X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_1%2Fty3%2Fpr3.ma;h=33f41ebe8f327abcdac98604bc35ffc91c75ff12;hb=57ae1762497a5f3ea75740e2908e04adb8642cc2;hp=8d184aa586604e8c2d0d853100107577c4edd182;hpb=88a68a9c334646bc17314d5327cd3b790202acd6;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/basic_1/ty3/pr3.ma b/matita/matita/contribs/lambdadelta/basic_1/ty3/pr3.ma
index 8d184aa58..33f41ebe8 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/ty3/pr3.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/ty3/pr3.ma
@@ -14,19 +14,19 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "Basic-1/csubt/ty3.ma".
+include "basic_1/csubt/ty3.ma".
 
-include "Basic-1/ty3/subst1.ma".
+include "basic_1/ty3/subst1.ma".
 
-include "Basic-1/ty3/fsubst0.ma".
+include "basic_1/ty3/fsubst0.ma".
 
-include "Basic-1/pc3/pc1.ma".
+include "basic_1/pc3/pc1.ma".
 
-include "Basic-1/pc3/wcpr0.ma".
+include "basic_1/pc3/wcpr0.ma".
 
-include "Basic-1/pc1/props.ma".
+include "basic_1/pc1/props.ma".
 
-theorem ty3_sred_wcpr0_pr0:
+lemma ty3_sred_wcpr0_pr0:
  \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t: T).((ty3 g c1 
 t1 t) \to (\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t2: T).((pr0 t1 t2) 
 \to (ty3 g c2 t2 t)))))))))
@@ -81,248 +81,225 @@ B).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H2: (ty3 g (CHead c (Bind b)
 u) t2 t3)).(\lambda (H3: ((\forall (c2: C).((wcpr0 (CHead c (Bind b) u) c2) 
 \to (\forall (t4: T).((pr0 t2 t4) \to (ty3 g c2 t4 t3))))))).(\lambda (c2: 
 C).(\lambda (H4: (wcpr0 c c2)).(\lambda (t4: T).(\lambda (H5: (pr0 (THead 
-(Bind b) u t2) t4)).(let H6 \def (match H5 in pr0 return (\lambda (t5: 
-T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 (THead (Bind b) u 
-t2)) \to ((eq T t6 t4) \to (ty3 g c2 t4 (THead (Bind b) u t3))))))) with 
-[(pr0_refl t5) \Rightarrow (\lambda (H6: (eq T t5 (THead (Bind b) u 
-t2))).(\lambda (H7: (eq T t5 t4)).(eq_ind T (THead (Bind b) u t2) (\lambda 
-(t6: T).((eq T t6 t4) \to (ty3 g c2 t4 (THead (Bind b) u t3)))) (\lambda (H8: 
-(eq T (THead (Bind b) u t2) t4)).(eq_ind T (THead (Bind b) u t2) (\lambda 
-(t6: T).(ty3 g c2 t6 (THead (Bind b) u t3))) (ty3_bind g c2 u t0 (H1 c2 H4 u 
-(pr0_refl u)) b t2 t3 (H3 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H4 u u 
-(pr0_refl u) (Bind b)) t2 (pr0_refl t2))) t4 H8)) t5 (sym_eq T t5 (THead 
-(Bind b) u t2) H6) H7))) | (pr0_comp u1 u2 H6 t5 t6 H7 k) \Rightarrow 
-(\lambda (H8: (eq T (THead k u1 t5) (THead (Bind b) u t2))).(\lambda (H9: (eq 
-T (THead k u2 t6) t4)).((let H10 \def (f_equal T T (\lambda (e: T).(match e 
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t5 | (TLRef _) 
-\Rightarrow t5 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t5) (THead 
-(Bind b) u t2) H8) in ((let H11 \def (f_equal T T (\lambda (e: T).(match e in 
-T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) 
-\Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t5) (THead 
-(Bind b) u t2) H8) in ((let H12 \def (f_equal T K (\lambda (e: T).(match e in 
-T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) 
-\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t5) (THead (Bind 
-b) u t2) H8) in (eq_ind K (Bind b) (\lambda (k0: K).((eq T u1 u) \to ((eq T 
-t5 t2) \to ((eq T (THead k0 u2 t6) t4) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to 
-(ty3 g c2 t4 (THead (Bind b) u t3)))))))) (\lambda (H13: (eq T u1 u)).(eq_ind 
-T u (\lambda (t7: T).((eq T t5 t2) \to ((eq T (THead (Bind b) u2 t6) t4) \to 
-((pr0 t7 u2) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3))))))) 
-(\lambda (H14: (eq T t5 t2)).(eq_ind T t2 (\lambda (t7: T).((eq T (THead 
-(Bind b) u2 t6) t4) \to ((pr0 u u2) \to ((pr0 t7 t6) \to (ty3 g c2 t4 (THead 
-(Bind b) u t3)))))) (\lambda (H15: (eq T (THead (Bind b) u2 t6) t4)).(eq_ind 
-T (THead (Bind b) u2 t6) (\lambda (t7: T).((pr0 u u2) \to ((pr0 t2 t6) \to 
-(ty3 g c2 t7 (THead (Bind b) u t3))))) (\lambda (H16: (pr0 u u2)).(\lambda 
-(H17: (pr0 t2 t6)).(ex_ind T (\lambda (t7: T).(ty3 g (CHead c2 (Bind b) u) t3 
-t7)) (ty3 g c2 (THead (Bind b) u2 t6) (THead (Bind b) u t3)) (\lambda (x: 
-T).(\lambda (H18: (ty3 g (CHead c2 (Bind b) u) t3 x)).(ex_ind T (\lambda (t7: 
-T).(ty3 g (CHead c2 (Bind b) u2) t3 t7)) (ty3 g c2 (THead (Bind b) u2 t6) 
-(THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (_: (ty3 g (CHead c2 (Bind 
-b) u2) t3 x0)).(ty3_conv g c2 (THead (Bind b) u t3) (THead (Bind b) u x) 
-(ty3_bind g c2 u t0 (H1 c2 H4 u (pr0_refl u)) b t3 x H18) (THead (Bind b) u2 
-t6) (THead (Bind b) u2 t3) (ty3_bind g c2 u2 t0 (H1 c2 H4 u2 H16) b t6 t3 (H3 
-(CHead c2 (Bind b) u2) (wcpr0_comp c c2 H4 u u2 H16 (Bind b)) t6 H17)) 
-(pc3_pr2_x c2 (THead (Bind b) u2 t3) (THead (Bind b) u t3) (pr2_head_1 c2 u 
-u2 (pr2_free c2 u u2 H16) (Bind b) t3))))) (ty3_correct g (CHead c2 (Bind b) 
-u2) t6 t3 (H3 (CHead c2 (Bind b) u2) (wcpr0_comp c c2 H4 u u2 H16 (Bind b)) 
-t6 H17))))) (ty3_correct g (CHead c2 (Bind b) u) t2 t3 (H3 (CHead c2 (Bind b) 
-u) (wcpr0_comp c c2 H4 u u (pr0_refl u) (Bind b)) t2 (pr0_refl t2)))))) t4 
-H15)) t5 (sym_eq T t5 t2 H14))) u1 (sym_eq T u1 u H13))) k (sym_eq K k (Bind 
-b) H12))) H11)) H10)) H9 H6 H7))) | (pr0_beta u0 v1 v2 H6 t5 t6 H7) 
-\Rightarrow (\lambda (H8: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u0 
-t5)) (THead (Bind b) u t2))).(\lambda (H9: (eq T (THead (Bind Abbr) v2 t6) 
-t4)).((let H10 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u0 
-t5)) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort 
-_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) 
-\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u t2) 
-H8) in (False_ind ((eq T (THead (Bind Abbr) v2 t6) t4) \to ((pr0 v1 v2) \to 
-((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3))))) H10)) H9 H6 H7))) | 
-(pr0_upsilon b0 H6 v1 v2 H7 u1 u2 H8 t5 t6 H9) \Rightarrow (\lambda (H10: (eq 
-T (THead (Flat Appl) v1 (THead (Bind b0) u1 t5)) (THead (Bind b) u 
-t2))).(\lambda (H11: (eq T (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) 
-O v2) t6)) t4)).((let H12 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind 
-b0) u1 t5)) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) 
-\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u t2) 
-H10) in (False_ind ((eq T (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) 
-O v2) t6)) t4) \to ((not (eq B b0 Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) 
-\to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3))))))) H12)) H11 H6 H7 
-H8 H9))) | (pr0_delta u1 u2 H6 t5 t6 H7 w H8) \Rightarrow (\lambda (H9: (eq T 
-(THead (Bind Abbr) u1 t5) (THead (Bind b) u t2))).(\lambda (H10: (eq T (THead 
-(Bind Abbr) u2 w) t4)).((let H11 \def (f_equal T T (\lambda (e: T).(match e 
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t5 | (TLRef _) 
-\Rightarrow t5 | (THead _ _ t7) \Rightarrow t7])) (THead (Bind Abbr) u1 t5) 
-(THead (Bind b) u t2) H9) in ((let H12 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | 
-(TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead (Bind 
-Abbr) u1 t5) (THead (Bind b) u t2) H9) in ((let H13 \def (f_equal T B 
-(\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) 
-\Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k _ _) \Rightarrow 
-(match k in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | 
-(Flat _) \Rightarrow Abbr])])) (THead (Bind Abbr) u1 t5) (THead (Bind b) u 
-t2) H9) in (eq_ind B Abbr (\lambda (b0: B).((eq T u1 u) \to ((eq T t5 t2) \to 
-((eq T (THead (Bind Abbr) u2 w) t4) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to 
-((subst0 O u2 t6 w) \to (ty3 g c2 t4 (THead (Bind b0) u t3))))))))) (\lambda 
-(H14: (eq T u1 u)).(eq_ind T u (\lambda (t7: T).((eq T t5 t2) \to ((eq T 
-(THead (Bind Abbr) u2 w) t4) \to ((pr0 t7 u2) \to ((pr0 t5 t6) \to ((subst0 O 
