LAMBDA-TYPES: some more generation lemmas and some proofs improved

[helm.git] / helm / software / matita / contribs / LAMBDA-TYPES / LambdaDelta-1 / arity / subst0.ma

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/subst0.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/subst0.ma
index 006b6c4937d0bee87d804103bb54e5b68c7226a5..d62f665f43deb3b9ab396fbdff1355ade3aa0248 100644 (file)
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/subst0.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/subst0.ma
@@ -270,153 +270,153 @@ d1 (Bind Abbr) u)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u c t c2
 t2) \to (arity g c2 t2 a0))))))))))) (\lambda (c: C).(\lambda (n: 
 nat).(\lambda (d1: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: (getl i 
 c (CHead d1 (Bind Abbr) u))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H1: 
-(fsubst0 i u c (TSort n) c2 t2)).(let H2 \def (fsubst0_gen_base c c2 (TSort 
-n) t2 u i H1) in (or3_ind (land (eq C c c2) (subst0 i u (TSort n) t2)) (land 
-(eq T (TSort n) t2) (csubst0 i u c c2)) (land (subst0 i u (TSort n) t2) 
-(csubst0 i u c c2)) (arity g c2 t2 (ASort O n)) (\lambda (H3: (land (eq C c 
-c2) (subst0 i u (TSort n) t2))).(and_ind (eq C c c2) (subst0 i u (TSort n) 
-t2) (arity g c2 t2 (ASort O n)) (\lambda (H4: (eq C c c2)).(\lambda (H5: 
-(subst0 i u (TSort n) t2)).(eq_ind C c (\lambda (c0: C).(arity g c0 t2 (ASort 
-O n))) (subst0_gen_sort u t2 i n H5 (arity g c t2 (ASort O n))) c2 H4))) H3)) 
-(\lambda (H3: (land (eq T (TSort n) t2) (csubst0 i u c c2))).(and_ind (eq T 
-(TSort n) t2) (csubst0 i u c c2) (arity g c2 t2 (ASort O n)) (\lambda (H4: 
-(eq T (TSort n) t2)).(\lambda (_: (csubst0 i u c c2)).(eq_ind T (TSort n) 
-(\lambda (t: T).(arity g c2 t (ASort O n))) (arity_sort g c2 n) t2 H4))) H3)) 
-(\lambda (H3: (land (subst0 i u (TSort n) t2) (csubst0 i u c c2))).(and_ind 
-(subst0 i u (TSort n) t2) (csubst0 i u c c2) (arity g c2 t2 (ASort O n)) 
-(\lambda (H4: (subst0 i u (TSort n) t2)).(\lambda (_: (csubst0 i u c 
-c2)).(subst0_gen_sort u t2 i n H4 (arity g c2 t2 (ASort O n))))) H3)) 
-H2))))))))))) (\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 (H1: (arity g d u a0)).(\lambda (H2: ((\forall (d1: C).(\forall 
-(u0: T).(\forall (i0: nat).((getl i0 d (CHead d1 (Bind Abbr) u0)) \to 
-(\forall (c2: C).(\forall (t2: T).((fsubst0 i0 u0 d u c2 t2) \to (arity g c2 
-t2 a0)))))))))).(\lambda (d1: C).(\lambda (u0: T).(\lambda (i0: nat).(\lambda 
-(H3: (getl i0 c (CHead d1 (Bind Abbr) u0))).(\lambda (c2: C).(\lambda (t2: 
-T).(\lambda (H4: (fsubst0 i0 u0 c (TLRef i) c2 t2)).(let H5 \def 
-(fsubst0_gen_base c c2 (TLRef i) t2 u0 i0 H4) in (or3_ind (land (eq C c c2) 
-(subst0 i0 u0 (TLRef i) t2)) (land (eq T (TLRef i) t2) (csubst0 i0 u0 c c2)) 
-(land (subst0 i0 u0 (TLRef i) t2) (csubst0 i0 u0 c c2)) (arity g c2 t2 a0) 
-(\lambda (H6: (land (eq C c c2) (subst0 i0 u0 (TLRef i) t2))).(and_ind (eq C 
-c c2) (subst0 i0 u0 (TLRef i) t2) (arity g c2 t2 a0) (\lambda (H7: (eq C c 
-c2)).(\lambda (H8: (subst0 i0 u0 (TLRef i) t2)).(eq_ind C c (\lambda (c0: 
-C).(arity g c0 t2 a0)) (and_ind (eq nat i i0) (eq T t2 (lift (S i) O u0)) 
-(arity g c t2 a0) (\lambda (H9: (eq nat i i0)).(\lambda (H10: (eq T t2 (lift 
-(S i) O u0))).(eq_ind_r T (lift (S i) O u0) (\lambda (t: T).(arity g c t a0)) 
-(let H11 \def (eq_ind_r nat i0 (\lambda (n: nat).(getl n c (CHead d1 (Bind 
-Abbr) u0))) H3 i H9) in (let H12 \def (eq_ind C (CHead d (Bind Abbr) u) 
-(\lambda (c0: C).(getl i c c0)) H0 (CHead d1 (Bind Abbr) u0) (getl_mono c 
-(CHead d (Bind Abbr) u) i H0 (CHead d1 (Bind Abbr) u0) H11)) in (let H13 \def 
-(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with 
-[(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind 
-Abbr) u) (CHead d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) i H0 
-(CHead d1 (Bind Abbr) u0) H11)) in ((let H14 \def (f_equal C T (\lambda (e: 
-C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | 
-(CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abbr) u) (CHead d1 (Bind Abbr) 
-u0) (getl_mono c (CHead d (Bind Abbr) u) i H0 (CHead d1 (Bind Abbr) u0) H11)) 
-in (\lambda (H15: (eq C d d1)).(let H16 \def (eq_ind_r T u0 (\lambda (t: 
-T).(getl i c (CHead d1 (Bind Abbr) t))) H12 u H14) in (eq_ind T u (\lambda 
-(t: T).(arity g c (lift (S i) O t) a0)) (let H17 \def (eq_ind_r C d1 (\lambda 
-(c0: C).(getl i c (CHead c0 (Bind Abbr) u))) H16 d H15) in (arity_lift g d u 
-a0 H1 c (S i) O (getl_drop Abbr c d u i H17))) u0 H14)))) H13)))) t2 H10))) 
-(subst0_gen_lref u0 t2 i0 i H8)) c2 H7))) H6)) (\lambda (H6: (land (eq T 
-(TLRef i) t2) (csubst0 i0 u0 c c2))).(and_ind (eq T (TLRef i) t2) (csubst0 i0 
-u0 c c2) (arity g c2 t2 a0) (\lambda (H7: (eq T (TLRef i) t2)).(\lambda (H8: 
-(csubst0 i0 u0 c c2)).(eq_ind T (TLRef i) (\lambda (t: T).(arity g c2 t a0)) 
-(lt_le_e i i0 (arity g c2 (TLRef i) a0) (\lambda (H9: (lt i i0)).(let H10 
-\def (csubst0_getl_lt i0 i H9 c c2 u0 H8 (CHead d (Bind Abbr) u) H0) in 
-(or4_ind (getl i c2 (CHead d (Bind Abbr) u)) (ex3_4 B C T T (\lambda (b: 
-B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind 
-Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(getl i c2 (CHead e0 (Bind b) w)))))) 
-(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
-(minus i0 (S i)) u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 
-(Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(u1: T).(getl i c2 (CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) 
-(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) 
-u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(getl i c2 (CHead e2 (Bind b) w))))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
-(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))) 
-(arity g c2 (TLRef i) a0) (\lambda (H11: (getl i c2 (CHead d (Bind Abbr) 
-u))).(let H12 \def (eq_ind nat (minus i0 i) (\lambda (n: nat).(getl n (CHead 
-d (Bind Abbr) u) (CHead d1 (Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind 
-Abbr) u0) c H3 (CHead d (Bind Abbr) u) i H0 (le_S_n i i0 (le_S (S i) i0 H9))) 
-(S (minus i0 (S i))) (minus_x_Sy i0 i H9)) in (arity_abbr g c2 d u i H11 a0 
-H1))) (\lambda (H11: (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda 
