(*#* #stop file *)

Require pr0_confluence.
Require pr1_defs.

(*#* #caption "main properties of predicate \\texttt{pr1}" *)
(*#* #clauses pr1_props *)

   Section pr1_confluence. (*************************************************)

(*#* #caption "confluence with single step reduction: strip lemma" *)
(*#* #cap #cap t0, t1, t2, t *)

      Theorem pr1_strip : (t0,t1:?) (pr1 t0 t1) -> (t2:?) (pr0 t0 t2) ->
                          (EX t | (pr1 t1 t) & (pr1 t2 t)).
      Intros until 1; XElim H; Intros.
(* case 1 : pr1_r *)
      XEAuto.
(* case 2 : pr1_u *)
      Pr0Confluence.
      LApply (H1 x); [ Clear H1 H2; Intros H1 | XAuto ].
      XElim H1; Intros; XEAuto.
      Qed.

(*#* #caption "confluence with itself: Church-Rosser property" *)
(*#* #cap #cap t0, t1, t2, t *)

      Theorem pr1_confluence : (t0,t1:?) (pr1 t0 t1) -> (t2:?) (pr1 t0 t2) ->
                               (EX t | (pr1 t1 t) & (pr1 t2 t)).
      Intros until 1; XElim H; Intros.
(* case 1 : pr1_r *)
      XEAuto.
(* case 2 : pr1_u *)
      LApply (pr1_strip t3 t5); [ Clear H2; Intros H2 | XAuto ].
      LApply (H2 t2); [ Clear H H2; Intros H | XAuto ].
      XElim H; Intros.
      LApply (H1 x); [ Clear H1 H2; Intros H1 | XAuto ].
      XElim H1; Intros; XEAuto.
      Qed.

   End pr1_confluence.

      Tactic Definition Pr1Confluence :=
         Match Context With
            | [ H1: (pr1 ?1 ?2); H2: (pr0 ?1 ?3) |-? ] ->
               LApply (pr1_strip ?1 ?2); [ Clear H1; Intros H1 | XAuto ];
               LApply (H1 ?3); [ Clear H1 H2; Intros H1 | XAuto ];
               XElim H1; Intros
            | [ H1: (pr1 ?1 ?2); H2: (pr1 ?1 ?3) |-? ] ->
               LApply (pr1_confluence ?1 ?2); [ Clear H1; Intros H1 | XAuto ];
               LApply (H1 ?3); [ Clear H1 H2; Intros H1 | XAuto ];
               XElim H1; Intros
            | _ -> Pr0Confluence.

(*#* #single *)