X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2FLAMBDA-TYPES%2FLambdaDelta-1%2Farity%2Faprem.ma;h=5045207ec1f78984da49648e2f5fedd70fc8f6f4;hb=89519c7b52e06304a94019dd528925300380cdc0;hp=6113a2a1139f7d14a4fb252b00bd6942584c86b3;hpb=e92710b1d9774a6491122668c8463b8658114610;p=helm.git

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/aprem.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/aprem.ma
index 6113a2a11..5045207ec 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/aprem.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/aprem.ma
@@ -33,37 +33,10 @@ A).(\forall (i: nat).(\forall (b: A).((aprem i a0 b) \to (ex2_3 C T nat
 (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) 
 (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g 
 b)))))))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (i: nat).(\lambda 
-(b: A).(\lambda (H0: (aprem i (ASort O n) b)).(let H1 \def (match H0 in aprem 
-return (\lambda (n0: nat).(\lambda (a0: A).(\lambda (a1: A).(\lambda (_: 
-(aprem n0 a0 a1)).((eq nat n0 i) \to ((eq A a0 (ASort O n)) \to ((eq A a1 b) 
-\to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop 
-(plus i j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: 
-nat).(arity g d u (asucc g b))))))))))))) with [(aprem_zero a1 a2) 
-\Rightarrow (\lambda (H1: (eq nat O i)).(\lambda (H2: (eq A (AHead a1 a2) 
-(ASort O n))).(\lambda (H3: (eq A a1 b)).(eq_ind nat O (\lambda (n0: 
-nat).((eq A (AHead a1 a2) (ASort O n)) \to ((eq A a1 b) \to (ex2_3 C T nat 
-(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus n0 j) O d 
-c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc 
-g b))))))))) (\lambda (H4: (eq A (AHead a1 a2) (ASort O n))).(let H5 \def 
-(eq_ind A (AHead a1 a2) (\lambda (e: A).(match e in A return (\lambda (_: 
-A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow 
-True])) I (ASort O n) H4) in (False_ind ((eq A a1 b) \to (ex2_3 C T nat 
-(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus O j) O d c0)))) 
-(\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g 
-b))))))) H5))) i H1 H2 H3)))) | (aprem_succ a2 a0 i0 H1 a1) \Rightarrow 
-(\lambda (H2: (eq nat (S i0) i)).(\lambda (H3: (eq A (AHead a1 a2) (ASort O 
-n))).(\lambda (H4: (eq A a0 b)).(eq_ind nat (S i0) (\lambda (n0: nat).((eq A 
-(AHead a1 a2) (ASort O n)) \to ((eq A a0 b) \to ((aprem i0 a2 a0) \to (ex2_3 
-C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus n0 j) O 
-d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u 
-(asucc g b)))))))))) (\lambda (H5: (eq A (AHead a1 a2) (ASort O n))).(let H6 
-\def (eq_ind A (AHead a1 a2) (\lambda (e: A).(match e in A return (\lambda 
-(_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow 
-True])) I (ASort O n) H5) in (False_ind ((eq A a0 b) \to ((aprem i0 a2 a0) 
-\to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop 
-(plus (S i0) j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: 
-nat).(arity g d u (asucc g b)))))))) H6))) i H2 H3 H4 H1))))]) in (H1 
-(refl_equal nat i) (refl_equal A (ASort O n)) (refl_equal A b)))))))) 
+(b: A).(\lambda (H0: (aprem i (ASort O n) b)).(let H_x \def (aprem_gen_sort b 
+i O n H0) in (let H1 \def H_x in (False_ind (ex2_3 C T nat (\lambda (d: 
+C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: 
+C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b)))))) H1)))))))) 
 (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda 
 (H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (a0: A).(\lambda (_: 
 (arity g d u a0)).(\lambda (H2: ((\forall (i0: nat).(\forall (b: A).((aprem 
@@ -151,164 +124,93 @@ A).(\lambda (H4: (aprem i (AHead a1 a2) b)).(nat_ind (\lambda (n:
 nat).((aprem n (AHead a1 a2) b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda 
 (_: T).(\lambda (j: nat).(drop (plus n j) O d c0)))) (\lambda (d: C).(\lambda 
 (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b)))))))) (\lambda (H5: 
-(aprem O (AHead a1 a2) b)).(let H6 \def (match H5 in aprem return (\lambda 
-(n: nat).(\lambda (a0: A).(\lambda (a3: A).(\lambda (_: (aprem n a0 a3)).((eq 
-nat n O) \to ((eq A a0 (AHead a1 a2)) \to ((eq A a3 b) \to (ex2_3 C T nat 
-(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus O j) O d c0)))) 
-(\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g 
-b))))))))))))) with [(aprem_zero a0 a3) \Rightarrow (\lambda (_: (eq nat O 
-O)).(\lambda (H7: (eq A (AHead a0 a3) (AHead a1 a2))).(\lambda (H8: (eq A a0 
-b)).((let H9 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda 
-(_: A).A) with [(ASort _ _) \Rightarrow a3 | (AHead _ a4) \Rightarrow a4])) 
-(AHead a0 a3) (AHead a1 a2) H7) in ((let H10 \def (f_equal A A (\lambda (e: 
-A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 | 
-(AHead a4 _) \Rightarrow a4])) (AHead a0 a3) (AHead a1 a2) H7) in (eq_ind A 
-a1 (\lambda (a4: A).((eq A a3 a2) \to ((eq A a4 b) \to (ex2_3 C T nat 
-(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus O j) O d c0)))) 
-(\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g 
-b))))))))) (\lambda (H11: (eq A a3 a2)).(eq_ind A a2 (\lambda (_: A).((eq A 
-a1 b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: 
-nat).(drop (plus O j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda 
-(_: nat).(arity g d u0 (asucc g b)))))))) (\lambda (H12: (eq A a1 b)).(eq_ind 
-A b (\lambda (_: A).(ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda 
-(j: nat).(drop (plus O j) O d c0)))) (\lambda (d: C).(\lambda (u0: 
-T).(\lambda (_: nat).(arity g d u0 (asucc g b))))))) (eq_ind A a1 (\lambda 
-(a4: A).(ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: 
-nat).(drop (plus O j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda 
-(_: nat).(arity g d u0 (asucc g a4))))))) (ex2_3_intro C T nat (\lambda (d: 
-C).(\lambda (_: T).(\lambda (j: nat).(drop (plus O j) O d c0)))) (\lambda (d: 
-C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g a1))))) c0 u O 
-(drop_refl c0) H0) b H12) a1 (sym_eq A a1 b H12))) a3 (sym_eq A a3 a2 H11))) 
-a0 (sym_eq A a0 a1 H10))) H9)) H8)))) | (aprem_succ a0 a3 i0 H6 a4) 
-\Rightarrow (\lambda (H7: (eq nat (S i0) O)).(\lambda (H8: (eq A (AHead a4 
-a0) (AHead a1 a2))).(\lambda (H9: (eq A a3 b)).((let H10 \def (eq_ind nat (S 
-i0) (\lambda (e: nat).(match e in nat return (\lambda (_: nat).Prop) with [O 
-\Rightarrow False | (S _) \Rightarrow True])) I O H7) in (False_ind ((eq A 
-(AHead a4 a0) (AHead a1 a2)) \to ((eq A a3 b) \to ((aprem i0 a0 a3) \to 
-(ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus 
-O j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d 
-u0 (asucc g b))))))))) H10)) H8 H9 H6))))]) in (H6 (refl_equal nat O) 
-(refl_equal A (AHead a1 a2)) (refl_equal A b)))) (\lambda (i0: nat).(\lambda 
-(_: (((aprem i0 (AHead a1 a2) b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda 
-(_: T).(\lambda (j: nat).(drop (plus i0 j) O d c0)))) (\lambda (d: 
-C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g 
-b))))))))).(\lambda (H5: (aprem (S i0) (AHead a1 a2) b)).(let H6 \def (match 
-H5 in aprem return (\lambda (n: nat).(\lambda (a0: A).(\lambda (a3: 
-A).(\lambda (_: (aprem n a0 a3)).((eq nat n (S i0)) \to ((eq A a0 (AHead a1 
-a2)) \to ((eq A a3 b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: 
