--- /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/csuba/getl.ma".
+
+include "Basic-1/csuba/props.ma".
+
+include "Basic-1/arity/props.ma".
+
+include "Basic-1/csubv/getl.ma".
+
+theorem csuba_arity:
+ \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1
+t a) \to (\forall (c2: C).((csuba g c1 c2) \to (arity g c2 t a)))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H:
+(arity g c1 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0:
+A).(\forall (c2: C).((csuba g c c2) \to (arity g c2 t0 a0)))))) (\lambda (c:
+C).(\lambda (n: nat).(\lambda (c2: C).(\lambda (_: (csuba g c
+c2)).(arity_sort g c2 n))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u:
+T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d (Bind Abbr)
+u))).(\lambda (a0: A).(\lambda (_: (arity g d u a0)).(\lambda (H2: ((\forall
+(c2: C).((csuba g d c2) \to (arity g c2 u a0))))).(\lambda (c2: C).(\lambda
+(H3: (csuba g c c2)).(let H4 \def (csuba_getl_abbr g c d u i H0 c2 H3) in
+(ex2_ind C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda
+(d2: C).(csuba g d d2)) (arity g c2 (TLRef i) a0) (\lambda (x: C).(\lambda
+(H5: (getl i c2 (CHead x (Bind Abbr) u))).(\lambda (H6: (csuba g d
+x)).(arity_abbr g c2 x u i H5 a0 (H2 x H6))))) H4)))))))))))) (\lambda (c:
+C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c
+(CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (H1: (arity g d u (asucc
+g a0))).(\lambda (H2: ((\forall (c2: C).((csuba g d c2) \to (arity g c2 u
+(asucc g a0)))))).(\lambda (c2: C).(\lambda (H3: (csuba g c c2)).(let H4 \def
+(csuba_getl_abst g c d u i H0 c2 H3) in (or_ind (ex2 C (\lambda (d2: C).(getl
+i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d d2))) (ex4_3 C T
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2
+(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d u (asucc
+g a1))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2
+a1))))) (arity g c2 (TLRef i) a0) (\lambda (H5: (ex2 C (\lambda (d2: C).(getl
+i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d d2)))).(ex2_ind C
+(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2:
+C).(csuba g d d2)) (arity g c2 (TLRef i) a0) (\lambda (x: C).(\lambda (H6:
+(getl i c2 (CHead x (Bind Abst) u))).(\lambda (H7: (csuba g d x)).(arity_abst
+g c2 x u i H6 a0 (H2 x H7))))) H5)) (\lambda (H5: (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d d2)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d u (asucc g a1)))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2
+a1)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
+A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a1: A).(arity g d u (asucc g a1))))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a1: A).(arity g d2 u2 a1)))) (arity g c2 (TLRef i) a0) (\lambda
+(x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H6: (getl i c2 (CHead x0
+(Bind Abbr) x1))).(\lambda (_: (csuba g d x0)).(\lambda (H8: (arity g d u
+(asucc g x2))).(\lambda (H9: (arity g x0 x1 x2)).(arity_repl g c2 (TLRef i)
+x2 (arity_abbr g c2 x0 x1 i H6 x2 H9) a0 (asucc_inj g x2 a0 (arity_mono g d u
+(asucc g x2) H8 (asucc g a0) H1)))))))))) H5)) H4)))))))))))) (\lambda (b:
+B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c: C).(\lambda (u:
+T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H2: ((\forall
+(c2: C).((csuba g c c2) \to (arity g c2 u a1))))).(\lambda (t0: T).(\lambda
+(a2: A).(\lambda (_: (arity g (CHead c (Bind b) u) t0 a2)).(\lambda (H4:
+((\forall (c2: C).((csuba g (CHead c (Bind b) u) c2) \to (arity g c2 t0
+a2))))).(\lambda (c2: C).(\lambda (H5: (csuba g c c2)).(arity_bind g b H0 c2
