+++ /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 "basic_1/arity/props.ma".
-
-include "basic_1/fsubst0/fwd.ma".
-
-include "basic_1/csubst0/getl.ma".
-
-include "basic_1/subst0/dec.ma".
-
-include "basic_1/subst0/fwd.ma".
-
-include "basic_1/getl/getl.ma".
-
-lemma arity_gen_cvoid_subst0:
- \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t
-a) \to (\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d
-(Bind Void) u)) \to (\forall (w: T).(\forall (v: T).((subst0 i w t v) \to
-(\forall (P: Prop).P))))))))))))
-\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 (t0: T).(\lambda (_:
-A).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c0 (CHead d
-(Bind Void) u)) \to (\forall (w: T).(\forall (v: T).((subst0 i w t0 v) \to
-(\forall (P: Prop).P))))))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda
-(d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d
-(Bind Void) u))).(\lambda (w: T).(\lambda (v: T).(\lambda (H1: (subst0 i w
-(TSort n) v)).(\lambda (P: Prop).(subst0_gen_sort w v i n H1 P)))))))))))
-(\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 (_: ((\forall (d0: C).(\forall (u0: T).(\forall
-(i0: nat).((getl i0 d (CHead d0 (Bind Void) u0)) \to (\forall (w: T).(\forall
-(v: T).((subst0 i0 w u v) \to (\forall (P: Prop).P)))))))))).(\lambda (d0:
-C).(\lambda (u0: T).(\lambda (i0: nat).(\lambda (H3: (getl i0 c0 (CHead d0
-(Bind Void) u0))).(\lambda (w: T).(\lambda (v: T).(\lambda (H4: (subst0 i0 w
-(TLRef i) v)).(\lambda (P: Prop).(land_ind (eq nat i i0) (eq T v (lift (S i)
-O w)) P (\lambda (H5: (eq nat i i0)).(\lambda (_: (eq T v (lift (S i) O
-w))).(let H7 \def (eq_ind_r nat i0 (\lambda (n: nat).(getl n c0 (CHead d0
-(Bind Void) u0))) H3 i H5) in (let H8 \def (eq_ind C (CHead d (Bind Abbr) u)
-(\lambda (c1: C).(getl i c0 c1)) H0 (CHead d0 (Bind Void) u0) (getl_mono c0
-(CHead d (Bind Abbr) u) i H0 (CHead d0 (Bind Void) u0) H7)) in (let H9 \def
-(eq_ind C (CHead d (Bind Abbr) u) (\lambda (ee: C).(match ee with [(CSort _)
-\Rightarrow False | (CHead _ k _) \Rightarrow (match k with [(Bind b)
-\Rightarrow (match b with [Abbr \Rightarrow True | Abst \Rightarrow False |
-Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead d0 (Bind
-Void) u0) (getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead d0 (Bind Void)
-u0) H7)) in (False_ind P H9)))))) (subst0_gen_lref w v i0 i
-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 (_:
-((\forall (d0: C).(\forall (u0: T).(\forall (i0: nat).((getl i0 d (CHead d0
-(Bind Void) u0)) \to (\forall (w: T).(\forall (v: T).((subst0 i0 w u v) \to
-(\forall (P: Prop).P)))))))))).(\lambda (d0: C).(\lambda (u0: T).(\lambda
-(i0: nat).(\lambda (H3: (getl i0 c0 (CHead d0 (Bind Void) u0))).(\lambda (w:
-T).(\lambda (v: T).(\lambda (H4: (subst0 i0 w (TLRef i) v)).(\lambda (P:
-Prop).(land_ind (eq nat i i0) (eq T v (lift (S i) O w)) P (\lambda (H5: (eq
-nat i i0)).(\lambda (_: (eq T v (lift (S i) O w))).(let H7 \def (eq_ind_r nat
-i0 (\lambda (n: nat).(getl n c0 (CHead d0 (Bind Void) u0))) H3 i H5) in (let
-H8 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (c1: C).(getl i c0 c1)) H0
-(CHead d0 (Bind Void) u0) (getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead
-d0 (Bind Void) u0) H7)) in (let H9 \def (eq_ind C (CHead d (Bind Abst) u)
-(\lambda (ee: C).(match ee with [(CSort _) \Rightarrow False | (CHead _ k _)
-\Rightarrow (match k with [(Bind b) \Rightarrow (match b with [Abbr
-\Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat
-_) \Rightarrow False])])) I (CHead d0 (Bind Void) u0) (getl_mono c0 (CHead d
-(Bind Abst) u) i H0 (CHead d0 (Bind Void) u0) H7)) in (False_ind P H9))))))
-(subst0_gen_lref w v i0 i 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 (H2: ((\forall (d: C).(\forall
-(u0: T).(\forall (i: nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall
-(w: T).(\forall (v: T).((subst0 i w u v) \to (\forall (P:
-Prop).P)))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g
-(CHead c0 (Bind b) u) t0 a2)).(\lambda (H4: ((\forall (d: C).(\forall (u0:
-T).(\forall (i: nat).((getl i (CHead c0 (Bind b) u) (CHead d (Bind Void) u0))
-\to (\forall (w: T).(\forall (v: T).((subst0 i w t0 v) \to (\forall (P:
-Prop).P)))))))))).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda
-(H5: (getl i c0 (CHead d (Bind Void) u0))).(\lambda (w: T).(\lambda (v:
-T).(\lambda (H6: (subst0 i w (THead (Bind b) u t0) v)).(\lambda (P:
-Prop).(or3_ind (ex2 T (\lambda (u2: T).(eq T v (THead (Bind b) u2 t0)))
-(\lambda (u2: T).(subst0 i w u u2))) (ex2 T (\lambda (t2: T).(eq T v (THead
-(Bind b) u t2))) (\lambda (t2: T).(subst0 (s (Bind b) i) w t0 t2))) (ex3_2 T
-T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Bind b) u2 t2))))
-(\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda
-(t2: T).(subst0 (s (Bind b) i) w t0 t2)))) P (\lambda (H7: (ex2 T (\lambda
-(u2: T).(eq T v (THead (Bind b) u2 t0))) (\lambda (u2: T).(subst0 i w u
-u2)))).(ex2_ind T (\lambda (u2: T).(eq T v (THead (Bind b) u2 t0))) (\lambda
-(u2: T).(subst0 i w u u2)) P (\lambda (x: T).(\lambda (_: (eq T v (THead
-(Bind b) x t0))).(\lambda (H9: (subst0 i w u x)).(H2 d u0 i H5 w x H9 P))))
-H7)) (\lambda (H7: (ex2 T (\lambda (t2: T).(eq T v (THead (Bind b) u t2)))
-(\lambda (t2: T).(subst0 (s (Bind b) i) w t0 t2)))).(ex2_ind T (\lambda (t2:
-T).(eq T v (THead (Bind b) u t2))) (\lambda (t2: T).(subst0 (s (Bind b) i) w
-t0 t2)) P (\lambda (x: T).(\lambda (_: (eq T v (THead (Bind b) u
-x))).(\lambda (H9: (subst0 (s (Bind b) i) w t0 x)).(H4 d u0 (S i)
-(getl_clear_bind b (CHead c0 (Bind b) u) c0 u (clear_bind b c0 u) (CHead d
-(Bind Void) u0) i H5) w x H9 P)))) H7)) (\lambda (H7: (ex3_2 T T (\lambda
-(u2: T).(\lambda (t2: T).(eq T v (THead (Bind b) u2 t2)))) (\lambda (u2:
-T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda (t2:
-T).(subst0 (s (Bind b) i) w t0 t2))))).(ex3_2_ind T T (\lambda (u2:
-T).(\lambda (t2: T).(eq T v (THead (Bind b) u2 t2)))) (\lambda (u2:
-T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda (t2:
-T).(subst0 (s (Bind b) i) w t0 t2))) P (\lambda (x0: T).(\lambda (x1:
-T).(\lambda (_: (eq T v (THead (Bind b) x0 x1))).(\lambda (H9: (subst0 i w u
-x0)).(\lambda (_: (subst0 (s (Bind b) i) w t0 x1)).(H2 d u0 i H5 w x0 H9
-P)))))) H7)) (subst0_gen_head (Bind b) w u t0 v i H6)))))))))))))))))))))
-(\lambda (c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u
-(asucc g a1))).(\lambda (H1: ((\forall (d: C).(\forall (u0: T).(\forall (i:
-nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall (w: T).(\forall (v:
-T).((subst0 i w u v) \to (\forall (P: Prop).P)))))))))).(\lambda (t0:
-T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t0
-a2)).(\lambda (H3: ((\forall (d: C).(\forall (u0: T).(\forall (i: nat).((getl
-i (CHead c0 (Bind Abst) u) (CHead d (Bind Void) u0)) \to (\forall (w:
-T).(\forall (v: T).((subst0 i w t0 v) \to (\forall (P:
-Prop).P)))))))))).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda
-(H4: (getl i c0 (CHead d (Bind Void) u0))).(\lambda (w: T).(\lambda (v:
-T).(\lambda (H5: (subst0 i w (THead (Bind Abst) u t0) v)).(\lambda (P:
-Prop).(or3_ind (ex2 T (\lambda (u2: T).(eq T v (THead (Bind Abst) u2 t0)))
-(\lambda (u2: T).(subst0 i w u u2))) (ex2 T (\lambda (t2: T).(eq T v (THead
-(Bind Abst) u t2))) (\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 t2)))
-(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Bind Abst) u2
-t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_:
-T).(\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 t2)))) P (\lambda (H6:
-(ex2 T (\lambda (u2: T).(eq T v (THead (Bind Abst) u2 t0))) (\lambda (u2:
-T).(subst0 i w u u2)))).(ex2_ind T (\lambda (u2: T).(eq T v (THead (Bind
-Abst) u2 t0))) (\lambda (u2: T).(subst0 i w u u2)) P (\lambda (x: T).(\lambda
-(_: (eq T v (THead (Bind Abst) x t0))).(\lambda (H8: (subst0 i w u x)).(H1 d
-u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex2 T (\lambda (t2: T).(eq T v
-(THead (Bind Abst) u t2))) (\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0
-t2)))).(ex2_ind T (\lambda (t2: T).(eq T v (THead (Bind Abst) u t2)))
