(* This file was automatically generated: do not edit *********************)
-set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/ty3/arity".
+include "LambdaDelta-1/ty3/pr3_props.ma".
-include "ty3/pr3_props.ma".
+include "LambdaDelta-1/arity/pr3.ma".
-include "arity/pr3.ma".
-
-include "asucc/fwd.ma".
+include "LambdaDelta-1/asucc/fwd.ma".
theorem ty3_arity:
\forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c
a1)))) (\lambda (x0: A).(\lambda (H7: (leq g x (asucc g x0))).(ex_intro2 A
(\lambda (a1: A).(arity g c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0
(lift (S n) O u) (asucc g a1))) x0 (arity_abst g c0 d u n H0 x0 (arity_repl g
-d u x H4 (asucc g x0) H7)) (arity_lift g d u (asucc g x0) (arity_repl g d u x
-H4 (asucc g x0) H7) c0 (S n) O (getl_drop Abst c0 d u n H0))))) H6))))))
-H3)))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (t: T).(\lambda (_:
-(ty3 g c0 u t)).(\lambda (H1: (ex2 A (\lambda (a1: A).(arity g c0 u a1))
-(\lambda (a1: A).(arity g c0 t (asucc g a1))))).(\lambda (b: B).(\lambda (t3:
-T).(\lambda (t4: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t3
-t4)).(\lambda (H3: (ex2 A (\lambda (a1: A).(arity g (CHead c0 (Bind b) u) t3
-a1)) (\lambda (a1: A).(arity g (CHead c0 (Bind b) u) t4 (asucc g
-a1))))).(\lambda (t0: T).(\lambda (H4: (ty3 g (CHead c0 (Bind b) u) t4
-t0)).(\lambda (H5: (ex2 A (\lambda (a1: A).(arity g (CHead c0 (Bind b) u) t4
-a1)) (\lambda (a1: A).(arity g (CHead c0 (Bind b) u) t0 (asucc g a1))))).(let
-H6 \def H1 in (ex2_ind A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1:
-A).(arity g c0 t (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g c0 (THead
-(Bind b) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind b) u t4) (asucc
-g a1)))) (\lambda (x: A).(\lambda (H7: (arity g c0 u x)).(\lambda (_: (arity
-g c0 t (asucc g x))).(let H9 \def H3 in (ex2_ind A (\lambda (a1: A).(arity g
-(CHead c0 (Bind b) u) t3 a1)) (\lambda (a1: A).(arity g (CHead c0 (Bind b) u)
-t4 (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Bind b) u t3)
-a1)) (\lambda (a1: A).(arity g c0 (THead (Bind b) u t4) (asucc g a1))))
-(\lambda (x0: A).(\lambda (H10: (arity g (CHead c0 (Bind b) u) t3
-x0)).(\lambda (H11: (arity g (CHead c0 (Bind b) u) t4 (asucc g x0))).(let H_x
-\def (leq_asucc g x) in (let H12 \def H_x in (ex_ind A (\lambda (a0: A).(leq
-g x (asucc g a0))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Bind b) u t3)
-a1)) (\lambda (a1: A).(arity g c0 (THead (Bind b) u t4) (asucc g a1))))
-(\lambda (x1: A).(\lambda (H13: (leq g x (asucc g x1))).(B_ind (\lambda (b0:
-B).((ty3 g (CHead c0 (Bind b0) u) t4 t0) \to ((ex2 A (\lambda (a1: A).(arity
-g (CHead c0 (Bind b0) u) t4 a1)) (\lambda (a1: A).(arity g (CHead c0 (Bind
-b0) u) t0 (asucc g a1)))) \to ((arity g (CHead c0 (Bind b0) u) t3 x0) \to
+d u x H4 (asucc g x0) H7)) (arity_repl g c0 (lift (S n) O u) x (arity_lift g
