More lemmas for `filter` w.r.t. maps

• filter P m ##ₘ filter (λ v, ¬ P v) m
• filter P m ∪ filter (λ v, ¬ P v) m = m
• (∀ i, i ∈ X ↔ ∃ x, m !! i = Some x ∧ P (i, x)) → dom D (filter P m) ≡ X

For the last one, I have repurposed the name dom_map_filter, which was previously used for dom D (filter P m) ⊆ dom D m. I have renamed the old lemma into dom_map_filter_subseteq. This follows current naming conventions.

This MR received partial contributions from @dfrumin.

theories/fin_map_dom.v

@@ -19,7 +19,14 @@ Class FinMapDom K M D `{∀ A, Dom (M A) D, FMap M,
 Section fin_map_dom.
 Context `{FinMapDom K M D}.
 
-Lemma dom_map_filter {A} (P : K * A → Prop) `{!∀ x, Decision (P x)} (m : M A):
+Lemma dom_map_filter {A} (P : K * A → Prop) `{!∀ x, Decision (P x)} (m : M A) X :
+  (∀ i, i ∈ X ↔ ∃ x, m !! i = Some x ∧ P (i, x)) →
+  dom D (filter P m) ≡ X.
+Proof.
+  intros HX i. rewrite elem_of_dom, HX.
+  unfold is_Some. by setoid_rewrite map_filter_lookup_Some.
+Qed.
+Lemma dom_map_filter_subseteq {A} (P : K * A → Prop) `{!∀ x, Decision (P x)} (m : M A):
   dom D (filter P m) ⊆ dom D m.
 Proof.
   intros ?. rewrite 2!elem_of_dom.
@@ -132,6 +139,10 @@ Global Instance dom_proper_L `{!Equiv A, !LeibnizEquiv D} :
 Proof. intros ???. unfold_leibniz. by apply dom_proper. Qed.
 
 Context `{!LeibnizEquiv D}.
+Lemma dom_map_filter_L {A} (P : K * A → Prop) `{!∀ x, Decision (P x)} (m : M A) X :
+  (∀ i, i ∈ X ↔ ∃ x, m !! i = Some x ∧ P (i, x)) →
+  dom D (filter P m) = X.
+Proof. unfold_leibniz. apply dom_map_filter. Qed.
 Lemma dom_empty_L {A} : dom D (@empty (M A) _) = ∅.
 Proof. unfold_leibniz; apply dom_empty. Qed.
 Lemma dom_empty_inv_L {A} (m : M A) : dom D m = ∅ → m = ∅.