X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=blobdiff_plain;f=matita%2Fcontribs%2FLAMBDA-TYPES%2FLambdaDelta-1%2Farity%2Fpr3.ma;fp=matita%2Fcontribs%2FLAMBDA-TYPES%2FLambdaDelta-1%2Farity%2Fpr3.ma;h=b56a8945a05500e3259140626a7a7eab992bb64a;hp=0000000000000000000000000000000000000000;hb=f61af501fb4608cc4fb062a0864c774e677f0d76;hpb=58ae1809c352e71e7b5530dc41e2bfc834e1aef1

diff --git a/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/pr3.ma b/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/pr3.ma
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+++ b/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/pr3.ma
@@ -0,0 +1,623 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/csuba/arity.ma".
+
+include "LambdaDelta-1/pr3/defs.ma".
+
+include "LambdaDelta-1/pr1/defs.ma".
+
+include "LambdaDelta-1/wcpr0/getl.ma".
+
+include "LambdaDelta-1/pr0/fwd.ma".
+
+include "LambdaDelta-1/arity/subst0.ma".
+
+theorem arity_sred_wcpr0_pr0:
+ \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (a: A).((arity g 
+c1 t1 a) \to (\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t2: T).((pr0 t1 
+t2) \to (arity g c2 t2 a)))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (a: A).(\lambda 
+(H: (arity g c1 t1 a)).(arity_ind g (\lambda (c: C).(\lambda (t: T).(\lambda 
+(a0: A).(\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 t t2) \to 
+(arity g c2 t2 a0)))))))) (\lambda (c: C).(\lambda (n: nat).(\lambda (c2: 
+C).(\lambda (_: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H1: (pr0 (TSort n) 
+t2)).(eq_ind_r T (TSort n) (\lambda (t: T).(arity g c2 t (ASort O n))) 
+(arity_sort g c2 n) t2 (pr0_gen_sort t2 n H1)))))))) (\lambda (c: C).(\lambda 
+(d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d 
+(Bind Abbr) u))).(\lambda (a0: A).(\lambda (_: (arity g d u a0)).(\lambda 
+(H2: ((\forall (c2: C).((wcpr0 d c2) \to (\forall (t2: T).((pr0 u t2) \to 
+(arity g c2 t2 a0))))))).(\lambda (c2: C).(\lambda (H3: (wcpr0 c 
+c2)).(\lambda (t2: T).(\lambda (H4: (pr0 (TLRef i) t2)).(eq_ind_r T (TLRef i) 
+(\lambda (t: T).(arity g c2 t a0)) (ex3_2_ind C T (\lambda (e2: C).(\lambda 
+(u2: T).(getl i c2 (CHead e2 (Bind Abbr) u2)))) (\lambda (e2: C).(\lambda (_: 
+T).(wcpr0 d e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u u2))) (arity g c2 
+(TLRef i) a0) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (getl i c2 
+(CHead x0 (Bind Abbr) x1))).(\lambda (H6: (wcpr0 d x0)).(\lambda (H7: (pr0 u 
+x1)).(arity_abbr g c2 x0 x1 i H5 a0 (H2 x0 H6 x1 H7))))))) (wcpr0_getl c c2 
+H3 i d u (Bind Abbr) H0)) t2 (pr0_gen_lref t2 i H4)))))))))))))) (\lambda (c: 
+C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c 
+(CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (_: (arity g d u (asucc g 
+a0))).(\lambda (H2: ((\forall (c2: C).((wcpr0 d c2) \to (\forall (t2: 
+T).((pr0 u t2) \to (arity g c2 t2 (asucc g a0)))))))).(\lambda (c2: 
+C).(\lambda (H3: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H4: (pr0 (TLRef i) 
+t2)).(eq_ind_r T (TLRef i) (\lambda (t: T).(arity g c2 t a0)) (ex3_2_ind C T 
+(\lambda (e2: C).(\lambda (u2: T).(getl i c2 (CHead e2 (Bind Abst) u2)))) 
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 d e2))) (\lambda (_: C).(\lambda (u2: 
+T).(pr0 u u2))) (arity g c2 (TLRef i) a0) (\lambda (x0: C).(\lambda (x1: 
+T).(\lambda (H5: (getl i c2 (CHead x0 (Bind Abst) x1))).(\lambda (H6: (wcpr0 
+d x0)).(\lambda (H7: (pr0 u x1)).(arity_abst g c2 x0 x1 i H5 a0 (H2 x0 H6 x1 
+H7))))))) (wcpr0_getl c c2 H3 i d u (Bind Abst) H0)) t2 (pr0_gen_lref t2 i 
+H4)))))))))))))) (\lambda (b: B).(\lambda (H0: (not (eq B b Abst))).(\lambda 
+(c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c u 
+a1)).(\lambda (H2: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 
+u t2) \to (arity g c2 t2 a1))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda 
+(H3: (arity g (CHead c (Bind b) u) t a2)).(\lambda (H4: ((\forall (c2: 
+C).((wcpr0 (CHead c (Bind b) u) c2) \to (\forall (t2: T).((pr0 t t2) \to 
+(arity g c2 t2 a2))))))).(\lambda (c2: C).(\lambda (H5: (wcpr0 c 
+c2)).(\lambda (t2: T).(\lambda (H6: (pr0 (THead (Bind b) u t) t2)).(insert_eq 
+T (THead (Bind b) u t) (\lambda (t0: T).(pr0 t0 t2)) (\lambda (_: T).(arity g 
+c2 t2 a2)) (\lambda (y: T).(\lambda (H7: (pr0 y t2)).(pr0_ind (\lambda (t0: 
+T).(\lambda (t3: T).((eq T t0 (THead (Bind b) u t)) \to (arity g c2 t3 a2)))) 
+(\lambda (t0: T).(\lambda (H8: (eq T t0 (THead (Bind b) u t))).(let H9 \def 
+(f_equal T T (\lambda (e: T).e) t0 (THead (Bind b) u t) H8) in (eq_ind_r T 
+(THead (Bind b) u t) (\lambda (t3: T).(arity g c2 t3 a2)) (arity_bind g b H0 
+c2 u a1 (H2 c2 H5 u (pr0_refl u)) t a2 (H4 (CHead c2 (Bind b) u) (wcpr0_comp 
+c c2 H5 u u (pr0_refl u) (Bind b)) t (pr0_refl t))) t0 H9)))) (\lambda (u1: 
+T).(\lambda (u2: T).(\lambda (H8: (pr0 u1 u2)).(\lambda (H9: (((eq T u1 
