(* 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)
(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:
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
(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:
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
(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
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)
(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
(\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)))))))).
-(* COMMENTS
-Initial nodes: 257
-END *)
theorem pr0_confluence:
\forall (t0: T).(\forall (t1: T).((pr0 t0 t1) \to (\forall (t2: T).((pr0 t0
(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
-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
(\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
-(\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
-(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
-(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
-(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:
-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
-(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) (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
-(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
-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
-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
-(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).(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
-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
-(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
-(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
-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
-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)
-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
-(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))
-(\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)
-\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:
-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:
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
-(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))
-(\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
-(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
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
-(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:
-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
-(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)
-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)
-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)
-(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
-(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)
-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
-(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
(\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
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)
(\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)
-(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
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)
(\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
(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)))
(_: (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
(\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
-(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
-\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)
-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
-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 _ _)
-\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).
projects / helm.git / blobdiff