-(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) 
-u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(getl i c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 
-w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: 
-T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) u1)))))) 
-(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl i c2 
-(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w))))) (arity g c2 (TLRef 
-i) a0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: 
-T).(\lambda (H12: (eq C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) 
-x2))).(\lambda (H13: (getl i c2 (CHead x1 (Bind x0) x3))).(\lambda (H14: 
-(subst0 (minus i0 (S i)) u0 x2 x3)).(let H15 \def (eq_ind nat (minus i0 i) 
-(\lambda (n: nat).(getl n (CHead d (Bind Abbr) u) (CHead d1 (Bind Abbr) u0))) 
-(getl_conf_le i0 (CHead d1 (Bind Abbr) u0) c H3 (CHead d (Bind Abbr) u) i H0 
-(le_S_n i i0 (le_S (S i) i0 H9))) (S (minus i0 (S i))) (minus_x_Sy i0 i H9)) 
-in (let H16 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda 
-(_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) 
-(CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H12) in ((let H17 \def 
-(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with 
-[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k in K return 
-(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow 
-Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H12) in ((let H18 
-\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) 
-with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind 
-Abbr) u) (CHead x1 (Bind x0) x2) H12) in (\lambda (H19: (eq B Abbr 
-x0)).(\lambda (H20: (eq C d x1)).(let H21 \def (eq_ind_r T x2 (\lambda (t: 
-T).(subst0 (minus i0 (S i)) u0 t x3)) H14 u H18) in (let H22 \def (eq_ind_r C 
-x1 (\lambda (c0: C).(getl i c2 (CHead c0 (Bind x0) x3))) H13 d H20) in (let 
-H23 \def (eq_ind_r B x0 (\lambda (b: B).(getl i c2 (CHead d (Bind b) x3))) 
-H22 Abbr H19) in (arity_abbr g c2 d x3 i H23 a0 (H2 d1 u0 (r (Bind Abbr) 
-(minus i0 (S i))) (getl_gen_S (Bind Abbr) d (CHead d1 (Bind Abbr) u0) u 
-(minus i0 (S i)) H15) d x3 (fsubst0_snd (r (Bind Abbr) (minus i0 (S i))) u0 d 
-u x3 H21))))))))) H17)) H16)))))))))) H11)) (\lambda (H11: (ex3_4 B C C T 
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C 
-(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c2 (CHead e2 (Bind b) 
-u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda (b: 
+(fsubst0 i u c (TSort n) c2 t2)).(let H_x \def (fsubst0_gen_base c c2 (TSort 
+n) t2 u i H1) in (let H2 \def H_x in (or3_ind (land (eq C c c2) (subst0 i u 
+(TSort n) t2)) (land (eq T (TSort n) t2) (csubst0 i u c c2)) (land (subst0 i 
+u (TSort n) t2) (csubst0 i u c c2)) (arity g c2 t2 (ASort O n)) (\lambda (H3: 
+(land (eq C c c2) (subst0 i u (TSort n) t2))).(and_ind (eq C c c2) (subst0 i 
+u (TSort n) t2) (arity g c2 t2 (ASort O n)) (\lambda (H4: (eq C c 
+c2)).(\lambda (H5: (subst0 i u (TSort n) t2)).(eq_ind C c (\lambda (c0: 
+C).(arity g c0 t2 (ASort O n))) (subst0_gen_sort u t2 i n H5 (arity g c t2 
+(ASort O n))) c2 H4))) H3)) (\lambda (H3: (land (eq T (TSort n) t2) (csubst0 
+i u c c2))).(and_ind (eq T (TSort n) t2) (csubst0 i u c c2) (arity g c2 t2 
+(ASort O n)) (\lambda (H4: (eq T (TSort n) t2)).(\lambda (_: (csubst0 i u c 
+c2)).(eq_ind T (TSort n) (\lambda (t: T).(arity g c2 t (ASort O n))) 
+(arity_sort g c2 n) t2 H4))) H3)) (\lambda (H3: (land (subst0 i u (TSort n) 
+t2) (csubst0 i u c c2))).(and_ind (subst0 i u (TSort n) t2) (csubst0 i u c 
+c2) (arity g c2 t2 (ASort O n)) (\lambda (H4: (subst0 i u (TSort n) 
+t2)).(\lambda (_: (csubst0 i u c c2)).(subst0_gen_sort u t2 i n H4 (arity g 
+c2 t2 (ASort O n))))) H3)) H2)))))))))))) (\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 (H1: (arity g d u a0)).(\lambda (H2: 
+((\forall (d1: C).(\forall (u0: T).(\forall (i0: nat).((getl i0 d (CHead d1 
+(Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i0 u0 d u c2 
+t2) \to (arity g c2 t2 a0)))))))))).(\lambda (d1: C).(\lambda (u0: 
+T).(\lambda (i0: nat).(\lambda (H3: (getl i0 c (CHead d1 (Bind Abbr) 
+u0))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H4: (fsubst0 i0 u0 c (TLRef 
+i) c2 t2)).(let H_x \def (fsubst0_gen_base c c2 (TLRef i) t2 u0 i0 H4) in 
+(let H5 \def H_x in (or3_ind (land (eq C c c2) (subst0 i0 u0 (TLRef i) t2)) 
+(land (eq T (TLRef i) t2) (csubst0 i0 u0 c c2)) (land (subst0 i0 u0 (TLRef i) 
+t2) (csubst0 i0 u0 c c2)) (arity g c2 t2 a0) (\lambda (H6: (land (eq C c c2) 
+(subst0 i0 u0 (TLRef i) t2))).(and_ind (eq C c c2) (subst0 i0 u0 (TLRef i) 
+t2) (arity g c2 t2 a0) (\lambda (H7: (eq C c c2)).(\lambda (H8: (subst0 i0 u0 
+(TLRef i) t2)).(eq_ind C c (\lambda (c0: C).(arity g c0 t2 a0)) (and_ind (eq 
+nat i i0) (eq T t2 (lift (S i) O u0)) (arity g c t2 a0) (\lambda (H9: (eq nat 
+i i0)).(\lambda (H10: (eq T t2 (lift (S i) O u0))).(eq_ind_r T (lift (S i) O 
+u0) (\lambda (t: T).(arity g c t a0)) (let H11 \def (eq_ind_r nat i0 (\lambda 
+(n: nat).(getl n c (CHead d1 (Bind Abbr) u0))) H3 i H9) in (let H12 \def 
+(eq_ind C (CHead d (Bind Abbr) u) (\lambda (c0: C).(getl i c c0)) H0 (CHead 
+d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) i H0 (CHead d1 (Bind 
+Abbr) u0) H11)) in (let H13 \def (f_equal C C (\lambda (e: C).(match e in C 
+return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) 
+\Rightarrow c0])) (CHead d (Bind Abbr) u) (CHead d1 (Bind Abbr) u0) 
+(getl_mono c (CHead d (Bind Abbr) u) i H0 (CHead d1 (Bind Abbr) u0) H11)) in 
+((let H14 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: 
+C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d 
+(Bind Abbr) u) (CHead d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) 
+i H0 (CHead d1 (Bind Abbr) u0) H11)) in (\lambda (H15: (eq C d d1)).(let H16 
+\def (eq_ind_r T u0 (\lambda (t: T).(getl i c (CHead d1 (Bind Abbr) t))) H12 
+u H14) in (eq_ind T u (\lambda (t: T).(arity g c (lift (S i) O t) a0)) (let 
+H17 \def (eq_ind_r C d1 (\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) u))) 
+H16 d H15) in (arity_lift g d u a0 H1 c (S i) O (getl_drop Abbr c d u i 
+H17))) u0 H14)))) H13)))) t2 H10))) (subst0_gen_lref u0 t2 i0 i H8)) c2 H7))) 