+(THead (Bind b) u t)) \to (arity g c2 u2 a2)))).(\lambda (t3: T).(\lambda 
+(t4: T).(\lambda (H10: (pr0 t3 t4)).(\lambda (H11: (((eq T t3 (THead (Bind b) 
+u t)) \to (arity g c2 t4 a2)))).(\lambda (k: K).(\lambda (H12: (eq T (THead k 
+u1 t3) (THead (Bind b) u t))).(let H13 \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 t) H12) in ((let H14 \def (f_equal T T (\lambda (e: 
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | 
+(TLRef _) \Rightarrow u1 | (THead _ t0 _) \Rightarrow t0])) (THead k u1 t3) 
+(THead (Bind b) u t) H12) in ((let H15 \def (f_equal T T (\lambda (e: 
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | 
+(TLRef _) \Rightarrow t3 | (THead _ _ t0) \Rightarrow t0])) (THead k u1 t3) 
+(THead (Bind b) u t) H12) in (\lambda (H16: (eq T u1 u)).(\lambda (H17: (eq K 
+k (Bind b))).(eq_ind_r K (Bind b) (\lambda (k0: K).(arity g c2 (THead k0 u2 
+t4) a2)) (let H18 \def (eq_ind T t3 (\lambda (t0: T).((eq T t0 (THead (Bind 
+b) u t)) \to (arity g c2 t4 a2))) H11 t H15) in (let H19 \def (eq_ind T t3 
+(\lambda (t0: T).(pr0 t0 t4)) H10 t H15) in (let H20 \def (eq_ind T u1 
+(\lambda (t0: T).((eq T t0 (THead (Bind b) u t)) \to (arity g c2 u2 a2))) H9 
+u H16) in (let H21 \def (eq_ind T u1 (\lambda (t0: T).(pr0 t0 u2)) H8 u H16) 
+in (arity_bind g b H0 c2 u2 a1 (H2 c2 H5 u2 H21) t4 a2 (H4 (CHead c2 (Bind b) 
+u2) (wcpr0_comp c c2 H5 u u2 H21 (Bind b)) t4 H19)))))) k H17)))) H14)) 
+H13)))))))))))) (\lambda (u0: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda 
+(_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind b) u t)) \to (arity g 
+c2 v2 a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 
+t4)).(\lambda (_: (((eq T t3 (THead (Bind b) u t)) \to (arity g c2 t4 
+a2)))).(\lambda (H12: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) 
+(THead (Bind b) u t))).(let H13 \def (eq_ind T (THead (Flat Appl) v1 (THead 
+(Bind Abst) u0 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) u t) H12) in (False_ind (arity g c2 (THead (Bind Abbr) v2 t4) a2) 
+H13)))))))))))) (\lambda (b0: B).(\lambda (_: (not (eq B b0 Abst))).(\lambda 
+(v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 
+(THead (Bind b) u t)) \to (arity g c2 v2 a2)))).(\lambda (u1: T).(\lambda 
+(u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 (THead (Bind b) u 
+t)) \to (arity g c2 u2 a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: 
+(pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead (Bind b) u t)) \to (arity g c2 t4 
+a2)))).(\lambda (H15: (eq T (THead (Flat Appl) v1 (THead (Bind b0) u1 t3)) 
+(THead (Bind b) u t))).(let H16 \def (eq_ind T (THead (Flat Appl) v1 (THead 
+(Bind b0) 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 
+b) u t) H15) in (False_ind (arity g c2 (THead (Bind b0) u2 (THead (Flat Appl) 
+(lift (S O) O v2) t4)) a2) H16))))))))))))))))) (\lambda (u1: T).(\lambda 
+(u2: T).(\lambda (H8: (pr0 u1 u2)).(\lambda (H9: (((eq T u1 (THead (Bind b) u 
+t)) \to (arity g c2 u2 a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda 
+(H10: (pr0 t3 t4)).(\lambda (H11: (((eq T t3 (THead (Bind b) u t)) \to (arity 
+g c2 t4 a2)))).(\lambda (w: T).(\lambda (H12: (subst0 O u2 t4 w)).(\lambda 
+(H13: (eq T (THead (Bind Abbr) u1 t3) (THead (Bind b) u t))).(let H14 \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 t) H13) in ((let H15 \def (f_equal T T (\lambda (e: 
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | 
+(TLRef _) \Rightarrow u1 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind 
+Abbr) u1 t3) (THead (Bind b) u t) H13) in ((let H16 \def (f_equal T T 
+(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
+\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t0) \Rightarrow t0])) 
+(THead (Bind Abbr) u1 t3) (THead (Bind b) u t) H13) in (\lambda (H17: (eq T 
+u1 u)).(\lambda (H18: (eq B Abbr b)).(let H19 \def (eq_ind T t3 (\lambda (t0: 
+T).((eq T t0 (THead (Bind b) u t)) \to (arity g c2 t4 a2))) H11 t H16) in 
+(let H20 \def (eq_ind T t3 (\lambda (t0: T).(pr0 t0 t4)) H10 t H16) in (let 
+H21 \def (eq_ind T u1 (\lambda (t0: T).((eq T t0 (THead (Bind b) u t)) \to 
+(arity g c2 u2 a2))) H9 u H17) in (let H22 \def (eq_ind T u1 (\lambda (t0: 
+T).(pr0 t0 u2)) H8 u H17) in (let H23 \def (eq_ind_r B b (\lambda (b0: 
+B).((eq T t (THead (Bind b0) u t)) \to (arity g c2 t4 a2))) H19 Abbr H18) in 
+(let H24 \def (eq_ind_r B b (\lambda (b0: B).((eq T u (THead (Bind b0) u t)) 
+\to (arity g c2 u2 a2))) H21 Abbr H18) in (let H25 \def (eq_ind_r B b 
+(\lambda (b0: B).(\forall (c3: C).((wcpr0 (CHead c (Bind b0) u) c3) \to 
+(\forall (t5: T).((pr0 t t5) \to (arity g c3 t5 a2)))))) H4 Abbr H18) in (let 
+H26 \def (eq_ind_r B b (\lambda (b0: B).(arity g (CHead c (Bind b0) u) t a2)) 
+H3 Abbr H18) in (let H27 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 
+Abst))) H0 Abbr H18) in (arity_bind g Abbr H27 c2 u2 a1 (H2 c2 H5 u2 H22) w 
+a2 (arity_subst0 g (CHead c2 (Bind Abbr) u2) t4 a2 (H25 (CHead c2 (Bind Abbr) 
