component: pr0

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr0/dec.ma b/matita/matita/contribs/lambdadelta/basic_1/pr0/dec.ma
index c28504beec6e582dd5eacce1a1939a48d5ddab6a..96d70c708f8402008a5027f9e8852946ef4910f4 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_1/pr0/dec.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/pr0/dec.ma
@@ -14,13 +14,13 @@
 
 (* This file was automatically generated: do not edit *********************)
 
 
 (* This file was automatically generated: do not edit *********************)
 

-include "Basic-1/pr0/fwd.ma".
+include "basic_1/pr0/props.ma".

 
 

-include "Basic-1/subst0/dec.ma".
+include "basic_1/subst0/dec.ma".

 
 

-include "Basic-1/T/dec.ma".
+include "basic_1/T/dec.ma".

 
 

-include "Basic-1/T/props.ma".
+include "basic_1/T/props.ma".

 
 theorem nf0_dec:
  \forall (t1: T).(or (\forall (t2: T).((pr0 t1 t2) \to (eq T t1 t2))) (ex2 T 
 
 theorem nf0_dec:
  \forall (t1: T).(or (\forall (t2: T).((pr0 t1 t2) \to (eq T t1 t2))) (ex2 T 


@@ -67,30 +67,30 @@ O) O x))).(ex_intro2 T (\lambda (t2: T).((eq T (THead (Bind Abbr) t t0) t2)
 \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abbr) t t0) 
 t2)) (THead (Bind Abbr) t (lift (S O) O x)) (\lambda (H4: (eq T (THead (Bind 
 Abbr) t t0) (THead (Bind Abbr) t (lift (S O) O x)))).(\lambda (P: Prop).(let 
 \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abbr) t t0) 
 t2)) (THead (Bind Abbr) t (lift (S O) O x)) (\lambda (H4: (eq T (THead (Bind 
 Abbr) t t0) (THead (Bind Abbr) t (lift (S O) O x)))).(\lambda (P: Prop).(let 

-H5 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) 
-with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t2) 
-\Rightarrow t2])) (THead (Bind Abbr) t t0) (THead (Bind Abbr) t (lift (S O) O 
-x)) H4) in (let H6 \def (eq_ind T t0 (\lambda (t2: T).(subst0 O t t2 (lift (S 
-O) O x))) H3 (lift (S O) O x) H5) in (subst0_refl t (lift (S O) O x) O H6 
-P))))) (pr0_delta t t (pr0_refl t) t0 t0 (pr0_refl t0) (lift (S O) O x) H3))) 
-(\lambda (H3: (eq T t0 (lift (S O) O x))).(eq_ind_r T (lift (S O) O x) 
-(\lambda (t2: T).(ex2 T (\lambda (t3: T).((eq T (THead (Bind Abbr) t t2) t3) 
-\to (\forall (P: Prop).P))) (\lambda (t3: T).(pr0 (THead (Bind Abbr) t t2) 
-t3)))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Bind Abbr) t (lift (S O) 
-O x)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind 
-Abbr) t (lift (S O) O x)) t2)) x (\lambda (H4: (eq T (THead (Bind Abbr) t 
-(lift (S O) O x)) x)).(\lambda (P: Prop).(thead_x_lift_y_y (Bind Abbr) x t (S 
-O) O H4 P))) (pr0_zeta Abbr not_abbr_abst x x (pr0_refl x) t)) t0 H3)) H2))) 
-H1)))) (let H1 \def H in (or_ind (\forall (t2: T).((pr0 t t2) \to (eq T t 
-t2))) (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) 
-(\lambda (t2: T).(pr0 t t2))) (or (\forall (t2: T).((pr0 (THead (Bind Abst) t 
-t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T (\lambda (t2: T).((eq 
-T (THead (Bind Abst) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
-T).(pr0 (THead (Bind Abst) t t0) t2)))) (\lambda (H2: ((\forall (t2: T).((pr0 
-t t2) \to (eq T t t2))))).(let H3 \def H0 in (or_ind (\forall (t2: T).((pr0 
-t0 t2) \to (eq T t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall 
-(P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2))) (or (\forall (t2: T).((pr0 
-(THead (Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T 
+H5 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t0 
+| (TLRef _) \Rightarrow t0 | (THead _ _ t2) \Rightarrow t2])) (THead (Bind 
+Abbr) t t0) (THead (Bind Abbr) t (lift (S O) O x)) H4) in (let H6 \def 
+(eq_ind T t0 (\lambda (t2: T).(subst0 O t t2 (lift (S O) O x))) H3 (lift (S 
+O) O x) H5) in (subst0_refl t (lift (S O) O x) O H6 P))))) (pr0_delta t t 
+(pr0_refl t) t0 t0 (pr0_refl t0) (lift (S O) O x) H3))) (\lambda (H3: (eq T 
+t0 (lift (S O) O x))).(eq_ind_r T (lift (S O) O x) (\lambda (t2: T).(ex2 T 
+(\lambda (t3: T).((eq T (THead (Bind Abbr) t t2) t3) \to (\forall (P: 
+Prop).P))) (\lambda (t3: T).(pr0 (THead (Bind Abbr) t t2) t3)))) (ex_intro2 T 
+(\lambda (t2: T).((eq T (THead (Bind Abbr) t (lift (S O) O x)) t2) \to 
+(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abbr) t (lift (S 
+O) O x)) t2)) x (\lambda (H4: (eq T (THead (Bind Abbr) t (lift (S O) O x)) 
+x)).(\lambda (P: Prop).(thead_x_lift_y_y (Bind Abbr) x t (S O) O H4 P))) 
+(pr0_zeta Abbr not_abbr_abst x x (pr0_refl x) t)) t0 H3)) H2))) H1)))) (let 
+H1 \def H in (or_ind (\forall (t2: T).((pr0 t t2) \to (eq T t t2))) (ex2 T 
+(\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
+T).(pr0 t t2))) (or (\forall (t2: T).((pr0 (THead (Bind Abst) t t0) t2) \to 
+(eq T (THead (Bind Abst) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead 
+(Bind Abst) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 
+(THead (Bind Abst) t t0) t2)))) (\lambda (H2: ((\forall (t2: T).((pr0 t t2) 
+\to (eq T t t2))))).(let H3 \def H0 in (or_ind (\forall (t2: T).((pr0 t0 t2) 
+\to (eq T t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: 
+Prop).P))) (\lambda (t2: T).(pr0 t0 t2))) (or (\forall (t2: T).((pr0 (THead 
+(Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T 

 (\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: 
 Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)))) (\lambda 
 (H4: ((\forall (t2: T).((pr0 t0 t2) \to (eq T t0 t2))))).(or_introl (\forall 
 (\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: 
 Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)))) (\lambda 
 (H4: ((\forall (t2: T).((pr0 t0 t2) \to (eq T t0 t2))))).(or_introl (\forall 


@@ -124,30 +124,29 @@ Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2))) (ex_intro2 T
 (\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: 
 Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)) (THead (Bind 
 Abst) t x) (\lambda (H7: (eq T (THead (Bind Abst) t t0) (THead (Bind Abst) t 
 (\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: 
 Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)) (THead (Bind 
 Abst) t x) (\lambda (H7: (eq T (THead (Bind Abst) t t0) (THead (Bind Abst) t 

-x))).(\lambda (P: Prop).(let H8 \def (f_equal T T (\lambda (e: T).(match e in 
-T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) 
-\Rightarrow t0 | (THead _ _ t2) \Rightarrow t2])) (THead (Bind Abst) t t0) 
-(THead (Bind Abst) t x) H7) in (let H9 \def (eq_ind_r T x (\lambda (t2: 
-T).(pr0 t0 t2)) H6 t0 H8) in (let H10 \def (eq_ind_r T x (\lambda (t2: 
-T).((eq T t0 t2) \to (\forall (P0: Prop).P0))) H5 t0 H8) in (H10 (refl_equal 
-T t0) P)))))) (pr0_comp t t (pr0_refl t) t0 x H6 (Bind Abst))))))) H4)) H3))) 
-(\lambda (H2: (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: 
-Prop).P))) (\lambda (t2: T).(pr0 t t2)))).(ex2_ind T (\lambda (t2: T).((eq T 
-t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2)) (or (\forall 
-(t2: T).((pr0 (THead (Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) 
-t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to 
-(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)))) 
-(\lambda (x: T).(\lambda (H3: (((eq T t x) \to (\forall (P: 
-Prop).P)))).(\lambda (H4: (pr0 t x)).(or_intror (\forall (t2: T).((pr0 (THead 
+x))).(\lambda (P: Prop).(let H8 \def (f_equal T T (\lambda (e: T).(match e 
+with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t2) 
+\Rightarrow t2])) (THead (Bind Abst) t t0) (THead (Bind Abst) t x) H7) in 
+(let H9 \def (eq_ind_r T x (\lambda (t2: T).(pr0 t0 t2)) H6 t0 H8) in (let 
+H10 \def (eq_ind_r T x (\lambda (t2: T).((eq T t0 t2) \to (\forall (P0: 
+Prop).P0))) H5 t0 H8) in (H10 (refl_equal T t0) P)))))) (pr0_comp t t 
+(pr0_refl t) t0 x H6 (Bind Abst))))))) H4)) H3))) (\lambda (H2: (ex2 T 
+(\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
+T).(pr0 t t2)))).(ex2_ind T (\lambda (t2: T).((eq T t t2) \to (\forall (P: 
+Prop).P))) (\lambda (t2: T).(pr0 t t2)) (or (\forall (t2: T).((pr0 (THead 

 (Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T 
 (\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: 
 (Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T 
 (\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: 

-Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2))) (ex_intro2 T 
-(\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: 
-Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)) (THead (Bind 
-Abst) x t0) (\lambda (H5: (eq T (THead (Bind Abst) t t0) (THead (Bind Abst) x 
-t0))).(\lambda (P: Prop).(let H6 \def (f_equal T T (\lambda (e: T).(match e 
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t | (TLRef _) 
-\Rightarrow t | (THead _ t2 _) \Rightarrow t2])) (THead (Bind Abst) t t0) 
+Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)))) (\lambda (x: 
+T).(\lambda (H3: (((eq T t x) \to (\forall (P: Prop).P)))).(\lambda (H4: (pr0 
+t x)).(or_intror (\forall (t2: T).((pr0 (THead (Bind Abst) t t0) t2) \to (eq 
+T (THead (Bind Abst) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind 
+Abst) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead 
+(Bind Abst) t t0) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Bind 
+Abst) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead 
+(Bind Abst) t t0) t2)) (THead (Bind Abst) x t0) (\lambda (H5: (eq T (THead 
+(Bind Abst) t t0) (THead (Bind Abst) x t0))).(\lambda (P: Prop).(let H6 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t | (TLRef 
+_) \Rightarrow t | (THead _ t2 _) \Rightarrow t2])) (THead (Bind Abst) t t0) 

 (THead (Bind Abst) x t0) H5) in (let H7 \def (eq_ind_r T x (\lambda (t2: 
 T).(pr0 t t2)) H4 t H6) in (let H8 \def (eq_ind_r T x (\lambda (t2: T).((eq T 
 t t2) \to (\forall (P0: Prop).P0))) H3 t H6) in (H8 (refl_equal T t) P)))))) 
 (THead (Bind Abst) x t0) H5) in (let H7 \def (eq_ind_r T x (\lambda (t2: 
 T).(pr0 t t2)) H4 t H6) in (let H8 \def (eq_ind_r T x (\lambda (t2: T).((eq T 
 t t2) \to (\forall (P0: Prop).P0))) H3 t H6) in (H8 (refl_equal T t) P)))))) 


@@ -202,7 +201,7 @@ T).(eq T t2 (THead (Bind Void) t3 t0))) H14 t H_y0) in (eq_ind_r T (THead
 H10 \def (eq_ind T t0 (\lambda (t3: T).(subst0 O t t3 (lift (S O) O x))) H3 
 (lift (S O) O t2) H_y) in (eq_ind_r T (lift (S O) O t2) (\lambda (t3: T).(eq 
 T (THead (Bind Void) t t3) t2)) (subst0_gen_lift_false t2 t (lift (S O) O x) 
 H10 \def (eq_ind T t0 (\lambda (t3: T).(subst0 O t t3 (lift (S O) O x))) H3 
 (lift (S O) O t2) H_y) in (eq_ind_r T (lift (S O) O t2) (\lambda (t3: T).(eq 
 T (THead (Bind Void) t t3) t2)) (subst0_gen_lift_false t2 t (lift (S O) O x) 

-(S O) O O (le_n O) (eq_ind_r nat (plus (S O) O) (\lambda (n: nat).(lt O n)) 
+(S O) O O (le_O_n O) (eq_ind_r nat (plus (S O) O) (\lambda (n: nat).(lt O n)) 

 (le_n (plus (S O) O)) (plus O (S O)) (plus_sym O (S O))) H10 (eq T (THead 
 (Bind Void) t (lift (S O) O t2)) t2)) t0 H_y)))) (pr0_gen_void t t0 t2 
 H8)))))) (\lambda (H7: (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: 
 (le_n (plus (S O) O)) (plus O (S O)) (plus_sym O (S O))) H10 (eq T (THead 
 (Bind Void) t (lift (S O) O t2)) t2)) t0 H_y)))) (pr0_gen_void t t0 t2 
 H8)))))) (\lambda (H7: (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: 


@@ -220,172 +219,228 @@ Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2))) (ex_intro2 T
 Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)) (THead (Bind 
 Void) t x0) (\lambda (H10: (eq T (THead (Bind Void) t t0) (THead (Bind Void) 
 t x0))).(\lambda (P: Prop).(let H11 \def (f_equal T T (\lambda (e: T).(match 
 Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)) (THead (Bind 
 Void) t x0) (\lambda (H10: (eq T (THead (Bind Void) t t0) (THead (Bind Void) 
 t x0))).(\lambda (P: Prop).(let H11 \def (f_equal T T (\lambda (e: T).(match 

-e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) 
-\Rightarrow t0 | (THead _ _ t2) \Rightarrow t2])) (THead (Bind Void) t t0) 
-(THead (Bind Void) t x0) H10) in (let H12 \def (eq_ind_r T x0 (\lambda (t2: 
-T).(pr0 t0 t2)) H9 t0 H11) in (let H13 \def (eq_ind_r T x0 (\lambda (t2: 
-T).((eq T t0 t2) \to (\forall (P0: Prop).P0))) H8 t0 H11) in (H13 (refl_equal 
-T t0) P)))))) (pr0_comp t t (pr0_refl t) t0 x0 H9 (Bind Void))))))) H7)) 
-H6))) (\lambda (H5: (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: 
-Prop).P))) (\lambda (t2: T).(pr0 t t2)))).(ex2_ind T (\lambda (t2: T).((eq T 
-t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2)) (or (\forall 
-(t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) 
-t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to 
-(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)))) 
-(\lambda (x0: T).(\lambda (H6: (((eq T t x0) \to (\forall (P: 
-Prop).P)))).(\lambda (H7: (pr0 t x0)).(or_intror (\forall (t2: T).((pr0 
-(THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T 
-(\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: 
-Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2))) (ex_intro2 T 
+e with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t2) 
+\Rightarrow t2])) (THead (Bind Void) t t0) (THead (Bind Void) t x0) H10) in 
+(let H12 \def (eq_ind_r T x0 (\lambda (t2: T).(pr0 t0 t2)) H9 t0 H11) in (let 
+H13 \def (eq_ind_r T x0 (\lambda (t2: T).((eq T t0 t2) \to (\forall (P0: 
+Prop).P0))) H8 t0 H11) in (H13 (refl_equal T t0) P)))))) (pr0_comp t t 
+(pr0_refl t) t0 x0 H9 (Bind Void))))))) H7)) H6))) (\lambda (H5: (ex2 T 
+(\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
+T).(pr0 t t2)))).(ex2_ind T (\lambda (t2: T).((eq T t t2) \to (\forall (P: 
+Prop).P))) (\lambda (t2: T).(pr0 t t2)) (or (\forall (t2: T).((pr0 (THead 
+(Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T 

 (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: 
 (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: 

-Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)) (THead (Bind 
-Void) x0 t0) (\lambda (H8: (eq T (THead (Bind Void) t t0) (THead (Bind Void) 
-x0 t0))).(\lambda (P: Prop).(let H9 \def (f_equal T T (\lambda (e: T).(match 
-e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t | (TLRef _) 
-\Rightarrow t | (THead _ t2 _) \Rightarrow t2])) (THead (Bind Void) t t0) 
-(THead (Bind Void) x0 t0) H8) in (let H10 \def (eq_ind_r T x0 (\lambda (t2: 
-T).(pr0 t t2)) H7 t H9) in (let H11 \def (eq_ind_r T x0 (\lambda (t2: T).((eq 
-T t t2) \to (\forall (P0: Prop).P0))) H6 t H9) in (H11 (refl_equal T t) 
-P)))))) (pr0_comp t x0 H7 t0 t0 (pr0_refl t0) (Bind Void))))))) H5)) H4))) 
-(\lambda (H3: (eq T t0 (lift (S O) O x))).(let H4 \def (eq_ind T t0 (\lambda 
-(t2: T).(or (\forall (t3: T).((pr0 t2 t3) \to (eq T t2 t3))) (ex2 T (\lambda 
-(t3: T).((eq T t2 t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(pr0 t2 
-t3))))) H0 (lift (S O) O x) H3) in (eq_ind_r T (lift (S O) O x) (\lambda (t2: 
-T).(or (\forall (t3: T).((pr0 (THead (Bind Void) t t2) t3) \to (eq T (THead 
-(Bind Void) t t2) t3))) (ex2 T (\lambda (t3: T).((eq T (THead (Bind Void) t 
-t2) t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(pr0 (THead (Bind Void) 
-t t2) t3))))) (or_intror (\forall (t2: T).((pr0 (THead (Bind Void) t (lift (S 
-O) O x)) t2) \to (eq T (THead (Bind Void) t (lift (S O) O x)) t2))) (ex2 T 
-(\lambda (t2: T).((eq T (THead (Bind Void) t (lift (S O) O x)) t2) \to 
+Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)))) (\lambda 
+(x0: T).(\lambda (H6: (((eq T t x0) \to (\forall (P: Prop).P)))).(\lambda 
+(H7: (pr0 t x0)).(or_intror (\forall (t2: T).((pr0 (THead (Bind Void) t t0) 
+t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T 
+(THead (Bind Void) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
+T).(pr0 (THead (Bind Void) t t0) t2))) (ex_intro2 T (\lambda (t2: T).((eq T 
+(THead (Bind Void) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
+T).(pr0 (THead (Bind Void) t t0) t2)) (THead (Bind Void) x0 t0) (\lambda (H8: 
+(eq T (THead (Bind Void) t t0) (THead (Bind Void) x0 t0))).(\lambda (P: 
+Prop).(let H9 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow t | (TLRef _) \Rightarrow t | (THead _ t2 _) \Rightarrow t2])) 
+(THead (Bind Void) t t0) (THead (Bind Void) x0 t0) H8) in (let H10 \def 
+(eq_ind_r T x0 (\lambda (t2: T).(pr0 t t2)) H7 t H9) in (let H11 \def 
+(eq_ind_r T x0 (\lambda (t2: T).((eq T t t2) \to (\forall (P0: Prop).P0))) H6 
+t H9) in (H11 (refl_equal T t) P)))))) (pr0_comp t x0 H7 t0 t0 (pr0_refl t0) 
+(Bind Void))))))) H5)) H4))) (\lambda (H3: (eq T t0 (lift (S O) O x))).(let 
+H4 \def (eq_ind T t0 (\lambda (t2: T).(or (\forall (t3: T).((pr0 t2 t3) \to 
+(eq T t2 t3))) (ex2 T (\lambda (t3: T).((eq T t2 t3) \to (\forall (P: 
+Prop).P))) (\lambda (t3: T).(pr0 t2 t3))))) H0 (lift (S O) O x) H3) in 
+(eq_ind_r T (lift (S O) O x) (\lambda (t2: T).(or (\forall (t3: T).((pr0 
+(THead (Bind Void) t t2) t3) \to (eq T (THead (Bind Void) t t2) t3))) (ex2 T 
+(\lambda (t3: T).((eq T (THead (Bind Void) t t2) t3) \to (\forall (P: 
+Prop).P))) (\lambda (t3: T).(pr0 (THead (Bind Void) t t2) t3))))) (or_intror 
+(\forall (t2: T).((pr0 (THead (Bind Void) t (lift (S O) O x)) t2) \to (eq T 
+(THead (Bind Void) t (lift (S O) O x)) t2))) (ex2 T (\lambda (t2: T).((eq T 
+(THead (Bind Void) t (lift (S O) O x)) t2) \to (\forall (P: Prop).P))) 
+(\lambda (t2: T).(pr0 (THead (Bind Void) t (lift (S O) O x)) t2))) (ex_intro2 
+T (\lambda (t2: T).((eq T (THead (Bind Void) t (lift (S O) O x)) t2) \to 

 (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t (lift (S 
 (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t (lift (S 

-O) O x)) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Bind Void) t 
-(lift (S O) O x)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 
-(THead (Bind Void) t (lift (S O) O x)) t2)) x (\lambda (H5: (eq T (THead 
-(Bind Void) t (lift (S O) O x)) x)).(\lambda (P: Prop).(thead_x_lift_y_y 
-(Bind Void) x t (S O) O H5 P))) (pr0_zeta Void (sym_not_eq B Abst Void 
-not_abst_void) x x (pr0_refl x) t))) t0 H3))) H2))) H1))) b)) (\lambda (f: 
-F).(F_ind (\lambda (f0: F).(or (\forall (t2: T).((pr0 (THead (Flat f0) t t0) 
-t2) \to (eq T (THead (Flat f0) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T 
-(THead (Flat f0) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 
-(THead (Flat f0) t t0) t2))))) (let H_x \def (binder_dec t0) in (let H1 \def 
-H_x in (or_ind (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: 
-T).(eq T t0 (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: 
-T).(\forall (u: T).((eq T t0 (THead (Bind b) w u)) \to (\forall (P: 
-Prop).P))))) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq 
-T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat 
-Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead 
-(Flat Appl) t t0) t2)))) (\lambda (H2: (ex_3 B T T (\lambda (b: B).(\lambda 
-(w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w u))))))).(ex_3_ind B T T 
+O) O x)) t2)) x (\lambda (H5: (eq T (THead (Bind Void) t (lift (S O) O x)) 
+x)).(\lambda (P: Prop).(thead_x_lift_y_y (Bind Void) x t (S O) O H5 P))) 
+(pr0_zeta Void not_void_abst x x (pr0_refl x) t))) t0 H3))) H2))) H1))) b)) 
+(\lambda (f: F).(F_ind (\lambda (f0: F).(or (\forall (t2: T).((pr0 (THead 
+(Flat f0) t t0) t2) \to (eq T (THead (Flat f0) t t0) t2))) (ex2 T (\lambda 
+(t2: T).((eq T (THead (Flat f0) t t0) t2) \to (\forall (P: Prop).P))) 
+(\lambda (t2: T).(pr0 (THead (Flat f0) t t0) t2))))) (let H_x \def 
+(binder_dec t0) in (let H1 \def H_x in (or_ind (ex_3 B T T (\lambda (b: 
+B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w u)))))) 
+(\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead (Bind b) w 
+u)) \to (\forall (P: Prop).P))))) (or (\forall (t2: T).((pr0 (THead (Flat 
+Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: 
+T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda 
+(t2: T).(pr0 (THead (Flat Appl) t t0) t2)))) (\lambda (H2: (ex_3 B T T 

 (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w 
 (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w 

-u))))) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T 
-(THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat 
-Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead 
-(Flat Appl) t t0) t2)))) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: 
-T).(\lambda (H3: (eq T t0 (THead (Bind x0) x1 x2))).(let H4 \def (eq_ind T t0 
-(\lambda (t2: T).(or (\forall (t3: T).((pr0 t2 t3) \to (eq T t2 t3))) (ex2 T 
-(\lambda (t3: T).((eq T t2 t3) \to (\forall (P: Prop).P))) (\lambda (t3: 
-T).(pr0 t2 t3))))) H0 (THead (Bind x0) x1 x2) H3) in (eq_ind_r T (THead (Bind 
-x0) x1 x2) (\lambda (t2: T).(or (\forall (t3: T).((pr0 (THead (Flat Appl) t 
-t2) t3) \to (eq T (THead (Flat Appl) t t2) t3))) (ex2 T (\lambda (t3: T).((eq 
-T (THead (Flat Appl) t t2) t3) \to (\forall (P: Prop).P))) (\lambda (t3: 
-T).(pr0 (THead (Flat Appl) t t2) t3))))) (B_ind (\lambda (b: B).((or (\forall 
-(t2: T).((pr0 (THead (Bind b) x1 x2) t2) \to (eq T (THead (Bind b) x1 x2) 
-t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind b) x1 x2) t2) \to (\forall 
-(P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind b) x1 x2) t2)))) \to (or 
-(\forall (t2: T).((pr0 (THead (Flat Appl) t (THead (Bind b) x1 x2)) t2) \to 
-(eq T (THead (Flat Appl) t (THead (Bind b) x1 x2)) t2))) (ex2 T (\lambda (t2: 
-T).((eq T (THead (Flat Appl) t (THead (Bind b) x1 x2)) t2) \to (\forall (P: 
-Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind b) x1 x2)) 
-t2)))))) (\lambda (_: (or (\forall (t2: T).((pr0 (THead (Bind Abbr) x1 x2) 
-t2) \to (eq T (THead (Bind Abbr) x1 x2) t2))) (ex2 T (\lambda (t2: T).((eq T 
-(THead (Bind Abbr) x1 x2) t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
-T).(pr0 (THead (Bind Abbr) x1 x2) t2))))).(or_intror (\forall (t2: T).((pr0 
-(THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) t2) \to (eq T (THead (Flat 
-Appl) t (THead (Bind Abbr) x1 x2)) t2))) (ex2 T (\lambda (t2: T).((eq T 
-(THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) t2) \to (\forall (P: 
-Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind Abbr) x1 
-x2)) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t (THead 
-(Bind Abbr) x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 
-(THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) t2)) (THead (Bind Abbr) x1 
-(THead (Flat Appl) (lift (S O) O t) x2)) (\lambda (H6: (eq T (THead (Flat 
-Appl) t (THead (Bind Abbr) x1 x2)) (THead (Bind Abbr) x1 (THead (Flat Appl) 
-(lift (S O) O t) x2)))).(\lambda (P: Prop).(let H7 \def (eq_ind T (THead 
-(Flat Appl) t (THead (Bind Abbr) x1 x2)) (\lambda (ee: T).(match ee in T 
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead _ _ t2) \Rightarrow (match t2 in T return (\lambda 
-(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
-| (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda (_: K).Prop) with 
-[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])])) I (THead (Bind 
-Abbr) x1 (THead (Flat Appl) (lift (S O) O t) x2)) H6) in (False_ind P H7)))) 
-(pr0_upsilon Abbr not_abbr_abst t t (pr0_refl t) x1 x1 (pr0_refl x1) x2 x2 
-(pr0_refl x2))))) (\lambda (_: (or (\forall (t2: T).((pr0 (THead (Bind Abst) 
-x1 x2) t2) \to (eq T (THead (Bind Abst) x1 x2) t2))) (ex2 T (\lambda (t2: 
-T).((eq T (THead (Bind Abst) x1 x2) t2) \to (\forall (P: Prop).P))) (\lambda 
-(t2: T).(pr0 (THead (Bind Abst) x1 x2) t2))))).(or_intror (\forall (t2: 
-T).((pr0 (THead (Flat Appl) t (THead (Bind Abst) x1 x2)) t2) \to (eq T (THead 
-(Flat Appl) t (THead (Bind Abst) x1 x2)) t2))) (ex2 T (\lambda (t2: T).((eq T 
-(THead (Flat Appl) t (THead (Bind Abst) x1 x2)) t2) \to (\forall (P: 
-Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind Abst) x1 
-x2)) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t (THead 
-(Bind Abst) x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 
-(THead (Flat Appl) t (THead (Bind Abst) x1 x2)) t2)) (THead (Bind Abbr) t x2) 
-(\lambda (H6: (eq T (THead (Flat Appl) t (THead (Bind Abst) x1 x2)) (THead 
-(Bind Abbr) t x2))).(\lambda (P: Prop).(let H7 \def (eq_ind T (THead (Flat 
-Appl) t (THead (Bind Abst) x1 x2)) (\lambda (ee: T).(match ee in T return 
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda 
-(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow 
-True])])) I (THead (Bind Abbr) t x2) H6) in (False_ind P H7)))) (pr0_beta x1 
-t t (pr0_refl t) x2 x2 (pr0_refl x2))))) (\lambda (_: (or (\forall (t2: 
-T).((pr0 (THead (Bind Void) x1 x2) t2) \to (eq T (THead (Bind Void) x1 x2) 
-t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) x1 x2) t2) \to 
-(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) x1 x2) 
+u))))))).(ex_3_ind B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq 
+T t0 (THead (Bind b) w u))))) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t 
+t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq 
+T (THead (Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
+T).(pr0 (THead (Flat Appl) t t0) t2)))) (\lambda (x0: B).(\lambda (x1: 
+T).(\lambda (x2: T).(\lambda (H3: (eq T t0 (THead (Bind x0) x1 x2))).(let H4 
+\def (eq_ind T t0 (\lambda (t2: T).(or (\forall (t3: T).((pr0 t2 t3) \to (eq 
+T t2 t3))) (ex2 T (\lambda (t3: T).((eq T t2 t3) \to (\forall (P: Prop).P))) 
+(\lambda (t3: T).(pr0 t2 t3))))) H0 (THead (Bind x0) x1 x2) H3) in (eq_ind_r 
+T (THead (Bind x0) x1 x2) (\lambda (t2: T).(or (\forall (t3: T).((pr0 (THead 
+(Flat Appl) t t2) t3) \to (eq T (THead (Flat Appl) t t2) t3))) (ex2 T 
+(\lambda (t3: T).((eq T (THead (Flat Appl) t t2) t3) \to (\forall (P: 
+Prop).P))) (\lambda (t3: T).(pr0 (THead (Flat Appl) t t2) t3))))) (B_ind 
+(\lambda (b: B).((or (\forall (t2: T).((pr0 (THead (Bind b) x1 x2) t2) \to 
+(eq T (THead (Bind b) x1 x2) t2))) (ex2 T (\lambda (t2: T).((eq T (THead 
+(Bind b) x1 x2) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead 
+(Bind b) x1 x2) t2)))) \to (or (\forall (t2: T).((pr0 (THead (Flat Appl) t 
+(THead (Bind b) x1 x2)) t2) \to (eq T (THead (Flat Appl) t (THead (Bind b) x1 
+x2)) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t (THead (Bind 
+b) x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat 
+Appl) t (THead (Bind b) x1 x2)) t2)))))) (\lambda (_: (or (\forall (t2: 
+T).((pr0 (THead (Bind Abbr) x1 x2) t2) \to (eq T (THead (Bind Abbr) x1 x2) 
+t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abbr) x1 x2) t2) \to 
+(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abbr) x1 x2) 

 t2))))).(or_intror (\forall (t2: T).((pr0 (THead (Flat Appl) t (THead (Bind 
 t2))))).(or_intror (\forall (t2: T).((pr0 (THead (Flat Appl) t (THead (Bind 

-Void) x1 x2)) t2) \to (eq T (THead (Flat Appl) t (THead (Bind Void) x1 x2)) 
-t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t (THead (Bind Void) 
+Abbr) x1 x2)) t2) \to (eq T (THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) 
+t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t (THead (Bind Abbr) 

 x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat 
 x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat 

-Appl) t (THead (Bind Void) x1 x2)) t2))) (ex_intro2 T (\lambda (t2: T).((eq T 
-(THead (Flat Appl) t (THead (Bind Void) x1 x2)) t2) \to (\forall (P: 
-Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind Void) x1 
-x2)) t2)) (THead (Bind Void) x1 (THead (Flat Appl) (lift (S O) O t) x2)) 
+Appl) t (THead (Bind Abbr) x1 x2)) t2))) (ex_intro2 T (\lambda (t2: T).((eq T 
+(THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) t2) \to (\forall (P: 
+Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind Abbr) x1 
+x2)) t2)) (THead (Bind Abbr) x1 (THead (Flat Appl) (lift (S O) O t) x2)) 
+(\lambda (H6: (eq T (THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) (THead 
+(Bind Abbr) x1 (THead (Flat Appl) (lift (S O) O t) x2)))).(\lambda (P: 
+Prop).(let H7 \def (eq_ind T (THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) 
+(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead _ _ t2) \Rightarrow (match t2 with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow 
+(match k0 with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])])) 
+I (THead (Bind Abbr) x1 (THead (Flat Appl) (lift (S O) O t) x2)) H6) in 
+(False_ind P H7)))) (pr0_upsilon Abbr not_abbr_abst t t (pr0_refl t) x1 x1 
+(pr0_refl x1) x2 x2 (pr0_refl x2))))) (\lambda (_: (or (\forall (t2: T).((pr0 
+(THead (Bind Abst) x1 x2) t2) \to (eq T (THead (Bind Abst) x1 x2) t2))) (ex2 
+T (\lambda (t2: T).((eq T (THead (Bind Abst) x1 x2) t2) \to (\forall (P: 
+Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) x1 x2) t2))))).(or_intror 
+(\forall (t2: T).((pr0 (THead (Flat Appl) t (THead (Bind Abst) x1 x2)) t2) 
+\to (eq T (THead (Flat Appl) t (THead (Bind Abst) x1 x2)) t2))) (ex2 T 
+(\lambda (t2: T).((eq T (THead (Flat Appl) t (THead (Bind Abst) x1 x2)) t2) 
+\to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead 
+(Bind Abst) x1 x2)) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat 
+Appl) t (THead (Bind Abst) x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda 
+(t2: T).(pr0 (THead (Flat Appl) t (THead (Bind Abst) x1 x2)) t2)) (THead 
+(Bind Abbr) t x2) (\lambda (H6: (eq T (THead (Flat Appl) t (THead (Bind Abst) 
+x1 x2)) (THead (Bind Abbr) t x2))).(\lambda (P: Prop).(let H7 \def (eq_ind T 
+(THead (Flat Appl) t (THead (Bind Abst) x1 x2)) (\lambda (ee: T).(match ee 
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ 
+_) \Rightarrow (match k0 with [(Bind _) \Rightarrow False | (Flat _) 
+\Rightarrow True])])) I (THead (Bind Abbr) t x2) H6) in (False_ind P H7)))) 
+(pr0_beta x1 t t (pr0_refl t) x2 x2 (pr0_refl x2))))) (\lambda (_: (or 
+(\forall (t2: T).((pr0 (THead (Bind Void) x1 x2) t2) \to (eq T (THead (Bind 
+Void) x1 x2) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) x1 x2) 
+t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) x1 
+x2) t2))))).(or_intror (\forall (t2: T).((pr0 (THead (Flat Appl) t (THead 
+(Bind Void) x1 x2)) t2) \to (eq T (THead (Flat Appl) t (THead (Bind Void) x1 
+x2)) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t (THead (Bind 
+Void) x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead 
+(Flat Appl) t (THead (Bind Void) x1 x2)) t2))) (ex_intro2 T (\lambda (t2: 
+T).((eq T (THead (Flat Appl) t (THead (Bind Void) x1 x2)) t2) \to (\forall 
+(P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind Void) 
+x1 x2)) t2)) (THead (Bind Void) x1 (THead (Flat Appl) (lift (S O) O t) x2)) 

 (\lambda (H6: (eq T (THead (Flat Appl) t (THead (Bind Void) x1 x2)) (THead 
 (Bind Void) x1 (THead (Flat Appl) (lift (S O) O t) x2)))).(\lambda (P: 
 Prop).(let H7 \def (eq_ind T (THead (Flat Appl) t (THead (Bind Void) x1 x2)) 
 (\lambda (H6: (eq T (THead (Flat Appl) t (THead (Bind Void) x1 x2)) (THead 
 (Bind Void) x1 (THead (Flat Appl) (lift (S O) O t) x2)))).(\lambda (P: 
 Prop).(let H7 \def (eq_ind T (THead (Flat Appl) t (THead (Bind Void) x1 x2)) 

-(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ t2) \Rightarrow 
-(match t2 in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False 
-| (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K 
-return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) 
-\Rightarrow False])])])) I (THead (Bind Void) x1 (THead (Flat Appl) (lift (S 
-O) O t) x2)) H6) in (False_ind P H7)))) (pr0_upsilon Void (sym_not_eq B Abst 
-Void not_abst_void) t t (pr0_refl t) x1 x1 (pr0_refl x1) x2 x2 (pr0_refl 
-x2))))) x0 H4) t0 H3)))))) H2)) (\lambda (H2: ((\forall (b: B).(\forall (w: 
-T).(\forall (u: T).((eq T t0 (THead (Bind b) w u)) \to (\forall (P: 
-Prop).P))))))).(let H3 \def H in (or_ind (\forall (t2: T).((pr0 t t2) \to (eq 
-T t t2))) (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) 
-(\lambda (t2: T).(pr0 t t2))) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t 
+(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead _ _ t2) \Rightarrow (match t2 with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow 
+(match k0 with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])])) 
+I (THead (Bind Void) x1 (THead (Flat Appl) (lift (S O) O t) x2)) H6) in 
+(False_ind P H7)))) (pr0_upsilon Void not_void_abst t t (pr0_refl t) x1 x1 
+(pr0_refl x1) x2 x2 (pr0_refl x2))))) x0 H4) t0 H3)))))) H2)) (\lambda (H2: 
+((\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead (Bind b) w 
+u)) \to (\forall (P: Prop).P))))))).(let H3 \def H in (or_ind (\forall (t2: 
+T).((pr0 t t2) \to (eq T t t2))) (ex2 T (\lambda (t2: T).((eq T t t2) \to 
+(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2))) (or (\forall (t2: 
+T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) 
+t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to 
+(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)))) 
+(\lambda (H4: ((\forall (t2: T).((pr0 t t2) \to (eq T t t2))))).(let H5 \def 
+H0 in (or_ind (\forall (t2: T).((pr0 t0 t2) \to (eq T t0 t2))) (ex2 T 
+(\lambda (t2: T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
+T).(pr0 t0 t2))) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to 
+(eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead 
+(Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 
+(THead (Flat Appl) t t0) t2)))) (\lambda (H6: ((\forall (t2: T).((pr0 t0 t2) 
+\to (eq T t0 t2))))).(or_introl (\forall (t2: T).((pr0 (THead (Flat Appl) t 

 t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq 
 T (THead (Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
 t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq 
 T (THead (Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: 

-T).(pr0 (THead (Flat Appl) t t0) t2)))) (\lambda (H4: ((\forall (t2: T).((pr0 
-t t2) \to (eq T t t2))))).(let H5 \def H0 in (or_ind (\forall (t2: T).((pr0 
-t0 t2) \to (eq T t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall 
-(P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2))) (or (\forall (t2: T).((pr0 
-(THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T 
-(\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: 
-Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)))) (\lambda 
-(H6: ((\forall (t2: T).((pr0 t0 t2) \to (eq T t0 t2))))).(or_introl (\forall 
-(t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) 
-t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to 
-(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2))) 
-(\lambda (t2: T).(\lambda (H7: (pr0 (THead (Flat Appl) t t0) t2)).(or3_ind 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda 
-(t3: T).(pr0 t0 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
+T).(pr0 (THead (Flat Appl) t t0) t2))) (\lambda (t2: T).(\lambda (H7: (pr0 
+(THead (Flat Appl) t t0) t2)).(or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr0 t u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t0 t3)))) (ex4_4 T T T 
+T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t0 
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 t u2))))) (\lambda 
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) 
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(eq T t0 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T 
+t2 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
+T).(\lambda (_: T).(pr0 t u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (t3: T).(pr0 z1 t3)))))))) (eq T (THead (Flat Appl) t t0) t2) 
+(\lambda (H8: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda 
+(_: T).(\lambda (t3: T).(pr0 t0 t3))))).(ex3_2_ind T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t0 
+t3))) (eq T (THead (Flat Appl) t t0) t2) (\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (H9: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H10: (pr0 t 
+x0)).(\lambda (H11: (pr0 t0 x1)).(let H_y \def (H6 x1 H11) in (let H_y0 \def 
+(H4 x0 H10) in (let H12 \def (eq_ind_r T x1 (\lambda (t3: T).(pr0 t0 t3)) H11 
+t0 H_y) in (let H13 \def (eq_ind_r T x1 (\lambda (t3: T).(eq T t2 (THead 
+(Flat Appl) x0 t3))) H9 t0 H_y) in (let H14 \def (eq_ind_r T x0 (\lambda (t3: 
+T).(pr0 t t3)) H10 t H_y0) in (let H15 \def (eq_ind_r T x0 (\lambda (t3: 
+T).(eq T t2 (THead (Flat Appl) t3 t0))) H13 t H_y0) in (eq_ind_r T (THead 
+(Flat Appl) t t0) (\lambda (t3: T).(eq T (THead (Flat Appl) t t0) t3)) 
+(refl_equal T (THead (Flat Appl) t t0)) t2 H15)))))))))))) H8)) (\lambda (H8: 
+(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(eq T t0 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 t 
+u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: 
+T).(pr0 z1 t3))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: 

 T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind Abst) y1 z1)))))) 
 (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 
 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
 T).(\lambda (_: T).(pr0 t u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda 
 T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind Abst) y1 z1)))))) 
 (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 
 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
 T).(\lambda (_: T).(pr0 t u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda 

-(_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: 
+(_: T).(\lambda (t3: T).(pr0 z1 t3))))) (eq T (THead (Flat Appl) t t0) t2) 
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda 
+(H9: (eq T t0 (THead (Bind Abst) x0 x1))).(\lambda (H10: (eq T t2 (THead 
+(Bind Abbr) x2 x3))).(\lambda (_: (pr0 t x2)).(\lambda (_: (pr0 x1 
+x3)).(eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda (t3: T).(eq T (THead 
+(Flat Appl) t t0) t3)) (let H13 \def (eq_ind T t0 (\lambda (t3: T).(\forall 
+(t4: T).((pr0 t3 t4) \to (eq T t3 t4)))) H6 (THead (Bind Abst) x0 x1) H9) in 
+(let H14 \def (eq_ind T t0 (\lambda (t3: T).(\forall (b: B).(\forall (w: 
+T).(\forall (u: T).((eq T t3 (THead (Bind b) w u)) \to (\forall (P: 
+Prop).P)))))) H2 (THead (Bind Abst) x0 x1) H9) in (eq_ind_r T (THead (Bind 
+Abst) x0 x1) (\lambda (t3: T).(eq T (THead (Flat Appl) t t3) (THead (Bind 
+Abbr) x2 x3))) (H14 Abst x0 x1 (H13 (THead (Bind Abst) x0 x1) (pr0_refl 
+(THead (Bind Abst) x0 x1))) (eq T (THead (Flat Appl) t (THead (Bind Abst) x0 
+x1)) (THead (Bind Abbr) x2 x3))) t0 H9))) t2 H10))))))))) H8)) (\lambda (H8: 
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(eq T t0 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T 
+t2 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
+T).(\lambda (_: T).(pr0 t u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (t3: T).(pr0 z1 t3))))))))).(ex6_6_ind B T T T T T (\lambda (b: 

 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
 (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
 (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind 
 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
 (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
 (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind 


@@ -396,121 +451,60 @@ T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 t
 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
 T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: 
 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
 T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: 
 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 

-(t3: T).(pr0 z1 t3)))))))) (eq T (THead (Flat Appl) t t0) t2) (\lambda (H8: 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda 
-(t3: T).(pr0 t0 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq 
-T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t 
-u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t0 t3))) (eq T (THead (Flat Appl) 
-t t0) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H9: (eq T t2 (THead 
-(Flat Appl) x0 x1))).(\lambda (H10: (pr0 t x0)).(\lambda (H11: (pr0 t0 
-x1)).(let H_y \def (H6 x1 H11) in (let H_y0 \def (H4 x0 H10) in (let H12 \def 
-(eq_ind_r T x1 (\lambda (t3: T).(pr0 t0 t3)) H11 t0 H_y) in (let H13 \def 
-(eq_ind_r T x1 (\lambda (t3: T).(eq T t2 (THead (Flat Appl) x0 t3))) H9 t0 
-H_y) in (let H14 \def (eq_ind_r T x0 (\lambda (t3: T).(pr0 t t3)) H10 t H_y0) 
-in (let H15 \def (eq_ind_r T x0 (\lambda (t3: T).(eq T t2 (THead (Flat Appl) 
-t3 t0))) H13 t H_y0) in (eq_ind_r T (THead (Flat Appl) t t0) (\lambda (t3: 
-T).(eq T (THead (Flat Appl) t t0) t3)) (refl_equal T (THead (Flat Appl) t 
-t0)) t2 H15)))))))))))) H8)) (\lambda (H8: (ex4_4 T T T T (\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind 
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr0 t u2))))) (\lambda (_: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))).(ex4_4_ind T T T T 
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t0 
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 t u2))))) (\lambda 
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))) (eq 
-T (THead (Flat Appl) t t0) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda 
-(x2: T).(\lambda (x3: T).(\lambda (H9: (eq T t0 (THead (Bind Abst) x0 
-x1))).(\lambda (H10: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (_: (pr0 t 
-x2)).(\lambda (_: (pr0 x1 x3)).(eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda 
-(t3: T).(eq T (THead (Flat Appl) t t0) t3)) (let H13 \def (eq_ind T t0 
-(\lambda (t3: T).(\forall (t4: T).((pr0 t3 t4) \to (eq T t3 t4)))) H6 (THead 
-(Bind Abst) x0 x1) H9) in (let H14 \def (eq_ind T t0 (\lambda (t3: 
-T).(\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t3 (THead (Bind b) 
-w u)) \to (\forall (P: Prop).P)))))) H2 (THead (Bind Abst) x0 x1) H9) in 
-(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t3: T).(eq T (THead (Flat 
-Appl) t t3) (THead (Bind Abbr) x2 x3))) (H14 Abst x0 x1 (H13 (THead (Bind 
-Abst) x0 x1) (pr0_refl (THead (Bind Abst) x0 x1))) (eq T (THead (Flat Appl) t 
-(THead (Bind Abst) x0 x1)) (THead (Bind Abbr) x2 x3))) t0 H9))) t2 
-H10))))))))) H8)) (\lambda (H8: (ex6_6 B T T T T T (\lambda (b: B).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not 
-(eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind b) 
-y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat 
-Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 t u2))))))) 
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 
-t3))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind b) y1 z1)))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat Appl) (lift 
-(S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 t u2))))))) (\lambda 
-(_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: 
-T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))) 
-(eq T (THead (Flat Appl) t t0) t2) (\lambda (x0: B).(\lambda (x1: T).(\lambda 
-(x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (not 
-(eq B x0 Abst))).(\lambda (H10: (eq T t0 (THead (Bind x0) x1 x2))).(\lambda 
-(H11: (eq T t2 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) 
-x5)))).(\lambda (_: (pr0 t x3)).(\lambda (_: (pr0 x1 x4)).(\lambda (_: (pr0 
-x2 x5)).(eq_ind_r T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) 
-x5)) (\lambda (t3: T).(eq T (THead (Flat Appl) t t0) t3)) (let H15 \def 
-(eq_ind T t0 (\lambda (t3: T).(\forall (t4: T).((pr0 t3 t4) \to (eq T t3 
-t4)))) H6 (THead (Bind x0) x1 x2) H10) in (let H16 \def (eq_ind T t0 (\lambda 
-(t3: T).(\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t3 (THead 
-(Bind b) w u)) \to (\forall (P: Prop).P)))))) H2 (THead (Bind x0) x1 x2) H10) 
-in (eq_ind_r T (THead (Bind x0) x1 x2) (\lambda (t3: T).(eq T (THead (Flat 
-Appl) t t3) (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)))) 
-(H16 x0 x1 x2 (H15 (THead (Bind x0) x1 x2) (pr0_refl (THead (Bind x0) x1 
-x2))) (eq T (THead (Flat Appl) t (THead (Bind x0) x1 x2)) (THead (Bind x0) x4 
-(THead (Flat Appl) (lift (S O) O x3) x5)))) t0 H10))) t2 H11))))))))))))) 
-H8)) (pr0_gen_appl t t0 t2 H7)))))) (\lambda (H6: (ex2 T (\lambda (t2: 
-T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 
-t2)))).(ex2_ind T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: Prop).P))) 
-(\lambda (t2: T).(pr0 t0 t2)) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t 
-t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq 
-T (THead (Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
-T).(pr0 (THead (Flat Appl) t t0) t2)))) (\lambda (x: T).(\lambda (H7: (((eq T 
-t0 x) \to (\forall (P: Prop).P)))).(\lambda (H8: (pr0 t0 x)).(or_intror 
-(\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat 
-Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) 
-\to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) 
-t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to 
-(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)) 
-(THead (Flat Appl) t x) (\lambda (H9: (eq T (THead (Flat Appl) t t0) (THead 
-(Flat Appl) t x))).(\lambda (P: Prop).(let H10 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | 
-(TLRef _) \Rightarrow t0 | (THead _ _ t2) \Rightarrow t2])) (THead (Flat 
-Appl) t t0) (THead (Flat Appl) t x) H9) in (let H11 \def (eq_ind_r T x 
-(\lambda (t2: T).(pr0 t0 t2)) H8 t0 H10) in (let H12 \def (eq_ind_r T x 
-(\lambda (t2: T).((eq T t0 t2) \to (\forall (P0: Prop).P0))) H7 t0 H10) in 
-(H12 (refl_equal T t0) P)))))) (pr0_comp t t (pr0_refl t) t0 x H8 (Flat 
-Appl))))))) H6)) H5))) (\lambda (H4: (ex2 T (\lambda (t2: T).((eq T t t2) \to 
-(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2)))).(ex2_ind T (\lambda 
-(t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2)) 
-(or (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead 
-(Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t 
-t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) 
-t t0) t2)))) (\lambda (x: T).(\lambda (H5: (((eq T t x) \to (\forall (P: 
-Prop).P)))).(\lambda (H6: (pr0 t x)).(or_intror (\forall (t2: T).((pr0 (THead 
-(Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T 
+(t3: T).(pr0 z1 t3))))))) (eq T (THead (Flat Appl) t t0) t2) (\lambda (x0: 
+B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: 
+T).(\lambda (x5: T).(\lambda (_: (not (eq B x0 Abst))).(\lambda (H10: (eq T 
+t0 (THead (Bind x0) x1 x2))).(\lambda (H11: (eq T t2 (THead (Bind x0) x4 
+(THead (Flat Appl) (lift (S O) O x3) x5)))).(\lambda (_: (pr0 t x3)).(\lambda 
+(_: (pr0 x1 x4)).(\lambda (_: (pr0 x2 x5)).(eq_ind_r T (THead (Bind x0) x4 
+(THead (Flat Appl) (lift (S O) O x3) x5)) (\lambda (t3: T).(eq T (THead (Flat 
+Appl) t t0) t3)) (let H15 \def (eq_ind T t0 (\lambda (t3: T).(\forall (t4: 
+T).((pr0 t3 t4) \to (eq T t3 t4)))) H6 (THead (Bind x0) x1 x2) H10) in (let 
+H16 \def (eq_ind T t0 (\lambda (t3: T).(\forall (b: B).(\forall (w: 
+T).(\forall (u: T).((eq T t3 (THead (Bind b) w u)) \to (\forall (P: 
+Prop).P)))))) H2 (THead (Bind x0) x1 x2) H10) in (eq_ind_r T (THead (Bind x0) 
+x1 x2) (\lambda (t3: T).(eq T (THead (Flat Appl) t t3) (THead (Bind x0) x4 
+(THead (Flat Appl) (lift (S O) O x3) x5)))) (H16 x0 x1 x2 (H15 (THead (Bind 
+x0) x1 x2) (pr0_refl (THead (Bind x0) x1 x2))) (eq T (THead (Flat Appl) t 
+(THead (Bind x0) x1 x2)) (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O 
+x3) x5)))) t0 H10))) t2 H11))))))))))))) H8)) (pr0_gen_appl t t0 t2 H7)))))) 
+(\lambda (H6: (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: 
+Prop).P))) (\lambda (t2: T).(pr0 t0 t2)))).(ex2_ind T (\lambda (t2: T).((eq T 
+t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2)) (or (\forall 
+(t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) 
+t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to 
+(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)))) 
+(\lambda (x: T).(\lambda (H7: (((eq T t0 x) \to (\forall (P: 
+Prop).P)))).(\lambda (H8: (pr0 t0 x)).(or_intror (\forall (t2: T).((pr0 
+(THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T 

 (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: 
 Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2))) (ex_intro2 T 
 (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: 
 Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)) (THead (Flat 
 (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: 
 Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2))) (ex_intro2 T 
 (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: 
 Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)) (THead (Flat 

-Appl) x t0) (\lambda (H7: (eq T (THead (Flat Appl) t t0) (THead (Flat Appl) x 
-t0))).(\lambda (P: Prop).(let H8 \def (f_equal T T (\lambda (e: T).(match e 
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t | (TLRef _) 
-\Rightarrow t | (THead _ t2 _) \Rightarrow t2])) (THead (Flat Appl) t t0) 
+Appl) t x) (\lambda (H9: (eq T (THead (Flat Appl) t t0) (THead (Flat Appl) t 
+x))).(\lambda (P: Prop).(let H10 \def (f_equal T T (\lambda (e: T).(match e 
+with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t2) 
+\Rightarrow t2])) (THead (Flat Appl) t t0) (THead (Flat Appl) t x) H9) in 
+(let H11 \def (eq_ind_r T x (\lambda (t2: T).(pr0 t0 t2)) H8 t0 H10) in (let 
+H12 \def (eq_ind_r T x (\lambda (t2: T).((eq T t0 t2) \to (\forall (P0: 
+Prop).P0))) H7 t0 H10) in (H12 (refl_equal T t0) P)))))) (pr0_comp t t 
+(pr0_refl t) t0 x H8 (Flat Appl))))))) H6)) H5))) (\lambda (H4: (ex2 T 
+(\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
+T).(pr0 t t2)))).(ex2_ind T (\lambda (t2: T).((eq T t t2) \to (\forall (P: 
+Prop).P))) (\lambda (t2: T).(pr0 t t2)) (or (\forall (t2: T).((pr0 (THead 
+(Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T 
+(\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: 
+Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)))) (\lambda (x: 
+T).(\lambda (H5: (((eq T t x) \to (\forall (P: Prop).P)))).(\lambda (H6: (pr0 
+t x)).(or_intror (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq 
+T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat 
+Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead 
+(Flat Appl) t t0) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat 
+Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead 
+(Flat Appl) t t0) t2)) (THead (Flat Appl) x t0) (\lambda (H7: (eq T (THead 
+(Flat Appl) t t0) (THead (Flat Appl) x t0))).(\lambda (P: Prop).(let H8 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t | (TLRef 
+_) \Rightarrow t | (THead _ t2 _) \Rightarrow t2])) (THead (Flat Appl) t t0) 

 (THead (Flat Appl) x t0) H7) in (let H9 \def (eq_ind_r T x (\lambda (t2: 
 T).(pr0 t t2)) H6 t H8) in (let H10 \def (eq_ind_r T x (\lambda (t2: T).((eq 
 T t t2) \to (\forall (P0: Prop).P0))) H5 t H8) in (H10 (refl_equal T t) 
 (THead (Flat Appl) x t0) H7) in (let H9 \def (eq_ind_r T x (\lambda (t2: 
 T).(pr0 t t2)) H6 t H8) in (let H10 \def (eq_ind_r T x (\lambda (t2: T).((eq 
 T t t2) \to (\forall (P0: Prop).P0))) H5 t H8) in (H10 (refl_equal T t) 


@@ -523,7 +517,4 @@ Cast) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead
 (Flat Cast) t t0) t2)) t0 (\lambda (H1: (eq T (THead (Flat Cast) t t0) 
 t0)).(\lambda (P: Prop).(thead_x_y_y (Flat Cast) t t0 H1 P))) (pr0_tau t0 t0 
 (pr0_refl t0) t))) f)) k)))))) t1).
 (Flat Cast) t t0) t2)) t0 (\lambda (H1: (eq T (THead (Flat Cast) t t0) 
 t0)).(\lambda (P: Prop).(thead_x_y_y (Flat Cast) t t0 H1 P))) (pr0_tau t0 t0 
 (pr0_refl t0) t))) f)) k)))))) t1).

-(* COMMENTS
-Initial nodes: 10459
-END *)

 
 

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr0/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/pr0/defs.ma
index 0568e070cc8ae3cacd346768f09698e1a5bc2f65..5f6bd58fb45d94f3c84441d3fcdf14d77c0e73bf 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_1/pr0/defs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/pr0/defs.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
 
 (* This file was automatically generated: do not edit *********************)
 

-include "Basic-1/subst0/defs.ma".
+include "basic_1/subst0/defs.ma".

 
 inductive pr0: T \to (T \to Prop) \def
 | pr0_refl: \forall (t: T).(pr0 t t)
 
 inductive pr0: T \to (T \to Prop) \def
 | pr0_refl: \forall (t: T).(pr0 t t)

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr0/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/pr0/fwd.ma
index 46caceab4fb19492e138f6dc1000d4bc3a3a67c0..a9898d7ffe06c27064ac2a0ab1cece4a455f8612 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_1/pr0/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/pr0/fwd.ma
@@ -14,7 +14,42 @@
 
 (* This file was automatically generated: do not edit *********************)
 
 
 (* This file was automatically generated: do not edit *********************)
 

-include "Basic-1/pr0/props.ma".
+include "basic_1/pr0/defs.ma".
+
+include "basic_1/subst0/fwd.ma".
+
+let rec pr0_ind (P: (T \to (T \to Prop))) (f: (\forall (t: T).(P t t))) (f0: 
+(\forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to ((P u1 u2) \to (\forall 
+(t1: T).(\forall (t2: T).((pr0 t1 t2) \to ((P t1 t2) \to (\forall (k: K).(P 
+(THead k u1 t1) (THead k u2 t2)))))))))))) (f1: (\forall (u: T).(\forall (v1: 
+T).(\forall (v2: T).((pr0 v1 v2) \to ((P v1 v2) \to (\forall (t1: T).(\forall 
+(t2: T).((pr0 t1 t2) \to ((P t1 t2) \to (P (THead (Flat Appl) v1 (THead (Bind 
+Abst) u t1)) (THead (Bind Abbr) v2 t2)))))))))))) (f2: (\forall (b: B).((not 
+(eq B b Abst)) \to (\forall (v1: T).(\forall (v2: T).((pr0 v1 v2) \to ((P v1 
+v2) \to (\forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to ((P u1 u2) \to 
+(\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to ((P t1 t2) \to (P (THead 
+(Flat Appl) v1 (THead (Bind b) u1 t1)) (THead (Bind b) u2 (THead (Flat Appl) 
+(lift (S O) O v2) t2)))))))))))))))))) (f3: (\forall (u1: T).(\forall (u2: 
+T).((pr0 u1 u2) \to ((P u1 u2) \to (\forall (t1: T).(\forall (t2: T).((pr0 t1 
+t2) \to ((P t1 t2) \to (\forall (w: T).((subst0 O u2 t2 w) \to (P (THead 
+(Bind Abbr) u1 t1) (THead (Bind Abbr) u2 w))))))))))))) (f4: (\forall (b: 
+B).((not (eq B b Abst)) \to (\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) 
+\to ((P t1 t2) \to (\forall (u: T).(P (THead (Bind b) u (lift (S O) O t1)) 
+t2))))))))) (f5: (\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to ((P t1 
+t2) \to (\forall (u: T).(P (THead (Flat Cast) u t1) t2))))))) (t: T) (t0: T) 
+(p: pr0 t t0) on p: P t t0 \def match p with [(pr0_refl t1) \Rightarrow (f 
+t1) | (pr0_comp u1 u2 p0 t1 t2 p1 k) \Rightarrow (f0 u1 u2 p0 ((pr0_ind P f 
+f0 f1 f2 f3 f4 f5) u1 u2 p0) t1 t2 p1 ((pr0_ind P f f0 f1 f2 f3 f4 f5) t1 t2 
+p1) k) | (pr0_beta u v1 v2 p0 t1 t2 p1) \Rightarrow (f1 u v1 v2 p0 ((pr0_ind 
+P f f0 f1 f2 f3 f4 f5) v1 v2 p0) t1 t2 p1 ((pr0_ind P f f0 f1 f2 f3 f4 f5) t1 
+t2 p1)) | (pr0_upsilon b n v1 v2 p0 u1 u2 p1 t1 t2 p2) \Rightarrow (f2 b n v1 
+v2 p0 ((pr0_ind P f f0 f1 f2 f3 f4 f5) v1 v2 p0) u1 u2 p1 ((pr0_ind P f f0 f1 
+f2 f3 f4 f5) u1 u2 p1) t1 t2 p2 ((pr0_ind P f f0 f1 f2 f3 f4 f5) t1 t2 p2)) | 
+(pr0_delta u1 u2 p0 t1 t2 p1 w s0) \Rightarrow (f3 u1 u2 p0 ((pr0_ind P f f0 
+f1 f2 f3 f4 f5) u1 u2 p0) t1 t2 p1 ((pr0_ind P f f0 f1 f2 f3 f4 f5) t1 t2 p1) 
+w s0) | (pr0_zeta b n t1 t2 p0 u) \Rightarrow (f4 b n t1 t2 p0 ((pr0_ind P f 
+f0 f1 f2 f3 f4 f5) t1 t2 p0) u) | (pr0_tau t1 t2 p0 u) \Rightarrow (f5 t1 t2 
+p0 ((pr0_ind P f f0 f1 f2 f3 f4 f5) t1 t2 p0) u)].

 
 theorem pr0_gen_sort:
  \forall (x: T).(\forall (n: nat).((pr0 (TSort n) x) \to (eq T x (TSort n))))
 
 theorem pr0_gen_sort:
  \forall (x: T).(\forall (n: nat).((pr0 (TSort n) x) \to (eq T x (TSort n))))


@@ -29,57 +64,51 @@ t H2)))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda
 (_: (((eq T u1 (TSort n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2: 
 T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 
 t1)))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u1 t1) (TSort n))).(let 
 (_: (((eq T u1 (TSort n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2: 
 T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 
 t1)))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u1 t1) (TSort n))).(let 

-H6 \def (eq_ind T (THead k u1 t1) (\lambda (ee: T).(match ee in T return 
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H5) in 
-(False_ind (eq T (THead k u2 t2) (THead k u1 t1)) H6)))))))))))) (\lambda (u: 
-T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: 
-(((eq T v1 (TSort n)) \to (eq T v2 v1)))).(\lambda (t1: T).(\lambda (t2: 
-T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 
-t1)))).(\lambda (H5: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t1)) 
-(TSort n))).(let H6 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u 
-t1)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) 
-\Rightarrow True])) I (TSort n) H5) in (False_ind (eq T (THead (Bind Abbr) v2 
-t2) (THead (Flat Appl) v1 (THead (Bind Abst) u t1))) H6)))))))))))) (\lambda 
-(b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: 
+H6 \def (eq_ind T (THead k u1 t1) (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
+True])) I (TSort n) H5) in (False_ind (eq T (THead k u2 t2) (THead k u1 t1)) 
+H6)))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: 
+(pr0 v1 v2)).(\lambda (_: (((eq T v1 (TSort n)) \to (eq T v2 v1)))).(\lambda 
+(t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 
+(TSort n)) \to (eq T t2 t1)))).(\lambda (H5: (eq T (THead (Flat Appl) v1 
+(THead (Bind Abst) u t1)) (TSort n))).(let H6 \def (eq_ind T (THead (Flat 
+Appl) v1 (THead (Bind Abst) u t1)) (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
+True])) I (TSort n) H5) in (False_ind (eq T (THead (Bind Abbr) v2 t2) (THead 
+(Flat Appl) v1 (THead (Bind Abst) u t1))) H6)))))))))))) (\lambda (b: 
+B).(\lambda (_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: 

 T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (TSort n)) \to (eq T v2 
 v1)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda 
 (_: (((eq T u1 (TSort n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2: 
 T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 
 t1)))).(\lambda (H8: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t1)) 
 (TSort n))).(let H9 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 
 T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (TSort n)) \to (eq T v2 
 v1)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda 
 (_: (((eq T u1 (TSort n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2: 
 T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 
 t1)))).(\lambda (H8: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t1)) 
 (TSort n))).(let H9 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 

-t1)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) 
-\Rightarrow True])) I (TSort n) H8) in (False_ind (eq T (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Flat Appl) v1 (THead (Bind 
-b) u1 t1))) H9))))))))))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda 
-(_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 (TSort n)) \to (eq T u2 
-u1)))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda 
-(_: (((eq T t1 (TSort n)) \to (eq T t2 t1)))).(\lambda (w: T).(\lambda (_: 
-(subst0 O u2 t2 w)).(\lambda (H6: (eq T (THead (Bind Abbr) u1 t1) (TSort 
-n))).(let H7 \def (eq_ind T (THead (Bind Abbr) u1 t1) (\lambda (ee: T).(match 
-ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | 
-(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) 
-H6) in (False_ind (eq T (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u1 t1)) 
+t1)) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H8) in 
+(False_ind (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) 
+(THead (Flat Appl) v1 (THead (Bind b) u1 t1))) H9))))))))))))))))) (\lambda 
+(u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 
+(TSort n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: 
+(pr0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 t1)))).(\lambda 
+(w: T).(\lambda (_: (subst0 O u2 t2 w)).(\lambda (H6: (eq T (THead (Bind 
+Abbr) u1 t1) (TSort n))).(let H7 \def (eq_ind T (THead (Bind Abbr) u1 t1) 
+(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H6) in 
+(False_ind (eq T (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u1 t1)) 

 H7))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda 
 (t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 
 (TSort n)) \to (eq T t2 t1)))).(\lambda (u: T).(\lambda (H4: (eq T (THead 
 (Bind b) u (lift (S O) O t1)) (TSort n))).(let H5 \def (eq_ind T (THead (Bind 
 H7))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda 
 (t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 
 (TSort n)) \to (eq T t2 t1)))).(\lambda (u: T).(\lambda (H4: (eq T (THead 
 (Bind b) u (lift (S O) O t1)) (TSort n))).(let H5 \def (eq_ind T (THead (Bind 

-b) u (lift (S O) O t1)) (\lambda (ee: T).(match ee in T return (\lambda (_: 
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
-(THead _ _ _) \Rightarrow True])) I (TSort n) H4) in (False_ind (eq T t2 
-(THead (Bind b) u (lift (S O) O t1))) H5)))))))))) (\lambda (t1: T).(\lambda 
-(t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq 
-T t2 t1)))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u t1) 
-(TSort n))).(let H4 \def (eq_ind T (THead (Flat Cast) u t1) (\lambda (ee: 
-T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I 
-(TSort n) H3) in (False_ind (eq T t2 (THead (Flat Cast) u t1)) H4)))))))) y x 
-H0))) H))).
-(* COMMENTS
-Initial nodes: 1045
-END *)
+b) u (lift (S O) O t1)) (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
+True])) I (TSort n) H4) in (False_ind (eq T t2 (THead (Bind b) u (lift (S O) 
+O t1))) H5)))))))))) (\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 
+t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 t1)))).(\lambda (u: 
+T).(\lambda (H3: (eq T (THead (Flat Cast) u t1) (TSort n))).(let H4 \def 
+(eq_ind T (THead (Flat Cast) u t1) (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
+True])) I (TSort n) H3) in (False_ind (eq T t2 (THead (Flat Cast) u t1)) 
+H4)))))))) y x H0))) H))).

 
 theorem pr0_gen_lref:
  \forall (x: T).(\forall (n: nat).((pr0 (TLRef n) x) \to (eq T x (TLRef n))))
 
 theorem pr0_gen_lref:
  \forall (x: T).(\forall (n: nat).((pr0 (TLRef n) x) \to (eq T x (TLRef n))))


@@ -94,57 +123,51 @@ t H2)))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda
 (_: (((eq T u1 (TLRef n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2: 
 T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (eq T t2 
 t1)))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u1 t1) (TLRef n))).(let 
 (_: (((eq T u1 (TLRef n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2: 
 T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (eq T t2 
 t1)))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u1 t1) (TLRef n))).(let 

-H6 \def (eq_ind T (THead k u1 t1) (\lambda (ee: T).(match ee in T return 
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H5) in 
-(False_ind (eq T (THead k u2 t2) (THead k u1 t1)) H6)))))))))))) (\lambda (u: 
-T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: 
-(((eq T v1 (TLRef n)) \to (eq T v2 v1)))).(\lambda (t1: T).(\lambda (t2: 
-T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (eq T t2 
-t1)))).(\lambda (H5: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t1)) 
-(TLRef n))).(let H6 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u 
-t1)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) 
-\Rightarrow True])) I (TLRef n) H5) in (False_ind (eq T (THead (Bind Abbr) v2 
-t2) (THead (Flat Appl) v1 (THead (Bind Abst) u t1))) H6)))))))))))) (\lambda 
-(b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: 
+H6 \def (eq_ind T (THead k u1 t1) (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
+True])) I (TLRef n) H5) in (False_ind (eq T (THead k u2 t2) (THead k u1 t1)) 
+H6)))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: 
+(pr0 v1 v2)).(\lambda (_: (((eq T v1 (TLRef n)) \to (eq T v2 v1)))).(\lambda 
+(t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 
+(TLRef n)) \to (eq T t2 t1)))).(\lambda (H5: (eq T (THead (Flat Appl) v1 
+(THead (Bind Abst) u t1)) (TLRef n))).(let H6 \def (eq_ind T (THead (Flat 
+Appl) v1 (THead (Bind Abst) u t1)) (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
+True])) I (TLRef n) H5) in (False_ind (eq T (THead (Bind Abbr) v2 t2) (THead 
+(Flat Appl) v1 (THead (Bind Abst) u t1))) H6)))))))))))) (\lambda (b: 
+B).(\lambda (_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: 

 T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (TLRef n)) \to (eq T v2 
 v1)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda 
 (_: (((eq T u1 (TLRef n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2: 
 T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (eq T t2 
 t1)))).(\lambda (H8: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t1)) 
 (TLRef n))).(let H9 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 
 T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (TLRef n)) \to (eq T v2 
 v1)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda 
 (_: (((eq T u1 (TLRef n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2: 
 T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (eq T t2 
 t1)))).(\lambda (H8: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t1)) 
 (TLRef n))).(let H9 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 

-t1)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) 
-\Rightarrow True])) I (TLRef n) H8) in (False_ind (eq T (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Flat Appl) v1 (THead (Bind 
-b) u1 t1))) H9))))))))))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda 
-(_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 (TLRef n)) \to (eq T u2 
-u1)))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda 
-(_: (((eq T t1 (TLRef n)) \to (eq T t2 t1)))).(\lambda (w: T).(\lambda (_: 
-(subst0 O u2 t2 w)).(\lambda (H6: (eq T (THead (Bind Abbr) u1 t1) (TLRef 
-n))).(let H7 \def (eq_ind T (THead (Bind Abbr) u1 t1) (\lambda (ee: T).(match 
-ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | 
-(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) 
-H6) in (False_ind (eq T (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u1 t1)) 
+t1)) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H8) in 
+(False_ind (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) 
+(THead (Flat Appl) v1 (THead (Bind b) u1 t1))) H9))))))))))))))))) (\lambda 
+(u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 
+(TLRef n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: 
+(pr0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (eq T t2 t1)))).(\lambda 
+(w: T).(\lambda (_: (subst0 O u2 t2 w)).(\lambda (H6: (eq T (THead (Bind 
+Abbr) u1 t1) (TLRef n))).(let H7 \def (eq_ind T (THead (Bind Abbr) u1 t1) 
+(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H6) in 
+(False_ind (eq T (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u1 t1)) 

 H7))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda 
 (t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 
 (TLRef n)) \to (eq T t2 t1)))).(\lambda (u: T).(\lambda (H4: (eq T (THead 
 (Bind b) u (lift (S O) O t1)) (TLRef n))).(let H5 \def (eq_ind T (THead (Bind 
 H7))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda 
 (t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 
 (TLRef n)) \to (eq T t2 t1)))).(\lambda (u: T).(\lambda (H4: (eq T (THead 
 (Bind b) u (lift (S O) O t1)) (TLRef n))).(let H5 \def (eq_ind T (THead (Bind 

-b) u (lift (S O) O t1)) (\lambda (ee: T).(match ee in T return (\lambda (_: 
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
-(THead _ _ _) \Rightarrow True])) I (TLRef n) H4) in (False_ind (eq T t2 
-(THead (Bind b) u (lift (S O) O t1))) H5)))))))))) (\lambda (t1: T).(\lambda 
-(t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (eq 
-T t2 t1)))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u t1) 
-(TLRef n))).(let H4 \def (eq_ind T (THead (Flat Cast) u t1) (\lambda (ee: 
-T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I 
-(TLRef n) H3) in (False_ind (eq T t2 (THead (Flat Cast) u t1)) H4)))))))) y x 
-H0))) H))).
-(* COMMENTS
-Initial nodes: 1045
-END *)
+b) u (lift (S O) O t1)) (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
+True])) I (TLRef n) H4) in (False_ind (eq T t2 (THead (Bind b) u (lift (S O) 
+O t1))) H5)))))))))) (\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 
+t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (eq T t2 t1)))).(\lambda (u: 
+T).(\lambda (H3: (eq T (THead (Flat Cast) u t1) (TLRef n))).(let H4 \def 
+(eq_ind T (THead (Flat Cast) u t1) (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
+True])) I (TLRef n) H3) in (False_ind (eq T t2 (THead (Flat Cast) u t1)) 
+H4)))))))) y x H0))) H))).

 
 theorem pr0_gen_abst:
  \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Abst) u1 
 
 theorem pr0_gen_abst:
  \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Abst) u1 


@@ -180,20 +203,19 @@ t2)).(\lambda (H4: (((eq T t0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T
 (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: 
 T).(pr0 t1 t3))))))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u0 t0) 
 (THead (Bind Abst) u1 t1))).(let H6 \def (f_equal T K (\lambda (e: T).(match 
 (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: 
 T).(pr0 t1 t3))))))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u0 t0) 
 (THead (Bind Abst) u1 t1))).(let H6 \def (f_equal T K (\lambda (e: T).(match 

-e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) 
-\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u0 t0) (THead (Bind 
-Abst) u1 t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T 
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) 
-\Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k u0 t0) (THead (Bind 
-Abst) u1 t1) H5) in ((let H8 \def (f_equal T T (\lambda (e: T).(match e in T 
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) 
-\Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Bind 
-Abst) u1 t1) H5) in (\lambda (H9: (eq T u0 u1)).(\lambda (H10: (eq K k (Bind 
-Abst))).(eq_ind_r K (Bind Abst) (\lambda (k0: K).(ex3_2 T T (\lambda (u3: 
-T).(\lambda (t3: T).(eq T (THead k0 u2 t2) (THead (Bind Abst) u3 t3)))) 
-(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: 
-T).(pr0 t1 t3))))) (let H11 \def (eq_ind T t0 (\lambda (t: T).((eq T t (THead 
-(Bind Abst) u1 t1)) \to (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 
+e with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) 
+\Rightarrow k0])) (THead k u0 t0) (THead (Bind Abst) u1 t1) H5) in ((let H7 
+\def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u0 | 
+(TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k u0 t0) 
+(THead (Bind Abst) u1 t1) H5) in ((let H8 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | 
+(THead _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Bind Abst) u1 t1) H5) 
+in (\lambda (H9: (eq T u0 u1)).(\lambda (H10: (eq K k (Bind Abst))).(eq_ind_r 
+K (Bind Abst) (\lambda (k0: K).(ex3_2 T T (\lambda (u3: T).(\lambda (t3: 
+T).(eq T (THead k0 u2 t2) (THead (Bind Abst) u3 t3)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
+t3))))) (let H11 \def (eq_ind T t0 (\lambda (t: T).((eq T t (THead (Bind 
+Abst) u1 t1)) \to (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 

 (THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) 
 (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))))) H4 t1 H8) in (let H12 \def 
 (eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H3 t1 H8) in (let H13 \def (eq_ind T 
 (THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) 
 (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))))) H4 t1 H8) in (let H12 \def 
 (eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H3 t1 H8) in (let H13 \def (eq_ind T 


@@ -216,31 +238,29 @@ Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
 T).(\lambda (t3: T).(pr0 t1 t3))))))).(\lambda (H5: (eq T (THead (Flat Appl) 
 v1 (THead (Bind Abst) u t0)) (THead (Bind Abst) u1 t1))).(let H6 \def (eq_ind 
 T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (ee: T).(match ee 
 T).(\lambda (t3: T).(pr0 t1 t3))))))).(\lambda (H5: (eq T (THead (Flat Appl) 
 v1 (THead (Bind Abst) u t0)) (THead (Bind Abst) u1 t1))).(let H6 \def (eq_ind 
 T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (ee: T).(match ee 

-in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef 
-_) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return 
-(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow 
-True])])) I (THead (Bind Abst) u1 t1) H5) in (False_ind (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) (THead (Bind Abst) 
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr0 t1 t3)))) H6)))))))))))) (\lambda (b: B).(\lambda 
-(_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 
-v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind Abst) u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: 
-T).(pr0 t1 t2))))))).(\lambda (u0: T).(\lambda (u2: T).(\lambda (_: (pr0 u0 
-u2)).(\lambda (_: (((eq T u0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Bind Abst) u3 t2)))) 
-(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: 
-T).(pr0 t1 t2))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 
-t2)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u3 t3)))) 
-(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: 
-T).(pr0 t1 t3))))))).(\lambda (H8: (eq T (THead (Flat Appl) v1 (THead (Bind 
-b) u0 t0)) (THead (Bind Abst) u1 t1))).(let H9 \def (eq_ind T (THead (Flat 
-Appl) v1 (THead (Bind b) u0 t0)) (\lambda (ee: T).(match ee in T return 
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda 
-(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow 
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ 
+_) \Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat _) 
+\Rightarrow True])])) I (THead (Bind Abst) u1 t1) H5) in (False_ind (ex3_2 T 
+T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) (THead 
+(Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
+(_: T).(\lambda (t3: T).(pr0 t1 t3)))) H6)))))))))))) (\lambda (b: 
+B).(\lambda (_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: 
+T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abst) u1 
+t1)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind 
+Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
+T).(\lambda (t2: T).(pr0 t1 t2))))))).(\lambda (u0: T).(\lambda (u2: 
+T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead (Bind Abst) u1 
+t1)) \to (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Bind 
+Abst) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
+T).(\lambda (t2: T).(pr0 t1 t2))))))).(\lambda (t0: T).(\lambda (t2: 
+T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u1 
+t1)) \to (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
+Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
+T).(\lambda (t3: T).(pr0 t1 t3))))))).(\lambda (H8: (eq T (THead (Flat Appl) 
+v1 (THead (Bind b) u0 t0)) (THead (Bind Abst) u1 t1))).(let H9 \def (eq_ind T 
+(THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (\lambda (ee: T).(match ee with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
+\Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow 

 True])])) I (THead (Bind Abst) u1 t1) H8) in (False_ind (ex3_2 T T (\lambda 
 (u3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
 O) O v2) t2)) (THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: 
 True])])) I (THead (Bind Abst) u1 t1) H8) in (False_ind (ex3_2 T T (\lambda 
 (u3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
 O) O v2) t2)) (THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: 


@@ -256,70 +276,52 @@ t2)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T
 T).(pr0 t1 t3))))))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t2 
 w)).(\lambda (H6: (eq T (THead (Bind Abbr) u0 t0) (THead (Bind Abst) u1 
 t1))).(let H7 \def (eq_ind T (THead (Bind Abbr) u0 t0) (\lambda (ee: 
 T).(pr0 t1 t3))))))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t2 
 w)).(\lambda (H6: (eq T (THead (Bind Abbr) u0 t0) (THead (Bind Abst) u1 
 t1))).(let H7 \def (eq_ind T (THead (Bind Abbr) u0 t0) (\lambda (ee: 

-T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K 
-return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B return 
-(\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | 
-Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (THead (Bind 
-Abst) u1 t1) H6) in (False_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t3: 
-T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind Abst) u3 t3)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
-t3)))) H7))))))))))))) (\lambda (b: B).(\lambda (H1: (not (eq B b 
-Abst))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda 
-(H3: (((eq T t0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
-t3))))))).(\lambda (u: T).(\lambda (H4: (eq T (THead (Bind b) u (lift (S O) O 
-t0)) (THead (Bind Abst) u1 t1))).(let H5 \def (f_equal T B (\lambda (e: 
-T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | 
-(TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return 
-(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow 
-b])])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abst) u1 t1) H4) in 
-((let H6 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: 
-T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) 
-\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abst) u1 
-t1) H4) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T return 
-(\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat 
-\to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow 
-(TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true 
-\Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t3) \Rightarrow 
-(THead k (lref_map f d u0) (lref_map f (s k d) t3))]) in lref_map) (\lambda 
-(x0: nat).(plus x0 (S O))) O t0) | (TLRef _) \Rightarrow ((let rec lref_map 
-(f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) 
-\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with 
-[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t3) 
-\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t3))]) in 
-lref_map) (\lambda (x0: nat).(plus x0 (S O))) O t0) | (THead _ _ t) 
-\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abst) u1 
-t1) H4) in (\lambda (_: (eq T u u1)).(\lambda (H9: (eq B b Abst)).(let H10 
-\def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H1 Abst H9) in (let 
-H11 \def (eq_ind_r T t1 (\lambda (t: T).((eq T t0 (THead (Bind Abst) u1 t)) 
-\to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) 
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr0 t t3)))))) H3 (lift (S O) O t0) H7) in (eq_ind T 
-(lift (S O) O t0) (\lambda (t: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t t3))))) (let H12 
-\def (match (H10 (refl_equal B Abst)) in False return (\lambda (_: 
-False).(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
+T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
+| (THead k _ _) \Rightarrow (match k with [(Bind b) \Rightarrow (match b with 
+[Abbr \Rightarrow True | Abst \Rightarrow False | Void \Rightarrow False]) | 
+(Flat _) \Rightarrow False])])) I (THead (Bind Abst) u1 t1) H6) in (False_ind 
+(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w) 
+(THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) 
+(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) H7))))))))))))) (\lambda (b: 
+B).(\lambda (H1: (not (eq B b Abst))).(\lambda (t0: T).(\lambda (t2: 
+T).(\lambda (_: (pr0 t0 t2)).(\lambda (H3: (((eq T t0 (THead (Bind Abst) u1 
+t1)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 

 Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
 Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 

-T).(\lambda (t3: T).(pr0 (lift (S O) O t0) t3))))) with []) in H12) t1 
-H7)))))) H6)) H5)))))))))) (\lambda (t0: T).(\lambda (t2: T).(\lambda (_: 
-(pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T 
-T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: 
-T).(pr0 t1 t3))))))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u 
-t0) (THead (Bind Abst) u1 t1))).(let H4 \def (eq_ind T (THead (Flat Cast) u 
-t0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort 
-_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) 
-\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u1 
-t1) H3) in (False_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 
+T).(\lambda (t3: T).(pr0 t1 t3))))))).(\lambda (u: T).(\lambda (H4: (eq T 
+(THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abst) u1 t1))).(let H5 \def 
+(f_equal T B (\lambda (e: T).(match e with [(TSort _) \Rightarrow b | (TLRef 
+_) \Rightarrow b | (THead k _ _) \Rightarrow (match k with [(Bind b0) 
+\Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead (Bind b) u (lift (S O) O 
+t0)) (THead (Bind Abst) u1 t1) H4) in ((let H6 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead 
+_ t _) \Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind 
+Abst) u1 t1) H4) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow (lref_map (\lambda (x0: nat).(plus x0 (S O))) O t0) | 
+(TLRef _) \Rightarrow (lref_map (\lambda (x0: nat).(plus x0 (S O))) O t0) | 
+(THead _ _ t) \Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead 
+(Bind Abst) u1 t1) H4) in (\lambda (_: (eq T u u1)).(\lambda (H9: (eq B b 
+Abst)).(let H10 \def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H1 
+Abst H9) in (let H11 \def (eq_ind_r T t1 (\lambda (t: T).((eq T t0 (THead 
+(Bind Abst) u1 t)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 

 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) 
 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) 

-(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) H4)))))))) y x H0))) H)))).
-(* COMMENTS
-Initial nodes: 2838
-END *)
+(\lambda (_: T).(\lambda (t3: T).(pr0 t t3)))))) H3 (lift (S O) O t0) H7) in 
+(eq_ind T (lift (S O) O t0) (\lambda (t: T).(ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t 
+t3))))) (let H12 \def (match (H10 (refl_equal B Abst)) in False with []) in 
+H12) t1 H7)))))) H6)) H5)))))))))) (\lambda (t0: T).(\lambda (t2: T).(\lambda 
+(_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u1 t1)) \to 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(pr0 t1 t3))))))).(\lambda (u: T).(\lambda (H3: (eq T 
+(THead (Flat Cast) u t0) (THead (Bind Abst) u1 t1))).(let H4 \def (eq_ind T 
+(THead (Flat Cast) u t0) (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
+(match k with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I 
+(THead (Bind Abst) u1 t1) H3) in (False_ind (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
+t3)))) H4)))))))) y x H0))) H)))).

 
 theorem pr0_gen_appl:
  \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Flat Appl) u1 
 
 theorem pr0_gen_appl:
  \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Flat Appl) u1 


@@ -470,42 +472,21 @@ T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2)))))))
 (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
 T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (k: K).(\lambda (H5: (eq 
 T (THead k u0 t0) (THead (Flat Appl) u1 t1))).(let H6 \def (f_equal T K 
 (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
 T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (k: K).(\lambda (H5: (eq 
 T (THead k u0 t0) (THead (Flat Appl) u1 t1))).(let H6 \def (f_equal T K 

-(\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) 
-\Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) 
-(THead k u0 t0) (THead (Flat Appl) u1 t1) H5) in ((let H7 \def (f_equal T T 
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) 
-(THead k u0 t0) (THead (Flat Appl) u1 t1) H5) in ((let H8 \def (f_equal T T 
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) 
-(THead k u0 t0) (THead (Flat Appl) u1 t1) H5) in (\lambda (H9: (eq T u0 
-u1)).(\lambda (H10: (eq K k (Flat Appl))).(eq_ind_r K (Flat Appl) (\lambda 
-(k0: K).(or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead k0 u2 
-t2) (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 
-u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda 
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead 
-(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
-T).(\lambda (t3: T).(eq T (THead k0 u2 t2) (THead (Bind Abbr) u3 t3)))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 
-u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: 
-T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B 
-b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
-(v2: T).(\lambda (t3: T).(eq T (THead k0 u2 t2) (THead (Bind b) v2 (THead 
-(Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 
-u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(t3: T).(pr0 z1 t3)))))))))) (let H11 \def (eq_ind T t0 (\lambda (t: T).((eq 
-T t (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda 
-(t3: T).(eq T t2 (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: 
-T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T 
-T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
-T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow k | (TLRef _) 
+\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u0 t0) (THead (Flat 
+Appl) u1 t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) 
+\Rightarrow t])) (THead k u0 t0) (THead (Flat Appl) u1 t1) H5) in ((let H8 
+\def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t0 | 
+(TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u0 t0) 
+(THead (Flat Appl) u1 t1) H5) in (\lambda (H9: (eq T u0 u1)).(\lambda (H10: 
+(eq K k (Flat Appl))).(eq_ind_r K (Flat Appl) (\lambda (k0: K).(or3 (ex3_2 T 
+T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead k0 u2 t2) (THead (Flat Appl) 
+u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
+T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: 
+T).(eq T (THead k0 u2 t2) (THead (Bind Abbr) u3 t3)))))) (\lambda (_: 

 T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda 
 (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) 
 (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
 T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda 
 (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) 
 (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 


@@ -513,36 +494,56 @@ T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda
 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
 (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
 T).(\lambda (_: T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T 
 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
 (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
 T).(\lambda (_: T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T 

-t2 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) 
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: 
-T).(\lambda (_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 
-v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))))))) H4 t1 H8) in (let 
-H12 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H3 t1 H8) in (let H13 \def 
-(eq_ind T u0 (\lambda (t: T).((eq T t (THead (Flat Appl) u1 t1)) \to (or3 
-(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead (Flat Appl) u3 
+(THead k0 u2 t2) (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u3) 
+t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
+T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda 
+(y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 
+y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))))) (let H11 \def 
+(eq_ind T t0 (\lambda (t: T).((eq T t (THead (Flat Appl) u1 t1)) \to (or3 
+(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u3 

 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
 T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
 (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 
 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: 
 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
 T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
 (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 
 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: 

-T).(eq T u2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: 
+T).(eq T t2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: 

 T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda 
 (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T 
 (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
 T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda 
 (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T 
 (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 

-T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T u2 (THead (Bind 
+T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 

 b) v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: 
 B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda 
 (_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
 T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) 
 (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
 b) v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: 
 B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda 
 (_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
 T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) 
 (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 

-T).(\lambda (t3: T).(pr0 z1 t3))))))))))) H2 u1 H9) in (let H14 \def (eq_ind 
-T u0 (\lambda (t: T).(pr0 t u2)) H1 u1 H9) in (or3_intro0 (ex3_2 T T (\lambda 
-(u3: T).(\lambda (t3: T).(eq T (THead (Flat Appl) u2 t2) (THead (Flat Appl) 
-u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
+T).(\lambda (t3: T).(pr0 z1 t3))))))))))) H4 t1 H8) in (let H12 \def (eq_ind 
+T t0 (\lambda (t: T).(pr0 t t2)) H3 t1 H8) in (let H13 \def (eq_ind T u0 
+(\lambda (t: T).((eq T t (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead (Flat Appl) u3 t3)))) 
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: 
+T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda 
+(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead (Bind 
+Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
+(_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T u2 (THead (Bind b) v2 (THead 
+(Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 
+u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(t3: T).(pr0 z1 t3))))))))))) H2 u1 H9) in (let H14 \def (eq_ind T u0 
+(\lambda (t: T).(pr0 t u2)) H1 u1 H9) in (or3_intro0 (ex3_2 T T (\lambda (u3: 
+T).(\lambda (t3: T).(eq T (THead (Flat Appl) u2 t2) (THead (Flat Appl) u3 
+t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 

 T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
 (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 
 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: 
 T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
 (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 
 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: 


@@ -606,15 +607,14 @@ T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3)))))))
 (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
 T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (H5: (eq T (THead (Flat 
 Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Appl) u1 t1))).(let H6 \def 
 (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
 T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (H5: (eq T (THead (Flat 
 Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Appl) u1 t1))).(let H6 \def 

-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t _) 
-\Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat 
-Appl) u1 t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T 
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow (THead (Bind Abst) u 
-t0) | (TLRef _) \Rightarrow (THead (Bind Abst) u t0) | (THead _ _ t) 
-\Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat 
-Appl) u1 t1) H5) in (\lambda (H8: (eq T v1 u1)).(let H9 \def (eq_ind T v1 
-(\lambda (t: T).((eq T t (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow v1 | (TLRef 
+_) \Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) v1 
+(THead (Bind Abst) u t0)) (THead (Flat Appl) u1 t1) H5) in ((let H7 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow (THead 
+(Bind Abst) u t0) | (TLRef _) \Rightarrow (THead (Bind Abst) u t0) | (THead _ 
+_ t) \Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead 
+(Flat Appl) u1 t1) H5) in (\lambda (H8: (eq T v1 u1)).(let H9 \def (eq_ind T 
+v1 (\lambda (t: T).((eq T t (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T 

 (\lambda (u2: T).(\lambda (t3: T).(eq T v2 (THead (Flat Appl) u2 t3)))) 
 (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: 
 T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda 
 (\lambda (u2: T).(\lambda (t3: T).(eq T v2 (THead (Flat Appl) u2 t3)))) 
 (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: 
 T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda 


@@ -786,19 +786,18 @@ T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3)))))))
 (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
 T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (H8: (eq T (THead (Flat 
 Appl) v1 (THead (Bind b) u0 t0)) (THead (Flat Appl) u1 t1))).(let H9 \def 
 (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
 T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (H8: (eq T (THead (Flat 
 Appl) v1 (THead (Bind b) u0 t0)) (THead (Flat Appl) u1 t1))).(let H9 \def 

-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t _) 
-\Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Flat 
-Appl) u1 t1) H8) in ((let H10 \def (f_equal T T (\lambda (e: T).(match e in T 
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow (THead (Bind b) u0 t0) 
-| (TLRef _) \Rightarrow (THead (Bind b) u0 t0) | (THead _ _ t) \Rightarrow 
-t])) (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Flat Appl) u1 t1) 
-H8) in (\lambda (H11: (eq T v1 u1)).(let H12 \def (eq_ind T v1 (\lambda (t: 
-T).((eq T t (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t3: T).(eq T v2 (THead (Flat Appl) u3 t3)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
-t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow v1 | (TLRef 
+_) \Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) v1 
+(THead (Bind b) u0 t0)) (THead (Flat Appl) u1 t1) H8) in ((let H10 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow (THead 
+(Bind b) u0 t0) | (TLRef _) \Rightarrow (THead (Bind b) u0 t0) | (THead _ _ 
+t) \Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead 
+(Flat Appl) u1 t1) H8) in (\lambda (H11: (eq T v1 u1)).(let H12 \def (eq_ind 
+T v1 (\lambda (t: T).((eq T t (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t3: T).(eq T v2 (THead (Flat Appl) u3 t3)))) 
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: 
+T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda 
+(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 

 T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T v2 (THead (Bind 
 Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
 (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
 T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T v2 (THead (Bind 
 Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
 (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 


@@ -975,78 +974,32 @@ T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2)))))))
 T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (w: T).(\lambda (_: 
 (subst0 O u2 t2 w)).(\lambda (H6: (eq T (THead (Bind Abbr) u0 t0) (THead 
 (Flat Appl) u1 t1))).(let H7 \def (eq_ind T (THead (Bind Abbr) u0 t0) 
 T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (w: T).(\lambda (_: 
 (subst0 O u2 t2 w)).(\lambda (H6: (eq T (THead (Bind Abbr) u0 t0) (THead 
 (Flat Appl) u1 t1))).(let H7 \def (eq_ind T (THead (Bind Abbr) u0 t0) 

-(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
-(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | 
-(Flat _) \Rightarrow False])])) I (THead (Flat Appl) u1 t1) H6) in (False_ind 
-(or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 
-w) (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 
-u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda 
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead 
-(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
-T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u3 
-t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: 
-T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
-(t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B 
-b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
-(v2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind b) v2 
-(THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 
-u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(t3: T).(pr0 z1 t3))))))))) H7))))))))))))) (\lambda (b: B).(\lambda (_: (not 
-(eq B b Abst))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 
-t2)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: 
-T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda 
-(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
-b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b0) v2 (THead 
-(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(t3: T).(pr0 z1 t3)))))))))))).(\lambda (u: T).(\lambda (H4: (eq T (THead 
-(Bind b) u (lift (S O) O t0)) (THead (Flat Appl) u1 t1))).(let H5 \def 
-(eq_ind T (THead (Bind b) u (lift (S O) O t0)) (\lambda (ee: T).(match ee in 
-T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda 
-(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow 
-False])])) I (THead (Flat Appl) u1 t1) H4) in (False_ind (or3 (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: 
-T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda 
-(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
-b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b0) v2 (THead 
-(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(t3: T).(pr0 z1 t3))))))))) H5)))))))))) (\lambda (t0: T).(\lambda (t2: 
-T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u1 
-t1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
+(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _) 
+\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) u1 
+t1) H6) in (False_ind (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T 
+(THead (Bind Abbr) u2 w) (THead (Flat Appl) u3 t3)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
+t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) 
+u2 w) (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T (THead (Bind 
+Abbr) u2 w) (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u3) 
+t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
+T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda 
+(y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 
+y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))))) H7))))))))))))) 
+(\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (t0: T).(\lambda 
+(t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Appl) 
+u1 t1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 

 (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
 (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: 
 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
 (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
 (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: 
 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 


@@ -1054,23 +1007,66 @@ Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
 (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
 T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda 
 (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T 
 (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
 T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda 
 (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T 

-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
+(\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: 

 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 

-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: 

 T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
 T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 

-b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: 
+b0) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: 

 B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda 
 (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
 T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) 
 (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
 B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda 
 (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
 T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) 
 (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 

-T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (u: T).(\lambda (H3: (eq 
-T (THead (Flat Cast) u t0) (THead (Flat Appl) u1 t1))).(let H4 \def (eq_ind T 
-(THead (Flat Cast) u t0) (\lambda (ee: T).(match ee in T return (\lambda (_: 
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
-(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with 
-[(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f in F return 
-(\lambda (_: F).Prop) with [Appl \Rightarrow False | Cast \Rightarrow 
+T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (u: T).(\lambda (H4: (eq 
+T (THead (Bind b) u (lift (S O) O t0)) (THead (Flat Appl) u1 t1))).(let H5 
+\def (eq_ind T (THead (Bind b) u (lift (S O) O t0)) (\lambda (ee: T).(match 
+ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k 
+_ _) \Rightarrow (match k with [(Bind _) \Rightarrow True | (Flat _) 
+\Rightarrow False])])) I (THead (Flat Appl) u1 t1) H4) in (False_ind (or3 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: 
+T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T 
+(\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
+(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
+b0) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda 
+(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (t3: T).(pr0 z1 t3))))))))) H5)))))))))) (\lambda (t0: 
+T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead 
+(Flat Appl) u1 t1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq 
+T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 
+u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda 
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead 
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda 
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) 
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T 
+t2 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
+T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 
+v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (u: 
+T).(\lambda (H3: (eq T (THead (Flat Cast) u t0) (THead (Flat Appl) u1 
+t1))).(let H4 \def (eq_ind T (THead (Flat Cast) u t0) (\lambda (ee: T).(match 
+ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k 
+_ _) \Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat f) 
+\Rightarrow (match f with [Appl \Rightarrow False | Cast \Rightarrow 

 True])])])) I (THead (Flat Appl) u1 t1) H3) in (False_ind (or3 (ex3_2 T T 
 (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) 
 (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: 
 True])])])) I (THead (Flat Appl) u1 t1) H3) in (False_ind (or3 (ex3_2 T T 
 (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) 
 (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: 


@@ -1091,9 +1087,6 @@ u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
 T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: 
 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
 (t3: T).(pr0 z1 t3))))))))) H4)))))))) y x H0))) H)))).
 T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: 
 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
 (t3: T).(pr0 z1 t3))))))))) H4)))))))) y x H0))) H)))).

