DIR=$(shell basename $$PWD)
+MATITAOPTIONS=-onepass
$(DIR) all:
- ../../../matitac
+ ../../../matitac $(MATITAOPTIONS)
$(DIR).opt opt all.opt:
- ../../../matitac.opt
+ ../../../matitac.opt $(MATITAOPTIONS)
clean:
../../../matitaclean
clean.opt:
../../../matitaclean.opt
depend:
- ../matitadep
+ ../../../matitadep
depend.opt:
- ../matitadep.opt
+ ../../../matitadep.opt
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Base-1/blt/defs".
include "preamble.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Base-1/blt/props".
include "blt/defs.ma".
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).
+\to (lt n0 (S n))))).(\lambda (H1: (eq bool (blt n0 n) true)).(lt_n_S n0 n (H
+n0 H1))))) y)))) x).
theorem bge_le:
\forall (x: nat).(\forall (y: nat).((eq bool (blt y x) false) \to (le x y)))
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Base-1/definitions".
include "types/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Base-1/ext/arith".
include "preamble.ma".
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_Sx_Sy:
+ \forall (x: nat).(\forall (y: nat).(eq nat (minus (S x) (S y)) (minus x y)))
+\def
+ \lambda (x: nat).(\lambda (y: nat).(refl_equal nat (minus x y))).
+
theorem minus_plus_r:
\forall (m: nat).(\forall (n: nat).(eq nat (minus (plus m n) n) m))
\def
(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))))))).
+(plus d2 h2) h1) (plus n h1) (le_n_S (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 (IH O O) in (let H2 \def (eq_ind_r nat (minus z0 O) (\lambda (n:
nat).(\forall (y3: nat).(\forall (y4: nat).((le O z0) \to ((le O z0) \to ((eq
nat (plus n y3) (plus n y4)) \to (eq nat y4 y3))))))) H_y z0 (minus_n_O z0))
-in (H2 y1 y2 (le_O_n z0) (le_O_n z0) (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_eq
-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:
+in (H2 y1 y2 (le_O_n z0) (le_O_n z0) (eq_add_S (plus z0 y1) (plus z0 y2)
+H1))))))))) (\lambda (x3: nat).(\lambda (_: ((\forall (y1: nat).(\forall (y2:
+nat).((le O (S z0)) \to ((le x3 (S z0)) \to ((eq nat (S (plus z0 y1)) (plus
+(match x3 with [O \Rightarrow (S z0) | (S l) \Rightarrow (minus z0 l)]) y2))
+\to (eq nat y2 (plus x3 y1))))))))).(\lambda (y1: nat).(\lambda (y2:
nat).(\lambda (_: (le O (S z0))).(\lambda (H0: (le (S x3) (S z0))).(\lambda
(H1: (eq nat (S (plus z0 y1)) (plus (minus z0 x3) y2))).(let H_y \def (IH O
x3 (S y1)) in (let H2 \def (eq_ind_r nat (minus z0 O) (\lambda (n:
\lambda (d: nat).(\lambda (h: nat).(\lambda (n: nat).(\lambda (H: (le (plus
d h) n)).(let H0 \def (le_trans d (plus d h) n (le_plus_l d h) H) in (let H1
\def (eq_ind nat n (\lambda (n0: nat).(le d n0)) H0 (plus (minus n h) h)
-(le_plus_minus_sym h n (le_trans_plus_r d h n H))) in (le_S d (minus n h)
-(le_minus d n h H))))))).
+(le_plus_minus_sym h n (le_trans h (plus d h) n (le_plus_r d h) H))) in (le_S
+d (minus n h) (le_minus d n h H))))))).
theorem lt_x_pred_y:
\forall (x: nat).(\forall (y: nat).((lt x (pred y)) \to (lt (S x) y)))
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Base-1/ext/tactics".
include "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)))))
+((S \to Prop))).(((\forall (y: S).((P y) \to ((eq S y x) \to (G y))))) \to
+((P x) \to (G x))))))
\def
\lambda (S: Set).(\lambda (x: S).(\lambda (P: ((S \to Prop))).(\lambda (G:
-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))))))).
+((S \to Prop))).(\lambda (H: ((\forall (y: S).((P y) \to ((eq S y x) \to (G
+y)))))).(\lambda (H0: (P x)).(H x H0 (refl_equal S x))))))).
theorem unintro:
\forall (A: Set).(\forall (a: A).(\forall (P: ((A \to Prop))).(((\forall (x:
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Base-1/plist/defs".
include "preamble.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Base-1/plist/props".
include "plist/defs.ma".
(* *)
(**************************************************************************)
-
-
include "coq.ma".
alias symbol "eq" = "Coq's leibnitz's equality".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Base-1/spare".
include "theory.ma".
-(*
-inductive pr: Set \def
-| pr_zero: pr
-| pr_succ: pr
-| pr_proj: nat \to pr
-| pr_comp: ((nat \to pr)) \to (pr \to pr)
-| pr_prec: pr \to (pr \to pr).
-
-definition pr_type:
- Set
-\def
- ((nat \to nat)) \to nat.
-
-definition prec_appl:
- pr_type \to (pr_type \to (nat \to pr_type))
-\def
- let rec prec_appl (f: pr_type) (g: pr_type) (n: nat) on n: pr_type \def
-(match n with [O \Rightarrow f | (S m) \Rightarrow (\lambda (ns: ((nat \to
-nat))).(g (\lambda (i: nat).(match i with [O \Rightarrow (prec_appl f g m ns)
-| (S n0) \Rightarrow (match n0 with [O \Rightarrow m | (S j) \Rightarrow (ns
-j)])]))))]) in prec_appl.
-
-definition pr_appl:
- pr \to pr_type
-\def
- let rec pr_appl (h: pr) on h: pr_type \def (match h with [pr_zero
-\Rightarrow (\lambda (_: ((nat \to nat))).O) | pr_succ \Rightarrow (\lambda
-(ns: ((nat \to nat))).(S (ns O))) | (pr_proj i) \Rightarrow (\lambda (ns:
-((nat \to nat))).(ns i)) | (pr_comp fs g) \Rightarrow (\lambda (ns: ((nat \to
-nat))).(pr_appl g (\lambda (i: nat).(pr_appl (fs i) ns)))) | (pr_prec f g)
-\Rightarrow (\lambda (ns: ((nat \to nat))).(prec_appl (pr_appl f) (pr_appl g)
-(ns O) (\lambda (i: nat).(ns (S i)))))]) in pr_appl.
-
-inductive pr_arity: pr \to (nat \to Prop) \def
-| pr_arity_zero: \forall (n: nat).(pr_arity pr_zero n)
-| pr_arity_succ: \forall (n: nat).((lt O n) \to (pr_arity pr_succ n))
-| pr_arity_proj: \forall (n: nat).(\forall (i: nat).((lt i n) \to (pr_arity
-(pr_proj i) n)))
-| pr_arity_comp: \forall (n: nat).(\forall (m: nat).(\forall (fs: ((nat \to
-pr))).(\forall (g: pr).((pr_arity g m) \to (((\forall (i: nat).((lt i m) \to
-(pr_arity (fs i) n)))) \to (pr_arity (pr_comp fs g) n))))))
-| pr_arity_prec: \forall (n: nat).(\forall (f: pr).(\forall (g: pr).((lt O n)
-\to ((pr_arity f (pred n)) \to ((pr_arity g (S n)) \to (pr_arity (pr_prec f
-g) n)))))).
-
-theorem pr_arity_le:
- \forall (h: pr).(\forall (m: nat).((pr_arity h m) \to (\forall (n: nat).((le
-m n) \to (pr_arity h n)))))
-\def
- \lambda (h: pr).(\lambda (m: nat).(\lambda (H: (pr_arity h m)).(pr_arity_ind
-(\lambda (p: pr).(\lambda (n: nat).(\forall (n0: nat).((le n n0) \to
-(pr_arity p n0))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (_: (le n
-n0)).(pr_arity_zero n0)))) (\lambda (n: nat).(\lambda (H0: (lt O n)).(\lambda
-(n0: nat).(\lambda (H1: (le n n0)).(pr_arity_succ n0 (lt_le_trans O n n0 H0
-H1)))))) (\lambda (n: nat).(\lambda (i: nat).(\lambda (H0: (lt i n)).(\lambda
-(n0: nat).(\lambda (H1: (le n n0)).(pr_arity_proj n0 i (lt_le_trans i n n0 H0
-H1))))))) (\lambda (n: nat).(\lambda (m0: nat).(\lambda (fs: ((nat \to
-pr))).(\lambda (g: pr).(\lambda (H0: (pr_arity g m0)).(\lambda (_: ((\forall
-(n0: nat).((le m0 n0) \to (pr_arity g n0))))).(\lambda (_: ((\forall (i:
-nat).((lt i m0) \to (pr_arity (fs i) n))))).(\lambda (H3: ((\forall (i:
-nat).((lt i m0) \to (\forall (n0: nat).((le n n0) \to (pr_arity (fs i)
-n0))))))).(\lambda (n0: nat).(\lambda (H4: (le n n0)).(pr_arity_comp n0 m0 fs
-g H0 (\lambda (i: nat).(\lambda (H5: (lt i m0)).(H3 i H5 n0 H4))))))))))))))
-(\lambda (n: nat).(\lambda (f: pr).(\lambda (g: pr).(\lambda (H0: (lt O
-n)).(\lambda (_: (pr_arity f (pred n))).(\lambda (H2: ((\forall (n0:
-nat).((le (pred n) n0) \to (pr_arity f n0))))).(\lambda (_: (pr_arity g (S
-n))).(\lambda (H4: ((\forall (n0: nat).((le (S n) n0) \to (pr_arity g
-n0))))).(\lambda (n0: nat).(\lambda (H5: (le n n0)).(pr_arity_prec n0 f g
-(lt_le_trans O n n0 H0 H5) (H2 (pred n0) (le_n_pred n n0 H5)) (H4 (S n0)
-(le_n_S n n0 H5))))))))))))) h m H))).
-
-theorem pr_arity_appl:
- \forall (h: pr).(\forall (n: nat).((pr_arity h n) \to (\forall (ns: ((nat
-\to nat))).(\forall (ms: ((nat \to nat))).(((\forall (i: nat).((lt i n) \to
-(eq nat (ns i) (ms i))))) \to (eq nat (pr_appl h ns) (pr_appl h ms)))))))
-\def
- \lambda (h: pr).(\lambda (n: nat).(\lambda (H: (pr_arity h n)).(pr_arity_ind
-(\lambda (p: pr).(\lambda (n0: nat).(\forall (ns: ((nat \to nat))).(\forall
-(ms: ((nat \to nat))).(((\forall (i: nat).((lt i n0) \to (eq nat (ns i) (ms
-i))))) \to (eq nat (pr_appl p ns) (pr_appl p ms))))))) (\lambda (n0:
-nat).(\lambda (ns: ((nat \to nat))).(\lambda (ms: ((nat \to nat))).(\lambda
-(_: ((\forall (i: nat).((lt i n0) \to (eq nat (ns i) (ms i)))))).(refl_equal
-nat O))))) (\lambda (n0: nat).(\lambda (H0: (lt O n0)).(\lambda (ns: ((nat
-\to nat))).(\lambda (ms: ((nat \to nat))).(\lambda (H1: ((\forall (i:
-nat).((lt i n0) \to (eq nat (ns i) (ms i)))))).(f_equal nat nat S (ns O) (ms
-O) (H1 O H0))))))) (\lambda (n0: nat).(\lambda (i: nat).(\lambda (H0: (lt i
-n0)).(\lambda (ns: ((nat \to nat))).(\lambda (ms: ((nat \to nat))).(\lambda
-(H1: ((\forall (i0: nat).((lt i0 n0) \to (eq nat (ns i0) (ms i0)))))).(H1 i
-H0))))))) (\lambda (n0: nat).(\lambda (m: nat).(\lambda (fs: ((nat \to
-pr))).(\lambda (g: pr).(\lambda (_: (pr_arity g m)).(\lambda (H1: ((\forall
-(ns: ((nat \to nat))).(\forall (ms: ((nat \to nat))).(((\forall (i: nat).((lt
-i m) \to (eq nat (ns i) (ms i))))) \to (eq nat (pr_appl g ns) (pr_appl g
-ms))))))).(\lambda (_: ((\forall (i: nat).((lt i m) \to (pr_arity (fs i)
-n0))))).(\lambda (H3: ((\forall (i: nat).((lt i m) \to (\forall (ns: ((nat
-\to nat))).(\forall (ms: ((nat \to nat))).(((\forall (i0: nat).((lt i0 n0)
-\to (eq nat (ns i0) (ms i0))))) \to (eq nat (pr_appl (fs i) ns) (pr_appl (fs
-i) ms))))))))).(\lambda (ns: ((nat \to nat))).(\lambda (ms: ((nat \to
-nat))).(\lambda (H4: ((\forall (i: nat).((lt i n0) \to (eq nat (ns i) (ms
-i)))))).(H1 (\lambda (i: nat).(pr_appl (fs i) ns)) (\lambda (i: nat).(pr_appl
-(fs i) ms)) (\lambda (i: nat).(\lambda (H5: (lt i m)).(H3 i H5 ns ms
-H4))))))))))))))) (\lambda (n0: nat).(\lambda (f: pr).(\lambda (g:
-pr).(\lambda (H0: (lt O n0)).(\lambda (_: (pr_arity f (pred n0))).(\lambda
-(H2: ((\forall (ns: ((nat \to nat))).(\forall (ms: ((nat \to
-nat))).(((\forall (i: nat).((lt i (pred n0)) \to (eq nat (ns i) (ms i)))))
-\to (eq nat (pr_appl f ns) (pr_appl f ms))))))).(\lambda (_: (pr_arity g (S
-n0))).(\lambda (H4: ((\forall (ns: ((nat \to nat))).(\forall (ms: ((nat \to
-nat))).(((\forall (i: nat).((lt i (S n0)) \to (eq nat (ns i) (ms i))))) \to
-(eq nat (pr_appl g ns) (pr_appl g ms))))))).(\lambda (ns: ((nat \to
-nat))).(\lambda (ms: ((nat \to nat))).(\lambda (H5: ((\forall (i: nat).((lt i
-n0) \to (eq nat (ns i) (ms i)))))).(eq_ind nat (ns O) (\lambda (n1: nat).(eq
-nat (prec_appl (pr_appl f) (pr_appl g) (ns O) (\lambda (i: nat).(ns (S i))))
-(prec_appl (pr_appl f) (pr_appl g) n1 (\lambda (i: nat).(ms (S i)))))) (let
-n1 \def (ns O) in (nat_ind (\lambda (n2: nat).(eq nat (prec_appl (pr_appl f)
-(pr_appl g) n2 (\lambda (i: nat).(ns (S i)))) (prec_appl (pr_appl f) (pr_appl
-g) n2 (\lambda (i: nat).(ms (S i)))))) (H2 (\lambda (i: nat).(ns (S i)))
-(\lambda (i: nat).(ms (S i))) (\lambda (i: nat).(\lambda (H6: (lt i (pred
-n0))).(H5 (S i) (lt_x_pred_y i n0 H6))))) (\lambda (n2: nat).(\lambda (IHn0:
-(eq nat (prec_appl (pr_appl f) (pr_appl g) n2 (\lambda (i: nat).(ns (S i))))
-(prec_appl (pr_appl f) (pr_appl g) n2 (\lambda (i: nat).(ms (S i)))))).(H4
-(\lambda (i: nat).(match i with [O \Rightarrow (prec_appl (pr_appl f)
-(pr_appl g) n2 (\lambda (i0: nat).(ns (S i0)))) | (S n3) \Rightarrow (match
-n3 with [O \Rightarrow n2 | (S j) \Rightarrow (ns (S j))])])) (\lambda (i:
-nat).(match i with [O \Rightarrow (prec_appl (pr_appl f) (pr_appl g) n2
-(\lambda (i0: nat).(ms (S i0)))) | (S n3) \Rightarrow (match n3 with [O
-\Rightarrow n2 | (S j) \Rightarrow (ms (S j))])])) (\lambda (i: nat).(\lambda
-(H6: (lt i (S n0))).(nat_ind (\lambda (n3: nat).((lt n3 (S n0)) \to (eq nat
-(match n3 with [O \Rightarrow (prec_appl (pr_appl f) (pr_appl g) n2 (\lambda
-(i0: nat).(ns (S i0)))) | (S n4) \Rightarrow (match n4 with [O \Rightarrow n2
-| (S j) \Rightarrow (ns (S j))])]) (match n3 with [O \Rightarrow (prec_appl
-(pr_appl f) (pr_appl g) n2 (\lambda (i0: nat).(ms (S i0)))) | (S n4)
-\Rightarrow (match n4 with [O \Rightarrow n2 | (S j) \Rightarrow (ms (S
-j))])])))) (\lambda (_: (lt O (S n0))).IHn0) (\lambda (i0: nat).(\lambda (_:
-(((lt i0 (S n0)) \to (eq nat (match i0 with [O \Rightarrow (prec_appl
-(pr_appl f) (pr_appl g) n2 (\lambda (i1: nat).(ns (S i1)))) | (S n3)
-\Rightarrow (match n3 with [O \Rightarrow n2 | (S j) \Rightarrow (ns (S
-j))])]) (match i0 with [O \Rightarrow (prec_appl (pr_appl f) (pr_appl g) n2
-(\lambda (i1: nat).(ms (S i1)))) | (S n3) \Rightarrow (match n3 with [O
-\Rightarrow n2 | (S j) \Rightarrow (ms (S j))])]))))).(\lambda (H7: (lt (S
-i0) (S n0))).(let H_y \def (H5 i0 (lt_S_n i0 n0 H7)) in (nat_ind (\lambda
-(n3: nat).((eq nat (ns n3) (ms n3)) \to (eq nat (match n3 with [O \Rightarrow
-n2 | (S j) \Rightarrow (ns (S j))]) (match n3 with [O \Rightarrow n2 | (S j)
-\Rightarrow (ms (S j))])))) (\lambda (_: (eq nat (ns O) (ms O))).(refl_equal
-nat n2)) (\lambda (i1: nat).(\lambda (_: (((eq nat (ns i1) (ms i1)) \to (eq
-nat (match i1 with [O \Rightarrow n2 | (S j) \Rightarrow (ns (S j))]) (match
-i1 with [O \Rightarrow n2 | (S j) \Rightarrow (ms (S j))]))))).(\lambda (H8:
-(eq nat (ns (S i1)) (ms (S i1)))).H8))) i0 H_y))))) i H6)))))) n1)) (ms O)
-(H5 O H0))))))))))))) h n H))).
-
-theorem pr_arity_comp_proj_zero:
- \forall (n: nat).(pr_arity (pr_comp pr_proj pr_zero) n)
-\def
- \lambda (n: nat).(pr_arity_comp n n pr_proj pr_zero (pr_arity_zero n)
-(\lambda (i: nat).(\lambda (H: (lt i n)).(pr_arity_proj n i H)))).
-*)
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Base-1/theory".
include "ext/tactics.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Base-1/types/defs".
include "preamble.ma".
inductive and4 (P0: Prop) (P1: Prop) (P2: Prop) (P3: Prop): Prop \def
| and4_intro: P0 \to (P1 \to (P2 \to (P3 \to (and4 P0 P1 P2 P3)))).
+inductive and5 (P0: Prop) (P1: Prop) (P2: Prop) (P3: Prop) (P4: Prop): Prop
+\def
+| and5_intro: P0 \to (P1 \to (P2 \to (P3 \to (P4 \to (and5 P0 P1 P2 P3
+P4))))).
+
inductive or3 (P0: Prop) (P1: Prop) (P2: Prop): Prop \def
| or3_intro0: P0 \to (or3 P0 P1 P2)
| or3_intro1: P1 \to (or3 P0 P1 P2)
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Base-1/types/props".
include "types/defs.ma".
(* This file was automatically generated: do not edit *********************)
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/A/defs".
-
-include "preamble3.ma".
+include "preamble.ma".
inductive A: Set \def
| ASort: nat \to (nat \to A)
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/C/defs".
include "T/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/C/props".
include "C/defs.ma".
(* This file was automatically generated: do not edit *********************)
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/G/defs".
-
-include "preamble3.ma".
+include "preamble.ma".
record G : Set \def {
next: (nat \to nat);
DIR=$(shell basename $$PWD)
+MATITAOPTIONS=-onepass
$(DIR) all:
- ../../../matitac
+ ../../../matitac $(MATITAOPTIONS)
$(DIR).opt opt all.opt:
- ../../../matitac.opt
+ ../../../matitac.opt $(MATITAOPTIONS)
clean:
../../../matitaclean
clean.opt:
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/T/dec".
include "T/defs.ma".
(* This file was automatically generated: do not edit *********************)
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/T/defs".
-
-include "preamble3.ma".
+include "preamble.ma".
inductive B: Set \def
| Abbr: B
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/T/props".
include "T/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/aplus/defs".
include "asucc/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/aplus/props".
include "aplus/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/aprem/defs".
include "A/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/aprem/props".
include "aprem/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/arity/aprem".
include "arity/props.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/arity/cimp".
include "arity/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/arity/defs".
include "leq/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/arity/fwd".
include "arity/defs.ma".
\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 in T return (\lambda (_: T).nat) with [(TSort n1)
-\Rightarrow n1 | (TLRef _) \Rightarrow n0 | (THead _ _ _) \Rightarrow n0]))
-(TSort n0) (TSort n) H1) in (eq_ind_r nat n (\lambda (n1: nat).(leq g (ASort
-O n1) (ASort O n))) (leq_refl g (ASort O n)) n0 H2))))) (\lambda (c0:
-C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: (getl i c0
-(CHead d (Bind Abbr) u))).(\lambda (a0: A).(\lambda (_: (arity g d u
-a0)).(\lambda (_: (((eq T u (TSort n)) \to (leq g a0 (ASort O n))))).(\lambda
-(H4: (eq T (TLRef i) (TSort n))).(let H5 \def (eq_ind T (TLRef i) (\lambda
-(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
-False])) I (TSort n) H4) in (False_ind (leq g a0 (ASort O n)) H5)))))))))))
+c t a)) (\lambda (_: T).(leq g a (ASort O n))) (\lambda (y: T).(\lambda (H0:
+(arity g c y a)).(arity_ind g (\lambda (_: C).(\lambda (t: T).(\lambda (a0:
+A).((eq T t (TSort n)) \to (leq g a0 (ASort O n)))))) (\lambda (_:
+C).(\lambda (n0: nat).(\lambda (H1: (eq T (TSort n0) (TSort n))).(let H2 \def
+(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with
+[(TSort n1) \Rightarrow n1 | (TLRef _) \Rightarrow n0 | (THead _ _ _)
+\Rightarrow n0])) (TSort n0) (TSort n) H1) in (eq_ind_r nat n (\lambda (n1:
+nat).(leq g (ASort O n1) (ASort O n))) (leq_refl g (ASort O n)) n0 H2)))))
(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
-(_: (getl i c0 (CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (_: (arity
-g d u (asucc g a0))).(\lambda (_: (((eq T u (TSort n)) \to (leq g (asucc g
-a0) (ASort O n))))).(\lambda (H4: (eq T (TLRef i) (TSort n))).(let H5 \def
-(eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
-(THead _ _ _) \Rightarrow False])) I (TSort n) H4) in (False_ind (leq g a0
-(ASort O n)) H5))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b
-Abst))).(\lambda (c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity
-g c0 u a1)).(\lambda (_: (((eq T u (TSort n)) \to (leq g a1 (ASort O
-n))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c0 (Bind
-b) u) t a2)).(\lambda (_: (((eq T t (TSort n)) \to (leq g a2 (ASort O
-n))))).(\lambda (H6: (eq T (THead (Bind b) u t) (TSort n))).(let H7 \def
-(eq_ind T (THead (Bind b) u t) (\lambda (ee: T).(match ee in T return
+(_: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (a0: A).(\lambda (_: (arity
+g d u a0)).(\lambda (_: (((eq T u (TSort n)) \to (leq g a0 (ASort O
+n))))).(\lambda (H4: (eq T (TLRef i) (TSort n))).(let H5 \def (eq_ind T
+(TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _)
+\Rightarrow False])) I (TSort n) H4) in (False_ind (leq g a0 (ASort O n))
+H5))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i:
+nat).(\lambda (_: (getl i c0 (CHead d (Bind Abst) u))).(\lambda (a0:
+A).(\lambda (_: (arity g d u (asucc g a0))).(\lambda (_: (((eq T u (TSort n))
+\to (leq g (asucc g a0) (ASort O n))))).(\lambda (H4: (eq T (TLRef i) (TSort
+n))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (TSort n) H4) in
+(False_ind (leq g a0 (ASort O n)) H5))))))))))) (\lambda (b: B).(\lambda (_:
+(not (eq B b Abst))).(\lambda (c0: C).(\lambda (u: T).(\lambda (a1:
+A).(\lambda (_: (arity g c0 u a1)).(\lambda (_: (((eq T u (TSort n)) \to (leq
+g a1 (ASort O n))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (_: (arity g
+(CHead c0 (Bind b) u) t a2)).(\lambda (_: (((eq T t (TSort n)) \to (leq g a2
+(ASort O n))))).(\lambda (H6: (eq T (THead (Bind b) u t) (TSort n))).(let H7
+\def (eq_ind T (THead (Bind b) u t) (\lambda (ee: T).(match ee in T return
(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H6) in
(False_ind (leq g a2 (ASort O n)) H7)))))))))))))) (\lambda (c0: C).(\lambda
\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 in T return (\lambda (_: T).Prop) with [(TSort
-_) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
-False])) I (TLRef i) H1) in (False_ind (or (ex2_2 C T (\lambda (d:
-C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d:
-C).(\lambda (u: T).(arity g d u (ASort O n))))) (ex2_2 C T (\lambda (d:
+c t a)) (\lambda (_: T).(or (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl
+i c (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u
+a)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c (CHead d (Bind
+Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a)))))))
+(\lambda (y: T).(\lambda (H0: (arity g c y a)).(arity_ind g (\lambda (c0:
+C).(\lambda (t: T).(\lambda (a0: A).((eq T t (TLRef i)) \to (or (ex2_2 C T
+(\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u))))
+(\lambda (d: C).(\lambda (u: T).(arity g d u a0)))) (ex2_2 C T (\lambda (d:
C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d:
-C).(\lambda (u: T).(arity g d u (asucc g (ASort O n))))))) H2))))) (\lambda
-(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i0: nat).(\lambda (H1:
-(getl i0 c0 (CHead d (Bind Abbr) u))).(\lambda (a0: A).(\lambda (H2: (arity g
-d u a0)).(\lambda (_: (((eq T u (TLRef i)) \to (or (ex2_2 C T (\lambda (d0:
-C).(\lambda (u0: T).(getl i d (CHead d0 (Bind Abbr) u0)))) (\lambda (d0:
-C).(\lambda (u0: T).(arity g d0 u0 a0)))) (ex2_2 C T (\lambda (d0:
-C).(\lambda (u0: T).(getl i d (CHead d0 (Bind Abst) u0)))) (\lambda (d0:
-C).(\lambda (u0: T).(arity g d0 u0 (asucc g a0))))))))).(\lambda (H4: (eq T
-(TLRef i0) (TLRef i))).(let H5 \def (f_equal T nat (\lambda (e: T).(match e
-in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow i0 | (TLRef n)
-\Rightarrow n | (THead _ _ _) \Rightarrow i0])) (TLRef i0) (TLRef i) H4) in
-(let H6 \def (eq_ind nat i0 (\lambda (n: nat).(getl n c0 (CHead d (Bind Abbr)
-u))) H1 i H5) in (or_introl (ex2_2 C T (\lambda (d0: C).(\lambda (u0:
-T).(getl i c0 (CHead d0 (Bind Abbr) u0)))) (\lambda (d0: C).(\lambda (u0:
-T).(arity g d0 u0 a0)))) (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i
-c0 (CHead d0 (Bind Abst) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0
-u0 (asucc g a0))))) (ex2_2_intro C T (\lambda (d0: C).(\lambda (u0: T).(getl
-i c0 (CHead d0 (Bind Abbr) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g
-d0 u0 a0))) d u H6 H2))))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda
-(u: T).(\lambda (i0: nat).(\lambda (H1: (getl i0 c0 (CHead d (Bind Abst)
-u))).(\lambda (a0: A).(\lambda (H2: (arity g d u (asucc g a0))).(\lambda (_:
-(((eq T u (TLRef i)) \to (or (ex2_2 C T (\lambda (d0: C).(\lambda (u0:
-T).(getl i d (CHead d0 (Bind Abbr) u0)))) (\lambda (d0: C).(\lambda (u0:
-T).(arity g d0 u0 (asucc g a0))))) (ex2_2 C T (\lambda (d0: C).(\lambda (u0:
-T).(getl i d (CHead d0 (Bind Abst) u0)))) (\lambda (d0: C).(\lambda (u0:
-T).(arity g d0 u0 (asucc g (asucc g a0)))))))))).(\lambda (H4: (eq T (TLRef
-i0) (TLRef i))).(let H5 \def (f_equal T nat (\lambda (e: T).(match e in T
-return (\lambda (_: T).nat) with [(TSort _) \Rightarrow i0 | (TLRef n)
-\Rightarrow n | (THead _ _ _) \Rightarrow i0])) (TLRef i0) (TLRef i) H4) in
-(let H6 \def (eq_ind nat i0 (\lambda (n: nat).(getl n c0 (CHead d (Bind Abst)
-u))) H1 i H5) in (or_intror (ex2_2 C T (\lambda (d0: C).(\lambda (u0:
-T).(getl i c0 (CHead d0 (Bind Abbr) u0)))) (\lambda (d0: C).(\lambda (u0:
-T).(arity g d0 u0 a0)))) (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i
-c0 (CHead d0 (Bind Abst) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0
-u0 (asucc g a0))))) (ex2_2_intro C T (\lambda (d0: C).(\lambda (u0: T).(getl
-i c0 (CHead d0 (Bind Abst) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g
-d0 u0 (asucc g a0)))) d u H6 H2))))))))))))) (\lambda (b: B).(\lambda (_:
-(not (eq B b Abst))).(\lambda (c0: C).(\lambda (u: T).(\lambda (a1:
-A).(\lambda (_: (arity g c0 u a1)).(\lambda (_: (((eq T u (TLRef i)) \to (or
-(ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr)
-u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 a1)))) (ex2_2 C T
-(\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abst) u0))))
-(\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g a1))))))))).(\lambda
-(t: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c0 (Bind b) u) t
-a2)).(\lambda (_: (((eq T t (TLRef i)) \to (or (ex2_2 C T (\lambda (d:
-C).(\lambda (u0: T).(getl i (CHead c0 (Bind b) u) (CHead d (Bind Abbr) u0))))
-(\lambda (d: C).(\lambda (u0: T).(arity g d u0 a2)))) (ex2_2 C T (\lambda (d:
-C).(\lambda (u0: T).(getl i (CHead c0 (Bind b) u) (CHead d (Bind Abst) u0))))
-(\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g a2))))))))).(\lambda
-(H6: (eq T (THead (Bind b) u t) (TLRef i))).(let H7 \def (eq_ind T (THead
-(Bind b) u t) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop)
-with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _
-_) \Rightarrow True])) I (TLRef i) H6) in (False_ind (or (ex2_2 C T (\lambda
-(d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d:
-C).(\lambda (u0: T).(arity g d u0 a2)))) (ex2_2 C T (\lambda (d: C).(\lambda
-(u0: T).(getl i c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0:
-T).(arity g d u0 (asucc g a2)))))) H7)))))))))))))) (\lambda (c0: C).(\lambda
-(u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u (asucc g a1))).(\lambda
-(_: (((eq T u (TLRef i)) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u0:
-T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0:
-T).(arity g d u0 (asucc g a1))))) (ex2_2 C T (\lambda (d: C).(\lambda (u0:
-T).(getl i c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0:
-T).(arity g d u0 (asucc g (asucc g a1)))))))))).(\lambda (t: T).(\lambda (a2:
-A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t a2)).(\lambda (_: (((eq T
-t (TLRef i)) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i
-(CHead c0 (Bind Abst) u) (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda
-(u0: T).(arity g d u0 a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u0:
-T).(getl i (CHead c0 (Bind Abst) u) (CHead d (Bind Abst) u0)))) (\lambda (d:
-C).(\lambda (u0: T).(arity g d u0 (asucc g a2))))))))).(\lambda (H5: (eq T
-(THead (Bind Abst) u t) (TLRef i))).(let H6 \def (eq_ind T (THead (Bind Abst)
-u t) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
-\Rightarrow True])) I (TLRef i) H5) in (False_ind (or (ex2_2 C T (\lambda (d:
-C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d:
-C).(\lambda (u0: T).(arity g d u0 (AHead a1 a2))))) (ex2_2 C T (\lambda (d:
-C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abst) u0)))) (\lambda (d:
-C).(\lambda (u0: T).(arity g d u0 (asucc g (AHead a1 a2))))))) H6))))))))))))
-(\lambda (c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u
+C).(\lambda (u: T).(arity g d u (asucc g a0)))))))))) (\lambda (c0:
+C).(\lambda (n: nat).(\lambda (H1: (eq T (TSort n) (TLRef i))).(let H2 \def
+(eq_ind T (TSort n) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False |
+(THead _ _ _) \Rightarrow False])) I (TLRef i) H1) in (False_ind (or (ex2_2 C
+T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u))))
+(\lambda (d: C).(\lambda (u: T).(arity g d u (ASort O n))))) (ex2_2 C T
+(\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u))))
+(\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g (ASort O n)))))))
+H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i0:
+nat).(\lambda (H1: (getl i0 c0 (CHead d (Bind Abbr) u))).(\lambda (a0:
+A).(\lambda (H2: (arity g d u a0)).(\lambda (_: (((eq T u (TLRef i)) \to (or
+(ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i d (CHead d0 (Bind Abbr)
+u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a0)))) (ex2_2 C T
+(\lambda (d0: C).(\lambda (u0: T).(getl i d (CHead d0 (Bind Abst) u0))))
+(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g
+a0))))))))).(\lambda (H4: (eq T (TLRef i0) (TLRef i))).(let H5 \def (f_equal
+T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort
+_) \Rightarrow i0 | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i0]))
+(TLRef i0) (TLRef i) H4) in (let H6 \def (eq_ind nat i0 (\lambda (n:
+nat).(getl n c0 (CHead d (Bind Abbr) u))) H1 i H5) in (or_introl (ex2_2 C T
+(\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abbr) u0))))
+(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a0)))) (ex2_2 C T (\lambda
+(d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda
+(d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a0))))) (ex2_2_intro C T
+(\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abbr) u0))))
+(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a0))) d u H6 H2)))))))))))))
+(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i0: nat).(\lambda
+(H1: (getl i0 c0 (CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (H2:
+(arity g d u (asucc g a0))).(\lambda (_: (((eq T u (TLRef i)) \to (or (ex2_2
+C T (\lambda (d0: C).(\lambda (u0: T).(getl i d (CHead d0 (Bind Abbr) u0))))
+(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a0))))) (ex2_2 C T
+(\lambda (d0: C).(\lambda (u0: T).(getl i d (CHead d0 (Bind Abst) u0))))
+(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g (asucc g
+a0)))))))))).(\lambda (H4: (eq T (TLRef i0) (TLRef i))).(let H5 \def (f_equal
+T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort
+_) \Rightarrow i0 | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i0]))
+(TLRef i0) (TLRef i) H4) in (let H6 \def (eq_ind nat i0 (\lambda (n:
+nat).(getl n c0 (CHead d (Bind Abst) u))) H1 i H5) in (or_intror (ex2_2 C T
+(\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abbr) u0))))
+(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a0)))) (ex2_2 C T (\lambda
+(d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda
+(d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a0))))) (ex2_2_intro C T
+(\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0))))
+(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a0)))) d u H6
+H2))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda
+(c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u
a1)).(\lambda (_: (((eq T u (TLRef i)) \to (or (ex2_2 C T (\lambda (d:
C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d:
C).(\lambda (u0: T).(arity g d u0 a1)))) (ex2_2 C T (\lambda (d: C).(\lambda
(u0: T).(getl i c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0:
T).(arity g d u0 (asucc g a1))))))))).(\lambda (t: T).(\lambda (a2:
-A).(\lambda (_: (arity g c0 t (AHead a1 a2))).(\lambda (_: (((eq T t (TLRef
-i)) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d
-(Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 (AHead a1
-a2))))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind
-Abst) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g (AHead
-a1 a2)))))))))).(\lambda (H5: (eq T (THead (Flat Appl) u t) (TLRef i))).(let
-H6 \def (eq_ind T (THead (Flat Appl) u t) (\lambda (ee: T).(match ee in T
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+A).(\lambda (_: (arity g (CHead c0 (Bind b) u) t a2)).(\lambda (_: (((eq T t
+(TLRef i)) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i (CHead
+c0 (Bind b) u) (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0:
+T).(arity g d u0 a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i
+(CHead c0 (Bind b) u) (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda
+(u0: T).(arity g d u0 (asucc g a2))))))))).(\lambda (H6: (eq T (THead (Bind
+b) u t) (TLRef i))).(let H7 \def (eq_ind T (THead (Bind b) u t) (\lambda (ee:
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I
+(TLRef i) H6) in (False_ind (or (ex2_2 C T (\lambda (d: C).(\lambda (u0:
+T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0:
+T).(arity g d u0 a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i
+c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0
+(asucc g a2)))))) H7)))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda
+(a1: A).(\lambda (_: (arity g c0 u (asucc g a1))).(\lambda (_: (((eq T u
+(TLRef i)) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0
+(CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0
+(asucc g a1))))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0
+(CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0
+(asucc g (asucc g a1)))))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (_:
+(arity g (CHead c0 (Bind Abst) u) t a2)).(\lambda (_: (((eq T t (TLRef i))
+\to (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i (CHead c0 (Bind
+Abst) u) (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity
+g d u0 a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i (CHead c0
+(Bind Abst) u) (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0:
+T).(arity g d u0 (asucc g a2))))))))).(\lambda (H5: (eq T (THead (Bind Abst)
+u t) (TLRef i))).(let H6 \def (eq_ind T (THead (Bind Abst) u t) (\lambda (ee:
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I
+(TLRef i) H5) in (False_ind (or (ex2_2 C T (\lambda (d: C).(\lambda (u0:
+T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0:
+T).(arity g d u0 (AHead a1 a2))))) (ex2_2 C T (\lambda (d: C).(\lambda (u0:
+T).(getl i c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0:
+T).(arity g d u0 (asucc g (AHead a1 a2))))))) H6)))))))))))) (\lambda (c0:
+C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u a1)).(\lambda
+(_: (((eq T u (TLRef i)) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u0:
+T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0:
+T).(arity g d u0 a1)))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i
+c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0
+(asucc g a1))))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (_: (arity g
+c0 t (AHead a1 a2))).(\lambda (_: (((eq T t (TLRef i)) \to (or (ex2_2 C T
+(\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr) u0))))
+(\lambda (d: C).(\lambda (u0: T).(arity g d u0 (AHead a1 a2))))) (ex2_2 C T
+(\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abst) u0))))
+(\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g (AHead a1
+a2)))))))))).(\lambda (H5: (eq T (THead (Flat Appl) u t) (TLRef i))).(let H6
+\def (eq_ind T (THead (Flat Appl) u t) (\lambda (ee: T).(match ee in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef i) H5) in
(False_ind (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead
d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 a2))))
\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 in T return
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _)
-\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t)
-H2) in (False_ind (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (_:
-A).(arity g (CHead c0 (Bind b) u) t (ASort O n)))) H3))))) (\lambda (c0:
-C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0
-(CHead d (Bind Abbr) u0))).(\lambda (a: A).(\lambda (_: (arity g d u0
-a)).(\lambda (_: (((eq T u0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a1:
-A).(arity g d u a1)) (\lambda (_: A).(arity g (CHead d (Bind b) u) t
+T).(arity g c t0 a2)) (\lambda (_: T).(ex2 A (\lambda (a1: A).(arity g c u
+a1)) (\lambda (_: A).(arity g (CHead c (Bind b) u) t a2)))) (\lambda (y:
+T).(\lambda (H1: (arity g c y a2)).(arity_ind g (\lambda (c0: C).(\lambda
+(t0: T).(\lambda (a: A).((eq T t0 (THead (Bind b) u t)) \to (ex2 A (\lambda
+(a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t
+a))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (H2: (eq T (TSort n)
+(THead (Bind b) u t))).(let H3 \def (eq_ind T (TSort n) (\lambda (ee:
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I
+(THead (Bind b) u t) H2) in (False_ind (ex2 A (\lambda (a1: A).(arity g c0 u
+a1)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t (ASort O n)))) H3)))))
+(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda
+(_: (getl i c0 (CHead d (Bind Abbr) u0))).(\lambda (a: A).(\lambda (_: (arity
+g d u0 a)).(\lambda (_: (((eq T u0 (THead (Bind b) u t)) \to (ex2 A (\lambda
+(a1: A).(arity g d u a1)) (\lambda (_: A).(arity g (CHead d (Bind b) u) t
a)))))).(\lambda (H5: (eq T (TLRef i) (THead (Bind b) u t))).(let H6 \def
(eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_:
T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
\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
+(Bind Abst) u t) (\lambda (t0: T).(arity g c t0 a)) (\lambda (_: T).(ex3_2 A
+A (\lambda (a1: A).(\lambda (a2: A).(eq A a (AHead a1 a2)))) (\lambda (a1:
+A).(\lambda (_: A).(arity g c u (asucc g a1)))) (\lambda (_: A).(\lambda (a2:
+A).(arity g (CHead c (Bind Abst) u) t a2))))) (\lambda (y: T).(\lambda (H0:
+(arity g c y a)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (a0:
+A).((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a1:
+A).(\lambda (a2: A).(eq A a0 (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_:
+A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g
+(CHead c0 (Bind Abst) u) t a2)))))))) (\lambda (c0: C).(\lambda (n:
+nat).(\lambda (H1: (eq T (TSort n) (THead (Bind Abst) u t))).(let H2 \def
+(eq_ind T (TSort n) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False |
+(THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) u t) H1) in
+(False_ind (ex3_2 A A (\lambda (a1: A).(\lambda (a2: A).(eq A (ASort O n)
+(AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u (asucc g
a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind Abst) u) t
-a2)))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (H1: (eq T (TSort n)
-(THead (Bind Abst) u t))).(let H2 \def (eq_ind T (TSort n) (\lambda (ee:
-T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I
-(THead (Bind Abst) u t) H1) in (False_ind (ex3_2 A A (\lambda (a1:
-A).(\lambda (a2: A).(eq A (ASort O n) (AHead a1 a2)))) (\lambda (a1:
-A).(\lambda (_: A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda
-(a2: A).(arity g (CHead c0 (Bind Abst) u) t a2)))) H2))))) (\lambda (c0:
-C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0
-(CHead d (Bind Abbr) u0))).(\lambda (a0: A).(\lambda (_: (arity g d u0
-a0)).(\lambda (_: (((eq T u0 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda
-(a1: A).(\lambda (a2: A).(eq A a0 (AHead a1 a2)))) (\lambda (a1: A).(\lambda
-(_: A).(arity g d u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g
-(CHead d (Bind Abst) u) t a2))))))).(\lambda (H4: (eq T (TLRef i) (THead
-(Bind Abst) u t))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match
-ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
-(TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead
-(Bind Abst) u t) H4) in (False_ind (ex3_2 A A (\lambda (a1: A).(\lambda (a2:
-A).(eq A a0 (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u
-(asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind
-Abst) u) t a2)))) H5))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda
-(u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind Abst)
-u0))).(\lambda (a0: A).(\lambda (_: (arity g d u0 (asucc g a0))).(\lambda (_:
-(((eq T u0 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a1: A).(\lambda
-(a2: A).(eq A (asucc g a0) (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_:
-A).(arity g d u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g
-(CHead d (Bind Abst) u) t a2))))))).(\lambda (H4: (eq T (TLRef i) (THead
-(Bind Abst) u t))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match
-ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
-(TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead
-(Bind Abst) u t) H4) in (False_ind (ex3_2 A A (\lambda (a1: A).(\lambda (a2:
-A).(eq A a0 (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u
-(asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind
-Abst) u) t a2)))) H5))))))))))) (\lambda (b: B).(\lambda (H1: (not (eq B b
-Abst))).(\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (H2:
-(arity g c0 u0 a1)).(\lambda (H3: (((eq T u0 (THead (Bind Abst) u t)) \to
-(ex3_2 A A (\lambda (a2: A).(\lambda (a3: A).(eq A a1 (AHead a2 a3))))
-(\lambda (a2: A).(\lambda (_: A).(arity g c0 u (asucc g a2)))) (\lambda (_:
-A).(\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u) t a3))))))).(\lambda
-(t0: T).(\lambda (a2: A).(\lambda (H4: (arity g (CHead c0 (Bind b) u0) t0
-a2)).(\lambda (H5: (((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A A
-(\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3:
-A).(\lambda (_: A).(arity g (CHead c0 (Bind b) u0) u (asucc g a3)))) (\lambda
-(_: A).(\lambda (a4: A).(arity g (CHead (CHead c0 (Bind b) u0) (Bind Abst) u)
-t a4))))))).(\lambda (H6: (eq T (THead (Bind b) u0 t0) (THead (Bind Abst) u
-t))).(let H7 \def (f_equal T B (\lambda (e: T).(match e in T return (\lambda
-(_: T).B) with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead k
-_ _) \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0)
-\Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead (Bind b) u0 t0) (THead
-(Bind Abst) u t) H6) in ((let H8 \def (f_equal T T (\lambda (e: T).(match e
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _)
-\Rightarrow u0 | (THead _ t1 _) \Rightarrow t1])) (THead (Bind b) u0 t0)
-(THead (Bind Abst) u t) H6) in ((let H9 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 |
-(TLRef _) \Rightarrow t0 | (THead _ _ t1) \Rightarrow t1])) (THead (Bind b)
-u0 t0) (THead (Bind Abst) u t) H6) in (\lambda (H10: (eq T u0 u)).(\lambda
-(H11: (eq B b Abst)).(let H12 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1
+a2)))) H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i:
+nat).(\lambda (_: (getl i c0 (CHead d (Bind Abbr) u0))).(\lambda (a0:
+A).(\lambda (_: (arity g d u0 a0)).(\lambda (_: (((eq T u0 (THead (Bind Abst)
+u t)) \to (ex3_2 A A (\lambda (a1: A).(\lambda (a2: A).(eq A a0 (AHead a1
+a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g d u (asucc g a1)))) (\lambda
+(_: A).(\lambda (a2: A).(arity g (CHead d (Bind Abst) u) t a2))))))).(\lambda
+(H4: (eq T (TLRef i) (THead (Bind Abst) u t))).(let H5 \def (eq_ind T (TLRef
+i) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
+_) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
+False])) I (THead (Bind Abst) u t) H4) in (False_ind (ex3_2 A A (\lambda (a1:
+A).(\lambda (a2: A).(eq A a0 (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_:
+A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g
+(CHead c0 (Bind Abst) u) t a2)))) H5))))))))))) (\lambda (c0: C).(\lambda (d:
+C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind
+Abst) u0))).(\lambda (a0: A).(\lambda (_: (arity g d u0 (asucc g
+a0))).(\lambda (_: (((eq T u0 (THead (Bind Abst) u t)) \to (ex3_2 A A
+(\lambda (a1: A).(\lambda (a2: A).(eq A (asucc g a0) (AHead a1 a2))))
+(\lambda (a1: A).(\lambda (_: A).(arity g d u (asucc g a1)))) (\lambda (_:
+A).(\lambda (a2: A).(arity g (CHead d (Bind Abst) u) t a2))))))).(\lambda
+(H4: (eq T (TLRef i) (THead (Bind Abst) u t))).(let H5 \def (eq_ind T (TLRef
+i) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
+_) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
+False])) I (THead (Bind Abst) u t) H4) in (False_ind (ex3_2 A A (\lambda (a1:
+A).(\lambda (a2: A).(eq A a0 (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_:
+A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g
+(CHead c0 (Bind Abst) u) t a2)))) H5))))))))))) (\lambda (b: B).(\lambda (H1:
+(not (eq B b Abst))).(\lambda (c0: C).(\lambda (u0: T).(\lambda (a1:
+A).(\lambda (H2: (arity g c0 u0 a1)).(\lambda (H3: (((eq T u0 (THead (Bind
+Abst) u t)) \to (ex3_2 A A (\lambda (a2: A).(\lambda (a3: A).(eq A a1 (AHead
+a2 a3)))) (\lambda (a2: A).(\lambda (_: A).(arity g c0 u (asucc g a2))))
+(\lambda (_: A).(\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u) t
+a3))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H4: (arity g (CHead c0
+(Bind b) u0) t0 a2)).(\lambda (H5: (((eq T t0 (THead (Bind Abst) u t)) \to
+(ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4))))
+(\lambda (a3: A).(\lambda (_: A).(arity g (CHead c0 (Bind b) u0) u (asucc g
+a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead (CHead c0 (Bind b)
+u0) (Bind Abst) u) t a4))))))).(\lambda (H6: (eq T (THead (Bind b) u0 t0)
+(THead (Bind Abst) u t))).(let H7 \def (f_equal T B (\lambda (e: T).(match e
+in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | (TLRef _)
+\Rightarrow b | (THead k _ _) \Rightarrow (match k in K return (\lambda (_:
+K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead
+(Bind b) u0 t0) (THead (Bind Abst) u t) H6) in ((let H8 \def (f_equal T T
+(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t1 _) \Rightarrow t1]))
+(THead (Bind b) u0 t0) (THead (Bind Abst) u t) H6) in ((let H9 \def (f_equal
+T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t1) \Rightarrow t1]))
+(THead (Bind b) u0 t0) (THead (Bind Abst) u t) H6) in (\lambda (H10: (eq T u0
+u)).(\lambda (H11: (eq B b Abst)).(let H12 \def (eq_ind T t0 (\lambda (t1:
+T).((eq T t1 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3:
+A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_:
+A).(arity g (CHead c0 (Bind b) u0) u (asucc g a3)))) (\lambda (_: A).(\lambda
+(a4: A).(arity g (CHead (CHead c0 (Bind b) u0) (Bind Abst) u) t a4)))))) H5 t
+H9) in (let H13 \def (eq_ind T t0 (\lambda (t1: T).(arity g (CHead c0 (Bind
+b) u0) t1 a2)) H4 t H9) in (let H14 \def (eq_ind T u0 (\lambda (t1: T).((eq T
+t (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4:
+A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g (CHead
+c0 (Bind b) t1) u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g
+(CHead (CHead c0 (Bind b) t1) (Bind Abst) u) t a4)))))) H12 u H10) in (let
+H15 \def (eq_ind T u0 (\lambda (t1: T).(arity g (CHead c0 (Bind b) t1) t a2))
+H13 u H10) in (let H16 \def (eq_ind T u0 (\lambda (t1: T).((eq T t1 (THead
+(Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a1
+(AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g
+a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t
+a4)))))) H3 u H10) in (let H17 \def (eq_ind T u0 (\lambda (t1: T).(arity g c0
+t1 a1)) H2 u H10) in (let H18 \def (eq_ind B b (\lambda (b0: B).((eq T t
(THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq
A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g (CHead c0
-(Bind b) u0) u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g
-(CHead (CHead c0 (Bind b) u0) (Bind Abst) u) t a4)))))) H5 t H9) in (let H13
-\def (eq_ind T t0 (\lambda (t1: T).(arity g (CHead c0 (Bind b) u0) t1 a2)) H4
-t H9) in (let H14 \def (eq_ind T u0 (\lambda (t1: T).((eq T t (THead (Bind
+(Bind b0) u) u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g
+(CHead (CHead c0 (Bind b0) u) (Bind Abst) u) t a4)))))) H14 Abst H11) in (let
+H19 \def (eq_ind B b (\lambda (b0: B).(arity g (CHead c0 (Bind b0) u) t a2))
+H15 Abst H11) in (let H20 \def (eq_ind B b (\lambda (b0: B).(not (eq B b0
+Abst))) H1 Abst H11) in (let H21 \def (match (H20 (refl_equal B Abst)) in
+False return (\lambda (_: False).(ex3_2 A A (\lambda (a3: A).(\lambda (a4:
+A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u
+(asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind
+Abst) u) t a4))))) with []) in H21))))))))))))) H8)) H7))))))))))))))
+(\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (H1: (arity g c0
+u0 (asucc g a1))).(\lambda (H2: (((eq T u0 (THead (Bind Abst) u t)) \to
+(ex3_2 A A (\lambda (a2: A).(\lambda (a3: A).(eq A (asucc g a1) (AHead a2
+a3)))) (\lambda (a2: A).(\lambda (_: A).(arity g c0 u (asucc g a2))))
+(\lambda (_: A).(\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u) t
+a3))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H3: (arity g (CHead c0
+(Bind Abst) u0) t0 a2)).(\lambda (H4: (((eq T t0 (THead (Bind Abst) u t)) \to
+(ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4))))
+(\lambda (a3: A).(\lambda (_: A).(arity g (CHead c0 (Bind Abst) u0) u (asucc
+g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead (CHead c0 (Bind
+Abst) u0) (Bind Abst) u) t a4))))))).(\lambda (H5: (eq T (THead (Bind Abst)
+u0 t0) (THead (Bind Abst) u t))).(let H6 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 |
+(TLRef _) \Rightarrow u0 | (THead _ t1 _) \Rightarrow t1])) (THead (Bind
+Abst) u0 t0) (THead (Bind Abst) u t) H5) in ((let H7 \def (f_equal T T
+(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t1) \Rightarrow t1]))
+(THead (Bind Abst) u0 t0) (THead (Bind Abst) u t) H5) in (\lambda (H8: (eq T
+u0 u)).(let H9 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1 (THead (Bind
Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead
-a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g (CHead c0 (Bind b) t1) u
-(asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead (CHead c0
-(Bind b) t1) (Bind Abst) u) t a4)))))) H12 u H10) in (let H15 \def (eq_ind T
-u0 (\lambda (t1: T).(arity g (CHead c0 (Bind b) t1) t a2)) H13 u H10) in (let
-H16 \def (eq_ind T u0 (\lambda (t1: T).((eq T t1 (THead (Bind Abst) u t)) \to
-(ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a1 (AHead a3 a4))))
-(\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) (\lambda (_:
-A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t a4)))))) H3 u H10) in
-(let H17 \def (eq_ind T u0 (\lambda (t1: T).(arity g c0 t1 a1)) H2 u H10) in
-(let H18 \def (eq_ind B b (\lambda (b0: B).((eq T t (THead (Bind Abst) u t))
+a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g (CHead c0 (Bind Abst) u0)
+u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead (CHead c0
+(Bind Abst) u0) (Bind Abst) u) t a4)))))) H4 t H7) in (let H10 \def (eq_ind T
+t0 (\lambda (t1: T).(arity g (CHead c0 (Bind Abst) u0) t1 a2)) H3 t H7) in
+(let H11 \def (eq_ind T u0 (\lambda (t1: T).((eq T t (THead (Bind Abst) u t))
\to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4))))
-(\lambda (a3: A).(\lambda (_: A).(arity g (CHead c0 (Bind b0) u) u (asucc g
-a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead (CHead c0 (Bind b0)
-u) (Bind Abst) u) t a4)))))) H14 Abst H11) in (let H19 \def (eq_ind B b
-(\lambda (b0: B).(arity g (CHead c0 (Bind b0) u) t a2)) H15 Abst H11) in (let
-H20 \def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H1 Abst H11) in
-(let H21 \def (match (H20 (refl_equal B Abst)) in False return (\lambda (_:
-False).(ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4))))
-(\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) (\lambda (_:
-A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t a4))))) with []) in
-H21))))))))))))) H8)) H7)))))))))))))) (\lambda (c0: C).(\lambda (u0:
-T).(\lambda (a1: A).(\lambda (H1: (arity g c0 u0 (asucc g a1))).(\lambda (H2:
-(((eq T u0 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a2: A).(\lambda
-(a3: A).(eq A (asucc g a1) (AHead a2 a3)))) (\lambda (a2: A).(\lambda (_:
-A).(arity g c0 u (asucc g a2)))) (\lambda (_: A).(\lambda (a3: A).(arity g
-(CHead c0 (Bind Abst) u) t a3))))))).(\lambda (t0: T).(\lambda (a2:
-A).(\lambda (H3: (arity g (CHead c0 (Bind Abst) u0) t0 a2)).(\lambda (H4:
-(((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda
-(a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g
-(CHead c0 (Bind Abst) u0) u (asucc g a3)))) (\lambda (_: A).(\lambda (a4:
-A).(arity g (CHead (CHead c0 (Bind Abst) u0) (Bind Abst) u) t
-a4))))))).(\lambda (H5: (eq T (THead (Bind Abst) u0 t0) (THead (Bind Abst) u
-t))).(let H6 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda
-(_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead
-_ t1 _) \Rightarrow t1])) (THead (Bind Abst) u0 t0) (THead (Bind Abst) u t)
-H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T return
-(\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0
-| (THead _ _ t1) \Rightarrow t1])) (THead (Bind Abst) u0 t0) (THead (Bind
-Abst) u t) H5) in (\lambda (H8: (eq T u0 u)).(let H9 \def (eq_ind T t0
-(\lambda (t1: T).((eq T t1 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda
-(a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda
-(_: A).(arity g (CHead c0 (Bind Abst) u0) u (asucc g a3)))) (\lambda (_:
-A).(\lambda (a4: A).(arity g (CHead (CHead c0 (Bind Abst) u0) (Bind Abst) u)
-t a4)))))) H4 t H7) in (let H10 \def (eq_ind T t0 (\lambda (t1: T).(arity g
-(CHead c0 (Bind Abst) u0) t1 a2)) H3 t H7) in (let H11 \def (eq_ind T u0
-(\lambda (t1: T).((eq T t (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda
-(a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda
-(_: A).(arity g (CHead c0 (Bind Abst) t1) u (asucc g a3)))) (\lambda (_:
-A).(\lambda (a4: A).(arity g (CHead (CHead c0 (Bind Abst) t1) (Bind Abst) u)
-t a4)))))) H9 u H8) in (let H12 \def (eq_ind T u0 (\lambda (t1: T).(arity g
-(CHead c0 (Bind Abst) t1) t a2)) H10 u H8) in (let H13 \def (eq_ind T u0
-(\lambda (t1: T).((eq T t1 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda
-(a3: A).(\lambda (a4: A).(eq A (asucc g a1) (AHead a3 a4)))) (\lambda (a3:
-A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda
-(a4: A).(arity g (CHead c0 (Bind Abst) u) t a4)))))) H2 u H8) in (let H14
-\def (eq_ind T u0 (\lambda (t1: T).(arity g c0 t1 (asucc g a1))) H1 u H8) in
-(ex3_2_intro A A (\lambda (a3: A).(\lambda (a4: A).(eq A (AHead a1 a2) (AHead
-a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g a3))))
-(\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t a4))) a1
-a2 (refl_equal A (AHead a1 a2)) H14 H12))))))))) H6)))))))))))) (\lambda (c0:
-C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u0
-a1)).(\lambda (_: (((eq T u0 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda
-(a2: A).(\lambda (a3: A).(eq A a1 (AHead a2 a3)))) (\lambda (a2: A).(\lambda
-(_: A).(arity g c0 u (asucc g a2)))) (\lambda (_: A).(\lambda (a3: A).(arity
-g (CHead c0 (Bind Abst) u) t a3))))))).(\lambda (t0: T).(\lambda (a2:
-A).(\lambda (_: (arity g c0 t0 (AHead a1 a2))).(\lambda (_: (((eq T t0 (THead
-(Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A
+(\lambda (a3: A).(\lambda (_: A).(arity g (CHead c0 (Bind Abst) t1) u (asucc
+g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead (CHead c0 (Bind
+Abst) t1) (Bind Abst) u) t a4)))))) H9 u H8) in (let H12 \def (eq_ind T u0
+(\lambda (t1: T).(arity g (CHead c0 (Bind Abst) t1) t a2)) H10 u H8) in (let
+H13 \def (eq_ind T u0 (\lambda (t1: T).((eq T t1 (THead (Bind Abst) u t)) \to
+(ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A (asucc g a1) (AHead a3
+a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g a3))))
+(\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t a4))))))
+H2 u H8) in (let H14 \def (eq_ind T u0 (\lambda (t1: T).(arity g c0 t1 (asucc
+g a1))) H1 u H8) in (ex3_2_intro A A (\lambda (a3: A).(\lambda (a4: A).(eq A
(AHead a1 a2) (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u
(asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind
-Abst) u) t a4))))))).(\lambda (H5: (eq T (THead (Flat Appl) u0 t0) (THead
-(Bind Abst) u t))).(let H6 \def (eq_ind T (THead (Flat Appl) u0 t0) (\lambda
-(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
-(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False |
-(Flat _) \Rightarrow True])])) I (THead (Bind Abst) u t) H5) in (False_ind
-(ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4))))
-(\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) (\lambda (_:
-A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t a4)))) H6))))))))))))
-(\lambda (c0: C).(\lambda (u0: T).(\lambda (a0: A).(\lambda (_: (arity g c0
-u0 (asucc g a0))).(\lambda (_: (((eq T u0 (THead (Bind Abst) u t)) \to (ex3_2
-A A (\lambda (a1: A).(\lambda (a2: A).(eq A (asucc g a0) (AHead a1 a2))))
-(\lambda (a1: A).(\lambda (_: A).(arity g c0 u (asucc g a1)))) (\lambda (_:
-A).(\lambda (a2: A).(arity g (CHead c0 (Bind Abst) u) t a2))))))).(\lambda
-(t0: T).(\lambda (_: (arity g c0 t0 a0)).(\lambda (_: (((eq T t0 (THead (Bind
-Abst) u t)) \to (ex3_2 A A (\lambda (a1: A).(\lambda (a2: A).(eq A a0 (AHead
-a1 a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u (asucc g a1))))
-(\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind Abst) u) t
-a2))))))).(\lambda (H5: (eq T (THead (Flat Cast) u0 t0) (THead (Bind Abst) u
-t))).(let H6 \def (eq_ind T (THead (Flat Cast) u0 t0) (\lambda (ee: T).(match
-ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
-(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return
-(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
-True])])) I (THead (Bind Abst) u t) H5) in (False_ind (ex3_2 A A (\lambda
-(a1: A).(\lambda (a2: A).(eq A a0 (AHead a1 a2)))) (\lambda (a1: A).(\lambda
-(_: A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity
-g (CHead c0 (Bind Abst) u) t a2)))) H6))))))))))) (\lambda (c0: C).(\lambda
-(t0: T).(\lambda (a1: A).(\lambda (H1: (arity g c0 t0 a1)).(\lambda (H2:
-(((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a2: A).(\lambda
-(a3: A).(eq A a1 (AHead a2 a3)))) (\lambda (a2: A).(\lambda (_: A).(arity g
-c0 u (asucc g a2)))) (\lambda (_: A).(\lambda (a3: A).(arity g (CHead c0
-(Bind Abst) u) t a3))))))).(\lambda (a2: A).(\lambda (H3: (leq g a1
-a2)).(\lambda (H4: (eq T t0 (THead (Bind Abst) u t))).(let H5 \def (f_equal T
-T (\lambda (e: T).e) t0 (THead (Bind Abst) u t) H4) in (let H6 \def (eq_ind T
-t0 (\lambda (t1: T).((eq T t1 (THead (Bind Abst) u t)) \to (ex3_2 A A
-(\lambda (a3: A).(\lambda (a4: A).(eq A a1 (AHead a3 a4)))) (\lambda (a3:
+Abst) u) t a4))) a1 a2 (refl_equal A (AHead a1 a2)) H14 H12)))))))))
+H6)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda
+(_: (arity g c0 u0 a1)).(\lambda (_: (((eq T u0 (THead (Bind Abst) u t)) \to
+(ex3_2 A A (\lambda (a2: A).(\lambda (a3: A).(eq A a1 (AHead a2 a3))))
+(\lambda (a2: A).(\lambda (_: A).(arity g c0 u (asucc g a2)))) (\lambda (_:
+A).(\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u) t a3))))))).(\lambda
+(t0: T).(\lambda (a2: A).(\lambda (_: (arity g c0 t0 (AHead a1 a2))).(\lambda
+(_: (((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3:
+A).(\lambda (a4: A).(eq A (AHead a1 a2) (AHead a3 a4)))) (\lambda (a3:
A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda
-(a4: A).(arity g (CHead c0 (Bind Abst) u) t a4)))))) H2 (THead (Bind Abst) u
-t) H5) in (let H7 \def (eq_ind T t0 (\lambda (t1: T).(arity g c0 t1 a1)) H1
-(THead (Bind Abst) u t) H5) in (let H8 \def (H6 (refl_equal T (THead (Bind
-Abst) u t))) in (ex3_2_ind A A (\lambda (a3: A).(\lambda (a4: A).(eq A a1
-(AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g
+(a4: A).(arity g (CHead c0 (Bind Abst) u) t a4))))))).(\lambda (H5: (eq T
+(THead (Flat Appl) u0 t0) (THead (Bind Abst) u t))).(let H6 \def (eq_ind T
+(THead (Flat Appl) u0 t0) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
+[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind
+Abst) u t) H5) in (False_ind (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq
+A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g
a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t
-a4))) (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4))))
+a4)))) H6)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a0:
+A).(\lambda (_: (arity g c0 u0 (asucc g a0))).(\lambda (_: (((eq T u0 (THead
+(Bind Abst) u t)) \to (ex3_2 A A (\lambda (a1: A).(\lambda (a2: A).(eq A
+(asucc g a0) (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u
+(asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind
+Abst) u) t a2))))))).(\lambda (t0: T).(\lambda (_: (arity g c0 t0
+a0)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda
+(a1: A).(\lambda (a2: A).(eq A a0 (AHead a1 a2)))) (\lambda (a1: A).(\lambda
+(_: A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity
+g (CHead c0 (Bind Abst) u) t a2))))))).(\lambda (H5: (eq T (THead (Flat Cast)
+u0 t0) (THead (Bind Abst) u t))).(let H6 \def (eq_ind T (THead (Flat Cast) u0
+t0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
+_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
+\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
+\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u t)
+H5) in (False_ind (ex3_2 A A (\lambda (a1: A).(\lambda (a2: A).(eq A a0
+(AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u (asucc g
+a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind Abst) u) t
+a2)))) H6))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (a1:
+A).(\lambda (H1: (arity g c0 t0 a1)).(\lambda (H2: (((eq T t0 (THead (Bind
+Abst) u t)) \to (ex3_2 A A (\lambda (a2: A).(\lambda (a3: A).(eq A a1 (AHead
+a2 a3)))) (\lambda (a2: A).(\lambda (_: A).(arity g c0 u (asucc g a2))))
+(\lambda (_: A).(\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u) t
+a3))))))).(\lambda (a2: A).(\lambda (H3: (leq g a1 a2)).(\lambda (H4: (eq T
+t0 (THead (Bind Abst) u t))).(let H5 \def (f_equal T T (\lambda (e: T).e) t0
+(THead (Bind Abst) u t) H4) in (let H6 \def (eq_ind T t0 (\lambda (t1:
+T).((eq T t1 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3:
+A).(\lambda (a4: A).(eq A a1 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_:
+A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g
+(CHead c0 (Bind Abst) u) t a4)))))) H2 (THead (Bind Abst) u t) H5) in (let H7
+\def (eq_ind T t0 (\lambda (t1: T).(arity g c0 t1 a1)) H1 (THead (Bind Abst)
+u t) H5) in (let H8 \def (H6 (refl_equal T (THead (Bind Abst) u t))) in
+(ex3_2_ind A A (\lambda (a3: A).(\lambda (a4: A).(eq A a1 (AHead a3 a4))))
(\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) (\lambda (_:
-A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t a4)))) (\lambda (x0:
-A).(\lambda (x1: A).(\lambda (H9: (eq A a1 (AHead x0 x1))).(\lambda (H10:
-(arity g c0 u (asucc g x0))).(\lambda (H11: (arity g (CHead c0 (Bind Abst) u)
-t x1)).(let H12 \def (eq_ind A a1 (\lambda (a0: A).(leq g a0 a2)) H3 (AHead
-x0 x1) H9) in (let H13 \def (eq_ind A a1 (\lambda (a0: A).(arity g c0 (THead
-(Bind Abst) u t) a0)) H7 (AHead x0 x1) H9) in (let H_x \def (leq_gen_head g
-x0 x1 a2 H12) in (let H14 \def H_x in (ex3_2_ind A A (\lambda (a3:
+A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t a4))) (ex3_2 A A
+(\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3:
+A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda
+(a4: A).(arity g (CHead c0 (Bind Abst) u) t a4)))) (\lambda (x0: A).(\lambda
+(x1: A).(\lambda (H9: (eq A a1 (AHead x0 x1))).(\lambda (H10: (arity g c0 u
+(asucc g x0))).(\lambda (H11: (arity g (CHead c0 (Bind Abst) u) t x1)).(let
+H12 \def (eq_ind A a1 (\lambda (a0: A).(leq g a0 a2)) H3 (AHead x0 x1) H9) in
+(let H13 \def (eq_ind A a1 (\lambda (a0: A).(arity g c0 (THead (Bind Abst) u
+t) a0)) H7 (AHead x0 x1) H9) in (let H_x \def (leq_gen_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).(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).(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)))) (\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)))))).
+(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:
\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 in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I
-(THead (Flat Appl) u t) H1) in (False_ind (ex2 A (\lambda (a1: A).(arity g c0
-u a1)) (\lambda (a1: A).(arity g c0 t (AHead a1 (ASort O n))))) H2)))))
-(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda
-(_: (getl i c0 (CHead d (Bind Abbr) u0))).(\lambda (a: A).(\lambda (_: (arity
-g d u0 a)).(\lambda (_: (((eq T u0 (THead (Flat Appl) u t)) \to (ex2 A
-(\lambda (a1: A).(arity g d u a1)) (\lambda (a1: A).(arity g d t (AHead a1
-a))))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat Appl) u t))).(let H5 \def
-(eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
-(THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) u t) H4) in
-(False_ind (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity
-g c0 t (AHead a1 a)))) H5))))))))))) (\lambda (c0: C).(\lambda (d:
+(Flat Appl) u t) (\lambda (t0: T).(arity g c t0 a2)) (\lambda (_: T).(ex2 A
+(\lambda (a1: A).(arity g c u a1)) (\lambda (a1: A).(arity g c t (AHead a1
+a2))))) (\lambda (y: T).(\lambda (H0: (arity g c y a2)).(arity_ind g (\lambda
+(c0: C).(\lambda (t0: T).(\lambda (a: A).((eq T t0 (THead (Flat Appl) u t))
+\to (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t
+(AHead a1 a)))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (H1: (eq T
+(TSort n) (THead (Flat Appl) u t))).(let H2 \def (eq_ind T (TSort n) (\lambda
+(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
+False])) I (THead (Flat Appl) u t) H1) in (False_ind (ex2 A (\lambda (a1:
+A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t (AHead a1 (ASort O
+n))))) H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i:
+nat).(\lambda (_: (getl i c0 (CHead d (Bind Abbr) u0))).(\lambda (a:
+A).(\lambda (_: (arity g d u0 a)).(\lambda (_: (((eq T u0 (THead (Flat Appl)
+u t)) \to (ex2 A (\lambda (a1: A).(arity g d u a1)) (\lambda (a1: A).(arity g
+d t (AHead a1 a))))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat Appl) u
+t))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) u
+t) H4) in (False_ind (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1:
+A).(arity g c0 t (AHead a1 a)))) H5))))))))))) (\lambda (c0: C).(\lambda (d:
C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind
Abst) u0))).(\lambda (a: A).(\lambda (_: (arity g d u0 (asucc g a))).(\lambda
(_: (((eq T u0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda (a1: A).(arity g
\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 in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I
-(THead (Flat Cast) u t) H1) in (False_ind (land (arity g c0 u (asucc g (ASort
-O n))) (arity g c0 t (ASort O n))) H2))))) (\lambda (c0: C).(\lambda (d:
-C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind
-Abbr) u0))).(\lambda (a0: A).(\lambda (_: (arity g d u0 a0)).(\lambda (_:
-(((eq T u0 (THead (Flat Cast) u t)) \to (land (arity g d u (asucc g a0))
-(arity g d t a0))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat Cast) u
-t))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+(Flat Cast) u t) (\lambda (t0: T).(arity g c t0 a)) (\lambda (_: T).(land
+(arity g c u (asucc g a)) (arity g c t a))) (\lambda (y: T).(\lambda (H0:
+(arity g c y a)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (a0:
+A).((eq T t0 (THead (Flat Cast) u t)) \to (land (arity g c0 u (asucc g a0))
+(arity g c0 t a0)))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (H1: (eq T
+(TSort n) (THead (Flat Cast) u t))).(let H2 \def (eq_ind T (TSort n) (\lambda
+(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
+False])) I (THead (Flat Cast) u t) H1) in (False_ind (land (arity g c0 u
+(asucc g (ASort O n))) (arity g c0 t (ASort O n))) H2))))) (\lambda (c0:
+C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0
+(CHead d (Bind Abbr) u0))).(\lambda (a0: A).(\lambda (_: (arity g d u0
+a0)).(\lambda (_: (((eq T u0 (THead (Flat Cast) u t)) \to (land (arity g d u
+(asucc g a0)) (arity g d t a0))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat
+Cast) u t))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) u
t) H4) in (False_ind (land (arity g c0 u (asucc g a0)) (arity g c0 t a0))
H5))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i:
\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
+(lift h d t) (\lambda (t0: T).(arity g c1 t0 a)) (\lambda (_: T).(\forall
+(c2: C).((drop h d c1 c2) \to (arity g c2 t a)))) (\lambda (y: T).(\lambda
+(H0: (arity g c1 y a)).(unintro T t (\lambda (t0: T).((eq T y (lift h d t0))
+\to (\forall (c2: C).((drop h d c1 c2) \to (arity g c2 t0 a))))) (unintro nat
+d (\lambda (n: nat).(\forall (x: T).((eq T y (lift h n x)) \to (\forall (c2:
+C).((drop h n c1 c2) \to (arity g c2 x a)))))) (arity_ind g (\lambda (c:
+C).(\lambda (t0: T).(\lambda (a0: A).(\forall (x: nat).(\forall (x0: T).((eq
+T t0 (lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to (arity g c2 x0
+a0))))))))) (\lambda (c: C).(\lambda (n: nat).(\lambda (x: nat).(\lambda (x0:
+T).(\lambda (H1: (eq T (TSort n) (lift h x x0))).(\lambda (c2: C).(\lambda
+(_: (drop h x c c2)).(eq_ind_r T (TSort n) (\lambda (t0: T).(arity g c2 t0
+(ASort O n))) (arity_sort g c2 n) x0 (lift_gen_sort h x n x0 H1)))))))))
+(\lambda (c: C).(\lambda (d0: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H1: (getl i c (CHead d0 (Bind Abbr) u))).(\lambda (a0: A).(\lambda (H2:
+(arity g d0 u a0)).(\lambda (H3: ((\forall (x: nat).(\forall (x0: T).((eq T u
+(lift h x x0)) \to (\forall (c2: C).((drop h x d0 c2) \to (arity g c2 x0
a0)))))))).(\lambda (x: nat).(\lambda (x0: T).(\lambda (H4: (eq T (TLRef i)
(lift h x x0))).(\lambda (c2: C).(\lambda (H5: (drop h x c c2)).(let H_x \def
(lift_gen_lref x0 x h i H4) in (let H6 \def H_x in (or_ind (land (lt i x) (eq
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/arity/lift1".
include "arity/props.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/arity/pr3".
include "csuba/arity.ma".
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 in T
-return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
-\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) (THead (Bind
-b) u t) H12) in ((let H14 \def (f_equal T T (\lambda (e: T).(match e in T
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _)
-\Rightarrow u1 | (THead _ t0 _) \Rightarrow t0])) (THead k u1 t3) (THead
-(Bind b) u t) H12) in ((let H15 \def (f_equal T T (\lambda (e: T).(match e in
-T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _)
-\Rightarrow t3 | (THead _ _ t0) \Rightarrow t0])) (THead k u1 t3) (THead
-(Bind b) u t) H12) in (\lambda (H16: (eq T u1 u)).(\lambda (H17: (eq K k
-(Bind b))).(eq_ind_r K (Bind b) (\lambda (k0: K).(arity g c2 (THead k0 u2 t4)
-a2)) (let H18 \def (eq_ind T t3 (\lambda (t0: T).((eq T t0 (THead (Bind b) u
-t)) \to (arity g c2 t4 a2))) H11 t H15) in (let H19 \def (eq_ind T t3
+T (THead (Bind b) u t) (\lambda (t0: T).(pr0 t0 t2)) (\lambda (_: T).(arity g
+c2 t2 a2)) (\lambda (y: T).(\lambda (H7: (pr0 y t2)).(pr0_ind (\lambda (t0:
+T).(\lambda (t3: T).((eq T t0 (THead (Bind b) u t)) \to (arity g c2 t3 a2))))
+(\lambda (t0: T).(\lambda (H8: (eq T t0 (THead (Bind b) u t))).(let H9 \def
+(f_equal T T (\lambda (e: T).e) t0 (THead (Bind b) u t) H8) in (eq_ind_r T
+(THead (Bind b) u t) (\lambda (t3: T).(arity g c2 t3 a2)) (arity_bind g b H0
+c2 u a1 (H2 c2 H5 u (pr0_refl u)) t a2 (H4 (CHead c2 (Bind b) u) (wcpr0_comp
+c c2 H5 u u (pr0_refl u) (Bind b)) t (pr0_refl t))) t0 H9)))) (\lambda (u1:
+T).(\lambda (u2: T).(\lambda (H8: (pr0 u1 u2)).(\lambda (H9: (((eq T u1
+(THead (Bind b) u t)) \to (arity g c2 u2 a2)))).(\lambda (t3: T).(\lambda
+(t4: T).(\lambda (H10: (pr0 t3 t4)).(\lambda (H11: (((eq T t3 (THead (Bind b)
+u t)) \to (arity g c2 t4 a2)))).(\lambda (k: K).(\lambda (H12: (eq T (THead k
+u1 t3) (THead (Bind b) u t))).(let H13 \def (f_equal T K (\lambda (e:
+T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k |
+(TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3)
+(THead (Bind b) u t) H12) in ((let H14 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 |
+(TLRef _) \Rightarrow u1 | (THead _ t0 _) \Rightarrow t0])) (THead k u1 t3)
+(THead (Bind b) u t) H12) in ((let H15 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 |
+(TLRef _) \Rightarrow t3 | (THead _ _ t0) \Rightarrow t0])) (THead k u1 t3)
+(THead (Bind b) u t) H12) in (\lambda (H16: (eq T u1 u)).(\lambda (H17: (eq K
+k (Bind b))).(eq_ind_r K (Bind b) (\lambda (k0: K).(arity g c2 (THead k0 u2
+t4) a2)) (let H18 \def (eq_ind T t3 (\lambda (t0: T).((eq T t0 (THead (Bind
+b) u t)) \to (arity g c2 t4 a2))) H11 t H15) in (let H19 \def (eq_ind T t3
(\lambda (t0: T).(pr0 t0 t4)) H10 t H15) in (let H20 \def (eq_ind T u1
(\lambda (t0: T).((eq T t0 (THead (Bind b) u t)) \to (arity g c2 u2 a2))) H9
u H16) in (let H21 \def (eq_ind T u1 (\lambda (t0: T).(pr0 t0 u2)) H8 u H16)
(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:
+(\lambda (_: T).(arity g c2 t2 (AHead a1 a2))) (\lambda (y: T).(\lambda (H6:
+(pr0 y t2)).(pr0_ind (\lambda (t0: T).(\lambda (t3: T).((eq T t0 (THead (Bind
+Abst) u t)) \to (arity g c2 t3 (AHead a1 a2))))) (\lambda (t0: T).(\lambda
+(H7: (eq T t0 (THead (Bind Abst) u t))).(let H8 \def (f_equal T T (\lambda
+(e: T).e) t0 (THead (Bind Abst) u t) H7) in (eq_ind_r T (THead (Bind Abst) u
+t) (\lambda (t3: T).(arity g c2 t3 (AHead a1 a2))) (arity_head g c2 u a1 (H1
+c2 H4 u (pr0_refl u)) t a2 (H3 (CHead c2 (Bind Abst) u) (wcpr0_comp c c2 H4 u
+u (pr0_refl u) (Bind Abst)) t (pr0_refl t))) t0 H8)))) (\lambda (u1:
T).(\lambda (u2: T).(\lambda (H7: (pr0 u1 u2)).(\lambda (H8: (((eq T u1
(THead (Bind Abst) u t)) \to (arity g c2 u2 (AHead a1 a2))))).(\lambda (t3:
T).(\lambda (t4: T).(\lambda (H9: (pr0 t3 t4)).(\lambda (H10: (((eq T t3
\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 in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k
-| (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3)
+Appl) u t) (\lambda (t0: T).(pr0 t0 t2)) (\lambda (_: T).(arity g c2 t2 a2))
+(\lambda (y: T).(\lambda (H6: (pr0 y t2)).(pr0_ind (\lambda (t0: T).(\lambda
+(t3: T).((eq T t0 (THead (Flat Appl) u t)) \to (arity g c2 t3 a2)))) (\lambda
+(t0: T).(\lambda (H7: (eq T t0 (THead (Flat Appl) u t))).(let H8 \def
+(f_equal T T (\lambda (e: T).e) t0 (THead (Flat Appl) u t) H7) in (eq_ind_r T
+(THead (Flat Appl) u t) (\lambda (t3: T).(arity g c2 t3 a2)) (arity_appl g c2
+u a1 (H1 c2 H4 u (pr0_refl u)) t a2 (H3 c2 H4 t (pr0_refl t))) t0 H8))))
+(\lambda (u1: T).(\lambda (u2: T).(\lambda (H7: (pr0 u1 u2)).(\lambda (H8:
+(((eq T u1 (THead (Flat Appl) u t)) \to (arity g c2 u2 a2)))).(\lambda (t3:
+T).(\lambda (t4: T).(\lambda (H9: (pr0 t3 t4)).(\lambda (H10: (((eq T t3
+(THead (Flat Appl) u t)) \to (arity g c2 t4 a2)))).(\lambda (k: K).(\lambda
+(H11: (eq T (THead k u1 t3) (THead (Flat Appl) u t))).(let H12 \def (f_equal
+T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _)
+\Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0]))
+(THead k u1 t3) (THead (Flat Appl) u t) H11) in ((let H13 \def (f_equal T T
+(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t0 _) \Rightarrow t0]))
+(THead k u1 t3) (THead (Flat Appl) u t) H11) in ((let H14 \def (f_equal T T
+(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t0) \Rightarrow t0]))
+(THead k u1 t3) (THead (Flat Appl) u t) H11) in (\lambda (H15: (eq T u1
+u)).(\lambda (H16: (eq K k (Flat Appl))).(eq_ind_r K (Flat Appl) (\lambda
+(k0: K).(arity g c2 (THead k0 u2 t4) a2)) (let H17 \def (eq_ind T t3 (\lambda
+(t0: T).((eq T t0 (THead (Flat Appl) u t)) \to (arity g c2 t4 a2))) H10 t
+H14) in (let H18 \def (eq_ind T t3 (\lambda (t0: T).(pr0 t0 t4)) H9 t H14) in
+(let H19 \def (eq_ind T u1 (\lambda (t0: T).((eq T t0 (THead (Flat Appl) u
+t)) \to (arity g c2 u2 a2))) H8 u H15) in (let H20 \def (eq_ind T u1 (\lambda
+(t0: T).(pr0 t0 u2)) H7 u H15) in (arity_appl g c2 u2 a1 (H1 c2 H4 u2 H20) t4
+a2 (H3 c2 H4 t4 H18)))))) k H16)))) H13)) H12)))))))))))) (\lambda (u0:
+T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H7: (pr0 v1 v2)).(\lambda (H8:
+(((eq T v1 (THead (Flat Appl) u t)) \to (arity g c2 v2 a2)))).(\lambda (t3:
+T).(\lambda (t4: T).(\lambda (H9: (pr0 t3 t4)).(\lambda (H10: (((eq T t3
+(THead (Flat Appl) u t)) \to (arity g c2 t4 a2)))).(\lambda (H11: (eq T
+(THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) (THead (Flat Appl) u
+t))).(let H12 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda
+(_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead
+_ t0 _) \Rightarrow t0])) (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3))
(THead (Flat Appl) u t) H11) in ((let H13 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 |
-(TLRef _) \Rightarrow u1 | (THead _ t0 _) \Rightarrow t0])) (THead k u1 t3)
-(THead (Flat Appl) u t) H11) in ((let H14 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 |
-(TLRef _) \Rightarrow t3 | (THead _ _ t0) \Rightarrow t0])) (THead k u1 t3)
-(THead (Flat Appl) u t) H11) in (\lambda (H15: (eq T u1 u)).(\lambda (H16:
-(eq K k (Flat Appl))).(eq_ind_r K (Flat Appl) (\lambda (k0: K).(arity g c2
-(THead k0 u2 t4) a2)) (let H17 \def (eq_ind T t3 (\lambda (t0: T).((eq T t0
-(THead (Flat Appl) u t)) \to (arity g c2 t4 a2))) H10 t H14) in (let H18 \def
-(eq_ind T t3 (\lambda (t0: T).(pr0 t0 t4)) H9 t H14) in (let H19 \def (eq_ind
-T u1 (\lambda (t0: T).((eq T t0 (THead (Flat Appl) u t)) \to (arity g c2 u2
-a2))) H8 u H15) in (let H20 \def (eq_ind T u1 (\lambda (t0: T).(pr0 t0 u2))
-H7 u H15) in (arity_appl g c2 u2 a1 (H1 c2 H4 u2 H20) t4 a2 (H3 c2 H4 t4
-H18)))))) k H16)))) H13)) H12)))))))))))) (\lambda (u0: T).(\lambda (v1:
-T).(\lambda (v2: T).(\lambda (H7: (pr0 v1 v2)).(\lambda (H8: (((eq T v1
-(THead (Flat Appl) u t)) \to (arity g c2 v2 a2)))).(\lambda (t3: T).(\lambda
-(t4: T).(\lambda (H9: (pr0 t3 t4)).(\lambda (H10: (((eq T t3 (THead (Flat
-Appl) u t)) \to (arity g c2 t4 a2)))).(\lambda (H11: (eq T (THead (Flat Appl)
-v1 (THead (Bind Abst) u0 t3)) (THead (Flat Appl) u t))).(let H12 \def
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
-[(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t0 _)
-\Rightarrow t0])) (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) (THead
-(Flat Appl) u t) H11) in ((let H13 \def (f_equal T T (\lambda (e: T).(match e
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow (THead (Bind Abst)
-u0 t3) | (TLRef _) \Rightarrow (THead (Bind Abst) u0 t3) | (THead _ _ t0)
-\Rightarrow t0])) (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) (THead
-(Flat Appl) u t) H11) in (\lambda (H14: (eq T v1 u)).(let H15 \def (eq_ind T
-v1 (\lambda (t0: T).((eq T t0 (THead (Flat Appl) u t)) \to (arity g c2 v2
-a2))) H8 u H14) in (let H16 \def (eq_ind T v1 (\lambda (t0: T).(pr0 t0 v2))
-H7 u H14) in (let H17 \def (eq_ind_r T t (\lambda (t0: T).((eq T t3 (THead
-(Flat Appl) u t0)) \to (arity g c2 t4 a2))) H10 (THead (Bind Abst) u0 t3)
-H13) in (let H18 \def (eq_ind_r T t (\lambda (t0: T).((eq T u (THead (Flat
-Appl) u t0)) \to (arity g c2 v2 a2))) H15 (THead (Bind Abst) u0 t3) H13) in
-(let H19 \def (eq_ind_r T t (\lambda (t0: T).(\forall (c3: C).((wcpr0 c c3)
-\to (\forall (t5: T).((pr0 t0 t5) \to (arity g c3 t5 (AHead a1 a2))))))) H3
-(THead (Bind Abst) u0 t3) H13) in (let H20 \def (eq_ind_r T t (\lambda (t0:
-T).(arity g c t0 (AHead a1 a2))) H2 (THead (Bind Abst) u0 t3) H13) in (let
-H21 \def (H1 c2 H4 v2 H16) in (let H22 \def (H19 c2 H4 (THead (Bind Abst) u0
-t4) (pr0_comp u0 u0 (pr0_refl u0) t3 t4 H9 (Bind Abst))) in (let H23 \def
-(arity_gen_abst g c2 u0 t4 (AHead a1 a2) H22) in (ex3_2_ind A A (\lambda (a3:
-A).(\lambda (a4: A).(eq A (AHead a1 a2) (AHead a3 a4)))) (\lambda (a3:
-A).(\lambda (_: A).(arity g c2 u0 (asucc g a3)))) (\lambda (_: A).(\lambda
-(a4: A).(arity g (CHead c2 (Bind Abst) u0) t4 a4))) (arity g c2 (THead (Bind
-Abbr) v2 t4) a2) (\lambda (x0: A).(\lambda (x1: A).(\lambda (H24: (eq A
-(AHead a1 a2) (AHead x0 x1))).(\lambda (H25: (arity g c2 u0 (asucc g
-x0))).(\lambda (H26: (arity g (CHead c2 (Bind Abst) u0) t4 x1)).(let H27 \def
-(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
-[(ASort _ _) \Rightarrow a1 | (AHead a0 _) \Rightarrow a0])) (AHead a1 a2)
-(AHead x0 x1) H24) in ((let H28 \def (f_equal A A (\lambda (e: A).(match e in
-A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a2 | (AHead _ a0)
-\Rightarrow a0])) (AHead a1 a2) (AHead x0 x1) H24) in (\lambda (H29: (eq A a1
-x0)).(let H30 \def (eq_ind_r A x1 (\lambda (a0: A).(arity g (CHead c2 (Bind
-Abst) u0) t4 a0)) H26 a2 H28) in (let H31 \def (eq_ind_r A x0 (\lambda (a0:
-A).(arity g c2 u0 (asucc g a0))) H25 a1 H29) in (arity_bind g Abbr
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow (THead
+(Bind Abst) u0 t3) | (TLRef _) \Rightarrow (THead (Bind Abst) u0 t3) | (THead
+_ _ t0) \Rightarrow t0])) (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3))
+(THead (Flat Appl) u t) H11) in (\lambda (H14: (eq T v1 u)).(let H15 \def
+(eq_ind T v1 (\lambda (t0: T).((eq T t0 (THead (Flat Appl) u t)) \to (arity g
+c2 v2 a2))) H8 u H14) in (let H16 \def (eq_ind T v1 (\lambda (t0: T).(pr0 t0
+v2)) H7 u H14) in (let H17 \def (eq_ind_r T t (\lambda (t0: T).((eq T t3
+(THead (Flat Appl) u t0)) \to (arity g c2 t4 a2))) H10 (THead (Bind Abst) u0
+t3) H13) in (let H18 \def (eq_ind_r T t (\lambda (t0: T).((eq T u (THead
+(Flat Appl) u t0)) \to (arity g c2 v2 a2))) H15 (THead (Bind Abst) u0 t3)
+H13) in (let H19 \def (eq_ind_r T t (\lambda (t0: T).(\forall (c3: C).((wcpr0
+c c3) \to (\forall (t5: T).((pr0 t0 t5) \to (arity g c3 t5 (AHead a1
+a2))))))) H3 (THead (Bind Abst) u0 t3) H13) in (let H20 \def (eq_ind_r T t
+(\lambda (t0: T).(arity g c t0 (AHead a1 a2))) H2 (THead (Bind Abst) u0 t3)
+H13) in (let H21 \def (H1 c2 H4 v2 H16) in (let H22 \def (H19 c2 H4 (THead
+(Bind Abst) u0 t4) (pr0_comp u0 u0 (pr0_refl u0) t3 t4 H9 (Bind Abst))) in
+(let H23 \def (arity_gen_abst g c2 u0 t4 (AHead a1 a2) H22) in (ex3_2_ind A A
+(\lambda (a3: A).(\lambda (a4: A).(eq A (AHead a1 a2) (AHead a3 a4))))
+(\lambda (a3: A).(\lambda (_: A).(arity g c2 u0 (asucc g a3)))) (\lambda (_:
+A).(\lambda (a4: A).(arity g (CHead c2 (Bind Abst) u0) t4 a4))) (arity g c2
+(THead (Bind Abbr) v2 t4) a2) (\lambda (x0: A).(\lambda (x1: A).(\lambda
+(H24: (eq A (AHead a1 a2) (AHead x0 x1))).(\lambda (H25: (arity g c2 u0
+(asucc g x0))).(\lambda (H26: (arity g (CHead c2 (Bind Abst) u0) t4 x1)).(let
+H27 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A)
+with [(ASort _ _) \Rightarrow a1 | (AHead a0 _) \Rightarrow a0])) (AHead a1
+a2) (AHead x0 x1) H24) in ((let H28 \def (f_equal A A (\lambda (e: A).(match
+e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a2 | (AHead _
+a0) \Rightarrow a0])) (AHead a1 a2) (AHead x0 x1) H24) in (\lambda (H29: (eq
+A a1 x0)).(let H30 \def (eq_ind_r A x1 (\lambda (a0: A).(arity g (CHead c2
+(Bind Abst) u0) t4 a0)) H26 a2 H28) in (let H31 \def (eq_ind_r A x0 (\lambda
+(a0: A).(arity g c2 u0 (asucc g a0))) H25 a1 H29) in (arity_bind g Abbr
not_abbr_abst c2 v2 a1 H21 t4 a2 (csuba_arity g (CHead c2 (Bind Abst) u0) t4
a2 H30 (CHead c2 (Bind Abbr) v2) (csuba_abst g c2 c2 (csuba_refl g c2) u0 a1
H31 v2 H21))))))) H27))))))) H23)))))))))))) H12)))))))))))) (\lambda (b:
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 in T
-return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
-\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) (THead (Flat
-Cast) u t) H11) in ((let H13 \def (f_equal T T (\lambda (e: T).(match e in T
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _)
-\Rightarrow u1 | (THead _ t0 _) \Rightarrow t0])) (THead k u1 t3) (THead
-(Flat Cast) u t) H11) in ((let H14 \def (f_equal T T (\lambda (e: T).(match e
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _)
-\Rightarrow t3 | (THead _ _ t0) \Rightarrow t0])) (THead k u1 t3) (THead
-(Flat Cast) u t) H11) in (\lambda (H15: (eq T u1 u)).(\lambda (H16: (eq K k
-(Flat Cast))).(eq_ind_r K (Flat Cast) (\lambda (k0: K).(arity g c2 (THead k0
-u2 t4) a0)) (let H17 \def (eq_ind T t3 (\lambda (t0: T).((eq T t0 (THead
-(Flat Cast) u t)) \to (arity g c2 t4 a0))) H10 t H14) in (let H18 \def
+(t0: T).(pr0 t0 t2)) (\lambda (_: T).(arity g c2 t2 a0)) (\lambda (y:
+T).(\lambda (H6: (pr0 y t2)).(pr0_ind (\lambda (t0: T).(\lambda (t3: T).((eq
+T t0 (THead (Flat Cast) u t)) \to (arity g c2 t3 a0)))) (\lambda (t0:
+T).(\lambda (H7: (eq T t0 (THead (Flat Cast) u t))).(let H8 \def (f_equal T T
+(\lambda (e: T).e) t0 (THead (Flat Cast) u t) H7) in (eq_ind_r T (THead (Flat
+Cast) u t) (\lambda (t3: T).(arity g c2 t3 a0)) (arity_cast g c2 u a0 (H1 c2
+H4 u (pr0_refl u)) t (H3 c2 H4 t (pr0_refl t))) t0 H8)))) (\lambda (u1:
+T).(\lambda (u2: T).(\lambda (H7: (pr0 u1 u2)).(\lambda (H8: (((eq T u1
+(THead (Flat Cast) u t)) \to (arity g c2 u2 a0)))).(\lambda (t3: T).(\lambda
+(t4: T).(\lambda (H9: (pr0 t3 t4)).(\lambda (H10: (((eq T t3 (THead (Flat
+Cast) u t)) \to (arity g c2 t4 a0)))).(\lambda (k: K).(\lambda (H11: (eq T
+(THead k u1 t3) (THead (Flat Cast) u t))).(let H12 \def (f_equal T K (\lambda
+(e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k
+| (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3)
+(THead (Flat Cast) u t) H11) in ((let H13 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 |
+(TLRef _) \Rightarrow u1 | (THead _ t0 _) \Rightarrow t0])) (THead k u1 t3)
+(THead (Flat Cast) u t) H11) in ((let H14 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 |
+(TLRef _) \Rightarrow t3 | (THead _ _ t0) \Rightarrow t0])) (THead k u1 t3)
+(THead (Flat Cast) u t) H11) in (\lambda (H15: (eq T u1 u)).(\lambda (H16:
+(eq K k (Flat Cast))).(eq_ind_r K (Flat Cast) (\lambda (k0: K).(arity g c2
+(THead k0 u2 t4) a0)) (let H17 \def (eq_ind T t3 (\lambda (t0: T).((eq T t0
+(THead (Flat Cast) u t)) \to (arity g c2 t4 a0))) H10 t H14) in (let H18 \def
(eq_ind T t3 (\lambda (t0: T).(pr0 t0 t4)) H9 t H14) in (let H19 \def (eq_ind
T u1 (\lambda (t0: T).((eq T t0 (THead (Flat Cast) u t)) \to (arity g c2 u2
a0))) H8 u H15) in (let H20 \def (eq_ind T u1 (\lambda (t0: T).(pr0 t0 u2))
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/arity/props".
include "arity/fwd.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/arity/subst0".
include "arity/props.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/asucc/defs".
include "A/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/asucc/fwd".
include "asucc/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/cimp/defs".
include "getl/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/cimp/props".
include "cimp/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/clear/defs".
include "C/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/clear/drop".
include "clear/fwd.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/clear/fwd".
include "clear/defs.ma".
\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 in C return
-(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _)
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
-\Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead e (Flat f) u) H1)
-in (False_ind P H2)))))) (\lambda (e0: C).(\lambda (c: C).(\lambda (H1:
-(clear e0 c)).(\lambda (H2: (((eq C c (CHead e (Flat f) u)) \to P))).(\lambda
-(_: F).(\lambda (_: T).(\lambda (H3: (eq C c (CHead e (Flat f) u))).(let H4
-\def (eq_ind C c (\lambda (c0: C).((eq C c0 (CHead e (Flat f) u)) \to P)) H2
-(CHead e (Flat f) u) H3) in (let H5 \def (eq_ind C c (\lambda (c0: C).(clear
-e0 c0)) H1 (CHead e (Flat f) u) H3) in (H4 (refl_equal C (CHead e (Flat f)
-u)))))))))))) x y H0))) H)))))).
+(Flat f) u) (\lambda (c: C).(clear x c)) (\lambda (_: C).P) (\lambda (y:
+C).(\lambda (H0: (clear x y)).(clear_ind (\lambda (_: C).(\lambda (c0:
+C).((eq C c0 (CHead e (Flat f) u)) \to P))) (\lambda (b: B).(\lambda (e0:
+C).(\lambda (u0: T).(\lambda (H1: (eq C (CHead e0 (Bind b) u0) (CHead e (Flat
+f) u))).(let H2 \def (eq_ind C (CHead e0 (Bind b) u0) (\lambda (ee: C).(match
+ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False |
+(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
+[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead e (Flat
+f) u) H1) in (False_ind P H2)))))) (\lambda (e0: C).(\lambda (c: C).(\lambda
+(H1: (clear e0 c)).(\lambda (H2: (((eq C c (CHead e (Flat f) u)) \to
+P))).(\lambda (_: F).(\lambda (_: T).(\lambda (H3: (eq C c (CHead e (Flat f)
+u))).(let H4 \def (eq_ind C c (\lambda (c0: C).((eq C c0 (CHead e (Flat f)
+u)) \to P)) H2 (CHead e (Flat f) u) H3) in (let H5 \def (eq_ind C c (\lambda
+(c0: C).(clear e0 c0)) H1 (CHead e (Flat f) u) H3) in (H4 (refl_equal C
+(CHead e (Flat f) u)))))))))))) x y H0))) H)))))).
theorem clear_gen_all:
\forall (c1: C).(\forall (c2: C).((clear c1 c2) \to (ex_3 B C T (\lambda (b:
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/clear/props".
include "clear/fwd.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/clen/defs".
include "C/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/clen/getl".
include "clen/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/cnt/defs".
include "T/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/cnt/props".
include "cnt/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/csuba/arity".
include "csuba/getl.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/csuba/clear".
include "csuba/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/csuba/defs".
include "arity/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/csuba/drop".
include "csuba/fwd.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/csuba/fwd".
include "csuba/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/csuba/getl".
include "csuba/drop.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/csuba/props".
include "csuba/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/csubc/arity".
include "csubc/csuba.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/csubc/clear".
include "csubc/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/csubc/csuba".
include "csubc/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/csubc/defs".
include "sc3/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/csubc/drop".
include "csubc/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/csubc/drop1".
include "csubc/drop.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/csubc/getl".
include "csubc/drop.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/csubc/props".
include "csubc/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/csubst0/clear".
include "csubst0/fwd.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/csubst0/defs".
include "subst0/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/csubst0/drop".
include "csubst0/fwd.ma".
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)
+(\lambda (n0: nat).(or4 (drop n0 n0 c2 c1) (ex3_4 F C T T (\lambda (f:
+F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c1 (CHead e0 (Flat
+f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(drop n0 n0 c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v u w)))))) (ex3_4 F C C T
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c1
(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(drop 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 in
-nat 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))))))
+C).(\lambda (u: T).(drop n0 n0 c2 (CHead e2 (Flat f) u)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v e1
+e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c1 (CHead e1 (Flat f) u)))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop n0 n0 c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v u w))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 n0 v e1 e2))))))))) (\lambda (y: nat).(\lambda (H1: (csubst0
+y v c1 c2)).(csubst0_ind (\lambda (n0: nat).(\lambda (t: T).(\lambda (c:
+C).(\lambda (c0: C).((eq nat n0 O) \to (or4 (drop n0 n0 c0 c) (ex3_4 F C T T
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c
+(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop n0 n0 c0 (CHead e0 (Flat f) w)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 t u w))))))
(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C (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))))))
+T).(eq C c (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop n0 n0 c0 (CHead e2 (Flat f) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0
+t e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f) u)))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop n0 n0 c0 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 t u w))))))
(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 O v0 e1 e2)))))))) H4)))))))))) (\lambda (f: F).(\lambda (i:
+(_: T).(csubst0 n0 t e1 e2))))))))))))) (\lambda (k: K).(K_ind (\lambda (k0:
+K).(\forall (i: nat).(\forall (v0: T).(\forall (u1: T).(\forall (u2:
+T).((subst0 i v0 u1 u2) \to (\forall (c: C).((eq nat (s k0 i) O) \to (or4
+(drop (s k0 i) (s k0 i) (CHead c k0 u2) (CHead c k0 u1)) (ex3_4 F C T T
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead
+c k0 u1) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda
+(_: T).(\lambda (w: T).(drop (s k0 i) (s k0 i) (CHead c k0 u2) (CHead e0
+(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (s k0 i) v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c k0 u1) (CHead e1 (Flat f)
+u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop (s k0 i) (s k0 i) (CHead c k0 u2) (CHead e2 (Flat f) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (s
+k0 i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c k0 u1)
+(CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) (s k0 i) (CHead c k0 u2)
+(CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (s k0 i) v0 u w)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(s k0 i) v0 e1 e2)))))))))))))))) (\lambda (b: B).(\lambda (i: nat).(\lambda
+(v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0 i v0 u1
+u2)).(\lambda (c: C).(\lambda (H3: (eq nat (S i) O)).(let H4 \def (eq_ind nat
+(S i) (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with
+[O \Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind (or4
+(drop (S i) (S i) (CHead c (Bind b) u2) (CHead c (Bind b) u1)) (ex3_4 F C T T
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead
+c (Bind b) u1) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S i) (S i) (CHead c (Bind b) u2)
+(CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (S i) v0 u w)))))) (ex3_4 F C C T (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c (Bind b)
+u1) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop (S i) (S i) (CHead c (Bind b) u2) (CHead e2 (Flat f)
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 (S i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c (Bind
+b) u1) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S i) (S i) (CHead c (Bind b)
+u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (S i) v0 u w)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(S i) v0 e1 e2)))))))) H4)))))))))) (\lambda (f: F).(\lambda (i:
nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0
i v0 u1 u2)).(\lambda (c: C).(\lambda (H3: (eq nat i O)).(let H4 \def (eq_ind
-nat i (\lambda (n0: nat).(subst0 n0 v0 u1 u2)) H2 O H3) in (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:
+nat i (\lambda (n0: nat).(subst0 n0 v0 u1 u2)) H2 O H3) in (eq_ind_r nat O
+(\lambda (n0: nat).(or4 (drop n0 n0 (CHead c (Flat f) u2) (CHead c (Flat f)
+u1)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u:
T).(\lambda (_: T).(eq C (CHead c (Flat f) u1) (CHead e0 (Flat f0) u))))))
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0
(CHead c (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 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))))))
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v0 u w)))))) (ex3_4 F C C T
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C
+(CHead c (Flat f) u1) (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 n0 (CHead c (Flat f) u2)
+(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
+(CHead c (Flat f) u1) (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 (CHead c
+(Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v0 u w))))))
(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 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:
+(_: T).(csubst0 n0 v0 e1 e2))))))))) (or4_intro1 (drop O O (CHead c (Flat f)
+u2) (CHead c (Flat f) u1)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c (Flat f) u1) (CHead e0
+(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(w: T).(drop O O (CHead c (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w))))))
+(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C (CHead c (Flat f) u1) (CHead e1 (Flat f0) u)))))) (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O (CHead c (Flat
+f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda
+(f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
+C (CHead c (Flat f) u1) (CHead e1 (Flat f0) u))))))) (\lambda (f0:
F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O
-(CHead 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))))))
+(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))))))))))))))))) (\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
+(_: T).(csubst0 O v0 e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0:
+F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c (Flat f)
+u1) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda
+(_: T).(\lambda (w: T).(drop O O (CHead c (Flat f) u2) (CHead e0 (Flat f0)
+w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v0 u w))))) f c u1 u2 (refl_equal C (CHead c (Flat f) u1))
+(drop_refl (CHead c (Flat f) u2)) H4)) i H3)))))))))) k)) (\lambda (k:
+K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (c3: C).(\forall (c4:
+C).(\forall (v0: T).((csubst0 i v0 c3 c4) \to ((((eq nat i O) \to (or4 (drop
+i i c4 c3) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop i i c4 (CHead e0
+(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 i v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Flat f) u))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop i i c4
+(CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 i v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3
+(CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop i i c4 (CHead e2 (Flat f) w)))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 i v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v0 e1 e2)))))))))) \to (\forall
+(u: T).((eq nat (s k0 i) O) \to (or4 (drop (s k0 i) (s k0 i) (CHead c4 k0 u)
+(CHead c3 k0 u)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda
+(u0: T).(\lambda (_: T).(eq C (CHead c3 k0 u) (CHead e0 (Flat f) u0))))))
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k0
+i) (s k0 i) (CHead c4 k0 u) (CHead e0 (Flat f) w)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (s k0 i) v0 u0
+w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u0: T).(eq C (CHead c3 k0 u) (CHead e1 (Flat f) u0))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s k0
+i) (s k0 i) (CHead c4 k0 u) (CHead e2 (Flat f) u0)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (s k0 i) v0 e1
+e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 k0 u) (CHead e1 (Flat f)
+u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (s k0 i) (s k0 i) (CHead c4 k0 u) (CHead e2 (Flat f)
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0:
+T).(\lambda (w: T).(subst0 (s k0 i) v0 u0 w)))))) (\lambda (_: F).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (s k0 i) v0
+e1 e2))))))))))))))))) (\lambda (b: B).(\lambda (i: nat).(\lambda (c3:
+C).(\lambda (c4: C).(\lambda (v0: T).(\lambda (_: (csubst0 i v0 c3
+c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop i i c4 c3) (ex3_4 F C T T
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3
(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop O O c4 (CHead e0 (Flat f) w)))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w))))))
+T).(\lambda (w: T).(drop i i c4 (CHead e0 (Flat f) w)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w))))))
(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
T).(eq C c3 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 (Flat f) u))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
+C).(\lambda (e2: C).(\lambda (u: T).(drop i i c4 (CHead e2 (Flat f) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f) u)))))))
(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(drop 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))))))
+T).(drop i i c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w))))))
(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 O v0 e1 e2))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat
+(_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat
(S i) O)).(let H5 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee in nat
return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow
-True])) I O H4) in (False_ind (or4 (drop 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 (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
-T).(eq C c3 (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0:
-C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (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 c3 (CHead e1 (Flat f0) u)))))) (\lambda (f0:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O c4 (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 c3 (CHead e1
-(Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (w: T).(drop O O c4 (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))))))))))).(\lambda (u: T).(\lambda
-(H4: (eq nat i O)).(let H5 \def (eq_ind nat i (\lambda (n0: nat).((eq nat n0
-O) \to (or4 (drop O O c4 c3) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
-C).(\lambda (u0: T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat f0) u0))))))
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O
-c4 (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 c3 (CHead e1 (Flat
-f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0:
-T).(drop O O c4 (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 c3 (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (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)))))))))) H3 O H4) in (let H6 \def (eq_ind nat i (\lambda (n0:
-nat).(csubst0 n0 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 in nat 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
+True])) I O H4) in (False_ind (or4 (drop (S i) (S i) (CHead c4 (Bind b) u)
+(CHead c3 (Bind b) u)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
+C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 (Bind b) u) (CHead e0
+(Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(w: T).(drop (S i) (S i) (CHead c4 (Bind b) u) (CHead e0 (Flat f) w))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (S
+i) v0 u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u0: T).(eq C (CHead c3 (Bind b) u) (CHead e1 (Flat f) u0))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S i)
+(S i) (CHead c4 (Bind b) u) (CHead e2 (Flat f) u0)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (S i) v0 e1
e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (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
+C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 (Bind b) u) (CHead e1
+(Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (w: T).(drop (S i) (S i) (CHead c4 (Bind b) u) (CHead e2
+(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u0: T).(\lambda (w: T).(subst0 (S i) v0 u0 w)))))) (\lambda (_: F).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (S i) v0 e1
+e2)))))))) H5))))))))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (c3:
+C).(\lambda (c4: C).(\lambda (v0: T).(\lambda (H2: (csubst0 i v0 c3
+c4)).(\lambda (H3: (((eq nat i O) \to (or4 (drop i i c4 c3) (ex3_4 F C T T
(\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3
(CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop O O c4 (CHead e0 (Flat f0) w)))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w))))))
+T).(\lambda (w: T).(drop i i c4 (CHead e0 (Flat f0) w)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w))))))
(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
T).(eq C c3 (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 (Flat f0) u))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
+C).(\lambda (e2: C).(\lambda (u: T).(drop i i c4 (CHead e2 (Flat f0) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda
(_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f0) u)))))))
(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(w: T).(drop O O c4 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w))))))
+(w: T).(drop i i c4 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w))))))
(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 O v0 e1 e2))))))))))).(\lambda (H5: (eq nat i O)).(let H6
-\def (eq_ind nat i (\lambda (n0: nat).((eq nat n0 O) \to (or4 (drop O O c4
-c3) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda
-(_: T).(eq C c3 (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0:
-C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (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 c3 (CHead e1 (Flat f0) u)))))) (\lambda (f0:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O c4 (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 c3 (CHead e1
-(Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (w: T).(drop O O c4 (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)))))))))) H4 O H5) in (let H7 \def
+(_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat i
+O)).(let H5 \def (eq_ind nat i (\lambda (n0: nat).((eq nat n0 O) \to (or4
+(drop n0 n0 c4 c3) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda
+(u0: T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat f0) u0)))))) (\lambda (f0:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 c4 (CHead e0
+(Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda
+(w: T).(subst0 n0 v0 u0 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u0: T).(eq C c3 (CHead e1 (Flat f0) u0))))))
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop n0
+n0 c4 (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda
+(f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq
+C c3 (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 c4 (CHead e2 (Flat f0)
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0:
+T).(\lambda (w: T).(subst0 n0 v0 u0 w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 v0 e1
+e2)))))))))) H3 O H4) in (let H6 \def (eq_ind nat i (\lambda (n0:
+nat).(csubst0 n0 v0 c3 c4)) H2 O H4) in (eq_ind_r nat O (\lambda (n0:
+nat).(or4 (drop n0 n0 (CHead c4 (Flat f) u) (CHead c3 (Flat f) u)) (ex3_4 F C
+T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C
+(CHead c3 (Flat f) u) (CHead e0 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 (CHead c4 (Flat f) u)
+(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0:
+T).(\lambda (w: T).(subst0 n0 v0 u0 w)))))) (ex3_4 F C C T (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 (Flat f)
+u) (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (u0: T).(drop n0 n0 (CHead c4 (Flat f) u) (CHead e2 (Flat
+f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 (Flat f)
+u) (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 (CHead c4 (Flat f) u)
+(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u0: T).(\lambda (w: T).(subst0 n0 v0 u0 w)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+n0 v0 e1 e2))))))))) (or4_intro2 (drop O O (CHead c4 (Flat f) u) (CHead c3
+(Flat f) u)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0:
+T).(\lambda (_: T).(eq C (CHead c3 (Flat f) u) (CHead e0 (Flat f0) u0))))))
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O
+(CHead c4 (Flat f) u) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda
+(_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v0 u0 w)))))) (ex3_4 F C C
+T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C
+(CHead c3 (Flat f) u) (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (u0: T).(drop O O (CHead c4 (Flat f) u)
+(CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C
+(CHead c3 (Flat f) u) (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4
+(Flat f) u) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v0 u0 w))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v0 e1 e2))))))) (ex3_4_intro F C C T (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 (Flat f)
+u) (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (u0: T).(drop O O (CHead c4 (Flat f) u) (CHead e2 (Flat f0)
+u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 O v0 e1 e2))))) f c3 c4 u (refl_equal C (CHead c3 (Flat f) u))
+(drop_refl (CHead c4 (Flat f) u)) H6)) i H4)))))))))))) k)) (\lambda (k:
+K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (v0: T).(\forall (u1:
+T).(\forall (u2: T).((subst0 i v0 u1 u2) \to (\forall (c3: C).(\forall (c4:
+C).((csubst0 i v0 c3 c4) \to ((((eq nat i O) \to (or4 (drop i i c4 c3) (ex3_4
+F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq
+C c3 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop i i c4 (CHead e0 (Flat f) w)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w))))))
+(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C c3 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop i i c4 (CHead e2 (Flat f) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
+v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f) u)))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop i i c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 i v0 e1 e2)))))))))) \to ((eq nat (s k0 i) O) \to (or4 (drop
+(s k0 i) (s k0 i) (CHead c4 k0 u2) (CHead c3 k0 u1)) (ex3_4 F C T T (\lambda
+(f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 k0
+u1) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (s k0 i) (s k0 i) (CHead c4 k0 u2) (CHead e0 (Flat
+f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (s k0 i) v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c3 k0 u1) (CHead e1 (Flat f)
+u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop (s k0 i) (s k0 i) (CHead c4 k0 u2) (CHead e2 (Flat f) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (s
+k0 i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 k0 u1)
+(CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) (s k0 i) (CHead c4 k0 u2)
+(CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (s k0 i) v0 u w)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(s k0 i) v0 e1 e2))))))))))))))))))) (\lambda (b: B).(\lambda (i:
+nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0
+i v0 u1 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (_: (csubst0 i v0 c3
+c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop i i c4 c3) (ex3_4 F C T T
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3
+(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop i i c4 (CHead e0 (Flat f) w)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w))))))
+(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C c3 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop i i c4 (CHead e2 (Flat f) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
+v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f) u)))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop i i c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (H5: (eq nat (S i) O)).(let H6
+\def (eq_ind nat (S i) (\lambda (ee: nat).(match ee in nat return (\lambda
+(_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H5)
+in (False_ind (or4 (drop (S i) (S i) (CHead c4 (Bind b) u2) (CHead c3 (Bind
+b) u1)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C (CHead c3 (Bind b) u1) (CHead e0 (Flat f) u))))))
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S i)
+(S i) (CHead c4 (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (S i) v0 u w))))))
+(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C (CHead c3 (Bind b) u1) (CHead e1 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S i) (S i) (CHead
+c4 (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (S i) v0 e1 e2)))))) (ex4_5 F C
+C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (_: T).(eq C (CHead c3 (Bind b) u1) (CHead e1 (Flat f) u)))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S i) (S i) (CHead c4 (Bind b) u2) (CHead e2 (Flat f) w)))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (S i) v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (S i) v0 e1 e2))))))))
+H6))))))))))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (v0: T).(\lambda
+(u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 i v0 u1 u2)).(\lambda (c3:
+C).(\lambda (c4: C).(\lambda (H3: (csubst0 i v0 c3 c4)).(\lambda (H4: (((eq
+nat i O) \to (or4 (drop i i c4 c3) (ex3_4 F C T T (\lambda (f0: F).(\lambda
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat f0) u))))))
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop i i
+c4 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 i v0 u w)))))) (ex3_4 F C C T (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Flat
+f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop i i c4 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v0 e1 e2)))))) (ex4_5 F C C T
+T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
+(_: T).(eq C c3 (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop i i c4 (CHead e2
+(Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (w: T).(subst0 i v0 u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v0 e1
+e2))))))))))).(\lambda (H5: (eq nat i O)).(let H6 \def (eq_ind nat i (\lambda
+(n0: nat).((eq nat n0 O) \to (or4 (drop n0 n0 c4 c3) (ex3_4 F C T T (\lambda
+(f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0
+(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(w: T).(drop n0 n0 c4 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v0 u w)))))) (ex3_4 F C C T
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3
+(CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop n0 n0 c4 (CHead e2 (Flat f0) u)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v0 e1
+e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f0) u)))))))
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(w: T).(drop n0 n0 c4 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda
+(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v0 u
+w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))))))) H4 O H5) in (let H7 \def
(eq_ind nat i (\lambda (n0: nat).(csubst0 n0 v0 c3 c4)) H3 O H5) in (let H8
\def (eq_ind nat i (\lambda (n0: nat).(subst0 n0 v0 u1 u2)) H2 O H5) in
-(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
+(eq_ind_r nat O (\lambda (n0: nat).(or4 (drop n0 n0 (CHead c4 (Flat f) u2)
+(CHead c3 (Flat f) u1)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 (Flat f) u1) (CHead e0
+(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(w: T).(drop n0 n0 (CHead c4 (Flat f) u2) (CHead e0 (Flat f0) w))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v0
+u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u))))))
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 n0
+(CHead c4 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F
+C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (_: T).(eq C (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u)))))))
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(w: T).(drop n0 n0 (CHead c4 (Flat f) u2) (CHead e2 (Flat f0) w)))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 n0 v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 v0 e1 e2))))))))) (or4_intro3
+(drop O O (CHead c4 (Flat f) u2) (CHead c3 (Flat f) u1)) (ex3_4 F C T T
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C
(CHead c3 (Flat f) u1) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda
(e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Flat f) u2)
(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
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:
+(drop_refl (CHead c4 (Flat f) u2)) H8 H7)) i H5))))))))))))))) k)) y v c1 c2
+H1))) H) e (drop_gen_refl c1 e H0)))))))) (\lambda (n0: nat).(\lambda (IHn:
((\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 n0 v c1 c2) \to
(\forall (e: C).((drop n0 O c1 e) \to (or4 (drop n0 O c2 e) (ex3_4 F C T T
(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
(_: 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 in
-nat 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 (n0: nat).(subst0 n0 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 (_:
+(\lambda (n0: nat).(or4 (drop n0 n0 c1 c2) (ex3_4 F C T T (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c2 (CHead e0 (Flat
+f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
+T).(drop n0 n0 c1 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 n0 v u1 u2)))))) (ex3_4 F C C T
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c2
+(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(drop n0 n0 c1 (CHead e1 (Flat f) u)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v e1
+e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C c2 (CHead e2 (Flat f) u2)))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(drop n0 n0 c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda
+(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 n0 v u1
+u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 n0 v e1 e2))))))))) (\lambda (y: nat).(\lambda
+(H1: (csubst0 y v c1 c2)).(csubst0_ind (\lambda (n0: nat).(\lambda (t:
+T).(\lambda (c: C).(\lambda (c0: C).((eq nat n0 O) \to (or4 (drop n0 n0 c c0)
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2:
+T).(eq C c0 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
+C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 c (CHead e0 (Flat f) u1))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 n0
+t u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 n0 c (CHead e1
+(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 n0 t e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e2
+(Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (_: T).(drop n0 n0 c (CHead e1 (Flat f) u1))))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 n0 t u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 t e1 e2))))))))))))) (\lambda
+(k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (v0: T).(\forall
+(u1: T).(\forall (u2: T).((subst0 i v0 u1 u2) \to (\forall (c: C).((eq nat (s
+k0 i) O) \to (or4 (drop (s k0 i) (s k0 i) (CHead c k0 u1) (CHead c k0 u2))
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4:
+T).(eq C (CHead c k0 u2) (CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda
+(e0: C).(\lambda (u3: T).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c k0
+u1) (CHead e0 (Flat f) u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3:
+T).(\lambda (u4: T).(subst0 (s k0 i) v0 u3 u4)))))) (ex3_4 F C C T (\lambda
+(f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c k0 u2)
+(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(drop (s k0 i) (s k0 i) (CHead c k0 u1) (CHead e1 (Flat f)
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 (s k0 i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c k0
+u2) (CHead e2 (Flat f) u4))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u3: T).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c k0
+u1) (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u3: T).(\lambda (u4: T).(subst0 (s k0 i) v0 u3 u4)))))) (\lambda
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_:
+T).(csubst0 (s k0 i) v0 e1 e2)))))))))))))))) (\lambda (b: B).(\lambda (i:
+nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0
+i v0 u1 u2)).(\lambda (c: C).(\lambda (H3: (eq nat (S i) O)).(let H4 \def
+(eq_ind nat (S i) (\lambda (ee: nat).(match ee in nat return (\lambda (_:
+nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H3) in
+(False_ind (or4 (drop (S i) (S i) (CHead c (Bind b) u1) (CHead c (Bind b)
+u2)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(u4: T).(eq C (CHead c (Bind b) u2) (CHead e0 (Flat f) u4)))))) (\lambda (f:
+F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop (S i) (S i) (CHead
+c (Bind b) u1) (CHead e0 (Flat f) u3)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u3: T).(\lambda (u4: T).(subst0 (S i) v0 u3 u4)))))) (ex3_4 F C
+C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C
+(CHead c (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S i) (S i) (CHead c (Bind b)
+u1) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 (S i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda
+(f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq
+C (CHead c (Bind b) u2) (CHead e2 (Flat f) u4))))))) (\lambda (f: F).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (_: T).(drop (S i) (S i)
+(CHead c (Bind b) u1) (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda
+(_: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 (S i) v0 u3
+u4)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (S i) v0 e1 e2)))))))) H4)))))))))) (\lambda (f:
+F).(\lambda (i: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2:
+T).(\lambda (H2: (subst0 i v0 u1 u2)).(\lambda (c: C).(\lambda (H3: (eq nat i
+O)).(let H4 \def (eq_ind nat i (\lambda (n0: nat).(subst0 n0 v0 u1 u2)) H2 O
+H3) in (eq_ind_r nat O (\lambda (n0: nat).(or4 (drop n0 n0 (CHead c (Flat f)
+u1) (CHead c (Flat f) u2)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c (Flat f) u2) (CHead e0
+(Flat f0) u4)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u3:
+T).(\lambda (_: T).(drop n0 n0 (CHead c (Flat f) u1) (CHead e0 (Flat f0)
+u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4:
+T).(subst0 n0 v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c (Flat f) u2) (CHead e2
+(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u: T).(drop n0 n0 (CHead c (Flat f) u1) (CHead e1 (Flat f0) u)))))) (\lambda
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v0 e1
+e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c (Flat f) u2) (CHead e2
+(Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u3: T).(\lambda (_: T).(drop n0 n0 (CHead c (Flat f) u1) (CHead
+e1 (Flat f0) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u3: T).(\lambda (u4: T).(subst0 n0 v0 u3 u4)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+n0 v0 e1 e2))))))))) (or4_intro1 (drop O O (CHead c (Flat f) u1) (CHead c
+(Flat f) u2)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_:
T).(\lambda (u4: T).(eq C (CHead c (Flat f) u2) (CHead e0 (Flat f0) u4))))))
(\lambda (f0: F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop O O
(CHead c (Flat f) u1) (CHead e0 (Flat f0) u3)))))) (\lambda (_: F).(\lambda
(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:
+(drop_refl (CHead c (Flat f) u1)) H4)) i H3)))))))))) k)) (\lambda (k:
+K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (c3: C).(\forall (c4:
+C).(\forall (v0: T).((csubst0 i v0 c3 c4) \to ((((eq nat i O) \to (or4 (drop
+i i c3 c4) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (u2: T).(eq C c4 (CHead e0 (Flat f) u2)))))) (\lambda (f:
+F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop i i c3 (CHead e0
+(Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 i v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f) u))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop i i c3
+(CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 i v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4
+(CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(drop i i c3 (CHead e1 (Flat f) u1)))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 i v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v0 e1 e2)))))))))) \to
+(\forall (u: T).((eq nat (s k0 i) O) \to (or4 (drop (s k0 i) (s k0 i) (CHead
+c3 k0 u) (CHead c4 k0 u)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 k0 u) (CHead e0 (Flat f)
+u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
+T).(drop (s k0 i) (s k0 i) (CHead c3 k0 u) (CHead e0 (Flat f) u1))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (s
+k0 i) v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (u0: T).(eq C (CHead c4 k0 u) (CHead e2 (Flat f) u0))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(drop (s k0
+i) (s k0 i) (CHead c3 k0 u) (CHead e1 (Flat f) u0)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (s k0 i) v0 e1
+e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 k0 u) (CHead e2 (Flat f)
u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (_: T).(drop 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 in nat 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 (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4
-(CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1:
-T).(\lambda (_: T).(drop O O c3 (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 (u:
-T).(eq C c4 (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(drop O O c3 (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 (u2: T).(eq C c4 (CHead e2 (Flat f0)
-u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (_: T).(drop O O c3 (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))))))))))).(\lambda (u:
-T).(\lambda (H4: (eq nat i O)).(let H5 \def (eq_ind nat i (\lambda (n0:
-nat).((eq nat n0 O) \to (or4 (drop O O c3 c4) (ex3_4 F C T T (\lambda (f0:
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e0 (Flat
-f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
-T).(drop O O c3 (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 c4
-(CHead e2 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u0: T).(drop O O c3 (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 c4 (CHead e2 (Flat f0) u2)))))))
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(_: T).(drop O O c3 (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)))))))))) H3 O H4) in (let H6 \def
-(eq_ind nat i (\lambda (n0: nat).(csubst0 n0 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).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c3 k0 u) (CHead e1 (Flat f)
+u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 (s k0 i) v0 u1 u2)))))) (\lambda (_: F).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (s k0 i) v0
+e1 e2))))))))))))))))) (\lambda (b: B).(\lambda (i: nat).(\lambda (c3:
+C).(\lambda (c4: C).(\lambda (v0: T).(\lambda (_: (csubst0 i v0 c3
+c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop i i c3 c4) (ex3_4 F C T T
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4
+(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
+T).(\lambda (_: T).(drop i i c3 (CHead e0 (Flat f) u1)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v0 u1 u2))))))
+(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 (CHead e1 (Flat f) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
+v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e2 (Flat f) u2)))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(drop i i c3 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v0 u1 u2))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat
+(S i) O)).(let H5 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee in nat
+return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow
+True])) I O H4) in (False_ind (or4 (drop (S i) (S i) (CHead c3 (Bind b) u)
+(CHead c4 (Bind b) u)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 (Bind b) u) (CHead e0
+(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda
+(_: T).(drop (S i) (S i) (CHead c3 (Bind b) u) (CHead e0 (Flat f) u1))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (S
+i) v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (u0: T).(eq C (CHead c4 (Bind b) u) (CHead e2 (Flat f)
+u0)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0:
+T).(drop (S i) (S i) (CHead c3 (Bind b) u) (CHead e1 (Flat f) u0))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (S
+i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 (Bind b) u) (CHead
+e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S i) (S i) (CHead c3 (Bind b) u)
+(CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (S i) v0 u1 u2)))))) (\lambda
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_:
+T).(csubst0 (S i) v0 e1 e2)))))))) H5))))))))))) (\lambda (f: F).(\lambda (i:
+nat).(\lambda (c3: C).(\lambda (c4: C).(\lambda (v0: T).(\lambda (H2:
+(csubst0 i v0 c3 c4)).(\lambda (H3: (((eq nat i O) \to (or4 (drop i i c3 c4)
+(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(u2: T).(eq C c4 (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (u1: T).(\lambda (_: T).(drop i i c3 (CHead e0 (Flat f0) u1))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i
+v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f0) u)))))) (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 (CHead e1
+(Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 i v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e2
+(Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(drop i i c3 (CHead e1 (Flat f0)
+u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 i v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v0 e1
+e2))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat i O)).(let H5 \def
+(eq_ind nat i (\lambda (n0: nat).((eq nat n0 O) \to (or4 (drop n0 n0 c3 c4)
+(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(u2: T).(eq C c4 (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 c3 (CHead e0 (Flat f0)
+u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 n0 v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u0: T).(eq C c4 (CHead e2 (Flat f0) u0))))))
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(drop n0
+n0 c3 (CHead e1 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda
+(f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq
+C c4 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 c3 (CHead e1 (Flat f0)
+u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 n0 v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 v0 e1
+e2)))))))))) H3 O H4) in (let H6 \def (eq_ind nat i (\lambda (n0:
+nat).(csubst0 n0 v0 c3 c4)) H2 O H4) in (eq_ind_r nat O (\lambda (n0:
+nat).(or4 (drop n0 n0 (CHead c3 (Flat f) u) (CHead c4 (Flat f) u)) (ex3_4 F C
T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C
(CHead c4 (Flat f) u) (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda
-(e0: C).(\lambda (u1: T).(\lambda (_: T).(drop O O (CHead c3 (Flat f) u)
+(e0: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 (CHead c3 (Flat f) u)
(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f0:
+T).(\lambda (u2: T).(subst0 n0 v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f0:
F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(eq C (CHead c4 (Flat f)
u) (CHead e2 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda
-(_: C).(\lambda (u0: T).(drop O O (CHead c3 (Flat f) u) (CHead e1 (Flat f0)
+(_: C).(\lambda (u0: T).(drop n0 n0 (CHead c3 (Flat f) u) (CHead e1 (Flat f0)
u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_:
+T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_:
C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 (Flat f)
u) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda
-(_: C).(\lambda (u1: T).(\lambda (_: T).(drop O O (CHead c3 (Flat f) u)
+(_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 (CHead c3 (Flat f) u)
(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 n0 v0 u1 u2)))))) (\lambda (_:
F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-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 (u4: T).(eq C c4 (CHead e0
-(Flat f) u4)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u3: T).(\lambda
-(_: T).(drop O O c3 (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 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:
+n0 v0 e1 e2))))))))) (or4_intro2 (drop O O (CHead c3 (Flat f) u) (CHead c4
+(Flat f) u)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (u2: T).(eq C (CHead c4 (Flat f) u) (CHead e0 (Flat f0) u2))))))
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop O O
+(CHead c3 (Flat f) u) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda
+(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (ex3_4 F C
+C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(eq C
+(CHead c4 (Flat f) u) (CHead e2 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u0: T).(drop O O (CHead c3 (Flat f) u)
+(CHead e1 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C
+(CHead c4 (Flat f) u) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O O (CHead c3
+(Flat f) u) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v0 e1 e2))))))) (ex3_4_intro F C C T (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(eq C (CHead c4 (Flat f)
+u) (CHead e2 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u0: T).(drop O O (CHead c3 (Flat f) u) (CHead e1 (Flat f0)
+u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 O v0 e1 e2))))) f c3 c4 u (refl_equal C (CHead c4 (Flat f) u))
+(drop_refl (CHead c3 (Flat f) u)) H6)) i H4)))))))))))) k)) (\lambda (k:
+K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (v0: T).(\forall (u1:
+T).(\forall (u2: T).((subst0 i v0 u1 u2) \to (\forall (c3: C).(\forall (c4:
+C).((csubst0 i v0 c3 c4) \to ((((eq nat i O) \to (or4 (drop i i c3 c4) (ex3_4
+F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq
+C c4 (CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda
+(u3: T).(\lambda (_: T).(drop i i c3 (CHead e0 (Flat f) u3)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3 u4))))))
+(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 (CHead e1 (Flat f) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
+v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 (CHead e2 (Flat f) u4)))))))
(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda
-(_: T).(drop O O c3 (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4))))))
+(_: T).(drop i i c3 (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3 u4))))))
(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 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
+(_: T).(csubst0 i v0 e1 e2)))))))))) \to ((eq nat (s k0 i) O) \to (or4 (drop
+(s k0 i) (s k0 i) (CHead c3 k0 u1) (CHead c4 k0 u2)) (ex3_4 F C T T (\lambda
+(f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 k0
+u2) (CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda
+(u3: T).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c3 k0 u1) (CHead e0
+(Flat f) u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda
+(u4: T).(subst0 (s k0 i) v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c4 k0 u2)
+(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(drop (s k0 i) (s k0 i) (CHead c3 k0 u1) (CHead e1 (Flat
+f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 (s k0 i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 k0
+u2) (CHead e2 (Flat f) u4))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u3: T).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c3 k0
u1) (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u3: T).(\lambda (u4: T).(subst0 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
+C).(\lambda (u3: T).(\lambda (u4: T).(subst0 (s k0 i) v0 u3 u4)))))) (\lambda
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_:
+T).(csubst0 (s k0 i) v0 e1 e2))))))))))))))))))) (\lambda (b: B).(\lambda (i:
+nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0
+i v0 u1 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (_: (csubst0 i v0 c3
+c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop i i c3 c4) (ex3_4 F C T T
(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C c4
(CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u3:
-T).(\lambda (_: T).(drop O O c3 (CHead e0 (Flat f) u3)))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4))))))
+T).(\lambda (_: T).(drop i i c3 (CHead e0 (Flat f) u3)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3 u4))))))
(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(drop O O c3 (CHead e1 (Flat f) u))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
+C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 (CHead e1 (Flat f) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 (CHead e2 (Flat f) u4)))))))
(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda
-(_: T).(drop O O c3 (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4))))))
+(_: T).(drop i i c3 (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3 u4))))))
(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 O v0 e1 e2))))))))))).(\lambda (H5: (eq nat (S i) O)).(let H6
+(_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (H5: (eq nat (S i) O)).(let H6
\def (eq_ind nat (S i) (\lambda (ee: nat).(match ee in nat return (\lambda
(_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H5)
-in (False_ind (or4 (drop 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:
+in (False_ind (or4 (drop (S i) (S i) (CHead c3 (Bind b) u1) (CHead c4 (Bind
+b) u2)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (u4: T).(eq C (CHead c4 (Bind b) u2) (CHead e0 (Flat f) u4))))))
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop (S i)
+(S i) (CHead c3 (Bind b) u1) (CHead e0 (Flat f) u3)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 (S i) v0 u3
+u4)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C (CHead c4 (Bind b) u2) (CHead e2 (Flat f) u))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S i)
+(S i) (CHead c3 (Bind b) u1) (CHead e1 (Flat f) u)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (S i) v0 e1
+e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Bind b) u2) (CHead e2
+(Flat f) u4))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u3: T).(\lambda (_: T).(drop (S i) (S i) (CHead c3 (Bind b) u1) (CHead e1
+(Flat f) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u3: T).(\lambda (u4: T).(subst0 (S i) v0 u3 u4)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(S i) v0 e1 e2)))))))) H6))))))))))))) (\lambda (f: F).(\lambda (i:
nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0
i v0 u1 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (H3: (csubst0 i v0 c3
-c4)).(\lambda (H4: (((eq nat i O) \to (or4 (drop O O c3 c4) (ex3_4 F C T T
+c4)).(\lambda (H4: (((eq nat i O) \to (or4 (drop i i c3 c4) (ex3_4 F C T T
(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C c4
(CHead e0 (Flat f0) u4)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u3:
-T).(\lambda (_: T).(drop O O c3 (CHead e0 (Flat f0) u3)))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4))))))
+T).(\lambda (_: T).(drop i i c3 (CHead e0 (Flat f0) u3)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3 u4))))))
(ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
T).(eq C c4 (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(drop O O c3 (CHead e1 (Flat f0) u))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
+C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 (CHead e1 (Flat f0) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda
(e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 (CHead e2 (Flat f0)
u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3:
-T).(\lambda (_: T).(drop O O c3 (CHead e1 (Flat f0) u3))))))) (\lambda (_:
+T).(\lambda (_: T).(drop i i c3 (CHead e1 (Flat f0) u3))))))) (\lambda (_:
F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0
-O v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (_: T).(csubst0 O v0 e1 e2))))))))))).(\lambda (H5: (eq nat i
+i v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (H5: (eq nat i
O)).(let H6 \def (eq_ind nat i (\lambda (n0: nat).((eq nat n0 O) \to (or4
-(drop O O c3 c4) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda
+(drop n0 n0 c3 c4) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda
(_: T).(\lambda (u4: T).(eq C c4 (CHead e0 (Flat f0) u4)))))) (\lambda (f0:
-F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop O O c3 (CHead e0
+F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop n0 n0 c3 (CHead e0
(Flat f0) u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda
-(u4: T).(subst0 O v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f0) u))))))
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop O O
+(u4: T).(subst0 n0 v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f0) u))))))
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 n0
c3 (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
+C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C c4
(CHead e2 (Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u3: T).(\lambda (_: T).(drop O O c3 (CHead e1 (Flat f0)
+C).(\lambda (u3: T).(\lambda (_: T).(drop n0 n0 c3 (CHead e1 (Flat f0)
u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3:
-T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1
+T).(\lambda (u4: T).(subst0 n0 v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 v0 e1
e2)))))))))) H4 O H5) in (let H7 \def (eq_ind nat i (\lambda (n0:
nat).(csubst0 n0 v0 c3 c4)) H3 O H5) in (let H8 \def (eq_ind nat i (\lambda
-(n0: nat).(subst0 n0 v0 u1 u2)) H2 O H5) in (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)
+(n0: nat).(subst0 n0 v0 u1 u2)) H2 O H5) in (eq_ind_r nat O (\lambda (n0:
+nat).(or4 (drop n0 n0 (CHead c3 (Flat f) u1) (CHead c4 (Flat f) u2)) (ex3_4 F
+C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq
+C (CHead c4 (Flat f) u2) (CHead e0 (Flat f0) u4)))))) (\lambda (f0:
+F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop n0 n0 (CHead c3
+(Flat f) u1) (CHead e0 (Flat f0) u3)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u3: T).(\lambda (u4: T).(subst0 n0 v0 u3 u4)))))) (ex3_4 F C C T
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C
+(CHead c4 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 n0 (CHead c3 (Flat f) u1)
+(CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C
+(CHead c4 (Flat f) u2) (CHead e2 (Flat f0) u4))))))) (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (_: T).(drop n0
+n0 (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u3))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0
+n0 v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 n0 v0 e1 e2))))))))) (or4_intro3 (drop O O
+(CHead c3 (Flat f) u1) (CHead c4 (Flat f) u2)) (ex3_4 F C T T (\lambda (f0:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Flat f)
+u2) (CHead e0 (Flat f0) u4)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda
+(u3: T).(\lambda (_: T).(drop O O (CHead c3 (Flat f) u1) (CHead e0 (Flat f0)
u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4:
T).(subst0 O v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_:
C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c4 (Flat f) u2) (CHead e2
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:
+(drop_refl (CHead c3 (Flat f) u1)) H8 H7)) i H5))))))))))))))) k)) y v c1 c2
+H1))) H) e (drop_gen_refl c2 e H0)))))))) (\lambda (n0: nat).(\lambda (IHn:
((\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 n0 v c1 c2) \to
(\forall (e: C).((drop n0 O c2 e) \to (or4 (drop n0 O c1 e) (ex3_4 F C T T
(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/csubst0/fwd".