+u2) (wcpr0_comp c c2 H5 u u2 H22 (Bind Abbr)) t4 H20) c2 u2 O (getl_refl Abbr 
+c2 u2) w H12)))))))))))))) H15)) H14))))))))))))) (\lambda (b0: B).(\lambda 
+(H8: (not (eq B b0 Abst))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H9: 
+(pr0 t3 t4)).(\lambda (H10: (((eq T t3 (THead (Bind b) u t)) \to (arity g c2 
+t4 a2)))).(\lambda (u0: T).(\lambda (H11: (eq T (THead (Bind b0) u0 (lift (S 
+O) O t3)) (THead (Bind b) u t))).(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 t3)) (THead (Bind b) u t) H11) in 
+((let H13 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: 
+T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t0 
+_) \Rightarrow t0])) (THead (Bind b0) u0 (lift (S O) O t3)) (THead (Bind b) u 
+t) H11) in ((let H14 \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) (t0: T) on t0: T \def (match t0 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 t5) 
+\Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d) t5))]) in 
+lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (TLRef _) \Rightarrow 
+((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match 
+t0 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 t5) \Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d) 
+t5))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (THead _ _ t0) 
+\Rightarrow t0])) (THead (Bind b0) u0 (lift (S O) O t3)) (THead (Bind b) u t) 
+H11) in (\lambda (_: (eq T u0 u)).(\lambda (H16: (eq B b0 b)).(let H17 \def 
+(eq_ind B b0 (\lambda (b1: B).(not (eq B b1 Abst))) H8 b H16) in (let H18 
+\def (eq_ind_r T t (\lambda (t0: T).((eq T t3 (THead (Bind b) u t0)) \to 
+(arity g c2 t4 a2))) H10 (lift (S O) O t3) H14) in (let H19 \def (eq_ind_r T 
+t (\lambda (t0: T).(\forall (c3: C).((wcpr0 (CHead c (Bind b) u) c3) \to 
+(\forall (t5: T).((pr0 t0 t5) \to (arity g c3 t5 a2)))))) H4 (lift (S O) O 
+t3) H14) in (let H20 \def (eq_ind_r T t (\lambda (t0: T).(arity g (CHead c 
+(Bind b) u) t0 a2)) H3 (lift (S O) O t3) H14) in (arity_gen_lift g (CHead c2 
+(Bind b) u) t4 a2 (S O) O (H19 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H5 u u 
+(pr0_refl u) (Bind b)) (lift (S O) O t4) (pr0_lift t3 t4 H9 (S O) O)) c2 
+(drop_drop (Bind b) O c2 c2 (drop_refl c2) u))))))))) H13)) H12)))))))))) 
+(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: 
+(((eq T t3 (THead (Bind b) u t)) \to (arity g c2 t4 a2)))).(\lambda (u0: 
+T).(\lambda (H10: (eq T (THead (Flat Cast) u0 t3) (THead (Bind b) u t))).(let 
+H11 \def (eq_ind T (THead (Flat Cast) u0 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) u t) H10) in (False_ind (arity g c2 t4 a2) 
+H11)))))))) y t2 H7))) H6)))))))))))))))) (\lambda (c: C).(\lambda (u: 
+T).(\lambda (a1: A).(\lambda (_: (arity g c u (asucc g a1))).(\lambda (H1: 
+((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 u t2) \to (arity g 
+c2 t2 (asucc g a1)))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (H2: 
+(arity g (CHead c (Bind Abst) u) t a2)).(\lambda (H3: ((\forall (c2: 
+C).((wcpr0 (CHead c (Bind Abst) u) c2) \to (\forall (t2: T).((pr0 t t2) \to 
+(arity g c2 t2 a2))))))).(\lambda (c2: C).(\lambda (H4: (wcpr0 c 
+c2)).(\lambda (t2: T).(\lambda (H5: (pr0 (THead (Bind Abst) u t) 
+t2)).(insert_eq T (THead (Bind Abst) u t) (\lambda (t0: T).(pr0 t0 t2)) 
+(\lambda (_: T).(arity g c2 t2 (AHead a1 a2))) (\lambda (y: T).(\lambda (H6: 
+(pr0 y t2)).(pr0_ind (\lambda (t0: T).(\lambda (t3: T).((eq T t0 (THead (Bind 
+Abst) u t)) \to (arity g c2 t3 (AHead a1 a2))))) (\lambda (t0: T).(\lambda 
+(H7: (eq T t0 (THead (Bind Abst) u t))).(let H8 \def (f_equal T T (\lambda 
+(e: T).e) t0 (THead (Bind Abst) u t) H7) in (eq_ind_r T (THead (Bind Abst) u 
+t) (\lambda (t3: T).(arity g c2 t3 (AHead a1 a2))) (arity_head g c2 u a1 (H1 
+c2 H4 u (pr0_refl u)) t a2 (H3 (CHead c2 (Bind Abst) u) (wcpr0_comp c c2 H4 u 
+u (pr0_refl u) (Bind Abst)) t (pr0_refl t))) t0 H8)))) (\lambda (u1: 
+T).(\lambda (u2: T).(\lambda (H7: (pr0 u1 u2)).(\lambda (H8: (((eq T u1 
+(THead (Bind Abst) u t)) \to (arity g c2 u2 (AHead a1 a2))))).(\lambda (t3: 
+T).(\lambda (t4: T).(\lambda (H9: (pr0 t3 t4)).(\lambda (H10: (((eq T t3 
+(THead (Bind Abst) u t)) \to (arity g c2 t4 (AHead a1 a2))))).(\lambda (k: 
+K).(\lambda (H11: (eq T (THead k u1 t3) (THead (Bind Abst) u t))).(let H12 
+\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) 
+with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) 
+\Rightarrow k0])) (THead k u1 t3) (THead (Bind Abst) u t) H11) in ((let H13 
+\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) 
+with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t0 _) 
+\Rightarrow t0])) (THead k u1 t3) (THead (Bind Abst) u t) H11) 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 _ _ t0) 
+\Rightarrow t0])) (THead k u1 t3) (THead (Bind Abst) u t) H11) in (\lambda 
+(H15: (eq T u1 u)).(\lambda (H16: (eq K k (Bind Abst))).(eq_ind_r K (Bind 
