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 *********************)
19 include "subst1/defs.ma".
21 include "subst0/props.ma".
23 theorem subst1_gen_sort:
24 \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst1
25 i v (TSort n) x) \to (eq T x (TSort n))))))
27 \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda
28 (H: (subst1 i v (TSort n) x)).(subst1_ind i v (TSort n) (\lambda (t: T).(eq T
29 t (TSort n))) (refl_equal T (TSort n)) (\lambda (t2: T).(\lambda (H0: (subst0
30 i v (TSort n) t2)).(subst0_gen_sort v t2 i n H0 (eq T t2 (TSort n))))) x
33 theorem subst1_gen_lref:
34 \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst1
35 i v (TLRef n) x) \to (or (eq T x (TLRef n)) (land (eq nat n i) (eq T x (lift
38 \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda
39 (H: (subst1 i v (TLRef n) x)).(subst1_ind i v (TLRef n) (\lambda (t: T).(or
40 (eq T t (TLRef n)) (land (eq nat n i) (eq T t (lift (S n) O v))))) (or_introl
41 (eq T (TLRef n) (TLRef n)) (land (eq nat n i) (eq T (TLRef n) (lift (S n) O
42 v))) (refl_equal T (TLRef n))) (\lambda (t2: T).(\lambda (H0: (subst0 i v
43 (TLRef n) t2)).(and_ind (eq nat n i) (eq T t2 (lift (S n) O v)) (or (eq T t2
44 (TLRef n)) (land (eq nat n i) (eq T t2 (lift (S n) O v)))) (\lambda (H1: (eq
45 nat n i)).(\lambda (H2: (eq T t2 (lift (S n) O v))).(or_intror (eq T t2
46 (TLRef n)) (land (eq nat n i) (eq T t2 (lift (S n) O v))) (conj (eq nat n i)
47 (eq T t2 (lift (S n) O v)) H1 H2)))) (subst0_gen_lref v t2 i n H0)))) x
50 theorem subst1_gen_head:
51 \forall (k: K).(\forall (v: T).(\forall (u1: T).(\forall (t1: T).(\forall
52 (x: T).(\forall (i: nat).((subst1 i v (THead k u1 t1) x) \to (ex3_2 T T
53 (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2:
54 T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (t2:
55 T).(subst1 (s k i) v t1 t2))))))))))
57 \lambda (k: K).(\lambda (v: T).(\lambda (u1: T).(\lambda (t1: T).(\lambda
58 (x: T).(\lambda (i: nat).(\lambda (H: (subst1 i v (THead k u1 t1)
59 x)).(subst1_ind i v (THead k u1 t1) (\lambda (t: T).(ex3_2 T T (\lambda (u2:
60 T).(\lambda (t2: T).(eq T t (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_:
61 T).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (t2: T).(subst1 (s k i) v t1
62 t2))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead k u1
63 t1) (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2)))
64 (\lambda (_: T).(\lambda (t2: T).(subst1 (s k i) v t1 t2))) u1 t1 (refl_equal
65 T (THead k u1 t1)) (subst1_refl i v u1) (subst1_refl (s k i) v t1)) (\lambda
66 (t2: T).(\lambda (H0: (subst0 i v (THead k u1 t1) t2)).(or3_ind (ex2 T
67 (\lambda (u2: T).(eq T t2 (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1
68 u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3:
69 T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
70 T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v
71 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3)))) (ex3_2
72 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda
73 (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (t3:
74 T).(subst1 (s k i) v t1 t3)))) (\lambda (H1: (ex2 T (\lambda (u2: T).(eq T t2
75 (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2)))).(ex2_ind T (\lambda
76 (u2: T).(eq T t2 (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2))
77 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3))))
78 (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_:
79 T).(\lambda (t3: T).(subst1 (s k i) v t1 t3)))) (\lambda (x0: T).(\lambda
80 (H2: (eq T t2 (THead k x0 t1))).(\lambda (H3: (subst0 i v u1
81 x0)).(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2
82 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_:
83 T).(\lambda (t3: T).(subst1 (s k i) v t1 t3))) x0 t1 H2 (subst1_single i v u1
84 x0 H3) (subst1_refl (s k i) v t1))))) H1)) (\lambda (H1: (ex2 T (\lambda (t3:
85 T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1
86 t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3:
87 T).(subst0 (s k i) v t1 t3)) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq
88 T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2)))
89 (\lambda (_: T).(\lambda (t3: T).(subst1 (s k i) v t1 t3)))) (\lambda (x0:
90 T).(\lambda (H2: (eq T t2 (THead k u1 x0))).(\lambda (H3: (subst0 (s k i) v
91 t1 x0)).(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k
92 u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_:
93 T).(\lambda (t3: T).(subst1 (s k i) v t1 t3))) u1 x0 H2 (subst1_refl i v u1)
94 (subst1_single (s k i) v t1 x0 H3))))) H1)) (\lambda (H1: (ex3_2 T T (\lambda
95 (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2:
96 T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t3:
97 T).(subst0 (s k i) v t1 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3:
98 T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v
99 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T
100 T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2:
101 T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (t3:
102 T).(subst1 (s k i) v t1 t3)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda
103 (H2: (eq T t2 (THead k x0 x1))).(\lambda (H3: (subst0 i v u1 x0)).(\lambda
104 (H4: (subst0 (s k i) v t1 x1)).(ex3_2_intro T T (\lambda (u2: T).(\lambda
105 (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1
106 i v u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst1 (s k i) v t1 t3))) x0
107 x1 H2 (subst1_single i v u1 x0 H3) (subst1_single (s k i) v t1 x1 H4)))))))
