(*#* #stop file *)
Require Export pr0_defs.
+Require Export pr1_defs.
- Inductive pc0 [t1,t2:T] : Prop :=
- | pc0_r : (pr0 t1 t2) -> (pc0 t1 t2)
- | pc0_x : (pr0 t2 t1) -> (pc0 t1 t2).
+ Definition pc1 := [t1,t2:?] (EX t | (pr1 t1 t) & (pr1 t2 t)).
- Hint pc0 : ltlc := Constructors pc0.
+ Hints Unfold pc1 : ltlc.
- 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.
+ 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).
+ 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).
+ 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.
+ Theorem pc1_pr0_u: (t2,t1:?) (pr0 t1 t2) ->
+ (t3:?) (pc1 t2 t3) -> (pc1 t1 t3).
+ Intros; Pc1Unfold; XEAuto.
Qed.
-
- Theorem pc1_t : (t2,t1:?) (pc1 t1 t2) ->
- (t3:?) (pc1 t2 t3) -> (pc1 t1 t3).
- Intros t2 t1 H; XElim H; XEAuto.
+
+ Theorem pc1_refl: (t:?) (pc1 t t).
+ XEAuto.
Qed.
- Hints Resolve pc1_t : ltlc.
-
- Theorem pc1_s : (t2,t1:?) (pc1 t1 t2) -> (pc1 t2 t1).
- Intros; XElim H; XEAuto.
+ 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; XElim H; Intros.
-(* case 1: pc1_r *)
- XAuto.
-(* case 2: pc1_u *)
- XElim H; Intros; XEAuto.
+ 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; 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.
+ Intros; Pc1Unfold; XEAuto.
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.
+ Hints Resolve pc1_refl pc1_pr0_u pc1_pr0_r pc1_pr0_x pc1_s
+ pc1_tail_1 pc1_tail_2 : ltlc.