+Abst) (\lambda (k0: K).(arity g c2 (THead k0 u2 t4) (AHead a1 a2))) (let H17 
+\def (eq_ind T t3 (\lambda (t0: T).((eq T t0 (THead (Bind Abst) u t)) \to 
+(arity g c2 t4 (AHead a1 a2)))) H10 t H14) in (let H18 \def (eq_ind T t3 
+(\lambda (t0: T).(pr0 t0 t4)) H9 t H14) in (let H19 \def (eq_ind T u1 
+(\lambda (t0: T).((eq T t0 (THead (Bind Abst) u t)) \to (arity g c2 u2 (AHead 
+a1 a2)))) H8 u H15) in (let H20 \def (eq_ind T u1 (\lambda (t0: T).(pr0 t0 
+u2)) H7 u H15) in (arity_head g c2 u2 a1 (H1 c2 H4 u2 H20) t4 a2 (H3 (CHead 
+c2 (Bind Abst) u2) (wcpr0_comp c c2 H4 u u2 H20 (Bind Abst)) t4 H18)))))) k 
+H16)))) H13)) H12)))))))))))) (\lambda (u0: T).(\lambda (v1: T).(\lambda (v2: 
+T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abst) u t)) 
+\to (arity g c2 v2 (AHead a1 a2))))).(\lambda (t3: T).(\lambda (t4: 
+T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead (Bind Abst) u t)) 
+\to (arity g c2 t4 (AHead a1 a2))))).(\lambda (H11: (eq T (THead (Flat Appl) 
+v1 (THead (Bind Abst) u0 t3)) (THead (Bind Abst) u t))).(let H12 \def (eq_ind 
+T (THead (Flat Appl) v1 (THead (Bind Abst) u0 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 Abst) u t) H11) in (False_ind (arity g c2 (THead 
+(Bind Abbr) v2 t4) (AHead a1 a2)) H12)))))))))))) (\lambda (b: B).(\lambda 
+(_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 
+v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abst) u t)) \to (arity g c2 v2 
+(AHead a1 a2))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 
+u2)).(\lambda (_: (((eq T u1 (THead (Bind Abst) u t)) \to (arity g c2 u2 
+(AHead a1 a2))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 
+t4)).(\lambda (_: (((eq T t3 (THead (Bind Abst) u t)) \to (arity g c2 t4 
+(AHead a1 a2))))).(\lambda (H14: (eq T (THead (Flat Appl) v1 (THead (Bind b) 
+u1 t3)) (THead (Bind Abst) u t))).(let H15 \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 
+Abst) u t) H14) in (False_ind (arity g c2 (THead (Bind b) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t4)) (AHead a1 a2)) H15))))))))))))))))) (\lambda 
+(u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 
+(THead (Bind Abst) u t)) \to (arity g c2 u2 (AHead a1 a2))))).(\lambda (t3: 
+T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead 
+(Bind Abst) u t)) \to (arity g c2 t4 (AHead a1 a2))))).(\lambda (w: 
+T).(\lambda (_: (subst0 O u2 t4 w)).(\lambda (H12: (eq T (THead (Bind Abbr) 
+u1 t3) (THead (Bind Abst) u t))).(let H13 \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 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 t) H12) in (False_ind (arity g 
+c2 (THead (Bind Abbr) u2 w) (AHead a1 a2)) H13))))))))))))) (\lambda (b: 
+B).(\lambda (H7: (not (eq B b Abst))).(\lambda (t3: T).(\lambda (t4: 
+T).(\lambda (_: (pr0 t3 t4)).(\lambda (H9: (((eq T t3 (THead (Bind Abst) u 
+t)) \to (arity g c2 t4 (AHead a1 a2))))).(\lambda (u0: T).(\lambda (H10: (eq 
+T (THead (Bind b) u0 (lift (S O) O t3)) (THead (Bind Abst) u t))).(let H11 
+\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) u0 (lift (S O) O 
+t3)) (THead (Bind Abst) u t) H10) in ((let H12 \def (f_equal T T (\lambda (e: 
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | 
+(TLRef _) \Rightarrow u0 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind b) 
+u0 (lift (S O) O t3)) (THead (Bind Abst) u t) H10) in ((let H13 \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) (t0: T) on t0: T 
+\def (match t0 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 t5) \Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d) 
+t5))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (TLRef _) 
+\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T 
+\def (match t0 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 t5) \Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d) 
+t5))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (THead _ _ t0) 
+\Rightarrow t0])) (THead (Bind b) u0 (lift (S O) O t3)) (THead (Bind Abst) u 
+t) H10) in (\lambda (_: (eq T u0 u)).(\lambda (H15: (eq B b Abst)).(let H16 
+\def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H7 Abst H15) in (let 
+H17 \def (eq_ind_r T t (\lambda (t0: T).((eq T t3 (THead (Bind Abst) u t0)) 
+\to (arity g c2 t4 (AHead a1 a2)))) H9 (lift (S O) O t3) H13) in (let H18 
+\def (eq_ind_r T t (\lambda (t0: T).(\forall (c3: C).((wcpr0 (CHead c (Bind 
+Abst) u) c3) \to (\forall (t5: T).((pr0 t0 t5) \to (arity g c3 t5 a2)))))) H3 
+(lift (S O) O t3) H13) in (let H19 \def (eq_ind_r T t (\lambda (t0: T).(arity 
+g (CHead c (Bind Abst) u) t0 a2)) H2 (lift (S O) O t3) H13) in (let H20 \def 
+(match (H16 (refl_equal B Abst)) in False return (\lambda (_: False).(arity g 
+c2 t4 (AHead a1 a2))) with []) in H20)))))))) H12)) H11)))))))))) (\lambda 
+(t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 
+(THead (Bind Abst) u t)) \to (arity g c2 t4 (AHead a1 a2))))).(\lambda (u0: 
