+++ /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/defs.ma".
-
-include "basic_1/leq/asucc.ma".
-
-include "basic_1/getl/drop.ma".
-
-implied rec lemma arity_ind (g: G) (P: (C \to (T \to (A \to Prop)))) (f:
-(\forall (c: C).(\forall (n: nat).(P c (TSort n) (ASort O n))))) (f0:
-(\forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c
-(CHead d (Bind Abbr) u)) \to (\forall (a: A).((arity g d u a) \to ((P d u a)
-\to (P c (TLRef i) a)))))))))) (f1: (\forall (c: C).(\forall (d: C).(\forall
-(u: T).(\forall (i: nat).((getl i c (CHead d (Bind Abst) u)) \to (\forall (a:
-A).((arity g d u (asucc g a)) \to ((P d u (asucc g a)) \to (P c (TLRef i)
-a)))))))))) (f2: (\forall (b: B).((not (eq B b Abst)) \to (\forall (c:
-C).(\forall (u: T).(\forall (a1: A).((arity g c u a1) \to ((P c u a1) \to
-(\forall (t: T).(\forall (a2: A).((arity g (CHead c (Bind b) u) t a2) \to ((P
-(CHead c (Bind b) u) t a2) \to (P c (THead (Bind b) u t) a2))))))))))))) (f3:
-(\forall (c: C).(\forall (u: T).(\forall (a1: A).((arity g c u (asucc g a1))
-\to ((P c u (asucc g a1)) \to (\forall (t: T).(\forall (a2: A).((arity g
-(CHead c (Bind Abst) u) t a2) \to ((P (CHead c (Bind Abst) u) t a2) \to (P c
-(THead (Bind Abst) u t) (AHead a1 a2)))))))))))) (f4: (\forall (c:
-C).(\forall (u: T).(\forall (a1: A).((arity g c u a1) \to ((P c u a1) \to
-(\forall (t: T).(\forall (a2: A).((arity g c t (AHead a1 a2)) \to ((P c t
-(AHead a1 a2)) \to (P c (THead (Flat Appl) u t) a2))))))))))) (f5: (\forall
-(c: C).(\forall (u: T).(\forall (a: A).((arity g c u (asucc g a)) \to ((P c u
-(asucc g a)) \to (\forall (t: T).((arity g c t a) \to ((P c t a) \to (P c
-(THead (Flat Cast) u t) a)))))))))) (f6: (\forall (c: C).(\forall (t:
-T).(\forall (a1: A).((arity g c t a1) \to ((P c t a1) \to (\forall (a2:
-A).((leq g a1 a2) \to (P c t a2))))))))) (c: C) (t: T) (a: A) (a0: arity g c
-t a) on a0: P c t a \def match a0 with [(arity_sort c0 n) \Rightarrow (f c0
-n) | (arity_abbr c0 d u i g0 a1 a2) \Rightarrow (f0 c0 d u i g0 a1 a2
-((arity_ind g P f f0 f1 f2 f3 f4 f5 f6) d u a1 a2)) | (arity_abst c0 d u i g0
-a1 a2) \Rightarrow (f1 c0 d u i g0 a1 a2 ((arity_ind g P f f0 f1 f2 f3 f4 f5
-f6) d u (asucc g a1) a2)) | (arity_bind b n c0 u a1 a2 t0 a3 a4) \Rightarrow
-(f2 b n c0 u a1 a2 ((arity_ind g P f f0 f1 f2 f3 f4 f5 f6) c0 u a1 a2) t0 a3
-a4 ((arity_ind g P f f0 f1 f2 f3 f4 f5 f6) (CHead c0 (Bind b) u) t0 a3 a4)) |
-(arity_head c0 u a1 a2 t0 a3 a4) \Rightarrow (f3 c0 u a1 a2 ((arity_ind g P f
-f0 f1 f2 f3 f4 f5 f6) c0 u (asucc g a1) a2) t0 a3 a4 ((arity_ind g P f f0 f1
-f2 f3 f4 f5 f6) (CHead c0 (Bind Abst) u) t0 a3 a4)) | (arity_appl c0 u a1 a2
-t0 a3 a4) \Rightarrow (f4 c0 u a1 a2 ((arity_ind g P f f0 f1 f2 f3 f4 f5 f6)
-c0 u a1 a2) t0 a3 a4 ((arity_ind g P f f0 f1 f2 f3 f4 f5 f6) c0 t0 (AHead a1
-a3) a4)) | (arity_cast c0 u a1 a2 t0 a3) \Rightarrow (f5 c0 u a1 a2
-((arity_ind g P f f0 f1 f2 f3 f4 f5 f6) c0 u (asucc g a1) a2) t0 a3
-((arity_ind g P f f0 f1 f2 f3 f4 f5 f6) c0 t0 a1 a3)) | (arity_repl c0 t0 a1
-a2 a3 l) \Rightarrow (f6 c0 t0 a1 a2 ((arity_ind g P f f0 f1 f2 f3 f4 f5 f6)
-c0 t0 a1 a2) a3 l)].
-
-lemma arity_gen_sort:
- \forall (g: G).(\forall (c: C).(\forall (n: nat).(\forall (a: A).((arity g c
-(TSort n) a) \to (leq g a (ASort O n))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (n: nat).(\lambda (a: A).(\lambda
-(H: (arity g c (TSort n) a)).(insert_eq T (TSort n) (\lambda (t: T).(arity g
-c t a)) (\lambda (_: T).(leq g a (ASort O n))) (\lambda (y: T).(\lambda (H0:
-(arity g c y a)).(arity_ind g (\lambda (_: C).(\lambda (t: T).(\lambda (a0:
-A).((eq T t (TSort n)) \to (leq g a0 (ASort O n)))))) (\lambda (_:
-C).(\lambda (n0: nat).(\lambda (H1: (eq T (TSort n0) (TSort n))).(let H2 \def
-(f_equal T nat (\lambda (e: T).(match e with [(TSort n1) \Rightarrow n1 |
-(TLRef _) \Rightarrow n0 | (THead _ _ _) \Rightarrow n0])) (TSort n0) (TSort
-n) H1) in (eq_ind_r nat n (\lambda (n1: nat).(leq g (ASort O n1) (ASort O
-n))) (leq_refl g (ASort O n)) n0 H2))))) (\lambda (c0: C).(\lambda (d:
-C).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind
-Abbr) u))).(\lambda (a0: A).(\lambda (_: (arity g d u a0)).(\lambda (_: (((eq
-T u (TSort n)) \to (leq g a0 (ASort O n))))).(\lambda (H4: (eq T (TLRef i)
-(TSort n))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee with
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _)
-\Rightarrow False])) I (TSort n) H4) in (False_ind (leq g a0 (ASort O n))
-H5))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i:
-nat).(\lambda (_: (getl i c0 (CHead d (Bind Abst) u))).(\lambda (a0:
-A).(\lambda (_: (arity g d u (asucc g a0))).(\lambda (_: (((eq T u (TSort n))
-\to (leq g (asucc g a0) (ASort O n))))).(\lambda (H4: (eq T (TLRef i) (TSort
-n))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee with [(TSort
-_) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
-False])) I (TSort n) H4) in (False_ind (leq g a0 (ASort O n)) H5)))))))))))
-(\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 (_: (((eq T
-u (TSort n)) \to (leq g a1 (ASort O n))))).(\lambda (t: T).(\lambda (a2:
-A).(\lambda (_: (arity g (CHead c0 (Bind b) u) t a2)).(\lambda (_: (((eq T t
-(TSort n)) \to (leq g a2 (ASort O n))))).(\lambda (H6: (eq T (THead (Bind b)
-u t) (TSort n))).(let H7 \def (eq_ind T (THead (Bind b) u t) (\lambda (ee:
-T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False
-| (THead _ _ _) \Rightarrow True])) I (TSort n) H6) in (False_ind (leq g a2
-(ASort O n)) H7)))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1:
-A).(\lambda (_: (arity g c0 u (asucc g a1))).(\lambda (_: (((eq T u (TSort
-n)) \to (leq g (asucc g a1) (ASort O n))))).(\lambda (t: T).(\lambda (a2:
-A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t a2)).(\lambda (_: (((eq T
-t (TSort n)) \to (leq g a2 (ASort O n))))).(\lambda (H5: (eq T (THead (Bind
-Abst) u t) (TSort n))).(let H6 \def (eq_ind T (THead (Bind Abst) u t)
-(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H5) in
-(False_ind (leq g (AHead a1 a2) (ASort O n)) H6)))))))))))) (\lambda (c0:
-C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u a1)).(\lambda
-(_: (((eq T u (TSort n)) \to (leq g a1 (ASort O n))))).(\lambda (t:
-T).(\lambda (a2: A).(\lambda (_: (arity g c0 t (AHead a1 a2))).(\lambda (_:
-(((eq T t (TSort n)) \to (leq g (AHead a1 a2) (ASort O n))))).(\lambda (H5:
-(eq T (THead (Flat Appl) u t) (TSort n))).(let H6 \def (eq_ind T (THead (Flat
-Appl) u t) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False |
-(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n)
-H5) in (False_ind (leq g a2 (ASort O n)) H6)))))))))))) (\lambda (c0:
-C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_: (arity g c0 u (asucc g
-a0))).(\lambda (_: (((eq T u (TSort n)) \to (leq g (asucc g a0) (ASort O
-n))))).(\lambda (t: T).(\lambda (_: (arity g c0 t a0)).(\lambda (_: (((eq T t
-(TSort n)) \to (leq g a0 (ASort O n))))).(\lambda (H5: (eq T (THead (Flat
-Cast) u t) (TSort n))).(let H6 \def (eq_ind T (THead (Flat Cast) u t)
-(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H5) in
-(False_ind (leq g a0 (ASort O n)) H6))))))))))) (\lambda (c0: C).(\lambda (t:
-T).(\lambda (a1: A).(\lambda (H1: (arity g c0 t a1)).(\lambda (H2: (((eq T t
-(TSort n)) \to (leq g a1 (ASort O n))))).(\lambda (a2: A).(\lambda (H3: (leq
-g a1 a2)).(\lambda (H4: (eq T t (TSort n))).(let H5 \def (f_equal T T
-(\lambda (e: T).e) t (TSort n) H4) in (let H6 \def (eq_ind T t (\lambda (t0:
-T).((eq T t0 (TSort n)) \to (leq g a1 (ASort O n)))) H2 (TSort n) H5) in (let
-H7 \def (eq_ind T t (\lambda (t0: T).(arity g c0 t0 a1)) H1 (TSort n) H5) in
-(leq_trans g a2 a1 (leq_sym g a1 a2 H3) (ASort O n) (H6 (refl_equal T (TSort
-n))))))))))))))) c y a H0))) H))))).
