightarrow (\lambda (H1: (eq nat i0 
-i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3: (eq T (THead k u1 t) (TLRef 
-n))).(\lambda (H4: (eq T (THead k u2 t) x)).(eq_ind nat i (\lambda (n0: 
-nat).((eq T v0 v) \to ((eq T (THead k u1 t) (TLRef n)) \to ((eq T (THead k u2 
-t) x) \to ((subst0 n0 v0 u1 u2) \to (land (eq nat n i) (eq T x (lift (S n) O 
-v)))))))) (\lambda (H5: (eq T v0 v)).(eq_ind T v (\lambda (t0: T).((eq T 
-(THead k u1 t) (TLRef n)) \to ((eq T (THead k u2 t) x) \to ((subst0 i t0 u1 
-u2) \to (land (eq nat n i) (eq T x (lift (S n) O v))))))) (\lambda (H6: (eq T 
-(THead k u1 t) (TLRef n))).(let H7 \def (eq_ind T (THead k u1 t) (\lambda (e: 
-T).(match e return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | 
-(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) 
-H6) in (False_ind ((eq T (THead k u2 t) x) \to ((subst0 i v u1 u2) \to (land 
-(eq nat n i) (eq T x (lift (S n) O v))))) H7))) v0 (sym_eq T v0 v H5))) i0 
-(sym_eq nat i0 i H1) H2 H3 H4 H0))))) | (subst0_snd k v0 t2 t1 i0 H0 u) 
-\Rightarrow (\lambda (H1: (eq nat i0 i)).(\lambda (H2: (eq T v0 v)).(\lambda 
-(H3: (eq T (THead k u t1) (TLRef n))).(\lambda (H4: (eq T (THead k u t2) 
-x)).(eq_ind nat i (\lambda (n0: nat).((eq T v0 v) \to ((eq T (THead k u t1) 
-(TLRef n)) \to ((eq T (THead k u t2) x) \to ((subst0 (s k n0) v0 t1 t2) \to 
-(land (eq nat n i) (eq T x (lift (S n) O v)))))))) (\lambda (H5: (eq T v0 
-v)).(eq_ind T v (\lambda (t: T).((eq T (THead k u t1) (TLRef n)) \to ((eq T 
-(THead k u t2) x) \to ((subst0 (s k i) t t1 t2) \to (land (eq nat n i) (eq T 
-x (lift (S n) O v))))))) (\lambda (H6: (eq T (THead k u t1) (TLRef n))).(let 
-H7 \def (eq_ind T (THead k u t1) (\lambda (e: T).(match e return (\lambda (_: 
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
-(THead _ _ _) \Rightarrow True])) I (TLRef n) H6) in (False_ind ((eq T (THead 
-k u t2) x) \to ((subst0 (s k i) v t1 t2) \to (land (eq nat n i) (eq T x (lift 
-(S n) O v))))) H7))) v0 (sym_eq T v0 v H5))) i0 (sym_eq nat i0 i H1) H2 H3 H4 
-H0))))) | (subst0_both v0 u1 u2 i0 H0 k t1 t2 H1) \Rightarrow (\lambda (H2: 
-(eq nat i0 i)).(\lambda (H3: (eq T v0 v)).(\lambda (H4: (eq T (THead k u1 t1) 
-(TLRef n))).(\lambda (H5: (eq T (THead k u2 t2) x)).(eq_ind nat i (\lambda 
-(n0: nat).((eq T v0 v) \to ((eq T (THead k u1 t1) (TLRef n)) \to ((eq T 
-(THead k u2 t2) x) \to ((subst0 n0 v0 u1 u2) \to ((subst0 (s k n0) v0 t1 t2) 
-\to (land (eq nat n i) (eq T x (lift (S n) O v))))))))) (\lambda (H6: (eq T 
-v0 v)).(eq_ind T v (\lambda (t: T).((eq T (THead k u1 t1) (TLRef n)) \to ((eq 
-T (THead k u2 t2) x) \to ((subst0 i t u1 u2) \to ((subst0 (s k i) t t1 t2) 
-\to (land (eq nat n i) (eq T x (lift (S n) O v)))))))) (\lambda (H7: (eq T 
-(THead k u1 t1) (TLRef n))).(let H8 \def (eq_ind T (THead k u1 t1) (\lambda 
-(e: T).(match e return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I 
-(TLRef n) H7) in (False_ind ((eq T (THead k u2 t2) x) \to ((subst0 i v u1 u2) 
-\to ((subst0 (s k i) v t1 t2) \to (land (eq nat n i) (eq T x (lift (S n) O 
-v)))))) H8))) v0 (sym_eq T v0 v H6))) i0 (sym_eq nat i0 i H2) H3 H4 H5 H0 
-H1)))))]) in (H0 (refl_equal nat i) (refl_equal T v) (refl_equal T (TLRef n)) 
-(refl_equal T x))))))).
-
-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)).(let H0 
-\def (match H return (\lambda (n: nat).(\lambda (t: T).(\lambda (t0: 
-T).(\lambda (t2: T).(\lambda (_: (subst0 n t t0 t2)).((eq nat n i) \to ((eq T 
-t v) \to ((eq T t0 (THead k u1 t1)) \to ((eq T t2 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 (t3: T).(eq T x (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 x (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)))))))))))))) with [(subst0_lref 
-v0 i0) \Rightarrow (\lambda (H0: (eq nat i0 i)).(\lambda (H1: (eq T v0 
-v)).(\lambda (H2: (eq T (TLRef i0) (THead k u1 t1))).(\lambda (H3: (eq T 
-(lift (S i0) O v0) x)).(eq_ind nat i (\lambda (n: nat).((eq T v0 v) \to ((eq 
-T (TLRef n) (THead k u1 t1)) \to ((eq T (lift (S n) O v0) 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))))))))) 
-(\lambda (H4: (eq T v0 v)).(eq_ind T v (\lambda (t: T).((eq T (TLRef i) 
-(THead k u1 t1)) \to ((eq T (lift (S i) O t) 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)))))))) (\lambda (H5: (eq T 
-(TLRef i) (THead k u1 t1))).(let H6 \def (eq_ind T (TLRef i) (\lambda (e: 
-T).(match e return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | 
-(TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead k u1 
-t1) H5) in (False_ind ((eq T (lift (S i) O v) 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)))))) H6))) v0 (sym_eq T v0 v 
-H4))) i0 (sym_eq nat i0 i H0) H1 H2 H3))))) | (subst0_fst v0 u2 u0 i0 H0 t 
-k0) \Rightarrow (\lambda (H1: (eq nat i0 i)).(\lambda (H2: (eq T v0 
-v)).(\lambda (H3: (eq T (THead k0 u0 t) (THead k u1 t1))).(\lambda (H4: (eq T 
-(THead k0 u2 t) x)).(eq_ind nat i (\lambda (n: nat).((eq T v0 v) \to ((eq T 
-(THead k0 u0 t) (THead k u1 t1)) \to ((eq T (THead k0 u2 t) x) \to ((subst0 n 
-v0 u0 u2) \to (or3 (ex2 T (\lambda (u3: T).(eq T x (THead k u3 t1))) (\lambda 
-(u3: T).(subst0 i v u1 u3))) (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 (u3: 
-T).(\lambda (t2: T).(eq T x (THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: 
-T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 
-t2)))))))))) (\lambda (H5: (eq T v0 v)).(eq_ind T v (\lambda (t0: T).((eq T 
-(THead k0 u0 t) (THead k u1 t1)) \to ((eq T (THead k0 u2 t) x) \to ((subst0 i 
-t0 u0 u2) \to (or3 (ex2 T (\lambda (u3: T).(eq T x (THead k u3 t1))) (\lambda 
-(u3: T).(subst0 i v u1 u3))) (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 (u3: 
-T).(\lambda (t2: T).(eq T x (THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: 
-T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 
-t2))))))))) (\lambda (H6: (eq T (THead k0 u0 t) (THead k u1 t1))).(let H7 
-\def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t) 
-\Rightarrow t])) (THead k0 u0 t) (THead k u1 t1) H6) in ((let H8 \def 
-(f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort 
-_) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) 
-(THead k0 u0 t) (THead k u1 t1) H6) in ((let H9 \def (f_equal T K (\lambda 
-(e: T).(match e return (\lambda (_: T).K) with [(TSort _) \Rightarrow k0 | 
-(TLRef _) \Rightarrow k0 | (THead k _ _) \Rightarrow k])) (THead k0 u0 t) 
-(THead k u1 t1) H6) in (eq_ind K k (\lambda (k1: K).((eq T u0 u1) \to ((eq T 
-t t1) \to ((eq T (THead k1 u2 t) x) \to ((subst0 i v u0 u2) \to (or3 (ex2 T 
-(\lambda (u3: T).(eq T x (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 
-u3))) (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 (u3: T).(\lambda (t2: 
-T).(eq T x (THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 
-u3))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 t2)))))))))) 
-(\lambda (H10: (eq T u0 u1)).(eq_ind T u1 (\lambda (t0: T).((eq T t t1) \to 
-((eq T (THead k u2 t) x) \to ((subst0 i v t0 u2) \to (or3 (ex2 T (\lambda 
-(u3: T).(eq T x (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (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 (u3: T).(\lambda (t2: T).(eq T x (THead k u3 
-t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: 
-T).(\lambda (t2: T).(subst0 (s k i) v t1 t2))))))))) (\lambda (H11: (eq T t 
-t1)).(eq_ind T t1 (\lambda (t0: T).((eq T (THead k u2 t0) x) \to ((subst0 i v 
-u1 u2) \to (or3 (ex2 T (\lambda (u3: T).(eq T x (THead k u3 t1))) (\lambda 
-(u3: T).(subst0 i v u1 u3))) (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 (u3: 
-T).(\lambda (t2: T).(eq T x (THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: 
-T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 
-t2)))))))) (\lambda (H12: (eq T (THead k u2 t1) x)).(eq_ind T (THead k u2 t1) 
-(\lambda (t0: T).((subst0 i v u1 u2) \to (or3 (ex2 T (\lambda (u3: T).(eq T 
-t0 (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda 
-(t2: T).(eq T t0 (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 
-t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T t0 (THead k u3 
-t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: 
-T).(\lambda (t2: T).(subst0 (s k i) v t1 t2))))))) (\lambda (H13: (subst0 i v 
-u1 u2)).(or3_intro0 (ex2 T (\lambda (u3: T).(eq T (THead k u2 t1) (THead k u3 
-t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t2: T).(eq T 
-(THead k u2 t1) (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v 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 i v u1 u3))) (\lambda (_: 
-T).(\lambda (t2: T).(subst0 (s k i) v t1 t2)))) (ex_intro2 T (\lambda (u3: 
-T).(eq T (THead k u2 t1) (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 
-u3)) u2 (refl_equal T (THead k u2 t1)) H13))) x H12)) t (sym_eq T t t1 H11))) 
-u0 (sym_eq T u0 u1 H10))) k0 (sym_eq K k0 k H9))) H8)) H7))) v0 (sym_eq T v0 
-v H5))) i0 (sym_eq nat i0 i H1) H2 H3 H4 H0))))) | (subst0_snd k0 v0 t2 t0 i0 
-H0 u) \Rightarrow (\lambda (H1: (eq nat i0 i)).(\lambda (H2: (eq T v0 
-v)).(\lambda (H3: (eq T (THead k0 u t0) (THead k u1 t1))).(\lambda (H4: (eq T 
-(THead k0 u t2) x)).(eq_ind nat i (\lambda (n: nat).((eq T v0 v) \to ((eq T 
-(THead k0 u t0) (THead k u1 t1)) \to ((eq T (THead k0 u t2) x) \to ((subst0 
-(s k0 n) v0 t0 t2) \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 (t3: T).(eq T x (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 x (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)))))))))) (\lambda (H5: (eq T v0 v)).(eq_ind T v (\lambda (t: T).((eq 
-T (THead k0 u t0) (THead k u1 t1)) \to ((eq T (THead k0 u t2) x) \to ((subst0 
-(s k0 i) t t0 t2) \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 (t3: T).(eq T x (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 x (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))))))))) (\lambda (H6: (eq T (THead k0 u t0) (THead k u1 t1))).(let 
-H7 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) 
-\Rightarrow t])) (THead k0 u t0) (THead k u1 t1) H6) in ((let H8 \def 
-(f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort 
-_) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) 
-(THead k0 u t0) (THead k u1 t1) H6) in ((let H9 \def (f_equal T K (\lambda 
-(e: T).(match e return (\lambda (_: T).K) with [(TSort _) \Rightarrow k0 | 
-(TLRef _) \Rightarrow k0 | (THead k _ _) \Rightarrow k])) (THead k0 u t0) 
-(THead k u1 t1) H6) in (eq_ind K k (\lambda (k1: K).((eq T u u1) \to ((eq T 
-t0 t1) \to ((eq T (THead k1 u t2) x) \to ((subst0 (s k1 i) v t0 t2) \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 (t3: T).(eq T x (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 x (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)))))))))) 
-(\lambda (H10: (eq T u u1)).(eq_ind T u1 (\lambda (t: T).((eq T t0 t1) \to 
-((eq T (THead k t t2) x) \to ((subst0 (s k i) v t0 t2) \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 (t3: T).(eq T x (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 x (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))))))))) 
-(\lambda (H11: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T (THead k u1 
-t2) x) \to ((subst0 (s k i) v t t2) \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 (t3: 
-T).(eq T x (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 x (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)))))))) (\lambda (H12: (eq T 
-(THead k u1 t2) x)).(eq_ind T (THead k u1 t2) (\lambda (t: T).((subst0 (s k 
-i) v t1 t2) \to (or3 (ex2 T (\lambda (u2: T).(eq T t (THead k u2 t1))) 
-(\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t3: T).(eq T t (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 t (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))))))) (\lambda (H13: (subst0 (s k i) v t1 t2)).(or3_intro1 (ex2 T 
-(\lambda (u2: T).(eq T (THead k u1 t2) (THead k u2 t1))) (\lambda (u2: 
-T).(subst0 i v u1 u2))) (ex2 T (\lambda (t3: T).(eq T (THead k u1 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 (THead k u1 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)))) (ex_intro2 T (\lambda (t3: T).(eq T (THead k 
-u1 t2) (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3)) t2 
-(refl_equal T (THead k u1 t2)) H13))) x H12)) t0 (sym_eq T t0 t1 H11))) u 
-(sym_eq T u u1 H10))) k0 (sym_eq K k0 k H9))) H8)) H7))) v0 (sym_eq T v0 v 
-H5))) i0 (sym_eq nat i0 i H1) H2 H3 H4 H0))))) | (subst0_both v0 u0 u2 i0 H0 
-k0 t0 t2 H1) \Rightarrow (\lambda (H2: (eq nat i0 i)).(\lambda (H3: (eq T v0 
-v)).(\lambda (H4: (eq T (THead k0 u0 t0) (THead k u1 t1))).(\lambda (H5: (eq 
-T (THead k0 u2 t2) x)).(eq_ind nat i (\lambda (n: nat).((eq T v0 v) \to ((eq 
-T (THead k0 u0 t0) (THead k u1 t1)) \to ((eq T (THead k0 u2 t2) x) \to 
-((subst0 n v0 u0 u2) \to ((subst0 (s k0 n) v0 t0 t2) \to (or3 (ex2 T (\lambda 
-(u3: T).(eq T x (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 
-T (\lambda (t3: T).(eq T x (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) 
-v t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead k u3 
-t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: 
-T).(\lambda (t3: T).(subst0 (s k i) v t1 t3))))))))))) (\lambda (H6: (eq T v0 
-v)).(eq_ind T v (\lambda (t: T).((eq T (THead k0 u0 t0) (THead k u1 t1)) \to 
-((eq T (THead k0 u2 t2) x) \to ((subst0 i t u0 u2) \to ((subst0 (s k0 i) t t0 
-t2) \to (or3 (ex2 T (\lambda (u3: T).(eq T x (THead k u3 t1))) (\lambda (u3: 
-T).(subst0 i v u1 u3))) (ex2 T (\lambda (t3: T).(eq T x (THead k u1 t3))) 
-(\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t3: T).(eq T x (THead k u3 t3)))) (\lambda (u3: T).(\lambda (_: 
-T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 
-t3)))))))))) (\lambda (H7: (eq T (THead k0 u0 t0) (THead k u1 t1))).(let H8 
-\def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) 
-\Rightarrow t])) (THead k0 u0 t0) (THead k u1 t1) H7) in ((let H9 \def 
-(f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort 
-_) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) 
-(THead k0 u0 t0) (THead k u1 t1) H7) in ((let H10 \def (f_equal T K (\lambda 
-(e: T).(match e return (\lambda (_: T).K) with [(TSort _) \Rightarrow k0 | 
-(TLRef _) \Rightarrow k0 | (THead k _ _) \Rightarrow k])) (THead k0 u0 t0) 
-(THead k u1 t1) H7) in (eq_ind K k (\lambda (k1: K).((eq T u0 u1) \to ((eq T 
-t0 t1) \to ((eq T (THead k1 u2 t2) x) \to ((subst0 i v u0 u2) \to ((subst0 (s 
-k1 i) v t0 t2) \to (or3 (ex2 T (\lambda (u3: T).(eq T x (THead k u3 t1))) 
-(\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t3: T).(eq T x (THead 
-k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda 
-(u3: T).(\lambda (t3: T).(eq T x (THead k u3 t3)))) (\lambda (u3: T).(\lambda 
-(_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) 
-v t1 t3))))))))))) (\lambda (H11: (eq T u0 u1)).(eq_ind T u1 (\lambda (t: 
-T).((eq T t0 t1) \to ((eq T (THead k u2 t2) x) \to ((subst0 i v t u2) \to 
-((subst0 (s k i) v t0 t2) \to (or3 (ex2 T (\lambda (u3: T).(eq T x (THead k 
-u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t3: T).(eq T 
-x (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead k u3 t3)))) (\lambda (u3: 
-T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t3: 
-T).(subst0 (s k i) v t1 t3)))))))))) (\lambda (H12: (eq T t0 t1)).(eq_ind T 
-t1 (\lambda (t: T).((eq T (THead k u2 t2) x) \to ((subst0 i v u1 u2) \to 
-((subst0 (s k i) v t t2) \to (or3 (ex2 T (\lambda (u3: T).(eq T x (THead k u3 
-t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t3: T).(eq T x 
-(THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead k u3 t3)))) (\lambda (u3: 
-T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t3: 
-T).(subst0 (s k i) v t1 t3))))))))) (\lambda (H13: (eq T (THead k u2 t2) 
-x)).(eq_ind T (THead k u2 t2) (\lambda (t: T).((subst0 i v u1 u2) \to 
-((subst0 (s k i) v t1 t2) \to (or3 (ex2 T (\lambda (u3: T).(eq T t (THead k 
-u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t3: T).(eq T 
-t (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t3: T).(eq T t (THead k u3 t3)))) (\lambda (u3: 
-T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t3: 
-T).(subst0 (s k i) v t1 t3)))))))) (\lambda (H14: (subst0 i v u1 
-u2)).(\lambda (H15: (subst0 (s k i) v t1 t2)).(or3_intro2 (ex2 T (\lambda 
-(u3: T).(eq T (THead k u2 t2) (THead k u3 t1))) (\lambda (u3: T).(subst0 i v 
-u1 u3))) (ex2 T (\lambda (t3: T).(eq T (THead k u2 t2) (THead k u1 t3))) 
-(\lambda (t3: T).(subst0 (s k i) v 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 i v u1 u3))) (\lambda (_: T).(\lambda (t3: 
-T).(subst0 (s k i) v 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 i v u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) 
-v t1 t3))) u2 t2 (refl_equal T (THead k u2 t2)) H14 H15)))) x H13)) t0 
-(sym_eq T t0 t1 H12))) u0 (sym_eq T u0 u1 H11))) k0 (sym_eq K k0 k H10))) 
-H9)) H8))) v0 (sym_eq T v0 v H6))) i0 (sym_eq nat i0 i H2) H3 H4 H5 H0 
-H1)))))]) in (H0 (refl_equal nat i) (refl_equal T v) (refl_equal T (THead k 
-u1 t1)) (refl_equal T x))))))))).
-
-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).(and_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 return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ t _) 
-\Rightarrow t])) (THead k t0 t1) (THead k x t1) H3) in (let H6 \def (eq_ind_r 
-T x (\lambda (t: T).(subst0 d u t0 t)) 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 return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ _ t) \Rightarrow t])) 
-(THead k t0 t1) (THead k t0 x) H3) in (let H6 \def (eq_ind_r T x (\lambda (t: 
-T).(subst0 (s k d) u t1 t)) 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 return (\lambda (_: T).T) with [(TSort 
-_) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ t _) \Rightarrow t])) 
-(THead k t0 t1) (THead k x0 x1) H3) in ((let H7 \def (f_equal T T (\lambda 
-(e: T).(match e return (\lambda (_: T).T) with [(TSort _) \Rightarrow t1 | 
-(TLRef _) \Rightarrow t1 | (THead _ _ t) \Rightarrow t])) (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 (t: T).(subst0 (s k d) u t1 t)) H5 t1 H7) in (let H10 \def 
-(eq_ind_r T x0 (\lambda (t: T).(subst0 d u t0 t)) 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_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 (and_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 (and_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 (t: T).(subst0 i (lift h d u) t 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 
-(t: T).(subst0 i u t 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 (t: T).(subst0 i u t x)) H1 
-(TLRef n) (lift_lref_lt n h d H2)) in (and_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 (n: nat).(lt n 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 (t: T).(subst0 i u t 
-x)) H1 (TLRef (plus n h)) (lift_lref_ge n h d H2)) in (and_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 (n: nat).(lt n (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 (t: T).(subst0 i u t 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 
-(and_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 (n: nat).(lt n 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 (and_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_comm n 
-h)) (lift h d (lift (S n) O u)) (lift_free u (S n) h O d (le_trans d (S n) 
-(plus O (S n)) (le_S d n H1) (le_n (plus O (S n)))) (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 (t: T).(subst0 
-i u t 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)).
-
-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 (d: nat).((lt (s k 
-i0) d) \to (\forall (h: nat).(subst0 (s k i0) (lift h (minus d n) v) (lift h 
-d t3) (lift h d 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 (d: nat).((lt (s k i0) d) \to (\forall (h: nat).(subst0 (s k 
-i0) (lift h (minus d n) v) (lift h d t0) (lift h d 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) (lt_le_S (s k i0) (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_comm 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 (d: nat).((le 
-d (s k i0)) \to (subst0 n v (lift h d t3) (lift h d 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 (d: nat).((le d (s k i0)) \to (subst0 n v (lift h d t0) (lift h 
-d 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_comm 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)))))))).
-
-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 (u2: T).(\forall (u: T).(\forall (i0: 
-nat).((subst0 i0 u u2 v) \to (ex2 T (\lambda (t: T).(subst0 i u2 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 (u2: 
-T).(\forall (u: T).(\forall (i0: nat).((subst0 i0 u u2 v) \to (ex2 T (\lambda 
-(t: T).(subst0 i u2 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 (u1: T).(\forall (u: 
-T).(\forall (i0: nat).((subst0 i0 u u1 v) \to (ex2 T (\lambda (t: T).(subst0 
-(s k i) u1 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 (u2: T).(\forall (u: T).(\forall (i0: nat).((subst0 i0 u v u2) \to 
-(ex2 T (\lambda (t: T).(subst0 i u2 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 (u2: T).(\forall (u: T).(\forall (i0: 
-nat).((subst0 i0 u v u2) \to (ex2 T (\lambda (t: T).(subst0 i u2 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 (u1: T).(\forall (u: T).(\forall (i0: 
-nat).((subst0 i0 u v u1) \to (ex2 T (\lambda (t: T).(subst0 (s k i) u1 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 (u2: T).(eq T t3 (THead k u2 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 (t2: T).(eq T t3 (THead k u2 t2))) (\lambda (t2: 
-T).(subst0 (s k i0) v0 t t2)))).(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 (u2: T).(\lambda (t2: T).(eq T 
-t3 (THead k u2 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) 
-(\lambda (_: T).(\lambda (t2: T).(subst0 (s k i0) v0 t t2))))).(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 (t2: T).(eq T t4 (THead k u 
-t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t0 t2)))).(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 (t2: T).(eq T t4 (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(subst0 i0 v0 u u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i0) v0 
-t0 t2))))).(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 (u2: T).(eq T t4 (THead k u2 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 (t2: T).(eq T t4 (THead k u2 t2))) 
-(\lambda (t2: T).(subst0 (s k i0) v0 t3 t2)))).(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 
-(u2: T).(\lambda (t2: T).(eq T t4 (THead k u2 t2)))) (\lambda (u3: 
-T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) (\lambda (_: T).(\lambda (t2: 
-T).(subst0 (s k i0) v0 t3 t2))))).(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))).(and_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)) 
-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 (u2: T).(eq T t2 (THead k u2 t))) (\lambda (u2: T).(subst0 i2 u3 u0 
-u2)))).(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 (t2: T).(subst0 (s k i2) u3 t t2)))).(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 (u2: T).(\lambda (t3: T).(eq 
-T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i2 u3 u0 
-u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i2) u3 t 
-t2))))).(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 (u2: T).(eq T t4 (THead k u2 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 (t2: T).(eq T t4 (THead k u t2))) (\lambda (t2: T).(subst0 (s k i2) 
-u2 t3 t2)))).(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 (\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 (u2: T).(\lambda (t2: T).(eq T t4 (THead k u2 t2)))) 
-(\lambda (u3: T).(\lambda (_: T).(subst0 i2 u2 u u3))) (\lambda (_: 
-T).(\lambda (t2: T).(subst0 (s k i2) u2 t3 t2))))).(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 (\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 (u2: T).(\forall (i2: nat).((subst0 i2 u2 t2 t4) \to ((not (eq 
-nat (s k i) i2)) \to (ex2 T (\lambda (t: T).(subst0 i2 u2 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 (u2: T).(eq T t4 (THead k u2 t2))) (\lambda 
-(u2: T).(subst0 i2 u3 u0 u2)))).(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 (t2: T).(eq T t4 (THead k u0 t2))) (\lambda (t3: T).(subst0 (s k i2) 
-u3 t2 t3)))).(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 
-(\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 (u2: T).(\lambda (t2: T).(eq T t4 
-(THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i2 u3 u0 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(subst0 (s k i2) u3 t2 t3))))).(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 (\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)).(and_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 (u2: T).(eq T t2 (THead k u2 
-t))) (\lambda (u2: T).(subst0 i0 v u1 u2)))).(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 (t: T).(subst0 i0 v u2 t)) 
-(\lambda (t: T).(subst0 i0 v x t)))).(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 
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-T (THead k u2 t3) (THead 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)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda 
-(t: T).(subst0 i0 v (THead k x0 x1) t)) (THead k x t3) (subst0_fst v x u2 i0 
-H11 t3 k) (subst0_both v x0 x i0 H12 k x1 t3 H9)))))) H10)) (\lambda (H10: 
-(subst0 i0 v u2 x0)).(or4_intro1 (eq T (THead k u2 t3) (THead 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)) (ex_intro2 T (\lambda (t: T).(subst0 i0 
-v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t)) (THead 
-k x0 t3) (subst0_fst v x0 u2 i0 H10 t3 k) (subst0_snd k v t3 x1 i0 H9 x0)))) 
-(\lambda (H10: (subst0 i0 v x0 u2)).(or4_intro3 (eq T (THead k u2 t3) (THead 
-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_equal 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_comm 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_comm 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_comm 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)))))))).
-
-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)) 
-(plus_le_compat (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)) (plus_le_compat (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)) (plus_le_compat (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))).(lt_le_S (plus (weight_map f0 u2) (weight_map f0 t0)) (S (plus 
-(weight_map g u1) (weight_map g t0))) (le_lt_n_Sm (plus (weight_map f0 u2) 
-(weight_map f0 t0)) (plus (weight_map g u1) (weight_map g t0)) 
-(plus_le_compat (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)) (plus_le_compat (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)) (lt_le_S (weight_map (wadd f (S (weight_map f u0))) (lift (S 
-(S i)) O v)) (wadd g (S (weight_map g u0)) (S i)) (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)) (plus_le_compat 
-(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)) (plus_le_compat (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 (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 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_le_S (plus (weight_map f0 u0) (weight_map f0 
-t2)) (S (plus (weight_map g u0) (weight_map g t1))) (le_lt_n_Sm (plus 
-(weight_map f0 u0) (weight_map f0 t2)) (plus (weight_map g u0) (weight_map g 
-t1)) (plus_le_compat (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)) (plus_le_compat (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)) (lt_le_S 
-(weight_map (wadd f (S (weight_map f u2))) (lift (S (S i)) O v)) (wadd g (S 
-(weight_map g u1)) (S i)) (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)) (plus_le_compat (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)) (plus_le_compat (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 
-(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 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_le_S (plus (weight_map f0 u2) (weight_map f0 t2)) (S (plus 
-(weight_map g u1) (weight_map g t1))) (le_lt_n_Sm (plus (weight_map f0 u2) 
-(weight_map f0 t2)) (plus (weight_map g u1) (weight_map g t1)) 
-(plus_le_compat (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)) 
-(plus_lt_le_compat (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_S (S (weight_map f u2)) (weight_map g u1) (lt_le_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)) (plus_lt_le_compat (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)) (plus_lt_le_compat (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)) (plus_lt_le_compat 
-(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)) (plus_le_lt_compat (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)) (lt_le_S (weight_map f u0) (S (weight_map g u0)) 
-(le_lt_n_Sm (weight_map f u0) (weight_map g u0) (weight_le u0 f g H2))) m)) 
-(lt_le_S (weight_map (wadd f (S (weight_map f u0))) (lift (S (S i)) O v)) 
-(wadd g (S (weight_map g u0)) (S i)) (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)) (plus_le_lt_compat (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)) (plus_le_lt_compat (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 (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 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)) (plus_le_lt_compat 
-(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)) (plus_lt_le_compat 
-(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_le f g H4 (S (weight_map f u2)) 
-(S (weight_map g u1)) (le_S (S (weight_map f u2)) (weight_map g u1) (lt_le_S 
-(weight_map f u2) (weight_map g u1) (H1 f g H4 H5))) m)) (lt_le_S (weight_map 
-(wadd f (S (weight_map f u2))) (lift (S (S i)) O v)) (wadd g (S (weight_map g 
-u1)) (S i)) (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)) (plus_lt_compat (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)))) (lt_le_S (weight_map (wadd f O) (lift (S (S i)) O v)) (wadd g O (S 
-i)) (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)) (plus_lt_compat (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)))) (lt_le_S (weight_map (wadd f 
-O) (lift (S (S i)) O v)) (wadd g O (S i)) (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 (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 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)) (plus_lt_compat (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)) (plus_le_lt_compat (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))))).
-
-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).(T_ind (\lambda (t0: T).(\forall (d: nat).(ex 
-T (\lambda (v: T).(or (subst0 d w t0 (lift (S O) d v)) (eq T t0 (lift (S O) d 
-v))))))) (\lambda (n: nat).(\lambda (d: nat).(ex_intro T (\lambda (v: T).(or 
-(subst0 d w (TSort n) (lift (S O) d v)) (eq T (TSort n) (lift (S O) d v)))) 
-(TSort n) (eq_ind_r T (TSort n) (\lambda (t0: T).(or (subst0 d w (TSort n) 
-t0) (eq T (TSort n) t0))) (or_intror (subst0 d w (TSort n) (TSort n)) (eq T 
-(TSort n) (TSort n)) (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 
-(ex T (\lambda (v: T).(or (subst0 d w (TLRef n) (lift (S O) d v)) (eq T 
-(TLRef n) (lift (S O) d v))))) (\lambda (H: (lt n d)).(ex_intro T (\lambda 
-(v: T).(or (subst0 d w (TLRef n) (lift (S O) d v)) (eq T (TLRef n) (lift (S 
-O) d v)))) (TLRef n) (eq_ind_r T (TLRef n) (\lambda (t0: T).(or (subst0 d w 
-(TLRef n) t0) (eq T (TLRef n) t0))) (or_intror (subst0 d w (TLRef n) (TLRef 
-n)) (eq T (TLRef n) (TLRef n)) (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).(ex T (\lambda (v: T).(or (subst0 n0 w (TLRef n) (lift (S 
-O) n0 v)) (eq T (TLRef n) (lift (S O) n0 v)))))) (ex_intro T (\lambda (v: 
-T).(or (subst0 n w (TLRef n) (lift (S O) n v)) (eq T (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).(or 
-(subst0 n w (TLRef n) t0) (eq T (TLRef n) t0))) (or_introl (subst0 n w (TLRef 
-n) (lift (S n) O w)) (eq T (TLRef n) (lift (S n) O w)) (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)).(ex_intro T (\lambda (v: T).(or 
-(subst0 d w (TLRef n) (lift (S O) d v)) (eq T (TLRef n) (lift (S O) d v)))) 
-(TLRef (pred n)) (eq_ind_r T (TLRef n) (\lambda (t0: T).(or (subst0 d w 
-(TLRef n) t0) (eq T (TLRef n) t0))) (or_intror (subst0 d w (TLRef n) (TLRef 
-n)) (eq T (TLRef n) (TLRef n)) (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).(ex T (\lambda (v: T).(or (subst0 d w t0 
-(lift (S O) d v)) (eq T t0 (lift (S O) d v)))))))).(\lambda (t1: T).(\lambda 
-(H0: ((\forall (d: nat).(ex T (\lambda (v: T).(or (subst0 d w t1 (lift (S O) 
-d v)) (eq T t1 (lift (S O) d v)))))))).(\lambda (d: nat).(let H_x \def (H d) 
-in (let H1 \def H_x in (ex_ind T (\lambda (v: T).(or (subst0 d w t0 (lift (S 
-O) d v)) (eq T t0 (lift (S O) d v)))) (ex T (\lambda (v: T).(or (subst0 d w 
-(THead k t0 t1) (lift (S O) d v)) (eq T (THead k t0 t1) (lift (S O) d v))))) 
-(\lambda (x: T).(\lambda (H2: (or (subst0 d w t0 (lift (S O) d x)) (eq T t0 
-(lift (S O) d x)))).(or_ind (subst0 d w t0 (lift (S O) d x)) (eq T t0 (lift 
-(S O) d x)) (ex T (\lambda (v: T).(or (subst0 d w (THead k t0 t1) (lift (S O) 
-d v)) (eq T (THead k t0 t1) (lift (S O) d v))))) (\lambda (H3: (subst0 d w t0 
-(lift (S O) d x))).(let H_x0 \def (H0 (s k d)) in (let H4 \def H_x0 in 
-(ex_ind T (\lambda (v: T).(or (subst0 (s k d) w t1 (lift (S O) (s k d) v)) 
-(eq T t1 (lift (S O) (s k d) v)))) (ex T (\lambda (v: T).(or (subst0 d w 
-(THead k t0 t1) (lift (S O) d v)) (eq T (THead k t0 t1) (lift (S O) d v))))) 
-(\lambda (x0: T).(\lambda (H5: (or (subst0 (s k d) w t1 (lift (S O) (s k d) 
-x0)) (eq T t1 (lift (S O) (s k d) x0)))).(or_ind (subst0 (s k d) w t1 (lift 
-(S O) (s k d) x0)) (eq T t1 (lift (S O) (s k d) x0)) (ex T (\lambda (v: 
-T).(or (subst0 d w (THead k t0 t1) (lift (S O) d v)) (eq T (THead k t0 t1) 
-(lift (S O) d v))))) (\lambda (H6: (subst0 (s k d) w t1 (lift (S O) (s k d) 
-x0))).(ex_intro T (\lambda (v: T).(or (subst0 d w (THead k t0 t1) (lift (S O) 
-d v)) (eq T (THead k t0 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).(or 
-(subst0 d w (THead k t0 t1) t2) (eq T (THead k t0 t1) t2))) (or_introl 
-(subst0 d w (THead k t0 t1) (THead k (lift (S O) d x) (lift (S O) (s k d) 
-x0))) (eq T (THead k t0 t1) (THead k (lift (S O) d x) (lift (S O) (s k d) 
-x0))) (subst0_both w t0 (lift (S O) d x) d H3 k t1 (lift (S O) (s k d) x0) 
-H6)) (lift (S O) d (THead k x x0)) (lift_head k x x0 (S O) d)))) (\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).(ex T (\lambda (v: T).(or (subst0 d w (THead k t0 t2) (lift 
-(S O) d v)) (eq T (THead k t0 t2) (lift (S O) d v)))))) (ex_intro T (\lambda 
-(v: T).(or (subst0 d w (THead k t0 (lift (S O) (s k d) x0)) (lift (S O) d v)) 
-(eq T (THead k t0 (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).(or (subst0 d w (THead k t0 (lift (S O) (s k d) x0)) t2) (eq T (THead k t0 
-(lift (S O) (s k d) x0)) t2))) (or_introl (subst0 d w (THead k t0 (lift (S O) 
-(s k d) x0)) (THead k (lift (S O) d x) (lift (S O) (s k d) x0))) (eq T (THead 
-k t0 (lift (S O) (s k d) x0)) (THead k (lift (S O) d x) (lift (S O) (s k d) 
-x0))) (subst0_fst w (lift (S O) d x) t0 d H3 (lift (S O) (s k d) x0) k)) 
-(lift (S O) d (THead k x x0)) (lift_head k x x0 (S O) d))) t1 H6)) H5))) 
-H4)))) (\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 (ex_ind T (\lambda (v: T).(or (subst0 (s k d) w t1 
-(lift (S O) (s k d) v)) (eq T t1 (lift (S O) (s k d) v)))) (ex T (\lambda (v: 
-T).(or (subst0 d w (THead k t0 t1) (lift (S O) d v)) (eq T (THead k t0 t1) 
-(lift (S O) d v))))) (\lambda (x0: T).(\lambda (H5: (or (subst0 (s k d) w t1 
-(lift (S O) (s k d) x0)) (eq T t1 (lift (S O) (s k d) x0)))).(or_ind (subst0 
-(s k d) w t1 (lift (S O) (s k d) x0)) (eq T t1 (lift (S O) (s k d) x0)) (ex T 
-(\lambda (v: T).(or (subst0 d w (THead k t0 t1) (lift (S O) d v)) (eq T 
-(THead k t0 t1) (lift (S O) d v))))) (\lambda (H6: (subst0 (s k d) w t1 (lift 
-(S O) (s k d) x0))).(eq_ind_r T (lift (S O) d x) (\lambda (t2: T).(ex T 
-(\lambda (v: T).(or (subst0 d w (THead k t2 t1) (lift (S O) d v)) (eq T 
-(THead k t2 t1) (lift (S O) d v)))))) (ex_intro T (\lambda (v: T).(or (subst0 
-d w (THead k (lift (S O) d x) t1) (lift (S O) d v)) (eq T (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).(or (subst0 d w (THead k 
-(lift (S O) d x) t1) t2) (eq T (THead k (lift (S O) d x) t1) t2))) (or_introl 
-(subst0 d w (THead k (lift (S O) d x) t1) (THead k (lift (S O) d x) (lift (S 
-O) (s k d) x0))) (eq T (THead k (lift (S O) d x) t1) (THead k (lift (S O) d 
-x) (lift (S O) (s k d) x0))) (subst0_snd k w (lift (S O) (s k d) x0) t1 d H6 
-(lift (S O) d x))) (lift (S O) d (THead k x x0)) (lift_head k x x0 (S O) d))) 
-t0 H3)) (\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).(ex T (\lambda (v: T).(or (subst0 d w (THead 
-k t0 t2) (lift (S O) d v)) (eq T (THead k t0 t2) (lift (S O) d v)))))) 
-(eq_ind_r T (lift (S O) d x) (\lambda (t2: T).(ex T (\lambda (v: T).(or 
-(subst0 d w (THead k t2 (lift (S O) (s k d) x0)) (lift (S O) d v)) (eq T 
-(THead k t2 (lift (S O) (s k d) x0)) (lift (S O) d v)))))) (ex_intro T 
-(\lambda (v: T).(or (subst0 d w (THead k (lift (S O) d x) (lift (S O) (s k d) 
-x0)) (lift (S O) d v)) (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).(or (subst0 d w (THead k (lift (S 
-O) d x) (lift (S O) (s k d) x0)) t2) (eq T (THead k (lift (S O) d x) (lift (S 
-O) (s k d) x0)) t2))) (or_intror (subst0 d w (THead k (lift (S O) d x) (lift 
-(S O) (s k d) x0)) (THead k (lift (S O) d x) (lift (S O) (s k d) x0))) (eq T 
-(THead k (lift (S O) d x) (lift (S O) (s k d) x0)) (THead k (lift (S O) d x) 
-(lift (S O) (s k d) x0))) (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)).
-
-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)).
-
-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_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)).(and_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 (t2: T).(subst0 (s k i) v t1 
-t2)))).(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 (t2: 
-T).(subst0 (s k i) v t1 t2))))).(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)))))))).
-
-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_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)) (ex2 T 
-(\lambda (t: T).(subst1 j u1 t1 t)) (\lambda (t: T).(subst1 (S (plus i j)) u 
-t 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)) (\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_equal 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_comm 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)) (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_comm 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))))).
-
-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)))))))))).
-
-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))))))))))).
-
-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).(let H0 \def (match H 
-return (\lambda (n0: nat).(\lambda (t: T).(\lambda (c: C).(\lambda (c0: 
-C).(\lambda (_: (csubst0 n0 t c c0)).((eq nat n0 i) \to ((eq T t v) \to ((eq 
-C c (CSort n)) \to ((eq C c0 x) \to P))))))))) with [(csubst0_snd k i0 v0 u1 
-u2 H0 c) \Rightarrow (\lambda (H1: (eq nat (s k i0) i)).(\lambda (H2: (eq T 
-v0 v)).(\lambda (H3: (eq C (CHead c k u1) (CSort n))).(\lambda (H4: (eq C 
-(CHead c k u2) x)).(eq_ind nat (s k i0) (\lambda (_: nat).((eq T v0 v) \to 
-((eq C (CHead c k u1) (CSort n)) \to ((eq C (CHead c k u2) x) \to ((subst0 i0 
-v0 u1 u2) \to P))))) (\lambda (H5: (eq T v0 v)).(eq_ind T v (\lambda (t: 
-T).((eq C (CHead c k u1) (CSort n)) \to ((eq C (CHead c k u2) x) \to ((subst0 
-i0 t u1 u2) \to P)))) (\lambda (H6: (eq C (CHead c k u1) (CSort n))).(let H7 
-\def (eq_ind C (CHead c k u1) (\lambda (e: C).(match e return (\lambda (_: 
-C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow 
-True])) I (CSort n) H6) in (False_ind ((eq C (CHead c k u2) x) \to ((subst0 
-i0 v u1 u2) \to P)) H7))) v0 (sym_eq T v0 v H5))) i H1 H2 H3 H4 H0))))) | 
-(csubst0_fst k i0 c1 c2 v0 H0 u) \Rightarrow (\lambda (H1: (eq nat (s k i0) 
-i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3: (eq C (CHead c1 k u) (CSort 
-n))).(\lambda (H4: (eq C (CHead c2 k u) x)).(eq_ind nat (s k i0) (\lambda (_: 
-nat).((eq T v0 v) \to ((eq C (CHead c1 k u) (CSort n)) \to ((eq C (CHead c2 k 
-u) x) \to ((csubst0 i0 v0 c1 c2) \to P))))) (\lambda (H5: (eq T v0 
-v)).(eq_ind T v (\lambda (t: T).((eq C (CHead c1 k u) (CSort n)) \to ((eq C 
-(CHead c2 k u) x) \to ((csubst0 i0 t c1 c2) \to P)))) (\lambda (H6: (eq C 
-(CHead c1 k u) (CSort n))).(let H7 \def (eq_ind C (CHead c1 k u) (\lambda (e: 
-C).(match e return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | 
-(CHead _ _ _) \Rightarrow True])) I (CSort n) H6) in (False_ind ((eq C (CHead 
-c2 k u) x) \to ((csubst0 i0 v c1 c2) \to P)) H7))) v0 (sym_eq T v0 v H5))) i 
-H1 H2 H3 H4 H0))))) | (csubst0_both k i0 v0 u1 u2 H0 c1 c2 H1) \Rightarrow 
-(\lambda (H2: (eq nat (s k i0) i)).(\lambda (H3: (eq T v0 v)).(\lambda (H4: 
-(eq C (CHead c1 k u1) (CSort n))).(\lambda (H5: (eq C (CHead c2 k u2) 
-x)).(eq_ind nat (s k i0) (\lambda (_: nat).((eq T v0 v) \to ((eq C (CHead c1 
-k u1) (CSort n)) \to ((eq C (CHead c2 k u2) x) \to ((subst0 i0 v0 u1 u2) \to 
-((csubst0 i0 v0 c1 c2) \to P)))))) (\lambda (H6: (eq T v0 v)).(eq_ind T v 
-(\lambda (t: T).((eq C (CHead c1 k u1) (CSort n)) \to ((eq C (CHead c2 k u2) 
-x) \to ((subst0 i0 t u1 u2) \to ((csubst0 i0 t c1 c2) \to P))))) (\lambda 
-(H7: (eq C (CHead c1 k u1) (CSort n))).(let H8 \def (eq_ind C (CHead c1 k u1) 
-(\lambda (e: C).(match e return (\lambda (_: C).Prop) with [(CSort _) 
-\Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H7) in 
-(False_ind ((eq C (CHead c2 k u2) x) \to ((subst0 i0 v u1 u2) \to ((csubst0 
-i0 v c1 c2) \to P))) H8))) v0 (sym_eq T v0 v H6))) i H2 H3 H4 H5 H0 H1)))))]) 
-in (H0 (refl_equal nat i) (refl_equal T v) (refl_equal C (CSort n)) 
-(refl_equal C x)))))))).
-
-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)).(let 
-H0 \def (match H return (\lambda (n: nat).(\lambda (t: T).(\lambda (c: 
-C).(\lambda (c0: C).(\lambda (_: (csubst0 n t c c0)).((eq nat n i) \to ((eq T 
-t v) \to ((eq C c (CHead c1 k u1)) \to ((eq C c0 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))))))))))))))) with [(csubst0_snd k0 i0 v0 u0 u2 H0 c) \Rightarrow 
-(\lambda (H1: (eq nat (s k0 i0) i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3: 
-(eq C (CHead c k0 u0) (CHead c1 k u1))).(\lambda (H4: (eq C (CHead c k0 u2) 
-x)).(eq_ind nat (s k0 i0) (\lambda (n: nat).((eq T v0 v) \to ((eq C (CHead c 
-k0 u0) (CHead c1 k u1)) \to ((eq C (CHead c k0 u2) x) \to ((subst0 i0 v0 u0 
-u2) \to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat n (s k 
-j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C x (CHead c1 k u3)))) (\lambda 
-(u3: T).(\lambda (j: nat).(subst0 j v u1 u3)))) (ex3_2 C nat (\lambda (_: 
-C).(\lambda (j: nat).(eq nat n (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 n (s k j))))) (\lambda (u3: T).(\lambda (c2: C).(\lambda (_: 
-nat).(eq C x (CHead c2 k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda 
-(j: nat).(subst0 j v u1 u3)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: 
-nat).(csubst0 j v c1 c2))))))))))) (\lambda (H5: (eq T v0 v)).(eq_ind T v 
-(\lambda (t: T).((eq C (CHead c k0 u0) (CHead c1 k u1)) \to ((eq C (CHead c 
-k0 u2) x) \to ((subst0 i0 t u0 u2) \to (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 x (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: 
-nat).(subst0 j v 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 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 (s k0 i0) (s k 
-j))))) (\lambda (u3: T).(\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 
-k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 
-u3)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 
-c2)))))))))) (\lambda (H6: (eq C (CHead c k0 u0) (CHead c1 k u1))).(let H7 
-\def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: C).T) with 
-[(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c k0 u0) 
-(CHead c1 k u1) H6) in ((let H8 \def (f_equal C K (\lambda (e: C).(match e 
-return (\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _) 
-\Rightarrow k])) (CHead c k0 u0) (CHead c1 k u1) H6) in ((let H9 \def 
-(f_equal C C (\lambda (e: C).(match e return (\lambda (_: C).C) with [(CSort 
-_) \Rightarrow c | (CHead c _ _) \Rightarrow c])) (CHead c k0 u0) (CHead c1 k 
-u1) H6) in (eq_ind C c1 (\lambda (c0: C).((eq K k0 k) \to ((eq T u0 u1) \to 
-((eq C (CHead c0 k0 u2) x) \to ((subst0 i0 v u0 u2) \to (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 x (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: 
-nat).(subst0 j v 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 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 (s k0 i0) (s k 
-j))))) (\lambda (u3: T).(\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 
-k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 
-u3)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 
-c2))))))))))) (\lambda (H10: (eq K k0 k)).(eq_ind K k (\lambda (k1: K).((eq T 
-u0 u1) \to ((eq C (CHead c1 k1 u2) x) \to ((subst0 i0 v u0 u2) \to (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 x (CHead c1 k u3)))) (\lambda (u3: 
-T).(\lambda (j: nat).(subst0 j v 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 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 (s k1 i0) (s k j))))) (\lambda (u3: T).(\lambda 
-(c2: C).(\lambda (_: nat).(eq C x (CHead c2 k u3))))) (\lambda (u3: 
-T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u3)))) (\lambda (_: 
-T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2)))))))))) (\lambda 
-(H11: (eq T u0 u1)).(eq_ind T u1 (\lambda (t: T).((eq C (CHead c1 k u2) x) 
-\to ((subst0 i0 v t u2) \to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: 
-nat).(eq nat (s k i0) (s k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C x 
-(CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v 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 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 (s k i0) (s k j))))) (\lambda 
-(u3: T).(\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k u3))))) 
-(\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u3)))) 
-(\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 
-c2))))))))) (\lambda (H12: (eq C (CHead c1 k u2) x)).(eq_ind C (CHead c1 k 
-u2) (\lambda (c0: C).((subst0 i0 v u1 u2) \to (or3 (ex3_2 T nat (\lambda (_: 
-T).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (u3: T).(\lambda 
-(_: nat).(eq C c0 (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: 
-nat).(subst0 j v 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 c0 (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 (s k i0) (s k 
-j))))) (\lambda (u3: T).(\lambda (c2: C).(\lambda (_: nat).(eq C c0 (CHead c2 
-k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 
-u3)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 
-c2)))))))) (\lambda (H13: (subst0 i0 v u1 u2)).(let H \def (eq_ind K k0 
-(\lambda (k: K).(eq nat (s k i0) i)) H1 k H10) in (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 v 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 v 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 v u1 u3)))) (\lambda (_: 
-T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v 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 v u1 u3))) u2 i0 (refl_equal nat (s k 
-i0)) (refl_equal C (CHead c1 k u2)) H13)))) x H12)) u0 (sym_eq T u0 u1 H11))) 
-k0 (sym_eq K k0 k H10))) c (sym_eq C c c1 H9))) H8)) H7))) v0 (sym_eq T v0 v 
-H5))) i H1 H2 H3 H4 H0))))) | (csubst0_fst k0 i0 c0 c2 v0 H0 u) \Rightarrow 
-(\lambda (H1: (eq nat (s k0 i0) i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3: 
-(eq C (CHead c0 k0 u) (CHead c1 k u1))).(\lambda (H4: (eq C (CHead c2 k0 u) 
-x)).(eq_ind nat (s k0 i0) (\lambda (n: nat).((eq T v0 v) \to ((eq C (CHead c0 
-k0 u) (CHead c1 k u1)) \to ((eq C (CHead c2 k0 u) x) \to ((csubst0 i0 v0 c0 
-c2) \to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat n (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 n (s k j)))) (\lambda (c3: C).(\lambda (_: 
-nat).(eq C x (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 n (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: 
-nat).(eq C x (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))))))))))) (\lambda (H5: (eq T v0 v)).(eq_ind T v 
-(\lambda (t: T).((eq C (CHead c0 k0 u) (CHead c1 k u1)) \to ((eq C (CHead c2 
-k0 u) x) \to ((csubst0 i0 t c0 c2) \to (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 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 (s k0 i0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C x (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 k0 i0) (s k 
-j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C x (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)))))))))) (\lambda (H6: (eq C (CHead c0 k0 u) (CHead c1 k u1))).(let H7 
-\def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: C).T) with 
-[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u) 
-(CHead c1 k u1) H6) in ((let H8 \def (f_equal C K (\lambda (e: C).(match e 
-return (\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _) 
-\Rightarrow k])) (CHead c0 k0 u) (CHead c1 k u1) H6) in ((let H9 \def 
-(f_equal C C (\lambda (e: C).(match e return (\lambda (_: C).C) with [(CSort 
-_) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k0 u) (CHead c1 
-k u1) H6) in (eq_ind C c1 (\lambda (c: C).((eq K k0 k) \to ((eq T u u1) \to 
-((eq C (CHead c2 k0 u) x) \to ((csubst0 i0 v c c2) \to (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 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 (s k0 i0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C x (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 k0 i0) (s k 
-j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C x (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))))))))))) (\lambda (H10: (eq K k0 k)).(eq_ind K k (\lambda (k1: K).((eq T 
-u u1) \to ((eq C (CHead c2 k1 u) x) \to ((csubst0 i0 v c1 c2) \to (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 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 (s k1 i0) (s k j)))) (\lambda (c3: C).(\lambda 
-(_: nat).(eq C x (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 k1 i0) (s k j))))) (\lambda (u2: T).(\lambda 
-(c3: C).(\lambda (_: nat).(eq C x (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)))))))))) (\lambda 
-(H11: (eq T u u1)).(eq_ind T u1 (\lambda (t: T).((eq C (CHead c2 k t) x) \to 
-((csubst0 i0 v c1 c2) \to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: 
-nat).(eq nat (s k i0) (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 (s k i0) (s k j)))) 
-(\lambda (c3: C).(\lambda (_: nat).(eq C x (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 i0) (s k j))))) (\lambda 
-(u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C x (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))))))))) (\lambda (H12: (eq C (CHead c2 k u1) x)).(eq_ind C (CHead c2 k 
-u1) (\lambda (c: C).((csubst0 i0 v c1 c2) \to (or3 (ex3_2 T nat (\lambda (_: 
-T).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (u2: T).(\lambda 
-(_: nat).(eq C c (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 i0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c (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 i0) (s k 
-j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c (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)))))))) (\lambda (H13: (csubst0 i0 v c1 c2)).(let H \def (eq_ind K k0 
-(\lambda (k: K).(eq nat (s k i0) i)) H1 k H10) in (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 v 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 v 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 v u1 u2)))) (\lambda (_: 
-T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v 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 v c1 c3))) c2 i0 (refl_equal nat (s k 
-i0)) (refl_equal C (CHead c2 k u1)) H13)))) x H12)) u (sym_eq T u u1 H11))) 
-k0 (sym_eq K k0 k H10))) c0 (sym_eq C c0 c1 H9))) H8)) H7))) v0 (sym_eq T v0 
-v H5))) i H1 H2 H3 H4 H0))))) | (csubst0_both k0 i0 v0 u0 u2 H0 c0 c2 H1) 
-\Rightarrow (\lambda (H2: (eq nat (s k0 i0) i)).(\lambda (H3: (eq T v0 
-v)).(\lambda (H4: (eq C (CHead c0 k0 u0) (CHead c1 k u1))).(\lambda (H5: (eq 
-C (CHead c2 k0 u2) x)).(eq_ind nat (s k0 i0) (\lambda (n: nat).((eq T v0 v) 
-\to ((eq C (CHead c0 k0 u0) (CHead c1 k u1)) \to ((eq C (CHead c2 k0 u2) x) 
-\to ((subst0 i0 v0 u0 u2) \to ((csubst0 i0 v0 c0 c2) \to (or3 (ex3_2 T nat 
-(\lambda (_: T).(\lambda (j: nat).(eq nat n (s k j)))) (\lambda (u3: 
-T).(\lambda (_: nat).(eq C x (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: 
-nat).(subst0 j v u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq 
-nat n (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C x (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 n (s k j))))) 
-(\lambda (u3: T).(\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k 
-u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 
-u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 
-c3)))))))))))) (\lambda (H6: (eq T v0 v)).(eq_ind T v (\lambda (t: T).((eq C 
-(CHead c0 k0 u0) (CHead c1 k u1)) \to ((eq C (CHead c2 k0 u2) x) \to ((subst0 
-i0 t u0 u2) \to ((csubst0 i0 t c0 c2) \to (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 x (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: 
-nat).(subst0 j v u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq 
-nat (s k0 i0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C x (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 k0 i0) (s k 
-j))))) (\lambda (u3: T).(\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 
-k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 
-u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 
-c3))))))))))) (\lambda (H7: (eq C (CHead c0 k0 u0) (CHead c1 k u1))).(let H8 
-\def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: C).T) with 
-[(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u0) 
-(CHead c1 k u1) H7) in ((let H9 \def (f_equal C K (\lambda (e: C).(match e 
-return (\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _) 
-\Rightarrow k])) (CHead c0 k0 u0) (CHead c1 k u1) H7) in ((let H10 \def 
-(f_equal C C (\lambda (e: C).(match e return (\lambda (_: C).C) with [(CSort 
-_) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k0 u0) (CHead c1 
-k u1) H7) in (eq_ind C c1 (\lambda (c: C).((eq K k0 k) \to ((eq T u0 u1) \to 
-((eq C (CHead c2 k0 u2) x) \to ((subst0 i0 v u0 u2) \to ((csubst0 i0 v c c2) 
-\to (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 x (CHead c1 k u3)))) 
-(\lambda (u3: T).(\lambda (j: nat).(subst0 j v u1 u3)))) (ex3_2 C nat 
-(\lambda (_: C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) (\lambda (c3: 
-C).(\lambda (_: nat).(eq C x (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 k0 i0) (s k j))))) (\lambda (u3: T).(\lambda 
-(c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u3))))) (\lambda (u3: 
-T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u3)))) (\lambda (_: 
-T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))))))))))) (\lambda 
-(H11: (eq K k0 k)).(eq_ind K k (\lambda (k1: K).((eq T u0 u1) \to ((eq C 
-(CHead c2 k1 u2) x) \to ((subst0 i0 v u0 u2) \to ((csubst0 i0 v c1 c2) \to 
-(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 x (CHead c1 k u3)))) (\lambda 
-(u3: T).(\lambda (j: nat).(subst0 j v 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 x (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 k1 i0) (s k j))))) (\lambda (u3: T).(\lambda 
-(c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u3))))) (\lambda (u3: 
-T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u3)))) (\lambda (_: 
-T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3))))))))))) (\lambda 
-(H12: (eq T u0 u1)).(eq_ind T u1 (\lambda (t: T).((eq C (CHead c2 k u2) x) 
-\to ((subst0 i0 v t u2) \to ((csubst0 i0 v c1 c2) \to (or3 (ex3_2 T nat 
-(\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (u3: 
-T).(\lambda (_: nat).(eq C x (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: 
-nat).(subst0 j v 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 x (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 i0) (s k 
-j))))) (\lambda (u3: T).(\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 
-k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 
-u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 
-c3)))))))))) (\lambda (H13: (eq C (CHead c2 k u2) x)).(eq_ind C (CHead c2 k 
-u2) (\lambda (c: C).((subst0 i0 v u1 u2) \to ((csubst0 i0 v c1 c2) \to (or3 
-(ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) 
-(\lambda (u3: T).(\lambda (_: nat).(eq C c (CHead c1 k u3)))) (\lambda (u3: 
-T).(\lambda (j: nat).(subst0 j v 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 c (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 i0) (s k j))))) (\lambda (u3: T).(\lambda 
-(c3: C).(\lambda (_: nat).(eq C c (CHead c3 k u3))))) (\lambda (u3: 
-T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u3)))) (\lambda (_: 
-T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3))))))))) (\lambda 
-(H14: (subst0 i0 v u1 u2)).(\lambda (H15: (csubst0 i0 v c1 c2)).(let H \def 
-(eq_ind K k0 (\lambda (k: K).(eq nat (s k i0) i)) H2 k H11) in (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 v 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 v 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 v u1 
-u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v 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 v u1 u3)))) (\lambda (_: T).(\lambda (c3: 
-C).(\lambda (j: nat).(csubst0 j v c1 c3)))) u2 c2 i0 (refl_equal nat (s k 
-i0)) (refl_equal C (CHead c2 k u2)) H14 H15))))) x H13)) u0 (sym_eq T u0 u1 
-H12))) k0 (sym_eq K k0 k H11))) c0 (sym_eq C c0 c1 H10))) H9)) H8))) v0 
-(sym_eq T v0 v H6))) i H2 H3 H4 H5 H0 H1)))))]) in (H0 (refl_equal nat i) 
-(refl_equal T v) (refl_equal C (CHead c1 k u1)) (refl_equal C x))))))))).
-
-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)).(let H2 \def (match H return (\lambda (n: nat).(\lambda (_: (le ? 
-n)).((eq nat n O) \to (drop O O c2 e)))) with [le_n \Rightarrow (\lambda (H2: 
-(eq nat (S i) O)).(let H3 \def (eq_ind nat (S i) (\lambda (e0: nat).(match e0 
-return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow 
-True])) I O H2) in (False_ind (drop O O c2 e) H3))) | (le_S m H2) \Rightarrow 
-(\lambda (H3: (eq nat (S m) O)).((let H4 \def (eq_ind nat (S m) (\lambda (e0: 
-nat).(match e0 return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S 
-_) \Rightarrow True])) I O H3) in (False_ind ((le (S i) m) \to (drop O O c2 
-e)) H4)) H2))]) in (H2 (refl_equal nat O))))))))))) (\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 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 (n: 
-nat).(\forall (c2: C).(\forall (v: T).((csubst0 n v c c2) \to (\forall (e: 
-C).((drop (S n0) O c e) \to (drop (S n0) O c2 e))))))) H1 (s k x1) H5) in 
-(let H9 \def (eq_ind nat i (\lambda (n: nat).(lt n (S n0))) H0 (s k x1) H5) 
-in ((match k return (\lambda (k0: K).((drop (r k0 n0) O c e) \to (((\forall 
-(c2: C).(\forall (v: T).((csubst0 (s k0 x1) v c c2) \to (\forall (e: 
-C).((drop (S n0) O c e) \to (drop (S n0) O c2 e))))))) \to ((lt (s k0 x1) (S 
-n0)) \to (drop (S n0) O (CHead c k0 x0) e))))) with [(Bind b) \Rightarrow 
-(\lambda (H10: (drop (r (Bind b) n0) O c e)).(\lambda (_: ((\forall (c2: 
-C).(\forall (v: T).((csubst0 (s (Bind b) x1) v c c2) \to (\forall (e: 
-C).((drop (S n0) O c e) \to (drop (S n0) O c2 e)))))))).(\lambda (_: (lt (s 
-(Bind b) x1) (S n0))).(drop_drop (Bind b) n0 c e H10 x0)))) | (Flat f) 
-\Rightarrow (\lambda (H10: (drop (r (Flat f) n0) O c e)).(\lambda (_: 
-((\forall (c2: C).(\forall (v: T).((csubst0 (s (Flat f) x1) v c c2) \to 
-(\forall (e: C).((drop (S n0) O c e) \to (drop (S n0) O c2 e)))))))).(\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)))))]) 
-(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 (c2: C).(\lambda (j: 
-nat).(csubst0 j v c c2))))).(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 (n: nat).(\forall (c2: C).(\forall (v: 
-T).((csubst0 n v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S 
-n0) O c2 e))))))) H1 (s k x1) H5) in (let H9 \def (eq_ind nat i (\lambda (n: 
-nat).(lt n (S n0))) H0 (s k x1) H5) in ((match k return (\lambda (k0: 
-K).((drop (r k0 n0) O c e) \to (((\forall (c2: C).(\forall (v: T).((csubst0 
-(s k0 x1) v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S n0) O 
-c2 e))))))) \to ((lt (s k0 x1) (S n0)) \to (drop (S n0) O (CHead x0 k0 t) 
-e))))) with [(Bind b) \Rightarrow (\lambda (H10: (drop (r (Bind b) n0) O c 
-e)).(\lambda (_: ((\forall (c2: C).(\forall (v: T).((csubst0 (s (Bind b) x1) 
-v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S n0) O c2 
-e)))))))).(\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)))) | (Flat f) \Rightarrow 
-(\lambda (H10: (drop (r (Flat f) n0) O c e)).(\lambda (H11: ((\forall (c2: 
-C).(\forall (v: T).((csubst0 (s (Flat f) x1) v c c2) \to (\forall (e: 
-C).((drop (S n0) O c e) \to (drop (S n0) O c2 e)))))))).(\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)))))]) (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 (c2: C).(\lambda (j: 
-nat).(csubst0 j v c c2)))))).(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 (n: 
-nat).(\forall (c2: C).(\forall (v: T).((csubst0 n v c c2) \to (\forall (e: 
-C).((drop (S n0) O c e) \to (drop (S n0) O c2 e))))))) H1 (s k x2) H5) in 
-(let H10 \def (eq_ind nat i (\lambda (n: nat).(lt n (S n0))) H0 (s k x2) H5) 
-in ((match k return (\lambda (k0: K).((drop (r k0 n0) O c e) \to (((\forall 
-(c2: C).(\forall (v: T).((csubst0 (s k0 x2) v c c2) \to (\forall (e: 
-C).((drop (S n0) O c e) \to (drop (S n0) O c2 e))))))) \to ((lt (s k0 x2) (S 
-n0)) \to (drop (S n0) O (CHead x1 k0 x0) e))))) with [(Bind b) \Rightarrow 
-(\lambda (H11: (drop (r (Bind b) n0) O c e)).(\lambda (_: ((\forall (c2: 
-C).(\forall (v: T).((csubst0 (s (Bind b) x2) v c c2) \to (\forall (e: 
-C).((drop (S n0) O c e) \to (drop (S n0) O c2 e)))))))).(\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)))) | (Flat f) \Rightarrow (\lambda (H11: (drop (r (Flat f) 
-n0) O c e)).(\lambda (H12: ((\forall (c2: C).(\forall (v: T).((csubst0 (s 
-(Flat f) x2) v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S 
-n0) O c2 e)))))))).(\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)))))]) 
-(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)).(let H2 \def (match H return (\lambda (n: nat).(\lambda (_: (le ? 
-n)).((eq nat n O) \to (drop O O c1 e)))) with [le_n \Rightarrow (\lambda (H2: 
-(eq nat (S i) O)).(let H3 \def (eq_ind nat (S i) (\lambda (e0: nat).(match e0 
-return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow 
-True])) I O H2) in (False_ind (drop O O c1 e) H3))) | (le_S m H2) \Rightarrow 
-(\lambda (H3: (eq nat (S m) O)).((let H4 \def (eq_ind nat (S m) (\lambda (e0: 
-nat).(match e0 return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S 
-_) \Rightarrow True])) I O H3) in (False_ind ((le (S i) m) \to (drop O O c1 
-e)) H4)) H2))]) in (H2 (refl_equal nat O))))))))))) (\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 (c: 
-C).(drop (S n0) O c e)) H3 (CHead c k x0) H6) in (let H9 \def (eq_ind nat i 
-(\lambda (n: nat).(\forall (c2: C).(\forall (v: T).((csubst0 n v c c2) \to 
-(\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c e))))))) H1 (s k 
-x1) H5) in (let H10 \def (eq_ind nat i (\lambda (n: nat).(lt n (S n0))) H0 (s 
-k x1) H5) in ((match k return (\lambda (k0: K).(((\forall (c2: C).(\forall 
-(v: T).((csubst0 (s k0 x1) v c c2) \to (\forall (e: C).((drop (S n0) O c2 e) 
-\to (drop (S n0) O c e))))))) \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))))) with [(Bind b) \Rightarrow 
-(\lambda (_: ((\forall (c2: C).(\forall (v: T).((csubst0 (s (Bind b) x1) v c 
-c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c 
-e)))))))).(\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)))) | (Flat f) 
-\Rightarrow (\lambda (_: ((\forall (c2: C).(\forall (v: T).((csubst0 (s (Flat 
-f) x1) v c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c 
-e)))))))).(\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)))))]) 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 (c2: C).(\lambda (j: nat).(csubst0 j 
-v c c2))))).(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 (c: C).(drop (S n0) O c e)) H3 (CHead 
-x0 k t) H6) in (let H9 \def (eq_ind nat i (\lambda (n: nat).(\forall (c2: 
-C).(\forall (v: T).((csubst0 n v c c2) \to (\forall (e: C).((drop (S n0) O c2 
-e) \to (drop (S n0) O c e))))))) H1 (s k x1) H5) in (let H10 \def (eq_ind nat 
-i (\lambda (n: nat).(lt n (S n0))) H0 (s k x1) H5) in ((match k return 
-(\lambda (k0: K).(((\forall (c2: C).(\forall (v: T).((csubst0 (s k0 x1) v c 
-c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c e))))))) 
-\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))))) with [(Bind b) \Rightarrow (\lambda (_: ((\forall (c2: 
-C).(\forall (v: T).((csubst0 (s (Bind b) x1) v c c2) \to (\forall (e: 
-C).((drop (S n0) O c2 e) \to (drop (S n0) O c e)))))))).(\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)))) | (Flat f) \Rightarrow (\lambda (H11: ((\forall (c2: C).(\forall (v: 
-T).((csubst0 (s (Flat f) x1) v c c2) \to (\forall (e: C).((drop (S n0) O c2 
-e) \to (drop (S n0) O c e)))))))).(\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)))))]) 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 (c2: 
-C).(\lambda (j: nat).(csubst0 j v c c2)))))).(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 (c: C).(drop (S n0) O c e)) H3 (CHead 
-x1 k x0) H6) in (let H10 \def (eq_ind nat i (\lambda (n: nat).(\forall (c2: 
-C).(\forall (v: T).((csubst0 n v c c2) \to (\forall (e: C).((drop (S n0) O c2 
-e) \to (drop (S n0) O c e))))))) H1 (s k x2) H5) in (let H11 \def (eq_ind nat 
-i (\lambda (n: nat).(lt n (S n0))) H0 (s k x2) H5) in ((match k return 
-(\lambda (k0: K).(((\forall (c2: C).(\forall (v: T).((csubst0 (s k0 x2) v c 
-c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c e))))))) 
-\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))))) with [(Bind b) \Rightarrow (\lambda (_: ((\forall (c2: 
-C).(\forall (v: T).((csubst0 (s (Bind b) x2) v c c2) \to (\forall (e: 
-C).((drop (S n0) O c2 e) \to (drop (S n0) O c e)))))))).(\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)))) | (Flat f) \Rightarrow (\lambda (H12: ((\forall (c2: C).(\forall (v: 
-T).((csubst0 (s (Flat f) x2) v c c2) \to (\forall (e: C).((drop (S n0) O c2 
-e) \to (drop (S n0) O c e)))))))).(\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)))))]) 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 (n: nat).(subst0 n 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 (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C c3 (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop O O c4 (CHead e0 k w)))))) (\lambda (k: K).(\lambda 
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k O)) v0 u 
-w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C c3 (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 k u)))))) (\lambda 
-(k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 
-(s k O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 k 
-u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop O O c4 (CHead e2 k w))))))) (\lambda (k: K).(\lambda 
-(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k 
-O)) v0 u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 (minus i0 (s k O)) v0 e1 e2))))))))).(\lambda 
-(u: T).(let H4 \def (eq_ind_r nat i0 (\lambda (n: nat).(csubst0 n 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 (n: nat).(or4 (drop O O c4 c3) (ex3_4 K C T T (\lambda (k: 
-K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0 k 
-u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop O O c4 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (w: T).(subst0 (minus n (s k O)) v0 u w)))))) (ex3_4 K C C T 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 
-(CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(u: T).(drop O O c4 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n (s k O)) v0 e1 e2)))))) 
-(ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (_: T).(eq C c3 (CHead e1 k u))))))) (\lambda (k: K).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead 
-e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 (minus n (s k O)) v0 u w)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus n (s k 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 (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C c3 (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop O O c4 (CHead e0 k w)))))) (\lambda (k: K).(\lambda 
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k O)) v0 u 
-w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C c3 (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 k u)))))) (\lambda 
-(k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 
-(s k O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 k 
-u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop O O c4 (CHead e2 k w))))))) (\lambda (k: K).(\lambda 
-(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k 
-O)) v0 u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 (minus i0 (s k O)) v0 e1 e2))))))))).(let H5 
-\def (eq_ind_r nat i0 (\lambda (n: nat).(subst0 n 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 (n: 
-nat).(csubst0 n 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 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)).(let H3 \def (match H1 return (\lambda 
-(n: nat).(\lambda (t0: T).(\lambda (c0: C).(\lambda (c1: C).(\lambda (_: 
-(csubst0 n t0 c0 c1)).((eq nat n i) \to ((eq T t0 v) \to ((eq C c0 (CHead c k 
-t)) \to ((eq C c1 c2) \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))))))))))))))))) with [(csubst0_snd k0 i0 v0 u1 u2 H3 c0) \Rightarrow 
-(\lambda (H4: (eq nat (s k0 i0) i)).(\lambda (H5: (eq T v0 v)).(\lambda (H6: 
-(eq C (CHead c0 k0 u1) (CHead c k t))).(\lambda (H7: (eq C (CHead c0 k0 u2) 
-c2)).(eq_ind nat (s k0 i0) (\lambda (n: nat).((eq T v0 v) \to ((eq C (CHead 
-c0 k0 u1) (CHead c k t)) \to ((eq C (CHead c0 k0 u2) c2) \to ((subst0 i0 v0 
-u1 u2) \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 n (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 n (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 n (s k (S n0))) v u w)))))) 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 (minus n (s k (S n0))) v e1 e2))))))))))))) (\lambda (H8: (eq 
-T v0 v)).(eq_ind T v (\lambda (t0: T).((eq C (CHead c0 k0 u1) (CHead c k t)) 
-\to ((eq C (CHead c0 k0 u2) c2) \to ((subst0 i0 t0 u1 u2) \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 (s k0 i0) (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 (s k0 i0) (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 (s k0 i0) (s k (S n0))) v u 
-w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 (minus (s k0 i0) (s k (S n0))) v e1 
-e2)))))))))))) (\lambda (H9: (eq C (CHead c0 k0 u1) (CHead c k t))).(let H10 
-\def (f_equal C T (\lambda (e0: C).(match e0 return (\lambda (_: C).T) with 
-[(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u1) 
-(CHead c k t) H9) in ((let H11 \def (f_equal C K (\lambda (e0: C).(match e0 
-return (\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _) 
-\Rightarrow k])) (CHead c0 k0 u1) (CHead c k t) H9) in ((let H12 \def 
-(f_equal C C (\lambda (e0: C).(match e0 return (\lambda (_: C).C) with 
-[(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k0 u1) 
-(CHead c k t) H9) in (eq_ind C c (\lambda (c: C).((eq K k0 k) \to ((eq T u1 
-t) \to ((eq C (CHead c k0 u2) c2) \to ((subst0 i0 v u1 u2) \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 (s k0 i0) (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 (s k0 i0) (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 (s k0 i0) (s k (S n0))) v u 
-w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 (minus (s k0 i0) (s k (S n0))) v e1 
-e2))))))))))))) (\lambda (H13: (eq K k0 k)).(eq_ind K k (\lambda (k: K).((eq 
-T u1 t) \to ((eq C (CHead c k u2) c2) \to ((subst0 i0 v u1 u2) \to (or4 (drop 
-(S n0) O c2 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 c2 (CHead e0 k1 
-w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus (s k i0) (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 c2 (CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (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 c2 (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s 
-k1 (S n0))) v u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v 
-e1 e2)))))))))))) (\lambda (H14: (eq T u1 t)).(eq_ind T t (\lambda (t: 
-T).((eq C (CHead c k u2) c2) \to ((subst0 i0 v t u2) \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 (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus (s k i0) (s k1 (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 (k1: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (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 (k1: K).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u 
-w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2))))))))))) 
-(\lambda (H15: (eq C (CHead c k u2) c2)).(eq_ind C (CHead c k u2) (\lambda 
-(c: C).((subst0 i0 v t u2) \to (or4 (drop (S n0) O c 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 c (CHead e0 k w)))))) (\lambda (k1: K).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (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 c (CHead e2 k u)))))) 
-(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
-(minus (s k i0) (s k1 (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 c (CHead e2 k w))))))) 
-(\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus (s k i0) (s k1 (S n0))) v u w)))))) (\lambda (k1: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s k i0) (s k1 (S n0))) v e1 e2)))))))))) (\lambda (_: (subst0 i0 v t 
-u2)).(let H1 \def (eq_ind K k0 (\lambda (k: K).(eq nat (s k i0) i)) H4 k H13) 
-in (let H17 \def (eq_ind_r nat i (\lambda (n: nat).(\forall (c2: C).(\forall 
-(v: T).((csubst0 n 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 n (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 n (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 n (s k (S n0))) v u w)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus n (s k (S n0))) v e1 e2)))))))))))))) H0 (s k i0) H1) in (let H18 \def 
-(eq_ind_r nat i (\lambda (n: nat).(lt (S n0) n)) H (s k i0) H1) in (K_ind 
-(\lambda (k: K).((drop (r k n0) O c e) \to (((\forall (c2: C).(\forall (v: 
-T).((csubst0 (s k i0) 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 (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus (s k i0) (s k0 (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 (k0: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 (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 (k0: K).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 (S n0))) v u 
-w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 (S n0))) v e1 
-e2)))))))))))))) \to ((lt (S n0) (s k i0)) \to (or4 (drop (S n0) O (CHead c k 
-u2) 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 k u2) (CHead 
-e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus (s k i0) (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 k u2) (CHead e2 k1 u)))))) (\lambda (k1: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) 
-(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 k u2) (CHead e2 k1 w))))))) 
-(\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus (s k i0) (s k1 (S n0))) v u w)))))) (\lambda (k1: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s k i0) (s k1 (S n0))) v e1 e2)))))))))))) (\lambda (b: B).(\lambda 
-(H2: (drop (r (Bind b) n0) O c e)).(\lambda (_: ((\forall (c2: C).(\forall 
-(v: T).((csubst0 (s (Bind b) i0) 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 (s (Bind b) i0) (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 (s (Bind b) i0) (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 (s (Bind b) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s 
-(Bind b) i0) (s k (S n0))) v e1 e2))))))))))))))).(\lambda (_: (lt (S n0) (s 
-(Bind b) i0))).(or4_intro0 (drop (S n0) O (CHead c (Bind b) u2) 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 (CHead c (Bind b) u2) (CHead e0 k w)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
-i0) (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 (CHead c 
-(Bind b) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (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 (CHead c (Bind b) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
-i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S 
-n0))) v e1 e2))))))) (drop_drop (Bind b) n0 c e H2 u2)))))) (\lambda (f: 
-F).(\lambda (H2: (drop (r (Flat f) n0) O c e)).(\lambda (_: ((\forall (c2: 
-C).(\forall (v: T).((csubst0 (s (Flat f) i0) 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 (s (Flat f) i0) (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 (s (Flat f) i0) (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 (s (Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))))))))))))).(\lambda (_: (lt 
-(S n0) (s (Flat f) i0))).(or4_intro0 (drop (S n0) O (CHead c (Flat f) u2) 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 (CHead c (Flat f) u2) (CHead e0 k w)))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus (s (Flat f) i0) (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 (CHead c (Flat f) u2) (CHead e2 k u)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat 
-f) i0) (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 (CHead c (Flat f) u2) (CHead e2 k w))))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))))) (drop_drop (Flat f) n0 c 
-e H2 u2)))))) k (drop_gen_drop k c e t n0 H2) H17 H18))))) c2 H15)) u1 
-(sym_eq T u1 t H14))) k0 (sym_eq K k0 k H13))) c0 (sym_eq C c0 c H12))) H11)) 
-H10))) v0 (sym_eq T v0 v H8))) i H4 H5 H6 H7 H3))))) | (csubst0_fst k0 i0 c1 
-c0 v0 H3 u) \Rightarrow (\lambda (H4: (eq nat (s k0 i0) i)).(\lambda (H5: (eq 
-T v0 v)).(\lambda (H6: (eq C (CHead c1 k0 u) (CHead c k t))).(\lambda (H7: 
-(eq C (CHead c0 k0 u) c2)).(eq_ind nat (s k0 i0) (\lambda (n: nat).((eq T v0 
-v) \to ((eq C (CHead c1 k0 u) (CHead c k t)) \to ((eq C (CHead c0 k0 u) c2) 
-\to ((csubst0 i0 v0 c1 c0) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T 
-(\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e 
-(CHead e0 k u0)))))) (\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 (u0: T).(\lambda (w: T).(subst0 (minus n (s k (S 
-n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u0: T).(eq C e (CHead e1 k u0)))))) (\lambda (k: K).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O c2 (CHead e2 k 
-u0)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 (minus n (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda 
-(k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq 
-C e (CHead e1 k u0))))))) (\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 (u0: T).(\lambda (w: 
-T).(subst0 (minus n (s k (S n0))) v u0 w)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n (s k (S 
-n0))) v e1 e2))))))))))))) (\lambda (H8: (eq T v0 v)).(eq_ind T v (\lambda 
-(t0: T).((eq C (CHead c1 k0 u) (CHead c k t)) \to ((eq C (CHead c0 k0 u) c2) 
-\to ((csubst0 i0 t0 c1 c0) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T 
-(\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e 
-(CHead e0 k u0)))))) (\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 (u0: T).(\lambda (w: T).(subst0 (minus (s k0 i0) 
-(s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k u0)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O c2 (CHead 
-e2 k u0)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 (minus (s k0 i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda 
-(_: T).(eq C e (CHead e1 k u0))))))) (\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 (u0: 
-T).(\lambda (w: T).(subst0 (minus (s k0 i0) (s k (S n0))) v u0 w)))))) 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 (minus (s k0 i0) (s k (S n0))) v e1 e2)))))))))))) (\lambda 
-(H9: (eq C (CHead c1 k0 u) (CHead c k t))).(let H10 \def (f_equal C T 
-(\lambda (e0: C).(match e0 return (\lambda (_: C).T) with [(CSort _) 
-\Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k0 u) (CHead c k t) 
-H9) in ((let H11 \def (f_equal C K (\lambda (e0: C).(match e0 return (\lambda 
-(_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _) \Rightarrow k])) 
-(CHead c1 k0 u) (CHead c k t) H9) in ((let H12 \def (f_equal C C (\lambda 
-(e0: C).(match e0 return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | 
-(CHead c _ _) \Rightarrow c])) (CHead c1 k0 u) (CHead c k t) H9) in (eq_ind C 
-c (\lambda (c: C).((eq K k0 k) \to ((eq T u t) \to ((eq C (CHead c0 k0 u) c2) 
-\to ((csubst0 i0 v c c0) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T 
-(\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e 
-(CHead e0 k u0)))))) (\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 (u0: T).(\lambda (w: T).(subst0 (minus (s k0 i0) 
-(s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k u0)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O c2 (CHead 
-e2 k u0)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 (minus (s k0 i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda 
-(_: T).(eq C e (CHead e1 k u0))))))) (\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 (u0: 
-T).(\lambda (w: T).(subst0 (minus (s k0 i0) (s k (S n0))) v u0 w)))))) 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 (minus (s k0 i0) (s k (S n0))) v e1 e2))))))))))))) (\lambda 
-(H13: (eq K k0 k)).(eq_ind K k (\lambda (k: K).((eq T u t) \to ((eq C (CHead 
-c0 k u) c2) \to ((csubst0 i0 v c c0) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C 
-T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C 
-e (CHead e0 k1 u0)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k1 w)))))) (\lambda (k1: 
-K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k i0) 
-(s k1 (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k1 u0)))))) (\lambda 
-(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O c2 
-(CHead e2 k1 u0)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2)))))) 
-(ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u0: T).(\lambda (_: T).(eq C e (CHead e1 k1 u0))))))) (\lambda (k1: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O c2 (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v 
-u0 w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 
-e2)))))))))))) (\lambda (H14: (eq T u t)).(eq_ind T t (\lambda (t: T).((eq C 
-(CHead c0 k t) c2) \to ((csubst0 i0 v c c0) \to (or4 (drop (S n0) O c2 e) 
-(ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: 
-T).(eq C e (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k1: 
-K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k i0) 
-(s k1 (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k u0)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O c2 (CHead 
-e2 k u0)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda 
-(_: T).(eq C e (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k 
-w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: 
-T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u0 w)))))) 
-(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2))))))))))) (\lambda 
-(H15: (eq C (CHead c0 k t) c2)).(eq_ind C (CHead c0 k t) (\lambda (c2: 
-C).((csubst0 i0 v c c0) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda 
-(k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 k 
-u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k1: K).(\lambda (_: 
-C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v 
-u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u0: T).(eq C e (CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O c2 (CHead e2 k u0)))))) 
-(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
-(minus (s k i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e 
-(CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) 
-(\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda 
-(w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u0 w)))))) (\lambda (k1: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s k i0) (s k1 (S n0))) v e1 e2)))))))))) (\lambda (H16: (csubst0 i0 v 
-c c0)).(let H1 \def (eq_ind K k0 (\lambda (k: K).(eq nat (s k i0) i)) H4 k 
-H13) in (let H17 \def (eq_ind_r nat i (\lambda (n: nat).(\forall (c2: 
-C).(\forall (v: T).((csubst0 n 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 n (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 n (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 n (s k (S n0))) v u w)))))) 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 (minus n (s k (S n0))) v e1 e2)))))))))))))) H0 (s k i0) H1) 
-in (let H18 \def (eq_ind_r nat i (\lambda (n: nat).(lt (S n0) n)) H (s k i0) 
-H1) in (K_ind (\lambda (k: K).((drop (r k n0) O c e) \to (((\forall (c2: 
-C).(\forall (v: T).((csubst0 (s k i0) 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 (k0: K).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 (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 (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
-(minus (s k i0) (s k0 (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 (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus (s k i0) (s k0 (S n0))) v u w)))))) (\lambda (k0: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s k i0) (s k0 (S n0))) v e1 e2)))))))))))))) \to ((lt (S n0) (s k 
-i0)) \to (or4 (drop (S n0) O (CHead c0 k t) e) (ex3_4 K C T T (\lambda (k1: 
-K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 k1 
-u0)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O (CHead c0 k t) (CHead e0 k1 w)))))) (\lambda (k1: 
-K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k i0) 
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-(minus (s (Bind b) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda 
-(k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq 
-C (CHead x1 x0 x3) (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 
-(Bind b) t) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S 
-n0))) v u0 w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 
-e2))))))) (ex3_4_intro K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u0: T).(eq C (CHead x1 x0 x3) (CHead e1 k u0)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O (CHead c0 
-(Bind b) t) (CHead e2 k u0)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 
-e2))))) x0 x1 x2 x3 (refl_equal C (CHead x1 x0 x3)) (drop_drop (Bind b) n0 c0 
-(CHead x2 x0 x3) H22 t) (eq_ind_r nat (S (s x0 n0)) (\lambda (n: 
-nat).(csubst0 (minus (s (Bind b) i0) n) v x1 x2)) H23 (s x0 (S n0)) (s_S x0 
-n0)))) e H21)))))))) H20)) (\lambda (H20: (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 c0 (CHead e2 k w))))))) 
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-T).(subst0 (minus i0 (s k n0)) v u w)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k 
-n0)) v e1 e2)))))))).(ex4_5_ind K C C T T (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k 
-u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
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-K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 
-(minus i0 (s k n0)) v u0 w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda 
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-e2)))))) (or4 (drop (S n0) O (CHead c0 (Bind b) t) e) (ex3_4 K C T T (\lambda 
-(k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 k 
-u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O (CHead c0 (Bind b) t) (CHead e0 k w)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Bind 
-b) i0) (s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k u0)))))) 
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-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k 
-(S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k 
-u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
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-T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u0 w)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
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-T).(\lambda (H21: (eq C e (CHead x1 x0 x3))).(\lambda (H22: (drop n0 O c0 
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-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C c (CHead e1 k 
-u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: 
-T).(drop (S n0) O (CHead c0 (Bind b) t) (CHead e2 k u0)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind 
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-(e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C c (CHead e1 k 
-u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
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-T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u0 w)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
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-n0) O (CHead c0 (Bind b) t) (CHead x1 x0 x3)) (ex3_4 K C T T (\lambda (k: 
-K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) 
-(CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Bind b) t) (CHead e0 k w)))))) 
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-(minus (s (Bind b) i0) (s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead x1 x0 x3) 
-(CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u0: T).(drop (S n0) O (CHead c0 (Bind b) t) (CHead e2 k u0)))))) 
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-(minus (s (Bind b) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda 
-(k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq 
-C (CHead x1 x0 x3) (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 
-(Bind b) t) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S 
-n0))) v u0 w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 
-e2))))))) (ex4_5_intro K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e1 k 
-u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Bind b) t) (CHead e2 k w))))))) 
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-T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u0 w)))))) (\lambda (k: 
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-(minus (s (Bind b) i0) (s k (S n0))) v e1 e2)))))) x0 x1 x2 x3 x4 (refl_equal 
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-(eq_ind_r nat (S (s x0 n0)) (\lambda (n: nat).(subst0 (minus (s (Bind b) i0) 
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-(\lambda (n: nat).(csubst0 (minus (s (Bind b) i0) n) v x1 x2)) H24 (s x0 (S 
-n0)) (s_S x0 n0)))) e H21)))))))))) H20)) H19)))))) (\lambda (f: F).(\lambda 
-(H2: (drop (r (Flat f) n0) O c e)).(\lambda (H0: ((\forall (c2: C).(\forall 
-(v: T).((csubst0 (s (Flat f) i0) v c c2) \to (\forall (e: C).((drop (S n0) O 
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-(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 (s (Flat f) i0) (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 (s (Flat f) i0) (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 (s (Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s 
-(Flat f) i0) (s k (S n0))) v e1 e2))))))))))))))).(\lambda (_: (lt (S n0) (s 
-(Flat f) i0))).(let H19 \def (H0 c0 v H16 e H2) in (or4_ind (drop (S n0) O c0 
-e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda 
-(_: T).(eq C e (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda 
-(_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 k w)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus i0 (s k (S 
-n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u0: T).(eq C e (CHead e1 k u0)))))) (\lambda (k: K).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O c0 (CHead e2 k 
-u0)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 (minus i0 (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda 
-(k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq 
-C e (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 k w))))))) 
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-T).(subst0 (minus i0 (s k (S n0))) v u0 w)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k 
-(S n0))) v e1 e2))))))) (or4 (drop (S n0) O (CHead c0 (Flat f) t) e) (ex3_4 K 
-C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C 
-e (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: 
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-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k 
-u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: 
-T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k u0)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat 
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-(e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k 
-u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k w))))))) 
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-T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))))) (\lambda (H20: (drop (S 
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-T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e 
-(CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: 
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-(minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k 
-u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: 
-T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k u0)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat 
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-(e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k 
-u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k w))))))) 
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-T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))))) (drop_drop (Flat f) n0 c0 
-e H20 t))) (\lambda (H20: (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 c0 (CHead 
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-T).(subst0 (minus i0 (s k (S n0))) v u w))))))).(ex3_4_ind K C T T (\lambda 
-(k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 k 
-u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O c0 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus i0 (s k (S n0))) v u0 
-w))))) (or4 (drop (S n0) O (CHead c0 (Flat f) t) e) (ex3_4 K C T T (\lambda 
-(k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 k 
-u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e0 k w)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Flat 
-f) i0) (s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k u0)))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S 
-n0) O (CHead c0 (Flat f) t) (CHead e2 k u0)))))) (\lambda (k: K).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k 
-(S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k 
-u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k w))))))) 
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-T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))))) (\lambda (x0: 
-K).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H21: (eq C e 
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-x3))).(\lambda (H23: (subst0 (minus i0 (s x0 (S n0))) v x2 x3)).(eq_ind_r C 
-(CHead x1 x0 x2) (\lambda (c: C).(or4 (drop (S n0) O (CHead c0 (Flat f) t) c) 
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-T).(eq C c (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: 
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-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C c (CHead e1 k 
-u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: 
-T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k u0)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat 
-f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C c (CHead e1 k 
-u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k w))))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: 
-T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))))))) (or4_intro1 (drop (S 
-n0) O (CHead c0 (Flat f) t) (CHead x1 x0 x2)) (ex3_4 K C T T (\lambda (k: 
-K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 x0 x2) 
-(CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e0 k w)))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 
-(minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead x1 x0 x2) 
-(CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u0: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k u0)))))) 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
-(minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda 
-(k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq 
-C (CHead x1 x0 x2) (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 
-(Flat f) t) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S 
-n0))) v u0 w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 
-e2))))))) (ex3_4_intro K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: 
-T).(\lambda (_: T).(eq C (CHead x1 x0 x2) (CHead e0 k u0)))))) (\lambda (k: 
-K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 
-(Flat f) t) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u0: 
-T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w))))) 
-x0 x1 x2 x3 (refl_equal C (CHead x1 x0 x2)) (drop_drop (Flat f) n0 c0 (CHead 
-x1 x0 x3) H22 t) H23)) e H21)))))))) H20)) (\lambda (H20: (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 c0 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (s k (S n0))) v e1 
-e2))))))).(ex3_4_ind K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u0: T).(eq C e (CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O c0 (CHead e2 k u0)))))) 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
-(minus i0 (s k (S n0))) v e1 e2))))) (or4 (drop (S n0) O (CHead c0 (Flat f) 
-t) e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: 
-T).(\lambda (_: T).(eq C e (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) 
-(CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda 
-(w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (ex3_4 K C C 
-T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e 
-(CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u0: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k u0)))))) 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
-(minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda 
-(k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq 
-C e (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) 
-(CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda 
-(u0: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 
-w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 
-e2)))))))) (\lambda (x0: K).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: 
-T).(\lambda (H21: (eq C e (CHead x1 x0 x3))).(\lambda (H22: (drop (S n0) O c0 
-(CHead x2 x0 x3))).(\lambda (H23: (csubst0 (minus i0 (s x0 (S n0))) v x1 
-x2)).(eq_ind_r C (CHead x1 x0 x3) (\lambda (c: C).(or4 (drop (S n0) O (CHead 
-c0 (Flat f) t) c) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda 
-(u0: T).(\lambda (_: T).(eq C c (CHead e0 k u0)))))) (\lambda (k: K).(\lambda 
-(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) 
-(CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda 
-(w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (ex3_4 K C C 
-T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C c 
-(CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u0: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k u0)))))) 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
-(minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda 
-(k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq 
-C c (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) 
-(CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda 
-(u0: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 
-w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 
-e2))))))))) (or4_intro2 (drop (S n0) O (CHead c0 (Flat f) t) (CHead x1 x0 
-x3)) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: 
-T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e0 k u0)))))) (\lambda (k: 
-K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 
-(Flat f) t) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u0: 
-T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) 
-(ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: 
-T).(eq C (CHead x1 x0 x3) (CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O (CHead c0 (Flat f) t) 
-(CHead e2 k u0)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e1 k 
-u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k w))))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: 
-T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))))) (ex3_4_intro K C C T 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C 
-(CHead x1 x0 x3) (CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O (CHead c0 (Flat f) t) 
-(CHead e2 k u0)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))) 
-x0 x1 x2 x3 (refl_equal C (CHead x1 x0 x3)) (drop_drop (Flat f) n0 c0 (CHead 
-x2 x0 x3) H22 t) H23)) e H21)))))))) H20)) (\lambda (H20: (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 c0 (CHead e2 k w))))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus i0 (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k 
-(S n0))) v e1 e2)))))))).(ex4_5_ind K C C T T (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k 
-u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 k w))))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 
-(minus i0 (s k (S n0))) v u0 w)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k 
-(S n0))) v e1 e2)))))) (or4 (drop (S n0) O (CHead c0 (Flat f) t) e) (ex3_4 K 
-C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C 
-e (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e0 k w)))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 
-(minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k 
-u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: 
-T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k u0)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat 
-f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k 
-u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k w))))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: 
-T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))))) (\lambda (x0: 
-K).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: 
-T).(\lambda (H21: (eq C e (CHead x1 x0 x3))).(\lambda (H22: (drop (S n0) O c0 
-(CHead x2 x0 x4))).(\lambda (H23: (subst0 (minus i0 (s x0 (S n0))) v x3 
-x4)).(\lambda (H24: (csubst0 (minus i0 (s x0 (S n0))) v x1 x2)).(eq_ind_r C 
-(CHead x1 x0 x3) (\lambda (c: C).(or4 (drop (S n0) O (CHead c0 (Flat f) t) c) 
-(ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: 
-T).(eq C c (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e0 k w)))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 
-(minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C c (CHead e1 k 
-u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: 
-T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k u0)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat 
-f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C c (CHead e1 k 
-u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k w))))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: 
-T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))))))) (or4_intro3 (drop (S 
-n0) O (CHead c0 (Flat f) t) (CHead x1 x0 x3)) (ex3_4 K C T T (\lambda (k: 
-K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) 
-(CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e0 k w)))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 
-(minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead x1 x0 x3) 
-(CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u0: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k u0)))))) 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
-(minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda 
-(k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq 
-C (CHead x1 x0 x3) (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 
-(Flat f) t) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S 
-n0))) v u0 w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 
-e2))))))) (ex4_5_intro K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e1 k 
-u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k w))))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: 
-T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))) x0 x1 x2 x3 x4 (refl_equal 
-C (CHead x1 x0 x3)) (drop_drop (Flat f) n0 c0 (CHead x2 x0 x4) H22 t) H23 
-H24)) e H21)))))))))) H20)) H19)))))) k (drop_gen_drop k c e t n0 H2) H17 
-H18))))) c2 H15)) u (sym_eq T u t H14))) k0 (sym_eq K k0 k H13))) c1 (sym_eq 
-C c1 c H12))) H11)) H10))) v0 (sym_eq T v0 v H8))) i H4 H5 H6 H7 H3))))) | 
-(csubst0_both k0 i0 v0 u1 u2 H3 c1 c0 H4) \Rightarrow (\lambda (H5: (eq nat 
-(s k0 i0) i)).(\lambda (H6: (eq T v0 v)).(\lambda (H7: (eq C (CHead c1 k0 u1) 
-(CHead c k t))).(\lambda (H8: (eq C (CHead c0 k0 u2) c2)).(eq_ind nat (s k0 
-i0) (\lambda (n: nat).((eq T v0 v) \to ((eq C (CHead c1 k0 u1) (CHead c k t)) 
-\to ((eq C (CHead c0 k0 u2) c2) \to ((subst0 i0 v0 u1 u2) \to ((csubst0 i0 v0 
-c1 c0) \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 n (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 n (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 n (s k (S n0))) v u w)))))) 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 (minus n (s k (S n0))) v e1 e2)))))))))))))) (\lambda (H9: 
-(eq T v0 v)).(eq_ind T v (\lambda (t0: T).((eq C (CHead c1 k0 u1) (CHead c k 
-t)) \to ((eq C (CHead c0 k0 u2) c2) \to ((subst0 i0 t0 u1 u2) \to ((csubst0 
-i0 t0 c1 c0) \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 (s k0 i0) (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 (s k0 i0) (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 (s k0 i0) (s k (S n0))) v u w)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s k0 i0) (s k (S n0))) v e1 e2))))))))))))) (\lambda (H10: (eq C 
-(CHead c1 k0 u1) (CHead c k t))).(let H11 \def (f_equal C T (\lambda (e0: 
-C).(match e0 return (\lambda (_: C).T) with [(CSort _) \Rightarrow u1 | 
-(CHead _ _ t) \Rightarrow t])) (CHead c1 k0 u1) (CHead c k t) H10) in ((let 
-H12 \def (f_equal C K (\lambda (e0: C).(match e0 return (\lambda (_: C).K) 
-with [(CSort _) \Rightarrow k0 | (CHead _ k _) \Rightarrow k])) (CHead c1 k0 
-u1) (CHead c k t) H10) in ((let H13 \def (f_equal C C (\lambda (e0: C).(match 
-e0 return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) 
-\Rightarrow c])) (CHead c1 k0 u1) (CHead c k t) H10) in (eq_ind C c (\lambda 
-(c: C).((eq K k0 k) \to ((eq T u1 t) \to ((eq C (CHead c0 k0 u2) c2) \to 
-((subst0 i0 v u1 u2) \to ((csubst0 i0 v c c0) \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 (s k0 i0) 
-(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 (s k0 i0) (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 (s k0 i0) (s k (S n0))) v u w)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s k0 i0) (s k (S n0))) v e1 e2)))))))))))))) (\lambda (H14: (eq K k0 
-k)).(eq_ind K k (\lambda (k: K).((eq T u1 t) \to ((eq C (CHead c0 k u2) c2) 
-\to ((subst0 i0 v u1 u2) \to ((csubst0 i0 v c c0) \to (or4 (drop (S n0) O c2 
-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 c2 (CHead e0 k1 w)))))) 
-(\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus (s k i0) (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 c2 (CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (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 c2 (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s 
-k1 (S n0))) v u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v 
-e1 e2))))))))))))) (\lambda (H15: (eq T u1 t)).(eq_ind T t (\lambda (t: 
-T).((eq C (CHead c0 k u2) c2) \to ((subst0 i0 v t u2) \to ((csubst0 i0 v c 
-c0) \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 (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda 
-(w: T).(subst0 (minus (s k i0) (s k1 (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 (k1: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (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 (k1: K).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u 
-w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 
-e2)))))))))))) (\lambda (H16: (eq C (CHead c0 k u2) c2)).(eq_ind C (CHead c0 
-k u2) (\lambda (c2: C).((subst0 i0 v t u2) \to ((csubst0 i0 v c c0) \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 (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus (s k i0) (s k1 (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 (k1: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (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 (k1: K).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u 
-w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2))))))))))) 
-(\lambda (_: (subst0 i0 v t u2)).(\lambda (H18: (csubst0 i0 v c c0)).(let H1 
-\def (eq_ind K k0 (\lambda (k: K).(eq nat (s k i0) i)) H5 k H14) in (let H19 
-\def (eq_ind_r nat i (\lambda (n: nat).(\forall (c2: C).(\forall (v: 
-T).((csubst0 n 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 n (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 n (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 n (s k (S n0))) v u w)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus n (s k (S n0))) v e1 e2)))))))))))))) H0 (s k i0) H1) in (let H20 \def 
-(eq_ind_r nat i (\lambda (n: nat).(lt (S n0) n)) H (s k i0) H1) in (K_ind 
-(\lambda (k: K).((drop (r k n0) O c e) \to (((\forall (c2: C).(\forall (v: 
-T).((csubst0 (s k i0) 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 (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus (s k i0) (s k0 (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 (k0: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 (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 (k0: K).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 (S n0))) v u 
-w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 (S n0))) v e1 
-e2)))))))))))))) \to ((lt (S n0) (s k i0)) \to (or4 (drop (S n0) O (CHead c0 
-k u2) 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 c0 k u2) (CHead 
-e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus (s k i0) (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 c0 k u2) (CHead e2 k1 u)))))) (\lambda (k1: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) 
-(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 c0 k u2) (CHead e2 k1 w))))))) 
-(\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus (s k i0) (s k1 (S n0))) v u w)))))) (\lambda (k1: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s k i0) (s k1 (S n0))) v e1 e2)))))))))))) (\lambda (b: B).(\lambda 
-(H2: (drop (r (Bind b) n0) O c e)).(\lambda (_: ((\forall (c2: C).(\forall 
-(v: T).((csubst0 (s (Bind b) i0) 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 (s (Bind b) i0) (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 (s (Bind b) i0) (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 (s (Bind b) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s 
-(Bind b) i0) (s k (S n0))) v e1 e2))))))))))))))).(\lambda (H: (lt (S n0) (s 
-(Bind b) i0))).(let H21 \def (IHn i0 (le_S_n (S n0) i0 H) c c0 v H18 e H2) in 
-(or4_ind (drop n0 O c0 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 c0 (CHead e0 k 
-w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus i0 (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 c0 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (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 c0 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k n0)) v u w)))))) 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 (minus i0 (s k n0)) v e1 e2))))))) (or4 (drop (S n0) O (CHead 
-c0 (Bind b) u2) 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 (CHead c0 (Bind b) u2) 
-(CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda 
-(w: T).(subst0 (minus (s (Bind b) i0) (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 (CHead c0 (Bind b) u2) (CHead e2 k u)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind 
-b) i0) (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 (CHead c0 (Bind b) u2) (CHead e2 k w))))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u w)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s (Bind b) i0) (s k (S n0))) v e1 e2)))))))) (\lambda (H22: (drop n0 
-O c0 e)).(or4_intro0 (drop (S n0) O (CHead c0 (Bind b) u2) 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 (CHead c0 (Bind b) u2) (CHead e0 k w)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
-i0) (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 (CHead c0 
-(Bind b) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (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 (CHead c0 (Bind b) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda 
-(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s 
-(Bind b) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind 
-b) i0) (s k (S n0))) v e1 e2))))))) (drop_drop (Bind b) n0 c0 e H22 u2))) 
-(\lambda (H22: (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 c0 (CHead e0 k w)))))) (\lambda 
-(k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k 
-n0)) v u w))))))).(ex3_4_ind 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 c0 (CHead e0 k 
-w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus i0 (s k n0)) v u w))))) (or4 (drop (S n0) O (CHead c0 (Bind 
-b) u2) 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 (CHead c0 (Bind b) u2) 
-(CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda 
-(w: T).(subst0 (minus (s (Bind b) i0) (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 (CHead c0 (Bind b) u2) (CHead e2 k u)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind 
-b) i0) (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 (CHead c0 (Bind b) u2) (CHead e2 k w))))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u w)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s (Bind b) i0) (s k (S n0))) v e1 e2)))))))) (\lambda (x0: 
-K).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H23: (eq C e 
-(CHead x1 x0 x2))).(\lambda (H24: (drop n0 O c0 (CHead x1 x0 x3))).(\lambda 
-(H25: (subst0 (minus i0 (s x0 n0)) v x2 x3)).(eq_ind_r C (CHead x1 x0 x2) 
-(\lambda (c: C).(or4 (drop (S n0) O (CHead c0 (Bind b) u2) 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 (CHead c0 (Bind b) u2) (CHead e0 k w)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
-i0) (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 (CHead c0 
-(Bind b) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (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 (CHead c0 (Bind b) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda 
-(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s 
-(Bind b) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind 
-b) i0) (s k (S n0))) v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead c0 
-(Bind b) u2) (CHead x1 x0 x2)) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 x0 x2) (CHead e0 k u)))))) 
-(\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) 
-O (CHead c0 (Bind b) u2) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (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 (CHead x1 x0 x2) (CHead e1 k u)))))) (\lambda 
-(k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead 
-c0 (Bind b) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (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 (CHead x1 x0 x2) (CHead e1 k 
-u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Bind b) u2) (CHead e2 k w))))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u w)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s (Bind b) i0) (s k (S n0))) v e1 e2))))))) (ex3_4_intro K C T T 
-(\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead 
-x1 x0 x2) (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Bind b) u2) (CHead e0 k w)))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus (s (Bind b) i0) (s k (S n0))) v u w))))) x0 x1 x2 x3 (refl_equal C 
-(CHead x1 x0 x2)) (drop_drop (Bind b) n0 c0 (CHead x1 x0 x3) H24 u2) 
-(eq_ind_r nat (S (s x0 n0)) (\lambda (n: nat).(subst0 (minus (s (Bind b) i0) 
-n) v x2 x3)) H25 (s x0 (S n0)) (s_S x0 n0)))) e H23)))))))) H22)) (\lambda 
-(H22: (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 c0 (CHead e2 k u)))))) 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
-(minus i0 (s k n0)) v e1 e2))))))).(ex3_4_ind 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 c0 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (s k n0)) v e1 e2))))) 
-(or4 (drop (S n0) O (CHead c0 (Bind b) u2) 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 (CHead c0 (Bind b) u2) (CHead e0 k w)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
-i0) (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 (CHead c0 
-(Bind b) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (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 (CHead c0 (Bind b) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda 
-(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s 
-(Bind b) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind 
-b) i0) (s k (S n0))) v e1 e2)))))))) (\lambda (x0: K).(\lambda (x1: 
-C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H23: (eq C e (CHead x1 x0 
-x3))).(\lambda (H24: (drop n0 O c0 (CHead x2 x0 x3))).(\lambda (H25: (csubst0 
-(minus i0 (s x0 n0)) v x1 x2)).(eq_ind_r C (CHead x1 x0 x3) (\lambda (c: 
-C).(or4 (drop (S n0) O (CHead c0 (Bind b) u2) 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 (CHead c0 (Bind b) u2) (CHead e0 k w)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
-i0) (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 (CHead c0 
-(Bind b) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (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 (CHead c0 (Bind b) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda 
-(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s 
-(Bind b) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind 
-b) i0) (s k (S n0))) v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead c0 
-(Bind b) u2) (CHead x1 x0 x3)) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e0 k u)))))) 
-(\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) 
-O (CHead c0 (Bind b) u2) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (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 (CHead x1 x0 x3) (CHead e1 k u)))))) (\lambda 
-(k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead 
-c0 (Bind b) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (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 (CHead x1 x0 x3) (CHead e1 k 
-u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Bind b) u2) (CHead e2 k w))))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u w)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s (Bind b) i0) (s k (S n0))) v e1 e2))))))) (ex3_4_intro K C C T 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead 
-x1 x0 x3) (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop (S n0) O (CHead c0 (Bind b) u2) (CHead e2 k u)))))) 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
-(minus (s (Bind b) i0) (s k (S n0))) v e1 e2))))) x0 x1 x2 x3 (refl_equal C 
-(CHead x1 x0 x3)) (drop_drop (Bind b) n0 c0 (CHead x2 x0 x3) H24 u2) 
-(eq_ind_r nat (S (s x0 n0)) (\lambda (n: nat).(csubst0 (minus (s (Bind b) i0) 
-n) v x1 x2)) H25 (s x0 (S n0)) (s_S x0 n0)))) e H23)))))))) H22)) (\lambda 
-(H22: (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 c0 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k n0)) v u w)))))) 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 (minus i0 (s k n0)) 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 e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c0 (CHead e2 k w))))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus i0 (s k n0)) v u w)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k 
-n0)) v e1 e2)))))) (or4 (drop (S n0) O (CHead c0 (Bind b) u2) 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 (CHead c0 (Bind b) u2) (CHead e0 k w)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
-i0) (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 (CHead c0 
-(Bind b) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (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 (CHead c0 (Bind b) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda 
-(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s 
-(Bind b) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind 
-b) i0) (s k (S n0))) v e1 e2)))))))) (\lambda (x0: K).(\lambda (x1: 
-C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H23: (eq C e 
-(CHead x1 x0 x3))).(\lambda (H24: (drop n0 O c0 (CHead x2 x0 x4))).(\lambda 
-(H25: (subst0 (minus i0 (s x0 n0)) v x3 x4)).(\lambda (H26: (csubst0 (minus 
-i0 (s x0 n0)) v x1 x2)).(eq_ind_r C (CHead x1 x0 x3) (\lambda (c: C).(or4 
-(drop (S n0) O (CHead c0 (Bind b) u2) 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 (CHead c0 (Bind b) u2) (CHead e0 k w)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
-i0) (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 (CHead c0 
-(Bind b) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (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 (CHead c0 (Bind b) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda 
-(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s 
-(Bind b) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind 
-b) i0) (s k (S n0))) v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c0 
-(Bind b) u2) (CHead x1 x0 x3)) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e0 k u)))))) 
-(\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) 
-O (CHead c0 (Bind b) u2) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (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 (CHead x1 x0 x3) (CHead e1 k u)))))) (\lambda 
-(k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead 
-c0 (Bind b) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (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 (CHead x1 x0 x3) (CHead e1 k 
-u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Bind b) u2) (CHead e2 k w))))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u w)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s (Bind b) i0) (s k (S n0))) v e1 e2))))))) (ex4_5_intro K C C T T 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C (CHead x1 x0 x3) (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 
-(Bind b) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda 
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S 
-n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 
-e2)))))) x0 x1 x2 x3 x4 (refl_equal C (CHead x1 x0 x3)) (drop_drop (Bind b) 
-n0 c0 (CHead x2 x0 x4) H24 u2) (eq_ind_r nat (S (s x0 n0)) (\lambda (n: 
-nat).(subst0 (minus (s (Bind b) i0) n) v x3 x4)) H25 (s x0 (S n0)) (s_S x0 
-n0)) (eq_ind_r nat (S (s x0 n0)) (\lambda (n: nat).(csubst0 (minus (s (Bind 
-b) i0) n) v x1 x2)) H26 (s x0 (S n0)) (s_S x0 n0)))) e H23)))))))))) H22)) 
-H21)))))) (\lambda (f: F).(\lambda (H2: (drop (r (Flat f) n0) O c 
-e)).(\lambda (H0: ((\forall (c2: C).(\forall (v: T).((csubst0 (s (Flat f) i0) 
-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 (s (Flat f) 
-i0) (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 (s (Flat f) i0) (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 (s (Flat f) i0) (s k (S n0))) v u w)))))) 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 
-e2))))))))))))))).(\lambda (_: (lt (S n0) (s (Flat f) i0))).(let H21 \def (H0 
-c0 v H18 e H2) in (or4_ind (drop (S n0) O c0 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 c0 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (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 c0 (CHead e2 k u)))))) 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
-(minus i0 (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 c0 (CHead e2 k w))))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus i0 (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k 
-(S n0))) v e1 e2))))))) (or4 (drop (S n0) O (CHead c0 (Flat f) u2) 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 (CHead c0 (Flat f) u2) (CHead e0 k w)))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus (s (Flat f) i0) (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 (CHead c0 (Flat f) u2) (CHead e2 k u)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat 
-f) i0) (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 (CHead c0 (Flat f) u2) (CHead e2 k w))))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))))) (\lambda (H22: (drop (S 
-n0) O c0 e)).(or4_intro0 (drop (S n0) O (CHead c0 (Flat f) u2) 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 (CHead c0 (Flat f) u2) (CHead e0 k w)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) 
-i0) (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 (CHead c0 
-(Flat f) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (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 (CHead c0 (Flat f) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda 
-(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s 
-(Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat 
-f) i0) (s k (S n0))) v e1 e2))))))) (drop_drop (Flat f) n0 c0 e H22 u2))) 
-(\lambda (H22: (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 c0 (CHead e0 k w)))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus i0 (s k (S n0))) v u w))))))).(ex3_4_ind 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 c0 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k (S n0))) v u w))))) 
-(or4 (drop (S n0) O (CHead c0 (Flat f) u2) 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 (CHead c0 (Flat f) u2) (CHead e0 k w)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) 
-i0) (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 (CHead c0 
-(Flat f) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (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 (CHead c0 (Flat f) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda 
-(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s 
-(Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat 
-f) i0) (s k (S n0))) v e1 e2)))))))) (\lambda (x0: K).(\lambda (x1: 
-C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H23: (eq C e (CHead x1 x0 
-x2))).(\lambda (H24: (drop (S n0) O c0 (CHead x1 x0 x3))).(\lambda (H25: 
-(subst0 (minus i0 (s x0 (S n0))) v x2 x3)).(eq_ind_r C (CHead x1 x0 x2) 
-(\lambda (c: C).(or4 (drop (S n0) O (CHead c0 (Flat f) u2) 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 (CHead c0 (Flat f) u2) (CHead e0 k w)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) 
-i0) (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 (CHead c0 
-(Flat f) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (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 (CHead c0 (Flat f) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda 
-(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s 
-(Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat 
-f) i0) (s k (S n0))) v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead c0 
-(Flat f) u2) (CHead x1 x0 x2)) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 x0 x2) (CHead e0 k u)))))) 
-(\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) 
-O (CHead c0 (Flat f) u2) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (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 (CHead x1 x0 x2) (CHead e1 k u)))))) (\lambda 
-(k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead 
-c0 (Flat f) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (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 (CHead x1 x0 x2) (CHead e1 k 
-u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e2 k w))))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))))) (ex3_4_intro K C T T 
-(\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead 
-x1 x0 x2) (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e0 k w)))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus (s (Flat f) i0) (s k (S n0))) v u w))))) x0 x1 x2 x3 (refl_equal C 
-(CHead x1 x0 x2)) (drop_drop (Flat f) n0 c0 (CHead x1 x0 x3) H24 u2) H25)) e 
-H23)))))))) H22)) (\lambda (H22: (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 c0 (CHead 
-e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 (minus i0 (s k (S n0))) v e1 e2))))))).(ex3_4_ind 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 c0 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (s k (S n0))) v e1 
-e2))))) (or4 (drop (S n0) O (CHead c0 (Flat f) u2) 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 (CHead c0 (Flat f) u2) (CHead e0 k w)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) 
-i0) (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 (CHead c0 
-(Flat f) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (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 (CHead c0 (Flat f) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda 
-(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s 
-(Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat 
-f) i0) (s k (S n0))) v e1 e2)))))))) (\lambda (x0: K).(\lambda (x1: 
-C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H23: (eq C e (CHead x1 x0 
-x3))).(\lambda (H24: (drop (S n0) O c0 (CHead x2 x0 x3))).(\lambda (H25: 
-(csubst0 (minus i0 (s x0 (S n0))) v x1 x2)).(eq_ind_r C (CHead x1 x0 x3) 
-(\lambda (c: C).(or4 (drop (S n0) O (CHead c0 (Flat f) u2) 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 (CHead c0 (Flat f) u2) (CHead e0 k w)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) 
-i0) (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 (CHead c0 
-(Flat f) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (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 (CHead c0 (Flat f) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda 
-(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s 
-(Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat 
-f) i0) (s k (S n0))) v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead c0 
-(Flat f) u2) (CHead x1 x0 x3)) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e0 k u)))))) 
-(\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) 
-O (CHead c0 (Flat f) u2) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (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 (CHead x1 x0 x3) (CHead e1 k u)))))) (\lambda 
-(k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead 
-c0 (Flat f) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (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 (CHead x1 x0 x3) (CHead e1 k 
-u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e2 k w))))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))))) (ex3_4_intro K C C T 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead 
-x1 x0 x3) (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e2 k u)))))) 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
-(minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))) x0 x1 x2 x3 (refl_equal C 
-(CHead x1 x0 x3)) (drop_drop (Flat f) n0 c0 (CHead x2 x0 x3) H24 u2) H25)) e 
-H23)))))))) H22)) (\lambda (H22: (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 c0 (CHead e2 k w))))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus i0 (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k (S n0))) 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 e (CHead e1 k u))))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O c0 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k (S n0))) v u 
-w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 (minus i0 (s k (S n0))) v e1 e2)))))) (or4 (drop 
-(S n0) O (CHead c0 (Flat f) u2) 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 (CHead 
-c0 (Flat f) u2) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (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 (CHead c0 (Flat f) u2) 
-(CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (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 (CHead c0 (Flat f) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda 
-(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s 
-(Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat 
-f) i0) (s k (S n0))) v e1 e2)))))))) (\lambda (x0: K).(\lambda (x1: 
-C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H23: (eq C e 
-(CHead x1 x0 x3))).(\lambda (H24: (drop (S n0) O c0 (CHead x2 x0 
-x4))).(\lambda (H25: (subst0 (minus i0 (s x0 (S n0))) v x3 x4)).(\lambda 
-(H26: (csubst0 (minus i0 (s x0 (S n0))) v x1 x2)).(eq_ind_r C (CHead x1 x0 
-x3) (\lambda (c: C).(or4 (drop (S n0) O (CHead c0 (Flat f) u2) 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 (CHead c0 (Flat f) u2) (CHead e0 k w)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) 
-i0) (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 (CHead c0 
-(Flat f) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (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 (CHead c0 (Flat f) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda 
-(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s 
-(Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat 
-f) i0) (s k (S n0))) v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c0 
-(Flat f) u2) (CHead x1 x0 x3)) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e0 k u)))))) 
-(\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) 
-O (CHead c0 (Flat f) u2) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (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 (CHead x1 x0 x3) (CHead e1 k u)))))) (\lambda 
-(k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead 
-c0 (Flat f) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (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 (CHead x1 x0 x3) (CHead e1 k 
-u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e2 k w))))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))))) (ex4_5_intro K C C T T 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C (CHead x1 x0 x3) (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 
-(Flat f) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda 
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S 
-n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 
-e2)))))) x0 x1 x2 x3 x4 (refl_equal C (CHead x1 x0 x3)) (drop_drop (Flat f) 
-n0 c0 (CHead x2 x0 x4) H24 u2) H25 H26)) e H23)))))))))) H22)) H21)))))) k 
-(drop_gen_drop k c e t n0 H2) H19 H20)))))) c2 H16)) u1 (sym_eq T u1 t H15))) 
-k0 (sym_eq K k0 k H14))) c1 (sym_eq C c1 c H13))) H12)) H11))) v0 (sym_eq T 
-v0 v H9))) i H5 H6 H7 H8 H3 H4)))))]) in (H3 (refl_equal nat i) (refl_equal T 
-v) (refl_equal C (CHead c k t)) (refl_equal C c2)))))))))))) 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)) 
-(or4 (drop O O 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 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 c1 (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 c1 (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)))))))) (\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 O O 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 O O c0 
-(CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 O 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 O O c0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 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 O O c0 (CHead e2 
-(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (w: T).(subst0 O t u w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O 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 O O (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 O O (CHead c k0 
-u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 O 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 O O (CHead c k0 u2) (CHead e2 (Flat f) u)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
-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 O O (CHead c k0 u2) (CHead e2 (Flat f) 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)))))))))))))))) 
-(\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 
-return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow 
-True])) I O H3) in (False_ind (or4 (drop O O (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 O O 
-(CHead c (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O 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 O O (CHead c (Bind b) u2) (CHead e2 
-(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(csubst0 O 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 O O (CHead c (Bind b) u2) 
-(CHead e2 (Flat f) 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)))))))) 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 (n: nat).(subst0 n v0 u1 u2)) H2 O H3) in (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))))))))))) 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 O O 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 O O c4 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O 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 O O c4 (CHead e2 (Flat f) u)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 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 O O c4 (CHead e2 (Flat f) 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)))))))))) \to (\forall (u: T).((eq nat (s k0 i) O) 
-\to (or4 (drop O O (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 O O (CHead c4 k0 u) (CHead e0 (Flat f) w)))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O 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 O O 
-(CHead c4 k0 u) (CHead e2 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 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 O O 
-(CHead c4 k0 u) (CHead e2 (Flat f) 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))))))))))))))))) (\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 O O 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 O O c4 (CHead e0 (Flat f) w)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O 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 O O c4 (CHead e2 (Flat f) u)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
-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 O O c4 (CHead e2 (Flat f) 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))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat 
-(S i) O)).(let H5 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee return 
-(\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) 
-I O H4) in (False_ind (or4 (drop O O (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 O O (CHead c4 (Bind 
-b) u) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda 
-(u0: T).(\lambda (w: T).(subst0 O 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 O O (CHead c4 (Bind b) u) (CHead e2 (Flat f) 
-u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 O 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 O O (CHead c4 (Bind b) u) (CHead e2 
-(Flat f) 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)))))))) 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 O O 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 O O c4 (CHead e0 (Flat f) w)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O 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 O O c4 (CHead e2 (Flat f) u)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
-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 O O c4 (CHead e2 (Flat f) 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))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat i 
-O)).(let H5 \def (eq_ind nat i (\lambda (n: nat).((eq nat n O) \to (or4 (drop 
-O O 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 O O c4 (CHead e0 
-(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 O 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 O O c4 
-(CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 O 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 O O c4 (CHead e2 (Flat f) 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)))))))))) H3 O H4) in 
-(let H6 \def (eq_ind nat i (\lambda (n: nat).(csubst0 n v0 c3 c4)) H2 O H4) 
-in (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))))))))))))) 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 O O 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 O O c4 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O 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 O O c4 (CHead e2 (Flat f) u)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 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 O O c4 (CHead e2 (Flat f) 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)))))))))) \to ((eq nat (s k0 i) O) \to (or4 (drop 
-O O (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 O O (CHead c4 k0 u2) (CHead e0 (Flat f) w)))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O 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 O O (CHead c4 
-k0 u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 O 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 O O (CHead c4 k0 
-u2) (CHead e2 (Flat f) 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))))))))))))))))))) (\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 O O 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 O O c4 (CHead e0 (Flat f) w)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O 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 O O c4 (CHead e2 (Flat f) u)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
-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 O O c4 (CHead e2 (Flat f) 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))))))))))).(\lambda (H5: (eq nat (S i) O)).(let H6 
-\def (eq_ind nat (S i) (\lambda (ee: nat).(match ee return (\lambda (_: 
-nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H5) in 
-(False_ind (or4 (drop O O (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 O O (CHead c4 (Bind 
-b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (w: T).(subst0 O 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 O O (CHead c4 (Bind b) u2) (CHead e2 (Flat f) 
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 O 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 O O (CHead c4 (Bind b) u2) (CHead e2 
-(Flat f) 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)))))))) 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 O O 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 O O c4 (CHead e0 (Flat f) w)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O 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 O O c4 (CHead e2 (Flat f) u)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
-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 O O c4 (CHead e2 (Flat f) 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))))))))))).(\lambda (H5: (eq nat i O)).(let H6 
-\def (eq_ind nat i (\lambda (n: nat).((eq nat n O) \to (or4 (drop O O 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 O O c4 (CHead e0 (Flat f) w)))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O 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 O O c4 (CHead e2 
-(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(csubst0 O 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 O O c4 (CHead e2 (Flat f) 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)))))))))) H4 O H5) in (let H7 \def 
-(eq_ind nat i (\lambda (n: nat).(csubst0 n v0 c3 c4)) H3 O H5) in (let H8 
-\def (eq_ind nat i (\lambda (n: nat).(subst0 n v0 u1 u2)) H2 O H5) in 
-(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)))))))))))))))) 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 
-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)).(let H2 \def (match H0 return (\lambda (n: nat).(\lambda (t0: T).(\lambda 
-(c0: C).(\lambda (c1: C).(\lambda (_: (csubst0 n t0 c0 c1)).((eq nat n (S 
-n0)) \to ((eq T t0 v) \to ((eq C c0 (CHead c k t)) \to ((eq C c1 c2) \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))))))))))))))))) with [(csubst0_snd k0 i v0 u1 u2 H2 c0) \Rightarrow 
-(\lambda (H3: (eq nat (s k0 i) (S n0))).(\lambda (H4: (eq T v0 v)).(\lambda 
-(H5: (eq C (CHead c0 k0 u1) (CHead c k t))).(\lambda (H6: (eq C (CHead c0 k0 
-u2) c2)).((let H7 \def (f_equal nat nat (\lambda (e0: nat).e0) (s k0 i) (S 
-n0) H3) in (eq_ind nat (s k0 i) (\lambda (n: nat).((eq T v0 v) \to ((eq C 
-(CHead c0 k0 u1) (CHead c k t)) \to ((eq C (CHead c0 k0 u2) c2) \to ((subst0 
-i v0 u1 u2) \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))))))))))))) (\lambda (H8: (eq T v0 v)).(eq_ind T v (\lambda (t0: T).((eq 
-C (CHead c0 k0 u1) (CHead c k t)) \to ((eq C (CHead c0 k0 u2) c2) \to 
-((subst0 i t0 u1 u2) \to (or4 (drop (s k0 i) 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 k0 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 e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop (s k0 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 e (CHead e1 (Flat f) 
-u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (s k0 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)))))))))))) (\lambda 
-(H9: (eq C (CHead c0 k0 u1) (CHead c k t))).(let H10 \def (f_equal C T 
-(\lambda (e0: C).(match e0 return (\lambda (_: C).T) with [(CSort _) 
-\Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u1) (CHead c k 
-t) H9) in ((let H11 \def (f_equal C K (\lambda (e0: C).(match e0 return 
-(\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _) \Rightarrow 
-k])) (CHead c0 k0 u1) (CHead c k t) H9) in ((let H12 \def (f_equal C C 
-(\lambda (e0: C).(match e0 return (\lambda (_: C).C) with [(CSort _) 
-\Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k0 u1) (CHead c k 
-t) H9) in (eq_ind C c (\lambda (c: C).((eq K k0 k) \to ((eq T u1 t) \to ((eq 
-C (CHead c k0 u2) c2) \to ((subst0 i v u1 u2) \to (or4 (drop (s k0 i) 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 k0 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 e (CHead e1 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s k0 
-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 
-e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 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))))))))))))) (\lambda (H13: (eq K k0 k)).(eq_ind K k (\lambda (k: K).((eq 
-T u1 t) \to ((eq C (CHead c k u2) c2) \to ((subst0 i v u1 u2) \to (or4 (drop 
-(s k i) 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 k 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 e (CHead e1 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s k 
-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 
-e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (s k 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)))))))))))) (\lambda (H14: (eq T u1 t)).(eq_ind T t (\lambda (t: T).((eq C 
-(CHead c k u2) c2) \to ((subst0 i v t u2) \to (or4 (drop (s k i) 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 k 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 e (CHead e1 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s k 
-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 
-e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (s k 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))))))))))) (\lambda (H15: (eq C (CHead c k u2) c2)).(eq_ind C (CHead c k 
-u2) (\lambda (c: C).((subst0 i v t u2) \to (or4 (drop (s k i) O c 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 k i) O c (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 k i) O c (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 k i) O c (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 (H16: (subst0 i v t 
-u2)).(let H0 \def (eq_ind K k0 (\lambda (k: K).(eq nat (s k i) (S n0))) H7 k 
-H13) in (K_ind (\lambda (k: K).((drop (r k n0) O c e) \to ((eq nat (s k i) (S 
-n0)) \to (or4 (drop (s k i) O (CHead c k u2) 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 k i) O (CHead c k u2) (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 k i) O (CHead c k u2) (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 k i) O (CHead c k u2) (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 (H1: (drop (r (Bind b) n0) O c 
-e)).(\lambda (H17: (eq nat (s (Bind b) i) (S n0))).(let H18 \def (f_equal nat 
-nat (\lambda (e0: nat).(match e0 return (\lambda (_: nat).nat) with [O 
-\Rightarrow i | (S n) \Rightarrow n])) (S i) (S n0) H17) in (let H19 \def 
-(eq_ind nat i (\lambda (n: nat).(subst0 n v t u2)) H16 n0 H18) in (eq_ind_r 
-nat n0 (\lambda (n: nat).(or4 (drop (s (Bind b) n) O (CHead c (Bind b) u2) 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 (Bind b) n) O (CHead c (Bind b) 
-u2) (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 (Bind b) n) O (CHead c (Bind b) u2) (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 (Bind b) n) O (CHead c (Bind b) u2) (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_intro0 
-(drop (s (Bind b) n0) O (CHead c (Bind b) u2) 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 (Bind b) n0) O (CHead c (Bind b) u2) (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 (Bind b) n0) O 
-(CHead c (Bind b) u2) (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 
-(Bind b) n0) O (CHead c (Bind b) u2) (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 H1 
-u2)) i H18)))))) (\lambda (f: F).(\lambda (H1: (drop (r (Flat f) n0) O c 
-e)).(\lambda (H17: (eq nat (s (Flat f) i) (S n0))).(let H18 \def (f_equal nat 
-nat (\lambda (e0: nat).e0) i (S n0) H17) in (let H19 \def (eq_ind nat i 
-(\lambda (n: nat).(subst0 n v t u2)) H16 (S n0) H18) in (eq_ind_r nat (S n0) 
-(\lambda (n: nat).(or4 (drop (s (Flat f) n) O (CHead c (Flat f) u2) 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 (Flat f) n) O (CHead c (Flat f) u2) (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 
-(Flat f) n) O (CHead c (Flat f) u2) (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 
-(Flat f) n) 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 
-v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro0 (drop (s (Flat f) 
-(S n0)) O (CHead c (Flat f) u2) 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 
-(Flat f) (S n0)) O (CHead c (Flat f) u2) (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 (Flat f) (S n0)) O (CHead c 
-(Flat f) u2) (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 (Flat f) (S n0)) 
-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 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 H1 u2)) i 
-H18)))))) k (drop_gen_drop k c e t n0 H1) H0))) c2 H15)) u1 (sym_eq T u1 t 
-H14))) k0 (sym_eq K k0 k H13))) c0 (sym_eq C c0 c H12))) H11)) H10))) v0 
-(sym_eq T v0 v H8))) (S n0) H7)) H4 H5 H6 H2))))) | (csubst0_fst k0 i c1 c0 
-v0 H2 u) \Rightarrow (\lambda (H3: (eq nat (s k0 i) (S n0))).(\lambda (H4: 
-(eq T v0 v)).(\lambda (H5: (eq C (CHead c1 k0 u) (CHead c k t))).(\lambda 
-(H6: (eq C (CHead c0 k0 u) c2)).((let H7 \def (f_equal nat nat (\lambda (e0: 
-nat).e0) (s k0 i) (S n0) H3) in (eq_ind nat (s k0 i) (\lambda (n: nat).((eq T 
-v0 v) \to ((eq C (CHead c1 k0 u) (CHead c k t)) \to ((eq C (CHead c0 k0 u) 
-c2) \to ((csubst0 i v0 c1 c0) \to (or4 (drop n O c2 e) (ex3_4 F C T T 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e 
-(CHead e0 (Flat f) u0)))))) (\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 (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) 
-(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: 
-T).(eq C e (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u0: T).(drop n O c2 (CHead e2 (Flat f) u0)))))) 
-(\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 (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u0))))))) 
-(\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 (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v e1 e2))))))))))))) (\lambda (H8: (eq T v0 v)).(eq_ind T v 
-(\lambda (t0: T).((eq C (CHead c1 k0 u) (CHead c k t)) \to ((eq C (CHead c0 
-k0 u) c2) \to ((csubst0 i t0 c1 c0) \to (or4 (drop (s k0 i) O c2 e) (ex3_4 F 
-C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C 
-e (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (s k0 i) O c2 (CHead e0 (Flat f) w)))))) (\lambda 
-(_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 
-w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f) u0)))))) (\lambda (f: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s k0 i) O c2 
-(CHead e2 (Flat f) u0)))))) (\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 (u0: T).(\lambda (_: T).(eq C e 
-(CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) O c2 (CHead e2 (Flat f) 
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: 
-T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2)))))))))))) (\lambda (H9: (eq C (CHead c1 k0 u) (CHead c k t))).(let H10 
-\def (f_equal C T (\lambda (e0: C).(match e0 return (\lambda (_: C).T) with 
-[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k0 u) 
-(CHead c k t) H9) in ((let H11 \def (f_equal C K (\lambda (e0: C).(match e0 
-return (\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _) 
-\Rightarrow k])) (CHead c1 k0 u) (CHead c k t) H9) in ((let H12 \def (f_equal 
-C C (\lambda (e0: C).(match e0 return (\lambda (_: C).C) with [(CSort _) 
-\Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k0 u) (CHead c k t) 
-H9) in (eq_ind C c (\lambda (c: C).((eq K k0 k) \to ((eq T u t) \to ((eq C 
-(CHead c0 k0 u) c2) \to ((csubst0 i v c c0) \to (or4 (drop (s k0 i) O c2 e) 
-(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: 
-T).(eq C e (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) O c2 (CHead e0 (Flat f) 
-w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: 
-T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f) u0)))))) 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s k0 
-i) O c2 (CHead e2 (Flat f) u0)))))) (\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 (u0: T).(\lambda (_: T).(eq 
-C e (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) O c2 (CHead e2 (Flat 
-f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: 
-T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2))))))))))))) (\lambda (H13: (eq K k0 k)).(eq_ind K k (\lambda (k: K).((eq 
-T u t) \to ((eq C (CHead c0 k u) c2) \to ((csubst0 i v c c0) \to (or4 (drop 
-(s k i) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: 
-T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u0)))))) (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c2 (CHead 
-e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda 
-(w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f) u0)))))) 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s k 
-i) O c2 (CHead e2 (Flat f) u0)))))) (\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 (u0: T).(\lambda (_: T).(eq 
-C e (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c2 (CHead e2 (Flat f) 
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: 
-T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2)))))))))))) (\lambda (H14: (eq T u t)).(eq_ind T t (\lambda (t: T).((eq C 
-(CHead c0 k t) c2) \to ((csubst0 i v c c0) \to (or4 (drop (s k i) O c2 e) 
-(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: 
-T).(eq C e (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c2 (CHead e0 (Flat f) 
-w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: 
-T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f) u0)))))) 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s k 
-i) O c2 (CHead e2 (Flat f) u0)))))) (\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 (u0: T).(\lambda (_: T).(eq 
-C e (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c2 (CHead e2 (Flat f) 
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: 
-T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2))))))))))) (\lambda (H15: (eq C (CHead c0 k t) c2)).(eq_ind C (CHead c0 k 
-t) (\lambda (c2: C).((csubst0 i v c c0) \to (or4 (drop (s k i) O c2 e) (ex3_4 
-F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq 
-C e (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (s k i) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) 
-(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: 
-T).(eq C e (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u0: T).(drop (s k i) O c2 (CHead e2 (Flat f) 
-u0)))))) (\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 (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat 
-f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (s k i) O c2 (CHead e2 (Flat f) w))))))) (\lambda 
-(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: 
-T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))) (\lambda 
-(H16: (csubst0 i v c c0)).(let H0 \def (eq_ind K k0 (\lambda (k: K).(eq nat 
-(s k i) (S n0))) H7 k H13) in (K_ind (\lambda (k: K).((drop (r k n0) O c e) 
-\to ((eq nat (s k i) (S n0)) \to (or4 (drop (s k i) O (CHead c0 k t) e) 
-(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: 
-T).(eq C e (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O (CHead c0 k t) (CHead e0 
-(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda 
-(w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f) u0)))))) 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s k 
-i) O (CHead c0 k t) (CHead e2 (Flat f) u0)))))) (\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 (u0: T).(\lambda 
-(_: T).(eq C e (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O (CHead c0 
-k t) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v e1 e2))))))))))) (\lambda (b: B).(\lambda (H1: (drop (r (Bind b) n0) O c 
-e)).(\lambda (H17: (eq nat (s (Bind b) i) (S n0))).(let H18 \def (f_equal nat 
-nat (\lambda (e0: nat).(match e0 return (\lambda (_: nat).nat) with [O 
-\Rightarrow i | (S n) \Rightarrow n])) (S i) (S n0) H17) in (let H19 \def 
-(eq_ind nat i (\lambda (n: nat).(csubst0 n v c c0)) H16 n0 H18) in (eq_ind_r 
-nat n0 (\lambda (n: nat).(or4 (drop (s (Bind b) n) O (CHead c0 (Bind b) t) e) 
-(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: 
-T).(eq C e (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n) O (CHead c0 (Bind b) 
-t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: 
-T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat 
-f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: 
-T).(drop (s (Bind b) n) O (CHead c0 (Bind b) t) (CHead e2 (Flat f) u0)))))) 
-(\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 (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u0))))))) 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (s (Bind b) n) O (CHead c0 (Bind b) t) (CHead e2 (Flat f) w))))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: 
-T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (let H \def 
-(IHn c c0 v H19 e H1) in (or4_ind (drop n0 O c0 e) (ex3_4 F C T T (\lambda 
-(f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 
-(Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(w: T).(drop n0 O c0 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e 
-(CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u0: T).(drop n0 O c0 (CHead e2 (Flat f) u0)))))) (\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 
-(u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u0))))))) (\lambda (f: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 
-O c0 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v e1 e2))))))) (or4 (drop (s (Bind b) n0) O (CHead c0 (Bind b) t) e) (ex3_4 
-F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq 
-C e (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e0 
-(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda 
-(w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f) u0)))))) 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s 
-(Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2 (Flat f) u0)))))) (\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 
-(u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u0))))))) (\lambda (f: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
-(Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O 
-v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H20: (drop n0 O c0 
-e)).(or4_intro0 (drop (s (Bind b) n0) O (CHead c0 (Bind b) t) e) (ex3_4 F C T 
-T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e 
-(CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e0 
-(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda 
-(w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f) u0)))))) 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s 
-(Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2 (Flat f) u0)))))) (\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 
-(u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u0))))))) (\lambda (f: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
-(Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O 
-v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Bind b) n0 c0 e H20 
-t))) (\lambda (H20: (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 c0 (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 (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u0)))))) 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O 
-c0 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: 
-T).(\lambda (w: T).(subst0 O v u0 w))))) (or4 (drop (s (Bind b) n0) O (CHead 
-c0 (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda 
-(u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u0)))))) (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O 
-(CHead c0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e 
-(CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u0: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2 
-(Flat f) u0)))))) (\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 (u0: T).(\lambda (_: T).(eq C e (CHead e1 
-(Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead 
-e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda 
-(u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2)))))))) (\lambda (x0: F).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: 
-T).(\lambda (H21: (eq C e (CHead x1 (Flat x0) x2))).(\lambda (H22: (drop n0 O 
-c0 (CHead x1 (Flat x0) x3))).(\lambda (H23: (subst0 O v x2 x3)).(eq_ind_r C 
-(CHead x1 (Flat x0) x2) (\lambda (c: C).(or4 (drop (s (Bind b) n0) O (CHead 
-c0 (Bind b) t) c) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda 
-(u0: T).(\lambda (_: T).(eq C c (CHead e0 (Flat f) u0)))))) (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O 
-(CHead c0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C c 
-(CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u0: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2 
-(Flat f) u0)))))) (\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 (u0: T).(\lambda (_: T).(eq C c (CHead e1 
-(Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead 
-e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda 
-(u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2))))))))) (or4_intro1 (drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead 
-x1 (Flat x0) x2)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda 
-(u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x2) (CHead e0 (Flat f) 
-u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e0 (Flat f) w)))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v 
-u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u0: T).(eq C (CHead x1 (Flat x0) x2) (CHead e1 (Flat f) u0)))))) 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s 
-(Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2 (Flat f) u0)))))) (\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 
-(u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x2) (CHead e1 (Flat f) 
-u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2 
-(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda 
-(u0: T).(\lambda (w: T).(subst0 O v u0 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 (u0: 
-T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x2) (CHead e0 (Flat f) u0)))))) 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
-(Bind b) n0) O (CHead c0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w))))) x0 
-x1 x2 x3 (refl_equal C (CHead x1 (Flat x0) x2)) (drop_drop (Bind b) n0 c0 
-(CHead x1 (Flat x0) x3) H22 t) H23)) e H21)))))))) H20)) (\lambda (H20: 
-(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 c0 (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 (u0: T).(eq C e (CHead e1 (Flat f) u0)))))) (\lambda (f: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop n0 O c0 (CHead e2 
-(Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(csubst0 O v e1 e2))))) (or4 (drop (s (Bind b) n0) O (CHead c0 (Bind 
-b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: 
-T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u0)))))) (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O 
-(CHead c0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e 
-(CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u0: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2 
-(Flat f) u0)))))) (\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 (u0: T).(\lambda (_: T).(eq C e (CHead e1 
-(Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead 
-e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda 
-(u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2)))))))) (\lambda (x0: F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: 
-T).(\lambda (H21: (eq C e (CHead x1 (Flat x0) x3))).(\lambda (H22: (drop n0 O 
-c0 (CHead x2 (Flat x0) x3))).(\lambda (H23: (csubst0 O v x1 x2)).(eq_ind_r C 
-(CHead x1 (Flat x0) x3) (\lambda (c: C).(or4 (drop (s (Bind b) n0) O (CHead 
-c0 (Bind b) t) c) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda 
-(u0: T).(\lambda (_: T).(eq C c (CHead e0 (Flat f) u0)))))) (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O 
-(CHead c0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C c 
-(CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u0: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2 
-(Flat f) u0)))))) (\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 (u0: T).(\lambda (_: T).(eq C c (CHead e1 
-(Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead 
-e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda 
-(u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2))))))))) (or4_intro2 (drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead 
-x1 (Flat x0) x3)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda 
-(u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e0 (Flat f) 
-u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e0 (Flat f) w)))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v 
-u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u0: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 (Flat f) u0)))))) 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s 
-(Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2 (Flat f) u0)))))) (\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 
-(u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 (Flat f) 
-u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2 
-(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda 
-(u0: T).(\lambda (w: T).(subst0 O v u0 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 (u0: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 (Flat f) u0)))))) 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s 
-(Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2 (Flat f) u0)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) 
-x0 x1 x2 x3 (refl_equal C (CHead x1 (Flat x0) x3)) (drop_drop (Bind b) n0 c0 
-(CHead x2 (Flat x0) x3) H22 t) H23)) e H21)))))))) H20)) (\lambda (H20: 
-(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 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)))))))).(ex4_5_ind F C C T T (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat 
-f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop n0 O c0 (CHead e2 (Flat f) w))))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O 
-v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (s (Bind b) n0) O 
-(CHead c0 (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
-C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u0)))))) 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
-(Bind b) n0) O (CHead c0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) 
-(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: 
-T).(eq C e (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Bind b) n0) O (CHead c0 (Bind 
-b) t) (CHead e2 (Flat f) u0)))))) (\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 (u0: T).(\lambda (_: T).(eq 
-C e (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 
-(Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: F).(\lambda (x1: C).(\lambda 
-(x2: C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H21: (eq C e (CHead x1 
-(Flat x0) x3))).(\lambda (H22: (drop n0 O c0 (CHead x2 (Flat x0) 
-x4))).(\lambda (H23: (subst0 O v x3 x4)).(\lambda (H24: (csubst0 O v x1 
-x2)).(eq_ind_r C (CHead x1 (Flat x0) x3) (\lambda (c: C).(or4 (drop (s (Bind 
-b) n0) O (CHead c0 (Bind b) t) c) (ex3_4 F C T T (\lambda (f: F).(\lambda 
-(e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C c (CHead e0 (Flat f) u0)))))) 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
-(Bind b) n0) O (CHead c0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) 
-(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: 
-T).(eq C c (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Bind b) n0) O (CHead c0 (Bind 
-b) t) (CHead e2 (Flat f) u0)))))) (\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 (u0: T).(\lambda (_: T).(eq 
-C c (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 
-(Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (s (Bind b) n0) O (CHead 
-c0 (Bind b) t) (CHead x1 (Flat x0) x3)) (ex3_4 F C T T (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat 
-x0) x3) (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda 
-(_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead 
-e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda 
-(w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 
-(Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(u0: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2 (Flat f) 
-u0)))))) (\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 (u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) 
-x3) (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 
-(Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 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 (u0: T).(\lambda (_: T).(eq C 
-(CHead x1 (Flat x0) x3) (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) 
-O (CHead c0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda 
-(_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 
-w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 O v e1 e2)))))) x0 x1 x2 x3 x4 (refl_equal C 
-(CHead x1 (Flat x0) x3)) (drop_drop (Bind b) n0 c0 (CHead x2 (Flat x0) x4) 
-H22 t) H23 H24)) e H21)))))))))) H20)) H)) i H18)))))) (\lambda (f: 
-F).(\lambda (H1: (drop (r (Flat f) n0) O c e)).(\lambda (H17: (eq nat (s 
-(Flat f) i) (S n0))).(let H18 \def (f_equal nat nat (\lambda (e0: nat).e0) i 
-(S n0) H17) in (let H19 \def (eq_ind nat i (\lambda (n: nat).(csubst0 n v c 
-c0)) H16 (S n0) H18) in (eq_ind_r nat (S n0) (\lambda (n: nat).(or4 (drop (s 
-(Flat f) n) O (CHead c0 (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: 
-F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat 
-f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (s (Flat f) n) O (CHead c0 (Flat f) t) (CHead e0 (Flat f0) w)))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v 
-u0 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f0) u0)))))) (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Flat f) n) O 
-(CHead c0 (Flat f) t) (CHead e2 (Flat f0) u0)))))) (\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 (u0: 
-T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u0))))))) (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
-(Flat f) n) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O 
-v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (let H20 \def (H c0 v H19 e 
-H1) in (or4_ind (drop (S n0) O c0 e) (ex3_4 F C T T (\lambda (f0: F).(\lambda 
-(e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u0)))))) 
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O c0 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda 
-(u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f0: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat 
-f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: 
-T).(drop (S n0) O c0 (CHead e2 (Flat f0) u0)))))) (\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 (u0: 
-T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u0))))))) (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O c0 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v e1 e2))))))) (or4 (drop (s (Flat f) (S n0)) O (CHead c0 
-(Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: 
-T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u0)))))) (\lambda (f0: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) 
-O (CHead c0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda 
-(_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T 
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e 
-(CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u0: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead 
-e2 (Flat f0) u0)))))) (\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 (u0: T).(\lambda (_: T).(eq C e 
-(CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 
-(Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v e1 e2)))))))) (\lambda (H21: (drop (S n0) O c0 
-e)).(or4_intro0 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) e) (ex3_4 F 
-C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq 
-C e (CHead e0 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda 
-(_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) 
-(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: 
-T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f0: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat 
-f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: 
-T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) 
-u0)))))) (\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 (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat 
-f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead 
-e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v e1 e2))))))) (drop_drop (Flat f) n0 c0 e H21 t))) (\lambda (H21: (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_ind F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: 
-T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u0)))))) (\lambda (f0: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead 
-e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: 
-T).(\lambda (w: T).(subst0 O v u0 w))))) (or4 (drop (s (Flat f) (S n0)) O 
-(CHead c0 (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
-C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u0)))))) 
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
-(Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda 
-(_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 
-w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f0) u0)))))) (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Flat f) (S 
-n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) u0)))))) (\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 
-(u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u0))))))) (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
-(Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) w))))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: 
-T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: 
-F).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H22: (eq C e 
-(CHead x1 (Flat x0) x2))).(\lambda (H23: (drop (S n0) O c0 (CHead x1 (Flat 
-x0) x3))).(\lambda (H24: (subst0 O v x2 x3)).(eq_ind_r C (CHead x1 (Flat x0) 
-x2) (\lambda (c: C).(or4 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) c) 
-(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda 
-(_: T).(eq C c (CHead e0 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 
-(Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T 
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C c 
-(CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u0: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead 
-e2 (Flat f0) u0)))))) (\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 (u0: T).(\lambda (_: T).(eq C c 
-(CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 
-(Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v e1 e2))))))))) (or4_intro1 (drop (s (Flat f) (S n0)) O 
-(CHead c0 (Flat f) t) (CHead x1 (Flat x0) x2)) (ex3_4 F C T T (\lambda (f0: 
-F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat 
-x0) x2) (CHead e0 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 
-(Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T 
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C 
-(CHead x1 (Flat x0) x2) (CHead e1 (Flat f0) u0)))))) (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Flat f) (S 
-n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) u0)))))) (\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 
-(u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x2) (CHead e1 (Flat f0) 
-u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead 
-e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 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 (u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x2) (CHead e0 
-(Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e0 (Flat f0) 
-w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: 
-T).(subst0 O v u0 w))))) x0 x1 x2 x3 (refl_equal C (CHead x1 (Flat x0) x2)) 
-(drop_drop (Flat f) n0 c0 (CHead x1 (Flat x0) x3) H23 t) H24)) e H22)))))))) 
-H21)) (\lambda (H21: (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))))))).(ex3_4_ind F C C T (\lambda (f0: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat 
-f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: 
-T).(drop (S n0) O c0 (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) (or4 (drop 
-(s (Flat f) (S n0)) O (CHead c0 (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: 
-F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat 
-f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e0 (Flat f0) 
-w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: 
-T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f0) u0)))))) 
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s 
-(Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) u0)))))) 
-(\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 (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u0))))))) 
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) 
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: 
-T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2)))))))) (\lambda (x0: F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: 
-T).(\lambda (H22: (eq C e (CHead x1 (Flat x0) x3))).(\lambda (H23: (drop (S 
-n0) O c0 (CHead x2 (Flat x0) x3))).(\lambda (H24: (csubst0 O v x1 
-x2)).(eq_ind_r C (CHead x1 (Flat x0) x3) (\lambda (c: C).(or4 (drop (s (Flat 
-f) (S n0)) O (CHead c0 (Flat f) t) c) (ex3_4 F C T T (\lambda (f0: 
-F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C c (CHead e0 (Flat 
-f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e0 (Flat f0) 
-w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: 
-T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u0: T).(eq C c (CHead e1 (Flat f0) u0)))))) 
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s 
-(Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) u0)))))) 
-(\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 (u0: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f0) u0))))))) 
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) 
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: 
-T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2))))))))) (or4_intro2 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) 
-(CHead x1 (Flat x0) x3)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
-C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e0 
-(Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e0 (Flat f0) 
-w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: 
-T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 
-(Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(u0: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) 
-u0)))))) (\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 (u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) 
-x3) (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 
-(Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 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 (u0: T).(eq C (CHead x1 (Flat 
-x0) x3) (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Flat f) (S n0)) O (CHead c0 
-(Flat f) t) (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) x0 x1 x2 x3 
-(refl_equal C (CHead x1 (Flat x0) x3)) (drop_drop (Flat f) n0 c0 (CHead x2 
-(Flat x0) x3) H23 t) H24)) e H22)))))))) H21)) (\lambda (H21: (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)))))))).(ex4_5_ind F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u0))))))) 
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(w: T).(drop (S n0) O c0 (CHead e2 (Flat f0) w))))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O 
-v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (s (Flat f) (S n0)) O 
-(CHead c0 (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
-C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u0)))))) 
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
-(Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda 
-(_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 
-w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f0) u0)))))) (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Flat f) (S 
-n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) u0)))))) (\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 
-(u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u0))))))) (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
-(Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) w))))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: 
-T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: 
-F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: 
-T).(\lambda (H22: (eq C e (CHead x1 (Flat x0) x3))).(\lambda (H23: (drop (S 
-n0) O c0 (CHead x2 (Flat x0) x4))).(\lambda (H24: (subst0 O v x3 
-x4)).(\lambda (H25: (csubst0 O v x1 x2)).(eq_ind_r C (CHead x1 (Flat x0) x3) 
-(\lambda (c: C).(or4 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) c) 
-(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda 
-(_: T).(eq C c (CHead e0 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 
-(Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T 
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C c 
-(CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u0: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead 
-e2 (Flat f0) u0)))))) (\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 (u0: T).(\lambda (_: T).(eq C c 
-(CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 
-(Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (s (Flat f) (S n0)) O 
-(CHead c0 (Flat f) t) (CHead x1 (Flat x0) x3)) (ex3_4 F C T T (\lambda (f0: 
-F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat 
-x0) x3) (CHead e0 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 
-(Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T 
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C 
-(CHead x1 (Flat x0) x3) (CHead e1 (Flat f0) u0)))))) (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Flat f) (S 
-n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) u0)))))) (\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 
-(u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 (Flat f0) 
-u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead 
-e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 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 (u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) 
-x3) (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 
-(Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v e1 e2)))))) x0 x1 x2 x3 x4 (refl_equal C (CHead x1 (Flat 
-x0) x3)) (drop_drop (Flat f) n0 c0 (CHead x2 (Flat x0) x4) H23 t) H24 H25)) e 
-H22)))))))))) H21)) H20)) i H18)))))) k (drop_gen_drop k c e t n0 H1) H0))) 
-c2 H15)) u (sym_eq T u t H14))) k0 (sym_eq K k0 k H13))) c1 (sym_eq C c1 c 
-H12))) H11)) H10))) v0 (sym_eq T v0 v H8))) (S n0) H7)) H4 H5 H6 H2))))) | 
-(csubst0_both k0 i v0 u1 u2 H2 c1 c0 H3) \Rightarrow (\lambda (H4: (eq nat (s 
-k0 i) (S n0))).(\lambda (H5: (eq T v0 v)).(\lambda (H6: (eq C (CHead c1 k0 
-u1) (CHead c k t))).(\lambda (H7: (eq C (CHead c0 k0 u2) c2)).((let H8 \def 
-(f_equal nat nat (\lambda (e0: nat).e0) (s k0 i) (S n0) H4) in (eq_ind nat (s 
-k0 i) (\lambda (n: nat).((eq T v0 v) \to ((eq C (CHead c1 k0 u1) (CHead c k 
-t)) \to ((eq C (CHead c0 k0 u2) c2) \to ((subst0 i v0 u1 u2) \to ((csubst0 i 
-v0 c1 c0) \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)))))))))))))) (\lambda (H9: (eq T v0 v)).(eq_ind T v (\lambda (t0: T).((eq 
-C (CHead c1 k0 u1) (CHead c k t)) \to ((eq C (CHead c0 k0 u2) c2) \to 
-((subst0 i t0 u1 u2) \to ((csubst0 i t0 c1 c0) \to (or4 (drop (s k0 i) 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 k0 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 e (CHead e1 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s k0 
-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 
-e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 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))))))))))))) (\lambda (H10: (eq C (CHead c1 k0 u1) (CHead c k t))).(let 
-H11 \def (f_equal C T (\lambda (e0: C).(match e0 return (\lambda (_: C).T) 
-with [(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c1 k0 
-u1) (CHead c k t) H10) in ((let H12 \def (f_equal C K (\lambda (e0: C).(match 
-e0 return (\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _) 
-\Rightarrow k])) (CHead c1 k0 u1) (CHead c k t) H10) in ((let H13 \def 
-(f_equal C C (\lambda (e0: C).(match e0 return (\lambda (_: C).C) with 
-[(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k0 u1) 
-(CHead c k t) H10) in (eq_ind C c (\lambda (c: C).((eq K k0 k) \to ((eq T u1 
-t) \to ((eq C (CHead c0 k0 u2) c2) \to ((subst0 i v u1 u2) \to ((csubst0 i v 
-c c0) \to (or4 (drop (s k0 i) 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 k0 
-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 e (CHead e1 (Flat 
-f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(drop (s k0 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 e (CHead e1 (Flat f) u))))))) (\lambda (f: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
-k0 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)))))))))))))) (\lambda (H14: (eq K k0 k)).(eq_ind K 
-k (\lambda (k: K).((eq T u1 t) \to ((eq C (CHead c0 k u2) c2) \to ((subst0 i 
-v u1 u2) \to ((csubst0 i v c c0) \to (or4 (drop (s k i) 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 k 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 e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop (s k 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 e (CHead e1 (Flat f) 
-u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (s k 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))))))))))))) (\lambda 
-(H15: (eq T u1 t)).(eq_ind T t (\lambda (t: T).((eq C (CHead c0 k u2) c2) \to 
-((subst0 i v t u2) \to ((csubst0 i v c c0) \to (or4 (drop (s k i) 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 k 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 e (CHead e1 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s k 
-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 
-e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (s k 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)))))))))))) (\lambda (H16: (eq C (CHead c0 k u2) c2)).(eq_ind C (CHead c0 
-k u2) (\lambda (c2: C).((subst0 i v t u2) \to ((csubst0 i v c c0) \to (or4 
-(drop (s k i) 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 k 
-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 e (CHead e1 (Flat 
-f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(drop (s k 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 e (CHead e1 (Flat f) u))))))) (\lambda (f: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
-k 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))))))))))) (\lambda (H17: (subst0 i v t 
-u2)).(\lambda (H18: (csubst0 i v c c0)).(let H0 \def (eq_ind K k0 (\lambda 
-(k: K).(eq nat (s k i) (S n0))) H8 k H14) in (K_ind (\lambda (k: K).((drop (r 
-k n0) O c e) \to ((eq nat (s k i) (S n0)) \to (or4 (drop (s k i) O (CHead c0 
-k u2) 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 k i) O (CHead c0 
-k u2) (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 k i) O (CHead c0 k u2) (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 
-k i) O (CHead c0 k u2) (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 (H1: (drop (r 
-(Bind b) n0) O c e)).(\lambda (H19: (eq nat (s (Bind b) i) (S n0))).(let H20 
-\def (f_equal nat nat (\lambda (e0: nat).(match e0 return (\lambda (_: 
-nat).nat) with [O \Rightarrow i | (S n) \Rightarrow n])) (S i) (S n0) H19) in 
-(let H21 \def (eq_ind nat i (\lambda (n: nat).(csubst0 n v c c0)) H18 n0 H20) 
-in (let H22 \def (eq_ind nat i (\lambda (n: nat).(subst0 n v t u2)) H17 n0 
-H20) in (eq_ind_r nat n0 (\lambda (n: nat).(or4 (drop (s (Bind b) n) O (CHead 
-c0 (Bind b) u2) 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 (Bind b) n) O 
-(CHead c0 (Bind b) u2) (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 (Bind b) n) O (CHead c0 (Bind b) u2) (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 (Bind b) n) O (CHead c0 (Bind b) u2) (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 H \def (IHn c c0 v H21 e H1) in (or4_ind (drop 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 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 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 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))))))) (or4 (drop (s (Bind b) n0) O 
-(CHead c0 (Bind b) u2) 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 
-(Bind b) n0) O (CHead c0 (Bind b) u2) (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 (Bind b) n0) O (CHead c0 (Bind 
-b) u2) (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 (Bind b) n0) O (CHead c0 (Bind b) 
-u2) (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 (H23: (drop n0 O c0 e)).(or4_intro0 (drop (s (Bind 
-b) n0) O (CHead c0 (Bind b) u2) 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 
-(Bind b) n0) O (CHead c0 (Bind b) u2) (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 (Bind b) n0) O (CHead c0 (Bind 
-b) u2) (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 (Bind b) n0) O (CHead c0 (Bind b) 
-u2) (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 c0 e H23 u2))) (\lambda (H23: (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 c0 (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 c0 (CHead e0 
-(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 O v u w))))) (or4 (drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) 
-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 (Bind b) n0) O (CHead c0 (Bind b) 
-u2) (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 (Bind b) n0) O (CHead c0 (Bind b) u2) (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 (Bind b) n0) O (CHead c0 (Bind b) u2) (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: 
-F).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H24: (eq C e 
-(CHead x1 (Flat x0) x2))).(\lambda (H25: (drop n0 O c0 (CHead x1 (Flat x0) 
-x3))).(\lambda (H26: (subst0 O v x2 x3)).(eq_ind_r C (CHead x1 (Flat x0) x2) 
-(\lambda (c: C).(or4 (drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) 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 (Bind b) n0) O (CHead c0 (Bind b) u2) (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 
-(Bind b) n0) O (CHead c0 (Bind b) u2) (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 
-(Bind b) n0) O (CHead c0 (Bind b) u2) (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 (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead x1 (Flat x0) x2)) 
-(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C (CHead x1 (Flat x0) x2) (CHead e0 (Flat f) u)))))) (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O 
-(CHead c0 (Bind b) u2) (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 
-x1 (Flat x0) x2) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Bind b) n0) O (CHead c0 (Bind 
-b) u2) (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 x1 (Flat x0) x2) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) 
-O (CHead c0 (Bind b) u2) (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 x1 (Flat x0) 
-x2) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e0 
-(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 O v u w))))) x0 x1 x2 x3 (refl_equal C (CHead x1 (Flat x0) x2)) 
-(drop_drop (Bind b) n0 c0 (CHead x1 (Flat x0) x3) H25 u2) H26)) e H24)))))))) 
-H23)) (\lambda (H23: (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 c0 (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 
-c0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 O v e1 e2))))) (or4 (drop (s (Bind b) n0) O 
-(CHead c0 (Bind b) u2) 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 
-(Bind b) n0) O (CHead c0 (Bind b) u2) (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 (Bind b) n0) O (CHead c0 (Bind 
-b) u2) (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 (Bind b) n0) O (CHead c0 (Bind b) 
-u2) (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: F).(\lambda (x1: C).(\lambda (x2: C).(\lambda 
-(x3: T).(\lambda (H24: (eq C e (CHead x1 (Flat x0) x3))).(\lambda (H25: (drop 
-n0 O c0 (CHead x2 (Flat x0) x3))).(\lambda (H26: (csubst0 O v x1 
-x2)).(eq_ind_r C (CHead x1 (Flat x0) x3) (\lambda (c: C).(or4 (drop (s (Bind 
-b) n0) O (CHead c0 (Bind b) u2) 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 
-(Bind b) n0) O (CHead c0 (Bind b) u2) (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 (Bind b) n0) O (CHead c0 (Bind 
-b) u2) (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 (Bind b) n0) O (CHead c0 (Bind b) 
-u2) (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 (Bind b) n0) O (CHead c0 (Bind b) u2) 
-(CHead x1 (Flat x0) x3)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e0 
-(Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (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 x1 (Flat x0) x3) (CHead e1 
-(Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(u: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (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 x1 (Flat x0) 
-x3) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) 
-u2) (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 x1 (Flat x0) x3) (CHead e1 
-(Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(u: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) 
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 O v e1 e2))))) x0 x1 x2 x3 (refl_equal C (CHead x1 (Flat x0) x3)) 
-(drop_drop (Bind b) n0 c0 (CHead x2 (Flat x0) x3) H25 u2) H26)) e H24)))))))) 
-H23)) (\lambda (H23: (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 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)))))))).(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 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)))))) (or4 (drop (s 
-(Bind b) n0) O (CHead c0 (Bind b) u2) 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 (Bind b) n0) O (CHead c0 (Bind b) u2) (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 (Bind b) n0) O 
-(CHead c0 (Bind b) u2) (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 
-(Bind b) n0) O (CHead c0 (Bind b) u2) (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: 
-F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: 
-T).(\lambda (H24: (eq C e (CHead x1 (Flat x0) x3))).(\lambda (H25: (drop n0 O 
-c0 (CHead x2 (Flat x0) x4))).(\lambda (H26: (subst0 O v x3 x4)).(\lambda 
-(H27: (csubst0 O v x1 x2)).(eq_ind_r C (CHead x1 (Flat x0) x3) (\lambda (c: 
-C).(or4 (drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) 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 (Bind b) n0) O (CHead c0 (Bind b) u2) (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 
-(Bind b) n0) O (CHead c0 (Bind b) u2) (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 
-(Bind b) n0) O (CHead c0 (Bind b) u2) (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 (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead x1 (Flat x0) x3)) 
-(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C (CHead x1 (Flat x0) x3) (CHead e0 (Flat f) u)))))) (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O 
-(CHead c0 (Bind b) u2) (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 
-x1 (Flat x0) x3) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Bind b) n0) O (CHead c0 (Bind 
-b) u2) (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 x1 (Flat x0) x3) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) 
-O (CHead c0 (Bind b) u2) (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 x1 (Flat x0) x3) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) 
-O (CHead c0 (Bind b) u2) (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)))))) x0 x1 x2 x3 x4 (refl_equal C (CHead x1 (Flat 
-x0) x3)) (drop_drop (Bind b) n0 c0 (CHead x2 (Flat x0) x4) H25 u2) H26 H27)) 
-e H24)))))))))) H23)) H)) i H20))))))) (\lambda (f: F).(\lambda (H1: (drop (r 
-(Flat f) n0) O c e)).(\lambda (H19: (eq nat (s (Flat f) i) (S n0))).(let H20 
-\def (f_equal nat nat (\lambda (e0: nat).e0) i (S n0) H19) in (let H21 \def 
-(eq_ind nat i (\lambda (n: nat).(csubst0 n v c c0)) H18 (S n0) H20) in (let 
-H22 \def (eq_ind nat i (\lambda (n: nat).(subst0 n v t u2)) H17 (S n0) H20) 
-in (eq_ind_r nat (S n0) (\lambda (n: nat).(or4 (drop (s (Flat f) n) O (CHead 
-c0 (Flat f) u2) 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 (Flat f) n) O 
-(CHead c0 (Flat f) u2) (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 (Flat f) n) O (CHead c0 (Flat f) u2) (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 (Flat f) n) O (CHead c0 (Flat f) u2) (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))))))))) (let H23 \def (H c0 v H21 e H1) in (or4_ind (drop (S n0) O 
-c0 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 c0 (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 c0 (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 c0 (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 (Flat f) (S n0)) O (CHead c0 (Flat f) u2) 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 (Flat f) (S n0)) O (CHead c0 
-(Flat f) u2) (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 (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (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 (Flat f) (S n0)) O (CHead c0 
-(Flat f) u2) (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 (H24: (drop (S n0) O c0 
-e)).(or4_intro0 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) 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 (Flat f) (S n0)) O (CHead c0 (Flat f) u2) 
-(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 (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (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 (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (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 c0 e H24 u2))) (\lambda (H24: (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_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 c0 (CHead 
-e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda 
-(w: T).(subst0 O v u w))))) (or4 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat 
-f) u2) 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 (Flat f) (S n0)) 
-O (CHead c0 (Flat f) u2) (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 (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (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 (Flat f) (S n0)) O (CHead c0 
-(Flat f) u2) (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 (x0: F).(\lambda (x1: C).(\lambda 
-(x2: T).(\lambda (x3: T).(\lambda (H25: (eq C e (CHead x1 (Flat x0) 
-x2))).(\lambda (H26: (drop (S n0) O c0 (CHead x1 (Flat x0) x3))).(\lambda 
-(H27: (subst0 O v x2 x3)).(eq_ind_r C (CHead x1 (Flat x0) x2) (\lambda (c: 
-C).(or4 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) c) (ex3_4 F C T T 
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c 
-(CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (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 c (CHead e1 (Flat f0) u)))))) 
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s 
-(Flat f) (S n0)) O (CHead c0 (Flat f) u2) (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 c (CHead e1 (Flat f0) u))))))) 
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (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 (Flat f) (S n0)) O (CHead c0 (Flat f) u2) 
-(CHead x1 (Flat x0) x2)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x2) (CHead e0 
-(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (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 x1 (Flat x0) x2) (CHead e1 
-(Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(u: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (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 x1 (Flat x0) 
-x2) (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 
-(Flat f) u2) (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 x1 (Flat x0) 
-x2) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda 
-(_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) 
-(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 O v u w))))) x0 x1 x2 x3 (refl_equal C (CHead x1 
-(Flat x0) x2)) (drop_drop (Flat f) n0 c0 (CHead x1 (Flat x0) x3) H26 u2) 
-H27)) e H25)))))))) H24)) (\lambda (H24: (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))))))).(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 c0 (CHead e2 (Flat f0) u)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) 
-(or4 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) 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 (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (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 
-(Flat f) (S n0)) O (CHead c0 (Flat f) u2) (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 (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (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 (x0: F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: 
-T).(\lambda (H25: (eq C e (CHead x1 (Flat x0) x3))).(\lambda (H26: (drop (S 
-n0) O c0 (CHead x2 (Flat x0) x3))).(\lambda (H27: (csubst0 O v x1 
-x2)).(eq_ind_r C (CHead x1 (Flat x0) x3) (\lambda (c: C).(or4 (drop (s (Flat 
-f) (S n0)) O (CHead c0 (Flat f) u2) c) (ex3_4 F C T T (\lambda (f0: 
-F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Flat 
-f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (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 c (CHead e1 (Flat f0) u)))))) 
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s 
-(Flat f) (S n0)) O (CHead c0 (Flat f) u2) (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 c (CHead e1 (Flat f0) u))))))) 
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (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 (Flat f) (S n0)) O (CHead c0 (Flat f) u2) 
-(CHead x1 (Flat x0) x3)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e0 
-(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (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 x1 (Flat x0) x3) (CHead e1 
-(Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(u: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (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 x1 (Flat x0) 
-x3) (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 
-(Flat f) u2) (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 x1 (Flat x0) 
-x3) (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (u: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) 
-(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 O v e1 e2))))) x0 x1 x2 x3 (refl_equal C (CHead 
-x1 (Flat x0) x3)) (drop_drop (Flat f) n0 c0 (CHead x2 (Flat x0) x3) H26 u2) 
-H27)) e H25)))))))) H24)) (\lambda (H24: (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)))))))).(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 c0 (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 (Flat f) (S n0)) O 
-(CHead c0 (Flat f) u2) 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 
-(Flat f) (S n0)) O (CHead c0 (Flat f) u2) (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 (Flat f) (S n0)) 
-O (CHead c0 (Flat f) u2) (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 
-(Flat f) (S n0)) O (CHead c0 (Flat f) u2) (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 (x0: 
-F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: 
-T).(\lambda (H25: (eq C e (CHead x1 (Flat x0) x3))).(\lambda (H26: (drop (S 
-n0) O c0 (CHead x2 (Flat x0) x4))).(\lambda (H27: (subst0 O v x3 
-x4)).(\lambda (H28: (csubst0 O v x1 x2)).(eq_ind_r C (CHead x1 (Flat x0) x3) 
-(\lambda (c: C).(or4 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) c) 
-(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C c (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 
-(Flat f) u2) (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 c 
-(CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (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 c 
-(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 
-(Flat f) u2) (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 (Flat f) (S n0)) O 
-(CHead c0 (Flat f) u2) (CHead x1 (Flat x0) x3)) (ex3_4 F C T T (\lambda (f0: 
-F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) 
-x3) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda 
-(_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) 
-(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 x1 (Flat x0) 
-x3) (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (u: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) 
-(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 x1 (Flat x0) x3) (CHead e1 (Flat f0) u))))))) (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
-(Flat f) (S n0)) O (CHead c0 (Flat f) u2) (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 x1 (Flat x0) x3) (CHead e1 (Flat f0) u))))))) 
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (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)))))) 
-x0 x1 x2 x3 x4 (refl_equal C (CHead x1 (Flat x0) x3)) (drop_drop (Flat f) n0 
-c0 (CHead x2 (Flat x0) x4) H26 u2) H27 H28)) e H25)))))))))) H24)) H23)) i 
-H20))))))) k (drop_gen_drop k c e t n0 H1) H0)))) c2 H16)) u1 (sym_eq T u1 t 
-H15))) k0 (sym_eq K k0 k H14))) c1 (sym_eq C c1 c H13))) H12)) H11))) v0 
-(sym_eq T v0 v H9))) (S n0) H8)) H5 H6 H7 H2 H3)))))]) in (H2 (refl_equal nat 
-(S n0)) (refl_equal T v) (refl_equal C (CHead c k t)) (refl_equal C 
-c2)))))))))))) 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)) 
-(or4 (drop O O 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 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 c2 (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 c2 (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)))))))) (\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 O O 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 O O c 
-(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 O 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 O O c (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 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 O O c (CHead e1 
-(Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda 
-(u1: T).(\lambda (u2: T).(subst0 O t u1 u2)))))) (\lambda (_: F).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O 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 O O (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 O O 
-(CHead c k0 u1) (CHead e0 (Flat f) u3)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O 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 O O (CHead c k0 u1) (CHead e1 (Flat f) u)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
-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 O O (CHead c k0 u1) (CHead e1 (Flat f) 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)))))))))))))))) 
-(\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 
-return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow 
-True])) I O H3) in (False_ind (or4 (drop O O (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 O O 
-(CHead c (Bind b) u1) (CHead e0 (Flat f) u3)))))) (\lambda (_: F).(\lambda 
-(_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O 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 O O (CHead c (Bind b) u1) 
-(CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 O 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 O O (CHead c 
-(Bind b) u1) (CHead e1 (Flat f) 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)))))))) 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 (n: nat).(subst0 n v0 u1 u2)) H2 O H3) in (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))))))))))) 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 O O 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 O O c3 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O 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 O O c3 (CHead e1 (Flat f) u)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 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 O O c3 (CHead e1 (Flat f) 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)))))))))) \to (\forall (u: T).((eq nat (s k0 i) O) 
-\to (or4 (drop O O (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 O O (CHead c3 k0 u) (CHead e0 (Flat f) u1)))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O 
-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 O O 
-(CHead c3 k0 u) (CHead e1 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 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 O 
-O (CHead c3 k0 u) (CHead e1 (Flat f) 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))))))))))))))))) (\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 O O 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 O O c3 (CHead e0 (Flat f) u1)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O 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 O O c3 (CHead e1 (Flat f) u)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
-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 O O c3 (CHead e1 (Flat f) 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))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat 
-(S i) O)).(let H5 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee return 
-(\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) 
-I O H4) in (False_ind (or4 (drop O O (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 O O (CHead c3 
-(Bind b) u) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O 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 O O (CHead c3 (Bind b) u) 
-(CHead e1 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 O 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 O O (CHead c3 
-(Bind b) u) (CHead e1 (Flat f) 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)))))))) 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 O O 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 O O c3 (CHead e0 (Flat f) u1)))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O 
-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 O O c3 (CHead e1 
-(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(csubst0 O 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 O O c3 (CHead e1 (Flat f) 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))))))))))).(\lambda 
-(u: T).(\lambda (H4: (eq nat i O)).(let H5 \def (eq_ind nat i (\lambda (n: 
-nat).((eq nat n O) \to (or4 (drop O O 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 O O c3 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O 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 O O c3 (CHead e1 (Flat f) u)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 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 O O c3 (CHead e1 (Flat f) 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)))))))))) H3 O H4) in (let H6 \def (eq_ind nat i 
-(\lambda (n: nat).(csubst0 n v0 c3 c4)) H2 O H4) in (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))))))))))))) 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 O O 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 O O 
-c3 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 O 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 O O c3 (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 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 O O c3 (CHead e1 
-(Flat f) 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)))))))))) \to ((eq nat (s k0 i) O) \to (or4 (drop O O (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 O O 
-(CHead c3 k0 u1) (CHead e0 (Flat f) u3)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O 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 O O (CHead c3 k0 u1) (CHead e1 (Flat 
-f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 O 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 O O (CHead c3 k0 u1) (CHead e1 
-(Flat f) 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))))))))))))))))))) (\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 O O 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 O O c3 (CHead e0 (Flat f) u1)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O 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 O O c3 (CHead e1 (Flat f) u)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
-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 O O c3 (CHead e1 (Flat f) 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))))))))))).(\lambda (H5: (eq nat (S i) O)).(let H6 
-\def (eq_ind nat (S i) (\lambda (ee: nat).(match ee return (\lambda (_: 
-nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H5) in 
-(False_ind (or4 (drop O O (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 O O (CHead c3 
-(Bind b) u1) (CHead e0 (Flat f) u3)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O 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 O O (CHead c3 (Bind b) u1) (CHead e1 
-(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(csubst0 O 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 O O (CHead c3 (Bind 
-b) u1) (CHead e1 (Flat f) 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)))))))) 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 O O 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 O O c3 (CHead e0 (Flat f) u1)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O 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 O O c3 (CHead e1 (Flat f) u)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
-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 O O c3 (CHead e1 (Flat f) 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))))))))))).(\lambda (H5: (eq nat i O)).(let H6 
-\def (eq_ind nat i (\lambda (n: nat).((eq nat n O) \to (or4 (drop O O 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 O O c3 (CHead e0 (Flat f) u1)))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O 
-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 O O c3 (CHead e1 
-(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(csubst0 O 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 O O c3 (CHead e1 (Flat f) 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)))))))))) H4 O H5) in 
-(let H7 \def (eq_ind nat i (\lambda (n: nat).(csubst0 n v0 c3 c4)) H3 O H5) 
-in (let H8 \def (eq_ind nat i (\lambda (n: nat).(subst0 n v0 u1 u2)) H2 O H5) 
-in (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)))))))))))))))) 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 (c: C).(drop (S n0) O c e)) H1 (CHead c k x0) H4) in 
-((match k return (\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))))))))))) with 
-[(Bind b) \Rightarrow (\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 return (\lambda (_: nat).nat) with [O \Rightarrow n0 | (S 
-n) \Rightarrow n])) (S n0) (S x1) H7) in (let H10 \def (eq_ind_r nat x1 
-(\lambda (n: nat).(subst0 n 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)))))) | (Flat f) \Rightarrow 
-(\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 (n: nat).(subst0 n 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))))))]) 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 (c2: 
-C).(\lambda (j: nat).(csubst0 j v c c2))))).(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 (c: C).(drop (S n0) O c e)) H1 (CHead x0 k t) H4) 
-in ((match k return (\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))))))))))) with [(Bind b) \Rightarrow (\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 return (\lambda (_: nat).nat) 
-with [O \Rightarrow n0 | (S n) \Rightarrow n])) (S n0) (S x1) H7) in (let H10 
-\def (eq_ind_r nat x1 (\lambda (n: nat).(csubst0 n 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)))))) | 
-(Flat f) \Rightarrow (\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 (n: 
-nat).(csubst0 n 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 (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_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 (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))))))).(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 (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)))))))).(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))))))]) 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 (c2: 
-C).(\lambda (j: nat).(csubst0 j v c c2)))))).(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 (c: C).(drop (S n0) O c e)) H1 
-(CHead x1 k x0) H4) in ((match k return (\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))))))))))) with [(Bind b) \Rightarrow (\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 return (\lambda (_: nat).nat) 
-with [O \Rightarrow n0 | (S n) \Rightarrow n])) (S n0) (S x2) H8) in (let H11 
-\def (eq_ind_r nat x2 (\lambda (n: nat).(csubst0 n v c x1)) H6 n0 H10) in 
-(let H12 \def (eq_ind_r nat x2 (\lambda (n: nat).(subst0 n 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))))))) | 
-(Flat f) \Rightarrow (\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 (n: 
-nat).(csubst0 n v c x1)) H6 (S n0) H10) in (let H12 \def (eq_ind_r nat x2 
-(\lambda (n: nat).(subst0 n 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 (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_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 (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))))))).(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 (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)))))))).(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)))))))]) 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_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)) ((match k return (\lambda (k0: K).((clear 
-(CHead c k0 t) c0) \to ((eq nat O (s k0 x1)) \to (clear (CHead c k0 x0) 
-c0)))) with [(Bind b) \Rightarrow (\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 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)))) | (Flat f) \Rightarrow (\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))))]) 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 (c2: C).(\lambda (j: nat).(csubst0 j v c 
-c2))))).(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)) ((match k return (\lambda (k0: 
-K).((clear (CHead c k0 t) c0) \to ((eq nat O (s k0 x1)) \to (clear (CHead x0 
-k0 t) c0)))) with [(Bind b) \Rightarrow (\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 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)))) | (Flat f) \Rightarrow (\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))))]) 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 (c2: C).(\lambda (j: nat).(csubst0 j v c c2)))))).(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)) ((match k 
-return (\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat O (s k0 x2)) 
-\to (clear (CHead x1 k0 x0) c0)))) with [(Bind b) \Rightarrow (\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 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)))) | (Flat f) 
-\Rightarrow (\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)))))]) 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 (c: C).(clear c c0)) H1 (CHead c k 
-x0) H4) in ((match k return (\lambda (k0: K).((eq nat O (s k0 x1)) \to 
-((clear (CHead c k0 x0) c0) \to (clear (CHead c k0 t) c0)))) with [(Bind b) 
-\Rightarrow (\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 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)))) | (Flat f) \Rightarrow (\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))))]) 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 (c2: 
-C).(\lambda (j: nat).(csubst0 j v c c2))))).(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 
-(c: C).(clear c c0)) H1 (CHead x0 k t) H4) in ((match k return (\lambda (k0: 
-K).((eq nat O (s k0 x1)) \to ((clear (CHead x0 k0 t) c0) \to (clear (CHead c 
-k0 t) c0)))) with [(Bind b) \Rightarrow (\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 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)))) | (Flat f) \Rightarrow (\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))))]) 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 (c2: C).(\lambda (j: nat).(csubst0 j v c c2)))))).(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 (c: C).(clear c c0)) H1 (CHead x1 k 
-x0) H4) in ((match k return (\lambda (k0: K).((eq nat O (s k0 x2)) \to 
-((clear (CHead x1 k0 x0) c0) \to (clear (CHead c k0 t) c0)))) with [(Bind b) 
-\Rightarrow (\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 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)))) | (Flat f) \Rightarrow (\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)))))]) 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))))))))) ((match k return (\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))))))))))) with [(Bind b) \Rightarrow (\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 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)))))) | (Flat f) 
-\Rightarrow (\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))))))]) 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 (c2: C).(\lambda (j: 
-nat).(csubst0 j v c c2))))).(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))))))))) ((match k return (\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))))))))))) with [(Bind b) \Rightarrow (\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 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)))))) | (Flat f) \Rightarrow (\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))))))]) 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 (c2: 
-C).(\lambda (j: nat).(csubst0 j v c c2)))))).(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))))))))) ((match k 
-return (\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))))))))))) with 
-[(Bind b) \Rightarrow (\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 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))))))) | 
-(Flat f) \Rightarrow (\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)))))))]) H1 
-H3) c2 H4)))))))) H2)) (csubst0_gen_head k c c2 t v (S i) H0)))))))))))) c1).