+H6)) (\lambda (H6: (land (eq T (TLRef i) t2) (csubst0 i0 u0 c c2))).(and_ind 
+(eq T (TLRef i) t2) (csubst0 i0 u0 c c2) (arity g c2 t2 a0) (\lambda (H7: (eq 
+T (TLRef i) t2)).(\lambda (H8: (csubst0 i0 u0 c c2)).(eq_ind T (TLRef i) 
+(\lambda (t: T).(arity g c2 t a0)) (lt_le_e i i0 (arity g c2 (TLRef i) a0) 
+(\lambda (H9: (lt i i0)).(let H10 \def (csubst0_getl_lt i0 i H9 c c2 u0 H8 
+(CHead d (Bind Abbr) u) H0) in (or4_ind (getl i c2 (CHead d (Bind Abbr) u)) 
+(ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: 
+T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: 
+B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl i c2 (CHead e0 
+(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(w: T).(subst0 (minus i0 (S i)) u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: 
 B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind 
 Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda 
 (e2: C).(\lambda (u1: T).(getl i c2 (CHead e2 (Bind b) u1)))))) (\lambda (_: 
 B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S 
-i)) u0 e1 e2))))) (arity g c2 (TLRef i) a0) (\lambda (x0: B).(\lambda (x1: 
-C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H12: (eq C (CHead d (Bind 
-Abbr) u) (CHead x1 (Bind x0) x3))).(\lambda (H13: (getl i c2 (CHead x2 (Bind 
-x0) x3))).(\lambda (H14: (csubst0 (minus i0 (S i)) u0 x1 x2)).(let H15 \def 
+i)) u0 e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda 
+(_: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead 
+e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(getl i c2 (CHead e2 (Bind b) w))))))) 
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: 
+T).(subst0 (minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) 
+u0 e1 e2))))))) (arity g c2 (TLRef i) a0) (\lambda (H11: (getl i c2 (CHead d 
+(Bind Abbr) u))).(let H12 \def (eq_ind nat (minus i0 i) (\lambda (n: 
+nat).(getl n (CHead d (Bind Abbr) u) (CHead d1 (Bind Abbr) u0))) 
+(getl_conf_le i0 (CHead d1 (Bind Abbr) u0) c H3 (CHead d (Bind Abbr) u) i H0 
+(le_S_n i i0 (le_S (S i) i0 H9))) (S (minus i0 (S i))) (minus_x_Sy i0 i H9)) 
+in (arity_abbr g c2 d u i H11 a0 H1))) (\lambda (H11: (ex3_4 B C T T (\lambda 
+(b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind 
+Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (w: T).(getl i c2 (CHead e0 (Bind b) w)))))) 
+(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
+(minus i0 (S i)) u0 u1 w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda 
+(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead 
+e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(getl i c2 (CHead e0 (Bind b) w)))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) 
+u0 u1 w))))) (arity g c2 (TLRef i) a0) (\lambda (x0: B).(\lambda (x1: 
+C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (eq C (CHead d (Bind 
+Abbr) u) (CHead x1 (Bind x0) x2))).(\lambda (H13: (getl i c2 (CHead x1 (Bind 
+x0) x3))).(\lambda (H14: (subst0 (minus i0 (S i)) u0 x2 x3)).(let H15 \def 
 (eq_ind nat (minus i0 i) (\lambda (n: nat).(getl n (CHead d (Bind Abbr) u) 
 (CHead d1 (Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind Abbr) u0) c H3 
 (CHead d (Bind Abbr) u) i H0 (le_S_n i i0 (le_S (S i) i0 H9))) (S (minus i0 
 (S i))) (minus_x_Sy i0 i H9)) in (let H16 \def (f_equal C C (\lambda (e: 
 C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | 
 (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) 
-x3) H12) in ((let H17 \def (f_equal C B (\lambda (e: C).(match e in C return 
+x2) H12) in ((let H17 \def (f_equal C B (\lambda (e: C).(match e in C return 
 (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) 
 \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) 
 \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead 
-x1 (Bind x0) x3) H12) in ((let H18 \def (f_equal C T (\lambda (e: C).(match e 
+x1 (Bind x0) x2) H12) in ((let H18 \def (f_equal C T (\lambda (e: C).(match e 
 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) 
-\Rightarrow t])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H12) in 
+\Rightarrow t])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H12) in 
 (\lambda (H19: (eq B Abbr x0)).(\lambda (H20: (eq C d x1)).(let H21 \def 
-(eq_ind_r T x3 (\lambda (t: T).(getl i c2 (CHead x2 (Bind x0) t))) H13 u H18) 
-in (let H22 \def (eq_ind_r C x1 (\lambda (c0: C).(csubst0 (minus i0 (S i)) u0 
-c0 x2)) H14 d H20) in (let H23 \def (eq_ind_r B x0 (\lambda (b: B).(getl i c2 
-(CHead x2 (Bind b) u))) H21 Abbr H19) in (arity_abbr g c2 x2 u i H23 a0 (H2 
-d1 u0 (r (Bind Abbr) (minus i0 (S i))) (getl_gen_S (Bind Abbr) d (CHead d1 
-(Bind Abbr) u0) u (minus i0 (S i)) H15) x2 u (fsubst0_fst (r (Bind Abbr) 
-(minus i0 (S i))) u0 d u x2 H22))))))))) H17)) H16)))))))))) H11)) (\lambda 
-(H11: (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+(eq_ind_r T x2 (\lambda (t: T).(subst0 (minus i0 (S i)) u0 t x3)) H14 u H18) 
+in (let H22 \def (eq_ind_r C x1 (\lambda (c0: C).(getl i c2 (CHead c0 (Bind 
+x0) x3))) H13 d H20) in (let H23 \def (eq_ind_r B x0 (\lambda (b: B).(getl i 
+c2 (CHead d (Bind b) x3))) H22 Abbr H19) in (arity_abbr g c2 d x3 i H23 a0 
+(H2 d1 u0 (r (Bind Abbr) (minus i0 (S i))) (getl_gen_S (Bind Abbr) d (CHead 
+d1 (Bind Abbr) u0) u (minus i0 (S i)) H15) d x3 (fsubst0_snd (r (Bind Abbr) 
+(minus i0 (S i))) u0 d u x3 H21))))))))) H17)) H16)))))))))) H11)) (\lambda 
+(H11: (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) 
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c2 
+(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C 
+T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C 
+(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c2 (CHead e2 (Bind b) 
+u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (minus i0 (S i)) u0 e1 e2))))) (arity g c2 (TLRef i) a0) (\lambda 
+(x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H12: (eq 
+C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3))).(\lambda (H13: (getl i c2 
+(CHead x2 (Bind x0) x3))).(\lambda (H14: (csubst0 (minus i0 (S i)) u0 x1 
+x2)).(let H15 \def (eq_ind nat (minus i0 i) (\lambda (n: nat).(getl n (CHead 
+d (Bind Abbr) u) (CHead d1 (Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind 
+Abbr) u0) c H3 (CHead d (Bind Abbr) u) i H0 (le_S_n i i0 (le_S (S i) i0 H9))) 