-
-lemma arity_gen_lref:
- \forall (g: G).(\forall (c: C).(\forall (i: nat).(\forall (a: A).((arity g c
-(TLRef i) a) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c
-(CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u a))))
-(ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c (CHead d (Bind Abst)
-u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (i: nat).(\lambda (a: A).(\lambda
-(H: (arity g c (TLRef i) a)).(insert_eq T (TLRef i) (\lambda (t: T).(arity g
-c t a)) (\lambda (_: T).(or (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl
-i c (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u
-a)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c (CHead d (Bind
-Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a)))))))
-(\lambda (y: T).(\lambda (H0: (arity g c y a)).(arity_ind g (\lambda (c0:
-C).(\lambda (t: T).(\lambda (a0: A).((eq T t (TLRef i)) \to (or (ex2_2 C T
-(\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u))))
-(\lambda (d: C).(\lambda (u: T).(arity g d u a0)))) (ex2_2 C T (\lambda (d:
-C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d:
-C).(\lambda (u: T).(arity g d u (asucc g a0)))))))))) (\lambda (c0:
-C).(\lambda (n: nat).(\lambda (H1: (eq T (TSort n) (TLRef i))).(let H2 \def
-(eq_ind T (TSort n) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow
-True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I
-(TLRef i) H1) in (False_ind (or (ex2_2 C T (\lambda (d: C).(\lambda (u:
-T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u:
-T).(arity g d u (ASort O n))))) (ex2_2 C T (\lambda (d: C).(\lambda (u:
-T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u:
-T).(arity g d u (asucc g (ASort O n))))))) H2))))) (\lambda (c0: C).(\lambda
-(d: C).(\lambda (u: T).(\lambda (i0: nat).(\lambda (H1: (getl i0 c0 (CHead d
-(Bind Abbr) u))).(\lambda (a0: A).(\lambda (H2: (arity g d u a0)).(\lambda
-(_: (((eq T u (TLRef i)) \to (or (ex2_2 C T (\lambda (d0: C).(\lambda (u0:
-T).(getl i d (CHead d0 (Bind Abbr) u0)))) (\lambda (d0: C).(\lambda (u0:
-T).(arity g d0 u0 a0)))) (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i
-d (CHead d0 (Bind Abst) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0
-u0 (asucc g a0))))))))).(\lambda (H4: (eq T (TLRef i0) (TLRef i))).(let H5
-\def (f_equal T nat (\lambda (e: T).(match e with [(TSort _) \Rightarrow i0 |
-(TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i0])) (TLRef i0) (TLRef
-i) H4) in (let H6 \def (eq_ind nat i0 (\lambda (n: nat).(getl n c0 (CHead d
-(Bind Abbr) u))) H1 i H5) in (or_introl (ex2_2 C T (\lambda (d0: C).(\lambda
-(u0: T).(getl i c0 (CHead d0 (Bind Abbr) u0)))) (\lambda (d0: C).(\lambda
-(u0: T).(arity g d0 u0 a0)))) (ex2_2 C T (\lambda (d0: C).(\lambda (u0:
-T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda (d0: C).(\lambda (u0:
-T).(arity g d0 u0 (asucc g a0))))) (ex2_2_intro C T (\lambda (d0: C).(\lambda
-(u0: T).(getl i c0 (CHead d0 (Bind Abbr) u0)))) (\lambda (d0: C).(\lambda
-(u0: T).(arity g d0 u0 a0))) d u H6 H2))))))))))))) (\lambda (c0: C).(\lambda
-(d: C).(\lambda (u: T).(\lambda (i0: nat).(\lambda (H1: (getl i0 c0 (CHead d
-(Bind Abst) u))).(\lambda (a0: A).(\lambda (H2: (arity g d u (asucc g
-a0))).(\lambda (_: (((eq T u (TLRef i)) \to (or (ex2_2 C T (\lambda (d0:
-C).(\lambda (u0: T).(getl i d (CHead d0 (Bind Abbr) u0)))) (\lambda (d0:
-C).(\lambda (u0: T).(arity g d0 u0 (asucc g a0))))) (ex2_2 C T (\lambda (d0:
-C).(\lambda (u0: T).(getl i d (CHead d0 (Bind Abst) u0)))) (\lambda (d0:
-C).(\lambda (u0: T).(arity g d0 u0 (asucc g (asucc g a0)))))))))).(\lambda
-(H4: (eq T (TLRef i0) (TLRef i))).(let H5 \def (f_equal T nat (\lambda (e:
-T).(match e with [(TSort _) \Rightarrow i0 | (TLRef n) \Rightarrow n | (THead
-_ _ _) \Rightarrow i0])) (TLRef i0) (TLRef i) H4) in (let H6 \def (eq_ind nat
-i0 (\lambda (n: nat).(getl n c0 (CHead d (Bind Abst) u))) H1 i H5) in
-(or_intror (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0
-(Bind Abbr) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a0))))
-(ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst)
-u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a0)))))
-(ex2_2_intro C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind
-Abst) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a0))))
-d u H6 H2))))))))))))) (\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 (_: (((eq T u (TLRef i)) \to (or (ex2_2 C T (\lambda (d:
-C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d:
-C).(\lambda (u0: T).(arity g d u0 a1)))) (ex2_2 C T (\lambda (d: C).(\lambda
-(u0: T).(getl i c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0:
-T).(arity g d u0 (asucc g a1))))))))).(\lambda (t: T).(\lambda (a2:
-A).(\lambda (_: (arity g (CHead c0 (Bind b) u) t a2)).(\lambda (_: (((eq T t
-(TLRef i)) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i (CHead
-c0 (Bind b) u) (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0:
-T).(arity g d u0 a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i
-(CHead c0 (Bind b) u) (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda
-(u0: T).(arity g d u0 (asucc g a2))))))))).(\lambda (H6: (eq T (THead (Bind
-b) u t) (TLRef i))).(let H7 \def (eq_ind T (THead (Bind b) u t) (\lambda (ee:
-T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False
-| (THead _ _ _) \Rightarrow True])) I (TLRef i) H6) in (False_ind (or (ex2_2
-C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr) u0))))
-(\lambda (d: C).(\lambda (u0: T).(arity g d u0 a2)))) (ex2_2 C T (\lambda (d:
-C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abst) u0)))) (\lambda (d:
-C).(\lambda (u0: T).(arity g d u0 (asucc g a2)))))) H7)))))))))))))) (\lambda
-(c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u (asucc g
-a1))).(\lambda (_: (((eq T u (TLRef i)) \to (or (ex2_2 C T (\lambda (d:
-C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d:
-C).(\lambda (u0: T).(arity g d u0 (asucc g a1))))) (ex2_2 C T (\lambda (d:
-C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abst) u0)))) (\lambda (d:
-C).(\lambda (u0: T).(arity g d u0 (asucc g (asucc g a1)))))))))).(\lambda (t:
-T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t
-a2)).(\lambda (_: (((eq T t (TLRef i)) \to (or (ex2_2 C T (\lambda (d:
-C).(\lambda (u0: T).(getl i (CHead c0 (Bind Abst) u) (CHead d (Bind Abbr)
-u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 a2)))) (ex2_2 C T
-(\lambda (d: C).(\lambda (u0: T).(getl i (CHead c0 (Bind Abst) u) (CHead d
-(Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g
-a2))))))))).(\lambda (H5: (eq T (THead (Bind Abst) u t) (TLRef i))).(let H6
-\def (eq_ind T (THead (Bind Abst) u t) (\lambda (ee: T).(match ee with
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
-\Rightarrow True])) I (TLRef i) H5) in (False_ind (or (ex2_2 C T (\lambda (d:
-C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d:
-C).(\lambda (u0: T).(arity g d u0 (AHead a1 a2))))) (ex2_2 C T (\lambda (d:
-C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abst) u0)))) (\lambda (d:
-C).(\lambda (u0: T).(arity g d u0 (asucc g (AHead a1 a2))))))) H6))))))))))))
-(\lambda (c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u
-a1)).(\lambda (_: (((eq T u (TLRef i)) \to (or (ex2_2 C T (\lambda (d:
-C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d:
-C).(\lambda (u0: T).(arity g d u0 a1)))) (ex2_2 C T (\lambda (d: C).(\lambda
-(u0: T).(getl i c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0:
-T).(arity g d u0 (asucc g a1))))))))).(\lambda (t: T).(\lambda (a2:
-A).(\lambda (_: (arity g c0 t (AHead a1 a2))).(\lambda (_: (((eq T t (TLRef
-i)) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d
-(Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 (AHead a1
-a2))))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind
-Abst) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g (AHead
-a1 a2)))))))))).(\lambda (H5: (eq T (THead (Flat Appl) u t) (TLRef i))).(let
-H6 \def (eq_ind T (THead (Flat Appl) u t) (\lambda (ee: T).(match ee with
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
-\Rightarrow True])) I (TLRef i) H5) in (False_ind (or (ex2_2 C T (\lambda (d:
-C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d:
-C).(\lambda (u0: T).(arity g d u0 a2)))) (ex2_2 C T (\lambda (d: C).(\lambda
-(u0: T).(getl i c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0:
-T).(arity g d u0 (asucc g a2)))))) H6)))))))))))) (\lambda (c0: C).(\lambda
-(u: T).(\lambda (a0: A).(\lambda (_: (arity g c0 u (asucc g a0))).(\lambda
-(_: (((eq T u (TLRef i)) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u0:
-T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0:
-T).(arity g d u0 (asucc g a0))))) (ex2_2 C T (\lambda (d: C).(\lambda (u0:
-T).(getl i c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0:
-T).(arity g d u0 (asucc g (asucc g a0)))))))))).(\lambda (t: T).(\lambda (_:
-(arity g c0 t a0)).(\lambda (_: (((eq T t (TLRef i)) \to (or (ex2_2 C T
-(\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr) u0))))
-(\lambda (d: C).(\lambda (u0: T).(arity g d u0 a0)))) (ex2_2 C T (\lambda (d:
-C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abst) u0)))) (\lambda (d:
-C).(\lambda (u0: T).(arity g d u0 (asucc g a0))))))))).(\lambda (H5: (eq T
-(THead (Flat Cast) u t) (TLRef i))).(let H6 \def (eq_ind T (THead (Flat Cast)
-u t) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef i) H5) in
-(False_ind (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead
-d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 a0))))
-(ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abst)
-u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g a0))))))
-H6))))))))))) (\lambda (c0: C).(\lambda (t: T).(\lambda (a1: A).(\lambda (H1:
-(arity g c0 t a1)).(\lambda (H2: (((eq T t (TLRef i)) \to (or (ex2_2 C T
-(\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u))))
-(\lambda (d: C).(\lambda (u: T).(arity g d u a1)))) (ex2_2 C T (\lambda (d:
-C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d:
-C).(\lambda (u: T).(arity g d u (asucc g a1))))))))).(\lambda (a2:
-A).(\lambda (H3: (leq g a1 a2)).(\lambda (H4: (eq T t (TLRef i))).(let H5
-\def (f_equal T T (\lambda (e: T).e) t (TLRef i) H4) in (let H6 \def (eq_ind
-T t (\lambda (t0: T).((eq T t0 (TLRef i)) \to (or (ex2_2 C T (\lambda (d:
-C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d:
-C).(\lambda (u: T).(arity g d u a1)))) (ex2_2 C T (\lambda (d: C).(\lambda
-(u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u:
-T).(arity g d u (asucc g a1)))))))) H2 (TLRef i) H5) in (let H7 \def (eq_ind
-T t (\lambda (t0: T).(arity g c0 t0 a1)) H1 (TLRef i) H5) in (let H8 \def (H6
-(refl_equal T (TLRef i))) in (or_ind (ex2_2 C T (\lambda (d: C).(\lambda (u:
-T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u:
-T).(arity g d u a1)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0
-(CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u
-(asucc g a1))))) (or (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0
-(CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u
-a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind
-Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a2))))))
-(\lambda (H9: (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d
-(Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u
-a1))))).(ex2_2_ind C T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d
-(Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u a1))) (or
-(ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr)
-u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u a2)))) (ex2_2 C T (\lambda
-(d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d:
-C).(\lambda (u: T).(arity g d u (asucc g a2)))))) (\lambda (x0: C).(\lambda
-(x1: T).(\lambda (H10: (getl i c0 (CHead x0 (Bind Abbr) x1))).(\lambda (H11:
-(arity g x0 x1 a1)).(or_introl (ex2_2 C T (\lambda (d: C).(\lambda (u:
-T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u:
-T).(arity g d u a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0
-(CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u
-(asucc g a2))))) (ex2_2_intro C T (\lambda (d: C).(\lambda (u: T).(getl i c0
-(CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u a2)))
-x0 x1 H10 (arity_repl g x0 x1 a1 H11 a2 H3))))))) H9)) (\lambda (H9: (ex2_2 C
-T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u))))
-(\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a1)))))).(ex2_2_ind C T
-(\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u))))
-(\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a1)))) (or (ex2_2 C T
-(\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u))))
-(\lambda (d: C).(\lambda (u: T).(arity g d u a2)))) (ex2_2 C T (\lambda (d:
-C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d:
-C).(\lambda (u: T).(arity g d u (asucc g a2)))))) (\lambda (x0: C).(\lambda
-(x1: T).(\lambda (H10: (getl i c0 (CHead x0 (Bind Abst) x1))).(\lambda (H11:
-(arity g x0 x1 (asucc g a1))).(or_intror (ex2_2 C T (\lambda (d: C).(\lambda
-(u: T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u:
-T).(arity g d u a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0
-(CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u
-(asucc g a2))))) (ex2_2_intro C T (\lambda (d: C).(\lambda (u: T).(getl i c0
-(CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u
-(asucc g a2)))) x0 x1 H10 (arity_repl g x0 x1 (asucc g a1) H11 (asucc g a2)
-(asucc_repl g a1 a2 H3)))))))) H9)) H8))))))))))))) c y a H0))) H))))).
