Commit a3dfa40e authored by Robbert Krebbers's avatar Robbert Krebbers

More properties and set_solver for filter.

parent 8fd8da22
...@@ -472,6 +472,19 @@ Section gset. ...@@ -472,6 +472,19 @@ Section gset.
Lemma big_sepS_singleton Φ x : ([ set] y {[ x ]}, Φ y) Φ x. Lemma big_sepS_singleton Φ x : ([ set] y {[ x ]}, Φ y) Φ x.
Proof. apply: big_opS_singleton. Qed. Proof. apply: big_opS_singleton. Qed.
Lemma big_sepS_filter (P : A Prop) `{ x, Decision (P x)} Φ X :
([ set] y filter P X, Φ y) ([ set] y X, P y Φ y).
Proof.
induction X as [|x X ? IH] using collection_ind_L.
{ by rewrite filter_empty_L !big_sepS_empty. }
destruct (decide (P x)).
- rewrite filter_union_L filter_singleton_L //.
rewrite !big_sepS_insert //; last set_solver.
by rewrite IH pure_True // left_id.
- rewrite filter_union_L filter_singleton_not_L // left_id_L.
by rewrite !big_sepS_insert // IH pure_False // False_impl left_id.
Qed.
Lemma big_sepS_sepS Φ Ψ X : Lemma big_sepS_sepS Φ Ψ X :
([ set] y X, Φ y Ψ y) ([ set] y X, Φ y) ([ set] y X, Ψ y). ([ set] y X, Φ y Ψ y) ([ set] y X, Φ y) ([ set] y X, Ψ y).
Proof. apply: big_opS_opS. Qed. Proof. apply: big_opS_opS. Qed.
......
...@@ -14,6 +14,7 @@ Definition collection_fold `{Elements A C} {B} ...@@ -14,6 +14,7 @@ Definition collection_fold `{Elements A C} {B}
Instance collection_filter Instance collection_filter
`{Elements A C, Empty C, Singleton A C, Union C} : Filter A C := λ P _ X, `{Elements A C, Empty C, Singleton A C, Union C} : Filter A C := λ P _ X,
of_list (filter P (elements X)). of_list (filter P (elements X)).
Typeclasses Opaque collection_filter.
Section fin_collection. Section fin_collection.
Context `{FinCollection A C}. Context `{FinCollection A C}.
...@@ -211,12 +212,41 @@ Lemma minimal_exists_L R `{!LeibnizEquiv C, !Transitive R, ...@@ -211,12 +212,41 @@ Lemma minimal_exists_L R `{!LeibnizEquiv C, !Transitive R,
Proof. unfold_leibniz. apply (minimal_exists R). Qed. Proof. unfold_leibniz. apply (minimal_exists R). Qed.
(** * Filter *) (** * Filter *)
Lemma elem_of_filter (P : A Prop) `{! x, Decision (P x)} X x : Section filter.
x filter P X P x x X. Context (P : A Prop) `{! x, Decision (P x)}.
Proof.
unfold filter, collection_filter. Lemma elem_of_filter X x : x filter P X P x x X.
by rewrite elem_of_of_list, elem_of_list_filter, elem_of_elements. Proof.
Qed. unfold filter, collection_filter.
by rewrite elem_of_of_list, elem_of_list_filter, elem_of_elements.
Qed.
Global Instance set_unfold_filter X Q :
SetUnfold (x X) Q SetUnfold (x filter P X) (P x Q).
Proof.
intros ??; constructor. by rewrite elem_of_filter, (set_unfold (x X) Q).
Qed.
Lemma filter_empty : filter P (:C) .
Proof. set_solver. Qed.
Lemma filter_union X Y : filter P (X Y) filter P X filter P Y.
Proof. set_solver. Qed.
Lemma filter_singleton x : P x filter P ({[ x ]} : C) {[ x ]}.
Proof. set_solver. Qed.
Lemma filter_singleton_not x : ¬P x filter P ({[ x ]} : C) .
Proof. set_solver. Qed.
Section leibniz_equiv.
Context `{!LeibnizEquiv C}.
Lemma filter_empty_L : filter P (:C) = .
Proof. set_solver. Qed.
Lemma filter_union_L X Y : filter P (X Y) = filter P X filter P Y.
Proof. set_solver. Qed.
Lemma filter_singleton_L x : P x filter P ({[ x ]} : C) = {[ x ]}.
Proof. set_solver. Qed.
Lemma filter_singleton_not_L x : ¬P x filter P ({[ x ]} : C) = .
Proof. set_solver. Qed.
End leibniz_equiv.
End filter.
(** * Decision procedures *) (** * Decision procedures *)
Global Instance set_Forall_dec `(P : A Prop) Global Instance set_Forall_dec `(P : A Prop)
......
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