+(S (minus i0 (S i))) (minus_x_Sy i0 i H9)) in (let H16 \def (f_equal C C 
+(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) 
+\Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abbr) u) 
+(CHead x1 (Bind x0) x3) H12) in ((let H17 \def (f_equal C B (\lambda (e: 
+C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | 
+(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind 
+b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) 
+(CHead x1 (Bind x0) x3) H12) in ((let H18 \def (f_equal C T (\lambda (e: 
+C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | 
+(CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) 
+x3) H12) in (\lambda (H19: (eq B Abbr x0)).(\lambda (H20: (eq C d x1)).(let 
+H21 \def (eq_ind_r T x3 (\lambda (t: T).(getl i c2 (CHead x2 (Bind x0) t))) 
+H13 u H18) in (let H22 \def (eq_ind_r C x1 (\lambda (c0: C).(csubst0 (minus 
+i0 (S i)) u0 c0 x2)) H14 d H20) in (let H23 \def (eq_ind_r B x0 (\lambda (b: 
+B).(getl i c2 (CHead x2 (Bind b) u))) H21 Abbr H19) in (arity_abbr g c2 x2 u 
+i H23 a0 (H2 d1 u0 (r (Bind Abbr) (minus i0 (S i))) (getl_gen_S (Bind Abbr) d 
+(CHead d1 (Bind Abbr) u0) u (minus i0 (S i)) H15) x2 u (fsubst0_fst (r (Bind 
+Abbr) (minus i0 (S i))) u0 d u x2 H22))))))))) H17)) H16)))))))))) H11)) 
+(\lambda (H11: (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
 C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 
 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda 
 (_: T).(\lambda (w: T).(getl i c2 (CHead e2 (Bind b) w))))))) (\lambda (_: 
@@ -483,30 +483,30 @@ H11 u H15) in (eq_ind T u (\lambda (t: T).(arity g c2 (lift (S i) O t) a0))
 u))) H17 d H16) in (arity_lift g d u a0 H1 c2 (S i) O (getl_drop Abbr c2 d u 
 i (csubst0_getl_ge i i (le_n i) c c2 u H18 (CHead d (Bind Abbr) u) H19)))) u0 
 H15))))) H14))))) t2 H10))) (subst0_gen_lref u0 t2 i0 i H7)))) H6)) 
-H5))))))))))))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda 
+H5)))))))))))))))))) (\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 (H1: (arity g d u (asucc g a0))).(\lambda (H2: ((\forall (d1: 
 C).(\forall (u0: T).(\forall (i0: nat).((getl i0 d (CHead d1 (Bind Abbr) u0)) 
 \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i0 u0 d u c2 t2) \to (arity g 
 c2 t2 (asucc g a0))))))))))).(\lambda (d1: C).(\lambda (u0: T).(\lambda (i0: 
 nat).(\lambda (H3: (getl i0 c (CHead d1 (Bind Abbr) u0))).(\lambda (c2: 
-C).(\lambda (t2: T).(\lambda (H4: (fsubst0 i0 u0 c (TLRef i) c2 t2)).(let H5 
-\def (fsubst0_gen_base c c2 (TLRef i) t2 u0 i0 H4) in (or3_ind (land (eq C c 
-c2) (subst0 i0 u0 (TLRef i) t2)) (land (eq T (TLRef i) t2) (csubst0 i0 u0 c 
-c2)) (land (subst0 i0 u0 (TLRef i) t2) (csubst0 i0 u0 c c2)) (arity g c2 t2 
-a0) (\lambda (H6: (land (eq C c c2) (subst0 i0 u0 (TLRef i) t2))).(and_ind 
-(eq C c c2) (subst0 i0 u0 (TLRef i) t2) (arity g c2 t2 a0) (\lambda (H7: (eq 
-C c c2)).(\lambda (H8: (subst0 i0 u0 (TLRef i) t2)).(eq_ind C c (\lambda (c0: 
-C).(arity g c0 t2 a0)) (and_ind (eq nat i i0) (eq T t2 (lift (S i) O u0)) 
-(arity g c t2 a0) (\lambda (H9: (eq nat i i0)).(\lambda (H10: (eq T t2 (lift 
-(S i) O u0))).(eq_ind_r T (lift (S i) O u0) (\lambda (t: T).(arity g c t a0)) 
-(let H11 \def (eq_ind_r nat i0 (\lambda (n: nat).(getl n c (CHead d1 (Bind 
-Abbr) u0))) H3 i H9) in (let H12 \def (eq_ind C (CHead d (Bind Abst) u) 
-(\lambda (c0: C).(getl i c c0)) H0 (CHead d1 (Bind Abbr) u0) (getl_mono c 
-(CHead d (Bind Abst) u) i H0 (CHead d1 (Bind Abbr) u0) H11)) in (let H13 \def 
-(eq_ind C (CHead d (Bind Abst) u) (\lambda (ee: C).(match ee in C return 
-(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) 
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) 
+C).(\lambda (t2: T).(\lambda (H4: (fsubst0 i0 u0 c (TLRef i) c2 t2)).(let H_x 
+\def (fsubst0_gen_base c c2 (TLRef i) t2 u0 i0 H4) in (let H5 \def H_x in 
+(or3_ind (land (eq C c c2) (subst0 i0 u0 (TLRef i) t2)) (land (eq T (TLRef i) 
+t2) (csubst0 i0 u0 c c2)) (land (subst0 i0 u0 (TLRef i) t2) (csubst0 i0 u0 c 
+c2)) (arity g c2 t2 a0) (\lambda (H6: (land (eq C c c2) (subst0 i0 u0 (TLRef 
+i) t2))).(and_ind (eq C c c2) (subst0 i0 u0 (TLRef i) t2) (arity g c2 t2 a0) 
+(\lambda (H7: (eq C c c2)).(\lambda (H8: (subst0 i0 u0 (TLRef i) t2)).(eq_ind 
+C c (\lambda (c0: C).(arity g c0 t2 a0)) (and_ind (eq nat i i0) (eq T t2 
+(lift (S i) O u0)) (arity g c t2 a0) (\lambda (H9: (eq nat i i0)).(\lambda 
+(H10: (eq T t2 (lift (S i) O u0))).(eq_ind_r T (lift (S i) O u0) (\lambda (t: 
+T).(arity g c t a0)) (let H11 \def (eq_ind_r nat i0 (\lambda (n: nat).(getl n 
+c (CHead d1 (Bind Abbr) u0))) H3 i H9) in (let H12 \def (eq_ind C (CHead d 
+(Bind Abst) u) (\lambda (c0: C).(getl i c c0)) H0 (CHead d1 (Bind Abbr) u0) 
+(getl_mono c (CHead d (Bind Abst) u) i H0 (CHead d1 (Bind Abbr) u0) H11)) in 
+(let H13 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (ee: C).(match ee in 
+C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ 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 
 False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) 
 \Rightarrow False])])) I (CHead d1 (Bind Abbr) u0) (getl_mono c (CHead d 
@@ -668,7 +668,7 @@ B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True | Void
 \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead d1 (Bind Abbr) 
 u0) (getl_mono c (CHead d (Bind Abst) u) i H0 (CHead d1 (Bind Abbr) u0) H12)) 
 in (False_ind (arity g c2 (lift (S i) O u0) a0) H14))))) t2 H10))) 
-(subst0_gen_lref u0 t2 i0 i H7)))) H6)) H5))))))))))))))))) (\lambda (b: 
+(subst0_gen_lref u0 t2 i0 i H7)))) H6)) H5)))))))))))))))))) (\lambda (b: 
 B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c: C).(\lambda (u: 
 T).(\lambda (a1: A).(\lambda (H1: (arity g c u a1)).(\lambda (H2: ((\forall 
 (d1: C).(\forall (u0: T).(\forall (i: nat).((getl i c (CHead d1 (Bind Abbr) 
@@ -680,19 +680,19 @@ u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 (CHead c (Bind b)
 u) t c2 t2) \to (arity g c2 t2 a2)))))))))).(\lambda (d1: C).(\lambda (u0: 
 T).(\lambda (i: nat).(\lambda (H5: (getl i c (CHead d1 (Bind Abbr) 
 u0))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H6: (fsubst0 i u0 c (THead 
-(Bind b) u t) c2 t2)).(let H7 \def (fsubst0_gen_base c c2 (THead (Bind b) u 