-
-lemma arity_gen_bind:
- \forall (b: B).((not (eq B b Abst)) \to (\forall (g: G).(\forall (c:
-C).(\forall (u: T).(\forall (t: T).(\forall (a2: A).((arity g c (THead (Bind
-b) u t) a2) \to (ex2 A (\lambda (a1: A).(arity g c u a1)) (\lambda (_:
-A).(arity g (CHead c (Bind b) u) t a2))))))))))
-\def
- \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (g: G).(\lambda
-(c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (a2: A).(\lambda (H0: (arity
-g c (THead (Bind b) u t) a2)).(insert_eq T (THead (Bind b) u t) (\lambda (t0:
-T).(arity g c t0 a2)) (\lambda (_: T).(ex2 A (\lambda (a1: A).(arity g c u
-a1)) (\lambda (_: A).(arity g (CHead c (Bind b) u) t a2)))) (\lambda (y:
-T).(\lambda (H1: (arity g c y a2)).(arity_ind g (\lambda (c0: C).(\lambda
-(t0: T).(\lambda (a: A).((eq T t0 (THead (Bind b) u t)) \to (ex2 A (\lambda
-(a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t
-a))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (H2: (eq T (TSort n)
-(THead (Bind b) u t))).(let H3 \def (eq_ind T (TSort n) (\lambda (ee:
-T).(match ee with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False |
-(THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t) H2) in (False_ind
-(ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0
-(Bind b) u) t (ASort O n)))) H3))))) (\lambda (c0: C).(\lambda (d:
-C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind
-Abbr) u0))).(\lambda (a: A).(\lambda (_: (arity g d u0 a)).(\lambda (_: (((eq
-T u0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a1: A).(arity g d u a1))
-(\lambda (_: A).(arity g (CHead d (Bind b) u) t a)))))).(\lambda (H5: (eq T
-(TLRef i) (THead (Bind b) u t))).(let H6 \def (eq_ind T (TLRef i) (\lambda
-(ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow
-True | (THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t) H5) in
-(False_ind (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity
-g (CHead c0 (Bind b) u) t a))) H6))))))))))) (\lambda (c0: C).(\lambda (d:
-C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind
-Abst) u0))).(\lambda (a: A).(\lambda (_: (arity g d u0 (asucc g a))).(\lambda
-(_: (((eq T u0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a1: A).(arity g d u
-a1)) (\lambda (_: A).(arity g (CHead d (Bind b) u) t (asucc g
-a))))))).(\lambda (H5: (eq T (TLRef i) (THead (Bind b) u t))).(let H6 \def
-(eq_ind T (TLRef i) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow
-False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I
-(THead (Bind b) u t) H5) in (False_ind (ex2 A (\lambda (a1: A).(arity g c0 u
-a1)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a))) H6)))))))))))
-(\lambda (b0: B).(\lambda (H2: (not (eq B b0 Abst))).(\lambda (c0:
-C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (H3: (arity g c0 u0
-a1)).(\lambda (H4: (((eq T u0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3:
-A).(arity g c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t
-a1)))))).(\lambda (t0: T).(\lambda (a0: A).(\lambda (H5: (arity g (CHead c0
-(Bind b0) u0) t0 a0)).(\lambda (H6: (((eq T t0 (THead (Bind b) u t)) \to (ex2
-A (\lambda (a3: A).(arity g (CHead c0 (Bind b0) u0) u a3)) (\lambda (_:
-A).(arity g (CHead (CHead c0 (Bind b0) u0) (Bind b) u) t a0)))))).(\lambda
-(H7: (eq T (THead (Bind b0) u0 t0) (THead (Bind b) u t))).(let H8 \def
-(f_equal T B (\lambda (e: T).(match e with [(TSort _) \Rightarrow b0 | (TLRef
-_) \Rightarrow b0 | (THead k _ _) \Rightarrow (match k with [(Bind b1)
-\Rightarrow b1 | (Flat _) \Rightarrow b0])])) (THead (Bind b0) u0 t0) (THead
-(Bind b) u t) H7) in ((let H9 \def (f_equal T T (\lambda (e: T).(match e with
-[(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t1 _)
-\Rightarrow t1])) (THead (Bind b0) u0 t0) (THead (Bind b) u t) H7) in ((let
-H10 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t0
-| (TLRef _) \Rightarrow t0 | (THead _ _ t1) \Rightarrow t1])) (THead (Bind
-b0) u0 t0) (THead (Bind b) u t) H7) in (\lambda (H11: (eq T u0 u)).(\lambda
-(H12: (eq B b0 b)).(let H13 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1
-(THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g (CHead c0 (Bind
-b0) u0) u a3)) (\lambda (_: A).(arity g (CHead (CHead c0 (Bind b0) u0) (Bind
-b) u) t a0))))) H6 t H10) in (let H14 \def (eq_ind T t0 (\lambda (t1:
-T).(arity g (CHead c0 (Bind b0) u0) t1 a0)) H5 t H10) in (let H15 \def
-(eq_ind T u0 (\lambda (t1: T).((eq T t (THead (Bind b) u t)) \to (ex2 A
-(\lambda (a3: A).(arity g (CHead c0 (Bind b0) t1) u a3)) (\lambda (_:
-A).(arity g (CHead (CHead c0 (Bind b0) t1) (Bind b) u) t a0))))) H13 u H11)
-in (let H16 \def (eq_ind T u0 (\lambda (t1: T).(arity g (CHead c0 (Bind b0)
-t1) t a0)) H14 u H11) in (let H17 \def (eq_ind T u0 (\lambda (t1: T).((eq T
-t1 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3))
-(\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a1))))) H4 u H11) in (let
-H18 \def (eq_ind T u0 (\lambda (t1: T).(arity g c0 t1 a1)) H3 u H11) in (let
-H19 \def (eq_ind B b0 (\lambda (b1: B).((eq T t (THead (Bind b) u t)) \to
-(ex2 A (\lambda (a3: A).(arity g (CHead c0 (Bind b1) u) u a3)) (\lambda (_:
-A).(arity g (CHead (CHead c0 (Bind b1) u) (Bind b) u) t a0))))) H15 b H12) in
-(let H20 \def (eq_ind B b0 (\lambda (b1: B).(arity g (CHead c0 (Bind b1) u) t
-a0)) H16 b H12) in (let H21 \def (eq_ind B b0 (\lambda (b1: B).(not (eq B b1
-Abst))) H2 b H12) in (ex_intro2 A (\lambda (a3: A).(arity g c0 u a3))
-(\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a0)) a1 H18 H20)))))))))))))
-H9)) H8)))))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a1:
-A).(\lambda (H2: (arity g c0 u0 (asucc g a1))).(\lambda (H3: (((eq T u0
-(THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda
-(_: A).(arity g (CHead c0 (Bind b) u) t (asucc g a1))))))).(\lambda (t0:
-T).(\lambda (a0: A).(\lambda (H4: (arity g (CHead c0 (Bind Abst) u0) t0
-a0)).(\lambda (H5: (((eq T t0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3:
-A).(arity g (CHead c0 (Bind Abst) u0) u a3)) (\lambda (_: A).(arity g (CHead
-(CHead c0 (Bind Abst) u0) (Bind b) u) t a0)))))).(\lambda (H6: (eq T (THead
-(Bind Abst) u0 t0) (THead (Bind b) u t))).(let H7 \def (f_equal T B (\lambda
-(e: T).(match e with [(TSort _) \Rightarrow Abst | (TLRef _) \Rightarrow Abst
-| (THead k _ _) \Rightarrow (match k with [(Bind b0) \Rightarrow b0 | (Flat
-_) \Rightarrow Abst])])) (THead (Bind Abst) u0 t0) (THead (Bind b) u t) H6)
-in ((let H8 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _)
-\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t1 _) \Rightarrow t1]))
-(THead (Bind Abst) u0 t0) (THead (Bind b) u t) H6) in ((let H9 \def (f_equal
-T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t0 | (TLRef _)
-\Rightarrow t0 | (THead _ _ t1) \Rightarrow t1])) (THead (Bind Abst) u0 t0)
-(THead (Bind b) u t) H6) in (\lambda (H10: (eq T u0 u)).(\lambda (H11: (eq B
-Abst b)).(let H12 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1 (THead (Bind
-b) u t)) \to (ex2 A (\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u0) u
-a3)) (\lambda (_: A).(arity g (CHead (CHead c0 (Bind Abst) u0) (Bind b) u) t
-a0))))) H5 t H9) in (let H13 \def (eq_ind T t0 (\lambda (t1: T).(arity g
-(CHead c0 (Bind Abst) u0) t1 a0)) H4 t H9) in (let H14 \def (eq_ind T u0
-(\lambda (t1: T).((eq T t (THead (Bind b) u t)) \to (ex2 A (\lambda (a3:
-A).(arity g (CHead c0 (Bind Abst) t1) u a3)) (\lambda (_: A).(arity g (CHead
-(CHead c0 (Bind Abst) t1) (Bind b) u) t a0))))) H12 u H10) in (let H15 \def
-(eq_ind T u0 (\lambda (t1: T).(arity g (CHead c0 (Bind Abst) t1) t a0)) H13 u
-H10) in (let H16 \def (eq_ind T u0 (\lambda (t1: T).((eq T t1 (THead (Bind b)
-u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (_: A).(arity g
-(CHead c0 (Bind b) u) t (asucc g a1)))))) H3 u H10) in (let H17 \def (eq_ind
-T u0 (\lambda (t1: T).(arity g c0 t1 (asucc g a1))) H2 u H10) in (let H18
-\def (eq_ind_r B b (\lambda (b0: B).((eq T t (THead (Bind b0) u t)) \to (ex2
-A (\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u) u a3)) (\lambda (_:
-A).(arity g (CHead (CHead c0 (Bind Abst) u) (Bind b0) u) t a0))))) H14 Abst
-H11) in (let H19 \def (eq_ind_r B b (\lambda (b0: B).((eq T u (THead (Bind
-b0) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (_:
-A).(arity g (CHead c0 (Bind b0) u) t (asucc g a1)))))) H16 Abst H11) in (let
-H20 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 Abst))) H Abst H11) in
-(eq_ind B Abst (\lambda (b0: B).(ex2 A (\lambda (a3: A).(arity g c0 u a3))
-(\lambda (_: A).(arity g (CHead c0 (Bind b0) u) t (AHead a1 a0))))) (let H21
-\def (match (H20 (refl_equal B Abst)) in False with []) in H21) b
-H11))))))))))))) H8)) H7)))))))))))) (\lambda (c0: C).(\lambda (u0:
-T).(\lambda (a1: A).(\lambda (_: (arity g c0 u0 a1)).(\lambda (_: (((eq T u0
-(THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda
-(_: A).(arity g (CHead c0 (Bind b) u) t a1)))))).(\lambda (t0: T).(\lambda
-(a0: A).(\lambda (_: (arity g c0 t0 (AHead a1 a0))).(\lambda (_: (((eq T t0
-(THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda
-(_: A).(arity g (CHead c0 (Bind b) u) t (AHead a1 a0))))))).(\lambda (H6: (eq
-T (THead (Flat Appl) u0 t0) (THead (Bind b) u t))).(let H7 \def (eq_ind T
-(THead (Flat Appl) u0 t0) (\lambda (ee: T).(match ee with [(TSort _)
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
-(match k with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I
-(THead (Bind b) u t) H6) in (False_ind (ex2 A (\lambda (a3: A).(arity g c0 u
-a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a0))) H7))))))))))))
-(\lambda (c0: C).(\lambda (u0: T).(\lambda (a: A).(\lambda (_: (arity g c0 u0
-(asucc g a))).(\lambda (_: (((eq T u0 (THead (Bind b) u t)) \to (ex2 A
-(\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0 (Bind
-b) u) t (asucc g a))))))).(\lambda (t0: T).(\lambda (_: (arity g c0 t0
-a)).(\lambda (_: (((eq T t0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a1:
-A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t
-a)))))).(\lambda (H6: (eq T (THead (Flat Cast) u0 t0) (THead (Bind b) u
-t))).(let H7 \def (eq_ind T (THead (Flat Cast) u0 t0) (\lambda (ee: T).(match
-ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k
-_ _) \Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat _)
-\Rightarrow True])])) I (THead (Bind b) u t) H6) in (False_ind (ex2 A
-(\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0 (Bind
-b) u) t a))) H7))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (a1:
-A).(\lambda (H2: (arity g c0 t0 a1)).(\lambda (H3: (((eq T t0 (THead (Bind b)
-u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (_: A).(arity g
-(CHead c0 (Bind b) u) t a1)))))).(\lambda (a0: A).(\lambda (H4: (leq g a1
-a0)).(\lambda (H5: (eq T t0 (THead (Bind b) u t))).(let H6 \def (f_equal T T
-(\lambda (e: T).e) t0 (THead (Bind b) u t) H5) in (let H7 \def (eq_ind T t0
-(\lambda (t1: T).((eq T t1 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3:
-A).(arity g c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t
-a1))))) H3 (THead (Bind b) u t) H6) in (let H8 \def (eq_ind T t0 (\lambda
-(t1: T).(arity g c0 t1 a1)) H2 (THead (Bind b) u t) H6) in (let H9 \def (H7
-(refl_equal T (THead (Bind b) u t))) in (ex2_ind A (\lambda (a3: A).(arity g
-c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a1)) (ex2 A
-(\lambda (a3: A).(arity g c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind
-b) u) t a0))) (\lambda (x: A).(\lambda (H10: (arity g c0 u x)).(\lambda (H11:
-(arity g (CHead c0 (Bind b) u) t a1)).(ex_intro2 A (\lambda (a3: A).(arity g
-c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a0)) x H10
-(arity_repl g (CHead c0 (Bind b) u) t a1 H11 a0 H4))))) H9))))))))))))) c y
-a2 H1))) H0)))))))).
