(*#* #stop file *)

Require Export pr0_defs.
Require Export pr1_defs.

      Definition pc1 := [t1,t2:?] (EX t | (pr1 t1 t) & (pr1 t2 t)). 

      Hints Unfold pc1 : ltlc.

      Tactic Definition Pc1Unfold :=
         Match Context With
            [ H: (pc1 ?2 ?3) |- ? ] -> Unfold pc1 in H; XDecompose H.

   Section pc1_props. (******************************************************)

      Theorem pc1_pr0_r: (t1,t2:?) (pr0 t1 t2) -> (pc1 t1 t2).
      XEAuto.
      Qed.

      Theorem pc1_pr0_x: (t1,t2:?) (pr0 t2 t1) -> (pc1 t1 t2).
      XEAuto.
      Qed.

      Theorem pc1_pr0_u: (t2,t1:?) (pr0 t1 t2) ->
                         (t3:?) (pc1 t2 t3) -> (pc1 t1 t3).
      Intros; Pc1Unfold; XEAuto.
      Qed.
      
      Theorem pc1_refl: (t:?) (pc1 t t).
      XEAuto.
      Qed.

      Theorem pc1_s: (t2,t1:?) (pc1 t1 t2) -> (pc1 t2 t1).
      Intros; Pc1Unfold; XEAuto.
      Qed.

      Theorem pc1_tail_1: (u1,u2:?) (pc1 u1 u2) ->
                          (t:?; k:?) (pc1 (TTail k u1 t) (TTail k u2 t)).
      Intros; Pc1Unfold; XEAuto.
      Qed.

      Theorem pc1_tail_2: (t1,t2:?) (pc1 t1 t2) ->
                          (u:?; k:?) (pc1 (TTail k u t1) (TTail k u t2)).
      Intros; Pc1Unfold; XEAuto.
      Qed.

   End pc1_props.

      Hints Resolve pc1_refl pc1_pr0_u pc1_pr0_r pc1_pr0_x pc1_s
                    pc1_tail_1 pc1_tail_2 : ltlc.