Commit cb9d659c by Joseph Tassarotti

### Remove more unneeded files; small clean-ups.

parent 0701bd20
 -Q theories fri -arg -w -arg -notation-overridden,-redundant-canonical-projection,-several-object-files -arg -w -arg -notation-overridden,-redundant-canonical-projection,-several-object-files,-projection-no-head-constant,-arguments-assert theories/prelude/list.v theories/prelude/compact.v theories/prelude/set_finite_setoid.v ... ... @@ -62,8 +62,6 @@ theories/program_logic/refine_raw_adequacy.v theories/program_logic/refine.v theories/program_logic/refine_ectx.v theories/program_logic/refine_ectx_delay.v theories/program_logic/boxes.v theories/proofmode/sts.v theories/heap_lang/lang.v theories/heap_lang/tactics.v theories/heap_lang/wp_tactics.v ... ...
 From fri.algebra Require Export upred list. From stdpp Require Import gmap fin_collections functions. Import uPred. (** We define the following big operators: - The operators [ [★] Ps ] and [ [∧] Ps ] fold [★] and [∧] over the list [Ps]. This operator is not a quantifier, so it binds strongly. - The operator [ [★ map] k ↦ x ∈ m, P ] asserts that [P] holds separately for each [k ↦ x] in the map [m]. This operator is a quantifier, and thus has the same precedence as [∀] and [∃]. - The operator [ [★ set] x ∈ X, P ] asserts that [P] holds separately for each [x] in the set [X]. This operator is a quantifier, and thus has the same precedence as [∀] and [∃]. *) (** * Big ops over lists *) (* These are the basic building blocks for other big ops *) Fixpoint uPred_big_and {M} (Ps : list (uPred M)) : uPred M := match Ps with [] => True | P :: Ps => P ∧ uPred_big_and Ps end%IP. Instance: Params (@uPred_big_and) 1. Notation "'[∧]' Ps" := (uPred_big_and Ps) (at level 20) : uPred_scope. Fixpoint uPred_big_sep {M} (Ps : list (uPred M)) : uPred M := match Ps with [] => Emp | P :: Ps => P ★ uPred_big_sep Ps end%IP. Instance: Params (@uPred_big_sep) 1. Notation "'[★]' Ps" := (uPred_big_sep Ps) (at level 20) : uPred_scope. (** * Other big ops *) (** We use a type class to obtain overloaded notations *) Definition uPred_big_sepM {M: ucmraT} `{Countable K} {A} (m : gmap K A) (Φ : K → A → uPred M) : uPred M := [★] (curry Φ <\$> map_to_list m). Instance: Params (@uPred_big_sepM) 6. Notation "'[★' 'map' ] k ↦ x ∈ m , P" := (uPred_big_sepM m (λ k x, P)) (at level 200, m at level 10, k, x at level 1, right associativity, format "[★ map ] k ↦ x ∈ m , P") : uPred_scope. Definition uPred_big_sepS {M} `{Countable A} (X : gset A) (Φ : A → uPred M) : uPred M := [★] (Φ <\$> elements X). Instance: Params (@uPred_big_sepS) 5. Notation "'[★' 'set' ] x ∈ X , P" := (uPred_big_sepS X (λ x, P)) (at level 200, X at level 10, x at level 1, right associativity, format "[★ set ] x ∈ X , P") : uPred_scope. (** * Persistence of lists of uPreds *) Class RelevantL {M} (Ps : list (uPred M)) := relevantL : Forall RelevantP Ps. Arguments relevantL {_} _ {_}. Class AffineL {M} (Ps : list (uPred M)) := affineL : Forall AffineP Ps. Arguments affineL {_} _ {_}. (** * Properties *) Section big_op. Context {M : ucmraT}. Implicit Types Ps Qs : list (uPred M). Implicit Types A : Type. (** ** Big ops over lists *) Global Instance big_and_proper : Proper ((≡) ==> (⊣⊢)) (@uPred_big_and M). Proof. by induction 1 as [|P Q Ps Qs HPQ ? IH]; rewrite /= ?HPQ ?IH. Qed. Global Instance big_sep_proper : Proper ((≡) ==> (⊣⊢)) (@uPred_big_sep M). Proof. by induction 1 as [|P Q Ps Qs HPQ ? IH]; rewrite /= ?HPQ ?IH. Qed. Global Instance big_and_ne n : Proper (dist n ==> dist n) (@uPred_big_and M). Proof. by induction 1 as [|P Q Ps Qs HPQ ? IH]; rewrite /= ?HPQ ?IH. Qed. Global Instance big_sep_ne n : Proper (dist n ==> dist n) (@uPred_big_sep M). Proof. by induction 1 as [|P Q Ps Qs HPQ ? IH]; rewrite /= ?HPQ ?IH. Qed. Global Instance big_and_mono' : Proper (Forall2 (⊢) ==> (⊢)) (@uPred_big_and M). Proof. by induction 1 as [|P Q Ps Qs HPQ ? IH]; rewrite /= ?HPQ ?IH. Qed. Global Instance big_sep_mono' : Proper (Forall2 (⊢) ==> (⊢)) (@uPred_big_sep M). Proof. by induction 1 as [|P Q Ps Qs HPQ ? IH]; rewrite /= ?HPQ ?IH. Qed. Global Instance big_and_perm : Proper ((≡ₚ) ==> (⊣⊢)) (@uPred_big_and M). Proof. induction 1 as [|P Ps Qs ? IH|P Q Ps|]; simpl; auto. - by rewrite IH. - by rewrite !assoc (comm _ P). - etrans; eauto. Qed. Global Instance big_sep_perm : Proper ((≡ₚ) ==> (⊣⊢)) (@uPred_big_sep M). Proof. induction 1 as [|P Ps Qs ? IH|P Q Ps|]; simpl; auto. - by rewrite IH. - by rewrite !assoc (comm _ P). - etrans; eauto. Qed. Lemma big_and_app Ps Qs : [∧] (Ps ++ Qs) ⊣⊢ [∧] Ps ∧ [∧] Qs. Proof. induction Ps as [|?? IH]; by rewrite /= ?left_id -?assoc ?IH. Qed. Lemma big_sep_app Ps Qs : [★] (Ps ++ Qs) ⊣⊢ [★] Ps ★ [★] Qs. Proof. by induction Ps as [|?? IH]; rewrite /= ?left_id -?assoc ?IH. Qed. Lemma big_and_contains Ps Qs : Qs ⊆+ Ps → [∧] Ps ⊢ [∧] Qs. Proof. intros [Ps' ->]%submseteq_Permutation. by rewrite big_and_app and_elim_l. Qed. Lemma big_sep_permutation Ps Qs : Permutation Qs Ps → [★] Ps ⊢ [★] Qs. Proof. by intros ->. Qed. Lemma big_and_elem_of Ps P : P ∈ Ps → [∧] Ps ⊢ P. Proof. induction 1; simpl; auto with I. Qed. (** ** Big ops over finite maps *) Section gmap. Context `{Countable K} {A : Type}. Implicit Types m : gmap K A. Implicit Types Φ Ψ : K → A → uPred M. Existing Instance gmap_finmap. (* Lemma big_sepM_mono Φ Ψ m1 m2 : m2 ⊆ m1 → (∀ k x, m2 !! k = Some x → Φ k x ⊢ Ψ k x) → ([★ map] k ↦ x ∈ m1, Φ k x) ⊢ [★ map] k ↦ x ∈ m2, Ψ k x. Proof. intros HX HΦ. trans ([★ map] k↦x ∈ m2, Φ k x)%IP. - by apply big_sep_contains, fmap_contains, map_to_list_contains. - apply big_sep_mono', Forall2_fmap, Forall_Forall2. apply Forall_forall=> -[i x] ? /=. by apply