+u a1 (H2 c2 H5) t0 a2 (H4 (CHead c2 (Bind b) u) (csuba_head g c c2 H5 (Bind
+b) u)))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a1:
+A).(\lambda (_: (arity g c u (asucc g a1))).(\lambda (H1: ((\forall (c2:
+C).((csuba g c c2) \to (arity g c2 u (asucc g a1)))))).(\lambda (t0:
+T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c (Bind Abst) u) t0
+a2)).(\lambda (H3: ((\forall (c2: C).((csuba g (CHead c (Bind Abst) u) c2)
+\to (arity g c2 t0 a2))))).(\lambda (c2: C).(\lambda (H4: (csuba g c
+c2)).(arity_head g c2 u a1 (H1 c2 H4) t0 a2 (H3 (CHead c2 (Bind Abst) u)
+(csuba_head g c c2 H4 (Bind Abst) u)))))))))))))) (\lambda (c: C).(\lambda
+(u: T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H1:
+((\forall (c2: C).((csuba g c c2) \to (arity g c2 u a1))))).(\lambda (t0:
+T).(\lambda (a2: A).(\lambda (_: (arity g c t0 (AHead a1 a2))).(\lambda (H3:
+((\forall (c2: C).((csuba g c c2) \to (arity g c2 t0 (AHead a1
+a2)))))).(\lambda (c2: C).(\lambda (H4: (csuba g c c2)).(arity_appl g c2 u a1
+(H1 c2 H4) t0 a2 (H3 c2 H4))))))))))))) (\lambda (c: C).(\lambda (u:
+T).(\lambda (a0: A).(\lambda (_: (arity g c u (asucc g a0))).(\lambda (H1:
+((\forall (c2: C).((csuba g c c2) \to (arity g c2 u (asucc g
+a0)))))).(\lambda (t0: T).(\lambda (_: (arity g c t0 a0)).(\lambda (H3:
+((\forall (c2: C).((csuba g c c2) \to (arity g c2 t0 a0))))).(\lambda (c2:
+C).(\lambda (H4: (csuba g c c2)).(arity_cast g c2 u a0 (H1 c2 H4) t0 (H3 c2
+H4)))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda (_:
+(arity g c t0 a1)).(\lambda (H1: ((\forall (c2: C).((csuba g c c2) \to (arity
+g c2 t0 a1))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda (c2:
+C).(\lambda (H3: (csuba g c c2)).(arity_repl g c2 t0 a1 (H1 c2 H3) a2
+H2)))))))))) c1 t a H))))).
+(* COMMENTS
+Initial nodes: 1505
+END *)
+
+theorem csuba_arity_rev:
+ \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1
+t a) \to (\forall (c2: C).((csuba g c2 c1) \to ((csubv c2 c1) \to (arity g c2
+t a))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H:
+(arity g c1 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0:
+A).(\forall (c2: C).((csuba g c2 c) \to ((csubv c2 c) \to (arity g c2 t0
+a0))))))) (\lambda (c: C).(\lambda (n: nat).(\lambda (c2: C).(\lambda (_:
+(csuba g c2 c)).(\lambda (_: (csubv c2 c)).(arity_sort g c2 n)))))) (\lambda
+(c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl
+i c (CHead d (Bind Abbr) u))).(\lambda (a0: A).(\lambda (H1: (arity g d u
+a0)).(\lambda (H2: ((\forall (c2: C).((csuba g c2 d) \to ((csubv c2 d) \to
+(arity g c2 u a0)))))).(\lambda (c2: C).(\lambda (H3: (csuba g c2
+c)).(\lambda (H4: (csubv c2 c)).(let H_x \def (csuba_getl_abbr_rev g c d u i
+H0 c2 H3) in (let H5 \def H_x in (or3_ind (ex2 C (\lambda (d2: C).(getl i c2
+(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d2 d))) (ex4_3 C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
+d)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2
+(asucc g a1))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d
+u a1))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d)))) (arity
+g c2 (TLRef i) a0) (\lambda (H6: (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2
+(Bind Abbr) u))) (\lambda (d2: C).(csuba g d2 d)))).(ex2_ind C (\lambda (d2:
+C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d2 d))
+(arity g c2 (TLRef i) a0) (\lambda (x: C).(\lambda (H7: (getl i c2 (CHead x
+(Bind Abbr) u))).(\lambda (H8: (csuba g x d)).(let H_x0 \def (csubv_getl_conf
+c2 c H4 Abbr x u i H7) in (let H9 \def H_x0 in (ex2_3_ind B C T (\lambda (_:
+B).(\lambda (d2: C).(\lambda (_: T).(csubv x d2)))) (\lambda (b2: B).(\lambda