-t) t2 u0 i H6) in (or3_ind (land (eq C c c2) (subst0 i u0 (THead (Bind b) u 
-t) t2)) (land (eq T (THead (Bind b) u t) t2) (csubst0 i u0 c c2)) (land 
-(subst0 i u0 (THead (Bind b) u t) t2) (csubst0 i u0 c c2)) (arity g c2 t2 a2) 
-(\lambda (H8: (land (eq C c c2) (subst0 i u0 (THead (Bind b) u t) 
-t2))).(and_ind (eq C c c2) (subst0 i u0 (THead (Bind b) u t) t2) (arity g c2 
-t2 a2) (\lambda (H9: (eq C c c2)).(\lambda (H10: (subst0 i u0 (THead (Bind b) 
-u t) t2)).(eq_ind C c (\lambda (c0: C).(arity g c0 t2 a2)) (or3_ind (ex2 T 
-(\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) (\lambda (u2: T).(subst0 i 
-u0 u u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))) (\lambda 
-(t3: T).(subst0 (s (Bind b) i) u0 t t3))) (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Bind b) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: 
+(Bind b) u t) c2 t2)).(let H_x \def (fsubst0_gen_base c c2 (THead (Bind b) u 
+t) t2 u0 i H6) in (let H7 \def H_x in (or3_ind (land (eq C c c2) (subst0 i u0 
+(THead (Bind b) u t) t2)) (land (eq T (THead (Bind b) u t) t2) (csubst0 i u0 
+c c2)) (land (subst0 i u0 (THead (Bind b) u t) t2) (csubst0 i u0 c c2)) 
+(arity g c2 t2 a2) (\lambda (H8: (land (eq C c c2) (subst0 i u0 (THead (Bind 
+b) u t) t2))).(and_ind (eq C c c2) (subst0 i u0 (THead (Bind b) u t) t2) 
+(arity g c2 t2 a2) (\lambda (H9: (eq C c c2)).(\lambda (H10: (subst0 i u0 
+(THead (Bind b) u t) t2)).(eq_ind C c (\lambda (c0: C).(arity g c0 t2 a2)) 
+(or3_ind (ex2 T (\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) (\lambda 
+(u2: T).(subst0 i u0 u u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead (Bind b) 
+u t3))) (\lambda (t3: T).(subst0 (s (Bind b) i) u0 t t3))) (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) u2 t3)))) (\lambda 
+(u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: 
 T).(subst0 (s (Bind b) i) u0 t t3)))) (arity g c t2 a2) (\lambda (H11: (ex2 T 
 (\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) (\lambda (u2: T).(subst0 i 
 u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) 
@@ -780,7 +780,7 @@ H13 c2 H10)) x1 a2 (H4 d1 u0 (S i) (getl_clear_bind b (CHead c (Bind b) u) c
 u (clear_bind b c u) (CHead d1 (Bind Abbr) u0) i H5) (CHead c2 (Bind b) x0) 
 x1 (fsubst0_both (S i) u0 (CHead c (Bind b) u) t x1 H14 (CHead c2 (Bind b) 
 x0) (csubst0_both_bind b i u0 u x0 H13 c c2 H10)))) t2 H12)))))) H11)) 
-(subst0_gen_head (Bind b) u0 u t t2 i H9)))) H8)) H7)))))))))))))))))))) 
+(subst0_gen_head (Bind b) u0 u t t2 i H9)))) H8)) H7))))))))))))))))))))) 
 (\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H0: (arity g c u 
 (asucc g a1))).(\lambda (H1: ((\forall (d1: C).(\forall (u0: T).(\forall (i: 
 nat).((getl i c (CHead d1 (Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: 
@@ -792,52 +792,52 @@ u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 (CHead c (Bind
 Abst) u) t c2 t2) \to (arity g c2 t2 a2)))))))))).(\lambda (d1: C).(\lambda 
 (u0: T).(\lambda (i: nat).(\lambda (H4: (getl i c (CHead d1 (Bind Abbr) 
 u0))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H5: (fsubst0 i u0 c (THead 
-(Bind Abst) u t) c2 t2)).(let H6 \def (fsubst0_gen_base c c2 (THead (Bind 
-Abst) u t) t2 u0 i H5) in (or3_ind (land (eq C c c2) (subst0 i u0 (THead 
-(Bind Abst) u t) t2)) (land (eq T (THead (Bind Abst) u t) t2) (csubst0 i u0 c 
-c2)) (land (subst0 i u0 (THead (Bind Abst) u t) t2) (csubst0 i u0 c c2)) 
-(arity g c2 t2 (AHead a1 a2)) (\lambda (H7: (land (eq C c c2) (subst0 i u0 
-(THead (Bind Abst) u t) t2))).(and_ind (eq C c c2) (subst0 i u0 (THead (Bind 
-Abst) u t) t2) (arity g c2 t2 (AHead a1 a2)) (\lambda (H8: (eq C c 
-c2)).(\lambda (H9: (subst0 i u0 (THead (Bind Abst) u t) t2)).(eq_ind C c 
-(\lambda (c0: C).(arity g c0 t2 (AHead a1 a2))) (or3_ind (ex2 T (\lambda (u2: 
-T).(eq T t2 (THead (Bind Abst) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2))) 
-(ex2 T (\lambda (t3: T).(eq T t2 (THead (Bind Abst) u t3))) (\lambda (t3: 
-T).(subst0 (s (Bind Abst) i) u0 t t3))) (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind 
-Abst) i) u0 t t3)))) (arity g c t2 (AHead a1 a2)) (\lambda (H10: (ex2 T 
-(\lambda (u2: T).(eq T t2 (THead (Bind Abst) u2 t))) (\lambda (u2: T).(subst0 
-i u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2 (THead (Bind Abst) u2 t))) 
-(\lambda (u2: T).(subst0 i u0 u u2)) (arity g c t2 (AHead a1 a2)) (\lambda 
-(x: T).(\lambda (H11: (eq T t2 (THead (Bind Abst) x t))).(\lambda (H12: 
-(subst0 i u0 u x)).(eq_ind_r T (THead (Bind Abst) x t) (\lambda (t0: 
-T).(arity g c t0 (AHead a1 a2))) (arity_head g c x a1 (H1 d1 u0 i H4 c x 
-(fsubst0_snd i u0 c u x H12)) t a2 (H3 d1 u0 (S i) (getl_clear_bind Abst 
-(CHead c (Bind Abst) u) c u (clear_bind Abst c u) (CHead d1 (Bind Abbr) u0) i 
-H4) (CHead c (Bind Abst) x) t (fsubst0_fst (S i) u0 (CHead c (Bind Abst) u) t 
-(CHead c (Bind Abst) x) (csubst0_snd_bind Abst i u0 u x H12 c)))) t2 H11)))) 
-H10)) (\lambda (H10: (ex2 T (\lambda (t3: T).(eq T t2 (THead (Bind Abst) u 
-t3))) (\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3)))).(ex2_ind T 
-(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u t3))) (\lambda (t3: T).(subst0 
-(s (Bind Abst) i) u0 t t3)) (arity g c t2 (AHead a1 a2)) (\lambda (x: 
-T).(\lambda (H11: (eq T t2 (THead (Bind Abst) u x))).(\lambda (H12: (subst0 
-(s (Bind Abst) i) u0 t x)).(eq_ind_r T (THead (Bind Abst) u x) (\lambda (t0: 
-T).(arity g c t0 (AHead a1 a2))) (arity_head g c u a1 H0 x a2 (H3 d1 u0 (S i) 
-(getl_clear_bind Abst (CHead c (Bind Abst) u) c u (clear_bind Abst c u) 
-(CHead d1 (Bind Abbr) u0) i H4) (CHead c (Bind Abst) u) x (fsubst0_snd (S i) 
-u0 (CHead c (Bind Abst) u) t x H12))) t2 H11)))) H10)) (\lambda (H10: (ex3_2 
-T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: 
-T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3))))).(ex3_2_ind T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: 
-T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3))) (arity g c t2 (AHead 
-a1 a2)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq T t2 (THead 
-(Bind Abst) x0 x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13: 
-(subst0 (s (Bind Abst) i) u0 t x1)).(eq_ind_r T (THead (Bind Abst) x0 x1) 
