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Rename [map] to [fmap]

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Adam Chlipala 2016-02-09 09:07:37 -05:00
parent 9720a6e0c6
commit 087e9334d8
3 changed files with 41 additions and 41 deletions
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2
Interpreters.v
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@ -26,7 +26,7 @@ Example ex2 := Plus (Var "y") (Times (Var "x") (Const 3)).
* We need to consider the meaning to be a function over a valuation * We need to consider the meaning to be a function over a valuation
* to the variables, which in turn is itself a finite map from variable * to the variables, which in turn is itself a finite map from variable
* names to numbers. We use the book library's [map] type family. *) * names to numbers. We use the book library's [map] type family. *)
Definition valuation := map var nat. Definition valuation := fmap var nat.
(* That is, the domain is [var] (a synonym for [string]) and the codomain/range (* That is, the domain is [var] (a synonym for [string]) and the codomain/range
* is [nat]. *) * is [nat]. *)

2
Interpreters_template.v
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@ -14,7 +14,7 @@ Inductive arith : Set :=
Example ex1 := Const 42. Example ex1 := Const 42.
Example ex2 := Plus (Var "y") (Times (Var "x") (Const 3)). Example ex2 := Plus (Var "y") (Times (Var "x") (Const 3)).
Definition valuation := map var nat. Definition valuation := fmap var nat.
(* A valuation is a finite map from [var] to [nat]. *) (* A valuation is a finite map from [var] to [nat]. *)
(* The interpreter is a fairly innocuous-looking recursive function. *) (* The interpreter is a fairly innocuous-looking recursive function. *)

78
Map.v
View file

@ -3,13 +3,13 @@ Require Import Classical Sets ClassicalEpsilon FunctionalExtensionality.
Set Implicit Arguments. Set Implicit Arguments.
Module Type S. Module Type S.
Parameter map : Type -> Type -> Type. Parameter fmap : Type -> Type -> Type.
Parameter empty : forall A B, map A B. Parameter empty : forall A B, fmap A B.
Parameter add : forall A B, map A B -> A -> B -> map A B. Parameter add : forall A B, fmap A B -> A -> B -> fmap A B.
Parameter join : forall A B, map A B -> map A B -> map A B. Parameter join : forall A B, fmap A B -> fmap A B -> fmap A B.
Parameter lookup : forall A B, map A B -> A -> option B. Parameter lookup : forall A B, fmap A B -> A -> option B.
Parameter includes : forall A B, map A B -> map 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 $+ ( k , v )" := (add m k v) (at level 50, left associativity).
@ -17,51 +17,51 @@ Module Type S.
Infix "$?" := lookup (at level 50, no associativity). Infix "$?" := lookup (at level 50, no associativity).
Infix "$<=" := includes (at level 90). Infix "$<=" := includes (at level 90).
Parameter dom : forall A B, map A B -> set A. Parameter dom : forall A B, fmap A B -> set A.
Axiom map_ext : forall A B (m1 m2 : map A B), Axiom fmap_ext : forall A B (m1 m2 : fmap A B),
(forall k, m1 $? k = m2 $? k) (forall k, m1 $? k = m2 $? k)
-> m1 = m2. -> m1 = m2.
Axiom lookup_empty : forall A B k, empty A B $? k = None. Axiom lookup_empty : forall A B k, empty A B $? k = None.
Axiom includes_lookup : forall A B (m m' : map A B) k v, Axiom includes_lookup : forall A B (m m' : fmap A B) k v,
m $? k = Some v m $? k = Some v
-> m $<= m' -> m $<= m'
-> lookup m' k = Some v. -> lookup m' k = Some v.
Axiom includes_add : forall A B (m m' : map A B) k v, Axiom includes_add : forall A B (m m' : fmap A B) k v,
m $<= m' m $<= m'
-> add m k v $<= add m' k v. -> add m k v $<= add m' k v.
Axiom lookup_add_eq : forall A B (m : map A B) k1 k2 v, Axiom lookup_add_eq : forall A B (m : fmap A B) k1 k2 v,
k1 = k2 k1 = k2
-> add m k1 v $? k2 = Some v. -> add m k1 v $? k2 = Some v.
