1 Require Export lift_defs.
3 (*#* #caption "axioms for strict substitution in terms",
4 "substituted local reference",
5 "substituted tail item: first operand",
6 "substituted tail item: second operand",
7 "substituted tail item: both operands"
9 (*#* #cap #cap t, t1, t2 #alpha v in W, u in V, u1 in V1, u2 in V2, k in z, s in p *)
11 Inductive subst0 : nat -> T -> T -> T -> Prop :=
12 | subst0_lref: (v:?; i:?) (subst0 i v (TLRef i) (lift (S i) (0) v))
13 | subst0_fst : (v,u2,u1:?; i:?) (subst0 i v u1 u2) ->
14 (t:?; k:?) (subst0 i v (TTail k u1 t) (TTail k u2 t))
15 | subst0_snd : (k:?; v,t2,t1:?; i:?) (subst0 (s k i) v t1 t2) -> (u:?)
16 (subst0 i v (TTail k u t1) (TTail k u t2))
17 | subst0_both: (v,u1,u2:?; i:?) (subst0 i v u1 u2) ->
18 (k:?; t1,t2:?) (subst0 (s k i) v t1 t2) ->
19 (subst0 i v (TTail k u1 t1) (TTail k u2 t2)).
23 Hint subst0 : ltlc := Constructors subst0.
25 Section subst0_gen_base. (************************************************)
27 Theorem subst0_gen_sort : (v,x:?; i,n:?) (subst0 i v (TSort n) x) ->
32 Theorem subst0_gen_lref : (v,x:?; i,n:?) (subst0 i v (TLRef n) x) ->
33 n = i /\ x = (lift (S n) (0) v).
34 Intros; Inversion H; XAuto.
37 Theorem subst0_gen_tail : (k:?; v,u1,t1,x:?; i:?)
38 (subst0 i v (TTail k u1 t1) x) -> (OR
39 (EX u2 | x = (TTail k u2 t1) &
41 (EX t2 | x = (TTail k u1 t2) &
42 (subst0 (s k i) v t1 t2)) |
43 (EX u2 t2 | x = (TTail k u2 t2) &
45 (subst0 (s k i) v t1 t2))
48 Intros; Inversion H; XEAuto.
53 Tactic Definition Subst0GenBase :=
55 | [ H: (subst0 ?1 ?2 (TSort ?3) ?4) |- ? ] ->
56 Apply (subst0_gen_sort ?2 ?4 ?1 ?3); Apply H
57 | [ H: (subst0 ?1 ?2 (TLRef ?3) ?4) |- ? ] ->
58 LApply (subst0_gen_lref ?2 ?4 ?1 ?3); [ Clear H; Intros H | XAuto ];
60 | [ H: (subst0 ?1 ?2 (TTail ?3 ?4 ?5) ?6) |- ? ] ->
61 LApply (subst0_gen_tail ?3 ?2 ?4 ?5 ?6 ?1); [ Clear H; Intros H | XAuto ];
62 XElim H; Intros H; XElim H; Intros.