include "csubst0/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/csubst0/getl".
include "csubst0/clear.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/csubst0/props".
include "csubst0/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/csubst1/defs".
include "csubst0/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/csubst1/fwd".
include "csubst1/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/csubst1/getl".
include "csubst1/props.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/csubst1/props".
include "csubst1/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/csubt/clear".
include "csubt/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/csubt/defs".
include "ty3/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/csubt/drop".
include "csubt/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/csubt/fwd".
include "csubt/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/csubt/getl".
include "csubt/fwd.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/csubt/pc3".
include "csubt/getl.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/csubt/props".
include "csubt/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/csubt/ty3".
include "csubt/pc3.ma".
c2) \to (ty3 g c2 u t))))).(\lambda (b: B).(\lambda (t0: T).(\lambda (t3:
T).(\lambda (_: (ty3 g (CHead c (Bind b) u) t0 t3)).(\lambda (H3: ((\forall
(c2: C).((csubt g (CHead c (Bind b) u) c2) \to (ty3 g c2 t0 t3))))).(\lambda
-(t4: T).(\lambda (_: (ty3 g (CHead c (Bind b) u) t3 t4)).(\lambda (H5:
-((\forall (c2: C).((csubt g (CHead c (Bind b) u) c2) \to (ty3 g c2 t3
-t4))))).(\lambda (c2: C).(\lambda (H6: (csubt g c c2)).(ty3_bind g c2 u t (H1
-c2 H6) b t0 t3 (H3 (CHead c2 (Bind b) u) (csubt_head g c c2 H6 (Bind b) u))
-t4 (H5 (CHead c2 (Bind b) u) (csubt_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).((csubt g c c2) \to (ty3 g
-c2 w u))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c v (THead
-(Bind Abst) u t))).(\lambda (H3: ((\forall (c2: C).((csubt g c c2) \to (ty3 g
-c2 v (THead (Bind Abst) u t)))))).(\lambda (c2: C).(\lambda (H4: (csubt g c
+(c2: C).(\lambda (H4: (csubt g c c2)).(ty3_bind g c2 u t (H1 c2 H4) b t0 t3
+(H3 (CHead c2 (Bind b) u) (csubt_head g c c2 H4 (Bind b) u)))))))))))))))
+(\lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c w
+u)).(\lambda (H1: ((\forall (c2: C).((csubt g c c2) \to (ty3 g c2 w
+u))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c v (THead (Bind
+Abst) u t))).(\lambda (H3: ((\forall (c2: C).((csubt g c c2) \to (ty3 g c2 v
+(THead (Bind Abst) u t)))))).(\lambda (c2: C).(\lambda (H4: (csubt g c
c2)).(ty3_appl g c2 w u (H1 c2 H4) v t (H3 c2 H4))))))))))))) (\lambda (c:
C).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (ty3 g c t0 t3)).(\lambda
(H1: ((\forall (c2: C).((csubt g c c2) \to (ty3 g c2 t0 t3))))).(\lambda (t4:
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / 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/LambdaDelta-1/definitions".
+
+include "tlt/defs.ma".
+
+include "iso/defs.ma".
+
+include "clen/defs.ma".
+
+include "flt/defs.ma".
+
+include "cnt/defs.ma".
+
+include "cimp/defs.ma".
+
+include "subst1/defs.ma".
+
+include "csubst1/defs.ma".
+
+include "fsubst0/defs.ma".
+
+include "next_plus/defs.ma".
+
+include "tau1/defs.ma".
+
+include "llt/defs.ma".
+
+include "aprem/defs.ma".
+
+include "ex0/defs.ma".
+
+include "wcpr0/defs.ma".
+
+include "csuba/defs.ma".
+
+include "nf2/defs.ma".
+
+include "ex2/defs.ma".
+
+include "csubc/defs.ma".
+
+include "pc1/defs.ma".
+
+include "ex1/defs.ma".
+
+include "csubt/defs.ma".
+
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-
-
-include "tlt/defs.ma".
-
-include "iso/defs.ma".
-
-include "clen/defs.ma".
-
-include "flt/defs.ma".
-
-include "cnt/defs.ma".
-
-include "cimp/defs.ma".
-
-include "subst1/defs.ma".
-
-include "csubst1/defs.ma".
-
-include "fsubst0/defs.ma".
-
-include "next_plus/defs.ma".
-
-include "tau1/defs.ma".
-
-include "llt/defs.ma".
-
-include "aprem/defs.ma".
-
-include "ex0/defs.ma".
-
-include "wcpr0/defs.ma".
-
-include "csuba/defs.ma".
-
-include "nf2/defs.ma".
-
-include "ex2/defs.ma".
-
-include "csubc/defs.ma".
-
-include "pc1/defs.ma".
-
-include "ex1/defs.ma".
-
-include "csubt/defs.ma".
-
-definitions3.ma aprem/defs.ma cimp/defs.ma clen/defs.ma cnt/defs.ma csuba/defs.ma csubc/defs.ma csubst1/defs.ma csubt/defs.ma ex0/defs.ma ex1/defs.ma ex2/defs.ma flt/defs.ma fsubst0/defs.ma iso/defs.ma llt/defs.ma next_plus/defs.ma nf2/defs.ma pc1/defs.ma subst1/defs.ma tau1/defs.ma tlt/defs.ma wcpr0/defs.ma
-preamble3.ma theory.ma
+theory.ma ex0/props.ma ex1/props.ma ex2/props.ma pr3/wcpr0.ma subst0/tlt.ma tau1/cnt.ma ty3/dec.ma ty3/nf2.ma ty3/tau0.ma wcpr0/fwd.ma
spare.ma theory.ma
-theory3.ma ex0/props.ma ex1/props.ma ex2/props.ma pr3/wcpr0.ma subst0/tlt.ma tau1/cnt.ma ty3/dec.ma ty3/nf2.ma ty3/tau0.ma wcpr0/fwd.ma
-tlist/props.ma tlist/defs.ma
+preamble.ma ../Base-1/theory.ma
+definitions.ma aprem/defs.ma cimp/defs.ma clen/defs.ma cnt/defs.ma csuba/defs.ma csubc/defs.ma csubst1/defs.ma csubt/defs.ma ex0/defs.ma ex1/defs.ma ex2/defs.ma flt/defs.ma fsubst0/defs.ma iso/defs.ma llt/defs.ma next_plus/defs.ma nf2/defs.ma pc1/defs.ma subst1/defs.ma tau1/defs.ma tlt/defs.ma wcpr0/defs.ma
+ex0/defs.ma A/defs.ma G/defs.ma
+ex0/props.ma aplus/props.ma ex0/defs.ma leq/defs.ma
+ex2/defs.ma C/defs.ma
+ex2/props.ma arity/fwd.ma ex2/defs.ma nf2/defs.ma pr2/fwd.ma
tlist/defs.ma T/defs.ma
-clen/defs.ma C/defs.ma s/defs.ma
-clen/getl.ma clen/defs.ma getl/props.ma
-leq/props.ma aplus/props.ma leq/defs.ma
-leq/asucc.ma aplus/props.ma leq/props.ma
-leq/defs.ma aplus/defs.ma
-leq/fwd.ma leq/defs.ma
-pr3/props.ma pr1/props.ma pr2/props.ma pr3/pr1.ma
-pr3/pr3.ma pr2/pr2.ma pr3/props.ma
-pr3/defs.ma pr2/defs.ma
-pr3/fwd.ma pr2/fwd.ma pr3/props.ma
-pr3/subst1.ma pr2/subst1.ma pr3/defs.ma
-pr3/iso.ma iso/props.ma pr3/fwd.ma tlist/props.ma
-pr3/wcpr0.ma pr3/props.ma wcpr0/getl.ma
-pr3/pr1.ma pr1/defs.ma pr3/defs.ma
+tlist/props.ma tlist/defs.ma
+csubt/defs.ma ty3/defs.ma
+csubt/fwd.ma csubt/defs.ma
+csubt/props.ma csubt/defs.ma
+csubt/clear.ma clear/fwd.ma csubt/defs.ma
+csubt/drop.ma csubt/defs.ma drop/fwd.ma
+csubt/getl.ma csubt/clear.ma csubt/drop.ma csubt/fwd.ma getl/clear.ma
+csubt/pc3.ma csubt/getl.ma pc3/left.ma
+csubt/ty3.ma csubt/pc3.ma csubt/props.ma
+csubc/defs.ma sc3/defs.ma
+csubc/props.ma csubc/defs.ma sc3/props.ma
+csubc/csuba.ma csuba/defs.ma csubc/defs.ma sc3/props.ma
+csubc/drop.ma csubc/defs.ma sc3/props.ma
+csubc/drop1.ma csubc/drop.ma
+csubc/clear.ma csubc/defs.ma
+csubc/getl.ma csubc/clear.ma csubc/drop.ma
+csubc/arity.ma arity/defs.ma csubc/csuba.ma
ty3/fsubst0.ma csubst0/props.ma getl/getl.ma pc3/fsubst0.ma ty3/props.ma
-ty3/nf2.ma nf2/arity.ma pc3/nf2.ma ty3/arity.ma
+ty3/defs.ma G/defs.ma pc3/defs.ma
+ty3/fwd.ma pc3/props.ma ty3/defs.ma
ty3/props.ma pc3/fwd.ma ty3/fwd.ma
-ty3/arity_props.ma sc3/arity.ma ty3/arity.ma ty3/fwd.ma
-ty3/arity.ma arity/pr3.ma asucc/fwd.ma ty3/pr3_props.ma
+ty3/subst1.ma csubst1/fwd.ma csubst1/getl.ma getl/getl.ma pc3/fwd.ma pc3/subst1.ma ty3/props.ma
ty3/pr3_props.ma ty3/pr3.ma
-ty3/tau0.ma tau0/defs.ma ty3/pr3_props.ma
ty3/pr3.ma csubt/ty3.ma pc1/props.ma pc3/pc1.ma pc3/wcpr0.ma ty3/fsubst0.ma ty3/subst1.ma
-ty3/defs.ma G/defs.ma pc3/defs.ma
-ty3/fwd.ma pc3/props.ma ty3/defs.ma
+ty3/tau0.ma tau0/defs.ma ty3/pr3_props.ma
+ty3/arity.ma arity/pr3.ma asucc/fwd.ma ty3/pr3_props.ma
ty3/dec.ma getl/dec.ma getl/flt.ma pc3/dec.ma ty3/pr3_props.ma
-ty3/subst1.ma csubst1/fwd.ma csubst1/getl.ma getl/getl.ma pc3/fwd.ma pc3/subst1.ma ty3/props.ma
-cnt/props.ma cnt/defs.ma lift/fwd.ma
-cnt/defs.ma T/defs.ma
-fsubst0/defs.ma csubst0/defs.ma subst0/defs.ma
-fsubst0/fwd.ma fsubst0/defs.ma
-iso/props.ma iso/fwd.ma
-iso/defs.ma T/defs.ma
-iso/fwd.ma iso/defs.ma tlist/defs.ma
-lift/props.ma lift/fwd.ma s/props.ma tlist/defs.ma
-lift/defs.ma T/defs.ma s/defs.ma tlist/defs.ma
-lift/tlt.ma lift/fwd.ma tlt/props.ma
-lift/fwd.ma lift/defs.ma
-flt/props.ma C/props.ma flt/defs.ma
-flt/defs.ma C/defs.ma
-A/defs.ma preamble3.ma
-subst0/props.ma lift/props.ma subst0/fwd.ma
-subst0/defs.ma lift/defs.ma
-subst0/tlt.ma lift/props.ma lift/tlt.ma subst0/defs.ma
-subst0/fwd.ma lift/props.ma subst0/defs.ma
-subst0/subst0.ma subst0/props.ma
-subst0/dec.ma lift/props.ma subst0/defs.ma
-pr1/props.ma T/props.ma pr0/subst1.ma pr1/defs.ma subst1/props.ma
-pr1/defs.ma pr0/defs.ma
-pr1/pr1.ma pr0/pr0.ma pr1/props.ma
-T/props.ma T/defs.ma
-T/defs.ma preamble3.ma
-T/dec.ma T/defs.ma
+ty3/arity_props.ma sc3/arity.ma ty3/arity.ma ty3/fwd.ma
+ty3/nf2.ma nf2/arity.ma pc3/nf2.ma ty3/arity.ma
+pc3/defs.ma pr3/defs.ma
+pc3/props.ma pc3/defs.ma pr3/pr3.ma
+pc3/fsubst0.ma csubst0/getl.ma fsubst0/defs.ma pc3/left.ma
+pc3/pc1.ma pc1/defs.ma pc3/defs.ma pr3/pr1.ma
+pc3/wcpr0.ma pc3/props.ma wcpr0/getl.ma
+pc3/left.ma pc3/props.ma
+pc3/fwd.ma pc3/props.ma pr3/fwd.ma
+pc3/nf2.ma nf2/pr3.ma pc3/defs.ma
+pc3/subst1.ma pc3/props.ma pr3/subst1.ma
+pc3/dec.ma nf2/fwd.ma ty3/arity_props.ma ty3/pr3.ma
+pc1/defs.ma pr1/defs.ma
+pc1/props.ma pc1/defs.ma pr1/pr1.ma
+ex1/defs.ma C/defs.ma
+ex1/props.ma arity/defs.ma ex1/defs.ma leq/props.ma nf2/pr3.ma nf2/props.ma pc3/fwd.ma ty3/fwd.ma
+sc3/defs.ma arity/defs.ma drop1/defs.ma sn3/defs.ma
sc3/props.ma arity/aprem.ma arity/lift1.ma csuba/arity.ma drop1/getl.ma drop1/props.ma lift1/props.ma llt/props.ma nf2/lift1.ma sc3/defs.ma sn3/lift1.ma
sc3/arity.ma csubc/arity.ma csubc/drop1.ma csubc/getl.ma csubc/props.ma
-sc3/defs.ma arity/defs.ma drop1/defs.ma sn3/defs.ma
-tau1/props.ma tau0/props.ma tau1/defs.ma
-tau1/defs.ma tau0/defs.ma
-tau1/cnt.ma cnt/props.ma tau1/props.ma
-aplus/props.ma aplus/defs.ma next_plus/props.ma
-aplus/defs.ma asucc/defs.ma
-asucc/defs.ma A/defs.ma G/defs.ma
-asucc/fwd.ma asucc/defs.ma
-aprem/props.ma aprem/defs.ma leq/defs.ma
-aprem/defs.ma A/defs.ma
+sn3/defs.ma pr3/defs.ma
+sn3/nf2.ma nf2/dec.ma nf2/pr3.ma sn3/defs.ma
+sn3/props.ma iso/props.ma nf2/iso.ma pr3/iso.ma sn3/fwd.ma sn3/nf2.ma
+sn3/lift1.ma drop1/defs.ma lift1/fwd.ma sn3/props.ma
+sn3/fwd.ma pr3/props.ma sn3/defs.ma
+nf2/defs.ma pr2/defs.ma
+nf2/fwd.ma T/props.ma nf2/defs.ma pr2/clen.ma subst0/dec.ma
nf2/props.ma nf2/defs.ma pr2/fwd.ma
nf2/arity.ma arity/subst0.ma nf2/fwd.ma
nf2/pr3.ma nf2/defs.ma pr3/pr3.ma
-nf2/defs.ma pr2/defs.ma
-nf2/fwd.ma T/props.ma nf2/defs.ma pr2/clen.ma subst0/dec.ma
-nf2/dec.ma C/props.ma nf2/defs.ma pr0/dec.ma pr2/clen.ma pr2/fwd.ma
nf2/lift1.ma drop1/defs.ma nf2/props.ma
nf2/iso.ma iso/props.ma nf2/pr3.ma pr3/fwd.ma
-drop/props.ma drop/fwd.ma lift/props.ma r/props.ma
-drop/defs.ma C/defs.ma lift/defs.ma r/defs.ma
-drop/fwd.ma drop/defs.ma
-csuba/drop.ma csuba/fwd.ma drop/fwd.ma
-csuba/clear.ma clear/fwd.ma csuba/defs.ma
-csuba/props.ma csuba/defs.ma
-csuba/arity.ma T/props.ma arity/props.ma csuba/getl.ma csuba/props.ma
+nf2/dec.ma C/props.ma nf2/defs.ma pr0/dec.ma pr2/clen.ma pr2/fwd.ma
csuba/defs.ma arity/defs.ma
csuba/fwd.ma csuba/defs.ma
+csuba/props.ma csuba/defs.ma
+csuba/arity.ma T/props.ma arity/props.ma csuba/getl.ma csuba/props.ma
+csuba/clear.ma clear/fwd.ma csuba/defs.ma
+csuba/drop.ma csuba/fwd.ma drop/fwd.ma
csuba/getl.ma csuba/clear.ma csuba/drop.ma getl/clear.ma
-C/props.ma C/defs.ma T/props.ma
-C/defs.ma T/defs.ma
-csubt/drop.ma csubt/defs.ma drop/fwd.ma
-csubt/clear.ma clear/fwd.ma csubt/defs.ma
-csubt/props.ma csubt/defs.ma
-csubt/pc3.ma csubt/getl.ma pc3/left.ma
-csubt/defs.ma ty3/defs.ma
-csubt/fwd.ma csubt/defs.ma
-csubt/ty3.ma csubt/pc3.ma csubt/props.ma
-csubt/getl.ma csubt/clear.ma csubt/drop.ma csubt/fwd.ma getl/clear.ma
-cimp/props.ma cimp/defs.ma getl/getl.ma
-cimp/defs.ma getl/defs.ma
-drop1/props.ma drop/props.ma drop1/defs.ma getl/defs.ma
-drop1/defs.ma drop/defs.ma lift1/defs.ma
-drop1/getl.ma drop1/defs.ma getl/drop.ma
-lift1/props.ma drop1/defs.ma lift/props.ma lift1/defs.ma
-lift1/defs.ma lift/defs.ma
-lift1/fwd.ma lift/fwd.ma lift1/defs.ma
-pr2/pr2.ma getl/props.ma pr0/pr0.ma pr2/defs.ma
-pr2/props.ma getl/clear.ma getl/drop.ma pr0/props.ma pr2/defs.ma
+wcpr0/getl.ma getl/props.ma wcpr0/defs.ma
+wcpr0/fwd.ma wcpr0/defs.ma
+wcpr0/defs.ma C/defs.ma pr0/defs.ma
+pr3/defs.ma pr2/defs.ma
+pr3/subst1.ma pr2/subst1.ma pr3/defs.ma
+pr3/props.ma pr1/props.ma pr2/props.ma pr3/pr1.ma
+pr3/pr1.ma pr1/defs.ma pr3/defs.ma
+pr3/wcpr0.ma pr3/props.ma wcpr0/getl.ma
+pr3/fwd.ma pr2/fwd.ma pr3/props.ma
+pr3/pr3.ma pr2/pr2.ma pr3/props.ma
+pr3/iso.ma iso/props.ma pr3/fwd.ma tlist/props.ma
pr2/defs.ma getl/defs.ma pr0/defs.ma
+pr2/props.ma getl/clear.ma getl/drop.ma pr0/props.ma pr2/defs.ma
pr2/fwd.ma getl/clear.ma getl/drop.ma pr0/fwd.ma pr2/defs.ma
pr2/subst1.ma csubst1/fwd.ma csubst1/getl.ma getl/drop.ma pr0/fwd.ma pr0/subst1.ma pr2/defs.ma subst1/subst1.ma
pr2/clen.ma clen/getl.ma pr2/props.ma
-pc3/fsubst0.ma csubst0/getl.ma fsubst0/defs.ma pc3/left.ma
-pc3/nf2.ma nf2/pr3.ma pc3/defs.ma
-pc3/pc1.ma pc1/defs.ma pc3/defs.ma pr3/pr1.ma
-pc3/props.ma pc3/defs.ma pr3/pr3.ma
-pc3/left.ma pc3/props.ma
-pc3/defs.ma pr3/defs.ma
-pc3/fwd.ma pc3/props.ma pr3/fwd.ma
-pc3/dec.ma nf2/fwd.ma ty3/arity_props.ma ty3/pr3.ma
-pc3/subst1.ma pc3/props.ma pr3/subst1.ma
-pc3/wcpr0.ma pc3/props.ma wcpr0/getl.ma
-pr0/props.ma pr0/defs.ma subst0/subst0.ma
-pr0/pr0.ma lift/tlt.ma pr0/fwd.ma
+pr2/pr2.ma getl/props.ma pr0/pr0.ma pr2/defs.ma
+pr1/defs.ma pr0/defs.ma
+pr1/props.ma T/props.ma pr0/subst1.ma pr1/defs.ma subst1/props.ma
+pr1/pr1.ma pr0/pr0.ma pr1/props.ma
+pr0/subst1.ma pr0/props.ma subst1/defs.ma
pr0/defs.ma subst0/defs.ma
pr0/fwd.ma pr0/props.ma
+pr0/props.ma pr0/defs.ma subst0/subst0.ma
+pr0/pr0.ma lift/tlt.ma pr0/fwd.ma
pr0/dec.ma T/dec.ma T/props.ma pr0/fwd.ma subst0/dec.ma
-pr0/subst1.ma pr0/props.ma subst1/defs.ma
-subst1/props.ma subst0/props.ma subst1/defs.ma
-subst1/defs.ma subst0/defs.ma
-subst1/fwd.ma subst0/props.ma subst1/defs.ma
-subst1/subst1.ma subst0/subst0.ma subst1/fwd.ma
-tlt/props.ma tlt/defs.ma
-tlt/defs.ma T/defs.ma
-r/props.ma r/defs.ma s/defs.ma
-r/defs.ma T/defs.ma
-wcpr0/defs.ma C/defs.ma pr0/defs.ma
-wcpr0/fwd.ma wcpr0/defs.ma
-wcpr0/getl.ma getl/props.ma wcpr0/defs.ma
-G/defs.ma preamble3.ma
-csubst0/drop.ma csubst0/fwd.ma drop/fwd.ma s/props.ma
-csubst0/clear.ma clear/fwd.ma csubst0/fwd.ma
-csubst0/props.ma csubst0/defs.ma
-csubst0/defs.ma C/defs.ma subst0/defs.ma
-csubst0/fwd.ma csubst0/defs.ma
-csubst0/getl.ma csubst0/clear.ma csubst0/drop.ma getl/fwd.ma
-next_plus/props.ma next_plus/defs.ma
-next_plus/defs.ma G/defs.ma
-tau0/props.ma getl/drop.ma tau0/defs.ma
-tau0/defs.ma G/defs.ma getl/defs.ma
-csubc/drop.ma csubc/defs.ma sc3/props.ma
-csubc/clear.ma csubc/defs.ma
-csubc/props.ma csubc/defs.ma sc3/props.ma
-csubc/arity.ma arity/defs.ma csubc/csuba.ma
-csubc/drop1.ma csubc/drop.ma
-csubc/defs.ma sc3/defs.ma
-csubc/csuba.ma csuba/defs.ma csubc/defs.ma sc3/props.ma
-csubc/getl.ma csubc/clear.ma csubc/drop.ma
-arity/props.ma arity/fwd.ma
-arity/aprem.ma aprem/props.ma arity/cimp.ma arity/props.ma
-arity/pr3.ma arity/subst0.ma csuba/arity.ma pr0/fwd.ma pr1/defs.ma pr3/defs.ma wcpr0/getl.ma
+tau1/defs.ma tau0/defs.ma
+tau1/props.ma tau0/props.ma tau1/defs.ma
+tau1/cnt.ma cnt/props.ma tau1/props.ma
arity/defs.ma getl/defs.ma leq/defs.ma
arity/fwd.ma arity/defs.ma getl/drop.ma leq/asucc.ma leq/fwd.ma
arity/subst0.ma arity/props.ma csubst0/getl.ma csubst0/props.ma fsubst0/fwd.ma getl/getl.ma subst0/dec.ma subst0/fwd.ma
+arity/props.ma arity/fwd.ma
+arity/pr3.ma arity/subst0.ma csuba/arity.ma pr0/fwd.ma pr1/defs.ma pr3/defs.ma wcpr0/getl.ma
arity/lift1.ma arity/props.ma drop1/defs.ma
arity/cimp.ma arity/defs.ma cimp/props.ma
-ex2/props.ma arity/fwd.ma ex2/defs.ma nf2/defs.ma pr2/fwd.ma
-ex2/defs.ma C/defs.ma
-getl/drop.ma clear/drop.ma getl/props.ma r/props.ma
-getl/clear.ma clear/drop.ma getl/props.ma
-getl/props.ma clear/props.ma drop/props.ma getl/fwd.ma
+arity/aprem.ma aprem/props.ma arity/cimp.ma arity/props.ma
+aprem/defs.ma A/defs.ma
+aprem/props.ma aprem/defs.ma leq/defs.ma
+llt/defs.ma A/defs.ma
+llt/props.ma leq/defs.ma llt/defs.ma
+leq/defs.ma aplus/defs.ma
+leq/asucc.ma aplus/props.ma leq/props.ma
+leq/props.ma aplus/props.ma leq/defs.ma
+leq/fwd.ma leq/defs.ma
+aplus/defs.ma asucc/defs.ma
+aplus/props.ma aplus/defs.ma next_plus/props.ma
+asucc/defs.ma A/defs.ma G/defs.ma
+asucc/fwd.ma asucc/defs.ma
+A/defs.ma preamble.ma
+fsubst0/defs.ma csubst0/defs.ma subst0/defs.ma
+fsubst0/fwd.ma fsubst0/defs.ma
+csubst1/defs.ma csubst0/defs.ma
+csubst1/props.ma csubst1/defs.ma subst1/defs.ma
+csubst1/fwd.ma csubst0/fwd.ma csubst1/defs.ma subst1/props.ma
+csubst1/getl.ma csubst0/getl.ma csubst0/props.ma csubst1/props.ma drop/props.ma subst1/props.ma
+csubst0/defs.ma C/defs.ma subst0/defs.ma
+csubst0/fwd.ma csubst0/defs.ma
+csubst0/props.ma csubst0/defs.ma
+csubst0/drop.ma csubst0/fwd.ma drop/fwd.ma s/props.ma
+csubst0/clear.ma clear/fwd.ma csubst0/fwd.ma
+csubst0/getl.ma csubst0/clear.ma csubst0/drop.ma getl/fwd.ma
+subst0/defs.ma lift/defs.ma
+subst0/fwd.ma lift/props.ma subst0/defs.ma
+subst0/subst0.ma subst0/props.ma
+subst0/props.ma lift/props.ma subst0/fwd.ma
+subst0/tlt.ma lift/props.ma lift/tlt.ma subst0/defs.ma
+subst0/dec.ma lift/props.ma subst0/defs.ma
+tau0/defs.ma G/defs.ma getl/defs.ma
+tau0/props.ma getl/drop.ma tau0/defs.ma
+subst1/subst1.ma subst0/subst0.ma subst1/fwd.ma
+subst1/defs.ma subst0/defs.ma
+subst1/fwd.ma subst0/props.ma subst1/defs.ma
+subst1/props.ma subst0/props.ma subst1/defs.ma
+cimp/defs.ma getl/defs.ma
+cimp/props.ma cimp/defs.ma getl/getl.ma
getl/defs.ma clear/defs.ma drop/defs.ma
+getl/props.ma clear/props.ma drop/props.ma getl/fwd.ma
getl/fwd.ma clear/fwd.ma drop/fwd.ma getl/defs.ma
+getl/drop.ma clear/drop.ma getl/props.ma r/props.ma
+getl/clear.ma clear/drop.ma getl/props.ma
+getl/getl.ma getl/clear.ma getl/drop.ma
getl/dec.ma getl/props.ma
getl/flt.ma clear/props.ma flt/props.ma getl/fwd.ma
-getl/getl.ma getl/clear.ma getl/drop.ma
-clear/drop.ma clear/fwd.ma drop/fwd.ma
-clear/props.ma clear/fwd.ma
clear/defs.ma C/defs.ma
clear/fwd.ma clear/defs.ma
+clear/props.ma clear/fwd.ma
+clear/drop.ma clear/fwd.ma drop/fwd.ma
+drop1/defs.ma drop/defs.ma lift1/defs.ma
+drop1/props.ma drop/props.ma drop1/defs.ma getl/defs.ma
+drop1/getl.ma drop1/defs.ma getl/drop.ma
+drop/defs.ma C/defs.ma lift/defs.ma r/defs.ma
+drop/fwd.ma drop/defs.ma
+drop/props.ma drop/fwd.ma lift/props.ma r/props.ma
+lift1/defs.ma lift/defs.ma
+lift1/fwd.ma lift/fwd.ma lift1/defs.ma
+lift1/props.ma drop1/defs.ma lift/props.ma lift1/defs.ma
+cnt/defs.ma T/defs.ma
+cnt/props.ma cnt/defs.ma lift/fwd.ma
+lift/defs.ma T/defs.ma s/defs.ma tlist/defs.ma
+lift/fwd.ma lift/defs.ma
+lift/props.ma lift/fwd.ma s/props.ma tlist/defs.ma
+lift/tlt.ma lift/fwd.ma tlt/props.ma
+iso/props.ma iso/fwd.ma
+iso/fwd.ma iso/defs.ma tlist/defs.ma
+iso/defs.ma T/defs.ma
+G/defs.ma preamble.ma
+flt/props.ma C/props.ma flt/defs.ma
+flt/defs.ma C/defs.ma
+tlt/props.ma tlt/defs.ma
+tlt/defs.ma T/defs.ma
+T/props.ma T/defs.ma
+T/dec.ma T/defs.ma
+T/defs.ma preamble.ma
s/props.ma s/defs.ma
s/defs.ma T/defs.ma
-pc1/props.ma pc1/defs.ma pr1/pr1.ma
-pc1/defs.ma pr1/defs.ma
-sn3/nf2.ma nf2/dec.ma nf2/pr3.ma sn3/defs.ma
-sn3/props.ma iso/props.ma nf2/iso.ma pr3/iso.ma sn3/fwd.ma sn3/nf2.ma
-sn3/defs.ma pr3/defs.ma
-sn3/fwd.ma pr3/props.ma sn3/defs.ma
-sn3/lift1.ma drop1/defs.ma lift1/fwd.ma sn3/props.ma
-llt/props.ma leq/defs.ma llt/defs.ma
-llt/defs.ma A/defs.ma
-ex1/props.ma arity/defs.ma ex1/defs.ma leq/props.ma nf2/pr3.ma nf2/props.ma pc3/fwd.ma ty3/fwd.ma
-ex1/defs.ma C/defs.ma
-ex0/props.ma aplus/props.ma ex0/defs.ma leq/defs.ma
-ex0/defs.ma A/defs.ma G/defs.ma
-csubst1/props.ma csubst1/defs.ma subst1/defs.ma
-csubst1/defs.ma csubst0/defs.ma
-csubst1/fwd.ma csubst0/fwd.ma csubst1/defs.ma subst1/props.ma
-csubst1/getl.ma csubst0/getl.ma csubst0/props.ma csubst1/props.ma drop/props.ma subst1/props.ma
-theory.ma
+C/props.ma C/defs.ma T/props.ma
+C/defs.ma T/defs.ma
+r/props.ma r/defs.ma s/defs.ma
+r/defs.ma T/defs.ma
+next_plus/props.ma next_plus/defs.ma
+next_plus/defs.ma G/defs.ma
+clen/getl.ma clen/defs.ma getl/props.ma
+clen/defs.ma C/defs.ma s/defs.ma
+../Base-1/theory.ma
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/drop/defs".
include "C/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/drop/fwd".
include "drop/defs.ma".
\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:
+C).(drop h d c x)) (\lambda (c: C).(and3 (eq C x c) (eq nat h O) (eq nat d
+O))) (\lambda (y: C).(\lambda (H0: (drop h d y x)).(drop_ind (\lambda (n0:
nat).(\lambda (n1: nat).(\lambda (c: C).(\lambda (c0: C).((eq C c (CSort n))
-\to (and3 (eq C c0 (CSort n)) (eq nat n0 O) (eq nat n1 O))))))) (\lambda (c:
+\to (and3 (eq C c0 c) (eq nat n0 O) (eq nat n1 O))))))) (\lambda (c:
C).(\lambda (H1: (eq C c (CSort n))).(let H2 \def (f_equal C C (\lambda (e:
C).e) c (CSort n) H1) in (eq_ind_r C (CSort n) (\lambda (c0: C).(and3 (eq C
-c0 (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 in C 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 in C return (\lambda (_: C).Prop)
+c0 c0) (eq nat O O) (eq nat O O))) (and3_intro (eq C (CSort n) (CSort n)) (eq
+nat O O) (eq nat O O) (refl_equal C (CSort n)) (refl_equal nat O) (refl_equal
+nat O)) c H2)))) (\lambda (k: K).(\lambda (h0: nat).(\lambda (c: C).(\lambda
+(e: C).(\lambda (_: (drop (r k h0) O c e)).(\lambda (_: (((eq C c (CSort n))
+\to (and3 (eq C e c) (eq nat (r k h0) O) (eq nat O O))))).(\lambda (u:
+T).(\lambda (H3: (eq C (CHead c k u) (CSort n))).(let H4 \def (eq_ind C
+(CHead c k u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop)
with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I
-(CSort n) H3) in (False_ind (and3 (eq C (CHead e k u) (CSort n)) (eq nat h0
-O) (eq nat (S d0) O)) H4))))))))))) h d y x H0))) H))))).
+(CSort n) H3) in (False_ind (and3 (eq C e (CHead c k u)) (eq nat (S h0) O)
+(eq nat O O)) H4)))))))))) (\lambda (k: K).(\lambda (h0: nat).(\lambda (d0:
+nat).(\lambda (c: C).(\lambda (e: C).(\lambda (_: (drop h0 (r k d0) c
+e)).(\lambda (_: (((eq C c (CSort n)) \to (and3 (eq C e c) (eq nat h0 O) (eq
+nat (r k d0) O))))).(\lambda (u: T).(\lambda (H3: (eq C (CHead c k (lift h0
+(r k d0) u)) (CSort n))).(let H4 \def (eq_ind C (CHead c k (lift h0 (r k d0)
+u)) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort
+_) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H3) in
+(False_ind (and3 (eq C (CHead e k u) (CHead c k (lift h0 (r k d0) u))) (eq
+nat h0 O) (eq nat (S d0) O)) H4))))))))))) h d y x H0))) H))))).
theorem drop_gen_refl:
\forall (x: C).(\forall (e: C).((drop O O x e) \to (eq C x e)))
\def
\lambda (x: C).(\lambda (e: C).(\lambda (H: (drop O O x e)).(insert_eq nat O
-(\lambda (n: nat).(drop n O x e)) (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 in nat return (\lambda (_: nat).Prop) with [O \Rightarrow
-False | (S _) \Rightarrow True])) I O H5) in (False_ind (eq C (CHead c k u)
-e0) H6))))))))))) (\lambda (k: K).(\lambda (h: nat).(\lambda (d:
-nat).(\lambda (c: C).(\lambda (e0: C).(\lambda (H2: (drop h (r k d) c
-e0)).(\lambda (H3: (((eq nat (r k d) O) \to ((eq nat h O) \to (eq C c
+(\lambda (n: nat).(drop n O x e)) (\lambda (_: nat).(eq C x e)) (\lambda (y:
+nat).(\lambda (H0: (drop y O x e)).(insert_eq nat O (\lambda (n: nat).(drop y
+n x e)) (\lambda (n: nat).((eq nat y n) \to (eq C x e))) (\lambda (y0:
+nat).(\lambda (H1: (drop y y0 x e)).(drop_ind (\lambda (n: nat).(\lambda (n0:
+nat).(\lambda (c: C).(\lambda (c0: C).((eq nat n0 O) \to ((eq nat n n0) \to
+(eq C c c0))))))) (\lambda (c: C).(\lambda (_: (eq nat O O)).(\lambda (_: (eq
+nat O O)).(refl_equal C c)))) (\lambda (k: K).(\lambda (h: nat).(\lambda (c:
+C).(\lambda (e0: C).(\lambda (_: (drop (r k h) O c e0)).(\lambda (_: (((eq
+nat O O) \to ((eq nat (r k h) O) \to (eq C c e0))))).(\lambda (u: T).(\lambda
+(_: (eq nat O O)).(\lambda (H5: (eq nat (S h) O)).(let H6 \def (eq_ind nat (S
+h) (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O
+\Rightarrow False | (S _) \Rightarrow True])) I O H5) in (False_ind (eq C
+(CHead c k u) e0) H6))))))))))) (\lambda (k: K).(\lambda (h: nat).(\lambda
+(d: nat).(\lambda (c: C).(\lambda (e0: C).(\lambda (H2: (drop h (r k d) c
+e0)).(\lambda (H3: (((eq nat (r k d) O) \to ((eq nat h (r k d)) \to (eq C c
e0))))).(\lambda (u: T).(\lambda (H4: (eq nat (S d) O)).(\lambda (H5: (eq nat
-h 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 in nat return (\lambda (_: nat).Prop) with [O
-\Rightarrow False | (S _) \Rightarrow True])) I O H4) in (False_ind (eq C
-(CHead c k (lift O (r k d) u)) (CHead e0 k u)) H9)) h H6)))))))))))))) y y0 x
-e H1))) H0))) H))).
+h (S d))).(let H6 \def (f_equal nat nat (\lambda (e1: nat).e1) h (S d) H5) in
+(let H7 \def (eq_ind nat h (\lambda (n: nat).((eq nat (r k d) O) \to ((eq nat
+n (r k d)) \to (eq C c e0)))) H3 (S d) H6) in (let H8 \def (eq_ind nat h
+(\lambda (n: nat).(drop n (r k d) c e0)) H2 (S d) H6) in (eq_ind_r nat (S d)
+(\lambda (n: nat).(eq C (CHead c k (lift n (r k d) u)) (CHead e0 k u))) (let
+H9 \def (eq_ind nat (S d) (\lambda (ee: nat).(match ee in nat return (\lambda
+(_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H4)
+in (False_ind (eq C (CHead c k (lift (S d) (r k d) u)) (CHead e0 k u)) H9)) h
+H6)))))))))))))) y y0 x e H1))) H0))) H))).
theorem drop_gen_drop:
\forall (k: K).(\forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h:
\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 in eq 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 (H6: (eq
-nat O (S h))).(let H7 \def (eq_ind nat O (\lambda (e: nat).(match e in nat
-return (\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow
-False])) I (S h) H6) in (False_ind (drop (r k h) O c c0) H7)))]) 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
+(\lambda (c0: C).(drop (S h) O c0 x)) (\lambda (_: C).(drop (r k h) O c x))
+(\lambda (y: C).(\lambda (H0: (drop (S h) O y x)).(insert_eq nat O (\lambda
+(n: nat).(drop (S h) n y x)) (\lambda (n: nat).((eq C y (CHead c k u)) \to
+(drop (r k h) n c x))) (\lambda (y0: nat).(\lambda (H1: (drop (S h) y0 y
+x)).(insert_eq nat (S h) (\lambda (n: nat).(drop n y0 y x)) (\lambda (_:
+nat).((eq nat y0 O) \to ((eq C y (CHead c k u)) \to (drop (r k h) y0 c x))))
+(\lambda (y1: nat).(\lambda (H2: (drop y1 y0 y x)).(drop_ind (\lambda (n:
+nat).(\lambda (n0: nat).(\lambda (c0: C).(\lambda (c1: C).((eq nat n (S h))
+\to ((eq nat n0 O) \to ((eq C c0 (CHead c k u)) \to (drop (r k h) n0 c
+c1)))))))) (\lambda (c0: C).(\lambda (H3: (eq nat O (S h))).(\lambda (_: (eq
+nat O O)).(\lambda (_: (eq C c0 (CHead c k u))).(let H6 \def (match H3 in eq
+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 (H6: (eq nat O (S
+h))).(let H7 \def (eq_ind nat O (\lambda (e: nat).(match e in nat return
+(\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False]))
+I (S h) H6) in (False_ind (drop (r k h) O c c0) H7)))]) 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 in eq return (\lambda (n: nat).(\lambda (_: (eq
? ? n)).((eq nat n (S h)) \to (drop (r k h) O c e)))) with [refl_equal
(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:
+((eq C c0 (CHead c k u)) \to (drop (r k h) (r k0 d) 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 in eq 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
+O) \to (drop (r k h) (S d) c (CHead e k0 u0))))) with [refl_equal \Rightarrow
(\lambda (H8: (eq nat (S d) O)).(let H9 \def (eq_ind nat (S d) (\lambda (e0:
nat).(match e0 in nat return (\lambda (_: nat).Prop) with [O \Rightarrow
-False | (S _) \Rightarrow True])) I O H8) in (False_ind (drop (r k h) O c
+False | (S _) \Rightarrow True])) I O H8) in (False_ind (drop (r k h) (S d) c
(CHead e k0 u0)) H9)))]) in (H8 (refl_equal nat O)))))))))))))) y1 y0 y x
H2))) H1))) H0))) H)))))).
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/drop/props".
include "drop/fwd.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/drop1/defs".
include "drop/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/drop1/getl".
include "drop1/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/drop1/props".
include "drop1/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/ex0/defs".
include "A/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/ex0/props".
include "ex0/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/ex1/defs".
include "C/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/ex1/props".
include "ex1/defs.ma".
Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S O) O
x4) x0)).(\lambda (H5: (getl O (CHead (CHead (CHead (CSort O) (Bind Abst)
(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead x2 (Bind
-Abbr) x3))).(\lambda (_: (ty3 g x2 x3 x4)).(ex4_3_ind T T T (\lambda (t2:
-T).(\lambda (_: T).(\lambda (_: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind
-Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (THead (Bind
-Abst) (TLRef (S (S O))) t2) (THead (Bind Abst) x0 x1))))) (\lambda (_:
-T).(\lambda (t: T).(\lambda (_: T).(ty3 g (CHead (CHead (CHead (CSort O)
-(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef
-(S (S O))) t)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g
-(CHead (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
-(TSort O)) (Bind Abst) (TLRef O)) (Bind Abst) (TLRef (S (S O)))) (TSort O)
-t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t0: T).(ty3 g (CHead (CHead
-(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind
-Abst) (TLRef O)) (Bind Abst) (TLRef (S (S O)))) t2 t0)))) P (\lambda (x5:
-T).(\lambda (x6: T).(\lambda (x7: T).(\lambda (_: (pc3 (CHead (CHead (CHead
-(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef
-O)) (THead (Bind Abst) (TLRef (S (S O))) x5) (THead (Bind Abst) x0
+Abbr) x3))).(\lambda (_: (ty3 g x2 x3 x4)).(ex3_2_ind T T (\lambda (t2:
+T).(\lambda (_: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O))
+(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (THead (Bind Abst) (TLRef (S (S
+O))) t2) (THead (Bind Abst) x0 x1)))) (\lambda (_: T).(\lambda (t: T).(ty3 g
+(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
+(Bind Abst) (TLRef O)) (TLRef (S (S O))) t))) (\lambda (t2: T).(\lambda (_:
+T).(ty3 g (CHead (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind
+Abst) (TSort O)) (Bind Abst) (TLRef O)) (Bind Abst) (TLRef (S (S O)))) (TSort
+O) t2))) P (\lambda (x5: T).(\lambda (x6: T).(\lambda (_: (pc3 (CHead (CHead
+(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst)
+(TLRef O)) (THead (Bind Abst) (TLRef (S (S O))) x5) (THead (Bind Abst) x0
x1))).(\lambda (H8: (ty3 g (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort
O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O)))
x6)).(\lambda (_: (ty3 g (CHead (CHead (CHead (CHead (CSort O) (Bind Abst)
(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (Bind Abst) (TLRef
-(S (S O)))) (TSort O) x5)).(\lambda (_: (ty3 g (CHead (CHead (CHead (CHead
-(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef
-O)) (Bind Abst) (TLRef (S (S O)))) x5 x7)).(or_ind (ex3_3 C T T (\lambda (_:
-C).(\lambda (_: T).(\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind
-Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S
-O))) O t) x6)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S
-O)) (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort
-O)) (Bind Abst) (TLRef O)) (CHead e (Bind Abbr) u0))))) (\lambda (e:
-C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t))))) (ex3_3 C T T (\lambda
-(_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 (CHead (CHead (CHead (CSort O)
-(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S
-(S (S O))) O u0) x6)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_:
-T).(getl (S (S O)) (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O))
-(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead e (Bind Abst) u0)))))
-(\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t))))) P
-(\lambda (H11: (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t:
-T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
-(TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O t) x6)))) (\lambda
-(e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) (CHead (CHead (CHead
-(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef
-O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda
-(t: T).(ty3 g e u0 t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (_:
+(S (S O)))) (TSort O) x5)).(or_ind (ex3_3 C T T (\lambda (_: C).(\lambda (_:
T).(\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O))
(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O t) x6))))
(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) (CHead
(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind
Abst) (TLRef O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0:
-T).(\lambda (t: T).(ty3 g e u0 t)))) P (\lambda (x8: C).(\lambda (x9:
-T).(\lambda (x10: T).(\lambda (_: (pc3 (CHead (CHead (CHead (CSort O) (Bind
-Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S
-O))) O x10) x6)).(\lambda (H13: (getl (S (S O)) (CHead (CHead (CHead (CSort
-O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O))
-(CHead x8 (Bind Abbr) x9))).(\lambda (_: (ty3 g x8 x9 x10)).(let H15 \def
-(getl_gen_all (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
-(TSort O)) (CHead x8 (Bind Abbr) x9) (r (Bind Abst) (S O)) (getl_gen_S (Bind
-Abst) (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
-(CHead x8 (Bind Abbr) x9) (TLRef O) (S O) H13)) in (ex2_ind C (\lambda (e:
-C).(drop (S O) O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
-(TSort O)) e)) (\lambda (e: C).(clear e (CHead x8 (Bind Abbr) x9))) P
-(\lambda (x: C).(\lambda (_: (drop (S O) O (CHead (CHead (CSort O) (Bind
-Abst) (TSort O)) (Bind Abst) (TSort O)) x)).(\lambda (_: (clear x (CHead x8
-(Bind Abbr) x9))).(let H18 \def (eq_ind C (CHead x2 (Bind Abbr) x3) (\lambda
-(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
-\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda
-(_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_:
-B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | Void
-\Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead (CHead (CHead
+T).(\lambda (t: T).(ty3 g e u0 t))))) (ex3_3 C T T (\lambda (_: C).(\lambda
+(u0: T).(\lambda (_: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst)
+(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O)))
+O u0) x6)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O))
+(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
+(Bind Abst) (TLRef O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda
+(u0: T).(\lambda (t: T).(ty3 g e u0 t))))) P (\lambda (H10: (ex3_3 C T T
+(\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 (CHead (CHead (CHead
(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef
-O)) (clear_gen_bind Abst (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind
-Abst) (TSort O)) (CHead x2 (Bind Abbr) x3) (TLRef O) (getl_gen_O (CHead
-(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind
-Abst) (TLRef O)) (CHead x2 (Bind Abbr) x3) H5))) in (False_ind P H18)))))
-H15)))))))) H11)) (\lambda (H11: (ex3_3 C T T (\lambda (_: C).(\lambda (u0:
+O)) (lift (S (S (S O))) O t) x6)))) (\lambda (e: C).(\lambda (u0: T).(\lambda
+(_: T).(getl (S (S O)) (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O))
+(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead e (Bind Abbr) u0)))))
+(\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0
+t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3
+(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
+(Bind Abst) (TLRef O)) (lift (S (S (S O))) O t) x6)))) (\lambda (e:
+C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) (CHead (CHead (CHead
+(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef
+O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda
+(t: T).(ty3 g e u0 t)))) P (\lambda (x7: C).(\lambda (x8: T).(\lambda (x9:
+T).(\lambda (_: (pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O))
+(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O x9)
+x6)).(\lambda (H12: (getl (S (S O)) (CHead (CHead (CHead (CSort O) (Bind
+Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead x7
+(Bind Abbr) x8))).(\lambda (_: (ty3 g x7 x8 x9)).(let H14 \def (getl_gen_all
+(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead
+x7 (Bind Abbr) x8) (r (Bind Abst) (S O)) (getl_gen_S (Bind Abst) (CHead
+(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead x7
+(Bind Abbr) x8) (TLRef O) (S O) H12)) in (ex2_ind C (\lambda (e: C).(drop (S
+O) O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
+e)) (\lambda (e: C).(clear e (CHead x7 (Bind Abbr) x8))) P (\lambda (x:
+C).(\lambda (_: (drop (S O) O (CHead (CHead (CSort O) (Bind Abst) (TSort O))
+(Bind Abst) (TSort O)) x)).(\lambda (_: (clear x (CHead x7 (Bind Abbr)
+x8))).(let H17 \def (eq_ind C (CHead x2 (Bind Abbr) x3) (\lambda (ee:
+C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow
+False | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop)
+with [(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with
+[Abbr \Rightarrow True | Abst \Rightarrow False | Void \Rightarrow False]) |
+(Flat _) \Rightarrow False])])) I (CHead (CHead (CHead (CSort O) (Bind Abst)
+(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (clear_gen_bind Abst
+(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead
+x2 (Bind Abbr) x3) (TLRef O) (getl_gen_O (CHead (CHead (CHead (CSort O) (Bind
+Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead x2
+(Bind Abbr) x3) H5))) in (False_ind P H17))))) H14)))))))) H10)) (\lambda
+(H10: (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3
+(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
+(Bind Abst) (TLRef O)) (lift (S (S (S O))) O u0) x6)))) (\lambda (e:
+C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) (CHead (CHead (CHead
+(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef
+O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda
+(t: T).(ty3 g e u0 t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (u0:
T).(\lambda (_: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O))
(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O u0)
x6)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O))
(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
(Bind Abst) (TLRef O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda
-(u0: T).(\lambda (t: T).(ty3 g e u0 t)))))).(ex3_3_ind C T T (\lambda (_:
-C).(\lambda (u0: T).(\lambda (_: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind
+(u0: T).(\lambda (t: T).(ty3 g e u0 t)))) P (\lambda (x7: C).(\lambda (x8:
+T).(\lambda (x9: T).(\lambda (_: (pc3 (CHead (CHead (CHead (CSort O) (Bind
Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S
-O))) O u0) x6)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S
-(S O)) (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
-(TSort O)) (Bind Abst) (TLRef O)) (CHead e (Bind Abst) u0))))) (\lambda (e:
-C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t)))) P (\lambda (x8:
-C).(\lambda (x9: T).(\lambda (x10: T).(\lambda (_: (pc3 (CHead (CHead (CHead
-(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef
-O)) (lift (S (S (S O))) O x9) x6)).(\lambda (H13: (getl (S (S O)) (CHead
-(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind
-Abst) (TLRef O)) (CHead x8 (Bind Abst) x9))).(\lambda (_: (ty3 g x8 x9
-x10)).(let H15 \def (getl_gen_all (CHead (CHead (CSort O) (Bind Abst) (TSort
-O)) (Bind Abst) (TSort O)) (CHead x8 (Bind Abst) x9) (r (Bind Abst) (S O))
-(getl_gen_S (Bind Abst) (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind
-Abst) (TSort O)) (CHead x8 (Bind Abst) x9) (TLRef O) (S O) H13)) in (ex2_ind
-C (\lambda (e: C).(drop (S O) O (CHead (CHead (CSort O) (Bind Abst) (TSort
-O)) (Bind Abst) (TSort O)) e)) (\lambda (e: C).(clear e (CHead x8 (Bind Abst)
-x9))) P (\lambda (x: C).(\lambda (_: (drop (S O) O (CHead (CHead (CSort O)
-(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) x)).(\lambda (_: (clear x
-(CHead x8 (Bind Abst) x9))).(let H18 \def (eq_ind C (CHead x2 (Bind Abbr) x3)
-(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
+O))) O x8) x6)).(\lambda (H12: (getl (S (S O)) (CHead (CHead (CHead (CSort O)
+(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead
+x7 (Bind Abst) x8))).(\lambda (_: (ty3 g x7 x8 x9)).(let H14 \def
+(getl_gen_all (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
+(TSort O)) (CHead x7 (Bind Abst) x8) (r (Bind Abst) (S O)) (getl_gen_S (Bind
+Abst) (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
+(CHead x7 (Bind Abst) x8) (TLRef O) (S O) H12)) in (ex2_ind C (\lambda (e:
+C).(drop (S O) O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
+(TSort O)) e)) (\lambda (e: C).(clear e (CHead x7 (Bind Abst) x8))) P
+(\lambda (x: C).(\lambda (_: (drop (S O) O (CHead (CHead (CSort O) (Bind
+Abst) (TSort O)) (Bind Abst) (TSort O)) x)).(\lambda (_: (clear x (CHead x7
+(Bind Abst) x8))).(let H17 \def (eq_ind C (CHead x2 (Bind Abbr) x3) (\lambda
+(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda
(_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_:
B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | Void
O)) (clear_gen_bind Abst (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind
Abst) (TSort O)) (CHead x2 (Bind Abbr) x3) (TLRef O) (getl_gen_O (CHead
(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind
-Abst) (TLRef O)) (CHead x2 (Bind Abbr) x3) H5))) in (False_ind P H18)))))
-H15)))))))) H11)) (ty3_gen_lref g (CHead (CHead (CHead (CSort O) (Bind Abst)
+Abst) (TLRef O)) (CHead x2 (Bind Abbr) x3) H5))) in (False_ind P H17)))))
+H14)))))))) H10)) (ty3_gen_lref g (CHead (CHead (CHead (CSort O) (Bind Abst)
(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) x6 (S (S O))
-H8))))))))) (ty3_gen_bind g Abst (CHead (CHead (CHead (CSort O) (Bind Abst)
+H8))))))) (ty3_gen_bind g Abst (CHead (CHead (CHead (CSort O) (Bind Abst)
(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O)))
(TSort O) (THead (Bind Abst) x0 x1) H1)))))))) H3)) (\lambda (H3: (ex3_3 C T
T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 (CHead (CHead (CHead
(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef
O)) (lift (S O) O x3) x0)).(\lambda (H5: (getl O (CHead (CHead (CHead (CSort
O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O))
-(CHead x2 (Bind Abst) x3))).(\lambda (H6: (ty3 g x2 x3 x4)).(ex4_3_ind T T T
-(\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 (CHead (CHead (CHead
-(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef
-O)) (THead (Bind Abst) (TLRef (S (S O))) t2) (THead (Bind Abst) x0 x1)))))
-(\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g (CHead (CHead (CHead
-(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef
-O)) (TLRef (S (S O))) t)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_:
-T).(ty3 g (CHead (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind
-Abst) (TSort O)) (Bind Abst) (TLRef O)) (Bind Abst) (TLRef (S (S O)))) (TSort
-O) t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t0: T).(ty3 g (CHead
-(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
-(Bind Abst) (TLRef O)) (Bind Abst) (TLRef (S (S O)))) t2 t0)))) P (\lambda
-(x5: T).(\lambda (x6: T).(\lambda (x7: T).(\lambda (H7: (pc3 (CHead (CHead
+(CHead x2 (Bind Abst) x3))).(\lambda (H6: (ty3 g x2 x3 x4)).(ex3_2_ind T T
+(\lambda (t2: T).(\lambda (_: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind
+Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (THead (Bind
+Abst) (TLRef (S (S O))) t2) (THead (Bind Abst) x0 x1)))) (\lambda (_:
+T).(\lambda (t: T).(ty3 g (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort
+O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) t)))
+(\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead (CHead (CHead (CHead (CSort O)
+(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (Bind
+Abst) (TLRef (S (S O)))) (TSort O) t2))) P (\lambda (x5: T).(\lambda (x6:
+T).(\lambda (H7: (pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O))
+(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (THead (Bind Abst) (TLRef (S (S
+O))) x5) (THead (Bind Abst) x0 x1))).(\lambda (H8: (ty3 g (CHead (CHead
(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst)
-(TLRef O)) (THead (Bind Abst) (TLRef (S (S O))) x5) (THead (Bind Abst) x0
-x1))).(\lambda (H8: (ty3 g (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort
-O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O)))
-x6)).(\lambda (_: (ty3 g (CHead (CHead (CHead (CHead (CSort O) (Bind Abst)
-(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (Bind Abst) (TLRef
-(S (S O)))) (TSort O) x5)).(\lambda (_: (ty3 g (CHead (CHead (CHead (CHead
+(TLRef O)) (TLRef (S (S O))) x6)).(\lambda (_: (ty3 g (CHead (CHead (CHead
+(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst)
+(TLRef O)) (Bind Abst) (TLRef (S (S O)))) (TSort O) x5)).(or_ind (ex3_3 C T T
+(\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 (CHead (CHead (CHead
(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef
-O)) (Bind Abst) (TLRef (S (S O)))) x5 x7)).(or_ind (ex3_3 C T T (\lambda (_:
-C).(\lambda (_: T).(\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind
-Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S
-O))) O t) x6)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S
-O)) (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort
-O)) (Bind Abst) (TLRef O)) (CHead e (Bind Abbr) u0))))) (\lambda (e:
-C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t))))) (ex3_3 C T T (\lambda
-(_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 (CHead (CHead (CHead (CSort O)
-(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S
-(S (S O))) O u0) x6)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_:
-T).(getl (S (S O)) (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O))
-(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead e (Bind Abst) u0)))))
-(\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t))))) P
-(\lambda (H11: (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t:
+O)) (lift (S (S (S O))) O t) x6)))) (\lambda (e: C).(\lambda (u0: T).(\lambda
+(_: T).(getl (S (S O)) (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O))
+(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead e (Bind Abbr) u0)))))
+(\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t))))) (ex3_3 C
+T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 (CHead (CHead
+(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst)
+(TLRef O)) (lift (S (S (S O))) O u0) x6)))) (\lambda (e: C).(\lambda (u0:
+T).(\lambda (_: T).(getl (S (S O)) (CHead (CHead (CHead (CSort O) (Bind Abst)
+(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead e (Bind Abst)
+u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t))))) P
+(\lambda (H10: (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t:
T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
(TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O t) x6)))) (\lambda
(e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) (CHead (CHead (CHead
(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) (CHead
(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind
Abst) (TLRef O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0:
-T).(\lambda (t: T).(ty3 g e u0 t)))) P (\lambda (x8: C).(\lambda (x9:
-T).(\lambda (x10: T).(\lambda (_: (pc3 (CHead (CHead (CHead (CSort O) (Bind
+T).(\lambda (t: T).(ty3 g e u0 t)))) P (\lambda (x7: C).(\lambda (x8:
+T).(\lambda (x9: T).(\lambda (_: (pc3 (CHead (CHead (CHead (CSort O) (Bind
Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S
-O))) O x10) x6)).(\lambda (H13: (getl (S (S O)) (CHead (CHead (CHead (CSort
-O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O))
-(CHead x8 (Bind Abbr) x9))).(\lambda (_: (ty3 g x8 x9 x10)).(let H15 \def
+O))) O x9) x6)).(\lambda (H12: (getl (S (S O)) (CHead (CHead (CHead (CSort O)
+(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead
+x7 (Bind Abbr) x8))).(\lambda (_: (ty3 g x7 x8 x9)).(let H14 \def
(getl_gen_all (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
-(TSort O)) (CHead x8 (Bind Abbr) x9) (r (Bind Abst) (S O)) (getl_gen_S (Bind
+(TSort O)) (CHead x7 (Bind Abbr) x8) (r (Bind Abst) (S O)) (getl_gen_S (Bind
Abst) (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
-(CHead x8 (Bind Abbr) x9) (TLRef O) (S O) H13)) in (ex2_ind C (\lambda (e:
+(CHead x7 (Bind Abbr) x8) (TLRef O) (S O) H12)) in (ex2_ind C (\lambda (e:
C).(drop (S O) O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
-(TSort O)) e)) (\lambda (e: C).(clear e (CHead x8 (Bind Abbr) x9))) P
-(\lambda (x: C).(\lambda (H16: (drop (S O) O (CHead (CHead (CSort O) (Bind
-Abst) (TSort O)) (Bind Abst) (TSort O)) x)).(\lambda (H17: (clear x (CHead x8
-(Bind Abbr) x9))).(let H18 \def (f_equal C C (\lambda (e: C).(match e in C
+(TSort O)) e)) (\lambda (e: C).(clear e (CHead x7 (Bind Abbr) x8))) P
+(\lambda (x: C).(\lambda (H15: (drop (S O) O (CHead (CHead (CSort O) (Bind
+Abst) (TSort O)) (Bind Abst) (TSort O)) x)).(\lambda (H16: (clear x (CHead x7
+(Bind Abbr) x8))).(let H17 \def (f_equal C C (\lambda (e: C).(match e in C
return (\lambda (_: C).C) with [(CSort _) \Rightarrow x2 | (CHead c _ _)
\Rightarrow c])) (CHead x2 (Bind Abst) x3) (CHead (CHead (CHead (CSort O)
(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O))
(clear_gen_bind Abst (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind
Abst) (TSort O)) (CHead x2 (Bind Abst) x3) (TLRef O) (getl_gen_O (CHead
(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind
-Abst) (TLRef O)) (CHead x2 (Bind Abst) x3) H5))) in ((let H19 \def (f_equal C
+Abst) (TLRef O)) (CHead x2 (Bind Abst) x3) H5))) in ((let H18 \def (f_equal C
T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _)
\Rightarrow x3 | (CHead _ _ t) \Rightarrow t])) (CHead x2 (Bind Abst) x3)
(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead x2 (Bind Abst) x3) (TLRef O)
(getl_gen_O (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
(TSort O)) (Bind Abst) (TLRef O)) (CHead x2 (Bind Abst) x3) H5))) in (\lambda
-(H20: (eq C x2 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
-(TSort O)))).(let H21 \def (eq_ind T x3 (\lambda (t: T).(ty3 g x2 t x4)) H6
-(TLRef O) H19) in (let H22 \def (eq_ind T x3 (\lambda (t: T).(pc3 (CHead
+(H19: (eq C x2 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
+(TSort O)))).(let H20 \def (eq_ind T x3 (\lambda (t: T).(ty3 g x2 t x4)) H6
+(TLRef O) H18) in (let H21 \def (eq_ind T x3 (\lambda (t: T).(pc3 (CHead
(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind
-Abst) (TLRef O)) (lift (S O) O t) x0)) H4 (TLRef O) H19) in (let H23 \def
-(eq_ind C x2 (\lambda (c: C).(ty3 g c (TLRef O) x4)) H21 (CHead (CHead (CSort
-O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) H20) in (let H24 \def
-(eq_ind_r C x (\lambda (c: C).(clear c (CHead x8 (Bind Abbr) x9))) H17 (CHead
+Abst) (TLRef O)) (lift (S O) O t) x0)) H4 (TLRef O) H18) in (let H22 \def
+(eq_ind C x2 (\lambda (c: C).(ty3 g c (TLRef O) x4)) H20 (CHead (CHead (CSort
+O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) H19) in (let H23 \def
+(eq_ind_r C x (\lambda (c: C).(clear c (CHead x7 (Bind Abbr) x8))) H16 (CHead
(CSort O) (Bind Abst) (TSort O)) (drop_gen_refl (CHead (CSort O) (Bind Abst)
(TSort O)) x (drop_gen_drop (Bind Abst) (CHead (CSort O) (Bind Abst) (TSort
-O)) x (TSort O) O H16))) in (let H25 \def (eq_ind C (CHead x8 (Bind Abbr) x9)
+O)) x (TSort O) O H15))) in (let H24 \def (eq_ind C (CHead x7 (Bind Abbr) x8)
(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda
(_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_:
B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | Void
\Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead (CSort O)
-(Bind Abst) (TSort O)) (clear_gen_bind Abst (CSort O) (CHead x8 (Bind Abbr)
-x9) (TSort O) H24)) in (False_ind P H25)))))))) H18))))) H15)))))))) H11))
-(\lambda (H11: (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_:
+(Bind Abst) (TSort O)) (clear_gen_bind Abst (CSort O) (CHead x7 (Bind Abbr)
+x8) (TSort O) H23)) in (False_ind P H24)))))))) H17))))) H14)))))))) H10))
+(\lambda (H10: (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_:
T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
(TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O u0) x6)))) (\lambda
(e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) (CHead (CHead (CHead
x6)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O))
(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
(Bind Abst) (TLRef O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda
-(u0: T).(\lambda (t: T).(ty3 g e u0 t)))) P (\lambda (x8: C).(\lambda (x9:
-T).(\lambda (x10: T).(\lambda (H12: (pc3 (CHead (CHead (CHead (CSort O) (Bind
+(u0: T).(\lambda (t: T).(ty3 g e u0 t)))) P (\lambda (x7: C).(\lambda (x8:
+T).(\lambda (x9: T).(\lambda (H11: (pc3 (CHead (CHead (CHead (CSort O) (Bind
Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S
-O))) O x9) x6)).(\lambda (H13: (getl (S (S O)) (CHead (CHead (CHead (CSort O)
+O))) O x8) x6)).(\lambda (H12: (getl (S (S O)) (CHead (CHead (CHead (CSort O)
(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead
-x8 (Bind Abst) x9))).(\lambda (H14: (ty3 g x8 x9 x10)).(let H15 \def
+x7 (Bind Abst) x8))).(\lambda (H13: (ty3 g x7 x8 x9)).(let H14 \def
(getl_gen_all (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
-(TSort O)) (CHead x8 (Bind Abst) x9) (r (Bind Abst) (S O)) (getl_gen_S (Bind
+(TSort O)) (CHead x7 (Bind Abst) x8) (r (Bind Abst) (S O)) (getl_gen_S (Bind
Abst) (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
-(CHead x8 (Bind Abst) x9) (TLRef O) (S O) H13)) in (ex2_ind C (\lambda (e:
+(CHead x7 (Bind Abst) x8) (TLRef O) (S O) H12)) in (ex2_ind C (\lambda (e:
C).(drop (S O) O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
-(TSort O)) e)) (\lambda (e: C).(clear e (CHead x8 (Bind Abst) x9))) P
-(\lambda (x: C).(\lambda (H16: (drop (S O) O (CHead (CHead (CSort O) (Bind
-Abst) (TSort O)) (Bind Abst) (TSort O)) x)).(\lambda (H17: (clear x (CHead x8
-(Bind Abst) x9))).(let H18 \def (f_equal C C (\lambda (e: C).(match e in C
+(TSort O)) e)) (\lambda (e: C).(clear e (CHead x7 (Bind Abst) x8))) P
+(\lambda (x: C).(\lambda (H15: (drop (S O) O (CHead (CHead (CSort O) (Bind
+Abst) (TSort O)) (Bind Abst) (TSort O)) x)).(\lambda (H16: (clear x (CHead x7
+(Bind Abst) x8))).(let H17 \def (f_equal C C (\lambda (e: C).(match e in C
return (\lambda (_: C).C) with [(CSort _) \Rightarrow x2 | (CHead c _ _)
\Rightarrow c])) (CHead x2 (Bind Abst) x3) (CHead (CHead (CHead (CSort O)
(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O))
(clear_gen_bind Abst (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind
Abst) (TSort O)) (CHead x2 (Bind Abst) x3) (TLRef O) (getl_gen_O (CHead
(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind
-Abst) (TLRef O)) (CHead x2 (Bind Abst) x3) H5))) in ((let H19 \def (f_equal C
+Abst) (TLRef O)) (CHead x2 (Bind Abst) x3) H5))) in ((let H18 \def (f_equal C
T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _)
\Rightarrow x3 | (CHead _ _ t) \Rightarrow t])) (CHead x2 (Bind Abst) x3)
(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead x2 (Bind Abst) x3) (TLRef O)
(getl_gen_O (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
(TSort O)) (Bind Abst) (TLRef O)) (CHead x2 (Bind Abst) x3) H5))) in (\lambda
-(H20: (eq C x2 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
-(TSort O)))).(let H21 \def (eq_ind T x3 (\lambda (t: T).(ty3 g x2 t x4)) H6
-(TLRef O) H19) in (let H22 \def (eq_ind T x3 (\lambda (t: T).(pc3 (CHead
+(H19: (eq C x2 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
+(TSort O)))).(let H20 \def (eq_ind T x3 (\lambda (t: T).(ty3 g x2 t x4)) H6
+(TLRef O) H18) in (let H21 \def (eq_ind T x3 (\lambda (t: T).(pc3 (CHead
(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind
-Abst) (TLRef O)) (lift (S O) O t) x0)) H4 (TLRef O) H19) in (let H23 \def
-(eq_ind C x2 (\lambda (c: C).(ty3 g c (TLRef O) x4)) H21 (CHead (CHead (CSort
-O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) H20) in (let H24 \def
-(eq_ind_r C x (\lambda (c: C).(clear c (CHead x8 (Bind Abst) x9))) H17 (CHead
+Abst) (TLRef O)) (lift (S O) O t) x0)) H4 (TLRef O) H18) in (let H22 \def
+(eq_ind C x2 (\lambda (c: C).(ty3 g c (TLRef O) x4)) H20 (CHead (CHead (CSort
+O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) H19) in (let H23 \def
+(eq_ind_r C x (\lambda (c: C).(clear c (CHead x7 (Bind Abst) x8))) H16 (CHead
(CSort O) (Bind Abst) (TSort O)) (drop_gen_refl (CHead (CSort O) (Bind Abst)
(TSort O)) x (drop_gen_drop (Bind Abst) (CHead (CSort O) (Bind Abst) (TSort
-O)) x (TSort O) O H16))) in (let H25 \def (f_equal C C (\lambda (e: C).(match
-e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow x8 | (CHead c _
-_) \Rightarrow c])) (CHead x8 (Bind Abst) x9) (CHead (CSort O) (Bind Abst)
-(TSort O)) (clear_gen_bind Abst (CSort O) (CHead x8 (Bind Abst) x9) (TSort O)
-H24)) in ((let H26 \def (f_equal C T (\lambda (e: C).(match e in C return
-(\lambda (_: C).T) with [(CSort _) \Rightarrow x9 | (CHead _ _ t) \Rightarrow
-t])) (CHead x8 (Bind Abst) x9) (CHead (CSort O) (Bind Abst) (TSort O))
-(clear_gen_bind Abst (CSort O) (CHead x8 (Bind Abst) x9) (TSort O) H24)) in
-(\lambda (H27: (eq C x8 (CSort O))).(let H28 \def (eq_ind T x9 (\lambda (t:
-T).(ty3 g x8 t x10)) H14 (TSort O) H26) in (let H29 \def (eq_ind T x9
-(\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O))
-(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O t) x6))
-H12 (TSort O) H26) in (let H30 \def (eq_ind C x8 (\lambda (c: C).(ty3 g c
-(TSort O) x10)) H28 (CSort O) H27) in (or_ind (ex3_3 C T T (\lambda (_:
-C).(\lambda (_: T).(\lambda (t: T).(pc3 (CHead (CHead (CSort O) (Bind Abst)
-(TSort O)) (Bind Abst) (TSort O)) (lift (S O) O t) x4)))) (\lambda (e:
-C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead (CHead (CSort O) (Bind
-Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead e (Bind Abbr) u0))))) (\lambda
-(e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t))))) (ex3_3 C T T
-(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 (CHead (CHead (CSort O)
-(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (lift (S O) O u0) x4))))
-(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead (CHead (CSort
-O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead e (Bind Abst) u0)))))
-(\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t))))) P
-(\lambda (H31: (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t:
-T).(pc3 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
-(lift (S O) O t) x4)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_:
-T).(getl O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort
-O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda
-(t: T).(ty3 g e u0 t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (_:
+O)) x (TSort O) O H15))) in (let H24 \def (f_equal C C (\lambda (e: C).(match
+e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow x7 | (CHead c _
+_) \Rightarrow c])) (CHead x7 (Bind Abst) x8) (CHead (CSort O) (Bind Abst)
+(TSort O)) (clear_gen_bind Abst (CSort O) (CHead x7 (Bind Abst) x8) (TSort O)
+H23)) in ((let H25 \def (f_equal C T (\lambda (e: C).(match e in C return
+(\lambda (_: C).T) with [(CSort _) \Rightarrow x8 | (CHead _ _ t) \Rightarrow
+t])) (CHead x7 (Bind Abst) x8) (CHead (CSort O) (Bind Abst) (TSort O))
+(clear_gen_bind Abst (CSort O) (CHead x7 (Bind Abst) x8) (TSort O) H23)) in
+(\lambda (H26: (eq C x7 (CSort O))).(let H27 \def (eq_ind T x8 (\lambda (t:
+T).(ty3 g x7 t x9)) H13 (TSort O) H25) in (let H28 \def (eq_ind T x8 (\lambda
+(t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
+(TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O t) x6)) H11 (TSort O)
+H25) in (let H29 \def (eq_ind C x7 (\lambda (c: C).(ty3 g c (TSort O) x9))
+H27 (CSort O) H26) in (or_ind (ex3_3 C T T (\lambda (_: C).(\lambda (_:
T).(\lambda (t: T).(pc3 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind
Abst) (TSort O)) (lift (S O) O t) x4)))) (\lambda (e: C).(\lambda (u0:
T).(\lambda (_: T).(getl O (CHead (CHead (CSort O) (Bind Abst) (TSort O))
(Bind Abst) (TSort O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda
-(u0: T).(\lambda (t: T).(ty3 g e u0 t)))) P (\lambda (x11: C).(\lambda (x12:
-T).(\lambda (x13: T).(\lambda (_: (pc3 (CHead (CHead (CSort O) (Bind Abst)
-(TSort O)) (Bind Abst) (TSort O)) (lift (S O) O x13) x4)).(\lambda (H33:
-(getl O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
-(CHead x11 (Bind Abbr) x12))).(\lambda (_: (ty3 g x11 x12 x13)).(let H35 \def
-(eq_ind C (CHead x11 (Bind Abbr) x12) (\lambda (ee: C).(match ee in C return
-(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _)
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b)
-\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow
-True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _)
-\Rightarrow False])])) I (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind
-Abst) (TSort O)) (clear_gen_bind Abst (CHead (CSort O) (Bind Abst) (TSort O))
-(CHead x11 (Bind Abbr) x12) (TSort O) (getl_gen_O (CHead (CHead (CSort O)
-(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead x11 (Bind Abbr) x12)
-H33))) in (False_ind P H35)))))))) H31)) (\lambda (H31: (ex3_3 C T T (\lambda
-(_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 (CHead (CHead (CSort O) (Bind
-Abst) (TSort O)) (Bind Abst) (TSort O)) (lift (S O) O u0) x4)))) (\lambda (e:
+(u0: T).(\lambda (t: T).(ty3 g e u0 t))))) (ex3_3 C T T (\lambda (_:
+C).(\lambda (u0: T).(\lambda (_: T).(pc3 (CHead (CHead (CSort O) (Bind Abst)
+(TSort O)) (Bind Abst) (TSort O)) (lift (S O) O u0) x4)))) (\lambda (e:
C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead (CHead (CSort O) (Bind
Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead e (Bind Abst) u0))))) (\lambda
-(e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t)))))).(ex3_3_ind C T T
-(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 (CHead (CHead (CSort O)
-(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (lift (S O) O u0) x4))))
-(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead (CHead (CSort
-O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead e (Bind Abst) u0)))))
-(\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t)))) P (\lambda
-(x11: C).(\lambda (x12: T).(\lambda (x13: T).(\lambda (H32: (pc3 (CHead
+(e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t))))) P (\lambda (H30:
+(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 (CHead
(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (lift (S O) O
-x12) x4)).(\lambda (H33: (getl O (CHead (CHead (CSort O) (Bind Abst) (TSort
-O)) (Bind Abst) (TSort O)) (CHead x11 (Bind Abst) x12))).(\lambda (H34: (ty3
-g x11 x12 x13)).(let H35 \def (f_equal C C (\lambda (e: C).(match e in C
-return (\lambda (_: C).C) with [(CSort _) \Rightarrow x11 | (CHead c _ _)
-\Rightarrow c])) (CHead x11 (Bind Abst) x12) (CHead (CHead (CSort O) (Bind
-Abst) (TSort O)) (Bind Abst) (TSort O)) (clear_gen_bind Abst (CHead (CSort O)
-(Bind Abst) (TSort O)) (CHead x11 (Bind Abst) x12) (TSort O) (getl_gen_O
+t) x4)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead
+(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead e (Bind
+Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0
+t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3
+(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (lift
+(S O) O t) x4)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl O
(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead
-x11 (Bind Abst) x12) H33))) in ((let H36 \def (f_equal C T (\lambda (e:
-C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow x12 |
-(CHead _ _ t) \Rightarrow t])) (CHead x11 (Bind Abst) x12) (CHead (CHead
+e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g
+e u0 t)))) P (\lambda (x10: C).(\lambda (x11: T).(\lambda (x12: T).(\lambda
+(_: (pc3 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort
+O)) (lift (S O) O x12) x4)).(\lambda (H32: (getl O (CHead (CHead (CSort O)
+(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead x10 (Bind Abbr)
+x11))).(\lambda (_: (ty3 g x10 x11 x12)).(let H34 \def (eq_ind C (CHead x10
+(Bind Abbr) x11) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop)
+with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K
+return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B return
+(\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False |
+Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead (CHead
(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (clear_gen_bind Abst
-(CHead (CSort O) (Bind Abst) (TSort O)) (CHead x11 (Bind Abst) x12) (TSort O)
+(CHead (CSort O) (Bind Abst) (TSort O)) (CHead x10 (Bind Abbr) x11) (TSort O)
(getl_gen_O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort
-O)) (CHead x11 (Bind Abst) x12) H33))) in (\lambda (H37: (eq C x11 (CHead
-(CSort O) (Bind Abst) (TSort O)))).(let H38 \def (eq_ind T x12 (\lambda (t:
-T).(ty3 g x11 t x13)) H34 (TSort O) H36) in (let H39 \def (eq_ind T x12
-(\lambda (t: T).(pc3 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind
-Abst) (TSort O)) (lift (S O) O t) x4)) H32 (TSort O) H36) in (let H40 \def
-(eq_ind C x11 (\lambda (c: C).(ty3 g c (TSort O) x13)) H38 (CHead (CSort O)
-(Bind Abst) (TSort O)) H37) in (and_ind (pc3 (CHead (CHead (CHead (CSort O)
-(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef
-(S (S O))) x0) (\forall (b: B).(\forall (u0: T).(pc3 (CHead (CHead (CHead
-(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst)
-(TLRef O)) (Bind b) u0) x5 x1))) P (\lambda (H41: (pc3 (CHead (CHead (CHead
-(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef
-O)) (TLRef (S (S O))) x0)).(\lambda (_: ((\forall (b: B).(\forall (u0:
-T).(pc3 (CHead (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind
-Abst) (TSort O)) (Bind Abst) (TLRef O)) (Bind b) u0) x5 x1))))).(let H43 \def
-(eq_ind T (lift (S O) O (TLRef O)) (\lambda (t: T).(pc3 (CHead (CHead (CHead
-(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef
-O)) (TLRef (S (S O))) t)) (pc3_t x0 (CHead (CHead (CHead (CSort O) (Bind
+O)) (CHead x10 (Bind Abbr) x11) H32))) in (False_ind P H34)))))))) H30))
+(\lambda (H30: (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_:
+T).(pc3 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
+(lift (S O) O u0) x4)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_:
+T).(getl O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort
+O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda
+(t: T).(ty3 g e u0 t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (u0:
+T).(\lambda (_: T).(pc3 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind
+Abst) (TSort O)) (lift (S O) O u0) x4)))) (\lambda (e: C).(\lambda (u0:
+T).(\lambda (_: T).(getl O (CHead (CHead (CSort O) (Bind Abst) (TSort O))
+(Bind Abst) (TSort O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda
+(u0: T).(\lambda (t: T).(ty3 g e u0 t)))) P (\lambda (x10: C).(\lambda (x11:
+T).(\lambda (x12: T).(\lambda (H31: (pc3 (CHead (CHead (CSort O) (Bind Abst)
+(TSort O)) (Bind Abst) (TSort O)) (lift (S O) O x11) x4)).(\lambda (H32:
+(getl O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
+(CHead x10 (Bind Abst) x11))).(\lambda (H33: (ty3 g x10 x11 x12)).(let H34
+\def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C)
+with [(CSort _) \Rightarrow x10 | (CHead c _ _) \Rightarrow c])) (CHead x10
+(Bind Abst) x11) (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
+(TSort O)) (clear_gen_bind Abst (CHead (CSort O) (Bind Abst) (TSort O))
+(CHead x10 (Bind Abst) x11) (TSort O) (getl_gen_O (CHead (CHead (CSort O)
+(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead x10 (Bind Abst) x11)
+H32))) in ((let H35 \def (f_equal C T (\lambda (e: C).(match e in C return
+(\lambda (_: C).T) with [(CSort _) \Rightarrow x11 | (CHead _ _ t)
+\Rightarrow t])) (CHead x10 (Bind Abst) x11) (CHead (CHead (CSort O) (Bind
+Abst) (TSort O)) (Bind Abst) (TSort O)) (clear_gen_bind Abst (CHead (CSort O)
+(Bind Abst) (TSort O)) (CHead x10 (Bind Abst) x11) (TSort O) (getl_gen_O
+(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead
+x10 (Bind Abst) x11) H32))) in (\lambda (H36: (eq C x10 (CHead (CSort O)
+(Bind Abst) (TSort O)))).(let H37 \def (eq_ind T x11 (\lambda (t: T).(ty3 g
+x10 t x12)) H33 (TSort O) H35) in (let H38 \def (eq_ind T x11 (\lambda (t:
+T).(pc3 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
+(lift (S O) O t) x4)) H31 (TSort O) H35) in (let H39 \def (eq_ind C x10
+(\lambda (c: C).(ty3 g c (TSort O) x12)) H37 (CHead (CSort O) (Bind Abst)
+(TSort O)) H36) in (and_ind (pc3 (CHead (CHead (CHead (CSort O) (Bind Abst)
+(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O)))
+x0) (\forall (b: B).(\forall (u0: T).(pc3 (CHead (CHead (CHead (CHead (CSort
+O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (Bind
+b) u0) x5 x1))) P (\lambda (H40: (pc3 (CHead (CHead (CHead (CSort O) (Bind
Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S
-O))) H41 (lift (S O) O (TLRef O)) (ex2_sym T (pr3 (CHead (CHead (CHead (CSort
-O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift
-(S O) O (TLRef O))) (pr3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort
-O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) x0) H22)) (TLRef (plus O (S
-O))) (lift_lref_ge O (S O) O (le_n O))) in (let H44 \def H43 in (ex2_ind T
-(\lambda (t: T).(pr3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O))
-(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) t)) (\lambda
-(t: T).(pr3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
-(TSort O)) (Bind Abst) (TLRef O)) (TLRef (S O)) t)) P (\lambda (x14:
-T).(\lambda (H45: (pr3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O))
-(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O)))
-x14)).(\lambda (H46: (pr3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort
-O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S O)) x14)).(let
-H47 \def (eq_ind_r T x14 (\lambda (t: T).(eq T (TLRef (S (S O))) t))
-(nf2_pr3_unfold (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind
-Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) x14 H45
-(nf2_lref_abst (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind
-Abst) (TSort O)) (Bind Abst) (TLRef O)) (CSort O) (TSort O) (S (S O))
-(getl_clear_bind Abst (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O))
-(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead (CHead (CSort O) (Bind
-Abst) (TSort O)) (Bind Abst) (TSort O)) (TLRef O) (clear_bind Abst (CHead
-(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (TLRef O))
-(CHead (CSort O) (Bind Abst) (TSort O)) (S O) (getl_head (Bind Abst) O (CHead
-(CSort O) (Bind Abst) (TSort O)) (CHead (CSort O) (Bind Abst) (TSort O))
-(getl_refl Abst (CSort O) (TSort O)) (TSort O))))) (TLRef (S O))
-(nf2_pr3_unfold (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind
-Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S O)) x14 H46 (nf2_lref_abst
+O))) x0)).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pc3 (CHead (CHead
+(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind
+Abst) (TLRef O)) (Bind b) u0) x5 x1))))).(let H42 \def (eq_ind T (lift (S O)
+O (TLRef O)) (\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst)
+(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O)))
+t)) (pc3_t x0 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind
+Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) H40 (lift (S O) O
+(TLRef O)) (ex2_sym T (pr3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort
+O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S O) O (TLRef O)))
+(pr3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort
+O)) (Bind Abst) (TLRef O)) x0) H21)) (TLRef (plus O (S O))) (lift_lref_ge O
+(S O) O (le_n O))) in (let H43 \def H42 in (ex2_ind T (\lambda (t: T).(pr3
(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
-(Bind Abst) (TLRef O)) (CHead (CSort O) (Bind Abst) (TSort O)) (TSort O) (S
-O) (getl_head (Bind Abst) O (CHead (CHead (CSort O) (Bind Abst) (TSort O))
-(Bind Abst) (TSort O)) (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind
-Abst) (TSort O)) (getl_refl Abst (CHead (CSort O) (Bind Abst) (TSort O))
-(TSort O)) (TLRef O))))) in (let H48 \def (eq_ind T (TLRef (S (S O)))
-(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow False | (TLRef n) \Rightarrow (match n in nat return (\lambda (_:
-nat).Prop) with [O \Rightarrow False | (S n0) \Rightarrow (match n0 in nat
-return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow
-True])]) | (THead _ _ _) \Rightarrow False])) I (TLRef (S O)) H47) in
-(False_ind P H48)))))) H44))))) (pc3_gen_abst (CHead (CHead (CHead (CSort O)
-(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef
-(S (S O))) x0 x5 x1 H7))))))) H35)))))))) H31)) (ty3_gen_lref g (CHead (CHead
-(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) x4 O H23)))))))
-H25)))))))) H18))))) H15)))))))) H11)) (ty3_gen_lref g (CHead (CHead (CHead
-(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef
-O)) x6 (S (S O)) H8))))))))) (ty3_gen_bind g Abst (CHead (CHead (CHead (CSort
-O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O))
-(TLRef (S (S O))) (TSort O) (THead (Bind Abst) x0 x1) H1)))))))) H3))
-(ty3_gen_lref g (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind
-Abst) (TSort O)) (Bind Abst) (TLRef O)) x0 O H2))))))) (ty3_gen_appl g (CHead
+(Bind Abst) (TLRef O)) (TLRef (S (S O))) t)) (\lambda (t: T).(pr3 (CHead
(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind
-Abst) (TLRef O)) (TLRef O) (THead (Bind Abst) (TLRef (S (S O))) (TSort O)) u
-H))))).
+Abst) (TLRef O)) (TLRef (S O)) t)) P (\lambda (x13: T).(\lambda (H44: (pr3
+(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
+(Bind Abst) (TLRef O)) (TLRef (S (S O))) x13)).(\lambda (H45: (pr3 (CHead
+(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind
+Abst) (TLRef O)) (TLRef (S O)) x13)).(let H46 \def (eq_ind_r T x13 (\lambda
+(t: T).(eq T (TLRef (S (S O))) t)) (nf2_pr3_unfold (CHead (CHead (CHead
+(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef
+O)) (TLRef (S (S O))) x13 H44 (nf2_lref_abst (CHead (CHead (CHead (CSort O)
+(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CSort
+O) (TSort O) (S (S O)) (getl_clear_bind Abst (CHead (CHead (CHead (CSort O)
+(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead
+(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (TLRef O)
+(clear_bind Abst (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
+(TSort O)) (TLRef O)) (CHead (CSort O) (Bind Abst) (TSort O)) (S O)
+(getl_head (Bind Abst) O (CHead (CSort O) (Bind Abst) (TSort O)) (CHead
+(CSort O) (Bind Abst) (TSort O)) (getl_refl Abst (CSort O) (TSort O)) (TSort
+O))))) (TLRef (S O)) (nf2_pr3_unfold (CHead (CHead (CHead (CSort O) (Bind
+Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S O))
+x13 H45 (nf2_lref_abst (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O))
+(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead (CSort O) (Bind Abst)
+(TSort O)) (TSort O) (S O) (getl_head (Bind Abst) O (CHead (CHead (CSort O)
+(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead (CHead (CSort O) (Bind
+Abst) (TSort O)) (Bind Abst) (TSort O)) (getl_refl Abst (CHead (CSort O)
+(Bind Abst) (TSort O)) (TSort O)) (TLRef O))))) in (let H47 \def (eq_ind T
+(TLRef (S (S O))) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef n) \Rightarrow (match n
+in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S n0)
+\Rightarrow (match n0 in nat return (\lambda (_: nat).Prop) with [O
+\Rightarrow False | (S _) \Rightarrow True])]) | (THead _ _ _) \Rightarrow
+False])) I (TLRef (S O)) H46) in (False_ind P H47)))))) H43)))))
+(pc3_gen_abst (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind
+Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) x0 x5 x1 H7)))))))
+H34)))))))) H30)) (ty3_gen_lref g (CHead (CHead (CSort O) (Bind Abst) (TSort
+O)) (Bind Abst) (TSort O)) x4 O H22))))))) H24)))))))) H17))))) H14))))))))
+H10)) (ty3_gen_lref g (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O))
+(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) x6 (S (S O)) H8)))))))
+(ty3_gen_bind g Abst (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O))
+(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) (TSort O)
+(THead (Bind Abst) x0 x1) H1)))))))) H3)) (ty3_gen_lref g (CHead (CHead
+(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst)
+(TLRef O)) x0 O H2))))))) (ty3_gen_appl g (CHead (CHead (CHead (CSort O)
+(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef
+O) (THead (Bind Abst) (TLRef (S (S O))) (TSort O)) u H))))).
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/ex2/defs".
include "C/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/ex2/props".
include "ex2/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/flt/defs".
include "C/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/flt/props".
include "flt/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/fsubst0/defs".
include "csubst0/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/fsubst0/fwd".
include "fsubst0/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/getl/clear".
include "getl/props.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/getl/dec".
include "getl/props.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/getl/defs".
include "drop/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/getl/drop".
include "getl/props.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/getl/flt".
include "getl/fwd.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/getl/fwd".
include "getl/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/getl/getl".
include "getl/drop.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/getl/props".
include "getl/fwd.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/iso/defs".
include "T/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/iso/fwd".
include "iso/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/iso/props".
include "iso/fwd.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/leq/asucc".
include "leq/props.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/leq/defs".
include "aplus/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/leq/fwd".
include "leq/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/leq/props".
include "leq/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/lift/defs".
include "T/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/lift/fwd".
include "lift/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/lift/props".
include "tlist/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/lift/tlt".
include "lift/fwd.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/lift1/defs".
include "lift/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/lift1/fwd".
include "lift1/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/lift1/props".
include "lift1/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/llt/defs".
include "A/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/llt/props".
include "llt/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/next_plus/defs".
include "G/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/next_plus/props".
include "next_plus/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/nf2/arity".
include "nf2/fwd.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/nf2/dec".
include "nf2/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/nf2/defs".
include "pr2/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/nf2/fwd".
include "nf2/defs.ma".
(THead (Flat f) u t) (THead (Flat f) u t2) (H (THead (Flat f) u t2)
(pr2_head_2 c u t t2 (Flat f) (pr2_cflat c t t2 H0 f u)))) in H1)))))))).
-theorem nf2_gen__aux:
+theorem nf2_gen__nf2_gen_aux:
\forall (b: B).(\forall (x: T).(\forall (u: T).(\forall (d: nat).((eq T
(THead (Bind b) u (lift (S O) d x)) x) \to (\forall (P: Prop).P)))))
\def
(subst0_refl u (lift (S O) O x) O H4 P)))) (\lambda (H2: (eq T t (lift (S O)
O x))).(let H3 \def (eq_ind T t (\lambda (t0: T).(\forall (t2: T).((pr2 c
(THead (Bind Abbr) u t0) t2) \to (eq T (THead (Bind Abbr) u t0) t2)))) H
-(lift (S O) O x) H2) in (nf2_gen__aux Abbr x u O (H3 x (pr2_free c (THead
-(Bind Abbr) u (lift (S O) O x)) x (pr0_zeta Abbr not_abbr_abst x x (pr0_refl
-x) u))) P))) H1))) H0))))))).
+(lift (S O) O x) H2) in (nf2_gen__nf2_gen_aux Abbr x u O (H3 x (pr2_free c
+(THead (Bind Abbr) u (lift (S O) O x)) x (pr0_zeta Abbr not_abbr_abst x x
+(pr0_refl x) u))) P))) H1))) H0))))))).
theorem nf2_gen_void:
\forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Bind Void) u
\def
\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: ((\forall (t2:
T).((pr2 c (THead (Bind Void) u (lift (S O) O t)) t2) \to (eq T (THead (Bind
-Void) u (lift (S O) O t)) t2))))).(\lambda (P: Prop).(nf2_gen__aux Void t u O
-(H t (pr2_free c (THead (Bind Void) u (lift (S O) O t)) t (pr0_zeta Void
-not_void_abst t t (pr0_refl t) u))) P))))).
+Void) u (lift (S O) O t)) t2))))).(\lambda (P: Prop).(nf2_gen__nf2_gen_aux
+Void t u O (H t (pr2_free c (THead (Bind Void) u (lift (S O) O t)) t
+(pr0_zeta Void not_void_abst t t (pr0_refl t) u))) P))))).
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/nf2/iso".
include "nf2/pr3.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/nf2/lift1".
include "nf2/props.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/nf2/pr3".
include "nf2/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/nf2/props".
include "nf2/defs.ma".
u x0 t x1 (refl_equal K (Bind Abst)) (H x0 H4) (H0 x1 (H5 Abst u))) t2
H3)))))) H2)))))))).
+theorem nf2_appls_lref:
+ \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (vs:
+TList).((nfs2 c vs) \to (nf2 c (THeads (Flat Appl) vs (TLRef i)))))))
+\def
+ \lambda (c: C).(\lambda (i: nat).(\lambda (H: (nf2 c (TLRef i))).(\lambda
+(vs: TList).(TList_ind (\lambda (t: TList).((nfs2 c t) \to (nf2 c (THeads
+(Flat Appl) t (TLRef i))))) (\lambda (_: True).H) (\lambda (t: T).(\lambda
+(t0: TList).(\lambda (H0: (((nfs2 c t0) \to (nf2 c (THeads (Flat Appl) t0
+(TLRef i)))))).(\lambda (H1: (land (nf2 c t) (nfs2 c t0))).(let H2 \def H1 in
+(and_ind (nf2 c t) (nfs2 c t0) (nf2 c (THead (Flat Appl) t (THeads (Flat
+Appl) t0 (TLRef i)))) (\lambda (H3: (nf2 c t)).(\lambda (H4: (nfs2 c
+t0)).(let H_y \def (H0 H4) in (\lambda (t2: T).(\lambda (H5: (pr2 c (THead
+(Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) t2)).(let H6 \def
+(pr2_gen_appl c t (THeads (Flat Appl) t0 (TLRef i)) t2 H5) in (or3_ind (ex3_2
+T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c t u2))) (\lambda (_: T).(\lambda (t3:
+T).(pr2 c (THeads (Flat Appl) t0 (TLRef i)) t3)))) (ex4_4 T T T T (\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THeads (Flat
+Appl) t0 (TLRef i)) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda
+(_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
+T).(pr2 c t u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
+(t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1
+t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(eq T (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (eq T (THead (Flat Appl) t
+(THeads (Flat Appl) t0 (TLRef i))) t2) (\lambda (H7: (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c t u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
+(THeads (Flat Appl) t0 (TLRef i)) t3))))).(ex3_2_ind T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c t u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
+(THeads (Flat Appl) t0 (TLRef i)) t3))) (eq T (THead (Flat Appl) t (THeads
+(Flat Appl) t0 (TLRef i))) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda
+(H8: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H9: (pr2 c t
+x0)).(\lambda (H10: (pr2 c (THeads (Flat Appl) t0 (TLRef i)) x1)).(eq_ind_r T
+(THead (Flat Appl) x0 x1) (\lambda (t1: T).(eq T (THead (Flat Appl) t (THeads
+(Flat Appl) t0 (TLRef i))) t1)) (let H11 \def (eq_ind_r T x1 (\lambda (t1:
+T).(pr2 c (THeads (Flat Appl) t0 (TLRef i)) t1)) H10 (THeads (Flat Appl) t0
+(TLRef i)) (H_y x1 H10)) in (eq_ind T (THeads (Flat Appl) t0 (TLRef i))
+(\lambda (t1: T).(eq T (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef
+i))) (THead (Flat Appl) x0 t1))) (let H12 \def (eq_ind_r T x0 (\lambda (t1:
+T).(pr2 c t t1)) H9 t (H3 x0 H9)) in (eq_ind T t (\lambda (t1: T).(eq T
+(THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) (THead (Flat Appl) t1
+(THeads (Flat Appl) t0 (TLRef i))))) (refl_equal T (THead (Flat Appl) t
+(THeads (Flat Appl) t0 (TLRef i)))) x0 (H3 x0 H9))) x1 (H_y x1 H10))) t2
+H8)))))) H7)) (\lambda (H7: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THeads (Flat Appl) t0 (TLRef i))
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t u2))))) (\lambda
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b:
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3))))))))).(ex4_4_ind T T T
+T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c t u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
+z1 t3))))))) (eq T (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i)))
+t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3:
+T).(\lambda (H8: (eq T (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst)
+x0 x1))).(\lambda (H9: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (_: (pr2
+c t x2)).(\lambda (_: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b)
+u) x1 x3))))).(eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda (t1: T).(eq T
+(THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) t1)) (TList_ind
+(\lambda (t1: TList).((nf2 c (THeads (Flat Appl) t1 (TLRef i))) \to ((eq T
+(THeads (Flat Appl) t1 (TLRef i)) (THead (Bind Abst) x0 x1)) \to (eq T (THead
+(Flat Appl) t (THeads (Flat Appl) t1 (TLRef i))) (THead (Bind Abbr) x2
+x3))))) (\lambda (_: (nf2 c (THeads (Flat Appl) TNil (TLRef i)))).(\lambda
+(H13: (eq T (THeads (Flat Appl) TNil (TLRef i)) (THead (Bind Abst) x0
+x1))).(let H14 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) x0
+x1) H13) in (False_ind (eq T (THead (Flat Appl) t (THeads (Flat Appl) TNil
+(TLRef i))) (THead (Bind Abbr) x2 x3)) H14)))) (\lambda (t1: T).(\lambda (t3:
+TList).(\lambda (_: (((nf2 c (THeads (Flat Appl) t3 (TLRef i))) \to ((eq T
+(THeads (Flat Appl) t3 (TLRef i)) (THead (Bind Abst) x0 x1)) \to (eq T (THead
+(Flat Appl) t (THeads (Flat Appl) t3 (TLRef i))) (THead (Bind Abbr) x2
+x3)))))).(\lambda (_: (nf2 c (THeads (Flat Appl) (TCons t1 t3) (TLRef
+i)))).(\lambda (H13: (eq T (THeads (Flat Appl) (TCons t1 t3) (TLRef i))
+(THead (Bind Abst) x0 x1))).(let H14 \def (eq_ind T (THead (Flat Appl) t1
+(THeads (Flat Appl) t3 (TLRef i))) (\lambda (ee: T).(match ee in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
+(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
+True])])) I (THead (Bind Abst) x0 x1) H13) in (False_ind (eq T (THead (Flat
+Appl) t (THeads (Flat Appl) (TCons t1 t3) (TLRef i))) (THead (Bind Abbr) x2
+x3)) H14))))))) t0 H_y H8) t2 H9))))))))) H7)) (\lambda (H7: (ex6_6 B T T T T
+T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq
+T (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind b) y1 z1)))))))) (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
+T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S
+O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
+y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
+b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THeads (Flat Appl) t0 (TLRef i))
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c t u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (eq T (THead
+(Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) t2) (\lambda (x0:
+B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4:
+T).(\lambda (x5: T).(\lambda (_: (not (eq B x0 Abst))).(\lambda (H9: (eq T
+(THeads (Flat Appl) t0 (TLRef i)) (THead (Bind x0) x1 x2))).(\lambda (H10:
+(eq T t2 (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4)
+x3)))).(\lambda (_: (pr2 c t x4)).(\lambda (_: (pr2 c x1 x5)).(\lambda (_:
+(pr2 (CHead c (Bind x0) x5) x2 x3)).(eq_ind_r T (THead (Bind x0) x5 (THead
+(Flat Appl) (lift (S O) O x4) x3)) (\lambda (t1: T).(eq T (THead (Flat Appl)
+t (THeads (Flat Appl) t0 (TLRef i))) t1)) (TList_ind (\lambda (t1:
+TList).((nf2 c (THeads (Flat Appl) t1 (TLRef i))) \to ((eq T (THeads (Flat
+Appl) t1 (TLRef i)) (THead (Bind x0) x1 x2)) \to (eq T (THead (Flat Appl) t
+(THeads (Flat Appl) t1 (TLRef i))) (THead (Bind x0) x5 (THead (Flat Appl)
+(lift (S O) O x4) x3)))))) (\lambda (_: (nf2 c (THeads (Flat Appl) TNil
+(TLRef i)))).(\lambda (H15: (eq T (THeads (Flat Appl) TNil (TLRef i)) (THead
+(Bind x0) x1 x2))).(let H16 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match
+ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
+(TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead
+(Bind x0) x1 x2) H15) in (False_ind (eq T (THead (Flat Appl) t (THeads (Flat
+Appl) TNil (TLRef i))) (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O
+x4) x3))) H16)))) (\lambda (t1: T).(\lambda (t3: TList).(\lambda (_: (((nf2 c
+(THeads (Flat Appl) t3 (TLRef i))) \to ((eq T (THeads (Flat Appl) t3 (TLRef
+i)) (THead (Bind x0) x1 x2)) \to (eq T (THead (Flat Appl) t (THeads (Flat
+Appl) t3 (TLRef i))) (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4)
+x3))))))).(\lambda (_: (nf2 c (THeads (Flat Appl) (TCons t1 t3) (TLRef
+i)))).(\lambda (H15: (eq T (THeads (Flat Appl) (TCons t1 t3) (TLRef i))
+(THead (Bind x0) x1 x2))).(let H16 \def (eq_ind T (THead (Flat Appl) t1
+(THeads (Flat Appl) t3 (TLRef i))) (\lambda (ee: T).(match ee in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
+(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
+True])])) I (THead (Bind x0) x1 x2) H15) in (False_ind (eq T (THead (Flat
+Appl) t (THeads (Flat Appl) (TCons t1 t3) (TLRef i))) (THead (Bind x0) x5
+(THead (Flat Appl) (lift (S O) O x4) x3))) H16))))))) t0 H_y H9) t2
+H10))))))))))))) H7)) H6))))))) H2)))))) vs)))).
+
theorem nf2_appl_lref:
\forall (c: C).(\forall (u: T).((nf2 c u) \to (\forall (i: nat).((nf2 c
(TLRef i)) \to (nf2 c (THead (Flat Appl) u (TLRef i)))))))
\def
- \lambda (c: C).(\lambda (u: T).(\lambda (H: ((\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 (t3: T).(pr2 c
-(TLRef i) t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2
-(THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c 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 (t3:
-T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1
-t3))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(eq T (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 (u0: T).(pr2 (CHead c (Bind b) u0) 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 in T 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 in T 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)))))))).
+ \lambda (c: C).(\lambda (u: T).(\lambda (H: (nf2 c u)).(\lambda (i:
+nat).(\lambda (H0: (nf2 c (TLRef i))).(let H_y \def (nf2_appls_lref c i H0
+(TCons u TNil)) in (H_y (conj (nf2 c u) True H I))))))).
theorem nf2_lref_abst:
\forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (i: nat).((getl i c
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/pc1/defs".
include "pr1/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/pc1/props".
include "pc1/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/pc3/dec".
include "ty3/arity_props.ma".
theorem pc3_dec:
\forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (t1: T).((ty3 g c
u1 t1) \to (\forall (u2: T).(\forall (t2: T).((ty3 g c u2 t2) \to (or (pc3 c
-u1 u2) ((pc3 c u1 u2) \to (\forall (P: Prop).P))))))))))
+u1 u2) ((pc3 c u1 u2) \to False)))))))))
\def
\lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda
(H: (ty3 g c u1 t1)).(\lambda (u2: T).(\lambda (t2: T).(\lambda (H0: (ty3 g c
u2 t2)).(let H_y \def (ty3_sn3 g c u1 t1 H) in (let H_y0 \def (ty3_sn3 g c u2
t2 H0) in (let H_x \def (nf2_sn3 c u1 H_y) in (let H1 \def H_x in (ex2_ind T
(\lambda (u: T).(pr3 c u1 u)) (\lambda (u: T).(nf2 c u)) (or (pc3 c u1 u2)
-((pc3 c u1 u2) \to (\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)))))))))))).
