update in basic_2

[helm.git] / matita / matita / contribs / lambdadelta / basic_1 / pr0 / pr0.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr0/pr0.ma b/matita/matita/contribs/lambdadelta/basic_1/pr0/pr0.ma
index 9a3b397fe9f10f9dfd52c60c7bfb989a059d84c1..d3a8f78c97f1d1245ef812708682570d673845c5 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_1/pr0/pr0.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/pr0/pr0.ma
@@ -14,11 +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/subst0.ma".

 
 

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

 
 

-theorem pr0_confluence__pr0_cong_upsilon_refl:
+include "basic_1/tlt/fwd.ma".
+
+fact pr0_confluence__pr0_cong_upsilon_refl:

  \forall (b: B).((not (eq B b Abst)) \to (\forall (u0: T).(\forall (u3: 
 T).((pr0 u0 u3) \to (\forall (t4: T).(\forall (t5: T).((pr0 t4 t5) \to 
 (\forall (u2: T).(\forall (v2: T).(\forall (x: T).((pr0 u2 x) \to ((pr0 v2 x) 
  \forall (b: B).((not (eq B b Abst)) \to (\forall (u0: T).(\forall (u3: 
 T).((pr0 u0 u3) \to (\forall (t4: T).(\forall (t5: T).((pr0 t4 t5) \to 
 (\forall (u2: T).(\forall (v2: T).(\forall (x: T).((pr0 u2 x) \to ((pr0 v2 x) 


@@ -37,11 +39,8 @@ 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:
+fact pr0_confluence__pr0_cong_upsilon_cong:

  \forall (b: B).((not (eq B b Abst)) \to (\forall (u2: T).(\forall (v2: 
 T).(\forall (x: T).((pr0 u2 x) \to ((pr0 v2 x) \to (\forall (t2: T).(\forall 
 (t5: T).(\forall (x0: T).((pr0 t2 x0) \to ((pr0 t5 x0) \to (\forall (u5: 
  \forall (b: B).((not (eq B b Abst)) \to (\forall (u2: T).(\forall (v2: 
 T).(\forall (x: T).((pr0 u2 x) \to ((pr0 v2 x) \to (\forall (t2: T).(\forall 
 (t5: T).(\forall (x0: T).((pr0 t2 x0) \to ((pr0 t5 x0) \to (\forall (u5: 


@@ -62,11 +61,8 @@ 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:
+fact pr0_confluence__pr0_cong_upsilon_delta:

  (not (eq B Abbr Abst)) \to (\forall (u5: T).(\forall (t2: T).(\forall (w: 
 T).((subst0 O u5 t2 w) \to (\forall (u2: T).(\forall (v2: T).(\forall (x: 
 T).((pr0 u2 x) \to ((pr0 v2 x) \to (\forall (t5: T).(\forall (x0: T).((pr0 t2 
  (not (eq B Abbr Abst)) \to (\forall (u5: T).(\forall (t2: T).(\forall (w: 
 T).((subst0 O u5 t2 w) \to (\forall (u2: T).(\forall (v2: T).(\forall (x: 
 T).((pr0 u2 x) \to ((pr0 v2 x) \to (\forall (t5: T).(\forall (x0: T).((pr0 t2 


@@ -106,11 +102,8 @@ 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:
+fact pr0_confluence__pr0_cong_upsilon_zeta:

  \forall (b: B).((not (eq B b Abst)) \to (\forall (u0: T).(\forall (u3: 
 T).((pr0 u0 u3) \to (\forall (u2: T).(\forall (v2: T).(\forall (x0: T).((pr0 
 u2 x0) \to ((pr0 v2 x0) \to (\forall (x: T).(\forall (t3: T).(\forall (x1: 
  \forall (b: B).((not (eq B b Abst)) \to (\forall (u0: T).(\forall (u3: 
 T).((pr0 u0 u3) \to (\forall (u2: T).(\forall (v2: T).(\forall (x0: T).((pr0 
 u2 x0) \to ((pr0 v2 x0) \to (\forall (x: T).(\forall (t3: T).(\forall (x1: 


@@ -131,11 +124,8 @@ 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:
+fact pr0_confluence__pr0_cong_delta:

  \forall (u3: T).(\forall (t5: T).(\forall (w: T).((subst0 O u3 t5 w) \to 
 (\forall (u2: T).(\forall (x: T).((pr0 u2 x) \to ((pr0 u3 x) \to (\forall 
 (t3: T).(\forall (x0: T).((pr0 t3 x0) \to ((pr0 t5 x0) \to (ex2 T (\lambda 
  \forall (u3: T).(\forall (t5: T).(\forall (w: T).((subst0 O u3 t5 w) \to 
 (\forall (u2: T).(\forall (x: T).((pr0 u2 x) \to ((pr0 u3 x) \to (\forall 
 (t3: T).(\forall (x0: T).((pr0 t3 x0) \to ((pr0 t5 x0) \to (ex2 T (\lambda 


@@ -160,11 +150,8 @@ 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:
+fact pr0_confluence__pr0_upsilon_upsilon:

  \forall (b: B).((not (eq B b Abst)) \to (\forall (v1: T).(\forall (v2: 
 T).(\forall (x0: T).((pr0 v1 x0) \to ((pr0 v2 x0) \to (\forall (u1: 
 T).(\forall (u2: T).(\forall (x1: T).((pr0 u1 x1) \to ((pr0 u2 x1) \to 
  \forall (b: B).((not (eq B b Abst)) \to (\forall (v1: T).(\forall (v2: 
 T).(\forall (x0: T).((pr0 v1 x0) \to ((pr0 v2 x0) \to (\forall (u1: 
 T).(\forall (u2: T).(\forall (x1: T).((pr0 u1 x1) \to ((pr0 u2 x1) \to 


@@ -187,11 +174,8 @@ 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:
+fact pr0_confluence__pr0_delta_delta:

  \forall (u2: T).(\forall (t3: T).(\forall (w: T).((subst0 O u2 t3 w) \to 
 (\forall (u3: T).(\forall (t5: T).(\forall (w0: T).((subst0 O u3 t5 w0) \to 
 (\forall (x: T).((pr0 u2 x) \to ((pr0 u3 x) \to (\forall (x0: T).((pr0 t3 x0) 
  \forall (u2: T).(\forall (t3: T).(\forall (w: T).((subst0 O u2 t3 w) \to 
 (\forall (u3: T).(\forall (t5: T).(\forall (w0: T).((subst0 O u3 t5 w0) \to 
 (\forall (x: T).((pr0 u2 x) \to ((pr0 u3 x) \to (\forall (x0: T).((pr0 t3 x0) 


@@ -259,11 +243,8 @@ 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:
+fact pr0_confluence__pr0_delta_tau:

  \forall (u2: T).(\forall (t3: T).(\forall (w: T).((subst0 O u2 t3 w) \to 
 (\forall (t4: T).((pr0 (lift (S O) O t4) t3) \to (\forall (t2: T).(ex2 T 
 (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 t2 
  \forall (u2: T).(\forall (t3: T).(\forall (w: T).((subst0 O u2 t3 w) \to 
 (\forall (t4: T).((pr0 (lift (S O) O t4) t3) \to (\forall (t2: T).(ex2 T 
 (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 t2 


@@ -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).