+++ /dev/null
-Require Export contexts_defs.
-Require Export lift_defs.
-
-(*#* #caption "current axioms for dropping",
- "base case", "untouched tail item",
- "dropped tail item", "updated tail item"
-*)
-(*#* #cap #alpha c in C1, e in C2, u in V, k in z, n in k, d in i, r in q *)
-
- Inductive drop: nat -> nat -> C -> C -> Prop :=
- | drop_sort: (h,d,n:?) (drop h d (CSort n) (CSort n))
- | drop_comp: (c,e:?) (drop (0) (0) c e) ->
- (k:?; u:?) (drop (0) (0) (CTail c k u) (CTail e k u))
- | drop_drop: (k:?; h:?; c,e:?) (drop (r k h) (0) c e) ->
- (u:?) (drop (S h) (0) (CTail c k u) e)
- | drop_skip: (k:?; h,d:?; c,e:?) (drop h (r k d) c e) -> (u:?)
- (drop h (S d) (CTail c k (lift h (r k d) u)) (CTail e k u)).
-
-(*#* #stop file *)
-
- Hint drop : ltlc := Constructors drop.
-
- Hint discr : ltlc := Extern 4 (drop ? ? ? ?) Simpl.
-
- Section drop_gen_base. (**************************************************)
-
- Theorem drop_gen_sort: (n,h,d:?; x:?)
- (drop h d (CSort n) x) -> x = (CSort n).
- Intros until 1; InsertEq H '(CSort n); XElim H; Intros;
- Try Inversion H1; XAuto.
- Qed.
-
- Theorem drop_gen_refl: (x,e:?) (drop (0) (0) x e) -> x = e.
- Intros until 1; Repeat InsertEq H '(0); XElim H; Intros.
-(* case 1: drop_sort *)
- XAuto.
-(* case 2: drop_comp *)
- Rewrite H0; XAuto.
-(* case 3: drop_drop *)
- Inversion H2.
-(* case 4: drop_skip *)
- Inversion H1.
- Qed.
-
- Theorem drop_gen_drop: (k:?; c,x:?; u:?; h:?)
- (drop (S h) (0) (CTail c k u) x) ->
- (drop (r k h) (0) c x).
- Intros until 1;
- InsertEq H '(CTail c k u); InsertEq H '(0); InsertEq H '(S h);
- XElim H; Intros.
-(* case 1: drop_sort *)
- Inversion H1.
-(* case 2: drop_comp *)
- Inversion H1.
-(* case 3: drop_drop *)
- Inversion H1; Inversion H3.
- Rewrite <- H5; Rewrite <- H6; Rewrite <- H7; XAuto.
-(* case 4: drop_skip *)
- Inversion H2.
- Qed.
-
- Theorem drop_gen_skip_r: (c,x:?; u:?; h,d:?; k:?)
- (drop h (S d) x (CTail c k u)) ->
- (EX e | x = (CTail e k (lift h (r k d) u)) & (drop h (r k d) e c)).
- Intros; Inversion_clear H; XEAuto.
- Qed.
-
- Theorem drop_gen_skip_l: (c,x:?; u:?; h,d:?; k:?)
- (drop h (S d) (CTail c k u) x) ->
- (EX e v | x = (CTail e k v) &
- u = (lift h (r k d) v) &
- (drop h (r k d) c e)
- ).
- Intros; Inversion_clear H; XEAuto.
- Qed.
-
- End drop_gen_base.
-
- Hints Resolve drop_gen_refl : ltlc.
-
- Tactic Definition DropGenBase :=
- Match Context With
- | [ H: (drop (0) (0) ?0 ?1) |- ? ] ->
- LApply (drop_gen_refl ?0 ?1); [ Clear H; Intros | XAuto ]
- | [ H: (drop ?0 ?1 (CSort ?2) ?3) |- ? ] ->
- LApply (drop_gen_sort ?2 ?0 ?1 ?3); [ Clear H; Intros | XAuto ]
- | [ H: (drop (S ?0) (0) (CTail ?1 ?2 ?3) ?4) |- ? ] ->
- LApply (drop_gen_drop ?2 ?1 ?4 ?3 ?0); [ Clear H; Intros | XAuto ]
- | [ H: (drop ?1 (S ?2) ?3 (CTail ?4 ?5 ?6)) |- ? ] ->
- LApply (drop_gen_skip_r ?4 ?3 ?6 ?1 ?2 ?5); [ Clear H; Intros H | XAuto ];
- XElim H; Intros
- | [ H: (drop ?1 (S ?2) (CTail ?4 ?5 ?6) ?3) |- ? ] ->
- LApply (drop_gen_skip_l ?4 ?3 ?6 ?1 ?2 ?5); [ Clear H; Intros H | XAuto ];
- XElim H; Intros.
-
- Section drop_props. (*****************************************************)
-
- Theorem drop_skip_bind: (h,d:?; c,e:?) (drop h d c e) -> (b:?; u:?)
- (drop h (S d) (CTail c (Bind b) (lift h d u)) (CTail e (Bind b) u)).
- Intros; Pattern 2 d; Replace d with (r (Bind b) d); XAuto.
- Qed.
-
- Theorem drop_refl: (c:?) (drop (0) (0) c c).
- XElim c; XAuto.
- Qed.
-
- Hints Resolve drop_refl : ltlc.
-
- Theorem drop_S: (b:?; c,e:?; u:?; h:?)
- (drop h (0) c (CTail e (Bind b) u)) ->
- (drop (S h) (0) c e).
- XElim c.
-(* case 1: CSort *)
- Intros; DropGenBase; Inversion H.
-(* case 2: CTail *)
- XElim h; Intros; DropGenBase.
-(* case 2.1: h = 0 *)
- Inversion H0; XAuto.
-(* case 2.1: h > 0 *)
- Apply drop_drop; RRw; XEAuto. (**) (* explicit constructor *)
- Qed.
-
- End drop_props.
-
- Hints Resolve drop_skip_bind drop_refl drop_S : ltlc.
-
- Tactic Definition DropS :=
- Match Context With
- [ _: (drop ?1 (0) ?2 (CTail ?3 (Bind ?4) ?5)) |- ? ] ->
- LApply (drop_S ?4 ?2 ?3 ?5 ?1); [ Intros | XAuto ].