+d u x H4 c0 (S n) O (getl_drop Abst c0 d u n H0)) (asucc g x0) H7))))
+H6)))))) H3)))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (t:
+T).(\lambda (_: (ty3 g c0 u t)).(\lambda (H1: (ex2 A (\lambda (a1: A).(arity
+g c0 u a1)) (\lambda (a1: A).(arity g c0 t (asucc g a1))))).(\lambda (b:
+B).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g (CHead c0 (Bind b)
+u) t3 t4)).(\lambda (H3: (ex2 A (\lambda (a1: A).(arity g (CHead c0 (Bind b)
+u) t3 a1)) (\lambda (a1: A).(arity g (CHead c0 (Bind b) u) t4 (asucc g
+a1))))).(let H4 \def H1 in (ex2_ind A (\lambda (a1: A).(arity g c0 u a1))
+(\lambda (a1: A).(arity g c0 t (asucc g a1))) (ex2 A (\lambda (a1: A).(arity
+g c0 (THead (Bind b) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind b)
+u t4) (asucc g a1)))) (\lambda (x: A).(\lambda (H5: (arity g c0 u
+x)).(\lambda (_: (arity g c0 t (asucc g x))).(let H7 \def H3 in (ex2_ind A
+(\lambda (a1: A).(arity g (CHead c0 (Bind b) u) t3 a1)) (\lambda (a1:
+A).(arity g (CHead c0 (Bind b) u) t4 (asucc g a1))) (ex2 A (\lambda (a1:
+A).(arity g c0 (THead (Bind b) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead
+(Bind b) u t4) (asucc g a1)))) (\lambda (x0: A).(\lambda (H8: (arity g (CHead
+c0 (Bind b) u) t3 x0)).(\lambda (H9: (arity g (CHead c0 (Bind b) u) t4 (asucc
+g x0))).(let H_x \def (leq_asucc g x) in (let H10 \def H_x in (ex_ind A
+(\lambda (a0: A).(leq g x (asucc g a0))) (ex2 A (\lambda (a1: A).(arity g c0
+(THead (Bind b) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind b) u t4)
+(asucc g a1)))) (\lambda (x1: A).(\lambda (H11: (leq g x (asucc g
+x1))).(B_ind (\lambda (b0: B).((arity g (CHead c0 (Bind b0) u) t3 x0) \to
((arity g (CHead c0 (Bind b0) u) t4 (asucc g x0)) \to (ex2 A (\lambda (a1:
A).(arity g c0 (THead (Bind b0) u t3) a1)) (\lambda (a1: A).(arity g c0
-(THead (Bind b0) u t4) (asucc g a1))))))))) (\lambda (_: (ty3 g (CHead c0
-(Bind Abbr) u) t4 t0)).(\lambda (_: (ex2 A (\lambda (a1: A).(arity g (CHead
-c0 (Bind Abbr) u) t4 a1)) (\lambda (a1: A).(arity g (CHead c0 (Bind Abbr) u)
-t0 (asucc g a1))))).(\lambda (H16: (arity g (CHead c0 (Bind Abbr) u) t3
-x0)).(\lambda (H17: (arity g (CHead c0 (Bind Abbr) u) t4 (asucc g
-x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Bind Abbr) u t3) a1))
-(\lambda (a1: A).(arity g c0 (THead (Bind Abbr) u t4) (asucc g a1))) x0
-(arity_bind g Abbr not_abbr_abst c0 u x H7 t3 x0 H16) (arity_bind g Abbr
-not_abbr_abst c0 u x H7 t4 (asucc g x0) H17)))))) (\lambda (_: (ty3 g (CHead
-c0 (Bind Abst) u) t4 t0)).(\lambda (_: (ex2 A (\lambda (a1: A).(arity g
-(CHead c0 (Bind Abst) u) t4 a1)) (\lambda (a1: A).(arity g (CHead c0 (Bind
-Abst) u) t0 (asucc g a1))))).(\lambda (H16: (arity g (CHead c0 (Bind Abst) u)
-t3 x0)).(\lambda (H17: (arity g (CHead c0 (Bind Abst) u) t4 (asucc g
-x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Bind Abst) u t3) a1))