+(d2: C).(\lambda (v2: T).(getl i c (CHead d2 (Bind b2) v2))))) (arity g c2
+(TLRef i) a0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda
+(H10: (csubv x x1)).(\lambda (H11: (getl i c (CHead x1 (Bind x0) x2))).(let
+H12 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c0: C).(getl i c c0)) H0
+(CHead x1 (Bind x0) x2) (getl_mono c (CHead d (Bind Abbr) u) i H0 (CHead x1
+(Bind x0) x2) H11)) in (let H13 \def (f_equal C C (\lambda (e: C).(match e in
+C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _)
+\Rightarrow c0])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) (getl_mono
+c (CHead d (Bind Abbr) u) i H0 (CHead x1 (Bind x0) x2) H11)) in ((let H14
+\def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B)
+with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k in K
+return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
+Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) (getl_mono c (CHead
+d (Bind Abbr) u) i H0 (CHead x1 (Bind x0) x2) H11)) in ((let H15 \def
+(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind
+Abbr) u) (CHead x1 (Bind x0) x2) (getl_mono c (CHead d (Bind Abbr) u) i H0
+(CHead x1 (Bind x0) x2) H11)) in (\lambda (H16: (eq B Abbr x0)).(\lambda
+(H17: (eq C d x1)).(let H18 \def (eq_ind_r T x2 (\lambda (t0: T).(getl i c
+(CHead x1 (Bind x0) t0))) H12 u H15) in (let H19 \def (eq_ind_r C x1 (\lambda
+(c0: C).(getl i c (CHead c0 (Bind x0) u))) H18 d H17) in (let H20 \def
+(eq_ind_r C x1 (\lambda (c0: C).(csubv x c0)) H10 d H17) in (let H21 \def
+(eq_ind_r B x0 (\lambda (b: B).(getl i c (CHead d (Bind b) u))) H19 Abbr H16)
+in (arity_abbr g c2 x u i H7 a0 (H2 x H8 H20))))))))) H14)) H13))))))))
+H9)))))) H6)) (\lambda (H6: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2 (asucc g a1))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d u a1)))))).(ex4_3_ind C T
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d2 d)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2
+(asucc g a1))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d
+u a1)))) (arity g c2 (TLRef i) a0) (\lambda (x0: C).(\lambda (x1: T).(\lambda
+(x2: A).(\lambda (H7: (getl i c2 (CHead x0 (Bind Abst) x1))).(\lambda (_:
+(csuba g x0 d)).(\lambda (H9: (arity g x0 x1 (asucc g x2))).(\lambda (H10:
+(arity g d u x2)).(arity_repl g c2 (TLRef i) x2 (arity_abst g c2 x0 x1 i H7
+x2 H9) a0 (arity_mono g d u x2 H10 a0 H1))))))))) H6)) (\lambda (H6: (ex2_2 C
+T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d))))).(ex2_2_ind C T (\lambda
+(d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda
+(d2: C).(\lambda (_: T).(csuba g d2 d))) (arity g c2 (TLRef i) a0) (\lambda
+(x0: C).(\lambda (x1: T).(\lambda (H7: (getl i c2 (CHead x0 (Bind Void)
+x1))).(\lambda (_: (csuba g x0 d)).(let H_x0 \def (csubv_getl_conf_void c2 c
+H4 x0 x1 i H7) in (let H9 \def H_x0 in (ex2_2_ind C T (\lambda (d2:
+C).(\lambda (_: T).(csubv x0 d2))) (\lambda (d2: C).(\lambda (v2: T).(getl i
+c (CHead d2 (Bind Void) v2)))) (arity g c2 (TLRef i) a0) (\lambda (x2:
+C).(\lambda (x3: T).(\lambda (_: (csubv x0 x2)).(\lambda (H11: (getl i c
+(CHead x2 (Bind Void) x3))).(let H12 \def (eq_ind C (CHead d (Bind Abbr) u)
+(\lambda (c0: C).(getl i c c0)) H0 (CHead x2 (Bind Void) x3) (getl_mono c
+(CHead d (Bind Abbr) u) i H0 (CHead x2 (Bind Void) x3) H11)) in (let H13 \def
+(eq_ind C (CHead d (Bind Abbr) u) (\lambda (ee: C).(match ee in C return
+(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _)
+\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b)
+\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow
+True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _)
