upred_big_op.v 26.1 KB
 Robbert Krebbers committed Mar 21, 2016 1 ``````From iris.algebra Require Export upred list. `````` Robbert Krebbers committed Apr 11, 2016 2 ``````From iris.prelude Require Import gmap fin_collections functions. `````` Ralf Jung committed Feb 17, 2016 3 ``````Import uPred. `````` Robbert Krebbers committed Feb 14, 2016 4 `````` `````` Robbert Krebbers committed May 24, 2016 5 6 7 8 ``````(** 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. `````` Robbert Krebbers committed Aug 24, 2016 9 10 11 ``````- The operator [ [★ list] k ↦ x ∈ l, P ] asserts that [P] holds separately for each element [x] at position [x] in the list [l]. This operator is a quantifier, and thus has the same precedence as [∀] and [∃]. `````` Robbert Krebbers committed May 24, 2016 12 13 14 15 16 17 18 ``````- 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 [∃]. *) `````` Robbert Krebbers committed Feb 16, 2016 19 20 ``````(** * Big ops over lists *) (* These are the basic building blocks for other big ops *) `````` Robbert Krebbers committed May 24, 2016 21 ``````Fixpoint uPred_big_and {M} (Ps : list (uPred M)) : uPred M := `````` Robbert Krebbers committed Feb 16, 2016 22 23 `````` match Ps with [] => True | P :: Ps => P ∧ uPred_big_and Ps end%I. Instance: Params (@uPred_big_and) 1. `````` Robbert Krebbers committed May 24, 2016 24 ``````Notation "'[∧]' Ps" := (uPred_big_and Ps) (at level 20) : uPred_scope. `````` Robbert Krebbers committed Feb 16, 2016 25 26 27 ``````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. `````` Robbert Krebbers committed May 24, 2016 28 ``````Notation "'[★]' Ps" := (uPred_big_sep Ps) (at level 20) : uPred_scope. `````` Robbert Krebbers committed Feb 14, 2016 29 `````` `````` Robbert Krebbers committed Feb 16, 2016 30 ``````(** * Other big ops *) `````` Robbert Krebbers committed Aug 24, 2016 31 32 33 34 35 36 37 38 39 40 41 ``````Definition uPred_big_sepL {M A} (l : list A) (Φ : nat → A → uPred M) : uPred M := [★] (imap Φ l). Instance: Params (@uPred_big_sepL) 2. Typeclasses Opaque uPred_big_sepL. Notation "'[★' 'list' ] k ↦ x ∈ l , P" := (uPred_big_sepL l (λ k x, P)) (at level 200, l at level 10, k, x at level 1, right associativity, format "[★ list ] k ↦ x ∈ l , P") : uPred_scope. Notation "'[★' 'list' ] x ∈ l , P" := (uPred_big_sepL l (λ _ x, P)) (at level 200, l at level 10, x at level 1, right associativity, format "[★ list ] x ∈ l , P") : uPred_scope. `````` Robbert Krebbers committed Feb 17, 2016 42 ``````Definition uPred_big_sepM {M} `{Countable K} {A} `````` Robbert Krebbers committed Feb 18, 2016 43 `````` (m : gmap K A) (Φ : K → A → uPred M) : uPred M := `````` Robbert Krebbers committed May 24, 2016 44 `````` [★] (curry Φ <\$> map_to_list m). `````` Robbert Krebbers committed Feb 17, 2016 45 ``````Instance: Params (@uPred_big_sepM) 6. `````` Robbert Krebbers committed Aug 24, 2016 46 ``````Typeclasses Opaque uPred_big_sepM. `````` Robbert Krebbers committed May 24, 2016 47 48 49 ``````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. `````` Robbert Krebbers committed Feb 14, 2016 50 `````` `````` Robbert Krebbers committed Feb 17, 2016 51 ``````Definition uPred_big_sepS {M} `{Countable A} `````` Robbert Krebbers committed May 24, 2016 52 `````` (X : gset A) (Φ : A → uPred M) : uPred M := [★] (Φ <\$> elements X). `````` Robbert Krebbers committed Feb 17, 2016 53 ``````Instance: Params (@uPred_big_sepS) 5. `````` Robbert Krebbers committed Aug 24, 2016 54 ``````Typeclasses Opaque uPred_big_sepS. `````` Robbert Krebbers committed May 24, 2016 55 56 57 ``````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. `````` Robbert Krebbers committed Feb 16, 2016 58 `````` `````` Robbert Krebbers committed Aug 24, 2016 59 ``````(** * Persistence and timelessness of lists of uPreds *) `````` Robbert Krebbers committed Mar 11, 2016 60 ``````Class PersistentL {M} (Ps : list (uPred M)) := `````` Robbert Krebbers committed Mar 15, 2016 61 `````` persistentL : Forall PersistentP Ps. `````` Robbert Krebbers committed Mar 11, 2016 62 ``````Arguments persistentL {_} _ {_}. `````` Robbert Krebbers committed Feb 14, 2016 63 `````` `````` Robbert Krebbers committed Aug 24, 2016 64 65 66 67 ``````Class TimelessL {M} (Ps : list (uPred M)) := timelessL : Forall TimelessP Ps. Arguments timelessL {_} _ {_}. `````` Robbert Krebbers committed Apr 08, 2016 68 ``````(** * Properties *) `````` Robbert Krebbers committed Feb 14, 2016 69 ``````Section big_op. `````` Robbert Krebbers committed May 27, 2016 70 ``````Context {M : ucmraT}. `````` Robbert Krebbers committed Feb 14, 2016 71 72 73 ``````Implicit Types Ps Qs : list (uPred M). Implicit Types A : Type. `````` Robbert Krebbers committed Aug 24, 2016 74 ``````(** ** Generic big ops over lists of upreds *) `````` Ralf Jung committed Mar 10, 2016 75 ``````Global Instance