+T).(\lambda (H9: (eq T (THead (Flat Cast) u0 t3) (THead (Bind Abst) u 
+t))).(let H10 \def (eq_ind T (THead (Flat Cast) u0 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 Abst) u t) H9) in (False_ind (arity g c2 
+t4 (AHead a1 a2)) H10)))))))) y t2 H6))) H5)))))))))))))) (\lambda (c: 
+C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda 
+(H1: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 u t2) \to 
+(arity g c2 t2 a1))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (H2: 
+(arity g c t (AHead a1 a2))).(\lambda (H3: ((\forall (c2: C).((wcpr0 c c2) 
+\to (\forall (t2: T).((pr0 t t2) \to (arity g c2 t2 (AHead a1 
+a2)))))))).(\lambda (c2: C).(\lambda (H4: (wcpr0 c c2)).(\lambda (t2: 
+T).(\lambda (H5: (pr0 (THead (Flat Appl) u t) t2)).(insert_eq T (THead (Flat 
+Appl) u t) (\lambda (t0: T).(pr0 t0 t2)) (\lambda (_: T).(arity g c2 t2 a2)) 
+(\lambda (y: T).(\lambda (H6: (pr0 y t2)).(pr0_ind (\lambda (t0: T).(\lambda 
+(t3: T).((eq T t0 (THead (Flat Appl) u t)) \to (arity g c2 t3 a2)))) (\lambda 
+(t0: T).(\lambda (H7: (eq T t0 (THead (Flat Appl) u t))).(let H8 \def 
+(f_equal T T (\lambda (e: T).e) t0 (THead (Flat Appl) u t) H7) in (eq_ind_r T 
+(THead (Flat Appl) u t) (\lambda (t3: T).(arity g c2 t3 a2)) (arity_appl g c2 
+u a1 (H1 c2 H4 u (pr0_refl u)) t a2 (H3 c2 H4 t (pr0_refl t))) t0 H8)))) 
+(\lambda (u1: T).(\lambda (u2: T).(\lambda (H7: (pr0 u1 u2)).(\lambda (H8: 
+(((eq T u1 (THead (Flat Appl) u t)) \to (arity g c2 u2 a2)))).(\lambda (t3: 
+T).(\lambda (t4: T).(\lambda (H9: (pr0 t3 t4)).(\lambda (H10: (((eq T t3 
+(THead (Flat Appl) u t)) \to (arity g c2 t4 a2)))).(\lambda (k: K).(\lambda 
+(H11: (eq T (THead k u1 t3) (THead (Flat Appl) u t))).(let H12 \def (f_equal 
+T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) 
+\Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) 
+(THead k u1 t3) (THead (Flat Appl) u t) H11) in ((let H13 \def (f_equal T T 
+(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
+\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t0 _) \Rightarrow t0])) 
+(THead k u1 t3) (THead (Flat Appl) u t) H11) 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 _ _ t0) \Rightarrow t0])) 
+(THead k u1 t3) (THead (Flat Appl) u t) H11) in (\lambda (H15: (eq T u1 
+u)).(\lambda (H16: (eq K k (Flat Appl))).(eq_ind_r K (Flat Appl) (\lambda 
+(k0: K).(arity g c2 (THead k0 u2 t4) a2)) (let H17 \def (eq_ind T t3 (\lambda 
+(t0: T).((eq T t0 (THead (Flat Appl) u t)) \to (arity g c2 t4 a2))) H10 t 
+H14) in (let H18 \def (eq_ind T t3 (\lambda (t0: T).(pr0 t0 t4)) H9 t H14) in 
+(let H19 \def (eq_ind T u1 (\lambda (t0: T).((eq T t0 (THead (Flat Appl) u 
+t)) \to (arity g c2 u2 a2))) H8 u H15) in (let H20 \def (eq_ind T u1 (\lambda 
+(t0: T).(pr0 t0 u2)) H7 u H15) in (arity_appl g c2 u2 a1 (H1 c2 H4 u2 H20) t4 
+a2 (H3 c2 H4 t4 H18)))))) k H16)))) H13)) H12)))))))))))) (\lambda (u0: 
+T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H7: (pr0 v1 v2)).(\lambda (H8: 
+(((eq T v1 (THead (Flat Appl) u t)) \to (arity g c2 v2 a2)))).(\lambda (t3: 
+T).(\lambda (t4: T).(\lambda (H9: (pr0 t3 t4)).(\lambda (H10: (((eq T t3 
+(THead (Flat Appl) u t)) \to (arity g c2 t4 a2)))).(\lambda (H11: (eq T 
+(THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) (THead (Flat Appl) u 
+t))).(let H12 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda 
+(_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead 
+_ t0 _) \Rightarrow t0])) (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) 
+(THead (Flat Appl) u t) H11) in ((let H13 \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 
+_ _ t0) \Rightarrow t0])) (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) 
+(THead (Flat Appl) u t) H11) in (\lambda (H14: (eq T v1 u)).(let H15 \def 
+(eq_ind T v1 (\lambda (t0: T).((eq T t0 (THead (Flat Appl) u t)) \to (arity g 
+c2 v2 a2))) H8 u H14) in (let H16 \def (eq_ind T v1 (\lambda (t0: T).(pr0 t0 
+v2)) H7 u H14) in (let H17 \def (eq_ind_r T t (\lambda (t0: T).((eq T t3 
+(THead (Flat Appl) u t0)) \to (arity g c2 t4 a2))) H10 (THead (Bind Abst) u0 
+t3) H13) in (let H18 \def (eq_ind_r T t (\lambda (t0: T).((eq T u (THead 
+(Flat Appl) u t0)) \to (arity g c2 v2 a2))) H15 (THead (Bind Abst) u0 t3) 
+H13) in (let H19 \def (eq_ind_r T t (\lambda (t0: T).(\forall (c3: C).((wcpr0 
+c c3) \to (\forall (t5: T).((pr0 t0 t5) \to (arity g c3 t5 (AHead a1 
+a2))))))) H3 (THead (Bind Abst) u0 t3) H13) in (let H20 \def (eq_ind_r T t 
+(\lambda (t0: T).(arity g c t0 (AHead a1 a2))) H2 (THead (Bind Abst) u0 t3) 
+H13) in (let H21 \def (H1 c2 H4 v2 H16) in (let H22 \def (H19 c2 H4 (THead 
+(Bind Abst) u0 t4) (pr0_comp u0 u0 (pr0_refl u0) t3 t4 H9 (Bind Abst))) in 
+(let H23 \def (arity_gen_abst g c2 u0 t4 (AHead a1 a2) H22) in (ex3_2_ind A A 
+(\lambda (a3: A).(\lambda (a4: A).(eq A (AHead a1 a2) (AHead a3 a4)))) 