+((pc3 c u1 u2) \to False)) (\lambda (x: T).(\lambda (H2: (pr3 c u1
+x)).(\lambda (H3: (nf2 c x)).(let H_x0 \def (nf2_sn3 c u2 H_y0) in (let H4
+\def H_x0 in (ex2_ind T (\lambda (u: T).(pr3 c u2 u)) (\lambda (u: T).(nf2 c
+u)) (or (pc3 c u1 u2) ((pc3 c u1 u2) \to False)) (\lambda (x0: T).(\lambda
+(H5: (pr3 c u2 x0)).(\lambda (H6: (nf2 c x0)).(let H_x1 \def (term_dec x x0)
+in (let H7 \def H_x1 in (or_ind (eq T x x0) ((eq T x x0) \to (\forall (P:
+Prop).P)) (or (pc3 c u1 u2) ((pc3 c u1 u2) \to False)) (\lambda (H8: (eq T x
+x0)).(let H9 \def (eq_ind_r T x0 (\lambda (t: T).(nf2 c t)) H6 x H8) in (let
+H10 \def (eq_ind_r T x0 (\lambda (t: T).(pr3 c u2 t)) H5 x H8) in (or_introl
+(pc3 c u1 u2) ((pc3 c u1 u2) \to False) (pc3_pr3_t c u1 x H2 u2 H10)))))
+(\lambda (H8: (((eq T x x0) \to (\forall (P: Prop).P)))).(or_intror (pc3 c u1
+u2) ((pc3 c u1 u2) \to False) (\lambda (H9: (pc3 c u1 u2)).(let H10 \def H9
+in (ex2_ind T (\lambda (t: T).(pr3 c u1 t)) (\lambda (t: T).(pr3 c u2 t))
+False (\lambda (x1: T).(\lambda (H11: (pr3 c u1 x1)).(\lambda (H12: (pr3 c u2
+x1)).(let H_x2 \def (pr3_confluence c u2 x0 H5 x1 H12) in (let H13 \def H_x2
+in (ex2_ind T (\lambda (t: T).(pr3 c x0 t)) (\lambda (t: T).(pr3 c x1 t))
+False (\lambda (x2: T).(\lambda (H14: (pr3 c x0 x2)).(\lambda (H15: (pr3 c x1
+x2)).(let H_y1 \def (nf2_pr3_unfold c x0 x2 H14 H6) in (let H16 \def
+(eq_ind_r T x2 (\lambda (t: T).(pr3 c x1 t)) H15 x0 H_y1) in (let H17 \def
+(nf2_pr3_confluence c x H3 x0 H6 u1 H2) in (H8 (H17 (pr3_t x1 u1 c H11 x0
+H16)) False))))))) H13)))))) H10))))) H7)))))) H4)))))) H1)))))))))))).
theorem pc3_abst_dec:
\forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (t1: T).((ty3 g c
(\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)))))))))))
+\to False))))))))))
\def
\lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda
(H: (ty3 g c u1 t1)).(\lambda (u2: T).(\lambda (t2: T).(\lambda (H0: (ty3 g c
(\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)
+\to False))) (\lambda (x: T).(\lambda (H4: (pr3 c u1 x)).(\lambda (H5: (nf2 c
+x)).(let H_x0 \def (nf2_sn3 c u2 H2) in (let H6 \def H_x0 in (ex2_ind T
+(\lambda (u: T).(pr3 c u2 u)) (\lambda (u: T).(nf2 c u)) (or (ex4_2 T T
+(\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2 u))))
+(\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) t1)))
+(\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: T).(\lambda
+(v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind Abst) u2 u))
+\to False))) (\lambda (x0: T).(\lambda (H7: (pr3 c u2 x0)).(\lambda (H8: (nf2
+c x0)).(let H_x1 \def (abst_dec x x0) in (let H9 \def H_x1 in (or_ind (ex T
+(\lambda (t: T).(eq T x (THead (Bind Abst) x0 t)))) (\forall (t: T).((eq T x
+(THead (Bind Abst) x0 t)) \to (\forall (P: Prop).P))) (or (ex4_2 T T (\lambda
+(u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2 u)))) (\lambda (u:
+T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) t1))) (\lambda (_:
+T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: T).(\lambda (v2: T).(nf2 c
+v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind Abst) u2 u)) \to False)))
+(\lambda (H10: (ex T (\lambda (t: T).(eq T x (THead (Bind Abst) x0
+t))))).(ex_ind T (\lambda (t: T).(eq T x (THead (Bind Abst) x0 t))) (or
+(ex4_2 T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2
+u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u)
t1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_:
T).(\lambda (v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind
-Abst) u2 u)) \to (\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
+Abst) u2 u)) \to False))) (\lambda (x1: T).(\lambda (H11: (eq T x (THead
+(Bind Abst) x0 x1))).(let H12 \def (eq_ind T x (\lambda (t: T).(nf2 c t)) H5
+(THead (Bind Abst) x0 x1) H11) in (let H13 \def (eq_ind T x (\lambda (t:
+T).(pr3 c u1 t)) H4 (THead (Bind Abst) x0 x1) H11) in (let H_y \def
+(ty3_sred_pr3 c u1 (THead (Bind Abst) x0 x1) H13 g t1 H) in (or_introl (ex4_2
+T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2 u))))
+(\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) t1)))
+(\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: T).(\lambda
+(v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind Abst) u2 u))
+\to False)) (ex4_2_intro T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead
(Bind Abst) u2 u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind
Abst) v2 u) t1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda
-(_: T).(\lambda (v2: T).(nf2 c v2)))) (\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:
+(_: T).(\lambda (v2: T).(nf2 c v2))) x1 x0 (pc3_pr3_t c u1 (THead (Bind Abst)
+x0 x1) H13 (THead (Bind Abst) u2 x1) (pr3_head_12 c u2 x0 H7 (Bind Abst) x1
+x1 (pr3_refl (CHead c (Bind Abst) x0) x1))) H_y H7 H8))))))) H10)) (\lambda
+(H10: ((\forall (t: T).((eq T x (THead (Bind Abst) x0 t)) \to (\forall (P:
Prop).P))))).(or_intror (ex4_2 T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1
(THead (Bind Abst) u2 u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead
(Bind Abst) v2 u) t1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2)))
(\lambda (_: T).(\lambda (v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1
-(THead (Bind Abst) u2 u)) \to (\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 (u0: T).(pr3 (CHead c (Bind b) u0) x4 x6))))).(let H27 \def
-(eq_ind T x (\lambda (t: T).(\forall (t0: T).((eq T t (THead (Bind Abst) x0
-t0)) \to (\forall (P0: Prop).P0)))) 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_eq T (THead (Bind Abst) x0 x6) (THead (Bind
-Abst) x5 x6) (f_equal3 K T T T THead (Bind Abst) (Bind Abst) x0 x5 x6 x6
-(refl_equal K (Bind Abst)) (H32 (pr3_t x3 u2 c H20 x5 H25)) (refl_equal T
-x6))) P)))) H29))))))))) H23)))))))) H18))))))) (pr3_confluence c u1 x1 H13 x
-H4))))) H12))))))) H9)))))) H6)))))) H3)))))))))))).
+(THead (Bind Abst) u2 u)) \to False)) (\lambda (u: T).(\lambda (H11: (pc3 c
+u1 (THead (Bind Abst) u2 u))).(let H12 \def H11 in (ex2_ind T (\lambda (t:
+T).(pr3 c u1 t)) (\lambda (t: T).(pr3 c (THead (Bind Abst) u2 u) t)) False
+(\lambda (x1: T).(\lambda (H13: (pr3 c u1 x1)).(\lambda (H14: (pr3 c (THead
+(Bind Abst) u2 u) x1)).(ex2_ind T (\lambda (t: T).(pr3 c x1 t)) (\lambda (t:
+T).(pr3 c x t)) False (\lambda (x2: T).(\lambda (H15: (pr3 c x1 x2)).(\lambda
+(H16: (pr3 c x x2)).(let H_y \def (nf2_pr3_unfold c x x2 H16 H5) in (let H17
+\def (eq_ind_r T x2 (\lambda (t: T).(pr3 c x1 t)) H15 x H_y) in (let H18 \def
+(pr3_gen_abst c u2 u x1 H14) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3:
+T).(eq T x1 (THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_:
+T).(pr3 c u2 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
+(u0: T).(pr3 (CHead c (Bind b) u0) u t3))))) False (\lambda (x3: T).(\lambda
+(x4: T).(\lambda (H19: (eq T x1 (THead (Bind Abst) x3 x4))).(\lambda (H20:
+(pr3 c u2 x3)).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pr3 (CHead c
+(Bind b) u0) u x4))))).(let H22 \def (eq_ind T x1 (\lambda (t: T).(pr3 c t
+x)) H17 (THead (Bind Abst) x3 x4) H19) in (let H23 \def (pr3_gen_abst c x3 x4
+x H22) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead
+(Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c x3 u3)))
+(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr3 (CHead
+c (Bind b) u0) x4 t3))))) False (\lambda (x5: T).(\lambda (x6: T).(\lambda
+(H24: (eq T x (THead (Bind Abst) x5 x6))).(\lambda (H25: (pr3 c x3
+x5)).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pr3 (CHead c (Bind b)
+u0) x4 x6))))).(let H27 \def (eq_ind T x (\lambda (t: T).(\forall (t0:
+T).((eq T t (THead (Bind Abst) x0 t0)) \to (\forall (P: Prop).P)))) H10
+(THead (Bind Abst) x5 x6) H24) in (let H28 \def (eq_ind T x (\lambda (t:
+T).(nf2 c t)) H5 (THead (Bind Abst) x5 x6) H24) in (let H29 \def
+(nf2_gen_abst c x5 x6 H28) in (and_ind (nf2 c x5) (nf2 (CHead c (Bind Abst)
+x5) x6) False (\lambda (H30: (nf2 c x5)).(\lambda (_: (nf2 (CHead c (Bind
+Abst) x5) x6)).(let H32 \def (nf2_pr3_confluence c x0 H8 x5 H30 u2 H7) in
+(H27 x6 (sym_eq T (THead (Bind Abst) x0 x6) (THead (Bind Abst) x5 x6)
+(f_equal3 K T T T THead (Bind Abst) (Bind Abst) x0 x5 x6 x6 (refl_equal K
+(Bind Abst)) (H32 (pr3_t x3 u2 c H20 x5 H25)) (refl_equal T x6))) False))))
+H29))))))))) H23)))))))) H18))))))) (pr3_confluence c u1 x1 H13 x H4)))))
+H12)))))) H9)))))) H6)))))) H3)))))))))))).
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/pc3/defs".
include "pr3/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/pc3/fsubst0".
include "pc3/left.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/pc3/fwd".
include "pc3/props.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/pc3/left".
include "pc3/props.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/pc3/nf2".
include "pc3/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/pc3/pc1".
include "pc3/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/pc3/props".
include "pc3/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/pc3/subst1".
include "pc3/props.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/pc3/wcpr0".
include "pc3/props.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/pr0/dec".
include "pr0/fwd.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/pr0/defs".
include "subst0/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/pr0/fwd".
include "pr0/props.ma".
\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:
-T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (H2: ((\forall (x0:
-T).(\forall (x1: nat).((eq T u1 (lift h x1 x0)) \to (ex2 T (\lambda (t2:
-T).(eq T u2 (lift h x1 t2))) (\lambda (t2: T).(pr0 x0 t2)))))))).(\lambda
-(t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 t3)).(\lambda (H4: ((\forall
-(x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to (ex2 T (\lambda (t4:
-T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4)))))))).(\lambda (k:
-K).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H5: (eq T (THead k u1 t2)
-(lift h x1 x0))).(K_ind (\lambda (k0: K).((eq T (THead k0 u1 t2) (lift h x1
-x0)) \to (ex2 T (\lambda (t4: T).(eq T (THead k0 u2 t3) (lift h x1 t4)))
-(\lambda (t4: T).(pr0 x0 t4))))) (\lambda (b: B).(\lambda (H6: (eq T (THead
-(Bind b) u1 t2) (lift h x1 x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z:
-T).(eq T x0 (THead (Bind b) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T
-u1 (lift h x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (S x1)
-z)))) (ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 t3) (lift h x1 t4)))
-(\lambda (t4: T).(pr0 x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda
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-(Flat Appl) x2 t) t4)))) (ex2_ind T (\lambda (t4: T).(eq T t3 (lift h (S x1)
-t4))) (\lambda (t4: T).(pr0 x5 t4)) (ex2 T (\lambda (t4: T).(eq T (THead
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-(lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind b)
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-t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind b) x4 x5))
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-(lift h x1 (THead (Bind b) x7 (THead (Flat Appl) (lift (S O) O x8) x6)))
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+h x1 t4))) (\lambda (t4: T).(pr0 t t4)))) (ex3_2_ind T T (\lambda (y0:
+T).(\lambda (z: T).(eq T x3 (THead (Bind Abst) y0 z)))) (\lambda (y0:
+T).(\lambda (_: T).(eq T u (lift h x1 y0)))) (\lambda (_: T).(\lambda (z:
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+t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind Abst) x4 x5))
+t4))) (\lambda (x7: T).(\lambda (H_x0: (eq T v2 (lift h x1 x7))).(\lambda
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+(t4: T).(eq T (THead (Bind Abbr) t (lift h (S x1) x6)) (lift h x1 t4)))
+(\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind Abst) x4 x5)) t4))))
+(ex_intro2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) (lift h x1 x7) (lift h
+(S x1) x6)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2
+(THead (Bind Abst) x4 x5)) t4)) (THead (Bind Abbr) x7 x6) (sym_eq T (lift h
+x1 (THead (Bind Abbr) x7 x6)) (THead (Bind Abbr) (lift h x1 x7) (lift h (S
+x1) x6)) (lift_bind Abbr x7 x6 h x1)) (pr0_beta x4 x2 x7 H13 x5 x6 H12)) v2
+H_x0)))) (H2 x2 x1 H7)) t3 H_x)))) (H4 x5 (S x1) H11)) x3 H9))))))
+(lift_gen_bind Abst u t2 x3 h x1 H8)) x0 H6)))))) (lift_gen_flat Appl v1
+(THead (Bind Abst) u t2) x0 h x1 H5)))))))))))))) (\lambda (b: B).(\lambda
+(H1: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0
+v1 v2)).(\lambda (H3: ((\forall (x0: T).(\forall (x1: nat).((eq T v1 (lift h
+x1 x0)) \to (ex2 T (\lambda (t2: T).(eq T v2 (lift h x1 t2))) (\lambda (t2:
+T).(pr0 x0 t2)))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1
+u2)).(\lambda (H5: ((\forall (x0: T).(\forall (x1: nat).((eq T u1 (lift h x1
x0)) \to (ex2 T (\lambda (t2: T).(eq T u2 (lift h x1 t2))) (\lambda (t2:
T).(pr0 x0 t2)))))))).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2
-t3)).(\lambda (H4: ((\forall (x0: T).(\forall (x1: nat).((eq T t2 (lift h x1
+t3)).(\lambda (H7: ((\forall (x0: T).(\forall (x1: nat).((eq T t2 (lift h x1
x0)) \to (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4:
-T).(pr0 x0 t4)))))))).(\lambda (w: T).(\lambda (H5: (subst0 O u2 t3
-w)).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H6: (eq T (THead (Bind
-Abbr) u1 t2) (lift h x1 x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z:
-T).(eq T x0 (THead (Bind Abbr) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq
-T u1 (lift h x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (S
-x1) z)))) (ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) u2 w) (lift h x1
-t4))) (\lambda (t4: T).(pr0 x0 t4))) (\lambda (x2: T).(\lambda (x3:
-T).(\lambda (H7: (eq T x0 (THead (Bind Abbr) x2 x3))).(\lambda (H8: (eq T u1
-(lift h x1 x2))).(\lambda (H9: (eq T t2 (lift h (S x1) x3))).(eq_ind_r T
-(THead (Bind Abbr) x2 x3) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T
-(THead (Bind Abbr) u2 w) (lift h x1 t4))) (\lambda (t4: T).(pr0 t t4))))
-(ex2_ind T (\lambda (t4: T).(eq T t3 (lift h (S x1) t4))) (\lambda (t4:
-T).(pr0 x3 t4)) (ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) u2 w) (lift
-h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind Abbr) x2 x3) t4))) (\lambda
-(x4: T).(\lambda (H_x: (eq T t3 (lift h (S x1) x4))).(\lambda (H10: (pr0 x3
-x4)).(let H11 \def (eq_ind T t3 (\lambda (t: T).(subst0 O u2 t w)) H5 (lift h
-(S x1) x4) H_x) in (ex2_ind T (\lambda (t4: T).(eq T u2 (lift h x1 t4)))
-(\lambda (t4: T).(pr0 x2 t4)) (ex2 T (\lambda (t4: T).(eq T (THead (Bind
-Abbr) u2 w) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind Abbr) x2 x3)
-t4))) (\lambda (x5: T).(\lambda (H_x0: (eq T u2 (lift h x1 x5))).(\lambda
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-(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
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-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
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-(t4: T).(pr0 (THead (Bind Abbr) x2 x3) t4)) (THead (Bind Abbr) x5 x6) (sym_eq
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-(lift h (S (plus O x1)) x6)) (lift_bind Abbr x5 x6 h (plus O x1))) (pr0_delta
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-(lift_gen_bind Abbr u1 t2 x0 h x1 H6))))))))))))))) (\lambda (b: B).(\lambda
-(H1: (not (eq B b Abst))).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (pr0
-t2 t3)).(\lambda (H3: ((\forall (x0: T).(\forall (x1: nat).((eq T t2 (lift h
-x1 x0)) \to (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4:
-T).(pr0 x0 t4)))))))).(\lambda (u: T).(\lambda (x0: T).(\lambda (x1:
-nat).(\lambda (H4: (eq T (THead (Bind b) u (lift (S O) O t2)) (lift h x1
+T).(pr0 x0 t4)))))))).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H8: (eq T
+(THead (Flat Appl) v1 (THead (Bind b) u1 t2)) (lift h x1 x0))).(ex3_2_ind T T
+(\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Flat Appl) y0 z))))
+(\lambda (y0: T).(\lambda (_: T).(eq T v1 (lift h x1 y0)))) (\lambda (_:
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+(\lambda (t4: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O
+v2) t3)) (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4))) (\lambda (x2:
+T).(\lambda (x3: T).(\lambda (H9: (eq T x0 (THead (Flat Appl) x2
+x3))).(\lambda (H10: (eq T v1 (lift h x1 x2))).(\lambda (H11: (eq T (THead
+(Bind b) u1 t2) (lift h x1 x3))).(eq_ind_r T (THead (Flat Appl) x2 x3)
+(\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 (THead (Flat
+Appl) (lift (S O) O v2) t3)) (lift h x1 t4))) (\lambda (t4: T).(pr0 t t4))))
+(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x3 (THead (Bind b) y0
+z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u1 (lift h x1 y0)))) (\lambda
+(_: T).(\lambda (z: T).(eq T t2 (lift h (S x1) z)))) (ex2 T (\lambda (t4:
+T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (lift h
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+(H13: (eq T u1 (lift h x1 x4))).(\lambda (H14: (eq T t2 (lift h (S x1)
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+T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (lift h
+x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 t) t4)))) (ex2_ind T
+(\lambda (t4: T).(eq T t3 (lift h (S x1) t4))) (\lambda (t4: T).(pr0 x5 t4))
+(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S
+O) O v2) t3)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2
+(THead (Bind b) x4 x5)) t4))) (\lambda (x6: T).(\lambda (H_x: (eq T t3 (lift
+h (S x1) x6))).(\lambda (H15: (pr0 x5 x6)).(eq_ind_r T (lift h (S x1) x6)
+(\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 (THead (Flat
+Appl) (lift (S O) O v2) t)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead
+(Flat Appl) x2 (THead (Bind b) x4 x5)) t4)))) (ex2_ind T (\lambda (t4: T).(eq
+T u2 (lift h x1 t4))) (\lambda (t4: T).(pr0 x4 t4)) (ex2 T (\lambda (t4:
+T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (lift h (S
+x1) x6))) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead
+(Bind b) x4 x5)) t4))) (\lambda (x7: T).(\lambda (H_x0: (eq T u2 (lift h x1
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+(THead (Flat Appl) x2 (THead (Bind b) x4 x5)) t4)))) (ex2_ind T (\lambda (t4:
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+(t4: T).(eq T (THead (Bind b) (lift h x1 x7) (THead (Flat Appl) (lift (S O) O
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+Appl) x2 (THead (Bind b) x4 x5)) t4))) (\lambda (x8: T).(\lambda (H_x1: (eq T
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+(\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) (lift h x1 x7)
+(THead (Flat Appl) (lift (S O) O t) (lift h (S x1) x6))) (lift h x1 t4)))
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+(eq_ind T (lift h (plus (S O) x1) (lift (S O) O x8)) (\lambda (t: T).(ex2 T
+(\lambda (t4: T).(eq T (THead (Bind b) (lift h x1 x7) (THead (Flat Appl) t
+(lift h (S x1) x6))) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat
+Appl) x2 (THead (Bind b) x4 x5)) t4)))) (eq_ind T (lift h (S x1) (THead (Flat
+Appl) (lift (S O) O x8) x6)) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T
+(THead (Bind b) (lift h x1 x7) t) (lift h x1 t4))) (\lambda (t4: T).(pr0
+(THead (Flat Appl) x2 (THead (Bind b) x4 x5)) t4)))) (ex_intro2 T (\lambda
+(t4: T).(eq T (THead (Bind b) (lift h x1 x7) (lift h (S x1) (THead (Flat
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+Appl) (lift (S O) O x8) x6)) (sym_eq T (lift h x1 (THead (Bind b) x7 (THead
+(Flat Appl) (lift (S O) O x8) x6))) (THead (Bind b) (lift h x1 x7) (lift h (S
+x1) (THead (Flat Appl) (lift (S O) O x8) x6))) (lift_bind b x7 (THead (Flat
+Appl) (lift (S O) O x8) x6) h x1)) (pr0_upsilon b H1 x2 x8 H17 x4 x7 H16 x5
+x6 H15)) (THead (Flat Appl) (lift h (S x1) (lift (S O) O x8)) (lift h (S x1)
+x6)) (lift_flat Appl (lift (S O) O x8) x6 h (S x1))) (lift (S O) O (lift h x1
+x8)) (lift_d x8 h (S O) x1 O (le_O_n x1))) v2 H_x1)))) (H3 x2 x1 H10)) u2
+H_x0)))) (H5 x4 x1 H13)) t3 H_x)))) (H7 x5 (S x1) H14)) x3 H12))))))
+(lift_gen_bind b u1 t2 x3 h x1 H11)) x0 H9)))))) (lift_gen_flat Appl v1
+(THead (Bind b) u1 t2) x0 h x1 H8))))))))))))))))))) (\lambda (u1:
+T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (H2: ((\forall (x0:
+T).(\forall (x1: nat).((eq T u1 (lift h x1 x0)) \to (ex2 T (\lambda (t2:
+T).(eq T u2 (lift h x1 t2))) (\lambda (t2: T).(pr0 x0 t2)))))))).(\lambda
+(t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 t3)).(\lambda (H4: ((\forall
+(x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to (ex2 T (\lambda (t4:
+T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4)))))))).(\lambda (w:
+T).(\lambda (H5: (subst0 O u2 t3 w)).(\lambda (x0: T).(\lambda (x1:
+nat).(\lambda (H6: (eq T (THead (Bind Abbr) u1 t2) (lift h x1
x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Bind
-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:
+Abbr) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u1 (lift h x1 y0))))
+(\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (S x1) z)))) (ex2 T (\lambda
+(t4: T).(eq T (THead (Bind Abbr) u2 w) (lift h x1 t4))) (\lambda (t4: T).(pr0
+x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq T x0 (THead
+(Bind Abbr) x2 x3))).(\lambda (H8: (eq T u1 (lift h x1 x2))).(\lambda (H9:
+(eq T t2 (lift h (S x1) x3))).(eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda
+(t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) u2 w) (lift h x1
+t4))) (\lambda (t4: T).(pr0 t t4)))) (ex2_ind T (\lambda (t4: T).(eq T t3
+(lift h (S x1) t4))) (\lambda (t4: T).(pr0 x3 t4)) (ex2 T (\lambda (t4:
+T).(eq T (THead (Bind Abbr) u2 w) (lift h x1 t4))) (\lambda (t4: T).(pr0
+(THead (Bind Abbr) x2 x3) t4))) (\lambda (x4: T).(\lambda (H_x: (eq T t3
+(lift h (S x1) x4))).(\lambda (H10: (pr0 x3 x4)).(let H11 \def (eq_ind T t3
+(\lambda (t: T).(subst0 O u2 t w)) H5 (lift h (S x1) x4) H_x) in (ex2_ind T
+(\lambda (t4: T).(eq T u2 (lift h x1 t4))) (\lambda (t4: T).(pr0 x2 t4)) (ex2
+T (\lambda (t4: T).(eq T (THead (Bind Abbr) u2 w) (lift h x1 t4))) (\lambda
+(t4: T).(pr0 (THead (Bind Abbr) x2 x3) t4))) (\lambda (x5: T).(\lambda (H_x0:
+(eq T u2 (lift h x1 x5))).(\lambda (H12: (pr0 x2 x5)).(eq_ind_r T (lift h x1
+x5) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) t w)
+(lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind Abbr) x2 x3) t4)))) (let
+H13 \def (eq_ind T u2 (\lambda (t: T).(subst0 O t (lift h (S x1) x4) w)) H11
+(lift h x1 x5) H_x0) in (let H14 \def (refl_equal nat (S (plus O x1))) in
+(let H15 \def (eq_ind nat (S x1) (\lambda (n: nat).(subst0 O (lift h x1 x5)
+(lift h n x4) w)) H13 (S (plus O x1)) H14) in (ex2_ind T (\lambda (t4: T).(eq
+T w (lift h (S (plus O x1)) t4))) (\lambda (t4: T).(subst0 O x5 x4 t4)) (ex2
+T (\lambda (t4: T).(eq T (THead (Bind Abbr) (lift h x1 x5) w) (lift h x1
+t4))) (\lambda (t4: T).(pr0 (THead (Bind Abbr) x2 x3) t4))) (\lambda (x6:
+T).(\lambda (H16: (eq T w (lift h (S (plus O x1)) x6))).(\lambda (H17:
+(subst0 O x5 x4 x6)).(eq_ind_r T (lift h (S (plus O x1)) x6) (\lambda (t:
+T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) (lift h x1 x5) t) (lift h
+x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind Abbr) x2 x3) t4)))) (ex_intro2 T
+(\lambda (t4: T).(eq T (THead (Bind Abbr) (lift h x1 x5) (lift h (S (plus O
+x1)) x6)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind Abbr) x2 x3)
+t4)) (THead (Bind Abbr) x5 x6) (sym_eq T (lift h x1 (THead (Bind Abbr) x5
+x6)) (THead (Bind Abbr) (lift h x1 x5) (lift h (S (plus O x1)) x6))
+(lift_bind Abbr x5 x6 h (plus O x1))) (pr0_delta x2 x5 H12 x3 x4 H10 x6 H17))
+w H16)))) (subst0_gen_lift_lt x5 x4 w O h x1 H15))))) u2 H_x0)))) (H2 x2 x1
+H8)))))) (H4 x3 (S x1) H9)) x0 H7)))))) (lift_gen_bind Abbr u1 t2 x0 h x1
+H6))))))))))))))) (\lambda (b: B).(\lambda (H1: (not (eq B b Abst))).(\lambda
+(t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 t3)).(\lambda (H3: ((\forall
+(x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to (ex2 T (\lambda (t4:
+T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4)))))))).(\lambda (u:
+T).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H4: (eq T (THead (Bind b) u
+(lift (S O) O t2)) (lift h x1 x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda
+(z: T).(eq T x0 (THead (Bind b) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq
+T u (lift h x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift (S O) O t2)
+(lift h (S x1) z)))) (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4)))
+(\lambda (t4: T).(pr0 x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda
+(H5: (eq T x0 (THead (Bind b) x2 x3))).(\lambda (_: (eq T u (lift h x1
+x2))).(\lambda (H7: (eq T (lift (S O) O t2) (lift h (S x1) x3))).(eq_ind_r T
+(THead (Bind b) x2 x3) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T t3 (lift
+h x1 t4))) (\lambda (t4: T).(pr0 t t4)))) (let H8 \def (eq_ind_r nat (plus (S
+O) x1) (\lambda (n: nat).(eq nat (S x1) n)) (refl_equal nat (plus (S O) x1))
+(plus x1 (S O)) (plus_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
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/pr0/pr0".
include "pr0/fwd.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/pr0/props".
include "pr0/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/pr0/subst1".
include "pr0/props.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/pr1/defs".
include "pr0/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/pr1/pr1".
include "pr1/props.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/pr1/props".
include "pr1/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/pr2/clen".
include "pr2/props.ma".
\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 (c1: C).(getl i c1 (CHead
-d (Bind Abbr) u0))) H1 (CTail k u c) H4) in (let H_x \def (getl_gen_tail k
-Abbr u u0 d c i H5) in (let H6 \def H_x in (or_ind (ex2 C (\lambda (e: C).(eq
-C d (CTail k u e))) (\lambda (e: C).(getl i c (CHead e (Bind Abbr) u0))))
-(ex4 nat (\lambda (_: nat).(eq nat i (clen c))) (\lambda (_: nat).(eq K k
-(Bind Abbr))) (\lambda (_: nat).(eq T u u0)) (\lambda (n: nat).(eq C d (CSort
-n)))) (or (pr2 c t3 t) (ex3 T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda
-(t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0 t)))) (\lambda
-(H7: (ex2 C (\lambda (e: C).(eq C d (CTail k u e))) (\lambda (e: C).(getl i c
-(CHead e (Bind Abbr) u0))))).(ex2_ind C (\lambda (e: C).(eq C d (CTail k u
-e))) (\lambda (e: C).(getl i c (CHead e (Bind Abbr) u0))) (or (pr2 c t3 t)
-(ex3 T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0))
-(\lambda (t0: T).(subst0 (clen c) u t0 t)))) (\lambda (x: C).(\lambda (_: (eq
-C d (CTail k u x))).(\lambda (H9: (getl i c (CHead x (Bind Abbr)
-u0))).(or_introl (pr2 c t3 t) (ex3 T (\lambda (_: T).(eq K k (Bind Abbr)))
-(\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0 t)))
-(pr2_delta c x u0 i H9 t3 t4 H2 t H3))))) H7)) (\lambda (H7: (ex4 nat
-(\lambda (_: nat).(eq nat i (clen c))) (\lambda (_: nat).(eq K k (Bind
-Abbr))) (\lambda (_: nat).(eq T u u0)) (\lambda (n: nat).(eq C d (CSort
-n))))).(ex4_ind nat (\lambda (_: nat).(eq nat i (clen c))) (\lambda (_:
-nat).(eq K k (Bind Abbr))) (\lambda (_: nat).(eq T u u0)) (\lambda (n:
-nat).(eq C d (CSort n))) (or (pr2 c t3 t) (ex3 T (\lambda (_: T).(eq K k
-(Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c)
-u t0 t)))) (\lambda (x0: nat).(\lambda (H8: (eq nat i (clen c))).(\lambda
-(H9: (eq K k (Bind Abbr))).(\lambda (H10: (eq T u u0)).(\lambda (_: (eq C d
-(CSort x0))).(let H12 \def (eq_ind nat i (\lambda (n: nat).(subst0 n u0 t4
-t)) H3 (clen c) H8) in (let H13 \def (eq_ind_r T u0 (\lambda (t0: T).(subst0
-(clen c) t0 t4 t)) H12 u H10) in (eq_ind_r K (Bind Abbr) (\lambda (k0: K).(or
-(pr2 c t3 t) (ex3 T (\lambda (_: T).(eq K k0 (Bind Abbr))) (\lambda (t0:
-T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0 t))))) (or_intror (pr2
-c t3 t) (ex3 T (\lambda (_: T).(eq K (Bind Abbr) (Bind Abbr))) (\lambda (t0:
-T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0 t))) (ex3_intro T
-(\lambda (_: T).(eq K (Bind Abbr) (Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0))
-(\lambda (t0: T).(subst0 (clen c) u t0 t)) t4 (refl_equal K (Bind Abbr)) H2
-H13)) k H9)))))))) H7)) H6))))))))))))))) y t1 t2 H0))) H)))))).
+(\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(or (pr2 c t1 t2) (ex3 T
+(\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t: T).(pr0 t1 t)) (\lambda
+(t: T).(subst0 (clen c) u t t2))))) (\lambda (y: C).(\lambda (H0: (pr2 y t1
+t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0
+(CTail k u c)) \to (or (pr2 c t t0) (ex3 T (\lambda (_: T).(eq K k (Bind
+Abbr))) (\lambda (t3: T).(pr0 t t3)) (\lambda (t3: T).(subst0 (clen c) u t3
+t0)))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1:
+(pr0 t3 t4)).(\lambda (_: (eq C c0 (CTail k u c))).(or_introl (pr2 c t3 t4)
+(ex3 T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t: T).(pr0 t3 t))
+(\lambda (t: T).(subst0 (clen c) u t t4))) (pr2_free c t3 t4 H1)))))))
+(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda
+(H1: (getl i c0 (CHead d (Bind Abbr) u0))).(\lambda (t3: T).(\lambda (t4:
+T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: (subst0 i u0 t4
+t)).(\lambda (H4: (eq C c0 (CTail k u c))).(let H5 \def (eq_ind C c0 (\lambda
+(c1: C).(getl i c1 (CHead d (Bind Abbr) u0))) H1 (CTail k u c) H4) in (let
+H_x \def (getl_gen_tail k Abbr u u0 d c i H5) in (let H6 \def H_x in (or_ind
+(ex2 C (\lambda (e: C).(eq C d (CTail k u e))) (\lambda (e: C).(getl i c
+(CHead e (Bind Abbr) u0)))) (ex4 nat (\lambda (_: nat).(eq nat i (clen c)))
+(\lambda (_: nat).(eq K k (Bind Abbr))) (\lambda (_: nat).(eq T u u0))
+(\lambda (n: nat).(eq C d (CSort n)))) (or (pr2 c t3 t) (ex3 T (\lambda (_:
+T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0:
+T).(subst0 (clen c) u t0 t)))) (\lambda (H7: (ex2 C (\lambda (e: C).(eq C d
+(CTail k u e))) (\lambda (e: C).(getl i c (CHead e (Bind Abbr)
+u0))))).(ex2_ind C (\lambda (e: C).(eq C d (CTail k u e))) (\lambda (e:
+C).(getl i c (CHead e (Bind Abbr) u0))) (or (pr2 c t3 t) (ex3 T (\lambda (_:
+T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0:
+T).(subst0 (clen c) u t0 t)))) (\lambda (x: C).(\lambda (_: (eq C d (CTail k
+u x))).(\lambda (H9: (getl i c (CHead x (Bind Abbr) u0))).(or_introl (pr2 c
+t3 t) (ex3 T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3
+t0)) (\lambda (t0: T).(subst0 (clen c) u t0 t))) (pr2_delta c x u0 i H9 t3 t4
+H2 t H3))))) H7)) (\lambda (H7: (ex4 nat (\lambda (_: nat).(eq nat i (clen
+c))) (\lambda (_: nat).(eq K k (Bind Abbr))) (\lambda (_: nat).(eq T u u0))
+(\lambda (n: nat).(eq C d (CSort n))))).(ex4_ind nat (\lambda (_: nat).(eq
+nat i (clen c))) (\lambda (_: nat).(eq K k (Bind Abbr))) (\lambda (_:
+nat).(eq T u u0)) (\lambda (n: nat).(eq C d (CSort n))) (or (pr2 c t3 t) (ex3
+T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0))
+(\lambda (t0: T).(subst0 (clen c) u t0 t)))) (\lambda (x0: nat).(\lambda (H8:
+(eq nat i (clen c))).(\lambda (H9: (eq K k (Bind Abbr))).(\lambda (H10: (eq T
+u u0)).(\lambda (_: (eq C d (CSort x0))).(let H12 \def (eq_ind nat i (\lambda
+(n: nat).(subst0 n u0 t4 t)) H3 (clen c) H8) in (let H13 \def (eq_ind_r T u0
+(\lambda (t0: T).(subst0 (clen c) t0 t4 t)) H12 u H10) in (eq_ind_r K (Bind
+Abbr) (\lambda (k0: K).(or (pr2 c t3 t) (ex3 T (\lambda (_: T).(eq K k0 (Bind
+Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0
+t))))) (or_intror (pr2 c t3 t) (ex3 T (\lambda (_: T).(eq K (Bind Abbr) (Bind
+Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0
+t))) (ex3_intro T (\lambda (_: T).(eq K (Bind Abbr) (Bind Abbr))) (\lambda
+(t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0 t)) t4
+(refl_equal K (Bind Abbr)) H2 H13)) k H9)))))))) H7)) H6))))))))))))))) y t1
+t2 H0))) H)))))).
theorem pr2_gen_cbind:
\forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/pr2/defs".
include "pr0/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/pr2/fwd".
include "pr2/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/pr2/pr2".
include "pr2/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/pr2/props".
include "pr2/defs.ma".
\lambda (b: B).(\lambda (H: (not (eq B b Abbr))).(\lambda (c: C).(\lambda
(v1: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr2 (CHead c (Bind
b) v1) t1 t2)).(\lambda (v2: T).(insert_eq C (CHead c (Bind b) v1) (\lambda
-(c0: C).(pr2 c0 t1 t2)) (pr2 (CHead c (Bind b) v2) t1 t2) (\lambda (y:
-C).(\lambda (H1: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: C).(\lambda (t:
-T).(\lambda (t0: T).((eq C c0 (CHead c (Bind b) v1)) \to (pr2 (CHead c (Bind
-b) v2) t t0))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda
-(H2: (pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c (Bind b) v1))).(pr2_free
-(CHead c (Bind b) v2) t3 t4 H2)))))) (\lambda (c0: C).(\lambda (d:
-C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H2: (getl i c0 (CHead d (Bind
-Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3
-t4)).(\lambda (t: T).(\lambda (H4: (subst0 i u t4 t)).(\lambda (H5: (eq C c0
-(CHead c (Bind b) v1))).(let H6 \def (eq_ind C c0 (\lambda (c1: C).(getl i c1
-(CHead d (Bind Abbr) u))) H2 (CHead c (Bind b) v1) H5) in (nat_ind (\lambda
-(n: nat).((getl n (CHead c (Bind b) v1) (CHead d (Bind Abbr) u)) \to ((subst0
-n u t4 t) \to (pr2 (CHead c (Bind b) v2) t3 t)))) (\lambda (H7: (getl O
-(CHead c (Bind b) v1) (CHead d (Bind Abbr) u))).(\lambda (H8: (subst0 O u t4
-t)).(let H9 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
-(_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1]))
-(CHead d (Bind Abbr) u) (CHead c (Bind b) v1) (clear_gen_bind b c (CHead d
-(Bind Abbr) u) v1 (getl_gen_O (CHead c (Bind b) v1) (CHead d (Bind Abbr) u)
-H7))) in ((let H10 \def (f_equal C B (\lambda (e: C).(match e in C return
-(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _)
-\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0)
-\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u)
-(CHead c (Bind b) v1) (clear_gen_bind b c (CHead d (Bind Abbr) u) v1
+(c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 (CHead c (Bind b) v2) t1 t2))
+(\lambda (y: C).(\lambda (H1: (pr2 y t1 t2)).(pr2_ind (\lambda (c0:
+C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c (Bind b) v1)) \to (pr2
+(CHead c (Bind b) v2) t t0))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda
+(t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c (Bind b)
+v1))).(pr2_free (CHead c (Bind b) v2) t3 t4 H2)))))) (\lambda (c0:
+C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H2: (getl i c0
+(CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H3:
+(pr0 t3 t4)).(\lambda (t: T).(\lambda (H4: (subst0 i u t4 t)).(\lambda (H5:
+(eq C c0 (CHead c (Bind b) v1))).(let H6 \def (eq_ind C c0 (\lambda (c1:
+C).(getl i c1 (CHead d (Bind Abbr) u))) H2 (CHead c (Bind b) v1) H5) in
+(nat_ind (\lambda (n: nat).((getl n (CHead c (Bind b) v1) (CHead d (Bind
+Abbr) u)) \to ((subst0 n u t4 t) \to (pr2 (CHead c (Bind b) v2) t3 t))))
+(\lambda (H7: (getl O (CHead c (Bind b) v1) (CHead d (Bind Abbr)
+u))).(\lambda (H8: (subst0 O u t4 t)).(let H9 \def (f_equal C C (\lambda (e:
+C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d |
+(CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u) (CHead c (Bind b)
+v1) (clear_gen_bind b c (CHead d (Bind Abbr) u) v1 (getl_gen_O (CHead c (Bind
+b) v1) (CHead d (Bind Abbr) u) H7))) in ((let H10 \def (f_equal C B (\lambda
+(e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow
+Abbr | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with
+[(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind
+Abbr) u) (CHead c (Bind b) v1) (clear_gen_bind b c (CHead d (Bind Abbr) u) v1
(getl_gen_O (CHead c (Bind b) v1) (CHead d (Bind Abbr) u) H7))) in ((let H11
\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/pr2/subst1".
include "pr2/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/pr3/defs".
include "pr2/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/pr3/fwd".
include "pr3/props.ma".
(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)))).
+n) x)).(insert_eq T (TSort n) (\lambda (t: T).(pr3 c t x)) (\lambda (t:
+T).(eq T x t)) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(pr3_ind c (\lambda
+(t: T).(\lambda (t0: T).((eq T t (TSort n)) \to (eq T t0 t)))) (\lambda (t:
+T).(\lambda (_: (eq T t (TSort n))).(refl_equal T t))) (\lambda (t2:
+T).(\lambda (t1: T).(\lambda (H1: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda
+(_: (pr3 c t2 t3)).(\lambda (H3: (((eq T t2 (TSort n)) \to (eq T t3
+t2)))).(\lambda (H4: (eq T t1 (TSort n))).(let H5 \def (eq_ind T t1 (\lambda
+(t: T).(pr2 c t t2)) H1 (TSort n) H4) in (eq_ind_r T (TSort n) (\lambda (t:
+T).(eq T t3 t)) (let H6 \def (eq_ind T t2 (\lambda (t: T).((eq T t (TSort n))
+\to (eq T t3 t))) H3 (TSort n) (pr2_gen_sort c t2 n H5)) in (H6 (refl_equal T
+(TSort n)))) t1 H4))))))))) y x H0))) H)))).
theorem pr3_gen_abst:
\forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c
\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)))
+t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(ex3_2 T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 t2))))))) (\lambda (y:
+T).(\lambda (H0: (pr3 c y x)).(unintro T t1 (\lambda (t: T).((eq T y (THead
+(Bind Abst) u1 t)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x
+(THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2)))
(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead
-c (Bind b) u) 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 (_:
+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))))))))))) (\lambda (t: T).(\lambda (x0: T).(\lambda (x1: T).(\lambda
-(H1: (eq T t (THead (Bind Abst) x0 x1))).(ex3_2_intro T T (\lambda (u2:
-T).(\lambda (t2: T).(eq T t (THead (Bind Abst) u2 t2)))) (\lambda (u2:
-T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t2))))) x0 x1 H1
-(pr3_refl c x0) (\lambda (b: B).(\lambda (u: T).(pr3_refl (CHead c (Bind b)
-u) x1)))))))) (\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c t3
-t2)).(\lambda (t4: T).(\lambda (_: (pr3 c t2 t4)).(\lambda (H3: ((\forall
-(x0: T).(\forall (x1: T).((eq T t2 (THead (Bind Abst) x0 x1)) \to (ex3_2 T T
-(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) u2 t5))))
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda
-(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1
-t5))))))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t3 (THead
-(Bind Abst) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c t t2))
-H1 (THead (Bind Abst) x0 x1) H4) in (let H6 \def (pr2_gen_abst c x0 x1 t2 H5)
-in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind
-Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_:
-T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
-x1 t5))))) (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind
-Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_:
+x1 t2))))) x0 x1 H1 (pr3_refl c x0) (\lambda (b: B).(\lambda (u: T).(pr3_refl
+(CHead c (Bind b) u) x1)))))))) (\lambda (t2: T).(\lambda (t3: T).(\lambda
+(H1: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda (_: (pr3 c t2 t4)).(\lambda
+(H3: ((\forall (x0: T).(\forall (x1: T).((eq T t2 (THead (Bind Abst) x0 x1))
+\to (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst)
+u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_:
T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
-x1 t5)))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq T t2 (THead
-(Bind Abst) x2 x3))).(\lambda (H8: (pr2 c x0 x2)).(\lambda (H9: ((\forall (b:
-B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 x3))))).(let H10 \def (eq_ind
-T t2 (\lambda (t: T).(\forall (x4: T).(\forall (x5: T).((eq T t (THead (Bind
-Abst) x4 x5)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4
-(THead (Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2)))
-(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead
-c (Bind b) u) x5 t5)))))))))) H3 (THead (Bind Abst) x2 x3) H7) in (let H11
-\def (H10 x2 x3 (refl_equal T (THead (Bind Abst) x2 x3))) in (ex3_2_ind T T
+x1 t5))))))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t3
+(THead (Bind Abst) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c
+t t2)) H1 (THead (Bind Abst) x0 x1) H4) in (let H6 \def (pr2_gen_abst c x0 x1
+t2 H5) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead
+(Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2)))
+(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead
+c (Bind b) u) x1 t5))))) (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T
+t4 (THead (Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0
+u2))) (\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3
+(CHead c (Bind b) u) x1 t5)))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda
+(H7: (eq T t2 (THead (Bind Abst) x2 x3))).(\lambda (H8: (pr2 c x0
+x2)).(\lambda (H9: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
+x1 x3))))).(let H10 \def (eq_ind T t2 (\lambda (t: T).(\forall (x4:
+T).(\forall (x5: T).((eq T t (THead (Bind Abst) x4 x5)) \to (ex3_2 T T
(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) u2 t5))))
-(\lambda (u2: T).(\lambda (_: T).(pr3 c 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 (_: T).(pr3 c x4 u2))) (\lambda (_: T).(\lambda
+(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x5
+t5)))))))))) H3 (THead (Bind Abst) x2 x3) H7) in (let H11 \def (H10 x2 x3
+(refl_equal T (THead (Bind Abst) x2 x3))) in (ex3_2_ind T T (\lambda (u2:
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5))))) (ex3_2 T T
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) u2 t5))))
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda
+(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5))))))
+(\lambda (x4: T).(\lambda (x5: T).(\lambda (H12: (eq T t4 (THead (Bind Abst)
+x4 x5))).(\lambda (H13: (pr3 c x2 x4)).(\lambda (H14: ((\forall (b:
+B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 x5))))).(ex3_2_intro T T
(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) u2 t5))))
(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda
(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))
\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:
+t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2:
T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2:
-T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t
-t2)))) (pr3 c t x)))) (unintro T u1 (\lambda (t: T).(\forall (x0: T).((eq T y
-(THead (Flat Cast) t x0)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
-T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3
-c t u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x0 t2)))) (pr3 c x0 x)))))
-(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1:
-T).((eq T t (THead (Flat Cast) x0 x1)) \to (or (ex3_2 T T (\lambda (u2:
-T).(\lambda (t2: T).(eq T t0 (THead (Flat Cast) u2 t2)))) (\lambda (u2:
-T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x1
-t2)))) (pr3 c x1 t0))))))) (\lambda (t: T).(\lambda (x0: T).(\lambda (x1:
-T).(\lambda (H1: (eq T t (THead (Flat Cast) x0 x1))).(eq_ind_r T (THead (Flat
-Cast) x0 x1) (\lambda (t0: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
-T).(eq T t0 (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_:
-T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2)))) (pr3 c
-x1 t0))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead
-(Flat Cast) x0 x1) (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_:
-T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2)))) (pr3 c
-x1 (THead (Flat Cast) x0 x1)) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2:
-T).(eq T (THead (Flat Cast) x0 x1) (THead (Flat Cast) u2 t2)))) (\lambda (u2:
-T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x1
-t2))) x0 x1 (refl_equal T (THead (Flat Cast) x0 x1)) (pr3_refl c x0)
-(pr3_refl c x1))) t H1))))) (\lambda (t2: T).(\lambda (t3: T).(\lambda (H1:
-(pr2 c t3 t2)).(\lambda (t4: T).(\lambda (H2: (pr3 c t2 t4)).(\lambda (H3:
-((\forall (x0: T).(\forall (x1: T).((eq T t2 (THead (Flat Cast) x0 x1)) \to
-(or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast)
-u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_:
-T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4))))))).(\lambda (x0:
-T).(\lambda (x1: T).(\lambda (H4: (eq T t3 (THead (Flat Cast) x0 x1))).(let
-H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c t t2)) H1 (THead (Flat Cast) x0
-x1) H4) in (let H6 \def (pr2_gen_cast c x0 x1 t2 H5) in (or_ind (ex3_2 T T
-(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Flat Cast) u2 t5))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda
-(t5: T).(pr2 c x1 t5)))) (pr2 c x1 t2) (or (ex3_2 T T (\lambda (u2:
+T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1
+t2)))) (pr3 c t1 x))) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T
+t1 (\lambda (t: T).((eq T y (THead (Flat Cast) u1 t)) \to (or (ex3_2 T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda
+(t2: T).(pr3 c t t2)))) (pr3 c t x)))) (unintro T u1 (\lambda (t: T).(\forall
+(x0: T).((eq T y (THead (Flat Cast) t x0)) \to (or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x0
+t2)))) (pr3 c x0 x))))) (pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall
+(x0: T).(\forall (x1: T).((eq T t (THead (Flat Cast) x0 x1)) \to (or (ex3_2 T
+T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Cast) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda
+(t2: T).(pr3 c x1 t2)))) (pr3 c x1 t0))))))) (\lambda (t: T).(\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H1: (eq T t (THead (Flat Cast) x0
+x1))).(eq_ind_r T (THead (Flat Cast) x0 x1) (\lambda (t0: T).(or (ex3_2 T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Cast) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda
+(t2: T).(pr3 c x1 t2)))) (pr3 c x1 t0))) (or_introl (ex3_2 T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T (THead (Flat Cast) x0 x1) (THead (Flat Cast) u2
+t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_:
+T).(\lambda (t2: T).(pr3 c x1 t2)))) (pr3 c x1 (THead (Flat Cast) x0 x1))
+(ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Cast)
+x0 x1) (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c
+x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2))) x0 x1 (refl_equal T
+(THead (Flat Cast) x0 x1)) (pr3_refl c x0) (pr3_refl c x1))) t H1)))))
+(\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4:
+T).(\lambda (H2: (pr3 c t2 t4)).(\lambda (H3: ((\forall (x0: T).(\forall (x1:
+T).((eq T t2 (THead (Flat Cast) x0 x1)) \to (or (ex3_2 T T (\lambda (u2:
T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5)))) (\lambda (u2:
T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1
-t5)))) (pr3 c x1 t4)) (\lambda (H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t5:
+t5)))) (pr3 c x1 t4))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4:
+(eq T t3 (THead (Flat Cast) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t:
+T).(pr2 c t t2)) H1 (THead (Flat Cast) x0 x1) H4) in (let H6 \def
+(pr2_gen_cast c x0 x1 t2 H5) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda
+(t5: T).(eq T t2 (THead (Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr2 c x1 t5)))) (pr2 c
+x1 t2) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat
+Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_:
+T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4)) (\lambda (H7: (ex3_2 T T
+(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Flat Cast) u2 t5))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda
+(t5: T).(pr2 c x1 t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5:
T).(eq T t2 (THead (Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_:
-T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr2 c x1
-t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead
-(Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2)))
-(\lambda (_: T).(\lambda (t5: T).(pr2 c x1 t5))) (or (ex3_2 T T (\lambda (u2:
-T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5)))) (\lambda (u2:
-T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1
-t5)))) (pr3 c x1 t4)) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H8: (eq T
-t2 (THead (Flat Cast) x2 x3))).(\lambda (H9: (pr2 c x0 x2)).(\lambda (H10:
-(pr2 c x1 x3)).(let H11 \def (eq_ind T t2 (\lambda (t: T).(\forall (x4:
-T).(\forall (x5: T).((eq T t (THead (Flat Cast) x4 x5)) \to (or (ex3_2 T T
-(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5))))
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2))) (\lambda (_: T).(\lambda
-(t5: T).(pr3 c x5 t5)))) (pr3 c x5 t4)))))) H3 (THead (Flat Cast) x2 x3) H8)
-in (let H12 \def (H11 x2 x3 (refl_equal T (THead (Flat Cast) x2 x3))) in
-(or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat
-Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_:
-T).(\lambda (t5: T).(pr3 c x3 t5)))) (pr3 c x3 t4) (or (ex3_2 T T (\lambda
+T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr2 c x1 t5))) (or
+(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2
+t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_:
+T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4)) (\lambda (x2: T).(\lambda
+(x3: T).(\lambda (H8: (eq T t2 (THead (Flat Cast) x2 x3))).(\lambda (H9: (pr2
+c x0 x2)).(\lambda (H10: (pr2 c x1 x3)).(let H11 \def (eq_ind T t2 (\lambda
+(t: T).(\forall (x4: T).(\forall (x5: T).((eq T t (THead (Flat Cast) x4 x5))
+\to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat
+Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2))) (\lambda (_:
+T).(\lambda (t5: T).(pr3 c x5 t5)))) (pr3 c x5 t4)))))) H3 (THead (Flat Cast)
+x2 x3) H8) in (let H12 \def (H11 x2 x3 (refl_equal T (THead (Flat Cast) x2
+x3))) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead
+(Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2)))
+(\lambda (_: T).(\lambda (t5: T).(pr3 c x3 t5)))) (pr3 c x3 t4) (or (ex3_2 T
+T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5))))
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda
+(t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4)) (\lambda (H13: (ex3_2 T T (\lambda
(u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5)))) (\lambda (u2:
-T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1
-t5)))) (pr3 c x1 t4)) (\lambda (H13: (ex3_2 T T (\lambda (u2: T).(\lambda
-(t5: T).(eq T t4 (THead (Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_:
-T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x3
+T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x3
t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead
(Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2)))
(\lambda (_: T).(\lambda (t5: T).(pr3 c x3 t5))) (or (ex3_2 T T (\lambda (u2:
\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:
+(\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(\forall (e: C).((drop h d c e)
+\to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr3 e
+t1 t2)))))) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T t1 (\lambda
+(t: T).((eq T y (lift h d t)) \to (\forall (e: C).((drop h d c e) \to (ex2 T
+(\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr3 e t t2)))))))
+(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (x0: T).((eq T t (lift h
+d x0)) \to (\forall (e: C).((drop h d c e) \to (ex2 T (\lambda (t2: T).(eq T
+t0 (lift h d t2))) (\lambda (t2: T).(pr3 e x0 t2))))))))) (\lambda (t:
T).(\lambda (x0: T).(\lambda (H1: (eq T t (lift h d x0))).(\lambda (e:
C).(\lambda (_: (drop h d c e)).(ex_intro2 T (\lambda (t2: T).(eq T t (lift h
d t2))) (\lambda (t2: T).(pr3 e x0 t2)) x0 H1 (pr3_refl e x0))))))) (\lambda
(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:
+n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(pr3 c t x)) (\lambda (t:
+T).(or (eq T x t) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_:
+T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u:
+T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(v: T).(eq T x (lift (S n) O v)))))))) (\lambda (y: T).(\lambda (H0: (pr3 c y
+x)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n)) \to (or
+(eq T t0 t) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_:
+T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u:
+T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(v: T).(eq T t0 (lift (S n) O v)))))))))) (\lambda (t: T).(\lambda (_: (eq T
+t (TLRef n))).(or_introl (eq T t t) (ex3_3 C T T (\lambda (d: C).(\lambda (u:
T).(\lambda (_: T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d:
C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (v: T).(eq T t (lift (S n) O v)))))) 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 (_:
+(_: T).(\lambda (v: T).(eq T t (lift (S n) O v)))))) (refl_equal T t))))
+(\lambda (t2: T).(\lambda (t1: T).(\lambda (H1: (pr2 c t1 t2)).(\lambda (t3:
+T).(\lambda (H2: (pr3 c t2 t3)).(\lambda (H3: (((eq T t2 (TLRef n)) \to (or
+(eq T t3 t2) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_:
+T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u:
+T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(v: T).(eq T t3 (lift (S n) O v)))))))))).(\lambda (H4: (eq T t1 (TLRef
+n))).(let H5 \def (eq_ind T t1 (\lambda (t: T).(pr2 c t t2)) H1 (TLRef n) H4)
+in (eq_ind_r T (TLRef n) (\lambda (t: T).(or (eq T t3 t) (ex3_3 C T T
+(\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead d (Bind
+Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T t3 (lift (S n) O
+v)))))))) (let H6 \def (pr2_gen_lref c t2 n H5) in (or_ind (eq T t2 (TLRef
+n)) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr)
+u)))) (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift (S n) O u))))) (or (eq T
+t3 (TLRef n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_:
+T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u:
+T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(v: T).(eq T t3 (lift (S n) O v))))))) (\lambda (H7: (eq T t2 (TLRef
+n))).(let H8 \def (eq_ind T t2 (\lambda (t: T).((eq T t (TLRef n)) \to (or
+(eq T t3 t) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_:
+T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u:
+T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(v: T).(eq T t3 (lift (S n) O v))))))))) H3 (TLRef n) H7) in (let H9 \def
+(eq_ind T t2 (\lambda (t: T).(pr3 c t t3)) H2 (TLRef n) H7) in (H8
+(refl_equal T (TLRef n)))))) (\lambda (H7: (ex2_2 C T (\lambda (d:
+C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_:
+C).(\lambda (u: T).(eq T t2 (lift (S n) O u)))))).(ex2_2_ind C T (\lambda (d:
+C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_:
C).(\lambda (u: T).(eq T t2 (lift (S n) O u)))) (or (eq T t3 (TLRef n))
(ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead
d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u
v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T t3 (lift (S n) O
v))))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H8: (getl n c (CHead x0
(Bind Abbr) x1))).(\lambda (H9: (eq T t2 (lift (S n) O x1))).(let H10 \def
-(eq_ind T t2 (\lambda (t: T).(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:
+(eq_ind T t2 (\lambda (t: T).((eq T t (TLRef n)) \to (or (eq T t3 t) (ex3_3 C
+T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead d (Bind
+Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T t3 (lift (S n) O
+v))))))))) H3 (lift (S n) O x1) H9) in (let H11 \def (eq_ind T t2 (\lambda
+(t: T).(pr3 c t t3)) H2 (lift (S n) O x1) H9) in (let H12 \def (pr3_gen_lift
+c x1 t3 (S n) O H11 x0 (getl_drop Abbr c x0 x1 n H8)) in (ex2_ind T (\lambda
+(t4: T).(eq T t3 (lift (S n) O t4))) (\lambda (t4: T).(pr3 x0 x1 t4)) (or (eq
+T t3 (TLRef n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_:
+T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u:
+T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(v: T).(eq T t3 (lift (S n) O v))))))) (\lambda (x2: T).(\lambda (H13: (eq T
+t3 (lift (S n) O x2))).(\lambda (H14: (pr3 x0 x1 x2)).(or_intror (eq T t3
+(TLRef n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl
+n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v:
+T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T t3
+(lift (S n) O v)))))) (ex3_3_intro C T T (\lambda (d: C).(\lambda (u:
T).(\lambda (_: T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d:
C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (v: T).(eq T t3 (lift (S n) O v)))))) (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)))).
+(_: T).(\lambda (v: T).(eq T t3 (lift (S n) O v))))) x0 x1 x2 H8 H14 H13)))))
+H12)))))))) H7)) H6)) t1 H4))))))))) y x H0))) H)))).
theorem pr3_gen_void:
\forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c
\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 (_:
+t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 t2)))))) (pr3 (CHead c
+(Bind Void) u1) t1 (lift (S O) O x)))) (\lambda (y: T).(\lambda (H0: (pr3 c y
+x)).(unintro T t1 (\lambda (t: T).((eq T y (THead (Bind Void) u1 t)) \to (or
+(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2
+t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_:
+T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
+t t2)))))) (pr3 (CHead c (Bind Void) u1) t (lift (S O) O x))))) (unintro T u1
+(\lambda (t: T).(\forall (x0: T).((eq T y (THead (Bind Void) t x0)) \to (or
+(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2
+t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_:
+T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
+x0 t2)))))) (pr3 (CHead c (Bind Void) t) x0 (lift (S O) O x)))))) (pr3_ind c
+(\lambda (t: T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1: T).((eq T t
+(THead (Bind Void) x0 x1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T t0 (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_:
T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall
(u: T).(pr3 (CHead c (Bind b) u) x1 t2)))))) (pr3 (CHead c (Bind Void) x0) x1
-(lift (S O) O (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
+(lift (S O) O t0)))))))) (\lambda (t: T).(\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H1: (eq T t (THead (Bind Void) x0 x1))).(eq_ind_r T (THead (Bind
+Void) x0 x1) (\lambda (t0: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T t0 (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall
+(u: T).(pr3 (CHead c (Bind b) u) x1 t2)))))) (pr3 (CHead c (Bind Void) x0) x1
+(lift (S O) O t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T (THead (Bind Void) x0 x1) (THead (Bind Void) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t2)))))) (pr3 (CHead c
+(Bind Void) x0) x1 (lift (S O) O (THead (Bind Void) x0 x1))) (ex3_2_intro T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind Void) x0 x1) (THead
+(Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))
+(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead
+c (Bind b) u) x1 t2))))) x0 x1 (refl_equal T (THead (Bind Void) x0 x1))
+(pr3_refl c x0) (\lambda (b: B).(\lambda (u: T).(pr3_refl (CHead c (Bind b)
+u) x1))))) t H1))))) (\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c
+t3 t2)).(\lambda (t4: T).(\lambda (H2: (pr3 c t2 t4)).(\lambda (H3: ((\forall
(x0: T).(\forall (x1: T).((eq T t2 (THead (Bind Void) x0 x1)) \to (or (ex3_2
T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5))))
(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda
\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
+t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3
+(CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S
+O) O x)))) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T t1 (\lambda
+(t: T).((eq T y (THead (Bind Abbr) u1 t)) \to (or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3
+(CHead c (Bind Abbr) u1) t t2)))) (pr3 (CHead c (Bind Abbr) u1) t (lift (S O)
+O x))))) (unintro T u1 (\lambda (t: T).(\forall (x0: T).((eq T y (THead (Bind
+Abbr) t x0)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x
+(THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2)))
+(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) t) x0 t2)))) (pr3
+(CHead c (Bind Abbr) t) x0 (lift (S O) O x)))))) (pr3_ind c (\lambda (t:
+T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1: T).((eq T t (THead (Bind
+Abbr) x0 x1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0
+(THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))
+(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) x0) x1 t2)))) (pr3
+(CHead c (Bind Abbr) x0) x1 (lift (S O) O t0)))))))) (\lambda (t: T).(\lambda
+(x0: T).(\lambda (x1: T).(\lambda (H1: (eq T t (THead (Bind Abbr) x0
x1))).(eq_ind_r T (THead (Bind Abbr) x0 x1) (\lambda (t0: T).(or (ex3_2 T T
(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2))))
(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda
\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:
+t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(or3 (ex3_2 T T (\lambda
+(u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1
+t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) x))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2)))))
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c t1
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
+z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(pr3 c t1 (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))
+x))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b)
+y2) z1 z2)))))))))) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T t1
+(\lambda (t: T).((eq T y (THead (Flat Appl) u1 t)) \to (or3 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda
+(t2: T).(pr3 c t t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) x)))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1
+u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(pr3 c t (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1:
T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3
(CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b:
B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x0 (THead
+(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)) 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 (_:
+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)))))))
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/pr3/iso".
include "pr3/fwd.ma".
(lifts (S O) O vs) t)) u2))))))))))
\def
\lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (vs:
-TList).(tlist_ind_rew (\lambda (t: TList).(\forall (u: T).(\forall (t0:
+TList).(tlist_ind_rev (\lambda (t: TList).(\forall (u: T).(\forall (t0:
T).(let u1 \def (THeads (Flat Appl) t (THead (Bind b) u t0)) in (\forall (c:
C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P:
Prop).P))) \to (pr3 c (THead (Bind b) u (THeads (Flat Appl) (lifts (S O) O t)
(lift (S O) O t) t0)) (\lambda (t1: T).(pr3 c (THead (Bind b) u t1) u2)) (let
H3 \def (eq_ind T (THeads (Flat Appl) (TApp ts t) (THead (Bind b) u t0))
(\lambda (t1: T).(pr3 c t1 u2)) H1 (THeads (Flat Appl) ts (THead (Flat Appl)
-t (THead (Bind b) u t0))) (theads_tapp (Flat Appl) ts t (THead (Bind b) u
-t0))) in (let H4 \def (eq_ind T (THeads (Flat Appl) (TApp ts t) (THead (Bind
+t (THead (Bind b) u t0))) (theads_tapp (Flat Appl) t (THead (Bind b) u t0)
+ts)) in (let H4 \def (eq_ind T (THeads (Flat Appl) (TApp ts t) (THead (Bind
b) u t0)) (\lambda (t1: T).((iso t1 u2) \to (\forall (P: Prop).P))) H2
(THeads (Flat Appl) ts (THead (Flat Appl) t (THead (Bind b) u t0)))
-(theads_tapp (Flat Appl) ts t (THead (Bind b) u t0))) in (TList_ind (\lambda
+(theads_tapp (Flat Appl) t (THead (Bind b) u t0) ts)) in (TList_ind (\lambda
(t1: TList).(((\forall (u0: T).(\forall (t2: T).(\forall (c0: C).(\forall
(u3: T).((pr3 c0 (THeads (Flat Appl) t1 (THead (Bind b) u0 t2)) u3) \to
((((iso (THeads (Flat Appl) t1 (THead (Bind b) u0 t2)) u3) \to (\forall (P:
(Bind b) u t0))) (THeads (Flat Appl) ts0 (THead (Bind b) u (THead (Flat Appl)
(lift (S O) O t) t0))) (Flat Appl)) u2 H8) P)))))))))) ts H0 H3 H4))) (THeads
(Flat Appl) (TApp (lifts (S O) O ts) (lift (S O) O t)) t0) (theads_tapp (Flat
-Appl) (lifts (S O) O ts) (lift (S O) O t) t0)) (lifts (S O) O (TApp ts t))
+Appl) (lift (S O) O t) t0 (lifts (S O) O ts))) (lifts (S O) O (TApp ts t))
(lifts_tapp (S O) O t ts))))))))))) vs))).
theorem pr3_iso_beta:
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/pr3/pr1".
include "pr3/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/pr3/pr3".
include "pr3/props.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/pr3/props".
include "pr3/pr1.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/pr3/subst1".
include "pr3/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/pr3/wcpr0".
include "pr3/props.ma".
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "../Base-1/theory.ma".
+
+alias id "and_ind" = "cic:/Coq/Init/Logic/and_ind.con".
+alias id "bool_ind" = "cic:/Coq/Init/Datatypes/bool_ind.con".
+alias id "ex" = "cic:/Coq/Init/Logic/ex.ind#xpointer(1/1)".
+alias id "ex_ind" = "cic:/Coq/Init/Logic/ex_ind.con".
+alias id "ex_intro" = "cic:/Coq/Init/Logic/ex.ind#xpointer(1/1/1)".
+alias id "f_equal2" = "cic:/Coq/Init/Logic/f_equal2.con".
+alias id "f_equal3" = "cic:/Coq/Init/Logic/f_equal3.con".
+alias id "le_antisym" = "cic:/Coq/Arith/Le/le_antisym.con".
+alias id "le_lt_trans" = "cic:/Coq/Arith/Lt/le_lt_trans.con".
+alias id "le_plus_trans" = "cic:/Coq/Arith/Plus/le_plus_trans.con".
+alias id "lt_le_trans" = "cic:/Coq/Arith/Lt/lt_le_trans.con".
+alias id "lt_n_Sn" = "cic:/Coq/Arith/Lt/lt_n_Sn.con".
+alias id "lt_S_n" = "cic:/Coq/Arith/Lt/lt_S_n.con".
+alias id "lt_trans" = "cic:/Coq/Arith/Lt/lt_trans.con".
+alias id "lt_wf_ind" = "cic:/Coq/Arith/Wf_nat/lt_wf_ind.con".
+alias id "minus_n_n" = "cic:/Coq/Arith/Minus/minus_n_n.con".
+alias id "minus_Sn_m" = "cic:/Coq/Arith/Minus/minus_Sn_m.con".
+alias id "plus_le_lt_compat" = "cic:/Coq/Arith/Plus/plus_le_lt_compat.con".
+alias id "plus_lt_compat" = "cic:/Coq/Arith/Plus/plus_lt_compat.con".
+alias id "plus_lt_compat_r" = "cic:/Coq/Arith/Plus/plus_lt_compat_r.con".
+alias id "plus_lt_le_compat" = "cic:/Coq/Arith/Plus/plus_lt_le_compat.con".
+alias id "plus_n_O" = "cic:/Coq/Init/Peano/plus_n_O.con".
+alias id "plus_Snm_nSm" = "cic:/Coq/Arith/Plus/plus_Snm_nSm.con".
+alias id "S_pred" = "cic:/Coq/Arith/Lt/S_pred.con".
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-
-
-include "theory.ma".
-
-alias id "and_ind" = "cic:/Coq/Init/Logic/and_ind.con".
-alias id "bool_ind" = "cic:/Coq/Init/Datatypes/bool_ind.con".
-alias id "ex" = "cic:/Coq/Init/Logic/ex.ind#xpointer(1/1)".
-alias id "ex_ind" = "cic:/Coq/Init/Logic/ex_ind.con".
-alias id "ex_intro" = "cic:/Coq/Init/Logic/ex.ind#xpointer(1/1/1)".
-alias id "f_equal2" = "cic:/Coq/Init/Logic/f_equal2.con".
-alias id "f_equal3" = "cic:/Coq/Init/Logic/f_equal3.con".
-alias id "le_antisym" = "cic:/Coq/Arith/Le/le_antisym.con".
-alias id "le_lt_trans" = "cic:/Coq/Arith/Lt/le_lt_trans.con".
-alias id "le_plus_trans" = "cic:/Coq/Arith/Plus/le_plus_trans.con".
-alias id "lt_le_trans" = "cic:/Coq/Arith/Lt/lt_le_trans.con".
-alias id "lt_n_Sn" = "cic:/Coq/Arith/Lt/lt_n_Sn.con".
-alias id "lt_S_n" = "cic:/Coq/Arith/Lt/lt_S_n.con".
-alias id "lt_trans" = "cic:/Coq/Arith/Lt/lt_trans.con".
-alias id "lt_wf_ind" = "cic:/Coq/Arith/Wf_nat/lt_wf_ind.con".
-alias id "minus_n_n" = "cic:/Coq/Arith/Minus/minus_n_n.con".
-alias id "minus_Sn_m" = "cic:/Coq/Arith/Minus/minus_Sn_m.con".
-alias id "plus_le_lt_compat" = "cic:/Coq/Arith/Plus/plus_le_lt_compat.con".
-alias id "plus_lt_compat" = "cic:/Coq/Arith/Plus/plus_lt_compat.con".
-alias id "plus_lt_compat_r" = "cic:/Coq/Arith/Plus/plus_lt_compat_r.con".
-alias id "plus_lt_le_compat" = "cic:/Coq/Arith/Plus/plus_lt_le_compat.con".
-alias id "plus_n_O" = "cic:/Coq/Init/Peano/plus_n_O.con".
-alias id "plus_Snm_nSm" = "cic:/Coq/Arith/Plus/plus_Snm_nSm.con".
-alias id "S_pred" = "cic:/Coq/Arith/Lt/S_pred.con".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/r/defs".
include "T/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/r/props".
include "r/defs.ma".
baseuri=cic:/matita/LAMBDA-TYPES/LambdaDelta-1
-include_paths= ../Base-1
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/s/defs".
include "T/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/s/props".
include "s/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/sc3/arity".
include "csubc/arity.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/sc3/defs".
include "sn3/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/sc3/props".
include "sc3/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/sn3/defs".
include "pr3/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/sn3/fwd".
include "sn3/defs.ma".
\def
\lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H:
(sn3 c (THead (Bind b) u t))).(insert_eq T (THead (Bind b) u t) (\lambda (t0:
-T).(sn3 c t0)) (land (sn3 c u) (sn3 (CHead c (Bind b) u) t)) (\lambda (y:
-T).(\lambda (H0: (sn3 c y)).(unintro T t (\lambda (t0: T).((eq T y (THead
-(Bind b) u t0)) \to (land (sn3 c u) (sn3 (CHead c (Bind b) u) t0)))) (unintro
-T u (\lambda (t0: T).(\forall (x: T).((eq T y (THead (Bind b) t0 x)) \to
-(land (sn3 c t0) (sn3 (CHead c (Bind b) t0) x))))) (sn3_ind c (\lambda (t0:
-T).(\forall (x: T).(\forall (x0: T).((eq T t0 (THead (Bind b) x x0)) \to
+T).(sn3 c t0)) (\lambda (_: T).(land (sn3 c u) (sn3 (CHead c (Bind b) u) t)))
+(\lambda (y: T).(\lambda (H0: (sn3 c y)).(unintro T t (\lambda (t0: T).((eq T
+y (THead (Bind b) u t0)) \to (land (sn3 c u) (sn3 (CHead c (Bind b) u) t0))))
+(unintro T u (\lambda (t0: T).(\forall (x: T).((eq T y (THead (Bind b) t0 x))
+\to (land (sn3 c t0) (sn3 (CHead c (Bind b) t0) x))))) (sn3_ind c (\lambda
+(t0: T).(\forall (x: T).(\forall (x0: T).((eq T t0 (THead (Bind b) x x0)) \to
(land (sn3 c x) (sn3 (CHead c (Bind b) x) x0)))))) (\lambda (t1: T).(\lambda
(H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3
c t1 t2) \to (sn3 c t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2)
\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 (t0: T).(\forall (t2:
-T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to
-(\forall (x1: T).(\forall (x2: T).((eq T t2 (THead (Flat f) x1 x2)) \to (land
-(sn3 c x1) (sn3 c x2))))))))) H2 (THead (Flat f) x x0) H3) in (let H5 \def
-(eq_ind T t1 (\lambda (t0: T).(\forall (t2: T).((((eq T t0 t2) \to (\forall
-(P: Prop).P))) \to ((pr3 c t0 t2) \to (sn3 c t2))))) H1 (THead (Flat f) x x0)
-H3) in (conj (sn3 c x) (sn3 c x0) (sn3_sing c x (\lambda (t2: T).(\lambda
-(H6: (((eq T x t2) \to (\forall (P: Prop).P)))).(\lambda (H7: (pr3 c x
-t2)).(let H8 \def (H4 (THead (Flat f) t2 x0) (\lambda (H8: (eq T (THead (Flat
-f) x x0) (THead (Flat f) t2 x0))).(\lambda (P: Prop).(let H9 \def (f_equal T
-T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t0 _) \Rightarrow t0]))
-(THead (Flat f) x x0) (THead (Flat f) t2 x0) H8) in (let H10 \def (eq_ind_r T
-t2 (\lambda (t0: T).(pr3 c x t0)) H7 x H9) in (let H11 \def (eq_ind_r T t2
-(\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0))) H6 x H9) in (H11
-(refl_equal T x) P)))))) (pr3_head_12 c x t2 H7 (Flat f) x0 x0 (pr3_refl
-(CHead c (Flat f) t2) x0)) t2 x0 (refl_equal T (THead (Flat f) t2 x0))) in
-(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 (H8: (eq T (THead (Flat f) x x0)
-(THead (Flat f) x t2))).(\lambda (P: Prop).(let H9 \def (f_equal T T (\lambda
-(e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0
-| (TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat f)
-x x0) (THead (Flat f) x t2) H8) in (let H10 \def (eq_ind_r T t2 (\lambda (t0:
+T).(sn3 c t0)) (\lambda (_: T).(land (sn3 c u) (sn3 c t))) (\lambda (y:
+T).(\lambda (H0: (sn3 c y)).(unintro T t (\lambda (t0: T).((eq T y (THead
+(Flat f) u t0)) \to (land (sn3 c u) (sn3 c t0)))) (unintro T u (\lambda (t0:
+T).(\forall (x: T).((eq T y (THead (Flat f) t0 x)) \to (land (sn3 c t0) (sn3
+c x))))) (sn3_ind c (\lambda (t0: T).(\forall (x: T).(\forall (x0: T).((eq T
+t0 (THead (Flat f) x x0)) \to (land (sn3 c x) (sn3 c x0)))))) (\lambda (t1:
+T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P:
+Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H2: ((\forall
+(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to
+(\forall (x: T).(\forall (x0: T).((eq T t2 (THead (Flat f) x x0)) \to (land
+(sn3 c x) (sn3 c x0)))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3:
+(eq T t1 (THead (Flat f) x x0))).(let H4 \def (eq_ind T t1 (\lambda (t0:
+T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c
+t0 t2) \to (\forall (x1: T).(\forall (x2: T).((eq T t2 (THead (Flat f) x1
+x2)) \to (land (sn3 c x1) (sn3 c x2))))))))) H2 (THead (Flat f) x x0) H3) in
+(let H5 \def (eq_ind T t1 (\lambda (t0: T).(\forall (t2: T).((((eq T t0 t2)
+\to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (sn3 c t2))))) H1 (THead
+(Flat f) x x0) H3) in (conj (sn3 c x) (sn3 c x0) (sn3_sing c x (\lambda (t2:
+T).(\lambda (H6: (((eq T x t2) \to (\forall (P: Prop).P)))).(\lambda (H7:
+(pr3 c x t2)).(let H8 \def (H4 (THead (Flat f) t2 x0) (\lambda (H8: (eq T
+(THead (Flat f) x x0) (THead (Flat f) t2 x0))).(\lambda (P: Prop).(let H9
+\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
+with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t0 _)
+\Rightarrow t0])) (THead (Flat f) x x0) (THead (Flat f) t2 x0) H8) in (let
+H10 \def (eq_ind_r T t2 (\lambda (t0: T).(pr3 c x t0)) H7 x H9) in (let H11
+\def (eq_ind_r T t2 (\lambda (t0: T).((eq T x t0) \to (\forall (P0:
+Prop).P0))) H6 x H9) in (H11 (refl_equal T x) P)))))) (pr3_head_12 c x t2 H7
+(Flat f) x0 x0 (pr3_refl (CHead c (Flat f) t2) x0)) t2 x0 (refl_equal T
+(THead (Flat f) t2 x0))) in (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 (H8: (eq T (THead (Flat f) x x0) (THead (Flat f) x
+t2))).(\lambda (P: Prop).(let H9 \def (f_equal T T (\lambda (e: T).(match e
+in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | (TLRef _)
+\Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat f) x x0)
+(THead (Flat f) x t2) H8) in (let H10 \def (eq_ind_r T t2 (\lambda (t0:
T).(pr3 c x0 t0)) H7 x0 H9) in (let H11 \def (eq_ind_r T t2 (\lambda (t0:
T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H6 x0 H9) in (H11 (refl_equal
T x0) P)))))) (pr3_thin_dx c x0 t2 H7 x f) x t2 (refl_equal T (THead (Flat f)
\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
-(t0: T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to
-((pr3 c1 t0 t2) \to (\forall (x0: T).((eq T t2 (lift h d x0)) \to (\forall
-(c3: C).((drop h d c1 c3) \to (sn3 c3 x0))))))))) H2 (lift h d x) H3) in (let
-H6 \def (eq_ind T t1 (\lambda (t0: T).(\forall (t2: T).((((eq T t0 t2) \to
-(\forall (P: Prop).P))) \to ((pr3 c1 t0 t2) \to (sn3 c1 t2))))) H1 (lift h d
-x) H3) in (sn3_sing c2 x (\lambda (t2: T).(\lambda (H7: (((eq T x t2) \to
-(\forall (P: Prop).P)))).(\lambda (H8: (pr3 c2 x t2)).(H5 (lift h d t2)
-(\lambda (H9: (eq T (lift h d x) (lift h d t2))).(\lambda (P: Prop).(let H10
-\def (eq_ind_r T t2 (\lambda (t0: T).(pr3 c2 x t0)) H8 x (lift_inj x t2 h d
-H9)) in (let H11 \def (eq_ind_r T t2 (\lambda (t0: T).((eq T x t0) \to
+t0)) (\lambda (_: T).(\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 t))))
+(\lambda (y: T).(\lambda (H0: (sn3 c1 y)).(unintro T t (\lambda (t0: T).((eq
+T y (lift h d t0)) \to (\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 t0)))))
+(sn3_ind c1 (\lambda (t0: T).(\forall (x: T).((eq T t0 (lift h d x)) \to
+(\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 x)))))) (\lambda (t1:
+T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P:
+Prop).P))) \to ((pr3 c1 t1 t2) \to (sn3 c1 t2)))))).(\lambda (H2: ((\forall
+(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c1 t1 t2) \to
+(\forall (x: T).((eq T t2 (lift h d x)) \to (\forall (c2: C).((drop h d c1
+c2) \to (sn3 c2 x)))))))))).(\lambda (x: T).(\lambda (H3: (eq T t1 (lift h d
+x))).(\lambda (c2: C).(\lambda (H4: (drop h d c1 c2)).(let H5 \def (eq_ind T
+t1 (\lambda (t0: T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P:
+Prop).P))) \to ((pr3 c1 t0 t2) \to (\forall (x0: T).((eq T t2 (lift h d x0))
+\to (\forall (c3: C).((drop h d c1 c3) \to (sn3 c3 x0))))))))) H2 (lift h d
+x) H3) in (let H6 \def (eq_ind T t1 (\lambda (t0: T).(\forall (t2: T).((((eq
+T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c1 t0 t2) \to (sn3 c1 t2)))))
+H1 (lift h d x) H3) in (sn3_sing c2 x (\lambda (t2: T).(\lambda (H7: (((eq T
+x t2) \to (\forall (P: Prop).P)))).(\lambda (H8: (pr3 c2 x t2)).(H5 (lift h d
+t2) (\lambda (H9: (eq T (lift h d x) (lift h d t2))).(\lambda (P: Prop).(let
+H10 \def (eq_ind_r T t2 (\lambda (t0: T).(pr3 c2 x t0)) H8 x (lift_inj x t2 h
+d H9)) in (let H11 \def (eq_ind_r T t2 (\lambda (t0: T).((eq T x t0) \to
(\forall (P0: Prop).P0))) H7 x (lift_inj x t2 h d H9)) in (H11 (refl_equal T
x) P))))) (pr3_lift c1 c2 h d H4 x t2 H8) t2 (refl_equal T (lift h d t2)) c2
H4)))))))))))))) y H0)))) H))))).
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/sn3/lift1".
include "sn3/props.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/sn3/nf2".
include "sn3/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/sn3/props".
include "sn3/nf2.ma".
\def
\lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (H: (sn3 c (THead
(Bind Abbr) v t))).(insert_eq T (THead (Bind Abbr) v t) (\lambda (t0: T).(sn3
-c t0)) (\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) v (THead
-(Bind Abst) w t))))) (\lambda (y: T).(\lambda (H0: (sn3 c y)).(unintro T t
-(\lambda (t0: T).((eq T y (THead (Bind Abbr) v t0)) \to (\forall (w: T).((sn3
-c w) \to (sn3 c (THead (Flat Appl) v (THead (Bind Abst) w t0))))))) (unintro
-T v (\lambda (t0: T).(\forall (x: T).((eq T y (THead (Bind Abbr) t0 x)) \to
-(\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) t0 (THead (Bind
-Abst) w x)))))))) (sn3_ind c (\lambda (t0: T).(\forall (x: T).(\forall (x0:
-T).((eq T t0 (THead (Bind Abbr) x x0)) \to (\forall (w: T).((sn3 c w) \to
-(sn3 c (THead (Flat Appl) x (THead (Bind Abst) w x0))))))))) (\lambda (t1:
-T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P:
-Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H2: ((\forall
-(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to
-(\forall (x: T).(\forall (x0: T).((eq T t2 (THead (Bind Abbr) x x0)) \to
-(\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) x (THead (Bind Abst)
-w x0))))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: (eq T t1
-(THead (Bind Abbr) x x0))).(\lambda (w: T).(\lambda (H4: (sn3 c w)).(let H5
-\def (eq_ind T t1 (\lambda (t0: T).(\forall (t2: T).((((eq T t0 t2) \to
-(\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (\forall (x1: T).(\forall (x2:
-T).((eq T t2 (THead (Bind Abbr) x1 x2)) \to (\forall (w0: T).((sn3 c w0) \to
-(sn3 c (THead (Flat Appl) x1 (THead (Bind Abst) w0 x2)))))))))))) H2 (THead
-(Bind Abbr) x x0) H3) in (let H6 \def (eq_ind T t1 (\lambda (t0: T).(\forall
-(t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to
-(sn3 c t2))))) H1 (THead (Bind Abbr) x x0) H3) in (sn3_ind c (\lambda (t0:
-T).(sn3 c (THead (Flat Appl) x (THead (Bind Abst) t0 x0)))) (\lambda (t2:
-T).(\lambda (H7: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P:
-Prop).P))) \to ((pr3 c t2 t3) \to (sn3 c t3)))))).(\lambda (H8: ((\forall
-(t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to
-(sn3 c (THead (Flat Appl) x (THead (Bind Abst) t3 x0)))))))).(sn3_pr2_intro c
-(THead (Flat Appl) x (THead (Bind Abst) t2 x0)) (\lambda (t3: T).(\lambda
-(H9: (((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t3) \to (\forall
-(P: Prop).P)))).(\lambda (H10: (pr2 c (THead (Flat Appl) x (THead (Bind Abst)
-t2 x0)) t3)).(let H11 \def (pr2_gen_appl c x (THead (Bind Abst) t2 x0) t3
-H10) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead
-(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2)))
-(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind Abst) t2 x0) t4))))
-(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
-T).(eq T (THead (Bind Abst) t2 x0) (THead (Bind Abst) y1 z1)))))) (\lambda
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead
-(Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda
-(_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind
-b) u) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
-b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) t2 x0) (THead
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
-(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (sn3 c t3)
-(\lambda (H12: (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead
-(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2)))
-(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind Abst) t2 x0)
-t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead
-(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2)))
-(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind Abst) t2 x0) t4))) (sn3
-c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (H13: (eq T t3 (THead (Flat
-Appl) x1 x2))).(\lambda (H14: (pr2 c x x1)).(\lambda (H15: (pr2 c (THead
-(Bind Abst) t2 x0) x2)).(let H16 \def (eq_ind T t3 (\lambda (t0: T).((eq T
-(THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t0) \to (\forall (P:
-Prop).P))) H9 (THead (Flat Appl) x1 x2) H13) in (eq_ind_r T (THead (Flat
-Appl) x1 x2) (\lambda (t0: T).(sn3 c t0)) (let H17 \def (pr2_gen_abst c t2 x0
-x2 H15) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead
-(Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c t2 u2)))
-(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead
-c (Bind b) u) x0 t4))))) (sn3 c (THead (Flat Appl) x1 x2)) (\lambda (x3:
-T).(\lambda (x4: T).(\lambda (H18: (eq T x2 (THead (Bind Abst) x3
-x4))).(\lambda (H19: (pr2 c t2 x3)).(\lambda (H20: ((\forall (b: B).(\forall
-(u: T).(pr2 (CHead c (Bind b) u) x0 x4))))).(let H21 \def (eq_ind T x2
-(\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0))
+c t0)) (\lambda (_: T).(\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat
+Appl) v (THead (Bind Abst) w t)))))) (\lambda (y: T).(\lambda (H0: (sn3 c
+y)).(unintro T t (\lambda (t0: T).((eq T y (THead (Bind Abbr) v t0)) \to
+(\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) v (THead (Bind Abst)
+w t0))))))) (unintro T v (\lambda (t0: T).(\forall (x: T).((eq T y (THead
+(Bind Abbr) t0 x)) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat
+Appl) t0 (THead (Bind Abst) w x)))))))) (sn3_ind c (\lambda (t0: T).(\forall
+(x: T).(\forall (x0: T).((eq T t0 (THead (Bind Abbr) x x0)) \to (\forall (w:
+T).((sn3 c w) \to (sn3 c (THead (Flat Appl) x (THead (Bind Abst) w
+x0))))))))) (\lambda (t1: T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2)
+\to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda
+(H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3
+c t1 t2) \to (\forall (x: T).(\forall (x0: T).((eq T t2 (THead (Bind Abbr) x
+x0)) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) x (THead
+(Bind Abst) w x0))))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3:
+(eq T t1 (THead (Bind Abbr) x x0))).(\lambda (w: T).(\lambda (H4: (sn3 c
+w)).(let H5 \def (eq_ind T t1 (\lambda (t0: T).(\forall (t2: T).((((eq T t0
+t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (\forall (x1:
+T).(\forall (x2: T).((eq T t2 (THead (Bind Abbr) x1 x2)) \to (\forall (w0:
+T).((sn3 c w0) \to (sn3 c (THead (Flat Appl) x1 (THead (Bind Abst) w0
+x2)))))))))))) H2 (THead (Bind Abbr) x x0) H3) in (let H6 \def (eq_ind T t1
+(\lambda (t0: T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P)))
+\to ((pr3 c t0 t2) \to (sn3 c t2))))) H1 (THead (Bind Abbr) x x0) H3) in
+(sn3_ind c (\lambda (t0: T).(sn3 c (THead (Flat Appl) x (THead (Bind Abst) t0
+x0)))) (\lambda (t2: T).(\lambda (H7: ((\forall (t3: T).((((eq T t2 t3) \to
+(\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to (sn3 c t3)))))).(\lambda (H8:
+((\forall (t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t2
+t3) \to (sn3 c (THead (Flat Appl) x (THead (Bind Abst) t3
+x0)))))))).(sn3_pr2_intro c (THead (Flat Appl) x (THead (Bind Abst) t2 x0))
+(\lambda (t3: T).(\lambda (H9: (((eq T (THead (Flat Appl) x (THead (Bind
+Abst) t2 x0)) t3) \to (\forall (P: Prop).P)))).(\lambda (H10: (pr2 c (THead
+(Flat Appl) x (THead (Bind Abst) t2 x0)) t3)).(let H11 \def (pr2_gen_appl c x
+(THead (Bind Abst) t2 x0) t3 H10) in (or3_ind (ex3_2 T T (\lambda (u2:
+T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c
+(THead (Bind Abst) t2 x0) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) t2 x0)
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b:
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4)))))))) (ex6_6 B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind Abst) t2 x0) (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
+y2) z1 z2)))))))) (sn3 c t3) (\lambda (H12: (ex3_2 T T (\lambda (u2:
+T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c
+(THead (Bind Abst) t2 x0) t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda
+(t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind Abst)
+t2 x0) t4))) (sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (H13: (eq
+T t3 (THead (Flat Appl) x1 x2))).(\lambda (H14: (pr2 c x x1)).(\lambda (H15:
+(pr2 c (THead (Bind Abst) t2 x0) x2)).(let H16 \def (eq_ind T t3 (\lambda
+(t0: T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t0) \to
+(\forall (P: Prop).P))) H9 (THead (Flat Appl) x1 x2) H13) in (eq_ind_r T
+(THead (Flat Appl) x1 x2) (\lambda (t0: T).(sn3 c t0)) (let H17 \def
+(pr2_gen_abst c t2 x0 x2 H15) in (ex3_2_ind T T (\lambda (u2: T).(\lambda
+(t4: T).(eq T x2 (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c t2 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
+(u: T).(pr2 (CHead c (Bind b) u) x0 t4))))) (sn3 c (THead (Flat Appl) x1 x2))
+(\lambda (x3: T).(\lambda (x4: T).(\lambda (H18: (eq T x2 (THead (Bind Abst)
+x3 x4))).(\lambda (H19: (pr2 c t2 x3)).(\lambda (H20: ((\forall (b:
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0 x4))))).(let H21 \def (eq_ind
+T x2 (\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0))
(THead (Flat Appl) x1 t0)) \to (\forall (P: Prop).P))) H16 (THead (Bind Abst)
x3 x4) H18) in (eq_ind_r T (THead (Bind Abst) x3 x4) (\lambda (t0: T).(sn3 c
(THead (Flat Appl) x1 t0))) (let H_x \def (term_dec t2 x3) in (let H22 \def
\lambda (c: C).(\lambda (d: C).(\lambda (w: T).(\lambda (i: nat).(\lambda
(H: (getl i c (CHead d (Bind Abbr) w))).(\lambda (v: T).(\lambda (H0: (sn3 c
(THead (Flat Appl) v (lift (S i) O w)))).(insert_eq T (THead (Flat Appl) v
-(lift (S i) O w)) (\lambda (t: T).(sn3 c t)) (sn3 c (THead (Flat Appl) v
-(TLRef i))) (\lambda (y: T).(\lambda (H1: (sn3 c y)).(unintro T v (\lambda
-(t: T).((eq T y (THead (Flat Appl) t (lift (S i) O w))) \to (sn3 c (THead
-(Flat Appl) t (TLRef i))))) (sn3_ind c (\lambda (t: T).(\forall (x: T).((eq T
-t (THead (Flat Appl) x (lift (S i) O w))) \to (sn3 c (THead (Flat Appl) x
-(TLRef i)))))) (\lambda (t1: T).(\lambda (H2: ((\forall (t2: T).((((eq T t1
-t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c
+(lift (S i) O w)) (\lambda (t: T).(sn3 c t)) (\lambda (_: T).(sn3 c (THead
+(Flat Appl) v (TLRef i)))) (\lambda (y: T).(\lambda (H1: (sn3 c y)).(unintro
+T v (\lambda (t: T).((eq T y (THead (Flat Appl) t (lift (S i) O w))) \to (sn3
+c (THead (Flat Appl) t (TLRef i))))) (sn3_ind c (\lambda (t: T).(\forall (x:
+T).((eq T t (THead (Flat Appl) x (lift (S i) O w))) \to (sn3 c (THead (Flat
+Appl) x (TLRef i)))))) (\lambda (t1: T).(\lambda (H2: ((\forall (t2:
+T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c
t2)))))).(\lambda (H3: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P:
Prop).P))) \to ((pr3 c t1 t2) \to (\forall (x: T).((eq T t2 (THead (Flat
Appl) x (lift (S i) O w))) \to (sn3 c (THead (Flat Appl) x (TLRef
\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
+c t)) (\lambda (_: T).(\forall (t0: T).((sn3 c (THead (Flat Appl) v t0)) \to
+(sn3 c (THead (Flat Appl) v (THead (Flat Cast) u t0)))))) (\lambda (y:
+T).(\lambda (H0: (sn3 c y)).(unintro T u (\lambda (t: T).((eq T y (THead
+(Flat Appl) v t)) \to (\forall (t0: T).((sn3 c (THead (Flat Appl) v t0)) \to
+(sn3 c (THead (Flat Appl) v (THead (Flat Cast) t t0))))))) (unintro T v
+(\lambda (t: T).(\forall (x: T).((eq T y (THead (Flat Appl) t x)) \to
+(\forall (t0: T).((sn3 c (THead (Flat Appl) t t0)) \to (sn3 c (THead (Flat
+Appl) t (THead (Flat Cast) x t0)))))))) (sn3_ind c (\lambda (t: T).(\forall
+(x: T).(\forall (x0: T).((eq T t (THead (Flat Appl) x x0)) \to (\forall (t0:
+T).((sn3 c (THead (Flat Appl) x t0)) \to (sn3 c (THead (Flat Appl) x (THead
+(Flat Cast) x0 t0))))))))) (\lambda (t1: T).(\lambda (H1: ((\forall (t2:
+T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c
+t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P:
+Prop).P))) \to ((pr3 c t1 t2) \to (\forall (x: T).(\forall (x0: T).((eq T t2
+(THead (Flat Appl) x x0)) \to (\forall (t: T).((sn3 c (THead (Flat Appl) x
+t)) \to (sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0
+t))))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: (eq T t1 (THead
+(Flat Appl) x x0))).(\lambda (t: T).(\lambda (H4: (sn3 c (THead (Flat Appl) x
+t))).(insert_eq T (THead (Flat Appl) x t) (\lambda (t0: T).(sn3 c t0))
+(\lambda (_: T).(sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0 t))))
+(\lambda (y0: T).(\lambda (H5: (sn3 c y0)).(unintro T t (\lambda (t0: T).((eq
+T y0 (THead (Flat Appl) x t0)) \to (sn3 c (THead (Flat Appl) x (THead (Flat
+Cast) x0 t0))))) (sn3_ind c (\lambda (t0: T).(\forall (x1: T).((eq T t0
(THead (Flat Appl) x x1)) \to (sn3 c (THead (Flat Appl) x (THead (Flat Cast)
x0 x1)))))) (\lambda (t0: T).(\lambda (H6: ((\forall (t2: T).((((eq T t0 t2)
\to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (sn3 c t2)))))).(\lambda
t))))))))))).(\lambda (t: T).(\lambda (v: T).(\lambda (H3: (sn3 (CHead c
(Bind b) t1) (THead (Flat Appl) (lift (S O) O v) t))).(insert_eq T (THead
(Flat Appl) (lift (S O) O v) t) (\lambda (t0: T).(sn3 (CHead c (Bind b) t1)
-t0)) (sn3 c (THead (Flat Appl) v (THead (Bind b) t1 t))) (\lambda (y:
-T).(\lambda (H4: (sn3 (CHead c (Bind b) t1) y)).(unintro T t (\lambda (t0:
-T).((eq T y (THead (Flat Appl) (lift (S O) O v) t0)) \to (sn3 c (THead (Flat
-Appl) v (THead (Bind b) t1 t0))))) (unintro T v (\lambda (t0: T).(\forall (x:
-T).((eq T y (THead (Flat Appl) (lift (S O) O t0) x)) \to (sn3 c (THead (Flat
-Appl) t0 (THead (Bind b) t1 x)))))) (sn3_ind (CHead c (Bind b) t1) (\lambda
-(t0: T).(\forall (x: T).(\forall (x0: T).((eq T t0 (THead (Flat Appl) (lift
-(S O) O x) x0)) \to (sn3 c (THead (Flat Appl) x (THead (Bind b) t1 x0)))))))
-(\lambda (t2: T).(\lambda (H5: ((\forall (t3: T).((((eq T t2 t3) \to (\forall
-(P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (sn3 (CHead c (Bind
-b) t1) t3)))))).(\lambda (H6: ((\forall (t3: T).((((eq T t2 t3) \to (\forall
-(P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (\forall (x:
-T).(\forall (x0: T).((eq T t3 (THead (Flat Appl) (lift (S O) O x) x0)) \to
-(sn3 c (THead (Flat Appl) x (THead (Bind b) t1 x0))))))))))).(\lambda (x:
-T).(\lambda (x0: T).(\lambda (H7: (eq T t2 (THead (Flat Appl) (lift (S O) O
-x) x0))).(let H8 \def (eq_ind T t2 (\lambda (t0: T).(\forall (t3: T).((((eq T
+t0)) (\lambda (_: T).(sn3 c (THead (Flat Appl) v (THead (Bind b) t1 t))))
+(\lambda (y: T).(\lambda (H4: (sn3 (CHead c (Bind b) t1) y)).(unintro T t
+(\lambda (t0: T).((eq T y (THead (Flat Appl) (lift (S O) O v) t0)) \to (sn3 c
+(THead (Flat Appl) v (THead (Bind b) t1 t0))))) (unintro T v (\lambda (t0:
+T).(\forall (x: T).((eq T y (THead (Flat Appl) (lift (S O) O t0) x)) \to (sn3
+c (THead (Flat Appl) t0 (THead (Bind b) t1 x)))))) (sn3_ind (CHead c (Bind b)
+t1) (\lambda (t0: T).(\forall (x: T).(\forall (x0: T).((eq T t0 (THead (Flat
+Appl) (lift (S O) O x) x0)) \to (sn3 c (THead (Flat Appl) x (THead (Bind b)
+t1 x0))))))) (\lambda (t2: T).(\lambda (H5: ((\forall (t3: T).((((eq T t2 t3)
+\to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (sn3
+(CHead c (Bind b) t1) t3)))))).(\lambda (H6: ((\forall (t3: T).((((eq T t2
+t3) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to
+(\forall (x: T).(\forall (x0: T).((eq T t3 (THead (Flat Appl) (lift (S O) O
+x) x0)) \to (sn3 c (THead (Flat Appl) x (THead (Bind b) t1
+x0))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H7: (eq T t2 (THead
+(Flat Appl) (lift (S O) O x) x0))).(let H8 \def (eq_ind T t2 (\lambda (t0:
+T).(\forall (t3: T).((((eq T t0 t3) \to (\forall (P: Prop).P))) \to ((pr3
+(CHead c (Bind b) t1) t0 t3) \to (\forall (x1: T).(\forall (x2: T).((eq T t3
+(THead (Flat Appl) (lift (S O) O x1) x2)) \to (sn3 c (THead (Flat Appl) x1
+(THead (Bind b) t1 x2)))))))))) H6 (THead (Flat Appl) (lift (S O) O x) x0)
+H7) in (let H9 \def (eq_ind T t2 (\lambda (t0: T).(\forall (t3: T).((((eq T
t0 t3) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t0 t3) \to
-(\forall (x1: T).(\forall (x2: T).((eq T t3 (THead (Flat Appl) (lift (S O) O
-x1) x2)) \to (sn3 c (THead (Flat Appl) x1 (THead (Bind b) t1 x2)))))))))) H6
-(THead (Flat Appl) (lift (S O) O x) x0) H7) in (let H9 \def (eq_ind T t2
-(\lambda (t0: T).(\forall (t3: T).((((eq T t0 t3) \to (\forall (P: Prop).P)))
-\to ((pr3 (CHead c (Bind b) t1) t0 t3) \to (sn3 (CHead c (Bind b) t1) t3)))))
-H5 (THead (Flat Appl) (lift (S O) O x) x0) H7) in (sn3_pr2_intro c (THead
-(Flat Appl) x (THead (Bind b) t1 x0)) (\lambda (t3: T).(\lambda (H10: (((eq T
-(THead (Flat Appl) x (THead (Bind b) t1 x0)) t3) \to (\forall (P:
-Prop).P)))).(\lambda (H11: (pr2 c (THead (Flat Appl) x (THead (Bind b) t1
-x0)) t3)).(let H12 \def (pr2_gen_appl c x (THead (Bind b) t1 x0) t3 H11) in
-(or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat
-Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_:
-T).(\lambda (t4: T).(pr2 c (THead (Bind b) t1 x0) t4)))) (ex4_4 T T T T
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
-(THead (Bind b) t1 x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+(sn3 (CHead c (Bind b) t1) t3))))) H5 (THead (Flat Appl) (lift (S O) O x) x0)
+H7) in (sn3_pr2_intro c (THead (Flat Appl) x (THead (Bind b) t1 x0)) (\lambda
+(t3: T).(\lambda (H10: (((eq T (THead (Flat Appl) x (THead (Bind b) t1 x0))
+t3) \to (\forall (P: Prop).P)))).(\lambda (H11: (pr2 c (THead (Flat Appl) x
+(THead (Bind b) t1 x0)) t3)).(let H12 \def (pr2_gen_appl c x (THead (Bind b)
+t1 x0) t3 H11) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T
+t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x
+u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind b) t1 x0) t4))))
+(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(eq T (THead (Bind b) t1 x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind
Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
(_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
\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) v1 t1) (\lambda (t: T).(sn3 c t)) (\lambda (t: T).(\forall
+(v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c t u2) \to ((((iso t u2)
+\to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to
+(sn3 c (THead (Flat Appl) v2 t)))))) (\lambda (y: T).(\lambda (H0: (sn3 c
+y)).(unintro T t1 (\lambda (t: T).((eq T y (THead (Flat Appl) v1 t)) \to
+(\forall (v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c y u2) \to ((((iso
+y u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2))))))
+\to (sn3 c (THead (Flat Appl) v2 y))))))) (unintro T v1 (\lambda (t:
+T).(\forall (x: T).((eq T y (THead (Flat Appl) t x)) \to (\forall (v2:
+T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c y u2) \to ((((iso y u2) \to
+(\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c
+(THead (Flat Appl) v2 y)))))))) (sn3_ind c (\lambda (t: T).(\forall (x:
+T).(\forall (x0: T).((eq T t (THead (Flat Appl) x x0)) \to (\forall (v2:
+T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c t u2) \to ((((iso t u2) \to
+(\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c
+(THead (Flat Appl) v2 t))))))))) (\lambda (t2: T).(\lambda (H1: ((\forall
+(t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to
+(sn3 c t3)))))).(\lambda (H2: ((\forall (t3: T).((((eq T t2 t3) \to (\forall
+(P: Prop).P))) \to ((pr3 c t2 t3) \to (\forall (x: T).(\forall (x0: T).((eq T
+t3 (THead (Flat Appl) x x0)) \to (\forall (v2: T).((sn3 c v2) \to (((\forall
+(u2: T).((pr3 c t3 u2) \to ((((iso t3 u2) \to (\forall (P: Prop).P))) \to
+(sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2
+t3))))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: (eq T t2
(THead (Flat Appl) x x0))).(\lambda (v2: T).(\lambda (H4: (sn3 c
-v2)).(sn3_ind c (\lambda (t: T).(((\forall (u2: T).((pr3 c (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
+v2)).(sn3_ind c (\lambda (t: T).(((\forall (u2: T).((pr3 c t2 u2) \to ((((iso
+t2 u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t u2))))))
+\to (sn3 c (THead (Flat Appl) t t2)))) (\lambda (t0: T).(\lambda (H5:
+((\forall (t3: T).((((eq T t0 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t0
+t3) \to (sn3 c t3)))))).(\lambda (H6: ((\forall (t3: T).((((eq T t0 t3) \to
+(\forall (P: Prop).P))) \to ((pr3 c t0 t3) \to (((\forall (u2: T).((pr3 c t2
+u2) \to ((((iso t2 u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat
+Appl) t3 u2)))))) \to (sn3 c (THead (Flat Appl) t3 t2)))))))).(\lambda (H7:
+((\forall (u2: T).((pr3 c t2 u2) \to ((((iso t2 u2) \to (\forall (P:
+Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2))))))).(let H8 \def (eq_ind T
+t2 (\lambda (t: T).(\forall (u2: T).((pr3 c t u2) \to ((((iso t u2) \to
+(\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2)))))) H7 (THead
+(Flat Appl) x x0) H3) in (let H9 \def (eq_ind T t2 (\lambda (t: T).(\forall
(t3: T).((((eq T t0 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t3) \to
-(sn3 c t3)))))).(\lambda (H6: ((\forall (t3: T).((((eq T t0 t3) \to (\forall
-(P: Prop).P))) \to ((pr3 c t0 t3) \to (((\forall (u2: T).((pr3 c (THead (Flat
-Appl) x x0) u2) \to ((((iso (THead (Flat Appl) x x0) u2) \to (\forall (P:
+(((\forall (u2: T).((pr3 c t u2) \to ((((iso t u2) \to (\forall (P:
Prop).P))) \to (sn3 c (THead (Flat Appl) t3 u2)))))) \to (sn3 c (THead (Flat
-Appl) t3 (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 (t3: T).((((eq T
-t t3) \to (\forall (P: Prop).P))) \to ((pr3 c t t3) \to (\forall (x1:
-T).(\forall (x2: T).((eq T t3 (THead (Flat Appl) x1 x2)) \to (\forall (v3:
-T).((sn3 c v3) \to (((\forall (u2: T).((pr3 c (THead (Flat Appl) x1 x2) u2)
-\to ((((iso (THead (Flat Appl) x1 x2) u2) \to (\forall (P: Prop).P))) \to
-(sn3 c (THead (Flat Appl) v3 u2)))))) \to (sn3 c (THead (Flat Appl) v3 (THead
-(Flat Appl) x1 x2))))))))))))) H2 (THead (Flat Appl) x x0) H3) in (let H9
-\def (eq_ind T t2 (\lambda (t: T).(\forall (t3: T).((((eq T t t3) \to
-(\forall (P: Prop).P))) \to ((pr3 c t t3) \to (sn3 c t3))))) H1 (THead (Flat
-Appl) x x0) H3) in (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
+Appl) t3 t))))))) H6 (THead (Flat Appl) x x0) H3) in (let H10 \def (eq_ind T
+t2 (\lambda (t: T).(\forall (t3: T).((((eq T t t3) \to (\forall (P:
+Prop).P))) \to ((pr3 c t t3) \to (\forall (x1: T).(\forall (x2: T).((eq T t3
+(THead (Flat Appl) x1 x2)) \to (\forall (v3: T).((sn3 c v3) \to (((\forall
+(u2: T).((pr3 c t3 u2) \to ((((iso t3 u2) \to (\forall (P: Prop).P))) \to
+(sn3 c (THead (Flat Appl) v3 u2)))))) \to (sn3 c (THead (Flat Appl) v3
+t3)))))))))))) H2 (THead (Flat Appl) x x0) H3) in (let H11 \def (eq_ind T t2
+(\lambda (t: T).(\forall (t3: T).((((eq T t t3) \to (\forall (P: Prop).P)))
+\to ((pr3 c t t3) \to (sn3 c t3))))) H1 (THead (Flat Appl) x x0) H3) in
+(eq_ind_r T (THead (Flat Appl) x x0) (\lambda (t: T).(sn3 c (THead (Flat
+Appl) t0 t))) (sn3_pr2_intro c (THead (Flat Appl) t0 (THead (Flat Appl) x
+x0)) (\lambda (t3: T).(\lambda (H12: (((eq T (THead (Flat Appl) t0 (THead
+(Flat Appl) x x0)) t3) \to (\forall (P: Prop).P)))).(\lambda (H13: (pr2 c
+(THead (Flat Appl) t0 (THead (Flat Appl) x x0)) t3)).(let H14 \def
+(pr2_gen_appl c t0 (THead (Flat Appl) x x0) t3 H13) in (or3_ind (ex3_2 T T
(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4))))
(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))) (\lambda (_: T).(\lambda
(t4: T).(pr2 c (THead (Flat Appl) x x0) t4)))) (ex4_4 T T T T (\lambda (y1:
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:
+y2) z1 z2)))))))) (sn3 c t3) (\lambda (H15: (ex3_2 T T (\lambda (u2:
T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2:
T).(\lambda (_: T).(pr2 c t0 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c
(THead (Flat Appl) x x0) t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda
(t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_:
T).(pr2 c t0 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Flat Appl)
-x x0) t4))) (sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (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:
+x x0) t4))) (sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (H16: (eq T
+t3 (THead (Flat Appl) x1 x2))).(\lambda (H17: (pr2 c t0 x1)).(\lambda (H18:
+(pr2 c (THead (Flat Appl) x x0) x2)).(let H19 \def (eq_ind T t3 (\lambda (t:
T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) t) \to (\forall (P:
-Prop).P))) 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
+Prop).P))) H12 (THead (Flat Appl) x1 x2) H16) in (eq_ind_r T (THead (Flat
+Appl) x1 x2) (\lambda (t: T).(sn3 c t)) (let H20 \def (pr2_gen_appl c x x0 x2
+H18) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead
(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2)))
(\lambda (_: T).(\lambda (t4: T).(pr2 c x0 t4)))) (ex4_4 T T T T (\lambda
(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 (THead
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:
+(THead (Flat Appl) x1 x2)) (\lambda (H21: (ex3_2 T T (\lambda (u2:
T).(\lambda (t4: T).(eq T x2 (THead (Flat Appl) u2 t4)))) (\lambda (u2:
T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c x0
t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead
(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2)))
(\lambda (_: T).(\lambda (t4: T).(pr2 c x0 t4))) (sn3 c (THead (Flat Appl) x1
-x2)) (\lambda (x3: T).(\lambda (x4: T).(\lambda (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
+x2)) (\lambda (x3: T).(\lambda (x4: T).(\lambda (H22: (eq T x2 (THead (Flat
+Appl) x3 x4))).(\lambda (H23: (pr2 c x x3)).(\lambda (H24: (pr2 c x0
+x4)).(let H25 \def (eq_ind T x2 (\lambda (t: T).((eq T (THead (Flat Appl) t0
(THead (Flat Appl) x x0)) (THead (Flat Appl) x1 t)) \to (\forall (P:
-Prop).P))) H17 (THead (Flat Appl) x3 x4) H20) in (eq_ind_r T (THead (Flat
+Prop).P))) H19 (THead (Flat Appl) x3 x4) H22) in (eq_ind_r T (THead (Flat
Appl) x3 x4) (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 t))) (let H_x \def
-(term_dec (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4)) in (let H24
+(term_dec (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4)) in (let H26
\def H_x in (or_ind (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4))
((eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4)) \to (\forall (P:
Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Flat Appl) x3 x4))) (\lambda
-(H25: (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4))).(let H26
+(H27: (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4))).(let H28
\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t _)
-\Rightarrow t])) (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4) H25) in
-((let H27 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
+\Rightarrow t])) (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4) H27) in
+((let H29 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _
-t) \Rightarrow t])) (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4) H25)
-in (\lambda (H28: (eq T x x3)).(let H29 \def (eq_ind_r T x4 (\lambda (t:
+t) \Rightarrow t])) (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4) H27)
+in (\lambda (H30: (eq T x x3)).(let H31 \def (eq_ind_r T x4 (\lambda (t:
T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) (THead (Flat Appl)
-x1 (THead (Flat Appl) x3 t))) \to (\forall (P: Prop).P))) 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
+x1 (THead (Flat Appl) x3 t))) \to (\forall (P: Prop).P))) H25 x0 H29) in (let
+H32 \def (eq_ind_r T x4 (\lambda (t: T).(pr2 c x0 t)) H24 x0 H29) in (eq_ind
T x0 (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 (THead (Flat Appl) x3 t))))
-(let H31 \def (eq_ind_r T x3 (\lambda (t: T).((eq T (THead (Flat Appl) t0
+(let H33 \def (eq_ind_r T x3 (\lambda (t: T).((eq T (THead (Flat Appl) t0
(THead (Flat Appl) x x0)) (THead (Flat Appl) x1 (THead (Flat Appl) t x0)))
-\to (\forall (P: Prop).P))) 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
+\to (\forall (P: Prop).P))) H31 x H30) in (let H34 \def (eq_ind_r T x3
+(\lambda (t: T).(pr2 c x t)) H23 x H30) in (eq_ind T x (\lambda (t: T).(sn3 c
(THead (Flat Appl) x1 (THead (Flat Appl) t x0)))) (let H_x0 \def (term_dec t0
-x1) in (let H33 \def H_x0 in (or_ind (eq T t0 x1) ((eq T t0 x1) \to (\forall
+x1) in (let H35 \def H_x0 in (or_ind (eq T t0 x1) ((eq T t0 x1) \to (\forall
(P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Flat Appl) x x0)))
-(\lambda (H34: (eq T t0 x1)).(let H35 \def (eq_ind_r T x1 (\lambda (t:
+(\lambda (H36: (eq T t0 x1)).(let H37 \def (eq_ind_r T x1 (\lambda (t:
T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) (THead (Flat Appl)
-t (THead (Flat Appl) x x0))) \to (\forall (P: Prop).P))) 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 (THead (Flat Appl) x x0))) \to (\forall (P: Prop).P))) H33 t0 H36) in (let
+H38 \def (eq_ind_r T x1 (\lambda (t: T).(pr2 c t0 t)) H17 t0 H36) in (eq_ind
T t0 (\lambda (t: T).(sn3 c (THead (Flat Appl) t (THead (Flat Appl) x x0))))
-(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
+(H37 (refl_equal T (THead (Flat Appl) t0 (THead (Flat Appl) x x0))) (sn3 c
+(THead (Flat Appl) t0 (THead (Flat Appl) x x0)))) x1 H36)))) (\lambda (H36:
+(((eq T t0 x1) \to (\forall (P: Prop).P)))).(H9 x1 H36 (pr3_pr2 c t0 x1 H17)
+(\lambda (u2: T).(\lambda (H37: (pr3 c (THead (Flat Appl) x x0) u2)).(\lambda
+(H38: (((iso (THead (Flat Appl) x x0) u2) \to (\forall (P:
+Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) t0 u2) (H8 u2 H37 H38) (THead
(Flat Appl) x1 u2) (pr3_pr2 c (THead (Flat Appl) t0 u2) (THead (Flat Appl) x1
-u2) (pr2_head_1 c t0 x1 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_flat c x x3 (pr3_pr2 c x x3 H21) x0 x4 (pr3_pr2 c x0 x4 H22) Appl) x3 x4
+u2) (pr2_head_1 c t0 x1 H17 (Flat Appl) u2)))))))) H35))) x3 H30))) x4
+H29))))) H28))) (\lambda (H27: (((eq T (THead (Flat Appl) x x0) (THead (Flat
+Appl) x3 x4)) \to (\forall (P: Prop).P)))).(H10 (THead (Flat Appl) x3 x4) H27
+(pr3_flat c x x3 (pr3_pr2 c x x3 H23) x0 x4 (pr3_pr2 c x0 x4 H24) Appl) x3 x4
(refl_equal T (THead (Flat Appl) x3 x4)) x1 (sn3_pr3_trans c t0 (sn3_sing c
-t0 H5) x1 (pr3_pr2 c t0 x1 H15)) (\lambda (u2: T).(\lambda (H26: (pr3 c
-(THead (Flat Appl) x3 x4) u2)).(\lambda (H27: (((iso (THead (Flat Appl) x3
+t0 H5) x1 (pr3_pr2 c t0 x1 H17)) (\lambda (u2: T).(\lambda (H28: (pr3 c
+(THead (Flat Appl) x3 x4) u2)).(\lambda (H29: (((iso (THead (Flat Appl) x3
x4) u2) \to (\forall (P: Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) t0
-u2) (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
+u2) (H8 u2 (pr3_sing c (THead (Flat Appl) x x4) (THead (Flat Appl) x x0)
+(pr2_thin_dx c x0 x4 H24 x Appl) u2 (pr3_sing c (THead (Flat Appl) x3 x4)
+(THead (Flat Appl) x x4) (pr2_head_1 c x x3 H23 (Flat Appl) x4) u2 H28))
+(\lambda (H30: (iso (THead (Flat Appl) x x0) u2)).(\lambda (P: Prop).(H29
(iso_trans (THead (Flat Appl) x3 x4) (THead (Flat Appl) x x0) (iso_head x3 x
-x4 x0 (Flat Appl)) u2 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
+x4 x0 (Flat Appl)) u2 H30) P)))) (THead (Flat Appl) x1 u2) (pr3_pr2 c (THead
+(Flat Appl) t0 u2) (THead (Flat Appl) x1 u2) (pr2_head_1 c t0 x1 H17 (Flat
+Appl) u2)))))))) H26))) x2 H22))))))) H21)) (\lambda (H21: (ex4_4 T T T T
(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x0
(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
T).(\lambda (t4: T).(eq T x2 (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
(_: T).(\lambda (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)
+(x6: T).(\lambda (H22: (eq T x0 (THead (Bind Abst) x3 x4))).(\lambda (H23:
+(eq T x2 (THead (Bind Abbr) x5 x6))).(\lambda (H24: (pr2 c x x5)).(\lambda
+(H25: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x4
+x6))))).(let H26 \def (eq_ind T x2 (\lambda (t: T).((eq T (THead (Flat Appl)
t0 (THead (Flat Appl) x x0)) (THead (Flat Appl) x1 t)) \to (\forall (P:
-Prop).P))) 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
+Prop).P))) H19 (THead (Bind Abbr) x5 x6) H23) in (eq_ind_r T (THead (Bind
+Abbr) x5 x6) (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 t))) (let H27 \def
(eq_ind T x0 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl)
x t)) (THead (Flat Appl) x1 (THead (Bind Abbr) x5 x6))) \to (\forall (P:
-Prop).P))) H24 (THead (Bind Abst) x3 x4) H20) in (let H26 \def (eq_ind T x0
+Prop).P))) H26 (THead (Bind Abst) x3 x4) H22) in (let H28 \def (eq_ind T x0
(\lambda (t: T).(\forall (t4: T).((((eq T (THead (Flat Appl) x t) t4) \to
(\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t4) \to (sn3 c
-t4))))) H9 (THead (Bind Abst) x3 x4) H20) in (let H27 \def (eq_ind T x0
+t4))))) H11 (THead (Bind Abst) x3 x4) H22) in (let H29 \def (eq_ind T x0
(\lambda (t: T).(\forall (t4: T).((((eq T (THead (Flat Appl) x t) t4) \to
(\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t4) \to (\forall
(x7: T).(\forall (x8: T).((eq T t4 (THead (Flat Appl) x7 x8)) \to (\forall
-(v3: T).((sn3 c v3) \to (((\forall (u2: T).((pr3 c (THead (Flat Appl) x7 x8)
-u2) \to ((((iso (THead (Flat Appl) x7 x8) u2) \to (\forall (P: Prop).P))) \to
-(sn3 c (THead (Flat Appl) v3 u2)))))) \to (sn3 c (THead (Flat Appl) v3 (THead
-(Flat Appl) x7 x8))))))))))))) 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 (t4: T).((((eq T t0
-t4) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t4) \to (((\forall (u2:
-T).((pr3 c (THead (Flat Appl) x t) u2) \to ((((iso (THead (Flat Appl) x t)
-u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t4 u2)))))) \to
-(sn3 c (THead (Flat Appl) t4 (THead (Flat Appl) x t)))))))) 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 in iso return (\lambda (t: T).(\lambda (t4: T).(\lambda (_: (iso t
-t4)).((eq T t (THead (Flat Appl) x (THead (Bind Abst) x3 x4))) \to ((eq T t4
-(THead (Bind Abbr) x5 x6)) \to P))))) with [(iso_sort n1 n2) \Rightarrow
-(\lambda (H31: (eq T (TSort n1) (THead (Flat Appl) x (THead (Bind Abst) x3
-x4)))).(\lambda (H32: (eq T (TSort n2) (THead (Bind Abbr) x5 x6))).((let H33
-\def (eq_ind T (TSort n1) (\lambda (e: T).(match e in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False |
-(THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) x (THead (Bind Abst)
-x3 x4)) H31) in (False_ind ((eq T (TSort n2) (THead (Bind Abbr) x5 x6)) \to
-P) H33)) H32))) | (iso_lref i1 i2) \Rightarrow (\lambda (H31: (eq T (TLRef
-i1) (THead (Flat Appl) x (THead (Bind Abst) x3 x4)))).(\lambda (H32: (eq T
-(TLRef i2) (THead (Bind Abbr) x5 x6))).((let H33 \def (eq_ind T (TLRef i1)
-(\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
-False])) I (THead (Flat Appl) x (THead (Bind Abst) x3 x4)) H31) in (False_ind
-((eq T (TLRef i2) (THead (Bind Abbr) x5 x6)) \to P) H33)) H32))) | (iso_head
-v4 v5 t4 t5 k) \Rightarrow (\lambda (H31: (eq T (THead k v4 t4) (THead (Flat
-Appl) x (THead (Bind Abst) x3 x4)))).(\lambda (H32: (eq T (THead k v5 t5)
-(THead (Bind Abbr) x5 x6))).((let H33 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t4 |
-(TLRef _) \Rightarrow t4 | (THead _ _ t) \Rightarrow t])) (THead k v4 t4)
-(THead (Flat Appl) x (THead (Bind Abst) x3 x4)) H31) in ((let H34 \def
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
-[(TSort _) \Rightarrow v4 | (TLRef _) \Rightarrow v4 | (THead _ t _)
-\Rightarrow t])) (THead k v4 t4) (THead (Flat Appl) x (THead (Bind Abst) x3
-x4)) H31) in ((let H35 \def (f_equal T K (\lambda (e: T).(match e in T return
-(\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k |
-(THead k0 _ _) \Rightarrow k0])) (THead k v4 t4) (THead (Flat Appl) x (THead
-(Bind Abst) x3 x4)) H31) in (eq_ind K (Flat Appl) (\lambda (k0: K).((eq T v4
-x) \to ((eq T t4 (THead (Bind Abst) x3 x4)) \to ((eq T (THead k0 v5 t5)
-(THead (Bind Abbr) x5 x6)) \to P)))) (\lambda (H36: (eq T v4 x)).(eq_ind T x
-(\lambda (_: T).((eq T t4 (THead (Bind Abst) x3 x4)) \to ((eq T (THead (Flat
-Appl) v5 t5) (THead (Bind Abbr) x5 x6)) \to P))) (\lambda (H37: (eq T t4
-(THead (Bind Abst) x3 x4))).(eq_ind T (THead (Bind Abst) x3 x4) (\lambda (_:
-T).((eq T (THead (Flat Appl) v5 t5) (THead (Bind Abbr) x5 x6)) \to P))
-(\lambda (H38: (eq T (THead (Flat Appl) v5 t5) (THead (Bind Abbr) x5
-x6))).(let H39 \def (eq_ind T (THead (Flat Appl) v5 t5) (\lambda (e:
-T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in
-K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _)
-\Rightarrow True])])) I (THead (Bind Abbr) x5 x6) H38) in (False_ind P H39)))
-t4 (sym_eq T t4 (THead (Bind Abst) x3 x4) H37))) v4 (sym_eq T v4 x H36))) k
-(sym_eq K k (Flat Appl) H35))) H34)) H33)) H32)))]) 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 (_:
+(v3: T).((sn3 c v3) \to (((\forall (u2: T).((pr3 c t4 u2) \to ((((iso t4 u2)
+\to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v3 u2)))))) \to
+(sn3 c (THead (Flat Appl) v3 t4)))))))))))) H10 (THead (Bind Abst) x3 x4)
+H22) in (let H30 \def (eq_ind T x0 (\lambda (t: T).(\forall (u2: T).((pr3 c
+(THead (Flat Appl) x t) u2) \to ((((iso (THead (Flat Appl) x t) u2) \to
+(\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2)))))) H8 (THead
+(Bind Abst) x3 x4) H22) in (let H31 \def (eq_ind T x0 (\lambda (t:
+T).(\forall (t4: T).((((eq T t0 t4) \to (\forall (P: Prop).P))) \to ((pr3 c
+t0 t4) \to (((\forall (u2: T).((pr3 c (THead (Flat Appl) x t) u2) \to ((((iso
+(THead (Flat Appl) x t) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead
+(Flat Appl) t4 u2)))))) \to (sn3 c (THead (Flat Appl) t4 (THead (Flat Appl) x
+t)))))))) H9 (THead (Bind Abst) x3 x4) H22) in (sn3_pr3_trans c (THead (Flat
+Appl) t0 (THead (Bind Abbr) x5 x6)) (H30 (THead (Bind Abbr) x5 x6) (pr3_sing
+c (THead (Bind Abbr) x x4) (THead (Flat Appl) x (THead (Bind Abst) x3 x4))
+(pr2_free c (THead (Flat Appl) x (THead (Bind Abst) x3 x4)) (THead (Bind
+Abbr) x x4) (pr0_beta x3 x x (pr0_refl x) x4 x4 (pr0_refl x4))) (THead (Bind
+Abbr) x5 x6) (pr3_head_12 c x x5 (pr3_pr2 c x x5 H24) (Bind Abbr) x4 x6
+(pr3_pr2 (CHead c (Bind Abbr) x5) x4 x6 (H25 Abbr x5)))) (\lambda (H32: (iso
+(THead (Flat Appl) x (THead (Bind Abst) x3 x4)) (THead (Bind Abbr) x5
+x6))).(\lambda (P: Prop).(let H33 \def (match H32 in iso return (\lambda (t:
+T).(\lambda (t4: T).(\lambda (_: (iso t t4)).((eq T t (THead (Flat Appl) x
+(THead (Bind Abst) x3 x4))) \to ((eq T t4 (THead (Bind Abbr) x5 x6)) \to
+P))))) with [(iso_sort n1 n2) \Rightarrow (\lambda (H33: (eq T (TSort n1)
+(THead (Flat Appl) x (THead (Bind Abst) x3 x4)))).(\lambda (H34: (eq T (TSort
+n2) (THead (Bind Abbr) x5 x6))).((let H35 \def (eq_ind T (TSort n1) (\lambda
+(e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I
+(THead (Flat Appl) x (THead (Bind Abst) x3 x4)) H33) in (False_ind ((eq T
+(TSort n2) (THead (Bind Abbr) x5 x6)) \to P) H35)) H34))) | (iso_lref i1 i2)
+\Rightarrow (\lambda (H33: (eq T (TLRef i1) (THead (Flat Appl) x (THead (Bind
+Abst) x3 x4)))).(\lambda (H34: (eq T (TLRef i2) (THead (Bind Abbr) x5
+x6))).((let H35 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e in T
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) x
+(THead (Bind Abst) x3 x4)) H33) in (False_ind ((eq T (TLRef i2) (THead (Bind
+Abbr) x5 x6)) \to P) H35)) H34))) | (iso_head v4 v5 t4 t5 k) \Rightarrow
+(\lambda (H33: (eq T (THead k v4 t4) (THead (Flat Appl) x (THead (Bind Abst)
+x3 x4)))).(\lambda (H34: (eq T (THead k v5 t5) (THead (Bind Abbr) x5
+x6))).((let H35 \def (f_equal T T (\lambda (e: T).(match e in T return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow t4 | (TLRef _) \Rightarrow t4
+| (THead _ _ t) \Rightarrow t])) (THead k v4 t4) (THead (Flat Appl) x (THead
+(Bind Abst) x3 x4)) H33) in ((let H36 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v4 |
+(TLRef _) \Rightarrow v4 | (THead _ t _) \Rightarrow t])) (THead k v4 t4)
+(THead (Flat Appl) x (THead (Bind Abst) x3 x4)) H33) in ((let H37 \def
+(f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with
+[(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _)
+\Rightarrow k0])) (THead k v4 t4) (THead (Flat Appl) x (THead (Bind Abst) x3
+x4)) H33) in (eq_ind K (Flat Appl) (\lambda (k0: K).((eq T v4 x) \to ((eq T
+t4 (THead (Bind Abst) x3 x4)) \to ((eq T (THead k0 v5 t5) (THead (Bind Abbr)
+x5 x6)) \to P)))) (\lambda (H38: (eq T v4 x)).(eq_ind T x (\lambda (_:
+T).((eq T t4 (THead (Bind Abst) x3 x4)) \to ((eq T (THead (Flat Appl) v5 t5)
+(THead (Bind Abbr) x5 x6)) \to P))) (\lambda (H39: (eq T t4 (THead (Bind
+Abst) x3 x4))).(eq_ind T (THead (Bind Abst) x3 x4) (\lambda (_: T).((eq T
+(THead (Flat Appl) v5 t5) (THead (Bind Abbr) x5 x6)) \to P)) (\lambda (H40:
+(eq T (THead (Flat Appl) v5 t5) (THead (Bind Abbr) x5 x6))).(let H41 \def
+(eq_ind T (THead (Flat Appl) v5 t5) (\lambda (e: T).(match e in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda
+(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
+True])])) I (THead (Bind Abbr) x5 x6) H40) in (False_ind P H41))) t4 (sym_eq
+T t4 (THead (Bind Abst) x3 x4) H39))) v4 (sym_eq T v4 x H38))) k (sym_eq K k
+(Flat Appl) H37))) H36)) H35)) H34)))]) in (H33 (refl_equal T (THead (Flat
+Appl) x (THead (Bind Abst) x3 x4))) (refl_equal T (THead (Bind Abbr) x5
+x6))))))) (THead (Flat Appl) x1 (THead (Bind Abbr) x5 x6)) (pr3_pr2 c (THead
+(Flat Appl) t0 (THead (Bind Abbr) x5 x6)) (THead (Flat Appl) x1 (THead (Bind
+Abbr) x5 x6)) (pr2_head_1 c t0 x1 H17 (Flat Appl) (THead (Bind Abbr) x5
+x6))))))))) x2 H23)))))))))) H21)) (\lambda (H21: (ex6_6 B T T T T T (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T x0
(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
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
+(x5: T).(\lambda (x6: T).(\lambda (x7: T).(\lambda (x8: T).(\lambda (H22:
+(not (eq B x3 Abst))).(\lambda (H23: (eq T x0 (THead (Bind x3) x4
+x5))).(\lambda (H24: (eq T x2 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S
+O) O x7) x6)))).(\lambda (H25: (pr2 c x x7)).(\lambda (H26: (pr2 c x4
+x8)).(\lambda (H27: (pr2 (CHead c (Bind x3) x8) x5 x6)).(let H28 \def (eq_ind
T x2 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0))
-(THead (Flat Appl) x1 t)) \to (\forall (P: Prop).P))) H17 (THead (Bind x3) x8
-(THead (Flat Appl) (lift (S O) O x7) x6)) H22) in (eq_ind_r T (THead (Bind
+(THead (Flat Appl) x1 t)) \to (\forall (P: Prop).P))) H19 (THead (Bind x3) x8
+(THead (Flat Appl) (lift (S O) O x7) x6)) H24) in (eq_ind_r T (THead (Bind
x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) (\lambda (t: T).(sn3 c
-(THead (Flat Appl) x1 t))) (let H27 \def (eq_ind T x0 (\lambda (t: T).((eq T
+(THead (Flat Appl) x1 t))) (let H29 \def (eq_ind T x0 (\lambda (t: T).((eq T
(THead (Flat Appl) t0 (THead (Flat Appl) x t)) (THead (Flat Appl) x1 (THead
(Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))) \to (\forall (P:
-Prop).P))) H26 (THead (Bind x3) x4 x5) H21) in (let H28 \def (eq_ind T x0
+Prop).P))) H28 (THead (Bind x3) x4 x5) H23) in (let H30 \def (eq_ind T x0
(\lambda (t: T).(\forall (t4: T).((((eq T (THead (Flat Appl) x t) t4) \to
(\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t4) \to (sn3 c
-t4))))) H9 (THead (Bind x3) x4 x5) H21) in (let H29 \def (eq_ind T x0
+t4))))) H11 (THead (Bind x3) x4 x5) H23) in (let H31 \def (eq_ind T x0
(\lambda (t: T).(\forall (t4: T).((((eq T (THead (Flat Appl) x t) t4) \to
(\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t4) \to (\forall
(x9: T).(\forall (x10: T).((eq T t4 (THead (Flat Appl) x9 x10)) \to (\forall
-(v3: T).((sn3 c v3) \to (((\forall (u2: T).((pr3 c (THead (Flat Appl) x9 x10)
-u2) \to ((((iso (THead (Flat Appl) x9 x10) u2) \to (\forall (P: Prop).P)))
-\to (sn3 c (THead (Flat Appl) v3 u2)))))) \to (sn3 c (THead (Flat Appl) v3
-(THead (Flat Appl) x9 x10))))))))))))) H8 (THead (Bind x3) x4 x5) H21) in
-(let H30 \def (eq_ind T x0 (\lambda (t: T).(\forall (u2: T).((pr3 c (THead
+(v3: T).((sn3 c v3) \to (((\forall (u2: T).((pr3 c t4 u2) \to ((((iso t4 u2)
+\to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v3 u2)))))) \to
+(sn3 c (THead (Flat Appl) v3 t4)))))))))))) H10 (THead (Bind x3) x4 x5) H23)
+in (let H32 \def (eq_ind T x0 (\lambda (t: T).(\forall (u2: T).((pr3 c (THead
(Flat Appl) x t) u2) \to ((((iso (THead (Flat Appl) x t) u2) \to (\forall (P:
-Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2)))))) H7 (THead (Bind x3) x4
-x5) H21) in (let H31 \def (eq_ind T x0 (\lambda (t: T).(\forall (t4:
+Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2)))))) H8 (THead (Bind x3) x4
+x5) H23) in (let H33 \def (eq_ind T x0 (\lambda (t: T).(\forall (t4:
T).((((eq T t0 t4) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t4) \to
(((\forall (u2: T).((pr3 c (THead (Flat Appl) x t) u2) \to ((((iso (THead
(Flat Appl) x t) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat
Appl) t4 u2)))))) \to (sn3 c (THead (Flat Appl) t4 (THead (Flat Appl) x
-t)))))))) 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
+t)))))))) H9 (THead (Bind x3) x4 x5) H23) in (sn3_pr3_trans c (THead (Flat
+Appl) t0 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) (H32
(THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) (pr3_sing c
(THead (Bind x3) x4 (THead (Flat Appl) (lift (S O) O x) x5)) (THead (Flat
Appl) x (THead (Bind x3) x4 x5)) (pr2_free c (THead (Flat Appl) x (THead
(Bind x3) x4 x5)) (THead (Bind x3) x4 (THead (Flat Appl) (lift (S O) O x)
-x5)) (pr0_upsilon x3 H20 x x (pr0_refl x) x4 x4 (pr0_refl x4) x5 x5 (pr0_refl
+x5)) (pr0_upsilon x3 H22 x x (pr0_refl x) x4 x4 (pr0_refl x4) x5 x5 (pr0_refl
x5))) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))
-(pr3_head_12 c x4 x8 (pr3_pr2 c x4 x8 H24) (Bind x3) (THead (Flat Appl) (lift
+(pr3_head_12 c x4 x8 (pr3_pr2 c x4 x8 H26) (Bind x3) (THead (Flat Appl) (lift
(S O) O x) x5) (THead (Flat Appl) (lift (S O) O x7) x6) (pr3_head_12 (CHead c
(Bind x3) x8) (lift (S O) O x) (lift (S O) O x7) (pr3_lift (CHead c (Bind x3)
x8) c (S O) O (drop_drop (Bind x3) O c c (drop_refl c) x8) x x7 (pr3_pr2 c x
-x7 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
+x7 H25)) (Flat Appl) x5 x6 (pr3_pr2 (CHead (CHead c (Bind x3) x8) (Flat Appl)
+(lift (S O) O x7)) x5 x6 (pr2_cflat (CHead c (Bind x3) x8) x5 x6 H27 Appl
+(lift (S O) O x7)))))) (\lambda (H34: (iso (THead (Flat Appl) x (THead (Bind
x3) x4 x5)) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7)
-x6)))).(\lambda (P: Prop).(let H33 \def (match H32 in iso return (\lambda (t:
+x6)))).(\lambda (P: Prop).(let H35 \def (match H34 in iso return (\lambda (t:
T).(\lambda (t4: T).(\lambda (_: (iso t t4)).((eq T t (THead (Flat Appl) x
(THead (Bind x3) x4 x5))) \to ((eq T t4 (THead (Bind x3) x8 (THead (Flat
Appl) (lift (S O) O x7) x6))) \to P))))) with [(iso_sort n1 n2) \Rightarrow
-(\lambda (H33: (eq T (TSort n1) (THead (Flat Appl) x (THead (Bind x3) x4
-x5)))).(\lambda (H34: (eq T (TSort n2) (THead (Bind x3) x8 (THead (Flat Appl)
-(lift (S O) O x7) x6)))).((let H35 \def (eq_ind T (TSort n1) (\lambda (e:
+(\lambda (H35: (eq T (TSort n1) (THead (Flat Appl) x (THead (Bind x3) x4
+x5)))).(\lambda (H36: (eq T (TSort n2) (THead (Bind x3) x8 (THead (Flat Appl)
+(lift (S O) O x7) x6)))).((let H37 \def (eq_ind T (TSort n1) (\lambda (e:
T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I
-(THead (Flat Appl) x (THead (Bind x3) x4 x5)) H33) in (False_ind ((eq T
+(THead (Flat Appl) x (THead (Bind x3) x4 x5)) H35) in (False_ind ((eq T
(TSort n2) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to
-P) H35)) H34))) | (iso_lref i1 i2) \Rightarrow (\lambda (H33: (eq T (TLRef
-i1) (THead (Flat Appl) x (THead (Bind x3) x4 x5)))).(\lambda (H34: (eq T
+P) H37)) H36))) | (iso_lref i1 i2) \Rightarrow (\lambda (H35: (eq T (TLRef
+i1) (THead (Flat Appl) x (THead (Bind x3) x4 x5)))).(\lambda (H36: (eq T
(TLRef i2) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7)
-x6)))).((let H35 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e in T
+x6)))).((let H37 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e in T
return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) x
-(THead (Bind x3) x4 x5)) H33) in (False_ind ((eq T (TLRef i2) (THead (Bind
-x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to P) H35)) H34))) |
-(iso_head v4 v5 t4 t5 k) \Rightarrow (\lambda (H33: (eq T (THead k v4 t4)
-(THead (Flat Appl) x (THead (Bind x3) x4 x5)))).(\lambda (H34: (eq T (THead k
+(THead (Bind x3) x4 x5)) H35) in (False_ind ((eq T (TLRef i2) (THead (Bind
+x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to P) H37)) H36))) |
+(iso_head v4 v5 t4 t5 k) \Rightarrow (\lambda (H35: (eq T (THead k v4 t4)
+(THead (Flat Appl) x (THead (Bind x3) x4 x5)))).(\lambda (H36: (eq T (THead k
v5 t5) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))).((let
-H35 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
+H37 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
with [(TSort _) \Rightarrow t4 | (TLRef _) \Rightarrow t4 | (THead _ _ t)
\Rightarrow t])) (THead k v4 t4) (THead (Flat Appl) x (THead (Bind x3) x4
-x5)) H33) in ((let H36 \def (f_equal T T (\lambda (e: T).(match e in T return
+x5)) H35) in ((let H38 \def (f_equal T T (\lambda (e: T).(match e in T return
(\lambda (_: T).T) with [(TSort _) \Rightarrow v4 | (TLRef _) \Rightarrow v4
| (THead _ t _) \Rightarrow t])) (THead k v4 t4) (THead (Flat Appl) x (THead
-(Bind x3) x4 x5)) H33) in ((let H37 \def (f_equal T K (\lambda (e: T).(match
+(Bind x3) x4 x5)) H35) in ((let H39 \def (f_equal T K (\lambda (e: T).(match
e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k v4 t4) (THead (Flat
-Appl) x (THead (Bind x3) x4 x5)) H33) in (eq_ind K (Flat Appl) (\lambda (k0:
+Appl) x (THead (Bind x3) x4 x5)) H35) in (eq_ind K (Flat Appl) (\lambda (k0:
K).((eq T v4 x) \to ((eq T t4 (THead (Bind x3) x4 x5)) \to ((eq T (THead k0
v5 t5) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to
-P)))) (\lambda (H38: (eq T v4 x)).(eq_ind T x (\lambda (_: T).((eq T t4
+P)))) (\lambda (H40: (eq T v4 x)).(eq_ind T x (\lambda (_: T).((eq T t4
(THead (Bind x3) x4 x5)) \to ((eq T (THead (Flat Appl) v5 t5) (THead (Bind
-x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to P))) (\lambda (H39: (eq
+x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to P))) (\lambda (H41: (eq
T t4 (THead (Bind x3) x4 x5))).(eq_ind T (THead (Bind x3) x4 x5) (\lambda (_:
T).((eq T (THead (Flat Appl) v5 t5) (THead (Bind x3) x8 (THead (Flat Appl)
-(lift (S O) O x7) x6))) \to P)) (\lambda (H40: (eq T (THead (Flat Appl) v5
-t5) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))).(let H41
+(lift (S O) O x7) x6))) \to P)) (\lambda (H42: (eq T (THead (Flat Appl) v5
+t5) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))).(let H43
\def (eq_ind T (THead (Flat Appl) v5 t5) (\lambda (e: T).(match e in T return
(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda
(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
True])])) I (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))
-H40) in (False_ind P H41))) t4 (sym_eq T t4 (THead (Bind x3) x4 x5) H39))) v4
-(sym_eq T v4 x H38))) k (sym_eq K k (Flat Appl) H37))) H36)) H35)) H34)))])
-in (H33 (refl_equal T (THead (Flat Appl) x (THead (Bind x3) x4 x5)))
+H42) in (False_ind P H43))) t4 (sym_eq T t4 (THead (Bind x3) x4 x5) H41))) v4
+(sym_eq T v4 x H40))) k (sym_eq K k (Flat Appl) H39))) H38)) H37)) H36)))])
+in (H35 (refl_equal T (THead (Flat Appl) x (THead (Bind x3) x4 x5)))
(refl_equal T (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7)
x6)))))))) (THead (Flat Appl) x1 (THead (Bind x3) x8 (THead (Flat Appl) (lift
(S O) O x7) x6))) (pr3_pr2 c (THead (Flat Appl) t0 (THead (Bind x3) x8 (THead
(Flat Appl) (lift (S O) O x7) x6))) (THead (Flat Appl) x1 (THead (Bind x3) x8
-(THead (Flat Appl) (lift (S O) O x7) x6))) (pr2_head_1 c t0 x1 H15 (Flat
+(THead (Flat Appl) (lift (S O) O x7) x6))) (pr2_head_1 c t0 x1 H17 (Flat
Appl) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))))))))))
-x2 H22)))))))))))))) H19)) H18)) t3 H14))))))) H13)) (\lambda (H13: (ex4_4 T
+x2 H24)))))))))))))) H21)) H20)) t3 H16))))))) H15)) (\lambda (H15: (ex4_4 T
T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
(THead (Flat Appl) x x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind
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
+T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H16: (eq T
+(THead (Flat Appl) x x0) (THead (Bind Abst) x1 x2))).(\lambda (H17: (eq T t3
(THead (Bind Abbr) x3 x4))).(\lambda (_: (pr2 c t0 x3)).(\lambda (_:
((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x2 x4))))).(let
-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
+H20 \def (eq_ind T t3 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead
+(Flat Appl) x x0)) t) \to (\forall (P: Prop).P))) H12 (THead (Bind Abbr) x3
+x4) H17) in (eq_ind_r T (THead (Bind Abbr) x3 x4) (\lambda (t: T).(sn3 c t))
+(let H21 \def (eq_ind T (THead (Flat Appl) x x0) (\lambda (ee: T).(match ee
in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef
_) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return
(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
-True])])) I (THead (Bind Abst) x1 x2) 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
+True])])) I (THead (Bind Abst) x1 x2) H16) in (False_ind (sn3 c (THead (Bind
+Abbr) x3 x4)) H21)) t3 H17)))))))))) H15)) (\lambda (H15: (ex6_6 B T T T T T
(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
(\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
+(x5: T).(\lambda (x6: T).(\lambda (_: (not (eq B x1 Abst))).(\lambda (H17:
+(eq T (THead (Flat Appl) x x0) (THead (Bind x1) x2 x3))).(\lambda (H18: (eq T
t3 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))).(\lambda
(_: (pr2 c t0 x5)).(\lambda (_: (pr2 c x2 x6)).(\lambda (_: (pr2 (CHead c
-(Bind x1) x6) x3 x4)).(let H20 \def (eq_ind T t3 (\lambda (t: T).((eq T
+(Bind x1) x6) x3 x4)).(let H22 \def (eq_ind T t3 (\lambda (t: T).((eq T
(THead (Flat Appl) t0 (THead (Flat Appl) x x0)) t) \to (\forall (P:
-Prop).P))) 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
+Prop).P))) H12 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4))
+H18) in (eq_ind_r T (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5)
+x4)) (\lambda (t: T).(sn3 c t)) (let H23 \def (eq_ind T (THead (Flat Appl) x
x0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind x1) x2 x3)
-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))))).