big_and_proper : Proper ((≡) ==> (⊣⊢)) (@uPred_big_and M). `````` Robbert Krebbers committed Feb 14, 2016 76 ``````Proof. by induction 1 as [|P Q Ps Qs HPQ ? IH]; rewrite /= ?HPQ ?IH. Qed. `````` Ralf Jung committed Mar 10, 2016 77 ``````Global Instance big_sep_proper : Proper ((≡) ==> (⊣⊢)) (@uPred_big_sep M). `````` Robbert Krebbers committed Feb 14, 2016 78 ``````Proof. by induction 1 as [|P Q Ps Qs HPQ ? IH]; rewrite /= ?HPQ ?IH. Qed. `````` Robbert Krebbers committed Feb 17, 2016 79 `````` `````` Robbert Krebbers committed Mar 21, 2016 80 ``````Global Instance big_and_ne n : Proper (dist n ==> dist n) (@uPred_big_and M). `````` Robbert Krebbers committed Feb 17, 2016 81 ``````Proof. by induction 1 as [|P Q Ps Qs HPQ ? IH]; rewrite /= ?HPQ ?IH. Qed. `````` Robbert Krebbers committed Mar 21, 2016 82 ``````Global Instance big_sep_ne n : Proper (dist n ==> dist n) (@uPred_big_sep M). `````` Robbert Krebbers committed Feb 17, 2016 83 84 ``````Proof. by induction 1 as [|P Q Ps Qs HPQ ? IH]; rewrite /= ?HPQ ?IH. Qed. `````` Ralf Jung committed Mar 10, 2016 85 ``````Global Instance big_and_mono' : Proper (Forall2 (⊢) ==> (⊢)) (@uPred_big_and M). `````` Robbert Krebbers committed Feb 17, 2016 86 ``````Proof. by induction 1 as [|P Q Ps Qs HPQ ? IH]; rewrite /= ?HPQ ?IH. Qed. `````` Ralf Jung committed Mar 10, 2016 87 ``````Global Instance big_sep_mono' : Proper (Forall2 (⊢) ==> (⊢)) (@uPred_big_sep M). `````` Robbert Krebbers committed Feb 17, 2016 88 89 ``````Proof. by induction 1 as [|P Q Ps Qs HPQ ? IH]; rewrite /= ?HPQ ?IH. Qed. `````` Ralf Jung committed Mar 10, 2016 90 ``````Global Instance big_and_perm : Proper ((≡ₚ) ==> (⊣⊢)) (@uPred_big_and M). `````` Robbert Krebbers committed Feb 14, 2016 91 92 ``````Proof. induction 1 as [|P Ps Qs ? IH|P Q Ps|]; simpl; auto. `````` Robbert Krebbers committed Feb 17, 2016 93 94 `````` - by rewrite IH. - by rewrite !assoc (comm _ P). `````` Ralf Jung committed Feb 20, 2016 95 `````` - etrans; eauto. `````` Robbert Krebbers committed Feb 14, 2016 96 ``````Qed. `````` Ralf Jung committed Mar 10, 2016 97 ``````Global Instance big_sep_perm : Proper ((≡ₚ) ==> (⊣⊢)) (@uPred_big_sep M). `````` Robbert Krebbers committed Feb 14, 2016 98 99 ``````Proof. induction 1 as [|P Ps Qs ? IH|P Q Ps|]; simpl; auto. `````` Robbert Krebbers committed Feb 17, 2016 100 101 `````` - by rewrite IH. - by rewrite !assoc (comm _ P). `````` Ralf Jung committed Feb 20, 2016 102 `````` - etrans; eauto. `````` Robbert Krebbers committed Feb 14, 2016 103 ``````Qed. `````` Robbert Krebbers committed Feb 17, 2016 104 `````` `````` Robbert Krebbers committed May 31, 2016 105 ``````Lemma big_and_app Ps Qs : [∧] (Ps ++ Qs) ⊣⊢ [∧] Ps ∧ [∧] Qs. `````` Robbert Krebbers committed May 24, 2016 106 ``````Proof. induction Ps as [|?? IH]; by rewrite /= ?left_id -?assoc ?IH. Qed. `````` Robbert Krebbers committed May 31, 2016 107 ``````Lemma big_sep_app Ps Qs : [★] (Ps ++ Qs) ⊣⊢ [★] Ps ★ [★] Qs. `````` Robbert Krebbers committed Feb 14, 2016 108 ``````Proof. by induction Ps as [|?? IH]; rewrite /= ?left_id -?assoc ?IH. Qed. `````` Robbert Krebbers committed Feb 17, 2016 109 `````` `````` Robbert Krebbers committed May 24, 2016 110 ``````Lemma big_and_contains Ps Qs : Qs `contains` Ps → [∧] Ps ⊢ [∧] Qs. `````` Robbert Krebbers committed Feb 17, 2016 111 ``````Proof. `````` Ralf Jung committed Feb 17, 2016 112 `````` intros [Ps' ->]%contains_Permutation. by rewrite big_and_app and_elim_l. `````` Robbert Krebbers committed Feb 17, 2016 113 ``````Qed. `````` Robbert Krebbers committed May 24, 2016 114 ``````Lemma big_sep_contains Ps Qs : Qs `contains` Ps → [★] Ps ⊢ [★] Qs. `````` Robbert Krebbers committed Feb 17, 2016 115 ``````Proof. `````` Ralf Jung committed Feb 17, 2016 116 `````` intros [Ps' ->]%contains_Permutation. by rewrite big_sep_app sep_elim_l. `````` Robbert Krebbers committed Feb 17, 2016 117 118 ``````Qed. `````` Robbert Krebbers committed May 24, 2016 119 ``````Lemma big_sep_and Ps : [★] Ps ⊢ [∧] Ps. `````` Robbert Krebbers committed Feb 14, 2016 120 ``````Proof. by induction Ps as [|P Ps IH]; simpl; auto with I. Qed. `````` Robbert Krebbers committed Feb 17, 2016 121 `````` `````` Robbert Krebbers committed May 24, 2016 122 ``````Lemma big_and_elem_of Ps P : P ∈ Ps → [∧] Ps ⊢ P. `````` Robbert Krebbers committed Feb 14, 2016 123 ``````Proof. induction 1; simpl; auto with I. Qed. `````` Robbert Krebbers committed May 24, 2016 124 ``````Lemma big_sep_elem_of Ps P : P ∈ Ps → [★] Ps ⊢ P. `````` Robbert Krebbers committed Feb 14, 2016 125 126 ``````Proof. induction 1; simpl; auto with I. Qed. `````` Robbert Krebbers committed Aug 24, 2016 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 ``````(** ** Persistence *) Global Instance big_and_persistent Ps : PersistentL Ps → PersistentP ([∧] Ps). Proof. induction 1; apply _. Qed. Global Instance