-
-lemma arity_gen_abst:
- \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (a:
-A).((arity g c (THead (Bind Abst) u t) a) \to (ex3_2 A A (\lambda (a1:
-A).(\lambda (a2: A).(eq A a (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_:
-A).(arity g c u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g
-(CHead c (Bind Abst) u) t a2)))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (a:
-A).(\lambda (H: (arity g c (THead (Bind Abst) u t) a)).(insert_eq T (THead
-(Bind Abst) u t) (\lambda (t0: T).(arity g c t0 a)) (\lambda (_: T).(ex3_2 A
-A (\lambda (a1: A).(\lambda (a2: A).(eq A a (AHead a1 a2)))) (\lambda (a1:
-A).(\lambda (_: A).(arity g c u (asucc g a1)))) (\lambda (_: A).(\lambda (a2:
-A).(arity g (CHead c (Bind Abst) u) t a2))))) (\lambda (y: T).(\lambda (H0:
-(arity g c y a)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (a0:
-A).((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a1:
-A).(\lambda (a2: A).(eq A a0 (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_:
-A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g
-(CHead c0 (Bind Abst) u) t a2)))))))) (\lambda (c0: C).(\lambda (n:
-nat).(\lambda (H1: (eq T (TSort n) (THead (Bind Abst) u t))).(let H2 \def
-(eq_ind T (TSort n) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow
-True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I
-(THead (Bind Abst) u t) H1) in (False_ind (ex3_2 A A (\lambda (a1:
-A).(\lambda (a2: A).(eq A (ASort O n) (AHead a1 a2)))) (\lambda (a1:
-A).(\lambda (_: A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda
-(a2: A).(arity g (CHead c0 (Bind Abst) u) t a2)))) H2))))) (\lambda (c0:
-C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0
-(CHead d (Bind Abbr) u0))).(\lambda (a0: A).(\lambda (_: (arity g d u0
-a0)).(\lambda (_: (((eq T u0 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda
-(a1: A).(\lambda (a2: A).(eq A a0 (AHead a1 a2)))) (\lambda (a1: A).(\lambda
-(_: A).(arity g d u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g
-(CHead d (Bind Abst) u) t a2))))))).(\lambda (H4: (eq T (TLRef i) (THead
-(Bind Abst) u t))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match
-ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _
-_ _) \Rightarrow False])) I (THead (Bind Abst) u t) H4) in (False_ind (ex3_2
-A A (\lambda (a1: A).(\lambda (a2: A).(eq A a0 (AHead a1 a2)))) (\lambda (a1:
-A).(\lambda (_: A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda
-(a2: A).(arity g (CHead c0 (Bind Abst) u) t a2)))) H5))))))))))) (\lambda
-(c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl
-i c0 (CHead d (Bind Abst) u0))).(\lambda (a0: A).(\lambda (_: (arity g d u0
-(asucc g a0))).(\lambda (_: (((eq T u0 (THead (Bind Abst) u t)) \to (ex3_2 A
-A (\lambda (a1: A).(\lambda (a2: A).(eq A (asucc g a0) (AHead a1 a2))))
-(\lambda (a1: A).(\lambda (_: A).(arity g d u (asucc g a1)))) (\lambda (_:
-A).(\lambda (a2: A).(arity g (CHead d (Bind Abst) u) t a2))))))).(\lambda
-(H4: (eq T (TLRef i) (THead (Bind Abst) u t))).(let H5 \def (eq_ind T (TLRef
-i) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) u
-t) H4) in (False_ind (ex3_2 A A (\lambda (a1: A).(\lambda (a2: A).(eq A a0
-(AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u (asucc g
-a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind Abst) u) t
-a2)))) H5))))))))))) (\lambda (b: B).(\lambda (H1: (not (eq B b
-Abst))).(\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (H2:
-(arity g c0 u0 a1)).(\lambda (H3: (((eq T u0 (THead (Bind Abst) u t)) \to
-(ex3_2 A A (\lambda (a2: A).(\lambda (a3: A).(eq A a1 (AHead a2 a3))))
-(\lambda (a2: A).(\lambda (_: A).(arity g c0 u (asucc g a2)))) (\lambda (_:
-A).(\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u) t a3))))))).(\lambda
-(t0: T).(\lambda (a2: A).(\lambda (H4: (arity g (CHead c0 (Bind b) u0) t0
-a2)).(\lambda (H5: (((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A A
-(\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3:
-A).(\lambda (_: A).(arity g (CHead c0 (Bind b) u0) u (asucc g a3)))) (\lambda
-(_: A).(\lambda (a4: A).(arity g (CHead (CHead c0 (Bind b) u0) (Bind Abst) u)
-t a4))))))).(\lambda (H6: (eq T (THead (Bind b) u0 t0) (THead (Bind Abst) u
-t))).(let H7 \def (f_equal T B (\lambda (e: T).(match e with [(TSort _)
-\Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k
-with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead (Bind b)
-u0 t0) (THead (Bind Abst) u t) H6) in ((let H8 \def (f_equal T T (\lambda (e:
-T).(match e with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 |
-(THead _ t1 _) \Rightarrow t1])) (THead (Bind b) u0 t0) (THead (Bind Abst) u
-t) H6) in ((let H9 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _)
-\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t1) \Rightarrow t1]))
-(THead (Bind b) u0 t0) (THead (Bind Abst) u t) H6) in (\lambda (H10: (eq T u0
-u)).(\lambda (H11: (eq B b Abst)).(let H12 \def (eq_ind T t0 (\lambda (t1:
-T).((eq T t1 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3:
-A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_:
-A).(arity g (CHead c0 (Bind b) u0) u (asucc g a3)))) (\lambda (_: A).(\lambda
-(a4: A).(arity g (CHead (CHead c0 (Bind b) u0) (Bind Abst) u) t a4)))))) H5 t
-H9) in (let H13 \def (eq_ind T t0 (\lambda (t1: T).(arity g (CHead c0 (Bind
-b) u0) t1 a2)) H4 t H9) in (let H14 \def (eq_ind T u0 (\lambda (t1: T).((eq T
-t (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4:
-A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g (CHead
-c0 (Bind b) t1) u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g
-(CHead (CHead c0 (Bind b) t1) (Bind Abst) u) t a4)))))) H12 u H10) in (let
-H15 \def (eq_ind T u0 (\lambda (t1: T).(arity g (CHead c0 (Bind b) t1) t a2))
-H13 u H10) in (let H16 \def (eq_ind T u0 (\lambda (t1: T).((eq T t1 (THead
-(Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a1
-(AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g
-a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t
-a4)))))) H3 u H10) in (let H17 \def (eq_ind T u0 (\lambda (t1: T).(arity g c0
-t1 a1)) H2 u H10) in (let H18 \def (eq_ind B b (\lambda (b0: B).((eq T t
-(THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq
-A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g (CHead c0
-(Bind b0) u) u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g
-(CHead (CHead c0 (Bind b0) u) (Bind Abst) u) t a4)))))) H14 Abst H11) in (let
-H19 \def (eq_ind B b (\lambda (b0: B).(arity g (CHead c0 (Bind b0) u) t a2))
-H15 Abst H11) in (let H20 \def (eq_ind B b (\lambda (b0: B).(not (eq B b0
-Abst))) H1 Abst H11) in (let H21 \def (match (H20 (refl_equal B Abst)) in
-False with []) in H21))))))))))))) H8)) H7)))))))))))))) (\lambda (c0:
-C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (H1: (arity g c0 u0 (asucc g
-a1))).(\lambda (H2: (((eq T u0 (THead (Bind Abst) u t)) \to (ex3_2 A A
-(\lambda (a2: A).(\lambda (a3: A).(eq A (asucc g a1) (AHead a2 a3))))
-(\lambda (a2: A).(\lambda (_: A).(arity g c0 u (asucc g a2)))) (\lambda (_:
-A).(\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u) t a3))))))).(\lambda
-(t0: T).(\lambda (a2: A).(\lambda (H3: (arity g (CHead c0 (Bind Abst) u0) t0
-a2)).(\lambda (H4: (((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A A
-(\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3:
-A).(\lambda (_: A).(arity g (CHead c0 (Bind Abst) u0) u (asucc g a3))))
-(\lambda (_: A).(\lambda (a4: A).(arity g (CHead (CHead c0 (Bind Abst) u0)
-(Bind Abst) u) t a4))))))).(\lambda (H5: (eq T (THead (Bind Abst) u0 t0)
-(THead (Bind Abst) u t))).(let H6 \def (f_equal T T (\lambda (e: T).(match e
-with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t1 _)
-\Rightarrow t1])) (THead (Bind Abst) u0 t0) (THead (Bind Abst) u t) H5) in
-((let H7 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _)
-\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t1) \Rightarrow t1]))
-(THead (Bind Abst) u0 t0) (THead (Bind Abst) u t) H5) in (\lambda (H8: (eq T
-u0 u)).(let H9 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1 (THead (Bind
-Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead
-a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g (CHead c0 (Bind Abst) u0)
-u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead (CHead c0
-(Bind Abst) u0) (Bind Abst) u) t a4)))))) H4 t H7) in (let H10 \def (eq_ind T
-t0 (\lambda (t1: T).(arity g (CHead c0 (Bind Abst) u0) t1 a2)) H3 t H7) in
-(let H11 \def (eq_ind T u0 (\lambda (t1: T).((eq T t (THead (Bind Abst) u t))
-\to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4))))
-(\lambda (a3: A).(\lambda (_: A).(arity g (CHead c0 (Bind Abst) t1) u (asucc
-g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead (CHead c0 (Bind
-Abst) t1) (Bind Abst) u) t a4)))))) H9 u H8) in (let H12 \def (eq_ind T u0
-(\lambda (t1: T).(arity g (CHead c0 (Bind Abst) t1) t a2)) H10 u H8) in (let
-H13 \def (eq_ind T u0 (\lambda (t1: T).((eq T t1 (THead (Bind Abst) u t)) \to
-(ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A (asucc g a1) (AHead a3
-a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g a3))))
-(\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t a4))))))
-H2 u H8) in (let H14 \def (eq_ind T u0 (\lambda (t1: T).(arity g c0 t1 (asucc
-g a1))) H1 u H8) in (ex3_2_intro A A (\lambda (a3: A).(\lambda (a4: A).(eq A
-(AHead a1 a2) (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u
-(asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind
-Abst) u) t a4))) a1 a2 (refl_equal A (AHead a1 a2)) H14 H12)))))))))
-H6)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda
-(_: (arity g c0 u0 a1)).(\lambda (_: (((eq T u0 (THead (Bind Abst) u t)) \to
-(ex3_2 A A (\lambda (a2: A).(\lambda (a3: A).(eq A a1 (AHead a2 a3))))
-(\lambda (a2: A).(\lambda (_: A).(arity g c0 u (asucc g a2)))) (\lambda (_:
-A).(\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u) t a3))))))).(\lambda
-(t0: T).(\lambda (a2: A).(\lambda (_: (arity g c0 t0 (AHead a1 a2))).(\lambda
-(_: (((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3:
-A).(\lambda (a4: A).(eq A (AHead a1 a2) (AHead a3 a4)))) (\lambda (a3:
-A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda
-(a4: A).(arity g (CHead c0 (Bind Abst) u) t a4))))))).(\lambda (H5: (eq T
-(THead (Flat Appl) u0 t0) (THead (Bind Abst) u t))).(let H6 \def (eq_ind T
-(THead (Flat Appl) u0 t0) (\lambda (ee: T).(match ee with [(TSort _)
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
-(match k with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I
-(THead (Bind Abst) u t) H5) in (False_ind (ex3_2 A A (\lambda (a3:
-A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_:
-A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g
-(CHead c0 (Bind Abst) u) t a4)))) H6)))))))))))) (\lambda (c0: C).(\lambda
-(u0: T).(\lambda (a0: A).(\lambda (_: (arity g c0 u0 (asucc g a0))).(\lambda
-(_: (((eq T u0 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a1:
-A).(\lambda (a2: A).(eq A (asucc g a0) (AHead a1 a2)))) (\lambda (a1:
-A).(\lambda (_: A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda
-(a2: A).(arity g (CHead c0 (Bind Abst) u) t a2))))))).(\lambda (t0:
-T).(\lambda (_: (arity g c0 t0 a0)).(\lambda (_: (((eq T t0 (THead (Bind
-Abst) u t)) \to (ex3_2 A A (\lambda (a1: A).(\lambda (a2: A).(eq A a0 (AHead
-a1 a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u (asucc g a1))))
-(\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind Abst) u) t
-a2))))))).(\lambda (H5: (eq T (THead (Flat Cast) u0 t0) (THead (Bind Abst) u
-t))).(let H6 \def (eq_ind T (THead (Flat Cast) u0 t0) (\lambda (ee: T).(match
-ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k
-_ _) \Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat _)
-\Rightarrow True])])) I (THead (Bind Abst) u t) H5) in (False_ind (ex3_2 A A
-(\lambda (a1: A).(\lambda (a2: A).(eq A a0 (AHead a1 a2)))) (\lambda (a1:
-A).(\lambda (_: A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda
-(a2: A).(arity g (CHead c0 (Bind Abst) u) t a2)))) H6))))))))))) (\lambda
-(c0: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda (H1: (arity g c0 t0
-a1)).(\lambda (H2: (((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A A
-(\lambda (a2: A).(\lambda (a3: A).(eq A a1 (AHead a2 a3)))) (\lambda (a2:
-A).(\lambda (_: A).(arity g c0 u (asucc g a2)))) (\lambda (_: A).(\lambda
-(a3: A).(arity g (CHead c0 (Bind Abst) u) t a3))))))).(\lambda (a2:
-A).(\lambda (H3: (leq g a1 a2)).(\lambda (H4: (eq T t0 (THead (Bind Abst) u
-t))).(let H5 \def (f_equal T T (\lambda (e: T).e) t0 (THead (Bind Abst) u t)
-H4) in (let H6 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1 (THead (Bind
-Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a1 (AHead
-a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g a3))))
-(\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t a4))))))
-H2 (THead (Bind Abst) u t) H5) in (let H7 \def (eq_ind T t0 (\lambda (t1:
-T).(arity g c0 t1 a1)) H1 (THead (Bind Abst) u t) H5) in (let H8 \def (H6
-(refl_equal T (THead (Bind Abst) u t))) in (ex3_2_ind A A (\lambda (a3:
-A).(\lambda (a4: A).(eq A a1 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_:
-A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g
-(CHead c0 (Bind Abst) u) t a4))) (ex3_2 A A (\lambda (a3: A).(\lambda (a4:
-A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u
-(asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind
-Abst) u) t a4)))) (\lambda (x0: A).(\lambda (x1: A).(\lambda (H9: (eq A a1
-(AHead x0 x1))).(\lambda (H10: (arity g c0 u (asucc g x0))).(\lambda (H11:
-(arity g (CHead c0 (Bind Abst) u) t x1)).(let H12 \def (eq_ind A a1 (\lambda
-(a0: A).(leq g a0 a2)) H3 (AHead x0 x1) H9) in (let H13 \def (eq_ind A a1
-(\lambda (a0: A).(arity g c0 (THead (Bind Abst) u t) a0)) H7 (AHead x0 x1)
-H9) in (let H_x \def (leq_gen_head1 g x0 x1 a2 H12) in (let H14 \def H_x in
-(ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq g x0 a3))) (\lambda (_:
-A).(\lambda (a4: A).(leq g x1 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A
-a2 (AHead a3 a4)))) (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2
-(AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g
-a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t
-a4)))) (\lambda (x2: A).(\lambda (x3: A).(\lambda (H15: (leq g x0
-x2)).(\lambda (H16: (leq g x1 x3)).(\lambda (H17: (eq A a2 (AHead x2
-x3))).(let H18 \def (f_equal A A (\lambda (e: A).e) a2 (AHead x2 x3) H17) in
-(eq_ind_r A (AHead x2 x3) (\lambda (a0: A).(ex3_2 A A (\lambda (a3:
-A).(\lambda (a4: A).(eq A a0 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_:
-A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g
-(CHead c0 (Bind Abst) u) t a4))))) (ex3_2_intro A A (\lambda (a3: A).(\lambda
-(a4: A).(eq A (AHead x2 x3) (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_:
-A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g
-(CHead c0 (Bind Abst) u) t a4))) x2 x3 (refl_equal A (AHead x2 x3))
-(arity_repl g c0 u (asucc g x0) H10 (asucc g x2) (asucc_repl g x0 x2 H15))
-(arity_repl g (CHead c0 (Bind Abst) u) t x1 H11 x3 H16)) a2 H18)))))))
-H14)))))))))) H8))))))))))))) c y a H0))) H)))))).
