1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 (* This file was automatically generated: do not edit *********************)
17 include "LambdaDelta-1/sn3/defs.ma".
19 include "LambdaDelta-1/pr3/props.ma".
22 \forall (b: B).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 c
23 (THead (Bind b) u t)) \to (land (sn3 c u) (sn3 (CHead c (Bind b) u) t))))))
25 \lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H:
26 (sn3 c (THead (Bind b) u t))).(insert_eq T (THead (Bind b) u t) (\lambda (t0:
27 T).(sn3 c t0)) (\lambda (_: T).(land (sn3 c u) (sn3 (CHead c (Bind b) u) t)))
28 (\lambda (y: T).(\lambda (H0: (sn3 c y)).(unintro T t (\lambda (t0: T).((eq T
29 y (THead (Bind b) u t0)) \to (land (sn3 c u) (sn3 (CHead c (Bind b) u) t0))))
30 (unintro T u (\lambda (t0: T).(\forall (x: T).((eq T y (THead (Bind b) t0 x))
31 \to (land (sn3 c t0) (sn3 (CHead c (Bind b) t0) x))))) (sn3_ind c (\lambda
32 (t0: T).(\forall (x: T).(\forall (x0: T).((eq T t0 (THead (Bind b) x x0)) \to
33 (land (sn3 c x) (sn3 (CHead c (Bind b) x) x0)))))) (\lambda (t1: T).(\lambda
34 (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3
35 c t1 t2) \to (sn3 c t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2)
36 \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (\forall (x: T).(\forall
37 (x0: T).((eq T t2 (THead (Bind b) x x0)) \to (land (sn3 c x) (sn3 (CHead c
38 (Bind b) x) x0)))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: (eq T
39 t1 (THead (Bind b) x x0))).(let H4 \def (eq_ind T t1 (\lambda (t0:
40 T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c
41 t0 t2) \to (\forall (x1: T).(\forall (x2: T).((eq T t2 (THead (Bind b) x1
42 x2)) \to (land (sn3 c x1) (sn3 (CHead c (Bind b) x1) x2))))))))) H2 (THead
43 (Bind b) x x0) H3) in (let H5 \def (eq_ind T t1 (\lambda (t0: T).(\forall
44 (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to
45 (sn3 c t2))))) H1 (THead (Bind b) x x0) H3) in (conj (sn3 c x) (sn3 (CHead c
46 (Bind b) x) x0) (sn3_sing c x (\lambda (t2: T).(\lambda (H6: (((eq T x t2)
47 \to (\forall (P: Prop).P)))).(\lambda (H7: (pr3 c x t2)).(let H8 \def (H4
48 (THead (Bind b) t2 x0) (\lambda (H8: (eq T (THead (Bind b) x x0) (THead (Bind
49 b) t2 x0))).(\lambda (P: Prop).(let H9 \def (f_equal T T (\lambda (e:
50 T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x |
51 (TLRef _) \Rightarrow x | (THead _ t0 _) \Rightarrow t0])) (THead (Bind b) x
52 x0) (THead (Bind b) t2 x0) H8) in (let H10 \def (eq_ind_r T t2 (\lambda (t0:
53 T).(pr3 c x t0)) H7 x H9) in (let H11 \def (eq_ind_r T t2 (\lambda (t0:
54 T).((eq T x t0) \to (\forall (P0: Prop).P0))) H6 x H9) in (H11 (refl_equal T
55 x) P)))))) (pr3_head_12 c x t2 H7 (Bind b) x0 x0 (pr3_refl (CHead c (Bind b)
56 t2) x0)) t2 x0 (refl_equal T (THead (Bind b) t2 x0))) in (land_ind (sn3 c t2)
57 (sn3 (CHead c (Bind b) t2) x0) (sn3 c t2) (\lambda (H9: (sn3 c t2)).(\lambda
58 (_: (sn3 (CHead c (Bind b) t2) x0)).H9)) H8)))))) (sn3_sing (CHead c (Bind b)
59 x) x0 (\lambda (t2: T).(\lambda (H6: (((eq T x0 t2) \to (\forall (P:
60 Prop).P)))).(\lambda (H7: (pr3 (CHead c (Bind b) x) x0 t2)).(let H8 \def (H4
61 (THead (Bind b) x t2) (\lambda (H8: (eq T (THead (Bind b) x x0) (THead (Bind
62 b) x t2))).(\lambda (P: Prop).(let H9 \def (f_equal T T (\lambda (e:
63 T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
64 (TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind b) x
65 x0) (THead (Bind b) x t2) H8) in (let H10 \def (eq_ind_r T t2 (\lambda (t0:
66 T).(pr3 (CHead c (Bind b) x) x0 t0)) H7 x0 H9) in (let H11 \def (eq_ind_r T
67 t2 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H6 x0 H9) in
68 (H11 (refl_equal T x0) P)))))) (pr3_head_12 c x x (pr3_refl c x) (Bind b) x0