-
-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 (n: 
-nat).(drop n O c1 x)) H3 i H5) in (let H7 \def (eq_ind_r nat n (\lambda (n: 
-nat).(le i n)) 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 ((match x0 return (\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)))))))))))) with [(Bind 
-b) \Rightarrow (\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))))) | (Flat f) \Rightarrow (\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))))))]) 
-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 ((match x0 return (\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)))))))))))) with [(Bind 
-b) \Rightarrow (\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))))) | (Flat 
-f) \Rightarrow (\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 (n: 
-nat).(csubst0 n 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 (e: C).(\lambda (_: 
-T).(\lambda (u2: T).(clear x2 (CHead e (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))))))]) 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 ((match x0 return (\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))))))))))))) with [(Bind b) \Rightarrow (\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)))))) | (Flat f) \Rightarrow (\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 (n: nat).(csubst0 n 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 (e: 
-C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e (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)))))))]) 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 (n: 
-nat).(drop n O c2 x)) H3 i H5) in (let H7 \def (eq_ind_r nat n (\lambda (n: 
-nat).(le i n)) 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)))))))))).
-
-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)).
-
-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)))))))))).
-
-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)).(let H0 \def (match H return (\lambda (c: C).(\lambda (_: (csubst1 ? ? ? 
-c)).((eq C c 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))))))) with [csubst1_refl 
-\Rightarrow (\lambda (H0: (eq C (CHead c1 k u1) x)).(eq_ind C (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)) x H0)) | (csubst1_sing c2 H0) \Rightarrow 
-(\lambda (H1: (eq C c2 x)).(eq_ind C x (\lambda (c: C).((csubst0 (s k i) v 
-(CHead c1 k u1) c) \to (ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C x 
-(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 (H2: 
-(csubst0 (s k i) v (CHead c1 k u1) x)).(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 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 (s k i) (s 
-k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C x (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 x (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 x (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 (H3: (ex3_2 T nat 
-(\lambda (_: T).(\lambda (j: nat).(eq nat (s k 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_ind T nat (\lambda (_: T).(\lambda (j: 
-nat).(eq nat (s k 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 T C (\lambda (u2: T).(\lambda (c3: C).(eq C x (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 (H: (eq nat (s k i) (s k x1))).(\lambda (H4: (eq C x (CHead c1 
-k x0))).(\lambda (H5: (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 H6 \def (eq_ind_r nat x1 
-(\lambda (n: nat).(subst0 n v u1 x0)) H5 i (s_inj k i x1 H)) 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 H6) (csubst1_refl i v c1))) x H4)))))) H3)) 
-(\lambda (H3: (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k 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))))).(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 x (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 
-x (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 (H: (eq nat (s k i) (s k x1))).(\lambda (H4: 
-(eq C x (CHead x0 k u1))).(\lambda (H5: (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 H6 \def 
-(eq_ind_r nat x1 (\lambda (n: nat).(csubst0 n v c1 x0)) H5 i (s_inj k i x1 
-H)) 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 H6))) x 
-H4)))))) H3)) (\lambda (H3: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: 
-C).(\lambda (j: nat).(eq nat (s k 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)))))).(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 x (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 x (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 (H: (eq nat (s k i) (s k x2))).(\lambda (H4: 
-(eq C x (CHead x1 k x0))).(\lambda (H5: (subst0 x2 v u1 x0)).(\lambda (H6: 
-(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 H7 \def (eq_ind_r nat x2 (\lambda (n: 
-nat).(csubst0 n v c1 x1)) H6 i (s_inj k i x2 H)) in (let H8 \def (eq_ind_r 
-nat x2 (\lambda (n: nat).(subst0 n v u1 x0)) H5 i (s_inj k i x2 H)) 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 H8) (csubst1_sing i v c1 x1 H7)))) 
-x H4)))))))) H3)) (csubst0_gen_head k c1 x u1 v (s k i) H2))) c2 (sym_eq C c2 
-x H1) H0))]) in (H0 (refl_equal C x))))))))).
-
-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 (e2: 
-C).(\lambda (u: T).(getl n c3 (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 (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 (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (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_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).(match k 
-return (\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))))))))) with [(Bind b) \Rightarrow 
-(\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 return (\lambda (_: C).C) with [(CSort _) \Rightarrow e | 
-(CHead c _ _) \Rightarrow c])) (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 return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | 
-(CHead _ k _) \Rightarrow (match k return (\lambda (_: K).B) with [(Bind b) 
-\Rightarrow b | (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 return (\lambda (_: C).T) with [(CSort _) 
-\Rightarrow u | (CHead _ _ t) \Rightarrow t])) (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)))))) | (Flat f) \Rightarrow (\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)))))))])))) 
-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).(match k return (\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))))))))) with [(Bind b) \Rightarrow (\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))))))) | (Flat f) \Rightarrow (\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)))))))])))) c)))) d).
-
-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)))).
-
-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)).(let H0 \def 
-(match H return (\lambda (c: C).(\lambda (t: T).(\lambda (_: (fsubst0 ? ? ? ? 
-c t)).((eq C c c2) \to ((eq T t 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)))))))) with [(fsubst0_snd t0 H0) \Rightarrow (\lambda 
-(H1: (eq C c1 c2)).(\lambda (H2: (eq T t0 t2)).(eq_ind C c2 (\lambda (c: 
-C).((eq T t0 t2) \to ((subst0 i v t1 t0) \to (or3 (land (eq C c c2) (subst0 i 
-v t1 t2)) (land (eq T t1 t2) (csubst0 i v c c2)) (land (subst0 i v t1 t2) 
-(csubst0 i v c c2)))))) (\lambda (H3: (eq T t0 t2)).(eq_ind T t2 (\lambda (t: 
-T).((subst0 i v t1 t) \to (or3 (land (eq C c2 c2) (subst0 i v t1 t2)) (land 
-(eq T t1 t2) (csubst0 i v c2 c2)) (land (subst0 i v t1 t2) (csubst0 i v c2 
-c2))))) (\lambda (H4: (subst0 i v t1 t2)).(or3_intro0 (land (eq C c2 c2) 
-(subst0 i v t1 t2)) (land (eq T t1 t2) (csubst0 i v c2 c2)) (land (subst0 i v 
-t1 t2) (csubst0 i v c2 c2)) (conj (eq C c2 c2) (subst0 i v t1 t2) (refl_equal 
-C c2) H4))) t0 (sym_eq T t0 t2 H3))) c1 (sym_eq C c1 c2 H1) H2 H0))) | 
-(fsubst0_fst c0 H0) \Rightarrow (\lambda (H1: (eq C c0 c2)).(\lambda (H2: (eq 
-T t1 t2)).(eq_ind C c2 (\lambda (c: C).((eq T t1 t2) \to ((csubst0 i v c1 c) 
-\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)))))) (\lambda (H3: (eq 
-T t1 t2)).(eq_ind T t2 (\lambda (t: T).((csubst0 i v c1 c2) \to (or3 (land 
-(eq C c1 c2) (subst0 i v t t2)) (land (eq T t t2) (csubst0 i v c1 c2)) (land 
-(subst0 i v t t2) (csubst0 i v c1 c2))))) (\lambda (H4: (csubst0 i v c1 
-c2)).(or3_intro1 (land (eq C c1 c2) (subst0 i v t2 t2)) (land (eq T t2 t2) 
-(csubst0 i v c1 c2)) (land (subst0 i v t2 t2) (csubst0 i v c1 c2)) (conj (eq 
-T t2 t2) (csubst0 i v c1 c2) (refl_equal T t2) H4))) t1 (sym_eq T t1 t2 H3))) 
-c0 (sym_eq C c0 c2 H1) H2 H0))) | (fsubst0_both t0 H0 c0 H1) \Rightarrow 
-(\lambda (H2: (eq C c0 c2)).(\lambda (H3: (eq T t0 t2)).(eq_ind C c2 (\lambda 
-(c: C).((eq T t0 t2) \to ((subst0 i v t1 t0) \to ((csubst0 i v c1 c) \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))))))) (\lambda (H4: (eq T 
-t0 t2)).(eq_ind T t2 (\lambda (t: T).((subst0 i v t1 t) \to ((csubst0 i v c1 
-c2) \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)))))) 
-(\lambda (H5: (subst0 i v t1 t2)).(\lambda (H6: (csubst0 i v c1 
-c2)).(or3_intro2 (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)) (conj 
-(subst0 i v t1 t2) (csubst0 i v c1 c2) H5 H6)))) t0 (sym_eq T t0 t2 H4))) c0 
-(sym_eq C c0 c2 H2) H3 H0 H1)))]) in (H0 (refl_equal C c2) (refl_equal T 
-t2))))))))).
-
-inductive tau0 (g:G): C \to (T \to (T \to Prop)) \def
-| tau0_sort: \forall (c: C).(\forall (n: nat).(tau0 g c (TSort n) (TSort 
-(next g n))))
-| tau0_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).((tau0 g d v w) 
-\to (tau0 g c (TLRef i) (lift (S i) O w))))))))
-| tau0_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).((tau0 g d v w) 
-\to (tau0 g c (TLRef i) (lift (S i) O v))))))))
-| tau0_bind: \forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t1: 
-T).(\forall (t2: T).((tau0 g (CHead c (Bind b) v) t1 t2) \to (tau0 g c (THead 
-(Bind b) v t1) (THead (Bind b) v t2)))))))
-| tau0_appl: \forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall (t2: 
-T).((tau0 g c t1 t2) \to (tau0 g c (THead (Flat Appl) v t1) (THead (Flat 
-Appl) v t2))))))
-| tau0_cast: \forall (c: C).(\forall (v1: T).(\forall (v2: T).((tau0 g c v1 
-v2) \to (\forall (t1: T).(\forall (t2: T).((tau0 g c t1 t2) \to (tau0 g c 
-(THead (Flat Cast) v1 t1) (THead (Flat Cast) v2 t2)))))))).
-
-theorem tau0_lift:
- \forall (g: G).(\forall (e: C).(\forall (t1: T).(\forall (t2: T).((tau0 g e 
-t1 t2) \to (\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c 
-e) \to (tau0 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: (tau0 g e t1 t2)).(tau0_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 (tau0 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).(tau0 g c0 t (lift h d 
-(TSort (next g n))))) (eq_ind_r T (TSort (next g n)) (\lambda (t: T).(tau0 g 
-c0 (TSort n) t)) (tau0_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: (tau0 g d v 
-w)).(\lambda (H2: ((\forall (c: C).(\forall (h: nat).(\forall (d0: 
-nat).((drop h d0 c d) \to (tau0 g c (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 (tau0 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)))) (tau0 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)) (tau0 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).(tau0 g c0 t (lift h d0 (lift (S i) O w)))) (eq_ind 
-nat (plus (S i) (minus d0 (S i))) (\lambda (n: nat).(tau0 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).(tau0 g c0 (TLRef i) t)) (eq_ind nat d0 (\lambda (_: 
-nat).(tau0 g c0 (TLRef i) (lift (S i) O (lift h (minus d0 (S i)) w)))) 
-(tau0_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).(tau0 g c0 t (lift h d0 (lift (S i) O w)))) (eq_ind nat 
-(S i) (\lambda (_: nat).(tau0 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).(tau0 g c0 
-(TLRef (plus i h)) t)) (eq_ind_r nat (plus (S i) h) (\lambda (n: nat).(tau0 g 
-c0 (TLRef (plus i h)) (lift n O w))) (tau0_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_comm 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_comm 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: (tau0 g d v w)).(\lambda (H2: ((\forall (c: C).(\forall (h: 
-nat).(\forall (d0: nat).((drop h d0 c d) \to (tau0 g c (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 (tau0 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)))) (tau0 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)) (tau0 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).(tau0 g c0 t (lift h d0 (lift (S i) 
-O v)))) (eq_ind nat (plus (S i) (minus d0 (S i))) (\lambda (n: nat).(tau0 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).(tau0 g c0 (TLRef i) t)) (eq_ind nat d0 
-(\lambda (_: nat).(tau0 g c0 (TLRef i) (lift (S i) O (lift h (minus d0 (S i)) 
-v)))) (tau0_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).(tau0 g c0 t (lift h d0 (lift (S i) O v)))) (eq_ind nat 
-(S i) (\lambda (_: nat).(tau0 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).(tau0 g c0 
-(TLRef (plus i h)) t)) (eq_ind_r nat (plus (S i) h) (\lambda (n: nat).(tau0 g 
-c0 (TLRef (plus i h)) (lift n O v))) (tau0_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_comm 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_comm 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 (_: (tau0 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 (tau0 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).(tau0 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).(tau0 g c0 (THead (Bind b) (lift h d v) (lift h (s (Bind b) 
-d) t3)) t)) (tau0_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 (_: (tau0 g 
-c t3 t4)).(\lambda (H1: ((\forall (c0: C).(\forall (h: nat).(\forall (d: 
-nat).((drop h d c0 c) \to (tau0 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).(tau0 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).(tau0 g c0 (THead (Flat Appl) (lift h d v) (lift h (s 
-(Flat Appl) d) t3)) t)) (tau0_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 (_: (tau0 g c v1 
-v2)).(\lambda (H1: ((\forall (c0: C).(\forall (h: nat).(\forall (d: 
-nat).((drop h d c0 c) \to (tau0 g c0 (lift h d v1) (lift h d 
-v2)))))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (tau0 g c t3 
-t4)).(\lambda (H3: ((\forall (c0: C).(\forall (h: nat).(\forall (d: 
-nat).((drop h d c0 c) \to (tau0 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).(tau0 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).(tau0 g c0 (THead (Flat Cast) (lift h d v1) (lift h 
-(s (Flat Cast) d) t3)) t)) (tau0_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 tau0_correct:
- \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((tau0 g c 
-t1 t) \to (ex T (\lambda (t2: T).(tau0 g c t t2)))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: 
-(tau0 g c t1 t)).(tau0_ind g (\lambda (c0: C).(\lambda (_: T).(\lambda (t2: 
-T).(ex T (\lambda (t3: T).(tau0 g c0 t2 t3)))))) (\lambda (c0: C).(\lambda 
-(n: nat).(ex_intro T (\lambda (t2: T).(tau0 g c0 (TSort (next g n)) t2)) 
-(TSort (next g (next g n))) (tau0_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 (_: (tau0 g d v 
-w)).(\lambda (H2: (ex T (\lambda (t2: T).(tau0 g d w t2)))).(let H3 \def H2 
-in (ex_ind T (\lambda (t2: T).(tau0 g d w t2)) (ex T (\lambda (t2: T).(tau0 g 
-c0 (lift (S i) O w) t2))) (\lambda (x: T).(\lambda (H4: (tau0 g d w 
-x)).(ex_intro T (\lambda (t2: T).(tau0 g c0 (lift (S i) O w) t2)) (lift (S i) 
-O x) (tau0_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: (tau0 g d v w)).(\lambda (H2: (ex T (\lambda (t2: T).(tau0 g 
-d w t2)))).(let H3 \def H2 in (ex_ind T (\lambda (t2: T).(tau0 g d w t2)) (ex 
-T (\lambda (t2: T).(tau0 g c0 (lift (S i) O v) t2))) (\lambda (x: T).(\lambda 
-(_: (tau0 g d w x)).(ex_intro T (\lambda (t2: T).(tau0 g c0 (lift (S i) O v) 
-t2)) (lift (S i) O w) (tau0_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 (_: (tau0 g (CHead c0 (Bind b) 
-v) t2 t3)).(\lambda (H1: (ex T (\lambda (t2: T).(tau0 g (CHead c0 (Bind b) v) 
-t3 t2)))).(let H2 \def H1 in (ex_ind T (\lambda (t4: T).(tau0 g (CHead c0 
-(Bind b) v) t3 t4)) (ex T (\lambda (t4: T).(tau0 g c0 (THead (Bind b) v t3) 
-t4))) (\lambda (x: T).(\lambda (H3: (tau0 g (CHead c0 (Bind b) v) t3 
-x)).(ex_intro T (\lambda (t4: T).(tau0 g c0 (THead (Bind b) v t3) t4)) (THead 
-(Bind b) v x) (tau0_bind g b c0 v t3 x H3)))) H2))))))))) (\lambda (c0: 
-C).(\lambda (v: T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (tau0 g c0 
-t2 t3)).(\lambda (H1: (ex T (\lambda (t2: T).(tau0 g c0 t3 t2)))).(let H2 
-\def H1 in (ex_ind T (\lambda (t4: T).(tau0 g c0 t3 t4)) (ex T (\lambda (t4: 
-T).(tau0 g c0 (THead (Flat Appl) v t3) t4))) (\lambda (x: T).(\lambda (H3: 
-(tau0 g c0 t3 x)).(ex_intro T (\lambda (t4: T).(tau0 g c0 (THead (Flat Appl) 
-v t3) t4)) (THead (Flat Appl) v x) (tau0_appl g c0 v t3 x H3)))) H2)))))))) 
-(\lambda (c0: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (tau0 g c0 v1 
-v2)).(\lambda (H1: (ex T (\lambda (t2: T).(tau0 g c0 v2 t2)))).(\lambda (t2: 
-T).(\lambda (t3: T).(\lambda (_: (tau0 g c0 t2 t3)).(\lambda (H3: (ex T 
-(\lambda (t2: T).(tau0 g c0 t3 t2)))).(let H4 \def H1 in (ex_ind T (\lambda 
-(t4: T).(tau0 g c0 v2 t4)) (ex T (\lambda (t4: T).(tau0 g c0 (THead (Flat 
-Cast) v2 t3) t4))) (\lambda (x: T).(\lambda (H5: (tau0 g c0 v2 x)).(let H6 
-\def H3 in (ex_ind T (\lambda (t4: T).(tau0 g c0 t3 t4)) (ex T (\lambda (t4: 
-T).(tau0 g c0 (THead (Flat Cast) v2 t3) t4))) (\lambda (x0: T).(\lambda (H7: 
-(tau0 g c0 t3 x0)).(ex_intro T (\lambda (t4: T).(tau0 g c0 (THead (Flat Cast) 
-v2 t3) t4)) (THead (Flat Cast) x x0) (tau0_cast g c0 v2 x H5 t3 x0 H7)))) 
-H6)))) H4))))))))))) c t1 t H))))).
-
-inductive tau1 (g:G) (c:C) (t1:T): T \to Prop \def
-| tau1_tau0: \forall (t2: T).((tau0 g c t1 t2) \to (tau1 g c t1 t2))
-| tau1_sing: \forall (t: T).((tau1 g c t1 t) \to (\forall (t2: T).((tau0 g c 
-t t2) \to (tau1 g c t1 t2)))).
-
-theorem tau1_trans:
- \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((tau1 g c 
-t1 t) \to (\forall (t2: T).((tau1 g c t t2) \to (tau1 g c t1 t2)))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: 
-(tau1 g c t1 t)).(\lambda (t2: T).(\lambda (H0: (tau1 g c t t2)).(tau1_ind g 
-c t (\lambda (t0: T).(tau1 g c t1 t0)) (\lambda (t3: T).(\lambda (H1: (tau0 g 
-c t t3)).(tau1_sing g c t1 t H t3 H1))) (\lambda (t0: T).(\lambda (_: (tau1 g 
-c t t0)).(\lambda (H2: (tau1 g c t1 t0)).(\lambda (t3: T).(\lambda (H3: (tau0 
-g c t0 t3)).(tau1_sing g c t1 t0 H2 t3 H3)))))) t2 H0))))))).
-
-theorem tau1_bind:
- \forall (g: G).(\forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t1: 
-T).(\forall (t2: T).((tau1 g (CHead c (Bind b) v) t1 t2) \to (tau1 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: (tau1 g (CHead c (Bind b) v) t1 
-t2)).(tau1_ind g (CHead c (Bind b) v) t1 (\lambda (t: T).(tau1 g c (THead 
-(Bind b) v t1) (THead (Bind b) v t))) (\lambda (t3: T).(\lambda (H0: (tau0 g 
-(CHead c (Bind b) v) t1 t3)).(tau1_tau0 g c (THead (Bind b) v t1) (THead 
-(Bind b) v t3) (tau0_bind g b c v t1 t3 H0)))) (\lambda (t: T).(\lambda (_: 
-(tau1 g (CHead c (Bind b) v) t1 t)).(\lambda (H1: (tau1 g c (THead (Bind b) v 
-t1) (THead (Bind b) v t))).(\lambda (t3: T).(\lambda (H2: (tau0 g (CHead c 
-(Bind b) v) t t3)).(tau1_sing g c (THead (Bind b) v t1) (THead (Bind b) v t) 
-H1 (THead (Bind b) v t3) (tau0_bind g b c v t t3 H2))))))) t2 H))))))).
-
-theorem tau1_appl:
- \forall (g: G).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall 
-(t2: T).((tau1 g c t1 t2) \to (tau1 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: (tau1 g c t1 t2)).(tau1_ind g c t1 (\lambda (t: T).(tau1 
-g c (THead (Flat Appl) v t1) (THead (Flat Appl) v t))) (\lambda (t3: 
-T).(\lambda (H0: (tau0 g c t1 t3)).(tau1_tau0 g c (THead (Flat Appl) v t1) 
-(THead (Flat Appl) v t3) (tau0_appl g c v t1 t3 H0)))) (\lambda (t: 
-T).(\lambda (_: (tau1 g c t1 t)).(\lambda (H1: (tau1 g c (THead (Flat Appl) v 
-t1) (THead (Flat Appl) v t))).(\lambda (t3: T).(\lambda (H2: (tau0 g c t 
-t3)).(tau1_sing g c (THead (Flat Appl) v t1) (THead (Flat Appl) v t) H1 
-(THead (Flat Appl) v t3) (tau0_appl g c v t t3 H2))))))) t2 H)))))).
-
-theorem tau1_lift:
- \forall (g: G).(\forall (e: C).(\forall (t1: T).(\forall (t2: T).((tau1 g e 
-t1 t2) \to (\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c 
-e) \to (tau1 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: (tau1 g e t1 t2)).(tau1_ind g e t1 (\lambda (t: T).(\forall (c: 
-C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to (tau1 g c (lift h 
-d t1) (lift h d t))))))) (\lambda (t3: T).(\lambda (H0: (tau0 g e t1 
-t3)).(\lambda (c: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (drop 
-h d c e)).(tau1_tau0 g c (lift h d t1) (lift h d t3) (tau0_lift g e t1 t3 H0 
-c h d H1)))))))) (\lambda (t: T).(\lambda (_: (tau1 g e t1 t)).(\lambda (H1: 
-((\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to 
-(tau1 g c (lift h d t1) (lift h d t)))))))).(\lambda (t3: T).(\lambda (H2: 
-(tau0 g e t t3)).(\lambda (c: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda 
-(H3: (drop h d c e)).(tau1_sing g c (lift h d t1) (lift h d t) (H1 c h d H3) 
-(lift h d t3) (tau0_lift g e t t3 H2 c h d H3))))))))))) t2 H))))).
-
-theorem tau1_correct:
- \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((tau1 g c 
-t1 t) \to (ex T (\lambda (t2: T).(tau0 g c t t2)))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: 
-(tau1 g c t1 t)).(tau1_ind g c t1 (\lambda (t0: T).(ex T (\lambda (t2: 
-T).(tau0 g c t0 t2)))) (\lambda (t2: T).(\lambda (H0: (tau0 g c t1 
-t2)).(tau0_correct g c t1 t2 H0))) (\lambda (t0: T).(\lambda (_: (tau1 g c t1 
-t0)).(\lambda (_: (ex T (\lambda (t2: T).(tau0 g c t0 t2)))).(\lambda (t2: 
-T).(\lambda (H2: (tau0 g c t0 t2)).(tau0_correct g c t0 t2 H2)))))) t H))))).
-
-theorem tau1_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).((tau1 g d v w) 
-\to (tau1 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: (tau1 g d v w)).(tau1_ind g d v (\lambda (t: T).(tau1 g c 
-(TLRef i) (lift (S i) O t))) (\lambda (t2: T).(\lambda (H1: (tau0 g d v 
-t2)).(tau1_tau0 g c (TLRef i) (lift (S i) O t2) (tau0_abbr g c d v i H t2 
-H1)))) (\lambda (t: T).(\lambda (_: (tau1 g d v t)).(\lambda (H2: (tau1 g c 
-(TLRef i) (lift (S i) O t))).(\lambda (t2: T).(\lambda (H3: (tau0 g d t 
-t2)).(tau1_sing g c (TLRef i) (lift (S i) O t) H2 (lift (S i) O t2) 
-(tau0_lift g d t t2 H3 c (S i) O (getl_drop Abbr c d v i H)))))))) w 
-H0)))))))).
-
-theorem tau1_cast2:
- \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((tau1 g c 
-t1 t2) \to (\forall (v1: T).(\forall (v2: T).((tau0 g c v1 v2) \to (ex2 T 
-(\lambda (v3: T).(tau1 g c v1 v3)) (\lambda (v3: T).(tau1 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: (tau1 g c t1 t2)).(tau1_ind g c t1 (\lambda (t: T).(\forall (v1: 
-T).(\forall (v2: T).((tau0 g c v1 v2) \to (ex2 T (\lambda (v3: T).(tau1 g c 
-v1 v3)) (\lambda (v3: T).(tau1 g c (THead (Flat Cast) v1 t1) (THead (Flat 
-Cast) v3 t)))))))) (\lambda (t3: T).(\lambda (H0: (tau0 g c t1 t3)).(\lambda 
-(v1: T).(\lambda (v2: T).(\lambda (H1: (tau0 g c v1 v2)).(ex_intro2 T 
-(\lambda (v3: T).(tau1 g c v1 v3)) (\lambda (v3: T).(tau1 g c (THead (Flat 
-Cast) v1 t1) (THead (Flat Cast) v3 t3))) v2 (tau1_tau0 g c v1 v2 H1) 
-(tau1_tau0 g c (THead (Flat Cast) v1 t1) (THead (Flat Cast) v2 t3) (tau0_cast 
-g c v1 v2 H1 t1 t3 H0)))))))) (\lambda (t: T).(\lambda (_: (tau1 g c t1 
-t)).(\lambda (H1: ((\forall (v1: T).(\forall (v2: T).((tau0 g c v1 v2) \to 
-(ex2 T (\lambda (v3: T).(tau1 g c v1 v3)) (\lambda (v3: T).(tau1 g c (THead 
-(Flat Cast) v1 t1) (THead (Flat Cast) v3 t))))))))).(\lambda (t3: T).(\lambda 
-(H2: (tau0 g c t t3)).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H3: (tau0 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).(tau1 g c v1 v3)) (\lambda (v3: T).(tau1 g c (THead (Flat 
-Cast) v1 t1) (THead (Flat Cast) v3 t))) (ex2 T (\lambda (v3: T).(tau1 g c v1 
-v3)) (\lambda (v3: T).(tau1 g c (THead (Flat Cast) v1 t1) (THead (Flat Cast) 
-v3 t3)))) (\lambda (x: T).(\lambda (H5: (tau1 g c v1 x)).(\lambda (H6: (tau1 
-g c (THead (Flat Cast) v1 t1) (THead (Flat Cast) x t))).(let H_x0 \def 
-(tau1_correct g c v1 x H5) in (let H7 \def H_x0 in (ex_ind T (\lambda (t4: 
-T).(tau0 g c x t4)) (ex2 T (\lambda (v3: T).(tau1 g c v1 v3)) (\lambda (v3: 
-T).(tau1 g c (THead (Flat Cast) v1 t1) (THead (Flat Cast) v3 t3)))) (\lambda 
-(x0: T).(\lambda (H8: (tau0 g c x x0)).(ex_intro2 T (\lambda (v3: T).(tau1 g 
-c v1 v3)) (\lambda (v3: T).(tau1 g c (THead (Flat Cast) v1 t1) (THead (Flat 
-Cast) v3 t3))) x0 (tau1_sing g c v1 x H5 x0 H8) (tau1_sing g c (THead (Flat 
-Cast) v1 t1) (THead (Flat Cast) x t) H6 (THead (Flat Cast) x0 t3) (tau0_cast 
-g c x x0 H8 t t3 H2))))) H7)))))) H4))))))))))) t2 H))))).
-
-theorem tau1_cnt:
- \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((tau0 g c 
-t1 t) \to (ex2 T (\lambda (t2: T).(tau1 g c t1 t2)) (\lambda (t2: T).(cnt 
-t2)))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: 
-(tau0 g c t1 t)).(tau0_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (_: 
-T).(ex2 T (\lambda (t3: T).(tau1 g c0 t0 t3)) (\lambda (t3: T).(cnt t3)))))) 
-(\lambda (c0: C).(\lambda (n: nat).(ex_intro2 T (\lambda (t2: T).(tau1 g c0 
-(TSort n) t2)) (\lambda (t2: T).(cnt t2)) (TSort (next g n)) (tau1_tau0 g c0 
-(TSort n) (TSort (next g n)) (tau0_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 (_: (tau0 
-g d v w)).(\lambda (H2: (ex2 T (\lambda (t2: T).(tau1 g d v t2)) (\lambda 
-(t2: T).(cnt t2)))).(let H3 \def H2 in (ex2_ind T (\lambda (t2: T).(tau1 g d 
-v t2)) (\lambda (t2: T).(cnt t2)) (ex2 T (\lambda (t2: T).(tau1 g c0 (TLRef 
-i) t2)) (\lambda (t2: T).(cnt t2))) (\lambda (x: T).(\lambda (H4: (tau1 g d v 
-x)).(\lambda (H5: (cnt x)).(ex_intro2 T (\lambda (t2: T).(tau1 g c0 (TLRef i) 
-t2)) (\lambda (t2: T).(cnt t2)) (lift (S i) O x) (tau1_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: (tau0 g d v w)).(\lambda (H2: (ex2 T 
-(\lambda (t2: T).(tau1 g d v t2)) (\lambda (t2: T).(cnt t2)))).(let H3 \def 
-H2 in (ex2_ind T (\lambda (t2: T).(tau1 g d v t2)) (\lambda (t2: T).(cnt t2)) 
-(ex2 T (\lambda (t2: T).(tau1 g c0 (TLRef i) t2)) (\lambda (t2: T).(cnt t2))) 
-(\lambda (x: T).(\lambda (H4: (tau1 g d v x)).(\lambda (H5: (cnt 
-x)).(ex_intro2 T (\lambda (t2: T).(tau1 g c0 (TLRef i) t2)) (\lambda (t2: 
-T).(cnt t2)) (lift (S i) O x) (tau1_trans g c0 (TLRef i) (lift (S i) O v) 
-(tau1_tau0 g c0 (TLRef i) (lift (S i) O v) (tau0_abst g c0 d v i H0 w H1)) 
-(lift (S i) O x) (tau1_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 (_: (tau0 g 
-(CHead c0 (Bind b) v) t2 t3)).(\lambda (H1: (ex2 T (\lambda (t3: T).(tau1 g 
-(CHead c0 (Bind b) v) t2 t3)) (\lambda (t2: T).(cnt t2)))).(let H2 \def H1 in 
-(ex2_ind T (\lambda (t4: T).(tau1 g (CHead c0 (Bind b) v) t2 t4)) (\lambda 
-(t4: T).(cnt t4)) (ex2 T (\lambda (t4: T).(tau1 g c0 (THead (Bind b) v t2) 
-t4)) (\lambda (t4: T).(cnt t4))) (\lambda (x: T).(\lambda (H3: (tau1 g (CHead 
-c0 (Bind b) v) t2 x)).(\lambda (H4: (cnt x)).(ex_intro2 T (\lambda (t4: 
-T).(tau1 g c0 (THead (Bind b) v t2) t4)) (\lambda (t4: T).(cnt t4)) (THead 
-(Bind b) v x) (tau1_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 (_: (tau0 g c0 t2 t3)).(\lambda (H1: (ex2 T (\lambda (t3: 
-T).(tau1 g c0 t2 t3)) (\lambda (t2: T).(cnt t2)))).(let H2 \def H1 in 
-(ex2_ind T (\lambda (t4: T).(tau1 g c0 t2 t4)) (\lambda (t4: T).(cnt t4)) 
-(ex2 T (\lambda (t4: T).(tau1 g c0 (THead (Flat Appl) v t2) t4)) (\lambda 
-(t4: T).(cnt t4))) (\lambda (x: T).(\lambda (H3: (tau1 g c0 t2 x)).(\lambda 
-(H4: (cnt x)).(ex_intro2 T (\lambda (t4: T).(tau1 g c0 (THead (Flat Appl) v 
-t2) t4)) (\lambda (t4: T).(cnt t4)) (THead (Flat Appl) v x) (tau1_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: (tau0 g c0 v1 
-v2)).(\lambda (_: (ex2 T (\lambda (t2: T).(tau1 g c0 v1 t2)) (\lambda (t2: 
-T).(cnt t2)))).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (tau0 g c0 t2 
-t3)).(\lambda (H3: (ex2 T (\lambda (t3: T).(tau1 g c0 t2 t3)) (\lambda (t2: 
-T).(cnt t2)))).(let H4 \def H3 in (ex2_ind T (\lambda (t4: T).(tau1 g c0 t2 
-t4)) (\lambda (t4: T).(cnt t4)) (ex2 T (\lambda (t4: T).(tau1 g c0 (THead 
-(Flat Cast) v1 t2) t4)) (\lambda (t4: T).(cnt t4))) (\lambda (x: T).(\lambda 
-(H5: (tau1 g c0 t2 x)).(\lambda (H6: (cnt x)).(let H_x \def (tau1_cast2 g c0 
-t2 x H5 v1 v2 H0) in (let H7 \def H_x in (ex2_ind T (\lambda (v3: T).(tau1 g 
-c0 v1 v3)) (\lambda (v3: T).(tau1 g c0 (THead (Flat Cast) v1 t2) (THead (Flat 
-Cast) v3 x))) (ex2 T (\lambda (t4: T).(tau1 g c0 (THead (Flat Cast) v1 t2) 
-t4)) (\lambda (t4: T).(cnt t4))) (\lambda (x0: T).(\lambda (_: (tau1 g c0 v1 
-x0)).(\lambda (H9: (tau1 g c0 (THead (Flat Cast) v1 t2) (THead (Flat Cast) x0 
-x))).(ex_intro2 T (\lambda (t4: T).(tau1 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))))).
-
-inductive A: Set \def
-| ASort: nat \to (nat \to A)
-| AHead: A \to (A \to A).
-
-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.
-
-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.
-
-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))))))).
-
-theorem leq_gen_sort:
- \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 (_: nat).(eq A a2 (ASort h2 n2))))) (\lambda (n2: nat).(\lambda 
-(h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort h1 n1) k) (aplus g (ASort 
-h2 n2) k))))))))))
-\def
- \lambda (g: G).(\lambda (h1: nat).(\lambda (n1: nat).(\lambda (a2: 
-A).(\lambda (H: (leq g (ASort h1 n1) a2)).(let H0 \def (match H return 
-(\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a a0)).((eq A a (ASort 
-h1 n1)) \to ((eq A a0 a2) \to (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda 
-(h2: nat).(\lambda (_: nat).(eq A a2 (ASort h2 n2))))) (\lambda (n2: 
-nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort h1 n1) k) 
-(aplus g (ASort h2 n2) k))))))))))) with [(leq_sort h0 h2 n0 n2 k H0) 
-\Rightarrow (\lambda (H1: (eq A (ASort h0 n0) (ASort h1 n1))).(\lambda (H2: 
-(eq A (ASort h2 n2) a2)).((let H3 \def (f_equal A nat (\lambda (e: A).(match 
-e return (\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) 
-\Rightarrow n0])) (ASort h0 n0) (ASort h1 n1) H1) in ((let H4 \def (f_equal A 
-nat (\lambda (e: A).(match e return (\lambda (_: A).nat) with [(ASort n _) 
-\Rightarrow n | (AHead _ _) \Rightarrow h0])) (ASort h0 n0) (ASort h1 n1) H1) 
-in (eq_ind nat h1 (\lambda (n: nat).((eq nat n0 n1) \to ((eq A (ASort h2 n2) 
-a2) \to ((eq A (aplus g (ASort n n0) k) (aplus g (ASort h2 n2) k)) \to (ex2_3 
-nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A a2 
-(ASort h3 n3))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (k0: 
-nat).(eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort h3 n3) k0)))))))))) 
-(\lambda (H5: (eq nat n0 n1)).(eq_ind nat n1 (\lambda (n: nat).((eq A (ASort 
-h2 n2) a2) \to ((eq A (aplus g (ASort h1 n) k) (aplus g (ASort h2 n2) k)) \to 
-(ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: 
-nat).(eq A a2 (ASort h3 n3))))) (\lambda (n3: nat).(\lambda (h3: 
-nat).(\lambda (k0: nat).(eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort h3 
-n3) k0))))))))) (\lambda (H6: (eq A (ASort h2 n2) a2)).(eq_ind A (ASort h2 
-n2) (\lambda (a: A).((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) 
-k)) \to (ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: 
-nat).(eq A a (ASort h3 n3))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda 
-(k0: nat).(eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort h3 n3) k0)))))))) 
-(\lambda (H7: (eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) 
-k))).(ex2_3_intro nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda 
-(_: nat).(eq A (ASort h2 n2) (ASort h3 n3))))) (\lambda (n3: nat).(\lambda 
-(h3: nat).(\lambda (k0: nat).(eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort 
-h3 n3) k0))))) n2 h2 k (refl_equal A (ASort h2 n2)) H7)) a2 H6)) n0 (sym_eq 
-nat n0 n1 H5))) h0 (sym_eq nat h0 h1 H4))) H3)) H2 H0))) | (leq_head a1 a0 H0 
-a3 a4 H1) \Rightarrow (\lambda (H2: (eq A (AHead a1 a3) (ASort h1 
-n1))).(\lambda (H3: (eq A (AHead a0 a4) a2)).((let H4 \def (eq_ind A (AHead 
-a1 a3) (\lambda (e: A).(match e return (\lambda (_: A).Prop) with [(ASort _ 
-_) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort h1 n1) H2) in 
-(False_ind ((eq A (AHead a0 a4) a2) \to ((leq g a1 a0) \to ((leq g a3 a4) \to 
-(ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: 
-nat).(eq A a2 (ASort h2 n2))))) (\lambda (n2: nat).(\lambda (h2: 
-nat).(\lambda (k: nat).(eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) 
-k))))))))) H4)) H3 H0 H1)))]) in (H0 (refl_equal A (ASort h1 n1)) (refl_equal 
-A a2))))))).
-
-theorem leq_gen_head:
- \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 (a4: A).(eq A a 
-(AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(leq g a1 a3))) (\lambda 
-(_: A).(\lambda (a4: A).(leq g a2 a4))))))))
-\def
- \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a: A).(\lambda 
-(H: (leq g (AHead a1 a2) a)).(let H0 \def (match H return (\lambda (a0: 
-A).(\lambda (a3: A).(\lambda (_: (leq ? a0 a3)).((eq A a0 (AHead a1 a2)) \to 
-((eq A a3 a) \to (ex3_2 A A (\lambda (a4: A).(\lambda (a5: A).(eq A a (AHead 
-a4 a5)))) (\lambda (a4: A).(\lambda (_: A).(leq g a1 a4))) (\lambda (_: 
-A).(\lambda (a5: A).(leq g a2 a5))))))))) with [(leq_sort h1 h2 n1 n2 k H0) 
-\Rightarrow (\lambda (H1: (eq A (ASort h1 n1) (AHead a1 a2))).(\lambda (H2: 
-(eq A (ASort h2 n2) a)).((let H3 \def (eq_ind A (ASort h1 n1) (\lambda (e: 
-A).(match e return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | 
-(AHead _ _) \Rightarrow False])) I (AHead a1 a2) H1) in (False_ind ((eq A 
-(ASort h2 n2) a) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) 
-k)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a (AHead a3 a4)))) 
-(\lambda (a3: A).(\lambda (_: A).(leq g a1 a3))) (\lambda (_: A).(\lambda 
-(a4: A).(leq g a2 a4)))))) H3)) H2 H0))) | (leq_head a0 a3 H0 a4 a5 H1) 
-\Rightarrow (\lambda (H2: (eq A (AHead a0 a4) (AHead a1 a2))).(\lambda (H3: 
-(eq A (AHead a3 a5) a)).((let H4 \def (f_equal A A (\lambda (e: A).(match e 
-return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a4 | (AHead _ a) 
-\Rightarrow a])) (AHead a0 a4) (AHead a1 a2) H2) in ((let H5 \def (f_equal A 
-A (\lambda (e: A).(match e return (\lambda (_: A).A) with [(ASort _ _) 
-\Rightarrow a0 | (AHead a _) \Rightarrow a])) (AHead a0 a4) (AHead a1 a2) H2) 
-in (eq_ind A a1 (\lambda (a6: A).((eq A a4 a2) \to ((eq A (AHead a3 a5) a) 
-\to ((leq g a6 a3) \to ((leq g a4 a5) \to (ex3_2 A A (\lambda (a7: 
-A).(\lambda (a8: A).(eq A a (AHead a7 a8)))) (\lambda (a7: A).(\lambda (_: 
-A).(leq g a1 a7))) (\lambda (_: A).(\lambda (a8: A).(leq g a2 a8))))))))) 
-(\lambda (H6: (eq A a4 a2)).(eq_ind A a2 (\lambda (a6: A).((eq A (AHead a3 
-a5) a) \to ((leq g a1 a3) \to ((leq g a6 a5) \to (ex3_2 A A (\lambda (a7: 
-A).(\lambda (a8: A).(eq A a (AHead a7 a8)))) (\lambda (a7: A).(\lambda (_: 
-A).(leq g a1 a7))) (\lambda (_: A).(\lambda (a8: A).(leq g a2 a8)))))))) 
-(\lambda (H7: (eq A (AHead a3 a5) a)).(eq_ind A (AHead a3 a5) (\lambda (a: 
-A).((leq g a1 a3) \to ((leq g a2 a5) \to (ex3_2 A A (\lambda (a6: A).(\lambda 
-(a7: A).(eq A a (AHead a6 a7)))) (\lambda (a6: A).(\lambda (_: A).(leq g a1 
-a6))) (\lambda (_: A).(\lambda (a7: A).(leq g a2 a7))))))) (\lambda (H8: (leq 
-g a1 a3)).(\lambda (H9: (leq g a2 a5)).(ex3_2_intro A A (\lambda (a6: 
-A).(\lambda (a7: A).(eq A (AHead a3 a5) (AHead a6 a7)))) (\lambda (a6: 
-A).(\lambda (_: A).(leq g a1 a6))) (\lambda (_: A).(\lambda (a7: A).(leq g a2 
-a7))) a3 a5 (refl_equal A (AHead a3 a5)) H8 H9))) a H7)) a4 (sym_eq A a4 a2 
-H6))) a0 (sym_eq A a0 a1 H5))) H4)) H3 H0 H1)))]) in (H0 (refl_equal A (AHead 
-a1 a2)) (refl_equal A a))))))).
-
-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 h) \Rightarrow (ASort h 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 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 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 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 (a2: A).(eq A a0 (AHead a1 a2))) (\lambda 
-(a0: A).(eq A a2 (asucc g a0))))))).(\lambda (a3: A).(\lambda (H0: (((eq A 
-(AHead a1 a2) (asucc g a3)) \to (ex2 A (\lambda (a0: A).(eq A a3 (AHead a1 
-a0))) (\lambda (a0: A).(eq A a2 (asucc g a0))))))).(\lambda (H1: (eq A (AHead 
-a1 a2) (asucc g (AHead a0 a3)))).(let H2 \def (f_equal A A (\lambda (e: 
-A).(match e return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a1 | 
-(AHead a _) \Rightarrow a])) (AHead a1 a2) (AHead a0 (asucc g a3)) H1) in 
-((let H3 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: A).A) 
-with [(ASort _ _) \Rightarrow a2 | (AHead _ a) \Rightarrow a])) (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 (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)))))) 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 (a: A).((eq A (AHead a1 a) (asucc 
-g a3)) \to (ex2 A (\lambda (a0: A).(eq A a3 (AHead a1 a0))) (\lambda (a0: 
-A).(eq A a (asucc g a0)))))) H0 (asucc g a3) H3) in (let H7 \def (eq_ind A a2 
-(\lambda (a: A).((eq A (AHead a1 a) (asucc g a1)) \to (ex2 A (\lambda (a0: 
-A).(eq A a1 (AHead a1 a0))) (\lambda (a0: A).(eq A a (asucc g a0)))))) 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)))).
-
-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)))).(sym_equal A (asucc g (aplus g a2 (plus n 
-h2))) (asucc g (aplus g a1 (plus n h1))) (sym_equal A (asucc g (aplus g a1 
-(plus n h1))) (asucc g (aplus g a2 (plus n h2))) (sym_equal A (asucc g (aplus 
-g a2 (plus n h2))) (asucc g (aplus g a1 (plus n h1))) (f_equal2 G A A asucc g 
-g (aplus g a2 (plus n h2)) (aplus g a1 (plus n h1)) (refl_equal G g) (sym_eq 
-A (aplus g a1 (plus n h1)) (aplus g a2 (plus n h2)) 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)))) (sym_equal A (asucc g (asucc g (aplus g a (plus n n0)))) 
-(asucc g (aplus g (asucc g (aplus g a n)) n0)) (sym_equal A (asucc g (aplus g 
-(asucc g (aplus g a n)) n0)) (asucc g (asucc g (aplus g a (plus n n0)))) 
-(sym_equal A (asucc g (asucc g (aplus g a (plus n n0)))) (asucc g (aplus g 
-(asucc g (aplus g a n)) n0)) (f_equal2 G A A asucc g g (asucc g (aplus g a 
-(plus n n0))) (aplus g (asucc g (aplus g a n)) n0) (refl_equal G g) (sym_eq A 
-(aplus g (asucc g (aplus g a n)) n0) (asucc g (aplus g a (plus n n0))) 
-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 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))).((match h1 return (\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)])))) with 
-[O \Rightarrow (\lambda (H1: (eq A (aplus g (ASort O n1) k) (aplus g (ASort 
-h2 n2) k))).((match h2 return (\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)])))) 
-with [O \Rightarrow (\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)))) | (S n) \Rightarrow (\lambda 
-(H2: (eq A (aplus g (ASort O n1) k) (aplus g (ASort (S n) n2) k))).(leq_sort 
-g O n (next g n1) n2 k (eq_ind A (aplus g (ASort O n1) (S k)) (\lambda (a: 
-A).(eq A a (aplus g (ASort n n2) k))) (eq_ind A (aplus g (ASort (S n) n2) (S 
-k)) (\lambda (a: A).(eq A (aplus g (ASort O n1) (S k)) a)) (eq_ind_r A (aplus 
-g (ASort (S n) n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g 
-(ASort (S n) n2) k)))) (refl_equal A (asucc g (aplus g (ASort (S n) n2) k))) 
-(aplus g (ASort O n1) k) H2) (aplus g (ASort n n2) k) (aplus_sort_S_S_simpl g 
-n2 n k)) (aplus g (ASort O (next g n1)) k) (aplus_sort_O_S_simpl g n1 k))))]) 
-H1)) | (S n) \Rightarrow (\lambda (H1: (eq A (aplus g (ASort (S n) n1) k) 
-(aplus g (ASort h2 n2) k))).((match h2 return (\lambda (n0: nat).((eq A 
-(aplus g (ASort (S n) n1) k) (aplus g (ASort n0 n2) k)) \to (leq g (ASort n 
-n1) (match n0 with [O \Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow 
-(ASort h n2)])))) with [O \Rightarrow (\lambda (H2: (eq A (aplus g (ASort (S 
-n) n1) k) (aplus g (ASort O n2) k))).(leq_sort g n O n1 (next g n2) k (eq_ind 
-A (aplus g (ASort O n2) (S k)) (\lambda (a: A).(eq A (aplus g (ASort n n1) k) 
-a)) (eq_ind A (aplus g (ASort (S n) 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 n) n1) k) H2) (aplus g 
-(ASort n n1) k) (aplus_sort_S_S_simpl g n1 n k)) (aplus g (ASort O (next g 
-n2)) k) (aplus_sort_O_S_simpl g n2 k)))) | (S n0) \Rightarrow (\lambda (H2: 
-(eq A (aplus g (ASort (S n) n1) k) (aplus g (ASort (S n0) n2) k))).(leq_sort 
-g n n0 n1 n2 k (eq_ind A (aplus g (ASort (S n) n1) (S k)) (\lambda (a: A).(eq 
-A a (aplus g (ASort n0 n2) k))) (eq_ind A (aplus g (ASort (S n0) n2) (S k)) 
-(\lambda (a: A).(eq A (aplus g (ASort (S n) n1) (S k)) a)) (eq_ind_r A (aplus 
-g (ASort (S n0) n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g 
-(ASort (S n0) n2) k)))) (refl_equal A (asucc g (aplus g (ASort (S n0) n2) 
-k))) (aplus g (ASort (S n) n1) k) H2) (aplus g (ASort n0 n2) k) 
-(aplus_sort_S_S_simpl g n2 n0 k)) (aplus g (ASort n n1) k) 
-(aplus_sort_S_S_simpl g n1 n k))))]) 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)))).((match n return (\lambda (n3: nat).((leq g (asucc 
-g (ASort n3 n0)) (asucc g (ASort n1 n2))) \to (leq g (ASort n3 n0) (ASort n1 
-n2)))) with [O \Rightarrow (\lambda (H0: (leq g (asucc g (ASort O n0)) (asucc 
-g (ASort n1 n2)))).((match n1 return (\lambda (n3: nat).((leq g (asucc g 
-(ASort O n0)) (asucc g (ASort n3 n2))) \to (leq g (ASort O n0) (ASort n3 
-n2)))) with [O \Rightarrow (\lambda (H1: (leq g (asucc g (ASort O n0)) (asucc 
-g (ASort O n2)))).(let H2 \def (match H1 return (\lambda (a: A).(\lambda (a0: 
-A).(\lambda (_: (leq ? a a0)).((eq A a (ASort O (next g n0))) \to ((eq A a0 
-(ASort O (next g n2))) \to (leq g (ASort O n0) (ASort O n2))))))) with 
-[(leq_sort h1 h2 n1 n3 k H0) \Rightarrow (\lambda (H1: (eq A (ASort h1 n1) 
-(ASort O (next g n0)))).(\lambda (H2: (eq A (ASort h2 n3) (ASort O (next g 
-n2)))).((let H3 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda 
-(_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n1])) 
-(ASort h1 n1) (ASort O (next g n0)) H1) in ((let H4 \def (f_equal A nat 
-(\lambda (e: A).(match e return (\lambda (_: A).nat) with [(ASort n _) 
-\Rightarrow n | (AHead _ _) \Rightarrow h1])) (ASort h1 n1) (ASort O (next g 
-n0)) H1) in (eq_ind nat O (\lambda (n: nat).((eq nat n1 (next g n0)) \to ((eq 
-A (ASort h2 n3) (ASort O (next g n2))) \to ((eq A (aplus g (ASort n n1) k) 
-(aplus g (ASort h2 n3) k)) \to (leq g (ASort O n0) (ASort O n2)))))) (\lambda 
-(H5: (eq nat n1 (next g n0))).(eq_ind nat (next g n0) (\lambda (n: nat).((eq 
-A (ASort h2 n3) (ASort O (next g n2))) \to ((eq A (aplus g (ASort O n) k) 
-(aplus g (ASort h2 n3) k)) \to (leq g (ASort O n0) (ASort O n2))))) (\lambda 
-(H6: (eq A (ASort h2 n3) (ASort O (next g n2)))).(let H7 \def (f_equal A nat 
-(\lambda (e: A).(match e return (\lambda (_: A).nat) with [(ASort _ n) 
-\Rightarrow n | (AHead _ _) \Rightarrow n3])) (ASort h2 n3) (ASort O (next g 
-n2)) H6) in ((let H8 \def (f_equal A nat (\lambda (e: A).(match e return 
-(\lambda (_: A).nat) with [(ASort n _) \Rightarrow n | (AHead _ _) 
-\Rightarrow h2])) (ASort h2 n3) (ASort O (next g n2)) H6) in (eq_ind nat O 
-(\lambda (n: nat).((eq nat n3 (next g n2)) \to ((eq A (aplus g (ASort O (next 
-g n0)) k) (aplus g (ASort n n3) k)) \to (leq g (ASort O n0) (ASort O n2))))) 
-(\lambda (H9: (eq nat n3 (next g n2))).(eq_ind nat (next g n2) (\lambda (n: 
-nat).((eq A (aplus g (ASort O (next g n0)) k) (aplus g (ASort O n) k)) \to 
-(leq g (ASort O n0) (ASort O n2)))) (\lambda (H10: (eq A (aplus g (ASort O 
-(next g n0)) k) (aplus g (ASort O (next g n2)) k))).(let H \def (eq_ind_r A 
-(aplus g (ASort O (next g n0)) k) (\lambda (a: A).(eq A a (aplus g (ASort O 
-(next g n2)) k))) H10 (aplus g (ASort O n0) (S k)) (aplus_sort_O_S_simpl g n0 
-k)) in (let H11 \def (eq_ind_r A (aplus g (ASort O (next g n2)) k) (\lambda 
-(a: A).(eq A (aplus g (ASort O n0) (S k)) a)) H (aplus g (ASort O n2) (S k)) 
-(aplus_sort_O_S_simpl g n2 k)) in (leq_sort g O O n0 n2 (S k) H11)))) n3 
-(sym_eq nat n3 (next g n2) H9))) h2 (sym_eq nat h2 O H8))) H7))) n1 (sym_eq 
-nat n1 (next g n0) H5))) h1 (sym_eq nat h1 O H4))) H3)) H2 H0))) | (leq_head 
-a1 a2 H0 a3 a4 H1) \Rightarrow (\lambda (H2: (eq A (AHead a1 a3) (ASort O 
-(next g n0)))).(\lambda (H3: (eq A (AHead a2 a4) (ASort O (next g 
-n2)))).((let H4 \def (eq_ind A (AHead a1 a3) (\lambda (e: A).(match e return 
-(\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) 
-\Rightarrow True])) I (ASort O (next g n0)) H2) in (False_ind ((eq A (AHead 
-a2 a4) (ASort O (next g n2))) \to ((leq g a1 a2) \to ((leq g a3 a4) \to (leq 
-g (ASort O n0) (ASort O n2))))) H4)) H3 H0 H1)))]) in (H2 (refl_equal A 
-(ASort O (next g n0))) (refl_equal A (ASort O (next g n2)))))) | (S n3) 
-\Rightarrow (\lambda (H1: (leq g (asucc g (ASort O n0)) (asucc g (ASort (S 
-n3) n2)))).(let H2 \def (match H1 return (\lambda (a: A).(\lambda (a0: 
-A).(\lambda (_: (leq ? a a0)).((eq A a (ASort O (next g n0))) \to ((eq A a0 
-(ASort n3 n2)) \to (leq g (ASort O n0) (ASort (S n3) n2))))))) with 
-[(leq_sort h1 h2 n1 n3 k H0) \Rightarrow (\lambda (H1: (eq A (ASort h1 n1) 
-(ASort O (next g n0)))).(\lambda (H2: (eq A (ASort h2 n3) (ASort n3 
-n2))).((let H3 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda 
-(_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n1])) 
-(ASort h1 n1) (ASort O (next g n0)) H1) in ((let H4 \def (f_equal A nat 
-(\lambda (e: A).(match e return (\lambda (_: A).nat) with [(ASort n _) 
-\Rightarrow n | (AHead _ _) \Rightarrow h1])) (ASort h1 n1) (ASort O (next g 
-n0)) H1) in (eq_ind nat O (\lambda (n: nat).((eq nat n1 (next g n0)) \to ((eq 
-A (ASort h2 n3) (ASort n3 n2)) \to ((eq A (aplus g (ASort n n1) k) (aplus g 
-(ASort h2 n3) k)) \to (leq g (ASort O n0) (ASort (S n3) n2)))))) (\lambda 
-(H5: (eq nat n1 (next g n0))).(eq_ind nat (next g n0) (\lambda (n: nat).((eq 
-A (ASort h2 n3) (ASort n3 n2)) \to ((eq A (aplus g (ASort O n) k) (aplus g 
-(ASort h2 n3) k)) \to (leq g (ASort O n0) (ASort (S n3) n2))))) (\lambda (H6: 
-(eq A (ASort h2 n3) (ASort n3 n2))).(let H7 \def (f_equal A nat (\lambda (e: 
-A).(match e return (\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | 
-(AHead _ _) \Rightarrow n3])) (ASort h2 n3) (ASort n3 n2) H6) in ((let H8 
-\def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: A).nat) with 
-[(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow h2])) (ASort h2 n3) 
-(ASort n3 n2) H6) in (eq_ind nat n3 (\lambda (n: nat).((eq nat n3 n2) \to 
-((eq A (aplus g (ASort O (next g n0)) k) (aplus g (ASort n n3) k)) \to (leq g 
-(ASort O n0) (ASort (S n3) n2))))) (\lambda (H9: (eq nat n3 n2)).(eq_ind nat 
-n2 (\lambda (n: nat).((eq A (aplus g (ASort O (next g n0)) k) (aplus g (ASort 
-n3 n) k)) \to (leq g (ASort O n0) (ASort (S n3) n2)))) (\lambda (H10: (eq A 
-(aplus g (ASort O (next g n0)) k) (aplus g (ASort n3 n2) k))).(let H \def 
-(eq_ind_r A (aplus g (ASort O (next g n0)) k) (\lambda (a: A).(eq A a (aplus 
-g (ASort n3 n2) k))) H10 (aplus g (ASort O n0) (S k)) (aplus_sort_O_S_simpl g 
-n0 k)) in (let H11 \def (eq_ind_r A (aplus g (ASort n3 n2) k) (\lambda (a: 
-A).(eq A (aplus g (ASort O n0) (S k)) a)) H (aplus g (ASort (S n3) n2) (S k)) 
-(aplus_sort_S_S_simpl g n2 n3 k)) in (leq_sort g O (S n3) n0 n2 (S k) H11)))) 
-n3 (sym_eq nat n3 n2 H9))) h2 (sym_eq nat h2 n3 H8))) H7))) n1 (sym_eq nat n1 
-(next g n0) H5))) h1 (sym_eq nat h1 O H4))) H3)) H2 H0))) | (leq_head a1 a2 
-H0 a3 a4 H1) \Rightarrow (\lambda (H2: (eq A (AHead a1 a3) (ASort O (next g 
-n0)))).(\lambda (H3: (eq A (AHead a2 a4) (ASort n3 n2))).((let H4 \def 
-(eq_ind A (AHead a1 a3) (\lambda (e: A).(match e return (\lambda (_: A).Prop) 
-with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I 
-(ASort O (next g n0)) H2) in (False_ind ((eq A (AHead a2 a4) (ASort n3 n2)) 
-\to ((leq g a1 a2) \to ((leq g a3 a4) \to (leq g (ASort O n0) (ASort (S n3) 
-n2))))) H4)) H3 H0 H1)))]) in (H2 (refl_equal A (ASort O (next g n0))) 
-(refl_equal A (ASort n3 n2)))))]) H0)) | (S n3) \Rightarrow (\lambda (H0: 
-(leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n1 n2)))).((match n1 
-return (\lambda (n4: nat).((leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort 
-n4 n2))) \to (leq g (ASort (S n3) n0) (ASort n4 n2)))) with [O \Rightarrow 
-(\lambda (H1: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort O 
-n2)))).(let H2 \def (match H1 return (\lambda (a: A).(\lambda (a0: 
-A).(\lambda (_: (leq ? a a0)).((eq A a (ASort n3 n0)) \to ((eq A a0 (ASort O 
-(next g n2))) \to (leq g (ASort (S n3) n0) (ASort O n2))))))) with [(leq_sort 
-h1 h2 n1 n3 k H0) \Rightarrow (\lambda (H1: (eq A (ASort h1 n1) (ASort n3 
-n0))).(\lambda (H2: (eq A (ASort h2 n3) (ASort O (next g n2)))).((let H3 \def 
-(f_equal A nat (\lambda (e: A).(match e return (\lambda (_: A).nat) with 
-[(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n1])) (ASort h1 n1) 
-(ASort n3 n0) H1) in ((let H4 \def (f_equal A nat (\lambda (e: A).(match e 
-return (\lambda (_: A).nat) with [(ASort n _) \Rightarrow n | (AHead _ _) 
-\Rightarrow h1])) (ASort h1 n1) (ASort n3 n0) H1) in (eq_ind nat n3 (\lambda 
-(n: nat).((eq nat n1 n0) \to ((eq A (ASort h2 n3) (ASort O (next g n2))) \to 
-((eq A (aplus g (ASort n n1) k) (aplus g (ASort h2 n3) k)) \to (leq g (ASort 
-(S n3) n0) (ASort O n2)))))) (\lambda (H5: (eq nat n1 n0)).(eq_ind nat n0 
-(\lambda (n: nat).((eq A (ASort h2 n3) (ASort O (next g n2))) \to ((eq A 
-(aplus g (ASort n3 n) k) (aplus g (ASort h2 n3) k)) \to (leq g (ASort (S n3) 
-n0) (ASort O n2))))) (\lambda (H6: (eq A (ASort h2 n3) (ASort O (next g 
-n2)))).(let H7 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda 
-(_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n3])) 
-(ASort h2 n3) (ASort O (next g n2)) H6) in ((let H8 \def (f_equal A nat 
-(\lambda (e: A).(match e return (\lambda (_: A).nat) with [(ASort n _) 
-\Rightarrow n | (AHead _ _) \Rightarrow h2])) (ASort h2 n3) (ASort O (next g 
-n2)) H6) in (eq_ind nat O (\lambda (n: nat).((eq nat n3 (next g n2)) \to ((eq 
-A (aplus g (ASort n3 n0) k) (aplus g (ASort n n3) k)) \to (leq g (ASort (S 
-n3) n0) (ASort O n2))))) (\lambda (H9: (eq nat n3 (next g n2))).(eq_ind nat 
-(next g n2) (\lambda (n: nat).((eq A (aplus g (ASort n3 n0) k) (aplus g 
-(ASort O n) k)) \to (leq g (ASort (S n3) n0) (ASort O n2)))) (\lambda (H10: 
-(eq A (aplus g (ASort n3 n0) k) (aplus g (ASort O (next g n2)) k))).(let H 
-\def (eq_ind_r A (aplus g (ASort n3 n0) k) (\lambda (a: A).(eq A a (aplus g 
-(ASort O (next g n2)) k))) H10 (aplus g (ASort (S n3) n0) (S k)) 
-(aplus_sort_S_S_simpl g n0 n3 k)) in (let H11 \def (eq_ind_r A (aplus g 
-(ASort O (next g n2)) k) (\lambda (a: A).(eq A (aplus g (ASort (S n3) n0) (S 
-k)) a)) H (aplus g (ASort O n2) (S k)) (aplus_sort_O_S_simpl g n2 k)) in 
-(leq_sort g (S n3) O n0 n2 (S k) H11)))) n3 (sym_eq nat n3 (next g n2) H9))) 
-h2 (sym_eq nat h2 O H8))) H7))) n1 (sym_eq nat n1 n0 H5))) h1 (sym_eq nat h1 
-n3 H4))) H3)) H2 H0))) | (leq_head a1 a2 H0 a3 a4 H1) \Rightarrow (\lambda 
-(H2: (eq A (AHead a1 a3) (ASort n3 n0))).(\lambda (H3: (eq A (AHead a2 a4) 
-(ASort O (next g n2)))).((let H4 \def (eq_ind A (AHead a1 a3) (\lambda (e: 
-A).(match e return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False 
-| (AHead _ _) \Rightarrow True])) I (ASort n3 n0) H2) in (False_ind ((eq A 
-(AHead a2 a4) (ASort O (next g n2))) \to ((leq g a1 a2) \to ((leq g a3 a4) 
-\to (leq g (ASort (S n3) n0) (ASort O n2))))) H4)) H3 H0 H1)))]) in (H2 
-(refl_equal A (ASort n3 n0)) (refl_equal A (ASort O (next g n2)))))) | (S n4) 
-\Rightarrow (\lambda (H1: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort 
-(S n4) n2)))).(let H2 \def (match H1 return (\lambda (a: A).(\lambda (a0: 
-A).(\lambda (_: (leq ? a a0)).((eq A a (ASort n3 n0)) \to ((eq A a0 (ASort n4 
-n2)) \to (leq g (ASort (S n3) n0) (ASort (S n4) n2))))))) with [(leq_sort h1 
-h2 n3 n4 k H0) \Rightarrow (\lambda (H1: (eq A (ASort h1 n3) (ASort n3 
-n0))).(\lambda (H2: (eq A (ASort h2 n4) (ASort n4 n2))).((let H3 \def 
-(f_equal A nat (\lambda (e: A).(match e return (\lambda (_: A).nat) with 
-[(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n3])) (ASort h1 n3) 
-(ASort n3 n0) H1) in ((let H4 \def (f_equal A nat (\lambda (e: A).(match e 
-return (\lambda (_: A).nat) with [(ASort n _) \Rightarrow n | (AHead _ _) 
-\Rightarrow h1])) (ASort h1 n3) (ASort n3 n0) H1) in (eq_ind nat n3 (\lambda 
-(n: nat).((eq nat n3 n0) \to ((eq A (ASort h2 n4) (ASort n4 n2)) \to ((eq A 
-(aplus g (ASort n n3) k) (aplus g (ASort h2 n4) k)) \to (leq g (ASort (S n3) 
-n0) (ASort (S n4) n2)))))) (\lambda (H5: (eq nat n3 n0)).(eq_ind nat n0 
-(\lambda (n: nat).((eq A (ASort h2 n4) (ASort n4 n2)) \to ((eq A (aplus g 
-(ASort n3 n) k) (aplus g (ASort h2 n4) k)) \to (leq g (ASort (S n3) n0) 
-(ASort (S n4) n2))))) (\lambda (H6: (eq A (ASort h2 n4) (ASort n4 n2))).(let 
-H7 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: A).nat) 
-with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n4])) (ASort h2 n4) 
-(ASort n4 n2) H6) in ((let H8 \def (f_equal A nat (\lambda (e: A).(match e 
-return (\lambda (_: A).nat) with [(ASort n _) \Rightarrow n | (AHead _ _) 
-\Rightarrow h2])) (ASort h2 n4) (ASort n4 n2) H6) in (eq_ind nat n4 (\lambda 
-(n: nat).((eq nat n4 n2) \to ((eq A (aplus g (ASort n3 n0) k) (aplus g (ASort 
-n n4) k)) \to (leq g (ASort (S n3) n0) (ASort (S n4) n2))))) (\lambda (H9: 
-(eq nat n4 n2)).(eq_ind nat n2 (\lambda (n: nat).((eq A (aplus g (ASort n3 
-n0) k) (aplus g (ASort n4 n) k)) \to (leq g (ASort (S n3) n0) (ASort (S n4) 
-n2)))) (\lambda (H10: (eq A (aplus g (ASort n3 n0) k) (aplus g (ASort n4 n2) 
-k))).(let H \def (eq_ind_r A (aplus g (ASort n3 n0) k) (\lambda (a: A).(eq A 
-a (aplus g (ASort n4 n2) k))) H10 (aplus g (ASort (S n3) n0) (S k)) 
-(aplus_sort_S_S_simpl g n0 n3 k)) in (let H11 \def (eq_ind_r A (aplus g 
-(ASort n4 n2) k) (\lambda (a: A).(eq A (aplus g (ASort (S n3) n0) (S k)) a)) 
-H (aplus g (ASort (S n4) n2) (S k)) (aplus_sort_S_S_simpl g n2 n4 k)) in 
-(leq_sort g (S n3) (S n4) n0 n2 (S k) H11)))) n4 (sym_eq nat n4 n2 H9))) h2 
-(sym_eq nat h2 n4 H8))) H7))) n3 (sym_eq nat n3 n0 H5))) h1 (sym_eq nat h1 n3 
-H4))) H3)) H2 H0))) | (leq_head a1 a2 H0 a3 a4 H1) \Rightarrow (\lambda (H2: 
-(eq A (AHead a1 a3) (ASort n3 n0))).(\lambda (H3: (eq A (AHead a2 a4) (ASort 
-n4 n2))).((let H4 \def (eq_ind A (AHead a1 a3) (\lambda (e: A).(match e 
-return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ 
-_) \Rightarrow True])) I (ASort n3 n0) H2) in (False_ind ((eq A (AHead a2 a4) 
-(ASort n4 n2)) \to ((leq g a1 a2) \to ((leq g a3 a4) \to (leq g (ASort (S n3) 
-n0) (ASort (S n4) n2))))) H4)) H3 H0 H1)))]) in (H2 (refl_equal A (ASort n3 
-n0)) (refl_equal A (ASort n4 n2)))))]) H0))]) 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)))).((match n return (\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)))))) with [O \Rightarrow (\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 H5 \def (match H4 return (\lambda (a1: A).(\lambda (a2: 
-A).(\lambda (_: (leq ? a1 a2)).((eq A a1 (ASort O (next g n0))) \to ((eq A a2 
-(AHead a (asucc g a0))) \to (leq g (ASort O n0) (AHead a a0))))))) with 
-[(leq_sort h1 h2 n1 n2 k H2) \Rightarrow (\lambda (H3: (eq A (ASort h1 n1) 
-(ASort O (next g n0)))).(\lambda (H4: (eq A (ASort h2 n2) (AHead a (asucc g 
-a0)))).((let H5 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda 
-(_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n1])) 
-(ASort h1 n1) (ASort O (next g n0)) H3) in ((let H6 \def (f_equal A nat 
-(\lambda (e: A).(match e return (\lambda (_: A).nat) with [(ASort n _) 
-\Rightarrow n | (AHead _ _) \Rightarrow h1])) (ASort h1 n1) (ASort O (next g 
-n0)) H3) in (eq_ind nat O (\lambda (n: nat).((eq nat n1 (next g n0)) \to ((eq 
-A (ASort h2 n2) (AHead a (asucc g a0))) \to ((eq A (aplus g (ASort n n1) k) 
-(aplus g (ASort h2 n2) k)) \to (leq g (ASort O n0) (AHead a a0)))))) (\lambda 
-(H7: (eq nat n1 (next g n0))).(eq_ind nat (next g n0) (\lambda (n: nat).((eq 
-A (ASort h2 n2) (AHead a (asucc g a0))) \to ((eq A (aplus g (ASort O n) k) 
-(aplus g (ASort h2 n2) k)) \to (leq g (ASort O n0) (AHead a a0))))) (\lambda 
-(H8: (eq A (ASort h2 n2) (AHead a (asucc g a0)))).(let H9 \def (eq_ind A 
-(ASort h2 n2) (\lambda (e: A).(match e return (\lambda (_: A).Prop) with 
-[(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead a 
-(asucc g a0)) H8) in (False_ind ((eq A (aplus g (ASort O (next g n0)) k) 
-(aplus g (ASort h2 n2) k)) \to (leq g (ASort O n0) (AHead a a0))) H9))) n1 
-(sym_eq nat n1 (next g n0) H7))) h1 (sym_eq nat h1 O H6))) H5)) H4 H2))) | 
-(leq_head a1 a2 H2 a3 a4 H3) \Rightarrow (\lambda (H4: (eq A (AHead a1 a3) 
-(ASort O (next g n0)))).(\lambda (H5: (eq A (AHead a2 a4) (AHead a (asucc g 
-a0)))).((let H6 \def (eq_ind A (AHead a1 a3) (\lambda (e: A).(match e return 
-(\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) 
-\Rightarrow True])) I (ASort O (next g n0)) H4) in (False_ind ((eq A (AHead 
-a2 a4) (AHead a (asucc g a0))) \to ((leq g a1 a2) \to ((leq g a3 a4) \to (leq 
-g (ASort O n0) (AHead a a0))))) H6)) H5 H2 H3)))]) in (H5 (refl_equal A 
-(ASort O (next g n0))) (refl_equal A (AHead a (asucc g a0)))))))) | (S n1) 
-\Rightarrow (\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 H5 \def (match H4 
-return (\lambda (a1: A).(\lambda (a2: A).(\lambda (_: (leq ? a1 a2)).((eq A 
-a1 (ASort n1 n0)) \to ((eq A a2 (AHead a (asucc g a0))) \to (leq g (ASort (S 
-n1) n0) (AHead a a0))))))) with [(leq_sort h1 h2 n1 n2 k H2) \Rightarrow 
-(\lambda (H3: (eq A (ASort h1 n1) (ASort n1 n0))).(\lambda (H4: (eq A (ASort 
-h2 n2) (AHead a (asucc g a0)))).((let H5 \def (f_equal A nat (\lambda (e: 
-A).(match e return (\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | 
-(AHead _ _) \Rightarrow n1])) (ASort h1 n1) (ASort n1 n0) H3) in ((let H6 
-\def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: A).nat) with 
-[(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow h1])) (ASort h1 n1) 
-(ASort n1 n0) H3) in (eq_ind nat n1 (\lambda (n: nat).((eq nat n1 n0) \to 
-((eq A (ASort h2 n2) (AHead a (asucc g a0))) \to ((eq A (aplus g (ASort n n1) 
-k) (aplus g (ASort h2 n2) k)) \to (leq g (ASort (S n1) n0) (AHead a a0)))))) 
-(\lambda (H7: (eq nat n1 n0)).(eq_ind nat n0 (\lambda (n: nat).((eq A (ASort 
-h2 n2) (AHead a (asucc g a0))) \to ((eq A (aplus g (ASort n1 n) k) (aplus g 
-(ASort h2 n2) k)) \to (leq g (ASort (S n1) n0) (AHead a a0))))) (\lambda (H8: 
-(eq A (ASort h2 n2) (AHead a (asucc g a0)))).(let H9 \def (eq_ind A (ASort h2 
-n2) (\lambda (e: A).(match e return (\lambda (_: A).Prop) with [(ASort _ _) 
-\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead a (asucc g a0)) 
-H8) in (False_ind ((eq A (aplus g (ASort n1 n0) k) (aplus g (ASort h2 n2) k)) 
-\to (leq g (ASort (S n1) n0) (AHead a a0))) H9))) n1 (sym_eq nat n1 n0 H7))) 
-h1 (sym_eq nat h1 n1 H6))) H5)) H4 H2))) | (leq_head a1 a2 H2 a3 a4 H3) 
-\Rightarrow (\lambda (H4: (eq A (AHead a1 a3) (ASort n1 n0))).(\lambda (H5: 
-(eq A (AHead a2 a4) (AHead a (asucc g a0)))).((let H6 \def (eq_ind A (AHead 
-a1 a3) (\lambda (e: A).(match e return (\lambda (_: A).Prop) with [(ASort _ 
-_) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort n1 n0) H4) in 
-(False_ind ((eq A (AHead a2 a4) (AHead a (asucc g a0))) \to ((leq g a1 a2) 
-\to ((leq g a3 a4) \to (leq g (ASort (S n1) n0) (AHead a a0))))) H6)) H5 H2 
-H3)))]) in (H5 (refl_equal A (ASort n1 n0)) (refl_equal A (AHead a (asucc g 
-a0))))))))]) 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)))).((match n return (\lambda (n1: nat).((leq g (asucc g (AHead a 
-a0)) (asucc g (ASort n1 n0))) \to (leq g (AHead a a0) (ASort n1 n0)))) with 
-[O \Rightarrow (\lambda (H2: (leq g (asucc g (AHead a a0)) (asucc g (ASort O 
-n0)))).(let H3 \def (match H2 return (\lambda (a1: A).(\lambda (a2: 
-A).(\lambda (_: (leq ? a1 a2)).((eq A a1 (AHead a (asucc g a0))) \to ((eq A 
-a2 (ASort O (next g n0))) \to (leq g (AHead a a0) (ASort O n0))))))) with 
-[(leq_sort h1 h2 n1 n2 k H2) \Rightarrow (\lambda (H3: (eq A (ASort h1 n1) 
-(AHead a (asucc g a0)))).(\lambda (H4: (eq A (ASort h2 n2) (ASort O (next g 
-n0)))).((let H5 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e return 
-(\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) 
-\Rightarrow False])) I (AHead a (asucc g a0)) H3) in (False_ind ((eq A (ASort 
-h2 n2) (ASort O (next g n0))) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g 
-(ASort h2 n2) k)) \to (leq g (AHead a a0) (ASort O n0)))) H5)) H4 H2))) | 
-(leq_head a1 a2 H2 a3 a4 H3) \Rightarrow (\lambda (H4: (eq A (AHead a1 a3) 
-(AHead a (asucc g a0)))).(\lambda (H5: (eq A (AHead a2 a4) (ASort O (next g 
-n0)))).((let H6 \def (f_equal A A (\lambda (e: A).(match e return (\lambda 
-(_: A).A) with [(ASort _ _) \Rightarrow a3 | (AHead _ a) \Rightarrow a])) 
-(AHead a1 a3) (AHead a (asucc g a0)) H4) in ((let H7 \def (f_equal A A 
-(\lambda (e: A).(match e return (\lambda (_: A).A) with [(ASort _ _) 
-\Rightarrow a1 | (AHead a _) \Rightarrow a])) (AHead a1 a3) (AHead a (asucc g 
-a0)) H4) in (eq_ind A a (\lambda (a5: A).((eq A a3 (asucc g a0)) \to ((eq A 
-(AHead a2 a4) (ASort O (next g n0))) \to ((leq g a5 a2) \to ((leq g a3 a4) 
-\to (leq g (AHead a a0) (ASort O n0))))))) (\lambda (H8: (eq A a3 (asucc g 
-a0))).(eq_ind A (asucc g a0) (\lambda (a5: A).((eq A (AHead a2 a4) (ASort O 
-(next g n0))) \to ((leq g a a2) \to ((leq g a5 a4) \to (leq g (AHead a a0) 
-(ASort O n0)))))) (\lambda (H9: (eq A (AHead a2 a4) (ASort O (next g 
-n0)))).(let H10 \def (eq_ind A (AHead a2 a4) (\lambda (e: A).(match e return 
-(\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) 
-\Rightarrow True])) I (ASort O (next g n0)) H9) in (False_ind ((leq g a a2) 
-\to ((leq g (asucc g a0) a4) \to (leq g (AHead a a0) (ASort O n0)))) H10))) 
-a3 (sym_eq A a3 (asucc g a0) H8))) a1 (sym_eq A a1 a H7))) H6)) H5 H2 H3)))]) 
-in (H3 (refl_equal A (AHead a (asucc g a0))) (refl_equal A (ASort O (next g 
-n0)))))) | (S n1) \Rightarrow (\lambda (H2: (leq g (asucc g (AHead a a0)) 
-(asucc g (ASort (S n1) n0)))).(let H3 \def (match H2 return (\lambda (a1: 
-A).(\lambda (a2: A).(\lambda (_: (leq ? a1 a2)).((eq A a1 (AHead a (asucc g 
-a0))) \to ((eq A a2 (ASort n1 n0)) \to (leq g (AHead a a0) (ASort (S n1) 
-n0))))))) with [(leq_sort h1 h2 n1 n2 k H2) \Rightarrow (\lambda (H3: (eq A 
-(ASort h1 n1) (AHead a (asucc g a0)))).(\lambda (H4: (eq A (ASort h2 n2) 
-(ASort n1 n0))).((let H5 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match 
-e return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ 
-_) \Rightarrow False])) I (AHead a (asucc g a0)) H3) in (False_ind ((eq A 
-(ASort h2 n2) (ASort n1 n0)) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g 
-(ASort h2 n2) k)) \to (leq g (AHead a a0) (ASort (S n1) n0)))) H5)) H4 H2))) 
-| (leq_head a1 a2 H2 a3 a4 H3) \Rightarrow (\lambda (H4: (eq A (AHead a1 a3) 
-(AHead a (asucc g a0)))).(\lambda (H5: (eq A (AHead a2 a4) (ASort n1 
-n0))).((let H6 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: 
-A).A) with [(ASort _ _) \Rightarrow a3 | (AHead _ a) \Rightarrow a])) (AHead 
-a1 a3) (AHead a (asucc g a0)) H4) in ((let H7 \def (f_equal A A (\lambda (e: 
-A).(match e return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a1 | 
-(AHead a _) \Rightarrow a])) (AHead a1 a3) (AHead a (asucc g a0)) H4) in 
-(eq_ind A a (\lambda (a5: A).((eq A a3 (asucc g a0)) \to ((eq A (AHead a2 a4) 
-(ASort n1 n0)) \to ((leq g a5 a2) \to ((leq g a3 a4) \to (leq g (AHead a a0) 
-(ASort (S n1) n0))))))) (\lambda (H8: (eq A a3 (asucc g a0))).(eq_ind A 
-(asucc g a0) (\lambda (a5: A).((eq A (AHead a2 a4) (ASort n1 n0)) \to ((leq g 
-a a2) \to ((leq g a5 a4) \to (leq g (AHead a a0) (ASort (S n1) n0)))))) 
-(\lambda (H9: (eq A (AHead a2 a4) (ASort n1 n0))).(let H10 \def (eq_ind A 
-(AHead a2 a4) (\lambda (e: A).(match e return (\lambda (_: A).Prop) with 
-[(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort n1 
-n0) H9) in (False_ind ((leq g a a2) \to ((leq g (asucc g a0) a4) \to (leq g 
-(AHead a a0) (ASort (S n1) n0)))) H10))) a3 (sym_eq A a3 (asucc g a0) H8))) 
-a1 (sym_eq A a1 a H7))) H6)) H5 H2 H3)))]) in (H3 (refl_equal A (AHead a 
-(asucc g a0))) (refl_equal A (ASort n1 n0)))))]) 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 H4 \def (match H3 return 
-(\lambda (a1: A).(\lambda (a2: A).(\lambda (_: (leq ? a1 a2)).((eq A a1 
-(AHead a (asucc g a0))) \to ((eq A a2 (AHead a3 (asucc g a4))) \to (leq g 
-(AHead a a0) (AHead a3 a4))))))) with [(leq_sort h1 h2 n1 n2 k H4) 
-\Rightarrow (\lambda (H5: (eq A (ASort h1 n1) (AHead a (asucc g 
-a0)))).(\lambda (H6: (eq A (ASort h2 n2) (AHead a3 (asucc g a4)))).((let H7 
-\def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e return (\lambda (_: 
-A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow 
-False])) I (AHead a (asucc g a0)) H5) in (False_ind ((eq A (ASort h2 n2) 
-(AHead a3 (asucc g a4))) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort 
-h2 n2) k)) \to (leq g (AHead a a0) (AHead a3 a4)))) H7)) H6 H4))) | (leq_head 
-a3 a4 H4 a5 a6 H5) \Rightarrow (\lambda (H6: (eq A (AHead a3 a5) (AHead a 
-(asucc g a0)))).(\lambda (H7: (eq A (AHead a4 a6) (AHead a3 (asucc g 
-a4)))).((let H8 \def (f_equal A A (\lambda (e: A).(match e return (\lambda 
-(_: A).A) with [(ASort _ _) \Rightarrow a5 | (AHead _ a) \Rightarrow a])) 
-(AHead a3 a5) (AHead a (asucc g a0)) H6) in ((let H9 \def (f_equal A A 
-(\lambda (e: A).(match e return (\lambda (_: A).A) with [(ASort _ _) 
-\Rightarrow a3 | (AHead a _) \Rightarrow a])) (AHead a3 a5) (AHead a (asucc g 
-a0)) H6) in (eq_ind A a (\lambda (a1: A).((eq A a5 (asucc g a0)) \to ((eq A 
-(AHead a4 a6) (AHead a3 (asucc g a4))) \to ((leq g a1 a4) \to ((leq g a5 a6) 
-\to (leq g (AHead a a0) (AHead a3 a4))))))) (\lambda (H10: (eq A a5 (asucc g 
-a0))).(eq_ind A (asucc g a0) (\lambda (a1: A).((eq A (AHead a4 a6) (AHead a3 
-(asucc g a4))) \to ((leq g a a4) \to ((leq g a1 a6) \to (leq g (AHead a a0) 
-(AHead a3 a4)))))) (\lambda (H11: (eq A (AHead a4 a6) (AHead a3 (asucc g 
-a4)))).(let H12 \def (f_equal A A (\lambda (e: A).(match e return (\lambda 
-(_: A).A) with [(ASort _ _) \Rightarrow a6 | (AHead _ a) \Rightarrow a])) 
-(AHead a4 a6) (AHead a3 (asucc g a4)) H11) in ((let H13 \def (f_equal A A 
-(\lambda (e: A).(match e return (\lambda (_: A).A) with [(ASort _ _) 
-\Rightarrow a4 | (AHead a _) \Rightarrow a])) (AHead a4 a6) (AHead a3 (asucc 
-g a4)) H11) in (eq_ind A a3 (\lambda (a1: A).((eq A a6 (asucc g a4)) \to 
-((leq g a a1) \to ((leq g (asucc g a0) a6) \to (leq g (AHead a a0) (AHead a3 
-a4)))))) (\lambda (H14: (eq A a6 (asucc g a4))).(eq_ind A (asucc g a4) 
-(\lambda (a1: A).((leq g a a3) \to ((leq g (asucc g a0) a1) \to (leq g (AHead 
-a a0) (AHead a3 a4))))) (\lambda (H15: (leq g a a3)).(\lambda (H16: (leq g 
-(asucc g a0) (asucc g a4))).(leq_head g a a3 H15 a0 a4 (H0 a4 H16)))) a6 
-(sym_eq A a6 (asucc g a4) H14))) a4 (sym_eq A a4 a3 H13))) H12))) a5 (sym_eq 
-A a5 (asucc g a0) H10))) a3 (sym_eq A a3 a H9))) H8)) H7 H4 H5)))]) in (H4 
-(refl_equal A (AHead a (asucc g a0))) (refl_equal A (AHead a3 (asucc g 
-a4)))))))))) a2)))))) a1)).
-
-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 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 h) 
-\Rightarrow (ASort h n0)]) h) (ASort n n0))).(\lambda (P: Prop).((match n 
-return (\lambda (n1: nat).((eq A (aplus g (match n1 with [O \Rightarrow 
-(ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]) h) (ASort n1 n0)) 
-\to P)) with [O \Rightarrow (\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 (a: A).(eq A a (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 return (\lambda (_: A).nat) with 
-[(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow ((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) 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 (n: nat).(eq nat 
-(next_plus g (next g n0) n) 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))))) | (S n1) \Rightarrow 
-(\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 (a: A).(eq A a (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 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 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 return (\lambda 
-(_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow ((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) 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))))]) 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 (a: A).(eq A a (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 return (\lambda (_: A).A) with [(ASort _ _) 
-\Rightarrow ((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) g 
-(asucc g a1) h) | (AHead _ a) \Rightarrow a])) (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)).
-
-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 H0 \def (match 
-H return (\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a a0)).((eq A a 
-(AHead a1 a2)) \to ((eq A a0 (AHead a3 a4)) \to (leq g a2 a4)))))) with 
-[(leq_sort h1 h2 n1 n2 k H0) \Rightarrow (\lambda (H1: (eq A (ASort h1 n1) 
-(AHead a1 a2))).(\lambda (H2: (eq A (ASort h2 n2) (AHead a3 a4))).((let H3 
-\def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e return (\lambda (_: 
-A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow 
-False])) I (AHead a1 a2) H1) in (False_ind ((eq A (ASort h2 n2) (AHead a3 
-a4)) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to (leq 
-g a2 a4))) H3)) H2 H0))) | (leq_head a0 a5 H0 a6 a7 H1) \Rightarrow (\lambda 
-(H2: (eq A (AHead a0 a6) (AHead a1 a2))).(\lambda (H3: (eq A (AHead a5 a7) 
-(AHead a3 a4))).((let H4 \def (f_equal A A (\lambda (e: A).(match e return 
-(\lambda (_: A).A) with [(ASort _ _) \Rightarrow a6 | (AHead _ a) \Rightarrow 
-a])) (AHead a0 a6) (AHead a1 a2) H2) in ((let H5 \def (f_equal A A (\lambda 
-(e: A).(match e return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 | 
-(AHead a _) \Rightarrow a])) (AHead a0 a6) (AHead a1 a2) H2) in (eq_ind A a1 
-(\lambda (a: A).((eq A a6 a2) \to ((eq A (AHead a5 a7) (AHead a3 a4)) \to 
-((leq g a a5) \to ((leq g a6 a7) \to (leq g a2 a4)))))) (\lambda (H6: (eq A 
-a6 a2)).(eq_ind A a2 (\lambda (a: A).((eq A (AHead a5 a7) (AHead a3 a4)) \to 
-((leq g a1 a5) \to ((leq g a a7) \to (leq g a2 a4))))) (\lambda (H7: (eq A 
-(AHead a5 a7) (AHead a3 a4))).(let H8 \def (f_equal A A (\lambda (e: 
-A).(match e return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a7 | 
-(AHead _ a) \Rightarrow a])) (AHead a5 a7) (AHead a3 a4) H7) in ((let H9 \def 
-(f_equal A A (\lambda (e: A).(match e return (\lambda (_: A).A) with [(ASort 
-_ _) \Rightarrow a5 | (AHead a _) \Rightarrow a])) (AHead a5 a7) (AHead a3 
-a4) H7) in (eq_ind A a3 (\lambda (a: A).((eq A a7 a4) \to ((leq g a1 a) \to 
-((leq g a2 a7) \to (leq g a2 a4))))) (\lambda (H10: (eq A a7 a4)).(eq_ind A 
-a4 (\lambda (a: A).((leq g a1 a3) \to ((leq g a2 a) \to (leq g a2 a4)))) 
-(\lambda (_: (leq g a1 a3)).(\lambda (H12: (leq g a2 a4)).H12)) a7 (sym_eq A 
-a7 a4 H10))) a5 (sym_eq A a5 a3 H9))) H8))) a6 (sym_eq A a6 a2 H6))) a0 
-(sym_eq A a0 a1 H5))) H4)) H3 H0 H1)))]) in (H0 (refl_equal A (AHead a1 a2)) 
-(refl_equal A (AHead a3 a4))))))))).
-
-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_r A a2 (\lambda (a: A).(leq g a a2)) (leq_refl g a2) a1 H)))).
-
-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 (a0: A).(leq g a1 (asucc g a0))))).(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_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 H2 \def (match H1 return (\lambda (a: A).(\lambda 
-(a0: A).(\lambda (_: (leq ? a a0)).((eq A a (ASort h2 n2)) \to ((eq A a0 a3) 
-\to (leq g (ASort h1 n1) a3)))))) with [(leq_sort h0 h3 n0 n3 k0 H1) 
-\Rightarrow (\lambda (H2: (eq A (ASort h0 n0) (ASort h2 n2))).(\lambda (H3: 
-(eq A (ASort h3 n3) a3)).((let H4 \def (f_equal A nat (\lambda (e: A).(match 
-e return (\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) 
-\Rightarrow n0])) (ASort h0 n0) (ASort h2 n2) H2) in ((let H5 \def (f_equal A 
-nat (\lambda (e: A).(match e return (\lambda (_: A).nat) with [(ASort n _) 
-\Rightarrow n | (AHead _ _) \Rightarrow h0])) (ASort h0 n0) (ASort h2 n2) H2) 
-in (eq_ind nat h2 (\lambda (n: nat).((eq nat n0 n2) \to ((eq A (ASort h3 n3) 
-a3) \to ((eq A (aplus g (ASort n n0) k0) (aplus g (ASort h3 n3) k0)) \to (leq 
-g (ASort h1 n1) a3))))) (\lambda (H6: (eq nat n0 n2)).(eq_ind nat n2 (\lambda 
-(n: nat).((eq A (ASort h3 n3) a3) \to ((eq A (aplus g (ASort h2 n) k0) (aplus 
-g (ASort h3 n3) k0)) \to (leq g (ASort h1 n1) a3)))) (\lambda (H7: (eq A 
-(ASort h3 n3) a3)).(eq_ind A (ASort h3 n3) (\lambda (a: A).((eq A (aplus g 
-(ASort h2 n2) k0) (aplus g (ASort h3 n3) k0)) \to (leq g (ASort h1 n1) a))) 
-(\lambda (H8: (eq A (aplus g (ASort h2 n2) k0) (aplus g (ASort h3 n3) 
-k0))).(lt_le_e k k0 (leq g (ASort h1 n1) (ASort h3 n3)) (\lambda (H9: (lt k 
-k0)).(let H_y \def (aplus_reg_r g (ASort h1 n1) (ASort h2 n2) k k H0 (minus 
-k0 k)) in (let H10 \def (eq_ind_r nat (plus (minus k0 k) k) (\lambda (n: 
-nat).(eq A (aplus g (ASort h1 n1) n) (aplus g (ASort h2 n2) n))) H_y k0 
-(le_plus_minus_sym k k0 (le_S_n k k0 (le_S (S k) k0 H9)))) in (leq_sort g h1 
-h3 n1 n3 k0 (trans_eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort h2 n2) k0) 
-(aplus g (ASort h3 n3) k0) H10 H8))))) (\lambda (H9: (le k0 k)).(let H_y \def 
-(aplus_reg_r g (ASort h2 n2) (ASort h3 n3) k0 k0 H8 (minus k k0)) in (let H10 
-\def (eq_ind_r nat (plus (minus k k0) k0) (\lambda (n: nat).(eq A (aplus g 
-(ASort h2 n2) n) (aplus g (ASort h3 n3) n))) H_y k (le_plus_minus_sym k0 k 
-H9)) in (leq_sort g h1 h3 n1 n3 k (trans_eq A (aplus g (ASort h1 n1) k) 
-(aplus g (ASort h2 n2) k) (aplus g (ASort h3 n3) k) H0 H10))))))) a3 H7)) n0 
-(sym_eq nat n0 n2 H6))) h0 (sym_eq nat h0 h2 H5))) H4)) H3 H1))) | (leq_head 
-a1 a2 H1 a0 a4 H2) \Rightarrow (\lambda (H3: (eq A (AHead a1 a0) (ASort h2 
-n2))).(\lambda (H4: (eq A (AHead a2 a4) a3)).((let H5 \def (eq_ind A (AHead 
-a1 a0) (\lambda (e: A).(match e return (\lambda (_: A).Prop) with [(ASort _ 
-_) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort h2 n2) H3) in 
-(False_ind ((eq A (AHead a2 a4) a3) \to ((leq g a1 a2) \to ((leq g a0 a4) \to 
-(leq g (ASort h1 n1) a3)))) H5)) H4 H1 H2)))]) in (H2 (refl_equal A (ASort h2 
-n2)) (refl_equal A a3))))))))))) (\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 (a3: A).((leq g a6 a3) \to (leq g a5 
-a3))))).(\lambda (a0: A).(\lambda (H4: (leq g (AHead a4 a6) a0)).(let H5 \def 
-(match H4 return (\lambda (a: A).(\lambda (a1: A).(\lambda (_: (leq ? a 
-a1)).((eq A a (AHead a4 a6)) \to ((eq A a1 a0) \to (leq g (AHead a3 a5) 
-a0)))))) with [(leq_sort h1 h2 n1 n2 k H4) \Rightarrow (\lambda (H5: (eq A 
-(ASort h1 n1) (AHead a4 a6))).(\lambda (H6: (eq A (ASort h2 n2) a0)).((let H7 
-\def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e return (\lambda (_: 
-A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow 
-False])) I (AHead a4 a6) H5) in (False_ind ((eq A (ASort h2 n2) a0) \to ((eq 
-A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to (leq g (AHead a3 
-a5) a0))) H7)) H6 H4))) | (leq_head a5 a6 H4 a7 a8 H5) \Rightarrow (\lambda 
-(H6: (eq A (AHead a5 a7) (AHead a4 a6))).(\lambda (H7: (eq A (AHead a6 a8) 
-a0)).((let H8 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: 
-A).A) with [(ASort _ _) \Rightarrow a7 | (AHead _ a) \Rightarrow a])) (AHead 
-a5 a7) (AHead a4 a6) H6) in ((let H9 \def (f_equal A A (\lambda (e: A).(match 
-e return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a5 | (AHead a _) 
-\Rightarrow a])) (AHead a5 a7) (AHead a4 a6) H6) in (eq_ind A a4 (\lambda (a: 
-A).((eq A a7 a6) \to ((eq A (AHead a6 a8) a0) \to ((leq g a a6) \to ((leq g 
-a7 a8) \to (leq g (AHead a3 a5) a0)))))) (\lambda (H10: (eq A a7 a6)).(eq_ind 
-A a6 (\lambda (a: A).((eq A (AHead a6 a8) a0) \to ((leq g a4 a6) \to ((leq g 
-a a8) \to (leq g (AHead a3 a5) a0))))) (\lambda (H11: (eq A (AHead a6 a8) 
-a0)).(eq_ind A (AHead a6 a8) (\lambda (a: A).((leq g a4 a6) \to ((leq g a6 
-a8) \to (leq g (AHead a3 a5) a)))) (\lambda (H12: (leq g a4 a6)).(\lambda 
-(H13: (leq g a6 a8)).(leq_head g a3 a6 (H1 a6 H12) a5 a8 (H3 a8 H13)))) a0 
-H11)) a7 (sym_eq A a7 a6 H10))) a5 (sym_eq A a5 a4 H9))) H8)) H7 H4 H5)))]) 
-in (H5 (refl_equal A (AHead a4 a6)) (refl_equal A a0))))))))))))) a1 a2 H)))).
-
-theorem leq_ahead_false:
- \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).((match n return (\lambda (n1: 
-nat).((leq g (AHead (ASort n1 n0) a2) (ASort n1 n0)) \to P)) with [O 
-\Rightarrow (\lambda (H0: (leq g (AHead (ASort O n0) a2) (ASort O n0))).(let 
-H1 \def (match H0 return (\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? 
-a a0)).((eq A a (AHead (ASort O n0) a2)) \to ((eq A a0 (ASort O n0)) \to 
-P))))) with [(leq_sort h1 h2 n1 n2 k H0) \Rightarrow (\lambda (H1: (eq A 
-(ASort h1 n1) (AHead (ASort O n0) a2))).(\lambda (H2: (eq A (ASort h2 n2) 
-(ASort O n0))).((let H3 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e 
-return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) 
-\Rightarrow False])) I (AHead (ASort O n0) a2) H1) in (False_ind ((eq A 
-(ASort h2 n2) (ASort O n0)) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g 
-(ASort h2 n2) k)) \to P)) H3)) H2 H0))) | (leq_head a1 a0 H0 a3 a4 H1) 
-\Rightarrow (\lambda (H2: (eq A (AHead a1 a3) (AHead (ASort O n0) 
-a2))).(\lambda (H3: (eq A (AHead a0 a4) (ASort O n0))).((let H4 \def (f_equal 
-A A (\lambda (e: A).(match e return (\lambda (_: A).A) with [(ASort _ _) 
-\Rightarrow a3 | (AHead _ a) \Rightarrow a])) (AHead a1 a3) (AHead (ASort O 
-n0) a2) H2) in ((let H5 \def (f_equal A A (\lambda (e: A).(match e return 
-(\lambda (_: A).A) with [(ASort _ _) \Rightarrow a1 | (AHead a _) \Rightarrow 
-a])) (AHead a1 a3) (AHead (ASort O n0) a2) H2) in (eq_ind A (ASort O n0) 
-(\lambda (a: A).((eq A a3 a2) \to ((eq A (AHead a0 a4) (ASort O n0)) \to 
-((leq g a a0) \to ((leq g a3 a4) \to P))))) (\lambda (H6: (eq A a3 
-a2)).(eq_ind A a2 (\lambda (a: A).((eq A (AHead a0 a4) (ASort O n0)) \to 
-((leq g (ASort O n0) a0) \to ((leq g a a4) \to P)))) (\lambda (H7: (eq A 
-(AHead a0 a4) (ASort O n0))).(let H8 \def (eq_ind A (AHead a0 a4) (\lambda 
-(e: A).(match e return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow 
-False | (AHead _ _) \Rightarrow True])) I (ASort O n0) H7) in (False_ind 
-((leq g (ASort O n0) a0) \to ((leq g a2 a4) \to P)) H8))) a3 (sym_eq A a3 a2 
-H6))) a1 (sym_eq A a1 (ASort O n0) H5))) H4)) H3 H0 H1)))]) in (H1 
-(refl_equal A (AHead (ASort O n0) a2)) (refl_equal A (ASort O n0))))) | (S 
-n1) \Rightarrow (\lambda (H0: (leq g (AHead (ASort (S n1) n0) a2) (ASort (S 
-n1) n0))).(let H1 \def (match H0 return (\lambda (a: A).(\lambda (a0: 
-A).(\lambda (_: (leq ? a a0)).((eq A a (AHead (ASort (S n1) n0) a2)) \to ((eq 
-A a0 (ASort (S n1) n0)) \to P))))) with [(leq_sort h1 h2 n1 n2 k H0) 
-\Rightarrow (\lambda (H1: (eq A (ASort h1 n1) (AHead (ASort (S n1) n0) 
-a2))).(\lambda (H2: (eq A (ASort h2 n2) (ASort (S n1) n0))).((let H3 \def 
-(eq_ind A (ASort h1 n1) (\lambda (e: A).(match e return (\lambda (_: A).Prop) 
-with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I 
-(AHead (ASort (S n1) n0) a2) H1) in (False_ind ((eq A (ASort h2 n2) (ASort (S 
-n1) n0)) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to 
-P)) H3)) H2 H0))) | (leq_head a1 a0 H0 a3 a4 H1) \Rightarrow (\lambda (H2: 
-(eq A (AHead a1 a3) (AHead (ASort (S n1) n0) a2))).(\lambda (H3: (eq A (AHead 
-a0 a4) (ASort (S n1) n0))).((let H4 \def (f_equal A A (\lambda (e: A).(match 
-e return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a3 | (AHead _ a) 
-\Rightarrow a])) (AHead a1 a3) (AHead (ASort (S n1) n0) a2) H2) in ((let H5 
-\def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: A).A) with 
-[(ASort _ _) \Rightarrow a1 | (AHead a _) \Rightarrow a])) (AHead a1 a3) 
-(AHead (ASort (S n1) n0) a2) H2) in (eq_ind A (ASort (S n1) n0) (\lambda (a: 
-A).((eq A a3 a2) \to ((eq A (AHead a0 a4) (ASort (S n1) n0)) \to ((leq g a 
-a0) \to ((leq g a3 a4) \to P))))) (\lambda (H6: (eq A a3 a2)).(eq_ind A a2 
-(\lambda (a: A).((eq A (AHead a0 a4) (ASort (S n1) n0)) \to ((leq g (ASort (S 
-n1) n0) a0) \to ((leq g a a4) \to P)))) (\lambda (H7: (eq A (AHead a0 a4) 
-(ASort (S n1) n0))).(let H8 \def (eq_ind A (AHead a0 a4) (\lambda (e: 
-A).(match e return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False 
-| (AHead _ _) \Rightarrow True])) I (ASort (S n1) n0) H7) in (False_ind ((leq 
-g (ASort (S n1) n0) a0) \to ((leq g a2 a4) \to P)) H8))) a3 (sym_eq A a3 a2 
-H6))) a1 (sym_eq A a1 (ASort (S n1) n0) H5))) H4)) H3 H0 H1)))]) in (H1 
-(refl_equal A (AHead (ASort (S n1) n0) a2)) (refl_equal A (ASort (S n1) 
-n0)))))]) 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 H2 \def (match H1 return (\lambda (a1: 
-A).(\lambda (a3: A).(\lambda (_: (leq ? a1 a3)).((eq A a1 (AHead (AHead a a0) 
-a2)) \to ((eq A a3 (AHead a a0)) \to P))))) with [(leq_sort h1 h2 n1 n2 k H2) 
-\Rightarrow (\lambda (H3: (eq A (ASort h1 n1) (AHead (AHead a a0) 
-a2))).(\lambda (H4: (eq A (ASort h2 n2) (AHead a a0))).((let H5 \def (eq_ind 
-A (ASort h1 n1) (\lambda (e: A).(match e return (\lambda (_: A).Prop) with 
-[(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead 
-(AHead a a0) a2) H3) in (False_ind ((eq A (ASort h2 n2) (AHead a a0)) \to 
-((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to P)) H5)) H4 
-H2))) | (leq_head a1 a3 H2 a4 a5 H3) \Rightarrow (\lambda (H4: (eq A (AHead 
-a1 a4) (AHead (AHead a a0) a2))).(\lambda (H5: (eq A (AHead a3 a5) (AHead a 
-a0))).((let H6 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: 
-A).A) with [(ASort _ _) \Rightarrow a4 | (AHead _ a) \Rightarrow a])) (AHead 
-a1 a4) (AHead (AHead a a0) a2) H4) in ((let H7 \def (f_equal A A (\lambda (e: 
-A).(match e return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a1 | 
-(AHead a _) \Rightarrow a])) (AHead a1 a4) (AHead (AHead a a0) a2) H4) in 
-(eq_ind A (AHead a a0) (\lambda (a6: A).((eq A a4 a2) \to ((eq A (AHead a3 
-a5) (AHead a a0)) \to ((leq g a6 a3) \to ((leq g a4 a5) \to P))))) (\lambda 
-(H8: (eq A a4 a2)).(eq_ind A a2 (\lambda (a2: A).((eq A (AHead a3 a5) (AHead 
-a a0)) \to ((leq g (AHead a a0) a3) \to ((leq g a2 a5) \to P)))) (\lambda 
-(H9: (eq A (AHead a3 a5) (AHead a a0))).(let H10 \def (f_equal A A (\lambda 
-(e: A).(match e return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a5 | 
-(AHead _ a) \Rightarrow a])) (AHead a3 a5) (AHead a a0) H9) in ((let H11 \def 
-(f_equal A A (\lambda (e: A).(match e return (\lambda (_: A).A) with [(ASort 
-_ _) \Rightarrow a3 | (AHead a _) \Rightarrow a])) (AHead a3 a5) (AHead a a0) 
-H9) in (eq_ind A a (\lambda (a6: A).((eq A a5 a0) \to ((leq g (AHead a a0) 
-a6) \to ((leq g a2 a5) \to P)))) (\lambda (H12: (eq A a5 a0)).(eq_ind A a0 
-(\lambda (a6: A).((leq g (AHead a a0) a) \to ((leq g a2 a6) \to P))) (\lambda 
-(H13: (leq g (AHead a a0) a)).(\lambda (_: (leq g a2 a0)).(H a0 H13 P))) a5 
-(sym_eq A a5 a0 H12))) a3 (sym_eq A a3 a H11))) H10))) a4 (sym_eq A a4 a2 
-H8))) a1 (sym_eq A a1 (AHead a a0) H7))) H6)) H5 H2 H3)))]) in (H2 
-(refl_equal A (AHead (AHead a a0) a2)) (refl_equal A (AHead a a0))))))))))) 
-a1)).
-
-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).((match n return (\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)) with [O 
-\Rightarrow (\lambda (H0: (leq g (AHead (ASort O n0) a2) (ASort O (next g 
-n0)))).(let H1 \def (match H0 return (\lambda (a: A).(\lambda (a0: 
-A).(\lambda (_: (leq ? a a0)).((eq A a (AHead (ASort O n0) a2)) \to ((eq A a0 
-(ASort O (next g n0))) \to P))))) with [(leq_sort h1 h2 n1 n2 k H0) 
-\Rightarrow (\lambda (H1: (eq A (ASort h1 n1) (AHead (ASort O n0) 
-a2))).(\lambda (H2: (eq A (ASort h2 n2) (ASort O (next g n0)))).((let H3 \def 
-(eq_ind A (ASort h1 n1) (\lambda (e: A).(match e return (\lambda (_: A).Prop) 
-with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I 
-(AHead (ASort O n0) a2) H1) in (False_ind ((eq A (ASort h2 n2) (ASort O (next 
-g n0))) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to 
-P)) H3)) H2 H0))) | (leq_head a1 a0 H0 a3 a4 H1) \Rightarrow (\lambda (H2: 
-(eq A (AHead a1 a3) (AHead (ASort O n0) a2))).(\lambda (H3: (eq A (AHead a0 
-a4) (ASort O (next g n0)))).((let H4 \def (f_equal A A (\lambda (e: A).(match 
-e return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a3 | (AHead _ a) 
-\Rightarrow a])) (AHead a1 a3) (AHead (ASort O n0) a2) H2) in ((let H5 \def 
-(f_equal A A (\lambda (e: A).(match e return (\lambda (_: A).A) with [(ASort 
-_ _) \Rightarrow a1 | (AHead a _) \Rightarrow a])) (AHead a1 a3) (AHead 
-(ASort O n0) a2) H2) in (eq_ind A (ASort O n0) (\lambda (a: A).((eq A a3 a2) 
-\to ((eq A (AHead a0 a4) (ASort O (next g n0))) \to ((leq g a a0) \to ((leq g 
-a3 a4) \to P))))) (\lambda (H6: (eq A a3 a2)).(eq_ind A a2 (\lambda (a: 
-A).((eq A (AHead a0 a4) (ASort O (next g n0))) \to ((leq g (ASort O n0) a0) 
-\to ((leq g a a4) \to P)))) (\lambda (H7: (eq A (AHead a0 a4) (ASort O (next 
-g n0)))).(let H8 \def (eq_ind A (AHead a0 a4) (\lambda (e: A).(match e return 
-(\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) 
-\Rightarrow True])) I (ASort O (next g n0)) H7) in (False_ind ((leq g (ASort 
-O n0) a0) \to ((leq g a2 a4) \to P)) H8))) a3 (sym_eq A a3 a2 H6))) a1 
-(sym_eq A a1 (ASort O n0) H5))) H4)) H3 H0 H1)))]) in (H1 (refl_equal A 
-(AHead (ASort O n0) a2)) (refl_equal A (ASort O (next g n0)))))) | (S n1) 
-\Rightarrow (\lambda (H0: (leq g (AHead (ASort (S n1) n0) a2) (ASort n1 
-n0))).(let H1 \def (match H0 return (\lambda (a: A).(\lambda (a0: A).(\lambda 
-(_: (leq ? a a0)).((eq A a (AHead (ASort (S n1) n0) a2)) \to ((eq A a0 (ASort 
-n1 n0)) \to P))))) with [(leq_sort h1 h2 n1 n2 k H0) \Rightarrow (\lambda 
-(H1: (eq A (ASort h1 n1) (AHead (ASort (S n1) n0) a2))).(\lambda (H2: (eq A 
-(ASort h2 n2) (ASort n1 n0))).((let H3 \def (eq_ind A (ASort h1 n1) (\lambda 
-(e: A).(match e return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow 
-True | (AHead _ _) \Rightarrow False])) I (AHead (ASort (S n1) n0) a2) H1) in 
-(False_ind ((eq A (ASort h2 n2) (ASort n1 n0)) \to ((eq A (aplus g (ASort h1 
-n1) k) (aplus g (ASort h2 n2) k)) \to P)) H3)) H2 H0))) | (leq_head a1 a0 H0 
-a3 a4 H1) \Rightarrow (\lambda (H2: (eq A (AHead a1 a3) (AHead (ASort (S n1) 
-n0) a2))).(\lambda (H3: (eq A (AHead a0 a4) (ASort n1 n0))).((let H4 \def 
-(f_equal A A (\lambda (e: A).(match e return (\lambda (_: A).A) with [(ASort 
-_ _) \Rightarrow a3 | (AHead _ a) \Rightarrow a])) (AHead a1 a3) (AHead 
-(ASort (S n1) n0) a2) H2) in ((let H5 \def (f_equal A A (\lambda (e: 
-A).(match e return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a1 | 
-(AHead a _) \Rightarrow a])) (AHead a1 a3) (AHead (ASort (S n1) n0) a2) H2) 
-in (eq_ind A (ASort (S n1) n0) (\lambda (a: A).((eq A a3 a2) \to ((eq A 
-(AHead a0 a4) (ASort n1 n0)) \to ((leq g a a0) \to ((leq g a3 a4) \to P))))) 
-(\lambda (H6: (eq A a3 a2)).(eq_ind A a2 (\lambda (a: A).((eq A (AHead a0 a4) 
-(ASort n1 n0)) \to ((leq g (ASort (S n1) n0) a0) \to ((leq g a a4) \to P)))) 
-(\lambda (H7: (eq A (AHead a0 a4) (ASort n1 n0))).(let H8 \def (eq_ind A 
-(AHead a0 a4) (\lambda (e: A).(match e return (\lambda (_: A).Prop) with 
-[(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort n1 
-n0) H7) in (False_ind ((leq g (ASort (S n1) n0) a0) \to ((leq g a2 a4) \to 
-P)) H8))) a3 (sym_eq A a3 a2 H6))) a1 (sym_eq A a1 (ASort (S n1) n0) H5))) 
-H4)) H3 H0 H1)))]) in (H1 (refl_equal A (AHead (ASort (S n1) n0) a2)) 
-(refl_equal A (ASort n1 n0)))))]) 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 H2 \def (match H1 return (\lambda (a1: A).(\lambda (a3: 
-A).(\lambda (_: (leq ? a1 a3)).((eq A a1 (AHead (AHead a a0) a2)) \to ((eq A 
-a3 (AHead a (asucc g a0))) \to P))))) with [(leq_sort h1 h2 n1 n2 k H2) 
-\Rightarrow (\lambda (H3: (eq A (ASort h1 n1) (AHead (AHead a a0) 
-a2))).(\lambda (H4: (eq A (ASort h2 n2) (AHead a (asucc g a0)))).((let H5 
-\def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e return (\lambda (_: 
-A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow 
-False])) I (AHead (AHead a a0) a2) H3) in (False_ind ((eq A (ASort h2 n2) 
-(AHead a (asucc g a0))) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort 
-h2 n2) k)) \to P)) H5)) H4 H2))) | (leq_head a1 a3 H2 a4 a5 H3) \Rightarrow 
-(\lambda (H4: (eq A (AHead a1 a4) (AHead (AHead a a0) a2))).(\lambda (H5: (eq 
-A (AHead a3 a5) (AHead a (asucc g a0)))).((let H6 \def (f_equal A A (\lambda 
-(e: A).(match e return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a4 | 
-(AHead _ a) \Rightarrow a])) (AHead a1 a4) (AHead (AHead a a0) a2) H4) in 
-((let H7 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: A).A) 
-with [(ASort _ _) \Rightarrow a1 | (AHead a _) \Rightarrow a])) (AHead a1 a4) 
-(AHead (AHead a a0) a2) H4) in (eq_ind A (AHead a a0) (\lambda (a6: A).((eq A 
-a4 a2) \to ((eq A (AHead a3 a5) (AHead a (asucc g a0))) \to ((leq g a6 a3) 
-\to ((leq g a4 a5) \to P))))) (\lambda (H8: (eq A a4 a2)).(eq_ind A a2 
-(\lambda (a2: A).((eq A (AHead a3 a5) (AHead a (asucc g a0))) \to ((leq g 
-(AHead a a0) a3) \to ((leq g a2 a5) \to P)))) (\lambda (H9: (eq A (AHead a3 
-a5) (AHead a (asucc g a0)))).(let H10 \def (f_equal A A (\lambda (e: 
-A).(match e return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a5 | 
-(AHead _ a) \Rightarrow a])) (AHead a3 a5) (AHead a (asucc g a0)) H9) in 
-((let H11 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: 
-A).A) with [(ASort _ _) \Rightarrow a3 | (AHead a _) \Rightarrow a])) (AHead 
-a3 a5) (AHead a (asucc g a0)) H9) in (eq_ind A a (\lambda (a6: A).((eq A a5 
-(asucc g a0)) \to ((leq g (AHead a a0) a6) \to ((leq g a2 a5) \to P)))) 
-(\lambda (H12: (eq A a5 (asucc g a0))).(eq_ind A (asucc g a0) (\lambda (a6: 
-A).((leq g (AHead a a0) a) \to ((leq g a2 a6) \to P))) (\lambda (H13: (leq g 
-(AHead a a0) a)).(\lambda (_: (leq g a2 (asucc g a0))).(leq_ahead_false g a 
-a0 H13 P))) a5 (sym_eq A a5 (asucc g a0) H12))) a3 (sym_eq A a3 a H11))) 
-H10))) a4 (sym_eq A a4 a2 H8))) a1 (sym_eq A a1 (AHead a a0) H7))) H6)) H5 H2 
-H3)))]) in (H2 (refl_equal A (AHead (AHead a a0) a2)) (refl_equal A (AHead a 
-(asucc g a0)))))))))))) 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).((match n return 
-(\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)) with [O 
-\Rightarrow (\lambda (H0: (leq g (ASort O (next g n0)) (ASort O n0))).(let H1 
-\def (match H0 return (\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a 
-a0)).((eq A a (ASort O (next g n0))) \to ((eq A a0 (ASort O n0)) \to P))))) 
-with [(leq_sort h1 h2 n1 n2 k H0) \Rightarrow (\lambda (H1: (eq A (ASort h1 
-n1) (ASort O (next g n0)))).(\lambda (H2: (eq A (ASort h2 n2) (ASort O 
-n0))).((let H3 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda 
-(_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n1])) 
-(ASort h1 n1) (ASort O (next g n0)) H1) in ((let H4 \def (f_equal A nat 
-(\lambda (e: A).(match e return (\lambda (_: A).nat) with [(ASort n _) 
-\Rightarrow n | (AHead _ _) \Rightarrow h1])) (ASort h1 n1) (ASort O (next g 
-n0)) H1) in (eq_ind nat O (\lambda (n: nat).((eq nat n1 (next g n0)) \to ((eq 
-A (ASort h2 n2) (ASort O n0)) \to ((eq A (aplus g (ASort n n1) k) (aplus g 
-(ASort h2 n2) k)) \to P)))) (\lambda (H5: (eq nat n1 (next g n0))).(eq_ind 
-nat (next g n0) (\lambda (n: nat).((eq A (ASort h2 n2) (ASort O n0)) \to ((eq 
-A (aplus g (ASort O n) k) (aplus g (ASort h2 n2) k)) \to P))) (\lambda (H6: 
-(eq A (ASort h2 n2) (ASort O n0))).(let H7 \def (f_equal A nat (\lambda (e: 
-A).(match e return (\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | 
-(AHead _ _) \Rightarrow n2])) (ASort h2 n2) (ASort O n0) H6) in ((let H8 \def 
-(f_equal A nat (\lambda (e: A).(match e return (\lambda (_: A).nat) with 
-[(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow h2])) (ASort h2 n2) 
-(ASort O n0) H6) in (eq_ind nat O (\lambda (n: nat).((eq nat n2 n0) \to ((eq 
-A (aplus g (ASort O (next g n0)) k) (aplus g (ASort n n2) k)) \to P))) 
-(\lambda (H9: (eq nat n2 n0)).(eq_ind nat n0 (\lambda (n: nat).((eq A (aplus 
-g (ASort O (next g n0)) k) (aplus g (ASort O n) k)) \to P)) (\lambda (H10: 
-(eq A (aplus g (ASort O (next g n0)) k) (aplus g (ASort O n0) k))).(let H 
-\def (eq_ind_r A (aplus g (ASort O (next g n0)) k) (\lambda (a: A).(eq A a 
-(aplus g (ASort O n0) k))) H10 (aplus g (ASort O n0) (S k)) 
-(aplus_sort_O_S_simpl g n0 k)) in (let H_y \def (aplus_inj g (S k) k (ASort O 
-n0) H) in (le_Sx_x k (eq_ind_r nat k (\lambda (n: nat).(le n k)) (le_n k) (S 
-k) H_y) P)))) n2 (sym_eq nat n2 n0 H9))) h2 (sym_eq nat h2 O H8))) H7))) n1 
-(sym_eq nat n1 (next g n0) H5))) h1 (sym_eq nat h1 O H4))) H3)) H2 H0))) | 
-(leq_head a1 a2 H0 a3 a4 H1) \Rightarrow (\lambda (H2: (eq A (AHead a1 a3) 
-(ASort O (next g n0)))).(\lambda (H3: (eq A (AHead a2 a4) (ASort O 
-n0))).((let H4 \def (eq_ind A (AHead a1 a3) (\lambda (e: A).(match e return 
-(\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) 
-\Rightarrow True])) I (ASort O (next g n0)) H2) in (False_ind ((eq A (AHead 
-a2 a4) (ASort O n0)) \to ((leq g a1 a2) \to ((leq g a3 a4) \to P))) H4)) H3 
-H0 H1)))]) in (H1 (refl_equal A (ASort O (next g n0))) (refl_equal A (ASort O 
-n0))))) | (S n1) \Rightarrow (\lambda (H0: (leq g (ASort n1 n0) (ASort (S n1) 
-n0))).(let H1 \def (match H0 return (\lambda (a: A).(\lambda (a0: A).(\lambda 
-(_: (leq ? a a0)).((eq A a (ASort n1 n0)) \to ((eq A a0 (ASort (S n1) n0)) 
-\to P))))) with [(leq_sort h1 h2 n1 n2 k H0) \Rightarrow (\lambda (H1: (eq A 
-(ASort h1 n1) (ASort n1 n0))).(\lambda (H2: (eq A (ASort h2 n2) (ASort (S n1) 
-n0))).((let H3 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda 
-(_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n1])) 
-(ASort h1 n1) (ASort n1 n0) H1) in ((let H4 \def (f_equal A nat (\lambda (e: 
-A).(match e return (\lambda (_: A).nat) with [(ASort n _) \Rightarrow n | 
-(AHead _ _) \Rightarrow h1])) (ASort h1 n1) (ASort n1 n0) H1) in (eq_ind nat 
-n1 (\lambda (n: nat).((eq nat n1 n0) \to ((eq A (ASort h2 n2) (ASort (S n1) 
-n0)) \to ((eq A (aplus g (ASort n n1) k) (aplus g (ASort h2 n2) k)) \to P)))) 
-(\lambda (H5: (eq nat n1 n0)).(eq_ind nat n0 (\lambda (n: nat).((eq A (ASort 
-h2 n2) (ASort (S n1) n0)) \to ((eq A (aplus g (ASort n1 n) k) (aplus g (ASort 
-h2 n2) k)) \to P))) (\lambda (H6: (eq A (ASort h2 n2) (ASort (S n1) 
-n0))).(let H7 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda 
-(_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n2])) 
-(ASort h2 n2) (ASort (S n1) n0) H6) in ((let H8 \def (f_equal A nat (\lambda 
-(e: A).(match e return (\lambda (_: A).nat) with [(ASort n _) \Rightarrow n | 
-(AHead _ _) \Rightarrow h2])) (ASort h2 n2) (ASort (S n1) n0) H6) in (eq_ind 
-nat (S n1) (\lambda (n: nat).((eq nat n2 n0) \to ((eq A (aplus g (ASort n1 
-n0) k) (aplus g (ASort n n2) k)) \to P))) (\lambda (H9: (eq nat n2 
-n0)).(eq_ind nat n0 (\lambda (n: nat).((eq A (aplus g (ASort n1 n0) k) (aplus 
-g (ASort (S n1) n) k)) \to P)) (\lambda (H10: (eq A (aplus g (ASort n1 n0) k) 
-(aplus g (ASort (S n1) n0) k))).(let H \def (eq_ind_r A (aplus g (ASort n1 
-n0) k) (\lambda (a: A).(eq A a (aplus g (ASort (S n1) n0) k))) H10 (aplus g 
-(ASort (S n1) n0) (S k)) (aplus_sort_S_S_simpl g n0 n1 k)) in (let H_y \def 
-(aplus_inj g (S k) k (ASort (S n1) n0) H) in (le_Sx_x k (eq_ind_r nat k 
-(\lambda (n: nat).(le n k)) (le_n k) (S k) H_y) P)))) n2 (sym_eq nat n2 n0 
-H9))) h2 (sym_eq nat h2 (S n1) H8))) H7))) n1 (sym_eq nat n1 n0 H5))) h1 
-(sym_eq nat h1 n1 H4))) H3)) H2 H0))) | (leq_head a1 a2 H0 a3 a4 H1) 
-\Rightarrow (\lambda (H2: (eq A (AHead a1 a3) (ASort n1 n0))).(\lambda (H3: 
-(eq A (AHead a2 a4) (ASort (S n1) n0))).((let H4 \def (eq_ind A (AHead a1 a3) 
-(\lambda (e: A).(match e return (\lambda (_: A).Prop) with [(ASort _ _) 
-\Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort n1 n0) H2) in 
-(False_ind ((eq A (AHead a2 a4) (ASort (S n1) n0)) \to ((leq g a1 a2) \to 
-((leq g a3 a4) \to P))) H4)) H3 H0 H1)))]) in (H1 (refl_equal A (ASort n1 
-n0)) (refl_equal A (ASort (S n1) n0)))))]) 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 H2 \def (match H1 return (\lambda (a: 
-A).(\lambda (a2: A).(\lambda (_: (leq ? a a2)).((eq A a (AHead a0 (asucc g 
-a1))) \to ((eq A a2 (AHead a0 a1)) \to P))))) with [(leq_sort h1 h2 n1 n2 k 
-H2) \Rightarrow (\lambda (H3: (eq A (ASort h1 n1) (AHead a0 (asucc g 
-a1)))).(\lambda (H4: (eq A (ASort h2 n2) (AHead a0 a1))).((let H5 \def 
-(eq_ind A (ASort h1 n1) (\lambda (e: A).(match e return (\lambda (_: A).Prop) 
-with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I 
-(AHead a0 (asucc g a1)) H3) in (False_ind ((eq A (ASort h2 n2) (AHead a0 a1)) 
-\to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to P)) H5)) 
-H4 H2))) | (leq_head a1 a2 H2 a3 a4 H3) \Rightarrow (\lambda (H4: (eq A 
-(AHead a1 a3) (AHead a0 (asucc g a1)))).(\lambda (H5: (eq A (AHead a2 a4) 
-(AHead a0 a1))).((let H6 \def (f_equal A A (\lambda (e: A).(match e return 
-(\lambda (_: A).A) with [(ASort _ _) \Rightarrow a3 | (AHead _ a) \Rightarrow 
-a])) (AHead a1 a3) (AHead a0 (asucc g a1)) H4) in ((let H7 \def (f_equal A A 
-(\lambda (e: A).(match e return (\lambda (_: A).A) with [(ASort _ _) 
-\Rightarrow a1 | (AHead a _) \Rightarrow a])) (AHead a1 a3) (AHead a0 (asucc 
-g a1)) H4) in (eq_ind A a0 (\lambda (a: A).((eq A a3 (asucc g a1)) \to ((eq A 
-(AHead a2 a4) (AHead a0 a1)) \to ((leq g a a2) \to ((leq g a3 a4) \to P))))) 
-(\lambda (H8: (eq A a3 (asucc g a1))).(eq_ind A (asucc g a1) (\lambda (a: 
-A).((eq A (AHead a2 a4) (AHead a0 a1)) \to ((leq g a0 a2) \to ((leq g a a4) 
-\to P)))) (\lambda (H9: (eq A (AHead a2 a4) (AHead a0 a1))).(let H10 \def 
-(f_equal A A (\lambda (e: A).(match e return (\lambda (_: A).A) with [(ASort 
-_ _) \Rightarrow a4 | (AHead _ a) \Rightarrow a])) (AHead a2 a4) (AHead a0 
-a1) H9) in ((let H11 \def (f_equal A A (\lambda (e: A).(match e return 
-(\lambda (_: A).A) with [(ASort _ _) \Rightarrow a2 | (AHead a _) \Rightarrow 
-a])) (AHead a2 a4) (AHead a0 a1) H9) in (eq_ind A a0 (\lambda (a: A).((eq A 
-a4 a1) \to ((leq g a0 a) \to ((leq g (asucc g a1) a4) \to P)))) (\lambda 
-(H12: (eq A a4 a1)).(eq_ind A a1 (\lambda (a: A).((leq g a0 a0) \to ((leq g 
-(asucc g a1) a) \to P))) (\lambda (_: (leq g a0 a0)).(\lambda (H14: (leq g 
-(asucc g a1) a1)).(H0 H14 P))) a4 (sym_eq A a4 a1 H12))) a2 (sym_eq A a2 a0 
-H11))) H10))) a3 (sym_eq A a3 (asucc g a1) H8))) a1 (sym_eq A a1 a0 H7))) 
-H6)) H5 H2 H3)))]) in (H2 (refl_equal A (AHead a0 (asucc g a1))) (refl_equal 
-A (AHead a0 a1)))))))))) a)).
-
-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))).
-
-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_S_n (S (lweight a1)) (S (plus (lweight 
-a1) (lweight a2))) (le_n_S (S (lweight a1)) (S (plus (lweight a1) (lweight 
-a2))) (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_S_n (S (lweight a2)) (S (plus (lweight 
-a1) (lweight a2))) (le_n_S (S (lweight a2)) (S (plus (lweight a1) (lweight 
-a2))) (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 (P: ((A \to 
-Prop))).(\lambda (n0: nat).(\forall (a: A).((eq nat (lweight a) n0) \to (P 
-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 (a: A).(P a)) m))))).(\lambda (a0: A).(\lambda (H1: (eq nat 
-(lweight a0) n0)).(let H2 \def (eq_ind_r nat n0 (\lambda (n: nat).(\forall 
-(m: nat).((lt m n) \to (\forall (a: A).((eq nat (lweight a) m) \to (P a)))))) 
-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)))).
-
-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))))).
-
-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 H2 \def (match H1 return 
-(\lambda (n: nat).(\lambda (a: A).(\lambda (a0: A).(\lambda (_: (aprem n a 
-a0)).((eq nat n i) \to ((eq A a (ASort h2 n2)) \to ((eq A a0 b2) \to (ex2 A 
-(\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem i (ASort h1 n1) 
-b1)))))))))) with [(aprem_zero a0 a3) \Rightarrow (\lambda (H1: (eq nat O 
-i)).(\lambda (H2: (eq A (AHead a0 a3) (ASort h2 n2))).(\lambda (H3: (eq A a0 
-b2)).(eq_ind nat O (\lambda (n: nat).((eq A (AHead a0 a3) (ASort h2 n2)) \to 
-((eq A a0 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: 
-A).(aprem n (ASort h1 n1) b1)))))) (\lambda (H4: (eq A (AHead a0 a3) (ASort 
-h2 n2))).(let H5 \def (eq_ind A (AHead a0 a3) (\lambda (e: A).(match e return 
-(\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) 
-\Rightarrow True])) I (ASort h2 n2) H4) in (False_ind ((eq A a0 b2) \to (ex2 
-A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem O (ASort h1 n1) 
-b1)))) H5))) i H1 H2 H3)))) | (aprem_succ a0 a i0 H1 a3) \Rightarrow (\lambda 
-(H2: (eq nat (S i0) i)).(\lambda (H3: (eq A (AHead a3 a0) (ASort h2 
-n2))).(\lambda (H4: (eq A a b2)).(eq_ind nat (S i0) (\lambda (n: nat).((eq A 
-(AHead a3 a0) (ASort h2 n2)) \to ((eq A a b2) \to ((aprem i0 a0 a) \to (ex2 A 
-(\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem n (ASort h1 n1) 
-b1))))))) (\lambda (H5: (eq A (AHead a3 a0) (ASort h2 n2))).(let H6 \def 
-(eq_ind A (AHead a3 a0) (\lambda (e: A).(match e return (\lambda (_: A).Prop) 
-with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I 
-(ASort h2 n2) H5) in (False_ind ((eq A a b2) \to ((aprem i0 a0 a) \to (ex2 A 
-(\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S i0) (ASort h1 n1) 
-b1))))) H6))) i H2 H3 H4 H1))))]) in (H2 (refl_equal nat i) (refl_equal A 
-(ASort h2 n2)) (refl_equal A b2)))))))))))) (\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)).((match i return (\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))))) with [O \Rightarrow (\lambda (H5: (aprem O (AHead a3 a5) 
-b2)).(let H6 \def (match H5 return (\lambda (n: nat).(\lambda (a: A).(\lambda 
-(a1: A).(\lambda (_: (aprem n a a1)).((eq nat n O) \to ((eq A a (AHead a3 
-a5)) \to ((eq A a1 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda 
-(b1: A).(aprem O (AHead a0 a4) b1)))))))))) with [(aprem_zero a6 a7) 
-\Rightarrow (\lambda (_: (eq nat O O)).(\lambda (H5: (eq A (AHead a6 a7) 
-(AHead a3 a5))).(\lambda (H6: (eq A a6 b2)).((let H7 \def (f_equal A A 
-(\lambda (e: A).(match e return (\lambda (_: A).A) with [(ASort _ _) 
-\Rightarrow a7 | (AHead _ a) \Rightarrow a])) (AHead a6 a7) (AHead a3 a5) H5) 
-in ((let H8 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: 
-A).A) with [(ASort _ _) \Rightarrow a6 | (AHead a _) \Rightarrow a])) (AHead 
-a6 a7) (AHead a3 a5) H5) in (eq_ind A a3 (\lambda (a: A).((eq A a7 a5) \to 
-((eq A a b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: 
-A).(aprem O (AHead a0 a4) b1)))))) (\lambda (H9: (eq A a7 a5)).(eq_ind A a5 
-(\lambda (_: A).((eq A a3 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) 
-(\lambda (b1: A).(aprem O (AHead a0 a4) b1))))) (\lambda (H10: (eq A a3 
-b2)).(eq_ind A b2 (\lambda (_: A).(ex2 A (\lambda (b1: A).(leq g b1 b2)) 
-(\lambda (b1: A).(aprem O (AHead a0 a4) b1)))) (eq_ind 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 H10) a3 (sym_eq A 
-a3 b2 H10))) a7 (sym_eq A a7 a5 H9))) a6 (sym_eq A a6 a3 H8))) H7)) H6)))) | 
-(aprem_succ a6 a i H4 a7) \Rightarrow (\lambda (H5: (eq nat (S i) 
-O)).(\lambda (H6: (eq A (AHead a7 a6) (AHead a3 a5))).(\lambda (H7: (eq A a 
-b2)).((let H8 \def (eq_ind nat (S i) (\lambda (e: nat).(match e return 
-(\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) 
-I O H5) in (False_ind ((eq A (AHead a7 a6) (AHead a3 a5)) \to ((eq A a b2) 
-\to ((aprem i a6 a) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: 
-A).(aprem O (AHead a0 a4) b1)))))) H8)) H6 H7 H4))))]) in (H6 (refl_equal nat 
-O) (refl_equal A (AHead a3 a5)) (refl_equal A b2)))) | (S n) \Rightarrow 
-(\lambda (H5: (aprem (S n) (AHead a3 a5) b2)).(let H6 \def (match H5 return 
-(\lambda (n0: nat).(\lambda (a: A).(\lambda (a1: A).(\lambda (_: (aprem n0 a 
-a1)).((eq nat n0 (S n)) \to ((eq A a (AHead a3 a5)) \to ((eq A a1 b2) \to 
-(ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S n) (AHead 
-a0 a4) b1)))))))))) with [(aprem_zero a6 a7) \Rightarrow (\lambda (H4: (eq 
-nat O (S n))).(\lambda (H5: (eq A (AHead a6 a7) (AHead a3 a5))).(\lambda (H6: 
-(eq A a6 b2)).((let H7 \def (eq_ind nat O (\lambda (e: nat).(match e return 
-(\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) 
-I (S n) H4) in (False_ind ((eq A (AHead a6 a7) (AHead a3 a5)) \to ((eq A a6 
-b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S n) 
-(AHead a0 a4) b1))))) H7)) H5 H6)))) | (aprem_succ a6 a i H4 a7) \Rightarrow 
-(\lambda (H5: (eq nat (S i) (S n))).(\lambda (H6: (eq A (AHead a7 a6) (AHead 
-a3 a5))).(\lambda (H7: (eq A a b2)).((let H8 \def (f_equal nat nat (\lambda 
-(e: nat).(match e return (\lambda (_: nat).nat) with [O \Rightarrow i | (S n) 
-\Rightarrow n])) (S i) (S n) H5) in (eq_ind nat n (\lambda (n0: nat).((eq A 
-(AHead a7 a6) (AHead a3 a5)) \to ((eq A a b2) \to ((aprem n0 a6 a) \to (ex2 A 
-(\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S n) (AHead a0 a4) 
-b1))))))) (\lambda (H9: (eq A (AHead a7 a6) (AHead a3 a5))).(let H10 \def 
-(f_equal A A (\lambda (e: A).(match e return (\lambda (_: A).A) with [(ASort 
-_ _) \Rightarrow a6 | (AHead _ a) \Rightarrow a])) (AHead a7 a6) (AHead a3 
-a5) H9) in ((let H11 \def (f_equal A A (\lambda (e: A).(match e return 
-(\lambda (_: A).A) with [(ASort _ _) \Rightarrow a7 | (AHead a _) \Rightarrow 
-a])) (AHead a7 a6) (AHead a3 a5) H9) in (eq_ind A a3 (\lambda (_: A).((eq A 
-a6 a5) \to ((eq A a b2) \to ((aprem n a6 a) \to (ex2 A (\lambda (b1: A).(leq 
-g b1 b2)) (\lambda (b1: A).(aprem (S n) (AHead a0 a4) b1))))))) (\lambda 
-(H12: (eq A a6 a5)).(eq_ind A a5 (\lambda (a1: A).((eq A a b2) \to ((aprem n 
-a1 a) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S 
-n) (AHead a0 a4) b1)))))) (\lambda (H13: (eq A a b2)).(eq_ind A b2 (\lambda 
-(a1: A).((aprem n a5 a1) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda 
-(b1: A).(aprem (S n) (AHead a0 a4) b1))))) (\lambda (H14: (aprem n a5 
-b2)).(let H_x \def (H3 n b2 H14) in (let H3 \def H_x in (ex2_ind A (\lambda 
-(b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem n a4 b1)) (ex2 A (\lambda (b1: 
-A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S n) (AHead a0 a4) b1))) (\lambda 
-(x: A).(\lambda (H15: (leq g x b2)).(\lambda (H16: (aprem n a4 x)).(ex_intro2 
-A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S n) (AHead a0 a4) 
-b1)) x H15 (aprem_succ a4 x n H16 a0))))) H3)))) a (sym_eq A a b2 H13))) a6 
-(sym_eq A a6 a5 H12))) a7 (sym_eq A a7 a3 H11))) H10))) i (sym_eq nat i n 
-H8))) H6 H7 H4))))]) in (H6 (refl_equal nat (S n)) (refl_equal A (AHead a3 
-a5)) (refl_equal A b2))))]) 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))))).
-
-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))))))).
-
-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))).(ex2_ind nat (\lambda (n1: nat).(eq nat (S k0) (S 
-n1))) (\lambda (n1: nat).(le n n1)) (eq A (asucc gz (aplus gz (ASort (S n) 
-n0) k0)) (ASort O (plus (minus k0 n) n0))) (\lambda (x: nat).(\lambda (H1: 
-(eq nat (S k0) (S x))).(\lambda (H2: (le n x)).(let H3 \def (f_equal nat nat 
-(\lambda (e: nat).(match e return (\lambda (_: nat).nat) with [O \Rightarrow 
-k0 | (S n) \Rightarrow n])) (S k0) (S x) H1) in (let H4 \def (eq_ind_r nat x 
-(\lambda (n0: nat).(le n n0)) H2 k0 H3) 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 H4))))))) (le_gen_S n (S k0) 
-H0)))))) 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 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))).(ex2_ind nat (\lambda (n1: nat).(eq 
-nat (S n0) (S n1))) (\lambda (n1: nat).(le k0 n1)) (eq A (asucc gz (aplus gz 
-(ASort (S n0) n) k0)) (ASort (minus n0 k0) n)) (\lambda (x: nat).(\lambda 
-(H1: (eq nat (S n0) (S x))).(\lambda (H2: (le k0 x)).(let H3 \def (f_equal 
-nat nat (\lambda (e: nat).(match e return (\lambda (_: nat).nat) with [O 
-\Rightarrow n0 | (S n) \Rightarrow n])) (S n0) (S x) H1) in (let H4 \def 
-(eq_ind_r nat x (\lambda (n: nat).(le k0 n)) H2 n0 H3) 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 H4))))))) (le_gen_S k0 (S n0) H0))))) 
-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 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 k) \Rightarrow (match m with [O \Rightarrow (S k) | (S l) \Rightarrow 
-(minus k 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 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 return (\lambda 
-(_: A).Prop) with [(ASort n _) \Rightarrow (match n 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_equal 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 return (\lambda (_: A).Prop) with [(ASort n _) 
-\Rightarrow (match n 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 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))).
-
-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)))))).
-
-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)) (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 return (\lambda (_: T).nat) with [(TSort n) 
-\Rightarrow n | (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 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 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 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 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 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 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 (t: T).((eq T t (TSort n)) \to (leq g a1 
-(ASort O n)))) H2 (TSort n) H5) in (let H7 \def (eq_ind T t (\lambda (t: 
-T).(arity g c0 t 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)) (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 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 (u: T).(getl i d (CHead d0 (Bind Abbr) u)))) (\lambda (d: 
-C).(\lambda (u: T).(arity g d u a0)))) (ex2_2 C T (\lambda (d0: C).(\lambda 
-(u: T).(getl i d (CHead d0 (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: 
-T).(arity g d u (asucc g a0))))))))).(\lambda (H4: (eq T (TLRef i0) (TLRef 
-i))).(let H5 \def (f_equal T nat (\lambda (e: T).(match e 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 (u: T).(getl i d (CHead d0 
-(Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g 
-a0))))) (ex2_2 C T (\lambda (d0: C).(\lambda (u: T).(getl i d (CHead d0 (Bind 
-Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u (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 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 (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 (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 (u: 
-T).(arity g d u 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 
-(u: T).(arity g d u (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 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 (u: T).(getl i c0 (CHead d (Bind Abbr) 
-u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g 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 (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 (u: T).(arity g d u 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 (u: T).(arity g d u (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 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 (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 (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 (u: T).(getl i c0 
-(CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u 
-(AHead a1 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 (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 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 (u: T).(getl i c0 (CHead d (Bind Abbr) u)))) 
-(\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g 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 (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 (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 (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 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 (t: T).((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)))))))) H2 (TLRef i) H5) in (let H7 \def (eq_ind 
-T t (\lambda (t: T).(arity g c0 t 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)) (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 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 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 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 (a1: A).(arity g c0 u a1)) (\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 (a1: A).(arity g (CHead c0 (Bind b0) u0) u a1)) 
-(\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 return (\lambda (_: 
-T).B) with [(TSort _) \Rightarrow b0 | (TLRef _) \Rightarrow b0 | (THead k _ 
-_) \Rightarrow (match k return (\lambda (_: K).B) with [(Bind b) \Rightarrow 
-b | (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 return (\lambda 
-(_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead 
-_ t _) \Rightarrow t])) (THead (Bind b0) u0 t0) (THead (Bind b) u t) H7) in 
-((let H10 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: 
-T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ 
-t) \Rightarrow t])) (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 (t0: T).((eq T t0 (THead (Bind b) u t)) \to (ex2 A (\lambda 
-(a1: A).(arity g (CHead c0 (Bind b0) u0) u a1)) (\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 (t: T).(arity g (CHead c0 (Bind b0) u0) t a0)) H5 
-t H10) in (let H15 \def (eq_ind T u0 (\lambda (t0: T).((eq T t (THead (Bind 
-b) u t)) \to (ex2 A (\lambda (a1: A).(arity g (CHead c0 (Bind b0) t0) u a1)) 
-(\lambda (_: A).(arity g (CHead (CHead c0 (Bind b0) t0) (Bind b) u) t a0))))) 
-H13 u H11) in (let H16 \def (eq_ind T u0 (\lambda (t0: T).(arity g (CHead c0 
-(Bind b0) t0) t a0)) H14 u H11) in (let H17 \def (eq_ind T u0 (\lambda (t0: 
-T).((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 a1))))) H4 u H11) in 
-(let H18 \def (eq_ind T u0 (\lambda (t: T).(arity g c0 t a1)) H3 u H11) in 
-(let H19 \def (eq_ind B b0 (\lambda (b0: B).((eq T t (THead (Bind b) u t)) 
-\to (ex2 A (\lambda (a1: A).(arity g (CHead c0 (Bind b0) u) u a1)) (\lambda 
-(_: A).(arity g (CHead (CHead c0 (Bind b0) u) (Bind b) u) t a0))))) H15 b 
-H12) in (let H20 \def (eq_ind B b0 (\lambda (b: B).(arity g (CHead c0 (Bind 
-b) u) t a0)) H16 b H12) in (let H21 \def (eq_ind B b0 (\lambda (b: B).(not 
-(eq B b 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 (a1: A).(arity g c0 u a1)) (\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 (a1: 
-A).(arity g (CHead c0 (Bind Abst) u0) u a1)) (\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 return (\lambda (_: T).B) with [(TSort _) \Rightarrow Abst | 
-(TLRef _) \Rightarrow Abst | (THead k _ _) \Rightarrow (match k return 
-(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (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 return (\lambda (_: T).T) with [(TSort 
-_) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) 
-(THead (Bind Abst) u0 t0) (THead (Bind b) u t) H6) in ((let H9 \def (f_equal 
-T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) 
-(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 (t0: 
-T).((eq T t0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a1: A).(arity g 
-(CHead c0 (Bind Abst) u0) u a1)) (\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 (t: T).(arity g (CHead c0 (Bind Abst) u0) t a0)) H4 t H9) in (let 
-H14 \def (eq_ind T u0 (\lambda (t0: T).((eq T t (THead (Bind b) u t)) \to 
-(ex2 A (\lambda (a1: A).(arity g (CHead c0 (Bind Abst) t0) u a1)) (\lambda 
-(_: A).(arity g (CHead (CHead c0 (Bind Abst) t0) (Bind b) u) t a0))))) H12 u 
-H10) in (let H15 \def (eq_ind T u0 (\lambda (t0: T).(arity g (CHead c0 (Bind 
-Abst) t0) t a0)) H13 u H10) in (let H16 \def (eq_ind T u0 (\lambda (t0: 
-T).((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 (asucc g a1)))))) H3 u 
-H10) in (let H17 \def (eq_ind T u0 (\lambda (t: T).(arity g c0 t (asucc g 
-a1))) H2 u H10) in (let H18 \def (eq_ind_r B b (\lambda (b: B).((eq T t 
-(THead (Bind b) u t)) \to (ex2 A (\lambda (a1: A).(arity g (CHead c0 (Bind 
-Abst) u) u a1)) (\lambda (_: A).(arity g (CHead (CHead c0 (Bind Abst) u) 
-(Bind b) u) t a0))))) H14 Abst H11) in (let H19 \def (eq_ind_r B b (\lambda 
-(b: B).((eq T u (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 a1)))))) H16 
-Abst H11) in (let H20 \def (eq_ind_r B b (\lambda (b: B).(not (eq B b 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)) return (\lambda (_: 
-False).(ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\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 (a1: A).(arity g c0 u a1)) (\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 (a1: A).(arity g c0 u a1)) (\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 return (\lambda (_: 
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
-(THead k _ _) \Rightarrow (match 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 return (\lambda (_: 
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
-(THead k _ _) \Rightarrow (match 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 (a1: 
-A).(arity g c0 u a1)) (\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 (t0: T).((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 a1))))) H3 (THead (Bind b) u t) H6) 
-in (let H8 \def (eq_ind T t0 (\lambda (t: T).(arity g c0 t 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)) (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 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 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 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 (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 (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 (a1: A).(\lambda (a3: A).(eq A a2 (AHead a1 a3)))) (\lambda (a1: 
-A).(\lambda (_: A).(arity g (CHead c0 (Bind b) u0) u (asucc g a1)))) (\lambda 
-(_: A).(\lambda (a2: A).(arity g (CHead (CHead c0 (Bind b) u0) (Bind Abst) u) 
-t a2))))))).(\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 return (\lambda (_: 
-T).B) with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) 
-\Rightarrow (match k return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | 
-(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 return (\lambda 
-(_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead 
-_ t _) \Rightarrow t])) (THead (Bind b) u0 t0) (THead (Bind Abst) u t) H6) in 
-((let H9 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) 
-with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) 
-\Rightarrow t])) (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 (t0: T).((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A 
-A (\lambda (a1: A).(\lambda (a3: A).(eq A a2 (AHead a1 a3)))) (\lambda (a1: 
-A).(\lambda (_: A).(arity g (CHead c0 (Bind b) u0) u (asucc g a1)))) (\lambda 
-(_: A).(\lambda (a2: A).(arity g (CHead (CHead c0 (Bind b) u0) (Bind Abst) u) 
-t a2)))))) H5 t H9) in (let H13 \def (eq_ind T t0 (\lambda (t: T).(arity g 
-(CHead c0 (Bind b) u0) t a2)) H4 t H9) in (let H14 \def (eq_ind T u0 (\lambda 
-(t0: T).((eq T t (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a1: 
-A).(\lambda (a3: A).(eq A a2 (AHead a1 a3)))) (\lambda (a1: A).(\lambda (_: 
-A).(arity g (CHead c0 (Bind b) t0) u (asucc g a1)))) (\lambda (_: A).(\lambda 
-(a2: A).(arity g (CHead (CHead c0 (Bind b) t0) (Bind Abst) u) t a2)))))) H12 
-u H10) in (let H15 \def (eq_ind T u0 (\lambda (t0: T).(arity g (CHead c0 
-(Bind b) t0) t a2)) H13 u H10) in (let H16 \def (eq_ind T u0 (\lambda (t0: 
-T).((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 (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)))))) H3 u H10) in (let H17 \def (eq_ind T u0 
-(\lambda (t: T).(arity g c0 t a1)) H2 u H10) in (let H18 \def (eq_ind B b 
-(\lambda (b: B).((eq T t (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda 
-(a1: A).(\lambda (a3: A).(eq A a2 (AHead a1 a3)))) (\lambda (a1: A).(\lambda 
-(_: A).(arity g (CHead c0 (Bind b) u) u (asucc g a1)))) (\lambda (_: 
-A).(\lambda (a2: A).(arity g (CHead (CHead c0 (Bind b) u) (Bind Abst) u) t 
-a2)))))) H14 Abst H11) in (let H19 \def (eq_ind B b (\lambda (b: B).(arity g 
-(CHead c0 (Bind b) u) t a2)) H15 Abst H11) in (let H20 \def (eq_ind B b 
-(\lambda (b: B).(not (eq B b Abst))) H1 Abst H11) in (let H21 \def (match 
-(H20 (refl_equal B Abst)) return (\lambda (_: False).(ex3_2 A A (\lambda (a1: 
-A).(\lambda (a3: A).(eq A a2 (AHead a1 a3)))) (\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))))) 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 (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 (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 (a1: A).(\lambda (a3: A).(eq A a2 (AHead a1 a3)))) 
-(\lambda (a1: A).(\lambda (_: A).(arity g (CHead c0 (Bind Abst) u0) u (asucc 
-g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g (CHead (CHead c0 (Bind 
-Abst) u0) (Bind Abst) u) t a2))))))).(\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 return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef 
-_) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead (Bind Abst) u0 t0) 
-(THead (Bind Abst) u t) H5) in ((let H7 \def (f_equal T T (\lambda (e: 
-T).(match e return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef 
-_) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (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 (t0: T).((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A 
-A (\lambda (a1: A).(\lambda (a3: A).(eq A a2 (AHead a1 a3)))) (\lambda (a1: 
-A).(\lambda (_: A).(arity g (CHead c0 (Bind Abst) u0) u (asucc g a1)))) 
-(\lambda (_: A).(\lambda (a2: A).(arity g (CHead (CHead c0 (Bind Abst) u0) 
-(Bind Abst) u) t a2)))))) H4 t H7) in (let H10 \def (eq_ind T t0 (\lambda (t: 
-T).(arity g (CHead c0 (Bind Abst) u0) t a2)) H3 t H7) in (let H11 \def 
-(eq_ind T u0 (\lambda (t0: T).((eq T t (THead (Bind Abst) u t)) \to (ex3_2 A 
-A (\lambda (a1: A).(\lambda (a3: A).(eq A a2 (AHead a1 a3)))) (\lambda (a1: 
-A).(\lambda (_: A).(arity g (CHead c0 (Bind Abst) t0) u (asucc g a1)))) 
-(\lambda (_: A).(\lambda (a2: A).(arity g (CHead (CHead c0 (Bind Abst) t0) 
-(Bind Abst) u) t a2)))))) H9 u H8) in (let H12 \def (eq_ind T u0 (\lambda 
-(t0: T).(arity g (CHead c0 (Bind Abst) t0) t a2)) H10 u H8) in (let H13 \def 
-(eq_ind T u0 (\lambda (t0: T).((eq T t0 (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 (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 u H8) in 
-(let H14 \def (eq_ind T u0 (\lambda (t: T).(arity g c0 t (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 (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 (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 (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 Appl) 
-u0 t0) (THead (Bind Abst) u t))).(let H6 \def (eq_ind T (THead (Flat Appl) u0 
-t0) (\lambda (ee: T).(match ee return (\lambda (_: T).Prop) with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
-(match 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 return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match 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 (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 (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 (t0: T).((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 (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 (THead (Bind Abst) u 
-t) H5) in (let H7 \def (eq_ind T t0 (\lambda (t: T).(arity g c0 t 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 (a: A).(leq g a a2)) H3 (AHead x0 
-x1) H9) in (let H13 \def (eq_ind A a1 (\lambda (a: A).(arity g c0 (THead 
-(Bind Abst) u t) a)) H7 (AHead x0 x1) H9) in (let H_x \def (leq_gen_head g x0 
-x1 a2 H12) in (let H14 \def H_x in (ex3_2_ind A A (\lambda (a3: A).(\lambda 
-(a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(leq g x0 
-a3))) (\lambda (_: A).(\lambda (a4: A).(leq g x1 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: (eq A a2 (AHead x2 x3))).(\lambda (H16: (leq g x0 
-x2)).(\lambda (H17: (leq g x1 x3)).(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 H16)) (arity_repl g (CHead c0 (Bind Abst) u) t x1 H11 x3 
-H17)) a2 H15)))))) 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)) (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 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 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 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 
-(a1: A).(arity g c0 u a1)) (\lambda (a2: A).(arity g c0 t (AHead a2 
-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 (a1: A).(arity g (CHead c0 (Bind b) u0) u a1)) (\lambda (a1: 
-A).(arity g (CHead c0 (Bind b) u0) t (AHead a1 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 return (\lambda (_: 
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
-(THead k _ _) \Rightarrow (match 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 (a1: A).(arity g 
-c0 u a1)) (\lambda (a2: A).(arity g c0 t (AHead a2 (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 (a1: A).(arity g (CHead c0 (Bind Abst) u0) u a1)) (\lambda 
-(a1: A).(arity g (CHead c0 (Bind Abst) u0) t (AHead a1 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 return 
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match 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 (a1: A).(arity g c0 u a1)) (\lambda (a2: A).(arity g c0 t 
-(AHead a2 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 (a1: A).(arity g c0 u a1)) (\lambda (a2: A).(arity g c0 t 
-(AHead a2 (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 
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) 
-\Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) u0 t0) 
-(THead (Flat Appl) u t) H5) in ((let H7 \def (f_equal T T (\lambda (e: 
-T).(match e return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef 
-_) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (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 (t0: T).((eq T t0 (THead (Flat Appl) u t)) \to (ex2 A 
-(\lambda (a1: A).(arity g c0 u a1)) (\lambda (a2: A).(arity g c0 t (AHead a2 
-(AHead a1 a0))))))) H4 t H7) in (let H10 \def (eq_ind T t0 (\lambda (t: 
-T).(arity g c0 t (AHead a1 a0))) H3 t H7) in (let H11 \def (eq_ind T u0 
-(\lambda (t0: T).((eq T t0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda (a1: 
-A).(arity g c0 u a1)) (\lambda (a2: A).(arity g c0 t (AHead a2 a1)))))) H2 u 
-H8) in (let H12 \def (eq_ind T u0 (\lambda (t: T).(arity g c0 t 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 return (\lambda (_: T).Prop) 
-with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ 
-_) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind _) 
-\Rightarrow False | (Flat f) \Rightarrow (match 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 
-(a1: A).(arity g c0 u a1)) (\lambda (a2: A).(arity g c0 t (AHead a2 
-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 (t0: 
-T).((eq T t0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda (a1: A).(arity g c0 
-u a1)) (\lambda (a2: A).(arity g c0 t (AHead a2 a1)))))) H2 (THead (Flat 
-Appl) u t) H5) in (let H7 \def (eq_ind T t0 (\lambda (t: T).(arity g c0 t 
-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)) (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 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 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 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 
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match 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 return (\lambda (_: 
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
-(THead k _ _) \Rightarrow (match 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 return 
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: 
-K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match 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 return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) 
-(THead (Flat Cast) u0 t0) (THead (Flat Cast) u t) H5) in ((let H7 \def 
-(f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort 
-_) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) 
-(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 (t0: T).((eq T t0 (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 (t: T).(arity g c0 t a0)) H3 t H7) 
-in (let H11 \def (eq_ind T u0 (\lambda (t0: T).((eq T t0 (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 (t: T).(arity g c0 t 
-(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 (t0: 
-T).((eq T t0 (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 (t: T).(arity g c0 t a1)) H1 (THead (Flat Cast) u t) H5) in (let 
-H8 \def (H6 (refl_equal T (THead (Flat Cast) u t))) in (and_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 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_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 
-(i: nat).((getl i d (CHead d0 (Bind Void) u0)) \to (\forall (w: T).(\forall 
-(v: T).((subst0 i 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).(and_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 (c: C).(getl i c0 c)) 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 return (\lambda 
-(_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow 
-(match k return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match 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 (i: nat).((getl i d (CHead d0 
-(Bind Void) u0)) \to (\forall (w: T).(\forall (v: T).((subst0 i 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).(and_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 (c: C).(getl i c0 c)) 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 return (\lambda (_: C).Prop) with [(CSort _) 
-\Rightarrow False | (CHead _ k _) \Rightarrow (match k return (\lambda (_: 
-K).Prop) with [(Bind b) \Rightarrow (match 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 (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 (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 
-(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 (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 
-(t: T).(arity g c t 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_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)) (\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)))).(and_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 (t: T).(\forall (x: nat).(\forall (x0: T).((eq T t 
-(lift h x x0)) \to (\forall (c2: C).((drop h x d0 c2) \to (arity g c2 x0 
-a0))))))) H3 (lift h (minus x (S i)) x1) H11) in (let H15 \def (eq_ind T u 
-(\lambda (t: T).(arity g d0 t 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))))).(and_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)))).(and_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 (t: T).(\forall (x: nat).(\forall (x0: T).((eq T t 
-(lift h x x0)) \to (\forall (c2: C).((drop h x d0 c2) \to (arity g c2 x0 
-(asucc g a0)))))))) H3 (lift h (minus x (S i)) x1) H11) in (let H15 \def 
-(eq_ind T u (\lambda (t: T).(arity g d0 t (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))))).(and_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 (t: T).(\forall (x: nat).(\forall (x0: T).((eq T t 
-(lift h x x0)) \to (\forall (c2: C).((drop h x (CHead c (Bind b) u) c2) \to 
-(arity g c2 x0 a2))))))) H5 (lift h (S x) x2) H10) in (let H12 \def (eq_ind T 
-t0 (\lambda (t: T).(arity g (CHead c (Bind b) u) t a2)) H4 (lift h (S x) x2) 
-H10) in (let H13 \def (eq_ind T u (\lambda (t: T).(arity g (CHead c (Bind b) 
-t) (lift h (S x) x2) a2)) H12 (lift h x x1) H9) in (let H14 \def (eq_ind T u 
-(\lambda (t: T).(\forall (x0: nat).(\forall (x1: T).((eq T (lift h (S x) x2) 
-(lift h x0 x1)) \to (\forall (c2: C).((drop h x0 (CHead c (Bind b) t) c2) \to 
-(arity g c2 x1 a2))))))) H11 (lift h x x1) H9) in (let H15 \def (eq_ind T u 
-(\lambda (t: T).(\forall (x: nat).(\forall (x0: T).((eq T t (lift h x x0)) 
-\to (\forall (c2: C).((drop h x c c2) \to (arity g c2 x0 a1))))))) H3 (lift h 
-x x1) H9) in (let H16 \def (eq_ind T u (\lambda (t: T).(arity g c t 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 (t: T).(\forall (x: nat).(\forall (x0: T).((eq T t (lift h x x0)) 
-\to (\forall (c2: C).((drop h x (CHead c (Bind Abst) u) c2) \to (arity g c2 
-x0 a2))))))) H4 (lift h (S x) x2) H9) in (let H11 \def (eq_ind T t0 (\lambda 
-(t: T).(arity g (CHead c (Bind Abst) u) t a2)) H3 (lift h (S x) x2) H9) in 
-(let H12 \def (eq_ind T u (\lambda (t: T).(arity g (CHead c (Bind Abst) t) 
-(lift h (S x) x2) a2)) H11 (lift h x x1) H8) in (let H13 \def (eq_ind T u 
-(\lambda (t: T).(\forall (x0: nat).(\forall (x1: T).((eq T (lift h (S x) x2) 
-(lift h x0 x1)) \to (\forall (c2: C).((drop h x0 (CHead c (Bind Abst) t) c2) 
-\to (arity g c2 x1 a2))))))) H10 (lift h x x1) H8) in (let H14 \def (eq_ind T 
-u (\lambda (t: T).(\forall (x: nat).(\forall (x0: T).((eq T t (lift h x x0)) 
-\to (\forall (c2: C).((drop h x c c2) \to (arity g c2 x0 (asucc g a1)))))))) 
-H2 (lift h x x1) H8) in (let H15 \def (eq_ind T u (\lambda (t: T).(arity g c 
-t (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 (t: T).(\forall (x: 
-nat).(\forall (x0: T).((eq T t (lift h x x0)) \to (\forall (c2: C).((drop h x 
-c c2) \to (arity g c2 x0 (AHead a1 a2)))))))) H4 (lift h x x2) H9) in (let 
-H11 \def (eq_ind T t0 (\lambda (t: T).(arity g c t (AHead a1 a2))) H3 (lift h 
-x x2) H9) in (let H12 \def (eq_ind T u (\lambda (t: T).(\forall (x: 
-nat).(\forall (x0: T).((eq T t (lift h x x0)) \to (\forall (c2: C).((drop h x 
-c c2) \to (arity g c2 x0 a1))))))) H2 (lift h x x1) H8) in (let H13 \def 
-(eq_ind T u (\lambda (t: T).(arity g c t 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 (t: T).(\forall (x: nat).(\forall 
-(x0: T).((eq T t (lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to 
-(arity g c2 x0 a0))))))) H4 (lift h x x2) H9) in (let H11 \def (eq_ind T t0 
-(\lambda (t: T).(arity g c t a0)) H3 (lift h x x2) H9) in (let H12 \def 
-(eq_ind T u (\lambda (t: T).(\forall (x: nat).(\forall (x0: T).((eq T t (lift 
-h x x0)) \to (\forall (c2: C).((drop h x c c2) \to (arity g c2 x0 (asucc g 
-a0)))))))) H2 (lift h x x1) H8) in (let H13 \def (eq_ind T u (\lambda (t: 
-T).(arity g c t (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))))))).
-
-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_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 H1 \def (match H return (\lambda (p: 
-PList).(\lambda (c: C).(\lambda (c0: C).(\lambda (_: (drop1 p c c0)).((eq 
-PList p PNil) \to ((eq C c c1) \to ((eq C c0 c2) \to (arity g c1 t a)))))))) 
-with [(drop1_nil c) \Rightarrow (\lambda (_: (eq PList PNil PNil)).(\lambda 
-(H2: (eq C c c1)).(\lambda (H3: (eq C c c2)).(eq_ind C c1 (\lambda (c0: 
-C).((eq C c0 c2) \to (arity g c1 t a))) (\lambda (H4: (eq C c1 c2)).(eq_ind C 
-c2 (\lambda (c0: C).(arity g c0 t a)) H0 c1 (sym_eq C c1 c2 H4))) c (sym_eq C 
-c c1 H2) H3)))) | (drop1_cons c0 c3 h d H1 c4 hds H2) \Rightarrow (\lambda 
-(H3: (eq PList (PCons h d hds) PNil)).(\lambda (H4: (eq C c0 c1)).(\lambda 
-(H5: (eq C c4 c2)).((let H6 \def (eq_ind PList (PCons h d hds) (\lambda (e: 
-PList).(match e return (\lambda (_: PList).Prop) with [PNil \Rightarrow False 
-| (PCons _ _ _) \Rightarrow True])) I PNil H3) in (False_ind ((eq C c0 c1) 
-\to ((eq C c4 c2) \to ((drop h d c0 c3) \to ((drop1 hds c3 c4) \to (arity g 
-c1 t a))))) H6)) H4 H5 H1 H2))))]) in (H1 (refl_equal PList PNil) (refl_equal 
-C c1) (refl_equal C c2))))))) (\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 H2 \def (match H0 return (\lambda (p0: PList).(\lambda 
-(c: C).(\lambda (c0: C).(\lambda (_: (drop1 p0 c c0)).((eq PList p0 (PCons n 
-n0 p)) \to ((eq C c c1) \to ((eq C c0 c2) \to (arity g c1 (lift n n0 (lift1 p 
-t)) a)))))))) with [(drop1_nil c) \Rightarrow (\lambda (H2: (eq PList PNil 
-(PCons n n0 p))).(\lambda (H3: (eq C c c1)).(\lambda (H4: (eq C c c2)).((let 
-H5 \def (eq_ind PList PNil (\lambda (e: PList).(match e return (\lambda (_: 
-PList).Prop) with [PNil \Rightarrow True | (PCons _ _ _) \Rightarrow False])) 
-I (PCons n n0 p) H2) in (False_ind ((eq C c c1) \to ((eq C c c2) \to (arity g 
-c1 (lift n n0 (lift1 p t)) a))) H5)) H3 H4)))) | (drop1_cons c0 c3 h d H2 c4 
-hds H3) \Rightarrow (\lambda (H4: (eq PList (PCons h d hds) (PCons n n0 
-p))).(\lambda (H5: (eq C c0 c1)).(\lambda (H6: (eq C c4 c2)).((let H7 \def 
-(f_equal PList PList (\lambda (e: PList).(match e return (\lambda (_: 
-PList).PList) with [PNil \Rightarrow hds | (PCons _ _ p) \Rightarrow p])) 
-(PCons h d hds) (PCons n n0 p) H4) in ((let H8 \def (f_equal PList nat 
-(\lambda (e: PList).(match e return (\lambda (_: PList).nat) with [PNil 
-\Rightarrow d | (PCons _ n _) \Rightarrow n])) (PCons h d hds) (PCons n n0 p) 
-H4) in ((let H9 \def (f_equal PList nat (\lambda (e: PList).(match e return 
-(\lambda (_: PList).nat) with [PNil \Rightarrow h | (PCons n _ _) \Rightarrow 
-n])) (PCons h d hds) (PCons n n0 p) H4) in (eq_ind nat n (\lambda (n1: 
-nat).((eq nat d n0) \to ((eq PList hds p) \to ((eq C c0 c1) \to ((eq C c4 c2) 
-\to ((drop n1 d c0 c3) \to ((drop1 hds c3 c4) \to (arity g c1 (lift n n0 
-(lift1 p t)) a)))))))) (\lambda (H10: (eq nat d n0)).(eq_ind nat n0 (\lambda 
-(n1: nat).((eq PList hds p) \to ((eq C c0 c1) \to ((eq C c4 c2) \to ((drop n 
-n1 c0 c3) \to ((drop1 hds c3 c4) \to (arity g c1 (lift n n0 (lift1 p t)) 
-a))))))) (\lambda (H11: (eq PList hds p)).(eq_ind PList p (\lambda (p0: 
-PList).((eq C c0 c1) \to ((eq C c4 c2) \to ((drop n n0 c0 c3) \to ((drop1 p0 
-c3 c4) \to (arity g c1 (lift n n0 (lift1 p t)) a)))))) (\lambda (H12: (eq C 
-c0 c1)).(eq_ind C c1 (\lambda (c: C).((eq C c4 c2) \to ((drop n n0 c c3) \to 
-((drop1 p c3 c4) \to (arity g c1 (lift n n0 (lift1 p t)) a))))) (\lambda 
-(H13: (eq C c4 c2)).(eq_ind C c2 (\lambda (c: C).((drop n n0 c1 c3) \to 
-((drop1 p c3 c) \to (arity g c1 (lift n n0 (lift1 p t)) a)))) (\lambda (H14: 
-(drop n n0 c1 c3)).(\lambda (H15: (drop1 p c3 c2)).(arity_lift g c3 (lift1 p 
-t) a (H c3 t H15 H1) c1 n n0 H14))) c4 (sym_eq C c4 c2 H13))) c0 (sym_eq C c0 
-c1 H12))) hds (sym_eq PList hds p H11))) d (sym_eq nat d n0 H10))) h (sym_eq 
-nat h n H9))) H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal PList (PCons n 
-n0 p)) (refl_equal C c1) (refl_equal C c2))))))))))) hds)))).
