diff --git a/CHANGELOG.md b/CHANGELOG.md index 9e159d300fec94aa91ed91de143368f487f328ce..2df67374d4cf24beb1b6e92157e3f4520050a96f 100644 --- a/CHANGELOG.md +++ b/CHANGELOG.md @@ -88,6 +88,8 @@ HeapLang, which is now in a separate package `coq-iris-heap-lang`. `big_sepL_lookup_acc_impl`, `big_sepL2_lookup_acc_impl`, `big_sepM_lookup_acc_impl`, `big_sepM2_lookup_acc_impl`, `big_sepS_elem_of_acc_impl`, `big_sepMS_elem_of_acc_impl`. +* Add lemmas `big_sepM_filter'` and `big_sepM_filter` matching the corresponding + `big_sepS` lemmas. **Changes in `proofmode`:** diff --git a/iris/algebra/big_op.v b/iris/algebra/big_op.v index 1c6aafe2c7c7ef87c7951f73a9a40c29abbb4316..a56da2ac8c8aee78070c5859a970549cf29407f6 100644 --- a/iris/algebra/big_op.v +++ b/iris/algebra/big_op.v @@ -396,6 +396,20 @@ Section gmap. ([^o map] k↦y ∈ <[i:=x]> m, <[i:=P]> f k) ≡ (P `o` [^o map] k↦y ∈ m, f k). Proof. apply (big_opM_fn_insert (λ _ _, id)). Qed. + Lemma big_opM_filter' (φ : K * A → Prop) `{∀ kx, Decision (φ kx)} f m : + ([^o map] k ↦ x ∈ filter φ m, f k x) + ≡ ([^o map] k ↦ x ∈ m, if decide (φ (k, x)) then f k x else monoid_unit). + Proof. + induction m as [|k v m ? IH] using map_ind. + { by rewrite map_filter_empty !big_opM_empty. } + destruct (decide (φ (k, v))). + - rewrite map_filter_insert //. + assert (filter φ m !! k = None) by (apply map_filter_lookup_None; eauto). + by rewrite !big_opM_insert // decide_True // IH. + - rewrite map_filter_insert_not' //; last by congruence. + rewrite !big_opM_insert // decide_False // IH. by rewrite left_id. + Qed. + Lemma big_opM_union f m1 m2 : m1 ##ₘ m2 → ([^o map] k↦y ∈ m1 ∪ m2, f k y) @@ -522,6 +536,20 @@ Section gset. by induction X using set_ind_L; rewrite /= ?big_opS_insert ?left_id // big_opS_eq. Qed. + Lemma big_opS_filter' (φ : A → Prop) `{∀ x, Decision (φ x)} f X : + ([^o set] y ∈ filter φ X, f y) + ≡ ([^o set] y ∈ X, if decide (φ y) then f y else monoid_unit). + Proof. + induction X as [|x X ? IH] using set_ind_L. + { by rewrite filter_empty_L !big_opS_empty. } + destruct (decide (φ x)). + - rewrite filter_union_L filter_singleton_L //. + rewrite !big_opS_insert //; last set_solver. + by rewrite decide_True // IH. + - rewrite filter_union_L filter_singleton_not_L // left_id_L. + by rewrite !big_opS_insert // decide_False // IH left_id. + Qed. + Lemma big_opS_op f g X : ([^o set] y ∈ X, f y `o` g y) ≡ ([^o set] y ∈ X, f y) `o` ([^o set] y ∈ X, g y). Proof. by rewrite big_opS_eq /big_opS_def -big_opL_op. Qed. diff --git a/iris/bi/big_op.v b/iris/bi/big_op.v index 323b2e2c22cb044e0092447781797b7ca1dc1452..680771920e617911b891047d3c2ec15f409f2d88 100644 --- a/iris/bi/big_op.v +++ b/iris/bi/big_op.v @@ -1137,6 +1137,16 @@ Section map. ([∗ map] k↦y ∈ <[i:=x]> m, <[i:=P]> Φ k) ⊣⊢ (P ∗ [∗ map] k↦y ∈ m, Φ k). Proof. apply big_opM_fn_insert'. Qed. + Lemma big_sepM_filter' (φ : K * A → Prop) `{∀ kx, Decision (φ kx)} Φ m : + ([∗ map] k ↦ x ∈ filter φ m, Φ k x) ⊣⊢ + ([∗ map] k ↦ x ∈ m, if decide (φ (k, x)) then Φ k x else emp). + Proof. apply: big_opM_filter'. Qed. + Lemma big_sepM_filter `{BiAffine PROP} + (φ : K * A → Prop) `{∀ kx, Decision (φ kx)} Φ m : + ([∗ map] k ↦ x ∈ filter φ m, Φ k x) ⊣⊢ + ([∗ map] k ↦ x ∈ m, ⌜φ (k, x)⌠→ Φ k x). + Proof. setoid_rewrite <-decide_emp. apply big_sepM_filter'. Qed. + Lemma big_sepM_union Φ m1 m2 : m1 ##ₘ m2 → ([∗ map] k↦y ∈ m1 ∪ m2, Φ k y) @@ -1809,43 +1819,32 @@ Section gset. Lemma big_sepS_singleton Φ x : ([∗ set] y ∈ {[ x ]}, Φ y) ⊣⊢ Φ x. 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, if decide (P y) then Φ y else emp). - Proof. - induction X as [|x X ? IH] using set_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 decide_True // IH. - - rewrite filter_union_L filter_singleton_not_L // left_id_L. - by rewrite !big_sepS_insert // decide_False // IH left_id. - Qed. + Lemma big_sepS_filter' (φ : A → Prop) `{∀ x, Decision (φ x)} Φ X : + ([∗ set] y ∈ filter φ X, Φ y) + ⊣⊢ ([∗ set] y ∈ X, if decide (φ y) then Φ y else emp). + Proof. apply: big_opS_filter'. Qed. + Lemma big_sepS_filter `{BiAffine PROP} + (φ : A → Prop) `{∀ x, Decision (φ x)} Φ X : + ([∗ set] y ∈ filter φ X, Φ y) ⊣⊢ ([∗ set] y ∈ X, ⌜φ y⌠→ Φ y). + Proof. setoid_rewrite <-decide_emp. apply big_sepS_filter'. Qed. - Lemma big_sepS_filter_acc' (P : A → Prop) `{∀ y, Decision (P y)} Φ X Y : - (∀ y, y ∈ Y → P y → y ∈ X) → + Lemma big_sepS_filter_acc' (φ : A → Prop) `{∀ y, Decision (φ y)} Φ X Y : + (∀ y, y ∈ Y → φ y → y ∈ X) → ([∗ set] y ∈ X, Φ y) -∗ - ([∗ set] y ∈ Y, if decide (P y) then Φ y else emp) ∗ - (([∗ set] y ∈ Y, if decide (P y) then Φ y else emp) -∗ [∗ set] y ∈ X, Φ y). + ([∗ set] y ∈ Y, if decide (φ y) then Φ y else emp) ∗ + (([∗ set] y ∈ Y, if decide (φ y) then Φ y else emp) -∗ [∗ set] y ∈ X, Φ y). Proof. - intros ?. destruct (proj1 (subseteq_disjoint_union_L (filter P Y) X)) + intros ?. destruct (proj1 (subseteq_disjoint_union_L (filter φ Y) X)) as (Z&->&?); first set_solver. rewrite big_sepS_union // big_sepS_filter'. by apply sep_mono_r, wand_intro_l. Qed. - - Lemma big_sepS_filter `{BiAffine PROP} - (P : A → Prop) `{∀ x, Decision (P x)} Φ X : - ([∗ set] y ∈ filter P X, Φ y) ⊣⊢ ([∗ set] y ∈ X, ⌜P y⌠→ Φ y). - Proof. setoid_rewrite <-decide_emp. apply big_sepS_filter'. Qed. - Lemma big_sepS_filter_acc `{BiAffine PROP} - (P : A → Prop) `{∀ y, Decision (P y)} Φ X Y : - (∀ y, y ∈ Y → P y → y ∈ X) → + (φ : A → Prop) `{∀ y, Decision (φ y)} Φ X Y : + (∀ y, y ∈ Y → φ y → y ∈ X) → ([∗ set] y ∈ X, Φ y) -∗ - ([∗ set] y ∈ Y, ⌜P y⌠→ Φ y) ∗ - (([∗ set] y ∈ Y, ⌜P y⌠→ Φ y) -∗ [∗ set] y ∈ X, Φ y). + ([∗ set] y ∈ Y, ⌜φ y⌠→ Φ y) ∗ + (([∗ set] y ∈ Y, ⌜φ y⌠→ Φ y) -∗ [∗ set] y ∈ X, Φ y). Proof. intros. setoid_rewrite <-decide_emp. by apply big_sepS_filter_acc'. Qed. Lemma big_sepS_list_to_set Φ (l : list A) :