-(\lambda (t0: T).(arity g c t0 (AHead a1 a2))) (arity_head g c x0 a1 (H1 d1 
-u0 i H4 c x0 (fsubst0_snd i u0 c u x0 H12)) x1 a2 (H3 d1 u0 (S i) 
+(Bind Abst) u t) c2 t2)).(let H_x \def (fsubst0_gen_base c c2 (THead (Bind 
+Abst) u t) t2 u0 i H5) in (let H6 \def H_x in (or3_ind (land (eq C c c2) 
+(subst0 i u0 (THead (Bind Abst) u t) t2)) (land (eq T (THead (Bind Abst) u t) 
+t2) (csubst0 i u0 c c2)) (land (subst0 i u0 (THead (Bind Abst) u t) t2) 
+(csubst0 i u0 c c2)) (arity g c2 t2 (AHead a1 a2)) (\lambda (H7: (land (eq C 
+c c2) (subst0 i u0 (THead (Bind Abst) u t) t2))).(and_ind (eq C c c2) (subst0 
+i u0 (THead (Bind Abst) u t) t2) (arity g c2 t2 (AHead a1 a2)) (\lambda (H8: 
+(eq C c c2)).(\lambda (H9: (subst0 i u0 (THead (Bind Abst) u t) t2)).(eq_ind 
+C c (\lambda (c0: C).(arity g c0 t2 (AHead a1 a2))) (or3_ind (ex2 T (\lambda 
+(u2: T).(eq T t2 (THead (Bind Abst) u2 t))) (\lambda (u2: T).(subst0 i u0 u 
+u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead (Bind Abst) u t3))) (\lambda 
+(t3: T).(subst0 (s (Bind Abst) i) u0 t t3))) (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: 
+T).(subst0 (s (Bind Abst) i) u0 t t3)))) (arity g c t2 (AHead a1 a2)) 
+(\lambda (H10: (ex2 T (\lambda (u2: T).(eq T t2 (THead (Bind Abst) u2 t))) 
+(\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2 
+(THead (Bind Abst) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)) (arity g c t2 
+(AHead a1 a2)) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Bind Abst) x 
+t))).(\lambda (H12: (subst0 i u0 u x)).(eq_ind_r T (THead (Bind Abst) x t) 
+(\lambda (t0: T).(arity g c t0 (AHead a1 a2))) (arity_head g c x a1 (H1 d1 u0 
+i H4 c x (fsubst0_snd i u0 c u x H12)) t a2 (H3 d1 u0 (S i) (getl_clear_bind 
+Abst (CHead c (Bind Abst) u) c u (clear_bind Abst c u) (CHead d1 (Bind Abbr) 
+u0) i H4) (CHead c (Bind Abst) x) t (fsubst0_fst (S i) u0 (CHead c (Bind 
+Abst) u) t (CHead c (Bind Abst) x) (csubst0_snd_bind Abst i u0 u x H12 c)))) 
+t2 H11)))) H10)) (\lambda (H10: (ex2 T (\lambda (t3: T).(eq T t2 (THead (Bind 
+Abst) u t3))) (\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3)))).(ex2_ind 
+T (\lambda (t3: T).(eq T t2 (THead (Bind Abst) u t3))) (\lambda (t3: 
+T).(subst0 (s (Bind Abst) i) u0 t t3)) (arity g c t2 (AHead a1 a2)) (\lambda 
+(x: T).(\lambda (H11: (eq T t2 (THead (Bind Abst) u x))).(\lambda (H12: 
+(subst0 (s (Bind Abst) i) u0 t x)).(eq_ind_r T (THead (Bind Abst) u x) 
+(\lambda (t0: T).(arity g c t0 (AHead a1 a2))) (arity_head g c u a1 H0 x a2 
+(H3 d1 u0 (S i) (getl_clear_bind Abst (CHead c (Bind Abst) u) c u (clear_bind 
+Abst c u) (CHead d1 (Bind Abbr) u0) i H4) (CHead c (Bind Abst) u) x 
+(fsubst0_snd (S i) u0 (CHead c (Bind Abst) u) t x H12))) t2 H11)))) H10)) 
+(\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
+(Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) 
+(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t 
+t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
+(Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) 
+(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3))) (arity 
+g c t2 (AHead a1 a2)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq T 
+t2 (THead (Bind Abst) x0 x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda 
+(H13: (subst0 (s (Bind Abst) i) u0 t x1)).(eq_ind_r T (THead (Bind Abst) x0 
+x1) (\lambda (t0: T).(arity g c t0 (AHead a1 a2))) (arity_head g c x0 a1 (H1 
+d1 u0 i H4 c x0 (fsubst0_snd i u0 c u x0 H12)) x1 a2 (H3 d1 u0 (S i) 
 (getl_clear_bind Abst (CHead c (Bind Abst) u) c u (clear_bind Abst c u) 
 (CHead d1 (Bind Abbr) u0) i H4) (CHead c (Bind Abst) x0) x1 (fsubst0_both (S 
 i) u0 (CHead c (Bind Abst) u) t x1 H13 (CHead c (Bind Abst) x0) 
@@ -899,7 +899,7 @@ u0 i H4 c2 x0 (fsubst0_both i u0 c u x0 H12 c2 H9)) x1 a2 (H3 d1 u0 (S i)
 (CHead d1 (Bind Abbr) u0) i H4) (CHead c2 (Bind Abst) x0) x1 (fsubst0_both (S 
 i) u0 (CHead c (Bind Abst) u) t x1 H13 (CHead c2 (Bind Abst) x0) 
 (csubst0_both_bind Abst i u0 u x0 H12 c c2 H9)))) t2 H11)))))) H10)) 
-(subst0_gen_head (Bind Abst) u0 u t t2 i H8)))) H7)) H6)))))))))))))))))) 
+(subst0_gen_head (Bind Abst) u0 u t t2 i H8)))) H7)) H6))))))))))))))))))) 
 (\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H0: (arity g c u 
 a1)).(\lambda (H1: ((\forall (d1: C).(\forall (u0: T).(\forall (i: 
 nat).((getl i c (CHead d1 (Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: 
@@ -910,34 +910,34 @@ T).(\lambda (a2: A).(\lambda (H2: (arity g c t (AHead a1 a2))).(\lambda (H3:
 t2) \to (arity g c2 t2 (AHead a1 a2))))))))))).(\lambda (d1: C).(\lambda (u0: 
 T).(\lambda (i: nat).(\lambda (H4: (getl i c (CHead d1 (Bind Abbr) 
 u0))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H5: (fsubst0 i u0 c (THead 
-(Flat Appl) u t) c2 t2)).(let H6 \def (fsubst0_gen_base c c2 (THead (Flat 
-Appl) u t) t2 u0 i H5) in (or3_ind (land (eq C c c2) (subst0 i u0 (THead 
-(Flat Appl) u t) t2)) (land (eq T (THead (Flat Appl) u t) t2) (csubst0 i u0 c 
-c2)) (land (subst0 i u0 (THead (Flat Appl) u t) t2) (csubst0 i u0 c c2)) 
-(arity g c2 t2 a2) (\lambda (H7: (land (eq C c c2) (subst0 i u0 (THead (Flat 
-Appl) u t) t2))).(and_ind (eq C c c2) (subst0 i u0 (THead (Flat Appl) u t) 
-t2) (arity g c2 t2 a2) (\lambda (H8: (eq C c c2)).(\lambda (H9: (subst0 i u0 
-(THead (Flat Appl) u t) t2)).(eq_ind C c (\lambda (c0: C).(arity g c0 t2 a2)) 
-(or3_ind (ex2 T (\lambda (u2: T).(eq T t2 (THead (Flat Appl) u2 t))) (\lambda 
-(u2: T).(subst0 i u0 u u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead (Flat 
-Appl) u t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t t3))) (ex3_2 T 
-T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: 
-T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t t3)))) (arity g c t2 a2) 
-(\lambda (H10: (ex2 T (\lambda (u2: T).(eq T t2 (THead (Flat Appl) u2 t))) 
-(\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2 
-(THead (Flat Appl) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)) (arity g c t2 
-a2) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat Appl) x 
-t))).(\lambda (H12: (subst0 i u0 u x)).(eq_ind_r T (THead (Flat Appl) x t) 
-(\lambda (t0: T).(arity g c t0 a2)) (arity_appl g c x a1 (H1 d1 u0 i H4 c x 
