(*#* #stop file *)

Require Export pr0_defs.

      Inductive pc0 [t1,t2:T] : Prop :=
         | pc0_r : (pr0 t1 t2) -> (pc0 t1 t2)
         | pc0_x : (pr0 t2 t1) -> (pc0 t1 t2).

      Hint pc0 : ltlc := Constructors pc0.

      Inductive pc1 : T -> T -> Prop :=
         | pc1_r : (t:?) (pc1 t t)
         | pc1_u : (t2,t1:?) (pc0 t1 t2) -> (t3:?) (pc1 t2 t3) -> (pc1 t1 t3).

      Hint pc1 : ltlc := Constructors pc1.

   Section pc0_props. (******************************************************)

      Theorem pc0_s : (t2,t1:?) (pc0 t1 t2) -> (pc0 t2 t1).
      Intros.
      Inversion H; XAuto.
      Qed.

   End pc0_props.

      Hints Resolve pc0_s : ltlc.

   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_pc0 : (t1,t2:?) (pc0 t1 t2) -> (pc1 t1 t2).
      XEAuto.
      Qed.

      Theorem pc1_t : (t2,t1:?) (pc1 t1 t2) ->
                      (t3:?) (pc1 t2 t3) -> (pc1 t1 t3).
      Intros t2 t1 H; XElim H; XEAuto.
      Qed.

      Hints Resolve pc1_t : ltlc.

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

      Theorem pc1_tail_1: (u1,u2:?) (pc1 u1 u2) ->
                          (t:?; k:?) (pc1 (TTail k u1 t) (TTail k u2 t)).
      Intros; XElim H; Intros.
(* case 1: pc1_r *)
      XAuto.
(* case 2: pc1_u *)
      XElim H; Intros; XEAuto.
      Qed.

      Theorem pc1_tail_2: (t1,t2:?) (pc1 t1 t2) ->
                          (u:?; k:?) (pc1 (TTail k u t1) (TTail k u t2)).
      Intros; XElim H; Intros.
(* case 1: pc1_r *)
      XAuto.
(* case 2: pc1_u *)
      XElim H; Intros; XEAuto.
      Qed.

      Theorem pc1_tail: (u1,u2:?) (pc1 u1 u2) -> (t1,t2:?) (pc1 t1 t2) ->
                        (k:?) (pc1 (TTail k u1 t1) (TTail k u2 t2)).
      Intros; EApply pc1_t; [ EApply pc1_tail_1 | EApply pc1_tail_2 ]; XAuto.
      Qed.

   End pc1_props.

      Hints Resolve pc1_pr0_r pc1_pr0_x pc1_pc0 pc1_t pc1_s
                    pc1_tail_1 pc1_tail_2 pc1_tail : ltlc.