Axiom lookup_add_ne : forall A B (m : map A B) k k' v, Axiom lookup_add_ne : forall A B (m : fmap A B) k k' v,
k' <> k k' <> k
-> add m k v $? k' = m $? k'. -> add m k v $? k' = m $? k'.
Axiom lookup_join1 : forall A B (m1 m2 : map 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.
Axiom lookup_join2 : forall A B (m1 m2 : map A B) k, Axiom lookup_join2 : forall A B (m1 m2 : fmap A B) k,
~k \in dom m1 ~k \in dom m1
-> (m1 $++ m2) $? k = m2 $? k. -> (m1 $++ m2) $? k = m2 $? k.
Axiom join_comm : forall A B (m1 m2 : map A B), Axiom join_comm : forall A B (m1 m2 : fmap A B),
dom m1 \cap dom m2 = {} dom m1 \cap dom m2 = {}
-> m1 $++ m2 = m2 $++ m1. -> m1 $++ m2 = m2 $++ m1.
Axiom join_assoc : forall A B (m1 m2 m3 : map A B), Axiom join_assoc : forall A B (m1 m2 m3 : fmap A B),
(m1 $++ m2) $++ m3 = m1 $++ (m2 $++ m3). (m1 $++ m2) $++ m3 = m1 $++ (m2 $++ m3).
Axiom empty_includes : forall A B (m : map A B), empty A B $<= m. Axiom empty_includes : forall A B (m : fmap A B), empty A B $<= m.
Axiom dom_empty : forall A B, dom (empty A B) = {}. Axiom dom_empty : forall A B, dom (empty A B) = {}.
Axiom dom_add : forall A B (m : map 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.
Hint Extern 1 => match goal with Hint Extern 1 => match goal with
@ -74,7 +74,7 @@ Module Type S.
Hint Rewrite lookup_add_eq lookup_add_ne using congruence. Hint Rewrite lookup_add_eq lookup_add_ne using congruence.
Ltac maps_equal := Ltac maps_equal :=
apply map_ext; intros; apply fmap_ext; intros;
repeat (subst; autorewrite with core; try reflexivity; repeat (subst; autorewrite with core; try reflexivity;
match goal with match goal with
| [ |- context[lookup (add _ ?k _) ?k' ] ] => destruct (classic (k = k')); subst | [ |- context[lookup (add _ ?k _) ?k' ] ] => destruct (classic (k = k')); subst
@ -84,9 +84,9 @@ Module Type S.
End S. End S.
Module M : S. Module M : S.
Definition map (A B : Type) := A -> option B. Definition fmap (A B : Type) := A -> option B.
Definition empty A B : map A B := fun _ => None. Definition empty A B : fmap A B := fun _ => None.
Section decide. Section decide.
Variable P : Prop. Variable P : Prop.
@ -102,20 +102,20 @@ Module M : S.
epsilon decided (fun _ => True). epsilon decided (fun _ => True).
End decide. End decide.
Definition add A B (m : map A B) (k : A) (v : B) : map 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 join A B (m1 m2 : map A B) : map 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
| x => x | x => x
end. end.
Definition lookup A B (m : map A B) (k : A) := m k. Definition lookup A B (m : fmap A B) (k : A) := m k.
Definition includes A B (m1 m2 : map 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.
Definition dom A B (m : map A B) : set A := fun x => m x <> None. Definition dom A B (m : fmap A B) : set A := fun x => m x <> None.
Theorem map_ext : forall A B (m1 m2 : map A B), Theorem fmap_ext : forall A B (m1 m2 : fmap A B),
(forall k, lookup m1 k = lookup m2 k) (forall k, lookup m1 k = lookup m2 k)
-> m1 = m2. -> m1 = m2.
Proof. Proof.
@ -127,7 +127,7 @@ Module M : S.
auto. auto.
Qed. Qed.
Theorem includes_lookup : forall A B (m m' : map A B) k v, Theorem includes_lookup : forall A B (m m' : fmap A B) k v,
lookup m k = Some v lookup m k = Some v
-> includes m m' -> includes m m'
-> lookup m' k = Some v. -> lookup m' k = Some v.
@ -135,7 +135,7 @@ Module M : S.
auto. auto.