-
-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 (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_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 (c: C).(getl i c0 c)) 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 return (\lambda (_: C).C) 
-with [(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow c])) (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 return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead 
-_ _ t) \Rightarrow t])) (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 (t: 
-T).(getl i c0 (CHead x0 (Bind Abbr) t))) H8 u H10) in (let H13 \def (eq_ind_r 
-T x1 (\lambda (t: T).(arity g x0 t a2)) H7 u H10) in (let H14 \def (eq_ind_r 
-C x0 (\lambda (c: C).(getl i c0 (CHead c (Bind Abbr) u))) H12 d H11) in (let 
-H15 \def (eq_ind_r C x0 (\lambda (c: C).(arity g c u a2)) H13 d H11) in (H2 
-a2 H15))))))) H9))))))) H5)) (\lambda (H5: (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_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 (c: C).(getl i c0 c)) 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 return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | 
-(CHead _ k _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind 
-b) \Rightarrow (match 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 (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_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 (c: 
-C).(getl i c0 c)) 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 return (\lambda (_: C).Prop) with 
-[(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k return 
-(\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match 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 
-(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_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 (c: C).(getl i c0 c)) 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 return 
-(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow 
-c])) (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 return (\lambda (_: C).T) with [(CSort _) 
-\Rightarrow u | (CHead _ _ t) \Rightarrow t])) (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 (t: T).(getl i c0 (CHead x0 (Bind Abst) t))) H8 u H10) in (let H13 
-\def (eq_ind_r T x1 (\lambda (t: T).(arity g x0 t (asucc g a2))) H7 u H10) in 
-(let H14 \def (eq_ind_r C x0 (\lambda (c: C).(getl i c0 (CHead c (Bind Abst) 
-u))) H12 d H11) in (let H15 \def (eq_ind_r C x0 (\lambda (c: C).(arity g c 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 (a2: A).((arity g (CHead c0 (Bind 
-b) u) t0 a2) \to (leq g a3 a2))))).(\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 (a2: A).((arity g (CHead c0 (Bind Abst) u) t0 
-a2) \to (leq g a3 a2))))).(\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 (and_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_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 (c: C).(getl i c2 c)) 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 return 
-(\lambda (_: C).C) with [(CSort _) \Rightarrow x | (CHead c _ _) \Rightarrow 
-c])) (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 (c: C).(getl i c2 (CHead c (Bind Abbr) u))) H9 x H10) in (let 
-H12 \def (eq_ind_r C x0 (\lambda (c: C).(cimp d c)) 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 (c: C).(getl i c2 c)) 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 return (\lambda (_: C).C) with [(CSort 
-_) \Rightarrow x | (CHead c _ _) \Rightarrow c])) (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 (c: C).(getl i 
-c2 (CHead c (Bind Abst) u))) H9 x H10) in (let H12 \def (eq_ind_r C x0 
-(\lambda (c: C).(cimp d c)) 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))))).
-
-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 H1 \def (match H0 return 
-(\lambda (n0: nat).(\lambda (a: A).(\lambda (a0: A).(\lambda (_: (aprem n0 a 
-a0)).((eq nat n0 i) \to ((eq A a (ASort O n)) \to ((eq A 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))))))))))))) with [(aprem_zero a1 a2) \Rightarrow (\lambda (H0: (eq nat O 
-i)).(\lambda (H1: (eq A (AHead a1 a2) (ASort O n))).(\lambda (H2: (eq A a1 
-b)).(eq_ind nat O (\lambda (n0: nat).((eq A (AHead a1 a2) (ASort O n)) \to 
-((eq A a1 b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: 
-nat).(drop (plus n0 j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda 
-(_: nat).(arity g d u (asucc g b))))))))) (\lambda (H3: (eq A (AHead a1 a2) 
-(ASort O n))).(let H4 \def (eq_ind A (AHead a1 a2) (\lambda (e: A).(match e 
-return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ 
-_) \Rightarrow True])) I (ASort O n) H3) in (False_ind ((eq A a1 b) \to 
-(ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus 
-O j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d 
-u (asucc g b))))))) H4))) i H0 H1 H2)))) | (aprem_succ a2 a i0 H0 a1) 
-\Rightarrow (\lambda (H1: (eq nat (S i0) i)).(\lambda (H2: (eq A (AHead a1 
-a2) (ASort O n))).(\lambda (H3: (eq A a b)).(eq_ind nat (S i0) (\lambda (n0: 
-nat).((eq A (AHead a1 a2) (ASort O n)) \to ((eq A a b) \to ((aprem i0 a2 a) 
-\to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop 
-(plus n0 j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: 
-nat).(arity g d u (asucc g b)))))))))) (\lambda (H4: (eq A (AHead a1 a2) 
-(ASort O n))).(let H5 \def (eq_ind A (AHead a1 a2) (\lambda (e: A).(match e 
-return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ 
-_) \Rightarrow True])) I (ASort O n) H4) in (False_ind ((eq A a b) \to 
-((aprem i0 a2 a) \to (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 (u: 
-T).(\lambda (_: nat).(arity g d u (asucc g b)))))))) H5))) i H1 H2 H3 
-H0))))]) in (H1 (refl_equal nat i) (refl_equal A (ASort O n)) (refl_equal A 
-b)))))))) (\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 (i: nat).(\forall 
-(b: A).((aprem i a0 b) \to (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: 
-T).(\lambda (j: nat).(drop (plus i j) O d0 d)))) (\lambda (d: C).(\lambda (u: 
-T).(\lambda (_: nat).(arity g d u (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 (i: nat).(\forall (b: 
-A).((aprem i (asucc g a0) b) \to (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: 
-T).(\lambda (j: nat).(drop (plus i j) O d0 d)))) (\lambda (d: C).(\lambda (u: 
-T).(\lambda (_: nat).(arity g d u (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 (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 (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 (u: T).(\lambda (_: nat).(arity g d u (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 (u: T).(\lambda (_: 
-nat).(arity g d u (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 (u: T).(\lambda (_: 
-nat).(arity g d u (asucc g b))))))))))).(\lambda (i: nat).(\lambda (b: 
-A).(\lambda (H4: (aprem i (AHead a1 a2) b)).((match i return (\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)))))))) with [O 
-\Rightarrow (\lambda (H5: (aprem O (AHead a1 a2) b)).(let H6 \def (match H5 
-return (\lambda (n: nat).(\lambda (a: A).(\lambda (a0: A).(\lambda (_: (aprem 
-n a a0)).((eq nat n O) \to ((eq A a (AHead a1 a2)) \to ((eq A a0 b) \to 
-(ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus 
-O j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d 
-u (asucc g b))))))))))))) with [(aprem_zero a0 a3) \Rightarrow (\lambda (_: 
-(eq nat O O)).(\lambda (H5: (eq A (AHead a0 a3) (AHead a1 a2))).(\lambda (H6: 
-(eq A a0 b)).((let H7 \def (f_equal A A (\lambda (e: A).(match e return 
-(\lambda (_: A).A) with [(ASort _ _) \Rightarrow a3 | (AHead _ a) \Rightarrow 
-a])) (AHead a0 a3) (AHead a1 a2) H5) in ((let H8 \def (f_equal A A (\lambda 
-(e: A).(match e return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 | 
-(AHead a _) \Rightarrow a])) (AHead a0 a3) (AHead a1 a2) H5) in (eq_ind A a1 
-(\lambda (a: A).((eq A a3 a2) \to ((eq A a b) \to (ex2_3 C T nat (\lambda (d: 
-C).(\lambda (_: T).(\lambda (j: nat).(drop (plus O j) O d c0)))) (\lambda (d: 
-C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b))))))))) 
-(\lambda (H9: (eq A a3 a2)).(eq_ind A a2 (\lambda (_: A).((eq A a1 b) \to 
-(ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus 
-O j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d 
-u (asucc g b)))))))) (\lambda (H10: (eq A a1 b)).(eq_ind A b (\lambda (_: 
-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 (u: T).(\lambda (_: 
-nat).(arity g d u (asucc g b))))))) (eq_ind A a1 (\lambda (a: 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 (u: T).(\lambda (_: nat).(arity g d u (asucc 
-g a))))))) (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 (u: T).(\lambda (_: 
-nat).(arity g d u (asucc g a1))))) c0 u O (drop_refl c0) H0) b H10) a1 
-(sym_eq A a1 b H10))) a3 (sym_eq A a3 a2 H9))) a0 (sym_eq A a0 a1 H8))) H7)) 
-H6)))) | (aprem_succ a0 a i H4 a3) \Rightarrow (\lambda (H5: (eq nat (S i) 
-O)).(\lambda (H6: (eq A (AHead a3 a0) (AHead a1 a2))).(\lambda (H7: (eq A a 
-b)).((let H8 \def (eq_ind nat (S i) (\lambda (e: nat).(match e return 
-(\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) 
-I O H5) in (False_ind ((eq A (AHead a3 a0) (AHead a1 a2)) \to ((eq A a b) \to 
-((aprem i a0 a) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda 
-(j: nat).(drop (plus O j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda 
-(_: nat).(arity g d u (asucc g b))))))))) H8)) H6 H7 H4))))]) in (H6 
-(refl_equal nat O) (refl_equal A (AHead a1 a2)) (refl_equal A b)))) | (S n) 
-\Rightarrow (\lambda (H5: (aprem (S n) (AHead a1 a2) b)).(let H6 \def (match 
-H5 return (\lambda (n0: nat).(\lambda (a: A).(\lambda (a0: A).(\lambda (_: 
-(aprem n0 a a0)).((eq nat n0 (S n)) \to ((eq A a (AHead a1 a2)) \to ((eq A a0 
-b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop 
-(plus (S n) j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: 
-nat).(arity g d u (asucc g b))))))))))))) with [(aprem_zero a0 a3) 
-\Rightarrow (\lambda (H4: (eq nat O (S n))).(\lambda (H5: (eq A (AHead a0 a3) 
-(AHead a1 a2))).(\lambda (H6: (eq A a0 b)).((let H7 \def (eq_ind nat O 
-(\lambda (e: nat).(match e return (\lambda (_: nat).Prop) with [O \Rightarrow 
-True | (S _) \Rightarrow False])) I (S n) H4) in (False_ind ((eq A (AHead a0 
-a3) (AHead a1 a2)) \to ((eq A a0 b) \to (ex2_3 C T nat (\lambda (d: 
-C).(\lambda (_: T).(\lambda (j: nat).(drop (plus (S n) j) O d c0)))) (\lambda 
-(d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b)))))))) H7)) 
-H5 H6)))) | (aprem_succ a0 a i H4 a3) \Rightarrow (\lambda (H5: (eq nat (S i) 
-(S n))).(\lambda (H6: (eq A (AHead a3 a0) (AHead a1 a2))).(\lambda (H7: (eq A 
-a b)).((let H8 \def (f_equal nat nat (\lambda (e: nat).(match e return 
-(\lambda (_: nat).nat) with [O \Rightarrow i | (S n) \Rightarrow n])) (S i) 
-(S n) H5) in (eq_ind nat n (\lambda (n0: nat).((eq A (AHead a3 a0) (AHead a1 
-a2)) \to ((eq A a b) \to ((aprem n0 a0 a) \to (ex2_3 C T nat (\lambda (d: 
-C).(\lambda (_: T).(\lambda (j: nat).(drop (plus (S n) j) O d c0)))) (\lambda 
-(d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b)))))))))) 
-(\lambda (H9: (eq A (AHead a3 a0) (AHead a1 a2))).(let H10 \def (f_equal A A 
-(\lambda (e: A).(match e return (\lambda (_: A).A) with [(ASort _ _) 
-\Rightarrow a0 | (AHead _ a) \Rightarrow a])) (AHead a3 a0) (AHead a1 a2) H9) 
-in ((let H11 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: 
-A).A) with [(ASort _ _) \Rightarrow a3 | (AHead a _) \Rightarrow a])) (AHead 
-a3 a0) (AHead a1 a2) H9) in (eq_ind A a1 (\lambda (_: A).((eq A a0 a2) \to 
-((eq A a b) \to ((aprem n a0 a) \to (ex2_3 C T nat (\lambda (d: C).(\lambda 
-(_: T).(\lambda (j: nat).(drop (plus (S n) j) O d c0)))) (\lambda (d: 
-C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b)))))))))) 
-(\lambda (H12: (eq A a0 a2)).(eq_ind A a2 (\lambda (a1: A).((eq A a b) \to 
-((aprem n a1 a) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda 
-(j: nat).(drop (plus (S n) j) O d c0)))) (\lambda (d: C).(\lambda (u: 
-T).(\lambda (_: nat).(arity g d u (asucc g b))))))))) (\lambda (H13: (eq A a 
-b)).(eq_ind A b (\lambda (a1: A).((aprem n a2 a1) \to (ex2_3 C T nat (\lambda 
-(d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus (S n) j) O d c0)))) 
-(\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g 
-b)))))))) (\lambda (H14: (aprem n a2 b)).(let H_x \def (H3 n b H14) in (let 
-H3 \def H_x in (ex2_3_ind C T nat (\lambda (d: C).(\lambda (_: T).(\lambda 
-(j: nat).(drop (plus n j) O d (CHead c0 (Bind Abst) u))))) (\lambda (d: 
-C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b))))) (ex2_3 C T 
-nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus (S n) 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 (H15: (drop (plus n x2) O x0 (CHead c0 (Bind Abst) 
-u))).(\lambda (H16: (arity g x0 x1 (asucc g b))).(ex2_3_intro C T nat 
-(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus (S n) j) O d 
-c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc 
-g b))))) x0 x1 x2 (drop_S Abst x0 c0 u (plus n x2) H15) H16)))))) H3)))) a 
-(sym_eq A a b H13))) a0 (sym_eq A a0 a2 H12))) a3 (sym_eq A a3 a1 H11))) 
-H10))) i (sym_eq nat i n H8))) H6 H7 H4))))]) in (H6 (refl_equal nat (S n)) 
-(refl_equal A (AHead a1 a2)) (refl_equal A b))))]) 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 (u: T).(\lambda (_: nat).(arity g d 
-u (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 (u: 
-T).(\lambda (_: nat).(arity g d u (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 
-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 (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 (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))).((match k return (\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))))))))) 
-with [(Bind b0) \Rightarrow (\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))) | (Flat f) \Rightarrow (\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))))]) 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 (u: T).(\lambda (_: nat).(arity g d u (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 (u: T).(\lambda (_: nat).(arity g d u (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))))).
-
-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))))))))).
-
-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 H2 \def (fsubst0_gen_base c c2 (TSort 
-n) t2 u i H1) 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))).(and_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))).(and_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))).(and_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 (i: nat).((getl i d (CHead d1 (Bind Abbr) u0)) \to (\forall 
-(c2: C).(\forall (t2: T).((fsubst0 i 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 H5 \def 
-(fsubst0_gen_base c c2 (TLRef i) t2 u0 i0 H4) 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))).(and_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)) (and_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 return (\lambda (_: C).C) with [(CSort 
-_) \Rightarrow d | (CHead c _ _) \Rightarrow c])) (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 
-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))).(and_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 (u0: T).(\lambda (_: 
-T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) u0)))))) (\lambda (b: 
-B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl i c2 (CHead e0 
-(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus i0 (S i)) u0 u 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 return (\lambda (_: C).C) with [(CSort _) 
-\Rightarrow d | (CHead c _ _) \Rightarrow c])) (CHead d (Bind Abbr) u) (CHead 
-x1 (Bind x0) x2) H12) in ((let H17 \def (f_equal C B (\lambda (e: C).(match e 
-return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) 
-\Rightarrow (match 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 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 (c: C).(getl i c2 (CHead c (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 (u0: T).(eq C 
-(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u0)))))) (\lambda (b: B).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (u: T).(getl i c2 (CHead e2 (Bind b) u)))))) 
-(\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 return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead 
-c _ _) \Rightarrow c])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H12) 
-in ((let H17 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: 
-C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match 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 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 (c: C).(csubst0 (minus i0 (S i)) u0 c 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 (u0: T).(\lambda 
-(_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u0))))))) (\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 (u: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u 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 return (\lambda (_: C).C) with [(CSort _) 
-\Rightarrow d | (CHead c _ _) \Rightarrow c])) (CHead d (Bind Abbr) u) (CHead 
-x1 (Bind x0) x3) H12) in ((let H18 \def (f_equal C B (\lambda (e: C).(match e 
-return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) 
-\Rightarrow (match 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 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 (c: C).(csubst0 (minus i0 (S i)) u0 c 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))).(and_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)).(and_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 return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead 
-c _ _) \Rightarrow c])) (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 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 (i: nat).((getl i d (CHead d1 (Bind Abbr) u0)) 
-\to (\forall (c2: C).(\forall (t2: T).((fsubst0 i 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 H5 
-\def (fsubst0_gen_base c c2 (TLRef i) t2 u0 i0 H4) 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))).(and_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)) (and_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 return (\lambda 
-(_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow 
-(match k return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match 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))).(and_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 
-(u0: T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead e0 (Bind b) 
-u0)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(getl i c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u 
-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 return (\lambda (_: 
-C).C) with [(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow c])) (CHead d 
-(Bind Abst) u) (CHead x1 (Bind x0) x2) H12) in ((let H17 \def (f_equal C B 
-(\lambda (e: C).(match e return (\lambda (_: C).B) with [(CSort _) 
-\Rightarrow Abst | (CHead _ k _) \Rightarrow (match 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 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 (c: C).(getl i 
-c2 (CHead c (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 (u0: T).(eq C (CHead d (Bind Abst) u) (CHead 
-e1 (Bind b) u0)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(getl i c2 (CHead e2 (Bind b) u)))))) (\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 return (\lambda 
-(_: C).C) with [(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow c])) 
-(CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H12) in ((let H17 \def 
-(f_equal C B (\lambda (e: C).(match e return (\lambda (_: C).B) with [(CSort 
-_) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match 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 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 (c: 
-C).(csubst0 (minus i0 (S i)) u0 c 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 (u0: T).(\lambda (_: T).(eq C 
-(CHead d (Bind Abst) u) (CHead e1 (Bind b) u0))))))) (\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 
-(u: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u 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 return (\lambda (_: C).C) with [(CSort _) 
-\Rightarrow d | (CHead c _ _) \Rightarrow c])) (CHead d (Bind Abst) u) (CHead 
-x1 (Bind x0) x3) H12) in ((let H18 \def (f_equal C B (\lambda (e: C).(match e 
-return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abst | (CHead _ k _) 
-\Rightarrow (match 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 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 (c: C).(csubst0 (minus i0 (S i)) u0 c 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))).(and_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)).(and_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 return (\lambda (_: C).Prop) with [(CSort 
-_) \Rightarrow False | (CHead _ k _) \Rightarrow (match k return (\lambda (_: 
-K).Prop) with [(Bind b) \Rightarrow (match 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 H7 \def (fsubst0_gen_base c c2 (THead (Bind b) u 
-t) t2 u0 i H6) 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))).(and_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 (t2: T).(subst0 (s (Bind b) i) u0 t 
-t2)))).(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 (t2: 
-T).(subst0 (s (Bind b) i) u0 t t2))))).(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))).(and_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))).(and_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 (t2: T).(subst0 (s (Bind b) i) u0 t 
-t2)))).(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 (t2: T).(subst0 (s (Bind b) i) u0 t t2))))).(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 H6 \def (fsubst0_gen_base c c2 (THead (Bind 
-Abst) u t) t2 u0 i H5) 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))).(and_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 (t2: T).(subst0 (s (Bind Abst) i) u0 t t2)))).(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 (t2: T).(subst0 (s (Bind Abst) i) u0 t 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).(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))).(and_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))).(and_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 (t2: T).(subst0 (s (Bind Abst) i) u0 t 
-t2)))).(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 (t2: T).(subst0 (s (Bind Abst) i) u0 t 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).(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 (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 (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 H6 \def (fsubst0_gen_base c c2 (THead (Flat 
-Appl) u t) t2 u0 i H5) 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))).(and_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 (t2: T).(subst0 
-(s (Flat Appl) i) u0 t t2)))).(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 (t2: T).(subst0 (s (Flat Appl) i) u0 t 
-t2))))).(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))).(and_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))).(and_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 (t2: T).(subst0 (s (Flat Appl) i) u0 t 
-t2)))).(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 (t2: 
-T).(subst0 (s (Flat Appl) i) u0 t t2))))).(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 (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 (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 H6 \def (fsubst0_gen_base c c2 (THead (Flat 
-Cast) u t) t2 u0 i H5) 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))).(and_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 (t2: T).(subst0 
-(s (Flat Cast) i) u0 t t2)))).(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 (t2: T).(subst0 (s (Flat Cast) i) u0 t 
-t2))))).(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))).(and_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))).(and_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 (t2: T).(subst0 (s (Flat Cast) i) u0 t 
-t2)))).(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 (t2: 
-T).(subst0 (s (Flat Cast) i) u0 t t2))))).(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 H5 \def (fsubst0_gen_base c c2 t t2 u i H4) 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))).(and_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))).(and_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))).(and_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)))))))))))).
-
-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_epsilon: \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (u: 
-T).(pr0 (THead (Flat Cast) u t1) t2)))).
-
-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)).(let H0 
-\def (match H return (\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr0 t 
-t0)).((eq T t (TSort n)) \to ((eq T t0 x) \to (eq T x (TSort n))))))) with 
-[(pr0_refl t) \Rightarrow (\lambda (H0: (eq T t (TSort n))).(\lambda (H1: (eq 
-T t x)).(eq_ind T (TSort n) (\lambda (t0: T).((eq T t0 x) \to (eq T x (TSort 
-n)))) (\lambda (H2: (eq T (TSort n) x)).(eq_ind T (TSort n) (\lambda (t0: 
-T).(eq T t0 (TSort n))) (refl_equal T (TSort n)) x H2)) t (sym_eq T t (TSort 
-n) H0) H1))) | (pr0_comp u1 u2 H0 t1 t2 H1 k) \Rightarrow (\lambda (H2: (eq T 
-(THead k u1 t1) (TSort n))).(\lambda (H3: (eq T (THead k u2 t2) x)).((let H4 
-\def (eq_ind T (THead k u1 t1) (\lambda (e: T).(match e return (\lambda (_: 
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
-(THead _ _ _) \Rightarrow True])) I (TSort n) H2) in (False_ind ((eq T (THead 
-k u2 t2) x) \to ((pr0 u1 u2) \to ((pr0 t1 t2) \to (eq T x (TSort n))))) H4)) 
-H3 H0 H1))) | (pr0_beta u v1 v2 H0 t1 t2 H1) \Rightarrow (\lambda (H2: (eq T 
-(THead (Flat Appl) v1 (THead (Bind Abst) u t1)) (TSort n))).(\lambda (H3: (eq 
-T (THead (Bind Abbr) v2 t2) x)).((let H4 \def (eq_ind T (THead (Flat Appl) v1 
-(THead (Bind Abst) u t1)) (\lambda (e: T).(match e return (\lambda (_: 
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
-(THead _ _ _) \Rightarrow True])) I (TSort n) H2) in (False_ind ((eq T (THead 
-(Bind Abbr) v2 t2) x) \to ((pr0 v1 v2) \to ((pr0 t1 t2) \to (eq T x (TSort 
-n))))) H4)) H3 H0 H1))) | (pr0_upsilon b H0 v1 v2 H1 u1 u2 H2 t1 t2 H3) 
-\Rightarrow (\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t1)) 
-(TSort n))).(\lambda (H5: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift 
-(S O) O v2) t2)) x)).((let H6 \def (eq_ind T (THead (Flat Appl) v1 (THead 
-(Bind b) u1 t1)) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) 
-\Rightarrow True])) I (TSort n) H4) in (False_ind ((eq T (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t2)) x) \to ((not (eq B b Abst)) \to 
-((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t1 t2) \to (eq T x (TSort n))))))) 
-H6)) H5 H0 H1 H2 H3))) | (pr0_delta u1 u2 H0 t1 t2 H1 w H2) \Rightarrow 
-(\lambda (H3: (eq T (THead (Bind Abbr) u1 t1) (TSort n))).(\lambda (H4: (eq T 
-(THead (Bind Abbr) u2 w) x)).((let H5 \def (eq_ind T (THead (Bind Abbr) u1 
-t1) (\lambda (e: T).(match e 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 Abbr) u2 w) x) \to 
-((pr0 u1 u2) \to ((pr0 t1 t2) \to ((subst0 O u2 t2 w) \to (eq T x (TSort 
-n)))))) H5)) H4 H0 H1 H2))) | (pr0_zeta b H0 t1 t2 H1 u) \Rightarrow (\lambda 
-(H2: (eq T (THead (Bind b) u (lift (S O) O t1)) (TSort n))).(\lambda (H3: (eq 
-T t2 x)).((let H4 \def (eq_ind T (THead (Bind b) u (lift (S O) O t1)) 
-(\lambda (e: T).(match e return (\lambda (_: T).Prop) with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
-True])) I (TSort n) H2) in (False_ind ((eq T t2 x) \to ((not (eq B b Abst)) 
-\to ((pr0 t1 t2) \to (eq T x (TSort n))))) H4)) H3 H0 H1))) | (pr0_epsilon t1 
-t2 H0 u) \Rightarrow (\lambda (H1: (eq T (THead (Flat Cast) u t1) (TSort 
-n))).(\lambda (H2: (eq T t2 x)).((let H3 \def (eq_ind T (THead (Flat Cast) u 
-t1) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
-True])) I (TSort n) H1) in (False_ind ((eq T t2 x) \to ((pr0 t1 t2) \to (eq T 
-x (TSort n)))) H3)) H2 H0)))]) in (H0 (refl_equal T (TSort n)) (refl_equal T 
-x))))).
-
-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)).(let H0 
-\def (match H return (\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr0 t 
-t0)).((eq T t (TLRef n)) \to ((eq T t0 x) \to (eq T x (TLRef n))))))) with 
-[(pr0_refl t) \Rightarrow (\lambda (H0: (eq T t (TLRef n))).(\lambda (H1: (eq 
-T t x)).(eq_ind T (TLRef n) (\lambda (t0: T).((eq T t0 x) \to (eq T x (TLRef 
-n)))) (\lambda (H2: (eq T (TLRef n) x)).(eq_ind T (TLRef n) (\lambda (t0: 
-T).(eq T t0 (TLRef n))) (refl_equal T (TLRef n)) x H2)) t (sym_eq T t (TLRef 
-n) H0) H1))) | (pr0_comp u1 u2 H0 t1 t2 H1 k) \Rightarrow (\lambda (H2: (eq T 
-(THead k u1 t1) (TLRef n))).(\lambda (H3: (eq T (THead k u2 t2) x)).((let H4 
-\def (eq_ind T (THead k u1 t1) (\lambda (e: T).(match e return (\lambda (_: 
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
-(THead _ _ _) \Rightarrow True])) I (TLRef n) H2) in (False_ind ((eq T (THead 
-k u2 t2) x) \to ((pr0 u1 u2) \to ((pr0 t1 t2) \to (eq T x (TLRef n))))) H4)) 
-H3 H0 H1))) | (pr0_beta u v1 v2 H0 t1 t2 H1) \Rightarrow (\lambda (H2: (eq T 
-(THead (Flat Appl) v1 (THead (Bind Abst) u t1)) (TLRef n))).(\lambda (H3: (eq 
-T (THead (Bind Abbr) v2 t2) x)).((let H4 \def (eq_ind T (THead (Flat Appl) v1 
-(THead (Bind Abst) u t1)) (\lambda (e: T).(match e return (\lambda (_: 
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
-(THead _ _ _) \Rightarrow True])) I (TLRef n) H2) in (False_ind ((eq T (THead 
-(Bind Abbr) v2 t2) x) \to ((pr0 v1 v2) \to ((pr0 t1 t2) \to (eq T x (TLRef 
-n))))) H4)) H3 H0 H1))) | (pr0_upsilon b H0 v1 v2 H1 u1 u2 H2 t1 t2 H3) 
-\Rightarrow (\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t1)) 
-(TLRef n))).(\lambda (H5: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift 
-(S O) O v2) t2)) x)).((let H6 \def (eq_ind T (THead (Flat Appl) v1 (THead 
-(Bind b) u1 t1)) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) 
-\Rightarrow True])) I (TLRef n) H4) in (False_ind ((eq T (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t2)) x) \to ((not (eq B b Abst)) \to 
-((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t1 t2) \to (eq T x (TLRef n))))))) 
-H6)) H5 H0 H1 H2 H3))) | (pr0_delta u1 u2 H0 t1 t2 H1 w H2) \Rightarrow 
-(\lambda (H3: (eq T (THead (Bind Abbr) u1 t1) (TLRef n))).(\lambda (H4: (eq T 
-(THead (Bind Abbr) u2 w) x)).((let H5 \def (eq_ind T (THead (Bind Abbr) u1 
-t1) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
-True])) I (TLRef n) H3) in (False_ind ((eq T (THead (Bind Abbr) u2 w) x) \to 
-((pr0 u1 u2) \to ((pr0 t1 t2) \to ((subst0 O u2 t2 w) \to (eq T x (TLRef 
-n)))))) H5)) H4 H0 H1 H2))) | (pr0_zeta b H0 t1 t2 H1 u) \Rightarrow (\lambda 
-(H2: (eq T (THead (Bind b) u (lift (S O) O t1)) (TLRef n))).(\lambda (H3: (eq 
-T t2 x)).((let H4 \def (eq_ind T (THead (Bind b) u (lift (S O) O t1)) 
-(\lambda (e: T).(match e return (\lambda (_: T).Prop) with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
-True])) I (TLRef n) H2) in (False_ind ((eq T t2 x) \to ((not (eq B b Abst)) 
-\to ((pr0 t1 t2) \to (eq T x (TLRef n))))) H4)) H3 H0 H1))) | (pr0_epsilon t1 
-t2 H0 u) \Rightarrow (\lambda (H1: (eq T (THead (Flat Cast) u t1) (TLRef 
-n))).(\lambda (H2: (eq T t2 x)).((let H3 \def (eq_ind T (THead (Flat Cast) u 
-t1) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
-True])) I (TLRef n) H1) in (False_ind ((eq T t2 x) \to ((pr0 t1 t2) \to (eq T 
-x (TLRef n)))) H3)) H2 H0)))]) in (H0 (refl_equal T (TLRef n)) (refl_equal T 
-x))))).
-
-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)).(let H0 \def (match H return (\lambda (t: T).(\lambda 
-(t0: T).(\lambda (_: (pr0 t t0)).((eq T t (THead (Bind Abst) u1 t1)) \to ((eq 
-T t0 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))))))))) with [(pr0_refl t) \Rightarrow 
-(\lambda (H0: (eq T t (THead (Bind Abst) u1 t1))).(\lambda (H1: (eq T t 
-x)).(eq_ind T (THead (Bind Abst) u1 t1) (\lambda (t0: T).((eq T t0 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)))))) (\lambda (H2: (eq T (THead (Bind Abst) 
-u1 t1) x)).(eq_ind 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)) x H2)) t (sym_eq T t (THead (Bind Abst) u1 t1) H0) H1))) | (pr0_comp u0 
-u2 H0 t0 t2 H1 k) \Rightarrow (\lambda (H2: (eq T (THead k u0 t0) (THead 
-(Bind Abst) u1 t1))).(\lambda (H3: (eq T (THead k u2 t2) x)).((let H4 \def 
-(f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort 
-_) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) 
-(THead k u0 t0) (THead (Bind Abst) u1 t1) H2) in ((let H5 \def (f_equal T T 
-(\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) 
-(THead k u0 t0) (THead (Bind Abst) u1 t1) H2) in ((let H6 \def (f_equal T K 
-(\lambda (e: T).(match e return (\lambda (_: T).K) with [(TSort _) 
-\Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) 
-(THead k u0 t0) (THead (Bind Abst) u1 t1) H2) in (eq_ind K (Bind Abst) 
-(\lambda (k0: K).((eq T u0 u1) \to ((eq T t0 t1) \to ((eq T (THead k0 u2 t2) 
-x) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (ex3_2 T T (\lambda (u3: T).(\lambda 
-(t3: T).(eq T x (THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: 
-T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))))))))) 
-(\lambda (H7: (eq T u0 u1)).(eq_ind T u1 (\lambda (t: T).((eq T t0 t1) \to 
-((eq T (THead (Bind Abst) u2 t2) x) \to ((pr0 t u2) \to ((pr0 t0 t2) \to 
-(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (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 t0 t1)).(eq_ind T 
-t1 (\lambda (t: T).((eq T (THead (Bind Abst) u2 t2) x) \to ((pr0 u1 u2) \to 
-((pr0 t t2) \to (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead 
-(Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda 
-(_: T).(\lambda (t3: T).(pr0 t1 t3)))))))) (\lambda (H9: (eq T (THead (Bind 
-Abst) u2 t2) x)).(eq_ind T (THead (Bind Abst) u2 t2) (\lambda (t: T).((pr0 u1 
-u2) \to ((pr0 t1 t2) \to (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t 
-(THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) 
-(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))))) (\lambda (H10: (pr0 u1 
-u2)).(\lambda (H11: (pr0 t1 t2)).(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)) H10 H11))) x H9)) t0 
-(sym_eq T t0 t1 H8))) u0 (sym_eq T u0 u1 H7))) k (sym_eq K k (Bind Abst) 
-H6))) H5)) H4)) H3 H0 H1))) | (pr0_beta u v1 v2 H0 t0 t2 H1) \Rightarrow 
-(\lambda (H2: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead 
-(Bind Abst) u1 t1))).(\lambda (H3: (eq T (THead (Bind Abbr) v2 t2) x)).((let 
-H4 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda 
-(e: T).(match e return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k 
-return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) 
-\Rightarrow True])])) I (THead (Bind Abst) u1 t1) H2) in (False_ind ((eq T 
-(THead (Bind Abbr) v2 t2) x) \to ((pr0 v1 v2) \to ((pr0 t0 t2) \to (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: 
-T).(pr0 t1 t3))))))) H4)) H3 H0 H1))) | (pr0_upsilon b H0 v1 v2 H1 u0 u2 H2 
-t0 t2 H3) \Rightarrow (\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind 
-b) u0 t0)) (THead (Bind Abst) u1 t1))).(\lambda (H5: (eq T (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t2)) x)).((let H6 \def (eq_ind T (THead 
-(Flat Appl) v1 (THead (Bind b) u0 t0)) (\lambda (e: T).(match e return 
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: 
-K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I 
-(THead (Bind Abst) u1 t1) H4) in (False_ind ((eq T (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t2)) x) \to ((not (eq B b Abst)) \to ((pr0 v1 
-v2) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u3 t3)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
-t3))))))))) H6)) H5 H0 H1 H2 H3))) | (pr0_delta u0 u2 H0 t0 t2 H1 w H2) 
-\Rightarrow (\lambda (H3: (eq T (THead (Bind Abbr) u0 t0) (THead (Bind Abst) 
-u1 t1))).(\lambda (H4: (eq T (THead (Bind Abbr) u2 w) x)).((let H5 \def 
-(eq_ind T (THead (Bind Abbr) u0 t0) (\lambda (e: T).(match e return (\lambda 
-(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
-| (THead k _ _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind 
-b) \Rightarrow (match b return (\lambda (_: B).Prop) with [Abbr \Rightarrow 
-True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) 
-\Rightarrow False])])) I (THead (Bind Abst) u1 t1) H3) in (False_ind ((eq T 
-(THead (Bind Abbr) u2 w) x) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to ((subst0 O 
-u2 t2 w) \to (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead 
-(Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda 
-(_: T).(\lambda (t3: T).(pr0 t1 t3)))))))) H5)) H4 H0 H1 H2))) | (pr0_zeta b 
-H0 t0 t2 H1 u) \Rightarrow (\lambda (H2: (eq T (THead (Bind b) u (lift (S O) 
-O t0)) (THead (Bind Abst) u1 t1))).(\lambda (H3: (eq T t2 x)).((let H4 \def 
-(f_equal T T (\lambda (e: T).(match e 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 u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) 
-t0))]) in lref_map) (\lambda (x: nat).(plus x (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 u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) 
-t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t0) | (THead _ _ t) 
-\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abst) u1 
-t1) H2) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e 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) H2) in ((let H6 \def (f_equal T B (\lambda (e: T).(match e 
-return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | (TLRef _) 
-\Rightarrow b | (THead k _ _) \Rightarrow (match k return (\lambda (_: K).B) 
-with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b])])) (THead (Bind b) u 
-(lift (S O) O t0)) (THead (Bind Abst) u1 t1) H2) in (eq_ind B Abst (\lambda 
-(b0: B).((eq T u u1) \to ((eq T (lift (S O) O t0) t1) \to ((eq T t2 x) \to 
-((not (eq B b0 Abst)) \to ((pr0 t0 t2) \to (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
-t3)))))))))) (\lambda (H7: (eq T u u1)).(eq_ind T u1 (\lambda (_: T).((eq T 
-(lift (S O) O t0) t1) \to ((eq T t2 x) \to ((not (eq B Abst Abst)) \to ((pr0 
-t0 t2) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind 
-Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr0 t1 t3))))))))) (\lambda (H8: (eq T (lift (S O) O t0) 
-t1)).(eq_ind T (lift (S O) O t0) (\lambda (t: T).((eq T t2 x) \to ((not (eq B 
-Abst Abst)) \to ((pr0 t0 t2) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 
-u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t t3)))))))) (\lambda (H9: (eq 
-T t2 x)).(eq_ind T x (\lambda (t: T).((not (eq B Abst Abst)) \to ((pr0 t0 t) 
-\to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (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))))))) (\lambda (H10: (not (eq 
-B Abst Abst))).(\lambda (_: (pr0 t0 x)).(False_ind (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T x (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)))) (H10 (refl_equal B Abst))))) t2 (sym_eq T t2 x H9))) t1 
-H8)) u (sym_eq T u u1 H7))) b (sym_eq B b Abst H6))) H5)) H4)) H3 H0 H1))) | 
-(pr0_epsilon t0 t2 H0 u) \Rightarrow (\lambda (H1: (eq T (THead (Flat Cast) u 
-t0) (THead (Bind Abst) u1 t1))).(\lambda (H2: (eq T t2 x)).((let H3 \def 
-(eq_ind T (THead (Flat Cast) u t0) (\lambda (e: T).(match e return (\lambda 
-(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
-| (THead k _ _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind 
-_) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u1 
-t1) H1) in (False_ind ((eq T t2 x) \to ((pr0 t0 t2) \to (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
-t3)))))) H3)) H2 H0)))]) in (H0 (refl_equal T (THead (Bind Abst) u1 t1)) 
-(refl_equal T x)))))).
-
-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)).(let H0 \def (match H return (\lambda (t: T).(\lambda 
-(t0: T).(\lambda (_: (pr0 t t0)).((eq T t (THead (Flat Appl) u1 t1)) \to ((eq 
-T t0 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)))))))))))))) with [(pr0_refl t) \Rightarrow (\lambda (H0: 
-(eq T t (THead (Flat Appl) u1 t1))).(\lambda (H1: (eq T t x)).(eq_ind T 
-(THead (Flat Appl) u1 t1) (\lambda (t0: T).((eq T t0 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))))))))))) (\lambda (H2: (eq T (THead (Flat Appl) u1 t1) 
-x)).(eq_ind 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))) x H2)) t (sym_eq T t (THead (Flat Appl) 
-u1 t1) H0) H1))) | (pr0_comp u0 u2 H0 t0 t2 H1 k) \Rightarrow (\lambda (H2: 
-(eq T (THead k u0 t0) (THead (Flat Appl) u1 t1))).(\lambda (H3: (eq T (THead 
-k u2 t2) x)).((let H4 \def (f_equal T T (\lambda (e: T).(match e return 
-(\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 
-| (THead _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Flat Appl) u1 t1) 
-H2) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e return (\lambda 
-(_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead 
-_ t _) \Rightarrow t])) (THead k u0 t0) (THead (Flat Appl) u1 t1) H2) in 
-((let H6 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: T).K) 
-with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) 
-\Rightarrow k])) (THead k u0 t0) (THead (Flat Appl) u1 t1) H2) in (eq_ind K 
-(Flat Appl) (\lambda (k0: K).((eq T u0 u1) \to ((eq T t0 t1) \to ((eq T 
-(THead k0 u2 t2) x) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (or3 (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t3: T).(eq T x (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 x (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 x (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 (H7: (eq T u0 u1)).(eq_ind T u1 
-(\lambda (t: T).((eq T t0 t1) \to ((eq T (THead (Flat Appl) u2 t2) x) \to 
-((pr0 t u2) \to ((pr0 t0 t2) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda 
-(t3: T).(eq T x (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 x (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 x 
-(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 (H8: 
-(eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T (THead (Flat Appl) u2 t2) 
-x) \to ((pr0 u1 u2) \to ((pr0 t t2) \to (or3 (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t3: T).(eq T x (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 x (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 x (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 (H9: (eq T (THead (Flat Appl) u2 t2) 
-x)).(eq_ind T (THead (Flat Appl) u2 t2) (\lambda (t: T).((pr0 u1 u2) \to 
-((pr0 t1 t2) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t 
-(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 t (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 t (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 (H10: (pr0 u1 u2)).(\lambda (H11: 
-(pr0 t1 t2)).(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)) H10 H11)))) x H9)) t0 (sym_eq T t0 t1 H8))) u0 (sym_eq T u0 u1 H7))) 
-k (sym_eq K k (Flat Appl) H6))) H5)) H4)) H3 H0 H1))) | (pr0_beta u v1 v2 H0 
-t0 t2 H1) \Rightarrow (\lambda (H2: (eq T (THead (Flat Appl) v1 (THead (Bind 
-Abst) u t0)) (THead (Flat Appl) u1 t1))).(\lambda (H3: (eq T (THead (Bind 
-Abbr) v2 t2) x)).((let H4 \def (f_equal T T (\lambda (e: T).(match e 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) H2) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e 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) H2) in (eq_ind T u1 (\lambda (t: T).((eq T 
-(THead (Bind Abst) u t0) t1) \to ((eq T (THead (Bind Abbr) v2 t2) x) \to 
-((pr0 t v2) \to ((pr0 t0 t2) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t3: T).(eq T x (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 x (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 x 
-(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 (H6: 
-(eq T (THead (Bind Abst) u t0) t1)).(eq_ind T (THead (Bind Abst) u t0) 
-(\lambda (t: T).((eq T (THead (Bind Abbr) v2 t2) x) \to ((pr0 u1 v2) \to 
-((pr0 t0 t2) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x 
-(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 x (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 x (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 (H7: (eq T (THead (Bind Abbr) v2 t2) 
-x)).(eq_ind T (THead (Bind Abbr) v2 t2) (\lambda (t: T).((pr0 u1 v2) \to 
-((pr0 t0 t2) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t 
-(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 t (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 t 
-(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 (H8: (pr0 
-u1 v2)).(\lambda (H9: (pr0 t0 t2)).(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)) H8 H9)))) x H7)) t1 H6)) v1 (sym_eq T v1 u1 H5))) 
-H4)) H3 H0 H1))) | (pr0_upsilon b H0 v1 v2 H1 u0 u2 H2 t0 t2 H3) \Rightarrow 
-(\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead 
-(Flat Appl) u1 t1))).(\lambda (H5: (eq T (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t2)) x)).((let H6 \def (f_equal T T (\lambda (e: 
-T).(match e 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) H4) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e 
-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) H4) in (eq_ind T u1 (\lambda (t: 
-T).((eq T (THead (Bind b) u0 t0) t1) \to ((eq T (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t2)) x) \to ((not (eq B b Abst)) \to ((pr0 t 
-v2) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (or3 (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t3: T).(eq T x (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 x (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 x (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 (Bind b) u0 t0) 
-t1)).(eq_ind T (THead (Bind b) u0 t0) (\lambda (t: T).((eq T (THead (Bind b) 
-u2 (THead (Flat Appl) (lift (S O) O v2) t2)) x) \to ((not (eq B b Abst)) \to 
-((pr0 u1 v2) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (or3 (ex3_2 T T (\lambda 
-(u3: T).(\lambda (t3: T).(eq T x (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 x (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 x (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 (H9: (eq T (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t2)) x)).(eq_ind T (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t2)) (\lambda (t: T).((not (eq B b 
-Abst)) \to ((pr0 u1 v2) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (or3 (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t3: T).(eq T t (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 t (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 t (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 (H10: (not (eq B b Abst))).(\lambda (H11: (pr0 u1 
-v2)).(\lambda (H12: (pr0 u0 u2)).(\lambda (H13: (pr0 t0 t2)).(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 H10 (refl_equal T (THead (Bind b) 
-u0 t0)) (refl_equal T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
-t2))) H11 H12 H13)))))) x H9)) t1 H8)) v1 (sym_eq T v1 u1 H7))) H6)) H5 H0 H1 
-H2 H3))) | (pr0_delta u0 u2 H0 t0 t2 H1 w H2) \Rightarrow (\lambda (H3: (eq T 
-(THead (Bind Abbr) u0 t0) (THead (Flat Appl) u1 t1))).(\lambda (H4: (eq T 
-(THead (Bind Abbr) u2 w) x)).((let H5 \def (eq_ind T (THead (Bind Abbr) u0 
-t0) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
-(match k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat 
-_) \Rightarrow False])])) I (THead (Flat Appl) u1 t1) H3) in (False_ind ((eq 
-T (THead (Bind Abbr) u2 w) x) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to ((subst0 
-O u2 t2 w) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x 
-(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 x (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 x (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))))))))))))) H5)) H4 H0 H1 H2))) | (pr0_zeta b H0 t0 t2 H1 
-u) \Rightarrow (\lambda (H2: (eq T (THead (Bind b) u (lift (S O) O t0)) 
-(THead (Flat Appl) u1 t1))).(\lambda (H3: (eq T t2 x)).((let H4 \def (eq_ind 
-T (THead (Bind b) u (lift (S O) O t0)) (\lambda (e: T).(match e return 
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: 
-K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I 
-(THead (Flat Appl) u1 t1) H2) in (False_ind ((eq T t2 x) \to ((not (eq B b 
-Abst)) \to ((pr0 t0 t2) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T x (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 x (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 x (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)))))))))))) H4)) H3 H0 H1))) | 
-(pr0_epsilon t0 t2 H0 u) \Rightarrow (\lambda (H1: (eq T (THead (Flat Cast) u 
-t0) (THead (Flat Appl) u1 t1))).(\lambda (H2: (eq T t2 x)).((let H3 \def 
-(eq_ind T (THead (Flat Cast) u t0) (\lambda (e: T).(match e return (\lambda 
-(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
-| (THead k _ _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind 
-_) \Rightarrow False | (Flat f) \Rightarrow (match f return (\lambda (_: 
-F).Prop) with [Appl \Rightarrow False | Cast \Rightarrow True])])])) I (THead 
-(Flat Appl) u1 t1) H1) in (False_ind ((eq T t2 x) \to ((pr0 t0 t2) \to (or3 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (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 x (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 x (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))))))))))) H3)) H2 H0)))]) in (H0 (refl_equal T (THead 
-(Flat Appl) u1 t1)) (refl_equal T x)))))).
-
-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)).(let H0 \def (match H return (\lambda (t: T).(\lambda 
-(t0: T).(\lambda (_: (pr0 t t0)).((eq T t (THead (Flat Cast) u1 t1)) \to ((eq 
-T t0 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))))))) with [(pr0_refl t) 
-\Rightarrow (\lambda (H0: (eq T t (THead (Flat Cast) u1 t1))).(\lambda (H1: 
-(eq T t x)).(eq_ind T (THead (Flat Cast) u1 t1) (\lambda (t0: T).((eq T t0 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)))) (\lambda (H2: (eq T (THead 
-(Flat Cast) u1 t1) x)).(eq_ind 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))) x H2)) t (sym_eq T t 
-(THead (Flat Cast) u1 t1) H0) H1))) | (pr0_comp u0 u2 H0 t0 t2 H1 k) 
-\Rightarrow (\lambda (H2: (eq T (THead k u0 t0) (THead (Flat Cast) u1 
-t1))).(\lambda (H3: (eq T (THead k u2 t2) x)).((let H4 \def (f_equal T T 
-(\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) 
-(THead k u0 t0) (THead (Flat Cast) u1 t1) H2) in ((let H5 \def (f_equal T T 
-(\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) 
-(THead k u0 t0) (THead (Flat Cast) u1 t1) H2) in ((let H6 \def (f_equal T K 
-(\lambda (e: T).(match e return (\lambda (_: T).K) with [(TSort _) 
-\Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) 
-(THead k u0 t0) (THead (Flat Cast) u1 t1) H2) in (eq_ind K (Flat Cast) 
-(\lambda (k0: K).((eq T u0 u1) \to ((eq T t0 t1) \to ((eq T (THead k0 u2 t2) 
-x) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u3 t3)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
-t3)))) (pr0 t1 x)))))))) (\lambda (H7: (eq T u0 u1)).(eq_ind T u1 (\lambda 
-(t: T).((eq T t0 t1) \to ((eq T (THead (Flat Cast) u2 t2) x) \to ((pr0 t u2) 
-\to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x 
-(THead (Flat Cast) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) 
-(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 x))))))) (\lambda 
-(H8: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T (THead (Flat Cast) u2 
-t2) x) \to ((pr0 u1 u2) \to ((pr0 t t2) \to (or (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u3 t3)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
-t3)))) (pr0 t1 x)))))) (\lambda (H9: (eq T (THead (Flat Cast) u2 t2) 
-x)).(eq_ind T (THead (Flat Cast) u2 t2) (\lambda (t: T).((pr0 u1 u2) \to 
-((pr0 t1 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t 
-(THead (Flat Cast) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) 
-(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t))))) (\lambda (H10: 
-(pr0 u1 u2)).(\lambda (H11: (pr0 t1 t2)).(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)) H10 H11)))) x H9)) t0 (sym_eq T t0 t1 H8))) u0 
-(sym_eq T u0 u1 H7))) k (sym_eq K k (Flat Cast) H6))) H5)) H4)) H3 H0 H1))) | 
-(pr0_beta u v1 v2 H0 t0 t2 H1) \Rightarrow (\lambda (H2: (eq T (THead (Flat 
-Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Cast) u1 t1))).(\lambda (H3: 
-(eq T (THead (Bind Abbr) v2 t2) x)).((let H4 \def (eq_ind T (THead (Flat 
-Appl) v1 (THead (Bind Abst) u t0)) (\lambda (e: T).(match e return (\lambda 
-(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
-| (THead k _ _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind 
-_) \Rightarrow False | (Flat f) \Rightarrow (match f return (\lambda (_: 
-F).Prop) with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead 
-(Flat Cast) u1 t1) H2) in (False_ind ((eq T (THead (Bind Abbr) v2 t2) x) \to 
-((pr0 v1 v2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 
-x))))) H4)) H3 H0 H1))) | (pr0_upsilon b H0 v1 v2 H1 u0 u2 H2 t0 t2 H3) 
-\Rightarrow (\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) 
-(THead (Flat Cast) u1 t1))).(\lambda (H5: (eq T (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t2)) x)).((let H6 \def (eq_ind T (THead (Flat 
-Appl) v1 (THead (Bind b) u0 t0)) (\lambda (e: T).(match e return (\lambda (_: 
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
-(THead k _ _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind 
-_) \Rightarrow False | (Flat f) \Rightarrow (match f return (\lambda (_: 
-F).Prop) with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead 
-(Flat Cast) u1 t1) H4) in (False_ind ((eq T (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t2)) x) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to 
-((pr0 u0 u2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda 
-(t3: T).(eq T x (THead (Flat Cast) u3 t3)))) (\lambda (u3: T).(\lambda (_: 
-T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 
-x))))))) H6)) H5 H0 H1 H2 H3))) | (pr0_delta u0 u2 H0 t0 t2 H1 w H2) 
-\Rightarrow (\lambda (H3: (eq T (THead (Bind Abbr) u0 t0) (THead (Flat Cast) 
-u1 t1))).(\lambda (H4: (eq T (THead (Bind Abbr) u2 w) x)).((let H5 \def 
-(eq_ind T (THead (Bind Abbr) u0 t0) (\lambda (e: T).(match e return (\lambda 
-(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
-| (THead k _ _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind 
-_) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Cast) u1 
-t1) H3) in (False_ind ((eq T (THead (Bind Abbr) u2 w) x) \to ((pr0 u0 u2) \to 
-((pr0 t0 t2) \to ((subst0 O u2 t2 w) \to (or (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u3 t3)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
-t3)))) (pr0 t1 x)))))) H5)) H4 H0 H1 H2))) | (pr0_zeta b H0 t0 t2 H1 u) 
-\Rightarrow (\lambda (H2: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead 
-(Flat Cast) u1 t1))).(\lambda (H3: (eq T t2 x)).((let H4 \def (eq_ind T 
-(THead (Bind b) u (lift (S O) O t0)) (\lambda (e: T).(match e return (\lambda 
-(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
-| (THead k _ _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind 
-_) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Cast) u1 
-t1) H2) in (False_ind ((eq T t2 x) \to ((not (eq B b Abst)) \to ((pr0 t0 t2) 
-\to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat 
-Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 x))))) H4)) H3 H0 H1))) | 
-(pr0_epsilon t0 t2 H0 u) \Rightarrow (\lambda (H1: (eq T (THead (Flat Cast) u 
-t0) (THead (Flat Cast) u1 t1))).(\lambda (H2: (eq T t2 x)).((let H3 \def 
-(f_equal T T (\lambda (e: T).(match e 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) H1) in ((let H4 \def 
-(f_equal T T (\lambda (e: T).(match e 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) H1) in (eq_ind T u1 
-(\lambda (_: T).((eq T t0 t1) \to ((eq T t2 x) \to ((pr0 t0 t2) \to (or 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 x)))))) (\lambda (H5: (eq T t0 
-t1)).(eq_ind T t1 (\lambda (t: T).((eq T t2 x) \to ((pr0 t t2) \to (or (ex3_2 
-T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: 
-T).(pr0 t1 t3)))) (pr0 t1 x))))) (\lambda (H6: (eq T t2 x)).(eq_ind T x 
-(\lambda (t: T).((pr0 t1 t) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 
-u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 x)))) 
-(\lambda (H7: (pr0 t1 x)).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 x) 
-H7)) t2 (sym_eq T t2 x H6))) t0 (sym_eq T t0 t1 H5))) u (sym_eq T u u1 H4))) 
-H3)) H2 H0)))]) in (H0 (refl_equal T (THead (Flat Cast) u1 t1)) (refl_equal T 
-x)))))).
-
-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 (t0: T).(\lambda (t3: T).(\lambda 
-(_: (pr0 t0 t3)).(\lambda (H3: ((\forall (h: nat).(\forall (d: nat).(pr0 
-(lift h d t0) (lift h d t3)))))).(\lambda (k: K).(\lambda (h: nat).(\lambda 
-(d: nat).(eq_ind_r T (THead k (lift h d u1) (lift h (s k d) t0)) (\lambda (t: 
-T).(pr0 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).(pr0 (THead k (lift h d u1) (lift h (s k 
-d) t0)) t)) (pr0_comp (lift h d u1) (lift h d u2) (H1 h d) (lift h (s k d) 
-t0) (lift h (s k d) t3) (H3 h (s k d)) k) (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))))))))))))) (\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 (t0: T).(\lambda (t3: T).(\lambda (_: (pr0 
-t0 t3)).(\lambda (H3: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t0) 
-(lift h d t3)))))).(\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 
-t0))) (\lambda (t: T).(pr0 t (lift h d (THead (Bind Abbr) v2 t3)))) (eq_ind_r 
-T (THead (Bind Abst) (lift h (s (Flat Appl) d) u) (lift h (s (Bind Abst) (s 
-(Flat Appl) d)) t0)) (\lambda (t: T).(pr0 (THead (Flat Appl) (lift h d v1) t) 
-(lift h d (THead (Bind Abbr) v2 t3)))) (eq_ind_r T (THead (Bind Abbr) (lift h 
-d v2) (lift h (s (Bind Abbr) d) t3)) (\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)) t0))) 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)) t0) (lift h (s (Bind Abbr) d) t3) (H3 h (s (Bind Abbr) d))) (lift h d 
-(THead (Bind Abbr) v2 t3)) (lift_head (Bind Abbr) v2 t3 h d)) (lift h (s 
-(Flat Appl) d) (THead (Bind Abst) u t0)) (lift_head (Bind Abst) u t0 h (s 
-(Flat Appl) d))) (lift h d (THead (Flat Appl) v1 (THead (Bind Abst) u t0))) 
-(lift_head (Flat Appl) v1 (THead (Bind Abst) u t0) 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 (t0: T).(\lambda (t3: 
-T).(\lambda (_: (pr0 t0 t3)).(\lambda (H6: ((\forall (h: nat).(\forall (d: 
-nat).(pr0 (lift h d t0) (lift h d t3)))))).(\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 t0))) (\lambda (t: T).(pr0 t (lift h d (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t3))))) (eq_ind_r T (THead (Bind b) 
-(lift h (s (Flat Appl) d) u1) (lift h (s (Bind b) (s (Flat Appl) d)) t0)) 
-(\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) t3))))) (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) t3))) (\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)) 
-t0))) 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)) t3)) (\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)) t0))) (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 t0))) (THead (Bind b) (lift h d 
-u2) (THead (Flat Appl) (lift h n (lift (S O) O v2)) (lift h n t3))))) 
-(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) 
-t0))) (THead (Bind b) (lift h d u2) (THead (Flat Appl) t (lift h (plus (S O) 
-d) t3))))) (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) t0) (lift h (plus (S O) d) 
-t3) (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) t3)) (lift_head (Flat Appl) (lift (S 
-O) O v2) t3 h (s (Bind b) d))) (lift h d (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t3))) (lift_head (Bind b) u2 (THead (Flat Appl) (lift 
-(S O) O v2) t3) h d)) (lift h (s (Flat Appl) d) (THead (Bind b) u1 t0)) 
-(lift_head (Bind b) u1 t0 h (s (Flat Appl) d))) (lift h d (THead (Flat Appl) 
-v1 (THead (Bind b) u1 t0))) (lift_head (Flat Appl) v1 (THead (Bind b) u1 t0) 
-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 (t0: T).(\lambda (t3: T).(\lambda (_: (pr0 t0 
-t3)).(\lambda (H3: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t0) 
-(lift h d t3)))))).(\lambda (w: T).(\lambda (H4: (subst0 O u2 t3 w)).(\lambda 
-(h: nat).(\lambda (d: nat).(eq_ind_r T (THead (Bind Abbr) (lift h d u1) (lift 
-h (s (Bind Abbr) d) t0)) (\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) t0)) t)) (pr0_delta (lift h d u1) (lift h d u2) (H1 h d) (lift h (S 
-d) t0) (lift h (S d) t3) (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' t3) (lift h d' w))) (subst0_lift_lt t3 w u2 O H4 (S d) (lt_le_S O 
-(S d) (le_lt_n_Sm 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 t0)) (lift_head (Bind Abbr) u1 t0 h d)))))))))))))) (\lambda (b: 
-B).(\lambda (H0: (not (eq B b Abst))).(\lambda (t0: T).(\lambda (t3: 
-T).(\lambda (_: (pr0 t0 t3)).(\lambda (H2: ((\forall (h: nat).(\forall (d: 
-nat).(pr0 (lift h d t0) (lift h d t3)))))).(\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 t0))) (\lambda (t: T).(pr0 t (lift h d t3))) 
-(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 t0))) (lift h d t3))) (eq_ind_r T (lift (S O) O 
-(lift h d t0)) (\lambda (t: T).(pr0 (THead (Bind b) (lift h d u) t) (lift h d 
-t3))) (pr0_zeta b H0 (lift h d t0) (lift h d t3) (H2 h d) (lift h d u)) (lift 
-h (plus (S O) d) (lift (S O) O t0)) (lift_d t0 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 t0))) 
-(lift_head (Bind b) u (lift (S O) O t0) h d))))))))))) (\lambda (t0: 
-T).(\lambda (t3: T).(\lambda (_: (pr0 t0 t3)).(\lambda (H1: ((\forall (h: 
-nat).(\forall (d: nat).(pr0 (lift h d t0) (lift h d t3)))))).(\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) t0)) (\lambda (t: T).(pr0 t (lift h d t3))) 
-(pr0_epsilon (lift h (s (Flat Cast) d) t0) (lift h d t3) (H1 h d) (lift h d 
-u)) (lift h d (THead (Flat Cast) u t0)) (lift_head (Flat Cast) u t0 h 
-d))))))))) t1 t2 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)).(let H0 \def (match H return (\lambda (t: T).(\lambda 
-(t0: T).(\lambda (_: (pr0 t t0)).((eq T t (THead (Bind Abbr) u1 t1)) \to ((eq 
-T t0 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)))))))) with 
-[(pr0_refl t) \Rightarrow (\lambda (H0: (eq T t (THead (Bind Abbr) u1 
-t1))).(\lambda (H1: (eq T t x)).(eq_ind T (THead (Bind Abbr) u1 t1) (\lambda 
-(t0: T).((eq T t0 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))))) (\lambda (H2: (eq T (THead (Bind Abbr) u1 t1) x)).(eq_ind 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 (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y 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 (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y 
-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 (y: T).(pr0 t1 y)) 
-(\lambda (y: T).(subst0 O u2 y t2)))))) u1 t1 (refl_equal T (THead (Bind 
-Abbr) u1 t1)) (pr0_refl u1) (or_introl (pr0 t1 t1) (ex2 T (\lambda (y: 
-T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u1 y t1))) (pr0_refl t1)))) x H2)) t 
-(sym_eq T t (THead (Bind Abbr) u1 t1) H0) H1))) | (pr0_comp u0 u2 H0 t0 t2 H1 
-k) \Rightarrow (\lambda (H2: (eq T (THead k u0 t0) (THead (Bind Abbr) u1 
-t1))).(\lambda (H3: (eq T (THead k u2 t2) x)).((let H4 \def (f_equal T T 
-(\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) 
-(THead k u0 t0) (THead (Bind Abbr) u1 t1) H2) in ((let H5 \def (f_equal T T 
-(\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) 
-(THead k u0 t0) (THead (Bind Abbr) u1 t1) H2) in ((let H6 \def (f_equal T K 
-(\lambda (e: T).(match e return (\lambda (_: T).K) with [(TSort _) 
-\Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) 
-(THead k u0 t0) (THead (Bind Abbr) u1 t1) H2) in (eq_ind K (Bind Abbr) 
-(\lambda (k0: K).((eq T u0 u1) \to ((eq T t0 t1) \to ((eq T (THead k0 u2 t2) 
-x) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t3: T).(eq T x (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 (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u3 y 
-t3))))))) (pr0 t1 (lift (S O) O x))))))))) (\lambda (H7: (eq T u0 
-u1)).(eq_ind T u1 (\lambda (t: T).((eq T t0 t1) \to ((eq T (THead (Bind Abbr) 
-u2 t2) x) \to ((pr0 t u2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t3: T).(eq T x (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 (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u3 y 
-t3))))))) (pr0 t1 (lift (S O) O x)))))))) (\lambda (H8: (eq T t0 t1)).(eq_ind 
-T t1 (\lambda (t: T).((eq T (THead (Bind Abbr) u2 t2) x) \to ((pr0 u1 u2) \to 
-((pr0 t t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x 
-(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 (y: T).(pr0 
-t1 y)) (\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O 
-x))))))) (\lambda (H9: (eq T (THead (Bind Abbr) u2 t2) x)).(eq_ind T (THead 
-(Bind Abbr) u2 t2) (\lambda (t: T).((pr0 u1 u2) \to ((pr0 t1 t2) \to (or 
-(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t (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 (y: T).(pr0 t1 y)) 
-(\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O t)))))) 
-(\lambda (H10: (pr0 u1 u2)).(\lambda (H11: (pr0 t1 t2)).(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 (y: T).(pr0 t1 y)) 
-(\lambda (y: T).(subst0 O u3 y 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 (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u3 y 
-t3)))))) u2 t2 (refl_equal T (THead (Bind Abbr) u2 t2)) H10 (or_introl (pr0 
-t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y 
-t2))) H11))))) x H9)) t0 (sym_eq T t0 t1 H8))) u0 (sym_eq T u0 u1 H7))) k 
-(sym_eq K k (Bind Abbr) H6))) H5)) H4)) H3 H0 H1))) | (pr0_beta u v1 v2 H0 t0 
-t2 H1) \Rightarrow (\lambda (H2: (eq T (THead (Flat Appl) v1 (THead (Bind 
-Abst) u t0)) (THead (Bind Abbr) u1 t1))).(\lambda (H3: (eq T (THead (Bind 
-Abbr) v2 t2) x)).((let H4 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind 
-Abst) u t0)) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
-\Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow 
-False | (Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1 t1) H2) in 
-(False_ind ((eq T (THead (Bind Abbr) v2 t2) x) \to ((pr0 v1 v2) \to ((pr0 t0 
-t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (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 (y: T).(pr0 t1 y)) 
-(\lambda (y: T).(subst0 O u2 y t3))))))) (pr0 t1 (lift (S O) O x)))))) H4)) 
-H3 H0 H1))) | (pr0_upsilon b H0 v1 v2 H1 u0 u2 H2 t0 t2 H3) \Rightarrow 
-(\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead 
-(Bind Abbr) u1 t1))).(\lambda (H5: (eq T (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t2)) x)).((let H6 \def (eq_ind T (THead (Flat Appl) 
-v1 (THead (Bind b) u0 t0)) (\lambda (e: T).(match e return (\lambda (_: 
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
-(THead k _ _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind 
-_) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1 
-t1) H4) in (False_ind ((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
-O) O v2) t2)) x) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u2) 
-\to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x 
-(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 (y: T).(pr0 
-t1 y)) (\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O 
-x)))))))) H6)) H5 H0 H1 H2 H3))) | (pr0_delta u0 u2 H0 t0 t2 H1 w H2) 
-\Rightarrow (\lambda (H3: (eq T (THead (Bind Abbr) u0 t0) (THead (Bind Abbr) 
-u1 t1))).(\lambda (H4: (eq T (THead (Bind Abbr) u2 w) x)).((let H5 \def 
-(f_equal T T (\lambda (e: T).(match e 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) H3) in ((let H6 \def 
-(f_equal T T (\lambda (e: T).(match e 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) H3) in (eq_ind T u1 
-(\lambda (t: T).((eq T t0 t1) \to ((eq T (THead (Bind Abbr) u2 w) x) \to 
-((pr0 t u2) \to ((pr0 t0 t2) \to ((subst0 O u2 t2 w) \to (or (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t3: T).(eq T x (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 (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 
-O u3 y t3))))))) (pr0 t1 (lift (S O) O x))))))))) (\lambda (H7: (eq T t0 
-t1)).(eq_ind T t1 (\lambda (t: T).((eq T (THead (Bind Abbr) u2 w) x) \to 
-((pr0 u1 u2) \to ((pr0 t t2) \to ((subst0 O u2 t2 w) \to (or (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t3: T).(eq T x (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 (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 
-O u3 y t3))))))) (pr0 t1 (lift (S O) O x)))))))) (\lambda (H8: (eq T (THead 
-(Bind Abbr) u2 w) x)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t: 
-T).((pr0 u1 u2) \to ((pr0 t1 t2) \to ((subst0 O u2 t2 w) \to (or (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t3: T).(eq T t (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 (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 
-O u3 y t3))))))) (pr0 t1 (lift (S O) O t))))))) (\lambda (H9: (pr0 u1 
-u2)).(\lambda (H10: (pr0 t1 t2)).(\lambda (H11: (subst0 O u2 t2 
-w)).(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 (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u3 y 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 (y: T).(pr0 t1 y)) 
-(\lambda (y: T).(subst0 O u3 y t3)))))) u2 w (refl_equal T (THead (Bind Abbr) 
-u2 w)) H9 (or_intror (pr0 t1 w) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda 
-(y: T).(subst0 O u2 y w))) (ex_intro2 T (\lambda (y: T).(pr0 t1 y)) (\lambda 
-(y: T).(subst0 O u2 y w)) t2 H10 H11))))))) x H8)) t0 (sym_eq T t0 t1 H7))) 
-u0 (sym_eq T u0 u1 H6))) H5)) H4 H0 H1 H2))) | (pr0_zeta b H0 t0 t2 H1 u) 
-\Rightarrow (\lambda (H2: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead 
-(Bind Abbr) u1 t1))).(\lambda (H3: (eq T t2 x)).((let H4 \def (f_equal T T 
-(\lambda (e: T).(match e 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 u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) 
-t0))]) in lref_map) (\lambda (x: nat).(plus x (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 u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) 
-t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t0) | (THead _ _ t) 
-\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1 
-t1) H2) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e 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) H2) in ((let H6 \def (f_equal T B (\lambda (e: T).(match e 
-return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | (TLRef _) 
-\Rightarrow b | (THead k _ _) \Rightarrow (match k return (\lambda (_: K).B) 
-with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b])])) (THead (Bind b) u 
-(lift (S O) O t0)) (THead (Bind Abbr) u1 t1) H2) in (eq_ind B Abbr (\lambda 
-(b0: B).((eq T u u1) \to ((eq T (lift (S O) O t0) t1) \to ((eq T t2 x) \to 
-((not (eq B b0 Abst)) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T x (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 (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y 
-t3))))))) (pr0 t1 (lift (S O) O x))))))))) (\lambda (H7: (eq T u u1)).(eq_ind 
-T u1 (\lambda (_: T).((eq T (lift (S O) O t0) t1) \to ((eq T t2 x) \to ((not 
-(eq B Abbr Abst)) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T x (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 (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y 
-t3))))))) (pr0 t1 (lift (S O) O x)))))))) (\lambda (H8: (eq T (lift (S O) O 
-t0) t1)).(eq_ind T (lift (S O) O t0) (\lambda (t: T).((eq T t2 x) \to ((not 
-(eq B Abbr Abst)) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T x (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 (y: T).(pr0 t y)) (\lambda (y: T).(subst0 O u2 y 
-t3))))))) (pr0 t (lift (S O) O x))))))) (\lambda (H9: (eq T t2 x)).(eq_ind T 
-x (\lambda (t: T).((not (eq B Abbr Abst)) \to ((pr0 t0 t) \to (or (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T x (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 (y: T).(pr0 (lift (S O) O 
-t0) y)) (\lambda (y: T).(subst0 O u2 y t3))))))) (pr0 (lift (S O) O t0) (lift 
-(S O) O x)))))) (\lambda (_: (not (eq B Abbr Abst))).(\lambda (H11: (pr0 t0 
-x)).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (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 (y: 
-T).(pr0 (lift (S O) O t0) y)) (\lambda (y: T).(subst0 O u2 y t3))))))) (pr0 
-(lift (S O) O t0) (lift (S O) O x)) (pr0_lift t0 x H11 (S O) O)))) t2 (sym_eq 
-T t2 x H9))) t1 H8)) u (sym_eq T u u1 H7))) b (sym_eq B b Abbr H6))) H5)) 
-H4)) H3 H0 H1))) | (pr0_epsilon t0 t2 H0 u) \Rightarrow (\lambda (H1: (eq T 
-(THead (Flat Cast) u t0) (THead (Bind Abbr) u1 t1))).(\lambda (H2: (eq T t2 
-x)).((let H3 \def (eq_ind T (THead (Flat Cast) u t0) (\lambda (e: T).(match e 
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: 
-K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I 
-(THead (Bind Abbr) u1 t1) H1) in (False_ind ((eq T t2 x) \to ((pr0 t0 t2) \to 
-(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (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 (y: T).(pr0 t1 y)) 
-(\lambda (y: T).(subst0 O u2 y t3))))))) (pr0 t1 (lift (S O) O x))))) H3)) H2 
-H0)))]) in (H0 (refl_equal T (THead (Bind Abbr) u1 t1)) (refl_equal T x)))))).
-
-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)).(let H0 \def (match H return (\lambda (t: T).(\lambda 
-(t0: T).(\lambda (_: (pr0 t t0)).((eq T t (THead (Bind Void) u1 t1)) \to ((eq 
-T t0 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)))))))) with 
-[(pr0_refl t) \Rightarrow (\lambda (H0: (eq T t (THead (Bind Void) u1 
-t1))).(\lambda (H1: (eq T t x)).(eq_ind T (THead (Bind Void) u1 t1) (\lambda 
-(t0: T).((eq T t0 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))))) (\lambda (H2: (eq T (THead (Bind Void) u1 t1) x)).(eq_ind 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))) x H2)) t (sym_eq T t (THead (Bind Void) 
-u1 t1) H0) H1))) | (pr0_comp u0 u2 H0 t0 t2 H1 k) \Rightarrow (\lambda (H2: 
-(eq T (THead k u0 t0) (THead (Bind Void) u1 t1))).(\lambda (H3: (eq T (THead 
-k u2 t2) x)).((let H4 \def (f_equal T T (\lambda (e: T).(match e return 
-(\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 
-| (THead _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Bind Void) u1 t1) 
-H2) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e return (\lambda 
-(_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead 
-_ t _) \Rightarrow t])) (THead k u0 t0) (THead (Bind Void) u1 t1) H2) in 
-((let H6 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: T).K) 
-with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) 
-\Rightarrow k])) (THead k u0 t0) (THead (Bind Void) u1 t1) H2) in (eq_ind K 
-(Bind Void) (\lambda (k0: K).((eq T u0 u1) \to ((eq T t0 t1) \to ((eq T 
-(THead k0 u2 t2) x) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (or (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t3: T).(eq T x (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 x))))))))) (\lambda (H7: (eq T u0 
-u1)).(eq_ind T u1 (\lambda (t: T).((eq T t0 t1) \to ((eq T (THead (Bind Void) 
-u2 t2) x) \to ((pr0 t u2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t3: T).(eq T x (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 x)))))))) (\lambda (H8: (eq T t0 t1)).(eq_ind T 
-t1 (\lambda (t: T).((eq T (THead (Bind Void) u2 t2) x) \to ((pr0 u1 u2) \to 
-((pr0 t t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x 
-(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 
-x))))))) (\lambda (H9: (eq T (THead (Bind Void) u2 t2) x)).(eq_ind T (THead 
-(Bind Void) u2 t2) (\lambda (t: T).((pr0 u1 u2) \to ((pr0 t1 t2) \to (or 
-(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t (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 t)))))) (\lambda 
-(H10: (pr0 u1 u2)).(\lambda (H11: (pr0 t1 t2)).(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)) H10 H11)))) x H9)) t0 (sym_eq T t0 
-t1 H8))) u0 (sym_eq T u0 u1 H7))) k (sym_eq K k (Bind Void) H6))) H5)) H4)) 
-H3 H0 H1))) | (pr0_beta u v1 v2 H0 t0 t2 H1) \Rightarrow (\lambda (H2: (eq T 
-(THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Bind Void) u1 
-t1))).(\lambda (H3: (eq T (THead (Bind Abbr) v2 t2) x)).((let H4 \def (eq_ind 
-T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (e: T).(match e 
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: 
-K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I 
-(THead (Bind Void) u1 t1) H2) in (False_ind ((eq T (THead (Bind Abbr) v2 t2) 
-x) \to ((pr0 v1 v2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T x (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 x)))))) H4)) H3 H0 H1))) | (pr0_upsilon b H0 v1 
-v2 H1 u0 u2 H2 t0 t2 H3) \Rightarrow (\lambda (H4: (eq T (THead (Flat Appl) 
-v1 (THead (Bind b) u0 t0)) (THead (Bind Void) u1 t1))).(\lambda (H5: (eq T 
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) x)).((let H6 
-\def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (\lambda (e: 
-T).(match e return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | 
-(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return 
-(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow 
-True])])) I (THead (Bind Void) u1 t1) H4) in (False_ind ((eq T (THead (Bind 
-b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) x) \to ((not (eq B b Abst)) 
-\to ((pr0 v1 v2) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (or (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t3: T).(eq T x (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 x)))))))) H6)) H5 H0 H1 H2 H3))) | 
-(pr0_delta u0 u2 H0 t0 t2 H1 w H2) \Rightarrow (\lambda (H3: (eq T (THead 
-(Bind Abbr) u0 t0) (THead (Bind Void) u1 t1))).(\lambda (H4: (eq T (THead 
-(Bind Abbr) u2 w) x)).((let H5 \def (eq_ind T (THead (Bind Abbr) u0 t0) 
-(\lambda (e: T).(match e return (\lambda (_: T).Prop) with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
-(match k return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b 
-return (\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow 
-False | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (THead 
-(Bind Void) u1 t1) H3) in (False_ind ((eq T (THead (Bind Abbr) u2 w) x) \to 
-((pr0 u0 u2) \to ((pr0 t0 t2) \to ((subst0 O u2 t2 w) \to (or (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t3: T).(eq T x (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 x))))))) H5)) H4 H0 H1 H2))) | 
-(pr0_zeta b H0 t0 t2 H1 u) \Rightarrow (\lambda (H2: (eq T (THead (Bind b) u 
-(lift (S O) O t0)) (THead (Bind Void) u1 t1))).(\lambda (H3: (eq T t2 
-x)).((let H4 \def (f_equal T T (\lambda (e: T).(match e 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 u t0) \Rightarrow (THead k (lref_map f 
-d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (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 u t0) \Rightarrow (THead k (lref_map f 
-d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S 
-O))) O t0) | (THead _ _ t) \Rightarrow t])) (THead (Bind b) u (lift (S O) O 
-t0)) (THead (Bind Void) u1 t1) H2) in ((let H5 \def (f_equal T T (\lambda (e: 
-T).(match e 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) H2) in ((let H6 \def (f_equal T B 
-(\lambda (e: T).(match e return (\lambda (_: T).B) with [(TSort _) 
-\Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k 
-return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow 
-b])])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Void) u1 t1) H2) in 
-(eq_ind B Void (\lambda (b0: B).((eq T u u1) \to ((eq T (lift (S O) O t0) t1) 
-\to ((eq T t2 x) \to ((not (eq B b0 Abst)) \to ((pr0 t0 t2) \to (or (ex3_2 T 
-T (\lambda (u2: T).(\lambda (t3: T).(eq T x (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 x))))))))) (\lambda (H7: (eq T u 
-u1)).(eq_ind T u1 (\lambda (_: T).((eq T (lift (S O) O t0) t1) \to ((eq T t2 
-x) \to ((not (eq B Void Abst)) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t3: T).(eq T x (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 x)))))))) (\lambda (H8: (eq T (lift (S O) O t0) 
-t1)).(eq_ind T (lift (S O) O t0) (\lambda (t: T).((eq T t2 x) \to ((not (eq B 
-Void Abst)) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t3: T).(eq T x (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 x))))))) (\lambda (H9: (eq T t2 x)).(eq_ind T x (\lambda (t: T).((not 
-(eq B Void Abst)) \to ((pr0 t0 t) \to (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T x (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 x)))))) (\lambda (_: 
-(not (eq B Void Abst))).(\lambda (H11: (pr0 t0 x)).(or_intror (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T x (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 x)) 
-(pr0_lift t0 x H11 (S O) O)))) t2 (sym_eq T t2 x H9))) t1 H8)) u (sym_eq T u 
-u1 H7))) b (sym_eq B b Void H6))) H5)) H4)) H3 H0 H1))) | (pr0_epsilon t0 t2 
-H0 u) \Rightarrow (\lambda (H1: (eq T (THead (Flat Cast) u t0) (THead (Bind 
-Void) u1 t1))).(\lambda (H2: (eq T t2 x)).((let H3 \def (eq_ind T (THead 
-(Flat Cast) u t0) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
-\Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow 
-False | (Flat _) \Rightarrow True])])) I (THead (Bind Void) u1 t1) H1) in 
-(False_ind ((eq T t2 x) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T x (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 x))))) H3)) H2 H0)))]) in (H0 (refl_equal T 
-(THead (Bind Void) u1 t1)) (refl_equal T x)))))).
-
-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)) (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: 
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-(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 
-(x: T).(\forall (x0: nat).((eq T t2 (lift h x0 x)) \to (ex2 T (\lambda (t2: 
-T).(eq T t3 (lift h x0 t2))) (\lambda (t2: T).(pr0 x t2)))))))).(\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_comm 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 
-(x: T).(\forall (x0: nat).((eq T t2 (lift h x0 x)) \to (ex2 T (\lambda (t2: 
-T).(eq T t3 (lift h x0 t2))) (\lambda (t2: T).(pr0 x t2)))))))).(\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_epsilon 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))))).
-
-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 (u0: T).(\lambda 
-(u1: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v u1 u0)).(\lambda (H1: 
-((\forall (u2: T).((pr0 u2 v) \to (ex2 T (\lambda (t: T).(subst0 i0 u2 u1 t)) 
-(\lambda (t: T).(pr0 t u0))))))).(\lambda (t: T).(\lambda (k: K).(\lambda 
-(u3: T).(\lambda (H2: (pr0 u3 v)).(ex2_ind T (\lambda (t0: T).(subst0 i0 u3 
-u1 t0)) (\lambda (t0: T).(pr0 t0 u0)) (ex2 T (\lambda (t0: T).(subst0 i0 u3 
-(THead k u1 t) t0)) (\lambda (t0: T).(pr0 t0 (THead k u0 t)))) (\lambda (x: 
-T).(\lambda (H3: (subst0 i0 u3 u1 x)).(\lambda (H4: (pr0 x u0)).(ex_intro2 T 
-(\lambda (t0: T).(subst0 i0 u3 (THead k u1 t) t0)) (\lambda (t0: T).(pr0 t0 
-(THead k u0 t))) (THead k x t) (subst0_fst u3 x u1 i0 H3 t k) (pr0_comp x u0 
-H4 t t (pr0_refl t) k))))) (H1 u3 H2)))))))))))) (\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 (u1: T).((pr0 u1 v) \to (ex2 T 
-(\lambda (t: T).(subst0 (s k i0) u1 t3 t)) (\lambda (t: T).(pr0 t 
-t0))))))).(\lambda (u: T).(\lambda (u1: T).(\lambda (H2: (pr0 u1 v)).(ex2_ind 
-T (\lambda (t: T).(subst0 (s k i0) u1 t3 t)) (\lambda (t: T).(pr0 t t0)) (ex2 
-T (\lambda (t: T).(subst0 i0 u1 (THead k u t3) t)) (\lambda (t: T).(pr0 t 
-(THead k u t0)))) (\lambda (x: T).(\lambda (H3: (subst0 (s k i0) u1 t3 
-x)).(\lambda (H4: (pr0 x t0)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 
-(THead k u t3) t)) (\lambda (t: T).(pr0 t (THead k u t0))) (THead k u x) 
-(subst0_snd k u1 x t3 i0 H3 u) (pr0_comp u u (pr0_refl u) x t0 H4 k))))) (H1 
-u1 H2)))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u0: T).(\lambda 
-(i0: nat).(\lambda (_: (subst0 i0 v u1 u0)).(\lambda (H1: ((\forall (u2: 
-T).((pr0 u2 v) \to (ex2 T (\lambda (t: T).(subst0 i0 u2 u1 t)) (\lambda (t: 
-T).(pr0 t u0))))))).(\lambda (k: K).(\lambda (t0: T).(\lambda (t3: 
-T).(\lambda (_: (subst0 (s k i0) v t0 t3)).(\lambda (H3: ((\forall (u1: 
-T).((pr0 u1 v) \to (ex2 T (\lambda (t: T).(subst0 (s k i0) u1 t0 t)) (\lambda 
-(t: T).(pr0 t t3))))))).(\lambda (u3: T).(\lambda (H4: (pr0 u3 v)).(ex2_ind T 
-(\lambda (t: T).(subst0 (s k i0) u3 t0 t)) (\lambda (t: T).(pr0 t t3)) (ex2 T 
-(\lambda (t: T).(subst0 i0 u3 (THead k u1 t0) t)) (\lambda (t: T).(pr0 t 
-(THead k u0 t3)))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i0) u3 t0 
-x)).(\lambda (H6: (pr0 x t3)).(ex2_ind T (\lambda (t: T).(subst0 i0 u3 u1 t)) 
-(\lambda (t: T).(pr0 t u0)) (ex2 T (\lambda (t: T).(subst0 i0 u3 (THead k u1 
-t0) t)) (\lambda (t: T).(pr0 t (THead k u0 t3)))) (\lambda (x0: T).(\lambda 
-(H7: (subst0 i0 u3 u1 x0)).(\lambda (H8: (pr0 x0 u0)).(ex_intro2 T (\lambda 
-(t: T).(subst0 i0 u3 (THead k u1 t0) t)) (\lambda (t: T).(pr0 t (THead k u0 
-t3))) (THead k x0 x) (subst0_both u3 u1 x0 i0 H7 k t0 x H5) (pr0_comp x0 u0 
-H8 x t3 H6 k))))) (H1 u3 H4))))) (H3 u3 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 (u0: T).(\lambda 
-(u1: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v u1 u0)).(\lambda (H1: 
-((\forall (u2: T).((pr0 v u2) \to (ex2 T (\lambda (t: T).(subst0 i0 u2 u1 t)) 
-(\lambda (t: T).(pr0 u0 t))))))).(\lambda (t: T).(\lambda (k: K).(\lambda 
-(u3: T).(\lambda (H2: (pr0 v u3)).(ex2_ind T (\lambda (t0: T).(subst0 i0 u3 
-u1 t0)) (\lambda (t0: T).(pr0 u0 t0)) (ex2 T (\lambda (t0: T).(subst0 i0 u3 
-(THead k u1 t) t0)) (\lambda (t0: T).(pr0 (THead k u0 t) t0))) (\lambda (x: 
-T).(\lambda (H3: (subst0 i0 u3 u1 x)).(\lambda (H4: (pr0 u0 x)).(ex_intro2 T 
-(\lambda (t0: T).(subst0 i0 u3 (THead k u1 t) t0)) (\lambda (t0: T).(pr0 
-(THead k u0 t) t0)) (THead k x t) (subst0_fst u3 x u1 i0 H3 t k) (pr0_comp u0 
-x H4 t t (pr0_refl t) k))))) (H1 u3 H2)))))))))))) (\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 (u1: T).((pr0 v u1) \to 
-(ex2 T (\lambda (t: T).(subst0 (s k i0) u1 t3 t)) (\lambda (t: T).(pr0 t0 
-t))))))).(\lambda (u: T).(\lambda (u1: T).(\lambda (H2: (pr0 v u1)).(ex2_ind 
-T (\lambda (t: T).(subst0 (s k i0) u1 t3 t)) (\lambda (t: T).(pr0 t0 t)) (ex2 
-T (\lambda (t: T).(subst0 i0 u1 (THead k u t3) t)) (\lambda (t: T).(pr0 
-(THead k u t0) t))) (\lambda (x: T).(\lambda (H3: (subst0 (s k i0) u1 t3 
-x)).(\lambda (H4: (pr0 t0 x)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 
-(THead k u t3) t)) (\lambda (t: T).(pr0 (THead k u t0) t)) (THead k u x) 
-(subst0_snd k u1 x t3 i0 H3 u) (pr0_comp u u (pr0_refl u) t0 x H4 k))))) (H1 
-u1 H2)))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u0: T).(\lambda 
-(i0: nat).(\lambda (_: (subst0 i0 v u1 u0)).(\lambda (H1: ((\forall (u2: 
-T).((pr0 v u2) \to (ex2 T (\lambda (t: T).(subst0 i0 u2 u1 t)) (\lambda (t: 
-T).(pr0 u0 t))))))).(\lambda (k: K).(\lambda (t0: T).(\lambda (t3: 
-T).(\lambda (_: (subst0 (s k i0) v t0 t3)).(\lambda (H3: ((\forall (u1: 
-T).((pr0 v u1) \to (ex2 T (\lambda (t: T).(subst0 (s k i0) u1 t0 t)) (\lambda 
-(t: T).(pr0 t3 t))))))).(\lambda (u3: T).(\lambda (H4: (pr0 v u3)).(ex2_ind T 
-(\lambda (t: T).(subst0 (s k i0) u3 t0 t)) (\lambda (t: T).(pr0 t3 t)) (ex2 T 
-(\lambda (t: T).(subst0 i0 u3 (THead k u1 t0) t)) (\lambda (t: T).(pr0 (THead 
-k u0 t3) t))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i0) u3 t0 
-x)).(\lambda (H6: (pr0 t3 x)).(ex2_ind T (\lambda (t: T).(subst0 i0 u3 u1 t)) 
-(\lambda (t: T).(pr0 u0 t)) (ex2 T (\lambda (t: T).(subst0 i0 u3 (THead k u1 
-t0) t)) (\lambda (t: T).(pr0 (THead k u0 t3) t))) (\lambda (x0: T).(\lambda 
-(H7: (subst0 i0 u3 u1 x0)).(\lambda (H8: (pr0 u0 x0)).(ex_intro2 T (\lambda 
-(t: T).(subst0 i0 u3 (THead k u1 t0) t)) (\lambda (t: T).(pr0 (THead k u0 t3) 
-t)) (THead k x0 x) (subst0_both u3 u1 x0 i0 H7 k t0 x H5) (pr0_comp u0 x0 H8 
-t3 x H6 k))))) (H1 u3 H4))))) (H3 u3 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: 
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-(f_equal nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O i))) in 
-(let H16 \def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n v2 w 
-x1)) H14 (S i) H15) in (or_intror (pr0 (THead (Bind Abbr) x t3) (THead (Bind 
-Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x t3) w2)) 
-(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))) (ex_intro2 T 
-(\lambda (w2: T).(pr0 (THead (Bind Abbr) x t3) w2)) (\lambda (w2: T).(subst0 
-i v2 (THead (Bind Abbr) u2 w) w2)) (THead (Bind Abbr) x0 x1) (pr0_delta x x0 
-H11 t3 t4 H2 x1 H13) (subst0_both v2 u2 x0 i H12 (Bind Abbr) w x1 H16)))))))) 
-(subst0_subst0_back t4 w u2 O H4 x0 v2 i H12))))) H10)) (H1 v1 x i H9 v2 H6)) 
-w1 H8)))) H7)) (\lambda (H7: (ex2 T (\lambda (t2: T).(eq T w1 (THead (Bind 
-Abbr) u1 t2))) (\lambda (t2: T).(subst0 (s (Bind Abbr) i) v1 t3 
-t2)))).(ex2_ind T (\lambda (t5: T).(eq T w1 (THead (Bind Abbr) u1 t5))) 
-(\lambda (t5: T).(subst0 (s (Bind Abbr) i) v1 t3 t5)) (or (pr0 w1 (THead 
-(Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: 
-T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x: T).(\lambda (H8: 
-(eq T w1 (THead (Bind Abbr) u1 x))).(\lambda (H9: (subst0 (s (Bind Abbr) i) 
-v1 t3 x)).(eq_ind_r T (THead (Bind Abbr) u1 x) (\lambda (t: T).(or (pr0 t 
-(THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: 
-T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))))) (or_ind (pr0 x t4) (ex2 T 
-(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind Abbr) i) v2 t4 
-w2))) (or (pr0 (THead (Bind Abbr) u1 x) (THead (Bind Abbr) u2 w)) (ex2 T 
-(\lambda (w2: T).(pr0 (THead (Bind Abbr) u1 x) w2)) (\lambda (w2: T).(subst0 
-i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (H10: (pr0 x t4)).(or_introl 
-(pr0 (THead (Bind Abbr) u1 x) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: 
-T).(pr0 (THead (Bind Abbr) u1 x) w2)) (\lambda (w2: T).(subst0 i v2 (THead 
-(Bind Abbr) u2 w) w2))) (pr0_delta u1 u2 H0 x t4 H10 w H4))) (\lambda (H10: 
-(ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind Abbr) 
-i) v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: 
-T).(subst0 (s (Bind Abbr) i) v2 t4 w2)) (or (pr0 (THead (Bind Abbr) u1 x) 
-(THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) u1 
-x) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) 
-(\lambda (x0: T).(\lambda (H11: (pr0 x x0)).(\lambda (H12: (subst0 (s (Bind 
-Abbr) i) v2 t4 x0)).(ex2_ind T (\lambda (t: T).(subst0 O u2 x0 t)) (\lambda 
-(t: T).(subst0 (s (Bind Abbr) i) v2 w t)) (or (pr0 (THead (Bind Abbr) u1 x) 
-(THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) u1 
-x) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) 
-(\lambda (x1: T).(\lambda (H13: (subst0 O u2 x0 x1)).(\lambda (H14: (subst0 
-(s (Bind Abbr) i) v2 w x1)).(or_intror (pr0 (THead (Bind Abbr) u1 x) (THead 
-(Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) u1 x) w2)) 
-(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))) (ex_intro2 T 
-(\lambda (w2: T).(pr0 (THead (Bind Abbr) u1 x) w2)) (\lambda (w2: T).(subst0 
-i v2 (THead (Bind Abbr) u2 w) w2)) (THead (Bind Abbr) u2 x1) (pr0_delta u1 u2 
-H0 x x0 H11 x1 H13) (subst0_snd (Bind Abbr) v2 x1 w i H14 u2)))))) 
-(subst0_confluence_neq t4 x0 v2 (s (Bind Abbr) i) H12 w u2 O H4 (sym_not_eq 
-nat O (S i) (O_S i))))))) H10)) (H3 v1 x (s (Bind Abbr) i) H9 v2 H6)) w1 
-H8)))) H7)) (\lambda (H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T 
-w1 (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v1 
-u1 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Bind Abbr) i) v1 t3 
-t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead 
-(Bind Abbr) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v1 u1 u3))) 
-(\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind Abbr) i) v1 t3 t5))) (or 
-(pr0 w1 (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) 
-(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x0: 
-T).(\lambda (x1: T).(\lambda (H8: (eq T w1 (THead (Bind Abbr) x0 
-x1))).(\lambda (H9: (subst0 i v1 u1 x0)).(\lambda (H10: (subst0 (s (Bind 
-Abbr) i) v1 t3 x1)).(eq_ind_r T (THead (Bind Abbr) x0 x1) (\lambda (t: T).(or 
-(pr0 t (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda 
-(w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))))) (or_ind (pr0 x1 t4) 
-(ex2 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 (H11: (pr0 x1 
-t4)).(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 (H12: 
-(pr0 x0 u2)).(or_introl (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))) (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 (t2: T).(eq T w1 (THead (Bind b) u t2))) (\lambda (t2: 
-T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t2)))).(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)).(eq_ind_r T (THead (Bind 
-b) 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))))) (eq_ind_r T (lift (S O) O x0) 
-(\lambda (t: T).(or (pr0 (THead (Bind b) u t) t4) (ex2 T (\lambda (w2: 
-T).(pr0 (THead (Bind b) u t) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))))) 
-(let H10 \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 (H11: (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 H11 u))) (\lambda (H11: (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 (H12: (pr0 x0 
-x1)).(\lambda (H13: (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 H12 u) H13))))) H11)) (H2 v1 
-x0 i H10 v2 H4))) x H8) w1 H6)))) (subst0_gen_lift_ge v1 t3 x (s (Bind b) i) 
-(S O) O H7 (le_S_n (S O) (S i) (lt_le_S (S O) (S (S i)) (lt_n_S O (S i) 
-(le_lt_n_Sm O i (le_O_n i)))))))))) H5)) (\lambda (H5: (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t2: T).(eq T w1 (THead (Bind b) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t2: 
-T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t2))))).(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)).(eq_ind_r 
-T (THead (Bind b) 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))))) (eq_ind_r T (lift (S 
-O) O x) (\lambda (t: T).(or (pr0 (THead (Bind b) x0 t) t4) (ex2 T (\lambda 
-(w2: T).(pr0 (THead (Bind b) x0 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 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 (H12: 
-(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 H12 x0))) (\lambda (H12: (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 (H13: (pr0 x x2)).(\lambda (H14: (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 H13 x0) H14))))) H12)) (H2 v1 x i H11 v2 H4))) x1 H9) w1 H6)))) 
-(subst0_gen_lift_ge v1 t3 x1 (s (Bind b) i) (S O) O H8 (le_S_n (S O) (S i) 
-(lt_le_S (S O) (S (S i)) (lt_n_S O (S i) (le_lt_n_Sm 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_epsilon t3 t4 H0 x)) w1 H5)))) H4)) (\lambda (H4: (ex2 T (\lambda (t2: 
-T).(eq T w1 (THead (Flat Cast) u t2))) (\lambda (t2: T).(subst0 (s (Flat 
-Cast) i) v1 t3 t2)))).(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_epsilon 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_epsilon 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 (t2: T).(eq T w1 (THead (Flat Cast) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t2: 
-T).(subst0 (s (Flat Cast) i) v1 t3 t2))))).(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_epsilon 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_epsilon 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))).
-
-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_epsilon:
- \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_comm 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 (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 
-t))))))))))).(\lambda (t1: T).(\lambda (H0: (pr0 t t1)).(\lambda (t2: 
-T).(\lambda (H1: (pr0 t t2)).(let H2 \def (match H0 return (\lambda (t0: 
-T).(\lambda (t3: T).(\lambda (_: (pr0 t0 t3)).((eq T t0 t) \to ((eq T t3 t1) 
-\to (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t)))))))) with 
-[(pr0_refl t0) \Rightarrow (\lambda (H2: (eq T t0 t)).(\lambda (H3: (eq T t0 
-t1)).(eq_ind T t (\lambda (t: T).((eq T t t1) \to (ex2 T (\lambda (t2: 
-T).(pr0 t1 t2)) (\lambda (t1: T).(pr0 t2 t1))))) (\lambda (H4: (eq T t 
-t1)).(eq_ind T t1 (\lambda (_: T).(ex2 T (\lambda (t2: T).(pr0 t1 t2)) 
-(\lambda (t1: T).(pr0 t2 t1)))) (let H5 \def (match H1 return (\lambda (t0: 
-T).(\lambda (t3: T).(\lambda (_: (pr0 t0 t3)).((eq T t0 t) \to ((eq T t3 t2) 
-\to (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t)))))))) with 
-[(pr0_refl t3) \Rightarrow (\lambda (H5: (eq T t3 t)).(\lambda (H6: (eq T t3 
-t2)).(eq_ind T t (\lambda (t: T).((eq T t t2) \to (ex2 T (\lambda (t0: 
-T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))) (\lambda (H7: (eq T t 
-t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 t1 t0)) 
-(\lambda (t0: T).(pr0 t2 t0)))) (let H0 \def (eq_ind T t (\lambda (t: T).(eq 
-T t3 t)) H5 t2 H7) in (let H1 \def (eq_ind T t (\lambda (t: T).(eq T t t1)) 
-H4 t2 H7) in (let H2 \def (eq_ind T t (\lambda (t: T).(eq T t0 t)) H2 t2 H7) 
-in (let H3 \def (eq_ind T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to 
-(\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T 
-(\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H t2 H7) 
-in (let H4 \def (eq_ind T t2 (\lambda (t: T).(\forall (v: T).((tlt v t) \to 
-(\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T 
-(\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H3 t1 H1) 
-in (eq_ind_r T t1 (\lambda (t: T).(ex2 T (\lambda (t0: T).(pr0 t1 t0)) 
-(\lambda (t0: T).(pr0 t t0)))) (let H8 \def (eq_ind T t2 (\lambda (t: T).(eq 
-T t0 t)) H2 t1 H1) in (ex_intro2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: 
-T).(pr0 t1 t)) t1 (pr0_refl t1) (pr0_refl t1))) t2 H1)))))) t (sym_eq T t t2 
-H7))) t3 (sym_eq T t3 t H5) H6))) | (pr0_comp u1 u2 H4 t3 t4 H5 k) 
-\Rightarrow (\lambda (H6: (eq T (THead k u1 t3) t)).(\lambda (H7: (eq T 
-(THead k u2 t4) t2)).(eq_ind T (THead k u1 t3) (\lambda (_: T).((eq T (THead 
-k u2 t4) t2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t0: 
-T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H8: (eq T (THead 
-k u2 t4) t2)).(eq_ind T (THead k u2 t4) (\lambda (t: T).((pr0 u1 u2) \to 
-((pr0 t3 t4) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t 
-t0)))))) (\lambda (H9: (pr0 u1 u2)).(\lambda (H10: (pr0 t3 t4)).(let H0 \def 
-(eq_ind_r T t (\lambda (t: T).(eq T t t1)) H4 (THead k u1 t3) H6) in (eq_ind 
-T (THead k u1 t3) (\lambda (t: T).(ex2 T (\lambda (t0: T).(pr0 t t0)) 
-(\lambda (t0: T).(pr0 (THead k u2 t4) t0)))) (let H1 \def (eq_ind_r T t 
-(\lambda (t: T).(eq T t0 t)) H2 (THead k u1 t3) H6) in (let H2 \def (eq_ind_r 
-T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v 
-t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) 
-(\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead k u1 t3) H6) in (ex_intro2 T 
-(\lambda (t: T).(pr0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 (THead k u2 t4) 
-t)) (THead k u2 t4) (pr0_comp u1 u2 H9 t3 t4 H10 k) (pr0_refl (THead k u2 
-t4))))) t1 H0)))) t2 H8)) t H6 H7 H4 H5))) | (pr0_beta u v1 v2 H4 t3 t4 H5) 
-\Rightarrow (\lambda (H6: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u 
-t3)) t)).(\lambda (H7: (eq T (THead (Bind Abbr) v2 t4) t2)).(eq_ind T (THead 
-(Flat Appl) v1 (THead (Bind Abst) u t3)) (\lambda (_: T).((eq T (THead (Bind 
-Abbr) v2 t4) t2) \to ((pr0 v1 v2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t0: 
-T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H8: (eq T (THead 
-(Bind Abbr) v2 t4) t2)).(eq_ind T (THead (Bind Abbr) v2 t4) (\lambda (t: 
-T).((pr0 v1 v2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) 
-(\lambda (t0: T).(pr0 t t0)))))) (\lambda (H9: (pr0 v1 v2)).(\lambda (H10: 
-(pr0 t3 t4)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T t t1)) H4 
-(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) H6) in (eq_ind T (THead (Flat 
-Appl) v1 (THead (Bind Abst) u t3)) (\lambda (t: T).(ex2 T (\lambda (t0: 
-T).(pr0 t t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t4) t0)))) (let H1 
-\def (eq_ind_r T t (\lambda (t: T).(eq T t0 t)) H2 (THead (Flat Appl) v1 
-(THead (Bind Abst) u t3)) H6) in (let H2 \def (eq_ind_r T t (\lambda (t: 
-T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall 
-(t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: 
-T).(pr0 t2 t0)))))))))) H (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) H6) 
-in (ex_intro2 T (\lambda (t: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) 
-u t3)) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t4) t)) (THead (Bind 
-Abbr) v2 t4) (pr0_beta u v1 v2 H9 t3 t4 H10) (pr0_refl (THead (Bind Abbr) v2 
-t4))))) t1 H0)))) t2 H8)) t H6 H7 H4 H5))) | (pr0_upsilon b H4 v1 v2 H5 u1 u2 
-H6 t3 t4 H7) \Rightarrow (\lambda (H8: (eq T (THead (Flat Appl) v1 (THead 
-(Bind b) u1 t3)) t)).(\lambda (H9: (eq T (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t4)) t2)).(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)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to 
-((pr0 u1 u2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) 
-(\lambda (t0: T).(pr0 t2 t0))))))))) (\lambda (H10: (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 (t: T).((not (eq B b 
-Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ex2 T (\lambda 
-(t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t t0)))))))) (\lambda (H11: (not 
-(eq B b Abst))).(\lambda (H12: (pr0 v1 v2)).(\lambda (H13: (pr0 u1 
-u2)).(\lambda (H14: (pr0 t3 t4)).(let H0 \def (eq_ind_r T t (\lambda (t: 
-T).(eq T t t1)) H4 (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) H8) in 
-(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (\lambda (t: T).(ex2 
-T (\lambda (t0: T).(pr0 t t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t4)) t0)))) (let H1 \def (eq_ind_r T t 
-(\lambda (t: T).(eq T t0 t)) H2 (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) 
-H8) in (let H2 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) 
-\to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T 
-(\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead 
-(Flat Appl) v1 (THead (Bind b) u1 t3)) H8) in 
-(pr0_confluence__pr0_cong_upsilon_refl b H11 u1 u2 H13 t3 t4 H14 v1 v2 v2 H12 
-(pr0_refl v2)))) t1 H0)))))) t2 H10)) t H8 H9 H4 H5 H6 H7))) | (pr0_delta u1 
-u2 H4 t3 t4 H5 w H6) \Rightarrow (\lambda (H7: (eq T (THead (Bind Abbr) u1 
-t3) t)).(\lambda (H8: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead 
-(Bind Abbr) u1 t3) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2) \to 
-((pr0 u1 u2) \to ((pr0 t3 t4) \to ((subst0 O u2 t4 w) \to (ex2 T (\lambda 
-(t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))) (\lambda (H9: (eq T 
-(THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t: 
-T).((pr0 u1 u2) \to ((pr0 t3 t4) \to ((subst0 O u2 t4 w) \to (ex2 T (\lambda 
-(t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t t0))))))) (\lambda (H10: (pr0 u1 
-u2)).(\lambda (H11: (pr0 t3 t4)).(\lambda (H12: (subst0 O u2 t4 w)).(let H0 
-\def (eq_ind_r T t (\lambda (t: T).(eq T t t1)) H4 (THead (Bind Abbr) u1 t3) 
-H7) in (eq_ind T (THead (Bind Abbr) u1 t3) (\lambda (t: T).(ex2 T (\lambda 
-(t0: T).(pr0 t t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)))) 
-(let H1 \def (eq_ind_r T t (\lambda (t: T).(eq T t0 t)) H2 (THead (Bind Abbr) 
-u1 t3) H7) in (let H2 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: 
-T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v 
-t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 
-t0)))))))))) H (THead (Bind Abbr) u1 t3) H7) in (ex_intro2 T (\lambda (t: 
-T).(pr0 (THead (Bind Abbr) u1 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) 
-u2 w) t)) (THead (Bind Abbr) u2 w) (pr0_delta u1 u2 H10 t3 t4 H11 w H12) 
-(pr0_refl (THead (Bind Abbr) u2 w))))) t1 H0))))) t2 H9)) t H7 H8 H4 H5 H6))) 
-| (pr0_zeta b H4 t3 t4 H5 u) \Rightarrow (\lambda (H6: (eq T (THead (Bind b) 
-u (lift (S O) O t3)) t)).(\lambda (H7: (eq T t4 t2)).(eq_ind T (THead (Bind 
-b) u (lift (S O) O t3)) (\lambda (_: T).((eq T t4 t2) \to ((not (eq B b 
-Abst)) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda 
-(t0: T).(pr0 t2 t0))))))) (\lambda (H8: (eq T t4 t2)).(eq_ind T t2 (\lambda 
-(t: T).((not (eq B b Abst)) \to ((pr0 t3 t) \to (ex2 T (\lambda (t0: T).(pr0 
-t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))) (\lambda (H9: (not (eq B b 
-Abst))).(\lambda (H10: (pr0 t3 t2)).(let H0 \def (eq_ind_r T t (\lambda (t: 
-T).(eq T t t1)) H4 (THead (Bind b) u (lift (S O) O t3)) H6) in (eq_ind T 
-(THead (Bind b) u (lift (S O) O t3)) (\lambda (t: T).(ex2 T (\lambda (t0: 
-T).(pr0 t t0)) (\lambda (t0: T).(pr0 t2 t0)))) (let H1 \def (eq_ind_r T t 
-(\lambda (t: T).(eq T t0 t)) H2 (THead (Bind b) u (lift (S O) O t3)) H6) in 
-(let H2 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to 
-(\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T 
-(\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead 
-(Bind b) u (lift (S O) O t3)) H6) in (ex_intro2 T (\lambda (t: T).(pr0 (THead 
-(Bind b) u (lift (S O) O t3)) t)) (\lambda (t: T).(pr0 t2 t)) t2 (pr0_zeta b 
-H9 t3 t2 H10 u) (pr0_refl t2)))) t1 H0)))) t4 (sym_eq T t4 t2 H8))) t H6 H7 
-H4 H5))) | (pr0_epsilon t3 t4 H4 u) \Rightarrow (\lambda (H5: (eq T (THead 
-(Flat Cast) u t3) t)).(\lambda (H6: (eq T t4 t2)).(eq_ind T (THead (Flat 
-Cast) u t3) (\lambda (_: T).((eq T t4 t2) \to ((pr0 t3 t4) \to (ex2 T 
-(\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))) (\lambda (H7: 
-(eq T t4 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t3 t) \to (ex2 T (\lambda 
-(t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))) (\lambda (H8: (pr0 t3 
-t2)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T t t1)) H4 (THead (Flat 
-Cast) u t3) H5) in (eq_ind T (THead (Flat Cast) u t3) (\lambda (t: T).(ex2 T 
-(\lambda (t0: T).(pr0 t t0)) (\lambda (t0: T).(pr0 t2 t0)))) (let H1 \def 
-(eq_ind_r T t (\lambda (t: T).(eq T t0 t)) H2 (THead (Flat Cast) u t3) H5) in 
-(let H2 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to 
-(\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T 
-(\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead 
-(Flat Cast) u t3) H5) in (ex_intro2 T (\lambda (t: T).(pr0 (THead (Flat Cast) 
-u t3) t)) (\lambda (t: T).(pr0 t2 t)) t2 (pr0_epsilon t3 t2 H8 u) (pr0_refl 
-t2)))) t1 H0))) t4 (sym_eq T t4 t2 H7))) t H5 H6 H4)))]) in (H5 (refl_equal T 
-t) (refl_equal T t2))) t (sym_eq T t t1 H4))) t0 (sym_eq T t0 t H2) H3))) | 
-(pr0_comp u1 u2 H2 t0 t3 H3 k) \Rightarrow (\lambda (H4: (eq T (THead k u1 
-t0) t)).(\lambda (H5: (eq T (THead k u2 t3) t1)).(eq_ind T (THead k u1 t0) 
-(\lambda (_: T).((eq T (THead k u2 t3) t1) \to ((pr0 u1 u2) \to ((pr0 t0 t3) 
-\to (ex2 T (\lambda (t2: T).(pr0 t1 t2)) (\lambda (t1: T).(pr0 t2 t1))))))) 
-(\lambda (H6: (eq T (THead k u2 t3) t1)).(eq_ind T (THead k u2 t3) (\lambda 
-(t: T).((pr0 u1 u2) \to ((pr0 t0 t3) \to (ex2 T (\lambda (t1: T).(pr0 t t1)) 
-(\lambda (t1: T).(pr0 t2 t1)))))) (\lambda (H7: (pr0 u1 u2)).(\lambda (H8: 
-(pr0 t0 t3)).(let H9 \def (match H1 return (\lambda (t0: T).(\lambda (t1: 
-T).(\lambda (_: (pr0 t0 t1)).((eq T t0 t) \to ((eq T t1 t2) \to (ex2 T 
-(\lambda (t: T).(pr0 (THead k u2 t3) t)) (\lambda (t: T).(pr0 t2 t)))))))) 
-with [(pr0_refl t4) \Rightarrow (\lambda (H6: (eq T t4 t)).(\lambda (H9: (eq 
-T t4 t2)).(eq_ind T t (\lambda (t: T).((eq T t t2) \to (ex2 T (\lambda (t0: 
-T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0))))) (\lambda (H10: 
-(eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 (THead 
-k u2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0)))) (let H0 \def (eq_ind_r T t 
-(\lambda (t: T).(eq T t t2)) H10 (THead k u1 t0) H4) in (eq_ind T (THead k u1 
-t0) (\lambda (t: T).(ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) 
-(\lambda (t0: T).(pr0 t t0)))) (let H1 \def (eq_ind_r T t (\lambda (t: T).(eq 
-T t4 t)) H6 (THead k u1 t0) H4) in (let H2 \def (eq_ind_r T t (\lambda (t: 
-T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall 
-(t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: 
-T).(pr0 t2 t0)))))))))) H (THead k u1 t0) H4) in (ex_intro2 T (\lambda (t: 
-T).(pr0 (THead k u2 t3) t)) (\lambda (t: T).(pr0 (THead k u1 t0) t)) (THead k 
-u2 t3) (pr0_refl (THead k u2 t3)) (pr0_comp u1 u2 H7 t0 t3 H8 k)))) t2 H0)) t 
-(sym_eq T t t2 H10))) t4 (sym_eq T t4 t H6) H9))) | (pr0_comp u0 u3 H6 t4 t5 
-H7 k0) \Rightarrow (\lambda (H9: (eq T (THead k0 u0 t4) t)).(\lambda (H10: 
-(eq T (THead k0 u3 t5) t2)).(eq_ind T (THead k0 u0 t4) (\lambda (_: T).((eq T 
-(THead k0 u3 t5) t2) \to ((pr0 u0 u3) \to ((pr0 t4 t5) \to (ex2 T (\lambda 
-(t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda 
-(H11: (eq T (THead k0 u3 t5) t2)).(eq_ind T (THead k0 u3 t5) (\lambda (t: 
-T).((pr0 u0 u3) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead k u2 
-t3) t0)) (\lambda (t0: T).(pr0 t t0)))))) (\lambda (H12: (pr0 u0 
-u3)).(\lambda (H13: (pr0 t4 t5)).(let H0 \def (eq_ind_r T t (\lambda (t: 
-T).(eq T (THead k u1 t0) t)) H4 (THead k0 u0 t4) H9) in (let H1 \def (match 
-H0 return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t (THead k0 u0 t4)) 
-\to (ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 
-(THead k0 u3 t5) t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T 
-(THead k u1 t0) (THead k0 u0 t4))).(let H1 \def (f_equal T T (\lambda (e: 
-T).(match e return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef 
-_) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u1 t0) (THead k0 
-u0 t4) H0) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return 
-(\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 
-| (THead _ t _) \Rightarrow t])) (THead k u1 t0) (THead k0 u0 t4) H0) in 
-((let H3 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: T).K) 
-with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) 
-\Rightarrow k])) (THead k u1 t0) (THead k0 u0 t4) H0) in (eq_ind K k0 
-(\lambda (k: K).((eq T u1 u0) \to ((eq T t0 t4) \to (ex2 T (\lambda (t: 
-T).(pr0 (THead k u2 t3) t)) (\lambda (t: T).(pr0 (THead k0 u3 t5) t)))))) 
-(\lambda (H10: (eq T u1 u0)).(eq_ind T u0 (\lambda (_: T).((eq T t0 t4) \to 
-(ex2 T (\lambda (t0: T).(pr0 (THead k0 u2 t3) t0)) (\lambda (t0: T).(pr0 
-(THead k0 u3 t5) t0))))) (\lambda (H11: (eq T t0 t4)).(eq_ind T t4 (\lambda 
-(_: T).(ex2 T (\lambda (t0: T).(pr0 (THead k0 u2 t3) t0)) (\lambda (t0: 
-T).(pr0 (THead k0 u3 t5) t0)))) (let H4 \def (eq_ind_r T t (\lambda (t: 
-T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall 
-(t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: 
-T).(pr0 t2 t0)))))))))) H (THead k0 u0 t4) H9) in (let H5 \def (eq_ind T t0 
-(\lambda (t: T).(pr0 t t3)) H8 t4 H11) in (let H6 \def (eq_ind T u1 (\lambda 
-(t: T).(pr0 t u2)) H7 u0 H10) in (ex2_ind T (\lambda (t: T).(pr0 u2 t)) 
-(\lambda (t: T).(pr0 u3 t)) (ex2 T (\lambda (t: T).(pr0 (THead k0 u2 t3) t)) 
-(\lambda (t: T).(pr0 (THead k0 u3 t5) t))) (\lambda (x: T).(\lambda (H7: (pr0 
-u2 x)).(\lambda (H8: (pr0 u3 x)).(ex2_ind T (\lambda (t: T).(pr0 t3 t)) 
-(\lambda (t: T).(pr0 t5 t)) (ex2 T (\lambda (t: T).(pr0 (THead k0 u2 t3) t)) 
-(\lambda (t: T).(pr0 (THead k0 u3 t5) t))) (\lambda (x0: T).(\lambda (H9: 
-(pr0 t3 x0)).(\lambda (H12: (pr0 t5 x0)).(ex_intro2 T (\lambda (t: T).(pr0 
-(THead k0 u2 t3) t)) (\lambda (t: T).(pr0 (THead k0 u3 t5) t)) (THead k0 x 
-x0) (pr0_comp u2 x H7 t3 x0 H9 k0) (pr0_comp u3 x H8 t5 x0 H12 k0))))) (H4 t4 
-(tlt_head_dx k0 u0 t4) t3 H5 t5 H13))))) (H4 u0 (tlt_head_sx k0 u0 t4) u2 H6 
-u3 H12))))) t0 (sym_eq T t0 t4 H11))) u1 (sym_eq T u1 u0 H10))) k (sym_eq K k 
-k0 H3))) H2)) H1)))]) in (H1 (refl_equal T (THead k0 u0 t4))))))) t2 H11)) t 
-H9 H10 H6 H7))) | (pr0_beta u v1 v2 H6 t4 t5 H7) \Rightarrow (\lambda (H9: 
-(eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) t)).(\lambda (H10: (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 (t0: T).(pr0 (THead k u2 t3) t0)) 
-(\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H11: (eq T (THead (Bind Abbr) v2 
-t5) t2)).(eq_ind T (THead (Bind Abbr) v2 t5) (\lambda (t: T).((pr0 v1 v2) \to 
-((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda 
-(t0: T).(pr0 t t0)))))) (\lambda (H12: (pr0 v1 v2)).(\lambda (H13: (pr0 t4 
-t5)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead k u1 t0) t)) H4 
-(THead (Flat Appl) v1 (THead (Bind Abst) u t4)) H9) in (let H1 \def (match H0 
-return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t (THead (Flat Appl) 
-v1 (THead (Bind Abst) u t4))) \to (ex2 T (\lambda (t0: T).(pr0 (THead k u2 
-t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t5) t0)))))) with 
-[refl_equal \Rightarrow (\lambda (H0: (eq T (THead k u1 t0) (THead (Flat 
-Appl) v1 (THead (Bind Abst) u t4)))).(let H1 \def (f_equal T T (\lambda (e: 
-T).(match e return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef 
-_) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u1 t0) (THead 
-(Flat Appl) v1 (THead (Bind Abst) u t4)) H0) in ((let H2 \def (f_equal T T 
-(\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t _) \Rightarrow t])) 
-(THead k u1 t0) (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) H0) in ((let 
-H3 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: T).K) with 
-[(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) 
-\Rightarrow k])) (THead k u1 t0) (THead (Flat Appl) v1 (THead (Bind Abst) u 
-t4)) H0) in (eq_ind K (Flat Appl) (\lambda (k: K).((eq T u1 v1) \to ((eq T t0 
-(THead (Bind Abst) u t4)) \to (ex2 T (\lambda (t: T).(pr0 (THead k u2 t3) t)) 
-(\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t5) t)))))) (\lambda (H10: (eq T 
-u1 v1)).(eq_ind T v1 (\lambda (_: T).((eq T t0 (THead (Bind Abst) u t4)) \to 
-(ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0: 
-T).(pr0 (THead (Bind Abbr) v2 t5) t0))))) (\lambda (H11: (eq T t0 (THead 
-(Bind Abst) u t4))).(eq_ind T (THead (Bind Abst) u t4) (\lambda (_: T).(ex2 T 
-(\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0: T).(pr0 
-(THead (Bind Abbr) v2 t5) t0)))) (let H4 \def (eq_ind_r T t (\lambda (t: 
-T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall 
-(t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: 
-T).(pr0 t2 t0)))))))))) H (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) H9) 
-in (let H5 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t3)) H8 (THead (Bind 
-Abst) u t4) H11) in (let H6 \def (match H5 return (\lambda (t: T).(\lambda 
-(t0: T).(\lambda (_: (pr0 t t0)).((eq T t (THead (Bind Abst) u t4)) \to ((eq 
-T t0 t3) \to (ex2 T (\lambda (t1: T).(pr0 (THead (Flat Appl) u2 t3) t1)) 
-(\lambda (t1: T).(pr0 (THead (Bind Abbr) v2 t5) t1)))))))) with [(pr0_refl t) 
-\Rightarrow (\lambda (H0: (eq T t (THead (Bind Abst) u t4))).(\lambda (H5: 
-(eq T t t3)).(eq_ind T (THead (Bind Abst) u t4) (\lambda (t0: T).((eq T t0 
-t3) \to (ex2 T (\lambda (t1: T).(pr0 (THead (Flat Appl) u2 t3) t1)) (\lambda 
-(t1: T).(pr0 (THead (Bind Abbr) v2 t5) t1))))) (\lambda (H6: (eq T (THead 
-(Bind Abst) u t4) t3)).(eq_ind T (THead (Bind Abst) u t4) (\lambda (t0: 
-T).(ex2 T (\lambda (t1: T).(pr0 (THead (Flat Appl) u2 t0) t1)) (\lambda (t1: 
-T).(pr0 (THead (Bind Abbr) v2 t5) t1)))) (let H1 \def (eq_ind T u1 (\lambda 
-(t: T).(pr0 t u2)) H7 v1 H10) in (ex2_ind T (\lambda (t0: T).(pr0 u2 t0)) 
-(\lambda (t0: T).(pr0 v2 t0)) (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) 
-u2 (THead (Bind Abst) u t4)) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 
-t5) t0))) (\lambda (x: T).(\lambda (H2: (pr0 u2 x)).(\lambda (H3: (pr0 v2 
-x)).(ex_intro2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 (THead (Bind 
-Abst) u t4)) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t5) t0)) (THead 
-(Bind Abbr) x t5) (pr0_beta u u2 x H2 t4 t5 H13) (pr0_comp v2 x H3 t5 t5 
-(pr0_refl t5) (Bind Abbr)))))) (H4 v1 (tlt_head_sx (Flat Appl) v1 (THead 
-(Bind Abst) u t4)) u2 H1 v2 H12))) t3 H6)) t (sym_eq T t (THead (Bind Abst) u 
-t4) H0) H5))) | (pr0_comp u0 u3 H0 t1 t2 H4 k0) \Rightarrow (\lambda (H5: (eq 
-T (THead k0 u0 t1) (THead (Bind Abst) u t4))).(\lambda (H8: (eq T (THead k0 
-u3 t2) t3)).((let H1 \def (f_equal T T (\lambda (e: T).(match e return 
-(\lambda (_: T).T) with [(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 
-| (THead _ _ t) \Rightarrow t])) (THead k0 u0 t1) (THead (Bind Abst) u t4) 
-H5) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda 
-(_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead 
-_ t _) \Rightarrow t])) (THead k0 u0 t1) (THead (Bind Abst) u t4) H5) in 
-((let H3 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: T).K) 
-with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k _ _) 
-\Rightarrow k])) (THead k0 u0 t1) (THead (Bind Abst) u t4) H5) in (eq_ind K 
-(Bind Abst) (\lambda (k: K).((eq T u0 u) \to ((eq T t1 t4) \to ((eq T (THead 
-k u3 t2) t3) \to ((pr0 u0 u3) \to ((pr0 t1 t2) \to (ex2 T (\lambda (t: 
-T).(pr0 (THead (Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) 
-v2 t5) t))))))))) (\lambda (H6: (eq T u0 u)).(eq_ind T u (\lambda (t: T).((eq 
-T t1 t4) \to ((eq T (THead (Bind Abst) u3 t2) t3) \to ((pr0 t u3) \to ((pr0 
-t1 t2) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) 
-(\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t5) t0)))))))) (\lambda (H9: (eq 
-T t1 t4)).(eq_ind T t4 (\lambda (t: T).((eq T (THead (Bind Abst) u3 t2) t3) 
-\to ((pr0 u u3) \to ((pr0 t t2) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Flat 
-Appl) u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t5) t0))))))) 
-(\lambda (H11: (eq T (THead (Bind Abst) u3 t2) t3)).(eq_ind T (THead (Bind 
-Abst) u3 t2) (\lambda (t: T).((pr0 u u3) \to ((pr0 t4 t2) \to (ex2 T (\lambda 
-(t0: T).(pr0 (THead (Flat Appl) u2 t) t0)) (\lambda (t0: T).(pr0 (THead (Bind 
-Abbr) v2 t5) t0)))))) (\lambda (_: (pr0 u u3)).(\lambda (H15: (pr0 t4 
-t2)).(let H7 \def (eq_ind T u1 (\lambda (t: T).(pr0 t u2)) H7 v1 H10) in 
-(ex2_ind T (\lambda (t: T).(pr0 u2 t)) (\lambda (t: T).(pr0 v2 t)) (ex2 T 
-(\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u3 t2)) t)) 
-(\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t5) t))) (\lambda (x: T).(\lambda 
-(H10: (pr0 u2 x)).(\lambda (H12: (pr0 v2 x)).(ex2_ind T (\lambda (t: T).(pr0 
-t2 t)) (\lambda (t: T).(pr0 t5 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Flat 
-Appl) u2 (THead (Bind Abst) u3 t2)) t)) (\lambda (t: T).(pr0 (THead (Bind 
-Abbr) v2 t5) t))) (\lambda (x0: T).(\lambda (H13: (pr0 t2 x0)).(\lambda (H16: 
-(pr0 t5 x0)).(ex_intro2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead 
-(Bind Abst) u3 t2)) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t5) t)) 
-(THead (Bind Abbr) x x0) (pr0_beta u3 u2 x H10 t2 x0 H13) (pr0_comp v2 x H12 
-t5 x0 H16 (Bind Abbr)))))) (H4 t4 (tlt_trans (THead (Bind Abst) u t4) t4 
-(THead (Flat Appl) v1 (THead (Bind Abst) u t4)) (tlt_head_dx (Bind Abst) u 
-t4) (tlt_head_dx (Flat Appl) v1 (THead (Bind Abst) u t4))) t2 H15 t5 H13))))) 
-(H4 v1 (tlt_head_sx (Flat Appl) v1 (THead (Bind Abst) u t4)) u2 H7 v2 
-H12))))) t3 H11)) t1 (sym_eq T t1 t4 H9))) u0 (sym_eq T u0 u H6))) k0 (sym_eq 
-K k0 (Bind Abst) H3))) H2)) H1)) H8 H0 H4))) | (pr0_beta u0 v0 v3 H0 t1 t2 
-H4) \Rightarrow (\lambda (H5: (eq T (THead (Flat Appl) v0 (THead (Bind Abst) 
-u0 t1)) (THead (Bind Abst) u t4))).(\lambda (H8: (eq T (THead (Bind Abbr) v3 
-t2) t3)).((let H1 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u0 
-t1)) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
-(match k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | 
-(Flat _) \Rightarrow True])])) I (THead (Bind Abst) u t4) H5) in (False_ind 
-((eq T (THead (Bind Abbr) v3 t2) t3) \to ((pr0 v0 v3) \to ((pr0 t1 t2) \to 
-(ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t3) t)) (\lambda (t: 
-T).(pr0 (THead (Bind Abbr) v2 t5) t)))))) H1)) H8 H0 H4))) | (pr0_upsilon b 
-H0 v0 v3 H4 u0 u3 H5 t1 t2 H8) \Rightarrow (\lambda (H11: (eq T (THead (Flat 
-Appl) v0 (THead (Bind b) u0 t1)) (THead (Bind Abst) u t4))).(\lambda (H12: 
-(eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v3) t2)) t3)).((let 
-H1 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind b) u0 t1)) (\lambda (e: 
-T).(match e return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | 
-(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return 
-(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow 
-True])])) I (THead (Bind Abst) u t4) H11) in (False_ind ((eq T (THead (Bind 
-b) u3 (THead (Flat Appl) (lift (S O) O v3) t2)) t3) \to ((not (eq B b Abst)) 
-\to ((pr0 v0 v3) \to ((pr0 u0 u3) \to ((pr0 t1 t2) \to (ex2 T (\lambda (t: 
-T).(pr0 (THead (Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) 
-v2 t5) t)))))))) H1)) H12 H0 H4 H5 H8))) | (pr0_delta u0 u3 H0 t1 t2 H4 w H5) 
-\Rightarrow (\lambda (H8: (eq T (THead (Bind Abbr) u0 t1) (THead (Bind Abst) 
-u t4))).(\lambda (H11: (eq T (THead (Bind Abbr) u3 w) t3)).((let H1 \def 
-(eq_ind T (THead (Bind Abbr) u0 t1) (\lambda (e: T).(match e return (\lambda 
-(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
-| (THead k _ _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind 
-b) \Rightarrow (match b return (\lambda (_: B).Prop) with [Abbr \Rightarrow 
-True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) 
-\Rightarrow False])])) I (THead (Bind Abst) u t4) H8) in (False_ind ((eq T 
-(THead (Bind Abbr) u3 w) t3) \to ((pr0 u0 u3) \to ((pr0 t1 t2) \to ((subst0 O 
-u3 t2 w) \to (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t3) t)) 
-(\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t5) t))))))) H1)) H11 H0 H4 H5))) 
-| (pr0_zeta b H0 t1 t2 H4 u0) \Rightarrow (\lambda (H5: (eq T (THead (Bind b) 
-u0 (lift (S O) O t1)) (THead (Bind Abst) u t4))).(\lambda (H8: (eq T t2 
-t3)).((let H1 \def (f_equal T T (\lambda (e: T).(match e 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 u t0) \Rightarrow (THead k (lref_map f 
-d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S 
-O))) O t1) | (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 u t0) \Rightarrow (THead k (lref_map f 
-d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S 
-O))) O t1) | (THead _ _ t) \Rightarrow t])) (THead (Bind b) u0 (lift (S O) O 
-t1)) (THead (Bind Abst) u t4) H5) in ((let H2 \def (f_equal T T (\lambda (e: 
-T).(match e return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef 
-_) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead (Bind b) u0 (lift 
-(S O) O t1)) (THead (Bind Abst) u t4) H5) in ((let H3 \def (f_equal T B 
-(\lambda (e: T).(match e return (\lambda (_: T).B) with [(TSort _) 
-\Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k 
-return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow 
-b])])) (THead (Bind b) u0 (lift (S O) O t1)) (THead (Bind Abst) u t4) H5) in 
-(eq_ind B Abst (\lambda (b0: B).((eq T u0 u) \to ((eq T (lift (S O) O t1) t4) 
-\to ((eq T t2 t3) \to ((not (eq B b0 Abst)) \to ((pr0 t1 t2) \to (ex2 T 
-(\lambda (t: T).(pr0 (THead (Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0 
-(THead (Bind Abbr) v2 t5) t))))))))) (\lambda (H6: (eq T u0 u)).(eq_ind T u 
-(\lambda (_: T).((eq T (lift (S O) O t1) t4) \to ((eq T t2 t3) \to ((not (eq 
-B Abst Abst)) \to ((pr0 t1 t2) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Flat 
-Appl) u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t5) t0)))))))) 
-(\lambda (H9: (eq T (lift (S O) O t1) t4)).(eq_ind T (lift (S O) O t1) 
-(\lambda (_: T).((eq T t2 t3) \to ((not (eq B Abst Abst)) \to ((pr0 t1 t2) 
-\to (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0: 
-T).(pr0 (THead (Bind Abbr) v2 t5) t0))))))) (\lambda (H7: (eq T t2 
-t3)).(eq_ind T t3 (\lambda (t: T).((not (eq B Abst Abst)) \to ((pr0 t1 t) \to 
-(ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0: 
-T).(pr0 (THead (Bind Abbr) v2 t5) t0)))))) (\lambda (H11: (not (eq B Abst 
-Abst))).(\lambda (_: (pr0 t1 t3)).(let H10 \def (match (H11 (refl_equal B 
-Abst)) return (\lambda (_: False).(ex2 T (\lambda (t: T).(pr0 (THead (Flat 
-Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t5) t)))) with 
-[]) in H10))) t2 (sym_eq T t2 t3 H7))) t4 H9)) u0 (sym_eq T u0 u H6))) b 
-(sym_eq B b Abst H3))) H2)) H1)) H8 H0 H4))) | (pr0_epsilon t1 t2 H0 u0) 
-\Rightarrow (\lambda (H4: (eq T (THead (Flat Cast) u0 t1) (THead (Bind Abst) 
-u t4))).(\lambda (H5: (eq T t2 t3)).((let H1 \def (eq_ind T (THead (Flat 
-Cast) u0 t1) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
-\Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow 
-False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u t4) H4) in 
-(False_ind ((eq T t2 t3) \to ((pr0 t1 t2) \to (ex2 T (\lambda (t: T).(pr0 
-(THead (Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t5) 
-t))))) H1)) H5 H0)))]) in (H6 (refl_equal T (THead (Bind Abst) u t4)) 
-(refl_equal T t3))))) t0 (sym_eq T t0 (THead (Bind Abst) u t4) H11))) u1 
-(sym_eq T u1 v1 H10))) k (sym_eq K k (Flat Appl) H3))) H2)) H1)))]) in (H1 
-(refl_equal T (THead (Flat Appl) v1 (THead (Bind Abst) u t4)))))))) t2 H11)) 
-t H9 H10 H6 H7))) | (pr0_upsilon b H6 v1 v2 H7 u0 u3 H8 t4 t5 H9) \Rightarrow 
-(\lambda (H10: (eq T (THead (Flat Appl) v1 (THead (Bind b) u0 t4)) 
-t)).(\lambda (H11: (eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O 
-v2) t5)) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t4)) 
-(\lambda (_: T).((eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O 
-v2) t5)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u3) \to 
-((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda 
-(t0: T).(pr0 t2 t0))))))))) (\lambda (H12: (eq T (THead (Bind b) u3 (THead 
-(Flat Appl) (lift (S O) O v2) t5)) t2)).(eq_ind T (THead (Bind b) u3 (THead 
-(Flat Appl) (lift (S O) O v2) t5)) (\lambda (t: T).((not (eq B b Abst)) \to 
-((pr0 v1 v2) \to ((pr0 u0 u3) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: 
-T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 t t0)))))))) (\lambda 
-(H13: (not (eq B b Abst))).(\lambda (H14: (pr0 v1 v2)).(\lambda (H15: (pr0 u0 
-u3)).(\lambda (H16: (pr0 t4 t5)).(let H0 \def (eq_ind_r T t (\lambda (t: 
-T).(eq T (THead k u1 t0) t)) H4 (THead (Flat Appl) v1 (THead (Bind b) u0 t4)) 
-H10) in (let H1 \def (match H0 return (\lambda (t: T).(\lambda (_: (eq ? ? 
-t)).((eq T t (THead (Flat Appl) v1 (THead (Bind b) u0 t4))) \to (ex2 T 
-(\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind 
-b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t0)))))) with [refl_equal 
-\Rightarrow (\lambda (H0: (eq T (THead k u1 t0) (THead (Flat Appl) v1 (THead 
-(Bind b) u0 t4)))).(let H1 \def (f_equal T T (\lambda (e: T).(match e return 
-(\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 
-| (THead _ _ t) \Rightarrow t])) (THead k u1 t0) (THead (Flat Appl) v1 (THead 
-(Bind b) u0 t4)) H0) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e 
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) 
-\Rightarrow u1 | (THead _ t _) \Rightarrow t])) (THead k u1 t0) (THead (Flat 
-Appl) v1 (THead (Bind b) u0 t4)) H0) in ((let H3 \def (f_equal T K (\lambda 
-(e: T).(match e return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | 
-(TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k u1 t0) 
-(THead (Flat Appl) v1 (THead (Bind b) u0 t4)) H0) in (eq_ind K (Flat Appl) 
-(\lambda (k: K).((eq T u1 v1) \to ((eq T t0 (THead (Bind b) u0 t4)) \to (ex2 
-T (\lambda (t: T).(pr0 (THead k u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind 
-b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t)))))) (\lambda (H11: (eq T 
-u1 v1)).(eq_ind T v1 (\lambda (_: T).((eq T t0 (THead (Bind b) u0 t4)) \to 
-(ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0: 
-T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t0))))) 
-(\lambda (H12: (eq T t0 (THead (Bind b) u0 t4))).(eq_ind T (THead (Bind b) u0 
-t4) (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) 
-t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) 
-O v2) t5)) t0)))) (let H4 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: 
-T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v 
-t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 
-t0)))))))))) H (THead (Flat Appl) v1 (THead (Bind b) u0 t4)) H10) in (let H5 
-\def (eq_ind T t0 (\lambda (t: T).(pr0 t t3)) H8 (THead (Bind b) u0 t4) H12) 
-in (let H6 \def (match H5 return (\lambda (t: T).(\lambda (t0: T).(\lambda 
-(_: (pr0 t t0)).((eq T t (THead (Bind b) u0 t4)) \to ((eq T t0 t3) \to (ex2 T 
-(\lambda (t1: T).(pr0 (THead (Flat Appl) u2 t3) t1)) (\lambda (t1: T).(pr0 
-(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t1)))))))) with 
-[(pr0_refl t) \Rightarrow (\lambda (H0: (eq T t (THead (Bind b) u0 
-t4))).(\lambda (H5: (eq T t t3)).(eq_ind T (THead (Bind b) u0 t4) (\lambda 
-(t0: T).((eq T t0 t3) \to (ex2 T (\lambda (t1: T).(pr0 (THead (Flat Appl) u2 
-t3) t1)) (\lambda (t1: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S 
-O) O v2) t5)) t1))))) (\lambda (H6: (eq T (THead (Bind b) u0 t4) t3)).(eq_ind 
-T (THead (Bind b) u0 t4) (\lambda (t0: T).(ex2 T (\lambda (t1: T).(pr0 (THead 
-(Flat Appl) u2 t0) t1)) (\lambda (t1: T).(pr0 (THead (Bind b) u3 (THead (Flat 
-Appl) (lift (S O) O v2) t5)) t1)))) (let H1 \def (eq_ind T u1 (\lambda (t: 
-T).(pr0 t u2)) H7 v1 H11) in (ex2_ind T (\lambda (t0: T).(pr0 u2 t0)) 
-(\lambda (t0: T).(pr0 v2 t0)) (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) 
-u2 (THead (Bind b) u0 t4)) t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u3 
-(THead (Flat Appl) (lift (S O) O v2) t5)) t0))) (\lambda (x: T).(\lambda (H2: 
-(pr0 u2 x)).(\lambda (H3: (pr0 v2 x)).(pr0_confluence__pr0_cong_upsilon_refl 
-b H13 u0 u3 H15 t4 t5 H16 u2 v2 x H2 H3)))) (H4 v1 (tlt_head_sx (Flat Appl) 
-v1 (THead (Bind b) u0 t4)) u2 H1 v2 H14))) t3 H6)) t (sym_eq T t (THead (Bind 
-b) u0 t4) H0) H5))) | (pr0_comp u4 u5 H0 t1 t2 H4 k0) \Rightarrow (\lambda 
-(H5: (eq T (THead k0 u4 t1) (THead (Bind b) u0 t4))).(\lambda (H10: (eq T 
-(THead k0 u5 t2) t3)).((let H1 \def (f_equal T T (\lambda (e: T).(match e 
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow t1 | (TLRef _) 
-\Rightarrow t1 | (THead _ _ t) \Rightarrow t])) (THead k0 u4 t1) (THead (Bind 
-b) u0 t4) H5) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return 
-(\lambda (_: T).T) with [(TSort _) \Rightarrow u4 | (TLRef _) \Rightarrow u4 
-| (THead _ t _) \Rightarrow t])) (THead k0 u4 t1) (THead (Bind b) u0 t4) H5) 
-in ((let H3 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: 
-T).K) with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k _ 
-_) \Rightarrow k])) (THead k0 u4 t1) (THead (Bind b) u0 t4) H5) in (eq_ind K 
-(Bind b) (\lambda (k: K).((eq T u4 u0) \to ((eq T t1 t4) \to ((eq T (THead k 
-u5 t2) t3) \to ((pr0 u4 u5) \to ((pr0 t1 t2) \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) t5)) t))))))))) (\lambda (H6: (eq T u4 
-u0)).(eq_ind T u0 (\lambda (t: T).((eq T t1 t4) \to ((eq T (THead (Bind b) u5 
-t2) t3) \to ((pr0 t u5) \to ((pr0 t1 t2) \to (ex2 T (\lambda (t0: T).(pr0 
-(THead (Flat Appl) u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u3 
-(THead (Flat Appl) (lift (S O) O v2) t5)) t0)))))))) (\lambda (H12: (eq T t1 
-t4)).(eq_ind T t4 (\lambda (t: T).((eq T (THead (Bind b) u5 t2) t3) \to ((pr0 
-u0 u5) \to ((pr0 t t2) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 
-t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S 
-O) O v2) t5)) t0))))))) (\lambda (H8: (eq T (THead (Bind b) u5 t2) 
-t3)).(eq_ind T (THead (Bind b) u5 t2) (\lambda (t: T).((pr0 u0 u5) \to ((pr0 
-t4 t2) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t) t0)) 
-(\lambda (t0: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) 
-t5)) t0)))))) (\lambda (H17: (pr0 u0 u5)).(\lambda (H18: (pr0 t4 t2)).(let H7 
-\def (eq_ind T u1 (\lambda (t: T).(pr0 t u2)) H7 v1 H11) in (ex2_ind T 
-(\lambda (t: T).(pr0 u2 t)) (\lambda (t: T).(pr0 v2 t)) (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))) 
-(\lambda (x: T).(\lambda (H9: (pr0 u2 x)).(\lambda (H11: (pr0 v2 x)).(ex2_ind 
-T (\lambda (t: T).(pr0 t2 t)) (\lambda (t: T).(pr0 t5 t)) (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))) 
-(\lambda (x0: T).(\lambda (H14: (pr0 t2 x0)).(\lambda (H16: (pr0 t5 
-x0)).(ex2_ind T (\lambda (t: T).(pr0 u5 t)) (\lambda (t: T).(pr0 u3 t)) (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))) (\lambda (x1: T).(\lambda (H15: (pr0 u5 x1)).(\lambda (H19: (pr0 u3 
-x1)).(pr0_confluence__pr0_cong_upsilon_cong b H13 u2 v2 x H9 H11 t2 t5 x0 H14 
-H16 u5 u3 x1 H15 H19)))) (H4 u0 (tlt_trans (THead (Bind b) u0 t4) u0 (THead 
-(Flat Appl) v1 (THead (Bind b) u0 t4)) (tlt_head_sx (Bind b) u0 t4) 
-(tlt_head_dx (Flat Appl) v1 (THead (Bind b) u0 t4))) u5 H17 u3 H15))))) (H4 
-t4 (tlt_trans (THead (Bind b) u0 t4) t4 (THead (Flat Appl) v1 (THead (Bind b) 
-u0 t4)) (tlt_head_dx (Bind b) u0 t4) (tlt_head_dx (Flat Appl) v1 (THead (Bind 
-b) u0 t4))) t2 H18 t5 H16))))) (H4 v1 (tlt_head_sx (Flat Appl) v1 (THead 
-(Bind b) u0 t4)) u2 H7 v2 H14))))) t3 H8)) t1 (sym_eq T t1 t4 H12))) u4 
-(sym_eq T u4 u0 H6))) k0 (sym_eq K k0 (Bind b) H3))) H2)) H1)) H10 H0 H4))) | 
-(pr0_beta u v0 v3 H0 t1 t2 H4) \Rightarrow (\lambda (H5: (eq T (THead (Flat 
-Appl) v0 (THead (Bind Abst) u t1)) (THead (Bind b) u0 t4))).(\lambda (H10: 
-(eq T (THead (Bind Abbr) v3 t2) t3)).((let H1 \def (eq_ind T (THead (Flat 
-Appl) v0 (THead (Bind Abst) u t1)) (\lambda (e: T).(match e return (\lambda 
-(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
-| (THead k _ _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind 
-_) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u0 
-t4) H5) in (False_ind ((eq T (THead (Bind Abbr) v3 t2) t3) \to ((pr0 v0 v3) 
-\to ((pr0 t1 t2) \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) t5)) t)))))) H1)) H10 H0 H4))) | (pr0_upsilon b0 H0 v0 v3 H4 u4 u5 H5 t1 
-t2 H10) \Rightarrow (\lambda (H13: (eq T (THead (Flat Appl) v0 (THead (Bind 
-b0) u4 t1)) (THead (Bind b) u0 t4))).(\lambda (H14: (eq T (THead (Bind b0) u5 
-(THead (Flat Appl) (lift (S O) O v3) t2)) t3)).((let H1 \def (eq_ind T (THead 
-(Flat Appl) v0 (THead (Bind b0) u4 t1)) (\lambda (e: T).(match e return 
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: 
-K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I 
-(THead (Bind b) u0 t4) H13) in (False_ind ((eq T (THead (Bind b0) u5 (THead 
-(Flat Appl) (lift (S O) O v3) t2)) t3) \to ((not (eq B b0 Abst)) \to ((pr0 v0 
-v3) \to ((pr0 u4 u5) \to ((pr0 t1 t2) \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) t5)) t)))))))) H1)) H14 H0 H4 H5 H10))) | (pr0_delta 
-u4 u5 H0 t1 t2 H4 w H5) \Rightarrow (\lambda (H10: (eq T (THead (Bind Abbr) 
-u4 t1) (THead (Bind b) u0 t4))).(\lambda (H17: (eq T (THead (Bind Abbr) u5 w) 
-t3)).((let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: 
-T).T) with [(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ _ 
-t) \Rightarrow t])) (THead (Bind Abbr) u4 t1) (THead (Bind b) u0 t4) H10) in 
-((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) 
-with [(TSort _) \Rightarrow u4 | (TLRef _) \Rightarrow u4 | (THead _ t _) 
-\Rightarrow t])) (THead (Bind Abbr) u4 t1) (THead (Bind b) u0 t4) H10) in 
-((let H3 \def (f_equal T B (\lambda (e: T).(match e return (\lambda (_: T).B) 
-with [(TSort _) \Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k _ _) 
-\Rightarrow (match k return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | 
-(Flat _) \Rightarrow Abbr])])) (THead (Bind Abbr) u4 t1) (THead (Bind b) u0 
-t4) H10) in (eq_ind B Abbr (\lambda (b: B).((eq T u4 u0) \to ((eq T t1 t4) 
-\to ((eq T (THead (Bind Abbr) u5 w) t3) \to ((pr0 u4 u5) \to ((pr0 t1 t2) \to 
-((subst0 O u5 t2 w) \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) t5)) t)))))))))) (\lambda (H6: (eq T u4 u0)).(eq_ind T u0 (\lambda (t: 
-T).((eq T t1 t4) \to ((eq T (THead (Bind Abbr) u5 w) t3) \to ((pr0 t u5) \to 
-((pr0 t1 t2) \to ((subst0 O u5 t2 w) \to (ex2 T (\lambda (t0: T).(pr0 (THead 
-(Flat Appl) u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) u3 (THead 
-(Flat Appl) (lift (S O) O v2) t5)) t0))))))))) (\lambda (H8: (eq T t1 
-t4)).(eq_ind T t4 (\lambda (t: T).((eq T (THead (Bind Abbr) u5 w) t3) \to 
-((pr0 u0 u5) \to ((pr0 t t2) \to ((subst0 O u5 t2 w) \to (ex2 T (\lambda (t0: 
-T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind 
-Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t0)))))))) (\lambda (H18: 
-(eq T (THead (Bind Abbr) u5 w) t3)).(eq_ind T (THead (Bind Abbr) u5 w) 
-(\lambda (t: T).((pr0 u0 u5) \to ((pr0 t4 t2) \to ((subst0 O u5 t2 w) \to 
-(ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t) t0)) (\lambda (t0: 
-T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) 
-t0))))))) (\lambda (H19: (pr0 u0 u5)).(\lambda (H20: (pr0 t4 t2)).(\lambda 
-(H21: (subst0 O u5 t2 w)).(let H9 \def (eq_ind_r B b (\lambda (b: B).(\forall 
-(v: T).((tlt v (THead (Flat Appl) v1 (THead (Bind b) u0 t4))) \to (\forall 
-(t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t: 
-T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t)))))))))) H4 Abbr H3) in (let H12 
-\def (eq_ind_r B b (\lambda (b: B).(eq T t0 (THead (Bind b) u0 t4))) H12 Abbr 
-H3) in (let H13 \def (eq_ind_r B b (\lambda (b: B).(not (eq B b Abst))) H13 
-Abbr H3) in (let H7 \def (eq_ind T u1 (\lambda (t: T).(pr0 t u2)) H7 v1 H11) 
-in (ex2_ind T (\lambda (t: T).(pr0 u2 t)) (\lambda (t: T).(pr0 v2 t)) (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 (x: T).(\lambda (H11: (pr0 u2 x)).(\lambda (H14: (pr0 
-v2 x)).(ex2_ind T (\lambda (t: T).(pr0 t2 t)) (\lambda (t: T).(pr0 t5 t)) 
-(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 (x0: T).(\lambda (H16: (pr0 t2 x0)).(\lambda (H22: 
-(pr0 t5 x0)).(ex2_ind T (\lambda (t: T).(pr0 u5 t)) (\lambda (t: T).(pr0 u3 
-t)) (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 (x1: T).(\lambda (H15: (pr0 u5 x1)).(\lambda 
-(H23: (pr0 u3 x1)).(pr0_confluence__pr0_cong_upsilon_delta H13 u5 t2 w H21 u2 
-v2 x H11 H14 t5 x0 H16 H22 u3 x1 H15 H23)))) (H9 u0 (tlt_trans (THead (Bind 
-Abbr) u0 t4) u0 (THead (Flat Appl) v1 (THead (Bind Abbr) u0 t4)) (tlt_head_sx 
-(Bind Abbr) u0 t4) (tlt_head_dx (Flat Appl) v1 (THead (Bind Abbr) u0 t4))) u5 
-H19 u3 H15))))) (H9 t4 (tlt_trans (THead (Bind Abbr) u0 t4) t4 (THead (Flat 
-Appl) v1 (THead (Bind Abbr) u0 t4)) (tlt_head_dx (Bind Abbr) u0 t4) 
-(tlt_head_dx (Flat Appl) v1 (THead (Bind Abbr) u0 t4))) t2 H20 t5 H16))))) 
-(H9 v1 (tlt_head_sx (Flat Appl) v1 (THead (Bind Abbr) u0 t4)) u2 H7 v2 
-H14))))))))) t3 H18)) t1 (sym_eq T t1 t4 H8))) u4 (sym_eq T u4 u0 H6))) b 
-H3)) H2)) H1)) H17 H0 H4 H5))) | (pr0_zeta b0 H0 t1 t2 H4 u) \Rightarrow 
-(\lambda (H5: (eq T (THead (Bind b0) u (lift (S O) O t1)) (THead (Bind b) u0 
-t4))).(\lambda (H10: (eq T t2 t3)).((let H1 \def (f_equal T T (\lambda (e: 
-T).(match e 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 u t0) 
-\Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) 
-(\lambda (x: nat).(plus x (S O))) O t1) | (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 u t0) 
-\Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) 
-(\lambda (x: nat).(plus x (S O))) O t1) | (THead _ _ t) \Rightarrow t])) 
-(THead (Bind b0) u (lift (S O) O t1)) (THead (Bind b) u0 t4) H5) in ((let H2 
-\def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) 
-\Rightarrow t])) (THead (Bind b0) u (lift (S O) O t1)) (THead (Bind b) u0 t4) 
-H5) in ((let H3 \def (f_equal T B (\lambda (e: T).(match e return (\lambda 
-(_: T).B) with [(TSort _) \Rightarrow b0 | (TLRef _) \Rightarrow b0 | (THead 
-k _ _) \Rightarrow (match k return (\lambda (_: K).B) with [(Bind b) 
-\Rightarrow b | (Flat _) \Rightarrow b0])])) (THead (Bind b0) u (lift (S O) O 
-t1)) (THead (Bind b) u0 t4) H5) in (eq_ind B b (\lambda (b1: B).((eq T u u0) 
-\to ((eq T (lift (S O) O t1) t4) \to ((eq T t2 t3) \to ((not (eq B b1 Abst)) 
-\to ((pr0 t1 t2) \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) t5)) t))))))))) (\lambda (H6: (eq T u u0)).(eq_ind T u0 (\lambda (_: 
-T).((eq T (lift (S O) O t1) t4) \to ((eq T t2 t3) \to ((not (eq B b Abst)) 
-\to ((pr0 t1 t2) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) 
-t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) 
-O v2) t5)) t0)))))))) (\lambda (H13: (eq T (lift (S O) O t1) t4)).(eq_ind T 
-(lift (S O) O t1) (\lambda (_: T).((eq T t2 t3) \to ((not (eq B b Abst)) \to 
-((pr0 t1 t2) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) 
-(\lambda (t0: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) 
-t5)) t0))))))) (\lambda (H8: (eq T t2 t3)).(eq_ind T t3 (\lambda (t: T).((not 
-(eq B b Abst)) \to ((pr0 t1 t) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Flat 
-Appl) u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) 
-(lift (S O) O v2) t5)) t0)))))) (\lambda (H17: (not (eq B b Abst))).(\lambda 
-(H18: (pr0 t1 t3)).(let H9 \def (eq_ind_r T t4 (\lambda (t: T).(\forall (v: 
-T).((tlt v (THead (Flat Appl) v1 (THead (Bind b) u0 t))) \to (\forall (t1: 
-T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: 
-T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H4 (lift (S O) O t1) 
-H13) in (let H12 \def (eq_ind_r T t4 (\lambda (t: T).(eq T t0 (THead (Bind b) 
-u0 t))) H12 (lift (S O) O t1) H13) in (let H16 \def (eq_ind_r T t4 (\lambda 
-(t: T).(pr0 t t5)) H16 (lift (S O) O t1) H13) in (ex2_ind T (\lambda (t3: 
-T).(eq T t5 (lift (S O) O t3))) (\lambda (t3: T).(pr0 t1 t3)) (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) t5)) t))) (\lambda (x: T).(\lambda 
-(H19: (eq T t5 (lift (S O) O x))).(\lambda (H20: (pr0 t1 x)).(eq_ind_r T 
-(lift (S O) O x) (\lambda (t: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Flat 
-Appl) u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) 
-(lift (S O) O v2) t)) t0)))) (let H7 \def (eq_ind T u1 (\lambda (t: T).(pr0 t 
-u2)) H7 v1 H11) in (ex2_ind T (\lambda (t: T).(pr0 u2 t)) (\lambda (t: 
-T).(pr0 v2 t)) (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))) (\lambda (x0: T).(\lambda (H11: (pr0 u2 x0)).(\lambda 
-(H14: (pr0 v2 x0)).(ex2_ind T (\lambda (t: T).(pr0 x t)) (\lambda (t: T).(pr0 
-t3 t)) (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))) (\lambda (x1: T).(\lambda (H21: (pr0 x x1)).(\lambda (H22: (pr0 t3 
-x1)).(pr0_confluence__pr0_cong_upsilon_zeta b H17 u0 u3 H15 u2 v2 x0 H11 H14 
-x t3 x1 H21 H22)))) (H9 t1 (tlt_trans (THead (Bind b) u0 (lift (S O) O t1)) 
-t1 (THead (Flat Appl) v1 (THead (Bind b) u0 (lift (S O) O t1))) (lift_tlt_dx 
-(Bind b) u0 t1 (S O) O) (tlt_head_dx (Flat Appl) v1 (THead (Bind b) u0 (lift 
-(S O) O t1)))) x H20 t3 H18))))) (H9 v1 (tlt_head_sx (Flat Appl) v1 (THead 
-(Bind b) u0 (lift (S O) O t1))) u2 H7 v2 H14))) t5 H19)))) (pr0_gen_lift t1 
-t5 (S O) O H16))))))) t2 (sym_eq T t2 t3 H8))) t4 H13)) u (sym_eq T u u0 
-H6))) b0 (sym_eq B b0 b H3))) H2)) H1)) H10 H0 H4))) | (pr0_epsilon t1 t2 H0 
-u) \Rightarrow (\lambda (H4: (eq T (THead (Flat Cast) u t1) (THead (Bind b) 
-u0 t4))).(\lambda (H5: (eq T t2 t3)).((let H1 \def (eq_ind T (THead (Flat 
-Cast) u t1) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
-\Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow 
-False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u0 t4) H4) in 
-(False_ind ((eq T t2 t3) \to ((pr0 t1 t2) \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) t5)) t))))) H1)) H5 H0)))]) in (H6 (refl_equal 
-T (THead (Bind b) u0 t4)) (refl_equal T t3))))) t0 (sym_eq T t0 (THead (Bind 
-b) u0 t4) H12))) u1 (sym_eq T u1 v1 H11))) k (sym_eq K k (Flat Appl) H3))) 
-H2)) H1)))]) in (H1 (refl_equal T (THead (Flat Appl) v1 (THead (Bind b) u0 
-t4)))))))))) t2 H12)) t H10 H11 H6 H7 H8 H9))) | (pr0_delta u0 u3 H6 t4 t5 H7 
-w H8) \Rightarrow (\lambda (H9: (eq T (THead (Bind Abbr) u0 t4) t)).(\lambda 
-(H10: (eq T (THead (Bind Abbr) u3 w) t2)).(eq_ind T (THead (Bind Abbr) u0 t4) 
-(\lambda (_: T).((eq T (THead (Bind Abbr) u3 w) t2) \to ((pr0 u0 u3) \to 
-((pr0 t4 t5) \to ((subst0 O u3 t5 w) \to (ex2 T (\lambda (t0: T).(pr0 (THead 
-k u2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0)))))))) (\lambda (H11: (eq T (THead 
-(Bind Abbr) u3 w) t2)).(eq_ind T (THead (Bind Abbr) u3 w) (\lambda (t: 
-T).((pr0 u0 u3) \to ((pr0 t4 t5) \to ((subst0 O u3 t5 w) \to (ex2 T (\lambda 
-(t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 t t0))))))) (\lambda 
-(H12: (pr0 u0 u3)).(\lambda (H13: (pr0 t4 t5)).(\lambda (H14: (subst0 O u3 t5 
-w)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead k u1 t0) t)) H4 
-(THead (Bind Abbr) u0 t4) H9) in (let H1 \def (match H0 return (\lambda (t: 
-T).(\lambda (_: (eq ? ? t)).((eq T t (THead (Bind Abbr) u0 t4)) \to (ex2 T 
-(\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind 
-Abbr) u3 w) t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead 
-k u1 t0) (THead (Bind Abbr) u0 t4))).(let H1 \def (f_equal T T (\lambda (e: 
-T).(match e return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef 
-_) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u1 t0) (THead 
-(Bind Abbr) u0 t4) H0) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e 
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) 
-\Rightarrow u1 | (THead _ t _) \Rightarrow t])) (THead k u1 t0) (THead (Bind 
-Abbr) u0 t4) H0) in ((let H3 \def (f_equal T K (\lambda (e: T).(match e 
-return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) 
-\Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k u1 t0) (THead (Bind 
-Abbr) u0 t4) H0) in (eq_ind K (Bind Abbr) (\lambda (k: K).((eq T u1 u0) \to 
-((eq T t0 t4) \to (ex2 T (\lambda (t: T).(pr0 (THead k u2 t3) t)) (\lambda 
-(t: T).(pr0 (THead (Bind Abbr) u3 w) t)))))) (\lambda (H10: (eq T u1 
-u0)).(eq_ind T u0 (\lambda (_: T).((eq T t0 t4) \to (ex2 T (\lambda (t0: 
-T).(pr0 (THead (Bind Abbr) u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind 
-Abbr) u3 w) t0))))) (\lambda (H11: (eq T t0 t4)).(eq_ind T t4 (\lambda (_: 
-T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 t3) t0)) (\lambda (t0: 
-T).(pr0 (THead (Bind Abbr) u3 w) t0)))) (let H4 \def (eq_ind_r T t (\lambda 
-(t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to 
-(\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) 
-(\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Bind Abbr) u0 t4) H9) in (let 
-H5 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t3)) H8 t4 H11) in (let H6 \def 
-(eq_ind T u1 (\lambda (t: T).(pr0 t u2)) H7 u0 H10) in (ex2_ind T (\lambda 
-(t: T).(pr0 u2 t)) (\lambda (t: T).(pr0 u3 t)) (ex2 T (\lambda (t: T).(pr0 
-(THead (Bind Abbr) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w) 
-t))) (\lambda (x: T).(\lambda (H7: (pr0 u2 x)).(\lambda (H8: (pr0 u3 
-x)).(ex2_ind T (\lambda (t: T).(pr0 t3 t)) (\lambda (t: T).(pr0 t5 t)) (ex2 T 
-(\lambda (t: T).(pr0 (THead (Bind Abbr) u2 t3) t)) (\lambda (t: T).(pr0 
-(THead (Bind Abbr) u3 w) t))) (\lambda (x0: T).(\lambda (H9: (pr0 t3 
-x0)).(\lambda (H12: (pr0 t5 x0)).(pr0_confluence__pr0_cong_delta u3 t5 w H14 
-u2 x H7 H8 t3 x0 H9 H12)))) (H4 t4 (tlt_head_dx (Bind Abbr) u0 t4) t3 H5 t5 
-H13))))) (H4 u0 (tlt_head_sx (Bind Abbr) u0 t4) u2 H6 u3 H12))))) t0 (sym_eq 
-T t0 t4 H11))) u1 (sym_eq T u1 u0 H10))) k (sym_eq K k (Bind Abbr) H3))) H2)) 
-H1)))]) in (H1 (refl_equal T (THead (Bind Abbr) u0 t4)))))))) t2 H11)) t H9 
-H10 H6 H7 H8))) | (pr0_zeta b H6 t4 t5 H7 u) \Rightarrow (\lambda (H9: (eq T 
-(THead (Bind b) u (lift (S O) O t4)) t)).(\lambda (H10: (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 (t0: T).(pr0 (THead 
-k u2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H11: (eq T t5 
-t2)).(eq_ind T t2 (\lambda (t: T).((not (eq B b Abst)) \to ((pr0 t4 t) \to 
-(ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 t2 
-t0)))))) (\lambda (H12: (not (eq B b Abst))).(\lambda (H13: (pr0 t4 t2)).(let 
-H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead k u1 t0) t)) H4 (THead 
-(Bind b) u (lift (S O) O t4)) H9) in (let H1 \def (match H0 return (\lambda 
-(t: T).(\lambda (_: (eq ? ? t)).((eq T t (THead (Bind b) u (lift (S O) O 
-t4))) \to (ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: 
-T).(pr0 t2 t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead k 
-u1 t0) (THead (Bind b) u (lift (S O) O t4)))).(let H1 \def (f_equal T T 
-(\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) 
-(THead k u1 t0) (THead (Bind b) u (lift (S O) O t4)) H0) in ((let H2 \def 
-(f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort 
-_) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t _) \Rightarrow t])) 
-(THead k u1 t0) (THead (Bind b) u (lift (S O) O t4)) H0) in ((let H3 \def 
-(f_equal T K (\lambda (e: T).(match e return (\lambda (_: T).K) with [(TSort 
-_) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) 
-(THead k u1 t0) (THead (Bind b) u (lift (S O) O t4)) H0) in (eq_ind K (Bind 
-b) (\lambda (k: K).((eq T u1 u) \to ((eq T t0 (lift (S O) O t4)) \to (ex2 T 
-(\lambda (t: T).(pr0 (THead k u2 t3) t)) (\lambda (t: T).(pr0 t2 t)))))) 
-(\lambda (H10: (eq T u1 u)).(eq_ind T u (\lambda (_: T).((eq T t0 (lift (S O) 
-O t4)) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 t3) t0)) (\lambda 
-(t0: T).(pr0 t2 t0))))) (\lambda (H11: (eq T t0 (lift (S O) O t4))).(eq_ind T 
-(lift (S O) O t4) (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind 
-b) u2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0)))) (let H4 \def (eq_ind_r T t 
-(\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) 
-\to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) 
-(\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Bind b) u (lift (S O) O t4)) 
-H9) in (let H5 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t3)) H8 (lift (S O) O 
-t4) H11) in (ex2_ind T (\lambda (t2: T).(eq T t3 (lift (S O) O t2))) (\lambda 
-(t2: T).(pr0 t4 t2)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 t3) t)) 
-(\lambda (t: T).(pr0 t2 t))) (\lambda (x: T).(\lambda (H6: (eq T t3 (lift (S 
-O) O x))).(\lambda (H8: (pr0 t4 x)).(eq_ind_r T (lift (S O) O x) (\lambda (t: 
-T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 t) t0)) (\lambda (t0: 
-T).(pr0 t2 t0)))) (let H7 \def (eq_ind T u1 (\lambda (t: T).(pr0 t u2)) H7 u 
-H10) in (ex2_ind T (\lambda (t: T).(pr0 x t)) (\lambda (t: T).(pr0 t2 t)) 
-(ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (lift (S O) O x)) t)) (\lambda 
-(t: T).(pr0 t2 t))) (\lambda (x0: T).(\lambda (H9: (pr0 x x0)).(\lambda (H13: 
-(pr0 t2 x0)).(ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (lift (S O) 
-O x)) t)) (\lambda (t: T).(pr0 t2 t)) x0 (pr0_zeta b H12 x x0 H9 u2) H13)))) 
-(H4 t4 (lift_tlt_dx (Bind b) u t4 (S O) O) x H8 t2 H13))) t3 H6)))) 
-(pr0_gen_lift t4 t3 (S O) O H5)))) t0 (sym_eq T t0 (lift (S O) O t4) H11))) 
-u1 (sym_eq T u1 u H10))) k (sym_eq K k (Bind b) H3))) H2)) H1)))]) in (H1 
-(refl_equal T (THead (Bind b) u (lift (S O) O t4)))))))) t5 (sym_eq T t5 t2 
-H11))) t H9 H10 H6 H7))) | (pr0_epsilon t4 t5 H6 u) \Rightarrow (\lambda (H9: 
-(eq T (THead (Flat Cast) u t4) t)).(\lambda (H10: (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 (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 t2 
-t0)))))) (\lambda (H11: (eq T t5 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t4 
-t) \to (ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: 
-T).(pr0 t2 t0))))) (\lambda (H12: (pr0 t4 t2)).(let H0 \def (eq_ind_r T t 
-(\lambda (t: T).(eq T (THead k u1 t0) t)) H4 (THead (Flat Cast) u t4) H9) in 
-(let H1 \def (match H0 return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T 
-t (THead (Flat Cast) u t4)) \to (ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) 
-t0)) (\lambda (t0: T).(pr0 t2 t0)))))) with [refl_equal \Rightarrow (\lambda 
-(H0: (eq T (THead k u1 t0) (THead (Flat Cast) u t4))).(let H1 \def (f_equal T 
-T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) 
-(THead k u1 t0) (THead (Flat Cast) u t4) H0) in ((let H2 \def (f_equal T T 
-(\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t _) \Rightarrow t])) 
-(THead k u1 t0) (THead (Flat Cast) u t4) H0) in ((let H3 \def (f_equal T K 
-(\lambda (e: T).(match e return (\lambda (_: T).K) with [(TSort _) 
-\Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) 
-(THead k u1 t0) (THead (Flat Cast) u t4) H0) in (eq_ind K (Flat Cast) 
-(\lambda (k: K).((eq T u1 u) \to ((eq T t0 t4) \to (ex2 T (\lambda (t: 
-T).(pr0 (THead k u2 t3) t)) (\lambda (t: T).(pr0 t2 t)))))) (\lambda (H10: 
-(eq T u1 u)).(eq_ind T u (\lambda (_: T).((eq T t0 t4) \to (ex2 T (\lambda 
-(t0: T).(pr0 (THead (Flat Cast) u2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0))))) 
-(\lambda (H11: (eq T t0 t4)).(eq_ind T t4 (\lambda (_: T).(ex2 T (\lambda 
-(t0: T).(pr0 (THead (Flat Cast) u2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0)))) 
-(let H4 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to 
-(\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T 
-(\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead 
-(Flat Cast) u t4) H9) in (let H5 \def (eq_ind T t0 (\lambda (t: T).(pr0 t 
-t3)) H8 t4 H11) in (let H6 \def (eq_ind T u1 (\lambda (t: T).(pr0 t u2)) H7 u 
-H10) in (ex2_ind T (\lambda (t: T).(pr0 t3 t)) (\lambda (t: T).(pr0 t2 t)) 
-(ex2 T (\lambda (t: T).(pr0 (THead (Flat Cast) u2 t3) t)) (\lambda (t: 
-T).(pr0 t2 t))) (\lambda (x: T).(\lambda (H7: (pr0 t3 x)).(\lambda (H8: (pr0 
-t2 x)).(ex_intro2 T (\lambda (t: T).(pr0 (THead (Flat Cast) u2 t3) t)) 
-(\lambda (t: T).(pr0 t2 t)) x (pr0_epsilon t3 x H7 u2) H8)))) (H4 t4 
-(tlt_head_dx (Flat Cast) u t4) t3 H5 t2 H12))))) t0 (sym_eq T t0 t4 H11))) u1 
-(sym_eq T u1 u H10))) k (sym_eq K k (Flat Cast) H3))) H2)) H1)))]) in (H1 
-(refl_equal T (THead (Flat Cast) u t4)))))) t5 (sym_eq T t5 t2 H11))) t H9 
-H10 H6)))]) in (H9 (refl_equal T t) (refl_equal T t2))))) t1 H6)) t H4 H5 H2 
-H3))) | (pr0_beta u v1 v2 H2 t0 t3 H3) \Rightarrow (\lambda (H4: (eq T (THead 
-(Flat Appl) v1 (THead (Bind Abst) u t0)) t)).(\lambda (H5: (eq T (THead (Bind 
-Abbr) v2 t3) t1)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) 
-(\lambda (_: T).((eq T (THead (Bind Abbr) v2 t3) t1) \to ((pr0 v1 v2) \to 
-((pr0 t0 t3) \to (ex2 T (\lambda (t2: T).(pr0 t1 t2)) (\lambda (t1: T).(pr0 
-t2 t1))))))) (\lambda (H6: (eq T (THead (Bind Abbr) v2 t3) t1)).(eq_ind T 
-(THead (Bind Abbr) v2 t3) (\lambda (t: T).((pr0 v1 v2) \to ((pr0 t0 t3) \to 
-(ex2 T (\lambda (t1: T).(pr0 t t1)) (\lambda (t1: T).(pr0 t2 t1)))))) 
-(\lambda (H7: (pr0 v1 v2)).(\lambda (H8: (pr0 t0 t3)).(let H9 \def (match H1 
-return (\lambda (t0: T).(\lambda (t1: T).(\lambda (_: (pr0 t0 t1)).((eq T t0 
-t) \to ((eq T t1 t2) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 
-t3) t)) (\lambda (t: T).(pr0 t2 t)))))))) with [(pr0_refl t4) \Rightarrow 
-(\lambda (H6: (eq T t4 t)).(\lambda (H9: (eq T t4 t2)).(eq_ind T t (\lambda 
-(t: T).((eq T t t2) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 
-t3) t0)) (\lambda (t0: T).(pr0 t2 t0))))) (\lambda (H10: (eq T t t2)).(eq_ind 
-T t2 (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) 
-t0)) (\lambda (t0: T).(pr0 t2 t0)))) (let H0 \def (eq_ind_r T t (\lambda (t: 
-T).(eq T t t2)) H10 (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) H4) in 
-(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (t: 
-T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: 
-T).(pr0 t t0)))) (let H1 \def (eq_ind_r T t (\lambda (t: T).(eq T t4 t)) H6 
-(THead (Flat Appl) v1 (THead (Bind Abst) u t0)) H4) in (let H2 \def (eq_ind_r 
-T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v 
-t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) 
-(\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Appl) v1 (THead (Bind 
-Abst) u t0)) H4) in (ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 
-t3) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) 
-t)) (THead (Bind Abbr) v2 t3) (pr0_refl (THead (Bind Abbr) v2 t3)) (pr0_beta 
-u v1 v2 H7 t0 t3 H8)))) t2 H0)) t (sym_eq T t t2 H10))) t4 (sym_eq T t4 t H6) 
-H9))) | (pr0_comp u1 u2 H6 t4 t5 H7 k) \Rightarrow (\lambda (H9: (eq T (THead 
-k u1 t4) t)).(\lambda (H10: (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 (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda 
-(t0: T).(pr0 t2 t0))))))) (\lambda (H11: (eq T (THead k u2 t5) t2)).(eq_ind T 
-(THead k u2 t5) (\lambda (t: T).((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T 
-(\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 t 
-t0)))))) (\lambda (H12: (pr0 u1 u2)).(\lambda (H13: (pr0 t4 t5)).(let H0 \def 
-(eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat Appl) v1 (THead (Bind Abst) 
-u t0)) t)) H4 (THead k u1 t4) H9) in (let H1 \def (match H0 return (\lambda 
-(t: T).(\lambda (_: (eq ? ? t)).((eq T t (THead k u1 t4)) \to (ex2 T (\lambda 
-(t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead k u2 
-t5) t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat 
-Appl) v1 (THead (Bind Abst) u t0)) (THead k u1 t4))).(let H1 \def (f_equal T 
-T (\lambda (e: T).(match e 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 k u1 t4) H0) in ((let H2 \def (f_equal T T (\lambda 
-(e: T).(match e 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 k u1 t4) H0) in ((let H3 \def (f_equal T 
-K (\lambda (e: T).(match e return (\lambda (_: T).K) with [(TSort _) 
-\Rightarrow (Flat Appl) | (TLRef _) \Rightarrow (Flat Appl) | (THead k _ _) 
-\Rightarrow k])) (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead k u1 
-t4) H0) in (eq_ind K (Flat Appl) (\lambda (k: K).((eq T v1 u1) \to ((eq T 
-(THead (Bind Abst) u t0) t4) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind 
-Abbr) v2 t3) t)) (\lambda (t: T).(pr0 (THead k u2 t5) t)))))) (\lambda (H10: 
-(eq T v1 u1)).(eq_ind T u1 (\lambda (_: T).((eq T (THead (Bind Abst) u t0) 
-t4) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda 
-(t0: T).(pr0 (THead (Flat Appl) u2 t5) t0))))) (\lambda (H11: (eq T (THead 
-(Bind Abst) u t0) t4)).(eq_ind T (THead (Bind Abst) u t0) (\lambda (_: 
-T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: 
-T).(pr0 (THead (Flat Appl) u2 t5) t0)))) (let H4 \def (eq_ind_r K k (\lambda 
-(k: K).(eq T (THead k u1 t4) t)) H9 (Flat Appl) H3) in (let H5 \def (eq_ind_r 
-T t4 (\lambda (t: T).(pr0 t t5)) H13 (THead (Bind Abst) u t0) H11) in (let H6 
-\def (match H5 return (\lambda (t: T).(\lambda (t1: T).(\lambda (_: (pr0 t 
-t1)).((eq T t (THead (Bind Abst) u t0)) \to ((eq T t1 t5) \to (ex2 T (\lambda 
-(t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead 
-(Flat Appl) u2 t5) t0)))))))) with [(pr0_refl t2) \Rightarrow (\lambda (H0: 
-(eq T t2 (THead (Bind Abst) u t0))).(\lambda (H1: (eq T t2 t5)).(eq_ind T 
-(THead (Bind Abst) u t0) (\lambda (t: T).((eq T t t5) \to (ex2 T (\lambda 
-(t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead 
-(Flat Appl) u2 t5) t0))))) (\lambda (H2: (eq T (THead (Bind Abst) u t0) 
-t5)).(eq_ind T (THead (Bind Abst) u t0) (\lambda (t: T).(ex2 T (\lambda (t0: 
-T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Flat 
-Appl) u2 t) t0)))) (let H3 \def (eq_ind_r T t4 (\lambda (t0: T).(eq T (THead 
-(Flat Appl) u1 t0) t)) H4 (THead (Bind Abst) u t0) H11) in (let H4 \def 
-(eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: 
-T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: 
-T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Appl) u1 
-(THead (Bind Abst) u t0)) H3) in (let H5 \def (eq_ind T v1 (\lambda (t: 
-T).(pr0 t v2)) H7 u1 H10) in (ex2_ind T (\lambda (t: T).(pr0 v2 t)) (\lambda 
-(t: T).(pr0 u2 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) 
-(\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u t0)) t))) 
-(\lambda (x: T).(\lambda (H6: (pr0 v2 x)).(\lambda (H7: (pr0 u2 
-x)).(ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda 
-(t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u t0)) t)) (THead (Bind 
-Abbr) x t3) (pr0_comp v2 x H6 t3 t3 (pr0_refl t3) (Bind Abbr)) (pr0_beta u u2 
-x H7 t0 t3 H8))))) (H4 u1 (tlt_head_sx (Flat Appl) u1 (THead (Bind Abst) u 
-t0)) v2 H5 u2 H12))))) t5 H2)) t2 (sym_eq T t2 (THead (Bind Abst) u t0) H0) 
-H1))) | (pr0_comp u0 u3 H0 t2 t6 H1 k) \Rightarrow (\lambda (H5: (eq T (THead 
-k u0 t2) (THead (Bind Abst) u t0))).(\lambda (H13: (eq T (THead k u3 t6) 
-t5)).((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: 
-T).T) with [(TSort _) \Rightarrow t2 | (TLRef _) \Rightarrow t2 | (THead _ _ 
-t) \Rightarrow t])) (THead k u0 t2) (THead (Bind Abst) u t0) H5) in ((let H3 
-\def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) 
-\Rightarrow t])) (THead k u0 t2) (THead (Bind Abst) u t0) H5) in ((let H6 
-\def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: T).K) with 
-[(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) 
-\Rightarrow k])) (THead k u0 t2) (THead (Bind Abst) u t0) H5) in (eq_ind K 
-(Bind Abst) (\lambda (k0: K).((eq T u0 u) \to ((eq T t2 t0) \to ((eq T (THead 
-k0 u3 t6) t5) \to ((pr0 u0 u3) \to ((pr0 t2 t6) \to (ex2 T (\lambda (t: 
-T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) 
-u2 t5) t))))))))) (\lambda (H9: (eq T u0 u)).(eq_ind T u (\lambda (t: T).((eq 
-T t2 t0) \to ((eq T (THead (Bind Abst) u3 t6) t5) \to ((pr0 t u3) \to ((pr0 
-t2 t6) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) 
-(\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t5) t0)))))))) (\lambda (H14: (eq 
-T t2 t0)).(eq_ind T t0 (\lambda (t: T).((eq T (THead (Bind Abst) u3 t6) t5) 
-\to ((pr0 u u3) \to ((pr0 t t6) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind 
-Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t5) t0))))))) 
-(\lambda (H15: (eq T (THead (Bind Abst) u3 t6) t5)).(eq_ind T (THead (Bind 
-Abst) u3 t6) (\lambda (t: T).((pr0 u u3) \to ((pr0 t0 t6) \to (ex2 T (\lambda 
-(t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead 
-(Flat Appl) u2 t) t0)))))) (\lambda (_: (pr0 u u3)).(\lambda (H17: (pr0 t0 
-t6)).(let H4 \def (eq_ind_r T t4 (\lambda (t0: T).(eq T (THead (Flat Appl) u1 
-t0) t)) H4 (THead (Bind Abst) u t0) H11) in (let H11 \def (eq_ind_r T t 
-(\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) 
-\to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) 
-(\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Appl) u1 (THead (Bind 
-Abst) u t0)) H4) in (let H7 \def (eq_ind T v1 (\lambda (t: T).(pr0 t v2)) H7 
-u1 H10) in (ex2_ind T (\lambda (t: T).(pr0 v2 t)) (\lambda (t: T).(pr0 u2 t)) 
-(ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: 
-T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u3 t6)) t))) (\lambda (x: 
-T).(\lambda (H10: (pr0 v2 x)).(\lambda (H12: (pr0 u2 x)).(ex2_ind T (\lambda 
-(t: T).(pr0 t6 t)) (\lambda (t: T).(pr0 t3 t)) (ex2 T (\lambda (t: T).(pr0 
-(THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u2 
-(THead (Bind Abst) u3 t6)) t))) (\lambda (x0: T).(\lambda (H8: (pr0 t6 
-x0)).(\lambda (H18: (pr0 t3 x0)).(ex_intro2 T (\lambda (t: T).(pr0 (THead 
-(Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead 
-(Bind Abst) u3 t6)) t)) (THead (Bind Abbr) x x0) (pr0_comp v2 x H10 t3 x0 H18 
-(Bind Abbr)) (pr0_beta u3 u2 x H12 t6 x0 H8))))) (H11 t0 (tlt_trans (THead 
-(Bind Abst) u t0) t0 (THead (Flat Appl) u1 (THead (Bind Abst) u t0)) 
-(tlt_head_dx (Bind Abst) u t0) (tlt_head_dx (Flat Appl) u1 (THead (Bind Abst) 
-u t0))) t6 H17 t3 H8))))) (H11 u1 (tlt_head_sx (Flat Appl) u1 (THead (Bind 
-Abst) u t0)) v2 H7 u2 H12))))))) t5 H15)) t2 (sym_eq T t2 t0 H14))) u0 
-(sym_eq T u0 u H9))) k (sym_eq K k (Bind Abst) H6))) H3)) H2)) H13 H0 H1))) | 
-(pr0_beta u0 v0 v3 H0 t2 t6 H1) \Rightarrow (\lambda (H4: (eq T (THead (Flat 
-Appl) v0 (THead (Bind Abst) u0 t2)) (THead (Bind Abst) u t0))).(\lambda (H11: 
-(eq T (THead (Bind Abbr) v3 t6) t5)).((let H2 \def (eq_ind T (THead (Flat 
-Appl) v0 (THead (Bind Abst) u0 t2)) (\lambda (e: T).(match e return (\lambda 
-(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
-| (THead k _ _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind 
-_) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u 
-t0) H4) in (False_ind ((eq T (THead (Bind Abbr) v3 t6) t5) \to ((pr0 v0 v3) 
-\to ((pr0 t2 t6) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) 
-t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t5) t)))))) H2)) H11 H0 H1))) 
-| (pr0_upsilon b H0 v0 v3 H1 u0 u3 H4 t2 t6 H11) \Rightarrow (\lambda (H12: 
-(eq T (THead (Flat Appl) v0 (THead (Bind b) u0 t2)) (THead (Bind Abst) u 
-t0))).(\lambda (H13: (eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) 
-O v3) t6)) t5)).((let H2 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind b) 
-u0 t2)) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
-(match k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | 
-(Flat _) \Rightarrow True])])) I (THead (Bind Abst) u t0) H12) in (False_ind 
-((eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) t5) \to 
-((not (eq B b Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to ((pr0 t2 t6) \to 
-(ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: 
-T).(pr0 (THead (Flat Appl) u2 t5) t)))))))) H2)) H13 H0 H1 H4 H11))) | 
-(pr0_delta u0 u3 H0 t2 t6 H1 w H4) \Rightarrow (\lambda (H11: (eq T (THead 
-(Bind Abbr) u0 t2) (THead (Bind Abst) u t0))).(\lambda (H12: (eq T (THead 
-(Bind Abbr) u3 w) t5)).((let H2 \def (eq_ind T (THead (Bind Abbr) u0 t2) 
-(\lambda (e: T).(match e return (\lambda (_: T).Prop) with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
-(match k return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b 
-return (\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow 
-False | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (THead 
-(Bind Abst) u t0) H11) in (False_ind ((eq T (THead (Bind Abbr) u3 w) t5) \to 
-((pr0 u0 u3) \to ((pr0 t2 t6) \to ((subst0 O u3 t6 w) \to (ex2 T (\lambda (t: 
-T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) 
-u2 t5) t))))))) H2)) H12 H0 H1 H4))) | (pr0_zeta b H0 t2 t6 H1 u0) 
-\Rightarrow (\lambda (H4: (eq T (THead (Bind b) u0 (lift (S O) O t2)) (THead 
-(Bind Abst) u t0))).(\lambda (H11: (eq T t6 t5)).((let H2 \def (f_equal T T 
-(\lambda (e: T).(match e 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 u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) 
-t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t2) | (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 u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) 
-t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t2) | (THead _ _ t) 
-\Rightarrow t])) (THead (Bind b) u0 (lift (S O) O t2)) (THead (Bind Abst) u 
-t0) H4) in ((let H3 \def (f_equal T T (\lambda (e: T).(match e return 
-(\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 
-| (THead _ t _) \Rightarrow t])) (THead (Bind b) u0 (lift (S O) O t2)) (THead 
-(Bind Abst) u t0) H4) in ((let H5 \def (f_equal T B (\lambda (e: T).(match e 
-return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | (TLRef _) 
-\Rightarrow b | (THead k _ _) \Rightarrow (match k return (\lambda (_: K).B) 
-with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b])])) (THead (Bind b) u0 
-(lift (S O) O t2)) (THead (Bind Abst) u t0) H4) in (eq_ind B Abst (\lambda 
-(b0: B).((eq T u0 u) \to ((eq T (lift (S O) O t2) t0) \to ((eq T t6 t5) \to 
-((not (eq B b0 Abst)) \to ((pr0 t2 t6) \to (ex2 T (\lambda (t: T).(pr0 (THead 
-(Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t5) 
-t))))))))) (\lambda (H6: (eq T u0 u)).(eq_ind T u (\lambda (_: T).((eq T 
-(lift (S O) O t2) t0) \to ((eq T t6 t5) \to ((not (eq B Abst Abst)) \to ((pr0 
-t2 t6) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) 
-(\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t5) t0)))))))) (\lambda (H12: (eq 
-T (lift (S O) O t2) t0)).(eq_ind T (lift (S O) O t2) (\lambda (_: T).((eq T 
-t6 t5) \to ((not (eq B Abst Abst)) \to ((pr0 t2 t6) \to (ex2 T (\lambda (t0: 
-T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Flat 
-Appl) u2 t5) t0))))))) (\lambda (H7: (eq T t6 t5)).(eq_ind T t5 (\lambda (t: 
-T).((not (eq B Abst Abst)) \to ((pr0 t2 t) \to (ex2 T (\lambda (t0: T).(pr0 
-(THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 
-t5) t0)))))) (\lambda (H13: (not (eq B Abst Abst))).(\lambda (_: (pr0 t2 
-t5)).(let H8 \def (match (H13 (refl_equal B Abst)) return (\lambda (_: 
-False).(ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: 
-T).(pr0 (THead (Flat Appl) u2 t5) t)))) with []) in H8))) t6 (sym_eq T t6 t5 
-H7))) t0 H12)) u0 (sym_eq T u0 u H6))) b (sym_eq B b Abst H5))) H3)) H2)) H11 
-H0 H1))) | (pr0_epsilon t2 t6 H0 u0) \Rightarrow (\lambda (H1: (eq T (THead 
-(Flat Cast) u0 t2) (THead (Bind Abst) u t0))).(\lambda (H4: (eq T t6 
-t5)).((let H2 \def (eq_ind T (THead (Flat Cast) u0 t2) (\lambda (e: T).(match 
-e return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: 
-K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I 
-(THead (Bind Abst) u t0) H1) in (False_ind ((eq T t6 t5) \to ((pr0 t2 t6) \to 
-(ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: 
-T).(pr0 (THead (Flat Appl) u2 t5) t))))) H2)) H4 H0)))]) in (H6 (refl_equal T 
-(THead (Bind Abst) u t0)) (refl_equal T t5))))) t4 H11)) v1 (sym_eq T v1 u1 
-H10))) k H3)) H2)) H1)))]) in (H1 (refl_equal T (THead k u1 t4))))))) t2 
-H11)) t H9 H10 H6 H7))) | (pr0_beta u0 v0 v3 H6 t4 t5 H7) \Rightarrow 
-(\lambda (H9: (eq T (THead (Flat Appl) v0 (THead (Bind Abst) u0 t4)) 
-t)).(\lambda (H10: (eq T (THead (Bind Abbr) v3 t5) t2)).(eq_ind T (THead 
-(Flat Appl) v0 (THead (Bind Abst) u0 t4)) (\lambda (_: T).((eq T (THead (Bind 
-Abbr) v3 t5) t2) \to ((pr0 v0 v3) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: 
-T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0))))))) 
-(\lambda (H11: (eq T (THead (Bind Abbr) v3 t5) t2)).(eq_ind T (THead (Bind 
-Abbr) v3 t5) (\lambda (t: T).((pr0 v0 v3) \to ((pr0 t4 t5) \to (ex2 T 
-(\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 t 
-t0)))))) (\lambda (H12: (pr0 v0 v3)).(\lambda (H13: (pr0 t4 t5)).(let H0 \def 
-(eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat Appl) v1 (THead (Bind Abst) 
-u t0)) t)) H4 (THead (Flat Appl) v0 (THead (Bind Abst) u0 t4)) H9) in (let H1 
-\def (match H0 return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t 
-(THead (Flat Appl) v0 (THead (Bind Abst) u0 t4))) \to (ex2 T (\lambda (t0: 
-T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind 
-Abbr) v3 t5) t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead 
-(Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Appl) v0 (THead (Bind 
-Abst) u0 t4)))).(let H1 \def (f_equal T T (\lambda (e: T).(match e return 
-(\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 
-| (THead _ _ t) \Rightarrow (match t return (\lambda (_: T).T) with [(TSort 
-_) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t0) \Rightarrow 
-t0])])) (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Appl) v0 
-(THead (Bind Abst) u0 t4)) H0) in ((let H2 \def (f_equal T T (\lambda (e: 
-T).(match e return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef 
-_) \Rightarrow u | (THead _ _ t) \Rightarrow (match t return (\lambda (_: 
-T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t0 
-_) \Rightarrow t0])])) (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead 
-(Flat Appl) v0 (THead (Bind Abst) u0 t4)) H0) in ((let H3 \def (f_equal T T 
-(\lambda (e: T).(match e 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) v0 (THead 
-(Bind Abst) u0 t4)) H0) in (eq_ind T v0 (\lambda (_: T).((eq T u u0) \to ((eq 
-T t0 t4) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) 
-(\lambda (t0: T).(pr0 (THead (Bind Abbr) v3 t5) t0)))))) (\lambda (H10: (eq T 
-u u0)).(eq_ind T u0 (\lambda (_: T).((eq T t0 t4) \to (ex2 T (\lambda (t0: 
-T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind 
-Abbr) v3 t5) t0))))) (\lambda (H11: (eq T t0 t4)).(eq_ind T t4 (\lambda (_: 
-T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: 
-T).(pr0 (THead (Bind Abbr) v3 t5) t0)))) (let H4 \def (eq_ind_r T t (\lambda 
-(t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to 
-(\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) 
-(\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Appl) v0 (THead (Bind 
-Abst) u0 t4)) H9) in (let H5 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t3)) H8 
-t4 H11) in (let H6 \def (eq_ind T v1 (\lambda (t: T).(pr0 t v2)) H7 v0 H3) in 
-(ex2_ind T (\lambda (t: T).(pr0 v2 t)) (\lambda (t: T).(pr0 v3 t)) (ex2 T 
-(\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 
-(THead (Bind Abbr) v3 t5) t))) (\lambda (x: T).(\lambda (H7: (pr0 v2 
-x)).(\lambda (H8: (pr0 v3 x)).(ex2_ind T (\lambda (t: T).(pr0 t3 t)) (\lambda 
-(t: T).(pr0 t5 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) 
-(\lambda (t: T).(pr0 (THead (Bind Abbr) v3 t5) t))) (\lambda (x0: T).(\lambda 
-(H9: (pr0 t3 x0)).(\lambda (H12: (pr0 t5 x0)).(ex_intro2 T (\lambda (t: 
-T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) 
-v3 t5) t)) (THead (Bind Abbr) x x0) (pr0_comp v2 x H7 t3 x0 H9 (Bind Abbr)) 
-(pr0_comp v3 x H8 t5 x0 H12 (Bind Abbr)))))) (H4 t4 (tlt_trans (THead (Bind 
-Abst) u0 t4) t4 (THead (Flat Appl) v0 (THead (Bind Abst) u0 t4)) (tlt_head_dx 
-(Bind Abst) u0 t4) (tlt_head_dx (Flat Appl) v0 (THead (Bind Abst) u0 t4))) t3 
-H5 t5 H13))))) (H4 v0 (tlt_head_sx (Flat Appl) v0 (THead (Bind Abst) u0 t4)) 
-v2 H6 v3 H12))))) t0 (sym_eq T t0 t4 H11))) u (sym_eq T u u0 H10))) v1 
-(sym_eq T v1 v0 H3))) H2)) H1)))]) in (H1 (refl_equal T (THead (Flat Appl) v0 
-(THead (Bind Abst) u0 t4)))))))) t2 H11)) t H9 H10 H6 H7))) | (pr0_upsilon b 
-H6 v0 v3 H7 u1 u2 H8 t4 t5 H9) \Rightarrow (\lambda (H10: (eq T (THead (Flat 
-Appl) v0 (THead (Bind b) u1 t4)) t)).(\lambda (H11: (eq T (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v3) t5)) t2)).(eq_ind T (THead (Flat Appl) 
-v0 (THead (Bind b) u1 t4)) (\lambda (_: T).((eq T (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v3) t5)) t2) \to ((not (eq B b Abst)) \to ((pr0 v0 
-v3) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead 
-(Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0))))))))) (\lambda (H12: 
-(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v3) t5)) 
-t2)).(eq_ind T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v3) t5)) 
-(\lambda (t: T).((not (eq B b Abst)) \to ((pr0 v0 v3) \to ((pr0 u1 u2) \to 
-((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) 
-(\lambda (t0: T).(pr0 t t0)))))))) (\lambda (H13: (not (eq B b 
-Abst))).(\lambda (_: (pr0 v0 v3)).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0 
-t4 t5)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat Appl) 
-v1 (THead (Bind Abst) u t0)) t)) H4 (THead (Flat Appl) v0 (THead (Bind b) u1 
-t4)) H10) in (let H1 \def (match H0 return (\lambda (t: T).(\lambda (_: (eq ? 
