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 Jacques-Henri Jourdan committed Dec 04, 2017 1 ``````From iris.bi Require Export derived_connectives. `````` Robbert Krebbers committed Oct 30, 2017 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 ``````From iris.algebra Require Import monoid. From stdpp Require Import hlist. Module bi. Import interface.bi. Section bi_derived. Context {PROP : bi}. Implicit Types φ : Prop. Implicit Types P Q R : PROP. Implicit Types Ps : list PROP. Implicit Types A : Type. Hint Extern 100 (NonExpansive _) => solve_proper. (* Force implicit argument PROP *) Notation "P ⊢ Q" := (@bi_entails PROP P%I Q%I). Notation "P ⊣⊢ Q" := (equiv (A:=bi_car PROP) P%I Q%I). (* Derived stuff about the entailment *) Global Instance entails_anti_sym : AntiSymm (⊣⊢) (@bi_entails PROP). Proof. intros P Q ??. by apply equiv_spec. Qed. Lemma equiv_entails P Q : (P ⊣⊢ Q) → (P ⊢ Q). Proof. apply equiv_spec. Qed. Lemma equiv_entails_sym P Q : (Q ⊣⊢ P) → (P ⊢ Q). Proof. apply equiv_spec. Qed. Global Instance entails_proper : Proper ((⊣⊢) ==> (⊣⊢) ==> iff) ((⊢) : relation PROP). Proof. move => P1 P2 /equiv_spec [HP1 HP2] Q1 Q2 /equiv_spec [HQ1 HQ2]; split=>?. - by trans P1; [|trans Q1]. - by trans P2; [|trans Q2]. Qed. Lemma entails_equiv_l P Q R : (P ⊣⊢ Q) → (Q ⊢ R) → (P ⊢ R). Proof. by intros ->. Qed. Lemma entails_equiv_r P Q R : (P ⊢ Q) → (Q ⊣⊢ R) → (P ⊢ R). Proof. by intros ? <-. Qed. Global Instance bi_valid_proper : Proper ((⊣⊢) ==> iff) (@bi_valid PROP). Proof. solve_proper. Qed. Global Instance bi_valid_mono : Proper ((⊢) ==> impl) (@bi_valid PROP). Proof. solve_proper. Qed. Global Instance bi_valid_flip_mono : Proper (flip (⊢) ==> flip impl) (@bi_valid PROP). Proof. solve_proper. Qed. (* Propers *) Global Instance pure_proper : Proper (iff ==> (⊣⊢)) (@bi_pure PROP) | 0. Proof. intros φ1 φ2 Hφ. apply equiv_dist=> n. by apply pure_ne. Qed. Global Instance and_proper : Proper ((⊣⊢) ==> (⊣⊢) ==> (⊣⊢)) (@bi_and PROP) := ne_proper_2 _. Global Instance or_proper : Proper ((⊣⊢) ==> (⊣⊢) ==> (⊣⊢)) (@bi_or PROP) := ne_proper_2 _. Global Instance impl_proper : Proper ((⊣⊢) ==> (⊣⊢) ==> (⊣⊢)) (@bi_impl PROP) := ne_proper_2 _. Global Instance sep_proper : Proper ((⊣⊢) ==> (⊣⊢) ==> (⊣⊢)) (@bi_sep PROP) := ne_proper_2 _. Global Instance wand_proper : Proper ((⊣⊢) ==> (⊣⊢) ==> (⊣⊢)) (@bi_wand PROP) := ne_proper_2 _. Global Instance forall_proper A : Proper (pointwise_relation _ (⊣⊢) ==> (⊣⊢)) (@bi_forall PROP A). Proof. intros Φ1 Φ2 HΦ. apply equiv_dist=> n. apply forall_ne=> x. apply equiv_dist, HΦ. Qed. Global Instance exist_proper A : Proper (pointwise_relation _ (⊣⊢) ==> (⊣⊢)) (@bi_exist PROP A). Proof. intros Φ1 Φ2 HΦ. apply equiv_dist=> n. apply exist_ne=> x. apply equiv_dist, HΦ. Qed. Global