69 t2 H7) x t2 (refl_equal T (THead (Bind b) x t2))) in (land_ind (sn3 c x) (sn3
70 (CHead c (Bind b) x) t2) (sn3 (CHead c (Bind b) x) t2) (\lambda (_: (sn3 c
71 x)).(\lambda (H10: (sn3 (CHead c (Bind b) x) t2)).H10)) H8))))))))))))))) y
75 \forall (f: F).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 c
76 (THead (Flat f) u t)) \to (land (sn3 c u) (sn3 c t))))))
78 \lambda (f: F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H:
79 (sn3 c (THead (Flat f) u t))).(insert_eq T (THead (Flat f) u t) (\lambda (t0:
80 T).(sn3 c t0)) (\lambda (_: T).(land (sn3 c u) (sn3 c t))) (\lambda (y:
81 T).(\lambda (H0: (sn3 c y)).(unintro T t (\lambda (t0: T).((eq T y (THead
82 (Flat f) u t0)) \to (land (sn3 c u) (sn3 c t0)))) (unintro T u (\lambda (t0:
83 T).(\forall (x: T).((eq T y (THead (Flat f) t0 x)) \to (land (sn3 c t0) (sn3
84 c x))))) (sn3_ind c (\lambda (t0: T).(\forall (x: T).(\forall (x0: T).((eq T
85 t0 (THead (Flat f) x x0)) \to (land (sn3 c x) (sn3 c x0)))))) (\lambda (t1:
86 T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P:
87 Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H2: ((\forall
88 (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to
89 (\forall (x: T).(\forall (x0: T).((eq T t2 (THead (Flat f) x x0)) \to (land
90 (sn3 c x) (sn3 c x0)))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3:
91 (eq T t1 (THead (Flat f) x x0))).(let H4 \def (eq_ind T t1 (\lambda (t0:
92 T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c
93 t0 t2) \to (\forall (x1: T).(\forall (x2: T).((eq T t2 (THead (Flat f) x1
94 x2)) \to (land (sn3 c x1) (sn3 c x2))))))))) H2 (THead (Flat f) x x0) H3) in
95 (let H5 \def (eq_ind T t1 (\lambda (t0: T).(\forall (t2: T).((((eq T t0 t2)
96 \to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (sn3 c t2))))) H1 (THead
97 (Flat f) x x0) H3) in (conj (sn3 c x) (sn3 c x0) (sn3_sing c x (\lambda (t2:
98 T).(\lambda (H6: (((eq T x t2) \to (\forall (P: Prop).P)))).(\lambda (H7:
99 (pr3 c x t2)).(let H8 \def (H4 (THead (Flat f) t2 x0) (\lambda (H8: (eq T
100 (THead (Flat f) x x0) (THead (Flat f) t2 x0))).(\lambda (P: Prop).(let H9
101 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
102 with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t0 _)
103 \Rightarrow t0])) (THead (Flat f) x x0) (THead (Flat f) t2 x0) H8) in (let
104 H10 \def (eq_ind_r T t2 (\lambda (t0: T).(pr3 c x t0)) H7 x H9) in (let H11
105 \def (eq_ind_r T t2 (\lambda (t0: T).((eq T x t0) \to (\forall (P0:
106 Prop).P0))) H6 x H9) in (H11 (refl_equal T x) P)))))) (pr3_head_12 c x t2 H7
107 (Flat f) x0 x0 (pr3_refl (CHead c (Flat f) t2) x0)) t2 x0 (refl_equal T
108 (THead (Flat f) t2 x0))) in (land_ind (sn3 c t2) (sn3 c x0) (sn3 c t2)
109 (\lambda (H9: (sn3 c t2)).(\lambda (_: (sn3 c x0)).H9)) H8)))))) (sn3_sing c
110 x0 (\lambda (t2: T).(\lambda (H6: (((eq T x0 t2) \to (\forall (P:
111 Prop).P)))).(\lambda (H7: (pr3 c x0 t2)).(let H8 \def (H4 (THead (Flat f) x
112 t2) (\lambda (H8: (eq T (THead (Flat f) x x0) (THead (Flat f) x
113 t2))).(\lambda (P: Prop).(let H9 \def (f_equal T T (\lambda (e: T).(match e
114 in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | (TLRef _)
115 \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat f) x x0)
116 (THead (Flat f) x t2) H8) in (let H10 \def (eq_ind_r T t2 (\lambda (t0:
117 T).(pr3 c x0 t0)) H7 x0 H9) in (let H11 \def (eq_ind_r T t2 (\lambda (t0:
118 T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H6 x0 H9) in (H11 (refl_equal
119 T x0) P)))))) (pr3_thin_dx c x0 t2 H7 x f) x t2 (refl_equal T (THead (Flat f)
120 x t2))) in (land_ind (sn3 c x) (sn3 c t2) (sn3 c t2) (\lambda (_: (sn3 c
121 x)).(\lambda (H10: (sn3 c t2)).H10)) H8))))))))))))))) y H0))))) H))))).