-? t)).((eq T t (THead (Flat Appl) v0 (THead (Bind b) u1 t4))) \to (ex2 T 
-(\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v3) t5)) t0)))))) with 
-[refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat Appl) v1 (THead 
-(Bind Abst) u t0)) (THead (Flat Appl) v0 (THead (Bind b) u1 t4)))).(let H1 
-\def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) 
-\Rightarrow (match t return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 
-| (TLRef _) \Rightarrow t0 | (THead _ _ t0) \Rightarrow t0])])) (THead (Flat 
-Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Appl) v0 (THead (Bind b) u1 
-t4)) H0) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return 
-(\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | 
-(THead _ _ t) \Rightarrow (match t return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t0 _) \Rightarrow t0])])) 
-(THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Appl) v0 (THead 
-(Bind b) u1 t4)) H0) in ((let H3 \def (f_equal T B (\lambda (e: T).(match e 
-return (\lambda (_: T).B) with [(TSort _) \Rightarrow Abst | (TLRef _) 
-\Rightarrow Abst | (THead _ _ t) \Rightarrow (match t return (\lambda (_: 
-T).B) with [(TSort _) \Rightarrow Abst | (TLRef _) \Rightarrow Abst | (THead 
-k _ _) \Rightarrow (match k return (\lambda (_: K).B) with [(Bind b) 
-\Rightarrow b | (Flat _) \Rightarrow Abst])])])) (THead (Flat Appl) v1 (THead 
-(Bind Abst) u t0)) (THead (Flat Appl) v0 (THead (Bind b) u1 t4)) H0) in ((let 
-H4 \def (f_equal T T (\lambda (e: T).(match e 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) v0 (THead (Bind b) u1 t4)) H0) in (eq_ind T v0 (\lambda (_: T).((eq B 
-Abst b) \to ((eq T u u1) \to ((eq T t0 t4) \to (ex2 T (\lambda (t0: T).(pr0 
-(THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v3) t5)) t0))))))) (\lambda (H11: (eq B Abst 
-b)).(eq_ind B Abst (\lambda (b: B).((eq T u u1) \to ((eq T t0 t4) \to (ex2 T 
-(\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v3) t5)) t)))))) (\lambda 
-(H12: (eq T u u1)).(eq_ind T u1 (\lambda (_: T).((eq T t0 t4) \to (ex2 T 
-(\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 
-(THead (Bind Abst) u2 (THead (Flat Appl) (lift (S O) O v3) t5)) t0))))) 
-(\lambda (H14: (eq T t0 t4)).(eq_ind T t4 (\lambda (_: T).(ex2 T (\lambda 
-(t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead 
-(Bind Abst) u2 (THead (Flat Appl) (lift (S O) O v3) t5)) t0)))) (let H5 \def 
-(eq_ind_r B b (\lambda (b: B).(not (eq B b Abst))) H13 Abst H11) in (let H6 
-\def (match (H5 (refl_equal B Abst)) return (\lambda (_: False).(ex2 T 
-(\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 
-(THead (Bind Abst) u2 (THead (Flat Appl) (lift (S O) O v3) t5)) t)))) with 
-[]) in H6)) t0 (sym_eq T t0 t4 H14))) u (sym_eq T u u1 H12))) b H11)) v1 
-(sym_eq T v1 v0 H4))) H3)) H2)) H1)))]) in (H1 (refl_equal T (THead (Flat 
-Appl) v0 (THead (Bind b) u1 t4)))))))))) t2 H12)) t H10 H11 H6 H7 H8 H9))) | 
-(pr0_delta u1 u2 H6 t4 t5 H7 w H8) \Rightarrow (\lambda (H9: (eq T (THead 
-(Bind Abbr) u1 t4) t)).(\lambda (H10: (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 (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: 
-T).(pr0 t2 t0)))))))) (\lambda (H11: (eq T (THead (Bind Abbr) u2 w) 
-t2)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t: T).((pr0 u1 u2) \to 
-((pr0 t4 t5) \to ((subst0 O u2 t5 w) \to (ex2 T (\lambda (t0: T).(pr0 (THead 
-(Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 t t0))))))) (\lambda (_: (pr0 
-u1 u2)).(\lambda (_: (pr0 t4 t5)).(\lambda (_: (subst0 O u2 t5 w)).(let H0 
-\def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat Appl) v1 (THead (Bind 
-Abst) u t0)) t)) H4 (THead (Bind Abbr) u1 t4) H9) in (let H1 \def (match H0 
-return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t (THead (Bind Abbr) 
-u1 t4)) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) 
-(\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)))))) with [refl_equal 
-\Rightarrow (\lambda (H0: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u 
-t0)) (THead (Bind Abbr) u1 t4))).(let H1 \def (eq_ind T (THead (Flat Appl) v1 
-(THead (Bind Abst) u t0)) (\lambda (e: T).(match e return (\lambda (_: 
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
-(THead k _ _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind 
-_) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1 
-t4) H0) in (False_ind (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) 
-t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t))) H1)))]) in (H1 
-(refl_equal T (THead (Bind Abbr) u1 t4)))))))) t2 H11)) t H9 H10 H6 H7 H8))) 
-| (pr0_zeta b H6 t4 t5 H7 u0) \Rightarrow (\lambda (H8: (eq T (THead (Bind b) 
-u0 (lift (S O) O t4)) t)).(\lambda (H9: (eq T t5 t2)).(eq_ind T (THead (Bind 
-b) u0 (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 (t0: T).(pr0 (THead (Bind Abbr) 
-v2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H10: (eq T t5 
-t2)).(eq_ind T t2 (\lambda (t: T).((not (eq B b Abst)) \to ((pr0 t4 t) \to 
-(ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: 
-T).(pr0 t2 t0)))))) (\lambda (_: (not (eq B b Abst))).(\lambda (_: (pr0 t4 
-t2)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat Appl) v1 
-(THead (Bind Abst) u t0)) t)) H4 (THead (Bind b) u0 (lift (S O) O t4)) H8) in 
-(let H1 \def (match H0 return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T 
-t (THead (Bind b) u0 (lift (S O) O t4))) \to (ex2 T (\lambda (t0: T).(pr0 
-(THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0)))))) with 
-[refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat Appl) v1 (THead 
-(Bind Abst) u t0)) (THead (Bind b) u0 (lift (S O) O t4)))).(let H1 \def 
-(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (e: 
-T).(match e return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | 
-(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return 
-(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow 
-True])])) I (THead (Bind b) u0 (lift (S O) O t4)) H0) in (False_ind (ex2 T 
-(\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 t2 
-t))) H1)))]) in (H1 (refl_equal T (THead (Bind b) u0 (lift (S O) O t4)))))))) 
-t5 (sym_eq T t5 t2 H10))) t H8 H9 H6 H7))) | (pr0_epsilon t4 t5 H6 u0) 
-\Rightarrow (\lambda (H7: (eq T (THead (Flat Cast) u0 t4) t)).(\lambda (H8: 
-(eq T t5 t2)).(eq_ind T (THead (Flat Cast) u0 t4) (\lambda (_: T).((eq T t5 
-t2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 
-t3) t0)) (\lambda (t0: T).(pr0 t2 t0)))))) (\lambda (H9: (eq T t5 
-t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t4 t) \to (ex2 T (\lambda (t0: 
-T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0))))) 
-(\lambda (_: (pr0 t4 t2)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T 
-(THead (Flat Appl) v1 (THead (Bind Abst) u t0)) t)) H4 (THead (Flat Cast) u0 
-t4) H7) in (let H1 \def (match H0 return (\lambda (t: T).(\lambda (_: (eq ? ? 
-t)).((eq T t (THead (Flat Cast) u0 t4)) \to (ex2 T (\lambda (t0: T).(pr0 
-(THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0)))))) with 
-[refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat Appl) v1 (THead 
-(Bind Abst) u t0)) (THead (Flat Cast) u0 t4))).(let H1 \def (eq_ind T (THead 
-(Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (e: T).(match e return 
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: 
-K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f 
-return (\lambda (_: F).Prop) with [Appl \Rightarrow True | Cast \Rightarrow 
-False])])])) I (THead (Flat Cast) u0 t4) H0) in (False_ind (ex2 T (\lambda 
-(t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 t2 t))) 
-H1)))]) in (H1 (refl_equal T (THead (Flat Cast) u0 t4)))))) t5 (sym_eq T t5 
-t2 H9))) t H7 H8 H6)))]) 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 t0 t3 H5) 
-\Rightarrow (\lambda (H6: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) 
-t)).(\lambda (H7: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O 
-v2) t3)) t1)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) 
-(\lambda (_: T).((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O 
-v2) t3)) t1) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to 
-((pr0 t0 t3) \to (ex2 T (\lambda (t2: T).(pr0 t1 t2)) (\lambda (t1: T).(pr0 
-t2 t1))))))))) (\lambda (H8: (eq T (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t3)) t1)).(eq_ind T (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t3)) (\lambda (t: T).((not (eq B b Abst)) \to ((pr0 v1 v2) 
-\to ((pr0 u1 u2) \to ((pr0 t0 t3) \to (ex2 T (\lambda (t1: T).(pr0 t t1)) 
-(\lambda (t1: T).(pr0 t2 t1)))))))) (\lambda (H9: (not (eq B b 
-Abst))).(\lambda (H10: (pr0 v1 v2)).(\lambda (H11: (pr0 u1 u2)).(\lambda 
-(H12: (pr0 t0 t3)).(let H13 \def (match H1 return (\lambda (t0: T).(\lambda 
-(t1: T).(\lambda (_: (pr0 t0 t1)).((eq T t0 t) \to ((eq T t1 t2) \to (ex2 T 
-(\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
-t3)) t)) (\lambda (t: T).(pr0 t2 t)))))))) with [(pr0_refl t4) \Rightarrow 
-(\lambda (H8: (eq T t4 t)).(\lambda (H13: (eq T t4 t2)).(eq_ind T t (\lambda 
-(t: T).((eq T t t2) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 t2 
-t0))))) (\lambda (H14: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T 
-(\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
-t3)) t0)) (\lambda (t0: T).(pr0 t2 t0)))) (let H0 \def (eq_ind_r T t (\lambda 
-(t: T).(eq T t t2)) H14 (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) H6) in 
-(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (\lambda (t: T).(ex2 
-T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O 
-v2) t3)) t0)) (\lambda (t0: T).(pr0 t t0)))) (let H1 \def (eq_ind_r T t 
-(\lambda (t: T).(eq T t4 t)) H8 (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) 
-H6) in (let H2 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) 
-\to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T 
-(\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead 
-(Flat Appl) v1 (THead (Bind b) u1 t0)) H6) in (ex2_sym T (pr0 (THead (Flat 
-Appl) v1 (THead (Bind b) u1 t0))) (pr0 (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t3))) (pr0_confluence__pr0_cong_upsilon_refl b H9 u1 u2 H11 
-t0 t3 H12 v1 v2 v2 H10 (pr0_refl v2))))) t2 H0)) t (sym_eq T t t2 H14))) t4 
-(sym_eq T t4 t H8) H13))) | (pr0_comp u0 u3 H8 t4 t5 H9 k) \Rightarrow 
-(\lambda (H13: (eq T (THead k u0 t4) t)).(\lambda (H14: (eq T (THead k u3 t5) 
-t2)).(eq_ind T (THead k u0 t4) (\lambda (_: T).((eq T (THead k u3 t5) t2) \to 
-((pr0 u0 u3) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind 
-b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 t2 
-t0))))))) (\lambda (H15: (eq T (THead k u3 t5) t2)).(eq_ind T (THead k u3 t5) 
-(\lambda (t: T).((pr0 u0 u3) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: 
-T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) 
-(\lambda (t0: T).(pr0 t t0)))))) (\lambda (H16: (pr0 u0 u3)).(\lambda (H17: 
-(pr0 t4 t5)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat 
-Appl) v1 (THead (Bind b) u1 t0)) t)) H6 (THead k u0 t4) H13) in (let H1 \def 
-(match H0 return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t (THead k 
-u0 t4)) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead k u3 t5) 
-t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat Appl) 
-v1 (THead (Bind b) u1 t0)) (THead k u0 t4))).(let H1 \def (f_equal T T 
-(\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow (THead (Bind b) u1 t0) | (TLRef _) \Rightarrow (THead (Bind b) u1 
-t0) | (THead _ _ t) \Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind b) u1 
-t0)) (THead k u0 t4) H0) in ((let H2 \def (f_equal T T (\lambda (e: T).(match 
-e return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) 
-\Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) v1 (THead 
-(Bind b) u1 t0)) (THead k u0 t4) H0) in ((let H3 \def (f_equal T K (\lambda 
-(e: T).(match e return (\lambda (_: T).K) with [(TSort _) \Rightarrow (Flat 
-Appl) | (TLRef _) \Rightarrow (Flat Appl) | (THead k _ _) \Rightarrow k])) 
-(THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (THead k u0 t4) H0) in (eq_ind 
-K (Flat Appl) (\lambda (k: K).((eq T v1 u0) \to ((eq T (THead (Bind b) u1 t0) 
-t4) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead k u3 t5) t)))))) 
-(\lambda (H14: (eq T v1 u0)).(eq_ind T u0 (\lambda (_: T).((eq T (THead (Bind 
-b) u1 t0) t4) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Flat 
-Appl) u3 t5) t0))))) (\lambda (H15: (eq T (THead (Bind b) u1 t0) t4)).(eq_ind 
-T (THead (Bind b) u1 t0) (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: 
-T).(pr0 (THead (Flat Appl) u3 t5) t0)))) (let H4 \def (eq_ind_r K k (\lambda 
-(k: K).(eq T (THead k u0 t4) t)) H13 (Flat Appl) H3) in (let H5 \def 
-(eq_ind_r T t4 (\lambda (t: T).(pr0 t t5)) H17 (THead (Bind b) u1 t0) H15) in 
-(let H6 \def (match H5 return (\lambda (t: T).(\lambda (t1: T).(\lambda (_: 
-(pr0 t t1)).((eq T t (THead (Bind b) u1 t0)) \to ((eq T t1 t5) \to (ex2 T 
-(\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
-t3)) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u3 t5) t0)))))))) with 
-[(pr0_refl t2) \Rightarrow (\lambda (H0: (eq T t2 (THead (Bind b) u1 
-t0))).(\lambda (H1: (eq T t2 t5)).(eq_ind T (THead (Bind b) u1 t0) (\lambda 
-(t: T).((eq T t t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead 
-(Flat Appl) u3 t5) t0))))) (\lambda (H2: (eq T (THead (Bind b) u1 t0) 
-t5)).(eq_ind T (THead (Bind b) u1 t0) (\lambda (t: T).(ex2 T (\lambda (t0: 
-T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) 
-(\lambda (t0: T).(pr0 (THead (Flat Appl) u3 t) t0)))) (let H3 \def (eq_ind_r 
-T t4 (\lambda (t0: T).(eq T (THead (Flat Appl) u0 t0) t)) H4 (THead (Bind b) 
-u1 t0) H15) in (let H4 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: 
-T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v 
-t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 
-t0)))))))))) H (THead (Flat Appl) u0 (THead (Bind b) u1 t0)) H3) in (let H5 
-\def (eq_ind T v1 (\lambda (t: T).(pr0 t v2)) H10 u0 H14) in (ex2_ind T 
-(\lambda (t: T).(pr0 v2 t)) (\lambda (t: T).(pr0 u3 t)) (ex2 T (\lambda (t: 
-T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) 
-(\lambda (t: T).(pr0 (THead (Flat Appl) u3 (THead (Bind b) u1 t0)) t))) 
-(\lambda (x: T).(\lambda (H6: (pr0 v2 x)).(\lambda (H7: (pr0 u3 x)).(ex2_sym 
-T (pr0 (THead (Flat Appl) u3 (THead (Bind b) u1 t0))) (pr0 (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t3))) 
-(pr0_confluence__pr0_cong_upsilon_refl b H9 u1 u2 H11 t0 t3 H12 u3 v2 x H7 
-H6))))) (H4 u0 (tlt_head_sx (Flat Appl) u0 (THead (Bind b) u1 t0)) v2 H5 u3 
-H16))))) t5 H2)) t2 (sym_eq T t2 (THead (Bind b) u1 t0) H0) H1))) | (pr0_comp 
-u4 u5 H0 t2 t6 H1 k) \Rightarrow (\lambda (H6: (eq T (THead k u4 t2) (THead 
-(Bind b) u1 t0))).(\lambda (H13: (eq T (THead k u5 t6) t5)).((let H2 \def 
-(f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort 
-_) \Rightarrow t2 | (TLRef _) \Rightarrow t2 | (THead _ _ t) \Rightarrow t])) 
-(THead k u4 t2) (THead (Bind b) u1 t0) H6) in ((let H3 \def (f_equal T T 
-(\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow u4 | (TLRef _) \Rightarrow u4 | (THead _ t _) \Rightarrow t])) 
-(THead k u4 t2) (THead (Bind b) u1 t0) H6) in ((let H5 \def (f_equal T K 
-(\lambda (e: T).(match e return (\lambda (_: T).K) with [(TSort _) 
-\Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) 
-(THead k u4 t2) (THead (Bind b) u1 t0) H6) in (eq_ind K (Bind b) (\lambda 
-(k0: K).((eq T u4 u1) \to ((eq T t2 t0) \to ((eq T (THead k0 u5 t6) t5) \to 
-((pr0 u4 u5) \to ((pr0 t2 t6) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) 
-u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead 
-(Flat Appl) u3 t5) t))))))))) (\lambda (H7: (eq T u4 u1)).(eq_ind T u1 
-(\lambda (t: T).((eq T t2 t0) \to ((eq T (THead (Bind b) u5 t6) t5) \to ((pr0 
-t u5) \to ((pr0 t2 t6) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead 
-(Flat Appl) u3 t5) t0)))))))) (\lambda (H17: (eq T t2 t0)).(eq_ind T t0 
-(\lambda (t: T).((eq T (THead (Bind b) u5 t6) t5) \to ((pr0 u1 u5) \to ((pr0 
-t t6) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u3 t5) 
-t0))))))) (\lambda (H8: (eq T (THead (Bind b) u5 t6) t5)).(eq_ind T (THead 
-(Bind b) u5 t6) (\lambda (t: T).((pr0 u1 u5) \to ((pr0 t0 t6) \to (ex2 T 
-(\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
-t3)) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u3 t) t0)))))) (\lambda 
-(H18: (pr0 u1 u5)).(\lambda (H19: (pr0 t0 t6)).(let H15 \def (eq_ind_r T t4 
-(\lambda (t0: T).(eq T (THead (Flat Appl) u0 t0) t)) H4 (THead (Bind b) u1 
-t0) H15) in (let H20 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt 
-v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to 
-(ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H 
-(THead (Flat Appl) u0 (THead (Bind b) u1 t0)) H15) in (let H4 \def (eq_ind T 
-v1 (\lambda (t: T).(pr0 t v2)) H10 u0 H14) in (ex2_ind T (\lambda (t: T).(pr0 
-v2 t)) (\lambda (t: T).(pr0 u3 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind 
-b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 
-(THead (Flat Appl) u3 (THead (Bind b) u5 t6)) t))) (\lambda (x: T).(\lambda 
-(H10: (pr0 v2 x)).(\lambda (H14: (pr0 u3 x)).(ex2_ind T (\lambda (t: T).(pr0 
-t6 t)) (\lambda (t: T).(pr0 t3 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind 
-b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 
-(THead (Flat Appl) u3 (THead (Bind b) u5 t6)) t))) (\lambda (x0: T).(\lambda 
-(H12: (pr0 t6 x0)).(\lambda (H16: (pr0 t3 x0)).(ex2_ind T (\lambda (t: 
-T).(pr0 u5 t)) (\lambda (t: T).(pr0 u2 t)) (ex2 T (\lambda (t: T).(pr0 (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: 
-T).(pr0 (THead (Flat Appl) u3 (THead (Bind b) u5 t6)) t))) (\lambda (x1: 
-T).(\lambda (H11: (pr0 u5 x1)).(\lambda (H21: (pr0 u2 x1)).(ex2_sym T (pr0 
-(THead (Flat Appl) u3 (THead (Bind b) u5 t6))) (pr0 (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t3))) (pr0_confluence__pr0_cong_upsilon_cong b 
-H9 u3 v2 x H14 H10 t6 t3 x0 H12 H16 u5 u2 x1 H11 H21))))) (H20 u1 (tlt_trans 
-(THead (Bind b) u1 t0) u1 (THead (Flat Appl) u0 (THead (Bind b) u1 t0)) 
-(tlt_head_sx (Bind b) u1 t0) (tlt_head_dx (Flat Appl) u0 (THead (Bind b) u1 
-t0))) u5 H18 u2 H11))))) (H20 t0 (tlt_trans (THead (Bind b) u1 t0) t0 (THead 
-(Flat Appl) u0 (THead (Bind b) u1 t0)) (tlt_head_dx (Bind b) u1 t0) 
-(tlt_head_dx (Flat Appl) u0 (THead (Bind b) u1 t0))) t6 H19 t3 H12))))) (H20 
-u0 (tlt_head_sx (Flat Appl) u0 (THead (Bind b) u1 t0)) v2 H4 u3 H16))))))) t5 
-H8)) t2 (sym_eq T t2 t0 H17))) u4 (sym_eq T u4 u1 H7))) k (sym_eq K k (Bind 
-b) H5))) H3)) H2)) H13 H0 H1))) | (pr0_beta u v0 v3 H0 t2 t6 H1) \Rightarrow 
-(\lambda (H6: (eq T (THead (Flat Appl) v0 (THead (Bind Abst) u t2)) (THead 
-(Bind b) u1 t0))).(\lambda (H13: (eq T (THead (Bind Abbr) v3 t6) t5)).((let 
-H2 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u t2)) (\lambda 
-(e: T).(match e return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k 
-return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) 
-\Rightarrow True])])) I (THead (Bind b) u1 t0) H6) in (False_ind ((eq T 
-(THead (Bind Abbr) v3 t6) t5) \to ((pr0 v0 v3) \to ((pr0 t2 t6) \to (ex2 T 
-(\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
-t3)) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u3 t5) t)))))) H2)) H13 H0 
-H1))) | (pr0_upsilon b0 H0 v0 v3 H1 u4 u5 H6 t2 t6 H13) \Rightarrow (\lambda 
-(H14: (eq T (THead (Flat Appl) v0 (THead (Bind b0) u4 t2)) (THead (Bind b) u1 
-t0))).(\lambda (H15: (eq T (THead (Bind b0) u5 (THead (Flat Appl) (lift (S O) 
-O v3) t6)) t5)).((let H2 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind 
-b0) u4 t2)) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
-\Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow 
-False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u1 t0) H14) in 
-(False_ind ((eq T (THead (Bind b0) u5 (THead (Flat Appl) (lift (S O) O v3) 
-t6)) t5) \to ((not (eq B b0 Abst)) \to ((pr0 v0 v3) \to ((pr0 u4 u5) \to 
-((pr0 t2 t6) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u3 
-t5) t)))))))) H2)) H15 H0 H1 H6 H13))) | (pr0_delta u4 u5 H0 t2 t6 H1 w H6) 
-\Rightarrow (\lambda (H13: (eq T (THead (Bind Abbr) u4 t2) (THead (Bind b) u1 
-t0))).(\lambda (H17: (eq T (THead (Bind Abbr) u5 w) t5)).((let H2 \def 
-(f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort 
-_) \Rightarrow t2 | (TLRef _) \Rightarrow t2 | (THead _ _ t) \Rightarrow t])) 
-(THead (Bind Abbr) u4 t2) (THead (Bind b) u1 t0) H13) in ((let H3 \def 
-(f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort 
-_) \Rightarrow u4 | (TLRef _) \Rightarrow u4 | (THead _ t _) \Rightarrow t])) 
-(THead (Bind Abbr) u4 t2) (THead (Bind b) u1 t0) H13) in ((let H5 \def 
-(f_equal T B (\lambda (e: T).(match e return (\lambda (_: T).B) with [(TSort 
-_) \Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k _ _) \Rightarrow 
-(match k return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) 
-\Rightarrow Abbr])])) (THead (Bind Abbr) u4 t2) (THead (Bind b) u1 t0) H13) 
-in (eq_ind B Abbr (\lambda (b: B).((eq T u4 u1) \to ((eq T t2 t0) \to ((eq T 
-(THead (Bind Abbr) u5 w) t5) \to ((pr0 u4 u5) \to ((pr0 t2 t6) \to ((subst0 O 
-u5 t6 w) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u3 
-t5) t)))))))))) (\lambda (H7: (eq T u4 u1)).(eq_ind T u1 (\lambda (t: T).((eq 
-T t2 t0) \to ((eq T (THead (Bind Abbr) u5 w) t5) \to ((pr0 t u5) \to ((pr0 t2 
-t6) \to ((subst0 O u5 t6 w) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind 
-Abbr) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 
-(THead (Flat Appl) u3 t5) t0))))))))) (\lambda (H18: (eq T t2 t0)).(eq_ind T 
-t0 (\lambda (t: T).((eq T (THead (Bind Abbr) u5 w) t5) \to ((pr0 u1 u5) \to 
-((pr0 t t6) \to ((subst0 O u5 t6 w) \to (ex2 T (\lambda (t0: T).(pr0 (THead 
-(Bind Abbr) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: 
-T).(pr0 (THead (Flat Appl) u3 t5) t0)))))))) (\lambda (H19: (eq T (THead 
-(Bind Abbr) u5 w) t5)).(eq_ind T (THead (Bind Abbr) u5 w) (\lambda (t: 
-T).((pr0 u1 u5) \to ((pr0 t0 t6) \to ((subst0 O u5 t6 w) \to (ex2 T (\lambda 
-(t0: T).(pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) 
-t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u3 t) t0))))))) (\lambda (H20: 
-(pr0 u1 u5)).(\lambda (H21: (pr0 t0 t6)).(\lambda (H22: (subst0 O u5 t6 
-w)).(let H15 \def (eq_ind_r B b (\lambda (b: B).(eq T (THead (Bind b) u1 t0) 
-t4)) H15 Abbr H5) in (let H9 \def (eq_ind_r B b (\lambda (b: B).(not (eq B b 
-Abst))) H9 Abbr H5) in (let H23 \def (eq_ind_r B b (\lambda (b: B).(eq T 
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t1)) H8 Abbr H5) 
-in (let H4 \def (eq_ind_r T t4 (\lambda (t0: T).(eq T (THead (Flat Appl) u0 
-t0) t)) H4 (THead (Bind Abbr) u1 t0) H15) in (let H8 \def (eq_ind_r T t 
-(\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) 
-\to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) 
-(\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Appl) u0 (THead (Bind 
-Abbr) u1 t0)) H4) in (let H10 \def (eq_ind T v1 (\lambda (t: T).(pr0 t v2)) 
-H10 u0 H14) in (ex2_ind T (\lambda (t: T).(pr0 v2 t)) (\lambda (t: T).(pr0 u3 
-t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u3 (THead 
-(Bind Abbr) u5 w)) t))) (\lambda (x: T).(\lambda (H14: (pr0 v2 x)).(\lambda 
-(H16: (pr0 u3 x)).(ex2_ind T (\lambda (t: T).(pr0 t6 t)) (\lambda (t: T).(pr0 
-t3 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u3 (THead 
-(Bind Abbr) u5 w)) t))) (\lambda (x0: T).(\lambda (H12: (pr0 t6 x0)).(\lambda 
-(H24: (pr0 t3 x0)).(ex2_ind T (\lambda (t: T).(pr0 u5 t)) (\lambda (t: 
-T).(pr0 u2 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u3 
-(THead (Bind Abbr) u5 w)) t))) (\lambda (x1: T).(\lambda (H11: (pr0 u5 
-x1)).(\lambda (H25: (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) t3))) (pr0_confluence__pr0_cong_upsilon_delta H9 u5 t6 w H22 u3 v2 x 
-H16 H14 t3 x0 H12 H24 u2 x1 H11 H25))))) (H8 u1 (tlt_trans (THead (Bind Abbr) 
-u1 t0) u1 (THead (Flat Appl) u0 (THead (Bind Abbr) u1 t0)) (tlt_head_sx (Bind 
-Abbr) u1 t0) (tlt_head_dx (Flat Appl) u0 (THead (Bind Abbr) u1 t0))) u5 H20 
-u2 H11))))) (H8 t0 (tlt_trans (THead (Bind Abbr) u1 t0) t0 (THead (Flat Appl) 
-u0 (THead (Bind Abbr) u1 t0)) (tlt_head_dx (Bind Abbr) u1 t0) (tlt_head_dx 
-(Flat Appl) u0 (THead (Bind Abbr) u1 t0))) t6 H21 t3 H12))))) (H8 u0 
-(tlt_head_sx (Flat Appl) u0 (THead (Bind Abbr) u1 t0)) v2 H10 u3 
-H16))))))))))) t5 H19)) t2 (sym_eq T t2 t0 H18))) u4 (sym_eq T u4 u1 H7))) b 
-H5)) H3)) H2)) H17 H0 H1 H6))) | (pr0_zeta b0 H0 t2 t6 H1 u) \Rightarrow 
-(\lambda (H6: (eq T (THead (Bind b0) u (lift (S O) O t2)) (THead (Bind b) u1 
-t0))).(\lambda (H13: (eq T t6 t5)).((let H2 \def (f_equal T T (\lambda (e: 
-T).(match e 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 u t0) 
-\Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) 
-(\lambda (x: nat).(plus x (S O))) O t2) | (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 u t0) 
-\Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) 
-(\lambda (x: nat).(plus x (S O))) O t2) | (THead _ _ t) \Rightarrow t])) 
-(THead (Bind b0) u (lift (S O) O t2)) (THead (Bind b) u1 t0) H6) in ((let H3 
-\def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) 
-\Rightarrow t])) (THead (Bind b0) u (lift (S O) O t2)) (THead (Bind b) u1 t0) 
-H6) in ((let H5 \def (f_equal T B (\lambda (e: T).(match e return (\lambda 
-(_: T).B) with [(TSort _) \Rightarrow b0 | (TLRef _) \Rightarrow b0 | (THead 
-k _ _) \Rightarrow (match k return (\lambda (_: K).B) with [(Bind b) 
-\Rightarrow b | (Flat _) \Rightarrow b0])])) (THead (Bind b0) u (lift (S O) O 
-t2)) (THead (Bind b) u1 t0) H6) in (eq_ind B b (\lambda (b1: B).((eq T u u1) 
-\to ((eq T (lift (S O) O t2) t0) \to ((eq T t6 t5) \to ((not (eq B b1 Abst)) 
-\to ((pr0 t2 t6) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Flat 
-Appl) u3 t5) t))))))))) (\lambda (H7: (eq T u u1)).(eq_ind T u1 (\lambda (_: 
-T).((eq T (lift (S O) O t2) t0) \to ((eq T t6 t5) \to ((not (eq B b Abst)) 
-\to ((pr0 t2 t6) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Flat 
-Appl) u3 t5) t0)))))))) (\lambda (H17: (eq T (lift (S O) O t2) t0)).(eq_ind T 
-(lift (S O) O t2) (\lambda (_: T).((eq T t6 t5) \to ((not (eq B b Abst)) \to 
-((pr0 t2 t6) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u3 
-t5) t0))))))) (\lambda (H8: (eq T t6 t5)).(eq_ind T t5 (\lambda (t: T).((not 
-(eq B b Abst)) \to ((pr0 t2 t) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind 
-b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 
-(THead (Flat Appl) u3 t5) t0)))))) (\lambda (H18: (not (eq B b 
-Abst))).(\lambda (H19: (pr0 t2 t5)).(let H9 \def (eq_ind_r T t0 (\lambda (t: 
-T).(eq T (THead (Bind b) u1 t) t4)) H15 (lift (S O) O t2) H17) in (let H15 
-\def (eq_ind_r T t4 (\lambda (t0: T).(eq T (THead (Flat Appl) u0 t0) t)) H4 
-(THead (Bind b) u1 (lift (S O) O t2)) H9) in (let H20 \def (eq_ind_r T t 
-(\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) 
-\to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) 
-(\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Appl) u0 (THead (Bind b) 
-u1 (lift (S O) O t2))) H15) in (let H12 \def (eq_ind_r T t0 (\lambda (t: 
-T).(pr0 t t3)) H12 (lift (S O) O t2) H17) in (ex2_ind T (\lambda (t4: T).(eq 
-T t3 (lift (S O) O t4))) (\lambda (t3: T).(pr0 t2 t3)) (ex2 T (\lambda (t: 
-T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) 
-(\lambda (t: T).(pr0 (THead (Flat Appl) u3 t5) t))) (\lambda (x: T).(\lambda 
-(H21: (eq T t3 (lift (S O) O x))).(\lambda (H22: (pr0 t2 x)).(eq_ind_r T 
-(lift (S O) O x) (\lambda (t: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) 
-u2 (THead (Flat Appl) (lift (S O) O v2) t)) t0)) (\lambda (t0: T).(pr0 (THead 
-(Flat Appl) u3 t5) t0)))) (let H4 \def (eq_ind T v1 (\lambda (t: T).(pr0 t 
-v2)) H10 u0 H14) in (ex2_ind T (\lambda (t: T).(pr0 v2 t)) (\lambda (t: 
-T).(pr0 u3 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) (lift (S O) O x))) t)) (\lambda (t: T).(pr0 (THead 
-(Flat Appl) u3 t5) t))) (\lambda (x0: T).(\lambda (H10: (pr0 v2 x0)).(\lambda 
-(H14: (pr0 u3 x0)).(ex2_ind T (\lambda (t: T).(pr0 x t)) (\lambda (t: T).(pr0 
-t5 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) (lift (S O) O x))) t)) (\lambda (t: T).(pr0 (THead (Flat 
-Appl) u3 t5) t))) (\lambda (x1: T).(\lambda (H16: (pr0 x x1)).(\lambda (H23: 
-(pr0 t5 x1)).(ex2_sym T (pr0 (THead (Flat Appl) u3 t5)) (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 H18 u1 u2 H11 u3 v2 x0 H14 H10 x t5 
-x1 H16 H23))))) (H20 t2 (tlt_trans (THead (Bind b) u1 (lift (S O) O t2)) t2 
-(THead (Flat Appl) u0 (THead (Bind b) u1 (lift (S O) O t2))) (lift_tlt_dx 
-(Bind b) u1 t2 (S O) O) (tlt_head_dx (Flat Appl) u0 (THead (Bind b) u1 (lift 
-(S O) O t2)))) x H22 t5 H19))))) (H20 u0 (tlt_head_sx (Flat Appl) u0 (THead 
-(Bind b) u1 (lift (S O) O t2))) v2 H4 u3 H16))) t3 H21)))) (pr0_gen_lift t2 
-t3 (S O) O H12)))))))) t6 (sym_eq T t6 t5 H8))) t0 H17)) u (sym_eq T u u1 
-H7))) b0 (sym_eq B b0 b H5))) H3)) H2)) H13 H0 H1))) | (pr0_epsilon t2 t6 H0 
-u) \Rightarrow (\lambda (H1: (eq T (THead (Flat Cast) u t2) (THead (Bind b) 
-u1 t0))).(\lambda (H6: (eq T t6 t5)).((let H2 \def (eq_ind T (THead (Flat 
-Cast) u t2) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
-\Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow 
-False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u1 t0) H1) in 
-(False_ind ((eq T t6 t5) \to ((pr0 t2 t6) \to (ex2 T (\lambda (t: T).(pr0 
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: 
-T).(pr0 (THead (Flat Appl) u3 t5) t))))) H2)) H6 H0)))]) in (H6 (refl_equal T 
-(THead (Bind b) u1 t0)) (refl_equal T t5))))) t4 H15)) v1 (sym_eq T v1 u0 
-H14))) k H3)) H2)) H1)))]) in (H1 (refl_equal T (THead k u0 t4))))))) t2 
-H15)) t H13 H14 H8 H9))) | (pr0_beta u v0 v3 H8 t4 t5 H9) \Rightarrow 
-(\lambda (H10: (eq T (THead (Flat Appl) v0 (THead (Bind Abst) u t4)) 
-t)).(\lambda (H13: (eq T (THead (Bind Abbr) v3 t5) t2)).(eq_ind T (THead 
-(Flat Appl) v0 (THead (Bind Abst) u t4)) (\lambda (_: T).((eq T (THead (Bind 
-Abbr) v3 t5) t2) \to ((pr0 v0 v3) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: 
-T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) 
-(\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H14: (eq T (THead (Bind Abbr) v3 
-t5) t2)).(eq_ind T (THead (Bind Abbr) v3 t5) (\lambda (t: T).((pr0 v0 v3) \to 
-((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 t t0)))))) (\lambda 
-(_: (pr0 v0 v3)).(\lambda (_: (pr0 t4 t5)).(let H0 \def (eq_ind_r T t 
-(\lambda (t: T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) t)) H6 
-(THead (Flat Appl) v0 (THead (Bind Abst) u t4)) H10) in (let H1 \def (match 
-H0 return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t (THead (Flat 
-Appl) v0 (THead (Bind Abst) u t4))) \to (ex2 T (\lambda (t0: T).(pr0 (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: 
-T).(pr0 (THead (Bind Abbr) v3 t5) t0)))))) with [refl_equal \Rightarrow 
-(\lambda (H0: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (THead 
-(Flat Appl) v0 (THead (Bind Abst) u t4)))).(let H1 \def (f_equal T T (\lambda 
-(e: T).(match e return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | 
-(TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow (match t return (\lambda 
-(_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead 
-_ _ t0) \Rightarrow t0])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) 
-(THead (Flat Appl) v0 (THead (Bind Abst) u t4)) H0) in ((let H2 \def (f_equal 
-T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ _ t) \Rightarrow (match 
-t return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) 
-\Rightarrow u1 | (THead _ t0 _) \Rightarrow t0])])) (THead (Flat Appl) v1 
-(THead (Bind b) u1 t0)) (THead (Flat Appl) v0 (THead (Bind Abst) u t4)) H0) 
-in ((let H3 \def (f_equal T B (\lambda (e: T).(match e return (\lambda (_: 
-T).B) with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead _ _ t) 
-\Rightarrow (match t return (\lambda (_: T).B) with [(TSort _) \Rightarrow b 
-| (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k return 
-(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow 
-b])])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (THead (Flat Appl) v0 
-(THead (Bind Abst) u t4)) H0) in ((let H4 \def (f_equal T T (\lambda (e: 
-T).(match e return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef 
-_) \Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) v1 
-(THead (Bind b) u1 t0)) (THead (Flat Appl) v0 (THead (Bind Abst) u t4)) H0) 
-in (eq_ind T v0 (\lambda (_: T).((eq B b Abst) \to ((eq T u1 u) \to ((eq T t0 
-t4) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v3 t5) 
-t0))))))) (\lambda (H13: (eq B b Abst)).(eq_ind B Abst (\lambda (b: B).((eq T 
-u1 u) \to ((eq T t0 t4) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead 
-(Bind Abbr) v3 t5) t)))))) (\lambda (H14: (eq T u1 u)).(eq_ind T u (\lambda 
-(_: T).((eq T t0 t4) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abst) u2 
-(THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead 
-(Bind Abbr) v3 t5) t0))))) (\lambda (H15: (eq T t0 t4)).(eq_ind T t4 (\lambda 
-(_: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abst) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v3 t5) 
-t0)))) (let H5 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) 
-\to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T 
-(\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead 
-(Flat Appl) v0 (THead (Bind Abst) u t4)) H10) in (let H6 \def (eq_ind T t0 
-(\lambda (t: T).(pr0 t t3)) H12 t4 H15) in (let H7 \def (eq_ind T u1 (\lambda 
-(t: T).(pr0 t u2)) H11 u H14) in (let H8 \def (eq_ind B b (\lambda (b: 
-B).(not (eq B b Abst))) H9 Abst H13) in (let H9 \def (match (H8 (refl_equal B 
-Abst)) return (\lambda (_: False).(ex2 T (\lambda (t: T).(pr0 (THead (Bind 
-Abst) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 
-(THead (Bind Abbr) v3 t5) t)))) with []) in H9))))) t0 (sym_eq T t0 t4 H15))) 
-u1 (sym_eq T u1 u H14))) b (sym_eq B b Abst H13))) v1 (sym_eq T v1 v0 H4))) 
-H3)) H2)) H1)))]) in (H1 (refl_equal T (THead (Flat Appl) v0 (THead (Bind 
-Abst) u t4)))))))) t2 H14)) t H10 H13 H8 H9))) | (pr0_upsilon b0 H8 v0 v3 H9 
-u0 u3 H10 t4 t5 H11) \Rightarrow (\lambda (H13: (eq T (THead (Flat Appl) v0 
-(THead (Bind b0) u0 t4)) t)).(\lambda (H14: (eq T (THead (Bind b0) u3 (THead 
-(Flat Appl) (lift (S O) O v3) t5)) t2)).(eq_ind T (THead (Flat Appl) v0 
-(THead (Bind b0) u0 t4)) (\lambda (_: T).((eq T (THead (Bind b0) u3 (THead 
-(Flat Appl) (lift (S O) O v3) t5)) t2) \to ((not (eq B b0 Abst)) \to ((pr0 v0 
-v3) \to ((pr0 u0 u3) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: 
-T).(pr0 t2 t0))))))))) (\lambda (H15: (eq T (THead (Bind b0) u3 (THead (Flat 
-Appl) (lift (S O) O v3) t5)) t2)).(eq_ind T (THead (Bind b0) u3 (THead (Flat 
-Appl) (lift (S O) O v3) t5)) (\lambda (t: T).((not (eq B b0 Abst)) \to ((pr0 
-v0 v3) \to ((pr0 u0 u3) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda 
-(t0: T).(pr0 t t0)))))))) (\lambda (_: (not (eq B b0 Abst))).(\lambda (H17: 
-(pr0 v0 v3)).(\lambda (H18: (pr0 u0 u3)).(\lambda (H19: (pr0 t4 t5)).(let H0 
-\def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat Appl) v1 (THead (Bind 
-b) u1 t0)) t)) H6 (THead (Flat Appl) v0 (THead (Bind b0) u0 t4)) H13) in (let 
-H1 \def (match H0 return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t 
-(THead (Flat Appl) v0 (THead (Bind b0) u0 t4))) \to (ex2 T (\lambda (t0: 
-T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) 
-(\lambda (t0: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O 
-v3) t5)) t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead 
-(Flat Appl) v1 (THead (Bind b) u1 t0)) (THead (Flat Appl) v0 (THead (Bind b0) 
-u0 t4)))).(let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda 
-(_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead 
-_ _ t) \Rightarrow (match t return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t0) \Rightarrow 
-t0])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (THead (Flat Appl) v0 
-(THead (Bind b0) u0 t4)) H0) in ((let H2 \def (f_equal T T (\lambda (e: 
-T).(match e return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef 
-_) \Rightarrow u1 | (THead _ _ t) \Rightarrow (match t return (\lambda (_: 
-T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t0 
-_) \Rightarrow t0])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (THead 
-(Flat Appl) v0 (THead (Bind b0) u0 t4)) H0) in ((let H3 \def (f_equal T B 
-(\lambda (e: T).(match e return (\lambda (_: T).B) with [(TSort _) 
-\Rightarrow b | (TLRef _) \Rightarrow b | (THead _ _ t) \Rightarrow (match t 
-return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | (TLRef _) 
-\Rightarrow b | (THead k _ _) \Rightarrow (match k return (\lambda (_: K).B) 
-with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b])])])) (THead (Flat 
-Appl) v1 (THead (Bind b) u1 t0)) (THead (Flat Appl) v0 (THead (Bind b0) u0 
-t4)) H0) in ((let H4 \def (f_equal T T (\lambda (e: T).(match e return 
-(\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 
-| (THead _ t _) \Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind b) u1 
-t0)) (THead (Flat Appl) v0 (THead (Bind b0) u0 t4)) H0) in (eq_ind T v0 
-(\lambda (_: T).((eq B b b0) \to ((eq T u1 u0) \to ((eq T t0 t4) \to (ex2 T 
-(\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
-t3)) t0)) (\lambda (t0: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift 
-(S O) O v3) t5)) t0))))))) (\lambda (H14: (eq B b b0)).(eq_ind B b0 (\lambda 
-(b: B).((eq T u1 u0) \to ((eq T t0 t4) \to (ex2 T (\lambda (t: T).(pr0 (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: 
-T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t5)) t)))))) 
-(\lambda (H15: (eq T u1 u0)).(eq_ind T u0 (\lambda (_: T).((eq T t0 t4) \to 
-(ex2 T (\lambda (t0: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S 
-O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Bind b0) u3 (THead (Flat 
-Appl) (lift (S O) O v3) t5)) t0))))) (\lambda (H16: (eq T t0 t4)).(eq_ind T 
-t4 (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind b0) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Bind 
-b0) u3 (THead (Flat Appl) (lift (S O) O v3) t5)) t0)))) (let H5 \def 
-(eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: 
-T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: 
-T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Appl) v0 
-(THead (Bind b0) u0 t4)) H13) in (let H6 \def (eq_ind T t0 (\lambda (t: 
-T).(pr0 t t3)) H12 t4 H16) in (let H7 \def (eq_ind T u1 (\lambda (t: T).(pr0 
-t u2)) H11 u0 H15) in (let H8 \def (eq_ind B b (\lambda (b: B).(not (eq B b 
-Abst))) H9 b0 H14) in (let H9 \def (eq_ind T v1 (\lambda (t: T).(pr0 t v2)) 
-H10 v0 H4) in (ex2_ind T (\lambda (t: T).(pr0 v2 t)) (\lambda (t: T).(pr0 v3 
-t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift 
-(S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Bind b0) u3 (THead (Flat 
-Appl) (lift (S O) O v3) t5)) t))) (\lambda (x: T).(\lambda (H10: (pr0 v2 
-x)).(\lambda (H11: (pr0 v3 x)).(ex2_ind T (\lambda (t: T).(pr0 u2 t)) 
-(\lambda (t: T).(pr0 u3 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind b0) u2 
-(THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead 
-(Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t5)) t))) (\lambda (x0: 
-T).(\lambda (H12: (pr0 u2 x0)).(\lambda (H13: (pr0 u3 x0)).(ex2_ind T 
-(\lambda (t: T).(pr0 t3 t)) (\lambda (t: T).(pr0 t5 t)) (ex2 T (\lambda (t: 
-T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) 
-(\lambda (t: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) 
-t5)) t))) (\lambda (x1: T).(\lambda (H17: (pr0 t3 x1)).(\lambda (H18: (pr0 t5 
-x1)).(pr0_confluence__pr0_upsilon_upsilon b0 H8 v2 v3 x H10 H11 u2 u3 x0 H12 
-H13 t3 t5 x1 H17 H18)))) (H5 t4 (tlt_trans (THead (Bind b0) u0 t4) t4 (THead 
-(Flat Appl) v0 (THead (Bind b0) u0 t4)) (tlt_head_dx (Bind b0) u0 t4) 
-(tlt_head_dx (Flat Appl) v0 (THead (Bind b0) u0 t4))) t3 H6 t5 H19))))) (H5 
-u0 (tlt_trans (THead (Bind b0) u0 t4) u0 (THead (Flat Appl) v0 (THead (Bind 
-b0) u0 t4)) (tlt_head_sx (Bind b0) u0 t4) (tlt_head_dx (Flat Appl) v0 (THead 
-(Bind b0) u0 t4))) u2 H7 u3 H18))))) (H5 v0 (tlt_head_sx (Flat Appl) v0 
-(THead (Bind b0) u0 t4)) v2 H9 v3 H17))))))) t0 (sym_eq T t0 t4 H16))) u1 
-(sym_eq T u1 u0 H15))) b (sym_eq B b b0 H14))) v1 (sym_eq T v1 v0 H4))) H3)) 
-H2)) H1)))]) in (H1 (refl_equal T (THead (Flat Appl) v0 (THead (Bind b0) u0 
-t4)))))))))) t2 H15)) t H13 H14 H8 H9 H10 H11))) | (pr0_delta u0 u3 H8 t4 t5 
-H9 w H10) \Rightarrow (\lambda (H11: (eq T (THead (Bind Abbr) u0 t4) 
-t)).(\lambda (H12: (eq T (THead (Bind Abbr) u3 w) t2)).(eq_ind T (THead (Bind 
-Abbr) u0 t4) (\lambda (_: T).((eq T (THead (Bind Abbr) u3 w) t2) \to ((pr0 u0 
-u3) \to ((pr0 t4 t5) \to ((subst0 O u3 t5 w) \to (ex2 T (\lambda (t0: T).(pr0 
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda 
-(t0: T).(pr0 t2 t0)))))))) (\lambda (H13: (eq T (THead (Bind Abbr) u3 w) 
-t2)).(eq_ind T (THead (Bind Abbr) u3 w) (\lambda (t: T).((pr0 u0 u3) \to 
-((pr0 t4 t5) \to ((subst0 O u3 t5 w) \to (ex2 T (\lambda (t0: T).(pr0 (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: 
-T).(pr0 t t0))))))) (\lambda (_: (pr0 u0 u3)).(\lambda (_: (pr0 t4 
-t5)).(\lambda (_: (subst0 O u3 t5 w)).(let H0 \def (eq_ind_r T t (\lambda (t: 
-T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) t)) H6 (THead (Bind 
-Abbr) u0 t4) H11) in (let H1 \def (match H0 return (\lambda (t: T).(\lambda 
-(_: (eq ? ? t)).((eq T t (THead (Bind Abbr) u0 t4)) \to (ex2 T (\lambda (t0: 
-T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) 
-(\lambda (t0: T).(pr0 (THead (Bind Abbr) u3 w) t0)))))) with [refl_equal 
-\Rightarrow (\lambda (H0: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) 
-(THead (Bind Abbr) u0 t4))).(let H1 \def (eq_ind T (THead (Flat Appl) v1 
-(THead (Bind b) u1 t0)) (\lambda (e: T).(match e return (\lambda (_: T).Prop) 
-with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ 
-_) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind _) 
-\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u0 
-t4) H0) in (False_ind (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Bind 
-Abbr) u3 w) t))) H1)))]) in (H1 (refl_equal T (THead (Bind Abbr) u0 
-t4)))))))) t2 H13)) t H11 H12 H8 H9 H10))) | (pr0_zeta b0 H8 t4 t5 H9 u) 
-\Rightarrow (\lambda (H10: (eq T (THead (Bind b0) u (lift (S O) O t4)) 
-t)).(\lambda (H11: (eq T t5 t2)).(eq_ind T (THead (Bind b0) u (lift (S O) O 
-t4)) (\lambda (_: T).((eq T t5 t2) \to ((not (eq B b0 Abst)) \to ((pr0 t4 t5) 
-\to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift 
-(S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H12: (eq T 
-t5 t2)).(eq_ind T t2 (\lambda (t: T).((not (eq B b0 Abst)) \to ((pr0 t4 t) 
-\to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift 
-(S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 t2 t0)))))) (\lambda (_: (not (eq 
-B b0 Abst))).(\lambda (_: (pr0 t4 t2)).(let H0 \def (eq_ind_r T t (\lambda 
-(t: T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) t)) H6 (THead 
-(Bind b0) u (lift (S O) O t4)) H10) in (let H1 \def (match H0 return (\lambda 
-(t: T).(\lambda (_: (eq ? ? t)).((eq T t (THead (Bind b0) u (lift (S O) O 
-t4))) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 t2 t0)))))) with 
-[refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat Appl) v1 (THead 
-(Bind b) u1 t0)) (THead (Bind b0) u (lift (S O) O t4)))).(let H1 \def (eq_ind 
-T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (\lambda (e: T).(match e 
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: 
-K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I 
-(THead (Bind b0) u (lift (S O) O t4)) H0) in (False_ind (ex2 T (\lambda (t: 
-T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) 
-(\lambda (t: T).(pr0 t2 t))) H1)))]) in (H1 (refl_equal T (THead (Bind b0) u 
-(lift (S O) O t4)))))))) t5 (sym_eq T t5 t2 H12))) t H10 H11 H8 H9))) | 
-(pr0_epsilon t4 t5 H8 u) \Rightarrow (\lambda (H9: (eq T (THead (Flat Cast) u 
-t4) t)).(\lambda (H10: (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 (t0: 
-T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) 
-(\lambda (t0: T).(pr0 t2 t0)))))) (\lambda (H11: (eq T t5 t2)).(eq_ind T t2 
-(\lambda (t: T).((pr0 t4 t) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) 
-u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 t2 
-t0))))) (\lambda (_: (pr0 t4 t2)).(let H0 \def (eq_ind_r T t (\lambda (t: 
-T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) t)) H6 (THead (Flat 
-Cast) u t4) H9) in (let H1 \def (match H0 return (\lambda (t: T).(\lambda (_: 
-(eq ? ? t)).((eq T t (THead (Flat Cast) u t4)) \to (ex2 T (\lambda (t0: 
-T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) 
-(\lambda (t0: T).(pr0 t2 t0)))))) with [refl_equal \Rightarrow (\lambda (H0: 
-(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (THead (Flat Cast) u 
-t4))).(let H1 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) 
-(\lambda (e: T).(match e return (\lambda (_: T).Prop) with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
-(match k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | 
-(Flat f) \Rightarrow (match f return (\lambda (_: F).Prop) with [Appl 
-\Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat Cast) u t4) 
-H0) in (False_ind (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 t2 t))) H1)))]) in (H1 
-(refl_equal T (THead (Flat Cast) u t4)))))) t5 (sym_eq T t5 t2 H11))) t H9 
-H10 H8)))]) in (H13 (refl_equal T t) (refl_equal T t2))))))) t1 H8)) t H6 H7 
-H2 H3 H4 H5))) | (pr0_delta u1 u2 H2 t0 t3 H3 w H4) \Rightarrow (\lambda (H5: 
-(eq T (THead (Bind Abbr) u1 t0) t)).(\lambda (H6: (eq T (THead (Bind Abbr) u2 
-w) t1)).(eq_ind T (THead (Bind Abbr) u1 t0) (\lambda (_: T).((eq T (THead 
-(Bind Abbr) u2 w) t1) \to ((pr0 u1 u2) \to ((pr0 t0 t3) \to ((subst0 O u2 t3 
-w) \to (ex2 T (\lambda (t2: T).(pr0 t1 t2)) (\lambda (t1: T).(pr0 t2 
-t1)))))))) (\lambda (H7: (eq T (THead (Bind Abbr) u2 w) t1)).(eq_ind T (THead 
-(Bind Abbr) u2 w) (\lambda (t: T).((pr0 u1 u2) \to ((pr0 t0 t3) \to ((subst0 
-O u2 t3 w) \to (ex2 T (\lambda (t1: T).(pr0 t t1)) (\lambda (t1: T).(pr0 t2 
-t1))))))) (\lambda (H8: (pr0 u1 u2)).(\lambda (H9: (pr0 t0 t3)).(\lambda 
-(H10: (subst0 O u2 t3 w)).(let H11 \def (match H1 return (\lambda (t0: 
-T).(\lambda (t1: T).(\lambda (_: (pr0 t0 t1)).((eq T t0 t) \to ((eq T t1 t2) 
-\to (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: 
-T).(pr0 t2 t)))))))) with [(pr0_refl t4) \Rightarrow (\lambda (H7: (eq T t4 
-t)).(\lambda (H11: (eq T t4 t2)).(eq_ind T t (\lambda (t: T).((eq T t t2) \to 
-(ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: 
-T).(pr0 t2 t0))))) (\lambda (H12: (eq T t t2)).(eq_ind T t2 (\lambda (_: 
-T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: 
-T).(pr0 t2 t0)))) (let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T t t2)) H12 
-(THead (Bind Abbr) u1 t0) H5) in (eq_ind T (THead (Bind Abbr) u1 t0) (\lambda 
-(t: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda 
-(t0: T).(pr0 t t0)))) (let H1 \def (eq_ind_r T t (\lambda (t: T).(eq T t4 t)) 
-H7 (THead (Bind Abbr) u1 t0) H5) in (let H2 \def (eq_ind_r T t (\lambda (t: 
-T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall 
-(t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: 
-T).(pr0 t2 t0)))))))))) H (THead (Bind Abbr) u1 t0) H5) in (ex_intro2 T 
-(\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead 
-(Bind Abbr) u1 t0) t)) (THead (Bind Abbr) u2 w) (pr0_refl (THead (Bind Abbr) 
-u2 w)) (pr0_delta u1 u2 H8 t0 t3 H9 w H10)))) t2 H0)) t (sym_eq T t t2 H12))) 
-t4 (sym_eq T t4 t H7) H11))) | (pr0_comp u0 u3 H7 t4 t5 H8 k) \Rightarrow 
-(\lambda (H11: (eq T (THead k u0 t4) t)).(\lambda (H12: (eq T (THead k u3 t5) 
-t2)).(eq_ind T (THead k u0 t4) (\lambda (_: T).((eq T (THead k u3 t5) t2) \to 
-((pr0 u0 u3) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind 
-Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H13: (eq T 
-(THead k u3 t5) t2)).(eq_ind T (THead k u3 t5) (\lambda (t: T).((pr0 u0 u3) 
-\to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) 
-t0)) (\lambda (t0: T).(pr0 t t0)))))) (\lambda (H14: (pr0 u0 u3)).(\lambda 
-(H15: (pr0 t4 t5)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead 
-(Bind Abbr) u1 t0) t)) H5 (THead k u0 t4) H11) in (let H1 \def (match H0 
-return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t (THead k u0 t4)) \to 
-(ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: 
-T).(pr0 (THead k u3 t5) t0)))))) with [refl_equal \Rightarrow (\lambda (H0: 
-(eq T (THead (Bind Abbr) u1 t0) (THead k u0 t4))).(let H1 \def (f_equal T T 
-(\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) 
-(THead (Bind Abbr) u1 t0) (THead k u0 t4) H0) in ((let H2 \def (f_equal T T 
-(\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t _) \Rightarrow t])) 
-(THead (Bind Abbr) u1 t0) (THead k u0 t4) H0) in ((let H3 \def (f_equal T K 
-(\lambda (e: T).(match e return (\lambda (_: T).K) with [(TSort _) 
-\Rightarrow (Bind Abbr) | (TLRef _) \Rightarrow (Bind Abbr) | (THead k _ _) 
-\Rightarrow k])) (THead (Bind Abbr) u1 t0) (THead k u0 t4) H0) in (eq_ind K 
-(Bind Abbr) (\lambda (k: K).((eq T u1 u0) \to ((eq T t0 t4) \to (ex2 T 
-(\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead 
-k u3 t5) t)))))) (\lambda (H12: (eq T u1 u0)).(eq_ind T u0 (\lambda (_: 
-T).((eq T t0 t4) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) 
-t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) u3 t5) t0))))) (\lambda (H13: 
-(eq T t0 t4)).(eq_ind T t4 (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 
-(THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) u3 t5) 
-t0)))) (let H4 \def (eq_ind_r K k (\lambda (k: K).(eq T (THead k u0 t4) t)) 
-H11 (Bind Abbr) H3) in (let H5 \def (eq_ind_r T t (\lambda (t: T).(\forall 
-(v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 
-v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 
-t0)))))))))) H (THead (Bind Abbr) u0 t4) H4) in (let H6 \def (eq_ind T t0 
-(\lambda (t: T).(pr0 t t3)) H9 t4 H13) in (let H7 \def (eq_ind T u1 (\lambda 
-(t: T).(pr0 t u2)) H8 u0 H12) in (ex2_ind T (\lambda (t: T).(pr0 u2 t)) 
-(\lambda (t: T).(pr0 u3 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 
-w) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 t5) t))) (\lambda (x: 
-T).(\lambda (H8: (pr0 u2 x)).(\lambda (H9: (pr0 u3 x)).(ex2_ind T (\lambda 
-(t: T).(pr0 t3 t)) (\lambda (t: T).(pr0 t5 t)) (ex2 T (\lambda (t: T).(pr0 
-(THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 t5) 
-t))) (\lambda (x0: T).(\lambda (H11: (pr0 t3 x0)).(\lambda (H14: (pr0 t5 
-x0)).(ex2_sym T (pr0 (THead (Bind Abbr) u3 t5)) (pr0 (THead (Bind Abbr) u2 
-w)) (pr0_confluence__pr0_cong_delta u2 t3 w H10 u3 x H9 H8 t5 x0 H14 H11))))) 
-(H5 t4 (tlt_head_dx (Bind Abbr) u0 t4) t3 H6 t5 H15))))) (H5 u0 (tlt_head_sx 
-(Bind Abbr) u0 t4) u2 H7 u3 H14)))))) t0 (sym_eq T t0 t4 H13))) u1 (sym_eq T 
-u1 u0 H12))) k H3)) H2)) H1)))]) in (H1 (refl_equal T (THead k u0 t4))))))) 
-t2 H13)) t H11 H12 H7 H8))) | (pr0_beta u v1 v2 H7 t4 t5 H8) \Rightarrow 
-(\lambda (H9: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) 
-t)).(\lambda (H10: (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 (t0: 
-T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t2 t0))))))) 
-(\lambda (H11: (eq T (THead (Bind Abbr) v2 t5) t2)).(eq_ind T (THead (Bind 
-Abbr) v2 t5) (\lambda (t: T).((pr0 v1 v2) \to ((pr0 t4 t5) \to (ex2 T 
-(\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t 
-t0)))))) (\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 t4 t5)).(let H0 \def 
-(eq_ind_r T t (\lambda (t: T).(eq T (THead (Bind Abbr) u1 t0) t)) H5 (THead 
-(Flat Appl) v1 (THead (Bind Abst) u t4)) H9) in (let H1 \def (match H0 return 
-(\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t (THead (Flat Appl) v1 
-(THead (Bind Abst) u t4))) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind 
-Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t5) t0)))))) 
-with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Bind Abbr) u1 t0) 
-(THead (Flat Appl) v1 (THead (Bind Abst) u t4)))).(let H1 \def (eq_ind T 
-(THead (Bind Abbr) u1 t0) (\lambda (e: T).(match e return (\lambda (_: 
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
-(THead k _ _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind 
-_) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) v1 
-(THead (Bind Abst) u t4)) H0) in (False_ind (ex2 T (\lambda (t: T).(pr0 
-(THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t5) 
-t))) H1)))]) in (H1 (refl_equal T (THead (Flat Appl) v1 (THead (Bind Abst) u 
-t4)))))))) t2 H11)) t H9 H10 H7 H8))) | (pr0_upsilon b H7 v1 v2 H8 u0 u3 H9 
-t4 t5 H10) \Rightarrow (\lambda (H11: (eq T (THead (Flat Appl) v1 (THead 
-(Bind b) u0 t4)) t)).(\lambda (H12: (eq T (THead (Bind b) u3 (THead (Flat 
-Appl) (lift (S O) O v2) t5)) t2)).(eq_ind T (THead (Flat Appl) v1 (THead 
-(Bind b) u0 t4)) (\lambda (_: T).((eq T (THead (Bind b) u3 (THead (Flat Appl) 
-(lift (S O) O v2) t5)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to 
-((pr0 u0 u3) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind 
-Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t2 t0))))))))) (\lambda (H13: (eq T 
-(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t2)).(eq_ind T 
-(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) (\lambda (t: 
-T).((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u3) \to ((pr0 t4 t5) 
-\to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: 
-T).(pr0 t t0)))))))) (\lambda (_: (not (eq B b Abst))).(\lambda (_: (pr0 v1 
-v2)).(\lambda (_: (pr0 u0 u3)).(\lambda (_: (pr0 t4 t5)).(let H0 \def 
-(eq_ind_r T t (\lambda (t: T).(eq T (THead (Bind Abbr) u1 t0) t)) H5 (THead 
-(Flat Appl) v1 (THead (Bind b) u0 t4)) H11) in (let H1 \def (match H0 return 
-(\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t (THead (Flat Appl) v1 
-(THead (Bind b) u0 t4))) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) 
-u2 w) t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift 
-(S O) O v2) t5)) t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T 
-(THead (Bind Abbr) u1 t0) (THead (Flat Appl) v1 (THead (Bind b) u0 
-t4)))).(let H1 \def (eq_ind T (THead (Bind Abbr) u1 t0) (\lambda (e: 
-T).(match e return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | 
-(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return 
-(\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow 
-False])])) I (THead (Flat Appl) v1 (THead (Bind b) u0 t4)) H0) in (False_ind 
-(ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 
-(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t))) H1)))]) in 
-(H1 (refl_equal T (THead (Flat Appl) v1 (THead (Bind b) u0 t4)))))))))) t2 
-H13)) t H11 H12 H7 H8 H9 H10))) | (pr0_delta u0 u3 H7 t4 t5 H8 w0 H9) 
-\Rightarrow (\lambda (H11: (eq T (THead (Bind Abbr) u0 t4) t)).(\lambda (H12: 
-(eq T (THead (Bind Abbr) u3 w0) t2)).(eq_ind T (THead (Bind Abbr) u0 t4) 
-(\lambda (_: T).((eq T (THead (Bind Abbr) u3 w0) t2) \to ((pr0 u0 u3) \to 
-((pr0 t4 t5) \to ((subst0 O u3 t5 w0) \to (ex2 T (\lambda (t0: T).(pr0 (THead 
-(Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t2 t0)))))))) (\lambda (H13: (eq 
-T (THead (Bind Abbr) u3 w0) t2)).(eq_ind T (THead (Bind Abbr) u3 w0) (\lambda 
-(t: T).((pr0 u0 u3) \to ((pr0 t4 t5) \to ((subst0 O u3 t5 w0) \to (ex2 T 
-(\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t 
-t0))))))) (\lambda (H14: (pr0 u0 u3)).(\lambda (H15: (pr0 t4 t5)).(\lambda 
-(H16: (subst0 O u3 t5 w0)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T 
-(THead (Bind Abbr) u1 t0) t)) H5 (THead (Bind Abbr) u0 t4) H11) in (let H1 
-\def (match H0 return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t 
-(THead (Bind Abbr) u0 t4)) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind 
-Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) u3 w0) t0)))))) 
-with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Bind Abbr) u1 t0) 
-(THead (Bind Abbr) u0 t4))).(let H1 \def (f_equal T T (\lambda (e: T).(match 
-e return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) 
-\Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Bind Abbr) u1 t0) 
-(THead (Bind Abbr) u0 t4) H0) in ((let H2 \def (f_equal T T (\lambda (e: 
-T).(match e return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef 
-_) \Rightarrow u1 | (THead _ t _) \Rightarrow t])) (THead (Bind Abbr) u1 t0) 
-(THead (Bind Abbr) u0 t4) H0) in (eq_ind T u0 (\lambda (_: T).((eq T t0 t4) 
-\to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: 
-T).(pr0 (THead (Bind Abbr) u3 w0) t0))))) (\lambda (H12: (eq T t0 
-t4)).(eq_ind T t4 (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind 
-Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) u3 w0) t0)))) (let 
-H3 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall 
-(t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: 
-T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Bind Abbr) u0 
-t4) H11) in (let H4 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t3)) H9 t4 H12) 
-in (let H5 \def (eq_ind T u1 (\lambda (t: T).(pr0 t u2)) H8 u0 H2) in 
-(ex2_ind T (\lambda (t: T).(pr0 u2 t)) (\lambda (t: T).(pr0 u3 t)) (ex2 T 
-(\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead 
-(Bind Abbr) u3 w0) t))) (\lambda (x: T).(\lambda (H6: (pr0 u2 x)).(\lambda 
-(H7: (pr0 u3 x)).(ex2_ind T (\lambda (t: T).(pr0 t3 t)) (\lambda (t: T).(pr0 
-t5 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: 
-T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (x0: T).(\lambda (H8: (pr0 t3 
-x0)).(\lambda (H9: (pr0 t5 x0)).(pr0_confluence__pr0_delta_delta u2 t3 w H10 
-u3 t5 w0 H16 x H6 H7 x0 H8 H9)))) (H3 t4 (tlt_head_dx (Bind Abbr) u0 t4) t3 
-H4 t5 H15))))) (H3 u0 (tlt_head_sx (Bind Abbr) u0 t4) u2 H5 u3 H14))))) t0 
-(sym_eq T t0 t4 H12))) u1 (sym_eq T u1 u0 H2))) H1)))]) in (H1 (refl_equal T 
-(THead (Bind Abbr) u0 t4)))))))) t2 H13)) t H11 H12 H7 H8 H9))) | (pr0_zeta b 
-H7 t4 t5 H8 u) \Rightarrow (\lambda (H11: (eq T (THead (Bind b) u (lift (S O) 
-O t4)) t)).(\lambda (H12: (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 (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda 
-(t0: T).(pr0 t2 t0))))))) (\lambda (H13: (eq T t5 t2)).(eq_ind T t2 (\lambda 
-(t: T).((not (eq B b Abst)) \to ((pr0 t4 t) \to (ex2 T (\lambda (t0: T).(pr0 
-(THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t2 t0)))))) (\lambda 
-(H14: (not (eq B b Abst))).(\lambda (H15: (pr0 t4 t2)).(let H0 \def (eq_ind_r 
-T t (\lambda (t: T).(eq T (THead (Bind Abbr) u1 t0) t)) H5 (THead (Bind b) u 
-(lift (S O) O t4)) H11) in (let H1 \def (match H0 return (\lambda (t: 
-T).(\lambda (_: (eq ? ? t)).((eq T t (THead (Bind b) u (lift (S O) O t4))) 
-\to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: 
-T).(pr0 t2 t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead 
-(Bind Abbr) u1 t0) (THead (Bind b) u (lift (S O) O t4)))).(let H1 \def 
-(f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort 
-_) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) 
-(THead (Bind Abbr) u1 t0) (THead (Bind b) u (lift (S O) O t4)) H0) in ((let 
-H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t _) 
-\Rightarrow t])) (THead (Bind Abbr) u1 t0) (THead (Bind b) u (lift (S O) O 
-t4)) H0) in ((let H3 \def (f_equal T B (\lambda (e: T).(match e return 
-(\lambda (_: T).B) with [(TSort _) \Rightarrow Abbr | (TLRef _) \Rightarrow 
-Abbr | (THead k _ _) \Rightarrow (match k return (\lambda (_: K).B) with 
-[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (THead (Bind Abbr) 
-u1 t0) (THead (Bind b) u (lift (S O) O t4)) H0) in (eq_ind B Abbr (\lambda 
-(_: B).((eq T u1 u) \to ((eq T t0 (lift (S O) O t4)) \to (ex2 T (\lambda (t: 
-T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 t2 t)))))) (\lambda 
-(H12: (eq T u1 u)).(eq_ind T u (\lambda (_: T).((eq T t0 (lift (S O) O t4)) 
-\to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: 
-T).(pr0 t2 t0))))) (\lambda (H13: (eq T t0 (lift (S O) O t4))).(eq_ind T 
-(lift (S O) O t4) (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind 
-Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t2 t0)))) (let H4 \def (eq_ind_r B b 
-(\lambda (b: B).(not (eq B b Abst))) H14 Abbr H3) in (let H5 \def (eq_ind_r B 
-b (\lambda (b: B).(eq T (THead (Bind b) u (lift (S O) O t4)) t)) H11 Abbr H3) 
-in (let H6 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to 
-(\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T 
-(\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead 
-(Bind Abbr) u (lift (S O) O t4)) H5) in (let H7 \def (eq_ind T t0 (\lambda 
-(t: T).(pr0 t t3)) H9 (lift (S O) O t4) H13) in (ex2_ind T (\lambda (t2: 
-T).(eq T t3 (lift (S O) O t2))) (\lambda (t2: T).(pr0 t4 t2)) (ex2 T (\lambda 
-(t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 t2 t))) 
-(\lambda (x: T).(\lambda (H9: (eq T t3 (lift (S O) O x))).(\lambda (H11: (pr0 
-t4 x)).(let H10 \def (eq_ind T t3 (\lambda (t: T).(subst0 O u2 t w)) H10 
-(lift (S O) O x) H9) in (let H8 \def (eq_ind T u1 (\lambda (t: T).(pr0 t u2)) 
-H8 u H12) in (ex2_ind T (\lambda (t: T).(pr0 x t)) (\lambda (t: T).(pr0 t2 
-t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: 
-T).(pr0 t2 t))) (\lambda (x0: T).(\lambda (_: (pr0 x x0)).(\lambda (_: (pr0 
-t2 x0)).(pr0_confluence__pr0_delta_epsilon u2 (lift (S O) O x) w H10 x 
-(pr0_refl (lift (S O) O x)) t2)))) (H6 t4 (lift_tlt_dx (Bind Abbr) u t4 (S O) 
-O) x H11 t2 H15))))))) (pr0_gen_lift t4 t3 (S O) O H7)))))) t0 (sym_eq T t0 
-(lift (S O) O t4) H13))) u1 (sym_eq T u1 u H12))) b H3)) H2)) H1)))]) in (H1 
-(refl_equal T (THead (Bind b) u (lift (S O) O t4)))))))) t5 (sym_eq T t5 t2 
-H13))) t H11 H12 H7 H8))) | (pr0_epsilon t4 t5 H7 u) \Rightarrow (\lambda 
-(H8: (eq T (THead (Flat Cast) u t4) t)).(\lambda (H9: (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 (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: 
-T).(pr0 t2 t0)))))) (\lambda (H10: (eq T t5 t2)).(eq_ind T t2 (\lambda (t: 
-T).((pr0 t4 t) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) 
-(\lambda (t0: T).(pr0 t2 t0))))) (\lambda (_: (pr0 t4 t2)).(let H0 \def 
-(eq_ind_r T t (\lambda (t: T).(eq T (THead (Bind Abbr) u1 t0) t)) H5 (THead 
-(Flat Cast) u t4) H8) in (let H1 \def (match H0 return (\lambda (t: 
-T).(\lambda (_: (eq ? ? t)).((eq T t (THead (Flat Cast) u t4)) \to (ex2 T 
-(\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t2 
-t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Bind Abbr) 
-u1 t0) (THead (Flat Cast) u t4))).(let H1 \def (eq_ind T (THead (Bind Abbr) 
-u1 t0) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
-(match k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat 
-_) \Rightarrow False])])) I (THead (Flat Cast) u t4) H0) in (False_ind (ex2 T 
-(\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 t2 
-t))) H1)))]) in (H1 (refl_equal T (THead (Flat Cast) u t4)))))) t5 (sym_eq T 
-t5 t2 H10))) t H8 H9 H7)))]) in (H11 (refl_equal T t) (refl_equal T t2)))))) 
-t1 H7)) t H5 H6 H2 H3 H4))) | (pr0_zeta b H2 t0 t3 H3 u) \Rightarrow (\lambda 
-(H4: (eq T (THead (Bind b) u (lift (S O) O t0)) t)).(\lambda (H5: (eq T t3 
-t1)).(eq_ind T (THead (Bind b) u (lift (S O) O t0)) (\lambda (_: T).((eq T t3 
-t1) \to ((not (eq B b Abst)) \to ((pr0 t0 t3) \to (ex2 T (\lambda (t2: 
-T).(pr0 t1 t2)) (\lambda (t1: T).(pr0 t2 t1))))))) (\lambda (H6: (eq T t3 
-t1)).(eq_ind T t1 (\lambda (t: T).((not (eq B b Abst)) \to ((pr0 t0 t) \to 
-(ex2 T (\lambda (t2: T).(pr0 t1 t2)) (\lambda (t1: T).(pr0 t2 t1)))))) 
-(\lambda (H7: (not (eq B b Abst))).(\lambda (H8: (pr0 t0 t1)).(let H9 \def 
-(match H1 return (\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (pr0 t0 
-t3)).((eq T t0 t) \to ((eq T t3 t2) \to (ex2 T (\lambda (t: T).(pr0 t1 t)) 
-(\lambda (t: T).(pr0 t2 t)))))))) with [(pr0_refl t4) \Rightarrow (\lambda 
-(H6: (eq T t4 t)).(\lambda (H9: (eq T t4 t2)).(eq_ind T t (\lambda (t: 
-T).((eq T t t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: 
-T).(pr0 t2 t0))))) (\lambda (H10: (eq T t t2)).(eq_ind T t2 (\lambda (_: 
-T).(ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))) (let 
-H0 \def (eq_ind_r T t (\lambda (t: T).(eq T t t2)) H10 (THead (Bind b) u 
-(lift (S O) O t0)) H4) in (eq_ind T (THead (Bind b) u (lift (S O) O t0)) 
-(\lambda (t: T).(ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t 
-t0)))) (let H1 \def (eq_ind_r T t (\lambda (t: T).(eq T t4 t)) H6 (THead 
-(Bind b) u (lift (S O) O t0)) H4) in (let H2 \def (eq_ind_r T t (\lambda (t: 
-T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall 
-(t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: 
-T).(pr0 t2 t0)))))))))) H (THead (Bind b) u (lift (S O) O t0)) H4) in 
-(ex_intro2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead (Bind b) 
-u (lift (S O) O t0)) t)) t1 (pr0_refl t1) (pr0_zeta b H7 t0 t1 H8 u)))) t2 
-H0)) t (sym_eq T t t2 H10))) t4 (sym_eq T t4 t H6) H9))) | (pr0_comp u1 u2 H6 
-t4 t5 H7 k) \Rightarrow (\lambda (H9: (eq T (THead k u1 t4) t)).(\lambda 
-(H10: (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 (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda 
-(H11: (eq T (THead k u2 t5) t2)).(eq_ind T (THead k u2 t5) (\lambda (t: 
-T).((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) 
-(\lambda (t0: T).(pr0 t t0)))))) (\lambda (_: (pr0 u1 u2)).(\lambda (H13: 
-(pr0 t4 t5)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Bind b) 
-u (lift (S O) O t0)) t)) H4 (THead k u1 t4) H9) in (let H1 \def (match H0 
-return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t (THead k u1 t4)) \to 
-(ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 (THead k u2 t5) 
-t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Bind b) u 
-(lift (S O) O t0)) (THead k u1 t4))).(let H1 \def (f_equal T T (\lambda (e: 
-T).(match e 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 u t0) 
-\Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) 
-(\lambda (x: nat).(plus x (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 u t0) 
-\Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) 
-(\lambda (x: nat).(plus x (S O))) O t0) | (THead _ _ t) \Rightarrow t])) 
-(THead (Bind b) u (lift (S O) O t0)) (THead k u1 t4) H0) in ((let H2 \def 
-(f_equal T T (\lambda (e: T).(match e 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 k u1 t4) H0) in ((let H3 \def 
-(f_equal T K (\lambda (e: T).(match e return (\lambda (_: T).K) with [(TSort 
-_) \Rightarrow (Bind b) | (TLRef _) \Rightarrow (Bind b) | (THead k _ _) 
-\Rightarrow k])) (THead (Bind b) u (lift (S O) O t0)) (THead k u1 t4) H0) in 
-(eq_ind K (Bind b) (\lambda (k: K).((eq T u u1) \to ((eq T (lift (S O) O t0) 
-t4) \to (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead k u2 
-t5) t)))))) (\lambda (H10: (eq T u u1)).(eq_ind T u1 (\lambda (_: T).((eq T 
-(lift (S O) O t0) t4) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: 
-T).(pr0 (THead (Bind b) u2 t5) t0))))) (\lambda (H11: (eq T (lift (S O) O t0) 
-t4)).(eq_ind T (lift (S O) O t0) (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 
-t1 t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u2 t5) t0)))) (let H4 \def 
-(eq_ind_r K k (\lambda (k: K).(eq T (THead k u1 t4) t)) H9 (Bind b) H3) in 
-(let H5 \def (eq_ind_r T t4 (\lambda (t: T).(pr0 t t5)) H13 (lift (S O) O t0) 
-H11) in (ex2_ind T (\lambda (t2: T).(eq T t5 (lift (S O) O t2))) (\lambda 
-(t2: T).(pr0 t0 t2)) (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 
-(THead (Bind b) u2 t5) t))) (\lambda (x: T).(\lambda (H6: (eq T t5 (lift (S 
-O) O x))).(\lambda (H9: (pr0 t0 x)).(let H12 \def (eq_ind_r T t4 (\lambda 
-(t0: T).(eq T (THead (Bind b) u1 t0) t)) H4 (lift (S O) O t0) H11) in (let 
-H13 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to 
-(\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T 
-(\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead 
-(Bind b) u1 (lift (S O) O t0)) H12) in (eq_ind_r T (lift (S O) O x) (\lambda 
-(t: T).(ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 (THead 
-(Bind b) u2 t) t0)))) (ex2_ind T (\lambda (t: T).(pr0 x t)) (\lambda (t: 
-T).(pr0 t1 t)) (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead 
-(Bind b) u2 (lift (S O) O x)) t))) (\lambda (x0: T).(\lambda (H8: (pr0 x 
-x0)).(\lambda (H14: (pr0 t1 x0)).(ex_intro2 T (\lambda (t: T).(pr0 t1 t)) 
-(\lambda (t: T).(pr0 (THead (Bind b) u2 (lift (S O) O x)) t)) x0 H14 
-(pr0_zeta b H7 x x0 H8 u2))))) (H13 t0 (lift_tlt_dx (Bind b) u1 t0 (S O) O) x 
-H9 t1 H8)) t5 H6)))))) (pr0_gen_lift t0 t5 (S O) O H5)))) t4 H11)) u (sym_eq 
-T u u1 H10))) k H3)) H2)) H1)))]) in (H1 (refl_equal T (THead k u1 t4))))))) 
-t2 H11)) t H9 H10 H6 H7))) | (pr0_beta u0 v1 v2 H6 t4 t5 H7) \Rightarrow 
-(\lambda (H8: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t4)) 
-t)).(\lambda (H9: (eq T (THead (Bind Abbr) v2 t5) t2)).(eq_ind T (THead (Flat 
-Appl) v1 (THead (Bind Abst) u0 t4)) (\lambda (_: T).((eq T (THead (Bind Abbr) 
-v2 t5) t2) \to ((pr0 v1 v2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 
-t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H10: (eq T (THead (Bind 
-Abbr) v2 t5) t2)).(eq_ind T (THead (Bind Abbr) v2 t5) (\lambda (t: T).((pr0 
-v1 v2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda 
-(t0: T).(pr0 t t0)))))) (\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 t4 
-t5)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Bind b) u (lift 
-(S O) O t0)) t)) H4 (THead (Flat Appl) v1 (THead (Bind Abst) u0 t4)) H8) in 
-(let H1 \def (match H0 return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T 
-t (THead (Flat Appl) v1 (THead (Bind Abst) u0 t4))) \to (ex2 T (\lambda (t0: 
-T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t5) t0)))))) with 
-[refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Bind b) u (lift (S O) O 
-t0)) (THead (Flat Appl) v1 (THead (Bind Abst) u0 t4)))).(let H1 \def (eq_ind 
-T (THead (Bind b) u (lift (S O) O t0)) (\lambda (e: T).(match e return 
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: 
-K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I 
-(THead (Flat Appl) v1 (THead (Bind Abst) u0 t4)) H0) in (False_ind (ex2 T 
-(\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t5) 
-t))) H1)))]) in (H1 (refl_equal T (THead (Flat Appl) v1 (THead (Bind Abst) u0 
-t4)))))))) t2 H10)) t H8 H9 H6 H7))) | (pr0_upsilon b0 H6 v1 v2 H7 u1 u2 H8 
-t4 t5 H9) \Rightarrow (\lambda (H10: (eq T (THead (Flat Appl) v1 (THead (Bind 
-b0) u1 t4)) t)).(\lambda (H11: (eq T (THead (Bind b0) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t5)) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind b0) 
-u1 t4)) (\lambda (_: T).((eq T (THead (Bind b0) u2 (THead (Flat Appl) (lift 
-(S O) O v2) t5)) t2) \to ((not (eq B b0 Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 
-u2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: 
-T).(pr0 t2 t0))))))))) (\lambda (H12: (eq T (THead (Bind b0) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t5)) t2)).(eq_ind T (THead (Bind b0) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t5)) (\lambda (t: T).((not (eq B b0 Abst)) \to ((pr0 
-v1 v2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 t1 
-t0)) (\lambda (t0: T).(pr0 t t0)))))))) (\lambda (_: (not (eq B b0 
-Abst))).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0 
-t4 t5)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Bind b) u 
-(lift (S O) O t0)) t)) H4 (THead (Flat Appl) v1 (THead (Bind b0) u1 t4)) H10) 
-in (let H1 \def (match H0 return (\lambda (t: T).(\lambda (_: (eq ? ? 
-t)).((eq T t (THead (Flat Appl) v1 (THead (Bind b0) u1 t4))) \to (ex2 T 
-(\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 (THead (Bind b0) u2 
-(THead (Flat Appl) (lift (S O) O v2) t5)) t0)))))) with [refl_equal 
-\Rightarrow (\lambda (H0: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead 
-(Flat Appl) v1 (THead (Bind b0) u1 t4)))).(let H1 \def (eq_ind T (THead (Bind 
-b) u (lift (S O) O t0)) (\lambda (e: T).(match e return (\lambda (_: T).Prop) 
-with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ 
-_) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind _) 
-\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) v1 
-(THead (Bind b0) u1 t4)) H0) in (False_ind (ex2 T (\lambda (t: T).(pr0 t1 t)) 
-(\lambda (t: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) 
-t5)) t))) H1)))]) in (H1 (refl_equal T (THead (Flat Appl) v1 (THead (Bind b0) 
-u1 t4)))))))))) t2 H12)) t H10 H11 H6 H7 H8 H9))) | (pr0_delta u1 u2 H6 t4 t5 
-H7 w H8) \Rightarrow (\lambda (H9: (eq T (THead (Bind Abbr) u1 t4) 
-t)).(\lambda (H10: (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 (t0: T).(pr0 
-t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))) (\lambda (H11: (eq T (THead (Bind 
-Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t: T).((pr0 u1 
-u2) \to ((pr0 t4 t5) \to ((subst0 O u2 t5 w) \to (ex2 T (\lambda (t0: T).(pr0 
-t1 t0)) (\lambda (t0: T).(pr0 t t0))))))) (\lambda (_: (pr0 u1 u2)).(\lambda 
-(H13: (pr0 t4 t5)).(\lambda (H14: (subst0 O u2 t5 w)).(let H0 \def (eq_ind_r 
-T t (\lambda (t: T).(eq T (THead (Bind b) u (lift (S O) O t0)) t)) H4 (THead 
-(Bind Abbr) u1 t4) H9) in (let H1 \def (match H0 return (\lambda (t: 
-T).(\lambda (_: (eq ? ? t)).((eq T t (THead (Bind Abbr) u1 t4)) \to (ex2 T 
-(\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) 
-t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Bind b) u 
-(lift (S O) O t0)) (THead (Bind Abbr) u1 t4))).(let H1 \def (f_equal T T 
-(\lambda (e: T).(match e 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 u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) 
-t0))]) in lref_map) (\lambda (x: nat).(plus x (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 u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) 
-t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t0) | (THead _ _ t) 
-\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1 
-t4) H0) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e 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 t4) H0) in ((let H3 \def (f_equal T B (\lambda (e: T).(match e 
-return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | (TLRef _) 
-\Rightarrow b | (THead k _ _) \Rightarrow (match k return (\lambda (_: K).B) 
-with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b])])) (THead (Bind b) u 
-(lift (S O) O t0)) (THead (Bind Abbr) u1 t4) H0) in (eq_ind B Abbr (\lambda 
-(_: B).((eq T u u1) \to ((eq T (lift (S O) O t0) t4) \to (ex2 T (\lambda (t: 
-T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)))))) (\lambda 
-(H10: (eq T u u1)).(eq_ind T u1 (\lambda (_: T).((eq T (lift (S O) O t0) t4) 
-\to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 (THead (Bind 
-Abbr) u2 w) t0))))) (\lambda (H11: (eq T (lift (S O) O t0) t4)).(eq_ind T 
-(lift (S O) O t0) (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 t1 t0)) 
-(\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)))) (let H4 \def (eq_ind_r 
-T t4 (\lambda (t: T).(pr0 t t5)) H13 (lift (S O) O t0) H11) in (ex2_ind T 
-(\lambda (t2: T).(eq T t5 (lift (S O) O t2))) (\lambda (t2: T).(pr0 t0 t2)) 
-(ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 
-w) t))) (\lambda (x: T).(\lambda (H5: (eq T t5 (lift (S O) O x))).(\lambda 
-(H6: (pr0 t0 x)).(let H9 \def (eq_ind_r T t4 (\lambda (t0: T).(eq T (THead 
-(Bind Abbr) u1 t0) t)) H9 (lift (S O) O t0) H11) in (let H12 \def (eq_ind_r T 
-t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) 
-\to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) 
-(\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Bind Abbr) u1 (lift (S O) O 
-t0)) H9) in (let H13 \def (eq_ind T t5 (\lambda (t: T).(subst0 O u2 t w)) H14 
-(lift (S O) O x) H5) in (let H7 \def (eq_ind B b (\lambda (b: B).(not (eq B b 
-Abst))) H7 Abbr H3) in (ex2_ind T (\lambda (t: T).(pr0 x t)) (\lambda (t: 
-T).(pr0 t1 t)) (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead 
-(Bind Abbr) u2 w) t))) (\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_epsilon u2 (lift (S O) O x) w H13 x (pr0_refl 
-(lift (S O) O x)) t1))))) (H12 t0 (lift_tlt_dx (Bind Abbr) u1 t0 (S O) O) x 
-H6 t1 H8))))))))) (pr0_gen_lift t0 t5 (S O) O H4))) t4 H11)) u (sym_eq T u u1 
-H10))) b (sym_eq B b Abbr H3))) H2)) H1)))]) in (H1 (refl_equal T (THead 
-(Bind Abbr) u1 t4)))))))) t2 H11)) t H9 H10 H6 H7 H8))) | (pr0_zeta b0 H6 t4 
-t5 H7 u0) \Rightarrow (\lambda (H9: (eq T (THead (Bind b0) u0 (lift (S O) O 
-t4)) t)).(\lambda (H10: (eq T t5 t2)).(eq_ind T (THead (Bind b0) u0 (lift (S 
-O) O t4)) (\lambda (_: T).((eq T t5 t2) \to ((not (eq B b0 Abst)) \to ((pr0 
-t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 
-t0))))))) (\lambda (H11: (eq T t5 t2)).(eq_ind T t2 (\lambda (t: T).((not (eq 
-B b0 Abst)) \to ((pr0 t4 t) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda 
-(t0: T).(pr0 t2 t0)))))) (\lambda (_: (not (eq B b0 Abst))).(\lambda (H13: 
-(pr0 t4 t2)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Bind b) 
-u (lift (S O) O t0)) t)) H4 (THead (Bind b0) u0 (lift (S O) O t4)) H9) in 
-(let H1 \def (match H0 return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T 
-t (THead (Bind b0) u0 (lift (S O) O t4))) \to (ex2 T (\lambda (t0: T).(pr0 t1 
-t0)) (\lambda (t0: T).(pr0 t2 t0)))))) with [refl_equal \Rightarrow (\lambda 
-(H0: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead (Bind b0) u0 (lift (S 
-O) O t4)))).(let H1 \def (f_equal T T (\lambda (e: T).(match e 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 u t0) \Rightarrow 
-(THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda 
-(x: nat).(plus x (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 u t0) \Rightarrow 
-(THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda 
-(x: nat).(plus x (S O))) O t0) | (THead _ _ t) \Rightarrow t])) (THead (Bind 
-b) u (lift (S O) O t0)) (THead (Bind b0) u0 (lift (S O) O t4)) H0) in ((let 
-H2 \def (f_equal T T (\lambda (e: T).(match e 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 b0) u0 
-(lift (S O) O t4)) H0) in ((let H3 \def (f_equal T B (\lambda (e: T).(match e 
-return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | (TLRef _) 
-\Rightarrow b | (THead k _ _) \Rightarrow (match k return (\lambda (_: K).B) 
-with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b])])) (THead (Bind b) u 
-(lift (S O) O t0)) (THead (Bind b0) u0 (lift (S O) O t4)) H0) in (eq_ind B b0 
-(\lambda (_: B).((eq T u u0) \to ((eq T (lift (S O) O t0) (lift (S O) O t4)) 
-\to (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t)))))) 
-(\lambda (H10: (eq T u u0)).(eq_ind T u0 (\lambda (_: T).((eq T (lift (S O) O 
-t0) (lift (S O) O t4)) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: 
-T).(pr0 t2 t0))))) (\lambda (H11: (eq T (lift (S O) O t0) (lift (S O) O 
-t4))).(eq_ind T (lift (S O) O t0) (\lambda (_: T).(ex2 T (\lambda (t0: 
-T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))) (let H4 \def (eq_ind_r T t 
-(\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) 
-\to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) 
-(\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Bind b0) u0 (lift (S O) O 
-t4)) H9) in (let H5 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t1)) H8 t4 
-(lift_inj t0 t4 (S O) O H11)) in (let H6 \def (eq_ind B b (\lambda (b: 
-B).(not (eq B b Abst))) H7 b0 H3) in (ex2_ind T (\lambda (t: T).(pr0 t1 t)) 
-(\lambda (t: T).(pr0 t2 t)) (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: 
-T).(pr0 t2 t))) (\lambda (x: T).(\lambda (H7: (pr0 t1 x)).(\lambda (H8: (pr0 
-t2 x)).(ex_intro2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t)) x 
-H7 H8)))) (H4 t4 (lift_tlt_dx (Bind b0) u0 t4 (S O) O) t1 H5 t2 H13))))) 
-(lift (S O) O t4) H11)) u (sym_eq T u u0 H10))) b (sym_eq B b b0 H3))) H2)) 
-H1)))]) in (H1 (refl_equal T (THead (Bind b0) u0 (lift (S O) O t4)))))))) t5 
-(sym_eq T t5 t2 H11))) t H9 H10 H6 H7))) | (pr0_epsilon t4 t5 H6 u0) 
-\Rightarrow (\lambda (H7: (eq T (THead (Flat Cast) u0 t4) t)).(\lambda (H8: 
-(eq T t5 t2)).(eq_ind T (THead (Flat Cast) u0 t4) (\lambda (_: T).((eq T t5 
-t2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: 
-T).(pr0 t2 t0)))))) (\lambda (H9: (eq T t5 t2)).(eq_ind T t2 (\lambda (t: 
-T).((pr0 t4 t) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 
-t2 t0))))) (\lambda (_: (pr0 t4 t2)).(let H0 \def (eq_ind_r T t (\lambda (t: 
-T).(eq T (THead (Bind b) u (lift (S O) O t0)) t)) H4 (THead (Flat Cast) u0 
-t4) H7) in (let H1 \def (match H0 return (\lambda (t: T).(\lambda (_: (eq ? ? 
-t)).((eq T t (THead (Flat Cast) u0 t4)) \to (ex2 T (\lambda (t0: T).(pr0 t1 
-t0)) (\lambda (t0: T).(pr0 t2 t0)))))) with [refl_equal \Rightarrow (\lambda 
-(H0: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead (Flat Cast) u0 
-t4))).(let H1 \def (eq_ind T (THead (Bind b) u (lift (S O) O t0)) (\lambda 
-(e: T).(match e return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k 
-return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) 
-\Rightarrow False])])) I (THead (Flat Cast) u0 t4) H0) in (False_ind (ex2 T 
-(\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t))) H1)))]) in (H1 
-(refl_equal T (THead (Flat Cast) u0 t4)))))) t5 (sym_eq T t5 t2 H9))) t H7 H8 
-H6)))]) in (H9 (refl_equal T t) (refl_equal T t2))))) t3 (sym_eq T t3 t1 
-H6))) t H4 H5 H2 H3))) | (pr0_epsilon t0 t3 H2 u) \Rightarrow (\lambda (H3: 
-(eq T (THead (Flat Cast) u t0) t)).(\lambda (H4: (eq T t3 t1)).(eq_ind T 
-(THead (Flat Cast) u t0) (\lambda (_: T).((eq T t3 t1) \to ((pr0 t0 t3) \to 
-(ex2 T (\lambda (t2: T).(pr0 t1 t2)) (\lambda (t1: T).(pr0 t2 t1)))))) 
-(\lambda (H5: (eq T t3 t1)).(eq_ind T t1 (\lambda (t: T).((pr0 t0 t) \to (ex2 
-T (\lambda (t2: T).(pr0 t1 t2)) (\lambda (t1: T).(pr0 t2 t1))))) (\lambda 
-(H6: (pr0 t0 t1)).(let H7 \def (match H1 return (\lambda (t0: T).(\lambda 
-(t3: T).(\lambda (_: (pr0 t0 t3)).((eq T t0 t) \to ((eq T t3 t2) \to (ex2 T 
-(\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t)))))))) with [(pr0_refl 
-t4) \Rightarrow (\lambda (H5: (eq T t4 t)).(\lambda (H7: (eq T t4 
-t2)).(eq_ind T t (\lambda (t: T).((eq T t t2) \to (ex2 T (\lambda (t0: 
-T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))) (\lambda (H8: (eq T t 
-t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 t1 t0)) 
-(\lambda (t0: T).(pr0 t2 t0)))) (let H0 \def (eq_ind_r T t (\lambda (t: 
-T).(eq T t t2)) H8 (THead (Flat Cast) u t0) H3) in (eq_ind T (THead (Flat 
-Cast) u t0) (\lambda (t: T).(ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda 
-(t0: T).(pr0 t t0)))) (let H1 \def (eq_ind_r T t (\lambda (t: T).(eq T t4 t)) 
-H5 (THead (Flat Cast) u t0) H3) in (let H2 \def (eq_ind_r T t (\lambda (t: 
-T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall 
-(t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: 
-T).(pr0 t2 t0)))))))))) H (THead (Flat Cast) u t0) H3) in (ex_intro2 T 
-(\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead (Flat Cast) u t0) t)) 
-t1 (pr0_refl t1) (pr0_epsilon t0 t1 H6 u)))) t2 H0)) t (sym_eq T t t2 H8))) 
-t4 (sym_eq T t4 t H5) H7))) | (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 (t0: T).(pr0 t1 t0)) 
-(\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H9: (eq T (THead k u2 t5) 
-t2)).(eq_ind T (THead k u2 t5) (\lambda (t: T).((pr0 u1 u2) \to ((pr0 t4 t5) 
-\to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t t0)))))) 
-(\lambda (_: (pr0 u1 u2)).(\lambda (H11: (pr0 t4 t5)).(let H0 \def (eq_ind_r 
-T t (\lambda (t: T).(eq T (THead (Flat Cast) u t0) t)) H3 (THead k u1 t4) H7) 
-in (let H1 \def (match H0 return (\lambda (t: T).(\lambda (_: (eq ? ? 
-t)).((eq T t (THead k u1 t4)) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) 
-(\lambda (t0: T).(pr0 (THead k u2 t5) t0)))))) with [refl_equal \Rightarrow 
-(\lambda (H0: (eq T (THead (Flat Cast) u t0) (THead k u1 t4))).(let H1 \def 
-(f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort 
-_) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) 
-(THead (Flat Cast) u t0) (THead k u1 t4) H0) in ((let H2 \def (f_equal T T 
-(\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) 
-(THead (Flat Cast) u t0) (THead k u1 t4) H0) in ((let H3 \def (f_equal T K 
-(\lambda (e: T).(match e return (\lambda (_: T).K) with [(TSort _) 
-\Rightarrow (Flat Cast) | (TLRef _) \Rightarrow (Flat Cast) | (THead k _ _) 
-\Rightarrow k])) (THead (Flat Cast) u t0) (THead k u1 t4) H0) in (eq_ind K 
-(Flat Cast) (\lambda (k: K).((eq T u u1) \to ((eq T t0 t4) \to (ex2 T 
-(\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead k u2 t5) t)))))) 
-(\lambda (H8: (eq T u u1)).(eq_ind T u1 (\lambda (_: T).((eq T t0 t4) \to 
-(ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 (THead (Flat Cast) 
-u2 t5) t0))))) (\lambda (H9: (eq T t0 t4)).(eq_ind T t4 (\lambda (_: T).(ex2 
-T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 (THead (Flat Cast) u2 
-t5) t0)))) (let H4 \def (eq_ind_r K k (\lambda (k: K).(eq T (THead k u1 t4) 
-t)) H7 (Flat Cast) H3) in (let H5 \def (eq_ind_r T t (\lambda (t: T).(\forall 
-(v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 
-v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 
-t0)))))))))) H (THead (Flat Cast) u1 t4) H4) in (let H6 \def (eq_ind T t0 
-(\lambda (t: T).(pr0 t t1)) H6 t4 H9) in (ex2_ind T (\lambda (t: T).(pr0 t1 
-t)) (\lambda (t: T).(pr0 t5 t)) (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda 
-(t: T).(pr0 (THead (Flat Cast) u2 t5) t))) (\lambda (x: T).(\lambda (H7: (pr0 
-t1 x)).(\lambda (H10: (pr0 t5 x)).(ex_intro2 T (\lambda (t: T).(pr0 t1 t)) 
-(\lambda (t: T).(pr0 (THead (Flat Cast) u2 t5) t)) x H7 (pr0_epsilon t5 x H10 
-u2))))) (H5 t4 (tlt_head_dx (Flat Cast) u1 t4) t1 H6 t5 H11))))) t0 (sym_eq T 
-t0 t4 H9))) u (sym_eq T u u1 H8))) k H3)) H2)) H1)))]) in (H1 (refl_equal T 
-(THead k u1 t4))))))) t2 H9)) t H7 H8 H5 H6))) | (pr0_beta u0 v1 v2 H5 t4 t5 
-H6) \Rightarrow (\lambda (H7: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) 
-u0 t4)) t)).(\lambda (H8: (eq T (THead (Bind Abbr) v2 t5) t2)).(eq_ind T 
-(THead (Flat Appl) v1 (THead (Bind Abst) u0 t4)) (\lambda (_: T).((eq T 
-(THead (Bind Abbr) v2 t5) t2) \to ((pr0 v1 v2) \to ((pr0 t4 t5) \to (ex2 T 
-(\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda 
-(H9: (eq T (THead (Bind Abbr) v2 t5) t2)).(eq_ind T (THead (Bind Abbr) v2 t5) 
-(\lambda (t: T).((pr0 v1 v2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: 
-T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t t0)))))) (\lambda (_: (pr0 v1 
-v2)).(\lambda (_: (pr0 t4 t5)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq 
-T (THead (Flat Cast) u t0) t)) H3 (THead (Flat Appl) v1 (THead (Bind Abst) u0 
-t4)) H7) in (let H1 \def (match H0 return (\lambda (t: T).(\lambda (_: (eq ? 
-? t)).((eq T t (THead (Flat Appl) v1 (THead (Bind Abst) u0 t4))) \to (ex2 T 
-(\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t5) 
-t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat Cast) 
-u t0) (THead (Flat Appl) v1 (THead (Bind Abst) u0 t4)))).(let H1 \def (eq_ind 
-T (THead (Flat Cast) u t0) (\lambda (e: T).(match e return (\lambda (_: 
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
-(THead k _ _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind 
-_) \Rightarrow False | (Flat f) \Rightarrow (match f return (\lambda (_: 
-F).Prop) with [Appl \Rightarrow False | Cast \Rightarrow True])])])) I (THead 
-(Flat Appl) v1 (THead (Bind Abst) u0 t4)) H0) in (False_ind (ex2 T (\lambda 
-(t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t5) t))) 
-H1)))]) in (H1 (refl_equal T (THead (Flat Appl) v1 (THead (Bind Abst) u0 
-t4)))))))) 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 (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 
-t2 t0))))))))) (\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 (t: T).((not (eq B b Abst)) \to ((pr0 v1 v2) 
-\to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) 
-(\lambda (t0: T).(pr0 t t0)))))))) (\lambda (_: (not (eq B b Abst))).(\lambda 
-(_: (pr0 v1 v2)).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0 t4 t5)).(let H0 
-\def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat Cast) u t0) t)) H3 
-(THead (Flat Appl) v1 (THead (Bind b) u1 t4)) H9) in (let H1 \def (match H0 
-return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t (THead (Flat Appl) 
-v1 (THead (Bind b) u1 t4))) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda 
-(t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) 
-t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat Cast) 
-u t0) (THead (Flat Appl) v1 (THead (Bind b) u1 t4)))).(let H1 \def (eq_ind T 
-(THead (Flat Cast) u t0) (\lambda (e: T).(match e return (\lambda (_: 
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
-(THead k _ _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind 
-_) \Rightarrow False | (Flat f) \Rightarrow (match f return (\lambda (_: 
-F).Prop) with [Appl \Rightarrow False | Cast \Rightarrow True])])])) I (THead 
-(Flat Appl) v1 (THead (Bind b) u1 t4)) H0) in (False_ind (ex2 T (\lambda (t: 
-T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t5)) t))) H1)))]) in (H1 (refl_equal T (THead (Flat Appl) 
-v1 (THead (Bind b) u1 t4)))))))))) 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 (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))) 
-(\lambda (H10: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind 
-Abbr) u2 w) (\lambda (t: T).((pr0 u1 u2) \to ((pr0 t4 t5) \to ((subst0 O u2 
-t5 w) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t 
-t0))))))) (\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0 t4 t5)).(\lambda (_: 
-(subst0 O u2 t5 w)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead 
-(Flat Cast) u t0) t)) H3 (THead (Bind Abbr) u1 t4) H8) in (let H1 \def (match 
-H0 return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t (THead (Bind 
-Abbr) u1 t4)) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 
-(THead (Bind Abbr) u2 w) t0)))))) with [refl_equal \Rightarrow (\lambda (H0: 
-(eq T (THead (Flat Cast) u t0) (THead (Bind Abbr) u1 t4))).(let H1 \def 
-(eq_ind T (THead (Flat Cast) u t0) (\lambda (e: T).(match e return (\lambda 
-(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
-| (THead k _ _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind 
-_) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1 
-t4) H0) in (False_ind (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 
-(THead (Bind Abbr) u2 w) t))) H1)))]) in (H1 (refl_equal T (THead (Bind Abbr) 
-u1 t4)))))))) t2 H10)) t H8 H9 H5 H6 H7))) | (pr0_zeta b H5 t4 t5 H6 u0) 
-\Rightarrow (\lambda (H7: (eq T (THead (Bind b) u0 (lift (S O) O t4)) 
-t)).(\lambda (H8: (eq T t5 t2)).(eq_ind T (THead (Bind b) u0 (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 (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))))) 
-(\lambda (H9: (eq T t5 t2)).(eq_ind T t2 (\lambda (t: T).((not (eq B b Abst)) 
-\to ((pr0 t4 t) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: 
-T).(pr0 t2 t0)))))) (\lambda (_: (not (eq B b Abst))).(\lambda (_: (pr0 t4 
-t2)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat Cast) u 
-t0) t)) H3 (THead (Bind b) u0 (lift (S O) O t4)) H7) in (let H1 \def (match 
-H0 return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t (THead (Bind b) 
-u0 (lift (S O) O t4))) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: 
-T).(pr0 t2 t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead 
-(Flat Cast) u t0) (THead (Bind b) u0 (lift (S O) O t4)))).(let H1 \def 
-(eq_ind T (THead (Flat Cast) u t0) (\lambda (e: T).(match e return (\lambda 
-(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
-| (THead k _ _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind 
-_) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u0 
-(lift (S O) O t4)) H0) in (False_ind (ex2 T (\lambda (t: T).(pr0 t1 t)) 
-(\lambda (t: T).(pr0 t2 t))) H1)))]) in (H1 (refl_equal T (THead (Bind b) u0 
-(lift (S O) O t4)))))))) t5 (sym_eq T t5 t2 H9))) t H7 H8 H5 H6))) | 
-(pr0_epsilon t4 t5 H5 u0) \Rightarrow (\lambda (H7: (eq T (THead (Flat Cast) 
-u0 t4) t)).(\lambda (H8: (eq T t5 t2)).(eq_ind T (THead (Flat Cast) u0 t4) 
-(\lambda (_: T).((eq T t5 t2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: 
-T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))) (\lambda (H9: (eq T t5 
-t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t4 t) \to (ex2 T (\lambda (t0: 
-T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))) (\lambda (H10: (pr0 t4 
-t2)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat Cast) u 
-t0) t)) H3 (THead (Flat Cast) u0 t4) H7) in (let H1 \def (match H0 return 
-(\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t (THead (Flat Cast) u0 t4)) 
-\to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))) 
-with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat Cast) u t0) 
-(THead (Flat Cast) u0 t4))).(let H1 \def (f_equal T T (\lambda (e: T).(match 
-e return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) 
-\Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Flat Cast) u t0) 
-(THead (Flat Cast) u0 t4) H0) in ((let H2 \def (f_equal T T (\lambda (e: 
-T).(match e return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef 
-_) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Flat Cast) u t0) 
-(THead (Flat Cast) u0 t4) H0) in (eq_ind T u0 (\lambda (_: T).((eq T t0 t4) 
-\to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))) 
-(\lambda (H8: (eq T t0 t4)).(eq_ind T t4 (\lambda (_: T).(ex2 T (\lambda (t0: 
-T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))) (let H3 \def (eq_ind_r T t 
-(\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) 
-\to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) 
-(\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Cast) u0 t4) H7) in (let 
-H4 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t1)) H6 t4 H8) in (ex2_ind T 
-(\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t)) (ex2 T (\lambda (t: 
-T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t))) (\lambda (x: T).(\lambda (H5: 
-(pr0 t1 x)).(\lambda (H6: (pr0 t2 x)).(ex_intro2 T (\lambda (t: T).(pr0 t1 
-t)) (\lambda (t: T).(pr0 t2 t)) x H5 H6)))) (H3 t4 (tlt_head_dx (Flat Cast) 
-u0 t4) t1 H4 t2 H10)))) t0 (sym_eq T t0 t4 H8))) u (sym_eq T u u0 H2))) 
-H1)))]) in (H1 (refl_equal T (THead (Flat Cast) u0 t4)))))) t5 (sym_eq T t5 
-t2 H9))) t H7 H8 H5)))]) in (H7 (refl_equal T t) (refl_equal T t2)))) t3 
-(sym_eq T t3 t1 H5))) t H3 H4 H2)))]) in (H2 (refl_equal T t) (refl_equal T 
-t1))))))))) t0).
-
-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))))))).
-
-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))))).(match k return (\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))))) with [(Bind b) 
-\Rightarrow (match b return (\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))))) with [Abbr 
-\Rightarrow (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 return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) 
-\Rightarrow t])) (THead (Bind Abbr) t t0) (THead (Bind Abbr) t (lift (S O) O 
-x)) H4) in (let H6 \def (eq_ind T t0 (\lambda (t0: T).(subst0 O t t0 (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)))) | Abst \Rightarrow (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 (t: T).(pr0 t0 t)) H8 t0 H_y) in (let H10 \def (eq_ind_r T x1 
-(\lambda (t: T).(eq T t2 (THead (Bind Abst) x0 t))) H6 t0 H_y) in (let H11 
-\def (eq_ind_r T x0 (\lambda (t0: T).(pr0 t t0)) H7 t H_y0) in (let H12 \def 
-(eq_ind_r T x0 (\lambda (t: T).(eq T t2 (THead (Bind Abst) t 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 return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef 
-_) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Bind Abst) t t0) 
-(THead (Bind Abst) t x) H7) in (let H9 \def (eq_ind_r T x (\lambda (t: 
-T).(pr0 t0 t)) H6 t0 H8) in (let H10 \def (eq_ind_r T x (\lambda (t: T).((eq 
-T t0 t) \to (\forall (P: Prop).P))) 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 
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow t | (TLRef _) 
-\Rightarrow t | (THead _ t _) \Rightarrow t])) (THead (Bind Abst) t t0) 
-(THead (Bind Abst) x t0) H5) in (let H7 \def (eq_ind_r T x (\lambda (t0: 
-T).(pr0 t t0)) H4 t H6) in (let H8 \def (eq_ind_r T x (\lambda (t0: T).((eq T 
-t t0) \to (\forall (P: Prop).P))) H3 t H6) in (H8 (refl_equal T t) P)))))) 
-(pr0_comp t x H4 t0 t0 (pr0_refl t0) (Bind Abst))))))) H2)) H1)) | Void 
-\Rightarrow (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 (t2: 
-T).(pr0 t0 t2))))).(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 (t: T).(pr0 t0 t)) H12 t0 H_y) in (let H14 \def (eq_ind_r T x1 
-(\lambda (t: T).(eq T t2 (THead (Bind Void) x0 t))) H10 t0 H_y) in (let H15 
-\def (eq_ind_r T x0 (\lambda (t0: T).(pr0 t t0)) H11 t H_y0) in (let H16 \def 
-(eq_ind_r T x0 (\lambda (t: T).(eq T t2 (THead (Bind Void) t 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 (t0: 
-T).(subst0 O t t0 (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_comm 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 return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | 
-(TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Bind Void) 
-t t0) (THead (Bind Void) t x0) H10) in (let H12 \def (eq_ind_r T x0 (\lambda 
-(t: T).(pr0 t0 t)) H9 t0 H11) in (let H13 \def (eq_ind_r T x0 (\lambda (t: 
-T).((eq T t0 t) \to (\forall (P: Prop).P))) 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 return (\lambda (_: T).T) with [(TSort _) \Rightarrow t | (TLRef _) 
-\Rightarrow t | (THead _ t _) \Rightarrow t])) (THead (Bind Void) t t0) 
-(THead (Bind Void) x0 t0) H8) in (let H10 \def (eq_ind_r T x0 (\lambda (t0: 
-T).(pr0 t t0)) H7 t H9) in (let H11 \def (eq_ind_r T x0 (\lambda (t0: T).((eq 
-T t t0) \to (\forall (P: Prop).P))) 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 (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))))) 
-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 not_void_abst x x (pr0_refl x) 
-t))) t0 H3))) H2))) H1)))]) | (Flat f) \Rightarrow (match f return (\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))))) with [Appl \Rightarrow (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 (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))))) 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))))) ((match x0 return (\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)))))) with [Abbr \Rightarrow (\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 return (\lambda (_: T).Prop) with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ t) \Rightarrow 
-(match t return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | 
-(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match 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))))) | Abst \Rightarrow (\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 return (\lambda (_: T).Prop) with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
-(match 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))))) | Void 
-\Rightarrow (\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 return 
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead _ _ t) \Rightarrow (match t return (\lambda (_: 
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
-(THead k _ _) \Rightarrow (match 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 not_void_abst t t (pr0_refl t) x1 x1 (pr0_refl x1) x2 x2 
-(pr0_refl x2)))))]) 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 (t2: T).(pr0 t0 t2))))).(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 (t: T).(pr0 t0 t)) H11 
-t0 H_y) in (let H13 \def (eq_ind_r T x1 (\lambda (t: T).(eq T t2 (THead (Flat 
-Appl) x0 t))) H9 t0 H_y) in (let H14 \def (eq_ind_r T x0 (\lambda (t0: 
-T).(pr0 t t0)) H10 t H_y0) in (let H15 \def (eq_ind_r T x0 (\lambda (t: 
-T).(eq T t2 (THead (Flat Appl) t 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 (t2: 
-T).(pr0 z1 t2))))))).(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 (t: T).(\forall 
-(t2: T).((pr0 t t2) \to (eq T t t2)))) H6 (THead (Bind Abst) x0 x1) H9) in 
-(let H14 \def (eq_ind T t0 (\lambda (t: T).(\forall (b: B).(\forall (w: 
-T).(\forall (u: T).((eq T t (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 (t2: T).(pr0 z1 t2))))))))).(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 (t: T).(\forall (t2: 
-T).((pr0 t t2) \to (eq T t t2)))) H6 (THead (Bind x0) x1 x2) H10) in (let H16 
-\def (eq_ind T t0 (\lambda (t: T).(\forall (b: B).(\forall (w: T).(\forall 
-(u: T).((eq T t (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 return (\lambda (_: 
-T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ 
-t) \Rightarrow t])) (THead (Flat Appl) t t0) (THead (Flat Appl) t x) H9) in 
-(let H11 \def (eq_ind_r T x (\lambda (t: T).(pr0 t0 t)) H8 t0 H10) in (let 
-H12 \def (eq_ind_r T x (\lambda (t: T).((eq T t0 t) \to (\forall (P: 
-Prop).P))) 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 return (\lambda (_: T).T) with [(TSort 
-_) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ t _) \Rightarrow t])) 
-(THead (Flat Appl) t t0) (THead (Flat Appl) x t0) H7) in (let H9 \def 
-(eq_ind_r T x (\lambda (t0: T).(pr0 t t0)) H6 t H8) in (let H10 \def 
-(eq_ind_r T x (\lambda (t0: T).((eq T t t0) \to (\forall (P: Prop).P))) 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))) | Cast \Rightarrow (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_epsilon t0 t0 (pr0_refl t0) 
-t)))])])))))) t1).
-
-inductive pr1: T \to (T \to Prop) \def
-| pr1_r: \forall (t: T).(pr1 t t)
-| pr1_u: \forall (t2: T).(\forall (t1: T).((pr0 t1 t2) \to (\forall (t3: 
-T).((pr1 t2 t3) \to (pr1 t1 t3))))).
-
-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_u t2 t1 H t2 
-(pr1_r 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 (t3: T).((pr1 t4 t3) \to (pr1 t0 
-t3))))).(\lambda (t5: T).(\lambda (H3: (pr1 t4 t5)).(pr1_u 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_r (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_u 
-(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_r (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_u (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_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_r t2))))) (\lambda 
-(t2: T).(\lambda (t3: T).(\lambda (H0: (pr0 t3 t2)).(\lambda (t4: T).(\lambda 
-(_: (pr1 t2 t4)).(\lambda (H2: ((\forall (t3: T).((pr0 t2 t3) \to (ex2 T 
-(\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t3 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)).(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 (H6: (pr1 t4 x0)).(\lambda (H7: (pr1 x x0)).(ex_intro2 T (\lambda 
-(t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t)) x0 H6 (pr1_u x t5 H4 x0 
-H7))))) (H2 x H5))))) (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_r t2))))) (\lambda (t2: T).(\lambda 
-(t3: T).(\lambda (H0: (pr0 t3 t2)).(\lambda (t4: T).(\lambda (_: (pr1 t2 
-t4)).(\lambda (H2: ((\forall (t3: T).((pr1 t2 t3) \to (ex2 T (\lambda (t: 
-T).(pr1 t4 t)) (\lambda (t: T).(pr1 t3 t))))))).(\lambda (t5: T).(\lambda 
-(H3: (pr1 t3 t5)).(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 (H4: (pr1 t5 x)).(\lambda (H5: (pr1 t2 x)).(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 (H6: 
-(pr1 t4 x0)).(\lambda (H7: (pr1 x x0)).(ex_intro2 T (\lambda (t: T).(pr1 t4 
-t)) (\lambda (t: T).(pr1 t5 t)) x0 H6 (pr1_t x t5 H4 x0 H7))))) (H2 x H5))))) 
-(pr1_strip t3 t5 H3 t2 H0)))))))))) t0 t1 H))).
-
-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)))))))).
-
-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)).(let H0 
-\def (match H return (\lambda (c: C).(\lambda (c0: C).(\lambda (_: (wcpr0 c 
-c0)).((eq C c (CSort n)) \to ((eq C c0 x) \to (eq C x (CSort n))))))) with 
-[(wcpr0_refl c) \Rightarrow (\lambda (H0: (eq C c (CSort n))).(\lambda (H1: 
-(eq C c x)).(eq_ind C (CSort n) (\lambda (c0: C).((eq C c0 x) \to (eq C x 
-(CSort n)))) (\lambda (H2: (eq C (CSort n) x)).(eq_ind C (CSort n) (\lambda 
-(c0: C).(eq C c0 (CSort n))) (refl_equal C (CSort n)) x H2)) c (sym_eq C c 
-(CSort n) H0) H1))) | (wcpr0_comp c1 c2 H0 u1 u2 H1 k) \Rightarrow (\lambda 
-(H2: (eq C (CHead c1 k u1) (CSort n))).(\lambda (H3: (eq C (CHead c2 k u2) 
-x)).((let H4 \def (eq_ind C (CHead c1 k u1) (\lambda (e: C).(match e return 
-(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) 
-\Rightarrow True])) I (CSort n) H2) in (False_ind ((eq C (CHead c2 k u2) x) 
-\to ((wcpr0 c1 c2) \to ((pr0 u1 u2) \to (eq C x (CSort n))))) H4)) H3 H0 
-H1)))]) in (H0 (refl_equal C (CSort n)) (refl_equal C x))))).
-
-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)).(let H0 \def (match H return (\lambda (c: 
-C).(\lambda (c0: C).(\lambda (_: (wcpr0 c c0)).((eq C c (CHead c1 k u1)) \to 
-((eq C c0 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)))))))))) with 
-[(wcpr0_refl c) \Rightarrow (\lambda (H0: (eq C c (CHead c1 k u1))).(\lambda 
-(H1: (eq C c x)).(eq_ind C (CHead c1 k u1) (\lambda (c0: C).((eq C c0 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))))))) (\lambda (H2: (eq C (CHead 
-c1 k u1) x)).(eq_ind C (CHead c1 k u1) (\lambda (c0: C).(or (eq C c0 (CHead 
-c1 k u1)) (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))) x H2)) 
-c (sym_eq C c (CHead c1 k u1) H0) H1))) | (wcpr0_comp c0 c2 H0 u0 u2 H1 k0) 
-\Rightarrow (\lambda (H2: (eq C (CHead c0 k0 u0) (CHead c1 k u1))).(\lambda 
-(H3: (eq C (CHead c2 k0 u2) x)).((let H4 \def (f_equal C T (\lambda (e: 
-C).(match e return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead 
-_ _ t) \Rightarrow t])) (CHead c0 k0 u0) (CHead c1 k u1) H2) in ((let H5 \def 
-(f_equal C K (\lambda (e: C).(match e return (\lambda (_: C).K) with [(CSort 
-_) \Rightarrow k0 | (CHead _ k _) \Rightarrow k])) (CHead c0 k0 u0) (CHead c1 
-k u1) H2) in ((let H6 \def (f_equal C C (\lambda (e: C).(match e return 
-(\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow 
-c])) (CHead c0 k0 u0) (CHead c1 k u1) H2) in (eq_ind C c1 (\lambda (c: 
-C).((eq K k0 k) \to ((eq T u0 u1) \to ((eq C (CHead c2 k0 u2) x) \to ((wcpr0 
-c c2) \to ((pr0 u0 u2) \to (or (eq C x (CHead c1 k u1)) (ex3_2 C T (\lambda 
-(c3: C).(\lambda (u3: T).(eq C x (CHead c3 k u3)))) (\lambda (c3: C).(\lambda 
-(_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))))))))))) 
-(\lambda (H7: (eq K k0 k)).(eq_ind K k (\lambda (k1: K).((eq T u0 u1) \to 
-((eq C (CHead c2 k1 u2) x) \to ((wcpr0 c1 c2) \to ((pr0 u0 u2) \to (or (eq C 
-x (CHead c1 k u1)) (ex3_2 C T (\lambda (c3: C).(\lambda (u3: T).(eq C x 
-(CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda 
-(_: C).(\lambda (u3: T).(pr0 u1 u3)))))))))) (\lambda (H8: (eq T u0 
-u1)).(eq_ind T u1 (\lambda (t: T).((eq C (CHead c2 k u2) x) \to ((wcpr0 c1 
-c2) \to ((pr0 t u2) \to (or (eq C x (CHead c1 k u1)) (ex3_2 C T (\lambda (c3: 
-C).(\lambda (u3: T).(eq C x (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: 
-T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))))))))) 
-(\lambda (H9: (eq C (CHead c2 k u2) x)).(eq_ind C (CHead c2 k u2) (\lambda 
-(c: C).((wcpr0 c1 c2) \to ((pr0 u1 u2) \to (or (eq C c (CHead c1 k u1)) 
-(ex3_2 C T (\lambda (c3: C).(\lambda (u3: T).(eq C c (CHead c3 k u3)))) 
-(\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda 
-(u3: T).(pr0 u1 u3)))))))) (\lambda (H10: (wcpr0 c1 c2)).(\lambda (H11: (pr0 
-u1 u2)).(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)) H10 H11)))) x H9)) u0 (sym_eq T u0 u1 H8))) k0 (sym_eq K k0 k H7))) c0 
-(sym_eq C c0 c1 H6))) H5)) H4)) H3 H0 H1)))]) in (H0 (refl_equal C (CHead c1 
-k u1)) (refl_equal C x))))))).
-
-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 (c0: C).(\lambda (c3: C).(\lambda (H0: (wcpr0 c0 
-c3)).(\lambda (H1: ((\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 c3 (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 c0 k u1) (CHead 
-e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead 
-c3 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 c0 k u1) 
-(CHead e1 k0 u0))).(let H4 \def (match (drop_gen_refl (CHead c0 k u1) (CHead 
-e1 k0 u0) H3) return (\lambda (c: C).(\lambda (_: (eq ? ? c)).((eq C c (CHead 
-e1 k0 u0)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead 
-c3 k u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) 
-(\lambda (_: C).(\lambda (u2: T).(pr0 u0 u2))))))) with [refl_equal 
-\Rightarrow (\lambda (H3: (eq C (CHead c0 k u1) (CHead e1 k0 u0))).(let H4 
-\def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: C).T) with 
-[(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c0 k u1) 
-(CHead e1 k0 u0) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e 
-return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k _) 
-\Rightarrow k])) (CHead c0 k u1) (CHead e1 k0 u0) H3) in ((let H6 \def 
-(f_equal C C (\lambda (e: C).(match e return (\lambda (_: C).C) with [(CSort 
-_) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k u1) (CHead e1 
-k0 u0) H3) in (eq_ind C e1 (\lambda (_: C).((eq K k k0) \to ((eq T u1 u0) \to 
-(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3 k u2) (CHead 
-e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: 
-C).(\lambda (u2: T).(pr0 u0 u2))))))) (\lambda (H7: (eq K k k0)).(eq_ind K k0 
-(\lambda (k: K).((eq T u1 u0) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: 
-T).(drop O O (CHead c3 k u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda 
-(_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u0 u2)))))) 
-(\lambda (H8: (eq T u1 u0)).(eq_ind T u0 (\lambda (_: T).(ex3_2 C T (\lambda 
-(e2: C).(\lambda (u3: T).(drop O O (CHead c3 k0 u2) (CHead e2 k0 u3)))) 
-(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda 
-(u2: T).(pr0 u0 u2))))) (let H9 \def (eq_ind T u1 (\lambda (t: T).(pr0 t u2)) 
-H2 u0 H8) in (let H10 \def (eq_ind C c0 (\lambda (c: C).(wcpr0 c c3)) H0 e1 
-H6) in (ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3 
-k0 u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) 
-(\lambda (_: C).(\lambda (u2: T).(pr0 u0 u2))) c3 u2 (drop_refl (CHead c3 k0 
-u2)) H10 H9))) u1 (sym_eq T u1 u0 H8))) k (sym_eq K k k0 H7))) c0 (sym_eq C 
-c0 e1 H6))) H5)) H4)))]) in (H4 (refl_equal C (CHead e1 k0 u0)))))))) (K_ind 
-(\lambda (k0: K).(\forall (n: nat).(((\forall (e1: C).(\forall (u3: 
-T).(\forall (k: K).((drop n O (CHead c0 k0 u1) (CHead e1 k u3)) \to (ex3_2 C 
-T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c3 k0 u2) (CHead e2 k 
-u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: 
-C).(\lambda (u2: T).(pr0 u3 u2))))))))) \to (\forall (e1: C).(\forall (u3: 
-T).(\forall (k1: K).((drop (S n) O (CHead c0 k0 u1) (CHead e1 k1 u3)) \to 
-(ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop (S n) O (CHead c3 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 (k: K).((drop n 
-O (CHead c0 (Bind b) u1) (CHead e1 k u3)) \to (ex3_2 C T (\lambda (e2: 
-C).(\lambda (u4: T).(drop n O (CHead c3 (Bind b) u2) (CHead e2 k u4)))) 
-(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda 
-(u2: T).(pr0 u3 u2)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: 
-K).(\lambda (H4: (drop (S n) O (CHead c0 (Bind b) u1) (CHead e1 k0 
-u0))).(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 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 c3 (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 (H5: (drop n O c3 (CHead x0 
-k0 x1))).(\lambda (H6: (wcpr0 e1 x0)).(\lambda (H7: (pr0 u0 x1)).(ex3_2_intro 
-C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead c3 (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 c3 (CHead 
-x0 k0 x1) H5 u2) H6 H7)))))) (H1 n e1 u0 k0 (drop_gen_drop (Bind b) c0 (CHead 
-e1 k0 u0) u1 n H4)))))))))) (\lambda (f: F).(\lambda (n: nat).(\lambda (_: 
-((\forall (e1: C).(\forall (u3: T).(\forall (k: K).((drop n O (CHead c0 (Flat 
-f) u1) (CHead e1 k u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: 
-T).(drop n O (CHead c3 (Flat f) u2) (CHead e2 k u4)))) (\lambda (e2: 
-C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u3 
-u2)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4: 
-(drop (S n) O (CHead c0 (Flat f) u1) (CHead e1 k0 u0))).(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 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 c3 (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 (H5: (drop (S n) O c3 (CHead x0 k0 
-x1))).(\lambda (H6: (wcpr0 e1 x0)).(\lambda (H7: (pr0 u0 x1)).(ex3_2_intro C 
-T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead c3 (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 c3 (CHead 
-x0 k0 x1) H5 u2) H6 H7)))))) (H1 (S n) e1 u0 k0 (drop_gen_drop (Flat f) c0 
-(CHead e1 k0 u0) u1 n H4)))))))))) 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 (c0: C).(\lambda (c3: C).(\lambda (H0: (wcpr0 c0 
-c3)).(\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 c0 (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 c3 k u2) (CHead 
-e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead 
-c0 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 c3 k u2) 
-(CHead e1 k0 u0))).(let H4 \def (match (drop_gen_refl (CHead c3 k u2) (CHead 
-e1 k0 u0) H3) return (\lambda (c: C).(\lambda (_: (eq ? ? c)).((eq C c (CHead 
-e1 k0 u0)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop O O (CHead 
-c0 k u1) (CHead e2 k0 u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) 
-(\lambda (_: C).(\lambda (u2: T).(pr0 u2 u0))))))) with [refl_equal 
-\Rightarrow (\lambda (H3: (eq C (CHead c3 k u2) (CHead e1 k0 u0))).(let H4 
-\def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: C).T) with 
-[(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c3 k u2) 
-(CHead e1 k0 u0) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e 
-return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k _) 
-\Rightarrow k])) (CHead c3 k u2) (CHead e1 k0 u0) H3) in ((let H6 \def 
-(f_equal C C (\lambda (e: C).(match e return (\lambda (_: C).C) with [(CSort 
-_) \Rightarrow c3 | (CHead c _ _) \Rightarrow c])) (CHead c3 k u2) (CHead e1 
-k0 u0) H3) in (eq_ind C e1 (\lambda (_: C).((eq K k k0) \to ((eq T u2 u0) \to 
-(ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop O O (CHead c0 k u1) (CHead 
-e2 k0 u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: 
-C).(\lambda (u2: T).(pr0 u2 u0))))))) (\lambda (H7: (eq K k k0)).(eq_ind K k0 
-(\lambda (k: K).((eq T u2 u0) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: 
-T).(drop O O (CHead c0 k u1) (CHead e2 k0 u2)))) (\lambda (e2: C).(\lambda 
-(_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 u0)))))) 
-(\lambda (H8: (eq T u2 u0)).(eq_ind T u0 (\lambda (_: T).(ex3_2 C T (\lambda 
-(e2: C).(\lambda (u2: T).(drop O O (CHead c0 k0 u1) (CHead e2 k0 u2)))) 
-(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda 
-(u2: T).(pr0 u2 u0))))) (let H9 \def (eq_ind T u2 (\lambda (t: T).(pr0 u1 t)) 
-H2 u0 H8) in (let H10 \def (eq_ind C c3 (\lambda (c: C).(wcpr0 c0 c)) H0 e1 
-H6) in (ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(drop O O (CHead c0 
-k0 u1) (CHead e2 k0 u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) 
-(\lambda (_: C).(\lambda (u2: T).(pr0 u2 u0))) c0 u1 (drop_refl (CHead c0 k0 
-u1)) H10 H9))) u2 (sym_eq T u2 u0 H8))) k (sym_eq K k k0 H7))) c3 (sym_eq C 
-c3 e1 H6))) H5)) H4)))]) in (H4 (refl_equal C (CHead e1 k0 u0)))))))) (K_ind 
-(\lambda (k0: K).(\forall (n: nat).(((\forall (e1: C).(\forall (u3: 
-T).(\forall (k: K).((drop n O (CHead c3 k0 u2) (CHead e1 k u3)) \to (ex3_2 C 
-T (\lambda (e2: C).(\lambda (u2: T).(drop n O (CHead c0 k0 u1) (CHead e2 k 
-u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: 
-C).(\lambda (u2: T).(pr0 u2 u3))))))))) \to (\forall (e1: C).(\forall (u3: 
-T).(\forall (k1: K).((drop (S n) O (CHead c3 k0 u2) (CHead e1 k1 u3)) \to 
-(ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop (S n) O (CHead c0 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 (k: K).((drop n 
-O (CHead c3 (Bind b) u2) (CHead e1 k u3)) \to (ex3_2 C T (\lambda (e2: 
-C).(\lambda (u2: T).(drop n O (CHead c0 (Bind b) u1) (CHead e2 k u2)))) 
-(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda 
-(u2: T).(pr0 u2 u3)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: 
-K).(\lambda (H4: (drop (S n) O (CHead c3 (Bind b) u2) (CHead e1 k0 
-u0))).(ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop n O c0 (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 c0 (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 (H5: (drop n O c0 (CHead x0 
-k0 x1))).(\lambda (H6: (wcpr0 x0 e1)).(\lambda (H7: (pr0 x1 u0)).(ex3_2_intro 
-C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead c0 (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 c0 (CHead 
-x0 k0 x1) H5 u1) H6 H7)))))) (H1 n e1 u0 k0 (drop_gen_drop (Bind b) c3 (CHead 
-e1 k0 u0) u2 n H4)))))))))) (\lambda (f: F).(\lambda (n: nat).(\lambda (_: 
-((\forall (e1: C).(\forall (u3: T).(\forall (k: K).((drop n O (CHead c3 (Flat 
-f) u2) (CHead e1 k u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: 
-T).(drop n O (CHead c0 (Flat f) u1) (CHead e2 k u2)))) (\lambda (e2: 
-C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 
-u3)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4: 
-(drop (S n) O (CHead c3 (Flat f) u2) (CHead e1 k0 u0))).(ex3_2_ind C T 
-(\lambda (e2: C).(\lambda (u3: T).(drop (S n) O c0 (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 c0 (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 (H5: (drop (S n) O c0 (CHead x0 k0 
-x1))).(\lambda (H6: (wcpr0 x0 e1)).(\lambda (H7: (pr0 x1 u0)).(ex3_2_intro C 
-T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead c0 (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 c0 (CHead 
-x0 k0 x1) H5 u1) H6 H7)))))) (H1 (S n) e1 u0 k0 (drop_gen_drop (Flat f) c3 
-(CHead e1 k0 u0) u2 n H4)))))))))) 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 (c0: C).(\lambda (c3: C).(\lambda (H0: (wcpr0 c0 
-c3)).(\lambda (H1: ((\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 c3 (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 c0 k u1) (CHead e1 
-k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c3 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 c0 k u1) 
-(CHead e1 k0 u0))).((match k return (\lambda (k1: K).((clear (CHead c0 k1 u1) 
-(CHead e1 k0 u0)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O 
-(CHead c3 k1 u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 
-e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))))) with [(Bind b) 
-\Rightarrow (\lambda (H4: (clear (CHead c0 (Bind b) u1) (CHead e1 k0 
-u0))).(let H5 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: 
-C).C) with [(CSort _) \Rightarrow e1 | (CHead c _ _) \Rightarrow c])) (CHead 
-e1 k0 u0) (CHead c0 (Bind b) u1) (clear_gen_bind b c0 (CHead e1 k0 u0) u1 
-H4)) in ((let H6 \def (f_equal C K (\lambda (e: C).(match e return (\lambda 
-(_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _) \Rightarrow k])) 
-(CHead e1 k0 u0) (CHead c0 (Bind b) u1) (clear_gen_bind b c0 (CHead e1 k0 u0) 
-u1 H4)) in ((let H7 \def (f_equal C T (\lambda (e: C).(match e return 
-(\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow 
-t])) (CHead e1 k0 u0) (CHead c0 (Bind b) u1) (clear_gen_bind b c0 (CHead e1 
-k0 u0) u1 H4)) in (\lambda (H8: (eq K k0 (Bind b))).(\lambda (H9: (eq C e1 
-c0)).(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) 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 c3 (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 c0 (\lambda (c: C).(ex3_2 C T 
-(\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 (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 c3 (Bind b) u2) (CHead e2 (Bind b) u3)))) 
-(\lambda (e2: C).(\lambda (_: T).(wcpr0 c0 e2))) (\lambda (_: C).(\lambda 
-(u3: T).(pr0 u1 u3))) c3 u2 (getl_refl b c3 u2) H0 H2) e1 H9) u0 H7) k0 
-H8)))) H6)) H5))) | (Flat f) \Rightarrow (\lambda (H4: (clear (CHead c0 (Flat 
-f) u1) (CHead e1 k0 u0))).(let H5 \def (H1 O e1 u0 k0 (getl_intro O c0 (CHead 
-e1 k0 u0) c0 (drop_refl c0) (clear_gen_flat f c0 (CHead e1 k0 u0) u1 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 e1 e2))) (\lambda (_: 
-C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: 
-T).(getl O (CHead c3 (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 c3 (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 c3 (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 c3 (CHead x0 k0 x1) O H6 f 
-u2) H7 H8)))))) H5)))]) (getl_gen_O (CHead c0 k u1) (CHead e1 k0 u0) H3)))))) 
-(K_ind (\lambda (k0: K).(\forall (n: nat).(((\forall (e1: C).(\forall (u3: 
-T).(\forall (k: K).((getl n (CHead c0 k0 u1) (CHead e1 k u3)) \to (ex3_2 C T 
-(\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c3 k0 u2) (CHead e2 k u4)))) 
-(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda 
-(u2: T).(pr0 u3 u2))))))))) \to (\forall (e1: C).(\forall (u3: T).(\forall 
-(k1: K).((getl (S n) (CHead c0 k0 u1) (CHead e1 k1 u3)) \to (ex3_2 C T 
-(\lambda (e2: C).(\lambda (u4: T).(getl (S n) (CHead c3 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 (k: K).((getl n 
-(CHead c0 (Bind b) u1) (CHead e1 k u3)) \to (ex3_2 C T (\lambda (e2: 
-C).(\lambda (u4: T).(getl n (CHead c3 (Bind b) u2) (CHead e2 k u4)))) 
-(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda 
-(u2: T).(pr0 u3 u2)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: 
-K).(\lambda (H4: (getl (S n) (CHead c0 (Bind b) u1) (CHead e1 k0 u0))).(let 
-H5 \def (H1 n e1 u0 k0 (getl_gen_S (Bind b) c0 (CHead e1 k0 u0) u1 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 e1 e2))) (\lambda (_: 
-C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: 
-T).(getl (S n) (CHead c3 (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 c3 (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 c3 (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 c3 (CHead 
-x0 k0 x1) H6 u2) H7 H8)))))) H5))))))))) (\lambda (f: F).(\lambda (n: 
-nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall (k: K).((getl n 
-(CHead c0 (Flat f) u1) (CHead e1 k u3)) \to (ex3_2 C T (\lambda (e2: 
-C).(\lambda (u4: T).(getl n (CHead c3 (Flat f) u2) (CHead e2 k u4)))) 
-(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda 
-(u2: T).(pr0 u3 u2)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: 
-K).(\lambda (H4: (getl (S n) (CHead c0 (Flat f) u1) (CHead e1 k0 u0))).(let 
-H5 \def (H1 (S n) e1 u0 k0 (getl_gen_S (Flat f) c0 (CHead e1 k0 u0) u1 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 e1 e2))) (\lambda (_: 
-C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: 
-T).(getl (S n) (CHead c3 (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) c3 (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 c3 
-(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 c3 (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 (c0: C).(\lambda (c3: C).(\lambda (H0: (wcpr0 c0 
-c3)).(\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 c0 (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 c3 k u2) (CHead e1 
-k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c0 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 c3 k u2) 
-(CHead e1 k0 u0))).((match k return (\lambda (k1: K).((clear (CHead c3 k1 u2) 
-(CHead e1 k0 u0)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O 
-(CHead c0 k1 u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 
-e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))))) with [(Bind b) 
-\Rightarrow (\lambda (H4: (clear (CHead c3 (Bind b) u2) (CHead e1 k0 
-u0))).(let H5 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: 
-C).C) with [(CSort _) \Rightarrow e1 | (CHead c _ _) \Rightarrow c])) (CHead 
-e1 k0 u0) (CHead c3 (Bind b) u2) (clear_gen_bind b c3 (CHead e1 k0 u0) u2 
-H4)) in ((let H6 \def (f_equal C K (\lambda (e: C).(match e return (\lambda 
-(_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _) \Rightarrow k])) 
-(CHead e1 k0 u0) (CHead c3 (Bind b) u2) (clear_gen_bind b c3 (CHead e1 k0 u0) 
-u2 H4)) in ((let H7 \def (f_equal C T (\lambda (e: C).(match e return 
-(\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow 
-t])) (CHead e1 k0 u0) (CHead c3 (Bind b) u2) (clear_gen_bind b c3 (CHead e1 
-k0 u0) u2 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 c0 (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 c0 (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 c3 (\lambda (c: C).(ex3_2 C T 
-(\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c0 (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 c0 (Bind b) u1) (CHead e2 (Bind b) u3)))) 
-(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 c3))) (\lambda (_: C).(\lambda 
-(u3: T).(pr0 u3 u2))) c0 u1 (getl_refl b c0 u1) H0 H2) e1 H9) u0 H7) k0 
-H8)))) H6)) H5))) | (Flat f) \Rightarrow (\lambda (H4: (clear (CHead c3 (Flat 
-f) u2) (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) u2 H4))) in 
-(ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(getl O c0 (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 c0 (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 c0 (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 c0 (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 c0 (CHead x0 k0 x1) O H6 f 
-u1) H7 H8)))))) H5)))]) (getl_gen_O (CHead c3 k u2) (CHead e1 k0 u0) H3)))))) 
-(K_ind (\lambda (k0: K).(\forall (n: nat).(((\forall (e1: C).(\forall (u3: 
-T).(\forall (k: K).((getl n (CHead c3 k0 u2) (CHead e1 k u3)) \to (ex3_2 C T 
-(\lambda (e2: C).(\lambda (u2: T).(getl n (CHead c0 k0 u1) (CHead e2 k u2)))) 
-(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda 
-(u2: T).(pr0 u2 u3))))))))) \to (\forall (e1: C).(\forall (u3: T).(\forall 
-(k1: K).((getl (S n) (CHead c3 k0 u2) (CHead e1 k1 u3)) \to (ex3_2 C T 
-(\lambda (e2: C).(\lambda (u4: T).(getl (S n) (CHead c0 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 (k: K).((getl n 
-(CHead c3 (Bind b) u2) (CHead e1 k u3)) \to (ex3_2 C T (\lambda (e2: 
-C).(\lambda (u2: T).(getl n (CHead c0 (Bind b) u1) (CHead e2 k u2)))) 
-(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda 
-(u2: T).(pr0 u2 u3)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: 
-K).(\lambda (H4: (getl (S n) (CHead c3 (Bind b) u2) (CHead e1 k0 u0))).(let 
-H5 \def (H1 n e1 u0 k0 (getl_gen_S (Bind b) c3 (CHead e1 k0 u0) u2 n H4)) in 
-(ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(getl n c0 (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 c0 (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 c0 (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 c0 (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 c0 (CHead 
-x0 k0 x1) H6 u1) H7 H8)))))) H5))))))))) (\lambda (f: F).(\lambda (n: 
-nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall (k: K).((getl n 
-(CHead c3 (Flat f) u2) (CHead e1 k u3)) \to (ex3_2 C T (\lambda (e2: 
-C).(\lambda (u2: T).(getl n (CHead c0 (Flat f) u1) (CHead e2 k u2)))) 
-(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda 
-(u2: T).(pr0 u2 u3)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: 
-K).(\lambda (H4: (getl (S n) (CHead c3 (Flat f) u2) (CHead e1 k0 u0))).(let 
-H5 \def (H1 (S n) e1 u0 k0 (getl_gen_S (Flat f) c3 (CHead e1 k0 u0) u2 n H4)) 
-in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) c0 (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 c0 (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) c0 (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 c0 
-(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 c0 (CHead x0 k0 x1) H6 u1) H7 H8)))))) H5))))))))) k) h)))))))))) c2 c1 
-H))).
-
-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)))))))))).
-
-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)).(let H0 \def (match H return (\lambda (c0: C).(\lambda (t: T).(\lambda 
-(t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 c) \to ((eq T t (TSort n)) \to 
-((eq T t0 x) \to (eq T x (TSort n))))))))) with [(pr2_free c0 t1 t2 H0) 
-\Rightarrow (\lambda (H1: (eq C c0 c)).(\lambda (H2: (eq T t1 (TSort 
-n))).(\lambda (H3: (eq T t2 x)).(eq_ind C c (\lambda (_: C).((eq T t1 (TSort 
-n)) \to ((eq T t2 x) \to ((pr0 t1 t2) \to (eq T x (TSort n)))))) (\lambda 
-(H4: (eq T t1 (TSort n))).(eq_ind T (TSort n) (\lambda (t: T).((eq T t2 x) 
-\to ((pr0 t t2) \to (eq T x (TSort n))))) (\lambda (H5: (eq T t2 x)).(eq_ind 
-T x (\lambda (t: T).((pr0 (TSort n) t) \to (eq T x (TSort n)))) (\lambda (H6: 
-(pr0 (TSort n) x)).(let H7 \def (eq_ind T x (\lambda (t: T).(pr2 c (TSort n) 
-t)) H (TSort n) (pr0_gen_sort x n H6)) in (eq_ind_r T (TSort n) (\lambda (t: 
-T).(eq T t (TSort n))) (refl_equal T (TSort n)) x (pr0_gen_sort x n H6)))) t2 
-(sym_eq T t2 x H5))) t1 (sym_eq T t1 (TSort n) H4))) c0 (sym_eq C c0 c H1) H2 
-H3 H0)))) | (pr2_delta c0 d u i H0 t1 t2 H1 t H2) \Rightarrow (\lambda (H3: 
-(eq C c0 c)).(\lambda (H4: (eq T t1 (TSort n))).(\lambda (H5: (eq T t 
-x)).(eq_ind C c (\lambda (c: C).((eq T t1 (TSort n)) \to ((eq T t x) \to 
-((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t1 t2) \to ((subst0 i u t2 t) 
-\to (eq T x (TSort n)))))))) (\lambda (H6: (eq T t1 (TSort n))).(eq_ind T 
-(TSort n) (\lambda (t0: T).((eq T t x) \to ((getl i c (CHead d (Bind Abbr) 
-u)) \to ((pr0 t0 t2) \to ((subst0 i u t2 t) \to (eq T x (TSort n))))))) 
-(\lambda (H7: (eq T t x)).(eq_ind T x (\lambda (t0: T).((getl i c (CHead d 
-(Bind Abbr) u)) \to ((pr0 (TSort n) t2) \to ((subst0 i u t2 t0) \to (eq T x 
-(TSort n)))))) (\lambda (_: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H9: 
-(pr0 (TSort n) t2)).(\lambda (H10: (subst0 i u t2 x)).(let H11 \def (eq_ind T 
-t2 (\lambda (t: T).(subst0 i u t x)) H10 (TSort n) (pr0_gen_sort t2 n H9)) in 
-(subst0_gen_sort u x i n H11 (eq T x (TSort n))))))) t (sym_eq T t x H7))) t1 
-(sym_eq T t1 (TSort n) H6))) c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in 
-(H0 (refl_equal C c) (refl_equal T (TSort n)) (refl_equal T x)))))).
-
-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)).(let H0 \def (match H return (\lambda (c0: C).(\lambda (t: T).(\lambda 
-(t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 c) \to ((eq T t (TLRef n)) \to 
-((eq T t0 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))))))))))))) with [(pr2_free c0 t1 t2 H0) 
-\Rightarrow (\lambda (H1: (eq C c0 c)).(\lambda (H2: (eq T t1 (TLRef 
-n))).(\lambda (H3: (eq T t2 x)).(eq_ind C c (\lambda (_: C).((eq T t1 (TLRef 
-n)) \to ((eq T t2 x) \to ((pr0 t1 t2) \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)))))))))) (\lambda (H4: (eq T 
-t1 (TLRef n))).(eq_ind T (TLRef n) (\lambda (t: T).((eq T t2 x) \to ((pr0 t 
-t2) \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))))))))) (\lambda (H5: (eq T t2 x)).(eq_ind T x (\lambda 
-(t: T).((pr0 (TLRef n) t) \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)))))))) (\lambda (H6: (pr0 (TLRef 
-n) x)).(let H7 \def (eq_ind T x (\lambda (t: T).(pr2 c (TLRef n) t)) H (TLRef 
-n) (pr0_gen_lref x n H6)) in (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 c (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 c (CHead d (Bind Abbr) u)))) (\lambda (_: 
-C).(\lambda (u: T).(eq T (TLRef n) (lift (S n) O u))))) (refl_equal T (TLRef 
-n))) x (pr0_gen_lref x n H6)))) t2 (sym_eq T t2 x H5))) t1 (sym_eq T t1 
-(TLRef n) H4))) c0 (sym_eq C c0 c H1) H2 H3 H0)))) | (pr2_delta c0 d u i H0 
-t1 t2 H1 t H2) \Rightarrow (\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq T t1 
-(TLRef n))).(\lambda (H5: (eq T t x)).(eq_ind C c (\lambda (c1: C).((eq T t1 
-(TLRef n)) \to ((eq T t x) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 
-t1 t2) \to ((subst0 i u t2 t) \to (or (eq T x (TLRef n)) (ex2_2 C T (\lambda 
-(d0: C).(\lambda (u0: T).(getl n c (CHead d0 (Bind Abbr) u0)))) (\lambda (_: 
-C).(\lambda (u0: T).(eq T x (lift (S n) O u0)))))))))))) (\lambda (H6: (eq T 
-t1 (TLRef n))).(eq_ind T (TLRef n) (\lambda (t0: T).((eq T t x) \to ((getl i 
-c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t2) \to ((subst0 i u t2 t) \to (or 
-(eq T x (TLRef n)) (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl n c 
-(CHead d0 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T x (lift 
-(S n) O u0))))))))))) (\lambda (H7: (eq T t x)).(eq_ind T x (\lambda (t0: 
-T).((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 (TLRef n) t2) \to ((subst0 i 
-u t2 t0) \to (or (eq T x (TLRef n)) (ex2_2 C T (\lambda (d0: C).(\lambda (u0: 
-T).(getl n c (CHead d0 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: 
-T).(eq T x (lift (S n) O u0)))))))))) (\lambda (H8: (getl i c (CHead d (Bind 
-Abbr) u))).(\lambda (H9: (pr0 (TLRef n) t2)).(\lambda (H10: (subst0 i u t2 
-x)).(let H11 \def (eq_ind T t2 (\lambda (t: T).(subst0 i u t x)) H10 (TLRef 
-n) (pr0_gen_lref t2 n H9)) in (and_ind (eq nat n i) (eq T x (lift (S n) O u)) 
-(or (eq T x (TLRef n)) (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl n c 
-(CHead d0 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T x (lift 
-(S n) O u0)))))) (\lambda (H12: (eq nat n i)).(\lambda (H13: (eq T x (lift (S 
-n) O u))).(let H14 \def (eq_ind_r nat i (\lambda (n: nat).(getl n c (CHead d 
-(Bind Abbr) u))) H8 n H12) in (let H15 \def (eq_ind T x (\lambda (t: T).(pr2 
-c (TLRef n) t)) H (lift (S n) O u) H13) in (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 c (CHead d0 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: 
-T).(eq T t0 (lift (S n) O u0))))))) (or_intror (eq T (lift (S n) O u) (TLRef 
-n)) (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl n c (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 c 
-(CHead d0 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T (lift (S 
-n) O u) (lift (S n) O u0)))) d u H14 (refl_equal T (lift (S n) O u)))) x 
-H13))))) (subst0_gen_lref u x i n H11)))))) t (sym_eq T t x H7))) t1 (sym_eq 
-T t1 (TLRef n) H6))) c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0 
-(refl_equal C c) (refl_equal T (TLRef n)) (refl_equal T x)))))).
