4 Require subst1_confluence.
9 Section pr2_subst1_props. (***********************************************)
11 Theorem pr2_delta1: (c,d:?; u:?; i:?) (drop i (0) c (CTail d (Bind Abbr) u)) ->
12 (t1,t2:?) (subst1 i u t1 t2) -> (pr2 c t1 t2).
13 Intros; XElim H0; Clear t2; XEAuto.
16 Hints Resolve pr2_delta1 : ltlc.
18 Theorem pr2_subst1: (c,e:?; v:?; i:?) (drop i (0) c (CTail e (Bind Abbr) v)) ->
19 (t1,t2:?) (pr2 c t1 t2) ->
20 (w1:?) (subst1 i v t1 w1) ->
21 (EX w2 | (pr2 c w1 w2) & (subst1 i v t2 w2)).
22 Intros until 2; XElim H0; Intros.
25 (* case 2: pr2_delta *)
26 Apply (neq_eq_e i i0); Intros.
27 (* case 2.1: i <> i0 *)
28 Subst1Confluence; XEAuto.
29 (* case 2.2: i = i0 *)
30 Rewrite <- H3 in H0; Rewrite <- H3 in H1; Clear H3 i0.
31 DropDis; Inversion H0; Rewrite H5 in H2; Clear H0 H4 H5 e v.
32 Subst1Confluence; XEAuto.
37 Hints Resolve pr2_delta1 : ltlc.
39 Tactic Definition Pr2Subst1 :=
41 | [ H0: (drop ?1 (0) ?2 (CTail ?3 (Bind Abbr) ?4));
42 H1: (pr2 ?2 ?5 ?6); H3: (subst1 ?1 ?4 ?5 ?7) |- ? ] ->
43 LApply (pr2_subst1 ?2 ?3 ?4 ?1); [ Intros H_x | XAuto ];
44 LApply (H_x ?5 ?6); [ Clear H_x H1; Intros H1 | XAuto ];
45 LApply (H1 ?7); [ Clear H1; Intros H1 | XAuto ];