+\Rightarrow False])])) I (CHead x2 (Bind Void) x3) (getl_mono c (CHead d
+(Bind Abbr) u) i H0 (CHead x2 (Bind Void) x3) H11)) in (False_ind (arity g c2
+(TLRef i) a0) H13))))))) H9))))))) H6)) H5)))))))))))))) (\lambda (c:
+C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c
+(CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (_: (arity g d u (asucc g
+a0))).(\lambda (H2: ((\forall (c2: C).((csuba g c2 d) \to ((csubv c2 d) \to
+(arity g c2 u (asucc g a0))))))).(\lambda (c2: C).(\lambda (H3: (csuba g c2
+c)).(\lambda (H4: (csubv c2 c)).(let H_x \def (csuba_getl_abst_rev g c d u i
+H0 c2 H3) in (let H5 \def H_x in (or_ind (ex2 C (\lambda (d2: C).(getl i c2
+(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d))) (ex2_2 C T
+(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d)))) (arity g c2 (TLRef i) a0)
+(\lambda (H6: (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u)))
+(\lambda (d2: C).(csuba g d2 d)))).(ex2_ind C (\lambda (d2: C).(getl i c2
+(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d)) (arity g c2
+(TLRef i) a0) (\lambda (x: C).(\lambda (H7: (getl i c2 (CHead x (Bind Abst)
+u))).(\lambda (H8: (csuba g x d)).(let H_x0 \def (csubv_getl_conf c2 c H4
+Abst x u i H7) in (let H9 \def H_x0 in (ex2_3_ind B C T (\lambda (_:
+B).(\lambda (d2: C).(\lambda (_: T).(csubv x d2)))) (\lambda (b2: B).(\lambda
+(d2: C).(\lambda (v2: T).(getl i c (CHead d2 (Bind b2) v2))))) (arity g c2
+(TLRef i) a0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda
+(H10: (csubv x x1)).(\lambda (H11: (getl i c (CHead x1 (Bind x0) x2))).(let
+H12 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (c0: C).(getl i c c0)) H0
+(CHead x1 (Bind x0) x2) (getl_mono c (CHead d (Bind Abst) u) i H0 (CHead x1
+(Bind x0) x2) H11)) in (let H13 \def (f_equal C C (\lambda (e: C).(match e in
+C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _)
+\Rightarrow c0])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x2) (getl_mono
+c (CHead d (Bind Abst) u) i H0 (CHead x1 (Bind x0) x2) H11)) in ((let H14
+\def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B)
+with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match k in K
+return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
+Abst])])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x2) (getl_mono c (CHead
+d (Bind Abst) u) i H0 (CHead x1 (Bind x0) x2) H11)) in ((let H15 \def
+(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind
+Abst) u) (CHead x1 (Bind x0) x2) (getl_mono c (CHead d (Bind Abst) u) i H0
+(CHead x1 (Bind x0) x2) H11)) in (\lambda (H16: (eq B Abst x0)).(\lambda
+(H17: (eq C d x1)).(let H18 \def (eq_ind_r T x2 (\lambda (t0: T).(getl i c
+(CHead x1 (Bind x0) t0))) H12 u H15) in (let H19 \def (eq_ind_r C x1 (\lambda
+(c0: C).(getl i c (CHead c0 (Bind x0) u))) H18 d H17) in (let H20 \def
+(eq_ind_r C x1 (\lambda (c0: C).(csubv x c0)) H10 d H17) in (let H21 \def
+(eq_ind_r B x0 (\lambda (b: B).(getl i c (CHead d (Bind b) u))) H19 Abst H16)
+in (arity_abst g c2 x u i H7 a0 (H2 x H8 H20))))))))) H14)) H13))))))))
+H9)))))) H6)) (\lambda (H6: (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g d2 d))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(getl
+i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g
+d2 d))) (arity g c2 (TLRef i) a0) (\lambda (x0: C).(\lambda (x1: T).(\lambda
+(H7: (getl i c2 (CHead x0 (Bind Void) x1))).(\lambda (_: (csuba g x0 d)).(let
+H_x0 \def (csubv_getl_conf_void c2 c H4 x0 x1 i H7) in (let H9 \def H_x0 in
+(ex2_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csubv x0 d2))) (\lambda (d2:
