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1 changed files with 16 additions and 3 deletions
19
Map.v
19
Map.v
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@ -6,11 +6,13 @@ Module Type S.
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Parameter fmap : Type -> Type -> Type.
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Parameter empty : forall A B, fmap A B.
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Parameter lookup : forall A B, fmap A B -> A -> option B.
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Parameter add : forall A B, fmap A B -> A -> B -> fmap A B.
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Parameter remove : forall A B, fmap A B -> A -> fmap A B.
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Parameter join : forall A B, fmap A B -> fmap A B -> fmap A B.
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Parameter merge : forall A B, (option B -> option B -> option B) -> fmap A B -> fmap A B -> fmap A B.
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Parameter lookup : forall A B, fmap A B -> A -> option B.
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Parameter restrict : forall A B, (A -> bool) -> fmap A B -> fmap A B.
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Parameter includes : forall A B, fmap A B -> fmap A B -> Prop.
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Notation "$0" := (empty _ _).
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@ -119,6 +121,9 @@ Module Type S.
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Axiom dom_add : forall A B (m : fmap A B) (k : A) (v : B),
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dom (add m k v) = {k} \cup dom m.
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Axiom lookup_restrict : forall A B (f : A -> bool) (m : fmap A B) k,
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lookup (restrict f m) k = if f k then lookup m k else None.
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Hint Extern 1 => match goal with
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| [ H : lookup (empty _ _) _ = Some _ |- _ ] =>
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rewrite lookup_empty in H; discriminate
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@ -127,7 +132,7 @@ Module Type S.
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Hint Resolve includes_lookup includes_add empty_includes.
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Hint Rewrite lookup_empty lookup_add_eq lookup_add_ne lookup_remove_eq lookup_remove_ne
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lookup_merge using congruence.
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lookup_merge lookup_restrict using congruence.
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Hint Rewrite dom_empty dom_add.
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@ -182,6 +187,8 @@ Module M : S.
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Definition merge A B f (m1 m2 : fmap A B) : fmap A B :=
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fun k => f (m1 k) (m2 k).
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Definition lookup A B (m : fmap A B) (k : A) := m k.
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Definition restrict A B (f : A -> bool) (m : fmap A B) : fmap A B :=
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fun k => if f k then m k else None.
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Definition includes A B (m1 m2 : fmap A B) :=
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forall k v, m1 k = Some v -> m2 k = Some v.
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@ -405,7 +412,13 @@ Module M : S.
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Proof.
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unfold lookup, add; simpl; intros.
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destruct (decide (k' = k)); intuition congruence.
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Qed.
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Qed.
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Theorem lookup_restrict : forall A B (f : A -> bool) (m : fmap A B) k,
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lookup (restrict f m) k = if f k then lookup m k else None.
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Proof.
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auto.
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Qed.
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End M.
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Export M.
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