+(\lambda (a3: A).(\lambda (_: A).(arity g c2 u0 (asucc g a3)))) (\lambda (_: 
+A).(\lambda (a4: A).(arity g (CHead c2 (Bind Abst) u0) t4 a4))) (arity g c2 
+(THead (Bind Abbr) v2 t4) a2) (\lambda (x0: A).(\lambda (x1: A).(\lambda 
+(H24: (eq A (AHead a1 a2) (AHead x0 x1))).(\lambda (H25: (arity g c2 u0 
+(asucc g x0))).(\lambda (H26: (arity g (CHead c2 (Bind Abst) u0) t4 x1)).(let 
+H27 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) 
+with [(ASort _ _) \Rightarrow a1 | (AHead a0 _) \Rightarrow a0])) (AHead a1 
+a2) (AHead x0 x1) H24) in ((let H28 \def (f_equal A A (\lambda (e: A).(match 
+e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a2 | (AHead _ 
+a0) \Rightarrow a0])) (AHead a1 a2) (AHead x0 x1) H24) in (\lambda (H29: (eq 
+A a1 x0)).(let H30 \def (eq_ind_r A x1 (\lambda (a0: A).(arity g (CHead c2 
+(Bind Abst) u0) t4 a0)) H26 a2 H28) in (let H31 \def (eq_ind_r A x0 (\lambda 
+(a0: A).(arity g c2 u0 (asucc g a0))) H25 a1 H29) in (arity_bind g Abbr 
+not_abbr_abst c2 v2 a1 H21 t4 a2 (csuba_arity g (CHead c2 (Bind Abst) u0) t4 
+a2 H30 (CHead c2 (Bind Abbr) v2) (csuba_abst g c2 c2 (csuba_refl g c2) u0 a1 
+H31 v2 H21))))))) H27))))))) H23)))))))))))) H12)))))))))))) (\lambda (b: 
+B).(\lambda (H7: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: 
+T).(\lambda (H8: (pr0 v1 v2)).(\lambda (H9: (((eq T v1 (THead (Flat Appl) u 
+t)) \to (arity g c2 v2 a2)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda 
+(H10: (pr0 u1 u2)).(\lambda (H11: (((eq T u1 (THead (Flat Appl) u t)) \to 
+(arity g c2 u2 a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H12: (pr0 
+t3 t4)).(\lambda (H13: (((eq T t3 (THead (Flat Appl) u t)) \to (arity g c2 t4 
+a2)))).(\lambda (H14: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) 
+(THead (Flat Appl) u t))).(let H15 \def (f_equal T T (\lambda (e: T).(match e 
+in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) 
+\Rightarrow v1 | (THead _ t0 _) \Rightarrow t0])) (THead (Flat Appl) v1 
+(THead (Bind b) u1 t3)) (THead (Flat Appl) u t) H14) in ((let H16 \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 _ _ t0) \Rightarrow t0])) (THead (Flat Appl) v1 
+(THead (Bind b) u1 t3)) (THead (Flat Appl) u t) H14) in (\lambda (H17: (eq T 
+v1 u)).(let H18 \def (eq_ind T v1 (\lambda (t0: T).((eq T t0 (THead (Flat 
+Appl) u t)) \to (arity g c2 v2 a2))) H9 u H17) in (let H19 \def (eq_ind T v1 
+(\lambda (t0: T).(pr0 t0 v2)) H8 u H17) in (let H20 \def (eq_ind_r T t 
+(\lambda (t0: T).((eq T t3 (THead (Flat Appl) u t0)) \to (arity g c2 t4 a2))) 
+H13 (THead (Bind b) u1 t3) H16) in (let H21 \def (eq_ind_r T t (\lambda (t0: 
+T).((eq T u1 (THead (Flat Appl) u t0)) \to (arity g c2 u2 a2))) H11 (THead 
+(Bind b) u1 t3) H16) in (let H22 \def (eq_ind_r T t (\lambda (t0: T).((eq T u 
+(THead (Flat Appl) u t0)) \to (arity g c2 v2 a2))) H18 (THead (Bind b) u1 t3) 
+H16) in (let H23 \def (eq_ind_r T t (\lambda (t0: T).(\forall (c3: C).((wcpr0 
+c c3) \to (\forall (t5: T).((pr0 t0 t5) \to (arity g c3 t5 (AHead a1 
+a2))))))) H3 (THead (Bind b) u1 t3) H16) in (let H24 \def (eq_ind_r T t 
+(\lambda (t0: T).(arity g c t0 (AHead a1 a2))) H2 (THead (Bind b) u1 t3) H16) 
+in (let H25 \def (H1 c2 H4 v2 H19) in (let H26 \def (H23 c2 H4 (THead (Bind 
+b) u2 t4) (pr0_comp u1 u2 H10 t3 t4 H12 (Bind b))) in (let H27 \def 
+(arity_gen_bind b H7 g c2 u2 t4 (AHead a1 a2) H26) in (ex2_ind A (\lambda 
+(a3: A).(arity g c2 u2 a3)) (\lambda (_: A).(arity g (CHead c2 (Bind b) u2) 
+t4 (AHead a1 a2))) (arity g c2 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
+O) O v2) t4)) a2) (\lambda (x: A).(\lambda (H28: (arity g c2 u2 x)).(\lambda 
+(H29: (arity g (CHead c2 (Bind b) u2) t4 (AHead a1 a2))).(arity_bind g b H7 
+c2 u2 x H28 (THead (Flat Appl) (lift (S O) O v2) t4) a2 (arity_appl g (CHead 
+c2 (Bind b) u2) (lift (S O) O v2) a1 (arity_lift g c2 v2 a1 H25 (CHead c2 
+(Bind b) u2) (S O) O (drop_drop (Bind b) O c2 c2 (drop_refl c2) u2)) t4 a2 
+H29))))) H27))))))))))))) H15))))))))))))))))) (\lambda (u1: T).(\lambda (u2: 
+T).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 (THead (Flat Appl) u t)) 
+\to (arity g c2 u2 a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 
+t3 t4)).(\lambda (_: (((eq T t3 (THead (Flat Appl) u t)) \to (arity g c2 t4 
+a2)))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t4 w)).(\lambda (H12: (eq T 
+(THead (Bind Abbr) u1 t3) (THead (Flat Appl) u t))).(let H13 \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 False])])) I (THead (Flat 
+Appl) u t) H12) in (False_ind (arity g c2 (THead (Bind Abbr) u2 w) a2) 
+H13))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda 
+(t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 
+(THead (Flat Appl) u t)) \to (arity g c2 t4 a2)))).(\lambda (u0: T).(\lambda 
+(H10: (eq T (THead (Bind b) u0 (lift (S O) O t3)) (THead (Flat Appl) u 
+t))).(let H11 \def (eq_ind T (THead (Bind b) u0 (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) u t) H10) in (False_ind 