-(* COMMENTS
-Initial nodes: 12299
-END *)

 
 theorem pr0_gen_cast:
  \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Flat Cast) u1 
 
 theorem pr0_gen_cast:
  \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Flat Cast) u1 


@@ -1133,32 +1126,31 @@ T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u3 t3)))) (\lambda (u3:
 T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
 t3)))) (pr0 t1 t2))))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u0 t0) 
 (THead (Flat Cast) u1 t1))).(let H6 \def (f_equal T K (\lambda (e: T).(match 
 T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
 t3)))) (pr0 t1 t2))))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u0 t0) 
 (THead (Flat Cast) u1 t1))).(let H6 \def (f_equal T K (\lambda (e: T).(match 

-e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) 
-\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u0 t0) (THead (Flat 
-Cast) u1 t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T 
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) 
-\Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k u0 t0) (THead (Flat 
-Cast) u1 t1) H5) in ((let H8 \def (f_equal T T (\lambda (e: T).(match e in T 
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) 
-\Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Flat 
-Cast) u1 t1) H5) in (\lambda (H9: (eq T u0 u1)).(\lambda (H10: (eq K k (Flat 
-Cast))).(eq_ind_r K (Flat Cast) (\lambda (k0: K).(or (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t3: T).(eq T (THead k0 u2 t2) (THead (Flat Cast) u3 t3)))) 
-(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: 
-T).(pr0 t1 t3)))) (pr0 t1 (THead k0 u2 t2)))) (let H11 \def (eq_ind T t0 
-(\lambda (t: T).((eq T t (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u3 t3)))) 
-(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: 
-T).(pr0 t1 t3)))) (pr0 t1 t2)))) H4 t1 H8) in (let H12 \def (eq_ind T t0 
-(\lambda (t: T).(pr0 t t2)) H3 t1 H8) in (let H13 \def (eq_ind T u0 (\lambda 
-(t: T).((eq T t (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t3: T).(eq T u2 (THead (Flat Cast) u3 t3)))) (\lambda (u3: 
+e with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) 
+\Rightarrow k0])) (THead k u0 t0) (THead (Flat Cast) u1 t1) H5) in ((let H7 
+\def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u0 | 
+(TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k u0 t0) 
+(THead (Flat Cast) u1 t1) H5) in ((let H8 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | 
+(THead _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Flat Cast) u1 t1) H5) 
+in (\lambda (H9: (eq T u0 u1)).(\lambda (H10: (eq K k (Flat Cast))).(eq_ind_r 
+K (Flat Cast) (\lambda (k0: K).(or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: 
+T).(eq T (THead k0 u2 t2) (THead (Flat Cast) u3 t3)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
+t3)))) (pr0 t1 (THead k0 u2 t2)))) (let H11 \def (eq_ind T t0 (\lambda (t: 
+T).((eq T t (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: 
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u3 t3)))) (\lambda (u3: 

 T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
 T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 

-t3)))) (pr0 t1 u2)))) H2 u1 H9) in (let H14 \def (eq_ind T u0 (\lambda (t: 
-T).(pr0 t u2)) H1 u1 H9) in (or_introl (ex3_2 T T (\lambda (u3: T).(\lambda 
-(t3: T).(eq T (THead (Flat Cast) u2 t2) (THead (Flat Cast) u3 t3)))) (\lambda 
-(u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 
-t1 t3)))) (pr0 t1 (THead (Flat Cast) u2 t2)) (ex3_2_intro T T (\lambda (u3: 
+t3)))) (pr0 t1 t2)))) H4 t1 H8) in (let H12 \def (eq_ind T t0 (\lambda (t: 
+T).(pr0 t t2)) H3 t1 H8) in (let H13 \def (eq_ind T u0 (\lambda (t: T).((eq T 
+t (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda 
+(t3: T).(eq T u2 (THead (Flat Cast) u3 t3)))) (\lambda (u3: T).(\lambda (_: 
+T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 
+u2)))) H2 u1 H9) in (let H14 \def (eq_ind T u0 (\lambda (t: T).(pr0 t u2)) H1 
+u1 H9) in (or_introl (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T 
+(THead (Flat Cast) u2 t2) (THead (Flat Cast) u3 t3)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
+t3)))) (pr0 t1 (THead (Flat Cast) u2 t2)) (ex3_2_intro T T (\lambda (u3: 

 T).(\lambda (t3: T).(eq T (THead (Flat Cast) u2 t2) (THead (Flat Cast) u3 
 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
 T).(\lambda (t3: T).(pr0 t1 t3))) u2 t2 (refl_equal T (THead (Flat Cast) u2 
 T).(\lambda (t3: T).(eq T (THead (Flat Cast) u2 t2) (THead (Flat Cast) u3 
 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
 T).(\lambda (t3: T).(pr0 t1 t3))) u2 t2 (refl_equal T (THead (Flat Cast) u2 


@@ -1173,535 +1165,88 @@ T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2:
 T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
 t3)))) (pr0 t1 t2))))).(\lambda (H5: (eq T (THead (Flat Appl) v1 (THead (Bind 
 Abst) u t0)) (THead (Flat Cast) u1 t1))).(let H6 \def (eq_ind T (THead (Flat 
 T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
 t3)))) (pr0 t1 t2))))).(\lambda (H5: (eq T (THead (Flat Appl) v1 (THead (Bind 
 Abst) u t0)) (THead (Flat Cast) u1 t1))).(let H6 \def (eq_ind T (THead (Flat 

-Appl) v1 (THead (Bind Abst) u t0)) (\lambda (ee: T).(match ee in T return 
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda 
-(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f 
-in F return (\lambda (_: F).Prop) with [Appl \Rightarrow True | Cast 
-\Rightarrow False])])])) I (THead (Flat Cast) u1 t1) H5) in (False_ind (or 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) 
-(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (THead (Bind Abbr) v2 
-t2))) H6)))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b 
-Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda 
-(_: (((eq T v1 (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T v2 (THead (Flat Cast) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 
-t2)))) (pr0 t1 v2))))).(\lambda (u0: T).(\lambda (u2: T).(\lambda (_: (pr0 u0 
-u2)).(\lambda (_: (((eq T u0 (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Cast) u3 t2)))) 
-(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: 
-T).(pr0 t1 t2)))) (pr0 t1 u2))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda 
-(_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) u1 t1)) \to (or 
-(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u3 
-t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
-T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2))))).(\lambda (H8: (eq T (THead 
-(Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Flat Cast) u1 t1))).(let H9 
-\def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (\lambda (ee: 
-T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K 
-return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) 
-\Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl \Rightarrow 
-True | Cast \Rightarrow False])])])) I (THead (Flat Cast) u1 t1) H8) in 
-(False_ind (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Flat Cast) u3 
-t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
-T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t2)))) H9))))))))))))))))) (\lambda (u0: T).(\lambda 
-(u2: T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead (Flat Cast) 
-u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead 
-(Flat Cast) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda 
-(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 u2))))).(\lambda (t0: 
-T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead 
-(Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq 
-T t2 (THead (Flat Cast) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 
-u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 
-t2))))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t2 w)).(\lambda (H6: (eq T 
-(THead (Bind Abbr) u0 t0) (THead (Flat Cast) u1 t1))).(let H7 \def (eq_ind T 
-(THead (Bind Abbr) u0 t0) (\lambda (ee: T).(match ee in T return (\lambda (_: 
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
-(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with 
-[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat 
-Cast) u1 t1) H6) in (False_ind (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: 
-T).(eq T (THead (Bind Abbr) u2 w) (THead (Flat Cast) u3 t3)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
-t3)))) (pr0 t1 (THead (Bind Abbr) u2 w))) H7))))))))))))) (\lambda (b: 
-B).(\lambda (_: (not (eq B b Abst))).(\lambda (t0: T).(\lambda (t2: 
-T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) u1 
-t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
-(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
-(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2))))).(\lambda (u: 
-T).(\lambda (H4: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead (Flat 
-Cast) u1 t1))).(let H5 \def (eq_ind T (THead (Bind b) u (lift (S O) O t0)) 
-(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) 
+Appl) v1 (THead (Bind Abst) u t0)) (\lambda (ee: T).(match ee with [(TSort _) 

 \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
 \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 

-(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | 
-(Flat _) \Rightarrow False])])) I (THead (Flat Cast) u1 t1) H4) in (False_ind 
-(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) 
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2)) H5)))))))))) (\lambda (t0: 
-T).(\lambda (t2: T).(\lambda (H1: (pr0 t0 t2)).(\lambda (H2: (((eq T t0 
-(THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 
-t2))))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u t0) (THead 
-(Flat Cast) u1 t1))).(let H4 \def (f_equal T T (\lambda (e: T).(match e in T 
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) 
-\Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Flat Cast) u t0) 
-(THead (Flat Cast) u1 t1) H3) in ((let H5 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | 
-(TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Flat Cast) 
-u t0) (THead (Flat Cast) u1 t1) H3) in (\lambda (_: (eq T u u1)).(let H7 \def 
-(eq_ind T t0 (\lambda (t: T).((eq T t (THead (Flat Cast) u1 t1)) \to (or 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2)))) H2 t1 H5) in (let H8 \def 
-(eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 t1 H5) in (or_intror (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: 
-T).(pr0 t1 t3)))) (pr0 t1 t2) H8))))) H4)))))))) y x H0))) H)))).
-(* COMMENTS
-Initial nodes: 2911
-END *)
-
-theorem pr0_gen_abbr:
- \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Abbr) u1 
-t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead 
-(Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
-(u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y)) 
-(\lambda (y: T).(subst0 O u2 y t2))))))) (pr0 t1 (lift (S O) O x))))))
-\def
- \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead 
-(Bind Abbr) u1 t1) x)).(insert_eq T (THead (Bind Abbr) u1 t1) (\lambda (t: 
-T).(pr0 t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
-T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 
-u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda 
-(y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y t2))))))) (pr0 t1 (lift (S 
-O) O x)))) (\lambda (y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: 
-T).(\lambda (t0: T).((eq T t (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: 
-T).(or (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: 
-T).(subst0 O u2 y0 t2))))))) (pr0 t1 (lift (S O) O t0)))))) (\lambda (t: 
-T).(\lambda (H1: (eq T t (THead (Bind Abbr) u1 t1))).(let H2 \def (f_equal T 
-T (\lambda (e: T).e) t (THead (Bind Abbr) u1 t1) H1) in (eq_ind_r T (THead 
-(Bind Abbr) u1 t1) (\lambda (t0: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T 
-(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 t2))))))) (pr0 
-t1 (lift (S O) O t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
-T).(eq T (THead (Bind Abbr) u1 t1) (THead (Bind Abbr) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0 
-t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 
-t2))))))) (pr0 t1 (lift (S O) O (THead (Bind Abbr) u1 t1))) (ex3_2_intro T T 
-(\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind Abbr) u1 t1) (THead 
-(Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
-(u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) 
-(\lambda (y0: T).(subst0 O u2 y0 t2)))))) u1 t1 (refl_equal T (THead (Bind 
-Abbr) u1 t1)) (pr0_refl u1) (or_introl (pr0 t1 t1) (ex2 T (\lambda (y0: 
-T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u1 y0 t1))) (pr0_refl t1)))) t 
-H2)))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 u2)).(\lambda 
-(H2: (((eq T u0 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t2: T).(eq T u2 (THead (Bind Abbr) u3 t2)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t2: T).(or (pr0 
-t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 
-t2))))))) (pr0 t1 (lift (S O) O u2)))))).(\lambda (t0: T).(\lambda (t2: 
-T).(\lambda (H3: (pr0 t0 t2)).(\lambda (H4: (((eq T t0 (THead (Bind Abbr) u1 
-t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead 
-(Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda 
-(u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) 
-(\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O 
-t2)))))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u0 t0) (THead (Bind 
-Abbr) u1 t1))).(let H6 \def (f_equal T K (\lambda (e: T).(match e in T return 
-(\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | 
-(THead k0 _ _) \Rightarrow k0])) (THead k u0 t0) (THead (Bind Abbr) u1 t1) 
-H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T return 
-(\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 
-| (THead _ t _) \Rightarrow t])) (THead k u0 t0) (THead (Bind Abbr) u1 t1) 
-H5) in ((let H8 \def (f_equal T T (\lambda (e: T).(match e in T return 
-(\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 
-| (THead _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Bind Abbr) u1 t1) 
-H5) in (\lambda (H9: (eq T u0 u1)).(\lambda (H10: (eq K k (Bind 
-Abbr))).(eq_ind_r K (Bind Abbr) (\lambda (k0: K).(or (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t3: T).(eq T (THead k0 u2 t2) (THead (Bind Abbr) u3 t3)))) 
-(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: 
-T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: 
-T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O (THead k0 u2 t2))))) (let 
-H11 \def (eq_ind T t0 (\lambda (t: T).((eq T t (THead (Bind Abbr) u1 t1)) \to 
-(or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) 
-u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: 
-T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) 
-(\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O t2))))) H4 
-t1 H8) in (let H12 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H3 t1 H8) in 
-(let H13 \def (eq_ind T u0 (\lambda (t: T).((eq T t (THead (Bind Abbr) u1 
-t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead 
-(Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda 
-(u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) 
-(\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O u2))))) H2 
-u1 H9) in (let H14 \def (eq_ind T u0 (\lambda (t: T).(pr0 t u2)) H1 u1 H9) in 
-(or_introl (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind 
-Abbr) u2 t2) (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: 
-T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T 
-(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 
-t1 (lift (S O) O (THead (Bind Abbr) u2 t2))) (ex3_2_intro T T (\lambda (u3: 
-T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 t2) (THead (Bind Abbr) u3 
-t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: 
-T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) 
-(\lambda (y0: T).(subst0 O u3 y0 t3)))))) u2 t2 (refl_equal T (THead (Bind 
-Abbr) u2 t2)) H14 (or_introl (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) 
-(\lambda (y0: T).(subst0 O u2 y0 t2))) H12))))))) k H10)))) H7)) 
-H6)))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: 
-(pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abbr) u1 t1)) \to (or 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind Abbr) u2 
-t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: 
-T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) 
-(\lambda (y0: T).(subst0 O u2 y0 t2))))))) (pr0 t1 (lift (S O) O 
-v2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda 
-(_: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0 
-t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 
-t3))))))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (H5: (eq T (THead (Flat 
-Appl) v1 (THead (Bind Abst) u t0)) (THead (Bind Abbr) u1 t1))).(let H6 \def 
-(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (ee: 
-T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K 
-return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) 
-\Rightarrow True])])) I (THead (Bind Abbr) u1 t1) H5) in (False_ind (or 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) 
-(THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) 
-(\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: 
-T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S 
-O) O (THead (Bind Abbr) v2 t2)))) H6)))))))))))) (\lambda (b: B).(\lambda (_: 
-(not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 
-v2)).(\lambda (_: (((eq T v1 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind Abbr) u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: 
-T).(or (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: 
-T).(subst0 O u2 y0 t2))))))) (pr0 t1 (lift (S O) O v2)))))).(\lambda (u0: 
+(match k with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f 
+with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat 
+Cast) u1 t1) H5) in (False_ind (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
+T).(eq T (THead (Bind Abbr) v2 t2) (THead (Flat Cast) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
+t3)))) (pr0 t1 (THead (Bind Abbr) v2 t2))) H6)))))))))))) (\lambda (b: 
+B).(\lambda (_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: 
+T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Flat Cast) u1 
+t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead 
+(Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
+(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 v2))))).(\lambda (u0: 

 T).(\lambda (u2: T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead 
 T).(\lambda (u2: T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead 

-(Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq 
-T u2 (THead (Bind Abbr) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 
-u3))) (\lambda (u3: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y0: 
-T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t2))))))) (pr0 t1 (lift (S 
-O) O u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 
-t2)).(\lambda (_: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u3 t3)))) 
-(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: 
-T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: 
-T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (H8: (eq 
-T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Bind Abbr) u1 
-t1))).(let H9 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) 
-(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
-(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | 
-(Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1 t1) H8) in (False_ind 
-(or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Bind Abbr) u3 t3)))) 
-(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: 
-T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: 
-T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t2))))) H9))))))))))))))))) (\lambda (u0: 
-T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 u2)).(\lambda (H2: (((eq T u0 
-(THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: 
-T).(eq T u2 (THead (Bind Abbr) u3 t2)))) (\lambda (u3: T).(\lambda (_: 
-T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T 
-(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t2))))))) (pr0 
-t1 (lift (S O) O u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H3: 
-(pr0 t0 t2)).(\lambda (H4: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or 
-(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u3 
-t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: 
-T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) 
-(\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O 
-t2)))))).(\lambda (w: T).(\lambda (H5: (subst0 O u2 t2 w)).(\lambda (H6: (eq 
-T (THead (Bind Abbr) u0 t0) (THead (Bind Abbr) u1 t1))).(let H7 \def (f_equal 
-T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) 
-(THead (Bind Abbr) u0 t0) (THead (Bind Abbr) u1 t1) H6) in ((let H8 \def 
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) 
-\Rightarrow t])) (THead (Bind Abbr) u0 t0) (THead (Bind Abbr) u1 t1) H6) in 
-(\lambda (H9: (eq T u0 u1)).(let H10 \def (eq_ind T t0 (\lambda (t: T).((eq T 
-t (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda 
-(t3: T).(eq T t2 (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: 
-T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T 
-(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 
-t1 (lift (S O) O t2))))) H4 t1 H8) in (let H11 \def (eq_ind T t0 (\lambda (t: 
-T).(pr0 t t2)) H3 t1 H8) in (let H12 \def (eq_ind T u0 (\lambda (t: T).((eq T 
-t (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda 
-(t3: T).(eq T u2 (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: 
-T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T 
-(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 
-t1 (lift (S O) O u2))))) H2 u1 H9) in (let H13 \def (eq_ind T u0 (\lambda (t: 
-T).(pr0 t u2)) H1 u1 H9) in (or_introl (ex3_2 T T (\lambda (u3: T).(\lambda 
-(t3: T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u3 t3)))) (\lambda 
-(u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or 
-(pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O 
-u3 y0 t3))))))) (pr0 t1 (lift (S O) O (THead (Bind Abbr) u2 w))) (ex3_2_intro 
-T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w) (THead 
-(Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda 
-(u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) 
-(\lambda (y0: T).(subst0 O u3 y0 t3)))))) u2 w (refl_equal T (THead (Bind 
-Abbr) u2 w)) H13 (or_intror (pr0 t1 w) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) 
-(\lambda (y0: T).(subst0 O u2 y0 w))) (ex_intro2 T (\lambda (y0: T).(pr0 t1 
-y0)) (\lambda (y0: T).(subst0 O u2 y0 w)) t2 H11 H5)))))))))) H7))))))))))))) 
-(\lambda (b: B).(\lambda (H1: (not (eq B b Abst))).(\lambda (t0: T).(\lambda 
-(t2: T).(\lambda (H2: (pr0 t0 t2)).(\lambda (H3: (((eq T t0 (THead (Bind 
-Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 
-(THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) 
-(\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: 
-T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S 
-O) O t2)))))).(\lambda (u: T).(\lambda (H4: (eq T (THead (Bind b) u (lift (S 
-O) O t0)) (THead (Bind Abbr) u1 t1))).(let H5 \def (f_equal T B (\lambda (e: 
-T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | 
-(TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return 
-(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow 
-b])])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1 t1) H4) in 
-((let H6 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: 
-T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) 
-\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1 
-t1) H4) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T return 
-(\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat 
-\to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow 
-(TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true 
-\Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t3) \Rightarrow 
-(THead k (lref_map f d u0) (lref_map f (s k d) t3))]) in lref_map) (\lambda 
-(x0: nat).(plus x0 (S O))) O t0) | (TLRef _) \Rightarrow ((let rec lref_map 
-(f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) 
-\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with 
-[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t3) 
-\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t3))]) in 
-lref_map) (\lambda (x0: nat).(plus x0 (S O))) O t0) | (THead _ _ t) 
-\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1 
-t1) H4) in (\lambda (_: (eq T u u1)).(\lambda (H9: (eq B b Abbr)).(let H10 
-\def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H1 Abbr H9) in (let 
-H11 \def (eq_ind_r T t1 (\lambda (t: T).((eq T t0 (THead (Bind Abbr) u1 t)) 
-\to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
-Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: 
-T).(\lambda (t3: T).(or (pr0 t t3) (ex2 T (\lambda (y0: T).(pr0 t y0)) 
-(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t (lift (S O) O t2))))) H3 
-(lift (S O) O t0) H7) in (eq_ind T (lift (S O) O t0) (\lambda (t: T).(or 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: 
-T).(\lambda (t3: T).(or (pr0 t t3) (ex2 T (\lambda (y0: T).(pr0 t y0)) 
-(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t (lift (S O) O t2)))) 
-(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
-Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: 
-T).(\lambda (t3: T).(or (pr0 (lift (S O) O t0) t3) (ex2 T (\lambda (y0: 
-T).(pr0 (lift (S O) O t0) y0)) (\lambda (y0: T).(subst0 O u2 y0 t3))))))) 
-(pr0 (lift (S O) O t0) (lift (S O) O t2)) (pr0_lift t0 t2 H2 (S O) O)) t1 
-H7)))))) H6)) H5)))))))))) (\lambda (t0: T).(\lambda (t2: T).(\lambda (_: 
-(pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: 
-T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) 
-(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S O) O 
-t2)))))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u t0) (THead 
-(Bind Abbr) u1 t1))).(let H4 \def (eq_ind T (THead (Flat Cast) u t0) (\lambda 
-(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) 
+(Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq 
+T u2 (THead (Flat Cast) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 
+u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 
+u2))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda 
+(_: (((eq T t0 (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: 
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u3 t3)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
+t3)))) (pr0 t1 t2))))).(\lambda (H8: (eq T (THead (Flat Appl) v1 (THead (Bind 
+b) u0 t0)) (THead (Flat Cast) u1 t1))).(let H9 \def (eq_ind T (THead (Flat 
+Appl) v1 (THead (Bind b) u0 t0)) (\lambda (ee: T).(match ee with [(TSort _) 

 \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
 \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 

-(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | 
-(Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1 t1) H3) in (False_ind 
-(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) 
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: 
-T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) 
-(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S O) O t2))) 
-H4)))))))) y x H0))) H)))).
-(* COMMENTS
-Initial nodes: 4711
-END *)
-
-theorem pr0_gen_void:
- \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Void) u1 
-t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead 
-(Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
-(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O x))))))
-\def
- \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead 
-(Bind Void) u1 t1) x)).(insert_eq T (THead (Bind Void) u1 t1) (\lambda (t: 
-T).(pr0 t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
-T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 
-u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) 
-O x)))) (\lambda (y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: 
-T).(\lambda (t0: T).((eq T t (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Void) u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: 
-T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O t0)))))) (\lambda (t: T).(\lambda 
-(H1: (eq T t (THead (Bind Void) u1 t1))).(let H2 \def (f_equal T T (\lambda 
-(e: T).e) t (THead (Bind Void) u1 t1) H1) in (eq_ind_r T (THead (Bind Void) 
-u1 t1) (\lambda (t0: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq 
-T t0 (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 
-u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O 
-t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead 
-(Bind Void) u1 t1) (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 
-(lift (S O) O (THead (Bind Void) u1 t1))) (ex3_2_intro T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T (THead (Bind Void) u1 t1) (THead (Bind Void) u2 
-t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
-T).(\lambda (t2: T).(pr0 t1 t2))) u1 t1 (refl_equal T (THead (Bind Void) u1 
-t1)) (pr0_refl u1) (pr0_refl t1))) t H2)))) (\lambda (u0: T).(\lambda (u2: 
-T).(\lambda (H1: (pr0 u0 u2)).(\lambda (H2: (((eq T u0 (THead (Bind Void) u1 
-t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead 
-(Bind Void) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda 
-(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O 
-u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H3: (pr0 t0 
-t2)).(\lambda (H4: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u3 t3)))) 
-(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: 
-T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (k: K).(\lambda 
-(H5: (eq T (THead k u0 t0) (THead (Bind Void) u1 t1))).(let H6 \def (f_equal 
-T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) 
-\Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) 
-(THead k u0 t0) (THead (Bind Void) u1 t1) H5) in ((let H7 \def (f_equal T T 
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) 
-(THead k u0 t0) (THead (Bind Void) u1 t1) H5) in ((let H8 \def (f_equal T T 
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) 
-(THead k u0 t0) (THead (Bind Void) u1 t1) H5) in (\lambda (H9: (eq T u0 
-u1)).(\lambda (H10: (eq K k (Bind Void))).(eq_ind_r K (Bind Void) (\lambda 
-(k0: K).(or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead k0 u2 
-t2) (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 
-u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O 
-(THead k0 u2 t2))))) (let H11 \def (eq_ind T t0 (\lambda (t: T).((eq T t 
-(THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: 
-T).(eq T t2 (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: 
-T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 
-(lift (S O) O t2))))) H4 t1 H8) in (let H12 \def (eq_ind T t0 (\lambda (t: 
-T).(pr0 t t2)) H3 t1 H8) in (let H13 \def (eq_ind T u0 (\lambda (t: T).((eq T 
-t (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda 
-(t3: T).(eq T u2 (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: 
-T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 
-(lift (S O) O u2))))) H2 u1 H9) in (let H14 \def (eq_ind T u0 (\lambda (t: 
-T).(pr0 t u2)) H1 u1 H9) in (or_introl (ex3_2 T T (\lambda (u3: T).(\lambda 
-(t3: T).(eq T (THead (Bind Void) u2 t2) (THead (Bind Void) u3 t3)))) (\lambda 
-(u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 
-t1 t3)))) (pr0 t1 (lift (S O) O (THead (Bind Void) u2 t2))) (ex3_2_intro T T 
-(\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Void) u2 t2) (THead 
-(Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda 
-(_: T).(\lambda (t3: T).(pr0 t1 t3))) u2 t2 (refl_equal T (THead (Bind Void) 
-u2 t2)) H14 H12)))))) k H10)))) H7)) H6)))))))))))) (\lambda (u: T).(\lambda 
-(v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 
-(THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
-T).(eq T v2 (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 
-(lift (S O) O v2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 
-t2)).(\lambda (_: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: 
-T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (H5: (eq T (THead 
-(Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Bind Void) u1 t1))).(let H6 
-\def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (ee: 
-T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K 
-return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) 
-\Rightarrow True])])) I (THead (Bind Void) u1 t1) H5) in (False_ind (or 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) 
-(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O (THead 
-(Bind Abbr) v2 t2)))) H6)))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B 
-b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 
-v2)).(\lambda (_: (((eq T v1 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind Void) u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: 
-T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O v2)))))).(\lambda (u0: T).(\lambda 
-(u2: T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead (Bind Void) 
-u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead 
-(Bind Void) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda 
-(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O 
-u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda 
-(_: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u3 t3)))) (\lambda (u3: 
+(match k with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f 
+with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat 
+Cast) u1 t1) H8) in (False_ind (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: 
+T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (THead 
+(Flat Cast) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda 
+(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (THead (Bind b) u2 (THead 
+(Flat Appl) (lift (S O) O v2) t2)))) H9))))))))))))))))) (\lambda (u0: 
+T).(\lambda (u2: T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead 
+(Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq 
+T u2 (THead (Flat Cast) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 
+u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 
+u2))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda 
+(_: (((eq T t0 (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: 
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u3 t3)))) (\lambda (u3: 

 T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
 T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 

-t3)))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (H8: (eq T (THead (Flat Appl) 
-v1 (THead (Bind b) u0 t0)) (THead (Bind Void) u1 t1))).(let H9 \def (eq_ind T 
-(THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (\lambda (ee: T).(match ee in T 
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda 
-(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow 
-True])])) I (THead (Bind Void) u1 t1) H8) in (False_ind (or (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t2)) (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda 
-(_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 
-(lift (S O) O (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2))))) 
-H9))))))))))))))))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (_: (pr0 u0 
-u2)).(\lambda (_: (((eq T u0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Bind Void) u3 t2)))) 
-(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: 
-T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O u2)))))).(\lambda (t0: T).(\lambda 
-(t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Void) 
-u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead 
-(Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda 
-(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O 
-t2)))))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t2 w)).(\lambda (H6: (eq T 
-(THead (Bind Abbr) u0 t0) (THead (Bind Void) u1 t1))).(let H7 \def (eq_ind T 
-(THead (Bind Abbr) u0 t0) (\lambda (ee: T).(match ee in T return (\lambda (_: 
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
-(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with 
-[(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr 
-\Rightarrow True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat 
-_) \Rightarrow False])])) I (THead (Bind Void) u1 t1) H6) in (False_ind (or 
+t3)))) (pr0 t1 t2))))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t2 
+w)).(\lambda (H6: (eq T (THead (Bind Abbr) u0 t0) (THead (Flat Cast) u1 
+t1))).(let H7 \def (eq_ind T (THead (Bind Abbr) u0 t0) (\lambda (ee: 
+T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
+| (THead k _ _) \Rightarrow (match k with [(Bind _) \Rightarrow True | (Flat 
+_) \Rightarrow False])])) I (THead (Flat Cast) u1 t1) H6) in (False_ind (or 

 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w) 
 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w) 

-(THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) 
-(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O (THead 
-(Bind Abbr) u2 w)))) H7))))))))))))) (\lambda (b: B).(\lambda (H1: (not (eq B 
-b Abst))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H2: (pr0 t0 
-t2)).(\lambda (H3: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: 
-T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (u: T).(\lambda 
-(H4: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Void) u1 
-t1))).(let H5 \def (f_equal T B (\lambda (e: T).(match e in T return (\lambda 
-(_: T).B) with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead k 
-_ _) \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0) 
-\Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead (Bind b) u (lift (S O) O 
-t0)) (THead (Bind Void) u1 t1) H4) in ((let H6 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | 
-(TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Bind b) u 
-(lift (S O) O t0)) (THead (Bind Void) u1 t1) H4) in ((let H7 \def (f_equal T 
-T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T 
-\def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow 
-(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) 
-| (THead k u0 t3) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) 
-t3))]) in lref_map) (\lambda (x0: nat).(plus x0 (S O))) O t0) | (TLRef _) 
-\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T 
-\def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow 
-(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) 
-| (THead k u0 t3) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) 
-t3))]) in lref_map) (\lambda (x0: nat).(plus x0 (S O))) O t0) | (THead _ _ t) 
-\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Void) u1 
-t1) H4) in (\lambda (_: (eq T u u1)).(\lambda (H9: (eq B b Void)).(let H10 
-\def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H1 Void H9) in (let 
-H11 \def (eq_ind_r T t1 (\lambda (t: T).((eq T t0 (THead (Bind Void) u1 t)) 
-\to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
-Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr0 t t3)))) (pr0 t (lift (S O) O t2))))) H3 (lift (S O) 
-O t0) H7) in (eq_ind T (lift (S O) O t0) (\lambda (t: T).(or (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: 
-T).(pr0 t t3)))) (pr0 t (lift (S O) O t2)))) (or_intror (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 (lift 
-(S O) O t0) t3)))) (pr0 (lift (S O) O t0) (lift (S O) O t2)) (pr0_lift t0 t2 
-H2 (S O) O)) t1 H7)))))) H6)) H5)))))))))) (\lambda (t0: T).(\lambda (t2: 
-T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Void) u1 
-t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
-(Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
-(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O 
-t2)))))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u t0) (THead 
-(Bind Void) u1 t1))).(let H4 \def (eq_ind T (THead (Flat Cast) u t0) (\lambda 
-(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) 
+(THead (Flat Cast) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) 
+(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (THead (Bind Abbr) u2 
+w))) H7))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b 
+Abst))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda 
+(_: (((eq T t0 (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
+t3)))) (pr0 t1 t2))))).(\lambda (u: T).(\lambda (H4: (eq T (THead (Bind b) u 
+(lift (S O) O t0)) (THead (Flat Cast) u1 t1))).(let H5 \def (eq_ind T (THead 
+(Bind b) u (lift (S O) O t0)) (\lambda (ee: T).(match ee with [(TSort _) 

 \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
 \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 

-(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | 
-(Flat _) \Rightarrow True])])) I (THead (Bind Void) u1 t1) H3) in (False_ind 
-(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) 
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2))) H4)))))))) y x 
-H0))) H)))).
-(* COMMENTS
-Initial nodes: 3436
-END *)
+(match k with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I 
+(THead (Flat Cast) u1 t1) H4) in (False_ind (or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
+t3)))) (pr0 t1 t2)) H5)))))))))) (\lambda (t0: T).(\lambda (t2: T).(\lambda 
+(H1: (pr0 t0 t2)).(\lambda (H2: (((eq T t0 (THead (Flat Cast) u1 t1)) \to (or 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2))))).(\lambda (u: T).(\lambda 
+(H3: (eq T (THead (Flat Cast) u t0) (THead (Flat Cast) u1 t1))).(let H4 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u | (TLRef 
+_) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Flat Cast) u t0) 
+(THead (Flat Cast) u1 t1) H3) in ((let H5 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | 
+(THead _ _ t) \Rightarrow t])) (THead (Flat Cast) u t0) (THead (Flat Cast) u1 
+t1) H3) in (\lambda (_: (eq T u u1)).(let H7 \def (eq_ind T t0 (\lambda (t: 
+T).((eq T t (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
+t3)))) (pr0 t1 t2)))) H2 t1 H5) in (let H8 \def (eq_ind T t0 (\lambda (t: 
+T).(pr0 t t2)) H1 t1 H5) in (or_intror (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2) 
+H8))))) H4)))))))) y x H0))) H)))).

 
 theorem pr0_gen_lift:
  \forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall (d: nat).((pr0 
 
 theorem pr0_gen_lift:
  \forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall (d: nat).((pr0 


@@ -1970,49 +1515,46 @@ T u (lift h x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift (S O) O t2)
 x2))).(\lambda (H7: (eq T (lift (S O) O t2) (lift h (S x1) x3))).(eq_ind_r T 
 (THead (Bind b) x2 x3) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T t3 (lift 
 h x1 t4))) (\lambda (t4: T).(pr0 t t4)))) (let H8 \def (eq_ind_r nat (plus (S 
 x2))).(\lambda (H7: (eq T (lift (S O) O t2) (lift h (S x1) x3))).(eq_ind_r T 
 (THead (Bind b) x2 x3) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T t3 (lift 
 h x1 t4))) (\lambda (t4: T).(pr0 t t4)))) (let H8 \def (eq_ind_r nat (plus (S 

-O) x1) (\lambda (n: nat).(eq nat (S x1) n)) (refl_equal nat (plus (S O) x1)) 
-(plus x1 (S O)) (plus_sym x1 (S O))) in (let H9 \def (eq_ind nat (S x1) 
-(\lambda (n: nat).(eq T (lift (S O) O t2) (lift h n x3))) H7 (plus x1 (S O)) 
-H8) in (ex2_ind T (\lambda (t4: T).(eq T x3 (lift (S O) O t4))) (\lambda (t4: 
-T).(eq T t2 (lift h x1 t4))) (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 
-t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 x3) t4))) (\lambda (x4: 
-T).(\lambda (H10: (eq T x3 (lift (S O) O x4))).(\lambda (H11: (eq T t2 (lift 
-h x1 x4))).(eq_ind_r T (lift (S O) O x4) (\lambda (t: T).(ex2 T (\lambda (t4: 
-T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 t) 
-t4)))) (ex2_ind T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: 
-T).(pr0 x4 t4)) (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda 
-(t4: T).(pr0 (THead (Bind b) x2 (lift (S O) O x4)) t4))) (\lambda (x5: 
-T).(\lambda (H_x: (eq T t3 (lift h x1 x5))).(\lambda (H12: (pr0 x4 
-x5)).(eq_ind_r T (lift h x1 x5) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T 
-t (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 (lift (S O) O 
-x4)) t4)))) (ex_intro2 T (\lambda (t4: T).(eq T (lift h x1 x5) (lift h x1 
-t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 (lift (S O) O x4)) t4)) x5 
-(refl_equal T (lift h x1 x5)) (pr0_zeta b H1 x4 x5 H12 x2)) t3 H_x)))) (H3 x4 
-x1 H11)) x3 H10)))) (lift_gen_lift t2 x3 (S O) h O x1 (le_O_n x1) H9)))) x0 
-H5)))))) (lift_gen_bind b u (lift (S O) O t2) x0 h x1 H4)))))))))))) (\lambda 
-(t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 t3)).(\lambda (H2: ((\forall 
-(x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to (ex2 T (\lambda (t4: 
-T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4)))))))).(\lambda (u: 
-T).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H3: (eq T (THead (Flat Cast) 
-u t2) (lift h x1 x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T 
-x0 (THead (Flat Cast) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u (lift 
-h x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h x1 z)))) (ex2 T 
-(\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4))) 
-(\lambda (x2: T).(\lambda (x3: T).(\lambda (H4: (eq T x0 (THead (Flat Cast) 
-x2 x3))).(\lambda (_: (eq T u (lift h x1 x2))).(\lambda (H6: (eq T t2 (lift h 
-x1 x3))).(eq_ind_r T (THead (Flat Cast) x2 x3) (\lambda (t: T).(ex2 T 
-(\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 t t4)))) 
-(ex2_ind T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 
-x3 t4)) (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: 
-T).(pr0 (THead (Flat Cast) x2 x3) t4))) (\lambda (x4: T).(\lambda (H_x: (eq T 
-t3 (lift h x1 x4))).(\lambda (H7: (pr0 x3 x4)).(eq_ind_r T (lift h x1 x4) 
-(\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T t (lift h x1 t4))) (\lambda 
-(t4: T).(pr0 (THead (Flat Cast) x2 x3) t4)))) (ex_intro2 T (\lambda (t4: 
-T).(eq T (lift h x1 x4) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat 
-Cast) x2 x3) t4)) x4 (refl_equal T (lift h x1 x4)) (pr0_tau x3 x4 H7 x2)) t3 
-H_x)))) (H2 x3 x1 H6)) x0 H4)))))) (lift_gen_flat Cast u t2 x0 h x1 
-H3)))))))))) y x H0))))) H))))).
-(* COMMENTS
-Initial nodes: 7569
-END *)
+O) x1) (\lambda (n: nat).(eq nat (S x1) n)) (le_antisym (S x1) (plus (S O) 
+x1) (le_n (plus (S O) x1)) (le_n (S x1))) (plus x1 (S O)) (plus_sym x1 (S 
+O))) in (let H9 \def (eq_ind nat (S x1) (\lambda (n: nat).(eq T (lift (S O) O 
+t2) (lift h n x3))) H7 (plus x1 (S O)) H8) in (ex2_ind T (\lambda (t4: T).(eq 
+T x3 (lift (S O) O t4))) (\lambda (t4: T).(eq T t2 (lift h x1 t4))) (ex2 T 
+(\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind 
+b) x2 x3) t4))) (\lambda (x4: T).(\lambda (H10: (eq T x3 (lift (S O) O 
+x4))).(\lambda (H11: (eq T t2 (lift h x1 x4))).(eq_ind_r T (lift (S O) O x4) 
+(\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda 
+(t4: T).(pr0 (THead (Bind b) x2 t) t4)))) (ex2_ind T (\lambda (t4: T).(eq T 
+t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x4 t4)) (ex2 T (\lambda (t4: T).(eq 
+T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 (lift (S O) O 
+x4)) t4))) (\lambda (x5: T).(\lambda (H_x: (eq T t3 (lift h x1 x5))).(\lambda 
+(H12: (pr0 x4 x5)).(eq_ind_r T (lift h x1 x5) (\lambda (t: T).(ex2 T (\lambda 
+(t4: T).(eq T t (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 
+(lift (S O) O x4)) t4)))) (ex_intro2 T (\lambda (t4: T).(eq T (lift h x1 x5) 
+(lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 (lift (S O) O x4)) 
+t4)) x5 (refl_equal T (lift h x1 x5)) (pr0_zeta b H1 x4 x5 H12 x2)) t3 
+H_x)))) (H3 x4 x1 H11)) x3 H10)))) (lift_gen_lift t2 x3 (S O) h O x1 (le_O_n 
+x1) H9)))) x0 H5)))))) (lift_gen_bind b u (lift (S O) O t2) x0 h x1 
+H4)))))))))))) (\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 
+t3)).(\lambda (H2: ((\forall (x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 
+x0)) \to (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: 
+T).(pr0 x0 t4)))))))).(\lambda (u: T).(\lambda (x0: T).(\lambda (x1: 
+nat).(\lambda (H3: (eq T (THead (Flat Cast) u t2) (lift h x1 x0))).(ex3_2_ind 
+T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Flat Cast) y0 z)))) 
+(\lambda (y0: T).(\lambda (_: T).(eq T u (lift h x1 y0)))) (\lambda (_: 
+T).(\lambda (z: T).(eq T t2 (lift h x1 z)))) (ex2 T (\lambda (t4: T).(eq T t3 
+(lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4))) (\lambda (x2: T).(\lambda 
+(x3: T).(\lambda (H4: (eq T x0 (THead (Flat Cast) x2 x3))).(\lambda (_: (eq T 
+u (lift h x1 x2))).(\lambda (H6: (eq T t2 (lift h x1 x3))).(eq_ind_r T (THead 
+(Flat Cast) x2 x3) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T t3 (lift h 
+x1 t4))) (\lambda (t4: T).(pr0 t t4)))) (ex2_ind T (\lambda (t4: T).(eq T t3 
+(lift h x1 t4))) (\lambda (t4: T).(pr0 x3 t4)) (ex2 T (\lambda (t4: T).(eq T 
+t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Cast) x2 x3) t4))) 
+(\lambda (x4: T).(\lambda (H_x: (eq T t3 (lift h x1 x4))).(\lambda (H7: (pr0 
+x3 x4)).(eq_ind_r T (lift h x1 x4) (\lambda (t: T).(ex2 T (\lambda (t4: 
+T).(eq T t (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Cast) x2 x3) 
+t4)))) (ex_intro2 T (\lambda (t4: T).(eq T (lift h x1 x4) (lift h x1 t4))) 
+(\lambda (t4: T).(pr0 (THead (Flat Cast) x2 x3) t4)) x4 (refl_equal T (lift h 
+x1 x4)) (pr0_tau x3 x4 H7 x2)) t3 H_x)))) (H2 x3 x1 H6)) x0 H4)))))) 
+(lift_gen_flat Cast u t2 x0 h x1 H3)))))))))) y x H0))))) H))))).

 
 

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr0/pr0.ma b/matita/matita/contribs/lambdadelta/basic_1/pr0/pr0.ma
index 9a3b397fe9f10f9dfd52c60c7bfb989a059d84c1..57f9498ce34d9d00b920fafccf2c54707d97fb1e 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_1/pr0/pr0.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/pr0/pr0.ma
@@ -14,9 +14,11 @@
 
 (* This file was automatically generated: do not edit *********************)
 
 
 (* This file was automatically generated: do not edit *********************)
 

-include "Basic-1/pr0/fwd.ma".
+include "basic_1/pr0/subst0.ma".

 
 

-include "Basic-1/lift/tlt.ma".
+include "basic_1/lift/tlt.ma".
+
+include "basic_1/tlt/fwd.ma".

 
 theorem pr0_confluence__pr0_cong_upsilon_refl:
  \forall (b: B).((not (eq B b Abst)) \to (\forall (u0: T).(\forall (u3: 
 
 theorem pr0_confluence__pr0_cong_upsilon_refl:
  \forall (b: B).((not (eq B b Abst)) \to (\forall (u0: T).(\forall (u3: 


@@ -37,9 +39,6 @@ t5 H1) (pr0_comp u3 u3 (pr0_refl u3) (THead (Flat Appl) (lift (S O) O v2) t5)
 (THead (Flat Appl) (lift (S O) O x) t5) (pr0_comp (lift (S O) O v2) (lift (S 
 O) O x) (pr0_lift v2 x H3 (S O) O) t5 t5 (pr0_refl t5) (Flat Appl)) (Bind 
 b))))))))))))))).
 (THead (Flat Appl) (lift (S O) O x) t5) (pr0_comp (lift (S O) O v2) (lift (S 
 O) O x) (pr0_lift v2 x H3 (S O) O) t5 t5 (pr0_refl t5) (Flat Appl)) (Bind 
 b))))))))))))))).

-(* COMMENTS
-Initial nodes: 257
-END *)

 
 theorem pr0_confluence__pr0_cong_upsilon_cong:
  \forall (b: B).((not (eq B b Abst)) \to (\forall (u2: T).(\forall (v2: 
 
 theorem pr0_confluence__pr0_cong_upsilon_cong:
  \forall (b: B).((not (eq B b Abst)) \to (\forall (u2: T).(\forall (v2: 


@@ -62,9 +61,6 @@ t5)) t)) (THead (Bind b) x1 (THead (Flat Appl) (lift (S O) O x) x0))
 Appl) (lift (S O) O v2) t5) (THead (Flat Appl) (lift (S O) O x) x0) (pr0_comp 
 (lift (S O) O v2) (lift (S O) O x) (pr0_lift v2 x H1 (S O) O) t5 x0 H3 (Flat 
 Appl)) (Bind b))))))))))))))))))).
 Appl) (lift (S O) O v2) t5) (THead (Flat Appl) (lift (S O) O x) x0) (pr0_comp 
 (lift (S O) O v2) (lift (S O) O x) (pr0_lift v2 x H1 (S O) O) t5 x0 H3 (Flat 
 Appl)) (Bind b))))))))))))))))))).

-(* COMMENTS
-Initial nodes: 269
-END *)

 
 theorem pr0_confluence__pr0_cong_upsilon_delta:
  (not (eq B Abbr Abst)) \to (\forall (u5: T).(\forall (t2: T).(\forall (w: 
 
 theorem pr0_confluence__pr0_cong_upsilon_delta:
  (not (eq B Abbr Abst)) \to (\forall (u5: T).(\forall (t2: T).(\forall (w: 


@@ -106,9 +102,6 @@ O v2) (lift (S O) O x) (pr0_lift v2 x H2 (S O) O) t5 x0 H4 (Flat Appl))
 (THead (Flat Appl) (lift (S O) O x) x2) (subst0_snd (Flat Appl) x1 x2 x0 O H9 
 (lift (S O) O x))))))) H7)) (pr0_subst0 t2 x0 H3 u5 w O H0 x1 
 H5))))))))))))))))))).
 (THead (Flat Appl) (lift (S O) O x) x2) (subst0_snd (Flat Appl) x1 x2 x0 O H9 
 (lift (S O) O x))))))) H7)) (pr0_subst0 t2 x0 H3 u5 w O H0 x1 
 H5))))))))))))))))))).

-(* COMMENTS
-Initial nodes: 769
-END *)

 
 theorem pr0_confluence__pr0_cong_upsilon_zeta:
  \forall (b: B).((not (eq B b Abst)) \to (\forall (u0: T).(\forall (u3: 
 
 theorem pr0_confluence__pr0_cong_upsilon_zeta:
  \forall (b: B).((not (eq B b Abst)) \to (\forall (u0: T).(\forall (u3: 


@@ -131,9 +124,6 @@ t3 x1 H4 (Flat Appl)) (pr0_zeta b H (THead (Flat Appl) v2 x) (THead (Flat
 Appl) x0 x1) (pr0_comp v2 x0 H2 x x1 H3 (Flat Appl)) u3)) (THead (Flat Appl) 
 (lift (S O) O v2) (lift (S O) O x)) (lift_flat Appl v2 x (S O) 
 O)))))))))))))))).
 Appl) x0 x1) (pr0_comp v2 x0 H2 x x1 H3 (Flat Appl)) u3)) (THead (Flat Appl) 
 (lift (S O) O v2) (lift (S O) O x)) (lift_flat Appl v2 x (S O) 
 O)))))))))))))))).

-(* COMMENTS
-Initial nodes: 283
-END *)

 
 theorem pr0_confluence__pr0_cong_delta:
  \forall (u3: T).(\forall (t5: T).(\forall (w: T).((subst0 O u3 t5 w) \to 
 
 theorem pr0_confluence__pr0_cong_delta:
  \forall (u3: T).(\forall (t5: T).(\forall (w: T).((subst0 O u3 t5 w) \to 


@@ -160,9 +150,6 @@ x1)).(ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 t3) t)) (\lambda
 (t: T).(pr0 (THead (Bind Abbr) u3 w) t)) (THead (Bind Abbr) x x1) (pr0_delta 
 u2 x H0 t3 x0 H2 x1 H6) (pr0_comp u3 x H1 w x1 H5 (Bind Abbr)))))) H4)) 
 (pr0_subst0 t5 x0 H3 u3 w O H x H1))))))))))))).
 (t: T).(pr0 (THead (Bind Abbr) u3 w) t)) (THead (Bind Abbr) x x1) (pr0_delta 
 u2 x H0 t3 x0 H2 x1 H6) (pr0_comp u3 x H1 w x1 H5 (Bind Abbr)))))) H4)) 
 (pr0_subst0 t5 x0 H3 u3 w O H x H1))))))))))))).

-(* COMMENTS
-Initial nodes: 409
-END *)

 
 theorem pr0_confluence__pr0_upsilon_upsilon:
  \forall (b: B).((not (eq B b Abst)) \to (\forall (v1: T).(\forall (v2: 
 
 theorem pr0_confluence__pr0_upsilon_upsilon:
  \forall (b: B).((not (eq B b Abst)) \to (\forall (v1: T).(\forall (v2: 


@@ -187,9 +174,6 @@ Appl) (lift (S O) O x0) x2) (pr0_comp (lift (S O) O v1) (lift (S O) O x0)
 H3 (THead (Flat Appl) (lift (S O) O v2) t2) (THead (Flat Appl) (lift (S O) O 
 x0) x2) (pr0_comp (lift (S O) O v2) (lift (S O) O x0) (pr0_lift v2 x0 H1 (S 
 O) O) t2 x2 H5 (Flat Appl)) (Bind b))))))))))))))))))).
 H3 (THead (Flat Appl) (lift (S O) O v2) t2) (THead (Flat Appl) (lift (S O) O 
 x0) x2) (pr0_comp (lift (S O) O v2) (lift (S O) O x0) (pr0_lift v2 x0 H1 (S 
 O) O) t2 x2 H5 (Flat Appl)) (Bind b))))))))))))))))))).

-(* COMMENTS
-Initial nodes: 347
-END *)

 
 theorem pr0_confluence__pr0_delta_delta:
  \forall (u2: T).(\forall (t3: T).(\forall (w: T).((subst0 O u2 t3 w) \to 
 
 theorem pr0_confluence__pr0_delta_delta:
  \forall (u2: T).(\forall (t3: T).(\forall (w: T).((subst0 O u2 t3 w) \to 


@@ -259,9 +243,6 @@ w0 x1 H6 (Bind Abbr)))) (\lambda (H11: (subst0 O x x1 x2)).(ex_intro2 T
 (Bind Abbr)) (pr0_delta u3 x H2 w0 x1 H6 x2 H11))) (subst0_confluence_eq x0 
 x2 x O H10 x1 H7))))) H8)) (pr0_subst0 t3 x0 H3 u2 w O H x H1))))) H5)) 
 (pr0_subst0 t5 x0 H4 u3 w0 O H0 x H2))))))))))))))).
 (Bind Abbr)) (pr0_delta u3 x H2 w0 x1 H6 x2 H11))) (subst0_confluence_eq x0 
 x2 x O H10 x1 H7))))) H8)) (pr0_subst0 t3 x0 H3 u2 w O H x H1))))) H5)) 
 (pr0_subst0 t5 x0 H4 u3 w0 O H0 x H2))))))))))))))).

-(* COMMENTS
-Initial nodes: 1501
-END *)

 
 theorem pr0_confluence__pr0_delta_tau:
  \forall (u2: T).(\forall (t3: T).(\forall (w: T).((subst0 O u2 t3 w) \to 
 
 theorem pr0_confluence__pr0_delta_tau:
  \forall (u2: T).(\forall (t3: T).(\forall (w: T).((subst0 O u2 t3 w) \to 


@@ -276,13 +257,10 @@ t3 w)).(\lambda (t4: T).(\lambda (H0: (pr0 (lift (S O) O t4) t3)).(\lambda
 (\lambda (t: T).(pr0 t2 t))) (\lambda (x: T).(\lambda (H1: (eq T t3 (lift (S 
 O) O x))).(\lambda (_: (pr0 t4 x)).(let H3 \def (eq_ind T t3 (\lambda (t: 
 T).(subst0 O u2 t w)) H (lift (S O) O x) H1) in (subst0_gen_lift_false x u2 w 
 (\lambda (t: T).(pr0 t2 t))) (\lambda (x: T).(\lambda (H1: (eq T t3 (lift (S 
 O) O x))).(\lambda (_: (pr0 t4 x)).(let H3 \def (eq_ind T t3 (\lambda (t: 
 T).(subst0 O u2 t w)) H (lift (S O) O x) H1) in (subst0_gen_lift_false x u2 w 

-(S O) O O (le_n O) (eq_ind_r nat (plus (S O) O) (\lambda (n: nat).(lt O n)) 
+(S O) O O (le_O_n O) (eq_ind_r nat (plus (S O) O) (\lambda (n: nat).(lt O n)) 

 (le_n (plus (S O) O)) (plus O (S O)) (plus_sym O (S O))) H3 (ex2 T (\lambda 
 (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 t2 t)))))))) 
 (pr0_gen_lift t4 t3 (S O) O H0)))))))).
 (le_n (plus (S O) O)) (plus O (S O)) (plus_sym O (S O))) H3 (ex2 T (\lambda 
 (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 t2 t)))))))) 
 (pr0_gen_lift t4 t3 (S O) O H0)))))))).

-(* COMMENTS
-Initial nodes: 257
-END *)

 
 theorem pr0_confluence:
  \forall (t0: T).(\forall (t1: T).((pr0 t0 t1) \to (\forall (t2: T).((pr0 t0 
 
 theorem pr0_confluence:
  \forall (t0: T).(\forall (t1: T).((pr0 t0 t1) \to (\forall (t2: T).((pr0 t0 


@@ -294,95 +272,90 @@ t2) \to (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t)))))))
 (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 
 v t2) \to (ex2 T (\lambda (t3: T).(pr0 t1 t3)) (\lambda (t3: T).(pr0 t2 
 t3))))))))))).(\lambda (t1: T).(\lambda (H0: (pr0 t t1)).(\lambda (t2: 
 (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 
 v t2) \to (ex2 T (\lambda (t3: T).(pr0 t1 t3)) (\lambda (t3: T).(pr0 t2 
 t3))))))))))).(\lambda (t1: T).(\lambda (H0: (pr0 t t1)).(\lambda (t2: 

-T).(\lambda (H1: (pr0 t t2)).(let H2 \def (match H0 in pr0 return (\lambda 
-(t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).((eq T t3 t) \to ((eq T t4 
-t1) \to (ex2 T (\lambda (t5: T).(pr0 t1 t5)) (\lambda (t5: T).(pr0 t2 
-t5)))))))) with [(pr0_refl t3) \Rightarrow (\lambda (H2: (eq T t3 
-t)).(\lambda (H3: (eq T t3 t1)).(eq_ind T t (\lambda (t4: T).((eq T t4 t1) 
-\to (ex2 T (\lambda (t5: T).(pr0 t1 t5)) (\lambda (t5: T).(pr0 t2 t5))))) 
-(\lambda (H4: (eq T t t1)).(eq_ind T t1 (\lambda (_: T).(ex2 T (\lambda (t5: 
-T).(pr0 t1 t5)) (\lambda (t5: T).(pr0 t2 t5)))) (let H5 \def (match H1 in pr0 
-return (\lambda (t4: T).(\lambda (t5: T).(\lambda (_: (pr0 t4 t5)).((eq T t4 
-t) \to ((eq T t5 t2) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: 
-T).(pr0 t2 t6)))))))) with [(pr0_refl t4) \Rightarrow (\lambda (H5: (eq T t4 
-t)).(\lambda (H6: (eq T t4 t2)).(eq_ind T t (\lambda (t5: T).((eq T t5 t2) 
-\to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6))))) 
-(\lambda (H7: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t6: 
-T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6)))) (let H8 \def (eq_ind T t 
-(\lambda (t5: T).(eq T t4 t5)) H5 t2 H7) in (let H9 \def (eq_ind T t (\lambda 
-(t5: T).(eq T t5 t1)) H4 t2 H7) in (let H10 \def (eq_ind T t (\lambda (t5: 
-T).(eq T t3 t5)) H2 t2 H7) in (let H11 \def (eq_ind T t (\lambda (t5: 
-T).(\forall (v: T).((tlt v t5) \to (\forall (t6: T).((pr0 v t6) \to (\forall 
-(t7: T).((pr0 v t7) \to (ex2 T (\lambda (t8: T).(pr0 t6 t8)) (\lambda (t8: 
-T).(pr0 t7 t8)))))))))) H t2 H7) in (let H12 \def (eq_ind T t2 (\lambda (t5: 
-T).(\forall (v: T).((tlt v t5) \to (\forall (t6: T).((pr0 v t6) \to (\forall 
-(t7: T).((pr0 v t7) \to (ex2 T (\lambda (t8: T).(pr0 t6 t8)) (\lambda (t8: 
-T).(pr0 t7 t8)))))))))) H11 t1 H9) in (eq_ind_r T t1 (\lambda (t5: T).(ex2 T 
-(\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t5 t6)))) (let H13 \def 
-(eq_ind T t2 (\lambda (t5: T).(eq T t3 t5)) H10 t1 H9) in (ex_intro2 T 
-(\lambda (t5: T).(pr0 t1 t5)) (\lambda (t5: T).(pr0 t1 t5)) t1 (pr0_refl t1) 
-(pr0_refl t1))) t2 H9)))))) t (sym_eq T t t2 H7))) t4 (sym_eq T t4 t H5) 
-H6))) | (pr0_comp u1 u2 H5 t4 t5 H6 k) \Rightarrow (\lambda (H7: (eq T (THead 
-k u1 t4) t)).(\lambda (H8: (eq T (THead k u2 t5) t2)).(eq_ind T (THead k u1 
-t4) (\lambda (_: T).((eq T (THead k u2 t5) t2) \to ((pr0 u1 u2) \to ((pr0 t4 
-t5) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 
-t7))))))) (\lambda (H9: (eq T (THead k u2 t5) t2)).(eq_ind T (THead k u2 t5) 
-(\lambda (t6: T).((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: 
-T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t6 t7)))))) (\lambda (H10: (pr0 u1 
-u2)).(\lambda (H11: (pr0 t4 t5)).(let H12 \def (eq_ind_r T t (\lambda (t6: 
-T).(eq T t6 t1)) H4 (THead k u1 t4) H7) in (eq_ind T (THead k u1 t4) (\lambda 
-(t6: T).(ex2 T (\lambda (t7: T).(pr0 t6 t7)) (\lambda (t7: T).(pr0 (THead k 
-u2 t5) t7)))) (let H13 \def (eq_ind_r T t (\lambda (t6: T).(eq T t3 t6)) H2 
-(THead k u1 t4) H7) in (let H14 \def (eq_ind_r T t (\lambda (t6: T).(\forall 
+T).(\lambda (H1: (pr0 t t2)).(let H2 \def (match H0 with [(pr0_refl t3) 
+\Rightarrow (\lambda (H2: (eq T t3 t)).(\lambda (H3: (eq T t3 t1)).(eq_ind T 
+t (\lambda (t4: T).((eq T t4 t1) \to (ex2 T (\lambda (t5: T).(pr0 t1 t5)) 
+(\lambda (t5: T).(pr0 t2 t5))))) (\lambda (H4: (eq T t t1)).(eq_ind T t1 
+(\lambda (_: T).(ex2 T (\lambda (t5: T).(pr0 t1 t5)) (\lambda (t5: T).(pr0 t2 
+t5)))) (let H5 \def (match H1 with [(pr0_refl t4) \Rightarrow (\lambda (H5: 
+(eq T t4 t)).(\lambda (H6: (eq T t4 t2)).(eq_ind T t (\lambda (t5: T).((eq T 
+t5 t2) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 
+t6))))) (\lambda (H7: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T 
+(\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6)))) (let H8 \def 
+(eq_ind T t (\lambda (t5: T).(eq T t4 t5)) H5 t2 H7) in (let H9 \def (eq_ind 
+T t (\lambda (t5: T).(eq T t5 t1)) H4 t2 H7) in (let H10 \def (eq_ind T t 
+(\lambda (t5: T).(eq T t3 t5)) H2 t2 H7) in (let H11 \def (eq_ind T t 
+(\lambda (t5: T).(\forall (v: T).((tlt v t5) \to (\forall (t6: T).((pr0 v t6) 
+\to (\forall (t7: T).((pr0 v t7) \to (ex2 T (\lambda (t8: T).(pr0 t6 t8)) 
+(\lambda (t8: T).(pr0 t7 t8)))))))))) H t2 H7) in (let H12 \def (eq_ind T t2 
+(\lambda (t5: T).(\forall (v: T).((tlt v t5) \to (\forall (t6: T).((pr0 v t6) 
+\to (\forall (t7: T).((pr0 v t7) \to (ex2 T (\lambda (t8: T).(pr0 t6 t8)) 
+(\lambda (t8: T).(pr0 t7 t8)))))))))) H11 t1 H9) in (eq_ind_r T t1 (\lambda 
+(t5: T).(ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t5 t6)))) 
+(let H13 \def (eq_ind T t2 (\lambda (t5: T).(eq T t3 t5)) H10 t1 H9) in 
+(ex_intro2 T (\lambda (t5: T).(pr0 t1 t5)) (\lambda (t5: T).(pr0 t1 t5)) t1 
+(pr0_refl t1) (pr0_refl t1))) t2 H9)))))) t (sym_eq T t t2 H7))) t4 (sym_eq T 
+t4 t H5) H6))) | (pr0_comp u1 u2 H5 t4 t5 H6 k) \Rightarrow (\lambda (H7: (eq 
+T (THead k u1 t4) t)).(\lambda (H8: (eq T (THead k u2 t5) t2)).(eq_ind T 
+(THead k u1 t4) (\lambda (_: T).((eq T (THead k u2 t5) t2) \to ((pr0 u1 u2) 
+\to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: 
+T).(pr0 t2 t7))))))) (\lambda (H9: (eq T (THead k u2 t5) t2)).(eq_ind T 
+(THead k u2 t5) (\lambda (t6: T).((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T 
+(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t6 t7)))))) (\lambda 
+(H10: (pr0 u1 u2)).(\lambda (H11: (pr0 t4 t5)).(let H12 \def (eq_ind_r T t 
+(\lambda (t6: T).(eq T t6 t1)) H4 (THead k u1 t4) H7) in (eq_ind T (THead k 
+u1 t4) (\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 t6 t7)) (\lambda (t7: 
+T).(pr0 (THead k u2 t5) t7)))) (let H13 \def (eq_ind_r T t (\lambda (t6: 
+T).(eq T t3 t6)) H2 (THead k u1 t4) H7) in (let H14 \def (eq_ind_r T t 
+(\lambda (t6: T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) 
+\to (\forall (t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) 
+(\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead k u1 t4) H7) in (ex_intro2 T 
+(\lambda (t6: T).(pr0 (THead k u1 t4) t6)) (\lambda (t6: T).(pr0 (THead k u2 
+t5) t6)) (THead k u2 t5) (pr0_comp u1 u2 H10 t4 t5 H11 k) (pr0_refl (THead k 
+u2 t5))))) t1 H12)))) t2 H9)) t H7 H8 H5 H6))) | (pr0_beta u v1 v2 H5 t4 t5 
+H6) \Rightarrow (\lambda (H7: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) 
+u t4)) t)).(\lambda (H8: (eq T (THead (Bind Abbr) v2 t5) t2)).(eq_ind T 
+(THead (Flat Appl) v1 (THead (Bind Abst) u t4)) (\lambda (_: T).((eq T (THead 
+(Bind Abbr) v2 t5) t2) \to ((pr0 v1 v2) \to ((pr0 t4 t5) \to (ex2 T (\lambda 
+(t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))))))) (\lambda (H9: (eq T 
+(THead (Bind Abbr) v2 t5) t2)).(eq_ind T (THead (Bind Abbr) v2 t5) (\lambda 
+(t6: T).((pr0 v1 v2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: T).(pr0 t1 
+t7)) (\lambda (t7: T).(pr0 t6 t7)))))) (\lambda (H10: (pr0 v1 v2)).(\lambda 
+(H11: (pr0 t4 t5)).(let H12 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t1)) 
+H4 (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) H7) in (eq_ind T (THead 
+(Flat Appl) v1 (THead (Bind Abst) u t4)) (\lambda (t6: T).(ex2 T (\lambda 
+(t7: T).(pr0 t6 t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t5) t7)))) 
+(let H13 \def (eq_ind_r T t (\lambda (t6: T).(eq T t3 t6)) H2 (THead (Flat 
+Appl) v1 (THead (Bind Abst) u t4)) H7) in (let H14 \def (eq_ind_r T t 
+(\lambda (t6: T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) 
+\to (\forall (t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) 
+(\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead (Flat Appl) v1 (THead (Bind 
+Abst) u t4)) H7) in (ex_intro2 T (\lambda (t6: T).(pr0 (THead (Flat Appl) v1 
+(THead (Bind Abst) u t4)) t6)) (\lambda (t6: T).(pr0 (THead (Bind Abbr) v2 
+t5) t6)) (THead (Bind Abbr) v2 t5) (pr0_beta u v1 v2 H10 t4 t5 H11) (pr0_refl 
+(THead (Bind Abbr) v2 t5))))) t1 H12)))) t2 H9)) t H7 H8 H5 H6))) | 
+(pr0_upsilon b H5 v1 v2 H6 u1 u2 H7 t4 t5 H8) \Rightarrow (\lambda (H9: (eq T 
+(THead (Flat Appl) v1 (THead (Bind b) u1 t4)) t)).(\lambda (H10: (eq T (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) t2)).(eq_ind T (THead 
+(Flat Appl) v1 (THead (Bind b) u1 t4)) (\lambda (_: T).((eq T (THead (Bind b) 
+u2 (THead (Flat Appl) (lift (S O) O v2) t5)) t2) \to ((not (eq B b Abst)) \to 
+((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: 
+T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))))))))) (\lambda (H11: (eq T 
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) t2)).(eq_ind T 
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) (\lambda (t6: 
+T).((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) 
+\to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t6 t7)))))))) 
+(\lambda (H12: (not (eq B b Abst))).(\lambda (H13: (pr0 v1 v2)).(\lambda 
+(H14: (pr0 u1 u2)).(\lambda (H15: (pr0 t4 t5)).(let H16 \def (eq_ind_r T t 
+(\lambda (t6: T).(eq T t6 t1)) H4 (THead (Flat Appl) v1 (THead (Bind b) u1 
+t4)) H9) in (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t4)) (\lambda 
+(t6: T).(ex2 T (\lambda (t7: T).(pr0 t6 t7)) (\lambda (t7: T).(pr0 (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) t7)))) (let H17 \def 
+(eq_ind_r T t (\lambda (t6: T).(eq T t3 t6)) H2 (THead (Flat Appl) v1 (THead 
+(Bind b) u1 t4)) H9) in (let H18 \def (eq_ind_r T t (\lambda (t6: T).(\forall 

 (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall (t8: 
 T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 
 (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall (t8: 
 T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 

-t8 t9)))))))))) H (THead k u1 t4) H7) in (ex_intro2 T (\lambda (t6: T).(pr0 
-(THead k u1 t4) t6)) (\lambda (t6: T).(pr0 (THead k u2 t5) t6)) (THead k u2 
-t5) (pr0_comp u1 u2 H10 t4 t5 H11 k) (pr0_refl (THead k u2 t5))))) t1 H12)))) 
-t2 H9)) t H7 H8 H5 H6))) | (pr0_beta u v1 v2 H5 t4 t5 H6) \Rightarrow 
-(\lambda (H7: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) 
-t)).(\lambda (H8: (eq T (THead (Bind Abbr) v2 t5) t2)).(eq_ind T (THead (Flat 
-Appl) v1 (THead (Bind Abst) u t4)) (\lambda (_: T).((eq T (THead (Bind Abbr) 
-v2 t5) t2) \to ((pr0 v1 v2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: T).(pr0 
-t1 t7)) (\lambda (t7: T).(pr0 t2 t7))))))) (\lambda (H9: (eq T (THead (Bind 
-Abbr) v2 t5) t2)).(eq_ind T (THead (Bind Abbr) v2 t5) (\lambda (t6: T).((pr0 
-v1 v2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda 
-(t7: T).(pr0 t6 t7)))))) (\lambda (H10: (pr0 v1 v2)).(\lambda (H11: (pr0 t4 
-t5)).(let H12 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t1)) H4 (THead 
-(Flat Appl) v1 (THead (Bind Abst) u t4)) H7) in (eq_ind T (THead (Flat Appl) 
-v1 (THead (Bind Abst) u t4)) (\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 t6 
-t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t5) t7)))) (let H13 \def 
-(eq_ind_r T t (\lambda (t6: T).(eq T t3 t6)) H2 (THead (Flat Appl) v1 (THead 
-(Bind Abst) u t4)) H7) in (let H14 \def (eq_ind_r T t (\lambda (t6: 
-T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall 
-(t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: 
-T).(pr0 t8 t9)))))))))) H (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) H7) 
-in (ex_intro2 T (\lambda (t6: T).(pr0 (THead (Flat Appl) v1 (THead (Bind 
-Abst) u t4)) t6)) (\lambda (t6: T).(pr0 (THead (Bind Abbr) v2 t5) t6)) (THead 
-(Bind Abbr) v2 t5) (pr0_beta u v1 v2 H10 t4 t5 H11) (pr0_refl (THead (Bind 
-Abbr) v2 t5))))) t1 H12)))) t2 H9)) t H7 H8 H5 H6))) | (pr0_upsilon b H5 v1 
-v2 H6 u1 u2 H7 t4 t5 H8) \Rightarrow (\lambda (H9: (eq T (THead (Flat Appl) 
-v1 (THead (Bind b) u1 t4)) t)).(\lambda (H10: (eq T (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t5)) t2)).(eq_ind T (THead (Flat Appl) v1 
-(THead (Bind b) u1 t4)) (\lambda (_: T).((eq T (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t5)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 
-v2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: T).(pr0 t1 
-t7)) (\lambda (t7: T).(pr0 t2 t7))))))))) (\lambda (H11: (eq T (THead (Bind 
-b) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) t2)).(eq_ind T (THead (Bind 
-b) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) (\lambda (t6: T).((not (eq B 
-b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T 
-(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t6 t7)))))))) (\lambda 
-(H12: (not (eq B b Abst))).(\lambda (H13: (pr0 v1 v2)).(\lambda (H14: (pr0 u1 
-u2)).(\lambda (H15: (pr0 t4 t5)).(let H16 \def (eq_ind_r T t (\lambda (t6: 
-T).(eq T t6 t1)) H4 (THead (Flat Appl) v1 (THead (Bind b) u1 t4)) H9) in 
-(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t4)) (\lambda (t6: T).(ex2 
-T (\lambda (t7: T).(pr0 t6 t7)) (\lambda (t7: T).(pr0 (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t5)) t7)))) (let H17 \def (eq_ind_r T t 
-(\lambda (t6: T).(eq T t3 t6)) H2 (THead (Flat Appl) v1 (THead (Bind b) u1 
-t4)) H9) in (let H18 \def (eq_ind_r T t (\lambda (t6: T).(\forall (v: 
-T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall (t8: T).((pr0 v 
-t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 
-t9)))))))))) H (THead (Flat Appl) v1 (THead (Bind b) u1 t4)) H9) in 
+t8 t9)))))))))) H (THead (Flat Appl) v1 (THead (Bind b) u1 t4)) H9) in 

 (pr0_confluence__pr0_cong_upsilon_refl b H12 u1 u2 H14 t4 t5 H15 v1 v2 v2 H13 
 (pr0_refl v2)))) t1 H16)))))) t2 H11)) t H9 H10 H5 H6 H7 H8))) | (pr0_delta 
 u1 u2 H5 t4 t5 H6 w H7) \Rightarrow (\lambda (H8: (eq T (THead (Bind Abbr) u1 
 (pr0_confluence__pr0_cong_upsilon_refl b H12 u1 u2 H14 t4 t5 H15 v1 v2 v2 H13 
 (pr0_refl v2)))) t1 H16)))))) t2 H11)) t H9 H10 H5 H6 H7 H8))) | (pr0_delta 
 u1 u2 H5 t4 t5 H6 w H7) \Rightarrow (\lambda (H8: (eq T (THead (Bind Abbr) u1 


@@ -448,995 +421,906 @@ t3) t)).(\lambda (H5: (eq T (THead k u2 t4) t1)).(eq_ind T (THead k u1 t3)
 (\lambda (H6: (eq T (THead k u2 t4) t1)).(eq_ind T (THead k u2 t4) (\lambda 
 (t5: T).((pr0 u1 u2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t5 
 t6)) (\lambda (t6: T).(pr0 t2 t6)))))) (\lambda (H7: (pr0 u1 u2)).(\lambda 
 (\lambda (H6: (eq T (THead k u2 t4) t1)).(eq_ind T (THead k u2 t4) (\lambda 
 (t5: T).((pr0 u1 u2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t5 
 t6)) (\lambda (t6: T).(pr0 t2 t6)))))) (\lambda (H7: (pr0 u1 u2)).(\lambda 

-(H8: (pr0 t3 t4)).(let H9 \def (match H1 in pr0 return (\lambda (t5: 
-T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2) 
-\to (ex2 T (\lambda (t7: T).(pr0 (THead k u2 t4) t7)) (\lambda (t7: T).(pr0 
-t2 t7)))))))) with [(pr0_refl t5) \Rightarrow (\lambda (H9: (eq T t5 
-t)).(\lambda (H10: (eq T t5 t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2) 
-\to (ex2 T (\lambda (t7: T).(pr0 (THead k u2 t4) t7)) (\lambda (t7: T).(pr0 
-t2 t7))))) (\lambda (H11: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T 
-(\lambda (t7: T).(pr0 (THead k u2 t4) t7)) (\lambda (t7: T).(pr0 t2 t7)))) 
-(let H12 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t2)) H11 (THead k u1 
-t3) H4) in (eq_ind T (THead k u1 t3) (\lambda (t6: T).(ex2 T (\lambda (t7: 
-T).(pr0 (THead k u2 t4) t7)) (\lambda (t7: T).(pr0 t6 t7)))) (let H13 \def 
-(eq_ind_r T t (\lambda (t6: T).(eq T t5 t6)) H9 (THead k u1 t3) H4) in (let 
-H14 \def (eq_ind_r T t (\lambda (t6: T).(\forall (v: T).((tlt v t6) \to 
-(\forall (t7: T).((pr0 v t7) \to (\forall (t8: T).((pr0 v t8) \to (ex2 T 
-(\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead 
-k u1 t3) H4) in (ex_intro2 T (\lambda (t6: T).(pr0 (THead k u2 t4) t6)) 
-(\lambda (t6: T).(pr0 (THead k u1 t3) t6)) (THead k u2 t4) (pr0_refl (THead k 
-u2 t4)) (pr0_comp u1 u2 H7 t3 t4 H8 k)))) t2 H12)) t (sym_eq T t t2 H11))) t5 
-(sym_eq T t5 t H9) H10))) | (pr0_comp u0 u3 H9 t5 t6 H10 k0) \Rightarrow 
-(\lambda (H11: (eq T (THead k0 u0 t5) t)).(\lambda (H12: (eq T (THead k0 u3 
-t6) t2)).(eq_ind T (THead k0 u0 t5) (\lambda (_: T).((eq T (THead k0 u3 t6) 
-t2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead 
-k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H13: (eq T (THead 
-k0 u3 t6) t2)).(eq_ind T (THead k0 u3 t6) (\lambda (t7: T).((pr0 u0 u3) \to 
-((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda 
-(t8: T).(pr0 t7 t8)))))) (\lambda (H14: (pr0 u0 u3)).(\lambda (H15: (pr0 t5 
-t6)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead k u1 t3) t7)) 
-H4 (THead k0 u0 t5) H11) in (let H17 \def (f_equal T K (\lambda (e: T).(match 
-e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) 
-\Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k u1 t3) (THead k0 u0 
-t5) H16) in ((let H18 \def (f_equal T T (\lambda (e: T).(match e in T return 
-(\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 
-| (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) (THead k0 u0 t5) H16) in 
-((let H19 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: 
-T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ 
-t7) \Rightarrow t7])) (THead k u1 t3) (THead k0 u0 t5) H16) in (\lambda (H20: 
-(eq T u1 u0)).(\lambda (H21: (eq K k k0)).(let H22 \def (eq_ind_r T t 
-(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) 
-\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) 
-(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead k0 u0 t5) H11) in (eq_ind_r 
-K k0 (\lambda (k1: K).(ex2 T (\lambda (t7: T).(pr0 (THead k1 u2 t4) t7)) 
-(\lambda (t7: T).(pr0 (THead k0 u3 t6) t7)))) (let H23 \def (eq_ind T u1 
-(\lambda (t7: T).(pr0 t7 u2)) H7 u0 H20) in (let H24 \def (eq_ind T t3 
-(\lambda (t7: T).(pr0 t7 t4)) H8 t5 H19) in (ex2_ind T (\lambda (t7: T).(pr0 
-t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead k0 
-u2 t4) t7)) (\lambda (t7: T).(pr0 (THead k0 u3 t6) t7))) (\lambda (x: 
-T).(\lambda (H25: (pr0 t4 x)).(\lambda (H26: (pr0 t6 x)).(ex2_ind T (\lambda 
-(t7: T).(pr0 u2 t7)) (\lambda (t7: T).(pr0 u3 t7)) (ex2 T (\lambda (t7: 
-T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7: T).(pr0 (THead k0 u3 t6) t7))) 
-(\lambda (x0: T).(\lambda (H27: (pr0 u2 x0)).(\lambda (H28: (pr0 u3 
-x0)).(ex_intro2 T (\lambda (t7: T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7: 
-T).(pr0 (THead k0 u3 t6) t7)) (THead k0 x0 x) (pr0_comp u2 x0 H27 t4 x H25 
-k0) (pr0_comp u3 x0 H28 t6 x H26 k0))))) (H22 u0 (tlt_head_sx k0 u0 t5) u2 
-H23 u3 H14))))) (H22 t5 (tlt_head_dx k0 u0 t5) t4 H24 t6 H15)))) k H21))))) 
-H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_beta u v1 v2 H9 t5 t6 H10) 
-\Rightarrow (\lambda (H11: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u 
-t5)) t)).(\lambda (H12: (eq T (THead (Bind Abbr) v2 t6) t2)).(eq_ind T (THead 
-(Flat Appl) v1 (THead (Bind Abst) u t5)) (\lambda (_: T).((eq T (THead (Bind 
-Abbr) v2 t6) t2) \to ((pr0 v1 v2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: 
-T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda 
-(H13: (eq T (THead (Bind Abbr) v2 t6) t2)).(eq_ind T (THead (Bind Abbr) v2 
-t6) (\lambda (t7: T).((pr0 v1 v2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: 
-T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (H14: 
-(pr0 v1 v2)).(\lambda (H15: (pr0 t5 t6)).(let H16 \def (eq_ind_r T t (\lambda 
-(t7: T).(eq T (THead k u1 t3) t7)) H4 (THead (Flat Appl) v1 (THead (Bind 
-Abst) u t5)) H11) in (let H17 \def (f_equal T K (\lambda (e: T).(match e in T 
-return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) 
+(H8: (pr0 t3 t4)).(let H9 \def (match H1 with [(pr0_refl t5) \Rightarrow 
+(\lambda (H9: (eq T t5 t)).(\lambda (H10: (eq T t5 t2)).(eq_ind T t (\lambda 
+(t6: T).((eq T t6 t2) \to (ex2 T (\lambda (t7: T).(pr0 (THead k u2 t4) t7)) 
+(\lambda (t7: T).(pr0 t2 t7))))) (\lambda (H11: (eq T t t2)).(eq_ind T t2 
+(\lambda (_: T).(ex2 T (\lambda (t7: T).(pr0 (THead k u2 t4) t7)) (\lambda 
+(t7: T).(pr0 t2 t7)))) (let H12 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 
+t2)) H11 (THead k u1 t3) H4) in (eq_ind T (THead k u1 t3) (\lambda (t6: 
+T).(ex2 T (\lambda (t7: T).(pr0 (THead k u2 t4) t7)) (\lambda (t7: T).(pr0 t6 
+t7)))) (let H13 \def (eq_ind_r T t (\lambda (t6: T).(eq T t5 t6)) H9 (THead k 
+u1 t3) H4) in (let H14 \def (eq_ind_r T t (\lambda (t6: T).(\forall (v: 
+T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall (t8: T).((pr0 v 
+t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 
+t9)))))))))) H (THead k u1 t3) H4) in (ex_intro2 T (\lambda (t6: T).(pr0 
+(THead k u2 t4) t6)) (\lambda (t6: T).(pr0 (THead k u1 t3) t6)) (THead k u2 
+t4) (pr0_refl (THead k u2 t4)) (pr0_comp u1 u2 H7 t3 t4 H8 k)))) t2 H12)) t 
+(sym_eq T t t2 H11))) t5 (sym_eq T t5 t H9) H10))) | (pr0_comp u0 u3 H9 t5 t6 
+H10 k0) \Rightarrow (\lambda (H11: (eq T (THead k0 u0 t5) t)).(\lambda (H12: 
+(eq T (THead k0 u3 t6) t2)).(eq_ind T (THead k0 u0 t5) (\lambda (_: T).((eq T 
+(THead k0 u3 t6) t2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda 
+(t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda 
+(H13: (eq T (THead k0 u3 t6) t2)).(eq_ind T (THead k0 u3 t6) (\lambda (t7: 
+T).((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 
+t4) t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (H14: (pr0 u0 
+u3)).(\lambda (H15: (pr0 t5 t6)).(let H16 \def (eq_ind_r T t (\lambda (t7: 
+T).(eq T (THead k u1 t3) t7)) H4 (THead k0 u0 t5) H11) in (let H17 \def 
+(f_equal T K (\lambda (e: T).(match e with [(TSort _) \Rightarrow k | (TLRef 
+_) \Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k u1 t3) (THead k0 
+u0 t5) H16) in ((let H18 \def (f_equal T T (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t7 _) 
+\Rightarrow t7])) (THead k u1 t3) (THead k0 u0 t5) H16) in ((let H19 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t3 | (TLRef 
+_) \Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t3) (THead 
+k0 u0 t5) H16) in (\lambda (H20: (eq T u1 u0)).(\lambda (H21: (eq K k 
+k0)).(let H22 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) 
+\to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T 
+(\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H 
+(THead k0 u0 t5) H11) in (eq_ind_r K k0 (\lambda (k1: K).(ex2 T (\lambda (t7: 
+T).(pr0 (THead k1 u2 t4) t7)) (\lambda (t7: T).(pr0 (THead k0 u3 t6) t7)))) 
+(let H23 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H7 u0 H20) in (let 
+H24 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H8 t5 H19) in (ex2_ind T 
+(\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda 
+(t7: T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7: T).(pr0 (THead k0 u3 t6) 
+t7))) (\lambda (x: T).(\lambda (H25: (pr0 t4 x)).(\lambda (H26: (pr0 t6 
+x)).(ex2_ind T (\lambda (t7: T).(pr0 u2 t7)) (\lambda (t7: T).(pr0 u3 t7)) 
+(ex2 T (\lambda (t7: T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7: T).(pr0 
+(THead k0 u3 t6) t7))) (\lambda (x0: T).(\lambda (H27: (pr0 u2 x0)).(\lambda 
+(H28: (pr0 u3 x0)).(ex_intro2 T (\lambda (t7: T).(pr0 (THead k0 u2 t4) t7)) 
+(\lambda (t7: T).(pr0 (THead k0 u3 t6) t7)) (THead k0 x0 x) (pr0_comp u2 x0 
+H27 t4 x H25 k0) (pr0_comp u3 x0 H28 t6 x H26 k0))))) (H22 u0 (tlt_head_sx k0 
+u0 t5) u2 H23 u3 H14))))) (H22 t5 (tlt_head_dx k0 u0 t5) t4 H24 t6 H15)))) k 
+H21))))) H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_beta u v1 v2 H9 
+t5 t6 H10) \Rightarrow (\lambda (H11: (eq T (THead (Flat Appl) v1 (THead 
+(Bind Abst) u t5)) t)).(\lambda (H12: (eq T (THead (Bind Abbr) v2 t6) 
+t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) (\lambda (_: 
+T).((eq T (THead (Bind Abbr) v2 t6) t2) \to ((pr0 v1 v2) \to ((pr0 t5 t6) \to 
+(ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 
+t8))))))) (\lambda (H13: (eq T (THead (Bind Abbr) v2 t6) t2)).(eq_ind T 
+(THead (Bind Abbr) v2 t6) (\lambda (t7: T).((pr0 v1 v2) \to ((pr0 t5 t6) \to 
+(ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t7 
+t8)))))) (\lambda (H14: (pr0 v1 v2)).(\lambda (H15: (pr0 t5 t6)).(let H16 
+\def (eq_ind_r T t (\lambda (t7: T).(eq T (THead k u1 t3) t7)) H4 (THead 
+(Flat Appl) v1 (THead (Bind Abst) u t5)) H11) in (let H17 \def (f_equal T K 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow k | (TLRef _) 

 \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) (THead (Flat 
 Appl) v1 (THead (Bind Abst) u t5)) H16) in ((let H18 \def (f_equal T T 
 \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) (THead (Flat 
 Appl) v1 (THead (Bind Abst) u t5)) H16) in ((let H18 \def (f_equal T T 

-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) 
-(THead k u1 t3) (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) H16) in ((let 
-H19 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) 
-with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7) 
-\Rightarrow t7])) (THead k u1 t3) (THead (Flat Appl) v1 (THead (Bind Abst) u 
-t5)) H16) in (\lambda (H20: (eq T u1 v1)).(\lambda (H21: (eq K k (Flat 
-Appl))).(let H22 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v 
-t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to 
-(ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 
-t10)))))))))) H (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) H11) in 
-(eq_ind_r K (Flat Appl) (\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 (THead 
-k0 u2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t6) t7)))) (let 
-H23 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H7 v1 H20) in (let H24 
-\def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H8 (THead (Bind Abst) u t5) 
-H19) in (let H25 \def (match H24 in pr0 return (\lambda (t7: T).(\lambda (t8: 
-T).(\lambda (_: (pr0 t7 t8)).((eq T t7 (THead (Bind Abst) u t5)) \to ((eq T 
-t8 t4) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) 
-(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9)))))))) with [(pr0_refl 
-t7) \Rightarrow (\lambda (H25: (eq T t7 (THead (Bind Abst) u t5))).(\lambda 
-(H26: (eq T t7 t4)).(eq_ind T (THead (Bind Abst) u t5) (\lambda (t8: T).((eq 
-T t8 t4) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) 
-(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9))))) (\lambda (H27: (eq T 
-(THead (Bind Abst) u t5) t4)).(eq_ind T (THead (Bind Abst) u t5) (\lambda 
-(t8: T).(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t8) t9)) (\lambda 
-(t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9)))) (ex2_ind T (\lambda (t8: 
-T).(pr0 u2 t8)) (\lambda (t8: T).(pr0 v2 t8)) (ex2 T (\lambda (t8: T).(pr0 
-(THead (Flat Appl) u2 (THead (Bind Abst) u t5)) t8)) (\lambda (t8: T).(pr0 
-(THead (Bind Abbr) v2 t6) t8))) (\lambda (x: T).(\lambda (H28: (pr0 u2 
-x)).(\lambda (H29: (pr0 v2 x)).(ex_intro2 T (\lambda (t8: T).(pr0 (THead 
-(Flat Appl) u2 (THead (Bind Abst) u t5)) t8)) (\lambda (t8: T).(pr0 (THead 
-(Bind Abbr) v2 t6) t8)) (THead (Bind Abbr) x t6) (pr0_beta u u2 x H28 t5 t6 
-H15) (pr0_comp v2 x H29 t6 t6 (pr0_refl t6) (Bind Abbr)))))) (H22 v1 
-(tlt_head_sx (Flat Appl) v1 (THead (Bind Abst) u t5)) u2 H23 v2 H14)) t4 
-H27)) t7 (sym_eq T t7 (THead (Bind Abst) u t5) H25) H26))) | (pr0_comp u0 u3 
-H25 t7 t8 H26 k0) \Rightarrow (\lambda (H27: (eq T (THead k0 u0 t7) (THead 
-(Bind Abst) u t5))).(\lambda (H28: (eq T (THead k0 u3 t8) t4)).((let H29 \def 
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow t7 | (TLRef _) \Rightarrow t7 | (THead _ _ t9) 
-\Rightarrow t9])) (THead k0 u0 t7) (THead (Bind Abst) u t5) H27) in ((let H30 
-\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) 
-with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t9 _) 
-\Rightarrow t9])) (THead k0 u0 t7) (THead (Bind Abst) u t5) H27) in ((let H31 
-\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) 
-with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _) 
-\Rightarrow k1])) (THead k0 u0 t7) (THead (Bind Abst) u t5) H27) in (eq_ind K 
-(Bind Abst) (\lambda (k1: K).((eq T u0 u) \to ((eq T t7 t5) \to ((eq T (THead 
-k1 u3 t8) t4) \to ((pr0 u0 u3) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: 
-T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind 
-Abbr) v2 t6) t9))))))))) (\lambda (H32: (eq T u0 u)).(eq_ind T u (\lambda 
-(t9: T).((eq T t7 t5) \to ((eq T (THead (Bind Abst) u3 t8) t4) \to ((pr0 t9 
-u3) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 
-t4) t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t6) t10)))))))) 
-(\lambda (H33: (eq T t7 t5)).(eq_ind T t5 (\lambda (t9: T).((eq T (THead 
-(Bind Abst) u3 t8) t4) \to ((pr0 u u3) \to ((pr0 t9 t8) \to (ex2 T (\lambda 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow u1 | (TLRef _) 
+\Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) (THead 
+(Flat Appl) v1 (THead (Bind Abst) u t5)) H16) in ((let H19 \def (f_equal T T 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow t3 | (TLRef _) 
+\Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t3) (THead 
+(Flat Appl) v1 (THead (Bind Abst) u t5)) H16) in (\lambda (H20: (eq T u1 
+v1)).(\lambda (H21: (eq K k (Flat Appl))).(let H22 \def (eq_ind_r T t 
+(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) 
+\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) 
+(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Appl) v1 (THead (Bind 
+Abst) u t5)) H11) in (eq_ind_r K (Flat Appl) (\lambda (k0: K).(ex2 T (\lambda 
+(t7: T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) 
+v2 t6) t7)))) (let H23 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H7 v1 
+H20) in (let H24 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H8 (THead 
+(Bind Abst) u t5) H19) in (let H25 \def (match H24 with [(pr0_refl t7) 
+\Rightarrow (\lambda (H25: (eq T t7 (THead (Bind Abst) u t5))).(\lambda (H26: 
+(eq T t7 t4)).(eq_ind T (THead (Bind Abst) u t5) (\lambda (t8: T).((eq T t8 
+t4) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda 
+(t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9))))) (\lambda (H27: (eq T (THead 
+(Bind Abst) u t5) t4)).(eq_ind T (THead (Bind Abst) u t5) (\lambda (t8: 
+T).(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t8) t9)) (\lambda (t9: 
+T).(pr0 (THead (Bind Abbr) v2 t6) t9)))) (ex2_ind T (\lambda (t8: T).(pr0 u2 
+t8)) (\lambda (t8: T).(pr0 v2 t8)) (ex2 T (\lambda (t8: T).(pr0 (THead (Flat 
+Appl) u2 (THead (Bind Abst) u t5)) t8)) (\lambda (t8: T).(pr0 (THead (Bind 
+Abbr) v2 t6) t8))) (\lambda (x: T).(\lambda (H28: (pr0 u2 x)).(\lambda (H29: 
+(pr0 v2 x)).(ex_intro2 T (\lambda (t8: T).(pr0 (THead (Flat Appl) u2 (THead 
+(Bind Abst) u t5)) t8)) (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t6) t8)) 
+(THead (Bind Abbr) x t6) (pr0_beta u u2 x H28 t5 t6 H15) (pr0_comp v2 x H29 
+t6 t6 (pr0_refl t6) (Bind Abbr)))))) (H22 v1 (tlt_head_sx (Flat Appl) v1 
+(THead (Bind Abst) u t5)) u2 H23 v2 H14)) t4 H27)) t7 (sym_eq T t7 (THead 
+(Bind Abst) u t5) H25) H26))) | (pr0_comp u0 u3 H25 t7 t8 H26 k0) \Rightarrow 
+(\lambda (H27: (eq T (THead k0 u0 t7) (THead (Bind Abst) u t5))).(\lambda 
+(H28: (eq T (THead k0 u3 t8) t4)).((let H29 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow t7 | (TLRef _) \Rightarrow t7 | 
+(THead _ _ t9) \Rightarrow t9])) (THead k0 u0 t7) (THead (Bind Abst) u t5) 
+H27) in ((let H30 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t9 _) \Rightarrow t9])) 
+(THead k0 u0 t7) (THead (Bind Abst) u t5) H27) in ((let H31 \def (f_equal T K 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow k0 | (TLRef _) 
+\Rightarrow k0 | (THead k1 _ _) \Rightarrow k1])) (THead k0 u0 t7) (THead 
+(Bind Abst) u t5) H27) in (eq_ind K (Bind Abst) (\lambda (k1: K).((eq T u0 u) 
+\to ((eq T t7 t5) \to ((eq T (THead k1 u3 t8) t4) \to ((pr0 u0 u3) \to ((pr0 
+t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) 
+(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9))))))))) (\lambda (H32: 
+(eq T u0 u)).(eq_ind T u (\lambda (t9: T).((eq T t7 t5) \to ((eq T (THead 
+(Bind Abst) u3 t8) t4) \to ((pr0 t9 u3) \to ((pr0 t7 t8) \to (ex2 T (\lambda 

 (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda (t10: T).(pr0 (THead 
 (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda (t10: T).(pr0 (THead 

-(Bind Abbr) v2 t6) t10))))))) (\lambda (H34: (eq T (THead (Bind Abst) u3 t8) 
-t4)).(eq_ind T (THead (Bind Abst) u3 t8) (\lambda (t9: T).((pr0 u u3) \to 
-((pr0 t5 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t9) 
-t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t6) t10)))))) (\lambda (_: 
-(pr0 u u3)).(\lambda (H36: (pr0 t5 t8)).(ex2_ind T (\lambda (t9: T).(pr0 t8 
-t9)) (\lambda (t9: T).(pr0 t6 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat 
-Appl) u2 (THead (Bind Abst) u3 t8)) t9)) (\lambda (t9: T).(pr0 (THead (Bind 
-Abbr) v2 t6) t9))) (\lambda (x: T).(\lambda (H37: (pr0 t8 x)).(\lambda (H38: 
-(pr0 t6 x)).(ex2_ind T (\lambda (t9: T).(pr0 u2 t9)) (\lambda (t9: T).(pr0 v2 
-t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u3 
-t8)) t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9))) (\lambda (x0: 
-T).(\lambda (H39: (pr0 u2 x0)).(\lambda (H40: (pr0 v2 x0)).(ex_intro2 T 
-(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u3 t8)) t9)) 
-(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9)) (THead (Bind Abbr) x0 x) 
-(pr0_beta u3 u2 x0 H39 t8 x H37) (pr0_comp v2 x0 H40 t6 x H38 (Bind 
-Abbr)))))) (H22 v1 (tlt_head_sx (Flat Appl) v1 (THead (Bind Abst) u t5)) u2 
-H23 v2 H14))))) (H22 t5 (tlt_trans (THead (Bind Abst) u t5) t5 (THead (Flat 
-Appl) v1 (THead (Bind Abst) u t5)) (tlt_head_dx (Bind Abst) u t5) 
-(tlt_head_dx (Flat Appl) v1 (THead (Bind Abst) u t5))) t8 H36 t6 H15)))) t4 
-H34)) t7 (sym_eq T t7 t5 H33))) u0 (sym_eq T u0 u H32))) k0 (sym_eq K k0 
-(Bind Abst) H31))) H30)) H29)) H28 H25 H26))) | (pr0_beta u0 v0 v3 H25 t7 t8 
-H26) \Rightarrow (\lambda (H27: (eq T (THead (Flat Appl) v0 (THead (Bind 
-Abst) u0 t7)) (THead (Bind Abst) u t5))).(\lambda (H28: (eq T (THead (Bind 
-Abbr) v3 t8) t4)).((let H29 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind 
-Abst) u0 t7)) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) 
-\Rightarrow (match k0 in K return (\lambda (_: K).Prop) with [(Bind _) 
-\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u t5) 
-H27) in (False_ind ((eq T (THead (Bind Abbr) v3 t8) t4) \to ((pr0 v0 v3) \to 
-((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) 
-(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9)))))) H29)) H28 H25 H26))) 
-| (pr0_upsilon b H25 v0 v3 H26 u0 u3 H27 t7 t8 H28) \Rightarrow (\lambda 
-(H29: (eq T (THead (Flat Appl) v0 (THead (Bind b) u0 t7)) (THead (Bind Abst) 
-u t5))).(\lambda (H30: (eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S 
-O) O v3) t8)) t4)).((let H31 \def (eq_ind T (THead (Flat Appl) v0 (THead 
-(Bind b) u0 t7)) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) 
-with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ 
-_) \Rightarrow (match k0 in K return (\lambda (_: K).Prop) with [(Bind _) 
-\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u t5) 
-H29) in (False_ind ((eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O 
-v3) t8)) t4) \to ((not (eq B b Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to 
-((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) 
-(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9)))))))) H31)) H30 H25 H26 
-H27 H28))) | (pr0_delta u0 u3 H25 t7 t8 H26 w H27) \Rightarrow (\lambda (H28: 
-(eq T (THead (Bind Abbr) u0 t7) (THead (Bind Abst) u t5))).(\lambda (H29: (eq 
-T (THead (Bind Abbr) u3 w) t4)).((let H30 \def (eq_ind T (THead (Bind Abbr) 
-u0 t7) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) 
-\Rightarrow (match k0 in K return (\lambda (_: K).Prop) with [(Bind b) 
-\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow 
-True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) 
-\Rightarrow False])])) I (THead (Bind Abst) u t5) H28) in (False_ind ((eq T 
-(THead (Bind Abbr) u3 w) t4) \to ((pr0 u0 u3) \to ((pr0 t7 t8) \to ((subst0 O 
-u3 t8 w) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) 
-(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9))))))) H30)) H29 H25 H26 
-H27))) | (pr0_zeta b H25 t7 t8 H26 u0) \Rightarrow (\lambda (H27: (eq T 
-(THead (Bind b) u0 (lift (S O) O t7)) (THead (Bind Abst) u t5))).(\lambda 
-(H28: (eq T t8 t4)).((let H29 \def (f_equal T T (\lambda (e: T).(match e in T 
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: 
-((nat \to nat))) (d: nat) (t9: T) on t9: T \def (match t9 with [(TSort n) 
-\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with 
-[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k0 u3 t10) 
-\Rightarrow (THead k0 (lref_map f d u3) (lref_map f (s k0 d) t10))]) in 
-lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (TLRef _) \Rightarrow 
-((let rec lref_map (f: ((nat \to nat))) (d: nat) (t9: T) on t9: T \def (match 
-t9 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef 
-(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | 
-(THead k0 u3 t10) \Rightarrow (THead k0 (lref_map f d u3) (lref_map f (s k0 
-d) t10))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (THead _ _ 
-t9) \Rightarrow t9])) (THead (Bind b) u0 (lift (S O) O t7)) (THead (Bind 
-Abst) u t5) H27) in ((let H30 \def (f_equal T T (\lambda (e: T).(match e in T 
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) 
-\Rightarrow u0 | (THead _ t9 _) \Rightarrow t9])) (THead (Bind b) u0 (lift (S 
-O) O t7)) (THead (Bind Abst) u t5) H27) in ((let H31 \def (f_equal T B 
-(\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) 
-\Rightarrow b | (TLRef _) \Rightarrow b | (THead k0 _ _) \Rightarrow (match 
-k0 in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) 
-\Rightarrow b])])) (THead (Bind b) u0 (lift (S O) O t7)) (THead (Bind Abst) u 
-t5) H27) in (eq_ind B Abst (\lambda (b0: B).((eq T u0 u) \to ((eq T (lift (S 
-O) O t7) t5) \to ((eq T t8 t4) \to ((not (eq B b0 Abst)) \to ((pr0 t7 t8) \to 
-(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: 
-T).(pr0 (THead (Bind Abbr) v2 t6) t9))))))))) (\lambda (H32: (eq T u0 
-u)).(eq_ind T u (\lambda (_: T).((eq T (lift (S O) O t7) t5) \to ((eq T t8 
-t4) \to ((not (eq B Abst Abst)) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: 
-T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda (t10: T).(pr0 (THead (Bind 
-Abbr) v2 t6) t10)))))))) (\lambda (H33: (eq T (lift (S O) O t7) t5)).(eq_ind 
-T (lift (S O) O t7) (\lambda (_: T).((eq T t8 t4) \to ((not (eq B Abst Abst)) 
-\to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) 
+(Bind Abbr) v2 t6) t10)))))))) (\lambda (H33: (eq T t7 t5)).(eq_ind T t5 
+(\lambda (t9: T).((eq T (THead (Bind Abst) u3 t8) t4) \to ((pr0 u u3) \to 
+((pr0 t9 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) 

 t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t6) t10))))))) (\lambda 
 t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t6) t10))))))) (\lambda 

