exportation completed!!

diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/Base/ext/arith.ma b/matita/contribs/LAMBDA-TYPES/Level-1/Base/ext/arith.ma
index 766a34d6feaa7e1dc23b6f5df84b098b8099399a..fc33cc52994649cb0e2cc37427fd0e0fbe74492c 100644 (file)
--- a/matita/contribs/LAMBDA-TYPES/Level-1/Base/ext/arith.ma
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/Base/ext/arith.ma
@@ -43,7 +43,7 @@ False | (S _) \Rightarrow True])) I O H0) in (False_ind P H1))))) (\lambda
 (P: Prop).P)))).(or_ind (eq nat n n0) ((eq nat n n0) \to (\forall (P: 
 Prop).P)) (or (eq nat (S n) (S n0)) ((eq nat (S n) (S n0)) \to (\forall (P: 
 Prop).P))) (\lambda (H1: (eq nat n n0)).(let H2 \def (eq_ind_r nat n0 
-(\lambda (n0: nat).(or (eq nat (S n) n0) ((eq nat (S n) n0) \to (\forall (P: 
+(\lambda (n3: nat).(or (eq nat (S n) n3) ((eq nat (S n) n3) \to (\forall (P: 
 Prop).P)))) H0 n H1) in (eq_ind nat n (\lambda (n3: nat).(or (eq nat (S n) (S 
 n3)) ((eq nat (S n) (S n3)) \to (\forall (P: Prop).P)))) (or_introl (eq nat 
 (S n) (S n)) ((eq nat (S n) (S n)) \to (\forall (P: Prop).P)) (refl_equal nat 
@@ -51,11 +51,12 @@ n3)) ((eq nat (S n) (S n3)) \to (\forall (P: Prop).P)))) (or_introl (eq nat
 Prop).P)))).(or_intror (eq nat (S n) (S n0)) ((eq nat (S n) (S n0)) \to 
 (\forall (P: Prop).P)) (\lambda (H2: (eq nat (S n) (S n0))).(\lambda (P: 
 Prop).(let H3 \def (f_equal nat nat (\lambda (e: nat).(match e in nat return 
-(\lambda (_: nat).nat) with [O \Rightarrow n | (S n) \Rightarrow n])) (S n) 
-(S n0) H2) in (let H4 \def (eq_ind_r nat n0 (\lambda (n0: nat).((eq nat n n0) 
-\to (\forall (P: Prop).P))) H1 n H3) in (let H5 \def (eq_ind_r nat n0 
-(\lambda (n0: nat).(or (eq nat (S n) n0) ((eq nat (S n) n0) \to (\forall (P: 
-Prop).P)))) H0 n H3) in (H4 (refl_equal nat n) P)))))))) (H n0)))) n2)))) n1).
+(\lambda (_: nat).nat) with [O \Rightarrow n | (S n3) \Rightarrow n3])) (S n) 
+(S n0) H2) in (let H4 \def (eq_ind_r nat n0 (\lambda (n3: nat).((eq nat n n3) 
+\to (\forall (P0: Prop).P0))) H1 n H3) in (let H5 \def (eq_ind_r nat n0 
+(\lambda (n3: nat).(or (eq nat (S n) n3) ((eq nat (S n) n3) \to (\forall (P0: 
+Prop).P0)))) H0 n H3) in (H4 (refl_equal nat n) P)))))))) (H n0)))) n2)))) 
+n1).
 
 theorem simpl_plus_r:
  \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((eq nat (plus m n) 
@@ -152,29 +153,29 @@ n) m) \to P))))
  \lambda (m: nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).(\forall (P: 
 Prop).((le n n0) \to ((le (S n0) n) \to P))))) (\lambda (n: nat).(\lambda (P: 
 Prop).(\lambda (_: (le O n)).(\lambda (H0: (le (S n) O)).(let H1 \def (match 
-H0 in le return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n O) \to 
+H0 in le return (\lambda (n0: nat).(\lambda (_: (le ? n0)).((eq nat n0 O) \to 
 P))) with [le_n \Rightarrow (\lambda (H1: (eq nat (S n) O)).(let H2 \def 
 (eq_ind nat (S n) (\lambda (e: nat).(match e in nat return (\lambda (_: 
 nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in 
-(False_ind P H2))) | (le_S m H1) \Rightarrow (\lambda (H2: (eq nat (S m) 
-O)).((let H3 \def (eq_ind nat (S m) (\lambda (e: nat).(match e in nat return 
+(False_ind P H2))) | (le_S m0 H1) \Rightarrow (\lambda (H2: (eq nat (S m0) 
+O)).((let H3 \def (eq_ind nat (S m0) (\lambda (e: nat).(match e in nat return 
 (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) 
-I O H2) in (False_ind ((le (S n) m) \to P) H3)) H1))]) in (H1 (refl_equal nat 
-O))))))) (\lambda (n: nat).(\lambda (H: ((\forall (n0: nat).(\forall (P: 
+I O H2) in (False_ind ((le (S n) m0) \to P) H3)) H1))]) in (H1 (refl_equal 
+nat O))))))) (\lambda (n: nat).(\lambda (H: ((\forall (n0: nat).(\forall (P: 
 Prop).((le n n0) \to ((le (S n0) n) \to P)))))).(\lambda (n0: nat).(nat_ind 
 (\lambda (n1: nat).(\forall (P: Prop).((le (S n) n1) \to ((le (S n1) (S n)) 
 \to P)))) (\lambda (P: Prop).(\lambda (H0: (le (S n) O)).(\lambda (_: (le (S 
-O) (S n))).(let H2 \def (match H0 in le return (\lambda (n: nat).(\lambda (_: 
-(le ? n)).((eq nat n O) \to P))) with [le_n \Rightarrow (\lambda (H2: (eq nat 
-(S n) O)).(let H3 \def (eq_ind nat (S n) (\lambda (e: nat).(match e in nat 
-return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow 
-True])) I O H2) in (False_ind P H3))) | (le_S m H2) \Rightarrow (\lambda (H3: 
-(eq nat (S m) O)).((let H4 \def (eq_ind nat (S m) (\lambda (e: nat).(match e 
+O) (S n))).(let H2 \def (match H0 in le return (\lambda (n1: nat).(\lambda 
+(_: (le ? n1)).((eq nat n1 O) \to P))) with [le_n \Rightarrow (\lambda (H2: 
+(eq nat (S n) O)).(let H3 \def (eq_ind nat (S n) (\lambda (e: nat).(match e 
 in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) 
-\Rightarrow True])) I O H3) in (False_ind ((le (S n) m) \to P) H4)) H2))]) in 
-(H2 (refl_equal nat O)))))) (\lambda (n1: nat).(\lambda (_: ((\forall (P: 
-Prop).((le (S n) n1) \to ((le (S n1) (S n)) \to P))))).(\lambda (P: 
-Prop).(\lambda (H1: (le (S n) (S n1))).(\lambda (H2: (le (S (S n1)) (S 
+\Rightarrow True])) I O H2) in (False_ind P H3))) | (le_S m0 H2) \Rightarrow 
+(\lambda (H3: (eq nat (S m0) O)).((let H4 \def (eq_ind nat (S m0) (\lambda 
+(e: nat).(match e in nat return (\lambda (_: nat).Prop) with [O \Rightarrow 
+False | (S _) \Rightarrow True])) I O H3) in (False_ind ((le (S n) m0) \to P) 
+H4)) H2))]) in (H2 (refl_equal nat O)))))) (\lambda (n1: nat).(\lambda (_: 
+((\forall (P: Prop).((le (S n) n1) \to ((le (S n1) (S n)) \to P))))).(\lambda 
+(P: Prop).(\lambda (H1: (le (S n) (S n1))).(\lambda (H2: (le (S (S n1)) (S 
 n))).(H n1 P (le_S_n n n1 H1) (le_S_n (S n1) n H2))))))) n0)))) m).
 
 theorem le_Sx_x:
@@ -293,9 +294,9 @@ p)) \to (lt n m))))
 \def
  \lambda (p: nat).(\lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt (plus 
 n p) (plus m p))).(plus_lt_reg_l n m p (let H0 \def (eq_ind nat (plus n p) 
-(\lambda (n: nat).(lt n (plus m p))) H (plus p n) (plus_comm n p)) in (let H1 
-\def (eq_ind nat (plus m p) (\lambda (n0: nat).(lt (plus p n) n0)) H0 (plus p 
-m) (plus_comm m p)) in H1)))))).
+(\lambda (n0: nat).(lt n0 (plus m p))) H (plus p n) (plus_comm n p)) in (let 
+H1 \def (eq_ind nat (plus m p) (\lambda (n0: nat).(lt (plus p n) n0)) H0 
+(plus p m) (plus_comm m p)) in H1)))))).
 
 theorem minus_x_Sy:
  \forall (x: nat).(\forall (y: nat).((lt y x) \to (eq nat (minus x y) (S 
@@ -525,8 +526,8 @@ nat).(nat_ind (\lambda (n: nat).(\forall (y1: nat).(\forall (y2: nat).((le O
 (y1: nat).(\lambda (y2: nat).(\lambda (_: (le O (S z0))).(\lambda (_: (le O 
 (S z0))).(\lambda (H1: (eq nat (S (plus z0 y1)) (S (plus z0 y2)))).(let H_y 
 \def (IH O O) in (let H2 \def (eq_ind_r nat (minus z0 O) (\lambda (n: 
-nat).(\forall (y1: nat).(\forall (y2: nat).((le O z0) \to ((le O z0) \to ((eq 
-nat (plus n y1) (plus n y2)) \to (eq nat y2 y1))))))) H_y z0 (minus_n_O z0)) 
+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_equal nat (plus z0 y1) (plus z0 y2) (eq_add_S (plus z0 y1) (plus z0 y2) 
@@ -537,14 +538,14 @@ y2)) \to (eq nat y2 (plus x3 y1))))))))).(\lambda (y1: nat).(\lambda (y2:
 nat).(\lambda (_: (le O (S z0))).(\lambda (H0: (le (S x3) (S z0))).(\lambda 
 (H1: (eq nat (S (plus z0 y1)) (plus (minus z0 x3) y2))).(let H_y \def (IH O 
 x3 (S y1)) in (let H2 \def (eq_ind_r nat (minus z0 O) (\lambda (n: 
-nat).(\forall (y2: nat).((le O z0) \to ((le x3 z0) \to ((eq nat (plus n (S 
-y1)) (plus (minus z0 x3) y2)) \to (eq nat y2 (plus x3 (S y1)))))))) H_y z0 
+nat).(\forall (y3: nat).((le O z0) \to ((le x3 z0) \to ((eq nat (plus n (S 
+y1)) (plus (minus z0 x3) y3)) \to (eq nat y3 (plus x3 (S y1)))))))) H_y z0 
 (minus_n_O z0)) in (let H3 \def (eq_ind_r nat (plus z0 (S y1)) (\lambda (n: 
-nat).(\forall (y2: nat).((le O z0) \to ((le x3 z0) \to ((eq nat n (plus 
-(minus z0 x3) y2)) \to (eq nat y2 (plus x3 (S y1)))))))) H2 (S (plus z0 y1)) 
+nat).(\forall (y3: nat).((le O z0) \to ((le x3 z0) \to ((eq nat n (plus 
+(minus z0 x3) y3)) \to (eq nat y3 (plus x3 (S y1)))))))) H2 (S (plus z0 y1)) 
 (plus_n_Sm z0 y1)) in (let H4 \def (eq_ind_r nat (plus x3 (S y1)) (\lambda 
-(n: nat).(\forall (y2: nat).((le O z0) \to ((le x3 z0) \to ((eq nat (S (plus 
-z0 y1)) (plus (minus z0 x3) y2)) \to (eq nat y2 n)))))) H3 (S (plus x3 y1)) 
+(n: nat).(\forall (y3: nat).((le O z0) \to ((le x3 z0) \to ((eq nat (S (plus 
+z0 y1)) (plus (minus z0 x3) y3)) \to (eq nat y3 n)))))) H3 (S (plus x3 y1)) 
 (plus_n_Sm x3 y1)) in (H4 y2 (le_O_n z0) (le_S_n x3 z0 H0) H1)))))))))))) 
 x2)) (\lambda (x2: nat).(\lambda (_: ((\forall (x3: nat).(\forall (y1: 
 nat).(\forall (y2: nat).((le x2 (S z0)) \to ((le x3 (S z0)) \to ((eq nat 
@@ -574,3 +575,13 @@ x2) y1) (plus (minus z0 x4) y2))).(f_equal nat nat S (plus x2 y2) (plus x4
 y1) (IH x2 x4 y1 y2 (le_S_n x2 z0 H) (le_S_n x4 z0 H0) H1))))))))) x3)))) 
 x1)))) z).
 
+theorem le_S_minus:
+ \forall (d: nat).(\forall (h: nat).(\forall (n: nat).((le (plus d h) n) \to 
+(le d (S (minus n h))))))
+\def
+ \lambda (d: nat).(\lambda (h: nat).(\lambda (n: nat).(\lambda (H: (le (plus 
+d h) n)).(let H0 \def (le_trans d (plus d h) n (le_plus_l d h) H) in (let H1 
+\def (eq_ind nat n (\lambda (n0: nat).(le d n0)) H0 (plus (minus n h) h) 
+(le_plus_minus_sym h n (le_trans_plus_r d h n H))) in (le_S d (minus n h) 
+(le_minus d n h H))))))).
+

diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/clear.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/clear.ma
new file mode 100644 (file)
index 0000000..70eeece
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/clear.ma
@@ -0,0 +1,73 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/clear".
+
+include "csub3/defs.ma".
+
+include "clear/fwd.ma".
+
+theorem csub3_clear_conf:
+ \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csub3 g c1 c2) \to 
+(\forall (e1: C).((clear c1 e1) \to (ex2 C (\lambda (e2: C).(csub3 g e1 e2)) 
+(\lambda (e2: C).(clear c2 e2))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csub3 g c1 
+c2)).(csub3_ind g (\lambda (c: C).(\lambda (c0: C).(\forall (e1: C).((clear c 
+e1) \to (ex2 C (\lambda (e2: C).(csub3 g e1 e2)) (\lambda (e2: C).(clear c0 
+e2))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (H0: (clear (CSort n) 
+e1)).(clear_gen_sort e1 n H0 (ex2 C (\lambda (e2: C).(csub3 g e1 e2)) 
+(\lambda (e2: C).(clear (CSort n) e2))))))) (\lambda (c3: C).(\lambda (c4: 
+C).(\lambda (H0: (csub3 g c3 c4)).(\lambda (H1: ((\forall (e1: C).((clear c3 
+e1) \to (ex2 C (\lambda (e2: C).(csub3 g e1 e2)) (\lambda (e2: C).(clear c4 
+e2))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (e1: C).(\lambda (H2: 
+(clear (CHead c3 k u) e1)).((match k in K return (\lambda (k0: K).((clear 
+(CHead c3 k0 u) e1) \to (ex2 C (\lambda (e2: C).(csub3 g e1 e2)) (\lambda 
+(e2: C).(clear (CHead c4 k0 u) e2))))) with [(Bind b) \Rightarrow (\lambda 
+(H3: (clear (CHead c3 (Bind b) u) e1)).(eq_ind_r C (CHead c3 (Bind b) u) 
+(\lambda (c: C).(ex2 C (\lambda (e2: C).(csub3 g c e2)) (\lambda (e2: 
+C).(clear (CHead c4 (Bind b) u) e2)))) (ex_intro2 C (\lambda (e2: C).(csub3 g 
+(CHead c3 (Bind b) u) e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) u) e2)) 
+(CHead c4 (Bind b) u) (csub3_head g c3 c4 H0 (Bind b) u) (clear_bind b c4 u)) 
+e1 (clear_gen_bind b c3 e1 u H3))) | (Flat f) \Rightarrow (\lambda (H3: 
+(clear (CHead c3 (Flat f) u) e1)).(let H4 \def (H1 e1 (clear_gen_flat f c3 e1 
+u H3)) in (ex2_ind C (\lambda (e2: C).(csub3 g e1 e2)) (\lambda (e2: 
+C).(clear c4 e2)) (ex2 C (\lambda (e2: C).(csub3 g e1 e2)) (\lambda (e2: 
+C).(clear (CHead c4 (Flat f) u) e2))) (\lambda (x: C).(\lambda (H5: (csub3 g 
+e1 x)).(\lambda (H6: (clear c4 x)).(ex_intro2 C (\lambda (e2: C).(csub3 g e1 
+e2)) (\lambda (e2: C).(clear (CHead c4 (Flat f) u) e2)) x H5 (clear_flat c4 x 
+H6 f u))))) H4)))]) H2))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda 
+(H0: (csub3 g c3 c4)).(\lambda (_: ((\forall (e1: C).((clear c3 e1) \to (ex2 
+C (\lambda (e2: C).(csub3 g e1 e2)) (\lambda (e2: C).(clear c4 
+e2))))))).(\lambda (b: B).(\lambda (H2: (not (eq B b Void))).(\lambda (u1: 
+T).(\lambda (u2: T).(\lambda (e1: C).(\lambda (H3: (clear (CHead c3 (Bind 
+Void) u1) e1)).(eq_ind_r C (CHead c3 (Bind Void) u1) (\lambda (c: C).(ex2 C 
+(\lambda (e2: C).(csub3 g c e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) 
+u2) e2)))) (ex_intro2 C (\lambda (e2: C).(csub3 g (CHead c3 (Bind Void) u1) 
+e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) u2) e2)) (CHead c4 (Bind b) 
+u2) (csub3_void g c3 c4 H0 b H2 u1 u2) (clear_bind b c4 u2)) e1 
+(clear_gen_bind Void c3 e1 u1 H3)))))))))))) (\lambda (c3: C).(\lambda (c4: 
+C).(\lambda (H0: (csub3 g c3 c4)).(\lambda (_: ((\forall (e1: C).((clear c3 
+e1) \to (ex2 C (\lambda (e2: C).(csub3 g e1 e2)) (\lambda (e2: C).(clear c4 
+e2))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H2: (ty3 g c4 u 
+t)).(\lambda (e1: C).(\lambda (H3: (clear (CHead c3 (Bind Abst) t) 
+e1)).(eq_ind_r C (CHead c3 (Bind Abst) t) (\lambda (c: C).(ex2 C (\lambda 
+(e2: C).(csub3 g c e2)) (\lambda (e2: C).(clear (CHead c4 (Bind Abbr) u) 
+e2)))) (ex_intro2 C (\lambda (e2: C).(csub3 g (CHead c3 (Bind Abst) t) e2)) 
+(\lambda (e2: C).(clear (CHead c4 (Bind Abbr) u) e2)) (CHead c4 (Bind Abbr) 
+u) (csub3_abst g c3 c4 H0 u t H2) (clear_bind Abbr c4 u)) e1 (clear_gen_bind 
+Abst c3 e1 t H3))))))))))) c1 c2 H)))).
+

diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/defs.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/defs.ma
new file mode 100644 (file)
index 0000000..df8b56a
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/defs.ma
@@ -0,0 +1,31 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/defs".
+
+include "ty3/defs.ma".
+
+inductive csub3 (g:G): C \to (C \to Prop) \def
+| csub3_sort: \forall (n: nat).(csub3 g (CSort n) (CSort n))
+| csub3_head: \forall (c1: C).(\forall (c2: C).((csub3 g c1 c2) \to (\forall 
+(k: K).(\forall (u: T).(csub3 g (CHead c1 k u) (CHead c2 k u))))))
+| csub3_void: \forall (c1: C).(\forall (c2: C).((csub3 g c1 c2) \to (\forall 
+(b: B).((not (eq B b Void)) \to (\forall (u1: T).(\forall (u2: T).(csub3 g 
+(CHead c1 (Bind Void) u1) (CHead c2 (Bind b) u2))))))))
+| csub3_abst: \forall (c1: C).(\forall (c2: C).((csub3 g c1 c2) \to (\forall 
+(u: T).(\forall (t: T).((ty3 g c2 u t) \to (csub3 g (CHead c1 (Bind Abst) t) 
+(CHead c2 (Bind Abbr) u))))))).
+

diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/drop.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/drop.ma
new file mode 100644 (file)
index 0000000..59141ad
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/drop.ma
@@ -0,0 +1,805 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/drop".
+
+include "csub3/defs.ma".
+
+include "drop/fwd.ma".
+
+theorem csub3_drop_flat:
+ \forall (g: G).(\forall (f: F).(\forall (n: nat).(\forall (c1: C).(\forall 
+(c2: C).((csub3 g c1 c2) \to (\forall (d1: C).(\forall (u: T).((drop n O c1 
+(CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
+(d2: C).(drop n O c2 (CHead d2 (Flat f) u))))))))))))
+\def
+ \lambda (g: G).(\lambda (f: F).(\lambda (n: nat).(nat_ind (\lambda (n0: 
+nat).(\forall (c1: C).(\forall (c2: C).((csub3 g c1 c2) \to (\forall (d1: 
+C).(\forall (u: T).((drop n0 O c1 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda 
+(d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Flat f) 
+u))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csub3 g c1 
+c2)).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop O O c1 (CHead d1 
+(Flat f) u))).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csub3 g c c2)) H 
+(CHead d1 (Flat f) u) (drop_gen_refl c1 (CHead d1 (Flat f) u) H0)) in (let H2 
+\def (match H1 in csub3 return (\lambda (c: C).(\lambda (c0: C).(\lambda (_: 
+(csub3 ? c c0)).((eq C c (CHead d1 (Flat f) u)) \to ((eq C c0 c2) \to (ex2 C 
+(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 
+(Flat f) u))))))))) with [(csub3_sort n0) \Rightarrow (\lambda (H2: (eq C 
+(CSort n0) (CHead d1 (Flat f) u))).(\lambda (H3: (eq C (CSort n0) c2)).((let 
+H4 \def (eq_ind C (CSort n0) (\lambda (e: C).(match e in C return (\lambda 
+(_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow 
+False])) I (CHead d1 (Flat f) u) H2) in (False_ind ((eq C (CSort n0) c2) \to 
+(ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead 
+d2 (Flat f) u))))) H4)) H3))) | (csub3_head c0 c3 H2 k u0) \Rightarrow 
+(\lambda (H3: (eq C (CHead c0 k u0) (CHead d1 (Flat f) u))).(\lambda (H4: (eq 
+C (CHead c3 k u0) c2)).((let H5 \def (f_equal C T (\lambda (e: C).(match e in 
+C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) 
+\Rightarrow t])) (CHead c0 k u0) (CHead d1 (Flat f) u) H3) in ((let H6 \def 
+(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with 
+[(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c0 k u0) 
+(CHead d1 (Flat f) u) H3) in ((let H7 \def (f_equal C C (\lambda (e: 
+C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | 
+(CHead c _ _) \Rightarrow c])) (CHead c0 k u0) (CHead d1 (Flat f) u) H3) in 
+(eq_ind C d1 (\lambda (c: C).((eq K k (Flat f)) \to ((eq T u0 u) \to ((eq C 
+(CHead c3 k u0) c2) \to ((csub3 g c c3) \to (ex2 C (\lambda (d2: C).(csub3 g 
+d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Flat f) u))))))))) (\lambda 
+(H8: (eq K k (Flat f))).(eq_ind K (Flat f) (\lambda (k0: K).((eq T u0 u) \to 
+((eq C (CHead c3 k0 u0) c2) \to ((csub3 g d1 c3) \to (ex2 C (\lambda (d2: 
+C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Flat f) 
+u)))))))) (\lambda (H9: (eq T u0 u)).(eq_ind T u (\lambda (t: T).((eq C 
+(CHead c3 (Flat f) t) c2) \to ((csub3 g d1 c3) \to (ex2 C (\lambda (d2: 
+C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Flat f) u))))))) 
+(\lambda (H10: (eq C (CHead c3 (Flat f) u) c2)).(eq_ind C (CHead c3 (Flat f) 
+u) (\lambda (c: C).((csub3 g d1 c3) \to (ex2 C (\lambda (d2: C).(csub3 g d1 
+d2)) (\lambda (d2: C).(drop O O c (CHead d2 (Flat f) u)))))) (\lambda (H11: 
+(csub3 g d1 c3)).(ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: 
+C).(drop O O (CHead c3 (Flat f) u) (CHead d2 (Flat f) u))) c3 H11 (drop_refl 
+(CHead c3 (Flat f) u)))) c2 H10)) u0 (sym_eq T u0 u H9))) k (sym_eq K k (Flat 
+f) H8))) c0 (sym_eq C c0 d1 H7))) H6)) H5)) H4 H2))) | (csub3_void c0 c3 H2 b 
+H3 u1 u2) \Rightarrow (\lambda (H4: (eq C (CHead c0 (Bind Void) u1) (CHead d1 
+(Flat f) u))).(\lambda (H5: (eq C (CHead c3 (Bind b) u2) c2)).((let H6 \def 
+(eq_ind C (CHead c0 (Bind Void) u1) (\lambda (e: C).(match e in C return 
+(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) 
+\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) 
+\Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead d1 (Flat f) u) 
+H4) in (False_ind ((eq C (CHead c3 (Bind b) u2) c2) \to ((csub3 g c0 c3) \to 
+((not (eq B b Void)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
+(d2: C).(drop O O c2 (CHead d2 (Flat f) u))))))) H6)) H5 H2 H3))) | 
+(csub3_abst c0 c3 H2 u0 t H3) \Rightarrow (\lambda (H4: (eq C (CHead c0 (Bind 
+Abst) t) (CHead d1 (Flat f) u))).(\lambda (H5: (eq C (CHead c3 (Bind Abbr) 
+u0) c2)).((let H6 \def (eq_ind C (CHead c0 (Bind Abst) t) (\lambda (e: 
+C).(match e in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow 
+False | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) 
+with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead d1 
+(Flat f) u) H4) in (False_ind ((eq C (CHead c3 (Bind Abbr) u0) c2) \to 
+((csub3 g c0 c3) \to ((ty3 g c3 u0 t) \to (ex2 C (\lambda (d2: C).(csub3 g d1 
+d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Flat f) u))))))) H6)) H5 H2 
+H3)))]) in (H2 (refl_equal C (CHead d1 (Flat f) u)) (refl_equal C 
+c2)))))))))) (\lambda (n0: nat).(\lambda (H: ((\forall (c1: C).(\forall (c2: 
+C).((csub3 g c1 c2) \to (\forall (d1: C).(\forall (u: T).((drop n0 O c1 
+(CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
+(d2: C).(drop n0 O c2 (CHead d2 (Flat f) u)))))))))))).(\lambda (c1: 
+C).(\lambda (c2: C).(\lambda (H0: (csub3 g c1 c2)).(csub3_ind g (\lambda (c: 
+C).(\lambda (c0: C).(\forall (d1: C).(\forall (u: T).((drop (S n0) O c (CHead 
+d1 (Flat f) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: 
+C).(drop (S n0) O c0 (CHead d2 (Flat f) u))))))))) (\lambda (n1: 
+nat).(\lambda (d1: C).(\lambda (u: T).(\lambda (H1: (drop (S n0) O (CSort n1) 
+(CHead d1 (Flat f) u))).(let H2 \def (match H1 in drop return (\lambda (n2: 
+nat).(\lambda (n3: nat).(\lambda (c: C).(\lambda (c0: C).(\lambda (_: (drop 
+n2 n3 c c0)).((eq nat n2 (S n0)) \to ((eq nat n3 O) \to ((eq C c (CSort n1)) 
+\to ((eq C c0 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 
+d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Flat f) 
+u))))))))))))) with [(drop_refl c) \Rightarrow (\lambda (H2: (eq nat O (S 
+n0))).(\lambda (H3: (eq nat O O)).(\lambda (H4: (eq C c (CSort n1))).(\lambda 
+(H5: (eq C c (CHead d1 (Flat f) u))).((let H6 \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 n0) H2) in (False_ind ((eq nat O O) \to 
+((eq C c (CSort n1)) \to ((eq C c (CHead d1 (Flat f) u)) \to (ex2 C (\lambda 
+(d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 
+(Flat f) u))))))) H6)) H3 H4 H5))))) | (drop_drop k h c e H2 u0) \Rightarrow 
+(\lambda (H3: (eq nat (S h) (S n0))).(\lambda (H4: (eq nat O O)).(\lambda 
+(H5: (eq C (CHead c k u0) (CSort n1))).(\lambda (H6: (eq C e (CHead d1 (Flat 
+f) u))).((let H7 \def (f_equal nat nat (\lambda (e0: nat).(match e0 in nat 
+return (\lambda (_: nat).nat) with [O \Rightarrow h | (S n2) \Rightarrow 
+n2])) (S h) (S n0) H3) in (eq_ind nat n0 (\lambda (n2: nat).((eq nat O O) \to 
+((eq C (CHead c k u0) (CSort n1)) \to ((eq C e (CHead d1 (Flat f) u)) \to 
+((drop (r k n2) O c e) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
+(d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Flat f) u))))))))) (\lambda (_: 
+(eq nat O O)).(\lambda (H9: (eq C (CHead c k u0) (CSort n1))).(let H10 \def 
+(eq_ind C (CHead c k u0) (\lambda (e0: C).(match e0 in C return (\lambda (_: 
+C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow 
+True])) I (CSort n1) H9) in (False_ind ((eq C e (CHead d1 (Flat f) u)) \to 
+((drop (r k n0) O c e) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
+(d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Flat f) u)))))) H10)))) h 
+(sym_eq nat h n0 H7))) H4 H5 H6 H2))))) | (drop_skip k h d c e H2 u0) 
+\Rightarrow (\lambda (H3: (eq nat h (S n0))).(\lambda (H4: (eq nat (S d) 
+O)).(\lambda (H5: (eq C (CHead c k (lift h (r k d) u0)) (CSort n1))).(\lambda 
+(H6: (eq C (CHead e k u0) (CHead d1 (Flat f) u))).(eq_ind nat (S n0) (\lambda 
+(n2: nat).((eq nat (S d) O) \to ((eq C (CHead c k (lift n2 (r k d) u0)) 
+(CSort n1)) \to ((eq C (CHead e k u0) (CHead d1 (Flat f) u)) \to ((drop n2 (r 
+k d) c e) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop 
+(S n0) O (CSort n1) (CHead d2 (Flat f) u))))))))) (\lambda (H7: (eq nat (S d) 
+O)).(let H8 \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 H7) in (False_ind ((eq C (CHead c k (lift (S n0) (r k d) u0)) (CSort n1)) 
+\to ((eq C (CHead e k u0) (CHead d1 (Flat f) u)) \to ((drop (S n0) (r k d) c 
+e) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) 
+O (CSort n1) (CHead d2 (Flat f) u))))))) H8))) h (sym_eq nat h (S n0) H3) H4 
+H5 H6 H2)))))]) in (H2 (refl_equal nat (S n0)) (refl_equal nat O) (refl_equal 
+C (CSort n1)) (refl_equal C (CHead d1 (Flat f) u)))))))) (\lambda (c0: 
+C).(\lambda (c3: C).(\lambda (H1: (csub3 g c0 c3)).(\lambda (H2: ((\forall 
+(d1: C).(\forall (u: T).((drop (S n0) O c0 (CHead d1 (Flat f) u)) \to (ex2 C 
+(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead 
+d2 (Flat f) u))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (u: 
+T).(\forall (d1: C).(\forall (u0: T).((drop (S n0) O (CHead c0 k0 u) (CHead 
+d1 (Flat f) u0)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: 
+C).(drop (S n0) O (CHead c3 k0 u) (CHead d2 (Flat f) u0))))))))) (\lambda (b: 
+B).(\lambda (u: T).(\lambda (d1: C).(\lambda (u0: T).(\lambda (H3: (drop (S 
+n0) O (CHead c0 (Bind b) u) (CHead d1 (Flat f) u0))).(ex2_ind C (\lambda (d2: 
+C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Flat f) u0))) 
+(ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O 
+(CHead c3 (Bind b) u) (CHead d2 (Flat f) u0)))) (\lambda (x: C).(\lambda (H4: 
+(csub3 g d1 x)).(\lambda (H5: (drop n0 O c3 (CHead x (Flat f) 
+u0))).(ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop 
+(S n0) O (CHead c3 (Bind b) u) (CHead d2 (Flat f) u0))) x H4 (drop_drop (Bind 
+b) n0 c3 (CHead x (Flat f) u0) H5 u))))) (H c0 c3 H1 d1 u0 (drop_gen_drop 
+(Bind b) c0 (CHead d1 (Flat f) u0) u n0 H3)))))))) (\lambda (f0: F).(\lambda 
+(u: T).(\lambda (d1: C).(\lambda (u0: T).(\lambda (H3: (drop (S n0) O (CHead 
+c0 (Flat f0) u) (CHead d1 (Flat f) u0))).(ex2_ind C (\lambda (d2: C).(csub3 g 
+d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f) u0))) (ex2 C 
+(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
+(Flat f0) u) (CHead d2 (Flat f) u0)))) (\lambda (x: C).(\lambda (H4: (csub3 g 
+d1 x)).(\lambda (H5: (drop (S n0) O c3 (CHead x (Flat f) u0))).(ex_intro2 C 
+(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
+(Flat f0) u) (CHead d2 (Flat f) u0))) x H4 (drop_drop (Flat f0) n0 c3 (CHead 
+x (Flat f) u0) H5 u))))) (H2 d1 u0 (drop_gen_drop (Flat f0) c0 (CHead d1 
+(Flat f) u0) u n0 H3)))))))) k)))))) (\lambda (c0: C).(\lambda (c3: 
+C).(\lambda (H1: (csub3 g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (u: 
+T).((drop (S n0) O c0 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: 
+C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f) 
+u))))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: 
+T).(\lambda (u2: T).(\lambda (d1: C).(\lambda (u: T).(\lambda (H4: (drop (S 
+n0) O (CHead c0 (Bind Void) u1) (CHead d1 (Flat f) u))).(ex2_ind C (\lambda 
+(d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Flat f) 
+u))) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O 
+(CHead c3 (Bind b) u2) (CHead d2 (Flat f) u)))) (\lambda (x: C).(\lambda (H5: 
+(csub3 g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x (Flat f) u))).(ex_intro2 
+C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
+(Bind b) u2) (CHead d2 (Flat f) u))) x H5 (drop_drop (Bind b) n0 c3 (CHead x 
+(Flat f) u) H6 u2))))) (H c0 c3 H1 d1 u (drop_gen_drop (Bind Void) c0 (CHead 
+d1 (Flat f) u) u1 n0 H4)))))))))))))) (\lambda (c0: C).(\lambda (c3: 
+C).(\lambda (H1: (csub3 g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (u: 
+T).((drop (S n0) O c0 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: 
+C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f) 
+u))))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g c3 u 
+t)).(\lambda (d1: C).(\lambda (u0: T).(\lambda (H4: (drop (S n0) O (CHead c0 
+(Bind Abst) t) (CHead d1 (Flat f) u0))).(ex2_ind C (\lambda (d2: C).(csub3 g 
+d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Flat f) u0))) (ex2 C 
+(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
+(Bind Abbr) u) (CHead d2 (Flat f) u0)))) (\lambda (x: C).(\lambda (H5: (csub3 
+g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x (Flat f) u0))).(ex_intro2 C 
+(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
+(Bind Abbr) u) (CHead d2 (Flat f) u0))) x H5 (drop_drop (Bind Abbr) n0 c3 
+(CHead x (Flat f) u0) H6 u))))) (H c0 c3 H1 d1 u0 (drop_gen_drop (Bind Abst) 
+c0 (CHead d1 (Flat f) u0) t n0 H4))))))))))))) c1 c2 H0)))))) n))).
+
+theorem csub3_drop_abbr:
+ \forall (g: G).(\forall (n: nat).(\forall (c1: C).(\forall (c2: C).((csub3 g 
+c1 c2) \to (\forall (d1: C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind 
+Abbr) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop 
+n O c2 (CHead d2 (Bind Abbr) u)))))))))))
+\def
+ \lambda (g: G).(\lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1: 
+C).(\forall (c2: C).((csub3 g c1 c2) \to (\forall (d1: C).(\forall (u: 
+T).((drop n0 O c1 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: 
+C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Bind Abbr) 
+u))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csub3 g c1 
+c2)).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop O O c1 (CHead d1 
+(Bind Abbr) u))).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csub3 g c c2)) H 
+(CHead d1 (Bind Abbr) u) (drop_gen_refl c1 (CHead d1 (Bind Abbr) u) H0)) in 
+(let H2 \def (match H1 in csub3 return (\lambda (c: C).(\lambda (c0: 
+C).(\lambda (_: (csub3 ? c c0)).((eq C c (CHead d1 (Bind Abbr) u)) \to ((eq C 
+c0 c2) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O 
+O c2 (CHead d2 (Bind Abbr) u))))))))) with [(csub3_sort n0) \Rightarrow 
+(\lambda (H2: (eq C (CSort n0) (CHead d1 (Bind Abbr) u))).(\lambda (H3: (eq C 
+(CSort n0) c2)).((let H4 \def (eq_ind C (CSort n0) (\lambda (e: C).(match e 
+in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ 
+_ _) \Rightarrow False])) I (CHead d1 (Bind Abbr) u) H2) in (False_ind ((eq C 
+(CSort n0) c2) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: 
+C).(drop O O c2 (CHead d2 (Bind Abbr) u))))) H4)) H3))) | (csub3_head c0 c3 
+H2 k u0) \Rightarrow (\lambda (H3: (eq C (CHead c0 k u0) (CHead d1 (Bind 
+Abbr) u))).(\lambda (H4: (eq C (CHead c3 k u0) c2)).((let H5 \def (f_equal C 
+T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) 
+\Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c0 k u0) (CHead d1 
+(Bind Abbr) u) H3) in ((let H6 \def (f_equal C K (\lambda (e: C).(match e in 
+C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) 
+\Rightarrow k0])) (CHead c0 k u0) (CHead d1 (Bind Abbr) u) H3) in ((let H7 
+\def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) 
+with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k 
+u0) (CHead d1 (Bind Abbr) u) H3) in (eq_ind C d1 (\lambda (c: C).((eq K k 
+(Bind Abbr)) \to ((eq T u0 u) \to ((eq C (CHead c3 k u0) c2) \to ((csub3 g c 
+c3) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O 
+c2 (CHead d2 (Bind Abbr) u))))))))) (\lambda (H8: (eq K k (Bind 
+Abbr))).(eq_ind K (Bind Abbr) (\lambda (k0: K).((eq T u0 u) \to ((eq C (CHead 
+c3 k0 u0) c2) \to ((csub3 g d1 c3) \to (ex2 C (\lambda (d2: C).(csub3 g d1 
+d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u)))))))) (\lambda 
+(H9: (eq T u0 u)).(eq_ind T u (\lambda (t: T).((eq C (CHead c3 (Bind Abbr) t) 
+c2) \to ((csub3 g d1 c3) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) 
+(\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u))))))) (\lambda (H10: 
+(eq C (CHead c3 (Bind Abbr) u) c2)).(eq_ind C (CHead c3 (Bind Abbr) u) 
+(\lambda (c: C).((csub3 g d1 c3) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) 
+(\lambda (d2: C).(drop O O c (CHead d2 (Bind Abbr) u)))))) (\lambda (H11: 
+(csub3 g d1 c3)).(ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: 
+C).(drop O O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u))) c3 H11 
+(drop_refl (CHead c3 (Bind Abbr) u)))) c2 H10)) u0 (sym_eq T u0 u H9))) k 
+(sym_eq K k (Bind Abbr) H8))) c0 (sym_eq C c0 d1 H7))) H6)) H5)) H4 H2))) | 
+(csub3_void c0 c3 H2 b H3 u1 u2) \Rightarrow (\lambda (H4: (eq C (CHead c0 
+(Bind Void) u1) (CHead d1 (Bind Abbr) u))).(\lambda (H5: (eq C (CHead c3 
+(Bind b) u2) c2)).((let H6 \def (eq_ind C (CHead c0 (Bind Void) u1) (\lambda 
+(e: C).(match e in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow 
+False | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) 
+with [(Bind b0) \Rightarrow (match b0 in B return (\lambda (_: B).Prop) with 
+[Abbr \Rightarrow False | Abst \Rightarrow False | Void \Rightarrow True]) | 
+(Flat _) \Rightarrow False])])) I (CHead d1 (Bind Abbr) u) H4) in (False_ind 
+((eq C (CHead c3 (Bind b) u2) c2) \to ((csub3 g c0 c3) \to ((not (eq B b 
+Void)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O 
+O c2 (CHead d2 (Bind Abbr) u))))))) H6)) H5 H2 H3))) | (csub3_abst c0 c3 H2 
+u0 t H3) \Rightarrow (\lambda (H4: (eq C (CHead c0 (Bind Abst) t) (CHead d1 
+(Bind Abbr) u))).(\lambda (H5: (eq C (CHead c3 (Bind Abbr) u0) c2)).((let H6 
+\def (eq_ind C (CHead c0 (Bind Abst) t) (\lambda (e: C).(match e in C return 
+(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) 
+\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) 
+\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow 
+False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) 
+\Rightarrow False])])) I (CHead d1 (Bind Abbr) u) H4) in (False_ind ((eq C 
+(CHead c3 (Bind Abbr) u0) c2) \to ((csub3 g c0 c3) \to ((ty3 g c3 u0 t) \to 
+(ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead 
+d2 (Bind Abbr) u))))))) H6)) H5 H2 H3)))]) in (H2 (refl_equal C (CHead d1 
+(Bind Abbr) u)) (refl_equal C c2)))))))))) (\lambda (n0: nat).(\lambda (H: 
+((\forall (c1: C).(\forall (c2: C).((csub3 g c1 c2) \to (\forall (d1: 
+C).(\forall (u: T).((drop n0 O c1 (CHead d1 (Bind Abbr) u)) \to (ex2 C 
+(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 
+(Bind Abbr) u)))))))))))).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H0: 
+(csub3 g c1 c2)).(csub3_ind g (\lambda (c: C).(\lambda (c0: C).(\forall (d1: 
+C).(\forall (u: T).((drop (S n0) O c (CHead d1 (Bind Abbr) u)) \to (ex2 C 
+(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c0 (CHead 
+d2 (Bind Abbr) u))))))))) (\lambda (n1: nat).(\lambda (d1: C).(\lambda (u: 
+T).(\lambda (H1: (drop (S n0) O (CSort n1) (CHead d1 (Bind Abbr) u))).(let H2 
+\def (match H1 in drop return (\lambda (n2: nat).(\lambda (n3: nat).(\lambda 
+(c: C).(\lambda (c0: C).(\lambda (_: (drop n2 n3 c c0)).((eq nat n2 (S n0)) 
+\to ((eq nat n3 O) \to ((eq C c (CSort n1)) \to ((eq C c0 (CHead d1 (Bind 
+Abbr) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop 
+(S n0) O (CSort n1) (CHead d2 (Bind Abbr) u))))))))))))) with [(drop_refl c) 
+\Rightarrow (\lambda (H2: (eq nat O (S n0))).(\lambda (H3: (eq nat O 
+O)).(\lambda (H4: (eq C c (CSort n1))).(\lambda (H5: (eq C c (CHead d1 (Bind 
+Abbr) u))).((let H6 \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 n0) H2) in (False_ind ((eq nat O O) \to ((eq C c (CSort n1)) 
+\to ((eq C c (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csub3 g 
+d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) 
+u))))))) H6)) H3 H4 H5))))) | (drop_drop k h c e H2 u0) \Rightarrow (\lambda 
+(H3: (eq nat (S h) (S n0))).(\lambda (H4: (eq nat O O)).(\lambda (H5: (eq C 
+(CHead c k u0) (CSort n1))).(\lambda (H6: (eq C e (CHead d1 (Bind Abbr) 
+u))).((let H7 \def (f_equal nat nat (\lambda (e0: nat).(match e0 in nat 
+return (\lambda (_: nat).nat) with [O \Rightarrow h | (S n2) \Rightarrow 
+n2])) (S h) (S n0) H3) in (eq_ind nat n0 (\lambda (n2: nat).((eq nat O O) \to 
+((eq C (CHead c k u0) (CSort n1)) \to ((eq C e (CHead d1 (Bind Abbr) u)) \to 
+((drop (r k n2) O c e) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
+(d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u))))))))) (\lambda 
+(_: (eq nat O O)).(\lambda (H9: (eq C (CHead c k u0) (CSort n1))).(let H10 
+\def (eq_ind C (CHead c k u0) (\lambda (e0: C).(match e0 in C return (\lambda 
+(_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow 
+True])) I (CSort n1) H9) in (False_ind ((eq C e (CHead d1 (Bind Abbr) u)) \to 
+((drop (r k n0) O c e) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
+(d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u)))))) H10)))) h 
+(sym_eq nat h n0 H7))) H4 H5 H6 H2))))) | (drop_skip k h d c e H2 u0) 
+\Rightarrow (\lambda (H3: (eq nat h (S n0))).(\lambda (H4: (eq nat (S d) 
+O)).(\lambda (H5: (eq C (CHead c k (lift h (r k d) u0)) (CSort n1))).(\lambda 
+(H6: (eq C (CHead e k u0) (CHead d1 (Bind Abbr) u))).(eq_ind nat (S n0) 
+(\lambda (n2: nat).((eq nat (S d) O) \to ((eq C (CHead c k (lift n2 (r k d) 
+u0)) (CSort n1)) \to ((eq C (CHead e k u0) (CHead d1 (Bind Abbr) u)) \to 
+((drop n2 (r k d) c e) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
+(d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u))))))))) (\lambda 
+(H7: (eq nat (S d) O)).(let H8 \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 H7) in (False_ind ((eq C (CHead c k 
+(lift (S n0) (r k d) u0)) (CSort n1)) \to ((eq C (CHead e k u0) (CHead d1 
+(Bind Abbr) u)) \to ((drop (S n0) (r k d) c e) \to (ex2 C (\lambda (d2: 
+C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 
+(Bind Abbr) u))))))) H8))) h (sym_eq nat h (S n0) H3) H4 H5 H6 H2)))))]) in 
+(H2 (refl_equal nat (S n0)) (refl_equal nat O) (refl_equal C (CSort n1)) 
+(refl_equal C (CHead d1 (Bind Abbr) u)))))))) (\lambda (c0: C).(\lambda (c3: 
+C).(\lambda (H1: (csub3 g c0 c3)).(\lambda (H2: ((\forall (d1: C).(\forall 
+(u: T).((drop (S n0) O c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: 
+C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abbr) 
+u))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (u: T).(\forall 
+(d1: C).(\forall (u0: T).((drop (S n0) O (CHead c0 k0 u) (CHead d1 (Bind 
+Abbr) u0)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: 
+C).(drop (S n0) O (CHead c3 k0 u) (CHead d2 (Bind Abbr) u0))))))))) (\lambda 
+(b: B).(\lambda (u: T).(\lambda (d1: C).(\lambda (u0: T).(\lambda (H3: (drop 
+(S n0) O (CHead c0 (Bind b) u) (CHead d1 (Bind Abbr) u0))).(ex2_ind C 
+(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 
+(Bind Abbr) u0))) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: 
+C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0)))) (\lambda 
+(x: C).(\lambda (H4: (csub3 g d1 x)).(\lambda (H5: (drop n0 O c3 (CHead x 
+(Bind Abbr) u0))).(ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
+(d2: C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0))) x H4 
+(drop_drop (Bind b) n0 c3 (CHead x (Bind Abbr) u0) H5 u))))) (H c0 c3 H1 d1 
+u0 (drop_gen_drop (Bind b) c0 (CHead d1 (Bind Abbr) u0) u n0 H3)))))))) 
+(\lambda (f: F).(\lambda (u: T).(\lambda (d1: C).(\lambda (u0: T).(\lambda 
+(H3: (drop (S n0) O (CHead c0 (Flat f) u) (CHead d1 (Bind Abbr) 
+u0))).(ex2_ind C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S 
+n0) O c3 (CHead d2 (Bind Abbr) u0))) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) 
+(\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abbr) 
+u0)))) (\lambda (x: C).(\lambda (H4: (csub3 g d1 x)).(\lambda (H5: (drop (S 
+n0) O c3 (CHead x (Bind Abbr) u0))).(ex_intro2 C (\lambda (d2: C).(csub3 g d1 
+d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind 
+Abbr) u0))) x H4 (drop_drop (Flat f) n0 c3 (CHead x (Bind Abbr) u0) H5 u))))) 
+(H2 d1 u0 (drop_gen_drop (Flat f) c0 (CHead d1 (Bind Abbr) u0) u n0 
+H3)))))))) k)))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H1: (csub3 g 
+c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (u: T).((drop (S n0) O c0 
+(CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) 
+(\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u))))))))).(\lambda 
+(b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: 
+T).(\lambda (d1: C).(\lambda (u: T).(\lambda (H4: (drop (S n0) O (CHead c0 
+(Bind Void) u1) (CHead d1 (Bind Abbr) u))).(ex2_ind C (\lambda (d2: C).(csub3 
+g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abbr) u))) (ex2 C 
+(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
+(Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda (x: C).(\lambda (H5: (csub3 
+g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x (Bind Abbr) u))).(ex_intro2 C 
+(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
+(Bind b) u2) (CHead d2 (Bind Abbr) u))) x H5 (drop_drop (Bind b) n0 c3 (CHead 
+x (Bind Abbr) u) H6 u2))))) (H c0 c3 H1 d1 u (drop_gen_drop (Bind Void) c0 
+(CHead d1 (Bind Abbr) u) u1 n0 H4)))))))))))))) (\lambda (c0: C).(\lambda 
+(c3: C).(\lambda (H1: (csub3 g c0 c3)).(\lambda (_: ((\forall (d1: 
+C).(\forall (u: T).((drop (S n0) O c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C 
+(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead 
+d2 (Bind Abbr) u))))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g 
+c3 u t)).(\lambda (d1: C).(\lambda (u0: T).(\lambda (H4: (drop (S n0) O 
+(CHead c0 (Bind Abst) t) (CHead d1 (Bind Abbr) u0))).(ex2_ind C (\lambda (d2: 
+C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abbr) 
+u0))) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) 
+O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (x: 
+C).(\lambda (H5: (csub3 g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x (Bind 
+Abbr) u0))).(ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: 
+C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0))) x H5 
+(drop_drop (Bind Abbr) n0 c3 (CHead x (Bind Abbr) u0) H6 u))))) (H c0 c3 H1 
+d1 u0 (drop_gen_drop (Bind Abst) c0 (CHead d1 (Bind Abbr) u0) t n0 
+H4))))))))))))) c1 c2 H0)))))) n)).
+
+theorem csub3_drop_abst:
+ \forall (g: G).(\forall (n: nat).(\forall (c1: C).(\forall (c2: C).((csub3 g 
+c1 c2) \to (\forall (d1: C).(\forall (t: T).((drop n O c1 (CHead d1 (Bind 
+Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: 
+C).(drop n O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
+C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop n 
+O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u 
+t))))))))))))
+\def
+ \lambda (g: G).(\lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1: 
+C).(\forall (c2: C).((csub3 g c1 c2) \to (\forall (d1: C).(\forall (t: 
+T).((drop n0 O c1 (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: 
+C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Bind Abst) 
+t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda 
+(d2: C).(\lambda (u: T).(drop n0 O c2 (CHead d2 (Bind Abbr) u)))) (\lambda 
+(d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))))) (\lambda (c1: C).(\lambda 
+(c2: C).(\lambda (H: (csub3 g c1 c2)).(\lambda (d1: C).(\lambda (t: 
+T).(\lambda (H0: (drop O O c1 (CHead d1 (Bind Abst) t))).(let H1 \def (eq_ind 
+C c1 (\lambda (c: C).(csub3 g c c2)) H (CHead d1 (Bind Abst) t) 
+(drop_gen_refl c1 (CHead d1 (Bind Abst) t) H0)) in (let H2 \def (match H1 in 
+csub3 return (\lambda (c: C).(\lambda (c0: C).(\lambda (_: (csub3 ? c 
+c0)).((eq C c (CHead d1 (Bind Abst) t)) \to ((eq C c0 c2) \to (or (ex2 C 
+(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 
+(Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 
+d2))) (\lambda (d2: C).(\lambda (u: T).(drop O O c2 (CHead d2 (Bind Abbr) 
+u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))) with 
+[(csub3_sort n0) \Rightarrow (\lambda (H2: (eq C (CSort n0) (CHead d1 (Bind 
+Abst) t))).(\lambda (H3: (eq C (CSort n0) c2)).((let H4 \def (eq_ind C (CSort 
+n0) (\lambda (e: C).(match e in C return (\lambda (_: C).Prop) with [(CSort 
+_) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 (Bind 
+Abst) t) H2) in (False_ind ((eq C (CSort n0) c2) \to (or (ex2 C (\lambda (d2: 
+C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t)))) 
+(ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: 
+C).(\lambda (u: T).(drop O O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: 
+C).(\lambda (u: T).(ty3 g d2 u t)))))) H4)) H3))) | (csub3_head c0 c3 H2 k u) 
+\Rightarrow (\lambda (H3: (eq C (CHead c0 k u) (CHead d1 (Bind Abst) 
+t))).(\lambda (H4: (eq C (CHead c3 k u) c2)).((let H5 \def (f_equal C T 
+(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) 
+\Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead c0 k u) (CHead d1 
+(Bind Abst) t) H3) in ((let H6 \def (f_equal C K (\lambda (e: C).(match e in 
+C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) 
+\Rightarrow k0])) (CHead c0 k u) (CHead d1 (Bind Abst) t) H3) in ((let H7 
+\def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) 
+with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k 
+u) (CHead d1 (Bind Abst) t) H3) in (eq_ind C d1 (\lambda (c: C).((eq K k 
+(Bind Abst)) \to ((eq T u t) \to ((eq C (CHead c3 k u) c2) \to ((csub3 g c 
+c3) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O 
+O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
+T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop O O c2 (CHead d2 
+(Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))))))))) 
+(\lambda (H8: (eq K k (Bind Abst))).(eq_ind K (Bind Abst) (\lambda (k0: 
+K).((eq T u t) \to ((eq C (CHead c3 k0 u) c2) \to ((csub3 g d1 c3) \to (or 
+(ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead 
+d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 
+d2))) (\lambda (d2: C).(\lambda (u0: T).(drop O O c2 (CHead d2 (Bind Abbr) 
+u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))))))))) (\lambda 
+(H9: (eq T u t)).(eq_ind T t (\lambda (t0: T).((eq C (CHead c3 (Bind Abst) 
+t0) c2) \to ((csub3 g d1 c3) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) 
+(\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda 
+(d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: 
+T).(drop O O c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: 
+T).(ty3 g d2 u0 t)))))))) (\lambda (H10: (eq C (CHead c3 (Bind Abst) t) 
+c2)).(eq_ind C (CHead c3 (Bind Abst) t) (\lambda (c: C).((csub3 g d1 c3) \to 
+(or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c 
+(CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
+T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop O O c (CHead d2 
+(Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))))))) 
+(\lambda (H11: (csub3 g d1 c3)).(or_introl (ex2 C (\lambda (d2: C).(csub3 g 
+d1 d2)) (\lambda (d2: C).(drop O O (CHead c3 (Bind Abst) t) (CHead d2 (Bind 
+Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) 
+(\lambda (d2: C).(\lambda (u0: T).(drop O O (CHead c3 (Bind Abst) t) (CHead 
+d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))) 
+(ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O 
+(CHead c3 (Bind Abst) t) (CHead d2 (Bind Abst) t))) c3 H11 (drop_refl (CHead 
+c3 (Bind Abst) t))))) c2 H10)) u (sym_eq T u t H9))) k (sym_eq K k (Bind 
+Abst) H8))) c0 (sym_eq C c0 d1 H7))) H6)) H5)) H4 H2))) | (csub3_void c0 c3 
+H2 b H3 u1 u2) \Rightarrow (\lambda (H4: (eq C (CHead c0 (Bind Void) u1) 
+(CHead d1 (Bind Abst) t))).(\lambda (H5: (eq C (CHead c3 (Bind b) u2) 
+c2)).((let H6 \def (eq_ind C (CHead c0 (Bind Void) u1) (\lambda (e: C).(match 
+e in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | 
+(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with 
+[(Bind b0) \Rightarrow (match b0 in B return (\lambda (_: B).Prop) with [Abbr 
+\Rightarrow False | Abst \Rightarrow False | Void \Rightarrow True]) | (Flat 
+_) \Rightarrow False])])) I (CHead d1 (Bind Abst) t) H4) in (False_ind ((eq C 
+(CHead c3 (Bind b) u2) c2) \to ((csub3 g c0 c3) \to ((not (eq B b Void)) \to 
+(or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 
+(CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
+T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop O O c2 (CHead d2 
+(Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))) H6)) 
+H5 H2 H3))) | (csub3_abst c0 c3 H2 u t0 H3) \Rightarrow (\lambda (H4: (eq C 
+(CHead c0 (Bind Abst) t0) (CHead d1 (Bind Abst) t))).(\lambda (H5: (eq C 
+(CHead c3 (Bind Abbr) u) c2)).((let H6 \def (f_equal C T (\lambda (e: 
+C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow t0 | 
+(CHead _ _ t1) \Rightarrow t1])) (CHead c0 (Bind Abst) t0) (CHead d1 (Bind 
+Abst) t) H4) in ((let H7 \def (f_equal C C (\lambda (e: C).(match e in C 
+return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) 
+\Rightarrow c])) (CHead c0 (Bind Abst) t0) (CHead d1 (Bind Abst) t) H4) in 
+(eq_ind C d1 (\lambda (c: C).((eq T t0 t) \to ((eq C (CHead c3 (Bind Abbr) u) 
+c2) \to ((csub3 g c c3) \to ((ty3 g c3 u t0) \to (or (ex2 C (\lambda (d2: 
+C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t)))) 
+(ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: 
+C).(\lambda (u0: T).(drop O O c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: 
+C).(\lambda (u0: T).(ty3 g d2 u0 t)))))))))) (\lambda (H8: (eq T t0 
+t)).(eq_ind T t (\lambda (t1: T).((eq C (CHead c3 (Bind Abbr) u) c2) \to 
+((csub3 g d1 c3) \to ((ty3 g c3 u t1) \to (or (ex2 C (\lambda (d2: C).(csub3 
+g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C 
+T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: 
+C).(\lambda (u0: T).(drop O O c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: 
+C).(\lambda (u0: T).(ty3 g d2 u0 t))))))))) (\lambda (H9: (eq C (CHead c3 
+(Bind Abbr) u) c2)).(eq_ind C (CHead c3 (Bind Abbr) u) (\lambda (c: 
+C).((csub3 g d1 c3) \to ((ty3 g c3 u t) \to (or (ex2 C (\lambda (d2: 
+C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c (CHead d2 (Bind Abst) t)))) 
+(ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: 
+C).(\lambda (u0: T).(drop O O c (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: 
+C).(\lambda (u0: T).(ty3 g d2 u0 t)))))))) (\lambda (H10: (csub3 g d1 
+c3)).(\lambda (H11: (ty3 g c3 u t)).(or_intror (ex2 C (\lambda (d2: C).(csub3 
+g d1 d2)) (\lambda (d2: C).(drop O O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind 
+Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) 
+(\lambda (d2: C).(\lambda (u0: T).(drop O O (CHead c3 (Bind Abbr) u) (CHead 
+d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))) 
+(ex3_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda 
+(d2: C).(\lambda (u0: T).(drop O O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind 
+Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))) c3 u H10 
+(drop_refl (CHead c3 (Bind Abbr) u)) H11)))) c2 H9)) t0 (sym_eq T t0 t H8))) 
+c0 (sym_eq C c0 d1 H7))) H6)) H5 H2 H3)))]) in (H2 (refl_equal C (CHead d1 
+(Bind Abst) t)) (refl_equal C c2)))))))))) (\lambda (n0: nat).(\lambda (H: 
+((\forall (c1: C).(\forall (c2: C).((csub3 g c1 c2) \to (\forall (d1: 
+C).(\forall (t: T).((drop n0 O c1 (CHead d1 (Bind Abst) t)) \to (or (ex2 C 
+(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 
+(Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 
+d2))) (\lambda (d2: C).(\lambda (u: T).(drop n0 O c2 (CHead d2 (Bind Abbr) 
+u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))))))))))).(\lambda 
+(c1: C).(\lambda (c2: C).(\lambda (H0: (csub3 g c1 c2)).(csub3_ind g (\lambda 
+(c: C).(\lambda (c0: C).(\forall (d1: C).(\forall (t: T).((drop (S n0) O c 
+(CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) 
+(\lambda (d2: C).(drop (S n0) O c0 (CHead d2 (Bind Abst) t)))) (ex3_2 C T 
+(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda 
+(u: T).(drop (S n0) O c0 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: 
+C).(\lambda (u: T).(ty3 g d2 u t)))))))))) (\lambda (n1: nat).(\lambda (d1: 
+C).(\lambda (t: T).(\lambda (H1: (drop (S n0) O (CSort n1) (CHead d1 (Bind 
+Abst) t))).(let H2 \def (match H1 in drop return (\lambda (n2: nat).(\lambda 
+(n3: nat).(\lambda (c: C).(\lambda (c0: C).(\lambda (_: (drop n2 n3 c 
+c0)).((eq nat n2 (S n0)) \to ((eq nat n3 O) \to ((eq C c (CSort n1)) \to ((eq 
+C c0 (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 
+d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abst) t)))) 
+(ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: 
+C).(\lambda (u: T).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u)))) 
+(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))))))) with [(drop_refl 
+c) \Rightarrow (\lambda (H2: (eq nat O (S n0))).(\lambda (H3: (eq nat O 
+O)).(\lambda (H4: (eq C c (CSort n1))).(\lambda (H5: (eq C c (CHead d1 (Bind 
+Abst) t))).((let H6 \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 n0) H2) in (False_ind ((eq nat O O) \to ((eq C c (CSort n1)) 
+\to ((eq C c (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csub3 
+g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abst) 
+t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda 
+(d2: C).(\lambda (u: T).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u)))) 
+(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))) H6)) H3 H4 H5))))) | 
+(drop_drop k h c e H2 u) \Rightarrow (\lambda (H3: (eq nat (S h) (S 
+n0))).(\lambda (H4: (eq nat O O)).(\lambda (H5: (eq C (CHead c k u) (CSort 
+n1))).(\lambda (H6: (eq C e (CHead d1 (Bind Abst) t))).((let H7 \def (f_equal 
+nat nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_: nat).nat) 
+with [O \Rightarrow h | (S n2) \Rightarrow n2])) (S h) (S n0) H3) in (eq_ind 
+nat n0 (\lambda (n2: nat).((eq nat O O) \to ((eq C (CHead c k u) (CSort n1)) 
+\to ((eq C e (CHead d1 (Bind Abst) t)) \to ((drop (r k n2) O c e) \to (or 
+(ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O 
+(CSort n1) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda 
+(_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O 
+(CSort n1) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: 
+T).(ty3 g d2 u0 t)))))))))) (\lambda (_: (eq nat O O)).(\lambda (H9: (eq C 
+(CHead c k u) (CSort n1))).(let H10 \def (eq_ind C (CHead c k u) (\lambda 
+(e0: C).(match e0 in C return (\lambda (_: C).Prop) with [(CSort _) 
+\Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n1) H9) in 
+(False_ind ((eq C e (CHead d1 (Bind Abst) t)) \to ((drop (r k n0) O c e) \to 
+(or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O 
+(CSort n1) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda 
+(_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O 
+(CSort n1) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: 
+T).(ty3 g d2 u0 t))))))) H10)))) h (sym_eq nat h n0 H7))) H4 H5 H6 H2))))) | 
+(drop_skip k h d c e H2 u) \Rightarrow (\lambda (H3: (eq nat h (S 
+n0))).(\lambda (H4: (eq nat (S d) O)).(\lambda (H5: (eq C (CHead c k (lift h 
+(r k d) u)) (CSort n1))).(\lambda (H6: (eq C (CHead e k u) (CHead d1 (Bind 
+Abst) t))).(eq_ind nat (S n0) (\lambda (n2: nat).((eq nat (S d) O) \to ((eq C 
+(CHead c k (lift n2 (r k d) u)) (CSort n1)) \to ((eq C (CHead e k u) (CHead 
+d1 (Bind Abst) t)) \to ((drop n2 (r k d) c e) \to (or (ex2 C (\lambda (d2: 
+C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 
+(Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 
+d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CSort n1) (CHead d2 
+(Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))))))))) 
+(\lambda (H7: (eq nat (S d) O)).(let H8 \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 H7) in (False_ind ((eq C (CHead c k 
+(lift (S n0) (r k d) u)) (CSort n1)) \to ((eq C (CHead e k u) (CHead d1 (Bind 
+Abst) t)) \to ((drop (S n0) (r k d) c e) \to (or (ex2 C (\lambda (d2: 
+C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 
+(Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 
+d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CSort n1) (CHead d2 
+(Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))))))) 
+H8))) h (sym_eq nat h (S n0) H3) H4 H5 H6 H2)))))]) in (H2 (refl_equal nat (S 
+n0)) (refl_equal nat O) (refl_equal C (CSort n1)) (refl_equal C (CHead d1 
+(Bind Abst) t)))))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H1: (csub3 
+g c0 c3)).(\lambda (H2: ((\forall (d1: C).(\forall (t: T).((drop (S n0) O c0 
+(CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) 
+(\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abst) t)))) (ex3_2 C T 
+(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda 
+(u: T).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: 
+C).(\lambda (u: T).(ty3 g d2 u t)))))))))).(\lambda (k: K).(K_ind (\lambda 
+(k0: K).(\forall (u: T).(\forall (d1: C).(\forall (t: T).((drop (S n0) O 
+(CHead c0 k0 u) (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: 
+C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 k0 u) (CHead d2 
+(Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 
+d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 k0 u) (CHead 
+d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 
+t)))))))))) (\lambda (b: B).(\lambda (u: T).(\lambda (d1: C).(\lambda (t: 
+T).(\lambda (H3: (drop (S n0) O (CHead c0 (Bind b) u) (CHead d1 (Bind Abst) 
+t))).(or_ind (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop 
+n0 O c3 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
+T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop n0 O c3 (CHead 
+d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))) 
+(or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O 
+(CHead c3 (Bind b) u) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
+C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop 
+(S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: 
+C).(\lambda (u0: T).(ty3 g d2 u0 t))))) (\lambda (H4: (ex2 C (\lambda (d2: 
+C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abst) 
+t))))).(ex2_ind C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 
+O c3 (CHead d2 (Bind Abst) t))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) 
+(\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abst) 
+t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda 
+(d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind 
+Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))))) (\lambda 
+(x: C).(\lambda (H5: (csub3 g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x 
+(Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
+(d2: C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abst) t)))) 
+(ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: 
+C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind 
+Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))) (ex_intro2 
+C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
+(Bind b) u) (CHead d2 (Bind Abst) t))) x H5 (drop_drop (Bind b) n0 c3 (CHead 
+x (Bind Abst) t) H6 u)))))) H4)) (\lambda (H4: (ex3_2 C T (\lambda (d2: 
+C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop 
+n0 O c3 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g 
+d2 u0 t))))).(ex3_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 
+d2))) (\lambda (d2: C).(\lambda (u0: T).(drop n0 O c3 (CHead d2 (Bind Abbr) 
+u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))) (or (ex2 C 
+(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
+(Bind b) u) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda 
+(_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O 
+(CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda 
+(u0: T).(ty3 g d2 u0 t))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: 
+(csub3 g d1 x0)).(\lambda (H6: (drop n0 O c3 (CHead x0 (Bind Abbr) 
+x1))).(\lambda (H7: (ty3 g x0 x1 t)).(or_intror (ex2 C (\lambda (d2: 
+C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u) 
+(CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
+T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead 
+c3 (Bind b) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: 
+T).(ty3 g d2 u0 t)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: 
+T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead 
+c3 (Bind b) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: 
+T).(ty3 g d2 u0 t))) x0 x1 H5 (drop_drop (Bind b) n0 c3 (CHead x0 (Bind Abbr) 
+x1) H6 u) H7))))))) H4)) (H c0 c3 H1 d1 t (drop_gen_drop (Bind b) c0 (CHead 
+d1 (Bind Abst) t) u n0 H3)))))))) (\lambda (f: F).(\lambda (u: T).(\lambda 
+(d1: C).(\lambda (t: T).(\lambda (H3: (drop (S n0) O (CHead c0 (Flat f) u) 
+(CHead d1 (Bind Abst) t))).(or_ind (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) 
+(\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abst) t)))) (ex3_2 C T 
+(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda 
+(u0: T).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: 
+C).(\lambda (u0: T).(ty3 g d2 u0 t)))) (or (ex2 C (\lambda (d2: C).(csub3 g 
+d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind 
+Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) 
+(\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Flat f) u) (CHead 
+d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))))) 
+(\lambda (H4: (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop 
+(S n0) O c3 (CHead d2 (Bind Abst) t))))).(ex2_ind C (\lambda (d2: C).(csub3 g 
+d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abst) t))) (or 
+(ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O 
+(CHead c3 (Flat f) u) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
+C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop 
+(S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: 
+C).(\lambda (u0: T).(ty3 g d2 u0 t))))) (\lambda (x: C).(\lambda (H5: (csub3 
+g d1 x)).(\lambda (H6: (drop (S n0) O c3 (CHead x (Bind Abst) t))).(or_introl 
+(ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O 
+(CHead c3 (Flat f) u) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
+C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop 
+(S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: 
+C).(\lambda (u0: T).(ty3 g d2 u0 t)))) (ex_intro2 C (\lambda (d2: C).(csub3 g 
+d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind 
+Abst) t))) x H5 (drop_drop (Flat f) n0 c3 (CHead x (Bind Abst) t) H6 u)))))) 
+H4)) (\lambda (H4: (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 
+d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O c3 (CHead d2 (Bind 
+Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))))).(ex3_2_ind 
+C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: 
+C).(\lambda (u0: T).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u0)))) (\lambda 
+(d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))) (or (ex2 C (\lambda (d2: C).(csub3 
+g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 
+(Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 
+d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Flat f) u) 
+(CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 
+t))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (csub3 g d1 
+x0)).(\lambda (H6: (drop (S n0) O c3 (CHead x0 (Bind Abbr) x1))).(\lambda 
+(H7: (ty3 g x0 x1 t)).(or_intror (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) 
+(\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abst) 
+t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda 
+(d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind 
+Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))) 
+(ex3_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda 
+(d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind 
+Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))) x0 x1 H5 
+(drop_drop (Flat f) n0 c3 (CHead x0 (Bind Abbr) x1) H6 u) H7))))))) H4)) (H2 
+d1 t (drop_gen_drop (Flat f) c0 (CHead d1 (Bind Abst) t) u n0 H3)))))))) 
+k)))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H1: (csub3 g c0 
+c3)).(\lambda (_: ((\forall (d1: C).(\forall (t: T).((drop (S n0) O c0 (CHead 
+d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
+(d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda 
+(d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: 
+T).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda 
+(u: T).(ty3 g d2 u t)))))))))).(\lambda (b: B).(\lambda (_: (not (eq B b 
+Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (d1: C).(\lambda (t: 
+T).(\lambda (H4: (drop (S n0) O (CHead c0 (Bind Void) u1) (CHead d1 (Bind 
+Abst) t))).(or_ind (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: 
+C).(drop n0 O c3 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
+C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop 
+n0 O c3 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g 
+d2 u t)))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: 
+C).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abst) t)))) (ex3_2 C 
+T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: 
+C).(\lambda (u: T).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind 
+Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (H5: 
+(ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 
+(CHead d2 (Bind Abst) t))))).(ex2_ind C (\lambda (d2: C).(csub3 g d1 d2)) 
+(\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abst) t))) (or (ex2 C (\lambda 
+(d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) 
+u2) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
+T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CHead 
+c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: 
+T).(ty3 g d2 u t))))) (\lambda (x: C).(\lambda (H6: (csub3 g d1 x)).(\lambda 
+(H7: (drop n0 O c3 (CHead x (Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: 
+C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u2) 
+(CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
+T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CHead 
+c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: 
+T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
+(d2: C).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abst) t))) x H6 
+(drop_drop (Bind b) n0 c3 (CHead x (Bind Abst) t) H7 u2)))))) H5)) (\lambda 
+(H5: (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda 
+(d2: C).(\lambda (u: T).(drop n0 O c3 (CHead d2 (Bind Abbr) u)))) (\lambda 
+(d2: C).(\lambda (u: T).(ty3 g d2 u t))))).(ex3_2_ind C T (\lambda (d2: 
+C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop 
+n0 O c3 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g 
+d2 u t))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop 
+(S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abst) t)))) (ex3_2 C T 
+(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda 
+(u: T).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) 
+(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x0: C).(\lambda 
+(x1: T).(\lambda (H6: (csub3 g d1 x0)).(\lambda (H7: (drop n0 O c3 (CHead x0 
+(Bind Abbr) x1))).(\lambda (H8: (ty3 g x0 x1 t)).(or_intror (ex2 C (\lambda 
+(d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) 
+u2) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
+T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CHead 
+c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: 
+T).(ty3 g d2 u t)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csub3 
+g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CHead c3 (Bind b) 
+u2) (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u 
+t))) x0 x1 H6 (drop_drop (Bind b) n0 c3 (CHead x0 (Bind Abbr) x1) H7 u2) 
+H8))))))) H5)) (H c0 c3 H1 d1 t (drop_gen_drop (Bind Void) c0 (CHead d1 (Bind 
+Abst) t) u1 n0 H4)))))))))))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda 
+(H1: (csub3 g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (t: T).((drop 
+(S n0) O c0 (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csub3 
+g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abst) t)))) 
+(ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: 
+C).(\lambda (u: T).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u)))) (\lambda 
+(d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))).(\lambda (u: T).(\lambda (t: 
+T).(\lambda (_: (ty3 g c3 u t)).(\lambda (d1: C).(\lambda (t0: T).(\lambda 
+(H4: (drop (S n0) O (CHead c0 (Bind Abst) t) (CHead d1 (Bind Abst) 
+t0))).(or_ind (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop 
+n0 O c3 (CHead d2 (Bind Abst) t0)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
+T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop n0 O c3 (CHead 
+d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0)))) 
+(or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O 
+(CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) (ex3_2 C T (\lambda 
+(d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: 
+T).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) 
+(\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0))))) (\lambda (H5: (ex2 C 
+(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 
+(Bind Abst) t0))))).(ex2_ind C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
+(d2: C).(drop n0 O c3 (CHead d2 (Bind Abst) t0))) (or (ex2 C (\lambda (d2: 
+C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) 
+(CHead d2 (Bind Abst) t0)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
+T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead 
+c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: 
+T).(ty3 g d2 u0 t0))))) (\lambda (x: C).(\lambda (H6: (csub3 g d1 
+x)).(\lambda (H7: (drop n0 O c3 (CHead x (Bind Abst) t0))).(or_introl (ex2 C 
+(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
+(Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) (ex3_2 C T (\lambda (d2: 
+C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop 
+(S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: 
+C).(\lambda (u0: T).(ty3 g d2 u0 t0)))) (ex_intro2 C (\lambda (d2: C).(csub3 
+g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 
+(Bind Abst) t0))) x H6 (drop_drop (Bind Abbr) n0 c3 (CHead x (Bind Abst) t0) 
+H7 u)))))) H5)) (\lambda (H5: (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
+T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop n0 O c3 (CHead 
+d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 
+t0))))).(ex3_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) 
+(\lambda (d2: C).(\lambda (u0: T).(drop n0 O c3 (CHead d2 (Bind Abbr) u0)))) 
+(\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0))) (or (ex2 C (\lambda (d2: 
+C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) 
+(CHead d2 (Bind Abst) t0)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
+T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead 
+c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: 
+T).(ty3 g d2 u0 t0))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: 
+(csub3 g d1 x0)).(\lambda (H7: (drop n0 O c3 (CHead x0 (Bind Abbr) 
+x1))).(\lambda (H8: (ty3 g x0 x1 t0)).(or_intror (ex2 C (\lambda (d2: 
+C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) 
+(CHead d2 (Bind Abst) t0)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
+T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead 
+c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: 
+T).(ty3 g d2 u0 t0)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: 
+T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead 
+c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: 
+T).(ty3 g d2 u0 t0))) x0 x1 H6 (drop_drop (Bind Abbr) n0 c3 (CHead x0 (Bind 
+Abbr) x1) H7 u) H8))))))) H5)) (H c0 c3 H1 d1 t0 (drop_gen_drop (Bind Abst) 
+c0 (CHead d1 (Bind Abst) t0) t n0 H4))))))))))))) c1 c2 H0)))))) n)).
+

diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/fwd.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/fwd.ma
new file mode 100644 (file)
index 0000000..99b9dae
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/fwd.ma
@@ -0,0 +1,336 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/fwd".
+
+include "csub3/defs.ma".
+
+theorem csub3_gen_abbr:
+ \forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v: T).((csub3 g 
+(CHead e1 (Bind Abbr) v) c2) \to (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 
+(Bind Abbr) v))) (\lambda (e2: C).(csub3 g e1 e2)))))))
+\def
+ \lambda (g: G).(\lambda (e1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda 
+(H: (csub3 g (CHead e1 (Bind Abbr) v) c2)).(let H0 \def (match H in csub3 
+return (\lambda (c: C).(\lambda (c0: C).(\lambda (_: (csub3 ? c c0)).((eq C c 
+(CHead e1 (Bind Abbr) v)) \to ((eq C c0 c2) \to (ex2 C (\lambda (e2: C).(eq C 
+c2 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csub3 g e1 e2)))))))) with 
+[(csub3_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) (CHead e1 (Bind 
+Abbr) v))).(\lambda (H1: (eq C (CSort n) c2)).((let H2 \def (eq_ind C (CSort 
+n) (\lambda (e: C).(match e in C return (\lambda (_: C).Prop) with [(CSort _) 
+\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind Abbr) 
+v) H0) in (False_ind ((eq C (CSort n) c2) \to (ex2 C (\lambda (e2: C).(eq C 
+c2 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csub3 g e1 e2)))) H2)) H1))) 
+| (csub3_head c1 c0 H0 k u) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u) 
+(CHead e1 (Bind Abbr) v))).(\lambda (H2: (eq C (CHead c0 k u) c2)).((let H3 
+\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) 
+with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u) 
+(CHead e1 (Bind Abbr) v) H1) in ((let H4 \def (f_equal C K (\lambda (e: 
+C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | 
+(CHead _ k0 _) \Rightarrow k0])) (CHead c1 k u) (CHead e1 (Bind Abbr) v) H1) 
+in ((let H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda 
+(_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) 
+(CHead c1 k u) (CHead e1 (Bind Abbr) v) H1) in (eq_ind C e1 (\lambda (c: 
+C).((eq K k (Bind Abbr)) \to ((eq T u v) \to ((eq C (CHead c0 k u) c2) \to 
+((csub3 g c c0) \to (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abbr) 
+v))) (\lambda (e2: C).(csub3 g e1 e2)))))))) (\lambda (H6: (eq K k (Bind 
+Abbr))).(eq_ind K (Bind Abbr) (\lambda (k0: K).((eq T u v) \to ((eq C (CHead 
+c0 k0 u) c2) \to ((csub3 g e1 c0) \to (ex2 C (\lambda (e2: C).(eq C c2 (CHead 
+e2 (Bind Abbr) v))) (\lambda (e2: C).(csub3 g e1 e2))))))) (\lambda (H7: (eq 
+T u v)).(eq_ind T v (\lambda (t: T).((eq C (CHead c0 (Bind Abbr) t) c2) \to 
+((csub3 g e1 c0) \to (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abbr) 
+v))) (\lambda (e2: C).(csub3 g e1 e2)))))) (\lambda (H8: (eq C (CHead c0 
+(Bind Abbr) v) c2)).(eq_ind C (CHead c0 (Bind Abbr) v) (\lambda (c: 
+C).((csub3 g e1 c0) \to (ex2 C (\lambda (e2: C).(eq C c (CHead e2 (Bind Abbr) 
+v))) (\lambda (e2: C).(csub3 g e1 e2))))) (\lambda (H9: (csub3 g e1 c0)).(let 
+H10 \def (eq_ind_r C c2 (\lambda (c: C).(csub3 g (CHead e1 (Bind Abbr) v) c)) 
+H (CHead c0 (Bind Abbr) v) H8) in (ex_intro2 C (\lambda (e2: C).(eq C (CHead 
+c0 (Bind Abbr) v) (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csub3 g e1 
+e2)) c0 (refl_equal C (CHead c0 (Bind Abbr) v)) H9))) c2 H8)) u (sym_eq T u v 
+H7))) k (sym_eq K k (Bind Abbr) H6))) c1 (sym_eq C c1 e1 H5))) H4)) H3)) H2 
+H0))) | (csub3_void c1 c0 H0 b H1 u1 u2) \Rightarrow (\lambda (H2: (eq C 
+(CHead c1 (Bind Void) u1) (CHead e1 (Bind Abbr) v))).(\lambda (H3: (eq C 
+(CHead c0 (Bind b) u2) c2)).((let H4 \def (eq_ind C (CHead c1 (Bind Void) u1) 
+(\lambda (e: C).(match e in C return (\lambda (_: C).Prop) with [(CSort _) 
+\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda 
+(_: K).Prop) with [(Bind b0) \Rightarrow (match b0 in B return (\lambda (_: 
+B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False | Void 
+\Rightarrow True]) | (Flat _) \Rightarrow False])])) I (CHead e1 (Bind Abbr) 
+v) H2) in (False_ind ((eq C (CHead c0 (Bind b) u2) c2) \to ((csub3 g c1 c0) 
+\to ((not (eq B b Void)) \to (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind 
+Abbr) v))) (\lambda (e2: C).(csub3 g e1 e2)))))) H4)) H3 H0 H1))) | 
+(csub3_abst c1 c0 H0 u t H1) \Rightarrow (\lambda (H2: (eq C (CHead c1 (Bind 
+Abst) t) (CHead e1 (Bind Abbr) v))).(\lambda (H3: (eq C (CHead c0 (Bind Abbr) 
+u) c2)).((let H4 \def (eq_ind C (CHead c1 (Bind Abst) t) (\lambda (e: 
+C).(match e in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow 
+False | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) 
+with [(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with 
+[Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | 
+(Flat _) \Rightarrow False])])) I (CHead e1 (Bind Abbr) v) H2) in (False_ind 
+((eq C (CHead c0 (Bind Abbr) u) c2) \to ((csub3 g c1 c0) \to ((ty3 g c0 u t) 
+\to (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abbr) v))) (\lambda (e2: 
+C).(csub3 g e1 e2)))))) H4)) H3 H0 H1)))]) in (H0 (refl_equal C (CHead e1 
+(Bind Abbr) v)) (refl_equal C c2))))))).
+
+theorem csub3_gen_abst:
+ \forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v1: T).((csub3 g 
+(CHead e1 (Bind Abst) v1) c2) \to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead 
+e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda 
+(e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: 
+C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g 
+e2 v2 v1)))))))))
+\def
+ \lambda (g: G).(\lambda (e1: C).(\lambda (c2: C).(\lambda (v1: T).(\lambda 
+(H: (csub3 g (CHead e1 (Bind Abst) v1) c2)).(let H0 \def (match H in csub3 
+return (\lambda (c: C).(\lambda (c0: C).(\lambda (_: (csub3 ? c c0)).((eq C c 
+(CHead e1 (Bind Abst) v1)) \to ((eq C c0 c2) \to (or (ex2 C (\lambda (e2: 
+C).(eq C c2 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) 
+(ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) 
+v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: 
+C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))))) with [(csub3_sort n) 
+\Rightarrow (\lambda (H0: (eq C (CSort n) (CHead e1 (Bind Abst) 
+v1))).(\lambda (H1: (eq C (CSort n) c2)).((let H2 \def (eq_ind C (CSort n) 
+(\lambda (e: C).(match e in C return (\lambda (_: C).Prop) with [(CSort _) 
+\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind Abst) 
+v1) H0) in (False_ind ((eq C (CSort n) c2) \to (or (ex2 C (\lambda (e2: 
+C).(eq C c2 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) 
+(ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) 
+v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: 
+C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))) H2)) H1))) | (csub3_head c1 c0 H0 k 
+u) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u) (CHead e1 (Bind Abst) 
+v1))).(\lambda (H2: (eq C (CHead c0 k u) c2)).((let H3 \def (f_equal C T 
+(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) 
+\Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u) (CHead e1 (Bind 
+Abst) v1) H1) in ((let H4 \def (f_equal C K (\lambda (e: C).(match e in C 
+return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) 
+\Rightarrow k0])) (CHead c1 k u) (CHead e1 (Bind Abst) v1) H1) in ((let H5 
+\def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) 
+with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k 
+u) (CHead e1 (Bind Abst) v1) H1) in (eq_ind C e1 (\lambda (c: C).((eq K k 
+(Bind Abst)) \to ((eq T u v1) \to ((eq C (CHead c0 k u) c2) \to ((csub3 g c 
+c0) \to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst) v1))) 
+(\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: 
+T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: 
+T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 
+v1)))))))))) (\lambda (H6: (eq K k (Bind Abst))).(eq_ind K (Bind Abst) 
+(\lambda (k0: K).((eq T u v1) \to ((eq C (CHead c0 k0 u) c2) \to ((csub3 g e1 
+c0) \to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst) v1))) 
+(\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: 
+T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: 
+T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 
+v1))))))))) (\lambda (H7: (eq T u v1)).(eq_ind T v1 (\lambda (t: T).((eq C 
+(CHead c0 (Bind Abst) t) c2) \to ((csub3 g e1 c0) \to (or (ex2 C (\lambda 
+(e2: C).(eq C c2 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1 
+e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind 
+Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: 
+C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))) (\lambda (H8: (eq C (CHead c0 
+(Bind Abst) v1) c2)).(eq_ind C (CHead c0 (Bind Abst) v1) (\lambda (c: 
+C).((csub3 g e1 c0) \to (or (ex2 C (\lambda (e2: C).(eq C c (CHead e2 (Bind 
+Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: 
+C).(\lambda (v2: T).(eq C c (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: 
+C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g 
+e2 v2 v1))))))) (\lambda (H9: (csub3 g e1 c0)).(let H10 \def (eq_ind_r C c2 
+(\lambda (c: C).(csub3 g (CHead e1 (Bind Abst) v1) c)) H (CHead c0 (Bind 
+Abst) v1) H8) in (or_introl (ex2 C (\lambda (e2: C).(eq C (CHead c0 (Bind 
+Abst) v1) (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) 
+(ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c0 (Bind Abst) v1) 
+(CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g e1 
+e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))) (ex_intro2 C 
+(\lambda (e2: C).(eq C (CHead c0 (Bind Abst) v1) (CHead e2 (Bind Abst) v1))) 
+(\lambda (e2: C).(csub3 g e1 e2)) c0 (refl_equal C (CHead c0 (Bind Abst) v1)) 
+H9)))) c2 H8)) u (sym_eq T u v1 H7))) k (sym_eq K k (Bind Abst) H6))) c1 
+(sym_eq C c1 e1 H5))) H4)) H3)) H2 H0))) | (csub3_void c1 c0 H0 b H1 u1 u2) 
+\Rightarrow (\lambda (H2: (eq C (CHead c1 (Bind Void) u1) (CHead e1 (Bind 
+Abst) v1))).(\lambda (H3: (eq C (CHead c0 (Bind b) u2) c2)).((let H4 \def 
+(eq_ind C (CHead c1 (Bind Void) u1) (\lambda (e: C).(match e in C return 
+(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) 
+\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b0) 
+\Rightarrow (match b0 in B return (\lambda (_: B).Prop) with [Abbr 
+\Rightarrow False | Abst \Rightarrow False | Void \Rightarrow True]) | (Flat 
+_) \Rightarrow False])])) I (CHead e1 (Bind Abst) v1) H2) in (False_ind ((eq 
+C (CHead c0 (Bind b) u2) c2) \to ((csub3 g c1 c0) \to ((not (eq B b Void)) 
+\to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst) v1))) (\lambda 
+(e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C 
+c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g e1 
+e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))) H4)) H3 H0 
+H1))) | (csub3_abst c1 c0 H0 u t H1) \Rightarrow (\lambda (H2: (eq C (CHead 
+c1 (Bind Abst) t) (CHead e1 (Bind Abst) v1))).(\lambda (H3: (eq C (CHead c0 
+(Bind Abbr) u) c2)).((let H4 \def (f_equal C T (\lambda (e: C).(match e in C 
+return (\lambda (_: C).T) with [(CSort _) \Rightarrow t | (CHead _ _ t0) 
+\Rightarrow t0])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind Abst) v1) H2) in 
+((let H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: 
+C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead 
+c1 (Bind Abst) t) (CHead e1 (Bind Abst) v1) H2) in (eq_ind C e1 (\lambda (c: 
+C).((eq T t v1) \to ((eq C (CHead c0 (Bind Abbr) u) c2) \to ((csub3 g c c0) 
+\to ((ty3 g c0 u t) \to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind 
+Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: 
+C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: 
+C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g 
+e2 v2 v1)))))))))) (\lambda (H6: (eq T t v1)).(eq_ind T v1 (\lambda (t0: 
+T).((eq C (CHead c0 (Bind Abbr) u) c2) \to ((csub3 g e1 c0) \to ((ty3 g c0 u 
+t0) \to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst) v1))) 
+(\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: 
+T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: 
+T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 
+v1))))))))) (\lambda (H7: (eq C (CHead c0 (Bind Abbr) u) c2)).(eq_ind C 
+(CHead c0 (Bind Abbr) u) (\lambda (c: C).((csub3 g e1 c0) \to ((ty3 g c0 u 
+v1) \to (or (ex2 C (\lambda (e2: C).(eq C c (CHead e2 (Bind Abst) v1))) 
+(\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: 
+T).(eq C c (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: 
+T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 
+v1)))))))) (\lambda (H8: (csub3 g e1 c0)).(\lambda (H9: (ty3 g c0 u v1)).(let 
+H10 \def (eq_ind_r C c2 (\lambda (c: C).(csub3 g (CHead e1 (Bind Abst) v1) 
+c)) H (CHead c0 (Bind Abbr) u) H7) in (or_intror (ex2 C (\lambda (e2: C).(eq 
+C (CHead c0 (Bind Abbr) u) (CHead e2 (Bind Abst) v1))) (\lambda (e2: 
+C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C 
+(CHead c0 (Bind Abbr) u) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: 
+C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g 
+e2 v2 v1)))) (ex3_2_intro C T (\lambda (e2: C).(\lambda (v2: T).(eq C (CHead 
+c0 (Bind Abbr) u) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: 
+T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))) c0 
+u (refl_equal C (CHead c0 (Bind Abbr) u)) H8 H9))))) c2 H7)) t (sym_eq T t v1 
+H6))) c1 (sym_eq C c1 e1 H5))) H4)) H3 H0 H1)))]) in (H0 (refl_equal C (CHead 
+e1 (Bind Abst) v1)) (refl_equal C c2))))))).
+
+theorem csub3_gen_bind:
+ \forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall 
+(v1: T).((csub3 g (CHead e1 (Bind b1) v1) c2) \to (ex2_3 B C T (\lambda (b2: 
+B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) 
+(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))))))))))
+\def
+ \lambda (g: G).(\lambda (b1: B).(\lambda (e1: C).(\lambda (c2: C).(\lambda 
+(v1: T).(\lambda (H: (csub3 g (CHead e1 (Bind b1) v1) c2)).(let H0 \def 
+(match H in csub3 return (\lambda (c: C).(\lambda (c0: C).(\lambda (_: (csub3 
+? c c0)).((eq C c (CHead e1 (Bind b1) v1)) \to ((eq C c0 c2) \to (ex2_3 B C T 
+(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind 
+b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 
+e2)))))))))) with [(csub3_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) 
+(CHead e1 (Bind b1) v1))).(\lambda (H1: (eq C (CSort n) c2)).((let H2 \def 
+(eq_ind C (CSort n) (\lambda (e: C).(match e in C return (\lambda (_: 
+C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow 
+False])) I (CHead e1 (Bind b1) v1) H0) in (False_ind ((eq C (CSort n) c2) \to 
+(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 
+(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: 
+T).(csub3 g e1 e2)))))) H2)) H1))) | (csub3_head c1 c0 H0 k u) \Rightarrow 
+(\lambda (H1: (eq C (CHead c1 k u) (CHead e1 (Bind b1) v1))).(\lambda (H2: 
+(eq C (CHead c0 k u) c2)).((let H3 \def (f_equal C T (\lambda (e: C).(match e 
+in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) 
+\Rightarrow t])) (CHead c1 k u) (CHead e1 (Bind b1) v1) H1) in ((let H4 \def 
+(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with 
+[(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k u) 
+(CHead e1 (Bind b1) v1) H1) in ((let H5 \def (f_equal C C (\lambda (e: 
+C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | 
+(CHead c _ _) \Rightarrow c])) (CHead c1 k u) (CHead e1 (Bind b1) v1) H1) in 
+(eq_ind C e1 (\lambda (c: C).((eq K k (Bind b1)) \to ((eq T u v1) \to ((eq C 
+(CHead c0 k u) c2) \to ((csub3 g c c0) \to (ex2_3 B C T (\lambda (b2: 
+B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) 
+(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2)))))))))) 
+(\lambda (H6: (eq K k (Bind b1))).(eq_ind K (Bind b1) (\lambda (k0: K).((eq T 
+u v1) \to ((eq C (CHead c0 k0 u) c2) \to ((csub3 g e1 c0) \to (ex2_3 B C T 
+(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind 
+b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 
+e2))))))))) (\lambda (H7: (eq T u v1)).(eq_ind T v1 (\lambda (t: T).((eq C 
+(CHead c0 (Bind b1) t) c2) \to ((csub3 g e1 c0) \to (ex2_3 B C T (\lambda 
+(b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) 
+v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 
+e2)))))))) (\lambda (H8: (eq C (CHead c0 (Bind b1) v1) c2)).(eq_ind C (CHead 
+c0 (Bind b1) v1) (\lambda (c: C).((csub3 g e1 c0) \to (ex2_3 B C T (\lambda 
+(b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c (CHead e2 (Bind b2) v2))))) 
+(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))))))) 
+(\lambda (H9: (csub3 g e1 c0)).(let H10 \def (eq_ind_r C c2 (\lambda (c: 
+C).(csub3 g (CHead e1 (Bind b1) v1) c)) H (CHead c0 (Bind b1) v1) H8) in 
+(ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C 
+(CHead c0 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda 
+(e2: C).(\lambda (_: T).(csub3 g e1 e2)))) b1 c0 v1 (refl_equal C (CHead c0 
+(Bind b1) v1)) H9))) c2 H8)) u (sym_eq T u v1 H7))) k (sym_eq K k (Bind b1) 
+H6))) c1 (sym_eq C c1 e1 H5))) H4)) H3)) H2 H0))) | (csub3_void c1 c0 H0 b H1 
+u1 u2) \Rightarrow (\lambda (H2: (eq C (CHead c1 (Bind Void) u1) (CHead e1 
+(Bind b1) v1))).(\lambda (H3: (eq C (CHead c0 (Bind b) u2) c2)).((let H4 \def 
+(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with 
+[(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c1 (Bind 
+Void) u1) (CHead e1 (Bind b1) v1) H2) in ((let H5 \def (f_equal C B (\lambda 
+(e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow 
+Void | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with 
+[(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Void])])) (CHead c1 (Bind 
+Void) u1) (CHead e1 (Bind b1) v1) H2) in ((let H6 \def (f_equal C C (\lambda 
+(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 
+| (CHead c _ _) \Rightarrow c])) (CHead c1 (Bind Void) u1) (CHead e1 (Bind 
+b1) v1) H2) in (eq_ind C e1 (\lambda (c: C).((eq B Void b1) \to ((eq T u1 v1) 
+\to ((eq C (CHead c0 (Bind b) u2) c2) \to ((csub3 g c c0) \to ((not (eq B b 
+Void)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: 
+T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
+C).(\lambda (_: T).(csub3 g e1 e2))))))))))) (\lambda (H7: (eq B Void 
+b1)).(eq_ind B Void (\lambda (_: B).((eq T u1 v1) \to ((eq C (CHead c0 (Bind 
+b) u2) c2) \to ((csub3 g e1 c0) \to ((not (eq B b Void)) \to (ex2_3 B C T 
+(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind 
+b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 
+e2)))))))))) (\lambda (H8: (eq T u1 v1)).(eq_ind T v1 (\lambda (_: T).((eq C 
+(CHead c0 (Bind b) u2) c2) \to ((csub3 g e1 c0) \to ((not (eq B b Void)) \to 
+(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 
+(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: 
+T).(csub3 g e1 e2))))))))) (\lambda (H9: (eq C (CHead c0 (Bind b) u2) 
+c2)).(eq_ind C (CHead c0 (Bind b) u2) (\lambda (c: C).((csub3 g e1 c0) \to 
+((not (eq B b Void)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: 
+C).(\lambda (v2: T).(eq C c (CHead e2 (Bind b2) v2))))) (\lambda (_: 
+B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2)))))))) (\lambda (H10: 
+(csub3 g e1 c0)).(\lambda (_: (not (eq B b Void))).(let H12 \def (eq_ind_r C 
+c2 (\lambda (c: C).(csub3 g (CHead e1 (Bind b1) v1) c)) H (CHead c0 (Bind b) 
+u2) H9) in (let H13 \def (eq_ind_r B b1 (\lambda (b0: B).(csub3 g (CHead e1 
+(Bind b0) v1) (CHead c0 (Bind b) u2))) H12 Void H7) in (ex2_3_intro B C T 
+(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c0 (Bind b) 
+u2) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: 
+T).(csub3 g e1 e2)))) b c0 u2 (refl_equal C (CHead c0 (Bind b) u2)) H10))))) 
+c2 H9)) u1 (sym_eq T u1 v1 H8))) b1 H7)) c1 (sym_eq C c1 e1 H6))) H5)) H4)) 
+H3 H0 H1))) | (csub3_abst c1 c0 H0 u t H1) \Rightarrow (\lambda (H2: (eq C 
+(CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1))).(\lambda (H3: (eq C (CHead 
+c0 (Bind Abbr) u) c2)).((let H4 \def (f_equal C T (\lambda (e: C).(match e in 
+C return (\lambda (_: C).T) with [(CSort _) \Rightarrow t | (CHead _ _ t0) 
+\Rightarrow t0])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H2) in 
+((let H5 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: 
+C).B) with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match k 
+in K return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) 
+\Rightarrow Abst])])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H2) in 
+((let H6 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: 
+C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead 
+c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H2) in (eq_ind C e1 (\lambda (c: 
+C).((eq B Abst b1) \to ((eq T t v1) \to ((eq C (CHead c0 (Bind Abbr) u) c2) 
+\to ((csub3 g c c0) \to ((ty3 g c0 u t) \to (ex2_3 B C T (\lambda (b2: 
+B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) 
+(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))))))))))) 
+(\lambda (H7: (eq B Abst b1)).(eq_ind B Abst (\lambda (_: B).((eq T t v1) \to 
+((eq C (CHead c0 (Bind Abbr) u) c2) \to ((csub3 g e1 c0) \to ((ty3 g c0 u t) 
+\to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 
+(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: 
+T).(csub3 g e1 e2)))))))))) (\lambda (H8: (eq T t v1)).(eq_ind T v1 (\lambda 
+(t0: T).((eq C (CHead c0 (Bind Abbr) u) c2) \to ((csub3 g e1 c0) \to ((ty3 g 
+c0 u t0) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: 
+T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
+C).(\lambda (_: T).(csub3 g e1 e2))))))))) (\lambda (H9: (eq C (CHead c0 
+(Bind Abbr) u) c2)).(eq_ind C (CHead c0 (Bind Abbr) u) (\lambda (c: 
+C).((csub3 g e1 c0) \to ((ty3 g c0 u v1) \to (ex2_3 B C T (\lambda (b2: 
+B).(\lambda (e2: C).(\lambda (v2: T).(eq C c (CHead e2 (Bind b2) v2))))) 
+(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2)))))))) 
+(\lambda (H10: (csub3 g e1 c0)).(\lambda (_: (ty3 g c0 u v1)).(let H12 \def 
+(eq_ind_r C c2 (\lambda (c: C).(csub3 g (CHead e1 (Bind b1) v1) c)) H (CHead 
+c0 (Bind Abbr) u) H9) in (let H13 \def (eq_ind_r B b1 (\lambda (b: B).(csub3 
+g (CHead e1 (Bind b) v1) (CHead c0 (Bind Abbr) u))) H12 Abst H7) in 
+(ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C 
+(CHead c0 (Bind Abbr) u) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda 
+(e2: C).(\lambda (_: T).(csub3 g e1 e2)))) Abbr c0 u (refl_equal C (CHead c0 
+(Bind Abbr) u)) H10))))) c2 H9)) t (sym_eq T t v1 H8))) b1 H7)) c1 (sym_eq C 
+c1 e1 H6))) H5)) H4)) H3 H0 H1)))]) in (H0 (refl_equal C (CHead e1 (Bind b1) 
+v1)) (refl_equal C c2)))))))).
+

diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/getl.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/getl.ma
new file mode 100644 (file)
index 0000000..68f1d82
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/getl.ma
@@ -0,0 +1,390 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/getl".
+
+include "csub3/fwd.ma".
+
+include "csub3/clear.ma".
+
+include "csub3/drop.ma".
+
+include "getl/clear.ma".
+
+theorem csub3_getl_abbr:
+ \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).(\forall 
+(n: nat).((getl n c1 (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csub3 g 
+c1 c2) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n 
+c2 (CHead d2 (Bind Abbr) u)))))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda 
+(n: nat).(\lambda (H: (getl n c1 (CHead d1 (Bind Abbr) u))).(let H0 \def 
+(getl_gen_all c1 (CHead d1 (Bind Abbr) u) n H) in (ex2_ind C (\lambda (e: 
+C).(drop n O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abbr) u))) 
+(\forall (c2: C).((csub3 g c1 c2) \to (ex2 C (\lambda (d2: C).(csub3 g d1 
+d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u)))))) (\lambda (x: 
+C).(\lambda (H1: (drop n O c1 x)).(\lambda (H2: (clear x (CHead d1 (Bind 
+Abbr) u))).((match x in C return (\lambda (c: C).((drop n O c1 c) \to ((clear 
+c (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csub3 g c1 c2) \to (ex2 C 
+(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind 
+Abbr) u))))))))) with [(CSort n0) \Rightarrow (\lambda (_: (drop n O c1 
+(CSort n0))).(\lambda (H4: (clear (CSort n0) (CHead d1 (Bind Abbr) 
+u))).(clear_gen_sort (CHead d1 (Bind Abbr) u) n0 H4 (\forall (c2: C).((csub3 
+g c1 c2) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl 
+n c2 (CHead d2 (Bind Abbr) u))))))))) | (CHead c k t) \Rightarrow (\lambda 
+(H3: (drop n O c1 (CHead c k t))).(\lambda (H4: (clear (CHead c k t) (CHead 
+d1 (Bind Abbr) u))).((match k in K return (\lambda (k0: K).((drop n O c1 
+(CHead c k0 t)) \to ((clear (CHead c k0 t) (CHead d1 (Bind Abbr) u)) \to 
+(\forall (c2: C).((csub3 g c1 c2) \to (ex2 C (\lambda (d2: C).(csub3 g d1 
+d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))))))))) with [(Bind 
+b) \Rightarrow (\lambda (H5: (drop n O c1 (CHead c (Bind b) t))).(\lambda 
+(H6: (clear (CHead c (Bind b) t) (CHead d1 (Bind Abbr) u))).(let H7 \def 
+(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with 
+[(CSort _) \Rightarrow d1 | (CHead c0 _ _) \Rightarrow c0])) (CHead d1 (Bind 
+Abbr) u) (CHead c (Bind b) t) (clear_gen_bind b c (CHead d1 (Bind Abbr) u) t 
+H6)) in ((let H8 \def (f_equal C B (\lambda (e: C).(match e in C return 
+(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) 
+\Rightarrow (match k0 in K return (\lambda (_: K).B) with [(Bind b0) 
+\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d1 (Bind Abbr) u) 
+(CHead c (Bind b) t) (clear_gen_bind b c (CHead d1 (Bind Abbr) u) t H6)) in 
+((let H9 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: 
+C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead 
+d1 (Bind Abbr) u) (CHead c (Bind b) t) (clear_gen_bind b c (CHead d1 (Bind 
+Abbr) u) t H6)) in (\lambda (H10: (eq B Abbr b)).(\lambda (H11: (eq C d1 
+c)).(\lambda (c2: C).(\lambda (H12: (csub3 g c1 c2)).(let H13 \def (eq_ind_r 
+T t (\lambda (t0: T).(drop n O c1 (CHead c (Bind b) t0))) H5 u H9) in (let 
+H14 \def (eq_ind_r B b (\lambda (b0: B).(drop n O c1 (CHead c (Bind b0) u))) 
+H13 Abbr H10) in (let H15 \def (eq_ind_r C c (\lambda (c0: C).(drop n O c1 
+(CHead c0 (Bind Abbr) u))) H14 d1 H11) in (ex2_ind C (\lambda (d2: C).(csub3 
+g d1 d2)) (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abbr) u))) (ex2 C 
+(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind 
+Abbr) u)))) (\lambda (x0: C).(\lambda (H16: (csub3 g d1 x0)).(\lambda (H17: 
+(drop n O c2 (CHead x0 (Bind Abbr) u))).(ex_intro2 C (\lambda (d2: C).(csub3 
+g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))) x0 H16 
+(getl_intro n c2 (CHead x0 (Bind Abbr) u) (CHead x0 (Bind Abbr) u) H17 
+(clear_bind Abbr x0 u)))))) (csub3_drop_abbr g n c1 c2 H12 d1 u H15)))))))))) 
+H8)) H7)))) | (Flat f) \Rightarrow (\lambda (H5: (drop n O c1 (CHead c (Flat 
+f) t))).(\lambda (H6: (clear (CHead c (Flat f) t) (CHead d1 (Bind Abbr) 
+u))).(let H7 \def H5 in (unintro C c1 (\lambda (c0: C).((drop n O c0 (CHead c 
+(Flat f) t)) \to (\forall (c2: C).((csub3 g c0 c2) \to (ex2 C (\lambda (d2: 
+C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) 
+u)))))))) (nat_ind (\lambda (n0: nat).(\forall (x0: C).((drop n0 O x0 (CHead 
+c (Flat f) t)) \to (\forall (c2: C).((csub3 g x0 c2) \to (ex2 C (\lambda (d2: 
+C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n0 c2 (CHead d2 (Bind Abbr) 
+u))))))))) (\lambda (x0: C).(\lambda (H8: (drop O O x0 (CHead c (Flat f) 
+t))).(\lambda (c2: C).(\lambda (H9: (csub3 g x0 c2)).(let H10 \def (eq_ind C 
+x0 (\lambda (c0: C).(csub3 g c0 c2)) H9 (CHead c (Flat f) t) (drop_gen_refl 
+x0 (CHead c (Flat f) t) H8)) in (let H_y \def (clear_flat c (CHead d1 (Bind 
+Abbr) u) (clear_gen_flat f c (CHead d1 (Bind Abbr) u) t H6) f t) in (let H11 
+\def (csub3_clear_conf g (CHead c (Flat f) t) c2 H10 (CHead d1 (Bind Abbr) u) 
+H_y) in (ex2_ind C (\lambda (e2: C).(csub3 g (CHead d1 (Bind Abbr) u) e2)) 
+(\lambda (e2: C).(clear c2 e2)) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) 
+(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (x1: 
+C).(\lambda (H12: (csub3 g (CHead d1 (Bind Abbr) u) x1)).(\lambda (H13: 
+(clear c2 x1)).(let H14 \def (csub3_gen_abbr g d1 x1 u H12) in (ex2_ind C 
+(\lambda (e2: C).(eq C x1 (CHead e2 (Bind Abbr) u))) (\lambda (e2: C).(csub3 
+g d1 e2)) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl O 
+c2 (CHead d2 (Bind Abbr) u)))) (\lambda (x2: C).(\lambda (H15: (eq C x1 
+(CHead x2 (Bind Abbr) u))).(\lambda (H16: (csub3 g d1 x2)).(let H17 \def 
+(eq_ind C x1 (\lambda (c0: C).(clear c2 c0)) H13 (CHead x2 (Bind Abbr) u) 
+H15) in (ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl 
+O c2 (CHead d2 (Bind Abbr) u))) x2 H16 (getl_intro O c2 (CHead x2 (Bind Abbr) 
+u) c2 (drop_refl c2) H17)))))) H14))))) H11)))))))) (\lambda (n0: 
+nat).(\lambda (H8: ((\forall (x0: C).((drop n0 O x0 (CHead c (Flat f) t)) \to 
+(\forall (c2: C).((csub3 g x0 c2) \to (ex2 C (\lambda (d2: C).(csub3 g d1 
+d2)) (\lambda (d2: C).(getl n0 c2 (CHead d2 (Bind Abbr) u)))))))))).(\lambda 
+(x0: C).(\lambda (H9: (drop (S n0) O x0 (CHead c (Flat f) t))).(\lambda (c2: 
+C).(\lambda (H10: (csub3 g x0 c2)).(let H11 \def (drop_clear x0 (CHead c 
+(Flat f) t) n0 H9) in (ex2_3_ind B C T (\lambda (b: B).(\lambda (e: 
+C).(\lambda (v: T).(clear x0 (CHead e (Bind b) v))))) (\lambda (_: 
+B).(\lambda (e: C).(\lambda (_: T).(drop n0 O e (CHead c (Flat f) t))))) (ex2 
+C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead 
+d2 (Bind Abbr) u)))) (\lambda (x1: B).(\lambda (x2: C).(\lambda (x3: 
+T).(\lambda (H12: (clear x0 (CHead x2 (Bind x1) x3))).(\lambda (H13: (drop n0 
+O x2 (CHead c (Flat f) t))).(let H14 \def (csub3_clear_conf g x0 c2 H10 
+(CHead x2 (Bind x1) x3) H12) in (ex2_ind C (\lambda (e2: C).(csub3 g (CHead 
+x2 (Bind x1) x3) e2)) (\lambda (e2: C).(clear c2 e2)) (ex2 C (\lambda (d2: 
+C).(csub3 g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) 
+u)))) (\lambda (x4: C).(\lambda (H15: (csub3 g (CHead x2 (Bind x1) x3) 
+x4)).(\lambda (H16: (clear c2 x4)).(let H17 \def (csub3_gen_bind g x1 x2 x4 
+x3 H15) in (ex2_3_ind B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: 
+T).(eq C x4 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
+C).(\lambda (_: T).(csub3 g x2 e2)))) (ex2 C (\lambda (d2: C).(csub3 g d1 
+d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda 
+(x5: B).(\lambda (x6: C).(\lambda (x7: T).(\lambda (H18: (eq C x4 (CHead x6 
+(Bind x5) x7))).(\lambda (H19: (csub3 g x2 x6)).(let H20 \def (eq_ind C x4 
+(\lambda (c0: C).(clear c2 c0)) H16 (CHead x6 (Bind x5) x7) H18) in (let H21 
+\def (H8 x2 H13 x6 H19) in (ex2_ind C (\lambda (d2: C).(csub3 g d1 d2)) 
+(\lambda (d2: C).(getl n0 x6 (CHead d2 (Bind Abbr) u))) (ex2 C (\lambda (d2: 
+C).(csub3 g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) 
+u)))) (\lambda (x8: C).(\lambda (H22: (csub3 g d1 x8)).(\lambda (H23: (getl 
+n0 x6 (CHead x8 (Bind Abbr) u))).(ex_intro2 C (\lambda (d2: C).(csub3 g d1 
+d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) u))) x8 H22 
+(getl_clear_bind x5 c2 x6 x7 H20 (CHead x8 (Bind Abbr) u) n0 H23))))) 
+H21)))))))) H17))))) H14))))))) H11)))))))) n) H7))))]) H3 H4)))]) H1 H2)))) 
+H0))))))).
+
+theorem csub3_getl_abst:
+ \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (t: T).(\forall 
+(n: nat).((getl n c1 (CHead d1 (Bind Abst) t)) \to (\forall (c2: C).((csub3 g 
+c1 c2) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: 
+C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
+C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n 
+c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u 
+t))))))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (t: T).(\lambda 
+(n: nat).(\lambda (H: (getl n c1 (CHead d1 (Bind Abst) t))).(let H0 \def 
+(getl_gen_all c1 (CHead d1 (Bind Abst) t) n H) in (ex2_ind C (\lambda (e: 
+C).(drop n O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abst) t))) 
+(\forall (c2: C).((csub3 g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 
+d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T 
+(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda 
+(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: 
+T).(ty3 g d2 u t))))))) (\lambda (x: C).(\lambda (H1: (drop n O c1 
+x)).(\lambda (H2: (clear x (CHead d1 (Bind Abst) t))).((match x in C return 
+(\lambda (c: C).((drop n O c1 c) \to ((clear c (CHead d1 (Bind Abst) t)) \to 
+(\forall (c2: C).((csub3 g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 
+d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T 
+(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda 
+(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: 
+T).(ty3 g d2 u t)))))))))) with [(CSort n0) \Rightarrow (\lambda (_: (drop n 
+O c1 (CSort n0))).(\lambda (H4: (clear (CSort n0) (CHead d1 (Bind Abst) 
+t))).(clear_gen_sort (CHead d1 (Bind Abst) t) n0 H4 (\forall (c2: C).((csub3 
+g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: 
+C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
+C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n 
+c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u 
+t)))))))))) | (CHead c k t0) \Rightarrow (\lambda (H3: (drop n O c1 (CHead c 
+k t0))).(\lambda (H4: (clear (CHead c k t0) (CHead d1 (Bind Abst) 
+t))).((match k in K return (\lambda (k0: K).((drop n O c1 (CHead c k0 t0)) 
+\to ((clear (CHead c k0 t0) (CHead d1 (Bind Abst) t)) \to (\forall (c2: 
+C).((csub3 g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
+(d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
+C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n 
+c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u 
+t)))))))))) with [(Bind b) \Rightarrow (\lambda (H5: (drop n O c1 (CHead c 
+(Bind b) t0))).(\lambda (H6: (clear (CHead c (Bind b) t0) (CHead d1 (Bind 
+Abst) t))).(let H7 \def (f_equal C C (\lambda (e: C).(match e in C return 
+(\lambda (_: C).C) with [(CSort _) \Rightarrow d1 | (CHead c0 _ _) 
+\Rightarrow c0])) (CHead d1 (Bind Abst) t) (CHead c (Bind b) t0) 
+(clear_gen_bind b c (CHead d1 (Bind Abst) t) t0 H6)) in ((let H8 \def 
+(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with 
+[(CSort _) \Rightarrow Abst | (CHead _ k0 _) \Rightarrow (match k0 in K 
+return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) 
+\Rightarrow Abst])])) (CHead d1 (Bind Abst) t) (CHead c (Bind b) t0) 
+(clear_gen_bind b c (CHead d1 (Bind Abst) t) t0 H6)) in ((let H9 \def 
+(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with 
+[(CSort _) \Rightarrow t | (CHead _ _ t1) \Rightarrow t1])) (CHead d1 (Bind 
+Abst) t) (CHead c (Bind b) t0) (clear_gen_bind b c (CHead d1 (Bind Abst) t) 
+t0 H6)) in (\lambda (H10: (eq B Abst b)).(\lambda (H11: (eq C d1 c)).(\lambda 
+(c2: C).(\lambda (H12: (csub3 g c1 c2)).(let H13 \def (eq_ind_r T t0 (\lambda 
+(t1: T).(drop n O c1 (CHead c (Bind b) t1))) H5 t H9) in (let H14 \def 
+(eq_ind_r B b (\lambda (b0: B).(drop n O c1 (CHead c (Bind b0) t))) H13 Abst 
+H10) in (let H15 \def (eq_ind_r C c (\lambda (c0: C).(drop n O c1 (CHead c0 
+(Bind Abst) t))) H14 d1 H11) in (or_ind (ex2 C (\lambda (d2: C).(csub3 g d1 
+d2)) (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T 
+(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda 
+(u: T).(drop n O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: 
+T).(ty3 g d2 u t)))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
+(d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
+C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n 
+c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u 
+t))))) (\lambda (H16: (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: 
+C).(drop n O c2 (CHead d2 (Bind Abst) t))))).(ex2_ind C (\lambda (d2: 
+C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abst) t))) 
+(or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 
+(CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
+T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 
+(Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) 
+(\lambda (x0: C).(\lambda (H17: (csub3 g d1 x0)).(\lambda (H18: (drop n O c2 
+(CHead x0 (Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: C).(csub3 g d1 
+d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T 
+(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda 
+(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: 
+T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
+(d2: C).(getl n c2 (CHead d2 (Bind Abst) t))) x0 H17 (getl_intro n c2 (CHead 
+x0 (Bind Abst) t) (CHead x0 (Bind Abst) t) H18 (clear_bind Abst x0 t))))))) 
+H16)) (\lambda (H16: (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 
+d2))) (\lambda (d2: C).(\lambda (u: T).(drop n O c2 (CHead d2 (Bind Abbr) 
+u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))).(ex3_2_ind C T 
+(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda 
+(u: T).(drop n O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: 
+T).(ty3 g d2 u t))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
+(d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
+C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n 
+c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u 
+t))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H17: (csub3 g d1 
+x0)).(\lambda (H18: (drop n O c2 (CHead x0 (Bind Abbr) x1))).(\lambda (H19: 
+(ty3 g x0 x1 t)).(or_intror (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
+(d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
+C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n 
+c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u 
+t)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) 
+(\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) 
+(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))) x0 x1 H17 (getl_intro n c2 
+(CHead x0 (Bind Abbr) x1) (CHead x0 (Bind Abbr) x1) H18 (clear_bind Abbr x0 
+x1)) H19))))))) H16)) (csub3_drop_abst g n c1 c2 H12 d1 t H15)))))))))) H8)) 
+H7)))) | (Flat f) \Rightarrow (\lambda (H5: (drop n O c1 (CHead c (Flat f) 
+t0))).(\lambda (H6: (clear (CHead c (Flat f) t0) (CHead d1 (Bind Abst) 
+t))).(let H7 \def H5 in (unintro C c1 (\lambda (c0: C).((drop n O c0 (CHead c 
+(Flat f) t0)) \to (\forall (c2: C).((csub3 g c0 c2) \to (or (ex2 C (\lambda 
+(d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) 
+t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda 
+(d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: 
+C).(\lambda (u: T).(ty3 g d2 u t))))))))) (nat_ind (\lambda (n0: 
+nat).(\forall (x0: C).((drop n0 O x0 (CHead c (Flat f) t0)) \to (\forall (c2: 
+C).((csub3 g x0 c2) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
+(d2: C).(getl n0 c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
+C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl 
+n0 c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 
+u t)))))))))) (\lambda (x0: C).(\lambda (H8: (drop O O x0 (CHead c (Flat f) 
+t0))).(\lambda (c2: C).(\lambda (H9: (csub3 g x0 c2)).(let H10 \def (eq_ind C 
+x0 (\lambda (c0: C).(csub3 g c0 c2)) H9 (CHead c (Flat f) t0) (drop_gen_refl 
+x0 (CHead c (Flat f) t0) H8)) in (let H_y \def (clear_flat c (CHead d1 (Bind 
+Abst) t) (clear_gen_flat f c (CHead d1 (Bind Abst) t) t0 H6) f t0) in (let 
+H11 \def (csub3_clear_conf g (CHead c (Flat f) t0) c2 H10 (CHead d1 (Bind 
+Abst) t) H_y) in (ex2_ind C (\lambda (e2: C).(csub3 g (CHead d1 (Bind Abst) 
+t) e2)) (\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(csub3 g 
+d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T 
+(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda 
+(u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: 
+T).(ty3 g d2 u t))))) (\lambda (x1: C).(\lambda (H12: (csub3 g (CHead d1 
+(Bind Abst) t) x1)).(\lambda (H13: (clear c2 x1)).(let H14 \def 
+(csub3_gen_abst g d1 x1 t H12) in (or_ind (ex2 C (\lambda (e2: C).(eq C x1 
+(CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csub3 g d1 e2))) (ex3_2 C T 
+(\lambda (e2: C).(\lambda (v2: T).(eq C x1 (CHead e2 (Bind Abbr) v2)))) 
+(\lambda (e2: C).(\lambda (_: T).(csub3 g d1 e2))) (\lambda (e2: C).(\lambda 
+(v2: T).(ty3 g e2 v2 t)))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) 
+(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda 
+(d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: 
+T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: 
+T).(ty3 g d2 u t))))) (\lambda (H15: (ex2 C (\lambda (e2: C).(eq C x1 (CHead 
+e2 (Bind Abst) t))) (\lambda (e2: C).(csub3 g d1 e2)))).(ex2_ind C (\lambda 
+(e2: C).(eq C x1 (CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csub3 g d1 e2)) 
+(or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl O c2 
+(CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
+T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 
+(Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) 
+(\lambda (x2: C).(\lambda (H16: (eq C x1 (CHead x2 (Bind Abst) t))).(\lambda 
+(H17: (csub3 g d1 x2)).(let H18 \def (eq_ind C x1 (\lambda (c0: C).(clear c2 
+c0)) H13 (CHead x2 (Bind Abst) t) H16) in (or_introl (ex2 C (\lambda (d2: 
+C).(csub3 g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) 
+(ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: 
+C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: 
+C).(\lambda (u: T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csub3 g 
+d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t))) x2 H17 
+(getl_intro O c2 (CHead x2 (Bind Abst) t) c2 (drop_refl c2) H18))))))) H15)) 
+(\lambda (H15: (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C x1 (CHead 
+e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g d1 e2))) 
+(\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 t))))).(ex3_2_ind C T (\lambda 
+(e2: C).(\lambda (v2: T).(eq C x1 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: 
+C).(\lambda (_: T).(csub3 g d1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g 
+e2 v2 t))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: 
+C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
+C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O 
+c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u 
+t))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (H16: (eq C x1 (CHead x2 
+(Bind Abbr) x3))).(\lambda (H17: (csub3 g d1 x2)).(\lambda (H18: (ty3 g x2 x3 
+t)).(let H19 \def (eq_ind C x1 (\lambda (c0: C).(clear c2 c0)) H13 (CHead x2 
+(Bind Abbr) x3) H16) in (or_intror (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) 
+(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda 
+(d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: 
+T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: 
+T).(ty3 g d2 u t)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csub3 
+g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) 
+u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))) x2 x3 H17 (getl_intro 
+O c2 (CHead x2 (Bind Abbr) x3) c2 (drop_refl c2) H19) H18)))))))) H15)) 
+H14))))) H11)))))))) (\lambda (n0: nat).(\lambda (H8: ((\forall (x0: 
+C).((drop n0 O x0 (CHead c (Flat f) t0)) \to (\forall (c2: C).((csub3 g x0 
+c2) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl 
+n0 c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
+T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n0 c2 (CHead d2 
+(Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u 
+t))))))))))).(\lambda (x0: C).(\lambda (H9: (drop (S n0) O x0 (CHead c (Flat 
+f) t0))).(\lambda (c2: C).(\lambda (H10: (csub3 g x0 c2)).(let H11 \def 
+(drop_clear x0 (CHead c (Flat f) t0) n0 H9) in (ex2_3_ind B C T (\lambda (b: 
+B).(\lambda (e: C).(\lambda (v: T).(clear x0 (CHead e (Bind b) v))))) 
+(\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop n0 O e (CHead c (Flat 
+f) t0))))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: 
+C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
+C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl 
+(S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g 
+d2 u t))))) (\lambda (x1: B).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H12: 
+(clear x0 (CHead x2 (Bind x1) x3))).(\lambda (H13: (drop n0 O x2 (CHead c 
+(Flat f) t0))).(let H14 \def (csub3_clear_conf g x0 c2 H10 (CHead x2 (Bind 
+x1) x3) H12) in (ex2_ind C (\lambda (e2: C).(csub3 g (CHead x2 (Bind x1) x3) 
+e2)) (\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(csub3 g d1 
+d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T 
+(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda 
+(u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda 
+(u: T).(ty3 g d2 u t))))) (\lambda (x4: C).(\lambda (H15: (csub3 g (CHead x2 
+(Bind x1) x3) x4)).(\lambda (H16: (clear c2 x4)).(let H17 \def 
+(csub3_gen_bind g x1 x2 x4 x3 H15) in (ex2_3_ind B C T (\lambda (b2: 
+B).(\lambda (e2: C).(\lambda (v2: T).(eq C x4 (CHead e2 (Bind b2) v2))))) 
+(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g x2 e2)))) (or (ex2 
+C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead 
+d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 
+d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) 
+u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x5: 
+B).(\lambda (x6: C).(\lambda (x7: T).(\lambda (H18: (eq C x4 (CHead x6 (Bind 
+x5) x7))).(\lambda (H19: (csub3 g x2 x6)).(let H20 \def (eq_ind C x4 (\lambda 
+(c0: C).(clear c2 c0)) H16 (CHead x6 (Bind x5) x7) H18) in (let H21 \def (H8 
+x2 H13 x6 H19) in (or_ind (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
+(d2: C).(getl n0 x6 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
+C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl 
+n0 x6 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 
+u t)))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl 
+(S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda 
+(_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 
+(CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u 
+t))))) (\lambda (H22: (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: 
+C).(getl n0 x6 (CHead d2 (Bind Abst) t))))).(ex2_ind C (\lambda (d2: 
+C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n0 x6 (CHead d2 (Bind Abst) t))) 
+(or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 
+(CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
+T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead 
+d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) 
+(\lambda (x8: C).(\lambda (H23: (csub3 g d1 x8)).(\lambda (H24: (getl n0 x6 
+(CHead x8 (Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: C).(csub3 g d1 
+d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T 
+(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda 
+(u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda 
+(u: T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) 
+(\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t))) x8 H23 
+(getl_clear_bind x5 c2 x6 x7 H20 (CHead x8 (Bind Abst) t) n0 H24)))))) H22)) 
+(\lambda (H22: (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) 
+(\lambda (d2: C).(\lambda (u: T).(getl n0 x6 (CHead d2 (Bind Abbr) u)))) 
+(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))).(ex3_2_ind C T (\lambda 
+(d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: 
+T).(getl n0 x6 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: 
+T).(ty3 g d2 u t))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
+(d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
+C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl 
+(S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g 
+d2 u t))))) (\lambda (x8: C).(\lambda (x9: T).(\lambda (H23: (csub3 g d1 
+x8)).(\lambda (H24: (getl n0 x6 (CHead x8 (Bind Abbr) x9))).(\lambda (H25: 
+(ty3 g x8 x9 t)).(or_intror (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
+(d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
+C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl 
+(S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g 
+d2 u t)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 
+d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) 
+u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))) x8 x9 H23 
+(getl_clear_bind x5 c2 x6 x7 H20 (CHead x8 (Bind Abbr) x9) n0 H24) H25))))))) 
+H22)) H21)))))))) H17))))) H14))))))) H11)))))))) n) H7))))]) H3 H4)))]) H1 
+H2)))) H0))))))).
+

diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/pc3.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/pc3.ma
new file mode 100644 (file)
index 0000000..9806725
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/pc3.ma
@@ -0,0 +1,58 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/pc3".
+
+include "csub3/getl.ma".
+
+include "pc3/left.ma".
+
+theorem csub3_pr2:
+ \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pr2 c1 
+t1 t2) \to (\forall (c2: C).((csub3 g c1 c2) \to (pr2 c2 t1 t2)))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
+(H: (pr2 c1 t1 t2)).(pr2_ind (\lambda (c: C).(\lambda (t: T).(\lambda (t0: 
+T).(\forall (c2: C).((csub3 g c c2) \to (pr2 c2 t t0)))))) (\lambda (c: 
+C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda (c2: 
+C).(\lambda (_: (csub3 g c c2)).(pr2_free c2 t3 t4 H0))))))) (\lambda (c: 
+C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c 
+(CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: 
+(pr0 t3 t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (c2: 
+C).(\lambda (H3: (csub3 g c c2)).(let H4 \def (csub3_getl_abbr g c d u i H0 
+c2 H3) in (ex2_ind C (\lambda (d2: C).(csub3 g d d2)) (\lambda (d2: C).(getl 
+i c2 (CHead d2 (Bind Abbr) u))) (pr2 c2 t3 t) (\lambda (x: C).(\lambda (_: 
+(csub3 g d x)).(\lambda (H6: (getl i c2 (CHead x (Bind Abbr) u))).(pr2_delta 
+c2 x u i H6 t3 t4 H1 t H2)))) H4)))))))))))))) c1 t1 t2 H))))).
+
+theorem csub3_pc3:
+ \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pc3 c1 
+t1 t2) \to (\forall (c2: C).((csub3 g c1 c2) \to (pc3 c2 t1 t2)))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
+(H: (pc3 c1 t1 t2)).(pc3_ind_left c1 (\lambda (t: T).(\lambda (t0: 
+T).(\forall (c2: C).((csub3 g c1 c2) \to (pc3 c2 t t0))))) (\lambda (t: 
+T).(\lambda (c2: C).(\lambda (_: (csub3 g c1 c2)).(pc3_refl c2 t)))) (\lambda 
+(t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c1 t0 t3)).(\lambda (t4: 
+T).(\lambda (_: (pc3 c1 t3 t4)).(\lambda (H2: ((\forall (c2: C).((csub3 g c1 
+c2) \to (pc3 c2 t3 t4))))).(\lambda (c2: C).(\lambda (H3: (csub3 g c1 
+c2)).(pc3_pr2_u c2 t3 t0 (csub3_pr2 g c1 t0 t3 H0 c2 H3) t4 (H2 c2 
+H3)))))))))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c1 t0 
+t3)).(\lambda (t4: T).(\lambda (_: (pc3 c1 t0 t4)).(\lambda (H2: ((\forall 
+(c2: C).((csub3 g c1 c2) \to (pc3 c2 t0 t4))))).(\lambda (c2: C).(\lambda 
+(H3: (csub3 g c1 c2)).(pc3_t t0 c2 t3 (pc3_pr2_x c2 t3 t0 (csub3_pr2 g c1 t0 
+t3 H0 c2 H3)) t4 (H2 c2 H3)))))))))) t1 t2 H))))).
+

diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/props.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/props.ma
new file mode 100644 (file)
index 0000000..3258cda
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/props.ma
@@ -0,0 +1,27 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/props".
+
+include "csub3/defs.ma".
+
+theorem csub3_refl:
+ \forall (g: G).(\forall (c: C).(csub3 g c c))
+\def
+ \lambda (g: G).(\lambda (c: C).(C_ind (\lambda (c0: C).(csub3 g c0 c0)) 
+(\lambda (n: nat).(csub3_sort g n)) (\lambda (c0: C).(\lambda (H: (csub3 g c0 
+c0)).(\lambda (k: K).(\lambda (t: T).(csub3_head g c0 c0 H k t))))) c)).
+

diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/ty3.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/ty3.ma
new file mode 100644 (file)
index 0000000..26ad14f
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/ty3.ma
@@ -0,0 +1,99 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/ty3".
+
+include "csub3/pc3.ma".
+
+include "csub3/props.ma".
+
+theorem csub3_ty3:
+ \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c1 
+t1 t2) \to (\forall (c2: C).((csub3 g c1 c2) \to (ty3 g c2 t1 t2)))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
+(H: (ty3 g c1 t1 t2)).(ty3_ind g (\lambda (c: C).(\lambda (t: T).(\lambda 
+(t0: T).(\forall (c2: C).((csub3 g c c2) \to (ty3 g c2 t t0)))))) (\lambda 
+(c: C).(\lambda (t0: T).(\lambda (t: T).(\lambda (_: (ty3 g c t0 t)).(\lambda 
+(H1: ((\forall (c2: C).((csub3 g c c2) \to (ty3 g c2 t0 t))))).(\lambda (u: 
+T).(\lambda (t3: T).(\lambda (_: (ty3 g c u t3)).(\lambda (H3: ((\forall (c2: 
+C).((csub3 g c c2) \to (ty3 g c2 u t3))))).(\lambda (H4: (pc3 c t3 
+t0)).(\lambda (c2: C).(\lambda (H5: (csub3 g c c2)).(ty3_conv g c2 t0 t (H1 
+c2 H5) u t3 (H3 c2 H5) (csub3_pc3 g c t3 t0 H4 c2 H5)))))))))))))) (\lambda 
+(c: C).(\lambda (m: nat).(\lambda (c2: C).(\lambda (_: (csub3 g c 
+c2)).(ty3_sort g c2 m))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d: 
+C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind Abbr) u))).(\lambda 
+(t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: ((\forall (c2: C).((csub3 g 
+d c2) \to (ty3 g c2 u t))))).(\lambda (c2: C).(\lambda (H3: (csub3 g c 
+c2)).(let H4 \def (csub3_getl_abbr g c d u n H0 c2 H3) in (ex2_ind C (\lambda 
+(d2: C).(csub3 g d d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) 
+u))) (ty3 g c2 (TLRef n) (lift (S n) O t)) (\lambda (x: C).(\lambda (H5: 
+(csub3 g d x)).(\lambda (H6: (getl n c2 (CHead x (Bind Abbr) u))).(ty3_abbr g 
+n c2 x u H6 t (H2 x H5))))) H4)))))))))))) (\lambda (n: nat).(\lambda (c: 
+C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind 
+Abst) u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: 
+((\forall (c2: C).((csub3 g d c2) \to (ty3 g c2 u t))))).(\lambda (c2: 
+C).(\lambda (H3: (csub3 g c c2)).(let H4 \def (csub3_getl_abst g c d u n H0 
+c2 H3) in (or_ind (ex2 C (\lambda (d2: C).(csub3 g d d2)) (\lambda (d2: 
+C).(getl n c2 (CHead d2 (Bind Abst) u)))) (ex3_2 C T (\lambda (d2: 
+C).(\lambda (_: T).(csub3 g d d2))) (\lambda (d2: C).(\lambda (u0: T).(getl n 
+c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 
+u0 u)))) (ty3 g c2 (TLRef n) (lift (S n) O u)) (\lambda (H5: (ex2 C (\lambda 
+(d2: C).(csub3 g d d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) 
+u))))).(ex2_ind C (\lambda (d2: C).(csub3 g d d2)) (\lambda (d2: C).(getl n 
+c2 (CHead d2 (Bind Abst) u))) (ty3 g c2 (TLRef n) (lift (S n) O u)) (\lambda 
+(x: C).(\lambda (H6: (csub3 g d x)).(\lambda (H7: (getl n c2 (CHead x (Bind 
+Abst) u))).(ty3_abst g n c2 x u H7 t (H2 x H6))))) H5)) (\lambda (H5: (ex3_2 
+C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d d2))) (\lambda (d2: 
+C).(\lambda (u0: T).(getl n c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: 
+C).(\lambda (u0: T).(ty3 g d2 u0 u))))).(ex3_2_ind C T (\lambda (d2: 
+C).(\lambda (_: T).(csub3 g d d2))) (\lambda (d2: C).(\lambda (u0: T).(getl n 
+c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 
+u0 u))) (ty3 g c2 (TLRef n) (lift (S n) O u)) (\lambda (x0: C).(\lambda (x1: 
+T).(\lambda (_: (csub3 g d x0)).(\lambda (H7: (getl n c2 (CHead x0 (Bind 
+Abbr) x1))).(\lambda (H8: (ty3 g x0 x1 u)).(ty3_abbr g n c2 x0 x1 H7 u 
+H8)))))) H5)) H4)))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (t: 
+T).(\lambda (_: (ty3 g c u t)).(\lambda (H1: ((\forall (c2: C).((csub3 g c 
+c2) \to (ty3 g c2 u t))))).(\lambda (b: B).(\lambda (t0: T).(\lambda (t3: 
+T).(\lambda (_: (ty3 g (CHead c (Bind b) u) t0 t3)).(\lambda (H3: ((\forall 
+(c2: C).((csub3 g (CHead c (Bind b) u) c2) \to (ty3 g c2 t0 t3))))).(\lambda 
+(t4: T).(\lambda (_: (ty3 g (CHead c (Bind b) u) t3 t4)).(\lambda (H5: 
+((\forall (c2: C).((csub3 g (CHead c (Bind b) u) c2) \to (ty3 g c2 t3 
+t4))))).(\lambda (c2: C).(\lambda (H6: (csub3 g c c2)).(ty3_bind g c2 u t (H1 
+c2 H6) b t0 t3 (H3 (CHead c2 (Bind b) u) (csub3_head g c c2 H6 (Bind b) u)) 
+t4 (H5 (CHead c2 (Bind b) u) (csub3_head g c c2 H6 (Bind b) 
+u)))))))))))))))))) (\lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\lambda 
+(_: (ty3 g c w u)).(\lambda (H1: ((\forall (c2: C).((csub3 g c c2) \to (ty3 g 
+c2 w u))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c v (THead 
+(Bind Abst) u t))).(\lambda (H3: ((\forall (c2: C).((csub3 g c c2) \to (ty3 g 
+c2 v (THead (Bind Abst) u t)))))).(\lambda (c2: C).(\lambda (H4: (csub3 g c 
+c2)).(ty3_appl g c2 w u (H1 c2 H4) v t (H3 c2 H4))))))))))))) (\lambda (c: 
+C).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (ty3 g c t0 t3)).(\lambda 
+(H1: ((\forall (c2: C).((csub3 g c c2) \to (ty3 g c2 t0 t3))))).(\lambda (t4: 
+T).(\lambda (_: (ty3 g c t3 t4)).(\lambda (H3: ((\forall (c2: C).((csub3 g c 
+c2) \to (ty3 g c2 t3 t4))))).(\lambda (c2: C).(\lambda (H4: (csub3 g c 
+c2)).(ty3_cast g c2 t0 t3 (H1 c2 H4) t4 (H3 c2 H4)))))))))))) c1 t1 t2 H))))).
+
+theorem csub3_ty3_ld:
+ \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (v: T).((ty3 g c u 
+v) \to (\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c (Bind Abst) v) t1 
+t2) \to (ty3 g (CHead c (Bind Abbr) u) t1 t2))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (v: T).(\lambda (H: 
+(ty3 g c u v)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (ty3 g (CHead 
+c (Bind Abst) v) t1 t2)).(csub3_ty3 g (CHead c (Bind Abst) v) t1 t2 H0 (CHead 
+c (Bind Abbr) u) (csub3_abst g c c (csub3_refl g c) u v H))))))))).
+

diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/nf2/fwd.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/nf2/fwd.ma
index a9ad986e1151db9bdf2151a7c98a3c6698dbed3e..d9081db2a5b979d9400a719eaa19a7b8654e86ee 100644 (file)
--- a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/nf2/fwd.ma
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/nf2/fwd.ma
@@ -20,38 +20,7 @@ include "nf2/defs.ma".
 
 include "pr2/clen.ma".
 
