1 Require pr2_confluence.
4 Section pr3_confluence. (*************************************************)
6 (*#* #caption "confluence with single step reduction: strip lemma" *)
7 (*#* #cap #cap c, t0, t1, t2, t *)
9 Theorem pr3_strip : (c:?; t0,t1:?) (pr3 c t0 t1) -> (t2:?) (pr2 c t0 t2) ->
10 (EX t | (pr3 c t1 t) & (pr3 c t2 t)).
14 Intros until 1; XElim H; Intros.
19 LApply (H1 x); [ Clear H1 H2; Intros H1 | XAuto ].
20 XElim H1; Intros; XEAuto.
25 (*#* #caption "confluence with itself: Church-Rosser property" *)
26 (*#* #cap #cap c, t0, t1, t2, t *)
28 Theorem pr3_confluence : (c:?; t0,t1:?) (pr3 c t0 t1) -> (t2:?) (pr3 c t0 t2) ->
29 (EX t | (pr3 c t1 t) & (pr3 c t2 t)).
33 Intros until 1; XElim H; Intros.
37 LApply (pr3_strip c t3 t5); [ Clear H2; Intros H2 | XAuto ].
38 LApply (H2 t2); [ Clear H H2; Intros H | XAuto ].
40 LApply (H1 x); [ Clear H1 H2; Intros H1 | XAuto ].
41 XElim H1; Intros; XEAuto.
46 Tactic Definition Pr3Confluence :=
48 | [ H1: (pr3 ?1 ?2 ?3); H2: (pr2 ?1 ?2 ?4) |-? ] ->
49 LApply (pr3_strip ?1 ?2 ?3); [ Clear H1; Intros H1 | XAuto ];
50 LApply (H1 ?4); [ Clear H1 H2; Intros H1 | XAuto ];
52 | [ H1: (pr3 ?1 ?2 ?3); H2: (pr3 ?1 ?2 ?4) |-? ] ->
53 LApply (pr3_confluence ?1 ?2 ?3); [ Clear H1; Intros H1 | XAuto ];
54 LApply (H1 ?4); [ Clear H1 H2; Intros H1 | XAuto ];