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Rename [map] to [fmap]
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Adam Chlipala
parent
9720a6e0c6
commit
087e9334d8
3 changed files with 41 additions and 41 deletions
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@ -26,7 +26,7 @@ Example ex2 := Plus (Var "y") (Times (Var "x") (Const 3)).
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* We need to consider the meaning to be a function over a valuation
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* to the variables, which in turn is itself a finite map from variable
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* names to numbers. We use the book library's [map] type family. *)
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Definition valuation := map var nat.
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Definition valuation := fmap var nat.
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(* That is, the domain is [var] (a synonym for [string]) and the codomain/range
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* is [nat]. *)
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@ -14,7 +14,7 @@ Inductive arith : Set :=
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Example ex1 := Const 42.
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Example ex2 := Plus (Var "y") (Times (Var "x") (Const 3)).
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Definition valuation := map var nat.
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Definition valuation := fmap var nat.
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(* A valuation is a finite map from [var] to [nat]. *)
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(* The interpreter is a fairly innocuous-looking recursive function. *)
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78
Map.v
78
Map.v
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@ -3,13 +3,13 @@ Require Import Classical Sets ClassicalEpsilon FunctionalExtensionality.
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Set Implicit Arguments.
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Module Type S.
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Parameter map : Type -> Type -> Type.
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Parameter fmap : Type -> Type -> Type.
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Parameter empty : forall A B, map A B.
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Parameter add : forall A B, map A B -> A -> B -> map A B.
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Parameter join : forall A B, map A B -> map A B -> map A B.
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Parameter lookup : forall A B, map A B -> A -> option B.
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Parameter includes : forall A B, map A B -> map A B -> Prop.
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Parameter empty : forall A B, fmap A B.
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Parameter add : forall A B, fmap A B -> A -> B -> 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 lookup : forall A B, fmap A B -> A -> option 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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Notation "m $+ ( k , v )" := (add m k v) (at level 50, left associativity).
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@ -17,51 +17,51 @@ Module Type S.
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Infix "$?" := lookup (at level 50, no associativity).
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Infix "$<=" := includes (at level 90).
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Parameter dom : forall A B, map A B -> set A.
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Parameter dom : forall A B, fmap A B -> set A.
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Axiom map_ext : forall A B (m1 m2 : map A B),
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Axiom fmap_ext : forall A B (m1 m2 : fmap A B),
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(forall k, m1 $? k = m2 $? k)
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-> m1 = m2.
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Axiom lookup_empty : forall A B k, empty A B $? k = None.
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Axiom includes_lookup : forall A B (m m' : map A B) k v,
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Axiom includes_lookup : forall A B (m m' : fmap A B) k v,
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m $? k = Some v
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-> m $<= m'
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-> lookup m' k = Some v.
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Axiom includes_add : forall A B (m m' : map A B) k v,
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Axiom includes_add : forall A B (m m' : fmap A B) k v,
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m $<= m'
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-> add m k v $<= add m' k v.
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Axiom lookup_add_eq : forall A B (m : map A B) k1 k2 v,
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Axiom lookup_add_eq : forall A B (m : fmap A B) k1 k2 v,
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k1 = k2
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-> add m k1 v $? k2 = Some v.
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Axiom lookup_add_ne : forall A B (m : map A B) k k' v,
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Axiom lookup_add_ne : forall A B (m : fmap A B) k k' v,
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k' <> k
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-> add m k v $? k' = m $? k'.
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Axiom lookup_join1 : forall A B (m1 m2 : map A B) k,
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Axiom lookup_join1 : forall A B (m1 m2 : fmap A B) k,
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k \in dom m1
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-> (m1 $++ m2) $? k = m1 $? k.
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Axiom lookup_join2 : forall A B (m1 m2 : map A B) k,
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Axiom lookup_join2 : forall A B (m1 m2 : fmap A B) k,
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~k \in dom m1
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-> (m1 $++ m2) $? k = m2 $? k.
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Axiom join_comm : forall A B (m1 m2 : map A B),
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Axiom join_comm : forall A B (m1 m2 : fmap A B),
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dom m1 \cap dom m2 = {}
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-> m1 $++ m2 = m2 $++ m1.
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Axiom join_assoc : forall A B (m1 m2 m3 : map A B),
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Axiom join_assoc : forall A B (m1 m2 m3 : fmap A B),
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(m1 $++ m2) $++ m3 = m1 $++ (m2 $++ m3).
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Axiom empty_includes : forall A B (m : map A B), empty A B $<= m.
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Axiom empty_includes : forall A B (m : fmap A B), empty A B $<= m.
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Axiom dom_empty : forall A B, dom (empty A B) = {}.
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Axiom dom_add : forall A B (m : map A B) (k : A) (v : B),
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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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Hint Extern 1 => match goal with
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@ -74,7 +74,7 @@ Module Type S.
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Hint Rewrite lookup_add_eq lookup_add_ne using congruence.
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Ltac maps_equal :=
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apply map_ext; intros;
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apply fmap_ext; intros;
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repeat (subst; autorewrite with core; try reflexivity;
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match goal with
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| [ |- context[lookup (add _ ?k _) ?k' ] ] => destruct (classic (k = k')); subst
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@ -84,9 +84,9 @@ Module Type S.
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End S.