-
-lemma arity_gen_appl:
- \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (a2:
-A).((arity g c (THead (Flat Appl) u t) a2) \to (ex2 A (\lambda (a1: A).(arity
-g c u a1)) (\lambda (a1: A).(arity g c t (AHead a1 a2)))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (a2:
-A).(\lambda (H: (arity g c (THead (Flat Appl) u t) a2)).(insert_eq T (THead
-(Flat Appl) u t) (\lambda (t0: T).(arity g c t0 a2)) (\lambda (_: T).(ex2 A
-(\lambda (a1: A).(arity g c u a1)) (\lambda (a1: A).(arity g c t (AHead a1
-a2))))) (\lambda (y: T).(\lambda (H0: (arity g c y a2)).(arity_ind g (\lambda
-(c0: C).(\lambda (t0: T).(\lambda (a: A).((eq T t0 (THead (Flat Appl) u t))
-\to (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t
-(AHead a1 a)))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (H1: (eq T
-(TSort n) (THead (Flat Appl) u t))).(let H2 \def (eq_ind T (TSort n) (\lambda
-(ee: T).(match ee with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow
-False | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) u t) H1) in
-(False_ind (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity
-g c0 t (AHead a1 (ASort O n))))) H2))))) (\lambda (c0: C).(\lambda (d:
-C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind
-Abbr) u0))).(\lambda (a: A).(\lambda (_: (arity g d u0 a)).(\lambda (_: (((eq
-T u0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda (a1: A).(arity g d u a1))
-(\lambda (a1: A).(arity g d t (AHead a1 a))))))).(\lambda (H4: (eq T (TLRef
-i) (THead (Flat Appl) u t))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee:
-T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
-(THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) u t) H4) in
-(False_ind (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity
-g c0 t (AHead a1 a)))) H5))))))))))) (\lambda (c0: C).(\lambda (d:
-C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind
-Abst) u0))).(\lambda (a: A).(\lambda (_: (arity g d u0 (asucc g a))).(\lambda
-(_: (((eq T u0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda (a1: A).(arity g
-d u a1)) (\lambda (a1: A).(arity g d t (AHead a1 (asucc g a)))))))).(\lambda
-(H4: (eq T (TLRef i) (THead (Flat Appl) u t))).(let H5 \def (eq_ind T (TLRef
-i) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) u
-t) H4) in (False_ind (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1:
-A).(arity g c0 t (AHead a1 a)))) H5))))))))))) (\lambda (b: B).(\lambda (_:
-(not (eq B b Abst))).(\lambda (c0: C).(\lambda (u0: T).(\lambda (a1:
-A).(\lambda (_: (arity g c0 u0 a1)).(\lambda (_: (((eq T u0 (THead (Flat
-Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3:
-A).(arity g c0 t (AHead a3 a1))))))).(\lambda (t0: T).(\lambda (a0:
-A).(\lambda (_: (arity g (CHead c0 (Bind b) u0) t0 a0)).(\lambda (_: (((eq T
-t0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g (CHead c0
-(Bind b) u0) u a3)) (\lambda (a3: A).(arity g (CHead c0 (Bind b) u0) t (AHead
-a3 a0))))))).(\lambda (H6: (eq T (THead (Bind b) u0 t0) (THead (Flat Appl) u
-t))).(let H7 \def (eq_ind T (THead (Bind b) u0 t0) (\lambda (ee: T).(match ee
-with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _
-_) \Rightarrow (match k with [(Bind _) \Rightarrow True | (Flat _)
-\Rightarrow False])])) I (THead (Flat Appl) u t) H6) in (False_ind (ex2 A
-(\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3
-a0)))) H7)))))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a1:
-A).(\lambda (_: (arity g c0 u0 (asucc g a1))).(\lambda (_: (((eq T u0 (THead
-(Flat Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda
-(a3: A).(arity g c0 t (AHead a3 (asucc g a1)))))))).(\lambda (t0: T).(\lambda
-(a0: A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u0) t0 a0)).(\lambda (_:
-(((eq T t0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g
-(CHead c0 (Bind Abst) u0) u a3)) (\lambda (a3: A).(arity g (CHead c0 (Bind
-Abst) u0) t (AHead a3 a0))))))).(\lambda (H5: (eq T (THead (Bind Abst) u0 t0)
-(THead (Flat Appl) u t))).(let H6 \def (eq_ind T (THead (Bind Abst) u0 t0)
-(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _)
-\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) u t)
-H5) in (False_ind (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3:
-A).(arity g c0 t (AHead a3 (AHead a1 a0))))) H6)))))))))))) (\lambda (c0:
-C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (H1: (arity g c0 u0
-a1)).(\lambda (H2: (((eq T u0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda
-(a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3
-a1))))))).(\lambda (t0: T).(\lambda (a0: A).(\lambda (H3: (arity g c0 t0
-(AHead a1 a0))).(\lambda (H4: (((eq T t0 (THead (Flat Appl) u t)) \to (ex2 A
-(\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3
-(AHead a1 a0)))))))).(\lambda (H5: (eq T (THead (Flat Appl) u0 t0) (THead
-(Flat Appl) u t))).(let H6 \def (f_equal T T (\lambda (e: T).(match e with
-[(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t1 _)
-\Rightarrow t1])) (THead (Flat Appl) u0 t0) (THead (Flat Appl) u t) H5) in
-((let H7 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _)
-\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t1) \Rightarrow t1]))
-(THead (Flat Appl) u0 t0) (THead (Flat Appl) u t) H5) in (\lambda (H8: (eq T
-u0 u)).(let H9 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1 (THead (Flat
-Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3:
-A).(arity g c0 t (AHead a3 (AHead a1 a0))))))) H4 t H7) in (let H10 \def
-(eq_ind T t0 (\lambda (t1: T).(arity g c0 t1 (AHead a1 a0))) H3 t H7) in (let
-H11 \def (eq_ind T u0 (\lambda (t1: T).((eq T t1 (THead (Flat Appl) u t)) \to
-(ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t
-(AHead a3 a1)))))) H2 u H8) in (let H12 \def (eq_ind T u0 (\lambda (t1:
-T).(arity g c0 t1 a1)) H1 u H8) in (ex_intro2 A (\lambda (a3: A).(arity g c0
-u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 a0))) a1 H12 H10)))))))
-H6)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a: A).(\lambda (_:
-(arity g c0 u0 (asucc g a))).(\lambda (_: (((eq T u0 (THead (Flat Appl) u t))
-\to (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t
-(AHead a1 (asucc g a)))))))).(\lambda (t0: T).(\lambda (_: (arity g c0 t0
-a)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda (a1:
-A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t (AHead a1
-a))))))).(\lambda (H5: (eq T (THead (Flat Cast) u0 t0) (THead (Flat Appl) u
-t))).(let H6 \def (eq_ind T (THead (Flat Cast) u0 t0) (\lambda (ee: T).(match
-ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k
-_ _) \Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat f)
-\Rightarrow (match f with [Appl \Rightarrow False | Cast \Rightarrow
-True])])])) I (THead (Flat Appl) u t) H5) in (False_ind (ex2 A (\lambda (a1:
-A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t (AHead a1 a))))
-H6))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda
-(H1: (arity g c0 t0 a1)).(\lambda (H2: (((eq T t0 (THead (Flat Appl) u t))
-\to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t
-(AHead a3 a1))))))).(\lambda (a0: A).(\lambda (H3: (leq g a1 a0)).(\lambda
-(H4: (eq T t0 (THead (Flat Appl) u t))).(let H5 \def (f_equal T T (\lambda
-(e: T).e) t0 (THead (Flat Appl) u t) H4) in (let H6 \def (eq_ind T t0
-(\lambda (t1: T).((eq T t1 (THead (Flat Appl) u t)) \to (ex2 A (\lambda (a3:
-A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 a1)))))) H2
-(THead (Flat Appl) u t) H5) in (let H7 \def (eq_ind T t0 (\lambda (t1:
-T).(arity g c0 t1 a1)) H1 (THead (Flat Appl) u t) H5) in (let H8 \def (H6
-(refl_equal T (THead (Flat Appl) u t))) in (ex2_ind A (\lambda (a3: A).(arity
-g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 a1))) (ex2 A (\lambda
-(a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 a0))))
-(\lambda (x: A).(\lambda (H9: (arity g c0 u x)).(\lambda (H10: (arity g c0 t
-(AHead x a1))).(ex_intro2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3:
-A).(arity g c0 t (AHead a3 a0))) x H9 (arity_repl g c0 t (AHead x a1) H10
-(AHead x a0) (leq_head g x x (leq_refl g x) a1 a0 H3)))))) H8))))))))))))) c
-y a2 H0))) H)))))).
