Commit 0e5917cf authored by Robbert Krebbers's avatar Robbert Krebbers
Browse files

Add `zip`/`zip_with` lemmas for big ops over maps.

parent ee535aa2
......@@ -417,6 +417,29 @@ Section gmap.
Qed.
End gmap.
Lemma big_opM_sep_zip_with `{Countable K} {A B C}
(f : A B C) (g1 : C A) (g2 : C B)
(h1 : K A M) (h2 : K B M) m1 m2 :
( x y, g1 (f x y) = x)
( x y, g2 (f x y) = y)
( k, is_Some (m1 !! k) is_Some (m2 !! k))
([^o map] kxy map_zip_with f m1 m2, h1 k (g1 xy) `o` h2 k (g2 xy))
([^o map] kx m1, h1 k x) `o` ([^o map] ky m2, h2 k y).
Proof.
intros Hdom Hg1 Hg2. rewrite big_opM_op.
rewrite -(big_opM_fmap g1) -(big_opM_fmap g2).
rewrite map_fmap_zip_with_r; [|naive_solver..].
by rewrite map_fmap_zip_with_l; [|naive_solver..].
Qed.
Lemma big_opM_sep_zip `{Countable K} {A B}
(h1 : K A M) (h2 : K B M) m1 m2 :
( k, is_Some (m1 !! k) is_Some (m2 !! k))
([^o map] kxy map_zip m1 m2, h1 k xy.1 `o` h2 k xy.2)
([^o map] kx m1, h1 k x) `o` ([^o map] ky m2, h2 k y).
Proof. intros. by apply big_opM_sep_zip_with. Qed.
(** ** Big ops over finite sets *)
Section gset.
Context `{Countable A}.
......
......@@ -1154,6 +1154,24 @@ Section map.
Proof. rewrite big_opM_eq. intros. apply big_sepL_timeless=> _ [??]; apply _. Qed.
End map.
(* Some lemmas depend on the generalized versions of the above ones. *)
Lemma big_sepM_sep_zip_with `{Countable K} {A B C}
(f : A B C) (g1 : C A) (g2 : C B)
(Φ1 : K A PROP) (Φ2 : K B PROP) m1 m2 :
( x y, g1 (f x y) = x)
( x y, g2 (f x y) = y)
( k, is_Some (m1 !! k) is_Some (m2 !! k))
([ map] kxy map_zip_with f m1 m2, Φ1 k (g1 xy) Φ2 k (g2 xy))
([ map] kx m1, Φ1 k x) ([ map] ky m2, Φ2 k y).
Proof. apply big_opM_sep_zip_with. Qed.
Lemma big_sepM_sep_zip `{Countable K} {A B}
(Φ1 : K A PROP) (Φ2 : K B PROP) m1 m2 :
( k, is_Some (m1 !! k) is_Some (m2 !! k))
([ map] kxy map_zip m1 m2, Φ1 k xy.1 Φ2 k xy.2)
([ map] kx m1, Φ1 k x) ([ map] ky m2, Φ2 k y).
Proof. apply big_opM_sep_zip. Qed.
(** ** Big ops over two maps *)
Section map2.
Context `{Countable K} {A B : Type}.
......
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