big_sep_persistent Ps : PersistentL Ps → PersistentP ([★] Ps). Proof. induction 1; apply _. Qed. Global Instance nil_persistent : PersistentL (@nil (uPred M)). Proof. constructor. Qed. Global Instance cons_persistent P Ps : PersistentP P → PersistentL Ps → PersistentL (P :: Ps). Proof. by constructor. Qed. Global Instance app_persistent Ps Ps' : PersistentL Ps → PersistentL Ps' → PersistentL (Ps ++ Ps'). Proof. apply Forall_app_2. Qed. Global Instance fmap_persistent {A} (f : A → uPred M) xs : (∀ x, PersistentP (f x)) → PersistentL (f <\$> xs). `````` Robbert Krebbers committed Aug 24, 2016 144 ``````Proof. intros. apply Forall_fmap, Forall_forall; auto. Qed. `````` Robbert Krebbers committed Aug 24, 2016 145 146 147 148 149 ``````Global Instance zip_with_persistent {A B} (f : A → B → uPred M) xs ys : (∀ x y, PersistentP (f x y)) → PersistentL (zip_with f xs ys). Proof. unfold PersistentL=> ?; revert ys; induction xs=> -[|??]; constructor; auto. Qed. `````` Robbert Krebbers committed Aug 24, 2016 150 151 152 153 154 ``````Global Instance imap_persistent {A} (f : nat → A → uPred M) xs : (∀ i x, PersistentP (f i x)) → PersistentL (imap f xs). Proof. rewrite /PersistentL /imap=> ?. generalize 0. induction xs; constructor; auto. Qed. `````` Robbert Krebbers committed Aug 24, 2016 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 `````` (** ** Timelessness *) Global Instance big_and_timeless Ps : TimelessL Ps → TimelessP ([∧] Ps). Proof. induction 1; apply _. Qed. Global Instance big_sep_timeless Ps : TimelessL Ps → TimelessP ([★] Ps). Proof. induction 1; apply _. Qed. Global Instance nil_timeless : TimelessL (@nil (uPred M)). Proof. constructor. Qed. Global Instance cons_timeless P Ps : TimelessP P → TimelessL Ps → TimelessL (P :: Ps). Proof. by constructor. Qed. Global Instance app_timeless Ps Ps' : TimelessL Ps → TimelessL Ps' → TimelessL (Ps ++ Ps'). Proof. apply Forall_app_2. Qed. Global Instance fmap_timeless {A} (f : A → uPred M) xs : (∀ x, TimelessP (f x)) → TimelessL (f <\$> xs). `````` Robbert Krebbers committed Aug 24, 2016 173 ``````Proof. intros. apply Forall_fmap, Forall_forall; auto. Qed. `````` Robbert Krebbers committed Aug 24, 2016 174 175 176 177 178 ``````Global Instance zip_with_timeless {A B} (f : A → B → uPred M) xs ys : (∀ x y, TimelessP (f x y)) → TimelessL (zip_with f xs ys). Proof. unfold TimelessL=> ?; revert ys; induction xs=> -[|??]; constructor; auto. Qed. `````` Robbert Krebbers committed Aug 24, 2016 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 ``````Global Instance imap_timeless {A} (f : nat → A → uPred M) xs : (∀ i x, TimelessP (f i x)) → TimelessL (imap f xs). Proof. rewrite /TimelessL /imap=> ?. generalize 0. induction xs; constructor; auto. Qed. (** ** Big ops over lists *) Section list. Context {A : Type}. Implicit Types l : list A. Implicit Types Φ Ψ : nat → A → uPred M. Lemma big_sepL_mono Φ Ψ l : (∀ k y, l !! k = Some y → Φ k y ⊢ Ψ k y) → ([★ list] k ↦ y ∈ l, Φ k y) ⊢ [★ list] k ↦ y ∈ l, Ψ k y. Proof. intros HΦ. apply big_sep_mono'. revert Φ Ψ HΦ. induction l as [|x l IH]=> Φ Ψ HΦ; first constructor. rewrite !imap_cons; constructor; eauto. Qed. Lemma big_sepL_proper Φ Ψ l : (∀ k y, l !! k = Some y → Φ k y ⊣⊢ Ψ k y) → ([★ list] k ↦ y ∈ l, Φ k y) ⊣⊢ ([★ list] k ↦ y ∈ l, Ψ k y). Proof. intros ?; apply (anti_symm (⊢)); apply big_sepL_mono; eauto using equiv_entails, equiv_entails_sym, lookup_weaken. Qed. Global Instance big_sepL_ne l n : Proper (pointwise_relation _ (pointwise_relation _ (dist n)) ==> (dist n)) (uPred_big_sepL (M:=M) l). Proof. intros Φ Ψ HΦ. apply big_sep_ne. revert Φ Ψ HΦ. induction l as [|x l IH]=> Φ Ψ HΦ; first constructor. rewrite !imap_cons; constructor. by apply HΦ. apply IH=> n'; apply HΦ. Qed. Global Instance big_sepL_proper' l : Proper (pointwise_relation _ (pointwise_relation _ (⊣⊢)) ==> (⊣⊢)) (uPred_big_sepL (M:=M) l). Proof. intros Φ1 Φ2 HΦ. by apply big_sepL_proper; intros; last apply HΦ. Qed. Global Instance big_sepL_mono' l : Proper (pointwise_relation _ (pointwise_relation _ (⊢)) ==> (⊢)) (uPred_big_sepL (M:=M) l). Proof. intros Φ1 Φ2 HΦ. by apply big_sepL_mono; intros; last apply HΦ. Qed. Lemma big_sepL_nil Φ : ([★ list] k↦y ∈ nil, Φ k y) ⊣⊢ True. Proof. done. Qed. Lemma big_sepL_cons Φ x l : ([★ list] k↦y ∈ x :: l, Φ k y) ⊣⊢ Φ 0 x ★ [★ list] k↦y ∈ l, Φ (S k) y. Proof. by rewrite /uPred_big_sepL imap_cons. Qed. Lemma big_sepL_singleton Φ x : ([★ list] k↦y ∈ [x], Φ k y) ⊣⊢ Φ 0 x. Proof. by rewrite big_sepL_cons big_sepL_nil right_id. Qed. Lemma big_sepL_app Φ l1 l2 : ([★ list] k↦y ∈ l1 ++ l2, Φ k y) ⊣⊢ ([★ list] k↦y ∈ l1, Φ k y) ★ ([★ list] k↦y ∈ l2, Φ (length l1 + k) y). Proof. by rewrite /uPred_big_sepL imap_app big_sep_app. Qed. Lemma big_sepL_lookup Φ l i x : l !! i = Some x → ([★ list] k↦y ∈ l, Φ k y) ⊢ Φ i x. Proof. intros. rewrite -(take_drop_middle l i x) // big_sepL_app big_sepL_cons. rewrite Nat.add_0_r take_length_le; eauto using lookup_lt_Some, Nat.lt_le_incl. by rewrite sep_elim_r sep_elim_l. Qed. `````` Robbert Krebbers committed Aug 28, 2016 247 248 249 250 251 252 `````` Lemma big_sepL_elem_of (Φ : A → uPred M) l x : x ∈ l → ([★ list] y ∈ l, Φ y) ⊢ Φ x. Proof. intros [i ?]%elem_of_list_lookup; eauto using (big_sepL_lookup (λ _, Φ)). Qed. `````` Robbert Krebbers committed Aug 24, 2016 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 `````` Lemma big_sepL_fmap {B} (f : A → B) (Φ : nat → B → uPred M) l : ([★ list] k↦y ∈ f <\$> l, Φ k y) ⊣⊢ ([★ list] k↦y ∈ l, Φ k (f y)). Proof. by rewrite /uPred_big_sepL imap_fmap. Qed. Lemma big_sepL_sepL Φ Ψ l : ([★ list] k↦x ∈ l, Φ k x ★ Ψ k x) ⊣⊢ ([★ list] k↦x ∈ l, Φ k x) ★ ([★ list] k↦x ∈ l, Ψ k x). Proof. revert Φ Ψ; induction l as [|x l IH]=> Φ Ψ. { by rewrite !big_sepL_nil left_id. } rewrite !big_sepL_cons IH. by rewrite -!assoc (assoc _ (Ψ _ _)) [(Ψ _ _ ★ _)%I]comm -!assoc. Qed. Lemma big_sepL_later Φ l : ▷ ([★ list] k↦x ∈ l, Φ k x) ⊣⊢ ([★ list] k↦x ∈ l, ▷ Φ k x). Proof. revert Φ. induction l as [|x l IH]=> Φ. { by rewrite !big_sepL_nil later_True. } by rewrite !big_sepL_cons later_sep IH. Qed. Lemma big_sepL_always Φ l : (□ [★ list] k↦x ∈ l, Φ k x) ⊣⊢ ([★ list] k↦x ∈ l, □ Φ k x). Proof. revert Φ. induction l as [|x l IH]=> Φ. { by rewrite !big_sepL_nil always_pure. } by rewrite !big_sepL_cons always_sep IH. Qed. Lemma big_sepL_always_if p Φ l : □?p ([★ list] k↦x ∈ l, Φ k x) ⊣⊢ ([★ list] k↦x ∈ l, □?p Φ k x). Proof. destruct p; simpl; auto using big_sepL_always. Qed. Lemma big_sepL_forall Φ l : (∀ k x, PersistentP (Φ k x)) → ([★ list] k↦x ∈ l, Φ k x) ⊣⊢ (∀ k x, l !! k = Some x → Φ k x). Proof. intros HΦ. apply (anti_symm _). { apply forall_intro=> k; apply forall_intro=> x. apply impl_intro_l, pure_elim_l=> ?; by apply big_sepL_lookup. } revert Φ HΦ. induction l as [|x l IH]=> Φ HΦ. { rewrite big_sepL_nil; auto with I. } rewrite big_sepL_cons. rewrite -always_and_sep_l; apply and_intro. - by rewrite (forall_elim 0) (forall_elim x) pure_equiv // True_impl. - rewrite -IH. apply forall_intro=> k; by rewrite (forall_elim (S k)). Qed. Lemma big_sepL_impl Φ Ψ l : □ (∀ k x, l !! k = Some x → Φ k x → Ψ k x) ∧ ([★ list] k↦x ∈ l, Φ k x) ⊢ [★ list] k↦x ∈ l, 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_sepL_forall -big_sepL_sepL. apply big_sepL_mono; auto=> k x ?. by rewrite -always_wand_impl always_elim wand_elim_l. Qed. `````` Robbert Krebbers committed Aug 28, 2016 311 312 313 314 315 `````` Global Instance big_sepL_nil_persistent Φ : PersistentP ([★ list] k↦x ∈ [], Φ k x). Proof. rewrite /uPred_big_sepL. apply _. Qed. Global Instance big_sepL_persistent Φ l : (∀ k x, PersistentP (Φ k x)) → PersistentP ([★ list] k↦x ∈ l, Φ k x). `````` Robbert Krebbers committed Aug 24, 2016 316 317 `````` Proof. rewrite /uPred_big_sepL. apply _. Qed. `````` Robbert Krebbers committed Aug 28, 2016 318 319 320 321 322 `````` Global Instance big_sepL_nil_timeless Φ : TimelessP ([★ list] k↦x ∈ [], Φ k x). Proof. rewrite /uPred_big_sepL. apply _. Qed. Global Instance big_sepL_timeless Φ l : (∀ k x, TimelessP (Φ k x)) → TimelessP ([★ list] k↦x ∈ l, Φ k x). `````` Robbert Krebbers committed Aug 24, 2016 323 324 325 `````` Proof. rewrite /uPred_big_sepL. apply _. Qed. End list. `````` Robbert Krebbers committed Aug 24, 2016 326 `````` `````` Robbert Krebbers committed Apr 08, 2016 327 ``````(** ** Big ops over finite maps *) `````` Robbert Krebbers committed Feb 17, 2016 328 329 330 ``````Section gmap. Context `{Countable K} {A : Type}. Implicit Types m : gmap K A. `````` Robbert Krebbers committed Feb 18, 2016 331 `````` Implicit Types Φ Ψ : K → A → uPred M. `````` Robbert Krebbers committed Feb 14, 2016 332 `````` `````` Robbert Krebbers committed Feb 18, 2016 333 `````` Lemma big_sepM_mono Φ Ψ m1 m2 : `````` Robbert Krebbers committed May 30, 2016 334 `````` m2 ⊆ m1 → (∀ k x, m2 !! k = Some x → Φ k x ⊢ Ψ k x) → `````` Robbert Krebbers committed May 31, 2016 335 `````` ([★ map] k ↦ x ∈ m1, Φ k x) ⊢ [★ map] k ↦ x ∈ m2, Ψ k x. `````` Robbert Krebbers committed Feb 16, 2016 336 `````` Proof. `````` Robbert Krebbers committed May 24, 2016 337 `````` intros HX HΦ. trans ([★ map] k↦x ∈ m2, Φ k x)%I. `````` Robbert Krebbers committed Feb 17, 2016 338 `````` - by apply big_sep_contains, fmap_contains, map_to_list_contains. `````` Robbert Krebbers committed Mar 21, 2016 339 `````` - apply big_sep_mono', Forall2_fmap, Forall_Forall2. `````` Robbert Krebbers committed Feb 18, 2016 340 `````` apply Forall_forall=> -[i x] ? /=. by apply HΦ, elem_of_map_to_list. `````` Robbert Krebbers committed Feb 16, 2016 341 `````` Qed. `````` Robbert Krebbers committed Jul 22, 2016 342 343 344 `````` 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). `````` Robbert Krebbers committed Jul 25, 2016 345 346 347 348 `````` Proof. intros ?; apply (anti_symm (⊢)); apply big_sepM_mono; eauto using equiv_entails, equiv_entails_sym, lookup_weaken. Qed. `````` Robbert Krebbers committed Feb 17, 2016 349 350 351 352 353 `````` Global Instance big_sepM_ne m n : Proper (pointwise_relation _ (pointwise_relation _ (dist n)) ==> (dist n)) (uPred_big_sepM (M:=M) m). Proof. `````` Robbert Krebbers committed Feb 18, 2016 354 `````` intros Φ1 Φ2 HΦ. apply big_sep_ne, Forall2_fmap. `````` Robbert Krebbers committed Mar 21, 2016 355 `````` apply Forall_Forall2, Forall_true=> -[i x]; apply HΦ. `````` Robbert Krebbers committed Feb 17, 2016 356 `````` Qed. `````` Robbert Krebbers committed Apr 11, 2016 357 `````` Global Instance big_sepM_proper' m : `````` Ralf Jung committed Mar 10, 2016 358 `````` Proper (pointwise_relation _ (pointwise_relation _ (⊣⊢)) ==> (⊣⊢)) `````` Robbert Krebbers committed Feb 17, 2016 359 `````` (uPred_big_sepM (M:=M) m). `````` Robbert Krebbers committed Apr 11, 2016 360 `````` Proof. intros Φ1 Φ2 HΦ. by apply big_sepM_proper; intros; last apply HΦ. Qed. `````` Robbert Krebbers committed Feb 17, 2016 361 `````` Global Instance big_sepM_mono' m : `````` Ralf Jung committed Mar 10, 2016 362 `````` Proper (pointwise_relation _ (pointwise_relation _ (⊢)) ==> (⊢)) `````` Robbert Krebbers committed Feb 17, 2016 363 `````` (uPred_big_sepM (M:=M) m). `````` Robbert Krebbers committed Apr 11, 2016 364 `````` Proof. intros Φ1 Φ2 HΦ. by apply big_sepM_mono; intros; last apply HΦ. Qed. `````` Robbert Krebbers committed Feb 17, 2016 365 `````` `````` Robbert Krebbers committed May 24, 2016 366 `````` Lemma big_sepM_empty Φ : ([★ map] k↦x ∈ ∅, Φ k x) ⊣⊢ True. `````` Robbert Krebbers committed Feb 17, 2016 367 `````` Proof. by rewrite /uPred_big_sepM map_to_list_empty. Qed. `````` Robbert Krebbers committed May 30, 2016 368 `````` `````` Robbert Krebbers committed May 31, 2016 369 `````` Lemma big_sepM_insert Φ m i x : `````` Robbert Krebbers committed May 24, 2016 370 `````` m !! i = None → `````` Robbert Krebbers committed May 31, 2016 371 `````` ([★ map] k↦y ∈ <[i:=x]> m, Φ k y) ⊣⊢ Φ i x ★ [★ map] k↦y ∈ m, Φ k y. `````` Robbert Krebbers committed Feb 17, 2016 372 `````` Proof. intros ?; by rewrite /uPred_big_sepM map_to_list_insert. Qed. `````` Robbert Krebbers committed May 30, 2016 373 `````` `````` Robbert Krebbers committed May 31, 2016 374 `````` Lemma big_sepM_delete Φ m i x : `````` Robbert Krebbers committed May 24, 2016 375 `````` m !! i = Some x → `````` Robbert Krebbers committed May 31, 2016 376 `````` ([★ map] k↦y ∈ m, Φ k y) ⊣⊢ Φ i x ★ [★ map] k↦y ∈ delete i m, Φ k y. `````` Robbert Krebbers committed May 31, 2016 377 378 379 380 `````` Proof. intros. rewrite -big_sepM_insert ?lookup_delete //. by rewrite insert_delete insert_id. Qed. `````` Robbert Krebbers committed May 30, 2016 381 `````` `````` Robbert Krebbers committed May 31, 2016 382 383 384 385 `````` Lemma big_sepM_lookup Φ m i x : m !! i = Some x → ([★ map] k↦y ∈ m, Φ k y) ⊢ Φ i x. Proof. intros. by rewrite big_sepM_delete // sep_elim_l. Qed. `````` Robbert Krebbers committed May 24, 2016 386 `````` Lemma big_sepM_singleton Φ i x : ([★ map] k↦y ∈ {[i:=x]}, Φ k y) ⊣⊢ Φ i x. `````` Robbert Krebbers committed Feb 14, 2016 387 388 389 390 `````` Proof. rewrite -insert_empty big_sepM_insert/=; last auto using lookup_empty. by rewrite big_sepM_empty right_id. Qed. `````` Ralf Jung committed Feb 17, 2016 391 `````` `````` Robbert Krebbers committed May 31, 2016 392 393 394 395 396 397 398 `````` 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. `````` Robbert Krebbers committed May 31, 2016 399 `````` Lemma big_sepM_insert_override (Φ : K → uPred M) m i x y : `````` Robbert Krebbers committed May 31, 2016 400 `````` m !! i = Some x → `````` Robbert Krebbers committed May 31, 2016 401 `````` ([★ map] k↦_ ∈ <[i:=y]> m, Φ k) ⊣⊢ ([★ map] k↦_ ∈ m, Φ k). `````` Robbert Krebbers committed May 31, 2016 402 403 404 405 406 `````` Proof. intros. rewrite -insert_delete big_sepM_insert ?lookup_delete //. by rewrite -big_sepM_delete. Qed. `````` Robbert Krebbers committed Jun 01, 2016 407 `````` Lemma big_sepM_fn_insert {B} (Ψ : K → A → B → uPred M) (f : K → B) m i x b : `````` Robbert Krebbers committed May 31, 2016 408 `````` m !! i = None → `````` Robbert Krebbers committed Jun 01, 2016 409 410 `````` ([★ 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)). `````` Robbert Krebbers committed May 31, 2016 411 412 `````` Proof. intros. rewrite big_sepM_insert // fn_lookup_insert. `````` Robbert Krebbers committed Jul 22, 2016 413 `````` apply sep_proper, big_sepM_proper; auto=> k y ?. `````` Robbert Krebbers committed May 31, 2016 414 415 416 417 418 419 420 `````` by rewrite fn_lookup_insert_ne; last set_solver. 