-(H34: (eq T t8 t4)).(eq_ind T t4 (\lambda (t9: T).((not (eq B Abst Abst)) \to 
-((pr0 t7 t9) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) 
-t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t6) t10)))))) (\lambda 
-(H35: (not (eq B Abst Abst))).(\lambda (_: (pr0 t7 t4)).(let H37 \def (match 
-(H35 (refl_equal B Abst)) in False return (\lambda (_: False).(ex2 T (\lambda 
-(t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead 
-(Bind Abbr) v2 t6) t9)))) with []) in H37))) t8 (sym_eq T t8 t4 H34))) t5 
-H33)) u0 (sym_eq T u0 u H32))) b (sym_eq B b Abst H31))) H30)) H29)) H28 H25 
-H26))) | (pr0_tau t7 t8 H25 u0) \Rightarrow (\lambda (H26: (eq T (THead (Flat 
-Cast) u0 t7) (THead (Bind Abst) u t5))).(\lambda (H27: (eq T t8 t4)).((let 
-H28 \def (eq_ind T (THead (Flat Cast) u0 t7) (\lambda (e: T).(match e in T 
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda 
-(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow 
-True])])) I (THead (Bind Abst) u t5) H26) in (False_ind ((eq T t8 t4) \to 
-((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) 
-(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9))))) H28)) H27 H25)))]) in 
-(H25 (refl_equal T (THead (Bind Abst) u t5)) (refl_equal T t4))))) k H21))))) 
-H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_upsilon b H9 v1 v2 H10 u0 
-u3 H11 t5 t6 H12) \Rightarrow (\lambda (H13: (eq T (THead (Flat Appl) v1 
-(THead (Bind b) u0 t5)) t)).(\lambda (H14: (eq T (THead (Bind b) u3 (THead 
-(Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Flat Appl) v1 
-(THead (Bind b) u0 t5)) (\lambda (_: T).((eq T (THead (Bind b) u3 (THead 
-(Flat Appl) (lift (S O) O v2) t6)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 
-v2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead 
-k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H15: (eq T 
-(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T 
-(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) (\lambda (t7: 
-T).((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) 
-\to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 
-t7 t8)))))))) (\lambda (H16: (not (eq B b Abst))).(\lambda (H17: (pr0 v1 
-v2)).(\lambda (H18: (pr0 u0 u3)).(\lambda (H19: (pr0 t5 t6)).(let H20 \def 
-(eq_ind_r T t (\lambda (t7: T).(eq T (THead k u1 t3) t7)) H4 (THead (Flat 
-Appl) v1 (THead (Bind b) u0 t5)) H13) in (let H21 \def (f_equal T K (\lambda 
-(e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k 
-| (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) 
-(THead (Flat Appl) v1 (THead (Bind b) u0 t5)) H20) in ((let H22 \def (f_equal 
-T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) 
-(THead k u1 t3) (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) H20) in ((let 
-H23 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) 
-with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7) 
-\Rightarrow t7])) (THead k u1 t3) (THead (Flat Appl) v1 (THead (Bind b) u0 
-t5)) H20) in (\lambda (H24: (eq T u1 v1)).(\lambda (H25: (eq K k (Flat 
-Appl))).(let H26 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v 
-t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to 
-(ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 
-t10)))))))))) H (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) H13) in 
-(eq_ind_r K (Flat Appl) (\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 (THead 
-k0 u2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) 
-(lift (S O) O v2) t6)) t7)))) (let H27 \def (eq_ind T u1 (\lambda (t7: 
-T).(pr0 t7 u2)) H7 v1 H24) in (let H28 \def (eq_ind T t3 (\lambda (t7: 
-T).(pr0 t7 t4)) H8 (THead (Bind b) u0 t5) H23) in (let H29 \def (match H28 in 
-pr0 return (\lambda (t7: T).(\lambda (t8: T).(\lambda (_: (pr0 t7 t8)).((eq T 
-t7 (THead (Bind b) u0 t5)) \to ((eq T t8 t4) \to (ex2 T (\lambda (t9: T).(pr0 
-(THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 
-(THead (Flat Appl) (lift (S O) O v2) t6)) t9)))))))) with [(pr0_refl t7) 
-\Rightarrow (\lambda (H29: (eq T t7 (THead (Bind b) u0 t5))).(\lambda (H30: 
-(eq T t7 t4)).(eq_ind T (THead (Bind b) u0 t5) (\lambda (t8: T).((eq T t8 t4) 
-\to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: 
-T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9))))) 
-(\lambda (H31: (eq T (THead (Bind b) u0 t5) t4)).(eq_ind T (THead (Bind b) u0 
-t5) (\lambda (t8: T).(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t8) 
-t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) 
-O v2) t6)) t9)))) (ex2_ind T (\lambda (t8: T).(pr0 u2 t8)) (\lambda (t8: 
-T).(pr0 v2 t8)) (ex2 T (\lambda (t8: T).(pr0 (THead (Flat Appl) u2 (THead 
-(Bind b) u0 t5)) t8)) (\lambda (t8: T).(pr0 (THead (Bind b) u3 (THead (Flat 
-Appl) (lift (S O) O v2) t6)) t8))) (\lambda (x: T).(\lambda (H32: (pr0 u2 
-x)).(\lambda (H33: (pr0 v2 x)).(pr0_confluence__pr0_cong_upsilon_refl b H16 
-u0 u3 H18 t5 t6 H19 u2 v2 x H32 H33)))) (H26 v1 (tlt_head_sx (Flat Appl) v1 
-(THead (Bind b) u0 t5)) u2 H27 v2 H17)) t4 H31)) t7 (sym_eq T t7 (THead (Bind 
-b) u0 t5) H29) H30))) | (pr0_comp u4 u5 H29 t7 t8 H30 k0) \Rightarrow 
-(\lambda (H31: (eq T (THead k0 u4 t7) (THead (Bind b) u0 t5))).(\lambda (H32: 
-(eq T (THead k0 u5 t8) t4)).((let H33 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t7 | 
-(TLRef _) \Rightarrow t7 | (THead _ _ t9) \Rightarrow t9])) (THead k0 u4 t7) 
-(THead (Bind b) u0 t5) H31) in ((let H34 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u4 | 
-(TLRef _) \Rightarrow u4 | (THead _ t9 _) \Rightarrow t9])) (THead k0 u4 t7) 
-(THead (Bind b) u0 t5) H31) in ((let H35 \def (f_equal T K (\lambda (e: 
-T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k0 | 
-(TLRef _) \Rightarrow k0 | (THead k1 _ _) \Rightarrow k1])) (THead k0 u4 t7) 
-(THead (Bind b) u0 t5) H31) in (eq_ind K (Bind b) (\lambda (k1: K).((eq T u4 
-u0) \to ((eq T t7 t5) \to ((eq T (THead k1 u5 t8) t4) \to ((pr0 u4 u5) \to 
-((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) 
-(\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) 
-t6)) t9))))))))) (\lambda (H36: (eq T u4 u0)).(eq_ind T u0 (\lambda (t9: 
-T).((eq T t7 t5) \to ((eq T (THead (Bind b) u5 t8) t4) \to ((pr0 t9 u5) \to 
-((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) 
-t10)) (\lambda (t10: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S 
-O) O v2) t6)) t10)))))))) (\lambda (H37: (eq T t7 t5)).(eq_ind T t5 (\lambda 
-(t9: T).((eq T (THead (Bind b) u5 t8) t4) \to ((pr0 u0 u5) \to ((pr0 t9 t8) 
-\to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda 
-(t10: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) 
-t10))))))) (\lambda (H38: (eq T (THead (Bind b) u5 t8) t4)).(eq_ind T (THead 
-(Bind b) u5 t8) (\lambda (t9: T).((pr0 u0 u5) \to ((pr0 t5 t8) \to (ex2 T 
-(\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t9) t10)) (\lambda (t10: T).(pr0 
-(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t10)))))) 
-(\lambda (H39: (pr0 u0 u5)).(\lambda (H40: (pr0 t5 t8)).(ex2_ind T (\lambda 
-(t9: T).(pr0 t8 t9)) (\lambda (t9: T).(pr0 t6 t9)) (ex2 T (\lambda (t9: 
-T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 t8)) t9)) (\lambda (t9: 
-T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9))) 
-(\lambda (x: T).(\lambda (H41: (pr0 t8 x)).(\lambda (H42: (pr0 t6 
-x)).(ex2_ind T (\lambda (t9: T).(pr0 u5 t9)) (\lambda (t9: T).(pr0 u3 t9)) 
-(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 t8)) 
-t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) 
-O v2) t6)) t9))) (\lambda (x0: T).(\lambda (H43: (pr0 u5 x0)).(\lambda (H44: 
-(pr0 u3 x0)).(ex2_ind T (\lambda (t9: T).(pr0 u2 t9)) (\lambda (t9: T).(pr0 
-v2 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 
-t8)) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift 
-(S O) O v2) t6)) t9))) (\lambda (x1: T).(\lambda (H45: (pr0 u2 x1)).(\lambda 
-(H46: (pr0 v2 x1)).(pr0_confluence__pr0_cong_upsilon_cong b H16 u2 v2 x1 H45 
-H46 t8 t6 x H41 H42 u5 u3 x0 H43 H44)))) (H26 v1 (tlt_head_sx (Flat Appl) v1 
-(THead (Bind b) u0 t5)) u2 H27 v2 H17))))) (H26 u0 (tlt_trans (THead (Bind b) 
-u0 t5) u0 (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) (tlt_head_sx (Bind b) 
-u0 t5) (tlt_head_dx (Flat Appl) v1 (THead (Bind b) u0 t5))) u5 H39 u3 
-H18))))) (H26 t5 (tlt_trans (THead (Bind b) u0 t5) t5 (THead (Flat Appl) v1 
-(THead (Bind b) u0 t5)) (tlt_head_dx (Bind b) u0 t5) (tlt_head_dx (Flat Appl) 
-v1 (THead (Bind b) u0 t5))) t8 H40 t6 H19)))) t4 H38)) t7 (sym_eq T t7 t5 
-H37))) u4 (sym_eq T u4 u0 H36))) k0 (sym_eq K k0 (Bind b) H35))) H34)) H33)) 
-H32 H29 H30))) | (pr0_beta u v0 v3 H29 t7 t8 H30) \Rightarrow (\lambda (H31: 
-(eq T (THead (Flat Appl) v0 (THead (Bind Abst) u t7)) (THead (Bind b) u0 
-t5))).(\lambda (H32: (eq T (THead (Bind Abbr) v3 t8) t4)).((let H33 \def 
-(eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u t7)) (\lambda (e: 
-T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in 
-K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) 
-\Rightarrow True])])) I (THead (Bind b) u0 t5) H31) in (False_ind ((eq T 
-(THead (Bind Abbr) v3 t8) t4) \to ((pr0 v0 v3) \to ((pr0 t7 t8) \to (ex2 T 
+(H34: (eq T (THead (Bind Abst) u3 t8) t4)).(eq_ind T (THead (Bind Abst) u3 
+t8) (\lambda (t9: T).((pr0 u u3) \to ((pr0 t5 t8) \to (ex2 T (\lambda (t10: 
+T).(pr0 (THead (Flat Appl) u2 t9) t10)) (\lambda (t10: T).(pr0 (THead (Bind 
+Abbr) v2 t6) t10)))))) (\lambda (_: (pr0 u u3)).(\lambda (H36: (pr0 t5 
+t8)).(ex2_ind T (\lambda (t9: T).(pr0 t8 t9)) (\lambda (t9: T).(pr0 t6 t9)) 
+(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u3 t8)) 
+t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9))) (\lambda (x: 
+T).(\lambda (H37: (pr0 t8 x)).(\lambda (H38: (pr0 t6 x)).(ex2_ind T (\lambda 
+(t9: T).(pr0 u2 t9)) (\lambda (t9: T).(pr0 v2 t9)) (ex2 T (\lambda (t9: 
+T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u3 t8)) t9)) (\lambda (t9: 
+T).(pr0 (THead (Bind Abbr) v2 t6) t9))) (\lambda (x0: T).(\lambda (H39: (pr0 
+u2 x0)).(\lambda (H40: (pr0 v2 x0)).(ex_intro2 T (\lambda (t9: T).(pr0 (THead 
+(Flat Appl) u2 (THead (Bind Abst) u3 t8)) t9)) (\lambda (t9: T).(pr0 (THead 
+(Bind Abbr) v2 t6) t9)) (THead (Bind Abbr) x0 x) (pr0_beta u3 u2 x0 H39 t8 x 
+H37) (pr0_comp v2 x0 H40 t6 x H38 (Bind Abbr)))))) (H22 v1 (tlt_head_sx (Flat 
+Appl) v1 (THead (Bind Abst) u t5)) u2 H23 v2 H14))))) (H22 t5 (tlt_trans 
+(THead (Bind Abst) u t5) t5 (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) 
+(tlt_head_dx (Bind Abst) u t5) (tlt_head_dx (Flat Appl) v1 (THead (Bind Abst) 
+u t5))) t8 H36 t6 H15)))) t4 H34)) t7 (sym_eq T t7 t5 H33))) u0 (sym_eq T u0 
+u H32))) k0 (sym_eq K k0 (Bind Abst) H31))) H30)) H29)) H28 H25 H26))) | 
+(pr0_beta u0 v0 v3 H25 t7 t8 H26) \Rightarrow (\lambda (H27: (eq T (THead 
+(Flat Appl) v0 (THead (Bind Abst) u0 t7)) (THead (Bind Abst) u t5))).(\lambda 
+(H28: (eq T (THead (Bind Abbr) v3 t8) t4)).((let H29 \def (eq_ind T (THead 
+(Flat Appl) v0 (THead (Bind Abst) u0 t7)) (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) 
+\Rightarrow (match k0 with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow 
+True])])) I (THead (Bind Abst) u t5) H27) in (False_ind ((eq T (THead (Bind 
+Abbr) v3 t8) t4) \to ((pr0 v0 v3) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: 
+T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind 
+Abbr) v2 t6) t9)))))) H29)) H28 H25 H26))) | (pr0_upsilon b H25 v0 v3 H26 u0 
+u3 H27 t7 t8 H28) \Rightarrow (\lambda (H29: (eq T (THead (Flat Appl) v0 
+(THead (Bind b) u0 t7)) (THead (Bind Abst) u t5))).(\lambda (H30: (eq T 
+(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v3) t8)) t4)).((let H31 
+\def (eq_ind T (THead (Flat Appl) v0 (THead (Bind b) u0 t7)) (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
+(THead k0 _ _) \Rightarrow (match k0 with [(Bind _) \Rightarrow False | (Flat 
+_) \Rightarrow True])])) I (THead (Bind Abst) u t5) H29) in (False_ind ((eq T 
+(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v3) t8)) t4) \to ((not 
+(eq B b Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to ((pr0 t7 t8) \to (ex2 T 

 (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 
 (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 

-(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9)))))) H33)) 
-H32 H29 H30))) | (pr0_upsilon b0 H29 v0 v3 H30 u4 u5 H31 t7 t8 H32) 
-\Rightarrow (\lambda (H33: (eq T (THead (Flat Appl) v0 (THead (Bind b0) u4 
-t7)) (THead (Bind b) u0 t5))).(\lambda (H34: (eq T (THead (Bind b0) u5 (THead 
-(Flat Appl) (lift (S O) O v3) t8)) t4)).((let H35 \def (eq_ind T (THead (Flat 
-Appl) v0 (THead (Bind b0) u4 t7)) (\lambda (e: T).(match e in T return 
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda 
-(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow 
-True])])) I (THead (Bind b) u0 t5) H33) in (False_ind ((eq T (THead (Bind b0) 
-u5 (THead (Flat Appl) (lift (S O) O v3) t8)) t4) \to ((not (eq B b0 Abst)) 
-\to ((pr0 v0 v3) \to ((pr0 u4 u5) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: 
-T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) 
-u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9)))))))) H35)) H34 H29 H30 H31 
-H32))) | (pr0_delta u4 u5 H29 t7 t8 H30 w H31) \Rightarrow (\lambda (H32: (eq 
-T (THead (Bind Abbr) u4 t7) (THead (Bind b) u0 t5))).(\lambda (H33: (eq T 
-(THead (Bind Abbr) u5 w) t4)).((let H34 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t7 | 
-(TLRef _) \Rightarrow t7 | (THead _ _ t9) \Rightarrow t9])) (THead (Bind 
-Abbr) u4 t7) (THead (Bind b) u0 t5) H32) in ((let H35 \def (f_equal T T 
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow u4 | (TLRef _) \Rightarrow u4 | (THead _ t9 _) \Rightarrow t9])) 
-(THead (Bind Abbr) u4 t7) (THead (Bind b) u0 t5) H32) in ((let H36 \def 
-(f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with 
-[(TSort _) \Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k0 _ _) 
-\Rightarrow (match k0 in K return (\lambda (_: K).B) with [(Bind b0) 
-\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (THead (Bind Abbr) u4 t7) 
-(THead (Bind b) u0 t5) H32) in (eq_ind B Abbr (\lambda (b0: B).((eq T u4 u0) 
-\to ((eq T t7 t5) \to ((eq T (THead (Bind Abbr) u5 w) t4) \to ((pr0 u4 u5) 
-\to ((pr0 t7 t8) \to ((subst0 O u5 t8 w) \to (ex2 T (\lambda (t9: T).(pr0 
-(THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b0) u3 
-(THead (Flat Appl) (lift (S O) O v2) t6)) t9)))))))))) (\lambda (H37: (eq T 
-u4 u0)).(eq_ind T u0 (\lambda (t9: T).((eq T t7 t5) \to ((eq T (THead (Bind 
-Abbr) u5 w) t4) \to ((pr0 t9 u5) \to ((pr0 t7 t8) \to ((subst0 O u5 t8 w) \to 
+(THead (Bind Abbr) v2 t6) t9)))))))) H31)) H30 H25 H26 H27 H28))) | 
+(pr0_delta u0 u3 H25 t7 t8 H26 w H27) \Rightarrow (\lambda (H28: (eq T (THead 
+(Bind Abbr) u0 t7) (THead (Bind Abst) u t5))).(\lambda (H29: (eq T (THead 
+(Bind Abbr) u3 w) t4)).((let H30 \def (eq_ind T (THead (Bind Abbr) u0 t7) 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 with [(Bind b) 
+\Rightarrow (match b with [Abbr \Rightarrow True | Abst \Rightarrow False | 
+Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (THead (Bind 
+Abst) u t5) H28) in (False_ind ((eq T (THead (Bind Abbr) u3 w) t4) \to ((pr0 
+u0 u3) \to ((pr0 t7 t8) \to ((subst0 O u3 t8 w) \to (ex2 T (\lambda (t9: 
+T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind 
+Abbr) v2 t6) t9))))))) H30)) H29 H25 H26 H27))) | (pr0_zeta b H25 t7 t8 H26 
+u0) \Rightarrow (\lambda (H27: (eq T (THead (Bind b) u0 (lift (S O) O t7)) 
+(THead (Bind Abst) u t5))).(\lambda (H28: (eq T t8 t4)).((let H29 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow (lref_map 
+(\lambda (x: nat).(plus x (S O))) O t7) | (TLRef _) \Rightarrow (lref_map 
+(\lambda (x: nat).(plus x (S O))) O t7) | (THead _ _ t9) \Rightarrow t9])) 
+(THead (Bind b) u0 (lift (S O) O t7)) (THead (Bind Abst) u t5) H27) in ((let 
+H30 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u0 
+| (TLRef _) \Rightarrow u0 | (THead _ t9 _) \Rightarrow t9])) (THead (Bind b) 
+u0 (lift (S O) O t7)) (THead (Bind Abst) u t5) H27) in ((let H31 \def 
+(f_equal T B (\lambda (e: T).(match e with [(TSort _) \Rightarrow b | (TLRef 
+_) \Rightarrow b | (THead k0 _ _) \Rightarrow (match k0 with [(Bind b0) 
+\Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead (Bind b) u0 (lift (S O) O 
+t7)) (THead (Bind Abst) u t5) H27) in (eq_ind B Abst (\lambda (b0: B).((eq T 
+u0 u) \to ((eq T (lift (S O) O t7) t5) \to ((eq T t8 t4) \to ((not (eq B b0 
+Abst)) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) 
+u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9))))))))) 
+(\lambda (H32: (eq T u0 u)).(eq_ind T u (\lambda (_: T).((eq T (lift (S O) O 
+t7) t5) \to ((eq T t8 t4) \to ((not (eq B Abst Abst)) \to ((pr0 t7 t8) \to 

 (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda (t10: 
 (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda (t10: 

-T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) 
-t10))))))))) (\lambda (H38: (eq T t7 t5)).(eq_ind T t5 (\lambda (t9: T).((eq 
-T (THead (Bind Abbr) u5 w) t4) \to ((pr0 u0 u5) \to ((pr0 t9 t8) \to ((subst0 
-O u5 t8 w) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) 
-(\lambda (t10: T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O 
-v2) t6)) t10)))))))) (\lambda (H39: (eq T (THead (Bind Abbr) u5 w) 
-t4)).(eq_ind T (THead (Bind Abbr) u5 w) (\lambda (t9: T).((pr0 u0 u5) \to 
-((pr0 t5 t8) \to ((subst0 O u5 t8 w) \to (ex2 T (\lambda (t10: T).(pr0 (THead 
-(Flat Appl) u2 t9) t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) u3 (THead 
-(Flat Appl) (lift (S O) O v2) t6)) t10))))))) (\lambda (H40: (pr0 u0 
-u5)).(\lambda (H41: (pr0 t5 t8)).(\lambda (H42: (subst0 O u5 t8 w)).(let H43 
-\def (eq_ind_r B b (\lambda (b0: B).(\forall (v: T).((tlt v (THead (Flat 
-Appl) v1 (THead (Bind b0) u0 t5))) \to (\forall (t9: T).((pr0 v t9) \to 
-(\forall (t10: T).((pr0 v t10) \to (ex2 T (\lambda (t11: T).(pr0 t9 t11)) 
-(\lambda (t11: T).(pr0 t10 t11)))))))))) H26 Abbr H36) in (let H44 \def 
-(eq_ind_r B b (\lambda (b0: B).(eq T t3 (THead (Bind b0) u0 t5))) H23 Abbr 
-H36) in (let H45 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 Abst))) 
-H16 Abbr H36) in (ex2_ind T (\lambda (t9: T).(pr0 t8 t9)) (\lambda (t9: 
-T).(pr0 t6 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead 
-(Bind Abbr) u5 w)) t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) u3 (THead 
-(Flat Appl) (lift (S O) O v2) t6)) t9))) (\lambda (x: T).(\lambda (H46: (pr0 
-t8 x)).(\lambda (H47: (pr0 t6 x)).(ex2_ind T (\lambda (t9: T).(pr0 u5 t9)) 
-(\lambda (t9: T).(pr0 u3 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) 
-u2 (THead (Bind Abbr) u5 w)) t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) u3 
-(THead (Flat Appl) (lift (S O) O v2) t6)) t9))) (\lambda (x0: T).(\lambda 
-(H48: (pr0 u5 x0)).(\lambda (H49: (pr0 u3 x0)).(ex2_ind T (\lambda (t9: 
-T).(pr0 u2 t9)) (\lambda (t9: T).(pr0 v2 t9)) (ex2 T (\lambda (t9: T).(pr0 
-(THead (Flat Appl) u2 (THead (Bind Abbr) u5 w)) t9)) (\lambda (t9: T).(pr0 
-(THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9))) 
-(\lambda (x1: T).(\lambda (H50: (pr0 u2 x1)).(\lambda (H51: (pr0 v2 
-x1)).(pr0_confluence__pr0_cong_upsilon_delta H45 u5 t8 w H42 u2 v2 x1 H50 H51 
-t6 x H46 H47 u3 x0 H48 H49)))) (H43 v1 (tlt_head_sx (Flat Appl) v1 (THead 
-(Bind Abbr) u0 t5)) u2 H27 v2 H17))))) (H43 u0 (tlt_trans (THead (Bind Abbr) 
-u0 t5) u0 (THead (Flat Appl) v1 (THead (Bind Abbr) u0 t5)) (tlt_head_sx (Bind 
-Abbr) u0 t5) (tlt_head_dx (Flat Appl) v1 (THead (Bind Abbr) u0 t5))) u5 H40 
-u3 H18))))) (H43 t5 (tlt_trans (THead (Bind Abbr) u0 t5) t5 (THead (Flat 
-Appl) v1 (THead (Bind Abbr) u0 t5)) (tlt_head_dx (Bind Abbr) u0 t5) 
-(tlt_head_dx (Flat Appl) v1 (THead (Bind Abbr) u0 t5))) t8 H41 t6 H19)))))))) 
-t4 H39)) t7 (sym_eq T t7 t5 H38))) u4 (sym_eq T u4 u0 H37))) b H36)) H35)) 
-H34)) H33 H29 H30 H31))) | (pr0_zeta b0 H29 t7 t8 H30 u) \Rightarrow (\lambda 
-(H31: (eq T (THead (Bind b0) u (lift (S O) O t7)) (THead (Bind b) u0 
-t5))).(\lambda (H32: (eq T t8 t4)).((let H33 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let 
-rec lref_map (f: ((nat \to nat))) (d: nat) (t9: T) on t9: T \def (match t9 
-with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match 
-(blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k0 
-u4 t10) \Rightarrow (THead k0 (lref_map f d u4) (lref_map f (s k0 d) t10))]) 
-in lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (TLRef _) \Rightarrow 
-((let rec lref_map (f: ((nat \to nat))) (d: nat) (t9: T) on t9: T \def (match 
-t9 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef 
-(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | 
-(THead k0 u4 t10) \Rightarrow (THead k0 (lref_map f d u4) (lref_map f (s k0 
-d) t10))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (THead _ _ 
-t9) \Rightarrow t9])) (THead (Bind b0) u (lift (S O) O t7)) (THead (Bind b) 
-u0 t5) H31) in ((let H34 \def (f_equal T T (\lambda (e: T).(match e in T 
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) 
-\Rightarrow u | (THead _ t9 _) \Rightarrow t9])) (THead (Bind b0) u (lift (S 
-O) O t7)) (THead (Bind b) u0 t5) H31) in ((let H35 \def (f_equal T B (\lambda 
-(e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b0 
-| (TLRef _) \Rightarrow b0 | (THead k0 _ _) \Rightarrow (match k0 in K return 
-(\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow 
-b0])])) (THead (Bind b0) u (lift (S O) O t7)) (THead (Bind b) u0 t5) H31) in 
-(eq_ind B b (\lambda (b1: B).((eq T u u0) \to ((eq T (lift (S O) O t7) t5) 
-\to ((eq T t8 t4) \to ((not (eq B b1 Abst)) \to ((pr0 t7 t8) \to (ex2 T 
+T).(pr0 (THead (Bind Abbr) v2 t6) t10)))))))) (\lambda (H33: (eq T (lift (S 
+O) O t7) t5)).(eq_ind T (lift (S O) O t7) (\lambda (_: T).((eq T t8 t4) \to 
+((not (eq B Abst Abst)) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 
+(THead (Flat Appl) u2 t4) t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 
+t6) t10))))))) (\lambda (H34: (eq T t8 t4)).(eq_ind T t4 (\lambda (t9: 
+T).((not (eq B Abst Abst)) \to ((pr0 t7 t9) \to (ex2 T (\lambda (t10: T).(pr0 
+(THead (Flat Appl) u2 t4) t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 
+t6) t10)))))) (\lambda (H35: (not (eq B Abst Abst))).(\lambda (_: (pr0 t7 
+t4)).(let H37 \def (match (H35 (refl_equal B Abst)) in False with []) in 
+H37))) t8 (sym_eq T t8 t4 H34))) t5 H33)) u0 (sym_eq T u0 u H32))) b (sym_eq 
+B b Abst H31))) H30)) H29)) H28 H25 H26))) | (pr0_tau t7 t8 H25 u0) 
+\Rightarrow (\lambda (H26: (eq T (THead (Flat Cast) u0 t7) (THead (Bind Abst) 
+u t5))).(\lambda (H27: (eq T t8 t4)).((let H28 \def (eq_ind T (THead (Flat 
+Cast) u0 t7) (\lambda (e: T).(match e with [(TSort _) \Rightarrow False | 
+(TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 with 
+[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind 
+Abst) u t5) H26) in (False_ind ((eq T t8 t4) \to ((pr0 t7 t8) \to (ex2 T 

 (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 
 (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 

-(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9))))))))) 
-(\lambda (H36: (eq T u u0)).(eq_ind T u0 (\lambda (_: T).((eq T (lift (S O) O 
-t7) t5) \to ((eq T t8 t4) \to ((not (eq B b Abst)) \to ((pr0 t7 t8) \to (ex2 
-T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda (t10: 
-T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) 
-t10)))))))) (\lambda (H37: (eq T (lift (S O) O t7) t5)).(eq_ind T (lift (S O) 
-O t7) (\lambda (_: T).((eq T t8 t4) \to ((not (eq B b Abst)) \to ((pr0 t7 t8) 
-\to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda 
-(t10: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) 
-t10))))))) (\lambda (H38: (eq T t8 t4)).(eq_ind T t4 (\lambda (t9: T).((not 
-(eq B b Abst)) \to ((pr0 t7 t9) \to (ex2 T (\lambda (t10: T).(pr0 (THead 
+(THead (Bind Abbr) v2 t6) t9))))) H28)) H27 H25)))]) in (H25 (refl_equal T 
+(THead (Bind Abst) u t5)) (refl_equal T t4))))) k H21))))) H18)) H17))))) t2 
+H13)) t H11 H12 H9 H10))) | (pr0_upsilon b H9 v1 v2 H10 u0 u3 H11 t5 t6 H12) 
+\Rightarrow (\lambda (H13: (eq T (THead (Flat Appl) v1 (THead (Bind b) u0 
+t5)) t)).(\lambda (H14: (eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S 
+O) O v2) t6)) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) 
+(\lambda (_: T).((eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O 
+v2) t6)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u3) \to 
+((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda 
+(t8: T).(pr0 t2 t8))))))))) (\lambda (H15: (eq T (THead (Bind b) u3 (THead 
+(Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Bind b) u3 (THead 
+(Flat Appl) (lift (S O) O v2) t6)) (\lambda (t7: T).((not (eq B b Abst)) \to 
+((pr0 v1 v2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: 
+T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda 
+(H16: (not (eq B b Abst))).(\lambda (H17: (pr0 v1 v2)).(\lambda (H18: (pr0 u0 
+u3)).(\lambda (H19: (pr0 t5 t6)).(let H20 \def (eq_ind_r T t (\lambda (t7: 
+T).(eq T (THead k u1 t3) t7)) H4 (THead (Flat Appl) v1 (THead (Bind b) u0 
+t5)) H13) in (let H21 \def (f_equal T K (\lambda (e: T).(match e with [(TSort 
+_) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) 
+(THead k u1 t3) (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) H20) in ((let 
+H22 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u1 
+| (TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) 
+(THead (Flat Appl) v1 (THead (Bind b) u0 t5)) H20) in ((let H23 \def (f_equal 
+T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t3 | (TLRef _) 
+\Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t3) (THead 
+(Flat Appl) v1 (THead (Bind b) u0 t5)) H20) in (\lambda (H24: (eq T u1 
+v1)).(\lambda (H25: (eq K k (Flat Appl))).(let H26 \def (eq_ind_r T t 
+(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) 
+\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) 
+(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Appl) v1 (THead (Bind 
+b) u0 t5)) H13) in (eq_ind_r K (Flat Appl) (\lambda (k0: K).(ex2 T (\lambda 
+(t7: T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind b) u3 
+(THead (Flat Appl) (lift (S O) O v2) t6)) t7)))) (let H27 \def (eq_ind T u1 
+(\lambda (t7: T).(pr0 t7 u2)) H7 v1 H24) in (let H28 \def (eq_ind T t3 
+(\lambda (t7: T).(pr0 t7 t4)) H8 (THead (Bind b) u0 t5) H23) in (let H29 \def 
+(match H28 with [(pr0_refl t7) \Rightarrow (\lambda (H29: (eq T t7 (THead 
+(Bind b) u0 t5))).(\lambda (H30: (eq T t7 t4)).(eq_ind T (THead (Bind b) u0 
+t5) (\lambda (t8: T).((eq T t8 t4) \to (ex2 T (\lambda (t9: T).(pr0 (THead 
+(Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat 
+Appl) (lift (S O) O v2) t6)) t9))))) (\lambda (H31: (eq T (THead (Bind b) u0 
+t5) t4)).(eq_ind T (THead (Bind b) u0 t5) (\lambda (t8: T).(ex2 T (\lambda 
+(t9: T).(pr0 (THead (Flat Appl) u2 t8) t9)) (\lambda (t9: T).(pr0 (THead 
+(Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9)))) (ex2_ind T 
+(\lambda (t8: T).(pr0 u2 t8)) (\lambda (t8: T).(pr0 v2 t8)) (ex2 T (\lambda 
+(t8: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u0 t5)) t8)) (\lambda (t8: 
+T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t8))) 
+(\lambda (x: T).(\lambda (H32: (pr0 u2 x)).(\lambda (H33: (pr0 v2 
+x)).(pr0_confluence__pr0_cong_upsilon_refl b H16 u0 u3 H18 t5 t6 H19 u2 v2 x 
+H32 H33)))) (H26 v1 (tlt_head_sx (Flat Appl) v1 (THead (Bind b) u0 t5)) u2 
+H27 v2 H17)) t4 H31)) t7 (sym_eq T t7 (THead (Bind b) u0 t5) H29) H30))) | 
+(pr0_comp u4 u5 H29 t7 t8 H30 k0) \Rightarrow (\lambda (H31: (eq T (THead k0 
+u4 t7) (THead (Bind b) u0 t5))).(\lambda (H32: (eq T (THead k0 u5 t8) 
+t4)).((let H33 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow t7 | (TLRef _) \Rightarrow t7 | (THead _ _ t9) \Rightarrow t9])) 
+(THead k0 u4 t7) (THead (Bind b) u0 t5) H31) in ((let H34 \def (f_equal T T 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow u4 | (TLRef _) 
+\Rightarrow u4 | (THead _ t9 _) \Rightarrow t9])) (THead k0 u4 t7) (THead 
+(Bind b) u0 t5) H31) in ((let H35 \def (f_equal T K (\lambda (e: T).(match e 
+with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _) 
+\Rightarrow k1])) (THead k0 u4 t7) (THead (Bind b) u0 t5) H31) in (eq_ind K 
+(Bind b) (\lambda (k1: K).((eq T u4 u0) \to ((eq T t7 t5) \to ((eq T (THead 
+k1 u5 t8) t4) \to ((pr0 u4 u5) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: 
+T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) 
+u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9))))))))) (\lambda (H36: (eq T 
+u4 u0)).(eq_ind T u0 (\lambda (t9: T).((eq T t7 t5) \to ((eq T (THead (Bind 
+b) u5 t8) t4) \to ((pr0 t9 u5) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: 
+T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda (t10: T).(pr0 (THead (Bind 
+b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t10)))))))) (\lambda (H37: 
+(eq T t7 t5)).(eq_ind T t5 (\lambda (t9: T).((eq T (THead (Bind b) u5 t8) t4) 
+\to ((pr0 u0 u5) \to ((pr0 t9 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead 

 (Flat Appl) u2 t4) t10)) (\lambda (t10: T).(pr0 (THead (Bind b) u3 (THead 
 (Flat Appl) u2 t4) t10)) (\lambda (t10: T).(pr0 (THead (Bind b) u3 (THead 

-(Flat Appl) (lift (S O) O v2) t6)) t10)))))) (\lambda (H39: (not (eq B b 
-Abst))).(\lambda (H40: (pr0 t7 t4)).(let H41 \def (eq_ind_r T t5 (\lambda 
-(t9: T).(\forall (v: T).((tlt v (THead (Flat Appl) v1 (THead (Bind b) u0 
-t9))) \to (\forall (t10: T).((pr0 v t10) \to (\forall (t11: T).((pr0 v t11) 
-\to (ex2 T (\lambda (t12: T).(pr0 t10 t12)) (\lambda (t12: T).(pr0 t11 
-t12)))))))))) H26 (lift (S O) O t7) H37) in (let H42 \def (eq_ind_r T t5 
-(\lambda (t9: T).(eq T t3 (THead (Bind b) u0 t9))) H23 (lift (S O) O t7) H37) 
-in (let H43 \def (eq_ind_r T t5 (\lambda (t9: T).(pr0 t9 t6)) H19 (lift (S O) 
-O t7) H37) in (ex2_ind T (\lambda (t9: T).(eq T t6 (lift (S O) O t9))) 
-(\lambda (t9: T).(pr0 t7 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) 
+(Flat Appl) (lift (S O) O v2) t6)) t10))))))) (\lambda (H38: (eq T (THead 
+(Bind b) u5 t8) t4)).(eq_ind T (THead (Bind b) u5 t8) (\lambda (t9: T).((pr0 
+u0 u5) \to ((pr0 t5 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) 
+u2 t9) t10)) (\lambda (t10: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) 
+(lift (S O) O v2) t6)) t10)))))) (\lambda (H39: (pr0 u0 u5)).(\lambda (H40: 
+(pr0 t5 t8)).(ex2_ind T (\lambda (t9: T).(pr0 t8 t9)) (\lambda (t9: T).(pr0 
+t6 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 
+t8)) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift 
+(S O) O v2) t6)) t9))) (\lambda (x: T).(\lambda (H41: (pr0 t8 x)).(\lambda 
+(H42: (pr0 t6 x)).(ex2_ind T (\lambda (t9: T).(pr0 u5 t9)) (\lambda (t9: 
+T).(pr0 u3 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead 
+(Bind b) u5 t8)) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat 
+Appl) (lift (S O) O v2) t6)) t9))) (\lambda (x0: T).(\lambda (H43: (pr0 u5 
+x0)).(\lambda (H44: (pr0 u3 x0)).(ex2_ind T (\lambda (t9: T).(pr0 u2 t9)) 
+(\lambda (t9: T).(pr0 v2 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) 
+u2 (THead (Bind b) u5 t8)) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 
+(THead (Flat Appl) (lift (S O) O v2) t6)) t9))) (\lambda (x1: T).(\lambda 
+(H45: (pr0 u2 x1)).(\lambda (H46: (pr0 v2 
+x1)).(pr0_confluence__pr0_cong_upsilon_cong b H16 u2 v2 x1 H45 H46 t8 t6 x 
+H41 H42 u5 u3 x0 H43 H44)))) (H26 v1 (tlt_head_sx (Flat Appl) v1 (THead (Bind 
+b) u0 t5)) u2 H27 v2 H17))))) (H26 u0 (tlt_trans (THead (Bind b) u0 t5) u0 
+(THead (Flat Appl) v1 (THead (Bind b) u0 t5)) (tlt_head_sx (Bind b) u0 t5) 
+(tlt_head_dx (Flat Appl) v1 (THead (Bind b) u0 t5))) u5 H39 u3 H18))))) (H26 
+t5 (tlt_trans (THead (Bind b) u0 t5) t5 (THead (Flat Appl) v1 (THead (Bind b) 
+u0 t5)) (tlt_head_dx (Bind b) u0 t5) (tlt_head_dx (Flat Appl) v1 (THead (Bind 
+b) u0 t5))) t8 H40 t6 H19)))) t4 H38)) t7 (sym_eq T t7 t5 H37))) u4 (sym_eq T 
+u4 u0 H36))) k0 (sym_eq K k0 (Bind b) H35))) H34)) H33)) H32 H29 H30))) | 
+(pr0_beta u v0 v3 H29 t7 t8 H30) \Rightarrow (\lambda (H31: (eq T (THead 
+(Flat Appl) v0 (THead (Bind Abst) u t7)) (THead (Bind b) u0 t5))).(\lambda 
+(H32: (eq T (THead (Bind Abbr) v3 t8) t4)).((let H33 \def (eq_ind T (THead 
+(Flat Appl) v0 (THead (Bind Abst) u t7)) (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) 
+\Rightarrow (match k0 with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow 
+True])])) I (THead (Bind b) u0 t5) H31) in (False_ind ((eq T (THead (Bind 
+Abbr) v3 t8) t4) \to ((pr0 v0 v3) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: 
+T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) 
+u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9)))))) H33)) H32 H29 H30))) | 
+(pr0_upsilon b0 H29 v0 v3 H30 u4 u5 H31 t7 t8 H32) \Rightarrow (\lambda (H33: 
+(eq T (THead (Flat Appl) v0 (THead (Bind b0) u4 t7)) (THead (Bind b) u0 
+t5))).(\lambda (H34: (eq T (THead (Bind b0) u5 (THead (Flat Appl) (lift (S O) 
+O v3) t8)) t4)).((let H35 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind 
+b0) u4 t7)) (\lambda (e: T).(match e with [(TSort _) \Rightarrow False | 
+(TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 with 
+[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind 
+b) u0 t5) H33) in (False_ind ((eq T (THead (Bind b0) u5 (THead (Flat Appl) 
+(lift (S O) O v3) t8)) t4) \to ((not (eq B b0 Abst)) \to ((pr0 v0 v3) \to 
+((pr0 u4 u5) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat 
+Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) 
+(lift (S O) O v2) t6)) t9)))))))) H35)) H34 H29 H30 H31 H32))) | (pr0_delta 
+u4 u5 H29 t7 t8 H30 w H31) \Rightarrow (\lambda (H32: (eq T (THead (Bind 
+Abbr) u4 t7) (THead (Bind b) u0 t5))).(\lambda (H33: (eq T (THead (Bind Abbr) 
+u5 w) t4)).((let H34 \def (f_equal T T (\lambda (e: T).(match e with [(TSort 
+_) \Rightarrow t7 | (TLRef _) \Rightarrow t7 | (THead _ _ t9) \Rightarrow 
+t9])) (THead (Bind Abbr) u4 t7) (THead (Bind b) u0 t5) H32) in ((let H35 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u4 | (TLRef 
+_) \Rightarrow u4 | (THead _ t9 _) \Rightarrow t9])) (THead (Bind Abbr) u4 
+t7) (THead (Bind b) u0 t5) H32) in ((let H36 \def (f_equal T B (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | 
+(THead k0 _ _) \Rightarrow (match k0 with [(Bind b0) \Rightarrow b0 | (Flat 
+_) \Rightarrow Abbr])])) (THead (Bind Abbr) u4 t7) (THead (Bind b) u0 t5) 
+H32) in (eq_ind B Abbr (\lambda (b0: B).((eq T u4 u0) \to ((eq T t7 t5) \to 
+((eq T (THead (Bind Abbr) u5 w) t4) \to ((pr0 u4 u5) \to ((pr0 t7 t8) \to 
+((subst0 O u5 t8 w) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 
+t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift 
+(S O) O v2) t6)) t9)))))))))) (\lambda (H37: (eq T u4 u0)).(eq_ind T u0 
+(\lambda (t9: T).((eq T t7 t5) \to ((eq T (THead (Bind Abbr) u5 w) t4) \to 
+((pr0 t9 u5) \to ((pr0 t7 t8) \to ((subst0 O u5 t8 w) \to (ex2 T (\lambda 
+(t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda (t10: T).(pr0 (THead 
+(Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t10))))))))) 
+(\lambda (H38: (eq T t7 t5)).(eq_ind T t5 (\lambda (t9: T).((eq T (THead 
+(Bind Abbr) u5 w) t4) \to ((pr0 u0 u5) \to ((pr0 t9 t8) \to ((subst0 O u5 t8 
+w) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda 
+(t10: T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) 
+t10)))))))) (\lambda (H39: (eq T (THead (Bind Abbr) u5 w) t4)).(eq_ind T 
+(THead (Bind Abbr) u5 w) (\lambda (t9: T).((pr0 u0 u5) \to ((pr0 t5 t8) \to 
+((subst0 O u5 t8 w) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 
+t9) t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) 
+(lift (S O) O v2) t6)) t10))))))) (\lambda (H40: (pr0 u0 u5)).(\lambda (H41: 
+(pr0 t5 t8)).(\lambda (H42: (subst0 O u5 t8 w)).(let H43 \def (eq_ind_r B b 
+(\lambda (b0: B).(\forall (v: T).((tlt v (THead (Flat Appl) v1 (THead (Bind 
+b0) u0 t5))) \to (\forall (t9: T).((pr0 v t9) \to (\forall (t10: T).((pr0 v 
+t10) \to (ex2 T (\lambda (t11: T).(pr0 t9 t11)) (\lambda (t11: T).(pr0 t10 
+t11)))))))))) H26 Abbr H36) in (let H44 \def (eq_ind_r B b (\lambda (b0: 
+B).(eq T t3 (THead (Bind b0) u0 t5))) H23 Abbr H36) in (let H45 \def 
+(eq_ind_r B b (\lambda (b0: B).(not (eq B b0 Abst))) H16 Abbr H36) in 
+(ex2_ind T (\lambda (t9: T).(pr0 t8 t9)) (\lambda (t9: T).(pr0 t6 t9)) (ex2 T 
+(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abbr) u5 w)) t9)) 
+(\lambda (t9: T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O 
+v2) t6)) t9))) (\lambda (x: T).(\lambda (H46: (pr0 t8 x)).(\lambda (H47: (pr0 
+t6 x)).(ex2_ind T (\lambda (t9: T).(pr0 u5 t9)) (\lambda (t9: T).(pr0 u3 t9)) 
+(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abbr) u5 w)) 
+t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S 
+O) O v2) t6)) t9))) (\lambda (x0: T).(\lambda (H48: (pr0 u5 x0)).(\lambda 
+(H49: (pr0 u3 x0)).(ex2_ind T (\lambda (t9: T).(pr0 u2 t9)) (\lambda (t9: 
+T).(pr0 v2 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead 
+(Bind Abbr) u5 w)) t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) u3 (THead 
+(Flat Appl) (lift (S O) O v2) t6)) t9))) (\lambda (x1: T).(\lambda (H50: (pr0 
+u2 x1)).(\lambda (H51: (pr0 v2 x1)).(pr0_confluence__pr0_cong_upsilon_delta 
+H45 u5 t8 w H42 u2 v2 x1 H50 H51 t6 x H46 H47 u3 x0 H48 H49)))) (H43 v1 
+(tlt_head_sx (Flat Appl) v1 (THead (Bind Abbr) u0 t5)) u2 H27 v2 H17))))) 
+(H43 u0 (tlt_trans (THead (Bind Abbr) u0 t5) u0 (THead (Flat Appl) v1 (THead 
+(Bind Abbr) u0 t5)) (tlt_head_sx (Bind Abbr) u0 t5) (tlt_head_dx (Flat Appl) 
+v1 (THead (Bind Abbr) u0 t5))) u5 H40 u3 H18))))) (H43 t5 (tlt_trans (THead 
+(Bind Abbr) u0 t5) t5 (THead (Flat Appl) v1 (THead (Bind Abbr) u0 t5)) 
+(tlt_head_dx (Bind Abbr) u0 t5) (tlt_head_dx (Flat Appl) v1 (THead (Bind 
+Abbr) u0 t5))) t8 H41 t6 H19)))))))) t4 H39)) t7 (sym_eq T t7 t5 H38))) u4 
+(sym_eq T u4 u0 H37))) b H36)) H35)) H34)) H33 H29 H30 H31))) | (pr0_zeta b0 
+H29 t7 t8 H30 u) \Rightarrow (\lambda (H31: (eq T (THead (Bind b0) u (lift (S 
+O) O t7)) (THead (Bind b) u0 t5))).(\lambda (H32: (eq T t8 t4)).((let H33 
+\def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow 
+(lref_map (\lambda (x: nat).(plus x (S O))) O t7) | (TLRef _) \Rightarrow 
+(lref_map (\lambda (x: nat).(plus x (S O))) O t7) | (THead _ _ t9) 
+\Rightarrow t9])) (THead (Bind b0) u (lift (S O) O t7)) (THead (Bind b) u0 
+t5) H31) in ((let H34 \def (f_equal T T (\lambda (e: T).(match e with [(TSort 
+_) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t9 _) \Rightarrow t9])) 
+(THead (Bind b0) u (lift (S O) O t7)) (THead (Bind b) u0 t5) H31) in ((let 
+H35 \def (f_equal T B (\lambda (e: T).(match e with [(TSort _) \Rightarrow b0 
+| (TLRef _) \Rightarrow b0 | (THead k0 _ _) \Rightarrow (match k0 with [(Bind 
+b1) \Rightarrow b1 | (Flat _) \Rightarrow b0])])) (THead (Bind b0) u (lift (S 
+O) O t7)) (THead (Bind b) u0 t5) H31) in (eq_ind B b (\lambda (b1: B).((eq T 
+u u0) \to ((eq T (lift (S O) O t7) t5) \to ((eq T t8 t4) \to ((not (eq B b1 
+Abst)) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) 

 u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift 
 u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift 

-(S O) O v2) t6)) t9))) (\lambda (x: T).(\lambda (H44: (eq T t6 (lift (S O) O 
-x))).(\lambda (H45: (pr0 t7 x)).(eq_ind_r T (lift (S O) O x) (\lambda (t9: 
-T).(ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda 
-(t10: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t9)) 
-t10)))) (ex2_ind T (\lambda (t9: T).(pr0 x t9)) (\lambda (t9: T).(pr0 t4 t9)) 
-(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: 
-T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) O 
-x))) t9))) (\lambda (x0: T).(\lambda (H46: (pr0 x x0)).(\lambda (H47: (pr0 t4 
-x0)).(ex2_ind T (\lambda (t9: T).(pr0 u2 t9)) (\lambda (t9: T).(pr0 v2 t9)) 
-(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: 
-T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) O 
-x))) t9))) (\lambda (x1: T).(\lambda (H48: (pr0 u2 x1)).(\lambda (H49: (pr0 
-v2 x1)).(pr0_confluence__pr0_cong_upsilon_zeta b H39 u0 u3 H18 u2 v2 x1 H48 
-H49 x t4 x0 H46 H47)))) (H41 v1 (tlt_head_sx (Flat Appl) v1 (THead (Bind b) 
-u0 (lift (S O) O t7))) u2 H27 v2 H17))))) (H41 t7 (tlt_trans (THead (Bind b) 
-u0 (lift (S O) O t7)) t7 (THead (Flat Appl) v1 (THead (Bind b) u0 (lift (S O) 
-O t7))) (lift_tlt_dx (Bind b) u0 t7 (S O) O) (tlt_head_dx (Flat Appl) v1 
-(THead (Bind b) u0 (lift (S O) O t7)))) x H45 t4 H40)) t6 H44)))) 
-(pr0_gen_lift t7 t6 (S O) O H43))))))) t8 (sym_eq T t8 t4 H38))) t5 H37)) u 
-(sym_eq T u u0 H36))) b0 (sym_eq B b0 b H35))) H34)) H33)) H32 H29 H30))) | 
-(pr0_tau t7 t8 H29 u) \Rightarrow (\lambda (H30: (eq T (THead (Flat Cast) u 
-t7) (THead (Bind b) u0 t5))).(\lambda (H31: (eq T t8 t4)).((let H32 \def 
-(eq_ind T (THead (Flat Cast) u t7) (\lambda (e: T).(match e in T return 
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda 
-(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow 
-True])])) I (THead (Bind b) u0 t5) H30) in (False_ind ((eq T t8 t4) \to ((pr0 
-t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) 
-(\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) 
-t6)) t9))))) H32)) H31 H29)))]) in (H29 (refl_equal T (THead (Bind b) u0 t5)) 
-(refl_equal T t4))))) k H25))))) H22)) H21))))))) t2 H15)) t H13 H14 H9 H10 
-H11 H12))) | (pr0_delta u0 u3 H9 t5 t6 H10 w H11) \Rightarrow (\lambda (H12: 
-(eq T (THead (Bind Abbr) u0 t5) t)).(\lambda (H13: (eq T (THead (Bind Abbr) 
-u3 w) t2)).(eq_ind T (THead (Bind Abbr) u0 t5) (\lambda (_: T).((eq T (THead 
-(Bind Abbr) u3 w) t2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to ((subst0 O u3 t6 
-w) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: 
-T).(pr0 t2 t8)))))))) (\lambda (H14: (eq T (THead (Bind Abbr) u3 w) 
-t2)).(eq_ind T (THead (Bind Abbr) u3 w) (\lambda (t7: T).((pr0 u0 u3) \to 
-((pr0 t5 t6) \to ((subst0 O u3 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 (THead 
-k u2 t4) t8)) (\lambda (t8: T).(pr0 t7 t8))))))) (\lambda (H15: (pr0 u0 
-u3)).(\lambda (H16: (pr0 t5 t6)).(\lambda (H17: (subst0 O u3 t6 w)).(let H18 
-\def (eq_ind_r T t (\lambda (t7: T).(eq T (THead k u1 t3) t7)) H4 (THead 
-(Bind Abbr) u0 t5) H12) in (let H19 \def (f_equal T K (\lambda (e: T).(match 
-e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) 
-\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) (THead (Bind 
-Abbr) u0 t5) H18) in ((let H20 \def (f_equal T T (\lambda (e: T).(match e in 
-T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) 
+(S O) O v2) t6)) t9))))))))) (\lambda (H36: (eq T u u0)).(eq_ind T u0 
+(\lambda (_: T).((eq T (lift (S O) O t7) t5) \to ((eq T t8 t4) \to ((not (eq 
+B b Abst)) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat 
+Appl) u2 t4) t10)) (\lambda (t10: T).(pr0 (THead (Bind b) u3 (THead (Flat 
+Appl) (lift (S O) O v2) t6)) t10)))))))) (\lambda (H37: (eq T (lift (S O) O 
+t7) t5)).(eq_ind T (lift (S O) O t7) (\lambda (_: T).((eq T t8 t4) \to ((not 
+(eq B b Abst)) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead 
+(Flat Appl) u2 t4) t10)) (\lambda (t10: T).(pr0 (THead (Bind b) u3 (THead 
+(Flat Appl) (lift (S O) O v2) t6)) t10))))))) (\lambda (H38: (eq T t8 
+t4)).(eq_ind T t4 (\lambda (t9: T).((not (eq B b Abst)) \to ((pr0 t7 t9) \to 
+(ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda (t10: 
+T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) 
+t10)))))) (\lambda (H39: (not (eq B b Abst))).(\lambda (H40: (pr0 t7 
+t4)).(let H41 \def (eq_ind_r T t5 (\lambda (t9: T).(\forall (v: T).((tlt v 
+(THead (Flat Appl) v1 (THead (Bind b) u0 t9))) \to (\forall (t10: T).((pr0 v 
+t10) \to (\forall (t11: T).((pr0 v t11) \to (ex2 T (\lambda (t12: T).(pr0 t10 
+t12)) (\lambda (t12: T).(pr0 t11 t12)))))))))) H26 (lift (S O) O t7) H37) in 
+(let H42 \def (eq_ind_r T t5 (\lambda (t9: T).(eq T t3 (THead (Bind b) u0 
+t9))) H23 (lift (S O) O t7) H37) in (let H43 \def (eq_ind_r T t5 (\lambda 
+(t9: T).(pr0 t9 t6)) H19 (lift (S O) O t7) H37) in (ex2_ind T (\lambda (t9: 
+T).(eq T t6 (lift (S O) O t9))) (\lambda (t9: T).(pr0 t7 t9)) (ex2 T (\lambda 
+(t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead 
+(Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9))) (\lambda (x: 
+T).(\lambda (H44: (eq T t6 (lift (S O) O x))).(\lambda (H45: (pr0 t7 
+x)).(eq_ind_r T (lift (S O) O x) (\lambda (t9: T).(ex2 T (\lambda (t10: 
+T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda (t10: T).(pr0 (THead (Bind 
+b) u3 (THead (Flat Appl) (lift (S O) O v2) t9)) t10)))) (ex2_ind T (\lambda 
+(t9: T).(pr0 x t9)) (\lambda (t9: T).(pr0 t4 t9)) (ex2 T (\lambda (t9: 
+T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) 
+u3 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) O x))) t9))) (\lambda 
+(x0: T).(\lambda (H46: (pr0 x x0)).(\lambda (H47: (pr0 t4 x0)).(ex2_ind T 
+(\lambda (t9: T).(pr0 u2 t9)) (\lambda (t9: T).(pr0 v2 t9)) (ex2 T (\lambda 
+(t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead 
+(Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) O x))) t9))) 
+(\lambda (x1: T).(\lambda (H48: (pr0 u2 x1)).(\lambda (H49: (pr0 v2 
+x1)).(pr0_confluence__pr0_cong_upsilon_zeta b H39 u0 u3 H18 u2 v2 x1 H48 H49 
+x t4 x0 H46 H47)))) (H41 v1 (tlt_head_sx (Flat Appl) v1 (THead (Bind b) u0 
+(lift (S O) O t7))) u2 H27 v2 H17))))) (H41 t7 (tlt_trans (THead (Bind b) u0 
+(lift (S O) O t7)) t7 (THead (Flat Appl) v1 (THead (Bind b) u0 (lift (S O) O 
+t7))) (lift_tlt_dx (Bind b) u0 t7 (S O) O) (tlt_head_dx (Flat Appl) v1 (THead 
+(Bind b) u0 (lift (S O) O t7)))) x H45 t4 H40)) t6 H44)))) (pr0_gen_lift t7 
+t6 (S O) O H43))))))) t8 (sym_eq T t8 t4 H38))) t5 H37)) u (sym_eq T u u0 
+H36))) b0 (sym_eq B b0 b H35))) H34)) H33)) H32 H29 H30))) | (pr0_tau t7 t8 
+H29 u) \Rightarrow (\lambda (H30: (eq T (THead (Flat Cast) u t7) (THead (Bind 
+b) u0 t5))).(\lambda (H31: (eq T t8 t4)).((let H32 \def (eq_ind T (THead 
+(Flat Cast) u t7) (\lambda (e: T).(match e with [(TSort _) \Rightarrow False 
+| (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 with 
+[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind 
+b) u0 t5) H30) in (False_ind ((eq T t8 t4) \to ((pr0 t7 t8) \to (ex2 T 
+(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 
+(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9))))) H32)) 
+H31 H29)))]) in (H29 (refl_equal T (THead (Bind b) u0 t5)) (refl_equal T 
+t4))))) k H25))))) H22)) H21))))))) t2 H15)) t H13 H14 H9 H10 H11 H12))) | 
+(pr0_delta u0 u3 H9 t5 t6 H10 w H11) \Rightarrow (\lambda (H12: (eq T (THead 
+(Bind Abbr) u0 t5) t)).(\lambda (H13: (eq T (THead (Bind Abbr) u3 w) 
+t2)).(eq_ind T (THead (Bind Abbr) u0 t5) (\lambda (_: T).((eq T (THead (Bind 
+Abbr) u3 w) t2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to ((subst0 O u3 t6 w) \to 
+(ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 
+t8)))))))) (\lambda (H14: (eq T (THead (Bind Abbr) u3 w) t2)).(eq_ind T 
+(THead (Bind Abbr) u3 w) (\lambda (t7: T).((pr0 u0 u3) \to ((pr0 t5 t6) \to 
+((subst0 O u3 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) 
+(\lambda (t8: T).(pr0 t7 t8))))))) (\lambda (H15: (pr0 u0 u3)).(\lambda (H16: 
+(pr0 t5 t6)).(\lambda (H17: (subst0 O u3 t6 w)).(let H18 \def (eq_ind_r T t 
+(\lambda (t7: T).(eq T (THead k u1 t3) t7)) H4 (THead (Bind Abbr) u0 t5) H12) 
+in (let H19 \def (f_equal T K (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) 
+(THead k u1 t3) (THead (Bind Abbr) u0 t5) H18) in ((let H20 \def (f_equal T T 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow u1 | (TLRef _) 

 \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) (THead 
 (Bind Abbr) u0 t5) H18) in ((let H21 \def (f_equal T T (\lambda (e: T).(match 
 \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) (THead 
 (Bind Abbr) u0 t5) H18) in ((let H21 \def (f_equal T T (\lambda (e: T).(match 

-e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) 
-\Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t3) (THead 
-(Bind Abbr) u0 t5) H18) in (\lambda (H22: (eq T u1 u0)).(\lambda (H23: (eq K 
-k (Bind Abbr))).(let H24 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: 
-T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v 
-t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 
-t10)))))))))) H (THead (Bind Abbr) u0 t5) H12) in (eq_ind_r K (Bind Abbr) 
-(\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 (THead k0 u2 t4) t7)) (\lambda 
-(t7: T).(pr0 (THead (Bind Abbr) u3 w) t7)))) (let H25 \def (eq_ind T u1 
-(\lambda (t7: T).(pr0 t7 u2)) H7 u0 H22) in (let H26 \def (eq_ind T t3 
-(\lambda (t7: T).(pr0 t7 t4)) H8 t5 H21) in (ex2_ind T (\lambda (t7: T).(pr0 
-t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead 
-(Bind Abbr) u2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u3 w) t7))) 
-(\lambda (x: T).(\lambda (H27: (pr0 t4 x)).(\lambda (H28: (pr0 t6 
-x)).(ex2_ind T (\lambda (t7: T).(pr0 u2 t7)) (\lambda (t7: T).(pr0 u3 t7)) 
-(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 t4) t7)) (\lambda (t7: 
-T).(pr0 (THead (Bind Abbr) u3 w) t7))) (\lambda (x0: T).(\lambda (H29: (pr0 
-u2 x0)).(\lambda (H30: (pr0 u3 x0)).(pr0_confluence__pr0_cong_delta u3 t6 w 
-H17 u2 x0 H29 H30 t4 x H27 H28)))) (H24 u0 (tlt_head_sx (Bind Abbr) u0 t5) u2 
-H25 u3 H15))))) (H24 t5 (tlt_head_dx (Bind Abbr) u0 t5) t4 H26 t6 H16)))) k 
-H23))))) H20)) H19)))))) t2 H14)) t H12 H13 H9 H10 H11))) | (pr0_zeta b H9 t5 
-t6 H10 u) \Rightarrow (\lambda (H11: (eq T (THead (Bind b) u (lift (S O) O 
-t5)) t)).(\lambda (H12: (eq T t6 t2)).(eq_ind T (THead (Bind b) u (lift (S O) 
-O t5)) (\lambda (_: T).((eq T t6 t2) \to ((not (eq B b Abst)) \to ((pr0 t5 
-t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: 
-T).(pr0 t2 t8))))))) (\lambda (H13: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: 
-T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 
-(THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H14: (not 
-(eq B b Abst))).(\lambda (H15: (pr0 t5 t2)).(let H16 \def (eq_ind_r T t 
-(\lambda (t7: T).(eq T (THead k u1 t3) t7)) H4 (THead (Bind b) u (lift (S O) 
-O t5)) H11) in (let H17 \def (f_equal T K (\lambda (e: T).(match e in T 
-return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) 
-\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) (THead (Bind 
-b) u (lift (S O) O t5)) H16) in ((let H18 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | 
-(TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) 
-(THead (Bind b) u (lift (S O) O t5)) H16) in ((let H19 \def (f_equal T T 
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) 
-(THead k u1 t3) (THead (Bind b) u (lift (S O) O t5)) H16) in (\lambda (H20: 
-(eq T u1 u)).(\lambda (H21: (eq K k (Bind b))).(let H22 \def (eq_ind_r T t 
-(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) 
-\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) 
-(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Bind b) u (lift (S O) O 
-t5)) H11) in (eq_ind_r K (Bind b) (\lambda (k0: K).(ex2 T (\lambda (t7: 
-T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let H23 \def 
-(eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H7 u H20) in (let H24 \def (eq_ind 
-T t3 (\lambda (t7: T).(pr0 t7 t4)) H8 (lift (S O) O t5) H19) in (ex2_ind T 
-(\lambda (t7: T).(eq T t4 (lift (S O) O t7))) (\lambda (t7: T).(pr0 t5 t7)) 
-(ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 t4) t7)) (\lambda (t7: 
-T).(pr0 t2 t7))) (\lambda (x: T).(\lambda (H25: (eq T t4 (lift (S O) O 
-x))).(\lambda (H26: (pr0 t5 x)).(eq_ind_r T (lift (S O) O x) (\lambda (t7: 
-T).(ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 t7) t8)) (\lambda (t8: 
-T).(pr0 t2 t8)))) (ex2_ind T (\lambda (t7: T).(pr0 x t7)) (\lambda (t7: 
-T).(pr0 t2 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (lift (S O) O 
-x)) t7)) (\lambda (t7: T).(pr0 t2 t7))) (\lambda (x0: T).(\lambda (H27: (pr0 
-x x0)).(\lambda (H28: (pr0 t2 x0)).(ex_intro2 T (\lambda (t7: T).(pr0 (THead 
-(Bind b) u2 (lift (S O) O x)) t7)) (\lambda (t7: T).(pr0 t2 t7)) x0 (pr0_zeta 
-b H14 x x0 H27 u2) H28)))) (H22 t5 (lift_tlt_dx (Bind b) u t5 (S O) O) x H26 
-t2 H15)) t4 H25)))) (pr0_gen_lift t5 t4 (S O) O H24)))) k H21))))) H18)) 
-H17))))) t6 (sym_eq T t6 t2 H13))) t H11 H12 H9 H10))) | (pr0_tau t5 t6 H9 u) 
-\Rightarrow (\lambda (H10: (eq T (THead (Flat Cast) u t5) t)).(\lambda (H11: 
-(eq T t6 t2)).(eq_ind T (THead (Flat Cast) u t5) (\lambda (_: T).((eq T t6 
-t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) 
-(\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H12: (eq T t6 t2)).(eq_ind T t2 
-(\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 
-t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))) (\lambda (H13: (pr0 t5 t2)).(let 
-H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead k u1 t3) t7)) H4 (THead 
-(Flat Cast) u t5) H10) in (let H15 \def (f_equal T K (\lambda (e: T).(match e 
-in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) 
+e with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7) 
+\Rightarrow t7])) (THead k u1 t3) (THead (Bind Abbr) u0 t5) H18) in (\lambda 
+(H22: (eq T u1 u0)).(\lambda (H23: (eq K k (Bind Abbr))).(let H24 \def 
+(eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: 
+T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: 
+T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Bind Abbr) 
+u0 t5) H12) in (eq_ind_r K (Bind Abbr) (\lambda (k0: K).(ex2 T (\lambda (t7: 
+T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u3 w) 
+t7)))) (let H25 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H7 u0 H22) in 
+(let H26 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H8 t5 H21) in 
+(ex2_ind T (\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T 
+(\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 t4) t7)) (\lambda (t7: T).(pr0 
+(THead (Bind Abbr) u3 w) t7))) (\lambda (x: T).(\lambda (H27: (pr0 t4 
+x)).(\lambda (H28: (pr0 t6 x)).(ex2_ind T (\lambda (t7: T).(pr0 u2 t7)) 
+(\lambda (t7: T).(pr0 u3 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) 
+u2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u3 w) t7))) (\lambda 
+(x0: T).(\lambda (H29: (pr0 u2 x0)).(\lambda (H30: (pr0 u3 
+x0)).(pr0_confluence__pr0_cong_delta u3 t6 w H17 u2 x0 H29 H30 t4 x H27 
+H28)))) (H24 u0 (tlt_head_sx (Bind Abbr) u0 t5) u2 H25 u3 H15))))) (H24 t5 
+(tlt_head_dx (Bind Abbr) u0 t5) t4 H26 t6 H16)))) k H23))))) H20)) H19)))))) 
+t2 H14)) t H12 H13 H9 H10 H11))) | (pr0_zeta b H9 t5 t6 H10 u) \Rightarrow 
+(\lambda (H11: (eq T (THead (Bind b) u (lift (S O) O t5)) t)).(\lambda (H12: 
+(eq T t6 t2)).(eq_ind T (THead (Bind b) u (lift (S O) O t5)) (\lambda (_: 
+T).((eq T t6 t2) \to ((not (eq B b Abst)) \to ((pr0 t5 t6) \to (ex2 T 
+(\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) 
+(\lambda (H13: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((not (eq B b 
+Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) 
+(\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H14: (not (eq B b 
+Abst))).(\lambda (H15: (pr0 t5 t2)).(let H16 \def (eq_ind_r T t (\lambda (t7: 
+T).(eq T (THead k u1 t3) t7)) H4 (THead (Bind b) u (lift (S O) O t5)) H11) in 
+(let H17 \def (f_equal T K (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) 
+(THead k u1 t3) (THead (Bind b) u (lift (S O) O t5)) H16) in ((let H18 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u1 | (TLRef 
+_) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) (THead 
+(Bind b) u (lift (S O) O t5)) H16) in ((let H19 \def (f_equal T T (\lambda 
+(e: T).(match e with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | 
+(THead _ _ t7) \Rightarrow t7])) (THead k u1 t3) (THead (Bind b) u (lift (S 
+O) O t5)) H16) in (\lambda (H20: (eq T u1 u)).(\lambda (H21: (eq K k (Bind 
+b))).(let H22 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) 
+\to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T 
+(\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H 
+(THead (Bind b) u (lift (S O) O t5)) H11) in (eq_ind_r K (Bind b) (\lambda 
+(k0: K).(ex2 T (\lambda (t7: T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7: 
+T).(pr0 t2 t7)))) (let H23 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H7 
+u H20) in (let H24 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H8 (lift 
+(S O) O t5) H19) in (ex2_ind T (\lambda (t7: T).(eq T t4 (lift (S O) O t7))) 
+(\lambda (t7: T).(pr0 t5 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 
+t4) t7)) (\lambda (t7: T).(pr0 t2 t7))) (\lambda (x: T).(\lambda (H25: (eq T 
+t4 (lift (S O) O x))).(\lambda (H26: (pr0 t5 x)).(eq_ind_r T (lift (S O) O x) 
+(\lambda (t7: T).(ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 t7) t8)) 
+(\lambda (t8: T).(pr0 t2 t8)))) (ex2_ind T (\lambda (t7: T).(pr0 x t7)) 
+(\lambda (t7: T).(pr0 t2 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 
+(lift (S O) O x)) t7)) (\lambda (t7: T).(pr0 t2 t7))) (\lambda (x0: 
+T).(\lambda (H27: (pr0 x x0)).(\lambda (H28: (pr0 t2 x0)).(ex_intro2 T 
+(\lambda (t7: T).(pr0 (THead (Bind b) u2 (lift (S O) O x)) t7)) (\lambda (t7: 
+T).(pr0 t2 t7)) x0 (pr0_zeta b H14 x x0 H27 u2) H28)))) (H22 t5 (lift_tlt_dx 
+(Bind b) u t5 (S O) O) x H26 t2 H15)) t4 H25)))) (pr0_gen_lift t5 t4 (S O) O 
+H24)))) k H21))))) H18)) H17))))) t6 (sym_eq T t6 t2 H13))) t H11 H12 H9 
+H10))) | (pr0_tau t5 t6 H9 u) \Rightarrow (\lambda (H10: (eq T (THead (Flat 
+Cast) u t5) t)).(\lambda (H11: (eq T t6 t2)).(eq_ind T (THead (Flat Cast) u 
+t5) (\lambda (_: T).((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: 
+T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H12: 
+(eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda 
+(t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))) (\lambda 
+(H13: (pr0 t5 t2)).(let H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead 
+k u1 t3) t7)) H4 (THead (Flat Cast) u t5) H10) in (let H15 \def (f_equal T K 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow k | (TLRef _) 

 \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) (THead (Flat 
 \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) (THead (Flat 

-Cast) u t5) H14) in ((let H16 \def (f_equal T T (\lambda (e: T).(match e in T 
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) 
-\Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) (THead 
-(Flat Cast) u t5) H14) in ((let H17 \def (f_equal T T (\lambda (e: T).(match 
-e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) 
-\Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t3) (THead 
-(Flat Cast) u t5) H14) in (\lambda (H18: (eq T u1 u)).(\lambda (H19: (eq K k 
-(Flat Cast))).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: 
-T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v 
-t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 
-t10)))))))))) H (THead (Flat Cast) u t5) H10) in (eq_ind_r K (Flat Cast) 
-(\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 (THead k0 u2 t4) t7)) (\lambda 
-(t7: T).(pr0 t2 t7)))) (let H21 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 
-u2)) H7 u H18) in (let H22 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H8 
-t5 H17) in (ex2_ind T (\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t2 
-t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Flat Cast) u2 t4) t7)) (\lambda 
-(t7: T).(pr0 t2 t7))) (\lambda (x: T).(\lambda (H23: (pr0 t4 x)).(\lambda 
-(H24: (pr0 t2 x)).(ex_intro2 T (\lambda (t7: T).(pr0 (THead (Flat Cast) u2 
-t4) t7)) (\lambda (t7: T).(pr0 t2 t7)) x (pr0_tau t4 x H23 u2) H24)))) (H20 
-t5 (tlt_head_dx (Flat Cast) u t5) t4 H22 t2 H13)))) k H19))))) H16)) H15)))) 
-t6 (sym_eq T t6 t2 H12))) t H10 H11 H9)))]) in (H9 (refl_equal T t) 
-(refl_equal T t2))))) t1 H6)) t H4 H5 H2 H3))) | (pr0_beta u v1 v2 H2 t3 t4 
-H3) \Rightarrow (\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) 
-u t3)) t)).(\lambda (H5: (eq T (THead (Bind Abbr) v2 t4) t1)).(eq_ind T 
+Cast) u t5) H14) in ((let H16 \def (f_equal T T (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t7 _) 
+\Rightarrow t7])) (THead k u1 t3) (THead (Flat Cast) u t5) H14) in ((let H17 
+\def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t3 | 
+(TLRef _) \Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t3) 
+(THead (Flat Cast) u t5) H14) in (\lambda (H18: (eq T u1 u)).(\lambda (H19: 
+(eq K k (Flat Cast))).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(\forall 
+(v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: 
+T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: 
+T).(pr0 t9 t10)))))))))) H (THead (Flat Cast) u t5) H10) in (eq_ind_r K (Flat 
+Cast) (\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 (THead k0 u2 t4) t7)) 
+(\lambda (t7: T).(pr0 t2 t7)))) (let H21 \def (eq_ind T u1 (\lambda (t7: 
+T).(pr0 t7 u2)) H7 u H18) in (let H22 \def (eq_ind T t3 (\lambda (t7: T).(pr0 
+t7 t4)) H8 t5 H17) in (ex2_ind T (\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: 
+T).(pr0 t2 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Flat Cast) u2 t4) t7)) 
+(\lambda (t7: T).(pr0 t2 t7))) (\lambda (x: T).(\lambda (H23: (pr0 t4 
+x)).(\lambda (H24: (pr0 t2 x)).(ex_intro2 T (\lambda (t7: T).(pr0 (THead 
+(Flat Cast) u2 t4) t7)) (\lambda (t7: T).(pr0 t2 t7)) x (pr0_tau t4 x H23 u2) 
+H24)))) (H20 t5 (tlt_head_dx (Flat Cast) u t5) t4 H22 t2 H13)))) k H19))))) 
+H16)) H15)))) t6 (sym_eq T t6 t2 H12))) t H10 H11 H9)))]) in (H9 (refl_equal 
+T t) (refl_equal T t2))))) t1 H6)) t H4 H5 H2 H3))) | (pr0_beta u v1 v2 H2 t3 
+t4 H3) \Rightarrow (\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind 
+Abst) u t3)) t)).(\lambda (H5: (eq T (THead (Bind Abbr) v2 t4) t1)).(eq_ind T 

 (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (\lambda (_: T).((eq T (THead 
 (Bind Abbr) v2 t4) t1) \to ((pr0 v1 v2) \to ((pr0 t3 t4) \to (ex2 T (\lambda 
 (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6))))))) (\lambda (H6: (eq T 
 (THead (Bind Abbr) v2 t4) t1)).(eq_ind T (THead (Bind Abbr) v2 t4) (\lambda 
 (t5: T).((pr0 v1 v2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t5 
 t6)) (\lambda (t6: T).(pr0 t2 t6)))))) (\lambda (H7: (pr0 v1 v2)).(\lambda 
 (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (\lambda (_: T).((eq T (THead 
 (Bind Abbr) v2 t4) t1) \to ((pr0 v1 v2) \to ((pr0 t3 t4) \to (ex2 T (\lambda 
 (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6))))))) (\lambda (H6: (eq T 
 (THead (Bind Abbr) v2 t4) t1)).(eq_ind T (THead (Bind Abbr) v2 t4) (\lambda 
 (t5: T).((pr0 v1 v2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t5 
 t6)) (\lambda (t6: T).(pr0 t2 t6)))))) (\lambda (H7: (pr0 v1 v2)).(\lambda 

-(H8: (pr0 t3 t4)).(let H9 \def (match H1 in pr0 return (\lambda (t5: 
-T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2) 
-\to (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: 
-T).(pr0 t2 t7)))))))) with [(pr0_refl t5) \Rightarrow (\lambda (H9: (eq T t5 
-t)).(\lambda (H10: (eq T t5 t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2) 
-\to (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: 
-T).(pr0 t2 t7))))) (\lambda (H11: (eq T t t2)).(eq_ind T t2 (\lambda (_: 
+(H8: (pr0 t3 t4)).(let H9 \def (match H1 with [(pr0_refl t5) \Rightarrow 
+(\lambda (H9: (eq T t5 t)).(\lambda (H10: (eq T t5 t2)).(eq_ind T t (\lambda 
+(t6: T).((eq T t6 t2) \to (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 
+t4) t7)) (\lambda (t7: T).(pr0 t2 t7))))) (\lambda (H11: (eq T t t2)).(eq_ind 
+T t2 (\lambda (_: T).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) 
+t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let H12 \def (eq_ind_r T t (\lambda 
+(t6: T).(eq T t6 t2)) H11 (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) H4) 
+in (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (\lambda (t6: 

 T).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: 
 T).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: 

-T).(pr0 t2 t7)))) (let H12 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t2)) 
-H11 (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) H4) in (eq_ind T (THead 
-(Flat Appl) v1 (THead (Bind Abst) u t3)) (\lambda (t6: T).(ex2 T (\lambda 
-(t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 t6 t7)))) 
-(let H13 \def (eq_ind_r T t (\lambda (t6: T).(eq T t5 t6)) H9 (THead (Flat 
-Appl) v1 (THead (Bind Abst) u t3)) H4) in (let H14 \def (eq_ind_r T t 
-(\lambda (t6: T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) 
-\to (\forall (t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) 
-(\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead (Flat Appl) v1 (THead (Bind 
-Abst) u t3)) H4) in (ex_intro2 T (\lambda (t6: T).(pr0 (THead (Bind Abbr) v2 
-t4) t6)) (\lambda (t6: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u 
-t3)) t6)) (THead (Bind Abbr) v2 t4) (pr0_refl (THead (Bind Abbr) v2 t4)) 
-(pr0_beta u v1 v2 H7 t3 t4 H8)))) t2 H12)) t (sym_eq T t t2 H11))) t5 (sym_eq 
-T t5 t H9) H10))) | (pr0_comp u1 u2 H9 t5 t6 H10 k) \Rightarrow (\lambda 
-(H11: (eq T (THead k u1 t5) t)).(\lambda (H12: (eq T (THead k u2 t6) 
-t2)).(eq_ind T (THead k u1 t5) (\lambda (_: T).((eq T (THead k u2 t6) t2) \to 
-((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind 
-Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H13: (eq T 
-(THead k u2 t6) t2)).(eq_ind T (THead k u2 t6) (\lambda (t7: T).((pr0 u1 u2) 
-\to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) 
-t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (H14: (pr0 u1 u2)).(\lambda 
-(H15: (pr0 t5 t6)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead 
-(Flat Appl) v1 (THead (Bind Abst) u t3)) t7)) H4 (THead k u1 t5) H11) in (let 
-H17 \def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) 
-with [(TSort _) \Rightarrow (Flat Appl) | (TLRef _) \Rightarrow (Flat Appl) | 
-(THead k0 _ _) \Rightarrow k0])) (THead (Flat Appl) v1 (THead (Bind Abst) u 
-t3)) (THead k u1 t5) H16) in ((let H18 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | 
-(TLRef _) \Rightarrow v1 | (THead _ t7 _) \Rightarrow t7])) (THead (Flat 
-Appl) v1 (THead (Bind Abst) u t3)) (THead k u1 t5) H16) in ((let H19 \def 
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow (THead (Bind Abst) u t3) | (TLRef _) \Rightarrow 
-(THead (Bind Abst) u t3) | (THead _ _ t7) \Rightarrow t7])) (THead (Flat 
-Appl) v1 (THead (Bind Abst) u t3)) (THead k u1 t5) H16) in (\lambda (H20: (eq 
-T v1 u1)).(\lambda (H21: (eq K (Flat Appl) k)).(eq_ind K (Flat Appl) (\lambda 
-(k0: K).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda 
-(t7: T).(pr0 (THead k0 u2 t6) t7)))) (let H22 \def (eq_ind_r K k (\lambda 
-(k0: K).(eq T (THead k0 u1 t5) t)) H11 (Flat Appl) H21) in (let H23 \def 
-(eq_ind_r T t5 (\lambda (t7: T).(pr0 t7 t6)) H15 (THead (Bind Abst) u t3) 
-H19) in (let H24 \def (match H23 in pr0 return (\lambda (t7: T).(\lambda (t8: 
-T).(\lambda (_: (pr0 t7 t8)).((eq T t7 (THead (Bind Abst) u t3)) \to ((eq T 
-t8 t6) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) 
-(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t6) t9)))))))) with [(pr0_refl 
-t7) \Rightarrow (\lambda (H24: (eq T t7 (THead (Bind Abst) u t3))).(\lambda 
-(H25: (eq T t7 t6)).(eq_ind T (THead (Bind Abst) u t3) (\lambda (t8: T).((eq 
-T t8 t6) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) 
-(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t6) t9))))) (\lambda (H26: (eq T 
-(THead (Bind Abst) u t3) t6)).(eq_ind T (THead (Bind Abst) u t3) (\lambda 
-(t8: T).(ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda 
-(t9: T).(pr0 (THead (Flat Appl) u2 t8) t9)))) (let H27 \def (eq_ind_r T t5 
-(\lambda (t8: T).(eq T (THead (Flat Appl) u1 t8) t)) H22 (THead (Bind Abst) u 
-t3) H19) in (let H28 \def (eq_ind_r T t (\lambda (t8: T).(\forall (v: 
-T).((tlt v t8) \to (\forall (t9: T).((pr0 v t9) \to (\forall (t10: T).((pr0 v 
-t10) \to (ex2 T (\lambda (t11: T).(pr0 t9 t11)) (\lambda (t11: T).(pr0 t10 
-t11)))))))))) H (THead (Flat Appl) u1 (THead (Bind Abst) u t3)) H27) in (let 
-H29 \def (eq_ind T v1 (\lambda (t8: T).(pr0 t8 v2)) H7 u1 H20) in (ex2_ind T 
-(\lambda (t8: T).(pr0 v2 t8)) (\lambda (t8: T).(pr0 u2 t8)) (ex2 T (\lambda 
-(t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 (THead 
-(Flat Appl) u2 (THead (Bind Abst) u t3)) t8))) (\lambda (x: T).(\lambda (H30: 
-(pr0 v2 x)).(\lambda (H31: (pr0 u2 x)).(ex_intro2 T (\lambda (t8: T).(pr0 
-(THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 (THead (Flat Appl) u2 
-(THead (Bind Abst) u t3)) t8)) (THead (Bind Abbr) x t4) (pr0_comp v2 x H30 t4 
-t4 (pr0_refl t4) (Bind Abbr)) (pr0_beta u u2 x H31 t3 t4 H8))))) (H28 u1 
-(tlt_head_sx (Flat Appl) u1 (THead (Bind Abst) u t3)) v2 H29 u2 H14))))) t6 
-H26)) t7 (sym_eq T t7 (THead (Bind Abst) u t3) H24) H25))) | (pr0_comp u0 u3 
-H24 t7 t8 H25 k0) \Rightarrow (\lambda (H26: (eq T (THead k0 u0 t7) (THead 
-(Bind Abst) u t3))).(\lambda (H27: (eq T (THead k0 u3 t8) t6)).((let H28 \def 
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow t7 | (TLRef _) \Rightarrow t7 | (THead _ _ t9) 
-\Rightarrow t9])) (THead k0 u0 t7) (THead (Bind Abst) u t3) H26) in ((let H29 
-\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) 
-with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t9 _) 
-\Rightarrow t9])) (THead k0 u0 t7) (THead (Bind Abst) u t3) H26) in ((let H30 
-\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) 
-with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _) 
-\Rightarrow k1])) (THead k0 u0 t7) (THead (Bind Abst) u t3) H26) in (eq_ind K 
-(Bind Abst) (\lambda (k1: K).((eq T u0 u) \to ((eq T t7 t3) \to ((eq T (THead 
-k1 u3 t8) t6) \to ((pr0 u0 u3) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: 
+T).(pr0 t6 t7)))) (let H13 \def (eq_ind_r T t (\lambda (t6: T).(eq T t5 t6)) 
+H9 (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) H4) in (let H14 \def 
+(eq_ind_r T t (\lambda (t6: T).(\forall (v: T).((tlt v t6) \to (\forall (t7: 
+T).((pr0 v t7) \to (\forall (t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: 
+T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead (Flat Appl) v1 
+(THead (Bind Abst) u t3)) H4) in (ex_intro2 T (\lambda (t6: T).(pr0 (THead 
+(Bind Abbr) v2 t4) t6)) (\lambda (t6: T).(pr0 (THead (Flat Appl) v1 (THead 
+(Bind Abst) u t3)) t6)) (THead (Bind Abbr) v2 t4) (pr0_refl (THead (Bind 
+Abbr) v2 t4)) (pr0_beta u v1 v2 H7 t3 t4 H8)))) t2 H12)) t (sym_eq T t t2 
+H11))) t5 (sym_eq T t5 t H9) H10))) | (pr0_comp u1 u2 H9 t5 t6 H10 k) 
+\Rightarrow (\lambda (H11: (eq T (THead k u1 t5) t)).(\lambda (H12: (eq T 
+(THead k u2 t6) t2)).(eq_ind T (THead k u1 t5) (\lambda (_: T).((eq T (THead 
+k u2 t6) t2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: 
+T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) 
+(\lambda (H13: (eq T (THead k u2 t6) t2)).(eq_ind T (THead k u2 t6) (\lambda 
+(t7: T).((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead 
+(Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (H14: (pr0 
+u1 u2)).(\lambda (H15: (pr0 t5 t6)).(let H16 \def (eq_ind_r T t (\lambda (t7: 
+T).(eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) t7)) H4 (THead k u1 
+t5) H11) in (let H17 \def (f_equal T K (\lambda (e: T).(match e with [(TSort 
+_) \Rightarrow (Flat Appl) | (TLRef _) \Rightarrow (Flat Appl) | (THead k0 _ 
+_) \Rightarrow k0])) (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead k 
+u1 t5) H16) in ((let H18 \def (f_equal T T (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t7 _) 
+\Rightarrow t7])) (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead k u1 
+t5) H16) in ((let H19 \def (f_equal T T (\lambda (e: T).(match e with [(TSort 
+_) \Rightarrow (THead (Bind Abst) u t3) | (TLRef _) \Rightarrow (THead (Bind 
+Abst) u t3) | (THead _ _ t7) \Rightarrow t7])) (THead (Flat Appl) v1 (THead 
+(Bind Abst) u t3)) (THead k u1 t5) H16) in (\lambda (H20: (eq T v1 
+u1)).(\lambda (H21: (eq K (Flat Appl) k)).(eq_ind K (Flat Appl) (\lambda (k0: 
+K).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: 
+T).(pr0 (THead k0 u2 t6) t7)))) (let H22 \def (eq_ind_r K k (\lambda (k0: 
+K).(eq T (THead k0 u1 t5) t)) H11 (Flat Appl) H21) in (let H23 \def (eq_ind_r 
+T t5 (\lambda (t7: T).(pr0 t7 t6)) H15 (THead (Bind Abst) u t3) H19) in (let 
+H24 \def (match H23 with [(pr0_refl t7) \Rightarrow (\lambda (H24: (eq T t7 
+(THead (Bind Abst) u t3))).(\lambda (H25: (eq T t7 t6)).(eq_ind T (THead 
+(Bind Abst) u t3) (\lambda (t8: T).((eq T t8 t6) \to (ex2 T (\lambda (t9: 

 T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Flat 
 T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Flat 

-Appl) u2 t6) t9))))))))) (\lambda (H31: (eq T u0 u)).(eq_ind T u (\lambda 
-(t9: T).((eq T t7 t3) \to ((eq T (THead (Bind Abst) u3 t8) t6) \to ((pr0 t9 
-u3) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 
-t4) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t6) t10)))))))) 
-(\lambda (H32: (eq T t7 t3)).(eq_ind T t3 (\lambda (t9: T).((eq T (THead 
-(Bind Abst) u3 t8) t6) \to ((pr0 u u3) \to ((pr0 t9 t8) \to (ex2 T (\lambda 
-(t10: T).(pr0 (THead (Bind Abbr) v2 t4) t10)) (\lambda (t10: T).(pr0 (THead 
-(Flat Appl) u2 t6) t10))))))) (\lambda (H33: (eq T (THead (Bind Abst) u3 t8) 
-t6)).(eq_ind T (THead (Bind Abst) u3 t8) (\lambda (t9: T).((pr0 u u3) \to 
-((pr0 t3 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t4) 
-t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t9) t10)))))) (\lambda (_: 
-(pr0 u u3)).(\lambda (H35: (pr0 t3 t8)).(let H36 \def (eq_ind_r T t5 (\lambda 
-(t9: T).(eq T (THead (Flat Appl) u1 t9) t)) H22 (THead (Bind Abst) u t3) H19) 
-in (let H37 \def (eq_ind_r T t (\lambda (t9: T).(\forall (v: T).((tlt v t9) 
-\to (\forall (t10: T).((pr0 v t10) \to (\forall (t11: T).((pr0 v t11) \to 
-(ex2 T (\lambda (t12: T).(pr0 t10 t12)) (\lambda (t12: T).(pr0 t11 
-t12)))))))))) H (THead (Flat Appl) u1 (THead (Bind Abst) u t3)) H36) in (let 
-H38 \def (eq_ind T v1 (\lambda (t9: T).(pr0 t9 v2)) H7 u1 H20) in (ex2_ind T 
-(\lambda (t9: T).(pr0 v2 t9)) (\lambda (t9: T).(pr0 u2 t9)) (ex2 T (\lambda 
-(t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead 
-(Flat Appl) u2 (THead (Bind Abst) u3 t8)) t9))) (\lambda (x: T).(\lambda 
-(H39: (pr0 v2 x)).(\lambda (H40: (pr0 u2 x)).(ex2_ind T (\lambda (t9: T).(pr0 
-t8 t9)) (\lambda (t9: T).(pr0 t4 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead 
-(Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead 
-(Bind Abst) u3 t8)) t9))) (\lambda (x0: T).(\lambda (H41: (pr0 t8 
-x0)).(\lambda (H42: (pr0 t4 x0)).(ex_intro2 T (\lambda (t9: T).(pr0 (THead 
-(Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead 
-(Bind Abst) u3 t8)) t9)) (THead (Bind Abbr) x x0) (pr0_comp v2 x H39 t4 x0 
-H42 (Bind Abbr)) (pr0_beta u3 u2 x H40 t8 x0 H41))))) (H37 t3 (tlt_trans 
-(THead (Bind Abst) u t3) t3 (THead (Flat Appl) u1 (THead (Bind Abst) u t3)) 
-(tlt_head_dx (Bind Abst) u t3) (tlt_head_dx (Flat Appl) u1 (THead (Bind Abst) 
-u t3))) t8 H35 t4 H8))))) (H37 u1 (tlt_head_sx (Flat Appl) u1 (THead (Bind 
-Abst) u t3)) v2 H38 u2 H14))))))) t6 H33)) t7 (sym_eq T t7 t3 H32))) u0 
-(sym_eq T u0 u H31))) k0 (sym_eq K k0 (Bind Abst) H30))) H29)) H28)) H27 H24 
-H25))) | (pr0_beta u0 v0 v3 H24 t7 t8 H25) \Rightarrow (\lambda (H26: (eq T 
-(THead (Flat Appl) v0 (THead (Bind Abst) u0 t7)) (THead (Bind Abst) u 
-t3))).(\lambda (H27: (eq T (THead (Bind Abbr) v3 t8) t6)).((let H28 \def 
-(eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u0 t7)) (\lambda (e: 
-T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in 
-K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) 
-\Rightarrow True])])) I (THead (Bind Abst) u t3) H26) in (False_ind ((eq T 
-(THead (Bind Abbr) v3 t8) t6) \to ((pr0 v0 v3) \to ((pr0 t7 t8) \to (ex2 T 
-(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 
-(THead (Flat Appl) u2 t6) t9)))))) H28)) H27 H24 H25))) | (pr0_upsilon b H24 
-v0 v3 H25 u0 u3 H26 t7 t8 H27) \Rightarrow (\lambda (H28: (eq T (THead (Flat 
-Appl) v0 (THead (Bind b) u0 t7)) (THead (Bind Abst) u t3))).(\lambda (H29: 
-(eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v3) t8)) t6)).((let 
-H30 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind b) u0 t7)) (\lambda (e: 
-T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in 
-K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) 
-\Rightarrow True])])) I (THead (Bind Abst) u t3) H28) in (False_ind ((eq T 
-(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v3) t8)) t6) \to ((not 
-(eq B b Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to ((pr0 t7 t8) \to (ex2 T 
+Appl) u2 t6) t9))))) (\lambda (H26: (eq T (THead (Bind Abst) u t3) 
+t6)).(eq_ind T (THead (Bind Abst) u t3) (\lambda (t8: T).(ex2 T (\lambda (t9: 
+T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Flat 
+Appl) u2 t8) t9)))) (let H27 \def (eq_ind_r T t5 (\lambda (t8: T).(eq T 
+(THead (Flat Appl) u1 t8) t)) H22 (THead (Bind Abst) u t3) H19) in (let H28 
+\def (eq_ind_r T t (\lambda (t8: T).(\forall (v: T).((tlt v t8) \to (\forall 
+(t9: T).((pr0 v t9) \to (\forall (t10: T).((pr0 v t10) \to (ex2 T (\lambda 
+(t11: T).(pr0 t9 t11)) (\lambda (t11: T).(pr0 t10 t11)))))))))) H (THead 
+(Flat Appl) u1 (THead (Bind Abst) u t3)) H27) in (let H29 \def (eq_ind T v1 
+(\lambda (t8: T).(pr0 t8 v2)) H7 u1 H20) in (ex2_ind T (\lambda (t8: T).(pr0 
+v2 t8)) (\lambda (t8: T).(pr0 u2 t8)) (ex2 T (\lambda (t8: T).(pr0 (THead 
+(Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 (THead (Flat Appl) u2 (THead 
+(Bind Abst) u t3)) t8))) (\lambda (x: T).(\lambda (H30: (pr0 v2 x)).(\lambda 
+(H31: (pr0 u2 x)).(ex_intro2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 
+t4) t8)) (\lambda (t8: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u 
+t3)) t8)) (THead (Bind Abbr) x t4) (pr0_comp v2 x H30 t4 t4 (pr0_refl t4) 
+(Bind Abbr)) (pr0_beta u u2 x H31 t3 t4 H8))))) (H28 u1 (tlt_head_sx (Flat 
+Appl) u1 (THead (Bind Abst) u t3)) v2 H29 u2 H14))))) t6 H26)) t7 (sym_eq T 
+t7 (THead (Bind Abst) u t3) H24) H25))) | (pr0_comp u0 u3 H24 t7 t8 H25 k0) 
+\Rightarrow (\lambda (H26: (eq T (THead k0 u0 t7) (THead (Bind Abst) u 
+t3))).(\lambda (H27: (eq T (THead k0 u3 t8) t6)).((let H28 \def (f_equal T T 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow t7 | (TLRef _) 
+\Rightarrow t7 | (THead _ _ t9) \Rightarrow t9])) (THead k0 u0 t7) (THead 
+(Bind Abst) u t3) H26) in ((let H29 \def (f_equal T T (\lambda (e: T).(match 
+e with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t9 _) 
+\Rightarrow t9])) (THead k0 u0 t7) (THead (Bind Abst) u t3) H26) in ((let H30 
+\def (f_equal T K (\lambda (e: T).(match e with [(TSort _) \Rightarrow k0 | 
+(TLRef _) \Rightarrow k0 | (THead k1 _ _) \Rightarrow k1])) (THead k0 u0 t7) 
+(THead (Bind Abst) u t3) H26) in (eq_ind K (Bind Abst) (\lambda (k1: K).((eq 
+T u0 u) \to ((eq T t7 t3) \to ((eq T (THead k1 u3 t8) t6) \to ((pr0 u0 u3) 
+\to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) 
+t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t6) t9))))))))) (\lambda 
+(H31: (eq T u0 u)).(eq_ind T u (\lambda (t9: T).((eq T t7 t3) \to ((eq T 
+(THead (Bind Abst) u3 t8) t6) \to ((pr0 t9 u3) \to ((pr0 t7 t8) \to (ex2 T 
+(\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t4) t10)) (\lambda (t10: T).(pr0 
+(THead (Flat Appl) u2 t6) t10)))))))) (\lambda (H32: (eq T t7 t3)).(eq_ind T 
+t3 (\lambda (t9: T).((eq T (THead (Bind Abst) u3 t8) t6) \to ((pr0 u u3) \to 
+((pr0 t9 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t4) 
+t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t6) t10))))))) (\lambda 
+(H33: (eq T (THead (Bind Abst) u3 t8) t6)).(eq_ind T (THead (Bind Abst) u3 
+t8) (\lambda (t9: T).((pr0 u u3) \to ((pr0 t3 t8) \to (ex2 T (\lambda (t10: 
+T).(pr0 (THead (Bind Abbr) v2 t4) t10)) (\lambda (t10: T).(pr0 (THead (Flat 
+Appl) u2 t9) t10)))))) (\lambda (_: (pr0 u u3)).(\lambda (H35: (pr0 t3 
+t8)).(let H36 \def (eq_ind_r T t5 (\lambda (t9: T).(eq T (THead (Flat Appl) 
+u1 t9) t)) H22 (THead (Bind Abst) u t3) H19) in (let H37 \def (eq_ind_r T t 
+(\lambda (t9: T).(\forall (v: T).((tlt v t9) \to (\forall (t10: T).((pr0 v 
+t10) \to (\forall (t11: T).((pr0 v t11) \to (ex2 T (\lambda (t12: T).(pr0 t10 
+t12)) (\lambda (t12: T).(pr0 t11 t12)))))))))) H (THead (Flat Appl) u1 (THead 
+(Bind Abst) u t3)) H36) in (let H38 \def (eq_ind T v1 (\lambda (t9: T).(pr0 
+t9 v2)) H7 u1 H20) in (ex2_ind T (\lambda (t9: T).(pr0 v2 t9)) (\lambda (t9: 
+T).(pr0 u2 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) 
+(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u3 t8)) t9))) 
+(\lambda (x: T).(\lambda (H39: (pr0 v2 x)).(\lambda (H40: (pr0 u2 
+x)).(ex2_ind T (\lambda (t9: T).(pr0 t8 t9)) (\lambda (t9: T).(pr0 t4 t9)) 
+(ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: 
+T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u3 t8)) t9))) (\lambda (x0: 
+T).(\lambda (H41: (pr0 t8 x0)).(\lambda (H42: (pr0 t4 x0)).(ex_intro2 T 

