-(*#* #stop file *)
-
-Require pr0_subst0.
-Require pr3_defs.
-Require pr3_props.
-Require cpr0_defs.
-
- 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_cont *)
- 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_cont *)
- 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.
-
- Section cpr0_pr3. (*******************************************************)
-
- Theorem cpr0_pr3_t : (c1,c2:?) (cpr0 c2 c1) -> (t1,t2:?) (pr3 c1 t1 t2) ->
- (pr3 c2 t1 t2).
- Intros until 1; XElim H; Intros.
-(* case 1 : cpr0_refl *)
- XAuto.
-(* case 2 : cpr0_cont *)
- Pr3Context.
- XElim H1; Intros.
-(* case 2.1 : pr3_r *)
- XAuto.
-(* case 2.2 : pr3_u *)
- EApply pr3_t; [ Idtac | XEAuto ]. Clear H2 H3 c1 c2 t1 t2 t4 u2.
- Inversion_clear H1.
-(* case 2.2.1 : pr2_pr0 *)
- XAuto.
-(* case 2.2.1 : pr2_delta *)
- Cpr0Drop; Pr0Subst0.
- EApply pr3_u; [ EApply pr2_delta; XEAuto | XAuto ].
- Qed.
-
- End cpr0_pr3.