-(*#* #stop file *)
-
Require Export pr2_defs.
Require Export pr3_defs.
Require Export pc1_defs.
- Inductive pc2 [c:C; t1,t2:T] : Prop :=
- | pc2_r : (pr2 c t1 t2) -> (pc2 c t1 t2)
- | pc2_x : (pr2 c t2 t1) -> (pc2 c t1 t2).
-
- Hint pc2 : ltlc := Constructors pc2.
-
- Inductive pc3 [c:C] : T -> T -> Prop :=
- | pc3_r : (t:?) (pc3 c t t)
- | pc3_u : (t2,t1:?) (pc2 c t1 t2) ->
- (t3:?) (pc3 c t2 t3) -> (pc3 c t1 t3).
+(*#* #caption "the relation $\\PcT{}{}{}$" *)
+(*#* #cap #cap c, t, t1, t2 *)
- Hint pc3 : ltlc := Constructors pc3.
+ Definition pc3 := [c:?; t1,t2:?] (EX t | (pr3 c t1 t) & (pr3 c t2 t)).
- Section pc2_props. (******************************************************)
-
- Theorem pc2_s : (c,t2,t1:?) (pc2 c t1 t2) -> (pc2 c t2 t1).
- Intros.
- Inversion H; XAuto.
- Qed.
-
- Theorem pc2_shift : (h:?; c,e:?) (drop h (0) c e) ->
- (t1,t2:?) (pc2 c t1 t2) ->
- (pc2 e (app c h t1) (app c h t2)).
- Intros until 2; XElim H0; Intros.
-(* case 1 : pc2_r *)
- XAuto.
-(* case 2 : pc2_x *)
- XEAuto.
- Qed.
+(*#* #stop file *)
- End pc2_props.
+ Hints Unfold pc3 : ltlc.
- Hints Resolve pc2_s pc2_shift : ltlc.
+ Tactic Definition Pc3Unfold :=
+ Match Context With
+ [ H: (pc3 ?1 ?2 ?3) |- ? ] -> Unfold pc3 in H; XDecompose H.
Section pc3_props. (******************************************************)
- Theorem pc3_pr2_r : (c,t1,t2:?) (pr2 c t1 t2) -> (pc3 c t1 t2).
+ Theorem pc3_pr2_r: (c,t1,t2:?) (pr2 c t1 t2) -> (pc3 c t1 t2).
XEAuto.
Qed.
- Theorem pc3_pr2_x : (c,t1,t2:?) (pr2 c t2 t1) -> (pc3 c t1 t2).
+ Theorem pc3_pr2_x: (c,t1,t2:?) (pr2 c t2 t1) -> (pc3 c t1 t2).
XEAuto.
Qed.
- Theorem pc3_pc2 : (c,t1,t2:?) (pc2 c t1 t2) -> (pc3 c t1 t2).
+ Theorem pc3_pr3_r: (c:?; t1,t2) (pr3 c t1 t2) -> (pc3 c t1 t2).
XEAuto.
Qed.
- Theorem pc3_t : (t2,c,t1:?) (pc3 c t1 t2) ->
- (t3:?) (pc3 c t2 t3) -> (pc3 c t1 t3).
- Intros t2 c t1 H; XElim H; XEAuto.
- Qed.
-
- Hints Resolve pc3_t : ltlc.
-
- Theorem pc3_s : (c,t2,t1:?) (pc3 c t1 t2) -> (pc3 c t2 t1).
- Intros; XElim H; [ XAuto | XEAuto ].
- Qed.
-
- Hints Resolve pc3_s : ltlc.
-
- Theorem pc3_pr3_r : (c:?; t1,t2) (pr3 c t1 t2) -> (pc3 c t1 t2).
- Intros; XElim H; XEAuto.
- Qed.
-
- Theorem pc3_pr3_x : (c:?; t1,t2) (pr3 c t2 t1) -> (pc3 c t1 t2).
- Intros; XElim H; XEAuto.
- Qed.
-
- Hints Resolve pc3_pr3_r pc3_pr3_x : ltlc.
-
- Theorem pc3_pr3_t : (c:?; t1,t0:?) (pr3 c t1 t0) ->
- (t2:?) (pr3 c t2 t0) -> (pc3 c t1 t2).
- Intros; Apply (pc3_t t0); XAuto.
+ Theorem pc3_pr3_x: (c:?; t1,t2) (pr3 c t2 t1) -> (pc3 c t1 t2).
+ XEAuto.
Qed.
- Theorem pc3_thin_dx : (c:? ;t1,t2:?) (pc3 c t1 t2) ->
- (u:?; f:?) (pc3 c (TTail (Flat f) u t1)
- (TTail (Flat f) u t2)).
- Intros; XElim H; [XAuto | Intros ].
- EApply pc3_u; [ Inversion H | Apply H1 ]; XAuto.
+ Theorem pc3_pr3_t: (c:?; t1,t0:?) (pr3 c t1 t0) ->
+ (t2:?) (pr3 c t2 t0) -> (pc3 c t1 t2).
+ XEAuto.
Qed.
- Theorem pc3_tail_1 : (c:?; u1,u2:?) (pc3 c u1 u2) ->
- (k:?; t:?) (pc3 c (TTail k u1 t) (TTail k u2 t)).
- Intros until 1; XElim H; Intros.
-(* case 1 : pc3_r *)
- XAuto.
-(* case 2 : pc3_u *)
- EApply pc3_u; [ Inversion H | Apply H1 ]; XAuto.
