+(*#* #stop file *)
+
+Require lift_gen.
+Require pr3_props.
+Require pr3_gen.
+Require pc3_defs.
+Require pc3_props.
+
+ Section pc3_gen. (********************************************************)
+
+ Theorem pc3_gen_abst : (c:?; u1,u2,t1,t2:?)
+ (pc3 c (TTail (Bind Abst) u1 t1)
+ (TTail (Bind Abst) u2 t2)
+ ) ->
+ (pc3 c u1 u2) /\
+ (b:?; u:?) (pc3 (CTail c (Bind b) u) t1 t2).
+ Intros.
+ Pc3Confluence; Pr3Gen; Pr3Gen; Rewrite H0 in H; Clear H0 x.
+ Inversion H; Rewrite H5 in H1; Rewrite H6 in H2.
+ Split; XEAuto.
+ Qed.
+
+ Theorem pc3_gen_lift: (c:?; t1,t2:?; h,d:?)
+ (pc3 c (lift h d t1) (lift h d t2)) ->
+ (e:?) (drop h d c e) ->
+ (pc3 e t1 t2).
+ Intros.
+ Pc3Confluence; Pr3Gen; Pr3Gen; Rewrite H1 in H; Clear H1 x.
+ LiftGen; Rewrite H in H2; XEAuto.
+ Qed.
+
+ Theorem pc3_gen_not_abst : (b:?) ~b=Abst -> (c:?; t1,t2,u1,u2:?)
+ (pc3 c (TTail (Bind b) u1 t1)
+ (TTail (Bind Abst) u2 t2)
+ ) ->
+ (pc3 (CTail c (Bind b) u1) t1
+ (lift (1) (0) (TTail (Bind Abst) u2 t2))
+ )
+ .
+ Intros b; XElim b; Intros;
+ Try EqFalse; Pc3Confluence; Pr3Gen; Pr3Gen;
+ Try (Rewrite H1 in H0; Inversion H0);
+ Rewrite H1 in H4; Pr3Context;
+ EApply pc3_pr3_t; XEAuto.
+ Qed.
+
+ Theorem pc3_gen_lift_abst : (c:?; t,t2,u2:?; h,d:?)
+ (pc3 c (lift h d t)
+ (TTail (Bind Abst) u2 t2)
+ ) ->
+ (e:?) (drop h d c e) ->
+ (EX u1 t1 | (pr3 e t (TTail (Bind Abst) u1 t1)) &
+ (pr3 c u2 (lift h d u1)) &
+ (b:B; u:T)(pr3 (CTail c (Bind b) u) t2 (lift h (S d) t1))
+ ).
+ Intros.
+ Pc3Confluence; Pr3Gen; Pr3Gen; Rewrite H1 in H; Clear H1 x.
+ LiftGenBase; Rewrite H in H4; Rewrite H1 in H2; Rewrite H5 in H3; XEAuto.
+ Qed.
+
+ End pc3_gen.
+
+ Tactic Definition Pc3Gen :=
+ Match Context With
+ | [ _: (pc3 ?1 (lift ?2 ?3 ?4) (lift ?2 ?3 ?5));
+ _: (drop ?2 ?3 ?1 ?6) |- ? ] ->
+ LApply (pc3_gen_lift ?1 ?4 ?5 ?2 ?3); [ Intros H_x | XAuto ];
+ LApply (H_x ?6); [ Clear H_x; Intros | XAuto ]
+ | [ H: (pc3 ?1 (TTail (Bind Abst) ?2 ?3) (TTail (Bind Abst) ?4 ?5)) |- ? ] ->
+ LApply (pc3_gen_abst ?1 ?2 ?4 ?3 ?5);[ Clear H; Intros H | XAuto ];
+ XElim H; Intros
+ | [ H: (pc3 ?1 (TTail (Bind ?2) ?3 ?4) (TTail (Bind Abst) ?5 ?6));
+ _: ~ ?2 = Abst |- ? ] ->
+ LApply (pc3_gen_not_abst ?2); [ Intros H_x | XAuto ];
+ LApply (H_x ?1 ?4 ?6 ?3 ?5); [ Clear H H_x; Intros | XAuto ]
+ | [ _: (pc3 ?1 (lift ?2 ?3 ?4) (TTail (Bind Abst) ?5 ?6));
+ _: (drop ?2 ?3 ?1 ?7) |- ? ] ->
+ LApply (pc3_gen_lift_abst ?1 ?4 ?6 ?5 ?2 ?3); [ Intros H_x | XAuto ];
+ LApply (H_x ?7); [ Clear H_x; Intros H_x | XAuto ];
+ XElim H_x; Intros.