 (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 
 (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 

-(THead (Flat Appl) u2 t6) t9)))))))) H30)) H29 H24 H25 H26 H27))) | 
-(pr0_delta u0 u3 H24 t7 t8 H25 w H26) \Rightarrow (\lambda (H27: (eq T (THead 
-(Bind Abbr) u0 t7) (THead (Bind Abst) u t3))).(\lambda (H28: (eq T (THead 
-(Bind Abbr) u3 w) t6)).((let H29 \def (eq_ind T (THead (Bind Abbr) u0 t7) 
-(\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) 
+(THead (Flat Appl) u2 (THead (Bind Abst) u3 t8)) t9)) (THead (Bind Abbr) x 
+x0) (pr0_comp v2 x H39 t4 x0 H42 (Bind Abbr)) (pr0_beta u3 u2 x H40 t8 x0 
+H41))))) (H37 t3 (tlt_trans (THead (Bind Abst) u t3) t3 (THead (Flat Appl) u1 
+(THead (Bind Abst) u t3)) (tlt_head_dx (Bind Abst) u t3) (tlt_head_dx (Flat 
+Appl) u1 (THead (Bind Abst) u t3))) t8 H35 t4 H8))))) (H37 u1 (tlt_head_sx 
+(Flat Appl) u1 (THead (Bind Abst) u t3)) v2 H38 u2 H14))))))) t6 H33)) t7 
+(sym_eq T t7 t3 H32))) u0 (sym_eq T u0 u H31))) k0 (sym_eq K k0 (Bind Abst) 
+H30))) H29)) H28)) H27 H24 H25))) | (pr0_beta u0 v0 v3 H24 t7 t8 H25) 
+\Rightarrow (\lambda (H26: (eq T (THead (Flat Appl) v0 (THead (Bind Abst) u0 
+t7)) (THead (Bind Abst) u t3))).(\lambda (H27: (eq T (THead (Bind Abbr) v3 
+t8) t6)).((let H28 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u0 
+t7)) (\lambda (e: T).(match e with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 with [(Bind _) 
+\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u t3) 
+H26) in (False_ind ((eq T (THead (Bind Abbr) v3 t8) t6) \to ((pr0 v0 v3) \to 
+((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) 
+(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t6) t9)))))) H28)) H27 H24 H25))) 
+| (pr0_upsilon b H24 v0 v3 H25 u0 u3 H26 t7 t8 H27) \Rightarrow (\lambda 
+(H28: (eq T (THead (Flat Appl) v0 (THead (Bind b) u0 t7)) (THead (Bind Abst) 
+u t3))).(\lambda (H29: (eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S 
+O) O v3) t8)) t6)).((let H30 \def (eq_ind T (THead (Flat Appl) v0 (THead 
+(Bind b) u0 t7)) (\lambda (e: T).(match e with [(TSort _) \Rightarrow False | 
+(TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 with 
+[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind 
+Abst) u t3) H28) in (False_ind ((eq T (THead (Bind b) u3 (THead (Flat Appl) 
+(lift (S O) O v3) t8)) t6) \to ((not (eq B b Abst)) \to ((pr0 v0 v3) \to 
+((pr0 u0 u3) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind 
+Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t6) t9)))))))) 
+H30)) H29 H24 H25 H26 H27))) | (pr0_delta u0 u3 H24 t7 t8 H25 w H26) 
+\Rightarrow (\lambda (H27: (eq T (THead (Bind Abbr) u0 t7) (THead (Bind Abst) 
+u t3))).(\lambda (H28: (eq T (THead (Bind Abbr) u3 w) t6)).((let H29 \def 
+(eq_ind T (THead (Bind Abbr) u0 t7) (\lambda (e: T).(match e with [(TSort _) 

 \Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow 
 \Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow 

-(match k0 in K return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match 
-b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst 
-\Rightarrow False | Void \Rightarrow False]) | (Flat _) \Rightarrow 
+(match k0 with [(Bind b) \Rightarrow (match b with [Abbr \Rightarrow True | 
+Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) \Rightarrow 

 False])])) I (THead (Bind Abst) u t3) H27) in (False_ind ((eq T (THead (Bind 
 Abbr) u3 w) t6) \to ((pr0 u0 u3) \to ((pr0 t7 t8) \to ((subst0 O u3 t8 w) \to 
 (ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: 
 T).(pr0 (THead (Flat Appl) u2 t6) t9))))))) H29)) H28 H24 H25 H26))) | 
 (pr0_zeta b H24 t7 t8 H25 u0) \Rightarrow (\lambda (H26: (eq T (THead (Bind 
 b) u0 (lift (S O) O t7)) (THead (Bind Abst) u t3))).(\lambda (H27: (eq T t8 
 False])])) I (THead (Bind Abst) u t3) H27) in (False_ind ((eq T (THead (Bind 
 Abbr) u3 w) t6) \to ((pr0 u0 u3) \to ((pr0 t7 t8) \to ((subst0 O u3 t8 w) \to 
 (ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: 
 T).(pr0 (THead (Flat Appl) u2 t6) t9))))))) H29)) H28 H24 H25 H26))) | 
 (pr0_zeta b H24 t7 t8 H25 u0) \Rightarrow (\lambda (H26: (eq T (THead (Bind 
 b) u0 (lift (S O) O t7)) (THead (Bind Abst) u t3))).(\lambda (H27: (eq T t8 

-t6)).((let H28 \def (f_equal T T (\lambda (e: T).(match e in T return 
-(\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat 
-\to nat))) (d: nat) (t9: T) on t9: T \def (match t9 with [(TSort n) 
-\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with 
-[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k0 u3 t10) 
-\Rightarrow (THead k0 (lref_map f d u3) (lref_map f (s k0 d) t10))]) in 
-lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (TLRef _) \Rightarrow 
-((let rec lref_map (f: ((nat \to nat))) (d: nat) (t9: T) on t9: T \def (match 
-t9 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef 
-(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | 
-(THead k0 u3 t10) \Rightarrow (THead k0 (lref_map f d u3) (lref_map f (s k0 
-d) t10))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (THead _ _ 
+t6)).((let H28 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow (lref_map (\lambda (x: nat).(plus x (S O))) O t7) | (TLRef _) 
+\Rightarrow (lref_map (\lambda (x: nat).(plus x (S O))) O t7) | (THead _ _ 

 t9) \Rightarrow t9])) (THead (Bind b) u0 (lift (S O) O t7)) (THead (Bind 
 t9) \Rightarrow t9])) (THead (Bind b) u0 (lift (S O) O t7)) (THead (Bind 

-Abst) u t3) H26) in ((let H29 \def (f_equal T T (\lambda (e: T).(match e in T 
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) 
-\Rightarrow u0 | (THead _ t9 _) \Rightarrow t9])) (THead (Bind b) u0 (lift (S 
-O) O t7)) (THead (Bind Abst) u t3) H26) in ((let H30 \def (f_equal T B 
-(\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) 
-\Rightarrow b | (TLRef _) \Rightarrow b | (THead k0 _ _) \Rightarrow (match 
-k0 in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) 
-\Rightarrow b])])) (THead (Bind b) u0 (lift (S O) O t7)) (THead (Bind Abst) u 
-t3) H26) in (eq_ind B Abst (\lambda (b0: B).((eq T u0 u) \to ((eq T (lift (S 
-O) O t7) t3) \to ((eq T t8 t6) \to ((not (eq B b0 Abst)) \to ((pr0 t7 t8) \to 
-(ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: 
-T).(pr0 (THead (Flat Appl) u2 t6) t9))))))))) (\lambda (H31: (eq T u0 
-u)).(eq_ind T u (\lambda (_: T).((eq T (lift (S O) O t7) t3) \to ((eq T t8 
-t6) \to ((not (eq B Abst Abst)) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: 
-T).(pr0 (THead (Bind Abbr) v2 t4) t10)) (\lambda (t10: T).(pr0 (THead (Flat 
-Appl) u2 t6) t10)))))))) (\lambda (H32: (eq T (lift (S O) O t7) t3)).(eq_ind 
-T (lift (S O) O t7) (\lambda (_: T).((eq T t8 t6) \to ((not (eq B Abst Abst)) 
+Abst) u t3) H26) in ((let H29 \def (f_equal T T (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t9 _) 
+\Rightarrow t9])) (THead (Bind b) u0 (lift (S O) O t7)) (THead (Bind Abst) u 
+t3) H26) in ((let H30 \def (f_equal T B (\lambda (e: T).(match e with [(TSort 
+_) \Rightarrow b | (TLRef _) \Rightarrow b | (THead k0 _ _) \Rightarrow 
+(match k0 with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead 
+(Bind b) u0 (lift (S O) O t7)) (THead (Bind Abst) u t3) H26) in (eq_ind B 
+Abst (\lambda (b0: B).((eq T u0 u) \to ((eq T (lift (S O) O t7) t3) \to ((eq 
+T t8 t6) \to ((not (eq B b0 Abst)) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: 
+T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Flat 
+Appl) u2 t6) t9))))))))) (\lambda (H31: (eq T u0 u)).(eq_ind T u (\lambda (_: 
+T).((eq T (lift (S O) O t7) t3) \to ((eq T t8 t6) \to ((not (eq B Abst Abst)) 

 \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t4) 
 \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t4) 

-t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t6) t10))))))) (\lambda 
-(H33: (eq T t8 t6)).(eq_ind T t6 (\lambda (t9: T).((not (eq B Abst Abst)) \to 
-((pr0 t7 t9) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t4) 
-t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t6) t10)))))) (\lambda 
-(H34: (not (eq B Abst Abst))).(\lambda (_: (pr0 t7 t6)).(let H36 \def (match 
-(H34 (refl_equal B Abst)) in False return (\lambda (_: False).(ex2 T (\lambda 
-(t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead 
-(Flat Appl) u2 t6) t9)))) with []) in H36))) t8 (sym_eq T t8 t6 H33))) t3 
-H32)) u0 (sym_eq T u0 u H31))) b (sym_eq B b Abst H30))) H29)) H28)) H27 H24 
-H25))) | (pr0_tau t7 t8 H24 u0) \Rightarrow (\lambda (H25: (eq T (THead (Flat 
-Cast) u0 t7) (THead (Bind Abst) u t3))).(\lambda (H26: (eq T t8 t6)).((let 
-H27 \def (eq_ind T (THead (Flat Cast) u0 t7) (\lambda (e: T).(match e in T 
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda 
-(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow 
-True])])) I (THead (Bind Abst) u t3) H25) in (False_ind ((eq T t8 t6) \to 
-((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) 
-(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t6) t9))))) H27)) H26 H24)))]) in 
-(H24 (refl_equal T (THead (Bind Abst) u t3)) (refl_equal T t6))))) k H21)))) 
-H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_beta u0 v0 v3 H9 t5 t6 
-H10) \Rightarrow (\lambda (H11: (eq T (THead (Flat Appl) v0 (THead (Bind 
-Abst) u0 t5)) t)).(\lambda (H12: (eq T (THead (Bind Abbr) v3 t6) t2)).(eq_ind 
-T (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5)) (\lambda (_: T).((eq T 
-(THead (Bind Abbr) v3 t6) t2) \to ((pr0 v0 v3) \to ((pr0 t5 t6) \to (ex2 T 
-(\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 
-t8))))))) (\lambda (H13: (eq T (THead (Bind Abbr) v3 t6) t2)).(eq_ind T 
-(THead (Bind Abbr) v3 t6) (\lambda (t7: T).((pr0 v0 v3) \to ((pr0 t5 t6) \to 
-(ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: 
-T).(pr0 t7 t8)))))) (\lambda (H14: (pr0 v0 v3)).(\lambda (H15: (pr0 t5 
-t6)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1 
-(THead (Bind Abst) u t3)) t7)) H4 (THead (Flat Appl) v0 (THead (Bind Abst) u0 
-t5)) H11) in (let H17 \def (f_equal T T (\lambda (e: T).(match e in T return 
-(\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 
-| (THead _ t7 _) \Rightarrow t7])) (THead (Flat Appl) v1 (THead (Bind Abst) u 
-t3)) (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5)) H16) in ((let H18 \def 
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ _ t7) 
-\Rightarrow (match t7 in T return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t8 _) \Rightarrow t8])])) 
+t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t6) t10)))))))) (\lambda 
+(H32: (eq T (lift (S O) O t7) t3)).(eq_ind T (lift (S O) O t7) (\lambda (_: 
+T).((eq T t8 t6) \to ((not (eq B Abst Abst)) \to ((pr0 t7 t8) \to (ex2 T 
+(\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t4) t10)) (\lambda (t10: T).(pr0 
+(THead (Flat Appl) u2 t6) t10))))))) (\lambda (H33: (eq T t8 t6)).(eq_ind T 
+t6 (\lambda (t9: T).((not (eq B Abst Abst)) \to ((pr0 t7 t9) \to (ex2 T 
+(\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t4) t10)) (\lambda (t10: T).(pr0 
+(THead (Flat Appl) u2 t6) t10)))))) (\lambda (H34: (not (eq B Abst 
+Abst))).(\lambda (_: (pr0 t7 t6)).(let H36 \def (match (H34 (refl_equal B 
+Abst)) in False with []) in H36))) t8 (sym_eq T t8 t6 H33))) t3 H32)) u0 
+(sym_eq T u0 u H31))) b (sym_eq B b Abst H30))) H29)) H28)) H27 H24 H25))) | 
+(pr0_tau t7 t8 H24 u0) \Rightarrow (\lambda (H25: (eq T (THead (Flat Cast) u0 
+t7) (THead (Bind Abst) u t3))).(\lambda (H26: (eq T t8 t6)).((let H27 \def 
+(eq_ind T (THead (Flat Cast) u0 t7) (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow 
+(match k0 with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I 
+(THead (Bind Abst) u t3) H25) in (False_ind ((eq T t8 t6) \to ((pr0 t7 t8) 
+\to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: 
+T).(pr0 (THead (Flat Appl) u2 t6) t9))))) H27)) H26 H24)))]) in (H24 
+(refl_equal T (THead (Bind Abst) u t3)) (refl_equal T t6))))) k H21)))) H18)) 
+H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_beta u0 v0 v3 H9 t5 t6 H10) 
+\Rightarrow (\lambda (H11: (eq T (THead (Flat Appl) v0 (THead (Bind Abst) u0 
+t5)) t)).(\lambda (H12: (eq T (THead (Bind Abbr) v3 t6) t2)).(eq_ind T (THead 
+(Flat Appl) v0 (THead (Bind Abst) u0 t5)) (\lambda (_: T).((eq T (THead (Bind 
+Abbr) v3 t6) t2) \to ((pr0 v0 v3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: 
+T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) 
+(\lambda (H13: (eq T (THead (Bind Abbr) v3 t6) t2)).(eq_ind T (THead (Bind 
+Abbr) v3 t6) (\lambda (t7: T).((pr0 v0 v3) \to ((pr0 t5 t6) \to (ex2 T 
+(\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t7 
+t8)))))) (\lambda (H14: (pr0 v0 v3)).(\lambda (H15: (pr0 t5 t6)).(let H16 
+\def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind 
+Abst) u t3)) t7)) H4 (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5)) H11) in 
+(let H17 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t7 _) \Rightarrow t7])) 

 (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead (Flat Appl) v0 (THead 
 (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead (Flat Appl) v0 (THead 

-(Bind Abst) u0 t5)) H16) in ((let H19 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | 
-(TLRef _) \Rightarrow t3 | (THead _ _ t7) \Rightarrow (match t7 in T return 
-(\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 
-| (THead _ _ t8) \Rightarrow t8])])) (THead (Flat Appl) v1 (THead (Bind Abst) 
-u t3)) (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5)) H16) in (\lambda (_: 
-(eq T u u0)).(\lambda (H21: (eq T v1 v0)).(let H22 \def (eq_ind_r T t 
-(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) 
-\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) 
-(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Appl) v0 (THead (Bind 
-Abst) u0 t5)) H11) in (let H23 \def (eq_ind T v1 (\lambda (t7: T).(pr0 t7 
-v2)) H7 v0 H21) in (let H24 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) 
-H8 t5 H19) in (ex2_ind T (\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 
-t6 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda 
-(t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7))) (\lambda (x: T).(\lambda (H25: 
-(pr0 t4 x)).(\lambda (H26: (pr0 t6 x)).(ex2_ind T (\lambda (t7: T).(pr0 v2 
-t7)) (\lambda (t7: T).(pr0 v3 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind 
-Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7))) 
-(\lambda (x0: T).(\lambda (H27: (pr0 v2 x0)).(\lambda (H28: (pr0 v3 
-x0)).(ex_intro2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) 
-(\lambda (t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7)) (THead (Bind Abbr) x0 x) 
-(pr0_comp v2 x0 H27 t4 x H25 (Bind Abbr)) (pr0_comp v3 x0 H28 t6 x H26 (Bind 
-Abbr)))))) (H22 v0 (tlt_head_sx (Flat Appl) v0 (THead (Bind Abst) u0 t5)) v2 
-H23 v3 H14))))) (H22 t5 (tlt_trans (THead (Bind Abst) u0 t5) t5 (THead (Flat 
-Appl) v0 (THead (Bind Abst) u0 t5)) (tlt_head_dx (Bind Abst) u0 t5) 
-(tlt_head_dx (Flat Appl) v0 (THead (Bind Abst) u0 t5))) t4 H24 t6 H15)))))))) 
-H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_upsilon b H9 v0 v3 H10 u1 
-u2 H11 t5 t6 H12) \Rightarrow (\lambda (H13: (eq T (THead (Flat Appl) v0 
-(THead (Bind b) u1 t5)) t)).(\lambda (H14: (eq T (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v3) t6)) t2)).(eq_ind T (THead (Flat Appl) v0 
-(THead (Bind b) u1 t5)) (\lambda (_: T).((eq T (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v3) t6)) t2) \to ((not (eq B b Abst)) \to ((pr0 v0 
-v3) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead 
-(Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H15: 
-(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v3) t6)) 
-t2)).(eq_ind T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v3) t6)) 
-(\lambda (t7: T).((not (eq B b Abst)) \to ((pr0 v0 v3) \to ((pr0 u1 u2) \to 
+(Bind Abst) u0 t5)) H16) in ((let H18 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead 
+_ _ t7) \Rightarrow (match t7 with [(TSort _) \Rightarrow u | (TLRef _) 
+\Rightarrow u | (THead _ t8 _) \Rightarrow t8])])) (THead (Flat Appl) v1 
+(THead (Bind Abst) u t3)) (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5)) 
+H16) in ((let H19 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7) \Rightarrow (match 
+t7 with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t8) 
+\Rightarrow t8])])) (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead 
+(Flat Appl) v0 (THead (Bind Abst) u0 t5)) H16) in (\lambda (_: (eq T u 
+u0)).(\lambda (H21: (eq T v1 v0)).(let H22 \def (eq_ind_r T t (\lambda (t7: 
+T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall 
+(t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: 
+T).(pr0 t9 t10)))))))))) H (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5)) 
+H11) in (let H23 \def (eq_ind T v1 (\lambda (t7: T).(pr0 t7 v2)) H7 v0 H21) 
+in (let H24 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H8 t5 H19) in 
+(ex2_ind T (\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T 
+(\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 
+(THead (Bind Abbr) v3 t6) t7))) (\lambda (x: T).(\lambda (H25: (pr0 t4 
+x)).(\lambda (H26: (pr0 t6 x)).(ex2_ind T (\lambda (t7: T).(pr0 v2 t7)) 
+(\lambda (t7: T).(pr0 v3 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) 
+v2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7))) (\lambda 
+(x0: T).(\lambda (H27: (pr0 v2 x0)).(\lambda (H28: (pr0 v3 x0)).(ex_intro2 T 
+(\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 
+(THead (Bind Abbr) v3 t6) t7)) (THead (Bind Abbr) x0 x) (pr0_comp v2 x0 H27 
+t4 x H25 (Bind Abbr)) (pr0_comp v3 x0 H28 t6 x H26 (Bind Abbr)))))) (H22 v0 
+(tlt_head_sx (Flat Appl) v0 (THead (Bind Abst) u0 t5)) v2 H23 v3 H14))))) 
+(H22 t5 (tlt_trans (THead (Bind Abst) u0 t5) t5 (THead (Flat Appl) v0 (THead 
+(Bind Abst) u0 t5)) (tlt_head_dx (Bind Abst) u0 t5) (tlt_head_dx (Flat Appl) 
+v0 (THead (Bind Abst) u0 t5))) t4 H24 t6 H15)))))))) H18)) H17))))) t2 H13)) 
+t H11 H12 H9 H10))) | (pr0_upsilon b H9 v0 v3 H10 u1 u2 H11 t5 t6 H12) 
+\Rightarrow (\lambda (H13: (eq T (THead (Flat Appl) v0 (THead (Bind b) u1 
+t5)) t)).(\lambda (H14: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
+O) O v3) t6)) t2)).(eq_ind T (THead (Flat Appl) v0 (THead (Bind b) u1 t5)) 
+(\lambda (_: T).((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O 
+v3) t6)) t2) \to ((not (eq B b Abst)) \to ((pr0 v0 v3) \to ((pr0 u1 u2) \to 

 ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) 
 ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) 

-(\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda (H16: (not (eq B b 
-Abst))).(\lambda (_: (pr0 v0 v3)).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0 
-t5 t6)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) 
-v1 (THead (Bind Abst) u t3)) t7)) H4 (THead (Flat Appl) v0 (THead (Bind b) u1 
-t5)) H13) in (let H21 \def (f_equal T T (\lambda (e: T).(match e in T return 
-(\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 
-| (THead _ t7 _) \Rightarrow t7])) (THead (Flat Appl) v1 (THead (Bind Abst) u 
-t3)) (THead (Flat Appl) v0 (THead (Bind b) u1 t5)) H20) in ((let H22 \def 
-(f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with 
+(\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H15: (eq T (THead (Bind b) u2 
+(THead (Flat Appl) (lift (S O) O v3) t6)) t2)).(eq_ind T (THead (Bind b) u2 
+(THead (Flat Appl) (lift (S O) O v3) t6)) (\lambda (t7: T).((not (eq B b 
+Abst)) \to ((pr0 v0 v3) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda 
+(t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t7 
+t8)))))))) (\lambda (H16: (not (eq B b Abst))).(\lambda (_: (pr0 v0 
+v3)).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0 t5 t6)).(let H20 \def 
+(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind Abst) 
+u t3)) t7)) H4 (THead (Flat Appl) v0 (THead (Bind b) u1 t5)) H13) in (let H21 
+\def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow v1 | 
+(TLRef _) \Rightarrow v1 | (THead _ t7 _) \Rightarrow t7])) (THead (Flat 
+Appl) v1 (THead (Bind Abst) u t3)) (THead (Flat Appl) v0 (THead (Bind b) u1 
+t5)) H20) in ((let H22 \def (f_equal T B (\lambda (e: T).(match e with 

 [(TSort _) \Rightarrow Abst | (TLRef _) \Rightarrow Abst | (THead _ _ t7) 
 [(TSort _) \Rightarrow Abst | (TLRef _) \Rightarrow Abst | (THead _ _ t7) 

-\Rightarrow (match t7 in T return (\lambda (_: T).B) with [(TSort _) 
-\Rightarrow Abst | (TLRef _) \Rightarrow Abst | (THead k _ _) \Rightarrow 
-(match k in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | 
-(Flat _) \Rightarrow Abst])])])) (THead (Flat Appl) v1 (THead (Bind Abst) u 
-t3)) (THead (Flat Appl) v0 (THead (Bind b) u1 t5)) H20) in ((let H23 \def 
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ _ t7) 
-\Rightarrow (match t7 in T return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t8 _) \Rightarrow t8])])) 
-(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead (Flat Appl) v0 (THead 
-(Bind b) u1 t5)) H20) in ((let H24 \def (f_equal T T (\lambda (e: T).(match e 
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) 
-\Rightarrow t3 | (THead _ _ t7) \Rightarrow (match t7 in T return (\lambda 
-(_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead 
-_ _ t8) \Rightarrow t8])])) (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) 
-(THead (Flat Appl) v0 (THead (Bind b) u1 t5)) H20) in (\lambda (_: (eq T u 
-u1)).(\lambda (H26: (eq B Abst b)).(\lambda (_: (eq T v1 v0)).(eq_ind B Abst 
-(\lambda (b0: B).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) 
-(\lambda (t7: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O 
-v3) t6)) t7)))) (let H28 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 
-Abst))) H16 Abst H26) in (let H29 \def (match (H28 (refl_equal B Abst)) in 
-False return (\lambda (_: False).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind 
-Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abst) u2 (THead (Flat 
-Appl) (lift (S O) O v3) t6)) t7)))) with []) in H29)) b H26))))) H23)) H22)) 
-H21))))))) t2 H15)) t H13 H14 H9 H10 H11 H12))) | (pr0_delta u1 u2 H9 t5 t6 
-H10 w H11) \Rightarrow (\lambda (H12: (eq T (THead (Bind Abbr) u1 t5) 
-t)).(\lambda (H13: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind 
-Abbr) u1 t5) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2) \to ((pr0 u1 
-u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 
-(THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda 
-(H14: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w) 
-(\lambda (t7: T).((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to 
+\Rightarrow (match t7 with [(TSort _) \Rightarrow Abst | (TLRef _) 
+\Rightarrow Abst | (THead k _ _) \Rightarrow (match k with [(Bind b0) 
+\Rightarrow b0 | (Flat _) \Rightarrow Abst])])])) (THead (Flat Appl) v1 
+(THead (Bind Abst) u t3)) (THead (Flat Appl) v0 (THead (Bind b) u1 t5)) H20) 
+in ((let H23 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ _ t7) \Rightarrow (match 
+t7 with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t8 _) 
+\Rightarrow t8])])) (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead 
+(Flat Appl) v0 (THead (Bind b) u1 t5)) H20) in ((let H24 \def (f_equal T T 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow t3 | (TLRef _) 
+\Rightarrow t3 | (THead _ _ t7) \Rightarrow (match t7 with [(TSort _) 
+\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t8) \Rightarrow 
+t8])])) (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead (Flat Appl) v0 
+(THead (Bind b) u1 t5)) H20) in (\lambda (_: (eq T u u1)).(\lambda (H26: (eq 
+B Abst b)).(\lambda (_: (eq T v1 v0)).(eq_ind B Abst (\lambda (b0: B).(ex2 T 
+(\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 
+(THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v3) t6)) t7)))) (let H28 
+\def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 Abst))) H16 Abst H26) in 
+(let H29 \def (match (H28 (refl_equal B Abst)) in False with []) in H29)) b 
+H26))))) H23)) H22)) H21))))))) t2 H15)) t H13 H14 H9 H10 H11 H12))) | 
+(pr0_delta u1 u2 H9 t5 t6 H10 w H11) \Rightarrow (\lambda (H12: (eq T (THead 
+(Bind Abbr) u1 t5) t)).(\lambda (H13: (eq T (THead (Bind Abbr) u2 w) 
+t2)).(eq_ind T (THead (Bind Abbr) u1 t5) (\lambda (_: T).((eq T (THead (Bind 
+Abbr) u2 w) t2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to 

 (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: 
 (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: 

-T).(pr0 t7 t8))))))) (\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0 t5 
-t6)).(\lambda (_: (subst0 O u2 t6 w)).(let H18 \def (eq_ind_r T t (\lambda 
-(t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) t7)) H4 (THead 
-(Bind Abbr) u1 t5) H12) in (let H19 \def (eq_ind T (THead (Flat Appl) v1 
-(THead (Bind Abst) u t3)) (\lambda (ee: T).(match ee in T return (\lambda (_: 
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
-(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with 
-[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind 
-Abbr) u1 t5) H18) in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind 
-Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7))) 
-H19)))))) t2 H14)) t H12 H13 H9 H10 H11))) | (pr0_zeta b H9 t5 t6 H10 u0) 
-\Rightarrow (\lambda (H11: (eq T (THead (Bind b) u0 (lift (S O) O t5)) 
-t)).(\lambda (H12: (eq T t6 t2)).(eq_ind T (THead (Bind b) u0 (lift (S O) O 
-t5)) (\lambda (_: T).((eq T t6 t2) \to ((not (eq B b Abst)) \to ((pr0 t5 t6) 
-\to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: 
-T).(pr0 t2 t8))))))) (\lambda (H13: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: 
-T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 
-(THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (_: 
-(not (eq B b Abst))).(\lambda (_: (pr0 t5 t2)).(let H16 \def (eq_ind_r T t 
-(\lambda (t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) t7)) 
-H4 (THead (Bind b) u0 (lift (S O) O t5)) H11) in (let H17 \def (eq_ind T 
-(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (\lambda (ee: T).(match ee in 
-T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda 
-(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow 
-True])])) I (THead (Bind b) u0 (lift (S O) O t5)) H16) in (False_ind (ex2 T 
-(\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 t2 
-t7))) H17))))) t6 (sym_eq T t6 t2 H13))) t H11 H12 H9 H10))) | (pr0_tau t5 t6 
-H9 u0) \Rightarrow (\lambda (H10: (eq T (THead (Flat Cast) u0 t5) 
-t)).(\lambda (H11: (eq T t6 t2)).(eq_ind T (THead (Flat Cast) u0 t5) (\lambda 
-(_: T).((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead 
-(Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H12: (eq 
-T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: 
-T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))) 
-(\lambda (_: (pr0 t5 t2)).(let H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T 
-(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) t7)) H4 (THead (Flat Cast) u0 
-t5) H10) in (let H15 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) 
-u t3)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) 
-\Rightarrow False | (Flat f) \Rightarrow (match f in F return (\lambda (_: 
-F).Prop) with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead 
-(Flat Cast) u0 t5) H14) in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead 
-(Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 t2 t7))) H15)))) t6 (sym_eq T 
-t6 t2 H12))) t H10 H11 H9)))]) in (H9 (refl_equal T t) (refl_equal T t2))))) 
-t1 H6)) t H4 H5 H2 H3))) | (pr0_upsilon b H2 v1 v2 H3 u1 u2 H4 t3 t4 H5) 
-\Rightarrow (\lambda (H6: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) 
-t)).(\lambda (H7: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O 
-v2) t4)) t1)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) 
-(\lambda (_: T).((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O 
-v2) t4)) t1) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to 
-((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 
-t2 t6))))))))) (\lambda (H8: (eq T (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t4)) t1)).(eq_ind T (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t4)) (\lambda (t5: T).((not (eq B b Abst)) \to ((pr0 v1 v2) 
-\to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t5 t6)) 
-(\lambda (t6: T).(pr0 t2 t6)))))))) (\lambda (H9: (not (eq B b 
-Abst))).(\lambda (H10: (pr0 v1 v2)).(\lambda (H11: (pr0 u1 u2)).(\lambda 
-(H12: (pr0 t3 t4)).(let H13 \def (match H1 in pr0 return (\lambda (t5: 
-T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2) 
-\to (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift 
-(S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t2 t7)))))))) with [(pr0_refl t5) 
+T).(pr0 t2 t8)))))))) (\lambda (H14: (eq T (THead (Bind Abbr) u2 w) 
+t2)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t7: T).((pr0 u1 u2) \to 
+((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 (THead 
+(Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t7 t8))))))) (\lambda (_: (pr0 
+u1 u2)).(\lambda (_: (pr0 t5 t6)).(\lambda (_: (subst0 O u2 t6 w)).(let H18 
+\def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind 
+Abst) u t3)) t7)) H4 (THead (Bind Abbr) u1 t5) H12) in (let H19 \def (eq_ind 
+T (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (\lambda (ee: T).(match ee 
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ 
+_) \Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat _) 
+\Rightarrow True])])) I (THead (Bind Abbr) u1 t5) H18) in (False_ind (ex2 T 
+(\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 
+(THead (Bind Abbr) u2 w) t7))) H19)))))) t2 H14)) t H12 H13 H9 H10 H11))) | 
+(pr0_zeta b H9 t5 t6 H10 u0) \Rightarrow (\lambda (H11: (eq T (THead (Bind b) 
+u0 (lift (S O) O t5)) t)).(\lambda (H12: (eq T t6 t2)).(eq_ind T (THead (Bind 
+b) u0 (lift (S O) O t5)) (\lambda (_: T).((eq T t6 t2) \to ((not (eq B b 
+Abst)) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) 
+v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H13: (eq T t6 
+t2)).(eq_ind T t2 (\lambda (t7: T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to 
+(ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: 
+T).(pr0 t2 t8)))))) (\lambda (_: (not (eq B b Abst))).(\lambda (_: (pr0 t5 
+t2)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1 
+(THead (Bind Abst) u t3)) t7)) H4 (THead (Bind b) u0 (lift (S O) O t5)) H11) 
+in (let H17 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) 
+(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _) 
+\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u0 (lift 
+(S O) O t5)) H16) in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind 
+Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 t2 t7))) H17))))) t6 (sym_eq T t6 t2 
+H13))) t H11 H12 H9 H10))) | (pr0_tau t5 t6 H9 u0) \Rightarrow (\lambda (H10: 
+(eq T (THead (Flat Cast) u0 t5) t)).(\lambda (H11: (eq T t6 t2)).(eq_ind T 
+(THead (Flat Cast) u0 t5) (\lambda (_: T).((eq T t6 t2) \to ((pr0 t5 t6) \to 
+(ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: 
+T).(pr0 t2 t8)))))) (\lambda (H12: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: 
+T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) 
+t8)) (\lambda (t8: T).(pr0 t2 t8))))) (\lambda (_: (pr0 t5 t2)).(let H14 \def 
+(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind Abst) 
+u t3)) t7)) H4 (THead (Flat Cast) u0 t5) H10) in (let H15 \def (eq_ind T 
+(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (\lambda (ee: T).(match ee 
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ 
+_) \Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat f) 
+\Rightarrow (match f with [Appl \Rightarrow True | Cast \Rightarrow 
+False])])])) I (THead (Flat Cast) u0 t5) H14) in (False_ind (ex2 T (\lambda 
+(t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 t2 t7))) 
+H15)))) t6 (sym_eq T t6 t2 H12))) t H10 H11 H9)))]) in (H9 (refl_equal T t) 
+(refl_equal T t2))))) t1 H6)) t H4 H5 H2 H3))) | (pr0_upsilon b H2 v1 v2 H3 
+u1 u2 H4 t3 t4 H5) \Rightarrow (\lambda (H6: (eq T (THead (Flat Appl) v1 
+(THead (Bind b) u1 t3)) t)).(\lambda (H7: (eq T (THead (Bind b) u2 (THead 
+(Flat Appl) (lift (S O) O v2) t4)) t1)).(eq_ind T (THead (Flat Appl) v1 
+(THead (Bind b) u1 t3)) (\lambda (_: T).((eq T (THead (Bind b) u2 (THead 
+(Flat Appl) (lift (S O) O v2) t4)) t1) \to ((not (eq B b Abst)) \to ((pr0 v1 
+v2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t1 
+t6)) (\lambda (t6: T).(pr0 t2 t6))))))))) (\lambda (H8: (eq T (THead (Bind b) 
+u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t1)).(eq_ind T (THead (Bind b) 
+u2 (THead (Flat Appl) (lift (S O) O v2) t4)) (\lambda (t5: T).((not (eq B b 
+Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ex2 T (\lambda 
+(t6: T).(pr0 t5 t6)) (\lambda (t6: T).(pr0 t2 t6)))))))) (\lambda (H9: (not 
+(eq B b Abst))).(\lambda (H10: (pr0 v1 v2)).(\lambda (H11: (pr0 u1 
+u2)).(\lambda (H12: (pr0 t3 t4)).(let H13 \def (match H1 with [(pr0_refl t5) 

 \Rightarrow (\lambda (H13: (eq T t5 t)).(\lambda (H14: (eq T t5 t2)).(eq_ind 
 T t (\lambda (t6: T).((eq T t6 t2) \to (ex2 T (\lambda (t7: T).(pr0 (THead 
 (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: 
 \Rightarrow (\lambda (H13: (eq T t5 t)).(\lambda (H14: (eq T t5 t2)).(eq_ind 
 T t (\lambda (t6: T).((eq T t6 t2) \to (ex2 T (\lambda (t7: T).(pr0 (THead 
 (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: 


@@ -1467,143 +1351,133 @@ H14))) | (pr0_comp u0 u3 H13 t5 t6 H14 k) \Rightarrow (\lambda (H15: (eq T
 T).(pr0 t7 t8)))))) (\lambda (H18: (pr0 u0 u3)).(\lambda (H19: (pr0 t5 
 t6)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1 
 (THead (Bind b) u1 t3)) t7)) H6 (THead k u0 t5) H15) in (let H21 \def 
 T).(pr0 t7 t8)))))) (\lambda (H18: (pr0 u0 u3)).(\lambda (H19: (pr0 t5 
 t6)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1 
 (THead (Bind b) u1 t3)) t7)) H6 (THead k u0 t5) H15) in (let H21 \def 

-(f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with 
-[(TSort _) \Rightarrow (Flat Appl) | (TLRef _) \Rightarrow (Flat Appl) | 
-(THead k0 _ _) \Rightarrow k0])) (THead (Flat Appl) v1 (THead (Bind b) u1 
-t3)) (THead k u0 t5) H20) in ((let H22 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | 
-(TLRef _) \Rightarrow v1 | (THead _ t7 _) \Rightarrow t7])) (THead (Flat 
-Appl) v1 (THead (Bind b) u1 t3)) (THead k u0 t5) H20) in ((let H23 \def 
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow (THead (Bind b) u1 t3) | (TLRef _) \Rightarrow (THead 
-(Bind b) u1 t3) | (THead _ _ t7) \Rightarrow t7])) (THead (Flat Appl) v1 
-(THead (Bind b) u1 t3)) (THead k u0 t5) H20) in (\lambda (H24: (eq T v1 
-u0)).(\lambda (H25: (eq K (Flat Appl) k)).(eq_ind K (Flat Appl) (\lambda (k0: 
-K).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift 
-(S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 (THead k0 u3 t6) t7)))) (let H26 
-\def (eq_ind_r K k (\lambda (k0: K).(eq T (THead k0 u0 t5) t)) H15 (Flat 
-Appl) H25) in (let H27 \def (eq_ind_r T t5 (\lambda (t7: T).(pr0 t7 t6)) H19 
-(THead (Bind b) u1 t3) H23) in (let H28 \def (match H27 in pr0 return 
-(\lambda (t7: T).(\lambda (t8: T).(\lambda (_: (pr0 t7 t8)).((eq T t7 (THead 
-(Bind b) u1 t3)) \to ((eq T t8 t6) \to (ex2 T (\lambda (t9: T).(pr0 (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: 
-T).(pr0 (THead (Flat Appl) u3 t6) t9)))))))) with [(pr0_refl t7) \Rightarrow 
-(\lambda (H28: (eq T t7 (THead (Bind b) u1 t3))).(\lambda (H29: (eq T t7 
-t6)).(eq_ind T (THead (Bind b) u1 t3) (\lambda (t8: T).((eq T t8 t6) \to (ex2 
-T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O 
-v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9))))) 
-(\lambda (H30: (eq T (THead (Bind b) u1 t3) t6)).(eq_ind T (THead (Bind b) u1 
-t3) (\lambda (t8: T).(ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat 
-Appl) u3 t8) t9)))) (let H31 \def (eq_ind_r T t5 (\lambda (t8: T).(eq T 
-(THead (Flat Appl) u0 t8) t)) H26 (THead (Bind b) u1 t3) H23) in (let H32 
-\def (eq_ind_r T t (\lambda (t8: T).(\forall (v: T).((tlt v t8) \to (\forall 
-(t9: T).((pr0 v t9) \to (\forall (t10: T).((pr0 v t10) \to (ex2 T (\lambda 
-(t11: T).(pr0 t9 t11)) (\lambda (t11: T).(pr0 t10 t11)))))))))) H (THead 
-(Flat Appl) u0 (THead (Bind b) u1 t3)) H31) in (let H33 \def (eq_ind T v1 
-(\lambda (t8: T).(pr0 t8 v2)) H10 u0 H24) in (ex2_ind T (\lambda (t8: T).(pr0 
-v2 t8)) (\lambda (t8: T).(pr0 u3 t8)) (ex2 T (\lambda (t8: T).(pr0 (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: 
-T).(pr0 (THead (Flat Appl) u3 (THead (Bind b) u1 t3)) t8))) (\lambda (x: 
-T).(\lambda (H34: (pr0 v2 x)).(\lambda (H35: (pr0 u3 x)).(ex2_sym T (pr0 
-(THead (Flat Appl) u3 (THead (Bind b) u1 t3))) (pr0 (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t4))) (pr0_confluence__pr0_cong_upsilon_refl b 
-H9 u1 u2 H11 t3 t4 H12 u3 v2 x H35 H34))))) (H32 u0 (tlt_head_sx (Flat Appl) 
-u0 (THead (Bind b) u1 t3)) v2 H33 u3 H18))))) t6 H30)) t7 (sym_eq T t7 (THead 
-(Bind b) u1 t3) H28) H29))) | (pr0_comp u4 u5 H28 t7 t8 H29 k0) \Rightarrow 
-(\lambda (H30: (eq T (THead k0 u4 t7) (THead (Bind b) u1 t3))).(\lambda (H31: 
-(eq T (THead k0 u5 t8) t6)).((let H32 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t7 | 
-(TLRef _) \Rightarrow t7 | (THead _ _ t9) \Rightarrow t9])) (THead k0 u4 t7) 
-(THead (Bind b) u1 t3) H30) in ((let H33 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u4 | 
-(TLRef _) \Rightarrow u4 | (THead _ t9 _) \Rightarrow t9])) (THead k0 u4 t7) 
-(THead (Bind b) u1 t3) H30) in ((let H34 \def (f_equal T K (\lambda (e: 
-T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k0 | 
-(TLRef _) \Rightarrow k0 | (THead k1 _ _) \Rightarrow k1])) (THead k0 u4 t7) 
-(THead (Bind b) u1 t3) H30) in (eq_ind K (Bind b) (\lambda (k1: K).((eq T u4 
-u1) \to ((eq T t7 t3) \to ((eq T (THead k1 u5 t8) t6) \to ((pr0 u4 u5) \to 
-((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 
-t6) t9))))))))) (\lambda (H35: (eq T u4 u1)).(eq_ind T u1 (\lambda (t9: 
-T).((eq T t7 t3) \to ((eq T (THead (Bind b) u5 t8) t6) \to ((pr0 t9 u5) \to 
-((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat 
-Appl) u3 t6) t10)))))))) (\lambda (H36: (eq T t7 t3)).(eq_ind T t3 (\lambda 
-(t9: T).((eq T (THead (Bind b) u5 t8) t6) \to ((pr0 u1 u5) \to ((pr0 t9 t8) 
-\to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift 
-(S O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6) 
-t10))))))) (\lambda (H37: (eq T (THead (Bind b) u5 t8) t6)).(eq_ind T (THead 
-(Bind b) u5 t8) (\lambda (t9: T).((pr0 u1 u5) \to ((pr0 t3 t8) \to (ex2 T 
-(\lambda (t10: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O 
-v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t9) t10)))))) 
-(\lambda (H38: (pr0 u1 u5)).(\lambda (H39: (pr0 t3 t8)).(let H40 \def 
-(eq_ind_r T t5 (\lambda (t9: T).(eq T (THead (Flat Appl) u0 t9) t)) H26 
-(THead (Bind b) u1 t3) H23) in (let H41 \def (eq_ind_r T t (\lambda (t9: 
-T).(\forall (v: T).((tlt v t9) \to (\forall (t10: T).((pr0 v t10) \to 
-(\forall (t11: T).((pr0 v t11) \to (ex2 T (\lambda (t12: T).(pr0 t10 t12)) 
-(\lambda (t12: T).(pr0 t11 t12)))))))))) H (THead (Flat Appl) u0 (THead (Bind 
-b) u1 t3)) H40) in (let H42 \def (eq_ind T v1 (\lambda (t9: T).(pr0 t9 v2)) 
-H10 u0 H24) in (ex2_ind T (\lambda (t9: T).(pr0 v2 t9)) (\lambda (t9: T).(pr0 
-u3 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 
-(THead (Bind b) u5 t8)) t9))) (\lambda (x: T).(\lambda (H43: (pr0 v2 
-x)).(\lambda (H44: (pr0 u3 x)).(ex2_ind T (\lambda (t9: T).(pr0 t8 t9)) 
-(\lambda (t9: T).(pr0 t4 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 
+(f_equal T K (\lambda (e: T).(match e with [(TSort _) \Rightarrow (Flat Appl) 
+| (TLRef _) \Rightarrow (Flat Appl) | (THead k0 _ _) \Rightarrow k0])) (THead 
+(Flat Appl) v1 (THead (Bind b) u1 t3)) (THead k u0 t5) H20) in ((let H22 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow v1 | (TLRef 
+_) \Rightarrow v1 | (THead _ t7 _) \Rightarrow t7])) (THead (Flat Appl) v1 
+(THead (Bind b) u1 t3)) (THead k u0 t5) H20) in ((let H23 \def (f_equal T T 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow (THead (Bind b) u1 t3) | 
+(TLRef _) \Rightarrow (THead (Bind b) u1 t3) | (THead _ _ t7) \Rightarrow 
+t7])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead k u0 t5) H20) in 
+(\lambda (H24: (eq T v1 u0)).(\lambda (H25: (eq K (Flat Appl) k)).(eq_ind K 
+(Flat Appl) (\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 
+(THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 (THead 
+k0 u3 t6) t7)))) (let H26 \def (eq_ind_r K k (\lambda (k0: K).(eq T (THead k0 
+u0 t5) t)) H15 (Flat Appl) H25) in (let H27 \def (eq_ind_r T t5 (\lambda (t7: 
+T).(pr0 t7 t6)) H19 (THead (Bind b) u1 t3) H23) in (let H28 \def (match H27 
+with [(pr0_refl t7) \Rightarrow (\lambda (H28: (eq T t7 (THead (Bind b) u1 
+t3))).(\lambda (H29: (eq T t7 t6)).(eq_ind T (THead (Bind b) u1 t3) (\lambda 
+(t8: T).((eq T t8 t6) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 

 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead 
 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead 

-(Flat Appl) u3 (THead (Bind b) u5 t8)) t9))) (\lambda (x0: T).(\lambda (H45: 
-(pr0 t8 x0)).(\lambda (H46: (pr0 t4 x0)).(ex2_ind T (\lambda (t9: T).(pr0 u5 
-t9)) (\lambda (t9: T).(pr0 u2 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Bind 
-b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 
-(THead (Flat Appl) u3 (THead (Bind b) u5 t8)) t9))) (\lambda (x1: T).(\lambda 
-(H47: (pr0 u5 x1)).(\lambda (H48: (pr0 u2 x1)).(ex2_sym T (pr0 (THead (Flat 
-Appl) u3 (THead (Bind b) u5 t8))) (pr0 (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t4))) (pr0_confluence__pr0_cong_upsilon_cong b H9 u3 v2 x 
-H44 H43 t8 t4 x0 H45 H46 u5 u2 x1 H47 H48))))) (H41 u1 (tlt_trans (THead 
-(Bind b) u1 t3) u1 (THead (Flat Appl) u0 (THead (Bind b) u1 t3)) (tlt_head_sx 
-(Bind b) u1 t3) (tlt_head_dx (Flat Appl) u0 (THead (Bind b) u1 t3))) u5 H38 
-u2 H11))))) (H41 t3 (tlt_trans (THead (Bind b) u1 t3) t3 (THead (Flat Appl) 
-u0 (THead (Bind b) u1 t3)) (tlt_head_dx (Bind b) u1 t3) (tlt_head_dx (Flat 
-Appl) u0 (THead (Bind b) u1 t3))) t8 H39 t4 H12))))) (H41 u0 (tlt_head_sx 
-(Flat Appl) u0 (THead (Bind b) u1 t3)) v2 H42 u3 H18))))))) t6 H37)) t7 
-(sym_eq T t7 t3 H36))) u4 (sym_eq T u4 u1 H35))) k0 (sym_eq K k0 (Bind b) 
-H34))) H33)) H32)) H31 H28 H29))) | (pr0_beta u v0 v3 H28 t7 t8 H29) 
-\Rightarrow (\lambda (H30: (eq T (THead (Flat Appl) v0 (THead (Bind Abst) u 
-t7)) (THead (Bind b) u1 t3))).(\lambda (H31: (eq T (THead (Bind Abbr) v3 t8) 
-t6)).((let H32 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u t7)) 
-(\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow 
-(match k0 in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False 
-| (Flat _) \Rightarrow True])])) I (THead (Bind b) u1 t3) H30) in (False_ind 
-((eq T (THead (Bind Abbr) v3 t8) t6) \to ((pr0 v0 v3) \to ((pr0 t7 t8) \to 
-(ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
-O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9)))))) 
-H32)) H31 H28 H29))) | (pr0_upsilon b0 H28 v0 v3 H29 u4 u5 H30 t7 t8 H31) 
-\Rightarrow (\lambda (H32: (eq T (THead (Flat Appl) v0 (THead (Bind b0) u4 
-t7)) (THead (Bind b) u1 t3))).(\lambda (H33: (eq T (THead (Bind b0) u5 (THead 
-(Flat Appl) (lift (S O) O v3) t8)) t6)).((let H34 \def (eq_ind T (THead (Flat 
-Appl) v0 (THead (Bind b0) u4 t7)) (\lambda (e: T).(match e in T return 
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda 
-(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow 
-True])])) I (THead (Bind b) u1 t3) H32) in (False_ind ((eq T (THead (Bind b0) 
-u5 (THead (Flat Appl) (lift (S O) O v3) t8)) t6) \to ((not (eq B b0 Abst)) 
-\to ((pr0 v0 v3) \to ((pr0 u4 u5) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: 
+(Flat Appl) u3 t6) t9))))) (\lambda (H30: (eq T (THead (Bind b) u1 t3) 
+t6)).(eq_ind T (THead (Bind b) u1 t3) (\lambda (t8: T).(ex2 T (\lambda (t9: 

 T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) 
 T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) 

-(\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9)))))))) H34)) H33 H28 H29 
-H30 H31))) | (pr0_delta u4 u5 H28 t7 t8 H29 w H30) \Rightarrow (\lambda (H31: 
-(eq T (THead (Bind Abbr) u4 t7) (THead (Bind b) u1 t3))).(\lambda (H32: (eq T 
-(THead (Bind Abbr) u5 w) t6)).((let H33 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t7 | 
-(TLRef _) \Rightarrow t7 | (THead _ _ t9) \Rightarrow t9])) (THead (Bind 
-Abbr) u4 t7) (THead (Bind b) u1 t3) H31) in ((let H34 \def (f_equal T T 
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
+(\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t8) t9)))) (let H31 \def 
+(eq_ind_r T t5 (\lambda (t8: T).(eq T (THead (Flat Appl) u0 t8) t)) H26 
+(THead (Bind b) u1 t3) H23) in (let H32 \def (eq_ind_r T t (\lambda (t8: 
+T).(\forall (v: T).((tlt v t8) \to (\forall (t9: T).((pr0 v t9) \to (\forall 
+(t10: T).((pr0 v t10) \to (ex2 T (\lambda (t11: T).(pr0 t9 t11)) (\lambda 
+(t11: T).(pr0 t10 t11)))))))))) H (THead (Flat Appl) u0 (THead (Bind b) u1 
+t3)) H31) in (let H33 \def (eq_ind T v1 (\lambda (t8: T).(pr0 t8 v2)) H10 u0 
+H24) in (ex2_ind T (\lambda (t8: T).(pr0 v2 t8)) (\lambda (t8: T).(pr0 u3 
+t8)) (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift 
+(S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 (THead (Flat Appl) u3 (THead 
+(Bind b) u1 t3)) t8))) (\lambda (x: T).(\lambda (H34: (pr0 v2 x)).(\lambda 
+(H35: (pr0 u3 x)).(ex2_sym T (pr0 (THead (Flat Appl) u3 (THead (Bind b) u1 
+t3))) (pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) 
+(pr0_confluence__pr0_cong_upsilon_refl b H9 u1 u2 H11 t3 t4 H12 u3 v2 x H35 
+H34))))) (H32 u0 (tlt_head_sx (Flat Appl) u0 (THead (Bind b) u1 t3)) v2 H33 
+u3 H18))))) t6 H30)) t7 (sym_eq T t7 (THead (Bind b) u1 t3) H28) H29))) | 
+(pr0_comp u4 u5 H28 t7 t8 H29 k0) \Rightarrow (\lambda (H30: (eq T (THead k0 
+u4 t7) (THead (Bind b) u1 t3))).(\lambda (H31: (eq T (THead k0 u5 t8) 
+t6)).((let H32 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow t7 | (TLRef _) \Rightarrow t7 | (THead _ _ t9) \Rightarrow t9])) 
+(THead k0 u4 t7) (THead (Bind b) u1 t3) H30) in ((let H33 \def (f_equal T T 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow u4 | (TLRef _) 
+\Rightarrow u4 | (THead _ t9 _) \Rightarrow t9])) (THead k0 u4 t7) (THead 
+(Bind b) u1 t3) H30) in ((let H34 \def (f_equal T K (\lambda (e: T).(match e 
+with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _) 
+\Rightarrow k1])) (THead k0 u4 t7) (THead (Bind b) u1 t3) H30) in (eq_ind K 
+(Bind b) (\lambda (k1: K).((eq T u4 u1) \to ((eq T t7 t3) \to ((eq T (THead 
+k1 u5 t8) t6) \to ((pr0 u4 u5) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: 
+T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) 
+(\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9))))))))) (\lambda (H35: 
+(eq T u4 u1)).(eq_ind T u1 (\lambda (t9: T).((eq T t7 t3) \to ((eq T (THead 
+(Bind b) u5 t8) t6) \to ((pr0 t9 u5) \to ((pr0 t7 t8) \to (ex2 T (\lambda 
+(t10: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
+t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6) t10)))))))) (\lambda 
+(H36: (eq T t7 t3)).(eq_ind T t3 (\lambda (t9: T).((eq T (THead (Bind b) u5 
+t8) t6) \to ((pr0 u1 u5) \to ((pr0 t9 t8) \to (ex2 T (\lambda (t10: T).(pr0 
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t10)) (\lambda 
+(t10: T).(pr0 (THead (Flat Appl) u3 t6) t10))))))) (\lambda (H37: (eq T 
+(THead (Bind b) u5 t8) t6)).(eq_ind T (THead (Bind b) u5 t8) (\lambda (t9: 
+T).((pr0 u1 u5) \to ((pr0 t3 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t10)) (\lambda (t10: 
+T).(pr0 (THead (Flat Appl) u3 t9) t10)))))) (\lambda (H38: (pr0 u1 
+u5)).(\lambda (H39: (pr0 t3 t8)).(let H40 \def (eq_ind_r T t5 (\lambda (t9: 
+T).(eq T (THead (Flat Appl) u0 t9) t)) H26 (THead (Bind b) u1 t3) H23) in 
+(let H41 \def (eq_ind_r T t (\lambda (t9: T).(\forall (v: T).((tlt v t9) \to 
+(\forall (t10: T).((pr0 v t10) \to (\forall (t11: T).((pr0 v t11) \to (ex2 T 
+(\lambda (t12: T).(pr0 t10 t12)) (\lambda (t12: T).(pr0 t11 t12)))))))))) H 
+(THead (Flat Appl) u0 (THead (Bind b) u1 t3)) H40) in (let H42 \def (eq_ind T 
+v1 (\lambda (t9: T).(pr0 t9 v2)) H10 u0 H24) in (ex2_ind T (\lambda (t9: 
+T).(pr0 v2 t9)) (\lambda (t9: T).(pr0 u3 t9)) (ex2 T (\lambda (t9: T).(pr0 
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda 
+(t9: T).(pr0 (THead (Flat Appl) u3 (THead (Bind b) u5 t8)) t9))) (\lambda (x: 
+T).(\lambda (H43: (pr0 v2 x)).(\lambda (H44: (pr0 u3 x)).(ex2_ind T (\lambda 
+(t9: T).(pr0 t8 t9)) (\lambda (t9: T).(pr0 t4 t9)) (ex2 T (\lambda (t9: 
+T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) 
+(\lambda (t9: T).(pr0 (THead (Flat Appl) u3 (THead (Bind b) u5 t8)) t9))) 
+(\lambda (x0: T).(\lambda (H45: (pr0 t8 x0)).(\lambda (H46: (pr0 t4 
+x0)).(ex2_ind T (\lambda (t9: T).(pr0 u5 t9)) (\lambda (t9: T).(pr0 u2 t9)) 
+(ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
+O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 (THead (Bind 
+b) u5 t8)) t9))) (\lambda (x1: T).(\lambda (H47: (pr0 u5 x1)).(\lambda (H48: 
+(pr0 u2 x1)).(ex2_sym T (pr0 (THead (Flat Appl) u3 (THead (Bind b) u5 t8))) 
+(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) 
+(pr0_confluence__pr0_cong_upsilon_cong b H9 u3 v2 x H44 H43 t8 t4 x0 H45 H46 
+u5 u2 x1 H47 H48))))) (H41 u1 (tlt_trans (THead (Bind b) u1 t3) u1 (THead 
+(Flat Appl) u0 (THead (Bind b) u1 t3)) (tlt_head_sx (Bind b) u1 t3) 
+(tlt_head_dx (Flat Appl) u0 (THead (Bind b) u1 t3))) u5 H38 u2 H11))))) (H41 
+t3 (tlt_trans (THead (Bind b) u1 t3) t3 (THead (Flat Appl) u0 (THead (Bind b) 
+u1 t3)) (tlt_head_dx (Bind b) u1 t3) (tlt_head_dx (Flat Appl) u0 (THead (Bind 
+b) u1 t3))) t8 H39 t4 H12))))) (H41 u0 (tlt_head_sx (Flat Appl) u0 (THead 
+(Bind b) u1 t3)) v2 H42 u3 H18))))))) t6 H37)) t7 (sym_eq T t7 t3 H36))) u4 
+(sym_eq T u4 u1 H35))) k0 (sym_eq K k0 (Bind b) H34))) H33)) H32)) H31 H28 
+H29))) | (pr0_beta u v0 v3 H28 t7 t8 H29) \Rightarrow (\lambda (H30: (eq T 
+(THead (Flat Appl) v0 (THead (Bind Abst) u t7)) (THead (Bind b) u1 
+t3))).(\lambda (H31: (eq T (THead (Bind Abbr) v3 t8) t6)).((let H32 \def 
+(eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u t7)) (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
+(THead k0 _ _) \Rightarrow (match k0 with [(Bind _) \Rightarrow False | (Flat 
+_) \Rightarrow True])])) I (THead (Bind b) u1 t3) H30) in (False_ind ((eq T 
+(THead (Bind Abbr) v3 t8) t6) \to ((pr0 v0 v3) \to ((pr0 t7 t8) \to (ex2 T 
+(\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
+t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9)))))) H32)) H31 
+H28 H29))) | (pr0_upsilon b0 H28 v0 v3 H29 u4 u5 H30 t7 t8 H31) \Rightarrow 
+(\lambda (H32: (eq T (THead (Flat Appl) v0 (THead (Bind b0) u4 t7)) (THead 
+(Bind b) u1 t3))).(\lambda (H33: (eq T (THead (Bind b0) u5 (THead (Flat Appl) 
+(lift (S O) O v3) t8)) t6)).((let H34 \def (eq_ind T (THead (Flat Appl) v0 
+(THead (Bind b0) u4 t7)) (\lambda (e: T).(match e with [(TSort _) \Rightarrow 
+False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 
+with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead 
+(Bind b) u1 t3) H32) in (False_ind ((eq T (THead (Bind b0) u5 (THead (Flat 
+Appl) (lift (S O) O v3) t8)) t6) \to ((not (eq B b0 Abst)) \to ((pr0 v0 v3) 
+\to ((pr0 u4 u5) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: 
+T).(pr0 (THead (Flat Appl) u3 t6) t9)))))))) H34)) H33 H28 H29 H30 H31))) | 
+(pr0_delta u4 u5 H28 t7 t8 H29 w H30) \Rightarrow (\lambda (H31: (eq T (THead 
+(Bind Abbr) u4 t7) (THead (Bind b) u1 t3))).(\lambda (H32: (eq T (THead (Bind 
+Abbr) u5 w) t6)).((let H33 \def (f_equal T T (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow t7 | (TLRef _) \Rightarrow t7 | (THead _ _ t9) 
+\Rightarrow t9])) (THead (Bind Abbr) u4 t7) (THead (Bind b) u1 t3) H31) in 
+((let H34 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 

 \Rightarrow u4 | (TLRef _) \Rightarrow u4 | (THead _ t9 _) \Rightarrow t9])) 
 (THead (Bind Abbr) u4 t7) (THead (Bind b) u1 t3) H31) in ((let H35 \def 
 \Rightarrow u4 | (TLRef _) \Rightarrow u4 | (THead _ t9 _) \Rightarrow t9])) 
 (THead (Bind Abbr) u4 t7) (THead (Bind b) u1 t3) H31) in ((let H35 \def 

-(f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with 
-[(TSort _) \Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k0 _ _) 
-\Rightarrow (match k0 in K return (\lambda (_: K).B) with [(Bind b0) 
-\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (THead (Bind Abbr) u4 t7) 
+(f_equal T B (\lambda (e: T).(match e with [(TSort _) \Rightarrow Abbr | 
+(TLRef _) \Rightarrow Abbr | (THead k0 _ _) \Rightarrow (match k0 with [(Bind 
+b0) \Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (THead (Bind Abbr) u4 t7) 

 (THead (Bind b) u1 t3) H31) in (eq_ind B Abbr (\lambda (b0: B).((eq T u4 u1) 
 \to ((eq T t7 t3) \to ((eq T (THead (Bind Abbr) u5 w) t6) \to ((pr0 u4 u5) 
 \to ((pr0 t7 t8) \to ((subst0 O u5 t8 w) \to (ex2 T (\lambda (t9: T).(pr0 
 (THead (Bind b) u1 t3) H31) in (eq_ind B Abbr (\lambda (b0: B).((eq T u4 u1) 
 \to ((eq T t7 t3) \to ((eq T (THead (Bind Abbr) u5 w) t6) \to ((pr0 u4 u5) 
 \to ((pr0 t7 t8) \to ((subst0 O u5 t8 w) \to (ex2 T (\lambda (t9: T).(pr0 


@@ -1660,326 +1534,299 @@ H18))))))))))) t6 H38)) t7 (sym_eq T t7 t3 H37))) u4 (sym_eq T u4 u1 H36))) b
 H35)) H34)) H33)) H32 H28 H29 H30))) | (pr0_zeta b0 H28 t7 t8 H29 u) 
 \Rightarrow (\lambda (H30: (eq T (THead (Bind b0) u (lift (S O) O t7)) (THead 
 (Bind b) u1 t3))).(\lambda (H31: (eq T t8 t6)).((let H32 \def (f_equal T T 
 H35)) H34)) H33)) H32 H28 H29 H30))) | (pr0_zeta b0 H28 t7 t8 H29 u) 
 \Rightarrow (\lambda (H30: (eq T (THead (Bind b0) u (lift (S O) O t7)) (THead 
 (Bind b) u1 t3))).(\lambda (H31: (eq T t8 t6)).((let H32 \def (f_equal T T 

-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t9: T) on t9: T 
-\def (match t9 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow 
-(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) 
-| (THead k0 u4 t10) \Rightarrow (THead k0 (lref_map f d u4) (lref_map f (s k0 
-d) t10))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (TLRef _) 
-\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t9: T) on t9: T 
-\def (match t9 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow 
-(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) 
-| (THead k0 u4 t10) \Rightarrow (THead k0 (lref_map f d u4) (lref_map f (s k0 
-d) t10))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (THead _ _ 
-t9) \Rightarrow t9])) (THead (Bind b0) u (lift (S O) O t7)) (THead (Bind b) 
-u1 t3) H30) in ((let H33 \def (f_equal T T (\lambda (e: T).(match e in T 
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) 
-\Rightarrow u | (THead _ t9 _) \Rightarrow t9])) (THead (Bind b0) u (lift (S 
-O) O t7)) (THead (Bind b) u1 t3) H30) in ((let H34 \def (f_equal T B (\lambda 
-(e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b0 
-| (TLRef _) \Rightarrow b0 | (THead k0 _ _) \Rightarrow (match k0 in K return 
-(\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow 
-b0])])) (THead (Bind b0) u (lift (S O) O t7)) (THead (Bind b) u1 t3) H30) in 
-(eq_ind B b (\lambda (b1: B).((eq T u u1) \to ((eq T (lift (S O) O t7) t3) 
-\to ((eq T t8 t6) \to ((not (eq B b1 Abst)) \to ((pr0 t7 t8) \to (ex2 T 
-(\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
-t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9))))))))) 
-(\lambda (H35: (eq T u u1)).(eq_ind T u1 (\lambda (_: T).((eq T (lift (S O) O 
-t7) t3) \to ((eq T t8 t6) \to ((not (eq B b Abst)) \to ((pr0 t7 t8) \to (ex2 
-T (\lambda (t10: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O 
-v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6) t10)))))))) 
-(\lambda (H36: (eq T (lift (S O) O t7) t3)).(eq_ind T (lift (S O) O t7) 
-(\lambda (_: T).((eq T t8 t6) \to ((not (eq B b Abst)) \to ((pr0 t7 t8) \to 
-(ex2 T (\lambda (t10: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
-O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6) 
-t10))))))) (\lambda (H37: (eq T t8 t6)).(eq_ind T t6 (\lambda (t9: T).((not 
-(eq B b Abst)) \to ((pr0 t7 t9) \to (ex2 T (\lambda (t10: T).(pr0 (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t10)) (\lambda (t10: 
-T).(pr0 (THead (Flat Appl) u3 t6) t10)))))) (\lambda (H38: (not (eq B b 
-Abst))).(\lambda (H39: (pr0 t7 t6)).(let H40 \def (eq_ind_r T t3 (\lambda 
-(t9: T).(eq T (THead (Bind b) u1 t9) t5)) H23 (lift (S O) O t7) H36) in (let 
-H41 \def (eq_ind_r T t5 (\lambda (t9: T).(eq T (THead (Flat Appl) u0 t9) t)) 
-H26 (THead (Bind b) u1 (lift (S O) O t7)) H40) in (let H42 \def (eq_ind_r T t 
-(\lambda (t9: T).(\forall (v: T).((tlt v t9) \to (\forall (t10: T).((pr0 v 
-t10) \to (\forall (t11: T).((pr0 v t11) \to (ex2 T (\lambda (t12: T).(pr0 t10 
-t12)) (\lambda (t12: T).(pr0 t11 t12)))))))))) H (THead (Flat Appl) u0 (THead 
-(Bind b) u1 (lift (S O) O t7))) H41) in (let H43 \def (eq_ind_r T t3 (\lambda 
-(t9: T).(pr0 t9 t4)) H12 (lift (S O) O t7) H36) in (ex2_ind T (\lambda (t9: 
-T).(eq T t4 (lift (S O) O t9))) (\lambda (t9: T).(pr0 t7 t9)) (ex2 T (\lambda 
-(t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
-t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9))) (\lambda (x: 
-T).(\lambda (H44: (eq T t4 (lift (S O) O x))).(\lambda (H45: (pr0 t7 
-x)).(eq_ind_r T (lift (S O) O x) (\lambda (t9: T).(ex2 T (\lambda (t10: 
-T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t9)) t10)) 
-(\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6) t10)))) (let H46 \def 
-(eq_ind T v1 (\lambda (t9: T).(pr0 t9 v2)) H10 u0 H24) in (ex2_ind T (\lambda 
-(t9: T).(pr0 v2 t9)) (\lambda (t9: T).(pr0 u3 t9)) (ex2 T (\lambda (t9: 
-T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) O 
-x))) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9))) (\lambda (x0: 
-T).(\lambda (H47: (pr0 v2 x0)).(\lambda (H48: (pr0 u3 x0)).(ex2_ind T 
-(\lambda (t9: T).(pr0 x t9)) (\lambda (t9: T).(pr0 t6 t9)) (ex2 T (\lambda 
-(t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (lift (S 
-O) O x))) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9))) (\lambda 
-(x1: T).(\lambda (H49: (pr0 x x1)).(\lambda (H50: (pr0 t6 x1)).(ex2_sym T 
-(pr0 (THead (Flat Appl) u3 t6)) (pr0 (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) (lift (S O) O x)))) (pr0_confluence__pr0_cong_upsilon_zeta 
-b H38 u1 u2 H11 u3 v2 x0 H48 H47 x t6 x1 H49 H50))))) (H42 t7 (tlt_trans 
-(THead (Bind b) u1 (lift (S O) O t7)) t7 (THead (Flat Appl) u0 (THead (Bind 
-b) u1 (lift (S O) O t7))) (lift_tlt_dx (Bind b) u1 t7 (S O) O) (tlt_head_dx 
-(Flat Appl) u0 (THead (Bind b) u1 (lift (S O) O t7)))) x H45 t6 H39))))) (H42 
-u0 (tlt_head_sx (Flat Appl) u0 (THead (Bind b) u1 (lift (S O) O t7))) v2 H46 
-u3 H18))) t4 H44)))) (pr0_gen_lift t7 t4 (S O) O H43)))))))) t8 (sym_eq T t8 
-t6 H37))) t3 H36)) u (sym_eq T u u1 H35))) b0 (sym_eq B b0 b H34))) H33)) 
-H32)) H31 H28 H29))) | (pr0_tau t7 t8 H28 u) \Rightarrow (\lambda (H29: (eq T 
-(THead (Flat Cast) u t7) (THead (Bind b) u1 t3))).(\lambda (H30: (eq T t8 
-t6)).((let H31 \def (eq_ind T (THead (Flat Cast) u t7) (\lambda (e: T).(match 
-e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | 
-(TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K 
-return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) 
-\Rightarrow True])])) I (THead (Bind b) u1 t3) H29) in (False_ind ((eq T t8 
-t6) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow (lref_map (\lambda (x: 
+nat).(plus x (S O))) O t7) | (TLRef _) \Rightarrow (lref_map (\lambda (x: 
+nat).(plus x (S O))) O t7) | (THead _ _ t9) \Rightarrow t9])) (THead (Bind 
+b0) u (lift (S O) O t7)) (THead (Bind b) u1 t3) H30) in ((let H33 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u | (TLRef 
+_) \Rightarrow u | (THead _ t9 _) \Rightarrow t9])) (THead (Bind b0) u (lift 
+(S O) O t7)) (THead (Bind b) u1 t3) H30) in ((let H34 \def (f_equal T B 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow b0 | (TLRef _) 
+\Rightarrow b0 | (THead k0 _ _) \Rightarrow (match k0 with [(Bind b1) 
+\Rightarrow b1 | (Flat _) \Rightarrow b0])])) (THead (Bind b0) u (lift (S O) 
+O t7)) (THead (Bind b) u1 t3) H30) in (eq_ind B b (\lambda (b1: B).((eq T u 
+u1) \to ((eq T (lift (S O) O t7) t3) \to ((eq T t8 t6) \to ((not (eq B b1 
+Abst)) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 
+(THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead 
+(Flat Appl) u3 t6) t9))))))))) (\lambda (H35: (eq T u u1)).(eq_ind T u1 
+(\lambda (_: T).((eq T (lift (S O) O t7) t3) \to ((eq T t8 t6) \to ((not (eq 
+B b Abst)) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind b) 
+u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 
+(THead (Flat Appl) u3 t6) t10)))))))) (\lambda (H36: (eq T (lift (S O) O t7) 
+t3)).(eq_ind T (lift (S O) O t7) (\lambda (_: T).((eq T t8 t6) \to ((not (eq 
+B b Abst)) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind b) 
+u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 
+(THead (Flat Appl) u3 t6) t10))))))) (\lambda (H37: (eq T t8 t6)).(eq_ind T 
+t6 (\lambda (t9: T).((not (eq B b Abst)) \to ((pr0 t7 t9) \to (ex2 T (\lambda 
+(t10: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
+t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6) t10)))))) (\lambda 
+(H38: (not (eq B b Abst))).(\lambda (H39: (pr0 t7 t6)).(let H40 \def 
+(eq_ind_r T t3 (\lambda (t9: T).(eq T (THead (Bind b) u1 t9) t5)) H23 (lift 
+(S O) O t7) H36) in (let H41 \def (eq_ind_r T t5 (\lambda (t9: T).(eq T 
+(THead (Flat Appl) u0 t9) t)) H26 (THead (Bind b) u1 (lift (S O) O t7)) H40) 
+in (let H42 \def (eq_ind_r T t (\lambda (t9: T).(\forall (v: T).((tlt v t9) 
+\to (\forall (t10: T).((pr0 v t10) \to (\forall (t11: T).((pr0 v t11) \to 
+(ex2 T (\lambda (t12: T).(pr0 t10 t12)) (\lambda (t12: T).(pr0 t11 
+t12)))))))))) H (THead (Flat Appl) u0 (THead (Bind b) u1 (lift (S O) O t7))) 
+H41) in (let H43 \def (eq_ind_r T t3 (\lambda (t9: T).(pr0 t9 t4)) H12 (lift 
+(S O) O t7) H36) in (ex2_ind T (\lambda (t9: T).(eq T t4 (lift (S O) O t9))) 
+(\lambda (t9: T).(pr0 t7 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 

 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead 
 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead 

-(Flat Appl) u3 t6) t9))))) H31)) H30 H28)))]) in (H28 (refl_equal T (THead 
-(Bind b) u1 t3)) (refl_equal T t6))))) k H25)))) H22)) H21))))) t2 H17)) t 
-H15 H16 H13 H14))) | (pr0_beta u v0 v3 H13 t5 t6 H14) \Rightarrow (\lambda 
-(H15: (eq T (THead (Flat Appl) v0 (THead (Bind Abst) u t5)) t)).(\lambda 
-(H16: (eq T (THead (Bind Abbr) v3 t6) t2)).(eq_ind T (THead (Flat Appl) v0 
-(THead (Bind Abst) u t5)) (\lambda (_: T).((eq T (THead (Bind Abbr) v3 t6) 
-t2) \to ((pr0 v0 v3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead 
+(Flat Appl) u3 t6) t9))) (\lambda (x: T).(\lambda (H44: (eq T t4 (lift (S O) 
+O x))).(\lambda (H45: (pr0 t7 x)).(eq_ind_r T (lift (S O) O x) (\lambda (t9: 
+T).(ex2 T (\lambda (t10: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift 
+(S O) O v2) t9)) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6) 
+t10)))) (let H46 \def (eq_ind T v1 (\lambda (t9: T).(pr0 t9 v2)) H10 u0 H24) 
+in (ex2_ind T (\lambda (t9: T).(pr0 v2 t9)) (\lambda (t9: T).(pr0 u3 t9)) 
+(ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
+O) O v2) (lift (S O) O x))) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 
+t6) t9))) (\lambda (x0: T).(\lambda (H47: (pr0 v2 x0)).(\lambda (H48: (pr0 u3 
+x0)).(ex2_ind T (\lambda (t9: T).(pr0 x t9)) (\lambda (t9: T).(pr0 t6 t9)) 
+(ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
+O) O v2) (lift (S O) O x))) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 
+t6) t9))) (\lambda (x1: T).(\lambda (H49: (pr0 x x1)).(\lambda (H50: (pr0 t6 
+x1)).(ex2_sym T (pr0 (THead (Flat Appl) u3 t6)) (pr0 (THead (Bind b) u2 
+(THead (Flat Appl) (lift (S O) O v2) (lift (S O) O x)))) 
+(pr0_confluence__pr0_cong_upsilon_zeta b H38 u1 u2 H11 u3 v2 x0 H48 H47 x t6 
+x1 H49 H50))))) (H42 t7 (tlt_trans (THead (Bind b) u1 (lift (S O) O t7)) t7 
+(THead (Flat Appl) u0 (THead (Bind b) u1 (lift (S O) O t7))) (lift_tlt_dx 
+(Bind b) u1 t7 (S O) O) (tlt_head_dx (Flat Appl) u0 (THead (Bind b) u1 (lift 
+(S O) O t7)))) x H45 t6 H39))))) (H42 u0 (tlt_head_sx (Flat Appl) u0 (THead 
+(Bind b) u1 (lift (S O) O t7))) v2 H46 u3 H18))) t4 H44)))) (pr0_gen_lift t7 
+t4 (S O) O H43)))))))) t8 (sym_eq T t8 t6 H37))) t3 H36)) u (sym_eq T u u1 
+H35))) b0 (sym_eq B b0 b H34))) H33)) H32)) H31 H28 H29))) | (pr0_tau t7 t8 
+H28 u) \Rightarrow (\lambda (H29: (eq T (THead (Flat Cast) u t7) (THead (Bind 
+b) u1 t3))).(\lambda (H30: (eq T t8 t6)).((let H31 \def (eq_ind T (THead 
+(Flat Cast) u t7) (\lambda (e: T).(match e with [(TSort _) \Rightarrow False 
+| (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 with 
+[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind 
+b) u1 t3) H29) in (False_ind ((eq T t8 t6) \to ((pr0 t7 t8) \to (ex2 T 
+(\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
+t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9))))) H31)) H30 
+H28)))]) in (H28 (refl_equal T (THead (Bind b) u1 t3)) (refl_equal T t6))))) 
+k H25)))) H22)) H21))))) t2 H17)) t H15 H16 H13 H14))) | (pr0_beta u v0 v3 
+H13 t5 t6 H14) \Rightarrow (\lambda (H15: (eq T (THead (Flat Appl) v0 (THead 
+(Bind Abst) u t5)) t)).(\lambda (H16: (eq T (THead (Bind Abbr) v3 t6) 
+t2)).(eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u t5)) (\lambda (_: 
+T).((eq T (THead (Bind Abbr) v3 t6) t2) \to ((pr0 v0 v3) \to ((pr0 t5 t6) \to 
+(ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
+O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H17: (eq T 
+(THead (Bind Abbr) v3 t6) t2)).(eq_ind T (THead (Bind Abbr) v3 t6) (\lambda 
+(t7: T).((pr0 v0 v3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead 

 (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: 
 (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: 

-T).(pr0 t2 t8))))))) (\lambda (H17: (eq T (THead (Bind Abbr) v3 t6) 
-t2)).(eq_ind T (THead (Bind Abbr) v3 t6) (\lambda (t7: T).((pr0 v0 v3) \to 
-((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda 
-(_: (pr0 v0 v3)).(\lambda (_: (pr0 t5 t6)).(let H20 \def (eq_ind_r T t 
-(\lambda (t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) t7)) H6 
-(THead (Flat Appl) v0 (THead (Bind Abst) u t5)) H15) in (let H21 \def 
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t7 _) 
-\Rightarrow t7])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat 
-Appl) v0 (THead (Bind Abst) u t5)) H20) in ((let H22 \def (f_equal T B 
-(\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) 
-\Rightarrow b | (TLRef _) \Rightarrow b | (THead _ _ t7) \Rightarrow (match 
-t7 in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | (TLRef _) 
-\Rightarrow b | (THead k _ _) \Rightarrow (match k in K return (\lambda (_: 
-K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b])])])) (THead 
-(Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) v0 (THead (Bind 
-Abst) u t5)) H20) in ((let H23 \def (f_equal T T (\lambda (e: T).(match e in 
-T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) 
-\Rightarrow u1 | (THead _ _ t7) \Rightarrow (match t7 in T return (\lambda 
-(_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead 
-_ t8 _) \Rightarrow t8])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) 
-(THead (Flat Appl) v0 (THead (Bind Abst) u t5)) H20) in ((let H24 \def 
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7) 
-\Rightarrow (match t7 in T return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t8) \Rightarrow 
+T).(pr0 t7 t8)))))) (\lambda (_: (pr0 v0 v3)).(\lambda (_: (pr0 t5 t6)).(let 
+H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1 (THead 
+(Bind b) u1 t3)) t7)) H6 (THead (Flat Appl) v0 (THead (Bind Abst) u t5)) H15) 
+in (let H21 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t7 _) \Rightarrow t7])) 
+(THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) v0 (THead 
+(Bind Abst) u t5)) H20) in ((let H22 \def (f_equal T B (\lambda (e: T).(match 
+e with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead _ _ t7) 
+\Rightarrow (match t7 with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b 
+| (THead k _ _) \Rightarrow (match k with [(Bind b0) \Rightarrow b0 | (Flat 
+_) \Rightarrow b])])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead 
+(Flat Appl) v0 (THead (Bind Abst) u t5)) H20) in ((let H23 \def (f_equal T T 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow u1 | (TLRef _) 
+\Rightarrow u1 | (THead _ _ t7) \Rightarrow (match t7 with [(TSort _) 
+\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t8 _) \Rightarrow 

 t8])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) v0 
 t8])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) v0 

-(THead (Bind Abst) u t5)) H20) in (\lambda (_: (eq T u1 u)).(\lambda (H26: 
-(eq B b Abst)).(\lambda (H27: (eq T v1 v0)).(let H28 \def (eq_ind_r T t 
-(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) 
-\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) 
-(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Appl) v0 (THead (Bind 
-Abst) u t5)) H15) in (let H29 \def (eq_ind T v1 (\lambda (t7: T).(pr0 t7 v2)) 
-H10 v0 H27) in (eq_ind_r B Abst (\lambda (b0: B).(ex2 T (\lambda (t7: T).(pr0 
-(THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda 
-(t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7)))) (let H30 \def (eq_ind B b 
-(\lambda (b0: B).(not (eq B b0 Abst))) H9 Abst H26) in (let H31 \def (match 
-(H30 (refl_equal B Abst)) in False return (\lambda (_: False).(ex2 T (\lambda 
-(t7: T).(pr0 (THead (Bind Abst) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
-t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7)))) with []) in H31)) 
-b H26))))))) H23)) H22)) H21))))) t2 H17)) t H15 H16 H13 H14))) | 
-(pr0_upsilon b0 H13 v0 v3 H14 u0 u3 H15 t5 t6 H16) \Rightarrow (\lambda (H17: 
-(eq T (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) t)).(\lambda (H18: (eq T 
-(THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) t2)).(eq_ind T 
-(THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) (\lambda (_: T).((eq T (THead 
-(Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) t2) \to ((not (eq B b0 
-Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda 
-(t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
-t8)) (\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H19: (eq T (THead (Bind 
-b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) t2)).(eq_ind T (THead (Bind 
-b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) (\lambda (t7: T).((not (eq B 
-b0 Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T 
-(\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
-t4)) t8)) (\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda (_: (not (eq B b0 
-Abst))).(\lambda (H21: (pr0 v0 v3)).(\lambda (H22: (pr0 u0 u3)).(\lambda 
-(H23: (pr0 t5 t6)).(let H24 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead 
-(Flat Appl) v1 (THead (Bind b) u1 t3)) t7)) H6 (THead (Flat Appl) v0 (THead 
-(Bind b0) u0 t5)) H17) in (let H25 \def (f_equal T T (\lambda (e: T).(match e 
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) 
+(THead (Bind Abst) u t5)) H20) in ((let H24 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | 
+(THead _ _ t7) \Rightarrow (match t7 with [(TSort _) \Rightarrow t3 | (TLRef 
+_) \Rightarrow t3 | (THead _ _ t8) \Rightarrow t8])])) (THead (Flat Appl) v1 
+(THead (Bind b) u1 t3)) (THead (Flat Appl) v0 (THead (Bind Abst) u t5)) H20) 
+in (\lambda (_: (eq T u1 u)).(\lambda (H26: (eq B b Abst)).(\lambda (H27: (eq 
+T v1 v0)).(let H28 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt 
+v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to 
+(ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 
+t10)))))))))) H (THead (Flat Appl) v0 (THead (Bind Abst) u t5)) H15) in (let 
+H29 \def (eq_ind T v1 (\lambda (t7: T).(pr0 t7 v2)) H10 v0 H27) in (eq_ind_r 
+B Abst (\lambda (b0: B).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind b0) u2 
+(THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 (THead 
+(Bind Abbr) v3 t6) t7)))) (let H30 \def (eq_ind B b (\lambda (b0: B).(not (eq 
+B b0 Abst))) H9 Abst H26) in (let H31 \def (match (H30 (refl_equal B Abst)) 
+in False with []) in H31)) b H26))))))) H23)) H22)) H21))))) t2 H17)) t H15 
+H16 H13 H14))) | (pr0_upsilon b0 H13 v0 v3 H14 u0 u3 H15 t5 t6 H16) 
+\Rightarrow (\lambda (H17: (eq T (THead (Flat Appl) v0 (THead (Bind b0) u0 
+t5)) t)).(\lambda (H18: (eq T (THead (Bind b0) u3 (THead (Flat Appl) (lift (S 
+O) O v3) t6)) t2)).(eq_ind T (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) 
+(\lambda (_: T).((eq T (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O 
+v3) t6)) t2) \to ((not (eq B b0 Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to 
+((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2 t8))))))))) 
+(\lambda (H19: (eq T (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) 
+t6)) t2)).(eq_ind T (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) 
+t6)) (\lambda (t7: T).((not (eq B b0 Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) 
+\to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead 
+(Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t7 t8)))))))) 
+(\lambda (_: (not (eq B b0 Abst))).(\lambda (H21: (pr0 v0 v3)).(\lambda (H22: 
+(pr0 u0 u3)).(\lambda (H23: (pr0 t5 t6)).(let H24 \def (eq_ind_r T t (\lambda 
+(t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) t7)) H6 (THead 
+(Flat Appl) v0 (THead (Bind b0) u0 t5)) H17) in (let H25 \def (f_equal T T 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow v1 | (TLRef _) 

 \Rightarrow v1 | (THead _ t7 _) \Rightarrow t7])) (THead (Flat Appl) v1 
 (THead (Bind b) u1 t3)) (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) H24) 
 \Rightarrow v1 | (THead _ t7 _) \Rightarrow t7])) (THead (Flat Appl) v1 
 (THead (Bind b) u1 t3)) (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) H24) 

-in ((let H26 \def (f_equal T B (\lambda (e: T).(match e in T return (\lambda 
-(_: T).B) with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead _ 
-_ t7) \Rightarrow (match t7 in T return (\lambda (_: T).B) with [(TSort _) 
-\Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k 
-in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) 
-\Rightarrow b])])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead 
-(Flat Appl) v0 (THead (Bind b0) u0 t5)) H24) in ((let H27 \def (f_equal T T 
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ _ t7) \Rightarrow (match 
-t7 in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) 
-\Rightarrow u1 | (THead _ t8 _) \Rightarrow t8])])) (THead (Flat Appl) v1 
+in ((let H26 \def (f_equal T B (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow b | (TLRef _) \Rightarrow b | (THead _ _ t7) \Rightarrow (match 
+t7 with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) 
+\Rightarrow (match k with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow 
+b])])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) v0 
+(THead (Bind b0) u0 t5)) H24) in ((let H27 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | 
+(THead _ _ t7) \Rightarrow (match t7 with [(TSort _) \Rightarrow u1 | (TLRef 
+_) \Rightarrow u1 | (THead _ t8 _) \Rightarrow t8])])) (THead (Flat Appl) v1 

 (THead (Bind b) u1 t3)) (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) H24) 
 (THead (Bind b) u1 t3)) (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) H24) 

-in ((let H28 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda 
-(_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead 
-_ _ t7) \Rightarrow (match t7 in T return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t8) \Rightarrow 
-t8])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) v0 
-(THead (Bind b0) u0 t5)) H24) in (\lambda (H29: (eq T u1 u0)).(\lambda (H30: 
-(eq B b b0)).(\lambda (H31: (eq T v1 v0)).(let H32 \def (eq_ind_r T t 
-(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) 
-\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) 
-(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Appl) v0 (THead (Bind 
-b0) u0 t5)) H17) in (let H33 \def (eq_ind T v1 (\lambda (t7: T).(pr0 t7 v2)) 
-H10 v0 H31) in (eq_ind_r B b0 (\lambda (b1: B).(ex2 T (\lambda (t7: T).(pr0 
-(THead (Bind b1) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda 
-(t7: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) 
-t7)))) (let H34 \def (eq_ind B b (\lambda (b1: B).(not (eq B b1 Abst))) H9 b0 
-H30) in (let H35 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H11 u0 H29) 
-in (let H36 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H12 t5 H28) in 
-(ex2_ind T (\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T 
-(\lambda (t7: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O 
+in ((let H28 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7) \Rightarrow (match 
+t7 with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t8) 
+\Rightarrow t8])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead 
+(Flat Appl) v0 (THead (Bind b0) u0 t5)) H24) in (\lambda (H29: (eq T u1 
+u0)).(\lambda (H30: (eq B b b0)).(\lambda (H31: (eq T v1 v0)).(let H32 \def 
+(eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: 
+T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: 
+T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Appl) 
+v0 (THead (Bind b0) u0 t5)) H17) in (let H33 \def (eq_ind T v1 (\lambda (t7: 
+T).(pr0 t7 v2)) H10 v0 H31) in (eq_ind_r B b0 (\lambda (b1: B).(ex2 T 
+(\lambda (t7: T).(pr0 (THead (Bind b1) u2 (THead (Flat Appl) (lift (S O) O 

 v2) t4)) t7)) (\lambda (t7: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) 
 v2) t4)) t7)) (\lambda (t7: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) 

-(lift (S O) O v3) t6)) t7))) (\lambda (x: T).(\lambda (H37: (pr0 t4 
-x)).(\lambda (H38: (pr0 t6 x)).(ex2_ind T (\lambda (t7: T).(pr0 u2 t7)) 
-(\lambda (t7: T).(pr0 u3 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b0) 
+(lift (S O) O v3) t6)) t7)))) (let H34 \def (eq_ind B b (\lambda (b1: B).(not 
+(eq B b1 Abst))) H9 b0 H30) in (let H35 \def (eq_ind T u1 (\lambda (t7: 
+T).(pr0 t7 u2)) H11 u0 H29) in (let H36 \def (eq_ind T t3 (\lambda (t7: 
+T).(pr0 t7 t4)) H12 t5 H28) in (ex2_ind T (\lambda (t7: T).(pr0 t4 t7)) 
+(\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b0) 

 u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 
 (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) t7))) (\lambda 
 u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 
 (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) t7))) (\lambda 

-(x0: T).(\lambda (H39: (pr0 u2 x0)).(\lambda (H40: (pr0 u3 x0)).(ex2_ind T 
-(\lambda (t7: T).(pr0 v2 t7)) (\lambda (t7: T).(pr0 v3 t7)) (ex2 T (\lambda 
+(x: T).(\lambda (H37: (pr0 t4 x)).(\lambda (H38: (pr0 t6 x)).(ex2_ind T 
+(\lambda (t7: T).(pr0 u2 t7)) (\lambda (t7: T).(pr0 u3 t7)) (ex2 T (\lambda 

 (t7: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
 t7)) (\lambda (t7: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) 
 (t7: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
 t7)) (\lambda (t7: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) 

-O v3) t6)) t7))) (\lambda (x1: T).(\lambda (H41: (pr0 v2 x1)).(\lambda (H42: 
-(pr0 v3 x1)).(pr0_confluence__pr0_upsilon_upsilon b0 H34 v2 v3 x1 H41 H42 u2 
-u3 x0 H39 H40 t4 t6 x H37 H38)))) (H32 v0 (tlt_head_sx (Flat Appl) v0 (THead 
-(Bind b0) u0 t5)) v2 H33 v3 H21))))) (H32 u0 (tlt_trans (THead (Bind b0) u0 
-t5) u0 (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) (tlt_head_sx (Bind b0) 
-u0 t5) (tlt_head_dx (Flat Appl) v0 (THead (Bind b0) u0 t5))) u2 H35 u3 
-H22))))) (H32 t5 (tlt_trans (THead (Bind b0) u0 t5) t5 (THead (Flat Appl) v0 
-(THead (Bind b0) u0 t5)) (tlt_head_dx (Bind b0) u0 t5) (tlt_head_dx (Flat 
-Appl) v0 (THead (Bind b0) u0 t5))) t4 H36 t6 H23))))) b H30))))))) H27)) 
-H26)) H25))))))) t2 H19)) t H17 H18 H13 H14 H15 H16))) | (pr0_delta u0 u3 H13 
-t5 t6 H14 w H15) \Rightarrow (\lambda (H16: (eq T (THead (Bind Abbr) u0 t5) 
-t)).(\lambda (H17: (eq T (THead (Bind Abbr) u3 w) t2)).(eq_ind T (THead (Bind 
-Abbr) u0 t5) (\lambda (_: T).((eq T (THead (Bind Abbr) u3 w) t2) \to ((pr0 u0 
-u3) \to ((pr0 t5 t6) \to ((subst0 O u3 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda 
-(t8: T).(pr0 t2 t8)))))))) (\lambda (H18: (eq T (THead (Bind Abbr) u3 w) 
-t2)).(eq_ind T (THead (Bind Abbr) u3 w) (\lambda (t7: T).((pr0 u0 u3) \to 
-((pr0 t5 t6) \to ((subst0 O u3 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: 
-T).(pr0 t7 t8))))))) (\lambda (_: (pr0 u0 u3)).(\lambda (_: (pr0 t5 
-t6)).(\lambda (_: (subst0 O u3 t6 w)).(let H22 \def (eq_ind_r T t (\lambda 
-(t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) t7)) H6 (THead 
-(Bind Abbr) u0 t5) H16) in (let H23 \def (eq_ind T (THead (Flat Appl) v1 
-(THead (Bind b) u1 t3)) (\lambda (ee: T).(match ee in T return (\lambda (_: 
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
-(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with 
-[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind 
-Abbr) u0 t5) H22) in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) 
-u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 
-(THead (Bind Abbr) u3 w) t7))) H23)))))) t2 H18)) t H16 H17 H13 H14 H15))) | 
-(pr0_zeta b0 H13 t5 t6 H14 u) \Rightarrow (\lambda (H15: (eq T (THead (Bind 
-b0) u (lift (S O) O t5)) t)).(\lambda (H16: (eq T t6 t2)).(eq_ind T (THead 
-(Bind b0) u (lift (S O) O t5)) (\lambda (_: T).((eq T t6 t2) \to ((not (eq B 
-b0 Abst)) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) 
-u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2 
-t8))))))) (\lambda (H17: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((not 
-(eq B b0 Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: 
-T).(pr0 t2 t8)))))) (\lambda (_: (not (eq B b0 Abst))).(\lambda (_: (pr0 t5 
-t2)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1 
-(THead (Bind b) u1 t3)) t7)) H6 (THead (Bind b0) u (lift (S O) O t5)) H15) in 
-(let H21 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) 
-(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
-(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | 
-(Flat _) \Rightarrow True])])) I (THead (Bind b0) u (lift (S O) O t5)) H20) 
-in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t2 t7))) H21))))) t6 
-(sym_eq T t6 t2 H17))) t H15 H16 H13 H14))) | (pr0_tau t5 t6 H13 u) 
-\Rightarrow (\lambda (H14: (eq T (THead (Flat Cast) u t5) t)).(\lambda (H15: 
-(eq T t6 t2)).(eq_ind T (THead (Flat Cast) u t5) (\lambda (_: T).((eq T t6 
-t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2 
-t8)))))) (\lambda (H16: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 
-t7) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2 t8))))) (\lambda (_: 
-(pr0 t5 t2)).(let H18 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat 
-Appl) v1 (THead (Bind b) u1 t3)) t7)) H6 (THead (Flat Cast) u t5) H14) in 
-(let H19 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) 
-(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) 
+O v3) t6)) t7))) (\lambda (x0: T).(\lambda (H39: (pr0 u2 x0)).(\lambda (H40: 
+(pr0 u3 x0)).(ex2_ind T (\lambda (t7: T).(pr0 v2 t7)) (\lambda (t7: T).(pr0 
+v3 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) 
+(lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 (THead (Bind b0) u3 (THead 
+(Flat Appl) (lift (S O) O v3) t6)) t7))) (\lambda (x1: T).(\lambda (H41: (pr0 
+v2 x1)).(\lambda (H42: (pr0 v3 x1)).(pr0_confluence__pr0_upsilon_upsilon b0 
+H34 v2 v3 x1 H41 H42 u2 u3 x0 H39 H40 t4 t6 x H37 H38)))) (H32 v0 
+(tlt_head_sx (Flat Appl) v0 (THead (Bind b0) u0 t5)) v2 H33 v3 H21))))) (H32 
+u0 (tlt_trans (THead (Bind b0) u0 t5) u0 (THead (Flat Appl) v0 (THead (Bind 
+b0) u0 t5)) (tlt_head_sx (Bind b0) u0 t5) (tlt_head_dx (Flat Appl) v0 (THead 
+(Bind b0) u0 t5))) u2 H35 u3 H22))))) (H32 t5 (tlt_trans (THead (Bind b0) u0 
+t5) t5 (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) (tlt_head_dx (Bind b0) 
+u0 t5) (tlt_head_dx (Flat Appl) v0 (THead (Bind b0) u0 t5))) t4 H36 t6 
+H23))))) b H30))))))) H27)) H26)) H25))))))) t2 H19)) t H17 H18 H13 H14 H15 
+H16))) | (pr0_delta u0 u3 H13 t5 t6 H14 w H15) \Rightarrow (\lambda (H16: (eq 
+T (THead (Bind Abbr) u0 t5) t)).(\lambda (H17: (eq T (THead (Bind Abbr) u3 w) 
+t2)).(eq_ind T (THead (Bind Abbr) u0 t5) (\lambda (_: T).((eq T (THead (Bind 
+Abbr) u3 w) t2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to ((subst0 O u3 t6 w) \to 
+(ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
+O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda (H18: (eq T 
+(THead (Bind Abbr) u3 w) t2)).(eq_ind T (THead (Bind Abbr) u3 w) (\lambda 
+(t7: T).((pr0 u0 u3) \to ((pr0 t5 t6) \to ((subst0 O u3 t6 w) \to (ex2 T 
+(\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
+t4)) t8)) (\lambda (t8: T).(pr0 t7 t8))))))) (\lambda (_: (pr0 u0 
+u3)).(\lambda (_: (pr0 t5 t6)).(\lambda (_: (subst0 O u3 t6 w)).(let H22 \def 
+(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 
+t3)) t7)) H6 (THead (Bind Abbr) u0 t5) H16) in (let H23 \def (eq_ind T (THead 
+(Flat Appl) v1 (THead (Bind b) u1 t3)) (\lambda (ee: T).(match ee with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
+\Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow 
+True])])) I (THead (Bind Abbr) u0 t5) H22) in (False_ind (ex2 T (\lambda (t7: 
+T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) 
+(\lambda (t7: T).(pr0 (THead (Bind Abbr) u3 w) t7))) H23)))))) t2 H18)) t H16 
+H17 H13 H14 H15))) | (pr0_zeta b0 H13 t5 t6 H14 u) \Rightarrow (\lambda (H15: 
+(eq T (THead (Bind b0) u (lift (S O) O t5)) t)).(\lambda (H16: (eq T t6 
+t2)).(eq_ind T (THead (Bind b0) u (lift (S O) O t5)) (\lambda (_: T).((eq T 
+t6 t2) \to ((not (eq B b0 Abst)) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: 
+T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) 
+(\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H17: (eq T t6 t2)).(eq_ind T t2 
+(\lambda (t7: T).((not (eq B b0 Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda 
+(t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
+t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (_: (not (eq B b0 
+Abst))).(\lambda (_: (pr0 t5 t2)).(let H20 \def (eq_ind_r T t (\lambda (t7: 
+T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) t7)) H6 (THead (Bind 
+b0) u (lift (S O) O t5)) H15) in (let H21 \def (eq_ind T (THead (Flat Appl) 
+v1 (THead (Bind b) u1 t3)) (\lambda (ee: T).(match ee with [(TSort _) 