+H17) in (False_ind (sn3 c (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O)
+O x5) x4))) H23)) t3 H18)))))))))))))) H15)) H14)))))) t2 H3))))))))) v2
+H4))))))))) y H0))))) H))))).
theorem sn3_appl_beta:
\forall (c: C).(\forall (u: T).(\forall (v: T).(\forall (t: T).((sn3 c
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / 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/LambdaDelta-1/spare".
+
+include "theory.ma".
+
+definition cbk:
+ C \to nat
+\def
+ let rec cbk (c: C) on c: nat \def (match c with [(CSort m) \Rightarrow m |
+(CHead c0 _ _) \Rightarrow (cbk c0)]) in cbk.
+
+definition app1:
+ C \to (T \to T)
+\def
+ let rec app1 (c: C) on c: (T \to T) \def (\lambda (t: T).(match c with
+[(CSort _) \Rightarrow t | (CHead c0 k u) \Rightarrow (app1 c0 (THead k u
+t))])) in app1.
+
+theorem lifts_inj:
+ \forall (xs: TList).(\forall (ts: TList).(\forall (h: nat).(\forall (d:
+nat).((eq TList (lifts h d xs) (lifts h d ts)) \to (eq TList xs ts)))))
+\def
+ \lambda (xs: TList).(TList_ind (\lambda (t: TList).(\forall (ts:
+TList).(\forall (h: nat).(\forall (d: nat).((eq TList (lifts h d t) (lifts h
+d ts)) \to (eq TList t ts)))))) (\lambda (ts: TList).(TList_ind (\lambda (t:
+TList).(\forall (h: nat).(\forall (d: nat).((eq TList (lifts h d TNil) (lifts
+h d t)) \to (eq TList TNil t))))) (\lambda (_: nat).(\lambda (_:
+nat).(\lambda (H: (eq TList TNil TNil)).H))) (\lambda (t: T).(\lambda (t0:
+TList).(\lambda (_: ((\forall (h: nat).(\forall (d: nat).((eq TList TNil
+(lifts h d t0)) \to (eq TList TNil t0)))))).(\lambda (h: nat).(\lambda (d:
+nat).(\lambda (H0: (eq TList TNil (TCons (lift h d t) (lifts h d t0)))).(let
+H1 \def (eq_ind TList TNil (\lambda (ee: TList).(match ee in TList return
+(\lambda (_: TList).Prop) with [TNil \Rightarrow True | (TCons _ _)
+\Rightarrow False])) I (TCons (lift h d t) (lifts h d t0)) H0) in (False_ind
+(eq TList TNil (TCons t t0)) H1)))))))) ts)) (\lambda (t: T).(\lambda (t0:
+TList).(\lambda (H: ((\forall (ts: TList).(\forall (h: nat).(\forall (d:
+nat).((eq TList (lifts h d t0) (lifts h d ts)) \to (eq TList t0
+ts))))))).(\lambda (ts: TList).(TList_ind (\lambda (t1: TList).(\forall (h:
+nat).(\forall (d: nat).((eq TList (lifts h d (TCons t t0)) (lifts h d t1))
+\to (eq TList (TCons t t0) t1))))) (\lambda (h: nat).(\lambda (d:
+nat).(\lambda (H0: (eq TList (TCons (lift h d t) (lifts h d t0)) TNil)).(let
+H1 \def (eq_ind TList (TCons (lift h d t) (lifts h d t0)) (\lambda (ee:
+TList).(match ee in TList return (\lambda (_: TList).Prop) with [TNil
+\Rightarrow False | (TCons _ _) \Rightarrow True])) I TNil H0) in (False_ind
+(eq TList (TCons t t0) TNil) H1))))) (\lambda (t1: T).(\lambda (t2:
+TList).(\lambda (_: ((\forall (h: nat).(\forall (d: nat).((eq TList (TCons
+(lift h d t) (lifts h d t0)) (lifts h d t2)) \to (eq TList (TCons t t0)
+t2)))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (eq TList (TCons
+(lift h d t) (lifts h d t0)) (TCons (lift h d t1) (lifts h d t2)))).(let H2
+\def (f_equal TList T (\lambda (e: TList).(match e in TList return (\lambda
+(_: TList).T) with [TNil \Rightarrow ((let rec lref_map (f: ((nat \to nat)))
+(d0: nat) (t3: T) on t3: T \def (match t3 with [(TSort n) \Rightarrow (TSort
+n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) with [true \Rightarrow i
+| false \Rightarrow (f i)])) | (THead k u t4) \Rightarrow (THead k (lref_map
+f d0 u) (lref_map f (s k d0) t4))]) in lref_map) (\lambda (x: nat).(plus x
+h)) d t) | (TCons t3 _) \Rightarrow t3])) (TCons (lift h d t) (lifts h d t0))
+(TCons (lift h d t1) (lifts h d t2)) H1) in ((let H3 \def (f_equal TList
+TList (\lambda (e: TList).(match e in TList return (\lambda (_: TList).TList)
+with [TNil \Rightarrow ((let rec lifts (h0: nat) (d0: nat) (ts0: TList) on
+ts0: TList \def (match ts0 with [TNil \Rightarrow TNil | (TCons t3 ts1)
+\Rightarrow (TCons (lift h0 d0 t3) (lifts h0 d0 ts1))]) in lifts) h d t0) |
+(TCons _ t3) \Rightarrow t3])) (TCons (lift h d t) (lifts h d t0)) (TCons
+(lift h d t1) (lifts h d t2)) H1) in (\lambda (H4: (eq T (lift h d t) (lift h
+d t1))).(eq_ind T t (\lambda (t3: T).(eq TList (TCons t t0) (TCons t3 t2)))
+(f_equal2 T TList TList TCons t t t0 t2 (refl_equal T t) (H t2 h d H3)) t1
+(lift_inj t t1 h d H4)))) H2)))))))) ts))))) xs).
+
+theorem nfs2_tapp:
+ \forall (c: C).(\forall (t: T).(\forall (ts: TList).((nfs2 c (TApp ts t))
+\to (land (nfs2 c ts) (nf2 c t)))))
+\def
+ \lambda (c: C).(\lambda (t: T).(\lambda (ts: TList).(TList_ind (\lambda (t0:
+TList).((nfs2 c (TApp t0 t)) \to (land (nfs2 c t0) (nf2 c t)))) (\lambda (H:
+(land (nf2 c t) True)).(let H0 \def H in (and_ind (nf2 c t) True (land True
+(nf2 c t)) (\lambda (H1: (nf2 c t)).(\lambda (_: True).(conj True (nf2 c t) I
+H1))) H0))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H: (((nfs2 c
+(TApp t1 t)) \to (land (nfs2 c t1) (nf2 c t))))).(\lambda (H0: (land (nf2 c
+t0) (nfs2 c (TApp t1 t)))).(let H1 \def H0 in (and_ind (nf2 c t0) (nfs2 c
+(TApp t1 t)) (land (land (nf2 c t0) (nfs2 c t1)) (nf2 c t)) (\lambda (H2:
+(nf2 c t0)).(\lambda (H3: (nfs2 c (TApp t1 t))).(let H_x \def (H H3) in (let
+H4 \def H_x in (and_ind (nfs2 c t1) (nf2 c t) (land (land (nf2 c t0) (nfs2 c
+t1)) (nf2 c t)) (\lambda (H5: (nfs2 c t1)).(\lambda (H6: (nf2 c t)).(conj
+(land (nf2 c t0) (nfs2 c t1)) (nf2 c t) (conj (nf2 c t0) (nfs2 c t1) H2 H5)
+H6))) H4))))) H1)))))) ts))).
+
+theorem pc3_nf2_unfold:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to ((nf2 c
+t2) \to (pr3 c t1 t2)))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1
+t2)).(\lambda (H0: (nf2 c t2)).(let H1 \def H in (ex2_ind T (\lambda (t:
+T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) (pr3 c t1 t2) (\lambda (x:
+T).(\lambda (H2: (pr3 c t1 x)).(\lambda (H3: (pr3 c t2 x)).(let H_y \def
+(nf2_pr3_unfold c t2 x H3 H0) in (let H4 \def (eq_ind_r T x (\lambda (t:
+T).(pr3 c t1 t)) H2 t2 H_y) in H4))))) H1)))))).
+
+theorem pc3_pr3_conf:
+ \forall (c: C).(\forall (t: T).(\forall (t1: T).((pc3 c t t1) \to (\forall
+(t2: T).((pr3 c t t2) \to (pc3 c t2 t1))))))
+\def
+ \lambda (c: C).(\lambda (t: T).(\lambda (t1: T).(\lambda (H: (pc3 c t
+t1)).(\lambda (t2: T).(\lambda (H0: (pr3 c t t2)).(pc3_t t c t2 (pc3_pr3_x c
+t2 t H0) t1 H)))))).
+
+axiom pc3_gen_appls_sort_abst:
+ \forall (c: C).(\forall (vs: TList).(\forall (w: T).(\forall (u: T).(\forall
+(n: nat).((pc3 c (THeads (Flat Appl) vs (TSort n)) (THead (Bind Abst) w u))
+\to False)))))
+.
+
+axiom pc3_gen_appls_lref_abst:
+ \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c
+(CHead d (Bind Abst) v)) \to (\forall (vs: TList).(\forall (w: T).(\forall
+(u: T).((pc3 c (THeads (Flat Appl) vs (TLRef i)) (THead (Bind Abst) w u)) \to
+False))))))))
+.
+
+axiom pc3_gen_appls_lref_sort:
+ \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c
+(CHead d (Bind Abst) v)) \to (\forall (vs: TList).(\forall (ws:
+TList).(\forall (n: nat).((pc3 c (THeads (Flat Appl) vs (TLRef i)) (THeads
+(Flat Appl) ws (TSort n))) \to False))))))))
+.
+
+inductive tys3 (g: G) (c: C): TList \to (T \to Prop) \def
+| tys3_nil: \forall (u: T).(\forall (u0: T).((ty3 g c u u0) \to (tys3 g c
+TNil u)))
+| tys3_cons: \forall (t: T).(\forall (u: T).((ty3 g c t u) \to (\forall (ts:
+TList).((tys3 g c ts u) \to (tys3 g c (TCons t ts) u))))).
+
+theorem tys3_inv_coq:
+ \forall (g: G).(\forall (c: C).(\forall (ts: TList).(\forall (u: T).(\forall
+(P: ((G \to (C \to (TList \to (T \to Prop)))))).((((tys3 g c ts u) \to
+(\forall (u0: T).(\forall (u1: T).((eq TList TNil ts) \to ((eq T u0 u) \to
+((ty3 g c u0 u1) \to (P g c ts u)))))))) \to ((((tys3 g c ts u) \to (\forall
+(t: T).(\forall (u0: T).(\forall (ts0: TList).((eq TList (TCons t ts0) ts)
+\to ((eq T u0 u) \to ((ty3 g c t u0) \to ((tys3 g c ts0 u0) \to (P g c ts
+u)))))))))) \to ((tys3 g c ts u) \to (P g c ts u))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (ts: TList).(\lambda (u: T).(\lambda
+(P: ((G \to (C \to (TList \to (T \to Prop)))))).(\lambda (H: (((tys3 g c ts
+u) \to (\forall (u0: T).(\forall (u1: T).((eq TList TNil ts) \to ((eq T u0 u)
+\to ((ty3 g c u0 u1) \to (P g c ts u))))))))).(\lambda (H0: (((tys3 g c ts u)
+\to (\forall (t: T).(\forall (u0: T).(\forall (ts0: TList).((eq TList (TCons
+t ts0) ts) \to ((eq T u0 u) \to ((ty3 g c t u0) \to ((tys3 g c ts0 u0) \to (P
+g c ts u))))))))))).(\lambda (H1: (tys3 g c ts u)).(let H2 \def (match H1 in
+tys3 return (\lambda (t: TList).(\lambda (t0: T).(\lambda (_: (tys3 ? ? t
+t0)).((eq TList t ts) \to ((eq T t0 u) \to (P g c ts u)))))) with [(tys3_nil
+u0 u1 H2) \Rightarrow (\lambda (H3: (eq TList TNil ts)).(\lambda (H4: (eq T
+u0 u)).(H H1 u0 u1 H3 H4 H2))) | (tys3_cons t u0 H2 ts0 H3) \Rightarrow
+(\lambda (H4: (eq TList (TCons t ts0) ts)).(\lambda (H5: (eq T u0 u)).(H0 H1
+t u0 ts0 H4 H5 H2 H3)))]) in (H2 (refl_equal TList ts) (refl_equal T
+u)))))))))).
+
+theorem tys3_gen_nil:
+ \forall (g: G).(\forall (c: C).(\forall (u: T).((tys3 g c TNil u) \to (ex T
+(\lambda (u0: T).(ty3 g c u u0))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (H: (tys3 g c TNil
+u)).(tys3_inv_coq g c TNil u (\lambda (g0: G).(\lambda (c0: C).(\lambda (_:
+TList).(\lambda (t0: T).(ex T (\lambda (u0: T).(ty3 g0 c0 t0 u0)))))))
+(\lambda (_: (tys3 g c TNil u)).(\lambda (u0: T).(\lambda (u1: T).(\lambda
+(_: (eq TList TNil TNil)).(\lambda (H2: (eq T u0 u)).(\lambda (H3: (ty3 g c
+u0 u1)).(let H4 \def (eq_ind T u0 (\lambda (t: T).(ty3 g c t u1)) H3 u H2) in
+(ex_intro T (\lambda (u2: T).(ty3 g c u u2)) u1 H4)))))))) (\lambda (_: (tys3
+g c TNil u)).(\lambda (t: T).(\lambda (u0: T).(\lambda (ts0: TList).(\lambda
+(H2: (eq TList (TCons t ts0) TNil)).(\lambda (H3: (eq T u0 u)).(\lambda (H1:
+(ty3 g c t u0)).(\lambda (H4: (tys3 g c ts0 u0)).(let H5 \def (eq_ind T u0
+(\lambda (t0: T).(tys3 g c ts0 t0)) H4 u H3) in (let H6 \def (eq_ind T u0
+(\lambda (t0: T).(ty3 g c t t0)) H1 u H3) in (let H7 \def (eq_ind TList
+(TCons t ts0) (\lambda (ee: TList).(match ee in TList return (\lambda (_:
+TList).Prop) with [TNil \Rightarrow False | (TCons _ _) \Rightarrow True])) I
+TNil H2) in (False_ind (ex T (\lambda (u1: T).(ty3 g c u u1))) H7))))))))))))
+H)))).
+
+theorem tys3_gen_cons:
+ \forall (g: G).(\forall (c: C).(\forall (ts: TList).(\forall (t: T).(\forall
+(u: T).((tys3 g c (TCons t ts) u) \to (land (ty3 g c t u) (tys3 g c ts
+u)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (ts: TList).(\lambda (t: T).(\lambda
+(u: T).(\lambda (H: (tys3 g c (TCons t ts) u)).(tys3_inv_coq g c (TCons t ts)
+u (\lambda (g0: G).(\lambda (c0: C).(\lambda (_: TList).(\lambda (t1:
+T).(land (ty3 g0 c0 t t1) (tys3 g0 c0 ts t1)))))) (\lambda (_: (tys3 g c
+(TCons t ts) u)).(\lambda (u0: T).(\lambda (u1: T).(\lambda (H1: (eq TList
+TNil (TCons t ts))).(\lambda (H2: (eq T u0 u)).(\lambda (H3: (ty3 g c u0
+u1)).(let H4 \def (eq_ind T u0 (\lambda (t0: T).(ty3 g c t0 u1)) H3 u H2) in
+(let H5 \def (eq_ind TList TNil (\lambda (ee: TList).(match ee in TList
+return (\lambda (_: TList).Prop) with [TNil \Rightarrow True | (TCons _ _)
+\Rightarrow False])) I (TCons t ts) H1) in (False_ind (land (ty3 g c t u)
+(tys3 g c ts u)) H5))))))))) (\lambda (_: (tys3 g c (TCons t ts) u)).(\lambda
+(t0: T).(\lambda (u0: T).(\lambda (ts0: TList).(\lambda (H2: (eq TList (TCons
+t0 ts0) (TCons t ts))).(\lambda (H3: (eq T u0 u)).(\lambda (H1: (ty3 g c t0
+u0)).(\lambda (H4: (tys3 g c ts0 u0)).(let H5 \def (eq_ind T u0 (\lambda (t1:
+T).(tys3 g c ts0 t1)) H4 u H3) in (let H6 \def (eq_ind T u0 (\lambda (t1:
+T).(ty3 g c t0 t1)) H1 u H3) in (let H7 \def (f_equal TList T (\lambda (e:
+TList).(match e in TList return (\lambda (_: TList).T) with [TNil \Rightarrow
+t0 | (TCons t1 _) \Rightarrow t1])) (TCons t0 ts0) (TCons t ts) H2) in ((let
+H8 \def (f_equal TList TList (\lambda (e: TList).(match e in TList return
+(\lambda (_: TList).TList) with [TNil \Rightarrow ts0 | (TCons _ t1)
+\Rightarrow t1])) (TCons t0 ts0) (TCons t ts) H2) in (\lambda (H9: (eq T t0
+t)).(let H10 \def (eq_ind TList ts0 (\lambda (t1: TList).(tys3 g c t1 u)) H5
+ts H8) in (let H11 \def (eq_ind T t0 (\lambda (t1: T).(ty3 g c t1 u)) H6 t
+H9) in (conj (ty3 g c t u) (tys3 g c ts u) H11 H10))))) H7))))))))))))
+H)))))).
+
+theorem ty3_getl_subst0:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t
+u) \to (\forall (v0: T).(\forall (t0: T).(\forall (i: nat).((subst0 i v0 t
+t0) \to (\forall (b: B).(\forall (d: C).(\forall (v: T).((getl i c (CHead d
+(Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v w)))))))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H:
+(ty3 g c t u)).(ty3_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (_:
+T).(\forall (v0: T).(\forall (t2: T).(\forall (i: nat).((subst0 i v0 t0 t2)
+\to (\forall (b: B).(\forall (d: C).(\forall (v: T).((getl i c0 (CHead d
+(Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v w)))))))))))))) (\lambda
+(c0: C).(\lambda (t2: T).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t2
+t0)).(\lambda (_: ((\forall (v0: T).(\forall (t1: T).(\forall (i:
+nat).((subst0 i v0 t2 t1) \to (\forall (b: B).(\forall (d: C).(\forall (v:
+T).((getl i c0 (CHead d (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v
+w))))))))))))).(\lambda (u0: T).(\lambda (t1: T).(\lambda (_: (ty3 g c0 u0
+t1)).(\lambda (H3: ((\forall (v0: T).(\forall (t3: T).(\forall (i:
+nat).((subst0 i v0 u0 t3) \to (\forall (b: B).(\forall (d: C).(\forall (v:
+T).((getl i c0 (CHead d (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v
+w))))))))))))).(\lambda (_: (pc3 c0 t1 t2)).(\lambda (v0: T).(\lambda (t3:
+T).(\lambda (i: nat).(\lambda (H5: (subst0 i v0 u0 t3)).(\lambda (b:
+B).(\lambda (d: C).(\lambda (v: T).(\lambda (H6: (getl i c0 (CHead d (Bind b)
+v))).(H3 v0 t3 i H5 b d v H6))))))))))))))))))) (\lambda (c0: C).(\lambda (m:
+nat).(\lambda (v0: T).(\lambda (t0: T).(\lambda (i: nat).(\lambda (H0:
+(subst0 i v0 (TSort m) t0)).(\lambda (b: B).(\lambda (d: C).(\lambda (v:
+T).(\lambda (_: (getl i c0 (CHead d (Bind b) v))).(subst0_gen_sort v0 t0 i m
+H0 (ex T (\lambda (w: T).(ty3 g d v w)))))))))))))) (\lambda (n:
+nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (H0: (getl n
+c0 (CHead d (Bind Abbr) u0))).(\lambda (t0: T).(\lambda (H1: (ty3 g d u0
+t0)).(\lambda (_: ((\forall (v0: T).(\forall (t1: T).(\forall (i:
+nat).((subst0 i v0 u0 t1) \to (\forall (b: B).(\forall (d0: C).(\forall (v:
+T).((getl i d (CHead d0 (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d0 v
+w))))))))))))).(\lambda (v0: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda
+(H3: (subst0 i v0 (TLRef n) t1)).(\lambda (b: B).(\lambda (d0: C).(\lambda
+(v: T).(\lambda (H4: (getl i c0 (CHead d0 (Bind b) v))).(and_ind (eq nat n i)
+(eq T t1 (lift (S n) O v0)) (ex T (\lambda (w: T).(ty3 g d0 v w))) (\lambda
+(H5: (eq nat n i)).(\lambda (_: (eq T t1 (lift (S n) O v0))).(let H7 \def
+(eq_ind_r nat i (\lambda (n0: nat).(getl n0 c0 (CHead d0 (Bind b) v))) H4 n
+H5) in (let H8 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda (c1: C).(getl
+n c0 c1)) H0 (CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind Abbr) u0) n
+H0 (CHead d0 (Bind b) v) H7)) in (let H9 \def (f_equal C C (\lambda (e:
+C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d |
+(CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u0) (CHead d0 (Bind b)
+v) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0 (Bind b) v) H7)) in
+((let H10 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_:
+C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k
+in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _)
+\Rightarrow Abbr])])) (CHead d (Bind Abbr) u0) (CHead d0 (Bind b) v)
+(getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0 (Bind b) v) H7)) in
+((let H11 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_:
+C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t2) \Rightarrow t2]))
+(CHead d (Bind Abbr) u0) (CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind
+Abbr) u0) n H0 (CHead d0 (Bind b) v) H7)) in (\lambda (H12: (eq B Abbr
+b)).(\lambda (H13: (eq C d d0)).(let H14 \def (eq_ind_r T v (\lambda (t2:
+T).(getl n c0 (CHead d0 (Bind b) t2))) H8 u0 H11) in (eq_ind T u0 (\lambda
+(t2: T).(ex T (\lambda (w: T).(ty3 g d0 t2 w)))) (let H15 \def (eq_ind_r C d0
+(\lambda (c1: C).(getl n c0 (CHead c1 (Bind b) u0))) H14 d H13) in (eq_ind C
+d (\lambda (c1: C).(ex T (\lambda (w: T).(ty3 g c1 u0 w)))) (let H16 \def
+(eq_ind_r B b (\lambda (b0: B).(getl n c0 (CHead d (Bind b0) u0))) H15 Abbr
+H12) in (ex_intro T (\lambda (w: T).(ty3 g d u0 w)) t0 H1)) d0 H13)) v
+H11))))) H10)) H9)))))) (subst0_gen_lref v0 t1 i n H3))))))))))))))))))
+(\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda
+(H0: (getl n c0 (CHead d (Bind Abst) u0))).(\lambda (t0: T).(\lambda (H1:
+(ty3 g d u0 t0)).(\lambda (_: ((\forall (v0: T).(\forall (t1: T).(\forall (i:
+nat).((subst0 i v0 u0 t1) \to (\forall (b: B).(\forall (d0: C).(\forall (v:
+T).((getl i d (CHead d0 (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d0 v
+w))))))))))))).(\lambda (v0: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda
+(H3: (subst0 i v0 (TLRef n) t1)).(\lambda (b: B).(\lambda (d0: C).(\lambda
+(v: T).(\lambda (H4: (getl i c0 (CHead d0 (Bind b) v))).(and_ind (eq nat n i)
+(eq T t1 (lift (S n) O v0)) (ex T (\lambda (w: T).(ty3 g d0 v w))) (\lambda
+(H5: (eq nat n i)).(\lambda (_: (eq T t1 (lift (S n) O v0))).(let H7 \def
+(eq_ind_r nat i (\lambda (n0: nat).(getl n0 c0 (CHead d0 (Bind b) v))) H4 n
+H5) in (let H8 \def (eq_ind C (CHead d (Bind Abst) u0) (\lambda (c1: C).(getl
+n c0 c1)) H0 (CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind Abst) u0) n
+H0 (CHead d0 (Bind b) v) H7)) in (let H9 \def (f_equal C C (\lambda (e:
+C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d |
+(CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abst) u0) (CHead d0 (Bind b)
+v) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead d0 (Bind b) v) H7)) in
+((let H10 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_:
+C).B) with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match k
+in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _)
+\Rightarrow Abst])])) (CHead d (Bind Abst) u0) (CHead d0 (Bind b) v)
+(getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead d0 (Bind b) v) H7)) in
+((let H11 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_:
+C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t2) \Rightarrow t2]))
+(CHead d (Bind Abst) u0) (CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind
+Abst) u0) n H0 (CHead d0 (Bind b) v) H7)) in (\lambda (H12: (eq B Abst
+b)).(\lambda (H13: (eq C d d0)).(let H14 \def (eq_ind_r T v (\lambda (t2:
+T).(getl n c0 (CHead d0 (Bind b) t2))) H8 u0 H11) in (eq_ind T u0 (\lambda
+(t2: T).(ex T (\lambda (w: T).(ty3 g d0 t2 w)))) (let H15 \def (eq_ind_r C d0
+(\lambda (c1: C).(getl n c0 (CHead c1 (Bind b) u0))) H14 d H13) in (eq_ind C
+d (\lambda (c1: C).(ex T (\lambda (w: T).(ty3 g c1 u0 w)))) (let H16 \def
+(eq_ind_r B b (\lambda (b0: B).(getl n c0 (CHead d (Bind b0) u0))) H15 Abst
+H12) in (ex_intro T (\lambda (w: T).(ty3 g d u0 w)) t0 H1)) d0 H13)) v
+H11))))) H10)) H9)))))) (subst0_gen_lref v0 t1 i n H3))))))))))))))))))
+(\lambda (c0: C).(\lambda (u0: T).(\lambda (t0: T).(\lambda (_: (ty3 g c0 u0
+t0)).(\lambda (H1: ((\forall (v0: T).(\forall (t1: T).(\forall (i:
+nat).((subst0 i v0 u0 t1) \to (\forall (b: B).(\forall (d: C).(\forall (v:
+T).((getl i c0 (CHead d (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v
+w))))))))))))).(\lambda (b: B).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_:
+(ty3 g (CHead c0 (Bind b) u0) t1 t2)).(\lambda (H3: ((\forall (v0:
+T).(\forall (t3: T).(\forall (i: nat).((subst0 i v0 t1 t3) \to (\forall (b0:
+B).(\forall (d: C).(\forall (v: T).((getl i (CHead c0 (Bind b) u0) (CHead d
+(Bind b0) v)) \to (ex T (\lambda (w: T).(ty3 g d v w))))))))))))).(\lambda
+(v0: T).(\lambda (t3: T).(\lambda (i: nat).(\lambda (H4: (subst0 i v0 (THead
+(Bind b) u0 t1) t3)).(\lambda (b0: B).(\lambda (d: C).(\lambda (v:
+T).(\lambda (H5: (getl i c0 (CHead d (Bind b0) v))).(or3_ind (ex2 T (\lambda
+(u2: T).(eq T t3 (THead (Bind b) u2 t1))) (\lambda (u2: T).(subst0 i v0 u0
+u2))) (ex2 T (\lambda (t4: T).(eq T t3 (THead (Bind b) u0 t4))) (\lambda (t4:
+T).(subst0 (s (Bind b) i) v0 t1 t4))) (ex3_2 T T (\lambda (u2: T).(\lambda
+(t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2: T).(\lambda (_:
+T).(subst0 i v0 u0 u2))) (\lambda (_: T).(\lambda (t4: T).(subst0 (s (Bind b)
+i) v0 t1 t4)))) (ex T (\lambda (w: T).(ty3 g d v w))) (\lambda (H6: (ex2 T
+(\lambda (u2: T).(eq T t3 (THead (Bind b) u2 t1))) (\lambda (u2: T).(subst0 i
+v0 u0 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t3 (THead (Bind b) u2 t1)))
+(\lambda (u2: T).(subst0 i v0 u0 u2)) (ex T (\lambda (w: T).(ty3 g d v w)))
+(\lambda (x: T).(\lambda (_: (eq T t3 (THead (Bind b) x t1))).(\lambda (H8:
+(subst0 i v0 u0 x)).(H1 v0 x i H8 b0 d v H5)))) H6)) (\lambda (H6: (ex2 T
+(\lambda (t4: T).(eq T t3 (THead (Bind b) u0 t4))) (\lambda (t4: T).(subst0
+(s (Bind b) i) v0 t1 t4)))).(ex2_ind T (\lambda (t4: T).(eq T t3 (THead (Bind
+b) u0 t4))) (\lambda (t4: T).(subst0 (s (Bind b) i) v0 t1 t4)) (ex T (\lambda
+(w: T).(ty3 g d v w))) (\lambda (x: T).(\lambda (_: (eq T t3 (THead (Bind b)
+u0 x))).(\lambda (H8: (subst0 (s (Bind b) i) v0 t1 x)).(H3 v0 x (S i) H8 b0 d
+v (getl_head (Bind b) i c0 (CHead d (Bind b0) v) H5 u0))))) H6)) (\lambda
+(H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind b) u2
+t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 u0 u2))) (\lambda (_:
+T).(\lambda (t4: T).(subst0 (s (Bind b) i) v0 t1 t4))))).(ex3_2_ind T T
+(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda
+(u2: T).(\lambda (_: T).(subst0 i v0 u0 u2))) (\lambda (_: T).(\lambda (t4:
+T).(subst0 (s (Bind b) i) v0 t1 t4))) (ex T (\lambda (w: T).(ty3 g d v w)))
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (eq T t3 (THead (Bind b) x0
+x1))).(\lambda (H8: (subst0 i v0 u0 x0)).(\lambda (_: (subst0 (s (Bind b) i)
+v0 t1 x1)).(H1 v0 x0 i H8 b0 d v H5)))))) H6)) (subst0_gen_head (Bind b) v0
+u0 t1 t3 i H4)))))))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda
+(u0: T).(\lambda (_: (ty3 g c0 w u0)).(\lambda (H1: ((\forall (v0:
+T).(\forall (t0: T).(\forall (i: nat).((subst0 i v0 w t0) \to (\forall (b:
+B).(\forall (d: C).(\forall (v: T).((getl i c0 (CHead d (Bind b) v)) \to (ex
+T (\lambda (w0: T).(ty3 g d v w0))))))))))))).(\lambda (v: T).(\lambda (t0:
+T).(\lambda (_: (ty3 g c0 v (THead (Bind Abst) u0 t0))).(\lambda (H3:
+((\forall (v0: T).(\forall (t1: T).(\forall (i: nat).((subst0 i v0 v t1) \to
+(\forall (b: B).(\forall (d: C).(\forall (v1: T).((getl i c0 (CHead d (Bind
+b) v1)) \to (ex T (\lambda (w0: T).(ty3 g d v1 w0))))))))))))).(\lambda (v0:
+T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (H4: (subst0 i v0 (THead (Flat
+Appl) w v) t1)).(\lambda (b: B).(\lambda (d: C).(\lambda (v1: T).(\lambda
+(H5: (getl i c0 (CHead d (Bind b) v1))).(or3_ind (ex2 T (\lambda (u2: T).(eq
+T t1 (THead (Flat Appl) u2 v))) (\lambda (u2: T).(subst0 i v0 w u2))) (ex2 T
+(\lambda (t2: T).(eq T t1 (THead (Flat Appl) w t2))) (\lambda (t2: T).(subst0
+(s (Flat Appl) i) v0 v t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq
+T t1 (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i
+v0 w u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) v0 v
+t2)))) (ex T (\lambda (w0: T).(ty3 g d v1 w0))) (\lambda (H6: (ex2 T (\lambda
+(u2: T).(eq T t1 (THead (Flat Appl) u2 v))) (\lambda (u2: T).(subst0 i v0 w
+u2)))).(ex2_ind T (\lambda (u2: T).(eq T t1 (THead (Flat Appl) u2 v)))
+(\lambda (u2: T).(subst0 i v0 w u2)) (ex T (\lambda (w0: T).(ty3 g d v1 w0)))
+(\lambda (x: T).(\lambda (_: (eq T t1 (THead (Flat Appl) x v))).(\lambda (H8:
+(subst0 i v0 w x)).(H1 v0 x i H8 b d v1 H5)))) H6)) (\lambda (H6: (ex2 T
+(\lambda (t2: T).(eq T t1 (THead (Flat Appl) w t2))) (\lambda (t2: T).(subst0
+(s (Flat Appl) i) v0 v t2)))).(ex2_ind T (\lambda (t2: T).(eq T t1 (THead
+(Flat Appl) w t2))) (\lambda (t2: T).(subst0 (s (Flat Appl) i) v0 v t2)) (ex
+T (\lambda (w0: T).(ty3 g d v1 w0))) (\lambda (x: T).(\lambda (_: (eq T t1
+(THead (Flat Appl) w x))).(\lambda (H8: (subst0 (s (Flat Appl) i) v0 v
+x)).(H3 v0 x (s (Flat Appl) i) H8 b d v1 H5)))) H6)) (\lambda (H6: (ex3_2 T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T t1 (THead (Flat Appl) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i v0 w u2))) (\lambda (_:
+T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) v0 v t2))))).(ex3_2_ind T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T t1 (THead (Flat Appl) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i v0 w u2))) (\lambda (_:
+T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) v0 v t2))) (ex T (\lambda (w0:
+T).(ty3 g d v1 w0))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (eq T t1
+(THead (Flat Appl) x0 x1))).(\lambda (_: (subst0 i v0 w x0)).(\lambda (H9:
+(subst0 (s (Flat Appl) i) v0 v x1)).(H3 v0 x1 (s (Flat Appl) i) H9 b d v1
+H5)))))) H6)) (subst0_gen_head (Flat Appl) v0 w v t1 i H4)))))))))))))))))))
+(\lambda (c0: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t1
+t2)).(\lambda (H1: ((\forall (v0: T).(\forall (t0: T).(\forall (i:
+nat).((subst0 i v0 t1 t0) \to (\forall (b: B).(\forall (d: C).(\forall (v:
+T).((getl i c0 (CHead d (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v
+w))))))))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t2 t0)).(\lambda (H3:
+((\forall (v0: T).(\forall (t3: T).(\forall (i: nat).((subst0 i v0 t2 t3) \to
+(\forall (b: B).(\forall (d: C).(\forall (v: T).((getl i c0 (CHead d (Bind b)
+v)) \to (ex T (\lambda (w: T).(ty3 g d v w))))))))))))).(\lambda (v0:
+T).(\lambda (t3: T).(\lambda (i: nat).(\lambda (H4: (subst0 i v0 (THead (Flat
+Cast) t2 t1) t3)).(\lambda (b: B).(\lambda (d: C).(\lambda (v: T).(\lambda
+(H5: (getl i c0 (CHead d (Bind b) v))).(or3_ind (ex2 T (\lambda (u2: T).(eq T
+t3 (THead (Flat Cast) u2 t1))) (\lambda (u2: T).(subst0 i v0 t2 u2))) (ex2 T
+(\lambda (t4: T).(eq T t3 (THead (Flat Cast) t2 t4))) (\lambda (t4:
+T).(subst0 (s (Flat Cast) i) v0 t1 t4))) (ex3_2 T T (\lambda (u2: T).(\lambda
+(t4: T).(eq T t3 (THead (Flat Cast) u2 t4)))) (\lambda (u2: T).(\lambda (_:
+T).(subst0 i v0 t2 u2))) (\lambda (_: T).(\lambda (t4: T).(subst0 (s (Flat
+Cast) i) v0 t1 t4)))) (ex T (\lambda (w: T).(ty3 g d v w))) (\lambda (H6:
+(ex2 T (\lambda (u2: T).(eq T t3 (THead (Flat Cast) u2 t1))) (\lambda (u2:
+T).(subst0 i v0 t2 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t3 (THead (Flat
+Cast) u2 t1))) (\lambda (u2: T).(subst0 i v0 t2 u2)) (ex T (\lambda (w:
+T).(ty3 g d v w))) (\lambda (x: T).(\lambda (_: (eq T t3 (THead (Flat Cast) x
+t1))).(\lambda (H8: (subst0 i v0 t2 x)).(H3 v0 x i H8 b d v H5)))) H6))
+(\lambda (H6: (ex2 T (\lambda (t4: T).(eq T t3 (THead (Flat Cast) t2 t4)))
+(\lambda (t4: T).(subst0 (s (Flat Cast) i) v0 t1 t4)))).(ex2_ind T (\lambda
+(t4: T).(eq T t3 (THead (Flat Cast) t2 t4))) (\lambda (t4: T).(subst0 (s
+(Flat Cast) i) v0 t1 t4)) (ex T (\lambda (w: T).(ty3 g d v w))) (\lambda (x:
+T).(\lambda (_: (eq T t3 (THead (Flat Cast) t2 x))).(\lambda (H8: (subst0 (s
+(Flat Cast) i) v0 t1 x)).(H1 v0 x (s (Flat Cast) i) H8 b d v H5)))) H6))
+(\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead
+(Flat Cast) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 t2 u2)))
+(\lambda (_: T).(\lambda (t4: T).(subst0 (s (Flat Cast) i) v0 t1
+t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead
+(Flat Cast) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 t2 u2)))
+(\lambda (_: T).(\lambda (t4: T).(subst0 (s (Flat Cast) i) v0 t1 t4))) (ex T
+(\lambda (w: T).(ty3 g d v w))) (\lambda (x0: T).(\lambda (x1: T).(\lambda
+(_: (eq T t3 (THead (Flat Cast) x0 x1))).(\lambda (H8: (subst0 i v0 t2
+x0)).(\lambda (_: (subst0 (s (Flat Cast) i) v0 t1 x1)).(H3 v0 x0 i H8 b d v
+H5)))))) H6)) (subst0_gen_head (Flat Cast) v0 t2 t1 t3 i H4))))))))))))))))))
+c t u H))))).
+
+theorem ty3_gen_appl_nf2:
+ \forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (v: T).(\forall (x:
+T).((ty3 g c (THead (Flat Appl) w v) x) \to (ex4_2 T T (\lambda (u:
+T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x)))
+(\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t))))
+(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t:
+T).(nf2 c (THead (Bind Abst) u t))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (w: T).(\lambda (v: T).(\lambda (x:
+T).(\lambda (H: (ty3 g c (THead (Flat Appl) w v) x)).(ex3_2_ind T T (\lambda
+(u: T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t))
+x))) (\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t))))
+(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (ex4_2 T T (\lambda (u:
+T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x)))
+(\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t))))
+(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t:
+T).(nf2 c (THead (Bind Abst) u t))))) (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H0: (pc3 c (THead (Flat Appl) w (THead (Bind Abst) x0 x1))
+x)).(\lambda (H1: (ty3 g c v (THead (Bind Abst) x0 x1))).(\lambda (H2: (ty3 g
+c w x0)).(let H_x \def (ty3_correct g c v (THead (Bind Abst) x0 x1) H1) in
+(let H3 \def H_x in (ex_ind T (\lambda (t: T).(ty3 g c (THead (Bind Abst) x0
+x1) t)) (ex4_2 T T (\lambda (u: T).(\lambda (t: T).(pc3 c (THead (Flat Appl)
+w (THead (Bind Abst) u t)) x))) (\lambda (u: T).(\lambda (t: T).(ty3 g c v
+(THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c w u)))
+(\lambda (u: T).(\lambda (t: T).(nf2 c (THead (Bind Abst) u t))))) (\lambda
+(x2: T).(\lambda (H4: (ty3 g c (THead (Bind Abst) x0 x1) x2)).(let H_x0 \def
+(ty3_correct g c w x0 H2) in (let H5 \def H_x0 in (ex_ind T (\lambda (t:
+T).(ty3 g c x0 t)) (ex4_2 T T (\lambda (u: T).(\lambda (t: T).(pc3 c (THead
+(Flat Appl) w (THead (Bind Abst) u t)) x))) (\lambda (u: T).(\lambda (t:
+T).(ty3 g c v (THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3
+g c w u))) (\lambda (u: T).(\lambda (t: T).(nf2 c (THead (Bind Abst) u t)))))
+(\lambda (x3: T).(\lambda (H6: (ty3 g c x0 x3)).(let H7 \def (ty3_sn3 g c
+(THead (Bind Abst) x0 x1) x2 H4) in (let H_x1 \def (nf2_sn3 c (THead (Bind
+Abst) x0 x1) H7) in (let H8 \def H_x1 in (ex2_ind T (\lambda (u: T).(pr3 c
+(THead (Bind Abst) x0 x1) u)) (\lambda (u: T).(nf2 c u)) (ex4_2 T T (\lambda
+(u: T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t))
+x))) (\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t))))
+(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t:
+T).(nf2 c (THead (Bind Abst) u t))))) (\lambda (x4: T).(\lambda (H9: (pr3 c
+(THead (Bind Abst) x0 x1) x4)).(\lambda (H10: (nf2 c x4)).(let H11 \def
+(pr3_gen_abst c x0 x1 x4 H9) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T x4 (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall
+(u: T).(pr3 (CHead c (Bind b) u) x1 t2))))) (ex4_2 T T (\lambda (u:
+T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x)))
+(\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t))))
+(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t:
+T).(nf2 c (THead (Bind Abst) u t))))) (\lambda (x5: T).(\lambda (x6:
+T).(\lambda (H12: (eq T x4 (THead (Bind Abst) x5 x6))).(\lambda (H13: (pr3 c
+x0 x5)).(\lambda (H14: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind
+b) u) x1 x6))))).(let H15 \def (eq_ind T x4 (\lambda (t: T).(nf2 c t)) H10
+(THead (Bind Abst) x5 x6) H12) in (let H16 \def (pr3_head_12 c x0 x5 H13
+(Bind Abst) x1 x6 (H14 Abst x5)) in (ex4_2_intro T T (\lambda (u: T).(\lambda
+(t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x))) (\lambda (u:
+T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) (\lambda (u:
+T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t: T).(nf2 c
+(THead (Bind Abst) u t)))) x5 x6 (pc3_pr3_conf c (THead (Flat Appl) w (THead
+(Bind Abst) x0 x1)) x H0 (THead (Flat Appl) w (THead (Bind Abst) x5 x6))
+(pr3_thin_dx c (THead (Bind Abst) x0 x1) (THead (Bind Abst) x5 x6) H16 w
+Appl)) (ty3_conv g c (THead (Bind Abst) x5 x6) x2 (ty3_sred_pr3 c (THead
+(Bind Abst) x0 x1) (THead (Bind Abst) x5 x6) H16 g x2 H4) v (THead (Bind
+Abst) x0 x1) H1 (pc3_pr3_r c (THead (Bind Abst) x0 x1) (THead (Bind Abst) x5
+x6) H16)) (ty3_conv g c x5 x3 (ty3_sred_pr3 c x0 x5 H13 g x3 H6) w x0 H2
+(pc3_pr3_r c x0 x5 H13)) H15)))))))) H11))))) H8)))))) H5))))) H3))))))))
+(ty3_gen_appl g c w v x H))))))).
+
+theorem ty3_inv_lref_nf2_pc3:
+ \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (i: nat).((ty3 g c
+(TLRef i) u1) \to ((nf2 c (TLRef i)) \to (\forall (u2: T).((nf2 c u2) \to
+((pc3 c u1 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u))))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (i: nat).(\lambda
+(H: (ty3 g c (TLRef i) u1)).(insert_eq T (TLRef i) (\lambda (t: T).(ty3 g c t
+u1)) (\lambda (t: T).((nf2 c t) \to (\forall (u2: T).((nf2 c u2) \to ((pc3 c
+u1 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u))))))))) (\lambda
+(y: T).(\lambda (H0: (ty3 g c y u1)).(ty3_ind g (\lambda (c0: C).(\lambda (t:
+T).(\lambda (t0: T).((eq T t (TLRef i)) \to ((nf2 c0 t) \to (\forall (u2:
+T).((nf2 c0 u2) \to ((pc3 c0 t0 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift
+(S i) O u)))))))))))) (\lambda (c0: C).(\lambda (t2: T).(\lambda (t:
+T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: (((eq T t2 (TLRef i)) \to ((nf2
+c0 t2) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t u2) \to (ex T
+(\lambda (u: T).(eq T u2 (lift (S i) O u))))))))))).(\lambda (u: T).(\lambda
+(t1: T).(\lambda (H3: (ty3 g c0 u t1)).(\lambda (H4: (((eq T u (TLRef i)) \to
+((nf2 c0 u) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t1 u2) \to (ex T
+(\lambda (u0: T).(eq T u2 (lift (S i) O u0))))))))))).(\lambda (H5: (pc3 c0
+t1 t2)).(\lambda (H6: (eq T u (TLRef i))).(\lambda (H7: (nf2 c0 u)).(\lambda
+(u2: T).(\lambda (H8: (nf2 c0 u2)).(\lambda (H9: (pc3 c0 t2 u2)).(let H10
+\def (eq_ind T u (\lambda (t0: T).(nf2 c0 t0)) H7 (TLRef i) H6) in (let H11
+\def (eq_ind T u (\lambda (t0: T).((eq T t0 (TLRef i)) \to ((nf2 c0 t0) \to
+(\forall (u3: T).((nf2 c0 u3) \to ((pc3 c0 t1 u3) \to (ex T (\lambda (u0:
+T).(eq T u3 (lift (S i) O u0)))))))))) H4 (TLRef i) H6) in (let H12 \def
+(eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t1)) H3 (TLRef i) H6) in (let H_y
+\def (H11 (refl_equal T (TLRef i)) H10 u2 H8) in (H_y (pc3_t t2 c0 t1 H5 u2
+H9))))))))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq
+T (TSort m) (TLRef i))).(\lambda (_: (nf2 c0 (TSort m))).(\lambda (u2:
+T).(\lambda (_: (nf2 c0 u2)).(\lambda (_: (pc3 c0 (TSort (next g m))
+u2)).(let H5 \def (eq_ind T (TSort m) (\lambda (ee: T).(match ee in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef i) H1) in
+(False_ind (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u)))) H5)))))))))
+(\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda
+(H1: (getl n c0 (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda (_: (ty3 g
+d u t)).(\lambda (_: (((eq T u (TLRef i)) \to ((nf2 d u) \to (\forall (u2:
+T).((nf2 d u2) \to ((pc3 d t u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S
+i) O u0))))))))))).(\lambda (H4: (eq T (TLRef n) (TLRef i))).(\lambda (H5:
+(nf2 c0 (TLRef n))).(\lambda (u2: T).(\lambda (_: (nf2 c0 u2)).(\lambda (H7:
+(pc3 c0 (lift (S n) O t) u2)).(let H8 \def (f_equal T nat (\lambda (e:
+T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow n |
+(TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef
+i) H4) in (let H9 \def (eq_ind nat n (\lambda (n0: nat).(pc3 c0 (lift (S n0)
+O t) u2)) H7 i H8) in (let H10 \def (eq_ind nat n (\lambda (n0: nat).(nf2 c0
+(TLRef n0))) H5 i H8) in (let H11 \def (eq_ind nat n (\lambda (n0: nat).(getl
+n0 c0 (CHead d (Bind Abbr) u))) H1 i H8) in (nf2_gen_lref c0 d u i H11 H10
+(ex T (\lambda (u0: T).(eq T u2 (lift (S i) O u0))))))))))))))))))))))
+(\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda
+(H1: (getl n c0 (CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (_: (ty3 g
+d u t)).(\lambda (_: (((eq T u (TLRef i)) \to ((nf2 d u) \to (\forall (u2:
+T).((nf2 d u2) \to ((pc3 d t u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S
+i) O u0))))))))))).(\lambda (H4: (eq T (TLRef n) (TLRef i))).(\lambda (H5:
+(nf2 c0 (TLRef n))).(\lambda (u2: T).(\lambda (H6: (nf2 c0 u2)).(\lambda (H7:
+(pc3 c0 (lift (S n) O u) u2)).(let H8 \def (f_equal T nat (\lambda (e:
+T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow n |
+(TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef
+i) H4) in (let H9 \def (eq_ind nat n (\lambda (n0: nat).(pc3 c0 (lift (S n0)
+O u) u2)) H7 i H8) in (let H10 \def (eq_ind nat n (\lambda (n0: nat).(nf2 c0
+(TLRef n0))) H5 i H8) in (let H11 \def (eq_ind nat n (\lambda (n0: nat).(getl
+n0 c0 (CHead d (Bind Abst) u))) H1 i H8) in (let H_y \def (pc3_nf2_unfold c0
+(lift (S i) O u) u2 H9 H6) in (let H12 \def (pr3_gen_lift c0 u u2 (S i) O H_y
+d (getl_drop Abst c0 d u i H11)) in (ex2_ind T (\lambda (t2: T).(eq T u2
+(lift (S i) O t2))) (\lambda (t2: T).(pr3 d u t2)) (ex T (\lambda (u0: T).(eq
+T u2 (lift (S i) O u0)))) (\lambda (x: T).(\lambda (H13: (eq T u2 (lift (S i)
+O x))).(\lambda (_: (pr3 d u x)).(eq_ind_r T (lift (S i) O x) (\lambda (t0:
+T).(ex T (\lambda (u0: T).(eq T t0 (lift (S i) O u0))))) (ex_intro T (\lambda
+(u0: T).(eq T (lift (S i) O x) (lift (S i) O u0))) x (refl_equal T (lift (S
+i) O x))) u2 H13)))) H12)))))))))))))))))))) (\lambda (c0: C).(\lambda (u:
+T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u (TLRef
+i)) \to ((nf2 c0 u) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t u2) \to
+(ex T (\lambda (u0: T).(eq T u2 (lift (S i) O u0))))))))))).(\lambda (b:
+B).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead c0 (Bind b)
+u) t1 t2)).(\lambda (_: (((eq T t1 (TLRef i)) \to ((nf2 (CHead c0 (Bind b) u)
+t1) \to (\forall (u2: T).((nf2 (CHead c0 (Bind b) u) u2) \to ((pc3 (CHead c0
+(Bind b) u) t2 u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S i) O
+u0))))))))))).(\lambda (H5: (eq T (THead (Bind b) u t1) (TLRef i))).(\lambda
+(_: (nf2 c0 (THead (Bind b) u t1))).(\lambda (u2: T).(\lambda (_: (nf2 c0
+u2)).(\lambda (_: (pc3 c0 (THead (Bind b) u t2) u2)).(let H9 \def (eq_ind T
+(THead (Bind b) u t1) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead _ _ _) \Rightarrow True])) I (TLRef i) H5) in (False_ind (ex T
+(\lambda (u0: T).(eq T u2 (lift (S i) O u0)))) H9))))))))))))))))) (\lambda
+(c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda
+(_: (((eq T w (TLRef i)) \to ((nf2 c0 w) \to (\forall (u2: T).((nf2 c0 u2)
+\to ((pc3 c0 u u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S i) O
+u0))))))))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead
+(Bind Abst) u t))).(\lambda (_: (((eq T v (TLRef i)) \to ((nf2 c0 v) \to
+(\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 (THead (Bind Abst) u t) u2) \to
+(ex T (\lambda (u0: T).(eq T u2 (lift (S i) O u0))))))))))).(\lambda (H5: (eq
+T (THead (Flat Appl) w v) (TLRef i))).(\lambda (_: (nf2 c0 (THead (Flat Appl)
+w v))).(\lambda (u2: T).(\lambda (_: (nf2 c0 u2)).(\lambda (_: (pc3 c0 (THead
+(Flat Appl) w (THead (Bind Abst) u t)) u2)).(let H9 \def (eq_ind T (THead
+(Flat Appl) w v) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop)
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _
+_) \Rightarrow True])) I (TLRef i) H5) in (False_ind (ex T (\lambda (u0:
+T).(eq T u2 (lift (S i) O u0)))) H9)))))))))))))))) (\lambda (c0: C).(\lambda
+(t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t1 t2)).(\lambda (_: (((eq T
+t1 (TLRef i)) \to ((nf2 c0 t1) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0
+t2 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u))))))))))).(\lambda
+(t0: T).(\lambda (_: (ty3 g c0 t2 t0)).(\lambda (_: (((eq T t2 (TLRef i)) \to
+((nf2 c0 t2) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t0 u2) \to (ex T
+(\lambda (u: T).(eq T u2 (lift (S i) O u))))))))))).(\lambda (H5: (eq T
+(THead (Flat Cast) t2 t1) (TLRef i))).(\lambda (_: (nf2 c0 (THead (Flat Cast)
+t2 t1))).(\lambda (u2: T).(\lambda (_: (nf2 c0 u2)).(\lambda (_: (pc3 c0
+(THead (Flat Cast) t0 t2) u2)).(let H9 \def (eq_ind T (THead (Flat Cast) t2
+t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
+_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
+\Rightarrow True])) I (TLRef i) H5) in (False_ind (ex T (\lambda (u: T).(eq T
+u2 (lift (S i) O u)))) H9))))))))))))))) c y u1 H0))) H))))).
+
+theorem ty3_inv_lref_nf2:
+ \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (i: nat).((ty3 g c
+(TLRef i) u) \to ((nf2 c (TLRef i)) \to ((nf2 c u) \to (ex T (\lambda (u0:
+T).(eq T u (lift (S i) O u0))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H: (ty3 g c (TLRef i) u)).(\lambda (H0: (nf2 c (TLRef i))).(\lambda (H1:
+(nf2 c u)).(ty3_inv_lref_nf2_pc3 g c u i H H0 u H1 (pc3_refl c u)))))))).
+
+theorem ty3_inv_appls_lref_nf2:
+ \forall (g: G).(\forall (c: C).(\forall (vs: TList).(\forall (u1:
+T).(\forall (i: nat).((ty3 g c (THeads (Flat Appl) vs (TLRef i)) u1) \to
+((nf2 c (TLRef i)) \to ((nf2 c u1) \to (ex2 T (\lambda (u: T).(nf2 c (lift (S
+i) O u))) (\lambda (u: T).(pc3 c (THeads (Flat Appl) vs (lift (S i) O u))
+u1))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (vs: TList).(TList_ind (\lambda (t:
+TList).(\forall (u1: T).(\forall (i: nat).((ty3 g c (THeads (Flat Appl) t
+(TLRef i)) u1) \to ((nf2 c (TLRef i)) \to ((nf2 c u1) \to (ex2 T (\lambda (u:
+T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3 c (THeads (Flat Appl) t
+(lift (S i) O u)) u1))))))))) (\lambda (u1: T).(\lambda (i: nat).(\lambda (H:
+(ty3 g c (TLRef i) u1)).(\lambda (H0: (nf2 c (TLRef i))).(\lambda (H1: (nf2 c
+u1)).(let H_x \def (ty3_inv_lref_nf2 g c u1 i H H0 H1) in (let H2 \def H_x in
+(ex_ind T (\lambda (u0: T).(eq T u1 (lift (S i) O u0))) (ex2 T (\lambda (u:
+T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3 c (lift (S i) O u) u1)))
+(\lambda (x: T).(\lambda (H3: (eq T u1 (lift (S i) O x))).(let H4 \def
+(eq_ind T u1 (\lambda (t: T).(nf2 c t)) H1 (lift (S i) O x) H3) in (eq_ind_r
+T (lift (S i) O x) (\lambda (t: T).(ex2 T (\lambda (u: T).(nf2 c (lift (S i)
+O u))) (\lambda (u: T).(pc3 c (lift (S i) O u) t)))) (ex_intro2 T (\lambda
+(u: T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3 c (lift (S i) O u)
+(lift (S i) O x))) x H4 (pc3_refl c (lift (S i) O x))) u1 H3)))) H2))))))))
+(\lambda (t: T).(\lambda (t0: TList).(\lambda (H: ((\forall (u1: T).(\forall
+(i: nat).((ty3 g c (THeads (Flat Appl) t0 (TLRef i)) u1) \to ((nf2 c (TLRef
+i)) \to ((nf2 c u1) \to (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O u)))
+(\lambda (u: T).(pc3 c (THeads (Flat Appl) t0 (lift (S i) O u))
+u1)))))))))).(\lambda (u1: T).(\lambda (i: nat).(\lambda (H0: (ty3 g c (THead
+(Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) u1)).(\lambda (H1: (nf2 c
+(TLRef i))).(\lambda (_: (nf2 c u1)).(let H_x \def (ty3_gen_appl_nf2 g c t
+(THeads (Flat Appl) t0 (TLRef i)) u1 H0) in (let H3 \def H_x in (ex4_2_ind T
+T (\lambda (u: T).(\lambda (t1: T).(pc3 c (THead (Flat Appl) t (THead (Bind
+Abst) u t1)) u1))) (\lambda (u: T).(\lambda (t1: T).(ty3 g c (THeads (Flat
+Appl) t0 (TLRef i)) (THead (Bind Abst) u t1)))) (\lambda (u: T).(\lambda (_:
+T).(ty3 g c t u))) (\lambda (u: T).(\lambda (t1: T).(nf2 c (THead (Bind Abst)
+u t1)))) (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O u))) (\lambda (u:
+T).(pc3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O u)))
+u1))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (pc3 c (THead (Flat
+Appl) t (THead (Bind Abst) x0 x1)) u1)).(\lambda (H5: (ty3 g c (THeads (Flat
+Appl) t0 (TLRef i)) (THead (Bind Abst) x0 x1))).(\lambda (_: (ty3 g c t
+x0)).(\lambda (H7: (nf2 c (THead (Bind Abst) x0 x1))).(let H8 \def
+(nf2_gen_abst c x0 x1 H7) in (and_ind (nf2 c x0) (nf2 (CHead c (Bind Abst)
+x0) x1) (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3
+c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O u))) u1)))
+(\lambda (H9: (nf2 c x0)).(\lambda (H10: (nf2 (CHead c (Bind Abst) x0)
+x1)).(let H_y \def (H (THead (Bind Abst) x0 x1) i H5 H1) in (let H11 \def
+(H_y (nf2_abst_shift c x0 H9 x1 H10)) in (ex2_ind T (\lambda (u: T).(nf2 c
+(lift (S i) O u))) (\lambda (u: T).(pc3 c (THeads (Flat Appl) t0 (lift (S i)
+O u)) (THead (Bind Abst) x0 x1))) (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O
+u))) (\lambda (u: T).(pc3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift
+(S i) O u))) u1))) (\lambda (x: T).(\lambda (H12: (nf2 c (lift (S i) O
+x))).(\lambda (H13: (pc3 c (THeads (Flat Appl) t0 (lift (S i) O x)) (THead
+(Bind Abst) x0 x1))).(ex_intro2 T (\lambda (u: T).(nf2 c (lift (S i) O u)))
+(\lambda (u: T).(pc3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S
+i) O u))) u1)) x H12 (pc3_t (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) c
+(THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O x))) (pc3_thin_dx c
+(THeads (Flat Appl) t0 (lift (S i) O x)) (THead (Bind Abst) x0 x1) H13 t
+Appl) u1 H4))))) H11))))) H8)))))))) H3))))))))))) vs))).
+
+theorem ty3_inv_lref_lref_nf2:
+ \forall (g: G).(\forall (c: C).(\forall (i: nat).(\forall (j: nat).((ty3 g c
+(TLRef i) (TLRef j)) \to ((nf2 c (TLRef i)) \to ((nf2 c (TLRef j)) \to (lt i
+j)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (i: nat).(\lambda (j: nat).(\lambda
+(H: (ty3 g c (TLRef i) (TLRef j))).(\lambda (H0: (nf2 c (TLRef i))).(\lambda
+(H1: (nf2 c (TLRef j))).(let H_x \def (ty3_inv_lref_nf2 g c (TLRef j) i H H0
+H1) in (let H2 \def H_x in (ex_ind T (\lambda (u0: T).(eq T (TLRef j) (lift
+(S i) O u0))) (lt i j) (\lambda (x: T).(\lambda (H3: (eq T (TLRef j) (lift (S
+i) O x))).(let H_x0 \def (lift_gen_lref x O (S i) j H3) in (let H4 \def H_x0
+in (or_ind (land (lt j O) (eq T x (TLRef j))) (land (le (plus O (S i)) j) (eq
+T x (TLRef (minus j (S i))))) (lt i j) (\lambda (H5: (land (lt j O) (eq T x
+(TLRef j)))).(and_ind (lt j O) (eq T x (TLRef j)) (lt i j) (\lambda (H6: (lt
+j O)).(\lambda (_: (eq T x (TLRef j))).(lt_x_O j H6 (lt i j)))) H5)) (\lambda
+(H5: (land (le (plus O (S i)) j) (eq T x (TLRef (minus j (S i)))))).(and_ind
+(le (plus O (S i)) j) (eq T x (TLRef (minus j (S i)))) (lt i j) (\lambda (H6:
+(le (plus O (S i)) j)).(\lambda (_: (eq T x (TLRef (minus j (S i))))).H6))
+H5)) H4))))) H2))))))))).
+
+inductive wf3 (g: G): C \to (C \to Prop) \def
+| wf3_sort: \forall (m: nat).(wf3 g (CSort m) (CSort m))
+| wf3_bind: \forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to (\forall (u:
+T).(\forall (t: T).((ty3 g c1 u t) \to (\forall (b: B).(wf3 g (CHead c1 (Bind
+b) u) (CHead c2 (Bind b) u))))))))
+| wf3_void: \forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to (\forall (u:
+T).(((\forall (t: T).((ty3 g c1 u t) \to False))) \to (\forall (b: B).(wf3 g
+(CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort O))))))))
+| wf3_flat: \forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to (\forall (u:
+T).(\forall (f: F).(wf3 g (CHead c1 (Flat f) u) c2))))).
+
+theorem wf3_gen_sort1:
+ \forall (g: G).(\forall (x: C).(\forall (m: nat).((wf3 g (CSort m) x) \to
+(eq C x (CSort m)))))
+\def
+ \lambda (g: G).(\lambda (x: C).(\lambda (m: nat).(\lambda (H: (wf3 g (CSort
+m) x)).(insert_eq C (CSort m) (\lambda (c: C).(wf3 g c x)) (\lambda (c:
+C).(eq C x c)) (\lambda (y: C).(\lambda (H0: (wf3 g y x)).(wf3_ind g (\lambda
+(c: C).(\lambda (c0: C).((eq C c (CSort m)) \to (eq C c0 c)))) (\lambda (m0:
+nat).(\lambda (H1: (eq C (CSort m0) (CSort m))).(let H2 \def (f_equal C nat
+(\lambda (e: C).(match e in C return (\lambda (_: C).nat) with [(CSort n)
+\Rightarrow n | (CHead _ _ _) \Rightarrow m0])) (CSort m0) (CSort m) H1) in
+(eq_ind_r nat m (\lambda (n: nat).(eq C (CSort n) (CSort n))) (refl_equal C
+(CSort m)) m0 H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (wf3 g c1
+c2)).(\lambda (_: (((eq C c1 (CSort m)) \to (eq C c2 c1)))).(\lambda (u:
+T).(\lambda (t: T).(\lambda (_: (ty3 g c1 u t)).(\lambda (b: B).(\lambda (H4:
+(eq C (CHead c1 (Bind b) u) (CSort m))).(let H5 \def (eq_ind C (CHead c1
+(Bind b) u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with
+[(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort m)
+H4) in (False_ind (eq C (CHead c2 (Bind b) u) (CHead c1 (Bind b) u))
+H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (wf3 g c1
+c2)).(\lambda (_: (((eq C c1 (CSort m)) \to (eq C c2 c1)))).(\lambda (u:
+T).(\lambda (_: ((\forall (t: T).((ty3 g c1 u t) \to False)))).(\lambda (b:
+B).(\lambda (H4: (eq C (CHead c1 (Bind b) u) (CSort m))).(let H5 \def (eq_ind
+C (CHead c1 (Bind b) u) (\lambda (ee: C).(match ee in C return (\lambda (_:
+C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow
+True])) I (CSort m) H4) in (False_ind (eq C (CHead c2 (Bind Void) (TSort O))
+(CHead c1 (Bind b) u)) H5)))))))))) (\lambda (c1: C).(\lambda (c2:
+C).(\lambda (_: (wf3 g c1 c2)).(\lambda (_: (((eq C c1 (CSort m)) \to (eq C
+c2 c1)))).(\lambda (u: T).(\lambda (f: F).(\lambda (H3: (eq C (CHead c1 (Flat
+f) u) (CSort m))).(let H4 \def (eq_ind C (CHead c1 (Flat f) u) (\lambda (ee:
+C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow
+False | (CHead _ _ _) \Rightarrow True])) I (CSort m) H3) in (False_ind (eq C
+c2 (CHead c1 (Flat f) u)) H4))))))))) y x H0))) H)))).
+
+theorem wf3_gen_bind1:
+ \forall (g: G).(\forall (c1: C).(\forall (x: C).(\forall (v: T).(\forall (b:
+B).((wf3 g (CHead c1 (Bind b) v) x) \to (or (ex3_2 C T (\lambda (c2:
+C).(\lambda (_: T).(eq C x (CHead c2 (Bind b) v)))) (\lambda (c2: C).(\lambda
+(_: T).(wf3 g c1 c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3
+C (\lambda (c2: C).(eq C x (CHead c2 (Bind Void) (TSort O)))) (\lambda (c2:
+C).(wf3 g c1 c2)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to
+False))))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (x: C).(\lambda (v: T).(\lambda (b:
+B).(\lambda (H: (wf3 g (CHead c1 (Bind b) v) x)).(insert_eq C (CHead c1 (Bind
+b) v) (\lambda (c: C).(wf3 g c x)) (\lambda (_: C).(or (ex3_2 C T (\lambda
+(c2: C).(\lambda (_: T).(eq C x (CHead c2 (Bind b) v)))) (\lambda (c2:
+C).(\lambda (_: T).(wf3 g c1 c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1
+v w)))) (ex3 C (\lambda (c2: C).(eq C x (CHead c2 (Bind Void) (TSort O))))
+(\lambda (c2: C).(wf3 g c1 c2)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v
+w) \to False)))))) (\lambda (y: C).(\lambda (H0: (wf3 g y x)).(wf3_ind g
+(\lambda (c: C).(\lambda (c0: C).((eq C c (CHead c1 (Bind b) v)) \to (or
+(ex3_2 C T (\lambda (c2: C).(\lambda (_: T).(eq C c0 (CHead c2 (Bind b) v))))
+(\lambda (c2: C).(\lambda (_: T).(wf3 g c1 c2))) (\lambda (_: C).(\lambda (w:
+T).(ty3 g c1 v w)))) (ex3 C (\lambda (c2: C).(eq C c0 (CHead c2 (Bind Void)
+(TSort O)))) (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (_: C).(\forall (w:
+T).((ty3 g c1 v w) \to False)))))))) (\lambda (m: nat).(\lambda (H1: (eq C
+(CSort m) (CHead c1 (Bind b) v))).(let H2 \def (eq_ind C (CSort m) (\lambda
+(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
+\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead c1 (Bind b) v)
+H1) in (False_ind (or (ex3_2 C T (\lambda (c2: C).(\lambda (_: T).(eq C
+(CSort m) (CHead c2 (Bind b) v)))) (\lambda (c2: C).(\lambda (_: T).(wf3 g c1
+c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c2:
+C).(eq C (CSort m) (CHead c2 (Bind Void) (TSort O)))) (\lambda (c2: C).(wf3 g
+c1 c2)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to False))))) H2))))
+(\lambda (c0: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c0 c2)).(\lambda (H2:
+(((eq C c0 (CHead c1 (Bind b) v)) \to (or (ex3_2 C T (\lambda (c3:
+C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda (c3:
+C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1
+v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort O))))
+(\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v
+w) \to False)))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H3: (ty3 g c0
+u t)).(\lambda (b0: B).(\lambda (H4: (eq C (CHead c0 (Bind b0) u) (CHead c1
+(Bind b) v))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return
+(\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow
+c])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) H4) in ((let H6 \def
+(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with
+[(CSort _) \Rightarrow b0 | (CHead _ k _) \Rightarrow (match k in K return
+(\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow
+b0])])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) H4) in ((let H7 \def
+(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead c0 (Bind
+b0) u) (CHead c1 (Bind b) v) H4) in (\lambda (H8: (eq B b0 b)).(\lambda (H9:
+(eq C c0 c1)).(eq_ind_r B b (\lambda (b1: B).(or (ex3_2 C T (\lambda (c3:
+C).(\lambda (_: T).(eq C (CHead c2 (Bind b1) u) (CHead c3 (Bind b) v))))
+(\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w:
+T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C (CHead c2 (Bind b1) u)
+(CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda
+(_: C).(\forall (w: T).((ty3 g c1 v w) \to False)))))) (let H10 \def (eq_ind
+T u (\lambda (t0: T).(ty3 g c0 t0 t)) H3 v H7) in (eq_ind_r T v (\lambda (t0:
+T).(or (ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C (CHead c2 (Bind b)
+t0) (CHead c3 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3)))
+(\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq
+C (CHead c2 (Bind b) t0) (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3:
+C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to
+False)))))) (let H11 \def (eq_ind C c0 (\lambda (c: C).(ty3 g c v t)) H10 c1
+H9) in (let H12 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 (Bind b)
+v)) \to (or (ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3
+(Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_:
+C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead
+c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_:
+C).(\forall (w: T).((ty3 g c1 v w) \to False))))))) H2 c1 H9) in (let H13
+\def (eq_ind C c0 (\lambda (c: C).(wf3 g c c2)) H1 c1 H9) in (or_introl
+(ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C (CHead c2 (Bind b) v)
+(CHead c3 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3)))
+(\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq
+C (CHead c2 (Bind b) v) (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3:
+C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to
+False)))) (ex3_2_intro C T (\lambda (c3: C).(\lambda (_: T).(eq C (CHead c2
+(Bind b) v) (CHead c3 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g
+c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w))) c2 t (refl_equal C
+(CHead c2 (Bind b) v)) H13 H11))))) u H7)) b0 H8)))) H6)) H5)))))))))))
+(\lambda (c0: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c0 c2)).(\lambda (H2:
+(((eq C c0 (CHead c1 (Bind b) v)) \to (or (ex3_2 C T (\lambda (c3:
+C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda (c3:
+C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1
+v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort O))))
+(\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v
+w) \to False)))))))).(\lambda (u: T).(\lambda (H3: ((\forall (t: T).((ty3 g
+c0 u t) \to False)))).(\lambda (b0: B).(\lambda (H4: (eq C (CHead c0 (Bind
+b0) u) (CHead c1 (Bind b) v))).(let H5 \def (f_equal C C (\lambda (e:
+C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 |
+(CHead c _ _) \Rightarrow c])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v)
+H4) in ((let H6 \def (f_equal C B (\lambda (e: C).(match e in C return
+(\lambda (_: C).B) with [(CSort _) \Rightarrow b0 | (CHead _ k _) \Rightarrow
+(match k in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 |
+(Flat _) \Rightarrow b0])])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) H4)
+in ((let H7 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda
+(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t]))
+(CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) H4) in (\lambda (_: (eq B b0
+b)).(\lambda (H9: (eq C c0 c1)).(let H10 \def (eq_ind T u (\lambda (t:
+T).(\forall (t0: T).((ty3 g c0 t t0) \to False))) H3 v H7) in (let H11 \def
+(eq_ind C c0 (\lambda (c: C).(\forall (t: T).((ty3 g c v t) \to False))) H10
+c1 H9) in (let H12 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 (Bind
+b) v)) \to (or (ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3
+(Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_:
+C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead
+c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_:
+C).(\forall (w: T).((ty3 g c1 v w) \to False))))))) H2 c1 H9) in (let H13
+\def (eq_ind C c0 (\lambda (c: C).(wf3 g c c2)) H1 c1 H9) in (or_intror
+(ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C (CHead c2 (Bind Void)
+(TSort O)) (CHead c3 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g
+c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda
+(c3: C).(eq C (CHead c2 (Bind Void) (TSort O)) (CHead c3 (Bind Void) (TSort
+O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g
+c1 v w) \to False)))) (ex3_intro C (\lambda (c3: C).(eq C (CHead c2 (Bind
+Void) (TSort O)) (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g
+c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to False))) c2
+(refl_equal C (CHead c2 (Bind Void) (TSort O))) H13 H11))))))))) H6))
+H5)))))))))) (\lambda (c0: C).(\lambda (c2: C).(\lambda (_: (wf3 g c0
+c2)).(\lambda (_: (((eq C c0 (CHead c1 (Bind b) v)) \to (or (ex3_2 C T
+(\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda
+(c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3
+g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort
+O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g
+c1 v w) \to False)))))))).(\lambda (u: T).(\lambda (f: F).(\lambda (H3: (eq C
+(CHead c0 (Flat f) u) (CHead c1 (Bind b) v))).(let H4 \def (eq_ind C (CHead
+c0 (Flat f) u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop)
+with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K
+return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _)
+\Rightarrow True])])) I (CHead c1 (Bind b) v) H3) in (False_ind (or (ex3_2 C
+T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda
+(c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3
+g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort
+O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g
+c1 v w) \to False))))) H4))))))))) y x H0))) H)))))).
+
+theorem wf3_gen_flat1:
+ \forall (g: G).(\forall (c1: C).(\forall (x: C).(\forall (v: T).(\forall (f:
+F).((wf3 g (CHead c1 (Flat f) v) x) \to (wf3 g c1 x))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (x: C).(\lambda (v: T).(\lambda (f:
+F).(\lambda (H: (wf3 g (CHead c1 (Flat f) v) x)).(insert_eq C (CHead c1 (Flat
+f) v) (\lambda (c: C).(wf3 g c x)) (\lambda (_: C).(wf3 g c1 x)) (\lambda (y:
+C).(\lambda (H0: (wf3 g y x)).(wf3_ind g (\lambda (c: C).(\lambda (c0:
+C).((eq C c (CHead c1 (Flat f) v)) \to (wf3 g c1 c0)))) (\lambda (m:
+nat).(\lambda (H1: (eq C (CSort m) (CHead c1 (Flat f) v))).(let H2 \def
+(eq_ind C (CSort m) (\lambda (ee: C).(match ee in C return (\lambda (_:
+C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow
+False])) I (CHead c1 (Flat f) v) H1) in (False_ind (wf3 g c1 (CSort m))
+H2)))) (\lambda (c0: C).(\lambda (c2: C).(\lambda (_: (wf3 g c0 c2)).(\lambda
+(_: (((eq C c0 (CHead c1 (Flat f) v)) \to (wf3 g c1 c2)))).(\lambda (u:
+T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (b: B).(\lambda (H4:
+(eq C (CHead c0 (Bind b) u) (CHead c1 (Flat f) v))).(let H5 \def (eq_ind C
+(CHead c0 (Bind b) u) (\lambda (ee: C).(match ee in C return (\lambda (_:
+C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match
+k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat
+_) \Rightarrow False])])) I (CHead c1 (Flat f) v) H4) in (False_ind (wf3 g c1
+(CHead c2 (Bind b) u)) H5))))))))))) (\lambda (c0: C).(\lambda (c2:
+C).(\lambda (_: (wf3 g c0 c2)).(\lambda (_: (((eq C c0 (CHead c1 (Flat f) v))
+\to (wf3 g c1 c2)))).(\lambda (u: T).(\lambda (_: ((\forall (t: T).((ty3 g c0
+u t) \to False)))).(\lambda (b: B).(\lambda (H4: (eq C (CHead c0 (Bind b) u)
+(CHead c1 (Flat f) v))).(let H5 \def (eq_ind C (CHead c0 (Bind b) u) (\lambda
+(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
+\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda
+(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow
+False])])) I (CHead c1 (Flat f) v) H4) in (False_ind (wf3 g c1 (CHead c2
+(Bind Void) (TSort O))) H5)))))))))) (\lambda (c0: C).(\lambda (c2:
+C).(\lambda (H1: (wf3 g c0 c2)).(\lambda (H2: (((eq C c0 (CHead c1 (Flat f)
+v)) \to (wf3 g c1 c2)))).(\lambda (u: T).(\lambda (f0: F).(\lambda (H3: (eq C
+(CHead c0 (Flat f0) u) (CHead c1 (Flat f) v))).(let H4 \def (f_equal C C
+(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 (Flat f0) u) (CHead
+c1 (Flat f) v) H3) in ((let H5 \def (f_equal C F (\lambda (e: C).(match e in
+C return (\lambda (_: C).F) with [(CSort _) \Rightarrow f0 | (CHead _ k _)
+\Rightarrow (match k in K return (\lambda (_: K).F) with [(Bind _)
+\Rightarrow f0 | (Flat f1) \Rightarrow f1])])) (CHead c0 (Flat f0) u) (CHead
+c1 (Flat f) v) H3) in ((let H6 \def (f_equal C T (\lambda (e: C).(match e in
+C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t)
+\Rightarrow t])) (CHead c0 (Flat f0) u) (CHead c1 (Flat f) v) H3) in (\lambda
+(_: (eq F f0 f)).(\lambda (H8: (eq C c0 c1)).(let H9 \def (eq_ind C c0
+(\lambda (c: C).((eq C c (CHead c1 (Flat f) v)) \to (wf3 g c1 c2))) H2 c1 H8)
+in (let H10 \def (eq_ind C c0 (\lambda (c: C).(wf3 g c c2)) H1 c1 H8) in
+H10))))) H5)) H4))))))))) y x H0))) H)))))).
+
+theorem wf3_gen_head2:
+ \forall (g: G).(\forall (x: C).(\forall (c: C).(\forall (v: T).(\forall (k:
+K).((wf3 g x (CHead c k v)) \to (ex B (\lambda (b: B).(eq K k (Bind b)))))))))
+\def
+ \lambda (g: G).(\lambda (x: C).(\lambda (c: C).(\lambda (v: T).(\lambda (k:
+K).(\lambda (H: (wf3 g x (CHead c k v))).(insert_eq C (CHead c k v) (\lambda
+(c0: C).(wf3 g x c0)) (\lambda (_: C).(ex B (\lambda (b: B).(eq K k (Bind
+b))))) (\lambda (y: C).(\lambda (H0: (wf3 g x y)).(wf3_ind g (\lambda (_:
+C).(\lambda (c1: C).((eq C c1 (CHead c k v)) \to (ex B (\lambda (b: B).(eq K
+k (Bind b))))))) (\lambda (m: nat).(\lambda (H1: (eq C (CSort m) (CHead c k
+v))).(let H2 \def (eq_ind C (CSort m) (\lambda (ee: C).(match ee in C return
+(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _)
+\Rightarrow False])) I (CHead c k v) H1) in (False_ind (ex B (\lambda (b:
+B).(eq K k (Bind b)))) H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1:
+(wf3 g c1 c2)).(\lambda (H2: (((eq C c2 (CHead c k v)) \to (ex B (\lambda (b:
+B).(eq K k (Bind b))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H3: (ty3
+g c1 u t)).(\lambda (b: B).(\lambda (H4: (eq C (CHead c2 (Bind b) u) (CHead c
+k v))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return
+(\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ _)
+\Rightarrow c0])) (CHead c2 (Bind b) u) (CHead c k v) H4) in ((let H6 \def
+(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with
+[(CSort _) \Rightarrow (Bind b) | (CHead _ k0 _) \Rightarrow k0])) (CHead c2
+(Bind b) u) (CHead c k v) H4) in ((let H7 \def (f_equal C T (\lambda (e:
+C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
+(CHead _ _ t0) \Rightarrow t0])) (CHead c2 (Bind b) u) (CHead c k v) H4) in
+(\lambda (H8: (eq K (Bind b) k)).(\lambda (H9: (eq C c2 c)).(let H10 \def
+(eq_ind T u (\lambda (t0: T).(ty3 g c1 t0 t)) H3 v H7) in (let H11 \def
+(eq_ind C c2 (\lambda (c0: C).((eq C c0 (CHead c k v)) \to (ex B (\lambda
+(b0: B).(eq K k (Bind b0)))))) H2 c H9) in (let H12 \def (eq_ind C c2
+(\lambda (c0: C).(wf3 g c1 c0)) H1 c H9) in (let H13 \def (eq_ind_r K k
+(\lambda (k0: K).((eq C c (CHead c k0 v)) \to (ex B (\lambda (b0: B).(eq K k0
+(Bind b0)))))) H11 (Bind b) H8) in (eq_ind K (Bind b) (\lambda (k0: K).(ex B
+(\lambda (b0: B).(eq K k0 (Bind b0))))) (ex_intro B (\lambda (b0: B).(eq K
+(Bind b) (Bind b0))) b (refl_equal K (Bind b))) k H8)))))))) H6))
+H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1
+c2)).(\lambda (H2: (((eq C c2 (CHead c k v)) \to (ex B (\lambda (b: B).(eq K
+k (Bind b))))))).(\lambda (u: T).(\lambda (_: ((\forall (t: T).((ty3 g c1 u
+t) \to False)))).(\lambda (_: B).(\lambda (H4: (eq C (CHead c2 (Bind Void)
+(TSort O)) (CHead c k v))).(let H5 \def (f_equal C C (\lambda (e: C).(match e
+in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _
+_) \Rightarrow c0])) (CHead c2 (Bind Void) (TSort O)) (CHead c k v) H4) in
+((let H6 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_:
+C).K) with [(CSort _) \Rightarrow (Bind Void) | (CHead _ k0 _) \Rightarrow
+k0])) (CHead c2 (Bind Void) (TSort O)) (CHead c k v) H4) in ((let H7 \def
+(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow (TSort O) | (CHead _ _ t) \Rightarrow t])) (CHead c2
+(Bind Void) (TSort O)) (CHead c k v) H4) in (\lambda (H8: (eq K (Bind Void)
+k)).(\lambda (H9: (eq C c2 c)).(let H10 \def (eq_ind C c2 (\lambda (c0:
+C).((eq C c0 (CHead c k v)) \to (ex B (\lambda (b0: B).(eq K k (Bind b0))))))
+H2 c H9) in (let H11 \def (eq_ind C c2 (\lambda (c0: C).(wf3 g c1 c0)) H1 c
+H9) in (let H12 \def (eq_ind_r K k (\lambda (k0: K).((eq C c (CHead c k0 v))
+\to (ex B (\lambda (b0: B).(eq K k0 (Bind b0)))))) H10 (Bind Void) H8) in
+(eq_ind K (Bind Void) (\lambda (k0: K).(ex B (\lambda (b0: B).(eq K k0 (Bind
+b0))))) (let H13 \def (eq_ind_r T v (\lambda (t: T).((eq C c (CHead c (Bind
+Void) t)) \to (ex B (\lambda (b0: B).(eq K (Bind Void) (Bind b0)))))) H12
+(TSort O) H7) in (ex_intro B (\lambda (b0: B).(eq K (Bind Void) (Bind b0)))
+Void (refl_equal K (Bind Void)))) k H8))))))) H6)) H5)))))))))) (\lambda (c1:
+C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1 c2)).(\lambda (H2: (((eq C c2
+(CHead c k v)) \to (ex B (\lambda (b: B).(eq K k (Bind b))))))).(\lambda (_:
+T).(\lambda (_: F).(\lambda (H3: (eq C c2 (CHead c k v))).(let H4 \def
+(f_equal C C (\lambda (e: C).e) c2 (CHead c k v) H3) in (let H5 \def (eq_ind
+C c2 (\lambda (c0: C).((eq C c0 (CHead c k v)) \to (ex B (\lambda (b: B).(eq
+K k (Bind b)))))) H2 (CHead c k v) H4) in (let H6 \def (eq_ind C c2 (\lambda
+(c0: C).(wf3 g c1 c0)) H1 (CHead c k v) H4) in (H5 (refl_equal C (CHead c k
+v))))))))))))) x y H0))) H)))))).
+
+theorem wf3_mono:
+ \forall (g: G).(\forall (c: C).(\forall (c1: C).((wf3 g c c1) \to (\forall
+(c2: C).((wf3 g c c2) \to (eq C c1 c2))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (c1: C).(\lambda (H: (wf3 g c
+c1)).(wf3_ind g (\lambda (c0: C).(\lambda (c2: C).(\forall (c3: C).((wf3 g c0
+c3) \to (eq C c2 c3))))) (\lambda (m: nat).(\lambda (c2: C).(\lambda (H0:
+(wf3 g (CSort m) c2)).(let H_y \def (wf3_gen_sort1 g c2 m H0) in (eq_ind_r C
+(CSort m) (\lambda (c0: C).(eq C (CSort m) c0)) (refl_equal C (CSort m)) c2
+H_y))))) (\lambda (c2: C).(\lambda (c3: C).(\lambda (_: (wf3 g c2
+c3)).(\lambda (H1: ((\forall (c4: C).((wf3 g c2 c4) \to (eq C c3
+c4))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H2: (ty3 g c2 u
+t)).(\lambda (b: B).(\lambda (c0: C).(\lambda (H3: (wf3 g (CHead c2 (Bind b)
+u) c0)).(let H_x \def (wf3_gen_bind1 g c2 c0 u b H3) in (let H4 \def H_x in
+(or_ind (ex3_2 C T (\lambda (c4: C).(\lambda (_: T).(eq C c0 (CHead c4 (Bind
+b) u)))) (\lambda (c4: C).(\lambda (_: T).(wf3 g c2 c4))) (\lambda (_:
+C).(\lambda (w: T).(ty3 g c2 u w)))) (ex3 C (\lambda (c4: C).(eq C c0 (CHead
+c4 (Bind Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda (_:
+C).(\forall (w: T).((ty3 g c2 u w) \to False)))) (eq C (CHead c3 (Bind b) u)
+c0) (\lambda (H5: (ex3_2 C T (\lambda (c4: C).(\lambda (_: T).(eq C c0 (CHead
+c4 (Bind b) u)))) (\lambda (c4: C).(\lambda (_: T).(wf3 g c2 c4))) (\lambda
+(_: C).(\lambda (w: T).(ty3 g c2 u w))))).(ex3_2_ind C T (\lambda (c4:
+C).(\lambda (_: T).(eq C c0 (CHead c4 (Bind b) u)))) (\lambda (c4:
+C).(\lambda (_: T).(wf3 g c2 c4))) (\lambda (_: C).(\lambda (w: T).(ty3 g c2
+u w))) (eq C (CHead c3 (Bind b) u) c0) (\lambda (x0: C).(\lambda (x1:
+T).(\lambda (H6: (eq C c0 (CHead x0 (Bind b) u))).(\lambda (H7: (wf3 g c2
+x0)).(\lambda (_: (ty3 g c2 u x1)).(eq_ind_r C (CHead x0 (Bind b) u) (\lambda
+(c4: C).(eq C (CHead c3 (Bind b) u) c4)) (f_equal3 C K T C CHead c3 x0 (Bind
+b) (Bind b) u u (H1 x0 H7) (refl_equal K (Bind b)) (refl_equal T u)) c0
+H6)))))) H5)) (\lambda (H5: (ex3 C (\lambda (c4: C).(eq C c0 (CHead c4 (Bind
+Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda (_: C).(\forall
+(w: T).((ty3 g c2 u w) \to False))))).(ex3_ind C (\lambda (c4: C).(eq C c0
+(CHead c4 (Bind Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda
+(_: C).(\forall (w: T).((ty3 g c2 u w) \to False))) (eq C (CHead c3 (Bind b)
+u) c0) (\lambda (x0: C).(\lambda (H6: (eq C c0 (CHead x0 (Bind Void) (TSort
+O)))).(\lambda (_: (wf3 g c2 x0)).(\lambda (H8: ((\forall (w: T).((ty3 g c2 u
+w) \to False)))).(eq_ind_r C (CHead x0 (Bind Void) (TSort O)) (\lambda (c4:
+C).(eq C (CHead c3 (Bind b) u) c4)) (let H_x0 \def (H8 t H2) in (let H9 \def
+H_x0 in (False_ind (eq C (CHead c3 (Bind b) u) (CHead x0 (Bind Void) (TSort
+O))) H9))) c0 H6))))) H5)) H4))))))))))))) (\lambda (c2: C).(\lambda (c3:
+C).(\lambda (_: (wf3 g c2 c3)).(\lambda (H1: ((\forall (c4: C).((wf3 g c2 c4)
+\to (eq C c3 c4))))).(\lambda (u: T).(\lambda (H2: ((\forall (t: T).((ty3 g
+c2 u t) \to False)))).(\lambda (b: B).(\lambda (c0: C).(\lambda (H3: (wf3 g
+(CHead c2 (Bind b) u) c0)).(let H_x \def (wf3_gen_bind1 g c2 c0 u b H3) in
+(let H4 \def H_x in (or_ind (ex3_2 C T (\lambda (c4: C).(\lambda (_: T).(eq C
+c0 (CHead c4 (Bind b) u)))) (\lambda (c4: C).(\lambda (_: T).(wf3 g c2 c4)))
+(\lambda (_: C).(\lambda (w: T).(ty3 g c2 u w)))) (ex3 C (\lambda (c4: C).(eq
+C c0 (CHead c4 (Bind Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4))
+(\lambda (_: C).(\forall (w: T).((ty3 g c2 u w) \to False)))) (eq C (CHead c3
+(Bind Void) (TSort O)) c0) (\lambda (H5: (ex3_2 C T (\lambda (c4: C).(\lambda
+(_: T).(eq C c0 (CHead c4 (Bind b) u)))) (\lambda (c4: C).(\lambda (_:
+T).(wf3 g c2 c4))) (\lambda (_: C).(\lambda (w: T).(ty3 g c2 u
+w))))).(ex3_2_ind C T (\lambda (c4: C).(\lambda (_: T).(eq C c0 (CHead c4
+(Bind b) u)))) (\lambda (c4: C).(\lambda (_: T).(wf3 g c2 c4))) (\lambda (_:
+C).(\lambda (w: T).(ty3 g c2 u w))) (eq C (CHead c3 (Bind Void) (TSort O))
+c0) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C c0 (CHead x0 (Bind
+b) u))).(\lambda (_: (wf3 g c2 x0)).(\lambda (H8: (ty3 g c2 u x1)).(eq_ind_r
+C (CHead x0 (Bind b) u) (\lambda (c4: C).(eq C (CHead c3 (Bind Void) (TSort
+O)) c4)) (let H_x0 \def (H2 x1 H8) in (let H9 \def H_x0 in (False_ind (eq C
+(CHead c3 (Bind Void) (TSort O)) (CHead x0 (Bind b) u)) H9))) c0 H6))))))
+H5)) (\lambda (H5: (ex3 C (\lambda (c4: C).(eq C c0 (CHead c4 (Bind Void)
+(TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda (_: C).(\forall (w:
+T).((ty3 g c2 u w) \to False))))).(ex3_ind C (\lambda (c4: C).(eq C c0 (CHead
+c4 (Bind Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda (_:
+C).(\forall (w: T).((ty3 g c2 u w) \to False))) (eq C (CHead c3 (Bind Void)
+(TSort O)) c0) (\lambda (x0: C).(\lambda (H6: (eq C c0 (CHead x0 (Bind Void)
+(TSort O)))).(\lambda (H7: (wf3 g c2 x0)).(\lambda (_: ((\forall (w: T).((ty3
+g c2 u w) \to False)))).(eq_ind_r C (CHead x0 (Bind Void) (TSort O)) (\lambda
+(c4: C).(eq C (CHead c3 (Bind Void) (TSort O)) c4)) (f_equal3 C K T C CHead
+c3 x0 (Bind Void) (Bind Void) (TSort O) (TSort O) (H1 x0 H7) (refl_equal K
+(Bind Void)) (refl_equal T (TSort O))) c0 H6))))) H5)) H4))))))))))))
+(\lambda (c2: C).(\lambda (c3: C).(\lambda (_: (wf3 g c2 c3)).(\lambda (H1:
+((\forall (c4: C).((wf3 g c2 c4) \to (eq C c3 c4))))).(\lambda (u:
+T).(\lambda (f: F).(\lambda (c0: C).(\lambda (H2: (wf3 g (CHead c2 (Flat f)
+u) c0)).(let H_y \def (wf3_gen_flat1 g c2 c0 u f H2) in (H1 c0 H_y))))))))))
+c c1 H)))).
+
+theorem wf3_clear_conf:
+ \forall (c1: C).(\forall (c: C).((clear c1 c) \to (\forall (g: G).(\forall
+(c2: C).((wf3 g c1 c2) \to (wf3 g c c2))))))
+\def
+ \lambda (c1: C).(\lambda (c: C).(\lambda (H: (clear c1 c)).(clear_ind
+(\lambda (c0: C).(\lambda (c2: C).(\forall (g: G).(\forall (c3: C).((wf3 g c0
+c3) \to (wf3 g c2 c3)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (u:
+T).(\lambda (g: G).(\lambda (c2: C).(\lambda (H0: (wf3 g (CHead e (Bind b) u)
+c2)).H0)))))) (\lambda (e: C).(\lambda (c0: C).(\lambda (_: (clear e
+c0)).(\lambda (H1: ((\forall (g: G).(\forall (c2: C).((wf3 g e c2) \to (wf3 g
+c0 c2)))))).(\lambda (f: F).(\lambda (u: T).(\lambda (g: G).(\lambda (c2:
+C).(\lambda (H2: (wf3 g (CHead e (Flat f) u) c2)).(let H_y \def
+(wf3_gen_flat1 g e c2 u f H2) in (H1 g c2 H_y))))))))))) c1 c H))).
+
+theorem clear_wf3_trans:
+ \forall (c1: C).(\forall (d1: C).((clear c1 d1) \to (\forall (g: G).(\forall
+(d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda
+(c2: C).(clear c2 d2))))))))
+\def
+ \lambda (c1: C).(\lambda (d1: C).(\lambda (H: (clear c1 d1)).(clear_ind
+(\lambda (c: C).(\lambda (c0: C).(\forall (g: G).(\forall (d2: C).((wf3 g c0
+d2) \to (ex2 C (\lambda (c2: C).(wf3 g c c2)) (\lambda (c2: C).(clear c2
+d2)))))))) (\lambda (b: B).(\lambda (e: C).(\lambda (u: T).(\lambda (g:
+G).(\lambda (d2: C).(\lambda (H0: (wf3 g (CHead e (Bind b) u) d2)).(let H_x
+\def (wf3_gen_bind1 g e d2 u b H0) in (let H1 \def H_x in (or_ind (ex3_2 C T
+(\lambda (c2: C).(\lambda (_: T).(eq C d2 (CHead c2 (Bind b) u)))) (\lambda
+(c2: C).(\lambda (_: T).(wf3 g e c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g
+e u w)))) (ex3 C (\lambda (c2: C).(eq C d2 (CHead c2 (Bind Void) (TSort O))))
+(\lambda (c2: C).(wf3 g e c2)) (\lambda (_: C).(\forall (w: T).((ty3 g e u w)
+\to False)))) (ex2 C (\lambda (c2: C).(wf3 g (CHead e (Bind b) u) c2))
+(\lambda (c2: C).(clear c2 d2))) (\lambda (H2: (ex3_2 C T (\lambda (c2:
+C).(\lambda (_: T).(eq C d2 (CHead c2 (Bind b) u)))) (\lambda (c2:
+C).(\lambda (_: T).(wf3 g e c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g e u
+w))))).(ex3_2_ind C T (\lambda (c2: C).(\lambda (_: T).(eq C d2 (CHead c2
+(Bind b) u)))) (\lambda (c2: C).(\lambda (_: T).(wf3 g e c2))) (\lambda (_:
+C).(\lambda (w: T).(ty3 g e u w))) (ex2 C (\lambda (c2: C).(wf3 g (CHead e
+(Bind b) u) c2)) (\lambda (c2: C).(clear c2 d2))) (\lambda (x0: C).(\lambda
+(x1: T).(\lambda (H3: (eq C d2 (CHead x0 (Bind b) u))).(\lambda (H4: (wf3 g e
+x0)).(\lambda (H5: (ty3 g e u x1)).(eq_ind_r C (CHead x0 (Bind b) u) (\lambda
+(c: C).(ex2 C (\lambda (c2: C).(wf3 g (CHead e (Bind b) u) c2)) (\lambda (c2:
+C).(clear c2 c)))) (ex_intro2 C (\lambda (c2: C).(wf3 g (CHead e (Bind b) u)
+c2)) (\lambda (c2: C).(clear c2 (CHead x0 (Bind b) u))) (CHead x0 (Bind b) u)
+(wf3_bind g e x0 H4 u x1 H5 b) (clear_bind b x0 u)) d2 H3)))))) H2)) (\lambda
+(H2: (ex3 C (\lambda (c2: C).(eq C d2 (CHead c2 (Bind Void) (TSort O))))
+(\lambda (c2: C).(wf3 g e c2)) (\lambda (_: C).(\forall (w: T).((ty3 g e u w)
+\to False))))).(ex3_ind C (\lambda (c2: C).(eq C d2 (CHead c2 (Bind Void)
+(TSort O)))) (\lambda (c2: C).(wf3 g e c2)) (\lambda (_: C).(\forall (w:
+T).((ty3 g e u w) \to False))) (ex2 C (\lambda (c2: C).(wf3 g (CHead e (Bind
+b) u) c2)) (\lambda (c2: C).(clear c2 d2))) (\lambda (x0: C).(\lambda (H3:
+(eq C d2 (CHead x0 (Bind Void) (TSort O)))).(\lambda (H4: (wf3 g e
+x0)).(\lambda (H5: ((\forall (w: T).((ty3 g e u w) \to False)))).(eq_ind_r C
+(CHead x0 (Bind Void) (TSort O)) (\lambda (c: C).(ex2 C (\lambda (c2: C).(wf3
+g (CHead e (Bind b) u) c2)) (\lambda (c2: C).(clear c2 c)))) (ex_intro2 C
+(\lambda (c2: C).(wf3 g (CHead e (Bind b) u) c2)) (\lambda (c2: C).(clear c2
+(CHead x0 (Bind Void) (TSort O)))) (CHead x0 (Bind Void) (TSort O)) (wf3_void
+g e x0 H4 u H5 b) (clear_bind Void x0 (TSort O))) d2 H3))))) H2)) H1)))))))))
+(\lambda (e: C).(\lambda (c: C).(\lambda (_: (clear e c)).(\lambda (H1:
+((\forall (g: G).(\forall (d2: C).((wf3 g c d2) \to (ex2 C (\lambda (c2:
+C).(wf3 g e c2)) (\lambda (c2: C).(clear c2 d2)))))))).(\lambda (f:
+F).(\lambda (u: T).(\lambda (g: G).(\lambda (d2: C).(\lambda (H2: (wf3 g c
+d2)).(let H_x \def (H1 g d2 H2) in (let H3 \def H_x in (ex2_ind C (\lambda
+(c2: C).(wf3 g e c2)) (\lambda (c2: C).(clear c2 d2)) (ex2 C (\lambda (c2:
+C).(wf3 g (CHead e (Flat f) u) c2)) (\lambda (c2: C).(clear c2 d2))) (\lambda
+(x: C).(\lambda (H4: (wf3 g e x)).(\lambda (H5: (clear x d2)).(ex_intro2 C
+(\lambda (c2: C).(wf3 g (CHead e (Flat f) u) c2)) (\lambda (c2: C).(clear c2
+d2)) x (wf3_flat g e x H4 u f) H5)))) H3)))))))))))) c1 d1 H))).