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Module M : S.
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Definition map (A B : Type) := A -> option B.
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Definition fmap (A B : Type) := A -> option B.
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Definition empty A B : map A B := fun _ => None.
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Definition empty A B : fmap A B := fun _ => None.
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Section decide.
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Variable P : Prop.
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@ -102,20 +102,20 @@ Module M : S.
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epsilon decided (fun _ => True).
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End decide.
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Definition add A B (m : map A B) (k : A) (v : B) : map A B :=
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Definition add A B (m : fmap A B) (k : A) (v : B) : fmap A B :=
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fun k' => if decide (k' = k) then Some v else m k'.
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Definition join A B (m1 m2 : map A B) : map A B :=
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Definition join A B (m1 m2 : fmap A B) : fmap A B :=
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fun k => match m1 k with
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| None => m2 k
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| x => x
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end.
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Definition lookup A B (m : map A B) (k : A) := m k.
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Definition includes A B (m1 m2 : map A B) :=
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Definition lookup A B (m : fmap A B) (k : A) := m k.
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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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Definition dom A B (m : map A B) : set A := fun x => m x <> None.
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Definition dom A B (m : fmap A B) : set A := fun x => m x <> None.
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Theorem map_ext : forall A B (m1 m2 : map A B),
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Theorem fmap_ext : forall A B (m1 m2 : fmap A B),
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(forall k, lookup m1 k = lookup m2 k)
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-> m1 = m2.
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Proof.
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@ -127,7 +127,7 @@ Module M : S.
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auto.
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Qed.
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Theorem includes_lookup : forall A B (m m' : map A B) k v,
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Theorem includes_lookup : forall A B (m m' : fmap A B) k v,
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lookup m k = Some v
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-> includes m m'
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-> lookup m' k = Some v.
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@ -135,7 +135,7 @@ Module M : S.
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auto.
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Qed.
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Theorem includes_add : forall A B (m m' : map A B) k v,
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Theorem includes_add : forall A B (m m' : fmap A B) k v,
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includes m m'
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-> includes (add m k v) (add m' k v).
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Proof.
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@ -143,7 +143,7 @@ Module M : S.
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destruct (decide (k0 = k)); auto.
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Qed.
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Theorem lookup_add_eq : forall A B (m : map A B) k1 k2 v,
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Theorem lookup_add_eq : forall A B (m : fmap A B) k1 k2 v,
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k1 = k2
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-> lookup (add m k1 v) k2 = Some v.
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Proof.
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@ -152,7 +152,7 @@ Module M : S.
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congruence.
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Qed.
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Theorem lookup_add_ne : forall A B (m : map A B) k k' v,
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Theorem lookup_add_ne : forall A B (m : fmap A B) k k' v,
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k' <> k
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-> lookup (add m k v) k' = lookup m k'.
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Proof.
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@ -160,7 +160,7 @@ Module M : S.
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destruct (decide (k' = k)); intuition.
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Qed.
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Theorem lookup_join1 : forall A B (m1 m2 : map A B) k,
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Theorem lookup_join1 : forall A B (m1 m2 : fmap A B) k,
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k \in dom m1
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-> lookup (join m1 m2) k = lookup m1 k.
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Proof.
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@ -168,7 +168,7 @@ Module M : S.
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destruct (m1 k); congruence.
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Qed.
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Theorem lookup_join2 : forall A B (m1 m2 : map A B) k,
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Theorem lookup_join2 : forall A B (m1 m2 : fmap A B) k,
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~k \in dom m1
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-> lookup (join m1 m2) k = lookup m2 k.
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Proof.
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@ -177,11 +177,11 @@ Module M : S.
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exfalso; apply H; congruence.
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Qed.
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Theorem join_comm : forall A B (m1 m2 : map A B),
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Theorem join_comm : forall A B (m1 m2 : fmap A B),
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dom m1 \cap dom m2 = {}
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-> join m1 m2 = join m2 m1.
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Proof.
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intros; apply map_ext; unfold join, lookup; intros.
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intros; apply fmap_ext; unfold join, lookup; intros.
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apply (f_equal (fun f => f k)) in H.
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unfold dom, intersection, constant in H; simpl in H.
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destruct (m1 k), (m2 k); auto.
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@ -189,14 +189,14 @@ Module M : S.
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intuition congruence.
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Qed.
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Theorem join_assoc : forall A B (m1 m2 m3 : map A B),
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Theorem join_assoc : forall A B (m1 m2 m3 : fmap A B),
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join (join m1 m2) m3 = join m1 (join m2 m3).
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Proof.
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intros; apply map_ext; unfold join, lookup; intros.
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intros; apply fmap_ext; unfold join, lookup; intros.
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destruct (m1 k); auto.
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Qed.
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Theorem empty_includes : forall A B (m : map A B), includes (empty (A := A) B) m.
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Theorem empty_includes : forall A B (m : fmap A B), includes (empty (A := A) B) m.
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Proof.
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unfold includes, empty; intuition congruence.
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Qed.
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@ -206,7 +206,7 @@ Module M : S.
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unfold dom, empty; intros; sets idtac.
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Qed.
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Theorem dom_add : forall A B (m : map A B) (k : A) (v : B),
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Theorem 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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Proof.
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unfold dom, add; simpl; intros.
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