-T).(\lambda (j: nat).(drop (plus (S i0) j) O d c0)))) (\lambda (d: 
-C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))))))))))) 
-with [(aprem_zero a0 a3) \Rightarrow (\lambda (H6: (eq nat O (S 
-i0))).(\lambda (H7: (eq A (AHead a0 a3) (AHead a1 a2))).(\lambda (H8: (eq A 
-a0 b)).((let H9 \def (eq_ind nat O (\lambda (e: nat).(match e in nat return 
-(\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) 
-I (S i0) H6) in (False_ind ((eq A (AHead a0 a3) (AHead a1 a2)) \to ((eq A a0 
-b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop 
-(plus (S i0) j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: 
-nat).(arity g d u0 (asucc g b)))))))) H9)) H7 H8)))) | (aprem_succ a0 a3 i1 
-H6 a4) \Rightarrow (\lambda (H7: (eq nat (S i1) (S i0))).(\lambda (H8: (eq A 
-(AHead a4 a0) (AHead a1 a2))).(\lambda (H9: (eq A a3 b)).((let H10 \def 
-(f_equal nat nat (\lambda (e: nat).(match e in nat return (\lambda (_: 
-nat).nat) with [O \Rightarrow i1 | (S n) \Rightarrow n])) (S i1) (S i0) H7) 
-in (eq_ind nat i0 (\lambda (n: nat).((eq A (AHead a4 a0) (AHead a1 a2)) \to 
-((eq A a3 b) \to ((aprem n a0 a3) \to (ex2_3 C T nat (\lambda (d: C).(\lambda 
-(_: T).(\lambda (j: nat).(drop (plus (S i0) j) O d c0)))) (\lambda (d: 
-C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b)))))))))) 
-(\lambda (H11: (eq A (AHead a4 a0) (AHead a1 a2))).(let H12 \def (f_equal A A 
-(\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) 
-\Rightarrow a0 | (AHead _ a5) \Rightarrow a5])) (AHead a4 a0) (AHead a1 a2) 
-H11) in ((let H13 \def (f_equal A A (\lambda (e: A).(match e in A return 
-(\lambda (_: A).A) with [(ASort _ _) \Rightarrow a4 | (AHead a5 _) 
-\Rightarrow a5])) (AHead a4 a0) (AHead a1 a2) H11) in (eq_ind A a1 (\lambda 
-(_: A).((eq A a0 a2) \to ((eq A a3 b) \to ((aprem i0 a0 a3) \to (ex2_3 C T 
-nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus (S i0) j) O 
-d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 
-(asucc g b)))))))))) (\lambda (H14: (eq A a0 a2)).(eq_ind A a2 (\lambda (a5: 
-A).((eq A a3 b) \to ((aprem i0 a5 a3) \to (ex2_3 C T nat (\lambda (d: 
-C).(\lambda (_: T).(\lambda (j: nat).(drop (plus (S i0) j) O d c0)))) 
+(aprem O (AHead a1 a2) b)).(let H_y \def (aprem_gen_head_O a1 a2 b H5) in 
+(eq_ind_r A a1 (\lambda (a0: A).(ex2_3 C T nat (\lambda (d: C).(\lambda (_: 
+T).(\lambda (j: nat).(drop (plus O j) O d c0)))) (\lambda (d: C).(\lambda 
+(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g a0))))))) (ex2_3_intro C T 
+nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus O j) O d 
+c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 
+(asucc g a1))))) c0 u O (drop_refl c0) H0) b H_y))) (\lambda (i0: 
+nat).(\lambda (_: (((aprem i0 (AHead a1 a2) b) \to (ex2_3 C T nat (\lambda 
+(d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d c0)))) 
 (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g 
-b))))))))) (\lambda (H15: (eq A a3 b)).(eq_ind A b (\lambda (a5: A).((aprem 
-i0 a2 a5) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: 
-nat).(drop (plus (S i0) j) O d c0)))) (\lambda (d: C).(\lambda (u0: 
-T).(\lambda (_: nat).(arity g d u0 (asucc g b)))))))) (\lambda (H16: (aprem 
-i0 a2 b)).(let H_x \def (H3 i0 b H16) in (let H17 \def H_x in (ex2_3_ind C T 
-nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d 
-(CHead c0 (Bind Abst) u))))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: 
-nat).(arity g d u0 (asucc g b))))) (ex2_3 C T nat (\lambda (d: C).(\lambda 
-(_: T).(\lambda (j: nat).(drop (plus (S i0) j) O d c0)))) (\lambda (d: 
-C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b)))))) (\lambda 