Instance internal_eq_proper (A : ofeT) : Proper ((≡) ==> (≡) ==> (⊣⊢)) (@bi_internal_eq PROP A) := ne_proper_2 _. `````` Jacques-Henri Jourdan committed Nov 03, 2017 73 74 ``````Global Instance plainly_proper : Proper ((⊣⊢) ==> (⊣⊢)) (@bi_plainly PROP) := ne_proper _. `````` Robbert Krebbers committed Oct 30, 2017 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 ``````Global Instance persistently_proper : Proper ((⊣⊢) ==> (⊣⊢)) (@bi_persistently PROP) := ne_proper _. (* Derived logical stuff *) Lemma and_elim_l' P Q R : (P ⊢ R) → P ∧ Q ⊢ R. Proof. by rewrite and_elim_l. Qed. Lemma and_elim_r' P Q R : (Q ⊢ R) → P ∧ Q ⊢ R. Proof. by rewrite and_elim_r. Qed. Lemma or_intro_l' P Q R : (P ⊢ Q) → P ⊢ Q ∨ R. Proof. intros ->; apply or_intro_l. Qed. Lemma or_intro_r' P Q R : (P ⊢ R) → P ⊢ Q ∨ R. Proof. intros ->; apply or_intro_r. Qed. Lemma exist_intro' {A} P (Ψ : A → PROP) a : (P ⊢ Ψ a) → P ⊢ ∃ a, Ψ a. Proof. intros ->; apply exist_intro. Qed. Lemma forall_elim' {A} P (Ψ : A → PROP) : (P ⊢ ∀ a, Ψ a) → ∀ a, P ⊢ Ψ a. Proof. move=> HP a. by rewrite HP forall_elim. Qed. Hint Resolve pure_intro forall_intro. Hint Resolve or_elim or_intro_l' or_intro_r'. Hint Resolve and_intro and_elim_l' and_elim_r'. Lemma impl_intro_l P Q R : (Q ∧ P ⊢ R) → P ⊢ Q → R. Proof. intros HR; apply impl_intro_r; rewrite -HR; auto. Qed. Lemma impl_elim P Q R : (P ⊢ Q → R) → (P ⊢ Q) → P ⊢ R. Proof. intros. rewrite -(impl_elim_l' P Q R); auto. Qed. Lemma impl_elim_r' P Q R : (Q ⊢ P → R) → P ∧ Q ⊢ R. Proof. intros; apply impl_elim with P; auto. Qed. Lemma impl_elim_l P Q : (P → Q) ∧ P ⊢ Q. Proof. by apply impl_elim_l'. Qed. Lemma impl_elim_r P Q : P ∧ (P → Q) ⊢ Q. Proof. by apply impl_elim_r'. Qed. Lemma False_elim P : False ⊢ P. Proof. by apply (pure_elim' False). Qed. Lemma True_intro P : P ⊢ True. Proof. by apply pure_intro. Qed. Hint Immediate False_elim. Lemma and_mono P P' Q Q' : (P ⊢ Q) → (P' ⊢ Q') → P ∧ P' ⊢ Q ∧ Q'. Proof. auto. Qed. Lemma and_mono_l P P' Q : (P ⊢ Q) → P ∧ P' ⊢ Q ∧ P'. Proof. by intros; apply and_mono. Qed. Lemma and_mono_r P P' Q' : (P' ⊢ Q') → P ∧ P' ⊢ P ∧ Q'. Proof. by apply and_mono. Qed. Lemma or_mono P P' Q Q' : (P ⊢ Q) → (P' ⊢ Q') → P ∨ P' ⊢ Q ∨ Q'. Proof. auto. Qed. Lemma or_mono_l P P' Q : (P ⊢ Q) → P ∨ P' ⊢ Q ∨ P'. Proof. by intros; apply or_mono. Qed. Lemma or_mono_r P P' Q' : (P' ⊢ Q') → P ∨ P' ⊢ P ∨ Q'. Proof. by apply or_mono. Qed. Lemma impl_mono P P' Q Q' : (Q ⊢ P) → (P' ⊢ Q') → (P → P') ⊢ Q → Q'. Proof. intros HP HQ'; apply impl_intro_l; rewrite -HQ'. apply impl_elim with P; eauto. Qed. Lemma forall_mono {A} (Φ Ψ : A → PROP) : (∀ a, Φ a ⊢ Ψ a) → (∀ a, Φ a) ⊢ ∀ a, Ψ a. Proof. intros HP. apply forall_intro=> a; rewrite -(HP a); apply forall_elim. Qed. Lemma exist_mono {A} (Φ Ψ : A → PROP) : (∀ a, Φ a ⊢ Ψ a) → (∃ a, Φ a) ⊢ ∃ a, Ψ a. Proof. intros HΦ. apply exist_elim=> a; rewrite (HΦ a); apply exist_intro. Qed. Global Instance and_mono' : Proper ((⊢) ==> (⊢) ==> (⊢)) (@bi_and PROP). Proof. by intros P P' HP Q Q' HQ; apply and_mono. Qed. Global Instance and_flip_mono' : Proper (flip (⊢) ==> flip (⊢) ==> flip (⊢)) (@bi_and PROP). Proof. by intros P P' HP Q Q' HQ; apply and_mono. Qed. Global Instance or_mono' : Proper ((⊢) ==> (⊢) ==> (⊢)) (@bi_or PROP). Proof. by intros P P' HP Q Q' HQ; apply or_mono. Qed. Global Instance or_flip_mono' : Proper (flip (⊢) ==> flip (⊢) ==> flip (⊢)) (@bi_or PROP). Proof. by intros P P' HP Q Q' HQ; apply or_mono. Qed. Global Instance impl_mono' : Proper (flip (⊢) ==> (⊢) ==> (⊢)) (@bi_impl PROP). Proof. by intros P P' HP Q Q' HQ; apply impl_mono. Qed. Global Instance impl_flip_mono' : Proper ((⊢) ==> flip (⊢) ==> flip (⊢)) (@bi_impl PROP). Proof. by intros P P' HP Q Q' HQ; apply impl_mono. Qed. Global Instance forall_mono' A : Proper (pointwise_relation _ (⊢) ==> (⊢)) (@bi_forall PROP A). Proof. intros P1 P2; apply forall_mono. Qed. Global Instance forall_flip_mono' A : Proper (pointwise_relation _ (flip (⊢)) ==> flip (⊢)) (@bi_forall PROP A). Proof. intros P1 P2; apply forall_mono. Qed. Global Instance exist_mono' A : Proper (pointwise_relation _ ((⊢)) ==> (⊢)) (@bi_exist PROP A). Proof. intros P1 P2; apply exist_mono. Qed. Global Instance exist_flip_mono' A : Proper (pointwise_relation _ (flip (⊢)) ==> flip (⊢)) (@bi_exist PROP A). Proof. intros P1 P2; apply exist_mono. Qed. Global Instance and_idem : IdemP (⊣⊢) (@bi_and PROP). Proof. intros P; apply (anti_symm (⊢)); auto. Qed. Global Instance or_idem : IdemP (⊣⊢) (@bi_or PROP). Proof. intros P; apply (anti_symm (⊢)); auto. Qed. Global Instance and_comm : Comm (⊣⊢) (@bi_and PROP). Proof. intros P Q; apply (anti_symm (⊢)); auto. Qed. Global Instance True_and : LeftId (⊣⊢) True%I (@bi_and PROP). Proof. intros P; apply (anti_symm (⊢)); auto. Qed. Global Instance and_True : RightId (⊣⊢) True%I (@bi_and PROP). Proof. intros P; apply (anti_symm (⊢)); auto. Qed. Global Instance False_and : LeftAbsorb (⊣⊢) False%I (@bi_and PROP). Proof. intros P; apply (anti_symm (⊢)); auto. Qed. Global Instance and_False : RightAbsorb (⊣⊢) False%I (@bi_and PROP). Proof. intros P; apply (anti_symm (⊢)); auto. Qed. Global Instance True_or : LeftAbsorb (⊣⊢) True%I (@bi_or PROP). Proof. intros P; apply (anti_symm (⊢)); auto. Qed. Global Instance or_True : RightAbsorb (⊣⊢) True%I (@bi_or PROP). Proof. intros P; apply (anti_symm (⊢)); auto. Qed. Global Instance False_or : LeftId (⊣⊢) False%I (@bi_or PROP). Proof. intros P; apply (anti_symm (⊢)); auto. Qed. Global Instance or_False : RightId (⊣⊢) False%I (@bi_or PROP). Proof. intros