+
+theorem wf3_getl_conf:
+ \forall (b: B).(\forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall
+(v: T).((getl i c1 (CHead d1 (Bind b) v)) \to (\forall (g: G).(\forall (c2:
+C).((wf3 g c1 c2) \to (\forall (w: T).((ty3 g d1 v w) \to (ex2 C (\lambda
+(d2: C).(getl i c2 (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1
+d2)))))))))))))
+\def
+ \lambda (b: B).(\lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1:
+C).(\forall (d1: C).(\forall (v: T).((getl n c1 (CHead d1 (Bind b) v)) \to
+(\forall (g: G).(\forall (c2: C).((wf3 g c1 c2) \to (\forall (w: T).((ty3 g
+d1 v w) \to (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind b) v)))
+(\lambda (d2: C).(wf3 g d1 d2))))))))))))) (\lambda (c1: C).(\lambda (d1:
+C).(\lambda (v: T).(\lambda (H: (getl O c1 (CHead d1 (Bind b) v))).(\lambda
+(g: G).(\lambda (c2: C).(\lambda (H0: (wf3 g c1 c2)).(\lambda (w: T).(\lambda
+(H1: (ty3 g d1 v w)).(let H_y \def (wf3_clear_conf c1 (CHead d1 (Bind b) v)
+(getl_gen_O c1 (CHead d1 (Bind b) v) H) g c2 H0) in (let H_x \def
+(wf3_gen_bind1 g d1 c2 v b H_y) in (let H2 \def H_x in (or_ind (ex3_2 C T
+(\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda
+(c3: C).(\lambda (_: T).(wf3 g d1 c3))) (\lambda (_: C).(\lambda (w0: T).(ty3
+g d1 v w0)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort
+O)))) (\lambda (c3: C).(wf3 g d1 c3)) (\lambda (_: C).(\forall (w0: T).((ty3
+g d1 v w0) \to False)))) (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind
+b) v))) (\lambda (d2: C).(wf3 g d1 d2))) (\lambda (H3: (ex3_2 C T (\lambda
+(c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda (c3:
+C).(\lambda (_: T).(wf3 g d1 c3))) (\lambda (_: C).(\lambda (w0: T).(ty3 g d1
+v w0))))).(ex3_2_ind C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3
+(Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g d1 c3))) (\lambda (_:
+C).(\lambda (w0: T).(ty3 g d1 v w0))) (ex2 C (\lambda (d2: C).(getl O c2
+(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2))) (\lambda (x0:
+C).(\lambda (x1: T).(\lambda (H4: (eq C c2 (CHead x0 (Bind b) v))).(\lambda
+(H5: (wf3 g d1 x0)).(\lambda (_: (ty3 g d1 v x1)).(eq_ind_r C (CHead x0 (Bind
+b) v) (\lambda (c: C).(ex2 C (\lambda (d2: C).(getl O c (CHead d2 (Bind b)
+v))) (\lambda (d2: C).(wf3 g d1 d2)))) (ex_intro2 C (\lambda (d2: C).(getl O
+(CHead x0 (Bind b) v) (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2))
+x0 (getl_refl b x0 v) H5) c2 H4)))))) H3)) (\lambda (H3: (ex3 C (\lambda (c3:
+C).(eq C c2 (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g d1
+c3)) (\lambda (_: C).(\forall (w0: T).((ty3 g d1 v w0) \to
+False))))).(ex3_ind C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort
+O)))) (\lambda (c3: C).(wf3 g d1 c3)) (\lambda (_: C).(\forall (w0: T).((ty3
+g d1 v w0) \to False))) (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind b)
+v))) (\lambda (d2: C).(wf3 g d1 d2))) (\lambda (x0: C).(\lambda (H4: (eq C c2
+(CHead x0 (Bind Void) (TSort O)))).(\lambda (_: (wf3 g d1 x0)).(\lambda (H6:
+((\forall (w0: T).((ty3 g d1 v w0) \to False)))).(eq_ind_r C (CHead x0 (Bind
+Void) (TSort O)) (\lambda (c: C).(ex2 C (\lambda (d2: C).(getl O c (CHead d2
+(Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)))) (let H_x0 \def (H6 w H1) in
+(let H7 \def H_x0 in (False_ind (ex2 C (\lambda (d2: C).(getl O (CHead x0
+(Bind Void) (TSort O)) (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1
+d2))) H7))) c2 H4))))) H3)) H2))))))))))))) (\lambda (n: nat).(\lambda (H:
+((\forall (c1: C).(\forall (d1: C).(\forall (v: T).((getl n c1 (CHead d1
+(Bind b) v)) \to (\forall (g: G).(\forall (c2: C).((wf3 g c1 c2) \to (\forall
+(w: T).((ty3 g d1 v w) \to (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind
+b) v))) (\lambda (d2: C).(wf3 g d1 d2)))))))))))))).(\lambda (c1: C).(C_ind
+(\lambda (c: C).(\forall (d1: C).(\forall (v: T).((getl (S n) c (CHead d1
+(Bind b) v)) \to (\forall (g: G).(\forall (c2: C).((wf3 g c c2) \to (\forall
+(w: T).((ty3 g d1 v w) \to (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2
+(Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)))))))))))) (\lambda (n0:
+nat).(\lambda (d1: C).(\lambda (v: T).(\lambda (H0: (getl (S n) (CSort n0)
+(CHead d1 (Bind b) v))).(\lambda (g: G).(\lambda (c2: C).(\lambda (_: (wf3 g
+(CSort n0) c2)).(\lambda (w: T).(\lambda (_: (ty3 g d1 v w)).(getl_gen_sort
+n0 (S n) (CHead d1 (Bind b) v) H0 (ex2 C (\lambda (d2: C).(getl (S n) c2
+(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2))))))))))))) (\lambda
+(c: C).(\lambda (H0: ((\forall (d1: C).(\forall (v: T).((getl (S n) c (CHead
+d1 (Bind b) v)) \to (\forall (g: G).(\forall (c2: C).((wf3 g c c2) \to
+(\forall (w: T).((ty3 g d1 v w) \to (ex2 C (\lambda (d2: C).(getl (S n) c2
+(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2))))))))))))).(\lambda
+(k: K).(\lambda (t: T).(\lambda (d1: C).(\lambda (v: T).(\lambda (H1: (getl
+(S n) (CHead c k t) (CHead d1 (Bind b) v))).(\lambda (g: G).(\lambda (c2:
+C).(\lambda (H2: (wf3 g (CHead c k t) c2)).(\lambda (w: T).(\lambda (H3: (ty3
+g d1 v w)).(K_ind (\lambda (k0: K).((wf3 g (CHead c k0 t) c2) \to ((getl (r
+k0 n) c (CHead d1 (Bind b) v)) \to (ex2 C (\lambda (d2: C).(getl (S n) c2
+(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)))))) (\lambda (b0:
+B).(\lambda (H4: (wf3 g (CHead c (Bind b0) t) c2)).(\lambda (H5: (getl (r
+(Bind b0) n) c (CHead d1 (Bind b) v))).(let H_x \def (wf3_gen_bind1 g c c2 t
+b0 H4) in (let H6 \def H_x in (or_ind (ex3_2 C T (\lambda (c3: C).(\lambda
+(_: T).(eq C c2 (CHead c3 (Bind b0) t)))) (\lambda (c3: C).(\lambda (_:
+T).(wf3 g c c3))) (\lambda (_: C).(\lambda (w0: T).(ty3 g c t w0)))) (ex3 C
+(\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3:
+C).(wf3 g c c3)) (\lambda (_: C).(\forall (w0: T).((ty3 g c t w0) \to
+False)))) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind b) v)))
+(\lambda (d2: C).(wf3 g d1 d2))) (\lambda (H7: (ex3_2 C T (\lambda (c3:
+C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b0) t)))) (\lambda (c3:
+C).(\lambda (_: T).(wf3 g c c3))) (\lambda (_: C).(\lambda (w0: T).(ty3 g c t
+w0))))).(ex3_2_ind C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3
+(Bind b0) t)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c c3))) (\lambda (_:
+C).(\lambda (w0: T).(ty3 g c t w0))) (ex2 C (\lambda (d2: C).(getl (S n) c2
+(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2))) (\lambda (x0:
+C).(\lambda (x1: T).(\lambda (H8: (eq C c2 (CHead x0 (Bind b0) t))).(\lambda
+(H9: (wf3 g c x0)).(\lambda (_: (ty3 g c t x1)).(eq_ind_r C (CHead x0 (Bind
+b0) t) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(getl (S n) c0 (CHead d2
+(Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)))) (let H_x0 \def (H c d1 v H5 g
+x0 H9 w H3) in (let H11 \def H_x0 in (ex2_ind C (\lambda (d2: C).(getl n x0
+(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)) (ex2 C (\lambda (d2:
+C).(getl (S n) (CHead x0 (Bind b0) t) (CHead d2 (Bind b) v))) (\lambda (d2:
+C).(wf3 g d1 d2))) (\lambda (x: C).(\lambda (H12: (getl n x0 (CHead x (Bind
+b) v))).(\lambda (H13: (wf3 g d1 x)).(ex_intro2 C (\lambda (d2: C).(getl (S
+n) (CHead x0 (Bind b0) t) (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1
+d2)) x (getl_head (Bind b0) n x0 (CHead x (Bind b) v) H12 t) H13)))) H11)))
+c2 H8)))))) H7)) (\lambda (H7: (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3
+(Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c c3)) (\lambda (_:
+C).(\forall (w0: T).((ty3 g c t w0) \to False))))).(ex3_ind C (\lambda (c3:
+C).(eq C c2 (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c c3))
+(\lambda (_: C).(\forall (w0: T).((ty3 g c t w0) \to False))) (ex2 C (\lambda
+(d2: C).(getl (S n) c2 (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1
+d2))) (\lambda (x0: C).(\lambda (H8: (eq C c2 (CHead x0 (Bind Void) (TSort
+O)))).(\lambda (H9: (wf3 g c x0)).(\lambda (_: ((\forall (w0: T).((ty3 g c t
+w0) \to False)))).(eq_ind_r C (CHead x0 (Bind Void) (TSort O)) (\lambda (c0:
+C).(ex2 C (\lambda (d2: C).(getl (S n) c0 (CHead d2 (Bind b) v))) (\lambda
+(d2: C).(wf3 g d1 d2)))) (let H_x0 \def (H c d1 v H5 g x0 H9 w H3) in (let
+H11 \def H_x0 in (ex2_ind C (\lambda (d2: C).(getl n x0 (CHead d2 (Bind b)
+v))) (\lambda (d2: C).(wf3 g d1 d2)) (ex2 C (\lambda (d2: C).(getl (S n)
+(CHead x0 (Bind Void) (TSort O)) (CHead d2 (Bind b) v))) (\lambda (d2:
+C).(wf3 g d1 d2))) (\lambda (x: C).(\lambda (H12: (getl n x0 (CHead x (Bind
+b) v))).(\lambda (H13: (wf3 g d1 x)).(ex_intro2 C (\lambda (d2: C).(getl (S
+n) (CHead x0 (Bind Void) (TSort O)) (CHead d2 (Bind b) v))) (\lambda (d2:
+C).(wf3 g d1 d2)) x (getl_head (Bind Void) n x0 (CHead x (Bind b) v) H12
+(TSort O)) H13)))) H11))) c2 H8))))) H7)) H6)))))) (\lambda (f: F).(\lambda
+(H4: (wf3 g (CHead c (Flat f) t) c2)).(\lambda (H5: (getl (r (Flat f) n) c
+(CHead d1 (Bind b) v))).(let H_y \def (wf3_gen_flat1 g c c2 t f H4) in (H0 d1
+v H5 g c2 H_y w H3))))) k H2 (getl_gen_S k c (CHead d1 (Bind b) v) t n
+H1)))))))))))))) c1)))) i)).
+
+theorem wf3_total:
+ \forall (g: G).(\forall (c1: C).(ex C (\lambda (c2: C).(wf3 g c1 c2))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(C_ind (\lambda (c: C).(ex C (\lambda (c2:
+C).(wf3 g c c2)))) (\lambda (n: nat).(ex_intro C (\lambda (c2: C).(wf3 g
+(CSort n) c2)) (CSort n) (wf3_sort g n))) (\lambda (c: C).(\lambda (H: (ex C
+(\lambda (c2: C).(wf3 g c c2)))).(\lambda (k: K).(\lambda (t: T).(let H0 \def
+H in (ex_ind C (\lambda (c2: C).(wf3 g c c2)) (ex C (\lambda (c2: C).(wf3 g
+(CHead c k t) c2))) (\lambda (x: C).(\lambda (H1: (wf3 g c x)).(K_ind
+(\lambda (k0: K).(ex C (\lambda (c2: C).(wf3 g (CHead c k0 t) c2)))) (\lambda
+(b: B).(let H_x \def (ty3_inference g c t) in (let H2 \def H_x in (or_ind (ex
+T (\lambda (t2: T).(ty3 g c t t2))) (\forall (t2: T).((ty3 g c t t2) \to
+False)) (ex C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2))) (\lambda
+(H3: (ex T (\lambda (t2: T).(ty3 g c t t2)))).(ex_ind T (\lambda (t2: T).(ty3
+g c t t2)) (ex C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2))) (\lambda
+(x0: T).(\lambda (H4: (ty3 g c t x0)).(ex_intro C (\lambda (c2: C).(wf3 g
+(CHead c (Bind b) t) c2)) (CHead x (Bind b) t) (wf3_bind g c x H1 t x0 H4
+b)))) H3)) (\lambda (H3: ((\forall (t2: T).((ty3 g c t t2) \to
+False)))).(ex_intro C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2))
+(CHead x (Bind Void) (TSort O)) (wf3_void g c x H1 t H3 b))) H2)))) (\lambda
+(f: F).(ex_intro C (\lambda (c2: C).(wf3 g (CHead c (Flat f) t) c2)) x
+(wf3_flat g c x H1 t f))) k))) H0)))))) c1)).
+
+theorem getl_wf3_trans:
+ \forall (i: nat).(\forall (c1: C).(\forall (d1: C).((getl i c1 d1) \to
+(\forall (g: G).(\forall (d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda (c2:
+C).(wf3 g c1 c2)) (\lambda (c2: C).(getl i c2 d2)))))))))
+\def
+ \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1:
+C).((getl n c1 d1) \to (\forall (g: G).(\forall (d2: C).((wf3 g d1 d2) \to
+(ex2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(getl n c2
+d2)))))))))) (\lambda (c1: C).(\lambda (d1: C).(\lambda (H: (getl O c1
+d1)).(\lambda (g: G).(\lambda (d2: C).(\lambda (H0: (wf3 g d1 d2)).(let H_x
+\def (clear_wf3_trans c1 d1 (getl_gen_O c1 d1 H) g d2 H0) in (let H1 \def H_x
+in (ex2_ind C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(clear c2 d2))
+(ex2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(getl O c2 d2)))
+(\lambda (x: C).(\lambda (H2: (wf3 g c1 x)).(\lambda (H3: (clear x
+d2)).(ex_intro2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(getl O c2
+d2)) x H2 (getl_intro O x d2 x (drop_refl x) H3))))) H1))))))))) (\lambda (n:
+nat).(\lambda (H: ((\forall (c1: C).(\forall (d1: C).((getl n c1 d1) \to
+(\forall (g: G).(\forall (d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda (c2:
+C).(wf3 g c1 c2)) (\lambda (c2: C).(getl n c2 d2))))))))))).(\lambda (c1:
+C).(C_ind (\lambda (c: C).(\forall (d1: C).((getl (S n) c d1) \to (\forall
+(g: G).(\forall (d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda (c2: C).(wf3 g c
+c2)) (\lambda (c2: C).(getl (S n) c2 d2))))))))) (\lambda (n0: nat).(\lambda
+(d1: C).(\lambda (H0: (getl (S n) (CSort n0) d1)).(\lambda (g: G).(\lambda
+(d2: C).(\lambda (_: (wf3 g d1 d2)).(getl_gen_sort n0 (S n) d1 H0 (ex2 C
+(\lambda (c2: C).(wf3 g (CSort n0) c2)) (\lambda (c2: C).(getl (S n) c2
+d2)))))))))) (\lambda (c: C).(\lambda (H0: ((\forall (d1: C).((getl (S n) c
+d1) \to (\forall (g: G).(\forall (d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda
+(c2: C).(wf3 g c c2)) (\lambda (c2: C).(getl (S n) c2 d2)))))))))).(\lambda
+(k: K).(\lambda (t: T).(\lambda (d1: C).(\lambda (H1: (getl (S n) (CHead c k
+t) d1)).(\lambda (g: G).(\lambda (d2: C).(\lambda (H2: (wf3 g d1 d2)).(K_ind
+(\lambda (k0: K).((getl (r k0 n) c d1) \to (ex2 C (\lambda (c2: C).(wf3 g
+(CHead c k0 t) c2)) (\lambda (c2: C).(getl (S n) c2 d2))))) (\lambda (b:
+B).(\lambda (H3: (getl (r (Bind b) n) c d1)).(let H_x \def (H c d1 H3 g d2
+H2) in (let H4 \def H_x in (ex2_ind C (\lambda (c2: C).(wf3 g c c2)) (\lambda
+(c2: C).(getl n c2 d2)) (ex2 C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t)
+c2)) (\lambda (c2: C).(getl (S n) c2 d2))) (\lambda (x: C).(\lambda (H5: (wf3
+g c x)).(\lambda (H6: (getl n x d2)).(let H_x0 \def (ty3_inference g c t) in
+(let H7 \def H_x0 in (or_ind (ex T (\lambda (t2: T).(ty3 g c t t2))) (\forall
+(t2: T).((ty3 g c t t2) \to False)) (ex2 C (\lambda (c2: C).(wf3 g (CHead c
+(Bind b) t) c2)) (\lambda (c2: C).(getl (S n) c2 d2))) (\lambda (H8: (ex T
+(\lambda (t2: T).(ty3 g c t t2)))).(ex_ind T (\lambda (t2: T).(ty3 g c t t2))
+(ex2 C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2)) (\lambda (c2:
+C).(getl (S n) c2 d2))) (\lambda (x0: T).(\lambda (H9: (ty3 g c t
+x0)).(ex_intro2 C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2)) (\lambda
+(c2: C).(getl (S n) c2 d2)) (CHead x (Bind b) t) (wf3_bind g c x H5 t x0 H9
+b) (getl_head (Bind b) n x d2 H6 t)))) H8)) (\lambda (H8: ((\forall (t2:
+T).((ty3 g c t t2) \to False)))).(ex_intro2 C (\lambda (c2: C).(wf3 g (CHead
+c (Bind b) t) c2)) (\lambda (c2: C).(getl (S n) c2 d2)) (CHead x (Bind Void)
+(TSort O)) (wf3_void g c x H5 t H8 b) (getl_head (Bind Void) n x d2 H6 (TSort
+O)))) H7)))))) H4))))) (\lambda (f: F).(\lambda (H3: (getl (r (Flat f) n) c
+d1)).(let H_x \def (H0 d1 H3 g d2 H2) in (let H4 \def H_x in (ex2_ind C
+(\lambda (c2: C).(wf3 g c c2)) (\lambda (c2: C).(getl (S n) c2 d2)) (ex2 C
+(\lambda (c2: C).(wf3 g (CHead c (Flat f) t) c2)) (\lambda (c2: C).(getl (S
+n) c2 d2))) (\lambda (x: C).(\lambda (H5: (wf3 g c x)).(\lambda (H6: (getl (S
+n) x d2)).(ex_intro2 C (\lambda (c2: C).(wf3 g (CHead c (Flat f) t) c2))
+(\lambda (c2: C).(getl (S n) c2 d2)) x (wf3_flat g c x H5 t f) H6)))) H4)))))
+k (getl_gen_S k c d1 t n H1))))))))))) c1)))) i).
+
+theorem ty3_shift1:
+ \forall (g: G).(\forall (c: C).((wf3 g c c) \to (\forall (t1: T).(\forall
+(t2: T).((ty3 g c t1 t2) \to (ty3 g (CSort (cbk c)) (app1 c t1) (app1 c
+t2)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (H: (wf3 g c c)).(insert_eq C c
+(\lambda (c0: C).(wf3 g c0 c)) (\lambda (c0: C).(\forall (t1: T).(\forall
+(t2: T).((ty3 g c0 t1 t2) \to (ty3 g (CSort (cbk c0)) (app1 c0 t1) (app1 c0
+t2)))))) (\lambda (y: C).(\lambda (H0: (wf3 g y c)).(wf3_ind g (\lambda (c0:
+C).(\lambda (c1: C).((eq C c0 c1) \to (\forall (t1: T).(\forall (t2: T).((ty3
+g c0 t1 t2) \to (ty3 g (CSort (cbk c0)) (app1 c0 t1) (app1 c0 t2))))))))
+(\lambda (m: nat).(\lambda (_: (eq C (CSort m) (CSort m))).(\lambda (t1:
+T).(\lambda (t2: T).(\lambda (H2: (ty3 g (CSort m) t1 t2)).H2))))) (\lambda
+(c1: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1 c2)).(\lambda (H2: (((eq C
+c1 c2) \to (\forall (t1: T).(\forall (t2: T).((ty3 g c1 t1 t2) \to (ty3 g
+(CSort (cbk c1)) (app1 c1 t1) (app1 c1 t2)))))))).(\lambda (u: T).(\lambda
+(t: T).(\lambda (H3: (ty3 g c1 u t)).(\lambda (b: B).(\lambda (H4: (eq C
+(CHead c1 (Bind b) u) (CHead c2 (Bind b) u))).(\lambda (t1: T).(\lambda (t2:
+T).(\lambda (H5: (ty3 g (CHead c1 (Bind b) u) t1 t2)).(let H6 \def (f_equal C
+C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow c1 | (CHead c0 _ _) \Rightarrow c0])) (CHead c1 (Bind b) u)
+(CHead c2 (Bind b) u) H4) in (let H7 \def (eq_ind_r C c2 (\lambda (c0:
+C).((eq C c1 c0) \to (\forall (t3: T).(\forall (t4: T).((ty3 g c1 t3 t4) \to
+(ty3 g (CSort (cbk c1)) (app1 c1 t3) (app1 c1 t4))))))) H2 c1 H6) in (let H8
+\def (eq_ind_r C c2 (\lambda (c0: C).(wf3 g c1 c0)) H1 c1 H6) in (ex_ind T
+(\lambda (t0: T).(ty3 g (CHead c1 (Bind b) u) t2 t0)) (ty3 g (CSort (cbk c1))
+(app1 c1 (THead (Bind b) u t1)) (app1 c1 (THead (Bind b) u t2))) (\lambda (x:
+T).(\lambda (_: (ty3 g (CHead c1 (Bind b) u) t2 x)).(H7 (refl_equal C c1)
+(THead (Bind b) u t1) (THead (Bind b) u t2) (ty3_bind g c1 u t H3 b t1 t2
+H5)))) (ty3_correct g (CHead c1 (Bind b) u) t1 t2 H5)))))))))))))))))
+(\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1 c2)).(\lambda (H2:
+(((eq C c1 c2) \to (\forall (t1: T).(\forall (t2: T).((ty3 g c1 t1 t2) \to
+(ty3 g (CSort (cbk c1)) (app1 c1 t1) (app1 c1 t2)))))))).(\lambda (u:
+T).(\lambda (H3: ((\forall (t: T).((ty3 g c1 u t) \to False)))).(\lambda (b:
+B).(\lambda (H4: (eq C (CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort
+O)))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H5: (ty3 g (CHead c1 (Bind
+b) u) t1 t2)).(let H6 \def (f_equal C C (\lambda (e: C).(match e in C return
+(\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _ _)
+\Rightarrow c0])) (CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort O)) H4)
+in ((let H7 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda
+(_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow (match k
+in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _)
+\Rightarrow b])])) (CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort O)) H4)
+in ((let H8 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda
+(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t]))
+(CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort O)) H4) in (\lambda (H9:
+(eq B b Void)).(\lambda (H10: (eq C c1 c2)).(let H11 \def (eq_ind B b
+(\lambda (b0: B).(ty3 g (CHead c1 (Bind b0) u) t1 t2)) H5 Void H9) in
+(eq_ind_r B Void (\lambda (b0: B).(ty3 g (CSort (cbk (CHead c1 (Bind b0) u)))
+(app1 (CHead c1 (Bind b0) u) t1) (app1 (CHead c1 (Bind b0) u) t2))) (let H12
+\def (eq_ind T u (\lambda (t: T).(ty3 g (CHead c1 (Bind Void) t) t1 t2)) H11
+(TSort O) H8) in (let H13 \def (eq_ind T u (\lambda (t: T).(\forall (t0:
+T).((ty3 g c1 t t0) \to False))) H3 (TSort O) H8) in (eq_ind_r T (TSort O)
+(\lambda (t: T).(ty3 g (CSort (cbk (CHead c1 (Bind Void) t))) (app1 (CHead c1
+(Bind Void) t) t1) (app1 (CHead c1 (Bind Void) t) t2))) (let H14 \def
+(eq_ind_r C c2 (\lambda (c0: C).((eq C c1 c0) \to (\forall (t3: T).(\forall
+(t4: T).((ty3 g c1 t3 t4) \to (ty3 g (CSort (cbk c1)) (app1 c1 t3) (app1 c1
+t4))))))) H2 c1 H10) in (let H15 \def (eq_ind_r C c2 (\lambda (c0: C).(wf3 g
+c1 c0)) H1 c1 H10) in (ex_ind T (\lambda (t: T).(ty3 g (CHead c1 (Bind Void)
+(TSort O)) t2 t)) (ty3 g (CSort (cbk c1)) (app1 c1 (THead (Bind Void) (TSort
+O) t1)) (app1 c1 (THead (Bind Void) (TSort O) t2))) (\lambda (x: T).(\lambda
+(_: (ty3 g (CHead c1 (Bind Void) (TSort O)) t2 x)).(H14 (refl_equal C c1)
+(THead (Bind Void) (TSort O) t1) (THead (Bind Void) (TSort O) t2) (ty3_bind g
+c1 (TSort O) (TSort (next g O)) (ty3_sort g c1 O) Void t1 t2 H12))))
+(ty3_correct g (CHead c1 (Bind Void) (TSort O)) t1 t2 H12)))) u H8))) b
+H9))))) H7)) H6))))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1:
+(wf3 g c1 c2)).(\lambda (H2: (((eq C c1 c2) \to (\forall (t1: T).(\forall
+(t2: T).((ty3 g c1 t1 t2) \to (ty3 g (CSort (cbk c1)) (app1 c1 t1) (app1 c1
+t2)))))))).(\lambda (u: T).(\lambda (f: F).(\lambda (H3: (eq C (CHead c1
+(Flat f) u) c2)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead
+c1 (Flat f) u) t1 t2)).(let H5 \def (f_equal C C (\lambda (e: C).e) (CHead c1
+(Flat f) u) c2 H3) in (let H6 \def (eq_ind_r C c2 (\lambda (c0: C).((eq C c1
+c0) \to (\forall (t3: T).(\forall (t4: T).((ty3 g c1 t3 t4) \to (ty3 g (CSort
+(cbk c1)) (app1 c1 t3) (app1 c1 t4))))))) H2 (CHead c1 (Flat f) u) H5) in
+(let H7 \def (eq_ind_r C c2 (\lambda (c0: C).(wf3 g c1 c0)) H1 (CHead c1
+(Flat f) u) H5) in (let H_x \def (wf3_gen_head2 g c1 c1 u (Flat f) H7) in
+(let H8 \def H_x in (ex_ind B (\lambda (b: B).(eq K (Flat f) (Bind b))) (ty3
+g (CSort (cbk c1)) (app1 c1 (THead (Flat f) u t1)) (app1 c1 (THead (Flat f) u
+t2))) (\lambda (x: B).(\lambda (H9: (eq K (Flat f) (Bind x))).(let H10 \def
+(eq_ind K (Flat f) (\lambda (ee: K).(match ee in K return (\lambda (_:
+K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])) I
+(Bind x) H9) in (False_ind (ty3 g (CSort (cbk c1)) (app1 c1 (THead (Flat f) u
+t1)) (app1 c1 (THead (Flat f) u t2))) H10)))) H8)))))))))))))))) y c H0)))
+H))).
+
+theorem wf3_pr2_conf:
+ \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pr2 c1
+t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (\forall (u: T).((ty3 g c1 t1
+u) \to (pr2 c2 t1 t2)))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(H: (pr2 c1 t1 t2)).(pr2_ind (\lambda (c: C).(\lambda (t: T).(\lambda (t0:
+T).(\forall (c2: C).((wf3 g c c2) \to (\forall (u: T).((ty3 g c t u) \to (pr2
+c2 t t0)))))))) (\lambda (c: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda
+(H0: (pr0 t3 t4)).(\lambda (c2: C).(\lambda (_: (wf3 g c c2)).(\lambda (u:
+T).(\lambda (_: (ty3 g c t3 u)).(pr2_free c2 t3 t4 H0))))))))) (\lambda (c:
+C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c
+(CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1:
+(pr0 t3 t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (c2:
+C).(\lambda (H3: (wf3 g c c2)).(\lambda (u0: T).(\lambda (H4: (ty3 g c t3
+u0)).(let H_y \def (ty3_sred_pr0 t3 t4 H1 g c u0 H4) in (let H_x \def
+(ty3_getl_subst0 g c t4 u0 H_y u t i H2 Abbr d u H0) in (let H5 \def H_x in
+(ex_ind T (\lambda (w: T).(ty3 g d u w)) (pr2 c2 t3 t) (\lambda (x:
+T).(\lambda (H6: (ty3 g d u x)).(let H_x0 \def (wf3_getl_conf Abbr i c d u H0
+g c2 H3 x H6) in (let H7 \def H_x0 in (ex2_ind C (\lambda (d2: C).(getl i c2
+(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(wf3 g d d2)) (pr2 c2 t3 t)
+(\lambda (x0: C).(\lambda (H8: (getl i c2 (CHead x0 (Bind Abbr) u))).(\lambda
+(_: (wf3 g d x0)).(pr2_delta c2 x0 u i H8 t3 t4 H1 t H2)))) H7)))))
+H5)))))))))))))))))) c1 t1 t2 H))))).
+
+theorem wf3_pr3_conf:
+ \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pr3 c1
+t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (\forall (u: T).((ty3 g c1 t1
+u) \to (pr3 c2 t1 t2)))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(H: (pr3 c1 t1 t2)).(pr3_ind c1 (\lambda (t: T).(\lambda (t0: T).(\forall
+(c2: C).((wf3 g c1 c2) \to (\forall (u: T).((ty3 g c1 t u) \to (pr3 c2 t
+t0))))))) (\lambda (t: T).(\lambda (c2: C).(\lambda (_: (wf3 g c1
+c2)).(\lambda (u: T).(\lambda (_: (ty3 g c1 t u)).(pr3_refl c2 t))))))
+(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr2 c1 t4 t3)).(\lambda (t5:
+T).(\lambda (_: (pr3 c1 t3 t5)).(\lambda (H2: ((\forall (c2: C).((wf3 g c1
+c2) \to (\forall (u: T).((ty3 g c1 t3 u) \to (pr3 c2 t3 t5))))))).(\lambda
+(c2: C).(\lambda (H3: (wf3 g c1 c2)).(\lambda (u: T).(\lambda (H4: (ty3 g c1
+t4 u)).(pr3_sing c2 t3 t4 (wf3_pr2_conf g c1 t4 t3 H0 c2 H3 u H4) t5 (H2 c2
+H3 u (ty3_sred_pr2 c1 t4 t3 H0 g u H4))))))))))))) t1 t2 H))))).
+
+theorem wf3_pc3_conf:
+ \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pc3 c1
+t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (\forall (u1: T).((ty3 g c1 t1
+u1) \to (\forall (u2: T).((ty3 g c1 t2 u2) \to (pc3 c2 t1 t2)))))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(H: (pc3 c1 t1 t2)).(\lambda (c2: C).(\lambda (H0: (wf3 g c1 c2)).(\lambda
+(u1: T).(\lambda (H1: (ty3 g c1 t1 u1)).(\lambda (u2: T).(\lambda (H2: (ty3 g
+c1 t2 u2)).(let H3 \def H in (ex2_ind T (\lambda (t: T).(pr3 c1 t1 t))
+(\lambda (t: T).(pr3 c1 t2 t)) (pc3 c2 t1 t2) (\lambda (x: T).(\lambda (H4:
+(pr3 c1 t1 x)).(\lambda (H5: (pr3 c1 t2 x)).(pc3_pr3_t c2 t1 x (wf3_pr3_conf
+g c1 t1 x H4 c2 H0 u1 H1) t2 (wf3_pr3_conf g c1 t2 x H5 c2 H0 u2 H2)))))
+H3)))))))))))).
+
+theorem wf3_ty3_conf:
+ \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c1
+t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (ty3 g c2 t1 t2)))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(H: (ty3 g c1 t1 t2)).(ty3_ind g (\lambda (c: C).(\lambda (t: T).(\lambda
+(t0: T).(\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 t t0)))))) (\lambda (c:
+C).(\lambda (t3: T).(\lambda (t: T).(\lambda (H0: (ty3 g c t3 t)).(\lambda
+(H1: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 t3 t))))).(\lambda (u:
+T).(\lambda (t4: T).(\lambda (H2: (ty3 g c u t4)).(\lambda (H3: ((\forall
+(c2: C).((wf3 g c c2) \to (ty3 g c2 u t4))))).(\lambda (H4: (pc3 c t4
+t3)).(\lambda (c2: C).(\lambda (H5: (wf3 g c c2)).(ex_ind T (\lambda (t0:
+T).(ty3 g c t4 t0)) (ty3 g c2 u t3) (\lambda (x: T).(\lambda (H6: (ty3 g c t4
+x)).(ty3_conv g c2 t3 t (H1 c2 H5) u t4 (H3 c2 H5) (wf3_pc3_conf g c t4 t3 H4
+c2 H5 x H6 t H0)))) (ty3_correct g c u t4 H2)))))))))))))) (\lambda (c:
+C).(\lambda (m: nat).(\lambda (c2: C).(\lambda (_: (wf3 g c c2)).(ty3_sort g
+c2 m))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d: C).(\lambda (u:
+T).(\lambda (H0: (getl n c (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda
+(H1: (ty3 g d u t)).(\lambda (H2: ((\forall (c2: C).((wf3 g d c2) \to (ty3 g
+c2 u t))))).(\lambda (c2: C).(\lambda (H3: (wf3 g c c2)).(let H_x \def
+(wf3_getl_conf Abbr n c d u H0 g c2 H3 t H1) in (let H4 \def H_x in (ex2_ind
+C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2:
+C).(wf3 g d d2)) (ty3 g c2 (TLRef n) (lift (S n) O t)) (\lambda (x:
+C).(\lambda (H5: (getl n c2 (CHead x (Bind Abbr) u))).(\lambda (H6: (wf3 g d
+x)).(ty3_abbr g n c2 x u H5 t (H2 x H6))))) H4))))))))))))) (\lambda (n:
+nat).(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c
+(CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u
+t)).(\lambda (H2: ((\forall (c2: C).((wf3 g d c2) \to (ty3 g c2 u
+t))))).(\lambda (c2: C).(\lambda (H3: (wf3 g c c2)).(let H_x \def
+(wf3_getl_conf Abst n c d u H0 g c2 H3 t H1) in (let H4 \def H_x in (ex2_ind
+C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u))) (\lambda (d2:
+C).(wf3 g d d2)) (ty3 g c2 (TLRef n) (lift (S n) O u)) (\lambda (x:
+C).(\lambda (H5: (getl n c2 (CHead x (Bind Abst) u))).(\lambda (H6: (wf3 g d
+x)).(ty3_abst g n c2 x u H5 t (H2 x H6))))) H4))))))))))))) (\lambda (c:
+C).(\lambda (u: T).(\lambda (t: T).(\lambda (H0: (ty3 g c u t)).(\lambda (H1:
+((\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 u t))))).(\lambda (b:
+B).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g (CHead c (Bind b) u)
+t3 t4)).(\lambda (H3: ((\forall (c2: C).((wf3 g (CHead c (Bind b) u) c2) \to
+(ty3 g c2 t3 t4))))).(\lambda (c2: C).(\lambda (H4: (wf3 g c c2)).(ty3_bind g
+c2 u t (H1 c2 H4) b t3 t4 (H3 (CHead c2 (Bind b) u) (wf3_bind g c c2 H4 u t
+H0 b))))))))))))))) (\lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\lambda
+(_: (ty3 g c w u)).(\lambda (H1: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g
+c2 w u))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c v (THead
+(Bind Abst) u t))).(\lambda (H3: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g
+c2 v (THead (Bind Abst) u t)))))).(\lambda (c2: C).(\lambda (H4: (wf3 g c
+c2)).(ty3_appl g c2 w u (H1 c2 H4) v t (H3 c2 H4))))))))))))) (\lambda (c:
+C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g c t3 t4)).(\lambda
+(H1: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 t3 t4))))).(\lambda (t0:
+T).(\lambda (_: (ty3 g c t4 t0)).(\lambda (H3: ((\forall (c2: C).((wf3 g c
+c2) \to (ty3 g c2 t4 t0))))).(\lambda (c2: C).(\lambda (H4: (wf3 g c
+c2)).(ty3_cast g c2 t3 t4 (H1 c2 H4) t0 (H3 c2 H4)))))))))))) c1 t1 t2 H))))).
+
+theorem wf3_idem:
+ \forall (g: G).(\forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to (wf3 g
+c2 c2))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wf3 g c1
+c2)).(wf3_ind g (\lambda (_: C).(\lambda (c0: C).(wf3 g c0 c0))) (\lambda (m:
+nat).(wf3_sort g m)) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wf3 g
+c3 c4)).(\lambda (H1: (wf3 g c4 c4)).(\lambda (u: T).(\lambda (t: T).(\lambda
+(H2: (ty3 g c3 u t)).(\lambda (b: B).(wf3_bind g c4 c4 H1 u t (wf3_ty3_conf g
+c3 u t H2 c4 H0) b))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (_:
+(wf3 g c3 c4)).(\lambda (H1: (wf3 g c4 c4)).(\lambda (u: T).(\lambda (_:
+((\forall (t: T).((ty3 g c3 u t) \to False)))).(\lambda (_: B).(wf3_bind g c4
+c4 H1 (TSort O) (TSort (next g O)) (ty3_sort g c4 O) Void)))))))) (\lambda
+(c3: C).(\lambda (c4: C).(\lambda (_: (wf3 g c3 c4)).(\lambda (H1: (wf3 g c4
+c4)).(\lambda (_: T).(\lambda (_: F).H1)))))) c1 c2 H)))).
+
+theorem wf3_ty3:
+ \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (u: T).((ty3 g c1 t
+u) \to (ex2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(ty3 g c2 t
+u)))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H:
+(ty3 g c1 t u)).(let H_x \def (wf3_total g c1) in (let H0 \def H_x in (ex_ind
+C (\lambda (c2: C).(wf3 g c1 c2)) (ex2 C (\lambda (c2: C).(wf3 g c1 c2))
+(\lambda (c2: C).(ty3 g c2 t u))) (\lambda (x: C).(\lambda (H1: (wf3 g c1
+x)).(ex_intro2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(ty3 g c2 t
+u)) x H1 (wf3_ty3_conf g c1 t u H x H1)))) H0))))))).
+
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-
-
-include "theory.ma".
-
-inductive sort: T \to Prop \def
-| sort_sort: \forall (n: nat).(sort (TSort n))
-| sort_abst: \forall (u: T).((sort u) \to (\forall (t: T).((sort t) \to (sort
-(THead (Bind Abst) u t))))).
-
-theorem sort_nf2:
- \forall (t: T).((sort t) \to (\forall (c: C).(nf2 c t)))
-\def
- \lambda (t: T).(\lambda (H: (sort t)).(sort_ind (\lambda (t0: T).(\forall
-(c: C).(nf2 c t0))) (\lambda (n: nat).(\lambda (c: C).(nf2_sort c n)))
-(\lambda (u: T).(\lambda (_: (sort u)).(\lambda (H1: ((\forall (c: C).(nf2 c
-u)))).(\lambda (t0: T).(\lambda (_: (sort t0)).(\lambda (H3: ((\forall (c:
-C).(nf2 c t0)))).(\lambda (c: C).(let H_y \def (H3 (CHead c (Bind Abst) u))
-in (nf2_abst_shift c u (H1 c) t0 H_y))))))))) t H)).
-
-theorem sort_pc3:
- \forall (t1: T).((sort t1) \to (\forall (t2: T).((sort t2) \to (\forall (c:
-C).((pc3 c t1 t2) \to (eq T t1 t2))))))
-\def
- \lambda (t1: T).(\lambda (H: (sort t1)).(sort_ind (\lambda (t: T).(\forall
-(t2: T).((sort t2) \to (\forall (c: C).((pc3 c t t2) \to (eq T t t2))))))
-(\lambda (n: nat).(\lambda (t2: T).(\lambda (H0: (sort t2)).(sort_ind
-(\lambda (t: T).(\forall (c: C).((pc3 c (TSort n) t) \to (eq T (TSort n)
-t)))) (\lambda (n0: nat).(\lambda (c: C).(\lambda (H1: (pc3 c (TSort n)
-(TSort n0))).(eq_ind nat n (\lambda (n1: nat).(eq T (TSort n) (TSort n1)))
-(refl_equal T (TSort n)) n0 (pc3_gen_sort c n n0 H1))))) (\lambda (u:
-T).(\lambda (_: (sort u)).(\lambda (_: ((\forall (c: C).((pc3 c (TSort n) u)
-\to (eq T (TSort n) u))))).(\lambda (t: T).(\lambda (_: (sort t)).(\lambda
-(_: ((\forall (c: C).((pc3 c (TSort n) t) \to (eq T (TSort n) t))))).(\lambda
-(c: C).(\lambda (H5: (pc3 c (TSort n) (THead (Bind Abst) u
-t))).(pc3_gen_sort_abst c u t n H5 (eq T (TSort n) (THead (Bind Abst) u
-t))))))))))) t2 H0)))) (\lambda (u: T).(\lambda (_: (sort u)).(\lambda (H1:
-((\forall (t2: T).((sort t2) \to (\forall (c: C).((pc3 c u t2) \to (eq T u
-t2))))))).(\lambda (t: T).(\lambda (_: (sort t)).(\lambda (H3: ((\forall (t2:
-T).((sort t2) \to (\forall (c: C).((pc3 c t t2) \to (eq T t
-t2))))))).(\lambda (t2: T).(\lambda (H4: (sort t2)).(sort_ind (\lambda (t0:
-T).(\forall (c: C).((pc3 c (THead (Bind Abst) u t) t0) \to (eq T (THead (Bind
-Abst) u t) t0)))) (\lambda (n: nat).(\lambda (c: C).(\lambda (H5: (pc3 c
-(THead (Bind Abst) u t) (TSort n))).(pc3_gen_sort_abst c u t n (pc3_s c
-(TSort n) (THead (Bind Abst) u t) H5) (eq T (THead (Bind Abst) u t) (TSort
-n)))))) (\lambda (u0: T).(\lambda (H5: (sort u0)).(\lambda (_: ((\forall (c:
-C).((pc3 c (THead (Bind Abst) u t) u0) \to (eq T (THead (Bind Abst) u t)
-u0))))).(\lambda (t0: T).(\lambda (H7: (sort t0)).(\lambda (_: ((\forall (c:
-C).((pc3 c (THead (Bind Abst) u t) t0) \to (eq T (THead (Bind Abst) u t)
-t0))))).(\lambda (c: C).(\lambda (H9: (pc3 c (THead (Bind Abst) u t) (THead
-(Bind Abst) u0 t0))).(and_ind (pc3 c u u0) (\forall (b: B).(\forall (u1:
-T).(pc3 (CHead c (Bind b) u1) t t0))) (eq T (THead (Bind Abst) u t) (THead
-(Bind Abst) u0 t0)) (\lambda (H10: (pc3 c u u0)).(\lambda (H11: ((\forall (b:
-B).(\forall (u1: T).(pc3 (CHead c (Bind b) u1) t t0))))).(let H_y \def (H11
-Abbr u) in (let H_y0 \def (H1 u0 H5 c H10) in (let H_y1 \def (H3 t0 H7 (CHead
-c (Bind Abbr) u) H_y) in (let H12 \def (eq_ind_r T t0 (\lambda (t3: T).(pc3
-(CHead c (Bind Abbr) u) t t3)) H_y t H_y1) in (let H13 \def (eq_ind_r T t0
-(\lambda (t3: T).(sort t3)) H7 t H_y1) in (eq_ind T t (\lambda (t3: T).(eq T
-(THead (Bind Abst) u t) (THead (Bind Abst) u0 t3))) (let H14 \def (eq_ind_r T
-u0 (\lambda (t3: T).(pc3 c u t3)) H10 u H_y0) in (let H15 \def (eq_ind_r T u0
-(\lambda (t3: T).(sort t3)) H5 u H_y0) in (eq_ind T u (\lambda (t3: T).(eq T
-(THead (Bind Abst) u t) (THead (Bind Abst) t3 t))) (refl_equal T (THead (Bind
-Abst) u t)) u0 H_y0))) t0 H_y1)))))))) (pc3_gen_abst c u u0 t t0 H9))))))))))
-t2 H4))))))))) t1 H)).
-
-theorem sort_correct:
- \forall (g: G).(\forall (t1: T).((sort t1) \to (\forall (c: C).(ex3 T
-(\lambda (t2: T).(tau0 g c t1 t2)) (\lambda (t2: T).(ty3 g c t1 t2)) (\lambda
-(t2: T).(sort t2))))))
-\def
- \lambda (g: G).(\lambda (t1: T).(\lambda (H: (sort t1)).(sort_ind (\lambda
-(t: T).(\forall (c: C).(ex3 T (\lambda (t2: T).(tau0 g c t t2)) (\lambda (t2:
-T).(ty3 g c t t2)) (\lambda (t2: T).(sort t2))))) (\lambda (n: nat).(\lambda
-(c: C).(ex3_intro T (\lambda (t2: T).(tau0 g c (TSort n) t2)) (\lambda (t2:
-T).(ty3 g c (TSort n) t2)) (\lambda (t2: T).(sort t2)) (TSort (next g n))
-(tau0_sort g c n) (ty3_sort g c n) (sort_sort (next g n))))) (\lambda (u:
-T).(\lambda (H0: (sort u)).(\lambda (H1: ((\forall (c: C).(ex3 T (\lambda
-(t2: T).(tau0 g c u t2)) (\lambda (t2: T).(ty3 g c u t2)) (\lambda (t2:
-T).(sort t2)))))).(\lambda (t: T).(\lambda (_: (sort t)).(\lambda (H3:
-((\forall (c: C).(ex3 T (\lambda (t2: T).(tau0 g c t t2)) (\lambda (t2:
-T).(ty3 g c t t2)) (\lambda (t2: T).(sort t2)))))).(\lambda (c: C).(let H_x
-\def (H1 c) in (let H4 \def H_x in (ex3_ind T (\lambda (t2: T).(tau0 g c u
-t2)) (\lambda (t2: T).(ty3 g c u t2)) (\lambda (t2: T).(sort t2)) (ex3 T
-(\lambda (t2: T).(tau0 g c (THead (Bind Abst) u t) t2)) (\lambda (t2: T).(ty3
-g c (THead (Bind Abst) u t) t2)) (\lambda (t2: T).(sort t2))) (\lambda (x0:
-T).(\lambda (_: (tau0 g c u x0)).(\lambda (H6: (ty3 g c u x0)).(\lambda (_:
-(sort x0)).(let H_x0 \def (H3 (CHead c (Bind Abst) u)) in (let H8 \def H_x0
-in (ex3_ind T (\lambda (t2: T).(tau0 g (CHead c (Bind Abst) u) t t2))
-(\lambda (t2: T).(ty3 g (CHead c (Bind Abst) u) t t2)) (\lambda (t2: T).(sort
-t2)) (ex3 T (\lambda (t2: T).(tau0 g c (THead (Bind Abst) u t) t2)) (\lambda
-(t2: T).(ty3 g c (THead (Bind Abst) u t) t2)) (\lambda (t2: T).(sort t2)))
-(\lambda (x1: T).(\lambda (H9: (tau0 g (CHead c (Bind Abst) u) t
-x1)).(\lambda (H10: (ty3 g (CHead c (Bind Abst) u) t x1)).(\lambda (H11:
-(sort x1)).(ex_ind T (\lambda (t0: T).(ty3 g (CHead c (Bind Abst) u) x1 t0))
-(ex3 T (\lambda (t2: T).(tau0 g c (THead (Bind Abst) u t) t2)) (\lambda (t2:
-T).(ty3 g c (THead (Bind Abst) u t) t2)) (\lambda (t2: T).(sort t2)))
-(\lambda (x: T).(\lambda (H12: (ty3 g (CHead c (Bind Abst) u) x1
-x)).(ex3_intro T (\lambda (t2: T).(tau0 g c (THead (Bind Abst) u t) t2))
-(\lambda (t2: T).(ty3 g c (THead (Bind Abst) u t) t2)) (\lambda (t2: T).(sort
-t2)) (THead (Bind Abst) u x1) (tau0_bind g Abst c u t x1 H9) (ty3_bind g c u
-x0 H6 Abst t x1 H10 x H12) (sort_abst u H0 x1 H11)))) (ty3_correct g (CHead c
-(Bind Abst) u) t x1 H10)))))) H8))))))) H4)))))))))) t1 H))).
-
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/subst0/dec".
include "subst0/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/subst0/defs".
include "lift/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/subst0/fwd".
include "subst0/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/subst0/props".
include "subst0/fwd.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/subst0/subst0".
include "subst0/props.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/subst0/tlt".
include "subst0/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/subst1/defs".
include "subst0/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/subst1/fwd".
include "subst1/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/subst1/props".
include "subst1/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/subst1/subst1".
include "subst1/fwd.ma".
(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
+i u u1 u2)).(insert_eq T u2 (\lambda (t: T).(subst1 i u u1 t)) (\lambda (_:
+T).(ex2 T (\lambda (t0: T).(subst1 j u1 t1 t0)) (\lambda (t0: T).(subst1 (S
+(plus i j)) u t0 t3)))) (\lambda (y: T).(\lambda (H2: (subst1 i u u1
+y)).(subst1_ind i u u1 (\lambda (t: T).((eq T t u2) \to (ex2 T (\lambda (t0:
+T).(subst1 j u1 t1 t0)) (\lambda (t0: T).(subst1 (S (plus i j)) u t0 t3)))))
+(\lambda (H3: (eq T u1 u2)).(eq_ind_r T u2 (\lambda (t: T).(ex2 T (\lambda
+(t0: T).(subst1 j t t1 t0)) (\lambda (t0: T).(subst1 (S (plus i j)) u t0
+t3)))) (ex_intro2 T (\lambda (t: T).(subst1 j u2 t1 t)) (\lambda (t:
+T).(subst1 (S (plus i j)) u t t3)) t3 (subst1_single j u2 t1 t3 H0)
+(subst1_refl (S (plus i j)) u t3)) u1 H3)) (\lambda (t0: T).(\lambda (H3:
+(subst0 i u u1 t0)).(\lambda (H4: (eq T t0 u2)).(let H5 \def (eq_ind T t0
+(\lambda (t: T).(subst0 i u u1 t)) H3 u2 H4) in (ex2_ind T (\lambda (t:
+T).(subst0 j u1 t1 t)) (\lambda (t: T).(subst0 (S (plus i j)) u t t3)) (ex2 T
(\lambda (t: T).(subst1 j u1 t1 t)) (\lambda (t: T).(subst1 (S (plus i j)) u
-t t3)) 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))))).
+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
\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_eq T (lift (S O) i t2) (lift (S O) i t1) (subst1_gen_lift_eq t1 u (lift
-(S O) i t2) (S O) i i (le_n i) (eq_ind_r nat (plus (S O) i) (\lambda (n:
-nat).(lt i n)) (le_n (plus (S O) i)) (plus i (S O)) (plus_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:
+(\lambda (t: T).(subst1 i u t0 t)) (\lambda (_: T).(\forall (t2: T).((subst1
+i u t0 (lift (S O) i t2)) \to (eq T t1 t2)))) (\lambda (y: T).(\lambda (H0:
+(subst1 i u t0 y)).(subst1_ind i u t0 (\lambda (t: T).((eq T t (lift (S O) i
+t1)) \to (\forall (t2: T).((subst1 i u t0 (lift (S O) i t2)) \to (eq T t1
+t2))))) (\lambda (H1: (eq T t0 (lift (S O) i t1))).(\lambda (t2: T).(\lambda
+(H2: (subst1 i u t0 (lift (S O) i t2))).(let H3 \def (eq_ind T t0 (\lambda
+(t: T).(subst1 i u t (lift (S O) i t2))) H2 (lift (S O) i t1) H1) in (let H4
+\def (sym_eq T (lift (S O) i t2) (lift (S O) i t1) (subst1_gen_lift_eq t1 u
+(lift (S O) i t2) (S O) i i (le_n i) (eq_ind_r nat (plus (S O) i) (\lambda
+(n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i (S O)) (plus_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))))).
+(lift (S O) i t3) (\lambda (t: T).(subst1 i u t0 t)) (\lambda (_: T).(eq T t1
+t3)) (\lambda (y0: T).(\lambda (H5: (subst1 i u t0 y0)).(subst1_ind i u t0
+(\lambda (t: T).((eq T t (lift (S O) i t3)) \to (eq T t1 t3))) (\lambda (H6:
+(eq T t0 (lift (S O) i t3))).(let H7 \def (eq_ind T t0 (\lambda (t:
+T).(subst0 i u t (lift (S O) i t1))) H4 (lift (S O) i t3) H6) in
+(subst0_gen_lift_false t3 u (lift (S O) i t1) (S O) i i (le_n i) (eq_ind_r
+nat (plus (S O) i) (\lambda (n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i
+(S O)) (plus_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))))).
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/tau0/defs".
include "G/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/tau0/props".
include "tau0/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/tau1/cnt".
include "tau1/props.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/tau1/defs".
include "tau0/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/tau1/props".
include "tau1/defs.ma".
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/theory".
+
+include "subst0/tlt.ma".
+
+include "tau1/cnt.ma".
+
+include "ex0/props.ma".
+
+include "wcpr0/fwd.ma".
+
+include "pr3/wcpr0.ma".
+
+include "ex2/props.ma".
+
+include "ex1/props.ma".
+
+include "ty3/tau0.ma".
+
+include "ty3/nf2.ma".
+
+include "ty3/dec.ma".
+
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-
-
-include "subst0/tlt.ma".
-
-include "tau1/cnt.ma".
-
-include "ex0/props.ma".
-
-include "wcpr0/fwd.ma".
-
-include "pr3/wcpr0.ma".
-
-include "ex2/props.ma".
-
-include "ex1/props.ma".
-
-include "ty3/tau0.ma".
-
-include "ty3/nf2.ma".
-
-include "ty3/dec.ma".
-
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/tlist/defs".
include "T/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/tlist/props".
include "tlist/defs.ma".
ts0))).(H2 (tslen ts1) H3 ts1 (refl_equal nat (tslen ts1))))))))))))) ts)))).
theorem theads_tapp:
- \forall (k: K).(\forall (vs: TList).(\forall (v: T).(\forall (t: T).(eq T
+ \forall (k: K).(\forall (v: T).(\forall (t: T).(\forall (vs: TList).(eq T
(THeads k (TApp vs v) t) (THeads k vs (THead k v t))))))
\def
- \lambda (k: K).(\lambda (vs: TList).(TList_ind (\lambda (t: TList).(\forall
-(v: T).(\forall (t0: T).(eq T (THeads k (TApp t v) t0) (THeads k t (THead k v
-t0)))))) (\lambda (v: T).(\lambda (t: T).(refl_equal T (THead k v t))))
-(\lambda (t: T).(\lambda (t0: TList).(\lambda (H: ((\forall (v: T).(\forall
-(t1: T).(eq T (THeads k (TApp t0 v) t1) (THeads k t0 (THead k v
-t1))))))).(\lambda (v: T).(\lambda (t1: T).(eq_ind_r T (THeads k t0 (THead k
-v t1)) (\lambda (t2: T).(eq T (THead k t t2) (THead k t (THeads k t0 (THead k
-v t1))))) (refl_equal T (THead k t (THeads k t0 (THead k v t1)))) (THeads k
-(TApp t0 v) t1) (H v t1))))))) vs)).
+ \lambda (k: K).(\lambda (v: T).(\lambda (t: T).(\lambda (vs:
+TList).(TList_ind (\lambda (t0: TList).(eq T (THeads k (TApp t0 v) t) (THeads
+k t0 (THead k v t)))) (refl_equal T (THead k v t)) (\lambda (t0: T).(\lambda
+(t1: TList).(\lambda (H: (eq T (THeads k (TApp t1 v) t) (THeads k t1 (THead k
+v t)))).(eq_ind T (THeads k (TApp t1 v) t) (\lambda (t2: T).(eq T (THead k t0
+(THeads k (TApp t1 v) t)) (THead k t0 t2))) (refl_equal T (THead k t0 (THeads
+k (TApp t1 v) t))) (THeads k t1 (THead k v t)) H)))) vs)))).
theorem tcons_tapp_ex:
\forall (ts1: TList).(\forall (t1: T).(ex2_2 TList T (\lambda (ts2:
(TCons t1 x0) x1 (refl_equal TList (TApp (TCons t1 x0) x1)) (refl_equal nat
(tslen (TCons t1 x0)))) (tslen t0) H2) (TCons t t0) H1))))) H0))))))) ts1).
-theorem tlist_ind_rew:
+theorem tlist_ind_rev:
\forall (P: ((TList \to Prop))).((P TNil) \to (((\forall (ts:
TList).(\forall (t: T).((P ts) \to (P (TApp ts t)))))) \to (\forall (ts:
TList).(P ts))))
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/tlt/defs".
include "T/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/tlt/props".
include "tlt/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/ty3/arity".
include "ty3/pr3_props.ma".
(\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:
+a1)) (\lambda (a1: A).(arity g (CHead c0 (Bind b) u) t4 (asucc g a1))))).(let
+H4 \def H1 in (ex2_ind A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1:
A).(arity g c0 t (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g c0 (THead
(Bind b) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind b) u t4) (asucc
-g a1)))) (\lambda (x: A).(\lambda (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
+g a1)))) (\lambda (x: A).(\lambda (H5: (arity g c0 u x)).(\lambda (_: (arity
+g c0 t (asucc g x))).(let H7 \def H3 in (ex2_ind A (\lambda (a1: A).(arity g
(CHead c0 (Bind b) u) t3 a1)) (\lambda (a1: A).(arity g (CHead c0 (Bind b) u)
t4 (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Bind b) u t3)
a1)) (\lambda (a1: A).(arity g c0 (THead (Bind b) u t4) (asucc g a1))))
-(\lambda (x0: A).(\lambda (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
+(\lambda (x0: A).(\lambda (H8: (arity g (CHead c0 (Bind b) u) t3
+x0)).(\lambda (H9: (arity g (CHead c0 (Bind b) u) t4 (asucc g x0))).(let H_x
+\def (leq_asucc g x) in (let H10 \def H_x in (ex_ind A (\lambda (a0: A).(leq
g x (asucc g a0))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Bind b) u t3)
a1)) (\lambda (a1: A).(arity g c0 (THead (Bind b) u t4) (asucc g a1))))
-(\lambda (x1: A).(\lambda (H13: (leq g x (asucc g x1))).(B_ind (\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))))))))) (\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)))))) (\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
+(\lambda (x1: A).(\lambda (H11: (leq g x (asucc g x1))).(B_ind (\lambda (b0:
+B).((arity g (CHead c0 (Bind b0) u) t3 x0) \to ((arity g (CHead c0 (Bind b0)
+u) t4 (asucc g x0)) \to (ex2 A (\lambda (a1: A).(arity g c0 (THead (Bind b0)
+u t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind b0) u t4) (asucc g
+a1))))))) (\lambda (H12: (arity g (CHead c0 (Bind Abbr) u) t3 x0)).(\lambda
+(H13: (arity g (CHead c0 (Bind Abbr) u) t4 (asucc g x0))).(ex_intro2 A
+(\lambda (a1: A).(arity g c0 (THead (Bind Abbr) u t3) a1)) (\lambda (a1:
+A).(arity g c0 (THead (Bind Abbr) u t4) (asucc g a1))) x0 (arity_bind g Abbr
+not_abbr_abst c0 u x H5 t3 x0 H12) (arity_bind g Abbr not_abbr_abst c0 u x H5
+t4 (asucc g x0) H13)))) (\lambda (H12: (arity g (CHead c0 (Bind Abst) u) t3
+x0)).(\lambda (H13: (arity g (CHead c0 (Bind Abst) u) t4 (asucc g
x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Bind Abst) u t3) a1))
(\lambda (a1: A).(arity g c0 (THead (Bind Abst) u t4) (asucc g a1))) (AHead
-x1 x0) (arity_head g c0 u x1 (arity_repl g c0 u x 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)))))))))
-(\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)))))) b 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_eq A (asucc g x0) (AHead
-x1 x2) H11) in (let H15 \def (asucc_gen_head g x1 x2 x0 H14) in (ex2_ind A
-(\lambda (a0: A).(eq A x0 (AHead x1 a0))) (\lambda (a0: A).(eq A x2 (asucc g
-a0))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v) a1))
-(\lambda (a1: A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u t))
-(asucc g a1)))) (\lambda (x3: A).(\lambda (H16: (eq A x0 (AHead x1
+x1 x0) (arity_head g c0 u x1 (arity_repl g c0 u x H5 (asucc g x1) H11) t3 x0
+H12) (arity_repl g c0 (THead (Bind Abst) u t4) (AHead x1 (asucc g x0))
+(arity_head g c0 u x1 (arity_repl g c0 u x H5 (asucc g x1) H11) t4 (asucc g
+x0) H13) (asucc g (AHead x1 x0)) (leq_refl g (asucc g (AHead x1 x0)))))))
+(\lambda (H12: (arity g (CHead c0 (Bind Void) u) t3 x0)).(\lambda (H13:
+(arity g (CHead c0 (Bind Void) u) t4 (asucc g x0))).(ex_intro2 A (\lambda
+(a1: A).(arity g c0 (THead (Bind Void) u t3) a1)) (\lambda (a1: A).(arity g
+c0 (THead (Bind Void) u t4) (asucc g a1))) x0 (arity_bind g Void
+not_void_abst c0 u x H5 t3 x0 H12) (arity_bind g Void not_void_abst c0 u x H5
+t4 (asucc g x0) H13)))) b H8 H9))) H10)))))) H7))))) H4)))))))))))) (\lambda
+(c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda
+(H1: (ex2 A (\lambda (a1: A).(arity g c0 w a1)) (\lambda (a1: A).(arity g c0
+u (asucc g a1))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v
+(THead (Bind Abst) u t))).(\lambda (H3: (ex2 A (\lambda (a1: A).(arity g c0 v
+a1)) (\lambda (a1: A).(arity g c0 (THead (Bind Abst) u t) (asucc g
+a1))))).(let H4 \def H1 in (ex2_ind A (\lambda (a1: A).(arity g c0 w a1))
+(\lambda (a1: A).(arity g c0 u (asucc g a1))) (ex2 A (\lambda (a1: A).(arity
+g c0 (THead (Flat Appl) w v) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat
+Appl) w (THead (Bind Abst) u t)) (asucc g a1)))) (\lambda (x: A).(\lambda
+(H5: (arity g c0 w x)).(\lambda (H6: (arity g c0 u (asucc g x))).(let H7 \def
+H3 in (ex2_ind A (\lambda (a1: A).(arity g c0 v a1)) (\lambda (a1: A).(arity
+g c0 (THead (Bind Abst) u t) (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g
+c0 (THead (Flat Appl) w v) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat
+Appl) w (THead (Bind Abst) u t)) (asucc g a1)))) (\lambda (x0: A).(\lambda
+(H8: (arity g c0 v x0)).(\lambda (H9: (arity g c0 (THead (Bind Abst) u t)
+(asucc g x0))).(let H10 \def (arity_gen_abst g c0 u t (asucc g x0) H9) in
+(ex3_2_ind A A (\lambda (a1: A).(\lambda (a2: A).(eq A (asucc g x0) (AHead a1
+a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u (asucc g a1))))
+(\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind Abst) u) t a2)))
+(ex2 A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v) a1)) (\lambda
+(a1: A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u t)) (asucc g
+a1)))) (\lambda (x1: A).(\lambda (x2: A).(\lambda (H11: (eq A (asucc g x0)
+(AHead x1 x2))).(\lambda (H12: (arity g c0 u (asucc g x1))).(\lambda (H13:
+(arity g (CHead c0 (Bind Abst) u) t x2)).(let H14 \def (sym_eq A (asucc g x0)
+(AHead x1 x2) H11) in (let H15 \def (asucc_gen_head g x1 x2 x0 H14) in
+(ex2_ind A (\lambda (a0: A).(eq A x0 (AHead x1 a0))) (\lambda (a0: A).(eq A
+x2 (asucc g a0))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v)
+a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u
+t)) (asucc g a1)))) (\lambda (x3: A).(\lambda (H16: (eq A x0 (AHead x1
x3))).(\lambda (H17: (eq A x2 (asucc g x3))).(let H18 \def (eq_ind A x2
(\lambda (a: A).(arity g (CHead c0 (Bind Abst) u) t a)) H13 (asucc g x3) H17)
in (let H19 \def (eq_ind A x0 (\lambda (a: A).(arity g c0 v a)) H8 (AHead x1
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/ty3/arity_props".
include "ty3/arity.ma".
\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)))))))))).
+v)).(\lambda (P: Prop).(let H1 \def H in (ex3_2_ind T T (\lambda (t2:
+T).(\lambda (_: T).(pc3 c (THead (Bind Abst) v t2) u))) (\lambda (_:
+T).(\lambda (t0: T).(ty3 g c v t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g
+(CHead c (Bind Abst) v) t t2))) P (\lambda (x0: T).(\lambda (x1: T).(\lambda
+(_: (pc3 c (THead (Bind Abst) v x0) u)).(\lambda (H3: (ty3 g c v
+x1)).(\lambda (_: (ty3 g (CHead c (Bind Abst) v) t x0)).(let H_y \def
+(ty3_conv g c v x1 H3 (THead (Bind Abst) v t) u H H0) in (let H_x \def
+(ty3_arity g c (THead (Bind Abst) v t) v H_y) in (let H5 \def H_x in (ex2_ind
+A (\lambda (a1: A).(arity g c (THead (Bind Abst) v t) a1)) (\lambda (a1:
+A).(arity g c v (asucc g a1))) P (\lambda (x: A).(\lambda (H6: (arity g c
+(THead (Bind Abst) v t) x)).(\lambda (H7: (arity g c v (asucc g x))).(let H8
+\def (arity_gen_abst g c v t x H6) in (ex3_2_ind A A (\lambda (a1:
+A).(\lambda (a2: A).(eq A x (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_:
+A).(arity g c v (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g
+(CHead c (Bind Abst) v) t a2))) P (\lambda (x2: A).(\lambda (x3: A).(\lambda
+(H9: (eq A x (AHead x2 x3))).(\lambda (H10: (arity g c v (asucc g
+x2))).(\lambda (_: (arity g (CHead c (Bind Abst) v) t x3)).(let H12 \def
+(eq_ind A x (\lambda (a: A).(arity g c v (asucc g a))) H7 (AHead x2 x3) H9)
+in (leq_ahead_asucc_false g x2 (asucc g x3) (arity_mono g c v (asucc g (AHead
+x2 x3)) H12 (asucc g x2) H10) P))))))) H8))))) H5))))))))) (ty3_gen_bind g
+Abst c v t u H1)))))))))).
theorem ty3_repellent:
\forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (t: T).(\forall (u1:
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/ty3/dec".
include "ty3/pr3_props.ma".
theorem ty3_inference:
\forall (g: G).(\forall (c: C).(\forall (t1: T).(or (ex T (\lambda (t2:
-T).(ty3 g c t1 t2))) (\forall (t2: T).((ty3 g c t1 t2) \to (\forall (P:
-Prop).P))))))
+T).(ty3 g c t1 t2))) (\forall (t2: T).((ty3 g c t1 t2) \to False)))))
\def
\lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(flt_wf_ind (\lambda (c0:
C).(\lambda (t: T).(or (ex T (\lambda (t2: T).(ty3 g c0 t t2))) (\forall (t2:
-T).((ty3 g c0 t t2) \to (\forall (P: Prop).P)))))) (\lambda (c2: C).(\lambda
-(t2: T).(T_ind (\lambda (t: T).(((\forall (c1: C).(\forall (t3: T).((flt c1
-t3 c2 t) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4:
-T).((ty3 g c1 t3 t4) \to (\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)))))) (\lambda (n: nat).(\lambda (_: ((\forall (c1: C).(\forall (t3:
-T).((flt c1 t3 c2 (TSort n)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3
-t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to (\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))))) (\lambda (n: nat).(\lambda (H: ((\forall (c1:
-C).(\forall (t3: T).((flt c1 t3 c2 (TLRef n)) \to (or (ex T (\lambda (t4:
-T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to (\forall (P:
-Prop).P))))))))).(let H_x \def (getl_dec c2 n) in (let H0 \def H_x in (or_ind
+T).((ty3 g c0 t t2) \to False))))) (\lambda (c2: C).(\lambda (t2: T).(T_ind
+(\lambda (t: T).(((\forall (c1: C).(\forall (t3: T).((flt c1 t3 c2 t) \to (or
+(ex T (\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4)
+\to False))))))) \to (or (ex T (\lambda (t3: T).(ty3 g c2 t t3))) (\forall
+(t3: T).((ty3 g c2 t t3) \to False))))) (\lambda (n: nat).(\lambda (_:
+((\forall (c1: C).(\forall (t3: T).((flt c1 t3 c2 (TSort n)) \to (or (ex T
+(\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to
+False)))))))).(or_introl (ex T (\lambda (t3: T).(ty3 g c2 (TSort n) t3)))
+(\forall (t3: T).((ty3 g c2 (TSort n) t3) \to False)) (ex_intro T (\lambda
+(t3: T).(ty3 g c2 (TSort n) t3)) (TSort (next g n)) (ty3_sort g c2 n)))))
+(\lambda (n: nat).(\lambda (H: ((\forall (c1: C).(\forall (t3: T).((flt c1 t3
+c2 (TLRef n)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4:
+T).((ty3 g c1 t3 t4) \to False)))))))).(let H_x \def (getl_dec c2 n) in (let
+H0 \def H_x in (or_ind (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda
+(v: T).(getl n c2 (CHead e (Bind b) v)))))) (\forall (d: C).((getl n c2 d)
+\to (\forall (P: Prop).P))) (or (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n)
+t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to False))) (\lambda (H1:
(ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl n c2 (CHead
-e (Bind b) v)))))) (\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 (t3: T).(ty3 g x0 x2
-t3)))).(ex_ind T (\lambda (t3: T).(ty3 g x0 x2 t3)) (or (ex T (\lambda (t3:
-T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to
-(\forall (P: Prop).P)))) (\lambda (x: T).(\lambda (H5: (ty3 g x0 x2
-x)).(B_ind (\lambda (b: B).((getl n c2 (CHead x0 (Bind b) x2)) \to (or (ex T
+e (Bind b) v))))))).(ex_3_ind C B T (\lambda (e: C).(\lambda (b: B).(\lambda
+(v: T).(getl n c2 (CHead e (Bind b) v))))) (or (ex T (\lambda (t3: T).(ty3 g
+c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to False)))
+(\lambda (x0: C).(\lambda (x1: B).(\lambda (x2: T).(\lambda (H2: (getl n c2
+(CHead x0 (Bind x1) x2))).(let H3 \def (H x0 x2 (getl_flt x1 c2 x0 x2 n H2))
+in (or_ind (ex T (\lambda (t3: T).(ty3 g x0 x2 t3))) (\forall (t3: T).((ty3 g
+x0 x2 t3) \to False)) (or (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3)))
+(\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to False))) (\lambda (H4: (ex T
+(\lambda (t3: T).(ty3 g x0 x2 t3)))).(ex_ind T (\lambda (t3: T).(ty3 g x0 x2
+t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3:
+T).((ty3 g c2 (TLRef n) t3) \to False))) (\lambda (x: T).(\lambda (H5: (ty3 g
+x0 x2 x)).(B_ind (\lambda (b: B).((getl n c2 (CHead x0 (Bind b) x2)) \to (or
+(ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2
+(TLRef n) t3) \to False))))) (\lambda (H6: (getl n c2 (CHead x0 (Bind Abbr)
+x2))).(or_introl (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall
+(t3: T).((ty3 g c2 (TLRef n) t3) \to False)) (ex_intro T (\lambda (t3:
+T).(ty3 g c2 (TLRef n) t3)) (lift (S n) O x) (ty3_abbr g n c2 x0 x2 H6 x
+H5)))) (\lambda (H6: (getl n c2 (CHead x0 (Bind Abst) x2))).(or_introl (ex T
(\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef
-n) t3) \to (\forall (P: Prop).P)))))) (\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)))) (\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)))) (\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 (c0:
-C).(getl n c2 c0)) H6 (CHead x3 (Bind Abbr) x4) (getl_mono c2 (CHead x0 (Bind
-Void) x2) n H6 (CHead x3 (Bind Abbr) x4) H10)) in (let H13 \def (eq_ind C
-(CHead x0 (Bind Void) x2) (\lambda (ee: C).(match ee in C return (\lambda (_:
-C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match
-k in K return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B
-return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow
-False | Void \Rightarrow True]) | (Flat _) \Rightarrow False])])) I (CHead x3
-(Bind Abbr) x4) (getl_mono c2 (CHead x0 (Bind Void) x2) n H6 (CHead x3 (Bind
-Abbr) x4) H10)) in (False_ind P H13))))))))) H8)) (\lambda (H8: (ex3_3 C T T
+n) t3) \to False)) (ex_intro T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))
+(lift (S n) O x2) (ty3_abst g n c2 x0 x2 H6 x H5)))) (\lambda (H6: (getl n c2
+(CHead x0 (Bind Void) x2))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2
+(TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to False))
+(\lambda (t3: T).(\lambda (H7: (ty3 g c2 (TLRef n) t3)).(or_ind (ex3_3 C T T
+(\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t)
+t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e
+(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u
+t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2
+(lift (S n) O u) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl
+n c2 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t:
+T).(ty3 g e u t))))) False (\lambda (H8: (ex3_3 C T T (\lambda (_:
+C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda
+(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u)))))
+(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind
+C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O
+t) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e
+(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u
+t)))) False (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_:
+(pc3 c2 (lift (S n) O x5) t3)).(\lambda (H10: (getl n c2 (CHead x3 (Bind
+Abbr) x4))).(\lambda (_: (ty3 g x3 x4 x5)).(let H12 \def (eq_ind C (CHead x0
+(Bind Void) x2) (\lambda (c0: C).(getl n c2 c0)) H6 (CHead x3 (Bind Abbr) x4)
+(getl_mono c2 (CHead x0 (Bind Void) x2) n H6 (CHead x3 (Bind Abbr) x4) H10))
+in (let H13 \def (eq_ind C (CHead x0 (Bind Void) x2) (\lambda (ee: C).(match
+ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False |
+(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
+[(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr
+\Rightarrow False | Abst \Rightarrow False | Void \Rightarrow True]) | (Flat
+_) \Rightarrow False])])) I (CHead x3 (Bind Abbr) x4) (getl_mono c2 (CHead x0
+(Bind Void) x2) n H6 (CHead x3 (Bind Abbr) x4) H10)) in (False_ind False
+H13))))))))) H8)) (\lambda (H8: (ex3_3 C T T (\lambda (_: C).(\lambda (u:
+T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: C).(\lambda
+(u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) (\lambda (e:
+C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T
(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u)
t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e
(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u
-t)))))).(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 (c0: C).(getl
-n c2 c0)) H6 (CHead x3 (Bind Abst) x4) (getl_mono c2 (CHead x0 (Bind Void)
-x2) n H6 (CHead x3 (Bind Abst) x4) H10)) in (let H13 \def (eq_ind C (CHead x0
-(Bind Void) x2) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop)
-with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K
-return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B return
-(\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False |
-Void \Rightarrow True]) | (Flat _) \Rightarrow False])])) I (CHead x3 (Bind
-Abst) x4) (getl_mono c2 (CHead x0 (Bind Void) x2) n H6 (CHead x3 (Bind Abst)
-x4) H10)) in (False_ind P H13))))))))) H8)) (ty3_gen_lref g c2 t3 n H7)))))))
-x1 H2))) H4)) (\lambda (H4: ((\forall (t3: T).((ty3 g x0 x2 t3) \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)))) False (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_:
+(pc3 c2 (lift (S n) O x4) t3)).(\lambda (H10: (getl n c2 (CHead x3 (Bind
+Abst) x4))).(\lambda (_: (ty3 g x3 x4 x5)).(let H12 \def (eq_ind C (CHead x0
+(Bind Void) x2) (\lambda (c0: C).(getl n c2 c0)) H6 (CHead x3 (Bind Abst) x4)
+(getl_mono c2 (CHead x0 (Bind Void) x2) n H6 (CHead x3 (Bind Abst) x4) H10))
+in (let H13 \def (eq_ind C (CHead x0 (Bind Void) x2) (\lambda (ee: C).(match
+ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False |
+(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
+[(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr
+\Rightarrow False | Abst \Rightarrow False | Void \Rightarrow True]) | (Flat
+_) \Rightarrow False])])) I (CHead x3 (Bind Abst) x4) (getl_mono c2 (CHead x0
+(Bind Void) x2) n H6 (CHead x3 (Bind Abst) x4) H10)) in (False_ind False
+H13))))))))) H8)) (ty3_gen_lref g c2 t3 n H7)))))) x1 H2))) H4)) (\lambda
+(H4: ((\forall (t3: T).((ty3 g x0 x2 t3) \to False)))).(or_intror (ex T
+(\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef
+n) t3) \to False)) (\lambda (t3: T).(\lambda (H5: (ty3 g c2 (TLRef n)
+t3)).(or_ind (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t:
T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda
(_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u:
T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda
(u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e:
C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u)))))
-(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) 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 (c0:
-C).(getl n c2 c0)) H2 (CHead x3 (Bind Abbr) x4) (getl_mono c2 (CHead x0 (Bind
-x1) x2) n H2 (CHead x3 (Bind Abbr) x4) H8)) in (let H11 \def (f_equal C C
-(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
-\Rightarrow x0 | (CHead c0 _ _) \Rightarrow c0])) (CHead x0 (Bind x1) x2)
-(CHead x3 (Bind Abbr) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead
-x3 (Bind Abbr) x4) H8)) in ((let H12 \def (f_equal C B (\lambda (e: C).(match
-e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow x1 | (CHead _ k
-_) \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b)
-\Rightarrow b | (Flat _) \Rightarrow x1])])) (CHead x0 (Bind x1) x2) (CHead
-x3 (Bind Abbr) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind
-Abbr) x4) H8)) in ((let H13 \def (f_equal C T (\lambda (e: C).(match e in C
-return (\lambda (_: C).T) with [(CSort _) \Rightarrow x2 | (CHead _ _ t)
-\Rightarrow t])) (CHead x0 (Bind x1) x2) (CHead x3 (Bind Abbr) x4) (getl_mono
-c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abbr) x4) H8)) in (\lambda
-(_: (eq B x1 Abbr)).(\lambda (H15: (eq C x0 x3)).(let H16 \def (eq_ind_r T x4
-(\lambda (t: T).(getl n c2 (CHead x3 (Bind Abbr) t))) H10 x2 H13) in (let H17
-\def (eq_ind_r T x4 (\lambda (t: T).(ty3 g x3 t x5)) H9 x2 H13) in (let H18
-\def (eq_ind_r C x3 (\lambda (c0: C).(getl n c2 (CHead c0 (Bind Abbr) x2)))
-H16 x0 H15) in (let H19 \def (eq_ind_r C x3 (\lambda (c0: C).(ty3 g c0 x2
-x5)) H17 x0 H15) in (H4 x5 H19 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:
+(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) False
+(\lambda (H6: (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t:
+T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda
+(_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u:
+T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T (\lambda (_:
+C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda
+(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u)))))
+(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))) False
+(\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (pc3 c2 (lift
+(S n) O x5) t3)).(\lambda (H8: (getl n c2 (CHead x3 (Bind Abbr)
+x4))).(\lambda (H9: (ty3 g x3 x4 x5)).(let H10 \def (eq_ind C (CHead x0 (Bind
+x1) x2) (\lambda (c0: C).(getl n c2 c0)) H2 (CHead x3 (Bind Abbr) x4)
+(getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abbr) x4) H8)) in
+(let H11 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_:
+C).C) with [(CSort _) \Rightarrow x0 | (CHead c0 _ _) \Rightarrow c0]))
+(CHead x0 (Bind x1) x2) (CHead x3 (Bind Abbr) x4) (getl_mono c2 (CHead x0
+(Bind x1) x2) n H2 (CHead x3 (Bind Abbr) x4) H8)) in ((let H12 \def (f_equal
+C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with [(CSort _)
+\Rightarrow x1 | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_:
+K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow x1])])) (CHead x0
+(Bind x1) x2) (CHead x3 (Bind Abbr) x4) (getl_mono c2 (CHead x0 (Bind x1) x2)
+n H2 (CHead x3 (Bind Abbr) x4) H8)) in ((let H13 \def (f_equal C T (\lambda
+(e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow x2
+| (CHead _ _ t) \Rightarrow t])) (CHead x0 (Bind x1) x2) (CHead x3 (Bind
+Abbr) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abbr)
+x4) H8)) in (\lambda (_: (eq B x1 Abbr)).(\lambda (H15: (eq C x0 x3)).(let
+H16 \def (eq_ind_r T x4 (\lambda (t: T).(getl n c2 (CHead x3 (Bind Abbr) t)))
+H10 x2 H13) in (let H17 \def (eq_ind_r T x4 (\lambda (t: T).(ty3 g x3 t x5))
+H9 x2 H13) in (let H18 \def (eq_ind_r C x3 (\lambda (c0: C).(getl n c2 (CHead
+c0 (Bind Abbr) x2))) H16 x0 H15) in (let H19 \def (eq_ind_r C x3 (\lambda
+(c0: C).(ty3 g c0 x2 x5)) H17 x0 H15) in (H4 x5 H19)))))))) H12))
+H11))))))))) H6)) (\lambda (H6: (ex3_3 C T T (\lambda (_: C).(\lambda (u:
T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: C).(\lambda
(u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) (\lambda (e:
-C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))) 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 (c0:
-C).(getl n c2 c0)) H2 (CHead x3 (Bind Abst) x4) (getl_mono c2 (CHead x0 (Bind
-x1) x2) n H2 (CHead x3 (Bind Abst) x4) H8)) in (let H11 \def (f_equal C C
-(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
-\Rightarrow x0 | (CHead c0 _ _) \Rightarrow c0])) (CHead x0 (Bind x1) x2)
-(CHead x3 (Bind Abst) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead
-x3 (Bind Abst) x4) H8)) in ((let H12 \def (f_equal C B (\lambda (e: C).(match
-e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow x1 | (CHead _ k
-_) \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b)
-\Rightarrow b | (Flat _) \Rightarrow x1])])) (CHead x0 (Bind x1) x2) (CHead
-x3 (Bind Abst) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind
-Abst) x4) H8)) in ((let H13 \def (f_equal C T (\lambda (e: C).(match e in C
-return (\lambda (_: C).T) with [(CSort _) \Rightarrow x2 | (CHead _ _ t)
-\Rightarrow t])) (CHead x0 (Bind x1) x2) (CHead x3 (Bind Abst) x4) (getl_mono
-c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abst) x4) H8)) in (\lambda
-(_: (eq B x1 Abst)).(\lambda (H15: (eq C x0 x3)).(let H16 \def (eq_ind_r T x4
-(\lambda (t: T).(getl n c2 (CHead x3 (Bind Abst) t))) H10 x2 H13) in (let H17
-\def (eq_ind_r T x4 (\lambda (t: T).(ty3 g x3 t x5)) H9 x2 H13) in (let H18
-\def (eq_ind_r T x4 (\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)) H7 x2 H13)
-in (let H19 \def (eq_ind_r C x3 (\lambda (c0: C).(getl n c2 (CHead c0 (Bind
-Abst) x2))) H16 x0 H15) in (let H20 \def (eq_ind_r C x3 (\lambda (c0: C).(ty3
-g c0 x2 x5)) H17 x0 H15) in (H4 x5 H20 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 (_:
+C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T
+(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u)
+t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e
+(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u
+t)))) False (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H7:
+(pc3 c2 (lift (S n) O x4) t3)).(\lambda (H8: (getl n c2 (CHead x3 (Bind Abst)
+x4))).(\lambda (H9: (ty3 g x3 x4 x5)).(let H10 \def (eq_ind C (CHead x0 (Bind
+x1) x2) (\lambda (c0: C).(getl n c2 c0)) H2 (CHead x3 (Bind Abst) x4)
+(getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abst) x4) H8)) in
+(let H11 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_:
+C).C) with [(CSort _) \Rightarrow x0 | (CHead c0 _ _) \Rightarrow c0]))
+(CHead x0 (Bind x1) x2) (CHead x3 (Bind Abst) x4) (getl_mono c2 (CHead x0
+(Bind x1) x2) n H2 (CHead x3 (Bind Abst) x4) H8)) in ((let H12 \def (f_equal
+C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with [(CSort _)
+\Rightarrow x1 | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_:
+K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow x1])])) (CHead x0
+(Bind x1) x2) (CHead x3 (Bind Abst) x4) (getl_mono c2 (CHead x0 (Bind x1) x2)
+n H2 (CHead x3 (Bind Abst) x4) H8)) in ((let H13 \def (f_equal C T (\lambda
+(e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow x2
+| (CHead _ _ t) \Rightarrow t])) (CHead x0 (Bind x1) x2) (CHead x3 (Bind
+Abst) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abst)
+x4) H8)) in (\lambda (_: (eq B x1 Abst)).(\lambda (H15: (eq C x0 x3)).(let
+H16 \def (eq_ind_r T x4 (\lambda (t: T).(getl n c2 (CHead x3 (Bind Abst) t)))
+H10 x2 H13) in (let H17 \def (eq_ind_r T x4 (\lambda (t: T).(ty3 g x3 t x5))
+H9 x2 H13) in (let H18 \def (eq_ind_r T x4 (\lambda (t: T).(pc3 c2 (lift (S
+n) O t) t3)) H7 x2 H13) in (let H19 \def (eq_ind_r C x3 (\lambda (c0:
+C).(getl n c2 (CHead c0 (Bind Abst) x2))) H16 x0 H15) in (let H20 \def
+(eq_ind_r C x3 (\lambda (c0: C).(ty3 g c0 x2 x5)) H17 x0 H15) in (H4 x5
+H20))))))))) H12)) H11))))))))) H6)) (ty3_gen_lref g c2 t3 n H5))))))
+H3)))))) H1)) (\lambda (H1: ((\forall (d: C).((getl n c2 d) \to (\forall (P:
+Prop).P))))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3)))
+(\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to False)) (\lambda (t3:
+T).(\lambda (H2: (ty3 g c2 (TLRef n) t3)).(or_ind (ex3_3 C T T (\lambda (_:
+C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda
+(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u)))))
+(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T
+T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u)
+t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e
+(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u
+t))))) False (\lambda (H3: (ex3_3 C T T (\lambda (_: C).(\lambda (_:
T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda
(u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e:
-C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 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)
+t)))) False (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_:
+(pc3 c2 (lift (S n) O x2) t3)).(\lambda (H5: (getl n c2 (CHead x0 (Bind Abbr)
x1))).(\lambda (_: (ty3 g x0 x1 x2)).(H1 (CHead x0 (Bind Abbr) x1) H5
-P))))))) H3)) (\lambda (H3: (ex3_3 C T T (\lambda (_: C).(\lambda (u:
+False))))))) H3)) (\lambda (H3: (ex3_3 C T T (\lambda (_: C).(\lambda (u:
T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: C).(\lambda
(u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) (\lambda (e:
C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T
(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u)
t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e
(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u
-t)))) 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)
+t)))) False (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_:
+(pc3 c2 (lift (S n) O x1) t3)).(\lambda (H5: (getl n c2 (CHead x0 (Bind Abst)
x1))).(\lambda (_: (ty3 g x0 x1 x2)).(H1 (CHead x0 (Bind Abst) x1) H5
-P))))))) H3)) (ty3_gen_lref g c2 t3 n H2))))))) H0))))) (\lambda (k:
+False))))))) H3)) (ty3_gen_lref g c2 t3 n H2)))))) H0))))) (\lambda (k:
K).(\lambda (t: T).(\lambda (_: ((((\forall (c1: C).(\forall (t3: T).((flt c1
t3 c2 t) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4:
-T).((ty3 g c1 t3 t4) \to (\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))))))).(\lambda (t0: T).(\lambda (_: ((((\forall (c1: C).(\forall
-(t3: T).((flt c1 t3 c2 t0) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3 t4)))
-(\forall (t4: T).((ty3 g c1 t3 t4) \to (\forall (P: Prop).P)))))))) \to (or
-(ex T (\lambda (t3: T).(ty3 g c2 t0 t3))) (\forall (t3: T).((ty3 g c2 t0 t3)
-\to (\forall (P: Prop).P))))))).(\lambda (H1: ((\forall (c1: C).(\forall (t3:
-T).((flt c1 t3 c2 (THead k t t0)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3
-t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to (\forall (P:
-Prop).P))))))))).(K_ind (\lambda (k0: K).(((\forall (c1: C).(\forall (t3:
+T).((ty3 g c1 t3 t4) \to False))))))) \to (or (ex T (\lambda (t3: T).(ty3 g
+c2 t t3))) (\forall (t3: T).((ty3 g c2 t t3) \to False)))))).(\lambda (t0:
+T).(\lambda (_: ((((\forall (c1: C).(\forall (t3: T).((flt c1 t3 c2 t0) \to
+(or (ex T (\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 g c1 t3
+t4) \to False))))))) \to (or (ex T (\lambda (t3: T).(ty3 g c2 t0 t3)))
+(\forall (t3: T).((ty3 g c2 t0 t3) \to False)))))).(\lambda (H1: ((\forall
+(c1: C).(\forall (t3: T).((flt c1 t3 c2 (THead k t t0)) \to (or (ex T
+(\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to
+False)))))))).(K_ind (\lambda (k0: K).(((\forall (c1: C).(\forall (t3:
T).((flt c1 t3 c2 (THead k0 t t0)) \to (or (ex T (\lambda (t4: T).(ty3 g c1
-t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to (\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)))))) (\lambda (b:
-B).(\lambda (H2: ((\forall (c1: C).(\forall (t3: T).((flt c1 t3 c2 (THead
-(Bind b) t t0)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall
-(t4: T).((ty3 g c1 t3 t4) \to (\forall (P: Prop).P))))))))).(let H3 \def (H2
-c2 t (flt_thead_sx (Bind b) c2 t t0)) in (or_ind (ex T (\lambda (t3: T).(ty3
-g c2 t t3))) (\forall (t3: T).((ty3 g c2 t t3) \to (\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))))
+t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to False))))))) \to (or (ex T
+(\lambda (t3: T).(ty3 g c2 (THead k0 t t0) t3))) (\forall (t3: T).((ty3 g c2
+(THead k0 t t0) t3) \to False))))) (\lambda (b: B).(\lambda (H2: ((\forall
+(c1: C).(\forall (t3: T).((flt c1 t3 c2 (THead (Bind b) t t0)) \to (or (ex T
+(\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to
+False)))))))).(let H3 \def (H2 c2 t (flt_thead_sx (Bind b) c2 t t0)) in
+(or_ind (ex T (\lambda (t3: T).(ty3 g c2 t t3))) (\forall (t3: T).((ty3 g c2
+t t3) \to False)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0)
+t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) \to False)))
(\lambda (H4: (ex T (\lambda (t3: T).(ty3 g c2 t t3)))).(ex_ind T (\lambda
(t3: T).(ty3 g c2 t t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Bind b)
t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) \to
-(\forall (P: Prop).P)))) (\lambda (x: T).(\lambda (H5: (ty3 g c2 t x)).(let
-H6 \def (H2 (CHead c2 (Bind b) t) t0 (flt_shift (Bind b) c2 t t0)) in (or_ind
-(ex T (\lambda (t3: T).(ty3 g (CHead c2 (Bind b) t) t0 t3))) (\forall (t3:
-T).((ty3 g (CHead c2 (Bind b) t) t0 t3) \to (\forall (P: Prop).P))) (or (ex T
+False))) (\lambda (x: T).(\lambda (H5: (ty3 g c2 t x)).(let H6 \def (H2
+(CHead c2 (Bind b) t) t0 (flt_shift (Bind b) c2 t t0)) in (or_ind (ex T
+(\lambda (t3: T).(ty3 g (CHead c2 (Bind b) t) t0 t3))) (\forall (t3: T).((ty3
+g (CHead c2 (Bind b) t) t0 t3) \to False)) (or (ex T (\lambda (t3: T).(ty3 g
+c2 (THead (Bind b) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t
+t0) t3) \to False))) (\lambda (H7: (ex T (\lambda (t3: T).(ty3 g (CHead c2
+(Bind b) t) t0 t3)))).(ex_ind T (\lambda (t3: T).(ty3 g (CHead c2 (Bind b) t)
+t0 t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0) t3)))
+(\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) \to False))) (\lambda
+(x0: T).(\lambda (H8: (ty3 g (CHead c2 (Bind b) t) t0 x0)).(or_introl (ex T
(\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0) t3))) (\forall (t3: T).((ty3
-g c2 (THead (Bind b) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (H7: (ex
-T (\lambda (t3: T).(ty3 g (CHead c2 (Bind b) t) t0 t3)))).(ex_ind T (\lambda
-(t3: T).(ty3 g (CHead c2 (Bind b) t) t0 t3)) (or (ex T (\lambda (t3: T).(ty3
-g c2 (THead (Bind b) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b)
-t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (x0: T).(\lambda (H8: (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 (H9: (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 H5 b t0 x0 H8 x1 H9)))))
-(ty3_correct g (CHead c2 (Bind b) t) t0 x0 H8)))) H7)) (\lambda (H7:
-((\forall (t3: T).((ty3 g (CHead c2 (Bind b) t) t0 t3) \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 (H8: (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
-(H11: (ty3 g (CHead c2 (Bind b) t) t0 x0)).(\lambda (_: (ty3 g (CHead c2
-(Bind b) t) x0 x2)).(H7 x0 H11 P)))))))) (ty3_gen_bind g b c2 t t0 t3
-H8))))))) H6)))) H4)) (\lambda (H4: ((\forall (t3: T).((ty3 g c2 t t3) \to
-(\forall (P: Prop).P))))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead
+g c2 (THead (Bind b) t t0) t3) \to False)) (ex_intro T (\lambda (t3: T).(ty3
+g c2 (THead (Bind b) t t0) t3)) (THead (Bind b) t x0) (ty3_bind g c2 t x H5 b
+t0 x0 H8))))) H7)) (\lambda (H7: ((\forall (t3: T).((ty3 g (CHead c2 (Bind b)
+t) t0 t3) \to False)))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead
(Bind b) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3)
-\to (\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H5: (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 (H7: (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)).(H4 x1 H7 P)))))))) (ty3_gen_bind g b c2 t t0
-t3 H5))))))) H3)))) (\lambda (f: F).(\lambda (H2: ((\forall (c1: C).(\forall
-(t3: T).((flt c1 t3 c2 (THead (Flat f) t t0)) \to (or (ex T (\lambda (t4:
-T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to (\forall (P:
-Prop).P))))))))).(F_ind (\lambda (f0: F).(((\forall (c1: C).(\forall (t3:
-T).((flt c1 t3 c2 (THead (Flat f0) t t0)) \to (or (ex T (\lambda (t4: T).(ty3
-g c1 t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to (\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)))))) (\lambda (H3: ((\forall (c1: C).(\forall (t3: T).((flt c1
-t3 c2 (THead (Flat Appl) t t0)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3
-t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to (\forall (P:
-Prop).P))))))))).(let H4 \def (H3 c2 t (flt_thead_sx (Flat Appl) c2 t t0)) in
-(or_ind (ex T (\lambda (t3: T).(ty3 g c2 t t3))) (\forall (t3: T).((ty3 g c2
-t t3) \to (\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 (H5: (ex T (\lambda (t3: T).(ty3 g
-c2 t t3)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t t3)) (or (ex T (\lambda
-(t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2
-(THead (Flat Appl) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (x:
-T).(\lambda (H6: (ty3 g c2 t x)).(let H7 \def (H3 c2 t0 (flt_thead_dx (Flat
-Appl) c2 t t0)) in (or_ind (ex T (\lambda (t3: T).(ty3 g c2 t0 t3))) (\forall
-(t3: T).((ty3 g c2 t0 t3) \to (\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 (H8: (ex T
-(\lambda (t3: T).(ty3 g c2 t0 t3)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t0
+\to False)) (\lambda (t3: T).(\lambda (H8: (ty3 g c2 (THead (Bind b) t t0)
+t3)).(ex3_2_ind T T (\lambda (t4: T).(\lambda (_: T).(pc3 c2 (THead (Bind b)
+t t4) t3))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c2 t t5))) (\lambda (t4:
+T).(\lambda (_: T).(ty3 g (CHead c2 (Bind b) t) t0 t4))) False (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead (Bind b) t x0) t3)).(\lambda
+(_: (ty3 g c2 t x1)).(\lambda (H11: (ty3 g (CHead c2 (Bind b) t) t0 x0)).(H7
+x0 H11)))))) (ty3_gen_bind g b c2 t t0 t3 H8)))))) H6)))) H4)) (\lambda (H4:
+((\forall (t3: T).((ty3 g c2 t t3) \to False)))).(or_intror (ex T (\lambda
+(t3: T).(ty3 g c2 (THead (Bind b) t t0) t3))) (\forall (t3: T).((ty3 g c2
+(THead (Bind b) t t0) t3) \to False)) (\lambda (t3: T).(\lambda (H5: (ty3 g
+c2 (THead (Bind b) t t0) t3)).(ex3_2_ind T T (\lambda (t4: T).(\lambda (_:
+T).(pc3 c2 (THead (Bind b) t t4) t3))) (\lambda (_: T).(\lambda (t5: T).(ty3
+g c2 t t5))) (\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind b) t) t0
+t4))) False (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead
+(Bind b) t x0) t3)).(\lambda (H7: (ty3 g c2 t x1)).(\lambda (_: (ty3 g (CHead
+c2 (Bind b) t) t0 x0)).(H4 x1 H7)))))) (ty3_gen_bind g b c2 t t0 t3 H5))))))
+H3)))) (\lambda (f: F).(\lambda (H2: ((\forall (c1: C).(\forall (t3: T).((flt
+c1 t3 c2 (THead (Flat f) t t0)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3
+t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to False)))))))).(F_ind (\lambda
+(f0: F).(((\forall (c1: C).(\forall (t3: T).((flt c1 t3 c2 (THead (Flat f0) t
+t0)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3
+g c1 t3 t4) \to False))))))) \to (or (ex T (\lambda (t3: T).(ty3 g c2 (THead
+(Flat f0) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat f0) t t0) t3)
+\to False))))) (\lambda (H3: ((\forall (c1: C).(\forall (t3: T).((flt c1 t3
+c2 (THead (Flat Appl) t t0)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3
+t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to False)))))))).(let H4 \def (H3
+c2 t (flt_thead_sx (Flat Appl) c2 t t0)) in (or_ind (ex T (\lambda (t3:
+T).(ty3 g c2 t t3))) (\forall (t3: T).((ty3 g c2 t t3) \to False)) (or (ex T
+(\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3:
+T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False))) (\lambda (H5: (ex T
+(\lambda (t3: T).(ty3 g c2 t t3)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t
t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3)))
-(\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to (\forall (P:
-Prop).P)))) (\lambda (x0: T).(\lambda (H9: (ty3 g c2 t0 x0)).(ex_ind T
-(\lambda (t3: T).(ty3 g c2 x0 t3)) (or (ex T (\lambda (t3: T).(ty3 g c2
-(THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl)
-t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (x1: T).(\lambda (H10: (ty3 g
-c2 x0 x1)).(ex_ind T (\lambda (t3: T).(ty3 g c2 x t3)) (or (ex T (\lambda
+(\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False)))
+(\lambda (x: T).(\lambda (H6: (ty3 g c2 t x)).(let H7 \def (H3 c2 t0
+(flt_thead_dx (Flat Appl) c2 t t0)) in (or_ind (ex T (\lambda (t3: T).(ty3 g
+c2 t0 t3))) (\forall (t3: T).((ty3 g c2 t0 t3) \to False)) (or (ex T (\lambda
(t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2
-(THead (Flat Appl) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (x2:
-T).(\lambda (H11: (ty3 g c2 x x2)).(let H12 \def (ty3_sn3 g c2 x x2 H11) in
-(let H_x \def (nf2_sn3 c2 x H12) in (let H13 \def H_x in (ex2_ind T (\lambda
-(u: T).(pr3 c2 x u)) (\lambda (u: T).(nf2 c2 u)) (or (ex T (\lambda (t3:
-T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2
-(THead (Flat Appl) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (x3:
+(THead (Flat Appl) t t0) t3) \to False))) (\lambda (H8: (ex T (\lambda (t3:
+T).(ty3 g c2 t0 t3)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t0 t3)) (or (ex T
+(\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3:
+T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False))) (\lambda (x0:
+T).(\lambda (H9: (ty3 g c2 t0 x0)).(ex_ind T (\lambda (t3: T).(ty3 g c2 x0
+t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3)))
+(\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False)))
+(\lambda (x1: T).(\lambda (H10: (ty3 g c2 x0 x1)).(ex_ind T (\lambda (t3:
+T).(ty3 g c2 x t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t
+t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to
+False))) (\lambda (x2: T).(\lambda (H11: (ty3 g c2 x x2)).(let H12 \def
+(ty3_sn3 g c2 x x2 H11) in (let H_x \def (nf2_sn3 c2 x H12) in (let H13 \def
+H_x in (ex2_ind T (\lambda (u: T).(pr3 c2 x u)) (\lambda (u: T).(nf2 c2 u))
+(or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall
+(t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False))) (\lambda (x3:
T).(\lambda (H14: (pr3 c2 x x3)).(\lambda (H15: (nf2 c2 x3)).(let H16 \def
(ty3_sred_pr3 c2 x x3 H14 g x2 H11) in (let H_x0 \def (pc3_abst_dec g c2 x0
x1 H10 x3 x2 H16) in (let H17 \def H_x0 in (or_ind (ex4_2 T T (\lambda (u:
T).(\lambda (_: T).(pc3 c2 x0 (THead (Bind Abst) x3 u)))) (\lambda (u:
T).(\lambda (v2: T).(ty3 g c2 (THead (Bind Abst) v2 u) x1))) (\lambda (_:
T).(\lambda (v2: T).(pr3 c2 x3 v2))) (\lambda (_: T).(\lambda (v2: T).(nf2 c2
-v2)))) (\forall (u: T).((pc3 c2 x0 (THead (Bind Abst) x3 u)) \to (\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 (H18: (ex4_2 T T (\lambda (u: T).(\lambda (_:
-T).(pc3 c2 x0 (THead (Bind Abst) x3 u)))) (\lambda (u: T).(\lambda (v2:
-T).(ty3 g c2 (THead (Bind Abst) v2 u) x1))) (\lambda (_: T).(\lambda (v2:
-T).(pr3 c2 x3 v2))) (\lambda (_: T).(\lambda (v2: T).(nf2 c2
-v2))))).(ex4_2_ind T T (\lambda (u: T).(\lambda (_: T).(pc3 c2 x0 (THead
-(Bind Abst) x3 u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c2 (THead (Bind
-Abst) v2 u) x1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c2 x3 v2))) (\lambda
-(_: T).(\lambda (v2: T).(nf2 c2 v2))) (or (ex T (\lambda (t3: T).(ty3 g c2
-(THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl)
-t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (x4: T).(\lambda (x5:
+v2)))) (\forall (u: T).((pc3 c2 x0 (THead (Bind Abst) x3 u)) \to False)) (or
+(ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3:
+T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False))) (\lambda (H18: (ex4_2
+T T (\lambda (u: T).(\lambda (_: T).(pc3 c2 x0 (THead (Bind Abst) x3 u))))
+(\lambda (u: T).(\lambda (v2: T).(ty3 g c2 (THead (Bind Abst) v2 u) x1)))
+(\lambda (_: T).(\lambda (v2: T).(pr3 c2 x3 v2))) (\lambda (_: T).(\lambda
+(v2: T).(nf2 c2 v2))))).(ex4_2_ind T T (\lambda (u: T).(\lambda (_: T).(pc3
+c2 x0 (THead (Bind Abst) x3 u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c2
+(THead (Bind Abst) v2 u) x1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c2 x3
+v2))) (\lambda (_: T).(\lambda (v2: T).(nf2 c2 v2))) (or (ex T (\lambda (t3:
+T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2
+(THead (Flat Appl) t t0) t3) \to False))) (\lambda (x4: T).(\lambda (x5:
T).(\lambda (H19: (pc3 c2 x0 (THead (Bind Abst) x3 x4))).(\lambda (H20: (ty3
g c2 (THead (Bind Abst) x5 x4) x1)).(\lambda (H21: (pr3 c2 x3 x5)).(\lambda
(_: (nf2 c2 x5)).(let H_y \def (nf2_pr3_unfold c2 x3 x5 H21 H15) in (let H23
\def (eq_ind_r T x5 (\lambda (t3: T).(ty3 g c2 (THead (Bind Abst) t3 x4) x1))
H20 x3 H_y) in (or_introl (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl)
t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to
-(\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 H6 x3 H14) t0 x4 (ty3_conv g c2 (THead (Bind Abst)
-x3 x4) x1 H24 t0 x0 H9 H19))))))))))))) H18)) (\lambda (H18: ((\forall (u:
-T).((pc3 c2 x0 (THead (Bind Abst) x3 u)) \to (\forall (P:
-Prop).P))))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t
+False)) (ex_intro T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))
+(THead (Flat Appl) t (THead (Bind Abst) x3 x4)) (ty3_appl g c2 t x3 (ty3_tred
+g c2 t x H6 x3 H14) t0 x4 (ty3_conv g c2 (THead (Bind Abst) x3 x4) x1 H24 t0
+x0 H9 H19))))))))))))) H18)) (\lambda (H18: ((\forall (u: T).((pc3 c2 x0
+(THead (Bind Abst) x3 u)) \to False)))).(or_intror (ex T (\lambda (t3:
+T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2
+(THead (Flat Appl) t t0) t3) \to False)) (\lambda (t3: T).(\lambda (H19: (ty3
+g c2 (THead (Flat Appl) t t0) t3)).(ex3_2_ind T T (\lambda (u: T).(\lambda
+(t4: T).(pc3 c2 (THead (Flat Appl) t (THead (Bind Abst) u t4)) t3))) (\lambda
+(u: T).(\lambda (t4: T).(ty3 g c2 t0 (THead (Bind Abst) u t4)))) (\lambda (u:
+T).(\lambda (_: T).(ty3 g c2 t u))) False (\lambda (x4: T).(\lambda (x5:
+T).(\lambda (_: (pc3 c2 (THead (Flat Appl) t (THead (Bind Abst) x4 x5))
+t3)).(\lambda (H21: (ty3 g c2 t0 (THead (Bind Abst) x4 x5))).(\lambda (H22:
+(ty3 g c2 t x4)).(let H_y \def (ty3_unique g c2 t x4 H22 x H6) in (let H_y0
+\def (ty3_unique g c2 t0 (THead (Bind Abst) x4 x5) H21 x0 H9) in (H18 x5
+(pc3_t (THead (Bind Abst) x4 x5) c2 x0 (pc3_s c2 x0 (THead (Bind Abst) x4 x5)
+H_y0) (THead (Bind Abst) x3 x5) (pc3_head_1 c2 x4 x3 (pc3_t x c2 x4 H_y x3
+(pc3_pr3_r c2 x x3 H14)) (Bind Abst) x5)))))))))) (ty3_gen_appl g c2 t t0 t3
+H19)))))) H17))))))) H13)))))) (ty3_correct g c2 t x H6)))) (ty3_correct g c2
+t0 x0 H9)))) H8)) (\lambda (H8: ((\forall (t3: T).((ty3 g c2 t0 t3) \to
+False)))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t
t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to
-(\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H19: (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 (H21: (ty3 g c2 t0 (THead (Bind Abst) x4
-x5))).(\lambda (H22: (ty3 g c2 t x4)).(let H_y \def (ty3_unique g c2 t x4 H22
-x H6) in (let H_y0 \def (ty3_unique g c2 t0 (THead (Bind Abst) x4 x5) H21 x0
-H9) in (H18 x5 (pc3_t (THead (Bind Abst) x4 x5) c2 x0 (pc3_s c2 x0 (THead
-(Bind Abst) x4 x5) H_y0) (THead (Bind Abst) x3 x5) (pc3_head_1 c2 x4 x3
-(pc3_t x c2 x4 H_y x3 (pc3_pr3_r c2 x x3 H14)) (Bind Abst) x5)) P))))))))
-(ty3_gen_appl g c2 t t0 t3 H19))))))) H17))))))) H13)))))) (ty3_correct g c2
-t x H6)))) (ty3_correct g c2 t0 x0 H9)))) H8)) (\lambda (H8: ((\forall (t3:
-T).((ty3 g c2 t0 t3) \to (\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 (H9: (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
+False)) (\lambda (t3: T).(\lambda (H9: (ty3 g c2 (THead (Flat Appl) t t0)
+t3)).(ex3_2_ind T T (\lambda (u: T).(\lambda (t4: T).(pc3 c2 (THead (Flat
Appl) t (THead (Bind Abst) u t4)) t3))) (\lambda (u: T).(\lambda (t4: T).(ty3
g c2 t0 (THead (Bind Abst) u t4)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c2
-t u))) P (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead (Flat
-Appl) t (THead (Bind Abst) x0 x1)) t3)).(\lambda (H11: (ty3 g c2 t0 (THead
-(Bind Abst) x0 x1))).(\lambda (_: (ty3 g c2 t x0)).(H8 (THead (Bind Abst) x0
-x1) H11 P)))))) (ty3_gen_appl g c2 t t0 t3 H9))))))) H7)))) H5)) (\lambda
-(H5: ((\forall (t3: T).((ty3 g c2 t t3) \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 (H6: (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 (H9: (ty3
-g c2 t x0)).(H5 x0 H9 P)))))) (ty3_gen_appl g c2 t t0 t3 H6))))))) H4)))
-(\lambda (H3: ((\forall (c1: C).(\forall (t3: T).((flt c1 t3 c2 (THead (Flat
-Cast) t t0)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4:
-T).((ty3 g c1 t3 t4) \to (\forall (P: Prop).P))))))))).(let H4 \def (H3 c2 t
-(flt_thead_sx (Flat Cast) c2 t t0)) in (or_ind (ex T (\lambda (t3: T).(ty3 g
-c2 t t3))) (\forall (t3: T).((ty3 g c2 t t3) \to (\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 (H5: (ex T (\lambda (t3: T).(ty3 g c2 t t3)))).(ex_ind T (\lambda
-(t3: T).(ty3 g c2 t t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat
-Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3)
-\to (\forall (P: Prop).P)))) (\lambda (x: T).(\lambda (H6: (ty3 g c2 t
-x)).(let H7 \def (H3 c2 t0 (flt_thead_dx (Flat Cast) c2 t t0)) in (or_ind (ex
-T (\lambda (t3: T).(ty3 g c2 t0 t3))) (\forall (t3: T).((ty3 g c2 t0 t3) \to
-(\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 (H8: (ex T (\lambda (t3: T).(ty3 g c2
-t0 t3)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t0 t3)) (or (ex T (\lambda
+t u))) False (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead
+(Flat Appl) t (THead (Bind Abst) x0 x1)) t3)).(\lambda (H11: (ty3 g c2 t0
+(THead (Bind Abst) x0 x1))).(\lambda (_: (ty3 g c2 t x0)).(H8 (THead (Bind
+Abst) x0 x1) H11)))))) (ty3_gen_appl g c2 t t0 t3 H9)))))) H7)))) H5))
+(\lambda (H5: ((\forall (t3: T).((ty3 g c2 t t3) \to False)))).(or_intror (ex
+T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3:
+T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False)) (\lambda (t3:
+T).(\lambda (H6: (ty3 g c2 (THead (Flat Appl) t t0) t3)).(ex3_2_ind T T
+(\lambda (u: T).(\lambda (t4: T).(pc3 c2 (THead (Flat Appl) t (THead (Bind
+Abst) u t4)) t3))) (\lambda (u: T).(\lambda (t4: T).(ty3 g c2 t0 (THead (Bind
+Abst) u t4)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c2 t u))) False
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead (Flat Appl) t
+(THead (Bind Abst) x0 x1)) t3)).(\lambda (_: (ty3 g c2 t0 (THead (Bind Abst)
+x0 x1))).(\lambda (H9: (ty3 g c2 t x0)).(H5 x0 H9)))))) (ty3_gen_appl g c2 t
+t0 t3 H6)))))) H4))) (\lambda (H3: ((\forall (c1: C).(\forall (t3: T).((flt
+c1 t3 c2 (THead (Flat Cast) t t0)) \to (or (ex T (\lambda (t4: T).(ty3 g c1
+t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to False)))))))).(let H4 \def
+(H3 c2 t (flt_thead_sx (Flat Cast) c2 t t0)) in (or_ind (ex T (\lambda (t3:
+T).(ty3 g c2 t t3))) (\forall (t3: T).((ty3 g c2 t t3) \to False)) (or (ex T
+(\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3:
+T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False))) (\lambda (H5: (ex T
+(\lambda (t3: T).(ty3 g c2 t t3)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t
+t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3)))
+(\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False)))
+(\lambda (x: T).(\lambda (H6: (ty3 g c2 t x)).(let H7 \def (H3 c2 t0
+(flt_thead_dx (Flat Cast) c2 t t0)) in (or_ind (ex T (\lambda (t3: T).(ty3 g
+c2 t0 t3))) (\forall (t3: T).((ty3 g c2 t0 t3) \to False)) (or (ex T (\lambda
(t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2
-(THead (Flat Cast) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (x0:
+(THead (Flat Cast) t t0) t3) \to False))) (\lambda (H8: (ex T (\lambda (t3:
+T).(ty3 g c2 t0 t3)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t0 t3)) (or (ex T
+(\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3:
+T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False))) (\lambda (x0:
T).(\lambda (H9: (ty3 g c2 t0 x0)).(ex_ind T (\lambda (t3: T).(ty3 g c2 x0
t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3)))
-(\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to (\forall (P:
-Prop).P)))) (\lambda (x1: T).(\lambda (H10: (ty3 g c2 x0 x1)).(let H_x \def
-(pc3_dec g c2 x0 x1 H10 t x H6) in (let H11 \def H_x in (or_ind (pc3 c2 x0 t)
-((pc3 c2 x0 t) \to (\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 (H12: (pc3 c2 x0
-t)).(or_introl (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0)
-t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to (\forall
-(P: Prop).P))) (ex_intro T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t
-t0) t3)) (THead (Flat Cast) x t) (ty3_cast g c2 t0 t (ty3_conv g c2 t x H6 t0
-x0 H9 H12) x H6)))) (\lambda (H12: (((pc3 c2 x0 t) \to (\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 (H13: (ty3 g c2 (THead
-(Flat Cast) t t0) t3)).(\lambda (P: Prop).(ex3_ind T (\lambda (t4: T).(pc3 c2
-(THead (Flat Cast) t4 t) t3)) (\lambda (_: T).(ty3 g c2 t0 t)) (\lambda (t4:
-T).(ty3 g c2 t t4)) P (\lambda (x2: T).(\lambda (_: (pc3 c2 (THead (Flat
-Cast) x2 t) t3)).(\lambda (H15: (ty3 g c2 t0 t)).(\lambda (H16: (ty3 g c2 t
-x2)).(let H_y \def (ty3_unique g c2 t x2 H16 x H6) in (let H_y0 \def
-(ty3_unique g c2 t0 t H15 x0 H9) in (H12 (ex2_sym T (pr3 c2 t) (pr3 c2 x0)
-H_y0) P))))))) (ty3_gen_cast g c2 t0 t t3 H13))))))) H11))))) (ty3_correct g
-c2 t0 x0 H9)))) H8)) (\lambda (H8: ((\forall (t3: T).((ty3 g c2 t0 t3) \to
-(\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 (H9: (ty3 g c2
-(THead (Flat Cast) t t0) t3)).(\lambda (P: Prop).(ex3_ind T (\lambda (t4:
-T).(pc3 c2 (THead (Flat Cast) t4 t) t3)) (\lambda (_: T).(ty3 g c2 t0 t))
-(\lambda (t4: T).(ty3 g c2 t t4)) P (\lambda (x0: T).(\lambda (_: (pc3 c2
-(THead (Flat Cast) x0 t) t3)).(\lambda (H11: (ty3 g c2 t0 t)).(\lambda (_:
-(ty3 g c2 t x0)).(H8 t H11 P))))) (ty3_gen_cast g c2 t0 t t3 H9))))))) H7))))
-H5)) (\lambda (H5: ((\forall (t3: T).((ty3 g c2 t t3) \to (\forall (P:
-Prop).P))))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t
+(\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False)))
+(\lambda (x1: T).(\lambda (H10: (ty3 g c2 x0 x1)).(let H_x \def (pc3_dec g c2
+x0 x1 H10 t x H6) in (let H11 \def H_x in (or_ind (pc3 c2 x0 t) ((pc3 c2 x0
+t) \to False) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0)
+t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False)))
+(\lambda (H12: (pc3 c2 x0 t)).(or_introl (ex T (\lambda (t3: T).(ty3 g c2
+(THead (Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast)
+t t0) t3) \to False)) (ex_intro T (\lambda (t3: T).(ty3 g c2 (THead (Flat
+Cast) t t0) t3)) (THead (Flat Cast) x t) (ty3_cast g c2 t0 t (ty3_conv g c2 t
+x H6 t0 x0 H9 H12) x H6)))) (\lambda (H12: (((pc3 c2 x0 t) \to
+False))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0)
+t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False))
+(\lambda (t3: T).(\lambda (H13: (ty3 g c2 (THead (Flat Cast) t t0)
+t3)).(ex3_ind T (\lambda (t4: T).(pc3 c2 (THead (Flat Cast) t4 t) t3))
+(\lambda (_: T).(ty3 g c2 t0 t)) (\lambda (t4: T).(ty3 g c2 t t4)) False
+(\lambda (x2: T).(\lambda (_: (pc3 c2 (THead (Flat Cast) x2 t) t3)).(\lambda
+(H15: (ty3 g c2 t0 t)).(\lambda (H16: (ty3 g c2 t x2)).(let H_y \def
+(ty3_unique g c2 t x2 H16 x H6) in (let H_y0 \def (ty3_unique g c2 t0 t H15
+x0 H9) in (H12 (ex2_sym T (pr3 c2 t) (pr3 c2 x0) H_y0)))))))) (ty3_gen_cast g
+c2 t0 t t3 H13)))))) H11))))) (ty3_correct g c2 t0 x0 H9)))) H8)) (\lambda
+(H8: ((\forall (t3: T).((ty3 g c2 t0 t3) \to False)))).(or_intror (ex T
+(\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3:
+T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False)) (\lambda (t3:
+T).(\lambda (H9: (ty3 g c2 (THead (Flat Cast) t t0) t3)).(ex3_ind T (\lambda
+(t4: T).(pc3 c2 (THead (Flat Cast) t4 t) t3)) (\lambda (_: T).(ty3 g c2 t0
+t)) (\lambda (t4: T).(ty3 g c2 t t4)) False (\lambda (x0: T).(\lambda (_:
+(pc3 c2 (THead (Flat Cast) x0 t) t3)).(\lambda (H11: (ty3 g c2 t0
+t)).(\lambda (_: (ty3 g c2 t x0)).(H8 t H11))))) (ty3_gen_cast g c2 t0 t t3
+H9)))))) H7)))) H5)) (\lambda (H5: ((\forall (t3: T).((ty3 g c2 t t3) \to
+False)))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t
t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to
-(\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H6: (ty3 g c2 (THead (Flat
-Cast) t t0) t3)).(\lambda (P: Prop).(ex3_ind T (\lambda (t4: T).(pc3 c2
-(THead (Flat Cast) t4 t) t3)) (\lambda (_: T).(ty3 g c2 t0 t)) (\lambda (t4:
-T).(ty3 g c2 t t4)) P (\lambda (x0: T).(\lambda (_: (pc3 c2 (THead (Flat
-Cast) x0 t) t3)).(\lambda (_: (ty3 g c2 t0 t)).(\lambda (H9: (ty3 g c2 t
-x0)).(ex_ind T (\lambda (t4: T).(ty3 g c2 x0 t4)) P (\lambda (x: T).(\lambda
-(_: (ty3 g c2 x0 x)).(H5 x0 H9 P))) (ty3_correct g c2 t x0 H9))))))
-(ty3_gen_cast g c2 t0 t t3 H6))))))) H4))) f H2))) k H1))))))) t2))) c t1))).
+False)) (\lambda (t3: T).(\lambda (H6: (ty3 g c2 (THead (Flat Cast) t t0)
+t3)).(ex3_ind T (\lambda (t4: T).(pc3 c2 (THead (Flat Cast) t4 t) t3))
+(\lambda (_: T).(ty3 g c2 t0 t)) (\lambda (t4: T).(ty3 g c2 t t4)) False
+(\lambda (x0: T).(\lambda (_: (pc3 c2 (THead (Flat Cast) x0 t) t3)).(\lambda
+(_: (ty3 g c2 t0 t)).(\lambda (H9: (ty3 g c2 t x0)).(ex_ind T (\lambda (t4:
+T).(ty3 g c2 x0 t4)) False (\lambda (x: T).(\lambda (_: (ty3 g c2 x0 x)).(H5
+x0 H9))) (ty3_correct g c2 t x0 H9)))))) (ty3_gen_cast g c2 t0 t t3 H6))))))
+H4))) f H2))) k H1))))))) t2))) c t1))).
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/ty3/defs".
include "G/defs.ma".
(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)))))))))))
+u) t1 t2) \to (ty3 g c (THead (Bind b) u t1) (THead (Bind b) u t2)))))))))
| ty3_appl: \forall (c: C).(\forall (w: T).(\forall (u: T).((ty3 g c w u) \to
(\forall (v: T).(\forall (t: T).((ty3 g c v (THead (Bind Abst) u t)) \to (ty3
g c (THead (Flat Appl) w v) (THead (Flat Appl) w (THead (Bind Abst) u
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/ty3/fsubst0".
include "ty3/props.ma".
(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 (_:
+t0)))))))))).(\lambda (b: B).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H2:
(ty3 g (CHead c (Bind b) u) t2 t3)).(\lambda (H3: ((\forall (i: nat).(\forall
(u0: T).(\forall (c2: C).(\forall (t4: T).((fsubst0 i u0 (CHead c (Bind b) u)
t2 c2 t4) \to (\forall (e: C).((getl i (CHead c (Bind b) u) (CHead e (Bind
-Abbr) u0)) \to (ty3 g c2 t4 t3)))))))))).(\lambda (t4: T).(\lambda (H4: (ty3
-g (CHead c (Bind b) u) t3 t4)).(\lambda (_: ((\forall (i: nat).(\forall (u0:
-T).(\forall (c2: C).(\forall (t5: T).((fsubst0 i u0 (CHead c (Bind b) u) t3
-c2 t5) \to (\forall (e: C).((getl i (CHead c (Bind b) u) (CHead e (Bind Abbr)
-u0)) \to (ty3 g c2 t5 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
+Abbr) u0)) \to (ty3 g c2 t4 t3)))))))))).(\lambda (i: nat).(\lambda (u0:
+T).(\lambda (c2: C).(\lambda (t4: T).(\lambda (H4: (fsubst0 i u0 c (THead
+(Bind b) u t2) c2 t4)).(fsubst0_ind i u0 c (THead (Bind b) u t2) (\lambda
+(c0: C).(\lambda (t5: T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u0))
+\to (ty3 g c0 t5 (THead (Bind b) u t3)))))) (\lambda (t5: T).(\lambda (H5:
+(subst0 i u0 (THead (Bind b) u t2) t5)).(\lambda (e: C).(\lambda (H6: (getl i
+c (CHead e (Bind Abbr) u0))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t5 (THead
+(Bind b) u2 t2))) (\lambda (u2: T).(subst0 i u0 u u2))) (ex2 T (\lambda (t6:
+T).(eq T t5 (THead (Bind b) u t6))) (\lambda (t6: T).(subst0 (s (Bind b) i)
+u0 t2 t6))) (ex3_2 T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead
+(Bind b) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2)))
+(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6)))) (ty3 g c
+t5 (THead (Bind b) u t3)) (\lambda (H7: (ex2 T (\lambda (u2: T).(eq T t5
(THead (Bind b) u2 t2))) (\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T
-(\lambda (u2: T).(eq T 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
-(t7: T).(eq T t6 (THead (Bind b) u t7))) (\lambda (t7: T).(subst0 (s (Bind b)
-i) u0 t2 t7)))).(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 (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))))).(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).(eq T t5 (THead (Bind b) u2 t2))) (\lambda (u2: T).(subst0 i
+u0 u u2)) (ty3 g c t5 (THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H8:
+(eq T t5 (THead (Bind b) x t2))).(\lambda (H9: (subst0 i u0 u x)).(eq_ind_r T
+(THead (Bind b) x t2) (\lambda (t6: T).(ty3 g c t6 (THead (Bind b) u t3)))
+(ex_ind T (\lambda (t6: T).(ty3 g (CHead c (Bind b) u) t3 t6)) (ty3 g c
+(THead (Bind b) x t2) (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (H10:
+(ty3 g (CHead c (Bind b) u) t3 x0)).(ex_ind T (\lambda (t6: T).(ty3 g (CHead
+c (Bind b) x) t3 t6)) (ty3 g c (THead (Bind b) x t2) (THead (Bind b) u t3))
+(\lambda (x1: T).(\lambda (_: (ty3 g (CHead c (Bind b) x) t3 x1)).(ty3_conv g
+c (THead (Bind b) u t3) (THead (Bind b) u x0) (ty3_bind g c u t0 H0 b t3 x0
+H10) (THead (Bind b) x t2) (THead (Bind b) x t3) (ty3_bind g c x t0 (H1 i u0
+c x (fsubst0_snd i u0 c u x H9) e H6) b t2 t3 (H3 (S i) u0 (CHead c (Bind b)
+x) t2 (fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2 (CHead c (Bind b) x)
+(csubst0_snd_bind b i u0 u x H9 c)) e (getl_head (Bind b) i c (CHead e (Bind
+Abbr) u0) H6 u))) (pc3_fsubst0 c (THead (Bind b) u t3) (THead (Bind b) u t3)
+(pc3_refl c (THead (Bind b) u t3)) i u0 c (THead (Bind b) x t3) (fsubst0_snd
+i u0 c (THead (Bind b) u t3) (THead (Bind b) x t3) (subst0_fst u0 x u i H9 t3
+(Bind b))) e H6)))) (ty3_correct g (CHead c (Bind b) x) t2 t3 (H3 (S i) u0
+(CHead c (Bind b) x) t2 (fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2 (CHead
+c (Bind b) x) (csubst0_snd_bind b i u0 u x H9 c)) e (getl_head (Bind b) i c
+(CHead e (Bind Abbr) u0) H6 u)))))) (ty3_correct g (CHead c (Bind b) u) t2 t3
+H2)) t5 H8)))) H7)) (\lambda (H7: (ex2 T (\lambda (t6: T).(eq T t5 (THead
+(Bind b) u t6))) (\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2
+t6)))).(ex2_ind T (\lambda (t6: T).(eq T t5 (THead (Bind b) u t6))) (\lambda
+(t6: T).(subst0 (s (Bind b) i) u0 t2 t6)) (ty3 g c t5 (THead (Bind b) u t3))
+(\lambda (x: T).(\lambda (H8: (eq T t5 (THead (Bind b) u x))).(\lambda (H9:
+(subst0 (s (Bind b) i) u0 t2 x)).(eq_ind_r T (THead (Bind b) u x) (\lambda
+(t6: T).(ty3 g c t6 (THead (Bind b) u t3))) (ex_ind T (\lambda (t6: T).(ty3 g
+(CHead c (Bind b) u) t3 t6)) (ty3 g c (THead (Bind b) u x) (THead (Bind b) u
+t3)) (\lambda (x0: T).(\lambda (_: (ty3 g (CHead c (Bind b) u) t3
+x0)).(ty3_bind g c u t0 H0 b x t3 (H3 (S i) u0 (CHead c (Bind b) u) x
+(fsubst0_snd (S i) u0 (CHead c (Bind b) u) t2 x H9) e (getl_head (Bind b) i c
+(CHead e (Bind Abbr) u0) H6 u))))) (ty3_correct g (CHead c (Bind b) u) x t3
+(H3 (S i) u0 (CHead c (Bind b) u) x (fsubst0_snd (S i) u0 (CHead c (Bind b)
+u) t2 x H9) e (getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H6 u)))) t5
+H8)))) H7)) (\lambda (H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t6: T).(eq T
+t5 (THead (Bind b) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u
+u2))) (\lambda (_: T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2
+t6))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead
+(Bind b) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2)))
+(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6))) (ty3 g c
+t5 (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H8: (eq
+T t5 (THead (Bind b) x0 x1))).(\lambda (H9: (subst0 i u0 u x0)).(\lambda
+(H10: (subst0 (s (Bind b) i) u0 t2 x1)).(eq_ind_r T (THead (Bind b) x0 x1)
+(\lambda (t6: T).(ty3 g c t6 (THead (Bind b) u t3))) (ex_ind T (\lambda (t6:
+T).(ty3 g (CHead c (Bind b) u) t3 t6)) (ty3 g c (THead (Bind b) x0 x1) (THead
+(Bind b) u t3)) (\lambda (x: T).(\lambda (H11: (ty3 g (CHead c (Bind b) u) t3
+x)).(ex_ind T (\lambda (t6: T).(ty3 g (CHead c (Bind b) x0) t3 t6)) (ty3 g c
+(THead (Bind b) x0 x1) (THead (Bind b) u t3)) (\lambda (x2: T).(\lambda (_:
+(ty3 g (CHead c (Bind b) x0) t3 x2)).(ty3_conv g c (THead (Bind b) u t3)
+(THead (Bind b) u x) (ty3_bind g c u t0 H0 b t3 x H11) (THead (Bind b) x0 x1)
+(THead (Bind b) x0 t3) (ty3_bind g c x0 t0 (H1 i u0 c x0 (fsubst0_snd i u0 c
+u x0 H9) e H6) b x1 t3 (H3 (S i) u0 (CHead c (Bind b) x0) x1 (fsubst0_both (S
+i) u0 (CHead c (Bind b) u) t2 x1 H10 (CHead c (Bind b) x0) (csubst0_snd_bind
+b i u0 u x0 H9 c)) e (getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H6 u)))
+(pc3_fsubst0 c (THead (Bind b) u t3) (THead (Bind b) u t3) (pc3_refl c (THead
+(Bind b) u t3)) i u0 c (THead (Bind b) x0 t3) (fsubst0_snd i u0 c (THead
+(Bind b) u t3) (THead (Bind b) x0 t3) (subst0_fst u0 x0 u i H9 t3 (Bind b)))
+e H6)))) (ty3_correct g (CHead c (Bind b) x0) x1 t3 (H3 (S i) u0 (CHead c
+(Bind b) x0) x1 (fsubst0_both (S i) u0 (CHead c (Bind b) u) t2 x1 H10 (CHead
+c (Bind b) x0) (csubst0_snd_bind b i u0 u x0 H9 c)) e (getl_head (Bind b) i c
+(CHead e (Bind Abbr) u0) H6 u)))))) (ty3_correct g (CHead c (Bind b) u) t2 t3
+H2)) t5 H8)))))) H7)) (subst0_gen_head (Bind b) u0 u t2 t5 i H5))))))
+(\lambda (c3: C).(\lambda (H5: (csubst0 i u0 c c3)).(\lambda (e: C).(\lambda
+(H6: (getl i c (CHead e (Bind Abbr) u0))).(ex_ind T (\lambda (t5: T).(ty3 g
+(CHead c3 (Bind b) u) t3 t5)) (ty3 g c3 (THead (Bind b) u t2) (THead (Bind b)
+u t3)) (\lambda (x: T).(\lambda (_: (ty3 g (CHead c3 (Bind b) u) t3
+x)).(ty3_bind g c3 u t0 (H1 i u0 c3 u (fsubst0_fst i u0 c u c3 H5) e H6) b t2
+t3 (H3 (S i) u0 (CHead c3 (Bind b) u) t2 (fsubst0_fst (S i) u0 (CHead c (Bind
+b) u) t2 (CHead c3 (Bind b) u) (csubst0_fst_bind b i c c3 u0 H5 u)) e
+(getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H6 u))))) (ty3_correct g
+(CHead c3 (Bind b) u) t2 t3 (H3 (S i) u0 (CHead c3 (Bind b) u) t2
+(fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2 (CHead c3 (Bind b) u)
+(csubst0_fst_bind b i c c3 u0 H5 u)) e (getl_head (Bind b) i c (CHead e (Bind
+Abbr) u0) H6 u)))))))) (\lambda (t5: T).(\lambda (H5: (subst0 i u0 (THead
+(Bind b) u t2) t5)).(\lambda (c3: C).(\lambda (H6: (csubst0 i u0 c
+c3)).(\lambda (e: C).(\lambda (H7: (getl i c (CHead e (Bind Abbr)
+u0))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t5 (THead (Bind b) u2 t2)))
+(\lambda (u2: T).(subst0 i u0 u u2))) (ex2 T (\lambda (t6: T).(eq T t5 (THead
+(Bind b) u t6))) (\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6))) (ex3_2 T
+T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead (Bind b) u2 t6))))
(\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_:
-T).(\lambda (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
+T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6)))) (ty3 g c3 t5 (THead
+(Bind b) u t3)) (\lambda (H8: (ex2 T (\lambda (u2: T).(eq T t5 (THead (Bind
b) u2 t2))) (\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T (\lambda (u2:
-T).(eq T 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
+T).(eq T t5 (THead (Bind b) u2 t2))) (\lambda (u2: T).(subst0 i u0 u u2))
+(ty3 g c3 t5 (THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H9: (eq T t5
+(THead (Bind b) x t2))).(\lambda (H10: (subst0 i u0 u x)).(eq_ind_r T (THead
+(Bind b) x t2) (\lambda (t6: T).(ty3 g c3 t6 (THead (Bind b) u t3))) (ex_ind
+T (\lambda (t6: T).(ty3 g (CHead c3 (Bind b) u) t3 t6)) (ty3 g c3 (THead
+(Bind b) x t2) (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (H11: (ty3 g
+(CHead c3 (Bind b) u) t3 x0)).(ex_ind T (\lambda (t6: T).(ty3 g (CHead c3
+(Bind b) u) x0 t6)) (ty3 g c3 (THead (Bind b) x t2) (THead (Bind b) u t3))
+(\lambda (x1: T).(\lambda (_: (ty3 g (CHead c3 (Bind b) u) x0 x1)).(ex_ind T
+(\lambda (t6: T).(ty3 g (CHead c3 (Bind b) x) t3 t6)) (ty3 g c3 (THead (Bind
+b) x t2) (THead (Bind b) u t3)) (\lambda (x2: T).(\lambda (_: (ty3 g (CHead
+c3 (Bind b) x) t3 x2)).(ty3_conv g c3 (THead (Bind b) u t3) (THead (Bind b) u
+x0) (ty3_bind g c3 u t0 (H1 i u0 c3 u (fsubst0_fst i u0 c u c3 H6) e H7) b t3
+x0 H11) (THead (Bind b) x t2) (THead (Bind b) x t3) (ty3_bind g c3 x t0 (H1 i
+u0 c3 x (fsubst0_both i u0 c u x H10 c3 H6) e H7) b t2 t3 (H3 (S i) u0 (CHead
c3 (Bind b) x) t2 (fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2 (CHead c3
-(Bind b) x) (csubst0_both_bind b i u0 u x 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 (t7: T).(eq T t6 (THead (Bind b) u t7))) (\lambda (t7: T).(subst0 (s
-(Bind b) i) u0 t2 t7)))).(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 (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))))).(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
+(Bind b) x) (csubst0_both_bind b i u0 u x H10 c c3 H6)) e (getl_head (Bind b)
+i c (CHead e (Bind Abbr) u0) H7 u))) (pc3_fsubst0 c (THead (Bind b) u t3)
+(THead (Bind b) u t3) (pc3_refl c (THead (Bind b) u t3)) i u0 c3 (THead (Bind
+b) x t3) (fsubst0_both i u0 c (THead (Bind b) u t3) (THead (Bind b) x t3)
+(subst0_fst u0 x u i H10 t3 (Bind b)) c3 H6) e H7)))) (ty3_correct g (CHead
+c3 (Bind b) x) t2 t3 (H3 (S i) u0 (CHead c3 (Bind b) x) t2 (fsubst0_fst (S i)
+u0 (CHead c (Bind b) u) t2 (CHead c3 (Bind b) x) (csubst0_both_bind b i u0 u
+x H10 c c3 H6)) e (getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H7 u))))))
+(ty3_correct g (CHead c3 (Bind b) u) t3 x0 H11)))) (ty3_correct g (CHead c3
+(Bind b) u) t2 t3 (H3 (S i) u0 (CHead c3 (Bind b) u) t2 (fsubst0_fst (S i) u0
+(CHead c (Bind b) u) t2 (CHead c3 (Bind b) u) (csubst0_fst_bind b i c c3 u0
+H6 u)) e (getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H7 u)))) t5 H9))))
+H8)) (\lambda (H8: (ex2 T (\lambda (t6: T).(eq T t5 (THead (Bind b) u t6)))
+(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6)))).(ex2_ind T (\lambda (t6:
+T).(eq T t5 (THead (Bind b) u t6))) (\lambda (t6: T).(subst0 (s (Bind b) i)
+u0 t2 t6)) (ty3 g c3 t5 (THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H9:
+(eq T t5 (THead (Bind b) u x))).(\lambda (H10: (subst0 (s (Bind b) i) u0 t2
+x)).(eq_ind_r T (THead (Bind b) u x) (\lambda (t6: T).(ty3 g c3 t6 (THead
+(Bind b) u t3))) (ex_ind T (\lambda (t6: T).(ty3 g (CHead c3 (Bind b) u) t3
+t6)) (ty3 g c3 (THead (Bind b) u x) (THead (Bind b) u t3)) (\lambda (x0:
+T).(\lambda (_: (ty3 g (CHead c3 (Bind b) u) t3 x0)).(ty3_bind g c3 u t0 (H1
+i u0 c3 u (fsubst0_fst i u0 c u c3 H6) e H7) b x t3 (H3 (S i) u0 (CHead c3
+(Bind b) u) x (fsubst0_both (S i) u0 (CHead c (Bind b) u) t2 x H10 (CHead c3
+(Bind b) u) (csubst0_fst_bind b i c c3 u0 H6 u)) e (getl_head (Bind b) i c
+(CHead e (Bind Abbr) u0) H7 u))))) (ty3_correct g (CHead c3 (Bind b) u) x t3
+(H3 (S i) u0 (CHead c3 (Bind b) u) x (fsubst0_both (S i) u0 (CHead c (Bind b)
+u) t2 x H10 (CHead c3 (Bind b) u) (csubst0_fst_bind b i c c3 u0 H6 u)) e
+(getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H7 u)))) t5 H9)))) H8))
+(\lambda (H8: (ex3_2 T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead
+(Bind b) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2)))
+(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2
+t6))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead
+(Bind b) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2)))
+(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6))) (ty3 g c3
+t5 (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H9: (eq
+T t5 (THead (Bind b) x0 x1))).(\lambda (H10: (subst0 i u0 u x0)).(\lambda
+(H11: (subst0 (s (Bind b) i) u0 t2 x1)).(eq_ind_r T (THead (Bind b) x0 x1)
+(\lambda (t6: T).(ty3 g c3 t6 (THead (Bind b) u t3))) (ex_ind T (\lambda (t6:
+T).(ty3 g (CHead c3 (Bind b) u) t3 t6)) (ty3 g c3 (THead (Bind b) x0 x1)
+(THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H12: (ty3 g (CHead c3 (Bind
+b) u) t3 x)).(ex_ind T (\lambda (t6: T).(ty3 g (CHead c3 (Bind b) u) x t6))
+(ty3 g c3 (THead (Bind b) x0 x1) (THead (Bind b) u t3)) (\lambda (x2:
+T).(\lambda (_: (ty3 g (CHead c3 (Bind b) u) x x2)).(ex_ind T (\lambda (t6:
+T).(ty3 g (CHead c3 (Bind b) x0) t3 t6)) (ty3 g c3 (THead (Bind b) x0 x1)
+(THead (Bind b) u t3)) (\lambda (x3: T).(\lambda (_: (ty3 g (CHead c3 (Bind
+b) x0) t3 x3)).(ty3_conv g c3 (THead (Bind b) u t3) (THead (Bind b) u x)
+(ty3_bind g c3 u t0 (H1 i u0 c3 u (fsubst0_fst i u0 c u c3 H6) e H7) b t3 x
+H12) (THead (Bind b) x0 x1) (THead (Bind b) x0 t3) (ty3_bind g c3 x0 t0 (H1 i
+u0 c3 x0 (fsubst0_both i u0 c u x0 H10 c3 H6) e H7) b x1 t3 (H3 (S i) u0
+(CHead c3 (Bind b) x0) x1 (fsubst0_both (S i) u0 (CHead c (Bind b) u) t2 x1
+H11 (CHead c3 (Bind b) x0) (csubst0_both_bind b i u0 u x0 H10 c c3 H6)) e
+(getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H7 u))) (pc3_fsubst0 c
+(THead (Bind b) u t3) (THead (Bind b) u t3) (pc3_refl c (THead (Bind b) u
t3)) i u0 c3 (THead (Bind b) x0 t3) (fsubst0_both i u0 c (THead (Bind b) u
-t3) (THead (Bind b) x0 t3) (subst0_fst u0 x0 u i 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)
+t3) (THead (Bind b) x0 t3) (subst0_fst u0 x0 u i H10 t3 (Bind b)) c3 H6) e
+H7)))) (ty3_correct g (CHead c3 (Bind b) x0) x1 t3 (H3 (S i) u0 (CHead c3
+(Bind b) x0) x1 (fsubst0_both (S i) u0 (CHead c (Bind b) u) t2 x1 H11 (CHead
+c3 (Bind b) x0) (csubst0_both_bind b i u0 u x0 H10 c c3 H6)) e (getl_head
+(Bind b) i c (CHead e (Bind Abbr) u0) H7 u)))))) (ty3_correct g (CHead c3
+(Bind b) u) t3 x H12)))) (ty3_correct g (CHead c3 (Bind b) u) t2 t3 (H3 (S i)
u0 (CHead c3 (Bind b) u) t2 (fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2
-(CHead c3 (Bind b) u) (csubst0_fst_bind b i c c3 u0 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:
+(CHead c3 (Bind b) u) (csubst0_fst_bind b i c c3 u0 H6 u)) e (getl_head (Bind
+b) i c (CHead e (Bind Abbr) u0) H7 u)))) t5 H9)))))) H8)) (subst0_gen_head
+(Bind b) u0 u t2 t5 i H5)))))))) c2 t4 H4)))))))))))))))) (\lambda (c:
C).(\lambda (w: T).(\lambda (u: T).(\lambda (H0: (ty3 g c w u)).(\lambda (H1:
((\forall (i: nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2:
T).((fsubst0 i u0 c w c2 t2) \to (\forall (e: C).((getl i c (CHead e (Bind
(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
+x0)).(ex3_2_ind T T (\lambda (t4: T).(\lambda (_: T).(pc3 c (THead (Bind
+Abst) u t4) x0))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c u t5))) (\lambda
+(t4: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t0 t4))) (ty3 g c
+(THead (Flat Appl) x v) (THead (Flat Appl) w (THead (Bind Abst) u t0)))
+(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 c (THead (Bind Abst) u
x1) x0)).(\lambda (_: (ty3 g c u x2)).(\lambda (H13: (ty3 g (CHead c (Bind
-Abst) u) t0 x1)).(\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
+Abst) u) t0 x1)).(ex_ind T (\lambda (t4: T).(ty3 g c u t4)) (ty3 g c (THead
+(Flat Appl) x v) (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda
+(x3: T).(\lambda (H14: (ty3 g c u x3)).(ty3_conv g c (THead (Flat Appl) w
+(THead (Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind Abst) u x1))
+(ty3_appl g c w u H0 (THead (Bind Abst) u t0) x1 (ty3_bind g c u x3 H14 Abst
+t0 x1 H13)) (THead (Flat Appl) x v) (THead (Flat Appl) x (THead (Bind Abst) u
+t0)) (ty3_appl g c x u (H1 i u0 c x (fsubst0_snd i u0 c w x H9) e H6) v t0
+H2) (pc3_fsubst0 c (THead (Flat Appl) w (THead (Bind Abst) u t0)) (THead
+(Flat Appl) w (THead (Bind Abst) u t0)) (pc3_refl c (THead (Flat Appl) w
+(THead (Bind Abst) u t0))) i u0 c (THead (Flat Appl) x (THead (Bind Abst) u
+t0)) (fsubst0_snd i u0 c (THead (Flat Appl) w (THead (Bind Abst) u t0))
+(THead (Flat Appl) x (THead (Bind Abst) u t0)) (subst0_fst u0 x w i H9 (THead
+(Bind Abst) u t0) (Flat Appl))) e H6)))) (ty3_correct g c x u (H1 i u0 c x
+(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 (t4: T).(eq T t3 (THead (Flat Appl) w t4)))
(\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4)))).(ex2_ind T (\lambda
(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) x0 x1) (THead (Flat Appl) w
(THead (Bind Abst) u t0))) (\lambda (x: T).(\lambda (H11: (ty3 g c (THead
-(Bind Abst) u t0) x)).(ex4_3_ind T T T (\lambda (t4: T).(\lambda (_:
-T).(\lambda (_: T).(pc3 c (THead (Bind Abst) u t4) x)))) (\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) x0 x1) (THead (Flat Appl) w (THead
-(Bind Abst) u t0))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (x4:
-T).(\lambda (_: (pc3 c (THead (Bind Abst) u x2) x)).(\lambda (_: (ty3 g c u
-x3)).(\lambda (H14: (ty3 g (CHead c (Bind Abst) u) t0 x2)).(\lambda (H15:
-(ty3 g (CHead c (Bind Abst) u) x2 x4)).(ex_ind T (\lambda (t4: T).(ty3 g c u
-t4)) (ty3 g c (THead (Flat Appl) x0 x1) (THead (Flat Appl) w (THead (Bind
-Abst) u t0))) (\lambda (x5: T).(\lambda (H16: (ty3 g c u x5)).(ty3_conv g c
-(THead (Flat Appl) w (THead (Bind Abst) u t0)) (THead (Flat Appl) w (THead
-(Bind Abst) u x2)) (ty3_appl g c w u H0 (THead (Bind Abst) u t0) x2 (ty3_bind
-g c u x5 H16 Abst t0 x2 H14 x4 H15)) (THead (Flat Appl) x0 x1) (THead (Flat
-Appl) x0 (THead (Bind Abst) u t0)) (ty3_appl g c x0 u (H1 i u0 c x0
-(fsubst0_snd i u0 c w x0 H9) e H6) x1 t0 (H3 (s (Flat Appl) i) u0 c x1
-(fsubst0_snd (s (Flat Appl) i) u0 c v x1 H10) e H6)) (pc3_fsubst0 c (THead
+(Bind Abst) u t0) x)).(ex3_2_ind T T (\lambda (t4: T).(\lambda (_: T).(pc3 c
+(THead (Bind Abst) u t4) x))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c u
+t5))) (\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t0
+t4))) (ty3 g c (THead (Flat Appl) x0 x1) (THead (Flat Appl) w (THead (Bind
+Abst) u t0))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (_: (pc3 c (THead
+(Bind Abst) u x2) x)).(\lambda (_: (ty3 g c u x3)).(\lambda (H14: (ty3 g
+(CHead c (Bind Abst) u) t0 x2)).(ex_ind T (\lambda (t4: T).(ty3 g c u t4))
+(ty3 g c (THead (Flat Appl) x0 x1) (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 t0)) (pc3_refl c (THead (Flat Appl) w (THead (Bind Abst) u t0))) i u0
-c (THead (Flat Appl) x0 (THead (Bind Abst) u t0)) (fsubst0_snd i u0 c (THead
-(Flat Appl) w (THead (Bind Abst) u t0)) (THead (Flat Appl) x0 (THead (Bind
-Abst) u t0)) (subst0_fst u0 x0 w i H9 (THead (Bind Abst) u t0) (Flat Appl)))
-e H6)))) (ty3_correct g c w u H0))))))))) (ty3_gen_bind g Abst c u t0 x
-H11)))) (ty3_correct g c v (THead (Bind Abst) u t0) H2)) t3 H8)))))) H7))
+Abst) u x2)) (ty3_appl g c w u H0 (THead (Bind Abst) u t0) x2 (ty3_bind g c u
+x4 H15 Abst t0 x2 H14)) (THead (Flat Appl) x0 x1) (THead (Flat Appl) x0
+(THead (Bind Abst) u t0)) (ty3_appl g c x0 u (H1 i u0 c x0 (fsubst0_snd i u0
+c w x0 H9) e H6) x1 t0 (H3 (s (Flat Appl) i) u0 c x1 (fsubst0_snd (s (Flat
+Appl) i) u0 c v x1 H10) e H6)) (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) x0
+(THead (Bind Abst) u t0)) (fsubst0_snd i u0 c (THead (Flat Appl) w (THead
+(Bind Abst) u t0)) (THead (Flat Appl) x0 (THead (Bind Abst) u t0))
+(subst0_fst u0 x0 w i H9 (THead (Bind Abst) u t0) (Flat Appl))) e H6))))
+(ty3_correct g c w u H0))))))) (ty3_gen_bind g Abst c u t0 x H11))))
+(ty3_correct g c v (THead (Bind Abst) u t0) H2)) t3 H8)))))) H7))
(subst0_gen_head (Flat Appl) u0 w v t3 i H5)))))) (\lambda (c3: C).(\lambda
(H5: (csubst0 i u0 c c3)).(\lambda (e: C).(\lambda (H6: (getl i c (CHead e
(Bind Abbr) u0))).(ty3_appl g c3 w u (H1 i u0 c3 w (fsubst0_fst i u0 c w c3
(THead (Bind Abst) u t0)))) (ex_ind T (\lambda (t4: T).(ty3 g c3 (THead (Bind
Abst) u t0) t4)) (ty3 g c3 (THead (Flat Appl) x v) (THead (Flat Appl) w
(THead (Bind Abst) u t0))) (\lambda (x0: T).(\lambda (H11: (ty3 g c3 (THead
-(Bind Abst) u t0) x0)).(ex4_3_ind T T T (\lambda (t4: T).(\lambda (_:
-T).(\lambda (_: T).(pc3 c3 (THead (Bind Abst) u t4) x0)))) (\lambda (_:
-T).(\lambda (t5: T).(\lambda (_: T).(ty3 g c3 u t5)))) (\lambda (t4:
-T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c3 (Bind Abst) u) t0 t4))))
-(\lambda (t4: T).(\lambda (_: T).(\lambda (t6: T).(ty3 g (CHead c3 (Bind
-Abst) u) t4 t6)))) (ty3 g c3 (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 c3 (THead (Bind Abst) u x1) x0)).(\lambda (H13: (ty3 g
-c3 u x2)).(\lambda (H14: (ty3 g (CHead c3 (Bind Abst) u) t0 x1)).(\lambda
-(H15: (ty3 g (CHead c3 (Bind Abst) u) x1 x3)).(ty3_conv g c3 (THead (Flat
-Appl) w (THead (Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind Abst) u
-x1)) (ty3_appl g c3 w u (H1 i u0 c3 w (fsubst0_fst i u0 c w c3 H6) e H7)
-(THead (Bind Abst) u t0) x1 (ty3_bind g c3 u x2 H13 Abst t0 x1 H14 x3 H15))
-(THead (Flat Appl) x v) (THead (Flat Appl) x (THead (Bind Abst) u t0))
-(ty3_appl g c3 x u (H1 i u0 c3 x (fsubst0_both i u0 c w x H10 c3 H6) e H7) v
-t0 (H3 i u0 c3 v (fsubst0_fst i u0 c v c3 H6) e H7)) (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
-c3 (THead (Flat Appl) x (THead (Bind Abst) u t0)) (fsubst0_both 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 H10 (THead (Bind Abst) u t0) (Flat Appl))
-c3 H6) e H7))))))))) (ty3_gen_bind g Abst c3 u t0 x0 H11)))) (ty3_correct g
-c3 v (THead (Bind Abst) u t0) (H3 i u0 c3 v (fsubst0_fst i u0 c v c3 H6) e
-H7))) t3 H9)))) H8)) (\lambda (H8: (ex2 T (\lambda (t4: T).(eq T t3 (THead
-(Flat Appl) w t4))) (\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v
-t4)))).(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 c3 t3 (THead
-(Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x: T).(\lambda (H9: (eq T
-t3 (THead (Flat Appl) w x))).(\lambda (H10: (subst0 (s (Flat Appl) i) u0 v
-x)).(eq_ind_r T (THead (Flat Appl) w x) (\lambda (t4: T).(ty3 g c3 t4 (THead
-(Flat Appl) w (THead (Bind Abst) u t0)))) (ty3_appl g c3 w u (H1 i u0 c3 w
-(fsubst0_fst i u0 c w c3 H6) e H7) x t0 (H3 i u0 c3 x (fsubst0_both i u0 c v
-x H10 c3 H6) e H7)) t3 H9)))) H8)) (\lambda (H8: (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))))).(ex3_2_ind 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 c3 t3 (THead (Flat Appl) w
-(THead (Bind Abst) u t0))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H9:
-(eq T t3 (THead (Flat Appl) x0 x1))).(\lambda (H10: (subst0 i u0 w
-x0)).(\lambda (H11: (subst0 (s (Flat Appl) i) u0 v x1)).(eq_ind_r T (THead
-(Flat Appl) x0 x1) (\lambda (t4: T).(ty3 g c3 t4 (THead (Flat Appl) w (THead
-(Bind Abst) u t0)))) (ex_ind T (\lambda (t4: T).(ty3 g c3 (THead (Bind Abst)
-u t0) t4)) (ty3 g c3 (THead (Flat Appl) x0 x1) (THead (Flat Appl) w (THead
-(Bind Abst) u t0))) (\lambda (x: T).(\lambda (H12: (ty3 g c3 (THead (Bind
-Abst) u t0) x)).(ex4_3_ind T T T (\lambda (t4: T).(\lambda (_: T).(\lambda
-(_: T).(pc3 c3 (THead (Bind Abst) u t4) x)))) (\lambda (_: T).(\lambda (t5:
-T).(\lambda (_: T).(ty3 g c3 u t5)))) (\lambda (t4: T).(\lambda (_:
-T).(\lambda (_: T).(ty3 g (CHead c3 (Bind Abst) u) t0 t4)))) (\lambda (t4:
-T).(\lambda (_: T).(\lambda (t6: T).(ty3 g (CHead c3 (Bind Abst) u) t4 t6))))
+(Bind Abst) u t0) x0)).(ex3_2_ind T T (\lambda (t4: T).(\lambda (_: T).(pc3
+c3 (THead (Bind Abst) u t4) x0))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c3
+u t5))) (\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c3 (Bind Abst) u) t0
+t4))) (ty3 g c3 (THead (Flat Appl) x v) (THead (Flat Appl) w (THead (Bind
+Abst) u t0))) (\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 c3 (THead
+(Bind Abst) u x1) x0)).(\lambda (H13: (ty3 g c3 u x2)).(\lambda (H14: (ty3 g
+(CHead c3 (Bind Abst) u) t0 x1)).(ty3_conv g c3 (THead (Flat Appl) w (THead
+(Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind Abst) u x1)) (ty3_appl g
+c3 w u (H1 i u0 c3 w (fsubst0_fst i u0 c w c3 H6) e H7) (THead (Bind Abst) u
+t0) x1 (ty3_bind g c3 u x2 H13 Abst t0 x1 H14)) (THead (Flat Appl) x v)
+(THead (Flat Appl) x (THead (Bind Abst) u t0)) (ty3_appl g c3 x u (H1 i u0 c3
+x (fsubst0_both i u0 c w x H10 c3 H6) e H7) v t0 (H3 i u0 c3 v (fsubst0_fst i
+u0 c v c3 H6) e H7)) (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 c3 (THead (Flat Appl) x (THead (Bind
+Abst) u t0)) (fsubst0_both 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 H10
+(THead (Bind Abst) u t0) (Flat Appl)) c3 H6) e H7))))))) (ty3_gen_bind g Abst
+c3 u t0 x0 H11)))) (ty3_correct g c3 v (THead (Bind Abst) u t0) (H3 i u0 c3 v
+(fsubst0_fst i u0 c v c3 H6) e H7))) t3 H9)))) H8)) (\lambda (H8: (ex2 T
+(\lambda (t4: T).(eq T t3 (THead (Flat Appl) w t4))) (\lambda (t4: T).(subst0
+(s (Flat Appl) i) u0 v t4)))).(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 c3 t3 (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x:
+T).(\lambda (H9: (eq T t3 (THead (Flat Appl) w x))).(\lambda (H10: (subst0 (s
+(Flat Appl) i) u0 v x)).(eq_ind_r T (THead (Flat Appl) w x) (\lambda (t4:
+T).(ty3 g c3 t4 (THead (Flat Appl) w (THead (Bind Abst) u t0)))) (ty3_appl g
+c3 w u (H1 i u0 c3 w (fsubst0_fst i u0 c w c3 H6) e H7) x t0 (H3 i u0 c3 x
+(fsubst0_both i u0 c v x H10 c3 H6) e H7)) t3 H9)))) H8)) (\lambda (H8:
+(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))))).(ex3_2_ind 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 c3 t3 (THead
+(Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H9: (eq T t3 (THead (Flat Appl) x0 x1))).(\lambda (H10: (subst0
+i u0 w x0)).(\lambda (H11: (subst0 (s (Flat Appl) i) u0 v x1)).(eq_ind_r T
+(THead (Flat Appl) x0 x1) (\lambda (t4: T).(ty3 g c3 t4 (THead (Flat Appl) w
+(THead (Bind Abst) u t0)))) (ex_ind T (\lambda (t4: T).(ty3 g c3 (THead (Bind
+Abst) u t0) t4)) (ty3 g c3 (THead (Flat Appl) x0 x1) (THead (Flat Appl) w
+(THead (Bind Abst) u t0))) (\lambda (x: T).(\lambda (H12: (ty3 g c3 (THead
+(Bind Abst) u t0) x)).(ex3_2_ind T T (\lambda (t4: T).(\lambda (_: T).(pc3 c3
+(THead (Bind Abst) u t4) x))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c3 u
+t5))) (\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c3 (Bind Abst) u) t0
+t4))) (ty3 g c3 (THead (Flat Appl) x0 x1) (THead (Flat Appl) w (THead (Bind
+Abst) u t0))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (_: (pc3 c3 (THead
+(Bind Abst) u x2) x)).(\lambda (_: (ty3 g c3 u x3)).(\lambda (H15: (ty3 g
+(CHead c3 (Bind Abst) u) t0 x2)).(ex_ind T (\lambda (t4: T).(ty3 g c3 u t4))
(ty3 g c3 (THead (Flat Appl) x0 x1) (THead (Flat Appl) w (THead (Bind Abst) u
-t0))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (_: (pc3 c3
-(THead (Bind Abst) u x2) x)).(\lambda (_: (ty3 g c3 u x3)).(\lambda (H15:
-(ty3 g (CHead c3 (Bind Abst) u) t0 x2)).(\lambda (H16: (ty3 g (CHead c3 (Bind
-Abst) u) x2 x4)).(ex_ind T (\lambda (t4: T).(ty3 g c3 u t4)) (ty3 g c3 (THead
-(Flat Appl) x0 x1) (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda
-(x5: T).(\lambda (H17: (ty3 g c3 u x5)).(ty3_conv g c3 (THead (Flat Appl) w
-(THead (Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind Abst) u x2))
-(ty3_appl g c3 w u (H1 i u0 c3 w (fsubst0_fst i u0 c w c3 H6) e H7) (THead
-(Bind Abst) u t0) x2 (ty3_bind g c3 u x5 H17 Abst t0 x2 H15 x4 H16)) (THead
-(Flat Appl) x0 x1) (THead (Flat Appl) x0 (THead (Bind Abst) u t0)) (ty3_appl
-g c3 x0 u (H1 i u0 c3 x0 (fsubst0_both i u0 c w x0 H10 c3 H6) e H7) x1 t0 (H3
-i u0 c3 x1 (fsubst0_both i u0 c v x1 H11 c3 H6) e H7)) (pc3_fsubst0 c (THead
+t0))) (\lambda (x4: T).(\lambda (H16: (ty3 g c3 u x4)).(ty3_conv g c3 (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
-c3 (THead (Flat Appl) x0 (THead (Bind Abst) u t0)) (fsubst0_both i u0 c
-(THead (Flat Appl) w (THead (Bind Abst) u t0)) (THead (Flat Appl) x0 (THead
-(Bind Abst) u t0)) (subst0_fst u0 x0 w i H10 (THead (Bind Abst) u t0) (Flat
-Appl)) c3 H6) e H7)))) (ty3_correct g c3 w u (H1 i u0 c3 w (fsubst0_fst i u0
-c w c3 H6) e H7)))))))))) (ty3_gen_bind g Abst c3 u t0 x H12)))) (ty3_correct
-g c3 v (THead (Bind Abst) u t0) (H3 i u0 c3 v (fsubst0_fst i u0 c v c3 H6) e
-H7))) t3 H9)))))) H8)) (subst0_gen_head (Flat Appl) u0 w v t3 i H5)))))))) c2
-t2 H4))))))))))))))) (\lambda (c: C).(\lambda (t2: T).(\lambda (t3:
-T).(\lambda (H0: (ty3 g c t2 t3)).(\lambda (H1: ((\forall (i: nat).(\forall
-(u: T).(\forall (c2: C).(\forall (t4: T).((fsubst0 i u c t2 c2 t4) \to
-(\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) \to (ty3 g c2 t4
+Abst) u x2)) (ty3_appl g c3 w u (H1 i u0 c3 w (fsubst0_fst i u0 c w c3 H6) e
+H7) (THead (Bind Abst) u t0) x2 (ty3_bind g c3 u x4 H16 Abst t0 x2 H15))
+(THead (Flat Appl) x0 x1) (THead (Flat Appl) x0 (THead (Bind Abst) u t0))
+(ty3_appl g c3 x0 u (H1 i u0 c3 x0 (fsubst0_both i u0 c w x0 H10 c3 H6) e H7)
+x1 t0 (H3 i u0 c3 x1 (fsubst0_both i u0 c v x1 H11 c3 H6) e H7)) (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 c3 (THead (Flat Appl) x0 (THead (Bind Abst) u t0)) (fsubst0_both i
+u0 c (THead (Flat Appl) w (THead (Bind Abst) u t0)) (THead (Flat Appl) x0
+(THead (Bind Abst) u t0)) (subst0_fst u0 x0 w i H10 (THead (Bind Abst) u t0)
+(Flat Appl)) c3 H6) e H7)))) (ty3_correct g c3 w u (H1 i u0 c3 w (fsubst0_fst
+i u0 c w c3 H6) e H7)))))))) (ty3_gen_bind g Abst c3 u t0 x H12))))
+(ty3_correct g c3 v (THead (Bind Abst) u t0) (H3 i u0 c3 v (fsubst0_fst i u0
+c v c3 H6) e H7))) t3 H9)))))) H8)) (subst0_gen_head (Flat Appl) u0 w v t3 i
+H5)))))))) c2 t2 H4))))))))))))))) (\lambda (c: C).(\lambda (t2: T).(\lambda
+(t3: T).(\lambda (H0: (ty3 g c t2 t3)).(\lambda (H1: ((\forall (i:
+nat).(\forall (u: T).(\forall (c2: C).(\forall (t4: T).((fsubst0 i u c t2 c2
+t4) \to (\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) \to (ty3 g c2 t4
t3)))))))))).(\lambda (t0: T).(\lambda (H2: (ty3 g c t3 t0)).(\lambda (H3:
((\forall (i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t4:
T).((fsubst0 i u c t3 c2 t4) \to (\forall (e: C).((getl i c (CHead e (Bind
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/ty3/fwd".
include "ty3/defs.ma".
\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 (t0: T).((eq T t0 (TSort n)) \to (pc3 c0
-(TSort (next g n)) t1))) H4 (TSort n) H7) in (let H9 \def (eq_ind T u
-(\lambda (t0: T).(ty3 g c0 t0 t1)) H3 (TSort n) H7) in (pc3_t t1 c0 (TSort
-(next g n)) (H8 (refl_equal T (TSort n))) t2 H5))))))))))))))) (\lambda (c0:
-C).(\lambda (m: nat).(\lambda (H1: (eq T (TSort m) (TSort n))).(let H2 \def
-(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with
-[(TSort n0) \Rightarrow n0 | (TLRef _) \Rightarrow m | (THead _ _ _)
-\Rightarrow m])) (TSort m) (TSort n) H1) in (eq_ind_r nat n (\lambda (n0:
-nat).(pc3 c0 (TSort (next g n)) (TSort (next g n0)))) (pc3_refl c0 (TSort
-(next g n))) m H2))))) (\lambda (n0: nat).(\lambda (c0: C).(\lambda (d:
-C).(\lambda (u: T).(\lambda (_: (getl n0 c0 (CHead d (Bind Abbr)
-u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (_: (((eq T u
-(TSort n)) \to (pc3 d (TSort (next g n)) t)))).(\lambda (H4: (eq T (TLRef n0)
-(TSort n))).(let H5 \def (eq_ind T (TLRef n0) (\lambda (ee: T).(match ee in T
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (TSort n) H4) in
-(False_ind (pc3 c0 (TSort (next g n)) (lift (S n0) O t)) H5)))))))))))
-(\lambda (n0: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda
-(_: (getl n0 c0 (CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (_: (ty3 g
-d u t)).(\lambda (_: (((eq T u (TSort n)) \to (pc3 d (TSort (next g n))
-t)))).(\lambda (H4: (eq T (TLRef n0) (TSort n))).(let H5 \def (eq_ind T
-(TLRef n0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _)
-\Rightarrow False])) I (TSort n) H4) in (False_ind (pc3 c0 (TSort (next g n))
-(lift (S n0) O u)) H5))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda
-(t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u (TSort n)) \to
-(pc3 c0 (TSort (next g n)) t)))).(\lambda (b: B).(\lambda (t1: T).(\lambda
-(t2: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t1 t2)).(\lambda (_: (((eq
-T t1 (TSort n)) \to (pc3 (CHead c0 (Bind b) u) (TSort (next g n))
-t2)))).(\lambda (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 in T
+x)) (\lambda (_: T).(pc3 c (TSort (next g n)) x)) (\lambda (y: T).(\lambda
+(H0: (ty3 g c y x)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0:
+T).((eq T t (TSort n)) \to (pc3 c0 (TSort (next g n)) t0))))) (\lambda (c0:
+C).(\lambda (t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda
+(_: (((eq T t2 (TSort n)) \to (pc3 c0 (TSort (next g n)) t)))).(\lambda (u:
+T).(\lambda (t1: T).(\lambda (H3: (ty3 g c0 u t1)).(\lambda (H4: (((eq T u
+(TSort n)) \to (pc3 c0 (TSort (next g n)) t1)))).(\lambda (H5: (pc3 c0 t1
+t2)).(\lambda (H6: (eq T u (TSort n))).(let H7 \def (f_equal T T (\lambda (e:
+T).e) u (TSort n) H6) in (let H8 \def (eq_ind T u (\lambda (t0: T).((eq T t0
+(TSort n)) \to (pc3 c0 (TSort (next g n)) t1))) H4 (TSort n) H7) in (let H9
+\def (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t1)) H3 (TSort n) H7) in
+(pc3_t t1 c0 (TSort (next g n)) (H8 (refl_equal T (TSort n))) t2
+H5))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq T
+(TSort m) (TSort n))).(let H2 \def (f_equal T nat (\lambda (e: T).(match e in
+T return (\lambda (_: T).nat) with [(TSort n0) \Rightarrow n0 | (TLRef _)
+\Rightarrow m | (THead _ _ _) \Rightarrow m])) (TSort m) (TSort n) H1) in
+(eq_ind_r nat n (\lambda (n0: nat).(pc3 c0 (TSort (next g n)) (TSort (next g
+n0)))) (pc3_refl c0 (TSort (next g n))) m H2))))) (\lambda (n0: nat).(\lambda
+(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (_: (getl n0 c0 (CHead d
+(Bind Abbr) u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (_:
+(((eq T u (TSort n)) \to (pc3 d (TSort (next g n)) t)))).(\lambda (H4: (eq T
+(TLRef n0) (TSort n))).(let H5 \def (eq_ind T (TLRef n0) (\lambda (ee:
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I
+(TSort n) H4) in (False_ind (pc3 c0 (TSort (next g n)) (lift (S n0) O t))
+H5))))))))))) (\lambda (n0: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda
+(u: T).(\lambda (_: (getl n0 c0 (CHead d (Bind Abst) u))).(\lambda (t:
+T).(\lambda (_: (ty3 g d u t)).(\lambda (_: (((eq T u (TSort n)) \to (pc3 d
+(TSort (next g n)) t)))).(\lambda (H4: (eq T (TLRef n0) (TSort n))).(let H5
+\def (eq_ind T (TLRef n0) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
+(THead _ _ _) \Rightarrow False])) I (TSort n) H4) in (False_ind (pc3 c0
+(TSort (next g n)) (lift (S n0) O u)) H5))))))))))) (\lambda (c0: C).(\lambda
+(u: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u
+(TSort n)) \to (pc3 c0 (TSort (next g n)) t)))).(\lambda (b: B).(\lambda (t1:
+T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t1
+t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (pc3 (CHead c0 (Bind b) u) (TSort
+(next g n)) t2)))).(\lambda (H5: (eq T (THead (Bind b) u t1) (TSort n))).(let
+H6 \def (eq_ind T (THead (Bind b) u t1) (\lambda (ee: T).(match ee in T
return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) 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 in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
-True])) I (TSort n) H5) in (False_ind (pc3 c0 (TSort (next g n)) (THead (Flat
-Appl) w (THead (Bind Abst) u t))) H6)))))))))))) (\lambda (c0: C).(\lambda
-(t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t1 t2)).(\lambda (_: (((eq T
-t1 (TSort n)) \to (pc3 c0 (TSort (next g n)) t2)))).(\lambda (t0: T).(\lambda
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H5) in
+(False_ind (pc3 c0 (TSort (next g n)) (THead (Bind b) u t2)) H6)))))))))))))
+(\lambda (c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w
+u)).(\lambda (_: (((eq T w (TSort n)) \to (pc3 c0 (TSort (next g n))
+u)))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead (Bind
+Abst) u t))).(\lambda (_: (((eq T v (TSort n)) \to (pc3 c0 (TSort (next g n))
+(THead (Bind Abst) u t))))).(\lambda (H5: (eq T (THead (Flat Appl) w v)
+(TSort n))).(let H6 \def (eq_ind T (THead (Flat Appl) w v) (\lambda (ee:
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I
+(TSort n) H5) in (False_ind (pc3 c0 (TSort (next g n)) (THead (Flat Appl) w
+(THead (Bind Abst) u t))) H6)))))))))))) (\lambda (c0: C).(\lambda (t1:
+T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t1 t2)).(\lambda (_: (((eq T t1
+(TSort n)) \to (pc3 c0 (TSort (next g n)) t2)))).(\lambda (t0: T).(\lambda
(_: (ty3 g c0 t2 t0)).(\lambda (_: (((eq T t2 (TSort n)) \to (pc3 c0 (TSort
(next g n)) t0)))).(\lambda (H5: (eq T (THead (Flat Cast) t2 t1) (TSort
n))).(let H6 \def (eq_ind T (THead (Flat Cast) t2 t1) (\lambda (ee: T).(match
\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
+x)) (\lambda (_: T).(or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda
+(t0: T).(pc3 c (lift (S n) O t0) x)))) (\lambda (e: C).(\lambda (u:
+T).(\lambda (_: T).(getl n c (CHead e (Bind Abbr) u))))) (\lambda (e:
+C).(\lambda (u: T).(\lambda (t0: T).(ty3 g e u t0))))) (ex3_3 C T T (\lambda
+(_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c (lift (S n) O u) x)))) (\lambda
+(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead e (Bind Abst) u)))))
+(\lambda (e: C).(\lambda (u: T).(\lambda (t0: T).(ty3 g e u t0)))))))
+(\lambda (y: T).(\lambda (H0: (ty3 g c y x)).(ty3_ind g (\lambda (c0:
+C).(\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n)) \to (or (ex3_3 C T T
+(\lambda (_: C).(\lambda (_: T).(\lambda (t1: T).(pc3 c0 (lift (S n) O t1)
+t0)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e
+(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t1: T).(ty3 g e
+u t1))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3
+c0 (lift (S n) O u) t0)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_:
+T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u:
+T).(\lambda (t1: T).(ty3 g e u t1)))))))))) (\lambda (c0: C).(\lambda (t2:
+T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: (((eq T t2
+(TLRef n)) \to (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0:
+T).(pc3 c0 (lift (S n) O t0) t)))) (\lambda (e: C).(\lambda (u: T).(\lambda
(_: T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u:
-T).(\lambda (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:
+T).(\lambda (t0: T).(ty3 g e u t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda
+(u: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u) t)))) (\lambda (e:
C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u)))))
-(\lambda (e: C).(\lambda (u: T).(\lambda (t1: T).(ty3 g e u t1))))))))))
-(\lambda (c0: C).(\lambda (t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2
-t)).(\lambda (_: (((eq T t2 (TLRef n)) \to (or (ex3_3 C T T (\lambda (_:
-C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t)))) (\lambda
-(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u)))))
-(\lambda (e: C).(\lambda (u: T).(\lambda (t0: T).(ty3 g e u t0))))) (ex3_3 C
-T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u)
-t)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e
-(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t0: T).(ty3 g e
-u t0))))))))).(\lambda (u: T).(\lambda (t1: T).(\lambda (H3: (ty3 g c0 u
+(\lambda (e: C).(\lambda (u: T).(\lambda (t0: T).(ty3 g e u
+t0))))))))).(\lambda (u: T).(\lambda (t1: T).(\lambda (H3: (ty3 g c0 u
t1)).(\lambda (H4: (((eq T u (TLRef n)) \to (or (ex3_3 C T T (\lambda (_:
C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t1)))) (\lambda
(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr)
C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 (CHead c0 (Bind
b) u) (lift (S n) O u0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_:
T).(getl n (CHead c0 (Bind b) u) (CHead e (Bind Abst) u0))))) (\lambda (e:
-C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))))))).(\lambda (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 (t3:
-T).(pc3 (CHead c0 (Bind b) u) (lift (S n) O t3) 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 (u0: T).(\lambda (t3: T).(ty3 g
-e u0 t3))))) (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 (u0: T).(\lambda (t3: T).(ty3 g
-e u0 t3))))))))).(\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 in T
-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
-(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e:
-C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t))))) (ex3_3 C T T (\lambda
-(_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) u))))
-(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind
-Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0
-t))))))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead
-(Bind Abst) u t))).(\lambda (_: (((eq T v (TLRef n)) \to (or (ex3_3 C T T
-(\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0)
-(THead (Bind Abst) u t))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_:
-T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0:
-T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda
-(u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) (THead (Bind Abst) u t)))))
+C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))))))).(\lambda (H5:
+(eq T (THead (Bind b) u t1) (TLRef n))).(let H6 \def (eq_ind T (THead (Bind
+b) u t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
+\Rightarrow True])) I (TLRef n) H5) in (False_ind (or (ex3_3 C T T (\lambda
+(_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) (THead
+(Bind b) u t2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n
+c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda
+(t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0:
+T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) (THead (Bind b) u t2)))))
(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind
Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0
+t0)))))) H6))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u:
+T).(\lambda (_: (ty3 g c0 w u)).(\lambda (_: (((eq T w (TLRef n)) \to (or
+(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c0 (lift (S
+n) O t) u)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0
+(CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t:
+T).(ty3 g e u0 t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda
+(_: T).(pc3 c0 (lift (S n) O u0) u)))) (\lambda (e: C).(\lambda (u0:
+T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e:
+C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t))))))))).(\lambda (v:
+T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead (Bind Abst) u
+t))).(\lambda (_: (((eq T v (TLRef n)) \to (or (ex3_3 C T T (\lambda (_:
+C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) (THead (Bind
+Abst) u t))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0
+(CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0:
+T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda
+(_: T).(pc3 c0 (lift (S n) O u0) (THead (Bind Abst) u t))))) (\lambda (e:
+C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0)))))
+(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0
t0))))))))).(\lambda (H5: (eq T (THead (Flat Appl) w v) (TLRef n))).(let H6
\def (eq_ind T (THead (Flat Appl) w v) (\lambda (ee: T).(match ee in T return
(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
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)))))))))))
+T).(\forall (x: T).((ty3 g c (THead (Bind b) u t1) x) \to (ex3_2 T T (\lambda
+(t2: T).(\lambda (_: T).(pc3 c (THead (Bind b) u t2) x))) (\lambda (_:
+T).(\lambda (t: T).(ty3 g c u t))) (\lambda (t2: T).(\lambda (_: T).(ty3 g
+(CHead c (Bind b) u) t1 t2))))))))))
\def
\lambda (g: G).(\lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (t1:
T).(\lambda (x: T).(\lambda (H: (ty3 g c (THead (Bind b) u t1) x)).(insert_eq
-T (THead (Bind b) u t1) (\lambda (t: T).(ty3 g c t x)) (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 (t3: T).(\lambda (_:
-T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t3) t)))) (\lambda (_:
-T).(\lambda (t0: T).(\lambda (_: T).(ty3 g c0 u t0)))) (\lambda (t3:
-T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3))))
-(\lambda (t3: T).(\lambda (_: T).(\lambda (t4: T).(ty3 g (CHead c0 (Bind b)
-u) t3 t4)))))))).(\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 (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)))))))).(\lambda (H5: (pc3 c0 t0 t2)).(\lambda
-(H6: (eq T u0 (THead (Bind b) u t1))).(let H7 \def (f_equal T T (\lambda (e:
-T).e) u0 (THead (Bind b) u t1) H6) in (let H8 \def (eq_ind T u0 (\lambda (t3:
-T).((eq T t3 (THead (Bind b) u t1)) \to (ex4_3 T T T (\lambda (t4:
-T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t4) t0))))
-(\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))))))) H4 (THead (Bind b) u t1) H7) in (let H9 \def (eq_ind
-T u0 (\lambda (t3: T).(ty3 g c0 t3 t0)) H3 (THead (Bind b) u t1) H7) in (let
-H10 \def (H8 (refl_equal T (THead (Bind b) u t1))) in (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 in T return (\lambda (_: T).Prop) with [(TSort _)
+T (THead (Bind b) u t1) (\lambda (t: T).(ty3 g c t x)) (\lambda (_: T).(ex3_2
+T T (\lambda (t2: T).(\lambda (_: T).(pc3 c (THead (Bind b) u t2) x)))
+(\lambda (_: T).(\lambda (t0: T).(ty3 g c u t0))) (\lambda (t2: T).(\lambda
+(_: T).(ty3 g (CHead c (Bind b) u) t1 t2))))) (\lambda (y: T).(\lambda (H0:
+(ty3 g c y x)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0:
+T).((eq T t (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda (t2: T).(\lambda
+(_: T).(pc3 c0 (THead (Bind b) u t2) t0))) (\lambda (_: T).(\lambda (t3:
+T).(ty3 g c0 u t3))) (\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind
+b) u) t1 t2)))))))) (\lambda (c0: C).(\lambda (t2: T).(\lambda (t:
+T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: (((eq T t2 (THead (Bind b) u
+t1)) \to (ex3_2 T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0 (THead (Bind b)
+u t3) t))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u t0))) (\lambda (t3:
+T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3))))))).(\lambda (u0:
+T).(\lambda (t0: T).(\lambda (H3: (ty3 g c0 u0 t0)).(\lambda (H4: (((eq T u0
+(THead (Bind b) u t1)) \to (ex3_2 T T (\lambda (t3: T).(\lambda (_: T).(pc3
+c0 (THead (Bind b) u t3) t0))) (\lambda (_: T).(\lambda (t4: T).(ty3 g c0 u
+t4))) (\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1
+t3))))))).(\lambda (H5: (pc3 c0 t0 t2)).(\lambda (H6: (eq T u0 (THead (Bind
+b) u t1))).(let H7 \def (f_equal T T (\lambda (e: T).e) u0 (THead (Bind b) u
+t1) H6) in (let H8 \def (eq_ind T u0 (\lambda (t3: T).((eq T t3 (THead (Bind
+b) u t1)) \to (ex3_2 T T (\lambda (t4: T).(\lambda (_: T).(pc3 c0 (THead
+(Bind b) u t4) t0))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c0 u t5)))
+(\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t4)))))) H4
+(THead (Bind b) u t1) H7) in (let H9 \def (eq_ind T u0 (\lambda (t3: T).(ty3
+g c0 t3 t0)) H3 (THead (Bind b) u t1) H7) in (let H10 \def (H8 (refl_equal T
+(THead (Bind b) u t1))) in (ex3_2_ind T T (\lambda (t3: T).(\lambda (_:
+T).(pc3 c0 (THead (Bind b) u t3) t0))) (\lambda (_: T).(\lambda (t4: T).(ty3
+g c0 u t4))) (\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1
+t3))) (ex3_2 T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u
+t3) t2))) (\lambda (_: T).(\lambda (t4: T).(ty3 g c0 u t4))) (\lambda (t3:
+T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3)))) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H11: (pc3 c0 (THead (Bind b) u x0)
+t0)).(\lambda (H12: (ty3 g c0 u x1)).(\lambda (H13: (ty3 g (CHead c0 (Bind b)
+u) t1 x0)).(ex3_2_intro T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0 (THead
+(Bind b) u t3) t2))) (\lambda (_: T).(\lambda (t4: T).(ty3 g c0 u t4)))
+(\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3))) x0 x1
+(pc3_t t0 c0 (THead (Bind b) u x0) H11 t2 H5) H12 H13))))))
+H10)))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq T
+(TSort m) (THead (Bind b) u t1))).(let H2 \def (eq_ind T (TSort m) (\lambda
+(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
-False])) I (THead (Bind b) u t1) H1) in (False_ind (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)
+False])) I (THead (Bind b) u t1) H1) in (False_ind (ex3_2 T T (\lambda (t2:
+T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2) (TSort (next g m)))))
+(\lambda (_: T).(\lambda (t: T).(ty3 g c0 u t))) (\lambda (t2: T).(\lambda
+(_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2)))) H2))))) (\lambda (n:
+nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (_: (getl n
+c0 (CHead d (Bind Abbr) u0))).(\lambda (t: T).(\lambda (_: (ty3 g d u0
+t)).(\lambda (_: (((eq T u0 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda
+(t2: T).(\lambda (_: T).(pc3 d (THead (Bind b) u t2) t))) (\lambda (_:
+T).(\lambda (t0: T).(ty3 g d u t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g
+(CHead d (Bind b) u) t1 t2))))))).(\lambda (H4: (eq T (TLRef n) (THead (Bind
+b) u t1))).(let H5 \def (eq_ind T (TLRef n) (\lambda (ee: T).(match ee in T
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t1)
+H4) in (False_ind (ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 c0 (THead
+(Bind b) u t2) (lift (S n) O t)))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c0
+u t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1
+t2)))) H5))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d:
+C).(\lambda (u0: T).(\lambda (_: (getl n c0 (CHead d (Bind Abst)
u0))).(\lambda (t: T).(\lambda (_: (ty3 g d u0 t)).(\lambda (_: (((eq T u0
-(THead (Bind b) u t1)) \to (ex4_3 T T T (\lambda (t2: T).(\lambda (_:
-T).(\lambda (_: T).(pc3 d (THead (Bind b) u t2) t)))) (\lambda (_:
-T).(\lambda (t0: T).(\lambda (_: T).(ty3 g d u t0)))) (\lambda (t2:
-T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead d (Bind b) u) t1 t2))))
-(\lambda (t2: T).(\lambda (_: T).(\lambda (t3: T).(ty3 g (CHead d (Bind b) u)
-t2 t3)))))))).(\lambda (H4: (eq T (TLRef n) (THead (Bind b) u t1))).(let H5
-\def (eq_ind T (TLRef n) (\lambda (ee: T).(match ee in T return (\lambda (_:
+(THead (Bind b) u t1)) \to (ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 d
+(THead (Bind b) u t2) t))) (\lambda (_: T).(\lambda (t0: T).(ty3 g d u t0)))
+(\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead d (Bind b) u) t1
+t2))))))).(\lambda (H4: (eq T (TLRef n) (THead (Bind b) u t1))).(let H5 \def
+(eq_ind T (TLRef n) (\lambda (ee: T).(match ee in T return (\lambda (_:
T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
(THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t1) H4) in (False_ind
-(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 (t0: T).(\lambda (_: T).(ty3 g d u t0))))
-(\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead d (Bind b) u)
-t1 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t3: T).(ty3 g (CHead d
-(Bind b) u) t2 t3)))))))).(\lambda (H4: (eq T (TLRef n) (THead (Bind b) u
-t1))).(let H5 \def (eq_ind T (TLRef n) (\lambda (ee: T).(match ee in T return
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t1)
-H4) in (False_ind (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 (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)))))))).(\lambda (b0: B).(\lambda
+(ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2)
+(lift (S n) O u0)))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u t0)))
+(\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2))))
+H5))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (t: T).(\lambda (H1:
+(ty3 g c0 u0 t)).(\lambda (H2: (((eq T u0 (THead (Bind b) u t1)) \to (ex3_2 T
+T (\lambda (t2: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2) t)))
+(\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u t0))) (\lambda (t2: T).(\lambda
+(_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2))))))).(\lambda (b0: B).(\lambda
(t0: T).(\lambda (t2: T).(\lambda (H3: (ty3 g (CHead c0 (Bind b0) u0) t0
-t2)).(\lambda (H4: (((eq T t0 (THead (Bind b) u t1)) \to (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 (t4: T).(\lambda (_:
-T).(ty3 g (CHead c0 (Bind b0) u0) u t4)))) (\lambda (t3: T).(\lambda (_:
-T).(\lambda (_: T).(ty3 g (CHead (CHead c0 (Bind b0) u0) (Bind b) u) t1
-t3)))) (\lambda (t3: T).(\lambda (_: T).(\lambda (t5: T).(ty3 g (CHead (CHead
-c0 (Bind b0) u0) (Bind b) u) t3 t5)))))))).(\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 (t4: T).(\lambda (_: T).(\lambda (_:
-T).(pc3 (CHead c0 (Bind b0) u0) (THead (Bind b) u t4) t3)))) (\lambda (_:
-T).(\lambda (t5: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b0) u0) u t5))))
-(\lambda (t4: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead (CHead c0
-(Bind b0) u0) (Bind b) u) t1 t4)))) (\lambda (t4: T).(\lambda (_: T).(\lambda
-(t6: T).(ty3 g (CHead (CHead c0 (Bind b0) u0) (Bind b) u) t4
-t6)))))))).(\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 in T return (\lambda
-(_: T).B) with [(TSort _) \Rightarrow b0 | (TLRef _) \Rightarrow b0 | (THead
-k _ _) \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b1)
-\Rightarrow b1 | (Flat _) \Rightarrow b0])])) (THead (Bind b0) u0 t0) (THead
-(Bind b) u t1) H7) in ((let H9 \def (f_equal T T (\lambda (e: T).(match e in
-T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _)
-\Rightarrow u0 | (THead _ t4 _) \Rightarrow t4])) (THead (Bind b0) u0 t0)
-(THead (Bind b) u t1) H7) in ((let H10 \def (f_equal T T (\lambda (e:
+t2)).(\lambda (H4: (((eq T t0 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda
+(t3: T).(\lambda (_: T).(pc3 (CHead c0 (Bind b0) u0) (THead (Bind b) u t3)
+t2))) (\lambda (_: T).(\lambda (t4: T).(ty3 g (CHead c0 (Bind b0) u0) u t4)))
+(\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead (CHead c0 (Bind b0) u0) (Bind
+b) u) t1 t3))))))).(\lambda (H5: (eq T (THead (Bind b0) u0 t0) (THead (Bind
+b) u t1))).(let H6 \def (f_equal T B (\lambda (e: T).(match e in T return
+(\lambda (_: T).B) with [(TSort _) \Rightarrow b0 | (TLRef _) \Rightarrow b0
+| (THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).B) with
+[(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b0])])) (THead (Bind b0) u0
+t0) (THead (Bind b) u t1) H5) in ((let H7 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 |
+(TLRef _) \Rightarrow u0 | (THead _ t3 _) \Rightarrow t3])) (THead (Bind b0)
+u0 t0) (THead (Bind b) u t1) H5) in ((let H8 \def (f_equal T T (\lambda (e:
T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 |
-(TLRef _) \Rightarrow t0 | (THead _ _ t4) \Rightarrow t4])) (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 (t4: T).((eq T t4
-(THead (Bind b) u t1)) \to (ex4_3 T T T (\lambda (t5: T).(\lambda (_:
-T).(\lambda (_: T).(pc3 (CHead c0 (Bind b0) u0) (THead (Bind b) u t5) t2))))
-(\lambda (_: T).(\lambda (t6: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b0)
-u0) u t6)))) (\lambda (t5: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead
-(CHead c0 (Bind b0) u0) (Bind b) u) t1 t5)))) (\lambda (t5: T).(\lambda (_:
-T).(\lambda (t7: T).(ty3 g (CHead (CHead c0 (Bind b0) u0) (Bind b) u) t5
-t7))))))) H4 t1 H10) in (let H14 \def (eq_ind T t0 (\lambda (t4: T).(ty3 g
-(CHead c0 (Bind b0) u0) t4 t2)) H3 t1 H10) in (let H15 \def (eq_ind B b0
-(\lambda (b1: B).((eq T t2 (THead (Bind b) u t1)) \to (ex4_3 T T T (\lambda
-(t4: T).(\lambda (_: T).(\lambda (_: T).(pc3 (CHead c0 (Bind b1) u0) (THead
-(Bind b) u t4) t3)))) (\lambda (_: T).(\lambda (t5: T).(\lambda (_: T).(ty3 g
-(CHead c0 (Bind b1) u0) u t5)))) (\lambda (t4: T).(\lambda (_: T).(\lambda
-(_: T).(ty3 g (CHead (CHead c0 (Bind b1) u0) (Bind b) u) t1 t4)))) (\lambda
-(t4: T).(\lambda (_: T).(\lambda (t6: T).(ty3 g (CHead (CHead c0 (Bind b1)
-u0) (Bind b) u) t4 t6))))))) H6 b H12) in (let H16 \def (eq_ind B b0 (\lambda
-(b1: B).(ty3 g (CHead c0 (Bind b1) u0) t2 t3)) H5 b H12) in (let H17 \def
-(eq_ind B b0 (\lambda (b1: B).((eq T t1 (THead (Bind b) u t1)) \to (ex4_3 T T
-T (\lambda (t4: T).(\lambda (_: T).(\lambda (_: T).(pc3 (CHead c0 (Bind b1)
-u0) (THead (Bind b) u t4) t2)))) (\lambda (_: T).(\lambda (t5: T).(\lambda
-(_: T).(ty3 g (CHead c0 (Bind b1) u0) u t5)))) (\lambda (t4: T).(\lambda (_:
-T).(\lambda (_: T).(ty3 g (CHead (CHead c0 (Bind b1) u0) (Bind b) u) t1
-t4)))) (\lambda (t4: T).(\lambda (_: T).(\lambda (t6: T).(ty3 g (CHead (CHead
-c0 (Bind b1) u0) (Bind b) u) t4 t6))))))) H13 b H12) in (let H18 \def (eq_ind
-B b0 (\lambda (b1: B).(ty3 g (CHead c0 (Bind b1) 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 (t4: T).((eq T t2
-(THead (Bind b) u t1)) \to (ex4_3 T T T (\lambda (t5: T).(\lambda (_:
-T).(\lambda (_: T).(pc3 (CHead c0 (Bind b) t4) (THead (Bind b) u t5) t3))))
-(\lambda (_: T).(\lambda (t6: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b)
-t4) u t6)))) (\lambda (t5: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead
-(CHead c0 (Bind b) t4) (Bind b) u) t1 t5)))) (\lambda (t5: T).(\lambda (_:
-T).(\lambda (t7: T).(ty3 g (CHead (CHead c0 (Bind b) t4) (Bind b) u) t5
-t7))))))) H15 u H11) in (let H20 \def (eq_ind T u0 (\lambda (t4: T).(ty3 g
-(CHead c0 (Bind b) t4) t2 t3)) H16 u H11) in (let H21 \def (eq_ind T u0
-(\lambda (t4: T).((eq T t1 (THead (Bind b) u t1)) \to (ex4_3 T T T (\lambda
-(t5: T).(\lambda (_: T).(\lambda (_: T).(pc3 (CHead c0 (Bind b) t4) (THead
-(Bind b) u t5) t2)))) (\lambda (_: T).(\lambda (t6: T).(\lambda (_: T).(ty3 g
-(CHead c0 (Bind b) t4) u t6)))) (\lambda (t5: T).(\lambda (_: T).(\lambda (_:
-T).(ty3 g (CHead (CHead c0 (Bind b) t4) (Bind b) u) t1 t5)))) (\lambda (t5:
-T).(\lambda (_: T).(\lambda (t7: T).(ty3 g (CHead (CHead c0 (Bind b) t4)
-(Bind b) u) t5 t7))))))) H17 u H11) in (let H22 \def (eq_ind T u0 (\lambda
-(t4: T).(ty3 g (CHead c0 (Bind b) t4) t1 t2)) H18 u H11) in (let H23 \def
-(eq_ind T u0 (\lambda (t4: T).((eq T t4 (THead (Bind b) u t1)) \to (ex4_3 T T
-T (\lambda (t5: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u
-t5) t)))) (\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))))))) H2 u H11) in (let H24 \def (eq_ind T u0 (\lambda
-(t4: T).(ty3 g c0 t4 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 (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)))))))).(\lambda (H5: (eq T (THead (Flat Appl) w v) (THead (Bind b)
-u t1))).(let H6 \def (eq_ind T (THead (Flat Appl) w v) (\lambda (ee:
-T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K
-return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _)
-\Rightarrow True])])) I (THead (Bind b) u t1) H5) in (False_ind (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 (t3: T).(\lambda (_: T).(\lambda (_: T).(ty3 g
-(CHead c0 (Bind b) u) t1 t3)))) (\lambda (t3: T).(\lambda (_: T).(\lambda
-(t4: T).(ty3 g (CHead c0 (Bind b) u) t3 t4)))))))).(\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 (t4: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead
-(Bind b) u t4) t3)))) (\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)))))))).(\lambda (H5: (eq T (THead (Flat Cast)
-t2 t0) (THead (Bind b) u t1))).(let H6 \def (eq_ind T (THead (Flat Cast) t2
-t0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
-_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
-\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u t1)
-H5) in (False_ind (ex4_3 T T T (\lambda (t4: T).(\lambda (_: T).(\lambda (_:
-T).(pc3 c0 (THead (Bind b) u t4) (THead (Flat Cast) t3 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))))))).
+(TLRef _) \Rightarrow t0 | (THead _ _ t3) \Rightarrow t3])) (THead (Bind b0)
+u0 t0) (THead (Bind b) u t1) H5) in (\lambda (H9: (eq T u0 u)).(\lambda (H10:
+(eq B b0 b)).(let H11 \def (eq_ind T t0 (\lambda (t3: T).((eq T t3 (THead
+(Bind b) u t1)) \to (ex3_2 T T (\lambda (t4: T).(\lambda (_: T).(pc3 (CHead
+c0 (Bind b0) u0) (THead (Bind b) u t4) t2))) (\lambda (_: T).(\lambda (t5:
+T).(ty3 g (CHead c0 (Bind b0) u0) u t5))) (\lambda (t4: T).(\lambda (_:
+T).(ty3 g (CHead (CHead c0 (Bind b0) u0) (Bind b) u) t1 t4)))))) H4 t1 H8) in
+(let H12 \def (eq_ind T t0 (\lambda (t3: T).(ty3 g (CHead c0 (Bind b0) u0) t3
+t2)) H3 t1 H8) in (let H13 \def (eq_ind B b0 (\lambda (b1: B).((eq T t1
+(THead (Bind b) u t1)) \to (ex3_2 T T (\lambda (t3: T).(\lambda (_: T).(pc3
+(CHead c0 (Bind b1) u0) (THead (Bind b) u t3) t2))) (\lambda (_: T).(\lambda
+(t4: T).(ty3 g (CHead c0 (Bind b1) u0) u t4))) (\lambda (t3: T).(\lambda (_:
+T).(ty3 g (CHead (CHead c0 (Bind b1) u0) (Bind b) u) t1 t3)))))) H11 b H10)
+in (let H14 \def (eq_ind B b0 (\lambda (b1: B).(ty3 g (CHead c0 (Bind b1) u0)
+t1 t2)) H12 b H10) in (eq_ind_r B b (\lambda (b1: B).(ex3_2 T T (\lambda (t3:
+T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t3) (THead (Bind b1) u0 t2))))
+(\lambda (_: T).(\lambda (t4: T).(ty3 g c0 u t4))) (\lambda (t3: T).(\lambda
+(_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3))))) (let H15 \def (eq_ind T u0
+(\lambda (t3: T).((eq T t1 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda
+(t4: T).(\lambda (_: T).(pc3 (CHead c0 (Bind b) t3) (THead (Bind b) u t4)
+t2))) (\lambda (_: T).(\lambda (t5: T).(ty3 g (CHead c0 (Bind b) t3) u t5)))
+(\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead (CHead c0 (Bind b) t3) (Bind
+b) u) t1 t4)))))) H13 u H9) in (let H16 \def (eq_ind T u0 (\lambda (t3:
+T).(ty3 g (CHead c0 (Bind b) t3) t1 t2)) H14 u H9) in (let H17 \def (eq_ind T
+u0 (\lambda (t3: T).((eq T t3 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda
+(t4: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t4) t))) (\lambda (_:
+T).(\lambda (t5: T).(ty3 g c0 u t5))) (\lambda (t4: T).(\lambda (_: T).(ty3 g
+(CHead c0 (Bind b) u) t1 t4)))))) H2 u H9) in (let H18 \def (eq_ind T u0
+(\lambda (t3: T).(ty3 g c0 t3 t)) H1 u H9) in (eq_ind_r T u (\lambda (t3:
+T).(ex3_2 T T (\lambda (t4: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t4)
+(THead (Bind b) t3 t2)))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c0 u t5)))
+(\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t4)))))
+(ex3_2_intro T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u
+t3) (THead (Bind b) u t2)))) (\lambda (_: T).(\lambda (t4: T).(ty3 g c0 u
+t4))) (\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3)))
+t2 t (pc3_refl c0 (THead (Bind b) u t2)) H18 H16) u0 H9))))) b0 H10))))))))
+H7)) H6))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u0:
+T).(\lambda (_: (ty3 g c0 w u0)).(\lambda (_: (((eq T w (THead (Bind b) u
+t1)) \to (ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 c0 (THead (Bind b)
+u t2) u0))) (\lambda (_: T).(\lambda (t: T).(ty3 g c0 u t))) (\lambda (t2:
+T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2))))))).(\lambda (v:
+T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead (Bind Abst) u0
+t))).(\lambda (_: (((eq T v (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda
+(t2: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2) (THead (Bind Abst) u0
+t)))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u t0))) (\lambda (t2:
+T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2))))))).(\lambda (H5:
+(eq T (THead (Flat Appl) w v) (THead (Bind b) u t1))).(let H6 \def (eq_ind T
+(THead (Flat Appl) w v) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
+[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind
+b) u t1) H5) in (False_ind (ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3
+c0 (THead (Bind b) u t2) (THead (Flat Appl) w (THead (Bind Abst) u0 t)))))
+(\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u t0))) (\lambda (t2: T).(\lambda
+(_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2)))) H6)))))))))))) (\lambda (c0:
+C).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t0 t2)).(\lambda
+(_: (((eq T t0 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda (t3:
+T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t3) t2))) (\lambda (_:
+T).(\lambda (t: T).(ty3 g c0 u t))) (\lambda (t3: T).(\lambda (_: T).(ty3 g
+(CHead c0 (Bind b) u) t1 t3))))))).(\lambda (t3: T).(\lambda (_: (ty3 g c0 t2
+t3)).(\lambda (_: (((eq T t2 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda
+(t4: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t4) t3))) (\lambda (_:
+T).(\lambda (t: T).(ty3 g c0 u t))) (\lambda (t4: T).(\lambda (_: T).(ty3 g
+(CHead c0 (Bind b) u) t1 t4))))))).(\lambda (H5: (eq T (THead (Flat Cast) t2
+t0) (THead (Bind b) u t1))).(let H6 \def (eq_ind T (THead (Flat Cast) t2 t0)
+(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
+(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False |
+(Flat _) \Rightarrow True])])) I (THead (Bind b) u t1) H5) in (False_ind
+(ex3_2 T T (\lambda (t4: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t4)
+(THead (Flat Cast) t3 t2)))) (\lambda (_: T).(\lambda (t: T).(ty3 g c0 u t)))
+(\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t4))))
+H6))))))))))) c y x H0))) H))))))).
theorem ty3_gen_appl:
\forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (v: T).(\forall (x:
\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 (t0: T).(ty3 g c0 v (THead (Bind Abst) u t0)))) (\lambda (u:
-T).(\lambda (_: T).(ty3 g c0 w u))))))).(\lambda (u: T).(\lambda (t1:
-T).(\lambda (H3: (ty3 g c0 u t1)).(\lambda (H4: (((eq T u (THead (Flat Appl)
-w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead (Flat
-Appl) w (THead (Bind Abst) u0 t0)) t1))) (\lambda (u0: T).(\lambda (t0:
+(Flat Appl) w v) (\lambda (t: T).(ty3 g c t x)) (\lambda (_: T).(ex3_2 T T
+(\lambda (u: T).(\lambda (t0: T).(pc3 c (THead (Flat Appl) w (THead (Bind
+Abst) u t0)) x))) (\lambda (u: T).(\lambda (t0: T).(ty3 g c v (THead (Bind
+Abst) u t0)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c w u))))) (\lambda (y:
+T).(\lambda (H0: (ty3 g c y x)).(ty3_ind g (\lambda (c0: C).(\lambda (t:
+T).(\lambda (t0: T).((eq T t (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda
+(u: T).(\lambda (t1: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u
+t1)) t0))) (\lambda (u: T).(\lambda (t1: T).(ty3 g c0 v (THead (Bind Abst) u
+t1)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u)))))))) (\lambda (c0:
+C).(\lambda (t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda
+(_: (((eq T t2 (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda (u:
+T).(\lambda (t0: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u t0))
+t))) (\lambda (u: T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u t0))))
+(\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u))))))).(\lambda (u: T).(\lambda
+(t1: T).(\lambda (H3: (ty3 g c0 u t1)).(\lambda (H4: (((eq T u (THead (Flat
+Appl) w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead
+(Flat Appl) w (THead (Bind Abst) u0 t0)) t1))) (\lambda (u0: T).(\lambda (t0:
T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_:
T).(ty3 g c0 w u0))))))).(\lambda (H5: (pc3 c0 t1 t2)).(\lambda (H6: (eq T u
(THead (Flat Appl) w v))).(let H7 \def (f_equal T T (\lambda (e: T).e) u
(u0: T).(\lambda (t0: T).(pc3 (CHead c0 (Bind b) u) (THead (Flat Appl) w
(THead (Bind Abst) u0 t0)) t2))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g
(CHead c0 (Bind b) u) v (THead (Bind Abst) u0 t0)))) (\lambda (u0:
-T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) w u0))))))).(\lambda (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 (t3:
-T).(pc3 (CHead c0 (Bind b) u) (THead (Flat Appl) w (THead (Bind Abst) u0 t3))
-t0))) (\lambda (u0: T).(\lambda (t3: T).(ty3 g (CHead c0 (Bind b) u) v (THead
-(Bind Abst) u0 t3)))) (\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 in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K
-return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _)
-\Rightarrow False])])) I (THead (Flat Appl) w v) 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 (u0: T).(\lambda (t: T).(ty3
-g c0 v (THead (Bind Abst) u0 t)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0
-w u0))))))).(\lambda (v0: T).(\lambda (t: T).(\lambda (H3: (ty3 g c0 v0
-(THead (Bind Abst) u t))).(\lambda (H4: (((eq T v0 (THead (Flat Appl) w v))
-\to (ex3_2 T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead (Flat Appl) w
-(THead (Bind Abst) u0 t0)) (THead (Bind Abst) u t)))) (\lambda (u0:
-T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0:
-T).(\lambda (_: T).(ty3 g c0 w u0))))))).(\lambda (H5: (eq T (THead (Flat
-Appl) w0 v0) (THead (Flat Appl) w v))).(let H6 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow w0 |
-(TLRef _) \Rightarrow w0 | (THead _ t0 _) \Rightarrow t0])) (THead (Flat
-Appl) w0 v0) (THead (Flat Appl) w v) H5) in ((let H7 \def (f_equal T T
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow v0 | (TLRef _) \Rightarrow v0 | (THead _ _ t0) \Rightarrow t0]))
-(THead (Flat Appl) w0 v0) (THead (Flat Appl) w v) H5) in (\lambda (H8: (eq T
-w0 w)).(let H9 \def (eq_ind T v0 (\lambda (t0: T).((eq T t0 (THead (Flat
-Appl) w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t1: T).(pc3 c0 (THead
-(Flat Appl) w (THead (Bind Abst) u0 t1)) (THead (Bind Abst) u t)))) (\lambda
-(u0: T).(\lambda (t1: T).(ty3 g c0 v (THead (Bind Abst) u0 t1)))) (\lambda
-(u0: T).(\lambda (_: T).(ty3 g c0 w u0)))))) H4 v H7) in (let H10 \def
-(eq_ind T v0 (\lambda (t0: T).(ty3 g c0 t0 (THead (Bind Abst) u t))) H3 v H7)
-in (let H11 \def (eq_ind T w0 (\lambda (t0: T).((eq T t0 (THead (Flat Appl) w
-v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t1: T).(pc3 c0 (THead (Flat
-Appl) w (THead (Bind Abst) u0 t1)) u))) (\lambda (u0: T).(\lambda (t1:
-T).(ty3 g c0 v (THead (Bind Abst) u0 t1)))) (\lambda (u0: T).(\lambda (_:
-T).(ty3 g c0 w u0)))))) H2 w H8) in (let H12 \def (eq_ind T w0 (\lambda (t0:
-T).(ty3 g c0 t0 u)) H1 w H8) in (eq_ind_r T w (\lambda (t0: T).(ex3_2 T T
-(\lambda (u0: T).(\lambda (t1: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind
-Abst) u0 t1)) (THead (Flat Appl) t0 (THead (Bind Abst) u t))))) (\lambda (u0:
+T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) w u0))))))).(\lambda (H5: (eq
+T (THead (Bind b) u t1) (THead (Flat Appl) w v))).(let H6 \def (eq_ind T
+(THead (Bind b) u t1) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
+[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat
+Appl) w v) H5) in (False_ind (ex3_2 T T (\lambda (u0: T).(\lambda (t0:
+T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) (THead (Bind b) u
+t2)))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0
+t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w u0)))) H6)))))))))))))
+(\lambda (c0: C).(\lambda (w0: T).(\lambda (u: T).(\lambda (H1: (ty3 g c0 w0
+u)).(\lambda (H2: (((eq T w0 (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda
+(u0: T).(\lambda (t: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0
+t)) u))) (\lambda (u0: T).(\lambda (t: T).(ty3 g c0 v (THead (Bind Abst) u0
+t)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w u0))))))).(\lambda (v0:
+T).(\lambda (t: T).(\lambda (H3: (ty3 g c0 v0 (THead (Bind Abst) u
+t))).(\lambda (H4: (((eq T v0 (THead (Flat Appl) w v)) \to (ex3_2 T T
+(\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind
+Abst) u0 t0)) (THead (Bind Abst) u t)))) (\lambda (u0: T).(\lambda (t0:
+T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_:
+T).(ty3 g c0 w u0))))))).(\lambda (H5: (eq T (THead (Flat Appl) w0 v0) (THead
+(Flat Appl) w v))).(let H6 \def (f_equal T T (\lambda (e: T).(match e in T
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow w0 | (TLRef _)
+\Rightarrow w0 | (THead _ t0 _) \Rightarrow t0])) (THead (Flat Appl) w0 v0)
+(THead (Flat Appl) w v) H5) in ((let H7 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v0 |
+(TLRef _) \Rightarrow v0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat
+Appl) w0 v0) (THead (Flat Appl) w v) H5) in (\lambda (H8: (eq T w0 w)).(let
+H9 \def (eq_ind T v0 (\lambda (t0: T).((eq T t0 (THead (Flat Appl) w v)) \to
+(ex3_2 T T (\lambda (u0: T).(\lambda (t1: T).(pc3 c0 (THead (Flat Appl) w
+(THead (Bind Abst) u0 t1)) (THead (Bind Abst) u t)))) (\lambda (u0:
T).(\lambda (t1: T).(ty3 g c0 v (THead (Bind Abst) u0 t1)))) (\lambda (u0:
-T).(\lambda (_: T).(ty3 g c0 w u0))))) (ex3_2_intro T T (\lambda (u0:
-T).(\lambda (t0: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t0))
-(THead (Flat Appl) w (THead (Bind Abst) u t))))) (\lambda (u0: T).(\lambda
-(t0: T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda
-(_: T).(ty3 g c0 w u0))) u t (pc3_refl c0 (THead (Flat Appl) w (THead (Bind
-Abst) u t))) H10 H12) w0 H8))))))) H6)))))))))))) (\lambda (c0: C).(\lambda
-(t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t1 t2)).(\lambda (_: (((eq T
-t1 (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda (u: T).(\lambda (t:
-T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u t)) t2))) (\lambda (u:
-T).(\lambda (t: T).(ty3 g c0 v (THead (Bind Abst) u t)))) (\lambda (u:
-T).(\lambda (_: T).(ty3 g c0 w u))))))).(\lambda (t0: T).(\lambda (_: (ty3 g
-c0 t2 t0)).(\lambda (_: (((eq T t2 (THead (Flat Appl) w v)) \to (ex3_2 T T
-(\lambda (u: T).(\lambda (t: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind
-Abst) u t)) t0))) (\lambda (u: T).(\lambda (t: T).(ty3 g c0 v (THead (Bind
-Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u))))))).(\lambda
-(H5: (eq T (THead (Flat Cast) t2 t1) (THead (Flat Appl) w v))).(let H6 \def
-(eq_ind T (THead (Flat Cast) t2 t1) (\lambda (ee: T).(match ee in T return
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
-(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f
-in F return (\lambda (_: F).Prop) with [Appl \Rightarrow False | Cast
-\Rightarrow True])])])) I (THead (Flat Appl) w v) H5) in (False_ind (ex3_2 T
-T (\lambda (u: T).(\lambda (t: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind
-Abst) u t)) (THead (Flat Cast) t0 t2)))) (\lambda (u: T).(\lambda (t: T).(ty3
-g c0 v (THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c0 w
-u)))) H6))))))))))) c y x H0))) H)))))).
+T).(\lambda (_: T).(ty3 g c0 w u0)))))) H4 v H7) in (let H10 \def (eq_ind T
+v0 (\lambda (t0: T).(ty3 g c0 t0 (THead (Bind Abst) u t))) H3 v H7) in (let
+H11 \def (eq_ind T w0 (\lambda (t0: T).((eq T t0 (THead (Flat Appl) w v)) \to
+(ex3_2 T T (\lambda (u0: T).(\lambda (t1: T).(pc3 c0 (THead (Flat Appl) w
+(THead (Bind Abst) u0 t1)) u))) (\lambda (u0: T).(\lambda (t1: T).(ty3 g c0 v
+(THead (Bind Abst) u0 t1)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w
+u0)))))) H2 w H8) in (let H12 \def (eq_ind T w0 (\lambda (t0: T).(ty3 g c0 t0
+u)) H1 w H8) in (eq_ind_r T w (\lambda (t0: T).(ex3_2 T T (\lambda (u0:
+T).(\lambda (t1: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t1))
+(THead (Flat Appl) t0 (THead (Bind Abst) u t))))) (\lambda (u0: T).(\lambda
+(t1: T).(ty3 g c0 v (THead (Bind Abst) u0 t1)))) (\lambda (u0: T).(\lambda
+(_: T).(ty3 g c0 w u0))))) (ex3_2_intro T T (\lambda (u0: T).(\lambda (t0:
+T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) (THead (Flat Appl)
+w (THead (Bind Abst) u t))))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g c0 v
+(THead (Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w
+u0))) u t (pc3_refl c0 (THead (Flat Appl) w (THead (Bind Abst) u t))) H10
+H12) w0 H8))))))) H6)))))))))))) (\lambda (c0: C).(\lambda (t1: T).(\lambda
+(t2: T).(\lambda (_: (ty3 g c0 t1 t2)).(\lambda (_: (((eq T t1 (THead (Flat
+Appl) w v)) \to (ex3_2 T T (\lambda (u: T).(\lambda (t: T).(pc3 c0 (THead
+(Flat Appl) w (THead (Bind Abst) u t)) t2))) (\lambda (u: T).(\lambda (t:
+T).(ty3 g c0 v (THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_:
+T).(ty3 g c0 w u))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t2
+t0)).(\lambda (_: (((eq T t2 (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda
+(u: T).(\lambda (t: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u t))
+t0))) (\lambda (u: T).(\lambda (t: T).(ty3 g c0 v (THead (Bind Abst) u t))))
+(\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u))))))).(\lambda (H5: (eq T
+(THead (Flat Cast) t2 t1) (THead (Flat Appl) w v))).(let H6 \def (eq_ind T
+(THead (Flat Cast) t2 t1) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
+[(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f in F return
+(\lambda (_: F).Prop) with [Appl \Rightarrow False | Cast \Rightarrow
+True])])])) I (THead (Flat Appl) w v) H5) in (False_ind (ex3_2 T T (\lambda
+(u: T).(\lambda (t: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u t))
+(THead (Flat Cast) t0 t2)))) (\lambda (u: T).(\lambda (t: T).(ty3 g c0 v
+(THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u))))
+H6))))))))))) c y x H0))) H)))))).
theorem ty3_gen_cast:
\forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).(\forall
\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)) (ex3 T (\lambda (t0:
-T).(pc3 c (THead (Flat Cast) t0 t2) x)) (\lambda (_: T).(ty3 g c t1 t2))
-(\lambda (t0: T).(ty3 g c t2 t0))) (\lambda (y: T).(\lambda (H0: (ty3 g c y
-x)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t
-(THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t3: T).(pc3 c0 (THead (Flat
-Cast) t3 t2) t0)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t3: T).(ty3 g
-c0 t2 t3))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t: T).(\lambda
-(_: (ty3 g c0 t0 t)).(\lambda (_: (((eq T t0 (THead (Flat Cast) t2 t1)) \to
-(ex3 T (\lambda (t3: T).(pc3 c0 (THead (Flat Cast) t3 t2) t)) (\lambda (_:
-T).(ty3 g c0 t1 t2)) (\lambda (t3: T).(ty3 g c0 t2 t3)))))).(\lambda (u:
-T).(\lambda (t3: T).(\lambda (H3: (ty3 g c0 u t3)).(\lambda (H4: (((eq T u
-(THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t4: T).(pc3 c0 (THead (Flat
-Cast) t4 t2) t3)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t4: T).(ty3 g
-c0 t2 t4)))))).(\lambda (H5: (pc3 c0 t3 t0)).(\lambda (H6: (eq T u (THead
-(Flat Cast) t2 t1))).(let H7 \def (f_equal T T (\lambda (e: T).e) u (THead
-(Flat Cast) t2 t1) H6) in (let H8 \def (eq_ind T u (\lambda (t4: T).((eq T t4
-(THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t5: T).(pc3 c0 (THead (Flat
-Cast) t5 t2) t3)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g
-c0 t2 t5))))) H4 (THead (Flat Cast) t2 t1) H7) in (let H9 \def (eq_ind T u
-(\lambda (t4: T).(ty3 g c0 t4 t3)) H3 (THead (Flat Cast) t2 t1) H7) in (let
-H10 \def (H8 (refl_equal T (THead (Flat Cast) t2 t1))) in (ex3_ind T (\lambda
+(THead (Flat Cast) t2 t1) (\lambda (t: T).(ty3 g c t x)) (\lambda (_: T).(ex3
+T (\lambda (t0: T).(pc3 c (THead (Flat Cast) t0 t2) x)) (\lambda (_: T).(ty3
+g c t1 t2)) (\lambda (t0: T).(ty3 g c t2 t0)))) (\lambda (y: T).(\lambda (H0:
+(ty3 g c y x)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0:
+T).((eq T t (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t3: T).(pc3 c0
+(THead (Flat Cast) t3 t2) t0)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda
+(t3: T).(ty3 g c0 t2 t3))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t:
+T).(\lambda (_: (ty3 g c0 t0 t)).(\lambda (_: (((eq T t0 (THead (Flat Cast)
+t2 t1)) \to (ex3 T (\lambda (t3: T).(pc3 c0 (THead (Flat Cast) t3 t2) t))
+(\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t3: T).(ty3 g c0 t2
+t3)))))).(\lambda (u: T).(\lambda (t3: T).(\lambda (H3: (ty3 g c0 u
+t3)).(\lambda (H4: (((eq T u (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda
(t4: T).(pc3 c0 (THead (Flat Cast) t4 t2) t3)) (\lambda (_: T).(ty3 g c0 t1
-t2)) (\lambda (t4: T).(ty3 g c0 t2 t4)) (ex3 T (\lambda (t4: T).(pc3 c0
-(THead (Flat Cast) t4 t2) t0)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda
-(t4: T).(ty3 g c0 t2 t4))) (\lambda (x0: T).(\lambda (H11: (pc3 c0 (THead
-(Flat Cast) x0 t2) t3)).(\lambda (H12: (ty3 g c0 t1 t2)).(\lambda (H13: (ty3
-g c0 t2 x0)).(ex3_intro T (\lambda (t4: T).(pc3 c0 (THead (Flat Cast) t4 t2)
-t0)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t4: T).(ty3 g c0 t2 t4)) x0
-(pc3_t t3 c0 (THead (Flat Cast) x0 t2) H11 t0 H5) H12 H13)))))
-H10)))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq T
-(TSort m) (THead (Flat Cast) t2 t1))).(let H2 \def (eq_ind T (TSort m)
-(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
-False])) I (THead (Flat Cast) t2 t1) H1) in (False_ind (ex3 T (\lambda (t0:
-T).(pc3 c0 (THead (Flat Cast) t0 t2) (TSort (next g m)))) (\lambda (_:
-T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0 t2 t0))) H2))))) (\lambda (n:
+t2)) (\lambda (t4: T).(ty3 g c0 t2 t4)))))).(\lambda (H5: (pc3 c0 t3
+t0)).(\lambda (H6: (eq T u (THead (Flat Cast) t2 t1))).(let H7 \def (f_equal
+T T (\lambda (e: T).e) u (THead (Flat Cast) t2 t1) H6) in (let H8 \def
+(eq_ind T u (\lambda (t4: T).((eq T t4 (THead (Flat Cast) t2 t1)) \to (ex3 T
+(\lambda (t5: T).(pc3 c0 (THead (Flat Cast) t5 t2) t3)) (\lambda (_: T).(ty3
+g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5))))) H4 (THead (Flat Cast) t2
+t1) H7) in (let H9 \def (eq_ind T u (\lambda (t4: T).(ty3 g c0 t4 t3)) H3
+(THead (Flat Cast) t2 t1) H7) in (let H10 \def (H8 (refl_equal T (THead (Flat
+Cast) t2 t1))) in (ex3_ind T (\lambda (t4: T).(pc3 c0 (THead (Flat Cast) t4
+t2) t3)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t4: T).(ty3 g c0 t2 t4))
+(ex3 T (\lambda (t4: T).(pc3 c0 (THead (Flat Cast) t4 t2) t0)) (\lambda (_:
+T).(ty3 g c0 t1 t2)) (\lambda (t4: T).(ty3 g c0 t2 t4))) (\lambda (x0:
+T).(\lambda (H11: (pc3 c0 (THead (Flat Cast) x0 t2) t3)).(\lambda (H12: (ty3
+g c0 t1 t2)).(\lambda (H13: (ty3 g c0 t2 x0)).(ex3_intro T (\lambda (t4:
+T).(pc3 c0 (THead (Flat Cast) t4 t2) t0)) (\lambda (_: T).(ty3 g c0 t1 t2))
+(\lambda (t4: T).(ty3 g c0 t2 t4)) x0 (pc3_t t3 c0 (THead (Flat Cast) x0 t2)
+H11 t0 H5) H12 H13))))) H10)))))))))))))))) (\lambda (c0: C).(\lambda (m:
+nat).(\lambda (H1: (eq T (TSort m) (THead (Flat Cast) t2 t1))).(let H2 \def
+(eq_ind T (TSort m) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False |
+(THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) t2 t1) H1) in
+(False_ind (ex3 T (\lambda (t0: T).(pc3 c0 (THead (Flat Cast) t0 t2) (TSort
+(next g m)))) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0 t2
+t0))) H2))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u:
+T).(\lambda (_: (getl n c0 (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda
+(_: (ty3 g d u t)).(\lambda (_: (((eq T u (THead (Flat Cast) t2 t1)) \to (ex3
+T (\lambda (t0: T).(pc3 d (THead (Flat Cast) t0 t2) t)) (\lambda (_: T).(ty3
+g d t1 t2)) (\lambda (t0: T).(ty3 g d t2 t0)))))).(\lambda (H4: (eq T (TLRef
+n) (THead (Flat Cast) t2 t1))).(let H5 \def (eq_ind T (TLRef n) (\lambda (ee:
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I
+(THead (Flat Cast) t2 t1) H4) in (False_ind (ex3 T (\lambda (t0: T).(pc3 c0
+(THead (Flat Cast) t0 t2) (lift (S n) O t))) (\lambda (_: T).(ty3 g c0 t1
+t2)) (\lambda (t0: T).(ty3 g c0 t2 t0))) H5))))))))))) (\lambda (n:
nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (_: (getl n c0
-(CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda (_: (ty3 g d u
+(CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (_: (ty3 g d u
t)).(\lambda (_: (((eq T u (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda
(t0: T).(pc3 d (THead (Flat Cast) t0 t2) t)) (\lambda (_: T).(ty3 g d t1 t2))
(\lambda (t0: T).(ty3 g d t2 t0)))))).(\lambda (H4: (eq T (TLRef n) (THead
ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
(TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead
(Flat Cast) t2 t1) H4) in (False_ind (ex3 T (\lambda (t0: T).(pc3 c0 (THead
-(Flat Cast) t0 t2) (lift (S n) O t))) (\lambda (_: T).(ty3 g c0 t1 t2))
-(\lambda (t0: T).(ty3 g c0 t2 t0))) H5))))))))))) (\lambda (n: nat).(\lambda
-(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (_: (getl n c0 (CHead d
-(Bind Abst) u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (_:
-(((eq T u (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t0: T).(pc3 d
-(THead (Flat Cast) t0 t2) t)) (\lambda (_: T).(ty3 g d t1 t2)) (\lambda (t0:
-T).(ty3 g d t2 t0)))))).(\lambda (H4: (eq T (TLRef n) (THead (Flat Cast) t2
-t1))).(let H5 \def (eq_ind T (TLRef n) (\lambda (ee: T).(match ee in T return
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) t2
-t1) H4) in (False_ind (ex3 T (\lambda (t0: T).(pc3 c0 (THead (Flat Cast) t0
-t2) (lift (S n) O u))) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0:
-T).(ty3 g c0 t2 t0))) H5))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda
-(t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u (THead (Flat Cast)
-t2 t1)) \to (ex3 T (\lambda (t0: T).(pc3 c0 (THead (Flat Cast) t0 t2) t))
-(\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0 t2
-t0)))))).(\lambda (b: B).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (ty3
-g (CHead c0 (Bind b) u) t0 t3)).(\lambda (_: (((eq T t0 (THead (Flat Cast) t2
-t1)) \to (ex3 T (\lambda (t4: T).(pc3 (CHead c0 (Bind b) u) (THead (Flat
-Cast) t4 t2) t3)) (\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2))
-(\lambda (t4: T).(ty3 g (CHead c0 (Bind b) u) t2 t4)))))).(\lambda (t4:
-T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t3 t4)).(\lambda (_: (((eq T t3
-(THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t5: T).(pc3 (CHead c0 (Bind
-b) u) (THead (Flat Cast) t5 t2) t4)) (\lambda (_: T).(ty3 g (CHead c0 (Bind
-b) u) t1 t2)) (\lambda (t5: T).(ty3 g (CHead c0 (Bind b) u) t2
-t5)))))).(\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
-in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef
-_) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return
-(\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow
-False])])) I (THead (Flat Cast) t2 t1) H7) in (False_ind (ex3 T (\lambda (t5:
-T).(pc3 c0 (THead (Flat Cast) t5 t2) (THead (Bind b) u t3))) (\lambda (_:
-T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5))) 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 (ex3 T (\lambda
-(t0: T).(pc3 c0 (THead (Flat Cast) t0 t2) u)) (\lambda (_: T).(ty3 g c0 t1
-t2)) (\lambda (t0: T).(ty3 g c0 t2 t0)))))).(\lambda (v: T).(\lambda (t:
-T).(\lambda (_: (ty3 g c0 v (THead (Bind Abst) u t))).(\lambda (_: (((eq T v
+(Flat Cast) t0 t2) (lift (S n) O u))) (\lambda (_: T).(ty3 g c0 t1 t2))
+(\lambda (t0: T).(ty3 g c0 t2 t0))) H5))))))))))) (\lambda (c0: C).(\lambda
+(u: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u
(THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t0: T).(pc3 c0 (THead (Flat
-Cast) t0 t2) (THead (Bind Abst) u t))) (\lambda (_: T).(ty3 g c0 t1 t2))
-(\lambda (t0: T).(ty3 g c0 t2 t0)))))).(\lambda (H5: (eq T (THead (Flat Appl)
-w v) (THead (Flat Cast) t2 t1))).(let H6 \def (eq_ind T (THead (Flat Appl) w
-v) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
-_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
-\Rightarrow False | (Flat f) \Rightarrow (match f in F return (\lambda (_:
-F).Prop) with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead
-(Flat Cast) t2 t1) H5) in (False_ind (ex3 T (\lambda (t0: T).(pc3 c0 (THead
-(Flat Cast) t0 t2) (THead (Flat Appl) w (THead (Bind Abst) u t)))) (\lambda
-(_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0 t2 t0))) H6))))))))))))
-(\lambda (c0: C).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H1: (ty3 g c0 t0
-t3)).(\lambda (H2: (((eq T t0 (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda
-(t4: T).(pc3 c0 (THead (Flat Cast) t4 t2) t3)) (\lambda (_: T).(ty3 g c0 t1
-t2)) (\lambda (t4: T).(ty3 g c0 t2 t4)))))).(\lambda (t4: T).(\lambda (H3:
-(ty3 g c0 t3 t4)).(\lambda (H4: (((eq T t3 (THead (Flat Cast) t2 t1)) \to
+Cast) t0 t2) t)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0
+t2 t0)))))).(\lambda (b: B).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_:
+(ty3 g (CHead c0 (Bind b) u) t0 t3)).(\lambda (_: (((eq T t0 (THead (Flat
+Cast) t2 t1)) \to (ex3 T (\lambda (t4: T).(pc3 (CHead c0 (Bind b) u) (THead
+(Flat Cast) t4 t2) t3)) (\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2))
+(\lambda (t4: T).(ty3 g (CHead c0 (Bind b) u) t2 t4)))))).(\lambda (H5: (eq T
+(THead (Bind b) u t0) (THead (Flat Cast) t2 t1))).(let H6 \def (eq_ind T
+(THead (Bind b) u t0) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
+[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat
+Cast) t2 t1) H5) in (False_ind (ex3 T (\lambda (t4: T).(pc3 c0 (THead (Flat
+Cast) t4 t2) (THead (Bind b) u t3))) (\lambda (_: T).(ty3 g c0 t1 t2))
+(\lambda (t4: T).(ty3 g c0 t2 t4))) H6))))))))))))) (\lambda (c0: C).(\lambda
+(w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda (_: (((eq T w
+(THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t0: T).(pc3 c0 (THead (Flat
+Cast) t0 t2) u)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0
+t2 t0)))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead
+(Bind Abst) u t))).(\lambda (_: (((eq T v (THead (Flat Cast) t2 t1)) \to (ex3
+T (\lambda (t0: T).(pc3 c0 (THead (Flat Cast) t0 t2) (THead (Bind Abst) u
+t))) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0 t2
+t0)))))).(\lambda (H5: (eq T (THead (Flat Appl) w v) (THead (Flat Cast) t2
+t1))).(let H6 \def (eq_ind T (THead (Flat Appl) w v) (\lambda (ee: T).(match
+ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
+(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return
+(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow
+(match f in F return (\lambda (_: F).Prop) with [Appl \Rightarrow True | Cast
+\Rightarrow False])])])) I (THead (Flat Cast) t2 t1) H5) in (False_ind (ex3 T
+(\lambda (t0: T).(pc3 c0 (THead (Flat Cast) t0 t2) (THead (Flat Appl) w
+(THead (Bind Abst) u t)))) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0:
+T).(ty3 g c0 t2 t0))) H6)))))))))))) (\lambda (c0: C).(\lambda (t0:
+T).(\lambda (t3: T).(\lambda (H1: (ty3 g c0 t0 t3)).(\lambda (H2: (((eq T t0
+(THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t4: T).(pc3 c0 (THead (Flat
+Cast) t4 t2) t3)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t4: T).(ty3 g
+c0 t2 t4)))))).(\lambda (t4: T).(\lambda (H3: (ty3 g c0 t3 t4)).(\lambda (H4:
+(((eq T t3 (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t5: T).(pc3 c0
+(THead (Flat Cast) t5 t2) t4)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda
+(t5: T).(ty3 g c0 t2 t5)))))).(\lambda (H5: (eq T (THead (Flat Cast) t3 t0)
+(THead (Flat Cast) t2 t1))).(let H6 \def (f_equal T T (\lambda (e: T).(match
+e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _)
+\Rightarrow t3 | (THead _ t _) \Rightarrow t])) (THead (Flat Cast) t3 t0)
+(THead (Flat Cast) t2 t1) H5) in ((let H7 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 |
+(TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Flat Cast)
+t3 t0) (THead (Flat Cast) t2 t1) H5) in (\lambda (H8: (eq T t3 t2)).(let H9
+\def (eq_ind T t3 (\lambda (t: T).((eq T t (THead (Flat Cast) t2 t1)) \to
(ex3 T (\lambda (t5: T).(pc3 c0 (THead (Flat Cast) t5 t2) t4)) (\lambda (_:
-T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5)))))).(\lambda (H5: (eq
-T (THead (Flat Cast) t3 t0) (THead (Flat Cast) t2 t1))).(let H6 \def (f_equal
-T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ t _) \Rightarrow t]))
-(THead (Flat Cast) t3 t0) (THead (Flat Cast) t2 t1) H5) in ((let H7 \def
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
-[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t)
-\Rightarrow t])) (THead (Flat Cast) t3 t0) (THead (Flat Cast) t2 t1) H5) in
-(\lambda (H8: (eq T t3 t2)).(let H9 \def (eq_ind T t3 (\lambda (t: T).((eq T
+T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5))))) H4 t2 H8) in (let
+H10 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t t4)) H3 t2 H8) in (let H11
+\def (eq_ind T t3 (\lambda (t: T).((eq T t0 (THead (Flat Cast) t2 t1)) \to
+(ex3 T (\lambda (t5: T).(pc3 c0 (THead (Flat Cast) t5 t2) t)) (\lambda (_:
+T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5))))) H2 t2 H8) in (let
+H12 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t0 t)) H1 t2 H8) in (eq_ind_r
+T t2 (\lambda (t: T).(ex3 T (\lambda (t5: T).(pc3 c0 (THead (Flat Cast) t5
+t2) (THead (Flat Cast) t4 t))) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda
+(t5: T).(ty3 g c0 t2 t5)))) (let H13 \def (eq_ind T t0 (\lambda (t: T).((eq T
t (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t5: T).(pc3 c0 (THead (Flat
-Cast) t5 t2) t4)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g
-c0 t2 t5))))) H4 t2 H8) in (let H10 \def (eq_ind T t3 (\lambda (t: T).(ty3 g
-c0 t t4)) H3 t2 H8) in (let H11 \def (eq_ind T t3 (\lambda (t: T).((eq T t0
-(THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t5: T).(pc3 c0 (THead (Flat
-Cast) t5 t2) t)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0
-t2 t5))))) H2 t2 H8) in (let H12 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0
-t0 t)) H1 t2 H8) in (eq_ind_r T t2 (\lambda (t: T).(ex3 T (\lambda (t5:
-T).(pc3 c0 (THead (Flat Cast) t5 t2) (THead (Flat Cast) t4 t))) (\lambda (_:
-T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5)))) (let H13 \def
-(eq_ind T t0 (\lambda (t: T).((eq T t (THead (Flat Cast) t2 t1)) \to (ex3 T
-(\lambda (t5: T).(pc3 c0 (THead (Flat Cast) t5 t2) t2)) (\lambda (_: T).(ty3
-g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5))))) H11 t1 H7) in (let H14
-\def (eq_ind T t0 (\lambda (t: T).(ty3 g c0 t t2)) H12 t1 H7) in (ex3_intro T
-(\lambda (t5: T).(pc3 c0 (THead (Flat Cast) t5 t2) (THead (Flat Cast) t4
-t2))) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5)) t4
-(pc3_refl c0 (THead (Flat Cast) t4 t2)) H14 H10))) t3 H8))))))) H6)))))))))))
-c y x H0))) H)))))).
+Cast) t5 t2) t2)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g
+c0 t2 t5))))) H11 t1 H7) in (let H14 \def (eq_ind T t0 (\lambda (t: T).(ty3 g
+c0 t t2)) H12 t1 H7) in (ex3_intro T (\lambda (t5: T).(pc3 c0 (THead (Flat
+Cast) t5 t2) (THead (Flat Cast) t4 t2))) (\lambda (_: T).(ty3 g c0 t1 t2))
+(\lambda (t5: T).(ty3 g c0 t2 t5)) t4 (pc3_refl c0 (THead (Flat Cast) t4 t2))
+H14 H10))) t3 H8))))))) H6))))))))))) c y x H0))) H)))))).
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/ty3/nf2".
include "ty3/arity.ma".
(n: nat).(eq nat m (next g n)))) (ex3_2 TList nat (\lambda (ws:
TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i)))))
(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_:
-TList).(\lambda (i: nat).(nf2 c (TLRef i))))))) (ex4_3_ind T T T (\lambda
-(t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c (THead (Bind Abst) x0 t2)
-(TSort m))))) (\lambda (_: T).(\lambda (t0: T).(\lambda (_: T).(ty3 g c x0
-t0)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c (Bind
-Abst) x0) x1 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t1: T).(ty3 g
-(CHead c (Bind Abst) x0) t2 t1)))) (or (ex2 nat (\lambda (n: nat).(eq T
-(THead (Bind Abst) x0 x1) (TSort n))) (\lambda (n: nat).(eq nat m (next g
-n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead
-(Bind Abst) x0 x1) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws:
+TList).(\lambda (i: nat).(nf2 c (TLRef i))))))) (ex3_2_ind T T (\lambda (t2:
+T).(\lambda (_: T).(pc3 c (THead (Bind Abst) x0 t2) (TSort m)))) (\lambda (_:
+T).(\lambda (t0: T).(ty3 g c x0 t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g
+(CHead c (Bind Abst) x0) x1 t2))) (or (ex2 nat (\lambda (n: nat).(eq T (THead
+(Bind Abst) x0 x1) (TSort n))) (\lambda (n: nat).(eq nat m (next g n))))
+(ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Bind
+Abst) x0 x1) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws:
TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i:
-nat).(nf2 c (TLRef i)))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (x4:
-T).(\lambda (H7: (pc3 c (THead (Bind Abst) x0 x2) (TSort m))).(\lambda (_:
-(ty3 g c x0 x3)).(\lambda (_: (ty3 g (CHead c (Bind Abst) x0) x1
-x2)).(\lambda (_: (ty3 g (CHead c (Bind Abst) x0) x2 x4)).(pc3_gen_sort_abst
+nat).(nf2 c (TLRef i)))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7:
+(pc3 c (THead (Bind Abst) x0 x2) (TSort m))).(\lambda (_: (ty3 g c x0
+x3)).(\lambda (_: (ty3 g (CHead c (Bind Abst) x0) x1 x2)).(pc3_gen_sort_abst
c x0 x2 m (pc3_s c (TSort m) (THead (Bind Abst) x0 x2) H7) (or (ex2 nat
(\lambda (n: nat).(eq T (THead (Bind Abst) x0 x1) (TSort n))) (\lambda (n:
nat).(eq nat m (next g n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda
(i: nat).(eq T (THead (Bind Abst) x0 x1) (THeads (Flat Appl) ws (TLRef i)))))
(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_:
-TList).(\lambda (i: nat).(nf2 c (TLRef i)))))))))))))) (ty3_gen_bind g Abst c
+TList).(\lambda (i: nat).(nf2 c (TLRef i)))))))))))) (ty3_gen_bind g Abst c
x0 x1 (TSort m) H6)) t H3))))))) H2)) (\lambda (H2: (ex nat (\lambda (n:
nat).(eq T t (TSort n))))).(ex_ind nat (\lambda (n: nat).(eq T t (TSort n)))
(or (ex2 nat (\lambda (n: nat).(eq T t (TSort n))) (\lambda (n: nat).(eq nat
x0 x1) (THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w
w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u)))
(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda
-(x: T).(\lambda (H9: (ty3 g c (THead (Bind Abst) w u) x)).(ex4_3_ind T T T
-(\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c (THead (Bind Abst) w
-t2) x)))) (\lambda (_: T).(\lambda (t0: T).(\lambda (_: T).(ty3 g c w t0))))
-(\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst)
-w) u t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t1: T).(ty3 g (CHead c
-(Bind Abst) w) t2 t1)))) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T
-(THead (Bind Abst) x0 x1) (THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda
-(w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c
-(Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind
-Abst) w) v)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda
-(_: (pc3 c (THead (Bind Abst) w x2) x)).(\lambda (H11: (ty3 g c w
-x3)).(\lambda (H12: (ty3 g (CHead c (Bind Abst) w) u x2)).(\lambda (_: (ty3 g
-(CHead c (Bind Abst) w) x2 x4)).(ex4_3_ind T T T (\lambda (t2: T).(\lambda
-(_: T).(\lambda (_: T).(pc3 c (THead (Bind Abst) x0 t2) (THead (Bind Abst) w
-u))))) (\lambda (_: T).(\lambda (t0: T).(\lambda (_: T).(ty3 g c x0 t0))))
-(\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst)
-x0) x1 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t1: T).(ty3 g (CHead
-c (Bind Abst) x0) t2 t1)))) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T
-(THead (Bind Abst) x0 x1) (THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda
-(w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c
-(Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind
-Abst) w) v)))) (\lambda (x5: T).(\lambda (x6: T).(\lambda (x7: T).(\lambda
-(H14: (pc3 c (THead (Bind Abst) x0 x5) (THead (Bind Abst) w u))).(\lambda (_:
-(ty3 g c x0 x6)).(\lambda (H16: (ty3 g (CHead c (Bind Abst) x0) x1
-x5)).(\lambda (_: (ty3 g (CHead c (Bind Abst) x0) x5 x7)).(and_ind (pc3 c x0
-w) (\forall (b: B).(\forall (u0: T).(pc3 (CHead c (Bind b) u0) x5 u))) (ex4_2
-T T (\lambda (v: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead
-(Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0)))
+(x: T).(\lambda (H9: (ty3 g c (THead (Bind Abst) w u) x)).(ex3_2_ind T T
+(\lambda (t2: T).(\lambda (_: T).(pc3 c (THead (Bind Abst) w t2) x)))
+(\lambda (_: T).(\lambda (t0: T).(ty3 g c w t0))) (\lambda (t2: T).(\lambda
+(_: T).(ty3 g (CHead c (Bind Abst) w) u t2))) (ex4_2 T T (\lambda (v:
+T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) w v))))
+(\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda
+(_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_:
+T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda (x2: T).(\lambda (x3:
+T).(\lambda (_: (pc3 c (THead (Bind Abst) w x2) x)).(\lambda (H11: (ty3 g c w
+x3)).(\lambda (H12: (ty3 g (CHead c (Bind Abst) w) u x2)).(ex3_2_ind T T
+(\lambda (t2: T).(\lambda (_: T).(pc3 c (THead (Bind Abst) x0 t2) (THead
+(Bind Abst) w u)))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c x0 t0)))
+(\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x0) x1 t2)))
+(ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1)
+(THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0)))
(\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u)))
(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda
-(H18: (pc3 c x0 w)).(\lambda (H19: ((\forall (b: B).(\forall (u0: T).(pc3
-(CHead c (Bind b) u0) x5 u))))).(let H_y \def (pc3_nf2 c x0 w H18 H6 H1) in
-(let H20 \def (eq_ind T x0 (\lambda (t0: T).(ty3 g (CHead c (Bind Abst) t0)
-x1 x5)) H16 w H_y) in (let H21 \def (eq_ind T x0 (\lambda (t0: T).(nf2 (CHead
-c (Bind Abst) t0) x1)) H7 w H_y) in (eq_ind_r T w (\lambda (t0: T).(ex4_2 T T
-(\lambda (v: T).(\lambda (_: T).(eq T (THead (Bind Abst) t0 x1) (THead (Bind
-Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v:
-T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v:
-T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v))))) (ex4_2_intro T T
-(\lambda (v: T).(\lambda (_: T).(eq T (THead (Bind Abst) w x1) (THead (Bind
-Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v:
-T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v:
-T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v))) x1 x3 (refl_equal T
-(THead (Bind Abst) w x1)) H11 (ty3_conv g (CHead c (Bind Abst) w) u x2 H12 x1
-x5 H20 (H19 Abst w)) H21) x0 H_y)))))) (pc3_gen_abst c x0 w x5 u H14)))))))))
-(ty3_gen_bind g Abst c x0 x1 (THead (Bind Abst) w u) H8)))))))))
-(ty3_gen_bind g Abst c w u x H9)))) (ty3_correct g c (THead (Bind Abst) x0
-x1) (THead (Bind Abst) w u) H8)) t H5))))))) H4)) (\lambda (H4: (ex nat
-(\lambda (n: nat).(eq T t (TSort n))))).(ex_ind nat (\lambda (n: nat).(eq T t
-(TSort n))) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T t (THead (Bind
-Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v:
-T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v:
-T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda (x:
-nat).(\lambda (H5: (eq T t (TSort x))).(let H6 \def (eq_ind T t (\lambda (t0:
-T).(ty3 g c t0 (THead (Bind Abst) w u))) H (TSort x) H5) in (eq_ind_r T
-(TSort x) (\lambda (t0: T).(ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T
-t0 (THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w
-w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u)))
-(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v)))))
-(pc3_gen_sort_abst c w u (next g x) (ty3_gen_sort g c (THead (Bind Abst) w u)
-x H6) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (TSort x) (THead (Bind
-Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v:
+(x4: T).(\lambda (x5: T).(\lambda (H13: (pc3 c (THead (Bind Abst) x0 x4)
+(THead (Bind Abst) w u))).(\lambda (_: (ty3 g c x0 x5)).(\lambda (H15: (ty3 g
+(CHead c (Bind Abst) x0) x1 x4)).(and_ind (pc3 c x0 w) (\forall (b:
+B).(\forall (u0: T).(pc3 (CHead c (Bind b) u0) x4 u))) (ex4_2 T T (\lambda
+(v: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) w
+v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v:
T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v:
-T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v))))) t H5)))) H4)) (\lambda
-(H4: (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t (THeads
-(Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c
-ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef i)))))).(ex3_2_ind
-TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl)
-ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws)))
-(\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef i)))) (ex4_2 T T (\lambda
+T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda (H16: (pc3 c
+x0 w)).(\lambda (H17: ((\forall (b: B).(\forall (u0: T).(pc3 (CHead c (Bind
+b) u0) x4 u))))).(let H_y \def (pc3_nf2 c x0 w H16 H6 H1) in (let H18 \def
+(eq_ind T x0 (\lambda (t0: T).(ty3 g (CHead c (Bind Abst) t0) x1 x4)) H15 w
+H_y) in (let H19 \def (eq_ind T x0 (\lambda (t0: T).(nf2 (CHead c (Bind Abst)
+t0) x1)) H7 w H_y) in (eq_ind_r T w (\lambda (t0: T).(ex4_2 T T (\lambda (v:
+T).(\lambda (_: T).(eq T (THead (Bind Abst) t0 x1) (THead (Bind Abst) w v))))
+(\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda
+(_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_:
+T).(nf2 (CHead c (Bind Abst) w) v))))) (ex4_2_intro T T (\lambda (v:
+T).(\lambda (_: T).(eq T (THead (Bind Abst) w x1) (THead (Bind Abst) w v))))
+(\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda
+(_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_:
+T).(nf2 (CHead c (Bind Abst) w) v))) x1 x3 (refl_equal T (THead (Bind Abst) w
+x1)) H11 (ty3_conv g (CHead c (Bind Abst) w) u x2 H12 x1 x4 H18 (H17 Abst w))
+H19) x0 H_y)))))) (pc3_gen_abst c x0 w x4 u H13))))))) (ty3_gen_bind g Abst c
+x0 x1 (THead (Bind Abst) w u) H8))))))) (ty3_gen_bind g Abst c w u x H9))))
+(ty3_correct g c (THead (Bind Abst) x0 x1) (THead (Bind Abst) w u) H8)) t
+H5))))))) H4)) (\lambda (H4: (ex nat (\lambda (n: nat).(eq T t (TSort
+n))))).(ex_ind nat (\lambda (n: nat).(eq T t (TSort n))) (ex4_2 T T (\lambda
(v: T).(\lambda (_: T).(eq T t (THead (Bind Abst) w v)))) (\lambda (_:
T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g
(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c
-(Bind Abst) w) v)))) (\lambda (x0: TList).(\lambda (x1: nat).(\lambda (H5:
-(eq T t (THeads (Flat Appl) x0 (TLRef x1)))).(\lambda (_: (nfs2 c
-x0)).(\lambda (H7: (nf2 c (TLRef x1))).(let H8 \def (eq_ind T t (\lambda (t0:
-T).(ty3 g c t0 (THead (Bind Abst) w u))) H (THeads (Flat Appl) x0 (TLRef x1))
-H5) in (eq_ind_r T (THeads (Flat Appl) x0 (TLRef x1)) (\lambda (t0: T).(ex4_2
-T T (\lambda (v: T).(\lambda (_: T).(eq T t0 (THead (Bind Abst) w v))))
-(\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda
-(_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_:
-T).(nf2 (CHead c (Bind Abst) w) v))))) (let H9 \def H2 in ((let H10 \def H8
-in (unintro T u (\lambda (t0: T).((ty3 g c (THeads (Flat Appl) x0 (TLRef x1))
-(THead (Bind Abst) w t0)) \to ((ty3_nf2_inv_abst_premise c w t0) \to (ex4_2 T
-T (\lambda (v: T).(\lambda (_: T).(eq T (THeads (Flat Appl) x0 (TLRef x1))
+(Bind Abst) w) v)))) (\lambda (x: nat).(\lambda (H5: (eq T t (TSort x))).(let
+H6 \def (eq_ind T t (\lambda (t0: T).(ty3 g c t0 (THead (Bind Abst) w u))) H
+(TSort x) H5) in (eq_ind_r T (TSort x) (\lambda (t0: T).(ex4_2 T T (\lambda
+(v: T).(\lambda (_: T).(eq T t0 (THead (Bind Abst) w v)))) (\lambda (_:
+T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g
+(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c
+(Bind Abst) w) v))))) (pc3_gen_sort_abst c w u (next g x) (ty3_gen_sort g c
+(THead (Bind Abst) w u) x H6) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq
+T (TSort x) (THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3
+g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v
+u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v))))) t
+H5)))) H4)) (\lambda (H4: (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i:
+nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws:
+TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i:
+nat).(nf2 c (TLRef i)))))).(ex3_2_ind TList nat (\lambda (ws: TList).(\lambda
+(i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws:
+TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i:
+nat).(nf2 c (TLRef i)))) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T t
(THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0)))
-(\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v t0)))
-(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v)))))))
-(unintro T w (\lambda (t0: T).(\forall (x: T).((ty3 g c (THeads (Flat Appl)
-x0 (TLRef x1)) (THead (Bind Abst) t0 x)) \to ((ty3_nf2_inv_abst_premise c t0
-x) \to (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (THeads (Flat Appl)
-x0 (TLRef x1)) (THead (Bind Abst) t0 v)))) (\lambda (_: T).(\lambda (w0:
-T).(ty3 g c t0 w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind
-Abst) t0) v x))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst)
-t0) v)))))))) (TList_ind (\lambda (t0: TList).(\forall (x: T).(\forall (x2:
-T).((ty3 g c (THeads (Flat Appl) t0 (TLRef x1)) (THead (Bind Abst) x x2)) \to
+(\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u)))
+(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda
+(x0: TList).(\lambda (x1: nat).(\lambda (H5: (eq T t (THeads (Flat Appl) x0
+(TLRef x1)))).(\lambda (_: (nfs2 c x0)).(\lambda (H7: (nf2 c (TLRef
+x1))).(let H8 \def (eq_ind T t (\lambda (t0: T).(ty3 g c t0 (THead (Bind
+Abst) w u))) H (THeads (Flat Appl) x0 (TLRef x1)) H5) in (eq_ind_r T (THeads
+(Flat Appl) x0 (TLRef x1)) (\lambda (t0: T).(ex4_2 T T (\lambda (v:
+T).(\lambda (_: T).(eq T t0 (THead (Bind Abst) w v)))) (\lambda (_:
+T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g
+(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c
+(Bind Abst) w) v))))) (let H9 \def H2 in ((let H10 \def H8 in (unintro T u
+(\lambda (t0: T).((ty3 g c (THeads (Flat Appl) x0 (TLRef x1)) (THead (Bind
+Abst) w t0)) \to ((ty3_nf2_inv_abst_premise c w t0) \to (ex4_2 T T (\lambda
+(v: T).(\lambda (_: T).(eq T (THeads (Flat Appl) x0 (TLRef x1)) (THead (Bind
+Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v:
+T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v t0))) (\lambda (v:
+T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v))))))) (unintro T w
+(\lambda (t0: T).(\forall (x: T).((ty3 g c (THeads (Flat Appl) x0 (TLRef x1))
+(THead (Bind Abst) t0 x)) \to ((ty3_nf2_inv_abst_premise c t0 x) \to (ex4_2 T
+T (\lambda (v: T).(\lambda (_: T).(eq T (THeads (Flat Appl) x0 (TLRef x1))
+(THead (Bind Abst) t0 v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c t0
+w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) t0) v x)))
+(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) t0) v))))))))
+(TList_ind (\lambda (t0: TList).(\forall (x: T).(\forall (x2: T).((ty3 g c
+(THeads (Flat Appl) t0 (TLRef x1)) (THead (Bind Abst) x x2)) \to
((ty3_nf2_inv_abst_premise c x x2) \to (ex4_2 T T (\lambda (v: T).(\lambda
(_: T).(eq T (THeads (Flat Appl) t0 (TLRef x1)) (THead (Bind Abst) x v))))
(\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/ty3/pr3".
include "csubt/ty3.ma".
