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).
+ Theorem pc3_gen_sort: (c:?; m,n:?) (pc3 c (TSort m) (TSort n)) -> m = n.
+ Intros; Pc3Unfold; Repeat Pr3GenBase.
+ Rewrite H0 in H; Clear H0 x c.
+ TGenBase; XAuto.
+ Qed.
+
+ 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.
+ Pc3Unfold; Repeat Pr3GenBase; Rewrite H1 in H; Clear H1 x.
+ TGenBase; Rewrite H1 in H4; Rewrite H6 in H5.
+ XEAuto.
Qed.
Theorem pc3_gen_lift: (c:?; t1,t2:?; h,d:?)
(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.
+ Pc3Unfold; Repeat Pr3Gen; Rewrite H2 in H; Clear H2 x.
+ LiftGen; Rewrite H in H4; 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;
+ 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))
+ ).
+ XElim b; Intros;
+ Try EqFalse; Pc3Unfold; Repeat Pr3Gen;
+ Try (Rewrite H0 in H3; TGenBase);
+ Rewrite H1 in H0; Clear H H1 x;
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))
- ).
+ 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.
+ Pc3Unfold; Repeat Pr3Gen; Rewrite H1 in H; Clear H1 x.
+ LiftGenBase; Rewrite H in H3; Rewrite H1 in H4; Rewrite H2 in H5; XEAuto.
Qed.
End pc3_gen.
Tactic Definition Pc3Gen :=
Match Context With
+ | [H: (pc3 ?1 (TSort ?2) (TSort ?3)) |- ? ] ->
+ LApply (pc3_gen_sort ?1 ?2 ?3); [ Clear H; Intros | XAuto ]
| [ _: (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 ];
_: (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.
+ XElim H_x; Intros
+ | _ -> Pr3Gen.