contribution about \lambda-\delta

[helm.git] / helm / coq-contribs / LAMBDA-TYPES / drop_defs.v

diff --git a/helm/coq-contribs/LAMBDA-TYPES/drop_defs.v b/helm/coq-contribs/LAMBDA-TYPES/drop_defs.v
new file mode 100644 (file)
index 0000000..e1576c5
--- /dev/null
+++ b/helm/coq-contribs/LAMBDA-TYPES/drop_defs.v
@@ -0,0 +1,125 @@
+(*#* #stop file *)
+
+Require Export contexts_defs.
+Require Export lift_defs.
+
+      Inductive drop : nat -> nat -> C -> C -> Prop :=
+         | drop_sort : (h,d,n:?) (drop h d (CSort n) (CSort n))
+         | drop_tail : (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)).
+
+      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_tail *)
+      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_tail *)
+      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 ].
+