-(\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 t2)) P (\lambda (x:
-T).(\lambda (_: (eq T v (THead (Bind Abst) u x))).(\lambda (H8: (subst0 (s
-(Bind Abst) i) w t0 x)).(H3 d u0 (S i) (getl_clear_bind Abst (CHead c0 (Bind
-Abst) u) c0 u (clear_bind Abst c0 u) (CHead d (Bind Void) u0) i H4) w x H8
-P)))) H6)) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v
-(THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w u
-u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0
-t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Bind
-Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda
-(_: T).(\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 t2))) P (\lambda (x0:
-T).(\lambda (x1: T).(\lambda (_: (eq T v (THead (Bind Abst) x0 x1))).(\lambda
-(H8: (subst0 i w u x0)).(\lambda (_: (subst0 (s (Bind Abst) i) w t0 x1)).(H1
-d u0 i H4 w x0 H8 P)))))) H6)) (subst0_gen_head (Bind Abst) w u t0 v i
-H5))))))))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1:
-A).(\lambda (_: (arity g c0 u a1)).(\lambda (H1: ((\forall (d: C).(\forall
-(u0: T).(\forall (i: nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall
-(w: T).(\forall (v: T).((subst0 i w u v) \to (\forall (P:
-Prop).P)))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g c0
-t0 (AHead a1 a2))).(\lambda (H3: ((\forall (d: C).(\forall (u0: T).(\forall
-(i: nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall (w: T).(\forall
-(v: T).((subst0 i w t0 v) \to (\forall (P: Prop).P)))))))))).(\lambda (d:
-C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H4: (getl i c0 (CHead d (Bind
-Void) u0))).(\lambda (w: T).(\lambda (v: T).(\lambda (H5: (subst0 i w (THead
-(Flat Appl) u t0) v)).(\lambda (P: Prop).(or3_ind (ex2 T (\lambda (u2: T).(eq
-T v (THead (Flat Appl) u2 t0))) (\lambda (u2: T).(subst0 i w u u2))) (ex2 T
-(\lambda (t2: T).(eq T v (THead (Flat Appl) u t2))) (\lambda (t2: T).(subst0
-(s (Flat Appl) i) w t0 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq
-T v (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w
-u u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) w t0
-t2)))) P (\lambda (H6: (ex2 T (\lambda (u2: T).(eq T v (THead (Flat Appl) u2
-t0))) (\lambda (u2: T).(subst0 i w u u2)))).(ex2_ind T (\lambda (u2: T).(eq T
-v (THead (Flat Appl) u2 t0))) (\lambda (u2: T).(subst0 i w u u2)) P (\lambda
-(x: T).(\lambda (_: (eq T v (THead (Flat Appl) x t0))).(\lambda (H8: (subst0
-i w u x)).(H1 d u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex2 T (\lambda (t2:
-T).(eq T v (THead (Flat Appl) u t2))) (\lambda (t2: T).(subst0 (s (Flat Appl)
-i) w t0 t2)))).(ex2_ind T (\lambda (t2: T).(eq T v (THead (Flat Appl) u t2)))
-(\lambda (t2: T).(subst0 (s (Flat Appl) i) w t0 t2)) P (\lambda (x:
-T).(\lambda (_: (eq T v (THead (Flat Appl) u x))).(\lambda (H8: (subst0 (s
-(Flat Appl) i) w t0 x)).(H3 d u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex3_2
-T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Flat Appl) u2 t2))))
-(\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda
-(t2: T).(subst0 (s (Flat Appl) i) w t0 t2))))).(ex3_2_ind T T (\lambda (u2:
-T).(\lambda (t2: T).(eq T v (THead (Flat Appl) u2 t2)))) (\lambda (u2:
-T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda (t2:
-T).(subst0 (s (Flat Appl) i) w t0 t2))) P (\lambda (x0: T).(\lambda (x1:
-T).(\lambda (_: (eq T v (THead (Flat Appl) x0 x1))).(\lambda (H8: (subst0 i w
-u x0)).(\lambda (_: (subst0 (s (Flat Appl) i) w t0 x1)).(H1 d u0 i H4 w x0 H8
-P)))))) H6)) (subst0_gen_head (Flat Appl) w u t0 v i H5)))))))))))))))))))
-(\lambda (c0: C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_: (arity g c0 u
-(asucc g a0))).(\lambda (H1: ((\forall (d: C).(\forall (u0: T).(\forall (i:
-nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall (w: T).(\forall (v:
-T).((subst0 i w u v) \to (\forall (P: Prop).P)))))))))).(\lambda (t0:
-T).(\lambda (_: (arity g c0 t0 a0)).(\lambda (H3: ((\forall (d: C).(\forall
-(u0: T).(\forall (i: nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall
-(w: T).(\forall (v: T).((subst0 i w t0 v) \to (\forall (P:
-Prop).P)))))))))).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda
-(H4: (getl i c0 (CHead d (Bind Void) u0))).(\lambda (w: T).(\lambda (v:
-T).(\lambda (H5: (subst0 i w (THead (Flat Cast) u t0) v)).(\lambda (P:
-Prop).(or3_ind (ex2 T (\lambda (u2: T).(eq T v (THead (Flat Cast) u2 t0)))
-(\lambda (u2: T).(subst0 i w u u2))) (ex2 T (\lambda (t2: T).(eq T v (THead
-(Flat Cast) u t2))) (\lambda (t2: T).(subst0 (s (Flat Cast) i) w t0 t2)))
-(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Flat Cast) u2
-t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_:
-T).(\lambda (t2: T).(subst0 (s (Flat Cast) i) w t0 t2)))) P (\lambda (H6:
-(ex2 T (\lambda (u2: T).(eq T v (THead (Flat Cast) u2 t0))) (\lambda (u2:
-T).(subst0 i w u u2)))).(ex2_ind T (\lambda (u2: T).(eq T v (THead (Flat
-Cast) u2 t0))) (\lambda (u2: T).(subst0 i w u u2)) P (\lambda (x: T).(\lambda
-(_: (eq T v (THead (Flat Cast) x t0))).(\lambda (H8: (subst0 i w u x)).(H1 d
-u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex2 T (\lambda (t2: T).(eq T v
-(THead (Flat Cast) u t2))) (\lambda (t2: T).(subst0 (s (Flat Cast) i) w t0
-t2)))).(ex2_ind T (\lambda (t2: T).(eq T v (THead (Flat Cast) u t2)))
-(\lambda (t2: T).(subst0 (s (Flat Cast) i) w t0 t2)) P (\lambda (x:
-T).(\lambda (_: (eq T v (THead (Flat Cast) u x))).(\lambda (H8: (subst0 (s
-(Flat Cast) i) w t0 x)).(H3 d u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex3_2
-T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Flat Cast) u2 t2))))
-(\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda
-(t2: T).(subst0 (s (Flat Cast) i) w t0 t2))))).(ex3_2_ind T T (\lambda (u2:
-T).(\lambda (t2: T).(eq T v (THead (Flat Cast) u2 t2)))) (\lambda (u2:
-T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda (t2:
-T).(subst0 (s (Flat Cast) i) w t0 t2))) P (\lambda (x0: T).(\lambda (x1:
-T).(\lambda (_: (eq T v (THead (Flat Cast) x0 x1))).(\lambda (H8: (subst0 i w
-u x0)).(\lambda (_: (subst0 (s (Flat Cast) i) w t0 x1)).(H1 d u0 i H4 w x0 H8
-P)))))) H6)) (subst0_gen_head (Flat Cast) w u t0 v i H5))))))))))))))))))
-(\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda (_: (arity g c0
-t0 a1)).(\lambda (H1: ((\forall (d: C).(\forall (u: T).(\forall (i:
-nat).((getl i c0 (CHead d (Bind Void) u)) \to (\forall (w: T).(\forall (v:
-T).((subst0 i w t0 v) \to (\forall (P: Prop).P)))))))))).(\lambda (a2:
-A).(\lambda (_: (leq g a1 a2)).(\lambda (d: C).(\lambda (u: T).(\lambda (i:
-nat).(\lambda (H3: (getl i c0 (CHead d (Bind Void) u))).(\lambda (w:
-T).(\lambda (v: T).(\lambda (H4: (subst0 i w t0 v)).(\lambda (P: Prop).(H1 d
-u i H3 w v H4 P)))))))))))))))) c t a H))))).
-
-lemma arity_gen_cvoid:
- \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t
-a) \to (\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d
-(Bind Void) u)) \to (ex T (\lambda (v: T).(eq T t (lift (S O) i v))))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H:
-(arity g c t a)).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
-(H0: (getl i c (CHead d (Bind Void) u))).(let H_x \def (dnf_dec u t i) in
-(let H1 \def H_x in (ex_ind T (\lambda (v: T).(or (subst0 i u t (lift (S O) i
-v)) (eq T t (lift (S O) i v)))) (ex T (\lambda (v: T).(eq T t (lift (S O) i
-v)))) (\lambda (x: T).(\lambda (H2: (or (subst0 i u t (lift (S O) i x)) (eq T
-t (lift (S O) i x)))).(or_ind (subst0 i u t (lift (S O) i x)) (eq T t (lift
-(S O) i x)) (ex T (\lambda (v: T).(eq T t (lift (S O) i v)))) (\lambda (H3:
-(subst0 i u t (lift (S O) i x))).(arity_gen_cvoid_subst0 g c t a H d u i H0 u
-(lift (S O) i x) H3 (ex T (\lambda (v: T).(eq T t (lift (S O) i v))))))
-(\lambda (H3: (eq T t (lift (S O) i x))).(let H4 \def (eq_ind T t (\lambda
-(t0: T).(arity g c t0 a)) H (lift (S O) i x) H3) in (eq_ind_r T (lift (S O) i
-x) (\lambda (t0: T).(ex T (\lambda (v: T).(eq T t0 (lift (S O) i v)))))
-(ex_intro T (\lambda (v: T).(eq T (lift (S O) i x) (lift (S O) i v))) x
-(refl_equal T (lift (S O) i x))) t H3))) H2))) H1))))))))))).