-
-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)).(let H0 \def (match H return 
-(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t 
-t0)).((eq C c0 c) \to ((eq T t (THead (Bind Abst) u1 t1)) \to ((eq T t0 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))))))))))))) with [(pr2_free c0 t0 t2 H0) \Rightarrow (\lambda (H1: (eq 
-C c0 c)).(\lambda (H2: (eq T t0 (THead (Bind Abst) u1 t1))).(\lambda (H3: (eq 
-T t2 x)).(eq_ind C c (\lambda (_: C).((eq T t0 (THead (Bind Abst) u1 t1)) \to 
-((eq T t2 x) \to ((pr0 t0 t2) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 
-c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: 
-T).(pr2 (CHead c (Bind b) u) t1 t3)))))))))) (\lambda (H4: (eq T t0 (THead 
-(Bind Abst) u1 t1))).(eq_ind T (THead (Bind Abst) u1 t1) (\lambda (t: T).((eq 
-T t2 x) \to ((pr0 t t2) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq 
-T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 
-u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 
-(CHead c (Bind b) u) t1 t3))))))))) (\lambda (H5: (eq T t2 x)).(eq_ind T x 
-(\lambda (t: T).((pr0 (THead (Bind Abst) u1 t1) t) \to (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall 
-(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3)))))))) (\lambda (H6: 
-(pr0 (THead (Bind Abst) u1 t1) x)).(ex3_2_ind T T (\lambda (u2: T).(\lambda 
-(t3: T).(eq T x (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 x (THead (Bind Abst) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3)))))) 
-(\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T x (THead (Bind Abst) x0 
-x1))).(\lambda (H8: (pr0 u1 x0)).(\lambda (H9: (pr0 t1 x1)).(let H10 \def 
-(eq_ind T x (\lambda (t: T).(pr2 c (THead (Bind Abst) u1 t1) t)) H (THead 
-(Bind Abst) x0 x1) H7) in (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 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (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 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
-(u: T).(pr2 (CHead c (Bind b) u) t1 t3))))) x0 x1 (refl_equal T (THead (Bind 
-Abst) x0 x1)) (pr2_free c u1 x0 H8) (\lambda (b: B).(\lambda (u: T).(pr2_free 
-(CHead c (Bind b) u) t1 x1 H9)))) x H7))))))) (pr0_gen_abst u1 t1 x H6))) t2 
-(sym_eq T t2 x H5))) t0 (sym_eq T t0 (THead (Bind Abst) u1 t1) H4))) c0 
-(sym_eq C c0 c H1) H2 H3 H0)))) | (pr2_delta c0 d u i H0 t0 t2 H1 t H2) 
-\Rightarrow (\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq T t0 (THead (Bind 
-Abst) u1 t1))).(\lambda (H5: (eq T t x)).(eq_ind C c (\lambda (c1: C).((eq T 
-t0 (THead (Bind Abst) u1 t1)) \to ((eq T t x) \to ((getl i c1 (CHead d (Bind 
-Abbr) u)) \to ((pr0 t0 t2) \to ((subst0 i u t2 t) \to (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))))))))) (\lambda 
-(H6: (eq T t0 (THead (Bind Abst) u1 t1))).(eq_ind T (THead (Bind Abst) u1 t1) 
-(\lambda (t3: T).((eq T t x) \to ((getl i c (CHead d (Bind Abbr) u)) \to 
-((pr0 t3 t2) \to ((subst0 i u t2 t) \to (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t4: T).(eq T x (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c (Bind b) u0) t1 t4))))))))))) (\lambda (H7: (eq T t 
-x)).(eq_ind T x (\lambda (t3: T).((getl i c (CHead d (Bind Abbr) u)) \to 
-((pr0 (THead (Bind Abst) u1 t1) t2) \to ((subst0 i u t2 t3) \to (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t4: T).(eq T x (THead (Bind Abst) u2 t4)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda 
-(t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 
-t4)))))))))) (\lambda (H8: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H9: 
-(pr0 (THead (Bind Abst) u1 t1) t2)).(\lambda (H10: (subst0 i u t2 
-x)).(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 x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 
-c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: 
-T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\lambda (x0: T).(\lambda (x1: 
-T).(\lambda (H11: (eq T t2 (THead (Bind Abst) x0 x1))).(\lambda (H12: (pr0 u1 
-x0)).(\lambda (H13: (pr0 t1 x1)).(let H14 \def (eq_ind T t2 (\lambda (t: 
-T).(subst0 i u t x)) H10 (THead (Bind Abst) x0 x1) H11) in (or3_ind (ex2 T 
-(\lambda (u2: T).(eq T x (THead (Bind Abst) u2 x1))) (\lambda (u2: T).(subst0 
-i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T x (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 x (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 x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\lambda (H15: (ex2 T (\lambda 
-(u2: T).(eq T x (THead (Bind Abst) u2 x1))) (\lambda (u2: T).(subst0 i u x0 
-u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (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 x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\lambda (x2: T).(\lambda 
-(H16: (eq T x (THead (Bind Abst) x2 x1))).(\lambda (H17: (subst0 i u x0 
-x2)).(let H18 \def (eq_ind T x (\lambda (t: T).(pr2 c (THead (Bind Abst) u1 
-t1) t)) H (THead (Bind Abst) x2 x1) H16) in (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 c u1 u2))) 
-(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead 
-c (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 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))))) x2 x1 
-(refl_equal T (THead (Bind Abst) x2 x1)) (pr2_delta c d u i H8 u1 x0 H12 x2 
-H17) (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c (Bind b) u0) t1 x1 
-H13)))) x H16))))) H15)) (\lambda (H15: (ex2 T (\lambda (t2: T).(eq T x 
-(THead (Bind Abst) x0 t2))) (\lambda (t2: T).(subst0 (s (Bind Abst) i) u x1 
-t2)))).(ex2_ind T (\lambda (t3: T).(eq T x (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 x (THead (Bind Abst) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\lambda (x2: 
-T).(\lambda (H16: (eq T x (THead (Bind Abst) x0 x2))).(\lambda (H17: (subst0 
-(s (Bind Abst) i) u x1 x2)).(let H18 \def (eq_ind T x (\lambda (t: T).(pr2 c 
-(THead (Bind Abst) u1 t1) t)) H (THead (Bind Abst) x0 x2) H16) in (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 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c (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 c u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead 
-c (Bind b) u0) t1 t3))))) x0 x2 (refl_equal T (THead (Bind Abst) x0 x2)) 
-(pr2_free c u1 x0 H12) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c 
-(Bind b) u0) d u (S i) (getl_head (Bind b) i c (CHead d (Bind Abbr) u) H8 u0) 
-t1 x1 H13 x2 H17)))) x H16))))) H15)) (\lambda (H15: (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t2: 
-T).(subst0 (s (Bind Abst) i) u x1 t2))))).(ex3_2_ind T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T x (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 x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\lambda (x2: T).(\lambda (x3: 
-T).(\lambda (H16: (eq T x (THead (Bind Abst) x2 x3))).(\lambda (H17: (subst0 
-i u x0 x2)).(\lambda (H18: (subst0 (s (Bind Abst) i) u x1 x3)).(let H19 \def 
-(eq_ind T x (\lambda (t: T).(pr2 c (THead (Bind Abst) u1 t1) t)) H (THead 
-(Bind Abst) x2 x3) H16) in (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 c u1 u2))) (\lambda (_: 
-T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (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 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))))) x2 x3 
-(refl_equal T (THead (Bind Abst) x2 x3)) (pr2_delta c d u i H8 u1 x0 H12 x2 
-H17) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c (Bind b) u0) d u (S 
-i) (getl_head (Bind b) i c (CHead d (Bind Abbr) u) H8 u0) t1 x1 H13 x3 
-H18)))) x H16))))))) H15)) (subst0_gen_head (Bind Abst) u x0 x1 x i 
-H14)))))))) (pr0_gen_abst u1 t1 t2 H9))))) t (sym_eq T t x H7))) t0 (sym_eq T 
-t0 (THead (Bind Abst) u1 t1) H6))) c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) 
-in (H0 (refl_equal C c) (refl_equal T (THead (Bind Abst) u1 t1)) (refl_equal 
-T x))))))).
-
-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)).(let H0 \def (match H return 
-(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t 
-t0)).((eq C c0 c) \to ((eq T t (THead (Flat Cast) u1 t1)) \to ((eq T t0 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))))))))) with [(pr2_free c0 
-t0 t2 H0) \Rightarrow (\lambda (H1: (eq C c0 c)).(\lambda (H2: (eq T t0 
-(THead (Flat Cast) u1 t1))).(\lambda (H3: (eq T t2 x)).(eq_ind C c (\lambda 
-(_: C).((eq T t0 (THead (Flat Cast) u1 t1)) \to ((eq T t2 x) \to ((pr0 t0 t2) 
-\to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat 
-Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)))))) (\lambda (H4: (eq T t0 
-(THead (Flat Cast) u1 t1))).(eq_ind T (THead (Flat Cast) u1 t1) (\lambda (t: 
-T).((eq T t2 x) \to ((pr0 t t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c 
-t1 x))))) (\lambda (H5: (eq T t2 x)).(eq_ind T x (\lambda (t: T).((pr0 (THead 
-(Flat Cast) u1 t1) t) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 
-c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)))) 
-(\lambda (H6: (pr0 (THead (Flat Cast) u1 t1) x)).(or_ind (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
-t3)))) (pr0 t1 x) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x 
-(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)) (\lambda (H7: 
-(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))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda 
-(t3: T).(eq T x (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 x (THead (Flat Cast) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)) (\lambda (x0: T).(\lambda (x1: 
-T).(\lambda (H8: (eq T x (THead (Flat Cast) x0 x1))).(\lambda (H9: (pr0 u1 
-x0)).(\lambda (H10: (pr0 t1 x1)).(let H11 \def (eq_ind T x (\lambda (t: 
-T).(pr2 c (THead (Flat Cast) u1 t1) t)) H (THead (Flat Cast) x0 x1) H8) in 
-(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 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 
-t3)))) (pr2 c 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 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 
-t3)))) (pr2 c 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 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c t1 t3))) x0 x1 (refl_equal T (THead (Flat Cast) x0 
-x1)) (pr2_free c u1 x0 H9) (pr2_free c t1 x1 H10))) x H8))))))) H7)) (\lambda 
-(H7: (pr0 t1 x)).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq 
-T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 
-u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x) 
-(pr2_free c t1 x H7))) (pr0_gen_cast u1 t1 x H6))) t2 (sym_eq T t2 x H5))) t0 
-(sym_eq T t0 (THead (Flat Cast) u1 t1) H4))) c0 (sym_eq C c0 c H1) H2 H3 
-H0)))) | (pr2_delta c0 d u i H0 t0 t2 H1 t H2) \Rightarrow (\lambda (H3: (eq 
-C c0 c)).(\lambda (H4: (eq T t0 (THead (Flat Cast) u1 t1))).(\lambda (H5: (eq 
-T t x)).(eq_ind C c (\lambda (c1: C).((eq T t0 (THead (Flat Cast) u1 t1)) \to 
-((eq T t x) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t2) \to 
-((subst0 i u t2 t) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
-x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 
-u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)))))))) 
-(\lambda (H6: (eq T t0 (THead (Flat Cast) u1 t1))).(eq_ind T (THead (Flat 
-Cast) u1 t1) (\lambda (t3: T).((eq T t x) \to ((getl i c (CHead d (Bind Abbr) 
-u)) \to ((pr0 t3 t2) \to ((subst0 i u t2 t) \to (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t4: T).(eq T x (THead (Flat Cast) u2 t4)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c t1 
-t4)))) (pr2 c t1 x))))))) (\lambda (H7: (eq T t x)).(eq_ind T x (\lambda (t3: 
-T).((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 (THead (Flat Cast) u1 t1) 
-t2) \to ((subst0 i u t2 t3) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t4: 
-T).(eq T x (THead (Flat Cast) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 
-c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c t1 t4)))) (pr2 c t1 
-x)))))) (\lambda (H8: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H9: (pr0 
-(THead (Flat Cast) u1 t1) t2)).(\lambda (H10: (subst0 i u t2 x)).(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 x (THead (Flat Cast) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 
-t3)))) (pr2 c t1 x)) (\lambda (H11: (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 (t2: T).(pr0 t1 t2))))).(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 x 
-(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)) (\lambda (x0: 
-T).(\lambda (x1: T).(\lambda (H12: (eq T t2 (THead (Flat Cast) x0 
-x1))).(\lambda (H13: (pr0 u1 x0)).(\lambda (H14: (pr0 t1 x1)).(let H15 \def 
-(eq_ind T t2 (\lambda (t: T).(subst0 i u t x)) H10 (THead (Flat Cast) x0 x1) 
-H12) in (or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead (Flat Cast) u2 x1))) 
-(\lambda (u2: T).(subst0 i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T x (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 x (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 x (THead (Flat Cast) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)) (\lambda (H16: (ex2 T (\lambda (u2: 
-T).(eq T x (THead (Flat Cast) u2 x1))) (\lambda (u2: T).(subst0 i u x0 
-u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (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 x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c 
-t1 x)) (\lambda (x2: T).(\lambda (H17: (eq T x (THead (Flat Cast) x2 
-x1))).(\lambda (H18: (subst0 i u x0 x2)).(or_introl (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 
-t3)))) (pr2 c t1 x) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
-x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 
-u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3))) x2 x1 H17 (pr2_delta c 
-d u i H8 u1 x0 H13 x2 H18) (pr2_free c t1 x1 H14)))))) H16)) (\lambda (H16: 
-(ex2 T (\lambda (t2: T).(eq T x (THead (Flat Cast) x0 t2))) (\lambda (t2: 
-T).(subst0 (s (Flat Cast) i) u x1 t2)))).(ex2_ind T (\lambda (t3: T).(eq T x 
-(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 x (THead (Flat 
-Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)) (\lambda (x2: T).(\lambda 
-(H17: (eq T x (THead (Flat Cast) x0 x2))).(\lambda (H18: (subst0 (s (Flat 
-Cast) i) u x1 x2)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 
-c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x) 
-(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) 
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c t1 t3))) x0 x2 H17 (pr2_free c u1 x0 H13) 
-(pr2_delta c d u i H8 t1 x1 H14 x2 H18)))))) H16)) (\lambda (H16: (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: 
-T).(\lambda (t2: T).(subst0 (s (Flat Cast) i) u x1 t2))))).(ex3_2_ind T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T x (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 x (THead (Flat Cast) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)) (\lambda (x2: T).(\lambda (x3: 
-T).(\lambda (H17: (eq T x (THead (Flat Cast) x2 x3))).(\lambda (H18: (subst0 
-i u x0 x2)).(\lambda (H19: (subst0 (s (Flat Cast) i) u x1 x3)).(or_introl 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x) (ex3_2_intro T T (\lambda 
-(u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 
-t3))) x2 x3 H17 (pr2_delta c d u i H8 u1 x0 H13 x2 H18) (pr2_delta c d u i H8 
-t1 x1 H14 x3 H19)))))))) H16)) (subst0_gen_head (Flat Cast) u x0 x1 x i 
-H15)))))))) H11)) (\lambda (H11: (pr0 t1 t2)).(or_intror (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 
-t3)))) (pr2 c t1 x) (pr2_delta c d u i H8 t1 t2 H11 x H10))) (pr0_gen_cast u1 
-t1 t2 H9))))) t (sym_eq T t x H7))) t0 (sym_eq T t0 (THead (Flat Cast) u1 t1) 
-H6))) c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C c) 
-(refl_equal T (THead (Flat Cast) u1 t1)) (refl_equal T x))))))).
-
-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)).(let H0 \def (match H return (\lambda (c: C).(\lambda (t: 
-T).(\lambda (t0: T).(\lambda (_: (pr2 c t t0)).((eq C c (CSort n)) \to ((eq T 
-t t1) \to ((eq T t0 t2) \to (pr0 t1 t2)))))))) with [(pr2_free c t0 t3 H0) 
-\Rightarrow (\lambda (H1: (eq C c (CSort n))).(\lambda (H2: (eq T t0 
-t1)).(\lambda (H3: (eq T t3 t2)).(eq_ind C (CSort n) (\lambda (_: C).((eq T 
-t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr0 t1 t2))))) (\lambda (H4: 
-(eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to 
-(pr0 t1 t2)))) (\lambda (H5: (eq T t3 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 
-t1 t) \to (pr0 t1 t2))) (\lambda (H6: (pr0 t1 t2)).H6) t3 (sym_eq T t3 t2 
-H5))) t0 (sym_eq T t0 t1 H4))) c (sym_eq C c (CSort n) H1) H2 H3 H0)))) | 
-(pr2_delta c d u i H0 t0 t3 H1 t H2) \Rightarrow (\lambda (H3: (eq C c (CSort 
-n))).(\lambda (H4: (eq T t0 t1)).(\lambda (H5: (eq T t t2)).(eq_ind C (CSort 
-n) (\lambda (c0: C).((eq T t0 t1) \to ((eq T t t2) \to ((getl i c0 (CHead d 
-(Bind Abbr) u)) \to ((pr0 t0 t3) \to ((subst0 i u t3 t) \to (pr0 t1 t2))))))) 
-(\lambda (H6: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to 
-((getl i (CSort n) (CHead d (Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u 
-t3 t) \to (pr0 t1 t2)))))) (\lambda (H7: (eq T t t2)).(eq_ind T t2 (\lambda 
-(t4: T).((getl i (CSort n) (CHead d (Bind Abbr) u)) \to ((pr0 t1 t3) \to 
-((subst0 i u t3 t4) \to (pr0 t1 t2))))) (\lambda (H8: (getl i (CSort n) 
-(CHead d (Bind Abbr) u))).(\lambda (_: (pr0 t1 t3)).(\lambda (_: (subst0 i u 
-t3 t2)).(getl_gen_sort n i (CHead d (Bind Abbr) u) H8 (pr0 t1 t2))))) t 
-(sym_eq T t t2 H7))) t0 (sym_eq T t0 t1 H6))) c (sym_eq C c (CSort n) H3) H4 
-H5 H0 H1 H2))))]) in (H0 (refl_equal C (CSort n)) (refl_equal T t1) 
-(refl_equal T t2)))))).
-
-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)) (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 (c: C).(getl i c (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_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).(K_ind (\lambda (k0: K).((pr2 (CHead c k0 u) t1 t2) \to (pr2 c (THead 
-k0 u t1) (THead k0 u t2)))) (\lambda (b: B).(\lambda (H: (pr2 (CHead c (Bind 
-b) u) t1 t2)).(let H0 \def (match H return (\lambda (c0: C).(\lambda (t: 
-T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 (CHead c (Bind b) 
-u)) \to ((eq T t t1) \to ((eq T t0 t2) \to (pr2 c (THead (Bind b) u t1) 
-(THead (Bind b) u t2))))))))) with [(pr2_free c0 t0 t3 H0) \Rightarrow 
-(\lambda (H1: (eq C c0 (CHead c (Bind b) u))).(\lambda (H2: (eq T t0 
-t1)).(\lambda (H3: (eq T t3 t2)).(eq_ind C (CHead c (Bind b) u) (\lambda (_: 
-C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr2 c (THead (Bind 
-b) u t1) (THead (Bind b) u t2)))))) (\lambda (H4: (eq T t0 t1)).(eq_ind T t1 
-(\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to (pr2 c (THead (Bind b) u 
-t1) (THead (Bind b) u t2))))) (\lambda (H5: (eq T t3 t2)).(eq_ind T t2 
-(\lambda (t: T).((pr0 t1 t) \to (pr2 c (THead (Bind b) u t1) (THead (Bind b) 
-u t2)))) (\lambda (H6: (pr0 t1 t2)).(pr2_free c (THead (Bind b) u t1) (THead 
-(Bind b) u t2) (pr0_comp u u (pr0_refl u) t1 t2 H6 (Bind b)))) t3 (sym_eq T 
-t3 t2 H5))) t0 (sym_eq T t0 t1 H4))) c0 (sym_eq C c0 (CHead c (Bind b) u) H1) 
-H2 H3 H0)))) | (pr2_delta c0 d u0 i H0 t0 t3 H1 t H2) \Rightarrow (\lambda 
-(H3: (eq C c0 (CHead c (Bind b) u))).(\lambda (H4: (eq T t0 t1)).(\lambda 
-(H5: (eq T t t2)).(eq_ind C (CHead c (Bind b) u) (\lambda (c1: C).((eq T t0 
-t1) \to ((eq T t t2) \to ((getl i c1 (CHead d (Bind Abbr) u0)) \to ((pr0 t0 
-t3) \to ((subst0 i u0 t3 t) \to (pr2 c (THead (Bind b) u t1) (THead (Bind b) 
-u t2)))))))) (\lambda (H6: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T 
-t t2) \to ((getl i (CHead c (Bind b) u) (CHead d (Bind Abbr) u0)) \to ((pr0 
-t4 t3) \to ((subst0 i u0 t3 t) \to (pr2 c (THead (Bind b) u t1) (THead (Bind 
-b) u t2))))))) (\lambda (H7: (eq T t t2)).(eq_ind T t2 (\lambda (t4: 
-T).((getl i (CHead c (Bind b) u) (CHead d (Bind Abbr) u0)) \to ((pr0 t1 t3) 
-\to ((subst0 i u0 t3 t4) \to (pr2 c (THead (Bind b) u t1) (THead (Bind b) u 
-t2)))))) (\lambda (H8: (getl i (CHead c (Bind b) u) (CHead d (Bind Abbr) 
-u0))).(\lambda (H9: (pr0 t1 t3)).(\lambda (H10: (subst0 i u0 t3 t2)).((match 
-i return (\lambda (n: nat).((getl n (CHead c (Bind b) u) (CHead d (Bind Abbr) 
-u0)) \to ((subst0 n u0 t3 t2) \to (pr2 c (THead (Bind b) u t1) (THead (Bind 
-b) u t2))))) with [O \Rightarrow (\lambda (H11: (getl O (CHead c (Bind b) u) 
-(CHead d (Bind Abbr) u0))).(\lambda (H12: (subst0 O u0 t3 t2)).(let H \def 
-(f_equal C C (\lambda (e: C).(match e return (\lambda (_: C).C) with [(CSort 
-_) \Rightarrow d | (CHead c _ _) \Rightarrow c])) (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) H11))) in ((let H13 
-\def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: C).B) with 
-[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k return 
-(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (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) H11))) in ((let H14 \def (f_equal C T (\lambda (e: C).(match e 
-return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) 
-\Rightarrow t])) (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) H11))) in (\lambda (H15: (eq B Abbr b)).(\lambda 
-(_: (eq C d c)).(let H17 \def (eq_ind T u0 (\lambda (t: T).(subst0 O t t3 
-t2)) H12 u H14) in (eq_ind B Abbr (\lambda (b: B).(pr2 c (THead (Bind b) u 
-t1) (THead (Bind b) u t2))) (pr2_free c (THead (Bind Abbr) u t1) (THead (Bind 
-Abbr) u t2) (pr0_delta u u (pr0_refl u) t1 t3 H9 t2 H17)) b H15))))) H13)) 
-H)))) | (S n) \Rightarrow (\lambda (H11: (getl (S n) (CHead c (Bind b) u) 
-(CHead d (Bind Abbr) u0))).(\lambda (H12: (subst0 (S n) u0 t3 t2)).(pr2_delta 
-c d u0 (r (Bind b) n) (getl_gen_S (Bind b) c (CHead d (Bind Abbr) u0) u n 
-H11) (THead (Bind b) u t1) (THead (Bind b) u t3) (pr0_comp u u (pr0_refl u) 
-t1 t3 H9 (Bind b)) (THead (Bind b) u t2) (subst0_snd (Bind b) u0 t2 t3 (r 
-(Bind b) n) H12 u))))]) H8 H10)))) t (sym_eq T t t2 H7))) t0 (sym_eq T t0 t1 
-H6))) c0 (sym_eq C c0 (CHead c (Bind b) u) H3) H4 H5 H0 H1 H2))))]) in (H0 
-(refl_equal C (CHead c (Bind b) u)) (refl_equal T t1) (refl_equal T t2))))) 
-(\lambda (f: F).(\lambda (H: (pr2 (CHead c (Flat f) u) t1 t2)).(let H0 \def 
-(match H return (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda 
-(_: (pr2 c0 t t0)).((eq C c0 (CHead c (Flat f) u)) \to ((eq T t t1) \to ((eq 
-T t0 t2) \to (pr2 c (THead (Flat f) u t1) (THead (Flat f) u t2))))))))) with 
-[(pr2_free c0 t0 t3 H0) \Rightarrow (\lambda (H1: (eq C c0 (CHead c (Flat f) 
-u))).(\lambda (H2: (eq T t0 t1)).(\lambda (H3: (eq T t3 t2)).(eq_ind C (CHead 
-c (Flat f) u) (\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 
-t3) \to (pr2 c (THead (Flat f) u t1) (THead (Flat f) u t2)))))) (\lambda (H4: 
-(eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to 
-(pr2 c (THead (Flat f) u t1) (THead (Flat f) u t2))))) (\lambda (H5: (eq T t3 
-t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t1 t) \to (pr2 c (THead (Flat f) u 
-t1) (THead (Flat f) u t2)))) (\lambda (H6: (pr0 t1 t2)).(pr2_free c (THead 
-(Flat f) u t1) (THead (Flat f) u t2) (pr0_comp u u (pr0_refl u) t1 t2 H6 
-(Flat f)))) t3 (sym_eq T t3 t2 H5))) t0 (sym_eq T t0 t1 H4))) c0 (sym_eq C c0 
-(CHead c (Flat f) u) H1) H2 H3 H0)))) | (pr2_delta c0 d u0 i H0 t0 t3 H1 t 
-H2) \Rightarrow (\lambda (H3: (eq C c0 (CHead c (Flat f) u))).(\lambda (H4: 
-(eq T t0 t1)).(\lambda (H5: (eq T t t2)).(eq_ind C (CHead c (Flat f) u) 
-(\lambda (c1: C).((eq T t0 t1) \to ((eq T t t2) \to ((getl i c1 (CHead d 
-(Bind Abbr) u0)) \to ((pr0 t0 t3) \to ((subst0 i u0 t3 t) \to (pr2 c (THead 
-(Flat f) u t1) (THead (Flat f) u t2)))))))) (\lambda (H6: (eq T t0 
-t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i (CHead c (Flat 
-f) u) (CHead d (Bind Abbr) u0)) \to ((pr0 t4 t3) \to ((subst0 i u0 t3 t) \to 
-(pr2 c (THead (Flat f) u t1) (THead (Flat f) u t2))))))) (\lambda (H7: (eq T 
-t t2)).(eq_ind T t2 (\lambda (t4: T).((getl i (CHead c (Flat f) u) (CHead d 
-(Bind Abbr) u0)) \to ((pr0 t1 t3) \to ((subst0 i u0 t3 t4) \to (pr2 c (THead 
-(Flat f) u t1) (THead (Flat f) u t2)))))) (\lambda (H8: (getl i (CHead c 
-(Flat f) u) (CHead d (Bind Abbr) u0))).(\lambda (H9: (pr0 t1 t3)).(\lambda 
-(H10: (subst0 i u0 t3 t2)).((match i return (\lambda (n: nat).((getl n (CHead 
-c (Flat f) u) (CHead d (Bind Abbr) u0)) \to ((subst0 n u0 t3 t2) \to (pr2 c 
-(THead (Flat f) u t1) (THead (Flat f) u t2))))) with [O \Rightarrow (\lambda 
-(H11: (getl O (CHead c (Flat f) u) (CHead d (Bind Abbr) u0))).(\lambda (H12: 
-(subst0 O u0 t3 t2)).(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) H11))) (THead (Flat 
-f) u t1) (THead (Flat f) u t3) (pr0_comp u u (pr0_refl u) t1 t3 H9 (Flat f)) 
-(THead (Flat f) u t2) (subst0_snd (Flat f) u0 t2 t3 O H12 u)))) | (S n) 
-\Rightarrow (\lambda (H11: (getl (S n) (CHead c (Flat f) u) (CHead d (Bind 
-Abbr) u0))).(\lambda (H12: (subst0 (S n) u0 t3 t2)).(pr2_delta c d u0 (r 
-(Flat f) n) (getl_gen_S (Flat f) c (CHead d (Bind Abbr) u0) u n H11) (THead 
-(Flat f) u t1) (THead (Flat f) u t3) (pr0_comp u u (pr0_refl u) t1 t3 H9 
-(Flat f)) (THead (Flat f) u t2) (subst0_snd (Flat f) u0 t2 t3 (r (Flat f) n) 
-H12 u))))]) H8 H10)))) t (sym_eq T t t2 H7))) t0 (sym_eq T t0 t1 H6))) c0 
-(sym_eq C c0 (CHead c (Flat f) u) H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal 
-C (CHead c (Flat f) u)) (refl_equal T t1) (refl_equal T t2))))) k))))).
-
-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)).(\lambda (c1: C).(\lambda (H0: (clear c1 c2)).(let H1 \def (match H 
-return (\lambda (c: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c t 
-t0)).((eq C c c2) \to ((eq T t t1) \to ((eq T t0 t2) \to (pr2 c1 t1 
-t2)))))))) with [(pr2_free c t0 t3 H1) \Rightarrow (\lambda (H2: (eq C c 
-c2)).(\lambda (H3: (eq T t0 t1)).(\lambda (H4: (eq T t3 t2)).(eq_ind C c2 
-(\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr2 c1 
-t1 t2))))) (\lambda (H5: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 
-t2) \to ((pr0 t t3) \to (pr2 c1 t1 t2)))) (\lambda (H6: (eq T t3 t2)).(eq_ind 
-T t2 (\lambda (t: T).((pr0 t1 t) \to (pr2 c1 t1 t2))) (\lambda (H7: (pr0 t1 
-t2)).(pr2_free c1 t1 t2 H7)) t3 (sym_eq T t3 t2 H6))) t0 (sym_eq T t0 t1 
-H5))) c (sym_eq C c c2 H2) H3 H4 H1)))) | (pr2_delta c d u i H1 t0 t3 H2 t 
-H3) \Rightarrow (\lambda (H4: (eq C c c2)).(\lambda (H5: (eq T t0 
-t1)).(\lambda (H6: (eq T t t2)).(eq_ind C c2 (\lambda (c0: C).((eq T t0 t1) 
-\to ((eq T t t2) \to ((getl i c0 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t3) 
-\to ((subst0 i u t3 t) \to (pr2 c1 t1 t2))))))) (\lambda (H7: (eq T t0 
-t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i c2 (CHead d 
-(Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u t3 t) \to (pr2 c1 t1 
-t2)))))) (\lambda (H8: (eq T t t2)).(eq_ind T t2 (\lambda (t4: T).((getl i c2 
-(CHead d (Bind Abbr) u)) \to ((pr0 t1 t3) \to ((subst0 i u t3 t4) \to (pr2 c1 
-t1 t2))))) (\lambda (H9: (getl i c2 (CHead d (Bind Abbr) u))).(\lambda (H10: 
-(pr0 t1 t3)).(\lambda (H11: (subst0 i u t3 t2)).(pr2_delta c1 d u i 
-(clear_getl_trans i c2 (CHead d (Bind Abbr) u) H9 c1 H0) t1 t3 H10 t2 H11)))) 
-t (sym_eq T t t2 H8))) t0 (sym_eq T t0 t1 H7))) c (sym_eq C c c2 H4) H5 H6 H1 
-H2 H3))))]) in (H1 (refl_equal C c2) (refl_equal T t1) (refl_equal T 
-t2)))))))).
-
-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)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (f: 
-F).(\forall (v: T).(pr2 (CHead c0 (Flat f) v) t t0)))))) (\lambda (c0: 
-C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda (f: 
-F).(\lambda (v: T).(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)).(\lambda 
-(f: F).(\lambda (v: 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_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)).(let H0 \def (match H 
-return (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 
-t t0)).((eq C c0 (CHead c (Bind b) v)) \to ((eq T t t1) \to ((eq T t0 t2) \to 
-(pr2 c (THead (Bind b) v t1) (THead (Bind b) v t2))))))))) with [(pr2_free c0 
-t0 t3 H0) \Rightarrow (\lambda (H1: (eq C c0 (CHead c (Bind b) v))).(\lambda 
-(H2: (eq T t0 t1)).(\lambda (H3: (eq T t3 t2)).(eq_ind C (CHead c (Bind b) v) 
-(\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr2 c 
-(THead (Bind b) v t1) (THead (Bind b) v t2)))))) (\lambda (H4: (eq T t0 
-t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to (pr2 c 
-(THead (Bind b) v t1) (THead (Bind b) v t2))))) (\lambda (H5: (eq T t3 
-t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t1 t) \to (pr2 c (THead (Bind b) v 
-t1) (THead (Bind b) v t2)))) (\lambda (H6: (pr0 t1 t2)).(pr2_free c (THead 
-(Bind b) v t1) (THead (Bind b) v t2) (pr0_comp v v (pr0_refl v) t1 t2 H6 
-(Bind b)))) t3 (sym_eq T t3 t2 H5))) t0 (sym_eq T t0 t1 H4))) c0 (sym_eq C c0 
-(CHead c (Bind b) v) H1) H2 H3 H0)))) | (pr2_delta c0 d u i H0 t0 t3 H1 t H2) 
-\Rightarrow (\lambda (H3: (eq C c0 (CHead c (Bind b) v))).(\lambda (H4: (eq T 
-t0 t1)).(\lambda (H5: (eq T t t2)).(eq_ind C (CHead c (Bind b) v) (\lambda 
-(c1: C).((eq T t0 t1) \to ((eq T t t2) \to ((getl i c1 (CHead d (Bind Abbr) 
-u)) \to ((pr0 t0 t3) \to ((subst0 i u t3 t) \to (pr2 c (THead (Bind b) v t1) 
-(THead (Bind b) v t2)))))))) (\lambda (H6: (eq T t0 t1)).(eq_ind T t1 
-(\lambda (t4: T).((eq T t t2) \to ((getl i (CHead c (Bind b) v) (CHead d 
-(Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u t3 t) \to (pr2 c (THead 
-(Bind b) v t1) (THead (Bind b) v t2))))))) (\lambda (H7: (eq T t t2)).(eq_ind 
-T t2 (\lambda (t4: T).((getl i (CHead c (Bind b) v) (CHead d (Bind Abbr) u)) 
-\to ((pr0 t1 t3) \to ((subst0 i u t3 t4) \to (pr2 c (THead (Bind b) v t1) 
-(THead (Bind b) v t2)))))) (\lambda (H8: (getl i (CHead c (Bind b) v) (CHead 
-d (Bind Abbr) u))).(\lambda (H9: (pr0 t1 t3)).(\lambda (H10: (subst0 i u t3 
-t2)).(let H_x \def (getl_gen_bind b c (CHead d (Bind Abbr) u) v i H8) in (let 
-H \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 t1) (THead 
-(Bind b) v t2)) (\lambda (H11: (land (eq nat i O) (eq C (CHead d (Bind Abbr) 
-u) (CHead c (Bind b) v)))).(and_ind (eq nat i O) (eq C (CHead d (Bind Abbr) 
-u) (CHead c (Bind b) v)) (pr2 c (THead (Bind b) v t1) (THead (Bind b) v t2)) 
-(\lambda (H12: (eq nat i O)).(\lambda (H13: (eq C (CHead d (Bind Abbr) u) 
-(CHead c (Bind b) v))).(let H14 \def (f_equal C C (\lambda (e: C).(match e 
-return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c _ _) 
-\Rightarrow c])) (CHead d (Bind Abbr) u) (CHead c (Bind b) v) H13) in ((let 
-H15 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: C).B) with 
-[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k return 
-(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow 
-Abbr])])) (CHead d (Bind Abbr) u) (CHead c (Bind b) v) H13) in ((let H16 \def 
-(f_equal C T (\lambda (e: C).(match e return (\lambda (_: C).T) with [(CSort 
-_) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abbr) u) 
-(CHead c (Bind b) v) H13) in (\lambda (H17: (eq B Abbr b)).(\lambda (_: (eq C 
-d c)).(let H19 \def (eq_ind nat i (\lambda (n: nat).(subst0 n u t3 t2)) H10 O 
-H12) in (let H20 \def (eq_ind T u (\lambda (t: T).(subst0 O t t3 t2)) H19 v 
-H16) in (eq_ind B Abbr (\lambda (b: B).(pr2 c (THead (Bind b) v t1) (THead 
-(Bind b) v t2))) (pr2_free c (THead (Bind Abbr) v t1) (THead (Bind Abbr) v 
-t2) (pr0_delta v v (pr0_refl v) t1 t3 H9 t2 H20)) b H17)))))) H15)) H14)))) 
-H11)) (\lambda (H11: (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 t1) (THead (Bind b) v t2)) (\lambda (x: 
-nat).(\lambda (H12: (eq nat i (S x))).(\lambda (H13: (getl x c (CHead d (Bind 
-Abbr) u))).(let H14 \def (f_equal nat nat (\lambda (e: nat).e) i (S x) H12) 
-in (let H15 \def (eq_ind nat i (\lambda (n: nat).(subst0 n u t3 t2)) H10 (S 
-x) H14) in (pr2_head_2 c v t1 t2 (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 H13) t1 t3 H9 t2 H15))))))) H11)) H)))))) t (sym_eq T t t2 
-H7))) t0 (sym_eq T t0 t1 H6))) c0 (sym_eq C c0 (CHead c (Bind b) v) H3) H4 H5 
-H0 H1 H2))))]) in (H0 (refl_equal C (CHead c (Bind b) v)) (refl_equal T t1) 
-(refl_equal T t2)))))))).
-
-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)).(let H0 \def (match H 
-return (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 
-t t0)).((eq C c0 (CHead c (Flat f) v)) \to ((eq T t t1) \to ((eq T t0 t2) \to 
-(pr2 c t1 t2)))))))) with [(pr2_free c0 t0 t3 H0) \Rightarrow (\lambda (H1: 
-(eq C c0 (CHead c (Flat f) v))).(\lambda (H2: (eq T t0 t1)).(\lambda (H3: (eq 
-T t3 t2)).(eq_ind C (CHead c (Flat f) v) (\lambda (_: C).((eq T t0 t1) \to 
-((eq T t3 t2) \to ((pr0 t0 t3) \to (pr2 c t1 t2))))) (\lambda (H4: (eq T t0 
-t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to (pr2 c t1 
-t2)))) (\lambda (H5: (eq T t3 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t1 t) 
-\to (pr2 c t1 t2))) (\lambda (H6: (pr0 t1 t2)).(pr2_free c t1 t2 H6)) t3 
-(sym_eq T t3 t2 H5))) t0 (sym_eq T t0 t1 H4))) c0 (sym_eq C c0 (CHead c (Flat 
-f) v) H1) H2 H3 H0)))) | (pr2_delta c0 d u i H0 t0 t3 H1 t H2) \Rightarrow 
-(\lambda (H3: (eq C c0 (CHead c (Flat f) v))).(\lambda (H4: (eq T t0 
-t1)).(\lambda (H5: (eq T t t2)).(eq_ind C (CHead c (Flat f) v) (\lambda (c1: 
-C).((eq T t0 t1) \to ((eq T t t2) \to ((getl i c1 (CHead d (Bind Abbr) u)) 
-\to ((pr0 t0 t3) \to ((subst0 i u t3 t) \to (pr2 c t1 t2))))))) (\lambda (H6: 
-(eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i (CHead 
-c (Flat f) v) (CHead d (Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u t3 
-t) \to (pr2 c t1 t2)))))) (\lambda (H7: (eq T t t2)).(eq_ind T t2 (\lambda 
-(t4: T).((getl i (CHead c (Flat f) v) (CHead d (Bind Abbr) u)) \to ((pr0 t1 
-t3) \to ((subst0 i u t3 t4) \to (pr2 c t1 t2))))) (\lambda (H8: (getl i 
-(CHead c (Flat f) v) (CHead d (Bind Abbr) u))).(\lambda (H9: (pr0 t1 
-t3)).(\lambda (H10: (subst0 i u t3 t2)).(let H_y \def (getl_gen_flat f c 
-(CHead d (Bind Abbr) u) v i H8) in (pr2_delta c d u i H_y t1 t3 H9 t2 
-H10))))) t (sym_eq T t t2 H7))) t0 (sym_eq T t0 t1 H6))) c0 (sym_eq C c0 
-(CHead c (Flat f) v) H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C (CHead c 
-(Flat f) v)) (refl_equal T t1) (refl_equal T t2)))))))).
-
-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)).(let H1 \def (match H0 return (\lambda (c0: C).(\lambda (t: T).(\lambda 
-(t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 e) \to ((eq T t t1) \to ((eq T 
-t0 t2) \to (pr2 c (lift h d t1) (lift h d t2))))))))) with [(pr2_free c0 t0 
-t3 H1) \Rightarrow (\lambda (H2: (eq C c0 e)).(\lambda (H3: (eq T t0 
-t1)).(\lambda (H4: (eq T t3 t2)).(eq_ind C e (\lambda (_: C).((eq T t0 t1) 
-\to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr2 c (lift h d t1) (lift h d 
-t2)))))) (\lambda (H5: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 
-t2) \to ((pr0 t t3) \to (pr2 c (lift h d t1) (lift h d t2))))) (\lambda (H6: 
-(eq T t3 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t1 t) \to (pr2 c (lift h d 
-t1) (lift h d t2)))) (\lambda (H7: (pr0 t1 t2)).(pr2_free c (lift h d t1) 
-(lift h d t2) (pr0_lift t1 t2 H7 h d))) t3 (sym_eq T t3 t2 H6))) t0 (sym_eq T 
-t0 t1 H5))) c0 (sym_eq C c0 e H2) H3 H4 H1)))) | (pr2_delta c0 d0 u i H1 t0 
-t3 H2 t H3) \Rightarrow (\lambda (H4: (eq C c0 e)).(\lambda (H5: (eq T t0 
-t1)).(\lambda (H6: (eq T t t2)).(eq_ind C e (\lambda (c1: C).((eq T t0 t1) 
-\to ((eq T t t2) \to ((getl i c1 (CHead d0 (Bind Abbr) u)) \to ((pr0 t0 t3) 
-\to ((subst0 i u t3 t) \to (pr2 c (lift h d t1) (lift h d t2)))))))) (\lambda 
-(H7: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i e 
-(CHead d0 (Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u t3 t) \to (pr2 c 
-(lift h d t1) (lift h d t2))))))) (\lambda (H8: (eq T t t2)).(eq_ind T t2 
-(\lambda (t4: T).((getl i e (CHead d0 (Bind Abbr) u)) \to ((pr0 t1 t3) \to 
-((subst0 i u t3 t4) \to (pr2 c (lift h d t1) (lift h d t2)))))) (\lambda (H9: 
-(getl i e (CHead d0 (Bind Abbr) u))).(\lambda (H10: (pr0 t1 t3)).(\lambda 
-(H11: (subst0 i u t3 t2)).(lt_le_e i d (pr2 c (lift h d t1) (lift h d t2)) 
-(\lambda (H0: (lt i d)).(let H \def (drop_getl_trans_le i d (le_S_n i d (le_S 
-(S i) d H0)) c e h H (CHead d0 (Bind Abbr) u) H9) 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 t1) (lift h d t2)) (\lambda (x0: 
-C).(\lambda (x1: C).(\lambda (H12: (drop i O c x0)).(\lambda (H13: (drop h 
-(minus d i) x0 x1)).(\lambda (H14: (clear x1 (CHead d0 (Bind Abbr) u))).(let 
-H15 \def (eq_ind nat (minus d i) (\lambda (n: nat).(drop h n x0 x1)) H13 (S 
-(minus d (S i))) (minus_x_Sy d i H0)) in (let H16 \def (drop_clear_S x1 x0 h 
-(minus d (S i)) H15 Abbr d0 u H14) 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 t1) (lift h d t2)) (\lambda (x: 
-C).(\lambda (H17: (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 H12 H17) (lift h d t1) (lift h d t3) (pr0_lift t1 t3 H10 h d) 
-(lift h d t2) (subst0_lift_lt t3 t2 u i H11 d H0 h))))) H16)))))))) H))) 
-(\lambda (H0: (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) H9 H0) (lift h d t1) (lift h d t3) (pr0_lift 
-t1 t3 H10 h d) (lift h d t2) (subst0_lift_ge t3 t2 u i h H11 d H0))))))) t 
-(sym_eq T t t2 H8))) t0 (sym_eq T t0 t1 H7))) c0 (sym_eq C c0 e H4) H5 H6 H1 
-H2 H3))))]) in (H1 (refl_equal C e) (refl_equal T t1) (refl_equal T 
-t2)))))))))).
-
-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)).(let H0 \def (match H return 
-(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t 
-t0)).((eq C c0 c) \to ((eq T t (THead (Flat Appl) u1 t1)) \to ((eq T t0 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)))))))))))))))) with [(pr2_free c0 
-t0 t2 H0) \Rightarrow (\lambda (H1: (eq C c0 c)).(\lambda (H2: (eq T t0 
-(THead (Flat Appl) u1 t1))).(\lambda (H3: (eq T t2 x)).(eq_ind C c (\lambda 
-(_: C).((eq T t0 (THead (Flat Appl) u1 t1)) \to ((eq T t2 x) \to ((pr0 t0 t2) 
-\to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat 
-Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c 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 x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 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 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 (H4: (eq T t0 
-(THead (Flat Appl) u1 t1))).(eq_ind T (THead (Flat Appl) u1 t1) (\lambda (t: 
-T).((eq T t2 x) \to ((pr0 t t2) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 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 x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 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 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 
-(H5: (eq T t2 x)).(eq_ind T x (\lambda (t: T).((pr0 (THead (Flat Appl) u1 t1) 
-t) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat 
-Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c 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 x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 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 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 (H6: (pr0 (THead 
-(Flat Appl) u1 t1) x)).(or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T x (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 x (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 x 
-(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 x (THead (Flat Appl) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(pr2 c 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 x 
-(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 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 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 (H7: (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))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x 
-(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 x (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 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 x (THead (Bind 
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c u1 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 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 (x0: T).(\lambda (x1: T).(\lambda (H: (eq T x 
-(THead (Flat Appl) x0 x1))).(\lambda (H8: (pr0 u1 x0)).(\lambda (H9: (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 c u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(pr2 c 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 c u1 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 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 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)))))))))) (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 c u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(pr2 c 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 c u1 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 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 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)))))))) (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 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c t1 t3))) x0 x1 (refl_equal T (THead (Flat Appl) x0 
-x1)) (pr2_free c u1 x0 H8) (pr2_free c t1 x1 H9))) x H)))))) H7)) (\lambda 
-(H7: (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))))))).(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 x (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 x (THead (Flat Appl) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(pr2 c 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 x 
-(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 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 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 (x0: T).(\lambda 
-(x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H: (eq T t1 (THead (Bind 
-Abst) x0 x1))).(\lambda (H8: (eq T x (THead (Bind Abbr) x2 x3))).(\lambda 
-(H9: (pr0 u1 x2)).(\lambda (H10: (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 c u1 
-u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 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 c u1 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 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 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)))))))))) (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 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c 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 c u1 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 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 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)))))))))) (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 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 
-(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 c u1 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 (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 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)))))))) (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 c u1 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))))))) x0 x1 x2 x3 (refl_equal T 
-(THead (Bind Abst) x0 x1)) (refl_equal T (THead (Bind Abbr) x2 x3)) (pr2_free 
-c u1 x2 H9) (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead c (Bind b) u) x1 
-x3 H10))))) t1 H) x 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 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))))))))).(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 x (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 x (THead 
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(pr2 c 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 x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 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 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 
-(x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: 
-T).(\lambda (x5: T).(\lambda (H: (not (eq B x0 Abst))).(\lambda (H8: (eq T t1 
-(THead (Bind x0) x1 x2))).(\lambda (H9: (eq T x (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)).(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 c u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(pr2 c 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 c u1 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 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 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)))))))))) (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 c u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(pr2 c 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 c u1 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 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 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)))))))))) (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 
-c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (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 c u1 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 (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 
-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)))))))) (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 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))))))) 
-x0 x1 x2 x5 x3 x4 H (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 c u1 
-x3 H10) (pr2_free c x1 x4 H11) (pr2_free (CHead c (Bind x0) x4) x2 x5 H12))) 
-t1 H8) x H9))))))))))))) H7)) (pr0_gen_appl u1 t1 x H6))) t2 (sym_eq T t2 x 
-H5))) t0 (sym_eq T t0 (THead (Flat Appl) u1 t1) H4))) c0 (sym_eq C c0 c H1) 
-H2 H3 H0)))) | (pr2_delta c0 d u i H0 t0 t2 H1 t H2) \Rightarrow (\lambda 
-(H3: (eq C c0 c)).(\lambda (H4: (eq T t0 (THead (Flat Appl) u1 t1))).(\lambda 
-(H5: (eq T t x)).(eq_ind C c (\lambda (c1: C).((eq T t0 (THead (Flat Appl) u1 
-t1)) \to ((eq T t x) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t0 
-t2) \to ((subst0 i u t2 t) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 
-c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 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 x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 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 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 (H6: (eq T t0 (THead (Flat Appl) u1 t1))).(eq_ind 
-T (THead (Flat Appl) u1 t1) (\lambda (t3: T).((eq T t x) \to ((getl i c 
-(CHead d (Bind Abbr) u)) \to ((pr0 t3 t2) \to ((subst0 i u t2 t) \to (or3 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x (THead (Flat Appl) u2 
-t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t4: T).(pr2 c 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 x (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c (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 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 (H7: (eq T t 
-x)).(eq_ind T x (\lambda (t3: T).((getl i c (CHead d (Bind Abbr) u)) \to 
-((pr0 (THead (Flat Appl) u1 t1) t2) \to ((subst0 i u t2 t3) \to (or3 (ex3_2 T 
-T (\lambda (u2: T).(\lambda (t4: T).(eq T x (THead (Flat Appl) u2 t4)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda 
-(t4: T).(pr2 c 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 x 
-(THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 
-(CHead c (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 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 (H8: (getl i c 
-(CHead d (Bind Abbr) u))).(\lambda (H9: (pr0 (THead (Flat Appl) u1 t1) 
-t2)).(\lambda (H10: (subst0 i u t2 x)).(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 x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 
-c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 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 x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 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 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 (H11: (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 (t2: T).(pr0 t1 t2))))).(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 x (THead (Flat 
-Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c 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 x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 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 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 (x0: T).(\lambda 
-(x1: T).(\lambda (H: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H12: (pr0 
-u1 x0)).(\lambda (H13: (pr0 t1 x1)).(let H14 \def (eq_ind T t2 (\lambda (t: 
-T).(subst0 i u t x)) H10 (THead (Flat Appl) x0 x1) H) in (or3_ind (ex2 T 
-(\lambda (u2: T).(eq T x (THead (Flat Appl) u2 x1))) (\lambda (u2: T).(subst0 
-i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T x (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 x (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 x (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 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 x (THead (Bind 
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c u1 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 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 (H15: (ex2 T (\lambda (u2: 
-T).(eq T x (THead (Flat Appl) u2 x1))) (\lambda (u2: T).(subst0 i u x0 
-u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (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 x (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 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 x (THead (Bind 
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c u1 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 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 (x2: T).(\lambda (H16: (eq T x 
-(THead (Flat Appl) x2 x1))).(\lambda (H17: (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 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c 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 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (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 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)))))))))) (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 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c 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 c u1 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 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 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)))))))) (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 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 
-t3))) x2 x1 (refl_equal T (THead (Flat Appl) x2 x1)) (pr2_delta c d u i H8 u1 
-x0 H12 x2 H17) (pr2_free c t1 x1 H13))) x H16)))) H15)) (\lambda (H15: (ex2 T 
-(\lambda (t2: T).(eq T x (THead (Flat Appl) x0 t2))) (\lambda (t2: T).(subst0 
-(s (Flat Appl) i) u x1 t2)))).(ex2_ind T (\lambda (t3: T).(eq T x (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 x (THead (Flat Appl) 
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c 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 x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 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 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 (x2: T).(\lambda 
-(H16: (eq T x (THead (Flat Appl) x0 x2))).(\lambda (H17: (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 c u1 u2))) (\lambda (_: 
-T).(\lambda (t4: T).(pr2 c 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 c u1 u2))))) (\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c (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 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)))))))))) 
-(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 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 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 c u1 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 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 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)))))))) (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 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c t1 t3))) x0 x2 (refl_equal T (THead (Flat Appl) x0 
-x2)) (pr2_free c u1 x0 H12) (pr2_delta c d u i H8 t1 x1 H13 x2 H17))) x 
-H16)))) H15)) (\lambda (H15: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq 
-T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u 
-x0 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) u x1 
-t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (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 x (THead (Flat Appl) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c 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 x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 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 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 (x2: T).(\lambda 
-(x3: T).(\lambda (H16: (eq T x (THead (Flat Appl) x2 x3))).(\lambda (H17: 
-(subst0 i u x0 x2)).(\lambda (H18: (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 c u1 u2))) (\lambda (_: T).(\lambda 
-(t4: T).(pr2 c 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 c u1 u2))))) (\lambda (_: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 
-(CHead c (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 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)))))))))) (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 c u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(pr2 c 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 c u1 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 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 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)))))))) (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 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c t1 t3))) x2 x3 (refl_equal T (THead (Flat Appl) x2 
-x3)) (pr2_delta c d u i H8 u1 x0 H12 x2 H17) (pr2_delta c d u i H8 t1 x1 H13 
-x3 H18))) x H16)))))) H15)) (subst0_gen_head (Flat Appl) u x0 x1 x i 
-H14)))))))) H11)) (\lambda (H11: (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 (t2: T).(pr0 z1 t2))))))).(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 x (THead (Flat Appl) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c 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 x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 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 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 (x0: T).(\lambda 
-(x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H: (eq T t1 (THead (Bind 
-Abst) x0 x1))).(\lambda (H12: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda 
-(H13: (pr0 u1 x2)).(\lambda (H14: (pr0 x1 x3)).(let H15 \def (eq_ind T t2 
-(\lambda (t: T).(subst0 i u t x)) H10 (THead (Bind Abbr) x2 x3) H12) in 
-(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t1: T).(or3 (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(pr2 c 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 x 
-(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 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 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)))))))))) (or3_ind (ex2 T (\lambda 
-(u2: T).(eq T x (THead (Bind Abbr) u2 x3))) (\lambda (u2: T).(subst0 i u x2 
-u2))) (ex2 T (\lambda (t3: T).(eq T x (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 x (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 x (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 
-(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 x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 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 (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 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 (H16: (ex2 T (\lambda (u2: T).(eq T x (THead 
-(Bind Abbr) u2 x3))) (\lambda (u2: T).(subst0 i u x2 u2)))).(ex2_ind T 
-(\lambda (u2: T).(eq T x (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 x (THead 
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(pr2 c (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 x (THead (Bind 
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c u1 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 (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 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 (x4: T).(\lambda 
-(H17: (eq T x (THead (Bind Abbr) x4 x3))).(\lambda (H18: (subst0 i u x2 
-x4)).(eq_ind_r T (THead (Bind Abbr) x4 x3) (\lambda (t1: T).(or3 (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T t1 (THead (Flat Appl) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(pr2 c (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 t1 (THead (Bind Abbr) u2 t3)))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 
-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 (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 t1 (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)))))))))) (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 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 
-(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 c u1 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 (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 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)))))))) (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 c u1 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))))))) x0 x1 x4 x3 (refl_equal T (THead (Bind Abst) x0 x1)) 
-(refl_equal T (THead (Bind Abbr) x4 x3)) (pr2_delta c d u i H8 u1 x2 H13 x4 
-H18) (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c (Bind b) u0) x1 x3 
-H14))))) x H17)))) H16)) (\lambda (H16: (ex2 T (\lambda (t2: T).(eq T x 
-(THead (Bind Abbr) x2 t2))) (\lambda (t2: T).(subst0 (s (Bind Abbr) i) u x3 
-t2)))).(ex2_ind T (\lambda (t3: T).(eq T x (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 x (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 
-(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 x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 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 (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 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 (x4: T).(\lambda (H17: (eq T x (THead (Bind Abbr) 
-x2 x4))).(\lambda (H18: (subst0 (s (Bind Abbr) i) u x3 x4)).(eq_ind_r T 
-(THead (Bind Abbr) x2 x4) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t1 (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 
-(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 t1 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 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 (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 t1 (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)))))))))) (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 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 
-(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 c u1 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 (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 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)))))))) (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 c u1 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))))))) x0 x1 x2 x4 (refl_equal T (THead (Bind Abst) x0 x1)) 
-(refl_equal T (THead (Bind Abbr) x2 x4)) (pr2_free c u1 x2 H13) (\lambda (b: 
-B).(\lambda (u0: T).(pr2_delta (CHead c (Bind b) u0) d u (S i) 
-(getl_clear_bind b (CHead c (Bind b) u0) c u0 (clear_bind b c u0) (CHead d 
-(Bind Abbr) u) i H8) x1 x3 H14 x4 H18))))) x H17)))) H16)) (\lambda (H16: 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 
-t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x2 u2))) (\lambda (_: 
-T).(\lambda (t2: T).(subst0 (s (Bind Abbr) i) u x3 t2))))).(ex3_2_ind T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T x (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 x (THead (Flat Appl) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(pr2 c (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 x (THead (Bind Abbr) u2 t3)))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 
-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 (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 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 (x4: T).(\lambda (x5: T).(\lambda (H17: (eq T x 
-(THead (Bind Abbr) x4 x5))).(\lambda (H18: (subst0 i u x2 x4)).(\lambda (H19: 
-(subst0 (s (Bind Abbr) i) u x3 x5)).(eq_ind_r T (THead (Bind Abbr) x4 x5) 
-(\lambda (t1: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t1 
-(THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(pr2 c (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 t1 (THead 
-(Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 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 (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 
-t1 (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)))))))))) (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 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 
-(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 c u1 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 (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 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)))))))) (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 c u1 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))))))) x0 x1 x4 x5 (refl_equal T (THead (Bind Abst) x0 x1)) 
-(refl_equal T (THead (Bind Abbr) x4 x5)) (pr2_delta c d u i H8 u1 x2 H13 x4 
-H18) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c (Bind b) u0) d u (S 
-i) (getl_clear_bind b (CHead c (Bind b) u0) c u0 (clear_bind b c u0) (CHead d 
-(Bind Abbr) u) i H8) x1 x3 H14 x5 H19))))) x H17)))))) H16)) (subst0_gen_head 
-(Bind Abbr) u x2 x3 x i H15)) t1 H)))))))))) H11)) (\lambda (H11: (ex6_6 B T 
-T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\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 (t2: T).(pr0 z1 t2))))))))).(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 x (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 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 x (THead (Bind 
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c u1 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 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 (x0: B).(\lambda (x1: 
-T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: 
-T).(\lambda (H: (not (eq B x0 Abst))).(\lambda (H12: (eq T t1 (THead (Bind 
-x0) x1 x2))).(\lambda (H13: (eq T t2 (THead (Bind x0) x4 (THead (Flat Appl) 
-(lift (S O) O x3) x5)))).(\lambda (H14: (pr0 u1 x3)).(\lambda (H15: (pr0 x1 
-x4)).(\lambda (H16: (pr0 x2 x5)).(let H17 \def (eq_ind T t2 (\lambda (t: 
-T).(subst0 i u t x)) H10 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O 
-x3) x5)) H13) in (eq_ind_r T (THead (Bind x0) x1 x2) (\lambda (t1: T).(or3 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c 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 x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 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 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)))))))))) (or3_ind (ex2 T (\lambda 
-(u2: T).(eq T x (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 x 
-(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 x (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 x (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 
-(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 x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 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 (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 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 (H18: (ex2 T (\lambda (u2: T).(eq T x (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 x (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 x (THead 
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(pr2 c (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 x (THead (Bind 
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c u1 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 (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 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 (x6: T).(\lambda 
-(H19: (eq T x (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) 
-x5)))).(\lambda (H20: (subst0 i u x4 x6)).(eq_ind_r T (THead (Bind x0) x6 
-(THead (Flat Appl) (lift (S O) O x3) x5)) (\lambda (t1: T).(or3 (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T t1 (THead (Flat Appl) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(pr2 c (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 t1 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 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 (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 t1 (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)))))))))) (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 c u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(pr2 c (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 c u1 
-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 (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 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)))))))) (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 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))))))) x0 x1 x2 x5 x3 x6 H (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 c u1 x3 H14) (pr2_delta c d u i H8 x1 x4 H15 x6 
-H20) (pr2_free (CHead c (Bind x0) x6) x2 x5 H16))) x H19)))) H18)) (\lambda 
-(H18: (ex2 T (\lambda (t2: T).(eq T x (THead (Bind x0) x4 t2))) (\lambda (t2: 
-T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5) 
-t2)))).(ex2_ind T (\lambda (t3: T).(eq T x (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 x (THead (Flat 
-Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c (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 x (THead (Bind 
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c u1 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 (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 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 (x6: T).(\lambda 
-(H19: (eq T x (THead (Bind x0) x4 x6))).(\lambda (H20: (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 (t1: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T t1 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (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 t1 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 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 (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 t1 (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)))))))))) (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 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 
-(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 c u1 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 (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 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 (H21: (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 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c (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 c u1 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 (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 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 (x: T).(\lambda (H22: (eq T x6 
-(THead (Flat Appl) x x5))).(\lambda (H23: (subst0 (s (Bind x0) i) u (lift (S 
-O) O x3) x)).(eq_ind_r T (THead (Flat Appl) x x5) (\lambda (t1: T).(or3 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 t1) 
-(THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(pr2 c (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 t1) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 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 (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 t1) (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)))))))))) (ex2_ind T (\lambda (t3: T).(eq T x 
-(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) x x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 
-(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) x x5)) (THead (Bind 
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c u1 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 (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) x 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 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 (x7: T).(\lambda (H24: (eq T x (lift (S O) O 
-x7))).(\lambda (H25: (subst0 (minus (s (Bind x0) i) (S O)) u x3 x7)).(let H26 
-\def (eq_ind nat (minus (s (Bind x0) i) (S O)) (\lambda (n: nat).(subst0 n u 
-x3 x7)) H25 i (s_arith1 x0 i)) in (eq_ind_r T (lift (S O) O x7) (\lambda (t1: 
-T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) 
-x4 (THead (Flat Appl) t1 x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 
-(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) t1 x5)) (THead (Bind 
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c u1 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 (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) t1 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 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)))))))))) (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 x7) x5)) (THead 
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(pr2 c (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 x7) x5)) (THead (Bind Abbr) u2 t3)))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 
-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 (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 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 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)))))))) (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 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 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))))))) x0 x1 x2 x5 x7 x4 H (refl_equal T (THead 
-(Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x4 (THead (Flat Appl) (lift 
-(S O) O x7) x5))) (pr2_delta c d u i H8 u1 x3 H14 x7 H26) (pr2_free c x1 x4 
-H15) (pr2_free (CHead c (Bind x0) x4) x2 x5 H16))) x H24))))) 
-(subst0_gen_lift_ge u x3 x (s (Bind x0) i) (S O) O H23 (le_S_n (S O) (S i) 
-(lt_le_S (S O) (S (S i)) (lt_n_S O (S i) (le_lt_n_Sm O i (le_O_n i))))))) x6 
-H22)))) H21)) (\lambda (H21: (ex2 T (\lambda (t2: T).(eq T x6 (THead (Flat 
-Appl) (lift (S O) O x3) t2))) (\lambda (t2: T).(subst0 (s (Flat Appl) (s 
-(Bind x0) i)) u x5 t2)))).(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 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 
-(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 c u1 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 (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 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 (x: T).(\lambda 
-(H22: (eq T x6 (THead (Flat Appl) (lift (S O) O x3) x))).(\lambda (H23: 
-(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 x)).(eq_ind_r T (THead (Flat 
-Appl) (lift (S O) O x3) x) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 t1) (THead (Flat Appl) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c (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 t1) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 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 (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 t1) (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)))))))))) (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) x)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 
-c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (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) x)) (THead 
-(Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 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 (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) 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)))))))) (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) 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))))))) 
-x0 x1 x2 x x3 x4 H (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T 
-(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x))) (pr2_free c u1 
-x3 H14) (pr2_free c x1 x4 H15) (pr2_delta (CHead c (Bind x0) x4) d u (S i) 
-(getl_clear_bind x0 (CHead c (Bind x0) x4) c x4 (clear_bind x0 c x4) (CHead d 
-(Bind Abbr) u) i H8) x2 x5 H16 x H23))) x6 H22)))) H21)) (\lambda (H21: 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x6 (THead (Flat Appl) u2 
-t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) i) u (lift (S O) 
-O x3) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Flat Appl) (s (Bind 
-x0) i)) u x5 t2))))).(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 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c (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 c u1 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 (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 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 (x7: T).(\lambda (x8: 
-T).(\lambda (H22: (eq T x6 (THead (Flat Appl) x7 x8))).(\lambda (H23: (subst0 
-(s (Bind x0) i) u (lift (S O) O x3) x7)).(\lambda (H24: (subst0 (s (Flat 
-Appl) (s (Bind x0) i)) u x5 x8)).(eq_ind_r T (THead (Flat Appl) x7 x8) 
-(\lambda (t1: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
-(THead (Bind x0) x4 t1) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 
-(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 t1) (THead (Bind Abbr) u2 t3)))))) (\lambda 
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 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 (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 t1) (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)))))))))) (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 c u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(pr2 c (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 c u1 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 (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 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 (x: T).(\lambda 
-(H25: (eq T x7 (lift (S O) O x))).(\lambda (H26: (subst0 (minus (s (Bind x0) 
-i) (S O)) u x3 x)).(let H27 \def (eq_ind nat (minus (s (Bind x0) i) (S O)) 
-(\lambda (n: nat).(subst0 n u x3 x)) H26 i (s_arith1 x0 i)) in (eq_ind_r T 
-(lift (S O) O x) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t1 x8)) (THead (Flat 
-Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c (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) t1 x8)) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 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 (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) t1 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 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)))))))))) (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 x) x8)) (THead (Flat 
-Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c (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 x) x8)) (THead (Bind Abbr) u2 t3)))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 
-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 (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 x) 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 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)))))))) 
-(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 x) 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 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))))))) x0 x1 x2 x8 x x4 H (refl_equal T (THead (Bind x0) x1 x2)) 
-(refl_equal T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x) x8))) 
-(pr2_delta c d u i H8 u1 x3 H14 x H27) (pr2_free c x1 x4 H15) (pr2_delta 
-(CHead c (Bind x0) x4) d u (S i) (getl_clear_bind x0 (CHead c (Bind x0) x4) c 
-x4 (clear_bind x0 c x4) (CHead d (Bind Abbr) u) i H8) x2 x5 H16 x8 H24))) x7 
-H25))))) (subst0_gen_lift_ge u x3 x7 (s (Bind x0) i) (S O) O H23 (le_S_n (S 
-O) (S i) (lt_le_S (S O) (S (S i)) (lt_n_S O (S i) (le_lt_n_Sm O i (le_O_n 
-i))))))) x6 H22)))))) H21)) (subst0_gen_head (Flat Appl) u (lift (S O) O x3) 
-x5 x6 (s (Bind x0) i) H20)) x H19)))) H18)) (\lambda (H18: (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind x0) u2 t2)))) (\lambda 
-(u2: T).(\lambda (_: T).(subst0 i u x4 u2))) (\lambda (_: T).(\lambda (t2: 
-T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5) 
-t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (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 x 
-(THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(pr2 c (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 x (THead (Bind 
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c u1 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 (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 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 (x6: T).(\lambda 
-(x7: T).(\lambda (H19: (eq T x (THead (Bind x0) x6 x7))).(\lambda (H20: 
-(subst0 i u x4 x6)).(\lambda (H21: (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 
-(t1: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t1 (THead 
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(pr2 c (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 t1 (THead (Bind 
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c u1 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 (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 t1 (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)))))))))) (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 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c (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 c u1 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 (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 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 (H22: (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 
-c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (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 c u1 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 (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 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 (x: T).(\lambda 
-(H23: (eq T x7 (THead (Flat Appl) x x5))).(\lambda (H24: (subst0 (s (Bind x0) 
-i) u (lift (S O) O x3) x)).(eq_ind_r T (THead (Flat Appl) x x5) (\lambda (t1: 
-T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) 
-x6 t1) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c 
-u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (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 t1) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 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 (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 t1) (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)))))))))) (ex2_ind T (\lambda (t3: 
-T).(eq T x (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) x x5)) (THead (Flat Appl) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(pr2 c (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) x x5)) 
-(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 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 (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) x 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 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 (x8: T).(\lambda (H25: (eq T x 
-(lift (S O) O x8))).(\lambda (H26: (subst0 (minus (s (Bind x0) i) (S O)) u x3 
-x8)).(let H27 \def (eq_ind nat (minus (s (Bind x0) i) (S O)) (\lambda (n: 
-nat).(subst0 n u x3 x8)) H26 i (s_arith1 x0 i)) in (eq_ind_r T (lift (S O) O 
-x8) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
-(THead (Bind x0) x6 (THead (Flat Appl) t1 x5)) (THead (Flat Appl) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(pr2 c (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) t1 x5)) 
-(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 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 (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) t1 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 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)))))))))) (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 x8) x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 
-(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 x8) x5)) 
-(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 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 (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 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 
-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)))))))) (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 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 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))))))) 
-x0 x1 x2 x5 x8 x6 H (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T 
-(THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x8) x5))) (pr2_delta c d 
-u i H8 u1 x3 H14 x8 H27) (pr2_delta c d u i H8 x1 x4 H15 x6 H20) (pr2_free 
-(CHead c (Bind x0) x6) x2 x5 H16))) x H25))))) (subst0_gen_lift_ge u x3 x (s 
-(Bind x0) i) (S O) O H24 (le_S_n (S O) (S i) (lt_le_S (S O) (S (S i)) (lt_n_S 
-O (S i) (le_lt_n_Sm O i (le_O_n i))))))) x7 H23)))) H22)) (\lambda (H22: (ex2 
-T (\lambda (t2: T).(eq T x7 (THead (Flat Appl) (lift (S O) O x3) t2))) 
-(\lambda (t2: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t2)))).(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 c u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(pr2 c (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 c u1 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 (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 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 (x: T).(\lambda (H23: (eq T x7 
-(THead (Flat Appl) (lift (S O) O x3) x))).(\lambda (H24: (subst0 (s (Flat 
-Appl) (s (Bind x0) i)) u x5 x)).(eq_ind_r T (THead (Flat Appl) (lift (S O) O 
-x3) x) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq 
-T (THead (Bind x0) x6 t1) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 
-(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 t1) (THead (Bind Abbr) u2 t3)))))) (\lambda 
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 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 (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 t1) (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)))))))))) (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) x)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 
-(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) x)) 
-(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 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 (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) 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)))))))) (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) 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))))))) 
-x0 x1 x2 x x3 x6 H (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T 
-(THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x))) (pr2_free c u1 
-x3 H14) (pr2_delta c d u i H8 x1 x4 H15 x6 H20) (pr2_delta (CHead c (Bind x0) 
-x6) d u (S i) (getl_clear_bind x0 (CHead c (Bind x0) x6) c x6 (clear_bind x0 
-c x6) (CHead d (Bind Abbr) u) i H8) x2 x5 H16 x H24))) x7 H23)))) H22)) 
-(\lambda (H22: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x7 (THead 
-(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) 
-i) u (lift (S O) O x3) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s 
-(Flat Appl) (s (Bind x0) i)) u x5 t2))))).(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 c u1 
-u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (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 c u1 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 (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 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 (x8: T).(\lambda (x9: 
-T).(\lambda (H23: (eq T x7 (THead (Flat Appl) x8 x9))).(\lambda (H24: (subst0 
-(s (Bind x0) i) u (lift (S O) O x3) x8)).(\lambda (H25: (subst0 (s (Flat 
-Appl) (s (Bind x0) i)) u x5 x9)).(eq_ind_r T (THead (Flat Appl) x8 x9) 
-(\lambda (t1: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
-(THead (Bind x0) x6 t1) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 
-(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 t1) (THead (Bind Abbr) u2 t3)))))) (\lambda 
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 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 (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 t1) (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)))))))))) (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 c u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(pr2 c (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 c u1 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 (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 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 (x: T).(\lambda 
-(H26: (eq T x8 (lift (S O) O x))).(\lambda (H27: (subst0 (minus (s (Bind x0) 
-i) (S O)) u x3 x)).(let H28 \def (eq_ind nat (minus (s (Bind x0) i) (S O)) 
-(\lambda (n: nat).(subst0 n u x3 x)) H27 i (s_arith1 x0 i)) in (eq_ind_r T 
-(lift (S O) O x) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t1 x9)) (THead (Flat 
-Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c (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) t1 x9)) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 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 (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) t1 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 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)))))))))) (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 x) x9)) (THead (Flat 
-Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c (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 x) x9)) (THead (Bind Abbr) u2 t3)))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 
-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 (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 x) 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 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)))))))) 
-(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 x) 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 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))))))) x0 x1 x2 x9 x x6 H (refl_equal T (THead (Bind x0) x1 x2)) 
-(refl_equal T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x) x9))) 
-(pr2_delta c d u i H8 u1 x3 H14 x H28) (pr2_delta c d u i H8 x1 x4 H15 x6 
-H20) (pr2_delta (CHead c (Bind x0) x6) d u (S i) (getl_clear_bind x0 (CHead c 
-(Bind x0) x6) c x6 (clear_bind x0 c x6) (CHead d (Bind Abbr) u) i H8) x2 x5 
-H16 x9 H25))) x8 H26))))) (subst0_gen_lift_ge u x3 x8 (s (Bind x0) i) (S O) O 
-H24 (le_S_n (S O) (S i) (lt_le_S (S O) (S (S i)) (lt_n_S O (S i) (le_lt_n_Sm 
-O i (le_O_n i))))))) x7 H23)))))) H22)) (subst0_gen_head (Flat Appl) u (lift 
-(S O) O x3) x5 x7 (s (Bind x0) i) H21)) x H19)))))) H18)) (subst0_gen_head 
-(Bind x0) u x4 (THead (Flat Appl) (lift (S O) O x3) x5) x i H17)) t1 
-H12)))))))))))))) H11)) (pr0_gen_appl u1 t1 t2 H9))))) t (sym_eq T t x H7))) 
-t0 (sym_eq T t0 (THead (Flat Appl) u1 t1) H6))) c0 (sym_eq C c0 c H3) H4 H5 
-H0 H1 H2))))]) in (H0 (refl_equal C c) (refl_equal T (THead (Flat Appl) u1 
-t1)) (refl_equal T x))))))).
-
-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)).(let H0 \def (match H return 
-(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t 
-t0)).((eq C c0 c) \to ((eq T t (THead (Bind Abbr) u1 t1)) \to ((eq T t0 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)))))))))))) with [(pr2_free c0 t0 t2 H0) \Rightarrow (\lambda (H1: (eq C c0 
-c)).(\lambda (H2: (eq T t0 (THead (Bind Abbr) u1 t1))).(\lambda (H3: (eq T t2 
-x)).(eq_ind C c (\lambda (_: C).((eq T t0 (THead (Bind Abbr) u1 t1)) \to ((eq 
-T t2 x) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 
-c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: 
-T).(pr2 (CHead c (Bind b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) 
-(\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t3))) (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 t3)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c 
-(Bind b) u) t1 (lift (S O) O x))))))))) (\lambda (H4: (eq T t0 (THead (Bind 
-Abbr) u1 t1))).(eq_ind T (THead (Bind Abbr) u1 t1) (\lambda (t: T).((eq T t2 
-x) \to ((pr0 t t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
-x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 
-u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: 
-T).(pr2 (CHead c (Bind b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) 
-(\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t3))) (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 t3)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c 
-(Bind b) u) t1 (lift (S O) O x)))))))) (\lambda (H5: (eq T t2 x)).(eq_ind T x 
-(\lambda (t: T).((pr0 (THead (Bind Abbr) u1 t1) t) \to (or (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 
-t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind 
-Abbr) u) t1 t3))) (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 t3)))))))) (\forall 
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-T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z x1)))) (ex_intro2 T 
-(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0) 
-t1 x1)) x0 H12 (pr2_delta (CHead c (Bind Abbr) x0) c x0 O (getl_refl Abbr c 
-x0) t1 x2 H13 x1 H14)))))))) (pr0_subst0_back x0 x2 x1 O H14 u1 H12))))) 
-H16)) (\lambda (H16: (ex2 T (\lambda (t2: T).(eq T x (THead (Bind Abbr) x0 
-t2))) (\lambda (t2: T).(subst0 (s (Bind Abbr) i) u x1 t2)))).(ex2_ind T 
-(\lambda (t3: T).(eq T x (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 x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 
-c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 
-u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0) t1 t3))) (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 t3)))))))) (\forall (b: B).(\forall (u0: 
-T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x))))) (\lambda (x3: 
-T).(\lambda (H17: (eq T x (THead (Bind Abbr) x0 x3))).(\lambda (H18: (subst0 
-(s (Bind Abbr) i) u x1 x3)).(ex2_ind T (\lambda (t1: T).(subst0 O u1 x2 t1)) 
-(\lambda (t1: T).(pr0 t1 x1)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 
-c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 
-u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0) t1 t3))) (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 t3)))))))) (\forall (b: B).(\forall (u0: 
-T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x))))) (\lambda (x4: 
-T).(\lambda (H19: (subst0 O u1 x2 x4)).(\lambda (H20: (pr0 x4 x1)).(or_introl 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind 
-b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 
-(CHead c (Bind Abbr) u0) t1 t3))) (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 
-t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 
-(lift (S O) O x)))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
-x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 
-u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: 
-T).(pr2 (CHead c (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) 
-(\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0) t1 t3))) (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 t3))))))) x0 x3 H17 (pr2_free c u1 x0 H12) (or3_intro2 
-(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 x3))) (ex2 T 
-(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0) 
-t1 x3))) (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 x3)))) (ex3_2_intro 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 x3))) x4 x1 (pr2_delta (CHead c (Bind 
-Abbr) u1) c u1 O (getl_refl Abbr c u1) t1 x2 H13 x4 H19) H20 (pr2_delta 
-(CHead c (Bind Abbr) u1) d u (S i) (getl_head (Bind Abbr) i c (CHead d (Bind 
-Abbr) u) H8 u1) x1 x1 (pr0_refl x1) x3 H18)))))))) (pr0_subst0_back x0 x2 x1 
-O H14 u1 H12))))) H16)) (\lambda (H16: (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Bind 
-Abbr) i) u x1 t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
-x (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 x (THead (Bind 
-Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind 
-b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 
-(CHead c (Bind Abbr) u0) t1 t3))) (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 
-t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 
-(lift (S O) O x))))) (\lambda (x3: T).(\lambda (x4: T).(\lambda (H17: (eq T x 
-(THead (Bind Abbr) x3 x4))).(\lambda (H18: (subst0 i u x0 x3)).(\lambda (H19: 
-(subst0 (s (Bind Abbr) i) u x1 x4)).(ex2_ind T (\lambda (t1: T).(subst0 O u1 
-x2 t1)) (\lambda (t1: T).(pr0 t1 x1)) (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 
-(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))) (ex2 T 
-(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0) 
-t1 t3))) (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 t3)))))))) (\forall (b: 
-B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x))))) 
-(\lambda (x5: T).(\lambda (H20: (subst0 O u1 x2 x5)).(\lambda (H21: (pr0 x5 
-x1)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead 
-(Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 
-(CHead c (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda 
-(u0: T).(pr2 (CHead c (Bind Abbr) u0) t1 t3))) (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 t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c 
-(Bind b) u0) t1 (lift (S O) O x)))) (ex3_2_intro T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 
-(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))) (ex2 T 
-(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0) 
-t1 t3))) (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 t3))))))) x3 x4 H17 
-(pr2_delta c d u i H8 u1 x0 H12 x3 H18) (or3_intro2 (\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c (Bind b) u0) t1 x4))) (ex2 T (\lambda (u0: T).(pr0 u1 
-u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0) t1 x4))) (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 x4)))) (ex3_2_intro 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 x4))) x5 x1 (pr2_delta (CHead c (Bind Abbr) u1) c u1 O 
-(getl_refl Abbr c u1) t1 x2 H13 x5 H20) H21 (pr2_delta (CHead c (Bind Abbr) 
-u1) d u (S i) (getl_head (Bind Abbr) i c (CHead d (Bind Abbr) u) H8 u1) x1 x1 
-(pr0_refl x1) x4 H19)))))))) (pr0_subst0_back x0 x2 x1 O H14 u1 H12))))))) 
-H16)) (subst0_gen_head (Bind Abbr) u x0 x1 x i H15)))))) H_x0)) H_x)))))) 
-H11)) (\lambda (H: (pr0 t1 (lift (S O) O t2))).(or_intror (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 
-(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))) (ex2 T 
-(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0) 
-t1 t3))) (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 t3)))))))) (\forall (b: 
-B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x)))) 
-(\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c (Bind b) u0) d u (S i) 
-(getl_head (Bind b) i c (CHead d (Bind Abbr) u) H8 u0) t1 (lift (S O) O t2) H 
-(lift (S O) O x) (subst0_lift_ge_S t2 x u i H10 O (le_O_n i))))))) 
-(pr0_gen_abbr u1 t1 t2 H9))))) t (sym_eq T t x H7))) t0 (sym_eq T t0 (THead 
-(Bind Abbr) u1 t1) H6))) c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0 
-(refl_equal C c) (refl_equal T (THead (Bind Abbr) u1 t1)) (refl_equal T 
-x))))))).
-
-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)).(let H0 \def (match H return 
-(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t 
-t0)).((eq C c0 c) \to ((eq T t (THead (Bind Void) u1 t1)) \to ((eq T t0 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)))))))))))) with [(pr2_free c0 t0 t2 H0) \Rightarrow (\lambda 
-(H1: (eq C c0 c)).(\lambda (H2: (eq T t0 (THead (Bind Void) u1 t1))).(\lambda 
-(H3: (eq T t2 x)).(eq_ind C c (\lambda (_: C).((eq T t0 (THead (Bind Void) u1 
-t1)) \to ((eq T t2 x) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall 
-(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3)))))) (\forall (b: 
-B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))))))) 
-(\lambda (H4: (eq T t0 (THead (Bind Void) u1 t1))).(eq_ind T (THead (Bind 
-Void) u1 t1) (\lambda (t: T).((eq T t2 x) \to ((pr0 t t2) \to (or (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3)))))) 
-(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O 
-x)))))))) (\lambda (H5: (eq T t2 x)).(eq_ind T x (\lambda (t: T).((pr0 (THead 
-(Bind Void) u1 t1) t) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 
-c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: 
-T).(pr2 (CHead c (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 
-(CHead c (Bind b) u) t1 (lift (S O) O x))))))) (\lambda (H6: (pr0 (THead 
-(Bind Void) u1 t1) x)).(or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T x (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 x)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind 
-Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
-t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 
-(lift (S O) O x))))) (\lambda (H7: (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))))).(ex3_2_ind T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T x (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 x 
-(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead 
-c (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind 
-b) u) t1 (lift (S O) O x))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H: 
-(eq T x (THead (Bind Void) x0 x1))).(\lambda (H8: (pr0 u1 x0)).(\lambda (H9: 
-(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 c u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3)))))) 
-(\forall (b: B).(\forall (u: T).(pr2 (CHead c (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 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
-(u: T).(pr2 (CHead c (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: 
-T).(pr2 (CHead c (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 c 
-u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: 
-T).(pr2 (CHead c (Bind b) u) t1 t3))))) x0 x1 (refl_equal T (THead (Bind 
-Void) x0 x1)) (pr2_free c u1 x0 H8) (\lambda (b: B).(\lambda (u: T).(pr2_free 
-(CHead c (Bind b) u) t1 x1 H9))))) x H)))))) H7)) (\lambda (H: (pr0 t1 (lift 
-(S O) O x))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x 
-(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead 
-c (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind 
-b) u) t1 (lift (S O) O x)))) (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead 
-c (Bind b) u) t1 (lift (S O) O x) H))))) (pr0_gen_void u1 t1 x H6))) t2 
-(sym_eq T t2 x H5))) t0 (sym_eq T t0 (THead (Bind Void) u1 t1) H4))) c0 
-(sym_eq C c0 c H1) H2 H3 H0)))) | (pr2_delta c0 d u i H0 t0 t2 H1 t H2) 
-\Rightarrow (\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq T t0 (THead (Bind 
-Void) u1 t1))).(\lambda (H5: (eq T t x)).(eq_ind C c (\lambda (c1: C).((eq T 
-t0 (THead (Bind Void) u1 t1)) \to ((eq T t x) \to ((getl i c1 (CHead d (Bind 
-Abbr) u)) \to ((pr0 t0 t2) \to ((subst0 i u t2 t) \to (or (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b: 
-B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x))))))))))) 
-(\lambda (H6: (eq T t0 (THead (Bind Void) u1 t1))).(eq_ind T (THead (Bind 
-Void) u1 t1) (\lambda (t3: T).((eq T t x) \to ((getl i c (CHead d (Bind Abbr) 
-u)) \to ((pr0 t3 t2) \to ((subst0 i u t2 t) \to (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t4: T).(eq T x (THead (Bind Void) u2 t4)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t4)))))) (\forall (b: 
-B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x)))))))))) 
-(\lambda (H7: (eq T t x)).(eq_ind T x (\lambda (t3: T).((getl i c (CHead d 
-(Bind Abbr) u)) \to ((pr0 (THead (Bind Void) u1 t1) t2) \to ((subst0 i u t2 
-t3) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x (THead (Bind 
-Void) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) 
-u0) t1 t4)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) 
-t1 (lift (S O) O x))))))))) (\lambda (H8: (getl i c (CHead d (Bind Abbr) 
-u))).(\lambda (H9: (pr0 (THead (Bind Void) u1 t1) t2)).(\lambda (H10: (subst0 
-i u t2 x)).(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 x (THead (Bind Void) 
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) 
-u0) t1 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) 
-t1 (lift (S O) O x))))) (\lambda (H11: (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 (t2: T).(pr0 t1 t2))))).(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 x 
-(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead 
-c (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c 
-(Bind b) u0) t1 (lift (S O) O x))))) (\lambda (x0: T).(\lambda (x1: 
-T).(\lambda (H: (eq T t2 (THead (Bind Void) x0 x1))).(\lambda (H12: (pr0 u1 
-x0)).(\lambda (H13: (pr0 t1 x1)).(let H14 \def (eq_ind T t2 (\lambda (t: 
-T).(subst0 i u t x)) H10 (THead (Bind Void) x0 x1) H) in (or3_ind (ex2 T 
-(\lambda (u2: T).(eq T x (THead (Bind Void) u2 x1))) (\lambda (u2: T).(subst0 
-i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T x (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 x (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 x (THead (Bind Void) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b: 
-B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x))))) 
-(\lambda (H15: (ex2 T (\lambda (u2: T).(eq T x (THead (Bind Void) u2 x1))) 
-(\lambda (u2: T).(subst0 i u x0 u2)))).(ex2_ind T (\lambda (u2: T).(eq T x 
-(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 x (THead (Bind Void) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 
-t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift 
-(S O) O x))))) (\lambda (x2: T).(\lambda (H16: (eq T x (THead (Bind Void) x2 
-x1))).(\lambda (H17: (subst0 i u x0 x2)).(or_introl (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b: 
-B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x)))) 
-(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) 
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) 
-u0) t1 t3))))) x2 x1 H16 (pr2_delta c d u i H8 u1 x0 H12 x2 H17) (\lambda (b: 
-B).(\lambda (u0: T).(pr2_free (CHead c (Bind b) u0) t1 x1 H13)))))))) H15)) 
-(\lambda (H15: (ex2 T (\lambda (t2: T).(eq T x (THead (Bind Void) x0 t2))) 
-(\lambda (t2: T).(subst0 (s (Bind Void) i) u x1 t2)))).(ex2_ind T (\lambda 
-(t3: T).(eq T x (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 x 
-(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead 
-c (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c 
-(Bind b) u0) t1 (lift (S O) O x))))) (\lambda (x2: T).(\lambda (H16: (eq T x 
-(THead (Bind Void) x0 x2))).(\lambda (H17: (subst0 (s (Bind Void) i) u x1 
-x2)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead 
-(Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead 
-c (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c 
-(Bind b) u0) t1 (lift (S O) O x)))) (ex3_2_intro T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))))) x0 x2 H16 
-(pr2_free c u1 x0 H12) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c 
-(Bind b) u0) d u (S i) (getl_head (Bind b) i c (CHead d (Bind Abbr) u) H8 u0) 
-t1 x1 H13 x2 H17)))))))) H15)) (\lambda (H15: (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t2: 
-T).(subst0 (s (Bind Void) i) u x1 t2))))).(ex3_2_ind T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T x (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 x (THead (Bind Void) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b: 
-B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x))))) 
-(\lambda (x2: T).(\lambda (x3: T).(\lambda (H16: (eq T x (THead (Bind Void) 
-x2 x3))).(\lambda (H17: (subst0 i u x0 x2)).(\lambda (H18: (subst0 (s (Bind 
-Void) i) u x1 x3)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 
-c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: 
-T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: 
-T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x)))) (ex3_2_intro T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 
-t3))))) x2 x3 H16 (pr2_delta c d u i H8 u1 x0 H12 x2 H17) (\lambda (b: 
-B).(\lambda (u0: T).(pr2_delta (CHead c (Bind b) u0) d u (S i) (getl_head 
-(Bind b) i c (CHead d (Bind Abbr) u) H8 u0) t1 x1 H13 x3 H18)))))))))) H15)) 
-(subst0_gen_head (Bind Void) u x0 x1 x i H14)))))))) H11)) (\lambda (H: (pr0 
-t1 (lift (S O) O t2))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 
-c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: 
-T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: 
-T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x)))) (\lambda (b: B).(\lambda 
-(u0: T).(pr2_delta (CHead c (Bind b) u0) d u (S i) (getl_head (Bind b) i c 
-(CHead d (Bind Abbr) u) H8 u0) t1 (lift (S O) O t2) H (lift (S O) O x) 
-(subst0_lift_ge_S t2 x u i H10 O (le_O_n i))))))) (pr0_gen_void u1 t1 t2 
-H9))))) t (sym_eq T t x H7))) t0 (sym_eq T t0 (THead (Bind Void) u1 t1) H6))) 
-c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C c) 
-(refl_equal T (THead (Bind Void) u1 t1)) (refl_equal T x))))))).
-
-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)).(\lambda (e: C).(\lambda (H0: 
-(drop h d c e)).(let H1 \def (match H return (\lambda (c0: C).(\lambda (t: 
-T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 c) \to ((eq T t 
-(lift h d t1)) \to ((eq T t0 x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d 
-t2))) (\lambda (t2: T).(pr2 e t1 t2)))))))))) with [(pr2_free c0 t0 t2 H1) 
-\Rightarrow (\lambda (H2: (eq C c0 c)).(\lambda (H3: (eq T t0 (lift h d 
-t1))).(\lambda (H4: (eq T t2 x)).(eq_ind C c (\lambda (_: C).((eq T t0 (lift 
-h d t1)) \to ((eq T t2 x) \to ((pr0 t0 t2) \to (ex2 T (\lambda (t3: T).(eq T 
-x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))))))) (\lambda (H5: (eq T t0 
-(lift h d t1))).(eq_ind T (lift h d t1) (\lambda (t: T).((eq T t2 x) \to 
-((pr0 t t2) \to (ex2 T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: 
-T).(pr2 e t1 t3)))))) (\lambda (H6: (eq T t2 x)).(eq_ind T x (\lambda (t: 
-T).((pr0 (lift h d t1) t) \to (ex2 T (\lambda (t3: T).(eq T x (lift h d t3))) 
-(\lambda (t3: T).(pr2 e t1 t3))))) (\lambda (H7: (pr0 (lift h d t1) 
-x)).(ex2_ind T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr0 
-t1 t3)) (ex2 T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 
-e t1 t3))) (\lambda (x0: T).(\lambda (H: (eq T x (lift h d x0))).(\lambda 
-(H8: (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 H8)) x H)))) 
-(pr0_gen_lift t1 x h d H7))) t2 (sym_eq T t2 x H6))) t0 (sym_eq T t0 (lift h 
-d t1) H5))) c0 (sym_eq C c0 c H2) H3 H4 H1)))) | (pr2_delta c0 d0 u i H1 t0 
-t2 H2 t H3) \Rightarrow (\lambda (H4: (eq C c0 c)).(\lambda (H5: (eq T t0 
-(lift h d t1))).(\lambda (H6: (eq T t x)).(eq_ind C c (\lambda (c: C).((eq T 
-t0 (lift h d t1)) \to ((eq T t x) \to ((getl i c (CHead d0 (Bind Abbr) u)) 
-\to ((pr0 t0 t2) \to ((subst0 i u t2 t) \to (ex2 T (\lambda (t3: T).(eq T x 
-(lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))))))))) (\lambda (H7: (eq T t0 
-(lift h d t1))).(eq_ind T (lift h d t1) (\lambda (t3: T).((eq T t x) \to 
-((getl i c (CHead d0 (Bind Abbr) u)) \to ((pr0 t3 t2) \to ((subst0 i u t2 t) 
-\to (ex2 T (\lambda (t4: T).(eq T x (lift h d t4))) (\lambda (t4: T).(pr2 e 
-t1 t4)))))))) (\lambda (H8: (eq T t x)).(eq_ind T x (\lambda (t3: T).((getl i 
-c (CHead d0 (Bind Abbr) u)) \to ((pr0 (lift h d t1) t2) \to ((subst0 i u t2 
-t3) \to (ex2 T (\lambda (t4: T).(eq T x (lift h d t4))) (\lambda (t4: T).(pr2 
-e t1 t4))))))) (\lambda (H9: (getl i c (CHead d0 (Bind Abbr) u))).(\lambda 
-(H10: (pr0 (lift h d t1) t2)).(\lambda (H11: (subst0 i u t2 x)).(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 x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))) 
-(\lambda (x0: T).(\lambda (H: (eq T t2 (lift h d x0))).(\lambda (H12: (pr0 t1 
-x0)).(let H13 \def (eq_ind T t2 (\lambda (t: T).(subst0 i u t x)) H11 (lift h 
-d x0) H) in (lt_le_e i d (ex2 T (\lambda (t3: T).(eq T x (lift h d t3))) 
-(\lambda (t3: T).(pr2 e t1 t3))) (\lambda (H14: (lt i d)).(let H15 \def 
-(eq_ind nat d (\lambda (n: nat).(drop h n c e)) H0 (S (plus i (minus d (S 
-i)))) (lt_plus_minus i d H14)) in (let H16 \def (eq_ind nat d (\lambda (n: 
-nat).(subst0 i u (lift h n x0) x)) H13 (S (plus i (minus d (S i)))) 
-(lt_plus_minus i d H14)) 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 x (lift h d 
-t3))) (\lambda (t3: T).(pr2 e t1 t3))) (\lambda (x1: T).(\lambda (x2: 
-C).(\lambda (H0: (eq T u (lift h (minus d (S i)) x1))).(\lambda (H17: (getl i 
-e (CHead x2 (Bind Abbr) x1))).(\lambda (_: (drop h (minus d (S i)) d0 
-x2)).(let H19 \def (eq_ind T u (\lambda (t: T).(subst0 i t (lift h (S (plus i 
-(minus d (S i)))) x0) x)) H16 (lift h (minus d (S i)) x1) H0) in (ex2_ind T 
-(\lambda (t3: T).(eq T x (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 x (lift h d t3))) 
-(\lambda (t3: T).(pr2 e t1 t3))) (\lambda (x3: T).(\lambda (H20: (eq T x 
-(lift h (S (plus i (minus d (S i)))) x3))).(\lambda (H21: (subst0 i x1 x0 
-x3)).(let H22 \def (eq_ind_r nat (S (plus i (minus d (S i)))) (\lambda (n: 
-nat).(eq T x (lift h n x3))) H20 d (lt_plus_minus i d H14)) in (ex_intro2 T 
-(\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)) x3 
-H22 (pr2_delta e x2 x1 i H17 t1 x0 H12 x3 H21)))))) (subst0_gen_lift_lt x1 x0 
-x i h (minus d (S i)) H19)))))))) (getl_drop_conf_lt Abbr c d0 u i H9 e h 
-(minus d (S i)) H15))))) (\lambda (H14: (le d i)).(lt_le_e i (plus d h) (ex2 
-T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))) 
-(\lambda (H15: (lt i (plus d h))).(subst0_gen_lift_false x0 u x h d i H14 H15 
-H13 (ex2 T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e 
-t1 t3))))) (\lambda (H15: (le (plus d h) i)).(ex2_ind T (\lambda (t3: T).(eq 
-T x (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u x0 t3)) (ex2 T 
-(\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))) 
-(\lambda (x1: T).(\lambda (H16: (eq T x (lift h d x1))).(\lambda (H17: 
-(subst0 (minus i h) u x0 x1)).(ex_intro2 T (\lambda (t3: T).(eq T x (lift h d 
-t3))) (\lambda (t3: T).(pr2 e t1 t3)) x1 H16 (pr2_delta e d0 u (minus i h) 
-(getl_drop_conf_ge i (CHead d0 (Bind Abbr) u) c H9 e h d H0 H15) t1 x0 H12 x1 
-H17))))) (subst0_gen_lift_ge u x0 x i h d H13 H15)))))))))) (pr0_gen_lift t1 
-t2 h d H10))))) t (sym_eq T t x H8))) t0 (sym_eq T t0 (lift h d t1) H7))) c0 
-(sym_eq C c0 c H4) H5 H6 H1 H2 H3))))]) in (H1 (refl_equal C c) (refl_equal T 
-(lift h d t1)) (refl_equal T x)))))))))).
-
-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 return (\lambda (_: C).C) with [(CSort _) 
-\Rightarrow d | (CHead c _ _) \Rightarrow c])) (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 
-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 
-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 
-(t0: T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2 t0)))))))))) 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 (t0: 
-T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2 t0)))))) (\lambda (H6: (eq T t4 
-t1)).(eq_ind T t1 (\lambda (t: T).((pr0 t0 t) \to (ex2 T (\lambda (t0: 
-T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2 t0))))) (\lambda (H7: (pr0 t0 
-t1)).(let H8 \def (match H0 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 t2) \to (ex2 T (\lambda (t0: T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2 
-t0)))))))))) with [(pr2_free c1 t5 t6 H5) \Rightarrow (\lambda (H6: (eq C c1 
-c)).(\lambda (H8: (eq T t5 t0)).(\lambda (H9: (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 
-(H10: (eq T t5 t0)).(eq_ind T t0 (\lambda (t: T).((eq T t6 t2) \to ((pr0 t 
-t6) \to (ex2 T (\lambda (t0: T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2 
-t0)))))) (\lambda (H11: (eq T t6 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t0 
-t) \to (ex2 T (\lambda (t0: T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2 
-t0))))) (\lambda (H12: (pr0 t0 t2)).(pr2_confluence__pr2_free_free c t0 t1 t2 
-H7 H12)) t6 (sym_eq T t6 t2 H11))) t5 (sym_eq T t5 t0 H10))) c1 (sym_eq C c1 
-c H6) H8 H9 H5)))) | (pr2_delta c1 d u i H5 t5 t6 H6 t H7) \Rightarrow 
-(\lambda (H8: (eq C c1 c)).(\lambda (H9: (eq T t5 t0)).(\lambda (H10: (eq T t 
-t2)).(eq_ind C c (\lambda (c0: C).((eq T t5 t0) \to ((eq T t t2) \to ((getl i 
-c0 (CHead d (Bind Abbr) u)) \to ((pr0 t5 t6) \to ((subst0 i u t6 t) \to (ex2 
-T (\lambda (t0: T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2 t0))))))))) 
-(\lambda (H11: (eq T t5 t0)).(eq_ind T t0 (\lambda (t0: T).((eq T t t2) \to 
-((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t6) \to ((subst0 i u t6 t) 
-\to (ex2 T (\lambda (t2: T).(pr2 c t1 t2)) (\lambda (t1: T).(pr2 c t2 
-t1)))))))) (\lambda (H12: (eq T t t2)).(eq_ind T t2 (\lambda (t3: T).((getl i 
-c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t6) \to ((subst0 i u t6 t3) \to (ex2 
-T (\lambda (t0: T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2 t0))))))) 
-(\lambda (H13: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H14: (pr0 t0 
-t6)).(\lambda (H15: (subst0 i u t6 t2)).(pr2_confluence__pr2_free_delta c d 
-t0 t1 t2 t6 u i H7 H13 H14 H15)))) t (sym_eq T t t2 H12))) t5 (sym_eq T t5 t0 
-H11))) c1 (sym_eq C c1 c H8) H9 H10 H5 H6 H7))))]) 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 (t0: T).(pr2 c t1 t0)) (\lambda 
-(t0: T).(pr2 c t2 t0))))))))) (\lambda (H7: (eq T t3 t0)).(eq_ind T t0 
-(\lambda (t0: T).((eq T t t1) \to ((getl i c (CHead d (Bind Abbr) u)) \to 
-((pr0 t0 t4) \to ((subst0 i u t4 t) \to (ex2 T (\lambda (t2: T).(pr2 c t1 
-t2)) (\lambda (t1: T).(pr2 c t2 t1)))))))) (\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 (t0: T).(pr2 c t1 t0)) (\lambda 
-(t0: T).(pr2 c t2 t0))))))) (\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 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 
-t2) \to (ex2 T (\lambda (t0: T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2 
-t0)))))))))) with [(pr2_free c1 t5 t6 H7) \Rightarrow (\lambda (H8: (eq C c1 
-c)).(\lambda (H12: (eq T t5 t0)).(\lambda (H13: (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 
-(H14: (eq T t5 t0)).(eq_ind T t0 (\lambda (t: T).((eq T t6 t2) \to ((pr0 t 
-t6) \to (ex2 T (\lambda (t0: T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2 
-t0)))))) (\lambda (H15: (eq T t6 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t0 
-t) \to (ex2 T (\lambda (t0: T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2 
-t0))))) (\lambda (H16: (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 H16 H9 H10 H11))) t6 
-(sym_eq T t6 t2 H15))) t5 (sym_eq T t5 t0 H14))) c1 (sym_eq C c1 c H8) H12 
-H13 H7)))) | (pr2_delta c1 d0 u0 i0 H7 t5 t6 H8 t7 H9) \Rightarrow (\lambda 
-(H12: (eq C c1 c)).(\lambda (H13: (eq T t5 t0)).(\lambda (H14: (eq T t7 
-t2)).(eq_ind C c (\lambda (c0: C).((eq T t5 t0) \to ((eq T t7 t2) \to ((getl 
-i0 c0 (CHead d0 (Bind Abbr) u0)) \to ((pr0 t5 t6) \to ((subst0 i0 u0 t6 t7) 
-\to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))))))))) 
-(\lambda (H15: (eq T t5 t0)).(eq_ind T t0 (\lambda (t: T).((eq T t7 t2) \to 
-((getl i0 c (CHead d0 (Bind Abbr) u0)) \to ((pr0 t t6) \to ((subst0 i0 u0 t6 
-t7) \to (ex2 T (\lambda (t0: T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2 
-t0)))))))) (\lambda (H16: (eq T t7 t2)).(eq_ind T t2 (\lambda (t: T).((getl 
-i0 c (CHead d0 (Bind Abbr) u0)) \to ((pr0 t0 t6) \to ((subst0 i0 u0 t6 t) \to 
-(ex2 T (\lambda (t0: T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2 t0))))))) 
-(\lambda (H17: (getl i0 c (CHead d0 (Bind Abbr) u0))).(\lambda (H18: (pr0 t0 
-t6)).(\lambda (H19: (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 H17 H18 H19)))) t7 (sym_eq T t7 t2 
-H16))) t5 (sym_eq T t5 t0 H15))) c1 (sym_eq C c1 c H12) H13 H14 H7 H8 
-H9))))]) 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)))))))).
-
-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)).(let H1 \def (match H0 return (\lambda (c0: 
-C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 c) 
-\to ((eq T t t1) \to ((eq T t0 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)))))))))))) with [(pr2_free c0 t0 t3 H1) \Rightarrow (\lambda (H2: (eq C 
-c0 c)).(\lambda (H3: (eq T t0 t1)).(\lambda (H4: (eq T t3 t2)).(eq_ind C c 
-(\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \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))))))))) (\lambda (H5: (eq T t0 
-t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \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)))))))) (\lambda (H6: (eq T t3 
-t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t1 t) \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))))))) (\lambda (H7: (pr0 t1 t2)).(\lambda (w1: T).(\lambda (H0: 
-(subst1 i v t1 w1)).(ex2_ind T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: 
-T).(subst1 i v t2 w2)) (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: 
-T).(subst1 i v t2 w2))) (\lambda (x: T).(\lambda (H8: (pr0 w1 x)).(\lambda 
-(H9: (subst1 i v t2 x)).(ex_intro2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda 
-(w2: T).(subst1 i v t2 w2)) x (pr2_free c w1 x H8) H9)))) (pr0_subst1 t1 t2 
-H7 v w1 i H0 v (pr0_refl v)))))) t3 (sym_eq T t3 t2 H6))) t0 (sym_eq T t0 t1 
-H5))) c0 (sym_eq C c0 c H2) H3 H4 H1)))) | (pr2_delta c0 d u i0 H1 t0 t3 H2 t 
-H3) \Rightarrow (\lambda (H4: (eq C c0 c)).(\lambda (H5: (eq T t0 
-t1)).(\lambda (H6: (eq T t t2)).(eq_ind C c (\lambda (c1: C).((eq T t0 t1) 
-\to ((eq T t t2) \to ((getl i0 c1 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t3) 
-\to ((subst0 i0 u t3 t) \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))))))))))) 
-(\lambda (H7: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to 
-((getl i0 c (CHead d (Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i0 u t3 t) 
-\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)))))))))) (\lambda (H8: (eq T t 
-t2)).(eq_ind T t2 (\lambda (t4: T).((getl i0 c (CHead d (Bind Abbr) u)) \to 
-((pr0 t1 t3) \to ((subst0 i0 u t3 t4) \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))))))))) (\lambda (H9: (getl i0 c (CHead d (Bind Abbr) u))).(\lambda 
-(H10: (pr0 t1 t3)).(\lambda (H11: (subst0 i0 u t3 t2)).(\lambda (w1: 
-T).(\lambda (H0: (subst1 i v t1 w1)).(ex2_ind T (\lambda (w2: T).(pr0 w1 w2)) 
-(\lambda (w2: T).(subst1 i v t3 w2)) (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) 
-(\lambda (w2: T).(subst1 i v t2 w2))) (\lambda (x: T).(\lambda (H12: (pr0 w1 
-x)).(\lambda (H13: (subst1 i v t3 x)).(neq_eq_e i i0 (ex2 T (\lambda (w2: 
-T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2))) (\lambda (H14: (not 
-(eq nat i i0))).(ex2_ind T (\lambda (t1: T).(subst1 i v t2 t1)) (\lambda (t1: 
-T).(subst1 i0 u x t1)) (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: 
-T).(subst1 i v t2 w2))) (\lambda (x0: T).(\lambda (H15: (subst1 i v t2 
-x0)).(\lambda (H16: (subst1 i0 u x x0)).(ex_intro2 T (\lambda (w2: T).(pr2 c 
-w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2)) x0 (pr2_delta1 c d u i0 H9 w1 x 
-H12 x0 H16) H15)))) (subst1_confluence_neq t3 t2 u i0 (subst1_single i0 u t3 
-t2 H11) x v i H13 (sym_not_eq nat i i0 H14)))) (\lambda (H14: (eq nat i 
-i0)).(let H15 \def (eq_ind_r nat i0 (\lambda (n: nat).(subst0 n u t3 t2)) H11 
-i H14) in (let H16 \def (eq_ind_r nat i0 (\lambda (n: nat).(getl n c (CHead d 
-(Bind Abbr) u))) H9 i H14) in (let H17 \def (eq_ind C (CHead e (Bind Abbr) v) 
-(\lambda (c0: C).(getl i c c0)) H (CHead d (Bind Abbr) u) (getl_mono c (CHead 
-e (Bind Abbr) v) i H (CHead d (Bind Abbr) u) H16)) in (let H18 \def (f_equal 
-C C (\lambda (e0: C).(match e0 return (\lambda (_: C).C) with [(CSort _) 
-\Rightarrow e | (CHead c _ _) \Rightarrow c])) (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) H16)) in ((let H19 \def (f_equal C T (\lambda (e0: C).(match e0 
-return (\lambda (_: C).T) with [(CSort _) \Rightarrow v | (CHead _ _ t) 
-\Rightarrow t])) (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) H16)) in (\lambda (H20: 
-(eq C e d)).(let H21 \def (eq_ind_r T u (\lambda (t: T).(getl i c (CHead d 
-(Bind Abbr) t))) H17 v H19) in (let H22 \def (eq_ind_r T u (\lambda (t: 
-T).(subst0 i t t3 t2)) H15 v H19) in (let H23 \def (eq_ind_r C d (\lambda 
-(c0: C).(getl i c (CHead c0 (Bind Abbr) v))) H21 e H20) in (ex2_ind T 
-(\lambda (t1: T).(subst1 i v t2 t1)) (\lambda (t1: T).(subst1 i v x t1)) (ex2 
-T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2))) 
-(\lambda (x0: T).(\lambda (H24: (subst1 i v t2 x0)).(\lambda (H25: (subst1 i 
-v x x0)).(ex_intro2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: 
-T).(subst1 i v t2 w2)) x0 (pr2_delta1 c e v i H23 w1 x H12 x0 H25) H24)))) 
-(subst1_confluence_eq t3 t2 v i (subst1_single i v t3 t2 H22) x H13))))))) 
-H18)))))))))) (pr0_subst1 t1 t3 H10 v w1 i H0 v (pr0_refl v)))))))) t (sym_eq 
-T t t2 H8))) t0 (sym_eq T t0 t1 H7))) c0 (sym_eq C c0 c H4) H5 H6 H1 H2 
-H3))))]) in (H1 (refl_equal C c) (refl_equal T t1) (refl_equal T t2)))))))))).
-
-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 (t: T).(subst1 d0 u0 t4 t)) 
-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 (c: C).(getl i 
-a0 c)) 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 (t: 
-T).(subst1 (minus d0 (S i)) u0 u t)) 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 (t: T).(subst1 (S (plus (minus d0 (S i)) i)) u0 x2 t)) 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 (c: C).(getl i c0 c)) 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 return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c _ _) 
-\Rightarrow c])) (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 return (\lambda (_: C).T) with 
-[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (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 (t: T).(getl i c0 (CHead e (Bind Abbr) t))) H17 u 
-H19) in (let H22 \def (eq_ind_r T u0 (\lambda (t: T).(subst1 i t t4 (lift (S 
-O) i x0))) H13 u H19) in (let H23 \def (eq_ind_r T u0 (\lambda (t: 
-T).(csubst1 i t 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 (c: C).(getl i c0 (CHead c 
-(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_comm 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_comm 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_comm 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_equal 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)))).
-
-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))))).
-
-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)) (eq T x (TSort n)) 
-(\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 (TSort n))))) (\lambda 
-(t: T).(\lambda (H1: (eq T t (TSort n))).H1)) (\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 (TSort n))))).(\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 (H3 (pr2_gen_sort c t2 n H5)))))))))) 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)) (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 (x: 
-T).(\forall (x0: T).((eq T t2 (THead (Bind Abst) x x0)) \to (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t2: T).(eq T t4 (THead (Bind Abst) u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x u2))) (\lambda (_: T).(\lambda (t2: 
-T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x0 
-t2))))))))))).(\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 (x: T).(\forall (x0: T).((eq T t (THead (Bind 
-Abst) x x0)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t4 (THead 
-(Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x u2))) 
-(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead 
-c (Bind b) u) x0 t2)))))))))) 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)) (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 (x: T).(\forall (x0: T).((eq T t2 (THead (Flat Cast) x x0)) \to (or 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t4 (THead (Flat Cast) u2 
-t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x u2))) (\lambda (_: 
-T).(\lambda (t2: T).(pr3 c x0 t2)))) (pr3 c x0 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 (t3: 
-T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c x1 
-t2))))).(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 (x: 
-T).(\forall (x0: T).((eq T t (THead (Flat Cast) x x0)) \to (or (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t2: T).(eq T t4 (THead (Flat Cast) u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x u2))) (\lambda (_: T).(\lambda (t2: 
-T).(pr3 c x0 t2)))) (pr3 c x0 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 
-(t2: T).(eq T t4 (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x3 
-t2))))).(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 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 
-(t3: T).((pr3 c t4 t3) \to (pr3 c t0 t3))))).(\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_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))).
-
-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_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)).(let H1 \def (match H0 return (\lambda (c0: 
-C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 
-(CHead c k u2)) \to ((eq T t t1) \to ((eq T t0 t2) \to (pr3 (CHead c k u1) t1 
-t2)))))))) with [(pr2_free c0 t0 t3 H1) \Rightarrow (\lambda (H2: (eq C c0 
-(CHead c k u2))).(\lambda (H3: (eq T t0 t1)).(\lambda (H4: (eq T t3 
-t2)).(eq_ind C (CHead c k u2) (\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2) 
-\to ((pr0 t0 t3) \to (pr3 (CHead c k u1) t1 t2))))) (\lambda (H5: (eq T t0 
-t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to (pr3 
-(CHead c k u1) t1 t2)))) (\lambda (H6: (eq T t3 t2)).(eq_ind T t2 (\lambda 
-(t: T).((pr0 t1 t) \to (pr3 (CHead c k u1) t1 t2))) (\lambda (H7: (pr0 t1 
-t2)).(pr3_pr2 (CHead c k u1) t1 t2 (pr2_free (CHead c k u1) t1 t2 H7))) t3 
-(sym_eq T t3 t2 H6))) t0 (sym_eq T t0 t1 H5))) c0 (sym_eq C c0 (CHead c k u2) 
-H2) H3 H4 H1)))) | (pr2_delta c0 d u i H1 t0 t3 H2 t H3) \Rightarrow (\lambda 
-(H4: (eq C c0 (CHead c k u2))).(\lambda (H5: (eq T t0 t1)).(\lambda (H6: (eq 
-T t t2)).(eq_ind C (CHead c k u2) (\lambda (c1: C).((eq T t0 t1) \to ((eq T t 
-t2) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t3) \to ((subst0 i 
-u t3 t) \to (pr3 (CHead c k u1) t1 t2))))))) (\lambda (H7: (eq T t0 
-t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i (CHead c k u2) 
-(CHead d (Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u t3 t) \to (pr3 
-(CHead c k u1) t1 t2)))))) (\lambda (H8: (eq T t t2)).(eq_ind T t2 (\lambda 
-(t4: T).((getl i (CHead c k u2) (CHead d (Bind Abbr) u)) \to ((pr0 t1 t3) \to 
-((subst0 i u t3 t4) \to (pr3 (CHead c k u1) t1 t2))))) (\lambda (H9: (getl i 
-(CHead c k u2) (CHead d (Bind Abbr) u))).(\lambda (H10: (pr0 t1 t3)).(\lambda 
-(H11: (subst0 i u t3 t2)).(nat_ind (\lambda (n: nat).((getl n (CHead c k u2) 
-(CHead d (Bind Abbr) u)) \to ((subst0 n u t3 t2) \to (pr3 (CHead c k u1) t1 
-t2)))) (\lambda (H12: (getl O (CHead c k u2) (CHead d (Bind Abbr) 
-u))).(\lambda (H13: (subst0 O u t3 t2)).(K_ind (\lambda (k: K).((getl O 
-(CHead c k u2) (CHead d (Bind Abbr) u)) \to (pr3 (CHead c k u1) t1 t2))) 
-(\lambda (b: B).(\lambda (H14: (getl O (CHead c (Bind b) u2) (CHead d (Bind 
-Abbr) u))).(let H0 \def (f_equal C C (\lambda (e: C).(match e return (\lambda 
-(_: C).C) with [(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow c])) 
-(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) 
-H14))) in ((let H15 \def (f_equal C B (\lambda (e: C).(match e return 
-(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) 
-\Rightarrow (match k return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | 
-(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) H14))) in ((let H16 \def (f_equal C T (\lambda 
-(e: C).(match e return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | 
-(CHead _ _ t) \Rightarrow t])) (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) H14))) in (\lambda (H17: (eq B Abbr b)).(\lambda 
-(_: (eq C d c)).(let H19 \def (eq_ind T u (\lambda (t: T).(subst0 O t t3 t2)) 
-H13 u2 H16) in (eq_ind B Abbr (\lambda (b0: B).(pr3 (CHead c (Bind b0) u1) t1 
-t2)) (ex2_ind T (\lambda (t1: T).(subst0 O u1 t3 t1)) (\lambda (t1: T).(pr0 
-t1 t2)) (pr3 (CHead c (Bind Abbr) u1) t1 t2) (\lambda (x: T).(\lambda (H20: 
-(subst0 O u1 t3 x)).(\lambda (H21: (pr0 x t2)).(pr3_sing (CHead c (Bind Abbr) 
-u1) x t1 (pr2_delta (CHead c (Bind Abbr) u1) c u1 O (getl_refl Abbr c u1) t1 
-t3 H10 x H20) t2 (pr3_pr2 (CHead c (Bind Abbr) u1) x t2 (pr2_free (CHead c 
-(Bind Abbr) u1) x t2 H21)))))) (pr0_subst0_back u2 t3 t2 O H19 u1 H)) b 
-H17))))) H15)) H0)))) (\lambda (f: F).(\lambda (H14: (getl O (CHead c (Flat 
-f) u2) (CHead d (Bind Abbr) u))).(pr3_pr2 (CHead c (Flat f) u1) t1 t2 
-(pr2_cflat c t1 t2 (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) H14))) t1 t3 H10 t2 H13) f 
-u1)))) k H12))) (\lambda (i0: nat).(\lambda (IHi: (((getl i0 (CHead c k u2) 
-(CHead d (Bind Abbr) u)) \to ((subst0 i0 u t3 t2) \to (pr3 (CHead c k u1) t1 
-t2))))).(\lambda (H12: (getl (S i0) (CHead c k u2) (CHead d (Bind Abbr) 
-u))).(\lambda (H13: (subst0 (S i0) u t3 t2)).(K_ind (\lambda (k: K).((getl (S 
-i0) (CHead c k u2) (CHead d (Bind Abbr) u)) \to ((((getl i0 (CHead c k u2) 
-(CHead d (Bind Abbr) u)) \to ((subst0 i0 u t3 t2) \to (pr3 (CHead c k u1) t1 
-t2)))) \to (pr3 (CHead c k u1) t1 t2)))) (\lambda (b: B).(\lambda (H14: (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 t3 t2) 
-\to (pr3 (CHead c (Bind b) u1) t1 t2))))).(pr3_pr2 (CHead c (Bind b) u1) t1 
-t2 (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 
-H14) u1) t1 t3 H10 t2 H13))))) (\lambda (f: F).(\lambda (H14: (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 t3 t2) \to 
-(pr3 (CHead c (Flat f) u1) t1 t2))))).(pr3_pr2 (CHead c (Flat f) u1) t1 t2 
-(pr2_cflat c t1 t2 (pr2_delta c d u (r (Flat f) i0) (getl_gen_S (Flat f) c 
-(CHead d (Bind Abbr) u) u2 i0 H14) t1 t3 H10 t2 H13) f u1))))) k H12 IHi))))) 
-i H9 H11)))) t (sym_eq T t t2 H8))) t0 (sym_eq T t0 t1 H7))) c0 (sym_eq C c0 
-(CHead c k u2) H4) H5 H6 H1 H2 H3))))]) in (H1 (refl_equal C (CHead c k u2)) 
-(refl_equal T t1) (refl_equal T t2)))))))))).
-
-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)).(let H0 \def (match H return (\lambda (c0: C).(\lambda (t: T).(\lambda 
-(t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 c) \to ((eq T t u1) \to ((eq T 
-t0 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)))))))))))) with [(pr2_free c0 t1 t2 
-H0) \Rightarrow (\lambda (H1: (eq C c0 c)).(\lambda (H2: (eq T t1 
-u1)).(\lambda (H3: (eq T t2 u2)).(eq_ind C c (\lambda (_: C).((eq T t1 u1) 
-\to ((eq T t2 u2) \to ((pr0 t1 t2) \to (\forall (t3: T).(\forall (t4: 
-T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pr3 (CHead c k u1) t3 
-t4))))))))) (\lambda (H4: (eq T t1 u1)).(eq_ind T u1 (\lambda (t: T).((eq T 
-t2 u2) \to ((pr0 t t2) \to (\forall (t3: T).(\forall (t4: T).(\forall (k: 
-K).((pr2 (CHead c k u2) t3 t4) \to (pr3 (CHead c k u1) t3 t4)))))))) (\lambda 
-(H5: (eq T t2 u2)).(eq_ind T u2 (\lambda (t: T).((pr0 u1 t) \to (\forall (t3: 
-T).(\forall (t4: T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pr3 
-(CHead c k u1) t3 t4))))))) (\lambda (H6: (pr0 u1 u2)).(\lambda (t3: 
-T).(\lambda (t4: T).(\lambda (k: K).(\lambda (H: (pr2 (CHead c k u2) t3 
-t4)).(pr3_pr0_pr2_t u1 u2 H6 c t3 t4 k H)))))) t2 (sym_eq T t2 u2 H5))) t1 
-(sym_eq T t1 u1 H4))) c0 (sym_eq C c0 c H1) H2 H3 H0)))) | (pr2_delta c0 d u 
-i H0 t1 t2 H1 t H2) \Rightarrow (\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq 
-T t1 u1)).(\lambda (H5: (eq T t u2)).(eq_ind C c (\lambda (c1: C).((eq T t1 
-u1) \to ((eq T t u2) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t1 
-t2) \to ((subst0 i u t2 t) \to (\forall (t3: T).(\forall (t4: T).(\forall (k: 
-K).((pr2 (CHead c k u2) t3 t4) \to (pr3 (CHead c k u1) t3 t4))))))))))) 
-(\lambda (H6: (eq T t1 u1)).(eq_ind T u1 (\lambda (t0: T).((eq T t u2) \to 
-((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t2) \to ((subst0 i u t2 t) 
-\to (\forall (t3: T).(\forall (t4: T).(\forall (k: K).((pr2 (CHead c k u2) t3 
-t4) \to (pr3 (CHead c k u1) t3 t4)))))))))) (\lambda (H7: (eq T t 
-u2)).(eq_ind T u2 (\lambda (t0: T).((getl i c (CHead d (Bind Abbr) u)) \to 
-((pr0 u1 t2) \to ((subst0 i u t2 t0) \to (\forall (t3: T).(\forall (t4: 
-T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pr3 (CHead c k u1) t3 
-t4))))))))) (\lambda (H8: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H9: 
-(pr0 u1 t2)).(\lambda (H10: (subst0 i u t2 u2)).(\lambda (t3: T).(\lambda 
-(t0: T).(\lambda (k: K).(\lambda (H: (pr2 (CHead c k u2) t3 t0)).(let H11 
-\def (match H return (\lambda (c0: C).(\lambda (t: T).(\lambda (t1: 
-T).(\lambda (_: (pr2 c0 t t1)).((eq C c0 (CHead c k u2)) \to ((eq T t t3) \to 
-((eq T t1 t0) \to (pr3 (CHead c k u1) t3 t0)))))))) with [(pr2_free c0 t3 t4 
-H3) \Rightarrow (\lambda (H4: (eq C c0 (CHead c k u2))).(\lambda (H5: (eq T 
-t3 t3)).(\lambda (H6: (eq T t4 t0)).(eq_ind C (CHead c k u2) (\lambda (_: 
-C).((eq T t3 t3) \to ((eq T t4 t0) \to ((pr0 t3 t4) \to (pr3 (CHead c k u1) 
-t3 t0))))) (\lambda (H7: (eq T t3 t3)).(eq_ind T t3 (\lambda (t: T).((eq T t4 
-t0) \to ((pr0 t t4) \to (pr3 (CHead c k u1) t3 t0)))) (\lambda (H8: (eq T t4 
-t0)).(eq_ind T t0 (\lambda (t: T).((pr0 t3 t) \to (pr3 (CHead c k u1) t3 
-t0))) (\lambda (H9: (pr0 t3 t0)).(pr3_pr2 (CHead c k u1) t3 t0 (pr2_free 
-(CHead c k u1) t3 t0 H9))) t4 (sym_eq T t4 t0 H8))) t3 (sym_eq T t3 t3 H7))) 
-c0 (sym_eq C c0 (CHead c k u2) H4) H5 H6 H3)))) | (pr2_delta c0 d0 u0 i0 H3 
-t3 t4 H4 t H5) \Rightarrow (\lambda (H6: (eq C c0 (CHead c k u2))).(\lambda 
-(H7: (eq T t3 t3)).(\lambda (H11: (eq T t t0)).(eq_ind C (CHead c k u2) 
-(\lambda (c1: C).((eq T t3 t3) \to ((eq T t t0) \to ((getl i0 c1 (CHead d0 
-(Bind Abbr) u0)) \to ((pr0 t3 t4) \to ((subst0 i0 u0 t4 t) \to (pr3 (CHead c 
-k u1) t3 t0))))))) (\lambda (H12: (eq T t3 t3)).(eq_ind T t3 (\lambda (t1: 
-T).((eq T t t0) \to ((getl i0 (CHead c k u2) (CHead d0 (Bind Abbr) u0)) \to 
-((pr0 t1 t4) \to ((subst0 i0 u0 t4 t) \to (pr3 (CHead c k u1) t3 t0)))))) 
-(\lambda (H13: (eq T t t0)).(eq_ind T t0 (\lambda (t1: T).((getl i0 (CHead c 
-k u2) (CHead d0 (Bind Abbr) u0)) \to ((pr0 t3 t4) \to ((subst0 i0 u0 t4 t1) 
-\to (pr3 (CHead c k u1) t3 t0))))) (\lambda (H14: (getl i0 (CHead c k u2) 
-(CHead d0 (Bind Abbr) u0))).(\lambda (H15: (pr0 t3 t4)).(\lambda (H16: 
-(subst0 i0 u0 t4 t0)).((match i0 return (\lambda (n: nat).((getl n (CHead c k 
-u2) (CHead d0 (Bind Abbr) u0)) \to ((subst0 n u0 t4 t0) \to (pr3 (CHead c k 
-u1) t3 t0)))) with [O \Rightarrow (\lambda (H17: (getl O (CHead c k u2) 
-(CHead d0 (Bind Abbr) u0))).(\lambda (H18: (subst0 O u0 t4 t0)).((match k 
-return (\lambda (k: K).((clear (CHead c k u2) (CHead d0 (Bind Abbr) u0)) \to 
-(pr3 (CHead c k u1) t3 t0))) with [(Bind b) \Rightarrow (\lambda (H19: (clear 
-(CHead c (Bind b) u2) (CHead d0 (Bind Abbr) u0))).(let H \def (f_equal C C 
-(\lambda (e: C).(match e return (\lambda (_: C).C) with [(CSort _) 
-\Rightarrow d0 | (CHead c _ _) \Rightarrow c])) (CHead d0 (Bind Abbr) u0) 
-(CHead c (Bind b) u2) (clear_gen_bind b c (CHead d0 (Bind Abbr) u0) u2 H19)) 
-in ((let H0 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: 
-C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k 
-return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow 
-Abbr])])) (CHead d0 (Bind Abbr) u0) (CHead c (Bind b) u2) (clear_gen_bind b c 
-(CHead d0 (Bind Abbr) u0) u2 H19)) in ((let H1 \def (f_equal C T (\lambda (e: 
-C).(match e return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead 
-_ _ t) \Rightarrow t])) (CHead d0 (Bind Abbr) u0) (CHead c (Bind b) u2) 
-(clear_gen_bind b c (CHead d0 (Bind Abbr) u0) u2 H19)) in (\lambda (H20: (eq 
-B Abbr b)).(\lambda (_: (eq C d0 c)).(let H22 \def (eq_ind T u0 (\lambda (t: 
-T).(subst0 O t t4 t0)) H18 u2 H1) in (eq_ind B Abbr (\lambda (b0: B).(pr3 
-(CHead c (Bind b0) u1) t3 t0)) (ex2_ind T (\lambda (t0: T).(subst0 O t2 t4 
-t0)) (\lambda (t1: T).(subst0 (S (plus i O)) u t1 t0)) (pr3 (CHead c (Bind 
-Abbr) u1) t3 t0) (\lambda (x: T).(\lambda (H2: (subst0 O t2 t4 x)).(\lambda 
-(H10: (subst0 (S (plus i O)) u x t0)).(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 t0)) H10 (S i) H23) in 
-(ex2_ind T (\lambda (t0: T).(subst0 O u1 t4 t0)) (\lambda (t0: T).(pr0 t0 x)) 
-(pr3 (CHead c (Bind Abbr) u1) t3 t0) (\lambda (x0: T).(\lambda (H9: (subst0 O 
-u1 t4 x0)).(\lambda (H25: (pr0 x0 x)).(pr3_sing (CHead c (Bind Abbr) u1) x0 
-t3 (pr2_delta (CHead c (Bind Abbr) u1) c u1 O (getl_refl Abbr c u1) t3 t4 H15 
-x0 H9) t0 (pr3_pr2 (CHead c (Bind Abbr) u1) x0 t0 (pr2_delta (CHead c (Bind 
-Abbr) u1) d u (S i) (getl_clear_bind Abbr (CHead c (Bind Abbr) u1) c u1 
-(clear_bind Abbr c u1) (CHead d (Bind Abbr) u) i H8) x0 x H25 t0 H24)))))) 
-(pr0_subst0_back t2 t4 x O H2 u1 H9))))))) (subst0_subst0 t4 t0 u2 O H22 t2 u 
-i H10)) b H20))))) H0)) H))) | (Flat f) \Rightarrow (\lambda (H8: (clear 
-(CHead c (Flat f) u2) (CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c (Flat f) 
-u1) t3 t0 (pr2_cflat c t3 t0 (pr2_delta c d0 u0 O (getl_intro O c (CHead d0 
-(Bind Abbr) u0) c (drop_refl c) (clear_gen_flat f c (CHead d0 (Bind Abbr) u0) 
-u2 H8)) t3 t4 H15 t0 H18) f u1)))]) (getl_gen_O (CHead c k u2) (CHead d0 
-(Bind Abbr) u0) H17)))) | (S n) \Rightarrow (\lambda (H8: (getl (S n) (CHead 
-c k u2) (CHead d0 (Bind Abbr) u0))).(\lambda (H9: (subst0 (S n) u0 t4 
-t0)).((match k return (\lambda (k: K).((getl (S n) (CHead c k u2) (CHead d0 
-(Bind Abbr) u0)) \to (pr3 (CHead c k u1) t3 t0))) with [(Bind b) \Rightarrow 
-(\lambda (H10: (getl (S n) (CHead c (Bind b) u2) (CHead d0 (Bind Abbr) 
-u0))).(pr3_pr2 (CHead c (Bind b) u1) t3 t0 (pr2_delta (CHead c (Bind b) u1) 
-d0 u0 (S n) (getl_head (Bind b) n c (CHead d0 (Bind Abbr) u0) (getl_gen_S 
-(Bind b) c (CHead d0 (Bind Abbr) u0) u2 n H10) u1) t3 t4 H15 t0 H9))) | (Flat 
-f) \Rightarrow (\lambda (H10: (getl (S n) (CHead c (Flat f) u2) (CHead d0 
-(Bind Abbr) u0))).(pr3_pr2 (CHead c (Flat f) u1) t3 t0 (pr2_cflat c t3 t0 
-(pr2_delta c d0 u0 (r (Flat f) n) (getl_gen_S (Flat f) c (CHead d0 (Bind 
-Abbr) u0) u2 n H10) t3 t4 H15 t0 H9) f u1)))]) H8)))]) H14 H16)))) t (sym_eq 
-T t t0 H13))) t3 (sym_eq T t3 t3 H12))) c0 (sym_eq C c0 (CHead c k u2) H6) H7 
-H11 H3 H4 H5))))]) in (H11 (refl_equal C (CHead c k u2)) (refl_equal T t3) 
-(refl_equal T t0)))))))))) t (sym_eq T t u2 H7))) t1 (sym_eq T t1 u1 H6))) c0 
-(sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C c) (refl_equal T 
-u1) (refl_equal T u2)))))).
-
-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 (t1: T).(\forall (t4: T).(\forall (k: K).((pr3 
-(CHead c k t3) t1 t4) \to (pr3 (CHead c k t2) t1 t4))))))).(\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_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) (let H7 \def (match H4 return (\lambda (c: C).(\lambda (t: 
-T).(\lambda (t1: T).(\lambda (_: (pr2 c t t1)).((eq C c (CHead c3 k u1)) \to 
-((eq T t t3) \to ((eq T t1 t0) \to (pr3 (CHead c0 k u1) t3 t0)))))))) with 
-[(pr2_free c t1 t2 H2) \Rightarrow (\lambda (H3: (eq C c (CHead c3 k 
-u1))).(\lambda (H4: (eq T t1 t3)).(\lambda (H5: (eq T t2 t0)).(eq_ind C 
-(CHead c3 k u1) (\lambda (_: C).((eq T t1 t3) \to ((eq T t2 t0) \to ((pr0 t1 
-t2) \to (pr3 (CHead c0 k u1) t3 t0))))) (\lambda (H6: (eq T t1 t3)).(eq_ind T 
-t3 (\lambda (t: T).((eq T t2 t0) \to ((pr0 t t2) \to (pr3 (CHead c0 k u1) t3 
-t0)))) (\lambda (H7: (eq T t2 t0)).(eq_ind T t0 (\lambda (t: T).((pr0 t3 t) 
-\to (pr3 (CHead c0 k u1) t3 t0))) (\lambda (H8: (pr0 t3 t0)).(pr3_pr2 (CHead 
-c0 k u1) t3 t0 (pr2_free (CHead c0 k u1) t3 t0 H8))) t2 (sym_eq T t2 t0 H7))) 
-t1 (sym_eq T t1 t3 H6))) c (sym_eq C c (CHead c3 k u1) H3) H4 H5 H2)))) | 
-(pr2_delta c d u i H2 t1 t2 H3 t H4) \Rightarrow (\lambda (H5: (eq C c (CHead 
-c3 k u1))).(\lambda (H6: (eq T t1 t3)).(\lambda (H7: (eq T t t0)).(eq_ind C 
-(CHead c3 k u1) (\lambda (c1: C).((eq T t1 t3) \to ((eq T t t0) \to ((getl i 
-c1 (CHead d (Bind Abbr) u)) \to ((pr0 t1 t2) \to ((subst0 i u t2 t) \to (pr3 
-(CHead c0 k u1) t3 t0))))))) (\lambda (H8: (eq T t1 t3)).(eq_ind T t3 
-(\lambda (t4: T).((eq T t t0) \to ((getl i (CHead c3 k u1) (CHead d (Bind 
-Abbr) u)) \to ((pr0 t4 t2) \to ((subst0 i u t2 t) \to (pr3 (CHead c0 k u1) t3 
-t0)))))) (\lambda (H9: (eq T t t0)).(eq_ind T t0 (\lambda (t4: T).((getl i 
-(CHead c3 k u1) (CHead d (Bind Abbr) u)) \to ((pr0 t3 t2) \to ((subst0 i u t2 
-t4) \to (pr3 (CHead c0 k u1) t3 t0))))) (\lambda (H10: (getl i (CHead c3 k 
-u1) (CHead d (Bind Abbr) u))).(\lambda (H11: (pr0 t3 t2)).(\lambda (H12: 
-(subst0 i u t2 t0)).(ex3_2_ind C T (\lambda (e2: C).(\lambda (u2: T).(getl i 
-(CHead c0 k u1) (CHead e2 (Bind Abbr) u2)))) (\lambda (e2: C).(\lambda (_: 
-T).(wcpr0 e2 d))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 u))) (pr3 (CHead 
-c0 k u1) t3 t0) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H1: (getl i 
-(CHead c0 k u1) (CHead x0 (Bind Abbr) x1))).(\lambda (_: (wcpr0 x0 
-d)).(\lambda (H14: (pr0 x1 u)).(ex2_ind T (\lambda (t0: T).(subst0 i x1 t2 
-t0)) (\lambda (t3: T).(pr0 t3 t0)) (pr3 (CHead c0 k u1) t3 t0) (\lambda (x: 
-T).(\lambda (H15: (subst0 i x1 t2 x)).(\lambda (H16: (pr0 x t0)).(pr3_sing 
-(CHead c0 k u1) x t3 (pr2_delta (CHead c0 k u1) x0 x1 i H1 t3 t2 H11 x H15) 
-t0 (pr3_pr2 (CHead c0 k u1) x t0 (pr2_free (CHead c0 k u1) x t0 H16)))))) 
-(pr0_subst0_back u t2 t0 i H12 x1 H14))))))) (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) 
-H10))))) t (sym_eq T t t0 H9))) t1 (sym_eq T t1 t3 H8))) c (sym_eq C c (CHead 
-c3 k u1) H5) H6 H7 H2 H3 H4))))]) in (H7 (refl_equal C (CHead c3 k u1)) 
-(refl_equal T t3) (refl_equal T t0))) t4 H6))))))) t1 t2 (pr3_pr2_pr3_t c3 u2 
-t1 t2 k H3 u1 (pr2_free c3 u1 u2 H2)))))))))))))) c2 c1 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)) (\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 (x: T).((eq T t2 (lift 
-h d x)) \to (\forall (e: C).((drop h d c e) \to (ex2 T (\lambda (t2: T).(eq T 
-t4 (lift h d t2))) (\lambda (t2: T).(pr3 e x t2))))))))).(\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)) (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))))))) (\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 (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 t0 
-(lift (S n) O v)))))))))) (\lambda (t: T).(\lambda (H1: (eq T t (TLRef 
-n))).(or_introl (eq T t (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 t (lift (S n) O v)))))) H1))) (\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 
-(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 (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 (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))).(H3 H7)) (\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).(pr3 c t t3)) H2 (lift (S n) O x1) H9) in (let 
-H11 \def (pr3_gen_lift c x1 t3 (S n) O H10 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 (H12: (eq T t3 (lift (S n) O x2))).(\lambda (H13: (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 H13 H12))))) H11))))))) H7)) H6)))))))))) 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)) (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 
-(x: T).(\forall (x0: T).((eq T t2 (THead (Bind Void) x x0)) \to (or (ex3_2 T 
-T (\lambda (u2: T).(\lambda (t2: T).(eq T t4 (THead (Bind Void) u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x 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) x) x0 (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 (t3: T).(eq T t2 (THead (Bind Void) 
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: 
-T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
-x1 t2))))))).(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 (x: T).(\forall (x0: T).((eq T t (THead (Bind 
-Void) x x0)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t4 
-(THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x 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) x) x0 (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 (t2: T).(eq T t4 (THead (Bind Void) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall 
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t2))))))).(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)) (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 (x: T).(\forall (x0: 
-T).((eq T t2 (THead (Bind Abbr) x x0)) \to (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T t4 (THead (Bind Abbr) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 
-(CHead c (Bind Abbr) x) x0 t2)))) (pr3 (CHead c (Bind Abbr) x) x0 (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 (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda 
-(t2: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 
-t2))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead c (Bind 
-Abbr) u) x1 t2))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c 
-(Bind Abbr) x0) x1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda 
-(_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) z t2))))))))).(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 (y: T).(\lambda (_: T).(pr2 
-(CHead c (Bind Abbr) x0) x1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y 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 (x: T).(\forall (x0: T).((eq T t (THead (Bind 
-Abbr) x x0)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t4 
-(THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x u2))) 
-(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) x) x0 t2)))) (pr3 
-(CHead c (Bind Abbr) x) x0 (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 (t2: T).(eq T 
-t4 (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 
-u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) x2) x3 
-t2))))).(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 (x: T).(\forall 
-(x0: T).((eq T t (THead (Bind Abbr) x x0)) \to (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T t4 (THead (Bind Abbr) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 
-(CHead c (Bind Abbr) x) x0 t2)))) (pr3 (CHead c (Bind Abbr) x) x0 (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 (t2: T).(eq T t4 (THead (Bind Abbr) u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda 
-(t2: T).(pr3 (CHead c (Bind Abbr) x2) x3 t2))))).(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 (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y))) 
-(\lambda (y: T).(\lambda (z: T).(pr0 y 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 (x: T).(\forall 
-(x0: T).((eq T t (THead (Bind Abbr) x x0)) \to (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T t4 (THead (Bind Abbr) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 
-(CHead c (Bind Abbr) x) x0 t2)))) (pr3 (CHead c (Bind Abbr) x) x0 (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 (t2: T).(eq T t4 (THead (Bind Abbr) u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda 
-(t2: T).(pr3 (CHead c (Bind Abbr) x2) x3 t2))))).(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)) (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 (x: T).(\forall (x0: T).((eq T t2 (THead (Flat 
-Appl) x x0)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t4 
-(THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x 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) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x 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)) t4))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr3 c x 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 (t3: T).(eq T t2 (THead (Flat Appl) 
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: 
-T).(\lambda (t2: T).(pr2 c x1 t2))))).(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 (x: T).(\forall (x0: T).((eq T t 
-(THead (Flat Appl) x x0)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
-T).(eq T t4 (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr3 c x 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) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u2: T).(\lambda (_: T).(pr3 c x 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)) t4))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr3 c x 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 (t2: T).(eq T t4 (THead (Flat 
-Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: 
-T).(\lambda (t2: T).(pr3 c x3 t2))))).(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 (t2: T).(pr3 c (THead (Bind 
-Abbr) u2 t2) 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 (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 (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 (t3: T).(eq T t2 (THead (Bind 
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c x0 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))))))))).(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 (x: T).(\forall (x0: T).((eq T t 
-(THead (Flat Appl) x x0)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
-T).(eq T t4 (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr3 c x 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) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u2: T).(\lambda (_: T).(pr3 c x 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)) t4))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr3 c x 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 (x: T).(\forall (x0: T).((eq T t (THead (Flat Appl) 
-x x0)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t4 (THead 
-(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x 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) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x 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)) t4))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr3 c x 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_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 (t3: T).((pr2 c t2 t3) \to (ex2 T (\lambda (t: 
-T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t3 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 (t3: T).((pr3 c t2 t3) \to (ex2 T (\lambda (t: T).(pr3 c t4 
-t)) (\lambda (t: T).(pr3 c t3 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)))).
-
-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)))).
-
-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 (u2: 
-T).(\lambda (_: T).(pr3 c v u2))) (\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 (Flat Cast) v x0 t 
-x1) (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 (u2: T).(\lambda (_: T).(pr3 c t0 u2))) 
-(\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 (Flat Appl) t0 x0 
-(THeads (Flat Appl) t1 (THead (Flat Cast) v t)) x1) (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 (u2: T).(\lambda (_: T).(pr3 c t0 u2))))) 
-(\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 (u2: T).(\lambda (_: T).(pr3 c t0 
-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) 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)))).
-
-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_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))))))))).
-
-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)).(let H0 \def (match H return (\lambda 
-(c0: C).(\lambda (c1: C).(\lambda (_: (csuba ? c0 c1)).((eq C c0 (CHead d1 
-(Bind Abbr) u)) \to ((eq C c1 c) \to (ex2 C (\lambda (d2: C).(eq C c (CHead 
-d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))))) with [(csuba_sort 
-n) \Rightarrow (\lambda (H0: (eq C (CSort n) (CHead d1 (Bind Abbr) 
-u))).(\lambda (H1: (eq C (CSort n) c)).((let H2 \def (eq_ind C (CSort n) 
-(\lambda (e: C).(match e return (\lambda (_: C).Prop) with [(CSort _) 
-\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 (Bind Abbr) 
-u) H0) in (False_ind ((eq C (CSort n) c) \to (ex2 C (\lambda (d2: C).(eq C c 
-(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) H2)) H1))) | 
-(csuba_head c1 c2 H0 k u0) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u0) 
-(CHead d1 (Bind Abbr) u))).(\lambda (H2: (eq C (CHead c2 k u0) c)).((let H3 
-\def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: C).T) with 
-[(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u0) 
-(CHead d1 (Bind Abbr) u) H1) in ((let H4 \def (f_equal C K (\lambda (e: 
-C).(match e return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead 
-_ k _) \Rightarrow k])) (CHead c1 k u0) (CHead d1 (Bind Abbr) u) H1) in ((let 
-H5 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: C).C) with 
-[(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u0) 
-(CHead d1 (Bind Abbr) u) H1) in (eq_ind C d1 (\lambda (c0: C).((eq K k (Bind 
-Abbr)) \to ((eq T u0 u) \to ((eq C (CHead c2 k u0) c) \to ((csuba g c0 c2) 
-\to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abbr) u))) (\lambda (d2: 
-C).(csuba g d1 d2)))))))) (\lambda (H6: (eq K k (Bind Abbr))).(eq_ind K (Bind 
-Abbr) (\lambda (k0: K).((eq T u0 u) \to ((eq C (CHead c2 k0 u0) c) \to 
-((csuba g d1 c2) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abbr) 
-u))) (\lambda (d2: C).(csuba g d1 d2))))))) (\lambda (H7: (eq T u0 
-u)).(eq_ind T u (\lambda (t: T).((eq C (CHead c2 (Bind Abbr) t) c) \to 
-((csuba g d1 c2) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abbr) 
-u))) (\lambda (d2: C).(csuba g d1 d2)))))) (\lambda (H8: (eq C (CHead c2 
-(Bind Abbr) u) c)).(eq_ind C (CHead c2 (Bind Abbr) u) (\lambda (c: C).((csuba 
-g d1 c2) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abbr) u))) 
-(\lambda (d2: C).(csuba g d1 d2))))) (\lambda (H9: (csuba g d1 
-c2)).(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)) H9)) c H8)) u0 (sym_eq T u0 u H7))) k (sym_eq K k (Bind Abbr) 
-H6))) c1 (sym_eq C c1 d1 H5))) H4)) H3)) H2 H0))) | (csuba_abst c1 c2 H0 t a 
-H1 u0 H2) \Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst) t) (CHead d1 
-(Bind Abbr) u))).(\lambda (H4: (eq C (CHead c2 (Bind Abbr) u0) c)).((let H5 
-\def (eq_ind C (CHead c1 (Bind Abst) t) (\lambda (e: C).(match e return 
-(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) 
-\Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow 
-(match b return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst 
-\Rightarrow True | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) 
-I (CHead d1 (Bind Abbr) u) H3) in (False_ind ((eq C (CHead c2 (Bind Abbr) u0) 
-c) \to ((csuba g c1 c2) \to ((arity g c1 t (asucc g a)) \to ((arity g c2 u0 
-a) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abbr) u))) (\lambda 
-(d2: C).(csuba g d1 d2))))))) H5)) H4 H0 H1 H2)))]) in (H0 (refl_equal C 
-(CHead d1 (Bind Abbr) u)) (refl_equal C c))))))).
-
-theorem csuba_gen_void:
- \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g 
-(CHead d1 (Bind Void) u) c) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 
-(Bind Void) 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 Void) u) c)).(let H0 \def (match H return (\lambda 
-(c0: C).(\lambda (c1: C).(\lambda (_: (csuba ? c0 c1)).((eq C c0 (CHead d1 
-(Bind Void) u)) \to ((eq C c1 c) \to (ex2 C (\lambda (d2: C).(eq C c (CHead 
-d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2)))))))) with [(csuba_sort 
-n) \Rightarrow (\lambda (H0: (eq C (CSort n) (CHead d1 (Bind Void) 
-u))).(\lambda (H1: (eq C (CSort n) c)).((let H2 \def (eq_ind C (CSort n) 
-(\lambda (e: C).(match e return (\lambda (_: C).Prop) with [(CSort _) 
-\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 (Bind Void) 
-u) H0) in (False_ind ((eq C (CSort n) c) \to (ex2 C (\lambda (d2: C).(eq C c 
-(CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2)))) H2)) H1))) | 
-(csuba_head c1 c2 H0 k u0) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u0) 
-(CHead d1 (Bind Void) u))).(\lambda (H2: (eq C (CHead c2 k u0) c)).((let H3 
-\def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: C).T) with 
-[(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u0) 
-(CHead d1 (Bind Void) u) H1) in ((let H4 \def (f_equal C K (\lambda (e: 
-C).(match e return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead 
-_ k _) \Rightarrow k])) (CHead c1 k u0) (CHead d1 (Bind Void) u) H1) in ((let 
-H5 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: C).C) with 
-[(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u0) 
-(CHead d1 (Bind Void) u) H1) in (eq_ind C d1 (\lambda (c0: C).((eq K k (Bind 
-Void)) \to ((eq T u0 u) \to ((eq C (CHead c2 k u0) c) \to ((csuba g c0 c2) 
-\to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Void) u))) (\lambda (d2: 
-C).(csuba g d1 d2)))))))) (\lambda (H6: (eq K k (Bind Void))).(eq_ind K (Bind 
-Void) (\lambda (k0: K).((eq T u0 u) \to ((eq C (CHead c2 k0 u0) c) \to 
-((csuba g d1 c2) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Void) 
-u))) (\lambda (d2: C).(csuba g d1 d2))))))) (\lambda (H7: (eq T u0 
-u)).(eq_ind T u (\lambda (t: T).((eq C (CHead c2 (Bind Void) t) c) \to 
-((csuba g d1 c2) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Void) 
-u))) (\lambda (d2: C).(csuba g d1 d2)))))) (\lambda (H8: (eq C (CHead c2 
-(Bind Void) u) c)).(eq_ind C (CHead c2 (Bind Void) u) (\lambda (c: C).((csuba 
-g d1 c2) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Void) u))) 
-(\lambda (d2: C).(csuba g d1 d2))))) (\lambda (H9: (csuba g d1 
-c2)).(ex_intro2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Void) u) (CHead d2 
-(Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2)) c2 (refl_equal C (CHead c2 
-(Bind Void) u)) H9)) c H8)) u0 (sym_eq T u0 u H7))) k (sym_eq K k (Bind Void) 
-H6))) c1 (sym_eq C c1 d1 H5))) H4)) H3)) H2 H0))) | (csuba_abst c1 c2 H0 t a 
-H1 u0 H2) \Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst) t) (CHead d1 
-(Bind Void) u))).(\lambda (H4: (eq C (CHead c2 (Bind Abbr) u0) c)).((let H5 
-\def (eq_ind C (CHead c1 (Bind Abst) t) (\lambda (e: C).(match e return 
-(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) 
-\Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow 
-(match b return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst 
-\Rightarrow True | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) 
-I (CHead d1 (Bind Void) u) H3) in (False_ind ((eq C (CHead c2 (Bind Abbr) u0) 
-c) \to ((csuba g c1 c2) \to ((arity g c1 t (asucc g a)) \to ((arity g c2 u0 
-a) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Void) u))) (\lambda 
-(d2: C).(csuba g d1 d2))))))) H5)) H4 H0 H1 H2)))]) in (H0 (refl_equal C 
-(CHead d1 (Bind Void) u)) (refl_equal C c))))))).
-
-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)).(let H0 \def (match H return 
-(\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csuba ? c0 c1)).((eq C c0 
-(CHead d1 (Bind Abst) u1)) \to ((eq C c1 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))))))))))) with [(csuba_sort n) \Rightarrow (\lambda (H0: 
-(eq C (CSort n) (CHead d1 (Bind Abst) u1))).(\lambda (H1: (eq C (CSort n) 
-c)).((let H2 \def (eq_ind C (CSort n) (\lambda (e: C).(match e return 
-(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) 
-\Rightarrow False])) I (CHead d1 (Bind Abst) u1) H0) in (False_ind ((eq C 
-(CSort n) 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))))))) 
-H2)) H1))) | (csuba_head c1 c2 H0 k u) \Rightarrow (\lambda (H1: (eq C (CHead 
-c1 k u) (CHead d1 (Bind Abst) u1))).(\lambda (H2: (eq C (CHead c2 k u) 
-c)).((let H3 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: 
-C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead 
-c1 k u) (CHead d1 (Bind Abst) u1) H1) in ((let H4 \def (f_equal C K (\lambda 
-(e: C).(match e return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | 
-(CHead _ k _) \Rightarrow k])) (CHead c1 k u) (CHead d1 (Bind Abst) u1) H1) 
-in ((let H5 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: 
-C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead 
-c1 k u) (CHead d1 (Bind Abst) u1) H1) in (eq_ind C d1 (\lambda (c0: C).((eq K 
-k (Bind Abst)) \to ((eq T u u1) \to ((eq C (CHead c2 k u) c) \to ((csuba g c0 
-c2) \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))))))))))) (\lambda 
-(H6: (eq K k (Bind Abst))).(eq_ind K (Bind Abst) (\lambda (k0: K).((eq T u 
-u1) \to ((eq C (CHead c2 k0 u) c) \to ((csuba g d1 c2) \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)))))))))) (\lambda (H7: (eq T u u1)).(eq_ind T u1 
-(\lambda (t: T).((eq C (CHead c2 (Bind Abst) t) c) \to ((csuba g d1 c2) \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))))))))) (\lambda (H8: (eq C (CHead 
-c2 (Bind Abst) u1) c)).(eq_ind C (CHead c2 (Bind Abst) u1) (\lambda (c: 
-C).((csuba g d1 c2) \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)))))))) 
-(\lambda (H9: (csuba g d1 c2)).(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)) 
-H9))) c H8)) u (sym_eq T u u1 H7))) k (sym_eq K k (Bind Abst) H6))) c1 
-(sym_eq C c1 d1 H5))) H4)) H3)) H2 H0))) | (csuba_abst c1 c2 H0 t a H1 u H2) 
-\Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst) t) (CHead d1 (Bind 
-Abst) u1))).(\lambda (H4: (eq C (CHead c2 (Bind Abbr) u) c)).((let H5 \def 
-(f_equal C T (\lambda (e: C).(match e return (\lambda (_: C).T) with [(CSort 
-_) \Rightarrow t | (CHead _ _ t) \Rightarrow t])) (CHead c1 (Bind Abst) t) 
-(CHead d1 (Bind Abst) u1) H3) in ((let H6 \def (f_equal C C (\lambda (e: 
-C).(match e return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead 
-c _ _) \Rightarrow c])) (CHead c1 (Bind Abst) t) (CHead d1 (Bind Abst) u1) 
-H3) in (eq_ind C d1 (\lambda (c0: C).((eq T t u1) \to ((eq C (CHead c2 (Bind 
-Abbr) u) c) \to ((csuba g c0 c2) \to ((arity g c0 t (asucc g a)) \to ((arity 
-g c2 u a) \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 (a0: A).(arity g d1 u1 (asucc g a0))))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 a0)))))))))))) 
-(\lambda (H7: (eq T t u1)).(eq_ind T u1 (\lambda (t0: T).((eq C (CHead c2 
-(Bind Abbr) u) c) \to ((csuba g d1 c2) \to ((arity g d1 t0 (asucc g a)) \to 
-((arity g c2 u a) \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 (a0: A).(arity g d1 u1 (asucc g a0))))) 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 
-a0))))))))))) (\lambda (H8: (eq C (CHead c2 (Bind Abbr) u) c)).(eq_ind C 
-(CHead c2 (Bind Abbr) u) (\lambda (c: C).((csuba g d1 c2) \to ((arity g d1 u1 
-(asucc g a)) \to ((arity g c2 u a) \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 (a0: A).(arity g d1 u1 (asucc 
-g a0))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 
-a0)))))))))) (\lambda (H9: (csuba g d1 c2)).(\lambda (H10: (arity g d1 u1 
-(asucc g a))).(\lambda (H11: (arity g c2 u a)).(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)) H9 H10 H11))))) c H8)) t (sym_eq T t 
-u1 H7))) c1 (sym_eq C c1 d1 H6))) H5)) H4 H0 H1 H2)))]) in (H0 (refl_equal C 
-(CHead d1 (Bind Abst) u1)) (refl_equal C c))))))).
-
-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)).(let H0 \def (match H 
-return (\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csuba ? c0 c1)).((eq C 
-c0 (CHead d1 (Flat f) u1)) \to ((eq C c1 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))))))))) with [(csuba_sort n) \Rightarrow 
-(\lambda (H0: (eq C (CSort n) (CHead d1 (Flat f) u1))).(\lambda (H1: (eq C 
-(CSort n) c)).((let H2 \def (eq_ind C (CSort n) (\lambda (e: C).(match e 
-return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) 
-\Rightarrow False])) I (CHead d1 (Flat f) u1) H0) in (False_ind ((eq C (CSort 
-n) 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))))) H2)) 
-H1))) | (csuba_head c1 c2 H0 k u) \Rightarrow (\lambda (H1: (eq C (CHead c1 k 
-u) (CHead d1 (Flat f) u1))).(\lambda (H2: (eq C (CHead c2 k u) c)).((let H3 
-\def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: C).T) with 
-[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u) 
-(CHead d1 (Flat f) u1) H1) in ((let H4 \def (f_equal C K (\lambda (e: 
-C).(match e return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead 
-_ k _) \Rightarrow k])) (CHead c1 k u) (CHead d1 (Flat f) u1) H1) in ((let H5 
-\def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: C).C) with 
-[(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u) 
-(CHead d1 (Flat f) u1) H1) in (eq_ind C d1 (\lambda (c0: C).((eq K k (Flat 
-f)) \to ((eq T u u1) \to ((eq C (CHead c2 k u) c) \to ((csuba g c0 c2) \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))))))))) (\lambda (H6: 
-(eq K k (Flat f))).(eq_ind K (Flat f) (\lambda (k0: K).((eq T u u1) \to ((eq 
-C (CHead c2 k0 u) c) \to ((csuba g d1 c2) \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)))))))) (\lambda (H7: (eq T u u1)).(eq_ind 
-T u1 (\lambda (t: T).((eq C (CHead c2 (Flat f) t) c) \to ((csuba g d1 c2) \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))))))) (\lambda (H8: 
-(eq C (CHead c2 (Flat f) u1) c)).(eq_ind C (CHead c2 (Flat f) u1) (\lambda 
-(c: C).((csuba g d1 c2) \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)))))) (\lambda (H9: (csuba g d1 c2)).(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)) H9)) c H8)) u (sym_eq T u u1 H7))) k (sym_eq K k (Flat f) 
-H6))) c1 (sym_eq C c1 d1 H5))) H4)) H3)) H2 H0))) | (csuba_abst c1 c2 H0 t a 
-H1 u H2) \Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst) t) (CHead d1 
-(Flat f) u1))).(\lambda (H4: (eq C (CHead c2 (Bind Abbr) u) c)).((let H5 \def 
-(eq_ind C (CHead c1 (Bind Abst) t) (\lambda (e: C).(match e return (\lambda 
-(_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow 
-(match k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat 
-_) \Rightarrow False])])) I (CHead d1 (Flat f) u1) H3) in (False_ind ((eq C 
-(CHead c2 (Bind Abbr) u) c) \to ((csuba g c1 c2) \to ((arity g c1 t (asucc g 
-a)) \to ((arity g c2 u a) \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)))))))) H5)) H4 H0 H1 H2)))]) in (H0 (refl_equal C (CHead d1 (Flat f) 
-u1)) (refl_equal C c)))))))).
-
-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)).(let H0 \def 
-(match H return (\lambda (c: C).(\lambda (c0: C).(\lambda (_: (csuba ? c 
-c0)).((eq C c (CHead e1 (Bind b1) v1)) \to ((eq C c0 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)))))))))) with [(csuba_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) 
-(CHead e1 (Bind b1) v1))).(\lambda (H1: (eq C (CSort n) c2)).((let H2 \def 
-(eq_ind C (CSort n) (\lambda (e: C).(match e return (\lambda (_: C).Prop) 
-with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I 
-(CHead e1 (Bind b1) v1) H0) in (False_ind ((eq C (CSort n) 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)))))) H2)) H1))) | (csuba_head c1 c0 H0 k u) \Rightarrow (\lambda (H1: (eq 
-C (CHead c1 k u) (CHead e1 (Bind b1) v1))).(\lambda (H2: (eq C (CHead c0 k u) 
-c2)).((let H3 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: 
-C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead 
-c1 k u) (CHead e1 (Bind b1) v1) H1) in ((let H4 \def (f_equal C K (\lambda 
-(e: C).(match e return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | 
-(CHead _ k _) \Rightarrow k])) (CHead c1 k u) (CHead e1 (Bind b1) v1) H1) in 
-((let H5 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: C).C) 
-with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k 
-u) (CHead e1 (Bind b1) v1) H1) in (eq_ind C e1 (\lambda (c: C).((eq K k (Bind 
-b1)) \to ((eq T u v1) \to ((eq C (CHead c0 k u) c2) \to ((csuba g c c0) \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)))))))))) (\lambda (H6: (eq K k (Bind b1))).(eq_ind K (Bind 
-b1) (\lambda (k0: K).((eq T u v1) \to ((eq C (CHead c0 k0 u) c2) \to ((csuba 
-g e1 c0) \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))))))))) (\lambda (H7: (eq T u v1)).(eq_ind 
-T v1 (\lambda (t: T).((eq C (CHead c0 (Bind b1) t) c2) \to ((csuba g e1 c0) 
-\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)))))))) (\lambda (H8: (eq C (CHead c0 (Bind b1) v1) 
-c2)).(eq_ind C (CHead c0 (Bind b1) v1) (\lambda (c: C).((csuba g e1 c0) \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 e1 e2))))))) (\lambda (H9: (csuba g e1 c0)).(let H10 \def 
-(eq_ind_r C c2 (\lambda (c: C).(csuba g (CHead e1 (Bind b1) v1) c)) H (CHead 
-c0 (Bind b1) v1) H8) in (ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: 
-C).(\lambda (v2: T).(eq C (CHead c0 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) 
-(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))) b1 c0 v1 
-(refl_equal C (CHead c0 (Bind b1) v1)) H9))) c2 H8)) u (sym_eq T u v1 H7))) k 
-(sym_eq K k (Bind b1) H6))) c1 (sym_eq C c1 e1 H5))) H4)) H3)) H2 H0))) | 
-(csuba_abst c1 c0 H0 t a H1 u H2) \Rightarrow (\lambda (H3: (eq C (CHead c1 
-(Bind Abst) t) (CHead e1 (Bind b1) v1))).(\lambda (H4: (eq C (CHead c0 (Bind 
-Abbr) u) c2)).((let H5 \def (f_equal C T (\lambda (e: C).(match e return 
-(\lambda (_: C).T) with [(CSort _) \Rightarrow t | (CHead _ _ t) \Rightarrow 
-t])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H3) in ((let H6 \def 
-(f_equal C B (\lambda (e: C).(match e return (\lambda (_: C).B) with [(CSort 
-_) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match k return (\lambda (_: 
-K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead c1 
-(Bind Abst) t) (CHead e1 (Bind b1) v1) H3) in ((let H7 \def (f_equal C C 
-(\lambda (e: C).(match e return (\lambda (_: C).C) with [(CSort _) 
-\Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 (Bind Abst) t) 
-(CHead e1 (Bind b1) v1) H3) in (eq_ind C e1 (\lambda (c: C).((eq B Abst b1) 
-\to ((eq T t v1) \to ((eq C (CHead c0 (Bind Abbr) u) c2) \to ((csuba g c c0) 
-\to ((arity g c t (asucc g a)) \to ((arity g c0 u a) \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)))))))))))) (\lambda (H8: (eq B Abst b1)).(eq_ind B Abst (\lambda (_: 
-B).((eq T t v1) \to ((eq C (CHead c0 (Bind Abbr) u) c2) \to ((csuba g e1 c0) 
-\to ((arity g e1 t (asucc g a)) \to ((arity g c0 u a) \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))))))))))) (\lambda (H9: (eq T t v1)).(eq_ind T v1 (\lambda (t0: T).((eq C 
-(CHead c0 (Bind Abbr) u) c2) \to ((csuba g e1 c0) \to ((arity g e1 t0 (asucc 
-g a)) \to ((arity g c0 u a) \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)))))))))) (\lambda (H10: 
-(eq C (CHead c0 (Bind Abbr) u) c2)).(eq_ind C (CHead c0 (Bind Abbr) u) 
-(\lambda (c: C).((csuba g e1 c0) \to ((arity g e1 v1 (asucc g a)) \to ((arity 
-g c0 u a) \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 e1 e2))))))))) (\lambda (H11: (csuba g e1 
-c0)).(\lambda (_: (arity g e1 v1 (asucc g a))).(\lambda (_: (arity g c0 u 
-a)).(let H14 \def (eq_ind_r C c2 (\lambda (c: C).(csuba g (CHead e1 (Bind b1) 
-v1) c)) H (CHead c0 (Bind Abbr) u) H10) in (let H15 \def (eq_ind_r B b1 
-(\lambda (b: B).(csuba g (CHead e1 (Bind b) v1) (CHead c0 (Bind Abbr) u))) 
-H14 Abst H8) in (ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda 
-(v2: T).(eq C (CHead c0 (Bind Abbr) u) (CHead e2 (Bind b2) v2))))) (\lambda 
-(_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))) Abbr c0 u 
-(refl_equal C (CHead c0 (Bind Abbr) u)) H11)))))) c2 H10)) t (sym_eq T t v1 
-H9))) b1 H8)) c1 (sym_eq C c1 e1 H7))) H6)) H5)) H4 H0 H1 H2)))]) in (H0 
-(refl_equal C (CHead e1 (Bind b1) v1)) (refl_equal C c2)))))))).
-
-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)).
-
-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)).((match k return (\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))))) with [(Bind b) \Rightarrow (\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))) | (Flat f) \Rightarrow (\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)))]) 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 (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_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 H2 \def (csuba_gen_abbr g d1 c2 u H1) 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 (H2: (eq C (CHead d1 (Bind Abbr) u) (CSort n0))).(\lambda (_: 
-(eq nat (S n) O)).(\lambda (_: (eq nat O O)).(let H5 \def (match H2 return 
-(\lambda (c: C).(\lambda (_: (eq ? ? c)).((eq C c (CSort n0)) \to (ex2 C 
-(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: 
-C).(csuba g d1 d2)))))) with [refl_equal \Rightarrow (\lambda (H4: (eq C 
-(CHead d1 (Bind Abbr) u) (CSort n0))).(let H5 \def (eq_ind C (CHead d1 (Bind 
-Abbr) u) (\lambda (e: C).(match e return (\lambda (_: C).Prop) with [(CSort 
-_) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n0) H4) 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)))]) in (H5 (refl_equal C (CSort 
-n0))))))) (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 H7 \def (csuba_gen_abbr g c c2 t H5) 
-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 (n: nat).(drop n 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 H7 \def 
-(csuba_gen_abst g c c2 t H5) 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 (n: nat).(drop 
-n 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 (n: nat).(drop n 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 H7 \def (csuba_gen_void g 
-c c2 t H5) in (ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Void) 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 Void) t))).(\lambda (H9: (csuba g c 
-x)).(eq_ind_r C (CHead x (Bind Void) 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 Void) 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 (n: nat).(drop n 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 Void) 
-t) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x0 (drop_drop 
-(Bind Void) n x (CHead x0 (Bind Abbr) u) H14 t) 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 H5 \def (csuba_gen_flat g c c2 t f H3) 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 H2 \def (csuba_gen_abst g d1 c2 u1 
-H1) 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 (H2: (eq C (CHead d1 (Bind Abst) u1) (CSort n0))).(\lambda 
-(_: (eq nat (S n) O)).(\lambda (_: (eq nat O O)).(let H5 \def (match H2 
-return (\lambda (c: C).(\lambda (_: (eq ? ? c)).((eq C c (CSort n0)) \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))))))))) with 
-[refl_equal \Rightarrow (\lambda (H4: (eq C (CHead d1 (Bind Abst) u1) (CSort 
-n0))).(let H5 \def (eq_ind C (CHead d1 (Bind Abst) u1) (\lambda (e: C).(match 
-e return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ 
-_) \Rightarrow True])) I (CSort n0) H4) 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)))]) in (H5 (refl_equal C 
-(CSort n0))))))) (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 H7 \def (csuba_gen_abbr g c c2 t H5) 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 
-(n: nat).(drop n 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 (n: nat).(drop n 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 H7 \def 
-(csuba_gen_abst g c c2 t H5) 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 (n: nat).(drop n 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 (n: nat).(drop n 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 (n: nat).(drop 
-n 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 (n: nat).(drop n 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 H7 \def (csuba_gen_void g c c2 t H5) in 
-(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Void) 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 Void) 
-t))).(\lambda (H9: (csuba g c x)).(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) 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 Void) 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 Void) 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 Void) 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 Void) 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 
-(n: nat).(drop n 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 Void) 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 Void) 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 Void) t) (CHead d2 (Bind Abst) u1))) (\lambda 
-(d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Void) 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 Void) 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 Void) 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 (n: nat).(drop n 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 Void) 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 Void) 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 Void) 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 Void) 
-n x (CHead x0 (Bind Abbr) x1) H17 t) 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 H5 \def (csuba_gen_flat g c c2 t f H3) 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_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))).((match x return (\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)))))))) with [(CSort n) \Rightarrow (\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)))))))) | (CHead c k t) \Rightarrow (\lambda 
-(H3: (drop i O c1 (CHead c k t))).(\lambda (H4: (clear (CHead c k t) (CHead 
-d1 (Bind Abbr) u))).((match k return (\lambda (k0: K).((drop i O c1 (CHead c 
-k0 t)) \to ((clear (CHead c 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)))))))) with [(Bind b) \Rightarrow 
-(\lambda (H5: (drop i O c1 (CHead c (Bind b) t))).(\lambda (H6: (clear (CHead 
-c (Bind b) t) (CHead d1 (Bind Abbr) u))).(let H7 \def (f_equal C C (\lambda 
-(e: C).(match e return (\lambda (_: C).C) with [(CSort _) \Rightarrow d1 | 
-(CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Abbr) u) (CHead c (Bind b) t) 
-(clear_gen_bind b c (CHead d1 (Bind Abbr) u) t H6)) in ((let H8 \def (f_equal 
-C B (\lambda (e: C).(match e return (\lambda (_: C).B) with [(CSort _) 
-\Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k return (\lambda (_: 
-K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d1 
-(Bind Abbr) u) (CHead c (Bind b) t) (clear_gen_bind b c (CHead d1 (Bind Abbr) 
-u) t H6)) in ((let H9 \def (f_equal C T (\lambda (e: C).(match e return 
-(\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow 
-t])) (CHead d1 (Bind Abbr) u) (CHead c (Bind b) t) (clear_gen_bind b c (CHead 
-d1 (Bind Abbr) u) t H6)) in (\lambda (H10: (eq B Abbr b)).(\lambda (H11: (eq 
-C d1 c)).(\lambda (c2: C).(\lambda (H12: (csuba g c1 c2)).(let H13 \def 
-(eq_ind_r T t (\lambda (t: T).(drop i O c1 (CHead c (Bind b) t))) H5 u H9) in 
-(let H14 \def (eq_ind_r B b (\lambda (b: B).(drop i O c1 (CHead c (Bind b) 
-u))) H13 Abbr H10) in (let H15 \def (eq_ind_r C c (\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 (x0: C).(\lambda (H17: (drop i O c2 (CHead x0 (Bind Abbr) 
-u))).(\lambda (H18: (csuba g d1 x0)).(ex_intro2 C (\lambda (d2: C).(getl i c2 
-(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x0 (getl_intro i 
-c2 (CHead x0 (Bind Abbr) u) (CHead x0 (Bind Abbr) u) H17 (clear_bind Abbr x0 
-u)) H18)))) H16)))))))))) H8)) H7)))) | (Flat f) \Rightarrow (\lambda (H5: 
-(drop i O c1 (CHead c (Flat f) t))).(\lambda (H6: (clear (CHead c (Flat f) t) 
-(CHead d1 (Bind Abbr) u))).(let H7 \def H5 in (unintro C c1 (\lambda (c0: 
-C).((drop i O c0 (CHead c (Flat f) t)) \to (\forall (c2: C).((csuba g c0 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 (x0: 
-C).((drop n O x0 (CHead c (Flat f) t)) \to (\forall (c2: C).((csuba g x0 c2) 
-\to (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))) (\lambda 
-(d2: C).(csuba g d1 d2)))))))) (\lambda (x0: C).(\lambda (H8: (drop O O x0 
-(CHead c (Flat f) t))).(\lambda (c2: C).(\lambda (H9: (csuba g x0 c2)).(let 
-H10 \def (eq_ind C x0 (\lambda (c: C).(csuba g c c2)) H9 (CHead c (Flat f) t) 
-(drop_gen_refl x0 (CHead c (Flat f) t) H8)) in (let H_y \def (clear_flat c 
-(CHead d1 (Bind Abbr) u) (clear_gen_flat f c (CHead d1 (Bind Abbr) u) t H6) f 
-t) in (let H11 \def (csuba_clear_conf g (CHead c (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 
-(x1: C).(\lambda (H12: (csuba g (CHead d1 (Bind Abbr) u) x1)).(\lambda (H13: 
-(clear c2 x1)).(let H14 \def (csuba_gen_abbr g d1 x1 u H12) in (ex2_ind C 
-(\lambda (d2: C).(eq C x1 (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 (x2: C).(\lambda (H15: (eq C x1 
-(CHead x2 (Bind Abbr) u))).(\lambda (H16: (csuba g d1 x2)).(let H17 \def 
-(eq_ind C x1 (\lambda (c: C).(clear c2 c)) H13 (CHead x2 (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)) x2 (getl_intro O c2 (CHead x2 (Bind Abbr) 
-u) c2 (drop_refl c2) H17) H16))))) H14))))) H11)))))))) (\lambda (n: 
-nat).(\lambda (H8: ((\forall (x: C).((drop n O x (CHead c (Flat f) t)) \to 
-(\forall (c2: C).((csuba g x c2) \to (ex2 C (\lambda (d2: C).(getl n c2 
-(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))))))).(\lambda 
-(x0: C).(\lambda (H9: (drop (S n) O x0 (CHead c (Flat f) t))).(\lambda (c2: 
-C).(\lambda (H10: (csuba g x0 c2)).(let H11 \def (drop_clear x0 (CHead c 
-(Flat f) t) n H9) in (ex2_3_ind B C T (\lambda (b: B).(\lambda (e: 
-C).(\lambda (v: T).(clear x0 (CHead e (Bind b) v))))) (\lambda (_: 
-B).(\lambda (e: C).(\lambda (_: T).(drop n O e (CHead c (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 (x1: B).(\lambda (x2: C).(\lambda (x3: 
-T).(\lambda (H12: (clear x0 (CHead x2 (Bind x1) x3))).(\lambda (H13: (drop n 
-O x2 (CHead c (Flat f) t))).(let H14 \def (csuba_clear_conf g x0 c2 H10 
-(CHead x2 (Bind x1) x3) H12) in (ex2_ind C (\lambda (e2: C).(csuba g (CHead 
-x2 (Bind x1) x3) 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 (x4: C).(\lambda (H15: (csuba g (CHead x2 (Bind x1) x3) 
-x4)).(\lambda (H16: (clear c2 x4)).(let H17 \def (csuba_gen_bind g x1 x2 x4 
-x3 H15) in (ex2_3_ind B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: 
-T).(eq C x4 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
-C).(\lambda (_: T).(csuba g x2 e2)))) (ex2 C (\lambda (d2: C).(getl (S n) c2 
-(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x5: 
-B).(\lambda (x6: C).(\lambda (x7: T).(\lambda (H18: (eq C x4 (CHead x6 (Bind 
-x5) x7))).(\lambda (H19: (csuba g x2 x6)).(let H20 \def (eq_ind C x4 (\lambda 
-(c: C).(clear c2 c)) H16 (CHead x6 (Bind x5) x7) H18) in (let H21 \def (H8 x2 
-H13 x6 H19) in (ex2_ind C (\lambda (d2: C).(getl n x6 (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 (x8: 
-C).(\lambda (H22: (getl n x6 (CHead x8 (Bind Abbr) u))).(\lambda (H23: (csuba 
-g d1 x8)).(ex_intro2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) 
-u))) (\lambda (d2: C).(csuba g d1 d2)) x8 (getl_clear_bind x5 c2 x6 x7 H20 
-(CHead x8 (Bind Abbr) u) n H22) H23)))) H21)))))))) H17))))) H14))))))) 
-H11)))))))) i) H7))))]) H3 H4)))]) 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))).((match x return (\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))))))))))) with 
-[(CSort n) \Rightarrow (\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))))))))))) | (CHead c k t) \Rightarrow (\lambda (H3: 
-(drop i O c1 (CHead c k t))).(\lambda (H4: (clear (CHead c k t) (CHead d1 
-(Bind Abst) u1))).((match k return (\lambda (k0: K).((drop i O c1 (CHead c k0 
-t)) \to ((clear (CHead c 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))))))))))) with [(Bind b) \Rightarrow (\lambda (H5: (drop i O c1 (CHead c 
-(Bind b) t))).(\lambda (H6: (clear (CHead c (Bind b) t) (CHead d1 (Bind Abst) 
-u1))).(let H7 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: 
-C).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _) \Rightarrow c])) (CHead 
-d1 (Bind Abst) u1) (CHead c (Bind b) t) (clear_gen_bind b c (CHead d1 (Bind 
-Abst) u1) t H6)) in ((let H8 \def (f_equal C B (\lambda (e: C).(match e 
-return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abst | (CHead _ k _) 
-\Rightarrow (match k return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | 
-(Flat _) \Rightarrow Abst])])) (CHead d1 (Bind Abst) u1) (CHead c (Bind b) t) 
-(clear_gen_bind b c (CHead d1 (Bind Abst) u1) t H6)) in ((let H9 \def 
-(f_equal C T (\lambda (e: C).(match e return (\lambda (_: C).T) with [(CSort 
-_) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead d1 (Bind Abst) u1) 
-(CHead c (Bind b) t) (clear_gen_bind b c (CHead d1 (Bind Abst) u1) t H6)) in 
-(\lambda (H10: (eq B Abst b)).(\lambda (H11: (eq C d1 c)).(\lambda (c2: 
-C).(\lambda (H12: (csuba g c1 c2)).(let H13 \def (eq_ind_r T t (\lambda (t: 
-T).(drop i O c1 (CHead c (Bind b) t))) H5 u1 H9) in (let H14 \def (eq_ind_r B 
-b (\lambda (b: B).(drop i O c1 (CHead c (Bind b) u1))) H13 Abst H10) in (let 
-H15 \def (eq_ind_r C c (\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 (x0: C).(\lambda (H18: (drop i O c2 (CHead x0 (Bind Abst) 
-u1))).(\lambda (H19: (csuba g d1 x0)).(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)) x0 (getl_intro i c2 
-(CHead x0 (Bind Abst) u1) (CHead x0 (Bind Abst) u1) H18 (clear_bind Abst x0 
-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 (x0: 
-C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H18: (drop i O c2 (CHead x0 
-(Bind Abbr) x1))).(\lambda (H19: (csuba g d1 x0)).(\lambda (H20: (arity g d1 
-u1 (asucc g x2))).(\lambda (H21: (arity g x0 x1 x2)).(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)))) 
-x0 x1 x2 (getl_intro i c2 (CHead x0 (Bind Abbr) x1) (CHead x0 (Bind Abbr) x1) 
-H18 (clear_bind Abbr x0 x1)) H19 H20 H21))))))))) H17)) H16)))))))))) H8)) 
-H7)))) | (Flat f) \Rightarrow (\lambda (H5: (drop i O c1 (CHead c (Flat f) 
-t))).(\lambda (H6: (clear (CHead c (Flat f) t) (CHead d1 (Bind Abst) 
-u1))).(let H7 \def H5 in (unintro C c1 (\lambda (c0: C).((drop i O c0 (CHead 
-c (Flat f) t)) \to (\forall (c2: C).((csuba g c0 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 (x0: 
-C).((drop n O x0 (CHead c (Flat f) t)) \to (\forall (c2: C).((csuba g x0 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 
-(x0: C).(\lambda (H8: (drop O O x0 (CHead c (Flat f) t))).(\lambda (c2: 
-C).(\lambda (H9: (csuba g x0 c2)).(let H10 \def (eq_ind C x0 (\lambda (c: 
-C).(csuba g c c2)) H9 (CHead c (Flat f) t) (drop_gen_refl x0 (CHead c (Flat 
-f) t) H8)) in (let H_y \def (clear_flat c (CHead d1 (Bind Abst) u1) 
-(clear_gen_flat f c (CHead d1 (Bind Abst) u1) t H6) f t) in (let H11 \def 
-(csuba_clear_conf g (CHead c (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 (x1: C).(\lambda (H12: (csuba g (CHead d1 (Bind Abst) u1) 
-x1)).(\lambda (H13: (clear c2 x1)).(let H14 \def (csuba_gen_abst g d1 x1 u1 
-H12) in (or_ind (ex2 C (\lambda (d2: C).(eq C 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).(eq C 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).(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 x1 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 
-d2)))).(ex2_ind C (\lambda (d2: C).(eq C x1 (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 (x2: C).(\lambda (H16: (eq C x1 (CHead x2 (Bind Abst) 
-u1))).(\lambda (H17: (csuba g d1 x2)).(let H18 \def (eq_ind C x1 (\lambda (c: 
-C).(clear c2 c)) H13 (CHead x2 (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)) 
-x2 (getl_intro O c2 (CHead x2 (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 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).(eq C 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).(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 (x3: T).(\lambda (x4: A).(\lambda (H16: (eq C x1 (CHead x2 (Bind 
-Abbr) x3))).(\lambda (H17: (csuba g d1 x2)).(\lambda (H18: (arity g d1 u1 
-(asucc g x4))).(\lambda (H19: (arity g x2 x3 x4)).(let H20 \def (eq_ind C x1 
-(\lambda (c: C).(clear c2 c)) H13 (CHead x2 (Bind Abbr) x3) 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)))) x2 x3 x4 (getl_intro O c2 (CHead x2 (Bind Abbr) x3) c2 (drop_refl 
-c2) H20) H17 H18 H19)))))))))) H15)) H14))))) H11)))))))) (\lambda (n: 
-nat).(\lambda (H8: ((\forall (x: C).((drop n O x (CHead c (Flat f) t)) \to 
-(\forall (c2: C).((csuba g x 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 (x0: C).(\lambda (H9: (drop (S n) O x0 (CHead c (Flat 
-f) t))).(\lambda (c2: C).(\lambda (H10: (csuba g x0 c2)).(let H11 \def 
-(drop_clear x0 (CHead c (Flat f) t) n H9) in (ex2_3_ind B C T (\lambda (b: 
-B).(\lambda (e: C).(\lambda (v: T).(clear x0 (CHead e (Bind b) v))))) 
-(\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop n O e (CHead c (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 (x1: 
-B).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H12: (clear x0 (CHead x2 (Bind 
-x1) x3))).(\lambda (H13: (drop n O x2 (CHead c (Flat f) t))).(let H14 \def 
-(csuba_clear_conf g x0 c2 H10 (CHead x2 (Bind x1) x3) H12) in (ex2_ind C 
-(\lambda (e2: C).(csuba g (CHead x2 (Bind x1) x3) 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 (x4: C).(\lambda (H15: (csuba g (CHead x2 (Bind x1) x3) 
-x4)).(\lambda (H16: (clear c2 x4)).(let H17 \def (csuba_gen_bind g x1 x2 x4 
-x3 H15) in (ex2_3_ind B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: 
-T).(eq C x4 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
-C).(\lambda (_: T).(csuba g x2 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: B).(\lambda (x6: C).(\lambda (x7: T).(\lambda (H18: 
-(eq C x4 (CHead x6 (Bind x5) x7))).(\lambda (H19: (csuba g x2 x6)).(let H20 
-\def (eq_ind C x4 (\lambda (c: C).(clear c2 c)) H16 (CHead x6 (Bind x5) x7) 
-H18) in (let H21 \def (H8 x2 H13 x6 H19) in (or_ind (ex2 C (\lambda (d2: 
-C).(getl n x6 (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 x6 
-(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 x6 (CHead d2 (Bind 
-Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: 
-C).(getl n x6 (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 (x8: 
-C).(\lambda (H23: (getl n x6 (CHead x8 (Bind Abst) u1))).(\lambda (H24: 
-(csuba g d1 x8)).(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)) x8 (getl_clear_bind x5 c2 x6 x7 H20 
-(CHead x8 (Bind Abst) u1) n H23) H24))))) H22)) (\lambda (H22: (ex4_3 C T A 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl n x6 (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 x6 (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 (x8: C).(\lambda (x9: T).(\lambda 
-(x10: A).(\lambda (H23: (getl n x6 (CHead x8 (Bind Abbr) x9))).(\lambda (H24: 
-(csuba g d1 x8)).(\lambda (H25: (arity g d1 u1 (asucc g x10))).(\lambda (H26: 
-(arity g x8 x9 x10)).(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)))) x8 x9 x10 (getl_clear_bind x5 c2 
-x6 x7 H20 (CHead x8 (Bind Abbr) x9) n H23) H24 H25 H26))))))))) H22)) 
-H21)))))))) H17))))) H14))))))) H11)))))))) i) H7))))]) H3 H4)))]) H1 H2)))) 
-H0))))))).