HΦ, elem_of_map_to_list. Qed. <<<<<<< HEAD *) Lemma big_sepM_proper Φ Ψ m : (∀ k x, m !! k = Some x → Φ k x ⊣⊢ Ψ k x) → ([★ map] k ↦ x ∈ m, Φ k x) ⊣⊢ ([★ map] k ↦ x ∈ m, Ψ k x). Proof. intros HΦ. apply (anti_symm (⊢)). - trans ([★ map] k ↦ x ∈ m, Φ k x)%IP. * apply big_sep_permutation, fmap_Permutation. apply (anti_symm (submseteq)); apply map_to_list_submseteq; set_solver. * apply big_sep_mono', Forall2_fmap, Forall_Forall2. apply Forall_forall=> -[i x] ? /=. rewrite -HΦ; auto; apply elem_of_map_to_list; set_solver. - trans ([★ map] k ↦ x ∈ m, Ψ k x)%IP. * apply big_sep_permutation, fmap_Permutation. apply (anti_symm (submseteq)); apply map_to_list_submseteq; set_solver. * apply big_sep_mono', Forall2_fmap, Forall_Forall2. apply Forall_forall=> -[i x] ? /=. rewrite -HΦ; auto; apply elem_of_map_to_list; set_solver. Qed. Global Instance big_sepM_ne m n : Proper (pointwise_relation _ (pointwise_relation _ (dist n)) ==> (dist n)) (uPred_big_sepM (M:=M) m). Proof. intros Φ1 Φ2 HΦ. apply big_sep_ne, Forall2_fmap. apply Forall_Forall2, Forall_true=> -[i x]; apply HΦ. Qed. Global Instance big_sepM_proper' m : Proper (pointwise_relation _ (pointwise_relation _ (⊣⊢)) ==> (⊣⊢)) (uPred_big_sepM (M:=M) m). Proof. intros Φ1 Φ2 HΦ. by apply big_sepM_proper; intros; last apply HΦ. Qed. Lemma big_sepM_empty Φ : ([★ map] k↦x ∈ ∅, Φ k x) ⊣⊢ Emp. Proof. by rewrite /uPred_big_sepM map_to_list_empty. Qed. Lemma big_sepM_insert Φ m i x : m !! i = None → ([★ map] k↦y ∈ <[i:=x]> m, Φ k y) ⊣⊢ Φ i x ★ [★ map] k↦y ∈ m, Φ k y. Proof. intros ?; by rewrite /uPred_big_sepM map_to_list_insert. Qed. Lemma big_sepM_delete Φ m i x : m !! i = Some x → ([★ map] k↦y ∈ m, Φ k y) ⊣⊢ Φ i x ★ [★ map] k↦y ∈ delete i m, Φ k y. Proof. intros. rewrite -big_sepM_insert ?lookup_delete //. by rewrite insert_delete insert_id. Qed. Lemma big_sepM_singleton Φ i x : ([★ map] k↦y ∈ {[i:=x]}, Φ k y) ⊣⊢ Φ i x. Proof. rewrite -insert_empty big_sepM_insert/=; last auto using lookup_empty. by rewrite big_sepM_empty right_id. Qed. Lemma big_sepM_fmap {B} (f : A → B) (Φ : K → B → uPred M) m : ([★ map] k↦y ∈ f <\$> m, Φ k y) ⊣⊢ ([★ map] k↦y ∈ m, Φ k (f y)). Proof. rewrite /uPred_big_sepM map_to_list_fmap -list_fmap_compose. f_equiv; apply reflexive_eq, list_fmap_ext. by intros []. done. Qed. Lemma big_sepM_insert_override (Φ : K → uPred M) m i x y : m !! i = Some x → ([★ map] k↦_ ∈ <[i:=y]> m, Φ k) ⊣⊢ ([★ map] k↦_ ∈ m, Φ k). Proof. intros. rewrite -insert_delete big_sepM_insert ?lookup_delete //. by rewrite -big_sepM_delete. Qed. Lemma big_sepM_fn_insert {B} (Ψ : K → A → B → uPred M) (f : K → B) m i x b : m !! i = None → ([★ map] k↦y ∈ <[i:=x]> m, Ψ k y (<[i:=b]> f k)) ⊣⊢ (Ψ i x b ★ [★ map] k↦y ∈ m, Ψ k y (f k)). Proof. intros. rewrite big_sepM_insert // fn_lookup_insert. apply sep_proper, big_sepM_proper; auto=> k y ?. rewrite fn_lookup_insert_ne; last