-
-lemma arity_fsubst0:
- \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (a: A).((arity g
-c1 t1 a) \to (\forall (d1: C).(\forall (u: T).(\forall (i: nat).((getl i c1
-(CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u
-c1 t1 c2 t2) \to (arity g c2 t2 a))))))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (a: A).(\lambda
-(H: (arity g c1 t1 a)).(arity_ind g (\lambda (c: C).(\lambda (t: T).(\lambda
-(a0: A).(\forall (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 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 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))).(land_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))).(land_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))).(land_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))).(land_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)) (land_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 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 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))).(land_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_n (S i) (S i0) (le_S (S (S i)) (S i0)
-(le_n_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_n (S i) (S i0) (le_S (S (S
-i)) (S i0) (le_n_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 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 with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow
-(match k 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 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: 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_n (S i) (S i0)
-(le_S (S (S i)) (S i0) (le_n_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
-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 with [(CSort _) \Rightarrow Abbr | (CHead _ k _)
-\Rightarrow (match k 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 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 (_:
-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)))))))).(ex4_5_ind 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 (x0: B).(\lambda (x1: C).(\lambda (x2:
-C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H12: (eq C (CHead d (Bind
-Abbr) u) (CHead x1 (Bind x0) x3))).(\lambda (H13: (getl i c2 (CHead x2 (Bind
-x0) x4))).(\lambda (H14: (subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H15:
-(csubst0 (minus i0 (S i)) u0 x1 x2)).(let H16 \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_n (S i) (S i0) (le_S (S (S i)) (S i0) (le_n_S (S i) i0
-H9))))) (S (minus i0 (S i))) (minus_x_Sy i0 i H9)) in (let H17 \def (f_equal
-C C (\lambda (e: C).(match e with [(CSort _) \Rightarrow d | (CHead c0 _ _)
-\Rightarrow c0])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H12) in
-((let H18 \def (f_equal C B (\lambda (e: C).(match e with [(CSort _)
-\Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k with [(Bind b)
-\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead
-x1 (Bind x0) x3) H12) in ((let H19 \def (f_equal C T (\lambda (e: C).(match e
-with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind
-Abbr) u) (CHead x1 (Bind x0) x3) H12) in (\lambda (H20: (eq B Abbr
-x0)).(\lambda (H21: (eq C d x1)).(let H22 \def (eq_ind_r T x3 (\lambda (t:
-T).(subst0 (minus i0 (S i)) u0 t x4)) H14 u H19) in (let H23 \def (eq_ind_r C
-x1 (\lambda (c0: C).(csubst0 (minus i0 (S i)) u0 c0 x2)) H15 d H21) in (let
-H24 \def (eq_ind_r B x0 (\lambda (b: B).(getl i c2 (CHead x2 (Bind b) x4)))
-H13 Abbr H20) in (arity_abbr g c2 x2 x4 i H24 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)) H16) x2 x4 (fsubst0_both (r (Bind Abbr) (minus i0 (S i))) u0
-d u x4 H22 x2 H23))))))))) H18)) H17)))))))))))) H11)) H10))) (\lambda (H9:
-(le i0 i)).(arity_abbr g c2 d u i (csubst0_getl_ge i0 i H9 c c2 u0 H8 (CHead
-d (Bind Abbr) u) H0) a0 H1))) t2 H7))) H6)) (\lambda (H6: (land (subst0 i0 u0
-(TLRef i) t2) (csubst0 i0 u0 c c2))).(land_ind (subst0 i0 u0 (TLRef i) t2)
-(csubst0 i0 u0 c c2) (arity g c2 t2 a0) (\lambda (H7: (subst0 i0 u0 (TLRef i)
-t2)).(\lambda (H8: (csubst0 i0 u0 c c2)).(land_ind (eq nat i i0) (eq T t2
-(lift (S i) O u0)) (arity g c2 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 c2 t a0)) (let H11 \def (eq_ind_r nat i0 (\lambda (n:
-nat).(csubst0 n u0 c c2)) H8 i H9) in (let H12 \def (eq_ind_r nat i0 (\lambda
-(n: nat).(getl n c (CHead d1 (Bind Abbr) u0))) H3 i H9) in (let H13 \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) H12)) in (let H14 \def (f_equal C C (\lambda (e: C).(match e 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) H12)) in ((let H15 \def (f_equal C T (\lambda (e:
-C).(match e 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) H12)) in (\lambda (H16: (eq C d
-d1)).(let H17 \def (eq_ind_r T u0 (\lambda (t: T).(getl i c (CHead d1 (Bind
