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 "Basic-1/subst1/defs.ma".
19 include "Basic-1/subst0/props.ma".
22 \forall (v: T).(\forall (u1: T).(\forall (u2: T).(\forall (i: nat).((subst1
23 i v u1 u2) \to (\forall (k: K).(\forall (t1: T).(\forall (t2: T).((subst1 (s
24 k i) v t1 t2) \to (subst1 i v (THead k u1 t1) (THead k u2 t2))))))))))
26 \lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i: nat).(\lambda
27 (H: (subst1 i v u1 u2)).(subst1_ind i v u1 (\lambda (t: T).(\forall (k:
28 K).(\forall (t1: T).(\forall (t2: T).((subst1 (s k i) v t1 t2) \to (subst1 i
29 v (THead k u1 t1) (THead k t t2))))))) (\lambda (k: K).(\lambda (t1:
30 T).(\lambda (t2: T).(\lambda (H0: (subst1 (s k i) v t1 t2)).(subst1_ind (s k
31 i) v t1 (\lambda (t: T).(subst1 i v (THead k u1 t1) (THead k u1 t)))
32 (subst1_refl i v (THead k u1 t1)) (\lambda (t3: T).(\lambda (H1: (subst0 (s k
33 i) v t1 t3)).(subst1_single i v (THead k u1 t1) (THead k u1 t3) (subst0_snd k
34 v t3 t1 i H1 u1)))) t2 H0))))) (\lambda (t2: T).(\lambda (H0: (subst0 i v u1
35 t2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t0: T).(\lambda (H1: (subst1
36 (s k i) v t1 t0)).(subst1_ind (s k i) v t1 (\lambda (t: T).(subst1 i v (THead
37 k u1 t1) (THead k t2 t))) (subst1_single i v (THead k u1 t1) (THead k t2 t1)
38 (subst0_fst v t2 u1 i H0 t1 k)) (\lambda (t3: T).(\lambda (H2: (subst0 (s k
39 i) v t1 t3)).(subst1_single i v (THead k u1 t1) (THead k t2 t3) (subst0_both
40 v u1 t2 i H0 k t1 t3 H2)))) t0 H1))))))) u2 H))))).
45 theorem subst1_lift_lt:
46 \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).((subst1
47 i u t1 t2) \to (\forall (d: nat).((lt i d) \to (\forall (h: nat).(subst1 i
48 (lift h (minus d (S i)) u) (lift h d t1) (lift h d t2)))))))))
50 \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda
51 (H: (subst1 i u t1 t2)).(subst1_ind i u t1 (\lambda (t: T).(\forall (d:
52 nat).((lt i d) \to (\forall (h: nat).(subst1 i (lift h (minus d (S i)) u)
53 (lift h d t1) (lift h d t)))))) (\lambda (d: nat).(\lambda (_: (lt i
54 d)).(\lambda (h: nat).(subst1_refl i (lift h (minus d (S i)) u) (lift h d
55 t1))))) (\lambda (t3: T).(\lambda (H0: (subst0 i u t1 t3)).(\lambda (d:
56 nat).(\lambda (H1: (lt i d)).(\lambda (h: nat).(subst1_single i (lift h
57 (minus d (S i)) u) (lift h d t1) (lift h d t3) (subst0_lift_lt t1 t3 u i H0 d
58 H1 h))))))) t2 H))))).
63 theorem subst1_lift_ge:
64 \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).(\forall
65 (h: nat).((subst1 i u t1 t2) \to (\forall (d: nat).((le d i) \to (subst1
66 (plus i h) u (lift h d t1) (lift h d t2)))))))))
68 \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda
69 (h: nat).(\lambda (H: (subst1 i u t1 t2)).(subst1_ind i u t1 (\lambda (t:
70 T).(\forall (d: nat).((le d i) \to (subst1 (plus i h) u (lift h d t1) (lift h
71 d t))))) (\lambda (d: nat).(\lambda (_: (le d i)).(subst1_refl (plus i h) u
72 (lift h d t1)))) (\lambda (t3: T).(\lambda (H0: (subst0 i u t1 t3)).(\lambda
73 (d: nat).(\lambda (H1: (le d i)).(subst1_single (plus i h) u (lift h d t1)
74 (lift h d t3) (subst0_lift_ge t1 t3 u i h H0 d H1)))))) t2 H)))))).