 \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
 \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 

-(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | 
-(Flat f) \Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl 
-\Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat Cast) u t5) 
-H18) in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t2 t7))) 
-H19)))) t6 (sym_eq T t6 t2 H16))) t H14 H15 H13)))]) in (H13 (refl_equal T t) 
-(refl_equal T t2))))))) t1 H8)) t H6 H7 H2 H3 H4 H5))) | (pr0_delta u1 u2 H2 
-t3 t4 H3 w H4) \Rightarrow (\lambda (H5: (eq T (THead (Bind Abbr) u1 t3) 
-t)).(\lambda (H6: (eq T (THead (Bind Abbr) u2 w) t1)).(eq_ind T (THead (Bind 
-Abbr) u1 t3) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t1) \to ((pr0 u1 
-u2) \to ((pr0 t3 t4) \to ((subst0 O u2 t4 w) \to (ex2 T (\lambda (t6: T).(pr0 
-t1 t6)) (\lambda (t6: T).(pr0 t2 t6)))))))) (\lambda (H7: (eq T (THead (Bind 
-Abbr) u2 w) t1)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t5: T).((pr0 u1 
-u2) \to ((pr0 t3 t4) \to ((subst0 O u2 t4 w) \to (ex2 T (\lambda (t6: T).(pr0 
-t5 t6)) (\lambda (t6: T).(pr0 t2 t6))))))) (\lambda (H8: (pr0 u1 
-u2)).(\lambda (H9: (pr0 t3 t4)).(\lambda (H10: (subst0 O u2 t4 w)).(let H11 
-\def (match H1 in pr0 return (\lambda (t5: T).(\lambda (t6: T).(\lambda (_: 
-(pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2) \to (ex2 T (\lambda (t7: T).(pr0 
-(THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7)))))))) with 
-[(pr0_refl t5) \Rightarrow (\lambda (H11: (eq T t5 t)).(\lambda (H12: (eq T 
-t5 t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2) \to (ex2 T (\lambda (t7: 
-T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7))))) 
-(\lambda (H13: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t7: 
-T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let 
-H14 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t2)) H13 (THead (Bind Abbr) 
-u1 t3) H5) in (eq_ind T (THead (Bind Abbr) u1 t3) (\lambda (t6: T).(ex2 T 
-(\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t6 
-t7)))) (let H15 \def (eq_ind_r T t (\lambda (t6: T).(eq T t5 t6)) H11 (THead 
-(Bind Abbr) u1 t3) H5) in (let H16 \def (eq_ind_r T t (\lambda (t6: 
-T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall 
-(t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: 
-T).(pr0 t8 t9)))))))))) H (THead (Bind Abbr) u1 t3) H5) in (ex_intro2 T 
-(\lambda (t6: T).(pr0 (THead (Bind Abbr) u2 w) t6)) (\lambda (t6: T).(pr0 
-(THead (Bind Abbr) u1 t3) t6)) (THead (Bind Abbr) u2 w) (pr0_refl (THead 
-(Bind Abbr) u2 w)) (pr0_delta u1 u2 H8 t3 t4 H9 w H10)))) t2 H14)) t (sym_eq 
-T t t2 H13))) t5 (sym_eq T t5 t H11) H12))) | (pr0_comp u0 u3 H11 t5 t6 H12 
-k) \Rightarrow (\lambda (H13: (eq T (THead k u0 t5) t)).(\lambda (H14: (eq T 
-(THead k u3 t6) t2)).(eq_ind T (THead k u0 t5) (\lambda (_: T).((eq T (THead 
-k u3 t6) t2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: 
-T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) 
-(\lambda (H15: (eq T (THead k u3 t6) t2)).(eq_ind T (THead k u3 t6) (\lambda 
-(t7: T).((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead 
-(Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (H16: (pr0 
-u0 u3)).(\lambda (H17: (pr0 t5 t6)).(let H18 \def (eq_ind_r T t (\lambda (t7: 
-T).(eq T (THead (Bind Abbr) u1 t3) t7)) H5 (THead k u0 t5) H13) in (let H19 
-\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) 
-with [(TSort _) \Rightarrow (Bind Abbr) | (TLRef _) \Rightarrow (Bind Abbr) | 
-(THead k0 _ _) \Rightarrow k0])) (THead (Bind Abbr) u1 t3) (THead k u0 t5) 
-H18) in ((let H20 \def (f_equal T T (\lambda (e: T).(match e in T return 
-(\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 
-| (THead _ t7 _) \Rightarrow t7])) (THead (Bind Abbr) u1 t3) (THead k u0 t5) 
-H18) in ((let H21 \def (f_equal T T (\lambda (e: T).(match e in T return 
-(\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 
-| (THead _ _ t7) \Rightarrow t7])) (THead (Bind Abbr) u1 t3) (THead k u0 t5) 
-H18) in (\lambda (H22: (eq T u1 u0)).(\lambda (H23: (eq K (Bind Abbr) 
-k)).(eq_ind K (Bind Abbr) (\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 
-(THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead k0 u3 t6) t7)))) 
-(let H24 \def (eq_ind_r K k (\lambda (k0: K).(eq T (THead k0 u0 t5) t)) H13 
-(Bind Abbr) H23) in (let H25 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: 
-T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v 
-t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 
-t10)))))))))) H (THead (Bind Abbr) u0 t5) H24) in (let H26 \def (eq_ind T u1 
-(\lambda (t7: T).(pr0 t7 u2)) H8 u0 H22) in (let H27 \def (eq_ind T t3 
-(\lambda (t7: T).(pr0 t7 t4)) H9 t5 H21) in (ex2_ind T (\lambda (t7: T).(pr0 
-t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead 
-(Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u3 t6) t7))) 
-(\lambda (x: T).(\lambda (H28: (pr0 t4 x)).(\lambda (H29: (pr0 t6 
-x)).(ex2_ind T (\lambda (t7: T).(pr0 u2 t7)) (\lambda (t7: T).(pr0 u3 t7)) 
-(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: 
+(match k with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I 
+(THead (Bind b0) u (lift (S O) O t5)) H20) in (False_ind (ex2 T (\lambda (t7: 
+T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) 
+(\lambda (t7: T).(pr0 t2 t7))) H21))))) t6 (sym_eq T t6 t2 H17))) t H15 H16 
+H13 H14))) | (pr0_tau t5 t6 H13 u) \Rightarrow (\lambda (H14: (eq T (THead 
+(Flat Cast) u t5) t)).(\lambda (H15: (eq T t6 t2)).(eq_ind T (THead (Flat 
+Cast) u t5) (\lambda (_: T).((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T 
+(\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
+t4)) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H16: (eq T t6 
+t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: 
+T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) 
+(\lambda (t8: T).(pr0 t2 t8))))) (\lambda (_: (pr0 t5 t2)).(let H18 \def 
+(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 
+t3)) t7)) H6 (THead (Flat Cast) u t5) H14) in (let H19 \def (eq_ind T (THead 
+(Flat Appl) v1 (THead (Bind b) u1 t3)) (\lambda (ee: T).(match ee with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
+\Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow 
+(match f with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead 
+(Flat Cast) u t5) H18) in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: 
+T).(pr0 t2 t7))) H19)))) t6 (sym_eq T t6 t2 H16))) t H14 H15 H13)))]) in (H13 
+(refl_equal T t) (refl_equal T t2))))))) t1 H8)) t H6 H7 H2 H3 H4 H5))) | 
+(pr0_delta u1 u2 H2 t3 t4 H3 w H4) \Rightarrow (\lambda (H5: (eq T (THead 
+(Bind Abbr) u1 t3) t)).(\lambda (H6: (eq T (THead (Bind Abbr) u2 w) 
+t1)).(eq_ind T (THead (Bind Abbr) u1 t3) (\lambda (_: T).((eq T (THead (Bind 
+Abbr) u2 w) t1) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to ((subst0 O u2 t4 w) \to 
+(ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6)))))))) 
+(\lambda (H7: (eq T (THead (Bind Abbr) u2 w) t1)).(eq_ind T (THead (Bind 
+Abbr) u2 w) (\lambda (t5: T).((pr0 u1 u2) \to ((pr0 t3 t4) \to ((subst0 O u2 
+t4 w) \to (ex2 T (\lambda (t6: T).(pr0 t5 t6)) (\lambda (t6: T).(pr0 t2 
+t6))))))) (\lambda (H8: (pr0 u1 u2)).(\lambda (H9: (pr0 t3 t4)).(\lambda 
+(H10: (subst0 O u2 t4 w)).(let H11 \def (match H1 with [(pr0_refl t5) 
+\Rightarrow (\lambda (H11: (eq T t5 t)).(\lambda (H12: (eq T t5 t2)).(eq_ind 
+T t (\lambda (t6: T).((eq T t6 t2) \to (ex2 T (\lambda (t7: T).(pr0 (THead 
+(Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7))))) (\lambda (H13: (eq T 
+t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind 
+Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let H14 \def (eq_ind_r T t 
+(\lambda (t6: T).(eq T t6 t2)) H13 (THead (Bind Abbr) u1 t3) H5) in (eq_ind T 
+(THead (Bind Abbr) u1 t3) (\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 
+(THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t6 t7)))) (let H15 \def 
+(eq_ind_r T t (\lambda (t6: T).(eq T t5 t6)) H11 (THead (Bind Abbr) u1 t3) 
+H5) in (let H16 \def (eq_ind_r T t (\lambda (t6: T).(\forall (v: T).((tlt v 
+t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall (t8: T).((pr0 v t8) \to 
+(ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 t9)))))))))) H 
+(THead (Bind Abbr) u1 t3) H5) in (ex_intro2 T (\lambda (t6: T).(pr0 (THead 
+(Bind Abbr) u2 w) t6)) (\lambda (t6: T).(pr0 (THead (Bind Abbr) u1 t3) t6)) 
+(THead (Bind Abbr) u2 w) (pr0_refl (THead (Bind Abbr) u2 w)) (pr0_delta u1 u2 
+H8 t3 t4 H9 w H10)))) t2 H14)) t (sym_eq T t t2 H13))) t5 (sym_eq T t5 t H11) 
+H12))) | (pr0_comp u0 u3 H11 t5 t6 H12 k) \Rightarrow (\lambda (H13: (eq T 
+(THead k u0 t5) t)).(\lambda (H14: (eq T (THead k u3 t6) t2)).(eq_ind T 
+(THead k u0 t5) (\lambda (_: T).((eq T (THead k u3 t6) t2) \to ((pr0 u0 u3) 
+\to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) 
+t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H15: (eq T (THead k u3 t6) 
+t2)).(eq_ind T (THead k u3 t6) (\lambda (t7: T).((pr0 u0 u3) \to ((pr0 t5 t6) 
+\to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: 
+T).(pr0 t7 t8)))))) (\lambda (H16: (pr0 u0 u3)).(\lambda (H17: (pr0 t5 
+t6)).(let H18 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind Abbr) u1 
+t3) t7)) H5 (THead k u0 t5) H13) in (let H19 \def (f_equal T K (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow (Bind Abbr) | (TLRef _) \Rightarrow 
+(Bind Abbr) | (THead k0 _ _) \Rightarrow k0])) (THead (Bind Abbr) u1 t3) 
+(THead k u0 t5) H18) in ((let H20 \def (f_equal T T (\lambda (e: T).(match e 
+with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t7 _) 
+\Rightarrow t7])) (THead (Bind Abbr) u1 t3) (THead k u0 t5) H18) in ((let H21 
+\def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t3 | 
+(TLRef _) \Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead (Bind 
+Abbr) u1 t3) (THead k u0 t5) H18) in (\lambda (H22: (eq T u1 u0)).(\lambda 
+(H23: (eq K (Bind Abbr) k)).(eq_ind K (Bind Abbr) (\lambda (k0: K).(ex2 T 
+(\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 
+(THead k0 u3 t6) t7)))) (let H24 \def (eq_ind_r K k (\lambda (k0: K).(eq T 
+(THead k0 u0 t5) t)) H13 (Bind Abbr) H23) in (let H25 \def (eq_ind_r T t 
+(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) 
+\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) 
+(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Bind Abbr) u0 t5) H24) in 
+(let H26 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H8 u0 H22) in (let 
+H27 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H9 t5 H21) in (ex2_ind T 
+(\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda 
+(t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead (Bind 
+Abbr) u3 t6) t7))) (\lambda (x: T).(\lambda (H28: (pr0 t4 x)).(\lambda (H29: 
+(pr0 t6 x)).(ex2_ind T (\lambda (t7: T).(pr0 u2 t7)) (\lambda (t7: T).(pr0 u3 
+t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: 

 T).(pr0 (THead (Bind Abbr) u3 t6) t7))) (\lambda (x0: T).(\lambda (H30: (pr0 
 u2 x0)).(\lambda (H31: (pr0 u3 x0)).(ex2_sym T (pr0 (THead (Bind Abbr) u3 
 t6)) (pr0 (THead (Bind Abbr) u2 w)) (pr0_confluence__pr0_cong_delta u2 t4 w 
 T).(pr0 (THead (Bind Abbr) u3 t6) t7))) (\lambda (x0: T).(\lambda (H30: (pr0 
 u2 x0)).(\lambda (H31: (pr0 u3 x0)).(ex2_sym T (pr0 (THead (Bind Abbr) u3 
 t6)) (pr0 (THead (Bind Abbr) u2 w)) (pr0_confluence__pr0_cong_delta u2 t4 w 


@@ -1997,10 +1844,9 @@ t2)).(eq_ind T (THead (Bind Abbr) v2 t6) (\lambda (t7: T).((pr0 v1 v2) \to
 (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 
 t5 t6)).(let H18 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind Abbr) 
 u1 t3) t7)) H5 (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) H13) in (let 
 (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 
 t5 t6)).(let H18 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind Abbr) 
 u1 t3) t7)) H5 (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) H13) in (let 

-H19 \def (eq_ind T (THead (Bind Abbr) u1 t3) (\lambda (ee: T).(match ee in T 
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda 
-(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow 
+H19 \def (eq_ind T (THead (Bind Abbr) u1 t3) (\lambda (ee: T).(match ee with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
+\Rightarrow (match k with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow 

 False])])) I (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) H18) in 
 (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) 
 (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t6) t7))) H19))))) t2 H15)) t H13 
 False])])) I (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) H18) in 
 (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) 
 (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t6) t7))) H19))))) t2 H15)) t H13 


@@ -2020,9 +1866,8 @@ t8)))))))) (\lambda (_: (not (eq B b Abst))).(\lambda (_: (pr0 v1
 v2)).(\lambda (_: (pr0 u0 u3)).(\lambda (_: (pr0 t5 t6)).(let H22 \def 
 (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind Abbr) u1 t3) t7)) H5 (THead 
 (Flat Appl) v1 (THead (Bind b) u0 t5)) H15) in (let H23 \def (eq_ind T (THead 
 v2)).(\lambda (_: (pr0 u0 u3)).(\lambda (_: (pr0 t5 t6)).(let H22 \def 
 (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind Abbr) u1 t3) t7)) H5 (THead 
 (Flat Appl) v1 (THead (Bind b) u0 t5)) H15) in (let H23 \def (eq_ind T (THead 

-(Bind Abbr) u1 t3) (\lambda (ee: T).(match ee in T return (\lambda (_: 
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
-(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with 
+(Bind Abbr) u1 t3) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow 
+False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k with 

 [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat 
 Appl) v1 (THead (Bind b) u0 t5)) H22) in (False_ind (ex2 T (\lambda (t7: 
 T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead (Bind b) 
 [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat 
 Appl) v1 (THead (Bind b) u0 t5)) H22) in (False_ind (ex2 T (\lambda (t7: 
 T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead (Bind b) 


@@ -2039,49 +1884,47 @@ T).((pr0 u0 u3) \to ((pr0 t5 t6) \to ((subst0 O u3 t6 w0) \to (ex2 T (\lambda
 (\lambda (H17: (pr0 u0 u3)).(\lambda (H18: (pr0 t5 t6)).(\lambda (H19: 
 (subst0 O u3 t6 w0)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T 
 (THead (Bind Abbr) u1 t3) t7)) H5 (THead (Bind Abbr) u0 t5) H14) in (let H21 
 (\lambda (H17: (pr0 u0 u3)).(\lambda (H18: (pr0 t5 t6)).(\lambda (H19: 
 (subst0 O u3 t6 w0)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T 
 (THead (Bind Abbr) u1 t3) t7)) H5 (THead (Bind Abbr) u0 t5) H14) in (let H21 

-\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) 
-with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t7 _) 
-\Rightarrow t7])) (THead (Bind Abbr) u1 t3) (THead (Bind Abbr) u0 t5) H20) in 
-((let H22 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: 
-T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ 
-t7) \Rightarrow t7])) (THead (Bind Abbr) u1 t3) (THead (Bind Abbr) u0 t5) 
-H20) in (\lambda (H23: (eq T u1 u0)).(let H24 \def (eq_ind_r T t (\lambda 
-(t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to 
-(\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) 
-(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Bind Abbr) u0 t5) H14) in 
-(let H25 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H8 u0 H23) in (let 
-H26 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H9 t5 H22) in (ex2_ind T 
-(\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda 
-(t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead (Bind 
-Abbr) u3 w0) t7))) (\lambda (x: T).(\lambda (H27: (pr0 t4 x)).(\lambda (H28: 
-(pr0 t6 x)).(ex2_ind T (\lambda (t7: T).(pr0 u2 t7)) (\lambda (t7: T).(pr0 u3 
-t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: 
-T).(pr0 (THead (Bind Abbr) u3 w0) t7))) (\lambda (x0: T).(\lambda (H29: (pr0 
-u2 x0)).(\lambda (H30: (pr0 u3 x0)).(pr0_confluence__pr0_delta_delta u2 t4 w 
-H10 u3 t6 w0 H19 x0 H29 H30 x H27 H28)))) (H24 u0 (tlt_head_sx (Bind Abbr) u0 
-t5) u2 H25 u3 H17))))) (H24 t5 (tlt_head_dx (Bind Abbr) u0 t5) t4 H26 t6 
-H18))))))) H21)))))) t2 H16)) t H14 H15 H11 H12 H13))) | (pr0_zeta b H11 t5 
-t6 H12 u) \Rightarrow (\lambda (H13: (eq T (THead (Bind b) u (lift (S O) O 
-t5)) t)).(\lambda (H14: (eq T t6 t2)).(eq_ind T (THead (Bind b) u (lift (S O) 
-O t5)) (\lambda (_: T).((eq T t6 t2) \to ((not (eq B b Abst)) \to ((pr0 t5 
-t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda 
-(t8: T).(pr0 t2 t8))))))) (\lambda (H15: (eq T t6 t2)).(eq_ind T t2 (\lambda 
-(t7: T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda (t8: 
-T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) 
-(\lambda (H16: (not (eq B b Abst))).(\lambda (H17: (pr0 t5 t2)).(let H18 \def 
-(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind Abbr) u1 t3) t7)) H5 (THead 
-(Bind b) u (lift (S O) O t5)) H13) in (let H19 \def (f_equal T B (\lambda (e: 
-T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow Abbr | 
-(TLRef _) \Rightarrow Abbr | (THead k _ _) \Rightarrow (match k in K return 
-(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow 
-Abbr])])) (THead (Bind Abbr) u1 t3) (THead (Bind b) u (lift (S O) O t5)) H18) 
-in ((let H20 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda 
-(_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead 
-_ t7 _) \Rightarrow t7])) (THead (Bind Abbr) u1 t3) (THead (Bind b) u (lift 
-(S O) O t5)) H18) in ((let H21 \def (f_equal T T (\lambda (e: T).(match e in 
-T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) 
+\def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u1 | 
+(TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead (Bind 
+Abbr) u1 t3) (THead (Bind Abbr) u0 t5) H20) in ((let H22 \def (f_equal T T 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow t3 | (TLRef _) 

 \Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead (Bind Abbr) u1 t3) 
 \Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead (Bind Abbr) u1 t3) 

-(THead (Bind b) u (lift (S O) O t5)) H18) in (\lambda (H22: (eq T u1 
+(THead (Bind Abbr) u0 t5) H20) in (\lambda (H23: (eq T u1 u0)).(let H24 \def 
+(eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: 
+T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: 
+T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Bind Abbr) 
+u0 t5) H14) in (let H25 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H8 u0 
+H23) in (let H26 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H9 t5 H22) 
+in (ex2_ind T (\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) 
+(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: 
+T).(pr0 (THead (Bind Abbr) u3 w0) t7))) (\lambda (x: T).(\lambda (H27: (pr0 
+t4 x)).(\lambda (H28: (pr0 t6 x)).(ex2_ind T (\lambda (t7: T).(pr0 u2 t7)) 
+(\lambda (t7: T).(pr0 u3 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) 
+u2 w) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u3 w0) t7))) (\lambda 
+(x0: T).(\lambda (H29: (pr0 u2 x0)).(\lambda (H30: (pr0 u3 
+x0)).(pr0_confluence__pr0_delta_delta u2 t4 w H10 u3 t6 w0 H19 x0 H29 H30 x 
+H27 H28)))) (H24 u0 (tlt_head_sx (Bind Abbr) u0 t5) u2 H25 u3 H17))))) (H24 
+t5 (tlt_head_dx (Bind Abbr) u0 t5) t4 H26 t6 H18))))))) H21)))))) t2 H16)) t 
+H14 H15 H11 H12 H13))) | (pr0_zeta b H11 t5 t6 H12 u) \Rightarrow (\lambda 
+(H13: (eq T (THead (Bind b) u (lift (S O) O t5)) t)).(\lambda (H14: (eq T t6 
+t2)).(eq_ind T (THead (Bind b) u (lift (S O) O t5)) (\lambda (_: T).((eq T t6 
+t2) \to ((not (eq B b Abst)) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: 
+T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) 
+(\lambda (H15: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((not (eq B b 
+Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) 
+u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H16: (not (eq B b 
+Abst))).(\lambda (H17: (pr0 t5 t2)).(let H18 \def (eq_ind_r T t (\lambda (t7: 
+T).(eq T (THead (Bind Abbr) u1 t3) t7)) H5 (THead (Bind b) u (lift (S O) O 
+t5)) H13) in (let H19 \def (f_equal T B (\lambda (e: T).(match e with [(TSort 
+_) \Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k _ _) \Rightarrow 
+(match k with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) 
+(THead (Bind Abbr) u1 t3) (THead (Bind b) u (lift (S O) O t5)) H18) in ((let 
+H20 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u1 
+| (TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead (Bind 
+Abbr) u1 t3) (THead (Bind b) u (lift (S O) O t5)) H18) in ((let H21 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t3 | (TLRef 
+_) \Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead (Bind Abbr) u1 
+t3) (THead (Bind b) u (lift (S O) O t5)) H18) in (\lambda (H22: (eq T u1 

 u)).(\lambda (H23: (eq B Abbr b)).(let H24 \def (eq_ind_r B b (\lambda (b0: 
 B).(not (eq B b0 Abst))) H16 Abbr H23) in (let H25 \def (eq_ind_r B b 
 (\lambda (b0: B).(eq T (THead (Bind b0) u (lift (S O) O t5)) t)) H13 Abbr 
 u)).(\lambda (H23: (eq B Abbr b)).(let H24 \def (eq_ind_r B b (\lambda (b0: 
 B).(not (eq B b0 Abst))) H16 Abbr H23) in (let H25 \def (eq_ind_r B b 
 (\lambda (b0: B).(eq T (THead (Bind b0) u (lift (S O) O t5)) t)) H13 Abbr 


@@ -2111,10 +1954,9 @@ t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8:
 T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8))))) 
 (\lambda (_: (pr0 t5 t2)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T 
 (THead (Bind Abbr) u1 t3) t7)) H5 (THead (Flat Cast) u t5) H12) in (let H17 
 T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8))))) 
 (\lambda (_: (pr0 t5 t2)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T 
 (THead (Bind Abbr) u1 t3) t7)) H5 (THead (Flat Cast) u t5) H12) in (let H17 

-\def (eq_ind T (THead (Bind Abbr) u1 t3) (\lambda (ee: T).(match ee in T 
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda 
-(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow 
+\def (eq_ind T (THead (Bind Abbr) u1 t3) (\lambda (ee: T).(match ee with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
+\Rightarrow (match k with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow 

 False])])) I (THead (Flat Cast) u t5) H16) in (False_ind (ex2 T (\lambda (t7: 
 T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7))) H17)))) 
 t6 (sym_eq T t6 t2 H14))) t H12 H13 H11)))]) in (H11 (refl_equal T t) 
 False])])) I (THead (Flat Cast) u t5) H16) in (False_ind (ex2 T (\lambda (t7: 
 T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7))) H17)))) 
 t6 (sym_eq T t6 t2 H14))) t H12 H13 H11)))]) in (H11 (refl_equal T t) 


@@ -2126,51 +1968,39 @@ t3)) (\lambda (_: T).((eq T t4 t1) \to ((not (eq B b Abst)) \to ((pr0 t3 t4)
 (\lambda (H6: (eq T t4 t1)).(eq_ind T t1 (\lambda (t5: T).((not (eq B b 
 Abst)) \to ((pr0 t3 t5) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda 
 (t6: T).(pr0 t2 t6)))))) (\lambda (H7: (not (eq B b Abst))).(\lambda (H8: 
 (\lambda (H6: (eq T t4 t1)).(eq_ind T t1 (\lambda (t5: T).((not (eq B b 
 Abst)) \to ((pr0 t3 t5) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda 
 (t6: T).(pr0 t2 t6)))))) (\lambda (H7: (not (eq B b Abst))).(\lambda (H8: 

-(pr0 t3 t1)).(let H9 \def (match H1 in pr0 return (\lambda (t5: T).(\lambda 
-(t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2) \to (ex2 T 
-(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)))))))) with 
-[(pr0_refl t5) \Rightarrow (\lambda (H9: (eq T t5 t)).(\lambda (H10: (eq T t5 
-t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2) \to (ex2 T (\lambda (t7: 
-T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))))) (\lambda (H11: (eq T t 
-t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t7: T).(pr0 t1 t7)) 
-(\lambda (t7: T).(pr0 t2 t7)))) (let H12 \def (eq_ind_r T t (\lambda (t6: 
-T).(eq T t6 t2)) H11 (THead (Bind b) u (lift (S O) O t3)) H4) in (eq_ind T 
-(THead (Bind b) u (lift (S O) O t3)) (\lambda (t6: T).(ex2 T (\lambda (t7: 
-T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t6 t7)))) (let H13 \def (eq_ind_r T t 
-(\lambda (t6: T).(eq T t5 t6)) H9 (THead (Bind b) u (lift (S O) O t3)) H4) in 
-(let H14 \def (eq_ind_r T t (\lambda (t6: T).(\forall (v: T).((tlt v t6) \to 
-(\forall (t7: T).((pr0 v t7) \to (\forall (t8: T).((pr0 v t8) \to (ex2 T 
-(\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead 
-(Bind b) u (lift (S O) O t3)) H4) in (ex_intro2 T (\lambda (t6: T).(pr0 t1 
-t6)) (\lambda (t6: T).(pr0 (THead (Bind b) u (lift (S O) O t3)) t6)) t1 
-(pr0_refl t1) (pr0_zeta b H7 t3 t1 H8 u)))) t2 H12)) t (sym_eq T t t2 H11))) 
-t5 (sym_eq T t5 t H9) H10))) | (pr0_comp u1 u2 H9 t5 t6 H10 k) \Rightarrow 
-(\lambda (H11: (eq T (THead k u1 t5) t)).(\lambda (H12: (eq T (THead k u2 t6) 
-t2)).(eq_ind T (THead k u1 t5) (\lambda (_: T).((eq T (THead k u2 t6) t2) \to 
-((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) 
-(\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H13: (eq T (THead k u2 t6) 
-t2)).(eq_ind T (THead k u2 t6) (\lambda (t7: T).((pr0 u1 u2) \to ((pr0 t5 t6) 
-\to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8)))))) 
-(\lambda (_: (pr0 u1 u2)).(\lambda (H15: (pr0 t5 t6)).(let H16 \def (eq_ind_r 
-T t (\lambda (t7: T).(eq T (THead (Bind b) u (lift (S O) O t3)) t7)) H4 
-(THead k u1 t5) H11) in (let H17 \def (f_equal T K (\lambda (e: T).(match e 
-in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow (Bind b) | (TLRef 
-_) \Rightarrow (Bind b) | (THead k0 _ _) \Rightarrow k0])) (THead (Bind b) u 
-(lift (S O) O t3)) (THead k u1 t5) H16) in ((let H18 \def (f_equal T T 
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t7 _) \Rightarrow t7])) 
-(THead (Bind b) u (lift (S O) O t3)) (THead k u1 t5) H16) in ((let H19 \def 
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t7: 
-T) on t7: T \def (match t7 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) 
-\Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false 
-\Rightarrow (f i)])) | (THead k0 u0 t8) \Rightarrow (THead k0 (lref_map f d 
-u0) (lref_map f (s k0 d) t8))]) in lref_map) (\lambda (x: nat).(plus x (S 
-O))) O t3) | (TLRef _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) 
-(d: nat) (t7: T) on t7: T \def (match t7 with [(TSort n) \Rightarrow (TSort 
-n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | 
-false \Rightarrow (f i)])) | (THead k0 u0 t8) \Rightarrow (THead k0 (lref_map 
-f d u0) (lref_map f (s k0 d) t8))]) in lref_map) (\lambda (x: nat).(plus x (S 
+(pr0 t3 t1)).(let H9 \def (match H1 with [(pr0_refl t5) \Rightarrow (\lambda 
+(H9: (eq T t5 t)).(\lambda (H10: (eq T t5 t2)).(eq_ind T t (\lambda (t6: 
+T).((eq T t6 t2) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: 
+T).(pr0 t2 t7))))) (\lambda (H11: (eq T t t2)).(eq_ind T t2 (\lambda (_: 
+T).(ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let 
+H12 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t2)) H11 (THead (Bind b) u 
+(lift (S O) O t3)) H4) in (eq_ind T (THead (Bind b) u (lift (S O) O t3)) 
+(\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 
+t6 t7)))) (let H13 \def (eq_ind_r T t (\lambda (t6: T).(eq T t5 t6)) H9 
+(THead (Bind b) u (lift (S O) O t3)) H4) in (let H14 \def (eq_ind_r T t 
+(\lambda (t6: T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) 
+\to (\forall (t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) 
+(\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead (Bind b) u (lift (S O) O t3)) 
+H4) in (ex_intro2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 
+(THead (Bind b) u (lift (S O) O t3)) t6)) t1 (pr0_refl t1) (pr0_zeta b H7 t3 
+t1 H8 u)))) t2 H12)) t (sym_eq T t t2 H11))) t5 (sym_eq T t5 t H9) H10))) | 
+(pr0_comp u1 u2 H9 t5 t6 H10 k) \Rightarrow (\lambda (H11: (eq T (THead k u1 
+t5) t)).(\lambda (H12: (eq T (THead k u2 t6) t2)).(eq_ind T (THead k u1 t5) 
+(\lambda (_: T).((eq T (THead k u2 t6) t2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) 
+\to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))) 
+(\lambda (H13: (eq T (THead k u2 t6) t2)).(eq_ind T (THead k u2 t6) (\lambda 
+(t7: T).((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 
+t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (_: (pr0 u1 u2)).(\lambda 
+(H15: (pr0 t5 t6)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead 
+(Bind b) u (lift (S O) O t3)) t7)) H4 (THead k u1 t5) H11) in (let H17 \def 
+(f_equal T K (\lambda (e: T).(match e with [(TSort _) \Rightarrow (Bind b) | 
+(TLRef _) \Rightarrow (Bind b) | (THead k0 _ _) \Rightarrow k0])) (THead 
+(Bind b) u (lift (S O) O t3)) (THead k u1 t5) H16) in ((let H18 \def (f_equal 
+T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u | (TLRef _) 
+\Rightarrow u | (THead _ t7 _) \Rightarrow t7])) (THead (Bind b) u (lift (S 
+O) O t3)) (THead k u1 t5) H16) in ((let H19 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow (lref_map (\lambda (x: nat).(plus x 
+(S O))) O t3) | (TLRef _) \Rightarrow (lref_map (\lambda (x: nat).(plus x (S 

 O))) O t3) | (THead _ _ t7) \Rightarrow t7])) (THead (Bind b) u (lift (S O) O 
 t3)) (THead k u1 t5) H16) in (\lambda (_: (eq T u u1)).(\lambda (H21: (eq K 
 (Bind b) k)).(eq_ind K (Bind b) (\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 
 O))) O t3) | (THead _ _ t7) \Rightarrow t7])) (THead (Bind b) u (lift (S O) O 
 t3)) (THead k u1 t5) H16) in (\lambda (_: (eq T u u1)).(\lambda (H21: (eq K 
 (Bind b) k)).(eq_ind K (Bind b) (\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 


@@ -2206,9 +2036,8 @@ Abbr) v2 t6) (\lambda (t7: T).((pr0 v1 v2) \to ((pr0 t5 t6) \to (ex2 T
 (pr0 v1 v2)).(\lambda (_: (pr0 t5 t6)).(let H16 \def (eq_ind_r T t (\lambda 
 (t7: T).(eq T (THead (Bind b) u (lift (S O) O t3)) t7)) H4 (THead (Flat Appl) 
 v1 (THead (Bind Abst) u0 t5)) H11) in (let H17 \def (eq_ind T (THead (Bind b) 
 (pr0 v1 v2)).(\lambda (_: (pr0 t5 t6)).(let H16 \def (eq_ind_r T t (\lambda 
 (t7: T).(eq T (THead (Bind b) u (lift (S O) O t3)) t7)) H4 (THead (Flat Appl) 
 v1 (THead (Bind Abst) u0 t5)) H11) in (let H17 \def (eq_ind T (THead (Bind b) 

-u (lift (S O) O t3)) (\lambda (ee: T).(match ee in T return (\lambda (_: 
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
-(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with 
+u (lift (S O) O t3)) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow 
+False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k with 

 [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat 
 Appl) v1 (THead (Bind Abst) u0 t5)) H16) in (False_ind (ex2 T (\lambda (t7: 
 T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t6) t7))) 
 [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat 
 Appl) v1 (THead (Bind Abst) u0 t5)) H16) in (False_ind (ex2 T (\lambda (t7: 
 T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t6) t7))) 


@@ -2228,61 +2057,50 @@ Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda
 (_: (pr0 t5 t6)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead 
 (Bind b) u (lift (S O) O t3)) t7)) H4 (THead (Flat Appl) v1 (THead (Bind b0) 
 u1 t5)) H13) in (let H21 \def (eq_ind T (THead (Bind b) u (lift (S O) O t3)) 
 (_: (pr0 t5 t6)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead 
 (Bind b) u (lift (S O) O t3)) t7)) H4 (THead (Flat Appl) v1 (THead (Bind b0) 
 u1 t5)) H13) in (let H21 \def (eq_ind T (THead (Bind b) u (lift (S O) O t3)) 

-(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
-(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | 
-(Flat _) \Rightarrow False])])) I (THead (Flat Appl) v1 (THead (Bind b0) u1 
-t5)) H20) in (False_ind (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: 
-T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t6)) t7))) 
-H21))))))) t2 H15)) t H13 H14 H9 H10 H11 H12))) | (pr0_delta u1 u2 H9 t5 t6 
-H10 w H11) \Rightarrow (\lambda (H12: (eq T (THead (Bind Abbr) u1 t5) 
-t)).(\lambda (H13: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind 
-Abbr) u1 t5) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2) \to ((pr0 u1 
-u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 
-t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda (H14: (eq T (THead (Bind 
-Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t7: T).((pr0 u1 
-u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 
-t1 t8)) (\lambda (t8: T).(pr0 t7 t8))))))) (\lambda (_: (pr0 u1 u2)).(\lambda 
-(H16: (pr0 t5 t6)).(\lambda (H17: (subst0 O u2 t6 w)).(let H18 \def (eq_ind_r 
-T t (\lambda (t7: T).(eq T (THead (Bind b) u (lift (S O) O t3)) t7)) H4 
-(THead (Bind Abbr) u1 t5) H12) in (let H19 \def (f_equal T B (\lambda (e: 
-T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | 
-(TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return 
-(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow 
-b])])) (THead (Bind b) u (lift (S O) O t3)) (THead (Bind Abbr) u1 t5) H18) in 
-((let H20 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: 
-T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t7 
-_) \Rightarrow t7])) (THead (Bind b) u (lift (S O) O t3)) (THead (Bind Abbr) 
-u1 t5) H18) in ((let H21 \def (f_equal T T (\lambda (e: T).(match e in T 
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: 
-((nat \to nat))) (d: nat) (t7: T) on t7: T \def (match t7 with [(TSort n) 
-\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with 
-[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t8) 
-\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t8))]) in 
-lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (TLRef _) \Rightarrow 
-((let rec lref_map (f: ((nat \to nat))) (d: nat) (t7: T) on t7: T \def (match 
-t7 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef 
-(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | 
-(THead k u0 t8) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) 
-t8))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (THead _ _ t7) 
-\Rightarrow t7])) (THead (Bind b) u (lift (S O) O t3)) (THead (Bind Abbr) u1 
-t5) H18) in (\lambda (_: (eq T u u1)).(\lambda (H23: (eq B b Abbr)).(let H24 
-\def (eq_ind_r T t5 (\lambda (t7: T).(pr0 t7 t6)) H16 (lift (S O) O t3) H21) 
-in (ex2_ind T (\lambda (t7: T).(eq T t6 (lift (S O) O t7))) (\lambda (t7: 
-T).(pr0 t3 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 
-(THead (Bind Abbr) u2 w) t7))) (\lambda (x: T).(\lambda (H25: (eq T t6 (lift 
-(S O) O x))).(\lambda (H26: (pr0 t3 x)).(let H27 \def (eq_ind_r T t5 (\lambda 
-(t7: T).(eq T (THead (Bind Abbr) u1 t7) t)) H12 (lift (S O) O t3) H21) in 
-(let H28 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to 
-(\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T 
-(\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H 
-(THead (Bind Abbr) u1 (lift (S O) O t3)) H27) in (let H29 \def (eq_ind T t6 
-(\lambda (t7: T).(subst0 O u2 t7 w)) H17 (lift (S O) O x) H25) in (let H30 
-\def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H7 Abbr H23) in 
-(ex2_ind T (\lambda (t7: T).(pr0 x t7)) (\lambda (t7: T).(pr0 t1 t7)) (ex2 T 
-(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) 
-t7))) (\lambda (x0: T).(\lambda (_: (pr0 x x0)).(\lambda (_: (pr0 t1 
-x0)).(ex2_sym T (pr0 (THead (Bind Abbr) u2 w)) (pr0 t1) 
+(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _) 
+\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) v1 
+(THead (Bind b0) u1 t5)) H20) in (False_ind (ex2 T (\lambda (t7: T).(pr0 t1 
+t7)) (\lambda (t7: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) 
+O v2) t6)) t7))) H21))))))) t2 H15)) t H13 H14 H9 H10 H11 H12))) | (pr0_delta 
+u1 u2 H9 t5 t6 H10 w H11) \Rightarrow (\lambda (H12: (eq T (THead (Bind Abbr) 
+u1 t5) t)).(\lambda (H13: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T 
+(THead (Bind Abbr) u1 t5) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2) 
+\to ((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T (\lambda 
+(t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda (H14: (eq T 
+(THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda 
+(t7: T).((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T 
+(\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8))))))) (\lambda (_: 
+(pr0 u1 u2)).(\lambda (H16: (pr0 t5 t6)).(\lambda (H17: (subst0 O u2 t6 
+w)).(let H18 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind b) u 
+(lift (S O) O t3)) t7)) H4 (THead (Bind Abbr) u1 t5) H12) in (let H19 \def 
+(f_equal T B (\lambda (e: T).(match e with [(TSort _) \Rightarrow b | (TLRef 
+_) \Rightarrow b | (THead k _ _) \Rightarrow (match k with [(Bind b0) 
+\Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead (Bind b) u (lift (S O) O 
+t3)) (THead (Bind Abbr) u1 t5) H18) in ((let H20 \def (f_equal T T (\lambda 
+(e: T).(match e with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | 
+(THead _ t7 _) \Rightarrow t7])) (THead (Bind b) u (lift (S O) O t3)) (THead 
+(Bind Abbr) u1 t5) H18) in ((let H21 \def (f_equal T T (\lambda (e: T).(match 
+e with [(TSort _) \Rightarrow (lref_map (\lambda (x: nat).(plus x (S O))) O 
+t3) | (TLRef _) \Rightarrow (lref_map (\lambda (x: nat).(plus x (S O))) O t3) 
+| (THead _ _ t7) \Rightarrow t7])) (THead (Bind b) u (lift (S O) O t3)) 
+(THead (Bind Abbr) u1 t5) H18) in (\lambda (_: (eq T u u1)).(\lambda (H23: 
+(eq B b Abbr)).(let H24 \def (eq_ind_r T t5 (\lambda (t7: T).(pr0 t7 t6)) H16 
+(lift (S O) O t3) H21) in (ex2_ind T (\lambda (t7: T).(eq T t6 (lift (S O) O 
+t7))) (\lambda (t7: T).(pr0 t3 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) 
+(\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7))) (\lambda (x: T).(\lambda 
+(H25: (eq T t6 (lift (S O) O x))).(\lambda (H26: (pr0 t3 x)).(let H27 \def 
+(eq_ind_r T t5 (\lambda (t7: T).(eq T (THead (Bind Abbr) u1 t7) t)) H12 (lift 
+(S O) O t3) H21) in (let H28 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: 
+T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v 
+t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 
+t10)))))))))) H (THead (Bind Abbr) u1 (lift (S O) O t3)) H27) in (let H29 
+\def (eq_ind T t6 (\lambda (t7: T).(subst0 O u2 t7 w)) H17 (lift (S O) O x) 
+H25) in (let H30 \def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H7 
+Abbr H23) in (ex2_ind T (\lambda (t7: T).(pr0 x t7)) (\lambda (t7: T).(pr0 t1 
+t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Bind 
+Abbr) u2 w) t7))) (\lambda (x0: T).(\lambda (_: (pr0 x x0)).(\lambda (_: (pr0 
+t1 x0)).(ex2_sym T (pr0 (THead (Bind Abbr) u2 w)) (pr0 t1) 

 (pr0_confluence__pr0_delta_tau u2 (lift (S O) O x) w H29 x (pr0_refl (lift (S 
 O) O x)) t1))))) (H28 t3 (lift_tlt_dx (Bind Abbr) u1 t3 (S O) O) x H26 t1 
 H8))))))))) (pr0_gen_lift t3 t6 (S O) O H24)))))) H20)) H19)))))) t2 H14)) t 
 (pr0_confluence__pr0_delta_tau u2 (lift (S O) O x) w H29 x (pr0_refl (lift (S 
 O) O x)) t1))))) (H28 t3 (lift_tlt_dx (Bind Abbr) u1 t3 (S O) O) x H26 t1 
 H8))))))))) (pr0_gen_lift t3 t6 (S O) O H24)))))) H20)) H19)))))) t2 H14)) t 


@@ -2296,212 +2114,190 @@ t2)).(eq_ind T t2 (\lambda (t7: T).((not (eq B b0 Abst)) \to ((pr0 t5 t7) \to
 (\lambda (_: (not (eq B b0 Abst))).(\lambda (H15: (pr0 t5 t2)).(let H16 \def 
 (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind b) u (lift (S O) O t3)) 
 t7)) H4 (THead (Bind b0) u0 (lift (S O) O t5)) H11) in (let H17 \def (f_equal 
 (\lambda (_: (not (eq B b0 Abst))).(\lambda (H15: (pr0 t5 t2)).(let H16 \def 
 (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind b) u (lift (S O) O t3)) 
 t7)) H4 (THead (Bind b0) u0 (lift (S O) O t5)) H11) in (let H17 \def (f_equal 

-T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) 
-\Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k 
-in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) 
-\Rightarrow b])])) (THead (Bind b) u (lift (S O) O t3)) (THead (Bind b0) u0 
-(lift (S O) O t5)) H16) in ((let H18 \def (f_equal T T (\lambda (e: T).(match 
-e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) 
+T B (\lambda (e: T).(match e with [(TSort _) \Rightarrow b | (TLRef _) 
+\Rightarrow b | (THead k _ _) \Rightarrow (match k with [(Bind b1) 
+\Rightarrow b1 | (Flat _) \Rightarrow b])])) (THead (Bind b) u (lift (S O) O 
+t3)) (THead (Bind b0) u0 (lift (S O) O t5)) H16) in ((let H18 \def (f_equal T 
+T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u | (TLRef _) 

 \Rightarrow u | (THead _ t7 _) \Rightarrow t7])) (THead (Bind b) u (lift (S 
 O) O t3)) (THead (Bind b0) u0 (lift (S O) O t5)) H16) in ((let H19 \def 
 \Rightarrow u | (THead _ t7 _) \Rightarrow t7])) (THead (Bind b) u (lift (S 
 O) O t3)) (THead (Bind b0) u0 (lift (S O) O t5)) H16) in ((let H19 \def 

-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t7: 
-T) on t7: T \def (match t7 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) 
-\Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false 
-\Rightarrow (f i)])) | (THead k u1 t8) \Rightarrow (THead k (lref_map f d u1) 
-(lref_map f (s k d) t8))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O 
-t3) | (TLRef _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) 
-(t7: T) on t7: T \def (match t7 with [(TSort n) \Rightarrow (TSort n) | 
-(TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | 
-false \Rightarrow (f i)])) | (THead k u1 t8) \Rightarrow (THead k (lref_map f 
-d u1) (lref_map f (s k d) t8))]) in lref_map) (\lambda (x: nat).(plus x (S 
-O))) O t3) | (THead _ _ t7) \Rightarrow t7])) (THead (Bind b) u (lift (S O) O 
-t3)) (THead (Bind b0) u0 (lift (S O) O t5)) H16) in (\lambda (_: (eq T u 
-u0)).(\lambda (H21: (eq B b b0)).(let H22 \def (eq_ind_r T t (\lambda (t7: 
-T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall 
-(t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: 
-T).(pr0 t9 t10)))))))))) H (THead (Bind b0) u0 (lift (S O) O t5)) H11) in 
-(let H23 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t1)) H8 t5 (lift_inj t3 
-t5 (S O) O H19)) in (let H24 \def (eq_ind B b (\lambda (b1: B).(not (eq B b1 
-Abst))) H7 b0 H21) in (ex2_ind T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: 
-T).(pr0 t2 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 
-t7))) (\lambda (x: T).(\lambda (H25: (pr0 t1 x)).(\lambda (H26: (pr0 t2 
-x)).(ex_intro2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)) 
-x H25 H26)))) (H22 t5 (lift_tlt_dx (Bind b0) u0 t5 (S O) O) t1 H23 t2 
-H15)))))))) H18)) H17))))) t6 (sym_eq T t6 t2 H13))) t H11 H12 H9 H10))) | 
-(pr0_tau t5 t6 H9 u0) \Rightarrow (\lambda (H10: (eq T (THead (Flat Cast) u0 
-t5) t)).(\lambda (H11: (eq T t6 t2)).(eq_ind T (THead (Flat Cast) u0 t5) 
-(\lambda (_: T).((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: 
-T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H12: (eq T t6 
-t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: 
-T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))) (\lambda (_: (pr0 t5 
-t2)).(let H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind b) u 
-(lift (S O) O t3)) t7)) H4 (THead (Flat Cast) u0 t5) H10) in (let H15 \def 
-(eq_ind T (THead (Bind b) u (lift (S O) O t3)) (\lambda (ee: T).(match ee in 
-T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda 
-(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow 
-False])])) I (THead (Flat Cast) u0 t5) H14) in (False_ind (ex2 T (\lambda 
-(t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))) H15)))) t6 (sym_eq T t6 
-t2 H12))) t H10 H11 H9)))]) in (H9 (refl_equal T t) (refl_equal T t2))))) t4 
-(sym_eq T t4 t1 H6))) t H4 H5 H2 H3))) | (pr0_tau t3 t4 H2 u) \Rightarrow 
-(\lambda (H3: (eq T (THead (Flat Cast) u t3) t)).(\lambda (H4: (eq T t4 
-t1)).(eq_ind T (THead (Flat Cast) u t3) (\lambda (_: T).((eq T t4 t1) \to 
-((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 
-t2 t6)))))) (\lambda (H5: (eq T t4 t1)).(eq_ind T t1 (\lambda (t5: T).((pr0 
-t3 t5) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 
-t6))))) (\lambda (H6: (pr0 t3 t1)).(let H7 \def (match H1 in pr0 return 
-(\lambda (t5: T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to 
-((eq T t6 t2) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 
-t2 t7)))))))) with [(pr0_refl t5) \Rightarrow (\lambda (H7: (eq T t5 
-t)).(\lambda (H8: (eq T t5 t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2) 
-\to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))))) 
-(\lambda (H9: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t7: 
-T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let H10 \def (eq_ind_r T t 
-(\lambda (t6: T).(eq T t6 t2)) H9 (THead (Flat Cast) u t3) H3) in (eq_ind T 
-(THead (Flat Cast) u t3) (\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 t1 
-t7)) (\lambda (t7: T).(pr0 t6 t7)))) (let H11 \def (eq_ind_r T t (\lambda 
-(t6: T).(eq T t5 t6)) H7 (THead (Flat Cast) u t3) H3) in (let H12 \def 
-(eq_ind_r T t (\lambda (t6: T).(\forall (v: T).((tlt v t6) \to (\forall (t7: 
-T).((pr0 v t7) \to (\forall (t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: 
-T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead (Flat Cast) u 
-t3) H3) in (ex_intro2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 
-(THead (Flat Cast) u t3) t6)) t1 (pr0_refl t1) (pr0_tau t3 t1 H6 u)))) t2 
-H10)) t (sym_eq T t t2 H9))) t5 (sym_eq T t5 t H7) H8))) | (pr0_comp u1 u2 H7 
-t5 t6 H8 k) \Rightarrow (\lambda (H9: (eq T (THead k u1 t5) t)).(\lambda 
-(H10: (eq T (THead k u2 t6) t2)).(eq_ind T (THead k u1 t5) (\lambda (_: 
-T).((eq T (THead k u2 t6) t2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T 
-(\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda 
-(H11: (eq T (THead k u2 t6) t2)).(eq_ind T (THead k u2 t6) (\lambda (t7: 
-T).((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) 
-(\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (_: (pr0 u1 u2)).(\lambda (H13: 
-(pr0 t5 t6)).(let H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat 
-Cast) u t3) t7)) H3 (THead k u1 t5) H9) in (let H15 \def (f_equal T K 
-(\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) 
-\Rightarrow (Flat Cast) | (TLRef _) \Rightarrow (Flat Cast) | (THead k0 _ _) 
-\Rightarrow k0])) (THead (Flat Cast) u t3) (THead k u1 t5) H14) in ((let H16 
-\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow (lref_map 
+(\lambda (x: nat).(plus x (S O))) O t3) | (TLRef _) \Rightarrow (lref_map 
+(\lambda (x: nat).(plus x (S O))) O t3) | (THead _ _ t7) \Rightarrow t7])) 
+(THead (Bind b) u (lift (S O) O t3)) (THead (Bind b0) u0 (lift (S O) O t5)) 
+H16) in (\lambda (_: (eq T u u0)).(\lambda (H21: (eq B b b0)).(let H22 \def 
+(eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: 
+T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: 
+T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Bind b0) 
+u0 (lift (S O) O t5)) H11) in (let H23 \def (eq_ind T t3 (\lambda (t7: 
+T).(pr0 t7 t1)) H8 t5 (lift_inj t3 t5 (S O) O H19)) in (let H24 \def (eq_ind 
+B b (\lambda (b1: B).(not (eq B b1 Abst))) H7 b0 H21) in (ex2_ind T (\lambda 
+(t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)) (ex2 T (\lambda (t7: 
+T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))) (\lambda (x: T).(\lambda (H25: 
+(pr0 t1 x)).(\lambda (H26: (pr0 t2 x)).(ex_intro2 T (\lambda (t7: T).(pr0 t1 
+t7)) (\lambda (t7: T).(pr0 t2 t7)) x H25 H26)))) (H22 t5 (lift_tlt_dx (Bind 
+b0) u0 t5 (S O) O) t1 H23 t2 H15)))))))) H18)) H17))))) t6 (sym_eq T t6 t2 
+H13))) t H11 H12 H9 H10))) | (pr0_tau t5 t6 H9 u0) \Rightarrow (\lambda (H10: 
+(eq T (THead (Flat Cast) u0 t5) t)).(\lambda (H11: (eq T t6 t2)).(eq_ind T 
+(THead (Flat Cast) u0 t5) (\lambda (_: T).((eq T t6 t2) \to ((pr0 t5 t6) \to 
+(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))) 
+(\lambda (H12: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 t7) \to 
+(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))) 
+(\lambda (_: (pr0 t5 t2)).(let H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T 
+(THead (Bind b) u (lift (S O) O t3)) t7)) H4 (THead (Flat Cast) u0 t5) H10) 
+in (let H15 \def (eq_ind T (THead (Bind b) u (lift (S O) O t3)) (\lambda (ee: 
+T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
+| (THead k _ _) \Rightarrow (match k with [(Bind _) \Rightarrow True | (Flat 
+_) \Rightarrow False])])) I (THead (Flat Cast) u0 t5) H14) in (False_ind (ex2 
+T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))) H15)))) t6 
+(sym_eq T t6 t2 H12))) t H10 H11 H9)))]) in (H9 (refl_equal T t) (refl_equal 
+T t2))))) t4 (sym_eq T t4 t1 H6))) t H4 H5 H2 H3))) | (pr0_tau t3 t4 H2 u) 
+\Rightarrow (\lambda (H3: (eq T (THead (Flat Cast) u t3) t)).(\lambda (H4: 
+(eq T t4 t1)).(eq_ind T (THead (Flat Cast) u t3) (\lambda (_: T).((eq T t4 
+t1) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: 
+T).(pr0 t2 t6)))))) (\lambda (H5: (eq T t4 t1)).(eq_ind T t1 (\lambda (t5: 
+T).((pr0 t3 t5) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: 
+T).(pr0 t2 t6))))) (\lambda (H6: (pr0 t3 t1)).(let H7 \def (match H1 with 
+[(pr0_refl t5) \Rightarrow (\lambda (H7: (eq T t5 t)).(\lambda (H8: (eq T t5 
+t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2) \to (ex2 T (\lambda (t7: 
+T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))))) (\lambda (H9: (eq T t 
+t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t7: T).(pr0 t1 t7)) 
+(\lambda (t7: T).(pr0 t2 t7)))) (let H10 \def (eq_ind_r T t (\lambda (t6: 
+T).(eq T t6 t2)) H9 (THead (Flat Cast) u t3) H3) in (eq_ind T (THead (Flat 
+Cast) u t3) (\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda 
+(t7: T).(pr0 t6 t7)))) (let H11 \def (eq_ind_r T t (\lambda (t6: T).(eq T t5 
+t6)) H7 (THead (Flat Cast) u t3) H3) in (let H12 \def (eq_ind_r T t (\lambda 
+(t6: T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to 
+(\forall (t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) 
+(\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead (Flat Cast) u t3) H3) in 
+(ex_intro2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 (THead (Flat 
+Cast) u t3) t6)) t1 (pr0_refl t1) (pr0_tau t3 t1 H6 u)))) t2 H10)) t (sym_eq 
+T t t2 H9))) t5 (sym_eq T t5 t H7) H8))) | (pr0_comp u1 u2 H7 t5 t6 H8 k) 
+\Rightarrow (\lambda (H9: (eq T (THead k u1 t5) t)).(\lambda (H10: (eq T 
+(THead k u2 t6) t2)).(eq_ind T (THead k u1 t5) (\lambda (_: T).((eq T (THead 
+k u2 t6) t2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: 
+T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H11: (eq T 
+(THead k u2 t6) t2)).(eq_ind T (THead k u2 t6) (\lambda (t7: T).((pr0 u1 u2) 
+\to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: 
+T).(pr0 t7 t8)))))) (\lambda (_: (pr0 u1 u2)).(\lambda (H13: (pr0 t5 
+t6)).(let H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Cast) u 
+t3) t7)) H3 (THead k u1 t5) H9) in (let H15 \def (f_equal T K (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow (Flat Cast) | (TLRef _) \Rightarrow 
+(Flat Cast) | (THead k0 _ _) \Rightarrow k0])) (THead (Flat Cast) u t3) 
+(THead k u1 t5) H14) in ((let H16 \def (f_equal T T (\lambda (e: T).(match e 

 with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t7 _) 
 \Rightarrow t7])) (THead (Flat Cast) u t3) (THead k u1 t5) H14) in ((let H17 
 with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t7 _) 
 \Rightarrow t7])) (THead (Flat Cast) u t3) (THead k u1 t5) H14) in ((let H17 

-\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) 
-with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7) 
-\Rightarrow t7])) (THead (Flat Cast) u t3) (THead k u1 t5) H14) in (\lambda 
-(_: (eq T u u1)).(\lambda (H19: (eq K (Flat Cast) k)).(eq_ind K (Flat Cast) 
-(\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 
-(THead k0 u2 t6) t7)))) (let H20 \def (eq_ind_r K k (\lambda (k0: K).(eq T 
-(THead k0 u1 t5) t)) H9 (Flat Cast) H19) in (let H21 \def (eq_ind_r T t 
-(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) 
-\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) 
-(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Cast) u1 t5) H20) in 
-(let H22 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t1)) H6 t5 H17) in 
-(ex2_ind T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T 
+\def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t3 | 
+(TLRef _) \Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead (Flat 
+Cast) u t3) (THead k u1 t5) H14) in (\lambda (_: (eq T u u1)).(\lambda (H19: 
+(eq K (Flat Cast) k)).(eq_ind K (Flat Cast) (\lambda (k0: K).(ex2 T (\lambda 
+(t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead k0 u2 t6) t7)))) (let H20 
+\def (eq_ind_r K k (\lambda (k0: K).(eq T (THead k0 u1 t5) t)) H9 (Flat Cast) 
+H19) in (let H21 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v 
+t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to 
+(ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 
+t10)))))))))) H (THead (Flat Cast) u1 t5) H20) in (let H22 \def (eq_ind T t3 
+(\lambda (t7: T).(pr0 t7 t1)) H6 t5 H17) in (ex2_ind T (\lambda (t7: T).(pr0 
+t1 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) 
+(\lambda (t7: T).(pr0 (THead (Flat Cast) u2 t6) t7))) (\lambda (x: 
+T).(\lambda (H23: (pr0 t1 x)).(\lambda (H24: (pr0 t6 x)).(ex_intro2 T 

 (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Flat Cast) u2 t6) 
 (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Flat Cast) u2 t6) 

-t7))) (\lambda (x: T).(\lambda (H23: (pr0 t1 x)).(\lambda (H24: (pr0 t6 
-x)).(ex_intro2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead 
-(Flat Cast) u2 t6) t7)) x H23 (pr0_tau t6 x H24 u2))))) (H21 t5 (tlt_head_dx 
-(Flat Cast) u1 t5) t1 H22 t6 H13))))) k H19)))) H16)) H15))))) t2 H11)) t H9 
-H10 H7 H8))) | (pr0_beta u0 v1 v2 H7 t5 t6 H8) \Rightarrow (\lambda (H9: (eq 
-T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t5)) t)).(\lambda (H10: (eq T 
-(THead (Bind Abbr) v2 t6) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind 
-Abst) u0 t5)) (\lambda (_: T).((eq T (THead (Bind Abbr) v2 t6) t2) \to ((pr0 
-v1 v2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda 
-(t8: T).(pr0 t2 t8))))))) (\lambda (H11: (eq T (THead (Bind Abbr) v2 t6) 
-t2)).(eq_ind T (THead (Bind Abbr) v2 t6) (\lambda (t7: T).((pr0 v1 v2) \to 
+t7)) x H23 (pr0_tau t6 x H24 u2))))) (H21 t5 (tlt_head_dx (Flat Cast) u1 t5) 
+t1 H22 t6 H13))))) k H19)))) H16)) H15))))) t2 H11)) t H9 H10 H7 H8))) | 
+(pr0_beta u0 v1 v2 H7 t5 t6 H8) \Rightarrow (\lambda (H9: (eq T (THead (Flat 
+Appl) v1 (THead (Bind Abst) u0 t5)) t)).(\lambda (H10: (eq T (THead (Bind 
+Abbr) v2 t6) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t5)) 
+(\lambda (_: T).((eq T (THead (Bind Abbr) v2 t6) t2) \to ((pr0 v1 v2) \to 

 ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 
 ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 

-t7 t8)))))) (\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 t5 t6)).(let H14 \def 
-(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Cast) u t3) t7)) H3 (THead 
-(Flat Appl) v1 (THead (Bind Abst) u0 t5)) H9) in (let H15 \def (eq_ind T 
-(THead (Flat Cast) u t3) (\lambda (ee: T).(match ee in T return (\lambda (_: 
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
-(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with 
-[(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f in F return 
-(\lambda (_: F).Prop) with [Appl \Rightarrow False | Cast \Rightarrow 
-True])])])) I (THead (Flat Appl) v1 (THead (Bind Abst) u0 t5)) H14) in 
-(False_ind (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead 
-(Bind Abbr) v2 t6) t7))) H15))))) t2 H11)) t H9 H10 H7 H8))) | (pr0_upsilon b 
-H7 v1 v2 H8 u1 u2 H9 t5 t6 H10) \Rightarrow (\lambda (H11: (eq T (THead (Flat 
-Appl) v1 (THead (Bind b) u1 t5)) t)).(\lambda (H12: (eq T (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Flat Appl) 
-v1 (THead (Bind b) u1 t5)) (\lambda (_: T).((eq T (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t6)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 
-v2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 
-t8)) (\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H13: (eq T (THead (Bind 
-b) u2 (THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Bind 
-b) u2 (THead (Flat Appl) (lift (S O) O v2) t6)) (\lambda (t7: T).((not (eq B 
-b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T 
-(\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda 
-(_: (not (eq B b Abst))).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 u1 
-u2)).(\lambda (_: (pr0 t5 t6)).(let H18 \def (eq_ind_r T t (\lambda (t7: 
-T).(eq T (THead (Flat Cast) u t3) t7)) H3 (THead (Flat Appl) v1 (THead (Bind 
-b) u1 t5)) H11) in (let H19 \def (eq_ind T (THead (Flat Cast) u t3) (\lambda 
-(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
-(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | 
-(Flat f) \Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl 
-\Rightarrow False | Cast \Rightarrow True])])])) I (THead (Flat Appl) v1 
-(THead (Bind b) u1 t5)) H18) in (False_ind (ex2 T (\lambda (t7: T).(pr0 t1 
-t7)) (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) 
-O v2) t6)) t7))) H19))))))) t2 H13)) t H11 H12 H7 H8 H9 H10))) | (pr0_delta 
-u1 u2 H7 t5 t6 H8 w H9) \Rightarrow (\lambda (H10: (eq T (THead (Bind Abbr) 
-u1 t5) t)).(\lambda (H11: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T 
-(THead (Bind Abbr) u1 t5) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2) 
-\to ((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T (\lambda 
-(t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda (H12: (eq T 
-(THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda 
-(t7: T).((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T 
-(\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8))))))) (\lambda (_: 
-(pr0 u1 u2)).(\lambda (_: (pr0 t5 t6)).(\lambda (_: (subst0 O u2 t6 w)).(let 
-H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Cast) u t3) t7)) 
-H3 (THead (Bind Abbr) u1 t5) H10) in (let H17 \def (eq_ind T (THead (Flat 
-Cast) u t3) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with 
+t2 t8))))))) (\lambda (H11: (eq T (THead (Bind Abbr) v2 t6) t2)).(eq_ind T 
+(THead (Bind Abbr) v2 t6) (\lambda (t7: T).((pr0 v1 v2) \to ((pr0 t5 t6) \to 
+(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8)))))) 
+(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 t5 t6)).(let H14 \def (eq_ind_r T 
+t (\lambda (t7: T).(eq T (THead (Flat Cast) u t3) t7)) H3 (THead (Flat Appl) 
+v1 (THead (Bind Abst) u0 t5)) H9) in (let H15 \def (eq_ind T (THead (Flat 
+Cast) u t3) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | 
+(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind 
+_) \Rightarrow False | (Flat f) \Rightarrow (match f with [Appl \Rightarrow 
+False | Cast \Rightarrow True])])])) I (THead (Flat Appl) v1 (THead (Bind 
+Abst) u0 t5)) H14) in (False_ind (ex2 T (\lambda (t7: T).(pr0 t1 t7)) 
+(\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t6) t7))) H15))))) t2 H11)) t H9 
+H10 H7 H8))) | (pr0_upsilon b H7 v1 v2 H8 u1 u2 H9 t5 t6 H10) \Rightarrow 
+(\lambda (H11: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t5)) 
+t)).(\lambda (H12: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O 
+v2) t6)) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t5)) 
+(\lambda (_: T).((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O 
+v2) t6)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to 
+((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 
+t2 t8))))))))) (\lambda (H13: (eq T (THead (Bind b) u2 (THead (Flat Appl) 
+(lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Bind b) u2 (THead (Flat Appl) 
+(lift (S O) O v2) t6)) (\lambda (t7: T).((not (eq B b Abst)) \to ((pr0 v1 v2) 
+\to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) 
+(\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda (_: (not (eq B b 
+Abst))).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0 
+t5 t6)).(let H18 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Cast) 
+u t3) t7)) H3 (THead (Flat Appl) v1 (THead (Bind b) u1 t5)) H11) in (let H19 
+\def (eq_ind T (THead (Flat Cast) u t3) (\lambda (ee: T).(match ee with 

 [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
 [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 

-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) 
-\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1 
-t5) H16) in (False_ind (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: 
-T).(pr0 (THead (Bind Abbr) u2 w) t7))) H17)))))) t2 H12)) t H10 H11 H7 H8 
-H9))) | (pr0_zeta b H7 t5 t6 H8 u0) \Rightarrow (\lambda (H9: (eq T (THead 
-(Bind b) u0 (lift (S O) O t5)) t)).(\lambda (H10: (eq T t6 t2)).(eq_ind T 
-(THead (Bind b) u0 (lift (S O) O t5)) (\lambda (_: T).((eq T t6 t2) \to ((not 
-(eq B b Abst)) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) 
-(\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H11: (eq T t6 t2)).(eq_ind T t2 
-(\lambda (t7: T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda 
-(t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (_: (not (eq 
-B b Abst))).(\lambda (_: (pr0 t5 t2)).(let H14 \def (eq_ind_r T t (\lambda 
-(t7: T).(eq T (THead (Flat Cast) u t3) t7)) H3 (THead (Bind b) u0 (lift (S O) 
-O t5)) H9) in (let H15 \def (eq_ind T (THead (Flat Cast) u t3) (\lambda (ee: 
-T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K 
-return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) 
-\Rightarrow True])])) I (THead (Bind b) u0 (lift (S O) O t5)) H14) in 
-(False_ind (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 
-t7))) H15))))) t6 (sym_eq T t6 t2 H11))) t H9 H10 H7 H8))) | (pr0_tau t5 t6 
-H7 u0) \Rightarrow (\lambda (H8: (eq T (THead (Flat Cast) u0 t5) t)).(\lambda 
-(H9: (eq T t6 t2)).(eq_ind T (THead (Flat Cast) u0 t5) (\lambda (_: T).((eq T 
-t6 t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda 
-(t8: T).(pr0 t2 t8)))))) (\lambda (H10: (eq T t6 t2)).(eq_ind T t2 (\lambda 
-(t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: 
-T).(pr0 t2 t8))))) (\lambda (H11: (pr0 t5 t2)).(let H12 \def (eq_ind_r T t 
-(\lambda (t7: T).(eq T (THead (Flat Cast) u t3) t7)) H3 (THead (Flat Cast) u0 
-t5) H8) in (let H13 \def (f_equal T T (\lambda (e: T).(match e in T return 
-(\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | 
-(THead _ t7 _) \Rightarrow t7])) (THead (Flat Cast) u t3) (THead (Flat Cast) 
-u0 t5) H12) in ((let H14 \def (f_equal T T (\lambda (e: T).(match e in T 
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) 
-\Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead (Flat Cast) u t3) 
-(THead (Flat Cast) u0 t5) H12) in (\lambda (_: (eq T u u0)).(let H16 \def 
-(eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: 
-T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: 
-T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Cast) 
-u0 t5) H8) in (let H17 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t1)) H6 t5 
-H14) in (ex2_ind T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 
-t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))) 
-(\lambda (x: T).(\lambda (H18: (pr0 t1 x)).(\lambda (H19: (pr0 t2 
-x)).(ex_intro2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)) 
-x H18 H19)))) (H16 t5 (tlt_head_dx (Flat Cast) u0 t5) t1 H17 t2 H11)))))) 
-H13)))) t6 (sym_eq T t6 t2 H10))) t H8 H9 H7)))]) in (H7 (refl_equal T t) 
-(refl_equal T t2)))) t4 (sym_eq T t4 t1 H5))) t H3 H4 H2)))]) in (H2 
-(refl_equal T t) (refl_equal T t1))))))))) t0).
-(* COMMENTS
-Initial nodes: 46103
-END *)
+\Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow 
+(match f with [Appl \Rightarrow False | Cast \Rightarrow True])])])) I (THead 
+(Flat Appl) v1 (THead (Bind b) u1 t5)) H18) in (False_ind (ex2 T (\lambda 
+(t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t6)) t7))) H19))))))) t2 H13)) t H11 H12 H7 H8 H9 
+H10))) | (pr0_delta u1 u2 H7 t5 t6 H8 w H9) \Rightarrow (\lambda (H10: (eq T 
+(THead (Bind Abbr) u1 t5) t)).(\lambda (H11: (eq T (THead (Bind Abbr) u2 w) 
+t2)).(eq_ind T (THead (Bind Abbr) u1 t5) (\lambda (_: T).((eq T (THead (Bind 
+Abbr) u2 w) t2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to 
+(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) 
+(\lambda (H12: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind 
+Abbr) u2 w) (\lambda (t7: T).((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 
+t6 w) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 
+t8))))))) (\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0 t5 t6)).(\lambda (_: 
+(subst0 O u2 t6 w)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead 
+(Flat Cast) u t3) t7)) H3 (THead (Bind Abbr) u1 t5) H10) in (let H17 \def 
+(eq_ind T (THead (Flat Cast) u t3) (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
+(match k with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I 
+(THead (Bind Abbr) u1 t5) H16) in (False_ind (ex2 T (\lambda (t7: T).(pr0 t1 
+t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7))) H17)))))) t2 H12)) 
+t H10 H11 H7 H8 H9))) | (pr0_zeta b H7 t5 t6 H8 u0) \Rightarrow (\lambda (H9: 
+(eq T (THead (Bind b) u0 (lift (S O) O t5)) t)).(\lambda (H10: (eq T t6 
+t2)).(eq_ind T (THead (Bind b) u0 (lift (S O) O t5)) (\lambda (_: T).((eq T 
+t6 t2) \to ((not (eq B b Abst)) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: 
+T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H11: (eq T t6 
+t2)).(eq_ind T t2 (\lambda (t7: T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to 
+(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))) 
+(\lambda (_: (not (eq B b Abst))).(\lambda (_: (pr0 t5 t2)).(let H14 \def 
+(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Cast) u t3) t7)) H3 (THead 
+(Bind b) u0 (lift (S O) O t5)) H9) in (let H15 \def (eq_ind T (THead (Flat 
+Cast) u t3) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | 
+(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind 
+_) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u0 
+(lift (S O) O t5)) H14) in (False_ind (ex2 T (\lambda (t7: T).(pr0 t1 t7)) 
+(\lambda (t7: T).(pr0 t2 t7))) H15))))) t6 (sym_eq T t6 t2 H11))) t H9 H10 H7 
+H8))) | (pr0_tau t5 t6 H7 u0) \Rightarrow (\lambda (H8: (eq T (THead (Flat 
+Cast) u0 t5) t)).(\lambda (H9: (eq T t6 t2)).(eq_ind T (THead (Flat Cast) u0 
+t5) (\lambda (_: T).((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: 
+T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H10: (eq T t6 
+t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: 
+T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))) (\lambda (H11: (pr0 t5 
+t2)).(let H12 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Cast) u 
+t3) t7)) H3 (THead (Flat Cast) u0 t5) H8) in (let H13 \def (f_equal T T 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow u | (TLRef _) 
+\Rightarrow u | (THead _ t7 _) \Rightarrow t7])) (THead (Flat Cast) u t3) 
+(THead (Flat Cast) u0 t5) H12) in ((let H14 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | 
+(THead _ _ t7) \Rightarrow t7])) (THead (Flat Cast) u t3) (THead (Flat Cast) 
+u0 t5) H12) in (\lambda (_: (eq T u u0)).(let H16 \def (eq_ind_r T t (\lambda 
+(t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to 
+(\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) 
+(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Cast) u0 t5) H8) in 
+(let H17 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t1)) H6 t5 H14) in 
+(ex2_ind T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)) (ex2 T 
+(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))) (\lambda (x: 
+T).(\lambda (H18: (pr0 t1 x)).(\lambda (H19: (pr0 t2 x)).(ex_intro2 T 
+(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)) x H18 H19)))) 
+(H16 t5 (tlt_head_dx (Flat Cast) u0 t5) t1 H17 t2 H11)))))) H13)))) t6 
+(sym_eq T t6 t2 H10))) t H8 H9 H7)))]) in (H7 (refl_equal T t) (refl_equal T 
+t2)))) t4 (sym_eq T t4 t1 H5))) t H3 H4 H2)))]) in (H2 (refl_equal T t) 
+(refl_equal T t1))))))))) t0).

 
 

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr0/props.ma b/matita/matita/contribs/lambdadelta/basic_1/pr0/props.ma
index d7f69d691abb71849d095ce10739f316e6dd9c78..a99bdf288154c419401c6c4fb59921443c5c4c98 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_1/pr0/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/pr0/props.ma
@@ -14,1745 +14,1002 @@
 
 (* This file was automatically generated: do not edit *********************)
 
 
 (* This file was automatically generated: do not edit *********************)
 

-include "Basic-1/pr0/defs.ma".
+include "basic_1/pr0/fwd.ma".

 
 

-include "Basic-1/subst0/subst0.ma".
+include "basic_1/subst0/props.ma".

 
 theorem pr0_lift:
  \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (h: nat).(\forall 
 (d: nat).(pr0 (lift h d t1) (lift h d t2))))))
 \def
 
 theorem pr0_lift:
  \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (h: nat).(\forall 
 (d: nat).(pr0 (lift h d t1) (lift h d t2))))))
 \def

- \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(pr0_ind (\lambda 
-(t: T).(\lambda (t0: T).(\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t) 
-(lift h d t0)))))) (\lambda (t: T).(\lambda (h: nat).(\lambda (d: 
-nat).(pr0_refl (lift h d t))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda 
-(_: (pr0 u1 u2)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 
-(lift h d u1) (lift h d u2)))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda 
-(_: (pr0 t3 t4)).(\lambda (H3: ((\forall (h: nat).(\forall (d: nat).(pr0 
-(lift h d t3) (lift h d t4)))))).(\lambda (k: K).(\lambda (h: nat).(\lambda 
-(d: nat).(eq_ind_r T (THead k (lift h d u1) (lift h (s k d) t3)) (\lambda (t: 
-T).(pr0 t (lift h d (THead k u2 t4)))) (eq_ind_r T (THead k (lift h d u2) 
-(lift h (s k d) t4)) (\lambda (t: T).(pr0 (THead k (lift h d u1) (lift h (s k 
-d) t3)) t)) (pr0_comp (lift h d u1) (lift h d u2) (H1 h d) (lift h (s k d) 
-t3) (lift h (s k d) t4) (H3 h (s k d)) k) (lift h d (THead k u2 t4)) 
-(lift_head k u2 t4 h d)) (lift h d (THead k u1 t3)) (lift_head k u1 t3 h 
-d))))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: 
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(let TMP_3 \def 
+(\lambda (t: T).(\lambda (t0: T).(\forall (h: nat).(\forall (d: nat).(let 
+TMP_1 \def (lift h d t) in (let TMP_2 \def (lift h d t0) in (pr0 TMP_1 
+TMP_2))))))) in (let TMP_5 \def (\lambda (t: T).(\lambda (h: nat).(\lambda 
+(d: nat).(let TMP_4 \def (lift h d t) in (pr0_refl TMP_4))))) in (let TMP_39 
+\def (\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda 
+(H1: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d u1) (lift h d 
+u2)))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda 
+(H3: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t3) (lift h d 
+t4)))))).(\lambda (k: K).(\lambda (h: nat).(\lambda (d: nat).(let TMP_6 \def 
+(lift h d u1) in (let TMP_7 \def (s k d) in (let TMP_8 \def (lift h TMP_7 t3) 
+in (let TMP_9 \def (THead k TMP_6 TMP_8) in (let TMP_12 \def (\lambda (t: 
+T).(let TMP_10 \def (THead k u2 t4) in (let TMP_11 \def (lift h d TMP_10) in 
+(pr0 t TMP_11)))) in (let TMP_13 \def (lift h d u2) in (let TMP_14 \def (s k 
+d) in (let TMP_15 \def (lift h TMP_14 t4) in (let TMP_16 \def (THead k TMP_13 
+TMP_15) in (let TMP_21 \def (\lambda (t: T).(let TMP_17 \def (lift h d u1) in 
+(let TMP_18 \def (s k d) in (let TMP_19 \def (lift h TMP_18 t3) in (let 
+TMP_20 \def (THead k TMP_17 TMP_19) in (pr0 TMP_20 t)))))) in (let TMP_22 
+\def (lift h d u1) in (let TMP_23 \def (lift h d u2) in (let TMP_24 \def (H1 
+h d) in (let TMP_25 \def (s k d) in (let TMP_26 \def (lift h TMP_25 t3) in 
+(let TMP_27 \def (s k d) in (let TMP_28 \def (lift h TMP_27 t4) in (let 
+TMP_29 \def (s k d) in (let TMP_30 \def (H3 h TMP_29) in (let TMP_31 \def 
+(pr0_comp TMP_22 TMP_23 TMP_24 TMP_26 TMP_28 TMP_30 k) in (let TMP_32 \def 
+(THead k u2 t4) in (let TMP_33 \def (lift h d TMP_32) in (let TMP_34 \def 
+(lift_head k u2 t4 h d) in (let TMP_35 \def (eq_ind_r T TMP_16 TMP_21 TMP_31 
+TMP_33 TMP_34) in (let TMP_36 \def (THead k u1 t3) in (let TMP_37 \def (lift 
+h d TMP_36) in (let TMP_38 \def (lift_head k u1 t3 h d) in (eq_ind_r T TMP_9 
+TMP_12 TMP_35 TMP_37 TMP_38))))))))))))))))))))))))))))))))))))))) in (let 
+TMP_132 \def (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: 

 (pr0 v1 v2)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h 
 d v1) (lift h d v2)))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 
 t3 t4)).(\lambda (H3: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t3) 
 (pr0 v1 v2)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h 
 d v1) (lift h d v2)))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 
 t3 t4)).(\lambda (H3: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t3) 

-(lift h d t4)))))).(\lambda (h: nat).(\lambda (d: nat).(eq_ind_r T (THead 
-(Flat Appl) (lift h d v1) (lift h (s (Flat Appl) d) (THead (Bind Abst) u 
-t3))) (\lambda (t: T).(pr0 t (lift h d (THead (Bind Abbr) v2 t4)))) (eq_ind_r 
-T (THead (Bind Abst) (lift h (s (Flat Appl) d) u) (lift h (s (Bind Abst) (s 
-(Flat Appl) d)) t3)) (\lambda (t: T).(pr0 (THead (Flat Appl) (lift h d v1) t) 
-(lift h d (THead (Bind Abbr) v2 t4)))) (eq_ind_r T (THead (Bind Abbr) (lift h 
-d v2) (lift h (s (Bind Abbr) d) t4)) (\lambda (t: T).(pr0 (THead (Flat Appl) 
-(lift h d v1) (THead (Bind Abst) (lift h (s (Flat Appl) d) u) (lift h (s 
-(Bind Abst) (s (Flat Appl) d)) t3))) t)) (pr0_beta (lift h (s (Flat Appl) d) 
-u) (lift h d v1) (lift h d v2) (H1 h d) (lift h (s (Bind Abst) (s (Flat Appl) 
-d)) t3) (lift h (s (Bind Abbr) d) t4) (H3 h (s (Bind Abbr) d))) (lift h d 
-(THead (Bind Abbr) v2 t4)) (lift_head (Bind Abbr) v2 t4 h d)) (lift h (s 
-(Flat Appl) d) (THead (Bind Abst) u t3)) (lift_head (Bind Abst) u t3 h (s 
-(Flat Appl) d))) (lift h d (THead (Flat Appl) v1 (THead (Bind Abst) u t3))) 
-(lift_head (Flat Appl) v1 (THead (Bind Abst) u t3) h d))))))))))))) (\lambda 
-(b: B).(\lambda (H0: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: 
-T).(\lambda (_: (pr0 v1 v2)).(\lambda (H2: ((\forall (h: nat).(\forall (d: 
-nat).(pr0 (lift h d v1) (lift h d v2)))))).(\lambda (u1: T).(\lambda (u2: 
-T).(\lambda (_: (pr0 u1 u2)).(\lambda (H4: ((\forall (h: nat).(\forall (d: 
-nat).(pr0 (lift h d u1) (lift h d u2)))))).(\lambda (t3: T).(\lambda (t4: 
-T).(\lambda (_: (pr0 t3 t4)).(\lambda (H6: ((\forall (h: nat).(\forall (d: 
-nat).(pr0 (lift h d t3) (lift h d t4)))))).(\lambda (h: nat).(\lambda (d: 
-nat).(eq_ind_r T (THead (Flat Appl) (lift h d v1) (lift h (s (Flat Appl) d) 
-(THead (Bind b) u1 t3))) (\lambda (t: T).(pr0 t (lift h d (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t4))))) (eq_ind_r T (THead (Bind b) 
-(lift h (s (Flat Appl) d) u1) (lift h (s (Bind b) (s (Flat Appl) d)) t3)) 
-(\lambda (t: T).(pr0 (THead (Flat Appl) (lift h d v1) t) (lift h d (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))))) (eq_ind_r T (THead 
-(Bind b) (lift h d u2) (lift h (s (Bind b) d) (THead (Flat Appl) (lift (S O) 
-O v2) t4))) (\lambda (t: T).(pr0 (THead (Flat Appl) (lift h d v1) (THead 
-(Bind b) (lift h (s (Flat Appl) d) u1) (lift h (s (Bind b) (s (Flat Appl) d)) 
-t3))) t)) (eq_ind_r T (THead (Flat Appl) (lift h (s (Bind b) d) (lift (S O) O 
-v2)) (lift h (s (Flat Appl) (s (Bind b) d)) t4)) (\lambda (t: T).(pr0 (THead 
-(Flat Appl) (lift h d v1) (THead (Bind b) (lift h (s (Flat Appl) d) u1) (lift 
-h (s (Bind b) (s (Flat Appl) d)) t3))) (THead (Bind b) (lift h d u2) t))) 
-(eq_ind nat (plus (S O) d) (\lambda (n: nat).(pr0 (THead (Flat Appl) (lift h 
-d v1) (THead (Bind b) (lift h d u1) (lift h n t3))) (THead (Bind b) (lift h d 
-u2) (THead (Flat Appl) (lift h n (lift (S O) O v2)) (lift h n t4))))) 
-(eq_ind_r T (lift (S O) O (lift h d v2)) (\lambda (t: T).(pr0 (THead (Flat 
-Appl) (lift h d v1) (THead (Bind b) (lift h d u1) (lift h (plus (S O) d) 
-t3))) (THead (Bind b) (lift h d u2) (THead (Flat Appl) t (lift h (plus (S O) 
-d) t4))))) (pr0_upsilon b H0 (lift h d v1) (lift h d v2) (H2 h d) (lift h d 
-u1) (lift h d u2) (H4 h d) (lift h (plus (S O) d) t3) (lift h (plus (S O) d) 
-t4) (H6 h (plus (S O) d))) (lift h (plus (S O) d) (lift (S O) O v2)) (lift_d 
-v2 h (S O) d O (le_O_n d))) (S d) (refl_equal nat (S d))) (lift h (s (Bind b) 
-d) (THead (Flat Appl) (lift (S O) O v2) t4)) (lift_head (Flat Appl) (lift (S 
-O) O v2) t4 h (s (Bind b) d))) (lift h d (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t4))) (lift_head (Bind b) u2 (THead (Flat Appl) (lift 
-(S O) O v2) t4) h d)) (lift h (s (Flat Appl) d) (THead (Bind b) u1 t3)) 
-(lift_head (Bind b) u1 t3 h (s (Flat Appl) d))) (lift h d (THead (Flat Appl) 
-v1 (THead (Bind b) u1 t3))) (lift_head (Flat Appl) v1 (THead (Bind b) u1 t3) 
-h d)))))))))))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 
+(lift h d t4)))))).(\lambda (h: nat).(\lambda (d: nat).(let TMP_40 \def (Flat 
+Appl) in (let TMP_41 \def (lift h d v1) in (let TMP_42 \def (Flat Appl) in 
+(let TMP_43 \def (s TMP_42 d) in (let TMP_44 \def (Bind Abst) in (let TMP_45 
+\def (THead TMP_44 u t3) in (let TMP_46 \def (lift h TMP_43 TMP_45) in (let 
+TMP_47 \def (THead TMP_40 TMP_41 TMP_46) in (let TMP_51 \def (\lambda (t: 
+T).(let TMP_48 \def (Bind Abbr) in (let TMP_49 \def (THead TMP_48 v2 t4) in 
+(let TMP_50 \def (lift h d TMP_49) in (pr0 t TMP_50))))) in (let TMP_52 \def 
+(Bind Abst) in (let TMP_53 \def (Flat Appl) in (let TMP_54 \def (s TMP_53 d) 
+in (let TMP_55 \def (lift h TMP_54 u) in (let TMP_56 \def (Bind Abst) in (let 
+TMP_57 \def (Flat Appl) in (let TMP_58 \def (s TMP_57 d) in (let TMP_59 \def 
+(s TMP_56 TMP_58) in (let TMP_60 \def (lift h TMP_59 t3) in (let TMP_61 \def 
+(THead TMP_52 TMP_55 TMP_60) in (let TMP_68 \def (\lambda (t: T).(let TMP_62 
+\def (Flat Appl) in (let TMP_63 \def (lift h d v1) in (let TMP_64 \def (THead 
+TMP_62 TMP_63 t) in (let TMP_65 \def (Bind Abbr) in (let TMP_66 \def (THead 
+TMP_65 v2 t4) in (let TMP_67 \def (lift h d TMP_66) in (pr0 TMP_64 
+TMP_67)))))))) in (let TMP_69 \def (Bind Abbr) in (let TMP_70 \def (lift h d 
+v2) in (let TMP_71 \def (Bind Abbr) in (let TMP_72 \def (s TMP_71 d) in (let 
+TMP_73 \def (lift h TMP_72 t4) in (let TMP_74 \def (THead TMP_69 TMP_70 
+TMP_73) in (let TMP_88 \def (\lambda (t: T).(let TMP_75 \def (Flat Appl) in 
+(let TMP_76 \def (lift h d v1) in (let TMP_77 \def (Bind Abst) in (let TMP_78 
+\def (Flat Appl) in (let TMP_79 \def (s TMP_78 d) in (let TMP_80 \def (lift h 
+TMP_79 u) in (let TMP_81 \def (Bind Abst) in (let TMP_82 \def (Flat Appl) in 
+(let TMP_83 \def (s TMP_82 d) in (let TMP_84 \def (s TMP_81 TMP_83) in (let 
+TMP_85 \def (lift h TMP_84 t3) in (let TMP_86 \def (THead TMP_77 TMP_80 
+TMP_85) in (let TMP_87 \def (THead TMP_75 TMP_76 TMP_86) in (pr0 TMP_87 
+t))))))))))))))) in (let TMP_89 \def (Flat Appl) in (let TMP_90 \def (s 
+TMP_89 d) in (let TMP_91 \def (lift h TMP_90 u) in (let TMP_92 \def (lift h d 
+v1) in (let TMP_93 \def (lift h d v2) in (let TMP_94 \def (H1 h d) in (let 
+TMP_95 \def (Bind Abst) in (let TMP_96 \def (Flat Appl) in (let TMP_97 \def 
+(s TMP_96 d) in (let TMP_98 \def (s TMP_95 TMP_97) in (let TMP_99 \def (lift 
+h TMP_98 t3) in (let TMP_100 \def (Bind Abbr) in (let TMP_101 \def (s TMP_100 
+d) in (let TMP_102 \def (lift h TMP_101 t4) in (let TMP_103 \def (Bind Abbr) 
+in (let TMP_104 \def (s TMP_103 d) in (let TMP_105 \def (H3 h TMP_104) in 
+(let TMP_106 \def (pr0_beta TMP_91 TMP_92 TMP_93 TMP_94 TMP_99 TMP_102 
+TMP_105) in (let TMP_107 \def (Bind Abbr) in (let TMP_108 \def (THead TMP_107 
+v2 t4) in (let TMP_109 \def (lift h d TMP_108) in (let TMP_110 \def (Bind 
+Abbr) in (let TMP_111 \def (lift_head TMP_110 v2 t4 h d) in (let TMP_112 \def 
+(eq_ind_r T TMP_74 TMP_88 TMP_106 TMP_109 TMP_111) in (let TMP_113 \def (Flat 
+Appl) in (let TMP_114 \def (s TMP_113 d) in (let TMP_115 \def (Bind Abst) in 
+(let TMP_116 \def (THead TMP_115 u t3) in (let TMP_117 \def (lift h TMP_114 
+TMP_116) in (let TMP_118 \def (Bind Abst) in (let TMP_119 \def (Flat Appl) in 
+(let TMP_120 \def (s TMP_119 d) in (let TMP_121 \def (lift_head TMP_118 u t3 
+h TMP_120) in (let TMP_122 \def (eq_ind_r T TMP_61 TMP_68 TMP_112 TMP_117 
+TMP_121) in (let TMP_123 \def (Flat Appl) in (let TMP_124 \def (Bind Abst) in 
+(let TMP_125 \def (THead TMP_124 u t3) in (let TMP_126 \def (THead TMP_123 v1 
+TMP_125) in (let TMP_127 \def (lift h d TMP_126) in (let TMP_128 \def (Flat 
+Appl) in (let TMP_129 \def (Bind Abst) in (let TMP_130 \def (THead TMP_129 u 
+t3) in (let TMP_131 \def (lift_head TMP_128 v1 TMP_130 h d) in (eq_ind_r T 
+TMP_47 TMP_51 TMP_122 TMP_127 
+TMP_131)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
+))))))))))) in (let TMP_339 \def (\lambda (b: B).(\lambda (H0: (not (eq B b 
+Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda 
+(H2: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d v1) (lift h d 
+v2)))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda 
+(H4: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d u1) (lift h d 
+u2)))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda 
+(H6: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t3) (lift h d 
+t4)))))).(\lambda (h: nat).(\lambda (d: nat).(let TMP_133 \def (Flat Appl) in 
+(let TMP_134 \def (lift h d v1) in (let TMP_135 \def (Flat Appl) in (let 
+TMP_136 \def (s TMP_135 d) in (let TMP_137 \def (Bind b) in (let TMP_138 \def 
+(THead TMP_137 u1 t3) in (let TMP_139 \def (lift h TMP_136 TMP_138) in (let 
+TMP_140 \def (THead TMP_133 TMP_134 TMP_139) in (let TMP_148 \def (\lambda 
+(t: T).(let TMP_141 \def (Bind b) in (let TMP_142 \def (Flat Appl) in (let 
+TMP_143 \def (S O) in (let TMP_144 \def (lift TMP_143 O v2) in (let TMP_145 
+\def (THead TMP_142 TMP_144 t4) in (let TMP_146 \def (THead TMP_141 u2 
+TMP_145) in (let TMP_147 \def (lift h d TMP_146) in (pr0 t TMP_147))))))))) 
+in (let TMP_149 \def (Bind b) in (let TMP_150 \def (Flat Appl) in (let 
+TMP_151 \def (s TMP_150 d) in (let TMP_152 \def (lift h TMP_151 u1) in (let 
+TMP_153 \def (Bind b) in (let TMP_154 \def (Flat Appl) in (let TMP_155 \def 
+(s TMP_154 d) in (let TMP_156 \def (s TMP_153 TMP_155) in (let TMP_157 \def 
+(lift h TMP_156 t3) in (let TMP_158 \def (THead TMP_149 TMP_152 TMP_157) in 
+(let TMP_169 \def (\lambda (t: T).(let TMP_159 \def (Flat Appl) in (let 
+TMP_160 \def (lift h d v1) in (let TMP_161 \def (THead TMP_159 TMP_160 t) in 
+(let TMP_162 \def (Bind b) in (let TMP_163 \def (Flat Appl) in (let TMP_164 
+\def (S O) in (let TMP_165 \def (lift TMP_164 O v2) in (let TMP_166 \def 
+(THead TMP_163 TMP_165 t4) in (let TMP_167 \def (THead TMP_162 u2 TMP_166) in 
+(let TMP_168 \def (lift h d TMP_167) in (pr0 TMP_161 TMP_168)))))))))))) in 
+(let TMP_170 \def (Bind b) in (let TMP_171 \def (lift h d u2) in (let TMP_172 
+\def (Bind b) in (let TMP_173 \def (s TMP_172 d) in (let TMP_174 \def (Flat 
+Appl) in (let TMP_175 \def (S O) in (let TMP_176 \def (lift TMP_175 O v2) in 
+(let TMP_177 \def (THead TMP_174 TMP_176 t4) in (let TMP_178 \def (lift h 
+TMP_173 TMP_177) in (let TMP_179 \def (THead TMP_170 TMP_171 TMP_178) in (let 
+TMP_193 \def (\lambda (t: T).(let TMP_180 \def (Flat Appl) in (let TMP_181 
+\def (lift h d v1) in (let TMP_182 \def (Bind b) in (let TMP_183 \def (Flat 
+Appl) in (let TMP_184 \def (s TMP_183 d) in (let TMP_185 \def (lift h TMP_184 
+u1) in (let TMP_186 \def (Bind b) in (let TMP_187 \def (Flat Appl) in (let 
+TMP_188 \def (s TMP_187 d) in (let TMP_189 \def (s TMP_186 TMP_188) in (let 
+TMP_190 \def (lift h TMP_189 t3) in (let TMP_191 \def (THead TMP_182 TMP_185 
+TMP_190) in (let TMP_192 \def (THead TMP_180 TMP_181 TMP_191) in (pr0 TMP_192 
+t))))))))))))))) in (let TMP_194 \def (Flat Appl) in (let TMP_195 \def (Bind 
+b) in (let TMP_196 \def (s TMP_195 d) in (let TMP_197 \def (S O) in (let 
+TMP_198 \def (lift TMP_197 O v2) in (let TMP_199 \def (lift h TMP_196 
+TMP_198) in (let TMP_200 \def (Flat Appl) in (let TMP_201 \def (Bind b) in 
+(let TMP_202 \def (s TMP_201 d) in (let TMP_203 \def (s TMP_200 TMP_202) in 
+(let TMP_204 \def (lift h TMP_203 t4) in (let TMP_205 \def (THead TMP_194 
+TMP_199 TMP_204) in (let TMP_222 \def (\lambda (t: T).(let TMP_206 \def (Flat 
+Appl) in (let TMP_207 \def (lift h d v1) in (let TMP_208 \def (Bind b) in 
+(let TMP_209 \def (Flat Appl) in (let TMP_210 \def (s TMP_209 d) in (let 
+TMP_211 \def (lift h TMP_210 u1) in (let TMP_212 \def (Bind b) in (let 
+TMP_213 \def (Flat Appl) in (let TMP_214 \def (s TMP_213 d) in (let TMP_215 
+\def (s TMP_212 TMP_214) in (let TMP_216 \def (lift h TMP_215 t3) in (let 
+TMP_217 \def (THead TMP_208 TMP_211 TMP_216) in (let TMP_218 \def (THead 
+TMP_206 TMP_207 TMP_217) in (let TMP_219 \def (Bind b) in (let TMP_220 \def 
+(lift h d u2) in (let TMP_221 \def (THead TMP_219 TMP_220 t) in (pr0 TMP_218 
+TMP_221)))))))))))))))))) in (let TMP_223 \def (S O) in (let TMP_224 \def 
+(plus TMP_223 d) in (let TMP_241 \def (\lambda (n: nat).(let TMP_225 \def 
+(Flat Appl) in (let TMP_226 \def (lift h d v1) in (let TMP_227 \def (Bind b) 
+in (let TMP_228 \def (lift h d u1) in (let TMP_229 \def (lift h n t3) in (let 
+TMP_230 \def (THead TMP_227 TMP_228 TMP_229) in (let TMP_231 \def (THead 
+TMP_225 TMP_226 TMP_230) in (let TMP_232 \def (Bind b) in (let TMP_233 \def 
+(lift h d u2) in (let TMP_234 \def (Flat Appl) in (let TMP_235 \def (S O) in 
+(let TMP_236 \def (lift TMP_235 O v2) in (let TMP_237 \def (lift h n TMP_236) 
+in (let TMP_238 \def (lift h n t4) in (let TMP_239 \def (THead TMP_234 
+TMP_237 TMP_238) in (let TMP_240 \def (THead TMP_232 TMP_233 TMP_239) in (pr0 
+TMP_231 TMP_240)))))))))))))))))) in (let TMP_242 \def (S O) in (let TMP_243 
+\def (lift h d v2) in (let TMP_244 \def (lift TMP_242 O TMP_243) in (let 
+TMP_262 \def (\lambda (t: T).(let TMP_245 \def (Flat Appl) in (let TMP_246 
+\def (lift h d v1) in (let TMP_247 \def (Bind b) in (let TMP_248 \def (lift h 
+d u1) in (let TMP_249 \def (S O) in (let TMP_250 \def (plus TMP_249 d) in 
+(let TMP_251 \def (lift h TMP_250 t3) in (let TMP_252 \def (THead TMP_247 
+TMP_248 TMP_251) in (let TMP_253 \def (THead TMP_245 TMP_246 TMP_252) in (let 
+TMP_254 \def (Bind b) in (let TMP_255 \def (lift h d u2) in (let TMP_256 \def 
+(Flat Appl) in (let TMP_257 \def (S O) in (let TMP_258 \def (plus TMP_257 d) 
+in (let TMP_259 \def (lift h TMP_258 t4) in (let TMP_260 \def (THead TMP_256 
+t TMP_259) in (let TMP_261 \def (THead TMP_254 TMP_255 TMP_260) in (pr0 
+TMP_253 TMP_261))))))))))))))))))) in (let TMP_263 \def (lift h d v1) in (let 
+TMP_264 \def (lift h d v2) in (let TMP_265 \def (H2 h d) in (let TMP_266 \def 
+(lift h d u1) in (let TMP_267 \def (lift h d u2) in (let TMP_268 \def (H4 h 
+d) in (let TMP_269 \def (S O) in (let TMP_270 \def (plus TMP_269 d) in (let 
+TMP_271 \def (lift h TMP_270 t3) in (let TMP_272 \def (S O) in (let TMP_273 
+\def (plus TMP_272 d) in (let TMP_274 \def (lift h TMP_273 t4) in (let 
+TMP_275 \def (S O) in (let TMP_276 \def (plus TMP_275 d) in (let TMP_277 \def 
+(H6 h TMP_276) in (let TMP_278 \def (pr0_upsilon b H0 TMP_263 TMP_264 TMP_265 
+TMP_266 TMP_267 TMP_268 TMP_271 TMP_274 TMP_277) in (let TMP_279 \def (S O) 
+in (let TMP_280 \def (plus TMP_279 d) in (let TMP_281 \def (S O) in (let 
+TMP_282 \def (lift TMP_281 O v2) in (let TMP_283 \def (lift h TMP_280 
+TMP_282) in (let TMP_284 \def (S O) in (let TMP_285 \def (le_O_n d) in (let 
+TMP_286 \def (lift_d v2 h TMP_284 d O TMP_285) in (let TMP_287 \def (eq_ind_r 
+T TMP_244 TMP_262 TMP_278 TMP_283 TMP_286) in (let TMP_288 \def (S d) in (let 
+TMP_289 \def (S d) in (let TMP_290 \def (refl_equal nat TMP_289) in (let 
+TMP_291 \def (eq_ind nat TMP_224 TMP_241 TMP_287 TMP_288 TMP_290) in (let 
+TMP_292 \def (Bind b) in (let TMP_293 \def (s TMP_292 d) in (let TMP_294 \def 
+(Flat Appl) in (let TMP_295 \def (S O) in (let TMP_296 \def (lift TMP_295 O 
+v2) in (let TMP_297 \def (THead TMP_294 TMP_296 t4) in (let TMP_298 \def 
+(lift h TMP_293 TMP_297) in (let TMP_299 \def (Flat Appl) in (let TMP_300 
+\def (S O) in (let TMP_301 \def (lift TMP_300 O v2) in (let TMP_302 \def 
+(Bind b) in (let TMP_303 \def (s TMP_302 d) in (let TMP_304 \def (lift_head 
+TMP_299 TMP_301 t4 h TMP_303) in (let TMP_305 \def (eq_ind_r T TMP_205 
+TMP_222 TMP_291 TMP_298 TMP_304) in (let TMP_306 \def (Bind b) in (let 
+TMP_307 \def (Flat Appl) in (let TMP_308 \def (S O) in (let TMP_309 \def 
+(lift TMP_308 O v2) in (let TMP_310 \def (THead TMP_307 TMP_309 t4) in (let 
+TMP_311 \def (THead TMP_306 u2 TMP_310) in (let TMP_312 \def (lift h d 
+TMP_311) in (let TMP_313 \def (Bind b) in (let TMP_314 \def (Flat Appl) in 
+(let TMP_315 \def (S O) in (let TMP_316 \def (lift TMP_315 O v2) in (let 
+TMP_317 \def (THead TMP_314 TMP_316 t4) in (let TMP_318 \def (lift_head 
+TMP_313 u2 TMP_317 h d) in (let TMP_319 \def (eq_ind_r T TMP_179 TMP_193 
+TMP_305 TMP_312 TMP_318) in (let TMP_320 \def (Flat Appl) in (let TMP_321 
+\def (s TMP_320 d) in (let TMP_322 \def (Bind b) in (let TMP_323 \def (THead 
+TMP_322 u1 t3) in (let TMP_324 \def (lift h TMP_321 TMP_323) in (let TMP_325 
+\def (Bind b) in (let TMP_326 \def (Flat Appl) in (let TMP_327 \def (s 
+TMP_326 d) in (let TMP_328 \def (lift_head TMP_325 u1 t3 h TMP_327) in (let 
+TMP_329 \def (eq_ind_r T TMP_158 TMP_169 TMP_319 TMP_324 TMP_328) in (let 
+TMP_330 \def (Flat Appl) in (let TMP_331 \def (Bind b) in (let TMP_332 \def 
+(THead TMP_331 u1 t3) in (let TMP_333 \def (THead TMP_330 v1 TMP_332) in (let 
+TMP_334 \def (lift h d TMP_333) in (let TMP_335 \def (Flat Appl) in (let 
+TMP_336 \def (Bind b) in (let TMP_337 \def (THead TMP_336 u1 t3) in (let 
+TMP_338 \def (lift_head TMP_335 v1 TMP_337 h d) in (eq_ind_r T TMP_140 
+TMP_148 TMP_329 TMP_334 
+TMP_338)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
+))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) in 
+(let TMP_405 \def (\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 

 u2)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d u1) 
 (lift h d u2)))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 
 t4)).(\lambda (H3: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t3) 
 (lift h d t4)))))).(\lambda (w: T).(\lambda (H4: (subst0 O u2 t4 w)).(\lambda 
 u2)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d u1) 
 (lift h d u2)))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 
 t4)).(\lambda (H3: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t3) 
 (lift h d t4)))))).(\lambda (w: T).(\lambda (H4: (subst0 O u2 t4 w)).(\lambda 