-(fsubst0_snd i u0 c u x H12)) t a2 H2) t2 H11)))) H10)) (\lambda (H10: (ex2 T 
+(Flat Appl) u t) c2 t2)).(let H_x \def (fsubst0_gen_base c c2 (THead (Flat 
+Appl) u t) t2 u0 i H5) in (let H6 \def H_x in (or3_ind (land (eq C c c2) 
+(subst0 i u0 (THead (Flat Appl) u t) t2)) (land (eq T (THead (Flat Appl) u t) 
+t2) (csubst0 i u0 c c2)) (land (subst0 i u0 (THead (Flat Appl) u t) t2) 
+(csubst0 i u0 c c2)) (arity g c2 t2 a2) (\lambda (H7: (land (eq C c c2) 
+(subst0 i u0 (THead (Flat Appl) u t) t2))).(and_ind (eq C c c2) (subst0 i u0 
+(THead (Flat Appl) u t) t2) (arity g c2 t2 a2) (\lambda (H8: (eq C c 
+c2)).(\lambda (H9: (subst0 i u0 (THead (Flat Appl) u t) t2)).(eq_ind C c 
+(\lambda (c0: C).(arity g c0 t2 a2)) (or3_ind (ex2 T (\lambda (u2: T).(eq T 
+t2 (THead (Flat Appl) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2))) (ex2 T 
 (\lambda (t3: T).(eq T t2 (THead (Flat Appl) u t3))) (\lambda (t3: T).(subst0 
-(s (Flat Appl) i) u0 t t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead 
-(Flat Appl) u t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t t3)) 
-(arity g c t2 a2) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat Appl) 
-u x))).(\lambda (H12: (subst0 (s (Flat Appl) i) u0 t x)).(eq_ind_r T (THead 
-(Flat Appl) u x) (\lambda (t0: T).(arity g c t0 a2)) (arity_appl g c u a1 H0 
-x a2 (H3 d1 u0 i H4 c x (fsubst0_snd i u0 c t x H12))) t2 H11)))) H10)) 
+(s (Flat Appl) i) u0 t t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq 
+T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i 
+u0 u u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t 
+t3)))) (arity g c t2 a2) (\lambda (H10: (ex2 T (\lambda (u2: T).(eq T t2 
+(THead (Flat Appl) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T 
+(\lambda (u2: T).(eq T t2 (THead (Flat Appl) u2 t))) (\lambda (u2: T).(subst0 
+i u0 u u2)) (arity g c t2 a2) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead 
+(Flat Appl) x t))).(\lambda (H12: (subst0 i u0 u x)).(eq_ind_r T (THead (Flat 
+Appl) x t) (\lambda (t0: T).(arity g c t0 a2)) (arity_appl g c x a1 (H1 d1 u0 
+i H4 c x (fsubst0_snd i u0 c u x H12)) t a2 H2) t2 H11)))) H10)) (\lambda 
+(H10: (ex2 T (\lambda (t3: T).(eq T t2 (THead (Flat Appl) u t3))) (\lambda 
+(t3: T).(subst0 (s (Flat Appl) i) u0 t t3)))).(ex2_ind T (\lambda (t3: T).(eq 
+T t2 (THead (Flat Appl) u t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 
+t t3)) (arity g c t2 a2) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat 
+Appl) u x))).(\lambda (H12: (subst0 (s (Flat Appl) i) u0 t x)).(eq_ind_r T 
+(THead (Flat Appl) u x) (\lambda (t0: T).(arity g c t0 a2)) (arity_appl g c u 
+a1 H0 x a2 (H3 d1 u0 i H4 c x (fsubst0_snd i u0 c t x H12))) t2 H11)))) H10)) 
 (\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
 (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) 
 (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t 
@@ -994,7 +994,7 @@ Appl) i) u0 t x1)).(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t0:
 T).(arity g c2 t0 a2)) (arity_appl g c2 x0 a1 (H1 d1 u0 i H4 c2 x0 
 (fsubst0_both i u0 c u x0 H12 c2 H9)) x1 a2 (H3 d1 u0 i H4 c2 x1 
 (fsubst0_both i u0 c t x1 H13 c2 H9))) t2 H11)))))) H10)) (subst0_gen_head 
-(Flat Appl) u0 u t t2 i H8)))) H7)) H6)))))))))))))))))) (\lambda (c: 
+(Flat Appl) u0 u t t2 i H8)))) H7)) H6))))))))))))))))))) (\lambda (c: 
 C).(\lambda (u: T).(\lambda (a0: A).(\lambda (H0: (arity g c u (asucc g 
 a0))).(\lambda (H1: ((\forall (d1: C).(\forall (u0: T).(\forall (i: 
 nat).((getl i c (CHead d1 (Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: 
@@ -1005,36 +1005,36 @@ a0))))))))))).(\lambda (t: T).(\lambda (H2: (arity g c t a0)).(\lambda (H3:
 t2) \to (arity g c2 t2 a0)))))))))).(\lambda (d1: C).(\lambda (u0: 
 T).(\lambda (i: nat).(\lambda (H4: (getl i c (CHead d1 (Bind Abbr) 
 u0))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H5: (fsubst0 i u0 c (THead 
-(Flat Cast) u t) c2 t2)).(let H6 \def (fsubst0_gen_base c c2 (THead (Flat 
-Cast) u t) t2 u0 i H5) in (or3_ind (land (eq C c c2) (subst0 i u0 (THead 
-(Flat Cast) u t) t2)) (land (eq T (THead (Flat Cast) u t) t2) (csubst0 i u0 c 
-c2)) (land (subst0 i u0 (THead (Flat Cast) u t) t2) (csubst0 i u0 c c2)) 
-(arity g c2 t2 a0) (\lambda (H7: (land (eq C c c2) (subst0 i u0 (THead (Flat 
-Cast) u t) t2))).(and_ind (eq C c c2) (subst0 i u0 (THead (Flat Cast) u t) 
-t2) (arity g c2 t2 a0) (\lambda (H8: (eq C c c2)).(\lambda (H9: (subst0 i u0 
-(THead (Flat Cast) u t) t2)).(eq_ind C c (\lambda (c0: C).(arity g c0 t2 a0)) 
-(or3_ind (ex2 T (\lambda (u2: T).(eq T t2 (THead (Flat Cast) u2 t))) (\lambda 
-(u2: T).(subst0 i u0 u u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead (Flat 
-Cast) u t3))) (\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t t3))) (ex3_2 T 
-T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: 
-T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t t3)))) (arity g c t2 a0) 
-(\lambda (H10: (ex2 T (\lambda (u2: T).(eq T t2 (THead (Flat Cast) u2 t))) 
-(\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2 
-(THead (Flat Cast) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)) (arity g c t2 
-a0) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat Cast) x 
-t))).(\lambda (H12: (subst0 i u0 u x)).(eq_ind_r T (THead (Flat Cast) x t) 
-(\lambda (t0: T).(arity g c t0 a0)) (arity_cast g c x a0 (H1 d1 u0 i H4 c x 
-(fsubst0_snd i u0 c u x H12)) t H2) t2 H11)))) H10)) (\lambda (H10: (ex2 T 
+(Flat Cast) u t) c2 t2)).(let H_x \def (fsubst0_gen_base c c2 (THead (Flat 
+Cast) u t) t2 u0 i H5) in (let H6 \def H_x in (or3_ind (land (eq C c c2) 
+(subst0 i u0 (THead (Flat Cast) u t) t2)) (land (eq T (THead (Flat Cast) u t) 
+t2) (csubst0 i u0 c c2)) (land (subst0 i u0 (THead (Flat Cast) u t) t2) 
+(csubst0 i u0 c c2)) (arity g c2 t2 a0) (\lambda (H7: (land (eq C c c2) 
+(subst0 i u0 (THead (Flat Cast) u t) t2))).(and_ind (eq C c c2) (subst0 i u0 
+(THead (Flat Cast) u t) t2) (arity g c2 t2 a0) (\lambda (H8: (eq C c 
+c2)).(\lambda (H9: (subst0 i u0 (THead (Flat Cast) u t) t2)).(eq_ind C c 
+(\lambda (c0: C).(arity g c0 t2 a0)) (or3_ind (ex2 T (\lambda (u2: T).(eq T 
+t2 (THead (Flat Cast) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2))) (ex2 T 
