-(*#* #stop file *)
-
Require Export contexts_defs.
+Require Export drop_defs.
Require Export pr0_defs.
+(*#* #caption "current axioms for the relation $\\CprZ{}{}$",
+ "reflexivity", "compatibility"
+*)
+(*#* #cap #cap c, c1, c2 #alpha u1 in V1, u2 in V2, k in z *)
+
Inductive cpr0 : C -> C -> Prop :=
| cpr0_refl : (c:?) (cpr0 c c)
- | cpr0_cont : (c1,c2:?) (cpr0 c1 c2) -> (u1,u2:?) (pr0 u1 u2) ->
+ | cpr0_comp : (c1,c2:?) (cpr0 c1 c2) -> (u1,u2:?) (pr0 u1 u2) ->
(k:?) (cpr0 (CTail c1 k u1) (CTail c2 k u2)).
+(*#* #stop file *)
+
Hint cpr0 : ltlc := Constructors cpr0.
+
+ Section cpr0_drop. (******************************************************)
+
+ Theorem cpr0_drop : (c1,c2:?) (cpr0 c1 c2) -> (h:?; e1:?; u1:?; k:?)
+ (drop h (0) c1 (CTail e1 k u1)) ->
+ (EX e2 u2 | (drop h (0) c2 (CTail e2 k u2)) &
+ (cpr0 e1 e2) & (pr0 u1 u2)
+ ).
+ Intros until 1; XElim H.
+(* case 1 : cpr0_refl *)
+ XEAuto.
+(* case 2 : cpr0_comp *)
+ XElim h.
+(* case 2.1 : h = 0 *)
+ Intros; DropGenBase.
+ Inversion H2; Rewrite H6 in H1; Rewrite H4 in H; XEAuto.
+(* case 2.2 : h > 0 *)
+ XElim k; Intros; DropGenBase.
+(* case 2.2.1 : Bind *)
+ LApply (H0 n e1 u0 k); [ Clear H0 H3; Intros H0 | XAuto ].
+ XElim H0; XEAuto.
+(* case 2.2.2 : Flat *)
+ LApply (H0 (S n) e1 u0 k); [ Clear H0 H3; Intros H0 | XAuto ].
+ XElim H0; XEAuto.
+ Qed.
+
+ Theorem cpr0_drop_back : (c1,c2:?) (cpr0 c2 c1) -> (h:?; e1:?; u1:?; k:?)
+ (drop h (0) c1 (CTail e1 k u1)) ->
+ (EX e2 u2 | (drop h (0) c2 (CTail e2 k u2)) &
+ (cpr0 e2 e1) & (pr0 u2 u1)
+ ).
+ Intros until 1; XElim H.
+(* case 1 : cpr0_refl *)
+ XEAuto.
+(* case 2 : cpr0_comp *)
+ XElim h.
+(* case 2.1 : h = 0 *)
+ Intros; DropGenBase.
+ Inversion H2; Rewrite H6 in H1; Rewrite H4 in H; XEAuto.
+(* case 2.2 : h > 0 *)
+ XElim k; Intros; DropGenBase.
+(* case 2.2.1 : Bind *)
+ LApply (H0 n e1 u0 k); [ Clear H0 H3; Intros H0 | XAuto ].
+ XElim H0; XEAuto.
+(* case 2.2.2 : Flat *)
+ LApply (H0 (S n) e1 u0 k); [ Clear H0 H3; Intros H0 | XAuto ].
+ XElim H0; XEAuto.
+ Qed.
+
+ End cpr0_drop.
+
+ Tactic Definition Cpr0Drop :=
+ Match Context With
+ | [ _: (drop ?1 (0) ?2 (CTail ?3 ?4 ?5));
+ _: (cpr0 ?2 ?6) |- ? ] ->
+ LApply (cpr0_drop ?2 ?6); [ Intros H_x | XAuto ];
+ LApply (H_x ?1 ?3 ?5 ?4); [ Clear H_x; Intros H_x | XAuto ];
+ XElim H_x; Intros
+ | [ _: (drop ?1 (0) ?2 (CTail ?3 ?4 ?5));
+ _: (cpr0 ?6 ?2) |- ? ] ->
+ LApply (cpr0_drop_back ?2 ?6); [ Intros H_x | XAuto ];
+ LApply (H_x ?1 ?3 ?5 ?4); [ Clear H_x; Intros H_x | XAuto ];
+ XElim H_x; Intros
+ | [ _: (drop ?1 (0) (CTail ?2 ?7 ?8) (CTail ?3 ?4 ?5));
+ _: (cpr0 ?2 ?6) |- ? ] ->
+ LApply (cpr0_drop (CTail ?2 ?7 ?8) (CTail ?6 ?7 ?8)); [ Intros H_x | XAuto ];
+ LApply (H_x ?1 ?3 ?5 ?4); [ Clear H_x; Intros H_x | XAuto ];
+ XElim H_x; Intros
+ | [ _: (drop ?1 (0) (CTail ?2 ?7 ?8) (CTail ?3 ?4 ?5));
+ _: (cpr0 ?6 ?2) |- ? ] ->
+ LApply (cpr0_drop_back (CTail ?2 ?7 ?8) (CTail ?6 ?7 ?8)); [ Intros H_x | XAuto ];
+ LApply (H_x ?1 ?3 ?5 ?4); [ Clear H_x; Intros H_x | XAuto ];
+ XElim H_x; Intros.