123 theorem sn3_gen_head:
124 \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 c
125 (THead k u t)) \to (sn3 c u)))))
127 \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (c: C).(\forall (u:
128 T).(\forall (t: T).((sn3 c (THead k0 u t)) \to (sn3 c u)))))) (\lambda (b:
129 B).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (sn3 c (THead
130 (Bind b) u t))).(let H_x \def (sn3_gen_bind b c u t H) in (let H0 \def H_x in
131 (land_ind (sn3 c u) (sn3 (CHead c (Bind b) u) t) (sn3 c u) (\lambda (H1: (sn3
132 c u)).(\lambda (_: (sn3 (CHead c (Bind b) u) t)).H1)) H0)))))))) (\lambda (f:
133 F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (sn3 c (THead
134 (Flat f) u t))).(let H_x \def (sn3_gen_flat f c u t H) in (let H0 \def H_x in
135 (land_ind (sn3 c u) (sn3 c t) (sn3 c u) (\lambda (H1: (sn3 c u)).(\lambda (_:
136 (sn3 c t)).H1)) H0)))))))) k).
138 theorem sn3_gen_cflat:
139 \forall (f: F).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 (CHead
140 c (Flat f) u) t) \to (sn3 c t)))))
142 \lambda (f: F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H:
143 (sn3 (CHead c (Flat f) u) t)).(sn3_ind (CHead c (Flat f) u) (\lambda (t0:
144 T).(sn3 c t0)) (\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1
145 t2) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Flat f) u) t1 t2) \to
146 (sn3 (CHead c (Flat f) u) t2)))))).(\lambda (H1: ((\forall (t2: T).((((eq T
147 t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Flat f) u) t1 t2) \to
148 (sn3 c t2)))))).(sn3_sing c t1 (\lambda (t2: T).(\lambda (H2: (((eq T t1 t2)
149 \to (\forall (P: Prop).P)))).(\lambda (H3: (pr3 c t1 t2)).(H1 t2 H2
150 (pr3_cflat c t1 t2 H3 f u))))))))) t H))))).
152 theorem sn3_gen_lift:
153 \forall (c1: C).(\forall (t: T).(\forall (h: nat).(\forall (d: nat).((sn3 c1
154 (lift h d t)) \to (\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 t)))))))
156 \lambda (c1: C).(\lambda (t: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda
157 (H: (sn3 c1 (lift h d t))).(insert_eq T (lift h d t) (\lambda (t0: T).(sn3 c1
158 t0)) (\lambda (_: T).(\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 t))))
159 (\lambda (y: T).(\lambda (H0: (sn3 c1 y)).(unintro T t (\lambda (t0: T).((eq
160 T y (lift h d t0)) \to (\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 t0)))))
161 (sn3_ind c1 (\lambda (t0: T).(\forall (x: T).((eq T t0 (lift h d x)) \to
162 (\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 x)))))) (\lambda (t1:
163 T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P:
164 Prop).P))) \to ((pr3 c1 t1 t2) \to (sn3 c1 t2)))))).(\lambda (H2: ((\forall
165 (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c1 t1 t2) \to
166 (\forall (x: T).((eq T t2 (lift h d x)) \to (\forall (c2: C).((drop h d c1
167 c2) \to (sn3 c2 x)))))))))).(\lambda (x: T).(\lambda (H3: (eq T t1 (lift h d
168 x))).(\lambda (c2: C).(\lambda (H4: (drop h d c1 c2)).(let H5 \def (eq_ind T
169 t1 (\lambda (t0: T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P:
170 Prop).P))) \to ((pr3 c1 t0 t2) \to (\forall (x0: T).((eq T t2 (lift h d x0))
171 \to (\forall (c3: C).((drop h d c1 c3) \to (sn3 c3 x0))))))))) H2 (lift h d
172 x) H3) in (let H6 \def (eq_ind T t1 (\lambda (t0: T).(\forall (t2: T).((((eq
173 T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c1 t0 t2) \to (sn3 c1 t2)))))
174 H1 (lift h d x) H3) in (sn3_sing c2 x (\lambda (t2: T).(\lambda (H7: (((eq T
175 x t2) \to (\forall (P: Prop).P)))).(\lambda (H8: (pr3 c2 x t2)).(H5 (lift h d
176 t2) (\lambda (H9: (eq T (lift h d x) (lift h d t2))).(\lambda (P: Prop).(let
177 H10 \def (eq_ind_r T t2 (\lambda (t0: T).(pr3 c2 x t0)) H8 x (lift_inj x t2 h
178 d H9)) in (let H11 \def (eq_ind_r T t2 (\lambda (t0: T).((eq T x t0) \to
179 (\forall (P0: Prop).P0))) H7 x (lift_inj x t2 h d H9)) in (H11 (refl_equal T
180 x) P))))) (pr3_lift c1 c2 h d H4 x t2 H8) t2 (refl_equal T (lift h d t2)) c2
181 H4)))))))))))))) y H0)))) H))))).