-Abbr) t))) H13 u H15) in (let H18 \def (eq_ind_r T u0 (\lambda (t:
-T).(csubst0 i t c c2)) H11 u H15) in (eq_ind T u (\lambda (t: T).(arity g c2
-(lift (S i) O t) a0)) (let H19 \def (eq_ind_r C d1 (\lambda (c0: C).(getl i c
-(CHead c0 (Bind Abbr) 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 (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 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))).(land_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)) (land_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
-with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k with
-[(Bind b) \Rightarrow (match b 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) H11)) in (False_ind (arity g c (lift (S i) O u0) a0)
-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))).(land_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
-Abst) u) H0) in (or4_ind (getl i c2 (CHead d (Bind Abst) u)) (ex3_4 B C T T
-(\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C
-(CHead d (Bind Abst) 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 Abst) 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 Abst) 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 Abst)
-u))).(let H12 \def (eq_ind nat (minus i0 i) (\lambda (n: nat).(getl n (CHead
-d (Bind Abst) u) (CHead d1 (Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind
-Abbr) u0) c H3 (CHead d (Bind Abst) u) i H0 (le_S_n i i0 (le_S_n (S i) (S i0)
-(le_S (S (S i)) (S i0) (le_n_S (S i) i0 H9))))) (S (minus i0 (S i)))
-(minus_x_Sy i0 i H9)) in (arity_abst 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 Abst) 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
-Abst) 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 Abst) 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 Abst) u) (CHead d1 (Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind
-Abbr) u0) c H3 (CHead d (Bind Abst) u) i H0 (le_S_n i i0 (le_S_n (S i) (S i0)
-(le_S (S (S i)) (S i0) (le_n_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
-with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d
-(Bind Abst) u) (CHead x1 (Bind x0) x2) H12) in ((let H17 \def (f_equal C B
-(\lambda (e: C).(match e with [(CSort _) \Rightarrow Abst | (CHead _ k _)
-\Rightarrow (match k with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
-Abst])])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x2) H12) in ((let H18
-\def (f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u |
-(CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0)
-x2) H12) in (\lambda (H19: (eq B Abst 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 Abst H19) in (arity_abst g c2 d x3
-i H23 a0 (H2 d1 u0 (r (Bind Abst) (minus i0 (S i))) (getl_gen_S (Bind Abst) d
-(CHead d1 (Bind Abbr) u0) u (minus i0 (S i)) H15) d x3 (fsubst0_snd (r (Bind
-Abst) (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 Abst) 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 Abst) 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 Abst) 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 Abst) u) (CHead d1 (Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind
-Abbr) u0) c H3 (CHead d (Bind Abst) u) i H0 (le_S_n i i0 (le_S_n (S i) (S i0)
-(le_S (S (S i)) (S i0) (le_n_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
-with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d
-(Bind Abst) u) (CHead x1 (Bind x0) x3) H12) in ((let H17 \def (f_equal C B
-(\lambda (e: C).(match e with [(CSort _) \Rightarrow Abst | (CHead _ k _)
-\Rightarrow (match k with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
-Abst])])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H12) in ((let H18
-\def (f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u |
-(CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0)
-x3) H12) in (\lambda (H19: (eq B Abst 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 Abst H19) in (arity_abst g c2 x2 u
-i H23 a0 (H2 d1 u0 (r (Bind Abst) (minus i0 (S i))) (getl_gen_S (Bind Abst) d
-(CHead d1 (Bind Abbr) u0) u (minus i0 (S i)) H15) x2 u (fsubst0_fst (r (Bind
-Abst) (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 Abst) 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)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abst) 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 (x0: B).(\lambda (x1: C).(\lambda (x2:
-C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H12: (eq C (CHead d (Bind
-Abst) u) (CHead x1 (Bind x0) x3))).(\lambda (H13: (getl i c2 (CHead x2 (Bind
-x0) x4))).(\lambda (H14: (subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H15:
-(csubst0 (minus i0 (S i)) u0 x1 x2)).(let H16 \def (eq_ind nat (minus i0 i)
-(\lambda (n: nat).(getl n (CHead d (Bind Abst) u) (CHead d1 (Bind Abbr) u0)))