-
-lemma arity_gen_cast:
- \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (a:
-A).((arity g c (THead (Flat Cast) u t) a) \to (land (arity g c u (asucc g a))
-(arity g c t a)))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (a:
-A).(\lambda (H: (arity g c (THead (Flat Cast) u t) a)).(insert_eq T (THead
-(Flat Cast) u t) (\lambda (t0: T).(arity g c t0 a)) (\lambda (_: T).(land
-(arity g c u (asucc g a)) (arity g c t a))) (\lambda (y: T).(\lambda (H0:
-(arity g c y a)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (a0:
-A).((eq T t0 (THead (Flat Cast) u t)) \to (land (arity g c0 u (asucc g a0))
-(arity g c0 t a0)))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (H1: (eq T
-(TSort n) (THead (Flat Cast) u t))).(let H2 \def (eq_ind T (TSort n) (\lambda
-(ee: T).(match ee with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow
-False | (THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) u t) H1) in
-(False_ind (land (arity g c0 u (asucc g (ASort O n))) (arity g c0 t (ASort O
-n))) H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i:
-nat).(\lambda (_: (getl i c0 (CHead d (Bind Abbr) u0))).(\lambda (a0:
-A).(\lambda (_: (arity g d u0 a0)).(\lambda (_: (((eq T u0 (THead (Flat Cast)
-u t)) \to (land (arity g d u (asucc g a0)) (arity g d t a0))))).(\lambda (H4:
-(eq T (TLRef i) (THead (Flat Cast) u t))).(let H5 \def (eq_ind T (TLRef i)
-(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) u
-t) H4) in (False_ind (land (arity g c0 u (asucc g a0)) (arity g c0 t a0))
-H5))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i:
-nat).(\lambda (_: (getl i c0 (CHead d (Bind Abst) u0))).(\lambda (a0:
-A).(\lambda (_: (arity g d u0 (asucc g a0))).(\lambda (_: (((eq T u0 (THead
-(Flat Cast) u t)) \to (land (arity g d u (asucc g (asucc g a0))) (arity g d t
-(asucc g a0)))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat Cast) u
-t))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee with [(TSort
-_) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
-False])) I (THead (Flat Cast) u t) H4) in (False_ind (land (arity g c0 u
-(asucc g a0)) (arity g c0 t a0)) H5))))))))))) (\lambda (b: B).(\lambda (_:
-(not (eq B b Abst))).(\lambda (c0: C).(\lambda (u0: T).(\lambda (a1:
-A).(\lambda (_: (arity g c0 u0 a1)).(\lambda (_: (((eq T u0 (THead (Flat
-Cast) u t)) \to (land (arity g c0 u (asucc g a1)) (arity g c0 t
-a1))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c0
-(Bind b) u0) t0 a2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) u t)) \to
-(land (arity g (CHead c0 (Bind b) u0) u (asucc g a2)) (arity g (CHead c0
-(Bind b) u0) t a2))))).(\lambda (H6: (eq T (THead (Bind b) u0 t0) (THead
-(Flat Cast) u t))).(let H7 \def (eq_ind T (THead (Bind b) u0 t0) (\lambda
-(ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow
-False | (THead k _ _) \Rightarrow (match k with [(Bind _) \Rightarrow True |
-(Flat _) \Rightarrow False])])) I (THead (Flat Cast) u t) H6) in (False_ind
-(land (arity g c0 u (asucc g a2)) (arity g c0 t a2)) H7))))))))))))))
-(\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (_: (arity g c0
-u0 (asucc g a1))).(\lambda (_: (((eq T u0 (THead (Flat Cast) u t)) \to (land
-(arity g c0 u (asucc g (asucc g a1))) (arity g c0 t (asucc g
-a1)))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c0
-(Bind Abst) u0) t0 a2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) u t)) \to
-(land (arity g (CHead c0 (Bind Abst) u0) u (asucc g a2)) (arity g (CHead c0
-(Bind Abst) u0) t a2))))).(\lambda (H5: (eq T (THead (Bind Abst) u0 t0)
-(THead (Flat Cast) u t))).(let H6 \def (eq_ind T (THead (Bind Abst) u0 t0)
-(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _)
-\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Cast) u t)
-H5) in (False_ind (land (arity g c0 u (asucc g (AHead a1 a2))) (arity g c0 t
-(AHead a1 a2))) H6)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda
-(a1: A).(\lambda (_: (arity g c0 u0 a1)).(\lambda (_: (((eq T u0 (THead (Flat
-Cast) u t)) \to (land (arity g c0 u (asucc g a1)) (arity g c0 t
-a1))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g c0 t0 (AHead
-a1 a2))).(\lambda (_: (((eq T t0 (THead (Flat Cast) u t)) \to (land (arity g
-c0 u (asucc g (AHead a1 a2))) (arity g c0 t (AHead a1 a2)))))).(\lambda (H5:
-(eq T (THead (Flat Appl) u0 t0) (THead (Flat Cast) u t))).(let H6 \def
-(eq_ind T (THead (Flat Appl) u0 t0) (\lambda (ee: T).(match ee with [(TSort
-_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
-\Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow
-(match f with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead
-(Flat Cast) u t) H5) in (False_ind (land (arity g c0 u (asucc g a2)) (arity g
-c0 t a2)) H6)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a0:
-A).(\lambda (H1: (arity g c0 u0 (asucc g a0))).(\lambda (H2: (((eq T u0
-(THead (Flat Cast) u t)) \to (land (arity g c0 u (asucc g (asucc g a0)))
-(arity g c0 t (asucc g a0)))))).(\lambda (t0: T).(\lambda (H3: (arity g c0 t0
-a0)).(\lambda (H4: (((eq T t0 (THead (Flat Cast) u t)) \to (land (arity g c0
-u (asucc g a0)) (arity g c0 t a0))))).(\lambda (H5: (eq T (THead (Flat Cast)
-u0 t0) (THead (Flat Cast) u t))).(let H6 \def (f_equal T T (\lambda (e:
-T).(match e with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 |
-(THead _ t1 _) \Rightarrow t1])) (THead (Flat Cast) u0 t0) (THead (Flat Cast)
-u t) H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e with [(TSort
-_) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t1) \Rightarrow
-t1])) (THead (Flat Cast) u0 t0) (THead (Flat Cast) u t) H5) in (\lambda (H8:
-(eq T u0 u)).(let H9 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1 (THead
-(Flat Cast) u t)) \to (land (arity g c0 u (asucc g a0)) (arity g c0 t a0))))
-H4 t H7) in (let H10 \def (eq_ind T t0 (\lambda (t1: T).(arity g c0 t1 a0))
-H3 t H7) in (let H11 \def (eq_ind T u0 (\lambda (t1: T).((eq T t1 (THead
-(Flat Cast) u t)) \to (land (arity g c0 u (asucc g (asucc g a0))) (arity g c0
-t (asucc g a0))))) H2 u H8) in (let H12 \def (eq_ind T u0 (\lambda (t1:
-T).(arity g c0 t1 (asucc g a0))) H1 u H8) in (conj (arity g c0 u (asucc g
-a0)) (arity g c0 t a0) H12 H10))))))) H6))))))))))) (\lambda (c0: C).(\lambda
-(t0: T).(\lambda (a1: A).(\lambda (H1: (arity g c0 t0 a1)).(\lambda (H2:
-(((eq T t0 (THead (Flat Cast) u t)) \to (land (arity g c0 u (asucc g a1))
-(arity g c0 t a1))))).(\lambda (a2: A).(\lambda (H3: (leq g a1 a2)).(\lambda
-(H4: (eq T t0 (THead (Flat Cast) u t))).(let H5 \def (f_equal T T (\lambda
-(e: T).e) t0 (THead (Flat Cast) u t) H4) in (let H6 \def (eq_ind T t0
-(\lambda (t1: T).((eq T t1 (THead (Flat Cast) u t)) \to (land (arity g c0 u
-(asucc g a1)) (arity g c0 t a1)))) H2 (THead (Flat Cast) u t) H5) in (let H7
-\def (eq_ind T t0 (\lambda (t1: T).(arity g c0 t1 a1)) H1 (THead (Flat Cast)
-u t) H5) in (let H8 \def (H6 (refl_equal T (THead (Flat Cast) u t))) in
-(land_ind (arity g c0 u (asucc g a1)) (arity g c0 t a1) (land (arity g c0 u
-(asucc g a2)) (arity g c0 t a2)) (\lambda (H9: (arity g c0 u (asucc g
-a1))).(\lambda (H10: (arity g c0 t a1)).(conj (arity g c0 u (asucc g a2))
-(arity g c0 t a2) (arity_repl g c0 u (asucc g a1) H9 (asucc g a2) (asucc_repl
-g a1 a2 H3)) (arity_repl g c0 t a1 H10 a2 H3)))) H8))))))))))))) c y a H0)))
-H)))))).
-
-lemma arity_gen_appls:
- \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (vs: TList).(\forall
-(a2: A).((arity g c (THeads (Flat Appl) vs t) a2) \to (ex A (\lambda (a:
-A).(arity g c t a))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (vs:
-TList).(TList_ind (\lambda (t0: TList).(\forall (a2: A).((arity g c (THeads
-(Flat Appl) t0 t) a2) \to (ex A (\lambda (a: A).(arity g c t a)))))) (\lambda
-(a2: A).(\lambda (H: (arity g c t a2)).(ex_intro A (\lambda (a: A).(arity g c
-t a)) a2 H))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H: ((\forall
-(a2: A).((arity g c (THeads (Flat Appl) t1 t) a2) \to (ex A (\lambda (a:
-A).(arity g c t a))))))).(\lambda (a2: A).(\lambda (H0: (arity g c (THead
-(Flat Appl) t0 (THeads (Flat Appl) t1 t)) a2)).(let H1 \def (arity_gen_appl g
-c t0 (THeads (Flat Appl) t1 t) a2 H0) in (ex2_ind A (\lambda (a1: A).(arity g
-c t0 a1)) (\lambda (a1: A).(arity g c (THeads (Flat Appl) t1 t) (AHead a1
-a2))) (ex A (\lambda (a: A).(arity g c t a))) (\lambda (x: A).(\lambda (_:
-(arity g c t0 x)).(\lambda (H3: (arity g c (THeads (Flat Appl) t1 t) (AHead x
-a2))).(let H_x \def (H (AHead x a2) H3) in (let H4 \def H_x in (ex_ind A
-(\lambda (a: A).(arity g c t a)) (ex A (\lambda (a: A).(arity g c t a)))
-(\lambda (x0: A).(\lambda (H5: (arity g c t x0)).(ex_intro A (\lambda (a:
-A).(arity g c t a)) x0 H5))) H4)))))) H1))))))) vs)))).
-
-lemma arity_gen_lift:
- \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).(\forall (h:
-nat).(\forall (d: nat).((arity g c1 (lift h d t) a) \to (\forall (c2:
-C).((drop h d c1 c2) \to (arity g c2 t a)))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (h:
-nat).(\lambda (d: nat).(\lambda (H: (arity g c1 (lift h d t) a)).(insert_eq T
-(lift h d t) (\lambda (t0: T).(arity g c1 t0 a)) (\lambda (_: T).(\forall
-(c2: C).((drop h d c1 c2) \to (arity g c2 t a)))) (\lambda (y: T).(\lambda
-(H0: (arity g c1 y a)).(unintro T t (\lambda (t0: T).((eq T y (lift h d t0))
-\to (\forall (c2: C).((drop h d c1 c2) \to (arity g c2 t0 a))))) (unintro nat
-d (\lambda (n: nat).(\forall (x: T).((eq T y (lift h n x)) \to (\forall (c2:
-C).((drop h n c1 c2) \to (arity g c2 x a)))))) (arity_ind g (\lambda (c:
-C).(\lambda (t0: T).(\lambda (a0: A).(\forall (x: nat).(\forall (x0: T).((eq
-T t0 (lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to (arity g c2 x0
-a0))))))))) (\lambda (c: C).(\lambda (n: nat).(\lambda (x: nat).(\lambda (x0:
-T).(\lambda (H1: (eq T (TSort n) (lift h x x0))).(\lambda (c2: C).(\lambda
-(_: (drop h x c c2)).(eq_ind_r T (TSort n) (\lambda (t0: T).(arity g c2 t0
-(ASort O n))) (arity_sort g c2 n) x0 (lift_gen_sort h x n x0 H1)))))))))
-(\lambda (c: C).(\lambda (d0: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
-(H1: (getl i c (CHead d0 (Bind Abbr) u))).(\lambda (a0: A).(\lambda (H2:
-(arity g d0 u a0)).(\lambda (H3: ((\forall (x: nat).(\forall (x0: T).((eq T u
-(lift h x x0)) \to (\forall (c2: C).((drop h x d0 c2) \to (arity g c2 x0
-a0)))))))).(\lambda (x: nat).(\lambda (x0: T).(\lambda (H4: (eq T (TLRef i)
-(lift h x x0))).(\lambda (c2: C).(\lambda (H5: (drop h x c c2)).(let H_x \def
-(lift_gen_lref x0 x h i H4) in (let H6 \def H_x in (or_ind (land (lt i x) (eq
-T x0 (TLRef i))) (land (le (plus x h) i) (eq T x0 (TLRef (minus i h))))
-(arity g c2 x0 a0) (\lambda (H7: (land (lt i x) (eq T x0 (TLRef
-i)))).(land_ind (lt i x) (eq T x0 (TLRef i)) (arity g c2 x0 a0) (\lambda (H8:
-(lt i x)).(\lambda (H9: (eq T x0 (TLRef i))).(eq_ind_r T (TLRef i) (\lambda
-(t0: T).(arity g c2 t0 a0)) (let H10 \def (eq_ind nat x (\lambda (n:
-nat).(drop h n c c2)) H5 (S (plus i (minus x (S i)))) (lt_plus_minus i x H8))
-in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (minus x (S
-i)) v)))) (\lambda (v: T).(\lambda (e0: C).(getl i c2 (CHead e0 (Bind Abbr)
-v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (minus x (S i)) d0 e0)))
-(arity g c2 (TLRef i) a0) (\lambda (x1: T).(\lambda (x2: C).(\lambda (H11:
-(eq T u (lift h (minus x (S i)) x1))).(\lambda (H12: (getl i c2 (CHead x2
-(Bind Abbr) x1))).(\lambda (H13: (drop h (minus x (S i)) d0 x2)).(let H14
-\def (eq_ind T u (\lambda (t0: T).(\forall (x3: nat).(\forall (x4: T).((eq T
-t0 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 d0 c3) \to (arity g c3 x4
-a0))))))) H3 (lift h (minus x (S i)) x1) H11) in (let H15 \def (eq_ind T u
-(\lambda (t0: T).(arity g d0 t0 a0)) H2 (lift h (minus x (S i)) x1) H11) in
-(arity_abbr g c2 x2 x1 i H12 a0 (H14 (minus x (S i)) x1 (refl_equal T (lift h
-(minus x (S i)) x1)) x2 H13))))))))) (getl_drop_conf_lt Abbr c d0 u i H1 c2 h