+ Theorem pc3_pr2_u: (c:?; t2,t1:?) (pr2 c t1 t2) ->
+ (t3:?) (pc3 c t2 t3) -> (pc3 c t1 t3).
+ Intros; Pc3Unfold; XEAuto.
Qed.
-
- Theorem pc3_tail_2 : (c:?; u,t1,t2:?; k:?) (pc3 (CTail c k u) t1 t2) ->
- (pc3 c (TTail k u t1) (TTail k u t2)).
- Intros.
- XElim H; [ Idtac | Intros; Inversion H ]; XEAuto.
+
+ Theorem pc3_refl: (c:?; t:?) (pc3 c t t).
+ XEAuto.
Qed.
- Theorem pc3_tail_12 : (c:?; u1,u2:?) (pc3 c u1 u2) ->
- (k:?; t1,t2:?) (pc3 (CTail c k u2) t1 t2) ->
- (pc3 c (TTail k u1 t1) (TTail k u2 t2)).
- Intros.
- EApply pc3_t; [ Apply pc3_tail_1 | Apply pc3_tail_2 ]; XAuto.
+ Theorem pc3_s: (c,t2,t1:?) (pc3 c t1 t2) -> (pc3 c t2 t1).
+ Intros; Pc3Unfold; XEAuto.
Qed.
- Theorem pc3_tail_21 : (c:?; u1,u2:?) (pc3 c u1 u2) ->
- (k:?; t1,t2:?) (pc3 (CTail c k u1) t1 t2) ->
- (pc3 c (TTail k u1 t1) (TTail k u2 t2)).
- Intros.
- EApply pc3_t; [ Apply pc3_tail_2 | Apply pc3_tail_1 ]; XAuto.
+ Theorem pc3_thin_dx: (c:? ;t1,t2:?) (pc3 c t1 t2) ->
+ (u:?; f:?) (pc3 c (TTail (Flat f) u t1)
+ (TTail (Flat f) u t2)).
+ Intros; Pc3Unfold; XEAuto.
Qed.
- Theorem pc3_pr3_u : (c:?; t2,t1:?) (pr2 c t1 t2) ->
- (t3:?) (pc3 c t2 t3) -> (pc3 c t1 t3).
- XEAuto.
+ Theorem pc3_tail_1: (c:?; u1,u2:?) (pc3 c u1 u2) ->
+ (k:?; t:?) (pc3 c (TTail k u1 t) (TTail k u2 t)).
+ Intros; Pc3Unfold; XEAuto.
Qed.
- Theorem pc3_pr3_u2 : (c:?; t0,t1:?) (pr2 c t0 t1) ->
- (t2:?) (pc3 c t0 t2) -> (pc3 c t1 t2).
- Intros; Apply (pc3_t t0); XAuto.
+ Theorem pc3_tail_2: (c:?; u,t1,t2:?; k:?) (pc3 (CTail c k u) t1 t2) ->
+ (pc3 c (TTail k u t1) (TTail k u t2)).
+ Intros; Pc3Unfold; XEAuto.
Qed.
- Theorem pc3_shift : (h:?; c,e:?) (drop h (0) c e) ->
- (t1,t2:?) (pc3 c t1 t2) ->
- (pc3 e (app c h t1) (app c h t2)).
- Intros until 2; XElim H0; Clear t1 t2; Intros.
-(* case 1 : pc3_r *)
- XAuto.
-(* case 2 : pc3_u *)
- XEAuto.
+ Theorem pc3_shift: (h:?; c,e:?) (drop h (0) c e) ->
+ (t1,t2:?) (pc3 c t1 t2) ->
+ (pc3 e (app c h t1) (app c h t2)).
+ Intros; Pc3Unfold; XEAuto.
Qed.
Theorem pc3_pc1: (t1,t2:?) (pc1 t1 t2) -> (c:?) (pc3 c t1 t2).
- Intros; XElim H; Intros.
-(* case 1: pc1_r *)
- XAuto.
-(* case 2 : pc1_u *)
- XElim H; XEAuto.
+ Intros; Pc1Unfold; XEAuto.
Qed.
-
+
End pc3_props.
- Hints Resolve pc3_pr2_r pc3_pr2_x pc3_pc2 pc3_pr3_r pc3_pr3_x
- pc3_t pc3_s pc3_pr3_t pc3_thin_dx pc3_tail_1 pc3_tail_2
- pc3_tail_12 pc3_tail_21 pc3_pr3_u pc3_shift pc3_pc1 : ltlc.
-
- Tactic Definition Pc3T :=
- Match Context With
- | [ _: (pr3 ?1 ?2 (TTail ?3 ?4 ?5)); _: (pc3 ?1 ?6 ?4) |- ? ] ->
- LApply (pc3_t (TTail ?3 ?4 ?5) ?1 ?2); [ Intros H_x | XAuto ];
- LApply (H_x (TTail ?3 ?6 ?5)); [ Clear H_x; Intros | Apply pc3_s; XAuto ]
- | [ _: (pc3 ?1 ?2 ?3); _: (pr3 ?1 ?3 ?4) |- ? ] ->
- LApply (pc3_t ?3 ?1 ?2); [ Intros H_x | XAuto ];
- LApply (H_x ?4); [ Clear H_x; Intros | XAuto ].
+ Hints Resolve pc3_refl pc3_pr2_r pc3_pr2_x pc3_pr3_r pc3_pr3_x
+ pc3_s pc3_pr3_t pc3_thin_dx pc3_tail_1 pc3_tail_2
+ pc3_pr2_u pc3_shift pc3_pc1 : ltlc.
projects / helm.git / blobdiff