set_solver. congruence. Qed. Lemma big_sepM_fn_insert' (Φ : K → uPred M) m i x P : m !! i = None → ([★ map] k↦y ∈ <[i:=x]> m, <[i:=P]> Φ k) ⊣⊢ (P ★ [★ map] k↦y ∈ m, Φ k). Proof. apply (big_sepM_fn_insert (λ _ _, id)). Qed. Lemma big_sepM_sepM Φ Ψ m : ([★ map] k↦x ∈ m, Φ k x ★ Ψ k x) ⊣⊢ ([★ map] k↦x ∈ m, Φ k x) ★ ([★ map] k↦x ∈ m, Ψ k x). Proof. rewrite /uPred_big_sepM. induction (map_to_list m) as [|[i x] l IH]; csimpl; rewrite ?right_id //. by rewrite IH -!assoc (assoc _ (Ψ _ _)) [(Ψ _ _ ★ _)%IP]comm -!assoc. Qed. (* Lemma big_sepM_later Φ m : ▷ ([★ map] k↦x ∈ m, Φ k x) ⊣⊢ ([★ map] k↦x ∈ m, ▷ Φ k x). Proof. rewrite /uPred_big_sepM. induction (map_to_list m) as [|[i x] l IH]; csimpl; rewrite ?later_True //. by rewrite later_sep IH. Qed. *) (* Lemma big_sepM_forall Φ m : (∀ k x, RelevantP (Φ k x)) → ([★ map] k↦x ∈ m, Φ k x) ⊣⊢ (∀ k x, m !! k = Some x → Φ k x). Proof. intros. apply (anti_symm _). { apply forall_intro=> k; apply forall_intro=> x. apply impl_intro_l, pure_elim_l=> ?; by apply big_sepM_lookup. } rewrite /uPred_big_sepM. setoid_rewrite <-elem_of_map_to_list. induction (map_to_list m) as [|[i x] l IH]; csimpl; auto. rewrite -always_and_sep_l; apply and_intro. - rewrite (forall_elim i) (forall_elim x) pure_equiv; last by left. by rewrite True_impl. - rewrite -IH. apply forall_mono=> k; apply forall_mono=> y. apply impl_intro_l, pure_elim_l=> ?. rewrite pure_equiv; last by right. by rewrite True_impl. Qed. *) (* Lemma big_sepM_impl Φ Ψ m : □ (∀ k x, m !! k = Some x → Φ k x → Ψ k x) ∧ ([★ map] k↦x ∈ m, Φ k x) ⊢ [★ map] k↦x ∈ m, Ψ k x. Proof. rewrite always_and_sep_l. do 2 setoid_rewrite always_forall. setoid_rewrite always_impl; setoid_rewrite always_pure. rewrite -big_sepM_forall -big_sepM_sepM. apply big_sepM_mono; auto=> k x ?. by rewrite -always_wand_impl always_elim wand_elim_l. Qed. *) Lemma map_to_list_union m1 m2: dom (gset K) m1 ∩ dom (gset K) m2 ≡ ∅ → Permutation (map_to_list (m1 ∪ m2)) (map_to_list m1 ++ map_to_list m2). Proof. intros Hdom. apply (anti_symm (submseteq)). - apply NoDup_submseteq; first by apply NoDup_map_to_list. intros (x, a). rewrite elem_of_map_to_list. rewrite lookup_union_Some_raw. set_unfold. intros [?|?]. * left. apply elem_of_map_to_list; naive_solver. * right. apply elem_of_map_to_list; naive_solver. - apply NoDup_submseteq. apply NoDup_app. * split_and!; try by apply NoDup_map_to_list. intros (x, a). rewrite ?elem_of_map_to_list. intros Hin1 Hin2. assert (x ∈ dom (gset K) m1). { apply elem_of_dom; eauto. } assert (x ∈ dom (gset K) m2). { apply elem_of_dom; eauto. } set_solver. * intros (x, a). set_unfold. rewrite ?elem_of_map_to_list. intros [Hleft|Hright]; apply lookup_union_Some_raw. ** naive_solver. ** specialize (Hdom x). rewrite ?elem_of_dom Hright in Hdom *=>Hdom'. assert (m1 !! x = None). { destruct (m1 !! x); auto. exfalso. naive_solver. } naive_solver. Qed. Lemma