-(x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H18: (drop (plus i0 x2) 
-O x0 (CHead c0 (Bind Abst) u))).(\lambda (H19: (arity g x0 x1 (asucc g 
-b))).(ex2_3_intro C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: 
-nat).(drop (plus (S i0) j) O d c0)))) (\lambda (d: C).(\lambda (u0: 
-T).(\lambda (_: nat).(arity g d u0 (asucc g b))))) x0 x1 x2 (drop_S Abst x0 
-c0 u (plus i0 x2) H18) H19)))))) H17)))) a3 (sym_eq A a3 b H15))) a0 (sym_eq 
-A a0 a2 H14))) a4 (sym_eq A a4 a1 H13))) H12))) i1 (sym_eq nat i1 i0 H10))) 
-H8 H9 H6))))]) in (H6 (refl_equal nat (S i0)) (refl_equal A (AHead a1 a2)) 
-(refl_equal A b)))))) i H4))))))))))))) (\lambda (c0: C).(\lambda (u: 
-T).(\lambda (a1: A).(\lambda (_: (arity g c0 u a1)).(\lambda (_: ((\forall 
-(i: nat).(\forall (b: A).((aprem i a1 b) \to (ex2_3 C T nat (\lambda (d: 
-C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: 
-C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g 
-b))))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g c0 t0 
-(AHead a1 a2))).(\lambda (H3: ((\forall (i: nat).(\forall (b: A).((aprem i 
-(AHead a1 a2) b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda 
-(j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0: 
-T).(\lambda (_: nat).(arity g d u0 (asucc g b))))))))))).(\lambda (i: 
-nat).(\lambda (b: A).(\lambda (H4: (aprem i a2 b)).(let H5 \def (H3 (S i) b 
-(aprem_succ a2 b i H4 a1)) in (ex2_3_ind C T nat (\lambda (d: C).(\lambda (_: 
-T).(\lambda (j: nat).(drop (S (plus i j)) O d c0)))) (\lambda (d: C).(\lambda 
-(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))) (ex2_3 C T nat 
-(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) 
+b))))))))).(\lambda (H5: (aprem (S i0) (AHead a1 a2) b)).(let H_y \def 
+(aprem_gen_head_S a1 a2 b i0 H5) in (let H_x \def (H3 i0 b H_y) in (let H6 
+\def H_x in (ex2_3_ind C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: 
+nat).(drop (plus i0 j) O d (CHead c0 (Bind Abst) u))))) (\lambda (d: 
+C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))) (ex2_3 C 
+T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus (S i0) j) 
+O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 
+(asucc g b)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: 
+nat).(\lambda (H7: (drop (plus i0 x2) O x0 (CHead c0 (Bind Abst) 
+u))).(\lambda (H8: (arity g x0 x1 (asucc g b))).(ex2_3_intro C T nat (\lambda 
+(d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus (S i0) j) O d c0)))) 
 (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g 
-b)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H6: 
-(drop (S (plus i x2)) O x0 c0)).(\lambda (H7: (arity g x0 x1 (asucc g 
-b))).(C_ind (\lambda (c1: C).((drop (S (plus i x2)) O c1 c0) \to ((arity g c1 
-x1 (asucc g b)) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda 
-(j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0: 
-T).(\lambda (_: nat).(arity g d u0 (asucc g b))))))))) (\lambda (n: 
-nat).(\lambda (H8: (drop (S (plus i x2)) O (CSort n) c0)).(\lambda (_: (arity 
-g (CSort n) x1 (asucc g b))).(and3_ind (eq C c0 (CSort n)) (eq nat (S (plus i 
-x2)) O) (eq nat O O) (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda 
-(j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0: 
-T).(\lambda (_: nat).(arity g d u0 (asucc g b)))))) (\lambda (_: (eq C c0 
-(CSort n))).(\lambda (H11: (eq nat (S (plus i x2)) O)).(\lambda (_: (eq nat O 
-O)).(let H13 \def (eq_ind nat (S (plus i x2)) (\lambda (ee: nat).(match ee in 
-nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) 
-\Rightarrow True])) I O H11) in (False_ind (ex2_3 C T nat (\lambda (d: 
