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Map.restrict

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Adam Chlipala 2016-03-24 08:28:53 -04:00
parent c279d3d610
commit 90e194c27e
1 changed files with 16 additions and 3 deletions
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17
Map.v
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@ -6,11 +6,13 @@ Module Type S.
Parameter fmap : Type -> Type -> Type. Parameter fmap : Type -> Type -> Type.
Parameter empty : forall A B, fmap A B. Parameter empty : forall A B, fmap A B.
Parameter lookup : forall A B, fmap A B -> A -> option B.
Parameter add : forall A B, fmap A B -> A -> B -> fmap A B. Parameter add : forall A B, fmap A B -> A -> B -> fmap A B.
Parameter remove : forall A B, fmap A B -> A -> fmap A B. Parameter remove : forall A B, fmap A B -> A -> fmap A B.
Parameter join : forall A B, fmap A B -> fmap A B -> fmap A B. Parameter join : forall A B, fmap A B -> fmap A B -> fmap A B.
Parameter merge : forall A B, (option B -> option B -> option B) -> fmap A B -> fmap A B -> fmap A B. Parameter merge : forall A B, (option B -> option B -> option B) -> fmap A B -> fmap A B -> fmap A B.
Parameter lookup : forall A B, fmap A B -> A -> option B. Parameter restrict : forall A B, (A -> bool) -> fmap A B -> fmap A B.
Parameter includes : forall A B, fmap A B -> fmap A B -> Prop. Parameter includes : forall A B, fmap A B -> fmap A B -> Prop.
Notation "$0" := (empty _ _). Notation "$0" := (empty _ _).
@ -119,6 +121,9 @@ Module Type S.
Axiom dom_add : forall A B (m : fmap A B) (k : A) (v : B), Axiom dom_add : forall A B (m : fmap A B) (k : A) (v : B),
dom (add m k v) = {k} \cup dom m. dom (add m k v) = {k} \cup dom m.
Axiom lookup_restrict : forall A B (f : A -> bool) (m : fmap A B) k,
lookup (restrict f m) k = if f k then lookup m k else None.
Hint Extern 1 => match goal with Hint Extern 1 => match goal with
| [ H : lookup (empty _ _) _ = Some _ |- _ ] => | [ H : lookup (empty _ _) _ = Some _ |- _ ] =>
rewrite lookup_empty in H; discriminate rewrite lookup_empty in H; discriminate
@ -127,7 +132,7 @@ Module Type S.
Hint Resolve includes_lookup includes_add empty_includes. Hint Resolve includes_lookup includes_add empty_includes.
Hint Rewrite lookup_empty lookup_add_eq lookup_add_ne lookup_remove_eq lookup_remove_ne Hint Rewrite lookup_empty lookup_add_eq lookup_add_ne lookup_remove_eq lookup_remove_ne
lookup_merge using congruence. lookup_merge lookup_restrict using congruence.
Hint Rewrite dom_empty dom_add. Hint Rewrite dom_empty dom_add.
@ -182,6 +187,8 @@ Module M : S.
Definition merge A B f (m1 m2 : fmap A B) : fmap A B := Definition merge A B f (m1 m2 : fmap A B) : fmap A B :=
fun k => f (m1 k) (m2 k). fun k => f (m1 k) (m2 k).
Definition lookup A B (m : fmap A B) (k : A) := m k. Definition lookup A B (m : fmap A B) (k : A) := m k.
Definition restrict A B (f : A -> bool) (m : fmap A B) : fmap A B :=
fun k => if f k then m k else None.
Definition includes A B (m1 m2 : fmap A B) := Definition includes A B (m1 m2 : fmap A B) :=
forall k v, m1 k = Some v -> m2 k = Some v. forall k v, m1 k = Some v -> m2 k = Some v.
@ -406,6 +413,12 @@ Module M : S.
unfold lookup, add; simpl; intros. unfold lookup, add; simpl; intros.
destruct (decide (k' = k)); intuition congruence. destruct (decide (k' = k)); intuition congruence.
Qed. Qed.
Theorem lookup_restrict : forall A B (f : A -> bool) (m : fmap A B) k,
lookup (restrict f m) k = if f k then lookup m k else None.
Proof.
auto.
Qed.
End M. End M.
Export M. Export M.