-(h: nat).(\lambda (d: nat).(eq_ind_r T (THead (Bind Abbr) (lift h d u1) (lift 
-h (s (Bind Abbr) d) t3)) (\lambda (t: T).(pr0 t (lift h d (THead (Bind Abbr) 
-u2 w)))) (eq_ind_r T (THead (Bind Abbr) (lift h d u2) (lift h (s (Bind Abbr) 
-d) w)) (\lambda (t: T).(pr0 (THead (Bind Abbr) (lift h d u1) (lift h (s (Bind 
-Abbr) d) t3)) t)) (pr0_delta (lift h d u1) (lift h d u2) (H1 h d) (lift h (S 
-d) t3) (lift h (S d) t4) (H3 h (S d)) (lift h (S d) w) (let d' \def (S d) in 
-(eq_ind nat (minus (S d) (S O)) (\lambda (n: nat).(subst0 O (lift h n u2) 
-(lift h d' t4) (lift h d' w))) (subst0_lift_lt t4 w u2 O H4 (S d) (le_n_S O d 
-(le_O_n d)) h) d (eq_ind nat d (\lambda (n: nat).(eq nat n d)) (refl_equal 
-nat d) (minus d O) (minus_n_O d))))) (lift h d (THead (Bind Abbr) u2 w)) 
-(lift_head (Bind Abbr) u2 w h d)) (lift h d (THead (Bind Abbr) u1 t3)) 
-(lift_head (Bind Abbr) u1 t3 h d)))))))))))))) (\lambda (b: B).(\lambda (H0: 
-(not (eq B b Abst))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 
-t4)).(\lambda (H2: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t3) 
-(lift h d t4)))))).(\lambda (u: T).(\lambda (h: nat).(\lambda (d: 
-nat).(eq_ind_r T (THead (Bind b) (lift h d u) (lift h (s (Bind b) d) (lift (S 
-O) O t3))) (\lambda (t: T).(pr0 t (lift h d t4))) (eq_ind nat (plus (S O) d) 
-(\lambda (n: nat).(pr0 (THead (Bind b) (lift h d u) (lift h n (lift (S O) O 
-t3))) (lift h d t4))) (eq_ind_r T (lift (S O) O (lift h d t3)) (\lambda (t: 
-T).(pr0 (THead (Bind b) (lift h d u) t) (lift h d t4))) (pr0_zeta b H0 (lift 
-h d t3) (lift h d t4) (H2 h d) (lift h d u)) (lift h (plus (S O) d) (lift (S 
-O) O t3)) (lift_d t3 h (S O) d O (le_O_n d))) (S d) (refl_equal nat (S d))) 
-(lift h d (THead (Bind b) u (lift (S O) O t3))) (lift_head (Bind b) u (lift 
-(S O) O t3) h d))))))))))) (\lambda (t3: T).(\lambda (t4: T).(\lambda (_: 
-(pr0 t3 t4)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h 
-d t3) (lift h d t4)))))).(\lambda (u: T).(\lambda (h: nat).(\lambda (d: 
-nat).(eq_ind_r T (THead (Flat Cast) (lift h d u) (lift h (s (Flat Cast) d) 
-t3)) (\lambda (t: T).(pr0 t (lift h d t4))) (pr0_tau (lift h (s (Flat Cast) 
-d) t3) (lift h d t4) (H1 h d) (lift h d u)) (lift h d (THead (Flat Cast) u 
-t3)) (lift_head (Flat Cast) u t3 h d))))))))) t1 t2 H))).
-(* COMMENTS
-Initial nodes: 2845
-END *)
+(h: nat).(\lambda (d: nat).(let TMP_340 \def (Bind Abbr) in (let TMP_341 \def 
+(lift h d u1) in (let TMP_342 \def (Bind Abbr) in (let TMP_343 \def (s 
+TMP_342 d) in (let TMP_344 \def (lift h TMP_343 t3) in (let TMP_345 \def 
+(THead TMP_340 TMP_341 TMP_344) in (let TMP_349 \def (\lambda (t: T).(let 
+TMP_346 \def (Bind Abbr) in (let TMP_347 \def (THead TMP_346 u2 w) in (let 
+TMP_348 \def (lift h d TMP_347) in (pr0 t TMP_348))))) in (let TMP_350 \def 
+(Bind Abbr) in (let TMP_351 \def (lift h d u2) in (let TMP_352 \def (Bind 
+Abbr) in (let TMP_353 \def (s TMP_352 d) in (let TMP_354 \def (lift h TMP_353 
+w) in (let TMP_355 \def (THead TMP_350 TMP_351 TMP_354) in (let TMP_362 \def 
+(\lambda (t: T).(let TMP_356 \def (Bind Abbr) in (let TMP_357 \def (lift h d 
+u1) in (let TMP_358 \def (Bind Abbr) in (let TMP_359 \def (s TMP_358 d) in 
+(let TMP_360 \def (lift h TMP_359 t3) in (let TMP_361 \def (THead TMP_356 
+TMP_357 TMP_360) in (pr0 TMP_361 t)))))))) in (let TMP_363 \def (lift h d u1) 
+in (let TMP_364 \def (lift h d u2) in (let TMP_365 \def (H1 h d) in (let 
+TMP_366 \def (S d) in (let TMP_367 \def (lift h TMP_366 t3) in (let TMP_368 
+\def (S d) in (let TMP_369 \def (lift h TMP_368 t4) in (let TMP_370 \def (S 
+d) in (let TMP_371 \def (H3 h TMP_370) in (let TMP_372 \def (S d) in (let 
+TMP_373 \def (lift h TMP_372 w) in (let d' \def (S d) in (let TMP_374 \def (S 
+d) in (let TMP_375 \def (S O) in (let TMP_376 \def (minus TMP_374 TMP_375) in 
+(let TMP_380 \def (\lambda (n: nat).(let TMP_377 \def (lift h n u2) in (let 
+TMP_378 \def (lift h d' t4) in (let TMP_379 \def (lift h d' w) in (subst0 O 
+TMP_377 TMP_378 TMP_379))))) in (let TMP_381 \def (S d) in (let TMP_382 \def 
+(le_O_n d) in (let TMP_383 \def (le_n_S O d TMP_382) in (let TMP_384 \def 
+(subst0_lift_lt t4 w u2 O H4 TMP_381 TMP_383 h) in (let TMP_385 \def (\lambda 
+(n: nat).(eq nat n d)) in (let TMP_386 \def (le_n d) in (let TMP_387 \def 
+(le_n d) in (let TMP_388 \def (le_antisym d d TMP_386 TMP_387) in (let 
+TMP_389 \def (minus d O) in (let TMP_390 \def (minus_n_O d) in (let TMP_391 
+\def (eq_ind nat d TMP_385 TMP_388 TMP_389 TMP_390) in (let TMP_392 \def 
+(eq_ind nat TMP_376 TMP_380 TMP_384 d TMP_391) in (let TMP_393 \def 
+(pr0_delta TMP_363 TMP_364 TMP_365 TMP_367 TMP_369 TMP_371 TMP_373 TMP_392) 
+in (let TMP_394 \def (Bind Abbr) in (let TMP_395 \def (THead TMP_394 u2 w) in 
+(let TMP_396 \def (lift h d TMP_395) in (let TMP_397 \def (Bind Abbr) in (let 
+TMP_398 \def (lift_head TMP_397 u2 w h d) in (let TMP_399 \def (eq_ind_r T 
+TMP_355 TMP_362 TMP_393 TMP_396 TMP_398) in (let TMP_400 \def (Bind Abbr) in 
+(let TMP_401 \def (THead TMP_400 u1 t3) in (let TMP_402 \def (lift h d 
+TMP_401) in (let TMP_403 \def (Bind Abbr) in (let TMP_404 \def (lift_head 
+TMP_403 u1 t3 h d) in (eq_ind_r T TMP_345 TMP_349 TMP_399 TMP_402 
+TMP_404))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) in 
+(let TMP_461 \def (\lambda (b: B).(\lambda (H0: (not (eq B b Abst))).(\lambda 
+(t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (H2: ((\forall 
+(h: nat).(\forall (d: nat).(pr0 (lift h d t3) (lift h d t4)))))).(\lambda (u: 
+T).(\lambda (h: nat).(\lambda (d: nat).(let TMP_406 \def (Bind b) in (let 
+TMP_407 \def (lift h d u) in (let TMP_408 \def (Bind b) in (let TMP_409 \def 
+(s TMP_408 d) in (let TMP_410 \def (S O) in (let TMP_411 \def (lift TMP_410 O 
+t3) in (let TMP_412 \def (lift h TMP_409 TMP_411) in (let TMP_413 \def (THead 
+TMP_406 TMP_407 TMP_412) in (let TMP_415 \def (\lambda (t: T).(let TMP_414 
+\def (lift h d t4) in (pr0 t TMP_414))) in (let TMP_416 \def (S O) in (let 
+TMP_417 \def (plus TMP_416 d) in (let TMP_425 \def (\lambda (n: nat).(let 
+TMP_418 \def (Bind b) in (let TMP_419 \def (lift h d u) in (let TMP_420 \def 
+(S O) in (let TMP_421 \def (lift TMP_420 O t3) in (let TMP_422 \def (lift h n 
+TMP_421) in (let TMP_423 \def (THead TMP_418 TMP_419 TMP_422) in (let TMP_424 
+\def (lift h d t4) in (pr0 TMP_423 TMP_424))))))))) in (let TMP_426 \def (S 
+O) in (let TMP_427 \def (lift h d t3) in (let TMP_428 \def (lift TMP_426 O 
+TMP_427) in (let TMP_433 \def (\lambda (t: T).(let TMP_429 \def (Bind b) in 
+(let TMP_430 \def (lift h d u) in (let TMP_431 \def (THead TMP_429 TMP_430 t) 
+in (let TMP_432 \def (lift h d t4) in (pr0 TMP_431 TMP_432)))))) in (let 
+TMP_434 \def (lift h d t3) in (let TMP_435 \def (lift h d t4) in (let TMP_436 
+\def (H2 h d) in (let TMP_437 \def (lift h d u) in (let TMP_438 \def 
+(pr0_zeta b H0 TMP_434 TMP_435 TMP_436 TMP_437) in (let TMP_439 \def (S O) in 
+(let TMP_440 \def (plus TMP_439 d) in (let TMP_441 \def (S O) in (let TMP_442 
+\def (lift TMP_441 O t3) in (let TMP_443 \def (lift h TMP_440 TMP_442) in 
+(let TMP_444 \def (S O) in (let TMP_445 \def (le_O_n d) in (let TMP_446 \def 
+(lift_d t3 h TMP_444 d O TMP_445) in (let TMP_447 \def (eq_ind_r T TMP_428 
+TMP_433 TMP_438 TMP_443 TMP_446) in (let TMP_448 \def (S d) in (let TMP_449 
+\def (S d) in (let TMP_450 \def (refl_equal nat TMP_449) in (let TMP_451 \def 
+(eq_ind nat TMP_417 TMP_425 TMP_447 TMP_448 TMP_450) in (let TMP_452 \def 
+(Bind b) in (let TMP_453 \def (S O) in (let TMP_454 \def (lift TMP_453 O t3) 
+in (let TMP_455 \def (THead TMP_452 u TMP_454) in (let TMP_456 \def (lift h d 
+TMP_455) in (let TMP_457 \def (Bind b) in (let TMP_458 \def (S O) in (let 
+TMP_459 \def (lift TMP_458 O t3) in (let TMP_460 \def (lift_head TMP_457 u 
+TMP_459 h d) in (eq_ind_r T TMP_413 TMP_415 TMP_451 TMP_456 
+TMP_460))))))))))))))))))))))))))))))))))))))))))))))))))))) in (let TMP_482 
+\def (\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda 
+(H1: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t3) (lift h d 
+t4)))))).(\lambda (u: T).(\lambda (h: nat).(\lambda (d: nat).(let TMP_462 
+\def (Flat Cast) in (let TMP_463 \def (lift h d u) in (let TMP_464 \def (Flat 
+Cast) in (let TMP_465 \def (s TMP_464 d) in (let TMP_466 \def (lift h TMP_465 
+t3) in (let TMP_467 \def (THead TMP_462 TMP_463 TMP_466) in (let TMP_469 \def 
+(\lambda (t: T).(let TMP_468 \def (lift h d t4) in (pr0 t TMP_468))) in (let 
+TMP_470 \def (Flat Cast) in (let TMP_471 \def (s TMP_470 d) in (let TMP_472 
+\def (lift h TMP_471 t3) in (let TMP_473 \def (lift h d t4) in (let TMP_474 
+\def (H1 h d) in (let TMP_475 \def (lift h d u) in (let TMP_476 \def (pr0_tau 
+TMP_472 TMP_473 TMP_474 TMP_475) in (let TMP_477 \def (Flat Cast) in (let 
+TMP_478 \def (THead TMP_477 u t3) in (let TMP_479 \def (lift h d TMP_478) in 
+(let TMP_480 \def (Flat Cast) in (let TMP_481 \def (lift_head TMP_480 u t3 h 
+d) in (eq_ind_r T TMP_467 TMP_469 TMP_476 TMP_479 
+TMP_481))))))))))))))))))))))))))) in (pr0_ind TMP_3 TMP_5 TMP_39 TMP_132 
+TMP_339 TMP_405 TMP_461 TMP_482 t1 t2 H))))))))))).

 
 

-theorem pr0_subst0_back:
- \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst0 
-i u2 t1 t2) \to (\forall (u1: T).((pr0 u1 u2) \to (ex2 T (\lambda (t: 
-T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t t2)))))))))
+theorem pr0_gen_abbr:
+ \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Abbr) u1 
+t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead 
+(Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
+(u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y)) 
+(\lambda (y: T).(subst0 O u2 y t2))))))) (pr0 t1 (lift (S O) O x))))))

 \def
 \def

- \lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (i: nat).(\lambda 
-(H: (subst0 i u2 t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t: 
-T).(\lambda (t0: T).(\lambda (t3: T).(\forall (u1: T).((pr0 u1 t) \to (ex2 T 
-(\lambda (t4: T).(subst0 n u1 t0 t4)) (\lambda (t4: T).(pr0 t4 t3))))))))) 
-(\lambda (v: T).(\lambda (i0: nat).(\lambda (u1: T).(\lambda (H0: (pr0 u1 
-v)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 (TLRef i0) t)) (\lambda (t: 
-T).(pr0 t (lift (S i0) O v))) (lift (S i0) O u1) (subst0_lref u1 i0) 
-(pr0_lift u1 v H0 (S i0) O)))))) (\lambda (v: T).(\lambda (u3: T).(\lambda 
-(u1: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v u1 u3)).(\lambda (H1: 
-((\forall (u4: T).((pr0 u4 v) \to (ex2 T (\lambda (t: T).(subst0 i0 u4 u1 t)) 
-(\lambda (t: T).(pr0 t u3))))))).(\lambda (t: T).(\lambda (k: K).(\lambda 
-(u0: T).(\lambda (H2: (pr0 u0 v)).(ex2_ind T (\lambda (t0: T).(subst0 i0 u0 
-u1 t0)) (\lambda (t0: T).(pr0 t0 u3)) (ex2 T (\lambda (t0: T).(subst0 i0 u0 
-(THead k u1 t) t0)) (\lambda (t0: T).(pr0 t0 (THead k u3 t)))) (\lambda (x: 
-T).(\lambda (H3: (subst0 i0 u0 u1 x)).(\lambda (H4: (pr0 x u3)).(ex_intro2 T 
-(\lambda (t0: T).(subst0 i0 u0 (THead k u1 t) t0)) (\lambda (t0: T).(pr0 t0 
-(THead k u3 t))) (THead k x t) (subst0_fst u0 x u1 i0 H3 t k) (pr0_comp x u3 
-H4 t t (pr0_refl t) k))))) (H1 u0 H2)))))))))))) (\lambda (k: K).(\lambda (v: 
-T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (i0: nat).(\lambda (_: (subst0 
-(s k i0) v t4 t3)).(\lambda (H1: ((\forall (u1: T).((pr0 u1 v) \to (ex2 T 
-(\lambda (t: T).(subst0 (s k i0) u1 t4 t)) (\lambda (t: T).(pr0 t 
-t3))))))).(\lambda (u: T).(\lambda (u1: T).(\lambda (H2: (pr0 u1 v)).(ex2_ind 
-T (\lambda (t: T).(subst0 (s k i0) u1 t4 t)) (\lambda (t: T).(pr0 t t3)) (ex2 
-T (\lambda (t: T).(subst0 i0 u1 (THead k u t4) t)) (\lambda (t: T).(pr0 t 
-(THead k u t3)))) (\lambda (x: T).(\lambda (H3: (subst0 (s k i0) u1 t4 
-x)).(\lambda (H4: (pr0 x t3)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 
-(THead k u t4) t)) (\lambda (t: T).(pr0 t (THead k u t3))) (THead k u x) 
-(subst0_snd k u1 x t4 i0 H3 u) (pr0_comp u u (pr0_refl u) x t3 H4 k))))) (H1 
-u1 H2)))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u3: T).(\lambda 
-(i0: nat).(\lambda (_: (subst0 i0 v u1 u3)).(\lambda (H1: ((\forall (u4: 
-T).((pr0 u4 v) \to (ex2 T (\lambda (t: T).(subst0 i0 u4 u1 t)) (\lambda (t: 
-T).(pr0 t u3))))))).(\lambda (k: K).(\lambda (t3: T).(\lambda (t4: 
-T).(\lambda (_: (subst0 (s k i0) v t3 t4)).(\lambda (H3: ((\forall (u4: 
-T).((pr0 u4 v) \to (ex2 T (\lambda (t: T).(subst0 (s k i0) u4 t3 t)) (\lambda 
-(t: T).(pr0 t t4))))))).(\lambda (u0: T).(\lambda (H4: (pr0 u0 v)).(ex2_ind T 
-(\lambda (t: T).(subst0 (s k i0) u0 t3 t)) (\lambda (t: T).(pr0 t t4)) (ex2 T 
-(\lambda (t: T).(subst0 i0 u0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 t 
-(THead k u3 t4)))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i0) u0 t3 
-x)).(\lambda (H6: (pr0 x t4)).(ex2_ind T (\lambda (t: T).(subst0 i0 u0 u1 t)) 
-(\lambda (t: T).(pr0 t u3)) (ex2 T (\lambda (t: T).(subst0 i0 u0 (THead k u1 
-t3) t)) (\lambda (t: T).(pr0 t (THead k u3 t4)))) (\lambda (x0: T).(\lambda 
-(H7: (subst0 i0 u0 u1 x0)).(\lambda (H8: (pr0 x0 u3)).(ex_intro2 T (\lambda 
-(t: T).(subst0 i0 u0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 t (THead k u3 
-t4))) (THead k x0 x) (subst0_both u0 u1 x0 i0 H7 k t3 x H5) (pr0_comp x0 u3 
-H8 x t4 H6 k))))) (H1 u0 H4))))) (H3 u0 H4))))))))))))))) i u2 t1 t2 H))))).
-(* COMMENTS
-Initial nodes: 979
-END *)
+ \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead 
+(Bind Abbr) u1 t1) x)).(let TMP_1 \def (Bind Abbr) in (let TMP_2 \def (THead 
+TMP_1 u1 t1) in (let TMP_3 \def (\lambda (t: T).(pr0 t x)) in (let TMP_17 
+\def (\lambda (_: T).(let TMP_6 \def (\lambda (u2: T).(\lambda (t2: T).(let 
+TMP_4 \def (Bind Abbr) in (let TMP_5 \def (THead TMP_4 u2 t2) in (eq T x 
+TMP_5))))) in (let TMP_7 \def (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) 
+in (let TMP_12 \def (\lambda (u2: T).(\lambda (t2: T).(let TMP_8 \def (pr0 t1 
+t2) in (let TMP_9 \def (\lambda (y: T).(pr0 t1 y)) in (let TMP_10 \def 
+(\lambda (y: T).(subst0 O u2 y t2)) in (let TMP_11 \def (ex2 T TMP_9 TMP_10) 
+in (or TMP_8 TMP_11))))))) in (let TMP_13 \def (ex3_2 T T TMP_6 TMP_7 TMP_12) 
+in (let TMP_14 \def (S O) in (let TMP_15 \def (lift TMP_14 O x) in (let 
+TMP_16 \def (pr0 t1 TMP_15) in (or TMP_13 TMP_16))))))))) in (let TMP_448 
+\def (\lambda (y: T).(\lambda (H0: (pr0 y x)).(let TMP_31 \def (\lambda (t: 
+T).(\lambda (t0: T).((eq T t (THead (Bind Abbr) u1 t1)) \to (let TMP_20 \def 
+(\lambda (u2: T).(\lambda (t2: T).(let TMP_18 \def (Bind Abbr) in (let TMP_19 
+\def (THead TMP_18 u2 t2) in (eq T t0 TMP_19))))) in (let TMP_21 \def 
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) in (let TMP_26 \def (\lambda 
+(u2: T).(\lambda (t2: T).(let TMP_22 \def (pr0 t1 t2) in (let TMP_23 \def 
+(\lambda (y0: T).(pr0 t1 y0)) in (let TMP_24 \def (\lambda (y0: T).(subst0 O 
+u2 y0 t2)) in (let TMP_25 \def (ex2 T TMP_23 TMP_24) in (or TMP_22 
+TMP_25))))))) in (let TMP_27 \def (ex3_2 T T TMP_20 TMP_21 TMP_26) in (let 
+TMP_28 \def (S O) in (let TMP_29 \def (lift TMP_28 O t0) in (let TMP_30 \def 
+(pr0 t1 TMP_29) in (or TMP_27 TMP_30))))))))))) in (let TMP_91 \def (\lambda 
+(t: T).(\lambda (H1: (eq T t (THead (Bind Abbr) u1 t1))).(let TMP_32 \def 
+(\lambda (e: T).e) in (let TMP_33 \def (Bind Abbr) in (let TMP_34 \def (THead 
+TMP_33 u1 t1) in (let H2 \def (f_equal T T TMP_32 t TMP_34 H1) in (let TMP_35 
+\def (Bind Abbr) in (let TMP_36 \def (THead TMP_35 u1 t1) in (let TMP_50 \def 
+(\lambda (t0: T).(let TMP_39 \def (\lambda (u2: T).(\lambda (t2: T).(let 
+TMP_37 \def (Bind Abbr) in (let TMP_38 \def (THead TMP_37 u2 t2) in (eq T t0 
+TMP_38))))) in (let TMP_40 \def (\lambda (u2: T).(\lambda (_: T).(pr0 u1 
+u2))) in (let TMP_45 \def (\lambda (u2: T).(\lambda (t2: T).(let TMP_41 \def 
+(pr0 t1 t2) in (let TMP_42 \def (\lambda (y0: T).(pr0 t1 y0)) in (let TMP_43 
+\def (\lambda (y0: T).(subst0 O u2 y0 t2)) in (let TMP_44 \def (ex2 T TMP_42 
+TMP_43) in (or TMP_41 TMP_44))))))) in (let TMP_46 \def (ex3_2 T T TMP_39 
+TMP_40 TMP_45) in (let TMP_47 \def (S O) in (let TMP_48 \def (lift TMP_47 O 
+t0) in (let TMP_49 \def (pr0 t1 TMP_48) in (or TMP_46 TMP_49))))))))) in (let 
+TMP_55 \def (\lambda (u2: T).(\lambda (t2: T).(let TMP_51 \def (Bind Abbr) in 
+(let TMP_52 \def (THead TMP_51 u1 t1) in (let TMP_53 \def (Bind Abbr) in (let 
+TMP_54 \def (THead TMP_53 u2 t2) in (eq T TMP_52 TMP_54))))))) in (let TMP_56 
+\def (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) in (let TMP_61 \def 
+(\lambda (u2: T).(\lambda (t2: T).(let TMP_57 \def (pr0 t1 t2) in (let TMP_58 
+\def (\lambda (y0: T).(pr0 t1 y0)) in (let TMP_59 \def (\lambda (y0: 
+T).(subst0 O u2 y0 t2)) in (let TMP_60 \def (ex2 T TMP_58 TMP_59) in (or 
+TMP_57 TMP_60))))))) in (let TMP_62 \def (ex3_2 T T TMP_55 TMP_56 TMP_61) in 
+(let TMP_63 \def (S O) in (let TMP_64 \def (Bind Abbr) in (let TMP_65 \def 
+(THead TMP_64 u1 t1) in (let TMP_66 \def (lift TMP_63 O TMP_65) in (let 
+TMP_67 \def (pr0 t1 TMP_66) in (let TMP_72 \def (\lambda (u2: T).(\lambda 
+(t2: T).(let TMP_68 \def (Bind Abbr) in (let TMP_69 \def (THead TMP_68 u1 t1) 
+in (let TMP_70 \def (Bind Abbr) in (let TMP_71 \def (THead TMP_70 u2 t2) in 
+(eq T TMP_69 TMP_71))))))) in (let TMP_73 \def (\lambda (u2: T).(\lambda (_: 
+T).(pr0 u1 u2))) in (let TMP_78 \def (\lambda (u2: T).(\lambda (t2: T).(let 
+TMP_74 \def (pr0 t1 t2) in (let TMP_75 \def (\lambda (y0: T).(pr0 t1 y0)) in 
+(let TMP_76 \def (\lambda (y0: T).(subst0 O u2 y0 t2)) in (let TMP_77 \def 
+(ex2 T TMP_75 TMP_76) in (or TMP_74 TMP_77))))))) in (let TMP_79 \def (Bind 
+Abbr) in (let TMP_80 \def (THead TMP_79 u1 t1) in (let TMP_81 \def 
+(refl_equal T TMP_80) in (let TMP_82 \def (pr0_refl u1) in (let TMP_83 \def 
+(pr0 t1 t1) in (let TMP_84 \def (\lambda (y0: T).(pr0 t1 y0)) in (let TMP_85 
+\def (\lambda (y0: T).(subst0 O u1 y0 t1)) in (let TMP_86 \def (ex2 T TMP_84 
+TMP_85) in (let TMP_87 \def (pr0_refl t1) in (let TMP_88 \def (or_introl 
+TMP_83 TMP_86 TMP_87) in (let TMP_89 \def (ex3_2_intro T T TMP_72 TMP_73 
+TMP_78 u1 t1 TMP_81 TMP_82 TMP_88) in (let TMP_90 \def (or_introl TMP_62 
+TMP_67 TMP_89) in (eq_ind_r T TMP_36 TMP_50 TMP_90 t 
+H2)))))))))))))))))))))))))))))))))) in (let TMP_191 \def (\lambda (u0: 
+T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 u2)).(\lambda (H2: (((eq T u0 
+(THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: 
+T).(eq T u2 (THead (Bind Abbr) u3 t2)))) (\lambda (u3: T).(\lambda (_: 
+T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T 
+(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t2))))))) (pr0 
+t1 (lift (S O) O u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H3: 
+(pr0 t0 t2)).(\lambda (H4: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or 
+(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u3 
+t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: 
+T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) 
+(\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O 
+t2)))))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u0 t0) (THead (Bind 
+Abbr) u1 t1))).(let TMP_92 \def (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) in 
+(let TMP_93 \def (THead k u0 t0) in (let TMP_94 \def (Bind Abbr) in (let 
+TMP_95 \def (THead TMP_94 u1 t1) in (let H6 \def (f_equal T K TMP_92 TMP_93 
+TMP_95 H5) in (let TMP_96 \def (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) in 
+(let TMP_97 \def (THead k u0 t0) in (let TMP_98 \def (Bind Abbr) in (let 
+TMP_99 \def (THead TMP_98 u1 t1) in (let H7 \def (f_equal T T TMP_96 TMP_97 
+TMP_99 H5) in (let TMP_100 \def (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) in 
+(let TMP_101 \def (THead k u0 t0) in (let TMP_102 \def (Bind Abbr) in (let 
+TMP_103 \def (THead TMP_102 u1 t1) in (let H8 \def (f_equal T T TMP_100 
+TMP_101 TMP_103 H5) in (let TMP_189 \def (\lambda (H9: (eq T u0 u1)).(\lambda 
+(H10: (eq K k (Bind Abbr))).(let TMP_104 \def (Bind Abbr) in (let TMP_120 
+\def (\lambda (k0: K).(let TMP_108 \def (\lambda (u3: T).(\lambda (t3: 
+T).(let TMP_105 \def (THead k0 u2 t2) in (let TMP_106 \def (Bind Abbr) in 
+(let TMP_107 \def (THead TMP_106 u3 t3) in (eq T TMP_105 TMP_107)))))) in 
+(let TMP_109 \def (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) in (let 
+TMP_114 \def (\lambda (u3: T).(\lambda (t3: T).(let TMP_110 \def (pr0 t1 t3) 
+in (let TMP_111 \def (\lambda (y0: T).(pr0 t1 y0)) in (let TMP_112 \def 
+(\lambda (y0: T).(subst0 O u3 y0 t3)) in (let TMP_113 \def (ex2 T TMP_111 
+TMP_112) in (or TMP_110 TMP_113))))))) in (let TMP_115 \def (ex3_2 T T 
+TMP_108 TMP_109 TMP_114) in (let TMP_116 \def (S O) in (let TMP_117 \def 
+(THead k0 u2 t2) in (let TMP_118 \def (lift TMP_116 O TMP_117) in (let 
+TMP_119 \def (pr0 t1 TMP_118) in (or TMP_115 TMP_119)))))))))) in (let 
+TMP_134 \def (\lambda (t: T).((eq T t (THead (Bind Abbr) u1 t1)) \to (let 
+TMP_123 \def (\lambda (u3: T).(\lambda (t3: T).(let TMP_121 \def (Bind Abbr) 
+in (let TMP_122 \def (THead TMP_121 u3 t3) in (eq T t2 TMP_122))))) in (let 
+TMP_124 \def (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) in (let TMP_129 
+\def (\lambda (u3: T).(\lambda (t3: T).(let TMP_125 \def (pr0 t1 t3) in (let 
+TMP_126 \def (\lambda (y0: T).(pr0 t1 y0)) in (let TMP_127 \def (\lambda (y0: 
+T).(subst0 O u3 y0 t3)) in (let TMP_128 \def (ex2 T TMP_126 TMP_127) in (or 
+TMP_125 TMP_128))))))) in (let TMP_130 \def (ex3_2 T T TMP_123 TMP_124 
+TMP_129) in (let TMP_131 \def (S O) in (let TMP_132 \def (lift TMP_131 O t2) 
+in (let TMP_133 \def (pr0 t1 TMP_132) in (or TMP_130 TMP_133)))))))))) in 
+(let H11 \def (eq_ind T t0 TMP_134 H4 t1 H8) in (let TMP_135 \def (\lambda 
+(t: T).(pr0 t t2)) in (let H12 \def (eq_ind T t0 TMP_135 H3 t1 H8) in (let 
+TMP_149 \def (\lambda (t: T).((eq T t (THead (Bind Abbr) u1 t1)) \to (let 
+TMP_138 \def (\lambda (u3: T).(\lambda (t3: T).(let TMP_136 \def (Bind Abbr) 
+in (let TMP_137 \def (THead TMP_136 u3 t3) in (eq T u2 TMP_137))))) in (let 
+TMP_139 \def (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) in (let TMP_144 
+\def (\lambda (u3: T).(\lambda (t3: T).(let TMP_140 \def (pr0 t1 t3) in (let 
+TMP_141 \def (\lambda (y0: T).(pr0 t1 y0)) in (let TMP_142 \def (\lambda (y0: 
+T).(subst0 O u3 y0 t3)) in (let TMP_143 \def (ex2 T TMP_141 TMP_142) in (or 
+TMP_140 TMP_143))))))) in (let TMP_145 \def (ex3_2 T T TMP_138 TMP_139 
+TMP_144) in (let TMP_146 \def (S O) in (let TMP_147 \def (lift TMP_146 O u2) 
+in (let TMP_148 \def (pr0 t1 TMP_147) in (or TMP_145 TMP_148)))))))))) in 
+(let H13 \def (eq_ind T u0 TMP_149 H2 u1 H9) in (let TMP_150 \def (\lambda 
+(t: T).(pr0 t u2)) in (let H14 \def (eq_ind T u0 TMP_150 H1 u1 H9) in (let 
+TMP_155 \def (\lambda (u3: T).(\lambda (t3: T).(let TMP_151 \def (Bind Abbr) 
+in (let TMP_152 \def (THead TMP_151 u2 t2) in (let TMP_153 \def (Bind Abbr) 
+in (let TMP_154 \def (THead TMP_153 u3 t3) in (eq T TMP_152 TMP_154))))))) in 
+(let TMP_156 \def (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) in (let 
+TMP_161 \def (\lambda (u3: T).(\lambda (t3: T).(let TMP_157 \def (pr0 t1 t3) 
+in (let TMP_158 \def (\lambda (y0: T).(pr0 t1 y0)) in (let TMP_159 \def 
+(\lambda (y0: T).(subst0 O u3 y0 t3)) in (let TMP_160 \def (ex2 T TMP_158 
+TMP_159) in (or TMP_157 TMP_160))))))) in (let TMP_162 \def (ex3_2 T T 
+TMP_155 TMP_156 TMP_161) in (let TMP_163 \def (S O) in (let TMP_164 \def 
+(Bind Abbr) in (let TMP_165 \def (THead TMP_164 u2 t2) in (let TMP_166 \def 
+(lift TMP_163 O TMP_165) in (let TMP_167 \def (pr0 t1 TMP_166) in (let 
+TMP_172 \def (\lambda (u3: T).(\lambda (t3: T).(let TMP_168 \def (Bind Abbr) 
+in (let TMP_169 \def (THead TMP_168 u2 t2) in (let TMP_170 \def (Bind Abbr) 
+in (let TMP_171 \def (THead TMP_170 u3 t3) in (eq T TMP_169 TMP_171))))))) in 
+(let TMP_173 \def (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) in (let 
+TMP_178 \def (\lambda (u3: T).(\lambda (t3: T).(let TMP_174 \def (pr0 t1 t3) 
+in (let TMP_175 \def (\lambda (y0: T).(pr0 t1 y0)) in (let TMP_176 \def 
+(\lambda (y0: T).(subst0 O u3 y0 t3)) in (let TMP_177 \def (ex2 T TMP_175 
+TMP_176) in (or TMP_174 TMP_177))))))) in (let TMP_179 \def (Bind Abbr) in 
+(let TMP_180 \def (THead TMP_179 u2 t2) in (let TMP_181 \def (refl_equal T 
+TMP_180) in (let TMP_182 \def (pr0 t1 t2) in (let TMP_183 \def (\lambda (y0: 
+T).(pr0 t1 y0)) in (let TMP_184 \def (\lambda (y0: T).(subst0 O u2 y0 t2)) in 
+(let TMP_185 \def (ex2 T TMP_183 TMP_184) in (let TMP_186 \def (or_introl 
+TMP_182 TMP_185 H12) in (let TMP_187 \def (ex3_2_intro T T TMP_172 TMP_173 
+TMP_178 u2 t2 TMP_181 H14 TMP_186) in (let TMP_188 \def (or_introl TMP_162 
+TMP_167 TMP_187) in (eq_ind_r K TMP_104 TMP_120 TMP_188 k 
+H10))))))))))))))))))))))))))))))))))) in (let TMP_190 \def (TMP_189 H7) in 
+(TMP_190 H6)))))))))))))))))))))))))))) in (let TMP_217 \def (\lambda (u: 
+T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: 
+(((eq T v1 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T v2 (THead (Bind Abbr) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0 
+t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 
+t2))))))) (pr0 t1 (lift (S O) O v2)))))).(\lambda (t0: T).(\lambda (t2: 
+T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Abbr) u1 
+t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
+(Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
+(u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) 
+(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S O) O 
+t2)))))).(\lambda (H5: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) 
+(THead (Bind Abbr) u1 t1))).(let TMP_192 \def (Flat Appl) in (let TMP_193 
+\def (Bind Abst) in (let TMP_194 \def (THead TMP_193 u t0) in (let TMP_195 
+\def (THead TMP_192 v1 TMP_194) in (let TMP_196 \def (\lambda (ee: T).(match 
+ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k 
+_ _) \Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat _) 
+\Rightarrow True])])) in (let TMP_197 \def (Bind Abbr) in (let TMP_198 \def 
+(THead TMP_197 u1 t1) in (let H6 \def (eq_ind T TMP_195 TMP_196 I TMP_198 H5) 
+in (let TMP_203 \def (\lambda (u2: T).(\lambda (t3: T).(let TMP_199 \def 
+(Bind Abbr) in (let TMP_200 \def (THead TMP_199 v2 t2) in (let TMP_201 \def 
+(Bind Abbr) in (let TMP_202 \def (THead TMP_201 u2 t3) in (eq T TMP_200 
+TMP_202))))))) in (let TMP_204 \def (\lambda (u2: T).(\lambda (_: T).(pr0 u1 
+u2))) in (let TMP_209 \def (\lambda (u2: T).(\lambda (t3: T).(let TMP_205 
+\def (pr0 t1 t3) in (let TMP_206 \def (\lambda (y0: T).(pr0 t1 y0)) in (let 
+TMP_207 \def (\lambda (y0: T).(subst0 O u2 y0 t3)) in (let TMP_208 \def (ex2 
+T TMP_206 TMP_207) in (or TMP_205 TMP_208))))))) in (let TMP_210 \def (ex3_2 
+T T TMP_203 TMP_204 TMP_209) in (let TMP_211 \def (S O) in (let TMP_212 \def 
+(Bind Abbr) in (let TMP_213 \def (THead TMP_212 v2 t2) in (let TMP_214 \def 
+(lift TMP_211 O TMP_213) in (let TMP_215 \def (pr0 t1 TMP_214) in (let 
+TMP_216 \def (or TMP_210 TMP_215) in (False_ind TMP_216 
+H6))))))))))))))))))))))))))))) in (let TMP_251 \def (\lambda (b: B).(\lambda 
+(_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 
+v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind Abbr) u2 t2)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: 
+T).(or (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: 
+T).(subst0 O u2 y0 t2))))))) (pr0 t1 (lift (S O) O v2)))))).(\lambda (u0: 
+T).(\lambda (u2: T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead 
+(Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq 
+T u2 (THead (Bind Abbr) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 
+u3))) (\lambda (u3: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y0: 
+T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t2))))))) (pr0 t1 (lift (S 
+O) O u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 
+t2)).(\lambda (_: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u3 t3)))) 
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: 
+T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: 
+T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (H8: (eq 
+T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Bind Abbr) u1 
+t1))).(let TMP_218 \def (Flat Appl) in (let TMP_219 \def (Bind b) in (let 
+TMP_220 \def (THead TMP_219 u0 t0) in (let TMP_221 \def (THead TMP_218 v1 
+TMP_220) in (let TMP_222 \def (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
+(match k with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) in 
+(let TMP_223 \def (Bind Abbr) in (let TMP_224 \def (THead TMP_223 u1 t1) in 
+(let H9 \def (eq_ind T TMP_221 TMP_222 I TMP_224 H8) in (let TMP_233 \def 
+(\lambda (u3: T).(\lambda (t3: T).(let TMP_225 \def (Bind b) in (let TMP_226 
+\def (Flat Appl) in (let TMP_227 \def (S O) in (let TMP_228 \def (lift 
+TMP_227 O v2) in (let TMP_229 \def (THead TMP_226 TMP_228 t2) in (let TMP_230 
+\def (THead TMP_225 u2 TMP_229) in (let TMP_231 \def (Bind Abbr) in (let 
+TMP_232 \def (THead TMP_231 u3 t3) in (eq T TMP_230 TMP_232))))))))))) in 
+(let TMP_234 \def (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) in (let 
+TMP_239 \def (\lambda (u3: T).(\lambda (t3: T).(let TMP_235 \def (pr0 t1 t3) 
+in (let TMP_236 \def (\lambda (y0: T).(pr0 t1 y0)) in (let TMP_237 \def 
+(\lambda (y0: T).(subst0 O u3 y0 t3)) in (let TMP_238 \def (ex2 T TMP_236 
+TMP_237) in (or TMP_235 TMP_238))))))) in (let TMP_240 \def (ex3_2 T T 
+TMP_233 TMP_234 TMP_239) in (let TMP_241 \def (S O) in (let TMP_242 \def 
+(Bind b) in (let TMP_243 \def (Flat Appl) in (let TMP_244 \def (S O) in (let 
+TMP_245 \def (lift TMP_244 O v2) in (let TMP_246 \def (THead TMP_243 TMP_245 
+t2) in (let TMP_247 \def (THead TMP_242 u2 TMP_246) in (let TMP_248 \def 
+(lift TMP_241 O TMP_247) in (let TMP_249 \def (pr0 t1 TMP_248) in (let 
+TMP_250 \def (or TMP_240 TMP_249) in (False_ind TMP_250 
+H9)))))))))))))))))))))))))))))))))))))) in (let TMP_333 \def (\lambda (u0: 
+T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 u2)).(\lambda (H2: (((eq T u0 
+(THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: 
+T).(eq T u2 (THead (Bind Abbr) u3 t2)))) (\lambda (u3: T).(\lambda (_: 
+T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T 
+(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t2))))))) (pr0 
+t1 (lift (S O) O u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H3: 
+(pr0 t0 t2)).(\lambda (H4: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or 
+(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u3 
+t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: 
+T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) 
+(\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O 
+t2)))))).(\lambda (w: T).(\lambda (H5: (subst0 O u2 t2 w)).(\lambda (H6: (eq 
+T (THead (Bind Abbr) u0 t0) (THead (Bind Abbr) u1 t1))).(let TMP_252 \def 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow u0 | (TLRef _) 
+\Rightarrow u0 | (THead _ t _) \Rightarrow t])) in (let TMP_253 \def (Bind 
+Abbr) in (let TMP_254 \def (THead TMP_253 u0 t0) in (let TMP_255 \def (Bind 
+Abbr) in (let TMP_256 \def (THead TMP_255 u1 t1) in (let H7 \def (f_equal T T 
+TMP_252 TMP_254 TMP_256 H6) in (let TMP_257 \def (\lambda (e: T).(match e 
+with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) 
+\Rightarrow t])) in (let TMP_258 \def (Bind Abbr) in (let TMP_259 \def (THead 
+TMP_258 u0 t0) in (let TMP_260 \def (Bind Abbr) in (let TMP_261 \def (THead 
+TMP_260 u1 t1) in (let H8 \def (f_equal T T TMP_257 TMP_259 TMP_261 H6) in 
+(let TMP_332 \def (\lambda (H9: (eq T u0 u1)).(let TMP_275 \def (\lambda (t: 
+T).((eq T t (THead (Bind Abbr) u1 t1)) \to (let TMP_264 \def (\lambda (u3: 
+T).(\lambda (t3: T).(let TMP_262 \def (Bind Abbr) in (let TMP_263 \def (THead 
+TMP_262 u3 t3) in (eq T t2 TMP_263))))) in (let TMP_265 \def (\lambda (u3: 
+T).(\lambda (_: T).(pr0 u1 u3))) in (let TMP_270 \def (\lambda (u3: 
+T).(\lambda (t3: T).(let TMP_266 \def (pr0 t1 t3) in (let TMP_267 \def 
+(\lambda (y0: T).(pr0 t1 y0)) in (let TMP_268 \def (\lambda (y0: T).(subst0 O 
+u3 y0 t3)) in (let TMP_269 \def (ex2 T TMP_267 TMP_268) in (or TMP_266 
+TMP_269))))))) in (let TMP_271 \def (ex3_2 T T TMP_264 TMP_265 TMP_270) in 
+(let TMP_272 \def (S O) in (let TMP_273 \def (lift TMP_272 O t2) in (let 
+TMP_274 \def (pr0 t1 TMP_273) in (or TMP_271 TMP_274)))))))))) in (let H10 
+\def (eq_ind T t0 TMP_275 H4 t1 H8) in (let TMP_276 \def (\lambda (t: T).(pr0 
+t t2)) in (let H11 \def (eq_ind T t0 TMP_276 H3 t1 H8) in (let TMP_290 \def 
+(\lambda (t: T).((eq T t (THead (Bind Abbr) u1 t1)) \to (let TMP_279 \def 
+(\lambda (u3: T).(\lambda (t3: T).(let TMP_277 \def (Bind Abbr) in (let 
+TMP_278 \def (THead TMP_277 u3 t3) in (eq T u2 TMP_278))))) in (let TMP_280 
+\def (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) in (let TMP_285 \def 
+(\lambda (u3: T).(\lambda (t3: T).(let TMP_281 \def (pr0 t1 t3) in (let 
+TMP_282 \def (\lambda (y0: T).(pr0 t1 y0)) in (let TMP_283 \def (\lambda (y0: 
+T).(subst0 O u3 y0 t3)) in (let TMP_284 \def (ex2 T TMP_282 TMP_283) in (or 
+TMP_281 TMP_284))))))) in (let TMP_286 \def (ex3_2 T T TMP_279 TMP_280 
+TMP_285) in (let TMP_287 \def (S O) in (let TMP_288 \def (lift TMP_287 O u2) 
+in (let TMP_289 \def (pr0 t1 TMP_288) in (or TMP_286 TMP_289)))))))))) in 
+(let H12 \def (eq_ind T u0 TMP_290 H2 u1 H9) in (let TMP_291 \def (\lambda 
+(t: T).(pr0 t u2)) in (let H13 \def (eq_ind T u0 TMP_291 H1 u1 H9) in (let 
+TMP_296 \def (\lambda (u3: T).(\lambda (t3: T).(let TMP_292 \def (Bind Abbr) 
+in (let TMP_293 \def (THead TMP_292 u2 w) in (let TMP_294 \def (Bind Abbr) in 
+(let TMP_295 \def (THead TMP_294 u3 t3) in (eq T TMP_293 TMP_295))))))) in 
+(let TMP_297 \def (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) in (let 
+TMP_302 \def (\lambda (u3: T).(\lambda (t3: T).(let TMP_298 \def (pr0 t1 t3) 
+in (let TMP_299 \def (\lambda (y0: T).(pr0 t1 y0)) in (let TMP_300 \def 
+(\lambda (y0: T).(subst0 O u3 y0 t3)) in (let TMP_301 \def (ex2 T TMP_299 
+TMP_300) in (or TMP_298 TMP_301))))))) in (let TMP_303 \def (ex3_2 T T 
+TMP_296 TMP_297 TMP_302) in (let TMP_304 \def (S O) in (let TMP_305 \def 
+(Bind Abbr) in (let TMP_306 \def (THead TMP_305 u2 w) in (let TMP_307 \def 
+(lift TMP_304 O TMP_306) in (let TMP_308 \def (pr0 t1 TMP_307) in (let 
+TMP_313 \def (\lambda (u3: T).(\lambda (t3: T).(let TMP_309 \def (Bind Abbr) 
+in (let TMP_310 \def (THead TMP_309 u2 w) in (let TMP_311 \def (Bind Abbr) in 
+(let TMP_312 \def (THead TMP_311 u3 t3) in (eq T TMP_310 TMP_312))))))) in 
+(let TMP_314 \def (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) in (let 
+TMP_319 \def (\lambda (u3: T).(\lambda (t3: T).(let TMP_315 \def (pr0 t1 t3) 
+in (let TMP_316 \def (\lambda (y0: T).(pr0 t1 y0)) in (let TMP_317 \def 
+(\lambda (y0: T).(subst0 O u3 y0 t3)) in (let TMP_318 \def (ex2 T TMP_316 
+TMP_317) in (or TMP_315 TMP_318))))))) in (let TMP_320 \def (Bind Abbr) in 
+(let TMP_321 \def (THead TMP_320 u2 w) in (let TMP_322 \def (refl_equal T 
+TMP_321) in (let TMP_323 \def (pr0 t1 w) in (let TMP_324 \def (\lambda (y0: 
+T).(pr0 t1 y0)) in (let TMP_325 \def (\lambda (y0: T).(subst0 O u2 y0 w)) in 
+(let TMP_326 \def (ex2 T TMP_324 TMP_325) in (let TMP_327 \def (\lambda (y0: 
+T).(pr0 t1 y0)) in (let TMP_328 \def (\lambda (y0: T).(subst0 O u2 y0 w)) in 
+(let TMP_329 \def (ex_intro2 T TMP_327 TMP_328 t2 H11 H5) in (let TMP_330 
+\def (or_intror TMP_323 TMP_326 TMP_329) in (let TMP_331 \def (ex3_2_intro T 
+T TMP_313 TMP_314 TMP_319 u2 w TMP_322 H13 TMP_330) in (or_introl TMP_303 
+TMP_308 TMP_331)))))))))))))))))))))))))))))))))) in (TMP_332 
+H7))))))))))))))))))))))))) in (let TMP_427 \def (\lambda (b: B).(\lambda 
+(H1: (not (eq B b Abst))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H2: 
+(pr0 t0 t2)).(\lambda (H3: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: 
+T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) 
+(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S O) O 
+t2)))))).(\lambda (u: T).(\lambda (H4: (eq T (THead (Bind b) u (lift (S O) O 
+t0)) (THead (Bind Abbr) u1 t1))).(let TMP_334 \def (\lambda (e: T).(match e 
+with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) 
+\Rightarrow (match k with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow 
+b])])) in (let TMP_335 \def (Bind b) in (let TMP_336 \def (S O) in (let 
+TMP_337 \def (lift TMP_336 O t0) in (let TMP_338 \def (THead TMP_335 u 
+TMP_337) in (let TMP_339 \def (Bind Abbr) in (let TMP_340 \def (THead TMP_339 
+u1 t1) in (let H5 \def (f_equal T B TMP_334 TMP_338 TMP_340 H4) in (let 
+TMP_341 \def (\lambda (e: T).(match e with [(TSort _) \Rightarrow u | (TLRef 
+_) \Rightarrow u | (THead _ t _) \Rightarrow t])) in (let TMP_342 \def (Bind 
+b) in (let TMP_343 \def (S O) in (let TMP_344 \def (lift TMP_343 O t0) in 
+(let TMP_345 \def (THead TMP_342 u TMP_344) in (let TMP_346 \def (Bind Abbr) 
+in (let TMP_347 \def (THead TMP_346 u1 t1) in (let H6 \def (f_equal T T 
+TMP_341 TMP_345 TMP_347 H4) in (let TMP_362 \def (\lambda (e: T).(match e 
+with [(TSort _) \Rightarrow (let TMP_361 \def (\lambda (x0: nat).(let TMP_360 
+\def (S O) in (plus x0 TMP_360))) in (lref_map TMP_361 O t0)) | (TLRef _) 
+\Rightarrow (let TMP_354 \def (\lambda (x0: nat).(let TMP_353 \def (S O) in 
+(plus x0 TMP_353))) in (lref_map TMP_354 O t0)) | (THead _ _ t) \Rightarrow 
+t])) in (let TMP_363 \def (Bind b) in (let TMP_364 \def (S O) in (let TMP_365 
+\def (lift TMP_364 O t0) in (let TMP_366 \def (THead TMP_363 u TMP_365) in 
+(let TMP_367 \def (Bind Abbr) in (let TMP_368 \def (THead TMP_367 u1 t1) in 
+(let H7 \def (f_equal T T TMP_362 TMP_366 TMP_368 H4) in (let TMP_425 \def 
+(\lambda (_: (eq T u u1)).(\lambda (H9: (eq B b Abbr)).(let TMP_370 \def 
+(\lambda (b0: B).(let TMP_369 \def (eq B b0 Abst) in (not TMP_369))) in (let 
+H10 \def (eq_ind B b TMP_370 H1 Abbr H9) in (let TMP_384 \def (\lambda (t: 
+T).((eq T t0 (THead (Bind Abbr) u1 t)) \to (let TMP_373 \def (\lambda (u2: 
+T).(\lambda (t3: T).(let TMP_371 \def (Bind Abbr) in (let TMP_372 \def (THead 
+TMP_371 u2 t3) in (eq T t2 TMP_372))))) in (let TMP_374 \def (\lambda (u2: 
+T).(\lambda (_: T).(pr0 u1 u2))) in (let TMP_379 \def (\lambda (u2: 
+T).(\lambda (t3: T).(let TMP_375 \def (pr0 t t3) in (let TMP_376 \def 
+(\lambda (y0: T).(pr0 t y0)) in (let TMP_377 \def (\lambda (y0: T).(subst0 O 
+u2 y0 t3)) in (let TMP_378 \def (ex2 T TMP_376 TMP_377) in (or TMP_375 
+TMP_378))))))) in (let TMP_380 \def (ex3_2 T T TMP_373 TMP_374 TMP_379) in 
+(let TMP_381 \def (S O) in (let TMP_382 \def (lift TMP_381 O t2) in (let 
+TMP_383 \def (pr0 t TMP_382) in (or TMP_380 TMP_383)))))))))) in (let TMP_385 
+\def (S O) in (let TMP_386 \def (lift TMP_385 O t0) in (let H11 \def 
+(eq_ind_r T t1 TMP_384 H3 TMP_386 H7) in (let TMP_387 \def (S O) in (let 
+TMP_388 \def (lift TMP_387 O t0) in (let TMP_402 \def (\lambda (t: T).(let 
+TMP_391 \def (\lambda (u2: T).(\lambda (t3: T).(let TMP_389 \def (Bind Abbr) 
+in (let TMP_390 \def (THead TMP_389 u2 t3) in (eq T t2 TMP_390))))) in (let 
+TMP_392 \def (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) in (let TMP_397 
+\def (\lambda (u2: T).(\lambda (t3: T).(let TMP_393 \def (pr0 t t3) in (let 
+TMP_394 \def (\lambda (y0: T).(pr0 t y0)) in (let TMP_395 \def (\lambda (y0: 
+T).(subst0 O u2 y0 t3)) in (let TMP_396 \def (ex2 T TMP_394 TMP_395) in (or 
+TMP_393 TMP_396))))))) in (let TMP_398 \def (ex3_2 T T TMP_391 TMP_392 
+TMP_397) in (let TMP_399 \def (S O) in (let TMP_400 \def (lift TMP_399 O t2) 
+in (let TMP_401 \def (pr0 t TMP_400) in (or TMP_398 TMP_401))))))))) in (let 
+TMP_405 \def (\lambda (u2: T).(\lambda (t3: T).(let TMP_403 \def (Bind Abbr) 
+in (let TMP_404 \def (THead TMP_403 u2 t3) in (eq T t2 TMP_404))))) in (let 
+TMP_406 \def (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) in (let TMP_415 
+\def (\lambda (u2: T).(\lambda (t3: T).(let TMP_407 \def (S O) in (let 
+TMP_408 \def (lift TMP_407 O t0) in (let TMP_409 \def (pr0 TMP_408 t3) in 
+(let TMP_412 \def (\lambda (y0: T).(let TMP_410 \def (S O) in (let TMP_411 
+\def (lift TMP_410 O t0) in (pr0 TMP_411 y0)))) in (let TMP_413 \def (\lambda 
+(y0: T).(subst0 O u2 y0 t3)) in (let TMP_414 \def (ex2 T TMP_412 TMP_413) in 
+(or TMP_409 TMP_414))))))))) in (let TMP_416 \def (ex3_2 T T TMP_405 TMP_406 
+TMP_415) in (let TMP_417 \def (S O) in (let TMP_418 \def (lift TMP_417 O t0) 
+in (let TMP_419 \def (S O) in (let TMP_420 \def (lift TMP_419 O t2) in (let 
+TMP_421 \def (pr0 TMP_418 TMP_420) in (let TMP_422 \def (S O) in (let TMP_423 
+\def (pr0_lift t0 t2 H2 TMP_422 O) in (let TMP_424 \def (or_intror TMP_416 
+TMP_421 TMP_423) in (eq_ind T TMP_388 TMP_402 TMP_424 t1 
+H7)))))))))))))))))))))))) in (let TMP_426 \def (TMP_425 H6) in (TMP_426 
+H5))))))))))))))))))))))))))))))))))) in (let TMP_447 \def (\lambda (t0: 
+T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead 
+(Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq 
+T t2 (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 
+u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: 
+T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S 
+O) O t2)))))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u t0) 
+(THead (Bind Abbr) u1 t1))).(let TMP_428 \def (Flat Cast) in (let TMP_429 
+\def (THead TMP_428 u t0) in (let TMP_430 \def (\lambda (ee: T).(match ee 
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ 
+_) \Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat _) 
+\Rightarrow True])])) in (let TMP_431 \def (Bind Abbr) in (let TMP_432 \def 
+(THead TMP_431 u1 t1) in (let H4 \def (eq_ind T TMP_429 TMP_430 I TMP_432 H3) 
+in (let TMP_435 \def (\lambda (u2: T).(\lambda (t3: T).(let TMP_433 \def 
+(Bind Abbr) in (let TMP_434 \def (THead TMP_433 u2 t3) in (eq T t2 
+TMP_434))))) in (let TMP_436 \def (\lambda (u2: T).(\lambda (_: T).(pr0 u1 
+u2))) in (let TMP_441 \def (\lambda (u2: T).(\lambda (t3: T).(let TMP_437 
+\def (pr0 t1 t3) in (let TMP_438 \def (\lambda (y0: T).(pr0 t1 y0)) in (let 
+TMP_439 \def (\lambda (y0: T).(subst0 O u2 y0 t3)) in (let TMP_440 \def (ex2 
+T TMP_438 TMP_439) in (or TMP_437 TMP_440))))))) in (let TMP_442 \def (ex3_2 
+T T TMP_435 TMP_436 TMP_441) in (let TMP_443 \def (S O) in (let TMP_444 \def 
+(lift TMP_443 O t2) in (let TMP_445 \def (pr0 t1 TMP_444) in (let TMP_446 
+\def (or TMP_442 TMP_445) in (False_ind TMP_446 H4))))))))))))))))))))) in 
+(pr0_ind TMP_31 TMP_91 TMP_191 TMP_217 TMP_251 TMP_333 TMP_427 TMP_447 y x 
+H0))))))))))) in (insert_eq T TMP_2 TMP_3 TMP_17 TMP_448 H))))))))).

 
 

-theorem pr0_subst0_fwd:
- \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst0 
-i u2 t1 t2) \to (\forall (u1: T).((pr0 u2 u1) \to (ex2 T (\lambda (t: 
-T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t2 t)))))))))
+theorem pr0_gen_void:
+ \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Void) u1 
+t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead 
+(Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
+(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O x))))))

