--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/arity/props.ma".
+
+include "LambdaDelta-1/arity/cimp.ma".
+
+include "LambdaDelta-1/aprem/props.ma".
+
+theorem arity_aprem:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t
+a) \to (\forall (i: nat).(\forall (b: A).((aprem i a b) \to (ex2_3 C T nat
+(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c))))
+(\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g
+b)))))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H:
+(arity g c t a)).(arity_ind g (\lambda (c0: C).(\lambda (_: T).(\lambda (a0:
+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 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
+i0 a0 b) \to (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j:
+nat).(drop (plus i0 j) O d0 d)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda
+(_: nat).(arity g d0 u0 (asucc g b))))))))))).(\lambda (i0: nat).(\lambda (b:
+A).(\lambda (H3: (aprem i0 a0 b)).(let H_x \def (H2 i0 b H3) in (let H4 \def
+H_x in (ex2_3_ind C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j:
+nat).(drop (plus i0 j) O d0 d)))) (\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 i0 j) O d0 c0)))) (\lambda
+(d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b))))))
+(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H5: (drop
+(plus i0 x2) O x0 d)).(\lambda (H6: (arity g x0 x1 (asucc g b))).(let H_x0
+\def (getl_drop_conf_rev (plus i0 x2) x0 d H5 Abbr c0 u i H0) in (let H7 \def
+H_x0 in (ex2_ind C (\lambda (c1: C).(drop (plus i0 x2) O c1 c0)) (\lambda
+(c1: C).(drop (S i) (plus i0 x2) c1 x0)) (ex2_3 C T nat (\lambda (d0:
+C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0 c0)))) (\lambda
+(d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b))))))
+(\lambda (x: C).(\lambda (H8: (drop (plus i0 x2) O x c0)).(\lambda (H9: (drop
+(S i) (plus i0 x2) x x0)).(ex2_3_intro C T nat (\lambda (d0: C).(\lambda (_:
+T).(\lambda (j: nat).(drop (plus i0 j) O d0 c0)))) (\lambda (d0: C).(\lambda
+(u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b))))) x (lift (S i) (plus
+i0 x2) x1) x2 H8 (arity_lift g x0 x1 (asucc g b) H6 x (S i) (plus i0 x2)
+H9))))) H7)))))))) H4)))))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda
+(u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abst)
+u))).(\lambda (a0: A).(\lambda (_: (arity g d u (asucc g a0))).(\lambda (H2:
+((\forall (i0: nat).(\forall (b: A).((aprem i0 (asucc g a0) b) \to (ex2_3 C T
+nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0
+d)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0
+(asucc g b))))))))))).(\lambda (i0: nat).(\lambda (b: A).(\lambda (H3: (aprem
+i0 a0 b)).(let H4 \def (H2 i0 b (aprem_asucc g a0 b i0 H3)) in (ex2_3_ind C T
+nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0
+d)))) (\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 i0 j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda
+(_: nat).(arity g d0 u0 (asucc g b)))))) (\lambda (x0: C).(\lambda (x1:
+T).(\lambda (x2: nat).(\lambda (H5: (drop (plus i0 x2) O x0 d)).(\lambda (H6:
+(arity g x0 x1 (asucc g b))).(let H_x \def (getl_drop_conf_rev (plus i0 x2)
+x0 d H5 Abst c0 u i H0) in (let H7 \def H_x in (ex2_ind C (\lambda (c1:
+C).(drop (plus i0 x2) O c1 c0)) (\lambda (c1: C).(drop (S i) (plus i0 x2) c1
+x0)) (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop
+(plus i0 j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_:
+nat).(arity g d0 u0 (asucc g b)))))) (\lambda (x: C).(\lambda (H8: (drop
+(plus i0 x2) O x c0)).(\lambda (H9: (drop (S i) (plus i0 x2) x
+x0)).(ex2_3_intro C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j:
+nat).(drop (plus i0 j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda
+(_: nat).(arity g d0 u0 (asucc g b))))) x (lift (S i) (plus i0 x2) x1) x2 H8
+(arity_lift g x0 x1 (asucc g b) H6 x (S i) (plus i0 x2) H9))))) H7))))))))
+H4))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda
+(c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u
+a1)).(\lambda (_: ((\forall (i: nat).(\forall (b0: A).((aprem i a1 b0) \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 b0))))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_:
+(arity g (CHead c0 (Bind b) u) t0 a2)).(\lambda (H4: ((\forall (i:
+nat).(\forall (b0: A).((aprem i a2 b0) \to (ex2_3 C T nat (\lambda (d:
+C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d (CHead c0 (Bind b)
+u))))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0
+(asucc g b0))))))))))).(\lambda (i: nat).(\lambda (b0: A).(\lambda (H5:
+(aprem i a2 b0)).(let H_x \def (H4 i b0 H5) in (let H6 \def H_x in (ex2_3_ind
+C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O
+d (CHead c0 (Bind b) u))))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_:
+nat).(arity g d u0 (asucc g b0))))) (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 b0)))))) (\lambda (x0:
+C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H7: (drop (plus i x2) O x0
+(CHead c0 (Bind b) u))).(\lambda (H8: (arity g x0 x1 (asucc g b0))).(let H9
+\def (eq_ind nat (S (plus i x2)) (\lambda (n: nat).(drop n O x0 c0)) (drop_S
+b x0 c0 u (plus i x2) H7) (plus i (S x2)) (plus_n_Sm i x2)) in (ex2_3_intro 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 b0))))) x0 x1 (S x2) H9 H8))))))) H6))))))))))))))))) (\lambda (c0:
+C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H0: (arity g c0 u (asucc g
+a1))).(\lambda (_: ((\forall (i: nat).(\forall (b: A).((aprem i (asucc g 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 (CHead c0 (Bind Abst) u) t0 a2)).(\lambda (H3:
+((\forall (i: nat).(\forall (b: A).((aprem i a2 b) \to (ex2_3 C T nat
+(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d (CHead
+c0 (Bind Abst) u))))) (\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 (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 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 (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))))) 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)))))))))).(\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 (_:
+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 (_: (arity g c0 t0 a0)).(\lambda (H3: ((\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
+(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))))))))).(\lambda (i:
+nat).(\lambda (b: A).(\lambda (H4: (aprem i a0 b)).(let H_x \def (H3 i b H4)
+in (let H5 \def H_x in (ex2_3_ind 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))))) (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 (plus i x2) O x0 c0)).(\lambda (H7: (arity g x0 x1 (asucc g
+b))).(ex2_3_intro 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))))) x0 x1 x2 H6 H7)))))) H5))))))))))))))
+(\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda (_: (arity g c0
+t0 a1)).(\lambda (H1: ((\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 (u: T).(\lambda (_: nat).(arity g d
+u (asucc g b))))))))))).(\lambda (a2: A).(\lambda (H2: (leq g a1
+a2)).(\lambda (i: nat).(\lambda (b: A).(\lambda (H3: (aprem i a2 b)).(let H_x
+\def (aprem_repl g a1 a2 H2 i b H3) in (let H4 \def H_x in (ex2_ind A
+(\lambda (b1: A).(leq g b1 b)) (\lambda (b1: A).(aprem i a1 b1)) (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 (x: A).(\lambda (H5: (leq g x b)).(\lambda (H6: (aprem i
+a1 x)).(let H_x0 \def (H1 i x H6) in (let H7 \def H_x0 in (ex2_3_ind 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
+x))))) (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 (x0: C).(\lambda (x1: T).(\lambda
+(x2: nat).(\lambda (H8: (drop (plus i x2) O x0 c0)).(\lambda (H9: (arity g x0
+x1 (asucc g x))).(ex2_3_intro 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))))) x0 x1 x2 H8 (arity_repl g
+x0 x1 (asucc g x) H9 (asucc g b) (asucc_repl g x b H5)))))))) H7))))))
+H4))))))))))))) c t a H))))).
+
projects / helm.git / blobdiff