-C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: 
-C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b)))))) H13))))) 
-(drop_gen_sort n (S (plus i x2)) O c0 H8))))) (\lambda (d: C).(\lambda (IHd: 
-(((drop (S (plus i x2)) O d c0) \to ((arity g d x1 (asucc g b)) \to (ex2_3 C 
-T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O 
-d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 
-(asucc g b)))))))))).(\lambda (k: K).(\lambda (t1: T).(\lambda (H8: (drop (S 
-(plus i x2)) O (CHead d k t1) c0)).(\lambda (H9: (arity g (CHead d k t1) x1 
-(asucc g b))).(K_ind (\lambda (k0: K).((arity g (CHead d k0 t1) x1 (asucc g 
-b)) \to ((drop (r k0 (plus i x2)) O d c0) \to (ex2_3 C T nat (\lambda (d0: 
-C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d0 c0)))) (\lambda 
-(d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b))))))))) 
-(\lambda (b0: B).(\lambda (H10: (arity g (CHead d (Bind b0) t1) x1 (asucc g 
-b))).(\lambda (H11: (drop (r (Bind b0) (plus i x2)) O d c0)).(ex2_3_intro C T 
-nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d0 
-c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 
-(asucc g b))))) (CHead d (Bind b0) t1) x1 (S x2) (eq_ind nat (S (plus i x2)) 
-(\lambda (n: nat).(drop n O (CHead d (Bind b0) t1) c0)) (drop_drop (Bind b0) 
-(plus i x2) d c0 H11 t1) (plus i (S x2)) (plus_n_Sm i x2)) H10)))) (\lambda 
-(f: F).(\lambda (H10: (arity g (CHead d (Flat f) t1) x1 (asucc g 
-b))).(\lambda (H11: (drop (r (Flat f) (plus i x2)) O d c0)).(let H12 \def 
-(IHd H11 (arity_cimp_conf g (CHead d (Flat f) t1) x1 (asucc g b) H10 d 
-(cimp_flat_sx f d t1))) in (ex2_3_ind C T nat (\lambda (d0: C).(\lambda (_: 
+b))))) x0 x1 x2 (drop_S Abst x0 c0 u (plus i0 x2) H7) H8)))))) H6))))))) i 
+H4))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda 
+(_: (arity g c0 u a1)).(\lambda (_: ((\forall (i: nat).(\forall (b: 
+A).((aprem i a1 b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: 
+T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda 
+(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))))))))).(\lambda (t0: 
+T).(\lambda (a2: A).(\lambda (_: (arity g c0 t0 (AHead a1 a2))).(\lambda (H3: 
+((\forall (i: nat).(\forall (b: A).((aprem i (AHead a1 a2) b) \to (ex2_3 C T 
+nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d 
+c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 
+(asucc g b))))))))))).(\lambda (i: nat).(\lambda (b: A).(\lambda (H4: (aprem 
+i a2 b)).(let H5 \def (H3 (S i) b (aprem_succ a2 b i H4 a1)) in (ex2_3_ind C 
+T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (S (plus i j)) 
+O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 
+(asucc g b))))) (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: 
+nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda 
+(_: nat).(arity g d u0 (asucc g b)))))) (\lambda (x0: C).(\lambda (x1: 
+T).(\lambda (x2: nat).(\lambda (H6: (drop (S (plus i x2)) O x0 c0)).(\lambda 
+(H7: (arity g x0 x1 (asucc g b))).(C_ind (\lambda (c1: C).((drop (S (plus i 
+x2)) O c1 c0) \to ((arity g c1 x1 (asucc g b)) \to (ex2_3 C T nat (\lambda 
+(d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda 
+(d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))))))) 
+(\lambda (n: nat).(\lambda (H8: (drop (S (plus i x2)) O (CSort n) 
+c0)).(\lambda (_: (arity g (CSort n) x1 (asucc g b))).(and3_ind (eq C c0 
+(CSort n)) (eq nat (S (plus i x2)) O) (eq nat O O) (ex2_3 C T nat (\lambda 
+(d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda 
+(d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b)))))) 
+(\lambda (_: (eq C c0 (CSort n))).(\lambda (H11: (eq nat (S (plus i x2)) 