Qed. Qed.
Theorem includes_add : forall A B (m m' : map A B) k v, Theorem includes_add : forall A B (m m' : fmap A B) k v,
includes m m' includes m m'
-> includes (add m k v) (add m' k v). -> includes (add m k v) (add m' k v).
Proof. Proof.
@ -143,7 +143,7 @@ Module M : S.
destruct (decide (k0 = k)); auto. destruct (decide (k0 = k)); auto.
Qed. Qed.
Theorem lookup_add_eq : forall A B (m : map A B) k1 k2 v, Theorem lookup_add_eq : forall A B (m : fmap A B) k1 k2 v,
k1 = k2 k1 = k2
-> lookup (add m k1 v) k2 = Some v. -> lookup (add m k1 v) k2 = Some v.
Proof. Proof.
@ -152,7 +152,7 @@ Module M : S.
congruence. congruence.
Qed. Qed.
Theorem lookup_add_ne : forall A B (m : map A B) k k' v, Theorem lookup_add_ne : forall A B (m : fmap A B) k k' v,
k' <> k k' <> k
-> lookup (add m k v) k' = lookup m k'. -> lookup (add m k v) k' = lookup m k'.
Proof. Proof.
@ -160,7 +160,7 @@ Module M : S.
destruct (decide (k' = k)); intuition. destruct (decide (k' = k)); intuition.
Qed. Qed.
Theorem lookup_join1 : forall A B (m1 m2 : map 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.
Proof. Proof.
@ -168,7 +168,7 @@ Module M : S.
destruct (m1 k); congruence. destruct (m1 k); congruence.
Qed. Qed.
Theorem lookup_join2 : forall A B (m1 m2 : map A B) k, Theorem lookup_join2 : forall A B (m1 m2 : fmap A B) k,
~k \in dom m1 ~k \in dom m1
-> lookup (join m1 m2) k = lookup m2 k. -> lookup (join m1 m2) k = lookup m2 k.
Proof. Proof.
@ -177,11 +177,11 @@ Module M : S.
exfalso; apply H; congruence. exfalso; apply H; congruence.
Qed. Qed.
Theorem join_comm : forall A B (m1 m2 : map A B), Theorem join_comm : forall A B (m1 m2 : fmap A B),
dom m1 \cap dom m2 = {} dom m1 \cap dom m2 = {}
-> join m1 m2 = join m2 m1. -> join m1 m2 = join m2 m1.
Proof. Proof.
intros; apply map_ext; unfold join, lookup; intros. intros; apply fmap_ext; unfold join, lookup; intros.
apply (f_equal (fun f => f k)) in H. apply (f_equal (fun f => f k)) in H.
unfold dom, intersection, constant in H; simpl in H. unfold dom, intersection, constant in H; simpl in H.
destruct (m1 k), (m2 k); auto. destruct (m1 k), (m2 k); auto.
@ -189,14 +189,14 @@ Module M : S.
intuition congruence. intuition congruence.
Qed. Qed.
Theorem join_assoc : forall A B (m1 m2 m3 : map A B), Theorem join_assoc : forall A B (m1 m2 m3 : fmap A B),
join (join m1 m2) m3 = join m1 (join m2 m3). join (join m1 m2) m3 = join m1 (join m2 m3).
Proof. Proof.
intros; apply map_ext; unfold join, lookup; intros. intros; apply fmap_ext; unfold join, lookup; intros.
destruct (m1 k); auto. destruct (m1 k); auto.
Qed. Qed.
Theorem empty_includes : forall A B (m : map A B), includes (empty (A := A) B) m. Theorem empty_includes : forall A B (m : fmap A B), includes (empty (A := A) B) m.
Proof. Proof.
unfold includes, empty; intuition congruence. unfold includes, empty; intuition congruence.
Qed. Qed.
@ -206,7 +206,7 @@ Module M : S.
unfold dom, empty; intros; sets idtac. unfold dom, empty; intros; sets idtac.
Qed. Qed.
Theorem dom_add : forall A B (m : map A B) (k : A) (v : B), Theorem 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.
Proof. Proof.
unfold dom, add; simpl; intros. unfold dom, add; simpl; intros.