+(arity g c2 t4 a2) H11)))))))))) (\lambda (t3: T).(\lambda (t4: T).(\lambda 
+(_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead (Flat Appl) u t)) \to (arity 
+g c2 t4 a2)))).(\lambda (u0: T).(\lambda (H9: (eq T (THead (Flat Cast) u0 t3) 
+(THead (Flat Appl) u t))).(let H10 \def (eq_ind T (THead (Flat Cast) u0 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) u t) 
+H9) in (False_ind (arity g c2 t4 a2) H10)))))))) y t2 H6))) H5)))))))))))))) 
+(\lambda (c: C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_: (arity g c u 
+(asucc g a0))).(\lambda (H1: ((\forall (c2: C).((wcpr0 c c2) \to (\forall 
+(t2: T).((pr0 u t2) \to (arity g c2 t2 (asucc g a0)))))))).(\lambda (t: 
+T).(\lambda (_: (arity g c t a0)).(\lambda (H3: ((\forall (c2: C).((wcpr0 c 
+c2) \to (\forall (t2: T).((pr0 t t2) \to (arity g c2 t2 a0))))))).(\lambda 
+(c2: C).(\lambda (H4: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H5: (pr0 
+(THead (Flat Cast) u t) t2)).(insert_eq T (THead (Flat Cast) u t) (\lambda 
+(t0: T).(pr0 t0 t2)) (\lambda (_: T).(arity g c2 t2 a0)) (\lambda (y: 
+T).(\lambda (H6: (pr0 y t2)).(pr0_ind (\lambda (t0: T).(\lambda (t3: T).((eq 
+T t0 (THead (Flat Cast) u t)) \to (arity g c2 t3 a0)))) (\lambda (t0: 
+T).(\lambda (H7: (eq T t0 (THead (Flat Cast) u t))).(let H8 \def (f_equal T T 
+(\lambda (e: T).e) t0 (THead (Flat Cast) u t) H7) in (eq_ind_r T (THead (Flat 
+Cast) u t) (\lambda (t3: T).(arity g c2 t3 a0)) (arity_cast g c2 u a0 (H1 c2 
+H4 u (pr0_refl u)) t (H3 c2 H4 t (pr0_refl t))) t0 H8)))) (\lambda (u1: 
+T).(\lambda (u2: T).(\lambda (H7: (pr0 u1 u2)).(\lambda (H8: (((eq T u1 
+(THead (Flat Cast) u t)) \to (arity g c2 u2 a0)))).(\lambda (t3: T).(\lambda 
+(t4: T).(\lambda (H9: (pr0 t3 t4)).(\lambda (H10: (((eq T t3 (THead (Flat 
+Cast) u t)) \to (arity g c2 t4 a0)))).(\lambda (k: K).(\lambda (H11: (eq T 
+(THead k u1 t3) (THead (Flat Cast) u t))).(let H12 \def (f_equal T K (\lambda 
+(e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k 
+| (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) 
+(THead (Flat Cast) u t) H11) in ((let H13 \def (f_equal T T (\lambda (e: 
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | 
+(TLRef _) \Rightarrow u1 | (THead _ t0 _) \Rightarrow t0])) (THead k u1 t3) 
+(THead (Flat Cast) u t) H11) 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 _ _ t0) \Rightarrow t0])) (THead k u1 t3) 
+(THead (Flat Cast) u t) H11) in (\lambda (H15: (eq T u1 u)).(\lambda (H16: 
+(eq K k (Flat Cast))).(eq_ind_r K (Flat Cast) (\lambda (k0: K).(arity g c2 
+(THead k0 u2 t4) a0)) (let H17 \def (eq_ind T t3 (\lambda (t0: T).((eq T t0 
+(THead (Flat Cast) u t)) \to (arity g c2 t4 a0))) H10 t H14) in (let H18 \def 
+(eq_ind T t3 (\lambda (t0: T).(pr0 t0 t4)) H9 t H14) in (let H19 \def (eq_ind 
+T u1 (\lambda (t0: T).((eq T t0 (THead (Flat Cast) u t)) \to (arity g c2 u2 
+a0))) H8 u H15) in (let H20 \def (eq_ind T u1 (\lambda (t0: T).(pr0 t0 u2)) 
+H7 u H15) in (arity_cast g c2 u2 a0 (H1 c2 H4 u2 H20) t4 (H3 c2 H4 t4 
+H18)))))) k H16)))) H13)) H12)))))))))))) (\lambda (u0: T).(\lambda (v1: 
+T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead 
+(Flat Cast) u t)) \to (arity g c2 v2 a0)))).(\lambda (t3: T).(\lambda (t4: 
+T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead (Flat Cast) u t)) 
+\to (arity g c2 t4 a0)))).(\lambda (H11: (eq T (THead (Flat Appl) v1 (THead 
+(Bind Abst) u0 t3)) (THead (Flat Cast) u t))).(let H12 \def (eq_ind T (THead 
+(Flat Appl) v1 (THead (Bind Abst) u0 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) u t) H11) in (False_ind (arity 
+g c2 (THead (Bind Abbr) v2 t4) a0) H12)))))))))))) (\lambda (b: B).(\lambda 
+(_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 
+v1 v2)).(\lambda (_: (((eq T v1 (THead (Flat Cast) u t)) \to (arity g c2 v2 
+a0)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda 
+(_: (((eq T u1 (THead (Flat Cast) u t)) \to (arity g c2 u2 a0)))).(\lambda 
+(t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 
+(THead (Flat Cast) u t)) \to (arity g c2 t4 a0)))).(\lambda (H14: (eq T 
+(THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Cast) u t))).(let 
+H15 \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 f) \Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl 
+\Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat Cast) u t) 
+H14) in (False_ind (arity g c2 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
+O) O v2) t4)) a0) H15))))))))))))))))) (\lambda (u1: T).(\lambda (u2: 
+T).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 (THead (Flat Cast) u t)) 
+\to (arity g c2 u2 a0)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 
+t3 t4)).(\lambda (_: (((eq T t3 (THead (Flat Cast) u t)) \to (arity g c2 t4 
+a0)))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t4 w)).(\lambda (H12: (eq T 