 (\lambda (t3: T).(eq T t2 (THead (Flat Cast) u t3))) (\lambda (t3: T).(subst0 
-(s (Flat Cast) i) u0 t t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead 
-(Flat Cast) u t3))) (\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t t3)) 
-(arity g c t2 a0) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat Cast) 
-u x))).(\lambda (H12: (subst0 (s (Flat Cast) i) u0 t x)).(eq_ind_r T (THead 
-(Flat Cast) u x) (\lambda (t0: T).(arity g c t0 a0)) (arity_cast g c u a0 H0 
-x (H3 d1 u0 i H4 c x (fsubst0_snd i u0 c t x H12))) t2 H11)))) H10)) (\lambda 
-(H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat 
-Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) 
+(s (Flat Cast) i) u0 t t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq 
+T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i 
+u0 u u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t 
+t3)))) (arity g c t2 a0) (\lambda (H10: (ex2 T (\lambda (u2: T).(eq T t2 
+(THead (Flat Cast) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T 
+(\lambda (u2: T).(eq T t2 (THead (Flat Cast) u2 t))) (\lambda (u2: T).(subst0 
+i u0 u u2)) (arity g c t2 a0) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead 
+(Flat Cast) x t))).(\lambda (H12: (subst0 i u0 u x)).(eq_ind_r T (THead (Flat 
+Cast) x t) (\lambda (t0: T).(arity g c t0 a0)) (arity_cast g c x a0 (H1 d1 u0 
+i H4 c x (fsubst0_snd i u0 c u x H12)) t H2) t2 H11)))) H10)) (\lambda (H10: 
+(ex2 T (\lambda (t3: T).(eq T t2 (THead (Flat Cast) u t3))) (\lambda (t3: 
+T).(subst0 (s (Flat Cast) i) u0 t t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 
+(THead (Flat Cast) u t3))) (\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t 
+t3)) (arity g c t2 a0) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat 
+Cast) u x))).(\lambda (H12: (subst0 (s (Flat Cast) i) u0 t x)).(eq_ind_r T 
+(THead (Flat Cast) u x) (\lambda (t0: T).(arity g c t0 a0)) (arity_cast g c u 
+a0 H0 x (H3 d1 u0 i H4 c x (fsubst0_snd i u0 c t x H12))) t2 H11)))) H10)) 
+(\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
+(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) 
 (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t 
 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
 (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) 
@@ -1089,29 +1089,29 @@ Cast) i) u0 t x1)).(eq_ind_r T (THead (Flat Cast) x0 x1) (\lambda (t0:
 T).(arity g c2 t0 a0)) (arity_cast g c2 x0 a0 (H1 d1 u0 i H4 c2 x0 
 (fsubst0_both i u0 c u x0 H12 c2 H9)) x1 (H3 d1 u0 i H4 c2 x1 (fsubst0_both i 
 u0 c t x1 H13 c2 H9))) t2 H11)))))) H10)) (subst0_gen_head (Flat Cast) u0 u t 
-t2 i H8)))) H7)) H6))))))))))))))))) (\lambda (c: C).(\lambda (t: T).(\lambda 
-(a1: A).(\lambda (_: (arity g c t a1)).(\lambda (H1: ((\forall (d1: 
-C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d1 (Bind Abbr) u)) \to 
-(\forall (c2: C).(\forall (t2: T).((fsubst0 i u c t c2 t2) \to (arity g c2 t2 
-a1)))))))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda (d1: 
-C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H3: (getl i c (CHead d1 (Bind 
-Abbr) u))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H4: (fsubst0 i u c t 
-c2 t2)).(let H5 \def (fsubst0_gen_base c c2 t t2 u i H4) in (or3_ind (land 
-(eq C c c2) (subst0 i u t t2)) (land (eq T t t2) (csubst0 i u c c2)) (land 
-(subst0 i u t t2) (csubst0 i u c c2)) (arity g c2 t2 a2) (\lambda (H6: (land 
-(eq C c c2) (subst0 i u t t2))).(and_ind (eq C c c2) (subst0 i u t t2) (arity 
-g c2 t2 a2) (\lambda (H7: (eq C c c2)).(\lambda (H8: (subst0 i u t 
-t2)).(eq_ind C c (\lambda (c0: C).(arity g c0 t2 a2)) (arity_repl g c t2 a1 
-(H1 d1 u i H3 c t2 (fsubst0_snd i u c t t2 H8)) a2 H2) c2 H7))) H6)) (\lambda 
-(H6: (land (eq T t t2) (csubst0 i u c c2))).(and_ind (eq T t t2) (csubst0 i u 
-c c2) (arity g c2 t2 a2) (\lambda (H7: (eq T t t2)).(\lambda (H8: (csubst0 i 
-u c c2)).(eq_ind T t (\lambda (t0: T).(arity g c2 t0 a2)) (arity_repl g c2 t 
-a1 (H1 d1 u i H3 c2 t (fsubst0_fst i u c t c2 H8)) a2 H2) t2 H7))) H6)) 
-(\lambda (H6: (land (subst0 i u t t2) (csubst0 i u c c2))).(and_ind (subst0 i 
-u t t2) (csubst0 i u c c2) (arity g c2 t2 a2) (\lambda (H7: (subst0 i u t 
-t2)).(\lambda (H8: (csubst0 i u c c2)).(arity_repl g c2 t2 a1 (H1 d1 u i H3 
-c2 t2 (fsubst0_both i u c t t2 H7 c2 H8)) a2 H2))) H6)) H5)))))))))))))))) c1 
-t1 a H))))).
+t2 i H8)))) H7)) H6)))))))))))))))))) (\lambda (c: C).(\lambda (t: 
+T).(\lambda (a1: A).(\lambda (_: (arity g c t a1)).(\lambda (H1: ((\forall 
+(d1: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d1 (Bind Abbr) 
+u)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u c t c2 t2) \to (arity 
+g c2 t2 a1)))))))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda 
+(d1: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H3: (getl i c (CHead d1 
+(Bind Abbr) u))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H4: (fsubst0 i u 
+c t c2 t2)).(let H_x \def (fsubst0_gen_base c c2 t t2 u i H4) in (let H5 \def 
+H_x in (or3_ind (land (eq C c c2) (subst0 i u t t2)) (land (eq T t t2) 
+(csubst0 i u c c2)) (land (subst0 i u t t2) (csubst0 i u c c2)) (arity g c2 
+t2 a2) (\lambda (H6: (land (eq C c c2) (subst0 i u t t2))).(and_ind (eq C c 
+c2) (subst0 i u t t2) (arity g c2 t2 a2) (\lambda (H7: (eq C c c2)).(\lambda 
+(H8: (subst0 i u t t2)).(eq_ind C c (\lambda (c0: C).(arity g c0 t2 a2)) 
+(arity_repl g c t2 a1 (H1 d1 u i H3 c t2 (fsubst0_snd i u c t t2 H8)) a2 H2) 
+c2 H7))) H6)) (\lambda (H6: (land (eq T t t2) (csubst0 i u c c2))).(and_ind 
+(eq T t t2) (csubst0 i u c c2) (arity g c2 t2 a2) (\lambda (H7: (eq T t 
+t2)).(\lambda (H8: (csubst0 i u c c2)).(eq_ind T t (\lambda (t0: T).(arity g 
+c2 t0 a2)) (arity_repl g c2 t a1 (H1 d1 u i H3 c2 t (fsubst0_fst i u c t c2 
+H8)) a2 H2) t2 H7))) H6)) (\lambda (H6: (land (subst0 i u t t2) (csubst0 i u 
+c c2))).(and_ind (subst0 i u t t2) (csubst0 i u c c2) (arity g c2 t2 a2) 
+(\lambda (H7: (subst0 i u t t2)).(\lambda (H8: (csubst0 i u c 
+c2)).(arity_repl g c2 t2 a1 (H1 d1 u i H3 c2 t2 (fsubst0_both i u c t t2 H7 
+c2 H8)) a2 H2))) H6)) H5))))))))))))))))) c1 t1 a H))))).
 
 theorem arity_subst0:
  \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (a: A).((arity g c