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15 (* This file was automatically generated: do not edit *********************)
17 set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/ty3/arity_props".
19 include "ty3/arity.ma".
23 include "sc3/arity.ma".
25 theorem ty3_predicative:
26 \forall (g: G).(\forall (c: C).(\forall (v: T).(\forall (t: T).(\forall (u:
27 T).((ty3 g c (THead (Bind Abst) v t) u) \to ((pc3 c u v) \to (\forall (P:
30 \lambda (g: G).(\lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (u:
31 T).(\lambda (H: (ty3 g c (THead (Bind Abst) v t) u)).(\lambda (H0: (pc3 c u
32 v)).(\lambda (P: Prop).(let H1 \def H in (ex3_2_ind T T (\lambda (t2:
33 T).(\lambda (_: T).(pc3 c (THead (Bind Abst) v t2) u))) (\lambda (_:
34 T).(\lambda (t0: T).(ty3 g c v t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g
35 (CHead c (Bind Abst) v) t t2))) P (\lambda (x0: T).(\lambda (x1: T).(\lambda
36 (_: (pc3 c (THead (Bind Abst) v x0) u)).(\lambda (H3: (ty3 g c v
37 x1)).(\lambda (_: (ty3 g (CHead c (Bind Abst) v) t x0)).(let H_y \def
38 (ty3_conv g c v x1 H3 (THead (Bind Abst) v t) u H H0) in (let H_x \def
39 (ty3_arity g c (THead (Bind Abst) v t) v H_y) in (let H5 \def H_x in (ex2_ind
40 A (\lambda (a1: A).(arity g c (THead (Bind Abst) v t) a1)) (\lambda (a1:
41 A).(arity g c v (asucc g a1))) P (\lambda (x: A).(\lambda (H6: (arity g c
42 (THead (Bind Abst) v t) x)).(\lambda (H7: (arity g c v (asucc g x))).(let H8
43 \def (arity_gen_abst g c v t x H6) in (ex3_2_ind A A (\lambda (a1:
44 A).(\lambda (a2: A).(eq A x (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_:
45 A).(arity g c v (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g
46 (CHead c (Bind Abst) v) t a2))) P (\lambda (x2: A).(\lambda (x3: A).(\lambda
47 (H9: (eq A x (AHead x2 x3))).(\lambda (H10: (arity g c v (asucc g
48 x2))).(\lambda (_: (arity g (CHead c (Bind Abst) v) t x3)).(let H12 \def
49 (eq_ind A x (\lambda (a: A).(arity g c v (asucc g a))) H7 (AHead x2 x3) H9)
50 in (leq_ahead_asucc_false g x2 (asucc g x3) (arity_mono g c v (asucc g (AHead
51 x2 x3)) H12 (asucc g x2) H10) P))))))) H8))))) H5))))))))) (ty3_gen_bind g
52 Abst c v t u H1)))))))))).
54 theorem ty3_repellent:
55 \forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (t: T).(\forall (u1:
56 T).((ty3 g c (THead (Bind Abst) w t) u1) \to (\forall (u2: T).((ty3 g (CHead
57 c (Bind Abst) w) t (lift (S O) O u2)) \to ((pc3 c u1 u2) \to (\forall (P:
60 \lambda (g: G).(\lambda (c: C).(\lambda (w: T).(\lambda (t: T).(\lambda (u1:
61 T).(\lambda (H: (ty3 g c (THead (Bind Abst) w t) u1)).(\lambda (u2:
62 T).(\lambda (H0: (ty3 g (CHead c (Bind Abst) w) t (lift (S O) O
63 u2))).(\lambda (H1: (pc3 c u1 u2)).(\lambda (P: Prop).(ex_ind T (\lambda (t0:
64 T).(ty3 g (CHead c (Bind Abst) w) (lift (S O) O u2) t0)) P (\lambda (x:
65 T).(\lambda (H2: (ty3 g (CHead c (Bind Abst) w) (lift (S O) O u2) x)).(let H3
66 \def (ty3_gen_lift g (CHead c (Bind Abst) w) u2 x (S O) O H2 c (drop_drop
67 (Bind Abst) O c c (drop_refl c) w)) in (ex2_ind T (\lambda (t2: T).(pc3
68 (CHead c (Bind Abst) w) (lift (S O) O t2) x)) (\lambda (t2: T).(ty3 g c u2
69 t2)) P (\lambda (x0: T).(\lambda (_: (pc3 (CHead c (Bind Abst) w) (lift (S O)
70 O x0) x)).(\lambda (H5: (ty3 g c u2 x0)).(let H_y \def (ty3_conv g c u2 x0 H5
71 (THead (Bind Abst) w t) u1 H H1) in (let H_x \def (ty3_arity g (CHead c (Bind
72 Abst) w) t (lift (S O) O u2) H0) in (let H6 \def H_x in (ex2_ind A (\lambda
73 (a1: A).(arity g (CHead c (Bind Abst) w) t a1)) (\lambda (a1: A).(arity g
74 (CHead c (Bind Abst) w) (lift (S O) O u2) (asucc g a1))) P (\lambda (x1:
75 A).(\lambda (H7: (arity g (CHead c (Bind Abst) w) t x1)).(\lambda (H8: (arity
76 g (CHead c (Bind Abst) w) (lift (S O) O u2) (asucc g x1))).(let H_x0 \def
77 (ty3_arity g c (THead (Bind Abst) w t) u2 H_y) in (let H9 \def H_x0 in
78 (ex2_ind A (\lambda (a1: A).(arity g c (THead (Bind Abst) w t) a1)) (\lambda
79 (a1: A).(arity g c u2 (asucc g a1))) P (\lambda (x2: A).(\lambda (H10: (arity
80 g c (THead (Bind Abst) w t) x2)).(\lambda (H11: (arity g c u2 (asucc g
81 x2))).(arity_repellent g c w t x1 H7 x2 H10 (asucc_inj g x1 x2 (arity_mono g
82 c u2 (asucc g x1) (arity_gen_lift g (CHead c (Bind Abst) w) u2 (asucc g x1)
83 (S O) O H8 c (drop_drop (Bind Abst) O c c (drop_refl c) w)) (asucc g x2)
84 H11)) P)))) H9)))))) H6))))))) H3)))) (ty3_correct g (CHead c (Bind Abst) w)
85 t (lift (S O) O u2) H0))))))))))).
88 \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t
89 u) \to ((pc3 c u t) \to (\forall (P: Prop).P))))))
91 \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H:
92 (ty3 g c t u)).(\lambda (H0: (pc3 c u t)).(\lambda (P: Prop).(let H_y \def
93 (ty3_conv g c t u H t u H H0) in (let H_x \def (ty3_arity g c t t H_y) in
94 (let H1 \def H_x in (ex2_ind A (\lambda (a1: A).(arity g c t a1)) (\lambda
95 (a1: A).(arity g c t (asucc g a1))) P (\lambda (x: A).(\lambda (H2: (arity g
96 c t x)).(\lambda (H3: (arity g c t (asucc g x))).(leq_asucc_false g x
97 (arity_mono g c t (asucc g x) H3 x H2) P)))) H1)))))))))).
100 \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t
103 \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H:
104 (ty3 g c t u)).(let H_x \def (ty3_arity g c t u H) in (let H0 \def H_x in
105 (ex2_ind A (\lambda (a1: A).(arity g c t a1)) (\lambda (a1: A).(arity g c u
106 (asucc g a1))) (sn3 c t) (\lambda (x: A).(\lambda (H1: (arity g c t
107 x)).(\lambda (_: (arity g c u (asucc g x))).(sc3_sn3 g x c t (sc3_arity g c t