P; apply (anti_symm (⊢)); auto. Qed. Global Instance and_assoc : Assoc (⊣⊢) (@bi_and PROP). Proof. intros P Q R; apply (anti_symm (⊢)); auto. Qed. Global Instance or_comm : Comm (⊣⊢) (@bi_or PROP). Proof. intros P Q; apply (anti_symm (⊢)); auto. Qed. Global Instance or_assoc : Assoc (⊣⊢) (@bi_or PROP). Proof. intros P Q R; apply (anti_symm (⊢)); auto. Qed. Global Instance True_impl : LeftId (⊣⊢) True%I (@bi_impl PROP). Proof. intros P; apply (anti_symm (⊢)). - by rewrite -(left_id True%I (∧)%I (_ → _)%I) impl_elim_r. - by apply impl_intro_l; rewrite left_id. Qed. Lemma False_impl P : (False → P) ⊣⊢ True. Proof. apply (anti_symm (⊢)); [by auto|]. apply impl_intro_l. rewrite left_absorb. auto. Qed. `````` Jacques-Henri Jourdan committed Dec 11, 2017 211 ``````Lemma exist_impl_forall {A} P (Ψ : A → PROP) : `````` Robbert Krebbers committed Oct 30, 2017 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 `````` ((∃ x : A, Ψ x) → P) ⊣⊢ ∀ x : A, Ψ x → P. Proof. apply equiv_spec; split. - apply forall_intro=>x. by rewrite -exist_intro. - apply impl_intro_r, impl_elim_r', exist_elim=>x. apply impl_intro_r. by rewrite (forall_elim x) impl_elim_r. Qed. Lemma or_and_l P Q R : P ∨ Q ∧ R ⊣⊢ (P ∨ Q) ∧ (P ∨ R). Proof. apply (anti_symm (⊢)); first auto. do 2 (apply impl_elim_l', or_elim; apply impl_intro_l); auto. Qed. Lemma or_and_r P Q R : P ∧ Q ∨ R ⊣⊢ (P ∨ R) ∧ (Q ∨ R). Proof. by rewrite -!(comm _ R) or_and_l. Qed. Lemma and_or_l P Q R : P ∧ (Q ∨ R) ⊣⊢ P ∧ Q ∨ P ∧ R. Proof. apply (anti_symm (⊢)); last auto. apply impl_elim_r', or_elim; apply impl_intro_l; auto. Qed. Lemma and_or_r P Q R : (P ∨ Q) ∧ R ⊣⊢ P ∧ R ∨ Q ∧ R. Proof. by rewrite -!(comm _ R) and_or_l. Qed. Lemma and_exist_l {A} P (Ψ : A → PROP) : P ∧ (∃ a, Ψ a) ⊣⊢ ∃ a, P ∧ Ψ a. Proof. apply (anti_symm (⊢)). - apply impl_elim_r'. apply exist_elim=>a. apply impl_intro_l. by rewrite -(exist_intro a). - apply exist_elim=>a. apply and_intro; first by rewrite and_elim_l. by rewrite -(exist_intro a) and_elim_r. Qed. Lemma and_exist_r {A} P (Φ: A → PROP) : (∃ a, Φ a) ∧ P ⊣⊢ ∃ a, Φ a ∧ P. Proof. rewrite -(comm _ P) and_exist_l. apply exist_proper=>a. by rewrite comm. Qed. Lemma or_exist {A} (Φ Ψ : A → PROP) : (∃ a, Φ a ∨ Ψ a) ⊣⊢ (∃ a, Φ a) ∨ (∃ a, Ψ a). Proof. apply (anti_symm (⊢)). - apply exist_elim=> a. by rewrite -!(exist_intro a). - apply or_elim; apply exist_elim=> a; rewrite -(exist_intro a); auto. Qed. Lemma and_alt P Q : P ∧ Q ⊣⊢ ∀ b : bool, if b then P else Q. Proof. apply (anti_symm _); first apply forall_intro=> -[]; auto. by apply and_intro; [rewrite (forall_elim true)|rewrite (forall_elim false)]. Qed. Lemma or_alt P Q : P ∨ Q ⊣⊢ ∃ b : bool, if b then P else Q. Proof. apply (anti_symm _); last apply exist_elim=> -[]; auto. by apply or_elim; [rewrite -(exist_intro true)|rewrite -(exist_intro false)]. Qed. Lemma entails_equiv_and P Q : (P ⊣⊢ Q ∧ P) ↔ (P ⊢ Q). Proof. split. by intros ->; auto. intros; apply (anti_symm _); auto. Qed. Global Instance iff_ne : NonExpansive2 (@bi_iff PROP). Proof. unfold bi_iff; solve_proper. Qed. Global Instance iff_proper : Proper ((⊣⊢) ==> (⊣⊢) ==> (⊣⊢)) (@bi_iff PROP) := ne_proper_2 _. Lemma iff_refl Q P : Q ⊢ P ↔ P. Proof. rewrite /bi_iff; apply and_intro; apply impl_intro_l; auto. Qed. (* Equality stuff *) Hint Resolve internal_eq_refl. Lemma equiv_internal_eq {A : ofeT} P (a b : A) : a ≡ b → P ⊢ a ≡ b. Proof. intros ->. auto. Qed. Lemma internal_eq_rewrite' {A : ofeT} a b (Ψ : A → PROP) P {HΨ : NonExpansive Ψ} : (P ⊢ a ≡ b) → (P ⊢ Ψ a) → P ⊢ Ψ b. Proof. intros Heq HΨa. rewrite -(idemp bi_and P) {1}Heq HΨa. apply impl_elim_l'. by apply internal_eq_rewrite. Qed. Lemma internal_eq_sym {A : ofeT} (a b : A) : a ≡ b ⊢ b ≡ a. Proof. apply (internal_eq_rewrite' a b (λ b, b ≡ a)%I); auto. Qed. Lemma internal_eq_iff P Q : P ≡ Q ⊢ P ↔ Q. Proof. apply (internal_eq_rewrite' P Q (λ Q, P ↔ Q))%I; auto using iff_refl. Qed. Lemma f_equiv {A B : ofeT} (f : A → B) `{!NonExpansive f} x y : x ≡ y ⊢ f x ≡ f y. Proof. apply (internal_eq_rewrite' x y (λ y, f x ≡ f y)%I); auto. Qed. Lemma prod_equivI {A B : ofeT} (x y : A * B) : x ≡ y ⊣⊢ x.1 ≡ y.1 ∧ x.2 ≡ y.2. Proof. apply (anti_symm _). - apply and_intro; apply f_equiv; apply _. - rewrite {3}(surjective_pairing x) {3}(surjective_pairing y). apply (internal_eq_rewrite' (x.1) (y.1) (λ a, (x.1,x.2) ≡ (a,y.2))%I); auto. apply (internal_eq_rewrite' (x.2) (y.2) (λ b, (x.1,x.2) ≡ (x.1,b))%I); auto. Qed. Lemma sum_equivI {A B : ofeT} (x y : A + B) : x ≡ y ⊣⊢ match x, y with | inl a, inl a' => a ≡ a' | inr b, inr b' => b ≡ b' | _, _ => False end. Proof. apply (anti_symm _). - apply (internal_eq_rewrite' x y (λ y, match x, y with | inl a, inl a' => a ≡ a' | inr b, inr b' => b ≡ b' | _, _ => False end)%I); auto. destruct x; auto. - destruct x as [a|b], y as [a'|b']; auto; apply f_equiv, _. Qed. Lemma option_equivI {A : ofeT} (x y : option A) : x ≡ y ⊣⊢ match x, y with | Some a, Some a' => a ≡ a' | None, None => True | _, _ => False end. Proof. apply (anti_symm _). - apply (internal_eq_rewrite' x y (λ y, match x, y with | Some a, Some a' => a ≡ a' | None, None => True | _, _ => False end)%I); auto. destruct x; auto. - destruct x as [a|], y as [a'|]; auto. apply f_equiv, _. Qed. Lemma sig_equivI {A : ofeT} (P : A → Prop) (x y : sig P) : `x ≡ `y ⊣⊢ x ≡ y. Proof. apply (anti_symm _). apply sig_eq. apply f_equiv, _. Qed. `````` Robbert Krebbers committed Dec 02, 2017 335 ``````Lemma ofe_fun_equivI {A} {B : A → ofeT} (f g : ofe_fun B) : f ≡ g ⊣⊢ ∀ x, f x ≡ g x. `````` Robbert Krebbers committed