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. `````` Robbert Krebbers committed Feb 18, 2016 421 `````` Lemma big_sepM_sepM Φ Ψ m : `````` Robbert Krebbers committed May 24, 2016 422 `````` ([★ map] k↦x ∈ m, Φ k x ★ Ψ k x) `````` Robbert Krebbers committed May 31, 2016 423 `````` ⊣⊢ ([★ map] k↦x ∈ m, Φ k x) ★ ([★ map] k↦x ∈ m, Ψ k x). `````` Ralf Jung committed Feb 17, 2016 424 `````` Proof. `````` Robbert Krebbers committed Feb 17, 2016 425 426 `````` rewrite /uPred_big_sepM. induction (map_to_list m) as [|[i x] l IH]; csimpl; rewrite ?right_id //. `````` Robbert Krebbers committed Feb 18, 2016 427 `````` by rewrite IH -!assoc (assoc _ (Ψ _ _)) [(Ψ _ _ ★ _)%I]comm -!assoc. `````` Ralf Jung committed Feb 17, 2016 428 `````` Qed. `````` Robbert Krebbers committed May 31, 2016 429 `````` `````` Robbert Krebbers committed May 24, 2016 430 `````` Lemma big_sepM_later Φ m : `````` Robbert Krebbers committed May 31, 2016 431 `````` ▷ ([★ map] k↦x ∈ m, Φ k x) ⊣⊢ ([★ map] k↦x ∈ m, ▷ Φ k x). `````` Ralf Jung committed Feb 17, 2016 432 `````` Proof. `````` Robbert Krebbers committed Feb 17, 2016 433 434 435 `````` rewrite /uPred_big_sepM. induction (map_to_list m) as [|[i x] l IH]; csimpl; rewrite ?later_True //. by rewrite later_sep IH. `````` Ralf Jung committed Feb 17, 2016 436 `````` Qed. `````` Robbert Krebbers committed May 31, 2016 437 438 439 440 441 `````` Lemma big_sepM_always Φ m : (□ [★ map] k↦x ∈ m, Φ k x) ⊣⊢ ([★ map] k↦x ∈ m, □ Φ k x). Proof. rewrite /uPred_big_sepM. `````` Robbert Krebbers committed Jun 24, 2016 442 `````` induction (map_to_list m) as [|[i x] l IH]; csimpl; rewrite ?always_pure //. `````` Robbert Krebbers committed May 31, 2016 443 444 445 446 `````` by rewrite always_sep IH. Qed. Lemma big_sepM_always_if p Φ m : `````` Robbert Krebbers committed May 31, 2016 447 `````` □?p ([★ map] k↦x ∈ m, Φ k x) ⊣⊢ ([★ map] k↦x ∈ m, □?p Φ k x). `````` Robbert Krebbers committed May 31, 2016 448 `````` Proof. destruct p; simpl; auto using big_sepM_always. Qed. `````` Robbert Krebbers committed May 31, 2016 449 450 451 452 453 454 455 `````` Lemma big_sepM_forall Φ m : (∀ k x, PersistentP (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. `````` Robbert Krebbers committed Jun 24, 2016 456 `````` apply impl_intro_l, pure_elim_l=> ?; by apply big_sepM_lookup. } `````` Robbert Krebbers committed May 31, 2016 457 458 459 `````` 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. `````` Robbert Krebbers committed Jun 24, 2016 460 `````` - rewrite (forall_elim i) (forall_elim x) pure_equiv; last by left. `````` Robbert Krebbers committed May 31, 2016 461 462 `````` by rewrite True_impl. - rewrite -IH. apply forall_mono=> k; apply forall_mono=> y. `````` Robbert Krebbers committed Jun 24, 2016 463 `````` apply impl_intro_l, pure_elim_l=> ?. rewrite pure_equiv; last by right. `````` Robbert Krebbers committed May 31, 2016 464 465 466 467 `````` by rewrite True_impl. Qed. Lemma big_sepM_impl Φ Ψ m : `````` Robbert Krebbers committed May 31, 2016 468 `````` □ (∀ k x, m !! k = Some x → Φ k x → Ψ k x) ∧ ([★ map] k↦x ∈ m, Φ k x) `````` Robbert Krebbers committed May 31, 2016 469 470 471 `````` ⊢ [★ map] k↦x ∈ m, Ψ k x. Proof. rewrite always_and_sep_l. do 2 setoid_rewrite always_forall. `````` Robbert Krebbers committed Jun 24, 2016 472 `````` setoid_rewrite always_impl; setoid_rewrite always_pure. `````` Robbert Krebbers committed May 31, 2016 473 474 475 `````` 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. `````` Robbert Krebbers committed Aug 24, 2016 476 `````` `````` Robbert Krebbers committed Aug 28, 2016 477 478 479 `````` Global Instance big_sepM_empty_persistent Φ : PersistentP ([★ map] k↦x ∈ ∅, Φ k x). Proof. rewrite /uPred_big_sepM map_to_list_empty. apply _. Qed. `````` Robbert Krebbers committed Aug 24, 2016 480 481 482 483 `````` Global Instance big_sepM_persistent Φ m : (∀ k x, PersistentP (Φ k x)) → PersistentP ([★ map] k↦x ∈ m, Φ k x). Proof. intros. apply big_sep_persistent, fmap_persistent=>-[??] /=; auto. Qed. `````` Robbert Krebbers committed Aug 28, 2016 484 485 486 `````` Global Instance big_sepM_nil_timeless Φ : TimelessP ([★ map] k↦x ∈ ∅, Φ k x). Proof. rewrite /uPred_big_sepM map_to_list_empty. apply _. Qed. `````` Robbert Krebbers committed Aug 24, 2016 487 488 489 `````` Global Instance big_sepM_timeless Φ m : (∀ k x, TimelessP (Φ k x)) → TimelessP ([★ map] k↦x ∈ m, Φ k x). Proof. intro. apply big_sep_timeless, fmap_timeless=> -[??] /=; auto. Qed. `````` Robbert Krebbers committed Feb 17, 2016 490 491 ``````End gmap. `````` Robbert Krebbers committed Aug 24, 2016 492 `````` `````` Robbert Krebbers committed Apr 08, 2016 493 ``````(** ** Big ops over finite sets *) `````` Robbert Krebbers committed Feb 17, 2016 494 495 496 ``````Section gset. Context `{Countable A}. Implicit Types X : gset A. `````` Robbert Krebbers committed Feb 18, 2016 497 `````` Implicit Types Φ : A → uPred M. `````` Robbert Krebbers committed Feb 17, 2016 498 `````` `````` Robbert Krebbers committed Feb 18, 2016 499 `````` Lemma big_sepS_mono Φ Ψ X Y : `````` Robbert Krebbers committed May 24, 2016 500 `````` Y ⊆ X → (∀ x, x ∈ Y → Φ x ⊢ Ψ x) → `````` Robbert Krebbers committed May 31, 2016 501 `````` ([★ set] x ∈ X, Φ x) ⊢ [★ set] x ∈ Y, Ψ x. `````` Robbert Krebbers committed Feb 17, 2016 502 `````` Proof. `````` Robbert Krebbers committed May 24, 2016 503 `````` intros HX HΦ. trans ([★ set] x ∈ Y, Φ x)%I. `````` Robbert Krebbers committed Feb 17, 2016 504 `````` - by apply big_sep_contains, fmap_contains, elements_contains. `````` Robbert Krebbers committed Mar 21, 2016 505 `````` - apply big_sep_mono', Forall2_fmap, Forall_Forall2. `````` Robbert Krebbers committed Feb 18, 2016 506 `````` apply Forall_forall=> x ? /=. by apply HΦ, elem_of_elements. `````` Robbert Krebbers committed Feb 17, 2016 507 `````` Qed. `````` Robbert Krebbers committed Apr 11, 2016 508 `````` Lemma big_sepS_proper Φ Ψ X Y : `````` Robbert Krebbers committed May 24, 2016 509 510 `````` X ≡ Y → (∀ x, x ∈ X → x ∈ Y → Φ x ⊣⊢ Ψ x) → ([★ set] x ∈ X, Φ x) ⊣⊢ ([★ set] x ∈ Y, Ψ x). `````` Robbert Krebbers committed Apr 11, 2016 511 `````` Proof. `````` Robbert Krebbers committed Jul 22, 2016 512 513 `````` move=> /collection_equiv_spec [??] ?; apply (anti_symm (⊢)); apply big_sepS_mono; eauto using equiv_entails, equiv_entails_sym. `````` Robbert Krebbers committed Apr 11, 2016 514 `````` Qed. `````` Robbert Krebbers committed Feb 17, 2016 515 516 517 518 `````` Lemma big_sepS_ne X n : Proper (pointwise_relation _ (dist n) ==> dist n) (uPred_big_sepS (M:=M) X). Proof. `````` Robbert Krebbers committed Feb 18, 2016 519 `````` intros Φ1 Φ2 HΦ. apply big_sep_ne, Forall2_fmap. `````` Robbert Krebbers committed Mar 21, 2016 520 `````` apply Forall_Forall2, Forall_true=> x; apply HΦ. `````` Robbert Krebbers committed Feb 17, 2016 521 `````` Qed. `````` Robbert Krebbers committed Apr 11, 2016 522 `````` Lemma big_sepS_proper' X : `````` Ralf Jung committed Mar 10, 2016 523 `````` Proper (pointwise_relation _ (⊣⊢) ==> (⊣⊢)) (uPred_big_sepS (M:=M) X). `````` Robbert Krebbers committed Apr 11, 2016 524 `````` Proof. intros Φ1 Φ2 HΦ. apply big_sepS_proper; naive_solver. Qed. `````` Robbert Krebbers committed Feb 17, 2016 525 `````` Lemma big_sepS_mono' X : `````` Ralf Jung committed Mar 10, 2016 526 `````` Proper (pointwise_relation _ (⊢) ==> (⊢)) (uPred_big_sepS (M:=M) X). `````` Robbert Krebbers committed Feb 18, 2016 527 `````` Proof. intros Φ1 Φ2 HΦ. apply big_sepS_mono; naive_solver. Qed. `````` Robbert Krebbers committed Feb 17, 2016 528 `````` `````` Robbert Krebbers committed May 24, 2016 529 `````` Lemma big_sepS_empty Φ : ([★ set] x ∈ ∅, Φ x) ⊣⊢ True. `````` Robbert Krebbers committed Feb 17, 2016 530 `````` Proof. by rewrite /uPred_big_sepS elements_empty. Qed. `````` Robbert Krebbers committed Apr 11, 2016 531 `````` `````` Robbert Krebbers committed Feb 18, 2016 532 `````` Lemma big_sepS_insert Φ X x : `````` Robbert Krebbers committed May 24, 2016 533 `````` x ∉ X → ([★ set] y ∈ {[ x ]} ∪ X, Φ y) ⊣⊢ (Φ x ★ [★ set] y ∈ X, Φ y). `````` Robbert Krebbers committed Feb 17, 2016 534 `````` Proof. intros. by rewrite /uPred_big_sepS elements_union_singleton. Qed. `````` Robbert Krebbers committed Jun 01, 2016 535 `````` Lemma big_sepS_fn_insert {B} (Ψ : A → B → uPred M) f X x b : `````` Robbert Krebbers committed Apr 11, 2016 536 `````` x ∉ X → `````` Robbert Krebbers committed Jun 01, 2016 537 538 `````` ([★ set] y ∈ {[ x ]} ∪ X, Ψ y (<[x:=b]> f y)) ⊣⊢ (Ψ x b ★ [★ set] y ∈ X, Ψ y (f y)). `````` Robbert Krebbers committed Apr 11, 2016 539 540 541 542 543 `````` Proof. intros. rewrite big_sepS_insert // fn_lookup_insert. apply sep_proper, big_sepS_proper; auto=> y ??. by rewrite fn_lookup_insert_ne; last set_solver. Qed. `````` Robbert Krebbers committed May 30, 2016 544 `````` Lemma big_sepS_fn_insert' Φ X x P : `````` Robbert Krebbers committed May 24, 2016 545 `````` x ∉ X → ([★ set] y ∈ {[ x ]} ∪ X, <[x:=P]> Φ y) ⊣⊢ (P ★ [★ set] y ∈ X, Φ y). `````` Robbert Krebbers committed May 30, 2016 546 `````` Proof. apply (big_sepS_fn_insert (λ y, id)). Qed. `````` Robbert Krebbers committed Apr 11, 2016 547 `````` `````` Robbert Krebbers committed Feb 18, 2016 548 `````` Lemma big_sepS_delete Φ X x : `````` Robbert Krebbers committed May 31, 2016 549 `````` x ∈ X → ([★ set] y ∈ X, Φ y) ⊣⊢ Φ x ★ [★ set] y ∈ X ∖ {[ x ]}, Φ y. `````` Robbert Krebbers committed Feb 17, 2016 550 `````` Proof. `````` Robbert Krebbers committed Feb 17, 2016 551 552 `````` intros. rewrite -big_sepS_insert; last set_solver. by rewrite -union_difference_L; last set_solver. `````` Robbert Krebbers committed Feb 17, 2016 553 `````` Qed. `````` Robbert Krebbers committed Apr 11, 2016 554 `````` `````` Robbert Krebbers committed May 31, 2016 555 556 