Commit 5e1645b4 by Robbert Krebbers

### Big op for uPred_sep for finite sets.

`With nicely overloaded notations for sets and maps.`
parent 1109ca07
 From algebra Require Export upred. From prelude Require Import fin_maps. From prelude Require Import fin_maps fin_collections. Fixpoint uPred_big_and {M} (Ps : list (uPred M)) : uPred M:= match Ps with [] => True | P :: Ps => P ∧ uPred_big_and Ps end%I. Instance: Params (@uPred_big_and) 1. Notation "'Π∧' Ps" := (uPred_big_and Ps) (at level 20) : uPred_scope. Fixpoint uPred_big_sep {M} (Ps : list (uPred M)) : uPred M := match Ps with [] => True | P :: Ps => P ★ uPred_big_sep Ps end%I. Instance: Params (@uPred_big_sep) 1. Notation "'Π★' Ps" := (uPred_big_sep Ps) (at level 20) : uPred_scope. (** * Big ops over lists *) (* These are the basic building blocks for other big ops *) Fixpoint uPred_list_and {M} (Ps : list (uPred M)) : uPred M:= match Ps with [] => True | P :: Ps => P ∧ uPred_list_and Ps end%I. Instance: Params (@uPred_list_and) 1. Notation "'Π∧' Ps" := (uPred_list_and Ps) (at level 20) : uPred_scope. Fixpoint uPred_list_sep {M} (Ps : list (uPred M)) : uPred M := match Ps with [] => True | P :: Ps => P ★ uPred_list_sep Ps end%I. Instance: Params (@uPred_list_sep) 1. Notation "'Π★' Ps" := (uPred_list_sep Ps) (at level 20) : uPred_scope. Definition uPred_big_sepM {M : cmraT} `{FinMapToList K A MA} (P : K → A → uPred M) (m : MA) : uPred M := uPred_big_sep (curry P <\$> map_to_list m). Instance: Params (@uPred_big_sepM) 5. Notation "'Π★{' P } m" := (uPred_big_sepM P m) (at level 20, P at level 10, m at level 20, format "Π★{ P } m") : uPred_scope. (** * Other big ops *) (** We use a type class to obtain overloaded notations *) Class UPredBigSep (M : cmraT) (A B : Type) := uPred_big_sep : A → B → uPred M. Instance: Params (@uPred_big_sep) 4. Notation "'Π★{' x } P" := (uPred_big_sep x P) (at level 20, x at level 10, format "Π★{ x } P") : uPred_scope. Instance uPred_big_sepM {M} `{FinMapToList K A MA} : UPredBigSep M MA (K → A → uPred M) := λ m P, uPred_list_sep (curry P <\$> map_to_list m). Instance uPred_big_sepC {M} `{Elements A C} : UPredBigSep M C (A → uPred M) := λ X P, uPred_list_sep (P <\$> elements X). (** * Always stability for lists *) Class AlwaysStableL {M} (Ps : list (uPred M)) := always_stableL : Forall AlwaysStable Ps. Arguments always_stableL {_} _ {_}. ... ... @@ -28,45 +38,47 @@ Implicit Types Ps Qs : list (uPred M). Implicit Types A : Type. (* Big ops *) Global Instance big_and_proper : Proper ((≡) ==> (≡)) (@uPred_big_and M). Global Instance list_and_proper : Proper ((≡) ==> (≡)) (@uPred_list_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). Global Instance list_sep_proper : Proper ((≡) ==> (≡)) (@uPred_list_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). Global Instance list_and_perm : Proper ((≡ₚ) ==> (≡)) (@uPred_list_and M). Proof. induction 1 as [|P Ps Qs ? IH|P Q Ps|]; simpl; auto. * by rewrite IH. * by rewrite !assoc (comm _ P). * etransitivity; eauto. Qed. Global Instance big_sep_perm : Proper ((≡ₚ) ==> (≡)) (@uPred_big_sep M). Global Instance list_sep_perm : Proper ((≡ₚ) ==> (≡)) (@uPred_list_sep M). Proof. induction 1 as [|P Ps Qs ? IH|P Q Ps|]; simpl; auto. * by rewrite IH. * by rewrite !assoc (comm _ P). * etransitivity; eauto. Qed. Lemma big_and_app Ps Qs : (Π∧ (Ps ++ Qs))%I ≡ (Π∧ Ps ∧ Π∧ Qs)%I. Lemma list_and_app Ps Qs : (Π∧ (Ps ++ Qs))%I ≡ (Π∧ Ps ∧ Π∧ Qs)%I. Proof. by induction Ps as [|?? IH]; rewrite /= ?left_id -?assoc ?IH. Qed. Lemma big_sep_app Ps Qs : (Π★ (Ps ++ Qs))%I ≡ (Π★ Ps ★ Π★ Qs)%I. Lemma list_sep_app Ps Qs : (Π★ (Ps ++ Qs))%I ≡ (Π★ Ps ★ Π★ Qs)%I. Proof. by induction Ps as [|?? IH]; rewrite /= ?left_id -?assoc ?IH. Qed. Lemma big_sep_and Ps : (Π★ Ps) ⊑ (Π∧ Ps). Lemma list_sep_and Ps : (Π★ Ps) ⊑ (Π∧ Ps). Proof. by induction Ps as [|P Ps IH]; simpl; auto with I. Qed. Lemma big_and_elem_of Ps P : P ∈ Ps → (Π∧ Ps) ⊑ P. Lemma list_and_elem_of Ps P : P ∈ Ps → (Π∧ Ps) ⊑ P. Proof. induction 1; simpl; auto with I. Qed. Lemma big_sep_elem_of Ps P : P ∈ Ps → (Π★ Ps) ⊑ P. Lemma list_sep_elem_of Ps P : P ∈ Ps → (Π★ Ps) ⊑ P. Proof. induction 1; simpl; auto with I. Qed. (* Big ops over finite maps *) Section fin_map. Context `{FinMap K Ma} {A} (P : K → A → uPred M). Lemma big_sepM_empty : (Π★{P} ∅)%I ≡ True%I. Proof. by rewrite /uPred_big_sepM map_to_list_empty. Qed. Lemma big_sepM_empty : (Π★{∅} P)%I ≡ True%I. Proof. by rewrite /uPred_big_sep /uPred_big_sepM map_to_list_empty. Qed. Lemma big_sepM_insert (m : Ma A) i x : m !! i = None → (Π★{P} (<[i:=x]> m))%I ≡ (P i x ★ Π★{P} m)%I. Proof. intros ?; by rewrite /uPred_big_sepM map_to_list_insert. Qed. Lemma big_sepM_singleton i x : (Π★{P} {[i ↦ x]})%I ≡ (P i x)%I. m !! i = None → (Π★{<[i:=x]> m} P)%I ≡ (P i x ★ Π★{m} P)%I. Proof. intros ?; by rewrite /uPred_big_sep /uPred_big_sepM map_to_list_insert. Qed. Lemma big_sepM_singleton i x : (Π★{{[i ↦ x]}} P)%I ≡ (P i x)%I. Proof. rewrite -insert_empty big_sepM_insert/=; last auto using lookup_empty. by rewrite big_sepM_empty right_id. ... ... @@ -76,9 +88,9 @@ End fin_map. (* Always stable *) Local Notation AS := AlwaysStable. Local Notation ASL := AlwaysStableL. Global Instance big_and_always_stable Ps : ASL Ps → AS (Π∧ Ps). Global Instance list_and_always_stable Ps : ASL Ps → AS (Π∧ Ps). Proof. induction 1; apply _. Qed. Global Instance big_sep_always_stable Ps : ASL Ps → AS (Π★ Ps). Global Instance list_sep_always_stable Ps : ASL Ps → AS (Π★ Ps). Proof. induction 1; apply _. Qed. Global Instance nil_always_stable : ASL (@nil (uPred M)). ... ... @@ -90,4 +102,4 @@ Proof. apply Forall_app_2. Qed. Global Instance zip_with_always_stable {A B} (f : A → B → uPred M) xs ys : (∀ x y, AS (f x y)) → ASL (zip_with f xs ys). Proof. unfold ASL=> ?; revert ys; induction xs=> -[|??]; constructor; auto. Qed. End big_op. \ No newline at end of file End big_op.
 ... ... @@ -72,7 +72,7 @@ Section heap. Qed. Lemma heap_alloc N σ : ownP σ ⊑ pvs N N (∃ γ, heap_ctx HeapI γ N ∧ Π★{heap_mapsto HeapI γ} σ). ownP σ ⊑ pvs N N (∃ γ, heap_ctx HeapI γ N ∧ Π★{σ} heap_mapsto HeapI γ). Proof. rewrite -{1}(from_to_heap σ); etransitivity; first apply (auth_alloc (ownP ∘ of_heap) N (to_heap σ)), to_heap_valid. ... ...
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