big_sepM_union Φ m1 m2 : dom (gset K) m1 ∩ dom (gset K) m2 ≡ ∅ → ([★ map] k↦x ∈ m1, Φ k x) ★ ([★ map] k↦x ∈ m2, Φ k x) ⊣⊢ [★ map] k↦x ∈ (m1 ∪ m2), Φ k x. Proof. intros Hdom. rewrite /uPred_big_sepM //. rewrite map_to_list_union; eauto. induction (map_to_list m1). - by rewrite //= left_id. - simpl. rewrite -assoc. apply sep_proper; auto. Qed. Lemma map_union_least m1 m2 m3: m1 ⊆ m3 → m2 ⊆ m3 → m1 ∪ m2 ⊆ m3. Proof. intros Hsub1 Hsub2. apply map_subseteq_spec. intros i x Hlook%lookup_union_Some_raw. destruct Hlook as [Hlook1|(?&Hlook2)]. - eapply (map_subseteq_spec m1); eauto. - eapply (map_subseteq_spec m2); eauto. Qed. (* This could probably be cleaned up... I later saw there is a difference operation on maps? *) Lemma big_sepM_split Φ m m1 m2 : m1 ⊆ m → m2 ⊆ m → dom (gset K) m1 ∩ dom (gset K) m2 ≡ ∅ → ∃ m3, ([★ map] k↦x ∈ m, Φ k x) ⊣⊢ ([★ map] k↦x ∈ m1, Φ k x) ★ ([★ map] k↦x ∈ m2, Φ k x) ★ ([★ map] k↦x ∈ m3, Φ k x). Proof. intros Hsub1 Hsub2 Hdom. pose (m3 := (map_of_list (filter (λ p, p.1 ∉ dom (gset K) m1 ∧ p.1 ∉ dom (gset K) m2) (map_to_list m)) : gmap K A)). exists m3. rewrite assoc. rewrite big_sepM_union; auto. rewrite big_sepM_union; auto. - cut (m = m1 ∪ m2 ∪ m3). { by intros <-. } assert (m3 ⊆ m) as Hsub3. { apply map_subseteq_spec=> x a. rewrite /m3. intros Hlook3. apply elem_of_map_of_list_2, elem_of_list_filter in Hlook3. rewrite elem_of_map_to_list in Hlook3 *. naive_solver. } apply (anti_symm (⊆)). * apply map_subseteq_spec=>x a Hlook. apply lookup_union_Some. ** apply (map_disjoint_dom (D:=gset K)). set_unfold=>x'. rewrite dom_union_L=>Hin_12. rewrite ?elem_of_dom //=. inversion 1 as (a'&Hlook3). rewrite /m3 in Hlook3. apply elem_of_map_of_list_2, elem_of_list_filter in Hlook3. set_unfold. naive_solver. ** rewrite lookup_union_Some; last first. { apply (map_disjoint_dom (D:= gset K)). set_unfold. set_solver. } case_eq (m1 !! x). { intros ? Hlook'. eapply map_subseteq_spec in Hlook'; eauto. rewrite Hlook in Hlook'. inversion Hlook'. naive_solver. } intros Hnone1. case_eq (m2 !! x). { intros ? Hlook'. eapply map_subseteq_spec in Hlook'; eauto. rewrite Hlook in Hlook'. inversion Hlook'. naive_solver. } intros Hnone2. assert (is_Some (m3 !! x)) as (a'&Hlook'). { rewrite /is_Some. case_eq (m3 !! x); first eauto. rewrite /m3. intros Hnot%not_elem_of_map_of_list_2. exfalso; apply Hnot. apply elem_of_list_fmap. exists (x, a); split; auto. apply elem_of_list_filter. rewrite elem_of_map_to_list; split; auto. rewrite ?not_elem_of_dom. naive_solver. } rewrite Hlook'. eapply map_subseteq_spec in Hlook'; eauto. rewrite Hlook in Hlook'. inversion Hlook'. naive_solver. * apply map_union_least; auto. apply map_union_least; auto. - set_unfold=>x'. rewrite dom_union_L. intros (Hin_12&Hin_3). rewrite elem_of_dom //= in Hin_3 *. inversion 1 as (a'&Hlook3). rewrite /m3 in Hlook3. apply elem_of_map_of_list_2, elem_of_list_filter in Hlook3. set_unfold. naive_solver. Qed. End gmap. (** ** Big ops over finite sets *) Section gset. Context `{Countable A}. Implicit Types X : gset A. Implicit Types Φ : A → uPred M. (* Lemma big_sepS_mono Φ Ψ X Y : Y ⊆ X → (∀ x, x ∈ Y → Φ x ⊢ Ψ x) → ([★ set] x ∈ X, Φ x) ⊢ [★ set] x ∈ Y, Ψ x. Proof. intros HX HΦ. trans ([★ set] x ∈ Y, Φ x)%IP. - by apply big_sep_contains, fmap_contains, elements_contains. - apply big_sep_mono', Forall2_fmap, Forall_Forall2. apply Forall_forall=> x ? /=. by apply HΦ, elem_of_elements. Qed. *) Lemma big_sepS_proper Φ Ψ X Y : X ≡ Y → (∀ x, x ∈ X → x ∈ Y → Φ x ⊣⊢ Ψ x) → ([★ set] x ∈ X, Φ x) ⊣⊢ ([★ set] x ∈ Y, Ψ x). Proof. intros HX HΦ. apply (anti_symm (⊢)). - trans ([★ set] x ∈ Y, Φ x)%IP. * apply big_sep_permutation, fmap_Permutation. apply (anti_symm submseteq); set_solver. * apply big_sep_mono', Forall2_fmap, Forall_Forall2. apply Forall_forall=> x ? /=. rewrite HΦ; eauto. ** rewrite HX; by apply elem_of_elements. ** by apply elem_of_elements. - trans ([★ set] x ∈ X, Ψ x)%IP. * apply big_sep_permutation, fmap_Permutation. apply (anti_symm submseteq); set_solver. * apply big_sep_mono', Forall2_fmap, Forall_Forall2. apply Forall_forall=> x ? /=. rewrite HΦ; eauto. ** by apply elem_of_elements. ** rewrite -HX; by apply elem_of_elements. Qed. Lemma big_sepS_ne X n : Proper (pointwise_relation _ (dist n) ==> dist n) (uPred_big_sepS (M:=M) X). Proof. intros Φ1 Φ2 HΦ. apply big_sep_ne, Forall2_fmap. apply Forall_Forall2, Forall_true=> x; apply HΦ. Qed. Lemma big_sepS_proper' X : Proper (pointwise_relation _ (⊣⊢) ==> (⊣⊢)) (uPred_big_sepS (M:=M) X). Proof. intros Φ1 Φ2 HΦ. apply big_sepS_proper; naive_solver. Qed. (* Lemma big_sepS_mono' X : Proper (pointwise_relation _ (⊢) ==> (⊢)) (uPred_big_sepS (M:=M) X). Proof. intros Φ1 Φ2 HΦ. apply big_sepS_mono; naive_solver. Qed. *) Lemma big_sepS_empty Φ : ([★ set] x ∈ ∅, Φ x) ⊣⊢ Emp. Proof. by rewrite /uPred_big_sepS elements_empty. Qed. Lemma big_sepS_insert Φ X x : x ∉ X → ([★ set] y ∈ {[ x ]} ∪ X, Φ y) ⊣⊢ (Φ x ★ [★ set] y ∈ X, Φ y). Proof. intros. by rewrite /uPred_big_sepS elements_union_singleton. Qed. Lemma big_sepS_delete Φ X x : x ∈ X → ([★ set] y ∈ X, Φ y) ⊣⊢ Φ x ★ [★ set] y ∈ X ∖ {[ x ]}, Φ y. Proof. intros. rewrite -big_sepS_insert; last set_solver. rewrite -union_difference_L; try set_solver. Qed. Lemma big_sepS_singleton Φ x : ([★ set] y ∈ {[ x ]}, Φ y) ⊣⊢ Φ x. Proof. intros. by rewrite /uPred_big_sepS elements_singleton /= right_id. Qed. Lemma big_sepS_sepS Φ Ψ X : ([★ set] y ∈ X, Φ y ★ Ψ y) ⊣⊢ ([★ set] y ∈ X, Φ y) ★ ([★ set] y ∈ X, Ψ y). Proof. rewrite /uPred_big_sepS. induction (elements X) as [|x l IH]; csimpl; first by rewrite ?right_id. by rewrite IH -!assoc (assoc _ (Ψ _)) [(Ψ _ ★ _)%IP]comm -!assoc. Qed. (* Lemma