108 H1)) (subst0_gen_head k v u1 t1 t2 i H0)))) x H))))))).
110 theorem subst1_gen_lift_lt:
111 \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall
112 (h: nat).(\forall (d: nat).((subst1 i (lift h d u) (lift h (S (plus i d)) t1)
113 x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda
114 (t2: T).(subst1 i u t1 t2)))))))))
116 \lambda (u: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (i: nat).(\lambda
117 (h: nat).(\lambda (d: nat).(\lambda (H: (subst1 i (lift h d u) (lift h (S
118 (plus i d)) t1) x)).(subst1_ind i (lift h d u) (lift h (S (plus i d)) t1)
119 (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h (S (plus i d)) t2)))
120 (\lambda (t2: T).(subst1 i u t1 t2)))) (ex_intro2 T (\lambda (t2: T).(eq T
121 (lift h (S (plus i d)) t1) (lift h (S (plus i d)) t2))) (\lambda (t2:
122 T).(subst1 i u t1 t2)) t1 (refl_equal T (lift h (S (plus i d)) t1))
123 (subst1_refl i u t1)) (\lambda (t2: T).(\lambda (H0: (subst0 i (lift h d u)
124 (lift h (S (plus i d)) t1) t2)).(ex2_ind T (\lambda (t3: T).(eq T t2 (lift h
125 (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u t1 t3)) (ex2 T (\lambda
126 (t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst1 i u t1
127 t3))) (\lambda (x0: T).(\lambda (H1: (eq T t2 (lift h (S (plus i d))
128 x0))).(\lambda (H2: (subst0 i u t1 x0)).(ex_intro2 T (\lambda (t3: T).(eq T
129 t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst1 i u t1 t3)) x0 H1
130 (subst1_single i u t1 x0 H2))))) (subst0_gen_lift_lt u t1 t2 i h d H0)))) x
133 theorem subst1_gen_lift_eq:
134 \forall (t: T).(\forall (u: T).(\forall (x: T).(\forall (h: nat).(\forall
135 (d: nat).(\forall (i: nat).((le d i) \to ((lt i (plus d h)) \to ((subst1 i u
136 (lift h d t) x) \to (eq T x (lift h d t))))))))))
138 \lambda (t: T).(\lambda (u: T).(\lambda (x: T).(\lambda (h: nat).(\lambda
139 (d: nat).(\lambda (i: nat).(\lambda (H: (le d i)).(\lambda (H0: (lt i (plus d
140 h))).(\lambda (H1: (subst1 i u (lift h d t) x)).(subst1_ind i u (lift h d t)
141 (\lambda (t0: T).(eq T t0 (lift h d t))) (refl_equal T (lift h d t)) (\lambda
142 (t2: T).(\lambda (H2: (subst0 i u (lift h d t) t2)).(subst0_gen_lift_false t
143 u t2 h d i H H0 H2 (eq T t2 (lift h d t))))) x H1))))))))).
145 theorem subst1_gen_lift_ge:
146 \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall
147 (h: nat).(\forall (d: nat).((subst1 i u (lift h d t1) x) \to ((le (plus d h)
148 i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2:
149 T).(subst1 (minus i h) u t1 t2))))))))))
151 \lambda (u: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (i: nat).(\lambda
152 (h: nat).(\lambda (d: nat).(\lambda (H: (subst1 i u (lift h d t1)
153 x)).(\lambda (H0: (le (plus d h) i)).(subst1_ind i u (lift h d t1) (\lambda
154 (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h d t2))) (\lambda (t2:
155 T).(subst1 (minus i h) u t1 t2)))) (ex_intro2 T (\lambda (t2: T).(eq T (lift
156 h d t1) (lift h d t2))) (\lambda (t2: T).(subst1 (minus i h) u t1 t2)) t1
157 (refl_equal T (lift h d t1)) (subst1_refl (minus i h) u t1)) (\lambda (t2:
158 T).(\lambda (H1: (subst0 i u (lift h d t1) t2)).(ex2_ind T (\lambda (t3:
159 T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u t1 t3))
160 (ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst1
161 (minus i h) u t1 t3))) (\lambda (x0: T).(\lambda (H2: (eq T t2 (lift h d
162 x0))).(\lambda (H3: (subst0 (minus i h) u t1 x0)).(ex_intro2 T (\lambda (t3:
163 T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst1 (minus i h) u t1 t3)) x0
164 H2 (subst1_single (minus i h) u t1 x0 H3))))) (subst0_gen_lift_ge u t1 t2 i h
165 d H1 H0)))) x H)))))))).