(* Copyright (c) 2012-2015, Robbert Krebbers. *) (* This file is distributed under the terms of the BSD license. *) (** This file provides an axiomatization of the domain function of finite maps. We provide such an axiomatization, instead of implementing the domain function in a generic way, to allow more efficient implementations. *) Require Export collections fin_maps. Class FinMapDom K M D `{FMap M, ∀ A, Lookup K A (M A), ∀ A, Empty (M A), ∀ A, PartialAlter K A (M A), OMap M, Merge M, ∀ A, FinMapToList K A (M A), ∀ i j : K, Decision (i = j), ∀ A, Dom (M A) D, ElemOf K D, Empty D, Singleton K D, Union D, Intersection D, Difference D} := { finmap_dom_map :>> FinMap K M; finmap_dom_collection :>> Collection K D; elem_of_dom {A} (m : M A) i : i ∈ dom D m ↔ is_Some (m !! i) }. Section fin_map_dom. Context `{FinMapDom K M D}. Lemma elem_of_dom_2 {A} (m : M A) i x : m !! i = Some x → i ∈ dom D m. Proof. rewrite elem_of_dom; eauto. Qed. Lemma not_elem_of_dom {A} (m : M A) i : i ∉ dom D m ↔ m !! i = None. Proof. by rewrite elem_of_dom, eq_None_not_Some. Qed. Lemma subseteq_dom {A} (m1 m2 : M A) : m1 ⊆ m2 → dom D m1 ⊆ dom D m2. Proof. rewrite map_subseteq_spec. intros ??. rewrite !elem_of_dom. inversion 1; eauto. Qed. Lemma subset_dom {A} (m1 m2 : M A) : m1 ⊂ m2 → dom D m1 ⊂ dom D m2. Proof. intros [Hss1 Hss2]; split; [by apply subseteq_dom |]. contradict Hss2. rewrite map_subseteq_spec. intros i x Hi. specialize (Hss2 i). rewrite !elem_of_dom in Hss2. destruct Hss2; eauto. by simplify_map_equality. Qed. Lemma dom_empty {A} : dom D (@empty (M A) _) ≡ ∅. Proof. split; intro; [|solve_elem_of]. rewrite elem_of_dom, lookup_empty. by inversion 1. Qed. Lemma dom_empty_inv {A} (m : M A) : dom D m ≡ ∅ → m = ∅. Proof. intros E. apply map_empty. intros. apply not_elem_of_dom. rewrite E. solve_elem_of. Qed. Lemma dom_alter {A} f (m : M A) i : dom D (alter f i m) ≡ dom D m. Proof. apply elem_of_equiv; intros j; rewrite !elem_of_dom; unfold is_Some. destruct (decide (i = j)); simplify_map_equality'; eauto. destruct (m !! j); naive_solver. Qed. Lemma dom_insert {A} (m : M A) i x : dom D (<[i:=x]>m) ≡ {[ i ]} ∪ dom D m. Proof. apply elem_of_equiv. intros j. rewrite elem_of_union, !elem_of_dom. unfold is_Some. setoid_rewrite lookup_insert_Some. destruct (decide (i = j)); esolve_elem_of. Qed. Lemma dom_insert_subseteq {A} (m : M A) i x : dom D m ⊆ dom D (<[i:=x]>m). Proof. rewrite (dom_insert _). solve_elem_of. Qed. Lemma dom_insert_subseteq_compat_l {A} (m : M A) i x X : X ⊆ dom D m → X ⊆ dom D (<[i:=x]>m). Proof. intros. transitivity (dom D m); eauto using dom_insert_subseteq. Qed. Lemma dom_singleton {A} (i : K) (x : A) : dom D {[(i, x)]} ≡ {[ i ]}. Proof. unfold singleton at 1, map_singleton. rewrite dom_insert, dom_empty. solve_elem_of. Qed. Lemma dom_delete {A} (m : M A) i : dom D (delete i m) ≡ dom D m ∖ {[ i ]}. Proof. apply elem_of_equiv. intros j. rewrite elem_of_difference, !elem_of_dom. unfold