--- /dev/null
+(*#* #stop file *)
+
+Require subst0_subst0.
+Require pr0_subst0.
+Require pr3_defs.
+Require pr3_props.
+Require pr3_confluence.
+Require cpr0_defs.
+Require cpr0_props.
+Require pc3_defs.
+
+ Section pc3_confluence. (*************************************************)
+
+ Theorem pc3_confluence : (c:?; t1,t2:?) (pc3 c t1 t2) ->
+ (EX t0 | (pr3 c t1 t0) & (pr3 c t2 t0)).
+ Intros; XElim H; Intros.
+(* case 1 : pc3_r *)
+ XEAuto.
+(* case 2 : pc3_u *)
+ Clear H0; XElim H1; Intros.
+ Inversion_clear H; [ XEAuto | Pr3Confluence; XEAuto ].
+ Qed.
+
+ End pc3_confluence.
+
+ Tactic Definition Pc3Confluence :=
+ Match Context With
+ [ H: (pc3 ?1 ?2 ?3) |- ? ] ->
+ LApply (pc3_confluence ?1 ?2 ?3); [ Clear H; Intros H | XAuto ];
+ XElim H; Intros.
+
+ Section pc3_context. (****************************************************)
+
+ Theorem pc3_pr0_pr2_t : (u1,u2:?) (pr0 u2 u1) ->
+ (c:?; t1,t2:?; k:?) (pr2 (CTail c k u2) t1 t2) ->
+ (pc3 (CTail c k u1) t1 t2).
+ Intros.
+ Inversion H0; Clear H0; [ XAuto | NewInduction i ].
+(* case 1 : pr2_delta i = 0 *)
+ DropGenBase; Inversion H0; Clear H0 H3 H4 c k.
+ Rewrite H5 in H; Clear H5 u2.
+ Pr0Subst0; XEAuto.
+(* case 2 : pr2_delta i > 0 *)
+ NewInduction k; DropGenBase; XEAuto.
+ Qed.
+
+ Theorem pc3_pr2_pr2_t : (c:?; u1,u2:?) (pr2 c u2 u1) ->
+ (t1,t2:?; k:?) (pr2 (CTail c k u2) t1 t2) ->
+ (pc3 (CTail c k u1) t1 t2).
+ Intros until 1; Inversion H; Clear H; Intros.
+(* case 1 : pr2_pr0 *)
+ EApply pc3_pr0_pr2_t; [ Apply H0 | XAuto ].
+(* case 2 : pr2_delta *)
+ Inversion H; [ XAuto | NewInduction i0 ].
+(* case 2.1 : i0 = 0 *)
+ DropGenBase; Inversion H2; Clear H2.
+ Rewrite <- H5; Rewrite H6 in H; Rewrite <- H7 in H3; Clear H5 H6 H7 d0 k u0.
+ Subst0Subst0; Arith9'In H4 i.
+ EApply pc3_pr3_u.
+ EApply pr2_delta; XEAuto.
+ Apply pc3_pr2_x; EApply pr2_delta; [ Idtac | XEAuto ]; XEAuto.
+(* case 2.2 : i0 > 0 *)
+ Clear IHi0; NewInduction k; DropGenBase; XEAuto.
+ Qed.
+
+ Theorem pc3_pr2_pr3_t : (c:?; u2,t1,t2:?; k:?)
+ (pr3 (CTail c k u2) t1 t2) ->
+ (u1:?) (pr2 c u2 u1) ->
+ (pc3 (CTail c k u1) t1 t2).
+ Intros until 1; XElim H; Intros.
+(* case 1 : pr3_r *)
+ XAuto.
+(* case 2 : pr3_u *)
+ EApply pc3_t.
+ EApply pc3_pr2_pr2_t; [ Apply H2 | Apply H ].
+ XAuto.
+ Qed.
+
+ Theorem pc3_pr3_pc3_t : (c:?; u1,u2:?) (pr3 c u2 u1) ->
+ (t1,t2:?; k:?) (pc3 (CTail c k u2) t1 t2) ->
+ (pc3 (CTail c k u1) t1 t2).
+ Intros until 1; XElim H; Intros.
+(* case 1 : pr3_r *)
+ XAuto.
+(* case 2 : pr3_u *)
+ Apply H1; Pc3Confluence.
+ EApply pc3_t; [ Idtac | Apply pc3_s ]; EApply pc3_pr2_pr3_t; XEAuto.
+ Qed.
+
+ End pc3_context.