80 \forall (u: T).(\forall (t1: T).(\forall (d: nat).(ex T (\lambda (t2:
81 T).(subst1 d u t1 (lift (S O) d t2))))))
83 \lambda (u: T).(\lambda (t1: T).(T_ind (\lambda (t: T).(\forall (d: nat).(ex
84 T (\lambda (t2: T).(subst1 d u t (lift (S O) d t2)))))) (\lambda (n:
85 nat).(\lambda (d: nat).(ex_intro T (\lambda (t2: T).(subst1 d u (TSort n)
86 (lift (S O) d t2))) (TSort n) (eq_ind_r T (TSort n) (\lambda (t: T).(subst1 d
87 u (TSort n) t)) (subst1_refl d u (TSort n)) (lift (S O) d (TSort n))
88 (lift_sort n (S O) d))))) (\lambda (n: nat).(\lambda (d: nat).(lt_eq_gt_e n d
89 (ex T (\lambda (t2: T).(subst1 d u (TLRef n) (lift (S O) d t2)))) (\lambda
90 (H: (lt n d)).(ex_intro T (\lambda (t2: T).(subst1 d u (TLRef n) (lift (S O)
91 d t2))) (TLRef n) (eq_ind_r T (TLRef n) (\lambda (t: T).(subst1 d u (TLRef n)
92 t)) (subst1_refl d u (TLRef n)) (lift (S O) d (TLRef n)) (lift_lref_lt n (S
93 O) d H)))) (\lambda (H: (eq nat n d)).(eq_ind nat n (\lambda (n0: nat).(ex T
94 (\lambda (t2: T).(subst1 n0 u (TLRef n) (lift (S O) n0 t2))))) (ex_intro T
95 (\lambda (t2: T).(subst1 n u (TLRef n) (lift (S O) n t2))) (lift n O u)
96 (eq_ind_r T (lift (plus (S O) n) O u) (\lambda (t: T).(subst1 n u (TLRef n)
97 t)) (subst1_single n u (TLRef n) (lift (S n) O u) (subst0_lref u n)) (lift (S
98 O) n (lift n O u)) (lift_free u n (S O) O n (le_n (plus O n)) (le_O_n n)))) d
99 H)) (\lambda (H: (lt d n)).(ex_intro T (\lambda (t2: T).(subst1 d u (TLRef n)
100 (lift (S O) d t2))) (TLRef (pred n)) (eq_ind_r T (TLRef n) (\lambda (t:
101 T).(subst1 d u (TLRef n) t)) (subst1_refl d u (TLRef n)) (lift (S O) d (TLRef
102 (pred n))) (lift_lref_gt d n H))))))) (\lambda (k: K).(\lambda (t:
103 T).(\lambda (H: ((\forall (d: nat).(ex T (\lambda (t2: T).(subst1 d u t (lift
104 (S O) d t2))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (d: nat).(ex T
105 (\lambda (t2: T).(subst1 d u t0 (lift (S O) d t2))))))).(\lambda (d:
106 nat).(let H_x \def (H d) in (let H1 \def H_x in (ex_ind T (\lambda (t2:
107 T).(subst1 d u t (lift (S O) d t2))) (ex T (\lambda (t2: T).(subst1 d u
108 (THead k t t0) (lift (S O) d t2)))) (\lambda (x: T).(\lambda (H2: (subst1 d u
109 t (lift (S O) d x))).(let H_x0 \def (H0 (s k d)) in (let H3 \def H_x0 in
110 (ex_ind T (\lambda (t2: T).(subst1 (s k d) u t0 (lift (S O) (s k d) t2))) (ex
111 T (\lambda (t2: T).(subst1 d u (THead k t t0) (lift (S O) d t2)))) (\lambda
112 (x0: T).(\lambda (H4: (subst1 (s k d) u t0 (lift (S O) (s k d)
113 x0))).(ex_intro T (\lambda (t2: T).(subst1 d u (THead k t t0) (lift (S O) d
114 t2))) (THead k x x0) (eq_ind_r T (THead k (lift (S O) d x) (lift (S O) (s k
115 d) x0)) (\lambda (t2: T).(subst1 d u (THead k t t0) t2)) (subst1_head u t
116 (lift (S O) d x) d H2 k t0 (lift (S O) (s k d) x0) H4) (lift (S O) d (THead k
117 x x0)) (lift_head k x x0 (S O) d))))) H3))))) H1))))))))) t1)).