-(\lambda (a1: A).(arity g c0 (THead (Bind Abst) u t4) (asucc g a1))) (AHead
-x1 x0) (arity_head g c0 u x1 (arity_repl g c0 u x H7 (asucc g x1) H13) t3 x0
-H16) (arity_repl g c0 (THead (Bind Abst) u t4) (AHead x1 (asucc g x0))
-(arity_head g c0 u x1 (arity_repl g c0 u x H7 (asucc g x1) H13) t4 (asucc g
-x0) H17) (asucc g (AHead x1 x0)) (leq_refl g (asucc g (AHead x1 x0)))))))))
-(\lambda (_: (ty3 g (CHead c0 (Bind Void) u) t4 t0)).(\lambda (_: (ex2 A
-(\lambda (a1: A).(arity g (CHead c0 (Bind Void) u) t4 a1)) (\lambda (a1:
-A).(arity g (CHead c0 (Bind Void) u) t0 (asucc g a1))))).(\lambda (H16:
-(arity g (CHead c0 (Bind Void) u) t3 x0)).(\lambda (H17: (arity g (CHead c0
-(Bind Void) u) t4 (asucc g x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0
-(THead (Bind Void) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind Void)
-u t4) (asucc g a1))) x0 (arity_bind g Void not_void_abst c0 u x H7 t3 x0 H16)
-(arity_bind g Void not_void_abst c0 u x H7 t4 (asucc g x0) H17)))))) b H4 H5
-H10 H11))) H12)))))) H9))))) H6))))))))))))))) (\lambda (c0: C).(\lambda (w:
-T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda (H1: (ex2 A (\lambda
-(a1: A).(arity g c0 w a1)) (\lambda (a1: A).(arity g c0 u (asucc g
-a1))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead (Bind
-Abst) u t))).(\lambda (H3: (ex2 A (\lambda (a1: A).(arity g c0 v a1))
-(\lambda (a1: A).(arity g c0 (THead (Bind Abst) u t) (asucc g a1))))).(let H4
-\def H1 in (ex2_ind A (\lambda (a1: A).(arity g c0 w a1)) (\lambda (a1:
-A).(arity g c0 u (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g c0 (THead
-(Flat Appl) w v) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w
-(THead (Bind Abst) u t)) (asucc g a1)))) (\lambda (x: A).(\lambda (H5: (arity
-g c0 w x)).(\lambda (H6: (arity g c0 u (asucc g x))).(let H7 \def H3 in
-(ex2_ind A (\lambda (a1: A).(arity g c0 v a1)) (\lambda (a1: A).(arity g c0
-(THead (Bind Abst) u t) (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g c0
-(THead (Flat Appl) w v) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Appl)
-w (THead (Bind Abst) u t)) (asucc g a1)))) (\lambda (x0: A).(\lambda (H8:
-(arity g c0 v x0)).(\lambda (H9: (arity g c0 (THead (Bind Abst) u t) (asucc g
-x0))).(let H10 \def (arity_gen_abst g c0 u t (asucc g x0) H9) in (ex3_2_ind A
-A (\lambda (a1: A).(\lambda (a2: A).(eq A (asucc g x0) (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))) (ex2 A (\lambda
-(a1: A).(arity g c0 (THead (Flat Appl) w v) a1)) (\lambda (a1: A).(arity g c0
-(THead (Flat Appl) w (THead (Bind Abst) u t)) (asucc g a1)))) (\lambda (x1:
-A).(\lambda (x2: A).(\lambda (H11: (eq A (asucc g x0) (AHead x1
-x2))).(\lambda (H12: (arity g c0 u (asucc g x1))).(\lambda (H13: (arity g
-(CHead c0 (Bind Abst) u) t x2)).(let H14 \def (sym_eq A (asucc g x0) (AHead
-x1 x2) H11) in (let H15 \def (asucc_gen_head g x1 x2 x0 H14) in (ex2_ind A
-(\lambda (a0: A).(eq A x0 (AHead x1 a0))) (\lambda (a0: A).(eq A x2 (asucc g