557 `````` Lemma big_sepS_elem_of Φ X x : x ∈ X → ([★ set] y ∈ X, Φ y) ⊢ Φ x. Proof. intros. by rewrite big_sepS_delete // sep_elim_l. Qed. `````` Robbert Krebbers committed May 24, 2016 558 `````` Lemma big_sepS_singleton Φ x : ([★ set] y ∈ {[ x ]}, Φ y) ⊣⊢ Φ x. `````` Robbert Krebbers committed Feb 17, 2016 559 `````` Proof. intros. by rewrite /uPred_big_sepS elements_singleton /= right_id. Qed. `````` Ralf Jung committed Feb 17, 2016 560 `````` `````` Robbert Krebbers committed Feb 18, 2016 561 `````` Lemma big_sepS_sepS Φ Ψ X : `````` Robbert Krebbers committed May 31, 2016 562 `````` ([★ set] y ∈ X, Φ y ★ Ψ y) ⊣⊢ ([★ set] y ∈ X, Φ y) ★ ([★ set] y ∈ X, Ψ y). `````` Ralf Jung committed Feb 17, 2016 563 `````` Proof. `````` Robbert Krebbers committed Feb 17, 2016 564 565 `````` rewrite /uPred_big_sepS. induction (elements X) as [|x l IH]; csimpl; first by rewrite ?right_id. `````` Robbert Krebbers committed Feb 18, 2016 566 `````` by rewrite IH -!assoc (assoc _ (Ψ _)) [(Ψ _ ★ _)%I]comm -!assoc. `````` Ralf Jung committed Feb 17, 2016 567 568 `````` Qed. `````` Robbert Krebbers committed May 31, 2016 569 `````` Lemma big_sepS_later Φ X : ▷ ([★ set] y ∈ X, Φ y) ⊣⊢ ([★ set] y ∈ X, ▷ Φ y). `````` Ralf Jung committed Feb 17, 2016 570 `````` Proof. `````` Robbert Krebbers committed Feb 17, 2016 571 572 573 `````` rewrite /uPred_big_sepS. induction (elements X) as [|x l IH]; csimpl; first by rewrite ?later_True. by rewrite later_sep IH. `````` Ralf Jung committed Feb 17, 2016 574 `````` Qed. `````` Robbert Krebbers committed May 31, 2016 575 `````` `````` Robbert Krebbers committed May 31, 2016 576 `````` Lemma big_sepS_always Φ X : □ ([★ set] y ∈ X, Φ y) ⊣⊢ ([★ set] y ∈ X, □ Φ y). `````` Robbert Krebbers committed May 31, 2016 577 578 `````` Proof. rewrite /uPred_big_sepS. `````` Robbert Krebbers committed Jun 24, 2016 579 `````` induction (elements X) as [|x l IH]; csimpl; first by rewrite ?always_pure. `````` Robbert Krebbers committed May 31, 2016 580 581 582 583 `````` by rewrite always_sep IH. Qed. Lemma big_sepS_always_if q Φ X : `````` Robbert Krebbers committed May 31, 2016 584 `````` □?q ([★ set] y ∈ X, Φ y) ⊣⊢ ([★ set] y ∈ X, □?q Φ y). `````` Robbert Krebbers committed May 31, 2016 585 `````` Proof. destruct q; simpl; auto using big_sepS_always. Qed. `````` Robbert Krebbers committed May 31, 2016 586 587 588 589 590 591 `````` Lemma big_sepS_forall Φ X : (∀ x, PersistentP (Φ x)) → ([★ set] x ∈ X, Φ x) ⊣⊢ (∀ x, ■ (x ∈ X) → Φ x). Proof. intros. apply (anti_symm _). { apply forall_intro=> x. `````` Robbert Krebbers committed Jun 24, 2016 592 `````` apply impl_intro_l, pure_elim_l=> ?; by apply big_sepS_elem_of. } `````` Robbert Krebbers committed May 31, 2016 593 594 595 `````` 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. `````` Robbert Krebbers committed Jun 24, 2016 596 `````` - rewrite (forall_elim x) pure_equiv; last by left. by rewrite True_impl. `````` Robbert Krebbers committed May 31, 2016 597 `````` - rewrite -IH. apply forall_mono=> y. `````` Robbert Krebbers committed Jun 24, 2016 598 `````` apply impl_intro_l, pure_elim_l=> ?. rewrite pure_equiv; last by right. `````` Robbert Krebbers committed May 31, 2016 599 600 601 602 `````` by rewrite True_impl. Qed. Lemma big_sepS_impl Φ Ψ X : `````` Robbert Krebbers committed May 31, 2016 603 `````` □ (∀ x, ■ (x ∈ X) → Φ x → Ψ x) ∧ ([★ set] x ∈ X, Φ x) ⊢ [★ set] x ∈ X, Ψ x. `````` Robbert Krebbers committed May 31, 2016 604 605 `````` Proof. rewrite always_and_sep_l always_forall. `````` Robbert Krebbers committed Jun 24, 2016 606 `````` setoid_rewrite always_impl; setoid_rewrite always_pure. `````` Robbert Krebbers committed May 31, 2016 607 608 609 `````` rewrite -big_sepS_forall -big_sepS_sepS. apply big_sepS_mono; auto=> x ?. by rewrite -always_wand_impl always_elim wand_elim_l. Qed. `````` Robbert Krebbers committed Feb 14, 2016 610 `````` `````` Robbert Krebbers committed Aug 28, 2016 611 612 `````` Global Instance big_sepS_empty_persistent Φ : PersistentP ([★ set] x ∈ ∅, Φ x). Proof. rewrite /uPred_big_sepS elements_empty. apply _. Qed. `````` Robbert Krebbers committed Aug 24, 2016 613 614 615 `````` Global Instance big_sepS_persistent Φ X : (∀ x, PersistentP (Φ x)) → PersistentP ([★ set] x ∈ X, Φ x). Proof. rewrite /uPred_big_sepS. apply _. Qed. `````` Robbert Krebbers committed Feb 14, 2016 616 `````` `````` Robbert Krebbers committed Aug 28, 2016 617 618 `````` Global Instance big_sepS_nil_timeless Φ : TimelessP ([★ set] x ∈ ∅, Φ x). Proof. rewrite /uPred_big_sepS elements_empty. apply _. Qed. `````` Robbert Krebbers committed Aug 24, 2016 619 620 621 622 `````` Global Instance big_sepS_timeless Φ X : (∀ x, TimelessP (Φ x)) → TimelessP ([★ set] x ∈ X, Φ x). Proof. rewrite /uPred_big_sepS. apply _. Qed. End gset. `````` Robbert Krebbers committed Feb 16, 2016 623 ``End big_op.``