-
-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 (a: A).(arity g d u (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 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))))).
-
-axiom 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 (arity g c2 t a)))))))
-.
-
-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))))))) 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))))))))).
-
-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)) (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 return 
-(\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | 
-(THead k _ _) \Rightarrow k])) (THead k u1 t3) (THead (Bind b) u t) H12) in 
-((let H14 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: 
-T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t 
-_) \Rightarrow t])) (THead k u1 t3) (THead (Bind b) u t) H12) in ((let H15 
-\def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t) 
-\Rightarrow t])) (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 (t: T).(pr0 t 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 (t: 
-T).(pr0 t 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 return 
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match 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 return (\lambda 
-(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
-| (THead k _ _) \Rightarrow (match 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 return (\lambda (_: T).B) with [(TSort _) 
-\Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k _ _) \Rightarrow 
-(match k return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (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 return (\lambda (_: 
-T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t 
-_) \Rightarrow t])) (THead (Bind Abbr) u1 t3) (THead (Bind b) u t) H13) in 
-((let H16 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: 
-T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ 
-t) \Rightarrow t])) (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 (t: T).(pr0 t 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 (t: T).(pr0 t u2)) H8 u H17) in (let H23 \def (eq_ind_r B b 
-(\lambda (b: B).((eq T t (THead (Bind b) u t)) \to (arity g c2 t4 a2))) H19 
-Abbr H18) in (let H24 \def (eq_ind_r B b (\lambda (b: B).((eq T u (THead 
-(Bind b) u t)) \to (arity g c2 u2 a2))) H21 Abbr H18) in (let H25 \def 
-(eq_ind_r B b (\lambda (b: B).(\forall (c2: C).((wcpr0 (CHead c (Bind b) u) 
-c2) \to (\forall (t2: T).((pr0 t t2) \to (arity g c2 t2 a2)))))) H4 Abbr H18) 
-in (let H26 \def (eq_ind_r B b (\lambda (b: B).(arity g (CHead c (Bind b) u) 
-t a2)) H3 Abbr H18) in (let H27 \def (eq_ind_r B b (\lambda (b: B).(not (eq B 
-b 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 return (\lambda (_: T).B) with [(TSort _) \Rightarrow b0 | (TLRef 
-_) \Rightarrow b0 | (THead k _ _) \Rightarrow (match k return (\lambda (_: 
-K).B) with [(Bind b) \Rightarrow b | (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 return (\lambda (_: T).T) with [(TSort 
-_) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) 
-(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 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 u t0) \Rightarrow (THead k (lref_map f d u) 
-(lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O 
-t3) | (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 u t0) \Rightarrow (THead k (lref_map f d u) 
-(lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O 
-t3) | (THead _ _ t) \Rightarrow t])) (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 (b: B).(not (eq B b Abst))) H8 b 
-H16) in (let H18 \def (eq_ind_r T t (\lambda (t: T).((eq T t3 (THead (Bind b) 
-u t)) \to (arity g c2 t4 a2))) H10 (lift (S O) O t3) H14) in (let H19 \def 
-(eq_ind_r T t (\lambda (t: T).(\forall (c2: C).((wcpr0 (CHead c (Bind b) u) 
-c2) \to (\forall (t2: T).((pr0 t t2) \to (arity g c2 t2 a2)))))) H4 (lift (S 
-O) O t3) H14) in (let H20 \def (eq_ind_r T t (\lambda (t: T).(arity g (CHead 
-c (Bind b) u) t 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 
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match 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)) (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 return 
-(\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | 
-(THead k _ _) \Rightarrow k])) (THead k u1 t3) (THead (Bind Abst) u t) H11) 
-in ((let H13 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: 
-T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t 
-_) \Rightarrow t])) (THead k u1 t3) (THead (Bind Abst) u t) H11) in ((let H14 
-\def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t) 
-\Rightarrow t])) (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 (t: T).(pr0 t 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 (t: T).(pr0 t 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 
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match 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 return (\lambda (_: 
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
-(THead k _ _) \Rightarrow (match 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 return (\lambda (_: T).Prop) with [(TSort 
-_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
-\Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow 
-(match 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 return (\lambda (_: T).B) with [(TSort _) 
-\Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k 
-return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (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 return (\lambda (_: 
-T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t 
-_) \Rightarrow t])) (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 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 u t0) \Rightarrow 
-(THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda 
-(x: nat).(plus x (S O))) O t3) | (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 u t0) \Rightarrow 
-(THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda 
-(x: nat).(plus x (S O))) O t3) | (THead _ _ t) \Rightarrow t])) (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 (b: 
-B).(not (eq B b Abst))) H7 Abst H15) in (let H17 \def (eq_ind_r T t (\lambda 
-(t: T).((eq T t3 (THead (Bind Abst) u t)) \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 (t: 
-T).(\forall (c2: C).((wcpr0 (CHead c (Bind Abst) u) c2) \to (\forall (t2: 
-T).((pr0 t t2) \to (arity g c2 t2 a2)))))) H3 (lift (S O) O t3) H13) in (let 
-H19 \def (eq_ind_r T t (\lambda (t: T).(arity g (CHead c (Bind Abst) u) t 
-a2)) H2 (lift (S O) O t3) H13) in (let H20 \def (match (H16 (refl_equal B 
-Abst)) 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 return (\lambda (_: T).Prop) with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
-\Rightarrow (match 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)) 
-(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 return (\lambda (_: T).K) with 
-[(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) 
-\Rightarrow k])) (THead k u1 t3) (THead (Flat Appl) u t) H11) in ((let H13 
-\def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t _) 
-\Rightarrow t])) (THead k u1 t3) (THead (Flat Appl) u t) H11) in ((let H14 
-\def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t) 
-\Rightarrow t])) (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 (t: T).(pr0 t 
-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 (t: T).(pr0 t 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 
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) 
-\Rightarrow v1 | (THead _ t _) \Rightarrow t])) (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 return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow (THead (Bind Abst) u0 t3) | (TLRef _) \Rightarrow (THead (Bind 
-Abst) u0 t3) | (THead _ _ t) \Rightarrow t])) (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 (t: T).(pr0 t v2)) H7 u H14) in (let H17 \def (eq_ind_r T t (\lambda 
-(t: T).((eq T t3 (THead (Flat Appl) u t)) \to (arity g c2 t4 a2))) H10 (THead 
-(Bind Abst) u0 t3) H13) in (let H18 \def (eq_ind_r T t (\lambda (t: T).((eq T 
-u (THead (Flat Appl) u t)) \to (arity g c2 v2 a2))) H15 (THead (Bind Abst) u0 
-t3) H13) in (let H19 \def (eq_ind_r T t (\lambda (t: T).(\forall (c2: 
-C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 t t2) \to (arity g c2 t2 (AHead 
-a1 a2))))))) H3 (THead (Bind Abst) u0 t3) H13) in (let H20 \def (eq_ind_r T t 
-(\lambda (t: T).(arity g c t (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 return (\lambda (_: A).A) with 
-[(ASort _ _) \Rightarrow a1 | (AHead a _) \Rightarrow a])) (AHead a1 a2) 
-(AHead x0 x1) H24) in ((let H28 \def (f_equal A A (\lambda (e: A).(match e 
-return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a2 | (AHead _ a) 
-\Rightarrow a])) (AHead a1 a2) (AHead x0 x1) H24) in (\lambda (H29: (eq A a1 
-x0)).(let H30 \def (eq_ind_r A x1 (\lambda (a: A).(arity g (CHead c2 (Bind 
-Abst) u0) t4 a)) H26 a2 H28) in (let H31 \def (eq_ind_r A x0 (\lambda (a: 
-A).(arity g c2 u0 (asucc g a))) 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 
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) 
-\Rightarrow v1 | (THead _ t _) \Rightarrow t])) (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 return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow (THead (Bind b) u1 t3) | (TLRef _) \Rightarrow (THead (Bind b) u1 
-t3) | (THead _ _ t) \Rightarrow t])) (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 (t: 
-T).(pr0 t v2)) H8 u H17) in (let H20 \def (eq_ind_r T t (\lambda (t: T).((eq 
-T t3 (THead (Flat Appl) u t)) \to (arity g c2 t4 a2))) H13 (THead (Bind b) u1 
-t3) H16) in (let H21 \def (eq_ind_r T t (\lambda (t: T).((eq T u1 (THead 
-(Flat Appl) u t)) \to (arity g c2 u2 a2))) H11 (THead (Bind b) u1 t3) H16) in 
-(let H22 \def (eq_ind_r T t (\lambda (t: T).((eq T u (THead (Flat Appl) u t)) 
-\to (arity g c2 v2 a2))) H18 (THead (Bind b) u1 t3) H16) in (let H23 \def 
-(eq_ind_r T t (\lambda (t: T).(\forall (c2: C).((wcpr0 c c2) \to (\forall 
-(t2: T).((pr0 t t2) \to (arity g c2 t2 (AHead a1 a2))))))) H3 (THead (Bind b) 
-u1 t3) H16) in (let H24 \def (eq_ind_r T t (\lambda (t: T).(arity g c t 
-(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 return (\lambda (_: T).Prop) with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
-\Rightarrow (match 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 return 
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match 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 return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | 
-(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return 
-(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow 
-(match 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)) 
-(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 return (\lambda (_: T).K) with 
-[(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) 
-\Rightarrow k])) (THead k u1 t3) (THead (Flat Cast) u t) H11) in ((let H13 
-\def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t _) 
-\Rightarrow t])) (THead k u1 t3) (THead (Flat Cast) u t) H11) in ((let H14 
-\def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t) 
-\Rightarrow t])) (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 (t: T).(pr0 t 
-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 (t: T).(pr0 t 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 return (\lambda (_: T).Prop) with [(TSort 
-_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
-\Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow 
-False | (Flat f) \Rightarrow (match 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 return (\lambda (_: 
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
-(THead k _ _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind 
-_) \Rightarrow False | (Flat f) \Rightarrow (match 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 return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | 
-(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match 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 return (\lambda (_: T).Prop) 
-with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ 
-_) \Rightarrow (match 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 return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) 
-(THead (Flat Cast) u0 t3) (THead (Flat Cast) u t) H9) in ((let H11 \def 
-(f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort 
-_) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t) \Rightarrow t])) 
-(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 (t: T).(pr0 t 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)))).
-
-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)))).
-
-theorem nf2_gen_base__aux:
- \forall (k: K).(\forall (t: T).(\forall (u: T).((eq T (THead k u t) t) \to 
-(\forall (P: Prop).P))))
-\def
- \lambda (k: K).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (u: T).((eq 
-T (THead k u t0) t0) \to (\forall (P: Prop).P)))) (\lambda (n: nat).(\lambda 
-(u: T).(\lambda (H: (eq T (THead k u (TSort n)) (TSort n))).(\lambda (P: 
-Prop).(let H0 \def (eq_ind T (THead k u (TSort n)) (\lambda (ee: T).(match ee 
-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 (H: (eq T 
-(THead k u (TLRef n)) (TLRef n))).(\lambda (P: Prop).(let H0 \def (eq_ind T 
-(THead k u (TLRef n)) (\lambda (ee: T).(match ee 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 (u: T).((eq T (THead k u t0) t0) 
-\to (\forall (P: Prop).P))))).(\lambda (t1: T).(\lambda (H0: ((\forall (u: 
-T).((eq T (THead k u t1) t1) \to (\forall (P: Prop).P))))).(\lambda (u: 
-T).(\lambda (H1: (eq T (THead k u (THead k0 t0 t1)) (THead k0 t0 
-t1))).(\lambda (P: Prop).(let H2 \def (f_equal T K (\lambda (e: T).(match e 
-return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) 
-\Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k u (THead k0 t0 t1)) 
-(THead k0 t0 t1) H1) in ((let H3 \def (f_equal T T (\lambda (e: T).(match e 
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) 
-\Rightarrow u | (THead _ t _) \Rightarrow t])) (THead k u (THead k0 t0 t1)) 
-(THead k0 t0 t1) H1) in ((let H4 \def (f_equal T T (\lambda (e: T).(match e 
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow (THead k0 t0 t1) | 
-(TLRef _) \Rightarrow (THead k0 t0 t1) | (THead _ _ t) \Rightarrow t])) 
-(THead k u (THead k0 t0 t1)) (THead k0 t0 t1) H1) in (\lambda (_: (eq T u 
-t0)).(\lambda (H6: (eq K k k0)).(let H7 \def (eq_ind K k (\lambda (k: 
-K).(\forall (u: T).((eq T (THead k u t1) t1) \to (\forall (P: Prop).P)))) H0 
-k0 H6) in (H7 t0 H4 P))))) H3)) H2)))))))))) t)).
-
-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 return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef 
-_) \Rightarrow u | (THead _ t _) \Rightarrow t])) (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 (t: T).(pr2 c u t)) 
-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 return (\lambda (_: 
-T).T) with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t) 
-\Rightarrow t])) (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).(nf2_gen_base__aux (Flat Cast) t u (H t 
-(pr2_free c (THead (Flat Cast) u t) t (pr0_epsilon t t (pr0_refl t) u))) 
-P))))).
-
-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 return (\lambda (_: 
-T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) 
-\Rightarrow t])) (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 return (\lambda (_: T).T) with [(TSort _) \Rightarrow t | (TLRef 
-_) \Rightarrow t | (THead _ _ t) \Rightarrow t])) (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_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_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 (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 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_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 (t: T).(pr3 c t1 t)) H4 t1 
-H_y) in (let H6 \def (eq_ind_r T x (\lambda (t: T).(pr3 c t2 t)) 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 (t: T).(pr3 c t 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))))))))).
-
-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: ((\forall (t2: T).((pr2 c u t2) 
-\to (eq T u t2))))).(\lambda (i: nat).(\lambda (H0: ((\forall (t2: T).((pr2 c 
-(TLRef i) t2) \to (eq T (TLRef i) t2))))).(\lambda (t2: T).(\lambda (H1: (pr2 
-c (THead (Flat Appl) u (TLRef i)) t2)).(let H2 \def (pr2_gen_appl c u (TLRef 
-i) t2 H1) 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 u 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 u 
-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 (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 u 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) u (TLRef i)) t2) (\lambda (H3: (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 u u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c 
-(TLRef i) t2))))).(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 u u2))) 
-(\lambda (_: T).(\lambda (t3: T).(pr2 c (TLRef i) t3))) (eq T (THead (Flat 
-Appl) u (TLRef i)) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T 
-t2 (THead (Flat Appl) x0 x1))).(\lambda (H5: (pr2 c u x0)).(\lambda (H6: (pr2 
-c (TLRef i) x1)).(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t: T).(eq T 
-(THead (Flat Appl) u (TLRef i)) t)) (let H7 \def (eq_ind_r T x1 (\lambda (t: 
-T).(pr2 c (TLRef i) t)) H6 (TLRef i) (H0 x1 H6)) in (eq_ind T (TLRef i) 
-(\lambda (t: T).(eq T (THead (Flat Appl) u (TLRef i)) (THead (Flat Appl) x0 
-t))) (let H8 \def (eq_ind_r T x0 (\lambda (t: T).(pr2 c u t)) H5 u (H x0 H5)) 
-in (eq_ind T u (\lambda (t: T).(eq T (THead (Flat Appl) u (TLRef i)) (THead 
-(Flat Appl) t (TLRef i)))) (refl_equal T (THead (Flat Appl) u (TLRef i))) x0 
-(H x0 H5))) x1 (H0 x1 H6))) t2 H4)))))) H3)) (\lambda (H3: (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 u 
-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))))))))).(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 u 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))))))) (eq T (THead (Flat Appl) u (TLRef i)) t2) (\lambda (x0: 
-T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H4: (eq T 
-(TLRef i) (THead (Bind Abst) x0 x1))).(\lambda (H5: (eq T t2 (THead (Bind 
-Abbr) x2 x3))).(\lambda (_: (pr2 c u 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 (t: T).(eq T (THead (Flat Appl) u (TLRef i)) t)) 
-(let H8 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee return (\lambda 
-(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | 
-(THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) x0 x1) H4) in 
-(False_ind (eq T (THead (Flat Appl) u (TLRef i)) (THead (Bind Abbr) x2 x3)) 
-H8)) t2 H5))))))))) 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).(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 u 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 u 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) u (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 (H5: (eq T (TLRef i) (THead 
-(Bind x0) x1 x2))).(\lambda (H6: (eq T t2 (THead (Bind x0) x5 (THead (Flat 
-Appl) (lift (S O) O x4) x3)))).(\lambda (_: (pr2 c u 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 (t: T).(eq T 
-(THead (Flat Appl) u (TLRef i)) t)) (let H10 \def (eq_ind T (TLRef i) 
-(\lambda (ee: T).(match ee return (\lambda (_: T).Prop) with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow 
-False])) I (THead (Bind x0) x1 x2) H5) in (False_ind (eq T (THead (Flat Appl) 
-u (TLRef i)) (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3))) 
-H10)) t2 H6))))))))))))) H3)) H2)))))))).
-
-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 (u: T).(getl i c 
-(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 (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 return (\lambda (_: C).Prop) with [(CSort _) 
-\Rightarrow False | (CHead _ k _) \Rightarrow (match k return (\lambda (_: 
-K).Prop) with [(Bind b) \Rightarrow (match 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)))))))))).
-
-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 H1 \def (match H return (\lambda (p: 
-PList).(\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (drop1 p c0 c1)).((eq 
-PList p PNil) \to ((eq C c0 c) \to ((eq C c1 e) \to (nf2 c t)))))))) with 
-[(drop1_nil c0) \Rightarrow (\lambda (_: (eq PList PNil PNil)).(\lambda (H2: 
-(eq C c0 c)).(\lambda (H3: (eq C c0 e)).(eq_ind C c (\lambda (c1: C).((eq C 
-c1 e) \to (nf2 c t))) (\lambda (H4: (eq C c e)).(eq_ind C e (\lambda (c: 
-C).(nf2 c t)) H0 c (sym_eq C c e H4))) c0 (sym_eq C c0 c H2) H3)))) | 
-(drop1_cons c1 c2 h d H1 c3 hds H2) \Rightarrow (\lambda (H3: (eq PList 
-(PCons h d hds) PNil)).(\lambda (H4: (eq C c1 c)).(\lambda (H5: (eq C c3 
-e)).((let H6 \def (eq_ind PList (PCons h d hds) (\lambda (e0: PList).(match 
-e0 return (\lambda (_: PList).Prop) with [PNil \Rightarrow False | (PCons _ _ 
-_) \Rightarrow True])) I PNil H3) in (False_ind ((eq C c1 c) \to ((eq C c3 e) 
-\to ((drop h d c1 c2) \to ((drop1 hds c2 c3) \to (nf2 c t))))) H6)) H4 H5 H1 
-H2))))]) in (H1 (refl_equal PList PNil) (refl_equal C c) (refl_equal C 
-e))))))) (\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 H2 \def (match H0 return 
-(\lambda (p0: PList).(\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (drop1 p0 
-c0 c1)).((eq PList p0 (PCons n n0 p)) \to ((eq C c0 c) \to ((eq C c1 e) \to 
-(nf2 c (lift n n0 (lift1 p t)))))))))) with [(drop1_nil c0) \Rightarrow 
-(\lambda (H2: (eq PList PNil (PCons n n0 p))).(\lambda (H3: (eq C c0 
-c)).(\lambda (H4: (eq C c0 e)).((let H5 \def (eq_ind PList PNil (\lambda (e0: 
-PList).(match e0 return (\lambda (_: PList).Prop) with [PNil \Rightarrow True 
-| (PCons _ _ _) \Rightarrow False])) I (PCons n n0 p) H2) in (False_ind ((eq 
-C c0 c) \to ((eq C c0 e) \to (nf2 c (lift n n0 (lift1 p t))))) H5)) H3 H4)))) 
-| (drop1_cons c1 c2 h d H2 c3 hds H3) \Rightarrow (\lambda (H4: (eq PList 
-(PCons h d hds) (PCons n n0 p))).(\lambda (H5: (eq C c1 c)).(\lambda (H6: (eq 
-C c3 e)).((let H7 \def (f_equal PList PList (\lambda (e0: PList).(match e0 
-return (\lambda (_: PList).PList) with [PNil \Rightarrow hds | (PCons _ _ p) 
-\Rightarrow p])) (PCons h d hds) (PCons n n0 p) H4) in ((let H8 \def (f_equal 
-PList nat (\lambda (e0: PList).(match e0 return (\lambda (_: PList).nat) with 
-[PNil \Rightarrow d | (PCons _ n _) \Rightarrow n])) (PCons h d hds) (PCons n 
-n0 p) H4) in ((let H9 \def (f_equal PList nat (\lambda (e0: PList).(match e0 
-return (\lambda (_: PList).nat) with [PNil \Rightarrow h | (PCons n _ _) 
-\Rightarrow n])) (PCons h d hds) (PCons n n0 p) H4) in (eq_ind nat n (\lambda 
-(n1: nat).((eq nat d n0) \to ((eq PList hds p) \to ((eq C c1 c) \to ((eq C c3 
-e) \to ((drop n1 d c1 c2) \to ((drop1 hds c2 c3) \to (nf2 c (lift n n0 (lift1 
-p t)))))))))) (\lambda (H10: (eq nat d n0)).(eq_ind nat n0 (\lambda (n1: 
-nat).((eq PList hds p) \to ((eq C c1 c) \to ((eq C c3 e) \to ((drop n n1 c1 
-c2) \to ((drop1 hds c2 c3) \to (nf2 c (lift n n0 (lift1 p t))))))))) (\lambda 
-(H11: (eq PList hds p)).(eq_ind PList p (\lambda (p0: PList).((eq C c1 c) \to 
-((eq C c3 e) \to ((drop n n0 c1 c2) \to ((drop1 p0 c2 c3) \to (nf2 c (lift n 
-n0 (lift1 p t)))))))) (\lambda (H12: (eq C c1 c)).(eq_ind C c (\lambda (c0: 
-C).((eq C c3 e) \to ((drop n n0 c0 c2) \to ((drop1 p c2 c3) \to (nf2 c (lift 
-n n0 (lift1 p t))))))) (\lambda (H13: (eq C c3 e)).(eq_ind C e (\lambda (c0: 
-C).((drop n n0 c c2) \to ((drop1 p c2 c0) \to (nf2 c (lift n n0 (lift1 p 
-t)))))) (\lambda (H14: (drop n n0 c c2)).(\lambda (H15: (drop1 p c2 
-e)).(nf2_lift c2 (lift1 p t) (H c2 t H15 H1) c n n0 H14))) c3 (sym_eq C c3 e 
-H13))) c1 (sym_eq C c1 c H12))) hds (sym_eq PList hds p H11))) d (sym_eq nat 
-d n0 H10))) h (sym_eq nat h n H9))) H8)) H7)) H5 H6 H2 H3))))]) in (H2 
-(refl_equal PList (PCons n n0 p)) (refl_equal C c) (refl_equal C e))))))))))) 
-hds)).
-
-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_lref i 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 (Flat 
-Appl) t x0 (THeads (Flat Appl) t0 (TLRef i)) x1) 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 (u: T).(pr3 
-(CHead c (Bind b) u) 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 (u: T).(pr3 (CHead c (Bind b) u) 
-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)))).
-
-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))))).(match k return 
-(\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))))) with [(Bind b) \Rightarrow 
-(match b return (\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))))) with [Abbr \Rightarrow (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_comm (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 (t: T).(\forall (t2: 
-T).((pr2 c0 t t2) \to (eq T t 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 (t: T).(pr0 
-(lift (S O) (clen c0) x) t)) (\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_comm (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))) | Abst \Rightarrow 
-(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 (t: T).(pr0 t1 t)) 
-(\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 return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match 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)))))) | Void \Rightarrow 
-(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 (t: T).(pr0 t1 t)) 
-(\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 return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match 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))))))]) | (Flat f) 
-\Rightarrow (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 (t: T).(pr0 t1 t)) (\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 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))))))])) (\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).
-
-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.
-
-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)) (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 (t: T).(\forall (t2: 
-T).((((eq T t t2) \to (\forall (P: Prop).P))) \to ((pr3 c t t2) \to (\forall 
-(x: T).(\forall (x0: T).((eq T t2 (THead (Flat f) x x0)) \to (land (sn3 c x) 
-(sn3 c x0))))))))) H2 (THead (Flat f) x x0) H3) in (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 (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 (H3: (eq T (THead (Flat f) x x0) (THead (Flat 
-f) t2 x0))).(\lambda (P: Prop).(let H4 \def (f_equal T T (\lambda (e: 
-T).(match e return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef 
-_) \Rightarrow x | (THead _ t _) \Rightarrow t])) (THead (Flat f) x x0) 
-(THead (Flat f) t2 x0) H3) in (let H5 \def (eq_ind_r T t2 (\lambda (t: 
-T).(pr3 c x t)) H7 x H4) in (let H6 \def (eq_ind_r T t2 (\lambda (t: T).((eq 
-T x t) \to (\forall (P: Prop).P))) H6 x H4) in (H6 (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 (and_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 (H3: (eq T (THead (Flat f) x x0) (THead (Flat f) x 
-t2))).(\lambda (P: Prop).(let H4 \def (f_equal T T (\lambda (e: T).(match e 
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) 
-\Rightarrow x0 | (THead _ _ t) \Rightarrow t])) (THead (Flat f) x x0) (THead 
-(Flat f) x t2) H3) in (let H5 \def (eq_ind_r T t2 (\lambda (t: T).(pr3 c x0 
-t)) H7 x0 H4) in (let H6 \def (eq_ind_r T t2 (\lambda (t: T).((eq T x0 t) \to 
-(\forall (P: Prop).P))) H6 x0 H4) in (H6 (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 
-(and_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_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 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 (t2: T).((pr3 c t3 t2) \to (sn3 c t2)))))))).(\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 
-(t2: T).((((eq T t t2) \to (\forall (P: Prop).P))) \to ((pr2 c t t2) \to (sn3 
-c t2))))) \to ((((eq T t t0) \to (\forall (P: Prop).P))) \to (sn3 c t0))))) 
-(\lambda (t: T).(\lambda (H4: ((\forall (t2: T).((((eq T t t2) \to (\forall 
-(P: Prop).P))) \to ((pr2 c t t2) \to (sn3 c t2)))))).(\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 (t2: T).((((eq T t3 
-t2) \to (\forall (P: Prop).P))) \to ((pr2 c t3 t2) \to (sn3 c t2))))) \to 
-((((eq T t3 t5) \to (\forall (P: Prop).P))) \to (sn3 c t5))))).(\lambda (H7: 
-((\forall (t2: T).((((eq T t4 t2) \to (\forall (P: Prop).P))) \to ((pr2 c t4 
-t2) \to (sn3 c t2)))))).(\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 (t2: T).((((eq T t t2) \to (\forall (P: Prop).P))) \to ((pr2 c t 
-t2) \to (sn3 c t2))))) 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 (t2: T).(pr2 c t0 
-t2))))).(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 (t: T).((eq T (THead (Flat Cast) t1 t0) t) \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 (t: T).((eq T (THead (Flat 
-Cast) t1 t0) (THead (Flat Cast) t x1)) \to (\forall (P: Prop).P))) H12 t1 
-H14) in (let H16 \def (eq_ind T x0 (\lambda (t: T).(pr2 c t1 t)) 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 (t: 
-T).((eq T (THead (Flat Cast) t1 t0) (THead (Flat Cast) t1 t)) \to (\forall 
-(P: Prop).P))) H15 t0 H18) in (let H20 \def (eq_ind_r T x1 (\lambda (t: 
-T).(pr2 c t0 t)) 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 
-(t: T).((eq T t t1) \to (\forall (P: Prop).P))) H14 t1 H15) in (let H17 \def 
-(eq_ind_r T x0 (\lambda (t: T).((eq T (THead (Flat Cast) t1 t0) (THead (Flat 
-Cast) t x1)) \to (\forall (P: Prop).P))) H12 t1 H15) in (let H18 \def 
-(eq_ind_r T x0 (\lambda (t: T).(pr2 c t1 t)) 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 (t: T).((eq T (THead (Flat Cast) t1 t0) (THead (Flat Cast) x0 t)) 
-\to (\forall (P: Prop).P))) H12 t0 H16) in (let H18 \def (eq_ind_r T x1 
-(\lambda (t: T).(pr2 c t0 t)) 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 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))).
-
-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 (t2: T).(pr2 c (TLRef i) t2))))).(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 (t2: T).(\forall 
-(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t2))))))))).(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 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 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_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)) (\forall (t: T).((sn3 c (THead (Flat Appl) v t)) \to (sn3 c (THead 
-(Flat Appl) v (THead (Flat Cast) u t))))) (\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)) (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 (t: T).(\forall (t2: T).((((eq T t t2) \to (\forall (P: 
-Prop).P))) \to ((pr3 c t 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))))))))) H7 (THead (Flat Appl) x x1) H8) in (let H10 \def (eq_ind T t0 
-(\lambda (t: T).(\forall (t2: T).((((eq T t t2) \to (\forall (P: Prop).P))) 
-\to ((pr3 c t t2) \to (sn3 c t2))))) H6 (THead (Flat Appl) x x1) H8) in (let 
-H11 \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 (x: T).(\forall (x0: 
-T).((eq T t2 (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)))))))))))) H2 (THead (Flat Appl) x x0) H3) in (let H12 \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))))) 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 (t2: T).(pr2 c 
-(THead (Flat Cast) x0 x1) t2))))).(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 (t: 
-T).((eq T (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) t) \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 (t2: 
-T).(eq T x3 (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c x1 
-t2))))).(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 (t: T).((eq T (THead (Flat Appl) x 
-(THead (Flat Cast) x0 x1)) (THead (Flat Appl) x2 t)) \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 return (\lambda (_: 
-T).T) with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t _) 
-\Rightarrow t])) (THead (Flat Appl) x x0) (THead (Flat Appl) x2 x4) H28) in 
-((let H30 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: 
-T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ 
-t) \Rightarrow t])) (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 (t: 
-T).((eq T (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) (THead (Flat Appl) 
-x2 (THead (Flat Cast) t x5))) \to (\forall (P: Prop).P))) H26 x0 H30) in (let 
-H33 \def (eq_ind_r T x4 (\lambda (t: T).(pr2 c x0 t)) 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 (t: T).((eq T (THead (Flat Appl) 
-x (THead (Flat Cast) x0 x1)) (THead (Flat Appl) t (THead (Flat Cast) x0 x5))) 
-\to (\forall (P: Prop).P))) H32 x H31) in (let H35 \def (eq_ind_r T x2 
-(\lambda (t: T).(pr2 c x t)) 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 return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ _ t) \Rightarrow t])) 
-(THead (Flat Appl) x x1) (THead (Flat Appl) x x5) H37) in (let H39 \def 
-(eq_ind_r T x5 (\lambda (t: T).((eq T (THead (Flat Appl) x (THead (Flat Cast) 
-x0 x1)) (THead (Flat Appl) x (THead (Flat Cast) x0 t))) \to (\forall (P: 
-Prop).P))) H34 x1 H38) in (let H40 \def (eq_ind_r T x5 (\lambda (t: T).(pr2 c 
-x1 t)) 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 return (\lambda (_: 
-T).T) with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t _) 
-\Rightarrow t])) (THead (Flat Appl) x x1) (THead (Flat Appl) x2 x5) H30) in 
-((let H32 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: 
-T).T) with [(TSort _) \Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ _ 
-t) \Rightarrow t])) (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 (t: 
-T).(pr2 c x1 t)) 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 (t: T).((eq T (THead (Flat Appl) x x0) (THead (Flat Appl) t x4)) \to 
-(\forall (P: Prop).P))) H28 x H33) in (let H36 \def (eq_ind_r T x2 (\lambda 
-(t: T).(pr2 c x t)) 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_head_12 c x x2 (pr3_pr2 c 
-x x2 H18) (Flat Appl) x0 x4 (pr3_pr2 (CHead c (Flat Appl) x2) x0 x4 
-(pr2_cflat c x0 x4 H24 Appl x2))) x2 x4 (refl_equal T (THead (Flat Appl) x2 
-x4)) x5 (H10 (THead (Flat Appl) x2 x5) H30 (pr3_head_12 c x x2 (pr3_pr2 c x 
-x2 H18) (Flat Appl) x1 x5 (pr3_pr2 (CHead c (Flat Appl) x2) x1 x5 (pr2_cflat 
-c x1 x5 H25 Appl x2)))))) 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 return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t _) 
-\Rightarrow t])) (THead (Flat Appl) x x1) (THead (Flat Appl) x2 x3) H24) in 
-((let H26 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: 
-T).T) with [(TSort _) \Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ _ 
-t) \Rightarrow t])) (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 (t: 
-T).(pr2 c x1 t)) H22 x1 H26) in (let H29 \def (eq_ind_r T x3 (\lambda (t: 
-T).((eq T (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) (THead (Flat Appl) 
-x2 t)) \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 
-(t: T).((eq T (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) (THead (Flat 
-Appl) t x1)) \to (\forall (P: Prop).P))) H29 x H27) in (let H31 \def 
-(eq_ind_r T x2 (\lambda (t: T).(pr2 c x t)) 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_head_12 c x x2 (pr3_pr2 c 
-x x2 H18) (Flat Appl) x1 x3 (pr3_pr2 (CHead c (Flat Appl) x2) x1 x3 
-(pr2_cflat c x1 x3 H22 Appl x2))))) 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 (t2: T).(\forall (b: B).(\forall (u: T).(pr2 
-(CHead c (Bind b) u) z1 t2))))))))).(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 (u: T).(pr2 (CHead c (Bind b) 
-u) x3 x5))))).(let H21 \def (eq_ind T t2 (\lambda (t: T).((eq T (THead (Flat 
-Appl) x (THead (Flat Cast) x0 x1)) t) \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 return (\lambda (_: T).Prop) with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
-(match 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 (t: T).((eq T 
-(THead (Flat Appl) x (THead (Flat Cast) x0 x1)) t) \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 return (\lambda (_: T).Prop) with [(TSort 
-_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
-\Rightarrow (match 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_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)) (\forall (v2: T).((sn3 c 
-v2) \to (((\forall (u2: T).((pr3 c (THead (Flat Appl) v1 t1) u2) \to ((((iso 
-(THead (Flat Appl) v1 t1) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead 
-(Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2 (THead (Flat Appl) 
-v1 t1)))))) (\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 (THead (Flat Appl) v1 t) u2) \to ((((iso 
-(THead (Flat Appl) v1 t) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead 
-(Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2 (THead (Flat Appl) 
-v1 t)))))))) (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 (THead (Flat Appl) t x) u2) \to ((((iso (THead (Flat Appl) t x) 
-u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to 
-(sn3 c (THead (Flat Appl) v2 (THead (Flat Appl) t x))))))))) (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 (THead 
-(Flat Appl) x x0) u2) \to ((((iso (THead (Flat Appl) x x0) u2) \to (\forall 
-(P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c (THead 
-(Flat Appl) v2 (THead (Flat Appl) x x0)))))))))) (\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 (THead (Flat Appl) x x0) u2) \to ((((iso 
-(THead (Flat Appl) x x0) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead 
-(Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2 (THead (Flat Appl) x 
-x0)))))))))))))).(\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 (THead (Flat Appl) 
-x x0) u2) \to ((((iso (THead (Flat Appl) x x0) u2) \to (\forall (P: 
-Prop).P))) \to (sn3 c (THead (Flat Appl) t u2)))))) \to (sn3 c (THead (Flat 
-Appl) t (THead (Flat Appl) x x0))))) (\lambda (t0: T).(\lambda (H5: ((\forall 
-(t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to 
-(sn3 c t2)))))).(\lambda (H6: ((\forall (t2: T).((((eq T t0 t2) \to (\forall 
-(P: Prop).P))) \to ((pr3 c t0 t2) \to (((\forall (u2: T).((pr3 c (THead (Flat 
-Appl) x x0) u2) \to ((((iso (THead (Flat Appl) x x0) u2) \to (\forall (P: 
-Prop).P))) \to (sn3 c (THead (Flat Appl) t2 u2)))))) \to (sn3 c (THead (Flat 
-Appl) t2 (THead (Flat Appl) x x0))))))))).(\lambda (H7: ((\forall (u2: 
-T).((pr3 c (THead (Flat Appl) x x0) u2) \to ((((iso (THead (Flat Appl) x x0) 
-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 (t2: T).((((eq T 
-t t2) \to (\forall (P: Prop).P))) \to ((pr3 c t t2) \to (\forall (x: 
-T).(\forall (x0: T).((eq T t2 (THead (Flat Appl) x x0)) \to (\forall (v2: 
-T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c (THead (Flat Appl) x x0) u2) 
-\to ((((iso (THead (Flat Appl) x x0) u2) \to (\forall (P: Prop).P))) \to (sn3 
-c (THead (Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2 (THead 
-(Flat Appl) x x0))))))))))))) H2 (THead (Flat Appl) x x0) H3) in (let H9 \def 
-(eq_ind T t2 (\lambda (t: T).(\forall (t2: T).((((eq T t t2) \to (\forall (P: 
-Prop).P))) \to ((pr3 c t t2) \to (sn3 c t2))))) H1 (THead (Flat Appl) x x0) 
-H3) in (sn3_pr2_intro c (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) 
-(\lambda (t3: T).(\lambda (H10: (((eq T (THead (Flat Appl) t0 (THead (Flat 
-Appl) x x0)) t3) \to (\forall (P: Prop).P)))).(\lambda (H11: (pr2 c (THead 
-(Flat Appl) t0 (THead (Flat Appl) x x0)) t3)).(let H12 \def (pr2_gen_appl c 
-t0 (THead (Flat Appl) x 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 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 (H13: (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T t3 (THead (Flat Appl) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c t0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c 
-(THead (Flat Appl) x x0) t2))))).(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 (H14: (eq T 
-t3 (THead (Flat Appl) x1 x2))).(\lambda (H15: (pr2 c t0 x1)).(\lambda (H16: 
-(pr2 c (THead (Flat Appl) x x0) x2)).(let H17 \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))) H10 (THead (Flat Appl) x1 x2) H14) in (eq_ind_r T (THead (Flat 
-Appl) x1 x2) (\lambda (t: T).(sn3 c t)) (let H18 \def (pr2_gen_appl c x x0 x2 
-H16) 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 (H19: (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T x2 (THead (Flat Appl) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c x0 
-t2))))).(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 (H20: (eq T x2 (THead (Flat 
-Appl) x3 x4))).(\lambda (H21: (pr2 c x x3)).(\lambda (H22: (pr2 c x0 
-x4)).(let H23 \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))) H17 (THead (Flat Appl) x3 x4) H20) 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 H24 
-\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 
-(H25: (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4))).(let H26 
-\def (f_equal T T (\lambda (e: T).(match e 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) H25) in 
-((let H27 \def (f_equal T T (\lambda (e: T).(match e 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) H25) 
-in (\lambda (H28: (eq T x x3)).(let H29 \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))) H23 x0 H27) in (let 
-H30 \def (eq_ind_r T x4 (\lambda (t: T).(pr2 c x0 t)) H22 x0 H27) in (eq_ind 
-T x0 (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 (THead (Flat Appl) x3 t)))) 
-(let H31 \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))) H29 x H28) in (let H32 \def (eq_ind_r T x3 
-(\lambda (t: T).(pr2 c x t)) H21 x H28) 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 H33 \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 (H34: (eq T t0 x1)).(let H35 \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))) H31 t0 H34) in (let 
-H36 \def (eq_ind_r T x1 (\lambda (t: T).(pr2 c t0 t)) H15 t0 H34) in (eq_ind 
-T t0 (\lambda (t: T).(sn3 c (THead (Flat Appl) t (THead (Flat Appl) x x0)))) 
-(H35 (refl_equal T (THead (Flat Appl) t0 (THead (Flat Appl) x x0))) (sn3 c 
-(THead (Flat Appl) t0 (THead (Flat Appl) x x0)))) x1 H34)))) (\lambda (H34: 
-(((eq T t0 x1) \to (\forall (P: Prop).P)))).(H6 x1 H34 (pr3_pr2 c t0 x1 H15) 
-(\lambda (u2: T).(\lambda (H35: (pr3 c (THead (Flat Appl) x x0) u2)).(\lambda 
-(H36: (((iso (THead (Flat Appl) x x0) u2) \to (\forall (P: 
-Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) t0 u2) (H7 u2 H35 H36) (THead 
-(Flat Appl) x1 u2) (pr3_pr2 c (THead (Flat Appl) t0 u2) (THead (Flat Appl) x1 
-u2) (pr2_head_1 c t0 x1 H15 (Flat Appl) u2)))))))) H33))) x3 H28))) x4 
-H27))))) H26))) (\lambda (H25: (((eq T (THead (Flat Appl) x x0) (THead (Flat 
-Appl) x3 x4)) \to (\forall (P: Prop).P)))).(H8 (THead (Flat Appl) x3 x4) H25 
-(pr3_head_12 c x x3 (pr3_pr2 c x x3 H21) (Flat Appl) x0 x4 (pr3_pr2 (CHead c 
-(Flat Appl) x3) x0 x4 (pr2_cflat c x0 x4 H22 Appl x3))) 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 H15)) (\lambda (u2: T).(\lambda (H26: (pr3 c (THead (Flat 
-Appl) x3 x4) u2)).(\lambda (H27: (((iso (THead (Flat Appl) x3 x4) u2) \to 
-(\forall (P: Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) t0 u2) (H7 u2 
-(pr3_sing c (THead (Flat Appl) x x4) (THead (Flat Appl) x x0) (pr2_thin_dx c 
-x0 x4 H22 x Appl) u2 (pr3_sing c (THead (Flat Appl) x3 x4) (THead (Flat Appl) 
-x x4) (pr2_head_1 c x x3 H21 (Flat Appl) x4) u2 H26)) (\lambda (H28: (iso 
-(THead (Flat Appl) x x0) u2)).(\lambda (P: Prop).(H27 (iso_trans (THead (Flat 
-Appl) x3 x4) (THead (Flat Appl) x x0) (iso_head (Flat Appl) x3 x x4 x0) u2 
-H28) 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 H15 (Flat Appl) u2)))))))) 
-H24))) x2 H20))))))) H19)) (\lambda (H19: (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 
-(t2: T).(eq T x2 (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x 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))))))))).(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 (H20: (eq 
-T x0 (THead (Bind Abst) x3 x4))).(\lambda (H21: (eq T x2 (THead (Bind Abbr) 
-x5 x6))).(\lambda (H22: (pr2 c x x5)).(\lambda (H23: ((\forall (b: 
-B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x4 x6))))).(let H24 \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))) H17 (THead (Bind Abbr) 
-x5 x6) H21) in (eq_ind_r T (THead (Bind Abbr) x5 x6) (\lambda (t: T).(sn3 c 
-(THead (Flat Appl) x1 t))) (let H25 \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))) H24 (THead (Bind Abst) x3 
-x4) H20) in (let H26 \def (eq_ind T x0 (\lambda (t: T).(\forall (t2: 
-T).((((eq T (THead (Flat Appl) x t) t2) \to (\forall (P: Prop).P))) \to ((pr3 
-c (THead (Flat Appl) x t) t2) \to (sn3 c t2))))) H9 (THead (Bind Abst) x3 x4) 
-H20) in (let H27 \def (eq_ind T x0 (\lambda (t: T).(\forall (t2: T).((((eq T 
-(THead (Flat Appl) x t) t2) \to (\forall (P: Prop).P))) \to ((pr3 c (THead 
-(Flat Appl) x t) t2) \to (\forall (x: T).(\forall (x0: T).((eq T t2 (THead 
-(Flat Appl) x x0)) \to (\forall (v2: T).((sn3 c v2) \to (((\forall (u2: 
-T).((pr3 c (THead (Flat Appl) x x0) u2) \to ((((iso (THead (Flat Appl) x x0) 
-u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to 
-(sn3 c (THead (Flat Appl) v2 (THead (Flat Appl) x x0))))))))))))) H8 (THead 
-(Bind Abst) x3 x4) H20) in (let H28 \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)))))) H7 (THead (Bind Abst) x3 x4) H20) in (let H29 \def (eq_ind 
-T x0 (\lambda (t: T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: 
-Prop).P))) \to ((pr3 c t0 t2) \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) t2 u2)))))) \to (sn3 c (THead (Flat 
-Appl) t2 (THead (Flat Appl) x t)))))))) H6 (THead (Bind Abst) x3 x4) H20) in 
-(sn3_pr3_trans c (THead (Flat Appl) t0 (THead (Bind Abbr) x5 x6)) (H28 (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 
-H22) (Bind Abbr) x4 x6 (pr3_pr2 (CHead c (Bind Abbr) x5) x4 x6 (H23 Abbr 
-x5)))) (\lambda (H30: (iso (THead (Flat Appl) x (THead (Bind Abst) x3 x4)) 
-(THead (Bind Abbr) x5 x6))).(\lambda (P: Prop).(let H31 \def (match H30 
-return (\lambda (t: T).(\lambda (t0: T).(\lambda (_: (iso t t0)).((eq T t 
-(THead (Flat Appl) x (THead (Bind Abst) x3 x4))) \to ((eq T t0 (THead (Bind 
-Abbr) x5 x6)) \to P))))) with [(iso_sort n1 n2) \Rightarrow (\lambda (H0: (eq 
-T (TSort n1) (THead (Flat Appl) x (THead (Bind Abst) x3 x4)))).(\lambda (H1: 
-(eq T (TSort n2) (THead (Bind Abbr) x5 x6))).((let H2 \def (eq_ind T (TSort 
-n1) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with [(TSort _) 
-\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
-False])) I (THead (Flat Appl) x (THead (Bind Abst) x3 x4)) H0) in (False_ind 
-((eq T (TSort n2) (THead (Bind Abbr) x5 x6)) \to P) H2)) H1))) | (iso_lref i1 
-i2) \Rightarrow (\lambda (H0: (eq T (TLRef i1) (THead (Flat Appl) x (THead 
-(Bind Abst) x3 x4)))).(\lambda (H1: (eq T (TLRef i2) (THead (Bind Abbr) x5 
-x6))).((let H2 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e return 
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) x 
-(THead (Bind Abst) x3 x4)) H0) in (False_ind ((eq T (TLRef i2) (THead (Bind 
-Abbr) x5 x6)) \to P) H2)) H1))) | (iso_head k v4 v5 t1 t2) \Rightarrow 
-(\lambda (H0: (eq T (THead k v4 t1) (THead (Flat Appl) x (THead (Bind Abst) 
-x3 x4)))).(\lambda (H1: (eq T (THead k v5 t2) (THead (Bind Abbr) x5 
-x6))).((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: 
-T).T) with [(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ _ 
-t) \Rightarrow t])) (THead k v4 t1) (THead (Flat Appl) x (THead (Bind Abst) 
-x3 x4)) H0) in ((let H3 \def (f_equal T T (\lambda (e: T).(match e return 
-(\lambda (_: T).T) with [(TSort _) \Rightarrow v4 | (TLRef _) \Rightarrow v4 
-| (THead _ t _) \Rightarrow t])) (THead k v4 t1) (THead (Flat Appl) x (THead 
-(Bind Abst) x3 x4)) H0) in ((let H4 \def (f_equal T K (\lambda (e: T).(match 
-e return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) 
-\Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k v4 t1) (THead (Flat 
-Appl) x (THead (Bind Abst) x3 x4)) H0) in (eq_ind K (Flat Appl) (\lambda (k0: 
-K).((eq T v4 x) \to ((eq T t1 (THead (Bind Abst) x3 x4)) \to ((eq T (THead k0 
-v5 t2) (THead (Bind Abbr) x5 x6)) \to P)))) (\lambda (H5: (eq T v4 
-x)).(eq_ind T x (\lambda (_: T).((eq T t1 (THead (Bind Abst) x3 x4)) \to ((eq 
-T (THead (Flat Appl) v5 t2) (THead (Bind Abbr) x5 x6)) \to P))) (\lambda (H6: 
-(eq T t1 (THead (Bind Abst) x3 x4))).(eq_ind T (THead (Bind Abst) x3 x4) 
-(\lambda (_: T).((eq T (THead (Flat Appl) v5 t2) (THead (Bind Abbr) x5 x6)) 
-\to P)) (\lambda (H7: (eq T (THead (Flat Appl) v5 t2) (THead (Bind Abbr) x5 
-x6))).(let H8 \def (eq_ind T (THead (Flat Appl) v5 t2) (\lambda (e: T).(match 
-e return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: 
-K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I 
-(THead (Bind Abbr) x5 x6) H7) in (False_ind P H8))) t1 (sym_eq T t1 (THead 
-(Bind Abst) x3 x4) H6))) v4 (sym_eq T v4 x H5))) k (sym_eq K k (Flat Appl) 
-H4))) H3)) H2)) H1)))]) in (H31 (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 H15 (Flat Appl) (THead (Bind Abbr) x5 x6))))))))) x2 
-H21)))))))))) H19)) (\lambda (H19: (ex6_6 B T T T T T (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\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 (H20: (not (eq 
-B x3 Abst))).(\lambda (H21: (eq T x0 (THead (Bind x3) x4 x5))).(\lambda (H22: 
-(eq T x2 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) 
-x6)))).(\lambda (H23: (pr2 c x x7)).(\lambda (H24: (pr2 c x4 x8)).(\lambda 
-(H25: (pr2 (CHead c (Bind x3) x8) x5 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))) H17 (THead (Bind x3) x8 (THead (Flat 
-Appl) (lift (S O) O x7) x6)) H22) 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 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 x3) x8 (THead 
-(Flat Appl) (lift (S O) O x7) x6)))) \to (\forall (P: Prop).P))) H26 (THead 
-(Bind x3) x4 x5) H21) in (let H28 \def (eq_ind T x0 (\lambda (t: T).(\forall 
-(t2: T).((((eq T (THead (Flat Appl) x t) t2) \to (\forall (P: Prop).P))) \to 
-((pr3 c (THead (Flat Appl) x t) t2) \to (sn3 c t2))))) H9 (THead (Bind x3) x4 
-x5) H21) in (let H29 \def (eq_ind T x0 (\lambda (t: T).(\forall (t2: 
-T).((((eq T (THead (Flat Appl) x t) t2) \to (\forall (P: Prop).P))) \to ((pr3 
-c (THead (Flat Appl) x t) t2) \to (\forall (x: T).(\forall (x0: T).((eq T t2 
-(THead (Flat Appl) x x0)) \to (\forall (v2: T).((sn3 c v2) \to (((\forall 
-(u2: T).((pr3 c (THead (Flat Appl) x x0) u2) \to ((((iso (THead (Flat Appl) x 
-x0) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) 
-\to (sn3 c (THead (Flat Appl) v2 (THead (Flat Appl) x x0))))))))))))) H8 
-(THead (Bind x3) x4 x5) H21) 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)))))) H7 (THead (Bind x3) x4 x5) H21) in (let H31 \def (eq_ind T 
-x0 (\lambda (t: T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: 
-Prop).P))) \to ((pr3 c t0 t2) \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) t2 u2)))))) \to (sn3 c (THead (Flat 
-Appl) t2 (THead (Flat Appl) x t)))))))) H6 (THead (Bind x3) x4 x5) H21) in 
-(sn3_pr3_trans c (THead (Flat Appl) t0 (THead (Bind x3) x8 (THead (Flat Appl) 
-(lift (S O) O x7) x6))) (H30 (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 H20 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 H24) (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 H23)) (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 H25 Appl (lift (S O) O x7)))))) (\lambda (H32: 
-(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 H33 \def (match 
-H32 return (\lambda (t: T).(\lambda (t0: T).(\lambda (_: (iso t t0)).((eq T t 
-(THead (Flat Appl) x (THead (Bind x3) x4 x5))) \to ((eq T t0 (THead (Bind x3) 
-x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to P))))) with [(iso_sort n1 
-n2) \Rightarrow (\lambda (H0: (eq T (TSort n1) (THead (Flat Appl) x (THead 
-(Bind x3) x4 x5)))).(\lambda (H1: (eq T (TSort n2) (THead (Bind x3) x8 (THead 
-(Flat Appl) (lift (S O) O x7) x6)))).((let H2 \def (eq_ind T (TSort n1) 
-(\lambda (e: T).(match e return (\lambda (_: T).Prop) with [(TSort _) 
-\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
-False])) I (THead (Flat Appl) x (THead (Bind x3) x4 x5)) H0) in (False_ind 
-((eq T (TSort n2) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) 
-x6))) \to P) H2)) H1))) | (iso_lref i1 i2) \Rightarrow (\lambda (H0: (eq T 
-(TLRef i1) (THead (Flat Appl) x (THead (Bind x3) x4 x5)))).(\lambda (H1: (eq 
-T (TLRef i2) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) 
-x6)))).((let H2 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e return 
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) x 
-(THead (Bind x3) x4 x5)) H0) in (False_ind ((eq T (TLRef i2) (THead (Bind x3) 
-x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to P) H2)) H1))) | (iso_head k 
-v4 v5 t1 t2) \Rightarrow (\lambda (H0: (eq T (THead k v4 t1) (THead (Flat 
-Appl) x (THead (Bind x3) x4 x5)))).(\lambda (H1: (eq T (THead k v5 t2) (THead 
-(Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))).((let H2 \def 
-(f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort 
-_) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ _ t) \Rightarrow t])) 
-(THead k v4 t1) (THead (Flat Appl) x (THead (Bind x3) x4 x5)) H0) in ((let H3 
-\def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow v4 | (TLRef _) \Rightarrow v4 | (THead _ t _) 
-\Rightarrow t])) (THead k v4 t1) (THead (Flat Appl) x (THead (Bind x3) x4 
-x5)) H0) in ((let H4 \def (f_equal T K (\lambda (e: T).(match e return 
-(\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | 
-(THead k _ _) \Rightarrow k])) (THead k v4 t1) (THead (Flat Appl) x (THead 
-(Bind x3) x4 x5)) H0) in (eq_ind K (Flat Appl) (\lambda (k0: K).((eq T v4 x) 
-\to ((eq T t1 (THead (Bind x3) x4 x5)) \to ((eq T (THead k0 v5 t2) (THead 
-(Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to P)))) (\lambda 
-(H5: (eq T v4 x)).(eq_ind T x (\lambda (_: T).((eq T t1 (THead (Bind x3) x4 
-x5)) \to ((eq T (THead (Flat Appl) v5 t2) (THead (Bind x3) x8 (THead (Flat 
-Appl) (lift (S O) O x7) x6))) \to P))) (\lambda (H6: (eq T t1 (THead (Bind 
-x3) x4 x5))).(eq_ind T (THead (Bind x3) x4 x5) (\lambda (_: T).((eq T (THead 
-(Flat Appl) v5 t2) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) 
-x6))) \to P)) (\lambda (H7: (eq T (THead (Flat Appl) v5 t2) (THead (Bind x3) 
-x8 (THead (Flat Appl) (lift (S O) O x7) x6)))).(let H8 \def (eq_ind T (THead 
-(Flat Appl) v5 t2) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
-\Rightarrow (match 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)) H7) in (False_ind P H8))) t1 (sym_eq T t1 (THead 
-(Bind x3) x4 x5) H6))) v4 (sym_eq T v4 x H5))) k (sym_eq K k (Flat Appl) 
-H4))) H3)) H2)) H1)))]) in (H33 (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 H15 (Flat Appl) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O 
-x7) x6)))))))))) x2 H22)))))))))))))) H19)) 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 (Flat Appl) x x0) (THead (Bind Abst) y1 
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: 
-T).(eq T t3 (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 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))))))))).(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 (H14: (eq T (THead (Flat Appl) x x0) (THead (Bind Abst) x1 
-x2))).(\lambda (H15: (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 H18 \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))) H10 (THead 
-(Bind Abbr) x3 x4) H15) in (eq_ind_r T (THead (Bind Abbr) x3 x4) (\lambda (t: 
-T).(sn3 c t)) (let H19 \def (eq_ind T (THead (Flat Appl) x x0) (\lambda (ee: 
-T).(match ee return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | 
-(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return 
-(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow 
-True])])) I (THead (Bind Abst) x1 x2) H14) in (False_ind (sn3 c (THead (Bind 
-Abbr) x3 x4)) H19)) t3 H15)))))))))) H13)) (\lambda (H13: (ex6_6 B T T T T T 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\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 (H15: 
-(eq T (THead (Flat Appl) x 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 
-(_: (pr2 c t0 x5)).(\lambda (_: (pr2 c x2 x6)).(\lambda (_: (pr2 (CHead c 
-(Bind x1) x6) x3 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))) 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 (t: T).(sn3 c t)) (let H21 \def (eq_ind T (THead (Flat Appl) x 
-x0) (\lambda (ee: T).(match ee return (\lambda (_: T).Prop) with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
-(match k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | 
-(Flat _) \Rightarrow True])])) I (THead (Bind x1) x2 x3) H15) in (False_ind 
-(sn3 c (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4))) H21)) 
-t3 H16)))))))))))))) H13)) H12)))))))))))) v2 H4))))))))) y H0))))) H))))).
-
-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 (and_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 (and_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))).(and_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 (t: T).((sn3 c (THeads (Flat Appl) t1 t)) \to 
-(sn3 c (THeads (Flat Appl) t1 (THead (Flat Cast) u t)))))))) \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 (t: T).((sn3 c (THeads (Flat Appl) TNil t)) \to (sn3 c (THeads (Flat 
-Appl) TNil (THead (Flat Cast) u t))))))))).(\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 (t: T).((sn3 c (THeads (Flat Appl) t2 t)) \to (sn3 c (THeads (Flat 
-Appl) t2 (THead (Flat Cast) u t)))))))) \to (\forall (u: T).((sn3 c (THead 
-(Flat Appl) t (THeads (Flat Appl) t2 u))) \to (\forall (t0: T).((sn3 c (THead 
-(Flat Appl) t (THeads (Flat Appl) t2 t0))) \to (sn3 c (THead (Flat Appl) t 
-(THeads (Flat Appl) t2 (THead (Flat Cast) u t0))))))))))).(\lambda (H0: 
-((\forall (u: T).((sn3 c (THeads (Flat Appl) (TCons t1 t2) u)) \to (\forall 
-(t: T).((sn3 c (THeads (Flat Appl) (TCons t1 t2) t)) \to (sn3 c (THeads (Flat 
-Appl) (TCons t1 t2) (THead (Flat Cast) u t))))))))).(\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 H3 \def (sn3_gen_flat Appl c t (THeads 
-(Flat Appl) (TCons t1 t2) t3) H2) in (and_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 H7 \def (sn3_gen_flat Appl c t (THeads (Flat 
-Appl) (TCons t1 t2) u) H1) in (and_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_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 (t: T).((eq T (lift h i 
-t1) t) \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 (t: T).((eq T (lift h i t1) 
-(lift h i t)) \to (\forall (P: Prop).P))) H8 t1 H9) in (let H11 \def 
-(eq_ind_r T x (\lambda (t: T).(pr2 d t1 t)) 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 (d: C).(\lambda 
-(u: T).(getl i c (CHead d (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 return (\lambda (_: C).C) 
-with [(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow c])) (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 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 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 (and_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)))))).
-
-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 H1 \def (match H return (\lambda 
-(p: PList).(\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (drop1 p c0 
-c1)).((eq PList p PNil) \to ((eq C c0 c) \to ((eq C c1 e) \to (sns3 c (lifts1 
-PNil ts))))))))) with [(drop1_nil c0) \Rightarrow (\lambda (_: (eq PList PNil 
-PNil)).(\lambda (H2: (eq C c0 c)).(\lambda (H3: (eq C c0 e)).(eq_ind C c 
-(\lambda (c1: C).((eq C c1 e) \to (sns3 c (lifts1 PNil ts)))) (\lambda (H4: 
-(eq C c e)).(eq_ind C e (\lambda (c: C).(sns3 c (lifts1 PNil ts))) (eq_ind_r 
-TList ts (\lambda (t: TList).(sns3 e t)) H0 (lifts1 PNil ts) (lifts1_nil ts)) 
-c (sym_eq C c e H4))) c0 (sym_eq C c0 c H2) H3)))) | (drop1_cons c1 c2 h d H1 
-c3 hds H2) \Rightarrow (\lambda (H3: (eq PList (PCons h d hds) 
-PNil)).(\lambda (H4: (eq C c1 c)).(\lambda (H5: (eq C c3 e)).((let H6 \def 
-(eq_ind PList (PCons h d hds) (\lambda (e0: PList).(match e0 return (\lambda 
-(_: PList).Prop) with [PNil \Rightarrow False | (PCons _ _ _) \Rightarrow 
-True])) I PNil H3) in (False_ind ((eq C c1 c) \to ((eq C c3 e) \to ((drop h d 
-c1 c2) \to ((drop1 hds c2 c3) \to (sns3 c (lifts1 PNil ts)))))) H6)) H4 H5 H1 
-H2))))]) in (H1 (refl_equal PList PNil) (refl_equal C c) (refl_equal C 
-e))))))) (\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 H2 \def (match 
-H0 return (\lambda (p0: PList).(\lambda (c0: C).(\lambda (c1: C).(\lambda (_: 
-(drop1 p0 c0 c1)).((eq PList p0 (PCons n n0 p)) \to ((eq C c0 c) \to ((eq C 
-c1 e) \to (sns3 c (lifts1 (PCons n n0 p) ts))))))))) with [(drop1_nil c0) 
-\Rightarrow (\lambda (H2: (eq PList PNil (PCons n n0 p))).(\lambda (H3: (eq C 
-c0 c)).(\lambda (H4: (eq C c0 e)).((let H5 \def (eq_ind PList PNil (\lambda 
-(e0: PList).(match e0 return (\lambda (_: PList).Prop) with [PNil \Rightarrow 
-True | (PCons _ _ _) \Rightarrow False])) I (PCons n n0 p) H2) in (False_ind 
-((eq C c0 c) \to ((eq C c0 e) \to (sns3 c (lifts1 (PCons n n0 p) ts)))) H5)) 
-H3 H4)))) | (drop1_cons c1 c2 h d H2 c3 hds H3) \Rightarrow (\lambda (H4: (eq 
-PList (PCons h d hds) (PCons n n0 p))).(\lambda (H5: (eq C c1 c)).(\lambda 
-(H6: (eq C c3 e)).((let H7 \def (f_equal PList PList (\lambda (e0: 
-PList).(match e0 return (\lambda (_: PList).PList) with [PNil \Rightarrow hds 
-| (PCons _ _ p) \Rightarrow p])) (PCons h d hds) (PCons n n0 p) H4) in ((let 
-H8 \def (f_equal PList nat (\lambda (e0: PList).(match e0 return (\lambda (_: 
-PList).nat) with [PNil \Rightarrow d | (PCons _ n _) \Rightarrow n])) (PCons 
-h d hds) (PCons n n0 p) H4) in ((let H9 \def (f_equal PList nat (\lambda (e0: 
-PList).(match e0 return (\lambda (_: PList).nat) with [PNil \Rightarrow h | 
-(PCons n _ _) \Rightarrow n])) (PCons h d hds) (PCons n n0 p) H4) in (eq_ind 
-nat n (\lambda (n1: nat).((eq nat d n0) \to ((eq PList hds p) \to ((eq C c1 
-c) \to ((eq C c3 e) \to ((drop n1 d c1 c2) \to ((drop1 hds c2 c3) \to (sns3 c 
-(lifts1 (PCons n n0 p) ts))))))))) (\lambda (H10: (eq nat d n0)).(eq_ind nat 
-n0 (\lambda (n1: nat).((eq PList hds p) \to ((eq C c1 c) \to ((eq C c3 e) \to 
-((drop n n1 c1 c2) \to ((drop1 hds c2 c3) \to (sns3 c (lifts1 (PCons n n0 p) 
-ts)))))))) (\lambda (H11: (eq PList hds p)).(eq_ind PList p (\lambda (p0: 
-PList).((eq C c1 c) \to ((eq C c3 e) \to ((drop n n0 c1 c2) \to ((drop1 p0 c2 
-c3) \to (sns3 c (lifts1 (PCons n n0 p) ts))))))) (\lambda (H12: (eq C c1 
-c)).(eq_ind C c (\lambda (c0: C).((eq C c3 e) \to ((drop n n0 c0 c2) \to 
-((drop1 p c2 c3) \to (sns3 c (lifts1 (PCons n n0 p) ts)))))) (\lambda (H13: 
-(eq C c3 e)).(eq_ind C e (\lambda (c0: C).((drop n n0 c c2) \to ((drop1 p c2 
-c0) \to (sns3 c (lifts1 (PCons n n0 p) ts))))) (\lambda (H14: (drop n n0 c 
-c2)).(\lambda (H15: (drop1 p c2 e)).(eq_ind_r TList (lifts n n0 (lifts1 p 
-ts)) (\lambda (t: TList).(sns3 c t)) (sns3_lifts c c2 n n0 H14 (lifts1 p ts) 
-(H c2 H15 ts H1)) (lifts1 (PCons n n0 p) ts) (lifts1_cons n n0 p ts)))) c3 
-(sym_eq C c3 e H13))) c1 (sym_eq C c1 c H12))) hds (sym_eq PList hds p H11))) 
-d (sym_eq nat d n0 H10))) h (sym_eq nat h n H9))) H8)) H7)) H5 H6 H2 H3))))]) 
-in (H2 (refl_equal PList (PCons n n0 p)) (refl_equal C c) (refl_equal C 
-e))))))))))) hds)).
-
-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)) (\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 
-(t: T).(\forall (t2: T).((((eq T t t2) \to (\forall (P: Prop).P))) \to ((pr3 
-c1 t 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))))))))) H2 (lift h d x) H3) 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 c1 t 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 (t: T).(pr3 c2 x t)) H8 x (lift_inj x t2 h d 
-H9)) in (let H11 \def (eq_ind_r T t2 (\lambda (t: T).((eq T x t) \to (\forall 
-(P: Prop).P))) 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))))).
-
-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.
-
-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 (and_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 
-(and_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 (a1: 
-A).((llt a1 a) \to (\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))))))))) \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 
-(a1: A).((llt a1 (ASort n n0)) \to (\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)))))))))).(\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 (and_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_sort 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 (_: nat).(eq A a3 (ASort h2 n2))))) (\lambda 
-(n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort n n0) k) 
-(aplus g (ASort h2 n2) k))))) (sc3 g a3 c t) (\lambda (x0: nat).(\lambda (x1: 
-nat).(\lambda (x2: nat).(\lambda (H6: (eq A a3 (ASort x1 x0))).(\lambda (_: 
-(eq A (aplus g (ASort n n0) x2) (aplus g (ASort x1 x0) x2))).(let H8 \def 
-(eq_ind A a3 (\lambda (a: A).(arity g c t a)) H_y (ASort x1 x0) H6) 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) H8 H4) a3 H6))))))) H5)))))) H2)))))))))) (\lambda 
-(a: A).(\lambda (_: ((((\forall (a1: A).((llt a1 a) \to (\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))))))))) \to (\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 (a0: 
-A).(\lambda (H0: ((((\forall (a1: A).((llt a1 a0) \to (\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))))))))) \to (\forall (c: C).(\forall (t: T).((sc3 g a0 c t) 
-\to (\forall (a2: A).((leq g a0 a2) \to (sc3 g a2 c t))))))))).(\lambda (H1: 
-((\forall (a1: A).((llt a1 (AHead a a0)) \to (\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)))))))))).(\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 (and_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_head g a a0 a3 H3) in 
-(let H7 \def H_x in (ex3_2_ind A A (\lambda (a4: A).(\lambda (a5: A).(eq A a3 
-(AHead a4 a5)))) (\lambda (a4: A).(\lambda (_: A).(leq g a a4))) (\lambda (_: 
-A).(\lambda (a5: A).(leq g a0 a5))) (sc3 g a3 c t) (\lambda (x0: A).(\lambda 
-(x1: A).(\lambda (H8: (eq A a3 (AHead x0 x1))).(\lambda (H9: (leq g a 
-x0)).(\lambda (H10: (leq g a0 x1)).(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 H9 a0 x1 H10)) (\lambda (d: C).(\lambda (w: 
-T).(\lambda (H11: (sc3 g x0 d w)).(\lambda (is: PList).(\lambda (H12: (drop1 
-is d c)).(H0 (\lambda (a4: A).(\lambda (H13: (llt a4 a0)).(\lambda (c0: 
-C).(\lambda (t0: T).(\lambda (H14: (sc3 g a4 c0 t0)).(\lambda (a5: 
-A).(\lambda (H15: (leq g a4 a5)).(H1 a4 (llt_trans a4 a0 (AHead a a0) H13 
-(llt_head_dx a a0)) c0 t0 H14 a5 H15)))))))) d (THead (Flat Appl) w (lift1 is 
-t)) (H6 d w (H1 x0 (llt_repl g a x0 H9 (AHead a a0) (llt_head_sx a a0)) d w 
-H11 a (leq_sym g a x0 H9)) is H12) x1 H10))))))) a3 H8)))))) 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 
-(and_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 (and_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 (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)))))))))).(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 H1 \def (match H0 return (\lambda (p: PList).(\lambda (c0: 
-C).(\lambda (c1: C).(\lambda (_: (drop1 p c0 c1)).((eq PList p PNil) \to ((eq 
-C c0 c) \to ((eq C c1 e) \to (sc3 g a c t)))))))) with [(drop1_nil c0) 
-\Rightarrow (\lambda (_: (eq PList PNil PNil)).(\lambda (H2: (eq C c0 
-c)).(\lambda (H3: (eq C c0 e)).(eq_ind C c (\lambda (c1: C).((eq C c1 e) \to 
-(sc3 g a c t))) (\lambda (H4: (eq C c e)).(eq_ind C e (\lambda (c: C).(sc3 g 
-a c t)) H c (sym_eq C c e H4))) c0 (sym_eq C c0 c H2) H3)))) | (drop1_cons c1 
-c2 h d H1 c3 hds H2) \Rightarrow (\lambda (H3: (eq PList (PCons h d hds) 
-PNil)).(\lambda (H4: (eq C c1 c)).(\lambda (H5: (eq C c3 e)).((let H6 \def 
-(eq_ind PList (PCons h d hds) (\lambda (e0: PList).(match e0 return (\lambda 
-(_: PList).Prop) with [PNil \Rightarrow False | (PCons _ _ _) \Rightarrow 
-True])) I PNil H3) in (False_ind ((eq C c1 c) \to ((eq C c3 e) \to ((drop h d 
-c1 c2) \to ((drop1 hds c2 c3) \to (sc3 g a c t))))) H6)) H4 H5 H1 H2))))]) in 
-(H1 (refl_equal PList PNil) (refl_equal C c) (refl_equal C e))))))) (\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 H2 \def (match H1 return 
-(\lambda (p0: PList).(\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (drop1 p0 
-c0 c1)).((eq PList p0 (PCons n n0 p)) \to ((eq C c0 c) \to ((eq C c1 e) \to 
-(sc3 g a c (lift n n0 (lift1 p t)))))))))) with [(drop1_nil c0) \Rightarrow 
-(\lambda (H2: (eq PList PNil (PCons n n0 p))).(\lambda (H3: (eq C c0 
-c)).(\lambda (H4: (eq C c0 e)).((let H5 \def (eq_ind PList PNil (\lambda (e0: 
-PList).(match e0 return (\lambda (_: PList).Prop) with [PNil \Rightarrow True 
-| (PCons _ _ _) \Rightarrow False])) I (PCons n n0 p) H2) in (False_ind ((eq 
-C c0 c) \to ((eq C c0 e) \to (sc3 g a c (lift n n0 (lift1 p t))))) H5)) H3 
-H4)))) | (drop1_cons c1 c2 h d H2 c3 hds H3) \Rightarrow (\lambda (H4: (eq 
-PList (PCons h d hds) (PCons n n0 p))).(\lambda (H5: (eq C c1 c)).(\lambda 
-(H6: (eq C c3 e)).((let H7 \def (f_equal PList PList (\lambda (e0: 
-PList).(match e0 return (\lambda (_: PList).PList) with [PNil \Rightarrow hds 
-| (PCons _ _ p) \Rightarrow p])) (PCons h d hds) (PCons n n0 p) H4) in ((let 
-H8 \def (f_equal PList nat (\lambda (e0: PList).(match e0 return (\lambda (_: 
-PList).nat) with [PNil \Rightarrow d | (PCons _ n _) \Rightarrow n])) (PCons 
-h d hds) (PCons n n0 p) H4) in ((let H9 \def (f_equal PList nat (\lambda (e0: 
-PList).(match e0 return (\lambda (_: PList).nat) with [PNil \Rightarrow h | 
-(PCons n _ _) \Rightarrow n])) (PCons h d hds) (PCons n n0 p) H4) in (eq_ind 
-nat n (\lambda (n1: nat).((eq nat d n0) \to ((eq PList hds p) \to ((eq C c1 
-c) \to ((eq C c3 e) \to ((drop n1 d c1 c2) \to ((drop1 hds c2 c3) \to (sc3 g 
-a c (lift n n0 (lift1 p t)))))))))) (\lambda (H10: (eq nat d n0)).(eq_ind nat 
-n0 (\lambda (n1: nat).((eq PList hds p) \to ((eq C c1 c) \to ((eq C c3 e) \to 
-((drop n n1 c1 c2) \to ((drop1 hds c2 c3) \to (sc3 g a c (lift n n0 (lift1 p 
-t))))))))) (\lambda (H11: (eq PList hds p)).(eq_ind PList p (\lambda (p0: 
-PList).((eq C c1 c) \to ((eq C c3 e) \to ((drop n n0 c1 c2) \to ((drop1 p0 c2 
-c3) \to (sc3 g a c (lift n n0 (lift1 p t)))))))) (\lambda (H12: (eq C c1 
-c)).(eq_ind C c (\lambda (c0: C).((eq C c3 e) \to ((drop n n0 c0 c2) \to 
-((drop1 p c2 c3) \to (sc3 g a c (lift n n0 (lift1 p t))))))) (\lambda (H13: 
-(eq C c3 e)).(eq_ind C e (\lambda (c0: C).((drop n n0 c c2) \to ((drop1 p c2 
-c0) \to (sc3 g a c (lift n n0 (lift1 p t)))))) (\lambda (H14: (drop n n0 c 
-c2)).(\lambda (H15: (drop1 p c2 e)).(sc3_lift g a c2 (lift1 p t) (H c2 t H0 
-H15) c n n0 H14))) c3 (sym_eq C c3 e H13))) c1 (sym_eq C c1 c H12))) hds 
-(sym_eq PList hds p H11))) d (sym_eq nat d n0 H10))) h (sym_eq nat h n H9))) 
-H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal PList (PCons n n0 p)) 
-(refl_equal C c) (refl_equal C e))))))))))) hds)))).
-
-axiom 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)))))))))))
-.
-
-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)))).((match n return (\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))))))) with [O 
-\Rightarrow (\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 (and_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 
-(and_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)))) | (S n1) \Rightarrow 
-(\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 (and_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 (and_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))))]) 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 (and_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 (and_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)).
-
-axiom 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)))))))))))))
-.
-
-axiom 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))))))))))))))
-.
-
-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 (and_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 (and_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 (c: C).(\forall (t: T).((sc3 g a0 c t) \to (sn3 c t)))))).(\lambda 
-(H4: ((\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))))))))))).(let H5 \def H0 in 
-(and_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 (c: C).(\forall (t: T).((sc3 g a1 c t) \to (sn3 c 
-t)))))).(\lambda (_: ((\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))))))))))).(let H8 \def H1 in (and_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 H14 \def 
-(sn3_gen_flat Appl (CHead x0 (Bind Abst) x1) (TLRef O) (lift (S x2) O t) 
-H_y1) in (and_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 (and_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 (c: 
-C).(\forall (t: T).((sc3 g a0 c t) \to (sn3 c t)))))).(\lambda (_: ((\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))))))))))).(let H9 \def H0 in (and_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 (c: C).(\forall (t: T).((sc3 g a1 c t) \to (sn3 c t)))))).(\lambda 
-(H11: ((\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))))))))))).(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 (and_ind (\forall (c0: C).(\forall (t0: T).((sc3 g a c0 t0) \to (sn3 
-c0 t0)))) (\forall (vs: TList).(\forall (i: nat).(let t0 \def (THeads (Flat 
-Appl) vs (TLRef i)) in (\forall (c0: C).((arity g c0 t0 a) \to ((nf2 c0 
-(TLRef i)) \to ((sns3 c0 vs) \to (sc3 g a c0 t0)))))))) (sn3 c t) (\lambda 
-(H1: ((\forall (c: C).(\forall (t: T).((sc3 g a c t) \to (sn3 c 
-t)))))).(\lambda (_: ((\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)))))))))).(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 (and_ind (\forall (c0: 
-C).(\forall (t: T).((sc3 g a c0 t) \to (sn3 c0 t)))) (\forall (vs0: 
-TList).(\forall (i0: nat).(let t \def (THeads (Flat Appl) vs0 (TLRef i0)) in 
-(\forall (c0: C).((arity g c0 t a) \to ((nf2 c0 (TLRef i0)) \to ((sns3 c0 
-vs0) \to (sc3 g a c0 t)))))))) (sc3 g a c (THeads (Flat Appl) vs (TLRef i))) 
-(\lambda (_: ((\forall (c: C).(\forall (t: T).((sc3 g a c t) \to (sn3 c 
-t)))))).(\lambda (H4: ((\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)))))))))).(H4 vs i c H 
-H0 H1))) H2)))))))))).
-
-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_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))))))))).
-
-definition ceqc:
- G \to (C \to (C \to Prop))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(or (csubc g c1 c2) (csubc 
-g c2 c1)))).
-
-theorem scubc_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)).
-
-theorem ceqc_sym:
- \forall (g: G).(\forall (c1: C).(\forall (c2: C).((ceqc g c1 c2) \to (ceqc g 
-c2 c1))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (ceqc g c1 
-c2)).(let H0 \def H in (or_ind (csubc g c1 c2) (csubc g c2 c1) (ceqc g c2 c1) 
-(\lambda (H1: (csubc g c1 c2)).(or_intror (csubc g c2 c1) (csubc g c1 c2) 
-H1)) (\lambda (H1: (csubc g c2 c1)).(or_introl (csubc g c2 c1) (csubc g c1 
-c2) H1)) H0))))).
-
-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 (c: C).(csubc g c 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 
-(c: C).(csubc g c e1)) H2 (CHead x0 k x1) H3) in (let H7 \def (eq_ind C e2 
-(\lambda (c0: C).(\forall (h: nat).((drop h n (CHead c k t) c0) \to (\forall 
-(e1: C).((csubc g c0 e1) \to (ex2 C (\lambda (c1: C).(drop h n c1 e1)) 
-(\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 (t: T).(\forall (h: nat).((drop h n (CHead 
-c k t) (CHead x0 k x1)) \to (\forall (e1: C).((csubc g (CHead x0 k x1) e1) 
-\to (ex2 C (\lambda (c1: C).(drop h n c1 e1)) (\lambda (c1: C).(csubc g 
-(CHead c k t) 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 H9 \def (match H6 return 
-(\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csubc ? c0 c1)).((eq C c0 
-(CHead x0 k x1)) \to ((eq C c1 e1) \to (ex2 C (\lambda (c2: C).(drop h (S n) 
-c2 e1)) (\lambda (c2: C).(csubc g (CHead c k (lift h (r k n) x1)) c2)))))))) 
-with [(csubc_sort n0) \Rightarrow (\lambda (H1: (eq C (CSort n0) (CHead x0 k 
-x1))).(\lambda (H3: (eq C (CSort n0) e1)).((let H4 \def (eq_ind C (CSort n0) 
-(\lambda (e: C).(match e return (\lambda (_: C).Prop) with [(CSort _) 
-\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead x0 k x1) H1) 
-in (False_ind ((eq C (CSort n0) e1) \to (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)))) H4)) 
-H3))) | (csubc_head c1 c2 H1 k0 v) \Rightarrow (\lambda (H3: (eq C (CHead c1 
-k0 v) (CHead x0 k x1))).(\lambda (H6: (eq C (CHead c2 k0 v) e1)).((let H2 
-\def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: C).T) with 
-[(CSort _) \Rightarrow v | (CHead _ _ t) \Rightarrow t])) (CHead c1 k0 v) 
-(CHead x0 k x1) H3) in ((let H4 \def (f_equal C K (\lambda (e: C).(match e 
-return (\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _) 
-\Rightarrow k])) (CHead c1 k0 v) (CHead x0 k x1) H3) in ((let H7 \def 
-(f_equal C C (\lambda (e: C).(match e return (\lambda (_: C).C) with [(CSort 
-_) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k0 v) (CHead x0 
-k x1) H3) in (eq_ind C x0 (\lambda (c0: C).((eq K k0 k) \to ((eq T v x1) \to 
-((eq C (CHead c2 k0 v) e1) \to ((csubc g c0 c2) \to (ex2 C (\lambda (c3: 
-C).(drop h (S n) c3 e1)) (\lambda (c3: C).(csubc g (CHead c k (lift h (r k n) 
-x1)) c3)))))))) (\lambda (H8: (eq K k0 k)).(eq_ind K k (\lambda (k1: K).((eq 
-T v x1) \to ((eq C (CHead c2 k1 v) e1) \to ((csubc g x0 c2) \to (ex2 C 
-(\lambda (c3: C).(drop h (S n) c3 e1)) (\lambda (c3: C).(csubc g (CHead c k 
-(lift h (r k n) x1)) c3))))))) (\lambda (H9: (eq T v x1)).(eq_ind T x1 
-(\lambda (t: T).((eq C (CHead c2 k t) e1) \to ((csubc g x0 c2) \to (ex2 C 
-(\lambda (c3: C).(drop h (S n) c3 e1)) (\lambda (c3: C).(csubc g (CHead c k 
-(lift h (r k n) x1)) c3)))))) (\lambda (H10: (eq C (CHead c2 k x1) 
-e1)).(eq_ind C (CHead c2 k x1) (\lambda (c0: C).((csubc g x0 c2) \to (ex2 C 
-(\lambda (c3: C).(drop h (S n) c3 c0)) (\lambda (c3: C).(csubc g (CHead c k 
-(lift h (r k n) x1)) c3))))) (\lambda (H11: (csubc g x0 c2)).(let H_x \def (H 
-x0 (r k n) h H5 c2 H11) in (let H5 \def H_x in (ex2_ind C (\lambda (c3: 
-C).(drop h (r k n) c3 c2)) (\lambda (c3: C).(csubc g c c3)) (ex2 C (\lambda 
-(c3: C).(drop h (S n) c3 (CHead c2 k x1))) (\lambda (c3: C).(csubc g (CHead c 
-k (lift h (r k n) x1)) c3))) (\lambda (x: C).(\lambda (H12: (drop h (r k n) x 
-c2)).(\lambda (H13: (csubc g c x)).(ex_intro2 C (\lambda (c3: C).(drop h (S 
-n) c3 (CHead c2 k x1))) (\lambda (c3: C).(csubc g (CHead c k (lift h (r k n) 
-x1)) c3)) (CHead x k (lift h (r k n) x1)) (drop_skip k h n x c2 H12 x1) 
-(csubc_head g c x H13 k (lift h (r k n) x1)))))) H5)))) e1 H10)) v (sym_eq T 
-v x1 H9))) k0 (sym_eq K k0 k H8))) c1 (sym_eq C c1 x0 H7))) H4)) H2)) H6 
-H1))) | (csubc_abst c1 c2 H1 v a H3 w H5) \Rightarrow (\lambda (H6: (eq C 
-(CHead c1 (Bind Abst) v) (CHead x0 k x1))).(\lambda (H7: (eq C (CHead c2 
-(Bind Abbr) w) e1)).((let H2 \def (f_equal C T (\lambda (e: C).(match e 
-return (\lambda (_: C).T) with [(CSort _) \Rightarrow v | (CHead _ _ t) 
-\Rightarrow t])) (CHead c1 (Bind Abst) v) (CHead x0 k x1) H6) in ((let H4 
-\def (f_equal C K (\lambda (e: C).(match e return (\lambda (_: C).K) with 
-[(CSort _) \Rightarrow (Bind Abst) | (CHead _ k _) \Rightarrow k])) (CHead c1 
-(Bind Abst) v) (CHead x0 k x1) H6) in ((let H9 \def (f_equal C C (\lambda (e: 
-C).(match e return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead 
-c _ _) \Rightarrow c])) (CHead c1 (Bind Abst) v) (CHead x0 k x1) H6) in 
-(eq_ind C x0 (\lambda (c0: C).((eq K (Bind Abst) k) \to ((eq T v x1) \to ((eq 
-C (CHead c2 (Bind Abbr) w) e1) \to ((csubc g c0 c2) \to ((sc3 g (asucc g a) 
-c0 v) \to ((sc3 g a c2 w) \to (ex2 C (\lambda (c3: C).(drop h (S n) c3 e1)) 
-(\lambda (c3: C).(csubc g (CHead c k (lift h (r k n) x1)) c3)))))))))) 
-(\lambda (H10: (eq K (Bind Abst) k)).(eq_ind K (Bind Abst) (\lambda (k: 
-K).((eq T v x1) \to ((eq C (CHead c2 (Bind Abbr) w) e1) \to ((csubc g x0 c2) 
-\to ((sc3 g (asucc g a) x0 v) \to ((sc3 g a c2 w) \to (ex2 C (\lambda (c3: 
-C).(drop h (S n) c3 e1)) (\lambda (c3: C).(csubc g (CHead c k (lift h (r k n) 
-x1)) c3))))))))) (\lambda (H11: (eq T v x1)).(eq_ind T x1 (\lambda (t: 
-T).((eq C (CHead c2 (Bind Abbr) w) e1) \to ((csubc g x0 c2) \to ((sc3 g 
-(asucc g a) x0 t) \to ((sc3 g a c2 w) \to (ex2 C (\lambda (c3: C).(drop h (S 
-n) c3 e1)) (\lambda (c3: C).(csubc g (CHead c (Bind Abst) (lift h (r (Bind 
-Abst) n) x1)) c3)))))))) (\lambda (H12: (eq C (CHead c2 (Bind Abbr) w) 
-e1)).(eq_ind C (CHead c2 (Bind Abbr) w) (\lambda (c0: C).((csubc g x0 c2) \to 
-((sc3 g (asucc g a) x0 x1) \to ((sc3 g a c2 w) \to (ex2 C (\lambda (c3: 
-C).(drop h (S n) c3 c0)) (\lambda (c3: C).(csubc g (CHead c (Bind Abst) (lift 
-h (r (Bind Abst) n) x1)) c3))))))) (\lambda (H13: (csubc g x0 c2)).(\lambda 
-(H14: (sc3 g (asucc g a) x0 x1)).(\lambda (H15: (sc3 g a c2 w)).(let H8 \def 
-(eq_ind_r K k (\lambda (k: K).(\forall (h0: nat).((drop h0 n (CHead c k (lift 
-h (r k n) x1)) (CHead x0 k x1)) \to (\forall (e1: C).((csubc g (CHead x0 k 
-x1) e1) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e1)) (\lambda (c1: 
-C).(csubc g (CHead c k (lift h (r k n) x1)) c1)))))))) H8 (Bind Abst) H10) in 
-(let H16 \def (eq_ind_r K k (\lambda (k: K).(drop h (r k n) c x0)) H5 (Bind 
-Abst) H10) in (let H_x \def (H x0 (r (Bind Abst) n) h H16 c2 H13) in (let H17 
-\def H_x in (ex2_ind C (\lambda (c3: C).(drop h (r (Bind Abst) n) c3 c2)) 
-(\lambda (c3: C).(csubc g c c3)) (ex2 C (\lambda (c3: C).(drop h (S n) c3 
-(CHead c2 (Bind Abbr) w))) (\lambda (c3: C).(csubc g (CHead c (Bind Abst) 
-(lift h (r (Bind Abst) n) x1)) c3))) (\lambda (x: C).(\lambda (H18: (drop h 
-(r (Bind Abst) n) x c2)).(\lambda (H19: (csubc g c x)).(ex_intro2 C (\lambda 
-(c3: C).(drop h (S n) c3 (CHead c2 (Bind Abbr) w))) (\lambda (c3: C).(csubc g 
-(CHead c (Bind Abst) (lift h (r (Bind Abst) n) x1)) c3)) (CHead x (Bind Abbr) 
-(lift h n w)) (drop_skip_bind h n x c2 H18 Abbr w) (csubc_abst g c x H19 
-(lift h (r (Bind Abst) n) x1) a (sc3_lift g (asucc g a) x0 x1 H14 c h (r 
-(Bind Abst) n) H16) (lift h n w) (sc3_lift g a c2 w H15 x h n H18)))))) 
-H17)))))))) e1 H12)) v (sym_eq T v x1 H11))) k H10)) c1 (sym_eq C c1 x0 H9))) 
-H4)) H2)) H7 H1 H3 H5)))]) in (H9 (refl_equal C (CHead x0 k x1)) (refl_equal 
-C e1))) 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 (c: C).(csubc g e1 c)) 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 
-(c: C).(csubc g e1 c)) H2 (CHead x0 k x1) H3) in (let H7 \def (eq_ind C e2 
-(\lambda (c0: C).(\forall (h: nat).((drop h n (CHead c k t) c0) \to (\forall 
-(e1: C).((csubc g e1 c0) \to (ex2 C (\lambda (c1: C).(drop h n c1 e1)) 
-(\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 (t: T).(\forall (h: nat).((drop h n (CHead 
-c k t) (CHead x0 k x1)) \to (\forall (e1: C).((csubc g e1 (CHead x0 k x1)) 
-\to (ex2 C (\lambda (c1: C).(drop h n c1 e1)) (\lambda (c1: C).(csubc g c1 
-(CHead c k t))))))))) 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 H9 \def (match H6 return 
-(\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csubc ? c0 c1)).((eq C c0 e1) 
-\to ((eq C c1 (CHead x0 k x1)) \to (ex2 C (\lambda (c2: C).(drop h (S n) c2 
-e1)) (\lambda (c2: C).(csubc g c2 (CHead c k (lift h (r k n) x1)))))))))) 
-with [(csubc_sort n0) \Rightarrow (\lambda (H1: (eq C (CSort n0) 
-e1)).(\lambda (H3: (eq C (CSort n0) (CHead x0 k x1))).(eq_ind C (CSort n0) 
-(\lambda (c0: C).((eq C (CSort n0) (CHead x0 k x1)) \to (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))))))) (\lambda (H4: (eq C (CSort n0) (CHead x0 k x1))).(let H5 \def 
-(eq_ind C (CSort n0) (\lambda (e: C).(match e return (\lambda (_: C).Prop) 
-with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I 
-(CHead x0 k x1) H4) in (False_ind (ex2 C (\lambda (c1: C).(drop h (S n) c1 
-(CSort n0))) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1))))) 
-H5))) e1 H1 H3))) | (csubc_head c1 c2 H1 k0 v) \Rightarrow (\lambda (H3: (eq 
-C (CHead c1 k0 v) e1)).(\lambda (H6: (eq C (CHead c2 k0 v) (CHead x0 k 
-x1))).(eq_ind C (CHead c1 k0 v) (\lambda (c0: C).((eq C (CHead c2 k0 v) 
-(CHead x0 k x1)) \to ((csubc g c1 c2) \to (ex2 C (\lambda (c3: C).(drop h (S 
-n) c3 c0)) (\lambda (c3: C).(csubc g c3 (CHead c k (lift h (r k n) x1)))))))) 
-(\lambda (H7: (eq C (CHead c2 k0 v) (CHead x0 k x1))).(let H2 \def (f_equal C 
-T (\lambda (e: C).(match e return (\lambda (_: C).T) with [(CSort _) 
-\Rightarrow v | (CHead _ _ t) \Rightarrow t])) (CHead c2 k0 v) (CHead x0 k 
-x1) H7) in ((let H4 \def (f_equal C K (\lambda (e: C).(match e return 
-(\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _) \Rightarrow 
-k])) (CHead c2 k0 v) (CHead x0 k x1) H7) in ((let H8 \def (f_equal C C 
-(\lambda (e: C).(match e return (\lambda (_: C).C) with [(CSort _) 
-\Rightarrow c2 | (CHead c _ _) \Rightarrow c])) (CHead c2 k0 v) (CHead x0 k 
-x1) H7) in (eq_ind C x0 (\lambda (c0: C).((eq K k0 k) \to ((eq T v x1) \to 
-((csubc g c1 c0) \to (ex2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c1 k0 
-v))) (\lambda (c3: C).(csubc g c3 (CHead c k (lift h (r k n) x1))))))))) 
-(\lambda (H9: (eq K k0 k)).(eq_ind K k (\lambda (k1: K).((eq T v x1) \to 
-((csubc g c1 x0) \to (ex2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c1 k1 
-v))) (\lambda (c3: C).(csubc g c3 (CHead c k (lift h (r k n) x1)))))))) 
-(\lambda (H10: (eq T v x1)).(eq_ind T x1 (\lambda (t: T).((csubc g c1 x0) \to 
-(ex2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c1 k t))) (\lambda (c3: 
-C).(csubc g c3 (CHead c k (lift h (r k n) x1))))))) (\lambda (H11: (csubc g 
-c1 x0)).(let H12 \def (eq_ind T v (\lambda (t: T).(eq C (CHead c1 k0 t) e1)) 
-H3 x1 H10) in (let H13 \def (eq_ind K k0 (\lambda (k: K).(eq C (CHead c1 k 
-x1) e1)) H12 k H9) in (let H_x \def (H x0 (r k n) h H5 c1 H11) in (let H5 
-\def H_x in (ex2_ind C (\lambda (c3: C).(drop h (r k n) c3 c1)) (\lambda (c3: 
-C).(csubc g c3 c)) (ex2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c1 k x1))) 
-(\lambda (c3: C).(csubc g c3 (CHead c k (lift h (r k n) x1))))) (\lambda (x: 
-C).(\lambda (H14: (drop h (r k n) x c1)).(\lambda (H15: (csubc g x 
-c)).(ex_intro2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c1 k x1))) (\lambda 
-(c3: C).(csubc g c3 (CHead c k (lift h (r k n) x1)))) (CHead x k (lift h (r k 
-n) x1)) (drop_skip k h n x c1 H14 x1) (csubc_head g x c H15 k (lift h (r k n) 
-x1)))))) H5)))))) v (sym_eq T v x1 H10))) k0 (sym_eq K k0 k H9))) c2 (sym_eq 
-C c2 x0 H8))) H4)) H2))) e1 H3 H6 H1))) | (csubc_abst c1 c2 H1 v a H3 w H5) 
-\Rightarrow (\lambda (H6: (eq C (CHead c1 (Bind Abst) v) e1)).(\lambda (H7: 
-(eq C (CHead c2 (Bind Abbr) w) (CHead x0 k x1))).(eq_ind C (CHead c1 (Bind 
-Abst) v) (\lambda (c0: C).((eq C (CHead c2 (Bind Abbr) w) (CHead x0 k x1)) 
-\to ((csubc g c1 c2) \to ((sc3 g (asucc g a) c1 v) \to ((sc3 g a c2 w) \to 
-(ex2 C (\lambda (c3: C).(drop h (S n) c3 c0)) (\lambda (c3: C).(csubc g c3 
-(CHead c k (lift h (r k n) x1)))))))))) (\lambda (H9: (eq C (CHead c2 (Bind 
-Abbr) w) (CHead x0 k x1))).(let H2 \def (f_equal C T (\lambda (e: C).(match e 
-return (\lambda (_: C).T) with [(CSort _) \Rightarrow w | (CHead _ _ t) 
-\Rightarrow t])) (CHead c2 (Bind Abbr) w) (CHead x0 k x1) H9) in ((let H4 
-\def (f_equal C K (\lambda (e: C).(match e return (\lambda (_: C).K) with 
-[(CSort _) \Rightarrow (Bind Abbr) | (CHead _ k _) \Rightarrow k])) (CHead c2 
-(Bind Abbr) w) (CHead x0 k x1) H9) in ((let H10 \def (f_equal C C (\lambda 
-(e: C).(match e return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | 
-(CHead c _ _) \Rightarrow c])) (CHead c2 (Bind Abbr) w) (CHead x0 k x1) H9) 
-in (eq_ind C x0 (\lambda (c0: C).((eq K (Bind Abbr) k) \to ((eq T w x1) \to 
-((csubc g c1 c0) \to ((sc3 g (asucc g a) c1 v) \to ((sc3 g a c0 w) \to (ex2 C 
-(\lambda (c3: C).(drop h (S n) c3 (CHead c1 (Bind Abst) v))) (\lambda (c3: 
-C).(csubc g c3 (CHead c k (lift h (r k n) x1))))))))))) (\lambda (H11: (eq K 
-(Bind Abbr) k)).(eq_ind K (Bind Abbr) (\lambda (k: K).((eq T w x1) \to 
-((csubc g c1 x0) \to ((sc3 g (asucc g a) c1 v) \to ((sc3 g a x0 w) \to (ex2 C 
-(\lambda (c3: C).(drop h (S n) c3 (CHead c1 (Bind Abst) v))) (\lambda (c3: 
-C).(csubc g c3 (CHead c k (lift h (r k n) x1)))))))))) (\lambda (H12: (eq T w 
-x1)).(eq_ind T x1 (\lambda (t: T).((csubc g c1 x0) \to ((sc3 g (asucc g a) c1 
-v) \to ((sc3 g a x0 t) \to (ex2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c1 
-(Bind Abst) v))) (\lambda (c3: C).(csubc g c3 (CHead c (Bind Abbr) (lift h (r 
-(Bind Abbr) n) x1))))))))) (\lambda (H13: (csubc g c1 x0)).(\lambda (H14: 
-(sc3 g (asucc g a) c1 v)).(\lambda (H15: (sc3 g a x0 x1)).(let H8 \def 
-(eq_ind_r K k (\lambda (k: K).(\forall (h0: nat).((drop h0 n (CHead c k (lift 
-h (r k n) x1)) (CHead x0 k x1)) \to (\forall (e1: C).((csubc g e1 (CHead x0 k 
-x1)) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e1)) (\lambda (c1: C).(csubc g 
-c1 (CHead c k (lift h (r k n) x1)))))))))) H8 (Bind Abbr) H11) in (let H16 
-\def (eq_ind_r K k (\lambda (k: K).(drop h (r k n) c x0)) H5 (Bind Abbr) H11) 
-in (let H_x \def (H x0 (r (Bind Abbr) n) h H16 c1 H13) in (let H17 \def H_x 
-in (ex2_ind C (\lambda (c3: C).(drop h (r (Bind Abbr) n) c3 c1)) (\lambda 
-(c3: C).(csubc g c3 c)) (ex2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c1 
-(Bind Abst) v))) (\lambda (c3: C).(csubc g c3 (CHead c (Bind Abbr) (lift h (r 
-(Bind Abbr) n) x1))))) (\lambda (x: C).(\lambda (H18: (drop h (r (Bind Abbr) 
-n) x c1)).(\lambda (H19: (csubc g x c)).(ex_intro2 C (\lambda (c3: C).(drop h 
-(S n) c3 (CHead c1 (Bind Abst) v))) (\lambda (c3: C).(csubc g c3 (CHead c 
-(Bind Abbr) (lift h (r (Bind Abbr) n) x1)))) (CHead x (Bind Abst) (lift h n 
-v)) (drop_skip_bind h n x c1 H18 Abst v) (csubc_abst g x c H19 (lift h n v) a 
-(sc3_lift g (asucc g a) c1 v H14 x h n H18) (lift h (r (Bind Abbr) n) x1) 
-(sc3_lift g a x0 x1 H15 c h (r (Bind Abbr) n) H16)))))) H17)))))))) w (sym_eq 
-T w x1 H12))) k H11)) c2 (sym_eq C c2 x0 H10))) H4)) H2))) e1 H6 H7 H1 H3 
-H5)))]) in (H9 (refl_equal C e1) (refl_equal C (CHead x0 k x1)))) t 
-H4))))))))) (drop_gen_skip_l c e2 t h n k H1)))))))) d))))))) c2)).
-
-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 H1 \def (match H 
-return (\lambda (p: PList).(\lambda (c: C).(\lambda (c0: C).(\lambda (_: 
-(drop1 p c c0)).((eq PList p PNil) \to ((eq C c c2) \to ((eq C c0 e2) \to 
-(ex2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g c2 
-c1)))))))))) with [(drop1_nil c) \Rightarrow (\lambda (_: (eq PList PNil 
-PNil)).(\lambda (H2: (eq C c c2)).(\lambda (H3: (eq C c e2)).(eq_ind C c2 
-(\lambda (c0: C).((eq C c0 e2) \to (ex2 C (\lambda (c1: C).(drop1 PNil c1 
-e1)) (\lambda (c1: C).(csubc g c2 c1))))) (\lambda (H4: (eq C c2 e2)).(eq_ind 
-C e2 (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda 
-(c1: C).(csubc g c0 c1)))) (let H \def (eq_ind_r C e2 (\lambda (c: C).(csubc 
-g c e1)) H0 c2 H4) in (eq_ind C c2 (\lambda (c0: C).(ex2 C (\lambda (c1: 
-C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g c0 c1)))) (ex_intro2 C 
-(\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g c2 c1)) e1 
-(drop1_nil e1) H) e2 H4)) c2 (sym_eq C c2 e2 H4))) c (sym_eq C c c2 H2) 
-H3)))) | (drop1_cons c1 c0 h d H1 c3 hds H2) \Rightarrow (\lambda (H3: (eq 
-PList (PCons h d hds) PNil)).(\lambda (H4: (eq C c1 c2)).(\lambda (H5: (eq C 
-c3 e2)).((let H6 \def (eq_ind PList (PCons h d hds) (\lambda (e: 
-PList).(match e return (\lambda (_: PList).Prop) with [PNil \Rightarrow False 
-| (PCons _ _ _) \Rightarrow True])) I PNil H3) in (False_ind ((eq C c1 c2) 
-\to ((eq C c3 e2) \to ((drop h d c1 c0) \to ((drop1 hds c0 c3) \to (ex2 C 
-(\lambda (c2: C).(drop1 PNil c2 e1)) (\lambda (c4: C).(csubc g c2 c4))))))) 
-H6)) H4 H5 H1 H2))))]) in (H1 (refl_equal PList PNil) (refl_equal C c2) 
-(refl_equal C e2)))))))) (\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 H2 \def (match H0 return (\lambda (p0: PList).(\lambda 
-(c: C).(\lambda (c0: C).(\lambda (_: (drop1 p0 c c0)).((eq PList p0 (PCons n 
-n0 p)) \to ((eq C c c2) \to ((eq C c0 e2) \to (ex2 C (\lambda (c1: C).(drop1 
-(PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c2 c1)))))))))) with 
-[(drop1_nil c) \Rightarrow (\lambda (H2: (eq PList PNil (PCons n n0 
-p))).(\lambda (H3: (eq C c c2)).(\lambda (H4: (eq C c e2)).((let H5 \def 
-(eq_ind PList PNil (\lambda (e: PList).(match e return (\lambda (_: 
-PList).Prop) with [PNil \Rightarrow True | (PCons _ _ _) \Rightarrow False])) 
-I (PCons n n0 p) H2) in (False_ind ((eq C c c2) \to ((eq C c e2) \to (ex2 C 
-(\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c2 
-c1))))) H5)) H3 H4)))) | (drop1_cons c1 c0 h d H2 c3 hds H3) \Rightarrow 
-(\lambda (H4: (eq PList (PCons h d hds) (PCons n n0 p))).(\lambda (H5: (eq C 
-c1 c2)).(\lambda (H6: (eq C c3 e2)).((let H7 \def (f_equal PList PList 
-(\lambda (e: PList).(match e return (\lambda (_: PList).PList) with [PNil 
-\Rightarrow hds | (PCons _ _ p) \Rightarrow p])) (PCons h d hds) (PCons n n0 
-p) H4) in ((let H8 \def (f_equal PList nat (\lambda (e: PList).(match e 
-return (\lambda (_: PList).nat) with [PNil \Rightarrow d | (PCons _ n _) 
-\Rightarrow n])) (PCons h d hds) (PCons n n0 p) H4) in ((let H9 \def (f_equal 
-PList nat (\lambda (e: PList).(match e return (\lambda (_: PList).nat) with 
-[PNil \Rightarrow h | (PCons n _ _) \Rightarrow n])) (PCons h d hds) (PCons n 
-n0 p) H4) in (eq_ind nat n (\lambda (n1: nat).((eq nat d n0) \to ((eq PList 
-hds p) \to ((eq C c1 c2) \to ((eq C c3 e2) \to ((drop n1 d c1 c0) \to ((drop1 
-hds c0 c3) \to (ex2 C (\lambda (c2: C).(drop1 (PCons n n0 p) c2 e1)) (\lambda 
-(c4: C).(csubc g c2 c4)))))))))) (\lambda (H10: (eq nat d n0)).(eq_ind nat n0 
-(\lambda (n1: nat).((eq PList hds p) \to ((eq C c1 c2) \to ((eq C c3 e2) \to 
-((drop n n1 c1 c0) \to ((drop1 hds c0 c3) \to (ex2 C (\lambda (c2: C).(drop1 
-(PCons n n0 p) c2 e1)) (\lambda (c4: C).(csubc g c2 c4))))))))) (\lambda 
-(H11: (eq PList hds p)).(eq_ind PList p (\lambda (p0: PList).((eq C c1 c2) 
-\to ((eq C c3 e2) \to ((drop n n0 c1 c0) \to ((drop1 p0 c0 c3) \to (ex2 C 
-(\lambda (c2: C).(drop1 (PCons n n0 p) c2 e1)) (\lambda (c4: C).(csubc g c2 
-c4)))))))) (\lambda (H12: (eq C c1 c2)).(eq_ind C c2 (\lambda (c: C).((eq C 
-c3 e2) \to ((drop n n0 c c0) \to ((drop1 p c0 c3) \to (ex2 C (\lambda (c2: 
-C).(drop1 (PCons n n0 p) c2 e1)) (\lambda (c4: C).(csubc g c2 c4))))))) 
-(\lambda (H13: (eq C c3 e2)).(eq_ind C e2 (\lambda (c: C).((drop n n0 c2 c0) 
-\to ((drop1 p c0 c) \to (ex2 C (\lambda (c2: C).(drop1 (PCons n n0 p) c2 e1)) 
-(\lambda (c4: C).(csubc g c2 c4)))))) (\lambda (H14: (drop n n0 c2 
-c0)).(\lambda (H15: (drop1 p c0 e2)).(let H_x \def (H c0 e2 H15 e1 H1) in 
-(let H0 \def H_x in (ex2_ind C (\lambda (c2: C).(drop1 p c2 e1)) (\lambda 
-(c2: C).(csubc g c0 c2)) (ex2 C (\lambda (c2: C).(drop1 (PCons n n0 p) c2 
-e1)) (\lambda (c4: C).(csubc g c2 c4))) (\lambda (x: C).(\lambda (H1: (drop1 
-p x e1)).(\lambda (H16: (csubc g c0 x)).(let H_x0 \def (drop_csubc_trans g c2 
-c0 n0 n H14 x H16) in (let H \def H_x0 in (ex2_ind C (\lambda (c2: C).(drop n 
-n0 c2 x)) (\lambda (c4: C).(csubc g c2 c4)) (ex2 C (\lambda (c2: C).(drop1 
-(PCons n n0 p) c2 e1)) (\lambda (c4: C).(csubc g c2 c4))) (\lambda (x0: 
-C).(\lambda (H17: (drop n n0 x0 x)).(\lambda (H18: (csubc g c2 
-x0)).(ex_intro2 C (\lambda (c2: C).(drop1 (PCons n n0 p) c2 e1)) (\lambda 
-(c4: C).(csubc g c2 c4)) x0 (drop1_cons x0 x n n0 H17 e1 p H1) H18)))) 
-H)))))) H0))))) c3 (sym_eq C c3 e2 H13))) c1 (sym_eq C c1 c2 H12))) hds 
-(sym_eq PList hds p H11))) d (sym_eq nat d n0 H10))) h (sym_eq nat h n H9))) 
-H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal PList (PCons n n0 p)) 
-(refl_equal C c2) (refl_equal C e2)))))))))))) 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 H1 \def (match H 
-return (\lambda (p: PList).(\lambda (c: C).(\lambda (c0: C).(\lambda (_: 
-(drop1 p c c0)).((eq PList p PNil) \to ((eq C c c2) \to ((eq C c0 e2) \to 
-(ex2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g c1 
-c2)))))))))) with [(drop1_nil c) \Rightarrow (\lambda (_: (eq PList PNil 
-PNil)).(\lambda (H2: (eq C c c2)).(\lambda (H3: (eq C c e2)).(eq_ind C c2 
-(\lambda (c0: C).((eq C c0 e2) \to (ex2 C (\lambda (c1: C).(drop1 PNil c1 
-e1)) (\lambda (c1: C).(csubc g c1 c2))))) (\lambda (H4: (eq C c2 e2)).(eq_ind 
-C e2 (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda 
-(c1: C).(csubc g c1 c0)))) (let H \def (eq_ind_r C e2 (\lambda (c: C).(csubc 
-g e1 c)) H0 c2 H4) in (eq_ind C c2 (\lambda (c0: C).(ex2 C (\lambda (c1: 
-C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g c1 c0)))) (ex_intro2 C 
-(\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g c1 c2)) e1 
-(drop1_nil e1) H) e2 H4)) c2 (sym_eq C c2 e2 H4))) c (sym_eq C c c2 H2) 
-H3)))) | (drop1_cons c1 c0 h d H1 c3 hds H2) \Rightarrow (\lambda (H3: (eq 
-PList (PCons h d hds) PNil)).(\lambda (H4: (eq C c1 c2)).(\lambda (H5: (eq C 
-c3 e2)).((let H6 \def (eq_ind PList (PCons h d hds) (\lambda (e: 
-PList).(match e return (\lambda (_: PList).Prop) with [PNil \Rightarrow False 
-| (PCons _ _ _) \Rightarrow True])) I PNil H3) in (False_ind ((eq C c1 c2) 
-\to ((eq C c3 e2) \to ((drop h d c1 c0) \to ((drop1 hds c0 c3) \to (ex2 C 
-(\lambda (c2: C).(drop1 PNil c2 e1)) (\lambda (c4: C).(csubc g c4 c2))))))) 
-H6)) H4 H5 H1 H2))))]) in (H1 (refl_equal PList PNil) (refl_equal C c2) 
-(refl_equal C e2)))))))) (\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 H2 \def (match H0 return (\lambda (p0: PList).(\lambda 
-(c: C).(\lambda (c0: C).(\lambda (_: (drop1 p0 c c0)).((eq PList p0 (PCons n 
-n0 p)) \to ((eq C c c2) \to ((eq C c0 e2) \to (ex2 C (\lambda (c1: C).(drop1 
-(PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c1 c2)))))))))) with 
-[(drop1_nil c) \Rightarrow (\lambda (H2: (eq PList PNil (PCons n n0 
-p))).(\lambda (H3: (eq C c c2)).(\lambda (H4: (eq C c e2)).((let H5 \def 
-(eq_ind PList PNil (\lambda (e: PList).(match e return (\lambda (_: 
-PList).Prop) with [PNil \Rightarrow True | (PCons _ _ _) \Rightarrow False])) 
-I (PCons n n0 p) H2) in (False_ind ((eq C c c2) \to ((eq C c e2) \to (ex2 C 
-(\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c1 
-c2))))) H5)) H3 H4)))) | (drop1_cons c1 c0 h d H2 c3 hds H3) \Rightarrow 
-(\lambda (H4: (eq PList (PCons h d hds) (PCons n n0 p))).(\lambda (H5: (eq C 
-c1 c2)).(\lambda (H6: (eq C c3 e2)).((let H7 \def (f_equal PList PList 
-(\lambda (e: PList).(match e return (\lambda (_: PList).PList) with [PNil 
-\Rightarrow hds | (PCons _ _ p) \Rightarrow p])) (PCons h d hds) (PCons n n0 
-p) H4) in ((let H8 \def (f_equal PList nat (\lambda (e: PList).(match e 
-return (\lambda (_: PList).nat) with [PNil \Rightarrow d | (PCons _ n _) 
-\Rightarrow n])) (PCons h d hds) (PCons n n0 p) H4) in ((let H9 \def (f_equal 
-PList nat (\lambda (e: PList).(match e return (\lambda (_: PList).nat) with 
-[PNil \Rightarrow h | (PCons n _ _) \Rightarrow n])) (PCons h d hds) (PCons n 
-n0 p) H4) in (eq_ind nat n (\lambda (n1: nat).((eq nat d n0) \to ((eq PList 
-hds p) \to ((eq C c1 c2) \to ((eq C c3 e2) \to ((drop n1 d c1 c0) \to ((drop1 
-hds c0 c3) \to (ex2 C (\lambda (c2: C).(drop1 (PCons n n0 p) c2 e1)) (\lambda 
-(c4: C).(csubc g c4 c2)))))))))) (\lambda (H10: (eq nat d n0)).(eq_ind nat n0 
-(\lambda (n1: nat).((eq PList hds p) \to ((eq C c1 c2) \to ((eq C c3 e2) \to 
-((drop n n1 c1 c0) \to ((drop1 hds c0 c3) \to (ex2 C (\lambda (c2: C).(drop1 
-(PCons n n0 p) c2 e1)) (\lambda (c4: C).(csubc g c4 c2))))))))) (\lambda 
-(H11: (eq PList hds p)).(eq_ind PList p (\lambda (p0: PList).((eq C c1 c2) 
-\to ((eq C c3 e2) \to ((drop n n0 c1 c0) \to ((drop1 p0 c0 c3) \to (ex2 C 
-(\lambda (c2: C).(drop1 (PCons n n0 p) c2 e1)) (\lambda (c4: C).(csubc g c4 
-c2)))))))) (\lambda (H12: (eq C c1 c2)).(eq_ind C c2 (\lambda (c: C).((eq C 
-c3 e2) \to ((drop n n0 c c0) \to ((drop1 p c0 c3) \to (ex2 C (\lambda (c2: 
-C).(drop1 (PCons n n0 p) c2 e1)) (\lambda (c4: C).(csubc g c4 c2))))))) 
-(\lambda (H13: (eq C c3 e2)).(eq_ind C e2 (\lambda (c: C).((drop n n0 c2 c0) 
-\to ((drop1 p c0 c) \to (ex2 C (\lambda (c2: C).(drop1 (PCons n n0 p) c2 e1)) 
-(\lambda (c4: C).(csubc g c4 c2)))))) (\lambda (H14: (drop n n0 c2 
-c0)).(\lambda (H15: (drop1 p c0 e2)).(let H_x \def (H c0 e2 H15 e1 H1) in 
-(let H0 \def H_x in (ex2_ind C (\lambda (c2: C).(drop1 p c2 e1)) (\lambda 
-(c2: C).(csubc g c2 c0)) (ex2 C (\lambda (c2: C).(drop1 (PCons n n0 p) c2 
-e1)) (\lambda (c4: C).(csubc g c4 c2))) (\lambda (x: C).(\lambda (H1: (drop1 
-p x e1)).(\lambda (H16: (csubc g x c0)).(let H_x0 \def (csubc_drop_conf_rev g 
-c2 c0 n0 n H14 x H16) in (let H \def H_x0 in (ex2_ind C (\lambda (c2: 
-C).(drop n n0 c2 x)) (\lambda (c4: C).(csubc g c4 c2)) (ex2 C (\lambda (c2: 
-C).(drop1 (PCons n n0 p) c2 e1)) (\lambda (c4: C).(csubc g c4 c2))) (\lambda 
-(x0: C).(\lambda (H17: (drop n n0 x0 x)).(\lambda (H18: (csubc g x0 
-c2)).(ex_intro2 C (\lambda (c2: C).(drop1 (PCons n n0 p) c2 e1)) (\lambda 
-(c4: C).(csubc g c4 c2)) x0 (drop1_cons x0 x n n0 H17 e1 p H1) H18)))) 
-H)))))) H0))))) c3 (sym_eq C c3 e2 H13))) c1 (sym_eq C c1 c2 H12))) hds 
-(sym_eq PList hds p H11))) d (sym_eq nat d n0 H10))) h (sym_eq nat h n H9))) 
-H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal PList (PCons n n0 p)) 
-(refl_equal C c2) (refl_equal C e2)))))))))))) hds)).
-
-theorem drop1_ceqc_trans:
- \forall (g: G).(\forall (hds: PList).(\forall (c2: C).(\forall (e2: 
-C).((drop1 hds c2 e2) \to (\forall (e1: C).((ceqc g e2 e1) \to (ex2 C 
-(\lambda (c1: C).(drop1 hds c1 e1)) (\lambda (c1: C).(ceqc g c2 c1)))))))))
-\def
- \lambda (g: G).(\lambda (hds: PList).(\lambda (c2: C).(\lambda (e2: 
-C).(\lambda (H: (drop1 hds c2 e2)).(\lambda (e1: C).(\lambda (H0: (ceqc g e2 
-e1)).(let H1 \def H0 in (or_ind (csubc g e2 e1) (csubc g e1 e2) (ex2 C 
-(\lambda (c1: C).(drop1 hds c1 e1)) (\lambda (c1: C).(ceqc g c2 c1))) 
-(\lambda (H2: (csubc g e2 e1)).(let H_x \def (drop1_csubc_trans g hds c2 e2 H 
-e1 H2) in (let H3 \def H_x in (ex2_ind C (\lambda (c1: C).(drop1 hds c1 e1)) 
-(\lambda (c1: C).(csubc g c2 c1)) (ex2 C (\lambda (c1: C).(drop1 hds c1 e1)) 
-(\lambda (c1: C).(ceqc g c2 c1))) (\lambda (x: C).(\lambda (H4: (drop1 hds x 
-e1)).(\lambda (H5: (csubc g c2 x)).(ex_intro2 C (\lambda (c1: C).(drop1 hds 
-c1 e1)) (\lambda (c1: C).(ceqc g c2 c1)) x H4 (or_introl (csubc g c2 x) 
-(csubc g x c2) H5))))) H3)))) (\lambda (H2: (csubc g e1 e2)).(let H_x \def 
-(csubc_drop1_conf_rev g hds c2 e2 H e1 H2) in (let H3 \def H_x in (ex2_ind C 
-(\lambda (c1: C).(drop1 hds c1 e1)) (\lambda (c1: C).(csubc g c1 c2)) (ex2 C 
-(\lambda (c1: C).(drop1 hds c1 e1)) (\lambda (c1: C).(ceqc g c2 c1))) 
-(\lambda (x: C).(\lambda (H4: (drop1 hds x e1)).(\lambda (H5: (csubc g x 
-c2)).(ex_intro2 C (\lambda (c1: C).(drop1 hds c1 e1)) (\lambda (c1: C).(ceqc 
-g c2 c1)) x H4 (or_intror (csubc g c2 x) (csubc g x c2) H5))))) H3)))) 
-H1)))))))).
-
-axiom sc3_ceqc_trans:
- \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (c1: 
-C).(\forall (t: T).((sc3 g a c1 (THeads (Flat Appl) vs t)) \to (\forall (c2: 
-C).((ceqc g c2 c1) \to (sc3 g a c2 (THeads (Flat Appl) vs t)))))))))
-.
-
-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)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (a0: 
-A).(sc3 g a0 c0 t0)))) (\lambda (c0: C).(\lambda (n: nat).(conj (arity g c0 
-(TSort n) (ASort O n)) (sn3 c0 (TSort n)) (arity_sort g c0 n) (sn3_nf2 c0 
-(TSort n) (nf2_sort c0 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 (H2: (sc3 g a0 
-d u)).(let H_y \def (sc3_abbr g a0 TNil) in (H_y i d u c0 (sc3_lift g a0 d u 
-H2 c0 (S i) O (getl_drop Abbr c0 d u i H0)) H0)))))))))) (\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 (H1: (arity g d u (asucc 
-g a0))).(\lambda (_: (sc3 g (asucc g a0) d u)).(let H3 \def (sc3_abst g a0 
-TNil) in (H3 c0 i (arity_abst g c0 d u i H0 a0 H1) (nf2_lref_abst c0 d u i 
-H0) I)))))))))) (\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 (H2: (sc3 g a1 c0 u)).(\lambda (t0: T).(\lambda (a2: 
-A).(\lambda (_: (arity g (CHead c0 (Bind b) u) t0 a2)).(\lambda (H4: (sc3 g 
-a2 (CHead c0 (Bind b) u) t0)).(let H_y \def (sc3_bind g b H0 a1 a2 TNil) in 
-(H_y c0 u t0 H4 H2))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda 
-(a1: A).(\lambda (H0: (arity g c0 u (asucc g a1))).(\lambda (H1: (sc3 g 
-(asucc g a1) c0 u)).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H2: (arity g 
-(CHead c0 (Bind Abst) u) t0 a2)).(\lambda (H3: (sc3 g a2 (CHead c0 (Bind 
-Abst) u) t0)).(conj (arity g c0 (THead (Bind Abst) u t0) (AHead a1 a2)) 
-(\forall (d: C).(\forall (w: T).((sc3 g a1 d w) \to (\forall (is: 
-PList).((drop1 is d c0) \to (sc3 g a2 d (THead (Flat Appl) w (lift1 is (THead 
-(Bind Abst) u t0))))))))) (arity_head g c0 u a1 H0 t0 a2 H2) (\lambda (d: 
-C).(\lambda (w: T).(\lambda (H4: (sc3 g a1 d w)).(\lambda (is: 
-PList).(\lambda (H5: (drop1 is d c0)).(let H6 \def (sc3_appl g a1 a2 TNil) in 
-(eq_ind_r T (THead (Bind Abst) (lift1 is u) (lift1 (Ss is) t0)) (\lambda (t1: 
-T).(sc3 g a2 d (THead (Flat Appl) w t1))) (H6 d w (lift1 (Ss is) t0) (let H_y 
-\def (sc3_bind g Abbr (\lambda (H3: (eq B Abbr Abst)).(not_abbr_abst H3)) a1 
-a2 TNil) in (H_y d w (lift1 (Ss is) t0) (let H7 \def (sc3_ceqc_trans g a2 
-TNil) in (H7 (CHead d (Bind Abst) (lift1 is u)) (lift1 (Ss is) t0) (sc3_lift1 
-g (CHead c0 (Bind Abst) u) a2 (Ss is) (CHead d (Bind Abst) (lift1 is u)) t0 
-H3 (drop1_skip_bind Abst c0 is d u H5)) (CHead d (Bind Abbr) w) (or_intror 
-(csubc g (CHead d (Bind Abbr) w) (CHead d (Bind Abst) (lift1 is u))) (csubc g 
-(CHead d (Bind Abst) (lift1 is u)) (CHead d (Bind Abbr) w)) (csubc_abst g d d 
-(scubc_refl g d) (lift1 is u) a1 (sc3_lift1 g c0 (asucc g a1) is d u H1 H5) w 
-H4)))) H4)) H4 (lift1 is u) (sc3_lift1 g c0 (asucc g a1) is d u H1 H5)) 
-(lift1 is (THead (Bind Abst) u t0)) (lift1_bind Abst is u 
-t0)))))))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1: 
-A).(\lambda (_: (arity g c0 u a1)).(\lambda (H1: (sc3 g a1 c0 u)).(\lambda 
-(t0: T).(\lambda (a2: A).(\lambda (_: (arity g c0 t0 (AHead a1 a2))).(\lambda 
-(H3: (sc3 g (AHead a1 a2) c0 t0)).(let H4 \def H3 in (and_ind (arity g c0 t0 
-(AHead a1 a2)) (\forall (d: C).(\forall (w: T).((sc3 g a1 d w) \to (\forall 
-(is: PList).((drop1 is d c0) \to (sc3 g a2 d (THead (Flat Appl) w (lift1 is 
-t0)))))))) (sc3 g a2 c0 (THead (Flat Appl) u t0)) (\lambda (_: (arity g c0 t0 
-(AHead a1 a2))).(\lambda (H6: ((\forall (d: C).(\forall (w: T).((sc3 g a1 d 
-w) \to (\forall (is: PList).((drop1 is d c0) \to (sc3 g a2 d (THead (Flat 
-Appl) w (lift1 is t0)))))))))).(let H_y \def (H6 c0 u H1 PNil) in (H_y 
-(drop1_nil c0))))) H4))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda 
-(a0: A).(\lambda (_: (arity g c0 u (asucc g a0))).(\lambda (H1: (sc3 g (asucc 
-g a0) c0 u)).(\lambda (t0: T).(\lambda (_: (arity g c0 t0 a0)).(\lambda (H3: 
-(sc3 g a0 c0 t0)).(let H_y \def (sc3_cast g a0 TNil) in (H_y c0 u H1 t0 
-H3)))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda (_: 
-(arity g c0 t0 a1)).(\lambda (H1: (sc3 g a1 c0 t0)).(\lambda (a2: A).(\lambda 
-(H2: (leq g a1 a2)).(sc3_repl g a1 c0 t0 H1 a2 H2)))))))) c t a H))))).
-
-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)))).
-
-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_r 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_r t1) 
-(pr1_pr0 t2 t1 H)))).
-
-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_u t2 t1 H x H2) H3)))) 
-H1)))))).
-
-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_r t) (pr1_r t)).
-
-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)))))))).
-
-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))))).
-
-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_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_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_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))))).
-
-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_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)).(let H1 \def (match H0 return (\lambda (c0: 
-C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 
-(CHead c k u2)) \to ((eq T t t1) \to ((eq T t0 t2) \to (pc3 (CHead c k u1) t1 
-t2)))))))) with [(pr2_free c0 t0 t3 H1) \Rightarrow (\lambda (H2: (eq C c0 
-(CHead c k u2))).(\lambda (H3: (eq T t0 t1)).(\lambda (H4: (eq T t3 
-t2)).(eq_ind C (CHead c k u2) (\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2) 
-\to ((pr0 t0 t3) \to (pc3 (CHead c k u1) t1 t2))))) (\lambda (H5: (eq T t0 
-t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to (pc3 
-(CHead c k u1) t1 t2)))) (\lambda (H6: (eq T t3 t2)).(eq_ind T t2 (\lambda 
-(t: T).((pr0 t1 t) \to (pc3 (CHead c k u1) t1 t2))) (\lambda (H7: (pr0 t1 
-t2)).(pc3_pr2_r (CHead c k u1) t1 t2 (pr2_free (CHead c k u1) t1 t2 H7))) t3 
-(sym_eq T t3 t2 H6))) t0 (sym_eq T t0 t1 H5))) c0 (sym_eq C c0 (CHead c k u2) 
-H2) H3 H4 H1)))) | (pr2_delta c0 d u i H1 t0 t3 H2 t H3) \Rightarrow (\lambda 
-(H4: (eq C c0 (CHead c k u2))).(\lambda (H5: (eq T t0 t1)).(\lambda (H6: (eq 
-T t t2)).(eq_ind C (CHead c k u2) (\lambda (c1: C).((eq T t0 t1) \to ((eq T t 
-t2) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t3) \to ((subst0 i 
-u t3 t) \to (pc3 (CHead c k u1) t1 t2))))))) (\lambda (H7: (eq T t0 
-t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i (CHead c k u2) 
-(CHead d (Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u t3 t) \to (pc3 
-(CHead c k u1) t1 t2)))))) (\lambda (H8: (eq T t t2)).(eq_ind T t2 (\lambda 
-(t4: T).((getl i (CHead c k u2) (CHead d (Bind Abbr) u)) \to ((pr0 t1 t3) \to 
-((subst0 i u t3 t4) \to (pc3 (CHead c k u1) t1 t2))))) (\lambda (H9: (getl i 
-(CHead c k u2) (CHead d (Bind Abbr) u))).(\lambda (H10: (pr0 t1 t3)).(\lambda 
-(H11: (subst0 i u t3 t2)).(nat_ind (\lambda (n: nat).((getl n (CHead c k u2) 
-(CHead d (Bind Abbr) u)) \to ((subst0 n u t3 t2) \to (pc3 (CHead c k u1) t1 
-t2)))) (\lambda (H12: (getl O (CHead c k u2) (CHead d (Bind Abbr) 
-u))).(\lambda (H13: (subst0 O u t3 t2)).(K_ind (\lambda (k: K).((clear (CHead 
-c k u2) (CHead d (Bind Abbr) u)) \to (pc3 (CHead c k u1) t1 t2))) (\lambda 
-(b: B).(\lambda (H14: (clear (CHead c (Bind b) u2) (CHead d (Bind Abbr) 
-u))).(let H0 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: 
-C).C) with [(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow c])) (CHead d 
-(Bind Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) 
-u) u2 H14)) in ((let H15 \def (f_equal C B (\lambda (e: C).(match e return 
-(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) 
-\Rightarrow (match k return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | 
-(Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead c (Bind b) u2) 
-(clear_gen_bind b c (CHead d (Bind Abbr) u) u2 H14)) in ((let H16 \def 
-(f_equal C T (\lambda (e: C).(match e return (\lambda (_: C).T) with [(CSort 
-_) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abbr) u) 
-(CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 H14)) in 
-(\lambda (H17: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H19 \def (eq_ind 
-T u (\lambda (t: T).(subst0 O t t3 t2)) H13 u2 H16) in (eq_ind B Abbr 
-(\lambda (b0: B).(pc3 (CHead c (Bind b0) u1) t1 t2)) (ex2_ind T (\lambda (t1: 
-T).(subst0 O u1 t3 t1)) (\lambda (t1: T).(pr0 t2 t1)) (pc3 (CHead c (Bind 
-Abbr) u1) t1 t2) (\lambda (x: T).(\lambda (H: (subst0 O u1 t3 x)).(\lambda 
-(H20: (pr0 t2 x)).(pc3_pr3_t (CHead c (Bind Abbr) u1) t1 x (pr3_pr2 (CHead c 
-(Bind Abbr) u1) t1 x (pr2_delta (CHead c (Bind Abbr) u1) c u1 O (getl_refl 
-Abbr c u1) t1 t3 H10 x H)) t2 (pr3_pr2 (CHead c (Bind Abbr) u1) t2 x 
-(pr2_free (CHead c (Bind Abbr) u1) t2 x H20)))))) (pr0_subst0_fwd u2 t3 t2 O 
-H19 u1 H)) b H17))))) H15)) H0)))) (\lambda (f: F).(\lambda (H14: (clear 
-(CHead c (Flat f) u2) (CHead d (Bind Abbr) u))).(clear_pc3_trans (CHead d 
-(Bind Abbr) u) t1 t2 (pc3_pr2_r (CHead d (Bind Abbr) u) t1 t2 (pr2_delta 
-(CHead d (Bind Abbr) u) d u O (getl_refl Abbr d u) t1 t3 H10 t2 H13)) (CHead 
-c (Flat f) u1) (clear_flat c (CHead d (Bind Abbr) u) (clear_gen_flat f c 
-(CHead d (Bind Abbr) u) u2 H14) f u1)))) k (getl_gen_O (CHead c k u2) (CHead 
-d (Bind Abbr) u) H12)))) (\lambda (i0: nat).(\lambda (IHi: (((getl i0 (CHead 
-c k u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t3 t2) \to (pc3 (CHead c k 
-u1) t1 t2))))).(\lambda (H12: (getl (S i0) (CHead c k u2) (CHead d (Bind 
-Abbr) u))).(\lambda (H13: (subst0 (S i0) u t3 t2)).(K_ind (\lambda (k: 
-K).((((getl i0 (CHead c k u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t3 
-t2) \to (pc3 (CHead c k u1) t1 t2)))) \to ((getl (r k i0) c (CHead d (Bind 
-Abbr) u)) \to (pc3 (CHead c k u1) t1 t2)))) (\lambda (b: B).(\lambda (_: 
-(((getl i0 (CHead c (Bind b) u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u 
-t3 t2) \to (pc3 (CHead c (Bind b) u1) t1 t2))))).(\lambda (H0: (getl (r (Bind 
-b) i0) c (CHead d (Bind Abbr) u))).(pc3_pr2_r (CHead c (Bind b) u1) t1 t2 
-(pr2_delta (CHead c (Bind b) u1) d u (S i0) (getl_head (Bind b) i0 c (CHead d 
-(Bind Abbr) u) H0 u1) t1 t3 H10 t2 H13))))) (\lambda (f: F).(\lambda (_: 
-(((getl i0 (CHead c (Flat f) u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u 
-t3 t2) \to (pc3 (CHead c (Flat f) u1) t1 t2))))).(\lambda (H0: (getl (r (Flat 
-f) i0) c (CHead d (Bind Abbr) u))).(pc3_pr2_r (CHead c (Flat f) u1) t1 t2 
-(pr2_cflat c t1 t2 (pr2_delta c d u (r (Flat f) i0) H0 t1 t3 H10 t2 H13) f 
-u1))))) k IHi (getl_gen_S k c (CHead d (Bind Abbr) u) u2 i0 H12)))))) i H9 
-H11)))) t (sym_eq T t t2 H8))) t0 (sym_eq T t0 t1 H7))) c0 (sym_eq C c0 
-(CHead c k u2) H4) H5 H6 H1 H2 H3))))]) in (H1 (refl_equal C (CHead c k u2)) 
-(refl_equal T t1) (refl_equal T t2)))))))))).
-
-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)).(let H0 \def (match H return (\lambda (c0: C).(\lambda (t: T).(\lambda 
-(t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 c) \to ((eq T t u2) \to ((eq T 
-t0 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)))))))))))) with [(pr2_free c0 t1 t2 
-H0) \Rightarrow (\lambda (H1: (eq C c0 c)).(\lambda (H2: (eq T t1 
-u2)).(\lambda (H3: (eq T t2 u1)).(eq_ind C c (\lambda (_: C).((eq T t1 u2) 
-\to ((eq T t2 u1) \to ((pr0 t1 t2) \to (\forall (t3: T).(\forall (t4: 
-T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pc3 (CHead c k u1) t3 
-t4))))))))) (\lambda (H4: (eq T t1 u2)).(eq_ind T u2 (\lambda (t: T).((eq T 
-t2 u1) \to ((pr0 t t2) \to (\forall (t3: T).(\forall (t4: T).(\forall (k: 
-K).((pr2 (CHead c k u2) t3 t4) \to (pc3 (CHead c k u1) t3 t4)))))))) (\lambda 
-(H5: (eq T t2 u1)).(eq_ind T u1 (\lambda (t: T).((pr0 u2 t) \to (\forall (t3: 
-T).(\forall (t4: T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pc3 
-(CHead c k u1) t3 t4))))))) (\lambda (H6: (pr0 u2 u1)).(\lambda (t0: 
-T).(\lambda (t3: T).(\lambda (k: K).(\lambda (H: (pr2 (CHead c k u2) t0 
-t3)).(pc3_pr0_pr2_t u1 u2 H6 c t0 t3 k H)))))) t2 (sym_eq T t2 u1 H5))) t1 
-(sym_eq T t1 u2 H4))) c0 (sym_eq C c0 c H1) H2 H3 H0)))) | (pr2_delta c0 d u 
-i H0 t1 t2 H1 t H2) \Rightarrow (\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq 
-T t1 u2)).(\lambda (H5: (eq T t u1)).(eq_ind C c (\lambda (c1: C).((eq T t1 
-u2) \to ((eq T t u1) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t1 
-t2) \to ((subst0 i u t2 t) \to (\forall (t3: T).(\forall (t4: T).(\forall (k: 
-K).((pr2 (CHead c k u2) t3 t4) \to (pc3 (CHead c k u1) t3 t4))))))))))) 
-(\lambda (H6: (eq T t1 u2)).(eq_ind T u2 (\lambda (t0: T).((eq T t u1) \to 
-((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t2) \to ((subst0 i u t2 t) 
-\to (\forall (t3: T).(\forall (t4: T).(\forall (k: K).((pr2 (CHead c k u2) t3 
-t4) \to (pc3 (CHead c k u1) t3 t4)))))))))) (\lambda (H7: (eq T t 
-u1)).(eq_ind T u1 (\lambda (t0: T).((getl i c (CHead d (Bind Abbr) u)) \to 
-((pr0 u2 t2) \to ((subst0 i u t2 t0) \to (\forall (t3: T).(\forall (t4: 
-T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pc3 (CHead c k u1) t3 
-t4))))))))) (\lambda (H8: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H9: 
-(pr0 u2 t2)).(\lambda (H10: (subst0 i u t2 u1)).(\lambda (t0: T).(\lambda 
-(t3: T).(\lambda (k: K).(\lambda (H: (pr2 (CHead c k u2) t0 t3)).(let H11 
-\def (match H return (\lambda (c0: C).(\lambda (t: T).(\lambda (t1: 
-T).(\lambda (_: (pr2 c0 t t1)).((eq C c0 (CHead c k u2)) \to ((eq T t t0) \to 
-((eq T t1 t3) \to (pc3 (CHead c k u1) t0 t3)))))))) with [(pr2_free c0 t1 t4 
-H3) \Rightarrow (\lambda (H4: (eq C c0 (CHead c k u2))).(\lambda (H5: (eq T 
-t1 t0)).(\lambda (H6: (eq T t4 t3)).(eq_ind C (CHead c k u2) (\lambda (_: 
-C).((eq T t1 t0) \to ((eq T t4 t3) \to ((pr0 t1 t4) \to (pc3 (CHead c k u1) 
-t0 t3))))) (\lambda (H7: (eq T t1 t0)).(eq_ind T t0 (\lambda (t: T).((eq T t4 
-t3) \to ((pr0 t t4) \to (pc3 (CHead c k u1) t0 t3)))) (\lambda (H8: (eq T t4 
-t3)).(eq_ind T t3 (\lambda (t: T).((pr0 t0 t) \to (pc3 (CHead c k u1) t0 
-t3))) (\lambda (H9: (pr0 t0 t3)).(pc3_pr2_r (CHead c k u1) t0 t3 (pr2_free 
-(CHead c k u1) t0 t3 H9))) t4 (sym_eq T t4 t3 H8))) t1 (sym_eq T t1 t0 H7))) 
-c0 (sym_eq C c0 (CHead c k u2) H4) H5 H6 H3)))) | (pr2_delta c0 d0 u0 i0 H3 
-t1 t4 H4 t H5) \Rightarrow (\lambda (H6: (eq C c0 (CHead c k u2))).(\lambda 
-(H7: (eq T t1 t0)).(\lambda (H11: (eq T t t3)).(eq_ind C (CHead c k u2) 
-(\lambda (c1: C).((eq T t1 t0) \to ((eq T t t3) \to ((getl i0 c1 (CHead d0 
-(Bind Abbr) u0)) \to ((pr0 t1 t4) \to ((subst0 i0 u0 t4 t) \to (pc3 (CHead c 
-k u1) t0 t3))))))) (\lambda (H12: (eq T t1 t0)).(eq_ind T t0 (\lambda (t2: 
-T).((eq T t t3) \to ((getl i0 (CHead c k u2) (CHead d0 (Bind Abbr) u0)) \to 
-((pr0 t2 t4) \to ((subst0 i0 u0 t4 t) \to (pc3 (CHead c k u1) t0 t3)))))) 
-(\lambda (H13: (eq T t t3)).(eq_ind T t3 (\lambda (t2: T).((getl i0 (CHead c 
-k u2) (CHead d0 (Bind Abbr) u0)) \to ((pr0 t0 t4) \to ((subst0 i0 u0 t4 t2) 
-\to (pc3 (CHead c k u1) t0 t3))))) (\lambda (H14: (getl i0 (CHead c k u2) 
-(CHead d0 (Bind Abbr) u0))).(\lambda (H15: (pr0 t0 t4)).(\lambda (H16: 
-(subst0 i0 u0 t4 t3)).(nat_ind (\lambda (n: nat).((getl n (CHead c k u2) 
-(CHead d0 (Bind Abbr) u0)) \to ((subst0 n u0 t4 t3) \to (pc3 (CHead c k u1) 
-t0 t3)))) (\lambda (H17: (getl O (CHead c k u2) (CHead d0 (Bind Abbr) 
-u0))).(\lambda (H18: (subst0 O u0 t4 t3)).((match k return (\lambda (k: 
-K).((clear (CHead c k u2) (CHead d0 (Bind Abbr) u0)) \to (pc3 (CHead c k u1) 
-t0 t3))) with [(Bind b) \Rightarrow (\lambda (H19: (clear (CHead c (Bind b) 
-u2) (CHead d0 (Bind Abbr) u0))).(let H \def (f_equal C C (\lambda (e: 
-C).(match e return (\lambda (_: C).C) with [(CSort _) \Rightarrow d0 | (CHead 
-c _ _) \Rightarrow c])) (CHead d0 (Bind Abbr) u0) (CHead c (Bind b) u2) 
-(clear_gen_bind b c (CHead d0 (Bind Abbr) u0) u2 H19)) in ((let H0 \def 
-(f_equal C B (\lambda (e: C).(match e return (\lambda (_: C).B) with [(CSort 
-_) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k return (\lambda (_: 
-K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d0 
-(Bind Abbr) u0) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d0 (Bind 
-Abbr) u0) u2 H19)) in ((let H1 \def (f_equal C T (\lambda (e: C).(match e 
-return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) 
-\Rightarrow t])) (CHead d0 (Bind Abbr) u0) (CHead c (Bind b) u2) 
-(clear_gen_bind b c (CHead d0 (Bind Abbr) u0) u2 H19)) in (\lambda (H20: (eq 
-B Abbr b)).(\lambda (_: (eq C d0 c)).(let H22 \def (eq_ind T u0 (\lambda (t: 
-T).(subst0 O t t4 t3)) H18 u2 H1) in (eq_ind B Abbr (\lambda (b0: B).(pc3 
-(CHead c (Bind b0) u1) t0 t3)) (ex2_ind T (\lambda (t0: T).(subst0 O t2 t4 
-t0)) (\lambda (t0: T).(pr0 t3 t0)) (pc3 (CHead c (Bind Abbr) u1) t0 t3) 
-(\lambda (x: T).(\lambda (H2: (subst0 O t2 t4 x)).(\lambda (H9: (pr0 t3 
-x)).(ex2_ind T (\lambda (t0: T).(subst0 O u1 t4 t0)) (\lambda (t0: T).(subst0 
-(S (plus i O)) u x t0)) (pc3 (CHead c (Bind Abbr) u1) t0 t3) (\lambda (x0: 
-T).(\lambda (H10: (subst0 O u1 t4 x0)).(\lambda (H23: (subst0 (S (plus i O)) 
-u x x0)).(let H24 \def (f_equal nat nat S (plus i O) i (sym_eq nat i (plus i 
-O) (plus_n_O i))) in (let H25 \def (eq_ind nat (S (plus i O)) (\lambda (n: 
-nat).(subst0 n u x x0)) H23 (S i) H24) in (pc3_pr2_u (CHead c (Bind Abbr) u1) 
-x0 t0 (pr2_delta (CHead c (Bind Abbr) u1) c u1 O (getl_refl Abbr c u1) t0 t4 
-H15 x0 H10) t3 (pc3_pr2_x (CHead c (Bind Abbr) u1) x0 t3 (pr2_delta (CHead c 
-(Bind Abbr) u1) d u (S i) (getl_head (Bind Abbr) i c (CHead d (Bind Abbr) u) 
-H8 u1) t3 x H9 x0 H25)))))))) (subst0_subst0_back t4 x t2 O H2 u1 u i 
-H10))))) (pr0_subst0_fwd u2 t4 t3 O H22 t2 H9)) b H20))))) H0)) H))) | (Flat 
-f) \Rightarrow (\lambda (H8: (clear (CHead c (Flat f) u2) (CHead d0 (Bind 
-Abbr) u0))).(clear_pc3_trans (CHead d0 (Bind Abbr) u0) t0 t3 (pc3_pr2_r 
-(CHead d0 (Bind Abbr) u0) t0 t3 (pr2_delta (CHead d0 (Bind Abbr) u0) d0 u0 O 
-(getl_refl Abbr d0 u0) t0 t4 H15 t3 H18)) (CHead c (Flat f) u1) (clear_flat c 
-(CHead d0 (Bind Abbr) u0) (clear_gen_flat f c (CHead d0 (Bind Abbr) u0) u2 
-H8) f u1)))]) (getl_gen_O (CHead c k u2) (CHead d0 (Bind Abbr) u0) H17)))) 
-(\lambda (i1: nat).(\lambda (_: (((getl i1 (CHead c k u2) (CHead d0 (Bind 
-Abbr) u0)) \to ((subst0 i1 u0 t4 t3) \to (pc3 (CHead c k u1) t0 
-t3))))).(\lambda (H8: (getl (S i1) (CHead c k u2) (CHead d0 (Bind Abbr) 
-u0))).(\lambda (H9: (subst0 (S i1) u0 t4 t3)).(K_ind (\lambda (k: K).((getl 
-(r k i1) c (CHead d0 (Bind Abbr) u0)) \to (pc3 (CHead c k u1) t0 t3))) 
-(\lambda (b: B).(\lambda (H: (getl (r (Bind b) i1) c (CHead d0 (Bind Abbr) 
-u0))).(pc3_pr2_r (CHead c (Bind b) u1) t0 t3 (pr2_delta (CHead c (Bind b) u1) 
-d0 u0 (S i1) (getl_head (Bind b) i1 c (CHead d0 (Bind Abbr) u0) H u1) t0 t4 
-H15 t3 H9)))) (\lambda (f: F).(\lambda (H: (getl (r (Flat f) i1) c (CHead d0 
-(Bind Abbr) u0))).(pc3_pr2_r (CHead c (Flat f) u1) t0 t3 (pr2_cflat c t0 t3 
-(pr2_delta c d0 u0 (r (Flat f) i1) H t0 t4 H15 t3 H9) f u1)))) k (getl_gen_S 
-k c (CHead d0 (Bind Abbr) u0) u2 i1 H8)))))) i0 H14 H16)))) t (sym_eq T t t3 
-H13))) t1 (sym_eq T t1 t0 H12))) c0 (sym_eq C c0 (CHead c k u2) H6) H7 H11 H3 
-H4 H5))))]) in (H11 (refl_equal C (CHead c k u2)) (refl_equal T t0) 
-(refl_equal T t3)))))))))) t (sym_eq T t u1 H7))) t1 (sym_eq T t1 u2 H6))) c0 
-(sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C c) (refl_equal T 
-u2) (refl_equal T u1)))))).
-
-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 (t1: T).(\forall (t4: T).(\forall (k: K).((pc3 
-(CHead c k t2) t1 t4) \to (pc3 (CHead c k t3) t1 t4))))))).(\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_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 (t0: T).(\lambda (t3: T).(\lambda (H1: (pr2 (CHead c1 k u) t3 
-t0)).(\lambda (t4: T).(\lambda (_: (pr3 (CHead c1 k u) t0 t4)).(\lambda (H3: 
-(pc3 (CHead c2 k u) t0 t4)).(pc3_t t0 (CHead c2 k u) t3 (let H4 \def (match 
-H1 return (\lambda (c: C).(\lambda (t: T).(\lambda (t1: T).(\lambda (_: (pr2 
-c t t1)).((eq C c (CHead c1 k u)) \to ((eq T t t3) \to ((eq T t1 t0) \to (pc3 
-(CHead c2 k u) t3 t0)))))))) with [(pr2_free c t1 t2 H2) \Rightarrow (\lambda 
-(H3: (eq C c (CHead c1 k u))).(\lambda (H4: (eq T t1 t3)).(\lambda (H5: (eq T 
-t2 t0)).(eq_ind C (CHead c1 k u) (\lambda (_: C).((eq T t1 t3) \to ((eq T t2 
-t0) \to ((pr0 t1 t2) \to (pc3 (CHead c2 k u) t3 t0))))) (\lambda (H6: (eq T 
-t1 t3)).(eq_ind T t3 (\lambda (t: T).((eq T t2 t0) \to ((pr0 t t2) \to (pc3 
-(CHead c2 k u) t3 t0)))) (\lambda (H7: (eq T t2 t0)).(eq_ind T t0 (\lambda 
-(t: T).((pr0 t3 t) \to (pc3 (CHead c2 k u) t3 t0))) (\lambda (H8: (pr0 t3 
-t0)).(pc3_pr2_r (CHead c2 k u) t3 t0 (pr2_free (CHead c2 k u) t3 t0 H8))) t2 
-(sym_eq T t2 t0 H7))) t1 (sym_eq T t1 t3 H6))) c (sym_eq C c (CHead c1 k u) 
-H3) H4 H5 H2)))) | (pr2_delta c d u0 i H2 t1 t2 H3 t H4) \Rightarrow (\lambda 
-(H5: (eq C c (CHead c1 k u))).(\lambda (H6: (eq T t1 t3)).(\lambda (H7: (eq T 
-t t0)).(eq_ind C (CHead c1 k u) (\lambda (c0: C).((eq T t1 t3) \to ((eq T t 
-t0) \to ((getl i c0 (CHead d (Bind Abbr) u0)) \to ((pr0 t1 t2) \to ((subst0 i 
-u0 t2 t) \to (pc3 (CHead c2 k u) t3 t0))))))) (\lambda (H8: (eq T t1 
-t3)).(eq_ind T t3 (\lambda (t4: T).((eq T t t0) \to ((getl i (CHead c1 k u) 
-(CHead d (Bind Abbr) u0)) \to ((pr0 t4 t2) \to ((subst0 i u0 t2 t) \to (pc3 
-(CHead c2 k u) t3 t0)))))) (\lambda (H9: (eq T t t0)).(eq_ind T t0 (\lambda 
-(t4: T).((getl i (CHead c1 k u) (CHead d (Bind Abbr) u0)) \to ((pr0 t3 t2) 
-\to ((subst0 i u0 t2 t4) \to (pc3 (CHead c2 k u) t3 t0))))) (\lambda (H10: 
-(getl i (CHead c1 k u) (CHead d (Bind Abbr) u0))).(\lambda (H11: (pr0 t3 
-t2)).(\lambda (H12: (subst0 i u0 t2 t0)).(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) t3 t0) (\lambda (x0: C).(\lambda (x1: 
-T).(\lambda (H0: (getl i (CHead c2 k u) (CHead x0 (Bind Abbr) x1))).(\lambda 
-(_: (wcpr0 d x0)).(\lambda (H14: (pr0 u0 x1)).(ex2_ind T (\lambda (t0: 
-T).(subst0 i x1 t2 t0)) (\lambda (t3: T).(pr0 t0 t3)) (pc3 (CHead c2 k u) t3 
-t0) (\lambda (x: T).(\lambda (H15: (subst0 i x1 t2 x)).(\lambda (H16: (pr0 t0 
-x)).(pc3_pr2_u (CHead c2 k u) x t3 (pr2_delta (CHead c2 k u) x0 x1 i H0 t3 t2 
-H11 x H15) t0 (pc3_pr2_x (CHead c2 k u) x t0 (pr2_free (CHead c2 k u) t0 x 
-H16)))))) (pr0_subst0_fwd u0 t2 t0 i H12 x1 H14))))))) (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) H10))))) t (sym_eq T t t0 H9))) t1 (sym_eq T t1 t3 H8))) c (sym_eq C c 
-(CHead c1 k u) H5) H6 H7 H2 H3 H4))))]) in (H4 (refl_equal C (CHead c1 k u)) 
-(refl_equal T t3) (refl_equal T t0))) t4 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))))))).
-
-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))))).
-
-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 (t3: T).((pc3_left c t4 t3) \to (pc3_left c t0 
-t3))))).(\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_pr2_u c t3 t0 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))))))))).
-
-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 return (\lambda (_: T).nat) with [(TSort n) 
-\Rightarrow n | (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 
-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 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_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 (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 (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 (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) 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 return (\lambda (_: 
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
-(THead k _ _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind 
-b) \Rightarrow (match 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 (b: 
-B).(\forall (u: T).(pr3 (CHead c (Bind b) 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)) 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 (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))))))).(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 (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) 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 (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) 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 return (\lambda (_: 
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
-(THead k _ _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind 
-b) \Rightarrow (match 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 (b: 
-B).(\forall (u: T).(pr3 (CHead c (Bind b) 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 (t: T).(eq T t (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 (t: 
-T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t2 t)))) H10 
-(lift h (S d) x4) H14) in (let H16 \def (eq_ind T x1 (\lambda (t: T).(pr3 c 
-u2 t)) 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_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 (u0: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead 
-e0 (Bind b) u0)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (S i)) 
-u0 u 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 return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead 
-c _ _) \Rightarrow c])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H9) 
-in ((let H13 \def (f_equal C B (\lambda (e0: C).(match e0 return (\lambda (_: 
-C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match 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 return (\lambda (_: C).T) with 
-[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (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 (t: 
-T).(subst0 (minus i0 (S i)) u0 t x3)) H11 u H14) in (let H18 \def (eq_ind_r C 
-x1 (\lambda (c: C).(getl i c0 (CHead c (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 (u0: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 
-(Bind b) u0)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(u: T).(getl i c0 (CHead e2 (Bind b) u)))))) (\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 return 
-(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow 
-c])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H13 \def 
-(f_equal C B (\lambda (e0: C).(match e0 return (\lambda (_: C).B) with 
-[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match 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 return (\lambda (_: C).T) with 
-[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (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 (t: 
-T).(getl i c0 (CHead x2 (Bind x0) t))) H10 u H14) in (let H18 \def (eq_ind_r 
-C x1 (\lambda (c: C).(csubst0 (minus i0 (S i)) u0 c 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 (u0: T).(\lambda (_: T).(eq 
-C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u0))))))) (\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 (u: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u 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 return 
-(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow 
-c])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H14 \def 
-(f_equal C B (\lambda (e0: C).(match e0 return (\lambda (_: C).B) with 
-[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match 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 return (\lambda (_: C).T) with 
-[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (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 (t: 
-T).(subst0 (minus i0 (S i)) u0 t x4)) H11 u H15) in (let H19 \def (eq_ind_r C 
-x1 (\lambda (c: C).(csubst0 (minus i0 (S i)) u0 c 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 (u0: T).(\lambda (_: 
-T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) u0)))))) (\lambda (b: 
-B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl i c0 (CHead e0 
-(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus i0 (S i)) u0 u 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 return (\lambda (_: C).C) with 
-[(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow c])) (CHead d (Bind 
-Abbr) u) (CHead x1 (Bind x0) x2) H10) in ((let H14 \def (f_equal C B (\lambda 
-(e0: C).(match e0 return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr 
-| (CHead _ k _) \Rightarrow (match 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 return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) 
-\Rightarrow t])) (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 (t: T).(subst0 (minus i0 (S i)) u0 t x3)) H12 u H15) 
-in (let H19 \def (eq_ind_r C x1 (\lambda (c: C).(getl i c0 (CHead c (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 (u0: T).(eq C 
-(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u0)))))) (\lambda (b: B).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (u: T).(getl i c0 (CHead e2 (Bind b) u)))))) 
-(\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 return (\lambda (_: C).C) with 
-[(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow c])) (CHead d (Bind 
-Abbr) u) (CHead x1 (Bind x0) x3) H10) in ((let H14 \def (f_equal C B (\lambda 
-(e0: C).(match e0 return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr 
-| (CHead _ k _) \Rightarrow (match 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 return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) 
-\Rightarrow t])) (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 (t: T).(getl i c0 (CHead x2 (Bind x0) t))) H11 u H15) 
-in (let H19 \def (eq_ind_r C x1 (\lambda (c: C).(csubst0 (minus i0 (S i)) u0 
-c 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 (u0: 
-T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u0))))))) 
-(\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 (u: T).(\lambda (w: T).(subst0 (minus i0 (S i)) 
-u0 u 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 return 
-(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow 
-c])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H10) in ((let H15 \def 
-(f_equal C B (\lambda (e0: C).(match e0 return (\lambda (_: C).B) with 
-[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match 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 return (\lambda (_: C).T) with 
-[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (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 (t: 
-T).(subst0 (minus i0 (S i)) u0 t x4)) H12 u H16) in (let H20 \def (eq_ind_r C 
-x1 (\lambda (c: C).(csubst0 (minus i0 (S i)) u0 c 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 (u0: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead 
-e0 (Bind b) u0)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (S i)) 
-u0 u 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 return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead 
-c _ _) \Rightarrow c])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H9) 
-in ((let H13 \def (f_equal C B (\lambda (e0: C).(match e0 return (\lambda (_: 
-C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match 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 return (\lambda (_: C).T) with 
-[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (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 (t: 
-T).(subst0 (minus i0 (S i)) u0 t x3)) H11 u H14) in (let H18 \def (eq_ind_r C 
-x1 (\lambda (c: C).(getl i c0 (CHead c (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 (u0: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 
-(Bind b) u0)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(u: T).(getl i c0 (CHead e2 (Bind b) u)))))) (\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 return 
-(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow 
-c])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H13 \def 
-(f_equal C B (\lambda (e0: C).(match e0 return (\lambda (_: C).B) with 
-[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match 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 return (\lambda (_: C).T) with 
-[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (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 (t: 
-T).(getl i c0 (CHead x2 (Bind x0) t))) H10 u H14) in (let H18 \def (eq_ind_r 
-C x1 (\lambda (c: C).(csubst0 (minus i0 (S i)) u0 c 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 (u0: T).(\lambda (_: T).(eq 
-C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u0))))))) (\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 (u: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u 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 return 
-(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow 
-c])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H14 \def 
-(f_equal C B (\lambda (e0: C).(match e0 return (\lambda (_: C).B) with 
-[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match 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 return (\lambda (_: C).T) with 
-[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (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 (t: 
-T).(subst0 (minus i0 (S i)) u0 t x4)) H11 u H15) in (let H19 \def (eq_ind_r C 
-x1 (\lambda (c: C).(csubst0 (minus i0 (S i)) u0 c 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 (u0: 
-T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) u0)))))) 
-(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl i c0 
-(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u 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 return (\lambda (_: C).C) with 
-[(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow c])) (CHead d (Bind 
-Abbr) u) (CHead x1 (Bind x0) x2) H10) in ((let H14 \def (f_equal C B (\lambda 
-(e0: C).(match e0 return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr 
-| (CHead _ k _) \Rightarrow (match 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 return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) 
-\Rightarrow t])) (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 (t: T).(subst0 (minus i0 (S i)) u0 t x3)) H12 u H15) 
-in (let H19 \def (eq_ind_r C x1 (\lambda (c: C).(getl i c0 (CHead c (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 (u0: T).(eq C 
-(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u0)))))) (\lambda (b: B).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (u: T).(getl i c0 (CHead e2 (Bind b) u)))))) 
-(\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 return (\lambda (_: C).C) with 
-[(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow c])) (CHead d (Bind 
-Abbr) u) (CHead x1 (Bind x0) x3) H10) in ((let H14 \def (f_equal C B (\lambda 
-(e0: C).(match e0 return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr 
-| (CHead _ k _) \Rightarrow (match 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 return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) 
-\Rightarrow t])) (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 (t: T).(getl i c0 (CHead x2 (Bind x0) t))) H11 u H15) 
-in (let H19 \def (eq_ind_r C x1 (\lambda (c: C).(csubst0 (minus i0 (S i)) u0 
-c 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 (u0: 
-T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u0))))))) 
-(\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 (u: T).(\lambda (w: T).(subst0 (minus i0 (S i)) 
-u0 u 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 return 
-(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow 
-c])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H10) in ((let H15 \def 
-(f_equal C B (\lambda (e0: C).(match e0 return (\lambda (_: C).B) with 
-[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match 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 return (\lambda (_: C).T) with 
-[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (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 (t: 
-T).(subst0 (minus i0 (S i)) u0 t x4)) H12 u H16) in (let H20 \def (eq_ind_r C 
-x1 (\lambda (c: C).(csubst0 (minus i0 (S i)) u0 c 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 (t2: T).((fsubst0 i u c1 t0 c2 t2) \to (\forall 
-(e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 c2 t2 
-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)))).
-
-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))))))))))))))))).
-
-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 (\forall (t0: T).((ty3 g (CHead c (Bind b) u) t2 t0) \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) 
-t2)))))).