 \def
 \def

- \lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (i: nat).(\lambda 
-(H: (subst0 i u2 t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t: 
-T).(\lambda (t0: T).(\lambda (t3: T).(\forall (u1: T).((pr0 t u1) \to (ex2 T 
-(\lambda (t4: T).(subst0 n u1 t0 t4)) (\lambda (t4: T).(pr0 t3 t4))))))))) 
-(\lambda (v: T).(\lambda (i0: nat).(\lambda (u1: T).(\lambda (H0: (pr0 v 
-u1)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 (TLRef i0) t)) (\lambda (t: 
-T).(pr0 (lift (S i0) O v) t)) (lift (S i0) O u1) (subst0_lref u1 i0) 
-(pr0_lift v u1 H0 (S i0) O)))))) (\lambda (v: T).(\lambda (u3: T).(\lambda 
-(u1: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v u1 u3)).(\lambda (H1: 
-((\forall (u4: T).((pr0 v u4) \to (ex2 T (\lambda (t: T).(subst0 i0 u4 u1 t)) 
-(\lambda (t: T).(pr0 u3 t))))))).(\lambda (t: T).(\lambda (k: K).(\lambda 
-(u0: T).(\lambda (H2: (pr0 v u0)).(ex2_ind T (\lambda (t0: T).(subst0 i0 u0 
-u1 t0)) (\lambda (t0: T).(pr0 u3 t0)) (ex2 T (\lambda (t0: T).(subst0 i0 u0 
-(THead k u1 t) t0)) (\lambda (t0: T).(pr0 (THead k u3 t) t0))) (\lambda (x: 
-T).(\lambda (H3: (subst0 i0 u0 u1 x)).(\lambda (H4: (pr0 u3 x)).(ex_intro2 T 
-(\lambda (t0: T).(subst0 i0 u0 (THead k u1 t) t0)) (\lambda (t0: T).(pr0 
-(THead k u3 t) t0)) (THead k x t) (subst0_fst u0 x u1 i0 H3 t k) (pr0_comp u3 
-x H4 t t (pr0_refl t) k))))) (H1 u0 H2)))))))))))) (\lambda (k: K).(\lambda 
-(v: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (i0: nat).(\lambda (_: 
-(subst0 (s k i0) v t4 t3)).(\lambda (H1: ((\forall (u1: T).((pr0 v u1) \to 
-(ex2 T (\lambda (t: T).(subst0 (s k i0) u1 t4 t)) (\lambda (t: T).(pr0 t3 
-t))))))).(\lambda (u: T).(\lambda (u1: T).(\lambda (H2: (pr0 v u1)).(ex2_ind 
-T (\lambda (t: T).(subst0 (s k i0) u1 t4 t)) (\lambda (t: T).(pr0 t3 t)) (ex2 
-T (\lambda (t: T).(subst0 i0 u1 (THead k u t4) t)) (\lambda (t: T).(pr0 
-(THead k u t3) t))) (\lambda (x: T).(\lambda (H3: (subst0 (s k i0) u1 t4 
-x)).(\lambda (H4: (pr0 t3 x)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 
-(THead k u t4) t)) (\lambda (t: T).(pr0 (THead k u t3) t)) (THead k u x) 
-(subst0_snd k u1 x t4 i0 H3 u) (pr0_comp u u (pr0_refl u) t3 x H4 k))))) (H1 
-u1 H2)))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u3: T).(\lambda 
-(i0: nat).(\lambda (_: (subst0 i0 v u1 u3)).(\lambda (H1: ((\forall (u4: 
-T).((pr0 v u4) \to (ex2 T (\lambda (t: T).(subst0 i0 u4 u1 t)) (\lambda (t: 
-T).(pr0 u3 t))))))).(\lambda (k: K).(\lambda (t3: T).(\lambda (t4: 
-T).(\lambda (_: (subst0 (s k i0) v t3 t4)).(\lambda (H3: ((\forall (u4: 
-T).((pr0 v u4) \to (ex2 T (\lambda (t: T).(subst0 (s k i0) u4 t3 t)) (\lambda 
-(t: T).(pr0 t4 t))))))).(\lambda (u0: T).(\lambda (H4: (pr0 v u0)).(ex2_ind T 
-(\lambda (t: T).(subst0 (s k i0) u0 t3 t)) (\lambda (t: T).(pr0 t4 t)) (ex2 T 
-(\lambda (t: T).(subst0 i0 u0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 (THead 
-k u3 t4) t))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i0) u0 t3 
-x)).(\lambda (H6: (pr0 t4 x)).(ex2_ind T (\lambda (t: T).(subst0 i0 u0 u1 t)) 
-(\lambda (t: T).(pr0 u3 t)) (ex2 T (\lambda (t: T).(subst0 i0 u0 (THead k u1 
-t3) t)) (\lambda (t: T).(pr0 (THead k u3 t4) t))) (\lambda (x0: T).(\lambda 
-(H7: (subst0 i0 u0 u1 x0)).(\lambda (H8: (pr0 u3 x0)).(ex_intro2 T (\lambda 
-(t: T).(subst0 i0 u0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 (THead k u3 t4) 
-t)) (THead k x0 x) (subst0_both u0 u1 x0 i0 H7 k t3 x H5) (pr0_comp u3 x0 H8 
-t4 x H6 k))))) (H1 u0 H4))))) (H3 u0 H4))))))))))))))) i u2 t1 t2 H))))).
-(* COMMENTS
-Initial nodes: 979
-END *)
-
-theorem pr0_subst0:
- \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (v1: T).(\forall 
-(w1: T).(\forall (i: nat).((subst0 i v1 t1 w1) \to (\forall (v2: T).((pr0 v1 
-v2) \to (or (pr0 w1 t2) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: 
-T).(subst0 i v2 t2 w2))))))))))))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(pr0_ind (\lambda 
-(t: T).(\lambda (t0: T).(\forall (v1: T).(\forall (w1: T).(\forall (i: 
-nat).((subst0 i v1 t w1) \to (\forall (v2: T).((pr0 v1 v2) \to (or (pr0 w1 
-t0) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t0 
-w2)))))))))))) (\lambda (t: T).(\lambda (v1: T).(\lambda (w1: T).(\lambda (i: 
-nat).(\lambda (H0: (subst0 i v1 t w1)).(\lambda (v2: T).(\lambda (H1: (pr0 v1 
-v2)).(or_intror (pr0 w1 t) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: 
-T).(subst0 i v2 t w2))) (ex2_sym T (subst0 i v2 t) (pr0 w1) (pr0_subst0_fwd 
-v1 t w1 i H0 v2 H1)))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (H0: 
-(pr0 u1 u2)).(\lambda (H1: ((\forall (v1: T).(\forall (w1: T).(\forall (i: 
-nat).((subst0 i v1 u1 w1) \to (\forall (v2: T).((pr0 v1 v2) \to (or (pr0 w1 
-u2) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 u2 
-w2)))))))))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 
-t4)).(\lambda (H3: ((\forall (v1: T).(\forall (w1: T).(\forall (i: 
-nat).((subst0 i v1 t3 w1) \to (\forall (v2: T).((pr0 v1 v2) \to (or (pr0 w1 
-t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 
-w2)))))))))))).(\lambda (k: K).(\lambda (v1: T).(\lambda (w1: T).(\lambda (i: 
-nat).(\lambda (H4: (subst0 i v1 (THead k u1 t3) w1)).(\lambda (v2: 
-T).(\lambda (H5: (pr0 v1 v2)).(or3_ind (ex2 T (\lambda (u3: T).(eq T w1 
-(THead k u3 t3))) (\lambda (u3: T).(subst0 i v1 u1 u3))) (ex2 T (\lambda (t5: 
-T).(eq T w1 (THead k u1 t5))) (\lambda (t5: T).(subst0 (s k i) v1 t3 t5))) 
-(ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead k u3 t5)))) 
-(\lambda (u3: T).(\lambda (_: T).(subst0 i v1 u1 u3))) (\lambda (_: 
-T).(\lambda (t5: T).(subst0 (s k i) v1 t3 t5)))) (or (pr0 w1 (THead k u2 t4)) 
-(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 (THead k 
-u2 t4) w2)))) (\lambda (H6: (ex2 T (\lambda (u3: T).(eq T w1 (THead k u3 
-t3))) (\lambda (u3: T).(subst0 i v1 u1 u3)))).(ex2_ind T (\lambda (u3: T).(eq 
-T w1 (THead k u3 t3))) (\lambda (u3: T).(subst0 i v1 u1 u3)) (or (pr0 w1 
-(THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: 
-T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x: T).(\lambda (H7: (eq T w1 
-(THead k x t3))).(\lambda (H8: (subst0 i v1 u1 x)).(eq_ind_r T (THead k x t3) 
-(\lambda (t: T).(or (pr0 t (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 t 
-w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))))) (or_ind (pr0 x u2) 
-(ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 u2 w2))) 
-(or (pr0 (THead k x t3) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead 
-k x t3) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda 
-(H9: (pr0 x u2)).(or_introl (pr0 (THead k x t3) (THead k u2 t4)) (ex2 T 
-(\lambda (w2: T).(pr0 (THead k x t3) w2)) (\lambda (w2: T).(subst0 i v2 
-(THead k u2 t4) w2))) (pr0_comp x u2 H9 t3 t4 H2 k))) (\lambda (H9: (ex2 T 
-(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 u2 w2)))).(ex2_ind 
-T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 u2 w2)) (or (pr0 
-(THead k x t3) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x t3) 
-w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x0: 
-T).(\lambda (H10: (pr0 x x0)).(\lambda (H11: (subst0 i v2 u2 x0)).(or_intror 
-(pr0 (THead k x t3) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x 
-t3) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))) (ex_intro2 T 
-(\lambda (w2: T).(pr0 (THead k x t3) w2)) (\lambda (w2: T).(subst0 i v2 
-(THead k u2 t4) w2)) (THead k x0 t4) (pr0_comp x x0 H10 t3 t4 H2 k) 
-(subst0_fst v2 x0 u2 i H11 t4 k)))))) H9)) (H1 v1 x i H8 v2 H5)) w1 H7)))) 
-H6)) (\lambda (H6: (ex2 T (\lambda (t5: T).(eq T w1 (THead k u1 t5))) 
-(\lambda (t5: T).(subst0 (s k i) v1 t3 t5)))).(ex2_ind T (\lambda (t5: T).(eq 
-T w1 (THead k u1 t5))) (\lambda (t5: T).(subst0 (s k i) v1 t3 t5)) (or (pr0 
-w1 (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: 
-T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x: T).(\lambda (H7: (eq T w1 
-(THead k u1 x))).(\lambda (H8: (subst0 (s k i) v1 t3 x)).(eq_ind_r T (THead k 
-u1 x) (\lambda (t: T).(or (pr0 t (THead k u2 t4)) (ex2 T (\lambda (w2: 
-T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))))) (or_ind 
-(pr0 x t4) (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s k 
-i) v2 t4 w2))) (or (pr0 (THead k u1 x) (THead k u2 t4)) (ex2 T (\lambda (w2: 
-T).(pr0 (THead k u1 x) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) 
-w2)))) (\lambda (H9: (pr0 x t4)).(or_introl (pr0 (THead k u1 x) (THead k u2 
-t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k u1 x) w2)) (\lambda (w2: 
-T).(subst0 i v2 (THead k u2 t4) w2))) (pr0_comp u1 u2 H0 x t4 H9 k))) 
-(\lambda (H9: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s 
-k i) v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: 
-T).(subst0 (s k i) v2 t4 w2)) (or (pr0 (THead k u1 x) (THead k u2 t4)) (ex2 T 
-(\lambda (w2: T).(pr0 (THead k u1 x) w2)) (\lambda (w2: T).(subst0 i v2 
-(THead k u2 t4) w2)))) (\lambda (x0: T).(\lambda (H10: (pr0 x x0)).(\lambda 
-(H11: (subst0 (s k i) v2 t4 x0)).(or_intror (pr0 (THead k u1 x) (THead k u2 
-t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k u1 x) w2)) (\lambda (w2: 
-T).(subst0 i v2 (THead k u2 t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 
-(THead k u1 x) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)) (THead 
-k u2 x0) (pr0_comp u1 u2 H0 x x0 H10 k) (subst0_snd k v2 x0 t4 i H11 u2)))))) 
-H9)) (H3 v1 x (s k i) H8 v2 H5)) w1 H7)))) H6)) (\lambda (H6: (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead k u3 t5)))) (\lambda (u3: 
-T).(\lambda (_: T).(subst0 i v1 u1 u3))) (\lambda (_: T).(\lambda (t5: 
-T).(subst0 (s k i) v1 t3 t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda 
-(t5: T).(eq T w1 (THead k u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 
-i v1 u1 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i) v1 t3 t5))) 
-(or (pr0 w1 (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda 
-(w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x0: T).(\lambda (x1: 
-T).(\lambda (H7: (eq T w1 (THead k x0 x1))).(\lambda (H8: (subst0 i v1 u1 
-x0)).(\lambda (H9: (subst0 (s k i) v1 t3 x1)).(eq_ind_r T (THead k x0 x1) 
-(\lambda (t: T).(or (pr0 t (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 t 
-w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))))) (or_ind (pr0 x1 
-t4) (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s k i) v2 
-t4 w2))) (or (pr0 (THead k x0 x1) (THead k u2 t4)) (ex2 T (\lambda (w2: 
-T).(pr0 (THead k x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) 
-w2)))) (\lambda (H10: (pr0 x1 t4)).(or_ind (pr0 x0 u2) (ex2 T (\lambda (w2: 
-T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 u2 w2))) (or (pr0 (THead k x0 
-x1) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x0 x1) w2)) 
-(\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (H11: (pr0 x0 
-u2)).(or_introl (pr0 (THead k x0 x1) (THead k u2 t4)) (ex2 T (\lambda (w2: 
-T).(pr0 (THead k x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) 
-w2))) (pr0_comp x0 u2 H11 x1 t4 H10 k))) (\lambda (H11: (ex2 T (\lambda (w2: 
-T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 u2 w2)))).(ex2_ind T (\lambda 
-(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 u2 w2)) (or (pr0 (THead k 
-x0 x1) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x0 x1) w2)) 
-(\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x: T).(\lambda 
-(H12: (pr0 x0 x)).(\lambda (H13: (subst0 i v2 u2 x)).(or_intror (pr0 (THead k 
-x0 x1) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x0 x1) w2)) 
-(\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))) (ex_intro2 T (\lambda 
-(w2: T).(pr0 (THead k x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 
-t4) w2)) (THead k x t4) (pr0_comp x0 x H12 x1 t4 H10 k) (subst0_fst v2 x u2 i 
-H13 t4 k)))))) H11)) (H1 v1 x0 i H8 v2 H5))) (\lambda (H10: (ex2 T (\lambda 
-(w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s k i) v2 t4 w2)))).(ex2_ind T 
-(\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s k i) v2 t4 w2)) (or 
-(pr0 (THead k x0 x1) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k 
-x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x: 
-T).(\lambda (H11: (pr0 x1 x)).(\lambda (H12: (subst0 (s k i) v2 t4 
-x)).(or_ind (pr0 x0 u2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: 
-T).(subst0 i v2 u2 w2))) (or (pr0 (THead k x0 x1) (THead k u2 t4)) (ex2 T 
-(\lambda (w2: T).(pr0 (THead k x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 
-(THead k u2 t4) w2)))) (\lambda (H13: (pr0 x0 u2)).(or_intror (pr0 (THead k 
-x0 x1) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x0 x1) w2)) 
-(\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))) (ex_intro2 T (\lambda 
-(w2: T).(pr0 (THead k x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 
-t4) w2)) (THead k u2 x) (pr0_comp x0 u2 H13 x1 x H11 k) (subst0_snd k v2 x t4 
-i H12 u2)))) (\lambda (H13: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda 
-(w2: T).(subst0 i v2 u2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) 
-(\lambda (w2: T).(subst0 i v2 u2 w2)) (or (pr0 (THead k x0 x1) (THead k u2 
-t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x0 x1) w2)) (\lambda (w2: 
-T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x2: T).(\lambda (H14: (pr0 
-x0 x2)).(\lambda (H15: (subst0 i v2 u2 x2)).(or_intror (pr0 (THead k x0 x1) 
-(THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x0 x1) w2)) (\lambda 
-(w2: T).(subst0 i v2 (THead k u2 t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 
-(THead k x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)) 
-(THead k x2 x) (pr0_comp x0 x2 H14 x1 x H11 k) (subst0_both v2 u2 x2 i H15 k 
-t4 x H12)))))) H13)) (H1 v1 x0 i H8 v2 H5))))) H10)) (H3 v1 x1 (s k i) H9 v2 
-H5)) w1 H7)))))) H6)) (subst0_gen_head k v1 u1 t3 w1 i H4))))))))))))))))) 
-(\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H0: (pr0 v1 
-v2)).(\lambda (H1: ((\forall (v3: T).(\forall (w1: T).(\forall (i: 
-nat).((subst0 i v3 v1 w1) \to (\forall (v4: T).((pr0 v3 v4) \to (or (pr0 w1 
-v2) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v4 v2 
-w2)))))))))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 
-t4)).(\lambda (H3: ((\forall (v3: T).(\forall (w1: T).(\forall (i: 
-nat).((subst0 i v3 t3 w1) \to (\forall (v4: T).((pr0 v3 v4) \to (or (pr0 w1 
-t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v4 t4 
-w2)))))))))))).(\lambda (v0: T).(\lambda (w1: T).(\lambda (i: nat).(\lambda 
-(H4: (subst0 i v0 (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) 
-w1)).(\lambda (v3: T).(\lambda (H5: (pr0 v0 v3)).(or3_ind (ex2 T (\lambda 
-(u2: T).(eq T w1 (THead (Flat Appl) u2 (THead (Bind Abst) u t3)))) (\lambda 
-(u2: T).(subst0 i v0 v1 u2))) (ex2 T (\lambda (t5: T).(eq T w1 (THead (Flat 
-Appl) v1 t5))) (\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 (THead (Bind 
-Abst) u t3) t5))) (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T w1 
-(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 v1 
-u2))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 (THead 
-(Bind Abst) u t3) t5)))) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T 
-(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind 
-Abbr) v2 t4) w2)))) (\lambda (H6: (ex2 T (\lambda (u2: T).(eq T w1 (THead 
-(Flat Appl) u2 (THead (Bind Abst) u t3)))) (\lambda (u2: T).(subst0 i v0 v1 
-u2)))).(ex2_ind T (\lambda (u2: T).(eq T w1 (THead (Flat Appl) u2 (THead 
-(Bind Abst) u t3)))) (\lambda (u2: T).(subst0 i v0 v1 u2)) (or (pr0 w1 (THead 
-(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (x: T).(\lambda 
-(H7: (eq T w1 (THead (Flat Appl) x (THead (Bind Abst) u t3)))).(\lambda (H8: 
-(subst0 i v0 v1 x)).(eq_ind_r T (THead (Flat Appl) x (THead (Bind Abst) u 
-t3)) (\lambda (t: T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda 
-(w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) 
-w2))))) (or_ind (pr0 x v2) (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: 
-T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat Appl) x (THead (Bind Abst) u 
-t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat 
-Appl) x (THead (Bind Abst) u t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead 
-(Bind Abbr) v2 t4) w2)))) (\lambda (H9: (pr0 x v2)).(or_introl (pr0 (THead 
-(Flat Appl) x (THead (Bind Abst) u t3)) (THead (Bind Abbr) v2 t4)) (ex2 T 
-(\lambda (w2: T).(pr0 (THead (Flat Appl) x (THead (Bind Abst) u t3)) w2)) 
-(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (pr0_beta u x 
-v2 H9 t3 t4 H2))) (\lambda (H9: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda 
-(w2: T).(subst0 i v3 v2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) 
-(\lambda (w2: T).(subst0 i v3 v2 w2)) (or (pr0 (THead (Flat Appl) x (THead 
-(Bind Abst) u t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 
-(THead (Flat Appl) x (THead (Bind Abst) u t3)) w2)) (\lambda (w2: T).(subst0 
-i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (x0: T).(\lambda (H10: (pr0 x 
-x0)).(\lambda (H11: (subst0 i v3 v2 x0)).(or_intror (pr0 (THead (Flat Appl) x 
-(THead (Bind Abst) u t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: 
-T).(pr0 (THead (Flat Appl) x (THead (Bind Abst) u t3)) w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: 
-T).(pr0 (THead (Flat Appl) x (THead (Bind Abst) u t3)) w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) x0 t4) 
-(pr0_beta u x x0 H10 t3 t4 H2) (subst0_fst v3 x0 v2 i H11 t4 (Bind 
-Abbr))))))) H9)) (H1 v0 x i H8 v3 H5)) w1 H7)))) H6)) (\lambda (H6: (ex2 T 
-(\lambda (t5: T).(eq T w1 (THead (Flat Appl) v1 t5))) (\lambda (t5: 
-T).(subst0 (s (Flat Appl) i) v0 (THead (Bind Abst) u t3) t5)))).(ex2_ind T 
-(\lambda (t5: T).(eq T w1 (THead (Flat Appl) v1 t5))) (\lambda (t5: 
-T).(subst0 (s (Flat Appl) i) v0 (THead (Bind Abst) u t3) t5)) (or (pr0 w1 
-(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (x: T).(\lambda 
-(H7: (eq T w1 (THead (Flat Appl) v1 x))).(\lambda (H8: (subst0 (s (Flat Appl) 
-i) v0 (THead (Bind Abst) u t3) x)).(or3_ind (ex2 T (\lambda (u2: T).(eq T x 
-(THead (Bind Abst) u2 t3))) (\lambda (u2: T).(subst0 (s (Flat Appl) i) v0 u 
-u2))) (ex2 T (\lambda (t5: T).(eq T x (THead (Bind Abst) u t5))) (\lambda 
-(t5: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5))) (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t5: T).(eq T x (THead (Bind Abst) u2 t5)))) 
-(\lambda (u2: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u u2))) 
-(\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 
-t3 t5)))) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 
-w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) 
-(\lambda (H9: (ex2 T (\lambda (u2: T).(eq T x (THead (Bind Abst) u2 t3))) 
-(\lambda (u2: T).(subst0 (s (Flat Appl) i) v0 u u2)))).(ex2_ind T (\lambda 
-(u2: T).(eq T x (THead (Bind Abst) u2 t3))) (\lambda (u2: T).(subst0 (s (Flat 
-Appl) i) v0 u u2)) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda 
-(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) 
-w2)))) (\lambda (x0: T).(\lambda (H10: (eq T x (THead (Bind Abst) x0 
-t3))).(\lambda (_: (subst0 (s (Flat Appl) i) v0 u x0)).(let H12 \def (eq_ind 
-T x (\lambda (t: T).(eq T w1 (THead (Flat Appl) v1 t))) H7 (THead (Bind Abst) 
-x0 t3) H10) in (eq_ind_r T (THead (Flat Appl) v1 (THead (Bind Abst) x0 t3)) 
-(\lambda (t: T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: 
-T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) 
-w2))))) (or_introl (pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 t3)) 
-(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 
-(THead (Bind Abst) x0 t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind 
-Abbr) v2 t4) w2))) (pr0_beta x0 v1 v2 H0 t3 t4 H2)) w1 H12))))) H9)) (\lambda 
-(H9: (ex2 T (\lambda (t5: T).(eq T x (THead (Bind Abst) u t5))) (\lambda (t5: 
-T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5)))).(ex2_ind T (\lambda 
-(t5: T).(eq T x (THead (Bind Abst) u t5))) (\lambda (t5: T).(subst0 (s (Bind 
-Abst) (s (Flat Appl) i)) v0 t3 t5)) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) 
-(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead 
-(Bind Abbr) v2 t4) w2)))) (\lambda (x0: T).(\lambda (H10: (eq T x (THead 
-(Bind Abst) u x0))).(\lambda (H11: (subst0 (s (Bind Abst) (s (Flat Appl) i)) 
-v0 t3 x0)).(let H12 \def (eq_ind T x (\lambda (t: T).(eq T w1 (THead (Flat 
-Appl) v1 t))) H7 (THead (Bind Abst) u x0) H10) in (eq_ind_r T (THead (Flat 
-Appl) v1 (THead (Bind Abst) u x0)) (\lambda (t: T).(or (pr0 t (THead (Bind 
-Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i 
-v3 (THead (Bind Abbr) v2 t4) w2))))) (or_ind (pr0 x0 t4) (ex2 T (\lambda (w2: 
-T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 
-t4 w2))) (or (pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) (THead 
-(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead 
-(Bind Abst) u x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
-t4) w2)))) (\lambda (H13: (pr0 x0 t4)).(or_introl (pr0 (THead (Flat Appl) v1 
-(THead (Bind Abst) u x0)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: 
-T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (pr0_beta u v1 v2 H0 x0 t4 
-H13))) (\lambda (H13: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: 
-T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda 
-(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) 
-i)) v3 t4 w2)) (or (pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) 
-(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 
-(THead (Bind Abst) u x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind 
-Abbr) v2 t4) w2)))) (\lambda (x1: T).(\lambda (H14: (pr0 x0 x1)).(\lambda 
-(H15: (subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 x1)).(or_intror (pr0 
-(THead (Flat Appl) v1 (THead (Bind Abst) u x0)) (THead (Bind Abbr) v2 t4)) 
-(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) 
-w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (ex_intro2 
-T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) w2)) 
-(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)) (THead (Bind 
-Abbr) v2 x1) (pr0_beta u v1 v2 H0 x0 x1 H14) (subst0_snd (Bind Abbr) v3 x1 t4 
-i H15 v2)))))) H13)) (H3 v0 x0 (s (Bind Abst) (s (Flat Appl) i)) H11 v3 H5)) 
-w1 H12))))) H9)) (\lambda (H9: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: 
-T).(eq T x (THead (Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
-T).(subst0 (s (Flat Appl) i) v0 u u2))) (\lambda (_: T).(\lambda (t5: 
-T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5))))).(ex3_2_ind T T 
-(\lambda (u2: T).(\lambda (t5: T).(eq T x (THead (Bind Abst) u2 t5)))) 
-(\lambda (u2: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u u2))) 
-(\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 
-t3 t5))) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 
-w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) 
-(\lambda (x0: T).(\lambda (x1: T).(\lambda (H10: (eq T x (THead (Bind Abst) 
-x0 x1))).(\lambda (_: (subst0 (s (Flat Appl) i) v0 u x0)).(\lambda (H12: 
-(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 x1)).(let H13 \def (eq_ind T 
-x (\lambda (t: T).(eq T w1 (THead (Flat Appl) v1 t))) H7 (THead (Bind Abst) 
-x0 x1) H10) in (eq_ind_r T (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) 
-(\lambda (t: T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: 
-T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) 
-w2))))) (or_ind (pr0 x1 t4) (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda 
-(w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2))) (or (pr0 (THead 
-(Flat Appl) v1 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) v2 t4)) (ex2 T 
-(\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) w2)) 
-(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H14: 
-(pr0 x1 t4)).(or_introl (pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) 
-(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 
-(THead (Bind Abst) x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind 
-Abbr) v2 t4) w2))) (pr0_beta x0 v1 v2 H0 x1 t4 H14))) (\lambda (H14: (ex2 T 
-(\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s 
-(Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x1 w2)) 
-(\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2)) (or 
-(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) v2 
-t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 
-x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) 
-(\lambda (x2: T).(\lambda (H15: (pr0 x1 x2)).(\lambda (H16: (subst0 (s (Bind 
-Abst) (s (Flat Appl) i)) v3 t4 x2)).(or_intror (pr0 (THead (Flat Appl) v1 
-(THead (Bind Abst) x0 x1)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: 
-T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: 
-T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) v2 x2) 
-(pr0_beta x0 v1 v2 H0 x1 x2 H15) (subst0_snd (Bind Abbr) v3 x2 t4 i H16 
-v2)))))) H14)) (H3 v0 x1 (s (Bind Abst) (s (Flat Appl) i)) H12 v3 H5)) w1 
-H13))))))) H9)) (subst0_gen_head (Bind Abst) v0 u t3 x (s (Flat Appl) i) 
-H8))))) H6)) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T 
-w1 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 
-v1 u2))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 (THead 
-(Bind Abst) u t3) t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: 
-T).(eq T w1 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
-T).(subst0 i v0 v1 u2))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Flat 
-Appl) i) v0 (THead (Bind Abst) u t3) t5))) (or (pr0 w1 (THead (Bind Abbr) v2 
-t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 
-(THead (Bind Abbr) v2 t4) w2)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda 
-(H7: (eq T w1 (THead (Flat Appl) x0 x1))).(\lambda (H8: (subst0 i v0 v1 
-x0)).(\lambda (H9: (subst0 (s (Flat Appl) i) v0 (THead (Bind Abst) u t3) 
-x1)).(or3_ind (ex2 T (\lambda (u2: T).(eq T x1 (THead (Bind Abst) u2 t3))) 
-(\lambda (u2: T).(subst0 (s (Flat Appl) i) v0 u u2))) (ex2 T (\lambda (t5: 
-T).(eq T x1 (THead (Bind Abst) u t5))) (\lambda (t5: T).(subst0 (s (Bind 
-Abst) (s (Flat Appl) i)) v0 t3 t5))) (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t5: T).(eq T x1 (THead (Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
-T).(subst0 (s (Flat Appl) i) v0 u u2))) (\lambda (_: T).(\lambda (t5: 
-T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5)))) (or (pr0 w1 (THead 
-(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H10: (ex2 T 
-(\lambda (u2: T).(eq T x1 (THead (Bind Abst) u2 t3))) (\lambda (u2: 
-T).(subst0 (s (Flat Appl) i) v0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T x1 
-(THead (Bind Abst) u2 t3))) (\lambda (u2: T).(subst0 (s (Flat Appl) i) v0 u 
-u2)) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 
-w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda 
-(x: T).(\lambda (H11: (eq T x1 (THead (Bind Abst) x t3))).(\lambda (_: 
-(subst0 (s (Flat Appl) i) v0 u x)).(let H13 \def (eq_ind T x1 (\lambda (t: 
-T).(eq T w1 (THead (Flat Appl) x0 t))) H7 (THead (Bind Abst) x t3) H11) in 
-(eq_ind_r T (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) (\lambda (t: 
-T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 t w2)) 
-(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))))) (or_ind (pr0 
-x0 v2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 
-w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) (THead (Bind 
-Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind 
-Abst) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) 
-w2)))) (\lambda (H14: (pr0 x0 v2)).(or_introl (pr0 (THead (Flat Appl) x0 
-(THead (Bind Abst) x t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: 
-T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (pr0_beta x x0 v2 H14 t3 t4 
-H2))) (\lambda (H14: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: 
-T).(subst0 i v3 v2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda 
-(w2: T).(subst0 i v3 v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind 
-Abst) x t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead 
-(Flat Appl) x0 (THead (Bind Abst) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 
-(THead (Bind Abbr) v2 t4) w2)))) (\lambda (x2: T).(\lambda (H15: (pr0 x0 
-x2)).(\lambda (H16: (subst0 i v3 v2 x2)).(or_intror (pr0 (THead (Flat Appl) 
-x0 (THead (Bind Abst) x t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: 
-T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: 
-T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) x2 t4) 
-(pr0_beta x x0 x2 H15 t3 t4 H2) (subst0_fst v3 x2 v2 i H16 t4 (Bind 
-Abbr))))))) H14)) (H1 v0 x0 i H8 v3 H5)) w1 H13))))) H10)) (\lambda (H10: 
-(ex2 T (\lambda (t5: T).(eq T x1 (THead (Bind Abst) u t5))) (\lambda (t5: 
-T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5)))).(ex2_ind T (\lambda 
-(t5: T).(eq T x1 (THead (Bind Abst) u t5))) (\lambda (t5: T).(subst0 (s (Bind 
-Abst) (s (Flat Appl) i)) v0 t3 t5)) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) 
-(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead 
-(Bind Abbr) v2 t4) w2)))) (\lambda (x: T).(\lambda (H11: (eq T x1 (THead 
-(Bind Abst) u x))).(\lambda (H12: (subst0 (s (Bind Abst) (s (Flat Appl) i)) 
-v0 t3 x)).(let H13 \def (eq_ind T x1 (\lambda (t: T).(eq T w1 (THead (Flat 
-Appl) x0 t))) H7 (THead (Bind Abst) u x) H11) in (eq_ind_r T (THead (Flat 
-Appl) x0 (THead (Bind Abst) u x)) (\lambda (t: T).(or (pr0 t (THead (Bind 
-Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i 
-v3 (THead (Bind Abbr) v2 t4) w2))))) (or_ind (pr0 x t4) (ex2 T (\lambda (w2: 
-T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 
-t4 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind 
-Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind 
-Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) 
-w2)))) (\lambda (H14: (pr0 x t4)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: 
-T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat 
-Appl) x0 (THead (Bind Abst) u x)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda 
-(w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) w2)) (\lambda 
-(w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H15: (pr0 x0 
-v2)).(or_introl (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead 
-(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
-(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
-t4) w2))) (pr0_beta u x0 v2 H15 x t4 H14))) (\lambda (H15: (ex2 T (\lambda 
-(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)))).(ex2_ind T 
-(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)) (or (pr0 
-(THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind Abbr) v2 t4)) 
-(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) 
-w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda 
-(x2: T).(\lambda (H16: (pr0 x0 x2)).(\lambda (H17: (subst0 i v3 v2 
-x2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead 
-(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
-(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
-t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
-(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
-t4) w2)) (THead (Bind Abbr) x2 t4) (pr0_beta u x0 x2 H16 x t4 H14) 
-(subst0_fst v3 x2 v2 i H17 t4 (Bind Abbr))))))) H15)) (H1 v0 x0 i H8 v3 H5))) 
-(\lambda (H14: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 
-(s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda (w2: 
-T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 
-t4 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind 
-Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind 
-Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) 
-w2)))) (\lambda (x2: T).(\lambda (H15: (pr0 x x2)).(\lambda (H16: (subst0 (s 
-(Bind Abst) (s (Flat Appl) i)) v3 t4 x2)).(or_ind (pr0 x0 v2) (ex2 T (\lambda 
-(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead 
-(Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind Abbr) v2 t4)) (ex2 T 
-(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) w2)) 
-(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H17: 
-(pr0 x0 v2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) 
-(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 
-(THead (Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind 
-Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 
-(THead (Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind 
-Abbr) v2 t4) w2)) (THead (Bind Abbr) v2 x2) (pr0_beta u x0 v2 H17 x x2 H15) 
-(subst0_snd (Bind Abbr) v3 x2 t4 i H16 v2)))) (\lambda (H17: (ex2 T (\lambda 
-(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)))).(ex2_ind T 
-(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)) (or (pr0 
-(THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind Abbr) v2 t4)) 
-(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) 
-w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda 
-(x3: T).(\lambda (H18: (pr0 x0 x3)).(\lambda (H19: (subst0 i v3 v2 
-x3)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead 
-(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
-(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
-t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
-(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
-t4) w2)) (THead (Bind Abbr) x3 x2) (pr0_beta u x0 x3 H18 x x2 H15) 
-(subst0_both v3 v2 x3 i H19 (Bind Abbr) t4 x2 H16)))))) H17)) (H1 v0 x0 i H8 
-v3 H5))))) H14)) (H3 v0 x (s (Bind Abst) (s (Flat Appl) i)) H12 v3 H5)) w1 
-H13))))) H10)) (\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: 
-T).(eq T x1 (THead (Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
-T).(subst0 (s (Flat Appl) i) v0 u u2))) (\lambda (_: T).(\lambda (t5: 
-T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5))))).(ex3_2_ind T T 
-(\lambda (u2: T).(\lambda (t5: T).(eq T x1 (THead (Bind Abst) u2 t5)))) 
-(\lambda (u2: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u u2))) 
-(\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 
-t3 t5))) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 
-w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) 
-(\lambda (x2: T).(\lambda (x3: T).(\lambda (H11: (eq T x1 (THead (Bind Abst) 
-x2 x3))).(\lambda (_: (subst0 (s (Flat Appl) i) v0 u x2)).(\lambda (H13: 
-(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 x3)).(let H14 \def (eq_ind T 
-x1 (\lambda (t: T).(eq T w1 (THead (Flat Appl) x0 t))) H7 (THead (Bind Abst) 
-x2 x3) H11) in (eq_ind_r T (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) 
-(\lambda (t: T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: 
-T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) 
-w2))))) (or_ind (pr0 x3 t4) (ex2 T (\lambda (w2: T).(pr0 x3 w2)) (\lambda 
-(w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2))) (or (pr0 (THead 
-(Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead (Bind Abbr) v2 t4)) (ex2 T 
-(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) w2)) 
-(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H15: 
-(pr0 x3 t4)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) 
-(\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat Appl) x0 (THead 
-(Bind Abst) x2 x3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 
-(THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H16: (pr0 x0 
-v2)).(or_introl (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead 
-(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
-(Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
-t4) w2))) (pr0_beta x2 x0 v2 H16 x3 t4 H15))) (\lambda (H16: (ex2 T (\lambda 
-(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)))).(ex2_ind T 
-(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)) (or (pr0 
-(THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead (Bind Abbr) v2 t4)) 
-(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) 
-w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda 
-(x: T).(\lambda (H17: (pr0 x0 x)).(\lambda (H18: (subst0 i v3 v2 
-x)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead 
-(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
-(Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
-t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
-(Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
-t4) w2)) (THead (Bind Abbr) x t4) (pr0_beta x2 x0 x H17 x3 t4 H15) 
-(subst0_fst v3 x v2 i H18 t4 (Bind Abbr))))))) H16)) (H1 v0 x0 i H8 v3 H5))) 
-(\lambda (H15: (ex2 T (\lambda (w2: T).(pr0 x3 w2)) (\lambda (w2: T).(subst0 
-(s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda (w2: 
-T).(pr0 x3 w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 
-t4 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead 
-(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
-(Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
-t4) w2)))) (\lambda (x: T).(\lambda (H16: (pr0 x3 x)).(\lambda (H17: (subst0 
-(s (Bind Abst) (s (Flat Appl) i)) v3 t4 x)).(or_ind (pr0 x0 v2) (ex2 T 
-(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 
-(THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead (Bind Abbr) v2 t4)) 
-(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) 
-w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda 
-(H18: (pr0 x0 v2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) 
-x2 x3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat 
-Appl) x0 (THead (Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead 
-(Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat 
-Appl) x0 (THead (Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead 
-(Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) v2 x) (pr0_beta x2 x0 v2 H18 x3 x 
-H16) (subst0_snd (Bind Abbr) v3 x t4 i H17 v2)))) (\lambda (H18: (ex2 T 
-(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 
-w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 
-v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead 
-(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
-(Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
-t4) w2)))) (\lambda (x4: T).(\lambda (H19: (pr0 x0 x4)).(\lambda (H20: 
-(subst0 i v3 v2 x4)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) 
-x2 x3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat 
-Appl) x0 (THead (Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead 
-(Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat 
-Appl) x0 (THead (Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead 
-(Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) x4 x) (pr0_beta x2 x0 x4 H19 x3 x 
-H16) (subst0_both v3 v2 x4 i H20 (Bind Abbr) t4 x H17)))))) H18)) (H1 v0 x0 i 
-H8 v3 H5))))) H15)) (H3 v0 x3 (s (Bind Abst) (s (Flat Appl) i)) H13 v3 H5)) 
-w1 H14))))))) H10)) (subst0_gen_head (Bind Abst) v0 u t3 x1 (s (Flat Appl) i) 
-H9))))))) H6)) (subst0_gen_head (Flat Appl) v0 v1 (THead (Bind Abst) u t3) w1 
-i H4))))))))))))))))) (\lambda (b: B).(\lambda (H0: (not (eq B b 
-Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H1: (pr0 v1 v2)).(\lambda 
-(H2: ((\forall (v3: T).(\forall (w1: T).(\forall (i: nat).((subst0 i v3 v1 
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-(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v4 v2 w2)))))))))))).(\lambda 
-(u1: T).(\lambda (u2: T).(\lambda (H3: (pr0 u1 u2)).(\lambda (H4: ((\forall 
-(v3: T).(\forall (w1: T).(\forall (i: nat).((subst0 i v3 u1 w1) \to (\forall 
-(v4: T).((pr0 v3 v4) \to (or (pr0 w1 u2) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) 
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-(t4: T).(\lambda (H5: (pr0 t3 t4)).(\lambda (H6: ((\forall (v3: T).(\forall 
-(w1: T).(\forall (i: nat).((subst0 i v3 t3 w1) \to (\forall (v4: T).((pr0 v3 
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-T).(subst0 i v4 t4 w2)))))))))))).(\lambda (v0: T).(\lambda (w1: T).(\lambda 
-(i: nat).(\lambda (H7: (subst0 i v0 (THead (Flat Appl) v1 (THead (Bind b) u1 
-t3)) w1)).(\lambda (v3: T).(\lambda (H8: (pr0 v0 v3)).(or3_ind (ex2 T 
-(\lambda (u3: T).(eq T w1 (THead (Flat Appl) u3 (THead (Bind b) u1 t3)))) 
-(\lambda (u3: T).(subst0 i v0 v1 u3))) (ex2 T (\lambda (t5: T).(eq T w1 
-(THead (Flat Appl) v1 t5))) (\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 
-(THead (Bind b) u1 t3) t5))) (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq 
-T w1 (THead (Flat Appl) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i 
-v0 v1 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 
-(THead (Bind b) u1 t3) t5)))) (or (pr0 w1 (THead (Bind b) u2 (THead (Flat 
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-(w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
-t4)) w2)))) (\lambda (H9: (ex2 T (\lambda (u3: T).(eq T w1 (THead (Flat Appl) 
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-(Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) 
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-x)).(eq_ind_r T (THead (Flat Appl) x (THead (Bind b) u1 t3)) (\lambda (t: 
-T).(or (pr0 t (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) 
-(ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2))))) (or_ind (pr0 x 
-v2) (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v3 v2 
-w2))) (or (pr0 (THead (Flat Appl) x (THead (Bind b) u1 t3)) (THead (Bind b) 
-u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 
-(THead (Flat Appl) x (THead (Bind b) u1 t3)) w2)) (\lambda (w2: T).(subst0 i 
-v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) 
-(\lambda (H12: (pr0 x v2)).(or_introl (pr0 (THead (Flat Appl) x (THead (Bind 
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-T (\lambda (w2: T).(pr0 (THead (Flat Appl) x (THead (Bind b) u1 t3)) w2)) 
-(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
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-(H12: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v3 v2 
-w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v3 
-v2 w2)) (or (pr0 (THead (Flat Appl) x (THead (Bind b) u1 t3)) (THead (Bind b) 
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-(THead (Flat Appl) x (THead (Bind b) u1 t3)) w2)) (\lambda (w2: T).(subst0 i 
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-x0)).(or_intror (pr0 (THead (Flat Appl) x (THead (Bind b) u1 t3)) (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
-T).(pr0 (THead (Flat Appl) x (THead (Bind b) u1 t3)) w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
-w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x (THead (Bind b) 
-u1 t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead (Flat Appl) (lift 
-(S O) O x0) t4)) (pr0_upsilon b H0 x x0 H13 u1 u2 H3 t3 t4 H5) (subst0_snd 
-(Bind b) v3 (THead (Flat Appl) (lift (S O) O x0) t4) (THead (Flat Appl) (lift 
-(S O) O v2) t4) i (subst0_fst v3 (lift (S O) O x0) (lift (S O) O v2) (s (Bind 
-b) i) (subst0_lift_ge_s v2 x0 v3 i H14 O (le_O_n i) b) t4 (Flat Appl)) 
-u2)))))) H12)) (H2 v0 x i H11 v3 H8)) w1 H10)))) H9)) (\lambda (H9: (ex2 T 
-(\lambda (t5: T).(eq T w1 (THead (Flat Appl) v1 t5))) (\lambda (t5: 
-T).(subst0 (s (Flat Appl) i) v0 (THead (Bind b) u1 t3) t5)))).(ex2_ind T 
-(\lambda (t5: T).(eq T w1 (THead (Flat Appl) v1 t5))) (\lambda (t5: 
-T).(subst0 (s (Flat Appl) i) v0 (THead (Bind b) u1 t3) t5)) (or (pr0 w1 
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda 
-(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x: T).(\lambda (H10: (eq 
-T w1 (THead (Flat Appl) v1 x))).(\lambda (H11: (subst0 (s (Flat Appl) i) v0 
-(THead (Bind b) u1 t3) x)).(or3_ind (ex2 T (\lambda (u3: T).(eq T x (THead 
-(Bind b) u3 t3))) (\lambda (u3: T).(subst0 (s (Flat Appl) i) v0 u1 u3))) (ex2 
-T (\lambda (t5: T).(eq T x (THead (Bind b) u1 t5))) (\lambda (t5: T).(subst0 
-(s (Bind b) (s (Flat Appl) i)) v0 t3 t5))) (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t5: T).(eq T x (THead (Bind b) u3 t5)))) (\lambda (u3: 
-T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 u3))) (\lambda (_: 
-T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5)))) (or 
-(pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T 
-(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) 
-u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H12: (ex2 T 
-(\lambda (u3: T).(eq T x (THead (Bind b) u3 t3))) (\lambda (u3: T).(subst0 (s 
-(Flat Appl) i) v0 u1 u3)))).(ex2_ind T (\lambda (u3: T).(eq T x (THead (Bind 
-b) u3 t3))) (\lambda (u3: T).(subst0 (s (Flat Appl) i) v0 u1 u3)) (or (pr0 w1 
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda 
-(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x0: T).(\lambda (H13: (eq 
-T x (THead (Bind b) x0 t3))).(\lambda (H14: (subst0 (s (Flat Appl) i) v0 u1 
-x0)).(let H15 \def (eq_ind T x (\lambda (t: T).(eq T w1 (THead (Flat Appl) v1 
-t))) H10 (THead (Bind b) x0 t3) H13) in (eq_ind_r T (THead (Flat Appl) v1 
-(THead (Bind b) x0 t3)) (\lambda (t: T).(or (pr0 t (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 t w2)) 
-(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
-O) O v2) t4)) w2))))) (or_ind (pr0 x0 u2) (ex2 T (\lambda (w2: T).(pr0 x0 
-w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 u2 w2))) (or (pr0 (THead 
-(Flat Appl) v1 (THead (Bind b) x0 t3)) (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 
-(THead (Bind b) x0 t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H16: (pr0 x0 
-u2)).(or_introl (pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 t3)) (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
-T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 t3)) w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
-w2))) (pr0_upsilon b H0 v1 v2 H1 x0 u2 H16 t3 t4 H5))) (\lambda (H16: (ex2 T 
-(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 
-u2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 
-(s (Flat Appl) i) v3 u2 w2)) (or (pr0 (THead (Flat Appl) v1 (THead (Bind b) 
-x0 t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T 
-(\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 t3)) w2)) 
-(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
-O) O v2) t4)) w2)))) (\lambda (x1: T).(\lambda (H17: (pr0 x0 x1)).(\lambda 
-(H18: (subst0 (s (Flat Appl) i) v3 u2 x1)).(or_intror (pr0 (THead (Flat Appl) 
-v1 (THead (Bind b) x0 t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) 
-O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind 
-b) x0 t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead 
-(Flat Appl) v1 (THead (Bind b) x0 t3)) w2)) (\lambda (w2: T).(subst0 i v3 
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead 
-(Bind b) x1 (THead (Flat Appl) (lift (S O) O v2) t4)) (pr0_upsilon b H0 v1 v2 
-H1 x0 x1 H17 t3 t4 H5) (subst0_fst v3 x1 u2 i H18 (THead (Flat Appl) (lift (S 
-O) O v2) t4) (Bind b))))))) H16)) (H4 v0 x0 (s (Flat Appl) i) H14 v3 H8)) w1 
-H15))))) H12)) (\lambda (H12: (ex2 T (\lambda (t5: T).(eq T x (THead (Bind b) 
-u1 t5))) (\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 
-t5)))).(ex2_ind T (\lambda (t5: T).(eq T x (THead (Bind b) u1 t5))) (\lambda 
-(t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5)) (or (pr0 w1 (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
-T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x0: T).(\lambda (H13: (eq T x 
-(THead (Bind b) u1 x0))).(\lambda (H14: (subst0 (s (Bind b) (s (Flat Appl) 
-i)) v0 t3 x0)).(let H15 \def (eq_ind T x (\lambda (t: T).(eq T w1 (THead 
-(Flat Appl) v1 t))) H10 (THead (Bind b) u1 x0) H13) in (eq_ind_r T (THead 
-(Flat Appl) v1 (THead (Bind b) u1 x0)) (\lambda (t: T).(or (pr0 t (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
-T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t4)) w2))))) (or_ind (pr0 x0 t4) (ex2 T (\lambda (w2: 
-T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 
-w2))) (or (pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 x0)) (THead (Bind b) 
-u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 
-(THead (Flat Appl) v1 (THead (Bind b) u1 x0)) w2)) (\lambda (w2: T).(subst0 i 
-v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) 
-(\lambda (H16: (pr0 x0 t4)).(or_introl (pr0 (THead (Flat Appl) v1 (THead 
-(Bind b) u1 x0)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
-t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 
-x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t4)) w2))) (pr0_upsilon b H0 v1 v2 H1 u1 u2 H3 x0 t4 H16))) 
-(\lambda (H16: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 
-(s (Bind b) (s (Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 
-x0 w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 w2)) 
-(or (pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 x0)) (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 
-(THead (Flat Appl) v1 (THead (Bind b) u1 x0)) w2)) (\lambda (w2: T).(subst0 i 
-v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) 
-(\lambda (x1: T).(\lambda (H17: (pr0 x0 x1)).(\lambda (H18: (subst0 (s (Bind 
-b) (s (Flat Appl) i)) v3 t4 x1)).(or_intror (pr0 (THead (Flat Appl) v1 (THead 
-(Bind b) u1 x0)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
-t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 
-x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat 
-Appl) v1 (THead (Bind b) u1 x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) x1)) (pr0_upsilon b H0 v1 v2 H1 u1 u2 H3 
-x0 x1 H17) (subst0_snd (Bind b) v3 (THead (Flat Appl) (lift (S O) O v2) x1) 
-(THead (Flat Appl) (lift (S O) O v2) t4) i (subst0_snd (Flat Appl) v3 x1 t4 
-(s (Bind b) i) H18 (lift (S O) O v2)) u2)))))) H16)) (H6 v0 x0 (s (Bind b) (s 
-(Flat Appl) i)) H14 v3 H8)) w1 H15))))) H12)) (\lambda (H12: (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t5: T).(eq T x (THead (Bind b) u3 t5)))) (\lambda 
-(u3: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 u3))) (\lambda (_: 
-T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 
-t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t5: T).(eq T x (THead (Bind 
-b) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 
-u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) 
-v0 t3 t5))) (or (pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O 
-v2) t4))) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda 
-(x0: T).(\lambda (x1: T).(\lambda (H13: (eq T x (THead (Bind b) x0 
-x1))).(\lambda (H14: (subst0 (s (Flat Appl) i) v0 u1 x0)).(\lambda (H15: 
-(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 x1)).(let H16 \def (eq_ind T x 
-(\lambda (t: T).(eq T w1 (THead (Flat Appl) v1 t))) H10 (THead (Bind b) x0 
-x1) H13) in (eq_ind_r T (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) 
-(\lambda (t: T).(or (pr0 t (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) 
-O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2))))) (or_ind 
-(pr0 x1 t4) (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s 
-(Bind b) (s (Flat Appl) i)) v3 t4 w2))) (or (pr0 (THead (Flat Appl) v1 (THead 
-(Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
-t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 
-x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t4)) w2)))) (\lambda (H17: (pr0 x1 t4)).(or_ind (pr0 x0 u2) 
-(ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) 
-i) v3 u2 w2))) (or (pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
-T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
-w2)))) (\lambda (H18: (pr0 x0 u2)).(or_introl (pr0 (THead (Flat Appl) v1 
-(THead (Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O 
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-x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t4)) w2))) (pr0_upsilon b H0 v1 v2 H1 x0 u2 H18 x1 t4 
-H17))) (\lambda (H18: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: 
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-w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 u2 w2)) (or (pr0 (THead 
-(Flat Appl) v1 (THead (Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 
-(THead (Bind b) x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda 
-(H19: (pr0 x0 x2)).(\lambda (H20: (subst0 (s (Flat Appl) i) v3 u2 
-x2)).(or_intror (pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
-T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
-w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind 
-b) x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) x2 (THead (Flat Appl) (lift 
-(S O) O v2) t4)) (pr0_upsilon b H0 v1 v2 H1 x0 x2 H19 x1 t4 H17) (subst0_fst 
-v3 x2 u2 i H20 (THead (Flat Appl) (lift (S O) O v2) t4) (Bind b))))))) H18)) 
-(H4 v0 x0 (s (Flat Appl) i) H14 v3 H8))) (\lambda (H17: (ex2 T (\lambda (w2: 
-T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 
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-(Bind b) (s (Flat Appl) i)) v3 t4 w2)) (or (pr0 (THead (Flat Appl) v1 (THead 
-(Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
-t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 
-x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda (H18: (pr0 x1 
-x2)).(\lambda (H19: (subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 x2)).(or_ind 
-(pr0 x0 u2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s 
-(Flat Appl) i) v3 u2 w2))) (or (pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 
-x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T 
-(\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) 
-(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
-O) O v2) t4)) w2)))) (\lambda (H20: (pr0 x0 u2)).(or_intror (pr0 (THead (Flat 
-Appl) v1 (THead (Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift 
-(S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead 
-(Bind b) x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 
-(THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: T).(subst0 i 
-v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) x2)) (pr0_upsilon b H0 v1 v2 
-H1 x0 u2 H20 x1 x2 H18) (subst0_snd (Bind b) v3 (THead (Flat Appl) (lift (S 
-O) O v2) x2) (THead (Flat Appl) (lift (S O) O v2) t4) i (subst0_snd (Flat 
-Appl) v3 x2 t4 (s (Bind b) i) H19 (lift (S O) O v2)) u2)))) (\lambda (H20: 
-(ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) 
-i) v3 u2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: 
-T).(subst0 (s (Flat Appl) i) v3 u2 w2)) (or (pr0 (THead (Flat Appl) v1 (THead 
-(Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
-t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 
-x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t4)) w2)))) (\lambda (x3: T).(\lambda (H21: (pr0 x0 
-x3)).(\lambda (H22: (subst0 (s (Flat Appl) i) v3 u2 x3)).(or_intror (pr0 
-(THead (Flat Appl) v1 (THead (Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) 
-v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) 
-u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: 
-T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
-w2)) (THead (Bind b) x3 (THead (Flat Appl) (lift (S O) O v2) x2)) 
-(pr0_upsilon b H0 v1 v2 H1 x0 x3 H21 x1 x2 H18) (subst0_both v3 u2 x3 i H22 
-(Bind b) (THead (Flat Appl) (lift (S O) O v2) t4) (THead (Flat Appl) (lift (S 
-O) O v2) x2) (subst0_snd (Flat Appl) v3 x2 t4 (s (Bind b) i) H19 (lift (S O) 
-O v2)))))))) H20)) (H4 v0 x0 (s (Flat Appl) i) H14 v3 H8))))) H17)) (H6 v0 x1 
-(s (Bind b) (s (Flat Appl) i)) H15 v3 H8)) w1 H16))))))) H12)) 
-(subst0_gen_head (Bind b) v0 u1 t3 x (s (Flat Appl) i) H11))))) H9)) (\lambda 
-(H9: (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead (Flat Appl) 
-u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v0 v1 u3))) (\lambda (_: 
-T).(\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 (THead (Bind b) u1 t3) 
-t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead 
-(Flat Appl) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v0 v1 u3))) 
-(\lambda (_: T).(\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 (THead (Bind b) 
-u1 t3) t5))) (or (pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O 
-v2) t4))) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda 
-(x0: T).(\lambda (x1: T).(\lambda (H10: (eq T w1 (THead (Flat Appl) x0 
-x1))).(\lambda (H11: (subst0 i v0 v1 x0)).(\lambda (H12: (subst0 (s (Flat 
-Appl) i) v0 (THead (Bind b) u1 t3) x1)).(or3_ind (ex2 T (\lambda (u3: T).(eq 
-T x1 (THead (Bind b) u3 t3))) (\lambda (u3: T).(subst0 (s (Flat Appl) i) v0 
-u1 u3))) (ex2 T (\lambda (t5: T).(eq T x1 (THead (Bind b) u1 t5))) (\lambda 
-(t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5))) (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t5: T).(eq T x1 (THead (Bind b) u3 t5)))) (\lambda 
-(u3: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 u3))) (\lambda (_: 
-T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5)))) (or 
-(pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T 
-(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) 
-u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H13: (ex2 T 
-(\lambda (u3: T).(eq T x1 (THead (Bind b) u3 t3))) (\lambda (u3: T).(subst0 
-(s (Flat Appl) i) v0 u1 u3)))).(ex2_ind T (\lambda (u3: T).(eq T x1 (THead 
-(Bind b) u3 t3))) (\lambda (u3: T).(subst0 (s (Flat Appl) i) v0 u1 u3)) (or 
-(pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T 
-(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) 
-u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x: T).(\lambda 
-(H14: (eq T x1 (THead (Bind b) x t3))).(\lambda (H15: (subst0 (s (Flat Appl) 
-i) v0 u1 x)).(let H16 \def (eq_ind T x1 (\lambda (t: T).(eq T w1 (THead (Flat 
-Appl) x0 t))) H10 (THead (Bind b) x t3) H14) in (eq_ind_r T (THead (Flat 
-Appl) x0 (THead (Bind b) x t3)) (\lambda (t: T).(or (pr0 t (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 t 
-w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t4)) w2))))) (or_ind (pr0 x u2) (ex2 T (\lambda (w2: 
-T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 u2 w2))) (or 
-(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat 
-Appl) x0 (THead (Bind b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H17: 
-(pr0 x u2)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda 
-(w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) 
-x t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T 
-(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) 
-(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
-O) O v2) t4)) w2)))) (\lambda (H18: (pr0 x0 v2)).(or_introl (pr0 (THead (Flat 
-Appl) x0 (THead (Bind b) x t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift 
-(S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
-(Bind b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t4)) w2))) (pr0_upsilon b H0 x0 v2 H18 x u2 H17 
-t3 t4 H5))) (\lambda (H18: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: 
-T).(subst0 i v3 v2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda 
-(w2: T).(subst0 i v3 v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x 
-t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T 
-(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) 
-(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
-O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda (H19: (pr0 x0 x2)).(\lambda 
-(H20: (subst0 i v3 v2 x2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind 
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-T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) 
-(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
-O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 
-(THead (Bind b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O x2) t4)) (pr0_upsilon b H0 x0 x2 H19 x u2 H17 t3 t4 
-H5) (subst0_snd (Bind b) v3 (THead (Flat Appl) (lift (S O) O x2) t4) (THead 
-(Flat Appl) (lift (S O) O v2) t4) i (subst0_fst v3 (lift (S O) O x2) (lift (S 
-O) O v2) (s (Bind b) i) (subst0_lift_ge_s v2 x2 v3 i H20 O (le_O_n i) b) t4 
-(Flat Appl)) u2)))))) H18)) (H2 v0 x0 i H11 v3 H8))) (\lambda (H17: (ex2 T 
-(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 u2 
-w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s 
-(Flat Appl) i) v3 u2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x 
-t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T 
-(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) 
-(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
-O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda (H18: (pr0 x x2)).(\lambda 
-(H19: (subst0 (s (Flat Appl) i) v3 u2 x2)).(or_ind (pr0 x0 v2) (ex2 T 
-(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 
-(THead (Flat Appl) x0 (THead (Bind b) x t3)) (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) 
-x0 (THead (Bind b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) 
-u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H20: (pr0 x0 
-v2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
-T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
-w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind 
-b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) x2 (THead (Flat Appl) (lift 
-(S O) O v2) t4)) (pr0_upsilon b H0 x0 v2 H20 x x2 H18 t3 t4 H5) (subst0_fst 
-v3 x2 u2 i H19 (THead (Flat Appl) (lift (S O) O v2) t4) (Bind b))))) (\lambda 
-(H20: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 
-w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 
-v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) (THead (Bind b) 
-u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 
-(THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) (\lambda (w2: T).(subst0 i 
-v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) 
-(\lambda (x3: T).(\lambda (H21: (pr0 x0 x3)).(\lambda (H22: (subst0 i v3 v2 
-x3)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
-T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
-w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind 
-b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) x2 (THead (Flat Appl) (lift 
-(S O) O x3) t4)) (pr0_upsilon b H0 x0 x3 H21 x x2 H18 t3 t4 H5) (subst0_both 
-v3 u2 x2 i H19 (Bind b) (THead (Flat Appl) (lift (S O) O v2) t4) (THead (Flat 
-Appl) (lift (S O) O x3) t4) (subst0_fst v3 (lift (S O) O x3) (lift (S O) O 
-v2) (s (Bind b) i) (subst0_lift_ge_s v2 x3 v3 i H22 O (le_O_n i) b) t4 (Flat 
-Appl)))))))) H20)) (H2 v0 x0 i H11 v3 H8))))) H17)) (H4 v0 x (s (Flat Appl) 
-i) H15 v3 H8)) w1 H16))))) H13)) (\lambda (H13: (ex2 T (\lambda (t5: T).(eq T 
-x1 (THead (Bind b) u1 t5))) (\lambda (t5: T).(subst0 (s (Bind b) (s (Flat 
-Appl) i)) v0 t3 t5)))).(ex2_ind T (\lambda (t5: T).(eq T x1 (THead (Bind b) 
-u1 t5))) (\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5)) 
-(or (pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) 
-(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x: 
-T).(\lambda (H14: (eq T x1 (THead (Bind b) u1 x))).(\lambda (H15: (subst0 (s 
-(Bind b) (s (Flat Appl) i)) v0 t3 x)).(let H16 \def (eq_ind T x1 (\lambda (t: 
-T).(eq T w1 (THead (Flat Appl) x0 t))) H10 (THead (Bind b) u1 x) H14) in 
-(eq_ind_r T (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (\lambda (t: T).(or 
-(pr0 t (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T 
-(\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t4)) w2))))) (or_ind (pr0 x t4) (ex2 T 
-(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat 
-Appl) i)) v3 t4 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) 
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda 
-(w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
-w2)))) (\lambda (H17: (pr0 x t4)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: 
-T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat 
-Appl) x0 (THead (Bind b) u1 x)) (THead (Bind b) u2 (THead (Flat Appl) (lift 
-(S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
-(Bind b) u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H18: (pr0 x0 
-v2)).(or_introl (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
-T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
-w2))) (pr0_upsilon b H0 x0 v2 H18 u1 u2 H3 x t4 H17))) (\lambda (H18: (ex2 T 
-(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 
-w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 
-v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead (Bind b) 
-u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 
-(THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: T).(subst0 i 
-v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) 
-(\lambda (x2: T).(\lambda (H19: (pr0 x0 x2)).(\lambda (H20: (subst0 i v3 v2 
-x2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
-T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
-w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind 
-b) u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead (Flat Appl) (lift 
-(S O) O x2) t4)) (pr0_upsilon b H0 x0 x2 H19 u1 u2 H3 x t4 H17) (subst0_snd 
-(Bind b) v3 (THead (Flat Appl) (lift (S O) O x2) t4) (THead (Flat Appl) (lift 
-(S O) O v2) t4) i (subst0_fst v3 (lift (S O) O x2) (lift (S O) O v2) (s (Bind 
-b) i) (subst0_lift_ge_s v2 x2 v3 i H20 O (le_O_n i) b) t4 (Flat Appl)) 
-u2)))))) H18)) (H2 v0 x0 i H11 v3 H8))) (\lambda (H17: (ex2 T (\lambda (w2: 
-T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 
-w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s 
-(Bind b) (s (Flat Appl) i)) v3 t4 w2)) (or (pr0 (THead (Flat Appl) x0 (THead 
-(Bind b) u1 x)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) 
-(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) 
-w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda (H18: (pr0 x 
-x2)).(\lambda (H19: (subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 x2)).(or_ind 
-(pr0 x0 v2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i 
-v3 v2 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
-T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
-w2)))) (\lambda (H20: (pr0 x0 v2)).(or_intror (pr0 (THead (Flat Appl) x0 
-(THead (Bind b) u1 x)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O 
-v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) 
-u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead 
-(Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) x2)) (pr0_upsilon b H0 x0 v2 
-H20 u1 u2 H3 x x2 H18) (subst0_snd (Bind b) v3 (THead (Flat Appl) (lift (S O) 
-O v2) x2) (THead (Flat Appl) (lift (S O) O v2) t4) i (subst0_snd (Flat Appl) 
-v3 x2 t4 (s (Bind b) i) H19 (lift (S O) O v2)) u2)))) (\lambda (H20: (ex2 T 
-(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 
-w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 
-v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead (Bind b) 
-u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 
-(THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: T).(subst0 i 
-v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) 
-(\lambda (x3: T).(\lambda (H21: (pr0 x0 x3)).(\lambda (H22: (subst0 i v3 v2 
-x3)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
-T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
-w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind 
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-(Flat Appl) i) H12))))))) H9)) (subst0_gen_head (Flat Appl) v0 v1 (THead 
-(Bind b) u1 t3) w1 i H7)))))))))))))))))))))) (\lambda (u1: T).(\lambda (u2: 
-T).(\lambda (H0: (pr0 u1 u2)).(\lambda (H1: ((\forall (v1: T).(\forall (w1: 
-T).(\forall (i: nat).((subst0 i v1 u1 w1) \to (\forall (v2: T).((pr0 v1 v2) 
-\to (or (pr0 w1 u2) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: 
-T).(subst0 i v2 u2 w2)))))))))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda 
-(H2: (pr0 t3 t4)).(\lambda (H3: ((\forall (v1: T).(\forall (w1: T).(\forall 
-(i: nat).((subst0 i v1 t3 w1) \to (\forall (v2: T).((pr0 v1 v2) \to (or (pr0 
-w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 
-w2)))))))))))).(\lambda (w: T).(\lambda (H4: (subst0 O u2 t4 w)).(\lambda 
-(v1: T).(\lambda (w1: T).(\lambda (i: nat).(\lambda (H5: (subst0 i v1 (THead 
-(Bind Abbr) u1 t3) w1)).(\lambda (v2: T).(\lambda (H6: (pr0 v1 v2)).(or3_ind 
-(ex2 T (\lambda (u3: T).(eq T w1 (THead (Bind Abbr) u3 t3))) (\lambda (u3: 
-T).(subst0 i v1 u1 u3))) (ex2 T (\lambda (t5: T).(eq T w1 (THead (Bind Abbr) 
-u1 t5))) (\lambda (t5: T).(subst0 (s (Bind Abbr) i) v1 t3 t5))) (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead (Bind Abbr) u3 t5)))) 
-(\lambda (u3: T).(\lambda (_: T).(subst0 i v1 u1 u3))) (\lambda (_: 
-T).(\lambda (t5: T).(subst0 (s (Bind Abbr) i) v1 t3 t5)))) (or (pr0 w1 (THead 
-(Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: 
-T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (H7: (ex2 T (\lambda 
-(u3: T).(eq T w1 (THead (Bind Abbr) u3 t3))) (\lambda (u3: T).(subst0 i v1 u1 
-u3)))).(ex2_ind T (\lambda (u3: T).(eq T w1 (THead (Bind Abbr) u3 t3))) 
-(\lambda (u3: T).(subst0 i v1 u1 u3)) (or (pr0 w1 (THead (Bind Abbr) u2 w)) 
-(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 (THead 
-(Bind Abbr) u2 w) w2)))) (\lambda (x: T).(\lambda (H8: (eq T w1 (THead (Bind 
-Abbr) x t3))).(\lambda (H9: (subst0 i v1 u1 x)).(eq_ind_r T (THead (Bind 
-Abbr) x t3) (\lambda (t: T).(or (pr0 t (THead (Bind Abbr) u2 w)) (ex2 T 
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-u2 w) w2))))) (or_ind (pr0 x u2) (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda 
-(w2: T).(subst0 i v2 u2 w2))) (or (pr0 (THead (Bind Abbr) x t3) (THead (Bind 
-Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x t3) w2)) 
-(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (H10: 
-(pr0 x u2)).(or_introl (pr0 (THead (Bind Abbr) x t3) (THead (Bind Abbr) u2 
-w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x t3) w2)) (\lambda (w2: 
-T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))) (pr0_delta x u2 H10 t3 t4 H2 w 
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-(w2: T).(subst0 i v2 u2 w2)) (or (pr0 (THead (Bind Abbr) x t3) (THead (Bind 
-Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x t3) w2)) 
-(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x0: 
-T).(\lambda (H11: (pr0 x x0)).(\lambda (H12: (subst0 i v2 u2 x0)).(ex2_ind T 
-(\lambda (t: T).(subst0 O x0 t4 t)) (\lambda (t: T).(subst0 (S (plus i O)) v2 
-w t)) (or (pr0 (THead (Bind Abbr) x t3) (THead (Bind Abbr) u2 w)) (ex2 T 
-(\lambda (w2: T).(pr0 (THead (Bind Abbr) x t3) w2)) (\lambda (w2: T).(subst0 
-i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x1: T).(\lambda (H13: (subst0 
-O x0 t4 x1)).(\lambda (H14: (subst0 (S (plus i O)) v2 w x1)).(let H15 \def 
-(f_equal nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O i))) in 
-(let H16 \def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n v2 w 
-x1)) H14 (S i) H15) in (or_intror (pr0 (THead (Bind Abbr) x t3) (THead (Bind 
-Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x t3) w2)) 
-(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))) (ex_intro2 T 
-(\lambda (w2: T).(pr0 (THead (Bind Abbr) x t3) w2)) (\lambda (w2: T).(subst0 
-i v2 (THead (Bind Abbr) u2 w) w2)) (THead (Bind Abbr) x0 x1) (pr0_delta x x0 
-H11 t3 t4 H2 x1 H13) (subst0_both v2 u2 x0 i H12 (Bind Abbr) w x1 H16)))))))) 
-(subst0_subst0_back t4 w u2 O H4 x0 v2 i H12))))) H10)) (H1 v1 x i H9 v2 H6)) 
-w1 H8)))) H7)) (\lambda (H7: (ex2 T (\lambda (t5: T).(eq T w1 (THead (Bind 
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-t5)))).(ex2_ind T (\lambda (t5: T).(eq T w1 (THead (Bind Abbr) u1 t5))) 
-(\lambda (t5: T).(subst0 (s (Bind Abbr) i) v1 t3 t5)) (or (pr0 w1 (THead 
-(Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: 
-T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x: T).(\lambda (H8: 
-(eq T w1 (THead (Bind Abbr) u1 x))).(\lambda (H9: (subst0 (s (Bind Abbr) i) 
-v1 t3 x)).(eq_ind_r T (THead (Bind Abbr) u1 x) (\lambda (t: T).(or (pr0 t 
-(THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: 
-T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))))) (or_ind (pr0 x t4) (ex2 T 
-(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind Abbr) i) v2 t4 
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-(\lambda (w2: T).(pr0 (THead (Bind Abbr) u1 x) w2)) (\lambda (w2: T).(subst0 
-i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (H10: (pr0 x t4)).(or_introl 
-(pr0 (THead (Bind Abbr) u1 x) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: 
-T).(pr0 (THead (Bind Abbr) u1 x) w2)) (\lambda (w2: T).(subst0 i v2 (THead 
-(Bind Abbr) u2 w) w2))) (pr0_delta u1 u2 H0 x t4 H10 w H4))) (\lambda (H10: 
-(ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind Abbr) 
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-T).(subst0 (s (Bind Abbr) i) v2 t4 w2)) (or (pr0 (THead (Bind Abbr) u1 x) 
-(THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) u1 
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-(\lambda (x0: T).(\lambda (H11: (pr0 x x0)).(\lambda (H12: (subst0 (s (Bind 
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-(t: T).(subst0 (s (Bind Abbr) i) v2 w t)) (or (pr0 (THead (Bind Abbr) u1 x) 
-(THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) u1 
-x) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) 
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-(s (Bind Abbr) i) v2 w x1)).(or_intror (pr0 (THead (Bind Abbr) u1 x) (THead 
-(Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) u1 x) w2)) 
-(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))) (ex_intro2 T 
-(\lambda (w2: T).(pr0 (THead (Bind Abbr) u1 x) w2)) (\lambda (w2: T).(subst0 
-i v2 (THead (Bind Abbr) u2 w) w2)) (THead (Bind Abbr) u2 x1) (pr0_delta u1 u2 
-H0 x x0 H11 x1 H13) (subst0_snd (Bind Abbr) v2 x1 w i H14 u2)))))) 
-(subst0_confluence_neq t4 x0 v2 (s (Bind Abbr) i) H12 w u2 O H4 (sym_not_eq 
-nat O (S i) (O_S i))))))) H10)) (H3 v1 x (s (Bind Abbr) i) H9 v2 H6)) w1 
-H8)))) H7)) (\lambda (H7: (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T 
-w1 (THead (Bind Abbr) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v1 
-u1 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind Abbr) i) v1 t3 
-t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead 
-(Bind Abbr) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v1 u1 u3))) 
-(\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind Abbr) i) v1 t3 t5))) (or 
-(pr0 w1 (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) 
-(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x0: 
-T).(\lambda (x1: T).(\lambda (H8: (eq T w1 (THead (Bind Abbr) x0 
-x1))).(\lambda (H9: (subst0 i v1 u1 x0)).(\lambda (H10: (subst0 (s (Bind 
-Abbr) i) v1 t3 x1)).(eq_ind_r T (THead (Bind Abbr) x0 x1) (\lambda (t: T).(or 
-(pr0 t (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda 
-(w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))))) (or_ind (pr0 x1 t4) 
-(ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s (Bind Abbr) 
-i) v2 t4 w2))) (or (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) 
-(ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: 
-T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (H11: (pr0 x1 
-t4)).(or_ind (pr0 x0 u2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: 
-T).(subst0 i v2 u2 w2))) (or (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind 
-Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) 
-(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (H12: 
-(pr0 x0 u2)).(or_introl (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 
-w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: 
-T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))) (pr0_delta x0 u2 H12 x1 t4 H11 
-w H4))) (\lambda (H12: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: 
-T).(subst0 i v2 u2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda 
-(w2: T).(subst0 i v2 u2 w2)) (or (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind 
-Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) 
-(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x: 
-T).(\lambda (H13: (pr0 x0 x)).(\lambda (H14: (subst0 i v2 u2 x)).(ex2_ind T 
-(\lambda (t: T).(subst0 O x t4 t)) (\lambda (t: T).(subst0 (S (plus i O)) v2 
-w t)) (or (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T 
-(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 
-i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x2: T).(\lambda (H15: (subst0 
-O x t4 x2)).(\lambda (H16: (subst0 (S (plus i O)) v2 w x2)).(let H17 \def 
-(f_equal nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O i))) in 
-(let H18 \def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n v2 w 
-x2)) H16 (S i) H17) in (or_intror (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind 
-Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) 
-(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))) (ex_intro2 T 
-(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 
-i v2 (THead (Bind Abbr) u2 w) w2)) (THead (Bind Abbr) x x2) (pr0_delta x0 x 
-H13 x1 t4 H11 x2 H15) (subst0_both v2 u2 x i H14 (Bind Abbr) w x2 H18)))))))) 
-(subst0_subst0_back t4 w u2 O H4 x v2 i H14))))) H12)) (H1 v1 x0 i H9 v2 
-H6))) (\lambda (H11: (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: 
-T).(subst0 (s (Bind Abbr) i) v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x1 
-w2)) (\lambda (w2: T).(subst0 (s (Bind Abbr) i) v2 t4 w2)) (or (pr0 (THead 
-(Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 
-(THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind 
-Abbr) u2 w) w2)))) (\lambda (x: T).(\lambda (H12: (pr0 x1 x)).(\lambda (H13: 
-(subst0 (s (Bind Abbr) i) v2 t4 x)).(or_ind (pr0 x0 u2) (ex2 T (\lambda (w2: 
-T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 u2 w2))) (or (pr0 (THead (Bind 
-Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead 
-(Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 
-w) w2)))) (\lambda (H14: (pr0 x0 u2)).(ex2_ind T (\lambda (t: T).(subst0 O u2 
-x t)) (\lambda (t: T).(subst0 (s (Bind Abbr) i) v2 w t)) (or (pr0 (THead 
-(Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 
-(THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind 
-Abbr) u2 w) w2)))) (\lambda (x2: T).(\lambda (H15: (subst0 O u2 x 
-x2)).(\lambda (H16: (subst0 (s (Bind Abbr) i) v2 w x2)).(or_intror (pr0 
-(THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: 
-T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead 
-(Bind Abbr) u2 w) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) 
-x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)) 
-(THead (Bind Abbr) u2 x2) (pr0_delta x0 u2 H14 x1 x H12 x2 H15) (subst0_snd 
-(Bind Abbr) v2 x2 w i H16 u2)))))) (subst0_confluence_neq t4 x v2 (s (Bind 
-Abbr) i) H13 w u2 O H4 (sym_not_eq nat O (S i) (O_S i))))) (\lambda (H14: 
-(ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 u2 
-w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 
-u2 w2)) (or (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T 
-(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 
-i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x2: T).(\lambda (H15: (pr0 x0 
-x2)).(\lambda (H16: (subst0 i v2 u2 x2)).(ex2_ind T (\lambda (t: T).(subst0 O 
-x2 t4 t)) (\lambda (t: T).(subst0 (S (plus i O)) v2 w t)) (or (pr0 (THead 
-(Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 
-(THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind 
-Abbr) u2 w) w2)))) (\lambda (x3: T).(\lambda (H17: (subst0 O x2 t4 
-x3)).(\lambda (H18: (subst0 (S (plus i O)) v2 w x3)).(let H19 \def (f_equal 
-nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O i))) in (let H20 
-\def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n v2 w x3)) H18 (S 
-i) H19) in (ex2_ind T (\lambda (t: T).(subst0 (s (Bind Abbr) i) v2 x3 t)) 
-(\lambda (t: T).(subst0 O x2 x t)) (or (pr0 (THead (Bind Abbr) x0 x1) (THead 
-(Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) 
-w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda 
-(x4: T).(\lambda (H21: (subst0 (s (Bind Abbr) i) v2 x3 x4)).(\lambda (H22: 
-(subst0 O x2 x x4)).(or_intror (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind 
-Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) 
-(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))) (ex_intro2 T 
-(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 
-i v2 (THead (Bind Abbr) u2 w) w2)) (THead (Bind Abbr) x2 x4) (pr0_delta x0 x2 
-H15 x1 x H12 x4 H22) (subst0_both v2 u2 x2 i H16 (Bind Abbr) w x4 
-(subst0_trans x3 w v2 (s (Bind Abbr) i) H20 x4 H21))))))) 
-(subst0_confluence_neq t4 x3 x2 O H17 x v2 (s (Bind Abbr) i) H13 (O_S 
-i)))))))) (subst0_subst0_back t4 w u2 O H4 x2 v2 i H16))))) H14)) (H1 v1 x0 i 
-H9 v2 H6))))) H11)) (H3 v1 x1 (s (Bind Abbr) i) H10 v2 H6)) w1 H8)))))) H7)) 
-(subst0_gen_head (Bind Abbr) v1 u1 t3 w1 i H5)))))))))))))))))) (\lambda (b: 
-B).(\lambda (H0: (not (eq B b Abst))).(\lambda (t3: T).(\lambda (t4: 
-T).(\lambda (H1: (pr0 t3 t4)).(\lambda (H2: ((\forall (v1: T).(\forall (w1: 
-T).(\forall (i: nat).((subst0 i v1 t3 w1) \to (\forall (v2: T).((pr0 v1 v2) 
-\to (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: 
-T).(subst0 i v2 t4 w2)))))))))))).(\lambda (u: T).(\lambda (v1: T).(\lambda 
-(w1: T).(\lambda (i: nat).(\lambda (H3: (subst0 i v1 (THead (Bind b) u (lift 
-(S O) O t3)) w1)).(\lambda (v2: T).(\lambda (H4: (pr0 v1 v2)).(or3_ind (ex2 T 
-(\lambda (u2: T).(eq T w1 (THead (Bind b) u2 (lift (S O) O t3)))) (\lambda 
-(u2: T).(subst0 i v1 u u2))) (ex2 T (\lambda (t5: T).(eq T w1 (THead (Bind b) 
-u t5))) (\lambda (t5: T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5))) 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T w1 (THead (Bind b) u2 
-t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: 
-T).(\lambda (t5: T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5)))) (or 
-(pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i 
-v2 t4 w2)))) (\lambda (H5: (ex2 T (\lambda (u2: T).(eq T w1 (THead (Bind b) 
-u2 (lift (S O) O t3)))) (\lambda (u2: T).(subst0 i v1 u u2)))).(ex2_ind T 
-(\lambda (u2: T).(eq T w1 (THead (Bind b) u2 (lift (S O) O t3)))) (\lambda 
-(u2: T).(subst0 i v1 u u2)) (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 
-w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H6: 
-(eq T w1 (THead (Bind b) x (lift (S O) O t3)))).(\lambda (_: (subst0 i v1 u 
-x)).(eq_ind_r T (THead (Bind b) x (lift (S O) O t3)) (\lambda (t: T).(or (pr0 
-t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 
-w2))))) (or_introl (pr0 (THead (Bind b) x (lift (S O) O t3)) t4) (ex2 T 
-(\lambda (w2: T).(pr0 (THead (Bind b) x (lift (S O) O t3)) w2)) (\lambda (w2: 
-T).(subst0 i v2 t4 w2))) (pr0_zeta b H0 t3 t4 H1 x)) w1 H6)))) H5)) (\lambda 
-(H5: (ex2 T (\lambda (t5: T).(eq T w1 (THead (Bind b) u t5))) (\lambda (t5: 
-T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5)))).(ex2_ind T (\lambda 
-(t5: T).(eq T w1 (THead (Bind b) u t5))) (\lambda (t5: T).(subst0 (s (Bind b) 
-i) v1 (lift (S O) O t3) t5)) (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 
-w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H6: 
-(eq T w1 (THead (Bind b) u x))).(\lambda (H7: (subst0 (s (Bind b) i) v1 (lift 
-(S O) O t3) x)).(ex2_ind T (\lambda (t5: T).(eq T x (lift (S O) O t5))) 
-(\lambda (t5: T).(subst0 (minus (s (Bind b) i) (S O)) v1 t3 t5)) (or (pr0 w1 
-t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 
-w2)))) (\lambda (x0: T).(\lambda (H8: (eq T x (lift (S O) O x0))).(\lambda 
-(H9: (subst0 (minus (s (Bind b) i) (S O)) v1 t3 x0)).(let H10 \def (eq_ind T 
-x (\lambda (t: T).(eq T w1 (THead (Bind b) u t))) H6 (lift (S O) O x0) H8) in 
-(eq_ind_r T (THead (Bind b) u (lift (S O) O x0)) (\lambda (t: T).(or (pr0 t 
-t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 
-w2))))) (let H11 \def (eq_ind_r nat (minus i O) (\lambda (n: nat).(subst0 n 
-v1 t3 x0)) H9 i (minus_n_O i)) in (or_ind (pr0 x0 t4) (ex2 T (\lambda (w2: 
-T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (or (pr0 (THead (Bind 
-b) u (lift (S O) O x0)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) u 
-(lift (S O) O x0)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda 
-(H12: (pr0 x0 t4)).(or_introl (pr0 (THead (Bind b) u (lift (S O) O x0)) t4) 
-(ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) u (lift (S O) O x0)) w2)) 
-(\lambda (w2: T).(subst0 i v2 t4 w2))) (pr0_zeta b H0 x0 t4 H12 u))) (\lambda 
-(H12: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 t4 
-w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 
-t4 w2)) (or (pr0 (THead (Bind b) u (lift (S O) O x0)) t4) (ex2 T (\lambda 
-(w2: T).(pr0 (THead (Bind b) u (lift (S O) O x0)) w2)) (\lambda (w2: 
-T).(subst0 i v2 t4 w2)))) (\lambda (x1: T).(\lambda (H13: (pr0 x0 
-x1)).(\lambda (H14: (subst0 i v2 t4 x1)).(or_intror (pr0 (THead (Bind b) u 
-(lift (S O) O x0)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) u (lift 
-(S O) O x0)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (ex_intro2 T 
-(\lambda (w2: T).(pr0 (THead (Bind b) u (lift (S O) O x0)) w2)) (\lambda (w2: 
-T).(subst0 i v2 t4 w2)) x1 (pr0_zeta b H0 x0 x1 H13 u) H14))))) H12)) (H2 v1 
-x0 i H11 v2 H4))) w1 H10))))) (subst0_gen_lift_ge v1 t3 x (s (Bind b) i) (S 
-O) O H7 (le_n_S O i (le_O_n i))))))) H5)) (\lambda (H5: (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t5: T).(eq T w1 (THead (Bind b) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: 
-T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5))))).(ex3_2_ind T T 
-(\lambda (u2: T).(\lambda (t5: T).(eq T w1 (THead (Bind b) u2 t5)))) (\lambda 
-(u2: T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: 
-T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5))) (or (pr0 w1 t4) (ex2 T 
-(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) 
-(\lambda (x0: T).(\lambda (x1: T).(\lambda (H6: (eq T w1 (THead (Bind b) x0 
-x1))).(\lambda (_: (subst0 i v1 u x0)).(\lambda (H8: (subst0 (s (Bind b) i) 
-v1 (lift (S O) O t3) x1)).(ex2_ind T (\lambda (t5: T).(eq T x1 (lift (S O) O 
-t5))) (\lambda (t5: T).(subst0 (minus (s (Bind b) i) (S O)) v1 t3 t5)) (or 
-(pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i 
-v2 t4 w2)))) (\lambda (x: T).(\lambda (H9: (eq T x1 (lift (S O) O 
-x))).(\lambda (H10: (subst0 (minus (s (Bind b) i) (S O)) v1 t3 x)).(let H11 
-\def (eq_ind T x1 (\lambda (t: T).(eq T w1 (THead (Bind b) x0 t))) H6 (lift 
-(S O) O x) H9) in (eq_ind_r T (THead (Bind b) x0 (lift (S O) O x)) (\lambda 
-(t: T).(or (pr0 t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: 
-T).(subst0 i v2 t4 w2))))) (let H12 \def (eq_ind_r nat (minus i O) (\lambda 
-(n: nat).(subst0 n v1 t3 x)) H10 i (minus_n_O i)) in (or_ind (pr0 x t4) (ex2 
-T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (or 
-(pr0 (THead (Bind b) x0 (lift (S O) O x)) t4) (ex2 T (\lambda (w2: T).(pr0 
-(THead (Bind b) x0 (lift (S O) O x)) w2)) (\lambda (w2: T).(subst0 i v2 t4 
-w2)))) (\lambda (H13: (pr0 x t4)).(or_introl (pr0 (THead (Bind b) x0 (lift (S 
-O) O x)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) x0 (lift (S O) O 
-x)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (pr0_zeta b H0 x t4 H13 x0))) 
-(\lambda (H13: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i 
-v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 
-i v2 t4 w2)) (or (pr0 (THead (Bind b) x0 (lift (S O) O x)) t4) (ex2 T 
-(\lambda (w2: T).(pr0 (THead (Bind b) x0 (lift (S O) O x)) w2)) (\lambda (w2: 
-T).(subst0 i v2 t4 w2)))) (\lambda (x2: T).(\lambda (H14: (pr0 x 
-x2)).(\lambda (H15: (subst0 i v2 t4 x2)).(or_intror (pr0 (THead (Bind b) x0 
-(lift (S O) O x)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) x0 (lift 
-(S O) O x)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (ex_intro2 T (\lambda 
-(w2: T).(pr0 (THead (Bind b) x0 (lift (S O) O x)) w2)) (\lambda (w2: 
-T).(subst0 i v2 t4 w2)) x2 (pr0_zeta b H0 x x2 H14 x0) H15))))) H13)) (H2 v1 
-x i H12 v2 H4))) w1 H11))))) (subst0_gen_lift_ge v1 t3 x1 (s (Bind b) i) (S 
-O) O H8 (le_n_S O i (le_O_n i))))))))) H5)) (subst0_gen_head (Bind b) v1 u 
-(lift (S O) O t3) w1 i H3))))))))))))))) (\lambda (t3: T).(\lambda (t4: 
-T).(\lambda (H0: (pr0 t3 t4)).(\lambda (H1: ((\forall (v1: T).(\forall (w1: 
-T).(\forall (i: nat).((subst0 i v1 t3 w1) \to (\forall (v2: T).((pr0 v1 v2) 
-\to (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: 
-T).(subst0 i v2 t4 w2)))))))))))).(\lambda (u: T).(\lambda (v1: T).(\lambda 
-(w1: T).(\lambda (i: nat).(\lambda (H2: (subst0 i v1 (THead (Flat Cast) u t3) 
-w1)).(\lambda (v2: T).(\lambda (H3: (pr0 v1 v2)).(or3_ind (ex2 T (\lambda 
-(u2: T).(eq T w1 (THead (Flat Cast) u2 t3))) (\lambda (u2: T).(subst0 i v1 u 
-u2))) (ex2 T (\lambda (t5: T).(eq T w1 (THead (Flat Cast) u t5))) (\lambda 
-(t5: T).(subst0 (s (Flat Cast) i) v1 t3 t5))) (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t5: T).(eq T w1 (THead (Flat Cast) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: 
-T).(subst0 (s (Flat Cast) i) v1 t3 t5)))) (or (pr0 w1 t4) (ex2 T (\lambda 
-(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (H4: 
-(ex2 T (\lambda (u2: T).(eq T w1 (THead (Flat Cast) u2 t3))) (\lambda (u2: 
-T).(subst0 i v1 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T w1 (THead (Flat 
-Cast) u2 t3))) (\lambda (u2: T).(subst0 i v1 u u2)) (or (pr0 w1 t4) (ex2 T 
-(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) 
-(\lambda (x: T).(\lambda (H5: (eq T w1 (THead (Flat Cast) x t3))).(\lambda 
-(_: (subst0 i v1 u x)).(eq_ind_r T (THead (Flat Cast) x t3) (\lambda (t: 
-T).(or (pr0 t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: 
-T).(subst0 i v2 t4 w2))))) (or_introl (pr0 (THead (Flat Cast) x t3) t4) (ex2 
-T (\lambda (w2: T).(pr0 (THead (Flat Cast) x t3) w2)) (\lambda (w2: 
-T).(subst0 i v2 t4 w2))) (pr0_tau t3 t4 H0 x)) w1 H5)))) H4)) (\lambda (H4: 
-(ex2 T (\lambda (t5: T).(eq T w1 (THead (Flat Cast) u t5))) (\lambda (t5: 
-T).(subst0 (s (Flat Cast) i) v1 t3 t5)))).(ex2_ind T (\lambda (t5: T).(eq T 
-w1 (THead (Flat Cast) u t5))) (\lambda (t5: T).(subst0 (s (Flat Cast) i) v1 
-t3 t5)) (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: 
-T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H5: (eq T w1 (THead (Flat 
-Cast) u x))).(\lambda (H6: (subst0 (s (Flat Cast) i) v1 t3 x)).(eq_ind_r T 
-(THead (Flat Cast) u x) (\lambda (t: T).(or (pr0 t t4) (ex2 T (\lambda (w2: 
-T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))))) (or_ind (pr0 x t4) 
-(ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Cast) 
-i) v2 t4 w2))) (or (pr0 (THead (Flat Cast) u x) t4) (ex2 T (\lambda (w2: 
-T).(pr0 (THead (Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) 
-(\lambda (H7: (pr0 x t4)).(or_introl (pr0 (THead (Flat Cast) u x) t4) (ex2 T 
-(\lambda (w2: T).(pr0 (THead (Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i 
-v2 t4 w2))) (pr0_tau x t4 H7 u))) (\lambda (H7: (ex2 T (\lambda (w2: T).(pr0 
-x w2)) (\lambda (w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2)))).(ex2_ind T 
-(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Cast) i) v2 t4 
-w2)) (or (pr0 (THead (Flat Cast) u x) t4) (ex2 T (\lambda (w2: T).(pr0 (THead 
-(Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x0: 
-T).(\lambda (H8: (pr0 x x0)).(\lambda (H9: (subst0 (s (Flat Cast) i) v2 t4 
-x0)).(or_intror (pr0 (THead (Flat Cast) u x) t4) (ex2 T (\lambda (w2: T).(pr0 
-(THead (Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) 
-(ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) u x) w2)) (\lambda (w2: 
-T).(subst0 i v2 t4 w2)) x0 (pr0_tau x x0 H8 u) H9))))) H7)) (H1 v1 x (s (Flat 
-Cast) i) H6 v2 H3)) w1 H5)))) H4)) (\lambda (H4: (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t5: T).(eq T w1 (THead (Flat Cast) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: 
-T).(subst0 (s (Flat Cast) i) v1 t3 t5))))).(ex3_2_ind T T (\lambda (u2: 
-T).(\lambda (t5: T).(eq T w1 (THead (Flat Cast) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: 
-T).(subst0 (s (Flat Cast) i) v1 t3 t5))) (or (pr0 w1 t4) (ex2 T (\lambda (w2: 
-T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x0: 
-T).(\lambda (x1: T).(\lambda (H5: (eq T w1 (THead (Flat Cast) x0 
-x1))).(\lambda (_: (subst0 i v1 u x0)).(\lambda (H7: (subst0 (s (Flat Cast) 
-i) v1 t3 x1)).(eq_ind_r T (THead (Flat Cast) x0 x1) (\lambda (t: T).(or (pr0 
-t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 
-w2))))) (or_ind (pr0 x1 t4) (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda 
-(w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2))) (or (pr0 (THead (Flat Cast) x0 
-x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) x0 x1) w2)) (\lambda 
-(w2: T).(subst0 i v2 t4 w2)))) (\lambda (H8: (pr0 x1 t4)).(or_introl (pr0 
-(THead (Flat Cast) x0 x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) 
-x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (pr0_tau x1 t4 H8 x0))) 
-(\lambda (H8: (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 
-(s (Flat Cast) i) v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x1 w2)) 
-(\lambda (w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2)) (or (pr0 (THead (Flat 
-Cast) x0 x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) x0 x1) w2)) 
-(\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H9: (pr0 x1 
-x)).(\lambda (H10: (subst0 (s (Flat Cast) i) v2 t4 x)).(or_intror (pr0 (THead 
-(Flat Cast) x0 x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) x0 x1) 
-w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (ex_intro2 T (\lambda (w2: 
-T).(pr0 (THead (Flat Cast) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)) 
-x (pr0_tau x1 x H9 x0) H10))))) H8)) (H1 v1 x1 (s (Flat Cast) i) H7 v2 H3)) 
-w1 H5)))))) H4)) (subst0_gen_head (Flat Cast) v1 u t3 w1 i H2))))))))))))) t1 
-t2 H))).
-(* COMMENTS
-Initial nodes: 38857
-END *)
+ \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead 
+(Bind Void) u1 t1) x)).(let TMP_1 \def (Bind Void) in (let TMP_2 \def (THead 
+TMP_1 u1 t1) in (let TMP_3 \def (\lambda (t: T).(pr0 t x)) in (let TMP_13 
+\def (\lambda (_: T).(let TMP_6 \def (\lambda (u2: T).(\lambda (t2: T).(let 
+TMP_4 \def (Bind Void) in (let TMP_5 \def (THead TMP_4 u2 t2) in (eq T x 
+TMP_5))))) in (let TMP_7 \def (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) 
+in (let TMP_8 \def (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))) in (let 
+TMP_9 \def (ex3_2 T T TMP_6 TMP_7 TMP_8) in (let TMP_10 \def (S O) in (let 
+TMP_11 \def (lift TMP_10 O x) in (let TMP_12 \def (pr0 t1 TMP_11) in (or 
+TMP_9 TMP_12))))))))) in (let TMP_310 \def (\lambda (y: T).(\lambda (H0: (pr0 
+y x)).(let TMP_23 \def (\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Bind 
+Void) u1 t1)) \to (let TMP_16 \def (\lambda (u2: T).(\lambda (t2: T).(let 
+TMP_14 \def (Bind Void) in (let TMP_15 \def (THead TMP_14 u2 t2) in (eq T t0 
+TMP_15))))) in (let TMP_17 \def (\lambda (u2: T).(\lambda (_: T).(pr0 u1 
+u2))) in (let TMP_18 \def (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))) in 
+(let TMP_19 \def (ex3_2 T T TMP_16 TMP_17 TMP_18) in (let TMP_20 \def (S O) 
+in (let TMP_21 \def (lift TMP_20 O t0) in (let TMP_22 \def (pr0 t1 TMP_21) in 
+(or TMP_19 TMP_22))))))))))) in (let TMP_66 \def (\lambda (t: T).(\lambda 
+(H1: (eq T t (THead (Bind Void) u1 t1))).(let TMP_24 \def (\lambda (e: T).e) 
+in (let TMP_25 \def (Bind Void) in (let TMP_26 \def (THead TMP_25 u1 t1) in 
+(let H2 \def (f_equal T T TMP_24 t TMP_26 H1) in (let TMP_27 \def (Bind Void) 
+in (let TMP_28 \def (THead TMP_27 u1 t1) in (let TMP_38 \def (\lambda (t0: 
+T).(let TMP_31 \def (\lambda (u2: T).(\lambda (t2: T).(let TMP_29 \def (Bind 
+Void) in (let TMP_30 \def (THead TMP_29 u2 t2) in (eq T t0 TMP_30))))) in 
+(let TMP_32 \def (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) in (let 
+TMP_33 \def (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))) in (let TMP_34 
+\def (ex3_2 T T TMP_31 TMP_32 TMP_33) in (let TMP_35 \def (S O) in (let 
+TMP_36 \def (lift TMP_35 O t0) in (let TMP_37 \def (pr0 t1 TMP_36) in (or 
+TMP_34 TMP_37))))))))) in (let TMP_43 \def (\lambda (u2: T).(\lambda (t2: 
+T).(let TMP_39 \def (Bind Void) in (let TMP_40 \def (THead TMP_39 u1 t1) in 
+(let TMP_41 \def (Bind Void) in (let TMP_42 \def (THead TMP_41 u2 t2) in (eq 
+T TMP_40 TMP_42))))))) in (let TMP_44 \def (\lambda (u2: T).(\lambda (_: 
+T).(pr0 u1 u2))) in (let TMP_45 \def (\lambda (_: T).(\lambda (t2: T).(pr0 t1 
+t2))) in (let TMP_46 \def (ex3_2 T T TMP_43 TMP_44 TMP_45) in (let TMP_47 
+\def (S O) in (let TMP_48 \def (Bind Void) in (let TMP_49 \def (THead TMP_48 
+u1 t1) in (let TMP_50 \def (lift TMP_47 O TMP_49) in (let TMP_51 \def (pr0 t1 
+TMP_50) in (let TMP_56 \def (\lambda (u2: T).(\lambda (t2: T).(let TMP_52 
+\def (Bind Void) in (let TMP_53 \def (THead TMP_52 u1 t1) in (let TMP_54 \def 
+(Bind Void) in (let TMP_55 \def (THead TMP_54 u2 t2) in (eq T TMP_53 
+TMP_55))))))) in (let TMP_57 \def (\lambda (u2: T).(\lambda (_: T).(pr0 u1 
+u2))) in (let TMP_58 \def (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))) in 
+(let TMP_59 \def (Bind Void) in (let TMP_60 \def (THead TMP_59 u1 t1) in (let 
+TMP_61 \def (refl_equal T TMP_60) in (let TMP_62 \def (pr0_refl u1) in (let 
+TMP_63 \def (pr0_refl t1) in (let TMP_64 \def (ex3_2_intro T T TMP_56 TMP_57 
+TMP_58 u1 t1 TMP_61 TMP_62 TMP_63) in (let TMP_65 \def (or_introl TMP_46 
+TMP_51 TMP_64) in (eq_ind_r T TMP_28 TMP_38 TMP_65 t 
+H2))))))))))))))))))))))))))))) in (let TMP_141 \def (\lambda (u0: 
+T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 u2)).(\lambda (H2: (((eq T u0 
+(THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: 
+T).(eq T u2 (THead (Bind Void) u3 t2)))) (\lambda (u3: T).(\lambda (_: 
+T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 
+(lift (S O) O u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H3: (pr0 
+t0 t2)).(\lambda (H4: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T 
+T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u3 t3)))) 
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: 
+T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (k: K).(\lambda 
+(H5: (eq T (THead k u0 t0) (THead (Bind Void) u1 t1))).(let TMP_67 \def 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow k | (TLRef _) 
+\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) in (let TMP_68 \def (THead k 
+u0 t0) in (let TMP_69 \def (Bind Void) in (let TMP_70 \def (THead TMP_69 u1 
+t1) in (let H6 \def (f_equal T K TMP_67 TMP_68 TMP_70 H5) in (let TMP_71 \def 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow u0 | (TLRef _) 
+\Rightarrow u0 | (THead _ t _) \Rightarrow t])) in (let TMP_72 \def (THead k 
+u0 t0) in (let TMP_73 \def (Bind Void) in (let TMP_74 \def (THead TMP_73 u1 
+t1) in (let H7 \def (f_equal T T TMP_71 TMP_72 TMP_74 H5) in (let TMP_75 \def 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow t0 | (TLRef _) 
+\Rightarrow t0 | (THead _ _ t) \Rightarrow t])) in (let TMP_76 \def (THead k 
+u0 t0) in (let TMP_77 \def (Bind Void) in (let TMP_78 \def (THead TMP_77 u1 
+t1) in (let H8 \def (f_equal T T TMP_75 TMP_76 TMP_78 H5) in (let TMP_139 
+\def (\lambda (H9: (eq T u0 u1)).(\lambda (H10: (eq K k (Bind Void))).(let 
+TMP_79 \def (Bind Void) in (let TMP_91 \def (\lambda (k0: K).(let TMP_83 \def 
+(\lambda (u3: T).(\lambda (t3: T).(let TMP_80 \def (THead k0 u2 t2) in (let 
+TMP_81 \def (Bind Void) in (let TMP_82 \def (THead TMP_81 u3 t3) in (eq T 
+TMP_80 TMP_82)))))) in (let TMP_84 \def (\lambda (u3: T).(\lambda (_: T).(pr0 
+u1 u3))) in (let TMP_85 \def (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) 
+in (let TMP_86 \def (ex3_2 T T TMP_83 TMP_84 TMP_85) in (let TMP_87 \def (S 
+O) in (let TMP_88 \def (THead k0 u2 t2) in (let TMP_89 \def (lift TMP_87 O 
+TMP_88) in (let TMP_90 \def (pr0 t1 TMP_89) in (or TMP_86 TMP_90)))))))))) in 
+(let TMP_101 \def (\lambda (t: T).((eq T t (THead (Bind Void) u1 t1)) \to 
+(let TMP_94 \def (\lambda (u3: T).(\lambda (t3: T).(let TMP_92 \def (Bind 
+Void) in (let TMP_93 \def (THead TMP_92 u3 t3) in (eq T t2 TMP_93))))) in 
+(let TMP_95 \def (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) in (let 
+TMP_96 \def (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) in (let TMP_97 
+\def (ex3_2 T T TMP_94 TMP_95 TMP_96) in (let TMP_98 \def (S O) in (let 
+TMP_99 \def (lift TMP_98 O t2) in (let TMP_100 \def (pr0 t1 TMP_99) in (or 
+TMP_97 TMP_100)))))))))) in (let H11 \def (eq_ind T t0 TMP_101 H4 t1 H8) in 
+(let TMP_102 \def (\lambda (t: T).(pr0 t t2)) in (let H12 \def (eq_ind T t0 
+TMP_102 H3 t1 H8) in (let TMP_112 \def (\lambda (t: T).((eq T t (THead (Bind 
+Void) u1 t1)) \to (let TMP_105 \def (\lambda (u3: T).(\lambda (t3: T).(let 
+TMP_103 \def (Bind Void) in (let TMP_104 \def (THead TMP_103 u3 t3) in (eq T 
+u2 TMP_104))))) in (let TMP_106 \def (\lambda (u3: T).(\lambda (_: T).(pr0 u1 
+u3))) in (let TMP_107 \def (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) in 
+(let TMP_108 \def (ex3_2 T T TMP_105 TMP_106 TMP_107) in (let TMP_109 \def (S 
+O) in (let TMP_110 \def (lift TMP_109 O u2) in (let TMP_111 \def (pr0 t1 
+TMP_110) in (or TMP_108 TMP_111)))))))))) in (let H13 \def (eq_ind T u0 
+TMP_112 H2 u1 H9) in (let TMP_113 \def (\lambda (t: T).(pr0 t u2)) in (let 
+H14 \def (eq_ind T u0 TMP_113 H1 u1 H9) in (let TMP_118 \def (\lambda (u3: 
+T).(\lambda (t3: T).(let TMP_114 \def (Bind Void) in (let TMP_115 \def (THead 
+TMP_114 u2 t2) in (let TMP_116 \def (Bind Void) in (let TMP_117 \def (THead 
+TMP_116 u3 t3) in (eq T TMP_115 TMP_117))))))) in (let TMP_119 \def (\lambda 
+(u3: T).(\lambda (_: T).(pr0 u1 u3))) in (let TMP_120 \def (\lambda (_: 
+T).(\lambda (t3: T).(pr0 t1 t3))) in (let TMP_121 \def (ex3_2 T T TMP_118 
+TMP_119 TMP_120) in (let TMP_122 \def (S O) in (let TMP_123 \def (Bind Void) 
+in (let TMP_124 \def (THead TMP_123 u2 t2) in (let TMP_125 \def (lift TMP_122 
+O TMP_124) in (let TMP_126 \def (pr0 t1 TMP_125) in (let TMP_131 \def 
+(\lambda (u3: T).(\lambda (t3: T).(let TMP_127 \def (Bind Void) in (let 
+TMP_128 \def (THead TMP_127 u2 t2) in (let TMP_129 \def (Bind Void) in (let 
+TMP_130 \def (THead TMP_129 u3 t3) in (eq T TMP_128 TMP_130))))))) in (let 
+TMP_132 \def (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) in (let TMP_133 
+\def (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) in (let TMP_134 \def 
+(Bind Void) in (let TMP_135 \def (THead TMP_134 u2 t2) in (let TMP_136 \def 
+(refl_equal T TMP_135) in (let TMP_137 \def (ex3_2_intro T T TMP_131 TMP_132 
+TMP_133 u2 t2 TMP_136 H14 H12) in (let TMP_138 \def (or_introl TMP_121 
+TMP_126 TMP_137) in (eq_ind_r K TMP_79 TMP_91 TMP_138 k 
+H10)))))))))))))))))))))))))))))) in (let TMP_140 \def (TMP_139 H7) in 
+(TMP_140 H6)))))))))))))))))))))))))))) in (let TMP_163 \def (\lambda (u: 
+T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: 
+(((eq T v1 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T v2 (THead (Bind Void) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 
+t2)))) (pr0 t1 (lift (S O) O v2)))))).(\lambda (t0: T).(\lambda (t2: 
+T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Void) u1 
+t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
+(Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
+(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O 
+t2)))))).(\lambda (H5: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) 
+(THead (Bind Void) u1 t1))).(let TMP_142 \def (Flat Appl) in (let TMP_143 
+\def (Bind Abst) in (let TMP_144 \def (THead TMP_143 u t0) in (let TMP_145 
+\def (THead TMP_142 v1 TMP_144) in (let TMP_146 \def (\lambda (ee: T).(match 
+ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k 
+_ _) \Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat _) 
+\Rightarrow True])])) in (let TMP_147 \def (Bind Void) in (let TMP_148 \def 
+(THead TMP_147 u1 t1) in (let H6 \def (eq_ind T TMP_145 TMP_146 I TMP_148 H5) 
+in (let TMP_153 \def (\lambda (u2: T).(\lambda (t3: T).(let TMP_149 \def 
+(Bind Abbr) in (let TMP_150 \def (THead TMP_149 v2 t2) in (let TMP_151 \def 
+(Bind Void) in (let TMP_152 \def (THead TMP_151 u2 t3) in (eq T TMP_150 
+TMP_152))))))) in (let TMP_154 \def (\lambda (u2: T).(\lambda (_: T).(pr0 u1 
+u2))) in (let TMP_155 \def (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) in 
+(let TMP_156 \def (ex3_2 T T TMP_153 TMP_154 TMP_155) in (let TMP_157 \def (S 
+O) in (let TMP_158 \def (Bind Abbr) in (let TMP_159 \def (THead TMP_158 v2 
+t2) in (let TMP_160 \def (lift TMP_157 O TMP_159) in (let TMP_161 \def (pr0 
+t1 TMP_160) in (let TMP_162 \def (or TMP_156 TMP_161) in (False_ind TMP_162 
+H6))))))))))))))))))))))))))))) in (let TMP_193 \def (\lambda (b: B).(\lambda 
+(_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 
+v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind Void) u2 t2)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: 
+T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O v2)))))).(\lambda (u0: T).(\lambda 
+(u2: T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead (Bind Void) 
+u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead 
+(Bind Void) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda 
+(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O 
+u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda 
+(_: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: 
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u3 t3)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
+t3)))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (H8: (eq T (THead (Flat Appl) 
+v1 (THead (Bind b) u0 t0)) (THead (Bind Void) u1 t1))).(let TMP_164 \def 
+(Flat Appl) in (let TMP_165 \def (Bind b) in (let TMP_166 \def (THead TMP_165 
+u0 t0) in (let TMP_167 \def (THead TMP_164 v1 TMP_166) in (let TMP_168 \def 
+(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _) 
+\Rightarrow False | (Flat _) \Rightarrow True])])) in (let TMP_169 \def (Bind 
+Void) in (let TMP_170 \def (THead TMP_169 u1 t1) in (let H9 \def (eq_ind T 
+TMP_167 TMP_168 I TMP_170 H8) in (let TMP_179 \def (\lambda (u3: T).(\lambda 
+(t3: T).(let TMP_171 \def (Bind b) in (let TMP_172 \def (Flat Appl) in (let 
+TMP_173 \def (S O) in (let TMP_174 \def (lift TMP_173 O v2) in (let TMP_175 
+\def (THead TMP_172 TMP_174 t2) in (let TMP_176 \def (THead TMP_171 u2 
+TMP_175) in (let TMP_177 \def (Bind Void) in (let TMP_178 \def (THead TMP_177 
+u3 t3) in (eq T TMP_176 TMP_178))))))))))) in (let TMP_180 \def (\lambda (u3: 
+T).(\lambda (_: T).(pr0 u1 u3))) in (let TMP_181 \def (\lambda (_: 
+T).(\lambda (t3: T).(pr0 t1 t3))) in (let TMP_182 \def (ex3_2 T T TMP_179 
+TMP_180 TMP_181) in (let TMP_183 \def (S O) in (let TMP_184 \def (Bind b) in 
+(let TMP_185 \def (Flat Appl) in (let TMP_186 \def (S O) in (let TMP_187 \def 
+(lift TMP_186 O v2) in (let TMP_188 \def (THead TMP_185 TMP_187 t2) in (let 
+TMP_189 \def (THead TMP_184 u2 TMP_188) in (let TMP_190 \def (lift TMP_183 O 
+TMP_189) in (let TMP_191 \def (pr0 t1 TMP_190) in (let TMP_192 \def (or 
+TMP_182 TMP_191) in (False_ind TMP_192 
+H9)))))))))))))))))))))))))))))))))))))) in (let TMP_213 \def (\lambda (u0: 
+T).(\lambda (u2: T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead 
+(Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq 
+T u2 (THead (Bind Void) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 
+u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O 
+u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda 
+(_: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: 
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u3 t3)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
+t3)))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (w: T).(\lambda (_: (subst0 O 
+u2 t2 w)).(\lambda (H6: (eq T (THead (Bind Abbr) u0 t0) (THead (Bind Void) u1 
+t1))).(let TMP_194 \def (Bind Abbr) in (let TMP_195 \def (THead TMP_194 u0 
+t0) in (let TMP_196 \def (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
+(match k with [(Bind b) \Rightarrow (match b with [Abbr \Rightarrow True | 
+Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) \Rightarrow 
+False])])) in (let TMP_197 \def (Bind Void) in (let TMP_198 \def (THead 
+TMP_197 u1 t1) in (let H7 \def (eq_ind T TMP_195 TMP_196 I TMP_198 H6) in 
+(let TMP_203 \def (\lambda (u3: T).(\lambda (t3: T).(let TMP_199 \def (Bind 
+Abbr) in (let TMP_200 \def (THead TMP_199 u2 w) in (let TMP_201 \def (Bind 
+Void) in (let TMP_202 \def (THead TMP_201 u3 t3) in (eq T TMP_200 
+TMP_202))))))) in (let TMP_204 \def (\lambda (u3: T).(\lambda (_: T).(pr0 u1 
+u3))) in (let TMP_205 \def (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) in 
+(let TMP_206 \def (ex3_2 T T TMP_203 TMP_204 TMP_205) in (let TMP_207 \def (S 
+O) in (let TMP_208 \def (Bind Abbr) in (let TMP_209 \def (THead TMP_208 u2 w) 
+in (let TMP_210 \def (lift TMP_207 O TMP_209) in (let TMP_211 \def (pr0 t1 
+TMP_210) in (let TMP_212 \def (or TMP_206 TMP_211) in (False_ind TMP_212 
+H7)))))))))))))))))))))))))))) in (let TMP_293 \def (\lambda (b: B).(\lambda 
+(H1: (not (eq B b Abst))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H2: 
+(pr0 t0 t2)).(\lambda (H3: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2)))))).(\lambda 
+(u: T).(\lambda (H4: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead (Bind 
+Void) u1 t1))).(let TMP_214 \def (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k 
+with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b])])) in (let TMP_215 
+\def (Bind b) in (let TMP_216 \def (S O) in (let TMP_217 \def (lift TMP_216 O 
+t0) in (let TMP_218 \def (THead TMP_215 u TMP_217) in (let TMP_219 \def (Bind 
+Void) in (let TMP_220 \def (THead TMP_219 u1 t1) in (let H5 \def (f_equal T B 
+TMP_214 TMP_218 TMP_220 H4) in (let TMP_221 \def (\lambda (e: T).(match e 
+with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) 
+\Rightarrow t])) in (let TMP_222 \def (Bind b) in (let TMP_223 \def (S O) in 
+(let TMP_224 \def (lift TMP_223 O t0) in (let TMP_225 \def (THead TMP_222 u 
+TMP_224) in (let TMP_226 \def (Bind Void) in (let TMP_227 \def (THead TMP_226 
+u1 t1) in (let H6 \def (f_equal T T TMP_221 TMP_225 TMP_227 H4) in (let 
+TMP_242 \def (\lambda (e: T).(match e with [(TSort _) \Rightarrow (let 
+TMP_241 \def (\lambda (x0: nat).(let TMP_240 \def (S O) in (plus x0 
+TMP_240))) in (lref_map TMP_241 O t0)) | (TLRef _) \Rightarrow (let TMP_234 
+\def (\lambda (x0: nat).(let TMP_233 \def (S O) in (plus x0 TMP_233))) in 
+(lref_map TMP_234 O t0)) | (THead _ _ t) \Rightarrow t])) in (let TMP_243 
+\def (Bind b) in (let TMP_244 \def (S O) in (let TMP_245 \def (lift TMP_244 O 
+t0) in (let TMP_246 \def (THead TMP_243 u TMP_245) in (let TMP_247 \def (Bind 
+Void) in (let TMP_248 \def (THead TMP_247 u1 t1) in (let H7 \def (f_equal T T 
+TMP_242 TMP_246 TMP_248 H4) in (let TMP_291 \def (\lambda (_: (eq T u 
+u1)).(\lambda (H9: (eq B b Void)).(let TMP_250 \def (\lambda (b0: B).(let 
+TMP_249 \def (eq B b0 Abst) in (not TMP_249))) in (let H10 \def (eq_ind B b 
+TMP_250 H1 Void H9) in (let TMP_260 \def (\lambda (t: T).((eq T t0 (THead 
+(Bind Void) u1 t)) \to (let TMP_253 \def (\lambda (u2: T).(\lambda (t3: 
+T).(let TMP_251 \def (Bind Void) in (let TMP_252 \def (THead TMP_251 u2 t3) 
+in (eq T t2 TMP_252))))) in (let TMP_254 \def (\lambda (u2: T).(\lambda (_: 
+T).(pr0 u1 u2))) in (let TMP_255 \def (\lambda (_: T).(\lambda (t3: T).(pr0 t 
+t3))) in (let TMP_256 \def (ex3_2 T T TMP_253 TMP_254 TMP_255) in (let 
+TMP_257 \def (S O) in (let TMP_258 \def (lift TMP_257 O t2) in (let TMP_259 
+\def (pr0 t TMP_258) in (or TMP_256 TMP_259)))))))))) in (let TMP_261 \def (S 
+O) in (let TMP_262 \def (lift TMP_261 O t0) in (let H11 \def (eq_ind_r T t1 
+TMP_260 H3 TMP_262 H7) in (let TMP_263 \def (S O) in (let TMP_264 \def (lift 
+TMP_263 O t0) in (let TMP_274 \def (\lambda (t: T).(let TMP_267 \def (\lambda 
+(u2: T).(\lambda (t3: T).(let TMP_265 \def (Bind Void) in (let TMP_266 \def 
+(THead TMP_265 u2 t3) in (eq T t2 TMP_266))))) in (let TMP_268 \def (\lambda 
+(u2: T).(\lambda (_: T).(pr0 u1 u2))) in (let TMP_269 \def (\lambda (_: 
+T).(\lambda (t3: T).(pr0 t t3))) in (let TMP_270 \def (ex3_2 T T TMP_267 
+TMP_268 TMP_269) in (let TMP_271 \def (S O) in (let TMP_272 \def (lift 
+TMP_271 O t2) in (let TMP_273 \def (pr0 t TMP_272) in (or TMP_270 
+TMP_273))))))))) in (let TMP_277 \def (\lambda (u2: T).(\lambda (t3: T).(let 
+TMP_275 \def (Bind Void) in (let TMP_276 \def (THead TMP_275 u2 t3) in (eq T 
+t2 TMP_276))))) in (let TMP_278 \def (\lambda (u2: T).(\lambda (_: T).(pr0 u1 
+u2))) in (let TMP_281 \def (\lambda (_: T).(\lambda (t3: T).(let TMP_279 \def 
+(S O) in (let TMP_280 \def (lift TMP_279 O t0) in (pr0 TMP_280 t3))))) in 
+(let TMP_282 \def (ex3_2 T T TMP_277 TMP_278 TMP_281) in (let TMP_283 \def (S 
+O) in (let TMP_284 \def (lift TMP_283 O t0) in (let TMP_285 \def (S O) in 
+(let TMP_286 \def (lift TMP_285 O t2) in (let TMP_287 \def (pr0 TMP_284 
+TMP_286) in (let TMP_288 \def (S O) in (let TMP_289 \def (pr0_lift t0 t2 H2 
+TMP_288 O) in (let TMP_290 \def (or_intror TMP_282 TMP_287 TMP_289) in 
+(eq_ind T TMP_264 TMP_274 TMP_290 t1 H7)))))))))))))))))))))))) in (let 
+TMP_292 \def (TMP_291 H6) in (TMP_292 H5))))))))))))))))))))))))))))))))))) 
+in (let TMP_309 \def (\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 
+t2)).(\lambda (_: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: 
+T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (u: T).(\lambda 
+(H3: (eq T (THead (Flat Cast) u t0) (THead (Bind Void) u1 t1))).(let TMP_294 
+\def (Flat Cast) in (let TMP_295 \def (THead TMP_294 u t0) in (let TMP_296 
+\def (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _) 
+\Rightarrow False | (Flat _) \Rightarrow True])])) in (let TMP_297 \def (Bind 
+Void) in (let TMP_298 \def (THead TMP_297 u1 t1) in (let H4 \def (eq_ind T 
+TMP_295 TMP_296 I TMP_298 H3) in (let TMP_301 \def (\lambda (u2: T).(\lambda 
+(t3: T).(let TMP_299 \def (Bind Void) in (let TMP_300 \def (THead TMP_299 u2 
+t3) in (eq T t2 TMP_300))))) in (let TMP_302 \def (\lambda (u2: T).(\lambda 
+(_: T).(pr0 u1 u2))) in (let TMP_303 \def (\lambda (_: T).(\lambda (t3: 
+T).(pr0 t1 t3))) in (let TMP_304 \def (ex3_2 T T TMP_301 TMP_302 TMP_303) in 
+(let TMP_305 \def (S O) in (let TMP_306 \def (lift TMP_305 O t2) in (let 
+TMP_307 \def (pr0 t1 TMP_306) in (let TMP_308 \def (or TMP_304 TMP_307) in 
+(False_ind TMP_308 H4))))))))))))))))))))) in (pr0_ind TMP_23 TMP_66 TMP_141 
+TMP_163 TMP_193 TMP_213 TMP_293 TMP_309 y x H0))))))))))) in (insert_eq T 
+TMP_2 TMP_3 TMP_13 TMP_310 H))))))))).