+O)).(\lambda (_: (eq nat O O)).(let H13 \def (eq_ind nat (S (plus i x2)) 
+(\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O 
+\Rightarrow False | (S _) \Rightarrow True])) I O H11) in (False_ind (ex2_3 C 
+T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d 
+c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 
+(asucc g b)))))) H13))))) (drop_gen_sort n (S (plus i x2)) O c0 H8))))) 
+(\lambda (d: C).(\lambda (IHd: (((drop (S (plus i x2)) O d c0) \to ((arity g 
+d x1 (asucc g b)) \to (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: 
 T).(\lambda (j: nat).(drop (plus i j) O d0 c0)))) (\lambda (d0: C).(\lambda 
-(u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b))))) (ex2_3 C T nat 
-(\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d0 
-c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 
-(asucc g b)))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: 
-nat).(\lambda (H13: (drop (plus i x5) O x3 c0)).(\lambda (H14: (arity g x3 x4 
-(asucc g b))).(ex2_3_intro C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda 
-(j: nat).(drop (plus i j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: 
-T).(\lambda (_: nat).(arity g d0 u0 (asucc g b))))) x3 x4 x5 H13 H14)))))) 
-H12))))) k H9 (drop_gen_drop k d c0 t1 (plus i x2) H8)))))))) x0 H6 H7)))))) 
+(u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b)))))))))).(\lambda (k: 
+K).(\lambda (t1: T).(\lambda (H8: (drop (S (plus i x2)) O (CHead d k t1) 
+c0)).(\lambda (H9: (arity g (CHead d k t1) x1 (asucc g b))).(K_ind (\lambda 
+(k0: K).((arity g (CHead d k0 t1) x1 (asucc g b)) \to ((drop (r k0 (plus i 
+x2)) O d c0) \to (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: 
+nat).(drop (plus i j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda 
+(_: nat).(arity g d0 u0 (asucc g b))))))))) (\lambda (b0: B).(\lambda (H10: 
+(arity g (CHead d (Bind b0) t1) x1 (asucc g b))).(\lambda (H11: (drop (r 
+(Bind b0) (plus i x2)) O d c0)).(ex2_3_intro C T nat (\lambda (d0: 
+C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d0 c0)))) (\lambda 
+(d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b))))) 
+(CHead d (Bind b0) t1) x1 (S x2) (eq_ind nat (S (plus i x2)) (\lambda (n: 
+nat).(drop n O (CHead d (Bind b0) t1) c0)) (drop_drop (Bind b0) (plus i x2) d 
+c0 H11 t1) (plus i (S x2)) (plus_n_Sm i x2)) H10)))) (\lambda (f: F).(\lambda 
+(H10: (arity g (CHead d (Flat f) t1) x1 (asucc g b))).(\lambda (H11: (drop (r 
+(Flat f) (plus i x2)) O d c0)).(let H12 \def (IHd H11 (arity_cimp_conf g 
+(CHead d (Flat f) t1) x1 (asucc g b) H10 d (cimp_flat_sx f d t1))) in 
+(ex2_3_ind C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop 
+(plus i j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: 
+nat).(arity g d0 u0 (asucc g b))))) (ex2_3 C T nat (\lambda (d0: C).(\lambda 
+(_: T).(\lambda (j: nat).(drop (plus i j) O d0 c0)))) (\lambda (d0: 
+C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b)))))) 
+(\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: nat).(\lambda (H13: (drop 
+(plus i x5) O x3 c0)).(\lambda (H14: (arity g x3 x4 (asucc g 
+b))).(ex2_3_intro C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: 
+nat).(drop (plus i j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda 
+(_: nat).(arity g d0 u0 (asucc g b))))) x3 x4 x5 H13 H14)))))) H12))))) k H9 
+(drop_gen_drop k d c0 t1 (plus i x2) H8)))))))) x0 H6 H7)))))) 
 H5)))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a0: A).(\lambda 
 (_: (arity g c0 u (asucc g a0))).(\lambda (_: ((\forall (i: nat).(\forall (b: 
 A).((aprem i (asucc g a0) b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: