(*#* #stop file *)
Require subst0_gen.
+Require subst0_lift.
Require drop_props.
Require pr0_gen.
+Require pr0_subst0.
Require pr2_defs.
Section pr2_gen. (********************************************************)
- Theorem pr2_gen_abbr : (c:?; u1,t1,x:?)
- (pr2 c (TTail (Bind Abbr) u1 t1) x) ->
- (EX u2 t2 | x = (TTail (Bind Abbr) u2 t2) &
- (pr2 c u1 u2) &
- ((b:?; u:?) (pr2 (CTail c (Bind b) u) t1 t2)) \/
- (EX y | (pr0 t1 y) & (subst0 (0) u2 y t2))
- ) \/
- (pr0 t1 (lift (1) (0) x)).
- Intros; Inversion H;
- Try Pr0Gen; Try Subst0Gen; XDEAuto 8.
+ Theorem pr2_gen_abbr: (c:?; u1,t1,x:?)
+ (pr2 c (TTail (Bind Abbr) u1 t1) x) ->
+ (EX u2 t2 | x = (TTail (Bind Abbr) u2 t2) &
+ (pr2 c u1 u2) & (OR
+ (b:?; u:?) (pr2 (CTail c (Bind b) u) t1 t2) |
+ (EX u | (pr0 u1 u) & (pr2 (CTail c (Bind Abbr) u) t1 t2)) |
+ (EX y z | (pr2 (CTail c (Bind Abbr) u1) t1 y) & (pr0 y z) & (pr2 (CTail c (Bind Abbr) u1) z t2))
+ )) \/ (b:?; u:?)
+ (pr2 (CTail c (Bind b) u) t1 (lift (1) (0) x)).
+ Intros; Inversion H;
+ Pr0Gen; Try Rewrite H1 in H2; Try Subst0Gen; Try Pr0Subst0; XDEAuto 10.
Qed.
- Theorem pr2_gen_void : (c:?; u1,t1,x:?)
- (pr2 c (TTail (Bind Void) u1 t1) x) ->
- (EX u2 t2 | x = (TTail (Bind Void) u2 t2) &
- (pr2 c u1 u2) & (b:?; u:?)
- (pr2 (CTail c (Bind b) u) t1 t2)
- ) \/
- (pr0 t1 (lift (1) (0) x)).
- Intros; Inversion H;
- Try Pr0Gen; Try Subst0Gen; XDEAuto 7.
+ Theorem pr2_gen_void: (c:?; u1,t1,x:?)
+ (pr2 c (TTail (Bind Void) u1 t1) x) ->
+ (EX u2 t2 | x = (TTail (Bind Void) u2 t2) &
+ (pr2 c u1 u2) & (b:?; u:?)
+ (pr2 (CTail c (Bind b) u) t1 t2)
+ ) \/ (b:?; u:?)
+ (pr2 (CTail c (Bind b) u) t1 (lift (1) (0) x)).
+ Intros; Inversion H;
+ Try Pr0Gen; Try Rewrite H1 in H2; Try Subst0Gen; XDEAuto 7.
Qed.
-(*#* #start file *)
-
(*#* #caption "generation lemma for lift" *)
(*#* #cap #cap c #alpha e in D, t1 in U1, t2 in U2, x in T, d in i *)
- Theorem pr2_gen_lift : (c:?; t1,x:?; h,d:?) (pr2 c (lift h d t1) x) ->
- (e:?) (drop h d c e) ->
- (EX t2 | x = (lift h d t2) & (pr2 e t1 t2)).
-
-(*#* #stop file *)
-
+ Theorem pr2_gen_lift: (c:?; t1,x:?; h,d:?) (pr2 c (lift h d t1) x) ->
+ (e:?) (drop h d c e) ->
+ (EX t2 | x = (lift h d t2) & (pr2 e t1 t2)).
Intros.
- Inversion H; Clear H.
-(* case 1 : pr2_pr0 *)
- Pr0Gen; XEAuto.
+ Inversion H; Clear H; Pr0Gen.
+(* case 1 : pr2_free *)
+ XEAuto.
(* case 2 : pr2_delta *)
+ Rewrite H in H3; Clear H H4 t t2.
Apply (lt_le_e i d); Intros.
(* case 2.1 : i < d *)
- Rewrite (lt_plus_minus i d) in H0; [ Idtac | XAuto ].
- Rewrite (lt_plus_minus i d) in H2; [ Idtac | XAuto ].
- DropDis; Rewrite H0 in H2; Clear H0 u.
+ Rewrite (lt_plus_minus i d) in H0; [ Idtac | XAuto ].
+ Rewrite (lt_plus_minus i d) in H3; [ Idtac | XAuto ].
+ DropDis; Rewrite H0 in H3; Clear H0 u.
Subst0Gen; Rewrite <- lt_plus_minus in H0; XEAuto.
(* case 2.2 : i >= d *)
Apply (lt_le_e i (plus d h)); Intros.
(* case 2.2.1 : i < d + h *)
- EApply subst0_gen_lift_false; [ Apply H | Apply H3 | XEAuto ].
+ EApply subst0_gen_lift_false; [ Apply H | Apply H4 | XEAuto ].
(* case 2.2.2 : i >= d + h *)
DropDis; Subst0Gen; XEAuto.
Qed.
Match Context With
| [ H: (pr2 ?1 (TTail (Bind Abbr) ?2 ?3) ?4) |- ? ] ->
LApply (pr2_gen_abbr ?1 ?2 ?3 ?4); [ Clear H; Intros H | XAuto ];
- XElim H;
- [ Intros H; XElim H; Do 4 Intro; Intros H_x; XElim H_x;
- [ Intros | Intros H_x; XElim H_x; Intros ]
- | Intros ]
+ XDecompose H
| [ H: (pr2 ?1 (TTail (Bind Void) ?2 ?3) ?4) |- ? ] ->
LApply (pr2_gen_void ?1 ?2 ?3 ?4); [ Clear H; Intros H | XAuto ];
- XElim H; [ Intros H; XElim H; Intros | Intros ]
+ XDecompose H
| [ H0: (pr2 ?1 (lift ?2 ?3 ?4) ?5);
H1: (drop ?2 ?3 ?1 ?6) |- ? ] ->
LApply (pr2_gen_lift ?1 ?4 ?5 ?2 ?3); [ Clear H0; Intros H0 | XAuto ];
LApply (H0 ?6); [ Clear H0; Intros H0 | XAuto ];
- XElim H0; Intros
+ XDecompose H0
| _ -> Pr2GenBase.
+