upred_big_op.v 8.36 KB
 Robbert Krebbers committed Feb 14, 2016 1 ``````From algebra Require Export upred. `````` Robbert Krebbers committed Feb 17, 2016 2 ``````From prelude Require Import gmap fin_collections. `````` Robbert Krebbers committed Feb 14, 2016 3 `````` `````` Robbert Krebbers committed Feb 16, 2016 4 5 ``````(** * Big ops over lists *) (* These are the basic building blocks for other big ops *) `````` Robbert Krebbers committed Feb 16, 2016 6 7 8 9 10 11 12 13 ``````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. `````` Robbert Krebbers committed Feb 14, 2016 14 `````` `````` Robbert Krebbers committed Feb 16, 2016 15 16 ``````(** * Other big ops *) (** We use a type class to obtain overloaded notations *) `````` Robbert Krebbers committed Feb 17, 2016 17 18 ``````Definition uPred_big_sepM {M} `{Countable K} {A} (m : gmap K A) (P : K → A → uPred M) : uPred M := `````` Robbert Krebbers committed Feb 16, 2016 19 `````` uPred_big_sep (curry P <\$> map_to_list m). `````` Robbert Krebbers committed Feb 17, 2016 20 ``````Instance: Params (@uPred_big_sepM) 6. `````` Robbert Krebbers committed Feb 16, 2016 21 ``````Notation "'Π★{map' m } P" := (uPred_big_sepM m P) `````` Robbert Krebbers committed Feb 16, 2016 22 `````` (at level 20, m at level 10, format "Π★{map m } P") : uPred_scope. `````` Robbert Krebbers committed Feb 14, 2016 23 `````` `````` Robbert Krebbers committed Feb 17, 2016 24 25 26 ``````Definition uPred_big_sepS {M} `{Countable A} (X : gset A) (P : A → uPred M) : uPred M := uPred_big_sep (P <\$> elements X). Instance: Params (@uPred_big_sepS) 5. `````` Robbert Krebbers committed Feb 16, 2016 27 ``````Notation "'Π★{set' X } P" := (uPred_big_sepS X P) `````` Robbert Krebbers committed Feb 16, 2016 28 `````` (at level 20, X at level 10, format "Π★{set X } P") : uPred_scope. `````` Robbert Krebbers committed Feb 16, 2016 29 30 `````` (** * Always stability for lists *) `````` Robbert Krebbers committed Feb 14, 2016 31 32 33 34 35 36 37 38 39 40 ``````Class AlwaysStableL {M} (Ps : list (uPred M)) := always_stableL : Forall AlwaysStable Ps. Arguments always_stableL {_} _ {_}. Section big_op. Context {M : cmraT}. Implicit Types Ps Qs : list (uPred M). Implicit Types A : Type. (* Big ops *) `````` Robbert Krebbers committed Feb 16, 2016 41 ``````Global Instance big_and_proper : Proper ((≡) ==> (≡)) (@uPred_big_and M). `````` Robbert Krebbers committed Feb 14, 2016 42 ``````Proof. by induction 1 as [|P Q Ps Qs HPQ ? IH]; rewrite /= ?HPQ ?IH. Qed. `````` Robbert Krebbers committed Feb 16, 2016 43 ``````Global Instance big_sep_proper : Proper ((≡) ==> (≡)) (@uPred_big_sep M). `````` Robbert Krebbers committed Feb 14, 2016 44 ``````Proof. by induction 1 as [|P Q Ps Qs HPQ ? IH]; rewrite /= ?HPQ ?IH. Qed. `````` Robbert Krebbers committed Feb 17, 2016 45 46 47 48 49 50 51 52 53 54 55 56 57 `````` Global Instance big_and_ne n : Proper (Forall2 (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 (Forall2 (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. `````` Robbert Krebbers committed Feb 16, 2016 58 ``````Global Instance big_and_perm : Proper ((≡ₚ) ==> (≡)) (@uPred_big_and M). `````` Robbert Krebbers committed Feb 14, 2016 59 60 ``````Proof. induction 1 as [|P Ps Qs ? IH|P Q Ps|]; simpl; auto. `````` Robbert Krebbers committed Feb 17, 2016 61 62 63 `````` - by rewrite IH. - by rewrite !assoc (comm _ P). - etransitivity; eauto. `````` Robbert Krebbers committed Feb 14, 2016 64 ``````Qed. `````` Robbert Krebbers committed Feb 16, 2016 65 ``````Global Instance big_sep_perm : Proper ((≡ₚ) ==> (≡)) (@uPred_big_sep M). `````` Robbert Krebbers committed Feb 14, 2016 66 67 ``````Proof. induction 1 as [|P Ps Qs ? IH|P Q Ps|]; simpl; auto. `````` Robbert Krebbers committed Feb 17, 2016 68 69 70 `````` - by rewrite IH. - by rewrite !assoc (comm _ P). - etransitivity; eauto. `````` Robbert Krebbers committed Feb 14, 2016 71 ``````Qed. `````` Robbert Krebbers committed Feb 17, 2016 72 `````` `````` Robbert Krebbers committed Feb 16, 2016 73 ``````Lemma big_and_app Ps Qs : (Π∧ (Ps ++ Qs))%I ≡ (Π∧ Ps ∧ Π∧ Qs)%I. `````` Robbert Krebbers committed Feb 14, 2016 74 ``````Proof. by induction Ps as [|?? IH]; rewrite /= ?left_id -?assoc ?IH. Qed. `````` Robbert Krebbers committed Feb 16, 2016 75 ``````Lemma big_sep_app Ps Qs : (Π★ (Ps ++ Qs))%I ≡ (Π★ Ps ★ Π★ Qs)%I. `````` Robbert Krebbers committed Feb 14, 2016 76 ``````Proof. by induction Ps as [|?? IH]; rewrite /= ?left_id -?assoc ?IH. Qed. `````` Robbert Krebbers committed Feb 17, 2016 77 78 79 80 81 82 83 84 85 86 `````` Lemma big_and_contains Ps Qs : Qs `contains` Ps → (Π∧ Ps)%I ⊑ (Π∧ Qs)%I. Proof. intros [Ps' ->]%contains_Permutation. by rewrite big_and_app uPred.and_elim_l. Qed. Lemma big_sep_contains Ps Qs : Qs `contains` Ps → (Π★ Ps)%I ⊑ (Π★ Qs)%I. Proof. intros [Ps' ->]%contains_Permutation. by rewrite big_sep_app uPred.sep_elim_l. Qed. `````` Robbert Krebbers committed Feb 16, 2016 87 ``````Lemma big_sep_and Ps : (Π★ Ps) ⊑ (Π∧ Ps). `````` Robbert Krebbers committed Feb 14, 2016 88 ``````Proof. by induction Ps as [|P Ps IH]; simpl; auto with I. Qed. `````` Robbert Krebbers committed Feb 17, 2016 89 `````` `````` Robbert Krebbers committed Feb 16, 2016 90 ``````Lemma big_and_elem_of Ps P : P ∈ Ps → (Π∧ Ps) ⊑ P. `````` Robbert Krebbers committed Feb 14, 2016 91 ``````Proof. induction 1; simpl; auto with I. Qed. `````` Robbert Krebbers committed Feb 16, 2016 92 ``````Lemma big_sep_elem_of Ps P : P ∈ Ps → (Π★ Ps) ⊑ P. `````` Robbert Krebbers committed Feb 14, 2016 93 94 ``````Proof. induction 1; simpl; auto with I. Qed. `````` Robbert Krebbers committed Feb 14, 2016 95 ``````(* Big ops over finite maps *) `````` Robbert Krebbers committed Feb 17, 2016 96 97 98 99 ``````Section gmap. Context `{Countable K} {A : Type}. Implicit Types m : gmap K A. Implicit Types P : K → A → uPred M. `````` Robbert Krebbers committed Feb 14, 2016 100 `````` `````` Robbert Krebbers committed Feb 17, 2016 101 102 103 `````` Lemma big_sepM_mono P Q m1 m2 : m2 ⊆ m1 → (∀ x k, m2 !! k = Some x → P k x ⊑ Q k x) → (Π★{map m1} P) ⊑ (Π★{map m2} Q). `````` Robbert Krebbers committed Feb 16, 2016 104 `````` Proof. `````` Robbert Krebbers committed Feb 17, 2016 105 106 107 108 `````` intros HX HP. transitivity (Π★{map m2} P)%I. - by apply big_sep_contains, fmap_contains, map_to_list_contains. - apply big_sep_mono', Forall2_fmap, Forall2_Forall. apply Forall_forall=> -[i x] ? /=. by apply HP, elem_of_map_to_list. `````` Robbert Krebbers committed Feb 16, 2016 109 `````` Qed. `````` Robbert Krebbers committed Feb 17, 2016 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 `````` Global Instance big_sepM_ne m n : Proper (pointwise_relation _ (pointwise_relation _ (dist n)) ==> (dist n)) (uPred_big_sepM (M:=M) m). Proof. intros P1 P2 HP. apply big_sep_ne, Forall2_fmap. apply Forall2_Forall, Forall_true=> -[i x]; apply HP. Qed. Global Instance big_sepM_proper m : Proper (pointwise_relation _ (pointwise_relation _ (≡)) ==> (≡)) (uPred_big_sepM (M:=M) m). Proof. intros P1 P2 HP; apply equiv_dist=> n. apply big_sepM_ne=> k x; apply equiv_dist, HP. Qed. Global Instance big_sepM_mono' m : Proper (pointwise_relation _ (pointwise_relation _ (⊑)) ==> (⊑)) (uPred_big_sepM (M:=M) m). Proof. intros P1 P2 HP. apply big_sepM_mono; intros; [done|apply HP]. Qed. Lemma big_sepM_empty P : (Π★{map ∅} P)%I ≡ True%I. Proof. by rewrite /uPred_big_sepM map_to_list_empty. Qed. Lemma big_sepM_insert P (m : gmap K A) i x : m !! i = None → (Π★{map <[i:=x]> m} P)%I ≡ (P i x ★ Π★{map m} P)%I. Proof. intros ?; by rewrite /uPred_big_sepM map_to_list_insert. Qed. Lemma big_sepM_singleton P i x : (Π★{map {[i := x]}} P)%I ≡ (P i x)%I. `````` Robbert Krebbers committed Feb 14, 2016 136 137 138 139 `````` Proof. rewrite -insert_empty big_sepM_insert/=; last auto using lookup_empty. by rewrite big_sepM_empty right_id. Qed. `````` Robbert Krebbers committed Feb 17, 2016 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 ``````End gmap. (* Big ops over finite sets *) Section gset. Context `{Countable A}. Implicit Types X : gset A. Implicit Types P : A → uPred M. Lemma big_sepS_mono P Q X Y : Y ⊆ X → (∀ x, x ∈ Y → P x ⊑ Q x) → (Π★{set X} P) ⊑ (Π★{set Y} Q). Proof. intros HX HP. transitivity (Π★{set Y} P)%I. - by apply big_sep_contains, fmap_contains, elements_contains. - apply big_sep_mono', Forall2_fmap, Forall2_Forall. apply Forall_forall=> x ? /=. by apply HP, 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 P1 P2 HP. apply big_sep_ne, Forall2_fmap. apply Forall2_Forall, Forall_true=> x; apply HP. Qed. Lemma big_sepS_proper X : Proper (pointwise_relation _ (≡) ==> (≡)) (uPred_big_sepS (M:=M) X). Proof. intros P1 P2 HP; apply equiv_dist=> n. apply big_sepS_ne=> x; apply equiv_dist, HP. Qed. Lemma big_sepS_mono' X : Proper (pointwise_relation _ (⊑) ==> (⊑)) (uPred_big_sepS (M:=M) X). Proof. intros P1 P2 HP. apply big_sepS_mono; naive_solver. Qed. Lemma big_sepS_empty P : (Π★{set ∅} P)%I ≡ True%I. Proof. by rewrite /uPred_big_sepS elements_empty. Qed. Lemma big_sepS_insert P X x : x ∉ X → (Π★{set {[ x ]} ∪ X} P)%I ≡ (P x ★ Π★{set X} P)%I. Proof. intros. by rewrite /uPred_big_sepS elements_union_singleton. Qed. Lemma big_sepS_delete P X x : x ∈ X → (Π★{set X} P)%I ≡ (P x ★ Π★{set X ∖ {[ x ]}} P)%I. Proof. `````` Robbert Krebbers committed Feb 17, 2016 181 182 `````` intros. rewrite -big_sepS_insert; last set_solver. by rewrite -union_difference_L; last set_solver. `````` Robbert Krebbers committed Feb 17, 2016 183 184 185 186 `````` Qed. Lemma big_sepS_singleton P x : (Π★{set {[ x ]}} P)%I ≡ (P x)%I. Proof. intros. by rewrite /uPred_big_sepS elements_singleton /= right_id. Qed. End gset. `````` Robbert Krebbers committed Feb 14, 2016 187 `````` `````` Robbert Krebbers committed Feb 14, 2016 188 189 190 ``````(* Always stable *) Local Notation AS := AlwaysStable. Local Notation ASL := AlwaysStableL. `````` Robbert Krebbers committed Feb 16, 2016 191 ``````Global Instance big_and_always_stable Ps : ASL Ps → AS (Π∧ Ps). `````` Robbert Krebbers committed Feb 14, 2016 192 ``````Proof. induction 1; apply _. Qed. `````` Robbert Krebbers committed Feb 16, 2016 193 ``````Global Instance big_sep_always_stable Ps : ASL Ps → AS (Π★ Ps). `````` Robbert Krebbers committed Feb 14, 2016 194 195 196 197 198 199 200 201 202 203 204 ``````Proof. induction 1; apply _. Qed. Global Instance nil_always_stable : ASL (@nil (uPred M)). Proof. constructor. Qed. Global Instance cons_always_stable P Ps : AS P → ASL Ps → ASL (P :: Ps). Proof. by constructor. Qed. Global Instance app_always_stable Ps Ps' : ASL Ps → ASL Ps' → ASL (Ps ++ Ps'). 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. `````` Robbert Krebbers committed Feb 16, 2016 205 ``End big_op.``