Oct 30, 2017 336 337 338 339 340 341 342 343 344 ``````Proof. apply (anti_symm _); auto using fun_ext. apply (internal_eq_rewrite' f g (λ g, ∀ x : A, f x ≡ g x)%I); auto. intros n h h' Hh; apply forall_ne=> x; apply internal_eq_ne; auto. Qed. Lemma ofe_morC_equivI {A B : ofeT} (f g : A -n> B) : f ≡ g ⊣⊢ ∀ x, f x ≡ g x. Proof. apply (anti_symm _). - apply (internal_eq_rewrite' f g (λ g, ∀ x : A, f x ≡ g x)%I); auto. `````` Robbert Krebbers committed Dec 02, 2017 345 `````` - rewrite -(ofe_fun_equivI (ofe_mor_car _ _ f) (ofe_mor_car _ _ g)). `````` Robbert Krebbers committed Oct 30, 2017 346 `````` set (h1 (f : A -n> B) := `````` Robbert Krebbers committed Dec 02, 2017 347 348 `````` exist (λ f : A -c> B, NonExpansive (f : A → B)) f (ofe_mor_ne A B f)). set (h2 (f : sigC (λ f : A -c> B, NonExpansive (f : A → B))) := `````` Robbert Krebbers committed Oct 30, 2017 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 `````` @CofeMor A B (`f) (proj2_sig f)). assert (∀ f, h2 (h1 f) = f) as Hh by (by intros []). assert (NonExpansive h2) by (intros ??? EQ; apply EQ). by rewrite -{2}[f]Hh -{2}[g]Hh -f_equiv -sig_equivI. Qed. (* BI Stuff *) Hint Resolve sep_mono. Lemma sep_mono_l P P' Q : (P ⊢ Q) → P ∗ P' ⊢ Q ∗ P'. Proof. by intros; apply sep_mono. Qed. Lemma sep_mono_r P P' Q' : (P' ⊢ Q') → P ∗ P' ⊢ P ∗ Q'. Proof. by apply sep_mono. Qed. Global Instance sep_mono' : Proper ((⊢) ==> (⊢) ==> (⊢)) (@bi_sep PROP). Proof. by intros P P' HP Q Q' HQ; apply sep_mono. Qed. Global Instance sep_flip_mono' : Proper (flip (⊢) ==> flip (⊢) ==> flip (⊢)) (@bi_sep PROP). Proof. by intros P P' HP Q Q' HQ; apply sep_mono. Qed. Lemma wand_mono P P' Q Q' : (Q ⊢ P) → (P' ⊢ Q') → (P -∗ P') ⊢ Q -∗ Q'. Proof. intros HP HQ; apply wand_intro_r. rewrite HP -HQ. by apply wand_elim_l'. Qed. Global Instance wand_mono' : Proper (flip (⊢) ==> (⊢) ==> (⊢)) (@bi_wand PROP). Proof. by intros P P' HP Q Q' HQ; apply wand_mono. Qed. Global Instance wand_flip_mono' : Proper ((⊢) ==> flip (⊢) ==> flip (⊢)) (@bi_wand PROP). Proof. by intros P P' HP Q Q' HQ; apply wand_mono. Qed. Global Instance sep_comm : Comm (⊣⊢) (@bi_sep PROP). Proof. intros P Q; apply (anti_symm _); auto using sep_comm'. Qed. Global Instance sep_assoc : Assoc (⊣⊢) (@bi_sep PROP). Proof. intros P Q R; apply (anti_symm _); auto using sep_assoc'. by rewrite !(comm _ P) !(comm _ _ R) sep_assoc'. Qed. Global Instance emp_sep : LeftId (⊣⊢) emp%I (@bi_sep PROP). Proof. intros P; apply (anti_symm _); auto using emp_sep_1, emp_sep_2. Qed. Global Instance sep_emp : RightId (⊣⊢) emp%I (@bi_sep PROP). Proof. by intros P; rewrite comm left_id. Qed. Global Instance sep_False : LeftAbsorb (⊣⊢) False%I (@bi_sep PROP). Proof. intros P; apply (anti_symm _); auto using wand_elim_l'. Qed. Global