+C).(\lambda (v2: T).(getl i c (CHead d2 (Bind Void) v2)))) (arity g c2 (TLRef
+i) a0) (\lambda (x2: C).(\lambda (x3: T).(\lambda (_: (csubv x0 x2)).(\lambda
+(H11: (getl i c (CHead x2 (Bind Void) x3))).(let H12 \def (eq_ind C (CHead d
+(Bind Abst) u) (\lambda (c0: C).(getl i c c0)) H0 (CHead x2 (Bind Void) x3)
+(getl_mono c (CHead d (Bind Abst) u) i H0 (CHead x2 (Bind Void) x3) H11)) in
+(let H13 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (ee: C).(match ee in
+C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k
+_) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b)
+\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow
+False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _)
+\Rightarrow False])])) I (CHead x2 (Bind Void) x3) (getl_mono c (CHead d
+(Bind Abst) u) i H0 (CHead x2 (Bind Void) x3) H11)) in (False_ind (arity g c2
+(TLRef i) a0) H13))))))) H9))))))) H6)) H5)))))))))))))) (\lambda (b:
+B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c: C).(\lambda (u:
+T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H2: ((\forall
+(c2: C).((csuba g c2 c) \to ((csubv c2 c) \to (arity g c2 u a1)))))).(\lambda
+(t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c (Bind b) u) t0
+a2)).(\lambda (H4: ((\forall (c2: C).((csuba g c2 (CHead c (Bind b) u)) \to
+((csubv c2 (CHead c (Bind b) u)) \to (arity g c2 t0 a2)))))).(\lambda (c2:
+C).(\lambda (H5: (csuba g c2 c)).(\lambda (H6: (csubv c2 c)).(arity_bind g b
+H0 c2 u a1 (H2 c2 H5 H6) t0 a2 (H4 (CHead c2 (Bind b) u) (csuba_head g c2 c
+H5 (Bind b) u) (csubv_bind_same c2 c H6 b u u))))))))))))))))) (\lambda (c:
+C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c u (asucc g
+a1))).(\lambda (H1: ((\forall (c2: C).((csuba g c2 c) \to ((csubv c2 c) \to
+(arity g c2 u (asucc g a1))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda
+(_: (arity g (CHead c (Bind Abst) u) t0 a2)).(\lambda (H3: ((\forall (c2:
+C).((csuba g c2 (CHead c (Bind Abst) u)) \to ((csubv c2 (CHead c (Bind Abst)
+u)) \to (arity g c2 t0 a2)))))).(\lambda (c2: C).(\lambda (H4: (csuba g c2
+c)).(\lambda (H5: (csubv c2 c)).(arity_head g c2 u a1 (H1 c2 H4 H5) t0 a2 (H3
+(CHead c2 (Bind Abst) u) (csuba_head g c2 c H4 (Bind Abst) u)
+(csubv_bind_same c2 c H5 Abst u u))))))))))))))) (\lambda (c: C).(\lambda (u:
+T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H1: ((\forall
+(c2: C).((csuba g c2 c) \to ((csubv c2 c) \to (arity g c2 u a1)))))).(\lambda
+(t0: T).(\lambda (a2: A).(\lambda (_: (arity g c t0 (AHead a1 a2))).(\lambda
+(H3: ((\forall (c2: C).((csuba g c2 c) \to ((csubv c2 c) \to (arity g c2 t0
+(AHead a1 a2))))))).(\lambda (c2: C).(\lambda (H4: (csuba g c2 c)).(\lambda
+(H5: (csubv c2 c)).(arity_appl g c2 u a1 (H1 c2 H4 H5) t0 a2 (H3 c2 H4
+H5)))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a0: A).(\lambda
+(_: (arity g c u (asucc g a0))).(\lambda (H1: ((\forall (c2: C).((csuba g c2
+c) \to ((csubv c2 c) \to (arity g c2 u (asucc g a0))))))).(\lambda (t0:
+T).(\lambda (_: (arity g c t0 a0)).(\lambda (H3: ((\forall (c2: C).((csuba g
+c2 c) \to ((csubv c2 c) \to (arity g c2 t0 a0)))))).(\lambda (c2: C).(\lambda
+(H4: (csuba g c2 c)).(\lambda (H5: (csubv c2 c)).(arity_cast g c2 u a0 (H1 c2
+H4 H5) t0 (H3 c2 H4 H5))))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda
+(a1: A).(\lambda (_: (arity g c t0 a1)).(\lambda (H1: ((\forall (c2:
+C).((csuba g c2 c) \to ((csubv c2 c) \to (arity g c2 t0 a1)))))).(\lambda
+(a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda (c2: C).(\lambda (H3: (csuba g
+c2 c)).(\lambda (H4: (csubv c2 c)).(arity_repl g c2 t0 a1 (H1 c2 H3 H4) a2
+H2))))))))))) c1 t a H))))).
