Commit 324c9c50 by Robbert Krebbers

### Add function `map_kmap` that transforms the keys of a finite map.

parent 558df4bc
 ... ... @@ -124,6 +124,15 @@ Definition map_imap `{∀ A, Insert K A (M A), ∀ A, Empty (M A), ∀ A, FinMapToList K A (M A)} {A B} (f : K → A → option B) (m : M A) : M B := list_to_map (omap (λ ix, (fst ix ,.) <\$> curry f ix) (map_to_list m)). (** Given a function [f : K1 → K2], the function [map_kmap f] turns a maps with keys of type [K1] into a map with keys of type [K2]. The function [map_kmap f] is only well-behaved if [f] is injective, as otherwise it could map multiple entries into the same entry. All lemmas about [map_kmap f] thus have the premise [Inj (=) (=) f]. *) Definition map_kmap `{∀ A, Insert K2 A (M2 A), ∀ A, Empty (M2 A), ∀ A, FinMapToList K1 A (M1 A)} {A} (f : K1 → K2) (m : M1 A) : M2 A := list_to_map (fmap (prod_map f id) (map_to_list m)). (* The zip operation on maps combines two maps key-wise. The keys of resulting map correspond to the keys that are in both maps. *) Definition map_zip_with `{Merge M} {A B C} (f : A → B → C) : M A → M B → M C := ... ... @@ -2524,6 +2533,87 @@ Section map_seq. Qed. End map_seq. Section map_kmap. Context `{FinMap K1 M1} `{FinMap K2 M2}. Context (f : K1 → K2) `{!Inj (=) (=) f}. Local Notation map_kmap := (map_kmap (M1:=M1) (M2:=M2)). Lemma lookup_map_kmap_Some {A} (m : M1 A) (j : K2) x : map_kmap f m !! j = Some x ↔ ∃ i, j = f i ∧ m !! i = Some x. Proof. assert (∀ x', (j, x) ∈ prod_map f id <\$> map_to_list m → (j, x') ∈ prod_map f id <\$> map_to_list m → x = x'). { intros x'. rewrite !elem_of_list_fmap. intros [[j' y1] [??]] [[? y2] [??]]; simplify_eq/=. by apply (map_to_list_unique m j'). } unfold map_kmap. rewrite <-elem_of_list_to_map', elem_of_list_fmap by done. setoid_rewrite elem_of_map_to_list'. split. - intros [[??] [??]]; naive_solver. - intros [? [??]]. eexists (_, _); naive_solver. Qed. Lemma lookup_map_kmap_is_Some {A} (m : M1 A) (j : K2) : is_Some (map_kmap f m !! j) ↔ ∃ i, j = f i ∧ is_Some (m !! i). Proof. unfold is_Some. setoid_rewrite lookup_map_kmap_Some. naive_solver. Qed. Lemma lookup_map_kmap_None {A} (m : M1 A) (j : K2) : map_kmap f m !! j = None ↔ ∀ i, j = f i → m !! i = None. Proof. setoid_rewrite eq_None_not_Some. rewrite lookup_map_kmap_is_Some. naive_solver. Qed. Lemma lookup_map_kmap {A} (m : M1 A) (i : K1) : map_kmap f m !! f i = m !! i. Proof. apply option_eq. setoid_rewrite lookup_map_kmap_Some. naive_solver. Qed. Lemma lookup_total_map_kmap `{Inhabited A} (m : M1 A) (i : K1) : map_kmap f m !!! f i = m !!! i. Proof. by rewrite !lookup_total_alt, lookup_map_kmap. Qed. Lemma map_kmap_empty {A} : map_kmap f ∅ =@{M2 A} ∅. Proof. unfold map_kmap. by rewrite map_to_list_empty. Qed. Lemma map_kmap_singleton {A} i (x : A) : map_kmap f {[ i := x ]} = {[ f i := x ]}. Proof. unfold map_kmap. by rewrite map_to_list_singleton. Qed. Lemma map_kmap_partial_alter {A} (g : option A → option A) (m : M1 A) i : map_kmap f (partial_alter g i m) = partial_alter g (f i) (map_kmap f m). Proof. apply map_eq; intros j. apply option_eq; intros y. destruct (decide (j = f i)) as [->|?]. { by rewrite lookup_partial_alter, !lookup_map_kmap, lookup_partial_alter. } rewrite lookup_partial_alter_ne, !lookup_map_kmap_Some by done. split. - intros [i' [? Hm]]; simplify_eq/=. rewrite lookup_partial_alter_ne in Hm by naive_solver. naive_solver. - intros [i' [? Hm]]; simplify_eq/=. exists i'. rewrite lookup_partial_alter_ne by naive_solver. naive_solver. Qed. Lemma map_kmap_insert {A} (m : M1 A) i x : map_kmap f (<[i:=x]> m) = <[f i:=x]> (map_kmap f m). Proof. apply map_kmap_partial_alter. Qed. Lemma map_kmap_delete {A} (m : M1 A) i : map_kmap f (delete i m) = delete (f i) (map_kmap f m). Proof. apply map_kmap_partial_alter. Qed. Lemma map_kmap_alter {A} (g : A → A) (m : M1 A) i : map_kmap f (alter g i m) = alter g (f i) (map_kmap f m). Proof. apply map_kmap_partial_alter. Qed. Lemma map_kmap_imap {A B} (g : K2 → A → option B) (m : M1 A) : map_kmap f (map_imap (g ∘ f) m) = map_imap g (map_kmap f m). Proof. apply map_eq; intros j. apply option_eq; intros y. rewrite map_lookup_imap, bind_Some. setoid_rewrite lookup_map_kmap_Some. setoid_rewrite map_lookup_imap. setoid_rewrite bind_Some. naive_solver. Qed. Lemma map_kmap_omap {A B} (g : A → option B) (m : M1 A) : map_kmap f (omap g m) = omap g (map_kmap f m). Proof. apply map_eq; intros j. apply option_eq; intros y. rewrite lookup_omap, bind_Some. setoid_rewrite lookup_map_kmap_Some. setoid_rewrite lookup_omap. setoid_rewrite bind_Some. naive_solver. Qed. Lemma map_kmap_fmap {A B} (g : A → B) (m : M1 A) : map_kmap f (g <\$> m) = g <\$> (map_kmap f m). Proof. by rewrite !map_fmap_alt, map_kmap_omap. Qed. End map_kmap. (** * Tactics *) (** The tactic [decompose_map_disjoint] simplifies occurrences of [disjoint] in the hypotheses that involve the empty map [∅], the union [(∪)] or insert ... ...
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