Commit 1b86cdf9 authored by Simon Friis Vindum's avatar Simon Friis Vindum
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Add generalized implication lemma for big_sepM

parent cf11af4b
Pipeline #47857 passed with stage
...@@ -1450,6 +1450,73 @@ Lemma big_sepM_sep_zip `{Countable K} {A B} ...@@ -1450,6 +1450,73 @@ Lemma big_sepM_sep_zip `{Countable K} {A B}
([ map] kx m1, Φ1 k x) ([ map] ky m2, Φ2 k y). ([ map] kx m1, Φ1 k x) ([ map] ky m2, Φ2 k y).
Proof. apply big_opM_sep_zip. Qed. Proof. apply big_opM_sep_zip. Qed.
Lemma big_sepM_impl_foo `{Countable K} {A B} Φ `{ k x, Affine (Φ k x)}
(Ψ : K B PROP) m1 (m2 : gmap K B) :
([ map] kx m1, Φ k x) -
( (k : K) (y : B), m2 !! k = Some y
(( (x : A), m1 !! k = Some x Φ k x) m1 !! k = None) - Ψ k y) -
[ map] ky m2, Ψ k y.
Proof.
revert m1. induction m2 as [|i y m ? IH] using map_ind=> m1.
- apply wand_intro_r. rewrite big_sepM_empty. apply: affine.
- apply wand_intro_r.
rewrite big_sepM_insert; last done.
rewrite intuitionistically_sep_dup.
rewrite assoc. rewrite (comm _ _ ( _))%I.
rewrite {1}intuitionistically_elim {1}(forall_elim i) {1}(forall_elim y).
rewrite lookup_insert pure_True // left_id.
destruct (m1 !! i) as [x|] eqn:Hl.
+ rewrite -or_intro_l -(exist_intro x).
rewrite pure_True // left_id.
rewrite big_sepM_delete; last apply Hl.
rewrite assoc assoc wand_elim_l -assoc.
apply sep_mono_r.
specialize (IH (delete i m1)). apply wand_elim_l' in IH. rewrite -IH.
apply sep_mono_r.
apply intuitionistically_intro'. rewrite intuitionistically_elim.
apply forall_intro=> k. apply forall_intro=> b.
rewrite (forall_elim k) (forall_elim b).
apply impl_intro_l, pure_elim_l=> ?.
assert (i k) by congruence.
rewrite lookup_insert_ne // pure_True // left_id.
rewrite lookup_delete_ne //.
+ rewrite -or_intro_r.
rewrite impl_wand_2.
rewrite pure_True // left_id.
apply sep_mono_r.
specialize (IH m1). apply wand_elim_l' in IH. rewrite -IH.
apply sep_mono_r.
apply intuitionistically_intro'. rewrite intuitionistically_elim.
apply forall_intro=> k. apply forall_intro=> b.
rewrite (forall_elim k) (forall_elim b).
apply impl_intro_l, pure_elim_l=> ?.
rewrite lookup_insert_ne; last congruence.
rewrite pure_True // left_id //.
Qed.
Lemma big_sepM_impl_dom_subseteq `{Countable K} {A B}
Φ `{ k x, Affine (Φ k x)} (Ψ : K B PROP) (m1 : gmap K A) (m2 : gmap K B) :
dom (gset _) m2 dom _ m1
([ map] kx m1, Φ k x) -
( (k : K) (x : A) (y : B),
m1 !! k = Some x m2 !! k = Some y Φ k x - Ψ k y) -
[ map] ky m2, Ψ k y.
Proof.
intros Hsub. rewrite big_sepM_impl_foo.
apply wand_mono; last done.
apply intuitionistically_intro'. rewrite intuitionistically_elim.
apply forall_intro=> k. apply forall_intro=> y.
apply impl_intro_l, pure_elim_l=> look.
apply wand_intro_r. apply wand_elim_r'.
apply or_elim.
- apply exist_elim=> x. apply pure_elim_l=> ?.
rewrite (forall_elim k) (forall_elim x) (forall_elim y).
do 2 rewrite pure_True // left_id.
apply wand_intro_l. apply wand_elim_l.
- rewrite -not_elem_of_dom. apply pure_elim'=> ?.
apply elem_of_dom_2 in look. set_solver.
Qed.
(** ** Big ops over two maps *) (** ** Big ops over two maps *)
Lemma big_sepM2_alt `{Countable K} {A B} (Φ : K A B PROP) m1 m2 : Lemma big_sepM2_alt `{Countable K} {A B} (Φ : K A B PROP) m1 m2 :
([ map] ky1;y2 m1; m2, Φ k y1 y2) ([ map] ky1;y2 m1; m2, Φ k y1 y2)
......
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