-(minus x (S i)) H10))) x0 H9))) H7)) (\lambda (H7: (land (le (plus x h) i)
-(eq T x0 (TLRef (minus i h))))).(land_ind (le (plus x h) i) (eq T x0 (TLRef
-(minus i h))) (arity g c2 x0 a0) (\lambda (H8: (le (plus x h) i)).(\lambda
-(H9: (eq T x0 (TLRef (minus i h)))).(eq_ind_r T (TLRef (minus i h)) (\lambda
-(t0: T).(arity g c2 t0 a0)) (arity_abbr g c2 d0 u (minus i h)
-(getl_drop_conf_ge i (CHead d0 (Bind Abbr) u) c H1 c2 h x H5 H8) a0 H2) x0
-H9))) H7)) H6)))))))))))))))) (\lambda (c: C).(\lambda (d0: C).(\lambda (u:
-T).(\lambda (i: nat).(\lambda (H1: (getl i c (CHead d0 (Bind Abst)
-u))).(\lambda (a0: A).(\lambda (H2: (arity g d0 u (asucc g a0))).(\lambda
-(H3: ((\forall (x: nat).(\forall (x0: T).((eq T u (lift h x x0)) \to (\forall
-(c2: C).((drop h x d0 c2) \to (arity g c2 x0 (asucc g a0))))))))).(\lambda
-(x: nat).(\lambda (x0: T).(\lambda (H4: (eq T (TLRef i) (lift h x
-x0))).(\lambda (c2: C).(\lambda (H5: (drop h x c c2)).(let H_x \def
-(lift_gen_lref x0 x h i H4) in (let H6 \def H_x in (or_ind (land (lt i x) (eq
-T x0 (TLRef i))) (land (le (plus x h) i) (eq T x0 (TLRef (minus i h))))
-(arity g c2 x0 a0) (\lambda (H7: (land (lt i x) (eq T x0 (TLRef
-i)))).(land_ind (lt i x) (eq T x0 (TLRef i)) (arity g c2 x0 a0) (\lambda (H8:
-(lt i x)).(\lambda (H9: (eq T x0 (TLRef i))).(eq_ind_r T (TLRef i) (\lambda
-(t0: T).(arity g c2 t0 a0)) (let H10 \def (eq_ind nat x (\lambda (n:
-nat).(drop h n c c2)) H5 (S (plus i (minus x (S i)))) (lt_plus_minus i x H8))
-in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (minus x (S
-i)) v)))) (\lambda (v: T).(\lambda (e0: C).(getl i c2 (CHead e0 (Bind Abst)
-v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (minus x (S i)) d0 e0)))
-(arity g c2 (TLRef i) a0) (\lambda (x1: T).(\lambda (x2: C).(\lambda (H11:
-(eq T u (lift h (minus x (S i)) x1))).(\lambda (H12: (getl i c2 (CHead x2
-(Bind Abst) x1))).(\lambda (H13: (drop h (minus x (S i)) d0 x2)).(let H14
-\def (eq_ind T u (\lambda (t0: T).(\forall (x3: nat).(\forall (x4: T).((eq T
-t0 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 d0 c3) \to (arity g c3 x4
-(asucc g a0)))))))) H3 (lift h (minus x (S i)) x1) H11) in (let H15 \def
-(eq_ind T u (\lambda (t0: T).(arity g d0 t0 (asucc g a0))) H2 (lift h (minus
-x (S i)) x1) H11) in (arity_abst g c2 x2 x1 i H12 a0 (H14 (minus x (S i)) x1
-(refl_equal T (lift h (minus x (S i)) x1)) x2 H13))))))))) (getl_drop_conf_lt
-Abst c d0 u i H1 c2 h (minus x (S i)) H10))) x0 H9))) H7)) (\lambda (H7:
-(land (le (plus x h) i) (eq T x0 (TLRef (minus i h))))).(land_ind (le (plus x
-h) i) (eq T x0 (TLRef (minus i h))) (arity g c2 x0 a0) (\lambda (H8: (le
-(plus x h) i)).(\lambda (H9: (eq T x0 (TLRef (minus i h)))).(eq_ind_r T
-(TLRef (minus i h)) (\lambda (t0: T).(arity g c2 t0 a0)) (arity_abst g c2 d0
-u (minus i h) (getl_drop_conf_ge i (CHead d0 (Bind Abst) u) c H1 c2 h x H5
-H8) a0 H2) x0 H9))) H7)) H6)))))))))))))))) (\lambda (b: B).(\lambda (H1:
-(not (eq B b Abst))).(\lambda (c: C).(\lambda (u: T).(\lambda (a1:
-A).(\lambda (H2: (arity g c u a1)).(\lambda (H3: ((\forall (x: nat).(\forall
-(x0: T).((eq T u (lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to
-(arity g c2 x0 a1)))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H4:
-(arity g (CHead c (Bind b) u) t0 a2)).(\lambda (H5: ((\forall (x:
-nat).(\forall (x0: T).((eq T t0 (lift h x x0)) \to (\forall (c2: C).((drop h
-x (CHead c (Bind b) u) c2) \to (arity g c2 x0 a2)))))))).(\lambda (x:
-nat).(\lambda (x0: T).(\lambda (H6: (eq T (THead (Bind b) u t0) (lift h x
-x0))).(\lambda (c2: C).(\lambda (H7: (drop h x c c2)).(ex3_2_ind T T (\lambda
-(y0: T).(\lambda (z: T).(eq T x0 (THead (Bind b) y0 z)))) (\lambda (y0:
-T).(\lambda (_: T).(eq T u (lift h x y0)))) (\lambda (_: T).(\lambda (z:
-T).(eq T t0 (lift h (S x) z)))) (arity g c2 x0 a2) (\lambda (x1: T).(\lambda
-(x2: T).(\lambda (H8: (eq T x0 (THead (Bind b) x1 x2))).(\lambda (H9: (eq T u
-(lift h x x1))).(\lambda (H10: (eq T t0 (lift h (S x) x2))).(eq_ind_r T
-(THead (Bind b) x1 x2) (\lambda (t1: T).(arity g c2 t1 a2)) (let H11 \def
-(eq_ind T t0 (\lambda (t1: T).(\forall (x3: nat).(\forall (x4: T).((eq T t1
-(lift h x3 x4)) \to (\forall (c3: C).((drop h x3 (CHead c (Bind b) u) c3) \to
-(arity g c3 x4 a2))))))) H5 (lift h (S x) x2) H10) in (let H12 \def (eq_ind T
-t0 (\lambda (t1: T).(arity g (CHead c (Bind b) u) t1 a2)) H4 (lift h (S x)
-x2) H10) in (let H13 \def (eq_ind T u (\lambda (t1: T).(arity g (CHead c
-(Bind b) t1) (lift h (S x) x2) a2)) H12 (lift h x x1) H9) in (let H14 \def
-(eq_ind T u (\lambda (t1: T).(\forall (x3: nat).(\forall (x4: T).((eq T (lift
-h (S x) x2) (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 (CHead c (Bind
-b) t1) c3) \to (arity g c3 x4 a2))))))) H11 (lift h x x1) H9) in (let H15
-\def (eq_ind T u (\lambda (t1: T).(\forall (x3: nat).(\forall (x4: T).((eq T
-t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to (arity g c3 x4
-a1))))))) H3 (lift h x x1) H9) in (let H16 \def (eq_ind T u (\lambda (t1:
-T).(arity g c t1 a1)) H2 (lift h x x1) H9) in (arity_bind g b H1 c2 x1 a1
-(H15 x x1 (refl_equal T (lift h x x1)) c2 H7) x2 a2 (H14 (S x) x2 (refl_equal
-T (lift h (S x) x2)) (CHead c2 (Bind b) x1) (drop_skip_bind h x c c2 H7 b
-x1))))))))) x0 H8)))))) (lift_gen_bind b u t0 x0 h x H6))))))))))))))))))
-(\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H1: (arity g c u
-(asucc g a1))).(\lambda (H2: ((\forall (x: nat).(\forall (x0: T).((eq T u
-(lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to (arity g c2 x0
-(asucc g a1))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H3: (arity g
-(CHead c (Bind Abst) u) t0 a2)).(\lambda (H4: ((\forall (x: nat).(\forall
-(x0: T).((eq T t0 (lift h x x0)) \to (\forall (c2: C).((drop h x (CHead c
-(Bind Abst) u) c2) \to (arity g c2 x0 a2)))))))).(\lambda (x: nat).(\lambda
-(x0: T).(\lambda (H5: (eq T (THead (Bind Abst) u t0) (lift h x x0))).(\lambda
-(c2: C).(\lambda (H6: (drop h x c c2)).(ex3_2_ind T T (\lambda (y0:
-T).(\lambda (z: T).(eq T x0 (THead (Bind Abst) y0 z)))) (\lambda (y0:
-T).(\lambda (_: T).(eq T u (lift h x y0)))) (\lambda (_: T).(\lambda (z:
-T).(eq T t0 (lift h (S x) z)))) (arity g c2 x0 (AHead a1 a2)) (\lambda (x1:
-T).(\lambda (x2: T).(\lambda (H7: (eq T x0 (THead (Bind Abst) x1
-x2))).(\lambda (H8: (eq T u (lift h x x1))).(\lambda (H9: (eq T t0 (lift h (S
-x) x2))).(eq_ind_r T (THead (Bind Abst) x1 x2) (\lambda (t1: T).(arity g c2
-t1 (AHead a1 a2))) (let H10 \def (eq_ind T t0 (\lambda (t1: T).(\forall (x3:
-nat).(\forall (x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h
-x3 (CHead c (Bind Abst) u) c3) \to (arity g c3 x4 a2))))))) H4 (lift h (S x)
-x2) H9) in (let H11 \def (eq_ind T t0 (\lambda (t1: T).(arity g (CHead c
-(Bind Abst) u) t1 a2)) H3 (lift h (S x) x2) H9) in (let H12 \def (eq_ind T u
-(\lambda (t1: T).(arity g (CHead c (Bind Abst) t1) (lift h (S x) x2) a2)) H11
-(lift h x x1) H8) in (let H13 \def (eq_ind T u (\lambda (t1: T).(\forall (x3:
-nat).(\forall (x4: T).((eq T (lift h (S x) x2) (lift h x3 x4)) \to (\forall
-(c3: C).((drop h x3 (CHead c (Bind Abst) t1) c3) \to (arity g c3 x4 a2)))))))
-H10 (lift h x x1) H8) in (let H14 \def (eq_ind T u (\lambda (t1: T).(\forall
-(x3: nat).(\forall (x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3:
-C).((drop h x3 c c3) \to (arity g c3 x4 (asucc g a1)))))))) H2 (lift h x x1)
-H8) in (let H15 \def (eq_ind T u (\lambda (t1: T).(arity g c t1 (asucc g
-a1))) H1 (lift h x x1) H8) in (arity_head g c2 x1 a1 (H14 x x1 (refl_equal T
-(lift h x x1)) c2 H6) x2 a2 (H13 (S x) x2 (refl_equal T (lift h (S x) x2))
-(CHead c2 (Bind Abst) x1) (drop_skip_bind h x c c2 H6 Abst x1))))))))) x0
-H7)))))) (lift_gen_bind Abst u t0 x0 h x H5)))))))))))))))) (\lambda (c:
-C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H1: (arity g c u a1)).(\lambda
-(H2: ((\forall (x: nat).(\forall (x0: T).((eq T u (lift h x x0)) \to (\forall
-(c2: C).((drop h x c c2) \to (arity g c2 x0 a1)))))))).(\lambda (t0:
-T).(\lambda (a2: A).(\lambda (H3: (arity g c t0 (AHead a1 a2))).(\lambda (H4:
-((\forall (x: nat).(\forall (x0: T).((eq T t0 (lift h x x0)) \to (\forall
-(c2: C).((drop h x c c2) \to (arity g c2 x0 (AHead a1 a2))))))))).(\lambda
-(x: nat).(\lambda (x0: T).(\lambda (H5: (eq T (THead (Flat Appl) u t0) (lift
-h x x0))).(\lambda (c2: C).(\lambda (H6: (drop h x c c2)).(ex3_2_ind T T
-(\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Flat Appl) y0 z))))
-(\lambda (y0: T).(\lambda (_: T).(eq T u (lift h x y0)))) (\lambda (_:
-T).(\lambda (z: T).(eq T t0 (lift h x z)))) (arity g c2 x0 a2) (\lambda (x1:
-T).(\lambda (x2: T).(\lambda (H7: (eq T x0 (THead (Flat Appl) x1
-x2))).(\lambda (H8: (eq T u (lift h x x1))).(\lambda (H9: (eq T t0 (lift h x
-x2))).(eq_ind_r T (THead (Flat Appl) x1 x2) (\lambda (t1: T).(arity g c2 t1
-a2)) (let H10 \def (eq_ind T t0 (\lambda (t1: T).(\forall (x3: nat).(\forall
-(x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to
-(arity g c3 x4 (AHead a1 a2)))))))) H4 (lift h x x2) H9) in (let H11 \def
-(eq_ind T t0 (\lambda (t1: T).(arity g c t1 (AHead a1 a2))) H3 (lift h x x2)
-H9) in (let H12 \def (eq_ind T u (\lambda (t1: T).(\forall (x3: nat).(\forall
-(x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to
-(arity g c3 x4 a1))))))) H2 (lift h x x1) H8) in (let H13 \def (eq_ind T u
-(\lambda (t1: T).(arity g c t1 a1)) H1 (lift h x x1) H8) in (arity_appl g c2
-x1 a1 (H12 x x1 (refl_equal T (lift h x x1)) c2 H6) x2 a2 (H10 x x2
-(refl_equal T (lift h x x2)) c2 H6)))))) x0 H7)))))) (lift_gen_flat Appl u t0
-x0 h x H5)))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a0:
-A).(\lambda (H1: (arity g c u (asucc g a0))).(\lambda (H2: ((\forall (x:
-nat).(\forall (x0: T).((eq T u (lift h x x0)) \to (\forall (c2: C).((drop h x
-c c2) \to (arity g c2 x0 (asucc g a0))))))))).(\lambda (t0: T).(\lambda (H3:
-(arity g c t0 a0)).(\lambda (H4: ((\forall (x: nat).(\forall (x0: T).((eq T
-t0 (lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to (arity g c2 x0
-a0)))))))).(\lambda (x: nat).(\lambda (x0: T).(\lambda (H5: (eq T (THead
-(Flat Cast) u t0) (lift h x x0))).(\lambda (c2: C).(\lambda (H6: (drop h x c
-c2)).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Flat
-Cast) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u (lift h x y0))))
-(\lambda (_: T).(\lambda (z: T).(eq T t0 (lift h x z)))) (arity g c2 x0 a0)
-(\lambda (x1: T).(\lambda (x2: T).(\lambda (H7: (eq T x0 (THead (Flat Cast)
-x1 x2))).(\lambda (H8: (eq T u (lift h x x1))).(\lambda (H9: (eq T t0 (lift h
-x x2))).(eq_ind_r T (THead (Flat Cast) x1 x2) (\lambda (t1: T).(arity g c2 t1
-a0)) (let H10 \def (eq_ind T t0 (\lambda (t1: T).(\forall (x3: nat).(\forall
-(x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to
-(arity g c3 x4 a0))))))) H4 (lift h x x2) H9) in (let H11 \def (eq_ind T t0
-(\lambda (t1: T).(arity g c t1 a0)) H3 (lift h x x2) H9) in (let H12 \def
-(eq_ind T u (\lambda (t1: T).(\forall (x3: nat).(\forall (x4: T).((eq T t1
-(lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to (arity g c3 x4
-(asucc g a0)))))))) H2 (lift h x x1) H8) in (let H13 \def (eq_ind T u
-(\lambda (t1: T).(arity g c t1 (asucc g a0))) H1 (lift h x x1) H8) in
-(arity_cast g c2 x1 a0 (H12 x x1 (refl_equal T (lift h x x1)) c2 H6) x2 (H10
-x x2 (refl_equal T (lift h x x2)) c2 H6)))))) x0 H7)))))) (lift_gen_flat Cast
-u t0 x0 h x H5))))))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda (a1:
-A).(\lambda (_: (arity g c t0 a1)).(\lambda (H2: ((\forall (x: nat).(\forall
-(x0: T).((eq T t0 (lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to
-(arity g c2 x0 a1)))))))).(\lambda (a2: A).(\lambda (H3: (leq g a1
-a2)).(\lambda (x: nat).(\lambda (x0: T).(\lambda (H4: (eq T t0 (lift h x
-x0))).(\lambda (c2: C).(\lambda (H5: (drop h x c c2)).(arity_repl g c2 x0 a1
-(H2 x x0 H4 c2 H5) a2 H3))))))))))))) c1 y a H0))))) H))))))).