-a0))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v) a1))
-(\lambda (a1: A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u t))
-(asucc g a1)))) (\lambda (x3: A).(\lambda (H16: (eq A x0 (AHead x1
-x3))).(\lambda (H17: (eq A x2 (asucc g x3))).(let H18 \def (eq_ind A x2
-(\lambda (a: A).(arity g (CHead c0 (Bind Abst) u) t a)) H13 (asucc g x3) H17)
-in (let H19 \def (eq_ind A x0 (\lambda (a: A).(arity g c0 v a)) H8 (AHead x1
-x3) H16) in (ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v)
+(THead (Bind b0) u t4) (asucc g a1))))))) (\lambda (H12: (arity g (CHead c0
+(Bind Abbr) u) t3 x0)).(\lambda (H13: (arity g (CHead c0 (Bind Abbr) u) t4
+(asucc g x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Bind Abbr) u
+t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind Abbr) u t4) (asucc g a1)))
+x0 (arity_bind g Abbr not_abbr_abst c0 u x H5 t3 x0 H12) (arity_bind g Abbr
+not_abbr_abst c0 u x H5 t4 (asucc g x0) H13)))) (\lambda (H12: (arity g
+(CHead c0 (Bind Abst) u) t3 x0)).(\lambda (H13: (arity g (CHead c0 (Bind
+Abst) u) t4 (asucc g x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead
+(Bind Abst) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind Abst) u t4)
+(asucc g a1))) (AHead x1 x0) (arity_head g c0 u x1 (arity_repl g c0 u x H5
+(asucc g x1) H11) t3 x0 H12) (arity_repl g c0 (THead (Bind Abst) u t4) (AHead
+x1 (asucc g x0)) (arity_head g c0 u x1 (arity_repl g c0 u x H5 (asucc g x1)
+H11) t4 (asucc g x0) H13) (asucc g (AHead x1 x0)) (leq_refl g (asucc g (AHead
+x1 x0))))))) (\lambda (H12: (arity g (CHead c0 (Bind Void) u) t3
+x0)).(\lambda (H13: (arity g (CHead c0 (Bind Void) u) t4 (asucc g
+x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Bind Void) u t3) a1))
+(\lambda (a1: A).(arity g c0 (THead (Bind Void) u t4) (asucc g a1))) x0
+(arity_bind g Void not_void_abst c0 u x H5 t3 x0 H12) (arity_bind g Void
+not_void_abst c0 u x H5 t4 (asucc g x0) H13)))) b H8 H9))) H10)))))) H7)))))
+H4)))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_:
+(ty3 g c0 w u)).(\lambda (H1: (ex2 A (\lambda (a1: A).(arity g c0 w a1))
+(\lambda (a1: A).(arity g c0 u (asucc g a1))))).(\lambda (v: T).(\lambda (t:
+T).(\lambda (_: (ty3 g c0 v (THead (Bind Abst) u t))).(\lambda (H3: (ex2 A
+(\lambda (a1: A).(arity g c0 v a1)) (\lambda (a1: A).(arity g c0 (THead (Bind
+Abst) u t) (asucc g a1))))).(let H4 \def H1 in (ex2_ind A (\lambda (a1:
+A).(arity g c0 w a1)) (\lambda (a1: A).(arity g c0 u (asucc g a1))) (ex2 A
+(\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v) a1)) (\lambda (a1:
+A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u t)) (asucc g a1))))
+(\lambda (x: A).(\lambda (H5: (arity g c0 w x)).(\lambda (H6: (arity g c0 u
+(asucc g x))).(let H7 \def H3 in (ex2_ind A (\lambda (a1: A).(arity g c0 v
+a1)) (\lambda (a1: A).(arity g c0 (THead (Bind Abst) u t) (asucc g a1))) (ex2