-u2 t6 w) \to (ty3 g c2 t4 (THead (Bind Abbr) u t3)))))))) (\lambda (H15: (eq 
-T t5 t2)).(eq_ind T t2 (\lambda (t7: T).((eq T (THead (Bind Abbr) u2 w) t4) 
-\to ((pr0 u u2) \to ((pr0 t7 t6) \to ((subst0 O u2 t6 w) \to (ty3 g c2 t4 
-(THead (Bind Abbr) u t3))))))) (\lambda (H16: (eq T (THead (Bind Abbr) u2 w) 
-t4)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t7: T).((pr0 u u2) \to 
-((pr0 t2 t6) \to ((subst0 O u2 t6 w) \to (ty3 g c2 t7 (THead (Bind Abbr) u 
-t3)))))) (\lambda (H17: (pr0 u u2)).(\lambda (H18: (pr0 t2 t6)).(\lambda 
-(H19: (subst0 O u2 t6 w)).(let H20 \def (eq_ind_r B b (\lambda (b0: 
-B).(\forall (c3: C).((wcpr0 (CHead c (Bind b0) u) c3) \to (\forall (t7: 
-T).((pr0 t2 t7) \to (ty3 g c3 t7 t3)))))) H3 Abbr H13) in (let H21 \def 
-(eq_ind_r B b (\lambda (b0: B).(ty3 g (CHead c (Bind b0) u) t2 t3)) H2 Abbr 
-H13) in (ex_ind T (\lambda (t7: T).(ty3 g (CHead c2 (Bind Abbr) u) t3 t7)) 
-(ty3 g c2 (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u t3)) (\lambda (x: 
-T).(\lambda (H22: (ty3 g (CHead c2 (Bind Abbr) u) t3 x)).(ex_ind T (\lambda 
-(t7: T).(ty3 g (CHead c2 (Bind Abbr) u2) t3 t7)) (ty3 g c2 (THead (Bind Abbr) 
-u2 w) (THead (Bind Abbr) u t3)) (\lambda (x0: T).(\lambda (_: (ty3 g (CHead 
-c2 (Bind Abbr) u2) t3 x0)).(ty3_conv g c2 (THead (Bind Abbr) u t3) (THead 
-(Bind Abbr) u x) (ty3_bind g c2 u t0 (H1 c2 H4 u (pr0_refl u)) Abbr t3 x H22) 
-(THead (Bind Abbr) u2 w) (THead (Bind Abbr) u2 t3) (ty3_bind g c2 u2 t0 (H1 
-c2 H4 u2 H17) Abbr w t3 (ty3_subst0 g (CHead c2 (Bind Abbr) u2) t6 t3 (H20 
-(CHead c2 (Bind Abbr) u2) (wcpr0_comp c c2 H4 u u2 H17 (Bind Abbr)) t6 H18) 
-c2 u2 O (getl_refl Abbr c2 u2) w H19)) (pc3_pr2_x c2 (THead (Bind Abbr) u2 
-t3) (THead (Bind Abbr) u t3) (pr2_head_1 c2 u u2 (pr2_free c2 u u2 H17) (Bind 
-Abbr) t3))))) (ty3_correct g (CHead c2 (Bind Abbr) u2) t6 t3 (H20 (CHead c2 
-(Bind Abbr) u2) (wcpr0_comp c c2 H4 u u2 H17 (Bind Abbr)) t6 H18))))) 
-(ty3_correct g (CHead c2 (Bind Abbr) u) t2 t3 (H20 (CHead c2 (Bind Abbr) u) 
-(wcpr0_comp c c2 H4 u u (pr0_refl u) (Bind Abbr)) t2 (pr0_refl t2))))))))) t4 
-H16)) t5 (sym_eq T t5 t2 H15))) u1 (sym_eq T u1 u H14))) b H13)) H12)) H11)) 
-H10 H6 H7 H8))) | (pr0_zeta b0 H6 t5 t6 H7 u0) \Rightarrow (\lambda (H8: (eq 
-T (THead (Bind b0) u0 (lift (S O) O t5)) (THead (Bind b) u t2))).(\lambda 
-(H9: (eq T t6 t4)).((let H10 \def (f_equal T T (\lambda (e: T).(match e in T 
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: 
-((nat \to nat))) (d: nat) (t7: T) on t7: T \def (match t7 with [(TSort n) 
-\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with 
-[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u1 t8) 
-\Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d) t8))]) in 
-lref_map) (\lambda (x: nat).(plus x (S O))) O t5) | (TLRef _) \Rightarrow 
-((let rec lref_map (f: ((nat \to nat))) (d: nat) (t7: T) on t7: T \def (match 
-t7 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef 
-(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | 
-(THead k u1 t8) \Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d) 
-t8))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t5) | (THead _ _ t7) 
-\Rightarrow t7])) (THead (Bind b0) u0 (lift (S O) O t5)) (THead (Bind b) u 
-t2) H8) in ((let H11 \def (f_equal T T (\lambda (e: T).(match e in T return 
-(\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 
-| (THead _ t7 _) \Rightarrow t7])) (THead (Bind b0) u0 (lift (S O) O t5)) 
-(THead (Bind b) u t2) H8) in ((let H12 \def (f_equal T B (\lambda (e: 
-T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b0 | 
-(TLRef _) \Rightarrow b0 | (THead k _ _) \Rightarrow (match k in K return 
-(\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow 
-b0])])) (THead (Bind b0) u0 (lift (S O) O t5)) (THead (Bind b) u t2) H8) in 
-(eq_ind B b (\lambda (b1: B).((eq T u0 u) \to ((eq T (lift (S O) O t5) t2) 
-\to ((eq T t6 t4) \to ((not (eq B b1 Abst)) \to ((pr0 t5 t6) \to (ty3 g c2 t4 
-(THead (Bind b) u t3)))))))) (\lambda (H13: (eq T u0 u)).(eq_ind T u (\lambda 
-(_: T).((eq T (lift (S O) O t5) t2) \to ((eq T t6 t4) \to ((not (eq B b 
-Abst)) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3))))))) (\lambda 
-(H14: (eq T (lift (S O) O t5) t2)).(eq_ind T (lift (S O) O t5) (\lambda (_: 
-T).((eq T t6 t4) \to ((not (eq B b Abst)) \to ((pr0 t5 t6) \to (ty3 g c2 t4 
-(THead (Bind b) u t3)))))) (\lambda (H15: (eq T t6 t4)).(eq_ind T t4 (\lambda 
-(t7: T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to (ty3 g c2 t4 (THead (Bind 
-b) u t3))))) (\lambda (H16: (not (eq B b Abst))).(\lambda (H17: (pr0 t5 
-t4)).(let H18 \def (eq_ind_r T t2 (\lambda (t7: T).(\forall (c3: C).((wcpr0 
-(CHead c (Bind b) u) c3) \to (\forall (t8: T).((pr0 t7 t8) \to (ty3 g c3 t8 
-t3)))))) H3 (lift (S O) O t5) H14) in (let H19 \def (eq_ind_r T t2 (\lambda 
-(t7: T).(ty3 g (CHead c (Bind b) u) t7 t3)) H2 (lift (S O) O t5) H14) in 
-(ex_ind T (\lambda (t7: T).(ty3 g (CHead c2 (Bind b) u) t3 t7)) (ty3 g c2 t4 
-(THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H20: (ty3 g (CHead c2 (Bind 
-b) u) t3 x)).(B_ind (\lambda (b1: B).((not (eq B b1 Abst)) \to ((ty3 g (CHead 
-c2 (Bind b1) u) t3 x) \to ((ty3 g (CHead c2 (Bind b1) u) (lift (S O) O t4) 
-t3) \to (ty3 g c2 t4 (THead (Bind b1) u t3)))))) (\lambda (H21: (not (eq B 
-Abbr Abst))).(\lambda (H22: (ty3 g (CHead c2 (Bind Abbr) u) t3 x)).(\lambda 
-(H23: (ty3 g (CHead c2 (Bind Abbr) u) (lift (S O) O t4) t3)).(let H24 \def 
-(ty3_gen_cabbr g (CHead c2 (Bind Abbr) u) (lift (S O) O t4) t3 H23 c2 u O 
-(getl_refl Abbr c2 u) (CHead c2 (Bind Abbr) u) (csubst1_refl O u (CHead c2 
-(Bind Abbr) u)) c2 (drop_drop (Bind Abbr) O c2 c2 (drop_refl c2) u)) in 
-(ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 O u (lift (S O) O t4) 
-(lift (S O) O y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 O u t3 (lift (S 
-O) O y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g c2 y1 y2))) (ty3 g c2 t4 
-(THead (Bind Abbr) u t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H25: 
-(subst1 O u (lift (S O) O t4) (lift (S O) O x0))).(\lambda (H26: (subst1 O u 
-t3 (lift (S O) O x1))).(\lambda (H27: (ty3 g c2 x0 x1)).(let H28 \def (eq_ind 
-T x0 (\lambda (t7: T).(ty3 g c2 t7 x1)) H27 t4 (lift_inj x0 t4 (S O) O 
-(subst1_gen_lift_eq t4 u (lift (S O) O x0) (S O) O O (le_n O) (eq_ind_r nat 
-(plus (S O) O) (\lambda (n: nat).(lt O n)) (le_n (plus (S O) O)) (plus O (S 
-O)) (plus_sym O (S O))) H25))) in (ty3_conv g c2 (THead (Bind Abbr) u t3) 
-(THead (Bind Abbr) u x) (ty3_bind g c2 u t0 (H1 c2 H4 u (pr0_refl u)) Abbr t3 
-x H22) t4 x1 H28 (pc3_pr3_x c2 x1 (THead (Bind Abbr) u t3) (pr3_t (THead 
-(Bind Abbr) u (lift (S O) O x1)) (THead (Bind Abbr) u t3) c2 (pr3_pr2 c2 
-(THead (Bind Abbr) u t3) (THead (Bind Abbr) u (lift (S O) O x1)) (pr2_free c2 
-(THead (Bind Abbr) u t3) (THead (Bind Abbr) u (lift (S O) O x1)) (pr0_delta1 
-u u (pr0_refl u) t3 t3 (pr0_refl t3) (lift (S O) O x1) H26))) x1 (pr3_pr2 c2 
-(THead (Bind Abbr) u (lift (S O) O x1)) x1 (pr2_free c2 (THead (Bind Abbr) u 
-(lift (S O) O x1)) x1 (pr0_zeta Abbr H21 x1 x1 (pr0_refl x1) u)))))))))))) 
-H24))))) (\lambda (H21: (not (eq B Abst Abst))).(\lambda (_: (ty3 g (CHead c2 
-(Bind Abst) u) t3 x)).(\lambda (_: (ty3 g (CHead c2 (Bind Abst) u) (lift (S 
-O) O t4) t3)).(let H24 \def (match (H21 (refl_equal B Abst)) in False return 
-(\lambda (_: False).(ty3 g c2 t4 (THead (Bind Abst) u t3))) with []) in 
-H24)))) (\lambda (H21: (not (eq B Void Abst))).(\lambda (H22: (ty3 g (CHead 
-c2 (Bind Void) u) t3 x)).(\lambda (H23: (ty3 g (CHead c2 (Bind Void) u) (lift 
-(S O) O t4) t3)).(let H24 \def (ty3_gen_cvoid g (CHead c2 (Bind Void) u) 
-(lift (S O) O t4) t3 H23 c2 u O (getl_refl Void c2 u) c2 (drop_drop (Bind 
-Void) O c2 c2 (drop_refl c2) u)) in (ex3_2_ind T T (\lambda (y1: T).(\lambda 