big_sepS_always Φ X : □ ([★ set] y ∈ X, Φ y) ⊣⊢ ([★ set] y ∈ X, □ Φ y). Proof. rewrite /uPred_big_sepS. induction (elements X) as [|x l IH]; csimpl; first by rewrite ?always_pure. by rewrite always_sep IH. Qed. Lemma big_sepS_always_if q Φ X : □?q ([★ set] y ∈ X, Φ y) ⊣⊢ ([★ set] y ∈ X, □?q Φ y). Proof. destruct q; simpl; auto using big_sepS_always. Qed. *) (* Lemma big_sepS_forall Φ X : (∀ x, RelevantP (Φ x)) → ([★ set] x ∈ X, Φ x) ⊣⊢ (∀ x, ■ (x ∈ X) → Φ x). Proof. intros. apply (anti_symm _). { apply forall_intro=> x. apply impl_intro_l, pure_elim_l=> ?; by apply big_sepS_elem_of. } rewrite /uPred_big_sepS. setoid_rewrite <-elem_of_elements. induction (elements X) as [|x l IH]; csimpl; auto. rewrite -always_and_sep_l; apply and_intro. - rewrite (forall_elim x) pure_equiv; last by left. by rewrite True_impl. - rewrite -IH. apply forall_mono=> y. apply impl_intro_l, pure_elim_l=> ?. rewrite pure_equiv; last by right. by rewrite True_impl. Qed. *) (* Lemma big_sepS_impl Φ Ψ X : □ (∀ x, ■ (x ∈ X) → Φ x → Ψ x) ∧ ([★ set] x ∈ X, Φ x) ⊢ [★ set] x ∈ X, Ψ x. Proof. rewrite relevant_and_sep_l_1 relevant_forall. setoid_rewrite always_impl; setoid_rewrite always_pure. rewrite -big_sepS_forall -big_sepS_sepS. apply big_sepS_mono; auto=> x ?. by rewrite -always_wand_impl always_elim wand_elim_l. Qed. *) End gset. (** ** Relevant *) Global Instance big_sep_relevant Ps : RelevantL Ps → RelevantP ([★] Ps). Proof. induction 1; apply _. Qed. Global Instance big_sep_affine Ps : AffineL Ps → AffineP ([★] Ps). Proof. induction 1; apply _. Qed. Global Instance nil_relevant : RelevantL (@nil (uPred M)). Proof. constructor. Qed. Global Instance cons_relevant P Ps : RelevantP P → RelevantL Ps → RelevantL (P :: Ps). Proof. by constructor. Qed. Global Instance app_relevant Ps Ps' : RelevantL Ps → RelevantL Ps' → RelevantL (Ps ++ Ps'). Proof. apply Forall_app_2. Qed. Global Instance zip_with_relevant {A B} (f : A → B → uPred M) xs ys : (∀ x y, RelevantP (f x y)) → RelevantL (zip_with f xs ys). Proof. unfold RelevantL=> ?; revert ys; induction xs=> -[|??]; constructor; auto. Qed. Global Instance nil_affine : AffineL (@nil (uPred M)). Proof. constructor. Qed. Global Instance cons_affine P Ps : AffineP P → AffineL Ps → AffineL (P :: Ps). Proof. by constructor. Qed. Global Instance app_affine Ps Ps' : AffineL Ps → AffineL Ps' → AffineL (Ps ++ Ps'). Proof. apply Forall_app_2. Qed. Global Instance zip_with_affine {A B} (f : A → B → uPred M) xs ys : (∀ x y, AffineP (f x y)) → AffineL (zip_with f xs ys). Proof. unfold AffineL=> ?; revert ys; induction xs=> -[|??]; constructor; auto. Qed. End big_op.
 From stdpp Require Import gmap. From fri.algebra Require Export upred. From fri.algebra Require Export upred_big_op. Import uPred. Module uPred_reflection. Section uPred_reflection. Context {M : ucmraT}. Inductive expr :=