-
-theorem arity_mono:
- \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a1: A).((arity g c
-t a1) \to (\forall (a2: A).((arity g c t a2) \to (leq g a1 a2)))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a1: A).(\lambda (H:
-(arity g c t a1)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (a:
-A).(\forall (a2: A).((arity g c0 t0 a2) \to (leq g a a2)))))) (\lambda (c0:
-C).(\lambda (n: nat).(\lambda (a2: A).(\lambda (H0: (arity g c0 (TSort n)
-a2)).(leq_sym g a2 (ASort O n) (arity_gen_sort g c0 n a2 H0)))))) (\lambda
-(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl
-i c0 (CHead d (Bind Abbr) u))).(\lambda (a: A).(\lambda (_: (arity g d u
-a)).(\lambda (H2: ((\forall (a2: A).((arity g d u a2) \to (leq g a
-a2))))).(\lambda (a2: A).(\lambda (H3: (arity g c0 (TLRef i) a2)).(let H4
-\def (arity_gen_lref g c0 i a2 H3) in (or_ind (ex2_2 C T (\lambda (d0:
-C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abbr) u0)))) (\lambda (d0:
-C).(\lambda (u0: T).(arity g d0 u0 a2)))) (ex2_2 C T (\lambda (d0:
-C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda (d0:
-C).(\lambda (u0: T).(arity g d0 u0 (asucc g a2))))) (leq g a a2) (\lambda
-(H5: (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind
-Abbr) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0
-a2))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0
-(Bind Abbr) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a2)))
-(leq g a a2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl i c0
-(CHead x0 (Bind Abbr) x1))).(\lambda (H7: (arity g x0 x1 a2)).(let H8 \def
-(eq_ind C (CHead d (Bind Abbr) u) (\lambda (c1: C).(getl i c0 c1)) H0 (CHead
-x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead x0 (Bind
-Abbr) x1) H6)) in (let H9 \def (f_equal C C (\lambda (e: C).(match e with
-[(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind
-Abbr) u) (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr) u) i H0
-(CHead x0 (Bind Abbr) x1) H6)) in ((let H10 \def (f_equal C T (\lambda (e:
-C).(match e with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0]))
-(CHead d (Bind Abbr) u) (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d
-(Bind Abbr) u) i H0 (CHead x0 (Bind Abbr) x1) H6)) in (\lambda (H11: (eq C d
-x0)).(let H12 \def (eq_ind_r T x1 (\lambda (t0: T).(getl i c0 (CHead x0 (Bind
-Abbr) t0))) H8 u H10) in (let H13 \def (eq_ind_r T x1 (\lambda (t0: T).(arity
-g x0 t0 a2)) H7 u H10) in (let H14 \def (eq_ind_r C x0 (\lambda (c1: C).(getl
-i c0 (CHead c1 (Bind Abbr) u))) H12 d H11) in (let H15 \def (eq_ind_r C x0
-(\lambda (c1: C).(arity g c1 u a2)) H13 d H11) in (H2 a2 H15))))))) H9)))))))
-H5)) (\lambda (H5: (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0
-(CHead d0 (Bind Abst) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0
-(asucc g a2)))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0
-(CHead d0 (Bind Abst) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0
-(asucc g a2)))) (leq g a a2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6:
-(getl i c0 (CHead x0 (Bind Abst) x1))).(\lambda (_: (arity g x0 x1 (asucc g
-a2))).(let H8 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c1: C).(getl i
-c0 c1)) H0 (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abbr) u) i
-H0 (CHead x0 (Bind Abst) x1) H6)) 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 x0 (Bind Abst) x1) (getl_mono c0
-(CHead d (Bind Abbr) u) i H0 (CHead x0 (Bind Abst) x1) H6)) in (False_ind
-(leq g a a2) H9))))))) H5)) H4)))))))))))) (\lambda (c0: C).(\lambda (d:
-C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind
-Abst) u))).(\lambda (a: A).(\lambda (_: (arity g d u (asucc g a))).(\lambda
-(H2: ((\forall (a2: A).((arity g d u a2) \to (leq g (asucc g a)
-a2))))).(\lambda (a2: A).(\lambda (H3: (arity g c0 (TLRef i) a2)).(let H4
-\def (arity_gen_lref g c0 i a2 H3) in (or_ind (ex2_2 C T (\lambda (d0:
-C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abbr) u0)))) (\lambda (d0:
-C).(\lambda (u0: T).(arity g d0 u0 a2)))) (ex2_2 C T (\lambda (d0:
-C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda (d0:
-C).(\lambda (u0: T).(arity g d0 u0 (asucc g a2))))) (leq g a a2) (\lambda
-(H5: (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind
-Abbr) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0
-a2))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0
-(Bind Abbr) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a2)))
-(leq g a a2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl i c0
-(CHead x0 (Bind Abbr) x1))).(\lambda (_: (arity g x0 x1 a2)).(let H8 \def
-(eq_ind C (CHead d (Bind Abst) u) (\lambda (c1: C).(getl i c0 c1)) H0 (CHead
-x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead x0 (Bind
-Abbr) x1) H6)) 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 x0 (Bind Abbr) x1) (getl_mono c0 (CHead d
-(Bind Abst) u) i H0 (CHead x0 (Bind Abbr) x1) H6)) in (False_ind (leq g a a2)
-H9))))))) H5)) (\lambda (H5: (ex2_2 C T (\lambda (d0: C).(\lambda (u0:
-T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda (d0: C).(\lambda (u0:
-T).(arity g d0 u0 (asucc g a2)))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda
-(u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda (d0: C).(\lambda
-(u0: T).(arity g d0 u0 (asucc g a2)))) (leq g a a2) (\lambda (x0: C).(\lambda
-(x1: T).(\lambda (H6: (getl i c0 (CHead x0 (Bind Abst) x1))).(\lambda (H7:
-(arity g x0 x1 (asucc g a2))).(let H8 \def (eq_ind C (CHead d (Bind Abst) u)
-(\lambda (c1: C).(getl i c0 c1)) H0 (CHead x0 (Bind Abst) x1) (getl_mono c0
-(CHead d (Bind Abst) u) i H0 (CHead x0 (Bind Abst) x1) H6)) in (let H9 \def
-(f_equal C C (\lambda (e: C).(match e with [(CSort _) \Rightarrow d | (CHead
-c1 _ _) \Rightarrow c1])) (CHead d (Bind Abst) u) (CHead x0 (Bind Abst) x1)
-(getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead x0 (Bind Abst) x1) H6)) in
-((let H10 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _)
-\Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abst) u)
-(CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead
-x0 (Bind Abst) x1) H6)) in (\lambda (H11: (eq C d x0)).(let H12 \def
-(eq_ind_r T x1 (\lambda (t0: T).(getl i c0 (CHead x0 (Bind Abst) t0))) H8 u
-H10) in (let H13 \def (eq_ind_r T x1 (\lambda (t0: T).(arity g x0 t0 (asucc g
-a2))) H7 u H10) in (let H14 \def (eq_ind_r C x0 (\lambda (c1: C).(getl i c0
-(CHead c1 (Bind Abst) u))) H12 d H11) in (let H15 \def (eq_ind_r C x0
-(\lambda (c1: C).(arity g c1 u (asucc g a2))) H13 d H11) in (asucc_inj g a a2
-(H2 (asucc g a2) H15)))))))) H9))))))) H5)) H4)))))))))))) (\lambda (b:
-B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c0: C).(\lambda (u:
-T).(\lambda (a2: A).(\lambda (_: (arity g c0 u a2)).(\lambda (_: ((\forall
-(a3: A).((arity g c0 u a3) \to (leq g a2 a3))))).(\lambda (t0: T).(\lambda
-(a3: A).(\lambda (_: (arity g (CHead c0 (Bind b) u) t0 a3)).(\lambda (H4:
-((\forall (a4: A).((arity g (CHead c0 (Bind b) u) t0 a4) \to (leq g a3
-a4))))).(\lambda (a0: A).(\lambda (H5: (arity g c0 (THead (Bind b) u t0)
-a0)).(let H6 \def (arity_gen_bind b H0 g c0 u t0 a0 H5) in (ex2_ind A
-(\lambda (a4: A).(arity g c0 u a4)) (\lambda (_: A).(arity g (CHead c0 (Bind
-b) u) t0 a0)) (leq g a3 a0) (\lambda (x: A).(\lambda (_: (arity g c0 u
-x)).(\lambda (H8: (arity g (CHead c0 (Bind b) u) t0 a0)).(H4 a0 H8))))
-H6))))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a2: A).(\lambda
-(_: (arity g c0 u (asucc g a2))).(\lambda (H1: ((\forall (a3: A).((arity g c0
-u a3) \to (leq g (asucc g a2) a3))))).(\lambda (t0: T).(\lambda (a3:
-A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t0 a3)).(\lambda (H3:
-((\forall (a4: A).((arity g (CHead c0 (Bind Abst) u) t0 a4) \to (leq g a3
-a4))))).(\lambda (a0: A).(\lambda (H4: (arity g c0 (THead (Bind Abst) u t0)
-a0)).(let H5 \def (arity_gen_abst g c0 u t0 a0 H4) in (ex3_2_ind A A (\lambda
-(a4: A).(\lambda (a5: A).(eq A a0 (AHead a4 a5)))) (\lambda (a4: A).(\lambda
-(_: A).(arity g c0 u (asucc g a4)))) (\lambda (_: A).(\lambda (a5: A).(arity
-g (CHead c0 (Bind Abst) u) t0 a5))) (leq g (AHead a2 a3) a0) (\lambda (x0:
-A).(\lambda (x1: A).(\lambda (H6: (eq A a0 (AHead x0 x1))).(\lambda (H7:
-(arity g c0 u (asucc g x0))).(\lambda (H8: (arity g (CHead c0 (Bind Abst) u)
-t0 x1)).(eq_ind_r A (AHead x0 x1) (\lambda (a: A).(leq g (AHead a2 a3) a))
-(leq_head g a2 x0 (asucc_inj g a2 x0 (H1 (asucc g x0) H7)) a3 x1 (H3 x1 H8))
-a0 H6)))))) H5))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a2:
-A).(\lambda (_: (arity g c0 u a2)).(\lambda (_: ((\forall (a3: A).((arity g
-c0 u a3) \to (leq g a2 a3))))).(\lambda (t0: T).(\lambda (a3: A).(\lambda (_:
-(arity g c0 t0 (AHead a2 a3))).(\lambda (H3: ((\forall (a4: A).((arity g c0
-t0 a4) \to (leq g (AHead a2 a3) a4))))).(\lambda (a0: A).(\lambda (H4: (arity
-g c0 (THead (Flat Appl) u t0) a0)).(let H5 \def (arity_gen_appl g c0 u t0 a0
-H4) in (ex2_ind A (\lambda (a4: A).(arity g c0 u a4)) (\lambda (a4: A).(arity
-g c0 t0 (AHead a4 a0))) (leq g a3 a0) (\lambda (x: A).(\lambda (_: (arity g
-c0 u x)).(\lambda (H7: (arity g c0 t0 (AHead x a0))).(ahead_inj_snd g a2 a3 x
-a0 (H3 (AHead x a0) H7))))) H5))))))))))))) (\lambda (c0: C).(\lambda (u:
-T).(\lambda (a: A).(\lambda (_: (arity g c0 u (asucc g a))).(\lambda (_:
-((\forall (a2: A).((arity g c0 u a2) \to (leq g (asucc g a) a2))))).(\lambda
-(t0: T).(\lambda (_: (arity g c0 t0 a)).(\lambda (H3: ((\forall (a2:
-A).((arity g c0 t0 a2) \to (leq g a a2))))).(\lambda (a2: A).(\lambda (H4:
-(arity g c0 (THead (Flat Cast) u t0) a2)).(let H5 \def (arity_gen_cast g c0 u
-t0 a2 H4) in (land_ind (arity g c0 u (asucc g a2)) (arity g c0 t0 a2) (leq g
-a a2) (\lambda (_: (arity g c0 u (asucc g a2))).(\lambda (H7: (arity g c0 t0
-a2)).(H3 a2 H7))) H5)))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda
-(a2: A).(\lambda (_: (arity g c0 t0 a2)).(\lambda (H1: ((\forall (a3:
-A).((arity g c0 t0 a3) \to (leq g a2 a3))))).(\lambda (a3: A).(\lambda (H2:
-(leq g a2 a3)).(\lambda (a0: A).(\lambda (H3: (arity g c0 t0 a0)).(leq_trans
-g a3 a2 (leq_sym g a2 a3 H2) a0 (H1 a0 H3))))))))))) c t a1 H))))).
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projects / helm.git / blobdiff