-
-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)) (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 (t: T).((eq T t (TSort n)) \to (pc3 c0 
-(TSort (next g n)) t1))) H4 (TSort n) H7) in (let H9 \def (eq_ind T u 
-(\lambda (t: T).(ty3 g c0 t 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 return (\lambda (_: T).nat) with 
-[(TSort n) \Rightarrow n | (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 
-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 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 (t0: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t2 
-t0)).(\lambda (_: (((eq T t2 (TSort n)) \to (pc3 (CHead c0 (Bind b) u) (TSort 
-(next g n)) t0)))).(\lambda (H7: (eq T (THead (Bind b) u t1) (TSort n))).(let 
-H8 \def (eq_ind T (THead (Bind b) u t1) (\lambda (ee: T).(match ee return 
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H7) in 
-(False_ind (pc3 c0 (TSort (next g n)) (THead (Bind b) u t2)) 
-H8)))))))))))))))) (\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 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 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)) 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)) (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)))))) (\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 (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) 
-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).(ty3 g e u 
-t))))))))).(\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 (t: T).(pc3 c0 (lift (S n) O t) t1)))) (\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) 
-t1)))) (\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: (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 (t: T).((eq T t (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 (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) t1)))) (\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)))))))) H4 (TLRef n) H7) in (let H9 \def 
-(eq_ind T u (\lambda (t: T).(ty3 g c0 t 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 (t: T).(pc3 c0 (lift (S n) O t) t1)))) (\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_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 (u: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u) 
-t1)))) (\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)))))).(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 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 (u: T).(\lambda (_: T).(getl n d (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 d (lift (S n) O u) 
-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).(ty3 g e u 
-t))))))))).(\lambda (H4: (eq T (TLRef n0) (TLRef n))).(let H5 \def (f_equal T 
-nat (\lambda (e: T).(match e return (\lambda (_: T).nat) with [(TSort _) 
-\Rightarrow n0 | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow n0])) 
-(TLRef n0) (TLRef n) H4) in (let H6 \def (eq_ind nat n0 (\lambda (n: 
-nat).(getl n 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 (u: T).(\lambda (_: T).(getl n d 
-(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 d (lift (S n) O u) 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).(ty3 g e u t))))))))).(\lambda (H4: (eq T (TLRef n0) 
-(TLRef n))).(let H5 \def (f_equal T nat (\lambda (e: T).(match e return 
-(\lambda (_: T).nat) with [(TSort _) \Rightarrow n0 | (TLRef n) \Rightarrow n 
-| (THead _ _ _) \Rightarrow n0])) (TLRef n0) (TLRef n) H4) in (let H6 \def 
-(eq_ind nat n0 (\lambda (n: nat).(getl n 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 (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) 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).(ty3 g e u 
-t))))))))).(\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 (t: T).(pc3 (CHead 
-c0 (Bind b) u) (lift (S n) O t) 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 (u: T).(\lambda (t: T).(ty3 g e u t))))) (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 (u: T).(\lambda (t: T).(ty3 g e u t))))))))).(\lambda (t0: 
-T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t2 t0)).(\lambda (_: (((eq T t2 
-(TLRef n)) \to (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: 
-T).(pc3 (CHead c0 (Bind b) u) (lift (S n) O t) t0)))) (\lambda (e: 
-C).(\lambda (u0: T).(\lambda (_: T).(getl n (CHead c0 (Bind b) u) (CHead e 
-(Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e 
-u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 
-(CHead c0 (Bind b) u) (lift (S n) O u0) t0)))) (\lambda (e: C).(\lambda (u0: 
-T).(\lambda (_: T).(getl n (CHead c0 (Bind b) u) (CHead e (Bind Abst) u0))))) 
-(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u 
-t))))))))).(\lambda (H7: (eq T (THead (Bind b) u t1) (TLRef n))).(let H8 \def 
-(eq_ind T (THead (Bind b) u t1) (\lambda (ee: T).(match ee return (\lambda 
-(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
-| (THead _ _ _) \Rightarrow True])) I (TLRef n) H7) in (False_ind (or (ex3_3 
-C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t3: T).(pc3 c0 (lift (S n) O 
-t3) (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 (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) (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 (t3: T).(ty3 g e u0 
-t3)))))) H8)))))))))))))))) (\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 (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 (u0: T).(\lambda 
-(_: T).(pc3 c0 (lift (S n) O u0) u)))) (\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 (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 (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 (u0: T).(\lambda 
-(_: T).(pc3 c0 (lift (S n) O u0) (THead (Bind Abst) 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 (t: T).(ty3 g e u 
-t))))))))).(\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 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 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) 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)))))) 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 (ex4_3 T T T 
-(\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c (THead (Bind b) u t2) 
-x)))) (\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g c u t)))) 
-(\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c (Bind b) u) 
-t1 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t0: T).(ty3 g (CHead c 
-(Bind b) u) t2 t0)))))))))))
-\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)) (ex4_3 T T T (\lambda 
-(t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c (THead (Bind b) u t2) x)))) 
-(\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g c u t)))) (\lambda 
-(t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c (Bind b) u) t1 t2)))) 
-(\lambda (t2: T).(\lambda (_: T).(\lambda (t0: T).(ty3 g (CHead c (Bind b) u) 
-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 (Bind b) u t1)) \to 
-(ex4_3 T T T (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead 
-(Bind b) u t2) t0)))) (\lambda (_: T).(\lambda (t3: T).(\lambda (_: T).(ty3 g 
-c0 u t3)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 
-(Bind b) u) t1 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t4: T).(ty3 
-g (CHead c0 (Bind b) u) t2 t4))))))))) (\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 (ex4_3 T T T (\lambda (t2: T).(\lambda (_: 
-T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2) t)))) (\lambda (_: 
-T).(\lambda (t: T).(\lambda (_: T).(ty3 g c0 u t)))) (\lambda (t2: 
-T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2)))) 
-(\lambda (t2: T).(\lambda (_: T).(\lambda (t0: T).(ty3 g (CHead c0 (Bind b) 
-u) t2 t0)))))))).(\lambda (u0: T).(\lambda (t0: T).(\lambda (H3: (ty3 g c0 u0 
-t0)).(\lambda (H4: (((eq T u0 (THead (Bind b) u t1)) \to (ex4_3 T T T 
-(\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u 
-t2) t0)))) (\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g c0 u t)))) 
-(\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) 
-t1 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t0: T).(ty3 g (CHead c0 
-(Bind b) u) t2 t0)))))))).(\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 (t: T).((eq T 
-t (THead (Bind b) u t1)) \to (ex4_3 T T T (\lambda (t2: T).(\lambda (_: 
-T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2) t0)))) (\lambda (_: 
-T).(\lambda (t0: T).(\lambda (_: T).(ty3 g c0 u t0)))) (\lambda (t2: 
-T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2)))) 
-(\lambda (t2: T).(\lambda (_: T).(\lambda (t1: T).(ty3 g (CHead c0 (Bind b) 
-u) t2 t1))))))) H4 (THead (Bind b) u t1) H7) in (let H9 \def (eq_ind T u0 
-(\lambda (t: T).(ty3 g c0 t t0)) H3 (THead (Bind b) u t1) H7) in (let H10 
-\def (H8 (refl_equal T (THead (Bind b) u t1))) in (ex4_3_ind T T T (\lambda 
-(t3: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t3) t0)))) 
-(\lambda (_: T).(\lambda (t4: T).(\lambda (_: T).(ty3 g c0 u t4)))) (\lambda 
-(t3: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 
-t3)))) (\lambda (t3: T).(\lambda (_: T).(\lambda (t5: T).(ty3 g (CHead c0 
-(Bind b) u) t3 t5)))) (ex4_3 T T T (\lambda (t3: T).(\lambda (_: T).(\lambda 
-(_: T).(pc3 c0 (THead (Bind b) u t3) t2)))) (\lambda (_: T).(\lambda (t4: 
-T).(\lambda (_: T).(ty3 g c0 u t4)))) (\lambda (t3: T).(\lambda (_: 
-T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3)))) (\lambda (t3: 
-T).(\lambda (_: T).(\lambda (t5: T).(ty3 g (CHead c0 (Bind b) u) t3 t5))))) 
-(\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: 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)).(\lambda (H14: (ty3 g (CHead c0 (Bind b) 
-u) x0 x2)).(ex4_3_intro T T T (\lambda (t3: T).(\lambda (_: T).(\lambda (_: 
-T).(pc3 c0 (THead (Bind b) u t3) t2)))) (\lambda (_: T).(\lambda (t4: 
-T).(\lambda (_: T).(ty3 g c0 u t4)))) (\lambda (t3: T).(\lambda (_: 
-T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3)))) (\lambda (t3: 
-T).(\lambda (_: T).(\lambda (t5: T).(ty3 g (CHead c0 (Bind b) u) t3 t5)))) x0 
-x1 x2 (pc3_t t0 c0 (THead (Bind b) u x0) H11 t2 H5) H12 H13 H14)))))))) 
-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 return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
-True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I 
-(THead (Bind b) u t1) H1) in (False_ind (ex4_3 T T T (\lambda (t2: 
-T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2) (TSort (next 
-g m)))))) (\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g c0 u t)))) 
-(\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) 
-t1 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t0: T).(ty3 g (CHead c0 
-(Bind b) u) t2 t0))))) 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 (ex4_3 T T T (\lambda (t2: T).(\lambda (_: 
-T).(\lambda (_: T).(pc3 d (THead (Bind b) u t2) t)))) (\lambda (_: 
-T).(\lambda (t: T).(\lambda (_: T).(ty3 g d u t)))) (\lambda (t2: T).(\lambda 
-(_: T).(\lambda (_: T).(ty3 g (CHead d (Bind b) u) t1 t2)))) (\lambda (t2: 
-T).(\lambda (_: T).(\lambda (t0: T).(ty3 g (CHead d (Bind b) u) t2 
-t0)))))))).(\lambda (H4: (eq T (TLRef n) (THead (Bind b) u t1))).(let H5 \def 
-(eq_ind T (TLRef n) (\lambda (ee: T).(match ee return (\lambda (_: T).Prop) 
-with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ 
-_) \Rightarrow False])) I (THead (Bind b) u t1) H4) in (False_ind (ex4_3 T T 
-T (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u 
-t2) (lift (S n) O t))))) (\lambda (_: T).(\lambda (t0: T).(\lambda (_: 
-T).(ty3 g c0 u t0)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g 
-(CHead c0 (Bind b) u) t1 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda 
-(t3: T).(ty3 g (CHead c0 (Bind b) u) t2 t3))))) 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 (ex4_3 T T T (\lambda 
-(t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 d (THead (Bind b) u t2) t)))) 
-(\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g d u t)))) (\lambda 
-(t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead d (Bind b) u) t1 t2)))) 
-(\lambda (t2: T).(\lambda (_: T).(\lambda (t0: T).(ty3 g (CHead d (Bind b) u) 
-t2 t0)))))))).(\lambda (H4: (eq T (TLRef n) (THead (Bind b) u t1))).(let H5 
-\def (eq_ind T (TLRef n) (\lambda (ee: T).(match ee return (\lambda (_: 
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | 
-(THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t1) H4) in (False_ind 
-(ex4_3 T T T (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead 
-(Bind b) u t2) (lift (S n) O u0))))) (\lambda (_: T).(\lambda (t0: 
-T).(\lambda (_: T).(ty3 g c0 u t0)))) (\lambda (t2: T).(\lambda (_: 
-T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2)))) (\lambda (t2: 
-T).(\lambda (_: T).(\lambda (t3: T).(ty3 g (CHead c0 (Bind b) u) t2 t3))))) 
-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 (ex4_3 T 
-T T (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) 
-u t2) t)))) (\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g c0 u t)))) 
-(\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) 
-t1 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t0: T).(ty3 g (CHead c0 
-(Bind b) u) t2 t0)))))))).(\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 (ex4_3 T T T (\lambda (t3: T).(\lambda (_: 
-T).(\lambda (_: T).(pc3 (CHead c0 (Bind b0) u0) (THead (Bind b) u t3) t2)))) 
-(\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b0) 
-u0) u t)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead 
-(CHead c0 (Bind b0) u0) (Bind b) u) t1 t2)))) (\lambda (t2: T).(\lambda (_: 
-T).(\lambda (t0: T).(ty3 g (CHead (CHead c0 (Bind b0) u0) (Bind b) u) t2 
-t0)))))))).(\lambda (t3: T).(\lambda (H5: (ty3 g (CHead c0 (Bind b0) u0) t2 
-t3)).(\lambda (H6: (((eq T t2 (THead (Bind b) u t1)) \to (ex4_3 T T T 
-(\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 (CHead c0 (Bind b0) u0) 
-(THead (Bind b) u t2) t3)))) (\lambda (_: T).(\lambda (t: T).(\lambda (_: 
-T).(ty3 g (CHead c0 (Bind b0) u0) u t)))) (\lambda (t2: T).(\lambda (_: 
-T).(\lambda (_: T).(ty3 g (CHead (CHead c0 (Bind b0) u0) (Bind b) u) t1 
-t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t0: T).(ty3 g (CHead (CHead 
-c0 (Bind b0) u0) (Bind b) u) t2 t0)))))))).(\lambda (H7: (eq T (THead (Bind 
-b0) u0 t0) (THead (Bind b) u t1))).(let H8 \def (f_equal T B (\lambda (e: 
-T).(match e return (\lambda (_: T).B) with [(TSort _) \Rightarrow b0 | (TLRef 
-_) \Rightarrow b0 | (THead k _ _) \Rightarrow (match k return (\lambda (_: 
-K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b0])])) (THead 
-(Bind b0) u0 t0) (THead (Bind b) u t1) H7) in ((let H9 \def (f_equal T T 
-(\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) 
-(THead (Bind b0) u0 t0) (THead (Bind b) u t1) H7) in ((let H10 \def (f_equal 
-T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) 
-(THead (Bind b0) u0 t0) (THead (Bind b) u t1) H7) in (\lambda (H11: (eq T u0 
-u)).(\lambda (H12: (eq B b0 b)).(let H13 \def (eq_ind T t0 (\lambda (t: 
-T).((eq T t (THead (Bind b) u t1)) \to (ex4_3 T T T (\lambda (t3: T).(\lambda 
-(_: T).(\lambda (_: T).(pc3 (CHead c0 (Bind b0) u0) (THead (Bind b) u t3) 
-t2)))) (\lambda (_: T).(\lambda (t0: T).(\lambda (_: T).(ty3 g (CHead c0 
-(Bind b0) u0) u t0)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 
-g (CHead (CHead c0 (Bind b0) u0) (Bind b) u) t1 t2)))) (\lambda (t2: 
-T).(\lambda (_: T).(\lambda (t1: T).(ty3 g (CHead (CHead c0 (Bind b0) u0) 
-(Bind b) u) t2 t1))))))) H4 t1 H10) in (let H14 \def (eq_ind T t0 (\lambda 
-(t: T).(ty3 g (CHead c0 (Bind b0) u0) t t2)) H3 t1 H10) in (let H15 \def 
-(eq_ind B b0 (\lambda (b0: B).((eq T t2 (THead (Bind b) u t1)) \to (ex4_3 T T 
-T (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 (CHead c0 (Bind b0) 
-u0) (THead (Bind b) u t2) t3)))) (\lambda (_: T).(\lambda (t: T).(\lambda (_: 
-T).(ty3 g (CHead c0 (Bind b0) u0) u t)))) (\lambda (t2: T).(\lambda (_: 
-T).(\lambda (_: T).(ty3 g (CHead (CHead c0 (Bind b0) u0) (Bind b) u) t1 
-t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t0: T).(ty3 g (CHead (CHead 
-c0 (Bind b0) u0) (Bind b) u) t2 t0))))))) H6 b H12) in (let H16 \def (eq_ind 
-B b0 (\lambda (b: B).(ty3 g (CHead c0 (Bind b) u0) t2 t3)) H5 b H12) in (let 
-H17 \def (eq_ind B b0 (\lambda (b0: B).((eq T t1 (THead (Bind b) u t1)) \to 
-(ex4_3 T T T (\lambda (t3: T).(\lambda (_: T).(\lambda (_: T).(pc3 (CHead c0 
-(Bind b0) u0) (THead (Bind b) u t3) t2)))) (\lambda (_: T).(\lambda (t: 
-T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b0) u0) u t)))) (\lambda (t2: 
-T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead (CHead c0 (Bind b0) u0) 
-(Bind b) u) t1 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t0: T).(ty3 
-g (CHead (CHead c0 (Bind b0) u0) (Bind b) u) t2 t0))))))) H13 b H12) in (let 
-H18 \def (eq_ind B b0 (\lambda (b: B).(ty3 g (CHead c0 (Bind b) u0) t1 t2)) 
-H14 b H12) in (eq_ind_r B b (\lambda (b1: B).(ex4_3 T T T (\lambda (t4: 
-T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t4) (THead (Bind 
-b1) u0 t2))))) (\lambda (_: T).(\lambda (t5: T).(\lambda (_: T).(ty3 g c0 u 
-t5)))) (\lambda (t4: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 
-(Bind b) u) t1 t4)))) (\lambda (t4: T).(\lambda (_: T).(\lambda (t6: T).(ty3 
-g (CHead c0 (Bind b) u) t4 t6)))))) (let H19 \def (eq_ind T u0 (\lambda (t: 
-T).((eq T t2 (THead (Bind b) u t1)) \to (ex4_3 T T T (\lambda (t2: 
-T).(\lambda (_: T).(\lambda (_: T).(pc3 (CHead c0 (Bind b) t) (THead (Bind b) 
-u t2) t3)))) (\lambda (_: T).(\lambda (t0: T).(\lambda (_: T).(ty3 g (CHead 
-c0 (Bind b) t) u t0)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 
-g (CHead (CHead c0 (Bind b) t) (Bind b) u) t1 t2)))) (\lambda (t2: 
-T).(\lambda (_: T).(\lambda (t1: T).(ty3 g (CHead (CHead c0 (Bind b) t) (Bind 
-b) u) t2 t1))))))) H15 u H11) in (let H20 \def (eq_ind T u0 (\lambda (t: 
-T).(ty3 g (CHead c0 (Bind b) t) t2 t3)) H16 u H11) in (let H21 \def (eq_ind T 
-u0 (\lambda (t: T).((eq T t1 (THead (Bind b) u t1)) \to (ex4_3 T T T (\lambda 
-(t3: T).(\lambda (_: T).(\lambda (_: T).(pc3 (CHead c0 (Bind b) t) (THead 
-(Bind b) u t3) t2)))) (\lambda (_: T).(\lambda (t0: T).(\lambda (_: T).(ty3 g 
-(CHead c0 (Bind b) t) u t0)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_: 
-T).(ty3 g (CHead (CHead c0 (Bind b) t) (Bind b) u) t1 t2)))) (\lambda (t2: 
-T).(\lambda (_: T).(\lambda (t1: T).(ty3 g (CHead (CHead c0 (Bind b) t) (Bind 
-b) u) t2 t1))))))) H17 u H11) in (let H22 \def (eq_ind T u0 (\lambda (t: 
-T).(ty3 g (CHead c0 (Bind b) t) t1 t2)) H18 u H11) in (let H23 \def (eq_ind T 
-u0 (\lambda (t0: T).((eq T t0 (THead (Bind b) u t1)) \to (ex4_3 T T T 
-(\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u 
-t2) t)))) (\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g c0 u t)))) 
-(\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) 
-t1 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t1: T).(ty3 g (CHead c0 
-(Bind b) u) t2 t1))))))) H2 u H11) in (let H24 \def (eq_ind T u0 (\lambda 
-(t0: T).(ty3 g c0 t0 t)) H1 u H11) in (eq_ind_r T u (\lambda (t4: T).(ex4_3 T 
-T T (\lambda (t5: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) 
-u t5) (THead (Bind b) t4 t2))))) (\lambda (_: T).(\lambda (t6: T).(\lambda 
-(_: T).(ty3 g c0 u t6)))) (\lambda (t5: T).(\lambda (_: T).(\lambda (_: 
-T).(ty3 g (CHead c0 (Bind b) u) t1 t5)))) (\lambda (t5: T).(\lambda (_: 
-T).(\lambda (t7: T).(ty3 g (CHead c0 (Bind b) u) t5 t7)))))) (ex4_3_intro T T 
-T (\lambda (t4: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u 
-t4) (THead (Bind b) u t2))))) (\lambda (_: T).(\lambda (t5: T).(\lambda (_: 
-T).(ty3 g c0 u t5)))) (\lambda (t4: T).(\lambda (_: T).(\lambda (_: T).(ty3 g 
-(CHead c0 (Bind b) u) t1 t4)))) (\lambda (t4: T).(\lambda (_: T).(\lambda 
-(t6: T).(ty3 g (CHead c0 (Bind b) u) t4 t6)))) t2 t t3 (pc3_refl c0 (THead 
-(Bind b) u t2)) H24 H22 H20) u0 H11))))))) b0 H12)))))))))) H9)) 
-H8)))))))))))))))) (\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 (ex4_3 
-T T T (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead (Bind 
-b) u t2) u0)))) (\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g c0 u 
-t)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind 
-b) u) t1 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t0: T).(ty3 g 
-(CHead c0 (Bind b) u) t2 t0)))))))).(\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 (ex4_3 T T T (\lambda (t2: T).(\lambda (_: T).(\lambda 
-(_: T).(pc3 c0 (THead (Bind b) u t2) (THead (Bind Abst) u0 t))))) (\lambda 
-(_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g c0 u t)))) (\lambda (t2: 
-T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2)))) 
-(\lambda (t2: T).(\lambda (_: T).(\lambda (t0: T).(ty3 g (CHead c0 (Bind b) 
-u) t2 t0)))))))).(\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 return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | 
-(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return 
-(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow 
-True])])) I (THead (Bind b) u t1) H5) in (False_ind (ex4_3 T T T (\lambda 
-(t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2) (THead 
-(Flat Appl) w (THead (Bind Abst) u0 t)))))) (\lambda (_: T).(\lambda (t0: 
-T).(\lambda (_: T).(ty3 g c0 u t0)))) (\lambda (t2: T).(\lambda (_: 
-T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2)))) (\lambda (t2: 
-T).(\lambda (_: T).(\lambda (t3: T).(ty3 g (CHead c0 (Bind b) u) t2 t3))))) 
-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 
-(ex4_3 T T T (\lambda (t3: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead 
-(Bind b) u t3) t2)))) (\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g 
-c0 u t)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 
-(Bind b) u) t1 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t0: T).(ty3 
-g (CHead c0 (Bind b) u) t2 t0)))))))).(\lambda (t3: T).(\lambda (_: (ty3 g c0 
-t2 t3)).(\lambda (_: (((eq T t2 (THead (Bind b) u t1)) \to (ex4_3 T T T 
-(\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u 
-t2) t3)))) (\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g c0 u t)))) 
-(\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) 
-t1 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t0: T).(ty3 g (CHead c0 
-(Bind b) u) t2 t0)))))))).(\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 return (\lambda (_: T).Prop) with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
-(match k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | 
-(Flat _) \Rightarrow True])])) I (THead (Bind b) u t1) H5) in (False_ind 
-(ex4_3 T T T (\lambda (t4: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead 
-(Bind b) u t4) t2)))) (\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g 
-c0 u t)))) (\lambda (t4: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 
-(Bind b) u) t1 t4)))) (\lambda (t4: T).(\lambda (_: T).(\lambda (t5: T).(ty3 
-g (CHead c0 (Bind b) u) t4 t5))))) 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)) (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)))) (\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 (t: T).(ty3 g c0 v (THead (Bind Abst) u t)))) (\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 (u: T).(\lambda (t: T).(pc3 c0 (THead (Flat 
-Appl) w (THead (Bind Abst) u t)) t1))) (\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: (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 (t: T).((eq T t (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)) t1))) (\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)))))) H4 (THead (Flat Appl) w v) H7) in (let H9 \def (eq_ind 
-T u (\lambda (t: T).(ty3 g c0 t 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 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 (u: T).(\lambda (t0: T).(pc3 d (THead (Flat Appl) w (THead (Bind 
-Abst) u t0)) t))) (\lambda (u: T).(\lambda (t: T).(ty3 g d v (THead (Bind 
-Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 g d w u))))))).(\lambda 
-(H4: (eq T (TLRef n) (THead (Flat Appl) w v))).(let H5 \def (eq_ind T (TLRef 
-n) (\lambda (ee: T).(match ee 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 (u: T).(\lambda (t0: T).(pc3 d (THead (Flat Appl) w (THead (Bind 
-Abst) u t0)) t))) (\lambda (u: T).(\lambda (t: T).(ty3 g d v (THead (Bind 
-Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 g d w u))))))).(\lambda 
-(H4: (eq T (TLRef n) (THead (Flat Appl) w v))).(let H5 \def (eq_ind T (TLRef 
-n) (\lambda (ee: T).(match ee 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 (u: T).(\lambda (t0: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind 
-Abst) u t0)) t))) (\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 
-(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 (t: T).(pc3 (CHead c0 (Bind b) u) (THead (Flat 
-Appl) w (THead (Bind Abst) u0 t)) t2))) (\lambda (u0: T).(\lambda (t: T).(ty3 
-g (CHead c0 (Bind b) u) v (THead (Bind Abst) u0 t)))) (\lambda (u0: 
-T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) w u0))))))).(\lambda (t0: 
-T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t2 t0)).(\lambda (_: (((eq T t2 
-(THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t: T).(pc3 
-(CHead c0 (Bind b) u) (THead (Flat Appl) w (THead (Bind Abst) u0 t)) t0))) 
-(\lambda (u0: T).(\lambda (t: T).(ty3 g (CHead c0 (Bind b) u) v (THead (Bind 
-Abst) u0 t)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) 
-w u0))))))).(\lambda (H7: (eq T (THead (Bind b) u t1) (THead (Flat Appl) w 
-v))).(let H8 \def (eq_ind T (THead (Bind b) u t1) (\lambda (ee: T).(match ee 
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: 
-K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I 
-(THead (Flat Appl) w v) H7) in (False_ind (ex3_2 T T (\lambda (u0: 
-T).(\lambda (t3: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t3)) 
-(THead (Bind b) u t2)))) (\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)))) 
-H8)))))))))))))))) (\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 (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 (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 (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 
-Appl) w0 v0) (THead (Flat Appl) w v))).(let H6 \def (f_equal T T (\lambda (e: 
-T).(match e return (\lambda (_: T).T) with [(TSort _) \Rightarrow w0 | (TLRef 
-_) \Rightarrow w0 | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) w0 v0) 
-(THead (Flat Appl) w v) H5) in ((let H7 \def (f_equal T T (\lambda (e: 
-T).(match e return (\lambda (_: T).T) with [(TSort _) \Rightarrow v0 | (TLRef 
-_) \Rightarrow v0 | (THead _ _ t) \Rightarrow t])) (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 (u: T).(\lambda (t: T).(ty3 
-g c0 v (THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c0 w 
-u)))))) 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 (t: 
-T).((eq T t (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)) u))) (\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)))))) H2 w H8) in (let H12 \def (eq_ind T w0 
-(\lambda (t: T).(ty3 g c0 t 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 return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: 
-K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match 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)) 
-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 (land (pc3 c t2 x) (ty3 g c 
-t1 t2)))))))
-\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)) (land (pc3 c t2 x) 
-(ty3 g c 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 (Flat Cast) 
-t2 t1)) \to (land (pc3 c0 t2 t0) (ty3 g c0 t1 t2)))))) (\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 (land (pc3 c0 t2 t) (ty3 g c0 
-t1 t2))))).(\lambda (u: T).(\lambda (t3: T).(\lambda (H3: (ty3 g c0 u 
-t3)).(\lambda (H4: (((eq T u (THead (Flat Cast) t2 t1)) \to (land (pc3 c0 t2 
-t3) (ty3 g c0 t1 t2))))).(\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 (t: 
-T).((eq T t (THead (Flat Cast) t2 t1)) \to (land (pc3 c0 t2 t3) (ty3 g c0 t1 
-t2)))) H4 (THead (Flat Cast) t2 t1) H7) in (let H9 \def (eq_ind T u (\lambda 
-(t: T).(ty3 g c0 t t3)) H3 (THead (Flat Cast) t2 t1) H7) in (let H10 \def (H8 
-(refl_equal T (THead (Flat Cast) t2 t1))) in (and_ind (pc3 c0 t2 t3) (ty3 g 
-c0 t1 t2) (land (pc3 c0 t2 t0) (ty3 g c0 t1 t2)) (\lambda (H11: (pc3 c0 t2 
-t3)).(\lambda (H12: (ty3 g c0 t1 t2)).(conj (pc3 c0 t2 t0) (ty3 g c0 t1 t2) 
-(pc3_t t3 c0 t2 H11 t0 H5) H12))) 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 return 
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) 
-\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) 
-t2 t1) H1) in (False_ind (land (pc3 c0 t2 (TSort (next g m))) (ty3 g c0 t1 
-t2)) 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 
-(land (pc3 d t2 t) (ty3 g d t1 t2))))).(\lambda (H4: (eq T (TLRef n) (THead 
-(Flat Cast) t2 t1))).(let H5 \def (eq_ind T (TLRef n) (\lambda (ee: T).(match 
-ee return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) t2 
-t1) H4) in (False_ind (land (pc3 c0 t2 (lift (S n) O t)) (ty3 g c0 t1 t2)) 
-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 
-(land (pc3 d t2 t) (ty3 g d t1 t2))))).(\lambda (H4: (eq T (TLRef n) (THead 
-(Flat Cast) t2 t1))).(let H5 \def (eq_ind T (TLRef n) (\lambda (ee: T).(match 
-ee return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) t2 
-t1) H4) in (False_ind (land (pc3 c0 t2 (lift (S n) O u)) (ty3 g c0 t1 t2)) 
-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 (land 
-(pc3 c0 t2 t) (ty3 g c0 t1 t2))))).(\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 (land (pc3 (CHead c0 (Bind b) u) t2 t3) 
-(ty3 g (CHead c0 (Bind b) u) t1 t2))))).(\lambda (t4: T).(\lambda (_: (ty3 g 
-(CHead c0 (Bind b) u) t3 t4)).(\lambda (_: (((eq T t3 (THead (Flat Cast) t2 
-t1)) \to (land (pc3 (CHead c0 (Bind b) u) t2 t4) (ty3 g (CHead c0 (Bind b) u) 
-t1 t2))))).(\lambda (H7: (eq T (THead (Bind b) u t0) (THead (Flat Cast) t2 
-t1))).(let H8 \def (eq_ind T (THead (Bind b) u t0) (\lambda (ee: T).(match ee 
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: 
-K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I 
-(THead (Flat Cast) t2 t1) H7) in (False_ind (land (pc3 c0 t2 (THead (Bind b) 
-u t3)) (ty3 g c0 t1 t2)) H8)))))))))))))))) (\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 (land (pc3 c0 t2 u) (ty3 g c0 t1 t2))))).(\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 (land (pc3 c0 t2 
-(THead (Bind Abst) u t)) (ty3 g c0 t1 t2))))).(\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 return (\lambda (_: T).Prop) with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
-\Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow 
-False | (Flat f) \Rightarrow (match f return (\lambda (_: F).Prop) with [Appl 
-\Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat Cast) t2 t1) 
-H5) in (False_ind (land (pc3 c0 t2 (THead (Flat Appl) w (THead (Bind Abst) u 
-t))) (ty3 g c0 t1 t2)) 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 (land (pc3 c0 t2 t3) (ty3 g c0 t1 
-t2))))).(\lambda (t4: T).(\lambda (H3: (ty3 g c0 t3 t4)).(\lambda (H4: (((eq 
-T t3 (THead (Flat Cast) t2 t1)) \to (land (pc3 c0 t2 t4) (ty3 g c0 t1 
-t2))))).(\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 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 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 (land (pc3 c0 t2 t4) (ty3 g c0 t1 t2)))) 
-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 (land (pc3 c0 t2 t) (ty3 g c0 t1 t2)))) 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).(land (pc3 c0 t2 t) (ty3 g c0 t1 t2))) (let H13 \def 
-(eq_ind T t0 (\lambda (t: T).((eq T t (THead (Flat Cast) t2 t1)) \to (land 
-(pc3 c0 t2 t2) (ty3 g c0 t1 t2)))) H11 t1 H7) in (let H14 \def (eq_ind T t0 
-(\lambda (t: T).(ty3 g c0 t t2)) H12 t1 H7) in (conj (pc3 c0 t2 t2) (ty3 g c0 
-t1 t2) (pc3_refl c0 t2) H14))) t3 H8))))))) H6))))))))))) c y x H0))) H)))))).
-
-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 (c: C).(\forall (d0: nat).(\forall (h: 
-nat).((drop h d0 c d) \to (ty3 g c (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 (n: nat).(drop h n 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_comm 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_comm 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 (c: C).(\forall 
-(d0: nat).(\forall (h: nat).((drop h d0 c d) \to (ty3 g c (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 (n: nat).(drop h n 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_comm 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_comm 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 (t4: T).(\lambda (_: (ty3 g 
-(CHead c (Bind b) u) t3 t4)).(\lambda (H5: ((\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 t3) (lift h d t4)))))))).(\lambda (c0: C).(\lambda (d: 
-nat).(\lambda (h: nat).(\lambda (H6: (drop h d c0 c)).(eq_ind_r T (THead 
-(Bind b) (lift h d u) (lift h (s (Bind b) d) t0)) (\lambda (t5: T).(ty3 g c0 
-t5 (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 (t5: T).(ty3 g c0 (THead (Bind b) 
-(lift h d u) (lift h (s (Bind b) d) t0)) t5)) (ty3_bind g c0 (lift h d u) 
-(lift h d t) (H1 c0 d h H6) 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 H6 b u)) (lift h 
-(S d) t4) (H5 (CHead c0 (Bind b) (lift h d u)) (S d) h (drop_skip_bind h d c0 
-c H6 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 t3))) (ty3_cast g 
-c0 (lift h (s (Flat Cast) d) t0) (lift h d t3) (H1 c0 d h H4) (lift h d t4) 
-(H3 c0 d h H4)) (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 (t0: T).(ty3 g c0 t t0)))).(\lambda (u: T).(\lambda (t3: 
-T).(\lambda (_: (ty3 g c0 u t3)).(\lambda (_: (ex T (\lambda (t: T).(ty3 g c0 
-t3 t)))).(\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 (_: (ex T 
-(\lambda (t: T).(ty3 g (CHead c0 (Bind b) u) t3 t)))).(\lambda (t4: 
-T).(\lambda (H4: (ty3 g (CHead c0 (Bind b) u) t3 t4)).(\lambda (H5: (ex T 
-(\lambda (t: T).(ty3 g (CHead c0 (Bind b) u) t4 t)))).(let H6 \def H5 in 
-(ex_ind T (\lambda (t5: T).(ty3 g (CHead c0 (Bind b) u) t4 t5)) (ex T 
-(\lambda (t5: T).(ty3 g c0 (THead (Bind b) u t3) t5))) (\lambda (x: 
-T).(\lambda (H7: (ty3 g (CHead c0 (Bind b) u) t4 x)).(ex_intro T (\lambda 
-(t5: T).(ty3 g c0 (THead (Bind b) u t3) t5)) (THead (Bind b) u t4) (ty3_bind 
-g c0 u t H0 b t3 t4 H4 x H7)))) H6))))))))))))))) (\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)).(ex4_3_ind T T T (\lambda (t3: T).(\lambda 
-(_: T).(\lambda (_: T).(pc3 c0 (THead (Bind Abst) u t3) x0)))) (\lambda (_: 
-T).(\lambda (t0: T).(\lambda (_: T).(ty3 g c0 u t0)))) (\lambda (t3: 
-T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind Abst) u) t t3)))) 
-(\lambda (t3: T).(\lambda (_: T).(\lambda (t4: T).(ty3 g (CHead c0 (Bind 
-Abst) u) t3 t4)))) (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 
-(x3: 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)).(\lambda 
-(H11: (ty3 g (CHead c0 (Bind Abst) u) x1 x3)).(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 x3 H11)))))))))) (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 (H1: (ex T 
-(\lambda (t: T).(ty3 g c0 t3 t)))).(\lambda (t4: T).(\lambda (_: (ty3 g c0 t3 
-t4)).(\lambda (_: (ex T (\lambda (t: T).(ty3 g c0 t4 t)))).H1)))))))) 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 
-(t2: T).((ty3 g c0 u0 t2) \to (pc3 c0 t0 t2))))).(\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 (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_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 (c: C).(getl n 
-c0 c)) 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 return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead 
-c _ _) \Rightarrow c])) (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 return (\lambda (_: 
-C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (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 (t: T).(getl n c0 (CHead x0 (Bind 
-Abbr) t))) H8 u0 H10) in (let H13 \def (eq_ind_r T x1 (\lambda (t: T).(ty3 g 
-x0 t x2)) H7 u0 H10) in (let H14 \def (eq_ind_r C x0 (\lambda (c: C).(getl n 
-c0 (CHead c (Bind Abbr) u0))) H12 d H11) in (let H15 \def (eq_ind_r C x0 
-(\lambda (c: C).(ty3 g c 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 (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)))))).(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 
-(c: C).(getl n c0 c)) 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 return (\lambda (_: 
-C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match 
-k return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match 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 (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_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 (c: C).(getl n c0 c)) 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 
-return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k 
-_) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind b) 
-\Rightarrow (match 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 (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)))))).(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 
-(c: C).(getl n c0 c)) 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 return (\lambda (_: C).C) with [(CSort _) 
-\Rightarrow d | (CHead c _ _) \Rightarrow c])) (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 return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) 
-\Rightarrow t])) (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 (t: 
-T).(getl n c0 (CHead x0 (Bind Abst) t))) H8 u0 H10) in (let H13 \def 
-(eq_ind_r T x1 (\lambda (t: T).(ty3 g x0 t x2)) H7 u0 H10) in (let H14 \def 
-(eq_ind_r T x1 (\lambda (t: T).(pc3 c0 (lift (S n) O t) t2)) H5 u0 H10) in 
-(let H15 \def (eq_ind_r C x0 (\lambda (c: C).(getl n c0 (CHead c (Bind Abst) 
-u0))) H12 d H11) in (let H16 \def (eq_ind_r C x0 (\lambda (c: C).(ty3 g c 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 (_: (ty3 g (CHead c0 (Bind b) u0) t2 
-t3)).(\lambda (_: ((\forall (t4: T).((ty3 g (CHead c0 (Bind b) u0) t2 t4) \to 
-(pc3 (CHead c0 (Bind b) u0) t3 t4))))).(\lambda (t4: T).(\lambda (H6: (ty3 g 
-c0 (THead (Bind b) u0 t0) t4)).(ex4_3_ind T T T (\lambda (t5: T).(\lambda (_: 
-T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u0 t5) t4)))) (\lambda (_: 
-T).(\lambda (t6: T).(\lambda (_: T).(ty3 g c0 u0 t6)))) (\lambda (t5: 
-T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u0) t0 t5)))) 
-(\lambda (t5: T).(\lambda (_: T).(\lambda (t7: T).(ty3 g (CHead c0 (Bind b) 
-u0) t5 t7)))) (pc3 c0 (THead (Bind b) u0 t2) t4) (\lambda (x0: T).(\lambda 
-(x1: T).(\lambda (x2: T).(\lambda (H7: (pc3 c0 (THead (Bind b) u0 x0) 
-t4)).(\lambda (_: (ty3 g c0 u0 x1)).(\lambda (H9: (ty3 g (CHead c0 (Bind b) 
-u0) t0 x0)).(\lambda (_: (ty3 g (CHead c0 (Bind b) u0) x0 x2)).(pc3_t (THead 
-(Bind b) u0 x0) c0 (THead (Bind b) u0 t2) (pc3_head_2 c0 u0 t2 x0 (Bind b) 
-(H3 x0 H9)) t4 H7)))))))) (ty3_gen_bind g b c0 u0 t0 t4 H6))))))))))))))))) 
-(\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 (_: ((\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)).(and_ind (pc3 c0 t2 t4) (ty3 g c0 t0 t2) (pc3 c0 t2 t4) (\lambda (H5: 
-(pc3 c0 t2 t4)).(\lambda (_: (ty3 g c0 t0 t2)).H5)) (ty3_gen_cast g c0 t0 t2 
-t4 H4)))))))))))) c u t1 H))))).
-
-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 (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 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))).(and_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 (n: 
-nat).(getl n 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 return (\lambda 
-(_: C).C) with [(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow c])) 
-(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 return (\lambda (_: C).T) with [(CSort _) 
-\Rightarrow u | (CHead _ _ t) \Rightarrow t])) (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 (t: T).(getl n c (CHead d (Bind Abbr) t))) 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 (u0: T).(\lambda (_: T).(eq C 
-(CHead d (Bind Abbr) u) (CHead e0 (Bind b) u0)))))) (\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)) u0 u 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 return (\lambda (_: C).C) with 
-[(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow c])) (CHead d (Bind 
-Abbr) u) (CHead x1 (Bind x0) x2) H9) in ((let H13 \def (f_equal C B (\lambda 
-(e0: C).(match e0 return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr 
-| (CHead _ k _) \Rightarrow (match 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 return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) 
-\Rightarrow t])) (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 (t: T).(subst0 (minus i (S n)) u0 t x3)) H11 u H14) 
-in (let H18 \def (eq_ind_r C x1 (\lambda (c: C).(getl n c3 (CHead c (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 (n: nat).(getl n (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 (u0: T).(eq 
-C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u0)))))) (\lambda (b: 
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c3 (CHead e2 
-(Bind b) u)))))) (\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 return (\lambda 
-(_: C).C) with [(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow c])) 
-(CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H13 \def 
-(f_equal C B (\lambda (e0: C).(match e0 return (\lambda (_: C).B) with 
-[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match 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 return (\lambda (_: C).T) with 
-[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (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 (t: 
-T).(getl n c3 (CHead x2 (Bind x0) t))) H10 u H14) in (let H18 \def (eq_ind_r 
-C x1 (\lambda (c: C).(csubst0 (minus i (S n)) u0 c 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 (n: 
-nat).(getl n (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 (u0: T).(\lambda (_: T).(eq C (CHead d (Bind 
-Abbr) u) (CHead e1 (Bind b) u0))))))) (\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 
-(u: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u 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 return (\lambda 
-(_: C).C) with [(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow c])) 
-(CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H14 \def 
-(f_equal C B (\lambda (e0: C).(match e0 return (\lambda (_: C).B) with 
-[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match 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 return (\lambda (_: C).T) with 
-[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (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 (t: 
-T).(subst0 (minus i (S n)) u0 t x4)) H11 u H15) in (let H19 \def (eq_ind_r C 
-x1 (\lambda (c: C).(csubst0 (minus i (S n)) u0 c 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 (n: nat).(getl n 
-(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))).(and_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 (n: nat).(getl n c (CHead e 
-(Bind Abbr) u0))) H6 n H7) in (let H10 \def (eq_ind_r nat i (\lambda (n: 
-nat).(csubst0 n 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 return (\lambda (_: 
-C).C) with [(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow c])) (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 return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | 
-(CHead _ _ t) \Rightarrow t])) (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 (t: T).(getl n c (CHead d (Bind Abbr) t))) H15 u H13) 
-in (let H17 \def (eq_ind_r T u0 (\lambda (t: T).(csubst0 n t 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))).(and_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 (n: 
-nat).(getl n 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 return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | 
-(CHead _ k _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind 
-b) \Rightarrow (match 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 (u0: T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead e0 
-(Bind b) u0)))))) (\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)) u0 u 
-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 return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead 
-c _ _) \Rightarrow c])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x2) H9) 
-in ((let H13 \def (f_equal C B (\lambda (e0: C).(match e0 return (\lambda (_: 
-C).B) with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match 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 return (\lambda (_: C).T) with 
-[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (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 (t: 
-T).(subst0 (minus i (S n)) u0 t x3)) H11 u H14) in (let H18 \def (eq_ind_r C 
-x1 (\lambda (c: C).(getl n c3 (CHead c (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 (n: nat).(getl n 
-(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 (u0: T).(eq C (CHead d (Bind Abst) u) (CHead e1 
-(Bind b) u0)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(u: T).(getl n c3 (CHead e2 (Bind b) u)))))) (\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 return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead 
-c _ _) \Rightarrow c])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H9) 
-in ((let H13 \def (f_equal C B (\lambda (e0: C).(match e0 return (\lambda (_: 
-C).B) with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match 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 return (\lambda (_: C).T) with 
-[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (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 (t: 
-T).(getl n c3 (CHead x2 (Bind x0) t))) H10 u H14) in (let H18 \def (eq_ind_r 
-C x1 (\lambda (c: C).(csubst0 (minus i (S n)) u0 c 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 (n: 
-nat).(getl n (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 (u0: T).(\lambda (_: T).(eq C (CHead d (Bind 
-Abst) u) (CHead e1 (Bind b) u0))))))) (\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 
-(u: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u 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 return (\lambda 
-(_: C).C) with [(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow c])) 
-(CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H9) in ((let H14 \def 
-(f_equal C B (\lambda (e0: C).(match e0 return (\lambda (_: C).B) with 
-[(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match 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 return (\lambda (_: C).T) with 
-[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (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 (t: 
-T).(subst0 (minus i (S n)) u0 t x4)) H11 u H15) in (let H19 \def (eq_ind_r C 
-x1 (\lambda (c: C).(csubst0 (minus i (S n)) u0 c 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 (n: nat).(getl n 
-(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))).(and_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 (n: nat).(getl n c (CHead e (Bind Abbr) u0))) H6 n H7) in (let 
-H10 \def (eq_ind_r nat i (\lambda (n: nat).(csubst0 n 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 return (\lambda (_: C).Prop) with [(CSort _) 
-\Rightarrow False | (CHead _ k _) \Rightarrow (match k return (\lambda (_: 
-K).Prop) with [(Bind b) \Rightarrow (match 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 (_: (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 (t4: T).(\lambda (H4: (ty3 g (CHead c (Bind b) u) t3 
-t4)).(\lambda (_: ((\forall (i: nat).(\forall (u0: T).(\forall (c2: 
-C).(\forall (t2: T).((fsubst0 i u0 (CHead c (Bind b) u) t3 c2 t2) \to 
-(\forall (e: C).((getl i (CHead c (Bind b) u) (CHead e (Bind Abbr) u0)) \to 
-(ty3 g c2 t2 t4)))))))))).(\lambda (i: nat).(\lambda (u0: T).(\lambda (c2: 
-C).(\lambda (t5: T).(\lambda (H6: (fsubst0 i u0 c (THead (Bind b) u t2) c2 
-t5)).(fsubst0_ind i u0 c (THead (Bind b) u t2) (\lambda (c0: C).(\lambda (t6: 
-T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u0)) \to (ty3 g c0 t6 
-(THead (Bind b) u t3)))))) (\lambda (t6: T).(\lambda (H7: (subst0 i u0 (THead 
-(Bind b) u t2) t6)).(\lambda (e: C).(\lambda (H8: (getl i c (CHead e (Bind 
-Abbr) u0))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t6 (THead (Bind b) u2 
-t2))) (\lambda (u2: T).(subst0 i u0 u u2))) (ex2 T (\lambda (t7: T).(eq T t6 
-(THead (Bind b) u t7))) (\lambda (t7: T).(subst0 (s (Bind b) i) u0 t2 t7))) 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t7: T).(eq T t6 (THead (Bind b) u2 
-t7)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: 
-T).(\lambda (t7: T).(subst0 (s (Bind b) i) u0 t2 t7)))) (ty3 g c t6 (THead 
-(Bind b) u t3)) (\lambda (H9: (ex2 T (\lambda (u2: T).(eq T t6 (THead (Bind 
-b) u2 t2))) (\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T (\lambda (u2: 
-T).(eq T t6 (THead (Bind b) u2 t2))) (\lambda (u2: T).(subst0 i u0 u u2)) 
-(ty3 g c t6 (THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H10: (eq T t6 
-(THead (Bind b) x t2))).(\lambda (H11: (subst0 i u0 u x)).(eq_ind_r T (THead 
-(Bind b) x t2) (\lambda (t7: T).(ty3 g c t7 (THead (Bind b) u t3))) (ex_ind T 
-(\lambda (t7: T).(ty3 g (CHead c (Bind b) u) t4 t7)) (ty3 g c (THead (Bind b) 
-x t2) (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (H12: (ty3 g (CHead c 
-(Bind b) u) t4 x0)).(ex_ind T (\lambda (t7: T).(ty3 g (CHead c (Bind b) x) t3 
-t7)) (ty3 g c (THead (Bind b) x t2) (THead (Bind b) u t3)) (\lambda (x1: 
-T).(\lambda (H13: (ty3 g (CHead c (Bind b) x) t3 x1)).(ty3_conv g c (THead 
-(Bind b) u t3) (THead (Bind b) u t4) (ty3_bind g c u t0 H0 b t3 t4 H4 x0 H12) 
-(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 H11) e H8) 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 H11 c)) e (getl_head (Bind b) i c (CHead e (Bind 
-Abbr) u0) H8 u)) x1 H13) (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 H11 t3 (Bind b))) e H8)))) (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 H11 c)) e (getl_head 
-(Bind b) i c (CHead e (Bind Abbr) u0) H8 u)))))) (ty3_correct g (CHead c 
-(Bind b) u) t3 t4 H4)) t6 H10)))) H9)) (\lambda (H9: (ex2 T (\lambda (t2: 
-T).(eq T t6 (THead (Bind b) u t2))) (\lambda (t3: T).(subst0 (s (Bind b) i) 
-u0 t2 t3)))).(ex2_ind T (\lambda (t7: T).(eq T t6 (THead (Bind b) u t7))) 
-(\lambda (t7: T).(subst0 (s (Bind b) i) u0 t2 t7)) (ty3 g c t6 (THead (Bind 
-b) u t3)) (\lambda (x: T).(\lambda (H10: (eq T t6 (THead (Bind b) u 
-x))).(\lambda (H11: (subst0 (s (Bind b) i) u0 t2 x)).(eq_ind_r T (THead (Bind 
-b) u x) (\lambda (t7: T).(ty3 g c t7 (THead (Bind b) u t3))) (ex_ind T 
-(\lambda (t7: T).(ty3 g (CHead c (Bind b) u) t3 t7)) (ty3 g c (THead (Bind b) 
-u x) (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (H12: (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 H11) e (getl_head 
-(Bind b) i c (CHead e (Bind Abbr) u0) H8 u)) x0 H12))) (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 H11) e (getl_head (Bind b) i c (CHead e (Bind Abbr) 
-u0) H8 u)))) t6 H10)))) H9)) (\lambda (H9: (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T t6 (THead (Bind b) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: 
-T).(subst0 (s (Bind b) i) u0 t2 t3))))).(ex3_2_ind T T (\lambda (u2: 
-T).(\lambda (t7: T).(eq T t6 (THead (Bind b) u2 t7)))) (\lambda (u2: 
-T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t7: 
-T).(subst0 (s (Bind b) i) u0 t2 t7))) (ty3 g c t6 (THead (Bind b) u t3)) 
-(\lambda (x0: T).(\lambda (x1: T).(\lambda (H10: (eq T t6 (THead (Bind b) x0 
-x1))).(\lambda (H11: (subst0 i u0 u x0)).(\lambda (H12: (subst0 (s (Bind b) 
-i) u0 t2 x1)).(eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t7: T).(ty3 g c t7 
-(THead (Bind b) u t3))) (ex_ind T (\lambda (t7: T).(ty3 g (CHead c (Bind b) 
-u) t4 t7)) (ty3 g c (THead (Bind b) x0 x1) (THead (Bind b) u t3)) (\lambda 
-(x: T).(\lambda (H13: (ty3 g (CHead c (Bind b) u) t4 x)).(ex_ind T (\lambda 
-(t7: T).(ty3 g (CHead c (Bind b) x0) t3 t7)) (ty3 g c (THead (Bind b) x0 x1) 
-(THead (Bind b) u t3)) (\lambda (x2: T).(\lambda (H14: (ty3 g (CHead c (Bind 
-b) x0) t3 x2)).(ty3_conv g c (THead (Bind b) u t3) (THead (Bind b) u t4) 
-(ty3_bind g c u t0 H0 b t3 t4 H4 x H13) (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 H11) e 
-H8) 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 H12 (CHead c (Bind b) x0) (csubst0_snd_bind b i u0 
-u x0 H11 c)) e (getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H8 u)) x2 
-H14) (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 H11 t3 
-(Bind b))) e H8)))) (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 
-H12 (CHead c (Bind b) x0) (csubst0_snd_bind b i u0 u x0 H11 c)) e (getl_head 
-(Bind b) i c (CHead e (Bind Abbr) u0) H8 u)))))) (ty3_correct g (CHead c 
-(Bind b) u) t3 t4 H4)) t6 H10)))))) H9)) (subst0_gen_head (Bind b) u0 u t2 t6 
-i H7)))))) (\lambda (c3: C).(\lambda (H7: (csubst0 i u0 c c3)).(\lambda (e: 
-C).(\lambda (H8: (getl i c (CHead e (Bind Abbr) u0))).(ex_ind T (\lambda (t6: 
-T).(ty3 g (CHead c3 (Bind b) u) t3 t6)) (ty3 g c3 (THead (Bind b) u t2) 
-(THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H9: (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 H7) e 
-H8) 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 
-H7 u)) e (getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H8 u)) x H9))) 
-(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 H7 u)) e (getl_head (Bind b) i c (CHead e (Bind 
-Abbr) u0) H8 u)))))))) (\lambda (t6: T).(\lambda (H7: (subst0 i u0 (THead 
-(Bind b) u t2) t6)).(\lambda (c3: C).(\lambda (H8: (csubst0 i u0 c 
-c3)).(\lambda (e: C).(\lambda (H9: (getl i c (CHead e (Bind Abbr) 
-u0))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t6 (THead (Bind b) u2 t2))) 
-(\lambda (u2: T).(subst0 i u0 u u2))) (ex2 T (\lambda (t7: T).(eq T t6 (THead 
-(Bind b) u t7))) (\lambda (t7: T).(subst0 (s (Bind b) i) u0 t2 t7))) (ex3_2 T 
-T (\lambda (u2: T).(\lambda (t7: T).(eq T t6 (THead (Bind b) u2 t7)))) 
-(\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: 
-T).(\lambda (t7: T).(subst0 (s (Bind b) i) u0 t2 t7)))) (ty3 g c3 t6 (THead 
-(Bind b) u t3)) (\lambda (H10: (ex2 T (\lambda (u2: T).(eq T t6 (THead (Bind 
-b) u2 t2))) (\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T (\lambda (u2: 
-T).(eq T t6 (THead (Bind b) u2 t2))) (\lambda (u2: T).(subst0 i u0 u u2)) 
-(ty3 g c3 t6 (THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H11: (eq T t6 
-(THead (Bind b) x t2))).(\lambda (H12: (subst0 i u0 u x)).(eq_ind_r T (THead 
-(Bind b) x t2) (\lambda (t7: T).(ty3 g c3 t7 (THead (Bind b) u t3))) (ex_ind 
-T (\lambda (t7: T).(ty3 g (CHead c3 (Bind b) u) t3 t7)) (ty3 g c3 (THead 
-(Bind b) x t2) (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (H13: (ty3 g 
-(CHead c3 (Bind b) u) t3 x0)).(ex_ind T (\lambda (t7: T).(ty3 g (CHead c3 
-(Bind b) u) x0 t7)) (ty3 g c3 (THead (Bind b) x t2) (THead (Bind b) u t3)) 
-(\lambda (x1: T).(\lambda (H14: (ty3 g (CHead c3 (Bind b) u) x0 x1)).(ex_ind 
-T (\lambda (t7: T).(ty3 g (CHead c3 (Bind b) x) t3 t7)) (ty3 g c3 (THead 
-(Bind b) x t2) (THead (Bind b) u t3)) (\lambda (x2: T).(\lambda (H15: (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 H8) 
-e H9) b t3 x0 H13 x1 H14) (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 H12 c3 H8) e H9) 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 H12 c c3 
-H8)) e (getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H9 u)) x2 H15) 
-(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 H12 t3 (Bind b)) c3 
-H8) e H9)))) (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 H12 c c3 H8)) e (getl_head (Bind b) 
-i c (CHead e (Bind Abbr) u0) H9 u)))))) (ty3_correct g (CHead c3 (Bind b) u) 
-t3 x0 H13)))) (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 H8 u)) e (getl_head (Bind b) i c 
-(CHead e (Bind Abbr) u0) H9 u)))) t6 H11)))) H10)) (\lambda (H10: (ex2 T 
-(\lambda (t2: T).(eq T t6 (THead (Bind b) u t2))) (\lambda (t3: T).(subst0 (s 
-(Bind b) i) u0 t2 t3)))).(ex2_ind T (\lambda (t7: T).(eq T t6 (THead (Bind b) 
-u t7))) (\lambda (t7: T).(subst0 (s (Bind b) i) u0 t2 t7)) (ty3 g c3 t6 
-(THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H11: (eq T t6 (THead (Bind 
-b) u x))).(\lambda (H12: (subst0 (s (Bind b) i) u0 t2 x)).(eq_ind_r T (THead 
-(Bind b) u x) (\lambda (t7: T).(ty3 g c3 t7 (THead (Bind b) u t3))) (ex_ind T 
-(\lambda (t7: T).(ty3 g (CHead c3 (Bind b) u) t3 t7)) (ty3 g c3 (THead (Bind 
-b) u x) (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (H13: (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 H8) e H9) 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 H12 (CHead c3 (Bind b) u) (csubst0_fst_bind b 
-i c c3 u0 H8 u)) e (getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H9 u)) x0 
-H13))) (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 H12 (CHead c3 (Bind 
-b) u) (csubst0_fst_bind b i c c3 u0 H8 u)) e (getl_head (Bind b) i c (CHead e 
-(Bind Abbr) u0) H9 u)))) t6 H11)))) H10)) (\lambda (H10: (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t2: T).(eq T t6 (THead (Bind b) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: 
-T).(subst0 (s (Bind b) i) u0 t2 t3))))).(ex3_2_ind T T (\lambda (u2: 
-T).(\lambda (t7: T).(eq T t6 (THead (Bind b) u2 t7)))) (\lambda (u2: 
-T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t7: 
-T).(subst0 (s (Bind b) i) u0 t2 t7))) (ty3 g c3 t6 (THead (Bind b) u t3)) 
-(\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq T t6 (THead (Bind b) x0 
-x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13: (subst0 (s (Bind b) 
-i) u0 t2 x1)).(eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t7: T).(ty3 g c3 
-t7 (THead (Bind b) u t3))) (ex_ind T (\lambda (t7: T).(ty3 g (CHead c3 (Bind 
-b) u) t3 t7)) (ty3 g c3 (THead (Bind b) x0 x1) (THead (Bind b) u t3)) 
-(\lambda (x: T).(\lambda (H14: (ty3 g (CHead c3 (Bind b) u) t3 x)).(ex_ind T 
-(\lambda (t7: T).(ty3 g (CHead c3 (Bind b) u) x t7)) (ty3 g c3 (THead (Bind 
-b) x0 x1) (THead (Bind b) u t3)) (\lambda (x2: T).(\lambda (H15: (ty3 g 
-(CHead c3 (Bind b) u) x x2)).(ex_ind T (\lambda (t7: T).(ty3 g (CHead c3 
-(Bind b) x0) t3 t7)) (ty3 g c3 (THead (Bind b) x0 x1) (THead (Bind b) u t3)) 
-(\lambda (x3: T).(\lambda (H16: (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 H8) e H9) b t3 x H14 x2 H15) 
-(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 H12 c3 H8) e H9) 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 H13 
-(CHead c3 (Bind b) x0) (csubst0_both_bind b i u0 u x0 H12 c c3 H8)) e 
-(getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H9 u)) x3 H16) (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 H12 t3 (Bind b)) c3 H8) e 
-H9)))) (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 H13 (CHead 
-c3 (Bind b) x0) (csubst0_both_bind b i u0 u x0 H12 c c3 H8)) e (getl_head 
-(Bind b) i c (CHead e (Bind Abbr) u0) H9 u)))))) (ty3_correct g (CHead c3 
-(Bind b) u) t3 x H14)))) (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 H8 u)) e (getl_head (Bind 
-b) i c (CHead e (Bind Abbr) u0) H9 u)))) t6 H11)))))) H10)) (subst0_gen_head 
-(Bind b) u0 u t2 t6 i H7)))))))) c2 t5 H6))))))))))))))))))) (\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)).(ex4_3_ind T T T (\lambda (t4: T).(\lambda (_: T).(\lambda (_: T).(pc3 c 
-(THead (Bind Abst) u t4) x0)))) (\lambda (_: T).(\lambda (t5: T).(\lambda (_: 
-T).(ty3 g c u t5)))) (\lambda (t4: T).(\lambda (_: T).(\lambda (_: T).(ty3 g 
-(CHead c (Bind Abst) u) t0 t4)))) (\lambda (t4: T).(\lambda (_: T).(\lambda 
-(t6: T).(ty3 g (CHead c (Bind Abst) u) t4 t6)))) (ty3 g c (THead (Flat Appl) 
-x v) (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x1: 
-T).(\lambda (x2: T).(\lambda (x3: 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)).(\lambda (H14: (ty3 g (CHead c (Bind Abst) u) x1 
-x3)).(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 (x4: T).(\lambda 
-(H15: (ty3 g c u x4)).(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 x4 H15 Abst t0 x1 H13 x3 H14)) 
-(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 
-(fsubst0_snd i u0 c w x H9) e H6)))))))))) (ty3_gen_bind g Abst c u t0 x0 
-H10)))) (ty3_correct g c v (THead (Bind Abst) u t0) H2)) t3 H8)))) H7)) 
-(\lambda (H7: (ex2 T (\lambda (t2: T).(eq T t3 (THead (Flat Appl) w t2))) 
-(\lambda (t2: T).(subst0 (s (Flat Appl) i) u0 v t2)))).(ex2_ind T (\lambda 
-(t4: T).(eq T t3 (THead (Flat Appl) w t4))) (\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 (x: T).(\lambda (H8: (eq T t3 (THead (Flat Appl) w 
-x))).(\lambda (H9: (subst0 (s (Flat Appl) i) u0 v x)).(eq_ind_r T (THead 
-(Flat Appl) w x) (\lambda (t4: T).(ty3 g c t4 (THead (Flat Appl) w (THead 
-(Bind Abst) u t0)))) (ty3_appl g c w u H0 x t0 (H3 (s (Flat Appl) i) u0 c x 
-(fsubst0_snd (s (Flat Appl) i) u0 c v x H9) e H6)) t3 H8)))) H7)) (\lambda 
-(H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t3 (THead (Flat Appl) 
-u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 w u2))) (\lambda (_: 
-T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) u0 v t2))))).(ex3_2_ind T T 
-(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) 
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-(fsubst0_fst i u c t3 c3 H6) e H7) (THead (Flat Cast) x t2) 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 t3 t3 (pc3_refl c t3) i u c3 x (fsubst0_both i u c t3 x H10 c3 
-H6) e H7)) t5 H9)))) H8)) (\lambda (H8: (ex2 T (\lambda (t2: T).(eq T t5 
-(THead (Flat Cast) t3 t2))) (\lambda (t3: T).(subst0 (s (Flat Cast) i) u t2 
-t3)))).(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 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 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 (t2: T).(eq T t5 (THead (Flat Cast) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(subst0 i u t3 u2))) (\lambda (_: T).(\lambda (t3: 
-T).(subst0 (s (Flat Cast) i) u t2 t3))))).(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 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 t3)) (ty3_conv g c3 t3 t0 (H3 i u c3 t3 (fsubst0_fst i u c t3 
-c3 H6) e H7) (THead (Flat Cast) x0 x1) 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 t3 t3 (pc3_refl c t3) i u c3 x0 (fsubst0_both i u c t3 x0 H10 
-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))))))))))).
-
-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 (n: 
-nat).(getl n (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 (n: nat).(csubst1 n 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 (c: 
-C).(getl n a0 c)) H9 (CHead x1 (Bind Abbr) x0) H12) in (let H16 \def (eq_ind 
-nat d0 (\lambda (n: nat).(drop (S O) n 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 (t: T).(subst1 (minus d0 (S 
-n)) u0 u t)) 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 (t: T).(ty3 g x3 t 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 (n: nat).(drop (S 
-O) n a0 a)) H5 n H6) in (let H8 \def (eq_ind_r nat d0 (\lambda (n: 
-nat).(csubst1 n u0 c0 a0)) H4 n H6) in (let H9 \def (eq_ind_r nat d0 (\lambda 
-(n: nat).(getl n 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 (c: C).(getl n c0 c)) 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 return (\lambda (_: C).C) with 
-[(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow c])) (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 return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead 
-_ _ t) \Rightarrow t])) (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 (t: T).(getl 
-n c0 (CHead e (Bind Abbr) t))) H10 u H12) in (let H15 \def (eq_ind_r T u0 
-(\lambda (t: T).(csubst1 n t 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 (c: C).(getl n c0 (CHead c (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_comm 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_comm (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 (n: nat).(getl n 
-(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 (n: nat).(csubst1 n 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 (c: 
-C).(getl n a0 c)) H9 (CHead x1 (Bind Abst) x0) H12) in (let H16 \def (eq_ind 
-nat d0 (\lambda (n: nat).(drop (S O) n 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 (t: T).(subst1 (minus d0 (S 
-n)) u0 u t)) 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 (t: T).(ty3 g x3 t 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 (n: nat).(drop (S 
-O) n a0 a)) H5 n H6) in (let H8 \def (eq_ind_r nat d0 (\lambda (n: 
-nat).(csubst1 n u0 c0 a0)) H4 n H6) in (let H9 \def (eq_ind_r nat d0 (\lambda 
-(n: nat).(getl n 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 (c: C).(getl n c0 c)) 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 return (\lambda 
-(_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow 
-(match k return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match 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_comm 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_comm (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 (t0: T).(\lambda 
-(_: (ty3 g (CHead c0 (Bind b) u) t4 t0)).(\lambda (H5: ((\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 t4 (lift (S O) d y1)))) (\lambda (_: 
-T).(\lambda (y2: T).(subst1 d u0 t0 (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 (H6: (getl d c0 (CHead e (Bind Abbr) 
-u0))).(\lambda (a0: C).(\lambda (H7: (csubst1 d u0 c0 a0)).(\lambda (a: 
-C).(\lambda (H8: (drop (S O) d a0 a)).(let H9 \def (H1 e u0 d H6 a0 H7 a H8) 
-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 (H10: (subst1 d u0 u (lift (S O) d 
-x0))).(\lambda (_: (subst1 d u0 t (lift (S O) d x1))).(\lambda (H12: (ty3 g a 
-x0 x1)).(let H13 \def (H5 e u0 (S d) (getl_head (Bind b) d c0 (CHead e (Bind 
-Abbr) u0) H6 u) (CHead a0 (Bind b) (lift (S O) d x0)) (csubst1_bind b d u0 u 
-(lift (S O) d x0) H10 c0 a0 H7) (CHead a (Bind b) x0) (drop_skip_bind (S O) d 
-a0 a H8 b x0)) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 (S 
-d) u0 t4 (lift (S O) (S d) y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 (S 
-d) u0 t0 (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 (H14: (subst1 (S d) u0 t4 (lift (S O) (S d) 
-x2))).(\lambda (_: (subst1 (S d) u0 t0 (lift (S O) (S d) x3))).(\lambda (H16: 
-(ty3 g (CHead a (Bind b) x0) x2 x3)).(let H17 \def (H3 e u0 (S d) (getl_head 
-(Bind b) d c0 (CHead e (Bind Abbr) u0) H6 u) (CHead a0 (Bind b) (lift (S O) d 
-x0)) (csubst1_bind b d u0 u (lift (S O) d x0) H10 c0 a0 H7) (CHead a (Bind b) 
-x0) (drop_skip_bind (S O) d a0 a H8 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 (x4: T).(\lambda (x5: T).(\lambda (H18: (subst1 (S d) u0 t3 (lift (S 
-O) (S d) x4))).(\lambda (H19: (subst1 (S d) u0 t4 (lift (S O) (S d) 
-x5))).(\lambda (H20: (ty3 g (CHead a (Bind b) x0) x4 x5)).(let H21 \def 
-(eq_ind T x5 (\lambda (t: T).(ty3 g (CHead a (Bind b) x0) x4 t)) H20 x2 
-(subst1_confluence_lift t4 x5 u0 (S d) H19 x2 H14)) in (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 x4) (THead (Bind b) x0 x2) (eq_ind_r T (THead (Bind b) 
-(lift (S O) d x0) (lift (S O) (S d) x4)) (\lambda (t5: T).(subst1 d u0 (THead 
-(Bind b) u t3) t5)) (subst1_head u0 u (lift (S O) d x0) d H10 (Bind b) t3 
-(lift (S O) (S d) x4) H18) (lift (S O) d (THead (Bind b) x0 x4)) (lift_bind b 
-x0 x4 (S O) d)) (eq_ind_r T (THead (Bind b) (lift (S O) d x0) (lift (S O) (S 
-d) x2)) (\lambda (t5: T).(subst1 d u0 (THead (Bind b) u t4) t5)) (subst1_head 
-u0 u (lift (S O) d x0) d H10 (Bind b) t4 (lift (S O) (S d) x2) H14) (lift (S 
-O) d (THead (Bind b) x0 x2)) (lift_bind b x0 x2 (S O) d)) (ty3_bind g a x0 x1 
-H12 b x4 x2 H21 x3 H16)))))))) H17))))))) H13))))))) 
-H9))))))))))))))))))))))) (\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)).(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 (Bind Abst) 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 (H15: (eq T (lift (S O) d x1) (THead (Bind Abst) x4 
-x5))).(\lambda (H16: (subst1 d u0 u x4)).(\lambda (H17: (subst1 (s (Bind 
-Abst) d) u0 t x5)).(let H18 \def (sym_equal T (lift (S O) d x1) (THead (Bind 
-Abst) x4 x5) H15) 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 (H19: (eq T x1 (THead (Bind Abst) x6 x7))).(\lambda (H20: (eq T 
-x4 (lift (S O) d x6))).(\lambda (H21: (eq T x5 (lift (S O) (S d) x7))).(let 
-H22 \def (eq_ind T x5 (\lambda (t0: T).(subst1 (s (Bind Abst) d) u0 t t0)) 
-H17 (lift (S O) (S d) x7) H21) in (let H23 \def (eq_ind T x4 (\lambda (t: 
-T).(subst1 d u0 u t)) H16 (lift (S O) d x6) H20) in (let H24 \def (eq_ind T 
-x1 (\lambda (t: T).(ty3 g a x0 t)) H10 (THead (Bind Abst) x6 x7) H19) in (let 
-H25 \def (eq_ind T x6 (\lambda (t: T).(ty3 g a x0 (THead (Bind Abst) t x7))) 
-H24 x3 (subst1_confluence_lift u x6 u0 d H23 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) H22) (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 H25))))))))))) (lift_gen_bind Abst x4 x5 x1 (S O) d H18)))))))) 
-(subst1_gen_head (Bind Abst) u0 u t (lift (S O) d x1) d H9))))))) 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 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 (_: (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 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 t4 (lift (S O) d y2)))) 
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (THead (Flat Cast) x0 x2) 
-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)) H8 (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 (t: T).(pc3 c0 t t3)) H4 (lift (S O) d x1) H9) in (let 
-H12 \def (eq_ind T t4 (\lambda (t: T).(ty3 g c0 u t)) H2 (lift (S O) d x1) 
-H9) in (let H13 \def (eq_ind T u (\lambda (t: T).(ty3 g c0 t (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 (t: 
-T).(ty3 g c0 t3 t)) H0 (lift (S O) d x3) H16) in (let H19 \def (eq_ind T t3 
-(\lambda (t: T).(ty3 g c0 t (lift (S O) d x3))) H18 (lift (S O) d x2) H15) in 
-(let H20 \def (eq_ind T t3 (\lambda (t: T).(pc3 c0 (lift (S O) d x1) t)) 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 (n: nat).(getl n (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 (n: nat).(drop (S O) n 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 (e: C).(\forall (u: T).(\forall 
-(d0: nat).((getl d0 d (CHead e (Bind Void) u)) \to (\forall (a: C).((drop (S 
-O) d0 d a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t0 (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))))))))))) 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 (t: 
-T).(ty3 g d (lift (S O) (minus d0 (S n)) x0) t)) 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 (t: T).(ty3 g x1 t 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 (n: nat).(drop (S O) n c0 a)) H4 n H5) in (let H7 \def (eq_ind_r nat 
-d0 (\lambda (n: nat).(getl n 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 (c: C).(getl n c0 c)) 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 return (\lambda 
-(_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow 
-(match k return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match 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_comm 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_comm (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 (n: 
-nat).(getl n (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 (n: nat).(drop (S O) n 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 (e: 
-C).(\forall (u: T).(\forall (d0: nat).((getl d0 d (CHead e (Bind Void) u)) 
-\to (\forall (a: C).((drop (S O) d0 d a) \to (ex3_2 T T (\lambda (y1: 
-T).(\lambda (_: T).(eq T t0 (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))))))))))) 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 (t: T).(ty3 g d (lift (S O) (minus d0 (S n)) 
-x0) t)) H12 (lift (S O) (minus d0 (S n)) x3) H15) in (let H18 \def (eq_ind_r 
-T x2 (\lambda (t: T).(ty3 g x1 t 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 (n: nat).(drop (S 
-O) n c0 a)) H4 n H5) in (let H7 \def (eq_ind_r nat d0 (\lambda (n: nat).(getl 
-n 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 (c: C).(getl n c0 c)) 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 return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow 
-False | (CHead _ k _) \Rightarrow (match k return (\lambda (_: K).Prop) with 
-[(Bind b) \Rightarrow (match 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_comm 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_comm (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 (t0: T).(\lambda (H4: 
-(ty3 g (CHead c0 (Bind b) u) t4 t0)).(\lambda (H5: ((\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 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 (u0: T).(\lambda 
-(d: nat).(\lambda (H6: (getl d c0 (CHead e (Bind Void) u0))).(\lambda (a: 
-C).(\lambda (H7: (drop (S O) d c0 a)).(let H8 \def (H1 e u0 d H6 a H7) 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 (H9: (eq T u (lift (S O) d x0))).(\lambda (H10: (eq T t (lift (S 
-O) d x1))).(\lambda (H11: (ty3 g a x0 x1)).(let H12 \def (eq_ind T t (\lambda 
-(t: T).(ty3 g c0 u t)) H0 (lift (S O) d x1) H10) in (let H13 \def (eq_ind T u 
-(\lambda (t: T).(ty3 g c0 t (lift (S O) d x1))) H12 (lift (S O) d x0) H9) in 
-(let H14 \def (eq_ind T u (\lambda (t: T).(\forall (e: C).(\forall (u: 
-T).(\forall (d: nat).((getl d (CHead c0 (Bind b) t) (CHead e (Bind Void) u)) 
-\to (\forall (a: C).((drop (S O) d (CHead c0 (Bind b) t) 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))))))))))) H5 (lift (S O) d x0) H9) in (let H15 \def 
-(eq_ind T u (\lambda (t: T).(ty3 g (CHead c0 (Bind b) t) t4 t0)) H4 (lift (S 
-O) d x0) H9) in (let H16 \def (eq_ind T u (\lambda (t: T).(\forall (e: 
-C).(\forall (u: T).(\forall (d: nat).((getl d (CHead c0 (Bind b) t) (CHead e 
-(Bind Void) u)) \to (\forall (a: C).((drop (S O) d (CHead c0 (Bind b) t) 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))))))))))) H3 (lift (S O) d x0) H9) in 
-(let H17 \def (eq_ind T u (\lambda (t: T).(ty3 g (CHead c0 (Bind b) t) t3 
-t4)) H2 (lift (S O) d x0) H9) in (eq_ind_r T (lift (S O) d x0) (\lambda (t5: 
-T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Bind b) t5 t3) 
-(lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Bind b) 
-t5 t4) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 
-y2))))) (let H18 \def (H16 e u0 (S d) (getl_head (Bind b) d c0 (CHead e (Bind 
-Void) u0) H6 (lift (S O) d x0)) (CHead a (Bind b) x0) (drop_skip_bind (S O) d 
-c0 a H7 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 (H19: (eq T t3 (lift (S O) (S d) x2))).(\lambda (H20: (eq T t4 
-(lift (S O) (S d) x3))).(\lambda (H21: (ty3 g (CHead a (Bind b) x0) x2 
-x3)).(let H22 \def (eq_ind T t4 (\lambda (t: T).(\forall (e: C).(\forall (u: 
-T).(\forall (d0: nat).((getl d0 (CHead c0 (Bind b) (lift (S O) d x0)) (CHead 
-e (Bind Void) u)) \to (\forall (a: C).((drop (S O) d0 (CHead c0 (Bind b) 
-(lift (S O) d x0)) a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t 
-(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))))))))))) H14 
-(lift (S O) (S d) x3) H20) in (eq_ind_r T (lift (S O) (S d) x3) (\lambda (t5: 
-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) t5) (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 (t5: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead 
-(Bind b) (lift (S O) d x0) t5) (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))))) (let H23 
-\def (H22 e u0 (S d) (getl_head (Bind b) d c0 (CHead e (Bind Void) u0) H6 
-(lift (S O) d x0)) (CHead a (Bind b) x0) (drop_skip_bind (S O) d c0 a H7 b 
-x0)) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T (lift (S O) (S 
-d) x3) (lift (S O) (S d) y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t0 
-(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) (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)))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H24: (eq T 
-(lift (S O) (S d) x3) (lift (S O) (S d) x4))).(\lambda (_: (eq T t0 (lift (S 
-O) (S d) x5))).(\lambda (H26: (ty3 g (CHead a (Bind b) x0) x4 x5)).(let H27 
-\def (eq_ind_r T x4 (\lambda (t: T).(ty3 g (CHead a (Bind b) x0) t x5)) H26 
-x3 (lift_inj x3 x4 (S O) (S d) H24)) in (eq_ind T (lift (S O) d (THead (Bind 
-b) x0 x2)) (\lambda (t5: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T 
-t5 (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))))) (eq_ind T (lift (S O) d (THead 
-(Bind b) x0 x3)) (\lambda (t5: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: 
-T).(eq T (lift (S O) d (THead (Bind b) x0 x2)) (lift (S O) d y1)))) (\lambda 
-(_: T).(\lambda (y2: T).(eq T t5 (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 (Bind b) x0 x2)) (lift (S O) d 
-y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d (THead (Bind b) 
-x0 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) (refl_equal T (lift (S O) 
-d (THead (Bind b) x0 x2))) (refl_equal T (lift (S O) d (THead (Bind b) x0 
-x3))) (ty3_bind g a x0 x1 H11 b x2 x3 H21 x5 H27)) (THead (Bind b) (lift (S 
-O) d x0) (lift (S O) (S d) x3)) (lift_bind b x0 x3 (S O) d)) (THead (Bind b) 
-(lift (S O) d x0) (lift (S O) (S d) x2)) (lift_bind b x0 x2 (S O) d)))))))) 
-H23)) t3 H19) t4 H20))))))) H18)) u H9)))))))))))) H8))))))))))))))))))))) 
-(\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 (t: T).(ty3 g a x0 t)) 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 
-(t: 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 w (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))))))))))) 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 (t: T).(ty3 g a x4 t)) 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 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 (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 (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))))))))))) 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 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 
-(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 (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 (lift (S O) d x0) (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 x0) (lift (S O) 
-d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (THead (Flat 
-Cast) x0 x2) x0 (refl_equal T (lift (S O) d (THead (Flat Cast) x0 x2))) 
-(refl_equal T (lift (S O) d x0)) (ty3_cast g a x2 x0 H19 x1 H9)) (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)))))))))) H6)))))))))))))))) c t1 t2 H))))).
-
-inductive csub3 (g:G): C \to (C \to Prop) \def
-| csub3_sort: \forall (n: nat).(csub3 g (CSort n) (CSort n))
-| csub3_head: \forall (c1: C).(\forall (c2: C).((csub3 g c1 c2) \to (\forall 
-(k: K).(\forall (u: T).(csub3 g (CHead c1 k u) (CHead c2 k u))))))
-| csub3_void: \forall (c1: C).(\forall (c2: C).((csub3 g c1 c2) \to (\forall 
-(b: B).((not (eq B b Void)) \to (\forall (u1: T).(\forall (u2: T).(csub3 g 
-(CHead c1 (Bind Void) u1) (CHead c2 (Bind b) u2))))))))
-| csub3_abst: \forall (c1: C).(\forall (c2: C).((csub3 g c1 c2) \to (\forall 
-(u: T).(\forall (t: T).((ty3 g c2 u t) \to (csub3 g (CHead c1 (Bind Abst) t) 
-(CHead c2 (Bind Abbr) u))))))).
-
-theorem csub3_gen_abbr:
- \forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v: T).((csub3 g 
-(CHead e1 (Bind Abbr) v) c2) \to (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 
-(Bind Abbr) v))) (\lambda (e2: C).(csub3 g e1 e2)))))))
-\def
- \lambda (g: G).(\lambda (e1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda 
-(H: (csub3 g (CHead e1 (Bind Abbr) v) c2)).(let H0 \def (match H return 
-(\lambda (c: C).(\lambda (c0: C).(\lambda (_: (csub3 ? c c0)).((eq C c (CHead 
-e1 (Bind Abbr) v)) \to ((eq C c0 c2) \to (ex2 C (\lambda (e2: C).(eq C c2 
-(CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csub3 g e1 e2)))))))) with 
-[(csub3_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) (CHead e1 (Bind 
-Abbr) v))).(\lambda (H1: (eq C (CSort n) c2)).((let H2 \def (eq_ind C (CSort 
-n) (\lambda (e: C).(match e return (\lambda (_: C).Prop) with [(CSort _) 
-\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind Abbr) 
-v) H0) in (False_ind ((eq C (CSort n) c2) \to (ex2 C (\lambda (e2: C).(eq C 
-c2 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csub3 g e1 e2)))) H2)) H1))) 
-| (csub3_head c1 c0 H0 k u) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u) 
-(CHead e1 (Bind Abbr) v))).(\lambda (H2: (eq C (CHead c0 k u) c2)).((let H3 
-\def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: C).T) with 
-[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u) 
-(CHead e1 (Bind Abbr) v) H1) in ((let H4 \def (f_equal C K (\lambda (e: 
-C).(match e return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead 
-_ k _) \Rightarrow k])) (CHead c1 k u) (CHead e1 (Bind Abbr) v) H1) in ((let 
-H5 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: C).C) with 
-[(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u) 
-(CHead e1 (Bind Abbr) v) H1) in (eq_ind C e1 (\lambda (c: C).((eq K k (Bind 
-Abbr)) \to ((eq T u v) \to ((eq C (CHead c0 k u) c2) \to ((csub3 g c c0) \to 
-(ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abbr) v))) (\lambda (e2: 
-C).(csub3 g e1 e2)))))))) (\lambda (H6: (eq K k (Bind Abbr))).(eq_ind K (Bind 
-Abbr) (\lambda (k0: K).((eq T u v) \to ((eq C (CHead c0 k0 u) c2) \to ((csub3 
-g e1 c0) \to (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abbr) v))) 
-(\lambda (e2: C).(csub3 g e1 e2))))))) (\lambda (H7: (eq T u v)).(eq_ind T v 
-(\lambda (t: T).((eq C (CHead c0 (Bind Abbr) t) c2) \to ((csub3 g e1 c0) \to 
-(ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abbr) v))) (\lambda (e2: 
-C).(csub3 g e1 e2)))))) (\lambda (H8: (eq C (CHead c0 (Bind Abbr) v) 
-c2)).(eq_ind C (CHead c0 (Bind Abbr) v) (\lambda (c: C).((csub3 g e1 c0) \to 
-(ex2 C (\lambda (e2: C).(eq C c (CHead e2 (Bind Abbr) v))) (\lambda (e2: 
-C).(csub3 g e1 e2))))) (\lambda (H9: (csub3 g e1 c0)).(let H10 \def (eq_ind_r 
-C c2 (\lambda (c: C).(csub3 g (CHead e1 (Bind Abbr) v) c)) H (CHead c0 (Bind 
-Abbr) v) H8) in (ex_intro2 C (\lambda (e2: C).(eq C (CHead c0 (Bind Abbr) v) 
-(CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csub3 g e1 e2)) c0 (refl_equal C 
-(CHead c0 (Bind Abbr) v)) H9))) c2 H8)) u (sym_eq T u v H7))) k (sym_eq K k 
-(Bind Abbr) H6))) c1 (sym_eq C c1 e1 H5))) H4)) H3)) H2 H0))) | (csub3_void 
-c1 c0 H0 b H1 u1 u2) \Rightarrow (\lambda (H2: (eq C (CHead c1 (Bind Void) 
-u1) (CHead e1 (Bind Abbr) v))).(\lambda (H3: (eq C (CHead c0 (Bind b) u2) 
-c2)).((let H4 \def (eq_ind C (CHead c1 (Bind Void) u1) (\lambda (e: C).(match 
-e return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k 
-_) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind b) 
-\Rightarrow (match b return (\lambda (_: B).Prop) with [Abbr \Rightarrow 
-False | Abst \Rightarrow False | Void \Rightarrow True]) | (Flat _) 
-\Rightarrow False])])) I (CHead e1 (Bind Abbr) v) H2) in (False_ind ((eq C 
-(CHead c0 (Bind b) u2) c2) \to ((csub3 g c1 c0) \to ((not (eq B b Void)) \to 
-(ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abbr) v))) (\lambda (e2: 
-C).(csub3 g e1 e2)))))) H4)) H3 H0 H1))) | (csub3_abst c1 c0 H0 u t H1) 
-\Rightarrow (\lambda (H2: (eq C (CHead c1 (Bind Abst) t) (CHead e1 (Bind 
-Abbr) v))).(\lambda (H3: (eq C (CHead c0 (Bind Abbr) u) c2)).((let H4 \def 
-(eq_ind C (CHead c1 (Bind Abst) t) (\lambda (e: C).(match e return (\lambda 
-(_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow 
-(match k return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b 
-return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow 
-True | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead e1 
-(Bind Abbr) v) H2) in (False_ind ((eq C (CHead c0 (Bind Abbr) u) c2) \to 
-((csub3 g c1 c0) \to ((ty3 g c0 u t) \to (ex2 C (\lambda (e2: C).(eq C c2 
-(CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csub3 g e1 e2)))))) H4)) H3 H0 
-H1)))]) in (H0 (refl_equal C (CHead e1 (Bind Abbr) v)) (refl_equal C 
-c2))))))).
-
-theorem csub3_gen_abst:
- \forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v1: T).((csub3 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).(csub3 g e1 e2))) (ex3_2 C T (\lambda 
-(e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: 
-C).(\lambda (_: T).(csub3 g e1 e2))) (\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: (csub3 g (CHead e1 (Bind Abst) v1) c2)).(let H0 \def (match H return 
-(\lambda (c: C).(\lambda (c0: C).(\lambda (_: (csub3 ? c c0)).((eq C c (CHead 
-e1 (Bind Abst) v1)) \to ((eq C c0 c2) \to (or (ex2 C (\lambda (e2: C).(eq C 
-c2 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T 
-(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) 
-(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda 
-(v2: T).(ty3 g e2 v2 v1)))))))))) with [(csub3_sort n) \Rightarrow (\lambda 
-(H0: (eq C (CSort n) (CHead e1 (Bind Abst) v1))).(\lambda (H1: (eq C (CSort 
-n) c2)).((let H2 \def (eq_ind C (CSort n) (\lambda (e: C).(match e return 
-(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) 
-\Rightarrow False])) I (CHead e1 (Bind Abst) v1) H0) in (False_ind ((eq C 
-(CSort n) c2) \to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst) 
-v1))) (\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda 
-(v2: T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: 
-T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))) 
-H2)) H1))) | (csub3_head c1 c0 H0 k u) \Rightarrow (\lambda (H1: (eq C (CHead 
-c1 k u) (CHead e1 (Bind Abst) v1))).(\lambda (H2: (eq C (CHead c0 k u) 
-c2)).((let H3 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: 
-C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead 
-c1 k u) (CHead e1 (Bind Abst) v1) H1) in ((let H4 \def (f_equal C K (\lambda 
-(e: C).(match e return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | 
-(CHead _ k _) \Rightarrow k])) (CHead c1 k u) (CHead e1 (Bind Abst) v1) H1) 
-in ((let H5 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: 
-C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead 
-c1 k u) (CHead e1 (Bind Abst) v1) H1) in (eq_ind C e1 (\lambda (c: C).((eq K 
-k (Bind Abst)) \to ((eq T u v1) \to ((eq C (CHead c0 k u) c2) \to ((csub3 g c 
-c0) \to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst) v1))) 
-(\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: 
-T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: 
-T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 
-v1)))))))))) (\lambda (H6: (eq K k (Bind Abst))).(eq_ind K (Bind Abst) 
-(\lambda (k0: K).((eq T u v1) \to ((eq C (CHead c0 k0 u) c2) \to ((csub3 g e1 
-c0) \to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst) v1))) 
-(\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: 
-T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: 
-T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 
-v1))))))))) (\lambda (H7: (eq T u v1)).(eq_ind T v1 (\lambda (t: T).((eq C 
-(CHead c0 (Bind Abst) t) c2) \to ((csub3 g e1 c0) \to (or (ex2 C (\lambda 
-(e2: C).(eq C c2 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1 
-e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind 
-Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: 
-C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))) (\lambda (H8: (eq C (CHead c0 
-(Bind Abst) v1) c2)).(eq_ind C (CHead c0 (Bind Abst) v1) (\lambda (c: 
-C).((csub3 g e1 c0) \to (or (ex2 C (\lambda (e2: C).(eq C c (CHead e2 (Bind 
-Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: 
-C).(\lambda (v2: T).(eq C c (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: 
-C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g 
-e2 v2 v1))))))) (\lambda (H9: (csub3 g e1 c0)).(let H10 \def (eq_ind_r C c2 
-(\lambda (c: C).(csub3 g (CHead e1 (Bind Abst) v1) c)) H (CHead c0 (Bind 
-Abst) v1) H8) in (or_introl (ex2 C (\lambda (e2: C).(eq C (CHead c0 (Bind 
-Abst) v1) (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) 
-(ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c0 (Bind Abst) v1) 
-(CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g e1 
-e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))) (ex_intro2 C 
-(\lambda (e2: C).(eq C (CHead c0 (Bind Abst) v1) (CHead e2 (Bind Abst) v1))) 
-(\lambda (e2: C).(csub3 g e1 e2)) c0 (refl_equal C (CHead c0 (Bind Abst) v1)) 
-H9)))) c2 H8)) u (sym_eq T u v1 H7))) k (sym_eq K k (Bind Abst) H6))) c1 
-(sym_eq C c1 e1 H5))) H4)) H3)) H2 H0))) | (csub3_void c1 c0 H0 b H1 u1 u2) 
-\Rightarrow (\lambda (H2: (eq C (CHead c1 (Bind Void) u1) (CHead e1 (Bind 
-Abst) v1))).(\lambda (H3: (eq C (CHead c0 (Bind b) u2) c2)).((let H4 \def 
-(eq_ind C (CHead c1 (Bind Void) u1) (\lambda (e: C).(match e return (\lambda 
-(_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow 
-(match k return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b 
-return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow 
-False | Void \Rightarrow True]) | (Flat _) \Rightarrow False])])) I (CHead e1 
-(Bind Abst) v1) H2) in (False_ind ((eq C (CHead c0 (Bind b) u2) c2) \to 
-((csub3 g c1 c0) \to ((not (eq B b Void)) \to (or (ex2 C (\lambda (e2: C).(eq 
-C c2 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C 
-T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) 
-(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda 
-(v2: T).(ty3 g e2 v2 v1)))))))) H4)) H3 H0 H1))) | (csub3_abst c1 c0 H0 u t 
-H1) \Rightarrow (\lambda (H2: (eq C (CHead c1 (Bind Abst) t) (CHead e1 (Bind 
-Abst) v1))).(\lambda (H3: (eq C (CHead c0 (Bind Abbr) u) c2)).((let H4 \def 
-(f_equal C T (\lambda (e: C).(match e return (\lambda (_: C).T) with [(CSort 
-_) \Rightarrow t | (CHead _ _ t) \Rightarrow t])) (CHead c1 (Bind Abst) t) 
-(CHead e1 (Bind Abst) v1) H2) in ((let H5 \def (f_equal C C (\lambda (e: 
-C).(match e return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead 
-c _ _) \Rightarrow c])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind Abst) v1) 
-H2) in (eq_ind C e1 (\lambda (c: C).((eq T t v1) \to ((eq C (CHead c0 (Bind 
-Abbr) u) c2) \to ((csub3 g c c0) \to ((ty3 g c0 u t) \to (or (ex2 C (\lambda 
-(e2: C).(eq C c2 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1 
-e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind 
-Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: 
-C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))))) (\lambda (H6: (eq T t 
-v1)).(eq_ind T v1 (\lambda (t0: T).((eq C (CHead c0 (Bind Abbr) u) c2) \to 
-((csub3 g e1 c0) \to ((ty3 g c0 u t0) \to (or (ex2 C (\lambda (e2: C).(eq C 
-c2 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T 
-(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) 
-(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda 
-(v2: T).(ty3 g e2 v2 v1))))))))) (\lambda (H7: (eq C (CHead c0 (Bind Abbr) u) 
-c2)).(eq_ind C (CHead c0 (Bind Abbr) u) (\lambda (c: C).((csub3 g e1 c0) \to 
-((ty3 g c0 u v1) \to (or (ex2 C (\lambda (e2: C).(eq C c (CHead e2 (Bind 
-Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: 
-C).(\lambda (v2: T).(eq C c (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: 
-C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g 
-e2 v2 v1)))))))) (\lambda (H8: (csub3 g e1 c0)).(\lambda (H9: (ty3 g c0 u 
-v1)).(let H10 \def (eq_ind_r C c2 (\lambda (c: C).(csub3 g (CHead e1 (Bind 
-Abst) v1) c)) H (CHead c0 (Bind Abbr) u) H7) in (or_intror (ex2 C (\lambda 
-(e2: C).(eq C (CHead c0 (Bind Abbr) u) (CHead e2 (Bind Abst) v1))) (\lambda 
-(e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C 
-(CHead c0 (Bind Abbr) u) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: 
-C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g 
-e2 v2 v1)))) (ex3_2_intro C T (\lambda (e2: C).(\lambda (v2: T).(eq C (CHead 
-c0 (Bind Abbr) u) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: 
-T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))) c0 
-u (refl_equal C (CHead c0 (Bind Abbr) u)) H8 H9))))) c2 H7)) t (sym_eq T t v1 
-H6))) c1 (sym_eq C c1 e1 H5))) H4)) H3 H0 H1)))]) in (H0 (refl_equal C (CHead 
-e1 (Bind Abst) v1)) (refl_equal C c2))))))).
-
-theorem csub3_gen_bind:
- \forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall 
-(v1: T).((csub3 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).(csub3 g e1 e2))))))))))
-\def
- \lambda (g: G).(\lambda (b1: B).(\lambda (e1: C).(\lambda (c2: C).(\lambda 
-(v1: T).(\lambda (H: (csub3 g (CHead e1 (Bind b1) v1) c2)).(let H0 \def 
-(match H return (\lambda (c: C).(\lambda (c0: C).(\lambda (_: (csub3 ? c 
-c0)).((eq C c (CHead e1 (Bind b1) v1)) \to ((eq C c0 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).(csub3 g e1 
-e2)))))))))) with [(csub3_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) 
-(CHead e1 (Bind b1) v1))).(\lambda (H1: (eq C (CSort n) c2)).((let H2 \def 
-(eq_ind C (CSort n) (\lambda (e: C).(match e return (\lambda (_: C).Prop) 
-with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I 
-(CHead e1 (Bind b1) v1) H0) in (False_ind ((eq C (CSort n) 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).(csub3 g e1 
-e2)))))) H2)) H1))) | (csub3_head c1 c0 H0 k u) \Rightarrow (\lambda (H1: (eq 
-C (CHead c1 k u) (CHead e1 (Bind b1) v1))).(\lambda (H2: (eq C (CHead c0 k u) 
-c2)).((let H3 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: 
-C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead 
-c1 k u) (CHead e1 (Bind b1) v1) H1) in ((let H4 \def (f_equal C K (\lambda 
-(e: C).(match e return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | 
-(CHead _ k _) \Rightarrow k])) (CHead c1 k u) (CHead e1 (Bind b1) v1) H1) in 
-((let H5 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: C).C) 
-with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k 
-u) (CHead e1 (Bind b1) v1) H1) in (eq_ind C e1 (\lambda (c: C).((eq K k (Bind 
-b1)) \to ((eq T u v1) \to ((eq C (CHead c0 k u) c2) \to ((csub3 g c c0) \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).(csub3 g e1 e2)))))))))) (\lambda (H6: (eq K k (Bind b1))).(eq_ind K (Bind 
-b1) (\lambda (k0: K).((eq T u v1) \to ((eq C (CHead c0 k0 u) c2) \to ((csub3 
-g e1 c0) \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).(csub3 g e1 e2))))))))) (\lambda (H7: (eq T u v1)).(eq_ind 
-T v1 (\lambda (t: T).((eq C (CHead c0 (Bind b1) t) c2) \to ((csub3 g e1 c0) 
-\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).(csub3 g e1 e2)))))))) (\lambda (H8: (eq C (CHead c0 (Bind b1) v1) 
-c2)).(eq_ind C (CHead c0 (Bind b1) v1) (\lambda (c: C).((csub3 g e1 c0) \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).(csub3 g e1 e2))))))) (\lambda (H9: (csub3 g e1 c0)).(let H10 \def 
-(eq_ind_r C c2 (\lambda (c: C).(csub3 g (CHead e1 (Bind b1) v1) c)) H (CHead 
-c0 (Bind b1) v1) H8) in (ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: 
-C).(\lambda (v2: T).(eq C (CHead c0 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) 
-(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2)))) b1 c0 v1 
-(refl_equal C (CHead c0 (Bind b1) v1)) H9))) c2 H8)) u (sym_eq T u v1 H7))) k 
-(sym_eq K k (Bind b1) H6))) c1 (sym_eq C c1 e1 H5))) H4)) H3)) H2 H0))) | 
-(csub3_void c1 c0 H0 b H1 u1 u2) \Rightarrow (\lambda (H2: (eq C (CHead c1 
-(Bind Void) u1) (CHead e1 (Bind b1) v1))).(\lambda (H3: (eq C (CHead c0 (Bind 
-b) u2) c2)).((let H4 \def (f_equal C T (\lambda (e: C).(match e return 
-(\lambda (_: C).T) with [(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow 
-t])) (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) v1) H2) in ((let H5 \def 
-(f_equal C B (\lambda (e: C).(match e return (\lambda (_: C).B) with [(CSort 
-_) \Rightarrow Void | (CHead _ k _) \Rightarrow (match k return (\lambda (_: 
-K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Void])])) (CHead c1 
-(Bind Void) u1) (CHead e1 (Bind b1) v1) H2) in ((let H6 \def (f_equal C C 
-(\lambda (e: C).(match e return (\lambda (_: C).C) with [(CSort _) 
-\Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 (Bind Void) u1) 
-(CHead e1 (Bind b1) v1) H2) in (eq_ind C e1 (\lambda (c: C).((eq B Void b1) 
-\to ((eq T u1 v1) \to ((eq C (CHead c0 (Bind b) u2) c2) \to ((csub3 g c c0) 
-\to ((not (eq B b Void)) \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).(csub3 g e1 e2))))))))))) (\lambda (H7: 
-(eq B Void b1)).(eq_ind B Void (\lambda (_: B).((eq T u1 v1) \to ((eq C 
-(CHead c0 (Bind b) u2) c2) \to ((csub3 g e1 c0) \to ((not (eq B b Void)) \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).(csub3 g e1 e2)))))))))) (\lambda (H8: (eq T u1 v1)).(eq_ind T v1 (\lambda 
-(_: T).((eq C (CHead c0 (Bind b) u2) c2) \to ((csub3 g e1 c0) \to ((not (eq B 
-b Void)) \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).(csub3 g e1 e2))))))))) (\lambda (H9: (eq C (CHead c0 
-(Bind b) u2) c2)).(eq_ind C (CHead c0 (Bind b) u2) (\lambda (c: C).((csub3 g 
-e1 c0) \to ((not (eq B b Void)) \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).(csub3 g e1 e2)))))))) (\lambda (H10: 
-(csub3 g e1 c0)).(\lambda (_: (not (eq B b Void))).(let H12 \def (eq_ind_r C 
-c2 (\lambda (c: C).(csub3 g (CHead e1 (Bind b1) v1) c)) H (CHead c0 (Bind b) 
-u2) H9) in (let H13 \def (eq_ind_r B b1 (\lambda (b0: B).(csub3 g (CHead e1 
-(Bind b0) v1) (CHead c0 (Bind b) u2))) H12 Void H7) in (ex2_3_intro B C T 
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c0 (Bind b) 
-u2) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: 
-T).(csub3 g e1 e2)))) b c0 u2 (refl_equal C (CHead c0 (Bind b) u2)) H10))))) 
-c2 H9)) u1 (sym_eq T u1 v1 H8))) b1 H7)) c1 (sym_eq C c1 e1 H6))) H5)) H4)) 
-H3 H0 H1))) | (csub3_abst c1 c0 H0 u t H1) \Rightarrow (\lambda (H2: (eq C 
-(CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1))).(\lambda (H3: (eq C (CHead 
-c0 (Bind Abbr) u) c2)).((let H4 \def (f_equal C T (\lambda (e: C).(match e 
-return (\lambda (_: C).T) with [(CSort _) \Rightarrow t | (CHead _ _ t) 
-\Rightarrow t])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H2) in 
-((let H5 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: C).B) 
-with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match k return 
-(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow 
-Abst])])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H2) in ((let H6 
-\def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: C).C) with 
-[(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 (Bind 
-Abst) t) (CHead e1 (Bind b1) v1) H2) in (eq_ind C e1 (\lambda (c: C).((eq B 
-Abst b1) \to ((eq T t v1) \to ((eq C (CHead c0 (Bind Abbr) u) c2) \to ((csub3 
-g c c0) \to ((ty3 g c0 u t) \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).(csub3 g e1 e2))))))))))) (\lambda (H7: 
-(eq B Abst b1)).(eq_ind B Abst (\lambda (_: B).((eq T t v1) \to ((eq C (CHead 
-c0 (Bind Abbr) u) c2) \to ((csub3 g e1 c0) \to ((ty3 g c0 u t) \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).(csub3 g e1 
-e2)))))))))) (\lambda (H8: (eq T t v1)).(eq_ind T v1 (\lambda (t0: T).((eq C 
-(CHead c0 (Bind Abbr) u) c2) \to ((csub3 g e1 c0) \to ((ty3 g c0 u t0) \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).(csub3 g e1 e2))))))))) (\lambda (H9: (eq C (CHead c0 (Bind Abbr) u) 
-c2)).(eq_ind C (CHead c0 (Bind Abbr) u) (\lambda (c: C).((csub3 g e1 c0) \to 
-((ty3 g c0 u 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).(csub3 g e1 e2)))))))) (\lambda (H10: (csub3 g e1 
-c0)).(\lambda (_: (ty3 g c0 u v1)).(let H12 \def (eq_ind_r C c2 (\lambda (c: 
-C).(csub3 g (CHead e1 (Bind b1) v1) c)) H (CHead c0 (Bind Abbr) u) H9) in 
-(let H13 \def (eq_ind_r B b1 (\lambda (b: B).(csub3 g (CHead e1 (Bind b) v1) 
-(CHead c0 (Bind Abbr) u))) H12 Abst H7) in (ex2_3_intro B C T (\lambda (b2: 
-B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c0 (Bind Abbr) u) (CHead e2 
-(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g 
-e1 e2)))) Abbr c0 u (refl_equal C (CHead c0 (Bind Abbr) u)) H10))))) c2 H9)) 
-t (sym_eq T t v1 H8))) b1 H7)) c1 (sym_eq C c1 e1 H6))) H5)) H4)) H3 H0 
-H1)))]) in (H0 (refl_equal C (CHead e1 (Bind b1) v1)) (refl_equal C 
-c2)))))))).
-
-theorem csub3_refl:
- \forall (g: G).(\forall (c: C).(csub3 g c c))
-\def
- \lambda (g: G).(\lambda (c: C).(C_ind (\lambda (c0: C).(csub3 g c0 c0)) 
-(\lambda (n: nat).(csub3_sort g n)) (\lambda (c0: C).(\lambda (H: (csub3 g c0 
-c0)).(\lambda (k: K).(\lambda (t: T).(csub3_head g c0 c0 H k t))))) c)).
-
-theorem csub3_clear_conf:
- \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csub3 g c1 c2) \to 
-(\forall (e1: C).((clear c1 e1) \to (ex2 C (\lambda (e2: C).(csub3 g e1 e2)) 
-(\lambda (e2: C).(clear c2 e2))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csub3 g c1 
-c2)).(csub3_ind g (\lambda (c: C).(\lambda (c0: C).(\forall (e1: C).((clear c 
-e1) \to (ex2 C (\lambda (e2: C).(csub3 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).(csub3 g e1 e2)) 
-(\lambda (e2: C).(clear (CSort n) e2))))))) (\lambda (c3: C).(\lambda (c4: 
-C).(\lambda (H0: (csub3 g c3 c4)).(\lambda (H1: ((\forall (e1: C).((clear c3 
-e1) \to (ex2 C (\lambda (e2: C).(csub3 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)).((match k return (\lambda (k0: K).((clear (CHead 
-c3 k0 u) e1) \to (ex2 C (\lambda (e2: C).(csub3 g e1 e2)) (\lambda (e2: 
-C).(clear (CHead c4 k0 u) e2))))) with [(Bind b) \Rightarrow (\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).(csub3 g c e2)) (\lambda (e2: C).(clear (CHead 
-c4 (Bind b) u) e2)))) (ex_intro2 C (\lambda (e2: C).(csub3 g (CHead c3 (Bind 
-b) u) e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) u) e2)) (CHead c4 (Bind 
-b) u) (csub3_head g c3 c4 H0 (Bind b) u) (clear_bind b c4 u)) e1 
-(clear_gen_bind b c3 e1 u H3))) | (Flat f) \Rightarrow (\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).(csub3 g e1 e2)) (\lambda (e2: C).(clear c4 
-e2)) (ex2 C (\lambda (e2: C).(csub3 g e1 e2)) (\lambda (e2: C).(clear (CHead 
-c4 (Flat f) u) e2))) (\lambda (x: C).(\lambda (H5: (csub3 g e1 x)).(\lambda 
-(H6: (clear c4 x)).(ex_intro2 C (\lambda (e2: C).(csub3 g e1 e2)) (\lambda 
-(e2: C).(clear (CHead c4 (Flat f) u) e2)) x H5 (clear_flat c4 x H6 f u))))) 
-H4)))]) H2))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csub3 g 
-c3 c4)).(\lambda (_: ((\forall (e1: C).((clear c3 e1) \to (ex2 C (\lambda 
-(e2: C).(csub3 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).(csub3 g c e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) u2) e2)))) 
-(ex_intro2 C (\lambda (e2: C).(csub3 g (CHead c3 (Bind Void) u1) e2)) 
-(\lambda (e2: C).(clear (CHead c4 (Bind b) u2) e2)) (CHead c4 (Bind b) u2) 
-(csub3_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: 
-(csub3 g c3 c4)).(\lambda (_: ((\forall (e1: C).((clear c3 e1) \to (ex2 C 
-(\lambda (e2: C).(csub3 g e1 e2)) (\lambda (e2: C).(clear c4 
-e2))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H2: (ty3 g c4 u 
-t)).(\lambda (e1: C).(\lambda (H3: (clear (CHead c3 (Bind Abst) t) 
-e1)).(eq_ind_r C (CHead c3 (Bind Abst) t) (\lambda (c: C).(ex2 C (\lambda 
-(e2: C).(csub3 g c e2)) (\lambda (e2: C).(clear (CHead c4 (Bind Abbr) u) 
-e2)))) (ex_intro2 C (\lambda (e2: C).(csub3 g (CHead c3 (Bind Abst) t) e2)) 
-(\lambda (e2: C).(clear (CHead c4 (Bind Abbr) u) e2)) (CHead c4 (Bind Abbr) 
-u) (csub3_abst g c3 c4 H0 u t H2) (clear_bind Abbr c4 u)) e1 (clear_gen_bind 
-Abst c3 e1 t H3))))))))))) c1 c2 H)))).
-
-theorem csub3_drop_flat:
- \forall (g: G).(\forall (f: F).(\forall (n: nat).(\forall (c1: C).(\forall 
-(c2: C).((csub3 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).(csub3 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).((csub3 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).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Flat f) 
-u))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csub3 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).(csub3 g c c2)) H 
-(CHead d1 (Flat f) u) (drop_gen_refl c1 (CHead d1 (Flat f) u) H0)) in (let H2 
-\def (match H1 return (\lambda (c: C).(\lambda (c0: C).(\lambda (_: (csub3 ? 
-c c0)).((eq C c (CHead d1 (Flat f) u)) \to ((eq C c0 c2) \to (ex2 C (\lambda 
-(d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Flat f) 
-u))))))))) with [(csub3_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) 
-(CHead d1 (Flat f) u))).(\lambda (H1: (eq C (CSort n) c2)).((let H2 \def 
-(eq_ind C (CSort n) (\lambda (e: C).(match e return (\lambda (_: C).Prop) 
-with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I 
-(CHead d1 (Flat f) u) H0) in (False_ind ((eq C (CSort n) c2) \to (ex2 C 
-(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 
-(Flat f) u))))) H2)) H1))) | (csub3_head c1 c0 H0 k u0) \Rightarrow (\lambda 
-(H1: (eq C (CHead c1 k u0) (CHead d1 (Flat f) u))).(\lambda (H2: (eq C (CHead 
-c0 k u0) c2)).((let H3 \def (f_equal C T (\lambda (e: C).(match e return 
-(\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow 
-t])) (CHead c1 k u0) (CHead d1 (Flat f) u) H1) in ((let H4 \def (f_equal C K 
-(\lambda (e: C).(match e return (\lambda (_: C).K) with [(CSort _) 
-\Rightarrow k | (CHead _ k _) \Rightarrow k])) (CHead c1 k u0) (CHead d1 
-(Flat f) u) H1) in ((let H5 \def (f_equal C C (\lambda (e: C).(match e return 
-(\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow 
-c])) (CHead c1 k u0) (CHead d1 (Flat f) u) H1) in (eq_ind C d1 (\lambda (c: 
-C).((eq K k (Flat f)) \to ((eq T u0 u) \to ((eq C (CHead c0 k u0) c2) \to 
-((csub3 g c c0) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: 
-C).(drop O O c2 (CHead d2 (Flat f) u))))))))) (\lambda (H6: (eq K k (Flat 
-f))).(eq_ind K (Flat f) (\lambda (k0: K).((eq T u0 u) \to ((eq C (CHead c0 k0 
-u0) c2) \to ((csub3 g d1 c0) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) 
-(\lambda (d2: C).(drop O O c2 (CHead d2 (Flat f) u)))))))) (\lambda (H7: (eq 
-T u0 u)).(eq_ind T u (\lambda (t: T).((eq C (CHead c0 (Flat f) t) c2) \to 
-((csub3 g d1 c0) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: 
-C).(drop O O c2 (CHead d2 (Flat f) u))))))) (\lambda (H8: (eq C (CHead c0 
-(Flat f) u) c2)).(eq_ind C (CHead c0 (Flat f) u) (\lambda (c: C).((csub3 g d1 
-c0) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c 
-(CHead d2 (Flat f) u)))))) (\lambda (H9: (csub3 g d1 c0)).(ex_intro2 C 
-(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O (CHead c0 (Flat 
-f) u) (CHead d2 (Flat f) u))) c0 H9 (drop_refl (CHead c0 (Flat f) u)))) c2 
-H8)) u0 (sym_eq T u0 u H7))) k (sym_eq K k (Flat f) H6))) c1 (sym_eq C c1 d1 
-H5))) H4)) H3)) H2 H0))) | (csub3_void c1 c0 H0 b H1 u1 u2) \Rightarrow 
-(\lambda (H2: (eq C (CHead c1 (Bind Void) u1) (CHead d1 (Flat f) 
-u))).(\lambda (H3: (eq C (CHead c0 (Bind b) u2) c2)).((let H4 \def (eq_ind C 
-(CHead c1 (Bind Void) u1) (\lambda (e: C).(match e return (\lambda (_: 
-C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match 
-k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) 
-\Rightarrow False])])) I (CHead d1 (Flat f) u) H2) in (False_ind ((eq C 
-(CHead c0 (Bind b) u2) c2) \to ((csub3 g c1 c0) \to ((not (eq B b Void)) \to 
-(ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead 
-d2 (Flat f) u))))))) H4)) H3 H0 H1))) | (csub3_abst c1 c0 H0 u0 t H1) 
-\Rightarrow (\lambda (H2: (eq C (CHead c1 (Bind Abst) t) (CHead d1 (Flat f) 
-u))).(\lambda (H3: (eq C (CHead c0 (Bind Abbr) u0) c2)).((let H4 \def (eq_ind 
-C (CHead c1 (Bind Abst) t) (\lambda (e: C).(match e return (\lambda (_: 
-C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match 
-k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) 
-\Rightarrow False])])) I (CHead d1 (Flat f) u) H2) in (False_ind ((eq C 
-(CHead c0 (Bind Abbr) u0) c2) \to ((csub3 g c1 c0) \to ((ty3 g c0 u0 t) \to 
-(ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead 
-d2 (Flat f) u))))))) H4)) H3 H0 H1)))]) in (H2 (refl_equal C (CHead d1 (Flat 
-f) u)) (refl_equal C c2)))))))))) (\lambda (n0: nat).(\lambda (H: ((\forall 
-(c1: C).(\forall (c2: C).((csub3 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).(csub3 g 
-d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Flat f) 
-u)))))))))))).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H0: (csub3 g c1 
-c2)).(csub3_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).(csub3 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))).(let H2 \def (match H1 
-return (\lambda (n: nat).(\lambda (n2: nat).(\lambda (c: C).(\lambda (c0: 
-C).(\lambda (_: (drop n n2 c c0)).((eq nat n (S n0)) \to ((eq nat n2 O) \to 
-((eq C c (CSort n1)) \to ((eq C c0 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda 
-(d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 
-(Flat f) u))))))))))))) with [(drop_refl c) \Rightarrow (\lambda (H1: (eq nat 
-O (S n0))).(\lambda (H2: (eq nat O O)).(\lambda (H3: (eq C c (CSort 
-n1))).(\lambda (H4: (eq C c (CHead d1 (Flat f) u))).((let H5 \def (eq_ind nat 
-O (\lambda (e: nat).(match e return (\lambda (_: nat).Prop) with [O 
-\Rightarrow True | (S _) \Rightarrow False])) I (S n0) H1) in (False_ind ((eq 
-nat O O) \to ((eq C c (CSort n1)) \to ((eq C c (CHead d1 (Flat f) u)) \to 
-(ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O 
-(CSort n1) (CHead d2 (Flat f) u))))))) H5)) H2 H3 H4))))) | (drop_drop k h c 
-e H1 u0) \Rightarrow (\lambda (H2: (eq nat (S h) (S n0))).(\lambda (H3: (eq 
-nat O O)).(\lambda (H4: (eq C (CHead c k u0) (CSort n1))).(\lambda (H5: (eq C 
-e (CHead d1 (Flat f) u))).((let H6 \def (f_equal nat nat (\lambda (e0: 
-nat).(match e0 return (\lambda (_: nat).nat) with [O \Rightarrow h | (S n) 
-\Rightarrow n])) (S h) (S n0) H2) in (eq_ind nat n0 (\lambda (n: nat).((eq 
-nat O O) \to ((eq C (CHead c k u0) (CSort n1)) \to ((eq C e (CHead d1 (Flat 
-f) u)) \to ((drop (r k n) O c e) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) 
-(\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Flat f) u))))))))) 
-(\lambda (_: (eq nat O O)).(\lambda (H8: (eq C (CHead c k u0) (CSort 
-n1))).(let H9 \def (eq_ind C (CHead c k u0) (\lambda (e0: C).(match e0 return 
-(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) 
-\Rightarrow True])) I (CSort n1) H8) in (False_ind ((eq C e (CHead d1 (Flat 
-f) u)) \to ((drop (r k n0) O c e) \to (ex2 C (\lambda (d2: C).(csub3 g d1 
-d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Flat f) u)))))) 
-H9)))) h (sym_eq nat h n0 H6))) H3 H4 H5 H1))))) | (drop_skip k h d c e H1 
-u0) \Rightarrow (\lambda (H2: (eq nat h (S n0))).(\lambda (H3: (eq nat (S d) 
-O)).(\lambda (H4: (eq C (CHead c k (lift h (r k d) u0)) (CSort n1))).(\lambda 
-(H5: (eq C (CHead e k u0) (CHead d1 (Flat f) u))).(eq_ind nat (S n0) (\lambda 
-(n: nat).((eq nat (S d) O) \to ((eq C (CHead c k (lift n (r k d) u0)) (CSort 
-n1)) \to ((eq C (CHead e k u0) (CHead d1 (Flat f) u)) \to ((drop n (r k d) c 
-e) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) 
-O (CSort n1) (CHead d2 (Flat f) u))))))))) (\lambda (H6: (eq nat (S d) 
-O)).(let H7 \def (eq_ind nat (S d) (\lambda (e0: nat).(match e0 return 
-(\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) 
-I O H6) in (False_ind ((eq C (CHead c k (lift (S n0) (r k d) u0)) (CSort n1)) 
-\to ((eq C (CHead e k u0) (CHead d1 (Flat f) u)) \to ((drop (S n0) (r k d) c 
-e) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) 
-O (CSort n1) (CHead d2 (Flat f) u))))))) H7))) h (sym_eq nat h (S n0) H2) H3 
-H4 H5 H1)))))]) in (H2 (refl_equal nat (S n0)) (refl_equal nat O) (refl_equal 
-C (CSort n1)) (refl_equal C (CHead d1 (Flat f) u)))))))) (\lambda (c0: 
-C).(\lambda (c3: C).(\lambda (H1: (csub3 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).(csub3 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).(csub3 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).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Flat f) u0))) 
-(ex2 C (\lambda (d2: C).(csub3 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: 
-(csub3 g d1 x)).(\lambda (H5: (drop n0 O c3 (CHead x (Flat f) 
-u0))).(ex_intro2 C (\lambda (d2: C).(csub3 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).(csub3 g 
-d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f) u0))) (ex2 C 
-(\lambda (d2: C).(csub3 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: (csub3 g 
-d1 x)).(\lambda (H5: (drop (S n0) O c3 (CHead x (Flat f) u0))).(ex_intro2 C 
-(\lambda (d2: C).(csub3 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: (csub3 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).(csub3 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).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Flat f) 
-u))) (ex2 C (\lambda (d2: C).(csub3 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: 
-(csub3 g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x (Flat f) u))).(ex_intro2 
-C (\lambda (d2: C).(csub3 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: (csub3 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).(csub3 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 c3 u 
-t)).(\lambda (d1: C).(\lambda (u0: T).(\lambda (H4: (drop (S n0) O (CHead c0 
-(Bind Abst) t) (CHead d1 (Flat f) u0))).(ex2_ind C (\lambda (d2: C).(csub3 g 
-d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Flat f) u0))) (ex2 C 
-(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
-(Bind Abbr) u) (CHead d2 (Flat f) u0)))) (\lambda (x: C).(\lambda (H5: (csub3 
-g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x (Flat f) u0))).(ex_intro2 C 
-(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
-(Bind Abbr) u) (CHead d2 (Flat f) u0))) x H5 (drop_drop (Bind Abbr) n0 c3 
-(CHead x (Flat f) u0) H6 u))))) (H c0 c3 H1 d1 u0 (drop_gen_drop (Bind Abst) 
-c0 (CHead d1 (Flat f) u0) t n0 H4))))))))))))) c1 c2 H0)))))) n))).
-
-theorem csub3_drop_abbr:
- \forall (g: G).(\forall (n: nat).(\forall (c1: C).(\forall (c2: C).((csub3 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).(csub3 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).((csub3 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).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Bind Abbr) 
-u))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csub3 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).(csub3 g c c2)) H 
-(CHead d1 (Bind Abbr) u) (drop_gen_refl c1 (CHead d1 (Bind Abbr) u) H0)) in 
-(let H2 \def (match H1 return (\lambda (c: C).(\lambda (c0: C).(\lambda (_: 
-(csub3 ? c c0)).((eq C c (CHead d1 (Bind Abbr) u)) \to ((eq C c0 c2) \to (ex2 
-C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 
-(Bind Abbr) u))))))))) with [(csub3_sort n) \Rightarrow (\lambda (H0: (eq C 
-(CSort n) (CHead d1 (Bind Abbr) u))).(\lambda (H1: (eq C (CSort n) c2)).((let 
-H2 \def (eq_ind C (CSort n) (\lambda (e: C).(match e return (\lambda (_: 
-C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow 
-False])) I (CHead d1 (Bind Abbr) u) H0) in (False_ind ((eq C (CSort n) c2) 
-\to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 
-(CHead d2 (Bind Abbr) u))))) H2)) H1))) | (csub3_head c1 c0 H0 k u0) 
-\Rightarrow (\lambda (H1: (eq C (CHead c1 k u0) (CHead d1 (Bind Abbr) 
-u))).(\lambda (H2: (eq C (CHead c0 k u0) c2)).((let H3 \def (f_equal C T 
-(\lambda (e: C).(match e return (\lambda (_: C).T) with [(CSort _) 
-\Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u0) (CHead d1 
-(Bind Abbr) u) H1) in ((let H4 \def (f_equal C K (\lambda (e: C).(match e 
-return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k _) 
-\Rightarrow k])) (CHead c1 k u0) (CHead d1 (Bind Abbr) u) H1) in ((let H5 
-\def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: C).C) with 
-[(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u0) 
-(CHead d1 (Bind Abbr) u) H1) in (eq_ind C d1 (\lambda (c: C).((eq K k (Bind 
-Abbr)) \to ((eq T u0 u) \to ((eq C (CHead c0 k u0) c2) \to ((csub3 g c c0) 
-\to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 
-(CHead d2 (Bind Abbr) u))))))))) (\lambda (H6: (eq K k (Bind Abbr))).(eq_ind 
-K (Bind Abbr) (\lambda (k0: K).((eq T u0 u) \to ((eq C (CHead c0 k0 u0) c2) 
-\to ((csub3 g d1 c0) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
-(d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u)))))))) (\lambda (H7: (eq T u0 
-u)).(eq_ind T u (\lambda (t: T).((eq C (CHead c0 (Bind Abbr) t) c2) \to 
-((csub3 g d1 c0) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: 
-C).(drop O O c2 (CHead d2 (Bind Abbr) u))))))) (\lambda (H8: (eq C (CHead c0 
-(Bind Abbr) u) c2)).(eq_ind C (CHead c0 (Bind Abbr) u) (\lambda (c: 
-C).((csub3 g d1 c0) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
-(d2: C).(drop O O c (CHead d2 (Bind Abbr) u)))))) (\lambda (H9: (csub3 g d1 
-c0)).(ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O 
-O (CHead c0 (Bind Abbr) u) (CHead d2 (Bind Abbr) u))) c0 H9 (drop_refl (CHead 
-c0 (Bind Abbr) u)))) c2 H8)) u0 (sym_eq T u0 u H7))) k (sym_eq K k (Bind 
-Abbr) H6))) c1 (sym_eq C c1 d1 H5))) H4)) H3)) H2 H0))) | (csub3_void c1 c0 
-H0 b H1 u1 u2) \Rightarrow (\lambda (H2: (eq C (CHead c1 (Bind Void) u1) 
-(CHead d1 (Bind Abbr) u))).(\lambda (H3: (eq C (CHead c0 (Bind b) u2) 
-c2)).((let H4 \def (eq_ind C (CHead c1 (Bind Void) u1) (\lambda (e: C).(match 
-e return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k 
-_) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind b) 
-\Rightarrow (match b return (\lambda (_: B).Prop) with [Abbr \Rightarrow 
-False | Abst \Rightarrow False | Void \Rightarrow True]) | (Flat _) 
-\Rightarrow False])])) I (CHead d1 (Bind Abbr) u) H2) in (False_ind ((eq C 
-(CHead c0 (Bind b) u2) c2) \to ((csub3 g c1 c0) \to ((not (eq B b Void)) \to 
-(ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead 
-d2 (Bind Abbr) u))))))) H4)) H3 H0 H1))) | (csub3_abst c1 c0 H0 u0 t H1) 
-\Rightarrow (\lambda (H2: (eq C (CHead c1 (Bind Abst) t) (CHead d1 (Bind 
-Abbr) u))).(\lambda (H3: (eq C (CHead c0 (Bind Abbr) u0) c2)).((let H4 \def 
-(eq_ind C (CHead c1 (Bind Abst) t) (\lambda (e: C).(match e return (\lambda 
-(_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow 
-(match k return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b 
-return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow 
-True | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead d1 
-(Bind Abbr) u) H2) in (False_ind ((eq C (CHead c0 (Bind Abbr) u0) c2) \to 
-((csub3 g c1 c0) \to ((ty3 g c0 u0 t) \to (ex2 C (\lambda (d2: C).(csub3 g d1 
-d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u))))))) H4)) H3 H0 
-H1)))]) in (H2 (refl_equal C (CHead d1 (Bind Abbr) u)) (refl_equal C 
-c2)))))))))) (\lambda (n0: nat).(\lambda (H: ((\forall (c1: C).(\forall (c2: 
-C).((csub3 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).(csub3 g d1 d2)) 
-(\lambda (d2: C).(drop n0 O c2 (CHead d2 (Bind Abbr) u)))))))))))).(\lambda 
-(c1: C).(\lambda (c2: C).(\lambda (H0: (csub3 g c1 c2)).(csub3_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).(csub3 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))).(let H2 \def (match H1 return (\lambda 
-(n: nat).(\lambda (n2: nat).(\lambda (c: C).(\lambda (c0: C).(\lambda (_: 
-(drop n n2 c c0)).((eq nat n (S n0)) \to ((eq nat n2 O) \to ((eq C c (CSort 
-n1)) \to ((eq C c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: 
-C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 
-(Bind Abbr) u))))))))))))) with [(drop_refl c) \Rightarrow (\lambda (H1: (eq 
-nat O (S n0))).(\lambda (H2: (eq nat O O)).(\lambda (H3: (eq C c (CSort 
-n1))).(\lambda (H4: (eq C c (CHead d1 (Bind Abbr) u))).((let H5 \def (eq_ind 
-nat O (\lambda (e: nat).(match e return (\lambda (_: nat).Prop) with [O 
-\Rightarrow True | (S _) \Rightarrow False])) I (S n0) H1) in (False_ind ((eq 
-nat O O) \to ((eq C c (CSort n1)) \to ((eq C c (CHead d1 (Bind Abbr) u)) \to 
-(ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O 
-(CSort n1) (CHead d2 (Bind Abbr) u))))))) H5)) H2 H3 H4))))) | (drop_drop k h 
-c e H1 u0) \Rightarrow (\lambda (H2: (eq nat (S h) (S n0))).(\lambda (H3: (eq 
-nat O O)).(\lambda (H4: (eq C (CHead c k u0) (CSort n1))).(\lambda (H5: (eq C 
-e (CHead d1 (Bind Abbr) u))).((let H6 \def (f_equal nat nat (\lambda (e0: 
-nat).(match e0 return (\lambda (_: nat).nat) with [O \Rightarrow h | (S n) 
-\Rightarrow n])) (S h) (S n0) H2) in (eq_ind nat n0 (\lambda (n: nat).((eq 
-nat O O) \to ((eq C (CHead c k u0) (CSort n1)) \to ((eq C e (CHead d1 (Bind 
-Abbr) u)) \to ((drop (r k n) O c e) \to (ex2 C (\lambda (d2: C).(csub3 g d1 
-d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) 
-u))))))))) (\lambda (_: (eq nat O O)).(\lambda (H8: (eq C (CHead c k u0) 
-(CSort n1))).(let H9 \def (eq_ind C (CHead c k u0) (\lambda (e0: C).(match e0 
-return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ 
-_) \Rightarrow True])) I (CSort n1) H8) in (False_ind ((eq C e (CHead d1 
-(Bind Abbr) u)) \to ((drop (r k n0) O c e) \to (ex2 C (\lambda (d2: C).(csub3 
-g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) 
-u)))))) H9)))) h (sym_eq nat h n0 H6))) H3 H4 H5 H1))))) | (drop_skip k h d c 
-e H1 u0) \Rightarrow (\lambda (H2: (eq nat h (S n0))).(\lambda (H3: (eq nat 
-(S d) O)).(\lambda (H4: (eq C (CHead c k (lift h (r k d) u0)) (CSort 
-n1))).(\lambda (H5: (eq C (CHead e k u0) (CHead d1 (Bind Abbr) u))).(eq_ind 
-nat (S n0) (\lambda (n: nat).((eq nat (S d) O) \to ((eq C (CHead c k (lift n 
-(r k d) u0)) (CSort n1)) \to ((eq C (CHead e k u0) (CHead d1 (Bind Abbr) u)) 
-\to ((drop n (r k d) c e) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) 
-(\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u))))))))) 
-(\lambda (H6: (eq nat (S d) O)).(let H7 \def (eq_ind nat (S d) (\lambda (e0: 
-nat).(match e0 return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S 
-_) \Rightarrow True])) I O H6) in (False_ind ((eq C (CHead c k (lift (S n0) 
-(r k d) u0)) (CSort n1)) \to ((eq C (CHead e k u0) (CHead d1 (Bind Abbr) u)) 
-\to ((drop (S n0) (r k d) c e) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) 
-(\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u))))))) 
-H7))) h (sym_eq nat h (S n0) H2) H3 H4 H5 H1)))))]) in (H2 (refl_equal nat (S 
-n0)) (refl_equal nat O) (refl_equal C (CSort n1)) (refl_equal C (CHead d1 
-(Bind Abbr) u)))))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H1: (csub3 
-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).(csub3 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).(csub3 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).(csub3 g 
-d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abbr) u0))) (ex2 C 
-(\lambda (d2: C).(csub3 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: (csub3 
-g d1 x)).(\lambda (H5: (drop n0 O c3 (CHead x (Bind Abbr) u0))).(ex_intro2 C 
-(\lambda (d2: C).(csub3 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).(csub3 g 
-d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u0))) (ex2 C 
-(\lambda (d2: C).(csub3 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: (csub3 
-g d1 x)).(\lambda (H5: (drop (S n0) O c3 (CHead x (Bind Abbr) 
-u0))).(ex_intro2 C (\lambda (d2: C).(csub3 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: (csub3 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).(csub3 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).(csub3 g d1 d2)) (\lambda (d2: 
-C).(drop n0 O c3 (CHead d2 (Bind Abbr) u))) (ex2 C (\lambda (d2: C).(csub3 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: (csub3 g d1 x)).(\lambda (H6: 
-(drop n0 O c3 (CHead x (Bind Abbr) u))).(ex_intro2 C (\lambda (d2: C).(csub3 
-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: 
-(csub3 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).(csub3 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 c3 u t)).(\lambda (d1: C).(\lambda 
-(u0: T).(\lambda (H4: (drop (S n0) O (CHead c0 (Bind Abst) t) (CHead d1 (Bind 
-Abbr) u0))).(ex2_ind C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: 
-C).(drop n0 O c3 (CHead d2 (Bind Abbr) u0))) (ex2 C (\lambda (d2: C).(csub3 g 
-d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 
-(Bind Abbr) u0)))) (\lambda (x: C).(\lambda (H5: (csub3 g d1 x)).(\lambda 
-(H6: (drop n0 O c3 (CHead x (Bind Abbr) u0))).(ex_intro2 C (\lambda (d2: 
-C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) 
-(CHead d2 (Bind Abbr) u0))) x H5 (drop_drop (Bind Abbr) n0 c3 (CHead x (Bind 
-Abbr) u0) H6 u))))) (H c0 c3 H1 d1 u0 (drop_gen_drop (Bind Abst) c0 (CHead d1 
-(Bind Abbr) u0) t n0 H4))))))))))))) c1 c2 H0)))))) n)).
-
-theorem csub3_drop_abst:
- \forall (g: G).(\forall (n: nat).(\forall (c1: C).(\forall (c2: C).((csub3 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).(csub3 g d1 d2)) (\lambda (d2: 
-C).(drop n O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop n 
-O c2 (CHead d2 (Bind Abbr) u)))) (\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).((csub3 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).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Bind Abst) 
-t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda 
-(d2: C).(\lambda (u: T).(drop n0 O c2 (CHead d2 (Bind Abbr) u)))) (\lambda 
-(d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))))) (\lambda (c1: C).(\lambda 
-(c2: C).(\lambda (H: (csub3 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).(csub3 g c c2)) H (CHead d1 (Bind Abst) t) 
-(drop_gen_refl c1 (CHead d1 (Bind Abst) t) H0)) in (let H2 \def (match H1 
-return (\lambda (c: C).(\lambda (c0: C).(\lambda (_: (csub3 ? c c0)).((eq C c 
-(CHead d1 (Bind Abst) t)) \to ((eq C c0 c2) \to (or (ex2 C (\lambda (d2: 
-C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t)))) 
-(ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: 
-C).(\lambda (u: T).(drop O O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: 
-C).(\lambda (u: T).(ty3 g d2 u t)))))))))) with [(csub3_sort n) \Rightarrow 
-(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Abst) t))).(\lambda (H2: (eq C 
-(CSort n) c2)).((let H3 \def (eq_ind C (CSort n) (\lambda (e: C).(match e 
-return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) 
-\Rightarrow False])) I (CHead d1 (Bind Abst) t) H1) in (False_ind ((eq C 
-(CSort n) c2) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: 
-C).(drop O O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop O 
-O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u 
-t)))))) H3)) H2))) | (csub3_head c0 c3 H1 k u) \Rightarrow (\lambda (H2: (eq 
-C (CHead c0 k u) (CHead d1 (Bind Abst) t))).(\lambda (H3: (eq C (CHead c3 k 
-u) c2)).((let H4 \def (f_equal C T (\lambda (e: C).(match e return (\lambda 
-(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) 
-(CHead c0 k u) (CHead d1 (Bind Abst) t) H2) in ((let H5 \def (f_equal C K 
-(\lambda (e: C).(match e return (\lambda (_: C).K) with [(CSort _) 
-\Rightarrow k | (CHead _ k _) \Rightarrow k])) (CHead c0 k u) (CHead d1 (Bind 
-Abst) t) H2) in ((let H6 \def (f_equal C C (\lambda (e: C).(match e return 
-(\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow 
-c])) (CHead c0 k u) (CHead d1 (Bind Abst) t) H2) in (eq_ind C d1 (\lambda (c: 
-C).((eq K k (Bind Abst)) \to ((eq T u t) \to ((eq C (CHead c3 k u) c2) \to 
-((csub3 g c c3) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
-(d2: C).(drop O O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop 
-O O c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g 
-d2 u0 t)))))))))) (\lambda (H7: (eq K k (Bind Abst))).(eq_ind K (Bind Abst) 
-(\lambda (k0: K).((eq T u t) \to ((eq C (CHead c3 k0 u) c2) \to ((csub3 g d1 
-c3) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O 
-O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
-T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop O O c2 (CHead d2 
-(Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))))))))) 
-(\lambda (H8: (eq T u t)).(eq_ind T t (\lambda (t0: T).((eq C (CHead c3 (Bind 
-Abst) t0) c2) \to ((csub3 g d1 c3) \to (or (ex2 C (\lambda (d2: C).(csub3 g 
-d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T 
-(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda 
-(u0: T).(drop O O c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda 
-(u0: T).(ty3 g d2 u0 t)))))))) (\lambda (H9: (eq C (CHead c3 (Bind Abst) t) 
-c2)).(eq_ind C (CHead c3 (Bind Abst) t) (\lambda (c: C).((csub3 g d1 c3) \to 
-(or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c 
-(CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
-T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop O O c (CHead d2 
-(Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))))))) 
-(\lambda (H10: (csub3 g d1 c3)).(or_introl (ex2 C (\lambda (d2: C).(csub3 g 
-d1 d2)) (\lambda (d2: C).(drop O O (CHead c3 (Bind Abst) t) (CHead d2 (Bind 
-Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) 
-(\lambda (d2: C).(\lambda (u0: T).(drop O O (CHead c3 (Bind Abst) t) (CHead 
-d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))) 
-(ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O 
-(CHead c3 (Bind Abst) t) (CHead d2 (Bind Abst) t))) c3 H10 (drop_refl (CHead 
-c3 (Bind Abst) t))))) c2 H9)) u (sym_eq T u t H8))) k (sym_eq K k (Bind Abst) 
-H7))) c0 (sym_eq C c0 d1 H6))) H5)) H4)) H3 H1))) | (csub3_void c0 c3 H1 b H2 
-u1 u2) \Rightarrow (\lambda (H3: (eq C (CHead c0 (Bind Void) u1) (CHead d1 
-(Bind Abst) t))).(\lambda (H4: (eq C (CHead c3 (Bind b) u2) c2)).((let H5 
-\def (eq_ind C (CHead c0 (Bind Void) u1) (\lambda (e: C).(match e return 
-(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) 
-\Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow 
-(match b return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst 
-\Rightarrow False | Void \Rightarrow True]) | (Flat _) \Rightarrow False])])) 
-I (CHead d1 (Bind Abst) t) H3) in (False_ind ((eq C (CHead c3 (Bind b) u2) 
-c2) \to ((csub3 g c0 c3) \to ((not (eq B b Void)) \to (or (ex2 C (\lambda 
-(d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) 
-t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda 
-(d2: C).(\lambda (u: T).(drop O O c2 (CHead d2 (Bind Abbr) u)))) (\lambda 
-(d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))) H5)) H4 H1 H2))) | (csub3_abst 
-c0 c3 H1 u t0 H2) \Rightarrow (\lambda (H3: (eq C (CHead c0 (Bind Abst) t0) 
-(CHead d1 (Bind Abst) t))).(\lambda (H4: (eq C (CHead c3 (Bind Abbr) u) 
-c2)).((let H5 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: 
-C).T) with [(CSort _) \Rightarrow t0 | (CHead _ _ t) \Rightarrow t])) (CHead 
-c0 (Bind Abst) t0) (CHead d1 (Bind Abst) t) H3) in ((let H6 \def (f_equal C C 
-(\lambda (e: C).(match e return (\lambda (_: C).C) with [(CSort _) 
-\Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 (Bind Abst) t0) 
-(CHead d1 (Bind Abst) t) H3) in (eq_ind C d1 (\lambda (c: C).((eq T t0 t) \to 
-((eq C (CHead c3 (Bind Abbr) u) c2) \to ((csub3 g c c3) \to ((ty3 g c3 u t0) 
-\to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O 
-c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
-T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop O O c2 (CHead d2 
-(Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))))))))) 
-(\lambda (H7: (eq T t0 t)).(eq_ind T t (\lambda (t1: T).((eq C (CHead c3 
-(Bind Abbr) u) c2) \to ((csub3 g d1 c3) \to ((ty3 g c3 u t1) \to (or (ex2 C 
-(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 
-(Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 
-d2))) (\lambda (d2: C).(\lambda (u0: T).(drop O O c2 (CHead d2 (Bind Abbr) 
-u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))))))))) (\lambda 
-(H8: (eq C (CHead c3 (Bind Abbr) u) c2)).(eq_ind C (CHead c3 (Bind Abbr) u) 
-(\lambda (c: C).((csub3 g d1 c3) \to ((ty3 g c3 u t) \to (or (ex2 C (\lambda 
-(d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c (CHead d2 (Bind Abst) 
-t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda 
-(d2: C).(\lambda (u0: T).(drop O O c (CHead d2 (Bind Abbr) u0)))) (\lambda 
-(d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))))))) (\lambda (H9: (csub3 g d1 
-c3)).(\lambda (H10: (ty3 g c3 u t)).(or_intror (ex2 C (\lambda (d2: C).(csub3 
-g d1 d2)) (\lambda (d2: C).(drop O O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind 
-Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) 
-(\lambda (d2: C).(\lambda (u0: T).(drop O O (CHead c3 (Bind Abbr) u) (CHead 
-d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))) 
-(ex3_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda 
-(d2: C).(\lambda (u0: T).(drop O O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind 
-Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))) c3 u H9 
-(drop_refl (CHead c3 (Bind Abbr) u)) H10)))) c2 H8)) t0 (sym_eq T t0 t H7))) 
-c0 (sym_eq C c0 d1 H6))) H5)) H4 H1 H2)))]) in (H2 (refl_equal C (CHead d1 
-(Bind Abst) t)) (refl_equal C c2)))))))))) (\lambda (n0: nat).(\lambda (H: 
-((\forall (c1: C).(\forall (c2: C).((csub3 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).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 
-(Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 
-d2))) (\lambda (d2: C).(\lambda (u: T).(drop n0 O c2 (CHead d2 (Bind Abbr) 
-u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))))))))))).(\lambda 
-(c1: C).(\lambda (c2: C).(\lambda (H0: (csub3 g c1 c2)).(csub3_ind g (\lambda 
-(c: C).(\lambda (c0: C).(\forall (d1: C).(\forall (t: T).((drop (S n0) O c 
-(CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) 
-(\lambda (d2: C).(drop (S n0) O c0 (CHead d2 (Bind Abst) t)))) (ex3_2 C T 
-(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda 
-(u: T).(drop (S n0) O c0 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: 
-C).(\lambda (u: T).(ty3 g d2 u t)))))))))) (\lambda (n1: nat).(\lambda (d1: 
-C).(\lambda (t: T).(\lambda (H1: (drop (S n0) O (CSort n1) (CHead d1 (Bind 
-Abst) t))).(let H2 \def (match H1 return (\lambda (n: nat).(\lambda (n2: 
-nat).(\lambda (c: C).(\lambda (c0: C).(\lambda (_: (drop n n2 c c0)).((eq nat 
-n (S n0)) \to ((eq nat n2 O) \to ((eq C c (CSort n1)) \to ((eq C c0 (CHead d1 
-(Bind Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
-(d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abst) t)))) (ex3_2 C T 
-(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda 
-(u: T).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u)))) (\lambda (d2: 
-C).(\lambda (u: T).(ty3 g d2 u t)))))))))))))) with [(drop_refl c) 
-\Rightarrow (\lambda (H1: (eq nat O (S n0))).(\lambda (H2: (eq nat O 
-O)).(\lambda (H3: (eq C c (CSort n1))).(\lambda (H4: (eq C c (CHead d1 (Bind 
-Abst) t))).((let H5 \def (eq_ind nat O (\lambda (e: nat).(match e return 
-(\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) 
-I (S n0) H1) in (False_ind ((eq nat O O) \to ((eq C c (CSort n1)) \to ((eq C 
-c (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) 
-(\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abst) t)))) (ex3_2 
-C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: 
-C).(\lambda (u: T).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u)))) 
-(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))) H5)) H2 H3 H4))))) | 
-(drop_drop k h c e H1 u) \Rightarrow (\lambda (H2: (eq nat (S h) (S 
-n0))).(\lambda (H3: (eq nat O O)).(\lambda (H4: (eq C (CHead c k u) (CSort 
-n1))).(\lambda (H5: (eq C e (CHead d1 (Bind Abst) t))).((let H6 \def (f_equal 
-nat nat (\lambda (e0: nat).(match e0 return (\lambda (_: nat).nat) with [O 
-\Rightarrow h | (S n) \Rightarrow n])) (S h) (S n0) H2) in (eq_ind nat n0 
-(\lambda (n: nat).((eq nat O O) \to ((eq C (CHead c k u) (CSort n1)) \to ((eq 
-C e (CHead d1 (Bind Abst) t)) \to ((drop (r k n) O c e) \to (or (ex2 C 
-(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) 
-(CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
-T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CSort 
-n1) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 
-u0 t)))))))))) (\lambda (_: (eq nat O O)).(\lambda (H8: (eq C (CHead c k u) 
-(CSort n1))).(let H9 \def (eq_ind C (CHead c k u) (\lambda (e0: C).(match e0 
-return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ 
-_) \Rightarrow True])) I (CSort n1) H8) in (False_ind ((eq C e (CHead d1 
-(Bind Abst) t)) \to ((drop (r k n0) O c e) \to (or (ex2 C (\lambda (d2: 
-C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 
-(Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 
-d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CSort n1) (CHead d2 
-(Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))))))) 
-H9)))) h (sym_eq nat h n0 H6))) H3 H4 H5 H1))))) | (drop_skip k h d c e H1 u) 
-\Rightarrow (\lambda (H2: (eq nat h (S n0))).(\lambda (H3: (eq nat (S d) 
-O)).(\lambda (H4: (eq C (CHead c k (lift h (r k d) u)) (CSort n1))).(\lambda 
-(H5: (eq C (CHead e k u) (CHead d1 (Bind Abst) t))).(eq_ind nat (S n0) 
-(\lambda (n: nat).((eq nat (S d) O) \to ((eq C (CHead c k (lift n (r k d) u)) 
-(CSort n1)) \to ((eq C (CHead e k u) (CHead d1 (Bind Abst) t)) \to ((drop n 
-(r k d) c e) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: 
-C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda 
-(d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: 
-T).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: 
-C).(\lambda (u0: T).(ty3 g d2 u0 t)))))))))) (\lambda (H6: (eq nat (S d) 
-O)).(let H7 \def (eq_ind nat (S d) (\lambda (e0: nat).(match e0 return 
-(\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) 
-I O H6) in (False_ind ((eq C (CHead c k (lift (S n0) (r k d) u)) (CSort n1)) 
-\to ((eq C (CHead e k u) (CHead d1 (Bind Abst) t)) \to ((drop (S n0) (r k d) 
-c e) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop 
-(S n0) O (CSort n1) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop 
-(S n0) O (CSort n1) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda 
-(u0: T).(ty3 g d2 u0 t)))))))) H7))) h (sym_eq nat h (S n0) H2) H3 H4 H5 
-H1)))))]) in (H2 (refl_equal nat (S n0)) (refl_equal nat O) (refl_equal C 
-(CSort n1)) (refl_equal C (CHead d1 (Bind Abst) t)))))))) (\lambda (c0: 
-C).(\lambda (c3: C).(\lambda (H1: (csub3 g c0 c3)).(\lambda (H2: ((\forall 
-(d1: C).(\forall (t: T).((drop (S n0) O c0 (CHead d1 (Bind Abst) t)) \to (or 
-(ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 
-(CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
-T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O c3 
-(CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u 
-t)))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (u: T).(\forall 
-(d1: C).(\forall (t: T).((drop (S n0) O (CHead c0 k0 u) (CHead d1 (Bind Abst) 
-t)) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop 
-(S n0) O (CHead c3 k0 u) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop 
-(S n0) O (CHead c3 k0 u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: 
-C).(\lambda (u0: T).(ty3 g d2 u0 t)))))))))) (\lambda (b: B).(\lambda (u: 
-T).(\lambda (d1: C).(\lambda (t: T).(\lambda (H3: (drop (S n0) O (CHead c0 
-(Bind b) u) (CHead d1 (Bind Abst) t))).(or_ind (ex2 C (\lambda (d2: C).(csub3 
-g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abst) t)))) (ex3_2 C 
-T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: 
-C).(\lambda (u0: T).(drop n0 O c3 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: 
-C).(\lambda (u0: T).(ty3 g d2 u0 t)))) (or (ex2 C (\lambda (d2: C).(csub3 g 
-d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind 
-Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) 
-(\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind b) u) (CHead 
-d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))))) 
-(\lambda (H4: (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop 
-n0 O c3 (CHead d2 (Bind Abst) t))))).(ex2_ind C (\lambda (d2: C).(csub3 g d1 
-d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abst) t))) (or (ex2 C 
-(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
-(Bind b) u) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda 
-(_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O 
-(CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda 
-(u0: T).(ty3 g d2 u0 t))))) (\lambda (x: C).(\lambda (H5: (csub3 g d1 
-x)).(\lambda (H6: (drop n0 O c3 (CHead x (Bind Abst) t))).(or_introl (ex2 C 
-(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
-(Bind b) u) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda 
-(_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O 
-(CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda 
-(u0: T).(ty3 g d2 u0 t)))) (ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) 
-(\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abst) 
-t))) x H5 (drop_drop (Bind b) n0 c3 (CHead x (Bind Abst) t) H6 u)))))) H4)) 
-(\lambda (H4: (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) 
-(\lambda (d2: C).(\lambda (u: T).(drop n0 O c3 (CHead d2 (Bind Abbr) u)))) 
-(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))).(ex3_2_ind C T (\lambda 
-(d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: 
-T).(drop n0 O c3 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: 
-T).(ty3 g d2 u0 t))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
-(d2: C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abst) t)))) 
-(ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: 
-C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind 
-Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))))) (\lambda 
-(x0: C).(\lambda (x1: T).(\lambda (H5: (csub3 g d1 x0)).(\lambda (H6: (drop 
-n0 O c3 (CHead x0 (Bind Abbr) x1))).(\lambda (H7: (ty3 g x0 x1 t)).(or_intror 
-(ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O 
-(CHead c3 (Bind b) u) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop 
-(S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: 
-C).(\lambda (u0: T).(ty3 g d2 u0 t)))) (ex3_2_intro C T (\lambda (d2: 
-C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop 
-(S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: 
-C).(\lambda (u0: T).(ty3 g d2 u0 t))) x0 x1 H5 (drop_drop (Bind b) n0 c3 
-(CHead x0 (Bind Abbr) x1) H6 u) H7))))))) H4)) (H c0 c3 H1 d1 t 
-(drop_gen_drop (Bind b) c0 (CHead d1 (Bind Abst) t) u n0 H3)))))))) (\lambda 
-(f: F).(\lambda (u: T).(\lambda (d1: C).(\lambda (t: T).(\lambda (H3: (drop 
-(S n0) O (CHead c0 (Flat f) u) (CHead d1 (Bind Abst) t))).(or_ind (ex2 C 
-(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead 
-d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 
-d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O c3 (CHead d2 (Bind 
-Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))) (or (ex2 C 
-(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
-(Flat f) u) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda 
-(_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O 
-(CHead c3 (Flat f) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda 
-(u0: T).(ty3 g d2 u0 t))))) (\lambda (H4: (ex2 C (\lambda (d2: C).(csub3 g d1 
-d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abst) t))))).(ex2_ind 
-C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead 
-d2 (Bind Abst) t))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
-(d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abst) t)))) 
-(ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: 
-C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind 
-Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))))) (\lambda 
-(x: C).(\lambda (H5: (csub3 g d1 x)).(\lambda (H6: (drop (S n0) O c3 (CHead x 
-(Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
-(d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abst) t)))) 
-(ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: 
-C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind 
-Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))) (ex_intro2 
-C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
-(Flat f) u) (CHead d2 (Bind Abst) t))) x H5 (drop_drop (Flat f) n0 c3 (CHead 
-x (Bind Abst) t) H6 u)))))) H4)) (\lambda (H4: (ex3_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop 
-(S n0) O c3 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 
-g d2 u t))))).(ex3_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 
-d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O c3 (CHead d2 (Bind 
-Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))) (or (ex2 C 
-(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
-(Flat f) u) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda 
-(_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O 
-(CHead c3 (Flat f) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda 
-(u0: T).(ty3 g d2 u0 t))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: 
-(csub3 g d1 x0)).(\lambda (H6: (drop (S n0) O c3 (CHead x0 (Bind Abbr) 
-x1))).(\lambda (H7: (ty3 g x0 x1 t)).(or_intror (ex2 C (\lambda (d2: 
-C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u) 
-(CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
-T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead 
-c3 (Flat f) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: 
-T).(ty3 g d2 u0 t)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: 
-T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead 
-c3 (Flat f) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: 
-T).(ty3 g d2 u0 t))) x0 x1 H5 (drop_drop (Flat f) n0 c3 (CHead x0 (Bind Abbr) 
-x1) H6 u) H7))))))) H4)) (H2 d1 t (drop_gen_drop (Flat f) c0 (CHead d1 (Bind 
-Abst) t) u n0 H3)))))))) k)))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda 
-(H1: (csub3 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).(csub3 
-g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abst) t)))) 
-(ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: 
-C).(\lambda (u: T).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u)))) (\lambda 
-(d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))).(\lambda (b: B).(\lambda (_: 
-(not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (d1: 
-C).(\lambda (t: T).(\lambda (H4: (drop (S n0) O (CHead c0 (Bind Void) u1) 
-(CHead d1 (Bind Abst) t))).(or_ind (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) 
-(\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abst) t)))) (ex3_2 C T 
-(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda 
-(u: T).(drop n0 O c3 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda 
-(u: T).(ty3 g d2 u t)))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) 
-(\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abst) 
-t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda 
-(d2: C).(\lambda (u: T).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind 
-Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (H5: 
-(ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 
-(CHead d2 (Bind Abst) t))))).(ex2_ind C (\lambda (d2: C).(csub3 g d1 d2)) 
-(\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abst) t))) (or (ex2 C (\lambda 
-(d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) 
-u2) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
-T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CHead 
-c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: 
-T).(ty3 g d2 u t))))) (\lambda (x: C).(\lambda (H6: (csub3 g d1 x)).(\lambda 
-(H7: (drop n0 O c3 (CHead x (Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: 
-C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u2) 
-(CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
-T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CHead 
-c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: 
-T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csub3 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: (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda 
-(d2: C).(\lambda (u: T).(drop n0 O c3 (CHead d2 (Bind Abbr) u)))) (\lambda 
-(d2: C).(\lambda (u: T).(ty3 g d2 u t))))).(ex3_2_ind C T (\lambda (d2: 
-C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop 
-n0 O c3 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g 
-d2 u t))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop 
-(S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abst) t)))) (ex3_2 C T 
-(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda 
-(u: T).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) 
-(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x0: C).(\lambda 
-(x1: T).(\lambda (H6: (csub3 g d1 x0)).(\lambda (H7: (drop n0 O c3 (CHead x0 
-(Bind Abbr) x1))).(\lambda (H8: (ty3 g x0 x1 t)).(or_intror (ex2 C (\lambda 
-(d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) 
-u2) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
-T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CHead 
-c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: 
-T).(ty3 g d2 u t)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csub3 
-g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CHead c3 (Bind b) 
-u2) (CHead d2 (Bind Abbr) u)))) (\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))))))) 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: (csub3 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).(csub3 
-g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abst) t)))) 
-(ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: 
-C).(\lambda (u: T).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u)))) (\lambda 
-(d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))).(\lambda (u: T).(\lambda (t: 
-T).(\lambda (_: (ty3 g c3 u t)).(\lambda (d1: C).(\lambda (t0: T).(\lambda 
-(H4: (drop (S n0) O (CHead c0 (Bind Abst) t) (CHead d1 (Bind Abst) 
-t0))).(or_ind (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop 
-n0 O c3 (CHead d2 (Bind Abst) t0)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
-T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop n0 O c3 (CHead 
-d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0)))) 
-(or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O 
-(CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) (ex3_2 C T (\lambda 
-(d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: 
-T).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) 
-(\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0))))) (\lambda (H5: (ex2 C 
-(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 
-(Bind Abst) t0))))).(ex2_ind C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
-(d2: C).(drop n0 O c3 (CHead d2 (Bind Abst) t0))) (or (ex2 C (\lambda (d2: 
-C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) 
-(CHead d2 (Bind Abst) t0)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
-T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead 
-c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: 
-T).(ty3 g d2 u0 t0))))) (\lambda (x: C).(\lambda (H6: (csub3 g d1 
-x)).(\lambda (H7: (drop n0 O c3 (CHead x (Bind Abst) t0))).(or_introl (ex2 C 
-(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
-(Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) (ex3_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop 
-(S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: 
-C).(\lambda (u0: T).(ty3 g d2 u0 t0)))) (ex_intro2 C (\lambda (d2: C).(csub3 
-g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 
-(Bind Abst) t0))) x H6 (drop_drop (Bind Abbr) n0 c3 (CHead x (Bind Abst) t0) 
-H7 u)))))) H5)) (\lambda (H5: (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
-T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop n0 O c3 (CHead d2 
-(Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u 
-t0))))).(ex3_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) 
-(\lambda (d2: C).(\lambda (u0: T).(drop n0 O c3 (CHead d2 (Bind Abbr) u0)))) 
-(\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0))) (or (ex2 C (\lambda (d2: 
-C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) 
-(CHead d2 (Bind Abst) t0)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
-T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead 
-c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: 
-T).(ty3 g d2 u0 t0))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: 
-(csub3 g d1 x0)).(\lambda (H7: (drop n0 O c3 (CHead x0 (Bind Abbr) 
-x1))).(\lambda (H8: (ty3 g x0 x1 t0)).(or_intror (ex2 C (\lambda (d2: 
-C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) 
-(CHead d2 (Bind Abst) t0)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
-T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead 
-c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: 
-T).(ty3 g d2 u0 t0)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: 
-T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead 
-c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: 
-T).(ty3 g d2 u0 t0))) x0 x1 H6 (drop_drop (Bind Abbr) n0 c3 (CHead x0 (Bind 
-Abbr) x1) H7 u) H8))))))) H5)) (H c0 c3 H1 d1 t0 (drop_gen_drop (Bind Abst) 
-c0 (CHead d1 (Bind Abst) t0) t n0 H4))))))))))))) c1 c2 H0)))))) n)).
-
-theorem csub3_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).((csub3 g 
-c1 c2) \to (ex2 C (\lambda (d2: C).(csub3 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).((csub3 g c1 c2) \to (ex2 C (\lambda (d2: C).(csub3 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))).((match x return (\lambda (c: C).((drop n O c1 c) \to ((clear c 
-(CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csub3 g c1 c2) \to (ex2 C 
-(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind 
-Abbr) u))))))))) with [(CSort n0) \Rightarrow (\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).((csub3 
-g c1 c2) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl 
-n c2 (CHead d2 (Bind Abbr) u))))))))) | (CHead c k t) \Rightarrow (\lambda 
-(H3: (drop n O c1 (CHead c k t))).(\lambda (H4: (clear (CHead c k t) (CHead 
-d1 (Bind Abbr) u))).((match k return (\lambda (k0: K).((drop n O c1 (CHead c 
-k0 t)) \to ((clear (CHead c k0 t) (CHead d1 (Bind Abbr) u)) \to (\forall (c2: 
-C).((csub3 g c1 c2) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
-(d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))))))))) with [(Bind b) 
-\Rightarrow (\lambda (H5: (drop n O c1 (CHead c (Bind b) t))).(\lambda (H6: 
-(clear (CHead c (Bind b) t) (CHead d1 (Bind Abbr) u))).(let H7 \def (f_equal 
-C C (\lambda (e: C).(match e return (\lambda (_: C).C) with [(CSort _) 
-\Rightarrow d1 | (CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Abbr) u) 
-(CHead c (Bind b) t) (clear_gen_bind b c (CHead d1 (Bind Abbr) u) t H6)) in 
-((let H8 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: C).B) 
-with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k return 
-(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow 
-Abbr])])) (CHead d1 (Bind Abbr) u) (CHead c (Bind b) t) (clear_gen_bind b c 
-(CHead d1 (Bind Abbr) u) t H6)) in ((let H9 \def (f_equal C T (\lambda (e: 
-C).(match e return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead 
-_ _ t) \Rightarrow t])) (CHead d1 (Bind Abbr) u) (CHead c (Bind b) t) 
-(clear_gen_bind b c (CHead d1 (Bind Abbr) u) t H6)) in (\lambda (H10: (eq B 
-Abbr b)).(\lambda (H11: (eq C d1 c)).(\lambda (c2: C).(\lambda (H12: (csub3 g 
-c1 c2)).(let H13 \def (eq_ind_r T t (\lambda (t: T).(drop n O c1 (CHead c 
-(Bind b) t))) H5 u H9) in (let H14 \def (eq_ind_r B b (\lambda (b: B).(drop n 
-O c1 (CHead c (Bind b) u))) H13 Abbr H10) in (let H15 \def (eq_ind_r C c 
-(\lambda (c: C).(drop n O c1 (CHead c (Bind Abbr) u))) H14 d1 H11) in 
-(ex2_ind C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n O c2 
-(CHead d2 (Bind Abbr) u))) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
-(d2: C).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (x0: C).(\lambda 
-(H16: (csub3 g d1 x0)).(\lambda (H17: (drop n O c2 (CHead x0 (Bind Abbr) 
-u))).(ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n 
-c2 (CHead d2 (Bind Abbr) u))) x0 H16 (getl_intro n c2 (CHead x0 (Bind Abbr) 
-u) (CHead x0 (Bind Abbr) u) H17 (clear_bind Abbr x0 u)))))) (csub3_drop_abbr 
-g n c1 c2 H12 d1 u H15)))))))))) H8)) H7)))) | (Flat f) \Rightarrow (\lambda 
-(H5: (drop n O c1 (CHead c (Flat f) t))).(\lambda (H6: (clear (CHead c (Flat 
-f) t) (CHead d1 (Bind Abbr) u))).(let H7 \def H5 in (unintro C c1 (\lambda 
-(c0: C).((drop n O c0 (CHead c (Flat f) t)) \to (\forall (c2: C).((csub3 g c0 
-c2) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 
-(CHead d2 (Bind Abbr) u)))))))) (nat_ind (\lambda (n0: nat).(\forall (x0: 
-C).((drop n0 O x0 (CHead c (Flat f) t)) \to (\forall (c2: C).((csub3 g x0 c2) 
-\to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n0 c2 
-(CHead d2 (Bind Abbr) u))))))))) (\lambda (x0: C).(\lambda (H8: (drop O O x0 
-(CHead c (Flat f) t))).(\lambda (c2: C).(\lambda (H9: (csub3 g x0 c2)).(let 
-H10 \def (eq_ind C x0 (\lambda (c: C).(csub3 g c c2)) H9 (CHead c (Flat f) t) 
-(drop_gen_refl x0 (CHead c (Flat f) t) H8)) in (let H_y \def (clear_flat c 
-(CHead d1 (Bind Abbr) u) (clear_gen_flat f c (CHead d1 (Bind Abbr) u) t H6) f 
-t) in (let H11 \def (csub3_clear_conf g (CHead c (Flat f) t) c2 H10 (CHead d1 
-(Bind Abbr) u) H_y) in (ex2_ind C (\lambda (e2: C).(csub3 g (CHead d1 (Bind 
-Abbr) u) e2)) (\lambda (e2: C).(clear c2 e2)) (ex2 C (\lambda (d2: C).(csub3 
-g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda 
-(x1: C).(\lambda (H12: (csub3 g (CHead d1 (Bind Abbr) u) x1)).(\lambda (H13: 
-(clear c2 x1)).(let H14 \def (csub3_gen_abbr g d1 x1 u H12) in (ex2_ind C 
-(\lambda (e2: C).(eq C x1 (CHead e2 (Bind Abbr) u))) (\lambda (e2: C).(csub3 
-g d1 e2)) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl O 
-c2 (CHead d2 (Bind Abbr) u)))) (\lambda (x2: C).(\lambda (H15: (eq C x1 
-(CHead x2 (Bind Abbr) u))).(\lambda (H16: (csub3 g d1 x2)).(let H17 \def 
-(eq_ind C x1 (\lambda (c: C).(clear c2 c)) H13 (CHead x2 (Bind Abbr) u) H15) 
-in (ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl O c2 
-(CHead d2 (Bind Abbr) u))) x2 H16 (getl_intro O c2 (CHead x2 (Bind Abbr) u) 
-c2 (drop_refl c2) H17)))))) H14))))) H11)))))))) (\lambda (n0: nat).(\lambda 
-(H8: ((\forall (x: C).((drop n0 O x (CHead c (Flat f) t)) \to (\forall (c2: 
-C).((csub3 g x c2) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: 
-C).(getl n0 c2 (CHead d2 (Bind Abbr) u)))))))))).(\lambda (x0: C).(\lambda 
-(H9: (drop (S n0) O x0 (CHead c (Flat f) t))).(\lambda (c2: C).(\lambda (H10: 
-(csub3 g x0 c2)).(let H11 \def (drop_clear x0 (CHead c (Flat f) t) n0 H9) in 
-(ex2_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear x0 
-(CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: 
-T).(drop n0 O e (CHead c (Flat f) t))))) (ex2 C (\lambda (d2: C).(csub3 g d1 
-d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda 
-(x1: B).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H12: (clear x0 (CHead x2 
-(Bind x1) x3))).(\lambda (H13: (drop n0 O x2 (CHead c (Flat f) t))).(let H14 
-\def (csub3_clear_conf g x0 c2 H10 (CHead x2 (Bind x1) x3) H12) in (ex2_ind C 
-(\lambda (e2: C).(csub3 g (CHead x2 (Bind x1) x3) e2)) (\lambda (e2: 
-C).(clear c2 e2)) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: 
-C).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (x4: C).(\lambda 
-(H15: (csub3 g (CHead x2 (Bind x1) x3) x4)).(\lambda (H16: (clear c2 
-x4)).(let H17 \def (csub3_gen_bind g x1 x2 x4 x3 H15) in (ex2_3_ind B C T 
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C x4 (CHead e2 (Bind 
-b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g x2 
-e2)))) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl (S n0) 
-c2 (CHead d2 (Bind Abbr) u)))) (\lambda (x5: B).(\lambda (x6: C).(\lambda 
-(x7: T).(\lambda (H18: (eq C x4 (CHead x6 (Bind x5) x7))).(\lambda (H19: 
-(csub3 g x2 x6)).(let H20 \def (eq_ind C x4 (\lambda (c: C).(clear c2 c)) H16 
-(CHead x6 (Bind x5) x7) H18) in (let H21 \def (H8 x2 H13 x6 H19) in (ex2_ind 
-C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n0 x6 (CHead d2 
-(Bind Abbr) u))) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: 
-C).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (x8: C).(\lambda 
-(H22: (csub3 g d1 x8)).(\lambda (H23: (getl n0 x6 (CHead x8 (Bind Abbr) 
-u))).(ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl (S 
-n0) c2 (CHead d2 (Bind Abbr) u))) x8 H22 (getl_clear_bind x5 c2 x6 x7 H20 
-(CHead x8 (Bind Abbr) u) n0 H23))))) H21)))))))) H17))))) H14))))))) 
-H11)))))))) n) H7))))]) H3 H4)))]) H1 H2)))) H0))))))).
-
-theorem csub3_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).((csub3 g 
-c1 c2) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: 
-C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n 
-c2 (CHead d2 (Bind Abbr) u)))) (\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).((csub3 g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 
-d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T 
-(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda 
-(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\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))).((match x return 
-(\lambda (c: C).((drop n O c1 c) \to ((clear c (CHead d1 (Bind Abst) t)) \to 
-(\forall (c2: C).((csub3 g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 
-d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T 
-(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda 
-(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: 
-T).(ty3 g d2 u t)))))))))) with [(CSort n0) \Rightarrow (\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).((csub3 
-g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: 
-C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n 
-c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u 
-t)))))))))) | (CHead c k t0) \Rightarrow (\lambda (H3: (drop n O c1 (CHead c 
-k t0))).(\lambda (H4: (clear (CHead c k t0) (CHead d1 (Bind Abst) 
-t))).((match k return (\lambda (k0: K).((drop n O c1 (CHead c k0 t0)) \to 
-((clear (CHead c k0 t0) (CHead d1 (Bind Abst) t)) \to (\forall (c2: 
-C).((csub3 g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
-(d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n 
-c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u 
-t)))))))))) with [(Bind b) \Rightarrow (\lambda (H5: (drop n O c1 (CHead c 
-(Bind b) t0))).(\lambda (H6: (clear (CHead c (Bind b) t0) (CHead d1 (Bind 
-Abst) t))).(let H7 \def (f_equal C C (\lambda (e: C).(match e return (\lambda 
-(_: C).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _) \Rightarrow c])) 
-(CHead d1 (Bind Abst) t) (CHead c (Bind b) t0) (clear_gen_bind b c (CHead d1 
-(Bind Abst) t) t0 H6)) in ((let H8 \def (f_equal C B (\lambda (e: C).(match e 
-return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abst | (CHead _ k _) 
-\Rightarrow (match k return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | 
-(Flat _) \Rightarrow Abst])])) (CHead d1 (Bind Abst) t) (CHead c (Bind b) t0) 
-(clear_gen_bind b c (CHead d1 (Bind Abst) t) t0 H6)) in ((let H9 \def 
-(f_equal C T (\lambda (e: C).(match e return (\lambda (_: C).T) with [(CSort 
-_) \Rightarrow t | (CHead _ _ t) \Rightarrow t])) (CHead d1 (Bind Abst) t) 
-(CHead c (Bind b) t0) (clear_gen_bind b c (CHead d1 (Bind Abst) t) t0 H6)) in 
-(\lambda (H10: (eq B Abst b)).(\lambda (H11: (eq C d1 c)).(\lambda (c2: 
-C).(\lambda (H12: (csub3 g c1 c2)).(let H13 \def (eq_ind_r T t0 (\lambda (t: 
-T).(drop n O c1 (CHead c (Bind b) t))) H5 t H9) in (let H14 \def (eq_ind_r B 
-b (\lambda (b: B).(drop n O c1 (CHead c (Bind b) t))) H13 Abst H10) in (let 
-H15 \def (eq_ind_r C c (\lambda (c: C).(drop n O c1 (CHead c (Bind Abst) t))) 
-H14 d1 H11) in (or_ind (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: 
-C).(drop n O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop n 
-O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u 
-t)))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n 
-c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
-T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 
-(Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) 
-(\lambda (H16: (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: 
-C).(drop n O c2 (CHead d2 (Bind Abst) t))))).(ex2_ind C (\lambda (d2: 
-C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abst) t))) 
-(or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 
-(CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
-T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 
-(Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) 
-(\lambda (x0: C).(\lambda (H17: (csub3 g d1 x0)).(\lambda (H18: (drop n O c2 
-(CHead x0 (Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: C).(csub3 g d1 
-d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T 
-(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda 
-(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: 
-T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
-(d2: C).(getl n c2 (CHead d2 (Bind Abst) t))) x0 H17 (getl_intro n c2 (CHead 
-x0 (Bind Abst) t) (CHead x0 (Bind Abst) t) H18 (clear_bind Abst x0 t))))))) 
-H16)) (\lambda (H16: (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 
-d2))) (\lambda (d2: C).(\lambda (u: T).(drop n O c2 (CHead d2 (Bind Abbr) 
-u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))).(ex3_2_ind C T 
-(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda 
-(u: T).(drop n O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: 
-T).(ty3 g d2 u t))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
-(d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n 
-c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u 
-t))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H17: (csub3 g d1 
-x0)).(\lambda (H18: (drop n O c2 (CHead x0 (Bind Abbr) x1))).(\lambda (H19: 
-(ty3 g x0 x1 t)).(or_intror (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
-(d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n 
-c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u 
-t)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) 
-(\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) 
-(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))) x0 x1 H17 (getl_intro n c2 
-(CHead x0 (Bind Abbr) x1) (CHead x0 (Bind Abbr) x1) H18 (clear_bind Abbr x0 
-x1)) H19))))))) H16)) (csub3_drop_abst g n c1 c2 H12 d1 t H15)))))))))) H8)) 
-H7)))) | (Flat f) \Rightarrow (\lambda (H5: (drop n O c1 (CHead c (Flat f) 
-t0))).(\lambda (H6: (clear (CHead c (Flat f) t0) (CHead d1 (Bind Abst) 
-t))).(let H7 \def H5 in (unintro C c1 (\lambda (c0: C).((drop n O c0 (CHead c 
-(Flat f) t0)) \to (\forall (c2: C).((csub3 g c0 c2) \to (or (ex2 C (\lambda 
-(d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) 
-t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda 
-(d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: 
-C).(\lambda (u: T).(ty3 g d2 u t))))))))) (nat_ind (\lambda (n0: 
-nat).(\forall (x0: C).((drop n0 O x0 (CHead c (Flat f) t0)) \to (\forall (c2: 
-C).((csub3 g x0 c2) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
-(d2: C).(getl n0 c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl 
-n0 c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 
-u t)))))))))) (\lambda (x0: C).(\lambda (H8: (drop O O x0 (CHead c (Flat f) 
-t0))).(\lambda (c2: C).(\lambda (H9: (csub3 g x0 c2)).(let H10 \def (eq_ind C 
-x0 (\lambda (c: C).(csub3 g c c2)) H9 (CHead c (Flat f) t0) (drop_gen_refl x0 
-(CHead c (Flat f) t0) H8)) in (let H_y \def (clear_flat c (CHead d1 (Bind 
-Abst) t) (clear_gen_flat f c (CHead d1 (Bind Abst) t) t0 H6) f t0) in (let 
-H11 \def (csub3_clear_conf g (CHead c (Flat f) t0) c2 H10 (CHead d1 (Bind 
-Abst) t) H_y) in (ex2_ind C (\lambda (e2: C).(csub3 g (CHead d1 (Bind Abst) 
-t) e2)) (\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(csub3 g 
-d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T 
-(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda 
-(u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: 
-T).(ty3 g d2 u t))))) (\lambda (x1: C).(\lambda (H12: (csub3 g (CHead d1 
-(Bind Abst) t) x1)).(\lambda (H13: (clear c2 x1)).(let H14 \def 
-(csub3_gen_abst g d1 x1 t H12) in (or_ind (ex2 C (\lambda (e2: C).(eq C x1 
-(CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csub3 g d1 e2))) (ex3_2 C T 
-(\lambda (e2: C).(\lambda (v2: T).(eq C x1 (CHead e2 (Bind Abbr) v2)))) 
-(\lambda (e2: C).(\lambda (_: T).(csub3 g d1 e2))) (\lambda (e2: C).(\lambda 
-(v2: T).(ty3 g e2 v2 t)))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) 
-(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda 
-(d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: 
-T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: 
-T).(ty3 g d2 u t))))) (\lambda (H15: (ex2 C (\lambda (e2: C).(eq C x1 (CHead 
-e2 (Bind Abst) t))) (\lambda (e2: C).(csub3 g d1 e2)))).(ex2_ind C (\lambda 
-(e2: C).(eq C x1 (CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csub3 g d1 e2)) 
-(or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl O c2 
-(CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
-T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 
-(Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) 
-(\lambda (x2: C).(\lambda (H16: (eq C x1 (CHead x2 (Bind Abst) t))).(\lambda 
-(H17: (csub3 g d1 x2)).(let H18 \def (eq_ind C x1 (\lambda (c: C).(clear c2 
-c)) H13 (CHead x2 (Bind Abst) t) H16) in (or_introl (ex2 C (\lambda (d2: 
-C).(csub3 g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) 
-(ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: 
-C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: 
-C).(\lambda (u: T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csub3 g 
-d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t))) x2 H17 
-(getl_intro O c2 (CHead x2 (Bind Abst) t) c2 (drop_refl c2) H18))))))) H15)) 
-(\lambda (H15: (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C x1 (CHead 
-e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g d1 e2))) 
-(\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 t))))).(ex3_2_ind C T (\lambda 
-(e2: C).(\lambda (v2: T).(eq C x1 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: 
-C).(\lambda (_: T).(csub3 g d1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g 
-e2 v2 t))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: 
-C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O 
-c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u 
-t))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (H16: (eq C x1 (CHead x2 
-(Bind Abbr) x3))).(\lambda (H17: (csub3 g d1 x2)).(\lambda (H18: (ty3 g x2 x3 
-t)).(let H19 \def (eq_ind C x1 (\lambda (c: C).(clear c2 c)) H13 (CHead x2 
-(Bind Abbr) x3) H16) in (or_intror (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) 
-(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda 
-(d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: 
-T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: 
-T).(ty3 g d2 u t)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csub3 
-g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) 
-u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))) x2 x3 H17 (getl_intro 
-O c2 (CHead x2 (Bind Abbr) x3) c2 (drop_refl c2) H19) H18)))))))) H15)) 
-H14))))) H11)))))))) (\lambda (n0: nat).(\lambda (H8: ((\forall (x: C).((drop 
-n0 O x (CHead c (Flat f) t0)) \to (\forall (c2: C).((csub3 g x c2) \to (or 
-(ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n0 c2 (CHead 
-d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 
-d2))) (\lambda (d2: C).(\lambda (u: T).(getl n0 c2 (CHead d2 (Bind Abbr) 
-u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))))))))).(\lambda (x0: 
-C).(\lambda (H9: (drop (S n0) O x0 (CHead c (Flat f) t0))).(\lambda (c2: 
-C).(\lambda (H10: (csub3 g x0 c2)).(let H11 \def (drop_clear x0 (CHead c 
-(Flat f) t0) n0 H9) in (ex2_3_ind B C T (\lambda (b: B).(\lambda (e: 
-C).(\lambda (v: T).(clear x0 (CHead e (Bind b) v))))) (\lambda (_: 
-B).(\lambda (e: C).(\lambda (_: T).(drop n0 O e (CHead c (Flat f) t0))))) (or 
-(ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 
-(CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
-T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead 
-d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) 
-(\lambda (x1: B).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H12: (clear x0 
-(CHead x2 (Bind x1) x3))).(\lambda (H13: (drop n0 O x2 (CHead c (Flat f) 
-t0))).(let H14 \def (csub3_clear_conf g x0 c2 H10 (CHead x2 (Bind x1) x3) 
-H12) in (ex2_ind C (\lambda (e2: C).(csub3 g (CHead x2 (Bind x1) x3) e2)) 
-(\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) 
-(\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T 
-(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda 
-(u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda 
-(u: T).(ty3 g d2 u t))))) (\lambda (x4: C).(\lambda (H15: (csub3 g (CHead x2 
-(Bind x1) x3) x4)).(\lambda (H16: (clear c2 x4)).(let H17 \def 
-(csub3_gen_bind g x1 x2 x4 x3 H15) in (ex2_3_ind B C T (\lambda (b2: 
-B).(\lambda (e2: C).(\lambda (v2: T).(eq C x4 (CHead e2 (Bind b2) v2))))) 
-(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g x2 e2)))) (or (ex2 
-C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead 
-d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 
-d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) 
-u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x5: 
-B).(\lambda (x6: C).(\lambda (x7: T).(\lambda (H18: (eq C x4 (CHead x6 (Bind 
-x5) x7))).(\lambda (H19: (csub3 g x2 x6)).(let H20 \def (eq_ind C x4 (\lambda 
-(c: C).(clear c2 c)) H16 (CHead x6 (Bind x5) x7) H18) in (let H21 \def (H8 x2 
-H13 x6 H19) in (or_ind (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: 
-C).(getl n0 x6 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl 
-n0 x6 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 
-u t)))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl 
-(S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda 
-(_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 
-(CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u 
-t))))) (\lambda (H22: (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: 
-C).(getl n0 x6 (CHead d2 (Bind Abst) t))))).(ex2_ind C (\lambda (d2: 
-C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n0 x6 (CHead d2 (Bind Abst) t))) 
-(or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 
-(CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
-T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead 
-d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) 
-(\lambda (x8: C).(\lambda (H23: (csub3 g d1 x8)).(\lambda (H24: (getl n0 x6 
-(CHead x8 (Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: C).(csub3 g d1 
-d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T 
-(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda 
-(u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda 
-(u: T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) 
-(\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t))) x8 H23 
-(getl_clear_bind x5 c2 x6 x7 H20 (CHead x8 (Bind Abst) t) n0 H24)))))) H22)) 
-(\lambda (H22: (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) 
-(\lambda (d2: C).(\lambda (u: T).(getl n0 x6 (CHead d2 (Bind Abbr) u)))) 
-(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))).(ex3_2_ind C T (\lambda 
-(d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: 
-T).(getl n0 x6 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: 
-T).(ty3 g d2 u t))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
-(d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl 
-(S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g 
-d2 u t))))) (\lambda (x8: C).(\lambda (x9: T).(\lambda (H23: (csub3 g d1 
-x8)).(\lambda (H24: (getl n0 x6 (CHead x8 (Bind Abbr) x9))).(\lambda (H25: 
-(ty3 g x8 x9 t)).(or_intror (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
-(d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl 
-(S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g 
-d2 u t)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 
-d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) 
-u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))) x8 x9 H23 
-(getl_clear_bind x5 c2 x6 x7 H20 (CHead x8 (Bind Abbr) x9) n0 H24) H25))))))) 
-H22)) H21)))))))) H17))))) H14))))))) H11)))))))) n) H7))))]) H3 H4)))]) H1 
-H2)))) H0))))))).
-
-theorem csub3_pr2:
- \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pr2 c1 
-t1 t2) \to (\forall (c2: C).((csub3 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).((csub3 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 (_: (csub3 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: (csub3 g c c2)).(let H4 \def (csub3_getl_abbr g c d u i H0 
-c2 H3) in (ex2_ind C (\lambda (d2: C).(csub3 g d d2)) (\lambda (d2: C).(getl 
-i c2 (CHead d2 (Bind Abbr) u))) (pr2 c2 t3 t) (\lambda (x: C).(\lambda (_: 
-(csub3 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 csub3_pc3:
- \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pc3 c1 
-t1 t2) \to (\forall (c2: C).((csub3 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).((csub3 g c1 c2) \to (pc3 c2 t t0))))) (\lambda (t: 
-T).(\lambda (c2: C).(\lambda (_: (csub3 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).((csub3 g c1 
-c2) \to (pc3 c2 t3 t4))))).(\lambda (c2: C).(\lambda (H3: (csub3 g c1 
-c2)).(pc3_pr2_u c2 t3 t0 (csub3_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).((csub3 g c1 c2) \to (pc3 c2 t0 t4))))).(\lambda (c2: C).(\lambda 
-(H3: (csub3 g c1 c2)).(pc3_t t0 c2 t3 (pc3_pr2_x c2 t3 t0 (csub3_pr2 g c1 t0 
-t3 H0 c2 H3)) t4 (H2 c2 H3)))))))))) t1 t2 H))))).