-theorem nf2_gen_base__aux:
- \forall (k: K).(\forall (t: T).(\forall (u: T).((eq T (THead k u t) t) \to 
-(\forall (P: Prop).P))))
-\def
- \lambda (k: K).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (u: T).((eq 
-T (THead k u t0) t0) \to (\forall (P: Prop).P)))) (\lambda (n: nat).(\lambda 
-(u: T).(\lambda (H: (eq T (THead k u (TSort n)) (TSort n))).(\lambda (P: 
-Prop).(let H0 \def (eq_ind T (THead k u (TSort n)) (\lambda (ee: T).(match ee 
-in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef 
-_) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H) in 
-(False_ind P H0)))))) (\lambda (n: nat).(\lambda (u: T).(\lambda (H: (eq T 
-(THead k u (TLRef n)) (TLRef n))).(\lambda (P: Prop).(let H0 \def (eq_ind T 
-(THead k u (TLRef n)) (\lambda (ee: T).(match ee in T return (\lambda (_: 
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
-(THead _ _ _) \Rightarrow True])) I (TLRef n) H) in (False_ind P H0)))))) 
-(\lambda (k0: K).(\lambda (t0: T).(\lambda (_: ((\forall (u: T).((eq T (THead 
-k u t0) t0) \to (\forall (P: Prop).P))))).(\lambda (t1: T).(\lambda (H0: 
-((\forall (u: T).((eq T (THead k u t1) t1) \to (\forall (P: 
-Prop).P))))).(\lambda (u: T).(\lambda (H1: (eq T (THead k u (THead k0 t0 t1)) 
-(THead k0 t0 t1))).(\lambda (P: Prop).(let H2 \def (f_equal T K (\lambda (e: 
-T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | 
-(TLRef _) \Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k u (THead 
-k0 t0 t1)) (THead k0 t0 t1) H1) in ((let H3 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | 
-(TLRef _) \Rightarrow u | (THead _ t2 _) \Rightarrow t2])) (THead k u (THead 
-k0 t0 t1)) (THead k0 t0 t1) H1) in ((let H4 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow (THead 
-k0 t0 t1) | (TLRef _) \Rightarrow (THead k0 t0 t1) | (THead _ _ t2) 
-\Rightarrow t2])) (THead k u (THead k0 t0 t1)) (THead k0 t0 t1) H1) in 
-(\lambda (_: (eq T u t0)).(\lambda (H6: (eq K k k0)).(let H7 \def (eq_ind K k 
-(\lambda (k1: K).(\forall (u0: T).((eq T (THead k1 u0 t1) t1) \to (\forall 
-(P0: Prop).P0)))) H0 k0 H6) in (H7 t0 H4 P))))) H3)) H2)))))))))) t)).
+include "T/props.ma".
 
 theorem nf2_gen_lref:
  \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c 
@@ -92,7 +61,7 @@ theorem nf2_gen_cast:
 t)) \to (\forall (P: Prop).P))))
 \def
  \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (nf2 c (THead 
-(Flat Cast) u t))).(\lambda (P: Prop).(nf2_gen_base__aux (Flat Cast) t u (H t 
+(Flat Cast) u t))).(\lambda (P: Prop).(thead_x_y_y (Flat Cast) u t (H t 
 (pr2_free c (THead (Flat Cast) u t) t (pr0_epsilon t t (pr0_refl t) u))) 
 P))))).
 

diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/dec.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/dec.ma
new file mode 100644 (file)
index 0000000..761030b
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/dec.ma
@@ -0,0 +1,153 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/dec".
+
+include "ty3/arity_props.ma".
+
+include "ty3/pr3.ma".
+
+include "nf2/fwd.ma".
+
+theorem pc3_dec:
+ \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (t1: T).((ty3 g c 
+u1 t1) \to (\forall (u2: T).(\forall (t2: T).((ty3 g c u2 t2) \to (or (pc3 c 
+u1 u2) ((pc3 c u1 u2) \to (\forall (P: Prop).P))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda 
+(H: (ty3 g c u1 t1)).(\lambda (u2: T).(\lambda (t2: T).(\lambda (H0: (ty3 g c 
+u2 t2)).(let H_y \def (ty3_sn3 g c u1 t1 H) in (let H_y0 \def (ty3_sn3 g c u2 
+t2 H0) in (let H_x \def (nf2_sn3 c u1 H_y) in (let H1 \def H_x in (ex2_ind T 
+(\lambda (u: T).(pr3 c u1 u)) (\lambda (u: T).(nf2 c u)) (or (pc3 c u1 u2) 
+((pc3 c u1 u2) \to (\forall (P: Prop).P))) (\lambda (x: T).(\lambda (H2: (pr3 
+c u1 x)).(\lambda (H3: (nf2 c x)).(let H_x0 \def (nf2_sn3 c u2 H_y0) in (let 
+H4 \def H_x0 in (ex2_ind T (\lambda (u: T).(pr3 c u2 u)) (\lambda (u: T).(nf2 
+c u)) (or (pc3 c u1 u2) ((pc3 c u1 u2) \to (\forall (P: Prop).P))) (\lambda 
+(x0: T).(\lambda (H5: (pr3 c u2 x0)).(\lambda (H6: (nf2 c x0)).(let H_x1 \def 
+(term_dec x x0) in (let H7 \def H_x1 in (or_ind (eq T x x0) ((eq T x x0) \to 
+(\forall (P: Prop).P)) (or (pc3 c u1 u2) ((pc3 c u1 u2) \to (\forall (P: 
+Prop).P))) (\lambda (H8: (eq T x x0)).(let H9 \def (eq_ind_r T x0 (\lambda 
+(t: T).(nf2 c t)) H6 x H8) in (let H10 \def (eq_ind_r T x0 (\lambda (t: 
+T).(pr3 c u2 t)) H5 x H8) in (or_introl (pc3 c u1 u2) ((pc3 c u1 u2) \to 
+(\forall (P: Prop).P)) (pc3_pr3_t c u1 x H2 u2 H10))))) (\lambda (H8: (((eq T 
+x x0) \to (\forall (P: Prop).P)))).(or_intror (pc3 c u1 u2) ((pc3 c u1 u2) 
+\to (\forall (P: Prop).P)) (\lambda (H9: (pc3 c u1 u2)).(\lambda (P: 
+Prop).(let H10 \def H9 in (ex2_ind T (\lambda (t: T).(pr3 c u1 t)) (\lambda 
+(t: T).(pr3 c u2 t)) P (\lambda (x1: T).(\lambda (H11: (pr3 c u1 
+x1)).(\lambda (H12: (pr3 c u2 x1)).(let H_x2 \def (pr3_confluence c u2 x0 H5 
+x1 H12) in (let H13 \def H_x2 in (ex2_ind T (\lambda (t: T).(pr3 c x0 t)) 
+(\lambda (t: T).(pr3 c x1 t)) P (\lambda (x2: T).(\lambda (H14: (pr3 c x0 
+x2)).(\lambda (H15: (pr3 c x1 x2)).(let H_y1 \def (nf2_pr3_unfold c x0 x2 H14 
+H6) in (let H16 \def (eq_ind_r T x2 (\lambda (t: T).(pr3 c x1 t)) H15 x0 
+H_y1) in (let H17 \def (nf2_pr3_confluence c x H3 x0 H6 u1 H2) in (H8 (H17 
+(pr3_t x1 u1 c H11 x0 H16)) P))))))) H13)))))) H10)))))) H7)))))) H4)))))) 
+H1)))))))))))).
+
+theorem pc3_abst_dec:
+ \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (t1: T).((ty3 g c 
+u1 t1) \to (\forall (u2: T).(\forall (t2: T).((ty3 g c u2 t2) \to (or (ex4_2 
+T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2 u)))) 
+(\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) t1))) 
+(\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: T).(\lambda 
+(v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind Abst) u2 u)) 
+\to (\forall (P: Prop).P)))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda 
+(H: (ty3 g c u1 t1)).(\lambda (u2: T).(\lambda (t2: T).(\lambda (H0: (ty3 g c 
+u2 t2)).(let H1 \def (ty3_sn3 g c u1 t1 H) in (let H2 \def (ty3_sn3 g c u2 t2 
+H0) in (let H_x \def (nf2_sn3 c u1 H1) in (let H3 \def H_x in (ex2_ind T 
+(\lambda (u: T).(pr3 c u1 u)) (\lambda (u: T).(nf2 c u)) (or (ex4_2 T T 
+(\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2 u)))) 
+(\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) t1))) 
+(\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: T).(\lambda 
+(v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind Abst) u2 u)) 
+\to (\forall (P: Prop).P)))) (\lambda (x: T).(\lambda (H4: (pr3 c u1 
+x)).(\lambda (H5: (nf2 c x)).(let H_x0 \def (nf2_sn3 c u2 H2) in (let H6 \def 
+H_x0 in (ex2_ind T (\lambda (u: T).(pr3 c u2 u)) (\lambda (u: T).(nf2 c u)) 
+(or (ex4_2 T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) 
+u2 u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) 
+t1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: 
+T).(\lambda (v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind 
+Abst) u2 u)) \to (\forall (P: Prop).P)))) (\lambda (x0: T).(\lambda (H7: (pr3 
+c u2 x0)).(\lambda (H8: (nf2 c x0)).(let H_x1 \def (abst_dec x x0) in (let H9 
+\def H_x1 in (or_ind (ex T (\lambda (t: T).(eq T x (THead (Bind Abst) x0 
+t)))) (\forall (t: T).((eq T x (THead (Bind Abst) x0 t)) \to (\forall (P: 
+Prop).P))) (or (ex4_2 T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead 
+(Bind Abst) u2 u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind 
+Abst) v2 u) t1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda 
+(_: T).(\lambda (v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind 
+Abst) u2 u)) \to (\forall (P: Prop).P)))) (\lambda (H10: (ex T (\lambda (t: 
+T).(eq T x (THead (Bind Abst) x0 t))))).(ex_ind T (\lambda (t: T).(eq T x 
+(THead (Bind Abst) x0 t))) (or (ex4_2 T T (\lambda (u: T).(\lambda (_: 
+T).(pc3 c u1 (THead (Bind Abst) u2 u)))) (\lambda (u: T).(\lambda (v2: 
+T).(ty3 g c (THead (Bind Abst) v2 u) t1))) (\lambda (_: T).(\lambda (v2: 
+T).(pr3 c u2 v2))) (\lambda (_: T).(\lambda (v2: T).(nf2 c v2)))) (\forall 
+(u: T).((pc3 c u1 (THead (Bind Abst) u2 u)) \to (\forall (P: Prop).P)))) 
+(\lambda (x1: T).(\lambda (H11: (eq T x (THead (Bind Abst) x0 x1))).(let H12 
+\def (eq_ind T x (\lambda (t: T).(nf2 c t)) H5 (THead (Bind Abst) x0 x1) H11) 
+in (let H13 \def (eq_ind T x (\lambda (t: T).(pr3 c u1 t)) H4 (THead (Bind 
+Abst) x0 x1) H11) in (or_introl (ex4_2 T T (\lambda (u: T).(\lambda (_: 
+T).(pc3 c u1 (THead (Bind Abst) u2 u)))) (\lambda (u: T).(\lambda (v2: 
+T).(ty3 g c (THead (Bind Abst) v2 u) t1))) (\lambda (_: T).(\lambda (v2: 
+T).(pr3 c u2 v2))) (\lambda (_: T).(\lambda (v2: T).(nf2 c v2)))) (\forall 
+(u: T).((pc3 c u1 (THead (Bind Abst) u2 u)) \to (\forall (P: Prop).P))) 
+(ex4_2_intro T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) 
+u2 u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) 
+t1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: 
+T).(\lambda (v2: T).(nf2 c v2))) x1 x0 (pc3_pr3_t c u1 (THead (Bind Abst) x0 
+x1) H13 (THead (Bind Abst) u2 x1) (pr3_head_12 c u2 x0 H7 (Bind Abst) x1 x1 
+(pr3_refl (CHead c (Bind Abst) x0) x1))) (ty3_sred_pr3 c u1 (THead (Bind 
+Abst) x0 x1) H13 g t1 H) H7 H8)))))) H10)) (\lambda (H10: ((\forall (t: 
+T).((eq T x (THead (Bind Abst) x0 t)) \to (\forall (P: 
+Prop).P))))).(or_intror (ex4_2 T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1 
+(THead (Bind Abst) u2 u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead 
+(Bind Abst) v2 u) t1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) 
+(\lambda (_: T).(\lambda (v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 
+(THead (Bind Abst) u2 u)) \to (\forall (P: Prop).P))) (\lambda (u: 
+T).(\lambda (H11: (pc3 c u1 (THead (Bind Abst) u2 u))).(\lambda (P: 
+Prop).(let H12 \def H11 in (ex2_ind T (\lambda (t: T).(pr3 c u1 t)) (\lambda 
+(t: T).(pr3 c (THead (Bind Abst) u2 u) t)) P (\lambda (x1: T).(\lambda (H13: 
+(pr3 c u1 x1)).(\lambda (H14: (pr3 c (THead (Bind Abst) u2 u) x1)).(ex2_ind T 
+(\lambda (t: T).(pr3 c x1 t)) (\lambda (t: T).(pr3 c x t)) P (\lambda (x2: 
+T).(\lambda (H15: (pr3 c x1 x2)).(\lambda (H16: (pr3 c x x2)).(let H_y \def 
+(nf2_pr3_unfold c x x2 H16 H5) in (let H17 \def (eq_ind_r T x2 (\lambda (t: 
+T).(pr3 c x1 t)) H15 x H_y) in (let H18 \def (pr3_gen_abst c u2 u x1 H14) in 
+(ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T x1 (THead (Bind Abst) 
+u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3))) (\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr3 (CHead c (Bind b) 
+u0) u t3))))) P (\lambda (x3: T).(\lambda (x4: T).(\lambda (H19: (eq T x1 
+(THead (Bind Abst) x3 x4))).(\lambda (H20: (pr3 c u2 x3)).(\lambda (_: 
+((\forall (b: B).(\forall (u0: T).(pr3 (CHead c (Bind b) u0) u x4))))).(let 
+H22 \def (eq_ind T x1 (\lambda (t: T).(pr3 c t x)) H17 (THead (Bind Abst) x3 
+x4) H19) in (let H23 \def (pr3_gen_abst c x3 x4 x H22) in (ex3_2_ind T T 
+(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u3 t3)))) 
+(\lambda (u3: T).(\lambda (_: T).(pr3 c x3 u3))) (\lambda (_: T).(\lambda 
+(t3: T).(\forall (b: B).(\forall (u0: T).(pr3 (CHead c (Bind b) u0) x4 
+t3))))) P (\lambda (x5: T).(\lambda (x6: T).(\lambda (H24: (eq T x (THead 
+(Bind Abst) x5 x6))).(\lambda (H25: (pr3 c x3 x5)).(\lambda (_: ((\forall (b: 
+B).(\forall (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_equal T (THead (Bind Abst) x0 x6) (THead (Bind 
+Abst) x5 x6) (f_equal3 K T T T THead (Bind Abst) (Bind Abst) x0 x5 x6 x6 
+(refl_equal K (Bind Abst)) (H32 (pr3_t x3 u2 c H20 x5 H25)) (refl_equal T 
+x6))) P)))) H29))))))))) H23)))))))) H18))))))) (pr3_confluence c u1 x1 H13 x 
+H4))))) H12))))))) H9)))))) H6)))))) H3)))))))))))).
+

diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/fsubst0.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/fsubst0.ma
new file mode 100644 (file)
index 0000000..675efe3
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/fsubst0.ma
@@ -0,0 +1,719 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/fsubst0".
+
+include "pc3/left.ma".
+
+include "fsubst0/defs.ma".
+
+include "csubst0/getl.ma".
+
+theorem pc3_pr2_fsubst0:
+ \forall (c1: C).(\forall (t1: T).(\forall (t: T).((pr2 c1 t1 t) \to (\forall 
+(i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u c1 
+t1 c2 t2) \to (\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 
+c2 t2 t)))))))))))
+\def
+ \lambda (c1: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: (pr2 c1 t1 
+t)).(pr2_ind (\lambda (c: C).(\lambda (t0: T).(\lambda (t2: T).(\forall (i: 
+nat).(\forall (u: T).(\forall (c2: C).(\forall (t3: T).((fsubst0 i u c t0 c2 
+t3) \to (\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) \to (pc3 c2 t3 
+t2))))))))))) (\lambda (c: C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H0: 
+(pr0 t2 t3)).(\lambda (i: nat).(\lambda (u: T).(\lambda (c2: C).(\lambda (t0: 
+T).(\lambda (H1: (fsubst0 i u c t2 c2 t0)).(fsubst0_ind i u c t2 (\lambda 
+(c0: C).(\lambda (t4: T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) 
+\to (pc3 c0 t4 t3))))) (\lambda (t4: T).(\lambda (H2: (subst0 i u t2 
+t4)).(\lambda (e: C).(\lambda (H3: (getl i c (CHead e (Bind Abbr) 
+u))).(or_ind (pr0 t4 t3) (ex2 T (\lambda (w2: T).(pr0 t4 w2)) (\lambda (w2: 
+T).(subst0 i u t3 w2))) (pc3 c t4 t3) (\lambda (H4: (pr0 t4 t3)).(pc3_pr2_r c 
+t4 t3 (pr2_free c t4 t3 H4))) (\lambda (H4: (ex2 T (\lambda (w2: T).(pr0 t4 
+w2)) (\lambda (w2: T).(subst0 i u t3 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 
+t4 w2)) (\lambda (w2: T).(subst0 i u t3 w2)) (pc3 c t4 t3) (\lambda (x: 
+T).(\lambda (H5: (pr0 t4 x)).(\lambda (H6: (subst0 i u t3 x)).(pc3_pr2_u c x 
+t4 (pr2_free c t4 x H5) t3 (pc3_pr2_x c x t3 (pr2_delta c e u i H3 t3 t3 
+(pr0_refl t3) x H6)))))) H4)) (pr0_subst0 t2 t3 H0 u t4 i H2 u (pr0_refl 
+u))))))) (\lambda (c0: C).(\lambda (_: (csubst0 i u c c0)).(\lambda (e: 
+C).(\lambda (_: (getl i c (CHead e (Bind Abbr) u))).(pc3_pr2_r c0 t2 t3 
+(pr2_free c0 t2 t3 H0)))))) (\lambda (t4: T).(\lambda (H2: (subst0 i u t2 
+t4)).(\lambda (c0: C).(\lambda (H3: (csubst0 i u c c0)).(\lambda (e: 
+C).(\lambda (H4: (getl i c (CHead e (Bind Abbr) u))).(or_ind (pr0 t4 t3) (ex2 
+T (\lambda (w2: T).(pr0 t4 w2)) (\lambda (w2: T).(subst0 i u t3 w2))) (pc3 c0 
+t4 t3) (\lambda (H5: (pr0 t4 t3)).(pc3_pr2_r c0 t4 t3 (pr2_free c0 t4 t3 
+H5))) (\lambda (H5: (ex2 T (\lambda (w2: T).(pr0 t4 w2)) (\lambda (w2: 
+T).(subst0 i u t3 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t4 w2)) (\lambda 
+(w2: T).(subst0 i u t3 w2)) (pc3 c0 t4 t3) (\lambda (x: T).(\lambda (H6: (pr0 
+t4 x)).(\lambda (H7: (subst0 i u t3 x)).(pc3_pr2_u c0 x t4 (pr2_free c0 t4 x 
+H6) t3 (pc3_pr2_x c0 x t3 (pr2_delta c0 e u i (csubst0_getl_ge i i (le_n i) c 
+c0 u H3 (CHead e (Bind Abbr) u) H4) t3 t3 (pr0_refl t3) x H7)))))) H5)) 
+(pr0_subst0 t2 t3 H0 u t4 i H2 u (pr0_refl u))))))))) c2 t0 H1)))))))))) 
+(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda 
+(H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (t2: T).(\lambda (t3: 
+T).(\lambda (H1: (pr0 t2 t3)).(\lambda (t0: T).(\lambda (H2: (subst0 i u t3 
+t0)).(\lambda (i0: nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda (t4: 
+T).(\lambda (H3: (fsubst0 i0 u0 c t2 c2 t4)).(fsubst0_ind i0 u0 c t2 (\lambda 
+(c0: C).(\lambda (t5: T).(\forall (e: C).((getl i0 c (CHead e (Bind Abbr) 
+u0)) \to (pc3 c0 t5 t0))))) (\lambda (t5: T).(\lambda (H4: (subst0 i0 u0 t2 
+t5)).(\lambda (e: C).(\lambda (H5: (getl i0 c (CHead e (Bind Abbr) 
+u0))).(pc3_t t2 c t5 (pc3_s c t5 t2 (pc3_pr2_r c t2 t5 (pr2_delta c e u0 i0 
+H5 t2 t2 (pr0_refl t2) t5 H4))) t0 (pc3_pr2_r c t2 t0 (pr2_delta c d u i H0 
+t2 t3 H1 t0 H2))))))) (\lambda (c0: C).(\lambda (H4: (csubst0 i0 u0 c 
+c0)).(\lambda (e: C).(\lambda (H5: (getl i0 c (CHead e (Bind Abbr) 
+u0))).(lt_le_e i i0 (pc3 c0 t2 t0) (\lambda (H6: (lt i i0)).(let H7 \def 
+(csubst0_getl_lt i0 i H6 c c0 u0 H4 (CHead d (Bind Abbr) u) H0) in (or4_ind 
+(getl i c0 (CHead d (Bind Abbr) u)) (ex3_4 B C T T (\lambda (b: B).(\lambda 
+(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead 
+e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) 
+u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) 
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0 
+(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) (ex4_5 B C C T T 
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda 
+(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl 
+i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w)))))) 
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))) (pc3 c0 t2 t0) (\lambda (H8: 
+(getl i c0 (CHead d (Bind Abbr) u))).(pc3_pr2_r c0 t2 t0 (pr2_delta c0 d u i 
+H8 t2 t3 H1 t0 H2))) (\lambda (H8: (ex3_4 B C T T (\lambda (b: B).(\lambda 
+(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead 
+e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) 
+u0 u1 w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda 
+(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) 
+u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w))))) 
+(pc3 c0 t2 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda 
+(x3: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) 
+x2))).(\lambda (H10: (getl i c0 (CHead x1 (Bind x0) x3))).(\lambda (H11: 
+(subst0 (minus i0 (S i)) u0 x2 x3)).(let H12 \def (f_equal C C (\lambda (e0: 
+C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | 
+(CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) 
+x2) H9) in ((let H13 \def (f_equal C B (\lambda (e0: C).(match e0 in C return 
+(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) 
+\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) 
+\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead 
+x1 (Bind x0) x2) H9) in ((let H14 \def (f_equal C T (\lambda (e0: C).(match 
+e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ 
+t5) \Rightarrow t5])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H9) in 
+(\lambda (H15: (eq B Abbr x0)).(\lambda (H16: (eq C d x1)).(let H17 \def 
+(eq_ind_r T x2 (\lambda (t5: T).(subst0 (minus i0 (S i)) u0 t5 x3)) H11 u 
+H14) in (let H18 \def (eq_ind_r C x1 (\lambda (c3: C).(getl i c0 (CHead c3 
+(Bind x0) x3))) H10 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b: 
+B).(getl i c0 (CHead d (Bind b) x3))) H18 Abbr H15) in (ex2_ind T (\lambda 
+(t5: T).(subst0 i x3 t3 t5)) (\lambda (t5: T).(subst0 (S (plus (minus i0 (S 
+i)) i)) u0 t0 t5)) (pc3 c0 t2 t0) (\lambda (x: T).(\lambda (H20: (subst0 i x3 
+t3 x)).(\lambda (H21: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let 
+H22 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n: 
+nat).(subst0 n u0 t0 x)) H21 i0 (lt_plus_minus_r i i0 H6)) in (pc3_pr2_u c0 x 
+t2 (pr2_delta c0 d x3 i H19 t2 t3 H1 x H20) t0 (pc3_pr2_x c0 x t0 (pr2_delta 
+c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H4 (CHead e (Bind Abbr) 
+u0) H5) t0 t0 (pr0_refl t0) x H22))))))) (subst0_subst0_back t3 t0 u i H2 x3 
+u0 (minus i0 (S i)) H17)))))))) H13)) H12))))))))) H8)) (\lambda (H8: (ex3_4 
+B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq 
+C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: 
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 
+(Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda 
+(b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind 
+Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda 
+(e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_: 
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S 
+i)) u0 e1 e2))))) (pc3 c0 t2 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda 
+(x2: C).(\lambda (x3: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead 
+x1 (Bind x0) x3))).(\lambda (H10: (getl i c0 (CHead x2 (Bind x0) 
+x3))).(\lambda (H11: (csubst0 (minus i0 (S i)) u0 x1 x2)).(let H12 \def 
+(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with 
+[(CSort _) \Rightarrow d | (CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind 
+Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H13 \def (f_equal C B (\lambda 
+(e0: C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow 
+Abbr | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with 
+[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) 
+u) (CHead x1 (Bind x0) x3) H9) in ((let H14 \def (f_equal C T (\lambda (e0: 
+C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | 
+(CHead _ _ t5) \Rightarrow t5])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) 
+x3) H9) in (\lambda (H15: (eq B Abbr x0)).(\lambda (H16: (eq C d x1)).(let 
+H17 \def (eq_ind_r T x3 (\lambda (t5: T).(getl i c0 (CHead x2 (Bind x0) t5))) 
+H10 u H14) in (let H18 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus 
+i0 (S i)) u0 c3 x2)) H11 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b: 
+B).(getl i c0 (CHead x2 (Bind b) u))) H17 Abbr H15) in (pc3_pr2_r c0 t2 t0 
+(pr2_delta c0 x2 u i H19 t2 t3 H1 t0 H2)))))))) H13)) H12))))))))) H8)) 
+(\lambda (H8: (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 
+(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
+(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 
+e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 
+(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
+(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) 
+(pc3 c0 t2 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda 
+(x3: T).(\lambda (x4: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead 
+x1 (Bind x0) x3))).(\lambda (H10: (getl i c0 (CHead x2 (Bind x0) 
+x4))).(\lambda (H11: (subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H12: 
+(csubst0 (minus i0 (S i)) u0 x1 x2)).(let H13 \def (f_equal C C (\lambda (e0: 
+C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | 
+(CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) 
+x3) H9) in ((let H14 \def (f_equal C B (\lambda (e0: C).(match e0 in C return 
+(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) 
+\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) 
+\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead 
+x1 (Bind x0) x3) H9) in ((let H15 \def (f_equal C T (\lambda (e0: C).(match 
+e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ 
+t5) \Rightarrow t5])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in 
+(\lambda (H16: (eq B Abbr x0)).(\lambda (H17: (eq C d x1)).(let H18 \def 
+(eq_ind_r T x3 (\lambda (t5: T).(subst0 (minus i0 (S i)) u0 t5 x4)) H11 u 
+H15) in (let H19 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus i0 (S 
+i)) u0 c3 x2)) H12 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b: 
+B).(getl i c0 (CHead x2 (Bind b) x4))) H10 Abbr H16) in (ex2_ind T (\lambda 
+(t5: T).(subst0 i x4 t3 t5)) (\lambda (t5: T).(subst0 (S (plus (minus i0 (S 
+i)) i)) u0 t0 t5)) (pc3 c0 t2 t0) (\lambda (x: T).(\lambda (H21: (subst0 i x4 
+t3 x)).(\lambda (H22: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let 
+H23 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n: 
+nat).(subst0 n u0 t0 x)) H22 i0 (lt_plus_minus_r i i0 H6)) in (pc3_pr2_u c0 x 
+t2 (pr2_delta c0 x2 x4 i H20 t2 t3 H1 x H21) t0 (pc3_pr2_x c0 x t0 (pr2_delta 
+c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H4 (CHead e (Bind Abbr) 
+u0) H5) t0 t0 (pr0_refl t0) x H23))))))) (subst0_subst0_back t3 t0 u i H2 x4 
+u0 (minus i0 (S i)) H18)))))))) H14)) H13))))))))))) H8)) H7))) (\lambda (H6: 
+(le i0 i)).(pc3_pr2_r c0 t2 t0 (pr2_delta c0 d u i (csubst0_getl_ge i0 i H6 c 
+c0 u0 H4 (CHead d (Bind Abbr) u) H0) t2 t3 H1 t0 H2)))))))) (\lambda (t5: 
+T).(\lambda (H4: (subst0 i0 u0 t2 t5)).(\lambda (c0: C).(\lambda (H5: 
+(csubst0 i0 u0 c c0)).(\lambda (e: C).(\lambda (H6: (getl i0 c (CHead e (Bind 
+Abbr) u0))).(lt_le_e i i0 (pc3 c0 t5 t0) (\lambda (H7: (lt i i0)).(let H8 
+\def (csubst0_getl_lt i0 i H7 c c0 u0 H5 (CHead d (Bind Abbr) u) H0) in 
+(or4_ind (getl i c0 (CHead d (Bind Abbr) u)) (ex3_4 B C T T (\lambda (b: 
+B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind 
+Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) 
+(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
+(minus i0 (S i)) u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 
+(Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) 
+(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) 
+u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
+(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))) 
+(pc3 c0 t5 t0) (\lambda (H9: (getl i c0 (CHead d (Bind Abbr) u))).(pc3_pr2_u2 
+c0 t2 t5 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 
+(CHead e (Bind Abbr) u0) H6) t2 t2 (pr0_refl t2) t5 H4) t0 (pc3_pr2_r c0 t2 
+t0 (pr2_delta c0 d u i H9 t2 t3 H1 t0 H2)))) (\lambda (H9: (ex3_4 B C T T 
+(\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C 
+(CHead d (Bind Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda 
+(e0: C).(\lambda (_: T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) 
+(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
+(minus i0 (S i)) u0 u1 w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda 
+(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead 
+e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) 
+u0 u1 w))))) (pc3 c0 t5 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: 
+T).(\lambda (x3: T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead x1 
+(Bind x0) x2))).(\lambda (H11: (getl i c0 (CHead x1 (Bind x0) x3))).(\lambda 
+(H12: (subst0 (minus i0 (S i)) u0 x2 x3)).(let H13 \def (f_equal C C (\lambda 
+(e0: C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow 
+d | (CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind 
+x0) x2) H10) in ((let H14 \def (f_equal C B (\lambda (e0: C).(match e0 in C 
+return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) 
+\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) 
+\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead 
+x1 (Bind x0) x2) H10) in ((let H15 \def (f_equal C T (\lambda (e0: C).(match 
+e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ 
+t6) \Rightarrow t6])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H10) in 
+(\lambda (H16: (eq B Abbr x0)).(\lambda (H17: (eq C d x1)).(let H18 \def 
+(eq_ind_r T x2 (\lambda (t6: T).(subst0 (minus i0 (S i)) u0 t6 x3)) H12 u 
+H15) in (let H19 \def (eq_ind_r C x1 (\lambda (c3: C).(getl i c0 (CHead c3 
+(Bind x0) x3))) H11 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b: 
+B).(getl i c0 (CHead d (Bind b) x3))) H19 Abbr H16) in (ex2_ind T (\lambda 
+(t6: T).(subst0 i x3 t3 t6)) (\lambda (t6: T).(subst0 (S (plus (minus i0 (S 
+i)) i)) u0 t0 t6)) (pc3 c0 t5 t0) (\lambda (x: T).(\lambda (H21: (subst0 i x3 
+t3 x)).(\lambda (H22: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let 
+H23 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n: 
+nat).(subst0 n u0 t0 x)) H22 i0 (lt_plus_minus_r i i0 H7)) in (pc3_pr2_u2 c0 
+t2 t5 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 
+(CHead e (Bind Abbr) u0) H6) t2 t2 (pr0_refl t2) t5 H4) t0 (pc3_pr2_u c0 x t2 
+(pr2_delta c0 d x3 i H20 t2 t3 H1 x H21) t0 (pc3_pr2_x c0 x t0 (pr2_delta c0 
+e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e (Bind Abbr) u0) 
+H6) t0 t0 (pr0_refl t0) x H23)))))))) (subst0_subst0_back t3 t0 u i H2 x3 u0 
+(minus i0 (S i)) H18)))))))) H14)) H13))))))))) H9)) (\lambda (H9: (ex3_4 B C 
+C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C 
+(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 (Bind b) 
+u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda (b: 
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind 
+Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda 
+(e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_: 
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S 
+i)) u0 e1 e2))))) (pc3 c0 t5 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda 
+(x2: C).(\lambda (x3: T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead 
+x1 (Bind x0) x3))).(\lambda (H11: (getl i c0 (CHead x2 (Bind x0) 
+x3))).(\lambda (H12: (csubst0 (minus i0 (S i)) u0 x1 x2)).(let H13 \def 
+(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with 
+[(CSort _) \Rightarrow d | (CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind 
+Abbr) u) (CHead x1 (Bind x0) x3) H10) in ((let H14 \def (f_equal C B (\lambda 
+(e0: C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow 
+Abbr | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with 
+[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) 
+u) (CHead x1 (Bind x0) x3) H10) in ((let H15 \def (f_equal C T (\lambda (e0: 
+C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | 
+(CHead _ _ t6) \Rightarrow t6])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) 
+x3) H10) in (\lambda (H16: (eq B Abbr x0)).(\lambda (H17: (eq C d x1)).(let 
+H18 \def (eq_ind_r T x3 (\lambda (t6: T).(getl i c0 (CHead x2 (Bind x0) t6))) 
+H11 u H15) in (let H19 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus 
+i0 (S i)) u0 c3 x2)) H12 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b: 
+B).(getl i c0 (CHead x2 (Bind b) u))) H18 Abbr H16) in (pc3_pr2_u2 c0 t2 t5 
+(pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e 
+(Bind Abbr) u0) H6) t2 t2 (pr0_refl t2) t5 H4) t0 (pc3_pr2_r c0 t2 t0 
+(pr2_delta c0 x2 u i H20 t2 t3 H1 t0 H2))))))))) H14)) H13))))))))) H9)) 
+(\lambda (H9: (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 
+(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
+(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 
+e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 
+(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
+(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) 
+(pc3 c0 t5 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda 
+(x3: T).(\lambda (x4: T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead 
+x1 (Bind x0) x3))).(\lambda (H11: (getl i c0 (CHead x2 (Bind x0) 
+x4))).(\lambda (H12: (subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H13: 
+(csubst0 (minus i0 (S i)) u0 x1 x2)).(let H14 \def (f_equal C C (\lambda (e0: 
+C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | 
+(CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) 
+x3) H10) in ((let H15 \def (f_equal C B (\lambda (e0: C).(match e0 in C 
+return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) 
+\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) 
+\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead 
+x1 (Bind x0) x3) H10) in ((let H16 \def (f_equal C T (\lambda (e0: C).(match 
+e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ 
+t6) \Rightarrow t6])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H10) in 
+(\lambda (H17: (eq B Abbr x0)).(\lambda (H18: (eq C d x1)).(let H19 \def 
+(eq_ind_r T x3 (\lambda (t6: T).(subst0 (minus i0 (S i)) u0 t6 x4)) H12 u 
+H16) in (let H20 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus i0 (S 
+i)) u0 c3 x2)) H13 d H18) in (let H21 \def (eq_ind_r B x0 (\lambda (b: 
+B).(getl i c0 (CHead x2 (Bind b) x4))) H11 Abbr H17) in (ex2_ind T (\lambda 
+(t6: T).(subst0 i x4 t3 t6)) (\lambda (t6: T).(subst0 (S (plus (minus i0 (S 
+i)) i)) u0 t0 t6)) (pc3 c0 t5 t0) (\lambda (x: T).(\lambda (H22: (subst0 i x4 
+t3 x)).(\lambda (H23: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let 
+H24 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n: 
+nat).(subst0 n u0 t0 x)) H23 i0 (lt_plus_minus_r i i0 H7)) in (pc3_pr2_u2 c0 
+t2 t5 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 
+(CHead e (Bind Abbr) u0) H6) t2 t2 (pr0_refl t2) t5 H4) t0 (pc3_pr2_u c0 x t2 
+(pr2_delta c0 x2 x4 i H21 t2 t3 H1 x H22) t0 (pc3_pr2_x c0 x t0 (pr2_delta c0 
+e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e (Bind Abbr) u0) 
+H6) t0 t0 (pr0_refl t0) x H24)))))))) (subst0_subst0_back t3 t0 u i H2 x4 u0 
+(minus i0 (S i)) H19)))))))) H15)) H14))))))))))) H9)) H8))) (\lambda (H7: 
+(le i0 i)).(pc3_pr2_u2 c0 t2 t5 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 
+(le_n i0) c c0 u0 H5 (CHead e (Bind Abbr) u0) H6) t2 t2 (pr0_refl t2) t5 H4) 
+t0 (pc3_pr2_r c0 t2 t0 (pr2_delta c0 d u i (csubst0_getl_ge i0 i H7 c c0 u0 
+H5 (CHead d (Bind Abbr) u) H0) t2 t3 H1 t0 H2))))))))))) c2 t4 
+H3)))))))))))))))) c1 t1 t H)))).
+
+theorem pc3_pr2_fsubst0_back:
+ \forall (c1: C).(\forall (t: T).(\forall (t1: T).((pr2 c1 t t1) \to (\forall 
+(i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u c1 
+t1 c2 t2) \to (\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 
+c2 t t2)))))))))))
+\def
+ \lambda (c1: C).(\lambda (t: T).(\lambda (t1: T).(\lambda (H: (pr2 c1 t 
+t1)).(pr2_ind (\lambda (c: C).(\lambda (t0: T).(\lambda (t2: T).(\forall (i: 
+nat).(\forall (u: T).(\forall (c2: C).(\forall (t3: T).((fsubst0 i u c t2 c2 
+t3) \to (\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) \to (pc3 c2 t0 
+t3))))))))))) (\lambda (c: C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H0: 
+(pr0 t2 t3)).(\lambda (i: nat).(\lambda (u: T).(\lambda (c2: C).(\lambda (t0: 
+T).(\lambda (H1: (fsubst0 i u c t3 c2 t0)).(fsubst0_ind i u c t3 (\lambda 
+(c0: C).(\lambda (t4: T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) 
+\to (pc3 c0 t2 t4))))) (\lambda (t4: T).(\lambda (H2: (subst0 i u t3 
+t4)).(\lambda (e: C).(\lambda (H3: (getl i c (CHead e (Bind Abbr) 
+u))).(pc3_pr2_u c t3 t2 (pr2_free c t2 t3 H0) t4 (pc3_pr2_r c t3 t4 
+(pr2_delta c e u i H3 t3 t3 (pr0_refl t3) t4 H2))))))) (\lambda (c0: 
+C).(\lambda (_: (csubst0 i u c c0)).(\lambda (e: C).(\lambda (_: (getl i c 
+(CHead e (Bind Abbr) u))).(pc3_pr2_r c0 t2 t3 (pr2_free c0 t2 t3 H0)))))) 
+(\lambda (t4: T).(\lambda (H2: (subst0 i u t3 t4)).(\lambda (c0: C).(\lambda 
+(H3: (csubst0 i u c c0)).(\lambda (e: C).(\lambda (H4: (getl i c (CHead e 
+(Bind Abbr) u))).(pc3_pr2_u c0 t3 t2 (pr2_free c0 t2 t3 H0) t4 (pc3_pr2_r c0 
+t3 t4 (pr2_delta c0 e u i (csubst0_getl_ge i i (le_n i) c c0 u H3 (CHead e 
+(Bind Abbr) u) H4) t3 t3 (pr0_refl t3) t4 H2))))))))) c2 t0 H1)))))))))) 
+(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda 
+(H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (t2: T).(\lambda (t3: 
+T).(\lambda (H1: (pr0 t2 t3)).(\lambda (t0: T).(\lambda (H2: (subst0 i u t3 
+t0)).(\lambda (i0: nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda (t4: 
+T).(\lambda (H3: (fsubst0 i0 u0 c t0 c2 t4)).(fsubst0_ind i0 u0 c t0 (\lambda 
+(c0: C).(\lambda (t5: T).(\forall (e: C).((getl i0 c (CHead e (Bind Abbr) 
+u0)) \to (pc3 c0 t2 t5))))) (\lambda (t5: T).(\lambda (H4: (subst0 i0 u0 t0 
+t5)).(\lambda (e: C).(\lambda (H5: (getl i0 c (CHead e (Bind Abbr) 
+u0))).(pc3_t t3 c t2 (pc3_pr3_r c t2 t3 (pr3_pr2 c t2 t3 (pr2_free c t2 t3 
+H1))) t5 (pc3_pr3_r c t3 t5 (pr3_sing c t0 t3 (pr2_delta c d u i H0 t3 t3 
+(pr0_refl t3) t0 H2) t5 (pr3_pr2 c t0 t5 (pr2_delta c e u0 i0 H5 t0 t0 
+(pr0_refl t0) t5 H4))))))))) (\lambda (c0: C).(\lambda (H4: (csubst0 i0 u0 c 
+c0)).(\lambda (e: C).(\lambda (H5: (getl i0 c (CHead e (Bind Abbr) 
+u0))).(lt_le_e i i0 (pc3 c0 t2 t0) (\lambda (H6: (lt i i0)).(let H7 \def 
+(csubst0_getl_lt i0 i H6 c c0 u0 H4 (CHead d (Bind Abbr) u) H0) in (or4_ind 
+(getl i c0 (CHead d (Bind Abbr) u)) (ex3_4 B C T T (\lambda (b: B).(\lambda 
+(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead 
+e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) 
+u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) 
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0 
+(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) (ex4_5 B C C T T 
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda 
+(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl 
+i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w)))))) 
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))) (pc3 c0 t2 t0) (\lambda (H8: 
+(getl i c0 (CHead d (Bind Abbr) u))).(pc3_pr2_r c0 t2 t0 (pr2_delta c0 d u i 
+H8 t2 t3 H1 t0 H2))) (\lambda (H8: (ex3_4 B C T T (\lambda (b: B).(\lambda 
+(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead 
+e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) 
+u0 u1 w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda 
+(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) 
+u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w))))) 
+(pc3 c0 t2 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda 
+(x3: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) 
+x2))).(\lambda (H10: (getl i c0 (CHead x1 (Bind x0) x3))).(\lambda (H11: 
+(subst0 (minus i0 (S i)) u0 x2 x3)).(let H12 \def (f_equal C C (\lambda (e0: 
+C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | 
+(CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) 
+x2) H9) in ((let H13 \def (f_equal C B (\lambda (e0: C).(match e0 in C return 
+(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) 
+\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) 
+\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead 
+x1 (Bind x0) x2) H9) in ((let H14 \def (f_equal C T (\lambda (e0: C).(match 
+e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ 
+t5) \Rightarrow t5])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H9) in 
+(\lambda (H15: (eq B Abbr x0)).(\lambda (H16: (eq C d x1)).(let H17 \def 
+(eq_ind_r T x2 (\lambda (t5: T).(subst0 (minus i0 (S i)) u0 t5 x3)) H11 u 
+H14) in (let H18 \def (eq_ind_r C x1 (\lambda (c3: C).(getl i c0 (CHead c3 
+(Bind x0) x3))) H10 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b: 
+B).(getl i c0 (CHead d (Bind b) x3))) H18 Abbr H15) in (ex2_ind T (\lambda 
+(t5: T).(subst0 i x3 t3 t5)) (\lambda (t5: T).(subst0 (S (plus (minus i0 (S 
+i)) i)) u0 t0 t5)) (pc3 c0 t2 t0) (\lambda (x: T).(\lambda (H20: (subst0 i x3 
+t3 x)).(\lambda (H21: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let 
+H22 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n: 
+nat).(subst0 n u0 t0 x)) H21 i0 (lt_plus_minus_r i i0 H6)) in (pc3_pr2_u c0 x 
+t2 (pr2_delta c0 d x3 i H19 t2 t3 H1 x H20) t0 (pc3_pr2_x c0 x t0 (pr2_delta 
+c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H4 (CHead e (Bind Abbr) 
+u0) H5) t0 t0 (pr0_refl t0) x H22))))))) (subst0_subst0_back t3 t0 u i H2 x3 
+u0 (minus i0 (S i)) H17)))))))) H13)) H12))))))))) H8)) (\lambda (H8: (ex3_4 
+B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq 
+C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: 
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 
+(Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda 
+(b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind 
+Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda 
+(e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_: 
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S 
+i)) u0 e1 e2))))) (pc3 c0 t2 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda 
+(x2: C).(\lambda (x3: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead 
+x1 (Bind x0) x3))).(\lambda (H10: (getl i c0 (CHead x2 (Bind x0) 
+x3))).(\lambda (H11: (csubst0 (minus i0 (S i)) u0 x1 x2)).(let H12 \def 
+(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with 
+[(CSort _) \Rightarrow d | (CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind 
+Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H13 \def (f_equal C B (\lambda 
+(e0: C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow 
+Abbr | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with 
+[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) 
+u) (CHead x1 (Bind x0) x3) H9) in ((let H14 \def (f_equal C T (\lambda (e0: 
+C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | 
+(CHead _ _ t5) \Rightarrow t5])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) 
+x3) H9) in (\lambda (H15: (eq B Abbr x0)).(\lambda (H16: (eq C d x1)).(let 
+H17 \def (eq_ind_r T x3 (\lambda (t5: T).(getl i c0 (CHead x2 (Bind x0) t5))) 
+H10 u H14) in (let H18 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus 
+i0 (S i)) u0 c3 x2)) H11 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b: 
+B).(getl i c0 (CHead x2 (Bind b) u))) H17 Abbr H15) in (pc3_pr2_r c0 t2 t0 
+(pr2_delta c0 x2 u i H19 t2 t3 H1 t0 H2)))))))) H13)) H12))))))))) H8)) 
+(\lambda (H8: (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 
+(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
+(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 
+e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 
+(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
+(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) 
+(pc3 c0 t2 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda 
+(x3: T).(\lambda (x4: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead 
+x1 (Bind x0) x3))).(\lambda (H10: (getl i c0 (CHead x2 (Bind x0) 
+x4))).(\lambda (H11: (subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H12: 
+(csubst0 (minus i0 (S i)) u0 x1 x2)).(let H13 \def (f_equal C C (\lambda (e0: 
+C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | 
+(CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) 
+x3) H9) in ((let H14 \def (f_equal C B (\lambda (e0: C).(match e0 in C return 
+(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) 
+\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) 
+\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead 
+x1 (Bind x0) x3) H9) in ((let H15 \def (f_equal C T (\lambda (e0: C).(match 
+e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ 
+t5) \Rightarrow t5])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in 
+(\lambda (H16: (eq B Abbr x0)).(\lambda (H17: (eq C d x1)).(let H18 \def 
+(eq_ind_r T x3 (\lambda (t5: T).(subst0 (minus i0 (S i)) u0 t5 x4)) H11 u 
+H15) in (let H19 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus i0 (S 
+i)) u0 c3 x2)) H12 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b: 
+B).(getl i c0 (CHead x2 (Bind b) x4))) H10 Abbr H16) in (ex2_ind T (\lambda 
+(t5: T).(subst0 i x4 t3 t5)) (\lambda (t5: T).(subst0 (S (plus (minus i0 (S 
+i)) i)) u0 t0 t5)) (pc3 c0 t2 t0) (\lambda (x: T).(\lambda (H21: (subst0 i x4 
+t3 x)).(\lambda (H22: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let 
+H23 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n: 
+nat).(subst0 n u0 t0 x)) H22 i0 (lt_plus_minus_r i i0 H6)) in (pc3_pr2_u c0 x 
+t2 (pr2_delta c0 x2 x4 i H20 t2 t3 H1 x H21) t0 (pc3_pr2_x c0 x t0 (pr2_delta 
+c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H4 (CHead e (Bind Abbr) 
+u0) H5) t0 t0 (pr0_refl t0) x H23))))))) (subst0_subst0_back t3 t0 u i H2 x4 
+u0 (minus i0 (S i)) H18)))))))) H14)) H13))))))))))) H8)) H7))) (\lambda (H6: 
+(le i0 i)).(pc3_pr2_r c0 t2 t0 (pr2_delta c0 d u i (csubst0_getl_ge i0 i H6 c 
+c0 u0 H4 (CHead d (Bind Abbr) u) H0) t2 t3 H1 t0 H2)))))))) (\lambda (t5: 
+T).(\lambda (H4: (subst0 i0 u0 t0 t5)).(\lambda (c0: C).(\lambda (H5: 
+(csubst0 i0 u0 c c0)).(\lambda (e: C).(\lambda (H6: (getl i0 c (CHead e (Bind 
+Abbr) u0))).(lt_le_e i i0 (pc3 c0 t2 t5) (\lambda (H7: (lt i i0)).(let H8 
+\def (csubst0_getl_lt i0 i H7 c c0 u0 H5 (CHead d (Bind Abbr) u) H0) in 
+(or4_ind (getl i c0 (CHead d (Bind Abbr) u)) (ex3_4 B C T T (\lambda (b: 
+B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind 
+Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) 
+(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
+(minus i0 (S i)) u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 
+(Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) 
+(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) 
+u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
+(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))) 
+(pc3 c0 t2 t5) (\lambda (H9: (getl i c0 (CHead d (Bind Abbr) u))).(pc3_pr2_u 
+c0 t3 t2 (pr2_free c0 t2 t3 H1) t5 (pc3_pr3_r c0 t3 t5 (pr3_sing c0 t0 t3 
+(pr2_delta c0 d u i H9 t3 t3 (pr0_refl t3) t0 H2) t5 (pr3_pr2 c0 t0 t5 
+(pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e 
+(Bind Abbr) u0) H6) t0 t0 (pr0_refl t0) t5 H4)))))) (\lambda (H9: (ex3_4 B C 
+T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C 
+(CHead d (Bind Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda 
+(e0: C).(\lambda (_: T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) 
+(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
+(minus i0 (S i)) u0 u1 w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda 
+(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead 
+e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) 
+u0 u1 w))))) (pc3 c0 t2 t5) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: 
+T).(\lambda (x3: T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead x1 
+(Bind x0) x2))).(\lambda (H11: (getl i c0 (CHead x1 (Bind x0) x3))).(\lambda 
+(H12: (subst0 (minus i0 (S i)) u0 x2 x3)).(let H13 \def (f_equal C C (\lambda 
+(e0: C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow 
+d | (CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind 
+x0) x2) H10) in ((let H14 \def (f_equal C B (\lambda (e0: C).(match e0 in C 
+return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) 
+\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) 
+\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead 
+x1 (Bind x0) x2) H10) in ((let H15 \def (f_equal C T (\lambda (e0: C).(match 
+e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ 
+t6) \Rightarrow t6])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H10) in 
+(\lambda (H16: (eq B Abbr x0)).(\lambda (H17: (eq C d x1)).(let H18 \def 
+(eq_ind_r T x2 (\lambda (t6: T).(subst0 (minus i0 (S i)) u0 t6 x3)) H12 u 
+H15) in (let H19 \def (eq_ind_r C x1 (\lambda (c3: C).(getl i c0 (CHead c3 
+(Bind x0) x3))) H11 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b: 
+B).(getl i c0 (CHead d (Bind b) x3))) H19 Abbr H16) in (ex2_ind T (\lambda 
+(t6: T).(subst0 i x3 t3 t6)) (\lambda (t6: T).(subst0 (S (plus (minus i0 (S 
+i)) i)) u0 t0 t6)) (pc3 c0 t2 t5) (\lambda (x: T).(\lambda (H21: (subst0 i x3 
+t3 x)).(\lambda (H22: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let 
+H23 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n: 
+nat).(subst0 n u0 t0 x)) H22 i0 (lt_plus_minus_r i i0 H7)) in (pc3_pr2_u c0 x 
+t2 (pr2_delta c0 d x3 i H20 t2 t3 H1 x H21) t5 (pc3_pr2_u2 c0 t0 x (pr2_delta 
+c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e (Bind Abbr) 
+u0) H6) t0 t0 (pr0_refl t0) x H23) t5 (pc3_pr2_r c0 t0 t5 (pr2_delta c0 e u0 
+i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e (Bind Abbr) u0) H6) 
+t0 t0 (pr0_refl t0) t5 H4)))))))) (subst0_subst0_back t3 t0 u i H2 x3 u0 
+(minus i0 (S i)) H18)))))))) H14)) H13))))))))) H9)) (\lambda (H9: (ex3_4 B C 
+C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C 
+(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 (Bind b) 
+u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda (b: 
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind 
+Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda 
+(e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_: 
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S 
+i)) u0 e1 e2))))) (pc3 c0 t2 t5) (\lambda (x0: B).(\lambda (x1: C).(\lambda 
+(x2: C).(\lambda (x3: T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead 
+x1 (Bind x0) x3))).(\lambda (H11: (getl i c0 (CHead x2 (Bind x0) 
+x3))).(\lambda (H12: (csubst0 (minus i0 (S i)) u0 x1 x2)).(let H13 \def 
+(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with 
+[(CSort _) \Rightarrow d | (CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind 
+Abbr) u) (CHead x1 (Bind x0) x3) H10) in ((let H14 \def (f_equal C B (\lambda 
+(e0: C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow 
+Abbr | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with 
+[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) 
+u) (CHead x1 (Bind x0) x3) H10) in ((let H15 \def (f_equal C T (\lambda (e0: 
+C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | 
+(CHead _ _ t6) \Rightarrow t6])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) 
+x3) H10) in (\lambda (H16: (eq B Abbr x0)).(\lambda (H17: (eq C d x1)).(let 
+H18 \def (eq_ind_r T x3 (\lambda (t6: T).(getl i c0 (CHead x2 (Bind x0) t6))) 
+H11 u H15) in (let H19 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus 
+i0 (S i)) u0 c3 x2)) H12 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b: 
+B).(getl i c0 (CHead x2 (Bind b) u))) H18 Abbr H16) in (pc3_pr2_u c0 t0 t2 
+(pr2_delta c0 x2 u i H20 t2 t3 H1 t0 H2) t5 (pc3_pr2_r c0 t0 t5 (pr2_delta c0 
+e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e (Bind Abbr) u0) 
+H6) t0 t0 (pr0_refl t0) t5 H4))))))))) H14)) H13))))))))) H9)) (\lambda (H9: 
+(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) 
+u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
+(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 
+e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 
+(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
+(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) 
+(pc3 c0 t2 t5) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda 
+(x3: T).(\lambda (x4: T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead 
+x1 (Bind x0) x3))).(\lambda (H11: (getl i c0 (CHead x2 (Bind x0) 
+x4))).(\lambda (H12: (subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H13: 
+(csubst0 (minus i0 (S i)) u0 x1 x2)).(let H14 \def (f_equal C C (\lambda (e0: 
+C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | 
+(CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) 
+x3) H10) in ((let H15 \def (f_equal C B (\lambda (e0: C).(match e0 in C 
+return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) 
+\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) 
+\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead 
+x1 (Bind x0) x3) H10) in ((let H16 \def (f_equal C T (\lambda (e0: C).(match 
+e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ 
+t6) \Rightarrow t6])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H10) in 
+(\lambda (H17: (eq B Abbr x0)).(\lambda (H18: (eq C d x1)).(let H19 \def 
+(eq_ind_r T x3 (\lambda (t6: T).(subst0 (minus i0 (S i)) u0 t6 x4)) H12 u 
+H16) in (let H20 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus i0 (S 
+i)) u0 c3 x2)) H13 d H18) in (let H21 \def (eq_ind_r B x0 (\lambda (b: 
+B).(getl i c0 (CHead x2 (Bind b) x4))) H11 Abbr H17) in (ex2_ind T (\lambda 
+(t6: T).(subst0 i x4 t3 t6)) (\lambda (t6: T).(subst0 (S (plus (minus i0 (S 
+i)) i)) u0 t0 t6)) (pc3 c0 t2 t5) (\lambda (x: T).(\lambda (H22: (subst0 i x4 
+t3 x)).(\lambda (H23: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let 
+H24 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n: 
+nat).(subst0 n u0 t0 x)) H23 i0 (lt_plus_minus_r i i0 H7)) in (pc3_pr2_u c0 x 
+t2 (pr2_delta c0 x2 x4 i H21 t2 t3 H1 x H22) t5 (pc3_pr2_u2 c0 t0 x 
+(pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e 
+(Bind Abbr) u0) H6) t0 t0 (pr0_refl t0) x H24) t5 (pc3_pr2_r c0 t0 t5 
+(pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e 
+(Bind Abbr) u0) H6) t0 t0 (pr0_refl t0) t5 H4)))))))) (subst0_subst0_back t3 
+t0 u i H2 x4 u0 (minus i0 (S i)) H19)))))))) H15)) H14))))))))))) H9)) H8))) 
+(\lambda (H7: (le i0 i)).(pc3_pr2_u c0 t0 t2 (pr2_delta c0 d u i 
+(csubst0_getl_ge i0 i H7 c c0 u0 H5 (CHead d (Bind Abbr) u) H0) t2 t3 H1 t0 
+H2) t5 (pc3_pr2_r c0 t0 t5 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n 
+i0) c c0 u0 H5 (CHead e (Bind Abbr) u0) H6) t0 t0 (pr0_refl t0) t5 
+H4))))))))))) c2 t4 H3)))))))))))))))) c1 t t1 H)))).
+
+theorem pc3_fsubst0:
+ \forall (c1: C).(\forall (t1: T).(\forall (t: T).((pc3 c1 t1 t) \to (\forall 
+(i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u c1 
+t1 c2 t2) \to (\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 
+c2 t2 t)))))))))))
+\def
+ \lambda (c1: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: (pc3 c1 t1 
+t)).(pc3_ind_left c1 (\lambda (t0: T).(\lambda (t2: T).(\forall (i: 
+nat).(\forall (u: T).(\forall (c2: C).(\forall (t3: T).((fsubst0 i u c1 t0 c2 
+t3) \to (\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 c2 t3 
+t2)))))))))) (\lambda (t0: T).(\lambda (i: nat).(\lambda (u: T).(\lambda (c2: 
+C).(\lambda (t2: T).(\lambda (H0: (fsubst0 i u c1 t0 c2 t2)).(fsubst0_ind i u 
+c1 t0 (\lambda (c: C).(\lambda (t3: T).(\forall (e: C).((getl i c1 (CHead e 
+(Bind Abbr) u)) \to (pc3 c t3 t0))))) (\lambda (t3: T).(\lambda (H1: (subst0 
+i u t0 t3)).(\lambda (e: C).(\lambda (H2: (getl i c1 (CHead e (Bind Abbr) 
+u))).(pc3_pr2_x c1 t3 t0 (pr2_delta c1 e u i H2 t0 t0 (pr0_refl t0) t3 
+H1)))))) (\lambda (c0: C).(\lambda (_: (csubst0 i u c1 c0)).(\lambda (e: 
+C).(\lambda (_: (getl i c1 (CHead e (Bind Abbr) u))).(pc3_refl c0 t0))))) 
+(\lambda (t3: T).(\lambda (H1: (subst0 i u t0 t3)).(\lambda (c0: C).(\lambda 
+(H2: (csubst0 i u c1 c0)).(\lambda (e: C).(\lambda (H3: (getl i c1 (CHead e 
+(Bind Abbr) u))).(pc3_pr2_x c0 t3 t0 (pr2_delta c0 e u i (csubst0_getl_ge i i 
+(le_n i) c1 c0 u H2 (CHead e (Bind Abbr) u) H3) t0 t0 (pr0_refl t0) t3 
+H1)))))))) c2 t2 H0))))))) (\lambda (t0: T).(\lambda (t2: T).(\lambda (H0: 
+(pr2 c1 t0 t2)).(\lambda (t3: T).(\lambda (H1: (pc3 c1 t2 t3)).(\lambda (H2: 
+((\forall (i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t4: 
+T).((fsubst0 i u c1 t2 c2 t4) \to (\forall (e: C).((getl i c1 (CHead e (Bind 
+Abbr) u)) \to (pc3 c2 t4 t3)))))))))).(\lambda (i: nat).(\lambda (u: 
+T).(\lambda (c2: C).(\lambda (t4: T).(\lambda (H3: (fsubst0 i u c1 t0 c2 
+t4)).(fsubst0_ind i u c1 t0 (\lambda (c: C).(\lambda (t5: T).(\forall (e: 
+C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 c t5 t3))))) (\lambda (t5: 
+T).(\lambda (H4: (subst0 i u t0 t5)).(\lambda (e: C).(\lambda (H5: (getl i c1 
+(CHead e (Bind Abbr) u))).(pc3_t t2 c1 t5 (pc3_pr2_fsubst0 c1 t0 t2 H0 i u c1 
+t5 (fsubst0_snd i u c1 t0 t5 H4) e H5) t3 H1))))) (\lambda (c0: C).(\lambda 
+(H4: (csubst0 i u c1 c0)).(\lambda (e: C).(\lambda (H5: (getl i c1 (CHead e 
+(Bind Abbr) u))).(pc3_t t2 c0 t0 (pc3_pr2_fsubst0 c1 t0 t2 H0 i u c0 t0 
+(fsubst0_fst i u c1 t0 c0 H4) e H5) t3 (H2 i u c0 t2 (fsubst0_fst i u c1 t2 
+c0 H4) e H5)))))) (\lambda (t5: T).(\lambda (H4: (subst0 i u t0 t5)).(\lambda 
+(c0: C).(\lambda (H5: (csubst0 i u c1 c0)).(\lambda (e: C).(\lambda (H6: 
+(getl i c1 (CHead e (Bind Abbr) u))).(pc3_t t2 c0 t5 (pc3_pr2_fsubst0 c1 t0 
+t2 H0 i u c0 t5 (fsubst0_both i u c1 t0 t5 H4 c0 H5) e H6) t3 (H2 i u c0 t2 
+(fsubst0_fst i u c1 t2 c0 H5) e H6)))))))) c2 t4 H3)))))))))))) (\lambda (t0: 
+T).(\lambda (t2: T).(\lambda (H0: (pr2 c1 t0 t2)).(\lambda (t3: T).(\lambda 
+(H1: (pc3 c1 t0 t3)).(\lambda (H2: ((\forall (i: nat).(\forall (u: 
+T).(\forall (c2: C).(\forall (t4: T).((fsubst0 i u c1 t0 c2 t4) \to (\forall 
+(e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 c2 t4 
+t3)))))))))).(\lambda (i: nat).(\lambda (u: T).(\lambda (c2: C).(\lambda (t4: 
+T).(\lambda (H3: (fsubst0 i u c1 t2 c2 t4)).(fsubst0_ind i u c1 t2 (\lambda 
+(c: C).(\lambda (t5: T).(\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u)) 
+\to (pc3 c t5 t3))))) (\lambda (t5: T).(\lambda (H4: (subst0 i u t2 
+t5)).(\lambda (e: C).(\lambda (H5: (getl i c1 (CHead e (Bind Abbr) 
+u))).(pc3_t t0 c1 t5 (pc3_s c1 t5 t0 (pc3_pr2_fsubst0_back c1 t0 t2 H0 i u c1 
+t5 (fsubst0_snd i u c1 t2 t5 H4) e H5)) t3 H1))))) (\lambda (c0: C).(\lambda 
+(H4: (csubst0 i u c1 c0)).(\lambda (e: C).(\lambda (H5: (getl i c1 (CHead e 
+(Bind Abbr) u))).(pc3_t t0 c0 t2 (pc3_s c0 t2 t0 (pc3_pr2_fsubst0_back c1 t0 
+t2 H0 i u c0 t2 (fsubst0_fst i u c1 t2 c0 H4) e H5)) t3 (H2 i u c0 t0 
+(fsubst0_fst i u c1 t0 c0 H4) e H5)))))) (\lambda (t5: T).(\lambda (H4: 
+(subst0 i u t2 t5)).(\lambda (c0: C).(\lambda (H5: (csubst0 i u c1 
+c0)).(\lambda (e: C).(\lambda (H6: (getl i c1 (CHead e (Bind Abbr) 
+u))).(pc3_t t0 c0 t5 (pc3_s c0 t5 t0 (pc3_pr2_fsubst0_back c1 t0 t2 H0 i u c0 
+t5 (fsubst0_both i u c1 t2 t5 H4 c0 H5) e H6)) t3 (H2 i u c0 t0 (fsubst0_fst 
+i u c1 t0 c0 H5) e H6)))))))) c2 t4 H3)))))))))))) t1 t H)))).
+

diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/fwd.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/fwd.ma
new file mode 100644 (file)
index 0000000..a393d0f
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/fwd.ma
@@ -0,0 +1,279 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/fwd".
+
+include "pc3/props.ma".
+
+include "pr3/fwd.ma".
+
+theorem pc3_gen_sort:
+ \forall (c: C).(\forall (m: nat).(\forall (n: nat).((pc3 c (TSort m) (TSort 
+n)) \to (eq nat m n))))
+\def
+ \lambda (c: C).(\lambda (m: nat).(\lambda (n: nat).(\lambda (H: (pc3 c 
+(TSort m) (TSort n))).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr3 c 
+(TSort m) t)) (\lambda (t: T).(pr3 c (TSort n) t)) (eq nat m n) (\lambda (x: 
+T).(\lambda (H1: (pr3 c (TSort m) x)).(\lambda (H2: (pr3 c (TSort n) x)).(let 
+H3 \def (eq_ind T x (\lambda (t: T).(eq T t (TSort n))) (pr3_gen_sort c x n 
+H2) (TSort m) (pr3_gen_sort c x m H1)) in (let H4 \def (f_equal T nat 
+(\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort n0) 
+\Rightarrow n0 | (TLRef _) \Rightarrow m | (THead _ _ _) \Rightarrow m])) 
+(TSort m) (TSort n) H3) in H4))))) H0))))).
+
+theorem pc3_gen_abst:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).(\forall (t1: T).(\forall 
+(t2: T).((pc3 c (THead (Bind Abst) u1 t1) (THead (Bind Abst) u2 t2)) \to 
+(land (pc3 c u1 u2) (\forall (b: B).(\forall (u: T).(pc3 (CHead c (Bind b) u) 
+t1 t2)))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (t1: T).(\lambda 
+(t2: T).(\lambda (H: (pc3 c (THead (Bind Abst) u1 t1) (THead (Bind Abst) u2 
+t2))).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr3 c (THead (Bind Abst) 
+u1 t1) t)) (\lambda (t: T).(pr3 c (THead (Bind Abst) u2 t2) t)) (land (pc3 c 
+u1 u2) (\forall (b: B).(\forall (u: T).(pc3 (CHead c (Bind b) u) t1 t2)))) 
+(\lambda (x: T).(\lambda (H1: (pr3 c (THead (Bind Abst) u1 t1) x)).(\lambda 
+(H2: (pr3 c (THead (Bind Abst) u2 t2) x)).(let H3 \def (pr3_gen_abst c u2 t2 
+x H2) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead 
+(Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3))) 
+(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead 
+c (Bind b) u) t2 t3))))) (land (pc3 c u1 u2) (\forall (b: B).(\forall (u: 
+T).(pc3 (CHead c (Bind b) u) t1 t2)))) (\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (H4: (eq T x (THead (Bind Abst) x0 x1))).(\lambda (H5: (pr3 c u2 
+x0)).(\lambda (H6: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+t2 x1))))).(let H7 \def (pr3_gen_abst c u1 t1 x H1) in (ex3_2_ind T T 
+(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u3 t3)))) 
+(\lambda (u3: T).(\lambda (_: T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda 
+(t3: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 t3))))) 
+(land (pc3 c u1 u2) (\forall (b: B).(\forall (u: T).(pc3 (CHead c (Bind b) u) 
+t1 t2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H8: (eq T x (THead 
+(Bind Abst) x2 x3))).(\lambda (H9: (pr3 c u1 x2)).(\lambda (H10: ((\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 x3))))).(let H11 \def 
+(eq_ind T x (\lambda (t: T).(eq T t (THead (Bind Abst) x0 x1))) H4 (THead 
+(Bind Abst) x2 x3) H8) in (let H12 \def (f_equal T T (\lambda (e: T).(match e 
+in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x2 | (TLRef _) 
+\Rightarrow x2 | (THead _ t _) \Rightarrow t])) (THead (Bind Abst) x2 x3) 
+(THead (Bind Abst) x0 x1) H11) in ((let H13 \def (f_equal T T (\lambda (e: 
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x3 | 
+(TLRef _) \Rightarrow x3 | (THead _ _ t) \Rightarrow t])) (THead (Bind Abst) 
+x2 x3) (THead (Bind Abst) x0 x1) H11) in (\lambda (H14: (eq T x2 x0)).(let 
+H15 \def (eq_ind T x3 (\lambda (t: T).(\forall (b: B).(\forall (u: T).(pr3 
+(CHead c (Bind b) u) t1 t)))) H10 x1 H13) in (let H16 \def (eq_ind T x2 
+(\lambda (t: T).(pr3 c u1 t)) H9 x0 H14) in (conj (pc3 c u1 u2) (\forall (b: 
+B).(\forall (u: T).(pc3 (CHead c (Bind b) u) t1 t2))) (pc3_pr3_t c u1 x0 H16 
+u2 H5) (\lambda (b: B).(\lambda (u: T).(pc3_pr3_t (CHead c (Bind b) u) t1 x1 
+(H15 b u) t2 (H6 b u))))))))) H12)))))))) H7))))))) H3))))) H0))))))).
+
+theorem pc3_gen_lift:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).(\forall (h: nat).(\forall 
+(d: nat).((pc3 c (lift h d t1) (lift h d t2)) \to (\forall (e: C).((drop h d 
+c e) \to (pc3 e t1 t2))))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (h: nat).(\lambda 
+(d: nat).(\lambda (H: (pc3 c (lift h d t1) (lift h d t2))).(\lambda (e: 
+C).(\lambda (H0: (drop h d c e)).(let H1 \def H in (ex2_ind T (\lambda (t: 
+T).(pr3 c (lift h d t1) t)) (\lambda (t: T).(pr3 c (lift h d t2) t)) (pc3 e 
+t1 t2) (\lambda (x: T).(\lambda (H2: (pr3 c (lift h d t1) x)).(\lambda (H3: 
+(pr3 c (lift h d t2) x)).(let H4 \def (pr3_gen_lift c t2 x h d H3 e H0) in 
+(ex2_ind T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr3 e 
+t2 t3)) (pc3 e t1 t2) (\lambda (x0: T).(\lambda (H5: (eq T x (lift h d 
+x0))).(\lambda (H6: (pr3 e t2 x0)).(let H7 \def (pr3_gen_lift c t1 x h d H2 e 
+H0) in (ex2_ind T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: 
+T).(pr3 e t1 t3)) (pc3 e t1 t2) (\lambda (x1: T).(\lambda (H8: (eq T x (lift 
+h d x1))).(\lambda (H9: (pr3 e t1 x1)).(let H10 \def (eq_ind T x (\lambda (t: 
+T).(eq T t (lift h d x0))) H5 (lift h d x1) H8) in (let H11 \def (eq_ind T x1 
+(\lambda (t: T).(pr3 e t1 t)) H9 x0 (lift_inj x1 x0 h d H10)) in (pc3_pr3_t e 
+t1 x0 H11 t2 H6)))))) H7))))) H4))))) H1))))))))).
+
+theorem pc3_gen_not_abst:
+ \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (t1: 
+T).(\forall (t2: T).(\forall (u1: T).(\forall (u2: T).((pc3 c (THead (Bind b) 
+u1 t1) (THead (Bind Abst) u2 t2)) \to (pc3 (CHead c (Bind b) u1) t1 (lift (S 
+O) O (THead (Bind Abst) u2 t2))))))))))
+\def
+ \lambda (b: B).(B_ind (\lambda (b0: B).((not (eq B b0 Abst)) \to (\forall 
+(c: C).(\forall (t1: T).(\forall (t2: T).(\forall (u1: T).(\forall (u2: 
+T).((pc3 c (THead (Bind b0) u1 t1) (THead (Bind Abst) u2 t2)) \to (pc3 (CHead 
+c (Bind b0) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2))))))))))) (\lambda 
+(_: (not (eq B Abbr Abst))).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: 
+T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H0: (pc3 c (THead (Bind Abbr) 
+u1 t1) (THead (Bind Abst) u2 t2))).(let H1 \def H0 in (ex2_ind T (\lambda (t: 
+T).(pr3 c (THead (Bind Abbr) u1 t1) t)) (\lambda (t: T).(pr3 c (THead (Bind 
+Abst) u2 t2) t)) (pc3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O (THead (Bind 
+Abst) u2 t2))) (\lambda (x: T).(\lambda (H2: (pr3 c (THead (Bind Abbr) u1 t1) 
+x)).(\lambda (H3: (pr3 c (THead (Bind Abst) u2 t2) x)).(let H4 \def 
+(pr3_gen_abbr c u1 t1 x H2) in (or_ind (ex3_2 T T (\lambda (u3: T).(\lambda 
+(t3: T).(eq T x (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: 
+T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr3 (CHead c (Bind Abbr) 
+u1) t1 t3)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O x)) (pc3 (CHead 
+c (Bind Abbr) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2))) (\lambda (H5: 
+(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 
+t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c u1 u3))) (\lambda (_: 
+T).(\lambda (t3: T).(pr3 (CHead c (Bind Abbr) u1) t1 t3))))).(ex3_2_ind T T 
+(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 t3)))) 
+(\lambda (u3: T).(\lambda (_: T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda 
+(t3: T).(pr3 (CHead c (Bind Abbr) u1) t1 t3))) (pc3 (CHead c (Bind Abbr) u1) 
+t1 (lift (S O) O (THead (Bind Abst) u2 t2))) (\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (H6: (eq T x (THead (Bind Abbr) x0 x1))).(\lambda (_: (pr3 c u1 
+x0)).(\lambda (_: (pr3 (CHead c (Bind Abbr) u1) t1 x1)).(let H9 \def 
+(pr3_gen_abst c u2 t2 x H3) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: 
+T).(eq T x (THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 
+c u2 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall (b0: B).(\forall (u: 
+T).(pr3 (CHead c (Bind b0) u) t2 t3))))) (pc3 (CHead c (Bind Abbr) u1) t1 
+(lift (S O) O (THead (Bind Abst) u2 t2))) (\lambda (x2: T).(\lambda (x3: 
+T).(\lambda (H10: (eq T x (THead (Bind Abst) x2 x3))).(\lambda (_: (pr3 c u2 
+x2)).(\lambda (_: ((\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) 
+u) t2 x3))))).(let H13 \def (eq_ind T x (\lambda (t: T).(eq T t (THead (Bind 
+Abbr) x0 x1))) H6 (THead (Bind Abst) x2 x3) H10) in (let H14 \def (eq_ind T 
+(THead (Bind Abst) x2 x3) (\lambda (ee: T).(match ee in T return (\lambda (_: 
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
+(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with 
+[(Bind b0) \Rightarrow (match b0 in B return (\lambda (_: B).Prop) with [Abbr 
+\Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat 
+_) \Rightarrow False])])) I (THead (Bind Abbr) x0 x1) H13) in (False_ind (pc3 
+(CHead c (Bind Abbr) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2))) 
+H14)))))))) H9))))))) H5)) (\lambda (H5: (pr3 (CHead c (Bind Abbr) u1) t1 
+(lift (S O) O x))).(let H6 \def (pr3_gen_abst c u2 t2 x H3) in (ex3_2_ind T T 
+(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u3 t3)))) 
+(\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3))) (\lambda (_: T).(\lambda 
+(t3: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t2 
+t3))))) (pc3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O (THead (Bind Abst) u2 
+t2))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T x (THead (Bind 
+Abst) x0 x1))).(\lambda (H8: (pr3 c u2 x0)).(\lambda (H9: ((\forall (b0: 
+B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t2 x1))))).(let H10 \def 
+(eq_ind T x (\lambda (t: T).(pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O 
+t))) H5 (THead (Bind Abst) x0 x1) H7) in (pc3_pr3_t (CHead c (Bind Abbr) u1) 
+t1 (lift (S O) O (THead (Bind Abst) x0 x1)) H10 (lift (S O) O (THead (Bind 
+Abst) u2 t2)) (pr3_lift (CHead c (Bind Abbr) u1) c (S O) O (drop_drop (Bind 
+Abbr) O c c (drop_refl c) u1) (THead (Bind Abst) u2 t2) (THead (Bind Abst) x0 
+x1) (pr3_head_12 c u2 x0 H8 (Bind Abst) t2 x1 (H9 Abst x0)))))))))) H6))) 
+H4))))) H1))))))))) (\lambda (H: (not (eq B Abst Abst))).(\lambda (c: 
+C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (u1: T).(\lambda (u2: 
+T).(\lambda (_: (pc3 c (THead (Bind Abst) u1 t1) (THead (Bind Abst) u2 
+t2))).(let H1 \def (match (H (refl_equal B Abst)) in False return (\lambda 
+(_: False).(pc3 (CHead c (Bind Abst) u1) t1 (lift (S O) O (THead (Bind Abst) 
+u2 t2)))) with []) in H1)))))))) (\lambda (_: (not (eq B Void 
+Abst))).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (u1: 
+T).(\lambda (u2: T).(\lambda (H0: (pc3 c (THead (Bind Void) u1 t1) (THead 
+(Bind Abst) u2 t2))).(let H1 \def H0 in (ex2_ind T (\lambda (t: T).(pr3 c 
+(THead (Bind Void) u1 t1) t)) (\lambda (t: T).(pr3 c (THead (Bind Abst) u2 
+t2) t)) (pc3 (CHead c (Bind Void) u1) t1 (lift (S O) O (THead (Bind Abst) u2 
+t2))) (\lambda (x: T).(\lambda (H2: (pr3 c (THead (Bind Void) u1 t1) 
+x)).(\lambda (H3: (pr3 c (THead (Bind Abst) u2 t2) x)).(let H4 \def 
+(pr3_gen_void c u1 t1 x H2) in (or_ind (ex3_2 T T (\lambda (u3: T).(\lambda 
+(t3: T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: 
+T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall (b0: B).(\forall 
+(u: T).(pr3 (CHead c (Bind b0) u) t1 t3)))))) (pr3 (CHead c (Bind Void) u1) 
+t1 (lift (S O) O x)) (pc3 (CHead c (Bind Void) u1) t1 (lift (S O) O (THead 
+(Bind Abst) u2 t2))) (\lambda (H5: (ex3_2 T T (\lambda (u3: T).(\lambda (t3: 
+T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 
+c u1 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall (b0: B).(\forall (u: 
+T).(pr3 (CHead c (Bind b0) u) t1 t3))))))).(ex3_2_ind T T (\lambda (u3: 
+T).(\lambda (t3: T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall 
+(b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t1 t3))))) (pc3 (CHead c 
+(Bind Void) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2))) (\lambda (x0: 
+T).(\lambda (x1: T).(\lambda (H6: (eq T x (THead (Bind Void) x0 
+x1))).(\lambda (_: (pr3 c u1 x0)).(\lambda (_: ((\forall (b0: B).(\forall (u: 
+T).(pr3 (CHead c (Bind b0) u) t1 x1))))).(let H9 \def (pr3_gen_abst c u2 t2 x 
+H3) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind 
+Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3))) (\lambda (_: 
+T).(\lambda (t3: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) 
+u) t2 t3))))) (pc3 (CHead c (Bind Void) u1) t1 (lift (S O) O (THead (Bind 
+Abst) u2 t2))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H10: (eq T x 
+(THead (Bind Abst) x2 x3))).(\lambda (_: (pr3 c u2 x2)).(\lambda (_: 
+((\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t2 x3))))).(let 
+H13 \def (eq_ind T x (\lambda (t: T).(eq T t (THead (Bind Void) x0 x1))) H6 
+(THead (Bind Abst) x2 x3) H10) in (let H14 \def (eq_ind T (THead (Bind Abst) 
+x2 x3) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
+\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b0) 
+\Rightarrow (match b0 in B return (\lambda (_: B).Prop) with [Abbr 
+\Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat 
+_) \Rightarrow False])])) I (THead (Bind Void) x0 x1) H13) in (False_ind (pc3 
+(CHead c (Bind Void) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2))) 
+H14)))))))) H9))))))) H5)) (\lambda (H5: (pr3 (CHead c (Bind Void) u1) t1 
+(lift (S O) O x))).(let H6 \def (pr3_gen_abst c u2 t2 x H3) in (ex3_2_ind T T 
+(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u3 t3)))) 
+(\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3))) (\lambda (_: T).(\lambda 
+(t3: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t2 
+t3))))) (pc3 (CHead c (Bind Void) u1) t1 (lift (S O) O (THead (Bind Abst) u2 
+t2))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T x (THead (Bind 
+Abst) x0 x1))).(\lambda (H8: (pr3 c u2 x0)).(\lambda (H9: ((\forall (b0: 
+B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t2 x1))))).(let H10 \def 
+(eq_ind T x (\lambda (t: T).(pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O 
+t))) H5 (THead (Bind Abst) x0 x1) H7) in (pc3_pr3_t (CHead c (Bind Void) u1) 
+t1 (lift (S O) O (THead (Bind Abst) x0 x1)) H10 (lift (S O) O (THead (Bind 
+Abst) u2 t2)) (pr3_lift (CHead c (Bind Void) u1) c (S O) O (drop_drop (Bind 
+Void) O c c (drop_refl c) u1) (THead (Bind Abst) u2 t2) (THead (Bind Abst) x0 
+x1) (pr3_head_12 c u2 x0 H8 (Bind Abst) t2 x1 (H9 Abst x0)))))))))) H6))) 
+H4))))) H1))))))))) b).
+
+theorem pc3_gen_lift_abst:
+ \forall (c: C).(\forall (t: T).(\forall (t2: T).(\forall (u2: T).(\forall 
+(h: nat).(\forall (d: nat).((pc3 c (lift h d t) (THead (Bind Abst) u2 t2)) 
+\to (\forall (e: C).((drop h d c e) \to (ex3_2 T T (\lambda (u1: T).(\lambda 
+(t1: T).(pr3 e t (THead (Bind Abst) u1 t1)))) (\lambda (u1: T).(\lambda (_: 
+T).(pr3 c u2 (lift h d u1)))) (\lambda (_: T).(\lambda (t1: T).(\forall (b: 
+B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t2 (lift h (S d) 
+t1)))))))))))))))
+\def
+ \lambda (c: C).(\lambda (t: T).(\lambda (t2: T).(\lambda (u2: T).(\lambda 
+(h: nat).(\lambda (d: nat).(\lambda (H: (pc3 c (lift h d t) (THead (Bind 
+Abst) u2 t2))).(\lambda (e: C).(\lambda (H0: (drop h d c e)).(let H1 \def H 
+in (ex2_ind T (\lambda (t0: T).(pr3 c (lift h d t) t0)) (\lambda (t0: T).(pr3 
+c (THead (Bind Abst) u2 t2) t0)) (ex3_2 T T (\lambda (u1: T).(\lambda (t1: 
+T).(pr3 e t (THead (Bind Abst) u1 t1)))) (\lambda (u1: T).(\lambda (_: 
+T).(pr3 c u2 (lift h d u1)))) (\lambda (_: T).(\lambda (t1: T).(\forall (b: 
+B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t2 (lift h (S d) t1))))))) 
+(\lambda (x: T).(\lambda (H2: (pr3 c (lift h d t) x)).(\lambda (H3: (pr3 c 
+(THead (Bind Abst) u2 t2) x)).(let H4 \def (pr3_gen_lift c t x h d H2 e H0) 
+in (ex2_ind T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr3 
+e t t3)) (ex3_2 T T (\lambda (u1: T).(\lambda (t1: T).(pr3 e t (THead (Bind 
+Abst) u1 t1)))) (\lambda (u1: T).(\lambda (_: T).(pr3 c u2 (lift h d u1)))) 
+(\lambda (_: T).(\lambda (t1: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead 
+c (Bind b) u) t2 (lift h (S d) t1))))))) (\lambda (x0: T).(\lambda (H5: (eq T 
+x (lift h d x0))).(\lambda (H6: (pr3 e t x0)).(let H7 \def (pr3_gen_abst c u2 
+t2 x H3) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead 
+(Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3))) 
+(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead 
+c (Bind b) u) t2 t3))))) (ex3_2 T T (\lambda (u1: T).(\lambda (t1: T).(pr3 e 
+t (THead (Bind Abst) u1 t1)))) (\lambda (u1: T).(\lambda (_: T).(pr3 c u2 
+(lift h d u1)))) (\lambda (_: T).(\lambda (t1: T).(\forall (b: B).(\forall 
+(u: T).(pr3 (CHead c (Bind b) u) t2 (lift h (S d) t1))))))) (\lambda (x1: 
+T).(\lambda (x2: T).(\lambda (H8: (eq T x (THead (Bind Abst) x1 
+x2))).(\lambda (H9: (pr3 c u2 x1)).(\lambda (H10: ((\forall (b: B).(\forall 
+(u: T).(pr3 (CHead c (Bind b) u) t2 x2))))).(let H11 \def (eq_ind T x 
+(\lambda (t0: T).(eq T t0 (lift h d x0))) H5 (THead (Bind Abst) x1 x2) H8) in 
+(ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x0 (THead (Bind Abst) y 
+z)))) (\lambda (y: T).(\lambda (_: T).(eq T x1 (lift h d y)))) (\lambda (_: 
+T).(\lambda (z: T).(eq T x2 (lift h (S d) z)))) (ex3_2 T T (\lambda (u1: 
+T).(\lambda (t1: T).(pr3 e t (THead (Bind Abst) u1 t1)))) (\lambda (u1: 
+T).(\lambda (_: T).(pr3 c u2 (lift h d u1)))) (\lambda (_: T).(\lambda (t1: 
+T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t2 (lift h (S d) 
+t1))))))) (\lambda (x3: T).(\lambda (x4: T).(\lambda (H12: (eq T x0 (THead 
+(Bind Abst) x3 x4))).(\lambda (H13: (eq T x1 (lift h d x3))).(\lambda (H14: 
+(eq T x2 (lift h (S d) x4))).(let H15 \def (eq_ind T x2 (\lambda (t0: 
+T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t2 t0)))) H10 
+(lift h (S d) x4) H14) in (let H16 \def (eq_ind T x1 (\lambda (t0: T).(pr3 c 
+u2 t0)) H9 (lift h d x3) H13) in (let H17 \def (eq_ind T x0 (\lambda (t0: 
+T).(pr3 e t t0)) H6 (THead (Bind Abst) x3 x4) H12) in (ex3_2_intro T T 
+(\lambda (u1: T).(\lambda (t1: T).(pr3 e t (THead (Bind Abst) u1 t1)))) 
+(\lambda (u1: T).(\lambda (_: T).(pr3 c u2 (lift h d u1)))) (\lambda (_: 
+T).(\lambda (t1: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+t2 (lift h (S d) t1)))))) x3 x4 H17 H16 H15))))))))) (lift_gen_bind Abst x1 
+x2 x0 h d H11)))))))) H7))))) H4))))) H1)))))))))).
+

diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/left.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/left.ma
new file mode 100644 (file)
index 0000000..5caeee0
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/left.ma
@@ -0,0 +1,109 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/left".
+
+include "pc3/props.ma".
+
+theorem pc3_ind_left__pc3_left_pr3:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to 
+(pc3_left c t1 t2))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 
+t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(pc3_left c t t0))) (\lambda 
+(t: T).(pc3_left_r c t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 
+c t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: 
+(pc3_left c t0 t4)).(pc3_left_ur c t3 t0 H0 t4 H2))))))) t1 t2 H)))).
+
+theorem pc3_ind_left__pc3_left_trans:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3_left c t1 t2) \to 
+(\forall (t3: T).((pc3_left c t2 t3) \to (pc3_left c t1 t3))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3_left c t1 
+t2)).(pc3_left_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (t3: 
+T).((pc3_left c t0 t3) \to (pc3_left c t t3))))) (\lambda (t: T).(\lambda 
+(t3: T).(\lambda (H0: (pc3_left c t t3)).H0))) (\lambda (t0: T).(\lambda (t3: 
+T).(\lambda (H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t3 
+t4)).(\lambda (H2: ((\forall (t5: T).((pc3_left c t4 t5) \to (pc3_left c t3 
+t5))))).(\lambda (t5: T).(\lambda (H3: (pc3_left c t4 t5)).(pc3_left_ur c t0 
+t3 H0 t5 (H2 t5 H3)))))))))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: 
+(pr2 c t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t0 t4)).(\lambda 
+(H2: ((\forall (t5: T).((pc3_left c t4 t5) \to (pc3_left c t0 
+t5))))).(\lambda (t5: T).(\lambda (H3: (pc3_left c t4 t5)).(pc3_left_ux c t0 
+t3 H0 t5 (H2 t5 H3)))))))))) t1 t2 H)))).
+
+theorem pc3_ind_left__pc3_left_sym:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3_left c t1 t2) \to 
+(pc3_left c t2 t1))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3_left c t1 
+t2)).(pc3_left_ind c (\lambda (t: T).(\lambda (t0: T).(pc3_left c t0 t))) 
+(\lambda (t: T).(pc3_left_r c t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda 
+(H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t3 
+t4)).(\lambda (H2: (pc3_left c t4 t3)).(pc3_ind_left__pc3_left_trans c t4 t3 
+H2 t0 (pc3_left_ux c t0 t3 H0 t0 (pc3_left_r c t0))))))))) (\lambda (t0: 
+T).(\lambda (t3: T).(\lambda (H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda 
+(_: (pc3_left c t0 t4)).(\lambda (H2: (pc3_left c t4 
+t0)).(pc3_ind_left__pc3_left_trans c t4 t0 H2 t3 (pc3_left_ur c t0 t3 H0 t3 
+(pc3_left_r c t3))))))))) t1 t2 H)))).
+
+theorem pc3_ind_left__pc3_left_pc3:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to 
+(pc3_left c t1 t2))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1 
+t2)).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: 
+T).(pr3 c t2 t)) (pc3_left c t1 t2) (\lambda (x: T).(\lambda (H1: (pr3 c t1 
+x)).(\lambda (H2: (pr3 c t2 x)).(pc3_ind_left__pc3_left_trans c t1 x 
+(pc3_ind_left__pc3_left_pr3 c t1 x H1) t2 (pc3_ind_left__pc3_left_sym c t2 x 
+(pc3_ind_left__pc3_left_pr3 c t2 x H2)))))) H0))))).
+
+theorem pc3_ind_left__pc3_pc3_left:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3_left c t1 t2) \to 
+(pc3 c t1 t2))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3_left c t1 
+t2)).(pc3_left_ind c (\lambda (t: T).(\lambda (t0: T).(pc3 c t t0))) (\lambda 
+(t: T).(pc3_refl c t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c 
+t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t3 t4)).(\lambda (H2: (pc3 
+c t3 t4)).(pc3_pr2_u c t3 t0 H0 t4 H2))))))) (\lambda (t0: T).(\lambda (t3: 
+T).(\lambda (H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t0 
+t4)).(\lambda (H2: (pc3 c t0 t4)).(pc3_t t0 c t3 (pc3_pr2_x c t3 t0 H0) t4 
+H2))))))) t1 t2 H)))).
+
+theorem pc3_ind_left:
+ \forall (c: C).(\forall (P: ((T \to (T \to Prop)))).(((\forall (t: T).(P t 
+t))) \to (((\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall (t3: 
+T).((pc3 c t2 t3) \to ((P t2 t3) \to (P t1 t3)))))))) \to (((\forall (t1: 
+T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall (t3: T).((pc3 c t1 t3) \to 
+((P t1 t3) \to (P t2 t3)))))))) \to (\forall (t: T).(\forall (t0: T).((pc3 c 
+t t0) \to (P t t0))))))))
+\def
+ \lambda (c: C).(\lambda (P: ((T \to (T \to Prop)))).(\lambda (H: ((\forall 
+(t: T).(P t t)))).(\lambda (H0: ((\forall (t1: T).(\forall (t2: T).((pr2 c t1 
+t2) \to (\forall (t3: T).((pc3 c t2 t3) \to ((P t2 t3) \to (P t1 
+t3))))))))).(\lambda (H1: ((\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) 
+\to (\forall (t3: T).((pc3 c t1 t3) \to ((P t1 t3) \to (P t2 
+t3))))))))).(\lambda (t: T).(\lambda (t0: T).(\lambda (H2: (pc3 c t 
+t0)).(pc3_left_ind c (\lambda (t1: T).(\lambda (t2: T).(P t1 t2))) H (\lambda 
+(t1: T).(\lambda (t2: T).(\lambda (H3: (pr2 c t1 t2)).(\lambda (t3: 
+T).(\lambda (H4: (pc3_left c t2 t3)).(\lambda (H5: (P t2 t3)).(H0 t1 t2 H3 t3 
+(pc3_ind_left__pc3_pc3_left c t2 t3 H4) H5))))))) (\lambda (t1: T).(\lambda 
+(t2: T).(\lambda (H3: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda (H4: (pc3_left 
+c t1 t3)).(\lambda (H5: (P t1 t3)).(H1 t1 t2 H3 t3 
+(pc3_ind_left__pc3_pc3_left c t1 t3 H4) H5))))))) t t0 
+(pc3_ind_left__pc3_left_pc3 c t t0 H2))))))))).
+

diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/pc1.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/pc1.ma
new file mode 100644 (file)
index 0000000..6bc1403
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/pc1.ma
@@ -0,0 +1,35 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/pc1".
+
+include "pc3/defs.ma".
+
+include "pc1/defs.ma".
+
+include "pr3/pr1.ma".
+
+theorem pc3_pc1:
+ \forall (t1: T).(\forall (t2: T).((pc1 t1 t2) \to (\forall (c: C).(pc3 c t1 
+t2))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc1 t1 t2)).(\lambda (c: 
+C).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: 
+T).(pr1 t2 t)) (pc3 c t1 t2) (\lambda (x: T).(\lambda (H1: (pr1 t1 
+x)).(\lambda (H2: (pr1 t2 x)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) 
+(\lambda (t: T).(pr3 c t2 t)) x (pr3_pr1 t1 x H1 c) (pr3_pr1 t2 x H2 c))))) 
+H0))))).
+

diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/subst1.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/subst1.ma
new file mode 100644 (file)
index 0000000..c2cbcd1
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/subst1.ma
@@ -0,0 +1,47 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/subst1".
+
+include "pc3/props.ma".
+
+include "pr3/subst1.ma".
+
+theorem pc3_gen_cabbr:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to (\forall 
+(e: C).(\forall (u: T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) 
+\to (\forall (a0: C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d 
+a0 a) \to (\forall (x1: T).((subst1 d u t1 (lift (S O) d x1)) \to (\forall 
+(x2: T).((subst1 d u t2 (lift (S O) d x2)) \to (pc3 a x1 x2))))))))))))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1 
+t2)).(\lambda (e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda (H0: (getl d 
+c (CHead e (Bind Abbr) u))).(\lambda (a0: C).(\lambda (H1: (csubst1 d u c 
+a0)).(\lambda (a: C).(\lambda (H2: (drop (S O) d a0 a)).(\lambda (x1: 
+T).(\lambda (H3: (subst1 d u t1 (lift (S O) d x1))).(\lambda (x2: T).(\lambda 
+(H4: (subst1 d u t2 (lift (S O) d x2))).(let H5 \def H in (ex2_ind T (\lambda 
+(t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) (pc3 a x1 x2) (\lambda (x: 
+T).(\lambda (H6: (pr3 c t1 x)).(\lambda (H7: (pr3 c t2 x)).(ex2_ind T 
+(\lambda (x3: T).(subst1 d u x (lift (S O) d x3))) (\lambda (x3: T).(pr3 a x2 
+x3)) (pc3 a x1 x2) (\lambda (x0: T).(\lambda (H8: (subst1 d u x (lift (S O) d 
+x0))).(\lambda (H9: (pr3 a x2 x0)).(ex2_ind T (\lambda (x3: T).(subst1 d u x 
+(lift (S O) d x3))) (\lambda (x3: T).(pr3 a x1 x3)) (pc3 a x1 x2) (\lambda 
+(x3: T).(\lambda (H10: (subst1 d u x (lift (S O) d x3))).(\lambda (H11: (pr3 
+a x1 x3)).(let H12 \def (eq_ind T x3 (\lambda (t: T).(pr3 a x1 t)) H11 x0 
+(subst1_confluence_lift x x3 u d H10 x0 H8)) in (pc3_pr3_t a x1 x0 H12 x2 
+H9))))) (pr3_gen_cabbr c t1 x H6 e u d H0 a0 H1 a H2 x1 H3))))) 
+(pr3_gen_cabbr c t2 x H7 e u d H0 a0 H1 a H2 x2 H4))))) H5))))))))))))))))).
+

diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/wcpr0.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/wcpr0.ma
new file mode 100644 (file)
index 0000000..5a8fb46
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/wcpr0.ma
@@ -0,0 +1,103 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/wcpr0".
+
+include "pc3/props.ma".
+
+include "wcpr0/getl.ma".
+
+theorem pc3_wcpr0__pc3_wcpr0_t_aux:
+ \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (k: K).(\forall 
+(u: T).(\forall (t1: T).(\forall (t2: T).((pr3 (CHead c1 k u) t1 t2) \to (pc3 
+(CHead c2 k u) t1 t2))))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(\lambda (k: 
+K).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 
+(CHead c1 k u) t1 t2)).(pr3_ind (CHead c1 k u) (\lambda (t: T).(\lambda (t0: 
+T).(pc3 (CHead c2 k u) t t0))) (\lambda (t: T).(pc3_refl (CHead c2 k u) t)) 
+(\lambda (t0: T).(\lambda (t3: T).(\lambda (H1: (pr2 (CHead c1 k u) t3 
+t0)).(\lambda (t4: T).(\lambda (_: (pr3 (CHead c1 k u) t0 t4)).(\lambda (H3: 
+(pc3 (CHead c2 k u) t0 t4)).(pc3_t t0 (CHead c2 k u) t3 (let H4 \def (match 
+H1 in pr2 return (\lambda (c: C).(\lambda (t: T).(\lambda (t5: T).(\lambda 
+(_: (pr2 c t t5)).((eq C c (CHead c1 k u)) \to ((eq T t t3) \to ((eq T t5 t0) 
+\to (pc3 (CHead c2 k u) t3 t0)))))))) with [(pr2_free c t5 t6 H4) \Rightarrow 
+(\lambda (H5: (eq C c (CHead c1 k u))).(\lambda (H6: (eq T t5 t3)).(\lambda 
+(H7: (eq T t6 t0)).(eq_ind C (CHead c1 k u) (\lambda (_: C).((eq T t5 t3) \to 
+((eq T t6 t0) \to ((pr0 t5 t6) \to (pc3 (CHead c2 k u) t3 t0))))) (\lambda 
+(H8: (eq T t5 t3)).(eq_ind T t3 (\lambda (t: T).((eq T t6 t0) \to ((pr0 t t6) 
+\to (pc3 (CHead c2 k u) t3 t0)))) (\lambda (H9: (eq T t6 t0)).(eq_ind T t0 
+(\lambda (t: T).((pr0 t3 t) \to (pc3 (CHead c2 k u) t3 t0))) (\lambda (H10: 
+(pr0 t3 t0)).(pc3_pr2_r (CHead c2 k u) t3 t0 (pr2_free (CHead c2 k u) t3 t0 
+H10))) t6 (sym_eq T t6 t0 H9))) t5 (sym_eq T t5 t3 H8))) c (sym_eq C c (CHead 
+c1 k u) H5) H6 H7 H4)))) | (pr2_delta c d u0 i H4 t5 t6 H5 t H6) \Rightarrow 
+(\lambda (H7: (eq C c (CHead c1 k u))).(\lambda (H8: (eq T t5 t3)).(\lambda 
+(H9: (eq T t t0)).(eq_ind C (CHead c1 k u) (\lambda (c0: C).((eq T t5 t3) \to 
+((eq T t t0) \to ((getl i c0 (CHead d (Bind Abbr) u0)) \to ((pr0 t5 t6) \to 
+((subst0 i u0 t6 t) \to (pc3 (CHead c2 k u) t3 t0))))))) (\lambda (H10: (eq T 
+t5 t3)).(eq_ind T t3 (\lambda (t7: T).((eq T t t0) \to ((getl i (CHead c1 k 
+u) (CHead d (Bind Abbr) u0)) \to ((pr0 t7 t6) \to ((subst0 i u0 t6 t) \to 
+(pc3 (CHead c2 k u) t3 t0)))))) (\lambda (H11: (eq T t t0)).(eq_ind T t0 
+(\lambda (t7: T).((getl i (CHead c1 k u) (CHead d (Bind Abbr) u0)) \to ((pr0 
+t3 t6) \to ((subst0 i u0 t6 t7) \to (pc3 (CHead c2 k u) t3 t0))))) (\lambda 
+(H12: (getl i (CHead c1 k u) (CHead d (Bind Abbr) u0))).(\lambda (H13: (pr0 
+t3 t6)).(\lambda (H14: (subst0 i u0 t6 t0)).(ex3_2_ind C T (\lambda (e2: 
+C).(\lambda (u2: T).(getl i (CHead c2 k u) (CHead e2 (Bind Abbr) u2)))) 
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 d e2))) (\lambda (_: C).(\lambda (u2: 
+T).(pr0 u0 u2))) (pc3 (CHead c2 k u) t3 t0) (\lambda (x0: C).(\lambda (x1: 
+T).(\lambda (H15: (getl i (CHead c2 k u) (CHead x0 (Bind Abbr) x1))).(\lambda 
+(_: (wcpr0 d x0)).(\lambda (H17: (pr0 u0 x1)).(ex2_ind T (\lambda (t7: 
+T).(subst0 i x1 t6 t7)) (\lambda (t7: T).(pr0 t0 t7)) (pc3 (CHead c2 k u) t3 
+t0) (\lambda (x: T).(\lambda (H18: (subst0 i x1 t6 x)).(\lambda (H19: (pr0 t0 
+x)).(pc3_pr2_u (CHead c2 k u) x t3 (pr2_delta (CHead c2 k u) x0 x1 i H15 t3 
+t6 H13 x H18) t0 (pc3_pr2_x (CHead c2 k u) x t0 (pr2_free (CHead c2 k u) t0 x 
+H19)))))) (pr0_subst0_fwd u0 t6 t0 i H14 x1 H17))))))) (wcpr0_getl (CHead c1 
+k u) (CHead c2 k u) (wcpr0_comp c1 c2 H u u (pr0_refl u) k) i d u0 (Bind 
+Abbr) H12))))) t (sym_eq T t t0 H11))) t5 (sym_eq T t5 t3 H10))) c (sym_eq C 
+c (CHead c1 k u) H7) H8 H9 H4 H5 H6))))]) in (H4 (refl_equal C (CHead c1 k 
+u)) (refl_equal T t3) (refl_equal T t0))) t4 H3))))))) t1 t2 H0)))))))).
+
+theorem pc3_wcpr0_t:
+ \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t1: 
+T).(\forall (t2: T).((pr3 c1 t1 t2) \to (pc3 c2 t1 t2))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(wcpr0_ind 
+(\lambda (c: C).(\lambda (c0: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 
+t2) \to (pc3 c0 t1 t2)))))) (\lambda (c: C).(\lambda (t1: T).(\lambda (t2: 
+T).(\lambda (H0: (pr3 c t1 t2)).(pc3_pr3_r c t1 t2 H0))))) (\lambda (c0: 
+C).(\lambda (c3: C).(\lambda (H0: (wcpr0 c0 c3)).(\lambda (_: ((\forall (t1: 
+T).(\forall (t2: T).((pr3 c0 t1 t2) \to (pc3 c3 t1 t2)))))).(\lambda (u1: 
+T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 u2)).(\lambda (k: K).(\lambda (t1: 
+T).(\lambda (t2: T).(\lambda (H3: (pr3 (CHead c0 k u1) t1 t2)).(let H4 \def 
+(pc3_pr2_pr3_t c0 u1 t1 t2 k H3 u2 (pr2_free c0 u1 u2 H2)) in (ex2_ind T 
+(\lambda (t: T).(pr3 (CHead c0 k u2) t1 t)) (\lambda (t: T).(pr3 (CHead c0 k 
+u2) t2 t)) (pc3 (CHead c3 k u2) t1 t2) (\lambda (x: T).(\lambda (H5: (pr3 
+(CHead c0 k u2) t1 x)).(\lambda (H6: (pr3 (CHead c0 k u2) t2 x)).(pc3_t x 
+(CHead c3 k u2) t1 (pc3_wcpr0__pc3_wcpr0_t_aux c0 c3 H0 k u2 t1 x H5) t2 
+(pc3_s (CHead c3 k u2) x t2 (pc3_wcpr0__pc3_wcpr0_t_aux c0 c3 H0 k u2 t2 x 
+H6)))))) H4))))))))))))) c1 c2 H))).
+
+theorem pc3_wcpr0:
+ \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t1: 
+T).(\forall (t2: T).((pc3 c1 t1 t2) \to (pc3 c2 t1 t2))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(\lambda (t1: 
+T).(\lambda (t2: T).(\lambda (H0: (pc3 c1 t1 t2)).(let H1 \def H0 in (ex2_ind 
+T (\lambda (t: T).(pr3 c1 t1 t)) (\lambda (t: T).(pr3 c1 t2 t)) (pc3 c2 t1 
+t2) (\lambda (x: T).(\lambda (H2: (pr3 c1 t1 x)).(\lambda (H3: (pr3 c1 t2 
+x)).(pc3_t x c2 t1 (pc3_wcpr0_t c1 c2 H t1 x H2) t2 (pc3_s c2 x t2 
+(pc3_wcpr0_t c1 c2 H t2 x H3)))))) H1))))))).
+

diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/theory.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/theory.ma
index 7f7761df1cd285e7806ce0cc5bed95571d6c9d2c..2c1ae46cce58b41dac53a8d5765b5049f16c8dfc 100644 (file)
--- a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/theory.ma
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/theory.ma
@@ -324,3 +324,55 @@ include "pc3/defs.ma".
 
 include "pc3/props.ma".
 
+include "pc3/pc1.ma".
+
+include "pc3/wcpr0.ma".
+
+include "pc3/left.ma".
+
+include "pc3/fwd.ma".
+
+include "pc3/fsubst0.ma".
+
+include "pc3/subst1.ma".
+
+include "ty3/defs.ma".
+
+include "ty3/fwd.ma".
+
+include "ty3/props.ma".
+
+include "ty3/fsubst0.ma".
+
+include "ty3/subst1.ma".
+
+include "csub3/defs.ma".
+
+include "csub3/fwd.ma".
+
+include "csub3/props.ma".
+
+include "csub3/clear.ma".
+
+include "csub3/drop.ma".
+
+include "csub3/getl.ma".
+
+include "csub3/pc3.ma".
+
+include "csub3/ty3.ma".
+
+include "ty3/pr3.ma".
+
+include "ty3/pr3_props.ma".
+
+include "ty3/tau0.ma".
+
+include "ty3/arity.ma".
+
+include "ty3/arity_props.ma".
+
+include "pc3/dec.ma".
+
+include "ty3/dec.ma".
+

diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/arity.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/arity.ma
new file mode 100644 (file)
index 0000000..23f81ba
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/arity.ma
@@ -0,0 +1,191 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/arity".
+
+include "ty3/defs.ma".
+
+include "arity/pr3.ma".
+
+include "asucc/fwd.ma".
+
+theorem ty3_arity:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c 
+t1 t2) \to (ex2 A (\lambda (a1: A).(arity g c t1 a1)) (\lambda (a1: A).(arity 
+g c t2 (asucc g a1))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
+(H: (ty3 g c t1 t2)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda 
+(t0: T).(ex2 A (\lambda (a1: A).(arity g c0 t a1)) (\lambda (a1: A).(arity g 
+c0 t0 (asucc g a1))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t: 
+T).(\lambda (_: (ty3 g c0 t3 t)).(\lambda (H1: (ex2 A (\lambda (a1: A).(arity 
+g c0 t3 a1)) (\lambda (a1: A).(arity g c0 t (asucc g a1))))).(\lambda (u: 
+T).(\lambda (t4: T).(\lambda (_: (ty3 g c0 u t4)).(\lambda (H3: (ex2 A 
+(\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g 
+a1))))).(\lambda (H4: (pc3 c0 t4 t3)).(let H5 \def H1 in (ex2_ind A (\lambda 
+(a1: A).(arity g c0 t3 a1)) (\lambda (a1: A).(arity g c0 t (asucc g a1))) 
+(ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t3 
+(asucc g a1)))) (\lambda (x: A).(\lambda (H6: (arity g c0 t3 x)).(\lambda (_: 
+(arity g c0 t (asucc g x))).(let H8 \def H3 in (ex2_ind A (\lambda (a1: 
+A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g a1))) (ex2 A 
+(\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t3 (asucc g 
+a1)))) (\lambda (x0: A).(\lambda (H9: (arity g c0 u x0)).(\lambda (H10: 
+(arity g c0 t4 (asucc g x0))).(let H11 \def H4 in (ex2_ind T (\lambda (t0: 
+T).(pr3 c0 t4 t0)) (\lambda (t0: T).(pr3 c0 t3 t0)) (ex2 A (\lambda (a1: 
+A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t3 (asucc g a1)))) 
+(\lambda (x1: T).(\lambda (H12: (pr3 c0 t4 x1)).(\lambda (H13: (pr3 c0 t3 
+x1)).(ex_intro2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity 
+g c0 t3 (asucc g a1))) x0 H9 (arity_repl g c0 t3 x H6 (asucc g x0) (leq_sym g 
+(asucc g x0) x (arity_mono g c0 x1 (asucc g x0) (arity_sred_pr3 c0 t4 x1 H12 
+g (asucc g x0) H10) x (arity_sred_pr3 c0 t3 x1 H13 g x H6)))))))) H11))))) 
+H8))))) H5)))))))))))) (\lambda (c0: C).(\lambda (m: nat).(ex_intro2 A 
+(\lambda (a1: A).(arity g c0 (TSort m) a1)) (\lambda (a1: A).(arity g c0 
+(TSort (next g m)) (asucc g a1))) (ASort O m) (arity_sort g c0 m) (arity_sort 
+g c0 (next g m))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: 
+C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr) 
+u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: (ex2 A 
+(\lambda (a1: A).(arity g d u a1)) (\lambda (a1: A).(arity g d t (asucc g 
+a1))))).(let H3 \def H2 in (ex2_ind A (\lambda (a1: A).(arity g d u a1)) 
+(\lambda (a1: A).(arity g d t (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g 
+c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0 (lift (S n) O t) (asucc g 
+a1)))) (\lambda (x: A).(\lambda (H4: (arity g d u x)).(\lambda (H5: (arity g 
+d t (asucc g x))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (TLRef n) a1)) 
+(\lambda (a1: A).(arity g c0 (lift (S n) O t) (asucc g a1))) x (arity_abbr g 
+c0 d u n H0 x H4) (arity_lift g d t (asucc g x) H5 c0 (S n) O (getl_drop Abbr 
+c0 d u n H0)))))) H3)))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda 
+(d: C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abst) 
+u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: (ex2 A 
+(\lambda (a1: A).(arity g d u a1)) (\lambda (a1: A).(arity g d t (asucc g 
+a1))))).(let H3 \def H2 in (ex2_ind A (\lambda (a1: A).(arity g d u a1)) 
+(\lambda (a1: A).(arity g d t (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g 
+c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0 (lift (S n) O u) (asucc g 
+a1)))) (\lambda (x: A).(\lambda (H4: (arity g d u x)).(\lambda (_: (arity g d 
+t (asucc g x))).(let H_x \def (leq_asucc g x) in (let H6 \def H_x in (ex_ind 
+A (\lambda (a0: A).(leq g x (asucc g a0))) (ex2 A (\lambda (a1: A).(arity g 
+c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0 (lift (S n) O u) (asucc g 
+a1)))) (\lambda (x0: A).(\lambda (H7: (leq g x (asucc g x0))).(ex_intro2 A 
+(\lambda (a1: A).(arity g c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0 
+(lift (S n) O u) (asucc g a1))) x0 (arity_abst g c0 d u n H0 x0 (arity_repl g 
+d u x H4 (asucc g x0) H7)) (arity_lift g d u (asucc g x0) (arity_repl g d u x 
+H4 (asucc g x0) H7) c0 (S n) O (getl_drop Abst c0 d u n H0))))) H6)))))) 
+H3)))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (t: T).(\lambda (_: 
+(ty3 g c0 u t)).(\lambda (H1: (ex2 A (\lambda (a1: A).(arity g c0 u a1)) 
+(\lambda (a1: A).(arity g c0 t (asucc g a1))))).(\lambda (b: B).(\lambda (t3: 
+T).(\lambda (t4: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t3 
+t4)).(\lambda (H3: (ex2 A (\lambda (a1: A).(arity g (CHead c0 (Bind b) u) t3 
+a1)) (\lambda (a1: A).(arity g (CHead c0 (Bind b) u) t4 (asucc g 
+a1))))).(\lambda (t0: T).(\lambda (H4: (ty3 g (CHead c0 (Bind b) u) t4 
+t0)).(\lambda (H5: (ex2 A (\lambda (a1: A).(arity g (CHead c0 (Bind b) u) t4 
+a1)) (\lambda (a1: A).(arity g (CHead c0 (Bind b) u) t0 (asucc g a1))))).(let 
+H6 \def H1 in (ex2_ind A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: 
+A).(arity g c0 t (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g c0 (THead 
+(Bind b) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind b) u t4) (asucc 
+g a1)))) (\lambda (x: A).(\lambda (H7: (arity g c0 u x)).(\lambda (_: (arity 
+g c0 t (asucc g x))).(let H9 \def H3 in (ex2_ind A (\lambda (a1: A).(arity g 
+(CHead c0 (Bind b) u) t3 a1)) (\lambda (a1: A).(arity g (CHead c0 (Bind b) u) 
+t4 (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Bind b) u t3) 
+a1)) (\lambda (a1: A).(arity g c0 (THead (Bind b) u t4) (asucc g a1)))) 
+(\lambda (x0: A).(\lambda (H10: (arity g (CHead c0 (Bind b) u) t3 
+x0)).(\lambda (H11: (arity g (CHead c0 (Bind b) u) t4 (asucc g x0))).(let H_x 
+\def (leq_asucc g x) in (let H12 \def H_x in (ex_ind A (\lambda (a0: A).(leq 
+g x (asucc g a0))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Bind b) u t3) 
+a1)) (\lambda (a1: A).(arity g c0 (THead (Bind b) u t4) (asucc g a1)))) 
+(\lambda (x1: A).(\lambda (H13: (leq g x (asucc g x1))).((match b in B return 
+(\lambda (b0: B).((ty3 g (CHead c0 (Bind b0) u) t4 t0) \to ((ex2 A (\lambda 
+(a1: A).(arity g (CHead c0 (Bind b0) u) t4 a1)) (\lambda (a1: A).(arity g 
+(CHead c0 (Bind b0) u) t0 (asucc g a1)))) \to ((arity g (CHead c0 (Bind b0) 
+u) t3 x0) \to ((arity g (CHead c0 (Bind b0) u) t4 (asucc g x0)) \to (ex2 A 
+(\lambda (a1: A).(arity g c0 (THead (Bind b0) u t3) a1)) (\lambda (a1: 
+A).(arity g c0 (THead (Bind b0) u t4) (asucc g a1))))))))) with [Abbr 
+\Rightarrow (\lambda (_: (ty3 g (CHead c0 (Bind Abbr) u) t4 t0)).(\lambda (_: 
+(ex2 A (\lambda (a1: A).(arity g (CHead c0 (Bind Abbr) u) t4 a1)) (\lambda 
+(a1: A).(arity g (CHead c0 (Bind Abbr) u) t0 (asucc g a1))))).(\lambda (H16: 
+(arity g (CHead c0 (Bind Abbr) u) t3 x0)).(\lambda (H17: (arity g (CHead c0 
+(Bind Abbr) u) t4 (asucc g x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0 
+(THead (Bind Abbr) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind Abbr) 
+u t4) (asucc g a1))) x0 (arity_bind g Abbr not_abbr_abst c0 u x H7 t3 x0 H16) 
+(arity_bind g Abbr not_abbr_abst c0 u x H7 t4 (asucc g x0) H17)))))) | Abst 
+\Rightarrow (\lambda (_: (ty3 g (CHead c0 (Bind Abst) u) t4 t0)).(\lambda (_: 
+(ex2 A (\lambda (a1: A).(arity g (CHead c0 (Bind Abst) u) t4 a1)) (\lambda 
+(a1: A).(arity g (CHead c0 (Bind Abst) u) t0 (asucc g a1))))).(\lambda (H16: 
+(arity g (CHead c0 (Bind Abst) u) t3 x0)).(\lambda (H17: (arity g (CHead c0 
+(Bind Abst) u) t4 (asucc g x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0 
+(THead (Bind Abst) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind Abst) 
+u t4) (asucc g a1))) (AHead x1 x0) (arity_head g c0 u x1 (arity_repl g c0 u x 
+H7 (asucc g x1) H13) t3 x0 H16) (arity_repl g c0 (THead (Bind Abst) u t4) 
+(AHead x1 (asucc g x0)) (arity_head g c0 u x1 (arity_repl g c0 u x H7 (asucc 
+g x1) H13) t4 (asucc g x0) H17) (asucc g (AHead x1 x0)) (leq_refl g (asucc g 
+(AHead x1 x0))))))))) | Void \Rightarrow (\lambda (_: (ty3 g (CHead c0 (Bind 
+Void) u) t4 t0)).(\lambda (_: (ex2 A (\lambda (a1: A).(arity g (CHead c0 
+(Bind Void) u) t4 a1)) (\lambda (a1: A).(arity g (CHead c0 (Bind Void) u) t0 
+(asucc g a1))))).(\lambda (H16: (arity g (CHead c0 (Bind Void) u) t3 
+x0)).(\lambda (H17: (arity g (CHead c0 (Bind Void) u) t4 (asucc g 
+x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Bind Void) u t3) a1)) 
+(\lambda (a1: A).(arity g c0 (THead (Bind Void) u t4) (asucc g a1))) x0 
+(arity_bind g Void not_void_abst c0 u x H7 t3 x0 H16) (arity_bind g Void 
+not_void_abst c0 u x H7 t4 (asucc g x0) H17))))))]) H4 H5 H10 H11))) 
+H12)))))) H9))))) H6))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda 
+(u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda (H1: (ex2 A (\lambda (a1: 
+A).(arity g c0 w a1)) (\lambda (a1: A).(arity g c0 u (asucc g 
+a1))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead (Bind 
+Abst) u t))).(\lambda (H3: (ex2 A (\lambda (a1: A).(arity g c0 v a1)) 
+(\lambda (a1: A).(arity g c0 (THead (Bind Abst) u t) (asucc g a1))))).(let H4 
+\def H1 in (ex2_ind A (\lambda (a1: A).(arity g c0 w a1)) (\lambda (a1: 
+A).(arity g c0 u (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g c0 (THead 
+(Flat Appl) w v) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w 
+(THead (Bind Abst) u t)) (asucc g a1)))) (\lambda (x: A).(\lambda (H5: (arity 
+g c0 w x)).(\lambda (H6: (arity g c0 u (asucc g x))).(let H7 \def H3 in 
+(ex2_ind A (\lambda (a1: A).(arity g c0 v a1)) (\lambda (a1: A).(arity g c0 
+(THead (Bind Abst) u t) (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g c0 
+(THead (Flat Appl) w v) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Appl) 
+w (THead (Bind Abst) u t)) (asucc g a1)))) (\lambda (x0: A).(\lambda (H8: 
+(arity g c0 v x0)).(\lambda (H9: (arity g c0 (THead (Bind Abst) u t) (asucc g 
+x0))).(let H10 \def (arity_gen_abst g c0 u t (asucc g x0) H9) in (ex3_2_ind A 
+A (\lambda (a1: A).(\lambda (a2: A).(eq A (asucc g x0) (AHead a1 a2)))) 
+(\lambda (a1: A).(\lambda (_: A).(arity g c0 u (asucc g a1)))) (\lambda (_: 
+A).(\lambda (a2: A).(arity g (CHead c0 (Bind Abst) u) t a2))) (ex2 A (\lambda 
+(a1: A).(arity g c0 (THead (Flat Appl) w v) a1)) (\lambda (a1: A).(arity g c0 
+(THead (Flat Appl) w (THead (Bind Abst) u t)) (asucc g a1)))) (\lambda (x1: 
+A).(\lambda (x2: A).(\lambda (H11: (eq A (asucc g x0) (AHead x1 
+x2))).(\lambda (H12: (arity g c0 u (asucc g x1))).(\lambda (H13: (arity g 
+(CHead c0 (Bind Abst) u) t x2)).(let H14 \def (sym_equal A (asucc g x0) 
+(AHead x1 x2) H11) in (let H15 \def (asucc_gen_head g x1 x2 x0 H14) in 
+(ex2_ind A (\lambda (a0: A).(eq A x0 (AHead x1 a0))) (\lambda (a0: A).(eq A 
+x2 (asucc g a0))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v) 
+a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u 
+t)) (asucc g a1)))) (\lambda (x3: A).(\lambda (H16: (eq A x0 (AHead x1 
+x3))).(\lambda (H17: (eq A x2 (asucc g x3))).(let H18 \def (eq_ind A x2 
+(\lambda (a: A).(arity g (CHead c0 (Bind Abst) u) t a)) H13 (asucc g x3) H17) 
+in (let H19 \def (eq_ind A x0 (\lambda (a: A).(arity g c0 v a)) H8 (AHead x1 
+x3) H16) in (ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v) 
+a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u 
+t)) (asucc g a1))) x3 (arity_appl g c0 w x1 (arity_repl g c0 w x H5 x1 
+(leq_sym g x1 x (asucc_inj g x1 x (arity_mono g c0 u (asucc g x1) H12 (asucc 
+g x) H6)))) v x3 H19) (arity_appl g c0 w x H5 (THead (Bind Abst) u t) (asucc 
+g x3) (arity_head g c0 u x H6 t (asucc g x3) H18)))))))) H15)))))))) H10))))) 
+H7))))) H4))))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: 
+T).(\lambda (_: (ty3 g c0 t3 t4)).(\lambda (H1: (ex2 A (\lambda (a1: 
+A).(arity g c0 t3 a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g 
+a1))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t4 t0)).(\lambda (_: (ex2 A 
+(\lambda (a1: A).(arity g c0 t4 a1)) (\lambda (a1: A).(arity g c0 t0 (asucc g 
+a1))))).(let H4 \def H1 in (ex2_ind A (\lambda (a1: A).(arity g c0 t3 a1)) 
+(\lambda (a1: A).(arity g c0 t4 (asucc g a1))) (ex2 A (\lambda (a1: A).(arity 
+g c0 (THead (Flat Cast) t4 t3) a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g 
+a1)))) (\lambda (x: A).(\lambda (H5: (arity g c0 t3 x)).(\lambda (H6: (arity 
+g c0 t4 (asucc g x))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Flat 
+Cast) t4 t3) a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g a1))) x 
+(arity_cast g c0 t4 x H6 t3 H5) H6)))) H4)))))))))) c t1 t2 H))))).
+

diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/arity_props.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/arity_props.ma
new file mode 100644 (file)
index 0000000..098a32c
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/arity_props.ma
@@ -0,0 +1,78 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/arity_props".
+
+include "ty3/arity.ma".
+
+include "ty3/fwd.ma".
+
+include "sc3/arity.ma".
+
+theorem ty3_predicative:
+ \forall (g: G).(\forall (c: C).(\forall (v: T).(\forall (t: T).(\forall (u: 
+T).((ty3 g c (THead (Bind Abst) v t) u) \to ((pc3 c u v) \to (\forall (P: 
+Prop).P)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (u: 
+T).(\lambda (H: (ty3 g c (THead (Bind Abst) v t) u)).(\lambda (H0: (pc3 c u 
+v)).(\lambda (P: Prop).(let H1 \def H in (ex4_3_ind T T T (\lambda (t2: 
+T).(\lambda (_: T).(\lambda (_: T).(pc3 c (THead (Bind Abst) v t2) u)))) 
+(\lambda (_: T).(\lambda (t0: T).(\lambda (_: T).(ty3 g c v t0)))) (\lambda 
+(t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) v) t 
+t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t1: T).(ty3 g (CHead c 
+(Bind Abst) v) t2 t1)))) P (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: 
+T).(\lambda (_: (pc3 c (THead (Bind Abst) v x0) u)).(\lambda (H3: (ty3 g c v 
+x1)).(\lambda (_: (ty3 g (CHead c (Bind Abst) v) t x0)).(\lambda (_: (ty3 g 
+(CHead c (Bind Abst) v) x0 x2)).(let H_y \def (ty3_conv g c v x1 H3 (THead 
+(Bind Abst) v t) u H H0) in (let H_x \def (ty3_arity g c (THead (Bind Abst) v 
+t) v H_y) in (let H6 \def H_x in (ex2_ind A (\lambda (a1: A).(arity g c 
+(THead (Bind Abst) v t) a1)) (\lambda (a1: A).(arity g c v (asucc g a1))) P 
+(\lambda (x: A).(\lambda (H7: (arity g c (THead (Bind Abst) v t) x)).(\lambda 
+(H8: (arity g c v (asucc g x))).(let H9 \def (arity_gen_abst g c v t x H7) in 
+(ex3_2_ind A A (\lambda (a1: A).(\lambda (a2: A).(eq A x (AHead a1 a2)))) 
+(\lambda (a1: A).(\lambda (_: A).(arity g c v (asucc g a1)))) (\lambda (_: 
+A).(\lambda (a2: A).(arity g (CHead c (Bind Abst) v) t a2))) P (\lambda (x3: 
+A).(\lambda (x4: A).(\lambda (H10: (eq A x (AHead x3 x4))).(\lambda (H11: 
+(arity g c v (asucc g x3))).(\lambda (_: (arity g (CHead c (Bind Abst) v) t 
+x4)).(let H13 \def (eq_ind A x (\lambda (a: A).(arity g c v (asucc g a))) H8 
+(AHead x3 x4) H10) in (leq_ahead_asucc_false g x3 (asucc g x4) (arity_mono g 
+c v (asucc g (AHead x3 x4)) H13 (asucc g x3) H11) P))))))) H9))))) 
+H6))))))))))) (ty3_gen_bind g Abst c v t u H1)))))))))).
+
+theorem ty3_acyclic:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t 
+u) \to ((pc3 c u t) \to (\forall (P: Prop).P))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H: 
+(ty3 g c t u)).(\lambda (H0: (pc3 c u t)).(\lambda (P: Prop).(let H_y \def 
+(ty3_conv g c t u H t u H H0) in (let H_x \def (ty3_arity g c t t H_y) in 
+(let H1 \def H_x in (ex2_ind A (\lambda (a1: A).(arity g c t a1)) (\lambda 
+(a1: A).(arity g c t (asucc g a1))) P (\lambda (x: A).(\lambda (H2: (arity g 
+c t x)).(\lambda (H3: (arity g c t (asucc g x))).(leq_asucc_false g x 
+(arity_mono g c t (asucc g x) H3 x H2) P)))) H1)))))))))).
+
+theorem ty3_sn3:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t 
+u) \to (sn3 c t)))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H: 
+(ty3 g c t u)).(let H_x \def (ty3_arity g c t u H) in (let H0 \def H_x in 
+(ex2_ind A (\lambda (a1: A).(arity g c t a1)) (\lambda (a1: A).(arity g c u 
+(asucc g a1))) (sn3 c t) (\lambda (x: A).(\lambda (H1: (arity g c t 
+x)).(\lambda (_: (arity g c u (asucc g x))).(sc3_sn3 g x c t (sc3_arity g c t 
+x H1))))) H0))))))).
+

diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/dec.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/dec.ma
new file mode 100644 (file)
index 0000000..efbd85e
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/dec.ma
@@ -0,0 +1,456 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/dec".
+
+include "ty3/pr3_props.ma".
+
+include "pc3/dec.ma".
+
+include "getl/flt.ma".
+
+include "getl/dec.ma".
+
+theorem ty3_inference:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(or (ex T (\lambda (t2: 
+T).(ty3 g c t1 t2))) (\forall (t2: T).((ty3 g c t1 t2) \to (\forall (P: 
+Prop).P))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(flt_wf_ind (\lambda (c0: 
+C).(\lambda (t: T).(or (ex T (\lambda (t2: T).(ty3 g c0 t t2))) (\forall (t2: 
+T).((ty3 g c0 t t2) \to (\forall (P: Prop).P)))))) (\lambda (c2: C).(\lambda 
+(t2: T).(match t2 in T return (\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)))))) with [(TSort n) \Rightarrow (\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)))) | (TLRef n) \Rightarrow (\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 (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)).((match x1 in B return (\lambda (b: B).((getl n c2 (CHead x0 (Bind b) 
+x2)) \to (or (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: 
+T).((ty3 g c2 (TLRef n) t3) \to (\forall (P: Prop).P)))))) with [Abbr 
+\Rightarrow (\lambda (H6: (getl n c2 (CHead x0 (Bind Abbr) x2))).(or_introl 
+(ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 
+(TLRef n) t3) \to (\forall (P: Prop).P))) (ex_intro T (\lambda (t3: T).(ty3 g 
+c2 (TLRef n) t3)) (lift (S n) O x) (ty3_abbr g n c2 x0 x2 H6 x H5)))) | Abst 
+\Rightarrow (\lambda (H6: (getl n c2 (CHead x0 (Bind Abst) x2))).(or_introl 
+(ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 
+(TLRef n) t3) \to (\forall (P: Prop).P))) (ex_intro T (\lambda (t3: T).(ty3 g 
+c2 (TLRef n) t3)) (lift (S n) O x2) (ty3_abst g n c2 x0 x2 H6 x H5)))) | Void 
+\Rightarrow (\lambda (H6: (getl n c2 (CHead x0 (Bind Void) x2))).(or_intror 
+(ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 
+(TLRef n) t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H7: (ty3 
+g c2 (TLRef n) t3)).(\lambda (P: Prop).(or_ind (ex3_3 C T T (\lambda (_: 
+C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda 
+(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) 
+(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T 
+T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) 
+t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e 
+(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u 
+t))))) P (\lambda (H8: (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda 
+(t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda (u: 
+T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e: 
+C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T 
+(\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) 
+t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e 
+(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u 
+t)))) P (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (pc3 
+c2 (lift (S n) O x5) t3)).(\lambda (H10: (getl n c2 (CHead x3 (Bind Abbr) 
+x4))).(\lambda (_: (ty3 g x3 x4 x5)).(let H12 \def (eq_ind C (CHead x0 (Bind 
+Void) x2) (\lambda (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 (\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)))))))]) 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).(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: 
+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 (_: 
+T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda 
+(u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e: 
+C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T T (\lambda 
+(_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) 
+(\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind 
+Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u 
+t))))) P (\lambda (H3: (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda 
+(t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda (u: 
+T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e: 
+C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T 
+(\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) 
+t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e 
+(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u 
+t)))) P (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 
+c2 (lift (S n) O x2) t3)).(\lambda (H5: (getl n c2 (CHead x0 (Bind Abbr) 
+x1))).(\lambda (_: (ty3 g x0 x1 x2)).(H1 (CHead x0 (Bind Abbr) x1) H5 
+P))))))) H3)) (\lambda (H3: (ex3_3 C T T (\lambda (_: C).(\lambda (u: 
+T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: C).(\lambda 
+(u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) (\lambda (e: 
+C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T 
+(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) 
+t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e 
+(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u 
+t)))) P (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 
+c2 (lift (S n) O x1) t3)).(\lambda (H5: (getl n c2 (CHead x0 (Bind Abst) 
+x1))).(\lambda (_: (ty3 g x0 x1 x2)).(H1 (CHead x0 (Bind Abst) x1) H5 
+P))))))) H3)) (ty3_gen_lref g c2 t3 n H2))))))) H0)))) | (THead k t t0) 
+\Rightarrow (\lambda (H: ((\forall (c1: C).(\forall (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))))))))).((match k in K 
+return (\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)))))) with [(Bind b) \Rightarrow 
+(\lambda (H0: ((\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 H1 \def (H0 c2 t 
+(flt_thead_sx (Bind b) c2 t t0)) in (or_ind (ex T (\lambda (t3: T).(ty3 g c2 
+t t3))) (\forall (t3: T).((ty3 g c2 t t3) \to (\forall (P: Prop).P))) (or (ex 
+T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0) t3))) (\forall (t3: 
+T).((ty3 g c2 (THead (Bind b) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda 
+(H2: (ex T (\lambda (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 (H3: (ty3 g c2 t x)).(let H4 \def 
+(H0 (CHead c2 (Bind b) t) t0 (flt_shift (Bind b) c2 t t0)) in (or_ind (ex T 
+(\lambda (t3: T).(ty3 g (CHead c2 (Bind b) t) t0 t3))) (\forall (t3: T).((ty3 
+g (CHead c2 (Bind b) t) t0 t3) \to (\forall (P: Prop).P))) (or (ex T (\lambda 
+(t3: T).(ty3 g c2 (THead (Bind b) t t0) t3))) (\forall (t3: T).((ty3 g c2 
+(THead (Bind b) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (H5: (ex T 
+(\lambda (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 (H6: (ty3 g 
+(CHead c2 (Bind b) t) t0 x0)).(ex_ind T (\lambda (t3: T).(ty3 g (CHead c2 
+(Bind b) t) x0 t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) t 
+t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) \to (\forall 
+(P: Prop).P)))) (\lambda (x1: T).(\lambda (H7: (ty3 g (CHead c2 (Bind b) t) 
+x0 x1)).(or_introl (ex T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0) 
+t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) \to (\forall (P: 
+Prop).P))) (ex_intro T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0) t3)) 
+(THead (Bind b) t x0) (ty3_bind g c2 t x H3 b t0 x0 H6 x1 H7))))) 
+(ty3_correct g (CHead c2 (Bind b) t) t0 x0 H6)))) H5)) (\lambda (H5: 
+((\forall (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 (H6: (ty3 g c2 (THead (Bind b) t t0) 
+t3)).(\lambda (P: Prop).(ex4_3_ind T T T (\lambda (t4: T).(\lambda (_: 
+T).(\lambda (_: T).(pc3 c2 (THead (Bind b) t t4) t3)))) (\lambda (_: 
+T).(\lambda (t5: T).(\lambda (_: T).(ty3 g c2 t t5)))) (\lambda (t4: 
+T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind b) t) t0 t4)))) 
+(\lambda (t4: T).(\lambda (_: T).(\lambda (t6: T).(ty3 g (CHead c2 (Bind b) 
+t) t4 t6)))) P (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda 
+(_: (pc3 c2 (THead (Bind b) t x0) t3)).(\lambda (_: (ty3 g c2 t x1)).(\lambda 
+(H9: (ty3 g (CHead c2 (Bind b) t) t0 x0)).(\lambda (_: (ty3 g (CHead c2 (Bind 
+b) t) x0 x2)).(H5 x0 H9 P)))))))) (ty3_gen_bind g b c2 t t0 t3 H6))))))) 
+H4)))) H2)) (\lambda (H2: ((\forall (t3: T).((ty3 g c2 t 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 (H3: (ty3 g c2 (THead (Bind b) t t0) 
+t3)).(\lambda (P: Prop).(ex4_3_ind T T T (\lambda (t4: T).(\lambda (_: 
+T).(\lambda (_: T).(pc3 c2 (THead (Bind b) t t4) t3)))) (\lambda (_: 
+T).(\lambda (t5: T).(\lambda (_: T).(ty3 g c2 t t5)))) (\lambda (t4: 
+T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind b) t) t0 t4)))) 
+(\lambda (t4: T).(\lambda (_: T).(\lambda (t6: T).(ty3 g (CHead c2 (Bind b) 
+t) t4 t6)))) P (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda 
+(_: (pc3 c2 (THead (Bind b) t x0) t3)).(\lambda (H5: (ty3 g c2 t 
+x1)).(\lambda (_: (ty3 g (CHead c2 (Bind b) t) t0 x0)).(\lambda (_: (ty3 g 
+(CHead c2 (Bind b) t) x0 x2)).(H2 x1 H5 P)))))))) (ty3_gen_bind g b c2 t t0 
+t3 H3))))))) H1))) | (Flat f) \Rightarrow (\lambda (H0: ((\forall (c1: 
+C).(\forall (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))))))))).((match f in F return (\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)))))) with [Appl \Rightarrow (\lambda (H1: 
+((\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 H2 \def (H1 c2 t (flt_thead_sx 
+(Flat Appl) c2 t t0)) in (or_ind (ex T (\lambda (t3: T).(ty3 g c2 t t3))) 
+(\forall (t3: T).((ty3 g c2 t t3) \to (\forall (P: Prop).P))) (or (ex T 
+(\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: 
+T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to (\forall (P: Prop).P)))) 
+(\lambda (H3: (ex T (\lambda (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 (H4: (ty3 g c2 t 
+x)).(let H5 \def (H1 c2 t0 (flt_thead_dx (Flat Appl) c2 t t0)) in (or_ind (ex 
+T (\lambda (t3: T).(ty3 g c2 t0 t3))) (\forall (t3: T).((ty3 g c2 t0 t3) \to 
+(\forall (P: Prop).P))) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat 
+Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) 
+\to (\forall (P: Prop).P)))) (\lambda (H6: (ex T (\lambda (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 (\forall (P: Prop).P)))) (\lambda (x0: 
+T).(\lambda (H7: (ty3 g c2 t0 x0)).(ex_ind T (\lambda (t3: T).(ty3 g c2 x0 
+t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) 
+(\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to (\forall (P: 
+Prop).P)))) (\lambda (x1: T).(\lambda (H8: (ty3 g c2 x0 x1)).(ex_ind T 
+(\lambda (t3: T).(ty3 g c2 x t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead 
+(Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) 
+t3) \to (\forall (P: Prop).P)))) (\lambda (x2: T).(\lambda (H9: (ty3 g c2 x 
+x2)).(let H10 \def (ty3_sn3 g c2 x x2 H9) in (let H_x \def (nf2_sn3 c2 x H10) 
+in (let H11 \def H_x in (ex2_ind T (\lambda (u: T).(pr3 c2 x u)) (\lambda (u: 
+T).(nf2 c2 u)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) 
+t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to (\forall 
+(P: Prop).P)))) (\lambda (x3: T).(\lambda (H12: (pr3 c2 x x3)).(\lambda (H13: 
+(nf2 c2 x3)).(let H14 \def (ty3_sred_pr3 c2 x x3 H12 g x2 H9) in (let H_x0 
+\def (pc3_abst_dec g c2 x0 x1 H8 x3 x2 H14) in (let H15 \def H_x0 in (or_ind 
+(ex4_2 T T (\lambda (u: T).(\lambda (_: T).(pc3 c2 x0 (THead (Bind Abst) x3 
+u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c2 (THead (Bind Abst) v2 u) 
+x1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c2 x3 v2))) (\lambda (_: 
+T).(\lambda (v2: T).(nf2 c2 v2)))) (\forall (u: T).((pc3 c2 x0 (THead (Bind 
+Abst) x3 u)) \to (\forall (P: Prop).P))) (or (ex T (\lambda (t3: T).(ty3 g c2 
+(THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) 
+t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (H16: (ex4_2 T T (\lambda (u: 
+T).(\lambda (_: T).(pc3 c2 x0 (THead (Bind Abst) x3 u)))) (\lambda (u: 
+T).(\lambda (v2: T).(ty3 g c2 (THead (Bind Abst) v2 u) x1))) (\lambda (_: 
+T).(\lambda (v2: T).(pr3 c2 x3 v2))) (\lambda (_: T).(\lambda (v2: T).(nf2 c2 
+v2))))).(ex4_2_ind T T (\lambda (u: T).(\lambda (_: T).(pc3 c2 x0 (THead 
+(Bind Abst) x3 u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c2 (THead (Bind 
+Abst) v2 u) x1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c2 x3 v2))) (\lambda 
+(_: T).(\lambda (v2: T).(nf2 c2 v2))) (or (ex T (\lambda (t3: T).(ty3 g c2 
+(THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) 
+t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (x4: T).(\lambda (x5: 
+T).(\lambda (H17: (pc3 c2 x0 (THead (Bind Abst) x3 x4))).(\lambda (H18: (ty3 
+g c2 (THead (Bind Abst) x5 x4) x1)).(\lambda (H19: (pr3 c2 x3 x5)).(\lambda 
+(_: (nf2 c2 x5)).(let H_y \def (nf2_pr3_unfold c2 x3 x5 H19 H13) in (let H21 
+\def (eq_ind_r T x5 (\lambda (t3: T).(pr3 c2 x3 t3)) H19 x3 H_y) in (let H22 
+\def (eq_ind_r T x5 (\lambda (t3: T).(ty3 g c2 (THead (Bind Abst) t3 x4) x1)) 
+H18 x3 H_y) in (or_introl (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) 
+t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to 
+(\forall (P: Prop).P))) (ex_intro T (\lambda (t3: T).(ty3 g c2 (THead (Flat 
+Appl) t t0) t3)) (THead (Flat Appl) t (THead (Bind Abst) x3 x4)) (ty3_appl g 
+c2 t x3 (ty3_tred g c2 t x H4 x3 H12) t0 x4 (ty3_conv g c2 (THead (Bind Abst) 
+x3 x4) x1 H22 t0 x0 H7 H17))))))))))))) H16)) (\lambda (H16: ((\forall (u: 
+T).((pc3 c2 x0 (THead (Bind Abst) x3 u)) \to (\forall (P: 
+Prop).P))))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t 
+t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to 
+(\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H17: (ty3 g c2 (THead 
+(Flat Appl) t t0) t3)).(\lambda (P: Prop).(ex3_2_ind T T (\lambda (u: 
+T).(\lambda (t4: T).(pc3 c2 (THead (Flat Appl) t (THead (Bind Abst) u t4)) 
+t3))) (\lambda (u: T).(\lambda (t4: T).(ty3 g c2 t0 (THead (Bind Abst) u 
+t4)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c2 t u))) P (\lambda (x4: 
+T).(\lambda (x5: T).(\lambda (_: (pc3 c2 (THead (Flat Appl) t (THead (Bind 
+Abst) x4 x5)) t3)).(\lambda (H19: (ty3 g c2 t0 (THead (Bind Abst) x4 
+x5))).(\lambda (H20: (ty3 g c2 t x4)).(let H_y \def (ty3_unique g c2 t x4 H20 
+x H4) in (let H_y0 \def (ty3_unique g c2 t0 (THead (Bind Abst) x4 x5) H19 x0 
+H7) in (H16 x5 (pc3_t (THead (Bind Abst) x4 x5) c2 x0 (pc3_s c2 x0 (THead 
+(Bind Abst) x4 x5) H_y0) (THead (Bind Abst) x3 x5) (pc3_head_1 c2 x4 x3 
+(pc3_t x c2 x4 H_y x3 (pc3_pr3_r c2 x x3 H12)) (Bind Abst) x5)) P)))))))) 
+(ty3_gen_appl g c2 t t0 t3 H17))))))) H15))))))) H11)))))) (ty3_correct g c2 
+t x H4)))) (ty3_correct g c2 t0 x0 H7)))) H6)) (\lambda (H6: ((\forall (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 (H7: (ty3 g c2 (THead (Flat Appl) t t0) t3)).(\lambda (P: 
+Prop).(ex3_2_ind T T (\lambda (u: T).(\lambda (t4: T).(pc3 c2 (THead (Flat 
+Appl) t (THead (Bind Abst) u t4)) t3))) (\lambda (u: T).(\lambda (t4: T).(ty3 
+g c2 t0 (THead (Bind Abst) u t4)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c2 
+t u))) P (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead (Flat 
+Appl) t (THead (Bind Abst) x0 x1)) t3)).(\lambda (H9: (ty3 g c2 t0 (THead 
+(Bind Abst) x0 x1))).(\lambda (_: (ty3 g c2 t x0)).(H6 (THead (Bind Abst) x0 
+x1) H9 P)))))) (ty3_gen_appl g c2 t t0 t3 H7))))))) H5)))) H3)) (\lambda (H3: 
+((\forall (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 (H4: (ty3 g c2 (THead (Flat Appl) t t0) 
+t3)).(\lambda (P: Prop).(ex3_2_ind T T (\lambda (u: T).(\lambda (t4: T).(pc3 
+c2 (THead (Flat Appl) t (THead (Bind Abst) u t4)) t3))) (\lambda (u: 
+T).(\lambda (t4: T).(ty3 g c2 t0 (THead (Bind Abst) u t4)))) (\lambda (u: 
+T).(\lambda (_: T).(ty3 g c2 t u))) P (\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (_: (pc3 c2 (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) 
+t3)).(\lambda (_: (ty3 g c2 t0 (THead (Bind Abst) x0 x1))).(\lambda (H7: (ty3 
+g c2 t x0)).(H3 x0 H7 P)))))) (ty3_gen_appl g c2 t t0 t3 H4))))))) H2))) | 
+Cast \Rightarrow (\lambda (H1: ((\forall (c1: C).(\forall (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 H2 \def (H1 c2 t (flt_thead_sx (Flat Cast) c2 t t0)) in 
+(or_ind (ex T (\lambda (t3: T).(ty3 g c2 t t3))) (\forall (t3: T).((ty3 g c2 
+t t3) \to (\forall (P: Prop).P))) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead 
+(Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) 
+t3) \to (\forall (P: Prop).P)))) (\lambda (H3: (ex T (\lambda (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 (H4: (ty3 g c2 t x)).(let H5 \def (H1 c2 t0 (flt_thead_dx (Flat 
+Cast) c2 t t0)) in (or_ind (ex T (\lambda (t3: T).(ty3 g c2 t0 t3))) (\forall 
+(t3: T).((ty3 g c2 t0 t3) \to (\forall (P: Prop).P))) (or (ex T (\lambda (t3: 
+T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2 
+(THead (Flat Cast) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (H6: (ex T 
+(\lambda (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 (\forall (P: 
+Prop).P)))) (\lambda (x0: T).(\lambda (H7: (ty3 g c2 t0 x0)).(ex_ind T 
+(\lambda (t3: T).(ty3 g c2 x0 t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 
+(THead (Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) 
+t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (x1: T).(\lambda (H8: (ty3 g 
+c2 x0 x1)).(let H_x \def (pc3_dec g c2 x0 x1 H8 t x H4) in (let H9 \def H_x 
+in (or_ind (pc3 c2 x0 t) ((pc3 c2 x0 t) \to (\forall (P: Prop).P)) (or (ex T 
+(\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: 
+T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to (\forall (P: Prop).P)))) 
+(\lambda (H10: (pc3 c2 x0 t)).(or_introl (ex T (\lambda (t3: T).(ty3 g c2 
+(THead (Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) 
+t t0) t3) \to (\forall (P: Prop).P))) (ex_intro T (\lambda (t3: T).(ty3 g c2 
+(THead (Flat Cast) t t0) t3)) t (ty3_cast g c2 t0 t (ty3_conv g c2 t x H4 t0 
+x0 H7 H10) x H4)))) (\lambda (H10: (((pc3 c2 x0 t) \to (\forall (P: 
+Prop).P)))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t 
+t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to 
+(\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H11: (ty3 g c2 (THead 
+(Flat Cast) t t0) t3)).(\lambda (P: Prop).(and_ind (pc3 c2 t t3) (ty3 g c2 t0 
+t) P (\lambda (_: (pc3 c2 t t3)).(\lambda (H13: (ty3 g c2 t0 t)).(let H_y 
+\def (ty3_unique g c2 t0 t H13 x0 H7) in (H10 (pc3_s c2 x0 t H_y) P)))) 
+(ty3_gen_cast g c2 t0 t t3 H11))))))) H9))))) (ty3_correct g c2 t0 x0 H7)))) 
+H6)) (\lambda (H6: ((\forall (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 (H7: (ty3 g c2 (THead (Flat 
+Cast) t t0) t3)).(\lambda (P: Prop).(and_ind (pc3 c2 t t3) (ty3 g c2 t0 t) P 
+(\lambda (_: (pc3 c2 t t3)).(\lambda (H9: (ty3 g c2 t0 t)).(H6 t H9 P))) 
+(ty3_gen_cast g c2 t0 t t3 H7))))))) H5)))) H3)) (\lambda (H3: ((\forall (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 t0) t3))) (\forall (t3: T).((ty3 g c2 
+(THead (Flat Cast) t t0) t3) \to (\forall (P: Prop).P))) (\lambda (t3: 
+T).(\lambda (H4: (ty3 g c2 (THead (Flat Cast) t t0) t3)).(\lambda (P: 
+Prop).(and_ind (pc3 c2 t t3) (ty3 g c2 t0 t) P (\lambda (_: (pc3 c2 t 
+t3)).(\lambda (H6: (ty3 g c2 t0 t)).(ex_ind T (\lambda (t4: T).(ty3 g c2 t 
+t4)) P (\lambda (x: T).(\lambda (H7: (ty3 g c2 t x)).(H3 x H7 P))) 
+(ty3_correct g c2 t0 t H6)))) (ty3_gen_cast g c2 t0 t t3 H4))))))) H2)))]) 
+H0))]) H))]))) c t1))).
+

diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/defs.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/defs.ma
new file mode 100644 (file)
index 0000000..1eecab2
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/defs.ma
@@ -0,0 +1,46 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/defs".
+
+include "G/defs.ma".
+
+include "pc3/defs.ma".
+
+inductive ty3 (g:G): C \to (T \to (T \to Prop)) \def
+| ty3_conv: \forall (c: C).(\forall (t2: T).(\forall (t: T).((ty3 g c t2 t) 
+\to (\forall (u: T).(\forall (t1: T).((ty3 g c u t1) \to ((pc3 c t1 t2) \to 
+(ty3 g c u t2))))))))
+| ty3_sort: \forall (c: C).(\forall (m: nat).(ty3 g c (TSort m) (TSort (next 
+g m))))
+| ty3_abbr: \forall (n: nat).(\forall (c: C).(\forall (d: C).(\forall (u: 
+T).((getl n c (CHead d (Bind Abbr) u)) \to (\forall (t: T).((ty3 g d u t) \to 
+(ty3 g c (TLRef n) (lift (S n) O t))))))))
+| ty3_abst: \forall (n: nat).(\forall (c: C).(\forall (d: C).(\forall (u: 
+T).((getl n c (CHead d (Bind Abst) u)) \to (\forall (t: T).((ty3 g d u t) \to 
+(ty3 g c (TLRef n) (lift (S n) O u))))))))
+| ty3_bind: \forall (c: C).(\forall (u: T).(\forall (t: T).((ty3 g c u t) \to 
+(\forall (b: B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c (Bind b) 
+u) t1 t2) \to (\forall (t0: T).((ty3 g (CHead c (Bind b) u) t2 t0) \to (ty3 g 
+c (THead (Bind b) u t1) (THead (Bind b) u t2)))))))))))
+| ty3_appl: \forall (c: C).(\forall (w: T).(\forall (u: T).((ty3 g c w u) \to 
+(\forall (v: T).(\forall (t: T).((ty3 g c v (THead (Bind Abst) u t)) \to (ty3 
+g c (THead (Flat Appl) w v) (THead (Flat Appl) w (THead (Bind Abst) u 
+t)))))))))
+| ty3_cast: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c t1 t2) 
+\to (\forall (t0: T).((ty3 g c t2 t0) \to (ty3 g c (THead (Flat Cast) t2 t1) 
+t2)))))).
+

diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/fsubst0.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/fsubst0.ma
new file mode 100644 (file)
index 0000000..3a7c41e
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/fsubst0.ma
@@ -0,0 +1,978 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/fsubst0".
+
+include "ty3/props.ma".
+
+include "pc3/fsubst0.ma".
+
+include "csubst0/props.ma".
+
+include "getl/getl.ma".
+
+theorem ty3_fsubst0:
+ \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t: T).((ty3 g c1 
+t1 t) \to (\forall (i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t2: 
+T).((fsubst0 i u c1 t1 c2 t2) \to (\forall (e: C).((getl i c1 (CHead e (Bind 
+Abbr) u)) \to (ty3 g c2 t2 t))))))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t: T).(\lambda 
+(H: (ty3 g c1 t1 t)).(ty3_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda 
+(t2: T).(\forall (i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t3: 
+T).((fsubst0 i u c t0 c2 t3) \to (\forall (e: C).((getl i c (CHead e (Bind 
+Abbr) u)) \to (ty3 g c2 t3 t2))))))))))) (\lambda (c: C).(\lambda (t2: 
+T).(\lambda (t0: T).(\lambda (H0: (ty3 g c t2 t0)).(\lambda (H1: ((\forall 
+(i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t3: T).((fsubst0 i u c t2 
+c2 t3) \to (\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) \to (ty3 g c2 
+t3 t0)))))))))).(\lambda (u: T).(\lambda (t3: T).(\lambda (_: (ty3 g c u 
+t3)).(\lambda (H3: ((\forall (i: nat).(\forall (u0: T).(\forall (c2: 
+C).(\forall (t4: T).((fsubst0 i u0 c u c2 t4) \to (\forall (e: C).((getl i c 
+(CHead e (Bind Abbr) u0)) \to (ty3 g c2 t4 t3)))))))))).(\lambda (H4: (pc3 c 
+t3 t2)).(\lambda (i: nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda (t4: 
+T).(\lambda (H5: (fsubst0 i u0 c u c2 t4)).(fsubst0_ind i u0 c u (\lambda 
+(c0: C).(\lambda (t5: T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u0)) 
+\to (ty3 g c0 t5 t2))))) (\lambda (t5: T).(\lambda (H6: (subst0 i u0 u 
+t5)).(\lambda (e: C).(\lambda (H7: (getl i c (CHead e (Bind Abbr) 
+u0))).(ty3_conv g c t2 t0 H0 t5 t3 (H3 i u0 c t5 (fsubst0_snd i u0 c u t5 H6) 
+e H7) H4))))) (\lambda (c3: C).(\lambda (H6: (csubst0 i u0 c c3)).(\lambda 
+(e: C).(\lambda (H7: (getl i c (CHead e (Bind Abbr) u0))).(ty3_conv g c3 t2 
+t0 (H1 i u0 c3 t2 (fsubst0_fst i u0 c t2 c3 H6) e H7) u t3 (H3 i u0 c3 u 
+(fsubst0_fst i u0 c u c3 H6) e H7) (pc3_fsubst0 c t3 t2 H4 i u0 c3 t3 
+(fsubst0_fst i u0 c t3 c3 H6) e H7)))))) (\lambda (t5: T).(\lambda (H6: 
+(subst0 i u0 u t5)).(\lambda (c3: C).(\lambda (H7: (csubst0 i u0 c 
+c3)).(\lambda (e: C).(\lambda (H8: (getl i c (CHead e (Bind Abbr) 
+u0))).(ty3_conv g c3 t2 t0 (H1 i u0 c3 t2 (fsubst0_fst i u0 c t2 c3 H7) e H8) 
+t5 t3 (H3 i u0 c3 t5 (fsubst0_both i u0 c u t5 H6 c3 H7) e H8) (pc3_fsubst0 c 
+t3 t2 H4 i u0 c3 t3 (fsubst0_fst i u0 c t3 c3 H7) e H8)))))))) c2 t4 
+H5)))))))))))))))) (\lambda (c: C).(\lambda (m: nat).(\lambda (i: 
+nat).(\lambda (u: T).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H0: (fsubst0 
+i u c (TSort m) c2 t2)).(fsubst0_ind i u c (TSort m) (\lambda (c0: 
+C).(\lambda (t0: T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) \to 
+(ty3 g c0 t0 (TSort (next g m))))))) (\lambda (t3: T).(\lambda (H1: (subst0 i 
+u (TSort m) t3)).(\lambda (e: C).(\lambda (_: (getl i c (CHead e (Bind Abbr) 
+u))).(subst0_gen_sort u t3 i m H1 (ty3 g c t3 (TSort (next g m)))))))) 
+(\lambda (c3: C).(\lambda (_: (csubst0 i u c c3)).(\lambda (e: C).(\lambda 
+(_: (getl i c (CHead e (Bind Abbr) u))).(ty3_sort g c3 m))))) (\lambda (t3: 
+T).(\lambda (H1: (subst0 i u (TSort m) t3)).(\lambda (c3: C).(\lambda (_: 
+(csubst0 i u c c3)).(\lambda (e: C).(\lambda (_: (getl i c (CHead e (Bind 
+Abbr) u))).(subst0_gen_sort u t3 i m H1 (ty3 g c3 t3 (TSort (next g 
+m)))))))))) c2 t2 H0)))))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d: 
+C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind Abbr) u))).(\lambda 
+(t0: T).(\lambda (H1: (ty3 g d u t0)).(\lambda (H2: ((\forall (i: 
+nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 d u c2 
+t2) \to (\forall (e: C).((getl i d (CHead e (Bind Abbr) u0)) \to (ty3 g c2 t2 
+t0)))))))))).(\lambda (i: nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda 
+(t2: T).(\lambda (H3: (fsubst0 i u0 c (TLRef n) c2 t2)).(fsubst0_ind i u0 c 
+(TLRef n) (\lambda (c0: C).(\lambda (t3: T).(\forall (e: C).((getl i c (CHead 
+e (Bind Abbr) u0)) \to (ty3 g c0 t3 (lift (S n) O t0)))))) (\lambda (t3: 
+T).(\lambda (H4: (subst0 i u0 (TLRef n) t3)).(\lambda (e: C).(\lambda (H5: 
+(getl i c (CHead e (Bind Abbr) u0))).(and_ind (eq nat n i) (eq T t3 (lift (S 
+n) O u0)) (ty3 g c t3 (lift (S n) O t0)) (\lambda (H6: (eq nat n i)).(\lambda 
+(H7: (eq T t3 (lift (S n) O u0))).(eq_ind_r T (lift (S n) O u0) (\lambda (t4: 
+T).(ty3 g c t4 (lift (S n) O t0))) (let H8 \def (eq_ind_r nat i (\lambda (n0: 
+nat).(getl n0 c (CHead e (Bind Abbr) u0))) H5 n H6) in (let H9 \def (eq_ind C 
+(CHead d (Bind Abbr) u) (\lambda (c0: C).(getl n c c0)) H0 (CHead e (Bind 
+Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) n H0 (CHead e (Bind Abbr) u0) 
+H8)) in (let H10 \def (f_equal C C (\lambda (e0: C).(match e0 in C return 
+(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow 
+c0])) (CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c (CHead d 
+(Bind Abbr) u) n H0 (CHead e (Bind Abbr) u0) H8)) in ((let H11 \def (f_equal 
+C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) 
+\Rightarrow u | (CHead _ _ t4) \Rightarrow t4])) (CHead d (Bind Abbr) u) 
+(CHead e (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) n H0 (CHead e 
+(Bind Abbr) u0) H8)) in (\lambda (H12: (eq C d e)).(let H13 \def (eq_ind_r C 
+e (\lambda (c0: C).(getl n c (CHead c0 (Bind Abbr) u0))) H9 d H12) in (let 
+H14 \def (eq_ind_r T u0 (\lambda (t4: T).(getl n c (CHead d (Bind Abbr) t4))) 
+H13 u H11) in (eq_ind T u (\lambda (t4: T).(ty3 g c (lift (S n) O t4) (lift 
+(S n) O t0))) (ty3_lift g d u t0 H1 c O (S n) (getl_drop Abbr c d u n H14)) 
+u0 H11))))) H10)))) t3 H7))) (subst0_gen_lref u0 t3 i n H4)))))) (\lambda 
+(c3: C).(\lambda (H4: (csubst0 i u0 c c3)).(\lambda (e: C).(\lambda (H5: 
+(getl i c (CHead e (Bind Abbr) u0))).(lt_le_e n i (ty3 g c3 (TLRef n) (lift 
+(S n) O t0)) (\lambda (H6: (lt n i)).(let H7 \def (csubst0_getl_lt i n H6 c 
+c3 u0 H4 (CHead d (Bind Abbr) u) H0) in (or4_ind (getl n c3 (CHead d (Bind 
+Abbr) u)) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: 
+T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) u1)))))) 
+(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 
+(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) (ex3_4 B C C T 
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C 
+(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (u1: T).(getl n c3 (CHead e2 (Bind b) 
+u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (minus i (S n)) u0 e1 e2)))))) (ex4_5 B C C T T (\lambda (b: 
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C 
+(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e2 
+(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda 
+(u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) (\lambda (_: 
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+(minus i (S n)) u0 e1 e2))))))) (ty3 g c3 (TLRef n) (lift (S n) O t0)) 
+(\lambda (H8: (getl n c3 (CHead d (Bind Abbr) u))).(ty3_abbr g n c3 d u H8 t0 
+H1)) (\lambda (H8: (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda 
+(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) 
+u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(getl n c3 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 
+w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: 
+T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) u1)))))) 
+(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 
+(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w))))) (ty3 g c3 (TLRef n) 
+(lift (S n) O t0)) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: 
+T).(\lambda (x3: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead x1 
+(Bind x0) x2))).(\lambda (H10: (getl n c3 (CHead x1 (Bind x0) x3))).(\lambda 
+(H11: (subst0 (minus i (S n)) u0 x2 x3)).(let H12 \def (f_equal C C (\lambda 
+(e0: C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow 
+d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abbr) u) (CHead x1 (Bind 
+x0) x2) H9) in ((let H13 \def (f_equal C B (\lambda (e0: C).(match e0 in C 
+return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) 
+\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) 
+\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead 
+x1 (Bind x0) x2) H9) in ((let H14 \def (f_equal C T (\lambda (e0: C).(match 
+e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ 
+t3) \Rightarrow t3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H9) in 
+(\lambda (H15: (eq B Abbr x0)).(\lambda (H16: (eq C d x1)).(let H17 \def 
+(eq_ind_r T x2 (\lambda (t3: T).(subst0 (minus i (S n)) u0 t3 x3)) H11 u H14) 
+in (let H18 \def (eq_ind_r C x1 (\lambda (c0: C).(getl n c3 (CHead c0 (Bind 
+x0) x3))) H10 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b: B).(getl n 
+c3 (CHead d (Bind b) x3))) H18 Abbr H15) in (let H20 \def (eq_ind nat (minus 
+i n) (\lambda (n0: nat).(getl n0 (CHead d (Bind Abbr) x3) (CHead e (Bind 
+Abbr) u0))) (getl_conf_le i (CHead e (Bind Abbr) u0) c3 (csubst0_getl_ge i i 
+(le_n i) c c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead d (Bind Abbr) x3) n 
+H19 (le_S_n n i (le_S (S n) i H6))) (S (minus i (S n))) (minus_x_Sy i n H6)) 
+in (ty3_abbr g n c3 d x3 H19 t0 (H2 (minus i (S n)) u0 d x3 (fsubst0_snd 
+(minus i (S n)) u0 d u x3 H17) e (getl_gen_S (Bind Abbr) d (CHead e (Bind 
+Abbr) u0) x3 (minus i (S n)) H20)))))))))) H13)) H12))))))))) H8)) (\lambda 
+(H8: (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda 
+(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl n c3 (CHead e2 
+(Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(csubst0 (minus i (S n)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda 
+(b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind 
+Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda 
+(e2: C).(\lambda (u1: T).(getl n c3 (CHead e2 (Bind b) u1)))))) (\lambda (_: 
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) 
+u0 e1 e2))))) (ty3 g c3 (TLRef n) (lift (S n) O t0)) (\lambda (x0: 
+B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H9: (eq C 
+(CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3))).(\lambda (H10: (getl n c3 
+(CHead x2 (Bind x0) x3))).(\lambda (H11: (csubst0 (minus i (S n)) u0 x1 
+x2)).(let H12 \def (f_equal C C (\lambda (e0: C).(match e0 in C return 
+(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow 
+c0])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H13 \def 
+(f_equal C B (\lambda (e0: C).(match e0 in C return (\lambda (_: C).B) with 
+[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k in K return 
+(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow 
+Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H14 
+\def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) 
+with [(CSort _) \Rightarrow u | (CHead _ _ t3) \Rightarrow t3])) (CHead d 
+(Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in (\lambda (H15: (eq B Abbr 
+x0)).(\lambda (H16: (eq C d x1)).(let H17 \def (eq_ind_r T x3 (\lambda (t3: 
+T).(getl n c3 (CHead x2 (Bind x0) t3))) H10 u H14) in (let H18 \def (eq_ind_r 
+C x1 (\lambda (c0: C).(csubst0 (minus i (S n)) u0 c0 x2)) H11 d H16) in (let 
+H19 \def (eq_ind_r B x0 (\lambda (b: B).(getl n c3 (CHead x2 (Bind b) u))) 
+H17 Abbr H15) in (let H20 \def (eq_ind nat (minus i n) (\lambda (n0: 
+nat).(getl n0 (CHead x2 (Bind Abbr) u) (CHead e (Bind Abbr) u0))) 
+(getl_conf_le i (CHead e (Bind Abbr) u0) c3 (csubst0_getl_ge i i (le_n i) c 
+c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead x2 (Bind Abbr) u) n H19 (le_S_n 
+n i (le_S (S n) i H6))) (S (minus i (S n))) (minus_x_Sy i n H6)) in (ty3_abbr 
+g n c3 x2 u H19 t0 (H2 (minus i (S n)) u0 x2 u (fsubst0_fst (minus i (S n)) 
+u0 d u x2 H18) e (csubst0_getl_ge_back (minus i (S n)) (minus i (S n)) (le_n 
+(minus i (S n))) d x2 u0 H18 (CHead e (Bind Abbr) u0) (getl_gen_S (Bind Abbr) 
+x2 (CHead e (Bind Abbr) u0) u (minus i (S n)) H20))))))))))) H13)) 
+H12))))))))) H8)) (\lambda (H8: (ex4_5 B C C T T (\lambda (b: B).(\lambda 
+(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind 
+Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e2 
+(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda 
+(u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) (\lambda (_: 
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+(minus i (S n)) u0 e1 e2)))))))).(ex4_5_ind B C C T T (\lambda (b: 
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C 
+(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e2 
+(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda 
+(u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) (\lambda (_: 
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+(minus i (S n)) u0 e1 e2)))))) (ty3 g c3 (TLRef n) (lift (S n) O t0)) 
+(\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda 
+(x4: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) 
+x3))).(\lambda (H10: (getl n c3 (CHead x2 (Bind x0) x4))).(\lambda (H11: 
+(subst0 (minus i (S n)) u0 x3 x4)).(\lambda (H12: (csubst0 (minus i (S n)) u0 
+x1 x2)).(let H13 \def (f_equal C C (\lambda (e0: C).(match e0 in C return 
+(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow 
+c0])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H14 \def 
+(f_equal C B (\lambda (e0: C).(match e0 in C return (\lambda (_: C).B) with 
+[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k in K return 
+(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow 
+Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H15 
+\def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) 
+with [(CSort _) \Rightarrow u | (CHead _ _ t3) \Rightarrow t3])) (CHead d 
+(Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in (\lambda (H16: (eq B Abbr 
+x0)).(\lambda (H17: (eq C d x1)).(let H18 \def (eq_ind_r T x3 (\lambda (t3: 
+T).(subst0 (minus i (S n)) u0 t3 x4)) H11 u H15) in (let H19 \def (eq_ind_r C 
+x1 (\lambda (c0: C).(csubst0 (minus i (S n)) u0 c0 x2)) H12 d H17) in (let 
+H20 \def (eq_ind_r B x0 (\lambda (b: B).(getl n c3 (CHead x2 (Bind b) x4))) 
+H10 Abbr H16) in (let H21 \def (eq_ind nat (minus i n) (\lambda (n0: 
+nat).(getl n0 (CHead x2 (Bind Abbr) x4) (CHead e (Bind Abbr) u0))) 
+(getl_conf_le i (CHead e (Bind Abbr) u0) c3 (csubst0_getl_ge i i (le_n i) c 
+c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead x2 (Bind Abbr) x4) n H20 (le_S_n 
+n i (le_S (S n) i H6))) (S (minus i (S n))) (minus_x_Sy i n H6)) in (ty3_abbr 
+g n c3 x2 x4 H20 t0 (H2 (minus i (S n)) u0 x2 x4 (fsubst0_both (minus i (S 
+n)) u0 d u x4 H18 x2 H19) e (csubst0_getl_ge_back (minus i (S n)) (minus i (S 
+n)) (le_n (minus i (S n))) d x2 u0 H19 (CHead e (Bind Abbr) u0) (getl_gen_S 
+(Bind Abbr) x2 (CHead e (Bind Abbr) u0) x4 (minus i (S n)) H21))))))))))) 
+H14)) H13))))))))))) H8)) H7))) (\lambda (H6: (le i n)).(ty3_abbr g n c3 d u 
+(csubst0_getl_ge i n H6 c c3 u0 H4 (CHead d (Bind Abbr) u) H0) t0 H1))))))) 
+(\lambda (t3: T).(\lambda (H4: (subst0 i u0 (TLRef n) t3)).(\lambda (c3: 
+C).(\lambda (H5: (csubst0 i u0 c c3)).(\lambda (e: C).(\lambda (H6: (getl i c 
+(CHead e (Bind Abbr) u0))).(and_ind (eq nat n i) (eq T t3 (lift (S n) O u0)) 
+(ty3 g c3 t3 (lift (S n) O t0)) (\lambda (H7: (eq nat n i)).(\lambda (H8: (eq 
+T t3 (lift (S n) O u0))).(eq_ind_r T (lift (S n) O u0) (\lambda (t4: T).(ty3 
+g c3 t4 (lift (S n) O t0))) (let H9 \def (eq_ind_r nat i (\lambda (n0: 
+nat).(getl n0 c (CHead e (Bind Abbr) u0))) H6 n H7) in (let H10 \def 
+(eq_ind_r nat i (\lambda (n0: nat).(csubst0 n0 u0 c c3)) H5 n H7) in (let H11 
+\def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c0: C).(getl n c c0)) H0 
+(CHead e (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) n H0 (CHead e 
+(Bind Abbr) u0) H9)) in (let H12 \def (f_equal C C (\lambda (e0: C).(match e0 
+in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) 
+\Rightarrow c0])) (CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono 
+c (CHead d (Bind Abbr) u) n H0 (CHead e (Bind Abbr) u0) H9)) in ((let H13 
+\def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) 
+with [(CSort _) \Rightarrow u | (CHead _ _ t4) \Rightarrow t4])) (CHead d 
+(Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) 
+n H0 (CHead e (Bind Abbr) u0) H9)) in (\lambda (H14: (eq C d e)).(let H15 
+\def (eq_ind_r C e (\lambda (c0: C).(getl n c (CHead c0 (Bind Abbr) u0))) H11 
+d H14) in (let H16 \def (eq_ind_r T u0 (\lambda (t4: T).(getl n c (CHead d 
+(Bind Abbr) t4))) H15 u H13) in (let H17 \def (eq_ind_r T u0 (\lambda (t4: 
+T).(csubst0 n t4 c c3)) H10 u H13) in (eq_ind T u (\lambda (t4: T).(ty3 g c3 
+(lift (S n) O t4) (lift (S n) O t0))) (ty3_lift g d u t0 H1 c3 O (S n) 
+(getl_drop Abbr c3 d u n (csubst0_getl_ge n n (le_n n) c c3 u H17 (CHead d 
+(Bind Abbr) u) H16))) u0 H13)))))) H12))))) t3 H8))) (subst0_gen_lref u0 t3 i 
+n H4)))))))) c2 t2 H3)))))))))))))) (\lambda (n: nat).(\lambda (c: 
+C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind 
+Abst) u))).(\lambda (t0: T).(\lambda (H1: (ty3 g d u t0)).(\lambda (H2: 
+((\forall (i: nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2: 
+T).((fsubst0 i u0 d u c2 t2) \to (\forall (e: C).((getl i d (CHead e (Bind 
+Abbr) u0)) \to (ty3 g c2 t2 t0)))))))))).(\lambda (i: nat).(\lambda (u0: 
+T).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H3: (fsubst0 i u0 c (TLRef n) 
+c2 t2)).(fsubst0_ind i u0 c (TLRef n) (\lambda (c0: C).(\lambda (t3: 
+T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u0)) \to (ty3 g c0 t3 
+(lift (S n) O u)))))) (\lambda (t3: T).(\lambda (H4: (subst0 i u0 (TLRef n) 
+t3)).(\lambda (e: C).(\lambda (H5: (getl i c (CHead e (Bind Abbr) 
+u0))).(and_ind (eq nat n i) (eq T t3 (lift (S n) O u0)) (ty3 g c t3 (lift (S 
+n) O u)) (\lambda (H6: (eq nat n i)).(\lambda (H7: (eq T t3 (lift (S n) O 
+u0))).(eq_ind_r T (lift (S n) O u0) (\lambda (t4: T).(ty3 g c t4 (lift (S n) 
+O u))) (let H8 \def (eq_ind_r nat i (\lambda (n0: nat).(getl n0 c (CHead e 
+(Bind Abbr) u0))) H5 n H6) in (let H9 \def (eq_ind C (CHead d (Bind Abst) u) 
+(\lambda (c0: C).(getl n c c0)) H0 (CHead e (Bind Abbr) u0) (getl_mono c 
+(CHead d (Bind Abst) u) n H0 (CHead e (Bind Abbr) u0) H8)) in (let H10 \def 
+(eq_ind C (CHead d (Bind Abst) u) (\lambda (ee: C).(match ee in C return 
+(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) 
+\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) 
+\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow 
+False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) 
+\Rightarrow False])])) I (CHead e (Bind Abbr) u0) (getl_mono c (CHead d (Bind 
+Abst) u) n H0 (CHead e (Bind Abbr) u0) H8)) in (False_ind (ty3 g c (lift (S 
+n) O u0) (lift (S n) O u)) H10)))) t3 H7))) (subst0_gen_lref u0 t3 i n 
+H4)))))) (\lambda (c3: C).(\lambda (H4: (csubst0 i u0 c c3)).(\lambda (e: 
+C).(\lambda (H5: (getl i c (CHead e (Bind Abbr) u0))).(lt_le_e n i (ty3 g c3 
+(TLRef n) (lift (S n) O u)) (\lambda (H6: (lt n i)).(let H7 \def 
+(csubst0_getl_lt i n H6 c c3 u0 H4 (CHead d (Bind Abst) u) H0) in (or4_ind 
+(getl n c3 (CHead d (Bind Abst) u)) (ex3_4 B C T T (\lambda (b: B).(\lambda 
+(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead 
+e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(getl n c3 (CHead e0 (Bind b) w)))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n)) 
+u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u1: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1)))))) 
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl n c3 
+(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 (minus i (S n)) u0 e1 e2)))))) (ex4_5 B C C T T 
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(_: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1))))))) (\lambda 
+(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl 
+n c3 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) 
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 (minus i (S n)) u0 e1 e2))))))) (ty3 g c3 (TLRef n) (lift (S 
+n) O u)) (\lambda (H8: (getl n c3 (CHead d (Bind Abst) u))).(ty3_abst g n c3 
+d u H8 t0 H1)) (\lambda (H8: (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: 
+C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead e0 
+(Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda 
+(w: T).(getl n c3 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 
+w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: 
+T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead e0 (Bind b) u1)))))) 
+(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 
+(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w))))) (ty3 g c3 (TLRef n) 
+(lift (S n) O u)) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda 
+(x3: T).(\lambda (H9: (eq C (CHead d (Bind Abst) u) (CHead x1 (Bind x0) 
+x2))).(\lambda (H10: (getl n c3 (CHead x1 (Bind x0) x3))).(\lambda (H11: 
+(subst0 (minus i (S n)) u0 x2 x3)).(let H12 \def (f_equal C C (\lambda (e0: 
+C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | 
+(CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) 
+x2) H9) in ((let H13 \def (f_equal C B (\lambda (e0: C).(match e0 in C return 
+(\lambda (_: C).B) with [(CSort _) \Rightarrow Abst | (CHead _ k _) 
+\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) 
+\Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead d (Bind Abst) u) (CHead 
+x1 (Bind x0) x2) H9) in ((let H14 \def (f_equal C T (\lambda (e0: C).(match 
+e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ 
+t3) \Rightarrow t3])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x2) H9) in 
+(\lambda (H15: (eq B Abst x0)).(\lambda (H16: (eq C d x1)).(let H17 \def 
+(eq_ind_r T x2 (\lambda (t3: T).(subst0 (minus i (S n)) u0 t3 x3)) H11 u H14) 
+in (let H18 \def (eq_ind_r C x1 (\lambda (c0: C).(getl n c3 (CHead c0 (Bind 
+x0) x3))) H10 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b: B).(getl n 
+c3 (CHead d (Bind b) x3))) H18 Abst H15) in (let H20 \def (eq_ind nat (minus 
+i n) (\lambda (n0: nat).(getl n0 (CHead d (Bind Abst) x3) (CHead e (Bind 
+Abbr) u0))) (getl_conf_le i (CHead e (Bind Abbr) u0) c3 (csubst0_getl_ge i i 
+(le_n i) c c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead d (Bind Abst) x3) n 
+H19 (le_S_n n i (le_S (S n) i H6))) (S (minus i (S n))) (minus_x_Sy i n H6)) 
+in (ty3_conv g c3 (lift (S n) O u) (lift (S n) O t0) (ty3_lift g d u t0 H1 c3 
+O (S n) (getl_drop Abst c3 d x3 n H19)) (TLRef n) (lift (S n) O x3) (ty3_abst 
+g n c3 d x3 H19 t0 (H2 (minus i (S n)) u0 d x3 (fsubst0_snd (minus i (S n)) 
+u0 d u x3 H17) e (getl_gen_S (Bind Abst) d (CHead e (Bind Abbr) u0) x3 (minus 
+i (S n)) H20))) (pc3_lift c3 d (S n) O (getl_drop Abst c3 d x3 n H19) x3 u 
+(pc3_pr2_x d x3 u (pr2_delta d e u0 (r (Bind Abst) (minus i (S n))) 
+(getl_gen_S (Bind Abst) d (CHead e (Bind Abbr) u0) x3 (minus i (S n)) H20) u 
+u (pr0_refl u) x3 H17))))))))))) H13)) H12))))))))) H8)) (\lambda (H8: (ex3_4 
+B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq 
+C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: 
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl n c3 (CHead e2 
+(Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(csubst0 (minus i (S n)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda 
+(b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind 
+Abst) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda 
+(e2: C).(\lambda (u1: T).(getl n c3 (CHead e2 (Bind b) u1)))))) (\lambda (_: 
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) 
+u0 e1 e2))))) (ty3 g c3 (TLRef n) (lift (S n) O u)) (\lambda (x0: B).(\lambda 
+(x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H9: (eq C (CHead d (Bind 
+Abst) u) (CHead x1 (Bind x0) x3))).(\lambda (H10: (getl n c3 (CHead x2 (Bind 
+x0) x3))).(\lambda (H11: (csubst0 (minus i (S n)) u0 x1 x2)).(let H12 \def 
+(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with 
+[(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind 
+Abst) u) (CHead x1 (Bind x0) x3) H9) in ((let H13 \def (f_equal C B (\lambda 
+(e0: C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow 
+Abst | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with 
+[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead d (Bind Abst) 
+u) (CHead x1 (Bind x0) x3) H9) in ((let H14 \def (f_equal C T (\lambda (e0: 
+C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | 
+(CHead _ _ t3) \Rightarrow t3])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) 
+x3) H9) in (\lambda (H15: (eq B Abst x0)).(\lambda (H16: (eq C d x1)).(let 
+H17 \def (eq_ind_r T x3 (\lambda (t3: T).(getl n c3 (CHead x2 (Bind x0) t3))) 
+H10 u H14) in (let H18 \def (eq_ind_r C x1 (\lambda (c0: C).(csubst0 (minus i 
+(S n)) u0 c0 x2)) H11 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b: 
+B).(getl n c3 (CHead x2 (Bind b) u))) H17 Abst H15) in (let H20 \def (eq_ind 
+nat (minus i n) (\lambda (n0: nat).(getl n0 (CHead x2 (Bind Abst) u) (CHead e 
+(Bind Abbr) u0))) (getl_conf_le i (CHead e (Bind Abbr) u0) c3 
+(csubst0_getl_ge i i (le_n i) c c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead 
+x2 (Bind Abst) u) n H19 (le_S_n n i (le_S (S n) i H6))) (S (minus i (S n))) 
+(minus_x_Sy i n H6)) in (ty3_abst g n c3 x2 u H19 t0 (H2 (minus i (S n)) u0 
+x2 u (fsubst0_fst (minus i (S n)) u0 d u x2 H18) e (csubst0_getl_ge_back 
+(minus i (S n)) (minus i (S n)) (le_n (minus i (S n))) d x2 u0 H18 (CHead e 
+(Bind Abbr) u0) (getl_gen_S (Bind Abst) x2 (CHead e (Bind Abbr) u0) u (minus 
+i (S n)) H20))))))))))) H13)) H12))))))))) H8)) (\lambda (H8: (ex4_5 B C C T 
+T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(_: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1))))))) (\lambda 
+(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl 
+n c3 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) 
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 (minus i (S n)) u0 e1 e2)))))))).(ex4_5_ind B C C T T 
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(_: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1))))))) (\lambda 
+(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl 
+n c3 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) 
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 (minus i (S n)) u0 e1 e2)))))) (ty3 g c3 (TLRef n) (lift (S 
+n) O u)) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: 
+T).(\lambda (x4: T).(\lambda (H9: (eq C (CHead d (Bind Abst) u) (CHead x1 
+(Bind x0) x3))).(\lambda (H10: (getl n c3 (CHead x2 (Bind x0) x4))).(\lambda 
+(H11: (subst0 (minus i (S n)) u0 x3 x4)).(\lambda (H12: (csubst0 (minus i (S 
+n)) u0 x1 x2)).(let H13 \def (f_equal C C (\lambda (e0: C).(match e0 in C 
+return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) 
+\Rightarrow c0])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H9) in 
+((let H14 \def (f_equal C B (\lambda (e0: C).(match e0 in C return (\lambda 
+(_: C).B) with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match 
+k in K return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) 
+\Rightarrow Abst])])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H9) in 
+((let H15 \def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda 
+(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t3) \Rightarrow t3])) 
+(CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H9) in (\lambda (H16: (eq B 
+Abst x0)).(\lambda (H17: (eq C d x1)).(let H18 \def (eq_ind_r T x3 (\lambda 
+(t3: T).(subst0 (minus i (S n)) u0 t3 x4)) H11 u H15) in (let H19 \def 
+(eq_ind_r C x1 (\lambda (c0: C).(csubst0 (minus i (S n)) u0 c0 x2)) H12 d 
+H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b: B).(getl n c3 (CHead x2 
+(Bind b) x4))) H10 Abst H16) in (let H21 \def (eq_ind nat (minus i n) 
+(\lambda (n0: nat).(getl n0 (CHead x2 (Bind Abst) x4) (CHead e (Bind Abbr) 
+u0))) (getl_conf_le i (CHead e (Bind Abbr) u0) c3 (csubst0_getl_ge i i (le_n 
+i) c c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead x2 (Bind Abst) x4) n H20 
+(le_S_n n i (le_S (S n) i H6))) (S (minus i (S n))) (minus_x_Sy i n H6)) in 
+(ty3_conv g c3 (lift (S n) O u) (lift (S n) O t0) (ty3_lift g x2 u t0 (H2 
+(minus i (S n)) u0 x2 u (fsubst0_fst (minus i (S n)) u0 d u x2 H19) e 
+(csubst0_getl_ge_back (minus i (S n)) (minus i (S n)) (le_n (minus i (S n))) 
+d x2 u0 H19 (CHead e (Bind Abbr) u0) (getl_gen_S (Bind Abst) x2 (CHead e 
+(Bind Abbr) u0) x4 (minus i (S n)) H21))) c3 O (S n) (getl_drop Abst c3 x2 x4 
+n H20)) (TLRef n) (lift (S n) O x4) (ty3_abst g n c3 x2 x4 H20 t0 (H2 (minus 
+i (S n)) u0 x2 x4 (fsubst0_both (minus i (S n)) u0 d u x4 H18 x2 H19) e 
+(csubst0_getl_ge_back (minus i (S n)) (minus i (S n)) (le_n (minus i (S n))) 
+d x2 u0 H19 (CHead e (Bind Abbr) u0) (getl_gen_S (Bind Abst) x2 (CHead e 
+(Bind Abbr) u0) x4 (minus i (S n)) H21)))) (pc3_lift c3 x2 (S n) O (getl_drop 
+Abst c3 x2 x4 n H20) x4 u (pc3_fsubst0 d u u (pc3_refl d u) (minus i (S n)) 
+u0 x2 x4 (fsubst0_both (minus i (S n)) u0 d u x4 H18 x2 H19) e 
+(csubst0_getl_ge_back (minus i (S n)) (minus i (S n)) (le_n (minus i (S n))) 
+d x2 u0 H19 (CHead e (Bind Abbr) u0) (getl_gen_S (Bind Abst) x2 (CHead e 
+(Bind Abbr) u0) x4 (minus i (S n)) H21)))))))))))) H14)) H13))))))))))) H8)) 
+H7))) (\lambda (H6: (le i n)).(ty3_abst g n c3 d u (csubst0_getl_ge i n H6 c 
+c3 u0 H4 (CHead d (Bind Abst) u) H0) t0 H1))))))) (\lambda (t3: T).(\lambda 
+(H4: (subst0 i u0 (TLRef n) t3)).(\lambda (c3: C).(\lambda (H5: (csubst0 i u0 
+c c3)).(\lambda (e: C).(\lambda (H6: (getl i c (CHead e (Bind Abbr) 
+u0))).(and_ind (eq nat n i) (eq T t3 (lift (S n) O u0)) (ty3 g c3 t3 (lift (S 
+n) O u)) (\lambda (H7: (eq nat n i)).(\lambda (H8: (eq T t3 (lift (S n) O 
+u0))).(eq_ind_r T (lift (S n) O u0) (\lambda (t4: T).(ty3 g c3 t4 (lift (S n) 
+O u))) (let H9 \def (eq_ind_r nat i (\lambda (n0: nat).(getl n0 c (CHead e 
+(Bind Abbr) u0))) H6 n H7) in (let H10 \def (eq_ind_r nat i (\lambda (n0: 
+nat).(csubst0 n0 u0 c c3)) H5 n H7) in (let H11 \def (eq_ind C (CHead d (Bind 
+Abst) u) (\lambda (c0: C).(getl n c c0)) H0 (CHead e (Bind Abbr) u0) 
+(getl_mono c (CHead d (Bind Abst) u) n H0 (CHead e (Bind Abbr) u0) H9)) in 
+(let H12 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (ee: C).(match ee in 
+C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k 
+_) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) 
+\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow 
+False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) 
+\Rightarrow False])])) I (CHead e (Bind Abbr) u0) (getl_mono c (CHead d (Bind 
+Abst) u) n H0 (CHead e (Bind Abbr) u0) H9)) in (False_ind (ty3 g c3 (lift (S 
+n) O u0) (lift (S n) O u)) H12))))) t3 H8))) (subst0_gen_lref u0 t3 i n 
+H4)))))))) c2 t2 H3)))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda 
+(t0: T).(\lambda (H0: (ty3 g c u t0)).(\lambda (H1: ((\forall (i: 
+nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 c u c2 
+t2) \to (\forall (e: C).((getl i c (CHead e (Bind Abbr) u0)) \to (ty3 g c2 t2 
+t0)))))))))).(\lambda (b: B).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: 
+(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 
+(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).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: 
+T).(\lambda (t7: T).(subst0 (s (Bind b) i) u0 t2 t7)))) (ty3 g c3 t6 (THead 
+(Bind b) u t3)) (\lambda (H10: (ex2 T (\lambda (u2: T).(eq T t6 (THead (Bind 
+b) u2 t2))) (\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T (\lambda (u2: 
+T).(eq T t6 (THead (Bind b) u2 t2))) (\lambda (u2: T).(subst0 i u0 u u2)) 
+(ty3 g c3 t6 (THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H11: (eq T t6 
+(THead (Bind b) x t2))).(\lambda (H12: (subst0 i u0 u x)).(eq_ind_r T (THead 
+(Bind b) x t2) (\lambda (t7: T).(ty3 g c3 t7 (THead (Bind b) u t3))) (ex_ind 
+T (\lambda (t7: T).(ty3 g (CHead c3 (Bind b) u) t3 t7)) (ty3 g c3 (THead 
+(Bind b) x t2) (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (H13: (ty3 g 
+(CHead c3 (Bind b) u) t3 x0)).(ex_ind T (\lambda (t7: T).(ty3 g (CHead c3 
+(Bind b) u) x0 t7)) (ty3 g c3 (THead (Bind b) x t2) (THead (Bind b) u t3)) 
+(\lambda (x1: T).(\lambda (H14: (ty3 g (CHead c3 (Bind b) u) x0 x1)).(ex_ind 
+T (\lambda (t7: T).(ty3 g (CHead c3 (Bind b) x) t3 t7)) (ty3 g c3 (THead 
+(Bind b) x t2) (THead (Bind b) u t3)) (\lambda (x2: T).(\lambda (H15: (ty3 g 
+(CHead c3 (Bind b) x) t3 x2)).(ty3_conv g c3 (THead (Bind b) u t3) (THead 
+(Bind b) u x0) (ty3_bind g c3 u t0 (H1 i u0 c3 u (fsubst0_fst i u0 c u c3 H8) 
+e H9) b t3 x0 H13 x1 H14) (THead (Bind b) x t2) (THead (Bind b) x t3) 
+(ty3_bind g c3 x t0 (H1 i u0 c3 x (fsubst0_both i u0 c u x H12 c3 H8) e H9) b 
+t2 t3 (H3 (S i) u0 (CHead c3 (Bind b) x) t2 (fsubst0_fst (S i) u0 (CHead c 
+(Bind b) u) t2 (CHead c3 (Bind b) x) (csubst0_both_bind b i u0 u x H12 c c3 
+H8)) e (getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H9 u)) x2 H15) 
+(pc3_fsubst0 c (THead (Bind b) u t3) (THead (Bind b) u t3) (pc3_refl c (THead 
+(Bind b) u t3)) i u0 c3 (THead (Bind b) x t3) (fsubst0_both i u0 c (THead 
+(Bind b) u t3) (THead (Bind b) x t3) (subst0_fst u0 x u i H12 t3 (Bind b)) c3 
+H8) e H9)))) (ty3_correct g (CHead c3 (Bind b) x) t2 t3 (H3 (S i) u0 (CHead 
+c3 (Bind b) x) t2 (fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2 (CHead c3 
+(Bind b) x) (csubst0_both_bind b i u0 u x H12 c c3 H8)) e (getl_head (Bind b) 
+i c (CHead e (Bind Abbr) u0) H9 u)))))) (ty3_correct g (CHead c3 (Bind b) u) 
+t3 x0 H13)))) (ty3_correct g (CHead c3 (Bind b) u) t2 t3 (H3 (S i) u0 (CHead 
+c3 (Bind b) u) t2 (fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2 (CHead c3 
+(Bind b) u) (csubst0_fst_bind b i c c3 u0 H8 u)) e (getl_head (Bind b) i c 
+(CHead e (Bind Abbr) u0) H9 u)))) t6 H11)))) H10)) (\lambda (H10: (ex2 T 
+(\lambda (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 
+t3)) i u0 c3 (THead (Bind b) x0 t3) (fsubst0_both i u0 c (THead (Bind b) u 
+t3) (THead (Bind b) x0 t3) (subst0_fst u0 x0 u i H12 t3 (Bind b)) c3 H8) e 
+H9)))) (ty3_correct g (CHead c3 (Bind b) x0) x1 t3 (H3 (S i) u0 (CHead c3 
+(Bind b) x0) x1 (fsubst0_both (S i) u0 (CHead c (Bind b) u) t2 x1 H13 (CHead 
+c3 (Bind b) x0) (csubst0_both_bind b i u0 u x0 H12 c c3 H8)) e (getl_head 
+(Bind b) i c (CHead e (Bind Abbr) u0) H9 u)))))) (ty3_correct g (CHead c3 
+(Bind b) u) t3 x H14)))) (ty3_correct g (CHead c3 (Bind b) u) t2 t3 (H3 (S i) 
+u0 (CHead c3 (Bind b) u) t2 (fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2 
+(CHead c3 (Bind b) u) (csubst0_fst_bind b i c c3 u0 H8 u)) e (getl_head (Bind 
+b) i c (CHead e (Bind Abbr) u0) H9 u)))) t6 H11)))))) H10)) (subst0_gen_head 
+(Bind b) u0 u t2 t6 i H7)))))))) c2 t5 H6))))))))))))))))))) (\lambda (c: 
+C).(\lambda (w: T).(\lambda (u: T).(\lambda (H0: (ty3 g c w u)).(\lambda (H1: 
+((\forall (i: nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2: 
+T).((fsubst0 i u0 c w c2 t2) \to (\forall (e: C).((getl i c (CHead e (Bind 
+Abbr) u0)) \to (ty3 g c2 t2 u)))))))))).(\lambda (v: T).(\lambda (t0: 
+T).(\lambda (H2: (ty3 g c v (THead (Bind Abst) u t0))).(\lambda (H3: 
+((\forall (i: nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2: 
+T).((fsubst0 i u0 c v c2 t2) \to (\forall (e: C).((getl i c (CHead e (Bind 
+Abbr) u0)) \to (ty3 g c2 t2 (THead (Bind Abst) u t0))))))))))).(\lambda (i: 
+nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H4: 
+(fsubst0 i u0 c (THead (Flat Appl) w v) c2 t2)).(fsubst0_ind i u0 c (THead 
+(Flat Appl) w v) (\lambda (c0: C).(\lambda (t3: T).(\forall (e: C).((getl i c 
+(CHead e (Bind Abbr) u0)) \to (ty3 g c0 t3 (THead (Flat Appl) w (THead (Bind 
+Abst) u t0))))))) (\lambda (t3: T).(\lambda (H5: (subst0 i u0 (THead (Flat 
+Appl) w v) t3)).(\lambda (e: C).(\lambda (H6: (getl i c (CHead e (Bind Abbr) 
+u0))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t3 (THead (Flat Appl) u2 v))) 
+(\lambda (u2: T).(subst0 i u0 w u2))) (ex2 T (\lambda (t4: T).(eq T t3 (THead 
+(Flat Appl) w t4))) (\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4))) 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 
+t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 w u2))) (\lambda (_: 
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+(Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (H7: (ex2 T (\lambda (u2: 
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+u2)))).(ex2_ind T (\lambda (u2: T).(eq T t3 (THead (Flat Appl) u2 v))) 
+(\lambda (u2: T).(subst0 i u0 w u2)) (ty3 g c t3 (THead (Flat Appl) w (THead 
+(Bind Abst) u t0))) (\lambda (x: T).(\lambda (H8: (eq T t3 (THead (Flat Appl) 
+x v))).(\lambda (H9: (subst0 i u0 w x)).(eq_ind_r T (THead (Flat Appl) x v) 
+(\lambda (t4: T).(ty3 g c t4 (THead (Flat Appl) w (THead (Bind Abst) u t0)))) 
+(ex_ind T (\lambda (t4: T).(ty3 g c (THead (Bind Abst) u t0) t4)) (ty3 g c 
+(THead (Flat Appl) x v) (THead (Flat Appl) w (THead (Bind Abst) u t0))) 
+(\lambda (x0: T).(\lambda (H10: (ty3 g c (THead (Bind Abst) u t0) 
+x0)).(ex4_3_ind T T T (\lambda (t4: T).(\lambda (_: T).(\lambda (_: T).(pc3 c 
+(THead (Bind Abst) u t4) x0)))) (\lambda (_: T).(\lambda (t5: T).(\lambda (_: 
+T).(ty3 g c u t5)))) (\lambda (t4: T).(\lambda (_: T).(\lambda (_: T).(ty3 g 
+(CHead c (Bind Abst) u) t0 t4)))) (\lambda (t4: T).(\lambda (_: T).(\lambda 
+(t6: T).(ty3 g (CHead c (Bind Abst) u) t4 t6)))) (ty3 g c (THead (Flat Appl) 
+x v) (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x1: 
+T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (_: (pc3 c (THead (Bind Abst) u 
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+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 
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+(H15: (ty3 g c u x4)).(ty3_conv g c (THead (Flat Appl) w (THead (Bind Abst) u 
+t0)) (THead (Flat Appl) w (THead (Bind Abst) u x1)) (ty3_appl g c w u H0 
+(THead (Bind Abst) u t0) x1 (ty3_bind g c u x4 H15 Abst t0 x1 H13 x3 H14)) 
+(THead (Flat Appl) x v) (THead (Flat Appl) x (THead (Bind Abst) u t0)) 
+(ty3_appl g c x u (H1 i u0 c x (fsubst0_snd i u0 c w x H9) e H6) v t0 H2) 
+(pc3_fsubst0 c (THead (Flat Appl) w (THead (Bind Abst) u t0)) (THead (Flat 
+Appl) w (THead (Bind Abst) u t0)) (pc3_refl c (THead (Flat Appl) w (THead 
+(Bind Abst) u t0))) i u0 c (THead (Flat Appl) x (THead (Bind Abst) u t0)) 
+(fsubst0_snd i u0 c (THead (Flat Appl) w (THead (Bind Abst) u t0)) (THead 
+(Flat Appl) x (THead (Bind Abst) u t0)) (subst0_fst u0 x w i H9 (THead (Bind 
+Abst) u t0) (Flat Appl))) e H6)))) (ty3_correct g c x u (H1 i u0 c x 
+(fsubst0_snd i u0 c w x H9) e H6)))))))))) (ty3_gen_bind g Abst c u t0 x0 
+H10)))) (ty3_correct g c v (THead (Bind Abst) u t0) H2)) t3 H8)))) H7)) 
+(\lambda (H7: (ex2 T (\lambda (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 c t3 (THead (Flat Appl) w (THead (Bind Abst) u 
+t0))) (\lambda (x: T).(\lambda (H8: (eq T t3 (THead (Flat Appl) w 
+x))).(\lambda (H9: (subst0 (s (Flat Appl) i) u0 v x)).(eq_ind_r T (THead 
+(Flat Appl) w x) (\lambda (t4: T).(ty3 g c t4 (THead (Flat Appl) w (THead 
+(Bind Abst) u t0)))) (ty3_appl g c w u H0 x t0 (H3 (s (Flat Appl) i) u0 c x 
+(fsubst0_snd (s (Flat Appl) i) u0 c v x H9) e H6)) t3 H8)))) H7)) (\lambda 
+(H7: (ex3_2 T T (\lambda (u2: T).(\lambda (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 c t3 (THead 
+(Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (H8: (eq T t3 (THead (Flat Appl) x0 x1))).(\lambda (H9: (subst0 i 
+u0 w x0)).(\lambda (H10: (subst0 (s (Flat Appl) i) u0 v x1)).(eq_ind_r T 
+(THead (Flat Appl) x0 x1) (\lambda (t4: T).(ty3 g c t4 (THead (Flat Appl) w 
+(THead (Bind Abst) u t0)))) (ex_ind T (\lambda (t4: T).(ty3 g c (THead (Bind 
+Abst) u t0) t4)) (ty3 g c (THead (Flat Appl) 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 
+(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 
+H5) e H6) v t0 (H3 i u0 c3 v (fsubst0_fst i u0 c v c3 H5) e H6)))))) (\lambda 
+(t3: T).(\lambda (H5: (subst0 i u0 (THead (Flat Appl) w v) t3)).(\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 t3 (THead 
+(Flat Appl) u2 v))) (\lambda (u2: T).(subst0 i u0 w u2))) (ex2 T (\lambda 
+(t4: T).(eq T t3 (THead (Flat Appl) w t4))) (\lambda (t4: T).(subst0 (s (Flat 
+Appl) i) u0 v t4))) (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 
+(THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 w 
+u2))) (\lambda (_: T).(\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4)))) 
+(ty3 g c3 t3 (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (H8: 
+(ex2 T (\lambda (u2: T).(eq T t3 (THead (Flat Appl) u2 v))) (\lambda (u2: 
+T).(subst0 i u0 w u2)))).(ex2_ind T (\lambda (u2: T).(eq T t3 (THead (Flat 
+Appl) u2 v))) (\lambda (u2: T).(subst0 i u0 w u2)) (ty3 g c3 t3 (THead (Flat 
+Appl) w (THead (Bind Abst) u t0))) (\lambda (x: T).(\lambda (H9: (eq T t3 
+(THead (Flat Appl) x v))).(\lambda (H10: (subst0 i u0 w x)).(eq_ind_r T 
+(THead (Flat Appl) x v) (\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) 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 
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+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 
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+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))) 
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+(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)))) 
+(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 
+(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 
+Abbr) u)) \to (ty3 g c2 t4 t0)))))))))).(\lambda (i: nat).(\lambda (u: 
+T).(\lambda (c2: C).(\lambda (t4: T).(\lambda (H4: (fsubst0 i u c (THead 
+(Flat Cast) t3 t2) c2 t4)).(fsubst0_ind i u c (THead (Flat Cast) t3 t2) 
+(\lambda (c0: C).(\lambda (t5: T).(\forall (e: C).((getl i c (CHead e (Bind 
+Abbr) u)) \to (ty3 g c0 t5 t3))))) (\lambda (t5: T).(\lambda (H5: (subst0 i u 
+(THead (Flat Cast) t3 t2) t5)).(\lambda (e: C).(\lambda (H6: (getl i c (CHead 
+e (Bind Abbr) u))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t5 (THead (Flat 
+Cast) u2 t2))) (\lambda (u2: T).(subst0 i u t3 u2))) (ex2 T (\lambda (t6: 
+T).(eq T t5 (THead (Flat Cast) t3 t6))) (\lambda (t6: T).(subst0 (s (Flat 
+Cast) i) u t2 t6))) (ex3_2 T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 
+(THead (Flat Cast) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u t3 
+u2))) (\lambda (_: T).(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 t6)))) 
+(ty3 g c t5 t3) (\lambda (H7: (ex2 T (\lambda (u2: T).(eq T t5 (THead (Flat 
+Cast) u2 t2))) (\lambda (u2: T).(subst0 i u t3 u2)))).(ex2_ind T (\lambda 
+(u2: T).(eq T t5 (THead (Flat Cast) u2 t2))) (\lambda (u2: T).(subst0 i u t3 
+u2)) (ty3 g c t5 t3) (\lambda (x: T).(\lambda (H8: (eq T t5 (THead (Flat 
+Cast) x t2))).(\lambda (H9: (subst0 i u t3 x)).(eq_ind_r T (THead (Flat Cast) 
+x t2) (\lambda (t6: T).(ty3 g c t6 t3)) (ty3_conv g c t3 t0 H2 (THead (Flat 
+Cast) x t2) x (ty3_cast g c t2 x (ty3_conv g c x t0 (H3 i u c x (fsubst0_snd 
+i u c t3 x H9) e H6) t2 t3 H0 (pc3_s c t3 x (pc3_fsubst0 c t3 t3 (pc3_refl c 
+t3) i u c x (fsubst0_snd i u c t3 x H9) e H6))) t0 (H3 i u c x (fsubst0_snd i 
+u c t3 x H9) e H6)) (pc3_fsubst0 c t3 t3 (pc3_refl c t3) i u c x (fsubst0_snd 
+i u c t3 x H9) e H6)) t5 H8)))) H7)) (\lambda (H7: (ex2 T (\lambda (t6: 
+T).(eq T t5 (THead (Flat Cast) t3 t6))) (\lambda (t6: T).(subst0 (s (Flat 
+Cast) i) u t2 t6)))).(ex2_ind T (\lambda (t6: T).(eq T t5 (THead (Flat Cast) 
+t3 t6))) (\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 t6)) (ty3 g c t5 t3) 
+(\lambda (x: T).(\lambda (H8: (eq T t5 (THead (Flat Cast) t3 x))).(\lambda 
+(H9: (subst0 (s (Flat Cast) i) u t2 x)).(eq_ind_r T (THead (Flat Cast) t3 x) 
+(\lambda (t6: T).(ty3 g c t6 t3)) (ty3_cast g c x t3 (H1 (s (Flat Cast) i) u 
+c x (fsubst0_snd (s (Flat Cast) i) u c t2 x H9) e H6) t0 H2) t5 H8)))) H7)) 
+(\lambda (H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead 
+(Flat Cast) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u t3 u2))) 
+(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 
+t6))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead 
+(Flat Cast) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u t3 u2))) 
+(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 t6))) (ty3 g 
+c t5 t3) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H8: (eq T t5 (THead 
+(Flat Cast) x0 x1))).(\lambda (H9: (subst0 i u t3 x0)).(\lambda (H10: (subst0 
+(s (Flat Cast) i) u t2 x1)).(eq_ind_r T (THead (Flat Cast) x0 x1) (\lambda 
+(t6: T).(ty3 g c t6 t3)) (ty3_conv g c t3 t0 H2 (THead (Flat Cast) x0 x1) x0 
+(ty3_cast g c x1 x0 (ty3_conv g c x0 t0 (H3 i u c x0 (fsubst0_snd i u c t3 x0 
+H9) e H6) x1 t3 (H1 (s (Flat Cast) i) u c x1 (fsubst0_snd (s (Flat Cast) i) u 
+c t2 x1 H10) e H6) (pc3_s c t3 x0 (pc3_fsubst0 c t3 t3 (pc3_refl c t3) i u c 
+x0 (fsubst0_snd i u c t3 x0 H9) e H6))) t0 (H3 i u c x0 (fsubst0_snd i u c t3 
+x0 H9) e H6)) (pc3_fsubst0 c t3 t3 (pc3_refl c t3) i u c x0 (fsubst0_snd i u 
+c t3 x0 H9) e H6)) t5 H8)))))) H7)) (subst0_gen_head (Flat Cast) u t3 t2 t5 i 
+H5)))))) (\lambda (c3: C).(\lambda (H5: (csubst0 i u c c3)).(\lambda (e: 
+C).(\lambda (H6: (getl i c (CHead e (Bind Abbr) u))).(ty3_cast g c3 t2 t3 (H1 
+i u c3 t2 (fsubst0_fst i u c t2 c3 H5) e H6) t0 (H3 i u c3 t3 (fsubst0_fst i 
+u c t3 c3 H5) e H6)))))) (\lambda (t5: T).(\lambda (H5: (subst0 i u (THead 
+(Flat Cast) t3 t2) t5)).(\lambda (c3: C).(\lambda (H6: (csubst0 i u c 
+c3)).(\lambda (e: C).(\lambda (H7: (getl i c (CHead e (Bind Abbr) 
+u))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t5 (THead (Flat Cast) u2 t2))) 
+(\lambda (u2: T).(subst0 i u t3 u2))) (ex2 T (\lambda (t6: T).(eq T t5 (THead 
+(Flat Cast) t3 t6))) (\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 t6))) 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead (Flat Cast) u2 
+t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u t3 u2))) (\lambda (_: 
+T).(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 t6)))) (ty3 g c3 t5 t3) 
+(\lambda (H8: (ex2 T (\lambda (u2: T).(eq T t5 (THead (Flat Cast) u2 t2))) 
+(\lambda (u2: T).(subst0 i u t3 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t5 
+(THead (Flat Cast) u2 t2))) (\lambda (u2: T).(subst0 i u t3 u2)) (ty3 g c3 t5 
+t3) (\lambda (x: T).(\lambda (H9: (eq T t5 (THead (Flat Cast) x 
+t2))).(\lambda (H10: (subst0 i u t3 x)).(eq_ind_r T (THead (Flat Cast) x t2) 
+(\lambda (t6: T).(ty3 g c3 t6 t3)) (ty3_conv g c3 t3 t0 (H3 i u c3 t3 
+(fsubst0_fst i u c t3 c3 H6) e H7) (THead (Flat Cast) x t2) x (ty3_cast g c3 
+t2 x (ty3_conv g c3 x t0 (H3 i u c3 x (fsubst0_both i u c t3 x H10 c3 H6) e 
+H7) t2 t3 (H1 i u c3 t2 (fsubst0_fst i u c t2 c3 H6) e H7) (pc3_s c3 t3 x 
+(pc3_fsubst0 c t3 t3 (pc3_refl c t3) i u c3 x (fsubst0_both i u c t3 x H10 c3 
+H6) e H7))) t0 (H3 i u c3 x (fsubst0_both i u c t3 x H10 c3 H6) e H7)) 
+(pc3_fsubst0 c t3 t3 (pc3_refl c t3) i u c3 x (fsubst0_both i u c t3 x H10 c3 
+H6) e H7)) t5 H9)))) H8)) (\lambda (H8: (ex2 T (\lambda (t6: T).(eq T t5 
+(THead (Flat Cast) t3 t6))) (\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 
+t6)))).(ex2_ind T (\lambda (t6: T).(eq T t5 (THead (Flat Cast) t3 t6))) 
+(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 t6)) (ty3 g c3 t5 t3) 
+(\lambda (x: T).(\lambda (H9: (eq T t5 (THead (Flat Cast) t3 x))).(\lambda 
+(H10: (subst0 (s (Flat Cast) i) u t2 x)).(eq_ind_r T (THead (Flat Cast) t3 x) 
+(\lambda (t6: T).(ty3 g c3 t6 t3)) (ty3_cast g c3 x t3 (H1 i u c3 x 
+(fsubst0_both i u c t2 x H10 c3 H6) e H7) t0 (H3 i u c3 t3 (fsubst0_fst i u c 
+t3 c3 H6) e H7)) t5 H9)))) H8)) (\lambda (H8: (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t6: T).(eq T t5 (THead (Flat Cast) u2 t6)))) (\lambda (u2: 
+T).(\lambda (_: T).(subst0 i u t3 u2))) (\lambda (_: T).(\lambda (t6: 
+T).(subst0 (s (Flat Cast) i) u t2 t6))))).(ex3_2_ind T T (\lambda (u2: 
+T).(\lambda (t6: T).(eq T t5 (THead (Flat Cast) u2 t6)))) (\lambda (u2: 
+T).(\lambda (_: T).(subst0 i u t3 u2))) (\lambda (_: T).(\lambda (t6: 
+T).(subst0 (s (Flat Cast) i) u t2 t6))) (ty3 g c3 t5 t3) (\lambda (x0: 
+T).(\lambda (x1: T).(\lambda (H9: (eq T t5 (THead (Flat Cast) x0 
+x1))).(\lambda (H10: (subst0 i u t3 x0)).(\lambda (H11: (subst0 (s (Flat 
+Cast) i) u t2 x1)).(eq_ind_r T (THead (Flat Cast) x0 x1) (\lambda (t6: 
+T).(ty3 g c3 t6 t3)) (ty3_conv g c3 t3 t0 (H3 i u c3 t3 (fsubst0_fst i u c t3 
+c3 H6) e H7) (THead (Flat Cast) x0 x1) x0 (ty3_cast g c3 x1 x0 (ty3_conv g c3 
+x0 t0 (H3 i u c3 x0 (fsubst0_both i u c t3 x0 H10 c3 H6) e H7) x1 t3 (H1 i u 
+c3 x1 (fsubst0_both i u c t2 x1 H11 c3 H6) e H7) (pc3_s c3 t3 x0 (pc3_fsubst0 
+c t3 t3 (pc3_refl c t3) i u c3 x0 (fsubst0_both i u c t3 x0 H10 c3 H6) e 
+H7))) t0 (H3 i u c3 x0 (fsubst0_both i u c t3 x0 H10 c3 H6) e H7)) 
+(pc3_fsubst0 c t3 t3 (pc3_refl c t3) i u c3 x0 (fsubst0_both i u c t3 x0 H10 
+c3 H6) e H7)) t5 H9)))))) H8)) (subst0_gen_head (Flat Cast) u t3 t2 t5 i 
+H5)))))))) c2 t4 H4)))))))))))))) c1 t1 t H))))).
+
+theorem ty3_csubst0:
+ \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c1 
+t1 t2) \to (\forall (e: C).(\forall (u: T).(\forall (i: nat).((getl i c1 
+(CHead e (Bind Abbr) u)) \to (\forall (c2: C).((csubst0 i u c1 c2) \to (ty3 g 
+c2 t1 t2)))))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
+(H: (ty3 g c1 t1 t2)).(\lambda (e: C).(\lambda (u: T).(\lambda (i: 
+nat).(\lambda (H0: (getl i c1 (CHead e (Bind Abbr) u))).(\lambda (c2: 
+C).(\lambda (H1: (csubst0 i u c1 c2)).(ty3_fsubst0 g c1 t1 t2 H i u c2 t1 
+(fsubst0_fst i u c1 t1 c2 H1) e H0))))))))))).
+
+theorem ty3_subst0:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((ty3 g c t1 
+t) \to (\forall (e: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead e 
+(Bind Abbr) u)) \to (\forall (t2: T).((subst0 i u t1 t2) \to (ty3 g c t2 
+t)))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: 
+(ty3 g c t1 t)).(\lambda (e: C).(\lambda (u: T).(\lambda (i: nat).(\lambda 
+(H0: (getl i c (CHead e (Bind Abbr) u))).(\lambda (t2: T).(\lambda (H1: 
+(subst0 i u t1 t2)).(ty3_fsubst0 g c t1 t H i u c t2 (fsubst0_snd i u c t1 t2 
+H1) e H0))))))))))).
+

diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/fwd.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/fwd.ma
new file mode 100644 (file)
index 0000000..a3ee2cb
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/fwd.ma
@@ -0,0 +1,914 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/fwd".
+
+include "ty3/defs.ma".
+
+include "pc3/props.ma".
+
+theorem ty3_gen_sort:
+ \forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((ty3 g c 
+(TSort n) x) \to (pc3 c (TSort (next g n)) x)))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda 
+(H: (ty3 g c (TSort n) x)).(insert_eq T (TSort n) (\lambda (t: T).(ty3 g c t 
+x)) (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 
+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 
+(_: (ty3 g c0 t2 t0)).(\lambda (_: (((eq T t2 (TSort n)) \to (pc3 c0 (TSort 
+(next g n)) t0)))).(\lambda (H5: (eq T (THead (Flat Cast) t2 t1) (TSort 
+n))).(let H6 \def (eq_ind T (THead (Flat Cast) t2 t1) (\lambda (ee: T).(match 
+ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | 
+(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) 
+H5) in (False_ind (pc3 c0 (TSort (next g n)) t2) H6))))))))))) c y x H0))) 
+H))))).
+
+theorem ty3_gen_lref:
+ \forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((ty3 g c 
+(TLRef n) x) \to (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda 
+(t: T).(pc3 c (lift (S n) O t) x)))) (\lambda (e: C).(\lambda (u: T).(\lambda 
+(_: T).(getl n c (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: 
+T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda 
+(u: T).(\lambda (_: T).(pc3 c (lift (S n) O u) x)))) (\lambda (e: C).(\lambda 
+(u: T).(\lambda (_: T).(getl n c (CHead e (Bind Abst) u))))) (\lambda (e: 
+C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda 
+(H: (ty3 g c (TLRef n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(ty3 g c t 
+x)) (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c 
+(lift (S n) O t) x)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl 
+n c (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: 
+T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda 
+(_: T).(pc3 c (lift (S n) O u) x)))) (\lambda (e: C).(\lambda (u: T).(\lambda 
+(_: T).(getl n c (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: 
+T).(\lambda (t: T).(ty3 g e u t)))))) (\lambda (y: T).(\lambda (H0: (ty3 g c 
+y x)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t 
+(TLRef n)) \to (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t1: 
+T).(pc3 c0 (lift (S n) O t1) t0)))) (\lambda (e: C).(\lambda (u: T).(\lambda 
+(_: T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: 
+T).(\lambda (t1: T).(ty3 g e u t1))))) (ex3_3 C T T (\lambda (_: C).(\lambda 
+(u: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u) t0)))) (\lambda (e: 
+C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) 
+(\lambda (e: C).(\lambda (u: T).(\lambda (t1: T).(ty3 g e u t1)))))))))) 
+(\lambda (c0: C).(\lambda (t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 
+t)).(\lambda (_: (((eq T t2 (TLRef n)) \to (or (ex3_3 C T T (\lambda (_: 
+C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t)))) (\lambda 
+(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u))))) 
+(\lambda (e: C).(\lambda (u: T).(\lambda (t0: T).(ty3 g e u t0))))) (ex3_3 C 
+T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u) 
+t)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e 
+(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t0: T).(ty3 g e 
+u t0))))))))).(\lambda (u: T).(\lambda (t1: T).(\lambda (H3: (ty3 g c0 u 
+t1)).(\lambda (H4: (((eq T u (TLRef n)) \to (or (ex3_3 C T T (\lambda (_: 
+C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t1)))) (\lambda 
+(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) 
+u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) 
+(ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift 
+(S n) O u0) t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n 
+c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda 
+(t0: T).(ty3 g e u0 t0))))))))).(\lambda (H5: (pc3 c0 t1 t2)).(\lambda (H6: 
+(eq T u (TLRef n))).(let H7 \def (f_equal T T (\lambda (e: T).e) u (TLRef n) 
+H6) in (let H8 \def (eq_ind T u (\lambda (t0: T).((eq T t0 (TLRef n)) \to (or 
+(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t3: T).(pc3 c0 (lift 
+(S n) O t3) t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n 
+c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda 
+(t3: T).(ty3 g e u0 t3))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: 
+T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t1)))) (\lambda (e: C).(\lambda 
+(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: 
+C).(\lambda (u0: T).(\lambda (t3: T).(ty3 g e u0 t3)))))))) H4 (TLRef n) H7) 
+in (let H9 \def (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t1)) H3 (TLRef n) 
+H7) in (let H10 \def (H8 (refl_equal T (TLRef n))) in (or_ind (ex3_3 C T T 
+(\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) 
+t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e 
+(Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g 
+e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: 
+T).(pc3 c0 (lift (S n) O u0) t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda 
+(_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: 
+T).(\lambda (t0: T).(ty3 g e u0 t0))))) (or (ex3_3 C T T (\lambda (_: 
+C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda 
+(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) 
+u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) 
+(ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift 
+(S n) O u0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n 
+c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda 
+(t0: T).(ty3 g e u0 t0)))))) (\lambda (H11: (ex3_3 C T T (\lambda (_: 
+C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t1)))) (\lambda 
+(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) 
+u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 
+t0)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: 
+T).(pc3 c0 (lift (S n) O t0) t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda 
+(_: T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: 
+T).(\lambda (t0: T).(ty3 g e u0 t0)))) (or (ex3_3 C T T (\lambda (_: 
+C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda 
+(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) 
+u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) 
+(ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift 
+(S n) O u0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n 
+c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda 
+(t0: T).(ty3 g e u0 t0)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: 
+T).(\lambda (H12: (pc3 c0 (lift (S n) O x2) t1)).(\lambda (H13: (getl n c0 
+(CHead x0 (Bind Abbr) x1))).(\lambda (H14: (ty3 g x0 x1 x2)).(or_introl 
+(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift 
+(S n) O t0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n 
+c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda 
+(t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: 
+T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t2)))) (\lambda (e: C).(\lambda 
+(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: 
+C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3_intro C T T 
+(\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) 
+t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e 
+(Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g 
+e u0 t0)))) x0 x1 x2 (pc3_t t1 c0 (lift (S n) O x2) H12 t2 H5) H13 
+H14)))))))) H11)) (\lambda (H11: (ex3_3 C T T (\lambda (_: C).(\lambda (u0: 
+T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t1)))) (\lambda (e: C).(\lambda 
+(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: 
+C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))))).(ex3_3_ind C T T 
+(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) 
+t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e 
+(Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g 
+e u0 t0)))) (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: 
+T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda 
+(_: T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: 
+T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda 
+(u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t2)))) (\lambda (e: 
+C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) 
+(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))))) 
+(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H12: (pc3 c0 
+(lift (S n) O x1) t1)).(\lambda (H13: (getl n c0 (CHead x0 (Bind Abst) 
+x1))).(\lambda (H14: (ty3 g x0 x1 x2)).(or_intror (ex3_3 C T T (\lambda (_: 
+C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda 
+(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) 
+u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) 
+(ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift 
+(S n) O u0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n 
+c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda 
+(t0: T).(ty3 g e u0 t0))))) (ex3_3_intro C T T (\lambda (_: C).(\lambda (u0: 
+T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t2)))) (\lambda (e: C).(\lambda 
+(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: 
+C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))) x0 x1 x2 (pc3_t t1 c0 
+(lift (S n) O x1) H12 t2 H5) H13 H14)))))))) H11)) H10)))))))))))))))) 
+(\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq T (TSort m) (TLRef 
+n))).(let H2 \def (eq_ind T (TSort m) (\lambda (ee: T).(match ee in T return 
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) 
+\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef n) H1) in 
+(False_ind (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: 
+T).(pc3 c0 (lift (S n) O t) (TSort (next g m)))))) (\lambda (e: C).(\lambda 
+(u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: 
+C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T T (\lambda 
+(_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u) (TSort (next 
+g m)))))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e 
+(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u 
+t)))))) H2))))) (\lambda (n0: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda 
+(u: T).(\lambda (H1: (getl n0 c0 (CHead d (Bind Abbr) u))).(\lambda (t: 
+T).(\lambda (H2: (ty3 g d u t)).(\lambda (_: (((eq T u (TLRef n)) \to (or 
+(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 d (lift (S 
+n) O t0) t)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n d 
+(CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: 
+T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda 
+(_: T).(pc3 d (lift (S n) O u0) t)))) (\lambda (e: C).(\lambda (u0: 
+T).(\lambda (_: T).(getl n d (CHead e (Bind Abst) u0))))) (\lambda (e: 
+C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))))))).(\lambda (H4: 
+(eq T (TLRef n0) (TLRef n))).(let H5 \def (f_equal T nat (\lambda (e: 
+T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow n0 | 
+(TLRef n1) \Rightarrow n1 | (THead _ _ _) \Rightarrow n0])) (TLRef n0) (TLRef 
+n) H4) in (let H6 \def (eq_ind nat n0 (\lambda (n1: nat).(getl n1 c0 (CHead d 
+(Bind Abbr) u))) H1 n H5) in (eq_ind_r nat n (\lambda (n1: nat).(or (ex3_3 C 
+T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O 
+t0) (lift (S n1) O t))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: 
+T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: 
+T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda 
+(u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) (lift (S n1) O t))))) 
+(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind 
+Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 
+t0))))))) (or_introl (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda 
+(t0: T).(pc3 c0 (lift (S n) O t0) (lift (S n) O t))))) (\lambda (e: 
+C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u0))))) 
+(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 
+C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O 
+u0) (lift (S n) O t))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: 
+T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: 
+T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3_intro C T T (\lambda (_: 
+C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) (lift (S n) O 
+t))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e 
+(Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g 
+e u0 t0)))) d u t (pc3_refl c0 (lift (S n) O t)) H6 H2)) n0 H5)))))))))))) 
+(\lambda (n0: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda 
+(H1: (getl n0 c0 (CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (H2: (ty3 
+g d u t)).(\lambda (_: (((eq T u (TLRef n)) \to (or (ex3_3 C T T (\lambda (_: 
+C).(\lambda (_: T).(\lambda (t0: T).(pc3 d (lift (S n) O t0) t)))) (\lambda 
+(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n d (CHead e (Bind Abbr) 
+u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) 
+(ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 d (lift (S 
+n) O u0) t)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n d 
+(CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: 
+T).(ty3 g e u0 t0))))))))).(\lambda (H4: (eq T (TLRef n0) (TLRef n))).(let H5 
+\def (f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) 
+with [(TSort _) \Rightarrow n0 | (TLRef n1) \Rightarrow n1 | (THead _ _ _) 
+\Rightarrow n0])) (TLRef n0) (TLRef n) H4) in (let H6 \def (eq_ind nat n0 
+(\lambda (n1: nat).(getl n1 c0 (CHead d (Bind Abst) u))) H1 n H5) in 
+(eq_ind_r nat n (\lambda (n1: nat).(or (ex3_3 C T T (\lambda (_: C).(\lambda 
+(_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) (lift (S n1) O u))))) 
+(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind 
+Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 
+t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 
+(lift (S n) O u0) (lift (S n1) O u))))) (\lambda (e: C).(\lambda (u0: 
+T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: 
+C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))))) (or_intror (ex3_3 
+C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O 
+t0) (lift (S n) O u))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: 
+T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: 
+T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda 
+(u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) (lift (S n) O u))))) 
+(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind 
+Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 
+t0))))) (ex3_3_intro C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: 
+T).(pc3 c0 (lift (S n) O u0) (lift (S n) O u))))) (\lambda (e: C).(\lambda 
+(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: 
+C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))) d u t (pc3_refl c0 
+(lift (S n) O u)) H6 H2)) n0 H5)))))))))))) (\lambda (c0: C).(\lambda (u: 
+T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u (TLRef 
+n)) \to (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: 
+T).(pc3 c0 (lift (S n) O t0) t)))) (\lambda (e: C).(\lambda (u0: T).(\lambda 
+(_: T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: 
+T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda 
+(u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t)))) (\lambda (e: 
+C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) 
+(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 
+t0))))))))).(\lambda (b: B).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: 
+(ty3 g (CHead c0 (Bind b) u) t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to 
+(or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 (CHead 
+c0 (Bind b) u) (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda (u0: 
+T).(\lambda (_: T).(getl n (CHead c0 (Bind b) u) (CHead e (Bind Abbr) u0))))) 
+(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 
+C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 (CHead c0 (Bind 
+b) u) (lift (S n) O u0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: 
+T).(getl n (CHead c0 (Bind b) u) (CHead e (Bind Abst) u0))))) (\lambda (e: 
+C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))))))).(\lambda (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))))) 
+(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind 
+Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 
+t0))))))))).(\lambda (H5: (eq T (THead (Flat Appl) w v) (TLRef n))).(let H6 
+\def (eq_ind T (THead (Flat Appl) w v) (\lambda (ee: T).(match ee in T return 
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H5) in 
+(False_ind (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: 
+T).(pc3 c0 (lift (S n) O t0) (THead (Flat Appl) w (THead (Bind Abst) u 
+t)))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e 
+(Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g 
+e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: 
+T).(pc3 c0 (lift (S n) O u0) (THead (Flat Appl) w (THead (Bind Abst) u 
+t)))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e 
+(Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g 
+e u0 t0)))))) H6)))))))))))) (\lambda (c0: C).(\lambda (t1: T).(\lambda (t2: 
+T).(\lambda (_: (ty3 g c0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (or 
+(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c0 (lift (S 
+n) O t) t2)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 
+(CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: 
+T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda 
+(_: T).(pc3 c0 (lift (S n) O u) t2)))) (\lambda (e: C).(\lambda (u: 
+T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: 
+C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))))))).(\lambda (t0: 
+T).(\lambda (_: (ty3 g c0 t2 t0)).(\lambda (_: (((eq T t2 (TLRef n)) \to (or 
+(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c0 (lift (S 
+n) O t) t0)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 
+(CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: 
+T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda 
+(_: T).(pc3 c0 (lift (S n) O u) t0)))) (\lambda (e: C).(\lambda (u: 
+T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: 
+C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))))))).(\lambda (H5: (eq T 
+(THead (Flat Cast) t2 t1) (TLRef n))).(let H6 \def (eq_ind T (THead (Flat 
+Cast) t2 t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) 
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ 
+_) \Rightarrow True])) I (TLRef n) H5) in (False_ind (or (ex3_3 C T T 
+(\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c0 (lift (S n) O t) 
+t2)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e 
+(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u 
+t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c0 
+(lift (S n) O u) t2)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl 
+n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: 
+T).(ty3 g e u t)))))) H6))))))))))) c y x H0))) H))))).
+
+theorem ty3_gen_bind:
+ \forall (g: G).(\forall (b: B).(\forall (c: C).(\forall (u: T).(\forall (t1: 
+T).(\forall (x: T).((ty3 g c (THead (Bind b) u t1) x) \to (ex4_3 T T T 
+(\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c (THead (Bind b) u t2) 
+x)))) (\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g c u t)))) 
+(\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c (Bind b) u) 
+t1 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t0: T).(ty3 g (CHead c 
+(Bind b) u) t2 t0)))))))))))
+\def
+ \lambda (g: G).(\lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: 
+T).(\lambda (x: T).(\lambda (H: (ty3 g c (THead (Bind b) u t1) x)).(insert_eq 
+T (THead (Bind b) u t1) (\lambda (t: T).(ty3 g c t x)) (ex4_3 T T T (\lambda 
+(t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c (THead (Bind b) u t2) x)))) 
+(\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g c u t)))) (\lambda 
+(t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c (Bind b) u) t1 t2)))) 
+(\lambda (t2: T).(\lambda (_: T).(\lambda (t0: T).(ty3 g (CHead c (Bind b) u) 
+t2 t0))))) (\lambda (y: T).(\lambda (H0: (ty3 g c y x)).(ty3_ind g (\lambda 
+(c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Bind b) u t1)) \to 
+(ex4_3 T T T (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead 
+(Bind b) u t2) t0)))) (\lambda (_: T).(\lambda (t3: T).(\lambda (_: T).(ty3 g 
+c0 u t3)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 
+(Bind b) u) t1 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t4: T).(ty3 
+g (CHead c0 (Bind b) u) t2 t4))))))))) (\lambda (c0: C).(\lambda (t2: 
+T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: (((eq T t2 
+(THead (Bind b) u t1)) \to (ex4_3 T T T (\lambda (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 _) 
+\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
+False])) I (THead (Bind b) u t1) H1) in (False_ind (ex4_3 T T T (\lambda (t2: 
+T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2) (TSort (next 
+g m)))))) (\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g c0 u t)))) 
+(\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) 
+t1 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t0: T).(ty3 g (CHead c0 
+(Bind b) u) t2 t0))))) H2))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda 
+(d: C).(\lambda (u0: T).(\lambda (_: (getl n c0 (CHead d (Bind Abbr) 
+u0))).(\lambda (t: T).(\lambda (_: (ty3 g d u0 t)).(\lambda (_: (((eq T u0 
+(THead (Bind b) u t1)) \to (ex4_3 T T T (\lambda (t2: T).(\lambda (_: 
+T).(\lambda (_: T).(pc3 d (THead (Bind b) u t2) t)))) (\lambda (_: 
+T).(\lambda (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 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 
+(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: 
+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) t2)))) (\lambda (_: T).(\lambda (t: 
+T).(\lambda (_: T).(ty3 g c0 u t)))) (\lambda (t4: T).(\lambda (_: 
+T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t4)))) (\lambda (t4: 
+T).(\lambda (_: T).(\lambda (t5: T).(ty3 g (CHead c0 (Bind b) u) t4 t5))))) 
+H6))))))))))) c y x H0))) H))))))).
+
+theorem ty3_gen_appl:
+ \forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (v: T).(\forall (x: 
+T).((ty3 g c (THead (Flat Appl) w v) x) \to (ex3_2 T T (\lambda (u: 
+T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x))) 
+(\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) 
+(\lambda (u: T).(\lambda (_: T).(ty3 g c w u)))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (w: T).(\lambda (v: T).(\lambda (x: 
+T).(\lambda (H: (ty3 g c (THead (Flat Appl) w v) x)).(insert_eq T (THead 
+(Flat Appl) w v) (\lambda (t: T).(ty3 g c t x)) (ex3_2 T T (\lambda (u: 
+T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x))) 
+(\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) 
+(\lambda (u: T).(\lambda (_: T).(ty3 g c w u)))) (\lambda (y: T).(\lambda 
+(H0: (ty3 g c y x)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: 
+T).((eq T t (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda (u: T).(\lambda 
+(t1: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u t1)) t0))) (\lambda 
+(u: T).(\lambda (t1: T).(ty3 g c0 v (THead (Bind Abst) u t1)))) (\lambda (u: 
+T).(\lambda (_: T).(ty3 g c0 w u)))))))) (\lambda (c0: C).(\lambda (t2: 
+T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: (((eq T t2 
+(THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda (u: T).(\lambda (t0: T).(pc3 
+c0 (THead (Flat Appl) w (THead (Bind Abst) u t0)) t))) (\lambda (u: 
+T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u t0)))) (\lambda (u: 
+T).(\lambda (_: T).(ty3 g c0 w u))))))).(\lambda (u: T).(\lambda (t1: 
+T).(\lambda (H3: (ty3 g c0 u t1)).(\lambda (H4: (((eq T u (THead (Flat Appl) 
+w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead (Flat 
+Appl) w (THead (Bind Abst) u0 t0)) t1))) (\lambda (u0: T).(\lambda (t0: 
+T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: 
+T).(ty3 g c0 w u0))))))).(\lambda (H5: (pc3 c0 t1 t2)).(\lambda (H6: (eq T u 
+(THead (Flat Appl) w v))).(let H7 \def (f_equal T T (\lambda (e: T).e) u 
+(THead (Flat Appl) w v) H6) in (let H8 \def (eq_ind T u (\lambda (t0: T).((eq 
+T t0 (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t3: 
+T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t3)) t1))) (\lambda 
+(u0: T).(\lambda (t3: T).(ty3 g c0 v (THead (Bind Abst) u0 t3)))) (\lambda 
+(u0: T).(\lambda (_: T).(ty3 g c0 w u0)))))) H4 (THead (Flat Appl) w v) H7) 
+in (let H9 \def (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t1)) H3 (THead 
+(Flat Appl) w v) H7) in (let H10 \def (H8 (refl_equal T (THead (Flat Appl) w 
+v))) in (ex3_2_ind T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead (Flat 
+Appl) w (THead (Bind Abst) u0 t0)) t1))) (\lambda (u0: T).(\lambda (t0: 
+T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: 
+T).(ty3 g c0 w u0))) (ex3_2 T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 
+(THead (Flat Appl) w (THead (Bind Abst) u0 t0)) t2))) (\lambda (u0: 
+T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: 
+T).(\lambda (_: T).(ty3 g c0 w u0)))) (\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (H11: (pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) x0 x1)) 
+t1)).(\lambda (H12: (ty3 g c0 v (THead (Bind Abst) x0 x1))).(\lambda (H13: 
+(ty3 g c0 w x0)).(ex3_2_intro T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 
+(THead (Flat Appl) w (THead (Bind Abst) u0 t0)) t2))) (\lambda (u0: 
+T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: 
+T).(\lambda (_: T).(ty3 g c0 w u0))) x0 x1 (pc3_t t1 c0 (THead (Flat Appl) w 
+(THead (Bind Abst) x0 x1)) H11 t2 H5) H12 H13)))))) H10)))))))))))))))) 
+(\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq T (TSort m) (THead (Flat 
+Appl) w v))).(let H2 \def (eq_ind T (TSort m) (\lambda (ee: T).(match ee in T 
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) 
+\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) w 
+v) H1) in (False_ind (ex3_2 T T (\lambda (u: T).(\lambda (t: T).(pc3 c0 
+(THead (Flat Appl) w (THead (Bind Abst) u t)) (TSort (next g m))))) (\lambda 
+(u: T).(\lambda (t: T).(ty3 g c0 v (THead (Bind Abst) u t)))) (\lambda (u: 
+T).(\lambda (_: T).(ty3 g c0 w u)))) H2))))) (\lambda (n: nat).(\lambda (c0: 
+C).(\lambda (d: C).(\lambda (u: T).(\lambda (_: (getl n c0 (CHead d (Bind 
+Abbr) u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (_: (((eq T u 
+(THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t0: 
+T).(pc3 d (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) t))) (\lambda (u0: 
+T).(\lambda (t0: T).(ty3 g d v (THead (Bind Abst) u0 t0)))) (\lambda (u0: 
+T).(\lambda (_: T).(ty3 g d w u0))))))).(\lambda (H4: (eq T (TLRef n) (THead 
+(Flat Appl) w v))).(let H5 \def (eq_ind T (TLRef n) (\lambda (ee: T).(match 
+ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | 
+(TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead 
+(Flat Appl) w v) H4) in (False_ind (ex3_2 T T (\lambda (u0: T).(\lambda (t0: 
+T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) (lift (S n) O 
+t)))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0 
+t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w u0)))) H5))))))))))) 
+(\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda 
+(_: (getl n c0 (CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (_: (ty3 g 
+d u t)).(\lambda (_: (((eq T u (THead (Flat Appl) w v)) \to (ex3_2 T T 
+(\lambda (u0: T).(\lambda (t0: T).(pc3 d (THead (Flat Appl) w (THead (Bind 
+Abst) u0 t0)) t))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g d v (THead (Bind 
+Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g d w 
+u0))))))).(\lambda (H4: (eq T (TLRef n) (THead (Flat Appl) w v))).(let H5 
+\def (eq_ind T (TLRef n) (\lambda (ee: T).(match ee in T return (\lambda (_: 
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | 
+(THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) w v) H4) in 
+(False_ind (ex3_2 T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead (Flat 
+Appl) w (THead (Bind Abst) u0 t0)) (lift (S n) O u)))) (\lambda (u0: 
+T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: 
+T).(\lambda (_: T).(ty3 g c0 w u0)))) H5))))))))))) (\lambda (c0: C).(\lambda 
+(u: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u 
+(THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t0: 
+T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) t))) (\lambda (u0: 
+T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: 
+T).(\lambda (_: T).(ty3 g c0 w u0))))))).(\lambda (b: B).(\lambda (t1: 
+T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t1 
+t2)).(\lambda (_: (((eq T t1 (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda 
+(u0: T).(\lambda (t0: T).(pc3 (CHead c0 (Bind b) u) (THead (Flat Appl) w 
+(THead (Bind Abst) u0 t0)) t2))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g 
+(CHead c0 (Bind b) u) v (THead (Bind Abst) u0 t0)))) (\lambda (u0: 
+T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) w u0))))))).(\lambda (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 (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)) t2))) (\lambda (u: T).(\lambda (t: T).(ty3 g c0 v (THead (Bind 
+Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u)))) H6))))))))))) 
+c y x H0))) H)))))).
+
+theorem ty3_gen_cast:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).(\forall 
+(x: T).((ty3 g c (THead (Flat Cast) t2 t1) x) \to (land (pc3 c t2 x) (ty3 g c 
+t1 t2)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
+(x: T).(\lambda (H: (ty3 g c (THead (Flat Cast) t2 t1) x)).(insert_eq T 
+(THead (Flat Cast) t2 t1) (\lambda (t: T).(ty3 g c t x)) (land (pc3 c t2 x) 
+(ty3 g c t1 t2)) (\lambda (y: T).(\lambda (H0: (ty3 g c y x)).(ty3_ind g 
+(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Flat Cast) 
+t2 t1)) \to (land (pc3 c0 t2 t0) (ty3 g c0 t1 t2)))))) (\lambda (c0: 
+C).(\lambda (t0: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t0 t)).(\lambda 
+(_: (((eq T t0 (THead (Flat Cast) t2 t1)) \to (land (pc3 c0 t2 t) (ty3 g c0 
+t1 t2))))).(\lambda (u: T).(\lambda (t3: T).(\lambda (H3: (ty3 g c0 u 
+t3)).(\lambda (H4: (((eq T u (THead (Flat Cast) t2 t1)) \to (land (pc3 c0 t2 
+t3) (ty3 g c0 t1 t2))))).(\lambda (H5: (pc3 c0 t3 t0)).(\lambda (H6: (eq T u 
+(THead (Flat Cast) t2 t1))).(let H7 \def (f_equal T T (\lambda (e: T).e) u 
+(THead (Flat Cast) t2 t1) H6) in (let H8 \def (eq_ind T u (\lambda (t4: 
+T).((eq T t4 (THead (Flat Cast) t2 t1)) \to (land (pc3 c0 t2 t3) (ty3 g c0 t1 
+t2)))) H4 (THead (Flat Cast) t2 t1) H7) in (let H9 \def (eq_ind T u (\lambda 
+(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 (and_ind (pc3 c0 t2 t3) (ty3 
+g c0 t1 t2) (land (pc3 c0 t2 t0) (ty3 g c0 t1 t2)) (\lambda (H11: (pc3 c0 t2 
+t3)).(\lambda (H12: (ty3 g c0 t1 t2)).(conj (pc3 c0 t2 t0) (ty3 g c0 t1 t2) 
+(pc3_t t3 c0 t2 H11 t0 H5) H12))) H10)))))))))))))))) (\lambda (c0: 
+C).(\lambda (m: nat).(\lambda (H1: (eq T (TSort m) (THead (Flat Cast) t2 
+t1))).(let H2 \def (eq_ind T (TSort m) (\lambda (ee: T).(match ee 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 (land (pc3 c0 t2 (TSort (next g m))) (ty3 g c0 t1 
+t2)) H2))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: 
+T).(\lambda (_: (getl n c0 (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda 
+(_: (ty3 g d u t)).(\lambda (_: (((eq T u (THead (Flat Cast) t2 t1)) \to 
+(land (pc3 d t2 t) (ty3 g d t1 t2))))).(\lambda (H4: (eq T (TLRef n) (THead 
+(Flat Cast) t2 t1))).(let H5 \def (eq_ind T (TLRef n) (\lambda (ee: T).(match 
+ee 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 (land (pc3 c0 t2 (lift (S n) O t)) (ty3 
+g c0 t1 t2)) H5))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: 
+C).(\lambda (u: T).(\lambda (_: (getl n c0 (CHead d (Bind Abst) u))).(\lambda 
+(t: T).(\lambda (_: (ty3 g d u t)).(\lambda (_: (((eq T u (THead (Flat Cast) 
+t2 t1)) \to (land (pc3 d t2 t) (ty3 g d t1 t2))))).(\lambda (H4: (eq T (TLRef 
+n) (THead (Flat Cast) t2 t1))).(let H5 \def (eq_ind T (TLRef n) (\lambda (ee: 
+T).(match ee 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 (land (pc3 c0 t2 (lift (S n) O 
+u)) (ty3 g c0 t1 t2)) H5))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda 
+(t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u (THead (Flat Cast) 
+t2 t1)) \to (land (pc3 c0 t2 t) (ty3 g c0 t1 t2))))).(\lambda (b: B).(\lambda 
+(t0: T).(\lambda (t3: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t0 
+t3)).(\lambda (_: (((eq T t0 (THead (Flat Cast) t2 t1)) \to (land (pc3 (CHead 
+c0 (Bind b) u) t2 t3) (ty3 g (CHead c0 (Bind b) u) t1 t2))))).(\lambda (t4: 
+T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t3 t4)).(\lambda (_: (((eq T t3 
+(THead (Flat Cast) t2 t1)) \to (land (pc3 (CHead c0 (Bind b) u) t2 t4) (ty3 g 
+(CHead c0 (Bind b) u) t1 t2))))).(\lambda (H7: (eq T (THead (Bind b) u t0) 
+(THead (Flat Cast) t2 t1))).(let H8 \def (eq_ind T (THead (Bind b) u t0) 
+(\lambda (ee: T).(match ee 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 
+(land (pc3 c0 t2 (THead (Bind b) u t3)) (ty3 g c0 t1 t2)) H8)))))))))))))))) 
+(\lambda (c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w 
+u)).(\lambda (_: (((eq T w (THead (Flat Cast) t2 t1)) \to (land (pc3 c0 t2 u) 
+(ty3 g c0 t1 t2))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v 
+(THead (Bind Abst) u t))).(\lambda (_: (((eq T v (THead (Flat Cast) t2 t1)) 
+\to (land (pc3 c0 t2 (THead (Bind Abst) u t)) (ty3 g c0 t1 t2))))).(\lambda 
+(H5: (eq T (THead (Flat Appl) w v) (THead (Flat Cast) t2 t1))).(let H6 \def 
+(eq_ind T (THead (Flat Appl) w v) (\lambda (ee: T).(match ee 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 (land 
+(pc3 c0 t2 (THead (Flat Appl) w (THead (Bind Abst) u t))) (ty3 g c0 t1 t2)) 
+H6)))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t3: T).(\lambda 
+(H1: (ty3 g c0 t0 t3)).(\lambda (H2: (((eq T t0 (THead (Flat Cast) t2 t1)) 
+\to (land (pc3 c0 t2 t3) (ty3 g c0 t1 t2))))).(\lambda (t4: T).(\lambda (H3: 
+(ty3 g c0 t3 t4)).(\lambda (H4: (((eq T t3 (THead (Flat Cast) t2 t1)) \to 
+(land (pc3 c0 t2 t4) (ty3 g c0 t1 t2))))).(\lambda (H5: (eq T (THead (Flat 
+Cast) t3 t0) (THead (Flat Cast) t2 t1))).(let H6 \def (f_equal T T (\lambda 
+(e: T).(match e 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 (land (pc3 c0 t2 t4) (ty3 g c0 t1 t2)))) H4 t2 H8) in (let 
+H10 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t t4)) H3 t2 H8) in (let H11 
+\def (eq_ind T t3 (\lambda (t: T).((eq T t0 (THead (Flat Cast) t2 t1)) \to 
+(land (pc3 c0 t2 t) (ty3 g c0 t1 t2)))) H2 t2 H8) in (let H12 \def (eq_ind T 
+t3 (\lambda (t: T).(ty3 g c0 t0 t)) H1 t2 H8) in (eq_ind_r T t2 (\lambda (t: 
+T).(land (pc3 c0 t2 t) (ty3 g c0 t1 t2))) (let H13 \def (eq_ind T t0 (\lambda 
+(t: T).((eq T t (THead (Flat Cast) t2 t1)) \to (land (pc3 c0 t2 t2) (ty3 g c0 
+t1 t2)))) H11 t1 H7) in (let H14 \def (eq_ind T t0 (\lambda (t: T).(ty3 g c0 
+t t2)) H12 t1 H7) in (conj (pc3 c0 t2 t2) (ty3 g c0 t1 t2) (pc3_refl c0 t2) 
+H14))) t3 H8))))))) H6))))))))))) c y x H0))) H)))))).
+

diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/pr3.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/pr3.ma
new file mode 100644 (file)
index 0000000..0c14d34
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/pr3.ma
@@ -0,0 +1,732 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/pr3".
+
+include "csub3/ty3.ma".
+
+include "ty3/subst1.ma".
+
+include "ty3/fsubst0.ma".
+
+include "pc3/pc1.ma".
+
+include "pc3/wcpr0.ma".
+
+include "pc1/props.ma".
+
+theorem ty3_sred_wcpr0_pr0:
+ \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t: T).((ty3 g c1 
+t1 t) \to (\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t2: T).((pr0 t1 t2) 
+\to (ty3 g c2 t2 t)))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t: T).(\lambda 
+(H: (ty3 g c1 t1 t)).(ty3_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda 
+(t2: T).(\forall (c2: C).((wcpr0 c c2) \to (\forall (t3: T).((pr0 t0 t3) \to 
+(ty3 g c2 t3 t2)))))))) (\lambda (c: C).(\lambda (t2: T).(\lambda (t0: 
+T).(\lambda (_: (ty3 g c t2 t0)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c 
+c2) \to (\forall (t3: T).((pr0 t2 t3) \to (ty3 g c2 t3 t0))))))).(\lambda (u: 
+T).(\lambda (t3: T).(\lambda (_: (ty3 g c u t3)).(\lambda (H3: ((\forall (c2: 
+C).((wcpr0 c c2) \to (\forall (t4: T).((pr0 u t4) \to (ty3 g c2 t4 
+t3))))))).(\lambda (H4: (pc3 c t3 t2)).(\lambda (c2: C).(\lambda (H5: (wcpr0 
+c c2)).(\lambda (t4: T).(\lambda (H6: (pr0 u t4)).(ty3_conv g c2 t2 t0 (H1 c2 
+H5 t2 (pr0_refl t2)) t4 t3 (H3 c2 H5 t4 H6) (pc3_wcpr0 c c2 H5 t3 t2 
+H4)))))))))))))))) (\lambda (c: C).(\lambda (m: nat).(\lambda (c2: 
+C).(\lambda (_: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H1: (pr0 (TSort m) 
+t2)).(eq_ind_r T (TSort m) (\lambda (t0: T).(ty3 g c2 t0 (TSort (next g m)))) 
+(ty3_sort g c2 m) t2 (pr0_gen_sort t2 m H1)))))))) (\lambda (n: nat).(\lambda 
+(c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind 
+Abbr) u))).(\lambda (t0: T).(\lambda (_: (ty3 g d u t0)).(\lambda (H2: 
+((\forall (c2: C).((wcpr0 d c2) \to (\forall (t2: T).((pr0 u t2) \to (ty3 g 
+c2 t2 t0))))))).(\lambda (c2: C).(\lambda (H3: (wcpr0 c c2)).(\lambda (t2: 
+T).(\lambda (H4: (pr0 (TLRef n) t2)).(eq_ind_r T (TLRef n) (\lambda (t3: 
+T).(ty3 g c2 t3 (lift (S n) O t0))) (ex3_2_ind C T (\lambda (e2: C).(\lambda 
+(u2: T).(getl n c2 (CHead e2 (Bind Abbr) u2)))) (\lambda (e2: C).(\lambda (_: 
+T).(wcpr0 d e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u u2))) (ty3 g c2 
+(TLRef n) (lift (S n) O t0)) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: 
+(getl n c2 (CHead x0 (Bind Abbr) x1))).(\lambda (H6: (wcpr0 d x0)).(\lambda 
+(H7: (pr0 u x1)).(ty3_abbr g n c2 x0 x1 H5 t0 (H2 x0 H6 x1 H7))))))) 
+(wcpr0_getl c c2 H3 n d u (Bind Abbr) H0)) t2 (pr0_gen_lref t2 n 
+H4)))))))))))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d: C).(\lambda 
+(u: T).(\lambda (H0: (getl n c (CHead d (Bind Abst) u))).(\lambda (t0: 
+T).(\lambda (_: (ty3 g d u t0)).(\lambda (H2: ((\forall (c2: C).((wcpr0 d c2) 
+\to (\forall (t2: T).((pr0 u t2) \to (ty3 g c2 t2 t0))))))).(\lambda (c2: 
+C).(\lambda (H3: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H4: (pr0 (TLRef n) 
+t2)).(eq_ind_r T (TLRef n) (\lambda (t3: T).(ty3 g c2 t3 (lift (S n) O u))) 
+(ex3_2_ind C T (\lambda (e2: C).(\lambda (u2: T).(getl n c2 (CHead e2 (Bind 
+Abst) u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 d e2))) (\lambda (_: 
+C).(\lambda (u2: T).(pr0 u u2))) (ty3 g c2 (TLRef n) (lift (S n) O u)) 
+(\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (getl n c2 (CHead x0 (Bind 
+Abst) x1))).(\lambda (H6: (wcpr0 d x0)).(\lambda (H7: (pr0 u x1)).(ty3_conv g 
+c2 (lift (S n) O u) (lift (S n) O t0) (ty3_lift g x0 u t0 (H2 x0 H6 u 
+(pr0_refl u)) c2 O (S n) (getl_drop Abst c2 x0 x1 n H5)) (TLRef n) (lift (S 
+n) O x1) (ty3_abst g n c2 x0 x1 H5 t0 (H2 x0 H6 x1 H7)) (pc3_lift c2 x0 (S n) 
+O (getl_drop Abst c2 x0 x1 n H5) x1 u (pc3_pr2_x x0 x1 u (pr2_free x0 u x1 
+H7))))))))) (wcpr0_getl c c2 H3 n d u (Bind Abst) H0)) t2 (pr0_gen_lref t2 n 
+H4)))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (t0: T).(\lambda 
+(_: (ty3 g c u t0)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c c2) \to 
+(\forall (t2: T).((pr0 u t2) \to (ty3 g c2 t2 t0))))))).(\lambda (b: 
+B).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H2: (ty3 g (CHead c (Bind b) 
+u) t2 t3)).(\lambda (H3: ((\forall (c2: C).((wcpr0 (CHead c (Bind b) u) c2) 
+\to (\forall (t4: T).((pr0 t2 t4) \to (ty3 g c2 t4 t3))))))).(\lambda (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 
+[(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 
+(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 
+(\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: 
+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_pr2_x c2 x1 (THead (Bind Void) u (lift (S O) O x1)) 
+(pr2_free c2 (THead (Bind Void) u (lift (S O) O x1)) x1 (pr0_zeta Void 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 
+(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 
+(\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 (csub3_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 
+(csub3_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) 
+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 (_: 
+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 
+(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_s 
+(THead (Flat Appl) v2 (THead (Bind b) u2 (lift (S O) O (THead (Bind Abst) u 
+t0)))) (THead (Flat Appl) w (THead (Bind Abst) u t0)) (pc1_head w v2 
+(pc1_pr0_r w v2 H17) (THead (Bind Abst) u t0) (THead (Bind b) u2 (lift (S O) 
+O (THead (Bind Abst) u t0))) (pc1_pr0_x (THead (Bind Abst) u t0) (THead (Bind 
+b) u2 (lift (S O) O (THead (Bind Abst) u t0))) (pr0_zeta b 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 (\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 (\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 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 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 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 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 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 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 t3)))) (\lambda (H16: (pr0 t3 u2)).(\lambda (H17: (pr0 t2 
+t6)).(ty3_conv g c2 t3 t0 (H3 c2 H4 t3 (pr0_refl t3)) (THead (Flat Cast) u2 
+t6) 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_pr2_x c2 u2 t3 (pr2_free c2 t3 u2 H16))))) 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 t3)))) H10)) H9 H6 H7))) | (pr0_upsilon b H6 v1 v2 H7 u1 
+u2 H8 t5 t6 H9) \Rightarrow (\lambda (H10: (eq T (THead (Flat Appl) v1 (THead 
+(Bind b) u1 t5)) (THead (Flat Cast) t3 t2))).(\lambda (H11: (eq T (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t6)) t4)).((let H12 \def 
+(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t5)) (\lambda (e: 
+T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
+False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K 
+return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) 
+\Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl \Rightarrow 
+True | Cast \Rightarrow False])])])) I (THead (Flat Cast) t3 t2) H10) in 
+(False_ind ((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
+t6)) t4) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 
+t5 t6) \to (ty3 g c2 t4 t3)))))) H12)) H11 H6 H7 H8 H9))) | (pr0_delta u1 u2 
+H6 t5 t6 H7 w H8) \Rightarrow (\lambda (H9: (eq T (THead (Bind Abbr) u1 t5) 
+(THead (Flat Cast) t3 t2))).(\lambda (H10: (eq T (THead (Bind Abbr) u2 w) 
+t4)).((let H11 \def (eq_ind T (THead (Bind Abbr) u1 t5) (\lambda (e: 
+T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
+False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K 
+return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) 
+\Rightarrow False])])) I (THead (Flat Cast) t3 t2) H9) in (False_ind ((eq T 
+(THead (Bind Abbr) u2 w) t4) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O 
+u2 t6 w) \to (ty3 g c2 t4 t3))))) H11)) H10 H6 H7 H8))) | (pr0_zeta b H6 t5 
+t6 H7 u) \Rightarrow (\lambda (H8: (eq T (THead (Bind b) u (lift (S O) O t5)) 
+(THead (Flat Cast) t3 t2))).(\lambda (H9: (eq T t6 t4)).((let H10 \def 
+(eq_ind T (THead (Bind b) u (lift (S O) O t5)) (\lambda (e: T).(match e in T 
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda 
+(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow 
+False])])) I (THead (Flat Cast) t3 t2) H8) in (False_ind ((eq T t6 t4) \to 
+((not (eq B b Abst)) \to ((pr0 t5 t6) \to (ty3 g c2 t4 t3)))) H10)) H9 H6 
+H7))) | (pr0_epsilon t5 t6 H6 u) \Rightarrow (\lambda (H7: (eq T (THead (Flat 
+Cast) u t5) (THead (Flat Cast) t3 t2))).(\lambda (H8: (eq T t6 t4)).((let H9 
+\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) 
+with [(TSort _) \Rightarrow t5 | (TLRef _) \Rightarrow t5 | (THead _ _ t7) 
+\Rightarrow t7])) (THead (Flat Cast) u t5) (THead (Flat Cast) t3 t2) H7) in 
+((let H10 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: 
+T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t7 
+_) \Rightarrow t7])) (THead (Flat Cast) u t5) (THead (Flat Cast) t3 t2) H7) 
+in (eq_ind T t3 (\lambda (_: T).((eq T t5 t2) \to ((eq T t6 t4) \to ((pr0 t5 
+t6) \to (ty3 g c2 t4 t3))))) (\lambda (H11: (eq T t5 t2)).(eq_ind T t2 
+(\lambda (t7: T).((eq T t6 t4) \to ((pr0 t7 t6) \to (ty3 g c2 t4 t3)))) 
+(\lambda (H12: (eq T t6 t4)).(eq_ind T t4 (\lambda (t7: T).((pr0 t2 t7) \to 
+(ty3 g c2 t4 t3))) (\lambda (H13: (pr0 t2 t4)).(H1 c2 H4 t4 H13)) t6 (sym_eq 
+T t6 t4 H12))) t5 (sym_eq T t5 t2 H11))) u (sym_eq T u t3 H10))) H9)) H8 
+H6)))]) in (H6 (refl_equal T (THead (Flat Cast) t3 t2)) (refl_equal T 
+t4))))))))))))))) c1 t1 t H))))).
+
+theorem ty3_sred_pr1:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall 
+(g: G).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t)))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr1 t1 
+t2)).(pr1_ind (\lambda (t: T).(\lambda (t0: T).(\forall (g: G).(\forall (t3: 
+T).((ty3 g c t t3) \to (ty3 g c t0 t3)))))) (\lambda (t: T).(\lambda (g: 
+G).(\lambda (t0: T).(\lambda (H0: (ty3 g c t t0)).H0)))) (\lambda (t3: 
+T).(\lambda (t4: T).(\lambda (H0: (pr0 t4 t3)).(\lambda (t5: T).(\lambda (_: 
+(pr1 t3 t5)).(\lambda (H2: ((\forall (g: G).(\forall (t: T).((ty3 g c t3 t) 
+\to (ty3 g c t5 t)))))).(\lambda (g: G).(\lambda (t: T).(\lambda (H3: (ty3 g 
+c t4 t)).(H2 g t (ty3_sred_wcpr0_pr0 g c t4 t H3 c (wcpr0_refl c) t3 
+H0))))))))))) t1 t2 H)))).
+
+theorem ty3_sred_pr2:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall 
+(g: G).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t)))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 
+t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (g: 
+G).(\forall (t3: T).((ty3 g c0 t t3) \to (ty3 g c0 t0 t3))))))) (\lambda (c0: 
+C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda (g: 
+G).(\lambda (t: T).(\lambda (H1: (ty3 g c0 t3 t)).(ty3_sred_wcpr0_pr0 g c0 t3 
+t H1 c0 (wcpr0_refl c0) t4 H0)))))))) (\lambda (c0: C).(\lambda (d: 
+C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind 
+Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 
+t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (g: 
+G).(\lambda (t0: T).(\lambda (H3: (ty3 g c0 t3 t0)).(ty3_subst0 g c0 t4 t0 
+(ty3_sred_wcpr0_pr0 g c0 t3 t0 H3 c0 (wcpr0_refl c0) t4 H1) d u i H0 t 
+H2)))))))))))))) c t1 t2 H)))).
+
+theorem ty3_sred_pr3:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall 
+(g: G).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t)))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 
+t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (g: G).(\forall 
+(t3: T).((ty3 g c t t3) \to (ty3 g c t0 t3)))))) (\lambda (t: T).(\lambda (g: 
+G).(\lambda (t0: T).(\lambda (H0: (ty3 g c t t0)).H0)))) (\lambda (t3: 
+T).(\lambda (t4: T).(\lambda (H0: (pr2 c t4 t3)).(\lambda (t5: T).(\lambda 
+(_: (pr3 c t3 t5)).(\lambda (H2: ((\forall (g: G).(\forall (t: T).((ty3 g c 
+t3 t) \to (ty3 g c t5 t)))))).(\lambda (g: G).(\lambda (t: T).(\lambda (H3: 
+(ty3 g c t4 t)).(H2 g t (ty3_sred_pr2 c t4 t3 H0 g t H3))))))))))) t1 t2 
+H)))).
+

diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/pr3_props.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/pr3_props.ma
new file mode 100644 (file)
index 0000000..f429f69
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/pr3_props.ma
@@ -0,0 +1,501 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/pr3_props".
+
+include "ty3/pr3.ma".
+
+theorem ty3_cred_pr2:
+ \forall (g: G).(\forall (c: C).(\forall (v1: T).(\forall (v2: T).((pr2 c v1 
+v2) \to (\forall (b: B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c 
+(Bind b) v1) t1 t2) \to (ty3 g (CHead c (Bind b) v2) t1 t2)))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda 
+(H: (pr2 c v1 v2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: 
+T).(\forall (b: B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c0 (Bind 
+b) t) t1 t2) \to (ty3 g (CHead c0 (Bind b) t0) t1 t2)))))))) (\lambda (c0: 
+C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr0 t1 t2)).(\lambda (b: 
+B).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H1: (ty3 g (CHead c0 (Bind b) 
+t1) t0 t3)).(ty3_sred_wcpr0_pr0 g (CHead c0 (Bind b) t1) t0 t3 H1 (CHead c0 
+(Bind b) t2) (wcpr0_comp c0 c0 (wcpr0_refl c0) t1 t2 H0 (Bind b)) t0 
+(pr0_refl t0)))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: 
+T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) 
+u))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H1: (pr0 t1 t2)).(\lambda 
+(t: T).(\lambda (H2: (subst0 i u t2 t)).(\lambda (b: B).(\lambda (t0: 
+T).(\lambda (t3: T).(\lambda (H3: (ty3 g (CHead c0 (Bind b) t1) t0 
+t3)).(ty3_csubst0 g (CHead c0 (Bind b) t2) t0 t3 (ty3_sred_wcpr0_pr0 g (CHead 
+c0 (Bind b) t1) t0 t3 H3 (CHead c0 (Bind b) t2) (wcpr0_comp c0 c0 (wcpr0_refl 
+c0) t1 t2 H1 (Bind b)) t0 (pr0_refl t0)) d u (S i) (getl_clear_bind b (CHead 
+c0 (Bind b) t2) c0 t2 (clear_bind b c0 t2) (CHead d (Bind Abbr) u) i H0) 
+(CHead c0 (Bind b) t) (csubst0_snd_bind b i u t2 t H2 c0)))))))))))))))) c v1 
+v2 H))))).
+
+theorem ty3_cred_pr3:
+ \forall (g: G).(\forall (c: C).(\forall (v1: T).(\forall (v2: T).((pr3 c v1 
+v2) \to (\forall (b: B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c 
+(Bind b) v1) t1 t2) \to (ty3 g (CHead c (Bind b) v2) t1 t2)))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda 
+(H: (pr3 c v1 v2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (b: 
+B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c (Bind b) t) t1 t2) \to 
+(ty3 g (CHead c (Bind b) t0) t1 t2))))))) (\lambda (t: T).(\lambda (b: 
+B).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (ty3 g (CHead c (Bind b) 
+t) t1 t2)).H0))))) (\lambda (t2: T).(\lambda (t1: T).(\lambda (H0: (pr2 c t1 
+t2)).(\lambda (t3: T).(\lambda (_: (pr3 c t2 t3)).(\lambda (H2: ((\forall (b: 
+B).(\forall (t4: T).(\forall (t5: T).((ty3 g (CHead c (Bind b) t2) t4 t5) \to 
+(ty3 g (CHead c (Bind b) t3) t4 t5))))))).(\lambda (b: B).(\lambda (t0: 
+T).(\lambda (t4: T).(\lambda (H3: (ty3 g (CHead c (Bind b) t1) t0 t4)).(H2 b 
+t0 t4 (ty3_cred_pr2 g c t1 t2 H0 b t0 t4 H3)))))))))))) v1 v2 H))))).
+
+theorem ty3_gen_lift:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (x: T).(\forall (h: 
+nat).(\forall (d: nat).((ty3 g c (lift h d t1) x) \to (\forall (e: C).((drop 
+h d c e) \to (ex2 T (\lambda (t2: T).(pc3 c (lift h d t2) x)) (\lambda (t2: 
+T).(ty3 g e t1 t2)))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (x: T).(\lambda (h: 
+nat).(\lambda (d: nat).(\lambda (H: (ty3 g c (lift h d t1) x)).(insert_eq T 
+(lift h d t1) (\lambda (t: T).(ty3 g c t x)) (\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 
+(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda (t2: 
+T).(ty3 g e (TLRef (minus n h)) t2)) (lift (S (minus n h)) O t) (eq_ind_r T 
+(lift (plus h (S (minus n h))) O t) (\lambda (t0: T).(pc3 c0 t0 (lift (S n) O 
+t))) (eq_ind nat (S (plus h (minus n h))) (\lambda (n0: nat).(pc3 c0 (lift n0 
+O t) (lift (S n) O t))) (eq_ind nat n (\lambda (n0: nat).(pc3 c0 (lift (S n0) 
+O t) (lift (S n) O t))) (pc3_refl c0 (lift (S n) O t)) (plus h (minus n h)) 
+(le_plus_minus h n (le_trans_plus_r x1 h n H8))) (plus h (S (minus n h))) 
+(plus_n_Sm h (minus n h))) (lift h x1 (lift (S (minus n h)) O t)) (lift_free 
+t (S (minus n h)) h O x1 (le_trans x1 (S (minus n h)) (plus O (S (minus n 
+h))) (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)).(\lambda (H18: 
+(ty3 g x3 x2 x4)).(eq_ind_r nat (plus (S n) (minus x1 (S n))) (\lambda (n0: 
+nat).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h n0 t2) (lift (S n) O (lift h 
+(minus n0 (S n)) x2)))) (\lambda (t2: T).(ty3 g e (TLRef n) t2)))) (ex_intro2 
+T (\lambda (t2: T).(pc3 c0 (lift h (plus (S n) (minus x1 (S n))) t2) (lift (S 
+n) O (lift h (minus (plus (S n) (minus x1 (S n))) (S n)) x2)))) (\lambda (t2: 
+T).(ty3 g e (TLRef n) t2)) (lift (S n) O x2) (eq_ind_r T (lift (S n) O (lift 
+h (minus x1 (S n)) x2)) (\lambda (t0: T).(pc3 c0 t0 (lift (S n) O (lift h 
+(minus (plus (S n) (minus x1 (S n))) (S n)) x2)))) (eq_ind nat x1 (\lambda 
+(n0: nat).(pc3 c0 (lift (S n) O (lift h (minus x1 (S n)) x2)) (lift (S n) O 
+(lift h (minus n0 (S n)) x2)))) (pc3_refl c0 (lift (S n) O (lift h (minus x1 
+(S n)) x2))) (plus (S n) (minus x1 (S n))) (le_plus_minus (S n) x1 H8)) (lift 
+h (plus (S n) (minus x1 (S n))) (lift (S n) O x2)) (lift_d x2 h (S n) (minus 
+x1 (S n)) O (le_O_n (minus x1 (S n))))) (ty3_abst g n e x3 x2 H12 x4 H18)) x1 
+(le_plus_minus (S n) x1 H8))))) H16)) u H11)))))))) (getl_drop_conf_lt Abst 
+c0 d0 u n H1 e h (minus x1 (S n)) H10))) x0 H9))) H7)) (\lambda (H7: (land 
+(le (plus x1 h) n) (eq T x0 (TLRef (minus n h))))).(and_ind (le (plus x1 h) 
+n) (eq T x0 (TLRef (minus n h))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 
+t2) (lift (S n) O u))) (\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (H8: (le 
+(plus x1 h) n)).(\lambda (H9: (eq T x0 (TLRef (minus n h)))).(eq_ind_r T 
+(TLRef (minus n h)) (\lambda (t0: T).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h 
+x1 t2) (lift (S n) O u))) (\lambda (t2: T).(ty3 g e t0 t2)))) (ex_intro2 T 
+(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O u))) (\lambda (t2: 
+T).(ty3 g e (TLRef (minus n h)) t2)) (lift (S (minus n h)) O u) (eq_ind_r T 
+(lift (plus h (S (minus n h))) O u) (\lambda (t0: T).(pc3 c0 t0 (lift (S n) O 
+u))) (eq_ind nat (S (plus h (minus n h))) (\lambda (n0: nat).(pc3 c0 (lift n0 
+O u) (lift (S n) O u))) (eq_ind nat n (\lambda (n0: nat).(pc3 c0 (lift (S n0) 
+O u) (lift (S n) O u))) (pc3_refl c0 (lift (S n) O u)) (plus h (minus n h)) 
+(le_plus_minus h n (le_trans_plus_r x1 h n H8))) (plus h (S (minus n h))) 
+(plus_n_Sm h (minus n h))) (lift h x1 (lift (S (minus n h)) O u)) (lift_free 
+u (S (minus n h)) h O x1 (le_trans x1 (S (minus n h)) (plus O (S (minus n 
+h))) (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 (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 
+(x4: T).(\forall (x5: nat).((eq T t0 (lift h x5 x4)) \to (\forall (e0: 
+C).((drop h x5 c0 e0) \to (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x5 t2) 
+(THead (Bind Abst) u t))) (\lambda (t2: T).(ty3 g e0 x4 t2))))))))) H4 (lift 
+h x1 x3) H9) in (let H11 \def (eq_ind T v (\lambda (t0: T).(ty3 g c0 t0 
+(THead (Bind Abst) u t))) H3 (lift h x1 x3) H9) in (let H12 \def (eq_ind T w 
+(\lambda (t0: T).(\forall (x4: T).(\forall (x5: nat).((eq T t0 (lift h x5 
+x4)) \to (\forall (e0: C).((drop h x5 c0 e0) \to (ex2 T (\lambda (t2: T).(pc3 
+c0 (lift h x5 t2) u)) (\lambda (t2: T).(ty3 g e0 x4 t2))))))))) H2 (lift h x1 
+x2) H8) in (let H13 \def (eq_ind T w (\lambda (t0: T).(ty3 g c0 t0 u)) H1 
+(lift h x1 x2) H8) in (eq_ind_r T (lift h x1 x2) (\lambda (t0: T).(ex2 T 
+(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) t0 (THead (Bind 
+Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2)))) (let 
+H14 \def (H12 x2 x1 (refl_equal T (lift h x1 x2)) e H6) in (ex2_ind T 
+(\lambda (t2: T).(pc3 c0 (lift h x1 t2) u)) (\lambda (t2: T).(ty3 g e x2 t2)) 
+(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) (lift h x1 
+x2) (THead (Bind Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead (Flat Appl) 
+x2 x3) t2))) (\lambda (x4: T).(\lambda (H15: (pc3 c0 (lift h x1 x4) 
+u)).(\lambda (H16: (ty3 g e x2 x4)).(let H17 \def (H10 x3 x1 (refl_equal T 
+(lift h x1 x3)) e H6) in (ex2_ind T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) 
+(THead (Bind Abst) u t))) (\lambda (t2: T).(ty3 g e x3 t2)) (ex2 T (\lambda 
+(t2: T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind 
+Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2))) 
+(\lambda (x5: T).(\lambda (H18: (pc3 c0 (lift h x1 x5) (THead (Bind Abst) u 
+t))).(\lambda (H19: (ty3 g e x3 x5)).(ex3_2_ind T T (\lambda (u1: T).(\lambda 
+(t2: T).(pr3 e x5 (THead (Bind Abst) u1 t2)))) (\lambda (u1: T).(\lambda (_: 
+T).(pr3 c0 u (lift h x1 u1)))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: 
+B).(\forall (u0: T).(pr3 (CHead c0 (Bind b) u0) t (lift h (S x1) t2)))))) 
+(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) (lift h x1 
+x2) (THead (Bind Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead (Flat Appl) 
+x2 x3) t2))) (\lambda (x6: T).(\lambda (x7: T).(\lambda (H20: (pr3 e x5 
+(THead (Bind Abst) x6 x7))).(\lambda (H21: (pr3 c0 u (lift h x1 
+x6))).(\lambda (H22: ((\forall (b: B).(\forall (u0: T).(pr3 (CHead c0 (Bind 
+b) u0) t (lift h (S x1) x7)))))).(ex_ind T (\lambda (t0: T).(ty3 g e x5 t0)) 
+(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) (lift h x1 
+x2) (THead (Bind Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead (Flat Appl) 
+x2 x3) t2))) (\lambda (x8: T).(\lambda (H23: (ty3 g e x5 x8)).(ex4_3_ind T T 
+T (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 e (THead (Bind Abst) 
+x6 t2) x8)))) (\lambda (_: T).(\lambda (t0: T).(\lambda (_: T).(ty3 g e x6 
+t0)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead e (Bind 
+Abst) x6) x7 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t3: T).(ty3 g 
+(CHead e (Bind Abst) x6) t2 t3)))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 
+t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind Abst) u t)))) (\lambda 
+(t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2))) (\lambda (x9: T).(\lambda 
+(x10: T).(\lambda (x11: T).(\lambda (_: (pc3 e (THead (Bind Abst) x6 x9) 
+x8)).(\lambda (H25: (ty3 g e x6 x10)).(\lambda (H26: (ty3 g (CHead e (Bind 
+Abst) x6) x7 x9)).(\lambda (H27: (ty3 g (CHead e (Bind Abst) x6) x9 
+x11)).(ex_intro2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) 
+(lift h x1 x2) (THead (Bind Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead 
+(Flat Appl) x2 x3) t2)) (THead (Flat Appl) x2 (THead (Bind Abst) x6 x7)) 
+(eq_ind_r T (THead (Flat Appl) (lift h x1 x2) (lift h x1 (THead (Bind Abst) 
+x6 x7))) (\lambda (t0: T).(pc3 c0 t0 (THead (Flat Appl) (lift h x1 x2) (THead 
+(Bind Abst) u t)))) (pc3_thin_dx c0 (lift h x1 (THead (Bind Abst) x6 x7)) 
+(THead (Bind Abst) u t) (eq_ind_r T (THead (Bind Abst) (lift h x1 x6) (lift h 
+(S x1) x7)) (\lambda (t0: T).(pc3 c0 t0 (THead (Bind Abst) u t))) 
+(pc3_head_21 c0 (lift h x1 x6) u (pc3_pr3_x c0 (lift h x1 x6) u H21) (Bind 
+Abst) (lift h (S x1) x7) t (pc3_pr3_x (CHead c0 (Bind Abst) (lift h x1 x6)) 
+(lift h (S x1) x7) t (H22 Abst (lift h x1 x6)))) (lift h x1 (THead (Bind 
+Abst) x6 x7)) (lift_bind Abst x6 x7 h x1)) (lift h x1 x2) Appl) (lift h x1 
+(THead (Flat Appl) x2 (THead (Bind Abst) x6 x7))) (lift_flat Appl x2 (THead 
+(Bind Abst) x6 x7) h x1)) (ty3_appl g e x2 x6 (ty3_conv g e x6 x10 H25 x2 x4 
+H16 (pc3_gen_lift c0 x4 x6 h x1 (pc3_t u c0 (lift h x1 x4) H15 (lift h x1 x6) 
+(pc3_pr3_r c0 u (lift h x1 x6) H21)) e H6)) x3 x7 (ty3_conv g e (THead (Bind 
+Abst) x6 x7) (THead (Bind Abst) x6 x9) (ty3_bind g e x6 x10 H25 Abst x7 x9 
+H26 x11 H27) x3 x5 H19 (pc3_pr3_r e x5 (THead (Bind Abst) x6 x7) 
+H20))))))))))) (ty3_gen_bind g Abst e x6 x7 x8 (ty3_sred_pr3 e x5 (THead 
+(Bind Abst) x6 x7) H20 g x8 H23))))) (ty3_correct g e x3 x5 H19))))))) 
+(pc3_gen_lift_abst c0 x5 t u h x1 H18 e H6))))) H17))))) H14)) w H8))))) x0 
+H7)))))) (lift_gen_flat Appl w v x0 h x1 H5)))))))))))))))) (\lambda (c0: 
+C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (ty3 g c0 t2 t3)).(\lambda 
+(H2: ((\forall (x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to 
+(\forall (e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t4: T).(pc3 c0 (lift h 
+x1 t4) t3)) (\lambda (t4: T).(ty3 g e x0 t4)))))))))).(\lambda (t0: 
+T).(\lambda (H3: (ty3 g c0 t3 t0)).(\lambda (H4: ((\forall (x0: T).(\forall 
+(x1: nat).((eq T t3 (lift h x1 x0)) \to (\forall (e: C).((drop h x1 c0 e) \to 
+(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) t0)) (\lambda (t4: T).(ty3 g e 
+x0 t4)))))))))).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H5: (eq T 
+(THead (Flat Cast) t3 t2) (lift h x1 x0))).(\lambda (e: C).(\lambda (H6: 
+(drop h x1 c0 e)).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 
+(THead (Flat Cast) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T t3 (lift h 
+x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h x1 z)))) (ex2 T 
+(\lambda (t4: T).(pc3 c0 (lift h x1 t4) t3)) (\lambda (t4: T).(ty3 g e x0 
+t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq T x0 (THead (Flat 
+Cast) x2 x3))).(\lambda (H8: (eq T t3 (lift h x1 x2))).(\lambda (H9: (eq T t2 
+(lift h x1 x3))).(eq_ind_r T (THead (Flat Cast) x2 x3) (\lambda (t: T).(ex2 T 
+(\lambda (t4: T).(pc3 c0 (lift h x1 t4) t3)) (\lambda (t4: T).(ty3 g e t 
+t4)))) (let H10 \def (eq_ind T t3 (\lambda (t: T).(\forall (x4: T).(\forall 
+(x5: nat).((eq T t (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 c0 e0) 
+\to (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x5 t4) t0)) (\lambda (t4: T).(ty3 
+g e0 x4 t4))))))))) H4 (lift h x1 x2) H8) in (let H11 \def (eq_ind T t3 
+(\lambda (t: T).(ty3 g c0 t t0)) H3 (lift h x1 x2) H8) in (let H12 \def 
+(eq_ind T t3 (\lambda (t: T).(\forall (x4: T).(\forall (x5: nat).((eq T t2 
+(lift h x5 x4)) \to (\forall (e0: C).((drop h x5 c0 e0) \to (ex2 T (\lambda 
+(t4: T).(pc3 c0 (lift h x5 t4) t)) (\lambda (t4: T).(ty3 g e0 x4 t4))))))))) 
+H2 (lift h x1 x2) H8) in (let H13 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 
+t2 t)) H1 (lift h x1 x2) H8) in (eq_ind_r T (lift h x1 x2) (\lambda (t: 
+T).(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) t)) (\lambda (t4: T).(ty3 g 
+e (THead (Flat Cast) x2 x3) t4)))) (let H14 \def (eq_ind T t2 (\lambda (t: 
+T).(ty3 g c0 t (lift h x1 x2))) H13 (lift h x1 x3) H9) in (let H15 \def 
+(eq_ind T t2 (\lambda (t: T).(\forall (x4: T).(\forall (x5: nat).((eq T t 
+(lift h x5 x4)) \to (\forall (e0: C).((drop h x5 c0 e0) \to (ex2 T (\lambda 
+(t4: T).(pc3 c0 (lift h x5 t4) (lift h x1 x2))) (\lambda (t4: T).(ty3 g e0 x4 
+t4))))))))) H12 (lift h x1 x3) H9) in (let H16 \def (H15 x3 x1 (refl_equal T 
+(lift h x1 x3)) e H6) in (ex2_ind T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) 
+(lift h x1 x2))) (\lambda (t4: T).(ty3 g e x3 t4)) (ex2 T (\lambda (t4: 
+T).(pc3 c0 (lift h x1 t4) (lift h x1 x2))) (\lambda (t4: T).(ty3 g e (THead 
+(Flat Cast) x2 x3) t4))) (\lambda (x4: T).(\lambda (H17: (pc3 c0 (lift h x1 
+x4) (lift h x1 x2))).(\lambda (H18: (ty3 g e x3 x4)).(let H19 \def (H10 x2 x1 
+(refl_equal T (lift h x1 x2)) e H6) in (ex2_ind T (\lambda (t4: T).(pc3 c0 
+(lift h x1 t4) t0)) (\lambda (t4: T).(ty3 g e x2 t4)) (ex2 T (\lambda (t4: 
+T).(pc3 c0 (lift h x1 t4) (lift h x1 x2))) (\lambda (t4: T).(ty3 g e (THead 
+(Flat Cast) x2 x3) t4))) (\lambda (x5: T).(\lambda (_: (pc3 c0 (lift h x1 x5) 
+t0)).(\lambda (H21: (ty3 g e x2 x5)).(ex_intro2 T (\lambda (t4: T).(pc3 c0 
+(lift h x1 t4) (lift h x1 x2))) (\lambda (t4: T).(ty3 g e (THead (Flat Cast) 
+x2 x3) t4)) x2 (pc3_refl c0 (lift h x1 x2)) (ty3_cast g e x3 x2 (ty3_conv g e 
+x2 x5 H21 x3 x4 H18 (pc3_gen_lift c0 x4 x2 h x1 H17 e H6)) x5 H21))))) 
+H19))))) H16)))) t3 H8))))) x0 H7)))))) (lift_gen_flat Cast t3 t2 x0 h x1 
+H5))))))))))))))) c y x H0))))) H))))))).
+
+theorem ty3_tred:
+ \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).((ty3 g c u 
+t1) \to (\forall (t2: T).((pr3 c t1 t2) \to (ty3 g c u t2)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (H: 
+(ty3 g c u t1)).(\lambda (t2: T).(\lambda (H0: (pr3 c t1 t2)).(ex_ind T 
+(\lambda (t: T).(ty3 g c t1 t)) (ty3 g c u t2) (\lambda (x: T).(\lambda (H1: 
+(ty3 g c t1 x)).(ty3_conv g c t2 x (ty3_sred_pr3 c t1 t2 H0 g x H1) u t1 H 
+(pc3_pr3_r c t1 t2 H0)))) (ty3_correct g c u t1 H)))))))).
+
+theorem ty3_sconv_pc3:
+ \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (t1: T).((ty3 g c 
+u1 t1) \to (\forall (u2: T).(\forall (t2: T).((ty3 g c u2 t2) \to ((pc3 c u1 
+u2) \to (pc3 c t1 t2)))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda 
+(H: (ty3 g c u1 t1)).(\lambda (u2: T).(\lambda (t2: T).(\lambda (H0: (ty3 g c 
+u2 t2)).(\lambda (H1: (pc3 c u1 u2)).(let H2 \def H1 in (ex2_ind T (\lambda 
+(t: T).(pr3 c u1 t)) (\lambda (t: T).(pr3 c u2 t)) (pc3 c t1 t2) (\lambda (x: 
+T).(\lambda (H3: (pr3 c u1 x)).(\lambda (H4: (pr3 c u2 x)).(ty3_unique g c x 
+t1 (ty3_sred_pr3 c u1 x H3 g t1 H) t2 (ty3_sred_pr3 c u2 x H4 g t2 H0))))) 
+H2)))))))))).
+
+theorem ty3_sred_back:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t0: T).((ty3 g c 
+t1 t0) \to (\forall (t2: T).((pr3 c t1 t2) \to (\forall (t: T).((ty3 g c t2 
+t) \to (ty3 g c t1 t)))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t0: T).(\lambda 
+(H: (ty3 g c t1 t0)).(\lambda (t2: T).(\lambda (H0: (pr3 c t1 t2)).(\lambda 
+(t: T).(\lambda (H1: (ty3 g c t2 t)).(ex_ind T (\lambda (t3: T).(ty3 g c t 
+t3)) (ty3 g c t1 t) (\lambda (x: T).(\lambda (H2: (ty3 g c t x)).(ty3_conv g 
+c t x H2 t1 t0 H (ty3_unique g c t2 t0 (ty3_sred_pr3 c t1 t2 H0 g t0 H) t 
+H1)))) (ty3_correct g c t2 t H1)))))))))).
+
+theorem ty3_sconv:
+ \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (t1: T).((ty3 g c 
+u1 t1) \to (\forall (u2: T).(\forall (t2: T).((ty3 g c u2 t2) \to ((pc3 c u1 
+u2) \to (ty3 g c u1 t2)))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda 
+(H: (ty3 g c u1 t1)).(\lambda (u2: T).(\lambda (t2: T).(\lambda (H0: (ty3 g c 
+u2 t2)).(\lambda (H1: (pc3 c u1 u2)).(let H2 \def H1 in (ex2_ind T (\lambda 
+(t: T).(pr3 c u1 t)) (\lambda (t: T).(pr3 c u2 t)) (ty3 g c u1 t2) (\lambda 
+(x: T).(\lambda (H3: (pr3 c u1 x)).(\lambda (H4: (pr3 c u2 x)).(ty3_sred_back 
+g c u1 t1 H x H3 t2 (ty3_sred_pr3 c u2 x H4 g t2 H0))))) H2)))))))))).
+

diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/props.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/props.ma
new file mode 100644 (file)
index 0000000..d7615f7
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/props.ma
@@ -0,0 +1,422 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/props".
+
+include "ty3/fwd.ma".
+
+theorem ty3_lift:
+ \forall (g: G).(\forall (e: C).(\forall (t1: T).(\forall (t2: T).((ty3 g e 
+t1 t2) \to (\forall (c: C).(\forall (d: nat).(\forall (h: nat).((drop h d c 
+e) \to (ty3 g c (lift h d t1) (lift h d t2))))))))))
+\def
+ \lambda (g: G).(\lambda (e: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
+(H: (ty3 g e t1 t2)).(ty3_ind g (\lambda (c: C).(\lambda (t: T).(\lambda (t0: 
+T).(\forall (c0: C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c) \to 
+(ty3 g c0 (lift h d t) (lift h d t0))))))))) (\lambda (c: C).(\lambda (t0: 
+T).(\lambda (t: T).(\lambda (_: (ty3 g c t0 t)).(\lambda (H1: ((\forall (c0: 
+C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c) \to (ty3 g c0 (lift h 
+d t0) (lift h d t)))))))).(\lambda (u: T).(\lambda (t3: T).(\lambda (_: (ty3 
+g c u t3)).(\lambda (H3: ((\forall (c0: C).(\forall (d: nat).(\forall (h: 
+nat).((drop h d c0 c) \to (ty3 g c0 (lift h d u) (lift h d 
+t3)))))))).(\lambda (H4: (pc3 c t3 t0)).(\lambda (c0: C).(\lambda (d: 
+nat).(\lambda (h: nat).(\lambda (H5: (drop h d c0 c)).(ty3_conv g c0 (lift h 
+d t0) (lift h d t) (H1 c0 d h H5) (lift h d u) (lift h d t3) (H3 c0 d h H5) 
+(pc3_lift c0 c h d H5 t3 t0 H4)))))))))))))))) (\lambda (c: C).(\lambda (m: 
+nat).(\lambda (c0: C).(\lambda (d: nat).(\lambda (h: nat).(\lambda (_: (drop 
+h d c0 c)).(eq_ind_r T (TSort m) (\lambda (t: T).(ty3 g c0 t (lift h d (TSort 
+(next g m))))) (eq_ind_r T (TSort (next g m)) (\lambda (t: T).(ty3 g c0 
+(TSort m) t)) (ty3_sort g c0 m) (lift h d (TSort (next g m))) (lift_sort 
+(next g m) h d)) (lift h d (TSort m)) (lift_sort m h d)))))))) (\lambda (n: 
+nat).(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c 
+(CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u 
+t)).(\lambda (H2: ((\forall (c0: C).(\forall (d0: nat).(\forall (h: 
+nat).((drop h d0 c0 d) \to (ty3 g c0 (lift h d0 u) (lift h d0 
+t)))))))).(\lambda (c0: C).(\lambda (d0: nat).(\lambda (h: nat).(\lambda (H3: 
+(drop h d0 c0 c)).(lt_le_e n d0 (ty3 g c0 (lift h d0 (TLRef n)) (lift h d0 
+(lift (S n) O t))) (\lambda (H4: (lt n d0)).(let H5 \def (drop_getl_trans_le 
+n d0 (le_S_n n d0 (le_S (S n) d0 H4)) c0 c h H3 (CHead d (Bind Abbr) u) H0) 
+in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop n O c0 e0))) 
+(\lambda (e0: C).(\lambda (e1: C).(drop h (minus d0 n) e0 e1))) (\lambda (_: 
+C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abbr) u)))) (ty3 g c0 (lift h d0 
+(TLRef n)) (lift h d0 (lift (S n) O t))) (\lambda (x0: C).(\lambda (x1: 
+C).(\lambda (H6: (drop n O c0 x0)).(\lambda (H7: (drop h (minus d0 n) x0 
+x1)).(\lambda (H8: (clear x1 (CHead d (Bind Abbr) u))).(let H9 \def (eq_ind 
+nat (minus d0 n) (\lambda (n0: nat).(drop h n0 x0 x1)) H7 (S (minus d0 (S 
+n))) (minus_x_Sy d0 n H4)) in (let H10 \def (drop_clear_S x1 x0 h (minus d0 
+(S n)) H9 Abbr d u H8) in (ex2_ind C (\lambda (c1: C).(clear x0 (CHead c1 
+(Bind Abbr) (lift h (minus d0 (S n)) u)))) (\lambda (c1: C).(drop h (minus d0 
+(S n)) c1 d)) (ty3 g c0 (lift h d0 (TLRef n)) (lift h d0 (lift (S n) O t))) 
+(\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abbr) (lift h (minus 
+d0 (S n)) u)))).(\lambda (H12: (drop h (minus d0 (S n)) x d)).(eq_ind_r T 
+(TLRef n) (\lambda (t0: T).(ty3 g c0 t0 (lift h d0 (lift (S n) O t)))) 
+(eq_ind nat (plus (S n) (minus d0 (S n))) (\lambda (n0: nat).(ty3 g c0 (TLRef 
+n) (lift h n0 (lift (S n) O t)))) (eq_ind_r T (lift (S n) O (lift h (minus d0 
+(S n)) t)) (\lambda (t0: T).(ty3 g c0 (TLRef n) t0)) (eq_ind nat d0 (\lambda 
+(_: nat).(ty3 g c0 (TLRef n) (lift (S n) O (lift h (minus d0 (S n)) t)))) 
+(ty3_abbr g n c0 x (lift h (minus d0 (S n)) u) (getl_intro n c0 (CHead x 
+(Bind Abbr) (lift h (minus d0 (S n)) u)) x0 H6 H11) (lift h (minus d0 (S n)) 
+t) (H2 x (minus d0 (S n)) h H12)) (plus (S n) (minus d0 (S n))) 
+(le_plus_minus (S n) d0 H4)) (lift h (plus (S n) (minus d0 (S n))) (lift (S 
+n) O t)) (lift_d t h (S n) (minus d0 (S n)) O (le_O_n (minus d0 (S n))))) d0 
+(le_plus_minus_r (S n) d0 H4)) (lift h d0 (TLRef n)) (lift_lref_lt n h d0 
+H4))))) H10)))))))) H5))) (\lambda (H4: (le d0 n)).(eq_ind_r T (TLRef (plus n 
+h)) (\lambda (t0: T).(ty3 g c0 t0 (lift h d0 (lift (S n) O t)))) (eq_ind nat 
+(S n) (\lambda (_: nat).(ty3 g c0 (TLRef (plus n h)) (lift h d0 (lift (S n) O 
+t)))) (eq_ind_r T (lift (plus h (S n)) O t) (\lambda (t0: T).(ty3 g c0 (TLRef 
+(plus n h)) t0)) (eq_ind_r nat (plus (S n) h) (\lambda (n0: nat).(ty3 g c0 
+(TLRef (plus n h)) (lift n0 O t))) (ty3_abbr g (plus n h) c0 d u 
+(drop_getl_trans_ge n c0 c d0 h H3 (CHead d (Bind Abbr) u) H0 H4) t H1) (plus 
+h (S n)) (plus_comm h (S n))) (lift h d0 (lift (S n) O t)) (lift_free t (S n) 
+h O d0 (le_S d0 n H4) (le_O_n d0))) (plus n (S O)) (eq_ind_r nat (plus (S O) 
+n) (\lambda (n0: nat).(eq nat (S n) n0)) (refl_equal nat (plus (S O) n)) 
+(plus n (S O)) (plus_comm n (S O)))) (lift h d0 (TLRef n)) (lift_lref_ge n h 
+d0 H4)))))))))))))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d: 
+C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind Abst) u))).(\lambda 
+(t: T).(\lambda (H1: (ty3 g d u t)).(\lambda (H2: ((\forall (c0: C).(\forall 
+(d0: nat).(\forall (h: nat).((drop h d0 c0 d) \to (ty3 g c0 (lift h d0 u) 
+(lift h d0 t)))))))).(\lambda (c0: C).(\lambda (d0: nat).(\lambda (h: 
+nat).(\lambda (H3: (drop h d0 c0 c)).(lt_le_e n d0 (ty3 g c0 (lift h d0 
+(TLRef n)) (lift h d0 (lift (S n) O u))) (\lambda (H4: (lt n d0)).(let H5 
+\def (drop_getl_trans_le n d0 (le_S_n n d0 (le_S (S n) d0 H4)) c0 c h H3 
+(CHead d (Bind Abst) u) H0) in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: 
+C).(drop n O c0 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d0 n) 
+e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abst) 
+u)))) (ty3 g c0 (lift h d0 (TLRef n)) (lift h d0 (lift (S n) O u))) (\lambda 
+(x0: C).(\lambda (x1: C).(\lambda (H6: (drop n O c0 x0)).(\lambda (H7: (drop 
+h (minus d0 n) x0 x1)).(\lambda (H8: (clear x1 (CHead d (Bind Abst) u))).(let 
+H9 \def (eq_ind nat (minus d0 n) (\lambda (n0: nat).(drop h n0 x0 x1)) H7 (S 
+(minus d0 (S n))) (minus_x_Sy d0 n H4)) in (let H10 \def (drop_clear_S x1 x0 
+h (minus d0 (S n)) H9 Abst d u H8) in (ex2_ind C (\lambda (c1: C).(clear x0 
+(CHead c1 (Bind Abst) (lift h (minus d0 (S n)) u)))) (\lambda (c1: C).(drop h 
+(minus d0 (S n)) c1 d)) (ty3 g c0 (lift h d0 (TLRef n)) (lift h d0 (lift (S 
+n) O u))) (\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abst) (lift 
+h (minus d0 (S n)) u)))).(\lambda (H12: (drop h (minus d0 (S n)) x 
+d)).(eq_ind_r T (TLRef n) (\lambda (t0: T).(ty3 g c0 t0 (lift h d0 (lift (S 
+n) O u)))) (eq_ind nat (plus (S n) (minus d0 (S n))) (\lambda (n0: nat).(ty3 
+g c0 (TLRef n) (lift h n0 (lift (S n) O u)))) (eq_ind_r T (lift (S n) O (lift 
+h (minus d0 (S n)) u)) (\lambda (t0: T).(ty3 g c0 (TLRef n) t0)) (eq_ind nat 
+d0 (\lambda (_: nat).(ty3 g c0 (TLRef n) (lift (S n) O (lift h (minus d0 (S 
+n)) u)))) (ty3_abst g n c0 x (lift h (minus d0 (S n)) u) (getl_intro n c0 
+(CHead x (Bind Abst) (lift h (minus d0 (S n)) u)) x0 H6 H11) (lift h (minus 
+d0 (S n)) t) (H2 x (minus d0 (S n)) h H12)) (plus (S n) (minus d0 (S n))) 
+(le_plus_minus (S n) d0 H4)) (lift h (plus (S n) (minus d0 (S n))) (lift (S 
+n) O u)) (lift_d u h (S n) (minus d0 (S n)) O (le_O_n (minus d0 (S n))))) d0 
+(le_plus_minus_r (S n) d0 H4)) (lift h d0 (TLRef n)) (lift_lref_lt n h d0 
+H4))))) H10)))))))) H5))) (\lambda (H4: (le d0 n)).(eq_ind_r T (TLRef (plus n 
+h)) (\lambda (t0: T).(ty3 g c0 t0 (lift h d0 (lift (S n) O u)))) (eq_ind nat 
+(S n) (\lambda (_: nat).(ty3 g c0 (TLRef (plus n h)) (lift h d0 (lift (S n) O 
+u)))) (eq_ind_r T (lift (plus h (S n)) O u) (\lambda (t0: T).(ty3 g c0 (TLRef 
+(plus n h)) t0)) (eq_ind_r nat (plus (S n) h) (\lambda (n0: nat).(ty3 g c0 
+(TLRef (plus n h)) (lift n0 O u))) (ty3_abst g (plus n h) c0 d u 
+(drop_getl_trans_ge n c0 c d0 h H3 (CHead d (Bind Abst) u) H0 H4) t H1) (plus 
+h (S n)) (plus_comm h (S n))) (lift h d0 (lift (S n) O u)) (lift_free u (S n) 
+h O d0 (le_S d0 n H4) (le_O_n d0))) (plus n (S O)) (eq_ind_r nat (plus (S O) 
+n) (\lambda (n0: nat).(eq nat (S n) n0)) (refl_equal nat (plus (S O) n)) 
+(plus n (S O)) (plus_comm n (S O)))) (lift h d0 (TLRef n)) (lift_lref_ge n h 
+d0 H4)))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (t: 
+T).(\lambda (_: (ty3 g c u t)).(\lambda (H1: ((\forall (c0: C).(\forall (d: 
+nat).(\forall (h: nat).((drop h d c0 c) \to (ty3 g c0 (lift h d u) (lift h d 
+t)))))))).(\lambda (b: B).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (ty3 
+g (CHead c (Bind b) u) t0 t3)).(\lambda (H3: ((\forall (c0: C).(\forall (d: 
+nat).(\forall (h: nat).((drop h d c0 (CHead c (Bind b) u)) \to (ty3 g c0 
+(lift h d t0) (lift h d t3)))))))).(\lambda (t4: T).(\lambda (_: (ty3 g 
+(CHead c (Bind b) u) t3 t4)).(\lambda (H5: ((\forall (c0: C).(\forall (d: 
+nat).(\forall (h: nat).((drop h d c0 (CHead c (Bind b) u)) \to (ty3 g c0 
+(lift h d t3) (lift h d t4)))))))).(\lambda (c0: C).(\lambda (d: 
+nat).(\lambda (h: nat).(\lambda (H6: (drop h d c0 c)).(eq_ind_r T (THead 
+(Bind b) (lift h d u) (lift h (s (Bind b) d) t0)) (\lambda (t5: T).(ty3 g c0 
+t5 (lift h d (THead (Bind b) u t3)))) (eq_ind_r T (THead (Bind b) (lift h d 
+u) (lift h (s (Bind b) d) t3)) (\lambda (t5: T).(ty3 g c0 (THead (Bind b) 
+(lift h d u) (lift h (s (Bind b) d) t0)) t5)) (ty3_bind g c0 (lift h d u) 
+(lift h d t) (H1 c0 d h H6) b (lift h (S d) t0) (lift h (S d) t3) (H3 (CHead 
+c0 (Bind b) (lift h d u)) (S d) h (drop_skip_bind h d c0 c H6 b u)) (lift h 
+(S d) t4) (H5 (CHead c0 (Bind b) (lift h d u)) (S d) h (drop_skip_bind h d c0 
+c H6 b u))) (lift h d (THead (Bind b) u t3)) (lift_head (Bind b) u t3 h d)) 
+(lift h d (THead (Bind b) u t0)) (lift_head (Bind b) u t0 h 
+d))))))))))))))))))) (\lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\lambda 
+(_: (ty3 g c w u)).(\lambda (H1: ((\forall (c0: C).(\forall (d: nat).(\forall 
+(h: nat).((drop h d c0 c) \to (ty3 g c0 (lift h d w) (lift h d 
+u)))))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c v (THead 
+(Bind Abst) u t))).(\lambda (H3: ((\forall (c0: C).(\forall (d: nat).(\forall 
+(h: nat).((drop h d c0 c) \to (ty3 g c0 (lift h d v) (lift h d (THead (Bind 
+Abst) u t))))))))).(\lambda (c0: C).(\lambda (d: nat).(\lambda (h: 
+nat).(\lambda (H4: (drop h d c0 c)).(eq_ind_r T (THead (Flat Appl) (lift h d 
+w) (lift h (s (Flat Appl) d) v)) (\lambda (t0: T).(ty3 g c0 t0 (lift h d 
+(THead (Flat Appl) w (THead (Bind Abst) u t))))) (eq_ind_r T (THead (Flat 
+Appl) (lift h d w) (lift h (s (Flat Appl) d) (THead (Bind Abst) u t))) 
+(\lambda (t0: T).(ty3 g c0 (THead (Flat Appl) (lift h d w) (lift h (s (Flat 
+Appl) d) v)) t0)) (eq_ind_r T (THead (Bind Abst) (lift h (s (Flat Appl) d) u) 
+(lift h (s (Bind Abst) (s (Flat Appl) d)) t)) (\lambda (t0: T).(ty3 g c0 
+(THead (Flat Appl) (lift h d w) (lift h (s (Flat Appl) d) v)) (THead (Flat 
+Appl) (lift h d w) t0))) (ty3_appl g c0 (lift h d w) (lift h d u) (H1 c0 d h 
+H4) (lift h d v) (lift h (S d) t) (eq_ind T (lift h d (THead (Bind Abst) u 
+t)) (\lambda (t0: T).(ty3 g c0 (lift h d v) t0)) (H3 c0 d h H4) (THead (Bind 
+Abst) (lift h d u) (lift h (S d) t)) (lift_bind Abst u t h d))) (lift h (s 
+(Flat Appl) d) (THead (Bind Abst) u t)) (lift_head (Bind Abst) u t h (s (Flat 
+Appl) d))) (lift h d (THead (Flat Appl) w (THead (Bind Abst) u t))) 
+(lift_head (Flat Appl) w (THead (Bind Abst) u t) h d)) (lift h d (THead (Flat 
+Appl) w v)) (lift_head (Flat Appl) w v h d))))))))))))))) (\lambda (c: 
+C).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (ty3 g c t0 t3)).(\lambda 
+(H1: ((\forall (c0: C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c) 
+\to (ty3 g c0 (lift h d t0) (lift h d t3)))))))).(\lambda (t4: T).(\lambda 
+(_: (ty3 g c t3 t4)).(\lambda (H3: ((\forall (c0: C).(\forall (d: 
+nat).(\forall (h: nat).((drop h d c0 c) \to (ty3 g c0 (lift h d t3) (lift h d 
+t4)))))))).(\lambda (c0: C).(\lambda (d: nat).(\lambda (h: nat).(\lambda (H4: 
+(drop h d c0 c)).(eq_ind_r T (THead (Flat Cast) (lift h d t3) (lift h (s 
+(Flat Cast) d) t0)) (\lambda (t: T).(ty3 g c0 t (lift h d t3))) (ty3_cast g 
+c0 (lift h (s (Flat Cast) d) t0) (lift h d t3) (H1 c0 d h H4) (lift h d t4) 
+(H3 c0 d h H4)) (lift h d (THead (Flat Cast) t3 t0)) (lift_head (Flat Cast) 
+t3 t0 h d)))))))))))))) e t1 t2 H))))).
+
+theorem ty3_correct:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c 
+t1 t2) \to (ex T (\lambda (t: T).(ty3 g c t2 t)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
+(H: (ty3 g c t1 t2)).(ty3_ind g (\lambda (c0: C).(\lambda (_: T).(\lambda 
+(t0: T).(ex T (\lambda (t3: T).(ty3 g c0 t0 t3)))))) (\lambda (c0: 
+C).(\lambda (t0: T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 t0 t)).(\lambda 
+(_: (ex T (\lambda (t3: T).(ty3 g c0 t t3)))).(\lambda (u: T).(\lambda (t3: 
+T).(\lambda (_: (ty3 g c0 u t3)).(\lambda (_: (ex T (\lambda (t4: T).(ty3 g 
+c0 t3 t4)))).(\lambda (_: (pc3 c0 t3 t0)).(ex_intro T (\lambda (t4: T).(ty3 g 
+c0 t0 t4)) t H0))))))))))) (\lambda (c0: C).(\lambda (m: nat).(ex_intro T 
+(\lambda (t: T).(ty3 g c0 (TSort (next g m)) t)) (TSort (next g (next g m))) 
+(ty3_sort g c0 (next g m))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: 
+C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr) 
+u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: (ex T (\lambda 
+(t0: T).(ty3 g d t t0)))).(let H3 \def H2 in (ex_ind T (\lambda (t0: T).(ty3 
+g d t t0)) (ex T (\lambda (t0: T).(ty3 g c0 (lift (S n) O t) t0))) (\lambda 
+(x: T).(\lambda (H4: (ty3 g d t x)).(ex_intro T (\lambda (t0: T).(ty3 g c0 
+(lift (S n) O t) t0)) (lift (S n) O x) (ty3_lift g d t x H4 c0 O (S n) 
+(getl_drop Abbr c0 d u n H0))))) H3)))))))))) (\lambda (n: nat).(\lambda (c0: 
+C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind 
+Abst) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u t)).(\lambda (_: (ex T 
+(\lambda (t0: T).(ty3 g d t t0)))).(ex_intro T (\lambda (t0: T).(ty3 g c0 
+(lift (S n) O u) t0)) (lift (S n) O t) (ty3_lift g d u t H1 c0 O (S n) 
+(getl_drop Abst c0 d u n H0))))))))))) (\lambda (c0: C).(\lambda (u: 
+T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 u t)).(\lambda (_: (ex T (\lambda 
+(t0: T).(ty3 g c0 t t0)))).(\lambda (b: B).(\lambda (t0: T).(\lambda (t3: 
+T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t0 t3)).(\lambda (_: (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 (H1: (ex T 
+(\lambda (t: T).(ty3 g c0 t3 t)))).(\lambda (t4: T).(\lambda (_: (ty3 g c0 t3 
+t4)).(\lambda (_: (ex T (\lambda (t: T).(ty3 g c0 t4 t)))).H1)))))))) c t1 t2 
+H))))).
+
+theorem ty3_unique:
+ \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).((ty3 g c u 
+t1) \to (\forall (t2: T).((ty3 g c u t2) \to (pc3 c t1 t2)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (H: 
+(ty3 g c u t1)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: 
+T).(\forall (t2: T).((ty3 g c0 t t2) \to (pc3 c0 t0 t2)))))) (\lambda (c0: 
+C).(\lambda (t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda 
+(_: ((\forall (t3: T).((ty3 g c0 t2 t3) \to (pc3 c0 t t3))))).(\lambda (u0: 
+T).(\lambda (t0: T).(\lambda (_: (ty3 g c0 u0 t0)).(\lambda (H3: ((\forall 
+(t3: T).((ty3 g c0 u0 t3) \to (pc3 c0 t0 t3))))).(\lambda (H4: (pc3 c0 t0 
+t2)).(\lambda (t3: T).(\lambda (H5: (ty3 g c0 u0 t3)).(pc3_t t0 c0 t2 (pc3_s 
+c0 t2 t0 H4) t3 (H3 t3 H5)))))))))))))) (\lambda (c0: C).(\lambda (m: 
+nat).(\lambda (t2: T).(\lambda (H0: (ty3 g c0 (TSort m) t2)).(ty3_gen_sort g 
+c0 t2 m H0))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda 
+(u0: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr) u0))).(\lambda (t: 
+T).(\lambda (_: (ty3 g d u0 t)).(\lambda (H2: ((\forall (t2: T).((ty3 g d u0 
+t2) \to (pc3 d t t2))))).(\lambda (t2: T).(\lambda (H3: (ty3 g c0 (TLRef n) 
+t2)).(or_ind (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: 
+T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda 
+(_: T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1: 
+T).(\lambda (t0: T).(ty3 g e u1 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda 
+(u1: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u1) t2)))) (\lambda (e: 
+C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u1))))) 
+(\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0))))) (pc3 c0 
+(lift (S n) O t) t2) (\lambda (H4: (ex3_3 C T T (\lambda (_: C).(\lambda (_: 
+T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda 
+(u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: 
+C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0)))))).(ex3_3_ind C T T 
+(\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) 
+t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e 
+(Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g 
+e u1 t0)))) (pc3 c0 (lift (S n) O t) t2) (\lambda (x0: C).(\lambda (x1: 
+T).(\lambda (x2: T).(\lambda (H5: (pc3 c0 (lift (S n) O x2) t2)).(\lambda 
+(H6: (getl n c0 (CHead x0 (Bind Abbr) x1))).(\lambda (H7: (ty3 g x0 x1 
+x2)).(let H8 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda (c1: C).(getl n 
+c0 c1)) H0 (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n 
+H0 (CHead x0 (Bind Abbr) x1) H6)) in (let H9 \def (f_equal C C (\lambda (e: 
+C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | 
+(CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u0) (CHead x0 (Bind 
+Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead x0 (Bind Abbr) 
+x1) H6)) in ((let H10 \def (f_equal C T (\lambda (e: C).(match e in C return 
+(\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t0) 
+\Rightarrow t0])) (CHead d (Bind Abbr) u0) (CHead x0 (Bind Abbr) x1) 
+(getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in 
+(\lambda (H11: (eq C d x0)).(let H12 \def (eq_ind_r T x1 (\lambda (t0: 
+T).(getl n c0 (CHead x0 (Bind Abbr) t0))) H8 u0 H10) in (let H13 \def 
+(eq_ind_r T x1 (\lambda (t0: T).(ty3 g x0 t0 x2)) H7 u0 H10) in (let H14 \def 
+(eq_ind_r C x0 (\lambda (c1: C).(getl n c0 (CHead c1 (Bind Abbr) u0))) H12 d 
+H11) in (let H15 \def (eq_ind_r C x0 (\lambda (c1: C).(ty3 g c1 u0 x2)) H13 d 
+H11) in (pc3_t (lift (S n) O x2) c0 (lift (S n) O t) (pc3_lift c0 d (S n) O 
+(getl_drop Abbr c0 d u0 n H14) t x2 (H2 x2 H15)) t2 H5))))))) H9))))))))) 
+H4)) (\lambda (H4: (ex3_3 C T T (\lambda (_: C).(\lambda (u1: T).(\lambda (_: 
+T).(pc3 c0 (lift (S n) O u1) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda 
+(_: T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: 
+T).(\lambda (t0: T).(ty3 g e u1 t0)))))).(ex3_3_ind C T T (\lambda (_: 
+C).(\lambda (u1: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u1) t2)))) (\lambda 
+(e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) 
+u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0)))) 
+(pc3 c0 (lift (S n) O t) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: 
+T).(\lambda (_: (pc3 c0 (lift (S n) O x1) t2)).(\lambda (H6: (getl n c0 
+(CHead x0 (Bind Abst) x1))).(\lambda (_: (ty3 g x0 x1 x2)).(let H8 \def 
+(eq_ind C (CHead d (Bind Abbr) u0) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead 
+x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead x0 
+(Bind Abst) x1) H6)) in (let H9 \def (eq_ind C (CHead d (Bind Abbr) u0) 
+(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) 
+\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda 
+(_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_: 
+B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | Void 
+\Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead x0 (Bind Abst) 
+x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead x0 (Bind Abst) x1) 
+H6)) in (False_ind (pc3 c0 (lift (S n) O t) t2) H9))))))))) H4)) 
+(ty3_gen_lref g c0 t2 n H3)))))))))))) (\lambda (n: nat).(\lambda (c0: 
+C).(\lambda (d: C).(\lambda (u0: T).(\lambda (H0: (getl n c0 (CHead d (Bind 
+Abst) u0))).(\lambda (t: T).(\lambda (_: (ty3 g d u0 t)).(\lambda (_: 
+((\forall (t2: T).((ty3 g d u0 t2) \to (pc3 d t t2))))).(\lambda (t2: 
+T).(\lambda (H3: (ty3 g c0 (TLRef n) t2)).(or_ind (ex3_3 C T T (\lambda (_: 
+C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda 
+(e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) 
+u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0))))) 
+(ex3_3 C T T (\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(pc3 c0 (lift 
+(S n) O u1) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n 
+c0 (CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda 
+(t0: T).(ty3 g e u1 t0))))) (pc3 c0 (lift (S n) O u0) t2) (\lambda (H4: 
+(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift 
+(S n) O t0) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n 
+c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda 
+(t0: T).(ty3 g e u1 t0)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (_: 
+T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda 
+(u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: 
+C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0)))) (pc3 c0 (lift (S n) O 
+u0) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 
+c0 (lift (S n) O x2) t2)).(\lambda (H6: (getl n c0 (CHead x0 (Bind Abbr) 
+x1))).(\lambda (_: (ty3 g x0 x1 x2)).(let H8 \def (eq_ind C (CHead d (Bind 
+Abst) u0) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind Abbr) x1) 
+(getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in 
+(let H9 \def (eq_ind C (CHead d (Bind Abst) u0) (\lambda (ee: C).(match ee in 
+C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k 
+_) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) 
+\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow 
+False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) 
+\Rightarrow False])])) I (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d 
+(Bind Abst) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in (False_ind (pc3 c0 
+(lift (S n) O u0) t2) H9))))))))) H4)) (\lambda (H4: (ex3_3 C T T (\lambda 
+(_: C).(\lambda (u1: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u1) t2)))) 
+(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind 
+Abst) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 
+t0)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (u1: T).(\lambda (_: 
+T).(pc3 c0 (lift (S n) O u1) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda 
+(_: T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: 
+T).(\lambda (t0: T).(ty3 g e u1 t0)))) (pc3 c0 (lift (S n) O u0) t2) (\lambda 
+(x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H5: (pc3 c0 (lift (S n) O 
+x1) t2)).(\lambda (H6: (getl n c0 (CHead x0 (Bind Abst) x1))).(\lambda (H7: 
+(ty3 g x0 x1 x2)).(let H8 \def (eq_ind C (CHead d (Bind Abst) u0) (\lambda 
+(c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d 
+(Bind Abst) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in (let H9 \def (f_equal 
+C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) 
+\Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abst) u0) 
+(CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead 
+x0 (Bind Abst) x1) H6)) in ((let H10 \def (f_equal C T (\lambda (e: C).(match 
+e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ 
+t0) \Rightarrow t0])) (CHead d (Bind Abst) u0) (CHead x0 (Bind Abst) x1) 
+(getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in 
+(\lambda (H11: (eq C d x0)).(let H12 \def (eq_ind_r T x1 (\lambda (t0: 
+T).(getl n c0 (CHead x0 (Bind Abst) t0))) H8 u0 H10) in (let H13 \def 
+(eq_ind_r T x1 (\lambda (t0: T).(ty3 g x0 t0 x2)) H7 u0 H10) in (let H14 \def 
+(eq_ind_r T x1 (\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)) H5 u0 H10) in 
+(let H15 \def (eq_ind_r C x0 (\lambda (c1: C).(getl n c0 (CHead c1 (Bind 
+Abst) u0))) H12 d H11) in (let H16 \def (eq_ind_r C x0 (\lambda (c1: C).(ty3 
+g c1 u0 x2)) H13 d H11) in H14))))))) H9))))))))) H4)) (ty3_gen_lref g c0 t2 
+n H3)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (t: T).(\lambda 
+(_: (ty3 g c0 u0 t)).(\lambda (_: ((\forall (t2: T).((ty3 g c0 u0 t2) \to 
+(pc3 c0 t t2))))).(\lambda (b: B).(\lambda (t0: T).(\lambda (t2: T).(\lambda 
+(_: (ty3 g (CHead c0 (Bind b) u0) t0 t2)).(\lambda (H3: ((\forall (t3: 
+T).((ty3 g (CHead c0 (Bind b) u0) t0 t3) \to (pc3 (CHead c0 (Bind b) u0) t2 
+t3))))).(\lambda (t3: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u0) t2 
+t3)).(\lambda (_: ((\forall (t4: T).((ty3 g (CHead c0 (Bind b) u0) t2 t4) \to 
+(pc3 (CHead c0 (Bind b) u0) t3 t4))))).(\lambda (t4: T).(\lambda (H6: (ty3 g 
+c0 (THead (Bind b) u0 t0) t4)).(ex4_3_ind T T T (\lambda (t5: T).(\lambda (_: 
+T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u0 t5) t4)))) (\lambda (_: 
+T).(\lambda (t6: T).(\lambda (_: T).(ty3 g c0 u0 t6)))) (\lambda (t5: 
+T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u0) t0 t5)))) 
+(\lambda (t5: T).(\lambda (_: T).(\lambda (t7: T).(ty3 g (CHead c0 (Bind b) 
+u0) t5 t7)))) (pc3 c0 (THead (Bind b) u0 t2) t4) (\lambda (x0: T).(\lambda 
+(x1: T).(\lambda (x2: T).(\lambda (H7: (pc3 c0 (THead (Bind b) u0 x0) 
+t4)).(\lambda (_: (ty3 g c0 u0 x1)).(\lambda (H9: (ty3 g (CHead c0 (Bind b) 
+u0) t0 x0)).(\lambda (_: (ty3 g (CHead c0 (Bind b) u0) x0 x2)).(pc3_t (THead 
+(Bind b) u0 x0) c0 (THead (Bind b) u0 t2) (pc3_head_2 c0 u0 t2 x0 (Bind b) 
+(H3 x0 H9)) t4 H7)))))))) (ty3_gen_bind g b c0 u0 t0 t4 H6))))))))))))))))) 
+(\lambda (c0: C).(\lambda (w: T).(\lambda (u0: T).(\lambda (_: (ty3 g c0 w 
+u0)).(\lambda (_: ((\forall (t2: T).((ty3 g c0 w t2) \to (pc3 c0 u0 
+t2))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead (Bind 
+Abst) u0 t))).(\lambda (H3: ((\forall (t2: T).((ty3 g c0 v t2) \to (pc3 c0 
+(THead (Bind Abst) u0 t) t2))))).(\lambda (t2: T).(\lambda (H4: (ty3 g c0 
+(THead (Flat Appl) w v) t2)).(ex3_2_ind T T (\lambda (u1: T).(\lambda (t0: 
+T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u1 t0)) t2))) (\lambda 
+(u1: T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u1 t0)))) (\lambda 
+(u1: T).(\lambda (_: T).(ty3 g c0 w u1))) (pc3 c0 (THead (Flat Appl) w (THead 
+(Bind Abst) u0 t)) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (pc3 
+c0 (THead (Flat Appl) w (THead (Bind Abst) x0 x1)) t2)).(\lambda (H6: (ty3 g 
+c0 v (THead (Bind Abst) x0 x1))).(\lambda (_: (ty3 g c0 w x0)).(pc3_t (THead 
+(Flat Appl) w (THead (Bind Abst) x0 x1)) c0 (THead (Flat Appl) w (THead (Bind 
+Abst) u0 t)) (pc3_thin_dx c0 (THead (Bind Abst) u0 t) (THead (Bind Abst) x0 
+x1) (H3 (THead (Bind Abst) x0 x1) H6) w Appl) t2 H5)))))) (ty3_gen_appl g c0 
+w v t2 H4))))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: 
+T).(\lambda (_: (ty3 g c0 t0 t2)).(\lambda (_: ((\forall (t3: T).((ty3 g c0 
+t0 t3) \to (pc3 c0 t2 t3))))).(\lambda (t3: T).(\lambda (_: (ty3 g c0 t2 
+t3)).(\lambda (_: ((\forall (t4: T).((ty3 g c0 t2 t4) \to (pc3 c0 t3 
+t4))))).(\lambda (t4: T).(\lambda (H4: (ty3 g c0 (THead (Flat Cast) t2 t0) 
+t4)).(and_ind (pc3 c0 t2 t4) (ty3 g c0 t0 t2) (pc3 c0 t2 t4) (\lambda (H5: 
+(pc3 c0 t2 t4)).(\lambda (_: (ty3 g c0 t0 t2)).H5)) (ty3_gen_cast g c0 t0 t2 
+t4 H4)))))))))))) c u t1 H))))).
+

diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/subst1.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/subst1.ma
new file mode 100644 (file)
index 0000000..eb09bb8
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/subst1.ma
@@ -0,0 +1,1149 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/subst1".
+
+include "ty3/props.ma".
+
+include "pc3/subst1.ma".
+
+include "pc3/fwd.ma".
+
+include "csubst1/getl.ma".
+
+include "csubst1/fwd.ma".
+
+include "getl/getl.ma".
+
+theorem ty3_gen_cabbr:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c 
+t1 t2) \to (\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c 
+(CHead e (Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c a0) \to 
+(\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda 
+(_: T).(subst1 d u t1 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: 
+T).(subst1 d u t2 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 
+g a y1 y2))))))))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
+(H: (ty3 g c t1 t2)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda 
+(t0: T).(\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead 
+e (Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c0 a0) \to (\forall (a: 
+C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: 
+T).(subst1 d u t (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: 
+T).(subst1 d u t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 
+g a y1 y2))))))))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t: 
+T).(\lambda (_: (ty3 g c0 t3 t)).(\lambda (H1: ((\forall (e: C).(\forall (u: 
+T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u)) \to (\forall (a0: 
+C).((csubst1 d u c0 a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 T 
+T (\lambda (y1: T).(\lambda (_: T).(subst1 d u t3 (lift (S O) d y1)))) 
+(\lambda (_: T).(\lambda (y2: T).(subst1 d u t (lift (S O) d y2)))) (\lambda 
+(y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))))).(\lambda (u: 
+T).(\lambda (t4: T).(\lambda (_: (ty3 g c0 u t4)).(\lambda (H3: ((\forall (e: 
+C).(\forall (u0: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u0)) 
+\to (\forall (a0: C).((csubst1 d u0 c0 a0) \to (\forall (a: C).((drop (S O) d 
+a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift (S 
+O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t4 (lift (S O) d 
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 
+y2)))))))))))))).(\lambda (H4: (pc3 c0 t4 t3)).(\lambda (e: C).(\lambda (u0: 
+T).(\lambda (d: nat).(\lambda (H5: (getl d c0 (CHead e (Bind Abbr) 
+u0))).(\lambda (a0: C).(\lambda (H6: (csubst1 d u0 c0 a0)).(\lambda (a: 
+C).(\lambda (H7: (drop (S O) d a0 a)).(let H8 \def (H3 e u0 d H5 a0 H6 a H7) 
+in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift (S O) 
+d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t4 (lift (S O) d 
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T 
+(\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift (S O) d y1)))) (\lambda 
+(_: T).(\lambda (y2: T).(subst1 d u0 t3 (lift (S O) d y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (H9: (subst1 d u0 u (lift (S O) d x0))).(\lambda (H10: (subst1 d 
+u0 t4 (lift (S O) d x1))).(\lambda (H11: (ty3 g a x0 x1)).(let H12 \def (H1 e 
+u0 d H5 a0 H6 a H7) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: 
+T).(subst1 d u0 t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: 
+T).(subst1 d u0 t (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 
+g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift 
+(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t3 (lift (S O) d 
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: 
+T).(\lambda (x3: T).(\lambda (H13: (subst1 d u0 t3 (lift (S O) d 
+x2))).(\lambda (_: (subst1 d u0 t (lift (S O) d x3))).(\lambda (H15: (ty3 g a 
+x2 x3)).(ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u 
+(lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t3 (lift 
+(S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) x0 x2 H9 
+H13 (ty3_conv g a x2 x3 H15 x0 x1 H11 (pc3_gen_cabbr c0 t4 t3 H4 e u0 d H5 a0 
+H6 a H7 x1 H10 x2 H13)))))))) H12))))))) H8)))))))))))))))))))) (\lambda (c0: 
+C).(\lambda (m: nat).(\lambda (e: C).(\lambda (u: T).(\lambda (d: 
+nat).(\lambda (_: (getl d c0 (CHead e (Bind Abbr) u))).(\lambda (a0: 
+C).(\lambda (_: (csubst1 d u c0 a0)).(\lambda (a: C).(\lambda (_: (drop (S O) 
+d a0 a)).(ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u (TSort 
+m) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u (TSort 
+(next g m)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a 
+y1 y2))) (TSort m) (TSort (next g m)) (eq_ind_r T (TSort m) (\lambda (t: 
+T).(subst1 d u (TSort m) t)) (subst1_refl d u (TSort m)) (lift (S O) d (TSort 
+m)) (lift_sort m (S O) d)) (eq_ind_r T (TSort (next g m)) (\lambda (t: 
+T).(subst1 d u (TSort (next g m)) t)) (subst1_refl d u (TSort (next g m))) 
+(lift (S O) d (TSort (next g m))) (lift_sort (next g m) (S O) d)) (ty3_sort g 
+a m)))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda 
+(u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr) u))).(\lambda (t: 
+T).(\lambda (H1: (ty3 g d u t)).(\lambda (H2: ((\forall (e: C).(\forall (u0: 
+T).(\forall (d0: nat).((getl d0 d (CHead e (Bind Abbr) u0)) \to (\forall (a0: 
+C).((csubst1 d0 u0 d a0) \to (\forall (a: C).((drop (S O) d0 a0 a) \to (ex3_2 
+T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 u (lift (S O) d0 y1)))) 
+(\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 t (lift (S O) d0 y2)))) 
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))))).(\lambda (e: 
+C).(\lambda (u0: T).(\lambda (d0: nat).(\lambda (H3: (getl d0 c0 (CHead e 
+(Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H4: (csubst1 d0 u0 c0 
+a0)).(\lambda (a: C).(\lambda (H5: (drop (S O) d0 a0 a)).(lt_eq_gt_e n d0 
+(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S 
+O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O t) 
+(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) 
+(\lambda (H6: (lt n d0)).(let H7 \def (eq_ind nat (minus d0 n) (\lambda (n0: 
+nat).(getl n0 (CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0))) 
+(getl_conf_le d0 (CHead e (Bind Abbr) u0) c0 H3 (CHead d (Bind Abbr) u) n H0 
+(le_S_n n d0 (le_S (S n) d0 H6))) (S (minus d0 (S n))) (minus_x_Sy d0 n H6)) 
+in (ex2_ind C (\lambda (e2: C).(csubst1 (minus d0 n) u0 (CHead d (Bind Abbr) 
+u) e2)) (\lambda (e2: C).(getl n a0 e2)) (ex3_2 T T (\lambda (y1: T).(\lambda 
+(_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda 
+(y2: T).(subst1 d0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x: C).(\lambda (H8: (csubst1 
+(minus d0 n) u0 (CHead d (Bind Abbr) u) x)).(\lambda (H9: (getl n a0 x)).(let 
+H10 \def (eq_ind nat (minus d0 n) (\lambda (n0: nat).(csubst1 n0 u0 (CHead d 
+(Bind Abbr) u) x)) H8 (S (minus d0 (S n))) (minus_x_Sy d0 n H6)) in (let H11 
+\def (csubst1_gen_head (Bind Abbr) d x u u0 (minus d0 (S n)) H10) in 
+(ex3_2_ind T C (\lambda (u2: T).(\lambda (c2: C).(eq C x (CHead c2 (Bind 
+Abbr) u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 (minus d0 (S n)) u0 u 
+u2))) (\lambda (_: T).(\lambda (c2: C).(csubst1 (minus d0 (S n)) u0 d c2))) 
+(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S 
+O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O t) 
+(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) 
+(\lambda (x0: T).(\lambda (x1: C).(\lambda (H12: (eq C x (CHead x1 (Bind 
+Abbr) x0))).(\lambda (H13: (subst1 (minus d0 (S n)) u0 u x0)).(\lambda (H14: 
+(csubst1 (minus d0 (S n)) u0 d x1)).(let H15 \def (eq_ind C x (\lambda (c1: 
+C).(getl n a0 c1)) H9 (CHead x1 (Bind Abbr) x0) H12) in (let H16 \def (eq_ind 
+nat d0 (\lambda (n0: nat).(drop (S O) n0 a0 a)) H5 (S (plus n (minus d0 (S 
+n)))) (lt_plus_minus n d0 H6)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: 
+C).(eq T x0 (lift (S O) (minus d0 (S n)) v)))) (\lambda (v: T).(\lambda (e0: 
+C).(getl n a (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0: 
+C).(drop (S O) (minus d0 (S n)) x1 e0))) (ex3_2 T T (\lambda (y1: T).(\lambda 
+(_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda 
+(y2: T).(subst1 d0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3: 
+C).(\lambda (H17: (eq T x0 (lift (S O) (minus d0 (S n)) x2))).(\lambda (H18: 
+(getl n a (CHead x3 (Bind Abbr) x2))).(\lambda (H19: (drop (S O) (minus d0 (S 
+n)) x1 x3)).(let H20 \def (eq_ind T x0 (\lambda (t0: T).(subst1 (minus d0 (S 
+n)) u0 u t0)) H13 (lift (S O) (minus d0 (S n)) x2) H17) in (let H21 \def (H2 
+e u0 (minus d0 (S n)) (getl_gen_S (Bind Abbr) d (CHead e (Bind Abbr) u0) u 
+(minus d0 (S n)) H7) x1 H14 x3 H19) in (ex3_2_ind T T (\lambda (y1: 
+T).(\lambda (_: T).(subst1 (minus d0 (S n)) u0 u (lift (S O) (minus d0 (S n)) 
+y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 (minus d0 (S n)) u0 t (lift 
+(S O) (minus d0 (S n)) y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g x3 y1 
+y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) 
+(lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S 
+n) O t) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 
+y2)))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H22: (subst1 (minus d0 (S 
+n)) u0 u (lift (S O) (minus d0 (S n)) x4))).(\lambda (H23: (subst1 (minus d0 
+(S n)) u0 t (lift (S O) (minus d0 (S n)) x5))).(\lambda (H24: (ty3 g x3 x4 
+x5)).(let H25 \def (eq_ind T x4 (\lambda (t0: T).(ty3 g x3 t0 x5)) H24 x2 
+(subst1_confluence_lift u x4 u0 (minus d0 (S n)) H22 x2 H20)) in (eq_ind_r 
+nat (plus (minus d0 (S n)) (S n)) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: 
+T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(subst1 n0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) 
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r nat (plus (S 
+n) (minus d0 (S n))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda 
+(_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda 
+(y2: T).(subst1 (plus (minus d0 (S n)) (S n)) u0 (lift (S n) O t) (lift (S O) 
+n0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro 
+T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 
+y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 (plus (minus d0 (S n)) (S n)) 
+u0 (lift (S n) O t) (lift (S O) (plus (S n) (minus d0 (S n))) y2)))) (\lambda 
+(y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (TLRef n) (lift (S n) O x5) 
+(eq_ind_r T (TLRef n) (\lambda (t0: T).(subst1 d0 u0 (TLRef n) t0)) 
+(subst1_refl d0 u0 (TLRef n)) (lift (S O) d0 (TLRef n)) (lift_lref_lt n (S O) 
+d0 H6)) (eq_ind_r T (lift (S n) O (lift (S O) (minus d0 (S n)) x5)) (\lambda 
+(t0: T).(subst1 (plus (minus d0 (S n)) (S n)) u0 (lift (S n) O t) t0)) 
+(subst1_lift_ge t (lift (S O) (minus d0 (S n)) x5) u0 (minus d0 (S n)) (S n) 
+H23 O (le_O_n (minus d0 (S n)))) (lift (S O) (plus (S n) (minus d0 (S n))) 
+(lift (S n) O x5)) (lift_d x5 (S O) (S n) (minus d0 (S n)) O (le_O_n (minus 
+d0 (S n))))) (ty3_abbr g n a x3 x2 H18 x5 H25)) d0 (le_plus_minus (S n) d0 
+H6)) d0 (le_plus_minus_sym (S n) d0 H6)))))))) H21)))))))) (getl_drop_conf_lt 
+Abbr a0 x1 x0 n H15 a (S O) (minus d0 (S n)) H16))))))))) H11)))))) 
+(csubst1_getl_lt d0 n H6 c0 a0 u0 H4 (CHead d (Bind Abbr) u) H0)))) (\lambda 
+(H6: (eq nat n d0)).(let H7 \def (eq_ind_r nat d0 (\lambda (n0: nat).(drop (S 
+O) n0 a0 a)) H5 n H6) in (let H8 \def (eq_ind_r nat d0 (\lambda (n0: 
+nat).(csubst1 n0 u0 c0 a0)) H4 n H6) in (let H9 \def (eq_ind_r nat d0 
+(\lambda (n0: nat).(getl n0 c0 (CHead e (Bind Abbr) u0))) H3 n H6) in (eq_ind 
+nat n (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 
+n0 u0 (TLRef n) (lift (S O) n0 y1)))) (\lambda (_: T).(\lambda (y2: 
+T).(subst1 n0 u0 (lift (S n) O t) (lift (S O) n0 y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H10 \def (eq_ind C (CHead d 
+(Bind Abbr) u) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead e (Bind Abbr) u0) 
+(getl_mono c0 (CHead d (Bind Abbr) u) n H0 (CHead e (Bind Abbr) u0) H9)) in 
+(let H11 \def (f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda 
+(_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) 
+(CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind 
+Abbr) u) n H0 (CHead e (Bind Abbr) u0) H9)) in ((let H12 \def (f_equal C T 
+(\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) 
+\Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abbr) u) 
+(CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind Abbr) u) n H0 (CHead e 
+(Bind Abbr) u0) H9)) in (\lambda (H13: (eq C d e)).(let H14 \def (eq_ind_r T 
+u0 (\lambda (t0: T).(getl n c0 (CHead e (Bind Abbr) t0))) H10 u H12) in (let 
+H15 \def (eq_ind_r T u0 (\lambda (t0: T).(csubst1 n t0 c0 a0)) H8 u H12) in 
+(eq_ind T u (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: 
+T).(subst1 n t0 (TLRef n) (lift (S O) n y1)))) (\lambda (_: T).(\lambda (y2: 
+T).(subst1 n t0 (lift (S n) O t) (lift (S O) n y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H16 \def (eq_ind_r C e (\lambda 
+(c1: C).(getl n c0 (CHead c1 (Bind Abbr) u))) H14 d H13) in (ex3_2_intro T T 
+(\lambda (y1: T).(\lambda (_: T).(subst1 n u (TLRef n) (lift (S O) n y1)))) 
+(\lambda (_: T).(\lambda (y2: T).(subst1 n u (lift (S n) O t) (lift (S O) n 
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (lift n O u) (lift 
+n O t) (subst1_single n u (TLRef n) (lift (S O) n (lift n O u)) (eq_ind_r T 
+(lift (plus (S O) n) O u) (\lambda (t0: T).(subst0 n u (TLRef n) t0)) 
+(subst0_lref u n) (lift (S O) n (lift n O u)) (lift_free u n (S O) O n (le_n 
+(plus O n)) (le_O_n n)))) (eq_ind_r T (lift (plus (S O) n) O t) (\lambda (t0: 
+T).(subst1 n u (lift (S n) O t) t0)) (subst1_refl n u (lift (S n) O t)) (lift 
+(S O) n (lift n O t)) (lift_free t n (S O) O n (le_n (plus O n)) (le_O_n n))) 
+(ty3_lift g d u t H1 a O n (getl_conf_ge_drop Abbr a0 d u n (csubst1_getl_ge 
+n n (le_n n) c0 a0 u H15 (CHead d (Bind Abbr) u) H16) a H7)))) u0 H12))))) 
+H11))) d0 H6))))) (\lambda (H6: (lt d0 n)).(eq_ind_r nat (S (plus O (minus n 
+(S O)))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: 
+T).(subst1 d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda 
+(y2: T).(subst1 d0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind nat (plus (S O) (minus n (S 
+O))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 
+d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: 
+T).(subst1 d0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r nat (plus (minus n (S O)) 
+(S O)) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 
+d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: 
+T).(subst1 d0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: 
+T).(\lambda (_: T).(subst1 d0 u0 (TLRef (plus (minus n (S O)) (S O))) (lift 
+(S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O 
+t) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) 
+(TLRef (minus n (S O))) (lift n O t) (eq_ind_r T (TLRef (plus (minus n (S O)) 
+(S O))) (\lambda (t0: T).(subst1 d0 u0 (TLRef (plus (minus n (S O)) (S O))) 
+t0)) (subst1_refl d0 u0 (TLRef (plus (minus n (S O)) (S O)))) (lift (S O) d0 
+(TLRef (minus n (S O)))) (lift_lref_ge (minus n (S O)) (S O) d0 (lt_le_minus 
+d0 n H6))) (eq_ind_r T (lift (plus (S O) n) O t) (\lambda (t0: T).(subst1 d0 
+u0 (lift (S n) O t) t0)) (subst1_refl d0 u0 (lift (S n) O t)) (lift (S O) d0 
+(lift n O t)) (lift_free t n (S O) O d0 (le_S_n d0 (plus O n) (le_S (S d0) 
+(plus O n) H6)) (le_O_n d0))) (eq_ind_r nat (S (minus n (S O))) (\lambda (n0: 
+nat).(ty3 g a (TLRef (minus n (S O))) (lift n0 O t))) (ty3_abbr g (minus n (S 
+O)) a d u (getl_drop_conf_ge n (CHead d (Bind Abbr) u) a0 (csubst1_getl_ge d0 
+n (le_S_n d0 n (le_S (S d0) n H6)) c0 a0 u0 H4 (CHead d (Bind Abbr) u) H0) a 
+(S O) d0 H5 (eq_ind_r nat (plus (S O) d0) (\lambda (n0: nat).(le n0 n)) H6 
+(plus d0 (S O)) (plus_comm d0 (S O)))) t H1) n (minus_x_SO n (le_lt_trans O 
+d0 n (le_O_n d0) H6)))) (plus (S O) (minus n (S O))) (plus_comm (S O) (minus 
+n (S O)))) (S (plus O (minus n (S O)))) (refl_equal nat (S (plus O (minus n 
+(S O)))))) n (lt_plus_minus O n (le_lt_trans O d0 n (le_O_n d0) 
+H6))))))))))))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: 
+C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abst) 
+u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u t)).(\lambda (H2: ((\forall (e: 
+C).(\forall (u0: T).(\forall (d0: nat).((getl d0 d (CHead e (Bind Abbr) u0)) 
+\to (\forall (a0: C).((csubst1 d0 u0 d a0) \to (\forall (a: C).((drop (S O) 
+d0 a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 u 
+(lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 t (lift 
+(S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 
+y2)))))))))))))).(\lambda (e: C).(\lambda (u0: T).(\lambda (d0: nat).(\lambda 
+(H3: (getl d0 c0 (CHead e (Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H4: 
+(csubst1 d0 u0 c0 a0)).(\lambda (a: C).(\lambda (H5: (drop (S O) d0 a0 
+a)).(lt_eq_gt_e n d0 (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 
+u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 
+d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: 
+T).(ty3 g a y1 y2)))) (\lambda (H6: (lt n d0)).(let H7 \def (eq_ind nat 
+(minus d0 n) (\lambda (n0: nat).(getl n0 (CHead d (Bind Abst) u) (CHead e 
+(Bind Abbr) u0))) (getl_conf_le d0 (CHead e (Bind Abbr) u0) c0 H3 (CHead d 
+(Bind Abst) u) n H0 (le_S_n n d0 (le_S (S n) d0 H6))) (S (minus d0 (S n))) 
+(minus_x_Sy d0 n H6)) in (ex2_ind C (\lambda (e2: C).(csubst1 (minus d0 n) u0 
+(CHead d (Bind Abst) u) e2)) (\lambda (e2: C).(getl n a0 e2)) (ex3_2 T T 
+(\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 
+y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O u) (lift 
+(S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda 
+(x: C).(\lambda (H8: (csubst1 (minus d0 n) u0 (CHead d (Bind Abst) u) 
+x)).(\lambda (H9: (getl n a0 x)).(let H10 \def (eq_ind nat (minus d0 n) 
+(\lambda (n0: nat).(csubst1 n0 u0 (CHead d (Bind Abst) u) x)) H8 (S (minus d0 
+(S n))) (minus_x_Sy d0 n H6)) in (let H11 \def (csubst1_gen_head (Bind Abst) 
+d x u u0 (minus d0 (S n)) H10) in (ex3_2_ind T C (\lambda (u2: T).(\lambda 
+(c2: C).(eq C x (CHead c2 (Bind Abst) u2)))) (\lambda (u2: T).(\lambda (_: 
+C).(subst1 (minus d0 (S n)) u0 u u2))) (\lambda (_: T).(\lambda (c2: 
+C).(csubst1 (minus d0 (S n)) u0 d c2))) (ex3_2 T T (\lambda (y1: T).(\lambda 
+(_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda 
+(y2: T).(subst1 d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1: 
+C).(\lambda (H12: (eq C x (CHead x1 (Bind Abst) x0))).(\lambda (H13: (subst1 
+(minus d0 (S n)) u0 u x0)).(\lambda (H14: (csubst1 (minus d0 (S n)) u0 d 
+x1)).(let H15 \def (eq_ind C x (\lambda (c1: C).(getl n a0 c1)) H9 (CHead x1 
+(Bind Abst) x0) H12) in (let H16 \def (eq_ind nat d0 (\lambda (n0: nat).(drop 
+(S O) n0 a0 a)) H5 (S (plus n (minus d0 (S n)))) (lt_plus_minus n d0 H6)) in 
+(ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T x0 (lift (S O) (minus d0 
+(S n)) v)))) (\lambda (v: T).(\lambda (e0: C).(getl n a (CHead e0 (Bind Abst) 
+v)))) (\lambda (_: T).(\lambda (e0: C).(drop (S O) (minus d0 (S n)) x1 e0))) 
+(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S 
+O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O u) 
+(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) 
+(\lambda (x2: T).(\lambda (x3: C).(\lambda (H17: (eq T x0 (lift (S O) (minus 
+d0 (S n)) x2))).(\lambda (H18: (getl n a (CHead x3 (Bind Abst) x2))).(\lambda 
+(H19: (drop (S O) (minus d0 (S n)) x1 x3)).(let H20 \def (eq_ind T x0 
+(\lambda (t0: T).(subst1 (minus d0 (S n)) u0 u t0)) H13 (lift (S O) (minus d0 
+(S n)) x2) H17) in (let H21 \def (H2 e u0 (minus d0 (S n)) (getl_gen_S (Bind 
+Abst) d (CHead e (Bind Abbr) u0) u (minus d0 (S n)) H7) x1 H14 x3 H19) in 
+(ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 (minus d0 (S n)) u0 u 
+(lift (S O) (minus d0 (S n)) y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 
+(minus d0 (S n)) u0 t (lift (S O) (minus d0 (S n)) y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g x3 y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda 
+(_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda 
+(y2: T).(subst1 d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x4: T).(\lambda (x5: 
+T).(\lambda (H22: (subst1 (minus d0 (S n)) u0 u (lift (S O) (minus d0 (S n)) 
+x4))).(\lambda (_: (subst1 (minus d0 (S n)) u0 t (lift (S O) (minus d0 (S n)) 
+x5))).(\lambda (H24: (ty3 g x3 x4 x5)).(let H25 \def (eq_ind T x4 (\lambda 
+(t0: T).(ty3 g x3 t0 x5)) H24 x2 (subst1_confluence_lift u x4 u0 (minus d0 (S 
+n)) H22 x2 H20)) in (eq_ind_r nat (plus (minus d0 (S n)) (S n)) (\lambda (n0: 
+nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) 
+(lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 n0 u0 (lift (S 
+n) O u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 
+y2))))) (eq_ind_r nat (plus (S n) (minus d0 (S n))) (\lambda (n0: nat).(ex3_2 
+T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 
+y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 (plus (minus d0 (S n)) (S n)) 
+u0 (lift (S n) O u) (lift (S O) n0 y2)))) (\lambda (y1: T).(\lambda (y2: 
+T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: 
+T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda 
+(y2: T).(subst1 (plus (minus d0 (S n)) (S n)) u0 (lift (S n) O u) (lift (S O) 
+(plus (S n) (minus d0 (S n))) y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g 
+a y1 y2))) (TLRef n) (lift (S n) O x2) (eq_ind_r T (TLRef n) (\lambda (t0: 
+T).(subst1 d0 u0 (TLRef n) t0)) (subst1_refl d0 u0 (TLRef n)) (lift (S O) d0 
+(TLRef n)) (lift_lref_lt n (S O) d0 H6)) (eq_ind_r T (lift (S n) O (lift (S 
+O) (minus d0 (S n)) x2)) (\lambda (t0: T).(subst1 (plus (minus d0 (S n)) (S 
+n)) u0 (lift (S n) O u) t0)) (subst1_lift_ge u (lift (S O) (minus d0 (S n)) 
+x2) u0 (minus d0 (S n)) (S n) H20 O (le_O_n (minus d0 (S n)))) (lift (S O) 
+(plus (S n) (minus d0 (S n))) (lift (S n) O x2)) (lift_d x2 (S O) (S n) 
+(minus d0 (S n)) O (le_O_n (minus d0 (S n))))) (ty3_abst g n a x3 x2 H18 x5 
+H25)) d0 (le_plus_minus (S n) d0 H6)) d0 (le_plus_minus_sym (S n) d0 
+H6)))))))) H21)))))))) (getl_drop_conf_lt Abst a0 x1 x0 n H15 a (S O) (minus 
+d0 (S n)) H16))))))))) H11)))))) (csubst1_getl_lt d0 n H6 c0 a0 u0 H4 (CHead 
+d (Bind Abst) u) H0)))) (\lambda (H6: (eq nat n d0)).(let H7 \def (eq_ind_r 
+nat d0 (\lambda (n0: nat).(drop (S O) n0 a0 a)) H5 n H6) in (let H8 \def 
+(eq_ind_r nat d0 (\lambda (n0: nat).(csubst1 n0 u0 c0 a0)) H4 n H6) in (let 
+H9 \def (eq_ind_r nat d0 (\lambda (n0: nat).(getl n0 c0 (CHead e (Bind Abbr) 
+u0))) H3 n H6) in (eq_ind nat n (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: 
+T).(\lambda (_: T).(subst1 n0 u0 (TLRef n) (lift (S O) n0 y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(subst1 n0 u0 (lift (S n) O u) (lift (S O) n0 y2)))) 
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H10 \def (eq_ind C 
+(CHead d (Bind Abst) u) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead e (Bind 
+Abbr) u0) (getl_mono c0 (CHead d (Bind Abst) u) n H0 (CHead e (Bind Abbr) u0) 
+H9)) in (let H11 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (ee: 
+C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow 
+False | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) 
+with [(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with 
+[Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | 
+(Flat _) \Rightarrow False])])) I (CHead e (Bind Abbr) u0) (getl_mono c0 
+(CHead d (Bind Abst) u) n H0 (CHead e (Bind Abbr) u0) H9)) in (False_ind 
+(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 n u0 (TLRef n) (lift (S 
+O) n y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 n u0 (lift (S n) O u) 
+(lift (S O) n y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) 
+H11))) d0 H6))))) (\lambda (H6: (lt d0 n)).(eq_ind_r nat (S (plus O (minus n 
+(S O)))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: 
+T).(subst1 d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda 
+(y2: T).(subst1 d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind nat (plus (S O) (minus n (S 
+O))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 
+d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: 
+T).(subst1 d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r nat (plus (minus n (S O)) 
+(S O)) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 
+d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: 
+T).(subst1 d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: 
+T).(\lambda (_: T).(subst1 d0 u0 (TLRef (plus (minus n (S O)) (S O))) (lift 
+(S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O 
+u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) 
+(TLRef (minus n (S O))) (lift n O u) (eq_ind_r T (TLRef (plus (minus n (S O)) 
+(S O))) (\lambda (t0: T).(subst1 d0 u0 (TLRef (plus (minus n (S O)) (S O))) 
+t0)) (subst1_refl d0 u0 (TLRef (plus (minus n (S O)) (S O)))) (lift (S O) d0 
+(TLRef (minus n (S O)))) (lift_lref_ge (minus n (S O)) (S O) d0 (lt_le_minus 
+d0 n H6))) (eq_ind_r T (lift (plus (S O) n) O u) (\lambda (t0: T).(subst1 d0 
+u0 (lift (S n) O u) t0)) (subst1_refl d0 u0 (lift (S n) O u)) (lift (S O) d0 
+(lift n O u)) (lift_free u n (S O) O d0 (le_S_n d0 (plus O n) (le_S (S d0) 
+(plus O n) H6)) (le_O_n d0))) (eq_ind_r nat (S (minus n (S O))) (\lambda (n0: 
+nat).(ty3 g a (TLRef (minus n (S O))) (lift n0 O u))) (ty3_abst g (minus n (S 
+O)) a d u (getl_drop_conf_ge n (CHead d (Bind Abst) u) a0 (csubst1_getl_ge d0 
+n (le_S_n d0 n (le_S (S d0) n H6)) c0 a0 u0 H4 (CHead d (Bind Abst) u) H0) a 
+(S O) d0 H5 (eq_ind_r nat (plus (S O) d0) (\lambda (n0: nat).(le n0 n)) H6 
+(plus d0 (S O)) (plus_comm d0 (S O)))) t H1) n (minus_x_SO n (le_lt_trans O 
+d0 n (le_O_n d0) H6)))) (plus (S O) (minus n (S O))) (plus_comm (S O) (minus 
+n (S O)))) (S (plus O (minus n (S O)))) (refl_equal nat (S (plus O (minus n 
+(S O)))))) n (lt_plus_minus O n (le_lt_trans O d0 n (le_O_n d0) 
+H6))))))))))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (t: 
+T).(\lambda (_: (ty3 g c0 u t)).(\lambda (H1: ((\forall (e: C).(\forall (u0: 
+T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u0)) \to (\forall (a0: 
+C).((csubst1 d u0 c0 a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 
+T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift (S O) d y1)))) 
+(\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t (lift (S O) d y2)))) (\lambda 
+(y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))))).(\lambda (b: 
+B).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) 
+u) t3 t4)).(\lambda (H3: ((\forall (e: C).(\forall (u0: T).(\forall (d: 
+nat).((getl d (CHead c0 (Bind b) u) (CHead e (Bind Abbr) u0)) \to (\forall 
+(a0: C).((csubst1 d u0 (CHead c0 (Bind b) u) a0) \to (\forall (a: C).((drop 
+(S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 t3 
+(lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t4 (lift 
+(S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 
+y2)))))))))))))).(\lambda (t0: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) 
+t4 t0)).(\lambda (H5: ((\forall (e: C).(\forall (u0: T).(\forall (d: 
+nat).((getl d (CHead c0 (Bind b) u) (CHead e (Bind Abbr) u0)) \to (\forall 
+(a0: C).((csubst1 d u0 (CHead c0 (Bind b) u) a0) \to (\forall (a: C).((drop 
+(S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 t4 
+(lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t0 (lift 
+(S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 
+y2)))))))))))))).(\lambda (e: C).(\lambda (u0: T).(\lambda (d: nat).(\lambda 
+(H6: (getl d c0 (CHead e (Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H7: 
+(csubst1 d u0 c0 a0)).(\lambda (a: C).(\lambda (H8: (drop (S O) d a0 a)).(let 
+H9 \def (H1 e u0 d H6 a0 H7 a H8) in (ex3_2_ind T T (\lambda (y1: T).(\lambda 
+(_: T).(subst1 d u0 u (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: 
+T).(subst1 d u0 t (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 
+g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead 
+(Bind b) u t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 
+d u0 (THead (Bind b) u t4) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda 
+(y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H10: 
+(subst1 d u0 u (lift (S O) d x0))).(\lambda (_: (subst1 d u0 t (lift (S O) d 
+x1))).(\lambda (H12: (ty3 g a x0 x1)).(let H13 \def (H5 e u0 (S d) (getl_head 
+(Bind b) d c0 (CHead e (Bind Abbr) u0) H6 u) (CHead a0 (Bind b) (lift (S O) d 
+x0)) (csubst1_bind b d u0 u (lift (S O) d x0) H10 c0 a0 H7) (CHead a (Bind b) 
+x0) (drop_skip_bind (S O) d a0 a H8 b x0)) in (ex3_2_ind T T (\lambda (y1: 
+T).(\lambda (_: T).(subst1 (S d) u0 t4 (lift (S O) (S d) y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(subst1 (S d) u0 t0 (lift (S O) (S d) y2)))) (\lambda 
+(y1: T).(\lambda (y2: T).(ty3 g (CHead a (Bind b) x0) y1 y2))) (ex3_2 T T 
+(\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead (Bind b) u t3) (lift (S 
+O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Bind b) u 
+t4) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) 
+(\lambda (x2: T).(\lambda (x3: T).(\lambda (H14: (subst1 (S d) u0 t4 (lift (S 
+O) (S d) x2))).(\lambda (_: (subst1 (S d) u0 t0 (lift (S O) (S d) 
+x3))).(\lambda (H16: (ty3 g (CHead a (Bind b) x0) x2 x3)).(let H17 \def (H3 e 
+u0 (S d) (getl_head (Bind b) d c0 (CHead e (Bind Abbr) u0) H6 u) (CHead a0 
+(Bind b) (lift (S O) d x0)) (csubst1_bind b d u0 u (lift (S O) d x0) H10 c0 
+a0 H7) (CHead a (Bind b) x0) (drop_skip_bind (S O) d a0 a H8 b x0)) in 
+(ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 (S d) u0 t3 (lift (S 
+O) (S d) y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 (S d) u0 t4 (lift (S 
+O) (S d) y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g (CHead a (Bind b) 
+x0) y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead 
+(Bind b) u t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 
+d u0 (THead (Bind b) u t4) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda 
+(y2: T).(ty3 g a y1 y2)))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H18: 
+(subst1 (S d) u0 t3 (lift (S O) (S d) x4))).(\lambda (H19: (subst1 (S d) u0 
+t4 (lift (S O) (S d) x5))).(\lambda (H20: (ty3 g (CHead a (Bind b) x0) x4 
+x5)).(let H21 \def (eq_ind T x5 (\lambda (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: 
+T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u0)) \to (\forall (a0: 
+C).((csubst1 d u0 c0 a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 
+T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 w (lift (S O) d y1)))) 
+(\lambda (_: T).(\lambda (y2: T).(subst1 d u0 u (lift (S O) d y2)))) (\lambda 
+(y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))))).(\lambda (v: 
+T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead (Bind Abst) u 
+t))).(\lambda (H3: ((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl 
+d c0 (CHead e (Bind Abbr) u0)) \to (\forall (a0: C).((csubst1 d u0 c0 a0) \to 
+(\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda 
+(_: T).(subst1 d u0 v (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: 
+T).(subst1 d u0 (THead (Bind Abst) u t) (lift (S O) d y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))))).(\lambda (e: C).(\lambda 
+(u0: T).(\lambda (d: nat).(\lambda (H4: (getl d c0 (CHead e (Bind Abbr) 
+u0))).(\lambda (a0: C).(\lambda (H5: (csubst1 d u0 c0 a0)).(\lambda (a: 
+C).(\lambda (H6: (drop (S O) d a0 a)).(let H7 \def (H3 e u0 d H4 a0 H5 a H6) 
+in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 v (lift (S O) 
+d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Bind Abst) u 
+t) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) 
+(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead (Flat Appl) w 
+v) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead 
+(Flat Appl) w (THead (Bind Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (H8: (subst1 d u0 v (lift (S O) d x0))).(\lambda (H9: (subst1 d 
+u0 (THead (Bind Abst) u t) (lift (S O) d x1))).(\lambda (H10: (ty3 g a x0 
+x1)).(let H11 \def (H1 e u0 d H4 a0 H5 a H6) in (ex3_2_ind T T (\lambda (y1: 
+T).(\lambda (_: T).(subst1 d u0 w (lift (S O) d y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(subst1 d u0 u (lift (S O) d y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda 
+(_: T).(subst1 d u0 (THead (Flat Appl) w v) (lift (S O) d y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(subst1 d u0 (THead (Flat Appl) w (THead (Bind Abst) u 
+t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) 
+(\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (subst1 d u0 w (lift (S O) d 
+x2))).(\lambda (H13: (subst1 d u0 u (lift (S O) d x3))).(\lambda (H14: (ty3 g 
+a x2 x3)).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T (lift (S O) 
+d x1) (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 d 
+u0 u u2))) (\lambda (_: T).(\lambda (t3: T).(subst1 (s (Bind Abst) d) u0 t 
+t3))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead (Flat 
+Appl) w v) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 
+(THead (Flat Appl) w (THead (Bind Abst) u t)) (lift (S O) d y2)))) (\lambda 
+(y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x4: T).(\lambda (x5: 
+T).(\lambda (H15: (eq T (lift (S O) d x1) (THead (Bind Abst) x4 
+x5))).(\lambda (H16: (subst1 d u0 u x4)).(\lambda (H17: (subst1 (s (Bind 
+Abst) d) u0 t x5)).(let H18 \def (sym_equal T (lift (S O) d x1) (THead (Bind 
+Abst) x4 x5) H15) in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x1 
+(THead (Bind Abst) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T x4 (lift (S 
+O) d y)))) (\lambda (_: T).(\lambda (z: T).(eq T x5 (lift (S O) (S d) z)))) 
+(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead (Flat Appl) w 
+v) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead 
+(Flat Appl) w (THead (Bind Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x6: T).(\lambda (x7: 
+T).(\lambda (H19: (eq T x1 (THead (Bind Abst) x6 x7))).(\lambda (H20: (eq T 
+x4 (lift (S O) d x6))).(\lambda (H21: (eq T x5 (lift (S O) (S d) x7))).(let 
+H22 \def (eq_ind T x5 (\lambda (t0: T).(subst1 (s (Bind Abst) d) u0 t t0)) 
+H17 (lift (S O) (S d) x7) H21) in (let H23 \def (eq_ind T x4 (\lambda (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 t4 (lift (S O) d y2)))) (\lambda 
+(y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (H8: (subst1 d u t4 (lift (S O) d x0))).(\lambda (_: (subst1 d u 
+t0 (lift (S O) d x1))).(\lambda (H10: (ty3 g a x0 x1)).(let H11 \def (H1 e u 
+d H4 a0 H5 a H6) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 d 
+u t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u t4 
+(lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) 
+(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u (THead (Flat Cast) t4 
+t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u t4 
+(lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) 
+(\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (subst1 d u t3 (lift (S O) d 
+x2))).(\lambda (H13: (subst1 d u t4 (lift (S O) d x3))).(\lambda (H14: (ty3 g 
+a x2 x3)).(let H15 \def (eq_ind T x3 (\lambda (t: T).(ty3 g a x2 t)) H14 x0 
+(subst1_confluence_lift t4 x3 u d H13 x0 H8)) in (ex3_2_intro T T (\lambda 
+(y1: T).(\lambda (_: T).(subst1 d u (THead (Flat Cast) t4 t3) (lift (S O) d 
+y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u t4 (lift (S O) d y2)))) 
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (THead (Flat Cast) x0 x2) 
+x0 (eq_ind_r T (THead (Flat Cast) (lift (S O) d x0) (lift (S O) d x2)) 
+(\lambda (t: T).(subst1 d u (THead (Flat Cast) t4 t3) t)) (subst1_head u t4 
+(lift (S O) d x0) d H8 (Flat Cast) t3 (lift (S O) d x2) H12) (lift (S O) d 
+(THead (Flat Cast) x0 x2)) (lift_flat Cast x0 x2 (S O) d)) H8 (ty3_cast g a 
+x2 x0 H15 x1 H10)))))))) H11))))))) H7)))))))))))))))))) c t1 t2 H))))).
+
+theorem ty3_gen_cvoid:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c 
+t1 t2) \to (\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c 
+(CHead e (Bind Void) u)) \to (\forall (a: C).((drop (S O) d c a) \to (ex3_2 T 
+T (\lambda (y1: T).(\lambda (_: T).(eq T t1 (lift (S O) d y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(eq T t2 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda 
+(y2: T).(ty3 g a y1 y2))))))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
+(H: (ty3 g c t1 t2)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda 
+(t0: T).(\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead 
+e (Bind Void) u)) \to (\forall (a: C).((drop (S O) d c0 a) \to (ex3_2 T T 
+(\lambda (y1: T).(\lambda (_: T).(eq T t (lift (S O) d y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda 
+(y2: T).(ty3 g a y1 y2))))))))))))) (\lambda (c0: C).(\lambda (t3: 
+T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 t3 t)).(\lambda (H1: ((\forall (e: 
+C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Void) u)) \to 
+(\forall (a: C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda 
+(_: T).(eq T t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t 
+(lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 
+y2)))))))))))).(\lambda (u: T).(\lambda (t4: T).(\lambda (H2: (ty3 g c0 u 
+t4)).(\lambda (H3: ((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl 
+d c0 (CHead e (Bind Void) u0)) \to (\forall (a: C).((drop (S O) d c0 a) \to 
+(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T u (lift (S O) d y1)))) 
+(\lambda (_: T).(\lambda (y2: T).(eq T t4 (lift (S O) d y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (H4: (pc3 c0 t4 
+t3)).(\lambda (e: C).(\lambda (u0: T).(\lambda (d: nat).(\lambda (H5: (getl d 
+c0 (CHead e (Bind Void) u0))).(\lambda (a: C).(\lambda (H6: (drop (S O) d c0 
+a)).(let H7 \def (H3 e u0 d H5 a H6) in (ex3_2_ind T T (\lambda (y1: 
+T).(\lambda (_: T).(eq T u (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: 
+T).(eq T t4 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a 
+y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T u (lift (S O) d 
+y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t3 (lift (S O) d y2)))) 
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: 
+T).(\lambda (x1: T).(\lambda (H8: (eq T u (lift (S O) d x0))).(\lambda (H9: 
+(eq T t4 (lift (S O) d x1))).(\lambda (H10: (ty3 g a x0 x1)).(let H11 \def 
+(eq_ind T t4 (\lambda (t0: T).(pc3 c0 t0 t3)) H4 (lift (S O) d x1) H9) in 
+(let H12 \def (eq_ind T t4 (\lambda (t0: T).(ty3 g c0 u t0)) H2 (lift (S O) d 
+x1) H9) in (let H13 \def (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 (lift (S 
+O) d x1))) H12 (lift (S O) d x0) H8) in (eq_ind_r T (lift (S O) d x0) 
+(\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t0 (lift 
+(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t3 (lift (S O) d y2)))) 
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H14 \def (H1 e u0 
+d H5 a H6) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T t3 (lift 
+(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d y2)))) 
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: 
+T).(\lambda (_: T).(eq T (lift (S O) d x0) (lift (S O) d y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(eq T t3 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda 
+(y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H15: 
+(eq T t3 (lift (S O) d x2))).(\lambda (H16: (eq T t (lift (S O) d 
+x3))).(\lambda (H17: (ty3 g a x2 x3)).(let H18 \def (eq_ind T t (\lambda (t0: 
+T).(ty3 g c0 t3 t0)) H0 (lift (S O) d x3) H16) in (let H19 \def (eq_ind T t3 
+(\lambda (t0: T).(ty3 g c0 t0 (lift (S O) d x3))) H18 (lift (S O) d x2) H15) 
+in (let H20 \def (eq_ind T t3 (\lambda (t0: T).(pc3 c0 (lift (S O) d x1) t0)) 
+H11 (lift (S O) d x2) H15) in (eq_ind_r T (lift (S O) d x2) (\lambda (t0: 
+T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (lift (S O) d x0) (lift 
+(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) 
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T 
+(\lambda (y1: T).(\lambda (_: T).(eq T (lift (S O) d x0) (lift (S O) d y1)))) 
+(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d x2) (lift (S O) d y2)))) 
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) x0 x2 (refl_equal T (lift 
+(S O) d x0)) (refl_equal T (lift (S O) d x2)) (ty3_conv g a x2 x3 H17 x0 x1 
+H10 (pc3_gen_lift c0 x1 x2 (S O) d H20 a H6))) t3 H15))))))))) H14)) u 
+H8))))))))) H7)))))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda 
+(e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda (_: (getl d c0 (CHead e 
+(Bind Void) u))).(\lambda (a: C).(\lambda (_: (drop (S O) d c0 
+a)).(ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (TSort m) (lift 
+(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (TSort (next g m)) 
+(lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) 
+(TSort m) (TSort (next g m)) (eq_ind_r T (TSort m) (\lambda (t: T).(eq T 
+(TSort m) t)) (refl_equal T (TSort m)) (lift (S O) d (TSort m)) (lift_sort m 
+(S O) d)) (eq_ind_r T (TSort (next g m)) (\lambda (t: T).(eq T (TSort (next g 
+m)) t)) (refl_equal T (TSort (next g m))) (lift (S O) d (TSort (next g m))) 
+(lift_sort (next g m) (S O) d)) (ty3_sort g a m)))))))))) (\lambda (n: 
+nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n 
+c0 (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u 
+t)).(\lambda (H2: ((\forall (e: C).(\forall (u0: T).(\forall (d0: nat).((getl 
+d0 d (CHead e (Bind Void) u0)) \to (\forall (a: C).((drop (S O) d0 d a) \to 
+(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T u (lift (S O) d0 y1)))) 
+(\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d0 y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (e: C).(\lambda (u0: 
+T).(\lambda (d0: nat).(\lambda (H3: (getl d0 c0 (CHead e (Bind Void) 
+u0))).(\lambda (a: C).(\lambda (H4: (drop (S O) d0 c0 a)).(lt_eq_gt_e n d0 
+(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 
+y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O) d0 
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (H5: (lt 
+n d0)).(let H6 \def (eq_ind nat (minus d0 n) (\lambda (n0: nat).(getl n0 
+(CHead d (Bind Abbr) u) (CHead e (Bind Void) u0))) (getl_conf_le d0 (CHead e 
+(Bind Void) u0) c0 H3 (CHead d (Bind Abbr) u) n H0 (le_S_n n d0 (le_S (S n) 
+d0 H5))) (S (minus d0 (S n))) (minus_x_Sy d0 n H5)) in (let H7 \def (eq_ind 
+nat d0 (\lambda (n0: nat).(drop (S O) n0 c0 a)) H4 (S (plus n (minus d0 (S 
+n)))) (lt_plus_minus n d0 H5)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: 
+C).(eq T u (lift (S O) (minus d0 (S n)) v)))) (\lambda (v: T).(\lambda (e0: 
+C).(getl n a (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0: 
+C).(drop (S O) (minus d0 (S n)) d e0))) (ex3_2 T T (\lambda (y1: T).(\lambda 
+(_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: 
+T).(eq T (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda 
+(y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1: C).(\lambda (H8: 
+(eq T u (lift (S O) (minus d0 (S n)) x0))).(\lambda (H9: (getl n a (CHead x1 
+(Bind Abbr) x0))).(\lambda (H10: (drop (S O) (minus d0 (S n)) d x1)).(let H11 
+\def (eq_ind T u (\lambda (t0: T).(\forall (e0: C).(\forall (u1: T).(\forall 
+(d1: nat).((getl d1 d (CHead e0 (Bind Void) u1)) \to (\forall (a0: C).((drop 
+(S O) d1 d a0) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t0 (lift 
+(S O) d1 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d1 
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a0 y1 y2))))))))))) H2 (lift 
+(S O) (minus d0 (S n)) x0) H8) in (let H12 \def (eq_ind T u (\lambda (t0: 
+T).(ty3 g d t0 t)) H1 (lift (S O) (minus d0 (S n)) x0) H8) in (let H13 \def 
+(H11 e u0 (minus d0 (S n)) (getl_gen_S (Bind Abbr) d (CHead e (Bind Void) u0) 
+u (minus d0 (S n)) H6) x1 H10) in (ex3_2_ind T T (\lambda (y1: T).(\lambda 
+(_: T).(eq T (lift (S O) (minus d0 (S n)) x0) (lift (S O) (minus d0 (S n)) 
+y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) (minus d0 (S n)) 
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g x1 y1 y2))) (ex3_2 T T 
+(\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) 
+(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O) d0 y2)))) 
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: 
+T).(\lambda (x3: T).(\lambda (H14: (eq T (lift (S O) (minus d0 (S n)) x0) 
+(lift (S O) (minus d0 (S n)) x2))).(\lambda (H15: (eq T t (lift (S O) (minus 
+d0 (S n)) x3))).(\lambda (H16: (ty3 g x1 x2 x3)).(let H17 \def (eq_ind T t 
+(\lambda (t0: T).(ty3 g d (lift (S O) (minus d0 (S n)) x0) t0)) H12 (lift (S 
+O) (minus d0 (S n)) x3) H15) in (eq_ind_r T (lift (S O) (minus d0 (S n)) x3) 
+(\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) 
+(lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O 
+t0) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 
+y2))))) (let H18 \def (eq_ind_r T x2 (\lambda (t0: T).(ty3 g x1 t0 x3)) H16 
+x0 (lift_inj x0 x2 (S O) (minus d0 (S n)) H14)) in (eq_ind T (lift (S O) 
+(plus (S n) (minus d0 (S n))) (lift (S n) O x3)) (\lambda (t0: T).(ex3_2 T T 
+(\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) 
+(\lambda (_: T).(\lambda (y2: T).(eq T t0 (lift (S O) d0 y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind nat d0 (\lambda (n0: 
+nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) 
+d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) n0 (lift (S n) O 
+x3)) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 
+y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) 
+(lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d0 
+(lift (S n) O x3)) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: 
+T).(ty3 g a y1 y2))) (TLRef n) (lift (S n) O x3) (eq_ind_r T (TLRef n) 
+(\lambda (t0: T).(eq T (TLRef n) t0)) (refl_equal T (TLRef n)) (lift (S O) d0 
+(TLRef n)) (lift_lref_lt n (S O) d0 H5)) (refl_equal T (lift (S O) d0 (lift 
+(S n) O x3))) (ty3_abbr g n a x1 x0 H9 x3 H18)) (plus (S n) (minus d0 (S n))) 
+(le_plus_minus (S n) d0 H5)) (lift (S n) O (lift (S O) (minus d0 (S n)) x3)) 
+(lift_d x3 (S O) (S n) (minus d0 (S n)) O (le_O_n (minus d0 (S n)))))) t 
+H15))))))) H13))))))))) (getl_drop_conf_lt Abbr c0 d u n H0 a (S O) (minus d0 
+(S n)) H7))))) (\lambda (H5: (eq nat n d0)).(let H6 \def (eq_ind_r nat d0 
+(\lambda (n0: nat).(drop (S O) n0 c0 a)) H4 n H5) in (let H7 \def (eq_ind_r 
+nat d0 (\lambda (n0: nat).(getl n0 c0 (CHead e (Bind Void) u0))) H3 n H5) in 
+(eq_ind nat n (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: 
+T).(eq T (TLRef n) (lift (S O) n0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq 
+T (lift (S n) O t) (lift (S O) n0 y2)))) (\lambda (y1: T).(\lambda (y2: 
+T).(ty3 g a y1 y2))))) (let H8 \def (eq_ind C (CHead d (Bind Abbr) u) 
+(\lambda (c1: C).(getl n c0 c1)) H0 (CHead e (Bind Void) u0) (getl_mono c0 
+(CHead d (Bind Abbr) u) n H0 (CHead e (Bind Void) u0) H7)) in (let H9 \def 
+(eq_ind C (CHead d (Bind Abbr) u) (\lambda (ee: C).(match ee in C return 
+(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) 
+\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) 
+\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow 
+True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) 
+\Rightarrow False])])) I (CHead e (Bind Void) u0) (getl_mono c0 (CHead d 
+(Bind Abbr) u) n H0 (CHead e (Bind Void) u0) H7)) in (False_ind (ex3_2 T T 
+(\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) n y1)))) 
+(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O) n y2)))) 
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) H9))) d0 H5)))) (\lambda 
+(H5: (lt d0 n)).(eq_ind_r nat (S (plus O (minus n (S O)))) (\lambda (n0: 
+nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n0) (lift (S O) 
+d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O) 
+d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind nat 
+(plus (S O) (minus n (S O))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: 
+T).(\lambda (_: T).(eq T (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda 
+(y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r nat (plus (minus n (S 
+O)) (S O)) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq 
+T (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T 
+(lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 
+g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef 
+(plus (minus n (S O)) (S O))) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda 
+(y2: T).(eq T (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g a y1 y2))) (TLRef (minus n (S O))) (lift n O t) 
+(eq_ind_r T (TLRef (plus (minus n (S O)) (S O))) (\lambda (t0: T).(eq T 
+(TLRef (plus (minus n (S O)) (S O))) t0)) (refl_equal T (TLRef (plus (minus n 
+(S O)) (S O)))) (lift (S O) d0 (TLRef (minus n (S O)))) (lift_lref_ge (minus 
+n (S O)) (S O) d0 (lt_le_minus d0 n H5))) (eq_ind_r T (lift (plus (S O) n) O 
+t) (\lambda (t0: T).(eq T (lift (S n) O t) t0)) (refl_equal T (lift (S n) O 
+t)) (lift (S O) d0 (lift n O t)) (lift_free t n (S O) O d0 (le_S_n d0 (plus O 
+n) (le_S (S d0) (plus O n) H5)) (le_O_n d0))) (eq_ind_r nat (S (minus n (S 
+O))) (\lambda (n0: nat).(ty3 g a (TLRef (minus n (S O))) (lift n0 O t))) 
+(ty3_abbr g (minus n (S O)) a d u (getl_drop_conf_ge n (CHead d (Bind Abbr) 
+u) c0 H0 a (S O) d0 H4 (eq_ind_r nat (plus (S O) d0) (\lambda (n0: nat).(le 
+n0 n)) H5 (plus d0 (S O)) (plus_comm d0 (S O)))) t H1) n (minus_x_SO n 
+(le_lt_trans O d0 n (le_O_n d0) H5)))) (plus (S O) (minus n (S O))) 
+(plus_comm (S O) (minus n (S O)))) (S (plus O (minus n (S O)))) (refl_equal 
+nat (S (plus O (minus n (S O)))))) n (lt_plus_minus O n (le_lt_trans O d0 n 
+(le_O_n d0) H5))))))))))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda 
+(d: C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abst) 
+u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u t)).(\lambda (H2: ((\forall (e: 
+C).(\forall (u0: T).(\forall (d0: nat).((getl d0 d (CHead e (Bind Void) u0)) 
+\to (\forall (a: C).((drop (S O) d0 d a) \to (ex3_2 T T (\lambda (y1: 
+T).(\lambda (_: T).(eq T u (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda 
+(y2: T).(eq T t (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 
+g a y1 y2)))))))))))).(\lambda (e: C).(\lambda (u0: T).(\lambda (d0: 
+nat).(\lambda (H3: (getl d0 c0 (CHead e (Bind Void) u0))).(\lambda (a: 
+C).(\lambda (H4: (drop (S O) d0 c0 a)).(lt_eq_gt_e n d0 (ex3_2 T T (\lambda 
+(y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(eq T (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda 
+(y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (H5: (lt n d0)).(let H6 
+\def (eq_ind nat (minus d0 n) (\lambda (n0: nat).(getl n0 (CHead d (Bind 
+Abst) u) (CHead e (Bind Void) u0))) (getl_conf_le d0 (CHead e (Bind Void) u0) 
+c0 H3 (CHead d (Bind Abst) u) n H0 (le_S_n n d0 (le_S (S n) d0 H5))) (S 
+(minus d0 (S n))) (minus_x_Sy d0 n H5)) in (let H7 \def (eq_ind nat d0 
+(\lambda (n0: nat).(drop (S O) n0 c0 a)) H4 (S (plus n (minus d0 (S n)))) 
+(lt_plus_minus n d0 H5)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: 
+C).(eq T u (lift (S O) (minus d0 (S n)) v)))) (\lambda (v: T).(\lambda (e0: 
+C).(getl n a (CHead e0 (Bind Abst) v)))) (\lambda (_: T).(\lambda (e0: 
+C).(drop (S O) (minus d0 (S n)) d e0))) (ex3_2 T T (\lambda (y1: T).(\lambda 
+(_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: 
+T).(eq T (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda 
+(y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1: C).(\lambda (H8: 
+(eq T u (lift (S O) (minus d0 (S n)) x0))).(\lambda (H9: (getl n a (CHead x1 
+(Bind Abst) x0))).(\lambda (H10: (drop (S O) (minus d0 (S n)) d x1)).(let H11 
+\def (eq_ind T u (\lambda (t0: T).(\forall (e0: C).(\forall (u1: T).(\forall 
+(d1: nat).((getl d1 d (CHead e0 (Bind Void) u1)) \to (\forall (a0: C).((drop 
+(S O) d1 d a0) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t0 (lift 
+(S O) d1 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d1 
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a0 y1 y2))))))))))) H2 (lift 
+(S O) (minus d0 (S n)) x0) H8) in (let H12 \def (eq_ind T u (\lambda (t0: 
+T).(ty3 g d t0 t)) H1 (lift (S O) (minus d0 (S n)) x0) H8) in (eq_ind_r T 
+(lift (S O) (minus d0 (S n)) x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: 
+T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(eq T (lift (S n) O t0) (lift (S O) d0 y2)))) (\lambda 
+(y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H13 \def (H11 e u0 (minus 
+d0 (S n)) (getl_gen_S (Bind Abst) d (CHead e (Bind Void) u0) u (minus d0 (S 
+n)) H6) x1 H10) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T 
+(lift (S O) (minus d0 (S n)) x0) (lift (S O) (minus d0 (S n)) y1)))) (\lambda 
+(_: T).(\lambda (y2: T).(eq T t (lift (S O) (minus d0 (S n)) y2)))) (\lambda 
+(y1: T).(\lambda (y2: T).(ty3 g x1 y1 y2))) (ex3_2 T T (\lambda (y1: 
+T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(eq T (lift (S n) O (lift (S O) (minus d0 (S n)) x0)) 
+(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) 
+(\lambda (x2: T).(\lambda (x3: T).(\lambda (H14: (eq T (lift (S O) (minus d0 
+(S n)) x0) (lift (S O) (minus d0 (S n)) x2))).(\lambda (H15: (eq T t (lift (S 
+O) (minus d0 (S n)) x3))).(\lambda (H16: (ty3 g x1 x2 x3)).(let H17 \def 
+(eq_ind T t (\lambda (t0: T).(ty3 g d (lift (S O) (minus d0 (S n)) x0) t0)) 
+H12 (lift (S O) (minus d0 (S n)) x3) H15) in (let H18 \def (eq_ind_r T x2 
+(\lambda (t0: T).(ty3 g x1 t0 x3)) H16 x0 (lift_inj x0 x2 (S O) (minus d0 (S 
+n)) H14)) in (eq_ind T (lift (S O) (plus (S n) (minus d0 (S n))) (lift (S n) 
+O x0)) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T 
+(TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t0 
+(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) 
+(eq_ind nat d0 (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: 
+T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq 
+T (lift (S O) n0 (lift (S n) O x0)) (lift (S O) d0 y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: 
+T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(eq T (lift (S O) d0 (lift (S n) O x0)) (lift (S O) d0 
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (TLRef n) (lift (S 
+n) O x0) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (TLRef n) t0)) 
+(refl_equal T (TLRef n)) (lift (S O) d0 (TLRef n)) (lift_lref_lt n (S O) d0 
+H5)) (refl_equal T (lift (S O) d0 (lift (S n) O x0))) (ty3_abst g n a x1 x0 
+H9 x3 H18)) (plus (S n) (minus d0 (S n))) (le_plus_minus (S n) d0 H5)) (lift 
+(S n) O (lift (S O) (minus d0 (S n)) x0)) (lift_d x0 (S O) (S n) (minus d0 (S 
+n)) O (le_O_n (minus d0 (S n)))))))))))) H13)) u H8)))))))) 
+(getl_drop_conf_lt Abst c0 d u n H0 a (S O) (minus d0 (S n)) H7))))) (\lambda 
+(H5: (eq nat n d0)).(let H6 \def (eq_ind_r nat d0 (\lambda (n0: nat).(drop (S 
+O) n0 c0 a)) H4 n H5) in (let H7 \def (eq_ind_r nat d0 (\lambda (n0: 
+nat).(getl n0 c0 (CHead e (Bind Void) u0))) H3 n H5) in (eq_ind nat n 
+(\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef 
+n) (lift (S O) n0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O 
+u) (lift (S O) n0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 
+y2))))) (let H8 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (c1: C).(getl 
+n c0 c1)) H0 (CHead e (Bind Void) u0) (getl_mono c0 (CHead d (Bind Abst) u) n 
+H0 (CHead e (Bind Void) u0) H7)) in (let H9 \def (eq_ind C (CHead d (Bind 
+Abst) u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with 
+[(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return 
+(\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B return 
+(\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True | 
+Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead e (Bind 
+Void) u0) (getl_mono c0 (CHead d (Bind Abst) u) n H0 (CHead e (Bind Void) u0) 
+H7)) in (False_ind (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef 
+n) (lift (S O) n y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O 
+u) (lift (S O) n y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) 
+H9))) d0 H5)))) (\lambda (H5: (lt d0 n)).(eq_ind_r nat (S (plus O (minus n (S 
+O)))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T 
+(TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift 
+(S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a 
+y1 y2))))) (eq_ind nat (plus (S O) (minus n (S O))) (\lambda (n0: nat).(ex3_2 
+T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n0) (lift (S O) d0 y1)))) 
+(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O u) (lift (S O) d0 y2)))) 
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r nat (plus 
+(minus n (S O)) (S O)) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: 
+T).(\lambda (_: T).(eq T (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(eq T (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda 
+(y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: 
+T).(\lambda (_: T).(eq T (TLRef (plus (minus n (S O)) (S O))) (lift (S O) d0 
+y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O u) (lift (S O) d0 
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (TLRef (minus n (S 
+O))) (lift n O u) (eq_ind_r T (TLRef (plus (minus n (S O)) (S O))) (\lambda 
+(t0: T).(eq T (TLRef (plus (minus n (S O)) (S O))) t0)) (refl_equal T (TLRef 
+(plus (minus n (S O)) (S O)))) (lift (S O) d0 (TLRef (minus n (S O)))) 
+(lift_lref_ge (minus n (S O)) (S O) d0 (lt_le_minus d0 n H5))) (eq_ind_r T 
+(lift (plus (S O) n) O u) (\lambda (t0: T).(eq T (lift (S n) O u) t0)) 
+(refl_equal T (lift (S n) O u)) (lift (S O) d0 (lift n O u)) (lift_free u n 
+(S O) O d0 (le_S_n d0 (plus O n) (le_S (S d0) (plus O n) H5)) (le_O_n d0))) 
+(eq_ind_r nat (S (minus n (S O))) (\lambda (n0: nat).(ty3 g a (TLRef (minus n 
+(S O))) (lift n0 O u))) (ty3_abst g (minus n (S O)) a d u (getl_drop_conf_ge 
+n (CHead d (Bind Abst) u) c0 H0 a (S O) d0 H4 (eq_ind_r nat (plus (S O) d0) 
+(\lambda (n0: nat).(le n0 n)) H5 (plus d0 (S O)) (plus_comm d0 (S O)))) t H1) 
+n (minus_x_SO n (le_lt_trans O d0 n (le_O_n d0) H5)))) (plus (S O) (minus n 
+(S O))) (plus_comm (S O) (minus n (S O)))) (S (plus O (minus n (S O)))) 
+(refl_equal nat (S (plus O (minus n (S O)))))) n (lt_plus_minus O n 
+(le_lt_trans O d0 n (le_O_n d0) H5))))))))))))))))))) (\lambda (c0: 
+C).(\lambda (u: T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 u t)).(\lambda 
+(H1: ((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl d c0 (CHead e 
+(Bind Void) u0)) \to (\forall (a: C).((drop (S O) d c0 a) \to (ex3_2 T T 
+(\lambda (y1: T).(\lambda (_: T).(eq T u (lift (S O) d y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(eq T t (lift (S O) d y2)))) (\lambda (y1: T).(\lambda 
+(y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (b: B).(\lambda (t3: T).(\lambda 
+(t4: T).(\lambda (H2: (ty3 g (CHead c0 (Bind b) u) t3 t4)).(\lambda (H3: 
+((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl d (CHead c0 (Bind 
+b) u) (CHead e (Bind Void) u0)) \to (\forall (a: C).((drop (S O) d (CHead c0 
+(Bind b) u) a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t3 (lift 
+(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t4 (lift (S O) d y2)))) 
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (t0: 
+T).(\lambda (H4: (ty3 g (CHead c0 (Bind b) u) t4 t0)).(\lambda (H5: ((\forall 
+(e: C).(\forall (u0: T).(\forall (d: nat).((getl d (CHead c0 (Bind b) u) 
+(CHead e (Bind Void) u0)) \to (\forall (a: C).((drop (S O) d (CHead c0 (Bind 
+b) u) a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t4 (lift (S O) 
+d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) 
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (e: 
+C).(\lambda (u0: T).(\lambda (d: nat).(\lambda (H6: (getl d c0 (CHead e (Bind 
+Void) u0))).(\lambda (a: C).(\lambda (H7: (drop (S O) d c0 a)).(let H8 \def 
+(H1 e u0 d H6 a H7) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T 
+u (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d 
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T 
+(\lambda (y1: T).(\lambda (_: T).(eq T (THead (Bind b) u t3) (lift (S O) d 
+y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Bind b) u t4) (lift (S 
+O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda 
+(x0: T).(\lambda (x1: T).(\lambda (H9: (eq T u (lift (S O) d x0))).(\lambda 
+(H10: (eq T t (lift (S O) d x1))).(\lambda (H11: (ty3 g a x0 x1)).(let H12 
+\def (eq_ind T t (\lambda (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: 
+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 
+(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 
+(Bind b) (lift (S O) d x0) t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda 
+(y2: T).(eq T (THead (Bind b) (lift (S O) d x0) t5) (lift (S O) d y2)))) 
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r T (lift (S O) 
+(S d) x2) (\lambda (t5: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T 
+(THead (Bind b) (lift (S O) d x0) t5) (lift (S O) d y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(eq T (THead (Bind b) (lift (S O) d x0) (lift (S O) (S d) 
+x3)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 
+y2))))) (let H23 \def (H22 e u0 (S d) (getl_head (Bind b) d c0 (CHead e (Bind 
+Void) u0) H6 (lift (S O) d x0)) (CHead a (Bind b) x0) (drop_skip_bind (S O) d 
+c0 a H7 b x0)) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T (lift 
+(S O) (S d) x3) (lift (S O) (S d) y1)))) (\lambda (_: T).(\lambda (y2: T).(eq 
+T t0 (lift (S O) (S d) y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g (CHead 
+a (Bind b) x0) y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T 
+(THead (Bind b) (lift (S O) d x0) (lift (S O) (S d) x2)) (lift (S O) d y1)))) 
+(\lambda (_: T).(\lambda (y2: T).(eq T (THead (Bind b) (lift (S O) d x0) 
+(lift (S O) (S d) x3)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: 
+T).(ty3 g a y1 y2)))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H24: (eq T 
+(lift (S O) (S d) x3) (lift (S O) (S d) x4))).(\lambda (_: (eq T t0 (lift (S 
+O) (S d) x5))).(\lambda (H26: (ty3 g (CHead a (Bind b) x0) x4 x5)).(let H27 
+\def (eq_ind_r T x4 (\lambda (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 
+v (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Bind 
+Abst) u t) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 
+y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Appl) w 
+v) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat 
+Appl) w (THead (Bind Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (H7: (eq T v (lift (S O) d x0))).(\lambda (H8: (eq T (THead (Bind 
+Abst) u t) (lift (S O) d x1))).(\lambda (H9: (ty3 g a x0 x1)).(let H10 \def 
+(eq_ind T v (\lambda (t0: T).(ty3 g c0 t0 (THead (Bind Abst) u t))) H2 (lift 
+(S O) d x0) H7) in (eq_ind_r T (lift (S O) d x0) (\lambda (t0: T).(ex3_2 T T 
+(\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Appl) w t0) (lift (S O) d 
+y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) w (THead 
+(Bind Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 
+g a y1 y2))))) (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x1 (THead 
+(Bind Abst) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift (S O) d 
+y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift (S O) (S d) z)))) (ex3_2 
+T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Appl) w (lift (S O) d 
+x0)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat 
+Appl) w (THead (Bind Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3: 
+T).(\lambda (H11: (eq T x1 (THead (Bind Abst) x2 x3))).(\lambda (H12: (eq T u 
+(lift (S O) d x2))).(\lambda (H13: (eq T t (lift (S O) (S d) x3))).(let H14 
+\def (eq_ind T x1 (\lambda (t0: T).(ty3 g a x0 t0)) H9 (THead (Bind Abst) x2 
+x3) H11) in (eq_ind_r T (lift (S O) (S d) x3) (\lambda (t0: T).(ex3_2 T T 
+(\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Appl) w (lift (S O) d 
+x0)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat 
+Appl) w (THead (Bind Abst) u t0)) (lift (S O) d y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H15 \def (eq_ind T u (\lambda 
+(t0: T).(\forall (e0: C).(\forall (u1: T).(\forall (d0: nat).((getl d0 c0 
+(CHead e0 (Bind Void) u1)) \to (\forall (a0: C).((drop (S O) d0 c0 a0) \to 
+(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T w (lift (S O) d0 y1)))) 
+(\lambda (_: T).(\lambda (y2: T).(eq T t0 (lift (S O) d0 y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g a0 y1 y2))))))))))) H1 (lift (S O) d x2) H12) in 
+(eq_ind_r T (lift (S O) d x2) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: 
+T).(\lambda (_: T).(eq T (THead (Flat Appl) w (lift (S O) d x0)) (lift (S O) 
+d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) w (THead 
+(Bind Abst) t0 (lift (S O) (S d) x3))) (lift (S O) d y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H16 \def (H15 e u0 d H4 a H5) in 
+(ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T w (lift (S O) d y1)))) 
+(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d x2) (lift (S O) d y2)))) 
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: 
+T).(\lambda (_: T).(eq T (THead (Flat Appl) w (lift (S O) d x0)) (lift (S O) 
+d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) w (THead 
+(Bind Abst) (lift (S O) d x2) (lift (S O) (S d) x3))) (lift (S O) d y2)))) 
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x4: 
+T).(\lambda (x5: T).(\lambda (H17: (eq T w (lift (S O) d x4))).(\lambda (H18: 
+(eq T (lift (S O) d x2) (lift (S O) d x5))).(\lambda (H19: (ty3 g a x4 
+x5)).(eq_ind_r T (lift (S O) d x4) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: 
+T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (lift (S O) d x0)) (lift (S O) 
+d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) t0 (THead 
+(Bind Abst) (lift (S O) d x2) (lift (S O) (S d) x3))) (lift (S O) d y2)))) 
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H20 \def (eq_ind_r 
+T x5 (\lambda (t0: T).(ty3 g a x4 t0)) H19 x2 (lift_inj x2 x5 (S O) d H18)) 
+in (eq_ind T (lift (S O) d (THead (Bind Abst) x2 x3)) (\lambda (t0: T).(ex3_2 
+T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Appl) (lift (S O) d 
+x4) (lift (S O) d x0)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: 
+T).(eq T (THead (Flat Appl) (lift (S O) d x4) t0) (lift (S O) d y2)))) 
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind T (lift (S O) d 
+(THead (Flat Appl) x4 x0)) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: 
+T).(\lambda (_: T).(eq T t0 (lift (S O) d y1)))) (\lambda (_: T).(\lambda 
+(y2: T).(eq T (THead (Flat Appl) (lift (S O) d x4) (lift (S O) d (THead (Bind 
+Abst) x2 x3))) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g 
+a y1 y2))))) (eq_ind T (lift (S O) d (THead (Flat Appl) x4 (THead (Bind Abst) 
+x2 x3))) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T 
+(lift (S O) d (THead (Flat Appl) x4 x0)) (lift (S O) d y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda 
+(y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: 
+T).(eq T (lift (S O) d (THead (Flat Appl) x4 x0)) (lift (S O) d y1)))) 
+(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d (THead (Flat Appl) x4 
+(THead (Bind Abst) x2 x3))) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda 
+(y2: T).(ty3 g a y1 y2))) (THead (Flat Appl) x4 x0) (THead (Flat Appl) x4 
+(THead (Bind Abst) x2 x3)) (refl_equal T (lift (S O) d (THead (Flat Appl) x4 
+x0))) (refl_equal T (lift (S O) d (THead (Flat Appl) x4 (THead (Bind Abst) x2 
+x3)))) (ty3_appl g a x4 x2 H20 x0 x3 H14)) (THead (Flat Appl) (lift (S O) d 
+x4) (lift (S O) d (THead (Bind Abst) x2 x3))) (lift_flat Appl x4 (THead (Bind 
+Abst) x2 x3) (S O) d)) (THead (Flat Appl) (lift (S O) d x4) (lift (S O) d 
+x0)) (lift_flat Appl x4 x0 (S O) d)) (THead (Bind Abst) (lift (S O) d x2) 
+(lift (S O) (S d) x3)) (lift_bind Abst x2 x3 (S O) d))) w H17)))))) H16)) u 
+H12)) t H13))))))) (lift_gen_bind Abst u t x1 (S O) d H8)) v H7))))))) 
+H6))))))))))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: 
+T).(\lambda (H0: (ty3 g c0 t3 t4)).(\lambda (H1: ((\forall (e: C).(\forall 
+(u: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Void) u)) \to (\forall 
+(a: C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: 
+T).(eq T t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t4 
+(lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 
+y2)))))))))))).(\lambda (t0: T).(\lambda (H2: (ty3 g c0 t4 t0)).(\lambda (H3: 
+((\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e (Bind 
+Void) u)) \to (\forall (a: C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda 
+(y1: T).(\lambda (_: T).(eq T t4 (lift (S O) d y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda 
+(y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (e: C).(\lambda (u: T).(\lambda 
+(d: nat).(\lambda (H4: (getl d c0 (CHead e (Bind Void) u))).(\lambda (a: 
+C).(\lambda (H5: (drop (S O) d c0 a)).(let H6 \def (H3 e u d H4 a H5) in 
+(ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T t4 (lift (S O) d y1)))) 
+(\lambda (_: T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda 
+(_: T).(eq T (THead (Flat Cast) t4 t3) (lift (S O) d y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(eq T t4 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda 
+(y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: 
+(eq T t4 (lift (S O) d x0))).(\lambda (H8: (eq T t0 (lift (S O) d 
+x1))).(\lambda (H9: (ty3 g a x0 x1)).(let H10 \def (eq_ind T t0 (\lambda (t: 
+T).(ty3 g c0 t4 t)) H2 (lift (S O) d x1) H8) in (let H11 \def (eq_ind T t4 
+(\lambda (t: T).(ty3 g c0 t (lift (S O) d x1))) H10 (lift (S O) d x0) H7) in 
+(let H12 \def (eq_ind T t4 (\lambda (t: T).(\forall (e0: C).(\forall (u0: 
+T).(\forall (d0: nat).((getl d0 c0 (CHead e0 (Bind Void) u0)) \to (\forall 
+(a0: C).((drop (S O) d0 c0 a0) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: 
+T).(eq T t3 (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t 
+(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a0 y1 
+y2))))))))))) H1 (lift (S O) d x0) H7) in (let H13 \def (eq_ind T t4 (\lambda 
+(t: T).(ty3 g c0 t3 t)) H0 (lift (S O) d x0) H7) in (eq_ind_r T (lift (S O) d 
+x0) (\lambda (t: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead 
+(Flat Cast) t t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T 
+t (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) 
+(let H14 \def (H12 e u d H4 a H5) in (ex3_2_ind T T (\lambda (y1: T).(\lambda 
+(_: T).(eq T t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T 
+(lift (S O) d x0) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 
+g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat 
+Cast) (lift (S O) d x0) t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda 
+(y2: T).(eq T (lift (S O) d x0) (lift (S O) d y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3: 
+T).(\lambda (H15: (eq T t3 (lift (S O) d x2))).(\lambda (H16: (eq T (lift (S 
+O) d x0) (lift (S O) d x3))).(\lambda (H17: (ty3 g a x2 x3)).(let H18 \def 
+(eq_ind T t3 (\lambda (t: T).(ty3 g c0 t (lift (S O) d x0))) H13 (lift (S O) 
+d x2) H15) in (eq_ind_r T (lift (S O) d x2) (\lambda (t: T).(ex3_2 T T 
+(\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Cast) (lift (S O) d x0) 
+t) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d 
+x0) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 
+y2))))) (let H19 \def (eq_ind_r T x3 (\lambda (t: T).(ty3 g a x2 t)) H17 x0 
+(lift_inj x0 x3 (S O) d H16)) in (eq_ind T (lift (S O) d (THead (Flat Cast) 
+x0 x2)) (\lambda (t: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t 
+(lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d x0) 
+(lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) 
+(ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (lift (S O) d (THead 
+(Flat Cast) x0 x2)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq 
+T (lift (S O) d x0) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: 
+T).(ty3 g a y1 y2))) (THead (Flat Cast) x0 x2) x0 (refl_equal T (lift (S O) d 
+(THead (Flat Cast) x0 x2))) (refl_equal T (lift (S O) d x0)) (ty3_cast g a x2 
+x0 H19 x1 H9)) (THead (Flat Cast) (lift (S O) d x0) (lift (S O) d x2)) 
+(lift_flat Cast x0 x2 (S O) d))) t3 H15))))))) H14)) t4 H7)))))))))) 
+H6)))))))))))))))) c t1 t2 H))))).
+

diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/tau0.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/tau0.ma
new file mode 100644 (file)
index 0000000..ff9b2da
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/tau0.ma
@@ -0,0 +1,634 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/tau0".
+
+include "ty3/pr3_props.ma".
+
+include "tau0/defs.ma".
+
+theorem ty3_tau0:
+ \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).((ty3 g c u 
+t1) \to (\forall (t2: T).((tau0 g c u t2) \to (ty3 g c u t2)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (H: 
+(ty3 g c u t1)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (_: 
+T).(\forall (t2: T).((tau0 g c0 t t2) \to (ty3 g c0 t t2)))))) (\lambda (c0: 
+C).(\lambda (t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda 
+(_: ((\forall (t3: T).((tau0 g c0 t2 t3) \to (ty3 g c0 t2 t3))))).(\lambda 
+(u0: T).(\lambda (t3: T).(\lambda (_: (ty3 g c0 u0 t3)).(\lambda (H3: 
+((\forall (t4: T).((tau0 g c0 u0 t4) \to (ty3 g c0 u0 t4))))).(\lambda (_: 
+(pc3 c0 t3 t2)).(\lambda (t0: T).(\lambda (H5: (tau0 g c0 u0 t0)).(H3 t0 
+H5))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (t2: T).(\lambda 
+(H0: (tau0 g c0 (TSort m) t2)).(let H1 \def (match H0 in tau0 return (\lambda 
+(c1: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (tau0 ? c1 t t0)).((eq 
+C c1 c0) \to ((eq T t (TSort m)) \to ((eq T t0 t2) \to (ty3 g c0 (TSort m) 
+t2)))))))) with [(tau0_sort c1 n) \Rightarrow (\lambda (H1: (eq C c1 
+c0)).(\lambda (H2: (eq T (TSort n) (TSort m))).(\lambda (H3: (eq T (TSort 
+(next g n)) t2)).(eq_ind C c0 (\lambda (_: C).((eq T (TSort n) (TSort m)) \to 
+((eq T (TSort (next g n)) t2) \to (ty3 g c0 (TSort m) t2)))) (\lambda (H4: 
+(eq T (TSort n) (TSort m))).(let H5 \def (f_equal T nat (\lambda (e: 
+T).(match e in T return (\lambda (_: T).nat) with [(TSort n0) \Rightarrow n0 
+| (TLRef _) \Rightarrow n | (THead _ _ _) \Rightarrow n])) (TSort n) (TSort 
+m) H4) in (eq_ind nat m (\lambda (n0: nat).((eq T (TSort (next g n0)) t2) \to 
+(ty3 g c0 (TSort m) t2))) (\lambda (H6: (eq T (TSort (next g m)) t2)).(eq_ind 
+T (TSort (next g m)) (\lambda (t: T).(ty3 g c0 (TSort m) t)) (ty3_sort g c0 
+m) t2 H6)) n (sym_eq nat n m H5)))) c1 (sym_eq C c1 c0 H1) H2 H3)))) | 
+(tau0_abbr c1 d v i H1 w H2) \Rightarrow (\lambda (H3: (eq C c1 c0)).(\lambda 
+(H4: (eq T (TLRef i) (TSort m))).(\lambda (H5: (eq T (lift (S i) O w) 
+t2)).(eq_ind C c0 (\lambda (c2: C).((eq T (TLRef i) (TSort m)) \to ((eq T 
+(lift (S i) O w) t2) \to ((getl i c2 (CHead d (Bind Abbr) v)) \to ((tau0 g d 
+v w) \to (ty3 g c0 (TSort m) t2)))))) (\lambda (H6: (eq T (TLRef i) (TSort 
+m))).(let H7 \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 (TSort m) H6) in 
+(False_ind ((eq T (lift (S i) O w) t2) \to ((getl i c0 (CHead d (Bind Abbr) 
+v)) \to ((tau0 g d v w) \to (ty3 g c0 (TSort m) t2)))) H7))) c1 (sym_eq C c1 
+c0 H3) H4 H5 H1 H2)))) | (tau0_abst c1 d v i H1 w H2) \Rightarrow (\lambda 
+(H3: (eq C c1 c0)).(\lambda (H4: (eq T (TLRef i) (TSort m))).(\lambda (H5: 
+(eq T (lift (S i) O v) t2)).(eq_ind C c0 (\lambda (c2: C).((eq T (TLRef i) 
+(TSort m)) \to ((eq T (lift (S i) O v) t2) \to ((getl i c2 (CHead d (Bind 
+Abst) v)) \to ((tau0 g d v w) \to (ty3 g c0 (TSort m) t2)))))) (\lambda (H6: 
+(eq T (TLRef i) (TSort m))).(let H7 \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 
+(TSort m) H6) in (False_ind ((eq T (lift (S i) O v) t2) \to ((getl i c0 
+(CHead d (Bind Abst) v)) \to ((tau0 g d v w) \to (ty3 g c0 (TSort m) t2)))) 
+H7))) c1 (sym_eq C c1 c0 H3) H4 H5 H1 H2)))) | (tau0_bind b c1 v t0 t3 H1) 
+\Rightarrow (\lambda (H2: (eq C c1 c0)).(\lambda (H3: (eq T (THead (Bind b) v 
+t0) (TSort m))).(\lambda (H4: (eq T (THead (Bind b) v t3) t2)).(eq_ind C c0 
+(\lambda (c2: C).((eq T (THead (Bind b) v t0) (TSort m)) \to ((eq T (THead 
+(Bind b) v t3) t2) \to ((tau0 g (CHead c2 (Bind b) v) t0 t3) \to (ty3 g c0 
+(TSort m) t2))))) (\lambda (H5: (eq T (THead (Bind b) v t0) (TSort m))).(let 
+H6 \def (eq_ind T (THead (Bind b) v t0) (\lambda (e: T).(match e in T return 
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort m) H5) in 
+(False_ind ((eq T (THead (Bind b) v t3) t2) \to ((tau0 g (CHead c0 (Bind b) 
+v) t0 t3) \to (ty3 g c0 (TSort m) t2))) H6))) c1 (sym_eq C c1 c0 H2) H3 H4 
+H1)))) | (tau0_appl c1 v t0 t3 H1) \Rightarrow (\lambda (H2: (eq C c1 
+c0)).(\lambda (H3: (eq T (THead (Flat Appl) v t0) (TSort m))).(\lambda (H4: 
+(eq T (THead (Flat Appl) v t3) t2)).(eq_ind C c0 (\lambda (c2: C).((eq T 
+(THead (Flat Appl) v t0) (TSort m)) \to ((eq T (THead (Flat Appl) v t3) t2) 
+\to ((tau0 g c2 t0 t3) \to (ty3 g c0 (TSort m) t2))))) (\lambda (H5: (eq T 
+(THead (Flat Appl) v t0) (TSort m))).(let H6 \def (eq_ind T (THead (Flat 
+Appl) v t0) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) 
+\Rightarrow True])) I (TSort m) H5) in (False_ind ((eq T (THead (Flat Appl) v 
+t3) t2) \to ((tau0 g c0 t0 t3) \to (ty3 g c0 (TSort m) t2))) H6))) c1 (sym_eq 
+C c1 c0 H2) H3 H4 H1)))) | (tau0_cast c1 v1 v2 H1 t0 t3 H2) \Rightarrow 
+(\lambda (H3: (eq C c1 c0)).(\lambda (H4: (eq T (THead (Flat Cast) v1 t0) 
+(TSort m))).(\lambda (H5: (eq T (THead (Flat Cast) v2 t3) t2)).(eq_ind C c0 
+(\lambda (c2: C).((eq T (THead (Flat Cast) v1 t0) (TSort m)) \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 (TSort m) t2)))))) (\lambda (H6: (eq T (THead (Flat Cast) v1 
+t0) (TSort m))).(let H7 \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 _ _ _) \Rightarrow True])) I 
+(TSort m) H6) 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 (TSort m) t2)))) H7))) c1 
+(sym_eq C c1 c0 H3) H4 H5 H1 H2))))]) in (H1 (refl_equal C c0) (refl_equal T 
+(TSort m)) (refl_equal T t2))))))) (\lambda (n: nat).(\lambda (c0: 
+C).(\lambda (d: C).(\lambda (u0: T).(\lambda (H0: (getl n c0 (CHead d (Bind 
+Abbr) u0))).(\lambda (t: T).(\lambda (_: (ty3 g d u0 t)).(\lambda (H2: 
+((\forall (t2: T).((tau0 g d u0 t2) \to (ty3 g d u0 t2))))).(\lambda (t2: 
+T).(\lambda (H3: (tau0 g c0 (TLRef n) t2)).(let H4 \def (match H3 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 (TLRef n)) \to ((eq T t3 t2) \to 
+(ty3 g c0 (TLRef n) t2)))))))) with [(tau0_sort c1 n0) \Rightarrow (\lambda 
+(H4: (eq C c1 c0)).(\lambda (H5: (eq T (TSort n0) (TLRef n))).(\lambda (H6: 
+(eq T (TSort (next g n0)) t2)).(eq_ind C c0 (\lambda (_: C).((eq T (TSort n0) 
+(TLRef n)) \to ((eq T (TSort (next g n0)) t2) \to (ty3 g c0 (TLRef n) t2)))) 
+(\lambda (H7: (eq T (TSort n0) (TLRef n))).(let H8 \def (eq_ind T (TSort n0) 
+(\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) 
+\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
+False])) I (TLRef n) H7) in (False_ind ((eq T (TSort (next g n0)) t2) \to 
+(ty3 g c0 (TLRef n) t2)) H8))) c1 (sym_eq C c1 c0 H4) H5 H6)))) | (tau0_abbr 
+c1 d0 v i H4 w H5) \Rightarrow (\lambda (H6: (eq C c1 c0)).(\lambda (H7: (eq 
+T (TLRef i) (TLRef n))).(\lambda (H8: (eq T (lift (S i) O w) t2)).(eq_ind C 
+c0 (\lambda (c2: C).((eq T (TLRef i) (TLRef n)) \to ((eq T (lift (S i) O w) 
+t2) \to ((getl i c2 (CHead d0 (Bind Abbr) v)) \to ((tau0 g d0 v w) \to (ty3 g 
+c0 (TLRef n) t2)))))) (\lambda (H9: (eq T (TLRef i) (TLRef n))).(let H10 \def 
+(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with 
+[(TSort _) \Rightarrow i | (TLRef n0) \Rightarrow n0 | (THead _ _ _) 
+\Rightarrow i])) (TLRef i) (TLRef n) H9) in (eq_ind nat n (\lambda (n0: 
+nat).((eq T (lift (S n0) O w) t2) \to ((getl n0 c0 (CHead d0 (Bind Abbr) v)) 
+\to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) t2))))) (\lambda (H11: (eq T 
+(lift (S n) O w) t2)).(eq_ind T (lift (S n) O w) (\lambda (t0: T).((getl n c0 
+(CHead d0 (Bind Abbr) v)) \to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) t0)))) 
+(\lambda (H12: (getl n c0 (CHead d0 (Bind Abbr) v))).(\lambda (H13: (tau0 g 
+d0 v w)).(let H14 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda (c2: 
+C).(getl n c0 c2)) H0 (CHead d0 (Bind Abbr) v) (getl_mono c0 (CHead d (Bind 
+Abbr) u0) n H0 (CHead d0 (Bind Abbr) v) H12)) in (let H15 \def (f_equal C C 
+(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) 
+\Rightarrow d | (CHead c2 _ _) \Rightarrow c2])) (CHead d (Bind Abbr) u0) 
+(CHead d0 (Bind Abbr) v) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead 
+d0 (Bind Abbr) v) H12)) in ((let H16 \def (f_equal C T (\lambda (e: C).(match 
+e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ 
+t0) \Rightarrow t0])) (CHead d (Bind Abbr) u0) (CHead d0 (Bind Abbr) v) 
+(getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0 (Bind Abbr) v) H12)) in 
+(\lambda (H17: (eq C d d0)).(let H18 \def (eq_ind_r T v (\lambda (t0: 
+T).(getl n c0 (CHead d0 (Bind Abbr) t0))) H14 u0 H16) in (let H19 \def 
+(eq_ind_r T v (\lambda (t0: T).(tau0 g d0 t0 w)) H13 u0 H16) in (let H20 \def 
+(eq_ind_r C d0 (\lambda (c2: C).(getl n c0 (CHead c2 (Bind Abbr) u0))) H18 d 
+H17) in (let H21 \def (eq_ind_r C d0 (\lambda (c2: C).(tau0 g c2 u0 w)) H19 d 
+H17) in (ty3_abbr g n c0 d u0 H20 w (H2 w H21)))))))) H15))))) t2 H11)) i 
+(sym_eq nat i n H10)))) c1 (sym_eq C c1 c0 H6) H7 H8 H4 H5)))) | (tau0_abst 
+c1 d0 v i H4 w H5) \Rightarrow (\lambda (H6: (eq C c1 c0)).(\lambda (H7: (eq 
+T (TLRef i) (TLRef n))).(\lambda (H8: (eq T (lift (S i) O v) t2)).(eq_ind C 
+c0 (\lambda (c2: C).((eq T (TLRef i) (TLRef n)) \to ((eq T (lift (S i) O v) 
+t2) \to ((getl i c2 (CHead d0 (Bind Abst) v)) \to ((tau0 g d0 v w) \to (ty3 g 
+c0 (TLRef n) t2)))))) (\lambda (H9: (eq T (TLRef i) (TLRef n))).(let H10 \def 
+(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with 
+[(TSort _) \Rightarrow i | (TLRef n0) \Rightarrow n0 | (THead _ _ _) 
+\Rightarrow i])) (TLRef i) (TLRef n) H9) in (eq_ind nat n (\lambda (n0: 
+nat).((eq T (lift (S n0) O v) t2) \to ((getl n0 c0 (CHead d0 (Bind Abst) v)) 
+\to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) t2))))) (\lambda (H11: (eq T 
+(lift (S n) O v) t2)).(eq_ind T (lift (S n) O v) (\lambda (t0: T).((getl n c0 
+(CHead d0 (Bind Abst) v)) \to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) t0)))) 
+(\lambda (H12: (getl n c0 (CHead d0 (Bind Abst) v))).(\lambda (_: (tau0 g d0 
+v w)).(let H14 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda (c2: C).(getl 
+n c0 c2)) H0 (CHead d0 (Bind Abst) v) (getl_mono c0 (CHead d (Bind Abbr) u0) 
+n H0 (CHead d0 (Bind Abst) v) H12)) in (let H15 \def (eq_ind C (CHead d (Bind 
+Abbr) u0) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with 
+[(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return 
+(\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B return 
+(\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | 
+Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead d0 (Bind 
+Abst) v) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0 (Bind Abst) v) 
+H12)) in (False_ind (ty3 g c0 (TLRef n) (lift (S n) O v)) H15))))) t2 H11)) i 
+(sym_eq nat i n H10)))) c1 (sym_eq C c1 c0 H6) H7 H8 H4 H5)))) | (tau0_bind b 
+c1 v t0 t3 H4) \Rightarrow (\lambda (H5: (eq C c1 c0)).(\lambda (H6: (eq T 
+(THead (Bind b) v t0) (TLRef n))).(\lambda (H7: (eq T (THead (Bind b) v t3) 
+t2)).(eq_ind C c0 (\lambda (c2: C).((eq T (THead (Bind b) v t0) (TLRef n)) 
+\to ((eq T (THead (Bind b) v t3) t2) \to ((tau0 g (CHead c2 (Bind b) v) t0 
+t3) \to (ty3 g c0 (TLRef n) t2))))) (\lambda (H8: (eq T (THead (Bind b) v t0) 
+(TLRef n))).(let H9 \def (eq_ind T (THead (Bind b) v t0) (\lambda (e: 
+T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
+False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I 
+(TLRef n) H8) in (False_ind ((eq T (THead (Bind b) v t3) t2) \to ((tau0 g 
+(CHead c0 (Bind b) v) t0 t3) \to (ty3 g c0 (TLRef n) t2))) H9))) c1 (sym_eq C 
+c1 c0 H5) H6 H7 H4)))) | (tau0_appl c1 v t0 t3 H4) \Rightarrow (\lambda (H5: 
+(eq C c1 c0)).(\lambda (H6: (eq T (THead (Flat Appl) v t0) (TLRef 
+n))).(\lambda (H7: (eq T (THead (Flat Appl) v t3) t2)).(eq_ind C c0 (\lambda 
+(c2: C).((eq T (THead (Flat Appl) v t0) (TLRef n)) \to ((eq T (THead (Flat 
+Appl) v t3) t2) \to ((tau0 g c2 t0 t3) \to (ty3 g c0 (TLRef n) t2))))) 
+(\lambda (H8: (eq T (THead (Flat Appl) v t0) (TLRef n))).(let H9 \def (eq_ind 
+T (THead (Flat Appl) v t0) (\lambda (e: T).(match e in T return (\lambda (_: 
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
+(THead _ _ _) \Rightarrow True])) I (TLRef n) H8) in (False_ind ((eq T (THead 
+(Flat Appl) v t3) t2) \to ((tau0 g c0 t0 t3) \to (ty3 g c0 (TLRef n) t2))) 
+H9))) c1 (sym_eq C c1 c0 H5) H6 H7 H4)))) | (tau0_cast c1 v1 v2 H4 t0 t3 H5) 
+\Rightarrow (\lambda (H6: (eq C c1 c0)).(\lambda (H7: (eq T (THead (Flat 
+Cast) v1 t0) (TLRef n))).(\lambda (H8: (eq T (THead (Flat Cast) v2 t3) 
+t2)).(eq_ind C c0 (\lambda (c2: C).((eq T (THead (Flat Cast) v1 t0) (TLRef 
+n)) \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 (TLRef n) t2)))))) (\lambda (H9: (eq T 
+(THead (Flat Cast) v1 t0) (TLRef n))).(let H10 \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 _ _ _) 
+\Rightarrow True])) I (TLRef n) H9) 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 (TLRef 
+n) t2)))) H10))) c1 (sym_eq C c1 c0 H6) H7 H8 H4 H5))))]) in (H4 (refl_equal 
+C c0) (refl_equal T (TLRef n)) (refl_equal T t2))))))))))))) (\lambda (n: 
+nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (H0: (getl n 
+c0 (CHead d (Bind Abst) u0))).(\lambda (t: T).(\lambda (H1: (ty3 g d u0 
+t)).(\lambda (_: ((\forall (t2: T).((tau0 g d u0 t2) \to (ty3 g d u0 
+t2))))).(\lambda (t2: T).(\lambda (H3: (tau0 g c0 (TLRef n) t2)).(let H4 \def 
+(match H3 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 (TLRef n)) \to 
+((eq T t3 t2) \to (ty3 g c0 (TLRef n) t2)))))))) with [(tau0_sort c1 n0) 
+\Rightarrow (\lambda (H4: (eq C c1 c0)).(\lambda (H5: (eq T (TSort n0) (TLRef 
+n))).(\lambda (H6: (eq T (TSort (next g n0)) t2)).(eq_ind C c0 (\lambda (_: 
+C).((eq T (TSort n0) (TLRef n)) \to ((eq T (TSort (next g n0)) t2) \to (ty3 g 
+c0 (TLRef n) t2)))) (\lambda (H7: (eq T (TSort n0) (TLRef n))).(let H8 \def 
+(eq_ind T (TSort n0) (\lambda (e: T).(match e in T return (\lambda (_: 
+T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | 
+(THead _ _ _) \Rightarrow False])) I (TLRef n) H7) in (False_ind ((eq T 
+(TSort (next g n0)) t2) \to (ty3 g c0 (TLRef n) t2)) H8))) c1 (sym_eq C c1 c0 
+H4) H5 H6)))) | (tau0_abbr c1 d0 v i H4 w H5) \Rightarrow (\lambda (H6: (eq C 
+c1 c0)).(\lambda (H7: (eq T (TLRef i) (TLRef n))).(\lambda (H8: (eq T (lift 
+(S i) O w) t2)).(eq_ind C c0 (\lambda (c2: C).((eq T (TLRef i) (TLRef n)) \to 
+((eq T (lift (S i) O w) t2) \to ((getl i c2 (CHead d0 (Bind Abbr) v)) \to 
+((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) t2)))))) (\lambda (H9: (eq T (TLRef 
+i) (TLRef n))).(let H10 \def (f_equal T nat (\lambda (e: T).(match e in T 
+return (\lambda (_: T).nat) with [(TSort _) \Rightarrow i | (TLRef n0) 
+\Rightarrow n0 | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef n) H9) in 
+(eq_ind nat n (\lambda (n0: nat).((eq T (lift (S n0) O w) t2) \to ((getl n0 
+c0 (CHead d0 (Bind Abbr) v)) \to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) 
+t2))))) (\lambda (H11: (eq T (lift (S n) O w) t2)).(eq_ind T (lift (S n) O w) 
+(\lambda (t0: T).((getl n c0 (CHead d0 (Bind Abbr) v)) \to ((tau0 g d0 v w) 
+\to (ty3 g c0 (TLRef n) t0)))) (\lambda (H12: (getl n c0 (CHead d0 (Bind 
+Abbr) v))).(\lambda (_: (tau0 g d0 v w)).(let H14 \def (eq_ind C (CHead d 
+(Bind Abst) u0) (\lambda (c2: C).(getl n c0 c2)) H0 (CHead d0 (Bind Abbr) v) 
+(getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead d0 (Bind Abbr) v) H12)) in 
+(let H15 \def (eq_ind C (CHead d (Bind Abst) u0) (\lambda (ee: C).(match ee 
+in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead 
+_ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) 
+\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow 
+False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) 
+\Rightarrow False])])) I (CHead d0 (Bind Abbr) v) (getl_mono c0 (CHead d 
+(Bind Abst) u0) n H0 (CHead d0 (Bind Abbr) v) H12)) in (False_ind (ty3 g c0 
+(TLRef n) (lift (S n) O w)) H15))))) t2 H11)) i (sym_eq nat i n H10)))) c1 
+(sym_eq C c1 c0 H6) H7 H8 H4 H5)))) | (tau0_abst c1 d0 v i H4 w H5) 
+\Rightarrow (\lambda (H6: (eq C c1 c0)).(\lambda (H7: (eq T (TLRef i) (TLRef 
+n))).(\lambda (H8: (eq T (lift (S i) O v) t2)).(eq_ind C c0 (\lambda (c2: 
+C).((eq T (TLRef i) (TLRef n)) \to ((eq T (lift (S i) O v) t2) \to ((getl i 
+c2 (CHead d0 (Bind Abst) v)) \to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) 
+t2)))))) (\lambda (H9: (eq T (TLRef i) (TLRef n))).(let H10 \def (f_equal T 
+nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort _) 
+\Rightarrow i | (TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow i])) 
+(TLRef i) (TLRef n) H9) in (eq_ind nat n (\lambda (n0: nat).((eq T (lift (S 
+n0) O v) t2) \to ((getl n0 c0 (CHead d0 (Bind Abst) v)) \to ((tau0 g d0 v w) 
+\to (ty3 g c0 (TLRef n) t2))))) (\lambda (H11: (eq T (lift (S n) O v) 
+t2)).(eq_ind T (lift (S n) O v) (\lambda (t0: T).((getl n c0 (CHead d0 (Bind 
+Abst) v)) \to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) t0)))) (\lambda (H12: 
+(getl n c0 (CHead d0 (Bind Abst) v))).(\lambda (H13: (tau0 g d0 v w)).(let 
+H14 \def (eq_ind C (CHead d (Bind Abst) u0) (\lambda (c2: C).(getl n c0 c2)) 
+H0 (CHead d0 (Bind Abst) v) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 
+(CHead d0 (Bind Abst) v) H12)) in (let H15 \def (f_equal C C (\lambda (e: 
+C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | 
+(CHead c2 _ _) \Rightarrow c2])) (CHead d (Bind Abst) u0) (CHead d0 (Bind 
+Abst) v) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead d0 (Bind Abst) v) 
+H12)) in ((let H16 \def (f_equal C T (\lambda (e: C).(match e in C return 
+(\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t0) 
+\Rightarrow t0])) (CHead d (Bind Abst) u0) (CHead d0 (Bind Abst) v) 
+(getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead d0 (Bind Abst) v) H12)) in 
+(\lambda (H17: (eq C d d0)).(let H18 \def (eq_ind_r T v (\lambda (t0: 
+T).(getl n c0 (CHead d0 (Bind Abst) t0))) H14 u0 H16) in (let H19 \def 
+(eq_ind_r T v (\lambda (t0: T).(tau0 g d0 t0 w)) H13 u0 H16) in (eq_ind T u0 
+(\lambda (t0: T).(ty3 g c0 (TLRef n) (lift (S n) O t0))) (let H20 \def 
+(eq_ind_r C d0 (\lambda (c2: C).(getl n c0 (CHead c2 (Bind Abst) u0))) H18 d 
+H17) in (let H21 \def (eq_ind_r C d0 (\lambda (c2: C).(tau0 g c2 u0 w)) H19 d 
+H17) in (ty3_abst g n c0 d u0 H20 t H1))) v H16))))) H15))))) t2 H11)) i 
+(sym_eq nat i n H10)))) c1 (sym_eq C c1 c0 H6) H7 H8 H4 H5)))) | (tau0_bind b 
+c1 v t0 t3 H4) \Rightarrow (\lambda (H5: (eq C c1 c0)).(\lambda (H6: (eq T 
+(THead (Bind b) v t0) (TLRef n))).(\lambda (H7: (eq T (THead (Bind b) v t3) 
+t2)).(eq_ind C c0 (\lambda (c2: C).((eq T (THead (Bind b) v t0) (TLRef n)) 
+\to ((eq T (THead (Bind b) v t3) t2) \to ((tau0 g (CHead c2 (Bind b) v) t0 
+t3) \to (ty3 g c0 (TLRef n) t2))))) (\lambda (H8: (eq T (THead (Bind b) v t0) 
+(TLRef n))).(let H9 \def (eq_ind T (THead (Bind b) v t0) (\lambda (e: 
+T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
+False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I 
+(TLRef n) H8) in (False_ind ((eq T (THead (Bind b) v t3) t2) \to ((tau0 g 
+(CHead c0 (Bind b) v) t0 t3) \to (ty3 g c0 (TLRef n) t2))) H9))) c1 (sym_eq C 
+c1 c0 H5) H6 H7 H4)))) | (tau0_appl c1 v t0 t3 H4) \Rightarrow (\lambda (H5: 
+(eq C c1 c0)).(\lambda (H6: (eq T (THead (Flat Appl) v t0) (TLRef 
+n))).(\lambda (H7: (eq T (THead (Flat Appl) v t3) t2)).(eq_ind C c0 (\lambda 
+(c2: C).((eq T (THead (Flat Appl) v t0) (TLRef n)) \to ((eq T (THead (Flat 
+Appl) v t3) t2) \to ((tau0 g c2 t0 t3) \to (ty3 g c0 (TLRef n) t2))))) 
+(\lambda (H8: (eq T (THead (Flat Appl) v t0) (TLRef n))).(let H9 \def (eq_ind 
+T (THead (Flat Appl) v t0) (\lambda (e: T).(match e in T return (\lambda (_: 
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
+(THead _ _ _) \Rightarrow True])) I (TLRef n) H8) in (False_ind ((eq T (THead 
+(Flat Appl) v t3) t2) \to ((tau0 g c0 t0 t3) \to (ty3 g c0 (TLRef n) t2))) 
+H9))) c1 (sym_eq C c1 c0 H5) H6 H7 H4)))) | (tau0_cast c1 v1 v2 H4 t0 t3 H5) 
+\Rightarrow (\lambda (H6: (eq C c1 c0)).(\lambda (H7: (eq T (THead (Flat 
+Cast) v1 t0) (TLRef n))).(\lambda (H8: (eq T (THead (Flat Cast) v2 t3) 
+t2)).(eq_ind C c0 (\lambda (c2: C).((eq T (THead (Flat Cast) v1 t0) (TLRef 
+n)) \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 (TLRef n) t2)))))) (\lambda (H9: (eq T 
+(THead (Flat Cast) v1 t0) (TLRef n))).(let H10 \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 _ _ _) 
+\Rightarrow True])) I (TLRef n) H9) 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 (TLRef 
+n) t2)))) H10))) c1 (sym_eq C c1 c0 H6) H7 H8 H4 H5))))]) in (H4 (refl_equal 
+C c0) (refl_equal T (TLRef n)) (refl_equal T t2))))))))))))) (\lambda (c0: 
+C).(\lambda (u0: T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 u0 t)).(\lambda 
+(_: ((\forall (t2: T).((tau0 g c0 u0 t2) \to (ty3 g c0 u0 t2))))).(\lambda 
+(b: B).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (ty3 g (CHead c0 (Bind 
+b) u0) t2 t3)).(\lambda (H3: ((\forall (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 
+\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 (_: 
+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 
+(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 
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) w 
+v) H10) in (False_ind ((eq T (lift (S i) O w0) t2) \to ((getl i c0 (CHead d 
+(Bind Abbr) v0)) \to ((tau0 g d v0 w0) \to (ty3 g c0 (THead (Flat Appl) w v) 
+t2)))) H11))) c1 (sym_eq C c1 c0 H7) H8 H9 H5 H6)))) | (tau0_abst 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 v0) 
+t2)).(eq_ind C c0 (\lambda (c2: C).((eq T (TLRef i) (THead (Flat Appl) w v)) 
+\to ((eq T (lift (S i) O v0) t2) \to ((getl i c2 (CHead d (Bind Abst) 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 (\lambda (_: T).Prop) with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) 
+\Rightarrow False])) I (THead (Flat Appl) w v) H10) in (False_ind ((eq T 
+(lift (S i) O v0) t2) \to ((getl i c0 (CHead d (Bind Abst) v0)) \to ((tau0 g 
+d v0 w0) \to (ty3 g c0 (THead (Flat Appl) w v) t2)))) H11))) c1 (sym_eq C c1 
+c0 H7) H8 H9 H5 H6)))) | (tau0_bind b c1 v0 t0 t3 H5) \Rightarrow (\lambda 
+(H6: (eq C c1 c0)).(\lambda (H7: (eq T (THead (Bind b) v0 t0) (THead (Flat 
+Appl) w v))).(\lambda (H8: (eq T (THead (Bind b) v0 t3) t2)).(eq_ind C c0 
+(\lambda (c2: C).((eq T (THead (Bind b) v0 t0) (THead (Flat Appl) w v)) \to 
+((eq T (THead (Bind b) v0 t3) t2) \to ((tau0 g (CHead c2 (Bind b) v0) t0 t3) 
+\to (ty3 g c0 (THead (Flat Appl) w v) t2))))) (\lambda (H9: (eq T (THead 
+(Bind b) v0 t0) (THead (Flat Appl) w v))).(let H10 \def (eq_ind T (THead 
+(Bind b) v0 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 True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) w v) 
+H9) in (False_ind ((eq T (THead (Bind b) v0 t3) t2) \to ((tau0 g (CHead c0 
+(Bind b) v0) t0 t3) \to (ty3 g c0 (THead (Flat Appl) w v) t2))) H10))) c1 
+(sym_eq C c1 c0 H6) H7 H8 H5)))) | (tau0_appl c1 v0 t0 t3 H5) \Rightarrow 
+(\lambda (H6: (eq C c1 c0)).(\lambda (H7: (eq T (THead (Flat Appl) v0 t0) 
+(THead (Flat Appl) w v))).(\lambda (H8: (eq T (THead (Flat Appl) v0 t3) 
+t2)).(eq_ind C c0 (\lambda (c2: C).((eq T (THead (Flat Appl) v0 t0) (THead 
+(Flat Appl) w v)) \to ((eq T (THead (Flat Appl) v0 t3) t2) \to ((tau0 g c2 t0 
+t3) \to (ty3 g c0 (THead (Flat Appl) w v) t2))))) (\lambda (H9: (eq T (THead 
+(Flat Appl) v0 t0) (THead (Flat Appl) w v))).(let H10 \def (f_equal T T 
+(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
+\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t4) \Rightarrow t4])) 
+(THead (Flat Appl) v0 t0) (THead (Flat Appl) w v) H9) in ((let H11 \def 
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with 
+[(TSort _) \Rightarrow v0 | (TLRef _) \Rightarrow v0 | (THead _ t4 _) 
+\Rightarrow t4])) (THead (Flat Appl) v0 t0) (THead (Flat Appl) w v) H9) in 
+(eq_ind T w (\lambda (t4: T).((eq T t0 v) \to ((eq T (THead (Flat Appl) t4 
+t3) t2) \to ((tau0 g c0 t0 t3) \to (ty3 g c0 (THead (Flat Appl) w v) t2))))) 
+(\lambda (H12: (eq T t0 v)).(eq_ind T v (\lambda (t4: T).((eq T (THead (Flat 
+Appl) w t3) t2) \to ((tau0 g c0 t4 t3) \to (ty3 g c0 (THead (Flat Appl) w v) 
+t2)))) (\lambda (H13: (eq T (THead (Flat Appl) w t3) t2)).(eq_ind T (THead 
+(Flat Appl) w t3) (\lambda (t4: T).((tau0 g c0 v t3) \to (ty3 g c0 (THead 
+(Flat Appl) w v) t4))) (\lambda (H14: (tau0 g c0 v t3)).(let H_y \def (H3 t3 
+H14) in (let H15 \def (ty3_unique g c0 v t3 H_y (THead (Bind Abst) u0 t) H2) 
+in (ex_ind T (\lambda (t4: T).(ty3 g c0 t3 t4)) (ty3 g c0 (THead (Flat Appl) 
+w v) (THead (Flat Appl) w t3)) (\lambda (x: T).(\lambda (H16: (ty3 g c0 t3 
+x)).(ex_ind T (\lambda (t4: T).(ty3 g c0 u0 t4)) (ty3 g c0 (THead (Flat Appl) 
+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_s c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t)) 
+(THead (Flat Appl) w t3) (pc3_thin_dx c0 t3 (THead (Bind Abst) u0 t) H15 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 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 
+(THead (Flat Appl) v t6) t4) \to ((tau0 g c2 t5 t6) \to (ty3 g c0 (THead 
+(Flat Cast) t3 t2) t4))))) (\lambda (H9: (eq T (THead (Flat Appl) v t5) 
+(THead (Flat Cast) t3 t2))).(let H10 \def (eq_ind T (THead (Flat Appl) 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 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) 
+H9) in (False_ind ((eq T (THead (Flat Appl) v t6) t4) \to ((tau0 g c0 t5 t6) 
+\to (ty3 g c0 (THead (Flat Cast) t3 t2) t4))) H10))) c1 (sym_eq C c1 c0 H6) 
+H7 H8 H5)))) | (tau0_cast c1 v1 v2 H5 t5 t6 H6) \Rightarrow (\lambda (H7: (eq 
+C c1 c0)).(\lambda (H8: (eq T (THead (Flat Cast) v1 t5) (THead (Flat Cast) t3 
+t2))).(\lambda (H9: (eq T (THead (Flat Cast) v2 t6) t4)).(eq_ind C c0 
+(\lambda (c2: C).((eq T (THead (Flat Cast) v1 t5) (THead (Flat Cast) t3 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 (Flat Cast) t3 t2) t4)))))) (\lambda (H10: (eq 
+T (THead (Flat Cast) v1 t5) (THead (Flat Cast) t3 t2))).(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 _ _ t) 
+\Rightarrow t])) (THead (Flat Cast) v1 t5) (THead (Flat Cast) t3 t2) H10) in 
+((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 _ t 
+_) \Rightarrow t])) (THead (Flat Cast) v1 t5) (THead (Flat Cast) t3 t2) H10) 
+in (eq_ind T t3 (\lambda (t: T).((eq T t5 t2) \to ((eq T (THead (Flat Cast) 
+v2 t6) t4) \to ((tau0 g c0 t v2) \to ((tau0 g c0 t5 t6) \to (ty3 g c0 (THead 
+(Flat Cast) t3 t2) t4)))))) (\lambda (H13: (eq T t5 t2)).(eq_ind T t2 
+(\lambda (t: T).((eq T (THead (Flat Cast) v2 t6) t4) \to ((tau0 g c0 t3 v2) 
+\to ((tau0 g c0 t t6) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4))))) 
+(\lambda (H14: (eq T (THead (Flat Cast) v2 t6) t4)).(eq_ind T (THead (Flat 
+Cast) v2 t6) (\lambda (t: T).((tau0 g c0 t3 v2) \to ((tau0 g c0 t2 t6) \to 
+(ty3 g c0 (THead (Flat Cast) t3 t2) t)))) (\lambda (H15: (tau0 g c0 t3 
+v2)).(\lambda (H16: (tau0 g c0 t2 t6)).(let H_y \def (H1 t6 H16) in (let H_y0 
+\def (H3 v2 H15) in (let H17 \def (ty3_unique g c0 t2 t6 H_y t3 H0) in 
+(ex_ind T (\lambda (t: T).(ty3 g c0 v2 t)) (ty3 g c0 (THead (Flat Cast) t3 
+t2) (THead (Flat Cast) v2 t6)) (\lambda (x: T).(\lambda (H18: (ty3 g c0 v2 
+x)).(ex_ind T (\lambda (t: T).(ty3 g c0 t6 t)) (ty3 g c0 (THead (Flat Cast) 
+t3 t2) (THead (Flat Cast) v2 t6)) (\lambda (x0: T).(\lambda (H19: (ty3 g c0 
+t6 x0)).(ty3_conv g c0 (THead (Flat Cast) v2 t6) v2 (ty3_cast g c0 t6 v2 
+(ty3_sconv g c0 t6 x0 H19 t3 v2 H_y0 H17) x H18) (THead (Flat Cast) t3 t2) t3 
+(ty3_cast g c0 t2 t3 H0 v2 H_y0) (pc3_s c0 t3 (THead (Flat Cast) v2 t6) 
+(pc3_pr2_u c0 t6 (THead (Flat Cast) v2 t6) (pr2_free c0 (THead (Flat Cast) v2 
+t6) t6 (pr0_epsilon t6 t6 (pr0_refl t6) v2)) t3 H17))))) (ty3_correct g c0 t2 
+t6 H_y)))) (ty3_correct g c0 t3 v2 H_y0))))))) t4 H14)) t5 (sym_eq T t5 t2 
+H13))) v1 (sym_eq T v1 t3 H12))) H11))) c1 (sym_eq C c1 c0 H7) H8 H9 H5 
+H6))))]) in (H5 (refl_equal C c0) (refl_equal T (THead (Flat Cast) t3 t2)) 
+(refl_equal T t4))))))))))))) c u t1 H))))).
+