 
 

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr0/subst0.ma b/matita/matita/contribs/lambdadelta/basic_1/pr0/subst0.ma
new file mode 100644 (file)
index 0000000..12f1733
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1/pr0/subst0.ma
@@ -0,0 +1,1647 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1/pr0/props.ma".
+
+include "basic_1/subst0/subst0.ma".
+
+theorem pr0_subst0_back:
+ \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst0 
+i u2 t1 t2) \to (\forall (u1: T).((pr0 u1 u2) \to (ex2 T (\lambda (t: 
+T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t t2)))))))))
+\def
+ \lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (i: nat).(\lambda 
+(H: (subst0 i u2 t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t: 
+T).(\lambda (t0: T).(\lambda (t3: T).(\forall (u1: T).((pr0 u1 t) \to (ex2 T 
+(\lambda (t4: T).(subst0 n u1 t0 t4)) (\lambda (t4: T).(pr0 t4 t3))))))))) 
+(\lambda (v: T).(\lambda (i0: nat).(\lambda (u1: T).(\lambda (H0: (pr0 u1 
+v)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 (TLRef i0) t)) (\lambda (t: 
+T).(pr0 t (lift (S i0) O v))) (lift (S i0) O u1) (subst0_lref u1 i0) 
+(pr0_lift u1 v H0 (S i0) O)))))) (\lambda (v: T).(\lambda (u3: T).(\lambda 
+(u1: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v u1 u3)).(\lambda (H1: 
+((\forall (u4: T).((pr0 u4 v) \to (ex2 T (\lambda (t: T).(subst0 i0 u4 u1 t)) 
+(\lambda (t: T).(pr0 t u3))))))).(\lambda (t: T).(\lambda (k: K).(\lambda 
+(u0: T).(\lambda (H2: (pr0 u0 v)).(ex2_ind T (\lambda (t0: T).(subst0 i0 u0 
+u1 t0)) (\lambda (t0: T).(pr0 t0 u3)) (ex2 T (\lambda (t0: T).(subst0 i0 u0 
+(THead k u1 t) t0)) (\lambda (t0: T).(pr0 t0 (THead k u3 t)))) (\lambda (x: 
+T).(\lambda (H3: (subst0 i0 u0 u1 x)).(\lambda (H4: (pr0 x u3)).(ex_intro2 T 
+(\lambda (t0: T).(subst0 i0 u0 (THead k u1 t) t0)) (\lambda (t0: T).(pr0 t0 
+(THead k u3 t))) (THead k x t) (subst0_fst u0 x u1 i0 H3 t k) (pr0_comp x u3 
+H4 t t (pr0_refl t) k))))) (H1 u0 H2)))))))))))) (\lambda (k: K).(\lambda (v: 
+T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (i0: nat).(\lambda (_: (subst0 
+(s k i0) v t4 t3)).(\lambda (H1: ((\forall (u1: T).((pr0 u1 v) \to (ex2 T 
+(\lambda (t: T).(subst0 (s k i0) u1 t4 t)) (\lambda (t: T).(pr0 t 
+t3))))))).(\lambda (u: T).(\lambda (u1: T).(\lambda (H2: (pr0 u1 v)).(ex2_ind 
+T (\lambda (t: T).(subst0 (s k i0) u1 t4 t)) (\lambda (t: T).(pr0 t t3)) (ex2 
+T (\lambda (t: T).(subst0 i0 u1 (THead k u t4) t)) (\lambda (t: T).(pr0 t 
+(THead k u t3)))) (\lambda (x: T).(\lambda (H3: (subst0 (s k i0) u1 t4 
+x)).(\lambda (H4: (pr0 x t3)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 
+(THead k u t4) t)) (\lambda (t: T).(pr0 t (THead k u t3))) (THead k u x) 
+(subst0_snd k u1 x t4 i0 H3 u) (pr0_comp u u (pr0_refl u) x t3 H4 k))))) (H1 
+u1 H2)))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u3: T).(\lambda 
+(i0: nat).(\lambda (_: (subst0 i0 v u1 u3)).(\lambda (H1: ((\forall (u4: 
+T).((pr0 u4 v) \to (ex2 T (\lambda (t: T).(subst0 i0 u4 u1 t)) (\lambda (t: 
+T).(pr0 t u3))))))).(\lambda (k: K).(\lambda (t3: T).(\lambda (t4: 
+T).(\lambda (_: (subst0 (s k i0) v t3 t4)).(\lambda (H3: ((\forall (u4: 
+T).((pr0 u4 v) \to (ex2 T (\lambda (t: T).(subst0 (s k i0) u4 t3 t)) (\lambda 
+(t: T).(pr0 t t4))))))).(\lambda (u0: T).(\lambda (H4: (pr0 u0 v)).(ex2_ind T 
+(\lambda (t: T).(subst0 (s k i0) u0 t3 t)) (\lambda (t: T).(pr0 t t4)) (ex2 T 
+(\lambda (t: T).(subst0 i0 u0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 t 
+(THead k u3 t4)))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i0) u0 t3 
+x)).(\lambda (H6: (pr0 x t4)).(ex2_ind T (\lambda (t: T).(subst0 i0 u0 u1 t)) 
+(\lambda (t: T).(pr0 t u3)) (ex2 T (\lambda (t: T).(subst0 i0 u0 (THead k u1 
+t3) t)) (\lambda (t: T).(pr0 t (THead k u3 t4)))) (\lambda (x0: T).(\lambda 
+(H7: (subst0 i0 u0 u1 x0)).(\lambda (H8: (pr0 x0 u3)).(ex_intro2 T (\lambda 
+(t: T).(subst0 i0 u0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 t (THead k u3 
+t4))) (THead k x0 x) (subst0_both u0 u1 x0 i0 H7 k t3 x H5) (pr0_comp x0 u3 
+H8 x t4 H6 k))))) (H1 u0 H4))))) (H3 u0 H4))))))))))))))) i u2 t1 t2 H))))).
+
+theorem pr0_subst0_fwd:
+ \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst0 
+i u2 t1 t2) \to (\forall (u1: T).((pr0 u2 u1) \to (ex2 T (\lambda (t: 
+T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t2 t)))))))))
+\def
+ \lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (i: nat).(\lambda 
+(H: (subst0 i u2 t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t: 
+T).(\lambda (t0: T).(\lambda (t3: T).(\forall (u1: T).((pr0 t u1) \to (ex2 T 
+(\lambda (t4: T).(subst0 n u1 t0 t4)) (\lambda (t4: T).(pr0 t3 t4))))))))) 
+(\lambda (v: T).(\lambda (i0: nat).(\lambda (u1: T).(\lambda (H0: (pr0 v 
+u1)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 (TLRef i0) t)) (\lambda (t: 
+T).(pr0 (lift (S i0) O v) t)) (lift (S i0) O u1) (subst0_lref u1 i0) 
+(pr0_lift v u1 H0 (S i0) O)))))) (\lambda (v: T).(\lambda (u3: T).(\lambda 
+(u1: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v u1 u3)).(\lambda (H1: 
+((\forall (u4: T).((pr0 v u4) \to (ex2 T (\lambda (t: T).(subst0 i0 u4 u1 t)) 
+(\lambda (t: T).(pr0 u3 t))))))).(\lambda (t: T).(\lambda (k: K).(\lambda 
+(u0: T).(\lambda (H2: (pr0 v u0)).(ex2_ind T (\lambda (t0: T).(subst0 i0 u0 
+u1 t0)) (\lambda (t0: T).(pr0 u3 t0)) (ex2 T (\lambda (t0: T).(subst0 i0 u0 
+(THead k u1 t) t0)) (\lambda (t0: T).(pr0 (THead k u3 t) t0))) (\lambda (x: 
+T).(\lambda (H3: (subst0 i0 u0 u1 x)).(\lambda (H4: (pr0 u3 x)).(ex_intro2 T 
+(\lambda (t0: T).(subst0 i0 u0 (THead k u1 t) t0)) (\lambda (t0: T).(pr0 
+(THead k u3 t) t0)) (THead k x t) (subst0_fst u0 x u1 i0 H3 t k) (pr0_comp u3 
+x H4 t t (pr0_refl t) k))))) (H1 u0 H2)))))))))))) (\lambda (k: K).(\lambda 
+(v: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (i0: nat).(\lambda (_: 
+(subst0 (s k i0) v t4 t3)).(\lambda (H1: ((\forall (u1: T).((pr0 v u1) \to 
+(ex2 T (\lambda (t: T).(subst0 (s k i0) u1 t4 t)) (\lambda (t: T).(pr0 t3 
+t))))))).(\lambda (u: T).(\lambda (u1: T).(\lambda (H2: (pr0 v u1)).(ex2_ind 
+T (\lambda (t: T).(subst0 (s k i0) u1 t4 t)) (\lambda (t: T).(pr0 t3 t)) (ex2 
+T (\lambda (t: T).(subst0 i0 u1 (THead k u t4) t)) (\lambda (t: T).(pr0 
+(THead k u t3) t))) (\lambda (x: T).(\lambda (H3: (subst0 (s k i0) u1 t4 
+x)).(\lambda (H4: (pr0 t3 x)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 
+(THead k u t4) t)) (\lambda (t: T).(pr0 (THead k u t3) t)) (THead k u x) 
+(subst0_snd k u1 x t4 i0 H3 u) (pr0_comp u u (pr0_refl u) t3 x H4 k))))) (H1 
+u1 H2)))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u3: T).(\lambda 
+(i0: nat).(\lambda (_: (subst0 i0 v u1 u3)).(\lambda (H1: ((\forall (u4: 
+T).((pr0 v u4) \to (ex2 T (\lambda (t: T).(subst0 i0 u4 u1 t)) (\lambda (t: 
+T).(pr0 u3 t))))))).(\lambda (k: K).(\lambda (t3: T).(\lambda (t4: 
+T).(\lambda (_: (subst0 (s k i0) v t3 t4)).(\lambda (H3: ((\forall (u4: 
+T).((pr0 v u4) \to (ex2 T (\lambda (t: T).(subst0 (s k i0) u4 t3 t)) (\lambda 
+(t: T).(pr0 t4 t))))))).(\lambda (u0: T).(\lambda (H4: (pr0 v u0)).(ex2_ind T 
+(\lambda (t: T).(subst0 (s k i0) u0 t3 t)) (\lambda (t: T).(pr0 t4 t)) (ex2 T 
+(\lambda (t: T).(subst0 i0 u0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 (THead 
+k u3 t4) t))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i0) u0 t3 
+x)).(\lambda (H6: (pr0 t4 x)).(ex2_ind T (\lambda (t: T).(subst0 i0 u0 u1 t)) 
+(\lambda (t: T).(pr0 u3 t)) (ex2 T (\lambda (t: T).(subst0 i0 u0 (THead k u1 
+t3) t)) (\lambda (t: T).(pr0 (THead k u3 t4) t))) (\lambda (x0: T).(\lambda 
+(H7: (subst0 i0 u0 u1 x0)).(\lambda (H8: (pr0 u3 x0)).(ex_intro2 T (\lambda 
+(t: T).(subst0 i0 u0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 (THead k u3 t4) 
+t)) (THead k x0 x) (subst0_both u0 u1 x0 i0 H7 k t3 x H5) (pr0_comp u3 x0 H8 
+t4 x H6 k))))) (H1 u0 H4))))) (H3 u0 H4))))))))))))))) i u2 t1 t2 H))))).
+
+theorem pr0_subst0:
+ \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (v1: T).(\forall 
+(w1: T).(\forall (i: nat).((subst0 i v1 t1 w1) \to (\forall (v2: T).((pr0 v1 
+v2) \to (or (pr0 w1 t2) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: 
+T).(subst0 i v2 t2 w2))))))))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(pr0_ind (\lambda 
+(t: T).(\lambda (t0: T).(\forall (v1: T).(\forall (w1: T).(\forall (i: 
+nat).((subst0 i v1 t w1) \to (\forall (v2: T).((pr0 v1 v2) \to (or (pr0 w1 
+t0) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t0 
+w2)))))))))))) (\lambda (t: T).(\lambda (v1: T).(\lambda (w1: T).(\lambda (i: 
+nat).(\lambda (H0: (subst0 i v1 t w1)).(\lambda (v2: T).(\lambda (H1: (pr0 v1 
+v2)).(or_intror (pr0 w1 t) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: 
+T).(subst0 i v2 t w2))) (ex2_sym T (subst0 i v2 t) (pr0 w1) (pr0_subst0_fwd 
+v1 t w1 i H0 v2 H1)))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (H0: 
+(pr0 u1 u2)).(\lambda (H1: ((\forall (v1: T).(\forall (w1: T).(\forall (i: 
+nat).((subst0 i v1 u1 w1) \to (\forall (v2: T).((pr0 v1 v2) \to (or (pr0 w1 
+u2) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 u2 
+w2)))))))))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 
+t4)).(\lambda (H3: ((\forall (v1: T).(\forall (w1: T).(\forall (i: 
+nat).((subst0 i v1 t3 w1) \to (\forall (v2: T).((pr0 v1 v2) \to (or (pr0 w1 
+t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 
+w2)))))))))))).(\lambda (k: K).(\lambda (v1: T).(\lambda (w1: T).(\lambda (i: 
+nat).(\lambda (H4: (subst0 i v1 (THead k u1 t3) w1)).(\lambda (v2: 
+T).(\lambda (H5: (pr0 v1 v2)).(or3_ind (ex2 T (\lambda (u3: T).(eq T w1 
+(THead k u3 t3))) (\lambda (u3: T).(subst0 i v1 u1 u3))) (ex2 T (\lambda (t5: 
+T).(eq T w1 (THead k u1 t5))) (\lambda (t5: T).(subst0 (s k i) v1 t3 t5))) 
+(ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead k u3 t5)))) 
+(\lambda (u3: T).(\lambda (_: T).(subst0 i v1 u1 u3))) (\lambda (_: 
+T).(\lambda (t5: T).(subst0 (s k i) v1 t3 t5)))) (or (pr0 w1 (THead k u2 t4)) 
+(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 (THead k 
+u2 t4) w2)))) (\lambda (H6: (ex2 T (\lambda (u3: T).(eq T w1 (THead k u3 
+t3))) (\lambda (u3: T).(subst0 i v1 u1 u3)))).(ex2_ind T (\lambda (u3: T).(eq 
+T w1 (THead k u3 t3))) (\lambda (u3: T).(subst0 i v1 u1 u3)) (or (pr0 w1 
+(THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: 
+T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x: T).(\lambda (H7: (eq T w1 
+(THead k x t3))).(\lambda (H8: (subst0 i v1 u1 x)).(eq_ind_r T (THead k x t3) 
+(\lambda (t: T).(or (pr0 t (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 t 
+w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))))) (or_ind (pr0 x u2) 
+(ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 u2 w2))) 
+(or (pr0 (THead k x t3) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead 
+k x t3) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda 
+(H9: (pr0 x u2)).(or_introl (pr0 (THead k x t3) (THead k u2 t4)) (ex2 T 
+(\lambda (w2: T).(pr0 (THead k x t3) w2)) (\lambda (w2: T).(subst0 i v2 
+(THead k u2 t4) w2))) (pr0_comp x u2 H9 t3 t4 H2 k))) (\lambda (H9: (ex2 T 
+(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 u2 w2)))).(ex2_ind 
+T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 u2 w2)) (or (pr0 
+(THead k x t3) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x t3) 
+w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x0: 
+T).(\lambda (H10: (pr0 x x0)).(\lambda (H11: (subst0 i v2 u2 x0)).(or_intror 
+(pr0 (THead k x t3) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x 
+t3) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))) (ex_intro2 T 
+(\lambda (w2: T).(pr0 (THead k x t3) w2)) (\lambda (w2: T).(subst0 i v2 
+(THead k u2 t4) w2)) (THead k x0 t4) (pr0_comp x x0 H10 t3 t4 H2 k) 
+(subst0_fst v2 x0 u2 i H11 t4 k)))))) H9)) (H1 v1 x i H8 v2 H5)) w1 H7)))) 
+H6)) (\lambda (H6: (ex2 T (\lambda (t5: T).(eq T w1 (THead k u1 t5))) 
+(\lambda (t5: T).(subst0 (s k i) v1 t3 t5)))).(ex2_ind T (\lambda (t5: T).(eq 
+T w1 (THead k u1 t5))) (\lambda (t5: T).(subst0 (s k i) v1 t3 t5)) (or (pr0 
+w1 (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: 
+T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x: T).(\lambda (H7: (eq T w1 
+(THead k u1 x))).(\lambda (H8: (subst0 (s k i) v1 t3 x)).(eq_ind_r T (THead k 
+u1 x) (\lambda (t: T).(or (pr0 t (THead k u2 t4)) (ex2 T (\lambda (w2: 
+T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))))) (or_ind 
+(pr0 x t4) (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s k 
+i) v2 t4 w2))) (or (pr0 (THead k u1 x) (THead k u2 t4)) (ex2 T (\lambda (w2: 
+T).(pr0 (THead k u1 x) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) 
+w2)))) (\lambda (H9: (pr0 x t4)).(or_introl (pr0 (THead k u1 x) (THead k u2 
+t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k u1 x) w2)) (\lambda (w2: 
+T).(subst0 i v2 (THead k u2 t4) w2))) (pr0_comp u1 u2 H0 x t4 H9 k))) 
+(\lambda (H9: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s 
+k i) v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: 
+T).(subst0 (s k i) v2 t4 w2)) (or (pr0 (THead k u1 x) (THead k u2 t4)) (ex2 T 
+(\lambda (w2: T).(pr0 (THead k u1 x) w2)) (\lambda (w2: T).(subst0 i v2 
+(THead k u2 t4) w2)))) (\lambda (x0: T).(\lambda (H10: (pr0 x x0)).(\lambda 
+(H11: (subst0 (s k i) v2 t4 x0)).(or_intror (pr0 (THead k u1 x) (THead k u2 
+t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k u1 x) w2)) (\lambda (w2: 
+T).(subst0 i v2 (THead k u2 t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 
+(THead k u1 x) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)) (THead 
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+H9)) (H3 v1 x (s k i) H8 v2 H5)) w1 H7)))) H6)) (\lambda (H6: (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead k u3 t5)))) (\lambda (u3: 
+T).(\lambda (_: T).(subst0 i v1 u1 u3))) (\lambda (_: T).(\lambda (t5: 
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+u2)).(or_introl (pr0 (THead k x0 x1) (THead k u2 t4)) (ex2 T (\lambda (w2: 
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+(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 u2 w2)) (or (pr0 (THead k 
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+(\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x: T).(\lambda 
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+(\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))) (ex_intro2 T (\lambda 
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+(w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s k i) v2 t4 w2)))).(ex2_ind T 
+(\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s k i) v2 t4 w2)) (or 
+(pr0 (THead k x0 x1) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k 
+x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x: 
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+T).(subst0 i v2 u2 w2))) (or (pr0 (THead k x0 x1) (THead k u2 t4)) (ex2 T 
+(\lambda (w2: T).(pr0 (THead k x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 
+(THead k u2 t4) w2)))) (\lambda (H13: (pr0 x0 u2)).(or_intror (pr0 (THead k 
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+(\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))) (ex_intro2 T (\lambda 
+(w2: T).(pr0 (THead k x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 
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+(\lambda (w2: T).(subst0 i v2 u2 w2)) (or (pr0 (THead k x0 x1) (THead k u2 
+t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x0 x1) w2)) (\lambda (w2: 
+T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x2: T).(\lambda (H14: (pr0 
+x0 x2)).(\lambda (H15: (subst0 i v2 u2 x2)).(or_intror (pr0 (THead k x0 x1) 
+(THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x0 x1) w2)) (\lambda 
+(w2: T).(subst0 i v2 (THead k u2 t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 
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+v2)).(\lambda (H1: ((\forall (v3: T).(\forall (w1: T).(\forall (i: 
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+v2) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v4 v2 
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+(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 v1 
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+(Bind Abst) u t3) t5)))) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T 
+(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind 
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+u2)))).(ex2_ind T (\lambda (u2: T).(eq T w1 (THead (Flat Appl) u2 (THead 
+(Bind Abst) u t3)))) (\lambda (u2: T).(subst0 i v0 v1 u2)) (or (pr0 w1 (THead 
+(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: 
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+t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat 
+Appl) x (THead (Bind Abst) u t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead 
+(Bind Abbr) v2 t4) w2)))) (\lambda (H9: (pr0 x v2)).(or_introl (pr0 (THead 
+(Flat Appl) x (THead (Bind Abst) u t3)) (THead (Bind Abbr) v2 t4)) (ex2 T 
+(\lambda (w2: T).(pr0 (THead (Flat Appl) x (THead (Bind Abst) u t3)) w2)) 
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+(Bind Abst) u t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 
+(THead (Flat Appl) x (THead (Bind Abst) u t3)) w2)) (\lambda (w2: T).(subst0 
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+(THead (Bind Abst) u t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: 
+T).(pr0 (THead (Flat Appl) x (THead (Bind Abst) u t3)) w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: 
+T).(pr0 (THead (Flat Appl) x (THead (Bind Abst) u t3)) w2)) (\lambda (w2: 
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+(pr0_beta u x x0 H10 t3 t4 H2) (subst0_fst v3 x0 v2 i H11 t4 (Bind 
+Abbr))))))) H9)) (H1 v0 x i H8 v3 H5)) w1 H7)))) H6)) (\lambda (H6: (ex2 T 
+(\lambda (t5: T).(eq T w1 (THead (Flat Appl) v1 t5))) (\lambda (t5: 
+T).(subst0 (s (Flat Appl) i) v0 (THead (Bind Abst) u t3) t5)))).(ex2_ind T 
+(\lambda (t5: T).(eq T w1 (THead (Flat Appl) v1 t5))) (\lambda (t5: 
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+(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: 
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+(THead (Bind Abst) u2 t3))) (\lambda (u2: T).(subst0 (s (Flat Appl) i) v0 u 
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+(\lambda (u2: T).(\lambda (t5: T).(eq T x (THead (Bind Abst) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u u2))) 
+(\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 
+t3 t5)))) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 
+w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) 
+(\lambda (H9: (ex2 T (\lambda (u2: T).(eq T x (THead (Bind Abst) u2 t3))) 
+(\lambda (u2: T).(subst0 (s (Flat Appl) i) v0 u u2)))).(ex2_ind T (\lambda 
+(u2: T).(eq T x (THead (Bind Abst) u2 t3))) (\lambda (u2: T).(subst0 (s (Flat 
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+(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) 
+w2)))) (\lambda (x0: T).(\lambda (H10: (eq T x (THead (Bind Abst) x0 
+t3))).(\lambda (_: (subst0 (s (Flat Appl) i) v0 u x0)).(let H12 \def (eq_ind 
+T x (\lambda (t: T).(eq T w1 (THead (Flat Appl) v1 t))) H7 (THead (Bind Abst) 
+x0 t3) H10) in (eq_ind_r T (THead (Flat Appl) v1 (THead (Bind Abst) x0 t3)) 
+(\lambda (t: T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: 
+T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) 
+w2))))) (or_introl (pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 t3)) 
+(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 
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+(H9: (ex2 T (\lambda (t5: T).(eq T x (THead (Bind Abst) u t5))) (\lambda (t5: 
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+(t5: T).(eq T x (THead (Bind Abst) u t5))) (\lambda (t5: T).(subst0 (s (Bind 
+Abst) (s (Flat Appl) i)) v0 t3 t5)) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) 
+(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead 
+(Bind Abbr) v2 t4) w2)))) (\lambda (x0: T).(\lambda (H10: (eq T x (THead 
+(Bind Abst) u x0))).(\lambda (H11: (subst0 (s (Bind Abst) (s (Flat Appl) i)) 
+v0 t3 x0)).(let H12 \def (eq_ind T x (\lambda (t: T).(eq T w1 (THead (Flat 
+Appl) v1 t))) H7 (THead (Bind Abst) u x0) H10) in (eq_ind_r T (THead (Flat 
+Appl) v1 (THead (Bind Abst) u x0)) (\lambda (t: T).(or (pr0 t (THead (Bind 
+Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i 
+v3 (THead (Bind Abbr) v2 t4) w2))))) (or_ind (pr0 x0 t4) (ex2 T (\lambda (w2: 
+T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 
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+(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead 
+(Bind Abst) u x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
+t4) w2)))) (\lambda (H13: (pr0 x0 t4)).(or_introl (pr0 (THead (Flat Appl) v1 
+(THead (Bind Abst) u x0)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: 
+T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (pr0_beta u v1 v2 H0 x0 t4 
+H13))) (\lambda (H13: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: 
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+(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) 
+i)) v3 t4 w2)) (or (pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) 
+(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 
+(THead (Bind Abst) u x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind 
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+(H15: (subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 x1)).(or_intror (pr0 
+(THead (Flat Appl) v1 (THead (Bind Abst) u x0)) (THead (Bind Abbr) v2 t4)) 
+(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) 
+w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (ex_intro2 
+T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) w2)) 
+(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)) (THead (Bind 
+Abbr) v2 x1) (pr0_beta u v1 v2 H0 x0 x1 H14) (subst0_snd (Bind Abbr) v3 x1 t4 
+i H15 v2)))))) H13)) (H3 v0 x0 (s (Bind Abst) (s (Flat Appl) i)) H11 v3 H5)) 
+w1 H12))))) H9)) (\lambda (H9: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: 
+T).(eq T x (THead (Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
+T).(subst0 (s (Flat Appl) i) v0 u u2))) (\lambda (_: T).(\lambda (t5: 
+T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5))))).(ex3_2_ind T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T x (THead (Bind Abst) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u u2))) 
+(\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 
+t3 t5))) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 
+w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) 
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (H10: (eq T x (THead (Bind Abst) 
+x0 x1))).(\lambda (_: (subst0 (s (Flat Appl) i) v0 u x0)).(\lambda (H12: 
+(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 x1)).(let H13 \def (eq_ind T 
+x (\lambda (t: T).(eq T w1 (THead (Flat Appl) v1 t))) H7 (THead (Bind Abst) 
+x0 x1) H10) in (eq_ind_r T (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) 
+(\lambda (t: T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: 
+T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) 
+w2))))) (or_ind (pr0 x1 t4) (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda 
+(w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2))) (or (pr0 (THead 
+(Flat Appl) v1 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) v2 t4)) (ex2 T 
+(\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) w2)) 
+(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H14: 
+(pr0 x1 t4)).(or_introl (pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) 
+(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 
+(THead (Bind Abst) x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind 
+Abbr) v2 t4) w2))) (pr0_beta x0 v1 v2 H0 x1 t4 H14))) (\lambda (H14: (ex2 T 
+(\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s 
+(Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x1 w2)) 
+(\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2)) (or 
+(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) v2 
+t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 
+x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) 
+(\lambda (x2: T).(\lambda (H15: (pr0 x1 x2)).(\lambda (H16: (subst0 (s (Bind 
+Abst) (s (Flat Appl) i)) v3 t4 x2)).(or_intror (pr0 (THead (Flat Appl) v1 
+(THead (Bind Abst) x0 x1)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: 
+T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: 
+T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) v2 x2) 
+(pr0_beta x0 v1 v2 H0 x1 x2 H15) (subst0_snd (Bind Abbr) v3 x2 t4 i H16 
+v2)))))) H14)) (H3 v0 x1 (s (Bind Abst) (s (Flat Appl) i)) H12 v3 H5)) w1 
+H13))))))) H9)) (subst0_gen_head (Bind Abst) v0 u t3 x (s (Flat Appl) i) 
+H8))))) H6)) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T 
+w1 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 
+v1 u2))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 (THead 
+(Bind Abst) u t3) t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: 
+T).(eq T w1 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
+T).(subst0 i v0 v1 u2))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Flat 
+Appl) i) v0 (THead (Bind Abst) u t3) t5))) (or (pr0 w1 (THead (Bind Abbr) v2 
+t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 
+(THead (Bind Abbr) v2 t4) w2)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda 
+(H7: (eq T w1 (THead (Flat Appl) x0 x1))).(\lambda (H8: (subst0 i v0 v1 
+x0)).(\lambda (H9: (subst0 (s (Flat Appl) i) v0 (THead (Bind Abst) u t3) 
+x1)).(or3_ind (ex2 T (\lambda (u2: T).(eq T x1 (THead (Bind Abst) u2 t3))) 
+(\lambda (u2: T).(subst0 (s (Flat Appl) i) v0 u u2))) (ex2 T (\lambda (t5: 
+T).(eq T x1 (THead (Bind Abst) u t5))) (\lambda (t5: T).(subst0 (s (Bind 
+Abst) (s (Flat Appl) i)) v0 t3 t5))) (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t5: T).(eq T x1 (THead (Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
+T).(subst0 (s (Flat Appl) i) v0 u u2))) (\lambda (_: T).(\lambda (t5: 
+T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5)))) (or (pr0 w1 (THead 
+(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H10: (ex2 T 
+(\lambda (u2: T).(eq T x1 (THead (Bind Abst) u2 t3))) (\lambda (u2: 
+T).(subst0 (s (Flat Appl) i) v0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T x1 
+(THead (Bind Abst) u2 t3))) (\lambda (u2: T).(subst0 (s (Flat Appl) i) v0 u 
+u2)) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 
+w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda 
+(x: T).(\lambda (H11: (eq T x1 (THead (Bind Abst) x t3))).(\lambda (_: 
+(subst0 (s (Flat Appl) i) v0 u x)).(let H13 \def (eq_ind T x1 (\lambda (t: 
+T).(eq T w1 (THead (Flat Appl) x0 t))) H7 (THead (Bind Abst) x t3) H11) in 
+(eq_ind_r T (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) (\lambda (t: 
+T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 t w2)) 
+(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))))) (or_ind (pr0 
+x0 v2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 
+w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) (THead (Bind 
+Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind 
+Abst) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) 
+w2)))) (\lambda (H14: (pr0 x0 v2)).(or_introl (pr0 (THead (Flat Appl) x0 
+(THead (Bind Abst) x t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: 
+T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (pr0_beta x x0 v2 H14 t3 t4 
+H2))) (\lambda (H14: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: 
+T).(subst0 i v3 v2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda 
+(w2: T).(subst0 i v3 v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind 
+Abst) x t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead 
+(Flat Appl) x0 (THead (Bind Abst) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 
+(THead (Bind Abbr) v2 t4) w2)))) (\lambda (x2: T).(\lambda (H15: (pr0 x0 
+x2)).(\lambda (H16: (subst0 i v3 v2 x2)).(or_intror (pr0 (THead (Flat Appl) 
+x0 (THead (Bind Abst) x t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: 
+T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: 
+T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) x2 t4) 
+(pr0_beta x x0 x2 H15 t3 t4 H2) (subst0_fst v3 x2 v2 i H16 t4 (Bind 
+Abbr))))))) H14)) (H1 v0 x0 i H8 v3 H5)) w1 H13))))) H10)) (\lambda (H10: 
+(ex2 T (\lambda (t5: T).(eq T x1 (THead (Bind Abst) u t5))) (\lambda (t5: 
+T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5)))).(ex2_ind T (\lambda 
+(t5: T).(eq T x1 (THead (Bind Abst) u t5))) (\lambda (t5: T).(subst0 (s (Bind 
+Abst) (s (Flat Appl) i)) v0 t3 t5)) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) 
+(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead 
+(Bind Abbr) v2 t4) w2)))) (\lambda (x: T).(\lambda (H11: (eq T x1 (THead 
+(Bind Abst) u x))).(\lambda (H12: (subst0 (s (Bind Abst) (s (Flat Appl) i)) 
+v0 t3 x)).(let H13 \def (eq_ind T x1 (\lambda (t: T).(eq T w1 (THead (Flat 
+Appl) x0 t))) H7 (THead (Bind Abst) u x) H11) in (eq_ind_r T (THead (Flat 
+Appl) x0 (THead (Bind Abst) u x)) (\lambda (t: T).(or (pr0 t (THead (Bind 
+Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i 
+v3 (THead (Bind Abbr) v2 t4) w2))))) (or_ind (pr0 x t4) (ex2 T (\lambda (w2: 
+T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 
+t4 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind 
+Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind 
+Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) 
+w2)))) (\lambda (H14: (pr0 x t4)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: 
+T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat 
+Appl) x0 (THead (Bind Abst) u x)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda 
+(w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) w2)) (\lambda 
+(w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H15: (pr0 x0 
+v2)).(or_introl (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead 
+(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
+(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
+t4) w2))) (pr0_beta u x0 v2 H15 x t4 H14))) (\lambda (H15: (ex2 T (\lambda 
+(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)))).(ex2_ind T 
+(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)) (or (pr0 
+(THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind Abbr) v2 t4)) 
+(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) 
+w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda 
+(x2: T).(\lambda (H16: (pr0 x0 x2)).(\lambda (H17: (subst0 i v3 v2 
+x2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead 
+(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
+(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
+t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
+(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
+t4) w2)) (THead (Bind Abbr) x2 t4) (pr0_beta u x0 x2 H16 x t4 H14) 
+(subst0_fst v3 x2 v2 i H17 t4 (Bind Abbr))))))) H15)) (H1 v0 x0 i H8 v3 H5))) 
+(\lambda (H14: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 
+(s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda (w2: 
+T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 
+t4 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind 
+Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind 
+Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) 
+w2)))) (\lambda (x2: T).(\lambda (H15: (pr0 x x2)).(\lambda (H16: (subst0 (s 
+(Bind Abst) (s (Flat Appl) i)) v3 t4 x2)).(or_ind (pr0 x0 v2) (ex2 T (\lambda 
+(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead 
+(Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind Abbr) v2 t4)) (ex2 T 
+(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) w2)) 
+(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H17: 
+(pr0 x0 v2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) 
+(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 
+(THead (Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind 
+Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 
+(THead (Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind 
+Abbr) v2 t4) w2)) (THead (Bind Abbr) v2 x2) (pr0_beta u x0 v2 H17 x x2 H15) 
+(subst0_snd (Bind Abbr) v3 x2 t4 i H16 v2)))) (\lambda (H17: (ex2 T (\lambda 
+(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)))).(ex2_ind T 
+(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)) (or (pr0 
+(THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind Abbr) v2 t4)) 
+(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) 
+w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda 
+(x3: T).(\lambda (H18: (pr0 x0 x3)).(\lambda (H19: (subst0 i v3 v2 
+x3)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead 
+(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
+(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
+t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
+(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
+t4) w2)) (THead (Bind Abbr) x3 x2) (pr0_beta u x0 x3 H18 x x2 H15) 
+(subst0_both v3 v2 x3 i H19 (Bind Abbr) t4 x2 H16)))))) H17)) (H1 v0 x0 i H8 
+v3 H5))))) H14)) (H3 v0 x (s (Bind Abst) (s (Flat Appl) i)) H12 v3 H5)) w1 
+H13))))) H10)) (\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: 
+T).(eq T x1 (THead (Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
+T).(subst0 (s (Flat Appl) i) v0 u u2))) (\lambda (_: T).(\lambda (t5: 
+T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5))))).(ex3_2_ind T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T x1 (THead (Bind Abst) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u u2))) 
+(\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 
+t3 t5))) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 
+w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) 
+(\lambda (x2: T).(\lambda (x3: T).(\lambda (H11: (eq T x1 (THead (Bind Abst) 
+x2 x3))).(\lambda (_: (subst0 (s (Flat Appl) i) v0 u x2)).(\lambda (H13: 
+(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 x3)).(let H14 \def (eq_ind T 
+x1 (\lambda (t: T).(eq T w1 (THead (Flat Appl) x0 t))) H7 (THead (Bind Abst) 
+x2 x3) H11) in (eq_ind_r T (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) 
+(\lambda (t: T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: 
+T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) 
+w2))))) (or_ind (pr0 x3 t4) (ex2 T (\lambda (w2: T).(pr0 x3 w2)) (\lambda 
+(w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2))) (or (pr0 (THead 
+(Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead (Bind Abbr) v2 t4)) (ex2 T 
+(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) w2)) 
+(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H15: 
+(pr0 x3 t4)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) 
+(\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat Appl) x0 (THead 
+(Bind Abst) x2 x3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 
+(THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H16: (pr0 x0 
+v2)).(or_introl (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead 
+(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
+(Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
+t4) w2))) (pr0_beta x2 x0 v2 H16 x3 t4 H15))) (\lambda (H16: (ex2 T (\lambda 
+(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)))).(ex2_ind T 
+(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)) (or (pr0 
+(THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead (Bind Abbr) v2 t4)) 
+(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) 
+w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda 
+(x: T).(\lambda (H17: (pr0 x0 x)).(\lambda (H18: (subst0 i v3 v2 
+x)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead 
+(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
+(Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
+t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
+(Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
+t4) w2)) (THead (Bind Abbr) x t4) (pr0_beta x2 x0 x H17 x3 t4 H15) 
+(subst0_fst v3 x v2 i H18 t4 (Bind Abbr))))))) H16)) (H1 v0 x0 i H8 v3 H5))) 
+(\lambda (H15: (ex2 T (\lambda (w2: T).(pr0 x3 w2)) (\lambda (w2: T).(subst0 
+(s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda (w2: 
+T).(pr0 x3 w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 
+t4 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead 
+(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
+(Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
+t4) w2)))) (\lambda (x: T).(\lambda (H16: (pr0 x3 x)).(\lambda (H17: (subst0 
+(s (Bind Abst) (s (Flat Appl) i)) v3 t4 x)).(or_ind (pr0 x0 v2) (ex2 T 
+(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 
+(THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead (Bind Abbr) v2 t4)) 
+(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) 
+w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda 
+(H18: (pr0 x0 v2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) 
+x2 x3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat 
+Appl) x0 (THead (Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead 
+(Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat 
+Appl) x0 (THead (Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead 
+(Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) v2 x) (pr0_beta x2 x0 v2 H18 x3 x 
+H16) (subst0_snd (Bind Abbr) v3 x t4 i H17 v2)))) (\lambda (H18: (ex2 T 
+(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 
+w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 
+v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead 
+(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
+(Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
+t4) w2)))) (\lambda (x4: T).(\lambda (H19: (pr0 x0 x4)).(\lambda (H20: 
+(subst0 i v3 v2 x4)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) 
+x2 x3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat 
+Appl) x0 (THead (Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead 
+(Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat 
+Appl) x0 (THead (Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead 
+(Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) x4 x) (pr0_beta x2 x0 x4 H19 x3 x 
+H16) (subst0_both v3 v2 x4 i H20 (Bind Abbr) t4 x H17)))))) H18)) (H1 v0 x0 i 
+H8 v3 H5))))) H15)) (H3 v0 x3 (s (Bind Abst) (s (Flat Appl) i)) H13 v3 H5)) 
+w1 H14))))))) H10)) (subst0_gen_head (Bind Abst) v0 u t3 x1 (s (Flat Appl) i) 
+H9))))))) H6)) (subst0_gen_head (Flat Appl) v0 v1 (THead (Bind Abst) u t3) w1 
+i H4))))))))))))))))) (\lambda (b: B).(\lambda (H0: (not (eq B b 
+Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H1: (pr0 v1 v2)).(\lambda 
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+(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v4 v2 w2)))))))))))).(\lambda 
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+(\lambda (w2: T).(subst0 i v4 u2 w2)))))))))))).(\lambda (t3: T).(\lambda 
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+T).(subst0 i v4 t4 w2)))))))))))).(\lambda (v0: T).(\lambda (w1: T).(\lambda 
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+t3)) w1)).(\lambda (v3: T).(\lambda (H8: (pr0 v0 v3)).(or3_ind (ex2 T 
+(\lambda (u3: T).(eq T w1 (THead (Flat Appl) u3 (THead (Bind b) u1 t3)))) 
+(\lambda (u3: T).(subst0 i v0 v1 u3))) (ex2 T (\lambda (t5: T).(eq T w1 
+(THead (Flat Appl) v1 t5))) (\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 
+(THead (Bind b) u1 t3) t5))) (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq 
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+v0 v1 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 
+(THead (Bind b) u1 t3) t5)))) (or (pr0 w1 (THead (Bind b) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda 
+(w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
+t4)) w2)))) (\lambda (H9: (ex2 T (\lambda (u3: T).(eq T w1 (THead (Flat Appl) 
+u3 (THead (Bind b) u1 t3)))) (\lambda (u3: T).(subst0 i v0 v1 u3)))).(ex2_ind 
+T (\lambda (u3: T).(eq T w1 (THead (Flat Appl) u3 (THead (Bind b) u1 t3)))) 
+(\lambda (u3: T).(subst0 i v0 v1 u3)) (or (pr0 w1 (THead (Bind b) u2 (THead 
+(Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) 
+(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
+O) O v2) t4)) w2)))) (\lambda (x: T).(\lambda (H10: (eq T w1 (THead (Flat 
+Appl) x (THead (Bind b) u1 t3)))).(\lambda (H11: (subst0 i v0 v1 
+x)).(eq_ind_r T (THead (Flat Appl) x (THead (Bind b) u1 t3)) (\lambda (t: 
+T).(or (pr0 t (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) 
+(ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2))))) (or_ind (pr0 x 
+v2) (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v3 v2 
+w2))) (or (pr0 (THead (Flat Appl) x (THead (Bind b) u1 t3)) (THead (Bind b) 
+u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 
+(THead (Flat Appl) x (THead (Bind b) u1 t3)) w2)) (\lambda (w2: T).(subst0 i 
+v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) 
+(\lambda (H12: (pr0 x v2)).(or_introl (pr0 (THead (Flat Appl) x (THead (Bind 
+b) u1 t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 
+T (\lambda (w2: T).(pr0 (THead (Flat Appl) x (THead (Bind b) u1 t3)) w2)) 
+(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
+O) O v2) t4)) w2))) (pr0_upsilon b H0 x v2 H12 u1 u2 H3 t3 t4 H5))) (\lambda 
+(H12: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v3 v2 
+w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v3 
+v2 w2)) (or (pr0 (THead (Flat Appl) x (THead (Bind b) u1 t3)) (THead (Bind b) 
+u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 
+(THead (Flat Appl) x (THead (Bind b) u1 t3)) w2)) (\lambda (w2: T).(subst0 i 
+v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) 
+(\lambda (x0: T).(\lambda (H13: (pr0 x x0)).(\lambda (H14: (subst0 i v3 v2 
+x0)).(or_intror (pr0 (THead (Flat Appl) x (THead (Bind b) u1 t3)) (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
+T).(pr0 (THead (Flat Appl) x (THead (Bind b) u1 t3)) w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
+w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x (THead (Bind b) 
+u1 t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead (Flat Appl) (lift 
+(S O) O x0) t4)) (pr0_upsilon b H0 x x0 H13 u1 u2 H3 t3 t4 H5) (subst0_snd 
+(Bind b) v3 (THead (Flat Appl) (lift (S O) O x0) t4) (THead (Flat Appl) (lift 
+(S O) O v2) t4) i (subst0_fst v3 (lift (S O) O x0) (lift (S O) O v2) (s (Bind 
+b) i) (subst0_lift_ge_s v2 x0 v3 i H14 O (le_O_n i) b) t4 (Flat Appl)) 
+u2)))))) H12)) (H2 v0 x i H11 v3 H8)) w1 H10)))) H9)) (\lambda (H9: (ex2 T 
+(\lambda (t5: T).(eq T w1 (THead (Flat Appl) v1 t5))) (\lambda (t5: 
+T).(subst0 (s (Flat Appl) i) v0 (THead (Bind b) u1 t3) t5)))).(ex2_ind T 
+(\lambda (t5: T).(eq T w1 (THead (Flat Appl) v1 t5))) (\lambda (t5: 
+T).(subst0 (s (Flat Appl) i) v0 (THead (Bind b) u1 t3) t5)) (or (pr0 w1 
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda 
+(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead 
+(Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x: T).(\lambda (H10: (eq 
+T w1 (THead (Flat Appl) v1 x))).(\lambda (H11: (subst0 (s (Flat Appl) i) v0 
+(THead (Bind b) u1 t3) x)).(or3_ind (ex2 T (\lambda (u3: T).(eq T x (THead 
+(Bind b) u3 t3))) (\lambda (u3: T).(subst0 (s (Flat Appl) i) v0 u1 u3))) (ex2 
+T (\lambda (t5: T).(eq T x (THead (Bind b) u1 t5))) (\lambda (t5: T).(subst0 
+(s (Bind b) (s (Flat Appl) i)) v0 t3 t5))) (ex3_2 T T (\lambda (u3: 
+T).(\lambda (t5: T).(eq T x (THead (Bind b) u3 t5)))) (\lambda (u3: 
+T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 u3))) (\lambda (_: 
+T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5)))) (or 
+(pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T 
+(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) 
+u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H12: (ex2 T 
+(\lambda (u3: T).(eq T x (THead (Bind b) u3 t3))) (\lambda (u3: T).(subst0 (s 
+(Flat Appl) i) v0 u1 u3)))).(ex2_ind T (\lambda (u3: T).(eq T x (THead (Bind 
+b) u3 t3))) (\lambda (u3: T).(subst0 (s (Flat Appl) i) v0 u1 u3)) (or (pr0 w1 
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda 
+(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead 
+(Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x0: T).(\lambda (H13: (eq 
+T x (THead (Bind b) x0 t3))).(\lambda (H14: (subst0 (s (Flat Appl) i) v0 u1 
+x0)).(let H15 \def (eq_ind T x (\lambda (t: T).(eq T w1 (THead (Flat Appl) v1 
+t))) H10 (THead (Bind b) x0 t3) H13) in (eq_ind_r T (THead (Flat Appl) v1 
+(THead (Bind b) x0 t3)) (\lambda (t: T).(or (pr0 t (THead (Bind b) u2 (THead 
+(Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 t w2)) 
+(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
+O) O v2) t4)) w2))))) (or_ind (pr0 x0 u2) (ex2 T (\lambda (w2: T).(pr0 x0 
+w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 u2 w2))) (or (pr0 (THead 
+(Flat Appl) v1 (THead (Bind b) x0 t3)) (THead (Bind b) u2 (THead (Flat Appl) 
+(lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 
+(THead (Bind b) x0 t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 
+(THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H16: (pr0 x0 
+u2)).(or_introl (pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 t3)) (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
+T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 t3)) w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
+w2))) (pr0_upsilon b H0 v1 v2 H1 x0 u2 H16 t3 t4 H5))) (\lambda (H16: (ex2 T 
+(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 
+u2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 
+(s (Flat Appl) i) v3 u2 w2)) (or (pr0 (THead (Flat Appl) v1 (THead (Bind b) 
+x0 t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T 
+(\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 t3)) w2)) 
+(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
+O) O v2) t4)) w2)))) (\lambda (x1: T).(\lambda (H17: (pr0 x0 x1)).(\lambda 
+(H18: (subst0 (s (Flat Appl) i) v3 u2 x1)).(or_intror (pr0 (THead (Flat Appl) 
+v1 (THead (Bind b) x0 t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) 
+O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind 
+b) x0 t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead 
+(Flat Appl) v1 (THead (Bind b) x0 t3)) w2)) (\lambda (w2: T).(subst0 i v3 
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead 
+(Bind b) x1 (THead (Flat Appl) (lift (S O) O v2) t4)) (pr0_upsilon b H0 v1 v2 
+H1 x0 x1 H17 t3 t4 H5) (subst0_fst v3 x1 u2 i H18 (THead (Flat Appl) (lift (S 
+O) O v2) t4) (Bind b))))))) H16)) (H4 v0 x0 (s (Flat Appl) i) H14 v3 H8)) w1 
+H15))))) H12)) (\lambda (H12: (ex2 T (\lambda (t5: T).(eq T x (THead (Bind b) 
+u1 t5))) (\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 
+t5)))).(ex2_ind T (\lambda (t5: T).(eq T x (THead (Bind b) u1 t5))) (\lambda 
+(t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5)) (or (pr0 w1 (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
+T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x0: T).(\lambda (H13: (eq T x 
+(THead (Bind b) u1 x0))).(\lambda (H14: (subst0 (s (Bind b) (s (Flat Appl) 
+i)) v0 t3 x0)).(let H15 \def (eq_ind T x (\lambda (t: T).(eq T w1 (THead 
+(Flat Appl) v1 t))) H10 (THead (Bind b) u1 x0) H13) in (eq_ind_r T (THead 
+(Flat Appl) v1 (THead (Bind b) u1 x0)) (\lambda (t: T).(or (pr0 t (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
+T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t4)) w2))))) (or_ind (pr0 x0 t4) (ex2 T (\lambda (w2: 
+T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 
+w2))) (or (pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 x0)) (THead (Bind b) 
+u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 
+(THead (Flat Appl) v1 (THead (Bind b) u1 x0)) w2)) (\lambda (w2: T).(subst0 i 
+v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) 
+(\lambda (H16: (pr0 x0 t4)).(or_introl (pr0 (THead (Flat Appl) v1 (THead 
+(Bind b) u1 x0)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
+t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 
+x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) 
+(lift (S O) O v2) t4)) w2))) (pr0_upsilon b H0 v1 v2 H1 u1 u2 H3 x0 t4 H16))) 
+(\lambda (H16: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 
+(s (Bind b) (s (Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 
+x0 w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 w2)) 
+(or (pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 x0)) (THead (Bind b) u2 
+(THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 
+(THead (Flat Appl) v1 (THead (Bind b) u1 x0)) w2)) (\lambda (w2: T).(subst0 i 
+v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) 
+(\lambda (x1: T).(\lambda (H17: (pr0 x0 x1)).(\lambda (H18: (subst0 (s (Bind 
+b) (s (Flat Appl) i)) v3 t4 x1)).(or_intror (pr0 (THead (Flat Appl) v1 (THead 
+(Bind b) u1 x0)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
+t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 
+x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) 
+(lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat 
+Appl) v1 (THead (Bind b) u1 x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 
+(THead (Flat Appl) (lift (S O) O v2) x1)) (pr0_upsilon b H0 v1 v2 H1 u1 u2 H3 
+x0 x1 H17) (subst0_snd (Bind b) v3 (THead (Flat Appl) (lift (S O) O v2) x1) 
+(THead (Flat Appl) (lift (S O) O v2) t4) i (subst0_snd (Flat Appl) v3 x1 t4 
+(s (Bind b) i) H18 (lift (S O) O v2)) u2)))))) H16)) (H6 v0 x0 (s (Bind b) (s 
+(Flat Appl) i)) H14 v3 H8)) w1 H15))))) H12)) (\lambda (H12: (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t5: T).(eq T x (THead (Bind b) u3 t5)))) (\lambda 
+(u3: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 u3))) (\lambda (_: 
+T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 
+t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t5: T).(eq T x (THead (Bind 
+b) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 
+u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) 
+v0 t3 t5))) (or (pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O 
+v2) t4))) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda 
+(x0: T).(\lambda (x1: T).(\lambda (H13: (eq T x (THead (Bind b) x0 
+x1))).(\lambda (H14: (subst0 (s (Flat Appl) i) v0 u1 x0)).(\lambda (H15: 
+(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 x1)).(let H16 \def (eq_ind T x 
+(\lambda (t: T).(eq T w1 (THead (Flat Appl) v1 t))) H10 (THead (Bind b) x0 
+x1) H13) in (eq_ind_r T (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) 
+(\lambda (t: T).(or (pr0 t (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) 
+O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2))))) (or_ind 
+(pr0 x1 t4) (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s 
+(Bind b) (s (Flat Appl) i)) v3 t4 w2))) (or (pr0 (THead (Flat Appl) v1 (THead 
+(Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
+t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 
+x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) 
+(lift (S O) O v2) t4)) w2)))) (\lambda (H17: (pr0 x1 t4)).(or_ind (pr0 x0 u2) 
+(ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) 
+i) v3 u2 w2))) (or (pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
+T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
+w2)))) (\lambda (H18: (pr0 x0 u2)).(or_introl (pr0 (THead (Flat Appl) v1 
+(THead (Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O 
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+x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t4)) w2))) (pr0_upsilon b H0 v1 v2 H1 x0 u2 H18 x1 t4 
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+w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 u2 w2)) (or (pr0 (THead 
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+(THead (Bind b) x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 
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+x2)).(or_intror (pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) (THead 
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+T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
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+(S O) O v2) t4)) (pr0_upsilon b H0 v1 v2 H1 x0 x2 H19 x1 t4 H17) (subst0_fst 
+v3 x2 u2 i H20 (THead (Flat Appl) (lift (S O) O v2) t4) (Bind b))))))) H18)) 
+(H4 v0 x0 (s (Flat Appl) i) H14 v3 H8))) (\lambda (H17: (ex2 T (\lambda (w2: 
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+(Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
+t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 
+x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) 
+(lift (S O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda (H18: (pr0 x1 
+x2)).(\lambda (H19: (subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 x2)).(or_ind 
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+(Flat Appl) i) v3 u2 w2))) (or (pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 
+x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T 
+(\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) 
+(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
+O) O v2) t4)) w2)))) (\lambda (H20: (pr0 x0 u2)).(or_intror (pr0 (THead (Flat 
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+(Flat Appl) (lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 
+(THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: T).(subst0 i 
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+Appl) v3 x2 t4 (s (Bind b) i) H19 (lift (S O) O v2)) u2)))) (\lambda (H20: 
+(ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) 
+i) v3 u2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: 
+T).(subst0 (s (Flat Appl) i) v3 u2 w2)) (or (pr0 (THead (Flat Appl) v1 (THead 
+(Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
+t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 
+x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) 
+(lift (S O) O v2) t4)) w2)))) (\lambda (x3: T).(\lambda (H21: (pr0 x0 
+x3)).(\lambda (H22: (subst0 (s (Flat Appl) i) v3 u2 x3)).(or_intror (pr0 
+(THead (Flat Appl) v1 (THead (Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) 
+v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) 
+u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: 
+T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
+w2)) (THead (Bind b) x3 (THead (Flat Appl) (lift (S O) O v2) x2)) 
+(pr0_upsilon b H0 v1 v2 H1 x0 x3 H21 x1 x2 H18) (subst0_both v3 u2 x3 i H22 
+(Bind b) (THead (Flat Appl) (lift (S O) O v2) t4) (THead (Flat Appl) (lift (S 
+O) O v2) x2) (subst0_snd (Flat Appl) v3 x2 t4 (s (Bind b) i) H19 (lift (S O) 
+O v2)))))))) H20)) (H4 v0 x0 (s (Flat Appl) i) H14 v3 H8))))) H17)) (H6 v0 x1 
+(s (Bind b) (s (Flat Appl) i)) H15 v3 H8)) w1 H16))))))) H12)) 
+(subst0_gen_head (Bind b) v0 u1 t3 x (s (Flat Appl) i) H11))))) H9)) (\lambda 
+(H9: (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead (Flat Appl) 
+u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v0 v1 u3))) (\lambda (_: 
+T).(\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 (THead (Bind b) u1 t3) 
+t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead 
+(Flat Appl) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v0 v1 u3))) 
+(\lambda (_: T).(\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 (THead (Bind b) 
+u1 t3) t5))) (or (pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O 
+v2) t4))) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda 
+(x0: T).(\lambda (x1: T).(\lambda (H10: (eq T w1 (THead (Flat Appl) x0 
+x1))).(\lambda (H11: (subst0 i v0 v1 x0)).(\lambda (H12: (subst0 (s (Flat 
+Appl) i) v0 (THead (Bind b) u1 t3) x1)).(or3_ind (ex2 T (\lambda (u3: T).(eq 
+T x1 (THead (Bind b) u3 t3))) (\lambda (u3: T).(subst0 (s (Flat Appl) i) v0 
+u1 u3))) (ex2 T (\lambda (t5: T).(eq T x1 (THead (Bind b) u1 t5))) (\lambda 
+(t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5))) (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t5: T).(eq T x1 (THead (Bind b) u3 t5)))) (\lambda 
+(u3: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 u3))) (\lambda (_: 
+T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5)))) (or 
+(pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T 
+(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) 
+u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H13: (ex2 T 
+(\lambda (u3: T).(eq T x1 (THead (Bind b) u3 t3))) (\lambda (u3: T).(subst0 
+(s (Flat Appl) i) v0 u1 u3)))).(ex2_ind T (\lambda (u3: T).(eq T x1 (THead 
+(Bind b) u3 t3))) (\lambda (u3: T).(subst0 (s (Flat Appl) i) v0 u1 u3)) (or 
+(pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T 
+(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) 
+u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x: T).(\lambda 
+(H14: (eq T x1 (THead (Bind b) x t3))).(\lambda (H15: (subst0 (s (Flat Appl) 
+i) v0 u1 x)).(let H16 \def (eq_ind T x1 (\lambda (t: T).(eq T w1 (THead (Flat 
+Appl) x0 t))) H10 (THead (Bind b) x t3) H14) in (eq_ind_r T (THead (Flat 
+Appl) x0 (THead (Bind b) x t3)) (\lambda (t: T).(or (pr0 t (THead (Bind b) u2 
+(THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 t 
+w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) 
+(lift (S O) O v2) t4)) w2))))) (or_ind (pr0 x u2) (ex2 T (\lambda (w2: 
+T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 u2 w2))) (or 
+(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) (THead (Bind b) u2 (THead 
+(Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat 
+Appl) x0 (THead (Bind b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H17: 
+(pr0 x u2)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda 
+(w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) 
+x t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T 
+(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) 
+(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
+O) O v2) t4)) w2)))) (\lambda (H18: (pr0 x0 v2)).(or_introl (pr0 (THead (Flat 
+Appl) x0 (THead (Bind b) x t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift 
+(S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
+(Bind b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead 
+(Flat Appl) (lift (S O) O v2) t4)) w2))) (pr0_upsilon b H0 x0 v2 H18 x u2 H17 
+t3 t4 H5))) (\lambda (H18: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: 
+T).(subst0 i v3 v2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda 
+(w2: T).(subst0 i v3 v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x 
+t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T 
+(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) 
+(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
+O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda (H19: (pr0 x0 x2)).(\lambda 
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+b) x t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 
+T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) 
+(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
+O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 
+(THead (Bind b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 
+(THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead 
+(Flat Appl) (lift (S O) O x2) t4)) (pr0_upsilon b H0 x0 x2 H19 x u2 H17 t3 t4 
+H5) (subst0_snd (Bind b) v3 (THead (Flat Appl) (lift (S O) O x2) t4) (THead 
+(Flat Appl) (lift (S O) O v2) t4) i (subst0_fst v3 (lift (S O) O x2) (lift (S 
+O) O v2) (s (Bind b) i) (subst0_lift_ge_s v2 x2 v3 i H20 O (le_O_n i) b) t4 
+(Flat Appl)) u2)))))) H18)) (H2 v0 x0 i H11 v3 H8))) (\lambda (H17: (ex2 T 
+(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 u2 
+w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s 
+(Flat Appl) i) v3 u2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x 
+t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T 
+(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) 
+(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
+O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda (H18: (pr0 x x2)).(\lambda 
+(H19: (subst0 (s (Flat Appl) i) v3 u2 x2)).(or_ind (pr0 x0 v2) (ex2 T 
+(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 
+(THead (Flat Appl) x0 (THead (Bind b) x t3)) (THead (Bind b) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) 
+x0 (THead (Bind b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) 
+u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H20: (pr0 x0 
+v2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
+T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
+w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind 
+b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) x2 (THead (Flat Appl) (lift 
+(S O) O v2) t4)) (pr0_upsilon b H0 x0 v2 H20 x x2 H18 t3 t4 H5) (subst0_fst 
+v3 x2 u2 i H19 (THead (Flat Appl) (lift (S O) O v2) t4) (Bind b))))) (\lambda 
+(H20: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 
+w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 
+v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) (THead (Bind b) 
+u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 
+(THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) (\lambda (w2: T).(subst0 i 
+v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) 
+(\lambda (x3: T).(\lambda (H21: (pr0 x0 x3)).(\lambda (H22: (subst0 i v3 v2 
+x3)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
+T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
+w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind 
+b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) x2 (THead (Flat Appl) (lift 
+(S O) O x3) t4)) (pr0_upsilon b H0 x0 x3 H21 x x2 H18 t3 t4 H5) (subst0_both 
+v3 u2 x2 i H19 (Bind b) (THead (Flat Appl) (lift (S O) O v2) t4) (THead (Flat 
+Appl) (lift (S O) O x3) t4) (subst0_fst v3 (lift (S O) O x3) (lift (S O) O 
+v2) (s (Bind b) i) (subst0_lift_ge_s v2 x3 v3 i H22 O (le_O_n i) b) t4 (Flat 
+Appl)))))))) H20)) (H2 v0 x0 i H11 v3 H8))))) H17)) (H4 v0 x (s (Flat Appl) 
+i) H15 v3 H8)) w1 H16))))) H13)) (\lambda (H13: (ex2 T (\lambda (t5: T).(eq T 
+x1 (THead (Bind b) u1 t5))) (\lambda (t5: T).(subst0 (s (Bind b) (s (Flat 
+Appl) i)) v0 t3 t5)))).(ex2_ind T (\lambda (t5: T).(eq T x1 (THead (Bind b) 
+u1 t5))) (\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5)) 
+(or (pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) 
+(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x: 
+T).(\lambda (H14: (eq T x1 (THead (Bind b) u1 x))).(\lambda (H15: (subst0 (s 
+(Bind b) (s (Flat Appl) i)) v0 t3 x)).(let H16 \def (eq_ind T x1 (\lambda (t: 
+T).(eq T w1 (THead (Flat Appl) x0 t))) H10 (THead (Bind b) u1 x) H14) in 
+(eq_ind_r T (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (\lambda (t: T).(or 
+(pr0 t (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T 
+(\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 
+(THead (Flat Appl) (lift (S O) O v2) t4)) w2))))) (or_ind (pr0 x t4) (ex2 T 
+(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat 
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+T x1 (\lambda (t: T).(eq T w1 (THead (Flat Appl) x0 t))) H10 (THead (Bind b) 
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+(Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat 
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+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2))) (ex_intro2 T 
+(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) 
+(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
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+u2)))) (\lambda (H22: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: 
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+(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) 
+(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
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+T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) 
+(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
+O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 
+(THead (Bind b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 
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+(S O) O x4) (s (Bind b) i) (subst0_lift_ge_s v2 x4 v3 i H24 O (le_O_n i) b) 
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+(THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i 
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+(Flat Appl) i) H15 v3 H8))))) H18)) (H6 v0 x3 (s (Bind b) (s (Flat Appl) i)) 
+H16 v3 H8)) w1 H17))))))) H13)) (subst0_gen_head (Bind b) v0 u1 t3 x1 (s 
+(Flat Appl) i) H12))))))) H9)) (subst0_gen_head (Flat Appl) v0 v1 (THead 
+(Bind b) u1 t3) w1 i H7)))))))))))))))))))))) (\lambda (u1: T).(\lambda (u2: 
+T).(\lambda (H0: (pr0 u1 u2)).(\lambda (H1: ((\forall (v1: T).(\forall (w1: 
+T).(\forall (i: nat).((subst0 i v1 u1 w1) \to (\forall (v2: T).((pr0 v1 v2) 
+\to (or (pr0 w1 u2) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: 
+T).(subst0 i v2 u2 w2)))))))))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda 
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+(i: nat).((subst0 i v1 t3 w1) \to (\forall (v2: T).((pr0 v1 v2) \to (or (pr0 
+w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 
+w2)))))))))))).(\lambda (w: T).(\lambda (H4: (subst0 O u2 t4 w)).(\lambda 
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+(Bind Abbr) u1 t3) w1)).(\lambda (v2: T).(\lambda (H6: (pr0 v1 v2)).(or3_ind 
+(ex2 T (\lambda (u3: T).(eq T w1 (THead (Bind Abbr) u3 t3))) (\lambda (u3: 
+T).(subst0 i v1 u1 u3))) (ex2 T (\lambda (t5: T).(eq T w1 (THead (Bind Abbr) 
+u1 t5))) (\lambda (t5: T).(subst0 (s (Bind Abbr) i) v1 t3 t5))) (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead (Bind Abbr) u3 t5)))) 
+(\lambda (u3: T).(\lambda (_: T).(subst0 i v1 u1 u3))) (\lambda (_: 
+T).(\lambda (t5: T).(subst0 (s (Bind Abbr) i) v1 t3 t5)))) (or (pr0 w1 (THead 
+(Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: 
+T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (H7: (ex2 T (\lambda 
+(u3: T).(eq T w1 (THead (Bind Abbr) u3 t3))) (\lambda (u3: T).(subst0 i v1 u1 
+u3)))).(ex2_ind T (\lambda (u3: T).(eq T w1 (THead (Bind Abbr) u3 t3))) 
+(\lambda (u3: T).(subst0 i v1 u1 u3)) (or (pr0 w1 (THead (Bind Abbr) u2 w)) 
+(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 (THead 
+(Bind Abbr) u2 w) w2)))) (\lambda (x: T).(\lambda (H8: (eq T w1 (THead (Bind 
+Abbr) x t3))).(\lambda (H9: (subst0 i v1 u1 x)).(eq_ind_r T (THead (Bind 
+Abbr) x t3) (\lambda (t: T).(or (pr0 t (THead (Bind Abbr) u2 w)) (ex2 T 
+(\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) 
+u2 w) w2))))) (or_ind (pr0 x u2) (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda 
+(w2: T).(subst0 i v2 u2 w2))) (or (pr0 (THead (Bind Abbr) x t3) (THead (Bind 
+Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x t3) w2)) 
+(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (H10: 
+(pr0 x u2)).(or_introl (pr0 (THead (Bind Abbr) x t3) (THead (Bind Abbr) u2 
+w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x t3) w2)) (\lambda (w2: 
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+H4))) (\lambda (H10: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: 
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+(w2: T).(subst0 i v2 u2 w2)) (or (pr0 (THead (Bind Abbr) x t3) (THead (Bind 
+Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x t3) w2)) 
+(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x0: 
+T).(\lambda (H11: (pr0 x x0)).(\lambda (H12: (subst0 i v2 u2 x0)).(ex2_ind T 
+(\lambda (t: T).(subst0 O x0 t4 t)) (\lambda (t: T).(subst0 (S (plus i O)) v2 
+w t)) (or (pr0 (THead (Bind Abbr) x t3) (THead (Bind Abbr) u2 w)) (ex2 T 
+(\lambda (w2: T).(pr0 (THead (Bind Abbr) x t3) w2)) (\lambda (w2: T).(subst0 
+i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x1: T).(\lambda (H13: (subst0 
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+(f_equal nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O i))) in 
+(let H16 \def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n v2 w 
+x1)) H14 (S i) H15) in (or_intror (pr0 (THead (Bind Abbr) x t3) (THead (Bind 
+Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x t3) w2)) 
+(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))) (ex_intro2 T 
+(\lambda (w2: T).(pr0 (THead (Bind Abbr) x t3) w2)) (\lambda (w2: T).(subst0 
+i v2 (THead (Bind Abbr) u2 w) w2)) (THead (Bind Abbr) x0 x1) (pr0_delta x x0 
+H11 t3 t4 H2 x1 H13) (subst0_both v2 u2 x0 i H12 (Bind Abbr) w x1 H16)))))))) 
+(subst0_subst0_back t4 w u2 O H4 x0 v2 i H12))))) H10)) (H1 v1 x i H9 v2 H6)) 
+w1 H8)))) H7)) (\lambda (H7: (ex2 T (\lambda (t5: T).(eq T w1 (THead (Bind 
+Abbr) u1 t5))) (\lambda (t5: T).(subst0 (s (Bind Abbr) i) v1 t3 
+t5)))).(ex2_ind T (\lambda (t5: T).(eq T w1 (THead (Bind Abbr) u1 t5))) 
+(\lambda (t5: T).(subst0 (s (Bind Abbr) i) v1 t3 t5)) (or (pr0 w1 (THead 
+(Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: 
+T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x: T).(\lambda (H8: 
+(eq T w1 (THead (Bind Abbr) u1 x))).(\lambda (H9: (subst0 (s (Bind Abbr) i) 
+v1 t3 x)).(eq_ind_r T (THead (Bind Abbr) u1 x) (\lambda (t: T).(or (pr0 t 
+(THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: 
+T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))))) (or_ind (pr0 x t4) (ex2 T 
+(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind Abbr) i) v2 t4 
+w2))) (or (pr0 (THead (Bind Abbr) u1 x) (THead (Bind Abbr) u2 w)) (ex2 T 
+(\lambda (w2: T).(pr0 (THead (Bind Abbr) u1 x) w2)) (\lambda (w2: T).(subst0 
+i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (H10: (pr0 x t4)).(or_introl 
+(pr0 (THead (Bind Abbr) u1 x) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: 
+T).(pr0 (THead (Bind Abbr) u1 x) w2)) (\lambda (w2: T).(subst0 i v2 (THead 
+(Bind Abbr) u2 w) w2))) (pr0_delta u1 u2 H0 x t4 H10 w H4))) (\lambda (H10: 
+(ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind Abbr) 
+i) v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: 
+T).(subst0 (s (Bind Abbr) i) v2 t4 w2)) (or (pr0 (THead (Bind Abbr) u1 x) 
+(THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) u1 
+x) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) 
+(\lambda (x0: T).(\lambda (H11: (pr0 x x0)).(\lambda (H12: (subst0 (s (Bind 
+Abbr) i) v2 t4 x0)).(ex2_ind T (\lambda (t: T).(subst0 O u2 x0 t)) (\lambda 
+(t: T).(subst0 (s (Bind Abbr) i) v2 w t)) (or (pr0 (THead (Bind Abbr) u1 x) 
+(THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) u1 
+x) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) 
+(\lambda (x1: T).(\lambda (H13: (subst0 O u2 x0 x1)).(\lambda (H14: (subst0 
+(s (Bind Abbr) i) v2 w x1)).(or_intror (pr0 (THead (Bind Abbr) u1 x) (THead 
+(Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) u1 x) w2)) 
+(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))) (ex_intro2 T 
+(\lambda (w2: T).(pr0 (THead (Bind Abbr) u1 x) w2)) (\lambda (w2: T).(subst0 
+i v2 (THead (Bind Abbr) u2 w) w2)) (THead (Bind Abbr) u2 x1) (pr0_delta u1 u2 
+H0 x x0 H11 x1 H13) (subst0_snd (Bind Abbr) v2 x1 w i H14 u2)))))) 
+(subst0_confluence_neq t4 x0 v2 (s (Bind Abbr) i) H12 w u2 O H4 (sym_not_eq 
+nat O (S i) (O_S i))))))) H10)) (H3 v1 x (s (Bind Abbr) i) H9 v2 H6)) w1 
+H8)))) H7)) (\lambda (H7: (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T 
+w1 (THead (Bind Abbr) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v1 
+u1 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind Abbr) i) v1 t3 
+t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead 
+(Bind Abbr) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v1 u1 u3))) 
+(\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind Abbr) i) v1 t3 t5))) (or 
+(pr0 w1 (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) 
+(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x0: 
+T).(\lambda (x1: T).(\lambda (H8: (eq T w1 (THead (Bind Abbr) x0 
+x1))).(\lambda (H9: (subst0 i v1 u1 x0)).(\lambda (H10: (subst0 (s (Bind 
+Abbr) i) v1 t3 x1)).(eq_ind_r T (THead (Bind Abbr) x0 x1) (\lambda (t: T).(or 
+(pr0 t (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda 
+(w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))))) (or_ind (pr0 x1 t4) 
+(ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s (Bind Abbr) 
+i) v2 t4 w2))) (or (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) 
+(ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: 
+T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (H11: (pr0 x1 
+t4)).(or_ind (pr0 x0 u2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: 
+T).(subst0 i v2 u2 w2))) (or (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind 
+Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) 
+(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (H12: 
+(pr0 x0 u2)).(or_introl (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 
+w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: 
+T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))) (pr0_delta x0 u2 H12 x1 t4 H11 
+w H4))) (\lambda (H12: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: 
+T).(subst0 i v2 u2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda 
+(w2: T).(subst0 i v2 u2 w2)) (or (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind 
+Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) 
+(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x: 
+T).(\lambda (H13: (pr0 x0 x)).(\lambda (H14: (subst0 i v2 u2 x)).(ex2_ind T 
+(\lambda (t: T).(subst0 O x t4 t)) (\lambda (t: T).(subst0 (S (plus i O)) v2 
+w t)) (or (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T 
+(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 
+i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x2: T).(\lambda (H15: (subst0 
+O x t4 x2)).(\lambda (H16: (subst0 (S (plus i O)) v2 w x2)).(let H17 \def 
+(f_equal nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O i))) in 
+(let H18 \def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n v2 w 
+x2)) H16 (S i) H17) in (or_intror (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind 
+Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) 
+(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))) (ex_intro2 T 
+(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 
+i v2 (THead (Bind Abbr) u2 w) w2)) (THead (Bind Abbr) x x2) (pr0_delta x0 x 
+H13 x1 t4 H11 x2 H15) (subst0_both v2 u2 x i H14 (Bind Abbr) w x2 H18)))))))) 
+(subst0_subst0_back t4 w u2 O H4 x v2 i H14))))) H12)) (H1 v1 x0 i H9 v2 
+H6))) (\lambda (H11: (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: 
+T).(subst0 (s (Bind Abbr) i) v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x1 
+w2)) (\lambda (w2: T).(subst0 (s (Bind Abbr) i) v2 t4 w2)) (or (pr0 (THead 
+(Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 
+(THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind 
+Abbr) u2 w) w2)))) (\lambda (x: T).(\lambda (H12: (pr0 x1 x)).(\lambda (H13: 
+(subst0 (s (Bind Abbr) i) v2 t4 x)).(or_ind (pr0 x0 u2) (ex2 T (\lambda (w2: 
+T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 u2 w2))) (or (pr0 (THead (Bind 
+Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead 
+(Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 
+w) w2)))) (\lambda (H14: (pr0 x0 u2)).(ex2_ind T (\lambda (t: T).(subst0 O u2 
+x t)) (\lambda (t: T).(subst0 (s (Bind Abbr) i) v2 w t)) (or (pr0 (THead 
+(Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 
+(THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind 
+Abbr) u2 w) w2)))) (\lambda (x2: T).(\lambda (H15: (subst0 O u2 x 
+x2)).(\lambda (H16: (subst0 (s (Bind Abbr) i) v2 w x2)).(or_intror (pr0 
+(THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: 
+T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead 
+(Bind Abbr) u2 w) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) 
+x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)) 
+(THead (Bind Abbr) u2 x2) (pr0_delta x0 u2 H14 x1 x H12 x2 H15) (subst0_snd 
+(Bind Abbr) v2 x2 w i H16 u2)))))) (subst0_confluence_neq t4 x v2 (s (Bind 
+Abbr) i) H13 w u2 O H4 (sym_not_eq nat O (S i) (O_S i))))) (\lambda (H14: 
+(ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 u2 
+w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 
+u2 w2)) (or (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T 
+(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 
+i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x2: T).(\lambda (H15: (pr0 x0 
+x2)).(\lambda (H16: (subst0 i v2 u2 x2)).(ex2_ind T (\lambda (t: T).(subst0 O 
+x2 t4 t)) (\lambda (t: T).(subst0 (S (plus i O)) v2 w t)) (or (pr0 (THead 
+(Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 
+(THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind 
+Abbr) u2 w) w2)))) (\lambda (x3: T).(\lambda (H17: (subst0 O x2 t4 
+x3)).(\lambda (H18: (subst0 (S (plus i O)) v2 w x3)).(let H19 \def (f_equal 
+nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O i))) in (let H20 
+\def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n v2 w x3)) H18 (S 
+i) H19) in (ex2_ind T (\lambda (t: T).(subst0 (s (Bind Abbr) i) v2 x3 t)) 
+(\lambda (t: T).(subst0 O x2 x t)) (or (pr0 (THead (Bind Abbr) x0 x1) (THead 
+(Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) 
+w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda 
+(x4: T).(\lambda (H21: (subst0 (s (Bind Abbr) i) v2 x3 x4)).(\lambda (H22: 
+(subst0 O x2 x x4)).(or_intror (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind 
+Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) 
+(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))) (ex_intro2 T 
+(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 
+i v2 (THead (Bind Abbr) u2 w) w2)) (THead (Bind Abbr) x2 x4) (pr0_delta x0 x2 
+H15 x1 x H12 x4 H22) (subst0_both v2 u2 x2 i H16 (Bind Abbr) w x4 
+(subst0_trans x3 w v2 (s (Bind Abbr) i) H20 x4 H21))))))) 
+(subst0_confluence_neq t4 x3 x2 O H17 x v2 (s (Bind Abbr) i) H13 (O_S 
+i)))))))) (subst0_subst0_back t4 w u2 O H4 x2 v2 i H16))))) H14)) (H1 v1 x0 i 
+H9 v2 H6))))) H11)) (H3 v1 x1 (s (Bind Abbr) i) H10 v2 H6)) w1 H8)))))) H7)) 
+(subst0_gen_head (Bind Abbr) v1 u1 t3 w1 i H5)))))))))))))))))) (\lambda (b: 
+B).(\lambda (H0: (not (eq B b Abst))).(\lambda (t3: T).(\lambda (t4: 
+T).(\lambda (H1: (pr0 t3 t4)).(\lambda (H2: ((\forall (v1: T).(\forall (w1: 
+T).(\forall (i: nat).((subst0 i v1 t3 w1) \to (\forall (v2: T).((pr0 v1 v2) 
+\to (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: 
+T).(subst0 i v2 t4 w2)))))))))))).(\lambda (u: T).(\lambda (v1: T).(\lambda 
+(w1: T).(\lambda (i: nat).(\lambda (H3: (subst0 i v1 (THead (Bind b) u (lift 
+(S O) O t3)) w1)).(\lambda (v2: T).(\lambda (H4: (pr0 v1 v2)).(or3_ind (ex2 T 
+(\lambda (u2: T).(eq T w1 (THead (Bind b) u2 (lift (S O) O t3)))) (\lambda 
+(u2: T).(subst0 i v1 u u2))) (ex2 T (\lambda (t5: T).(eq T w1 (THead (Bind b) 
+u t5))) (\lambda (t5: T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5))) 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T w1 (THead (Bind b) u2 
+t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: 
+T).(\lambda (t5: T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5)))) (or 
+(pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i 
+v2 t4 w2)))) (\lambda (H5: (ex2 T (\lambda (u2: T).(eq T w1 (THead (Bind b) 
+u2 (lift (S O) O t3)))) (\lambda (u2: T).(subst0 i v1 u u2)))).(ex2_ind T 
+(\lambda (u2: T).(eq T w1 (THead (Bind b) u2 (lift (S O) O t3)))) (\lambda 
+(u2: T).(subst0 i v1 u u2)) (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 
+w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H6: 
+(eq T w1 (THead (Bind b) x (lift (S O) O t3)))).(\lambda (_: (subst0 i v1 u 
+x)).(eq_ind_r T (THead (Bind b) x (lift (S O) O t3)) (\lambda (t: T).(or (pr0 
+t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 
+w2))))) (or_introl (pr0 (THead (Bind b) x (lift (S O) O t3)) t4) (ex2 T 
+(\lambda (w2: T).(pr0 (THead (Bind b) x (lift (S O) O t3)) w2)) (\lambda (w2: 
+T).(subst0 i v2 t4 w2))) (pr0_zeta b H0 t3 t4 H1 x)) w1 H6)))) H5)) (\lambda 
+(H5: (ex2 T (\lambda (t5: T).(eq T w1 (THead (Bind b) u t5))) (\lambda (t5: 
+T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5)))).(ex2_ind T (\lambda 
+(t5: T).(eq T w1 (THead (Bind b) u t5))) (\lambda (t5: T).(subst0 (s (Bind b) 
+i) v1 (lift (S O) O t3) t5)) (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 
+w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H6: 
+(eq T w1 (THead (Bind b) u x))).(\lambda (H7: (subst0 (s (Bind b) i) v1 (lift 
+(S O) O t3) x)).(lt_le_e (s (Bind b) i) (S O) (or (pr0 w1 t4) (ex2 T (\lambda 
+(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (H8: 
+(lt (s (Bind b) i) (S O))).(subst0_gen_lift_false t3 v1 x (S O) O (s (Bind b) 
+i) (le_O_n (s (Bind b) i)) H8 H7 (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 
+w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))))) (\lambda (Hle: (le (S O) (s 
+(Bind b) i))).(let H_x \def (subst0_gen_lift_ge v1 t3 x (s (Bind b) i) (S O) 
+O H7 Hle) in (let H8 \def H_x in (ex2_ind T (\lambda (t5: T).(eq T x (lift (S 
+O) O t5))) (\lambda (t5: T).(subst0 (minus i O) v1 t3 t5)) (or (pr0 w1 t4) 
+(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) 
+(\lambda (x0: T).(\lambda (H9: (eq T x (lift (S O) O x0))).(\lambda (H10: 
+(subst0 (minus i O) v1 t3 x0)).(let H11 \def (eq_ind T x (\lambda (t: T).(eq 
+T w1 (THead (Bind b) u t))) H6 (lift (S O) O x0) H9) in (eq_ind_r T (THead 
+(Bind b) u (lift (S O) O x0)) (\lambda (t: T).(or (pr0 t t4) (ex2 T (\lambda 
+(w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))))) (let H12 \def 
+(eq_ind_r nat (minus i O) (\lambda (n: nat).(subst0 n v1 t3 x0)) H10 i 
+(minus_n_O i)) in (or_ind (pr0 x0 t4) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) 
+(\lambda (w2: T).(subst0 i v2 t4 w2))) (or (pr0 (THead (Bind b) u (lift (S O) 
+O x0)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) u (lift (S O) O x0)) 
+w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (H13: (pr0 x0 
+t4)).(or_introl (pr0 (THead (Bind b) u (lift (S O) O x0)) t4) (ex2 T (\lambda 
+(w2: T).(pr0 (THead (Bind b) u (lift (S O) O x0)) w2)) (\lambda (w2: 
+T).(subst0 i v2 t4 w2))) (pr0_zeta b H0 x0 t4 H13 u))) (\lambda (H13: (ex2 T 
+(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 t4 
+w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 
+t4 w2)) (or (pr0 (THead (Bind b) u (lift (S O) O x0)) t4) (ex2 T (\lambda 
+(w2: T).(pr0 (THead (Bind b) u (lift (S O) O x0)) w2)) (\lambda (w2: 
+T).(subst0 i v2 t4 w2)))) (\lambda (x1: T).(\lambda (H14: (pr0 x0 
+x1)).(\lambda (H15: (subst0 i v2 t4 x1)).(or_intror (pr0 (THead (Bind b) u 
+(lift (S O) O x0)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) u (lift 
+(S O) O x0)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (ex_intro2 T 
+(\lambda (w2: T).(pr0 (THead (Bind b) u (lift (S O) O x0)) w2)) (\lambda (w2: 
+T).(subst0 i v2 t4 w2)) x1 (pr0_zeta b H0 x0 x1 H14 u) H15))))) H13)) (H2 v1 
+x0 i H12 v2 H4))) w1 H11))))) H8)))))))) H5)) (\lambda (H5: (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T w1 (THead (Bind b) u2 t5)))) (\lambda 
+(u2: T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: 
+T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5))))).(ex3_2_ind T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T w1 (THead (Bind b) u2 t5)))) (\lambda 
+(u2: T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: 
+T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5))) (or (pr0 w1 t4) (ex2 T 
+(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) 
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (H6: (eq T w1 (THead (Bind b) x0 
+x1))).(\lambda (_: (subst0 i v1 u x0)).(\lambda (H8: (subst0 (s (Bind b) i) 
+v1 (lift (S O) O t3) x1)).(lt_le_e (s (Bind b) i) (S O) (or (pr0 w1 t4) (ex2 
+T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) 
+(\lambda (H9: (lt (s (Bind b) i) (S O))).(subst0_gen_lift_false t3 v1 x1 (S 
+O) O (s (Bind b) i) (le_O_n (s (Bind b) i)) H9 H8 (or (pr0 w1 t4) (ex2 T 
+(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))))) 
+(\lambda (Hle: (le (S O) (s (Bind b) i))).(let H_x \def (subst0_gen_lift_ge 
+v1 t3 x1 (s (Bind b) i) (S O) O H8 Hle) in (let H9 \def H_x in (ex2_ind T 
+(\lambda (t5: T).(eq T x1 (lift (S O) O t5))) (\lambda (t5: T).(subst0 (minus 
+i O) v1 t3 t5)) (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda 
+(w2: T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H10: (eq T x1 (lift 
+(S O) O x))).(\lambda (H11: (subst0 (minus i O) v1 t3 x)).(let H12 \def 
+(eq_ind T x1 (\lambda (t: T).(eq T w1 (THead (Bind b) x0 t))) H6 (lift (S O) 
+O x) H10) in (eq_ind_r T (THead (Bind b) x0 (lift (S O) O x)) (\lambda (t: 
+T).(or (pr0 t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: 
+T).(subst0 i v2 t4 w2))))) (let H13 \def (eq_ind_r nat (minus i O) (\lambda 
+(n: nat).(subst0 n v1 t3 x)) H11 i (minus_n_O i)) in (or_ind (pr0 x t4) (ex2 
+T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (or 
+(pr0 (THead (Bind b) x0 (lift (S O) O x)) t4) (ex2 T (\lambda (w2: T).(pr0 
+(THead (Bind b) x0 (lift (S O) O x)) w2)) (\lambda (w2: T).(subst0 i v2 t4 
+w2)))) (\lambda (H14: (pr0 x t4)).(or_introl (pr0 (THead (Bind b) x0 (lift (S 
+O) O x)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) x0 (lift (S O) O 
+x)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (pr0_zeta b H0 x t4 H14 x0))) 
+(\lambda (H14: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i 
+v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 
+i v2 t4 w2)) (or (pr0 (THead (Bind b) x0 (lift (S O) O x)) t4) (ex2 T 
+(\lambda (w2: T).(pr0 (THead (Bind b) x0 (lift (S O) O x)) w2)) (\lambda (w2: 
+T).(subst0 i v2 t4 w2)))) (\lambda (x2: T).(\lambda (H15: (pr0 x 
+x2)).(\lambda (H16: (subst0 i v2 t4 x2)).(or_intror (pr0 (THead (Bind b) x0 
+(lift (S O) O x)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) x0 (lift 
+(S O) O x)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (ex_intro2 T (\lambda 
+(w2: T).(pr0 (THead (Bind b) x0 (lift (S O) O x)) w2)) (\lambda (w2: 
+T).(subst0 i v2 t4 w2)) x2 (pr0_zeta b H0 x x2 H15 x0) H16))))) H14)) (H2 v1 
+x i H13 v2 H4))) w1 H12))))) H9)))))))))) H5)) (subst0_gen_head (Bind b) v1 u 
+(lift (S O) O t3) w1 i H3))))))))))))))) (\lambda (t3: T).(\lambda (t4: 
+T).(\lambda (H0: (pr0 t3 t4)).(\lambda (H1: ((\forall (v1: T).(\forall (w1: 
+T).(\forall (i: nat).((subst0 i v1 t3 w1) \to (\forall (v2: T).((pr0 v1 v2) 
+\to (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: 
+T).(subst0 i v2 t4 w2)))))))))))).(\lambda (u: T).(\lambda (v1: T).(\lambda 
+(w1: T).(\lambda (i: nat).(\lambda (H2: (subst0 i v1 (THead (Flat Cast) u t3) 
+w1)).(\lambda (v2: T).(\lambda (H3: (pr0 v1 v2)).(or3_ind (ex2 T (\lambda 
+(u2: T).(eq T w1 (THead (Flat Cast) u2 t3))) (\lambda (u2: T).(subst0 i v1 u 
+u2))) (ex2 T (\lambda (t5: T).(eq T w1 (THead (Flat Cast) u t5))) (\lambda 
+(t5: T).(subst0 (s (Flat Cast) i) v1 t3 t5))) (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T w1 (THead (Flat Cast) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: 
+T).(subst0 (s (Flat Cast) i) v1 t3 t5)))) (or (pr0 w1 t4) (ex2 T (\lambda 
+(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (H4: 
+(ex2 T (\lambda (u2: T).(eq T w1 (THead (Flat Cast) u2 t3))) (\lambda (u2: 
+T).(subst0 i v1 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T w1 (THead (Flat 
+Cast) u2 t3))) (\lambda (u2: T).(subst0 i v1 u u2)) (or (pr0 w1 t4) (ex2 T 
+(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) 
+(\lambda (x: T).(\lambda (H5: (eq T w1 (THead (Flat Cast) x t3))).(\lambda 
+(_: (subst0 i v1 u x)).(eq_ind_r T (THead (Flat Cast) x t3) (\lambda (t: 
+T).(or (pr0 t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: 
+T).(subst0 i v2 t4 w2))))) (or_introl (pr0 (THead (Flat Cast) x t3) t4) (ex2 
+T (\lambda (w2: T).(pr0 (THead (Flat Cast) x t3) w2)) (\lambda (w2: 
+T).(subst0 i v2 t4 w2))) (pr0_tau t3 t4 H0 x)) w1 H5)))) H4)) (\lambda (H4: 
+(ex2 T (\lambda (t5: T).(eq T w1 (THead (Flat Cast) u t5))) (\lambda (t5: 
+T).(subst0 (s (Flat Cast) i) v1 t3 t5)))).(ex2_ind T (\lambda (t5: T).(eq T 
+w1 (THead (Flat Cast) u t5))) (\lambda (t5: T).(subst0 (s (Flat Cast) i) v1 
+t3 t5)) (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: 
+T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H5: (eq T w1 (THead (Flat 
+Cast) u x))).(\lambda (H6: (subst0 (s (Flat Cast) i) v1 t3 x)).(eq_ind_r T 
+(THead (Flat Cast) u x) (\lambda (t: T).(or (pr0 t t4) (ex2 T (\lambda (w2: 
+T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))))) (or_ind (pr0 x t4) 
+(ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Cast) 
+i) v2 t4 w2))) (or (pr0 (THead (Flat Cast) u x) t4) (ex2 T (\lambda (w2: 
+T).(pr0 (THead (Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) 
+(\lambda (H7: (pr0 x t4)).(or_introl (pr0 (THead (Flat Cast) u x) t4) (ex2 T 
+(\lambda (w2: T).(pr0 (THead (Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i 
+v2 t4 w2))) (pr0_tau x t4 H7 u))) (\lambda (H7: (ex2 T (\lambda (w2: T).(pr0 
+x w2)) (\lambda (w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2)))).(ex2_ind T 
+(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Cast) i) v2 t4 
+w2)) (or (pr0 (THead (Flat Cast) u x) t4) (ex2 T (\lambda (w2: T).(pr0 (THead 
+(Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x0: 
+T).(\lambda (H8: (pr0 x x0)).(\lambda (H9: (subst0 (s (Flat Cast) i) v2 t4 
+x0)).(or_intror (pr0 (THead (Flat Cast) u x) t4) (ex2 T (\lambda (w2: T).(pr0 
+(THead (Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) 
+(ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) u x) w2)) (\lambda (w2: 
+T).(subst0 i v2 t4 w2)) x0 (pr0_tau x x0 H8 u) H9))))) H7)) (H1 v1 x (s (Flat 
+Cast) i) H6 v2 H3)) w1 H5)))) H4)) (\lambda (H4: (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T w1 (THead (Flat Cast) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: 
+T).(subst0 (s (Flat Cast) i) v1 t3 t5))))).(ex3_2_ind T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T w1 (THead (Flat Cast) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: 
+T).(subst0 (s (Flat Cast) i) v1 t3 t5))) (or (pr0 w1 t4) (ex2 T (\lambda (w2: 
+T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x0: 
+T).(\lambda (x1: T).(\lambda (H5: (eq T w1 (THead (Flat Cast) x0 
+x1))).(\lambda (_: (subst0 i v1 u x0)).(\lambda (H7: (subst0 (s (Flat Cast) 
+i) v1 t3 x1)).(eq_ind_r T (THead (Flat Cast) x0 x1) (\lambda (t: T).(or (pr0 
+t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 
+w2))))) (or_ind (pr0 x1 t4) (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda 
+(w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2))) (or (pr0 (THead (Flat Cast) x0 
+x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) x0 x1) w2)) (\lambda 
+(w2: T).(subst0 i v2 t4 w2)))) (\lambda (H8: (pr0 x1 t4)).(or_introl (pr0 
+(THead (Flat Cast) x0 x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) 
+x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (pr0_tau x1 t4 H8 x0))) 
+(\lambda (H8: (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 
+(s (Flat Cast) i) v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x1 w2)) 
+(\lambda (w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2)) (or (pr0 (THead (Flat 
+Cast) x0 x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) x0 x1) w2)) 
+(\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H9: (pr0 x1 
+x)).(\lambda (H10: (subst0 (s (Flat Cast) i) v2 t4 x)).(or_intror (pr0 (THead 
+(Flat Cast) x0 x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) x0 x1) 
+w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (ex_intro2 T (\lambda (w2: 
+T).(pr0 (THead (Flat Cast) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)) 
+x (pr0_tau x1 x H9 x0) H10))))) H8)) (H1 v1 x1 (s (Flat Cast) i) H7 v2 H3)) 
+w1 H5)))))) H4)) (subst0_gen_head (Flat Cast) v1 u t3 w1 i H2))))))))))))) t1 
+t2 H))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr0/subst1.ma b/matita/matita/contribs/lambdadelta/basic_1/pr0/subst1.ma
index 877f87f0173178efff7dd6b792787dd45e958cf9..5a67308c4aa025d1681612275a6ec09e78cdac6c 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_1/pr0/subst1.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/pr0/subst1.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
 
 (* This file was automatically generated: do not edit *********************)
 

-include "Basic-1/pr0/props.ma".
+include "basic_1/pr0/subst0.ma".

 
 

-include "Basic-1/subst1/defs.ma".
+include "basic_1/subst1/fwd.ma".

 
 theorem pr0_delta1:
  \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (t1: T).(\forall 
 
 theorem pr0_delta1:
  \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (t1: T).(\forall 


@@ -25,13 +25,13 @@ theorem pr0_delta1:
 \def
  \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr0 u1 u2)).(\lambda (t1: 
 T).(\lambda (t2: T).(\lambda (H0: (pr0 t1 t2)).(\lambda (w: T).(\lambda (H1: 
 \def
  \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr0 u1 u2)).(\lambda (t1: 
 T).(\lambda (t2: T).(\lambda (H0: (pr0 t1 t2)).(\lambda (w: T).(\lambda (H1: 

-(subst1 O u2 t2 w)).(subst1_ind O u2 t2 (\lambda (t: T).(pr0 (THead (Bind 
-Abbr) u1 t1) (THead (Bind Abbr) u2 t))) (pr0_comp u1 u2 H t1 t2 H0 (Bind 
-Abbr)) (\lambda (t0: T).(\lambda (H2: (subst0 O u2 t2 t0)).(pr0_delta u1 u2 H 
-t1 t2 H0 t0 H2))) w H1)))))))).
-(* COMMENTS
-Initial nodes: 115
-END *)
+(subst1 O u2 t2 w)).(let TMP_5 \def (\lambda (t: T).(let TMP_1 \def (Bind 
+Abbr) in (let TMP_2 \def (THead TMP_1 u1 t1) in (let TMP_3 \def (Bind Abbr) 
+in (let TMP_4 \def (THead TMP_3 u2 t) in (pr0 TMP_2 TMP_4)))))) in (let TMP_6 
+\def (Bind Abbr) in (let TMP_7 \def (pr0_comp u1 u2 H t1 t2 H0 TMP_6) in (let 
+TMP_8 \def (\lambda (t0: T).(\lambda (H2: (subst0 O u2 t2 t0)).(pr0_delta u1 
+u2 H t1 t2 H0 t0 H2))) in (subst1_ind O u2 t2 TMP_5 TMP_7 TMP_8 w 
+H1)))))))))))).

 
 theorem pr0_subst1_back:
  \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst1 
 
 theorem pr0_subst1_back:
  \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst1 


@@ -39,20 +39,24 @@ i u2 t1 t2) \to (\forall (u1: T).((pr0 u1 u2) \to (ex2 T (\lambda (t:
 T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t t2)))))))))
 \def
  \lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (i: nat).(\lambda 
 T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t t2)))))))))
 \def
  \lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (i: nat).(\lambda 

-(H: (subst1 i u2 t1 t2)).(subst1_ind i u2 t1 (\lambda (t: T).(\forall (u1: 
-T).((pr0 u1 u2) \to (ex2 T (\lambda (t0: T).(subst1 i u1 t1 t0)) (\lambda 
-(t0: T).(pr0 t0 t)))))) (\lambda (u1: T).(\lambda (_: (pr0 u1 u2)).(ex_intro2 
-T (\lambda (t: T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t t1)) t1 
-(subst1_refl i u1 t1) (pr0_refl t1)))) (\lambda (t0: T).(\lambda (H0: (subst0 
-i u2 t1 t0)).(\lambda (u1: T).(\lambda (H1: (pr0 u1 u2)).(ex2_ind T (\lambda 
-(t: T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t t0)) (ex2 T (\lambda (t: 
-T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t t0))) (\lambda (x: T).(\lambda 
-(H2: (subst0 i u1 t1 x)).(\lambda (H3: (pr0 x t0)).(ex_intro2 T (\lambda (t: 
-T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t t0)) x (subst1_single i u1 t1 x 
-H2) H3)))) (pr0_subst0_back u2 t1 t0 i H0 u1 H1)))))) t2 H))))).
-(* COMMENTS
-Initial nodes: 251
-END *)
+(H: (subst1 i u2 t1 t2)).(let TMP_3 \def (\lambda (t: T).(\forall (u1: 
+T).((pr0 u1 u2) \to (let TMP_1 \def (\lambda (t0: T).(subst1 i u1 t1 t0)) in 
+(let TMP_2 \def (\lambda (t0: T).(pr0 t0 t)) in (ex2 T TMP_1 TMP_2)))))) in 
+(let TMP_8 \def (\lambda (u1: T).(\lambda (_: (pr0 u1 u2)).(let TMP_4 \def 
+(\lambda (t: T).(subst1 i u1 t1 t)) in (let TMP_5 \def (\lambda (t: T).(pr0 t 
+t1)) in (let TMP_6 \def (subst1_refl i u1 t1) in (let TMP_7 \def (pr0_refl 
+t1) in (ex_intro2 T TMP_4 TMP_5 t1 TMP_6 TMP_7))))))) in (let TMP_19 \def 
+(\lambda (t0: T).(\lambda (H0: (subst0 i u2 t1 t0)).(\lambda (u1: T).(\lambda 
+(H1: (pr0 u1 u2)).(let TMP_9 \def (\lambda (t: T).(subst0 i u1 t1 t)) in (let 
+TMP_10 \def (\lambda (t: T).(pr0 t t0)) in (let TMP_11 \def (\lambda (t: 
+T).(subst1 i u1 t1 t)) in (let TMP_12 \def (\lambda (t: T).(pr0 t t0)) in 
+(let TMP_13 \def (ex2 T TMP_11 TMP_12) in (let TMP_17 \def (\lambda (x: 
+T).(\lambda (H2: (subst0 i u1 t1 x)).(\lambda (H3: (pr0 x t0)).(let TMP_14 
+\def (\lambda (t: T).(subst1 i u1 t1 t)) in (let TMP_15 \def (\lambda (t: 
+T).(pr0 t t0)) in (let TMP_16 \def (subst1_single i u1 t1 x H2) in (ex_intro2 
+T TMP_14 TMP_15 x TMP_16 H3))))))) in (let TMP_18 \def (pr0_subst0_back u2 t1 
+t0 i H0 u1 H1) in (ex2_ind T TMP_9 TMP_10 TMP_13 TMP_17 TMP_18)))))))))))) in 
+(subst1_ind i u2 t1 TMP_3 TMP_8 TMP_19 t2 H)))))))).

 
 theorem pr0_subst1_fwd:
  \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst1 
 
 theorem pr0_subst1_fwd:
  \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst1 


@@ -60,20 +64,24 @@ i u2 t1 t2) \to (\forall (u1: T).((pr0 u2 u1) \to (ex2 T (\lambda (t:
 T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t2 t)))))))))
 \def
  \lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (i: nat).(\lambda 
 T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t2 t)))))))))
 \def
  \lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (i: nat).(\lambda 

-(H: (subst1 i u2 t1 t2)).(subst1_ind i u2 t1 (\lambda (t: T).(\forall (u1: 
-T).((pr0 u2 u1) \to (ex2 T (\lambda (t0: T).(subst1 i u1 t1 t0)) (\lambda 
-(t0: T).(pr0 t t0)))))) (\lambda (u1: T).(\lambda (_: (pr0 u2 u1)).(ex_intro2 
-T (\lambda (t: T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t1 t)) t1 
-(subst1_refl i u1 t1) (pr0_refl t1)))) (\lambda (t0: T).(\lambda (H0: (subst0 
-i u2 t1 t0)).(\lambda (u1: T).(\lambda (H1: (pr0 u2 u1)).(ex2_ind T (\lambda 
-(t: T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t0 t)) (ex2 T (\lambda (t: 
-T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t0 t))) (\lambda (x: T).(\lambda 
-(H2: (subst0 i u1 t1 x)).(\lambda (H3: (pr0 t0 x)).(ex_intro2 T (\lambda (t: 
-T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t0 t)) x (subst1_single i u1 t1 x 
-H2) H3)))) (pr0_subst0_fwd u2 t1 t0 i H0 u1 H1)))))) t2 H))))).
-(* COMMENTS
-Initial nodes: 251
-END *)
+(H: (subst1 i u2 t1 t2)).(let TMP_3 \def (\lambda (t: T).(\forall (u1: 
+T).((pr0 u2 u1) \to (let TMP_1 \def (\lambda (t0: T).(subst1 i u1 t1 t0)) in 
+(let TMP_2 \def (\lambda (t0: T).(pr0 t t0)) in (ex2 T TMP_1 TMP_2)))))) in 
+(let TMP_8 \def (\lambda (u1: T).(\lambda (_: (pr0 u2 u1)).(let TMP_4 \def 
+(\lambda (t: T).(subst1 i u1 t1 t)) in (let TMP_5 \def (\lambda (t: T).(pr0 
+t1 t)) in (let TMP_6 \def (subst1_refl i u1 t1) in (let TMP_7 \def (pr0_refl 
+t1) in (ex_intro2 T TMP_4 TMP_5 t1 TMP_6 TMP_7))))))) in (let TMP_19 \def 
+(\lambda (t0: T).(\lambda (H0: (subst0 i u2 t1 t0)).(\lambda (u1: T).(\lambda 
+(H1: (pr0 u2 u1)).(let TMP_9 \def (\lambda (t: T).(subst0 i u1 t1 t)) in (let 
+TMP_10 \def (\lambda (t: T).(pr0 t0 t)) in (let TMP_11 \def (\lambda (t: 
+T).(subst1 i u1 t1 t)) in (let TMP_12 \def (\lambda (t: T).(pr0 t0 t)) in 
+(let TMP_13 \def (ex2 T TMP_11 TMP_12) in (let TMP_17 \def (\lambda (x: 
+T).(\lambda (H2: (subst0 i u1 t1 x)).(\lambda (H3: (pr0 t0 x)).(let TMP_14 
+\def (\lambda (t: T).(subst1 i u1 t1 t)) in (let TMP_15 \def (\lambda (t: 
+T).(pr0 t0 t)) in (let TMP_16 \def (subst1_single i u1 t1 x H2) in (ex_intro2 
+T TMP_14 TMP_15 x TMP_16 H3))))))) in (let TMP_18 \def (pr0_subst0_fwd u2 t1 
+t0 i H0 u1 H1) in (ex2_ind T TMP_9 TMP_10 TMP_13 TMP_17 TMP_18)))))))))))) in 
+(subst1_ind i u2 t1 TMP_3 TMP_8 TMP_19 t2 H)))))))).

 
 theorem pr0_subst1:
  \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (v1: T).(\forall 
 
 theorem pr0_subst1:
  \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (v1: T).(\forall 


@@ -82,24 +90,33 @@ v2) \to (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst1 i v2 t2
 w2)))))))))))
 \def
  \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(\lambda (v1: 
 w2)))))))))))
 \def
  \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(\lambda (v1: 

-T).(\lambda (w1: T).(\lambda (i: nat).(\lambda (H0: (subst1 i v1 t1 
-w1)).(subst1_ind i v1 t1 (\lambda (t: T).(\forall (v2: T).((pr0 v1 v2) \to 
-(ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst1 i v2 t2 w2)))))) 
-(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(ex_intro2 T (\lambda (w2: T).(pr0 
-t1 w2)) (\lambda (w2: T).(subst1 i v2 t2 w2)) t2 H (subst1_refl i v2 t2)))) 
-(\lambda (t0: T).(\lambda (H1: (subst0 i v1 t1 t0)).(\lambda (v2: T).(\lambda 
-(H2: (pr0 v1 v2)).(or_ind (pr0 t0 t2) (ex2 T (\lambda (w2: T).(pr0 t0 w2)) 
-(\lambda (w2: T).(subst0 i v2 t2 w2))) (ex2 T (\lambda (w2: T).(pr0 t0 w2)) 
-(\lambda (w2: T).(subst1 i v2 t2 w2))) (\lambda (H3: (pr0 t0 t2)).(ex_intro2 
-T (\lambda (w2: T).(pr0 t0 w2)) (\lambda (w2: T).(subst1 i v2 t2 w2)) t2 H3 
-(subst1_refl i v2 t2))) (\lambda (H3: (ex2 T (\lambda (w2: T).(pr0 t0 w2)) 
-(\lambda (w2: T).(subst0 i v2 t2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t0 
-w2)) (\lambda (w2: T).(subst0 i v2 t2 w2)) (ex2 T (\lambda (w2: T).(pr0 t0 
-w2)) (\lambda (w2: T).(subst1 i v2 t2 w2))) (\lambda (x: T).(\lambda (H4: 
-(pr0 t0 x)).(\lambda (H5: (subst0 i v2 t2 x)).(ex_intro2 T (\lambda (w2: 
-T).(pr0 t0 w2)) (\lambda (w2: T).(subst1 i v2 t2 w2)) x H4 (subst1_single i 
-v2 t2 x H5))))) H3)) (pr0_subst0 t1 t2 H v1 t0 i H1 v2 H2)))))) w1 H0))))))).
-(* COMMENTS
-Initial nodes: 385
-END *)
+T).(\lambda (w1: T).(\lambda (i: nat).(\lambda (H0: (subst1 i v1 t1 w1)).(let 
+TMP_3 \def (\lambda (t: T).(\forall (v2: T).((pr0 v1 v2) \to (let TMP_1 \def 
+(\lambda (w2: T).(pr0 t w2)) in (let TMP_2 \def (\lambda (w2: T).(subst1 i v2 
+t2 w2)) in (ex2 T TMP_1 TMP_2)))))) in (let TMP_7 \def (\lambda (v2: 
+T).(\lambda (_: (pr0 v1 v2)).(let TMP_4 \def (\lambda (w2: T).(pr0 t1 w2)) in 
+(let TMP_5 \def (\lambda (w2: T).(subst1 i v2 t2 w2)) in (let TMP_6 \def 
+(subst1_refl i v2 t2) in (ex_intro2 T TMP_4 TMP_5 t2 H TMP_6)))))) in (let 
+TMP_30 \def (\lambda (t0: T).(\lambda (H1: (subst0 i v1 t1 t0)).(\lambda (v2: 
+T).(\lambda (H2: (pr0 v1 v2)).(let TMP_8 \def (pr0 t0 t2) in (let TMP_9 \def 
+(\lambda (w2: T).(pr0 t0 w2)) in (let TMP_10 \def (\lambda (w2: T).(subst0 i 
+v2 t2 w2)) in (let TMP_11 \def (ex2 T TMP_9 TMP_10) in (let TMP_12 \def 
+(\lambda (w2: T).(pr0 t0 w2)) in (let TMP_13 \def (\lambda (w2: T).(subst1 i 
+v2 t2 w2)) in (let TMP_14 \def (ex2 T TMP_12 TMP_13) in (let TMP_18 \def 
+(\lambda (H3: (pr0 t0 t2)).(let TMP_15 \def (\lambda (w2: T).(pr0 t0 w2)) in 
+(let TMP_16 \def (\lambda (w2: T).(subst1 i v2 t2 w2)) in (let TMP_17 \def 
+(subst1_refl i v2 t2) in (ex_intro2 T TMP_15 TMP_16 t2 H3 TMP_17))))) in (let 
+TMP_28 \def (\lambda (H3: (ex2 T (\lambda (w2: T).(pr0 t0 w2)) (\lambda (w2: 
+T).(subst0 i v2 t2 w2)))).(let TMP_19 \def (\lambda (w2: T).(pr0 t0 w2)) in 
+(let TMP_20 \def (\lambda (w2: T).(subst0 i v2 t2 w2)) in (let TMP_21 \def 
+(\lambda (w2: T).(pr0 t0 w2)) in (let TMP_22 \def (\lambda (w2: T).(subst1 i 
+v2 t2 w2)) in (let TMP_23 \def (ex2 T TMP_21 TMP_22) in (let TMP_27 \def 
+(\lambda (x: T).(\lambda (H4: (pr0 t0 x)).(\lambda (H5: (subst0 i v2 t2 
+x)).(let TMP_24 \def (\lambda (w2: T).(pr0 t0 w2)) in (let TMP_25 \def 
+(\lambda (w2: T).(subst1 i v2 t2 w2)) in (let TMP_26 \def (subst1_single i v2 
+t2 x H5) in (ex_intro2 T TMP_24 TMP_25 x H4 TMP_26))))))) in (ex2_ind T 
+TMP_19 TMP_20 TMP_23 TMP_27 H3)))))))) in (let TMP_29 \def (pr0_subst0 t1 t2 
+H v1 t0 i H1 v2 H2) in (or_ind TMP_8 TMP_11 TMP_14 TMP_18 TMP_28 
+TMP_29))))))))))))))) in (subst1_ind i v1 t1 TMP_3 TMP_7 TMP_30 w1 
+H0)))))))))).