-(getl_conf_le i0 (CHead d1 (Bind Abbr) u0) c H3 (CHead d (Bind Abst) u) i H0
-(le_S_n i i0 (le_S_n (S i) (S i0) (le_S (S (S i)) (S i0) (le_n_S (S i) i0
-H9))))) (S (minus i0 (S i))) (minus_x_Sy i0 i H9)) in (let H17 \def (f_equal
-C C (\lambda (e: C).(match e with [(CSort _) \Rightarrow d | (CHead c0 _ _)
-\Rightarrow c0])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H12) in
-((let H18 \def (f_equal C B (\lambda (e: C).(match e with [(CSort _)
-\Rightarrow Abst | (CHead _ k _) \Rightarrow (match k with [(Bind b)
-\Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead d (Bind Abst) u) (CHead
-x1 (Bind x0) x3) H12) in ((let H19 \def (f_equal C T (\lambda (e: C).(match e
-with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind
-Abst) u) (CHead x1 (Bind x0) x3) H12) in (\lambda (H20: (eq B Abst
-x0)).(\lambda (H21: (eq C d x1)).(let H22 \def (eq_ind_r T x3 (\lambda (t:
-T).(subst0 (minus i0 (S i)) u0 t x4)) H14 u H19) in (let H23 \def (eq_ind_r C
-x1 (\lambda (c0: C).(csubst0 (minus i0 (S i)) u0 c0 x2)) H15 d H21) in (let
-H24 \def (eq_ind_r B x0 (\lambda (b: B).(getl i c2 (CHead x2 (Bind b) x4)))
-H13 Abst H20) in (arity_abst g c2 x2 x4 i H24 a0 (H2 d1 u0 (r (Bind Abst)
-(minus i0 (S i))) (getl_gen_S (Bind Abst) d (CHead d1 (Bind Abbr) u0) u
-(minus i0 (S i)) H16) x2 x4 (fsubst0_both (r (Bind Abst) (minus i0 (S i))) u0
-d u x4 H22 x2 H23))))))))) H18)) H17)))))))))))) H11)) H10))) (\lambda (H9:
-(le i0 i)).(arity_abst g c2 d u i (csubst0_getl_ge i0 i H9 c c2 u0 H8 (CHead
-d (Bind Abst) u) H0) a0 H1))) t2 H7))) H6)) (\lambda (H6: (land (subst0 i0 u0
-(TLRef i) t2) (csubst0 i0 u0 c c2))).(land_ind (subst0 i0 u0 (TLRef i) t2)
-(csubst0 i0 u0 c c2) (arity g c2 t2 a0) (\lambda (H7: (subst0 i0 u0 (TLRef i)
-t2)).(\lambda (H8: (csubst0 i0 u0 c c2)).(land_ind (eq nat i i0) (eq T t2
-(lift (S i) O u0)) (arity g c2 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 c2 t a0)) (let H11 \def (eq_ind_r nat i0 (\lambda (n:
-nat).(csubst0 n u0 c c2)) H8 i H9) in (let H12 \def (eq_ind_r nat i0 (\lambda
-(n: nat).(getl n c (CHead d1 (Bind Abbr) u0))) H3 i H9) in (let H13 \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) H12)) in (let H14 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda
-(ee: C).(match ee with [(CSort _) \Rightarrow False | (CHead _ k _)
-\Rightarrow (match k with [(Bind b) \Rightarrow (match b 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: 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) u0)) \to (\forall (c2: C).(\forall (t2:
-T).((fsubst0 i u0 c u c2 t2) \to (arity g c2 t2 a1)))))))))).(\lambda (t:
-T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c (Bind b) u) t
-a2)).(\lambda (H4: ((\forall (d1: C).(\forall (u0: T).(\forall (i:
-nat).((getl i (CHead c (Bind b) u) (CHead d1 (Bind Abbr) 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 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))).(land_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
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-u0 i H4 c2 x0 (fsubst0_both i u0 c u x0 H12 c2 H9)) 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 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)))))))))))))))))))
-(\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:
-T).((fsubst0 i u0 c u c2 t2) \to (arity g c2 t2 a1)))))))))).(\lambda (t:
-T).(\lambda (a2: A).(\lambda (H2: (arity g c t (AHead a1 a2))).(\lambda (H3:
-((\forall (d1: C).(\forall (u0: T).(\forall (i: nat).((getl i c (CHead d1
-(Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 c t c2
-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 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))).(land_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 (\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
-t3))))).(ex3_2_ind 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 (x0: T).(\lambda (x1: T).(\lambda (H11: (eq T t2 (THead
-(Flat Appl) x0 x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13:
-(subst0 (s (Flat Appl) i) u0 t x1)).(eq_ind_r T (THead (Flat Appl) x0 x1)
-(\lambda (t0: T).(arity g c t0 a2)) (arity_appl g c x0 a1 (H1 d1 u0 i H4 c x0
-(fsubst0_snd i u0 c u x0 H12)) x1 a2 (H3 d1 u0 i H4 c x1 (fsubst0_snd i u0 c
-t x1 H13))) t2 H11)))))) H10)) (subst0_gen_head (Flat Appl) u0 u t t2 i H9))
-c2 H8))) H7)) (\lambda (H7: (land (eq T (THead (Flat Appl) u t) t2) (csubst0
-i u0 c c2))).(land_ind (eq T (THead (Flat Appl) u t) t2) (csubst0 i u0 c c2)
-(arity g c2 t2 a2) (\lambda (H8: (eq T (THead (Flat Appl) u t) t2)).(\lambda
-(H9: (csubst0 i u0 c c2)).(eq_ind T (THead (Flat Appl) u t) (\lambda (t0:
-T).(arity g c2 t0 a2)) (arity_appl g c2 u a1 (H1 d1 u0 i H4 c2 u (fsubst0_fst
-i u0 c u c2 H9)) t a2 (H3 d1 u0 i H4 c2 t (fsubst0_fst i u0 c t c2 H9))) t2
