-(*#* #stop file *)
-
Require Export lift_defs.
+(*#* #caption "axioms for strict substitution in terms",
+ "substituted local reference",
+ "substituted tail item: first operand",
+ "substituted tail item: second operand",
+ "substituted tail item: both operands"
+*)
+(*#* #cap #cap t, t1, t2 #alpha v in W, u in V, u1 in V1, u2 in V2, k in z, s in p *)
+
Inductive subst0 : nat -> T -> T -> T -> Prop :=
- | subst0_bref : (v:?; i:?) (subst0 i v (TBRef i) (lift (S i) (0) v))
- | subst0_fst : (v,w,u:?; i:?) (subst0 i v u w) ->
- (t:?; k:?) (subst0 i v (TTail k u t) (TTail k w t))
- | subst0_snd : (k:?; v,w,t:?; i:?) (subst0 (s k i) v t w) -> (u:?)
- (subst0 i v (TTail k u t) (TTail k u w))
- | subst0_both : (v,u1,u2:?; i:?) (subst0 i v u1 u2) ->
- (k:?; t1,t2:?) (subst0 (s k i) v t1 t2) ->
- (subst0 i v (TTail k u1 t1) (TTail k u2 t2)).
+ | subst0_lref: (v:?; i:?) (subst0 i v (TLRef i) (lift (S i) (0) v))
+ | subst0_fst : (v,u2,u1:?; i:?) (subst0 i v u1 u2) ->
+ (t:?; k:?) (subst0 i v (TTail k u1 t) (TTail k u2 t))
+ | subst0_snd : (k:?; v,t2,t1:?; i:?) (subst0 (s k i) v t1 t2) -> (u:?)
+ (subst0 i v (TTail k u t1) (TTail k u t2))
+ | subst0_both: (v,u1,u2:?; i:?) (subst0 i v u1 u2) ->
+ (k:?; t1,t2:?) (subst0 (s k i) v t1 t2) ->
+ (subst0 i v (TTail k u1 t1) (TTail k u2 t2)).
+
+(*#* #stop file *)
Hint subst0 : ltlc := Constructors subst0.
Intros; Inversion H.
Qed.
- Theorem subst0_gen_bref : (v,x:?; i,n:?) (subst0 i v (TBRef n) x) ->
+ Theorem subst0_gen_lref : (v,x:?; i,n:?) (subst0 i v (TLRef n) x) ->
n = i /\ x = (lift (S n) (0) v).
Intros; Inversion H; XAuto.
Qed.
Match Context With
| [ H: (subst0 ?1 ?2 (TSort ?3) ?4) |- ? ] ->
Apply (subst0_gen_sort ?2 ?4 ?1 ?3); Apply H
- | [ H: (subst0 ?1 ?2 (TBRef ?3) ?4) |- ? ] ->
- LApply (subst0_gen_bref ?2 ?4 ?1 ?3); [ Clear H; Intros H | XAuto ];
+ | [ H: (subst0 ?1 ?2 (TLRef ?3) ?4) |- ? ] ->
+ LApply (subst0_gen_lref ?2 ?4 ?1 ?3); [ Clear H; Intros H | XAuto ];
XElim H; Intros
| [ H: (subst0 ?1 ?2 (TTail ?3 ?4 ?5) ?6) |- ? ] ->
LApply (subst0_gen_tail ?3 ?2 ?4 ?5 ?6 ?1); [ Clear H; Intros H | XAuto ];
XElim H; Intros H; XElim H; Intros.
-