(\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 (t5: T).((pr0 t3 t5)
-\to (ty3 g c2 t5 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 in pr0 return (\lambda (t6: T).(\lambda (t7: T).(\lambda (_:
-(pr0 t6 t7)).((eq T t6 (THead (Bind b) u t2)) \to ((eq T t7 t5) \to (ty3 g c2
-t5 (THead (Bind b) u t3))))))) with [(pr0_refl t6) \Rightarrow (\lambda (H8:
-(eq T t6 (THead (Bind b) u t2))).(\lambda (H9: (eq T t6 t5)).(eq_ind T (THead
-(Bind b) u t2) (\lambda (t7: T).((eq T t7 t5) \to (ty3 g c2 t5 (THead (Bind
-b) u t3)))) (\lambda (H10: (eq T (THead (Bind b) u t2) t5)).(eq_ind T (THead
-(Bind b) u t2) (\lambda (t7: T).(ty3 g c2 t7 (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 H10)) t6 (sym_eq T t6 (THead (Bind b) u t2) H8) H9))) |
-(pr0_comp u1 u2 H8 t6 t7 H9 k) \Rightarrow (\lambda (H10: (eq T (THead k u1
-t6) (THead (Bind b) u t2))).(\lambda (H11: (eq T (THead k u2 t7) t5)).((let
-H12 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
-with [(TSort _) \Rightarrow t6 | (TLRef _) \Rightarrow t6 | (THead _ _ t8)
-\Rightarrow t8])) (THead k u1 t6) (THead (Bind b) u t2) H10) in ((let H13
-\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
-with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t8 _)
-\Rightarrow t8])) (THead k u1 t6) (THead (Bind b) u t2) H10) in ((let H14
-\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K)
-with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _)
-\Rightarrow k0])) (THead k u1 t6) (THead (Bind b) u t2) H10) in (eq_ind K
-(Bind b) (\lambda (k0: K).((eq T u1 u) \to ((eq T t6 t2) \to ((eq T (THead k0
-u2 t7) t5) \to ((pr0 u1 u2) \to ((pr0 t6 t7) \to (ty3 g c2 t5 (THead (Bind b)
-u t3)))))))) (\lambda (H15: (eq T u1 u)).(eq_ind T u (\lambda (t8: T).((eq T
-t6 t2) \to ((eq T (THead (Bind b) u2 t7) t5) \to ((pr0 t8 u2) \to ((pr0 t6
-t7) \to (ty3 g c2 t5 (THead (Bind b) u t3))))))) (\lambda (H16: (eq T t6
-t2)).(eq_ind T t2 (\lambda (t8: T).((eq T (THead (Bind b) u2 t7) t5) \to
-((pr0 u u2) \to ((pr0 t8 t7) \to (ty3 g c2 t5 (THead (Bind b) u t3))))))
-(\lambda (H17: (eq T (THead (Bind b) u2 t7) t5)).(eq_ind T (THead (Bind b) u2
-t7) (\lambda (t8: T).((pr0 u u2) \to ((pr0 t2 t7) \to (ty3 g c2 t8 (THead
-(Bind b) u t3))))) (\lambda (H18: (pr0 u u2)).(\lambda (H19: (pr0 t2
-t7)).(ex_ind T (\lambda (t8: T).(ty3 g (CHead c2 (Bind b) u) t4 t8)) (ty3 g
-c2 (THead (Bind b) u2 t7) (THead (Bind b) u t3)) (\lambda (x: T).(\lambda
-(H20: (ty3 g (CHead c2 (Bind b) u) t4 x)).(ex_ind T (\lambda (t8: T).(ty3 g
-(CHead c2 (Bind b) u2) t3 t8)) (ty3 g c2 (THead (Bind b) u2 t7) (THead (Bind
-b) u t3)) (\lambda (x0: T).(\lambda (H21: (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 H20)
-(THead (Bind b) u2 t7) (THead (Bind b) u2 t3) (ty3_bind g c2 u2 t0 (H1 c2 H6
-u2 H18) b t7 t3 (H3 (CHead c2 (Bind b) u2) (wcpr0_comp c c2 H6 u u2 H18 (Bind
-b)) t7 H19) x0 H21) (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 H18) (Bind b) t3))))) (ty3_correct
-g (CHead c2 (Bind b) u2) t7 t3 (H3 (CHead c2 (Bind b) u2) (wcpr0_comp c c2 H6
-u u2 H18 (Bind b)) t7 H19))))) (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 H17)) t6 (sym_eq T t6 t2 H16))) u1 (sym_eq T u1 u
-H15))) k (sym_eq K k (Bind b) H14))) H13)) H12)) H11 H8 H9))) | (pr0_beta u0
-v1 v2 H8 t6 t7 H9) \Rightarrow (\lambda (H10: (eq T (THead (Flat Appl) v1
-(THead (Bind Abst) u0 t6)) (THead (Bind b) u t2))).(\lambda (H11: (eq T
-(THead (Bind Abbr) v2 t7) t5)).((let H12 \def (eq_ind T (THead (Flat Appl) v1
-(THead (Bind Abst) u0 t6)) (\lambda (e: T).(match e in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
-(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
-[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind
-b) u t2) H10) in (False_ind ((eq T (THead (Bind Abbr) v2 t7) t5) \to ((pr0 v1
-v2) \to ((pr0 t6 t7) \to (ty3 g c2 t5 (THead (Bind b) u t3))))) H12)) H11 H8
-H9))) | (pr0_upsilon b0 H8 v1 v2 H9 u1 u2 H10 t6 t7 H11) \Rightarrow (\lambda
-(H12: (eq T (THead (Flat Appl) v1 (THead (Bind b0) u1 t6)) (THead (Bind b) u
-t2))).(\lambda (H13: (eq T (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O)
-O v2) t7)) t5)).((let H14 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind
-b0) u1 t6)) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with
+\to (\forall (t4: T).((pr0 t2 t4) \to (ty3 g c2 t4 t3))))))).(\lambda (c2:
+C).(\lambda (H4: (wcpr0 c c2)).(\lambda (t4: T).(\lambda (H5: (pr0 (THead
+(Bind b) u t2) t4)).(let H6 \def (match H5 in pr0 return (\lambda (t5:
+T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 (THead (Bind b) u
+t2)) \to ((eq T t6 t4) \to (ty3 g c2 t4 (THead (Bind b) u t3))))))) with
+[(pr0_refl t5) \Rightarrow (\lambda (H6: (eq T t5 (THead (Bind b) u
+t2))).(\lambda (H7: (eq T t5 t4)).(eq_ind T (THead (Bind b) u t2) (\lambda
+(t6: T).((eq T t6 t4) \to (ty3 g c2 t4 (THead (Bind b) u t3)))) (\lambda (H8:
+(eq T (THead (Bind b) u t2) t4)).(eq_ind T (THead (Bind b) u t2) (\lambda
+(t6: T).(ty3 g c2 t6 (THead (Bind b) u t3))) (ty3_bind g c2 u t0 (H1 c2 H4 u
+(pr0_refl u)) b t2 t3 (H3 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H4 u u
+(pr0_refl u) (Bind b)) t2 (pr0_refl t2))) t4 H8)) t5 (sym_eq T t5 (THead
+(Bind b) u t2) H6) H7))) | (pr0_comp u1 u2 H6 t5 t6 H7 k) \Rightarrow
+(\lambda (H8: (eq T (THead k u1 t5) (THead (Bind b) u t2))).(\lambda (H9: (eq
+T (THead k u2 t6) t4)).((let H10 \def (f_equal T T (\lambda (e: T).(match e
+in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t5 | (TLRef _)
+\Rightarrow t5 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t5) (THead
+(Bind b) u t2) H8) in ((let H11 \def (f_equal T T (\lambda (e: T).(match e in
+T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _)
+\Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t5) (THead
+(Bind b) u t2) H8) in ((let H12 \def (f_equal T K (\lambda (e: T).(match e in
+T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
+\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t5) (THead (Bind
+b) u t2) H8) in (eq_ind K (Bind b) (\lambda (k0: K).((eq T u1 u) \to ((eq T
+t5 t2) \to ((eq T (THead k0 u2 t6) t4) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to
+(ty3 g c2 t4 (THead (Bind b) u t3)))))))) (\lambda (H13: (eq T u1 u)).(eq_ind
+T u (\lambda (t7: T).((eq T t5 t2) \to ((eq T (THead (Bind b) u2 t6) t4) \to
+((pr0 t7 u2) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3)))))))
+(\lambda (H14: (eq T t5 t2)).(eq_ind T t2 (\lambda (t7: T).((eq T (THead
+(Bind b) u2 t6) t4) \to ((pr0 u u2) \to ((pr0 t7 t6) \to (ty3 g c2 t4 (THead
+(Bind b) u t3)))))) (\lambda (H15: (eq T (THead (Bind b) u2 t6) t4)).(eq_ind
+T (THead (Bind b) u2 t6) (\lambda (t7: T).((pr0 u u2) \to ((pr0 t2 t6) \to
+(ty3 g c2 t7 (THead (Bind b) u t3))))) (\lambda (H16: (pr0 u u2)).(\lambda
+(H17: (pr0 t2 t6)).(ex_ind T (\lambda (t7: T).(ty3 g (CHead c2 (Bind b) u) t3
+t7)) (ty3 g c2 (THead (Bind b) u2 t6) (THead (Bind b) u t3)) (\lambda (x:
+T).(\lambda (H18: (ty3 g (CHead c2 (Bind b) u) t3 x)).(ex_ind T (\lambda (t7:
+T).(ty3 g (CHead c2 (Bind b) u2) t3 t7)) (ty3 g c2 (THead (Bind b) u2 t6)
+(THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (_: (ty3 g (CHead c2 (Bind
+b) u2) t3 x0)).(ty3_conv g c2 (THead (Bind b) u t3) (THead (Bind b) u x)
+(ty3_bind g c2 u t0 (H1 c2 H4 u (pr0_refl u)) b t3 x H18) (THead (Bind b) u2
+t6) (THead (Bind b) u2 t3) (ty3_bind g c2 u2 t0 (H1 c2 H4 u2 H16) b t6 t3 (H3
+(CHead c2 (Bind b) u2) (wcpr0_comp c c2 H4 u u2 H16 (Bind b)) t6 H17))
+(pc3_pr2_x c2 (THead (Bind b) u2 t3) (THead (Bind b) u t3) (pr2_head_1 c2 u
+u2 (pr2_free c2 u u2 H16) (Bind b) t3))))) (ty3_correct g (CHead c2 (Bind b)
+u2) t6 t3 (H3 (CHead c2 (Bind b) u2) (wcpr0_comp c c2 H4 u u2 H16 (Bind b))
+t6 H17))))) (ty3_correct g (CHead c2 (Bind b) u) t2 t3 (H3 (CHead c2 (Bind b)
+u) (wcpr0_comp c c2 H4 u u (pr0_refl u) (Bind b)) t2 (pr0_refl t2)))))) t4
+H15)) t5 (sym_eq T t5 t2 H14))) u1 (sym_eq T u1 u H13))) k (sym_eq K k (Bind
+b) H12))) H11)) H10)) H9 H6 H7))) | (pr0_beta u0 v1 v2 H6 t5 t6 H7)
+\Rightarrow (\lambda (H8: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u0
+t5)) (THead (Bind b) u t2))).(\lambda (H9: (eq T (THead (Bind Abbr) v2 t6)
+t4)).((let H10 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u0
+t5)) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort
+_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
+\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
+\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u t2)
+H8) in (False_ind ((eq T (THead (Bind Abbr) v2 t6) t4) \to ((pr0 v1 v2) \to
+((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3))))) H10)) H9 H6 H7))) |
+(pr0_upsilon b0 H6 v1 v2 H7 u1 u2 H8 t5 t6 H9) \Rightarrow (\lambda (H10: (eq
+T (THead (Flat Appl) v1 (THead (Bind b0) u1 t5)) (THead (Bind b) u
+t2))).(\lambda (H11: (eq T (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O)
+O v2) t6)) t4)).((let H12 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind
+b0) u1 t5)) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with
[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u t2)
-H12) in (False_ind ((eq T (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O)
-O v2) t7)) t5) \to ((not (eq B b0 Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2)
-\to ((pr0 t6 t7) \to (ty3 g c2 t5 (THead (Bind b) u t3))))))) H14)) H13 H8 H9
-H10 H11))) | (pr0_delta u1 u2 H8 t6 t7 H9 w H10) \Rightarrow (\lambda (H11:
-(eq T (THead (Bind Abbr) u1 t6) (THead (Bind b) u t2))).(\lambda (H12: (eq T
-(THead (Bind Abbr) u2 w) t5)).((let H13 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t6 |
-(TLRef _) \Rightarrow t6 | (THead _ _ t8) \Rightarrow t8])) (THead (Bind
-Abbr) u1 t6) (THead (Bind b) u t2) H11) in ((let H14 \def (f_equal T T
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t8 _) \Rightarrow t8]))
-(THead (Bind Abbr) u1 t6) (THead (Bind b) u t2) H11) in ((let H15 \def
-(f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with
-[(TSort _) \Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k _ _)
-\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0)
-\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (THead (Bind Abbr) u1 t6)
-(THead (Bind b) u t2) H11) in (eq_ind B Abbr (\lambda (b0: B).((eq T u1 u)
-\to ((eq T t6 t2) \to ((eq T (THead (Bind Abbr) u2 w) t5) \to ((pr0 u1 u2)
-\to ((pr0 t6 t7) \to ((subst0 O u2 t7 w) \to (ty3 g c2 t5 (THead (Bind b0) u
-t3))))))))) (\lambda (H16: (eq T u1 u)).(eq_ind T u (\lambda (t8: T).((eq T
-t6 t2) \to ((eq T (THead (Bind Abbr) u2 w) t5) \to ((pr0 t8 u2) \to ((pr0 t6
-t7) \to ((subst0 O u2 t7 w) \to (ty3 g c2 t5 (THead (Bind Abbr) u t3))))))))
-(\lambda (H17: (eq T t6 t2)).(eq_ind T t2 (\lambda (t8: T).((eq T (THead
-(Bind Abbr) u2 w) t5) \to ((pr0 u u2) \to ((pr0 t8 t7) \to ((subst0 O u2 t7
-w) \to (ty3 g c2 t5 (THead (Bind Abbr) u t3))))))) (\lambda (H18: (eq T
-(THead (Bind Abbr) u2 w) t5)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda
-(t8: T).((pr0 u u2) \to ((pr0 t2 t7) \to ((subst0 O u2 t7 w) \to (ty3 g c2 t8
-(THead (Bind Abbr) u t3)))))) (\lambda (H19: (pr0 u u2)).(\lambda (H20: (pr0
-t2 t7)).(\lambda (H21: (subst0 O u2 t7 w)).(let H22 \def (eq_ind_r B b
-(\lambda (b0: B).(\forall (c3: C).((wcpr0 (CHead c (Bind b0) u) c3) \to
-(\forall (t8: T).((pr0 t3 t8) \to (ty3 g c3 t8 t4)))))) H5 Abbr H15) in (let
-H23 \def (eq_ind_r B b (\lambda (b0: B).(ty3 g (CHead c (Bind b0) u) t3 t4))
-H4 Abbr H15) in (let H24 \def (eq_ind_r B b (\lambda (b0: B).(\forall (c3:
-C).((wcpr0 (CHead c (Bind b0) u) c3) \to (\forall (t8: T).((pr0 t2 t8) \to
-(ty3 g c3 t8 t3)))))) H3 Abbr H15) in (let H25 \def (eq_ind_r B b (\lambda
-(b0: B).(ty3 g (CHead c (Bind b0) u) t2 t3)) H2 Abbr H15) in (ex_ind T
-(\lambda (t8: T).(ty3 g (CHead c2 (Bind Abbr) u) t4 t8)) (ty3 g c2 (THead
-(Bind Abbr) u2 w) (THead (Bind Abbr) u t3)) (\lambda (x: T).(\lambda (H26:
-(ty3 g (CHead c2 (Bind Abbr) u) t4 x)).(ex_ind T (\lambda (t8: T).(ty3 g
-(CHead c2 (Bind Abbr) u2) t3 t8)) (ty3 g c2 (THead (Bind Abbr) u2 w) (THead
-(Bind Abbr) u t3)) (\lambda (x0: T).(\lambda (H27: (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 (H22 (CHead c2
-(Bind Abbr) u) (wcpr0_comp c c2 H6 u u (pr0_refl u) (Bind Abbr)) t3 (pr0_refl
-t3)) x H26) (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u2 t3) (ty3_bind g c2
-u2 t0 (H1 c2 H6 u2 H19) Abbr w t3 (ty3_subst0 g (CHead c2 (Bind Abbr) u2) t7
-t3 (H24 (CHead c2 (Bind Abbr) u2) (wcpr0_comp c c2 H6 u u2 H19 (Bind Abbr))
-t7 H20) c2 u2 O (getl_refl Abbr c2 u2) w H21) x0 H27) (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 H19) (Bind Abbr) t3))))) (ty3_correct g (CHead c2 (Bind Abbr) u2) t7 t3
-(H24 (CHead c2 (Bind Abbr) u2) (wcpr0_comp c c2 H6 u u2 H19 (Bind Abbr)) t7
-H20))))) (ty3_correct g (CHead c2 (Bind Abbr) u) t3 t4 (H22 (CHead c2 (Bind
-Abbr) u) (wcpr0_comp c c2 H6 u u (pr0_refl u) (Bind Abbr)) t3 (pr0_refl
-t3))))))))))) t5 H18)) t6 (sym_eq T t6 t2 H17))) u1 (sym_eq T u1 u H16))) b
-H15)) H14)) H13)) H12 H8 H9 H10))) | (pr0_zeta b0 H8 t6 t7 H9 u0) \Rightarrow
-(\lambda (H10: (eq T (THead (Bind b0) u0 (lift (S O) O t6)) (THead (Bind b) u
-t2))).(\lambda (H11: (eq T t7 t5)).((let H12 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let
-rec lref_map (f: ((nat \to nat))) (d: nat) (t8: T) on t8: T \def (match t8
-with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match
-(blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u1
-t9) \Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d) t9))]) in
-lref_map) (\lambda (x: nat).(plus x (S O))) O t6) | (TLRef _) \Rightarrow
-((let rec lref_map (f: ((nat \to nat))) (d: nat) (t8: T) on t8: T \def (match
-t8 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
+H10) in (False_ind ((eq T (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O)
+O v2) t6)) t4) \to ((not (eq B b0 Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2)
+\to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3))))))) H12)) H11 H6 H7
+H8 H9))) | (pr0_delta u1 u2 H6 t5 t6 H7 w H8) \Rightarrow (\lambda (H9: (eq T
+(THead (Bind Abbr) u1 t5) (THead (Bind b) u t2))).(\lambda (H10: (eq T (THead
+(Bind Abbr) u2 w) t4)).((let H11 \def (f_equal T T (\lambda (e: T).(match e
+in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t5 | (TLRef _)
+\Rightarrow t5 | (THead _ _ t7) \Rightarrow t7])) (THead (Bind Abbr) u1 t5)
+(THead (Bind b) u t2) H9) in ((let H12 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 |
+(TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead (Bind
+Abbr) u1 t5) (THead (Bind b) u t2) H9) in ((let H13 \def (f_equal T B
+(\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _)
+\Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k _ _) \Rightarrow
+(match k in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 |
+(Flat _) \Rightarrow Abbr])])) (THead (Bind Abbr) u1 t5) (THead (Bind b) u
+t2) H9) in (eq_ind B Abbr (\lambda (b0: B).((eq T u1 u) \to ((eq T t5 t2) \to
+((eq T (THead (Bind Abbr) u2 w) t4) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to
+((subst0 O u2 t6 w) \to (ty3 g c2 t4 (THead (Bind b0) u t3))))))))) (\lambda
+(H14: (eq T u1 u)).(eq_ind T u (\lambda (t7: T).((eq T t5 t2) \to ((eq T
+(THead (Bind Abbr) u2 w) t4) \to ((pr0 t7 u2) \to ((pr0 t5 t6) \to ((subst0 O
+u2 t6 w) \to (ty3 g c2 t4 (THead (Bind Abbr) u t3)))))))) (\lambda (H15: (eq
+T t5 t2)).(eq_ind T t2 (\lambda (t7: T).((eq T (THead (Bind Abbr) u2 w) t4)
+\to ((pr0 u u2) \to ((pr0 t7 t6) \to ((subst0 O u2 t6 w) \to (ty3 g c2 t4
+(THead (Bind Abbr) u t3))))))) (\lambda (H16: (eq T (THead (Bind Abbr) u2 w)
+t4)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t7: T).((pr0 u u2) \to
+((pr0 t2 t6) \to ((subst0 O u2 t6 w) \to (ty3 g c2 t7 (THead (Bind Abbr) u
+t3)))))) (\lambda (H17: (pr0 u u2)).(\lambda (H18: (pr0 t2 t6)).(\lambda
+(H19: (subst0 O u2 t6 w)).(let H20 \def (eq_ind_r B b (\lambda (b0:
+B).(\forall (c3: C).((wcpr0 (CHead c (Bind b0) u) c3) \to (\forall (t7:
+T).((pr0 t2 t7) \to (ty3 g c3 t7 t3)))))) H3 Abbr H13) in (let H21 \def
+(eq_ind_r B b (\lambda (b0: B).(ty3 g (CHead c (Bind b0) u) t2 t3)) H2 Abbr
+H13) in (ex_ind T (\lambda (t7: T).(ty3 g (CHead c2 (Bind Abbr) u) t3 t7))
+(ty3 g c2 (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u t3)) (\lambda (x:
+T).(\lambda (H22: (ty3 g (CHead c2 (Bind Abbr) u) t3 x)).(ex_ind T (\lambda
+(t7: T).(ty3 g (CHead c2 (Bind Abbr) u2) t3 t7)) (ty3 g c2 (THead (Bind Abbr)
+u2 w) (THead (Bind Abbr) u t3)) (\lambda (x0: T).(\lambda (_: (ty3 g (CHead
+c2 (Bind Abbr) u2) t3 x0)).(ty3_conv g c2 (THead (Bind Abbr) u t3) (THead
+(Bind Abbr) u x) (ty3_bind g c2 u t0 (H1 c2 H4 u (pr0_refl u)) Abbr t3 x H22)
+(THead (Bind Abbr) u2 w) (THead (Bind Abbr) u2 t3) (ty3_bind g c2 u2 t0 (H1
+c2 H4 u2 H17) Abbr w t3 (ty3_subst0 g (CHead c2 (Bind Abbr) u2) t6 t3 (H20
+(CHead c2 (Bind Abbr) u2) (wcpr0_comp c c2 H4 u u2 H17 (Bind Abbr)) t6 H18)
+c2 u2 O (getl_refl Abbr c2 u2) w H19)) (pc3_pr2_x c2 (THead (Bind Abbr) u2
+t3) (THead (Bind Abbr) u t3) (pr2_head_1 c2 u u2 (pr2_free c2 u u2 H17) (Bind
+Abbr) t3))))) (ty3_correct g (CHead c2 (Bind Abbr) u2) t6 t3 (H20 (CHead c2
+(Bind Abbr) u2) (wcpr0_comp c c2 H4 u u2 H17 (Bind Abbr)) t6 H18)))))
+(ty3_correct g (CHead c2 (Bind Abbr) u) t2 t3 (H20 (CHead c2 (Bind Abbr) u)
+(wcpr0_comp c c2 H4 u u (pr0_refl u) (Bind Abbr)) t2 (pr0_refl t2))))))))) t4
+H16)) t5 (sym_eq T t5 t2 H15))) u1 (sym_eq T u1 u H14))) b H13)) H12)) H11))
+H10 H6 H7 H8))) | (pr0_zeta b0 H6 t5 t6 H7 u0) \Rightarrow (\lambda (H8: (eq
+T (THead (Bind b0) u0 (lift (S O) O t5)) (THead (Bind b) u t2))).(\lambda
+(H9: (eq T t6 t4)).((let H10 \def (f_equal T T (\lambda (e: T).(match e in T
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f:
+((nat \to nat))) (d: nat) (t7: T) on t7: T \def (match t7 with [(TSort n)
+\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
+[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u1 t8)
+\Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d) t8))]) in
+lref_map) (\lambda (x: nat).(plus x (S O))) O t5) | (TLRef _) \Rightarrow
+((let rec lref_map (f: ((nat \to nat))) (d: nat) (t7: T) on t7: T \def (match
+t7 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
-(THead k u1 t9) \Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d)
-t9))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t6) | (THead _ _ t8)
-\Rightarrow t8])) (THead (Bind b0) u0 (lift (S O) O t6)) (THead (Bind b) u
-t2) H10) in ((let H13 \def (f_equal T T (\lambda (e: T).(match e in T return
+(THead k u1 t8) \Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d)
+t8))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t5) | (THead _ _ t7)
+\Rightarrow t7])) (THead (Bind b0) u0 (lift (S O) O t5)) (THead (Bind b) u
+t2) H8) in ((let H11 \def (f_equal T T (\lambda (e: T).(match e in T return
(\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0
-| (THead _ t8 _) \Rightarrow t8])) (THead (Bind b0) u0 (lift (S O) O t6))
-(THead (Bind b) u t2) H10) in ((let H14 \def (f_equal T B (\lambda (e:
+| (THead _ t7 _) \Rightarrow t7])) (THead (Bind b0) u0 (lift (S O) O t5))
+(THead (Bind b) u t2) H8) in ((let H12 \def (f_equal T B (\lambda (e:
T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b0 |
(TLRef _) \Rightarrow b0 | (THead k _ _) \Rightarrow (match k in K return
(\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow
-b0])])) (THead (Bind b0) u0 (lift (S O) O t6)) (THead (Bind b) u t2) H10) in
-(eq_ind B b (\lambda (b1: B).((eq T u0 u) \to ((eq T (lift (S O) O t6) t2)
-\to ((eq T t7 t5) \to ((not (eq B b1 Abst)) \to ((pr0 t6 t7) \to (ty3 g c2 t5
-(THead (Bind b) u t3)))))))) (\lambda (H15: (eq T u0 u)).(eq_ind T u (\lambda
-(_: T).((eq T (lift (S O) O t6) t2) \to ((eq T t7 t5) \to ((not (eq B b
-Abst)) \to ((pr0 t6 t7) \to (ty3 g c2 t5 (THead (Bind b) u t3))))))) (\lambda
-(H16: (eq T (lift (S O) O t6) t2)).(eq_ind T (lift (S O) O t6) (\lambda (_:
-T).((eq T t7 t5) \to ((not (eq B b Abst)) \to ((pr0 t6 t7) \to (ty3 g c2 t5
-(THead (Bind b) u t3)))))) (\lambda (H17: (eq T t7 t5)).(eq_ind T t5 (\lambda
-(t8: T).((not (eq B b Abst)) \to ((pr0 t6 t8) \to (ty3 g c2 t5 (THead (Bind
-b) u t3))))) (\lambda (H18: (not (eq B b Abst))).(\lambda (H19: (pr0 t6
-t5)).(let H20 \def (eq_ind_r T t2 (\lambda (t8: T).(\forall (c3: C).((wcpr0
-(CHead c (Bind b) u) c3) \to (\forall (t9: T).((pr0 t8 t9) \to (ty3 g c3 t9
-t3)))))) H3 (lift (S O) O t6) H16) in (let H21 \def (eq_ind_r T t2 (\lambda
-(t8: T).(ty3 g (CHead c (Bind b) u) t8 t3)) H2 (lift (S O) O t6) H16) in
-(ex_ind T (\lambda (t8: T).(ty3 g (CHead c2 (Bind b) u) t4 t8)) (ty3 g c2 t5
-(THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H22: (ty3 g (CHead c2 (Bind
-b) u) t4 x)).(B_ind (\lambda (b1: B).((not (eq B b1 Abst)) \to ((ty3 g (CHead
-c2 (Bind b1) u) t3 t4) \to ((ty3 g (CHead c2 (Bind b1) u) t4 x) \to ((ty3 g
-(CHead c2 (Bind b1) u) (lift (S O) O t5) t3) \to (ty3 g c2 t5 (THead (Bind
-b1) u t3))))))) (\lambda (H23: (not (eq B Abbr Abst))).(\lambda (H24: (ty3 g
-(CHead c2 (Bind Abbr) u) t3 t4)).(\lambda (H25: (ty3 g (CHead c2 (Bind Abbr)
-u) t4 x)).(\lambda (H26: (ty3 g (CHead c2 (Bind Abbr) u) (lift (S O) O t5)
-t3)).(let H27 \def (ty3_gen_cabbr g (CHead c2 (Bind Abbr) u) (lift (S O) O
-t5) t3 H26 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 (H28: (subst1 O u (lift (S O) O t5) (lift (S O)
-O x0))).(\lambda (H29: (subst1 O u t3 (lift (S O) O x1))).(\lambda (H30: (ty3
-g c2 x0 x1)).(let H31 \def (eq_ind T x0 (\lambda (t8: T).(ty3 g c2 t8 x1))
-H30 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))) H28))) 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 H24 x H25) t5 x1 H31 (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) H29))) 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 H23 x1 x1 (pr0_refl x1) u)))))))))))) H27)))))) (\lambda (H23: (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 H27 \def (match (H23 (refl_equal B
-Abst)) in False return (\lambda (_: False).(ty3 g c2 t5 (THead (Bind Abst) u
-t3))) with []) in H27))))) (\lambda (H23: (not (eq B Void Abst))).(\lambda
-(H24: (ty3 g (CHead c2 (Bind Void) u) t3 t4)).(\lambda (H25: (ty3 g (CHead c2
-(Bind Void) u) t4 x)).(\lambda (H26: (ty3 g (CHead c2 (Bind Void) u) (lift (S
-O) O t5) t3)).(let H27 \def (ty3_gen_cvoid g (CHead c2 (Bind Void) u) (lift
-(S O) O t5) t3 H26 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 (H28: (eq T (lift (S O) O t5) (lift (S O) O x0))).(\lambda
-(H29: (eq T t3 (lift (S O) O x1))).(\lambda (H30: (ty3 g c2 x0 x1)).(let H31
-\def (eq_ind T t3 (\lambda (t8: T).(ty3 g (CHead c2 (Bind Void) u) t8 t4))
-H24 (lift (S O) O x1) H29) in (eq_ind_r T (lift (S O) O x1) (\lambda (t8:
-T).(ty3 g c2 t5 (THead (Bind Void) u t8))) (let H32 \def (eq_ind_r T x0
-(\lambda (t8: T).(ty3 g c2 t8 x1)) H30 t5 (lift_inj t5 x0 (S O) O H28)) 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
-H31 x H25) t5 x1 H32 (pc3_s c2 x1 (THead (Bind Void) u (lift (S O) O x1))
-(pc3_pr2_r c2 (THead (Bind Void) u (lift (S O) O x1)) x1 (pr2_free c2 (THead
-(Bind Void) u (lift (S O) O x1)) x1 (pr0_zeta Void H23 x1 x1 (pr0_refl x1)
-u)))))) t3 H29))))))) H27)))))) b H18 (H5 (CHead c2 (Bind b) u) (wcpr0_comp c
-c2 H6 u u (pr0_refl u) (Bind b)) t3 (pr0_refl t3)) H22 (H20 (CHead c2 (Bind
-b) u) (wcpr0_comp c c2 H6 u u (pr0_refl u) (Bind b)) (lift (S O) O t5)
-(pr0_lift t6 t5 H19 (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)))))))) t7 (sym_eq T t7 t5 H17))) t2 H16)) u0 (sym_eq T u0 u
-H15))) b0 (sym_eq B b0 b H14))) H13)) H12)) H11 H8 H9))) | (pr0_epsilon t6 t7
-H8 u0) \Rightarrow (\lambda (H9: (eq T (THead (Flat Cast) u0 t6) (THead (Bind
-b) u t2))).(\lambda (H10: (eq T t7 t5)).((let H11 \def (eq_ind T (THead (Flat
-Cast) u0 t6) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
-\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u t2)
-H9) in (False_ind ((eq T t7 t5) \to ((pr0 t6 t7) \to (ty3 g c2 t5 (THead
-(Bind b) u t3)))) H11)) H10 H8)))]) 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 in
-pr0 return (\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).((eq T
-t3 (THead (Flat Appl) w v)) \to ((eq T t4 t2) \to (ty3 g c2 t2 (THead (Flat
-Appl) w (THead (Bind Abst) u t0)))))))) with [(pr0_refl t3) \Rightarrow
-(\lambda (H6: (eq T t3 (THead (Flat Appl) w v))).(\lambda (H7: (eq T t3
-t2)).(eq_ind T (THead (Flat Appl) w v) (\lambda (t4: T).((eq T t4 t2) \to
-(ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0))))) (\lambda (H8:
-(eq T (THead (Flat Appl) w v) t2)).(eq_ind T (THead (Flat Appl) w v) (\lambda
-(t4: T).(ty3 g c2 t4 (THead (Flat Appl) w (THead (Bind Abst) u t0))))
-(ty3_appl g c2 w u (H1 c2 H4 w (pr0_refl w)) v t0 (H3 c2 H4 v (pr0_refl v)))
-t2 H8)) t3 (sym_eq T t3 (THead (Flat Appl) w v) H6) H7))) | (pr0_comp u1 u2
-H6 t3 t4 H7 k) \Rightarrow (\lambda (H8: (eq T (THead k u1 t3) (THead (Flat
-Appl) w v))).(\lambda (H9: (eq T (THead k u2 t4) t2)).((let H10 \def (f_equal
-T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t5) \Rightarrow t5]))
-(THead k u1 t3) (THead (Flat Appl) w v) H8) in ((let H11 \def (f_equal T T
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t5 _) \Rightarrow t5]))
-(THead k u1 t3) (THead (Flat Appl) w v) H8) in ((let H12 \def (f_equal T K
-(\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _)
-\Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0]))
-(THead k u1 t3) (THead (Flat Appl) w v) H8) in (eq_ind K (Flat Appl) (\lambda
-(k0: K).((eq T u1 w) \to ((eq T t3 v) \to ((eq T (THead k0 u2 t4) t2) \to
-((pr0 u1 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead
-(Bind Abst) u t0))))))))) (\lambda (H13: (eq T u1 w)).(eq_ind T w (\lambda
-(t5: T).((eq T t3 v) \to ((eq T (THead (Flat Appl) u2 t4) t2) \to ((pr0 t5
-u2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst)
-u t0)))))))) (\lambda (H14: (eq T t3 v)).(eq_ind T v (\lambda (t5: T).((eq T
-(THead (Flat Appl) u2 t4) t2) \to ((pr0 w u2) \to ((pr0 t5 t4) \to (ty3 g c2
-t2 (THead (Flat Appl) w (THead (Bind Abst) u t0))))))) (\lambda (H15: (eq T
-(THead (Flat Appl) u2 t4) t2)).(eq_ind T (THead (Flat Appl) u2 t4) (\lambda
-(t5: T).((pr0 w u2) \to ((pr0 v t4) \to (ty3 g c2 t5 (THead (Flat Appl) w
-(THead (Bind Abst) u t0)))))) (\lambda (H16: (pr0 w u2)).(\lambda (H17: (pr0
-v t4)).(ex_ind T (\lambda (t5: T).(ty3 g c2 (THead (Bind Abst) u t0) t5))
-(ty3 g c2 (THead (Flat Appl) u2 t4) (THead (Flat Appl) w (THead (Bind Abst) u
-t0))) (\lambda (x: T).(\lambda (H18: (ty3 g c2 (THead (Bind Abst) u t0)
-x)).(ex4_3_ind T T T (\lambda (t5: T).(\lambda (_: T).(\lambda (_: T).(pc3 c2
-(THead (Bind Abst) u t5) x)))) (\lambda (_: T).(\lambda (t6: T).(\lambda (_:
-T).(ty3 g c2 u t6)))) (\lambda (t5: T).(\lambda (_: T).(\lambda (_: T).(ty3 g
-(CHead c2 (Bind Abst) u) t0 t5)))) (\lambda (t5: T).(\lambda (_: T).(\lambda
-(t7: T).(ty3 g (CHead c2 (Bind Abst) u) t5 t7)))) (ty3 g c2 (THead (Flat
-Appl) u2 t4) (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 (H20: (ty3 g c2 u x1)).(\lambda (H21: (ty3 g (CHead c2
-(Bind Abst) u) t0 x0)).(\lambda (H22: (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
+b0])])) (THead (Bind b0) u0 (lift (S O) O t5)) (THead (Bind b) u t2) H8) in
+(eq_ind B b (\lambda (b1: B).((eq T u0 u) \to ((eq T (lift (S O) O t5) t2)
+\to ((eq T t6 t4) \to ((not (eq B b1 Abst)) \to ((pr0 t5 t6) \to (ty3 g c2 t4
+(THead (Bind b) u t3)))))))) (\lambda (H13: (eq T u0 u)).(eq_ind T u (\lambda
+(_: T).((eq T (lift (S O) O t5) t2) \to ((eq T t6 t4) \to ((not (eq B b
+Abst)) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3))))))) (\lambda
+(H14: (eq T (lift (S O) O t5) t2)).(eq_ind T (lift (S O) O t5) (\lambda (_:
+T).((eq T t6 t4) \to ((not (eq B b Abst)) \to ((pr0 t5 t6) \to (ty3 g c2 t4
+(THead (Bind b) u t3)))))) (\lambda (H15: (eq T t6 t4)).(eq_ind T t4 (\lambda
+(t7: T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to (ty3 g c2 t4 (THead (Bind
+b) u t3))))) (\lambda (H16: (not (eq B b Abst))).(\lambda (H17: (pr0 t5
+t4)).(let H18 \def (eq_ind_r T t2 (\lambda (t7: T).(\forall (c3: C).((wcpr0
+(CHead c (Bind b) u) c3) \to (\forall (t8: T).((pr0 t7 t8) \to (ty3 g c3 t8
+t3)))))) H3 (lift (S O) O t5) H14) in (let H19 \def (eq_ind_r T t2 (\lambda
+(t7: T).(ty3 g (CHead c (Bind b) u) t7 t3)) H2 (lift (S O) O t5) H14) in
+(ex_ind T (\lambda (t7: T).(ty3 g (CHead c2 (Bind b) u) t3 t7)) (ty3 g c2 t4
+(THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H20: (ty3 g (CHead c2 (Bind
+b) u) t3 x)).(B_ind (\lambda (b1: B).((not (eq B b1 Abst)) \to ((ty3 g (CHead
+c2 (Bind b1) u) t3 x) \to ((ty3 g (CHead c2 (Bind b1) u) (lift (S O) O t4)
+t3) \to (ty3 g c2 t4 (THead (Bind b1) u t3)))))) (\lambda (H21: (not (eq B
+Abbr Abst))).(\lambda (H22: (ty3 g (CHead c2 (Bind Abbr) u) t3 x)).(\lambda
+(H23: (ty3 g (CHead c2 (Bind Abbr) u) (lift (S O) O t4) t3)).(let H24 \def
+(ty3_gen_cabbr g (CHead c2 (Bind Abbr) u) (lift (S O) O t4) t3 H23 c2 u O
+(getl_refl Abbr c2 u) (CHead c2 (Bind Abbr) u) (csubst1_refl O u (CHead c2
+(Bind Abbr) u)) c2 (drop_drop (Bind Abbr) O c2 c2 (drop_refl c2) u)) in
+(ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 O u (lift (S O) O t4)
+(lift (S O) O y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 O u t3 (lift (S
+O) O y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g c2 y1 y2))) (ty3 g c2 t4
+(THead (Bind Abbr) u t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H25:
+(subst1 O u (lift (S O) O t4) (lift (S O) O x0))).(\lambda (H26: (subst1 O u
+t3 (lift (S O) O x1))).(\lambda (H27: (ty3 g c2 x0 x1)).(let H28 \def (eq_ind
+T x0 (\lambda (t7: T).(ty3 g c2 t7 x1)) H27 t4 (lift_inj x0 t4 (S O) O
+(subst1_gen_lift_eq t4 u (lift (S O) O x0) (S O) O O (le_n O) (eq_ind_r nat
+(plus (S O) O) (\lambda (n: nat).(lt O n)) (le_n (plus (S O) O)) (plus O (S
+O)) (plus_comm O (S O))) H25))) in (ty3_conv g c2 (THead (Bind Abbr) u t3)
+(THead (Bind Abbr) u x) (ty3_bind g c2 u t0 (H1 c2 H4 u (pr0_refl u)) Abbr t3
+x H22) t4 x1 H28 (pc3_pr3_x c2 x1 (THead (Bind Abbr) u t3) (pr3_t (THead
+(Bind Abbr) u (lift (S O) O x1)) (THead (Bind Abbr) u t3) c2 (pr3_pr2 c2
+(THead (Bind Abbr) u t3) (THead (Bind Abbr) u (lift (S O) O x1)) (pr2_free c2
+(THead (Bind Abbr) u t3) (THead (Bind Abbr) u (lift (S O) O x1)) (pr0_delta1
+u u (pr0_refl u) t3 t3 (pr0_refl t3) (lift (S O) O x1) H26))) x1 (pr3_pr2 c2
+(THead (Bind Abbr) u (lift (S O) O x1)) x1 (pr2_free c2 (THead (Bind Abbr) u
+(lift (S O) O x1)) x1 (pr0_zeta Abbr H21 x1 x1 (pr0_refl x1) u))))))))))))
+H24))))) (\lambda (H21: (not (eq B Abst Abst))).(\lambda (_: (ty3 g (CHead c2
+(Bind Abst) u) t3 x)).(\lambda (_: (ty3 g (CHead c2 (Bind Abst) u) (lift (S
+O) O t4) t3)).(let H24 \def (match (H21 (refl_equal B Abst)) in False return
+(\lambda (_: False).(ty3 g c2 t4 (THead (Bind Abst) u t3))) with []) in
+H24)))) (\lambda (H21: (not (eq B Void Abst))).(\lambda (H22: (ty3 g (CHead
+c2 (Bind Void) u) t3 x)).(\lambda (H23: (ty3 g (CHead c2 (Bind Void) u) (lift
+(S O) O t4) t3)).(let H24 \def (ty3_gen_cvoid g (CHead c2 (Bind Void) u)
+(lift (S O) O t4) t3 H23 c2 u O (getl_refl Void c2 u) c2 (drop_drop (Bind
+Void) O c2 c2 (drop_refl c2) u)) in (ex3_2_ind T T (\lambda (y1: T).(\lambda
+(_: T).(eq T (lift (S O) O t4) (lift (S O) O y1)))) (\lambda (_: T).(\lambda
+(y2: T).(eq T t3 (lift (S O) O y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
+g c2 y1 y2))) (ty3 g c2 t4 (THead (Bind Void) u t3)) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H25: (eq T (lift (S O) O t4) (lift (S O) O
+x0))).(\lambda (H26: (eq T t3 (lift (S O) O x1))).(\lambda (H27: (ty3 g c2 x0
+x1)).(let H28 \def (eq_ind T t3 (\lambda (t7: T).(ty3 g (CHead c2 (Bind Void)
+u) t7 x)) H22 (lift (S O) O x1) H26) in (eq_ind_r T (lift (S O) O x1)
+(\lambda (t7: T).(ty3 g c2 t4 (THead (Bind Void) u t7))) (let H29 \def
+(eq_ind_r T x0 (\lambda (t7: T).(ty3 g c2 t7 x1)) H27 t4 (lift_inj t4 x0 (S
+O) O H25)) in (ty3_conv g c2 (THead (Bind Void) u (lift (S O) O x1)) (THead
+(Bind Void) u x) (ty3_bind g c2 u t0 (H1 c2 H4 u (pr0_refl u)) Void (lift (S
+O) O x1) x H28) t4 x1 H29 (pc3_s c2 x1 (THead (Bind Void) u (lift (S O) O
+x1)) (pc3_pr2_r c2 (THead (Bind Void) u (lift (S O) O x1)) x1 (pr2_free c2
+(THead (Bind Void) u (lift (S O) O x1)) x1 (pr0_zeta Void H21 x1 x1 (pr0_refl
+x1) u)))))) t3 H26))))))) H24))))) b H16 H20 (H18 (CHead c2 (Bind b) u)
+(wcpr0_comp c c2 H4 u u (pr0_refl u) (Bind b)) (lift (S O) O t4) (pr0_lift t5
+t4 H17 (S O) O))))) (ty3_correct g (CHead c2 (Bind b) u) (lift (S O) O t4) t3
+(H18 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H4 u u (pr0_refl u) (Bind b))
+(lift (S O) O t4) (pr0_lift t5 t4 H17 (S O) O)))))))) t6 (sym_eq T t6 t4
+H15))) t2 H14)) u0 (sym_eq T u0 u H13))) b0 (sym_eq B b0 b H12))) H11)) H10))
+H9 H6 H7))) | (pr0_epsilon t5 t6 H6 u0) \Rightarrow (\lambda (H7: (eq T
+(THead (Flat Cast) u0 t5) (THead (Bind b) u t2))).(\lambda (H8: (eq T t6
+t4)).((let H9 \def (eq_ind T (THead (Flat Cast) u0 t5) (\lambda (e: T).(match
+e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
+(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return
+(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
+True])])) I (THead (Bind b) u t2) H7) in (False_ind ((eq T t6 t4) \to ((pr0
+t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3)))) H9)) H8 H6)))]) in (H6
+(refl_equal T (THead (Bind b) u t2)) (refl_equal T t4)))))))))))))))))
+(\lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c w
+u)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0
+w t2) \to (ty3 g c2 t2 u))))))).(\lambda (v: T).(\lambda (t0: T).(\lambda
+(H2: (ty3 g c v (THead (Bind Abst) u t0))).(\lambda (H3: ((\forall (c2:
+C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 v t2) \to (ty3 g c2 t2 (THead
+(Bind Abst) u t0)))))))).(\lambda (c2: C).(\lambda (H4: (wcpr0 c
+c2)).(\lambda (t2: T).(\lambda (H5: (pr0 (THead (Flat Appl) w v) t2)).(let H6
+\def (match H5 in pr0 return (\lambda (t3: T).(\lambda (t4: T).(\lambda (_:
+(pr0 t3 t4)).((eq T t3 (THead (Flat Appl) w v)) \to ((eq T t4 t2) \to (ty3 g
+c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0)))))))) with [(pr0_refl
+t3) \Rightarrow (\lambda (H6: (eq T t3 (THead (Flat Appl) w v))).(\lambda
+(H7: (eq T t3 t2)).(eq_ind T (THead (Flat Appl) w v) (\lambda (t4: T).((eq T
+t4 t2) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0)))))
+(\lambda (H8: (eq T (THead (Flat Appl) w v) t2)).(eq_ind T (THead (Flat Appl)
+w v) (\lambda (t4: T).(ty3 g c2 t4 (THead (Flat Appl) w (THead (Bind Abst) u
+t0)))) (ty3_appl g c2 w u (H1 c2 H4 w (pr0_refl w)) v t0 (H3 c2 H4 v
+(pr0_refl v))) t2 H8)) t3 (sym_eq T t3 (THead (Flat Appl) w v) H6) H7))) |
+(pr0_comp u1 u2 H6 t3 t4 H7 k) \Rightarrow (\lambda (H8: (eq T (THead k u1
+t3) (THead (Flat Appl) w v))).(\lambda (H9: (eq T (THead k u2 t4) t2)).((let
+H10 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
+with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t5)
+\Rightarrow t5])) (THead k u1 t3) (THead (Flat Appl) w v) H8) in ((let H11
+\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
+with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t5 _)
+\Rightarrow t5])) (THead k u1 t3) (THead (Flat Appl) w v) H8) in ((let H12
+\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K)
+with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _)
+\Rightarrow k0])) (THead k u1 t3) (THead (Flat Appl) w v) H8) in (eq_ind K
+(Flat Appl) (\lambda (k0: K).((eq T u1 w) \to ((eq T t3 v) \to ((eq T (THead
+k0 u2 t4) t2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat
+Appl) w (THead (Bind Abst) u t0))))))))) (\lambda (H13: (eq T u1 w)).(eq_ind
+T w (\lambda (t5: T).((eq T t3 v) \to ((eq T (THead (Flat Appl) u2 t4) t2)
+\to ((pr0 t5 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w
+(THead (Bind Abst) u t0)))))))) (\lambda (H14: (eq T t3 v)).(eq_ind T v
+(\lambda (t5: T).((eq T (THead (Flat Appl) u2 t4) t2) \to ((pr0 w u2) \to
+((pr0 t5 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u
+t0))))))) (\lambda (H15: (eq T (THead (Flat Appl) u2 t4) t2)).(eq_ind T
+(THead (Flat Appl) u2 t4) (\lambda (t5: T).((pr0 w u2) \to ((pr0 v t4) \to
+(ty3 g c2 t5 (THead (Flat Appl) w (THead (Bind Abst) u t0)))))) (\lambda
+(H16: (pr0 w u2)).(\lambda (H17: (pr0 v t4)).(ex_ind T (\lambda (t5: T).(ty3
+g c2 (THead (Bind Abst) u t0) t5)) (ty3 g c2 (THead (Flat Appl) u2 t4) (THead
+(Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x: T).(\lambda (H18: (ty3
+g c2 (THead (Bind Abst) u t0) x)).(ex3_2_ind T T (\lambda (t5: T).(\lambda
+(_: T).(pc3 c2 (THead (Bind Abst) u t5) x))) (\lambda (_: T).(\lambda (t6:
+T).(ty3 g c2 u t6))) (\lambda (t5: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind
+Abst) u) t0 t5))) (ty3 g c2 (THead (Flat Appl) u2 t4) (THead (Flat Appl) w
+(THead (Bind Abst) u t0))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (_:
+(pc3 c2 (THead (Bind Abst) u x0) x)).(\lambda (H20: (ty3 g c2 u x1)).(\lambda
+(H21: (ty3 g (CHead c2 (Bind Abst) u) t0 x0)).(ty3_conv g c2 (THead (Flat
+Appl) w (THead (Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind Abst) u
+x0)) (ty3_appl g c2 w u (H1 c2 H4 w (pr0_refl w)) (THead (Bind Abst) u t0) x0
+(ty3_bind g c2 u x1 H20 Abst t0 x0 H21)) (THead (Flat Appl) u2 t4) (THead
+(Flat Appl) u2 (THead (Bind Abst) u t0)) (ty3_appl g c2 u2 u (H1 c2 H4 u2
+H16) t4 t0 (H3 c2 H4 t4 H17)) (pc3_pr2_x c2 (THead (Flat Appl) u2 (THead
+(Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind Abst) u t0)) (pr2_head_1
+c2 w u2 (pr2_free c2 w u2 H16) (Flat Appl) (THead (Bind Abst) u t0)))))))))
+(ty3_gen_bind g Abst c2 u t0 x H18)))) (ty3_correct g c2 v (THead (Bind Abst)
+u t0) (H3 c2 H4 v (pr0_refl v)))))) t2 H15)) t3 (sym_eq T t3 v H14))) u1
+(sym_eq T u1 w H13))) k (sym_eq K k (Flat Appl) H12))) H11)) H10)) H9 H6
+H7))) | (pr0_beta u0 v1 v2 H6 t3 t4 H7) \Rightarrow (\lambda (H8: (eq T
+(THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) (THead (Flat Appl) w
+v))).(\lambda (H9: (eq T (THead (Bind Abbr) v2 t4) t2)).((let H10 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow (THead (Bind Abst) u0 t3) | (TLRef _) \Rightarrow
+(THead (Bind Abst) u0 t3) | (THead _ _ t5) \Rightarrow t5])) (THead (Flat
+Appl) v1 (THead (Bind Abst) u0 t3)) (THead (Flat Appl) w v) H8) in ((let H11
+\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
+with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t5 _)
+\Rightarrow t5])) (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) (THead
+(Flat Appl) w v) H8) in (eq_ind T w (\lambda (t5: T).((eq T (THead (Bind
+Abst) u0 t3) v) \to ((eq T (THead (Bind Abbr) v2 t4) t2) \to ((pr0 t5 v2) \to
+((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u
+t0)))))))) (\lambda (H12: (eq T (THead (Bind Abst) u0 t3) v)).(eq_ind T
+(THead (Bind Abst) u0 t3) (\lambda (_: T).((eq T (THead (Bind Abbr) v2 t4)
+t2) \to ((pr0 w v2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w
+(THead (Bind Abst) u t0))))))) (\lambda (H13: (eq T (THead (Bind Abbr) v2 t4)
+t2)).(eq_ind T (THead (Bind Abbr) v2 t4) (\lambda (t5: T).((pr0 w v2) \to
+((pr0 t3 t4) \to (ty3 g c2 t5 (THead (Flat Appl) w (THead (Bind Abst) u
+t0)))))) (\lambda (H14: (pr0 w v2)).(\lambda (H15: (pr0 t3 t4)).(let H16 \def
+(eq_ind_r T v (\lambda (t5: T).(\forall (c3: C).((wcpr0 c c3) \to (\forall
+(t6: T).((pr0 t5 t6) \to (ty3 g c3 t6 (THead (Bind Abst) u t0))))))) H3
+(THead (Bind Abst) u0 t3) H12) in (let H17 \def (eq_ind_r T v (\lambda (t5:
+T).(ty3 g c t5 (THead (Bind Abst) u t0))) H2 (THead (Bind Abst) u0 t3) H12)
+in (ex_ind T (\lambda (t5: T).(ty3 g c2 (THead (Bind Abst) u t0) t5)) (ty3 g
+c2 (THead (Bind Abbr) v2 t4) (THead (Flat Appl) w (THead (Bind Abst) u t0)))
+(\lambda (x: T).(\lambda (H18: (ty3 g c2 (THead (Bind Abst) u t0)
+x)).(ex3_2_ind T T (\lambda (t5: T).(\lambda (_: T).(pc3 c2 (THead (Bind
+Abst) u t5) x))) (\lambda (_: T).(\lambda (t6: T).(ty3 g c2 u t6))) (\lambda
+(t5: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind Abst) u) t0 t5))) (ty3 g c2
+(THead (Bind Abbr) v2 t4) (THead (Flat Appl) w (THead (Bind Abst) u t0)))
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead (Bind Abst) u
+x0) x)).(\lambda (H20: (ty3 g c2 u x1)).(\lambda (H21: (ty3 g (CHead c2 (Bind
+Abst) u) t0 x0)).(ex3_2_ind T T (\lambda (t5: T).(\lambda (_: T).(pc3 c2
+(THead (Bind Abst) u0 t5) (THead (Bind Abst) u t0)))) (\lambda (_:
+T).(\lambda (t6: T).(ty3 g c2 u0 t6))) (\lambda (t5: T).(\lambda (_: T).(ty3
+g (CHead c2 (Bind Abst) u0) t4 t5))) (ty3 g c2 (THead (Bind Abbr) v2 t4)
+(THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x2: T).(\lambda
+(x3: T).(\lambda (H22: (pc3 c2 (THead (Bind Abst) u0 x2) (THead (Bind Abst) u
+t0))).(\lambda (H23: (ty3 g c2 u0 x3)).(\lambda (H24: (ty3 g (CHead c2 (Bind
+Abst) u0) t4 x2)).(and_ind (pc3 c2 u0 u) (\forall (b: B).(\forall (u1:
+T).(pc3 (CHead c2 (Bind b) u1) x2 t0))) (ty3 g c2 (THead (Bind Abbr) v2 t4)
+(THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (H25: (pc3 c2 u0
+u)).(\lambda (H26: ((\forall (b: B).(\forall (u1: T).(pc3 (CHead c2 (Bind b)
+u1) x2 t0))))).(ty3_conv g c2 (THead (Flat Appl) w (THead (Bind Abst) u t0))
+(THead (Flat Appl) w (THead (Bind Abst) u x0)) (ty3_appl g c2 w u (H1 c2 H4 w
(pr0_refl w)) (THead (Bind Abst) u t0) x0 (ty3_bind g c2 u x1 H20 Abst t0 x0
-H21 x2 H22)) (THead (Flat Appl) u2 t4) (THead (Flat Appl) u2 (THead (Bind
-Abst) u t0)) (ty3_appl g c2 u2 u (H1 c2 H4 u2 H16) t4 t0 (H3 c2 H4 t4 H17))
-(pc3_pr2_x c2 (THead (Flat Appl) u2 (THead (Bind Abst) u t0)) (THead (Flat
-Appl) w (THead (Bind Abst) u t0)) (pr2_head_1 c2 w u2 (pr2_free c2 w u2 H16)
-(Flat Appl) (THead (Bind Abst) u t0))))))))))) (ty3_gen_bind g Abst c2 u t0 x
-H18)))) (ty3_correct g c2 v (THead (Bind Abst) u t0) (H3 c2 H4 v (pr0_refl
-v)))))) t2 H15)) t3 (sym_eq T t3 v H14))) u1 (sym_eq T u1 w H13))) k (sym_eq
-K k (Flat Appl) H12))) H11)) H10)) H9 H6 H7))) | (pr0_beta u0 v1 v2 H6 t3 t4
-H7) \Rightarrow (\lambda (H8: (eq T (THead (Flat Appl) v1 (THead (Bind Abst)
-u0 t3)) (THead (Flat Appl) w v))).(\lambda (H9: (eq T (THead (Bind Abbr) v2
-t4) t2)).((let H10 \def (f_equal T T (\lambda (e: T).(match e in T return
-(\lambda (_: T).T) with [(TSort _) \Rightarrow (THead (Bind Abst) u0 t3) |
-(TLRef _) \Rightarrow (THead (Bind Abst) u0 t3) | (THead _ _ t5) \Rightarrow
-t5])) (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) (THead (Flat Appl) w
-v) H8) in ((let H11 \def (f_equal T T (\lambda (e: T).(match e in T return
+H21)) (THead (Bind Abbr) v2 t4) (THead (Bind Abbr) v2 x2) (ty3_bind g c2 v2 u
+(H1 c2 H4 v2 H14) Abbr t4 x2 (csubt_ty3_ld g c2 v2 u0 (ty3_conv g c2 u0 x3
+H23 v2 u (H1 c2 H4 v2 H14) (pc3_s c2 u u0 H25)) t4 x2 H24)) (pc3_t (THead
+(Bind Abbr) v2 t0) c2 (THead (Bind Abbr) v2 x2) (pc3_head_2 c2 v2 x2 t0 (Bind
+Abbr) (H26 Abbr v2)) (THead (Flat Appl) w (THead (Bind Abst) u t0))
+(pc3_pr2_x c2 (THead (Bind Abbr) v2 t0) (THead (Flat Appl) w (THead (Bind
+Abst) u t0)) (pr2_free c2 (THead (Flat Appl) w (THead (Bind Abst) u t0))
+(THead (Bind Abbr) v2 t0) (pr0_beta u w v2 H14 t0 t0 (pr0_refl t0))))))))
+(pc3_gen_abst c2 u0 u x2 t0 H22))))))) (ty3_gen_bind g Abst c2 u0 t4 (THead
+(Bind Abst) u t0) (H16 c2 H4 (THead (Bind Abst) u0 t4) (pr0_comp u0 u0
+(pr0_refl u0) t3 t4 H15 (Bind Abst)))))))))) (ty3_gen_bind g Abst c2 u t0 x
+H18)))) (ty3_correct g c2 (THead (Bind Abst) u0 t3) (THead (Bind Abst) u t0)
+(H16 c2 H4 (THead (Bind Abst) u0 t3) (pr0_refl (THead (Bind Abst) u0
+t3))))))))) t2 H13)) v H12)) v1 (sym_eq T v1 w H11))) H10)) H9 H6 H7))) |
+(pr0_upsilon b H6 v1 v2 H7 u1 u2 H8 t3 t4 H9) \Rightarrow (\lambda (H10: (eq
+T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) w
+v))).(\lambda (H11: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O
+v2) t4)) t2)).((let H12 \def (f_equal T T (\lambda (e: T).(match e in T
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow (THead (Bind b) u1 t3)
+| (TLRef _) \Rightarrow (THead (Bind b) u1 t3) | (THead _ _ t5) \Rightarrow
+t5])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) w v)
+H10) in ((let H13 \def (f_equal T T (\lambda (e: T).(match e in T return
(\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1
-| (THead _ t5 _) \Rightarrow t5])) (THead (Flat Appl) v1 (THead (Bind Abst)
-u0 t3)) (THead (Flat Appl) w v) H8) in (eq_ind T w (\lambda (t5: T).((eq T
-(THead (Bind Abst) u0 t3) v) \to ((eq T (THead (Bind Abbr) v2 t4) t2) \to
-((pr0 t5 v2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead
-(Bind Abst) u t0)))))))) (\lambda (H12: (eq T (THead (Bind Abst) u0 t3)
-v)).(eq_ind T (THead (Bind Abst) u0 t3) (\lambda (_: T).((eq T (THead (Bind
-Abbr) v2 t4) t2) \to ((pr0 w v2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead
-(Flat Appl) w (THead (Bind Abst) u t0))))))) (\lambda (H13: (eq T (THead
-(Bind Abbr) v2 t4) t2)).(eq_ind T (THead (Bind Abbr) v2 t4) (\lambda (t5:
-T).((pr0 w v2) \to ((pr0 t3 t4) \to (ty3 g c2 t5 (THead (Flat Appl) w (THead
-(Bind Abst) u t0)))))) (\lambda (H14: (pr0 w v2)).(\lambda (H15: (pr0 t3
-t4)).(let H16 \def (eq_ind_r T v (\lambda (t5: T).(\forall (c3: C).((wcpr0 c
-c3) \to (\forall (t6: T).((pr0 t5 t6) \to (ty3 g c3 t6 (THead (Bind Abst) u
-t0))))))) H3 (THead (Bind Abst) u0 t3) H12) in (let H17 \def (eq_ind_r T v
-(\lambda (t5: T).(ty3 g c t5 (THead (Bind Abst) u t0))) H2 (THead (Bind Abst)
-u0 t3) H12) in (ex_ind T (\lambda (t5: T).(ty3 g c2 (THead (Bind Abst) u t0)
-t5)) (ty3 g c2 (THead (Bind Abbr) v2 t4) (THead (Flat Appl) w (THead (Bind
-Abst) u t0))) (\lambda (x: T).(\lambda (H18: (ty3 g c2 (THead (Bind Abst) u
-t0) x)).(ex4_3_ind T T T (\lambda (t5: T).(\lambda (_: T).(\lambda (_:
-T).(pc3 c2 (THead (Bind Abst) u t5) x)))) (\lambda (_: T).(\lambda (t6:
-T).(\lambda (_: T).(ty3 g c2 u t6)))) (\lambda (t5: T).(\lambda (_:
-T).(\lambda (_: T).(ty3 g (CHead c2 (Bind Abst) u) t0 t5)))) (\lambda (t5:
-T).(\lambda (_: T).(\lambda (t7: T).(ty3 g (CHead c2 (Bind Abst) u) t5 t7))))
-(ty3 g c2 (THead (Bind Abbr) v2 t4) (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 (H20: (ty3 g c2 u x1)).(\lambda (H21:
-(ty3 g (CHead c2 (Bind Abst) u) t0 x0)).(\lambda (H22: (ty3 g (CHead c2 (Bind
-Abst) u) x0 x2)).(ex4_3_ind T T T (\lambda (t5: T).(\lambda (_: T).(\lambda
-(_: T).(pc3 c2 (THead (Bind Abst) u0 t5) (THead (Bind Abst) u t0)))))
-(\lambda (_: T).(\lambda (t6: T).(\lambda (_: T).(ty3 g c2 u0 t6)))) (\lambda
-(t5: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind Abst) u0) t4
-t5)))) (\lambda (t5: T).(\lambda (_: T).(\lambda (t7: T).(ty3 g (CHead c2
-(Bind Abst) u0) t5 t7)))) (ty3 g c2 (THead (Bind Abbr) v2 t4) (THead (Flat
-Appl) w (THead (Bind Abst) u t0))) (\lambda (x3: T).(\lambda (x4: T).(\lambda
-(x5: T).(\lambda (H23: (pc3 c2 (THead (Bind Abst) u0 x3) (THead (Bind Abst) u
-t0))).(\lambda (H24: (ty3 g c2 u0 x4)).(\lambda (H25: (ty3 g (CHead c2 (Bind
-Abst) u0) t4 x3)).(\lambda (H26: (ty3 g (CHead c2 (Bind Abst) u0) x3
-x5)).(and_ind (pc3 c2 u0 u) (\forall (b: B).(\forall (u1: T).(pc3 (CHead c2
-(Bind b) u1) x3 t0))) (ty3 g c2 (THead (Bind Abbr) v2 t4) (THead (Flat Appl)
-w (THead (Bind Abst) u t0))) (\lambda (H27: (pc3 c2 u0 u)).(\lambda (H28:
-((\forall (b: B).(\forall (u1: T).(pc3 (CHead c2 (Bind b) u1) 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 H20 Abst t0 x0
-H21 x2 H22)) (THead (Bind Abbr) v2 t4) (THead (Bind Abbr) v2 x3) (ty3_bind g
-c2 v2 u (H1 c2 H4 v2 H14) Abbr t4 x3 (csubt_ty3_ld g c2 v2 u0 (ty3_conv g c2
-u0 x4 H24 v2 u (H1 c2 H4 v2 H14) (pc3_s c2 u u0 H27)) t4 x3 H25) x5
-(csubt_ty3_ld g c2 v2 u0 (ty3_conv g c2 u0 x4 H24 v2 u (H1 c2 H4 v2 H14)
-(pc3_s c2 u u0 H27)) x3 x5 H26)) (pc3_t (THead (Bind Abbr) v2 t0) c2 (THead
-(Bind Abbr) v2 x3) (pc3_head_2 c2 v2 x3 t0 (Bind Abbr) (H28 Abbr v2)) (THead
-(Flat Appl) w (THead (Bind Abst) u t0)) (pc3_pr2_x c2 (THead (Bind Abbr) v2
-t0) (THead (Flat Appl) w (THead (Bind Abst) u t0)) (pr2_free c2 (THead (Flat
-Appl) w (THead (Bind Abst) u t0)) (THead (Bind Abbr) v2 t0) (pr0_beta u w v2
-H14 t0 t0 (pr0_refl t0)))))))) (pc3_gen_abst c2 u0 u x3 t0 H23)))))))))
-(ty3_gen_bind g Abst c2 u0 t4 (THead (Bind Abst) u t0) (H16 c2 H4 (THead
-(Bind Abst) u0 t4) (pr0_comp u0 u0 (pr0_refl u0) t3 t4 H15 (Bind
-Abst)))))))))))) (ty3_gen_bind g Abst c2 u t0 x H18)))) (ty3_correct g c2
-(THead (Bind Abst) u0 t3) (THead (Bind Abst) u t0) (H16 c2 H4 (THead (Bind
-Abst) u0 t3) (pr0_refl (THead (Bind Abst) u0 t3))))))))) t2 H13)) v H12)) v1
-(sym_eq T v1 w H11))) H10)) H9 H6 H7))) | (pr0_upsilon b H6 v1 v2 H7 u1 u2 H8
-t3 t4 H9) \Rightarrow (\lambda (H10: (eq T (THead (Flat Appl) v1 (THead (Bind
-b) u1 t3)) (THead (Flat Appl) w v))).(\lambda (H11: (eq T (THead (Bind b) u2
-(THead (Flat Appl) (lift (S O) O v2) t4)) t2)).((let H12 \def (f_equal T T
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow (THead (Bind b) u1 t3) | (TLRef _) \Rightarrow (THead (Bind b) u1
-t3) | (THead _ _ t5) \Rightarrow t5])) (THead (Flat Appl) v1 (THead (Bind b)
-u1 t3)) (THead (Flat Appl) w v) H10) in ((let H13 \def (f_equal T T (\lambda
-(e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1
-| (TLRef _) \Rightarrow v1 | (THead _ t5 _) \Rightarrow t5])) (THead (Flat
-Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) w v) H10) in (eq_ind T w
-(\lambda (t5: T).((eq T (THead (Bind b) u1 t3) v) \to ((eq T (THead (Bind b)
+| (THead _ t5 _) \Rightarrow t5])) (THead (Flat Appl) v1 (THead (Bind b) u1
+t3)) (THead (Flat Appl) w v) H10) in (eq_ind T w (\lambda (t5: T).((eq T
+(THead (Bind b) u1 t3) v) \to ((eq T (THead (Bind b) u2 (THead (Flat Appl)
+(lift (S O) O v2) t4)) t2) \to ((not (eq B b Abst)) \to ((pr0 t5 v2) \to
+((pr0 u1 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead
+(Bind Abst) u t0)))))))))) (\lambda (H14: (eq T (THead (Bind b) u1 t3)
+v)).(eq_ind T (THead (Bind b) u1 t3) (\lambda (_: T).((eq T (THead (Bind b)
u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t2) \to ((not (eq B b Abst)) \to
-((pr0 t5 v2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat
-Appl) w (THead (Bind Abst) u t0)))))))))) (\lambda (H14: (eq T (THead (Bind
-b) u1 t3) v)).(eq_ind T (THead (Bind b) u1 t3) (\lambda (_: T).((eq T (THead
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t2) \to ((not (eq B b
-Abst)) \to ((pr0 w v2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t2
-(THead (Flat Appl) w (THead (Bind Abst) u t0))))))))) (\lambda (H15: (eq T
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t2)).(eq_ind T
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) (\lambda (t5:
-T).((not (eq B b Abst)) \to ((pr0 w v2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to
-(ty3 g c2 t5 (THead (Flat Appl) w (THead (Bind Abst) u t0)))))))) (\lambda
-(H16: (not (eq B b Abst))).(\lambda (H17: (pr0 w v2)).(\lambda (H18: (pr0 u1
-u2)).(\lambda (H19: (pr0 t3 t4)).(let H20 \def (eq_ind_r T v (\lambda (t5:
-T).(\forall (c3: C).((wcpr0 c c3) \to (\forall (t6: T).((pr0 t5 t6) \to (ty3
-g c3 t6 (THead (Bind Abst) u t0))))))) H3 (THead (Bind b) u1 t3) H14) in (let
-H21 \def (eq_ind_r T v (\lambda (t5: T).(ty3 g c t5 (THead (Bind Abst) u
-t0))) H2 (THead (Bind b) u1 t3) H14) in (ex_ind T (\lambda (t5: T).(ty3 g c2
-(THead (Bind Abst) u t0) t5)) (ty3 g c2 (THead (Bind b) u2 (THead (Flat Appl)
-(lift (S O) O v2) t4)) (THead (Flat Appl) w (THead (Bind Abst) u t0)))
-(\lambda (x: T).(\lambda (H22: (ty3 g c2 (THead (Bind Abst) u t0) x)).(let
-H23 \def H22 in (ex4_3_ind T T T (\lambda (t5: T).(\lambda (_: T).(\lambda
-(_: T).(pc3 c2 (THead (Bind Abst) u t5) x)))) (\lambda (_: T).(\lambda (t6:
-T).(\lambda (_: T).(ty3 g c2 u t6)))) (\lambda (t5: T).(\lambda (_:
-T).(\lambda (_: T).(ty3 g (CHead c2 (Bind Abst) u) t0 t5)))) (\lambda (t5:
-T).(\lambda (_: T).(\lambda (t7: T).(ty3 g (CHead c2 (Bind Abst) u) t5 t7))))
-(ty3 g c2 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) (THead
-(Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x0: T).(\lambda (x1:
-T).(\lambda (x2: T).(\lambda (_: (pc3 c2 (THead (Bind Abst) u x0)
-x)).(\lambda (H25: (ty3 g c2 u x1)).(\lambda (H26: (ty3 g (CHead c2 (Bind
-Abst) u) t0 x0)).(\lambda (H27: (ty3 g (CHead c2 (Bind Abst) u) x0
-x2)).(ex4_3_ind T T T (\lambda (t5: T).(\lambda (_: T).(\lambda (_: T).(pc3
-c2 (THead (Bind b) u2 t5) (THead (Bind Abst) u t0))))) (\lambda (_:
-T).(\lambda (t6: T).(\lambda (_: T).(ty3 g c2 u2 t6)))) (\lambda (t5:
-T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind b) u2) t4 t5))))
-(\lambda (t5: T).(\lambda (_: T).(\lambda (t7: T).(ty3 g (CHead c2 (Bind b)
-u2) t5 t7)))) (ty3 g c2 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O
-v2) t4)) (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x3:
-T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H28: (pc3 c2 (THead (Bind b)
-u2 x3) (THead (Bind Abst) u t0))).(\lambda (H29: (ty3 g c2 u2 x4)).(\lambda
-(H30: (ty3 g (CHead c2 (Bind b) u2) t4 x3)).(\lambda (_: (ty3 g (CHead c2
-(Bind b) u2) x3 x5)).(let H32 \def (eq_ind T (lift (S O) O (THead (Bind Abst)
-u t0)) (\lambda (t5: T).(pc3 (CHead c2 (Bind b) u2) x3 t5)) (pc3_gen_not_abst
-b H16 c2 x3 t0 u2 u H28) (THead (Bind Abst) (lift (S O) O u) (lift (S O) (S
-O) t0)) (lift_bind Abst u t0 (S O) O)) in (let H33 \def (eq_ind T (lift (S O)
-O (THead (Bind Abst) u t0)) (\lambda (t5: T).(ty3 g (CHead c2 (Bind b) u2) t5
-(lift (S O) O x))) (ty3_lift g c2 (THead (Bind Abst) u t0) x H22 (CHead c2
-(Bind b) u2) O (S O) (drop_drop (Bind b) O c2 c2 (drop_refl c2) u2)) (THead
-(Bind Abst) (lift (S O) O u) (lift (S O) (S O) t0)) (lift_bind Abst u t0 (S
-O) O)) in (ex4_3_ind T T T (\lambda (t5: T).(\lambda (_: T).(\lambda (_:
-T).(pc3 (CHead c2 (Bind b) u2) (THead (Bind Abst) (lift (S O) O u) t5) (lift
-(S O) O x))))) (\lambda (_: T).(\lambda (t6: T).(\lambda (_: T).(ty3 g (CHead
-c2 (Bind b) u2) (lift (S O) O u) t6)))) (\lambda (t5: T).(\lambda (_:
+((pr0 w v2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat
+Appl) w (THead (Bind Abst) u t0))))))))) (\lambda (H15: (eq T (THead (Bind b)
+u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t2)).(eq_ind T (THead (Bind b)
+u2 (THead (Flat Appl) (lift (S O) O v2) t4)) (\lambda (t5: T).((not (eq B b
+Abst)) \to ((pr0 w v2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t5
+(THead (Flat Appl) w (THead (Bind Abst) u t0)))))))) (\lambda (H16: (not (eq
+B b Abst))).(\lambda (H17: (pr0 w v2)).(\lambda (H18: (pr0 u1 u2)).(\lambda
+(H19: (pr0 t3 t4)).(let H20 \def (eq_ind_r T v (\lambda (t5: T).(\forall (c3:
+C).((wcpr0 c c3) \to (\forall (t6: T).((pr0 t5 t6) \to (ty3 g c3 t6 (THead
+(Bind Abst) u t0))))))) H3 (THead (Bind b) u1 t3) H14) in (let H21 \def
+(eq_ind_r T v (\lambda (t5: T).(ty3 g c t5 (THead (Bind Abst) u t0))) H2
+(THead (Bind b) u1 t3) H14) in (ex_ind T (\lambda (t5: T).(ty3 g c2 (THead
+(Bind Abst) u t0) t5)) (ty3 g c2 (THead (Bind b) u2 (THead (Flat Appl) (lift
+(S O) O v2) t4)) (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x:
+T).(\lambda (H22: (ty3 g c2 (THead (Bind Abst) u t0) x)).(let H23 \def H22 in
+(ex3_2_ind T T (\lambda (t5: T).(\lambda (_: T).(pc3 c2 (THead (Bind Abst) u
+t5) x))) (\lambda (_: T).(\lambda (t6: T).(ty3 g c2 u t6))) (\lambda (t5:
+T).(\lambda (_: T).(ty3 g (CHead c2 (Bind Abst) u) t0 t5))) (ty3 g c2 (THead
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) (THead (Flat Appl) w
+(THead (Bind Abst) u t0))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (_:
+(pc3 c2 (THead (Bind Abst) u x0) x)).(\lambda (H25: (ty3 g c2 u x1)).(\lambda
+(H26: (ty3 g (CHead c2 (Bind Abst) u) t0 x0)).(ex3_2_ind T T (\lambda (t5:
+T).(\lambda (_: T).(pc3 c2 (THead (Bind b) u2 t5) (THead (Bind Abst) u t0))))
+(\lambda (_: T).(\lambda (t6: T).(ty3 g c2 u2 t6))) (\lambda (t5: T).(\lambda
+(_: T).(ty3 g (CHead c2 (Bind b) u2) t4 t5))) (ty3 g c2 (THead (Bind b) u2
+(THead (Flat Appl) (lift (S O) O v2) t4)) (THead (Flat Appl) w (THead (Bind
+Abst) u t0))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H27: (pc3 c2 (THead
+(Bind b) u2 x2) (THead (Bind Abst) u t0))).(\lambda (H28: (ty3 g c2 u2
+x3)).(\lambda (H29: (ty3 g (CHead c2 (Bind b) u2) t4 x2)).(let H30 \def
+(eq_ind T (lift (S O) O (THead (Bind Abst) u t0)) (\lambda (t5: T).(pc3
+(CHead c2 (Bind b) u2) x2 t5)) (pc3_gen_not_abst b H16 c2 x2 t0 u2 u H27)
+(THead (Bind Abst) (lift (S O) O u) (lift (S O) (S O) t0)) (lift_bind Abst u
+t0 (S O) O)) in (let H31 \def (eq_ind T (lift (S O) O (THead (Bind Abst) u
+t0)) (\lambda (t5: T).(ty3 g (CHead c2 (Bind b) u2) t5 (lift (S O) O x)))
+(ty3_lift g c2 (THead (Bind Abst) u t0) x H22 (CHead c2 (Bind b) u2) O (S O)
+(drop_drop (Bind b) O c2 c2 (drop_refl c2) u2)) (THead (Bind Abst) (lift (S
+O) O u) (lift (S O) (S O) t0)) (lift_bind Abst u t0 (S O) O)) in (ex3_2_ind T
+T (\lambda (t5: T).(\lambda (_: T).(pc3 (CHead c2 (Bind b) u2) (THead (Bind
+Abst) (lift (S O) O u) t5) (lift (S O) O x)))) (\lambda (_: T).(\lambda (t6:
+T).(ty3 g (CHead c2 (Bind b) u2) (lift (S O) O u) t6))) (\lambda (t5:
T).(\lambda (_: T).(ty3 g (CHead (CHead c2 (Bind b) u2) (Bind Abst) (lift (S
-O) O u)) (lift (S O) (S O) t0) t5)))) (\lambda (t5: T).(\lambda (_:
-T).(\lambda (t7: T).(ty3 g (CHead (CHead c2 (Bind b) u2) (Bind Abst) (lift (S
-O) O u)) t5 t7)))) (ty3 g c2 (THead (Bind b) u2 (THead (Flat Appl) (lift (S
-O) O v2) t4)) (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (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 (H35:
-(ty3 g (CHead c2 (Bind b) u2) (lift (S O) O u) x7)).(\lambda (H36: (ty3 g
+O) O u)) (lift (S O) (S O) t0) t5))) (ty3 g c2 (THead (Bind b) u2 (THead
+(Flat Appl) (lift (S O) O v2) t4)) (THead (Flat Appl) w (THead (Bind Abst) u
+t0))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (pc3 (CHead c2 (Bind b)
+u2) (THead (Bind Abst) (lift (S O) O u) x4) (lift (S O) O x))).(\lambda (H33:
+(ty3 g (CHead c2 (Bind b) u2) (lift (S O) O u) x5)).(\lambda (H34: (ty3 g
(CHead (CHead c2 (Bind b) u2) (Bind Abst) (lift (S O) O u)) (lift (S O) (S O)
-t0) x6)).(\lambda (H37: (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
-H25 Abst t0 x0 H26 x2 H27)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O)
-O v2) t4)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (THead
-(Bind Abst) (lift (S O) O u) (lift (S O) (S O) t0)))) (ty3_bind g c2 u2 x4
-H29 b (THead (Flat Appl) (lift (S O) O v2) t4) (THead (Flat Appl) (lift (S O)
-O v2) (THead (Bind Abst) (lift (S O) O u) (lift (S O) (S O) t0))) (ty3_appl g
-(CHead c2 (Bind b) u2) (lift (S O) O v2) (lift (S O) O u) (ty3_lift g c2 v2 u
-(H1 c2 H4 v2 H17) (CHead c2 (Bind b) u2) O (S O) (drop_drop (Bind b) O c2 c2
-(drop_refl c2) u2)) t4 (lift (S O) (S O) t0) (ty3_conv g (CHead c2 (Bind b)
-u2) (THead (Bind Abst) (lift (S O) O u) (lift (S O) (S O) t0)) (THead (Bind
-Abst) (lift (S O) O u) x6) (ty3_bind g (CHead c2 (Bind b) u2) (lift (S O) O
-u) x7 H35 Abst (lift (S O) (S O) t0) x6 H36 x8 H37) t4 x3 H30 H32)) (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 H17) (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 H35 Abst (lift (S O) (S O) t0) x6 H36 x8 H37))) (eq_ind T (lift (S
-O) O (THead (Bind Abst) u t0)) (\lambda (t5: T).(pc3 c2 (THead (Bind b) u2
-(THead (Flat Appl) (lift (S O) O v2) t5)) (THead (Flat Appl) w (THead (Bind
-Abst) u t0)))) (pc3_pc1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O
-v2) (lift (S O) O (THead (Bind Abst) u t0)))) (THead (Flat Appl) w (THead
-(Bind Abst) u t0)) (pc1_pr0_u2 (THead (Flat Appl) v2 (THead (Bind b) u2 (lift
-(S O) O (THead (Bind Abst) u t0)))) (THead (Bind b) u2 (THead (Flat Appl)
-(lift (S O) O v2) (lift (S O) O (THead (Bind Abst) u t0)))) (pr0_upsilon b
-H16 v2 v2 (pr0_refl v2) u2 u2 (pr0_refl u2) (lift (S O) O (THead (Bind Abst)
-u t0)) (lift (S O) O (THead (Bind Abst) u t0)) (pr0_refl (lift (S O) O (THead
-(Bind Abst) u t0)))) (THead (Flat Appl) w (THead (Bind Abst) u t0)) (pc1_head
-v2 w (pc1_pr0_x v2 w H17) (THead (Bind b) u2 (lift (S O) O (THead (Bind Abst)
-u t0))) (THead (Bind Abst) u t0) (pc1_pr0_r (THead (Bind b) u2 (lift (S O) O
-(THead (Bind Abst) u t0))) (THead (Bind Abst) u t0) (pr0_zeta b H16 (THead
-(Bind Abst) u t0) (THead (Bind Abst) u t0) (pr0_refl (THead (Bind Abst) u
-t0)) u2)) (Flat Appl))) c2) (THead (Bind Abst) (lift (S O) O u) (lift (S O)
-(S O) t0)) (lift_bind Abst u t0 (S O) O)))))))))) (ty3_gen_bind g Abst (CHead
-c2 (Bind b) u2) (lift (S O) O u) (lift (S O) (S O) t0) (lift (S O) O x)
-H33))))))))))) (ty3_gen_bind g b c2 u2 t4 (THead (Bind Abst) u t0) (H20 c2 H4
-(THead (Bind b) u2 t4) (pr0_comp u1 u2 H18 t3 t4 H19 (Bind b))))))))))))
-(ty3_gen_bind g Abst c2 u t0 x H23))))) (ty3_correct g c2 (THead (Bind b) u2
-t4) (THead (Bind Abst) u t0) (H20 c2 H4 (THead (Bind b) u2 t4) (pr0_comp u1
-u2 H18 t3 t4 H19 (Bind b))))))))))) t2 H15)) v H14)) v1 (sym_eq T v1 w H13)))
-H12)) H11 H6 H7 H8 H9))) | (pr0_delta u1 u2 H6 t3 t4 H7 w0 H8) \Rightarrow
-(\lambda (H9: (eq T (THead (Bind Abbr) u1 t3) (THead (Flat Appl) w
-v))).(\lambda (H10: (eq T (THead (Bind Abbr) u2 w0) t2)).((let H11 \def
-(eq_ind T (THead (Bind Abbr) u1 t3) (\lambda (e: T).(match e in T return
+t0) x4)).(ty3_conv g c2 (THead (Flat Appl) w (THead (Bind Abst) u t0)) (THead
+(Flat Appl) w (THead (Bind Abst) u x0)) (ty3_appl g c2 w u (H1 c2 H4 w
+(pr0_refl w)) (THead (Bind Abst) u t0) x0 (ty3_bind g c2 u x1 H25 Abst t0 x0
+H26)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) (THead
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (THead (Bind Abst) (lift (S
+O) O u) (lift (S O) (S O) t0)))) (ty3_bind g c2 u2 x3 H28 b (THead (Flat
+Appl) (lift (S O) O v2) t4) (THead (Flat Appl) (lift (S O) O v2) (THead (Bind
+Abst) (lift (S O) O u) (lift (S O) (S O) t0))) (ty3_appl g (CHead c2 (Bind b)
+u2) (lift (S O) O v2) (lift (S O) O u) (ty3_lift g c2 v2 u (H1 c2 H4 v2 H17)
+(CHead c2 (Bind b) u2) O (S O) (drop_drop (Bind b) O c2 c2 (drop_refl c2)
+u2)) t4 (lift (S O) (S O) t0) (ty3_conv g (CHead c2 (Bind b) u2) (THead (Bind
+Abst) (lift (S O) O u) (lift (S O) (S O) t0)) (THead (Bind Abst) (lift (S O)
+O u) x4) (ty3_bind g (CHead c2 (Bind b) u2) (lift (S O) O u) x5 H33 Abst
+(lift (S O) (S O) t0) x4 H34) t4 x2 H29 H30))) (eq_ind T (lift (S O) O (THead
+(Bind Abst) u t0)) (\lambda (t5: T).(pc3 c2 (THead (Bind b) u2 (THead (Flat
+Appl) (lift (S O) O v2) t5)) (THead (Flat Appl) w (THead (Bind Abst) u t0))))
+(pc3_pc1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (lift (S O)
+O (THead (Bind Abst) u t0)))) (THead (Flat Appl) w (THead (Bind Abst) u t0))
+(pc1_pr0_u2 (THead (Flat Appl) v2 (THead (Bind b) u2 (lift (S O) O (THead
+(Bind Abst) u t0)))) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2)
+(lift (S O) O (THead (Bind Abst) u t0)))) (pr0_upsilon b H16 v2 v2 (pr0_refl
+v2) u2 u2 (pr0_refl u2) (lift (S O) O (THead (Bind Abst) u t0)) (lift (S O) O
+(THead (Bind Abst) u t0)) (pr0_refl (lift (S O) O (THead (Bind Abst) u t0))))
+(THead (Flat Appl) w (THead (Bind Abst) u t0)) (pc1_head v2 w (pc1_pr0_x v2 w
+H17) (THead (Bind b) u2 (lift (S O) O (THead (Bind Abst) u t0))) (THead (Bind
+Abst) u t0) (pc1_pr0_r (THead (Bind b) u2 (lift (S O) O (THead (Bind Abst) u
+t0))) (THead (Bind Abst) u t0) (pr0_zeta b H16 (THead (Bind Abst) u t0)
+(THead (Bind Abst) u t0) (pr0_refl (THead (Bind Abst) u t0)) u2)) (Flat
+Appl))) c2) (THead (Bind Abst) (lift (S O) O u) (lift (S O) (S O) t0))
+(lift_bind Abst u t0 (S O) O)))))))) (ty3_gen_bind g Abst (CHead c2 (Bind b)
+u2) (lift (S O) O u) (lift (S O) (S O) t0) (lift (S O) O x) H31)))))))))
+(ty3_gen_bind g b c2 u2 t4 (THead (Bind Abst) u t0) (H20 c2 H4 (THead (Bind
+b) u2 t4) (pr0_comp u1 u2 H18 t3 t4 H19 (Bind b)))))))))) (ty3_gen_bind g
+Abst c2 u t0 x H23))))) (ty3_correct g c2 (THead (Bind b) u2 t4) (THead (Bind
+Abst) u t0) (H20 c2 H4 (THead (Bind b) u2 t4) (pr0_comp u1 u2 H18 t3 t4 H19
+(Bind b))))))))))) t2 H15)) v H14)) v1 (sym_eq T v1 w H13))) H12)) H11 H6 H7
+H8 H9))) | (pr0_delta u1 u2 H6 t3 t4 H7 w0 H8) \Rightarrow (\lambda (H9: (eq
+T (THead (Bind Abbr) u1 t3) (THead (Flat Appl) w v))).(\lambda (H10: (eq T
+(THead (Bind Abbr) u2 w0) t2)).((let H11 \def (eq_ind T (THead (Bind Abbr) u1
+t3) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort
+_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
+\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
+\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) w v)
+H9) in (False_ind ((eq T (THead (Bind Abbr) u2 w0) t2) \to ((pr0 u1 u2) \to
+((pr0 t3 t4) \to ((subst0 O u2 t4 w0) \to (ty3 g c2 t2 (THead (Flat Appl) w
+(THead (Bind Abst) u t0))))))) H11)) H10 H6 H7 H8))) | (pr0_zeta b H6 t3 t4
+H7 u0) \Rightarrow (\lambda (H8: (eq T (THead (Bind b) u0 (lift (S O) O t3))
+(THead (Flat Appl) w v))).(\lambda (H9: (eq T t4 t2)).((let H10 \def (eq_ind
+T (THead (Bind b) u0 (lift (S O) O t3)) (\lambda (e: T).(match e in T return
(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow
-False])])) I (THead (Flat Appl) w v) H9) in (False_ind ((eq T (THead (Bind
-Abbr) u2 w0) t2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to ((subst0 O u2 t4 w0)
-\to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0))))))) H11))
-H10 H6 H7 H8))) | (pr0_zeta b H6 t3 t4 H7 u0) \Rightarrow (\lambda (H8: (eq T
-(THead (Bind b) u0 (lift (S O) O t3)) (THead (Flat Appl) w v))).(\lambda (H9:
-(eq T t4 t2)).((let H10 \def (eq_ind T (THead (Bind b) u0 (lift (S O) O t3))
-(\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
-(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True |
-(Flat _) \Rightarrow False])])) I (THead (Flat Appl) w v) H8) in (False_ind
-((eq T t4 t2) \to ((not (eq B b Abst)) \to ((pr0 t3 t4) \to (ty3 g c2 t2
-(THead (Flat Appl) w (THead (Bind Abst) u t0)))))) H10)) H9 H6 H7))) |
-(pr0_epsilon t3 t4 H6 u0) \Rightarrow (\lambda (H7: (eq T (THead (Flat Cast)
-u0 t3) (THead (Flat Appl) w v))).(\lambda (H8: (eq T t4 t2)).((let H9 \def
-(eq_ind T (THead (Flat Cast) u0 t3) (\lambda (e: T).(match e in T return
+False])])) I (THead (Flat Appl) w v) H8) in (False_ind ((eq T t4 t2) \to
+((not (eq B b Abst)) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w
+(THead (Bind Abst) u t0)))))) H10)) H9 H6 H7))) | (pr0_epsilon t3 t4 H6 u0)
+\Rightarrow (\lambda (H7: (eq T (THead (Flat Cast) u0 t3) (THead (Flat Appl)
+w v))).(\lambda (H8: (eq T t4 t2)).((let H9 \def (eq_ind T (THead (Flat Cast)
+u0 t3) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
+\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
+\Rightarrow False | (Flat f) \Rightarrow (match f in F return (\lambda (_:
+F).Prop) with [Appl \Rightarrow False | Cast \Rightarrow True])])])) I (THead
+(Flat Appl) w v) H7) in (False_ind ((eq T t4 t2) \to ((pr0 t3 t4) \to (ty3 g
+c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0))))) H9)) H8 H6)))]) in
+(H6 (refl_equal T (THead (Flat Appl) w v)) (refl_equal T t2))))))))))))))))
+(\lambda (c: C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (ty3 g c t2
+t3)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t4: T).((pr0
+t2 t4) \to (ty3 g c2 t4 t3))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c t3
+t0)).(\lambda (H3: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t4: T).((pr0
+t3 t4) \to (ty3 g c2 t4 t0))))))).(\lambda (c2: C).(\lambda (H4: (wcpr0 c
+c2)).(\lambda (t4: T).(\lambda (H5: (pr0 (THead (Flat Cast) t3 t2) t4)).(let
+H6 \def (match H5 in pr0 return (\lambda (t5: T).(\lambda (t6: T).(\lambda
+(_: (pr0 t5 t6)).((eq T t5 (THead (Flat Cast) t3 t2)) \to ((eq T t6 t4) \to
+(ty3 g c2 t4 (THead (Flat Cast) t0 t3))))))) with [(pr0_refl t5) \Rightarrow
+(\lambda (H6: (eq T t5 (THead (Flat Cast) t3 t2))).(\lambda (H7: (eq T t5
+t4)).(eq_ind T (THead (Flat Cast) t3 t2) (\lambda (t6: T).((eq T t6 t4) \to
+(ty3 g c2 t4 (THead (Flat Cast) t0 t3)))) (\lambda (H8: (eq T (THead (Flat
+Cast) t3 t2) t4)).(eq_ind T (THead (Flat Cast) t3 t2) (\lambda (t6: T).(ty3 g
+c2 t6 (THead (Flat Cast) t0 t3))) (ty3_cast g c2 t2 t3 (H1 c2 H4 t2 (pr0_refl
+t2)) t0 (H3 c2 H4 t3 (pr0_refl t3))) t4 H8)) t5 (sym_eq T t5 (THead (Flat
+Cast) t3 t2) H6) H7))) | (pr0_comp u1 u2 H6 t5 t6 H7 k) \Rightarrow (\lambda
+(H8: (eq T (THead k u1 t5) (THead (Flat Cast) t3 t2))).(\lambda (H9: (eq T
+(THead k u2 t6) t4)).((let H10 \def (f_equal T T (\lambda (e: T).(match e in
+T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t5 | (TLRef _)
+\Rightarrow t5 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t5) (THead
+(Flat Cast) t3 t2) H8) in ((let H11 \def (f_equal T T (\lambda (e: T).(match
+e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _)
+\Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t5) (THead
+(Flat Cast) t3 t2) H8) in ((let H12 \def (f_equal T K (\lambda (e: T).(match
+e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
+\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t5) (THead (Flat
+Cast) t3 t2) H8) in (eq_ind K (Flat Cast) (\lambda (k0: K).((eq T u1 t3) \to
+((eq T t5 t2) \to ((eq T (THead k0 u2 t6) t4) \to ((pr0 u1 u2) \to ((pr0 t5
+t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3)))))))) (\lambda (H13: (eq T u1
+t3)).(eq_ind T t3 (\lambda (t7: T).((eq T t5 t2) \to ((eq T (THead (Flat
+Cast) u2 t6) t4) \to ((pr0 t7 u2) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead
+(Flat Cast) t0 t3))))))) (\lambda (H14: (eq T t5 t2)).(eq_ind T t2 (\lambda
+(t7: T).((eq T (THead (Flat Cast) u2 t6) t4) \to ((pr0 t3 u2) \to ((pr0 t7
+t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3)))))) (\lambda (H15: (eq T
+(THead (Flat Cast) u2 t6) t4)).(eq_ind T (THead (Flat Cast) u2 t6) (\lambda
+(t7: T).((pr0 t3 u2) \to ((pr0 t2 t6) \to (ty3 g c2 t7 (THead (Flat Cast) t0
+t3))))) (\lambda (H16: (pr0 t3 u2)).(\lambda (H17: (pr0 t2 t6)).(ex_ind T
+(\lambda (t7: T).(ty3 g c2 t0 t7)) (ty3 g c2 (THead (Flat Cast) u2 t6) (THead
+(Flat Cast) t0 t3)) (\lambda (x: T).(\lambda (H18: (ty3 g c2 t0 x)).(ty3_conv
+g c2 (THead (Flat Cast) t0 t3) (THead (Flat Cast) x t0) (ty3_cast g c2 t3 t0
+(H3 c2 H4 t3 (pr0_refl t3)) x H18) (THead (Flat Cast) u2 t6) (THead (Flat
+Cast) t0 u2) (ty3_cast g c2 t6 u2 (ty3_conv g c2 u2 t0 (H3 c2 H4 u2 H16) t6
+t3 (H1 c2 H4 t6 H17) (pc3_pr2_r c2 t3 u2 (pr2_free c2 t3 u2 H16))) t0 (H3 c2
+H4 u2 H16)) (pc3_s c2 (THead (Flat Cast) t0 u2) (THead (Flat Cast) t0 t3)
+(pc3_pr2_r c2 (THead (Flat Cast) t0 t3) (THead (Flat Cast) t0 u2)
+(pr2_thin_dx c2 t3 u2 (pr2_free c2 t3 u2 H16) t0 Cast)))))) (ty3_correct g c2
+t3 t0 (H3 c2 H4 t3 (pr0_refl t3)))))) t4 H15)) t5 (sym_eq T t5 t2 H14))) u1
+(sym_eq T u1 t3 H13))) k (sym_eq K k (Flat Cast) H12))) H11)) H10)) H9 H6
+H7))) | (pr0_beta u v1 v2 H6 t5 t6 H7) \Rightarrow (\lambda (H8: (eq T (THead
+(Flat Appl) v1 (THead (Bind Abst) u t5)) (THead (Flat Cast) t3 t2))).(\lambda
+(H9: (eq T (THead (Bind Abbr) v2 t6) t4)).((let H10 \def (eq_ind T (THead
+(Flat Appl) v1 (THead (Bind Abst) u t5)) (\lambda (e: T).(match e in T return
(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f
-in F return (\lambda (_: F).Prop) with [Appl \Rightarrow False | Cast
-\Rightarrow True])])])) I (THead (Flat Appl) w v) H7) in (False_ind ((eq T t4
-t2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst)
-u t0))))) H9)) H8 H6)))]) in (H6 (refl_equal T (THead (Flat Appl) w v))
-(refl_equal T t2)))))))))))))))) (\lambda (c: C).(\lambda (t2: T).(\lambda
-(t3: T).(\lambda (_: (ty3 g c t2 t3)).(\lambda (H1: ((\forall (c2: C).((wcpr0
-c c2) \to (\forall (t4: T).((pr0 t2 t4) \to (ty3 g c2 t4 t3))))))).(\lambda
-(t0: T).(\lambda (_: (ty3 g c t3 t0)).(\lambda (H3: ((\forall (c2: C).((wcpr0
-c c2) \to (\forall (t4: T).((pr0 t3 t4) \to (ty3 g c2 t4 t0))))))).(\lambda
-(c2: C).(\lambda (H4: (wcpr0 c c2)).(\lambda (t4: T).(\lambda (H5: (pr0
-(THead (Flat Cast) t3 t2) t4)).(let H6 \def (match H5 in pr0 return (\lambda
-(t5: T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 (THead (Flat
-Cast) t3 t2)) \to ((eq T t6 t4) \to (ty3 g c2 t4 (THead (Flat Cast) t0
-t3))))))) with [(pr0_refl t5) \Rightarrow (\lambda (H6: (eq T t5 (THead (Flat
-Cast) t3 t2))).(\lambda (H7: (eq T t5 t4)).(eq_ind T (THead (Flat Cast) t3
-t2) (\lambda (t6: T).((eq T t6 t4) \to (ty3 g c2 t4 (THead (Flat Cast) t0
-t3)))) (\lambda (H8: (eq T (THead (Flat Cast) t3 t2) t4)).(eq_ind T (THead
-(Flat Cast) t3 t2) (\lambda (t6: T).(ty3 g c2 t6 (THead (Flat Cast) t0 t3)))
-(ty3_cast g c2 t2 t3 (H1 c2 H4 t2 (pr0_refl t2)) t0 (H3 c2 H4 t3 (pr0_refl
-t3))) t4 H8)) t5 (sym_eq T t5 (THead (Flat Cast) t3 t2) H6) H7))) | (pr0_comp
-u1 u2 H6 t5 t6 H7 k) \Rightarrow (\lambda (H8: (eq T (THead k u1 t5) (THead
-(Flat Cast) t3 t2))).(\lambda (H9: (eq T (THead k u2 t6) t4)).((let H10 \def
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
-[(TSort _) \Rightarrow t5 | (TLRef _) \Rightarrow t5 | (THead _ _ t7)
-\Rightarrow t7])) (THead k u1 t5) (THead (Flat Cast) t3 t2) H8) in ((let H11
-\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
-with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t7 _)
-\Rightarrow t7])) (THead k u1 t5) (THead (Flat Cast) t3 t2) H8) in ((let H12
-\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K)
-with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _)
-\Rightarrow k0])) (THead k u1 t5) (THead (Flat Cast) t3 t2) H8) in (eq_ind K
-(Flat Cast) (\lambda (k0: K).((eq T u1 t3) \to ((eq T t5 t2) \to ((eq T
-(THead k0 u2 t6) t4) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ty3 g c2 t4
-(THead (Flat Cast) t0 t3)))))))) (\lambda (H13: (eq T u1 t3)).(eq_ind T t3
-(\lambda (t7: T).((eq T t5 t2) \to ((eq T (THead (Flat Cast) u2 t6) t4) \to
-((pr0 t7 u2) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0
-t3))))))) (\lambda (H14: (eq T t5 t2)).(eq_ind T t2 (\lambda (t7: T).((eq T
-(THead (Flat Cast) u2 t6) t4) \to ((pr0 t3 u2) \to ((pr0 t7 t6) \to (ty3 g c2
-t4 (THead (Flat Cast) t0 t3)))))) (\lambda (H15: (eq T (THead (Flat Cast) u2
-t6) t4)).(eq_ind T (THead (Flat Cast) u2 t6) (\lambda (t7: T).((pr0 t3 u2)
-\to ((pr0 t2 t6) \to (ty3 g c2 t7 (THead (Flat Cast) t0 t3))))) (\lambda
-(H16: (pr0 t3 u2)).(\lambda (H17: (pr0 t2 t6)).(ex_ind T (\lambda (t7:
-T).(ty3 g c2 t0 t7)) (ty3 g c2 (THead (Flat Cast) u2 t6) (THead (Flat Cast)
-t0 t3)) (\lambda (x: T).(\lambda (H18: (ty3 g c2 t0 x)).(ty3_conv g c2 (THead
-(Flat Cast) t0 t3) (THead (Flat Cast) x t0) (ty3_cast g c2 t3 t0 (H3 c2 H4 t3
-(pr0_refl t3)) x H18) (THead (Flat Cast) u2 t6) (THead (Flat Cast) t0 u2)
-(ty3_cast g c2 t6 u2 (ty3_conv g c2 u2 t0 (H3 c2 H4 u2 H16) t6 t3 (H1 c2 H4
-t6 H17) (pc3_pr2_r c2 t3 u2 (pr2_free c2 t3 u2 H16))) t0 (H3 c2 H4 u2 H16))
-(pc3_s c2 (THead (Flat Cast) t0 u2) (THead (Flat Cast) t0 t3) (pc3_pr2_r c2
-(THead (Flat Cast) t0 t3) (THead (Flat Cast) t0 u2) (pr2_thin_dx c2 t3 u2
-(pr2_free c2 t3 u2 H16) t0 Cast)))))) (ty3_correct g c2 t3 t0 (H3 c2 H4 t3
-(pr0_refl t3)))))) t4 H15)) t5 (sym_eq T t5 t2 H14))) u1 (sym_eq T u1 t3
-H13))) k (sym_eq K k (Flat Cast) H12))) H11)) H10)) H9 H6 H7))) | (pr0_beta u
-v1 v2 H6 t5 t6 H7) \Rightarrow (\lambda (H8: (eq T (THead (Flat Appl) v1
-(THead (Bind Abst) u t5)) (THead (Flat Cast) t3 t2))).(\lambda (H9: (eq T
-(THead (Bind Abbr) v2 t6) t4)).((let H10 \def (eq_ind T (THead (Flat Appl) v1
-(THead (Bind Abst) u t5)) (\lambda (e: T).(match e in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
-(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
-[(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f in F return
-(\lambda (_: F).Prop) with [Appl \Rightarrow True | Cast \Rightarrow
-False])])])) I (THead (Flat Cast) t3 t2) H8) in (False_ind ((eq T (THead
-(Bind Abbr) v2 t6) t4) \to ((pr0 v1 v2) \to ((pr0 t5 t6) \to (ty3 g c2 t4
-(THead (Flat Cast) t0 t3))))) H10)) H9 H6 H7))) | (pr0_upsilon b H6 v1 v2 H7
-u1 u2 H8 t5 t6 H9) \Rightarrow (\lambda (H10: (eq T (THead (Flat Appl) v1
+in F return (\lambda (_: F).Prop) with [Appl \Rightarrow True | Cast
+\Rightarrow False])])])) I (THead (Flat Cast) t3 t2) H8) in (False_ind ((eq T
+(THead (Bind Abbr) v2 t6) t4) \to ((pr0 v1 v2) \to ((pr0 t5 t6) \to (ty3 g c2
+t4 (THead (Flat Cast) t0 t3))))) H10)) H9 H6 H7))) | (pr0_upsilon b H6 v1 v2
+H7 u1 u2 H8 t5 t6 H9) \Rightarrow (\lambda (H10: (eq T (THead (Flat Appl) v1
(THead (Bind b) u1 t5)) (THead (Flat Cast) t3 t2))).(\lambda (H11: (eq T
(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t6)) t4)).((let H12
\def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t5)) (\lambda (e:
t5 t2 H11))) u (sym_eq T u t3 H10))) H9)) H8 H6)))]) in (H6 (refl_equal T
(THead (Flat Cast) t3 t2)) (refl_equal T t4))))))))))))))) c1 t1 t H))))).
+theorem ty3_sred_pr0:
+ \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (g: G).(\forall
+(c: C).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t)))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(\lambda (g:
+G).(\lambda (c: C).(\lambda (t: T).(\lambda (H0: (ty3 g c t1
+t)).(ty3_sred_wcpr0_pr0 g c t1 t H0 c (wcpr0_refl c) t2 H))))))).
+
theorem ty3_sred_pr1:
- \forall (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)))))))
+ \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (g: G).(\forall
+(c: C).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t)))))))
\def
- \lambda (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)))).
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr1 t1 t2)).(pr1_ind (\lambda
+(t: T).(\lambda (t0: T).(\forall (g: G).(\forall (c: C).(\forall (t3:
+T).((ty3 g c t t3) \to (ty3 g c t0 t3))))))) (\lambda (t: T).(\lambda (g:
+G).(\lambda (c: C).(\lambda (t0: T).(\lambda (H0: (ty3 g c t t0)).H0)))))
+(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t4 t3)).(\lambda (t5:
+T).(\lambda (_: (pr1 t3 t5)).(\lambda (H2: ((\forall (g: G).(\forall (c:
+C).(\forall (t: T).((ty3 g c t3 t) \to (ty3 g c t5 t))))))).(\lambda (g:
+G).(\lambda (c: C).(\lambda (t: T).(\lambda (H3: (ty3 g c t4 t)).(H2 g c t
+(ty3_sred_pr0 t4 t3 H0 g c t H3)))))))))))) t1 t2 H))).
theorem ty3_sred_pr2:
\forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/ty3/pr3_props".
include "ty3/pr3.ma".