122 theorem subst1_lift_S:
123 \forall (u: T).(\forall (i: nat).(\forall (h: nat).((le h i) \to (subst1 i
124 (TLRef h) (lift (S h) (S i) u) (lift (S h) i u)))))
126 \lambda (u: T).(T_ind (\lambda (t: T).(\forall (i: nat).(\forall (h:
127 nat).((le h i) \to (subst1 i (TLRef h) (lift (S h) (S i) t) (lift (S h) i
128 t)))))) (\lambda (n: nat).(\lambda (i: nat).(\lambda (h: nat).(\lambda (_:
129 (le h i)).(eq_ind_r T (TSort n) (\lambda (t: T).(subst1 i (TLRef h) t (lift
130 (S h) i (TSort n)))) (eq_ind_r T (TSort n) (\lambda (t: T).(subst1 i (TLRef
131 h) (TSort n) t)) (subst1_refl i (TLRef h) (TSort n)) (lift (S h) i (TSort n))
132 (lift_sort n (S h) i)) (lift (S h) (S i) (TSort n)) (lift_sort n (S h) (S
133 i))))))) (\lambda (n: nat).(\lambda (i: nat).(\lambda (h: nat).(\lambda (H:
134 (le h i)).(lt_eq_gt_e n i (subst1 i (TLRef h) (lift (S h) (S i) (TLRef n))
135 (lift (S h) i (TLRef n))) (\lambda (H0: (lt n i)).(eq_ind_r T (TLRef n)
136 (\lambda (t: T).(subst1 i (TLRef h) t (lift (S h) i (TLRef n)))) (eq_ind_r T
137 (TLRef n) (\lambda (t: T).(subst1 i (TLRef h) (TLRef n) t)) (subst1_refl i
138 (TLRef h) (TLRef n)) (lift (S h) i (TLRef n)) (lift_lref_lt n (S h) i H0))
139 (lift (S h) (S i) (TLRef n)) (lift_lref_lt n (S h) (S i) (le_S (S n) i H0))))
140 (\lambda (H0: (eq nat n i)).(let H1 \def (eq_ind_r nat i (\lambda (n0:
141 nat).(le h n0)) H n H0) in (eq_ind nat n (\lambda (n0: nat).(subst1 n0 (TLRef
142 h) (lift (S h) (S n0) (TLRef n)) (lift (S h) n0 (TLRef n)))) (eq_ind_r T
143 (TLRef n) (\lambda (t: T).(subst1 n (TLRef h) t (lift (S h) n (TLRef n))))
144 (eq_ind_r T (TLRef (plus n (S h))) (\lambda (t: T).(subst1 n (TLRef h) (TLRef
145 n) t)) (eq_ind nat (S (plus n h)) (\lambda (n0: nat).(subst1 n (TLRef h)
146 (TLRef n) (TLRef n0))) (eq_ind_r nat (plus h n) (\lambda (n0: nat).(subst1 n
147 (TLRef h) (TLRef n) (TLRef (S n0)))) (eq_ind nat (plus h (S n)) (\lambda (n0:
148 nat).(subst1 n (TLRef h) (TLRef n) (TLRef n0))) (eq_ind T (lift (S n) O
149 (TLRef h)) (\lambda (t: T).(subst1 n (TLRef h) (TLRef n) t)) (subst1_single n
150 (TLRef h) (TLRef n) (lift (S n) O (TLRef h)) (subst0_lref (TLRef h) n))
151 (TLRef (plus h (S n))) (lift_lref_ge h (S n) O (le_O_n h))) (S (plus h n))
152 (sym_eq nat (S (plus h n)) (plus h (S n)) (plus_n_Sm h n))) (plus n h)
153 (plus_sym n h)) (plus n (S h)) (plus_n_Sm n h)) (lift (S h) n (TLRef n))
154 (lift_lref_ge n (S h) n (le_n n))) (lift (S h) (S n) (TLRef n)) (lift_lref_lt
155 n (S h) (S n) (le_n (S n)))) i H0))) (\lambda (H0: (lt i n)).(eq_ind_r T
156 (TLRef (plus n (S h))) (\lambda (t: T).(subst1 i (TLRef h) t (lift (S h) i
157 (TLRef n)))) (eq_ind_r T (TLRef (plus n (S h))) (\lambda (t: T).(subst1 i
158 (TLRef h) (TLRef (plus n (S h))) t)) (subst1_refl i (TLRef h) (TLRef (plus n
159 (S h)))) (lift (S h) i (TLRef n)) (lift_lref_ge n (S h) i (le_S_n i n (le_S
160 (S i) n H0)))) (lift (S h) (S i) (TLRef n)) (lift_lref_ge n (S h) (S i)
161 H0)))))))) (\lambda (k: K).(\lambda (t: T).(\lambda (H: ((\forall (i:
162 nat).(\forall (h: nat).((le h i) \to (subst1 i (TLRef h) (lift (S h) (S i) t)
163 (lift (S h) i t))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (i:
164 nat).(\forall (h: nat).((le h i) \to (subst1 i (TLRef h) (lift (S h) (S i)
165 t0) (lift (S h) i t0))))))).(\lambda (i: nat).(\lambda (h: nat).(\lambda (H1:
166 (le h i)).(eq_ind_r T (THead k (lift (S h) (S i) t) (lift (S h) (s k (S i))
167 t0)) (\lambda (t1: T).(subst1 i (TLRef h) t1 (lift (S h) i (THead k t t0))))
168 (eq_ind_r T (THead k (lift (S h) i t) (lift (S h) (s k i) t0)) (\lambda (t1:
169 T).(subst1 i (TLRef h) (THead k (lift (S h) (S i) t) (lift (S h) (s k (S i))
170 t0)) t1)) (subst1_head (TLRef h) (lift (S h) (S i) t) (lift (S h) i t) i (H i
171 h H1) k (lift (S h) (s k (S i)) t0) (lift (S h) (s k i) t0) (eq_ind_r nat (S
172 (s k i)) (\lambda (n: nat).(subst1 (s k i) (TLRef h) (lift (S h) n t0) (lift
173 (S h) (s k i) t0))) (H0 (s k i) h (le_trans h i (s k i) H1 (s_inc k i))) (s k
174 (S i)) (s_S k i))) (lift (S h) i (THead k t t0)) (lift_head k t t0 (S h) i))
175 (lift (S h) (S i) (THead k t t0)) (lift_head k t t0 (S h) (S i))))))))))) u).