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Add Map remove

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Adam Chlipala 2016-02-09 22:44:03 -05:00
parent 19b98288ca
commit a0487bc153
1 changed files with 30 additions and 1 deletions
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31
Map.v
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@ -7,12 +7,14 @@ Module Type S.
Parameter empty : forall A B, fmap A B. Parameter empty : forall A B, fmap A 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 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 lookup : forall A B, fmap A B -> A -> option B. Parameter lookup : forall A B, fmap A B -> A -> option 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 _ _).
Notation "m $+ ( k , v )" := (add m k v) (at level 50, left associativity). Notation "m $+ ( k , v )" := (add m k v) (at level 50, left associativity).
Infix "$-" := remove (at level 50, left associativity).
Infix "$++" := join (at level 50, left associativity). Infix "$++" := join (at level 50, left associativity).
Infix "$?" := lookup (at level 50, no associativity). Infix "$?" := lookup (at level 50, no associativity).
Infix "$<=" := includes (at level 90). Infix "$<=" := includes (at level 90).
@ -42,6 +44,14 @@ Module Type S.
k' <> k k' <> k
-> add m k v $? k' = m $? k'. -> add m k v $? k' = m $? k'.
Axiom lookup_remove_eq : forall A B (m : fmap A B) k1 k2,
k1 = k2
-> remove m k1 $? k2 = None.
Axiom lookup_remove_ne : forall A B (m : fmap A B) k k',
k' <> k
-> remove m k $? k' = m $? k'.
Axiom lookup_join1 : forall A B (m1 m2 : fmap A B) k, Axiom lookup_join1 : forall A B (m1 m2 : fmap A B) k,
k \in dom m1 k \in dom m1
-> (m1 $++ m2) $? k = m1 $? k. -> (m1 $++ m2) $? k = m1 $? k.
@ -71,7 +81,7 @@ Module Type S.
Hint Resolve includes_lookup includes_add empty_includes. Hint Resolve includes_lookup includes_add empty_includes.
Hint Rewrite lookup_add_eq lookup_add_ne using congruence. Hint Rewrite lookup_empty lookup_add_eq lookup_add_ne lookup_remove_eq lookup_remove_ne using congruence.
Ltac maps_equal := Ltac maps_equal :=
apply fmap_ext; intros; apply fmap_ext; intros;
@ -104,6 +114,8 @@ Module M : S.
Definition add A B (m : fmap A B) (k : A) (v : B) : fmap A B := Definition add A B (m : fmap A B) (k : A) (v : B) : fmap A B :=
fun k' => if decide (k' = k) then Some v else m k'. fun k' => if decide (k' = k) then Some v else m k'.
Definition remove A B (m : fmap A B) (k : A) : fmap A B :=
fun k' => if decide (k' = k) then None else m k'.
Definition join A B (m1 m2 : fmap A B) : fmap A B := Definition join A B (m1 m2 : fmap A B) : fmap A B :=
fun k => match m1 k with fun k => match m1 k with
| None => m2 k | None => m2 k
@ -160,6 +172,23 @@ Module M : S.
destruct (decide (k' = k)); intuition. destruct (decide (k' = k)); intuition.
Qed. Qed.
Theorem lookup_remove_eq : forall A B (m : fmap A B) k1 k2,
k1 = k2
-> lookup (remove m k1) k2 = None.
Proof.
unfold lookup, remove; intuition.
destruct (decide (k2 = k1)); try tauto.
congruence.
Qed.
Theorem lookup_remove_ne : forall A B (m : fmap A B) k k',
k' <> k
-> lookup (remove m k) k' = lookup m k'.
Proof.
unfold lookup, remove; intuition.
destruct (decide (k' = k)); try tauto.
Qed.
Theorem lookup_join1 : forall A B (m1 m2 : fmap A B) k, Theorem lookup_join1 : forall A B (m1 m2 : fmap A B) k,
k \in dom m1 k \in dom m1
-> lookup (join m1 m2) k = lookup m1 k. -> lookup (join m1 m2) k = lookup m1 k.