-H8))) H7)) (\lambda (H7: (land (subst0 i u0 (THead (Flat Appl) u t) t2)
-(csubst0 i u0 c c2))).(land_ind (subst0 i u0 (THead (Flat Appl) u t) t2)
-(csubst0 i u0 c c2) (arity g c2 t2 a2) (\lambda (H8: (subst0 i u0 (THead
-(Flat Appl) u t) t2)).(\lambda (H9: (csubst0 i u0 c c2)).(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 c2 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 c2 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 c2 t0 a2)) (arity_appl g c2 x a1 (H1 d1 u0 i H4 c2 x
-(fsubst0_both i u0 c u x H12 c2 H9)) t a2 (H3 d1 u0 i H4 c2 t (fsubst0_fst i
-u0 c t c2 H9))) 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 c2 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 c2 t0 a2)) (arity_appl g c2 u a1 (H1 d1 u0 i H4 c2
-u (fsubst0_fst i u0 c u c2 H9)) x a2 (H3 d1 u0 i H4 c2 x (fsubst0_both i u0 c
-t x H12 c2 H9))) 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 t3))))).(ex3_2_ind 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 c2 t2 a2) (\lambda (x0:
-T).(\lambda (x1: T).(\lambda (H11: (eq T t2 (THead (Flat Appl) x0
-x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13: (subst0 (s (Flat
-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:
-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:
-T).((fsubst0 i u0 c u c2 t2) \to (arity g c2 t2 (asucc g
-a0))))))))))).(\lambda (t: T).(\lambda (H2: (arity g c t a0)).(\lambda (H3:
-((\forall (d1: C).(\forall (u0: T).(\forall (i: nat).((getl i c (CHead d1
-(Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 c t c2
-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 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))).(land_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 (\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)))
-(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t t3))) (arity
-g c t2 a0) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq T t2 (THead
-(Flat Cast) x0 x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13:
-(subst0 (s (Flat Cast) i) u0 t x1)).(eq_ind_r T (THead (Flat Cast) x0 x1)
-(\lambda (t0: T).(arity g c t0 a0)) (arity_cast g c x0 a0 (H1 d1 u0 i H4 c x0
-(fsubst0_snd i u0 c u x0 H12)) x1 (H3 d1 u0 i H4 c x1 (fsubst0_snd i u0 c t
-x1 H13))) t2 H11)))))) H10)) (subst0_gen_head (Flat Cast) u0 u t t2 i H9)) c2
-H8))) H7)) (\lambda (H7: (land (eq T (THead (Flat Cast) u t) t2) (csubst0 i
-u0 c c2))).(land_ind (eq T (THead (Flat Cast) u t) t2) (csubst0 i u0 c c2)
-(arity g c2 t2 a0) (\lambda (H8: (eq T (THead (Flat Cast) u t) t2)).(\lambda
-(H9: (csubst0 i u0 c c2)).(eq_ind T (THead (Flat Cast) u t) (\lambda (t0:
-T).(arity g c2 t0 a0)) (arity_cast g c2 u a0 (H1 d1 u0 i H4 c2 u (fsubst0_fst
-i u0 c u c2 H9)) t (H3 d1 u0 i H4 c2 t (fsubst0_fst i u0 c t c2 H9))) t2
-H8))) H7)) (\lambda (H7: (land (subst0 i u0 (THead (Flat Cast) u t) t2)
-(csubst0 i u0 c c2))).(land_ind (subst0 i u0 (THead (Flat Cast) u t) t2)
-(csubst0 i u0 c c2) (arity g c2 t2 a0) (\lambda (H8: (subst0 i u0 (THead
-(Flat Cast) u t) t2)).(\lambda (H9: (csubst0 i u0 c c2)).(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 c2 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 c2 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 c2 t0 a0)) (arity_cast g c2 x a0 (H1 d1 u0 i H4 c2 x
-(fsubst0_both i u0 c u x H12 c2 H9)) t (H3 d1 u0 i H4 c2 t (fsubst0_fst i u0
-c t c2 H9))) 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 c2 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 c2 t0 a0)) (arity_cast g c2 u a0 (H1 d1 u0 i H4 c2
-u (fsubst0_fst i u0 c u c2 H9)) x (H3 d1 u0 i H4 c2 x (fsubst0_both i u0 c t
-x H12 c2 H9))) 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))) (\lambda (_: T).(\lambda (t3:
-T).(subst0 (s (Flat Cast) i) u0 t t3))) (arity g c2 t2 a0) (\lambda (x0:
-T).(\lambda (x1: T).(\lambda (H11: (eq T t2 (THead (Flat Cast) x0
-x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13: (subst0 (s (Flat
-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 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))).(land_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))).(land_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))).(land_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))))).
-
-lemma arity_subst0:
- \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (a: A).((arity g c
-t1 a) \to (\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead
-d (Bind Abbr) u)) \to (\forall (t2: T).((subst0 i u t1 t2) \to (arity g c t2
-a)))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (a: A).(\lambda (H:
-(arity g c t1 a)).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
-(H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (t2: T).(\lambda (H1:
-(subst0 i u t1 t2)).(arity_fsubst0 g c t1 a H d u i H0 c t2 (fsubst0_snd i u
-c t1 t2 H1)))))))))))).
-