-(_: T).(eq T (lift (S O) O t4) (lift (S O) O y1)))) (\lambda (_: T).(\lambda 
-(y2: T).(eq T t3 (lift (S O) O y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 
-g c2 y1 y2))) (ty3 g c2 t4 (THead (Bind Void) u t3)) (\lambda (x0: 
-T).(\lambda (x1: T).(\lambda (H25: (eq T (lift (S O) O t4) (lift (S O) O 
-x0))).(\lambda (H26: (eq T t3 (lift (S O) O x1))).(\lambda (H27: (ty3 g c2 x0 
-x1)).(let H28 \def (eq_ind T t3 (\lambda (t7: T).(ty3 g (CHead c2 (Bind Void) 
-u) t7 x)) H22 (lift (S O) O x1) H26) in (eq_ind_r T (lift (S O) O x1) 
-(\lambda (t7: T).(ty3 g c2 t4 (THead (Bind Void) u t7))) (let H29 \def 
-(eq_ind_r T x0 (\lambda (t7: T).(ty3 g c2 t7 x1)) H27 t4 (lift_inj t4 x0 (S 
-O) O H25)) in (ty3_conv g c2 (THead (Bind Void) u (lift (S O) O x1)) (THead 
-(Bind Void) u x) (ty3_bind g c2 u t0 (H1 c2 H4 u (pr0_refl u)) Void (lift (S 
-O) O x1) x H28) t4 x1 H29 (pc3_s c2 x1 (THead (Bind Void) u (lift (S O) O 
-x1)) (pc3_pr2_r c2 (THead (Bind Void) u (lift (S O) O x1)) x1 (pr2_free c2 
-(THead (Bind Void) u (lift (S O) O x1)) x1 (pr0_zeta Void H21 x1 x1 (pr0_refl 
-x1) u)))))) t3 H26))))))) H24))))) b H16 H20 (H18 (CHead c2 (Bind b) u) 
-(wcpr0_comp c c2 H4 u u (pr0_refl u) (Bind b)) (lift (S O) O t4) (pr0_lift t5 
-t4 H17 (S O) O))))) (ty3_correct g (CHead c2 (Bind b) u) (lift (S O) O t4) t3 
-(H18 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H4 u u (pr0_refl u) (Bind b)) 
-(lift (S O) O t4) (pr0_lift t5 t4 H17 (S O) O)))))))) t6 (sym_eq T t6 t4 
-H15))) t2 H14)) u0 (sym_eq T u0 u H13))) b0 (sym_eq B b0 b H12))) H11)) H10)) 
-H9 H6 H7))) | (pr0_tau t5 t6 H6 u0) \Rightarrow (\lambda (H7: (eq T (THead 
-(Flat Cast) u0 t5) (THead (Bind b) u t2))).(\lambda (H8: (eq T t6 t4)).((let 
-H9 \def (eq_ind T (THead (Flat Cast) u0 t5) (\lambda (e: T).(match e in T 
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda 
-(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow 
-True])])) I (THead (Bind b) u t2) H7) in (False_ind ((eq T t6 t4) \to ((pr0 
-t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3)))) H9)) H8 H6)))]) in (H6 
-(refl_equal T (THead (Bind b) u t2)) (refl_equal T t4))))))))))))))))) 
-(\lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c w 
-u)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 
-w t2) \to (ty3 g c2 t2 u))))))).(\lambda (v: T).(\lambda (t0: T).(\lambda 
-(H2: (ty3 g c v (THead (Bind Abst) u t0))).(\lambda (H3: ((\forall (c2: 
-C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 v t2) \to (ty3 g c2 t2 (THead 
-(Bind Abst) u t0)))))))).(\lambda (c2: C).(\lambda (H4: (wcpr0 c 
-c2)).(\lambda (t2: T).(\lambda (H5: (pr0 (THead (Flat Appl) w v) t2)).(let H6 
-\def (match H5 in pr0 return (\lambda (t3: T).(\lambda (t4: T).(\lambda (_: 
-(pr0 t3 t4)).((eq T t3 (THead (Flat Appl) w v)) \to ((eq T t4 t2) \to (ty3 g 
-c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0)))))))) with [(pr0_refl 
-t3) \Rightarrow (\lambda (H6: (eq T t3 (THead (Flat Appl) w v))).(\lambda 
-(H7: (eq T t3 t2)).(eq_ind T (THead (Flat Appl) w v) (\lambda (t4: T).((eq T 
-t4 t2) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0))))) 
-(\lambda (H8: (eq T (THead (Flat Appl) w v) t2)).(eq_ind T (THead (Flat Appl) 
-w v) (\lambda (t4: T).(ty3 g c2 t4 (THead (Flat Appl) w (THead (Bind Abst) u 
-t0)))) (ty3_appl g c2 w u (H1 c2 H4 w (pr0_refl w)) v t0 (H3 c2 H4 v 
-(pr0_refl v))) t2 H8)) t3 (sym_eq T t3 (THead (Flat Appl) w v) H6) H7))) | 
-(pr0_comp u1 u2 H6 t3 t4 H7 k) \Rightarrow (\lambda (H8: (eq T (THead k u1 
-t3) (THead (Flat Appl) w v))).(\lambda (H9: (eq T (THead k u2 t4) t2)).((let 
-H10 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) 
-with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t5) 
-\Rightarrow t5])) (THead k u1 t3) (THead (Flat Appl) w v) H8) in ((let H11 
-\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) 
-with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t5 _) 
-\Rightarrow t5])) (THead k u1 t3) (THead (Flat Appl) w v) H8) in ((let H12 
-\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) 
+(Bind b) u t2) t4)).(let H6 \def (match H5 with [(pr0_refl t5) \Rightarrow 
+(\lambda (H6: (eq T t5 (THead (Bind b) u t2))).(\lambda (H7: (eq T t5 
+t4)).(eq_ind T (THead (Bind b) u t2) (\lambda (t6: T).((eq T t6 t4) \to (ty3 
+g c2 t4 (THead (Bind b) u t3)))) (\lambda (H8: (eq T (THead (Bind b) u t2) 
+t4)).(eq_ind T (THead (Bind b) u t2) (\lambda (t6: T).(ty3 g c2 t6 (THead 
+(Bind b) u t3))) (ty3_bind g c2 u t0 (H1 c2 H4 u (pr0_refl u)) b t2 t3 (H3 
+(CHead c2 (Bind b) u) (wcpr0_comp c c2 H4 u u (pr0_refl u) (Bind b)) t2 
+(pr0_refl t2))) t4 H8)) t5 (sym_eq T t5 (THead (Bind b) u t2) H6) H7))) | 
+(pr0_comp u1 u2 H6 t5 t6 H7 k) \Rightarrow (\lambda (H8: (eq T (THead k u1 
+t5) (THead (Bind b) u t2))).(\lambda (H9: (eq T (THead k u2 t6) t4)).((let 
+H10 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t5 
+| (TLRef _) \Rightarrow t5 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t5) 
+(THead (Bind b) u t2) H8) in ((let H11 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | 
+(THead _ t7 _) \Rightarrow t7])) (THead k u1 t5) (THead (Bind b) u t2) H8) in 
+((let H12 \def (f_equal T K (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) 
+(THead k u1 t5) (THead (Bind b) u t2) H8) in (eq_ind K (Bind b) (\lambda (k0: 
+K).((eq T u1 u) \to ((eq T t5 t2) \to ((eq T (THead k0 u2 t6) t4) \to ((pr0 
+u1 u2) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3)))))))) 
+(\lambda (H13: (eq T u1 u)).(eq_ind T u (\lambda (t7: T).((eq T t5 t2) \to 
+((eq T (THead (Bind b) u2 t6) t4) \to ((pr0 t7 u2) \to ((pr0 t5 t6) \to (ty3 
+g c2 t4 (THead (Bind b) u t3))))))) (\lambda (H14: (eq T t5 t2)).(eq_ind T t2 
+(\lambda (t7: T).((eq T (THead (Bind b) u2 t6) t4) \to ((pr0 u u2) \to ((pr0 
+t7 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3)))))) (\lambda (H15: (eq T 
+(THead (Bind b) u2 t6) t4)).(eq_ind T (THead (Bind b) u2 t6) (\lambda (t7: 
+T).((pr0 u u2) \to ((pr0 t2 t6) \to (ty3 g c2 t7 (THead (Bind b) u t3))))) 
+(\lambda (H16: (pr0 u u2)).(\lambda (H17: (pr0 t2 t6)).(ex_ind T (\lambda 
+(t7: T).(ty3 g (CHead c2 (Bind b) u) t3 t7)) (ty3 g c2 (THead (Bind b) u2 t6) 
+(THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H18: (ty3 g (CHead c2 (Bind 
+b) u) t3 x)).(ex_ind T (\lambda (t7: T).(ty3 g (CHead c2 (Bind b) u2) t3 t7)) 
+(ty3 g c2 (THead (Bind b) u2 t6) (THead (Bind b) u t3)) (\lambda (x0: 
+T).(\lambda (_: (ty3 g (CHead c2 (Bind b) u2) t3 x0)).(ty3_conv g c2 (THead 
+(Bind b) u t3) (THead (Bind b) u x) (ty3_bind g c2 u t0 (H1 c2 H4 u (pr0_refl 
+u)) b t3 x H18) (THead (Bind b) u2 t6) (THead (Bind b) u2 t3) (ty3_bind g c2 
+u2 t0 (H1 c2 H4 u2 H16) b t6 t3 (H3 (CHead c2 (Bind b) u2) (wcpr0_comp c c2 
+H4 u u2 H16 (Bind b)) t6 H17)) (pc3_pr2_x c2 (THead (Bind b) u2 t3) (THead 
+(Bind b) u t3) (pr2_head_1 c2 u u2 (pr2_free c2 u u2 H16) (Bind b) t3))))) 
+(ty3_correct g (CHead c2 (Bind b) u2) t6 t3 (H3 (CHead c2 (Bind b) u2) 
+(wcpr0_comp c c2 H4 u u2 H16 (Bind b)) t6 H17))))) (ty3_correct g (CHead c2 
+(Bind b) u) t2 t3 (H3 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H4 u u (pr0_refl 
+u) (Bind b)) t2 (pr0_refl t2)))))) t4 H15)) t5 (sym_eq T t5 t2 H14))) u1 
+(sym_eq T u1 u H13))) k (sym_eq K k (Bind b) H12))) H11)) H10)) H9 H6 H7))) | 
+(pr0_beta u0 v1 v2 H6 t5 t6 H7) \Rightarrow (\lambda (H8: (eq T (THead (Flat 
+Appl) v1 (THead (Bind Abst) u0 t5)) (THead (Bind b) u t2))).(\lambda (H9: (eq 
+T (THead (Bind Abbr) v2 t6) t4)).((let H10 \def (eq_ind T (THead (Flat Appl) 
+v1 (THead (Bind Abst) u0 t5)) (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
+(match k with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I 
+(THead (Bind b) u t2) H8) in (False_ind ((eq T (THead (Bind Abbr) v2 t6) t4) 
+\to ((pr0 v1 v2) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3))))) 
+H10)) H9 H6 H7))) | (pr0_upsilon b0 H6 v1 v2 H7 u1 u2 H8 t5 t6 H9) 
+\Rightarrow (\lambda (H10: (eq T (THead (Flat Appl) v1 (THead (Bind b0) u1 
+t5)) (THead (Bind b) u t2))).(\lambda (H11: (eq T (THead (Bind b0) u2 (THead 
+(Flat Appl) (lift (S O) O v2) t6)) t4)).((let H12 \def (eq_ind T (THead (Flat 
+Appl) v1 (THead (Bind b0) u1 t5)) (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
+(match k with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I 
+(THead (Bind b) u t2) H10) in (False_ind ((eq T (THead (Bind b0) u2 (THead 
+(Flat Appl) (lift (S O) O v2) t6)) t4) \to ((not (eq B b0 Abst)) \to ((pr0 v1 
+v2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u 
+t3))))))) H12)) H11 H6 H7 H8 H9))) | (pr0_delta u1 u2 H6 t5 t6 H7 w H8) 
+\Rightarrow (\lambda (H9: (eq T (THead (Bind Abbr) u1 t5) (THead (Bind b) u 
+t2))).(\lambda (H10: (eq T (THead (Bind Abbr) u2 w) t4)).((let H11 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t5 | (TLRef 
+_) \Rightarrow t5 | (THead _ _ t7) \Rightarrow t7])) (THead (Bind Abbr) u1 
+t5) (THead (Bind b) u t2) H9) in ((let H12 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | 
+(THead _ t7 _) \Rightarrow t7])) (THead (Bind Abbr) u1 t5) (THead (Bind b) u 
+t2) H9) in ((let H13 \def (f_equal T B (\lambda (e: T).(match e with [(TSort 
+_) \Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k _ _) \Rightarrow 
+(match k with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) 
+(THead (Bind Abbr) u1 t5) (THead (Bind b) u t2) H9) in (eq_ind B Abbr 
+(\lambda (b0: B).((eq T u1 u) \to ((eq T t5 t2) \to ((eq T (THead (Bind Abbr) 
+u2 w) t4) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ty3 
+g c2 t4 (THead (Bind b0) u t3))))))))) (\lambda (H14: (eq T u1 u)).(eq_ind T 
+u (\lambda (t7: T).((eq T t5 t2) \to ((eq T (THead (Bind Abbr) u2 w) t4) \to 
+((pr0 t7 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ty3 g c2 t4 (THead 
+(Bind Abbr) u t3)))))))) (\lambda (H15: (eq T t5 t2)).(eq_ind T t2 (\lambda 
+(t7: T).((eq T (THead (Bind Abbr) u2 w) t4) \to ((pr0 u u2) \to ((pr0 t7 t6) 
+\to ((subst0 O u2 t6 w) \to (ty3 g c2 t4 (THead (Bind Abbr) u t3))))))) 
+(\lambda (H16: (eq T (THead (Bind Abbr) u2 w) t4)).(eq_ind T (THead (Bind 
+Abbr) u2 w) (\lambda (t7: T).((pr0 u u2) \to ((pr0 t2 t6) \to ((subst0 O u2 
+t6 w) \to (ty3 g c2 t7 (THead (Bind Abbr) u t3)))))) (\lambda (H17: (pr0 u 
+u2)).(\lambda (H18: (pr0 t2 t6)).(\lambda (H19: (subst0 O u2 t6 w)).(let H20 
+\def (eq_ind_r B b (\lambda (b0: B).(\forall (c3: C).((wcpr0 (CHead c (Bind 
+b0) u) c3) \to (\forall (t7: T).((pr0 t2 t7) \to (ty3 g c3 t7 t3)))))) H3 
+Abbr H13) in (let H21 \def (eq_ind_r B b (\lambda (b0: B).(ty3 g (CHead c 
+(Bind b0) u) t2 t3)) H2 Abbr H13) in (ex_ind T (\lambda (t7: T).(ty3 g (CHead 
+c2 (Bind Abbr) u) t3 t7)) (ty3 g c2 (THead (Bind Abbr) u2 w) (THead (Bind 
+Abbr) u t3)) (\lambda (x: T).(\lambda (H22: (ty3 g (CHead c2 (Bind Abbr) u) 
+t3 x)).(ex_ind T (\lambda (t7: T).(ty3 g (CHead c2 (Bind Abbr) u2) t3 t7)) 
+(ty3 g c2 (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u t3)) (\lambda (x0: 
+T).(\lambda (_: (ty3 g (CHead c2 (Bind Abbr) u2) t3 x0)).(ty3_conv g c2 
+(THead (Bind Abbr) u t3) (THead (Bind Abbr) u x) (ty3_bind g c2 u t0 (H1 c2 
+H4 u (pr0_refl u)) Abbr t3 x H22) (THead (Bind Abbr) u2 w) (THead (Bind Abbr) 
+u2 t3) (ty3_bind g c2 u2 t0 (H1 c2 H4 u2 H17) Abbr w t3 (ty3_subst0 g (CHead 
+c2 (Bind Abbr) u2) t6 t3 (H20 (CHead c2 (Bind Abbr) u2) (wcpr0_comp c c2 H4 u 
+u2 H17 (Bind Abbr)) t6 H18) c2 u2 O (getl_refl Abbr c2 u2) w H19)) (pc3_pr2_x 
+c2 (THead (Bind Abbr) u2 t3) (THead (Bind Abbr) u t3) (pr2_head_1 c2 u u2 
+(pr2_free c2 u u2 H17) (Bind Abbr) t3))))) (ty3_correct g (CHead c2 (Bind 
+Abbr) u2) t6 t3 (H20 (CHead c2 (Bind Abbr) u2) (wcpr0_comp c c2 H4 u u2 H17 
+(Bind Abbr)) t6 H18))))) (ty3_correct g (CHead c2 (Bind Abbr) u) t2 t3 (H20 
+(CHead c2 (Bind Abbr) u) (wcpr0_comp c c2 H4 u u (pr0_refl u) (Bind Abbr)) t2 
+(pr0_refl t2))))))))) t4 H16)) t5 (sym_eq T t5 t2 H15))) u1 (sym_eq T u1 u 
+H14))) b H13)) H12)) H11)) H10 H6 H7 H8))) | (pr0_zeta b0 H6 t5 t6 H7 u0) 
+\Rightarrow (\lambda (H8: (eq T (THead (Bind b0) u0 (lift (S O) O t5)) (THead 
+(Bind b) u t2))).(\lambda (H9: (eq T t6 t4)).((let H10 \def (f_equal T T 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow (lref_map (\lambda (x: 
+nat).(plus x (S O))) O t5) | (TLRef _) \Rightarrow (lref_map (\lambda (x: 
+nat).(plus x (S O))) O t5) | (THead _ _ t7) \Rightarrow t7])) (THead (Bind 
+b0) u0 (lift (S O) O t5)) (THead (Bind b) u t2) H8) in ((let H11 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u0 | (TLRef 
+_) \Rightarrow u0 | (THead _ t7 _) \Rightarrow t7])) (THead (Bind b0) u0 
+(lift (S O) O t5)) (THead (Bind b) u t2) H8) in ((let H12 \def (f_equal T B 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow b0 | (TLRef _) 
+\Rightarrow b0 | (THead k _ _) \Rightarrow (match k with [(Bind b1) 
+\Rightarrow b1 | (Flat _) \Rightarrow b0])])) (THead (Bind b0) u0 (lift (S O) 
+O t5)) (THead (Bind b) u t2) H8) in (eq_ind B b (\lambda (b1: B).((eq T u0 u) 
+\to ((eq T (lift (S O) O t5) t2) \to ((eq T t6 t4) \to ((not (eq B b1 Abst)) 
+\to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3)))))))) (\lambda (H13: 
+(eq T u0 u)).(eq_ind T u (\lambda (_: T).((eq T (lift (S O) O t5) t2) \to 
+((eq T t6 t4) \to ((not (eq B b Abst)) \to ((pr0 t5 t6) \to (ty3 g c2 t4 
+(THead (Bind b) u t3))))))) (\lambda (H14: (eq T (lift (S O) O t5) 
+t2)).(eq_ind T (lift (S O) O t5) (\lambda (_: T).((eq T t6 t4) \to ((not (eq 
+B b Abst)) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3)))))) 
+(\lambda (H15: (eq T t6 t4)).(eq_ind T t4 (\lambda (t7: T).((not (eq B b 
+Abst)) \to ((pr0 t5 t7) \to (ty3 g c2 t4 (THead (Bind b) u t3))))) (\lambda 
+(H16: (not (eq B b Abst))).(\lambda (H17: (pr0 t5 t4)).(let H18 \def 
+(eq_ind_r T t2 (\lambda (t7: T).(\forall (c3: C).((wcpr0 (CHead c (Bind b) u) 
+c3) \to (\forall (t8: T).((pr0 t7 t8) \to (ty3 g c3 t8 t3)))))) H3 (lift (S 
+O) O t5) H14) in (let H19 \def (eq_ind_r T t2 (\lambda (t7: T).(ty3 g (CHead 
+c (Bind b) u) t7 t3)) H2 (lift (S O) O t5) H14) in (ex_ind T (\lambda (t7: 
+T).(ty3 g (CHead c2 (Bind b) u) t3 t7)) (ty3 g c2 t4 (THead (Bind b) u t3)) 
+(\lambda (x: T).(\lambda (H20: (ty3 g (CHead c2 (Bind b) u) t3 x)).(B_ind 
+(\lambda (b1: B).((not (eq B b1 Abst)) \to ((ty3 g (CHead c2 (Bind b1) u) t3 
+x) \to ((ty3 g (CHead c2 (Bind b1) u) (lift (S O) O t4) t3) \to (ty3 g c2 t4 
+(THead (Bind b1) u t3)))))) (\lambda (H21: (not (eq B Abbr Abst))).(\lambda 
+(H22: (ty3 g (CHead c2 (Bind Abbr) u) t3 x)).(\lambda (H23: (ty3 g (CHead c2 
+(Bind Abbr) u) (lift (S O) O t4) t3)).(let H24 \def (ty3_gen_cabbr g (CHead 
+c2 (Bind Abbr) u) (lift (S O) O t4) t3 H23 c2 u O (getl_refl Abbr c2 u) 
+(CHead c2 (Bind Abbr) u) (csubst1_refl O u (CHead c2 (Bind Abbr) u)) c2 
+(drop_drop (Bind Abbr) O c2 c2 (drop_refl c2) u)) in (ex3_2_ind T T (\lambda 
+(y1: T).(\lambda (_: T).(subst1 O u (lift (S O) O t4) (lift (S O) O y1)))) 
+(\lambda (_: T).(\lambda (y2: T).(subst1 O u t3 (lift (S O) O y2)))) (\lambda 
+(y1: T).(\lambda (y2: T).(ty3 g c2 y1 y2))) (ty3 g c2 t4 (THead (Bind Abbr) u 
+t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H25: (subst1 O u (lift (S O) 
+O t4) (lift (S O) O x0))).(\lambda (H26: (subst1 O u t3 (lift (S O) O 
+x1))).(\lambda (H27: (ty3 g c2 x0 x1)).(let H28 \def (eq_ind T x0 (\lambda 
+(t7: T).(ty3 g c2 t7 x1)) H27 t4 (lift_inj x0 t4 (S O) O (subst1_gen_lift_eq 
+t4 u (lift (S O) O x0) (S O) O O (le_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))) H25))) in (ty3_conv g c2 (THead (Bind Abbr) u t3) (THead (Bind Abbr) 
+u x) (ty3_bind g c2 u t0 (H1 c2 H4 u (pr0_refl u)) Abbr t3 x H22) t4 x1 H28 
+(pc3_pr3_x c2 x1 (THead (Bind Abbr) u t3) (pr3_t (THead (Bind Abbr) u (lift 
+(S O) O x1)) (THead (Bind Abbr) u t3) c2 (pr3_pr2 c2 (THead (Bind Abbr) u t3) 
+(THead (Bind Abbr) u (lift (S O) O x1)) (pr2_free c2 (THead (Bind Abbr) u t3) 
+(THead (Bind Abbr) u (lift (S O) O x1)) (pr0_delta1 u u (pr0_refl u) t3 t3 
+(pr0_refl t3) (lift (S O) O x1) H26))) x1 (pr3_pr2 c2 (THead (Bind Abbr) u 
+(lift (S O) O x1)) x1 (pr2_free c2 (THead (Bind Abbr) u (lift (S O) O x1)) x1 
+(pr0_zeta Abbr H21 x1 x1 (pr0_refl x1) u)))))))))))) H24))))) (\lambda (H21: 
+(not (eq B Abst Abst))).(\lambda (_: (ty3 g (CHead c2 (Bind Abst) u) t3 
+x)).(\lambda (_: (ty3 g (CHead c2 (Bind Abst) u) (lift (S O) O t4) t3)).(let 
+H24 \def (match (H21 (refl_equal B Abst)) in False with []) in H24)))) 
+(\lambda (H21: (not (eq B Void Abst))).(\lambda (H22: (ty3 g (CHead c2 (Bind 
+Void) u) t3 x)).(\lambda (H23: (ty3 g (CHead c2 (Bind Void) u) (lift (S O) O 
+t4) t3)).(let H24 \def (ty3_gen_cvoid g (CHead c2 (Bind Void) u) (lift (S O) 
+O t4) t3 H23 c2 u O (getl_refl Void c2 u) c2 (drop_drop (Bind Void) O c2 c2 
+(drop_refl c2) u)) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T 
+(lift (S O) O t4) (lift (S O) O y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T 
+t3 (lift (S O) O y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g c2 y1 y2))) 
+(ty3 g c2 t4 (THead (Bind Void) u t3)) (\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (H25: (eq T (lift (S O) O t4) (lift (S O) O x0))).(\lambda (H26: 
+(eq T t3 (lift (S O) O x1))).(\lambda (H27: (ty3 g c2 x0 x1)).(let H28 \def 
+(eq_ind T t3 (\lambda (t7: T).(ty3 g (CHead c2 (Bind Void) u) t7 x)) H22 
+(lift (S O) O x1) H26) in (eq_ind_r T (lift (S O) O x1) (\lambda (t7: T).(ty3 
+g c2 t4 (THead (Bind Void) u t7))) (let H29 \def (eq_ind_r T x0 (\lambda (t7: 
+T).(ty3 g c2 t7 x1)) H27 t4 (lift_inj t4 x0 (S O) O H25)) in (ty3_conv g c2 
+(THead (Bind Void) u (lift (S O) O x1)) (THead (Bind Void) u x) (ty3_bind g 
+c2 u t0 (H1 c2 H4 u (pr0_refl u)) Void (lift (S O) O x1) x H28) t4 x1 H29 
+(pc3_s c2 x1 (THead (Bind Void) u (lift (S O) O x1)) (pc3_pr2_r c2 (THead 
+(Bind Void) u (lift (S O) O x1)) x1 (pr2_free c2 (THead (Bind Void) u (lift 
+(S O) O x1)) x1 (pr0_zeta Void H21 x1 x1 (pr0_refl x1) u)))))) t3 H26))))))) 
+H24))))) b H16 H20 (H18 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H4 u u 
+(pr0_refl u) (Bind b)) (lift (S O) O t4) (pr0_lift t5 t4 H17 (S O) O))))) 
+(ty3_correct g (CHead c2 (Bind b) u) (lift (S O) O t4) t3 (H18 (CHead c2 
+(Bind b) u) (wcpr0_comp c c2 H4 u u (pr0_refl u) (Bind b)) (lift (S O) O t4) 
+(pr0_lift t5 t4 H17 (S O) O)))))))) t6 (sym_eq T t6 t4 H15))) t2 H14)) u0 
+(sym_eq T u0 u H13))) b0 (sym_eq B b0 b H12))) H11)) H10)) H9 H6 H7))) | 
+(pr0_tau t5 t6 H6 u0) \Rightarrow (\lambda (H7: (eq T (THead (Flat Cast) u0 
+t5) (THead (Bind b) u t2))).(\lambda (H8: (eq T t6 t4)).((let H9 \def (eq_ind 
+T (THead (Flat Cast) u0 t5) (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
+(match k with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I 
+(THead (Bind b) u t2) H7) in (False_ind ((eq T t6 t4) \to ((pr0 t5 t6) \to 
+(ty3 g c2 t4 (THead (Bind b) u t3)))) H9)) H8 H6)))]) in (H6 (refl_equal T 
+(THead (Bind b) u t2)) (refl_equal T t4))))))))))))))))) (\lambda (c: 
+C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c w u)).(\lambda (H1: 
+((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 w t2) \to (ty3 g 
+c2 t2 u))))))).(\lambda (v: T).(\lambda (t0: T).(\lambda (H2: (ty3 g c v 
+(THead (Bind Abst) u t0))).(\lambda (H3: ((\forall (c2: C).((wcpr0 c c2) \to 
+(\forall (t2: T).((pr0 v t2) \to (ty3 g c2 t2 (THead (Bind Abst) u 
+t0)))))))).(\lambda (c2: C).(\lambda (H4: (wcpr0 c c2)).(\lambda (t2: 
+T).(\lambda (H5: (pr0 (THead (Flat Appl) w v) t2)).(let H6 \def (match H5 
+with [(pr0_refl t3) \Rightarrow (\lambda (H6: (eq T t3 (THead (Flat Appl) w 
+v))).(\lambda (H7: (eq T t3 t2)).(eq_ind T (THead (Flat Appl) w v) (\lambda 
+(t4: T).((eq T t4 t2) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind 
+Abst) u t0))))) (\lambda (H8: (eq T (THead (Flat Appl) w v) t2)).(eq_ind T 
+(THead (Flat Appl) w v) (\lambda (t4: T).(ty3 g c2 t4 (THead (Flat Appl) w 
+(THead (Bind Abst) u t0)))) (ty3_appl g c2 w u (H1 c2 H4 w (pr0_refl w)) v t0 
+(H3 c2 H4 v (pr0_refl v))) t2 H8)) t3 (sym_eq T t3 (THead (Flat Appl) w v) 
+H6) H7))) | (pr0_comp u1 u2 H6 t3 t4 H7 k) \Rightarrow (\lambda (H8: (eq T 
+(THead k u1 t3) (THead (Flat Appl) w v))).(\lambda (H9: (eq T (THead k u2 t4) 
+t2)).((let H10 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t5) \Rightarrow t5])) 
+(THead k u1 t3) (THead (Flat Appl) w v) H8) in ((let H11 \def (f_equal T T 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow u1 | (TLRef _) 
+\Rightarrow u1 | (THead _ t5 _) \Rightarrow t5])) (THead k u1 t3) (THead 
+(Flat Appl) w v) H8) in ((let H12 \def (f_equal T K (\lambda (e: T).(match e 
 with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) 
 \Rightarrow k0])) (THead k u1 t3) (THead (Flat Appl) w v) H8) in (eq_ind K 
 (Flat Appl) (\lambda (k0: K).((eq T u1 w) \to ((eq T t3 v) \to ((eq T (THead 
@@ -359,31 +336,30 @@ u t0) (H3 c2 H4 v (pr0_refl v)))))) t2 H15)) t3 (sym_eq T t3 v H14))) u1
 H7))) | (pr0_beta u0 v1 v2 H6 t3 t4 H7) \Rightarrow (\lambda (H8: (eq T 
 (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) (THead (Flat Appl) w 
 v))).(\lambda (H9: (eq T (THead (Bind Abbr) v2 t4) t2)).((let H10 \def 
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow (THead (Bind Abst) u0 t3) | (TLRef _) \Rightarrow 
-(THead (Bind Abst) u0 t3) | (THead _ _ t5) \Rightarrow t5])) (THead (Flat 
-Appl) v1 (THead (Bind Abst) u0 t3)) (THead (Flat Appl) w v) H8) in ((let H11 
-\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) 
-with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t5 _) 
-\Rightarrow t5])) (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) (THead 
-(Flat Appl) w v) H8) in (eq_ind T w (\lambda (t5: T).((eq T (THead (Bind 
-Abst) u0 t3) v) \to ((eq T (THead (Bind Abbr) v2 t4) t2) \to ((pr0 t5 v2) \to 
-((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u 
-t0)))))))) (\lambda (H12: (eq T (THead (Bind Abst) u0 t3) v)).(eq_ind T 
-(THead (Bind Abst) u0 t3) (\lambda (_: T).((eq T (THead (Bind Abbr) v2 t4) 
-t2) \to ((pr0 w v2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w 
-(THead (Bind Abst) u t0))))))) (\lambda (H13: (eq T (THead (Bind Abbr) v2 t4) 
-t2)).(eq_ind T (THead (Bind Abbr) v2 t4) (\lambda (t5: T).((pr0 w v2) \to 
-((pr0 t3 t4) \to (ty3 g c2 t5 (THead (Flat Appl) w (THead (Bind Abst) u 
-t0)))))) (\lambda (H14: (pr0 w v2)).(\lambda (H15: (pr0 t3 t4)).(let H16 \def 
-(eq_ind_r T v (\lambda (t5: T).(\forall (c3: C).((wcpr0 c c3) \to (\forall 
-(t6: T).((pr0 t5 t6) \to (ty3 g c3 t6 (THead (Bind Abst) u t0))))))) H3 
-(THead (Bind Abst) u0 t3) H12) in (let H17 \def (eq_ind_r T v (\lambda (t5: 
-T).(ty3 g c t5 (THead (Bind Abst) u t0))) H2 (THead (Bind Abst) u0 t3) H12) 
-in (ex_ind T (\lambda (t5: T).(ty3 g c2 (THead (Bind Abst) u t0) t5)) (ty3 g 
-c2 (THead (Bind Abbr) v2 t4) (THead (Flat Appl) w (THead (Bind Abst) u t0))) 
-(\lambda (x: T).(\lambda (H18: (ty3 g c2 (THead (Bind Abst) u t0) 
-x)).(ex3_2_ind T T (\lambda (t5: T).(\lambda (_: T).(pc3 c2 (THead (Bind 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow (THead 
+(Bind Abst) u0 t3) | (TLRef _) \Rightarrow (THead (Bind Abst) u0 t3) | (THead 
+_ _ t5) \Rightarrow t5])) (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) 
+(THead (Flat Appl) w v) H8) in ((let H11 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | 
+(THead _ t5 _) \Rightarrow t5])) (THead (Flat Appl) v1 (THead (Bind Abst) u0 
+t3)) (THead (Flat Appl) w v) H8) in (eq_ind T w (\lambda (t5: T).((eq T 
+(THead (Bind Abst) u0 t3) v) \to ((eq T (THead (Bind Abbr) v2 t4) t2) \to 
+((pr0 t5 v2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead 
+(Bind Abst) u t0)))))))) (\lambda (H12: (eq T (THead (Bind Abst) u0 t3) 
+v)).(eq_ind T (THead (Bind Abst) u0 t3) (\lambda (_: T).((eq T (THead (Bind 
+Abbr) v2 t4) t2) \to ((pr0 w v2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead 
+(Flat Appl) w (THead (Bind Abst) u t0))))))) (\lambda (H13: (eq T (THead 
+(Bind Abbr) v2 t4) t2)).(eq_ind T (THead (Bind Abbr) v2 t4) (\lambda (t5: 
+T).((pr0 w v2) \to ((pr0 t3 t4) \to (ty3 g c2 t5 (THead (Flat Appl) w (THead 
+(Bind Abst) u t0)))))) (\lambda (H14: (pr0 w v2)).(\lambda (H15: (pr0 t3 
+t4)).(let H16 \def (eq_ind_r T v (\lambda (t5: T).(\forall (c3: C).((wcpr0 c 
+c3) \to (\forall (t6: T).((pr0 t5 t6) \to (ty3 g c3 t6 (THead (Bind Abst) u 
+t0))))))) H3 (THead (Bind Abst) u0 t3) H12) in (let H17 \def (eq_ind_r T v 
+(\lambda (t5: T).(ty3 g c t5 (THead (Bind Abst) u t0))) H2 (THead (Bind Abst) 
+u0 t3) H12) in (ex_ind T (\lambda (t5: T).(ty3 g c2 (THead (Bind Abst) u t0) 
+t5)) (ty3 g c2 (THead (Bind Abbr) v2 t4) (THead (Flat Appl) w (THead (Bind 
+Abst) u t0))) (\lambda (x: T).(\lambda (H18: (ty3 g c2 (THead (Bind Abst) u 
+t0) x)).(ex3_2_ind T T (\lambda (t5: T).(\lambda (_: T).(pc3 c2 (THead (Bind 
 Abst) u t5) x))) (\lambda (_: T).(\lambda (t6: T).(ty3 g c2 u t6))) (\lambda 
 (t5: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind Abst) u) t0 t5))) (ty3 g c2 
 (THead (Bind Abbr) v2 t4) (THead (Flat Appl) w (THead (Bind Abst) u t0))) 
@@ -420,18 +396,17 @@ t3))))))))) t2 H13)) v H12)) v1 (sym_eq T v1 w H11))) H10)) H9 H6 H7))) |
 (pr0_upsilon b H6 v1 v2 H7 u1 u2 H8 t3 t4 H9) \Rightarrow (\lambda (H10: (eq 
 T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) w 
 v))).(\lambda (H11: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O 
-v2) t4)) t2)).((let H12 \def (f_equal T T (\lambda (e: T).(match e in T 
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow (THead (Bind b) u1 t3) 
-| (TLRef _) \Rightarrow (THead (Bind b) u1 t3) | (THead _ _ t5) \Rightarrow 
-t5])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) w v) 
-H10) in ((let H13 \def (f_equal T T (\lambda (e: T).(match e in T return 
-(\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 
-| (THead _ t5 _) \Rightarrow t5])) (THead (Flat Appl) v1 (THead (Bind b) u1 
-t3)) (THead (Flat Appl) w v) H10) in (eq_ind T w (\lambda (t5: T).((eq T 
-(THead (Bind b) u1 t3) v) \to ((eq T (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t4)) t2) \to ((not (eq B b Abst)) \to ((pr0 t5 v2) \to 
-((pr0 u1 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead 
-(Bind Abst) u t0)))))))))) (\lambda (H14: (eq T (THead (Bind b) u1 t3) 
+v2) t4)) t2)).((let H12 \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 _ _ t5) \Rightarrow t5])) (THead (Flat Appl) v1 
+(THead (Bind b) u1 t3)) (THead (Flat Appl) w v) H10) in ((let H13 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow v1 | (TLRef 
+_) \Rightarrow v1 | (THead _ t5 _) \Rightarrow t5])) (THead (Flat Appl) v1 
+(THead (Bind b) u1 t3)) (THead (Flat Appl) w v) H10) in (eq_ind T w (\lambda 
+(t5: T).((eq T (THead (Bind b) u1 t3) v) \to ((eq T (THead (Bind b) u2 (THead 
+(Flat Appl) (lift (S O) O v2) t4)) t2) \to ((not (eq B b Abst)) \to ((pr0 t5 
+v2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w 
+(THead (Bind Abst) u t0)))))))))) (\lambda (H14: (eq T (THead (Bind b) u1 t3) 
 v)).(eq_ind T (THead (Bind b) u1 t3) (\lambda (_: T).((eq T (THead (Bind b) 
 u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t2) \to ((not (eq B b Abst)) \to 
 ((pr0 w v2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat 
@@ -520,159 +495,141 @@ Abst) u t0) (H20 c2 H4 (THead (Bind b) u2 t4) (pr0_comp u1 u2 H18 t3 t4 H19
 H8 H9))) | (pr0_delta u1 u2 H6 t3 t4 H7 w0 H8) \Rightarrow (\lambda (H9: (eq 
 T (THead (Bind Abbr) u1 t3) (THead (Flat Appl) w v))).(\lambda (H10: (eq T 
 (THead (Bind Abbr) u2 w0) t2)).((let H11 \def (eq_ind T (THead (Bind Abbr) u1 
-t3) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort 
-_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) 
+t3) (\lambda (e: T).(match e with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _) 
 \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) w v) 
 H9) in (False_ind ((eq T (THead (Bind Abbr) u2 w0) t2) \to ((pr0 u1 u2) \to 
 ((pr0 t3 t4) \to ((subst0 O u2 t4 w0) \to (ty3 g c2 t2 (THead (Flat Appl) w 
 (THead (Bind Abst) u t0))))))) H11)) H10 H6 H7 H8))) | (pr0_zeta b H6 t3 t4 
 H7 u0) \Rightarrow (\lambda (H8: (eq T (THead (Bind b) u0 (lift (S O) O t3)) 
 (THead (Flat Appl) w v))).(\lambda (H9: (eq T t4 t2)).((let H10 \def (eq_ind 
-T (THead (Bind b) u0 (lift (S O) O t3)) (\lambda (e: T).(match e in T return 
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda 
-(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow 
+T (THead (Bind b) u0 (lift (S O) O t3)) (\lambda (e: T).(match e with [(TSort 
+_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
+\Rightarrow (match k with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow 
 False])])) I (THead (Flat Appl) w v) H8) in (False_ind ((eq T t4 t2) \to 
 ((not (eq B b Abst)) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w 
 (THead (Bind Abst) u t0)))))) H10)) H9 H6 H7))) | (pr0_tau t3 t4 H6 u0) 
 \Rightarrow (\lambda (H7: (eq T (THead (Flat Cast) u0 t3) (THead (Flat Appl) 
 w v))).(\lambda (H8: (eq T t4 t2)).((let H9 \def (eq_ind T (THead (Flat Cast) 
-u0 t3) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) 
-\Rightarrow False | (Flat f) \Rightarrow (match f in F return (\lambda (_: 
-F).Prop) with [Appl \Rightarrow False | Cast \Rightarrow True])])])) I (THead 
-(Flat Appl) w v) H7) in (False_ind ((eq T t4 t2) \to ((pr0 t3 t4) \to (ty3 g 
-c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0))))) H9)) H8 H6)))]) in 
-(H6 (refl_equal T (THead (Flat Appl) w v)) (refl_equal T t2)))))))))))))))) 
-(\lambda (c: C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (ty3 g c t2 
-t3)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t4: T).((pr0 
-t2 t4) \to (ty3 g c2 t4 t3))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c t3 
-t0)).(\lambda (H3: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t4: T).((pr0 
-t3 t4) \to (ty3 g c2 t4 t0))))))).(\lambda (c2: C).(\lambda (H4: (wcpr0 c 
-c2)).(\lambda (t4: T).(\lambda (H5: (pr0 (THead (Flat Cast) t3 t2) t4)).(let 
-H6 \def (match H5 in pr0 return (\lambda (t5: T).(\lambda (t6: T).(\lambda 
-(_: (pr0 t5 t6)).((eq T t5 (THead (Flat Cast) t3 t2)) \to ((eq T t6 t4) \to 
-(ty3 g c2 t4 (THead (Flat Cast) t0 t3))))))) with [(pr0_refl t5) \Rightarrow 
-(\lambda (H6: (eq T t5 (THead (Flat Cast) t3 t2))).(\lambda (H7: (eq T t5 
-t4)).(eq_ind T (THead (Flat Cast) t3 t2) (\lambda (t6: T).((eq T t6 t4) \to 
-(ty3 g c2 t4 (THead (Flat Cast) t0 t3)))) (\lambda (H8: (eq T (THead (Flat 
-Cast) t3 t2) t4)).(eq_ind T (THead (Flat Cast) t3 t2) (\lambda (t6: T).(ty3 g 
-c2 t6 (THead (Flat Cast) t0 t3))) (ty3_cast g c2 t2 t3 (H1 c2 H4 t2 (pr0_refl 
-t2)) t0 (H3 c2 H4 t3 (pr0_refl t3))) t4 H8)) t5 (sym_eq T t5 (THead (Flat 
-Cast) t3 t2) H6) H7))) | (pr0_comp u1 u2 H6 t5 t6 H7 k) \Rightarrow (\lambda 
-(H8: (eq T (THead k u1 t5) (THead (Flat Cast) t3 t2))).(\lambda (H9: (eq T 
-(THead k u2 t6) t4)).((let H10 \def (f_equal T T (\lambda (e: T).(match e in 
-T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t5 | (TLRef _) 
-\Rightarrow t5 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t5) (THead 
+u0 t3) (\lambda (e: T).(match e 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) w v) H7) in 
+(False_ind ((eq T t4 t2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) 
+w (THead (Bind Abst) u t0))))) H9)) H8 H6)))]) in (H6 (refl_equal T (THead 
+(Flat Appl) w v)) (refl_equal T t2)))))))))))))))) (\lambda (c: C).(\lambda 
+(t2: T).(\lambda (t3: T).(\lambda (_: (ty3 g c t2 t3)).(\lambda (H1: 
+((\forall (c2: C).((wcpr0 c c2) \to (\forall (t4: T).((pr0 t2 t4) \to (ty3 g 
+c2 t4 t3))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c t3 t0)).(\lambda (H3: 
+((\forall (c2: C).((wcpr0 c c2) \to (\forall (t4: T).((pr0 t3 t4) \to (ty3 g 
+c2 t4 t0))))))).(\lambda (c2: C).(\lambda (H4: (wcpr0 c c2)).(\lambda (t4: 
+T).(\lambda (H5: (pr0 (THead (Flat Cast) t3 t2) t4)).(let H6 \def (match H5 
+with [(pr0_refl t5) \Rightarrow (\lambda (H6: (eq T t5 (THead (Flat Cast) t3 
+t2))).(\lambda (H7: (eq T t5 t4)).(eq_ind T (THead (Flat Cast) t3 t2) 
+(\lambda (t6: T).((eq T t6 t4) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3)))) 
+(\lambda (H8: (eq T (THead (Flat Cast) t3 t2) t4)).(eq_ind T (THead (Flat 
+Cast) t3 t2) (\lambda (t6: T).(ty3 g c2 t6 (THead (Flat Cast) t0 t3))) 
+(ty3_cast g c2 t2 t3 (H1 c2 H4 t2 (pr0_refl t2)) t0 (H3 c2 H4 t3 (pr0_refl 
+t3))) t4 H8)) t5 (sym_eq T t5 (THead (Flat Cast) t3 t2) H6) H7))) | (pr0_comp 
+u1 u2 H6 t5 t6 H7 k) \Rightarrow (\lambda (H8: (eq T (THead k u1 t5) (THead 
+(Flat Cast) t3 t2))).(\lambda (H9: (eq T (THead k u2 t6) t4)).((let H10 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t5 | (TLRef 
+_) \Rightarrow t5 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t5) (THead 
 (Flat Cast) t3 t2) H8) in ((let H11 \def (f_equal T T (\lambda (e: T).(match 
-e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) 
-\Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t5) (THead 
-(Flat Cast) t3 t2) H8) in ((let H12 \def (f_equal T K (\lambda (e: T).(match 
-e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) 
-\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t5) (THead (Flat 
-Cast) t3 t2) H8) in (eq_ind K (Flat Cast) (\lambda (k0: K).((eq T u1 t3) \to 
-((eq T t5 t2) \to ((eq T (THead k0 u2 t6) t4) \to ((pr0 u1 u2) \to ((pr0 t5 
-t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3)))))))) (\lambda (H13: (eq T u1 
-t3)).(eq_ind T t3 (\lambda (t7: T).((eq T t5 t2) \to ((eq T (THead (Flat 
-Cast) u2 t6) t4) \to ((pr0 t7 u2) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead 
-(Flat Cast) t0 t3))))))) (\lambda (H14: (eq T t5 t2)).(eq_ind T t2 (\lambda 
-(t7: T).((eq T (THead (Flat Cast) u2 t6) t4) \to ((pr0 t3 u2) \to ((pr0 t7 
-t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3)))))) (\lambda (H15: (eq T 
-(THead (Flat Cast) u2 t6) t4)).(eq_ind T (THead (Flat Cast) u2 t6) (\lambda 
-(t7: T).((pr0 t3 u2) \to ((pr0 t2 t6) \to (ty3 g c2 t7 (THead (Flat Cast) t0 
-t3))))) (\lambda (H16: (pr0 t3 u2)).(\lambda (H17: (pr0 t2 t6)).(ex_ind T 
-(\lambda (t7: T).(ty3 g c2 t0 t7)) (ty3 g c2 (THead (Flat Cast) u2 t6) (THead 
-(Flat Cast) t0 t3)) (\lambda (x: T).(\lambda (H18: (ty3 g c2 t0 x)).(ty3_conv 
-g c2 (THead (Flat Cast) t0 t3) (THead (Flat Cast) x t0) (ty3_cast g c2 t3 t0 
-(H3 c2 H4 t3 (pr0_refl t3)) x H18) (THead (Flat Cast) u2 t6) (THead (Flat 
-Cast) t0 u2) (ty3_cast g c2 t6 u2 (ty3_conv g c2 u2 t0 (H3 c2 H4 u2 H16) t6 
-t3 (H1 c2 H4 t6 H17) (pc3_pr2_r c2 t3 u2 (pr2_free c2 t3 u2 H16))) t0 (H3 c2 
-H4 u2 H16)) (pc3_s c2 (THead (Flat Cast) t0 u2) (THead (Flat Cast) t0 t3) 
-(pc3_pr2_r c2 (THead (Flat Cast) t0 t3) (THead (Flat Cast) t0 u2) 
-(pr2_thin_dx c2 t3 u2 (pr2_free c2 t3 u2 H16) t0 Cast)))))) (ty3_correct g c2 
-t3 t0 (H3 c2 H4 t3 (pr0_refl t3)))))) t4 H15)) t5 (sym_eq T t5 t2 H14))) u1 
-(sym_eq T u1 t3 H13))) k (sym_eq K k (Flat Cast) H12))) H11)) H10)) H9 H6 
+e with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t7 _) 
+\Rightarrow t7])) (THead k u1 t5) (THead (Flat Cast) t3 t2) H8) in ((let H12 
+\def (f_equal T K (\lambda (e: T).(match e with [(TSort _) \Rightarrow k | 
+(TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t5) 
+(THead (Flat Cast) t3 t2) H8) in (eq_ind K (Flat Cast) (\lambda (k0: K).((eq 
+T u1 t3) \to ((eq T t5 t2) \to ((eq T (THead k0 u2 t6) t4) \to ((pr0 u1 u2) 
+\to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3)))))))) (\lambda 
+(H13: (eq T u1 t3)).(eq_ind T t3 (\lambda (t7: T).((eq T t5 t2) \to ((eq T 
+(THead (Flat Cast) u2 t6) t4) \to ((pr0 t7 u2) \to ((pr0 t5 t6) \to (ty3 g c2 
+t4 (THead (Flat Cast) t0 t3))))))) (\lambda (H14: (eq T t5 t2)).(eq_ind T t2 
+(\lambda (t7: T).((eq T (THead (Flat Cast) u2 t6) t4) \to ((pr0 t3 u2) \to 
+((pr0 t7 t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3)))))) (\lambda (H15: 
+(eq T (THead (Flat Cast) u2 t6) t4)).(eq_ind T (THead (Flat Cast) u2 t6) 
+(\lambda (t7: T).((pr0 t3 u2) \to ((pr0 t2 t6) \to (ty3 g c2 t7 (THead (Flat 
+Cast) t0 t3))))) (\lambda (H16: (pr0 t3 u2)).(\lambda (H17: (pr0 t2 
+t6)).(ex_ind T (\lambda (t7: T).(ty3 g c2 t0 t7)) (ty3 g c2 (THead (Flat 
+Cast) u2 t6) (THead (Flat Cast) t0 t3)) (\lambda (x: T).(\lambda (H18: (ty3 g 
+c2 t0 x)).(ty3_conv g c2 (THead (Flat Cast) t0 t3) (THead (Flat Cast) x t0) 
+(ty3_cast g c2 t3 t0 (H3 c2 H4 t3 (pr0_refl t3)) x H18) (THead (Flat Cast) u2 
+t6) (THead (Flat Cast) t0 u2) (ty3_cast g c2 t6 u2 (ty3_conv g c2 u2 t0 (H3 
+c2 H4 u2 H16) t6 t3 (H1 c2 H4 t6 H17) (pc3_pr2_r c2 t3 u2 (pr2_free c2 t3 u2 
+H16))) t0 (H3 c2 H4 u2 H16)) (pc3_s c2 (THead (Flat Cast) t0 u2) (THead (Flat 
+Cast) t0 t3) (pc3_pr2_r c2 (THead (Flat Cast) t0 t3) (THead (Flat Cast) t0 
+u2) (pr2_thin_dx c2 t3 u2 (pr2_free c2 t3 u2 H16) t0 Cast)))))) (ty3_correct 
+g c2 t3 t0 (H3 c2 H4 t3 (pr0_refl t3)))))) t4 H15)) t5 (sym_eq T t5 t2 H14))) 
+u1 (sym_eq T u1 t3 H13))) k (sym_eq K k (Flat Cast) H12))) H11)) H10)) H9 H6 
 H7))) | (pr0_beta u v1 v2 H6 t5 t6 H7) \Rightarrow (\lambda (H8: (eq T (THead 
 (Flat Appl) v1 (THead (Bind Abst) u t5)) (THead (Flat Cast) t3 t2))).(\lambda 
 (H9: (eq T (THead (Bind Abbr) v2 t6) t4)).((let H10 \def (eq_ind T (THead 
-(Flat Appl) v1 (THead (Bind Abst) u t5)) (\lambda (e: T).(match e in T return 
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda 
-(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f 
-in F return (\lambda (_: F).Prop) with [Appl \Rightarrow True | Cast 
-\Rightarrow False])])])) I (THead (Flat Cast) t3 t2) H8) in (False_ind ((eq T 
-(THead (Bind Abbr) v2 t6) t4) \to ((pr0 v1 v2) \to ((pr0 t5 t6) \to (ty3 g c2 
-t4 (THead (Flat Cast) t0 t3))))) H10)) H9 H6 H7))) | (pr0_upsilon b H6 v1 v2 
-H7 u1 u2 H8 t5 t6 H9) \Rightarrow (\lambda (H10: (eq T (THead (Flat Appl) v1 
-(THead (Bind b) u1 t5)) (THead (Flat Cast) t3 t2))).(\lambda (H11: (eq T 
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t6)) t4)).((let H12 
-\def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t5)) (\lambda (e: 
-T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K 
-return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) 
-\Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl \Rightarrow 
-True | Cast \Rightarrow False])])])) I (THead (Flat Cast) t3 t2) H10) in 
-(False_ind ((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
-t6)) t4) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 
-t5 t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3))))))) H12)) H11 H6 H7 H8 
-H9))) | (pr0_delta u1 u2 H6 t5 t6 H7 w H8) \Rightarrow (\lambda (H9: (eq T 
-(THead (Bind Abbr) u1 t5) (THead (Flat Cast) t3 t2))).(\lambda (H10: (eq T 
-(THead (Bind Abbr) u2 w) t4)).((let H11 \def (eq_ind T (THead (Bind Abbr) u1 
-t5) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort 
-_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) 
-\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Cast) t3 
-t2) H9) in (False_ind ((eq T (THead (Bind Abbr) u2 w) t4) \to ((pr0 u1 u2) 
-\to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ty3 g c2 t4 (THead (Flat Cast) 
-t0 t3)))))) H11)) H10 H6 H7 H8))) | (pr0_zeta b H6 t5 t6 H7 u) \Rightarrow 
-(\lambda (H8: (eq T (THead (Bind b) u (lift (S O) O t5)) (THead (Flat Cast) 
-t3 t2))).(\lambda (H9: (eq T t6 t4)).((let H10 \def (eq_ind T (THead (Bind b) 
-u (lift (S O) O t5)) (\lambda (e: T).(match e in T return (\lambda (_: 
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
-(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with 
-[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat 
-Cast) t3 t2) H8) in (False_ind ((eq T t6 t4) \to ((not (eq B b Abst)) \to 
-((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3))))) H10)) H9 H6 H7))) 
-| (pr0_tau t5 t6 H6 u) \Rightarrow (\lambda (H7: (eq T (THead (Flat Cast) u 
-t5) (THead (Flat Cast) t3 t2))).(\lambda (H8: (eq T t6 t4)).((let H9 \def 
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow t5 | (TLRef _) \Rightarrow t5 | (THead _ _ t7) 
-\Rightarrow t7])) (THead (Flat Cast) u t5) (THead (Flat Cast) t3 t2) H7) in 
-((let H10 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: 
-T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t7 
-_) \Rightarrow t7])) (THead (Flat Cast) u t5) (THead (Flat Cast) t3 t2) H7) 
-in (eq_ind T t3 (\lambda (_: T).((eq T t5 t2) \to ((eq T t6 t4) \to ((pr0 t5 
-t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3)))))) (\lambda (H11: (eq T t5 
-t2)).(eq_ind T t2 (\lambda (t7: T).((eq T t6 t4) \to ((pr0 t7 t6) \to (ty3 g 
-c2 t4 (THead (Flat Cast) t0 t3))))) (\lambda (H12: (eq T t6 t4)).(eq_ind T t4 
-(\lambda (t7: T).((pr0 t2 t7) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3)))) 
-(\lambda (H13: (pr0 t2 t4)).(ex_ind T (\lambda (t7: T).(ty3 g c2 t0 t7)) (ty3 
-g c2 t4 (THead (Flat Cast) t0 t3)) (\lambda (x: T).(\lambda (H14: (ty3 g c2 
-t0 x)).(ty3_conv g c2 (THead (Flat Cast) t0 t3) (THead (Flat Cast) x t0) 
-(ty3_cast g c2 t3 t0 (H3 c2 H4 t3 (pr0_refl t3)) x H14) t4 t3 (H1 c2 H4 t4 
-H13) (pc3_pr2_x c2 t3 (THead (Flat Cast) t0 t3) (pr2_free c2 (THead (Flat 
-Cast) t0 t3) t3 (pr0_tau t3 t3 (pr0_refl t3) t0)))))) (ty3_correct g c2 t3 t0 
-(H3 c2 H4 t3 (pr0_refl t3))))) t6 (sym_eq T t6 t4 H12))) t5 (sym_eq T t5 t2 
-H11))) u (sym_eq T u t3 H10))) H9)) H8 H6)))]) in (H6 (refl_equal T (THead 
-(Flat Cast) t3 t2)) (refl_equal T t4))))))))))))))) c1 t1 t H))))).
-(* COMMENTS
-Initial nodes: 14710
-END *)
+(Flat Appl) v1 (THead (Bind Abst) u t5)) (\lambda (e: T).(match e 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) t3 t2) H8) in (False_ind ((eq T (THead (Bind Abbr) v2 t6) t4) \to 
+((pr0 v1 v2) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3))))) 
+H10)) H9 H6 H7))) | (pr0_upsilon b H6 v1 v2 H7 u1 u2 H8 t5 t6 H9) \Rightarrow 
+(\lambda (H10: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t5)) (THead 
+(Flat Cast) t3 t2))).(\lambda (H11: (eq T (THead (Bind b) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t6)) t4)).((let H12 \def (eq_ind T (THead (Flat Appl) 
+v1 (THead (Bind b) u1 t5)) (\lambda (e: T).(match e 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) t3 t2) H10) in (False_ind ((eq T (THead (Bind b) u2 (THead (Flat Appl) 
+(lift (S O) O v2) t6)) t4) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to 
+((pr0 u1 u2) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 
+t3))))))) H12)) H11 H6 H7 H8 H9))) | (pr0_delta u1 u2 H6 t5 t6 H7 w H8) 
+\Rightarrow (\lambda (H9: (eq T (THead (Bind Abbr) u1 t5) (THead (Flat Cast) 
+t3 t2))).(\lambda (H10: (eq T (THead (Bind Abbr) u2 w) t4)).((let H11 \def 
+(eq_ind T (THead (Bind Abbr) u1 t5) (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
+(match k with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I 
+(THead (Flat Cast) t3 t2) H9) in (False_ind ((eq T (THead (Bind Abbr) u2 w) 
+t4) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ty3 g c2 
+t4 (THead (Flat Cast) t0 t3)))))) H11)) H10 H6 H7 H8))) | (pr0_zeta b H6 t5 
+t6 H7 u) \Rightarrow (\lambda (H8: (eq T (THead (Bind b) u (lift (S O) O t5)) 
+(THead (Flat Cast) t3 t2))).(\lambda (H9: (eq T t6 t4)).((let H10 \def 
+(eq_ind T (THead (Bind b) u (lift (S O) O t5)) (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
+\Rightarrow (match k with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow 
+False])])) I (THead (Flat Cast) t3 t2) H8) in (False_ind ((eq T t6 t4) \to 
+((not (eq B b Abst)) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 
+t3))))) H10)) H9 H6 H7))) | (pr0_tau t5 t6 H6 u) \Rightarrow (\lambda (H7: 
+(eq T (THead (Flat Cast) u t5) (THead (Flat Cast) t3 t2))).(\lambda (H8: (eq 
+T t6 t4)).((let H9 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow t5 | (TLRef _) \Rightarrow t5 | (THead _ _ t7) \Rightarrow t7])) 
+(THead (Flat Cast) u t5) (THead (Flat Cast) t3 t2) H7) in ((let H10 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u | (TLRef 
+_) \Rightarrow u | (THead _ t7 _) \Rightarrow t7])) (THead (Flat Cast) u t5) 
+(THead (Flat Cast) t3 t2) H7) in (eq_ind T t3 (\lambda (_: T).((eq T t5 t2) 
+\to ((eq T t6 t4) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 
+t3)))))) (\lambda (H11: (eq T t5 t2)).(eq_ind T t2 (\lambda (t7: T).((eq T t6 
+t4) \to ((pr0 t7 t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3))))) (\lambda 
+(H12: (eq T t6 t4)).(eq_ind T t4 (\lambda (t7: T).((pr0 t2 t7) \to (ty3 g c2 
+t4 (THead (Flat Cast) t0 t3)))) (\lambda (H13: (pr0 t2 t4)).(ex_ind T 
+(\lambda (t7: T).(ty3 g c2 t0 t7)) (ty3 g c2 t4 (THead (Flat Cast) t0 t3)) 
+(\lambda (x: T).(\lambda (H14: (ty3 g c2 t0 x)).(ty3_conv g c2 (THead (Flat 
+Cast) t0 t3) (THead (Flat Cast) x t0) (ty3_cast g c2 t3 t0 (H3 c2 H4 t3 
+(pr0_refl t3)) x H14) t4 t3 (H1 c2 H4 t4 H13) (pc3_pr2_x c2 t3 (THead (Flat 
+Cast) t0 t3) (pr2_free c2 (THead (Flat Cast) t0 t3) t3 (pr0_tau t3 t3 
+(pr0_refl t3) t0)))))) (ty3_correct g c2 t3 t0 (H3 c2 H4 t3 (pr0_refl t3))))) 
+t6 (sym_eq T t6 t4 H12))) t5 (sym_eq T t5 t2 H11))) u (sym_eq T u t3 H10))) 
+H9)) H8 H6)))]) in (H6 (refl_equal T (THead (Flat Cast) t3 t2)) (refl_equal T 
+t4))))))))))))))) c1 t1 t H))))).
 
-theorem ty3_sred_pr0:
+lemma ty3_sred_pr0:
  \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (g: G).(\forall 
 (c: C).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t)))))))
 \def
  \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(\lambda (g: 
 G).(\lambda (c: C).(\lambda (t: T).(\lambda (H0: (ty3 g c t1 
 t)).(ty3_sred_wcpr0_pr0 g c t1 t H0 c (wcpr0_refl c) t2 H))))))).
-(* COMMENTS
-Initial nodes: 47
-END *)
 
-theorem ty3_sred_pr1:
+lemma ty3_sred_pr1:
  \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (g: G).(\forall 
 (c: C).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t)))))))
 \def
@@ -685,11 +642,8 @@ T).(\lambda (_: (pr1 t3 t5)).(\lambda (H2: ((\forall (g: G).(\forall (c:
 C).(\forall (t: T).((ty3 g c t3 t) \to (ty3 g c t5 t))))))).(\lambda (g: 
 G).(\lambda (c: C).(\lambda (t: T).(\lambda (H3: (ty3 g c t4 t)).(H2 g c t 
 (ty3_sred_pr0 t4 t3 H0 g c t H3)))))))))))) t1 t2 H))).
-(* COMMENTS
-Initial nodes: 151
-END *)
 
-theorem ty3_sred_pr2:
+lemma ty3_sred_pr2:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall 
 (g: G).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t)))))))
 \def
@@ -705,11 +659,8 @@ t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (g:
 G).(\lambda (t0: T).(\lambda (H3: (ty3 g c0 t3 t0)).(ty3_subst0 g c0 t4 t0 
 (ty3_sred_wcpr0_pr0 g c0 t3 t0 H3 c0 (wcpr0_refl c0) t4 H1) d u i H0 t 
 H2)))))))))))))) c t1 t2 H)))).
-(* COMMENTS
-Initial nodes: 205
-END *)
 
-theorem ty3_sred_pr3:
+lemma ty3_sred_pr3:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall 
 (g: G).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t)))))))
 \def
@@ -722,7 +673,4 @@ T).(\lambda (t4: T).(\lambda (H0: (pr2 c t4 t3)).(\lambda (t5: T).(\lambda
 t3 t) \to (ty3 g c t5 t)))))).(\lambda (g: G).(\lambda (t: T).(\lambda (H3: 
 (ty3 g c t4 t)).(H2 g t (ty3_sred_pr2 c t4 t3 H0 g t H3))))))))))) t1 t2 
 H)))).
-(* COMMENTS
-Initial nodes: 151
-END *)