+(THead (Bind Abbr) u1 t3) (THead (Flat Cast) u t))).(let H13 \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 False])])) I (THead (Flat 
+Cast) u t) H12) in (False_ind (arity g c2 (THead (Bind Abbr) u2 w) a0) 
+H13))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda 
+(t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 
+(THead (Flat Cast) u t)) \to (arity g c2 t4 a0)))).(\lambda (u0: T).(\lambda 
+(H10: (eq T (THead (Bind b) u0 (lift (S O) O t3)) (THead (Flat Cast) u 
+t))).(let H11 \def (eq_ind T (THead (Bind b) u0 (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) u t) H10) in (False_ind 
+(arity g c2 t4 a0) H11)))))))))) (\lambda (t3: T).(\lambda (t4: T).(\lambda 
+(H7: (pr0 t3 t4)).(\lambda (H8: (((eq T t3 (THead (Flat Cast) u t)) \to 
+(arity g c2 t4 a0)))).(\lambda (u0: T).(\lambda (H9: (eq T (THead (Flat Cast) 
+u0 t3) (THead (Flat Cast) u t))).(let H10 \def (f_equal T T (\lambda (e: 
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | 
+(TLRef _) \Rightarrow u0 | (THead _ t0 _) \Rightarrow t0])) (THead (Flat 
+Cast) u0 t3) (THead (Flat Cast) u t) H9) in ((let H11 \def (f_equal T T 
+(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
+\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t0) \Rightarrow t0])) 
+(THead (Flat Cast) u0 t3) (THead (Flat Cast) u t) H9) in (\lambda (_: (eq T 
+u0 u)).(let H13 \def (eq_ind T t3 (\lambda (t0: T).((eq T t0 (THead (Flat 
+Cast) u t)) \to (arity g c2 t4 a0))) H8 t H11) in (let H14 \def (eq_ind T t3 
+(\lambda (t0: T).(pr0 t0 t4)) H7 t H11) in (H3 c2 H4 t4 H14))))) H10)))))))) 
+y t2 H6))) H5))))))))))))) (\lambda (c: C).(\lambda (t: T).(\lambda (a1: 
+A).(\lambda (_: (arity g c t a1)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c 
+c2) \to (\forall (t2: T).((pr0 t t2) \to (arity g c2 t2 a1))))))).(\lambda 
+(a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda (c2: C).(\lambda (H3: (wcpr0 c 
+c2)).(\lambda (t2: T).(\lambda (H4: (pr0 t t2)).(arity_repl g c2 t2 a1 (H1 c2 
+H3 t2 H4) a2 H2)))))))))))) c1 t1 a H))))).
+
+theorem arity_sred_wcpr0_pr1:
+ \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (g: G).(\forall 
+(c1: C).(\forall (a: A).((arity g c1 t1 a) \to (\forall (c2: C).((wcpr0 c1 
+c2) \to (arity g c2 t2 a)))))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr1 t1 t2)).(pr1_ind (\lambda 
+(t: T).(\lambda (t0: T).(\forall (g: G).(\forall (c1: C).(\forall (a: 
+A).((arity g c1 t a) \to (\forall (c2: C).((wcpr0 c1 c2) \to (arity g c2 t0 
+a))))))))) (\lambda (t: T).(\lambda (g: G).(\lambda (c1: C).(\lambda (a: 
+A).(\lambda (H0: (arity g c1 t a)).(\lambda (c2: C).(\lambda (H1: (wcpr0 c1 
+c2)).(arity_sred_wcpr0_pr0 g c1 t a H0 c2 H1 t (pr0_refl t))))))))) (\lambda 
+(t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t4 t3)).(\lambda (t5: T).(\lambda 
+(_: (pr1 t3 t5)).(\lambda (H2: ((\forall (g: G).(\forall (c1: C).(\forall (a: 
+A).((arity g c1 t3 a) \to (\forall (c2: C).((wcpr0 c1 c2) \to (arity g c2 t5 
+a))))))))).(\lambda (g: G).(\lambda (c1: C).(\lambda (a: A).(\lambda (H3: 
+(arity g c1 t4 a)).(\lambda (c2: C).(\lambda (H4: (wcpr0 c1 c2)).(H2 g c2 a 
+(arity_sred_wcpr0_pr0 g c1 t4 a H3 c2 H4 t3 H0) c2 (wcpr0_refl 
+c2)))))))))))))) t1 t2 H))).
+
+theorem arity_sred_pr2:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall 
+(g: G).(\forall (a: A).((arity g c t1 a) \to (arity g c t2 a)))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 
+t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (g: 
+G).(\forall (a: A).((arity g c0 t a) \to (arity g c0 t0 a))))))) (\lambda 
+(c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda 
+(g: G).(\lambda (a: A).(\lambda (H1: (arity g c0 t3 a)).(arity_sred_wcpr0_pr0 
+g c0 t3 a H1 c0 (wcpr0_refl c0) t4 H0)))))))) (\lambda (c0: C).(\lambda (d: 
+C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind 
+Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 
+t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (g: 
+G).(\lambda (a: A).(\lambda (H3: (arity g c0 t3 a)).(arity_subst0 g c0 t4 a 
+(arity_sred_wcpr0_pr0 g c0 t3 a H3 c0 (wcpr0_refl c0) t4 H1) d u i H0 t 
+H2)))))))))))))) c t1 t2 H)))).
+
+theorem arity_sred_pr3:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall 
+(g: G).(\forall (a: A).((arity g c t1 a) \to (arity g c t2 a)))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 
+t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (g: G).(\forall (a: 
+A).((arity g c t a) \to (arity g c t0 a)))))) (\lambda (t: T).(\lambda (g: 
+G).(\lambda (a: A).(\lambda (H0: (arity g c t a)).H0)))) (\lambda (t3: 
+T).(\lambda (t4: T).(\lambda (H0: (pr2 c t4 t3)).(\lambda (t5: T).(\lambda 
+(_: (pr3 c t3 t5)).(\lambda (H2: ((\forall (g: G).(\forall (a: A).((arity g c 
+t3 a) \to (arity g c t5 a)))))).(\lambda (g: G).(\lambda (a: A).(\lambda (H3: 
+(arity g c t4 a)).(H2 g a (arity_sred_pr2 c t4 t3 H0 g a H3))))))))))) t1 t2 
+H)))).
+