+(* COMMENTS
+Initial nodes: 3597
+END *)
+
+theorem arity_appls_appl:
+ \forall (g: G).(\forall (c: C).(\forall (v: T).(\forall (a1: A).((arity g c
+v a1) \to (\forall (u: T).((arity g c u (asucc g a1)) \to (\forall (t:
+T).(\forall (vs: TList).(\forall (a2: A).((arity g c (THeads (Flat Appl) vs
+(THead (Bind Abbr) v t)) a2) \to (arity g c (THeads (Flat Appl) vs (THead
+(Flat Appl) v (THead (Bind Abst) u t))) a2)))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (v: T).(\lambda (a1: A).(\lambda (H:
+(arity g c v a1)).(\lambda (u: T).(\lambda (H0: (arity g c u (asucc g
+a1))).(\lambda (t: T).(\lambda (vs: TList).(TList_ind (\lambda (t0:
+TList).(\forall (a2: A).((arity g c (THeads (Flat Appl) t0 (THead (Bind Abbr)
+v t)) a2) \to (arity g c (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead
+(Bind Abst) u t))) a2)))) (\lambda (a2: A).(\lambda (H1: (arity g c (THead
+(Bind Abbr) v t) a2)).(let H_x \def (arity_gen_bind Abbr (\lambda (H2: (eq B
+Abbr Abst)).(not_abbr_abst H2)) g c v t a2 H1) in (let H2 \def H_x in
+(ex2_ind A (\lambda (a3: A).(arity g c v a3)) (\lambda (_: A).(arity g (CHead
+c (Bind Abbr) v) t a2)) (arity g c (THead (Flat Appl) v (THead (Bind Abst) u
+t)) a2) (\lambda (x: A).(\lambda (_: (arity g c v x)).(\lambda (H4: (arity g
+(CHead c (Bind Abbr) v) t a2)).(arity_appl g c v a1 H (THead (Bind Abst) u t)
+a2 (arity_head g c u a1 H0 t a2 (csuba_arity_rev g (CHead c (Bind Abbr) v) t
+a2 H4 (CHead c (Bind Abst) u) (csuba_abst g c c (csuba_refl g c) u a1 H0 v H)
+(csubv_bind c c (csubv_refl c) Abst (sym_not_eq B Void Abst not_void_abst)
+Abbr u v))))))) H2))))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H1:
+((\forall (a2: A).((arity g c (THeads (Flat Appl) t1 (THead (Bind Abbr) v t))
+a2) \to (arity g c (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind
+Abst) u t))) a2))))).(\lambda (a2: A).(\lambda (H2: (arity g c (THead (Flat
+Appl) t0 (THeads (Flat Appl) t1 (THead (Bind Abbr) v t))) a2)).(let H3 \def
+(arity_gen_appl g c t0 (THeads (Flat Appl) t1 (THead (Bind Abbr) v t)) a2 H2)
+in (ex2_ind A (\lambda (a3: A).(arity g c t0 a3)) (\lambda (a3: A).(arity g c
+(THeads (Flat Appl) t1 (THead (Bind Abbr) v t)) (AHead a3 a2))) (arity g c
+(THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead
+(Bind Abst) u t)))) a2) (\lambda (x: A).(\lambda (H4: (arity g c t0
+x)).(\lambda (H5: (arity g c (THeads (Flat Appl) t1 (THead (Bind Abbr) v t))
+(AHead x a2))).(arity_appl g c t0 x H4 (THeads (Flat Appl) t1 (THead (Flat
+Appl) v (THead (Bind Abst) u t))) a2 (H1 (AHead x a2) H5))))) H3)))))))
+vs))))))))).
+(* COMMENTS
+Initial nodes: 687
+END *)
+
projects / helm.git / blobdiff