+A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v) a1)) (\lambda (a1:
+A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u t)) (asucc g a1))))
+(\lambda (x0: A).(\lambda (H8: (arity g c0 v x0)).(\lambda (H9: (arity g c0
+(THead (Bind Abst) u t) (asucc g x0))).(let H10 \def (arity_gen_abst g c0 u t
+(asucc g x0) H9) in (ex3_2_ind A A (\lambda (a1: A).(\lambda (a2: A).(eq A
+(asucc g x0) (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))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v)
a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u
-t)) (asucc g a1))) x3 (arity_appl g c0 w x1 (arity_repl g c0 w x H5 x1
-(leq_sym g x1 x (asucc_inj g x1 x (arity_mono g c0 u (asucc g x1) H12 (asucc
-g x) H6)))) v x3 H19) (arity_appl g c0 w x H5 (THead (Bind Abst) u t) (asucc
-g x3) (arity_head g c0 u x H6 t (asucc g x3) H18)))))))) H15)))))))) H10)))))
-H7))))) H4))))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4:
-T).(\lambda (_: (ty3 g c0 t3 t4)).(\lambda (H1: (ex2 A (\lambda (a1:
-A).(arity g c0 t3 a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g
+t)) (asucc g a1)))) (\lambda (x1: A).(\lambda (x2: A).(\lambda (H11: (eq A
+(asucc g x0) (AHead x1 x2))).(\lambda (H12: (arity g c0 u (asucc g
+x1))).(\lambda (H13: (arity g (CHead c0 (Bind Abst) u) t x2)).(let H14 \def
+(sym_eq A (asucc g x0) (AHead x1 x2) H11) in (let H15 \def (asucc_gen_head g
+x1 x2 x0 H14) in (ex2_ind A (\lambda (a0: A).(eq A x0 (AHead x1 a0)))
+(\lambda (a0: A).(eq A x2 (asucc g a0))) (ex2 A (\lambda (a1: A).(arity g c0
+(THead (Flat Appl) w v) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Appl)
+w (THead (Bind Abst) u t)) (asucc g a1)))) (\lambda (x3: A).(\lambda (H16:
+(eq A x0 (AHead x1 x3))).(\lambda (H17: (eq A x2 (asucc g x3))).(let H18 \def
+(eq_ind A x2 (\lambda (a: A).(arity g (CHead c0 (Bind Abst) u) t a)) H13
+(asucc g x3) H17) in (let H19 \def (eq_ind A x0 (\lambda (a: A).(arity g c0 v
+a)) H8 (AHead x1 x3) H16) in (ex_intro2 A (\lambda (a1: A).(arity g c0 (THead
+(Flat Appl) w v) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w
+(THead (Bind Abst) u t)) (asucc g a1))) x3 (arity_appl g c0 w x1 (arity_repl
+g c0 w x H5 x1 (leq_sym g x1 x (asucc_inj g x1 x (arity_mono g c0 u (asucc g
+x1) H12 (asucc g x) H6)))) v x3 H19) (arity_appl g c0 w x H5 (THead (Bind
+Abst) u t) (asucc g x3) (arity_head g c0 u x H6 t (asucc g x3) H18))))))))
+H15)))))))) H10))))) H7))))) H4))))))))))) (\lambda (c0: C).(\lambda (t3:
+T).(\lambda (t4: T).(\lambda (_: (ty3 g c0 t3 t4)).(\lambda (H1: (ex2 A
+(\lambda (a1: A).(arity g c0 t3 a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g
a1))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t4 t0)).(\lambda (H3: (ex2 A
(\lambda (a1: A).(arity g c0 t4 a1)) (\lambda (a1: A).(arity g c0 t0 (asucc g
a1))))).(let H4 \def H1 in (ex2_ind A (\lambda (a1: A).(arity g c0 t3 a1))
projects / helm.git / blobdiff