diff --git a/theories/fin_maps.v b/theories/fin_maps.v index fc357d44bc8b938c3d1c61d512c0565bbbb09e53..447b6751a052cb5e272fea91da58f8d08ef662b6 100644 --- a/theories/fin_maps.v +++ b/theories/fin_maps.v @@ -1132,6 +1132,17 @@ Section map_filter. + rewrite Eq, Hm, lookup_insert. naive_solver. + by rewrite lookup_insert_ne. Qed. + Lemma map_filter_lookup_Some_11 m i x : + filter P m !! i = Some x → m !! i = Some x. + Proof. apply map_filter_lookup_Some. Qed. + Lemma map_filter_lookup_Some_12 m i x : + filter P m !! i = Some x → P (i,x). + Proof. apply map_filter_lookup_Some. Qed. + Lemma map_filter_lookup_Some_2 m i x : + m !! i = Some x → + P (i, x) → + filter P m !! i = Some x. + Proof. intros. by apply map_filter_lookup_Some. Qed. Lemma map_filter_lookup_None m i : filter P m !! i = None ↔ m !! i = None ∨ ∀ x, m !! i = Some x → ¬ P (i,x). @@ -1139,6 +1150,14 @@ Section map_filter. rewrite eq_None_not_Some. unfold is_Some. setoid_rewrite map_filter_lookup_Some. naive_solver. Qed. + Lemma map_filter_lookup_None_1 m i : + filter P m !! i = None → + m !! i = None ∨ ∀ x, m !! i = Some x → ¬ P (i,x). + Proof. apply map_filter_lookup_None. Qed. + Lemma map_filter_lookup_None_2 m i : + m !! i = None ∨ (∀ x : A, m !! i = Some x → ¬ P (i, x)) → + filter P m !! i = None. + Proof. apply map_filter_lookup_None. Qed. Lemma map_filter_lookup_eq m1 m2 : (∀ k x, P (k,x) → m1 !! k = Some x ↔ m2 !! k = Some x) → @@ -1170,6 +1189,14 @@ Section map_filter. (∀ y, ¬ P (i, y)) → filter P (<[i:=x]> m) = filter P m. Proof. intros HP. by apply map_filter_insert_not'. Qed. + Lemma map_filter_insert_not_delete m i x : + ¬ P (i, x) → filter P (<[i:=x]> m) = filter P (delete i m). + Proof. + intros. rewrite <-insert_delete by done. + rewrite map_filter_insert_not'; [done..|]. + rewrite lookup_delete; done. + Qed. + Lemma map_filter_delete m i : filter P (delete i m) = delete i (filter P m). Proof. apply map_eq. intros j. apply option_eq; intros y. @@ -1200,6 +1227,24 @@ Section map_filter. Qed. End map_filter. +Lemma map_filter_fmap {A A2} (P : K * A → Prop) `{!∀ x, Decision (P x)} + (f : A2 → A) (m : M A2) : + filter P (f <\$> m) = f <\$> filter (λ '(k, v), P (k, (f v))) m. +Proof. + apply map_eq. intros i. apply option_eq; intros x. + repeat (rewrite lookup_fmap, fmap_Some || setoid_rewrite map_filter_lookup_Some). + naive_solver. +Qed. + +Lemma map_filter_iff {A} (P1 P2 : K * A → Prop) + `{!∀ x, Decision (P1 x), !∀ x, Decision (P2 x)} (m : M A) : + (∀ x, P1 x ↔ P2 x) → + filter P1 m = filter P2 m. +Proof. + intros HPiff. rewrite !map_filter_alt. + f_equal. apply list_filter_iff. done. +Qed. + (** ** Properties of the [map_Forall] predicate *) Section map_Forall. Context {A} (P : K → A → Prop). diff --git a/theories/list.v b/theories/list.v index c0f62879df4aedf8645ebfc599f86f2f4c2daad8..71af4c0f63b8b09a14fef3ac208c8e8f2c5da905 100644 --- a/theories/list.v +++ b/theories/list.v @@ -1791,6 +1791,17 @@ Section filter. Qed. End filter. +Lemma list_filter_iff (P1 P2 : A → Prop) + `{!∀ x, Decision (P1 x), !∀ x, Decision (P2 x)} (l : list A) : + (∀ x, P1 x ↔ P2 x) → + filter P1 l = filter P2 l. +Proof. + intros HPiff. induction l as [|a l IH]; [done|]. + destruct (decide (P1 a)). + - rewrite !filter_cons_True by naive_solver. by rewrite IH. + - rewrite !filter_cons_False by naive_solver. by rewrite IH. +Qed. + (** ** Properties of the [prefix] and [suffix] predicates *) Global Instance: PreOrder (@prefix A). Proof.