Commit b778f3eb authored by Robbert Krebbers's avatar Robbert Krebbers

Merge branch 'dfrumin/coq-stdpp-map_properties'

parents 9b01e3d8 cd0b7f48
......@@ -843,14 +843,6 @@ Proof.
rewrite elem_of_map_to_list in Hj; simplify_option_eq.
Qed.
(** Properties of the zip_with function *)
Lemma map_lookup_zip_with {A B C} (f : A B C) m1 m2 i :
map_zip_with f m1 m2 !! i = x m1 !! i; y m2 !! i; Some (f x y).
Proof.
unfold map_zip_with. rewrite lookup_merge by done.
by destruct (m1 !! i), (m2 !! i).
Qed.
(** ** Properties of conversion from collections *)
Section map_of_to_collection.
Context {A : Type} `{FinCollection B C}.
......@@ -1136,6 +1128,79 @@ Lemma insert_merge_r m1 m2 i x z :
Proof. by intros; apply partial_alter_merge_r. Qed.
End more_merge.
(** Properties of the zip_with function *)
Lemma map_lookup_zip_with {A B C} (f : A B C) m1 m2 i :
map_zip_with f m1 m2 !! i = x m1 !! i; y m2 !! i; Some (f x y).
Proof.
unfold map_zip_with. rewrite lookup_merge by done.
by destruct (m1 !! i), (m2 !! i).
Qed.
Lemma map_zip_with_empty {A B C} (f : A B C) :
map_zip_with f = .
Proof.
unfold map_zip_with. by rewrite merge_empty by done.
Qed.
Lemma map_insert_zip_with {A B C} (f : A B C) m1 m2 i x y z :
f y z = x
<[i:=x]>(map_zip_with f m1 m2) = map_zip_with f (<[i:=y]>m1) (<[i:=z]>m2).
Proof.
intros Hf. unfold map_zip_with.
erewrite insert_merge; [ auto | by compute | by rewrite Hf ].
Qed.
Lemma map_zip_with_fmap {A' A B' B C} (f : A B C)
(g1 : A' A) (g2 : B' B) m1 m2 :
map_zip_with f (g1 <$> m1) (g2 <$> m2) = map_zip_with (λ x y, f (g1 x) (g2 y)) m1 m2.
Proof.
apply map_eq; intro i.
rewrite ?map_lookup_zip_with. rewrite ?lookup_fmap.
by destruct (m1 !! i), (m2 !! i).
Qed.
Lemma map_zip_with_fmap_1 {A' A B C} (f : A B C)
(g : A' A) m1 m2 :
map_zip_with f (g <$> m1) m2 = map_zip_with (λ x y, f (g x) y) m1 m2.
Proof.
rewrite <- (map_fmap_id m2) at 1.
by rewrite map_zip_with_fmap; simpl.
Qed.
Lemma map_zip_with_fmap_2 {A B' B C} (f : A B C)
(g : B' B) m1 m2 :
map_zip_with f m1 (g <$> m2) = map_zip_with (λ x y, f x (g y)) m1 m2.
Proof.
rewrite <- (map_fmap_id m1) at 1.
by rewrite map_zip_with_fmap; simpl.
Qed.
Lemma map_fmap_zip_with {A B C D} (f : A B C) (g : C D) m1 m2 :
g <$> map_zip_with f m1 m2 = map_zip_with (λ x y, g (f x y)) m1 m2.
Proof.
apply map_eq; intro i.
rewrite lookup_fmap. rewrite ?map_lookup_zip_with.
by destruct (m1 !! i), (m2 !! i).
Qed.
Lemma map_zip_with_map_zip {A B C} (f : A B C) m1 m2 :
map_zip_with f m1 m2 = curry f <$> map_zip m1 m2.
Proof.
apply map_eq; intro i.
rewrite lookup_fmap; rewrite ?map_lookup_zip_with; rewrite ?lookup_fmap.
by destruct (m1 !! i), (m2 !! i).
Qed.
Lemma map_fmap_zip {A' A B' B} (g1 : A' A) (g2 : B' B) m1 m2 :
map_zip (fmap g1 m1) (fmap g2 m2)
= prod_map g1 g2 <$> map_zip m1 m2.
Proof.
rewrite map_zip_with_fmap.
rewrite map_zip_with_map_zip.
generalize (map_zip m1 m2); intro m. apply map_eq; intro i.
by rewrite ?lookup_fmap; destruct (m !! i) as [[x1 x2]|].
Qed.
(** ** Properties on the [map_relation] relation *)
Section Forall2.
Context {A B} (R : A B Prop) (P : A Prop) (Q : B Prop).
......
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