is_Some. setoid_rewrite lookup_delete_Some. esolve_elem_of. Qed. Lemma delete_partial_alter_dom {A} (m : M A) i f : i ∉ dom D m → delete i (partial_alter f i m) = m. Proof. rewrite not_elem_of_dom. apply delete_partial_alter. Qed. Lemma delete_insert_dom {A} (m : M A) i x : i ∉ dom D m → delete i (<[i:=x]>m) = m. Proof. rewrite not_elem_of_dom. apply delete_insert. Qed. Lemma map_disjoint_dom {A} (m1 m2 : M A) : m1 ⊥ m2 ↔ dom D m1 ∩ dom D m2 ≡ ∅. Proof. rewrite map_disjoint_spec, elem_of_equiv_empty. setoid_rewrite elem_of_intersection. setoid_rewrite elem_of_dom. unfold is_Some. naive_solver. Qed. Lemma map_disjoint_dom_1 {A} (m1 m2 : M A) : m1 ⊥ m2 → dom D m1 ∩ dom D m2 ≡ ∅. Proof. apply map_disjoint_dom. Qed. Lemma map_disjoint_dom_2 {A} (m1 m2 : M A) : dom D m1 ∩ dom D m2 ≡ ∅ → m1 ⊥ m2. Proof. apply map_disjoint_dom. Qed. Lemma dom_union {A} (m1 m2 : M A) : dom D (m1 ∪ m2) ≡ dom D m1 ∪ dom D m2. Proof. apply elem_of_equiv. intros i. rewrite elem_of_union, !elem_of_dom. unfold is_Some. setoid_rewrite lookup_union_Some_raw. destruct (m1 !! i); naive_solver. Qed. Lemma dom_intersection {A} (m1 m2 : M A) : dom D (m1 ∩ m2) ≡ dom D m1 ∩ dom D m2. Proof. apply elem_of_equiv. intros i. rewrite elem_of_intersection, !elem_of_dom. unfold is_Some. setoid_rewrite lookup_intersection_Some. naive_solver. Qed. Lemma dom_difference {A} (m1 m2 : M A) : dom D (m1 ∖ m2) ≡ dom D m1 ∖ dom D m2. Proof. apply elem_of_equiv. intros i. rewrite elem_of_difference, !elem_of_dom. unfold is_Some. setoid_rewrite lookup_difference_Some. destruct (m2 !! i); naive_solver. Qed. Lemma dom_fmap {A B} (f : A → B) m : dom D (f <\$> m) ≡ dom D m. Proof. apply elem_of_equiv. intros i. rewrite !elem_of_dom, lookup_fmap, <-!not_eq_None_Some. destruct (m !! i); naive_solver. Qed. Context `{!LeibnizEquiv D}. 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 = ∅. Proof. by intros; apply dom_empty_inv; unfold_leibniz. Qed. Lemma dom_alter_L {A} f (m : M A) i : dom D (alter f i m) = dom D m. Proof. unfold_leibniz; apply dom_alter. Qed. Lemma dom_insert_L {A} (m : M A) i x : dom D (<[i:=x]>m) = {[ i ]} ∪ dom D m. Proof. unfold_leibniz; apply dom_insert. Qed. Lemma dom_singleton_L {A} (i : K) (x : A) : dom D {[(i, x)]} = {[ i ]}. Proof. unfold_leibniz; apply dom_singleton. Qed. Lemma dom_delete_L {A} (m : M A) i : dom D (delete i m) = dom D m ∖ {[ i ]}. Proof. unfold_leibniz; apply dom_delete. Qed. Lemma dom_union_L {A} (m1 m2 : M A) : dom D (m1 ∪ m2) = dom D m1 ∪ dom D m2. Proof. unfold_leibniz; apply dom_union. Qed. Lemma dom_intersection_L {A} (m1 m2 : M A) : dom D (m1 ∩ m2) = dom D m1 ∩ dom D m2. Proof. unfold_leibniz; apply dom_intersection. Qed. Lemma dom_difference_L {A} (m1 m2 : M A) : dom D (m1 ∖ m2) = dom D m1 ∖ dom D m2. Proof. unfold_leibniz; apply dom_difference. Qed. Lemma dom_fmap_L {A B} (f : A → B) m : dom D (f <\$> m) = dom D m. Proof. unfold_leibniz; apply dom_fmap. Qed. End fin_map_dom.