\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 (x0: T).(\forall (x1: nat).((eq T
-t2 (lift h x1 x0)) \to (\forall (e: C).((drop h x1 c0 e) \to (ex2 T (\lambda
-(t3: T).(pc3 c0 (lift h x1 t3) t)) (\lambda (t3: T).(ty3 g e x0
-t3)))))))))).(\lambda (u: T).(\lambda (t3: T).(\lambda (H3: (ty3 g c0 u
-t3)).(\lambda (H4: ((\forall (x0: T).(\forall (x1: nat).((eq T u (lift h x1
-x0)) \to (\forall (e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t4: T).(pc3
-c0 (lift h x1 t4) t3)) (\lambda (t4: T).(ty3 g e x0 t4)))))))))).(\lambda
-(H5: (pc3 c0 t3 t2)).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H6: (eq T
-u (lift h x1 x0))).(\lambda (e: C).(\lambda (H7: (drop h x1 c0 e)).(let H8
-\def (eq_ind T u (\lambda (t0: T).(\forall (x2: T).(\forall (x3: nat).((eq T
-t0 (lift h x3 x2)) \to (\forall (e0: C).((drop h x3 c0 e0) \to (ex2 T
-(\lambda (t4: T).(pc3 c0 (lift h x3 t4) t3)) (\lambda (t4: T).(ty3 g e0 x2
-t4))))))))) H4 (lift h x1 x0) H6) in (let H9 \def (eq_ind T u (\lambda (t0:
-T).(ty3 g c0 t0 t3)) H3 (lift h x1 x0) H6) in (let H10 \def (H8 x0 x1
-(refl_equal T (lift h x1 x0)) e H7) in (ex2_ind T (\lambda (t4: T).(pc3 c0
-(lift h x1 t4) t3)) (\lambda (t4: T).(ty3 g e x0 t4)) (ex2 T (\lambda (t4:
-T).(pc3 c0 (lift h x1 t4) t2)) (\lambda (t4: T).(ty3 g e x0 t4))) (\lambda
-(x2: T).(\lambda (H11: (pc3 c0 (lift h x1 x2) t3)).(\lambda (H12: (ty3 g e x0
-x2)).(ex_intro2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) t2)) (\lambda (t4:
-T).(ty3 g e x0 t4)) x2 (pc3_t t3 c0 (lift h x1 x2) H11 t2 H5) H12))))
-H10))))))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (x0:
-T).(\lambda (x1: nat).(\lambda (H1: (eq T (TSort m) (lift h x1 x0))).(\lambda
-(e: C).(\lambda (_: (drop h x1 c0 e)).(eq_ind_r T (TSort m) (\lambda (t:
-T).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (TSort (next g m))))
-(\lambda (t2: T).(ty3 g e t t2)))) (ex_intro2 T (\lambda (t2: T).(pc3 c0
-(lift h x1 t2) (TSort (next g m)))) (\lambda (t2: T).(ty3 g e (TSort m) t2))
-(TSort (next g m)) (eq_ind_r T (TSort (next g m)) (\lambda (t: T).(pc3 c0 t
-(TSort (next g m)))) (pc3_refl c0 (TSort (next g m))) (lift h x1 (TSort (next
-g m))) (lift_sort (next g m) h x1)) (ty3_sort g e m)) x0 (lift_gen_sort h x1
-m x0 H1))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d0: C).(\lambda
-(u: T).(\lambda (H1: (getl n c0 (CHead d0 (Bind Abbr) u))).(\lambda (t:
-T).(\lambda (H2: (ty3 g d0 u t)).(\lambda (H3: ((\forall (x0: T).(\forall
-(x1: nat).((eq T u (lift h x1 x0)) \to (\forall (e: C).((drop h x1 d0 e) \to
-(ex2 T (\lambda (t2: T).(pc3 d0 (lift h x1 t2) t)) (\lambda (t2: T).(ty3 g e
-x0 t2)))))))))).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H4: (eq T
-(TLRef n) (lift h x1 x0))).(\lambda (e: C).(\lambda (H5: (drop h x1 c0
-e)).(let H_x \def (lift_gen_lref x0 x1 h n H4) in (let H6 \def H_x in (or_ind
-(land (lt n x1) (eq T x0 (TLRef n))) (land (le (plus x1 h) n) (eq T x0 (TLRef
-(minus n h)))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O
-t))) (\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (H7: (land (lt n x1) (eq T
-x0 (TLRef n)))).(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 (n0: nat).(drop h n0 c0 e)) H5 (S (plus n (minus x1 (S n))))
-(lt_plus_minus n x1 H8)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_:
-C).(eq T u (lift h (minus x1 (S n)) v)))) (\lambda (v: T).(\lambda (e0:
-C).(getl n e (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0:
-C).(drop h (minus x1 (S n)) d0 e0))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h
-x1 t2) (lift (S n) O t))) (\lambda (t2: T).(ty3 g e (TLRef n) t2))) (\lambda
-(x2: T).(\lambda (x3: C).(\lambda (H11: (eq T u (lift h (minus x1 (S n))
-x2))).(\lambda (H12: (getl n e (CHead x3 (Bind Abbr) x2))).(\lambda (H13:
-(drop h (minus x1 (S n)) d0 x3)).(let H14 \def (eq_ind T u (\lambda (t0:
-T).(\forall (x4: T).(\forall (x5: nat).((eq T t0 (lift h x5 x4)) \to (\forall
-(e0: C).((drop h x5 d0 e0) \to (ex2 T (\lambda (t2: T).(pc3 d0 (lift h x5 t2)
-t)) (\lambda (t2: T).(ty3 g e0 x4 t2))))))))) H3 (lift h (minus x1 (S n)) x2)
-H11) in (let H15 \def (eq_ind T u (\lambda (t0: T).(ty3 g d0 t0 t)) H2 (lift
-h (minus x1 (S n)) x2) H11) in (let H16 \def (H14 x2 (minus x1 (S n))
-(refl_equal T (lift h (minus x1 (S n)) x2)) x3 H13) in (ex2_ind T (\lambda
-(t2: T).(pc3 d0 (lift h (minus x1 (S n)) t2) t)) (\lambda (t2: T).(ty3 g x3
-x2 t2)) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t)))
-(\lambda (t2: T).(ty3 g e (TLRef n) t2))) (\lambda (x4: T).(\lambda (H17:
-(pc3 d0 (lift h (minus x1 (S n)) x4) t)).(\lambda (H18: (ty3 g x3 x2
-x4)).(eq_ind_r nat (plus (S n) (minus x1 (S n))) (\lambda (n0: nat).(ex2 T
-(\lambda (t2: T).(pc3 c0 (lift h n0 t2) (lift (S n) O t))) (\lambda (t2:
-T).(ty3 g e (TLRef n) t2)))) (ex_intro2 T (\lambda (t2: T).(pc3 c0 (lift h
-(plus (S n) (minus x1 (S n))) t2) (lift (S n) O t))) (\lambda (t2: T).(ty3 g
-e (TLRef n) t2)) (lift (S n) O x4) (eq_ind_r T (lift (S n) O (lift h (minus
-x1 (S n)) x4)) (\lambda (t0: T).(pc3 c0 t0 (lift (S n) O t))) (pc3_lift c0 d0
-(S n) O (getl_drop Abbr c0 d0 u n H1) (lift h (minus x1 (S n)) x4) t H17)
-(lift h (plus (S n) (minus x1 (S n))) (lift (S n) O x4)) (lift_d x4 h (S n)
-(minus x1 (S n)) O (le_O_n (minus x1 (S n))))) (ty3_abbr g n e x3 x2 H12 x4
-H18)) x1 (le_plus_minus (S n) x1 H8))))) H16))))))))) (getl_drop_conf_lt Abbr
-c0 d0 u n H1 e h (minus x1 (S n)) H10))) x0 H9))) H7)) (\lambda (H7: (land
-(le (plus x1 h) n) (eq T x0 (TLRef (minus n h))))).(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
+(lift h d t1) (\lambda (t: T).(ty3 g c t x)) (\lambda (_: T).(\forall (e:
+C).((drop h d c e) \to (ex2 T (\lambda (t2: T).(pc3 c (lift h d t2) x))
+(\lambda (t2: T).(ty3 g e t1 t2)))))) (\lambda (y: T).(\lambda (H0: (ty3 g c
+y x)).(unintro nat d (\lambda (n: nat).((eq T y (lift h n t1)) \to (\forall
+(e: C).((drop h n c e) \to (ex2 T (\lambda (t2: T).(pc3 c (lift h n t2) x))
+(\lambda (t2: T).(ty3 g e t1 t2))))))) (unintro T t1 (\lambda (t: T).(\forall
+(x0: nat).((eq T y (lift h x0 t)) \to (\forall (e: C).((drop h x0 c e) \to
+(ex2 T (\lambda (t2: T).(pc3 c (lift h x0 t2) x)) (\lambda (t2: T).(ty3 g e t
+t2)))))))) (ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0:
+T).(\forall (x0: T).(\forall (x1: nat).((eq T t (lift h x1 x0)) \to (\forall
+(e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2)
+t0)) (\lambda (t2: T).(ty3 g e x0 t2))))))))))) (\lambda (c0: C).(\lambda
+(t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: ((\forall
+(x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to (\forall (e:
+C).((drop h x1 c0 e) \to (ex2 T (\lambda (t3: T).(pc3 c0 (lift h x1 t3) t))
+(\lambda (t3: T).(ty3 g e x0 t3)))))))))).(\lambda (u: T).(\lambda (t3:
+T).(\lambda (H3: (ty3 g c0 u t3)).(\lambda (H4: ((\forall (x0: T).(\forall
+(x1: nat).((eq T u (lift h x1 x0)) \to (\forall (e: C).((drop h x1 c0 e) \to
+(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) t3)) (\lambda (t4: T).(ty3 g e
+x0 t4)))))))))).(\lambda (H5: (pc3 c0 t3 t2)).(\lambda (x0: T).(\lambda (x1:
+nat).(\lambda (H6: (eq T u (lift h x1 x0))).(\lambda (e: C).(\lambda (H7:
+(drop h x1 c0 e)).(let H8 \def (eq_ind T u (\lambda (t0: T).(\forall (x2:
+T).(\forall (x3: nat).((eq T t0 (lift h x3 x2)) \to (\forall (e0: C).((drop h
+x3 c0 e0) \to (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x3 t4) t3)) (\lambda
+(t4: T).(ty3 g e0 x2 t4))))))))) H4 (lift h x1 x0) H6) in (let H9 \def
+(eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t3)) H3 (lift h x1 x0) H6) in (let
+H10 \def (H8 x0 x1 (refl_equal T (lift h x1 x0)) e H7) in (ex2_ind T (\lambda
+(t4: T).(pc3 c0 (lift h x1 t4) t3)) (\lambda (t4: T).(ty3 g e x0 t4)) (ex2 T
+(\lambda (t4: T).(pc3 c0 (lift h x1 t4) t2)) (\lambda (t4: T).(ty3 g e x0
+t4))) (\lambda (x2: T).(\lambda (H11: (pc3 c0 (lift h x1 x2) t3)).(\lambda
+(H12: (ty3 g e x0 x2)).(ex_intro2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4)
+t2)) (\lambda (t4: T).(ty3 g e x0 t4)) x2 (pc3_t t3 c0 (lift h x1 x2) H11 t2
+H5) H12)))) H10))))))))))))))))))) (\lambda (c0: C).(\lambda (m:
+nat).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H1: (eq T (TSort m) (lift
+h x1 x0))).(\lambda (e: C).(\lambda (_: (drop h x1 c0 e)).(eq_ind_r T (TSort
+m) (\lambda (t: T).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (TSort
+(next g m)))) (\lambda (t2: T).(ty3 g e t t2)))) (ex_intro2 T (\lambda (t2:
+T).(pc3 c0 (lift h x1 t2) (TSort (next g m)))) (\lambda (t2: T).(ty3 g e
+(TSort m) t2)) (TSort (next g m)) (eq_ind_r T (TSort (next g m)) (\lambda (t:
+T).(pc3 c0 t (TSort (next g m)))) (pc3_refl c0 (TSort (next g m))) (lift h x1
+(TSort (next g m))) (lift_sort (next g m) h x1)) (ty3_sort g e m)) x0
+(lift_gen_sort h x1 m x0 H1))))))))) (\lambda (n: nat).(\lambda (c0:
+C).(\lambda (d0: C).(\lambda (u: T).(\lambda (H1: (getl n c0 (CHead d0 (Bind
+Abbr) u))).(\lambda (t: T).(\lambda (H2: (ty3 g d0 u t)).(\lambda (H3:
+((\forall (x0: T).(\forall (x1: nat).((eq T u (lift h x1 x0)) \to (\forall
+(e: C).((drop h x1 d0 e) \to (ex2 T (\lambda (t2: T).(pc3 d0 (lift h x1 t2)
+t)) (\lambda (t2: T).(ty3 g e x0 t2)))))))))).(\lambda (x0: T).(\lambda (x1:
+nat).(\lambda (H4: (eq T (TLRef n) (lift h x1 x0))).(\lambda (e: C).(\lambda
+(H5: (drop h x1 c0 e)).(let H_x \def (lift_gen_lref x0 x1 h n H4) in (let H6
+\def H_x in (or_ind (land (lt n x1) (eq T x0 (TLRef n))) (land (le (plus x1
+h) n) (eq T x0 (TLRef (minus n h)))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h
+x1 t2) (lift (S n) O t))) (\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (H7:
+(land (lt n x1) (eq T x0 (TLRef n)))).(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 (n0: nat).(drop h n0 c0 e)) H5
+(S (plus n (minus x1 (S n)))) (lt_plus_minus n x1 H8)) in (ex3_2_ind T C
+(\lambda (v: T).(\lambda (_: C).(eq T u (lift h (minus x1 (S n)) v))))
+(\lambda (v: T).(\lambda (e0: C).(getl n e (CHead e0 (Bind Abbr) v))))
+(\lambda (_: T).(\lambda (e0: C).(drop h (minus x1 (S n)) d0 e0))) (ex2 T
(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda (t2:
-T).(ty3 g e (TLRef (minus n h)) t2)) (lift (S (minus n h)) O t) (eq_ind_r T
-(lift (plus h (S (minus n h))) O t) (\lambda (t0: T).(pc3 c0 t0 (lift (S n) O
-t))) (eq_ind nat (S (plus h (minus n h))) (\lambda (n0: nat).(pc3 c0 (lift n0
-O t) (lift (S n) O t))) (eq_ind nat n (\lambda (n0: nat).(pc3 c0 (lift (S n0)
-O t) (lift (S n) O t))) (pc3_refl c0 (lift (S n) O t)) (plus h (minus n h))
-(le_plus_minus h n (le_trans h (plus x1 h) n (le_plus_r x1 h) H8))) (plus h
-(S (minus n h))) (plus_n_Sm h (minus n h))) (lift h x1 (lift (S (minus n h))
-O t)) (lift_free t (S (minus n h)) h O x1 (le_trans x1 (S (minus n h)) (plus
-O (S (minus n h))) (le_S_minus x1 h n H8) (le_n (plus O (S (minus n h)))))
-(le_O_n x1))) (ty3_abbr g (minus n h) e d0 u (getl_drop_conf_ge n (CHead d0
-(Bind Abbr) u) c0 H1 e h x1 H5 H8) t H2)) x0 H9))) H7)) H6))))))))))))))))
-(\lambda (n: nat).(\lambda (c0: C).(\lambda (d0: C).(\lambda (u: T).(\lambda
-(H1: (getl n c0 (CHead d0 (Bind Abst) u))).(\lambda (t: T).(\lambda (H2: (ty3
-g d0 u t)).(\lambda (H3: ((\forall (x0: T).(\forall (x1: nat).((eq T u (lift
-h x1 x0)) \to (\forall (e: C).((drop h x1 d0 e) \to (ex2 T (\lambda (t2:
-T).(pc3 d0 (lift h x1 t2) t)) (\lambda (t2: T).(ty3 g e x0
-t2)))))))))).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H4: (eq T (TLRef
-n) (lift h x1 x0))).(\lambda (e: C).(\lambda (H5: (drop h x1 c0 e)).(let H_x
-\def (lift_gen_lref x0 x1 h n H4) in (let H6 \def H_x in (or_ind (land (lt n
-x1) (eq T x0 (TLRef n))) (land (le (plus x1 h) n) (eq T x0 (TLRef (minus n
-h)))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O u)))
-(\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (H7: (land (lt n x1) (eq T x0
-(TLRef n)))).(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 (n0: nat).(drop h n0 c0 e)) H5 (S (plus n (minus x1 (S n))))
-(lt_plus_minus n x1 H8)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_:
-C).(eq T u (lift h (minus x1 (S n)) v)))) (\lambda (v: T).(\lambda (e0:
-C).(getl n e (CHead e0 (Bind Abst) v)))) (\lambda (_: T).(\lambda (e0:
-C).(drop h (minus x1 (S n)) d0 e0))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h
-x1 t2) (lift (S n) O u))) (\lambda (t2: T).(ty3 g e (TLRef n) t2))) (\lambda
-(x2: T).(\lambda (x3: C).(\lambda (H11: (eq T u (lift h (minus x1 (S n))
-x2))).(\lambda (H12: (getl n e (CHead x3 (Bind Abst) x2))).(\lambda (H13:
-(drop h (minus x1 (S n)) d0 x3)).(let H14 \def (eq_ind T u (\lambda (t0:
-T).(\forall (x4: T).(\forall (x5: nat).((eq T t0 (lift h x5 x4)) \to (\forall
-(e0: C).((drop h x5 d0 e0) \to (ex2 T (\lambda (t2: T).(pc3 d0 (lift h x5 t2)
-t)) (\lambda (t2: T).(ty3 g e0 x4 t2))))))))) H3 (lift h (minus x1 (S n)) x2)
-H11) in (let H15 \def (eq_ind T u (\lambda (t0: T).(ty3 g d0 t0 t)) H2 (lift
-h (minus x1 (S n)) x2) H11) in (eq_ind_r T (lift h (minus x1 (S n)) x2)
-(\lambda (t0: T).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O
-t0))) (\lambda (t2: T).(ty3 g e (TLRef n) t2)))) (let H16 \def (H14 x2 (minus
-x1 (S n)) (refl_equal T (lift h (minus x1 (S n)) x2)) x3 H13) in (ex2_ind T
-(\lambda (t2: T).(pc3 d0 (lift h (minus x1 (S n)) t2) t)) (\lambda (t2:
-T).(ty3 g x3 x2 t2)) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S
-n) O (lift h (minus x1 (S n)) x2)))) (\lambda (t2: T).(ty3 g e (TLRef n)
-t2))) (\lambda (x4: T).(\lambda (_: (pc3 d0 (lift h (minus x1 (S n)) x4)
+T).(ty3 g e (TLRef n) t2))) (\lambda (x2: T).(\lambda (x3: C).(\lambda (H11:
+(eq T u (lift h (minus x1 (S n)) x2))).(\lambda (H12: (getl n e (CHead x3
+(Bind Abbr) x2))).(\lambda (H13: (drop h (minus x1 (S n)) d0 x3)).(let H14
+\def (eq_ind T u (\lambda (t0: T).(\forall (x4: T).(\forall (x5: nat).((eq T
+t0 (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 d0 e0) \to (ex2 T
+(\lambda (t2: T).(pc3 d0 (lift h x5 t2) t)) (\lambda (t2: T).(ty3 g e0 x4
+t2))))))))) H3 (lift h (minus x1 (S n)) x2) H11) in (let H15 \def (eq_ind T u
+(\lambda (t0: T).(ty3 g d0 t0 t)) H2 (lift h (minus x1 (S n)) x2) H11) in
+(let H16 \def (H14 x2 (minus x1 (S n)) (refl_equal T (lift h (minus x1 (S n))
+x2)) x3 H13) in (ex2_ind T (\lambda (t2: T).(pc3 d0 (lift h (minus x1 (S n))
+t2) t)) (\lambda (t2: T).(ty3 g x3 x2 t2)) (ex2 T (\lambda (t2: T).(pc3 c0
+(lift h x1 t2) (lift (S n) O t))) (\lambda (t2: T).(ty3 g e (TLRef n) t2)))
+(\lambda (x4: T).(\lambda (H17: (pc3 d0 (lift h (minus x1 (S n)) x4)
t)).(\lambda (H18: (ty3 g x3 x2 x4)).(eq_ind_r nat (plus (S n) (minus x1 (S
n))) (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h n0 t2) (lift
-(S n) O (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)))
+(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 u))) (\lambda (t2:
+(\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 u))) (\lambda (t2:
+(\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 u))) (\lambda (t2: T).(ty3 g e (TLRef (minus n h)) t2)) (lift
-(S (minus n h)) O u) (eq_ind_r T (lift (plus h (S (minus n h))) O u) (\lambda
-(t0: T).(pc3 c0 t0 (lift (S n) O u))) (eq_ind nat (S (plus h (minus n h)))
-(\lambda (n0: nat).(pc3 c0 (lift n0 O u) (lift (S n) O u))) (eq_ind nat n
-(\lambda (n0: nat).(pc3 c0 (lift (S n0) O u) (lift (S n) O u))) (pc3_refl c0
-(lift (S n) O u)) (plus h (minus n h)) (le_plus_minus h n (le_trans h (plus
+(lift (S n) O t))) (\lambda (t2: T).(ty3 g e (TLRef (minus n h)) t2)) (lift
+(S (minus n h)) O t) (eq_ind_r T (lift (plus h (S (minus n h))) O t) (\lambda
+(t0: T).(pc3 c0 t0 (lift (S n) O t))) (eq_ind nat (S (plus h (minus n h)))
+(\lambda (n0: nat).(pc3 c0 (lift n0 O t) (lift (S n) O t))) (eq_ind nat n
+(\lambda (n0: nat).(pc3 c0 (lift (S n0) O t) (lift (S n) O t))) (pc3_refl c0
+(lift (S n) O t)) (plus h (minus n h)) (le_plus_minus h n (le_trans h (plus
x1 h) n (le_plus_r x1 h) H8))) (plus h (S (minus n h))) (plus_n_Sm h (minus n
-h))) (lift h x1 (lift (S (minus n h)) O u)) (lift_free u (S (minus n h)) h O
+h))) (lift h x1 (lift (S (minus n h)) O t)) (lift_free t (S (minus n h)) h O
x1 (le_trans x1 (S (minus n h)) (plus O (S (minus n h))) (le_S_minus x1 h n
-H8) (le_n (plus O (S (minus n h))))) (le_O_n x1))) (ty3_abst g (minus n h) e
-d0 u (getl_drop_conf_ge n (CHead d0 (Bind Abst) u) c0 H1 e h x1 H5 H8) t H2))
-x0 H9))) H7)) H6)))))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda
-(t: T).(\lambda (H1: (ty3 g c0 u t)).(\lambda (H2: ((\forall (x0: T).(\forall
-(x1: nat).((eq T u (lift h x1 x0)) \to (\forall (e: C).((drop h x1 c0 e) \to
-(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) t)) (\lambda (t2: T).(ty3 g e
-x0 t2)))))))))).(\lambda (b: B).(\lambda (t2: T).(\lambda (t3: T).(\lambda
-(H3: (ty3 g (CHead c0 (Bind b) u) t2 t3)).(\lambda (H4: ((\forall (x0:
-T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to (\forall (e: C).((drop h
-x1 (CHead c0 (Bind b) u) e) \to (ex2 T (\lambda (t4: T).(pc3 (CHead c0 (Bind
-b) u) (lift h x1 t4) t3)) (\lambda (t4: T).(ty3 g e x0 t4)))))))))).(\lambda
-(t0: T).(\lambda (H5: (ty3 g (CHead c0 (Bind b) u) t3 t0)).(\lambda (H6:
-((\forall (x0: T).(\forall (x1: nat).((eq T t3 (lift h x1 x0)) \to (\forall
-(e: C).((drop h x1 (CHead c0 (Bind b) u) e) \to (ex2 T (\lambda (t4: T).(pc3
-(CHead c0 (Bind b) u) (lift h x1 t4) t0)) (\lambda (t4: T).(ty3 g e x0
-t4)))))))))).(\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 (t4:
-T).(\forall (x4: T).(\forall (x5: nat).((eq T t4 (lift h x5 x4)) \to (\forall
-(e0: C).((drop h x5 (CHead c0 (Bind b) u) e0) \to (ex2 T (\lambda (t5:
-T).(pc3 (CHead c0 (Bind b) u) (lift h x5 t5) t3)) (\lambda (t5: T).(ty3 g e0
-x4 t5))))))))) H4 (lift h (S x1) x3) H11) in (let H13 \def (eq_ind T t2
-(\lambda (t4: T).(ty3 g (CHead c0 (Bind b) u) t4 t3)) H3 (lift h (S x1) x3)
-H11) in (let H14 \def (eq_ind T u (\lambda (t4: T).(ty3 g (CHead c0 (Bind b)
-t4) (lift h (S x1) x3) t3)) H13 (lift h x1 x2) H10) in (let H15 \def (eq_ind
-T u (\lambda (t4: T).(\forall (x4: T).(\forall (x5: nat).((eq T (lift h (S
-x1) x3) (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 (CHead c0 (Bind b)
-t4) e0) \to (ex2 T (\lambda (t5: T).(pc3 (CHead c0 (Bind b) t4) (lift h x5
-t5) t3)) (\lambda (t5: T).(ty3 g e0 x4 t5))))))))) H12 (lift h x1 x2) H10) in
-(let H16 \def (eq_ind T u (\lambda (t4: T).(\forall (x4: T).(\forall (x5:
-nat).((eq T t3 (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 (CHead c0
-(Bind b) t4) e0) \to (ex2 T (\lambda (t5: T).(pc3 (CHead c0 (Bind b) t4)
-(lift h x5 t5) t0)) (\lambda (t5: T).(ty3 g e0 x4 t5))))))))) H6 (lift h x1
-x2) H10) in (let H17 \def (eq_ind T u (\lambda (t4: T).(ty3 g (CHead c0 (Bind
-b) t4) t3 t0)) H5 (lift h x1 x2) H10) in (let H18 \def (eq_ind T u (\lambda
-(t4: T).(\forall (x4: T).(\forall (x5: nat).((eq T t4 (lift h x5 x4)) \to
-(\forall (e0: C).((drop h x5 c0 e0) \to (ex2 T (\lambda (t5: T).(pc3 c0 (lift
-h x5 t5) t)) (\lambda (t5: T).(ty3 g e0 x4 t5))))))))) H2 (lift h x1 x2) H10)
-in (let H19 \def (eq_ind T u (\lambda (t4: T).(ty3 g c0 t4 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
+H8) (le_n (plus O (S (minus n h))))) (le_O_n x1))) (ty3_abbr g (minus n h) e
+d0 u (getl_drop_conf_ge n (CHead d0 (Bind Abbr) u) c0 H1 e h x1 H5 H8) t H2))
+x0 H9))) H7)) H6)))))))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda
+(d0: C).(\lambda (u: T).(\lambda (H1: (getl n c0 (CHead d0 (Bind Abst)
+u))).(\lambda (t: T).(\lambda (H2: (ty3 g d0 u t)).(\lambda (H3: ((\forall
+(x0: T).(\forall (x1: nat).((eq T u (lift h x1 x0)) \to (\forall (e:
+C).((drop h x1 d0 e) \to (ex2 T (\lambda (t2: T).(pc3 d0 (lift h x1 t2) t))
+(\lambda (t2: T).(ty3 g e x0 t2)))))))))).(\lambda (x0: T).(\lambda (x1:
+nat).(\lambda (H4: (eq T (TLRef n) (lift h x1 x0))).(\lambda (e: C).(\lambda
+(H5: (drop h x1 c0 e)).(let H_x \def (lift_gen_lref x0 x1 h n H4) in (let H6
+\def H_x in (or_ind (land (lt n x1) (eq T x0 (TLRef n))) (land (le (plus x1
+h) n) (eq T x0 (TLRef (minus n h)))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h
+x1 t2) (lift (S n) O u))) (\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (H7:
+(land (lt n x1) (eq T x0 (TLRef n)))).(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 (n0: nat).(drop h n0 c0 e)) H5
+(S (plus n (minus x1 (S n)))) (lt_plus_minus n x1 H8)) in (ex3_2_ind T C
+(\lambda (v: T).(\lambda (_: C).(eq T u (lift h (minus x1 (S n)) v))))
+(\lambda (v: T).(\lambda (e0: C).(getl n e (CHead e0 (Bind Abst) v))))
+(\lambda (_: T).(\lambda (e0: C).(drop h (minus x1 (S n)) d0 e0))) (ex2 T
+(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O u))) (\lambda (t2:
+T).(ty3 g e (TLRef n) t2))) (\lambda (x2: T).(\lambda (x3: C).(\lambda (H11:
+(eq T u (lift h (minus x1 (S n)) x2))).(\lambda (H12: (getl n e (CHead x3
+(Bind Abst) x2))).(\lambda (H13: (drop h (minus x1 (S n)) d0 x3)).(let H14
+\def (eq_ind T u (\lambda (t0: T).(\forall (x4: T).(\forall (x5: nat).((eq T
+t0 (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 d0 e0) \to (ex2 T
+(\lambda (t2: T).(pc3 d0 (lift h x5 t2) t)) (\lambda (t2: T).(ty3 g e0 x4
+t2))))))))) H3 (lift h (minus x1 (S n)) x2) H11) in (let H15 \def (eq_ind T u
+(\lambda (t0: T).(ty3 g d0 t0 t)) H2 (lift h (minus x1 (S n)) x2) H11) in
+(eq_ind_r T (lift h (minus x1 (S n)) x2) (\lambda (t0: T).(ex2 T (\lambda
+(t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t0))) (\lambda (t2: T).(ty3 g e
+(TLRef n) t2)))) (let H16 \def (H14 x2 (minus x1 (S n)) (refl_equal T (lift h
+(minus x1 (S n)) x2)) x3 H13) in (ex2_ind T (\lambda (t2: T).(pc3 d0 (lift h
+(minus x1 (S n)) t2) t)) (\lambda (t2: T).(ty3 g x3 x2 t2)) (ex2 T (\lambda
+(t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O (lift h (minus x1 (S n)) x2))))
+(\lambda (t2: T).(ty3 g e (TLRef n) t2))) (\lambda (x4: T).(\lambda (_: (pc3
+d0 (lift h (minus x1 (S n)) x4) t)).(\lambda (H18: (ty3 g x3 x2
+x4)).(eq_ind_r nat (plus (S n) (minus x1 (S n))) (\lambda (n0: nat).(ex2 T
+(\lambda (t2: T).(pc3 c0 (lift h n0 t2) (lift (S n) O (lift h (minus n0 (S
+n)) x2)))) (\lambda (t2: T).(ty3 g e (TLRef n) t2)))) (ex_intro2 T (\lambda
+(t2: T).(pc3 c0 (lift h (plus (S n) (minus x1 (S n))) t2) (lift (S n) O (lift
+h (minus (plus (S n) (minus x1 (S n))) (S n)) x2)))) (\lambda (t2: T).(ty3 g
+e (TLRef n) t2)) (lift (S n) O x2) (eq_ind_r T (lift (S n) O (lift h (minus
+x1 (S n)) x2)) (\lambda (t0: T).(pc3 c0 t0 (lift (S n) O (lift h (minus (plus
+(S n) (minus x1 (S n))) (S n)) x2)))) (eq_ind nat x1 (\lambda (n0: nat).(pc3
+c0 (lift (S n) O (lift h (minus x1 (S n)) x2)) (lift (S n) O (lift h (minus
+n0 (S n)) x2)))) (pc3_refl c0 (lift (S n) O (lift h (minus x1 (S n)) x2)))
+(plus (S n) (minus x1 (S n))) (le_plus_minus (S n) x1 H8)) (lift h (plus (S
+n) (minus x1 (S n))) (lift (S n) O x2)) (lift_d x2 h (S n) (minus x1 (S n)) O
+(le_O_n (minus x1 (S n))))) (ty3_abst g n e x3 x2 H12 x4 H18)) x1
+(le_plus_minus (S n) x1 H8))))) H16)) u H11)))))))) (getl_drop_conf_lt Abst
+c0 d0 u n H1 e h (minus x1 (S n)) H10))) x0 H9))) H7)) (\lambda (H7: (land
+(le (plus x1 h) n) (eq T x0 (TLRef (minus n h))))).(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 h (plus x1 h) n (le_plus_r x1 h) H8))) (plus h
+(S (minus n h))) (plus_n_Sm h (minus n h))) (lift h x1 (lift (S (minus n h))
+O u)) (lift_free u (S (minus n h)) h O x1 (le_trans x1 (S (minus n h)) (plus
+O (S (minus n h))) (le_S_minus x1 h n H8) (le_n (plus O (S (minus n h)))))
+(le_O_n x1))) (ty3_abst g (minus n h) e d0 u (getl_drop_conf_ge n (CHead d0
+(Bind Abst) u) c0 H1 e h x1 H5 H8) t H2)) x0 H9))) H7)) H6))))))))))))))))
+(\lambda (c0: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H1: (ty3 g c0 u
+t)).(\lambda (H2: ((\forall (x0: T).(\forall (x1: nat).((eq T u (lift h x1
+x0)) \to (\forall (e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t2: T).(pc3
+c0 (lift h x1 t2) t)) (\lambda (t2: T).(ty3 g e x0 t2)))))))))).(\lambda (b:
+B).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H3: (ty3 g (CHead c0 (Bind b)
+u) t2 t3)).(\lambda (H4: ((\forall (x0: T).(\forall (x1: nat).((eq T t2 (lift
+h x1 x0)) \to (\forall (e: C).((drop h x1 (CHead c0 (Bind b) u) e) \to (ex2 T
+(\lambda (t4: T).(pc3 (CHead c0 (Bind b) u) (lift h x1 t4) t3)) (\lambda (t4:
+T).(ty3 g e x0 t4)))))))))).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H5:
+(eq T (THead (Bind b) u t2) (lift h x1 x0))).(\lambda (e: C).(\lambda (H6:
+(drop h x1 c0 e)).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0
+(THead (Bind b) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u (lift h x1
+y0)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (S x1) z)))) (ex2 T
+(\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) u t3))) (\lambda (t4:
+T).(ty3 g e x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq T x0
+(THead (Bind b) x2 x3))).(\lambda (H8: (eq T u (lift h x1 x2))).(\lambda (H9:
+(eq T t2 (lift h (S x1) x3))).(eq_ind_r T (THead (Bind b) x2 x3) (\lambda
+(t0: T).(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) u
+t3))) (\lambda (t4: T).(ty3 g e t0 t4)))) (let H10 \def (eq_ind T t2 (\lambda
+(t0: T).(\forall (x4: T).(\forall (x5: nat).((eq T t0 (lift h x5 x4)) \to
+(\forall (e0: C).((drop h x5 (CHead c0 (Bind b) u) e0) \to (ex2 T (\lambda
+(t4: T).(pc3 (CHead c0 (Bind b) u) (lift h x5 t4) t3)) (\lambda (t4: T).(ty3
+g e0 x4 t4))))))))) H4 (lift h (S x1) x3) H9) in (let H11 \def (eq_ind T t2
+(\lambda (t0: T).(ty3 g (CHead c0 (Bind b) u) t0 t3)) H3 (lift h (S x1) x3)
+H9) in (let H12 \def (eq_ind T u (\lambda (t0: T).(ty3 g (CHead c0 (Bind b)
+t0) (lift h (S x1) x3) t3)) H11 (lift h x1 x2) H8) in (let H13 \def (eq_ind T
+u (\lambda (t0: T).(\forall (x4: T).(\forall (x5: nat).((eq T (lift h (S x1)
+x3) (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 (CHead c0 (Bind b) t0)
+e0) \to (ex2 T (\lambda (t4: T).(pc3 (CHead c0 (Bind b) t0) (lift h x5 t4)
+t3)) (\lambda (t4: T).(ty3 g e0 x4 t4))))))))) H10 (lift h x1 x2) H8) in (let
+H14 \def (eq_ind T u (\lambda (t0: T).(\forall (x4: T).(\forall (x5:
+nat).((eq T t0 (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 c0 e0) \to
+(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x5 t4) t)) (\lambda (t4: T).(ty3 g e0
+x4 t4))))))))) H2 (lift h x1 x2) H8) in (let H15 \def (eq_ind T u (\lambda
+(t0: T).(ty3 g c0 t0 t)) H1 (lift h x1 x2) H8) in (eq_ind_r T (lift h x1 x2)
+(\lambda (t0: T).(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind
+b) t0 t3))) (\lambda (t4: T).(ty3 g e (THead (Bind b) x2 x3) t4)))) (let H16
+\def (H14 x2 x1 (refl_equal T (lift h x1 x2)) e H6) in (ex2_ind T (\lambda
+(t4: T).(pc3 c0 (lift h x1 t4) t)) (\lambda (t4: T).(ty3 g e x2 t4)) (ex2 T
+(\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) (lift h x1 x2) t3)))
+(\lambda (t4: T).(ty3 g e (THead (Bind b) x2 x3) t4))) (\lambda (x4:
+T).(\lambda (_: (pc3 c0 (lift h x1 x4) t)).(\lambda (H18: (ty3 g e x2
+x4)).(let H19 \def (H13 x3 (S x1) (refl_equal T (lift h (S x1) x3)) (CHead e
+(Bind b) x2) (drop_skip_bind h x1 c0 e H6 b x2)) in (ex2_ind T (\lambda (t4:
+T).(pc3 (CHead c0 (Bind b) (lift h x1 x2)) (lift h (S x1) t4) t3)) (\lambda
+(t4: T).(ty3 g (CHead e (Bind b) x2) x3 t4)) (ex2 T (\lambda (t4: T).(pc3 c0
(lift h x1 t4) (THead (Bind b) (lift h x1 x2) t3))) (\lambda (t4: T).(ty3 g e
-(THead (Bind b) x2 x3) t4))) (\lambda (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 (x0: T).(\forall
-(x1: nat).((eq T w (lift h x1 x0)) \to (\forall (e: C).((drop h x1 c0 e) \to
-(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) u)) (\lambda (t2: T).(ty3 g e
-x0 t2)))))))))).(\lambda (v: T).(\lambda (t: T).(\lambda (H3: (ty3 g c0 v
-(THead (Bind Abst) u t))).(\lambda (H4: ((\forall (x0: T).(\forall (x1:
-nat).((eq T v (lift h x1 x0)) \to (\forall (e: C).((drop h x1 c0 e) \to (ex2
-T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead (Bind Abst) u t))) (\lambda
-(t2: T).(ty3 g e x0 t2)))))))))).(\lambda (x0: T).(\lambda (x1: nat).(\lambda
-(H5: (eq T (THead (Flat Appl) w v) (lift h x1 x0))).(\lambda (e: C).(\lambda
-(H6: (drop h x1 c0 e)).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T
-x0 (THead (Flat Appl) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T w (lift
-h x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T v (lift h x1 z)))) (ex2 T
-(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) w (THead (Bind
-Abst) u t)))) (\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (x2: T).(\lambda
-(x3: T).(\lambda (H7: (eq T x0 (THead (Flat Appl) x2 x3))).(\lambda (H8: (eq
-T w (lift h x1 x2))).(\lambda (H9: (eq T v (lift h x1 x3))).(eq_ind_r T
-(THead (Flat Appl) x2 x3) (\lambda (t0: T).(ex2 T (\lambda (t2: T).(pc3 c0
-(lift h x1 t2) (THead (Flat Appl) w (THead (Bind Abst) u t)))) (\lambda (t2:
-T).(ty3 g e t0 t2)))) (let H10 \def (eq_ind T v (\lambda (t0: T).(\forall
+(THead (Bind b) x2 x3) t4))) (\lambda (x5: T).(\lambda (H20: (pc3 (CHead c0
+(Bind b) (lift h x1 x2)) (lift h (S x1) x5) t3)).(\lambda (H21: (ty3 g (CHead
+e (Bind b) x2) x3 x5)).(ex_ind T (\lambda (t0: T).(ty3 g (CHead e (Bind b)
+x2) x5 t0)) (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b)
+(lift h x1 x2) t3))) (\lambda (t4: T).(ty3 g e (THead (Bind b) x2 x3) t4)))
+(\lambda (x6: T).(\lambda (_: (ty3 g (CHead e (Bind b) x2) x5 x6)).(ex_intro2
+T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) (lift h x1 x2)
+t3))) (\lambda (t4: T).(ty3 g e (THead (Bind b) x2 x3) t4)) (THead (Bind b)
+x2 x5) (eq_ind_r T (THead (Bind b) (lift h x1 x2) (lift h (S x1) x5))
+(\lambda (t0: T).(pc3 c0 t0 (THead (Bind b) (lift h x1 x2) t3))) (pc3_head_2
+c0 (lift h x1 x2) (lift h (S x1) x5) t3 (Bind b) H20) (lift h x1 (THead (Bind
+b) x2 x5)) (lift_bind b x2 x5 h x1)) (ty3_bind g e x2 x4 H18 b x3 x5 H21))))
+(ty3_correct g (CHead e (Bind b) x2) x3 x5 H21))))) H19))))) H16)) u
+H8))))))) x0 H7)))))) (lift_gen_bind b u t2 x0 h x1 H5)))))))))))))))))
+(\lambda (c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda (H1: (ty3 g c0 w
+u)).(\lambda (H2: ((\forall (x0: T).(\forall (x1: nat).((eq T w (lift h x1
+x0)) \to (\forall (e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t2: T).(pc3
+c0 (lift h x1 t2) u)) (\lambda (t2: T).(ty3 g e x0 t2)))))))))).(\lambda (v:
+T).(\lambda (t: T).(\lambda (H3: (ty3 g c0 v (THead (Bind Abst) u
+t))).(\lambda (H4: ((\forall (x0: T).(\forall (x1: nat).((eq T v (lift h x1
+x0)) \to (\forall (e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t2: T).(pc3
+c0 (lift h x1 t2) (THead (Bind Abst) u t))) (\lambda (t2: T).(ty3 g e x0
+t2)))))))))).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H5: (eq T (THead
+(Flat Appl) w v) (lift h x1 x0))).(\lambda (e: C).(\lambda (H6: (drop h x1 c0
+e)).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Flat
+Appl) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T w (lift h x1 y0))))
+(\lambda (_: T).(\lambda (z: T).(eq T v (lift h x1 z)))) (ex2 T (\lambda (t2:
+T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) w (THead (Bind Abst) u t))))
+(\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (x2: T).(\lambda (x3: T).(\lambda
+(H7: (eq T x0 (THead (Flat Appl) x2 x3))).(\lambda (H8: (eq T w (lift h x1
+x2))).(\lambda (H9: (eq T v (lift h x1 x3))).(eq_ind_r T (THead (Flat Appl)
+x2 x3) (\lambda (t0: T).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead
+(Flat Appl) w (THead (Bind Abst) u t)))) (\lambda (t2: T).(ty3 g e t0 t2))))
+(let H10 \def (eq_ind T v (\lambda (t0: T).(\forall (x4: T).(\forall (x5:
+nat).((eq T t0 (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 c0 e0) \to
+(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x5 t2) (THead (Bind Abst) u t)))
+(\lambda (t2: T).(ty3 g e0 x4 t2))))))))) H4 (lift h x1 x3) H9) in (let H11
+\def (eq_ind T v (\lambda (t0: T).(ty3 g c0 t0 (THead (Bind Abst) u t))) H3
+(lift h x1 x3) H9) in (let H12 \def (eq_ind T w (\lambda (t0: T).(\forall
(x4: T).(\forall (x5: nat).((eq T t0 (lift h x5 x4)) \to (\forall (e0:
-C).((drop h x5 c0 e0) \to (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x5 t2)
-(THead (Bind Abst) u t))) (\lambda (t2: T).(ty3 g e0 x4 t2))))))))) H4 (lift
-h x1 x3) H9) in (let H11 \def (eq_ind T v (\lambda (t0: T).(ty3 g c0 t0
-(THead (Bind Abst) u t))) H3 (lift h x1 x3) H9) in (let H12 \def (eq_ind T w
-(\lambda (t0: T).(\forall (x4: T).(\forall (x5: nat).((eq T t0 (lift h x5
-x4)) \to (\forall (e0: C).((drop h x5 c0 e0) \to (ex2 T (\lambda (t2: T).(pc3
-c0 (lift h x5 t2) u)) (\lambda (t2: T).(ty3 g e0 x4 t2))))))))) H2 (lift h x1
-x2) H8) in (let H13 \def (eq_ind T w (\lambda (t0: T).(ty3 g c0 t0 u)) H1
-(lift h x1 x2) H8) in (eq_ind_r T (lift h x1 x2) (\lambda (t0: T).(ex2 T
-(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) t0 (THead (Bind
-Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2)))) (let
-H14 \def (H12 x2 x1 (refl_equal T (lift h x1 x2)) e H6) in (ex2_ind T
-(\lambda (t2: T).(pc3 c0 (lift h x1 t2) u)) (\lambda (t2: T).(ty3 g e x2 t2))
-(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) (lift h x1
-x2) (THead (Bind Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead (Flat Appl)
-x2 x3) t2))) (\lambda (x4: T).(\lambda (H15: (pc3 c0 (lift h x1 x4)
-u)).(\lambda (H16: (ty3 g e x2 x4)).(let H17 \def (H10 x3 x1 (refl_equal T
-(lift h x1 x3)) e H6) in (ex2_ind T (\lambda (t2: T).(pc3 c0 (lift h x1 t2)
-(THead (Bind Abst) u t))) (\lambda (t2: T).(ty3 g e x3 t2)) (ex2 T (\lambda
+C).((drop h x5 c0 e0) \to (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x5 t2) u))
+(\lambda (t2: T).(ty3 g e0 x4 t2))))))))) H2 (lift h x1 x2) H8) in (let H13
+\def (eq_ind T w (\lambda (t0: T).(ty3 g c0 t0 u)) H1 (lift h x1 x2) H8) in
+(eq_ind_r T (lift h x1 x2) (\lambda (t0: T).(ex2 T (\lambda (t2: T).(pc3 c0
+(lift h x1 t2) (THead (Flat Appl) t0 (THead (Bind Abst) u t)))) (\lambda (t2:
+T).(ty3 g e (THead (Flat Appl) x2 x3) t2)))) (let H14 \def (H12 x2 x1
+(refl_equal T (lift h x1 x2)) e H6) in (ex2_ind T (\lambda (t2: T).(pc3 c0
+(lift h x1 t2) u)) (\lambda (t2: T).(ty3 g e x2 t2)) (ex2 T (\lambda (t2:
+T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind
+Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2)))
+(\lambda (x4: T).(\lambda (H15: (pc3 c0 (lift h x1 x4) u)).(\lambda (H16:
+(ty3 g e x2 x4)).(let H17 \def (H10 x3 x1 (refl_equal T (lift h x1 x3)) e H6)
+in (ex2_ind T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead (Bind Abst) u
+t))) (\lambda (t2: T).(ty3 g e x3 t2)) (ex2 T (\lambda (t2: T).(pc3 c0 (lift
+h x1 t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind Abst) u t))))
+(\lambda (t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2))) (\lambda (x5:
+T).(\lambda (H18: (pc3 c0 (lift h x1 x5) (THead (Bind Abst) u t))).(\lambda
+(H19: (ty3 g e x3 x5)).(ex3_2_ind T T (\lambda (u1: T).(\lambda (t2: T).(pr3
+e x5 (THead (Bind Abst) u1 t2)))) (\lambda (u1: T).(\lambda (_: T).(pr3 c0 u
+(lift h x1 u1)))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall
+(u0: T).(pr3 (CHead c0 (Bind b) u0) t (lift h (S x1) t2)))))) (ex2 T (\lambda
(t2: T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind
Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2)))
-(\lambda (x5: T).(\lambda (H18: (pc3 c0 (lift h x1 x5) (THead (Bind Abst) u
-t))).(\lambda (H19: (ty3 g e x3 x5)).(ex3_2_ind T T (\lambda (u1: T).(\lambda
-(t2: T).(pr3 e x5 (THead (Bind Abst) u1 t2)))) (\lambda (u1: T).(\lambda (_:
-T).(pr3 c0 u (lift h x1 u1)))) (\lambda (_: T).(\lambda (t2: T).(\forall (b:
-B).(\forall (u0: T).(pr3 (CHead c0 (Bind b) u0) t (lift h (S x1) t2))))))
-(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) (lift h x1
-x2) (THead (Bind Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead (Flat Appl)
-x2 x3) t2))) (\lambda (x6: T).(\lambda (x7: T).(\lambda (H20: (pr3 e x5
-(THead (Bind Abst) x6 x7))).(\lambda (H21: (pr3 c0 u (lift h x1
-x6))).(\lambda (H22: ((\forall (b: B).(\forall (u0: T).(pr3 (CHead c0 (Bind
-b) u0) t (lift h (S x1) x7)))))).(ex_ind T (\lambda (t0: T).(ty3 g e x5 t0))
-(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) (lift h x1
-x2) (THead (Bind Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead (Flat Appl)
-x2 x3) t2))) (\lambda (x8: T).(\lambda (H23: (ty3 g e x5 x8)).(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
+(\lambda (x6: T).(\lambda (x7: T).(\lambda (H20: (pr3 e x5 (THead (Bind Abst)
+x6 x7))).(\lambda (H21: (pr3 c0 u (lift h x1 x6))).(\lambda (H22: ((\forall
+(b: B).(\forall (u0: T).(pr3 (CHead c0 (Bind b) u0) t (lift h (S x1)
+x7)))))).(ex_ind T (\lambda (t0: T).(ty3 g e x5 t0)) (ex2 T (\lambda (t2:
+T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind
+Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2)))
+(\lambda (x8: T).(\lambda (H23: (ty3 g e x5 x8)).(let H_y \def (ty3_sred_pr3
+e x5 (THead (Bind Abst) x6 x7) H20 g x8 H23) in (ex3_2_ind T T (\lambda (t2:
+T).(\lambda (_: T).(pc3 e (THead (Bind Abst) x6 t2) x8))) (\lambda (_:
+T).(\lambda (t0: T).(ty3 g e x6 t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g
+(CHead e (Bind Abst) x6) x7 t2))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1
t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind Abst) u t)))) (\lambda
(t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2))) (\lambda (x9: T).(\lambda
-(x10: T).(\lambda (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)
+(x10: T).(\lambda (_: (pc3 e (THead (Bind Abst) x6 x9) x8)).(\lambda (H25:
+(ty3 g e x6 x10)).(\lambda (H26: (ty3 g (CHead e (Bind Abst) x6) x7
+x9)).(ex_intro2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl)
(lift h x1 x2) (THead (Bind Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead
(Flat Appl) x2 x3) t2)) (THead (Flat Appl) x2 (THead (Bind Abst) x6 x7))
(eq_ind_r T (THead (Flat Appl) (lift h x1 x2) (lift h x1 (THead (Bind Abst)
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)))))))
+H26) x3 x5 H19 (pc3_pr3_r e x5 (THead (Bind Abst) x6 x7) H20)))))))))
+(ty3_gen_bind g Abst e x6 x7 x8 H_y))))) (ty3_correct g e x3 x5 H19)))))))
(pc3_gen_lift_abst c0 x5 t u h x1 H18 e H6))))) H17))))) H14)) w H8))))) x0
H7)))))) (lift_gen_flat Appl w v x0 h x1 H5)))))))))))))))) (\lambda (c0:
C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (ty3 g c0 t2 t3)).(\lambda
\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)))))))).
+(ty3 g c t1 x)).(let H_y \def (ty3_sred_pr3 c t1 t2 H0 g x H1) in (ty3_conv g
+c t2 x H_y u t1 H (pc3_pr3_r c t1 t2 H0))))) (ty3_correct g c u t1 H)))))))).
theorem ty3_sconv_pc3:
\forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (t1: T).((ty3 g c
(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)))))))))).
+T).(\lambda (H3: (pr3 c u1 x)).(\lambda (H4: (pr3 c u2 x)).(let H_y \def
+(ty3_sred_pr3 c u2 x H4 g t2 H0) in (let H_y0 \def (ty3_sred_pr3 c u1 x H3 g
+t1 H) in (ty3_unique g c x t1 H_y0 t2 H_y)))))) H2)))))))))).
theorem ty3_sred_back:
\forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t0: T).((ty3 g c
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/ty3/props".
include "ty3/fwd.ma".
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 (THead (Flat Cast)
-t4 t3)))) (eq_ind_r T (THead (Flat Cast) (lift h d t4) (lift h (s (Flat Cast)
-d) t3)) (\lambda (t: T).(ty3 g c0 (THead (Flat Cast) (lift h d t3) (lift h (s
-(Flat Cast) d) t0)) t)) (ty3_cast g c0 (lift h (s (Flat Cast) d) t0) (lift h
-(s (Flat Cast) d) t3) (H1 c0 (s (Flat Cast) d) h H4) (lift h d t4) (H3 c0 d h
-H4)) (lift h d (THead (Flat Cast) t4 t3)) (lift_head (Flat Cast) t4 t3 h d))
-(lift h d (THead (Flat Cast) t3 t0)) (lift_head (Flat Cast) t3 t0 h
-d)))))))))))))) e t1 t2 H))))).
+(lift h d t0) (lift h d t3)))))))).(\lambda (c0: C).(\lambda (d:
+nat).(\lambda (h: nat).(\lambda (H4: (drop h d c0 c)).(eq_ind_r T (THead
+(Bind b) (lift h d u) (lift h (s (Bind b) d) t0)) (\lambda (t4: T).(ty3 g c0
+t4 (lift h d (THead (Bind b) u t3)))) (eq_ind_r T (THead (Bind b) (lift h d
+u) (lift h (s (Bind b) d) t3)) (\lambda (t4: T).(ty3 g c0 (THead (Bind b)
+(lift h d u) (lift h (s (Bind b) d) t0)) t4)) (ty3_bind g c0 (lift h d u)
+(lift h d t) (H1 c0 d h H4) b (lift h (S d) t0) (lift h (S d) t3) (H3 (CHead
+c0 (Bind b) (lift h d u)) (S d) h (drop_skip_bind h d c0 c H4 b u))) (lift h
+d (THead (Bind b) u t3)) (lift_head (Bind b) u t3 h d)) (lift h d (THead
+(Bind b) u t0)) (lift_head (Bind b) u t0 h d)))))))))))))))) (\lambda (c:
+C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c w u)).(\lambda (H1:
+((\forall (c0: C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c) \to
+(ty3 g c0 (lift h d w) (lift h d u)))))))).(\lambda (v: T).(\lambda (t:
+T).(\lambda (_: (ty3 g c v (THead (Bind Abst) u t))).(\lambda (H3: ((\forall
+(c0: C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c) \to (ty3 g c0
+(lift h d v) (lift h d (THead (Bind Abst) u t))))))))).(\lambda (c0:
+C).(\lambda (d: nat).(\lambda (h: nat).(\lambda (H4: (drop h d c0
+c)).(eq_ind_r T (THead (Flat Appl) (lift h d w) (lift h (s (Flat Appl) d) v))
+(\lambda (t0: T).(ty3 g c0 t0 (lift h d (THead (Flat Appl) w (THead (Bind
+Abst) u t))))) (eq_ind_r T (THead (Flat Appl) (lift h d w) (lift h (s (Flat
+Appl) d) (THead (Bind Abst) u t))) (\lambda (t0: T).(ty3 g c0 (THead (Flat
+Appl) (lift h d w) (lift h (s (Flat Appl) d) v)) t0)) (eq_ind_r T (THead
+(Bind Abst) (lift h (s (Flat Appl) d) u) (lift h (s (Bind Abst) (s (Flat
+Appl) d)) t)) (\lambda (t0: T).(ty3 g c0 (THead (Flat Appl) (lift h d w)
+(lift h (s (Flat Appl) d) v)) (THead (Flat Appl) (lift h d w) t0))) (ty3_appl
+g c0 (lift h d w) (lift h d u) (H1 c0 d h H4) (lift h d v) (lift h (S d) t)
+(eq_ind T (lift h d (THead (Bind Abst) u t)) (\lambda (t0: T).(ty3 g c0 (lift
+h d v) t0)) (H3 c0 d h H4) (THead (Bind Abst) (lift h d u) (lift h (S d) t))
+(lift_bind Abst u t h d))) (lift h (s (Flat Appl) d) (THead (Bind Abst) u t))
+(lift_head (Bind Abst) u t h (s (Flat Appl) d))) (lift h d (THead (Flat Appl)
+w (THead (Bind Abst) u t))) (lift_head (Flat Appl) w (THead (Bind Abst) u t)
+h d)) (lift h d (THead (Flat Appl) w v)) (lift_head (Flat Appl) w v h
+d))))))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda (t3: T).(\lambda
+(_: (ty3 g c t0 t3)).(\lambda (H1: ((\forall (c0: C).(\forall (d:
+nat).(\forall (h: nat).((drop h d c0 c) \to (ty3 g c0 (lift h d t0) (lift h d
+t3)))))))).(\lambda (t4: T).(\lambda (_: (ty3 g c t3 t4)).(\lambda (H3:
+((\forall (c0: C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c) \to
+(ty3 g c0 (lift h d t3) (lift h d t4)))))))).(\lambda (c0: C).(\lambda (d:
+nat).(\lambda (h: nat).(\lambda (H4: (drop h d c0 c)).(eq_ind_r T (THead
+(Flat Cast) (lift h d t3) (lift h (s (Flat Cast) d) t0)) (\lambda (t: T).(ty3
+g c0 t (lift h d (THead (Flat Cast) t4 t3)))) (eq_ind_r T (THead (Flat Cast)
+(lift h d t4) (lift h (s (Flat Cast) d) t3)) (\lambda (t: T).(ty3 g c0 (THead
+(Flat Cast) (lift h d t3) (lift h (s (Flat Cast) d) t0)) t)) (ty3_cast g c0
+(lift h (s (Flat Cast) d) t0) (lift h (s (Flat Cast) d) t3) (H1 c0 (s (Flat
+Cast) d) h H4) (lift h d t4) (H3 c0 d h H4)) (lift h d (THead (Flat Cast) t4
+t3)) (lift_head (Flat Cast) t4 t3 h d)) (lift h d (THead (Flat Cast) t3 t0))
+(lift_head (Flat Cast) t3 t0 h d)))))))))))))) e t1 t2 H))))).
theorem ty3_correct:
\forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c
(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 (t4: T).(ty3 g (CHead c0 (Bind b) u) t3 t4)))).(\lambda (t4:
-T).(\lambda (H4: (ty3 g (CHead c0 (Bind b) u) t3 t4)).(\lambda (H5: (ex T
-(\lambda (t5: T).(ty3 g (CHead c0 (Bind b) u) t4 t5)))).(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 (_: (ex T
-(\lambda (t: T).(ty3 g c0 t3 t)))).(\lambda (t4: T).(\lambda (H2: (ty3 g c0
-t3 t4)).(\lambda (H3: (ex T (\lambda (t: T).(ty3 g c0 t4 t)))).(let H4 \def
-H3 in (ex_ind T (\lambda (t: T).(ty3 g c0 t4 t)) (ex T (\lambda (t: T).(ty3 g
-c0 (THead (Flat Cast) t4 t3) t))) (\lambda (x: T).(\lambda (H5: (ty3 g c0 t4
-x)).(ex_intro T (\lambda (t: T).(ty3 g c0 (THead (Flat Cast) t4 t3) t))
-(THead (Flat Cast) x t4) (ty3_cast g c0 t3 t4 H2 x H5)))) H4)))))))))) c t1
-t2 H))))).
+T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t0 t3)).(\lambda (H3: (ex T
+(\lambda (t4: T).(ty3 g (CHead c0 (Bind b) u) t3 t4)))).(let H4 \def H3 in
+(ex_ind T (\lambda (t4: T).(ty3 g (CHead c0 (Bind b) u) t3 t4)) (ex T
+(\lambda (t4: T).(ty3 g c0 (THead (Bind b) u t3) t4))) (\lambda (x:
+T).(\lambda (H5: (ty3 g (CHead c0 (Bind b) u) t3 x)).(ex_intro T (\lambda
+(t4: T).(ty3 g c0 (THead (Bind b) u t3) t4)) (THead (Bind b) u x) (ty3_bind g
+c0 u t H0 b t3 x H5)))) H4)))))))))))) (\lambda (c0: C).(\lambda (w:
+T).(\lambda (u: T).(\lambda (H0: (ty3 g c0 w u)).(\lambda (H1: (ex T (\lambda
+(t: T).(ty3 g c0 u t)))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g
+c0 v (THead (Bind Abst) u t))).(\lambda (H3: (ex T (\lambda (t0: T).(ty3 g c0
+(THead (Bind Abst) u t) t0)))).(let H4 \def H1 in (ex_ind T (\lambda (t0:
+T).(ty3 g c0 u t0)) (ex T (\lambda (t0: T).(ty3 g c0 (THead (Flat Appl) w
+(THead (Bind Abst) u t)) t0))) (\lambda (x: T).(\lambda (_: (ty3 g c0 u
+x)).(let H6 \def H3 in (ex_ind T (\lambda (t0: T).(ty3 g c0 (THead (Bind
+Abst) u t) t0)) (ex T (\lambda (t0: T).(ty3 g c0 (THead (Flat Appl) w (THead
+(Bind Abst) u t)) t0))) (\lambda (x0: T).(\lambda (H7: (ty3 g c0 (THead (Bind
+Abst) u t) x0)).(ex3_2_ind T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0
+(THead (Bind Abst) u t3) x0))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u
+t0))) (\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind Abst) u) t
+t3))) (ex T (\lambda (t0: T).(ty3 g c0 (THead (Flat Appl) w (THead (Bind
+Abst) u t)) t0))) (\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 c0
+(THead (Bind Abst) u x1) x0)).(\lambda (H9: (ty3 g c0 u x2)).(\lambda (H10:
+(ty3 g (CHead c0 (Bind Abst) u) t x1)).(ex_intro T (\lambda (t0: T).(ty3 g c0
+(THead (Flat Appl) w (THead (Bind Abst) u t)) t0)) (THead (Flat Appl) w
+(THead (Bind Abst) u x1)) (ty3_appl g c0 w u H0 (THead (Bind Abst) u t) x1
+(ty3_bind g c0 u x2 H9 Abst t x1 H10)))))))) (ty3_gen_bind g Abst c0 u t x0
+H7)))) H6)))) H4))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t3:
+T).(\lambda (_: (ty3 g c0 t0 t3)).(\lambda (_: (ex T (\lambda (t: T).(ty3 g
+c0 t3 t)))).(\lambda (t4: T).(\lambda (H2: (ty3 g c0 t3 t4)).(\lambda (H3:
+(ex T (\lambda (t: T).(ty3 g c0 t4 t)))).(let H4 \def H3 in (ex_ind T
+(\lambda (t: T).(ty3 g c0 t4 t)) (ex T (\lambda (t: T).(ty3 g c0 (THead (Flat
+Cast) t4 t3) t))) (\lambda (x: T).(\lambda (H5: (ty3 g c0 t4 x)).(ex_intro T
+(\lambda (t: T).(ty3 g c0 (THead (Flat Cast) t4 t3) t)) (THead (Flat Cast) x
+t4) (ty3_cast g c0 t3 t4 H2 x H5)))) H4)))))))))) c t1 t2 H))))).
theorem ty3_unique:
\forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).((ty3 g c u
(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 (H3: ((\forall (t4: T).((ty3 g c0 t2 t4) \to (pc3 c0 t3
-t4))))).(\lambda (t4: T).(\lambda (H4: (ty3 g c0 (THead (Flat Cast) t2 t0)
-t4)).(ex3_ind T (\lambda (t5: T).(pc3 c0 (THead (Flat Cast) t5 t2) t4))
-(\lambda (_: T).(ty3 g c0 t0 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5)) (pc3 c0
-(THead (Flat Cast) t3 t2) t4) (\lambda (x0: T).(\lambda (H5: (pc3 c0 (THead
-(Flat Cast) x0 t2) t4)).(\lambda (_: (ty3 g c0 t0 t2)).(\lambda (H7: (ty3 g
-c0 t2 x0)).(pc3_t (THead (Flat Cast) x0 t2) c0 (THead (Flat Cast) t3 t2)
-(pc3_head_1 c0 t3 x0 (H3 x0 H7) (Flat Cast) t2) t4 H5))))) (ty3_gen_cast g c0
-t0 t2 t4 H4)))))))))))) c u t1 H))))).
+t3))))).(\lambda (t3: T).(\lambda (H4: (ty3 g c0 (THead (Bind b) u0 t0)
+t3)).(ex3_2_ind T T (\lambda (t4: T).(\lambda (_: T).(pc3 c0 (THead (Bind b)
+u0 t4) t3))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c0 u0 t5))) (\lambda
+(t4: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u0) t0 t4))) (pc3 c0 (THead
+(Bind b) u0 t2) t3) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (pc3 c0
+(THead (Bind b) u0 x0) t3)).(\lambda (_: (ty3 g c0 u0 x1)).(\lambda (H7: (ty3
+g (CHead c0 (Bind b) u0) t0 x0)).(pc3_t (THead (Bind b) u0 x0) c0 (THead
+(Bind b) u0 t2) (pc3_head_2 c0 u0 t2 x0 (Bind b) (H3 x0 H7)) t3 H5))))))
+(ty3_gen_bind g b c0 u0 t0 t3 H4)))))))))))))) (\lambda (c0: C).(\lambda (w:
+T).(\lambda (u0: T).(\lambda (_: (ty3 g c0 w u0)).(\lambda (_: ((\forall (t2:
+T).((ty3 g c0 w t2) \to (pc3 c0 u0 t2))))).(\lambda (v: T).(\lambda (t:
+T).(\lambda (_: (ty3 g c0 v (THead (Bind Abst) u0 t))).(\lambda (H3:
+((\forall (t2: T).((ty3 g c0 v t2) \to (pc3 c0 (THead (Bind Abst) u0 t)
+t2))))).(\lambda (t2: T).(\lambda (H4: (ty3 g c0 (THead (Flat Appl) w v)
+t2)).(ex3_2_ind T T (\lambda (u1: T).(\lambda (t0: T).(pc3 c0 (THead (Flat
+Appl) w (THead (Bind Abst) u1 t0)) t2))) (\lambda (u1: T).(\lambda (t0:
+T).(ty3 g c0 v (THead (Bind Abst) u1 t0)))) (\lambda (u1: T).(\lambda (_:
+T).(ty3 g c0 w u1))) (pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t))
+t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (pc3 c0 (THead (Flat
+Appl) w (THead (Bind Abst) x0 x1)) t2)).(\lambda (H6: (ty3 g c0 v (THead
+(Bind Abst) x0 x1))).(\lambda (_: (ty3 g c0 w x0)).(pc3_t (THead (Flat Appl)
+w (THead (Bind Abst) x0 x1)) c0 (THead (Flat Appl) w (THead (Bind Abst) u0
+t)) (pc3_thin_dx c0 (THead (Bind Abst) u0 t) (THead (Bind Abst) x0 x1) (H3
+(THead (Bind Abst) x0 x1) H6) w Appl) t2 H5)))))) (ty3_gen_appl g c0 w v t2
+H4))))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda
+(_: (ty3 g c0 t0 t2)).(\lambda (_: ((\forall (t3: T).((ty3 g c0 t0 t3) \to
+(pc3 c0 t2 t3))))).(\lambda (t3: T).(\lambda (_: (ty3 g c0 t2 t3)).(\lambda
+(H3: ((\forall (t4: T).((ty3 g c0 t2 t4) \to (pc3 c0 t3 t4))))).(\lambda (t4:
+T).(\lambda (H4: (ty3 g c0 (THead (Flat Cast) t2 t0) t4)).(ex3_ind T (\lambda
+(t5: T).(pc3 c0 (THead (Flat Cast) t5 t2) t4)) (\lambda (_: T).(ty3 g c0 t0
+t2)) (\lambda (t5: T).(ty3 g c0 t2 t5)) (pc3 c0 (THead (Flat Cast) t3 t2) t4)
+(\lambda (x0: T).(\lambda (H5: (pc3 c0 (THead (Flat Cast) x0 t2)
+t4)).(\lambda (_: (ty3 g c0 t0 t2)).(\lambda (H7: (ty3 g c0 t2 x0)).(pc3_t
+(THead (Flat Cast) x0 t2) c0 (THead (Flat Cast) t3 t2) (pc3_head_1 c0 t3 x0
+(H3 x0 H7) (Flat Cast) t2) t4 H5))))) (ty3_gen_cast g c0 t0 t2 t4
+H4)))))))))))) c u t1 H))))).
theorem ty3_gen_abst_abst:
\forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall
t2))).(ex_ind T (\lambda (t: T).(ty3 g c (THead (Bind Abst) u t2) t)) (ex2 T
(\lambda (w: T).(ty3 g c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) u)
t1 t2))) (\lambda (x: T).(\lambda (H0: (ty3 g c (THead (Bind Abst) u t2)
-x)).(ex4_3_ind T T T (\lambda (t3: T).(\lambda (_: T).(\lambda (_: T).(pc3 c
-(THead (Bind Abst) u t3) x)))) (\lambda (_: T).(\lambda (t: T).(\lambda (_:
-T).(ty3 g c u t)))) (\lambda (t3: T).(\lambda (_: T).(\lambda (_: T).(ty3 g
-(CHead c (Bind Abst) u) t2 t3)))) (\lambda (t3: T).(\lambda (_: T).(\lambda
-(t0: T).(ty3 g (CHead c (Bind Abst) u) t3 t0)))) (ex2 T (\lambda (w: T).(ty3
-g c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t1 t2))) (\lambda
-(x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 c (THead (Bind
-Abst) u x0) x)).(\lambda (H2: (ty3 g c u x1)).(\lambda (H3: (ty3 g (CHead c
-(Bind Abst) u) t2 x0)).(\lambda (_: (ty3 g (CHead c (Bind Abst) u) x0
-x2)).(ex4_3_ind T T T (\lambda (t3: T).(\lambda (_: T).(\lambda (_: T).(pc3 c
-(THead (Bind Abst) u t3) (THead (Bind Abst) u t2))))) (\lambda (_:
-T).(\lambda (t: T).(\lambda (_: T).(ty3 g c u t)))) (\lambda (t3: T).(\lambda
-(_: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t1 t3)))) (\lambda (t3:
-T).(\lambda (_: T).(\lambda (t0: T).(ty3 g (CHead c (Bind Abst) u) t3 t0))))
-(ex2 T (\lambda (w: T).(ty3 g c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind
-Abst) u) t1 t2))) (\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda
-(H5: (pc3 c (THead (Bind Abst) u x3) (THead (Bind Abst) u t2))).(\lambda (_:
-(ty3 g c u x4)).(\lambda (H7: (ty3 g (CHead c (Bind Abst) u) t1 x3)).(\lambda
-(_: (ty3 g (CHead c (Bind Abst) u) x3 x5)).(let H_y \def (pc3_gen_abst_shift
-c u x3 t2 H5) in (ex_intro2 T (\lambda (w: T).(ty3 g c u w)) (\lambda (_:
-T).(ty3 g (CHead c (Bind Abst) u) t1 t2)) x1 H2 (ty3_conv g (CHead c (Bind
-Abst) u) t2 x0 H3 t1 x3 H7 H_y)))))))))) (ty3_gen_bind g Abst c u t1 (THead
-(Bind Abst) u t2) H))))))))) (ty3_gen_bind g Abst c u t2 x H0))))
-(ty3_correct g c (THead (Bind Abst) u t1) (THead (Bind Abst) u t2) H))))))).
+x)).(ex3_2_ind T T (\lambda (t3: T).(\lambda (_: T).(pc3 c (THead (Bind Abst)
+u t3) x))) (\lambda (_: T).(\lambda (t: T).(ty3 g c u t))) (\lambda (t3:
+T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t2 t3))) (ex2 T (\lambda
+(w: T).(ty3 g c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t1 t2)))
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c (THead (Bind Abst) u
+x0) x)).(\lambda (_: (ty3 g c u x1)).(\lambda (H3: (ty3 g (CHead c (Bind
+Abst) u) t2 x0)).(ex3_2_ind T T (\lambda (t3: T).(\lambda (_: T).(pc3 c
+(THead (Bind Abst) u t3) (THead (Bind Abst) u t2)))) (\lambda (_: T).(\lambda
+(t: T).(ty3 g c u t))) (\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c (Bind
+Abst) u) t1 t3))) (ex2 T (\lambda (w: T).(ty3 g c u w)) (\lambda (_: T).(ty3
+g (CHead c (Bind Abst) u) t1 t2))) (\lambda (x2: T).(\lambda (x3: T).(\lambda
+(H4: (pc3 c (THead (Bind Abst) u x2) (THead (Bind Abst) u t2))).(\lambda (H5:
+(ty3 g c u x3)).(\lambda (H6: (ty3 g (CHead c (Bind Abst) u) t1 x2)).(let H_y
+\def (pc3_gen_abst_shift c u x2 t2 H4) in (ex_intro2 T (\lambda (w: T).(ty3 g
+c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t1 t2)) x3 H5
+(ty3_conv g (CHead c (Bind Abst) u) t2 x0 H3 t1 x2 H6 H_y))))))))
+(ty3_gen_bind g Abst c u t1 (THead (Bind Abst) u t2) H))))))) (ty3_gen_bind g
+Abst c u t2 x H0)))) (ty3_correct g c (THead (Bind Abst) u t1) (THead (Bind
+Abst) u t2) H))))))).
theorem ty3_typecheck:
\forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (v: T).((ty3 g c t
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/ty3/subst1".
include "ty3/props.ma".
(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
+(H4: (getl d c0 (CHead e (Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H5:
+(csubst1 d u0 c0 a0)).(\lambda (a: C).(\lambda (H6: (drop (S O) d a0 a)).(let
+H7 \def (H1 e u0 d H4 a0 H5 a H6) in (ex3_2_ind T T (\lambda (y1: T).(\lambda
(_: T).(subst1 d u0 u (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2:
T).(subst1 d u0 t (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead
(Bind b) u t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1
d u0 (THead (Bind b) u t4) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda
-(y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H10:
+(y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H8:
(subst1 d u0 u (lift (S O) d x0))).(\lambda (_: (subst1 d u0 t (lift (S O) d
-x1))).(\lambda (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
+x1))).(\lambda (H10: (ty3 g a x0 x1)).(let H11 \def (H3 e u0 (S d) (getl_head
+(Bind b) d c0 (CHead e (Bind Abbr) u0) H4 u) (CHead a0 (Bind b) (lift (S O) d
+x0)) (csubst1_bind b d u0 u (lift (S O) d x0) H8 c0 a0 H5) (CHead a (Bind b)
+x0) (drop_skip_bind (S O) d a0 a H6 b x0)) in (ex3_2_ind T T (\lambda (y1:
+T).(\lambda (_: T).(subst1 (S d) u0 t3 (lift (S O) (S d) y1)))) (\lambda (_:
+T).(\lambda (y2: T).(subst1 (S d) u0 t4 (lift (S O) (S d) y2)))) (\lambda
(y1: T).(\lambda (y2: T).(ty3 g (CHead a (Bind b) x0) y1 y2))) (ex3_2 T T
(\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead (Bind b) u t3) (lift (S
O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Bind b) u
t4) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))
-(\lambda (x2: T).(\lambda (x3: T).(\lambda (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 (t5: T).(ty3 g (CHead a (Bind b) x0)
-x4 t5)) 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:
+(\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (subst1 (S d) u0 t3 (lift (S
+O) (S d) x2))).(\lambda (H13: (subst1 (S d) u0 t4 (lift (S O) (S d)
+x3))).(\lambda (H14: (ty3 g (CHead a (Bind b) x0) x2 x3)).(ex3_2_intro T T
+(\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead (Bind b) u t3) (lift (S
+O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Bind b) u
+t4) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))
+(THead (Bind b) x0 x2) (THead (Bind b) x0 x3) (eq_ind_r T (THead (Bind b)
+(lift (S O) d x0) (lift (S O) (S d) x2)) (\lambda (t0: T).(subst1 d u0 (THead
+(Bind b) u t3) t0)) (subst1_head u0 u (lift (S O) d x0) d H8 (Bind b) t3
+(lift (S O) (S d) x2) H12) (lift (S O) d (THead (Bind b) x0 x2)) (lift_bind b
+x0 x2 (S O) d)) (eq_ind_r T (THead (Bind b) (lift (S O) d x0) (lift (S O) (S
+d) x3)) (\lambda (t0: T).(subst1 d u0 (THead (Bind b) u t4) t0)) (subst1_head
+u0 u (lift (S O) d x0) d H8 (Bind b) t4 (lift (S O) (S d) x3) H13) (lift (S
+O) d (THead (Bind b) x0 x3)) (lift_bind b x0 x3 (S O) d)) (ty3_bind g a x0 x1
+H10 b x2 x3 H14))))))) H11))))))) H7)))))))))))))))))))) (\lambda (c0:
+C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda (H1:
+((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl d c0 (CHead e
+(Bind Abbr) u0)) \to (\forall (a0: C).((csubst1 d u0 c0 a0) \to (\forall (a:
+C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_:
+T).(subst1 d u0 w (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2:
+T).(subst1 d u0 u (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
+g a y1 y2)))))))))))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g
+c0 v (THead (Bind Abst) u t))).(\lambda (H3: ((\forall (e: C).(\forall (u0:
T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u0)) \to (\forall (a0:
C).((csubst1 d u0 c0 a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (ex3_2
-T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 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 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)))))))))))))).(\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_eq 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 (t0:
-T).(subst1 d u0 u t0)) H16 (lift (S O) d x6) H20) in (let H24 \def (eq_ind T
-x1 (\lambda (t0: T).(ty3 g a x0 t0)) H10 (THead (Bind Abst) x6 x7) H19) in
-(let H25 \def (eq_ind T x6 (\lambda (t0: T).(ty3 g a x0 (THead (Bind Abst) t0
-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 (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 u (THead (Flat Cast) t4 t3) (lift (S O) d y1))))
-(\lambda (_: T).(\lambda (y2: T).(subst1 d u (THead (Flat Cast) t0 t4) (lift
-(S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda
-(x0: T).(\lambda (x1: T).(\lambda (H8: (subst1 d u t4 (lift (S O) d
-x0))).(\lambda (H9: (subst1 d u t0 (lift (S O) d x1))).(\lambda (H10: (ty3 g
-a x0 x1)).(let H11 \def (H1 e u d H4 a0 H5 a H6) in (ex3_2_ind T T (\lambda
-(y1: T).(\lambda (_: T).(subst1 d u t3 (lift (S O) d y1)))) (\lambda (_:
-T).(\lambda (y2: T).(subst1 d u t4 (lift (S O) d y2)))) (\lambda (y1:
+T).(\lambda (_: 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_eq
+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 (t0: T).(subst1 d u0 u t0)) H16 (lift (S O) d x6) H20)
+in (let H24 \def (eq_ind T x1 (\lambda (t0: T).(ty3 g a x0 t0)) H10 (THead
+(Bind Abst) x6 x7) H19) in (let H25 \def (eq_ind T x6 (\lambda (t0: T).(ty3 g
+a x0 (THead (Bind Abst) t0 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 (THead (Flat Cast) t0 t4) (lift (S O) d
-y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2:
-T).(\lambda (x3: T).(\lambda (H12: (subst1 d u t3 (lift (S O) d
-x2))).(\lambda (H13: (subst1 d u t4 (lift (S O) d x3))).(\lambda (H14: (ty3 g
-a x2 x3)).(let H15 \def (eq_ind T x3 (\lambda (t: T).(ty3 g a x2 t)) H14 x0
-(subst1_confluence_lift t4 x3 u d H13 x0 H8)) in (ex3_2_intro T T (\lambda
-(y1: T).(\lambda (_: T).(subst1 d u (THead (Flat Cast) t4 t3) (lift (S O) d
-y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u (THead (Flat Cast) t0 t4)
-(lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))
-(THead (Flat Cast) x0 x2) (THead (Flat Cast) x1 x0) (eq_ind_r T (THead (Flat
-Cast) (lift (S O) d x0) (lift (S O) d x2)) (\lambda (t: T).(subst1 d u (THead
-(Flat Cast) t4 t3) t)) (subst1_head u t4 (lift (S O) d x0) d H8 (Flat Cast)
-t3 (lift (S O) d x2) H12) (lift (S O) d (THead (Flat Cast) x0 x2)) (lift_flat
-Cast x0 x2 (S O) d)) (eq_ind_r T (THead (Flat Cast) (lift (S O) d x1) (lift
-(S O) d x0)) (\lambda (t: T).(subst1 d u (THead (Flat Cast) t0 t4) t))
-(subst1_head u t0 (lift (S O) d x1) d H9 (Flat Cast) t4 (lift (S O) d x0) H8)
-(lift (S O) d (THead (Flat Cast) x1 x0)) (lift_flat Cast x1 x0 (S O) d))
-(ty3_cast g a x2 x0 H15 x1 H10)))))))) H11))))))) H7)))))))))))))))))) c t1
-t2 H))))).
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H8: (subst1 d u t4 (lift (S O) d x0))).(\lambda
+(H9: (subst1 d u t0 (lift (S O) d x1))).(\lambda (H10: (ty3 g a x0 x1)).(let
+H11 \def (H1 e u d H4 a0 H5 a H6) in (ex3_2_ind T T (\lambda (y1: T).(\lambda
+(_: T).(subst1 d u t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2:
+T).(subst1 d u t4 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
+g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u (THead
+(Flat Cast) t4 t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2:
+T).(subst1 d u (THead (Flat Cast) t0 t4) (lift (S O) d y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3:
+T).(\lambda (H12: (subst1 d u t3 (lift (S O) d x2))).(\lambda (H13: (subst1 d
+u t4 (lift (S O) d x3))).(\lambda (H14: (ty3 g a x2 x3)).(let H15 \def
+(eq_ind T x3 (\lambda (t: T).(ty3 g a x2 t)) H14 x0 (subst1_confluence_lift
+t4 x3 u d H13 x0 H8)) in (ex3_2_intro T T (\lambda (y1: T).(\lambda (_:
+T).(subst1 d u (THead (Flat Cast) t4 t3) (lift (S O) d y1)))) (\lambda (_:
+T).(\lambda (y2: T).(subst1 d u (THead (Flat Cast) t0 t4) (lift (S O) d
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (THead (Flat Cast)
+x0 x2) (THead (Flat Cast) x1 x0) (eq_ind_r T (THead (Flat Cast) (lift (S O) d
+x0) (lift (S O) d x2)) (\lambda (t: T).(subst1 d u (THead (Flat Cast) t4 t3)
+t)) (subst1_head u t4 (lift (S O) d x0) d H8 (Flat Cast) t3 (lift (S O) d x2)
+H12) (lift (S O) d (THead (Flat Cast) x0 x2)) (lift_flat Cast x0 x2 (S O) d))
+(eq_ind_r T (THead (Flat Cast) (lift (S O) d x1) (lift (S O) d x0)) (\lambda
+(t: T).(subst1 d u (THead (Flat Cast) t0 t4) t)) (subst1_head u t0 (lift (S
+O) d x1) d H9 (Flat Cast) t4 (lift (S O) d x0) H8) (lift (S O) d (THead (Flat
+Cast) x1 x0)) (lift_flat Cast x1 x0 (S O) d)) (ty3_cast g a x2 x0 H15 x1
+H10)))))))) H11))))))) H7)))))))))))))))))) c t1 t2 H))))).
theorem ty3_gen_cvoid:
\forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c
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
+C).(\lambda (u0: T).(\lambda (d: nat).(\lambda (H4: (getl d c0 (CHead e (Bind
+Void) u0))).(\lambda (a: C).(\lambda (H5: (drop (S O) d c0 a)).(let H6 \def
+(H1 e u0 d H4 a H5) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T
u (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d
y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T
(\lambda (y1: T).(\lambda (_: T).(eq T (THead (Bind b) u t3) (lift (S O) d
y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Bind b) u t4) (lift (S
O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda
-(x0: T).(\lambda (x1: T).(\lambda (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 (t5: T).(ty3 g c0 u t5)) H0 (lift (S O) d x1) H10)
-in (let H13 \def (eq_ind T u (\lambda (t5: T).(ty3 g c0 t5 (lift (S O) d
-x1))) H12 (lift (S O) d x0) H9) in (let H14 \def (eq_ind T u (\lambda (t5:
+(x0: T).(\lambda (x1: T).(\lambda (H7: (eq T u (lift (S O) d x0))).(\lambda
+(H8: (eq T t (lift (S O) d x1))).(\lambda (H9: (ty3 g a x0 x1)).(let H10 \def
+(eq_ind T t (\lambda (t0: T).(ty3 g c0 u t0)) H0 (lift (S O) d x1) H8) in
+(let H11 \def (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 (lift (S O) d x1)))
+H10 (lift (S O) d x0) H7) in (let H12 \def (eq_ind T u (\lambda (t0:
T).(\forall (e0: C).(\forall (u1: T).(\forall (d0: nat).((getl d0 (CHead c0
-(Bind b) t5) (CHead e0 (Bind Void) u1)) \to (\forall (a0: C).((drop (S O) d0
-(CHead c0 (Bind b) t5) a0) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_:
-T).(eq T t4 (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t0
-(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a0 y1
-y2))))))))))) H5 (lift (S O) d x0) H9) in (let H15 \def (eq_ind T u (\lambda
-(t5: T).(ty3 g (CHead c0 (Bind b) t5) t4 t0)) H4 (lift (S O) d x0) H9) in
-(let H16 \def (eq_ind T u (\lambda (t5: T).(\forall (e0: C).(\forall (u1:
-T).(\forall (d0: nat).((getl d0 (CHead c0 (Bind b) t5) (CHead e0 (Bind Void)
-u1)) \to (\forall (a0: C).((drop (S O) d0 (CHead c0 (Bind b) t5) a0) \to
-(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t3 (lift (S O) d0 y1))))
-(\lambda (_: T).(\lambda (y2: T).(eq T t4 (lift (S O) d0 y2)))) (\lambda (y1:
-T).(\lambda (y2: T).(ty3 g a0 y1 y2))))))))))) H3 (lift (S O) d x0) H9) in
-(let H17 \def (eq_ind T u (\lambda (t5: T).(ty3 g (CHead c0 (Bind b) t5) 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 (t5: T).(\forall (e0: C).(\forall
-(u1: T).(\forall (d0: nat).((getl d0 (CHead c0 (Bind b) (lift (S O) d x0))
-(CHead e0 (Bind Void) u1)) \to (\forall (a0: C).((drop (S O) d0 (CHead c0
-(Bind b) (lift (S O) d x0)) a0) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_:
-T).(eq T t5 (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t0
+(Bind b) t0) (CHead e0 (Bind Void) u1)) \to (\forall (a0: C).((drop (S O) d0
+(CHead c0 (Bind b) t0) a0) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_:
+T).(eq T t3 (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t4
(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a0 y1
-y2))))))))))) 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
+y2))))))))))) H3 (lift (S O) d x0) H7) in (let H13 \def (eq_ind T u (\lambda
+(t0: T).(ty3 g (CHead c0 (Bind b) t0) t3 t4)) H2 (lift (S O) d x0) H7) in
+(eq_ind_r T (lift (S O) d x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y1:
+T).(\lambda (_: T).(eq T (THead (Bind b) t0 t3) (lift (S O) d y1)))) (\lambda
+(_: T).(\lambda (y2: T).(eq T (THead (Bind b) t0 t4) (lift (S O) d y2))))
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H14 \def (H12 e u0
+(S d) (getl_head (Bind b) d c0 (CHead e (Bind Void) u0) H4 (lift (S O) d x0))
+(CHead a (Bind b) x0) (drop_skip_bind (S O) d c0 a H5 b x0)) in (ex3_2_ind T
+T (\lambda (y1: T).(\lambda (_: T).(eq T t3 (lift (S O) (S d) y1)))) (\lambda
+(_: T).(\lambda (y2: T).(eq T t4 (lift (S O) (S d) y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g (CHead a (Bind b) x0) y1 y2))) (ex3_2 T T (\lambda
+(y1: T).(\lambda (_: T).(eq T (THead (Bind b) (lift (S O) d x0) t3) (lift (S
+O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Bind b) (lift (S
+O) d x0) t4) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a
+y1 y2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H15: (eq T t3 (lift (S
+O) (S d) x2))).(\lambda (H16: (eq T t4 (lift (S O) (S d) x3))).(\lambda (H17:
+(ty3 g (CHead a (Bind b) x0) x2 x3)).(eq_ind_r T (lift (S O) (S d) x3)
+(\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead
(Bind b) (lift (S O) d x0) t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda
-(y2: T).(eq T (THead (Bind b) (lift (S O) d x0) t5) (lift (S O) d y2))))
+(y2: T).(eq T (THead (Bind b) (lift (S O) d x0) t0) (lift (S O) d y2))))
(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r T (lift (S O)
-(S d) x2) (\lambda (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 (_:
+(S d) x2) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T
+(THead (Bind b) (lift (S O) d x0) t0) (lift (S O) d y1)))) (\lambda (_:
T).(\lambda (y2: T).(eq T (THead (Bind b) (lift (S O) d x0) (lift (S O) (S d)
x3)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
-y2))))) (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 (t5: T).(ty3 g (CHead a (Bind b) x0) t5 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
+y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Bind
+b) (lift (S O) d x0) (lift (S O) (S d) x2)) (lift (S O) d y1)))) (\lambda (_:
+T).(\lambda (y2: T).(eq T (THead (Bind b) (lift (S O) d x0) (lift (S O) (S d)
+x3)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))
+(THead (Bind b) x0 x2) (THead (Bind b) x0 x3) (sym_eq T (lift (S O) d (THead
+(Bind b) x0 x2)) (THead (Bind b) (lift (S O) d x0) (lift (S O) (S d) x2))
+(lift_bind b x0 x2 (S O) d)) (sym_eq T (lift (S O) d (THead (Bind b) x0 x3))
+(THead (Bind b) (lift (S O) d x0) (lift (S O) (S d) x3)) (lift_bind b x0 x3
+(S O) d)) (ty3_bind g a x0 x1 H9 b x2 x3 H17)) t3 H15) t4 H16)))))) H14)) u
+H7)))))))))) H6)))))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda
+(u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda (H1: ((\forall (e: C).(\forall
+(u0: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Void) u0)) \to (\forall
+(a: C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_:
+T).(eq T w (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T u
+(lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
+y2)))))))))))).(\lambda (v: T).(\lambda (t: T).(\lambda (H2: (ty3 g c0 v
+(THead (Bind Abst) u t))).(\lambda (H3: ((\forall (e: C).(\forall (u0:
+T).(\forall (d: nat).((getl d c0 (CHead e (Bind Void) u0)) \to (\forall (a:
+C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T
v (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Bind
Abst) u t) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
-y2))) (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
+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
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/ty3/tau0".
include "ty3/pr3_props.ma".
(b: B).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (ty3 g (CHead c0 (Bind
b) u0) t2 t3)).(\lambda (H3: ((\forall (t4: T).((tau0 g (CHead c0 (Bind b)
u0) t2 t4) \to (ty3 g (CHead c0 (Bind b) u0) t2 t4))))).(\lambda (t0:
-T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u0) t3 t0)).(\lambda (_: ((\forall
-(t4: T).((tau0 g (CHead c0 (Bind b) u0) t3 t4) \to (ty3 g (CHead c0 (Bind b)
-u0) t3 t4))))).(\lambda (t4: T).(\lambda (H6: (tau0 g c0 (THead (Bind b) u0
-t2) t4)).(let H7 \def (match H6 in tau0 return (\lambda (c1: C).(\lambda (t5:
-T).(\lambda (t6: T).(\lambda (_: (tau0 ? c1 t5 t6)).((eq C c1 c0) \to ((eq T
-t5 (THead (Bind b) u0 t2)) \to ((eq T t6 t4) \to (ty3 g c0 (THead (Bind b) u0
-t2) t4)))))))) with [(tau0_sort c1 n) \Rightarrow (\lambda (H7: (eq C c1
-c0)).(\lambda (H8: (eq T (TSort n) (THead (Bind b) u0 t2))).(\lambda (H9: (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 (H10: (eq T (TSort n) (THead (Bind b)
-u0 t2))).(let H11 \def (eq_ind T (TSort n) (\lambda (e: T).(match e in T
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _)
-\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Bind b) u0
-t2) H10) in (False_ind ((eq T (TSort (next g n)) t4) \to (ty3 g c0 (THead
-(Bind b) u0 t2) t4)) H11))) c1 (sym_eq C c1 c0 H7) H8 H9)))) | (tau0_abbr c1
-d v i H7 w H8) \Rightarrow (\lambda (H9: (eq C c1 c0)).(\lambda (H10: (eq T
-(TLRef i) (THead (Bind b) u0 t2))).(\lambda (H11: (eq T (lift (S i) O w)
-t4)).(eq_ind C c0 (\lambda (c2: C).((eq T (TLRef i) (THead (Bind b) u0 t2))
-\to ((eq T (lift (S i) O w) t4) \to ((getl i c2 (CHead d (Bind Abbr) v)) \to
-((tau0 g d v w) \to (ty3 g c0 (THead (Bind b) u0 t2) t4)))))) (\lambda (H12:
-(eq T (TLRef i) (THead (Bind b) u0 t2))).(let H13 \def (eq_ind T (TLRef i)
-(\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
-False])) I (THead (Bind b) u0 t2) H12) 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)))) H13))) c1 (sym_eq C c1 c0 H9) H10 H11 H7
-H8)))) | (tau0_abst c1 d v i H7 w H8) \Rightarrow (\lambda (H9: (eq C c1
-c0)).(\lambda (H10: (eq T (TLRef i) (THead (Bind b) u0 t2))).(\lambda (H11:
-(eq T (lift (S i) O v) t4)).(eq_ind C c0 (\lambda (c2: C).((eq T (TLRef i)
-(THead (Bind b) u0 t2)) \to ((eq T (lift (S i) O v) t4) \to ((getl i c2
-(CHead d (Bind Abst) v)) \to ((tau0 g d v w) \to (ty3 g c0 (THead (Bind b) u0
-t2) t4)))))) (\lambda (H12: (eq T (TLRef i) (THead (Bind b) u0 t2))).(let H13
+T).(\lambda (H4: (tau0 g c0 (THead (Bind b) u0 t2) t0)).(let H5 \def (match
+H4 in tau0 return (\lambda (c1: C).(\lambda (t4: T).(\lambda (t5: T).(\lambda
+(_: (tau0 ? c1 t4 t5)).((eq C c1 c0) \to ((eq T t4 (THead (Bind b) u0 t2))
+\to ((eq T t5 t0) \to (ty3 g c0 (THead (Bind b) u0 t2) t0)))))))) with
+[(tau0_sort c1 n) \Rightarrow (\lambda (H5: (eq C c1 c0)).(\lambda (H6: (eq T
+(TSort n) (THead (Bind b) u0 t2))).(\lambda (H7: (eq T (TSort (next g n))
+t0)).(eq_ind C c0 (\lambda (_: C).((eq T (TSort n) (THead (Bind b) u0 t2))
+\to ((eq T (TSort (next g n)) t0) \to (ty3 g c0 (THead (Bind b) u0 t2) t0))))
+(\lambda (H8: (eq T (TSort n) (THead (Bind b) u0 t2))).(let H9 \def (eq_ind T
+(TSort n) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _)
+\Rightarrow False])) I (THead (Bind b) u0 t2) H8) in (False_ind ((eq T (TSort
+(next g n)) t0) \to (ty3 g c0 (THead (Bind b) u0 t2) t0)) H9))) c1 (sym_eq C
+c1 c0 H5) H6 H7)))) | (tau0_abbr c1 d v i H5 w H6) \Rightarrow (\lambda (H7:
+(eq C c1 c0)).(\lambda (H8: (eq T (TLRef i) (THead (Bind b) u0 t2))).(\lambda
+(H9: (eq T (lift (S i) O w) t0)).(eq_ind C c0 (\lambda (c2: C).((eq T (TLRef
+i) (THead (Bind b) u0 t2)) \to ((eq T (lift (S i) O w) t0) \to ((getl i c2
+(CHead d (Bind Abbr) v)) \to ((tau0 g d v w) \to (ty3 g c0 (THead (Bind b) u0
+t2) t0)))))) (\lambda (H10: (eq T (TLRef i) (THead (Bind b) u0 t2))).(let H11
\def (eq_ind T (TLRef i) (\lambda (e: T).(match e in T return (\lambda (_:
T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
-(THead _ _ _) \Rightarrow False])) I (THead (Bind b) u0 t2) H12) 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)))) H13))) c1
-(sym_eq C c1 c0 H9) H10 H11 H7 H8)))) | (tau0_bind b0 c1 v t5 t6 H7)
-\Rightarrow (\lambda (H8: (eq C c1 c0)).(\lambda (H9: (eq T (THead (Bind b0)
-v t5) (THead (Bind b) u0 t2))).(\lambda (H10: (eq T (THead (Bind b0) v t6)
-t4)).(eq_ind C c0 (\lambda (c2: C).((eq T (THead (Bind b0) v t5) (THead (Bind
-b) u0 t2)) \to ((eq T (THead (Bind b0) v t6) t4) \to ((tau0 g (CHead c2 (Bind
-b0) v) t5 t6) \to (ty3 g c0 (THead (Bind b) u0 t2) t4))))) (\lambda (H11: (eq
-T (THead (Bind b0) v t5) (THead (Bind b) u0 t2))).(let H12 \def (f_equal T T
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow t5 | (TLRef _) \Rightarrow t5 | (THead _ _ t7) \Rightarrow t7]))
-(THead (Bind b0) v t5) (THead (Bind b) u0 t2) H11) in ((let H13 \def (f_equal
-T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow v | (TLRef _) \Rightarrow v | (THead _ t7 _) \Rightarrow t7]))
-(THead (Bind b0) v t5) (THead (Bind b) u0 t2) H11) in ((let H14 \def (f_equal
-T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _)
-\Rightarrow b0 | (TLRef _) \Rightarrow b0 | (THead k _ _) \Rightarrow (match
-k in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _)
-\Rightarrow b0])])) (THead (Bind b0) v t5) (THead (Bind b) u0 t2) H11) in
-(eq_ind B b (\lambda (b1: B).((eq T v u0) \to ((eq T t5 t2) \to ((eq T (THead
-(Bind b1) v t6) t4) \to ((tau0 g (CHead c0 (Bind b1) v) t5 t6) \to (ty3 g c0
-(THead (Bind b) u0 t2) t4)))))) (\lambda (H15: (eq T v u0)).(eq_ind T u0
-(\lambda (t7: T).((eq T t5 t2) \to ((eq T (THead (Bind b) t7 t6) t4) \to
-((tau0 g (CHead c0 (Bind b) t7) t5 t6) \to (ty3 g c0 (THead (Bind b) u0 t2)
-t4))))) (\lambda (H16: (eq T t5 t2)).(eq_ind T t2 (\lambda (t7: T).((eq T
-(THead (Bind b) u0 t6) t4) \to ((tau0 g (CHead c0 (Bind b) u0) t7 t6) \to
-(ty3 g c0 (THead (Bind b) u0 t2) t4)))) (\lambda (H17: (eq T (THead (Bind b)
-u0 t6) t4)).(eq_ind T (THead (Bind b) u0 t6) (\lambda (t7: T).((tau0 g (CHead
-c0 (Bind b) u0) t2 t6) \to (ty3 g c0 (THead (Bind b) u0 t2) t7))) (\lambda
-(H18: (tau0 g (CHead c0 (Bind b) u0) t2 t6)).(let H_y \def (H3 t6 H18) in
-(ex_ind T (\lambda (t7: T).(ty3 g (CHead c0 (Bind b) u0) t6 t7)) (ty3 g c0
-(THead (Bind b) u0 t2) (THead (Bind b) u0 t6)) (\lambda (x: T).(\lambda (H19:
-(ty3 g (CHead c0 (Bind b) u0) t6 x)).(ty3_bind g c0 u0 t H0 b t2 t6 H_y x
-H19))) (ty3_correct g (CHead c0 (Bind b) u0) t2 t6 H_y)))) t4 H17)) t5
-(sym_eq T t5 t2 H16))) v (sym_eq T v u0 H15))) b0 (sym_eq B b0 b H14))) H13))
-H12))) c1 (sym_eq C c1 c0 H8) H9 H10 H7)))) | (tau0_appl c1 v t5 t6 H7)
-\Rightarrow (\lambda (H8: (eq C c1 c0)).(\lambda (H9: (eq T (THead (Flat
-Appl) v t5) (THead (Bind b) u0 t2))).(\lambda (H10: (eq T (THead (Flat Appl)
-v t6) t4)).(eq_ind C c0 (\lambda (c2: C).((eq T (THead (Flat Appl) v t5)
-(THead (Bind b) u0 t2)) \to ((eq T (THead (Flat Appl) v t6) t4) \to ((tau0 g
-c2 t5 t6) \to (ty3 g c0 (THead (Bind b) u0 t2) t4))))) (\lambda (H11: (eq T
-(THead (Flat Appl) v t5) (THead (Bind b) u0 t2))).(let H12 \def (eq_ind T
-(THead (Flat Appl) v t5) (\lambda (e: T).(match e in T return (\lambda (_:
+(THead _ _ _) \Rightarrow False])) I (THead (Bind b) u0 t2) H10) in
+(False_ind ((eq T (lift (S i) O w) t0) \to ((getl i c0 (CHead d (Bind Abbr)
+v)) \to ((tau0 g d v w) \to (ty3 g c0 (THead (Bind b) u0 t2) t0)))) H11))) c1
+(sym_eq C c1 c0 H7) H8 H9 H5 H6)))) | (tau0_abst c1 d v i H5 w H6)
+\Rightarrow (\lambda (H7: (eq C c1 c0)).(\lambda (H8: (eq T (TLRef i) (THead
+(Bind b) u0 t2))).(\lambda (H9: (eq T (lift (S i) O v) t0)).(eq_ind C c0
+(\lambda (c2: C).((eq T (TLRef i) (THead (Bind b) u0 t2)) \to ((eq T (lift (S
+i) O v) t0) \to ((getl i c2 (CHead d (Bind Abst) v)) \to ((tau0 g d v w) \to
+(ty3 g c0 (THead (Bind b) u0 t2) t0)))))) (\lambda (H10: (eq T (TLRef i)
+(THead (Bind b) u0 t2))).(let H11 \def (eq_ind T (TLRef i) (\lambda (e:
+T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I
+(THead (Bind b) u0 t2) H10) in (False_ind ((eq T (lift (S i) O v) t0) \to
+((getl i c0 (CHead d (Bind Abst) v)) \to ((tau0 g d v w) \to (ty3 g c0 (THead
+(Bind b) u0 t2) t0)))) H11))) c1 (sym_eq C c1 c0 H7) H8 H9 H5 H6)))) |
+(tau0_bind b0 c1 v t4 t5 H5) \Rightarrow (\lambda (H6: (eq C c1 c0)).(\lambda
+(H7: (eq T (THead (Bind b0) v t4) (THead (Bind b) u0 t2))).(\lambda (H8: (eq
+T (THead (Bind b0) v t5) t0)).(eq_ind C c0 (\lambda (c2: C).((eq T (THead
+(Bind b0) v t4) (THead (Bind b) u0 t2)) \to ((eq T (THead (Bind b0) v t5) t0)
+\to ((tau0 g (CHead c2 (Bind b0) v) t4 t5) \to (ty3 g c0 (THead (Bind b) u0
+t2) t0))))) (\lambda (H9: (eq T (THead (Bind b0) v t4) (THead (Bind b) u0
+t2))).(let H10 \def (f_equal T T (\lambda (e: T).(match e in T return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow t4 | (TLRef _) \Rightarrow t4
+| (THead _ _ t6) \Rightarrow t6])) (THead (Bind b0) v t4) (THead (Bind b) u0
+t2) H9) in ((let H11 \def (f_equal T T (\lambda (e: T).(match e in T return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow v | (TLRef _) \Rightarrow v |
+(THead _ t6 _) \Rightarrow t6])) (THead (Bind b0) v t4) (THead (Bind b) u0
+t2) H9) in ((let H12 \def (f_equal T B (\lambda (e: T).(match e in T return
+(\lambda (_: T).B) with [(TSort _) \Rightarrow b0 | (TLRef _) \Rightarrow b0
+| (THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).B) with
+[(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b0])])) (THead (Bind b0) v
+t4) (THead (Bind b) u0 t2) H9) 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) t0) \to ((tau0 g (CHead
+c0 (Bind b1) v) t4 t5) \to (ty3 g c0 (THead (Bind b) u0 t2) t0)))))) (\lambda
+(H13: (eq T v u0)).(eq_ind T u0 (\lambda (t6: T).((eq T t4 t2) \to ((eq T
+(THead (Bind b) t6 t5) t0) \to ((tau0 g (CHead c0 (Bind b) t6) t4 t5) \to
+(ty3 g c0 (THead (Bind b) u0 t2) t0))))) (\lambda (H14: (eq T t4 t2)).(eq_ind
+T t2 (\lambda (t6: T).((eq T (THead (Bind b) u0 t5) t0) \to ((tau0 g (CHead
+c0 (Bind b) u0) t6 t5) \to (ty3 g c0 (THead (Bind b) u0 t2) t0)))) (\lambda
+(H15: (eq T (THead (Bind b) u0 t5) t0)).(eq_ind T (THead (Bind b) u0 t5)
+(\lambda (t6: T).((tau0 g (CHead c0 (Bind b) u0) t2 t5) \to (ty3 g c0 (THead
+(Bind b) u0 t2) t6))) (\lambda (H16: (tau0 g (CHead c0 (Bind b) u0) t2
+t5)).(ty3_bind g c0 u0 t H0 b t2 t5 (H3 t5 H16))) t0 H15)) t4 (sym_eq T t4 t2
+H14))) v (sym_eq T v u0 H13))) b0 (sym_eq B b0 b H12))) H11)) H10))) c1
+(sym_eq C c1 c0 H6) H7 H8 H5)))) | (tau0_appl c1 v t4 t5 H5) \Rightarrow
+(\lambda (H6: (eq C c1 c0)).(\lambda (H7: (eq T (THead (Flat Appl) v t4)
+(THead (Bind b) u0 t2))).(\lambda (H8: (eq T (THead (Flat Appl) v t5)
+t0)).(eq_ind C c0 (\lambda (c2: C).((eq T (THead (Flat Appl) v t4) (THead
+(Bind b) u0 t2)) \to ((eq T (THead (Flat Appl) v t5) t0) \to ((tau0 g c2 t4
+t5) \to (ty3 g c0 (THead (Bind b) u0 t2) t0))))) (\lambda (H9: (eq T (THead
+(Flat Appl) v t4) (THead (Bind b) u0 t2))).(let H10 \def (eq_ind T (THead
+(Flat Appl) v t4) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop)
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _
+_) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
+\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u0 t2)
+H9) in (False_ind ((eq T (THead (Flat Appl) v t5) t0) \to ((tau0 g c0 t4 t5)
+\to (ty3 g c0 (THead (Bind b) u0 t2) t0))) H10))) c1 (sym_eq C c1 c0 H6) H7
+H8 H5)))) | (tau0_cast c1 v1 v2 H5 t4 t5 H6) \Rightarrow (\lambda (H7: (eq C
+c1 c0)).(\lambda (H8: (eq T (THead (Flat Cast) v1 t4) (THead (Bind b) u0
+t2))).(\lambda (H9: (eq T (THead (Flat Cast) v2 t5) t0)).(eq_ind C c0
+(\lambda (c2: C).((eq T (THead (Flat Cast) v1 t4) (THead (Bind b) u0 t2)) \to
+((eq T (THead (Flat Cast) v2 t5) t0) \to ((tau0 g c2 v1 v2) \to ((tau0 g c2
+t4 t5) \to (ty3 g c0 (THead (Bind b) u0 t2) t0)))))) (\lambda (H10: (eq T
+(THead (Flat Cast) v1 t4) (THead (Bind b) u0 t2))).(let H11 \def (eq_ind T
+(THead (Flat Cast) v1 t4) (\lambda (e: T).(match e in T return (\lambda (_:
T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind
-b) u0 t2) H11) in (False_ind ((eq T (THead (Flat Appl) v t6) t4) \to ((tau0 g
-c0 t5 t6) \to (ty3 g c0 (THead (Bind b) u0 t2) t4))) H12))) c1 (sym_eq C c1
-c0 H8) H9 H10 H7)))) | (tau0_cast c1 v1 v2 H7 t5 t6 H8) \Rightarrow (\lambda
-(H9: (eq C c1 c0)).(\lambda (H10: (eq T (THead (Flat Cast) v1 t5) (THead
-(Bind b) u0 t2))).(\lambda (H11: (eq T (THead (Flat Cast) v2 t6) t4)).(eq_ind
-C c0 (\lambda (c2: C).((eq T (THead (Flat Cast) v1 t5) (THead (Bind b) u0
-t2)) \to ((eq T (THead (Flat Cast) v2 t6) t4) \to ((tau0 g c2 v1 v2) \to
-((tau0 g c2 t5 t6) \to (ty3 g c0 (THead (Bind b) u0 t2) t4)))))) (\lambda
-(H12: (eq T (THead (Flat Cast) v1 t5) (THead (Bind b) u0 t2))).(let H13 \def
-(eq_ind T (THead (Flat Cast) v1 t5) (\lambda (e: T).(match e in T return
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
-(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
-True])])) I (THead (Bind b) u0 t2) H12) in (False_ind ((eq T (THead (Flat
-Cast) v2 t6) t4) \to ((tau0 g c0 v1 v2) \to ((tau0 g c0 t5 t6) \to (ty3 g c0
-(THead (Bind b) u0 t2) t4)))) H13))) c1 (sym_eq C c1 c0 H9) H10 H11 H7
-H8))))]) 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 in tau0 return
-(\lambda (c1: C).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (tau0 ? c1 t0
-t3)).((eq C c1 c0) \to ((eq T t0 (THead (Flat Appl) w v)) \to ((eq T t3 t2)
-\to (ty3 g c0 (THead (Flat Appl) w v) t2)))))))) with [(tau0_sort c1 n)
-\Rightarrow (\lambda (H5: (eq C c1 c0)).(\lambda (H6: (eq T (TSort n) (THead
-(Flat Appl) w v))).(\lambda (H7: (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 (H8:
-(eq T (TSort n) (THead (Flat Appl) w v))).(let H9 \def (eq_ind T (TSort n)
-(\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
-False])) I (THead (Flat Appl) w v) H8) in (False_ind ((eq T (TSort (next g
-n)) t2) \to (ty3 g c0 (THead (Flat Appl) w v) t2)) H9))) c1 (sym_eq C c1 c0
-H5) H6 H7)))) | (tau0_abbr c1 d v0 i H5 w0 H6) \Rightarrow (\lambda (H7: (eq
-C c1 c0)).(\lambda (H8: (eq T (TLRef i) (THead (Flat Appl) w v))).(\lambda
-(H9: (eq T (lift (S i) O w0) t2)).(eq_ind C c0 (\lambda (c2: C).((eq T (TLRef
-i) (THead (Flat Appl) w v)) \to ((eq T (lift (S i) O w0) t2) \to ((getl i c2
+b) u0 t2) H10) in (False_ind ((eq T (THead (Flat Cast) v2 t5) t0) \to ((tau0
+g c0 v1 v2) \to ((tau0 g c0 t4 t5) \to (ty3 g c0 (THead (Bind b) u0 t2)
+t0)))) H11))) c1 (sym_eq C c1 c0 H7) H8 H9 H5 H6))))]) in (H5 (refl_equal C
+c0) (refl_equal T (THead (Bind b) u0 t2)) (refl_equal T t0)))))))))))))))
+(\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 in tau0 return (\lambda (c1:
+C).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (tau0 ? c1 t0 t3)).((eq C
+c1 c0) \to ((eq T t0 (THead (Flat Appl) w v)) \to ((eq T t3 t2) \to (ty3 g c0
+(THead (Flat Appl) w v) t2)))))))) with [(tau0_sort c1 n) \Rightarrow
+(\lambda (H5: (eq C c1 c0)).(\lambda (H6: (eq T (TSort n) (THead (Flat Appl)
+w v))).(\lambda (H7: (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 (H8: (eq T (TSort
+n) (THead (Flat Appl) w v))).(let H9 \def (eq_ind T (TSort n) (\lambda (e:
+T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I
+(THead (Flat Appl) w v) H8) in (False_ind ((eq T (TSort (next g n)) t2) \to
+(ty3 g c0 (THead (Flat Appl) w v) t2)) H9))) c1 (sym_eq C c1 c0 H5) H6 H7))))
+| (tau0_abbr c1 d v0 i H5 w0 H6) \Rightarrow (\lambda (H7: (eq C c1
+c0)).(\lambda (H8: (eq T (TLRef i) (THead (Flat Appl) w v))).(\lambda (H9:
+(eq T (lift (S i) O w0) t2)).(eq_ind C c0 (\lambda (c2: C).((eq T (TLRef i)
+(THead (Flat Appl) w v)) \to ((eq T (lift (S i) O w0) t2) \to ((getl i c2
(CHead d (Bind Abbr) v0)) \to ((tau0 g d v0 w0) \to (ty3 g c0 (THead (Flat
Appl) w v) t2)))))) (\lambda (H10: (eq T (TLRef i) (THead (Flat Appl) w
v))).(let H11 \def (eq_ind T (TLRef i) (\lambda (e: T).(match e in T return
w v) (THead (Flat Appl) w t3)) (\lambda (x0: T).(\lambda (_: (ty3 g c0 u0
x0)).(ex_ind T (\lambda (t4: T).(ty3 g c0 (THead (Bind Abst) u0 t) t4)) (ty3
g c0 (THead (Flat Appl) w v) (THead (Flat Appl) w t3)) (\lambda (x1:
-T).(\lambda (H18: (ty3 g c0 (THead (Bind Abst) u0 t) x1)).(ex4_3_ind T T T
-(\lambda (t4: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead (Bind Abst)
-u0 t4) x1)))) (\lambda (_: T).(\lambda (t5: T).(\lambda (_: T).(ty3 g c0 u0
-t5)))) (\lambda (t4: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0
-(Bind Abst) u0) t t4)))) (\lambda (t4: T).(\lambda (_: T).(\lambda (t6:
-T).(ty3 g (CHead c0 (Bind Abst) u0) t4 t6)))) (ty3 g c0 (THead (Flat Appl) w
-v) (THead (Flat Appl) w t3)) (\lambda (x2: T).(\lambda (x3: T).(\lambda (x4:
-T).(\lambda (_: (pc3 c0 (THead (Bind Abst) u0 x2) x1)).(\lambda (H20: (ty3 g
-c0 u0 x3)).(\lambda (H21: (ty3 g (CHead c0 (Bind Abst) u0) t x2)).(\lambda
-(H22: (ty3 g (CHead c0 (Bind Abst) u0) x2 x4)).(ty3_conv g c0 (THead (Flat
-Appl) w t3) (THead (Flat Appl) w (THead (Bind Abst) u0 x2)) (ty3_appl g c0 w
-u0 H0 t3 x2 (ty3_sconv g c0 t3 x H16 (THead (Bind Abst) u0 t) (THead (Bind
-Abst) u0 x2) (ty3_bind g c0 u0 x3 H20 Abst t x2 H21 x4 H22) H15)) (THead
-(Flat Appl) w v) (THead (Flat Appl) w (THead (Bind Abst) u0 t)) (ty3_appl g
-c0 w u0 H0 v t H2) (pc3_thin_dx c0 (THead (Bind Abst) u0 t) t3 (ty3_unique g
-c0 v (THead (Bind Abst) u0 t) H2 t3 H_y) w Appl))))))))) (ty3_gen_bind g Abst
-c0 u0 t x1 H18)))) (ty3_correct g c0 v (THead (Bind Abst) u0 t) H2))))
-(ty3_correct g c0 w u0 H0)))) (ty3_correct g c0 v t3 H_y))))) t2 H13)) t0
-(sym_eq T t0 v H12))) v0 (sym_eq T v0 w H11))) H10))) c1 (sym_eq C c1 c0 H6)
-H7 H8 H5)))) | (tau0_cast c1 v1 v2 H5 t0 t3 H6) \Rightarrow (\lambda (H7: (eq
-C c1 c0)).(\lambda (H8: (eq T (THead (Flat Cast) v1 t0) (THead (Flat Appl) w
-v))).(\lambda (H9: (eq T (THead (Flat Cast) v2 t3) t2)).(eq_ind C c0 (\lambda
-(c2: C).((eq T (THead (Flat Cast) v1 t0) (THead (Flat Appl) w v)) \to ((eq T
-(THead (Flat Cast) v2 t3) t2) \to ((tau0 g c2 v1 v2) \to ((tau0 g c2 t0 t3)
-\to (ty3 g c0 (THead (Flat Appl) w v) t2)))))) (\lambda (H10: (eq T (THead
-(Flat Cast) v1 t0) (THead (Flat Appl) w v))).(let H11 \def (eq_ind T (THead
-(Flat Cast) v1 t0) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop)
-with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _
-_) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
-\Rightarrow False | (Flat f) \Rightarrow (match f in F return (\lambda (_:
-F).Prop) with [Appl \Rightarrow False | Cast \Rightarrow True])])])) I (THead
-(Flat Appl) w v) H10) in (False_ind ((eq T (THead (Flat Cast) v2 t3) t2) \to
-((tau0 g c0 v1 v2) \to ((tau0 g c0 t0 t3) \to (ty3 g c0 (THead (Flat Appl) w
-v) t2)))) H11))) c1 (sym_eq C c1 c0 H7) H8 H9 H5 H6))))]) 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 (t4: T).((tau0 g c0 t2 t4) \to (ty3 g c0 t2
-t4))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t3 t0)).(\lambda (H3:
-((\forall (t4: T).((tau0 g c0 t3 t4) \to (ty3 g c0 t3 t4))))).(\lambda (t4:
-T).(\lambda (H4: (tau0 g c0 (THead (Flat Cast) t3 t2) t4)).(let H5 \def
-(match H4 in tau0 return (\lambda (c1: C).(\lambda (t: T).(\lambda (t5:
-T).(\lambda (_: (tau0 ? c1 t t5)).((eq C c1 c0) \to ((eq T t (THead (Flat
-Cast) t3 t2)) \to ((eq T t5 t4) \to (ty3 g c0 (THead (Flat Cast) t3 t2)
-t4)))))))) with [(tau0_sort c1 n) \Rightarrow (\lambda (H5: (eq C c1
-c0)).(\lambda (H6: (eq T (TSort n) (THead (Flat Cast) t3 t2))).(\lambda (H7:
-(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 (H8: (eq T (TSort n) (THead (Flat
-Cast) t3 t2))).(let H9 \def (eq_ind T (TSort n) (\lambda (e: T).(match e in T
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _)
-\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Flat Cast)
-t3 t2) H8) in (False_ind ((eq T (TSort (next g n)) t4) \to (ty3 g c0 (THead
-(Flat Cast) t3 t2) t4)) H9))) c1 (sym_eq C c1 c0 H5) H6 H7)))) | (tau0_abbr
-c1 d v i H5 w H6) \Rightarrow (\lambda (H7: (eq C c1 c0)).(\lambda (H8: (eq T
-(TLRef i) (THead (Flat Cast) t3 t2))).(\lambda (H9: (eq T (lift (S i) O w)
-t4)).(eq_ind C c0 (\lambda (c2: C).((eq T (TLRef i) (THead (Flat Cast) t3
-t2)) \to ((eq T (lift (S i) O w) t4) \to ((getl i c2 (CHead d (Bind Abbr) v))
-\to ((tau0 g d v w) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4)))))) (\lambda
-(H10: (eq T (TLRef i) (THead (Flat Cast) t3 t2))).(let H11 \def (eq_ind T
-(TLRef i) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _)
-\Rightarrow False])) I (THead (Flat Cast) t3 t2) H10) 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)))) H11))) c1 (sym_eq C c1 c0
-H7) H8 H9 H5 H6)))) | (tau0_abst c1 d v i H5 w H6) \Rightarrow (\lambda (H7:
-(eq C c1 c0)).(\lambda (H8: (eq T (TLRef i) (THead (Flat Cast) t3
-t2))).(\lambda (H9: (eq T (lift (S i) O v) t4)).(eq_ind C c0 (\lambda (c2:
-C).((eq T (TLRef i) (THead (Flat Cast) t3 t2)) \to ((eq T (lift (S i) O v)
-t4) \to ((getl i c2 (CHead d (Bind Abst) v)) \to ((tau0 g d v w) \to (ty3 g
-c0 (THead (Flat Cast) t3 t2) t4)))))) (\lambda (H10: (eq T (TLRef i) (THead
-(Flat Cast) t3 t2))).(let H11 \def (eq_ind T (TLRef i) (\lambda (e: T).(match
+T).(\lambda (H18: (ty3 g c0 (THead (Bind Abst) u0 t) x1)).(ex3_2_ind T T
+(\lambda (t4: T).(\lambda (_: T).(pc3 c0 (THead (Bind Abst) u0 t4) x1)))
+(\lambda (_: T).(\lambda (t5: T).(ty3 g c0 u0 t5))) (\lambda (t4: T).(\lambda
+(_: T).(ty3 g (CHead c0 (Bind Abst) u0) t t4))) (ty3 g c0 (THead (Flat Appl)
+w v) (THead (Flat Appl) w t3)) (\lambda (x2: T).(\lambda (x3: T).(\lambda (_:
+(pc3 c0 (THead (Bind Abst) u0 x2) x1)).(\lambda (H20: (ty3 g c0 u0
+x3)).(\lambda (H21: (ty3 g (CHead c0 (Bind Abst) u0) t x2)).(ty3_conv g c0
+(THead (Flat Appl) w t3) (THead (Flat Appl) w (THead (Bind Abst) u0 x2))
+(ty3_appl g c0 w u0 H0 t3 x2 (ty3_sconv g c0 t3 x H16 (THead (Bind Abst) u0
+t) (THead (Bind Abst) u0 x2) (ty3_bind g c0 u0 x3 H20 Abst t x2 H21) H15))
+(THead (Flat Appl) w v) (THead (Flat Appl) w (THead (Bind Abst) u0 t))
+(ty3_appl g c0 w u0 H0 v t H2) (pc3_thin_dx c0 (THead (Bind Abst) u0 t) t3
+(ty3_unique g c0 v (THead (Bind Abst) u0 t) H2 t3 H_y) w Appl)))))))
+(ty3_gen_bind g Abst c0 u0 t x1 H18)))) (ty3_correct g c0 v (THead (Bind
+Abst) u0 t) H2)))) (ty3_correct g c0 w u0 H0)))) (ty3_correct g c0 v t3
+H_y))))) t2 H13)) t0 (sym_eq T t0 v H12))) v0 (sym_eq T v0 w H11))) H10))) c1
+(sym_eq C c1 c0 H6) H7 H8 H5)))) | (tau0_cast c1 v1 v2 H5 t0 t3 H6)
+\Rightarrow (\lambda (H7: (eq C c1 c0)).(\lambda (H8: (eq T (THead (Flat
+Cast) v1 t0) (THead (Flat Appl) w v))).(\lambda (H9: (eq T (THead (Flat Cast)
+v2 t3) t2)).(eq_ind C c0 (\lambda (c2: C).((eq T (THead (Flat Cast) v1 t0)
+(THead (Flat Appl) w v)) \to ((eq T (THead (Flat Cast) v2 t3) t2) \to ((tau0
+g c2 v1 v2) \to ((tau0 g c2 t0 t3) \to (ty3 g c0 (THead (Flat Appl) w v)
+t2)))))) (\lambda (H10: (eq T (THead (Flat Cast) v1 t0) (THead (Flat Appl) w
+v))).(let H11 \def (eq_ind T (THead (Flat Cast) v1 t0) (\lambda (e: T).(match
e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
-(TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead
-(Flat Cast) t3 t2) H10) 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)))) H11))) c1 (sym_eq C c1 c0 H7) H8 H9 H5 H6)))) |
-(tau0_bind b c1 v t5 t6 H5) \Rightarrow (\lambda (H6: (eq C c1 c0)).(\lambda
-(H7: (eq T (THead (Bind b) v t5) (THead (Flat Cast) t3 t2))).(\lambda (H8:
-(eq T (THead (Bind b) v t6) t4)).(eq_ind C c0 (\lambda (c2: C).((eq T (THead
-(Bind b) v t5) (THead (Flat Cast) t3 t2)) \to ((eq T (THead (Bind b) v t6)
-t4) \to ((tau0 g (CHead c2 (Bind b) v) t5 t6) \to (ty3 g c0 (THead (Flat
-Cast) t3 t2) t4))))) (\lambda (H9: (eq T (THead (Bind b) v t5) (THead (Flat
-Cast) t3 t2))).(let H10 \def (eq_ind T (THead (Bind b) v t5) (\lambda (e:
-T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K
-return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _)
-\Rightarrow False])])) I (THead (Flat Cast) t3 t2) H9) in (False_ind ((eq T
-(THead (Bind b) v t6) t4) \to ((tau0 g (CHead c0 (Bind b) v) t5 t6) \to (ty3
-g c0 (THead (Flat Cast) t3 t2) t4))) H10))) c1 (sym_eq C c1 c0 H6) H7 H8
-H5)))) | (tau0_appl c1 v t5 t6 H5) \Rightarrow (\lambda (H6: (eq C c1
+(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return
+(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow
+(match f in F return (\lambda (_: F).Prop) with [Appl \Rightarrow False |
+Cast \Rightarrow True])])])) I (THead (Flat Appl) w v) H10) in (False_ind
+((eq T (THead (Flat Cast) v2 t3) t2) \to ((tau0 g c0 v1 v2) \to ((tau0 g c0
+t0 t3) \to (ty3 g c0 (THead (Flat Appl) w v) t2)))) H11))) c1 (sym_eq C c1 c0
+H7) H8 H9 H5 H6))))]) 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
+(t4: T).((tau0 g c0 t2 t4) \to (ty3 g c0 t2 t4))))).(\lambda (t0: T).(\lambda
+(_: (ty3 g c0 t3 t0)).(\lambda (H3: ((\forall (t4: T).((tau0 g c0 t3 t4) \to
+(ty3 g c0 t3 t4))))).(\lambda (t4: T).(\lambda (H4: (tau0 g c0 (THead (Flat
+Cast) t3 t2) t4)).(let H5 \def (match H4 in tau0 return (\lambda (c1:
+C).(\lambda (t: T).(\lambda (t5: T).(\lambda (_: (tau0 ? c1 t t5)).((eq C c1
+c0) \to ((eq T t (THead (Flat Cast) t3 t2)) \to ((eq T t5 t4) \to (ty3 g c0
+(THead (Flat Cast) t3 t2) t4)))))))) with [(tau0_sort c1 n) \Rightarrow
+(\lambda (H5: (eq C c1 c0)).(\lambda (H6: (eq T (TSort n) (THead (Flat Cast)
+t3 t2))).(\lambda (H7: (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 (H8: (eq T
+(TSort n) (THead (Flat Cast) t3 t2))).(let H9 \def (eq_ind T (TSort n)
+(\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
+False])) I (THead (Flat Cast) t3 t2) H8) in (False_ind ((eq T (TSort (next g
+n)) t4) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4)) H9))) c1 (sym_eq C c1 c0
+H5) H6 H7)))) | (tau0_abbr c1 d v i H5 w H6) \Rightarrow (\lambda (H7: (eq C
+c1 c0)).(\lambda (H8: (eq T (TLRef i) (THead (Flat Cast) t3 t2))).(\lambda
+(H9: (eq T (lift (S i) O w) t4)).(eq_ind C c0 (\lambda (c2: C).((eq T (TLRef
+i) (THead (Flat Cast) t3 t2)) \to ((eq T (lift (S i) O w) t4) \to ((getl i c2
+(CHead d (Bind Abbr) v)) \to ((tau0 g d v w) \to (ty3 g c0 (THead (Flat Cast)
+t3 t2) t4)))))) (\lambda (H10: (eq T (TLRef i) (THead (Flat Cast) t3
+t2))).(let H11 \def (eq_ind T (TLRef i) (\lambda (e: T).(match e in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) t3
+t2) H10) 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)))) H11))) c1 (sym_eq C c1 c0 H7) H8 H9 H5 H6)))) | (tau0_abst c1 d v i H5
+w H6) \Rightarrow (\lambda (H7: (eq C c1 c0)).(\lambda (H8: (eq T (TLRef i)
+(THead (Flat Cast) t3 t2))).(\lambda (H9: (eq T (lift (S i) O v) t4)).(eq_ind
+C c0 (\lambda (c2: C).((eq T (TLRef i) (THead (Flat Cast) t3 t2)) \to ((eq T
+(lift (S i) O v) t4) \to ((getl i c2 (CHead d (Bind Abst) v)) \to ((tau0 g d
+v w) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4)))))) (\lambda (H10: (eq T
+(TLRef i) (THead (Flat Cast) t3 t2))).(let H11 \def (eq_ind T (TLRef i)
+(\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
+False])) I (THead (Flat Cast) t3 t2) H10) 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)))) H11))) c1 (sym_eq C c1 c0 H7) H8 H9 H5
+H6)))) | (tau0_bind b c1 v t5 t6 H5) \Rightarrow (\lambda (H6: (eq C c1
+c0)).(\lambda (H7: (eq T (THead (Bind b) v t5) (THead (Flat Cast) t3
+t2))).(\lambda (H8: (eq T (THead (Bind b) v t6) t4)).(eq_ind C c0 (\lambda
+(c2: C).((eq T (THead (Bind b) v t5) (THead (Flat Cast) t3 t2)) \to ((eq T
+(THead (Bind b) v t6) t4) \to ((tau0 g (CHead c2 (Bind b) v) t5 t6) \to (ty3
+g c0 (THead (Flat Cast) t3 t2) t4))))) (\lambda (H9: (eq T (THead (Bind b) v
+t5) (THead (Flat Cast) t3 t2))).(let H10 \def (eq_ind T (THead (Bind b) v t5)
+(\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
+(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True |
+(Flat _) \Rightarrow False])])) I (THead (Flat Cast) t3 t2) H9) in (False_ind
+((eq T (THead (Bind b) v t6) t4) \to ((tau0 g (CHead c0 (Bind b) v) t5 t6)
+\to (ty3 g c0 (THead (Flat Cast) t3 t2) t4))) H10))) c1 (sym_eq C c1 c0 H6)
+H7 H8 H5)))) | (tau0_appl c1 v t5 t6 H5) \Rightarrow (\lambda (H6: (eq C c1
c0)).(\lambda (H7: (eq T (THead (Flat Appl) v t5) (THead (Flat Cast) t3
t2))).(\lambda (H8: (eq T (THead (Flat Appl) v t6) t4)).(eq_ind C c0 (\lambda
(c2: C).((eq T (THead (Flat Appl) v t5) (THead (Flat Cast) t3 t2)) \to ((eq T
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/wcpr0/defs".
include "pr0/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/wcpr0/fwd".
include "wcpr0/defs.ma".
(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/wcpr0/getl".
include "wcpr0/defs.ma".
projects / helm.git / commitdiff