-
-theorem csub3_ty3:
- \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c1 
-t1 t2) \to (\forall (c2: C).((csub3 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).((csub3 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).((csub3 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).((csub3 g c c2) \to (ty3 g c2 u t3))))).(\lambda (H4: (pc3 c t3 
-t0)).(\lambda (c2: C).(\lambda (H5: (csub3 g c c2)).(ty3_conv g c2 t0 t (H1 
-c2 H5) u t3 (H3 c2 H5) (csub3_pc3 g c t3 t0 H4 c2 H5)))))))))))))) (\lambda 
-(c: C).(\lambda (m: nat).(\lambda (c2: C).(\lambda (_: (csub3 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).((csub3 g 
-d c2) \to (ty3 g c2 u t))))).(\lambda (c2: C).(\lambda (H3: (csub3 g c 
-c2)).(let H4 \def (csub3_getl_abbr g c d u n H0 c2 H3) in (ex2_ind C (\lambda 
-(d2: C).(csub3 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: 
-(csub3 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).((csub3 g d c2) \to (ty3 g c2 u t))))).(\lambda (c2: 
-C).(\lambda (H3: (csub3 g c c2)).(let H4 \def (csub3_getl_abst g c d u n H0 
-c2 H3) in (or_ind (ex2 C (\lambda (d2: C).(csub3 g d d2)) (\lambda (d2: 
-C).(getl n c2 (CHead d2 (Bind Abst) u)))) (ex3_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csub3 g d d2))) (\lambda (d2: C).(\lambda (u0: T).(getl n 
-c2 (CHead d2 (Bind Abbr) u0)))) (\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).(csub3 g d d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) 
-u))))).(ex2_ind C (\lambda (d2: C).(csub3 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: (csub3 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: (ex3_2 
-C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d d2))) (\lambda (d2: 
-C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: 
-C).(\lambda (u0: T).(ty3 g d2 u0 u))))).(ex3_2_ind C T (\lambda (d2: 
-C).(\lambda (_: T).(csub3 g d d2))) (\lambda (d2: C).(\lambda (u0: T).(getl n 
-c2 (CHead d2 (Bind Abbr) u0)))) (\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 (_: (csub3 g d x0)).(\lambda (H7: (getl n c2 (CHead x0 (Bind 
-Abbr) x1))).(\lambda (H8: (ty3 g x0 x1 u)).(ty3_abbr g n c2 x0 x1 H7 u 
-H8)))))) H5)) H4)))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (t: 
-T).(\lambda (_: (ty3 g c u t)).(\lambda (H1: ((\forall (c2: C).((csub3 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).((csub3 g (CHead c (Bind b) u) c2) \to (ty3 g c2 t0 t3))))).(\lambda 
-(t4: T).(\lambda (_: (ty3 g (CHead c (Bind b) u) t3 t4)).(\lambda (H5: 
-((\forall (c2: C).((csub3 g (CHead c (Bind b) u) c2) \to (ty3 g c2 t3 
-t4))))).(\lambda (c2: C).(\lambda (H6: (csub3 g c c2)).(ty3_bind g c2 u t (H1 
-c2 H6) b t0 t3 (H3 (CHead c2 (Bind b) u) (csub3_head g c c2 H6 (Bind b) u)) 
-t4 (H5 (CHead c2 (Bind b) u) (csub3_head g c c2 H6 (Bind b) 
-u)))))))))))))))))) (\lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\lambda 
-(_: (ty3 g c w u)).(\lambda (H1: ((\forall (c2: C).((csub3 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).((csub3 g c c2) \to (ty3 g 
-c2 v (THead (Bind Abst) u t)))))).(\lambda (c2: C).(\lambda (H4: (csub3 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).((csub3 g c c2) \to (ty3 g c2 t0 t3))))).(\lambda (t4: 
-T).(\lambda (_: (ty3 g c t3 t4)).(\lambda (H3: ((\forall (c2: C).((csub3 g c 
-c2) \to (ty3 g c2 t3 t4))))).(\lambda (c2: C).(\lambda (H4: (csub3 g c 
-c2)).(ty3_cast g c2 t0 t3 (H1 c2 H4) t4 (H3 c2 H4)))))))))))) c1 t1 t2 H))))).
-
-theorem csub3_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)).(csub3_ty3 g (CHead c (Bind Abst) v) t1 t2 H0 (CHead 
-c (Bind Abbr) u) (csub3_abst g c c (csub3_refl g c) u v H))))))))).
-
-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 (t2: T).((pr0 u t2) \to (ty3 g c2 t2 
-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 (t4: 
-T).(\lambda (H4: (ty3 g (CHead c (Bind b) u) t3 t4)).(\lambda (H5: ((\forall 
-(c2: C).((wcpr0 (CHead c (Bind b) u) c2) \to (\forall (t2: T).((pr0 t3 t2) 
-\to (ty3 g c2 t2 t4))))))).(\lambda (c2: C).(\lambda (H6: (wcpr0 c 
-c2)).(\lambda (t5: T).(\lambda (H7: (pr0 (THead (Bind b) u t2) t5)).(let H8 
-\def (match H7 return (\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr0 t 
-t0)).((eq T t (THead (Bind b) u t2)) \to ((eq T t0 t5) \to (ty3 g c2 t5 
-(THead (Bind b) u t3))))))) with [(pr0_refl t4) \Rightarrow (\lambda (H7: (eq 
-T t4 (THead (Bind b) u t2))).(\lambda (H8: (eq T t4 t5)).(eq_ind T (THead 
-(Bind b) u t2) (\lambda (t: T).((eq T t t5) \to (ty3 g c2 t5 (THead (Bind b) 
-u t3)))) (\lambda (H9: (eq T (THead (Bind b) u t2) t5)).(eq_ind T (THead 
-(Bind b) u t2) (\lambda (t: T).(ty3 g c2 t (THead (Bind b) u t3))) (ty3_bind 
-g c2 u t0 (H1 c2 H6 u (pr0_refl u)) b t2 t3 (H3 (CHead c2 (Bind b) u) 
-(wcpr0_comp c c2 H6 u u (pr0_refl u) (Bind b)) t2 (pr0_refl t2)) t4 (H5 
-(CHead c2 (Bind b) u) (wcpr0_comp c c2 H6 u u (pr0_refl u) (Bind b)) t3 
-(pr0_refl t3))) t5 H9)) t4 (sym_eq T t4 (THead (Bind b) u t2) H7) H8))) | 
-(pr0_comp u1 u2 H7 t4 t5 H8 k) \Rightarrow (\lambda (H9: (eq T (THead k u1 
-t4) (THead (Bind b) u t2))).(\lambda (H10: (eq T (THead k u2 t5) t5)).((let 
-H11 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow t4 | (TLRef _) \Rightarrow t4 | (THead _ _ t) 
-\Rightarrow t])) (THead k u1 t4) (THead (Bind b) u t2) H9) in ((let H12 \def 
-(f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort 
-_) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t _) \Rightarrow t])) 
-(THead k u1 t4) (THead (Bind b) u t2) H9) in ((let H13 \def (f_equal T K 
-(\lambda (e: T).(match e return (\lambda (_: T).K) with [(TSort _) 
-\Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) 
-(THead k u1 t4) (THead (Bind b) u t2) H9) in (eq_ind K (Bind b) (\lambda (k0: 
-K).((eq T u1 u) \to ((eq T t4 t2) \to ((eq T (THead k0 u2 t5) t5) \to ((pr0 
-u1 u2) \to ((pr0 t4 t5) \to (ty3 g c2 t5 (THead (Bind b) u t3)))))))) 
-(\lambda (H14: (eq T u1 u)).(eq_ind T u (\lambda (t: T).((eq T t4 t2) \to 
-((eq T (THead (Bind b) u2 t5) t5) \to ((pr0 t u2) \to ((pr0 t4 t5) \to (ty3 g 
-c2 t5 (THead (Bind b) u t3))))))) (\lambda (H15: (eq T t4 t2)).(eq_ind T t2 
-(\lambda (t: T).((eq T (THead (Bind b) u2 t5) t5) \to ((pr0 u u2) \to ((pr0 t 
-t5) \to (ty3 g c2 t5 (THead (Bind b) u t3)))))) (\lambda (H16: (eq T (THead 
-(Bind b) u2 t5) t5)).(eq_ind T (THead (Bind b) u2 t5) (\lambda (t: T).((pr0 u 
-u2) \to ((pr0 t2 t5) \to (ty3 g c2 t (THead (Bind b) u t3))))) (\lambda (H17: 
-(pr0 u u2)).(\lambda (H18: (pr0 t2 t5)).(ex_ind T (\lambda (t: T).(ty3 g 
-(CHead c2 (Bind b) u) t4 t)) (ty3 g c2 (THead (Bind b) u2 t5) (THead (Bind b) 
-u t3)) (\lambda (x: T).(\lambda (H19: (ty3 g (CHead c2 (Bind b) u) t4 
-x)).(ex_ind T (\lambda (t: T).(ty3 g (CHead c2 (Bind b) u2) t3 t)) (ty3 g c2 
-(THead (Bind b) u2 t5) (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (H20: 
-(ty3 g (CHead c2 (Bind b) u2) t3 x0)).(ty3_conv g c2 (THead (Bind b) u t3) 
-(THead (Bind b) u t4) (ty3_bind g c2 u t0 (H1 c2 H6 u (pr0_refl u)) b t3 t4 
-(H5 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H6 u u (pr0_refl u) (Bind b)) t3 
-(pr0_refl t3)) x H19) (THead (Bind b) u2 t5) (THead (Bind b) u2 t3) (ty3_bind 
-g c2 u2 t0 (H1 c2 H6 u2 H17) b t5 t3 (H3 (CHead c2 (Bind b) u2) (wcpr0_comp c 
-c2 H6 u u2 H17 (Bind b)) t5 H18) x0 H20) (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 H17) (Bind b) 
-t3))))) (ty3_correct g (CHead c2 (Bind b) u2) t5 t3 (H3 (CHead c2 (Bind b) 
-u2) (wcpr0_comp c c2 H6 u u2 H17 (Bind b)) t5 H18))))) (ty3_correct g (CHead 
-c2 (Bind b) u) t3 t4 (H5 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H6 u u 
-(pr0_refl u) (Bind b)) t3 (pr0_refl t3)))))) t5 H16)) t4 (sym_eq T t4 t2 
-H15))) u1 (sym_eq T u1 u H14))) k (sym_eq K k (Bind b) H13))) H12)) H11)) H10 
-H7 H8))) | (pr0_beta u0 v1 v2 H7 t4 t5 H8) \Rightarrow (\lambda (H9: (eq T 
-(THead (Flat Appl) v1 (THead (Bind Abst) u0 t4)) (THead (Bind b) u 
-t2))).(\lambda (H10: (eq T (THead (Bind Abbr) v2 t5) t5)).((let H11 \def 
-(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t4)) (\lambda (e: 
-T).(match e return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | 
-(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return 
-(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow 
-True])])) I (THead (Bind b) u t2) H9) in (False_ind ((eq T (THead (Bind Abbr) 
-v2 t5) t5) \to ((pr0 v1 v2) \to ((pr0 t4 t5) \to (ty3 g c2 t5 (THead (Bind b) 
-u t3))))) H11)) H10 H7 H8))) | (pr0_upsilon b0 H7 v1 v2 H8 u1 u2 H9 t4 t5 
-H10) \Rightarrow (\lambda (H11: (eq T (THead (Flat Appl) v1 (THead (Bind b0) 
-u1 t4)) (THead (Bind b) u t2))).(\lambda (H12: (eq T (THead (Bind b0) u2 
-(THead (Flat Appl) (lift (S O) O v2) t5)) t5)).((let H13 \def (eq_ind T 
-(THead (Flat Appl) v1 (THead (Bind b0) u1 t4)) (\lambda (e: T).(match e 
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: 
-K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I 
-(THead (Bind b) u t2) H11) in (False_ind ((eq T (THead (Bind b0) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t5)) t5) \to ((not (eq B b0 Abst)) \to ((pr0 v1 
-v2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ty3 g c2 t5 (THead (Bind b) u 
-t3))))))) H13)) H12 H7 H8 H9 H10))) | (pr0_delta u1 u2 H7 t4 t5 H8 w H9) 
-\Rightarrow (\lambda (H10: (eq T (THead (Bind Abbr) u1 t4) (THead (Bind b) u 
-t2))).(\lambda (H11: (eq T (THead (Bind Abbr) u2 w) t5)).((let H12 \def 
-(f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort 
-_) \Rightarrow t4 | (TLRef _) \Rightarrow t4 | (THead _ _ t) \Rightarrow t])) 
-(THead (Bind Abbr) u1 t4) (THead (Bind b) u t2) H10) in ((let H13 \def 
-(f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort 
-_) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t _) \Rightarrow t])) 
-(THead (Bind Abbr) u1 t4) (THead (Bind b) u t2) H10) in ((let H14 \def 
-(f_equal T B (\lambda (e: T).(match e return (\lambda (_: T).B) with [(TSort 
-_) \Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k _ _) \Rightarrow 
-(match k return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) 
-\Rightarrow Abbr])])) (THead (Bind Abbr) u1 t4) (THead (Bind b) u t2) H10) in 
-(eq_ind B Abbr (\lambda (b: B).((eq T u1 u) \to ((eq T t4 t2) \to ((eq T 
-(THead (Bind Abbr) u2 w) t5) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to ((subst0 O 
-u2 t5 w) \to (ty3 g c2 t5 (THead (Bind b) u t3))))))))) (\lambda (H15: (eq T 
-u1 u)).(eq_ind T u (\lambda (t: T).((eq T t4 t2) \to ((eq T (THead (Bind 
-Abbr) u2 w) t5) \to ((pr0 t u2) \to ((pr0 t4 t5) \to ((subst0 O u2 t5 w) \to 
-(ty3 g c2 t5 (THead (Bind Abbr) u t3)))))))) (\lambda (H16: (eq T t4 
-t2)).(eq_ind T t2 (\lambda (t: T).((eq T (THead (Bind Abbr) u2 w) t5) \to 
-((pr0 u u2) \to ((pr0 t t5) \to ((subst0 O u2 t5 w) \to (ty3 g c2 t5 (THead 
-(Bind Abbr) u t3))))))) (\lambda (H17: (eq T (THead (Bind Abbr) u2 w) 
-t5)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t: T).((pr0 u u2) \to ((pr0 
-t2 t5) \to ((subst0 O u2 t5 w) \to (ty3 g c2 t (THead (Bind Abbr) u t3)))))) 
-(\lambda (H18: (pr0 u u2)).(\lambda (H19: (pr0 t2 t5)).(\lambda (H20: (subst0 
-O u2 t5 w)).(let H21 \def (eq_ind_r B b (\lambda (b: B).(\forall (c2: 
-C).((wcpr0 (CHead c (Bind b) u) c2) \to (\forall (t2: T).((pr0 t3 t2) \to 
-(ty3 g c2 t2 t4)))))) H5 Abbr H14) in (let H22 \def (eq_ind_r B b (\lambda 
-(b: B).(ty3 g (CHead c (Bind b) u) t3 t4)) H4 Abbr H14) in (let H23 \def 
-(eq_ind_r B b (\lambda (b: B).(\forall (c2: C).((wcpr0 (CHead c (Bind b) u) 
-c2) \to (\forall (t4: T).((pr0 t2 t4) \to (ty3 g c2 t4 t3)))))) H3 Abbr H14) 
-in (let H24 \def (eq_ind_r B b (\lambda (b: B).(ty3 g (CHead c (Bind b) u) t2 
-t3)) H2 Abbr H14) in (ex_ind T (\lambda (t: T).(ty3 g (CHead c2 (Bind Abbr) 
-u) t4 t)) (ty3 g c2 (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u t3)) 
-(\lambda (x: T).(\lambda (H25: (ty3 g (CHead c2 (Bind Abbr) u) t4 x)).(ex_ind 
-T (\lambda (t: T).(ty3 g (CHead c2 (Bind Abbr) u2) t3 t)) (ty3 g c2 (THead 
-(Bind Abbr) u2 w) (THead (Bind Abbr) u t3)) (\lambda (x0: T).(\lambda (H26: 
-(ty3 g (CHead c2 (Bind Abbr) u2) t3 x0)).(ty3_conv g c2 (THead (Bind Abbr) u 
-t3) (THead (Bind Abbr) u t4) (ty3_bind g c2 u t0 (H1 c2 H6 u (pr0_refl u)) 
-Abbr t3 t4 (H21 (CHead c2 (Bind Abbr) u) (wcpr0_comp c c2 H6 u u (pr0_refl u) 
-(Bind Abbr)) t3 (pr0_refl t3)) x H25) (THead (Bind Abbr) u2 w) (THead (Bind 
-Abbr) u2 t3) (ty3_bind g c2 u2 t0 (H1 c2 H6 u2 H18) Abbr w t3 (ty3_subst0 g 
-(CHead c2 (Bind Abbr) u2) t5 t3 (H23 (CHead c2 (Bind Abbr) u2) (wcpr0_comp c 
-c2 H6 u u2 H18 (Bind Abbr)) t5 H19) c2 u2 O (getl_refl Abbr c2 u2) w H20) x0 
-H26) (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 H18) (Bind Abbr) t3))))) (ty3_correct g 
-(CHead c2 (Bind Abbr) u2) t5 t3 (H23 (CHead c2 (Bind Abbr) u2) (wcpr0_comp c 
-c2 H6 u u2 H18 (Bind Abbr)) t5 H19))))) (ty3_correct g (CHead c2 (Bind Abbr) 
-u) t3 t4 (H21 (CHead c2 (Bind Abbr) u) (wcpr0_comp c c2 H6 u u (pr0_refl u) 
-(Bind Abbr)) t3 (pr0_refl t3))))))))))) t5 H17)) t4 (sym_eq T t4 t2 H16))) u1 
-(sym_eq T u1 u H15))) b H14)) H13)) H12)) H11 H7 H8 H9))) | (pr0_zeta b0 H7 
-t4 t5 H8 u0) \Rightarrow (\lambda (H9: (eq T (THead (Bind b0) u0 (lift (S O) 
-O t4)) (THead (Bind b) u t2))).(\lambda (H10: (eq T t5 t5)).((let H11 \def 
-(f_equal T T (\lambda (e: T).(match e 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 u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) 
-t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t4) | (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 u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) 
-t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t4) | (THead _ _ t) 
-\Rightarrow t])) (THead (Bind b0) u0 (lift (S O) O t4)) (THead (Bind b) u t2) 
-H9) in ((let H12 \def (f_equal T T (\lambda (e: T).(match e return (\lambda 
-(_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead 
-_ t _) \Rightarrow t])) (THead (Bind b0) u0 (lift (S O) O t4)) (THead (Bind 
-b) u t2) H9) in ((let H13 \def (f_equal T B (\lambda (e: T).(match e return 
-(\lambda (_: T).B) with [(TSort _) \Rightarrow b0 | (TLRef _) \Rightarrow b0 
-| (THead k _ _) \Rightarrow (match k return (\lambda (_: K).B) with [(Bind b) 
-\Rightarrow b | (Flat _) \Rightarrow b0])])) (THead (Bind b0) u0 (lift (S O) 
-O t4)) (THead (Bind b) u t2) H9) in (eq_ind B b (\lambda (b1: B).((eq T u0 u) 
-\to ((eq T (lift (S O) O t4) t2) \to ((eq T t5 t5) \to ((not (eq B b1 Abst)) 
-\to ((pr0 t4 t5) \to (ty3 g c2 t5 (THead (Bind b) u t3)))))))) (\lambda (H14: 
-(eq T u0 u)).(eq_ind T u (\lambda (_: T).((eq T (lift (S O) O t4) t2) \to 
-((eq T t5 t5) \to ((not (eq B b Abst)) \to ((pr0 t4 t5) \to (ty3 g c2 t5 
-(THead (Bind b) u t3))))))) (\lambda (H15: (eq T (lift (S O) O t4) 
-t2)).(eq_ind T (lift (S O) O t4) (\lambda (_: T).((eq T t5 t5) \to ((not (eq 
-B b Abst)) \to ((pr0 t4 t5) \to (ty3 g c2 t5 (THead (Bind b) u t3)))))) 
-(\lambda (H16: (eq T t5 t5)).(eq_ind T t5 (\lambda (t: T).((not (eq B b 
-Abst)) \to ((pr0 t4 t) \to (ty3 g c2 t5 (THead (Bind b) u t3))))) (\lambda 
-(H17: (not (eq B b Abst))).(\lambda (H18: (pr0 t4 t5)).(let H19 \def 
-(eq_ind_r T t2 (\lambda (t: T).(\forall (c2: C).((wcpr0 (CHead c (Bind b) u) 
-c2) \to (\forall (t2: T).((pr0 t t2) \to (ty3 g c2 t2 t3)))))) H3 (lift (S O) 
-O t4) H15) in (let H20 \def (eq_ind_r T t2 (\lambda (t: T).(ty3 g (CHead c 
-(Bind b) u) t t3)) H2 (lift (S O) O t4) H15) in (ex_ind T (\lambda (t: 
-T).(ty3 g (CHead c2 (Bind b) u) t4 t)) (ty3 g c2 t5 (THead (Bind b) u t3)) 
-(\lambda (x: T).(\lambda (H4: (ty3 g (CHead c2 (Bind b) u) t4 x)).(B_ind 
-(\lambda (b: B).((not (eq B b Abst)) \to ((ty3 g (CHead c2 (Bind b) u) t3 t4) 
-\to ((ty3 g (CHead c2 (Bind b) u) t4 x) \to ((ty3 g (CHead c2 (Bind b) u) 
-(lift (S O) O t5) t3) \to (ty3 g c2 t5 (THead (Bind b) u t3))))))) (\lambda 
-(H21: (not (eq B Abbr Abst))).(\lambda (H2: (ty3 g (CHead c2 (Bind Abbr) u) 
-t3 t4)).(\lambda (H5: (ty3 g (CHead c2 (Bind Abbr) u) t4 x)).(\lambda (H22: 
-(ty3 g (CHead c2 (Bind Abbr) u) (lift (S O) O t5) t3)).(let H \def 
-(ty3_gen_cabbr g (CHead c2 (Bind Abbr) u) (lift (S O) O t5) t3 H22 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 t5) 
-(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 t5 
-(THead (Bind Abbr) u t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H0: 
-(subst1 O u (lift (S O) O t5) (lift (S O) O x0))).(\lambda (H3: (subst1 O u 
-t3 (lift (S O) O x1))).(\lambda (H23: (ty3 g c2 x0 x1)).(let H24 \def (eq_ind 
-T x0 (\lambda (t: T).(ty3 g c2 t x1)) H23 t5 (lift_inj x0 t5 (S O) O 
-(subst1_gen_lift_eq t5 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_comm O (S O))) H0))) in (ty3_conv g c2 (THead (Bind Abbr) u t3) 
-(THead (Bind Abbr) u t4) (ty3_bind g c2 u t0 (H1 c2 H6 u (pr0_refl u)) Abbr 
-t3 t4 H2 x H5) t5 x1 H24 (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) H3))) 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)))))))))))) H)))))) (\lambda (H21: (not (eq B Abst Abst))).(\lambda (_: 
-(ty3 g (CHead c2 (Bind Abst) u) t3 t4)).(\lambda (_: (ty3 g (CHead c2 (Bind 
-Abst) u) t4 x)).(\lambda (_: (ty3 g (CHead c2 (Bind Abst) u) (lift (S O) O 
-t5) t3)).(let H \def (match (H21 (refl_equal B Abst)) return (\lambda (_: 
-False).(ty3 g c2 t5 (THead (Bind Abst) u t3))) with []) in H))))) (\lambda 
-(H21: (not (eq B Void Abst))).(\lambda (H2: (ty3 g (CHead c2 (Bind Void) u) 
-t3 t4)).(\lambda (H5: (ty3 g (CHead c2 (Bind Void) u) t4 x)).(\lambda (H22: 
-(ty3 g (CHead c2 (Bind Void) u) (lift (S O) O t5) t3)).(let H \def 
-(ty3_gen_cvoid g (CHead c2 (Bind Void) u) (lift (S O) O t5) t3 H22 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 t5) (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 t5 (THead 
-(Bind Void) u t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H0: (eq T 
-(lift (S O) O t5) (lift (S O) O x0))).(\lambda (H3: (eq T t3 (lift (S O) O 
-x1))).(\lambda (H23: (ty3 g c2 x0 x1)).(let H24 \def (eq_ind T t3 (\lambda 
-(t: T).(ty3 g (CHead c2 (Bind Void) u) t t4)) H2 (lift (S O) O x1) H3) in 
-(eq_ind_r T (lift (S O) O x1) (\lambda (t: T).(ty3 g c2 t5 (THead (Bind Void) 
-u t))) (let H25 \def (eq_ind_r T x0 (\lambda (t: T).(ty3 g c2 t x1)) H23 t5 
-(lift_inj t5 x0 (S O) O H0)) in (ty3_conv g c2 (THead (Bind Void) u (lift (S 
-O) O x1)) (THead (Bind Void) u t4) (ty3_bind g c2 u t0 (H1 c2 H6 u (pr0_refl 
-u)) Void (lift (S O) O x1) t4 H24 x H5) t5 x1 H25 (pc3_pr2_x c2 x1 (THead 
-(Bind Void) u (lift (S O) O 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 H3))))))) 
-H)))))) b H17 (H5 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H6 u u (pr0_refl u) 
-(Bind b)) t3 (pr0_refl t3)) H4 (H19 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H6 
-u u (pr0_refl u) (Bind b)) (lift (S O) O t5) (pr0_lift t4 t5 H18 (S O) O))))) 
-(ty3_correct g (CHead c2 (Bind b) u) t3 t4 (H5 (CHead c2 (Bind b) u) 
-(wcpr0_comp c c2 H6 u u (pr0_refl u) (Bind b)) t3 (pr0_refl t3)))))))) t5 
-(sym_eq T t5 t5 H16))) t2 H15)) u0 (sym_eq T u0 u H14))) b0 (sym_eq B b0 b 
-H13))) H12)) H11)) H10 H7 H8))) | (pr0_epsilon t4 t5 H7 u0) \Rightarrow 
-(\lambda (H8: (eq T (THead (Flat Cast) u0 t4) (THead (Bind b) u 
-t2))).(\lambda (H9: (eq T t5 t5)).((let H10 \def (eq_ind T (THead (Flat Cast) 
-u0 t4) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
-(match k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | 
-(Flat _) \Rightarrow True])])) I (THead (Bind b) u t2) H8) in (False_ind ((eq 
-T t5 t5) \to ((pr0 t4 t5) \to (ty3 g c2 t5 (THead (Bind b) u t3)))) H10)) H9 
-H7)))]) in (H8 (refl_equal T (THead (Bind b) u t2)) (refl_equal T 
-t5)))))))))))))))))))) (\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 return (\lambda (t: T).(\lambda 
-(t1: T).(\lambda (_: (pr0 t t1)).((eq T t (THead (Flat Appl) w v)) \to ((eq T 
-t1 t2) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0)))))))) 
-with [(pr0_refl t0) \Rightarrow (\lambda (H5: (eq T t0 (THead (Flat Appl) w 
-v))).(\lambda (H6: (eq T t0 t2)).(eq_ind T (THead (Flat Appl) w v) (\lambda 
-(t: T).((eq T t t2) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) 
-u t0))))) (\lambda (H7: (eq T (THead (Flat Appl) w v) t2)).(eq_ind T (THead 
-(Flat Appl) w v) (\lambda (t: T).(ty3 g c2 t (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 H7)) t0 (sym_eq T t0 (THead (Flat Appl) w v) H5) H6))) 
-| (pr0_comp u1 u2 H5 t1 t0 H6 k) \Rightarrow (\lambda (H7: (eq T (THead k u1 
-t1) (THead (Flat Appl) w v))).(\lambda (H8: (eq T (THead k u2 t0) t2)).((let 
-H9 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ _ t) 
-\Rightarrow t])) (THead k u1 t1) (THead (Flat Appl) w v) H7) in ((let H10 
-\def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t _) 
-\Rightarrow t])) (THead k u1 t1) (THead (Flat Appl) w v) H7) in ((let H11 
-\def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: T).K) with 
-[(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) 
-\Rightarrow k])) (THead k u1 t1) (THead (Flat Appl) w v) H7) in (eq_ind K 
-(Flat Appl) (\lambda (k0: K).((eq T u1 w) \to ((eq T t1 v) \to ((eq T (THead 
-k0 u2 t0) t2) \to ((pr0 u1 u2) \to ((pr0 t1 t0) \to (ty3 g c2 t2 (THead (Flat 
-Appl) w (THead (Bind Abst) u t0))))))))) (\lambda (H12: (eq T u1 w)).(eq_ind 
-T w (\lambda (t: T).((eq T t1 v) \to ((eq T (THead (Flat Appl) u2 t0) t2) \to 
-((pr0 t u2) \to ((pr0 t1 t0) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead 
-(Bind Abst) u t0)))))))) (\lambda (H13: (eq T t1 v)).(eq_ind T v (\lambda (t: 
-T).((eq T (THead (Flat Appl) u2 t0) t2) \to ((pr0 w u2) \to ((pr0 t t0) \to 
-(ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0))))))) (\lambda 
-(H14: (eq T (THead (Flat Appl) u2 t0) t2)).(eq_ind T (THead (Flat Appl) u2 
-t0) (\lambda (t: T).((pr0 w u2) \to ((pr0 v t0) \to (ty3 g c2 t (THead (Flat 
-Appl) w (THead (Bind Abst) u t0)))))) (\lambda (H15: (pr0 w u2)).(\lambda 
-(H16: (pr0 v t0)).(ex_ind T (\lambda (t: T).(ty3 g c2 (THead (Bind Abst) u 
-t0) t)) (ty3 g c2 (THead (Flat Appl) u2 t0) (THead (Flat Appl) w (THead (Bind 
-Abst) u t0))) (\lambda (x: T).(\lambda (H17: (ty3 g c2 (THead (Bind Abst) u 
-t0) x)).(ex4_3_ind T T T (\lambda (t2: T).(\lambda (_: T).(\lambda (_: 
-T).(pc3 c2 (THead (Bind Abst) u t2) x)))) (\lambda (_: T).(\lambda (t: 
-T).(\lambda (_: T).(ty3 g c2 u t)))) (\lambda (t2: T).(\lambda (_: 
-T).(\lambda (_: T).(ty3 g (CHead c2 (Bind Abst) u) t0 t2)))) (\lambda (t2: 
-T).(\lambda (_: T).(\lambda (t3: T).(ty3 g (CHead c2 (Bind Abst) u) t2 t3)))) 
-(ty3 g c2 (THead (Flat Appl) u2 t0) (THead (Flat Appl) w (THead (Bind Abst) u 
-t0))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 c2 
-(THead (Bind Abst) u x0) x)).(\lambda (H19: (ty3 g c2 u x1)).(\lambda (H20: 
-(ty3 g (CHead c2 (Bind Abst) u) t0 x0)).(\lambda (H21: (ty3 g (CHead c2 (Bind 
-Abst) u) x0 x2)).(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 H19 Abst 
-t0 x0 H20 x2 H21)) (THead (Flat Appl) u2 t0) (THead (Flat Appl) u2 (THead 
-(Bind Abst) u t0)) (ty3_appl g c2 u2 u (H1 c2 H4 u2 H15) t0 t0 (H3 c2 H4 t0 
-H16)) (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 
-H15) (Flat Appl) (THead (Bind Abst) u t0))))))))))) (ty3_gen_bind g Abst c2 u 
-t0 x H17)))) (ty3_correct g c2 v (THead (Bind Abst) u t0) (H3 c2 H4 v 
-(pr0_refl v)))))) t2 H14)) t1 (sym_eq T t1 v H13))) u1 (sym_eq T u1 w H12))) 
-k (sym_eq K k (Flat Appl) H11))) H10)) H9)) H8 H5 H6))) | (pr0_beta u0 v1 v2 
-H5 t1 t0 H6) \Rightarrow (\lambda (H7: (eq T (THead (Flat Appl) v1 (THead 
-(Bind Abst) u0 t1)) (THead (Flat Appl) w v))).(\lambda (H8: (eq T (THead 
-(Bind Abbr) v2 t0) t2)).((let H9 \def (f_equal T T (\lambda (e: T).(match e 
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow (THead (Bind Abst) u0 
-t1) | (TLRef _) \Rightarrow (THead (Bind Abst) u0 t1) | (THead _ _ t) 
-\Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind Abst) u0 t1)) (THead 
-(Flat Appl) w v) H7) in ((let H10 \def (f_equal T T (\lambda (e: T).(match e 
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) 
-\Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) v1 (THead 
-(Bind Abst) u0 t1)) (THead (Flat Appl) w v) H7) in (eq_ind T w (\lambda (t: 
-T).((eq T (THead (Bind Abst) u0 t1) v) \to ((eq T (THead (Bind Abbr) v2 t0) 
-t2) \to ((pr0 t v2) \to ((pr0 t1 t0) \to (ty3 g c2 t2 (THead (Flat Appl) w 
-(THead (Bind Abst) u t0)))))))) (\lambda (H11: (eq T (THead (Bind Abst) u0 
-t1) v)).(eq_ind T (THead (Bind Abst) u0 t1) (\lambda (_: T).((eq T (THead 
-(Bind Abbr) v2 t0) t2) \to ((pr0 w v2) \to ((pr0 t1 t0) \to (ty3 g c2 t2 
-(THead (Flat Appl) w (THead (Bind Abst) u t0))))))) (\lambda (H12: (eq T 
-(THead (Bind Abbr) v2 t0) t2)).(eq_ind T (THead (Bind Abbr) v2 t0) (\lambda 
-(t: T).((pr0 w v2) \to ((pr0 t1 t0) \to (ty3 g c2 t (THead (Flat Appl) w 
-(THead (Bind Abst) u t0)))))) (\lambda (H13: (pr0 w v2)).(\lambda (H14: (pr0 
-t1 t0)).(let H15 \def (eq_ind_r T v (\lambda (t: T).(\forall (c2: C).((wcpr0 
-c c2) \to (\forall (t2: T).((pr0 t t2) \to (ty3 g c2 t2 (THead (Bind Abst) u 
-t0))))))) H3 (THead (Bind Abst) u0 t1) H11) in (let H16 \def (eq_ind_r T v 
-(\lambda (t: T).(ty3 g c t (THead (Bind Abst) u t0))) H2 (THead (Bind Abst) 
-u0 t1) H11) in (ex_ind T (\lambda (t: T).(ty3 g c2 (THead (Bind Abst) u t0) 
-t)) (ty3 g c2 (THead (Bind Abbr) v2 t0) (THead (Flat Appl) w (THead (Bind 
-Abst) u t0))) (\lambda (x: T).(\lambda (H2: (ty3 g c2 (THead (Bind Abst) u 
-t0) x)).(ex4_3_ind T T T (\lambda (t2: T).(\lambda (_: T).(\lambda (_: 
-T).(pc3 c2 (THead (Bind Abst) u t2) x)))) (\lambda (_: T).(\lambda (t: 
-T).(\lambda (_: T).(ty3 g c2 u t)))) (\lambda (t2: T).(\lambda (_: 
-T).(\lambda (_: T).(ty3 g (CHead c2 (Bind Abst) u) t0 t2)))) (\lambda (t2: 
-T).(\lambda (_: T).(\lambda (t3: T).(ty3 g (CHead c2 (Bind Abst) u) t2 t3)))) 
-(ty3 g c2 (THead (Bind Abbr) v2 t0) (THead (Flat Appl) w (THead (Bind Abst) u 
-t0))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 c2 
-(THead (Bind Abst) u x0) x)).(\lambda (H17: (ty3 g c2 u x1)).(\lambda (H18: 
-(ty3 g (CHead c2 (Bind Abst) u) t0 x0)).(\lambda (H19: (ty3 g (CHead c2 (Bind 
-Abst) u) x0 x2)).(ex4_3_ind T T T (\lambda (t2: T).(\lambda (_: T).(\lambda 
-(_: T).(pc3 c2 (THead (Bind Abst) u0 t2) (THead (Bind Abst) u t0))))) 
-(\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g c2 u0 t)))) (\lambda 
-(t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind Abst) u0) t0 
-t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t3: T).(ty3 g (CHead c2 
-(Bind Abst) u0) t2 t3)))) (ty3 g c2 (THead (Bind Abbr) v2 t0) (THead (Flat 
-Appl) w (THead (Bind Abst) u t0))) (\lambda (x3: T).(\lambda (x4: T).(\lambda 
-(x5: T).(\lambda (H0: (pc3 c2 (THead (Bind Abst) u0 x3) (THead (Bind Abst) u 
-t0))).(\lambda (H20: (ty3 g c2 u0 x4)).(\lambda (H21: (ty3 g (CHead c2 (Bind 
-Abst) u0) t0 x3)).(\lambda (H22: (ty3 g (CHead c2 (Bind Abst) u0) x3 
-x5)).(and_ind (pc3 c2 u0 u) (\forall (b: B).(\forall (u: T).(pc3 (CHead c2 
-(Bind b) u) x3 t0))) (ty3 g c2 (THead (Bind Abbr) v2 t0) (THead (Flat Appl) w 
-(THead (Bind Abst) u t0))) (\lambda (H23: (pc3 c2 u0 u)).(\lambda (H24: 
-((\forall (b: B).(\forall (u: T).(pc3 (CHead c2 (Bind b) u) x3 
-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 H17 Abst t0 x0 
-H18 x2 H19)) (THead (Bind Abbr) v2 t0) (THead (Bind Abbr) v2 x3) (ty3_bind g 
-c2 v2 u (H1 c2 H4 v2 H13) Abbr t0 x3 (csub3_ty3_ld g c2 v2 u0 (ty3_conv g c2 
-u0 x4 H20 v2 u (H1 c2 H4 v2 H13) (pc3_s c2 u u0 H23)) t0 x3 H21) x5 
-(csub3_ty3_ld g c2 v2 u0 (ty3_conv g c2 u0 x4 H20 v2 u (H1 c2 H4 v2 H13) 
-(pc3_s c2 u u0 H23)) x3 x5 H22)) (pc3_t (THead (Bind Abbr) v2 t0) c2 (THead 
-(Bind Abbr) v2 x3) (pc3_head_2 c2 v2 x3 t0 (Bind Abbr) (H24 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 
-H13 t0 t0 (pr0_refl t0)))))))) (pc3_gen_abst c2 u0 u x3 t0 H0))))))))) 
-(ty3_gen_bind g Abst c2 u0 t0 (THead (Bind Abst) u t0) (H15 c2 H4 (THead 
-(Bind Abst) u0 t0) (pr0_comp u0 u0 (pr0_refl u0) t1 t0 H14 (Bind 
-Abst)))))))))))) (ty3_gen_bind g Abst c2 u t0 x H2)))) (ty3_correct g c2 
-(THead (Bind Abst) u0 t1) (THead (Bind Abst) u t0) (H15 c2 H4 (THead (Bind 
-Abst) u0 t1) (pr0_refl (THead (Bind Abst) u0 t1))))))))) t2 H12)) v H11)) v1 
-(sym_eq T v1 w H10))) H9)) H8 H5 H6))) | (pr0_upsilon b H5 v1 v2 H6 u1 u2 H7 
-t1 t0 H8) \Rightarrow (\lambda (H9: (eq T (THead (Flat Appl) v1 (THead (Bind 
-b) u1 t1)) (THead (Flat Appl) w v))).(\lambda (H10: (eq T (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t0)) t2)).((let H11 \def (f_equal T T 
-(\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow (THead (Bind b) u1 t1) | (TLRef _) \Rightarrow (THead (Bind b) u1 
-t1) | (THead _ _ t) \Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind b) u1 
-t1)) (THead (Flat Appl) w v) H9) in ((let H12 \def (f_equal T T (\lambda (e: 
-T).(match e return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef 
-_) \Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) v1 
-(THead (Bind b) u1 t1)) (THead (Flat Appl) w v) H9) in (eq_ind T w (\lambda 
-(t: T).((eq T (THead (Bind b) u1 t1) v) \to ((eq T (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t0)) t2) \to ((not (eq B b Abst)) \to ((pr0 t 
-v2) \to ((pr0 u1 u2) \to ((pr0 t1 t0) \to (ty3 g c2 t2 (THead (Flat Appl) w 
-(THead (Bind Abst) u t0)))))))))) (\lambda (H13: (eq T (THead (Bind b) u1 t1) 
-v)).(eq_ind T (THead (Bind b) u1 t1) (\lambda (_: T).((eq T (THead (Bind b) 
-u2 (THead (Flat Appl) (lift (S O) O v2) t0)) t2) \to ((not (eq B b Abst)) \to 
-((pr0 w v2) \to ((pr0 u1 u2) \to ((pr0 t1 t0) \to (ty3 g c2 t2 (THead (Flat 
-Appl) w (THead (Bind Abst) u t0))))))))) (\lambda (H14: (eq T (THead (Bind b) 
-u2 (THead (Flat Appl) (lift (S O) O v2) t0)) t2)).(eq_ind T (THead (Bind b) 
-u2 (THead (Flat Appl) (lift (S O) O v2) t0)) (\lambda (t: T).((not (eq B b 
-Abst)) \to ((pr0 w v2) \to ((pr0 u1 u2) \to ((pr0 t1 t0) \to (ty3 g c2 t 
-(THead (Flat Appl) w (THead (Bind Abst) u t0)))))))) (\lambda (H15: (not (eq 
-B b Abst))).(\lambda (H16: (pr0 w v2)).(\lambda (H17: (pr0 u1 u2)).(\lambda 
-(H18: (pr0 t1 t0)).(let H19 \def (eq_ind_r T v (\lambda (t: T).(\forall (c2: 
-C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 t t2) \to (ty3 g c2 t2 (THead 
-(Bind Abst) u t0))))))) H3 (THead (Bind b) u1 t1) H13) in (let H20 \def 
-(eq_ind_r T v (\lambda (t: T).(ty3 g c t (THead (Bind Abst) u t0))) H2 (THead 
-(Bind b) u1 t1) H13) in (ex_ind T (\lambda (t: T).(ty3 g c2 (THead (Bind 
-Abst) u t0) t)) (ty3 g c2 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O 
-v2) t0)) (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x: 
-T).(\lambda (H2: (ty3 g c2 (THead (Bind Abst) u t0) x)).(let H3 \def H2 in 
-(ex4_3_ind T T T (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c2 
-(THead (Bind Abst) u t2) x)))) (\lambda (_: T).(\lambda (t: T).(\lambda (_: 
-T).(ty3 g c2 u t)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g 
-(CHead c2 (Bind Abst) u) t0 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda 
-(t3: T).(ty3 g (CHead c2 (Bind Abst) u) t2 t3)))) (ty3 g c2 (THead (Bind b) 
-u2 (THead (Flat Appl) (lift (S O) O v2) t0)) (THead (Flat Appl) w (THead 
-(Bind Abst) u t0))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: 
-T).(\lambda (_: (pc3 c2 (THead (Bind Abst) u x0) x)).(\lambda (H22: (ty3 g c2 
-u x1)).(\lambda (H23: (ty3 g (CHead c2 (Bind Abst) u) t0 x0)).(\lambda (H24: 
-(ty3 g (CHead c2 (Bind Abst) u) x0 x2)).(ex4_3_ind T T T (\lambda (t2: 
-T).(\lambda (_: T).(\lambda (_: T).(pc3 c2 (THead (Bind b) u2 t2) (THead 
-(Bind Abst) u t0))))) (\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g 
-c2 u2 t)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c2 
-(Bind b) u2) t0 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t3: T).(ty3 
-g (CHead c2 (Bind b) u2) t2 t3)))) (ty3 g c2 (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t0)) (THead (Flat Appl) w (THead (Bind Abst) u t0))) 
-(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H0: (pc3 c2 
-(THead (Bind b) u2 x3) (THead (Bind Abst) u t0))).(\lambda (H25: (ty3 g c2 u2 
-x4)).(\lambda (H26: (ty3 g (CHead c2 (Bind b) u2) t0 x3)).(\lambda (_: (ty3 g 
-(CHead c2 (Bind b) u2) x3 x5)).(let H28 \def (eq_ind T (lift (S O) O (THead 
-(Bind Abst) u t0)) (\lambda (t: T).(pc3 (CHead c2 (Bind b) u2) x3 t)) 
-(pc3_gen_not_abst b H15 c2 x3 t0 u2 u H0) (THead (Bind Abst) (lift (S O) O u) 
-(lift (S O) (S O) t0)) (lift_bind Abst u t0 (S O) O)) in (let H29 \def 
-(eq_ind T (lift (S O) O (THead (Bind Abst) u t0)) (\lambda (t: T).(ty3 g 
-(CHead c2 (Bind b) u2) t (lift (S O) O x))) (ty3_lift g c2 (THead (Bind Abst) 
-u t0) x H2 (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 (ex4_3_ind T T T (\lambda (t2: 
-T).(\lambda (_: T).(\lambda (_: T).(pc3 (CHead c2 (Bind b) u2) (THead (Bind 
-Abst) (lift (S O) O u) t2) (lift (S O) O x))))) (\lambda (_: T).(\lambda (t: 
-T).(\lambda (_: T).(ty3 g (CHead c2 (Bind b) u2) (lift (S O) O u) t)))) 
-(\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead (CHead c2 
-(Bind b) u2) (Bind Abst) (lift (S O) O u)) (lift (S O) (S O) t0) t2)))) 
-(\lambda (t2: T).(\lambda (_: T).(\lambda (t3: T).(ty3 g (CHead (CHead c2 
-(Bind b) u2) (Bind Abst) (lift (S O) O u)) t2 t3)))) (ty3 g c2 (THead (Bind 
-b) u2 (THead (Flat Appl) (lift (S O) O v2) t0)) (THead (Flat Appl) w (THead 
-(Bind Abst) u t0))) (\lambda (x6: T).(\lambda (x7: T).(\lambda (x8: 
-T).(\lambda (_: (pc3 (CHead c2 (Bind b) u2) (THead (Bind Abst) (lift (S O) O 
-u) x6) (lift (S O) O x))).(\lambda (H31: (ty3 g (CHead c2 (Bind b) u2) (lift 
-(S O) O u) x7)).(\lambda (H32: (ty3 g (CHead (CHead c2 (Bind b) u2) (Bind 
-Abst) (lift (S O) O u)) (lift (S O) (S O) t0) x6)).(\lambda (H33: (ty3 g 
-(CHead (CHead c2 (Bind b) u2) (Bind Abst) (lift (S O) O u)) x6 x8)).(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 H22 Abst t0 x0 H23 x2 H24)) (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t0)) (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 x4 H25 b (THead (Flat Appl) (lift 
-(S O) O v2) t0) (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 H16) (CHead c2 
-(Bind b) u2) O (S O) (drop_drop (Bind b) O c2 c2 (drop_refl c2) u2)) t0 (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) x6) 
-(ty3_bind g (CHead c2 (Bind b) u2) (lift (S O) O u) x7 H31 Abst (lift (S O) 
-(S O) t0) x6 H32 x8 H33) t0 x3 H26 H28)) (THead (Flat Appl) (lift (S O) O v2) 
-(THead (Bind Abst) (lift (S O) O u) x6)) (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 H16) 
-(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)) x6 (ty3_bind 
-g (CHead c2 (Bind b) u2) (lift (S O) O u) x7 H31 Abst (lift (S O) (S O) t0) 
-x6 H32 x8 H33))) (eq_ind T (lift (S O) O (THead (Bind Abst) u t0)) (\lambda 
-(t: T).(pc3 c2 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t)) 
-(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 H15 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_s (THead (Flat Appl) v2 (THead (Bind b) u2 
-(lift (S O) O (THead (Bind Abst) u t0)))) (THead (Flat Appl) w (THead (Bind 
-Abst) u t0)) (pc1_head w v2 (pc1_pr0_r w v2 H16) (THead (Bind Abst) u t0) 
-(THead (Bind b) u2 (lift (S O) O (THead (Bind Abst) u t0))) (pc1_pr0_x (THead 
-(Bind Abst) u t0) (THead (Bind b) u2 (lift (S O) O (THead (Bind Abst) u t0))) 
-(pr0_zeta b H15 (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) H29))))))))))) (ty3_gen_bind g b c2 u2 t0 (THead 
-(Bind Abst) u t0) (H19 c2 H4 (THead (Bind b) u2 t0) (pr0_comp u1 u2 H17 t1 t0 
-H18 (Bind b)))))))))))) (ty3_gen_bind g Abst c2 u t0 x H3))))) (ty3_correct g 
-c2 (THead (Bind b) u2 t0) (THead (Bind Abst) u t0) (H19 c2 H4 (THead (Bind b) 
-u2 t0) (pr0_comp u1 u2 H17 t1 t0 H18 (Bind b))))))))))) t2 H14)) v H13)) v1 
-(sym_eq T v1 w H12))) H11)) H10 H5 H6 H7 H8))) | (pr0_delta u1 u2 H5 t1 t0 H6 
-w0 H7) \Rightarrow (\lambda (H8: (eq T (THead (Bind Abbr) u1 t1) (THead (Flat 
-Appl) w v))).(\lambda (H9: (eq T (THead (Bind Abbr) u2 w0) t2)).((let H10 
-\def (eq_ind T (THead (Bind Abbr) u1 t1) (\lambda (e: T).(match e return 
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: 
-K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I 
-(THead (Flat Appl) w v) H8) in (False_ind ((eq T (THead (Bind Abbr) u2 w0) 
-t2) \to ((pr0 u1 u2) \to ((pr0 t1 t0) \to ((subst0 O u2 t0 w0) \to (ty3 g c2 
-t2 (THead (Flat Appl) w (THead (Bind Abst) u t0))))))) H10)) H9 H5 H6 H7))) | 
-(pr0_zeta b H5 t1 t0 H6 u0) \Rightarrow (\lambda (H7: (eq T (THead (Bind b) 
-u0 (lift (S O) O t1)) (THead (Flat Appl) w v))).(\lambda (H8: (eq T t0 
-t2)).((let H9 \def (eq_ind T (THead (Bind b) u0 (lift (S O) O t1)) (\lambda 
-(e: T).(match e return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k 
-return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) 
-\Rightarrow False])])) I (THead (Flat Appl) w v) H7) in (False_ind ((eq T t0 
-t2) \to ((not (eq B b Abst)) \to ((pr0 t1 t0) \to (ty3 g c2 t2 (THead (Flat 
-Appl) w (THead (Bind Abst) u t0)))))) H9)) H8 H5 H6))) | (pr0_epsilon t1 t0 
-H5 u0) \Rightarrow (\lambda (H6: (eq T (THead (Flat Cast) u0 t1) (THead (Flat 
-Appl) w v))).(\lambda (H7: (eq T t0 t2)).((let H8 \def (eq_ind T (THead (Flat 
-Cast) u0 t1) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
-\Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow 
-False | (Flat f) \Rightarrow (match f return (\lambda (_: F).Prop) with [Appl 
-\Rightarrow False | Cast \Rightarrow True])])])) I (THead (Flat Appl) w v) 
-H6) in (False_ind ((eq T t0 t2) \to ((pr0 t1 t0) \to (ty3 g c2 t2 (THead 
-(Flat Appl) w (THead (Bind Abst) u t0))))) H8)) H7 H5)))]) 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 (t2: T).((pr0 
-t3 t2) \to (ty3 g c2 t2 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 return (\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr0 t 
-t0)).((eq T t (THead (Flat Cast) t3 t2)) \to ((eq T t0 t4) \to (ty3 g c2 t4 
-t3)))))) with [(pr0_refl t) \Rightarrow (\lambda (H5: (eq T t (THead (Flat 
-Cast) t3 t2))).(\lambda (H6: (eq T t t4)).(eq_ind T (THead (Flat Cast) t3 t2) 
-(\lambda (t0: T).((eq T t0 t4) \to (ty3 g c2 t4 t3))) (\lambda (H7: (eq T 
-(THead (Flat Cast) t3 t2) t4)).(eq_ind T (THead (Flat Cast) t3 t2) (\lambda 
-(t0: T).(ty3 g c2 t0 t3)) (ty3_cast g c2 t2 t3 (H1 c2 H4 t2 (pr0_refl t2)) t0 
-(H3 c2 H4 t3 (pr0_refl t3))) t4 H7)) t (sym_eq T t (THead (Flat Cast) t3 t2) 
-H5) H6))) | (pr0_comp u1 u2 H5 t4 t5 H6 k) \Rightarrow (\lambda (H7: (eq T 
-(THead k u1 t4) (THead (Flat Cast) t3 t2))).(\lambda (H8: (eq T (THead k u2 
-t5) t4)).((let H9 \def (f_equal T T (\lambda (e: T).(match e return (\lambda 
-(_: T).T) with [(TSort _) \Rightarrow t4 | (TLRef _) \Rightarrow t4 | (THead 
-_ _ t) \Rightarrow t])) (THead k u1 t4) (THead (Flat Cast) t3 t2) H7) in 
-((let H10 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: 
-T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t 
-_) \Rightarrow t])) (THead k u1 t4) (THead (Flat Cast) t3 t2) H7) in ((let 
-H11 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: T).K) with 
-[(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) 
-\Rightarrow k])) (THead k u1 t4) (THead (Flat Cast) t3 t2) H7) in (eq_ind K 
-(Flat Cast) (\lambda (k0: K).((eq T u1 t3) \to ((eq T t4 t2) \to ((eq T 
-(THead k0 u2 t5) t4) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ty3 g c2 t4 
-t3))))))) (\lambda (H12: (eq T u1 t3)).(eq_ind T t3 (\lambda (t: T).((eq T t4 
-t2) \to ((eq T (THead (Flat Cast) u2 t5) t4) \to ((pr0 t u2) \to ((pr0 t4 t5) 
-\to (ty3 g c2 t4 t3)))))) (\lambda (H13: (eq T t4 t2)).(eq_ind T t2 (\lambda 
-(t: T).((eq T (THead (Flat Cast) u2 t5) t4) \to ((pr0 t3 u2) \to ((pr0 t t5) 
-\to (ty3 g c2 t4 t3))))) (\lambda (H14: (eq T (THead (Flat Cast) u2 t5) 
-t4)).(eq_ind T (THead (Flat Cast) u2 t5) (\lambda (t: T).((pr0 t3 u2) \to 
-((pr0 t2 t5) \to (ty3 g c2 t t3)))) (\lambda (H15: (pr0 t3 u2)).(\lambda 
-(H16: (pr0 t2 t5)).(ty3_conv g c2 t3 t0 (H3 c2 H4 t3 (pr0_refl t3)) (THead 
-(Flat Cast) u2 t5) u2 (ty3_cast g c2 t5 u2 (ty3_conv g c2 u2 t0 (H3 c2 H4 u2 
-H15) t5 t3 (H1 c2 H4 t5 H16) (pc3_pr2_r c2 t3 u2 (pr2_free c2 t3 u2 H15))) t0 
-(H3 c2 H4 u2 H15)) (pc3_pr2_x c2 u2 t3 (pr2_free c2 t3 u2 H15))))) t4 H14)) 
-t4 (sym_eq T t4 t2 H13))) u1 (sym_eq T u1 t3 H12))) k (sym_eq K k (Flat Cast) 
-H11))) H10)) H9)) 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)) (THead 
-(Flat Cast) t3 t2))).(\lambda (H8: (eq T (THead (Bind Abbr) v2 t5) t4)).((let 
-H9 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) (\lambda 
-(e: T).(match e return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k 
-return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) 
-\Rightarrow (match f return (\lambda (_: F).Prop) with [Appl \Rightarrow True 
-| Cast \Rightarrow False])])])) I (THead (Flat Cast) t3 t2) H7) in (False_ind 
-((eq T (THead (Bind Abbr) v2 t5) t4) \to ((pr0 v1 v2) \to ((pr0 t4 t5) \to 
-(ty3 g c2 t4 t3)))) H9)) 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)) (THead (Flat Cast) t3 t2))).(\lambda (H10: (eq T (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t5)) t4)).((let H11 \def (eq_ind T 
-(THead (Flat Appl) v1 (THead (Bind b) u1 t4)) (\lambda (e: T).(match e return 
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: 
-K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f 
-return (\lambda (_: F).Prop) with [Appl \Rightarrow True | Cast \Rightarrow 
-False])])])) I (THead (Flat Cast) t3 t2) H9) in (False_ind ((eq T (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) t4) \to ((not (eq B b 
-Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ty3 g c2 t4 
-t3)))))) H11)) 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) (THead (Flat Cast) 
-t3 t2))).(\lambda (H9: (eq T (THead (Bind Abbr) u2 w) t4)).((let H10 \def 
-(eq_ind T (THead (Bind Abbr) u1 t4) (\lambda (e: T).(match e return (\lambda 
-(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
-| (THead k _ _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind 
-_) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Cast) t3 
-t2) H8) in (False_ind ((eq T (THead (Bind Abbr) u2 w) t4) \to ((pr0 u1 u2) 
-\to ((pr0 t4 t5) \to ((subst0 O u2 t5 w) \to (ty3 g c2 t4 t3))))) H10)) 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)) (THead (Flat Cast) t3 t2))).(\lambda (H8: (eq T 
-t5 t4)).((let H9 \def (eq_ind T (THead (Bind b) u (lift (S O) O t4)) (\lambda 
-(e: T).(match e return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k 
-return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) 
-\Rightarrow False])])) I (THead (Flat Cast) t3 t2) H7) in (False_ind ((eq T 
-t5 t4) \to ((not (eq B b Abst)) \to ((pr0 t4 t5) \to (ty3 g c2 t4 t3)))) H9)) 
-H8 H5 H6))) | (pr0_epsilon t4 t5 H5 u) \Rightarrow (\lambda (H6: (eq T (THead 
-(Flat Cast) u t4) (THead (Flat Cast) t3 t2))).(\lambda (H7: (eq T t5 
-t4)).((let H8 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: 
-T).T) with [(TSort _) \Rightarrow t4 | (TLRef _) \Rightarrow t4 | (THead _ _ 
-t) \Rightarrow t])) (THead (Flat Cast) u t4) (THead (Flat Cast) t3 t2) H6) in 
-((let H9 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) 
-with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) 
-\Rightarrow t])) (THead (Flat Cast) u t4) (THead (Flat Cast) t3 t2) H6) in 
-(eq_ind T t3 (\lambda (_: T).((eq T t4 t2) \to ((eq T t5 t4) \to ((pr0 t4 t5) 
-\to (ty3 g c2 t4 t3))))) (\lambda (H10: (eq T t4 t2)).(eq_ind T t2 (\lambda 
-(t: T).((eq T t5 t4) \to ((pr0 t t5) \to (ty3 g c2 t4 t3)))) (\lambda (H11: 
-(eq T t5 t4)).(eq_ind T t4 (\lambda (t: T).((pr0 t2 t) \to (ty3 g c2 t4 t3))) 
-(\lambda (H12: (pr0 t2 t4)).(H1 c2 H4 t4 H12)) t5 (sym_eq T t5 t4 H11))) t4 
-(sym_eq T t4 t2 H10))) u (sym_eq T u t3 H9))) H8)) H7 H5)))]) in (H6 
-(refl_equal T (THead (Flat Cast) t3 t2)) (refl_equal T t4))))))))))))))) c1 
-t1 t H))))).
-
-theorem ty3_sred_pr1:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr1 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: (pr1 t1 
-t2)).(pr1_ind (\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: (pr0 t4 t3)).(\lambda (t5: T).(\lambda (_: 
-(pr1 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_wcpr0_pr0 g c t4 t H3 c (wcpr0_refl c) t3 
-H0))))))))))) 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)))).
-
-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 (t1: T).(\forall (t4: T).((ty3 g (CHead c (Bind b) t2) t1 t4) \to 
-(ty3 g (CHead c (Bind b) t3) t1 t4))))))).(\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__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 (n: nat).(le d n)) H0 (plus (minus n h) h) 
-(le_plus_minus_sym h n (le_trans_plus_r d h n H))) in (le_S d (minus n h) 
-(le_minus d n h 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)) (\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 (x: T).(\forall (x0: nat).((eq T 
-t2 (lift h x0 x)) \to (\forall (e: C).((drop h x0 c0 e) \to (ex2 T (\lambda 
-(t2: T).(pc3 c0 (lift h x0 t2) t)) (\lambda (t2: T).(ty3 g e x 
-t2)))))))))).(\lambda (u: T).(\lambda (t3: T).(\lambda (H3: (ty3 g c0 u 
-t3)).(\lambda (H4: ((\forall (x: T).(\forall (x0: nat).((eq T u (lift h x0 
-x)) \to (\forall (e: C).((drop h x0 c0 e) \to (ex2 T (\lambda (t2: T).(pc3 c0 
-(lift h x0 t2) t3)) (\lambda (t2: T).(ty3 g e x t2)))))))))).(\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 (t: T).(\forall (x: T).(\forall (x0: nat).((eq T t (lift 
-h x0 x)) \to (\forall (e: C).((drop h x0 c0 e) \to (ex2 T (\lambda (t2: 
-T).(pc3 c0 (lift h x0 t2) t3)) (\lambda (t2: T).(ty3 g e x t2))))))))) H4 
-(lift h x1 x0) H6) in (let H9 \def (eq_ind T u (\lambda (t: T).(ty3 g c0 t 
-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 (x: T).(\forall (x0: nat).((eq T u (lift h x0 x)) 
-\to (\forall (e: C).((drop h x0 d0 e) \to (ex2 T (\lambda (t2: T).(pc3 d0 
-(lift h x0 t2) t)) (\lambda (t2: T).(ty3 g e x 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)))).(and_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 (n: 
-nat).(drop h n 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 (x: T).(\forall (x0: 
-nat).((eq T t0 (lift h x0 x)) \to (\forall (e: C).((drop h x0 d0 e) \to (ex2 
-T (\lambda (t2: T).(pc3 d0 (lift h x0 t2) t)) (\lambda (t2: T).(ty3 g e x 
-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))))).(and_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_plus_r x1 
-h n 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))) (ty3_gen__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 
-(x: T).(\forall (x0: nat).((eq T u (lift h x0 x)) \to (\forall (e: C).((drop 
-h x0 d0 e) \to (ex2 T (\lambda (t2: T).(pc3 d0 (lift h x0 t2) t)) (\lambda 
-(t2: T).(ty3 g e x 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)))).(and_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 (n: nat).(drop h n 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 (x: T).(\forall (x0: nat).((eq T t0 (lift h x0 
-x)) \to (\forall (e: C).((drop h x0 d0 e) \to (ex2 T (\lambda (t2: T).(pc3 d0 
-(lift h x0 t2) t)) (\lambda (t2: T).(ty3 g e x 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))))).(and_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_plus_r x1 
-h n 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))) (ty3_gen__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 (x: T).(\forall (x0: 
-nat).((eq T u (lift h x0 x)) \to (\forall (e: C).((drop h x0 c0 e) \to (ex2 T 
-(\lambda (t2: T).(pc3 c0 (lift h x0 t2) t)) (\lambda (t2: T).(ty3 g e x 
-t2)))))))))).(\lambda (b: B).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H3: 
-(ty3 g (CHead c0 (Bind b) u) t2 t3)).(\lambda (H4: ((\forall (x: T).(\forall 
-(x0: nat).((eq T t2 (lift h x0 x)) \to (\forall (e: C).((drop h x0 (CHead c0 
-(Bind b) u) e) \to (ex2 T (\lambda (t2: T).(pc3 (CHead c0 (Bind b) u) (lift h 
-x0 t2) t3)) (\lambda (t2: T).(ty3 g e x t2)))))))))).(\lambda (t0: 
-T).(\lambda (H5: (ty3 g (CHead c0 (Bind b) u) t3 t0)).(\lambda (H6: ((\forall 
-(x: T).(\forall (x0: nat).((eq T t3 (lift h x0 x)) \to (\forall (e: C).((drop 
-h x0 (CHead c0 (Bind b) u) e) \to (ex2 T (\lambda (t2: T).(pc3 (CHead c0 
-(Bind b) u) (lift h x0 t2) t0)) (\lambda (t2: T).(ty3 g e x 
-t2)))))))))).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H7: (eq T (THead 
-(Bind b) u t2) (lift h x1 x0))).(\lambda (e: C).(\lambda (H8: (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 (H9: (eq T x0 (THead 
-(Bind b) x2 x3))).(\lambda (H10: (eq T u (lift h x1 x2))).(\lambda (H11: (eq 
-T t2 (lift h (S x1) x3))).(eq_ind_r T (THead (Bind b) x2 x3) (\lambda (t4: 
-T).(ex2 T (\lambda (t5: T).(pc3 c0 (lift h x1 t5) (THead (Bind b) u t3))) 
-(\lambda (t5: T).(ty3 g e t4 t5)))) (let H12 \def (eq_ind T t2 (\lambda (t: 
-T).(\forall (x: T).(\forall (x0: nat).((eq T t (lift h x0 x)) \to (\forall 
-(e: C).((drop h x0 (CHead c0 (Bind b) u) e) \to (ex2 T (\lambda (t2: T).(pc3 
-(CHead c0 (Bind b) u) (lift h x0 t2) t3)) (\lambda (t2: T).(ty3 g e x 
-t2))))))))) H4 (lift h (S x1) x3) H11) in (let H13 \def (eq_ind T t2 (\lambda 
-(t: T).(ty3 g (CHead c0 (Bind b) u) t t3)) H3 (lift h (S x1) x3) H11) in (let 
-H14 \def (eq_ind T u (\lambda (t: T).(ty3 g (CHead c0 (Bind b) t) (lift h (S 
-x1) x3) t3)) H13 (lift h x1 x2) H10) in (let H15 \def (eq_ind T u (\lambda 
-(t: T).(\forall (x: T).(\forall (x0: nat).((eq T (lift h (S x1) x3) (lift h 
-x0 x)) \to (\forall (e: C).((drop h x0 (CHead c0 (Bind b) t) e) \to (ex2 T 
-(\lambda (t2: T).(pc3 (CHead c0 (Bind b) t) (lift h x0 t2) t3)) (\lambda (t2: 
-T).(ty3 g e x t2))))))))) H12 (lift h x1 x2) H10) in (let H16 \def (eq_ind T 
-u (\lambda (t: T).(\forall (x: T).(\forall (x0: nat).((eq T t3 (lift h x0 x)) 
-\to (\forall (e: C).((drop h x0 (CHead c0 (Bind b) t) e) \to (ex2 T (\lambda 
-(t2: T).(pc3 (CHead c0 (Bind b) t) (lift h x0 t2) t0)) (\lambda (t2: T).(ty3 
-g e x t2))))))))) H6 (lift h x1 x2) H10) in (let H17 \def (eq_ind T u 
-(\lambda (t: T).(ty3 g (CHead c0 (Bind b) t) t3 t0)) H5 (lift h x1 x2) H10) 
-in (let H18 \def (eq_ind T u (\lambda (t0: T).(\forall (x: T).(\forall (x0: 
-nat).((eq T t0 (lift h x0 x)) \to (\forall (e: C).((drop h x0 c0 e) \to (ex2 
-T (\lambda (t2: T).(pc3 c0 (lift h x0 t2) t)) (\lambda (t2: T).(ty3 g e x 
-t2))))))))) H2 (lift h x1 x2) H10) in (let H19 \def (eq_ind T u (\lambda (t0: 
-T).(ty3 g c0 t0 t)) H1 (lift h x1 x2) H10) in (eq_ind_r T (lift h x1 x2) 
-(\lambda (t4: T).(ex2 T (\lambda (t5: T).(pc3 c0 (lift h x1 t5) (THead (Bind 
-b) t4 t3))) (\lambda (t5: T).(ty3 g e (THead (Bind b) x2 x3) t5)))) (let H20 
-\def (H18 x2 x1 (refl_equal T (lift h x1 x2)) e H8) 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 (H22: (ty3 g e x2 
-x4)).(let H23 \def (H15 x3 (S x1) (refl_equal T (lift h (S x1) x3)) (CHead e 
-(Bind b) x2) (drop_skip_bind h x1 c0 e H8 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 (H24: (pc3 (CHead c0 
-(Bind b) (lift h x1 x2)) (lift h (S x1) x5) t3)).(\lambda (H25: (ty3 g (CHead 
-e (Bind b) x2) x3 x5)).(ex_ind T (\lambda (t4: T).(ty3 g (CHead e (Bind b) 
-x2) x5 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 (x6: T).(\lambda (H26: (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 (t4: T).(pc3 c0 t4 (THead (Bind b) (lift h x1 x2) t3))) 
-(pc3_head_2 c0 (lift h x1 x2) (lift h (S x1) x5) t3 (Bind b) H24) (lift h x1 
-(THead (Bind b) x2 x5)) (lift_bind b x2 x5 h x1)) (ty3_bind g e x2 x4 H22 b 
-x3 x5 H25 x6 H26)))) (ty3_correct g (CHead e (Bind b) x2) x3 x5 H25))))) 
-H23))))) H20)) u H10))))))))) x0 H9)))))) (lift_gen_bind b u t2 x0 h x1 
-H7)))))))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u: 
-T).(\lambda (H1: (ty3 g c0 w u)).(\lambda (H2: ((\forall (x: T).(\forall (x0: 
-nat).((eq T w (lift h x0 x)) \to (\forall (e: C).((drop h x0 c0 e) \to (ex2 T 
-(\lambda (t2: T).(pc3 c0 (lift h x0 t2) u)) (\lambda (t2: T).(ty3 g e x 
-t2)))))))))).(\lambda (v: T).(\lambda (t: T).(\lambda (H3: (ty3 g c0 v (THead 
-(Bind Abst) u t))).(\lambda (H4: ((\forall (x: T).(\forall (x0: nat).((eq T v 
-(lift h x0 x)) \to (\forall (e: C).((drop h x0 c0 e) \to (ex2 T (\lambda (t2: 
-T).(pc3 c0 (lift h x0 t2) (THead (Bind Abst) u t))) (\lambda (t2: T).(ty3 g e 
-x 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 (x: T).(\forall (x0: 
-nat).((eq T t0 (lift h x0 x)) \to (\forall (e: C).((drop h x0 c0 e) \to (ex2 
-T (\lambda (t2: T).(pc3 c0 (lift h x0 t2) (THead (Bind Abst) u t))) (\lambda 
-(t2: T).(ty3 g e x 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 (t: T).(\forall (x: T).(\forall 
-(x0: nat).((eq T t (lift h x0 x)) \to (\forall (e: C).((drop h x0 c0 e) \to 
-(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x0 t2) u)) (\lambda (t2: T).(ty3 g e 
-x t2))))))))) H2 (lift h x1 x2) H8) in (let H13 \def (eq_ind T w (\lambda (t: 
-T).(ty3 g c0 t 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 (u: T).(pr3 (CHead c0 (Bind 
-b) u) 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)).(ex4_3_ind T T 
-T (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 e (THead (Bind Abst) 
-x6 t2) x8)))) (\lambda (_: T).(\lambda (t0: T).(\lambda (_: T).(ty3 g e x6 
-t0)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead e (Bind 
-Abst) x6) x7 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t3: T).(ty3 g 
-(CHead e (Bind Abst) x6) t2 t3)))) (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 (x11: 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)).(\lambda (H27: (ty3 g (CHead e (Bind Abst) x6) x9 
-x11)).(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 x11 H27) x3 x5 H19 (pc3_pr3_r e x5 (THead (Bind Abst) x6 x7) 
-H20))))))))))) (ty3_gen_bind g Abst e x6 x7 x8 (ty3_sred_pr3 e x5 (THead 
-(Bind Abst) x6 x7) H20 g x8 H23))))) (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 (x: T).(\forall (x0: nat).((eq T t2 (lift h x0 x)) \to 
-(\forall (e: C).((drop h x0 c0 e) \to (ex2 T (\lambda (t2: T).(pc3 c0 (lift h 
-x0 t2) t3)) (\lambda (t2: T).(ty3 g e x t2)))))))))).(\lambda (t0: 
-T).(\lambda (H3: (ty3 g c0 t3 t0)).(\lambda (H4: ((\forall (x: T).(\forall 
-(x0: nat).((eq T t3 (lift h x0 x)) \to (\forall (e: C).((drop h x0 c0 e) \to 
-(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x0 t2) t0)) (\lambda (t2: T).(ty3 g e 
-x t2)))))))))).(\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) 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) t3)) (\lambda (t4: T).(ty3 g e t 
-t4)))) (let H10 \def (eq_ind T t3 (\lambda (t: T).(\forall (x: T).(\forall 
-(x0: nat).((eq T t (lift h x0 x)) \to (\forall (e: C).((drop h x0 c0 e) \to 
-(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x0 t2) t0)) (\lambda (t2: T).(ty3 g e 
-x t2))))))))) 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 (x: T).(\forall (x0: nat).((eq T t2 (lift h x0 x)) 
-\to (\forall (e: C).((drop h x0 c0 e) \to (ex2 T (\lambda (t2: T).(pc3 c0 
-(lift h x0 t2) t)) (\lambda (t2: T).(ty3 g e x t2))))))))) 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) 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 (x: T).(\forall (x0: nat).((eq T t (lift h x0 x)) \to (\forall 
-(e: C).((drop h x0 c0 e) \to (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x0 t2) 
-(lift h x1 x2))) (\lambda (t2: T).(ty3 g e x t2))))))))) 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) (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) (lift h x1 
-x2))) (\lambda (t4: T).(ty3 g e (THead (Flat Cast) x2 x3) t4))) (\lambda (x5: 
-T).(\lambda (_: (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) (lift h x1 x2))) 
-(\lambda (t4: T).(ty3 g e (THead (Flat Cast) x2 x3) t4)) x2 (pc3_refl c0 
-(lift h x1 x2)) (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)).(ty3_conv g c t2 x (ty3_sred_pr3 c t1 t2 H0 g x H1) 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)).(ty3_unique g c x 
-t1 (ty3_sred_pr3 c u1 x H3 g t1 H) t2 (ty3_sred_pr3 c u2 x H4 g t2 H0))))) 
-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)))))))))).
-
-theorem ty3_tau0:
- \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).((ty3 g c u 
-t1) \to (\forall (t2: T).((tau0 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).((tau0 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).((tau0 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 (t2: T).((tau0 g c0 u0 t2) \to (ty3 g c0 u0 t2))))).(\lambda (_: 
-(pc3 c0 t3 t2)).(\lambda (t0: T).(\lambda (H5: (tau0 g c0 u0 t0)).(H3 t0 
-H5))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (t2: T).(\lambda 
-(H0: (tau0 g c0 (TSort m) t2)).(let H1 \def (match H0 return (\lambda (c: 
-C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (tau0 ? c t t0)).((eq C c 
-c0) \to ((eq T t (TSort m)) \to ((eq T t0 t2) \to (ty3 g c0 (TSort m) 
-t2)))))))) with [(tau0_sort c0 n) \Rightarrow (\lambda (H0: (eq C c0 
-c0)).(\lambda (H1: (eq T (TSort n) (TSort m))).(\lambda (H2: (eq T (TSort 
-(next g n)) t2)).(eq_ind C c0 (\lambda (_: C).((eq T (TSort n) (TSort m)) \to 
-((eq T (TSort (next g n)) t2) \to (ty3 g c0 (TSort m) t2)))) (\lambda (H3: 
-(eq T (TSort n) (TSort m))).(let H4 \def (f_equal T nat (\lambda (e: 
-T).(match e return (\lambda (_: T).nat) with [(TSort n) \Rightarrow n | 
-(TLRef _) \Rightarrow n | (THead _ _ _) \Rightarrow n])) (TSort n) (TSort m) 
-H3) in (eq_ind nat m (\lambda (n0: nat).((eq T (TSort (next g n0)) t2) \to 
-(ty3 g c0 (TSort m) t2))) (\lambda (H5: (eq T (TSort (next g m)) t2)).(eq_ind 
-T (TSort (next g m)) (\lambda (t: T).(ty3 g c0 (TSort m) t)) (ty3_sort g c0 
-m) t2 H5)) n (sym_eq nat n m H4)))) c0 (sym_eq C c0 c0 H0) H1 H2)))) | 
-(tau0_abbr c0 d v i H0 w H1) \Rightarrow (\lambda (H2: (eq C c0 c0)).(\lambda 
-(H3: (eq T (TLRef i) (TSort m))).(\lambda (H4: (eq T (lift (S i) O w) 
-t2)).(eq_ind C c0 (\lambda (c: C).((eq T (TLRef i) (TSort m)) \to ((eq T 
-(lift (S i) O w) t2) \to ((getl i c (CHead d (Bind Abbr) v)) \to ((tau0 g d v 
-w) \to (ty3 g c0 (TSort m) t2)))))) (\lambda (H5: (eq T (TLRef i) (TSort 
-m))).(let H6 \def (eq_ind T (TLRef i) (\lambda (e: T).(match e return 
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (TSort m) H5) in 
-(False_ind ((eq T (lift (S i) O w) t2) \to ((getl i c0 (CHead d (Bind Abbr) 
-v)) \to ((tau0 g d v w) \to (ty3 g c0 (TSort m) t2)))) H6))) c0 (sym_eq C c0 
-c0 H2) H3 H4 H0 H1)))) | (tau0_abst c0 d v i H0 w H1) \Rightarrow (\lambda 
-(H2: (eq C c0 c0)).(\lambda (H3: (eq T (TLRef i) (TSort m))).(\lambda (H4: 
-(eq T (lift (S i) O v) t2)).(eq_ind C c0 (\lambda (c: C).((eq T (TLRef i) 
-(TSort m)) \to ((eq T (lift (S i) O v) t2) \to ((getl i c (CHead d (Bind 
-Abst) v)) \to ((tau0 g d v w) \to (ty3 g c0 (TSort m) t2)))))) (\lambda (H5: 
-(eq T (TLRef i) (TSort m))).(let H6 \def (eq_ind T (TLRef i) (\lambda (e: 
-T).(match e return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | 
-(TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (TSort m) 
-H5) in (False_ind ((eq T (lift (S i) O v) t2) \to ((getl i c0 (CHead d (Bind 
-Abst) v)) \to ((tau0 g d v w) \to (ty3 g c0 (TSort m) t2)))) H6))) c0 (sym_eq 
-C c0 c0 H2) H3 H4 H0 H1)))) | (tau0_bind b c0 v t1 t0 H0) \Rightarrow 
-(\lambda (H1: (eq C c0 c0)).(\lambda (H2: (eq T (THead (Bind b) v t1) (TSort 
-m))).(\lambda (H3: (eq T (THead (Bind b) v t0) t2)).(eq_ind C c0 (\lambda (c: 
-C).((eq T (THead (Bind b) v t1) (TSort m)) \to ((eq T (THead (Bind b) v t0) 
-t2) \to ((tau0 g (CHead c (Bind b) v) t1 t0) \to (ty3 g c0 (TSort m) t2))))) 
-(\lambda (H4: (eq T (THead (Bind b) v t1) (TSort m))).(let H5 \def (eq_ind T 
-(THead (Bind b) v t1) (\lambda (e: T).(match e return (\lambda (_: T).Prop) 
-with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ 
-_) \Rightarrow True])) I (TSort m) H4) in (False_ind ((eq T (THead (Bind b) v 
-t0) t2) \to ((tau0 g (CHead c0 (Bind b) v) t1 t0) \to (ty3 g c0 (TSort m) 
-t2))) H5))) c0 (sym_eq C c0 c0 H1) H2 H3 H0)))) | (tau0_appl c0 v t1 t0 H0) 
-\Rightarrow (\lambda (H1: (eq C c0 c0)).(\lambda (H2: (eq T (THead (Flat 
-Appl) v t1) (TSort m))).(\lambda (H3: (eq T (THead (Flat Appl) v t0) 
-t2)).(eq_ind C c0 (\lambda (c: C).((eq T (THead (Flat Appl) v t1) (TSort m)) 
-\to ((eq T (THead (Flat Appl) v t0) t2) \to ((tau0 g c t1 t0) \to (ty3 g c0 
-(TSort m) t2))))) (\lambda (H4: (eq T (THead (Flat Appl) v t1) (TSort 
-m))).(let H5 \def (eq_ind T (THead (Flat Appl) v t1) (\lambda (e: T).(match e 
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort m) H4) in 
-(False_ind ((eq T (THead (Flat Appl) v t0) t2) \to ((tau0 g c0 t1 t0) \to 
-(ty3 g c0 (TSort m) t2))) H5))) c0 (sym_eq C c0 c0 H1) H2 H3 H0)))) | 
-(tau0_cast c0 v1 v2 H0 t1 t0 H1) \Rightarrow (\lambda (H2: (eq C c0 
-c0)).(\lambda (H3: (eq T (THead (Flat Cast) v1 t1) (TSort m))).(\lambda (H4: 
-(eq T (THead (Flat Cast) v2 t0) t2)).(eq_ind C c0 (\lambda (c: C).((eq T 
-(THead (Flat Cast) v1 t1) (TSort m)) \to ((eq T (THead (Flat Cast) v2 t0) t2) 
-\to ((tau0 g c v1 v2) \to ((tau0 g c t1 t0) \to (ty3 g c0 (TSort m) t2)))))) 
-(\lambda (H5: (eq T (THead (Flat Cast) v1 t1) (TSort m))).(let H6 \def 
-(eq_ind T (THead (Flat Cast) v1 t1) (\lambda (e: T).(match e return (\lambda 
-(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
-| (THead _ _ _) \Rightarrow True])) I (TSort m) H5) in (False_ind ((eq T 
-(THead (Flat Cast) v2 t0) t2) \to ((tau0 g c0 v1 v2) \to ((tau0 g c0 t1 t0) 
-\to (ty3 g c0 (TSort m) t2)))) H6))) c0 (sym_eq C c0 c0 H2) H3 H4 H0 H1))))]) 
-in (H1 (refl_equal C c0) (refl_equal T (TSort m)) (refl_equal T t2))))))) 
-(\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).((tau0 g d u0 t2) \to (ty3 g d u0 
-t2))))).(\lambda (t2: T).(\lambda (H3: (tau0 g c0 (TLRef n) t2)).(let H4 \def 
-(match H3 return (\lambda (c: C).(\lambda (t: T).(\lambda (t0: T).(\lambda 
-(_: (tau0 ? c t t0)).((eq C c c0) \to ((eq T t (TLRef n)) \to ((eq T t0 t2) 
-\to (ty3 g c0 (TLRef n) t2)))))))) with [(tau0_sort c0 n0) \Rightarrow 
-(\lambda (H3: (eq C c0 c0)).(\lambda (H4: (eq T (TSort n0) (TLRef 
-n))).(\lambda (H5: (eq T (TSort (next g n0)) t2)).(eq_ind C c0 (\lambda (_: 
-C).((eq T (TSort n0) (TLRef n)) \to ((eq T (TSort (next g n0)) t2) \to (ty3 g 
-c0 (TLRef n) t2)))) (\lambda (H6: (eq T (TSort n0) (TLRef n))).(let H7 \def 
-(eq_ind T (TSort n0) (\lambda (e: T).(match e return (\lambda (_: T).Prop) 
-with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ 
-_) \Rightarrow False])) I (TLRef n) H6) in (False_ind ((eq T (TSort (next g 
-n0)) t2) \to (ty3 g c0 (TLRef n) t2)) H7))) c0 (sym_eq C c0 c0 H3) H4 H5)))) 
-| (tau0_abbr c0 d0 v i H3 w H4) \Rightarrow (\lambda (H5: (eq C c0 
-c0)).(\lambda (H6: (eq T (TLRef i) (TLRef n))).(\lambda (H7: (eq T (lift (S 
-i) O w) t2)).(eq_ind C c0 (\lambda (c: C).((eq T (TLRef i) (TLRef n)) \to 
-((eq T (lift (S i) O w) t2) \to ((getl i c (CHead d0 (Bind Abbr) v)) \to 
-((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) t2)))))) (\lambda (H8: (eq T (TLRef 
-i) (TLRef n))).(let H9 \def (f_equal T nat (\lambda (e: T).(match e return 
-(\lambda (_: T).nat) with [(TSort _) \Rightarrow i | (TLRef n) \Rightarrow n 
-| (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef n) H8) in (eq_ind nat n 
-(\lambda (n0: nat).((eq T (lift (S n0) O w) t2) \to ((getl n0 c0 (CHead d0 
-(Bind Abbr) v)) \to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) t2))))) (\lambda 
-(H10: (eq T (lift (S n) O w) t2)).(eq_ind T (lift (S n) O w) (\lambda (t: 
-T).((getl n c0 (CHead d0 (Bind Abbr) v)) \to ((tau0 g d0 v w) \to (ty3 g c0 
-(TLRef n) t)))) (\lambda (H11: (getl n c0 (CHead d0 (Bind Abbr) v))).(\lambda 
-(H12: (tau0 g d0 v w)).(let H13 \def (eq_ind C (CHead d (Bind Abbr) u0) 
-(\lambda (c: C).(getl n c0 c)) H0 (CHead d0 (Bind Abbr) v) (getl_mono c0 
-(CHead d (Bind Abbr) u0) n H0 (CHead d0 (Bind Abbr) v) H11)) in (let H14 \def 
-(f_equal C C (\lambda (e: C).(match e return (\lambda (_: C).C) with [(CSort 
-_) \Rightarrow d | (CHead c _ _) \Rightarrow c])) (CHead d (Bind Abbr) u0) 
-(CHead d0 (Bind Abbr) v) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead 
-d0 (Bind Abbr) v) H11)) in ((let H15 \def (f_equal C T (\lambda (e: C).(match 
-e return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) 
-\Rightarrow t])) (CHead d (Bind Abbr) u0) (CHead d0 (Bind Abbr) v) (getl_mono 
-c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0 (Bind Abbr) v) H11)) in (\lambda 
-(H16: (eq C d d0)).(let H17 \def (eq_ind_r T v (\lambda (t: T).(getl n c0 
-(CHead d0 (Bind Abbr) t))) H13 u0 H15) in (let H18 \def (eq_ind_r T v 
-(\lambda (t: T).(tau0 g d0 t w)) H12 u0 H15) in (let H19 \def (eq_ind_r C d0 
-(\lambda (c: C).(getl n c0 (CHead c (Bind Abbr) u0))) H17 d H16) in (let H20 
-\def (eq_ind_r C d0 (\lambda (c: C).(tau0 g c u0 w)) H18 d H16) in (ty3_abbr 
-g n c0 d u0 H19 w (H2 w H20)))))))) H14))))) t2 H10)) i (sym_eq nat i n 
-H9)))) c0 (sym_eq C c0 c0 H5) H6 H7 H3 H4)))) | (tau0_abst c0 d0 v i H3 w H4) 
-\Rightarrow (\lambda (H5: (eq C c0 c0)).(\lambda (H6: (eq T (TLRef i) (TLRef 
-n))).(\lambda (H7: (eq T (lift (S i) O v) t2)).(eq_ind C c0 (\lambda (c: 
-C).((eq T (TLRef i) (TLRef n)) \to ((eq T (lift (S i) O v) t2) \to ((getl i c 
-(CHead d0 (Bind Abst) v)) \to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) 
-t2)))))) (\lambda (H8: (eq T (TLRef i) (TLRef n))).(let H9 \def (f_equal T 
-nat (\lambda (e: T).(match e return (\lambda (_: T).nat) with [(TSort _) 
-\Rightarrow i | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i])) 
-(TLRef i) (TLRef n) H8) in (eq_ind nat n (\lambda (n0: nat).((eq T (lift (S 
-n0) O v) t2) \to ((getl n0 c0 (CHead d0 (Bind Abst) v)) \to ((tau0 g d0 v w) 
-\to (ty3 g c0 (TLRef n) t2))))) (\lambda (H10: (eq T (lift (S n) O v) 
-t2)).(eq_ind T (lift (S n) O v) (\lambda (t: T).((getl n c0 (CHead d0 (Bind 
-Abst) v)) \to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) t)))) (\lambda (H11: 
-(getl n c0 (CHead d0 (Bind Abst) v))).(\lambda (_: (tau0 g d0 v w)).(let H2 
-\def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda (c: C).(getl n c0 c)) H0 
-(CHead d0 (Bind Abst) v) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead 
-d0 (Bind Abst) v) H11)) in (let H13 \def (eq_ind C (CHead d (Bind Abbr) u0) 
-(\lambda (ee: C).(match ee return (\lambda (_: C).Prop) with [(CSort _) 
-\Rightarrow False | (CHead _ k _) \Rightarrow (match k return (\lambda (_: 
-K).Prop) with [(Bind b) \Rightarrow (match b return (\lambda (_: B).Prop) 
-with [Abbr \Rightarrow True | Abst \Rightarrow False | Void \Rightarrow 
-False]) | (Flat _) \Rightarrow False])])) I (CHead d0 (Bind Abst) v) 
-(getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0 (Bind Abst) v) H11)) in 
-(False_ind (ty3 g c0 (TLRef n) (lift (S n) O v)) H13))))) t2 H10)) i (sym_eq 
-nat i n H9)))) c0 (sym_eq C c0 c0 H5) H6 H7 H3 H4)))) | (tau0_bind b c0 v t1 
-t0 H3) \Rightarrow (\lambda (H4: (eq C c0 c0)).(\lambda (H5: (eq T (THead 
-(Bind b) v t1) (TLRef n))).(\lambda (H6: (eq T (THead (Bind b) v t0) 
-t2)).(eq_ind C c0 (\lambda (c: C).((eq T (THead (Bind b) v t1) (TLRef n)) \to 
-((eq T (THead (Bind b) v t0) t2) \to ((tau0 g (CHead c (Bind b) v) t1 t0) \to 
-(ty3 g c0 (TLRef n) t2))))) (\lambda (H7: (eq T (THead (Bind b) v t1) (TLRef 
-n))).(let H8 \def (eq_ind T (THead (Bind b) v t1) (\lambda (e: T).(match e 
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H7) in 
-(False_ind ((eq T (THead (Bind b) v t0) t2) \to ((tau0 g (CHead c0 (Bind b) 
-v) t1 t0) \to (ty3 g c0 (TLRef n) t2))) H8))) c0 (sym_eq C c0 c0 H4) H5 H6 
-H3)))) | (tau0_appl c0 v t1 t0 H3) \Rightarrow (\lambda (H4: (eq C c0 
-c0)).(\lambda (H5: (eq T (THead (Flat Appl) v t1) (TLRef n))).(\lambda (H6: 
-(eq T (THead (Flat Appl) v t0) t2)).(eq_ind C c0 (\lambda (c: C).((eq T 
-(THead (Flat Appl) v t1) (TLRef n)) \to ((eq T (THead (Flat Appl) v t0) t2) 
-\to ((tau0 g c t1 t0) \to (ty3 g c0 (TLRef n) t2))))) (\lambda (H7: (eq T 
-(THead (Flat Appl) v t1) (TLRef n))).(let H8 \def (eq_ind T (THead (Flat 
-Appl) v t1) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) 
-\Rightarrow True])) I (TLRef n) H7) in (False_ind ((eq T (THead (Flat Appl) v 
-t0) t2) \to ((tau0 g c0 t1 t0) \to (ty3 g c0 (TLRef n) t2))) H8))) c0 (sym_eq 
-C c0 c0 H4) H5 H6 H3)))) | (tau0_cast c0 v1 v2 H3 t1 t0 H4) \Rightarrow 
-(\lambda (H5: (eq C c0 c0)).(\lambda (H6: (eq T (THead (Flat Cast) v1 t1) 
-(TLRef n))).(\lambda (H7: (eq T (THead (Flat Cast) v2 t0) t2)).(eq_ind C c0 
-(\lambda (c: C).((eq T (THead (Flat Cast) v1 t1) (TLRef n)) \to ((eq T (THead 
-(Flat Cast) v2 t0) t2) \to ((tau0 g c v1 v2) \to ((tau0 g c t1 t0) \to (ty3 g 
-c0 (TLRef n) t2)))))) (\lambda (H8: (eq T (THead (Flat Cast) v1 t1) (TLRef 
-n))).(let H9 \def (eq_ind T (THead (Flat Cast) v1 t1) (\lambda (e: T).(match 
-e return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H8) in 
-(False_ind ((eq T (THead (Flat Cast) v2 t0) t2) \to ((tau0 g c0 v1 v2) \to 
-((tau0 g c0 t1 t0) \to (ty3 g c0 (TLRef n) t2)))) H9))) c0 (sym_eq C c0 c0 
-H5) H6 H7 H3 H4))))]) in (H4 (refl_equal C c0) (refl_equal T (TLRef n)) 
-(refl_equal T t2))))))))))))) (\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).((tau0 g d u0 t2) \to (ty3 g d u0 t2))))).(\lambda (t2: T).(\lambda 
-(H3: (tau0 g c0 (TLRef n) t2)).(let H4 \def (match H3 return (\lambda (c: 
-C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (tau0 ? c t t0)).((eq C c 
-c0) \to ((eq T t (TLRef n)) \to ((eq T t0 t2) \to (ty3 g c0 (TLRef n) 
-t2)))))))) with [(tau0_sort c0 n0) \Rightarrow (\lambda (H3: (eq C c0 
-c0)).(\lambda (H4: (eq T (TSort n0) (TLRef n))).(\lambda (H5: (eq T (TSort 
-(next g n0)) t2)).(eq_ind C c0 (\lambda (_: C).((eq T (TSort n0) (TLRef n)) 
-\to ((eq T (TSort (next g n0)) t2) \to (ty3 g c0 (TLRef n) t2)))) (\lambda 
-(H6: (eq T (TSort n0) (TLRef n))).(let H7 \def (eq_ind T (TSort n0) (\lambda 
-(e: T).(match e return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True 
-| (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef 
-n) H6) in (False_ind ((eq T (TSort (next g n0)) t2) \to (ty3 g c0 (TLRef n) 
-t2)) H7))) c0 (sym_eq C c0 c0 H3) H4 H5)))) | (tau0_abbr c0 d0 v i H3 w H4) 
-\Rightarrow (\lambda (H5: (eq C c0 c0)).(\lambda (H6: (eq T (TLRef i) (TLRef 
-n))).(\lambda (H7: (eq T (lift (S i) O w) t2)).(eq_ind C c0 (\lambda (c: 
-C).((eq T (TLRef i) (TLRef n)) \to ((eq T (lift (S i) O w) t2) \to ((getl i c 
-(CHead d0 (Bind Abbr) v)) \to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) 
-t2)))))) (\lambda (H8: (eq T (TLRef i) (TLRef n))).(let H9 \def (f_equal T 
-nat (\lambda (e: T).(match e return (\lambda (_: T).nat) with [(TSort _) 
-\Rightarrow i | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i])) 
-(TLRef i) (TLRef n) H8) in (eq_ind nat n (\lambda (n0: nat).((eq T (lift (S 
-n0) O w) t2) \to ((getl n0 c0 (CHead d0 (Bind Abbr) v)) \to ((tau0 g d0 v w) 
-\to (ty3 g c0 (TLRef n) t2))))) (\lambda (H10: (eq T (lift (S n) O w) 
-t2)).(eq_ind T (lift (S n) O w) (\lambda (t: T).((getl n c0 (CHead d0 (Bind 
-Abbr) v)) \to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) t)))) (\lambda (H11: 
-(getl n c0 (CHead d0 (Bind Abbr) v))).(\lambda (_: (tau0 g d0 v w)).(let H2 
-\def (eq_ind C (CHead d (Bind Abst) u0) (\lambda (c: C).(getl n c0 c)) H0 
-(CHead d0 (Bind Abbr) v) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead 
-d0 (Bind Abbr) v) H11)) in (let H13 \def (eq_ind C (CHead d (Bind Abst) u0) 
-(\lambda (ee: C).(match ee return (\lambda (_: C).Prop) with [(CSort _) 
-\Rightarrow False | (CHead _ k _) \Rightarrow (match k return (\lambda (_: 
-K).Prop) with [(Bind b) \Rightarrow (match b return (\lambda (_: B).Prop) 
-with [Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow 
-False]) | (Flat _) \Rightarrow False])])) I (CHead d0 (Bind Abbr) v) 
-(getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead d0 (Bind Abbr) v) H11)) in 
-(False_ind (ty3 g c0 (TLRef n) (lift (S n) O w)) H13))))) t2 H10)) i (sym_eq 
-nat i n H9)))) c0 (sym_eq C c0 c0 H5) H6 H7 H3 H4)))) | (tau0_abst c0 d0 v i 
-H3 w H4) \Rightarrow (\lambda (H5: (eq C c0 c0)).(\lambda (H6: (eq T (TLRef 
-i) (TLRef n))).(\lambda (H7: (eq T (lift (S i) O v) t2)).(eq_ind C c0 
-(\lambda (c: C).((eq T (TLRef i) (TLRef n)) \to ((eq T (lift (S i) O v) t2) 
-\to ((getl i c (CHead d0 (Bind Abst) v)) \to ((tau0 g d0 v w) \to (ty3 g c0 
-(TLRef n) t2)))))) (\lambda (H8: (eq T (TLRef i) (TLRef n))).(let H9 \def 
-(f_equal T nat (\lambda (e: T).(match e return (\lambda (_: T).nat) with 
-[(TSort _) \Rightarrow i | (TLRef n) \Rightarrow n | (THead _ _ _) 
-\Rightarrow i])) (TLRef i) (TLRef n) H8) in (eq_ind nat n (\lambda (n0: 
-nat).((eq T (lift (S n0) O v) t2) \to ((getl n0 c0 (CHead d0 (Bind Abst) v)) 
-\to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) t2))))) (\lambda (H10: (eq T 
-(lift (S n) O v) t2)).(eq_ind T (lift (S n) O v) (\lambda (t: T).((getl n c0 
-(CHead d0 (Bind Abst) v)) \to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) t)))) 
-(\lambda (H11: (getl n c0 (CHead d0 (Bind Abst) v))).(\lambda (H12: (tau0 g 
-d0 v w)).(let H2 \def (eq_ind C (CHead d (Bind Abst) u0) (\lambda (c: 
-C).(getl n c0 c)) H0 (CHead d0 (Bind Abst) v) (getl_mono c0 (CHead d (Bind 
-Abst) u0) n H0 (CHead d0 (Bind Abst) v) H11)) in (let H13 \def (f_equal C C 
-(\lambda (e: C).(match e return (\lambda (_: C).C) with [(CSort _) 
-\Rightarrow d | (CHead c _ _) \Rightarrow c])) (CHead d (Bind Abst) u0) 
-(CHead d0 (Bind Abst) v) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead 
-d0 (Bind Abst) v) H11)) in ((let H14 \def (f_equal C T (\lambda (e: C).(match 
-e return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) 
-\Rightarrow t])) (CHead d (Bind Abst) u0) (CHead d0 (Bind Abst) v) (getl_mono 
-c0 (CHead d (Bind Abst) u0) n H0 (CHead d0 (Bind Abst) v) H11)) in (\lambda 
-(H15: (eq C d d0)).(let H16 \def (eq_ind_r T v (\lambda (t: T).(getl n c0 
-(CHead d0 (Bind Abst) t))) H2 u0 H14) in (let H17 \def (eq_ind_r T v (\lambda 
-(t: T).(tau0 g d0 t w)) H12 u0 H14) in (eq_ind T u0 (\lambda (t: T).(ty3 g c0 
-(TLRef n) (lift (S n) O t))) (let H18 \def (eq_ind_r C d0 (\lambda (c: 
-C).(getl n c0 (CHead c (Bind Abst) u0))) H16 d H15) in (let H19 \def 
-(eq_ind_r C d0 (\lambda (c: C).(tau0 g c u0 w)) H17 d H15) in (ty3_abst g n 
-c0 d u0 H18 t H1))) v H14))))) H13))))) t2 H10)) i (sym_eq nat i n H9)))) c0 
-(sym_eq C c0 c0 H5) H6 H7 H3 H4)))) | (tau0_bind b c0 v t1 t0 H3) \Rightarrow 
-(\lambda (H4: (eq C c0 c0)).(\lambda (H5: (eq T (THead (Bind b) v t1) (TLRef 
-n))).(\lambda (H6: (eq T (THead (Bind b) v t0) t2)).(eq_ind C c0 (\lambda (c: 
-C).((eq T (THead (Bind b) v t1) (TLRef n)) \to ((eq T (THead (Bind b) v t0) 
-t2) \to ((tau0 g (CHead c (Bind b) v) t1 t0) \to (ty3 g c0 (TLRef n) t2))))) 
-(\lambda (H7: (eq T (THead (Bind b) v t1) (TLRef n))).(let H8 \def (eq_ind T 
-(THead (Bind b) v t1) (\lambda (e: T).(match e return (\lambda (_: T).Prop) 
-with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ 
-_) \Rightarrow True])) I (TLRef n) H7) in (False_ind ((eq T (THead (Bind b) v 
-t0) t2) \to ((tau0 g (CHead c0 (Bind b) v) t1 t0) \to (ty3 g c0 (TLRef n) 
-t2))) H8))) c0 (sym_eq C c0 c0 H4) H5 H6 H3)))) | (tau0_appl c0 v t1 t0 H3) 
-\Rightarrow (\lambda (H4: (eq C c0 c0)).(\lambda (H5: (eq T (THead (Flat 
-Appl) v t1) (TLRef n))).(\lambda (H6: (eq T (THead (Flat Appl) v t0) 
-t2)).(eq_ind C c0 (\lambda (c: C).((eq T (THead (Flat Appl) v t1) (TLRef n)) 
-\to ((eq T (THead (Flat Appl) v t0) t2) \to ((tau0 g c t1 t0) \to (ty3 g c0 
-(TLRef n) t2))))) (\lambda (H7: (eq T (THead (Flat Appl) v t1) (TLRef 
-n))).(let H8 \def (eq_ind T (THead (Flat Appl) v t1) (\lambda (e: T).(match e 
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H7) in 
-(False_ind ((eq T (THead (Flat Appl) v t0) t2) \to ((tau0 g c0 t1 t0) \to 
-(ty3 g c0 (TLRef n) t2))) H8))) c0 (sym_eq C c0 c0 H4) H5 H6 H3)))) | 
-(tau0_cast c0 v1 v2 H3 t1 t0 H4) \Rightarrow (\lambda (H5: (eq C c0 
-c0)).(\lambda (H6: (eq T (THead (Flat Cast) v1 t1) (TLRef n))).(\lambda (H7: 
-(eq T (THead (Flat Cast) v2 t0) t2)).(eq_ind C c0 (\lambda (c: C).((eq T 
-(THead (Flat Cast) v1 t1) (TLRef n)) \to ((eq T (THead (Flat Cast) v2 t0) t2) 
-\to ((tau0 g c v1 v2) \to ((tau0 g c t1 t0) \to (ty3 g c0 (TLRef n) t2)))))) 
-(\lambda (H8: (eq T (THead (Flat Cast) v1 t1) (TLRef n))).(let H9 \def 
-(eq_ind T (THead (Flat Cast) v1 t1) (\lambda (e: T).(match e return (\lambda 
-(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
-| (THead _ _ _) \Rightarrow True])) I (TLRef n) H8) in (False_ind ((eq T 
-(THead (Flat Cast) v2 t0) t2) \to ((tau0 g c0 v1 v2) \to ((tau0 g c0 t1 t0) 
-\to (ty3 g c0 (TLRef n) t2)))) H9))) c0 (sym_eq C c0 c0 H5) H6 H7 H3 H4))))]) 
-in (H4 (refl_equal C c0) (refl_equal T (TLRef n)) (refl_equal T 
-t2))))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (t: T).(\lambda 
-(H0: (ty3 g c0 u0 t)).(\lambda (_: ((\forall (t2: T).((tau0 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 
-(t3: T).((tau0 g (CHead c0 (Bind b) u0) t2 t3) \to (ty3 g (CHead c0 (Bind b) 
-u0) t2 t3))))).(\lambda (t0: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u0) t3 
-t0)).(\lambda (_: ((\forall (t2: T).((tau0 g (CHead c0 (Bind b) u0) t3 t2) 
-\to (ty3 g (CHead c0 (Bind b) u0) t3 t2))))).(\lambda (t4: T).(\lambda (H6: 
-(tau0 g c0 (THead (Bind b) u0 t2) t4)).(let H7 \def (match H6 return (\lambda 
-(c: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (tau0 ? c t t0)).((eq C 
-c c0) \to ((eq T t (THead (Bind b) u0 t2)) \to ((eq T t0 t4) \to (ty3 g c0 
-(THead (Bind b) u0 t2) t4)))))))) with [(tau0_sort c0 n) \Rightarrow (\lambda 
-(H6: (eq C c0 c0)).(\lambda (H7: (eq T (TSort n) (THead (Bind b) u0 
-t2))).(\lambda (H8: (eq T (TSort (next g n)) t4)).(eq_ind C c0 (\lambda (_: 
-C).((eq T (TSort n) (THead (Bind b) u0 t2)) \to ((eq T (TSort (next g n)) t4) 
-\to (ty3 g c0 (THead (Bind b) u0 t2) t4)))) (\lambda (H9: (eq T (TSort n) 
-(THead (Bind b) u0 t2))).(let H10 \def (eq_ind T (TSort n) (\lambda (e: 
-T).(match e return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | 
-(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead 
-(Bind b) u0 t2) H9) in (False_ind ((eq T (TSort (next g n)) t4) \to (ty3 g c0 
-(THead (Bind b) u0 t2) t4)) H10))) c0 (sym_eq C c0 c0 H6) H7 H8)))) | 
-(tau0_abbr c0 d v i H6 w H7) \Rightarrow (\lambda (H8: (eq C c0 c0)).(\lambda 
-(H9: (eq T (TLRef i) (THead (Bind b) u0 t2))).(\lambda (H10: (eq T (lift (S 
-i) O w) t4)).(eq_ind C c0 (\lambda (c: C).((eq T (TLRef i) (THead (Bind b) u0 
-t2)) \to ((eq T (lift (S i) O w) t4) \to ((getl i c (CHead d (Bind Abbr) v)) 
-\to ((tau0 g d v w) \to (ty3 g c0 (THead (Bind b) u0 t2) t4)))))) (\lambda 
-(H11: (eq T (TLRef i) (THead (Bind b) u0 t2))).(let H12 \def (eq_ind T (TLRef 
-i) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow 
-False])) I (THead (Bind b) u0 t2) H11) in (False_ind ((eq T (lift (S i) O w) 
-t4) \to ((getl i c0 (CHead d (Bind Abbr) v)) \to ((tau0 g d v w) \to (ty3 g 
-c0 (THead (Bind b) u0 t2) t4)))) H12))) c0 (sym_eq C c0 c0 H8) H9 H10 H6 
-H7)))) | (tau0_abst c0 d v i H6 w H7) \Rightarrow (\lambda (H8: (eq C c0 
-c0)).(\lambda (H9: (eq T (TLRef i) (THead (Bind b) u0 t2))).(\lambda (H10: 
-(eq T (lift (S i) O v) t4)).(eq_ind C c0 (\lambda (c: C).((eq T (TLRef i) 
-(THead (Bind b) u0 t2)) \to ((eq T (lift (S i) O v) t4) \to ((getl i c (CHead 
-d (Bind Abst) v)) \to ((tau0 g d v w) \to (ty3 g c0 (THead (Bind b) u0 t2) 
-t4)))))) (\lambda (H11: (eq T (TLRef i) (THead (Bind b) u0 t2))).(let H12 
-\def (eq_ind T (TLRef i) (\lambda (e: T).(match e return (\lambda (_: 
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | 
-(THead _ _ _) \Rightarrow False])) I (THead (Bind b) u0 t2) H11) in 
-(False_ind ((eq T (lift (S i) O v) t4) \to ((getl i c0 (CHead d (Bind Abst) 
-v)) \to ((tau0 g d v w) \to (ty3 g c0 (THead (Bind b) u0 t2) t4)))) H12))) c0 
-(sym_eq C c0 c0 H8) H9 H10 H6 H7)))) | (tau0_bind b0 c0 v t4 t5 H6) 
-\Rightarrow (\lambda (H7: (eq C c0 c0)).(\lambda (H8: (eq T (THead (Bind b0) 
-v t4) (THead (Bind b) u0 t2))).(\lambda (H9: (eq T (THead (Bind b0) v t5) 
-t4)).(eq_ind C c0 (\lambda (c: C).((eq T (THead (Bind b0) v t4) (THead (Bind 
-b) u0 t2)) \to ((eq T (THead (Bind b0) v t5) t4) \to ((tau0 g (CHead c (Bind 
-b0) v) t4 t5) \to (ty3 g c0 (THead (Bind b) u0 t2) t4))))) (\lambda (H10: (eq 
-T (THead (Bind b0) v t4) (THead (Bind b) u0 t2))).(let H11 \def (f_equal T T 
-(\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow t4 | (TLRef _) \Rightarrow t4 | (THead _ _ t) \Rightarrow t])) 
-(THead (Bind b0) v t4) (THead (Bind b) u0 t2) H10) in ((let H12 \def (f_equal 
-T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow v | (TLRef _) \Rightarrow v | (THead _ t _) \Rightarrow t])) 
-(THead (Bind b0) v t4) (THead (Bind b) u0 t2) H10) in ((let H13 \def (f_equal 
-T B (\lambda (e: T).(match e return (\lambda (_: T).B) with [(TSort _) 
-\Rightarrow b0 | (TLRef _) \Rightarrow b0 | (THead k _ _) \Rightarrow (match 
-k return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) 
-\Rightarrow b0])])) (THead (Bind b0) v t4) (THead (Bind b) u0 t2) H10) in 
-(eq_ind B b (\lambda (b1: B).((eq T v u0) \to ((eq T t4 t2) \to ((eq T (THead 
-(Bind b1) v t5) t4) \to ((tau0 g (CHead c0 (Bind b1) v) t4 t5) \to (ty3 g c0 
-(THead (Bind b) u0 t2) t4)))))) (\lambda (H14: (eq T v u0)).(eq_ind T u0 
-(\lambda (t: T).((eq T t4 t2) \to ((eq T (THead (Bind b) t t5) t4) \to ((tau0 
-g (CHead c0 (Bind b) t) t4 t5) \to (ty3 g c0 (THead (Bind b) u0 t2) t4))))) 
-(\lambda (H15: (eq T t4 t2)).(eq_ind T t2 (\lambda (t: T).((eq T (THead (Bind 
-b) u0 t5) t4) \to ((tau0 g (CHead c0 (Bind b) u0) t t5) \to (ty3 g c0 (THead 
-(Bind b) u0 t2) t4)))) (\lambda (H16: (eq T (THead (Bind b) u0 t5) 
-t4)).(eq_ind T (THead (Bind b) u0 t5) (\lambda (t: T).((tau0 g (CHead c0 
-(Bind b) u0) t2 t5) \to (ty3 g c0 (THead (Bind b) u0 t2) t))) (\lambda (H17: 
-(tau0 g (CHead c0 (Bind b) u0) t2 t5)).(let H_y \def (H3 t5 H17) in (ex_ind T 
-(\lambda (t: T).(ty3 g (CHead c0 (Bind b) u0) t5 t)) (ty3 g c0 (THead (Bind 
-b) u0 t2) (THead (Bind b) u0 t5)) (\lambda (x: T).(\lambda (H1: (ty3 g (CHead 
-c0 (Bind b) u0) t5 x)).(ty3_bind g c0 u0 t H0 b t2 t5 H_y x H1))) 
-(ty3_correct g (CHead c0 (Bind b) u0) t2 t5 H_y)))) t4 H16)) t4 (sym_eq T t4 
-t2 H15))) v (sym_eq T v u0 H14))) b0 (sym_eq B b0 b H13))) H12)) H11))) c0 
-(sym_eq C c0 c0 H7) H8 H9 H6)))) | (tau0_appl c0 v t4 t5 H6) \Rightarrow 
-(\lambda (H7: (eq C c0 c0)).(\lambda (H8: (eq T (THead (Flat Appl) v t4) 
-(THead (Bind b) u0 t2))).(\lambda (H9: (eq T (THead (Flat Appl) v t5) 
-t4)).(eq_ind C c0 (\lambda (c: C).((eq T (THead (Flat Appl) v t4) (THead 
-(Bind b) u0 t2)) \to ((eq T (THead (Flat Appl) v t5) t4) \to ((tau0 g c t4 
-t5) \to (ty3 g c0 (THead (Bind b) u0 t2) t4))))) (\lambda (H10: (eq T (THead 
-(Flat Appl) v t4) (THead (Bind b) u0 t2))).(let H11 \def (eq_ind T (THead 
-(Flat Appl) v t4) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
-\Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow 
-False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u0 t2) H10) in 
-(False_ind ((eq T (THead (Flat Appl) v t5) t4) \to ((tau0 g c0 t4 t5) \to 
-(ty3 g c0 (THead (Bind b) u0 t2) t4))) H11))) c0 (sym_eq C c0 c0 H7) H8 H9 
-H6)))) | (tau0_cast c0 v1 v2 H6 t4 t5 H7) \Rightarrow (\lambda (H8: (eq C c0 
-c0)).(\lambda (H9: (eq T (THead (Flat Cast) v1 t4) (THead (Bind b) u0 
-t2))).(\lambda (H10: (eq T (THead (Flat Cast) v2 t5) t4)).(eq_ind C c0 
-(\lambda (c: C).((eq T (THead (Flat Cast) v1 t4) (THead (Bind b) u0 t2)) \to 
-((eq T (THead (Flat Cast) v2 t5) t4) \to ((tau0 g c v1 v2) \to ((tau0 g c t4 
-t5) \to (ty3 g c0 (THead (Bind b) u0 t2) t4)))))) (\lambda (H11: (eq T (THead 
-(Flat Cast) v1 t4) (THead (Bind b) u0 t2))).(let H12 \def (eq_ind T (THead 
-(Flat Cast) v1 t4) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
-\Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow 
-False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u0 t2) H11) in 
-(False_ind ((eq T (THead (Flat Cast) v2 t5) t4) \to ((tau0 g c0 v1 v2) \to 
-((tau0 g c0 t4 t5) \to (ty3 g c0 (THead (Bind b) u0 t2) t4)))) H12))) c0 
-(sym_eq C c0 c0 H8) H9 H10 H6 H7))))]) in (H7 (refl_equal C c0) (refl_equal T 
-(THead (Bind b) u0 t2)) (refl_equal T t4)))))))))))))))))) (\lambda (c0: 
-C).(\lambda (w: T).(\lambda (u0: T).(\lambda (H0: (ty3 g c0 w u0)).(\lambda 
-(_: ((\forall (t2: T).((tau0 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).((tau0 g c0 v t2) \to (ty3 g c0 v 
-t2))))).(\lambda (t2: T).(\lambda (H4: (tau0 g c0 (THead (Flat Appl) w v) 
-t2)).(let H5 \def (match H4 return (\lambda (c: C).(\lambda (t: T).(\lambda 
-(t0: T).(\lambda (_: (tau0 ? c t t0)).((eq C c c0) \to ((eq T t (THead (Flat 
-Appl) w v)) \to ((eq T t0 t2) \to (ty3 g c0 (THead (Flat Appl) w v) 
-t2)))))))) with [(tau0_sort c0 n) \Rightarrow (\lambda (H4: (eq C c0 
-c0)).(\lambda (H5: (eq T (TSort n) (THead (Flat Appl) w v))).(\lambda (H6: 
-(eq T (TSort (next g n)) t2)).(eq_ind C c0 (\lambda (_: C).((eq T (TSort n) 
-(THead (Flat Appl) w v)) \to ((eq T (TSort (next g n)) t2) \to (ty3 g c0 
-(THead (Flat Appl) w v) t2)))) (\lambda (H7: (eq T (TSort n) (THead (Flat 
-Appl) w v))).(let H8 \def (eq_ind T (TSort n) (\lambda (e: T).(match e return 
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) 
-\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) w 
-v) H7) in (False_ind ((eq T (TSort (next g n)) t2) \to (ty3 g c0 (THead (Flat 
-Appl) w v) t2)) H8))) c0 (sym_eq C c0 c0 H4) H5 H6)))) | (tau0_abbr c0 d v0 i 
-H4 w0 H5) \Rightarrow (\lambda (H6: (eq C c0 c0)).(\lambda (H7: (eq T (TLRef 
-i) (THead (Flat Appl) w v))).(\lambda (H8: (eq T (lift (S i) O w0) 
-t2)).(eq_ind C c0 (\lambda (c: C).((eq T (TLRef i) (THead (Flat Appl) w v)) 
-\to ((eq T (lift (S i) O w0) t2) \to ((getl i c (CHead d (Bind Abbr) v0)) \to 
-((tau0 g d v0 w0) \to (ty3 g c0 (THead (Flat Appl) w v) t2)))))) (\lambda 
-(H9: (eq T (TLRef i) (THead (Flat Appl) w v))).(let H10 \def (eq_ind T (TLRef 
-i) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow 
-False])) I (THead (Flat Appl) w v) H9) in (False_ind ((eq T (lift (S i) O w0) 
-t2) \to ((getl i c0 (CHead d (Bind Abbr) v0)) \to ((tau0 g d v0 w0) \to (ty3 
-g c0 (THead (Flat Appl) w v) t2)))) H10))) c0 (sym_eq C c0 c0 H6) H7 H8 H4 
-H5)))) | (tau0_abst c0 d v0 i H4 w0 H5) \Rightarrow (\lambda (H6: (eq C c0 
-c0)).(\lambda (H7: (eq T (TLRef i) (THead (Flat Appl) w v))).(\lambda (H8: 
-(eq T (lift (S i) O v0) t2)).(eq_ind C c0 (\lambda (c: C).((eq T (TLRef i) 
-(THead (Flat Appl) w v)) \to ((eq T (lift (S i) O v0) t2) \to ((getl i c 
-(CHead d (Bind Abst) v0)) \to ((tau0 g d v0 w0) \to (ty3 g c0 (THead (Flat 
-Appl) w v) t2)))))) (\lambda (H9: (eq T (TLRef i) (THead (Flat Appl) w 
-v))).(let H10 \def (eq_ind T (TLRef i) (\lambda (e: T).(match e return 
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) w 
-v) H9) in (False_ind ((eq T (lift (S i) O v0) t2) \to ((getl i c0 (CHead d 
-(Bind Abst) v0)) \to ((tau0 g d v0 w0) \to (ty3 g c0 (THead (Flat Appl) w v) 
-t2)))) H10))) c0 (sym_eq C c0 c0 H6) H7 H8 H4 H5)))) | (tau0_bind b c0 v0 t1 
-t0 H4) \Rightarrow (\lambda (H5: (eq C c0 c0)).(\lambda (H6: (eq T (THead 
-(Bind b) v0 t1) (THead (Flat Appl) w v))).(\lambda (H7: (eq T (THead (Bind b) 
-v0 t0) t2)).(eq_ind C c0 (\lambda (c: C).((eq T (THead (Bind b) v0 t1) (THead 
-(Flat Appl) w v)) \to ((eq T (THead (Bind b) v0 t0) t2) \to ((tau0 g (CHead c 
-(Bind b) v0) t1 t0) \to (ty3 g c0 (THead (Flat Appl) w v) t2))))) (\lambda 
-(H8: (eq T (THead (Bind b) v0 t1) (THead (Flat Appl) w v))).(let H9 \def 
-(eq_ind T (THead (Bind b) v0 t1) (\lambda (e: T).(match e return (\lambda (_: 
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
-(THead k _ _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind 
-_) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) w 
-v) H8) in (False_ind ((eq T (THead (Bind b) v0 t0) t2) \to ((tau0 g (CHead c0 
-(Bind b) v0) t1 t0) \to (ty3 g c0 (THead (Flat Appl) w v) t2))) H9))) c0 
-(sym_eq C c0 c0 H5) H6 H7 H4)))) | (tau0_appl c0 v0 t1 t0 H4) \Rightarrow 
-(\lambda (H5: (eq C c0 c0)).(\lambda (H6: (eq T (THead (Flat Appl) v0 t1) 
-(THead (Flat Appl) w v))).(\lambda (H7: (eq T (THead (Flat Appl) v0 t0) 
-t2)).(eq_ind C c0 (\lambda (c: C).((eq T (THead (Flat Appl) v0 t1) (THead 
-(Flat Appl) w v)) \to ((eq T (THead (Flat Appl) v0 t0) t2) \to ((tau0 g c t1 
-t0) \to (ty3 g c0 (THead (Flat Appl) w v) t2))))) (\lambda (H8: (eq T (THead 
-(Flat Appl) v0 t1) (THead (Flat Appl) w v))).(let H9 \def (f_equal T T 
-(\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ _ t) \Rightarrow t])) 
-(THead (Flat Appl) v0 t1) (THead (Flat Appl) w v) H8) in ((let H10 \def 
-(f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort 
-_) \Rightarrow v0 | (TLRef _) \Rightarrow v0 | (THead _ t _) \Rightarrow t])) 
-(THead (Flat Appl) v0 t1) (THead (Flat Appl) w v) H8) in (eq_ind T w (\lambda 
-(t: T).((eq T t1 v) \to ((eq T (THead (Flat Appl) t t0) t2) \to ((tau0 g c0 
-t1 t0) \to (ty3 g c0 (THead (Flat Appl) w v) t2))))) (\lambda (H11: (eq T t1 
-v)).(eq_ind T v (\lambda (t: T).((eq T (THead (Flat Appl) w t0) t2) \to 
-((tau0 g c0 t t0) \to (ty3 g c0 (THead (Flat Appl) w v) t2)))) (\lambda (H12: 
-(eq T (THead (Flat Appl) w t0) t2)).(eq_ind T (THead (Flat Appl) w t0) 
-(\lambda (t: T).((tau0 g c0 v t0) \to (ty3 g c0 (THead (Flat Appl) w v) t))) 
-(\lambda (H13: (tau0 g c0 v t0)).(let H_y \def (H3 t0 H13) in (let H1 \def 
-(ty3_unique g c0 v t0 H_y (THead (Bind Abst) u0 t) H2) in (ex_ind T (\lambda 
-(t: T).(ty3 g c0 t0 t)) (ty3 g c0 (THead (Flat Appl) w v) (THead (Flat Appl) 
-w t0)) (\lambda (x: T).(\lambda (H3: (ty3 g c0 t0 x)).(ex_ind T (\lambda (t: 
-T).(ty3 g c0 u0 t)) (ty3 g c0 (THead (Flat Appl) w v) (THead (Flat Appl) w 
-t0)) (\lambda (x0: T).(\lambda (_: (ty3 g c0 u0 x0)).(ex_ind T (\lambda (t2: 
-T).(ty3 g c0 (THead (Bind Abst) u0 t) t2)) (ty3 g c0 (THead (Flat Appl) w v) 
-(THead (Flat Appl) w t0)) (\lambda (x1: T).(\lambda (H15: (ty3 g c0 (THead 
-(Bind Abst) u0 t) x1)).(ex4_3_ind T T T (\lambda (t2: T).(\lambda (_: 
-T).(\lambda (_: T).(pc3 c0 (THead (Bind Abst) u0 t2) x1)))) (\lambda (_: 
-T).(\lambda (t: T).(\lambda (_: T).(ty3 g c0 u0 t)))) (\lambda (t2: 
-T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind Abst) u0) t t2)))) 
-(\lambda (t2: T).(\lambda (_: T).(\lambda (t3: T).(ty3 g (CHead c0 (Bind 
-Abst) u0) t2 t3)))) (ty3 g c0 (THead (Flat Appl) w v) (THead (Flat Appl) w 
-t0)) (\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (_: (pc3 c0 
-(THead (Bind Abst) u0 x2) x1)).(\lambda (H17: (ty3 g c0 u0 x3)).(\lambda 
-(H18: (ty3 g (CHead c0 (Bind Abst) u0) t x2)).(\lambda (H19: (ty3 g (CHead c0 
-(Bind Abst) u0) x2 x4)).(ty3_conv g c0 (THead (Flat Appl) w t0) (THead (Flat 
-Appl) w (THead (Bind Abst) u0 x2)) (ty3_appl g c0 w u0 H0 t0 x2 (ty3_sconv g 
-c0 t0 x H3 (THead (Bind Abst) u0 t) (THead (Bind Abst) u0 x2) (ty3_bind g c0 
-u0 x3 H17 Abst t x2 H18 x4 H19) H1)) (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_s c0 
-(THead (Flat Appl) w (THead (Bind Abst) u0 t)) (THead (Flat Appl) w t0) 
-(pc3_thin_dx c0 t0 (THead (Bind Abst) u0 t) H1 w Appl)))))))))) (ty3_gen_bind 
-g Abst c0 u0 t x1 H15)))) (ty3_correct g c0 v (THead (Bind Abst) u0 t) H2)))) 
-(ty3_correct g c0 w u0 H0)))) (ty3_correct g c0 v t0 H_y))))) t2 H12)) t1 
-(sym_eq T t1 v H11))) v0 (sym_eq T v0 w H10))) H9))) c0 (sym_eq C c0 c0 H5) 
-H6 H7 H4)))) | (tau0_cast c0 v1 v2 H4 t1 t0 H5) \Rightarrow (\lambda (H6: (eq 
-C c0 c0)).(\lambda (H7: (eq T (THead (Flat Cast) v1 t1) (THead (Flat Appl) w 
-v))).(\lambda (H8: (eq T (THead (Flat Cast) v2 t0) t2)).(eq_ind C c0 (\lambda 
-(c: C).((eq T (THead (Flat Cast) v1 t1) (THead (Flat Appl) w v)) \to ((eq T 
-(THead (Flat Cast) v2 t0) t2) \to ((tau0 g c v1 v2) \to ((tau0 g c t1 t0) \to 
-(ty3 g c0 (THead (Flat Appl) w v) t2)))))) (\lambda (H9: (eq T (THead (Flat 
-Cast) v1 t1) (THead (Flat Appl) w v))).(let H10 \def (eq_ind T (THead (Flat 
-Cast) v1 t1) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
-\Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow 
-False | (Flat f) \Rightarrow (match f return (\lambda (_: F).Prop) with [Appl 
-\Rightarrow False | Cast \Rightarrow True])])])) I (THead (Flat Appl) w v) 
-H9) in (False_ind ((eq T (THead (Flat Cast) v2 t0) t2) \to ((tau0 g c0 v1 v2) 
-\to ((tau0 g c0 t1 t0) \to (ty3 g c0 (THead (Flat Appl) w v) t2)))) H10))) c0 
-(sym_eq C c0 c0 H6) H7 H8 H4 H5))))]) in (H5 (refl_equal C c0) (refl_equal T 
-(THead (Flat Appl) w v)) (refl_equal T t2)))))))))))))) (\lambda (c0: 
-C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H0: (ty3 g c0 t2 t3)).(\lambda 
-(H1: ((\forall (t3: T).((tau0 g c0 t2 t3) \to (ty3 g c0 t2 t3))))).(\lambda 
-(t0: T).(\lambda (_: (ty3 g c0 t3 t0)).(\lambda (H3: ((\forall (t2: T).((tau0 
-g c0 t3 t2) \to (ty3 g c0 t3 t2))))).(\lambda (t4: T).(\lambda (H4: (tau0 g 
-c0 (THead (Flat Cast) t3 t2) t4)).(let H5 \def (match H4 return (\lambda (c: 
-C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (tau0 ? c t t0)).((eq C c 
-c0) \to ((eq T t (THead (Flat Cast) t3 t2)) \to ((eq T t0 t4) \to (ty3 g c0 
-(THead (Flat Cast) t3 t2) t4)))))))) with [(tau0_sort c0 n) \Rightarrow 
-(\lambda (H4: (eq C c0 c0)).(\lambda (H5: (eq T (TSort n) (THead (Flat Cast) 
-t3 t2))).(\lambda (H6: (eq T (TSort (next g n)) t4)).(eq_ind C c0 (\lambda 
-(_: C).((eq T (TSort n) (THead (Flat Cast) t3 t2)) \to ((eq T (TSort (next g 
-n)) t4) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4)))) (\lambda (H7: (eq T 
-(TSort n) (THead (Flat Cast) t3 t2))).(let H8 \def (eq_ind T (TSort n) 
-(\lambda (e: T).(match e return (\lambda (_: T).Prop) with [(TSort _) 
-\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
-False])) I (THead (Flat Cast) t3 t2) H7) in (False_ind ((eq T (TSort (next g 
-n)) t4) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4)) H8))) c0 (sym_eq C c0 c0 
-H4) H5 H6)))) | (tau0_abbr c0 d v i H4 w H5) \Rightarrow (\lambda (H6: (eq C 
-c0 c0)).(\lambda (H7: (eq T (TLRef i) (THead (Flat Cast) t3 t2))).(\lambda 
-(H8: (eq T (lift (S i) O w) t4)).(eq_ind C c0 (\lambda (c: C).((eq T (TLRef 
-i) (THead (Flat Cast) t3 t2)) \to ((eq T (lift (S i) O w) t4) \to ((getl i c 
-(CHead d (Bind Abbr) v)) \to ((tau0 g d v w) \to (ty3 g c0 (THead (Flat Cast) 
-t3 t2) t4)))))) (\lambda (H9: (eq T (TLRef i) (THead (Flat Cast) t3 
-t2))).(let H10 \def (eq_ind T (TLRef i) (\lambda (e: T).(match e return 
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) t3 
-t2) H9) in (False_ind ((eq T (lift (S i) O w) t4) \to ((getl i c0 (CHead d 
-(Bind Abbr) v)) \to ((tau0 g d v w) \to (ty3 g c0 (THead (Flat Cast) t3 t2) 
-t4)))) H10))) c0 (sym_eq C c0 c0 H6) H7 H8 H4 H5)))) | (tau0_abst c0 d v i H4 
-w H5) \Rightarrow (\lambda (H6: (eq C c0 c0)).(\lambda (H7: (eq T (TLRef i) 
-(THead (Flat Cast) t3 t2))).(\lambda (H8: (eq T (lift (S i) O v) t4)).(eq_ind 
-C c0 (\lambda (c: C).((eq T (TLRef i) (THead (Flat Cast) t3 t2)) \to ((eq T 
-(lift (S i) O v) t4) \to ((getl i c (CHead d (Bind Abst) v)) \to ((tau0 g d v 
-w) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4)))))) (\lambda (H9: (eq T 
-(TLRef i) (THead (Flat Cast) t3 t2))).(let H10 \def (eq_ind T (TLRef i) 
-(\lambda (e: T).(match e return (\lambda (_: T).Prop) with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow 
-False])) I (THead (Flat Cast) t3 t2) H9) in (False_ind ((eq T (lift (S i) O 
-v) t4) \to ((getl i c0 (CHead d (Bind Abst) v)) \to ((tau0 g d v w) \to (ty3 
-g c0 (THead (Flat Cast) t3 t2) t4)))) H10))) c0 (sym_eq C c0 c0 H6) H7 H8 H4 
-H5)))) | (tau0_bind b c0 v t4 t5 H4) \Rightarrow (\lambda (H5: (eq C c0 
-c0)).(\lambda (H6: (eq T (THead (Bind b) v t4) (THead (Flat Cast) t3 
-t2))).(\lambda (H7: (eq T (THead (Bind b) v t5) t4)).(eq_ind C c0 (\lambda 
-(c: C).((eq T (THead (Bind b) v t4) (THead (Flat Cast) t3 t2)) \to ((eq T 
-(THead (Bind b) v t5) t4) \to ((tau0 g (CHead c (Bind b) v) t4 t5) \to (ty3 g 
-c0 (THead (Flat Cast) t3 t2) t4))))) (\lambda (H8: (eq T (THead (Bind b) v 
-t4) (THead (Flat Cast) t3 t2))).(let H9 \def (eq_ind T (THead (Bind b) v t4) 
-(\lambda (e: T).(match e return (\lambda (_: T).Prop) with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
-(match k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat 
-_) \Rightarrow False])])) I (THead (Flat Cast) t3 t2) H8) in (False_ind ((eq 
-T (THead (Bind b) v t5) t4) \to ((tau0 g (CHead c0 (Bind b) v) t4 t5) \to 
-(ty3 g c0 (THead (Flat Cast) t3 t2) t4))) H9))) c0 (sym_eq C c0 c0 H5) H6 H7 
-H4)))) | (tau0_appl c0 v t4 t5 H4) \Rightarrow (\lambda (H5: (eq C c0 
-c0)).(\lambda (H6: (eq T (THead (Flat Appl) v t4) (THead (Flat Cast) t3 
-t2))).(\lambda (H7: (eq T (THead (Flat Appl) v t5) t4)).(eq_ind C c0 (\lambda 
-(c: C).((eq T (THead (Flat Appl) v t4) (THead (Flat Cast) t3 t2)) \to ((eq T 
-(THead (Flat Appl) v t5) t4) \to ((tau0 g c t4 t5) \to (ty3 g c0 (THead (Flat 
-Cast) t3 t2) t4))))) (\lambda (H8: (eq T (THead (Flat Appl) v t4) (THead 
-(Flat Cast) t3 t2))).(let H9 \def (eq_ind T (THead (Flat Appl) v t4) (\lambda 
-(e: T).(match e return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k 
-return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) 
-\Rightarrow (match 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 (Flat Appl) v t5) t4) \to ((tau0 g c0 t4 t5) \to (ty3 g c0 
-(THead (Flat Cast) t3 t2) t4))) H9))) c0 (sym_eq C c0 c0 H5) H6 H7 H4)))) | 
-(tau0_cast c0 v1 v2 H4 t4 t5 H5) \Rightarrow (\lambda (H6: (eq C c0 
-c0)).(\lambda (H7: (eq T (THead (Flat Cast) v1 t4) (THead (Flat Cast) t3 
-t2))).(\lambda (H8: (eq T (THead (Flat Cast) v2 t5) t4)).(eq_ind C c0 
-(\lambda (c: C).((eq T (THead (Flat Cast) v1 t4) (THead (Flat Cast) t3 t2)) 
-\to ((eq T (THead (Flat Cast) v2 t5) t4) \to ((tau0 g c v1 v2) \to ((tau0 g c 
-t4 t5) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4)))))) (\lambda (H9: (eq T 
-(THead (Flat Cast) v1 t4) (THead (Flat Cast) t3 t2))).(let H10 \def (f_equal 
-T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow t4 | (TLRef _) \Rightarrow t4 | (THead _ _ t) \Rightarrow t])) 
-(THead (Flat Cast) v1 t4) (THead (Flat Cast) t3 t2) H9) in ((let H11 \def 
-(f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort 
-_) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t _) \Rightarrow t])) 
-(THead (Flat Cast) v1 t4) (THead (Flat Cast) t3 t2) H9) in (eq_ind T t3 
-(\lambda (t: T).((eq T t4 t2) \to ((eq T (THead (Flat Cast) v2 t5) t4) \to 
-((tau0 g c0 t v2) \to ((tau0 g c0 t4 t5) \to (ty3 g c0 (THead (Flat Cast) t3 
-t2) t4)))))) (\lambda (H12: (eq T t4 t2)).(eq_ind T t2 (\lambda (t: T).((eq T 
-(THead (Flat Cast) v2 t5) t4) \to ((tau0 g c0 t3 v2) \to ((tau0 g c0 t t5) 
-\to (ty3 g c0 (THead (Flat Cast) t3 t2) t4))))) (\lambda (H13: (eq T (THead 
-(Flat Cast) v2 t5) t4)).(eq_ind T (THead (Flat Cast) v2 t5) (\lambda (t: 
-T).((tau0 g c0 t3 v2) \to ((tau0 g c0 t2 t5) \to (ty3 g c0 (THead (Flat Cast) 
-t3 t2) t)))) (\lambda (H14: (tau0 g c0 t3 v2)).(\lambda (H15: (tau0 g c0 t2 
-t5)).(let H_y \def (H1 t5 H15) in (let H_y0 \def (H3 v2 H14) in (let H3 \def 
-(ty3_unique g c0 t2 t5 H_y t3 H0) in (ex_ind T (\lambda (t: T).(ty3 g c0 v2 
-t)) (ty3 g c0 (THead (Flat Cast) t3 t2) (THead (Flat Cast) v2 t5)) (\lambda 
-(x: T).(\lambda (H16: (ty3 g c0 v2 x)).(ex_ind T (\lambda (t: T).(ty3 g c0 t5 
-t)) (ty3 g c0 (THead (Flat Cast) t3 t2) (THead (Flat Cast) v2 t5)) (\lambda 
-(x0: T).(\lambda (H17: (ty3 g c0 t5 x0)).(ty3_conv g c0 (THead (Flat Cast) v2 
-t5) v2 (ty3_cast g c0 t5 v2 (ty3_sconv g c0 t5 x0 H17 t3 v2 H_y0 H3) x H16) 
-(THead (Flat Cast) t3 t2) t3 (ty3_cast g c0 t2 t3 H0 v2 H_y0) (pc3_s c0 t3 
-(THead (Flat Cast) v2 t5) (pc3_pr2_u c0 t5 (THead (Flat Cast) v2 t5) 
-(pr2_free c0 (THead (Flat Cast) v2 t5) t5 (pr0_epsilon t5 t5 (pr0_refl t5) 
-v2)) t3 H3))))) (ty3_correct g c0 t2 t5 H_y)))) (ty3_correct g c0 t3 v2 
-H_y0))))))) t4 H13)) t4 (sym_eq T t4 t2 H12))) v1 (sym_eq T v1 t3 H11))) 
-H10))) c0 (sym_eq C c0 c0 H6) H7 H8 H4 H5))))]) in (H5 (refl_equal C c0) 
-(refl_equal T (THead (Flat Cast) t3 t2)) (refl_equal T t4))))))))))))) c u t1 
-H))))).
-
-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_lift g d u (asucc g x0) (arity_repl g d u x 
-H4 (asucc g x0) H7) c0 (S n) O (getl_drop Abst c0 d u n H0))))) 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))))).(\lambda (t0: T).(\lambda (H4: (ty3 g (CHead c0 (Bind b) u) t4 
-t0)).(\lambda (H5: (ex2 A (\lambda (a1: A).(arity g (CHead c0 (Bind b) u) t4 
-a1)) (\lambda (a1: A).(arity g (CHead c0 (Bind b) u) t0 (asucc g a1))))).(let 
-H6 \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 (H7: (arity g c0 u x)).(\lambda (_: (arity 
-g c0 t (asucc g x))).(let H9 \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 (H10: (arity g (CHead c0 (Bind b) u) t3 
-x0)).(\lambda (H11: (arity g (CHead c0 (Bind b) u) t4 (asucc g x0))).(let H_x 
-\def (leq_asucc g x) in (let H12 \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 (H13: (leq g x (asucc g x1))).((match b return 
-(\lambda (b0: B).((ty3 g (CHead c0 (Bind b0) u) t4 t0) \to ((ex2 A (\lambda 
-(a1: A).(arity g (CHead c0 (Bind b0) u) t4 a1)) (\lambda (a1: A).(arity g 
-(CHead c0 (Bind b0) u) t0 (asucc g a1)))) \to ((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))))))))) with [Abbr 
-\Rightarrow (\lambda (_: (ty3 g (CHead c0 (Bind Abbr) u) t4 t0)).(\lambda (_: 
-(ex2 A (\lambda (a1: A).(arity g (CHead c0 (Bind Abbr) u) t4 a1)) (\lambda 
-(a1: A).(arity g (CHead c0 (Bind Abbr) u) t0 (asucc g a1))))).(\lambda (H16: 
-(arity g (CHead c0 (Bind Abbr) u) t3 x0)).(\lambda (H17: (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 H7 t3 x0 H16) 
-(arity_bind g Abbr not_abbr_abst c0 u x H7 t4 (asucc g x0) H17)))))) | Abst 
-\Rightarrow (\lambda (_: (ty3 g (CHead c0 (Bind Abst) u) t4 t0)).(\lambda (_: 
-(ex2 A (\lambda (a1: A).(arity g (CHead c0 (Bind Abst) u) t4 a1)) (\lambda 
-(a1: A).(arity g (CHead c0 (Bind Abst) u) t0 (asucc g a1))))).(\lambda (H16: 
-(arity g (CHead c0 (Bind Abst) u) t3 x0)).(\lambda (H17: (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 
-H7 (asucc g x1) H13) t3 x0 H16) (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 H7 (asucc 
-g x1) H13) t4 (asucc g x0) H17) (asucc g (AHead x1 x0)) (leq_refl g (asucc g 
-(AHead x1 x0))))))))) | Void \Rightarrow (\lambda (_: (ty3 g (CHead c0 (Bind 
-Void) u) t4 t0)).(\lambda (_: (ex2 A (\lambda (a1: A).(arity g (CHead c0 
-(Bind Void) u) t4 a1)) (\lambda (a1: A).(arity g (CHead c0 (Bind Void) u) t0 
-(asucc g a1))))).(\lambda (H16: (arity g (CHead c0 (Bind Void) u) t3 
-x0)).(\lambda (H17: (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 not_void_abst c0 u x H7 t3 x0 H16) (arity_bind g Void 
-not_void_abst c0 u x H7 t4 (asucc g x0) H17))))))]) H4 H5 H10 H11))) 
-H12)))))) H9))))) H6))))))))))))))) (\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_equal 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 (_: (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 t4 (asucc g 
-a1)))) (\lambda (x: A).(\lambda (H5: (arity g c0 t3 x)).(\lambda (H6: (arity 
-g c0 t4 (asucc g x))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Flat 
-Cast) t4 t3) a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g a1))) x 
-(arity_cast g c0 t4 x H6 t3 H5) H6)))) H4)))))))))) c t1 t2 H))))).
-
-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 (ex4_3_ind T T T (\lambda (t2: 
-T).(\lambda (_: T).(\lambda (_: T).(pc3 c (THead (Bind Abst) v t2) u)))) 
-(\lambda (_: T).(\lambda (t0: T).(\lambda (_: T).(ty3 g c v t0)))) (\lambda 
-(t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) v) t 
-t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t1: T).(ty3 g (CHead c 
-(Bind Abst) v) t2 t1)))) P (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: 
-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)).(\lambda (_: (ty3 g 
-(CHead c (Bind Abst) v) x0 x2)).(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 H6 \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 (H7: (arity g c (THead (Bind Abst) v t) x)).(\lambda 
-(H8: (arity g c v (asucc g x))).(let H9 \def (arity_gen_abst g c v t x H7) 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 (x3: 
-A).(\lambda (x4: A).(\lambda (H10: (eq A x (AHead x3 x4))).(\lambda (H11: 
-(arity g c v (asucc g x3))).(\lambda (_: (arity g (CHead c (Bind Abst) v) t 
-x4)).(let H13 \def (eq_ind A x (\lambda (a: A).(arity g c v (asucc g a))) H8 
-(AHead x3 x4) H10) in (leq_ahead_asucc_false g x3 (asucc g x4) (arity_mono g 
-c v (asucc g (AHead x3 x4)) H13 (asucc g x3) H11) P))))))) H9))))) 
-H6))))))))))) (ty3_gen_bind g Abst c v t u H1)))))))))).
-
-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))))))).
-
-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 (\forall (P: Prop).P))))))))))
-\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 (\forall (P: Prop).P))) (\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 (\forall (P: Prop).P))) (\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 (\forall (P: 
-Prop).P))) (\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 
-(\forall (P: Prop).P)) (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 (\forall (P: Prop).P)) (\lambda (H9: (pc3 c u1 u2)).(\lambda (P: 
-Prop).(let H10 \def H9 in (ex2_ind T (\lambda (t: T).(pr3 c u1 t)) (\lambda 
-(t: T).(pr3 c u2 t)) P (\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)) P (\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)) P))))))) 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 (\forall (P: Prop).P)))))))))))
-\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 (\forall (P: Prop).P)))) (\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 (\forall (P: Prop).P)))) (\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 (\forall (P: Prop).P)))) (\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 (\forall (P: Prop).P)))) 
-(\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 (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 (\forall (P: Prop).P))) 
-(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))) (ty3_sred_pr3 c u1 (THead (Bind 
-Abst) x0 x1) H13 g t1 H) 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 (\forall (P: Prop).P))) (\lambda (u: 
-T).(\lambda (H11: (pc3 c u1 (THead (Bind Abst) u2 u))).(\lambda (P: 
-Prop).(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)) P (\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)) P (\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))))) P (\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))))) P (\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 (u: T).(pr3 (CHead c (Bind b) u) 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 (and_ind (nf2 c x5) (nf2 (CHead c 
-(Bind Abst) x5) x6) P (\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_equal 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))) P)))) 
-H29))))))))) H23)))))))) H18))))))) (pr3_confluence c u1 x1 H13 x H4))))) 
-H12))))))) H9)))))) H6)))))) H3)))))))))))).
-
-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 (\forall (P: 
-Prop).P))))))
-\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 (\forall (P: Prop).P)))))) (\lambda (c2: C).(\lambda 
-(t2: T).(match t2 return (\lambda (t: T).(((\forall (c1: C).(\forall (t1: 
-T).((flt c1 t1 c2 t) \to (or (ex T (\lambda (t2: T).(ty3 g c1 t1 t2))) 
-(\forall (t2: T).((ty3 g c1 t1 t2) \to (\forall (P: Prop).P)))))))) \to (or 
-(ex T (\lambda (t3: T).(ty3 g c2 t t3))) (\forall (t3: T).((ty3 g c2 t t3) 
-\to (\forall (P: Prop).P)))))) with [(TSort n) \Rightarrow (\lambda (_: 
-((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 (TSort n)) \to (or (ex T 
-(\lambda (t2: T).(ty3 g c1 t1 t2))) (\forall (t2: T).((ty3 g c1 t1 t2) \to 
-(\forall (P: Prop).P))))))))).(or_introl (ex T (\lambda (t3: T).(ty3 g c2 
-(TSort n) t3))) (\forall (t3: T).((ty3 g c2 (TSort n) t3) \to (\forall (P: 
-Prop).P))) (ex_intro T (\lambda (t3: T).(ty3 g c2 (TSort n) t3)) (TSort (next 
-g n)) (ty3_sort g c2 n)))) | (TLRef n) \Rightarrow (\lambda (H: ((\forall 
-(c1: C).(\forall (t1: T).((flt c1 t1 c2 (TLRef n)) \to (or (ex T (\lambda 
-(t2: T).(ty3 g c1 t1 t2))) (\forall (t2: T).((ty3 g c1 t1 t2) \to (\forall 
-(P: Prop).P))))))))).(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 (\forall (P: Prop).P)))) (\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 (\forall (P: 
-Prop).P)))) (\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 (\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 
-(\forall (P: Prop).P)))) (\lambda (H4: (ex T (\lambda (t2: T).(ty3 g x0 x2 
-t2)))).(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 
-(\forall (P: Prop).P)))) (\lambda (x: T).(\lambda (H5: (ty3 g x0 x2 
-x)).((match x1 return (\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 (\forall (P: Prop).P)))))) with [Abbr \Rightarrow 
-(\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 (\forall (P: Prop).P))) (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)))) | Abst 
-\Rightarrow (\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 (\forall (P: Prop).P))) (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)))) | Void 
-\Rightarrow (\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 (\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H7: (ty3 
-g c2 (TLRef n) t3)).(\lambda (P: Prop).(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))))) P (\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)))) P (\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 (c: C).(getl n c2 c)) 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 return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ 
-k _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind b) 
-\Rightarrow (match 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 P 
-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)))) P (\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 (c: C).(getl n c2 c)) 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 return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ 
-k _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind b) 
-\Rightarrow (match 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 P 
-H13))))))))) H8)) (ty3_gen_lref g c2 t3 n H7)))))))]) H2))) H4)) (\lambda 
-(H4: ((\forall (t2: T).((ty3 g x0 x2 t2) \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 (\forall (P: Prop).P))) 
-(\lambda (t3: T).(\lambda (H5: (ty3 g c2 (TLRef n) t3)).(\lambda (P: 
-Prop).(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))))) P (\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)))) P (\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 (c: 
-C).(getl n c2 c)) 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 return (\lambda (_: C).C) with [(CSort _) 
-\Rightarrow x0 | (CHead c _ _) \Rightarrow c])) (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 return (\lambda (_: C).B) with [(CSort _) \Rightarrow x1 | (CHead _ k _) 
-\Rightarrow (match 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 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 (c: C).(getl n c2 (CHead c (Bind Abbr) x2))) H16 x0 
-H15) in (let H19 \def (eq_ind_r C x3 (\lambda (c: C).(ty3 g c x2 x5)) H17 x0 
-H15) in (H4 x5 H19 P)))))))) 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)))) P (\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 (c: C).(getl n 
-c2 c)) 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 return (\lambda (_: C).C) with [(CSort _) \Rightarrow x0 | (CHead 
-c _ _) \Rightarrow c])) (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 return (\lambda (_: 
-C).B) with [(CSort _) \Rightarrow x1 | (CHead _ k _) \Rightarrow (match 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 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 (c: C).(getl n c2 (CHead c (Bind Abst) x2))) H16 x0 H15) in (let H20 
-\def (eq_ind_r C x3 (\lambda (c: C).(ty3 g c x2 x5)) H17 x0 H15) in (H4 x5 
-H20 P))))))))) 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 (\forall (P: Prop).P))) 
-(\lambda (t3: T).(\lambda (H2: (ty3 g c2 (TLRef n) t3)).(\lambda (P: 
-Prop).(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))))) P (\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)))) P (\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 P))))))) 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)))) P (\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 P))))))) H3)) (ty3_gen_lref g c2 
-t3 n H2))))))) H0)))) | (THead k t t0) \Rightarrow (\lambda (H: ((\forall 
-(c1: C).(\forall (t1: T).((flt c1 t1 c2 (THead k t t0)) \to (or (ex T 
-(\lambda (t2: T).(ty3 g c1 t1 t2))) (\forall (t2: T).((ty3 g c1 t1 t2) \to 
-(\forall (P: Prop).P))))))))).((match k return (\lambda (k0: K).(((\forall 
-(c1: C).(\forall (t1: T).((flt c1 t1 c2 (THead k0 t t0)) \to (or (ex T 
-(\lambda (t2: T).(ty3 g c1 t1 t2))) (\forall (t2: T).((ty3 g c1 t1 t2) \to 
-(\forall (P: Prop).P)))))))) \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 (\forall 
-(P: Prop).P)))))) with [(Bind b) \Rightarrow (\lambda (H0: ((\forall (c1: 
-C).(\forall (t1: T).((flt c1 t1 c2 (THead (Bind b) t t0)) \to (or (ex T 
-(\lambda (t2: T).(ty3 g c1 t1 t2))) (\forall (t2: T).((ty3 g c1 t1 t2) \to 
-(\forall (P: Prop).P))))))))).(let H1 \def (H0 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 (\forall (P: Prop).P))) (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 (\forall (P: Prop).P)))) (\lambda (H2: (ex T (\lambda 
-(t2: T).(ty3 g c2 t t2)))).(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 (\forall (P: Prop).P)))) (\lambda 
-(x: T).(\lambda (H3: (ty3 g c2 t x)).(let H4 \def (H0 (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 (\forall (P: Prop).P))) (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 (\forall (P: Prop).P)))) (\lambda (H5: (ex T (\lambda (t2: T).(ty3 g 
-(CHead c2 (Bind b) t) t0 t2)))).(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 (\forall 
-(P: Prop).P)))) (\lambda (x0: T).(\lambda (H6: (ty3 g (CHead c2 (Bind b) t) 
-t0 x0)).(ex_ind T (\lambda (t3: T).(ty3 g (CHead c2 (Bind b) t) x0 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 (\forall (P: Prop).P)))) (\lambda 
-(x1: T).(\lambda (H7: (ty3 g (CHead c2 (Bind b) t) x0 x1)).(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 (\forall (P: Prop).P))) (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 H3 b t0 x0 H6 x1 H7))))) (ty3_correct g (CHead c2 (Bind b) 
-t) t0 x0 H6)))) H5)) (\lambda (H5: ((\forall (t2: T).((ty3 g (CHead c2 (Bind 
-b) t) t0 t2) \to (\forall (P: Prop).P))))).(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 (\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H6: 
-(ty3 g c2 (THead (Bind b) t t0) t3)).(\lambda (P: Prop).(ex4_3_ind T T T 
-(\lambda (t4: T).(\lambda (_: T).(\lambda (_: T).(pc3 c2 (THead (Bind b) t 
-t4) t3)))) (\lambda (_: T).(\lambda (t5: T).(\lambda (_: T).(ty3 g c2 t 
-t5)))) (\lambda (t4: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c2 
-(Bind b) t) t0 t4)))) (\lambda (t4: T).(\lambda (_: T).(\lambda (t6: T).(ty3 
-g (CHead c2 (Bind b) t) t4 t6)))) P (\lambda (x0: T).(\lambda (x1: 
-T).(\lambda (x2: T).(\lambda (_: (pc3 c2 (THead (Bind b) t x0) t3)).(\lambda 
-(_: (ty3 g c2 t x1)).(\lambda (H9: (ty3 g (CHead c2 (Bind b) t) t0 
-x0)).(\lambda (_: (ty3 g (CHead c2 (Bind b) t) x0 x2)).(H5 x0 H9 P)))))))) 
-(ty3_gen_bind g b c2 t t0 t3 H6))))))) H4)))) H2)) (\lambda (H2: ((\forall 
-(t2: T).((ty3 g c2 t t2) \to (\forall (P: Prop).P))))).(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 (\forall (P: Prop).P))) (\lambda (t3: 
-T).(\lambda (H3: (ty3 g c2 (THead (Bind b) t t0) t3)).(\lambda (P: 
-Prop).(ex4_3_ind T T T (\lambda (t4: T).(\lambda (_: T).(\lambda (_: T).(pc3 
-c2 (THead (Bind b) t t4) t3)))) (\lambda (_: T).(\lambda (t5: T).(\lambda (_: 
-T).(ty3 g c2 t t5)))) (\lambda (t4: T).(\lambda (_: T).(\lambda (_: T).(ty3 g 
-(CHead c2 (Bind b) t) t0 t4)))) (\lambda (t4: T).(\lambda (_: T).(\lambda 
-(t6: T).(ty3 g (CHead c2 (Bind b) t) t4 t6)))) P (\lambda (x0: T).(\lambda 
-(x1: T).(\lambda (x2: T).(\lambda (_: (pc3 c2 (THead (Bind b) t x0) 
-t3)).(\lambda (H5: (ty3 g c2 t x1)).(\lambda (_: (ty3 g (CHead c2 (Bind b) t) 
-t0 x0)).(\lambda (_: (ty3 g (CHead c2 (Bind b) t) x0 x2)).(H2 x1 H5 P)))))))) 
-(ty3_gen_bind g b c2 t t0 t3 H3))))))) H1))) | (Flat f) \Rightarrow (\lambda 
-(H0: ((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 (THead (Flat f) t t0)) 
-\to (or (ex T (\lambda (t2: T).(ty3 g c1 t1 t2))) (\forall (t2: T).((ty3 g c1 
-t1 t2) \to (\forall (P: Prop).P))))))))).((match f return (\lambda (f0: 
-F).(((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 (THead (Flat f0) t t0)) 
-\to (or (ex T (\lambda (t2: T).(ty3 g c1 t1 t2))) (\forall (t2: T).((ty3 g c1 
-t1 t2) \to (\forall (P: Prop).P)))))))) \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 (\forall (P: Prop).P)))))) with [Appl \Rightarrow (\lambda (H1: 
-((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 (THead (Flat Appl) t t0)) 
-\to (or (ex T (\lambda (t2: T).(ty3 g c1 t1 t2))) (\forall (t2: T).((ty3 g c1 
-t1 t2) \to (\forall (P: Prop).P))))))))).(let H2 \def (H1 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 (\forall (P: Prop).P))) (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 (\forall (P: Prop).P)))) 
-(\lambda (H3: (ex T (\lambda (t2: T).(ty3 g c2 t t2)))).(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 (\forall (P: Prop).P)))) (\lambda (x: T).(\lambda (H4: (ty3 g c2 t 
-x)).(let H5 \def (H1 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 
-(\forall (P: Prop).P))) (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 (\forall (P: Prop).P)))) (\lambda (H6: (ex T (\lambda (t2: T).(ty3 g c2 
-t0 t2)))).(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 (\forall (P: Prop).P)))) (\lambda (x0: 
-T).(\lambda (H7: (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 (\forall (P: 
-Prop).P)))) (\lambda (x1: T).(\lambda (H8: (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 (\forall (P: Prop).P)))) (\lambda (x2: T).(\lambda (H9: (ty3 g c2 x 
-x2)).(let H10 \def (ty3_sn3 g c2 x x2 H9) in (let H_x \def (nf2_sn3 c2 x H10) 
-in (let H11 \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 (\forall 
-(P: Prop).P)))) (\lambda (x3: T).(\lambda (H12: (pr3 c2 x x3)).(\lambda (H13: 
-(nf2 c2 x3)).(let H14 \def (ty3_sred_pr3 c2 x x3 H12 g x2 H9) in (let H_x0 
-\def (pc3_abst_dec g c2 x0 x1 H8 x3 x2 H14) in (let H15 \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 (\forall (P: Prop).P))) (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 (\forall (P: Prop).P)))) (\lambda (H16: (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 (\forall (P: Prop).P)))) (\lambda (x4: T).(\lambda (x5: 
-T).(\lambda (H17: (pc3 c2 x0 (THead (Bind Abst) x3 x4))).(\lambda (H18: (ty3 
-g c2 (THead (Bind Abst) x5 x4) x1)).(\lambda (H19: (pr3 c2 x3 x5)).(\lambda 
-(_: (nf2 c2 x5)).(let H_y \def (nf2_pr3_unfold c2 x3 x5 H19 H13) in (let H21 
-\def (eq_ind_r T x5 (\lambda (t: T).(pr3 c2 x3 t)) H19 x3 H_y) in (let H22 
-\def (eq_ind_r T x5 (\lambda (t: T).(ty3 g c2 (THead (Bind Abst) t x4) x1)) 
-H18 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 
-(\forall (P: Prop).P))) (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 H4 x3 H12) t0 x4 (ty3_conv g c2 (THead (Bind Abst) 
-x3 x4) x1 H22 t0 x0 H7 H17))))))))))))) H16)) (\lambda (H16: ((\forall (u: 
-T).((pc3 c2 x0 (THead (Bind Abst) x3 u)) \to (\forall (P: 
-Prop).P))))).(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 
-(\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H17: (ty3 g c2 (THead 
-(Flat Appl) t t0) t3)).(\lambda (P: Prop).(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))) P (\lambda (x4: 
-T).(\lambda (x5: T).(\lambda (_: (pc3 c2 (THead (Flat Appl) t (THead (Bind 
-Abst) x4 x5)) t3)).(\lambda (H19: (ty3 g c2 t0 (THead (Bind Abst) x4 
-x5))).(\lambda (H20: (ty3 g c2 t x4)).(let H_y \def (ty3_unique g c2 t x4 H20 
-x H4) in (let H_y0 \def (ty3_unique g c2 t0 (THead (Bind Abst) x4 x5) H19 x0 
-H7) in (H16 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 H12)) (Bind Abst) x5)) P)))))))) 
-(ty3_gen_appl g c2 t t0 t3 H17))))))) H15))))))) H11)))))) (ty3_correct g c2 
-t x H4)))) (ty3_correct g c2 t0 x0 H7)))) H6)) (\lambda (H6: ((\forall (t2: 
-T).((ty3 g c2 t0 t2) \to (\forall (P: Prop).P))))).(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 (\forall (P: Prop).P))) (\lambda (t3: 
-T).(\lambda (H7: (ty3 g c2 (THead (Flat Appl) t t0) t3)).(\lambda (P: 
-Prop).(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))) P (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead (Flat 
-Appl) t (THead (Bind Abst) x0 x1)) t3)).(\lambda (H9: (ty3 g c2 t0 (THead 
-(Bind Abst) x0 x1))).(\lambda (_: (ty3 g c2 t x0)).(H6 (THead (Bind Abst) x0 
-x1) H9 P)))))) (ty3_gen_appl g c2 t t0 t3 H7))))))) H5)))) H3)) (\lambda (H3: 
-((\forall (t2: T).((ty3 g c2 t t2) \to (\forall (P: Prop).P))))).(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 (\forall (P: Prop).P))) 
-(\lambda (t3: T).(\lambda (H4: (ty3 g c2 (THead (Flat Appl) t t0) 
-t3)).(\lambda (P: Prop).(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))) P (\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 (H7: (ty3 
-g c2 t x0)).(H3 x0 H7 P)))))) (ty3_gen_appl g c2 t t0 t3 H4))))))) H2))) | 
-Cast \Rightarrow (\lambda (H1: ((\forall (c1: C).(\forall (t1: T).((flt c1 t1 
-c2 (THead (Flat Cast) t t0)) \to (or (ex T (\lambda (t2: T).(ty3 g c1 t1 
-t2))) (\forall (t2: T).((ty3 g c1 t1 t2) \to (\forall (P: 
-Prop).P))))))))).(let H2 \def (H1 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 (\forall (P: Prop).P))) (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 (\forall (P: Prop).P)))) (\lambda (H3: (ex T (\lambda (t2: T).(ty3 g 
-c2 t t2)))).(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 (\forall (P: Prop).P)))) (\lambda (x: 
-T).(\lambda (H4: (ty3 g c2 t x)).(let H5 \def (H1 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 (\forall (P: Prop).P))) (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 (\forall (P: Prop).P)))) (\lambda (H6: (ex T 
-(\lambda (t2: T).(ty3 g c2 t0 t2)))).(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 (\forall (P: 
-Prop).P)))) (\lambda (x0: T).(\lambda (H7: (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 (\forall (P: Prop).P)))) (\lambda (x1: T).(\lambda (H8: (ty3 g 
-c2 x0 x1)).(let H_x \def (pc3_dec g c2 x0 x1 H8 t x H4) in (let H9 \def H_x 
-in (or_ind (pc3 c2 x0 t) ((pc3 c2 x0 t) \to (\forall (P: Prop).P)) (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 (\forall (P: Prop).P)))) 
-(\lambda (H10: (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 (\forall (P: Prop).P))) (ex_intro T (\lambda (t3: T).(ty3 g c2 
-(THead (Flat Cast) t t0) t3)) t (ty3_cast g c2 t0 t (ty3_conv g c2 t x H4 t0 
-x0 H7 H10) x H4)))) (\lambda (H10: (((pc3 c2 x0 t) \to (\forall (P: 
-Prop).P)))).(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 
-(\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H11: (ty3 g c2 (THead 
-(Flat Cast) t t0) t3)).(\lambda (P: Prop).(and_ind (pc3 c2 t t3) (ty3 g c2 t0 
-t) P (\lambda (_: (pc3 c2 t t3)).(\lambda (H13: (ty3 g c2 t0 t)).(let H_y 
-\def (ty3_unique g c2 t0 t H13 x0 H7) in (H10 (pc3_s c2 x0 t H_y) P)))) 
-(ty3_gen_cast g c2 t0 t t3 H11))))))) H9))))) (ty3_correct g c2 t0 x0 H7)))) 
-H6)) (\lambda (H6: ((\forall (t2: T).((ty3 g c2 t0 t2) \to (\forall (P: 
-Prop).P))))).(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 
-(\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H7: (ty3 g c2 (THead (Flat 
-Cast) t t0) t3)).(\lambda (P: Prop).(and_ind (pc3 c2 t t3) (ty3 g c2 t0 t) P 
-(\lambda (_: (pc3 c2 t t3)).(\lambda (H9: (ty3 g c2 t0 t)).(H6 t H9 P))) 
-(ty3_gen_cast g c2 t0 t t3 H7))))))) H5)))) H3)) (\lambda (H3: ((\forall (t2: 
-T).((ty3 g c2 t t2) \to (\forall (P: Prop).P))))).(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 (\forall (P: Prop).P))) (\lambda (t3: 
-T).(\lambda (H4: (ty3 g c2 (THead (Flat Cast) t t0) t3)).(\lambda (P: 
-Prop).(and_ind (pc3 c2 t t3) (ty3 g c2 t0 t) P (\lambda (_: (pc3 c2 t 
-t3)).(\lambda (H6: (ty3 g c2 t0 t)).(ex_ind T (\lambda (t4: T).(ty3 g c2 t 
-t4)) P (\lambda (x: T).(\lambda (H7: (ty3 g c2 t x)).(H3 x H7 P))) 
-(ty3_correct g c2 t0 t H6)))) (ty3_gen_cast g c2 t0 t t3 H4))))))) H2)))]) 
-H0))]) H))]))) c t1))).
-

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/G/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/G/defs.ma
new file mode 100644 (file)
index 0000000..c493641
--- /dev/null
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/G/defs.ma
@@ -0,0 +1,25 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/G/defs".
+
+include "../Base/theory.ma".
+
+record G : Set \def {
+ next: (nat \to nat);
+ next_lt: (\forall (n: nat).(lt n (next n)))
+}.
+

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/T/dec.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/T/dec.ma
new file mode 100644 (file)
index 0000000..bad18aa
--- /dev/null
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/T/dec.ma
@@ -0,0 +1,413 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/T/dec".
+
+include "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 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 b) \Rightarrow b | 
+(Flat _) \Rightarrow b])) (Bind b) (Bind b0) H1) in (let H3 \def (eq_ind_r B 
+b0 (\lambda (b0: B).((eq B b b0) \to (\forall (P: Prop).P))) 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 f) \Rightarrow f])) (Flat f) (Flat f0) H1) in 
+(let H3 \def (eq_ind_r F f0 (\lambda (f0: F).((eq F f f0) \to (\forall (P: 
+Prop).P))) 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 n) \Rightarrow n | (TLRef _) \Rightarrow n | (THead _ _ _) 
+\Rightarrow n])) (TSort n) (TSort n0) H1) in (let H3 \def (eq_ind_r nat n0 
+(\lambda (n0: nat).((eq nat n n0) \to (\forall (P: Prop).P))) 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 n) \Rightarrow n | (THead _ _ _) \Rightarrow n])) 
+(TLRef n) (TLRef n0) H1) in (let H3 \def (eq_ind_r nat n0 (\lambda (n0: 
+nat).((eq nat n n0) \to (\forall (P: Prop).P))) 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 (t1: T).(or (eq T (THead k t t0) t1) ((eq T 
+(THead k t t0) t1) \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 (t1: T).(or (eq T (THead k t t0) t1) ((eq T 
+(THead k t t0) t1) \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 k _ _) \Rightarrow k])) (THead k t t0) (THead k0 t t0) 
+H11) in (let H13 \def (eq_ind_r K k0 (\lambda (k0: K).((eq K k k0) \to 
+(\forall (P: Prop).P))) 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 k _ _) \Rightarrow k])) (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 _ _ t) \Rightarrow t])) (THead k t t0) (THead k0 t t4) H8) in 
+(\lambda (_: (eq K k k0)).(let H12 \def (eq_ind_r T t4 (\lambda (t: T).((eq T 
+t0 t) \to (\forall (P: Prop).P))) H7 t0 H10) in (let H13 \def (eq_ind_r T t4 
+(\lambda (t1: T).(or (eq T (THead k t t0) t1) ((eq T (THead k t t0) t1) \to 
+(\forall (P: Prop).P)))) 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 k _ _) \Rightarrow k])) 
+(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 _ t _) \Rightarrow t])) (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 _ _ t) \Rightarrow t])) (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 (t1: T).(or (eq T (THead k t t0) t1) ((eq T (THead k 
+t t0) t1) \to (\forall (P: Prop).P)))) H2 t0 H8) in (let H12 \def (eq_ind_r T 
+t3 (\lambda (t0: T).((eq T t t0) \to (\forall (P: Prop).P))) H4 t H9) in (let 
+H13 \def (eq_ind_r T t3 (\lambda (t1: T).(or (eq T (THead k t t0) t1) ((eq T 
+(THead k t t0) t1) \to (\forall (P: Prop).P)))) 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).(match k in K return (\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))))))))))) with [(Bind b) \Rightarrow (\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_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)))))))) 
+| (Flat f) \Rightarrow (\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 k _ _) \Rightarrow (match k 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))))))))))))])) 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).(match u in T return (\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)))))) with [(TSort n) 
+\Rightarrow (\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))))))) | 
+(TLRef n) \Rightarrow (\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))))))) | (THead k t t0) \Rightarrow (\lambda (v: T).(let H_x \def 
+(terms_props__kind_dec k (Bind Abst)) in (let H \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 (H0: (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 H1 \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 (H2: (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 H2)) (\lambda (H2: (((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 (H3: (eq T (THead (Bind Abst) t t0) (THead (Bind Abst) v 
+t1))).(\lambda (P: Prop).(let H4 \def (f_equal T T (\lambda (e: T).(match e 
+in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t | (TLRef _) 
+\Rightarrow t | (THead _ t _) \Rightarrow t])) (THead (Bind Abst) 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 t0 | 
+(TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Bind Abst) 
+t t0) (THead (Bind Abst) v t1) H3) in (\lambda (H6: (eq T t v)).(H2 H6 P))) 
+H4))))))) H1))) k H0)) (\lambda (H0: (((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 (H1: (eq T 
+(THead k t t0) (THead (Bind Abst) v 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 k _ _) 
+\Rightarrow k])) (THead k t t0) (THead (Bind Abst) v t1) H1) in ((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 _ t _) 
+\Rightarrow t])) (THead k t t0) (THead (Bind Abst) v 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 k t t0) (THead (Bind Abst) v t1) H1) in (\lambda (_: 
+(eq T t v)).(\lambda (H6: (eq K k (Bind Abst))).(H0 H6 P)))) H3)) H2))))))) 
+H))))]).
+

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/T/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/T/defs.ma
new file mode 100644 (file)
index 0000000..42d3dcb
--- /dev/null
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/T/defs.ma
@@ -0,0 +1,49 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/T/defs".
+
+include "../Base/theory.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)).
+
+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.
+

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/T/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/T/props.ma
new file mode 100644 (file)
index 0000000..a9c293f
--- /dev/null
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/T/props.ma
@@ -0,0 +1,69 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/T/props".
+
+include "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 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 k _ _) \Rightarrow k])) (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 _ t _) \Rightarrow t])) (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 _ _ t) 
+\Rightarrow t])) (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 (t: T).((eq T (THead k t t1) t1) \to (\forall (P: Prop).P))) H0 t0 
+H5) in (let H8 \def (eq_ind K k (\lambda (k: K).((eq T (THead k t0 t1) t1) 
+\to (\forall (P: Prop).P))) H7 k0 H6) in (H8 H4 P)))))) H3)) H2))))))))) t))).
+

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/makefile b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/makefile
new file mode 100644 (file)
index 0000000..a9ac218
--- /dev/null
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/makefile
@@ -0,0 +1,33 @@
+H=@
+
+RT_BASEDIR=/home/fguidi/svn/software/matita/
+OPTIONS=-bench
+MMAKE=$(RT_BASEDIR)matitamake $(OPTIONS)
+CLEAN=$(RT_BASEDIR)matitaclean $(OPTIONS) 
+MMAKEO=$(RT_BASEDIR)matitamake.opt $(OPTIONS)
+CLEANO=$(RT_BASEDIR)matitaclean.opt $(OPTIONS) 
+
+devel:=$(shell basename `pwd`)
+
+all: preall
+       $(H)MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKE) build $(devel)
+clean: preall
+       $(H)MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKE) clean $(devel)
+cleanall: preall
+       $(H)MATITA_FLAGS=$(MATITA_FLAGS) $(MCLEAN) all
+
+all.opt opt: preall
+       $(H)MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKEO) build $(devel)
+clean.opt: preall
+       $(H)MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKEO) clean $(devel)
+cleanall.opt: preall
+       $(H)MATITA_FLAGS=$(MATITA_FLAGS) $(MCLEANO) all
+
+%.mo: preall
+       $(H)MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKE) $@
+%.mo.opt: preall
+       $(H)MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKEO) $@
+       
+preall:
+       $(H)MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKE) init $(devel)
+

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/next_plus/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/next_plus/defs.ma
new file mode 100644 (file)
index 0000000..48e79bd
--- /dev/null
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/next_plus/defs.ma
@@ -0,0 +1,26 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/next_plus/defs".
+
+include "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.
+

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/next_plus/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/next_plus/props.ma
new file mode 100644 (file)
index 0000000..7cbabaf
--- /dev/null
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/next_plus/props.ma
@@ -0,0 +1,62 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/next_plus/props".
+
+include "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).(le_S_n (S n) 
+(next g n) (lt_le_S (S n) (S (next g n)) (lt_n_S n (next g n) (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)).
+

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/s/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/s/defs.ma
new file mode 100644 (file)
index 0000000..477e4a4
--- /dev/null
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/s/defs.ma
@@ -0,0 +1,26 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/s/defs".
+
+include "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])).
+

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/s/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/s/props.ma
new file mode 100644 (file)
index 0000000..1c062bd
--- /dev/null
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/s/props.ma
@@ -0,0 +1,114 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/s/props".
+
+include "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_S_n (S i) (S j) (lt_le_S (S 
+i) (S (S j)) (lt_n_S i (S j) (le_lt_n_Sm 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_S_n (S (S i)) (S j) (le_n_S 
+(S (S i)) (S 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_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))).
+

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/theory.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/theory.ma
new file mode 100644 (file)
index 0000000..5e6b423
--- /dev/null
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/theory.ma
@@ -0,0 +1,34 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/theory".
+
+include "T/defs.ma".
+
+include "T/props.ma".
+
+include "T/dec.ma".
+
+include "s/defs.ma".
+
+include "s/props.ma".
+
+include "G/defs.ma".
+
+include "next_plus/defs.ma".
+
+include "next_plus/props.ma".
+

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/makefile b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/makefile
deleted file mode 100644 (file)
index a9ac218..0000000
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/makefile
+++ /dev/null
@@ -1,33 +0,0 @@
-H=@
-
-RT_BASEDIR=/home/fguidi/svn/software/matita/
-OPTIONS=-bench
-MMAKE=$(RT_BASEDIR)matitamake $(OPTIONS)
-CLEAN=$(RT_BASEDIR)matitaclean $(OPTIONS) 
-MMAKEO=$(RT_BASEDIR)matitamake.opt $(OPTIONS)
-CLEANO=$(RT_BASEDIR)matitaclean.opt $(OPTIONS) 
-
-devel:=$(shell basename `pwd`)
-
-all: preall
-       $(H)MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKE) build $(devel)
-clean: preall
-       $(H)MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKE) clean $(devel)
-cleanall: preall
-       $(H)MATITA_FLAGS=$(MATITA_FLAGS) $(MCLEAN) all
-
-all.opt opt: preall
-       $(H)MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKEO) build $(devel)
-clean.opt: preall
-       $(H)MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKEO) clean $(devel)
-cleanall.opt: preall
-       $(H)MATITA_FLAGS=$(MATITA_FLAGS) $(MCLEANO) all
-
-%.mo: preall
-       $(H)MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKE) $@
-%.mo.opt: preall
-       $(H)MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKEO) $@
-       
-preall:
-       $(H)MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKE) init $(devel)
-