+
+ Tactic Definition Pc3Context :=
+ Match Context With
+ | [ H1: (pr0 ?3 ?2); H2: (pr2 (CTail ?1 ?4 ?3) ?5 ?6) |- ? ] ->
+ LApply (pc3_pr0_pr2_t ?2 ?3); [ Clear H1; Intros H1 | XAuto ];
+ LApply (H1 ?1 ?5 ?6 ?4); [ Clear H1 H2; Intros | XAuto ]
+ | [ H1: (pr0 ?3 ?2); H2: (pr3 (CTail ?1 ?4 ?3) ?5 ?6) |- ? ] ->
+ LApply (pc3_pr2_pr3_t ?1 ?3 ?5 ?6 ?4); [ Clear H2; Intros H2 | XAuto ];
+ LApply (H2 ?2); [ Clear H1 H2; Intros | XAuto ]
+ | [ H1: (pr2 ?1 ?3 ?2); H2: (pr2 (CTail ?1 ?4 ?3) ?5 ?6) |- ? ] ->
+ LApply (pc3_pr2_pr2_t ?1 ?2 ?3); [ Clear H1; Intros H1 | XAuto ];
+ LApply (H1 ?5 ?6 ?4); [ Clear H1 H2; Intros | XAuto ]
+ | [ H1: (pr2 ?1 ?3 ?2); H2: (pr3 (CTail ?1 ?4 ?3) ?5 ?6) |- ? ] ->
+ LApply (pc3_pr2_pr3_t ?1 ?3 ?5 ?6 ?4); [ Clear H2; Intros H2 | XAuto ];
+ LApply (H2 ?2); [ Clear H1 H2; Intros | XAuto ]
+ | [ H1: (pr3 ?1 ?3 ?2); H2: (pc3 (CTail ?1 ?4 ?3) ?5 ?6) |- ? ] ->
+ LApply (pc3_pr3_pc3_t ?1 ?2 ?3); [ Clear H1; Intros H1 | XAuto ];
+ LApply (H1 ?5 ?6 ?4); [ Clear H1 H2; Intros | XAuto ]
+ | _ -> Pr3Context.
+
+ Section pc3_lift. (*******************************************************)
+
+ Theorem pc3_lift : (c,e:?; h,d:?) (drop h d c e) ->
+ (t1,t2:?) (pc3 e t1 t2) ->
+ (pc3 c (lift h d t1) (lift h d t2)).
+
+ Intros.
+ Pc3Confluence.
+ EApply pc3_pr3_t; (EApply pr3_lift; [ XEAuto | Apply H0 Orelse Apply H1 ]).
+ Qed.
+
+ End pc3_lift.
+
+ Hints Resolve pc3_lift : ltlc.
+
+ Section pc3_cpr0. (*******************************************************)
+
+ Remark pc3_cpr0_t_aux : (c1,c2:?) (cpr0 c1 c2) ->
+ (k:?; u,t1,t2:?) (pr3 (CTail c1 k u) t1 t2) ->
+ (pc3 (CTail c2 k u) t1 t2).
+ Intros; XElim H0; Intros.
+(* case 1.1 : pr3_r *)
+ XAuto.
+(* case 1.2 : pr3_u *)
+ EApply pc3_t; [ Idtac | XEAuto ]. Clear H2 t1 t2.
+ Inversion_clear H0.
+(* case 1.2.1 : pr2_pr0 *)
+ XAuto.
+(* case 1.2.2 : pr2_delta *)
+ Cpr0Drop; Pr0Subst0.
+ EApply pc3_pr3_u; [ EApply pr2_delta; XEAuto | XAuto ].
+ Qed.
+
+ Theorem pc3_cpr0_t : (c1,c2:?) (cpr0 c1 c2) ->
+ (t1,t2:?) (pr3 c1 t1 t2) ->
+ (pc3 c2 t1 t2).
+ Intros until 1; XElim H; Intros.
+(* case 1 : cpr0_refl *)
+ XAuto.
+(* case 2 : cpr0_cont *)
+ Pc3Context; Pc3Confluence.
+ EApply pc3_t; [ Idtac | Apply pc3_s ]; EApply pc3_cpr0_t_aux; XEAuto.
+ Qed.
+
+ Theorem pc3_cpr0 : (c1,c2:?) (cpr0 c1 c2) -> (t1,t2:?) (pc3 c1 t1 t2) ->
+ (pc3 c2 t1 t2).
+ Intros; Pc3Confluence.
+ EApply pc3_t; [ Idtac | Apply pc3_s ]; EApply pc3_cpr0_t; XEAuto.
+ Qed.
+
+ End pc3_cpr0.
+
+ Hints Resolve pc3_cpr0 : ltlc.
projects / helm.git / blobdiff