Instance False_sep : RightAbsorb (⊣⊢) False%I (@bi_sep PROP). Proof. intros P. by rewrite comm left_absorb. Qed. Lemma True_sep_2 P : P ⊢ True ∗ P. Proof. rewrite -{1}[P](left_id emp%I bi_sep). auto using sep_mono. Qed. Lemma sep_True_2 P : P ⊢ P ∗ True. Proof. by rewrite comm -True_sep_2. Qed. Lemma sep_intro_valid_l P Q R : P → (R ⊢ Q) → R ⊢ P ∗ Q. Proof. intros ? ->. rewrite -{1}(left_id emp%I _ Q). by apply sep_mono. Qed. Lemma sep_intro_valid_r P Q R : (R ⊢ P) → Q → R ⊢ P ∗ Q. Proof. intros -> ?. rewrite comm. by apply sep_intro_valid_l. Qed. Lemma sep_elim_valid_l P Q R : P → (P ∗ R ⊢ Q) → R ⊢ Q. Proof. intros <- <-. by rewrite left_id. Qed. Lemma sep_elim_valid_r P Q R : P → (R ∗ P ⊢ Q) → R ⊢ Q. Proof. intros <- <-. by rewrite right_id. Qed. Lemma wand_intro_l P Q R : (Q ∗ P ⊢ R) → P ⊢ Q -∗ R. Proof. rewrite comm; apply wand_intro_r. Qed. Lemma wand_elim_l P Q : (P -∗ Q) ∗ P ⊢ Q. Proof. by apply wand_elim_l'. Qed. Lemma wand_elim_r P Q : P ∗ (P -∗ Q) ⊢ Q. Proof. rewrite (comm _ P); apply wand_elim_l. Qed. Lemma wand_elim_r' P Q R : (Q ⊢ P -∗ R) → P ∗ Q ⊢ R. Proof. intros ->; apply wand_elim_r. Qed. Lemma wand_apply P Q R S : (P ⊢ Q -∗ R) → (S ⊢ P ∗ Q) → S ⊢ R. Proof. intros HR%wand_elim_l' HQ. by rewrite HQ. Qed. Lemma wand_frame_l P Q R : (Q -∗ R) ⊢ P ∗ Q -∗ P ∗ R. Proof. apply wand_intro_l. rewrite -assoc. apply sep_mono_r, wand_elim_r. Qed. Lemma wand_frame_r P Q R : (Q -∗ R) ⊢ Q ∗ P -∗ R ∗ P. Proof. apply wand_intro_l. rewrite ![(_ ∗ P)%I]comm -assoc. apply sep_mono_r, wand_elim_r. Qed. Lemma emp_wand P : (emp -∗ P) ⊣⊢ P. Proof. apply (anti_symm _). - by rewrite -[(emp -∗ P)%I]left_id wand_elim_r. - apply wand_intro_l. by rewrite left_id. Qed. Lemma False_wand P : (False -∗ P) ⊣⊢ True. Proof. apply (anti_symm (⊢)); [by auto|]. apply wand_intro_l. rewrite left_absorb. auto. Qed. Lemma wand_curry P Q R : (P -∗ Q -∗ R) ⊣⊢ (P ∗ Q -∗ R). Proof. apply (anti_symm _). - apply wand_intro_l. by rewrite (comm _ P) -assoc !wand_elim_r. - do 2 apply wand_intro_l. by rewrite assoc (comm _ Q) wand_elim_r. Qed. Lemma sep_and_l P Q R : P ∗ (Q ∧ R) ⊢ (P ∗ Q) ∧ (P ∗ R). Proof. auto. Qed. Lemma sep_and_r P Q R : (P ∧ Q) ∗ R ⊢ (P ∗ R) ∧ (Q ∗ R). Proof. auto. Qed. Lemma sep_or_l P Q R : P ∗ (Q ∨ R) ⊣⊢ (P ∗ Q) ∨ (P ∗ R). Proof. apply (anti_symm (⊢)); last by eauto 8. apply wand_elim_r', or_elim; apply wand_intro_l; auto. Qed. Lemma sep_or_r P Q R : (P ∨ Q) ∗ R ⊣⊢ (P ∗ R) ∨ (Q ∗ R). Proof. by rewrite -!(comm _ R) sep_or_l. Qed. Lemma sep_exist_l {A} P (Ψ : A → PROP) : P ∗ (∃ a, Ψ a) ⊣⊢ ∃ a, P ∗ Ψ a. Proof. intros; apply (anti_symm (⊢)). - apply wand_elim_r', exist_elim=>a. apply wand_intro_l. by rewrite -(exist_intro a). - apply exist_elim=> a; apply sep_mono; auto using exist_intro. Qed. Lemma sep_exist_r {A} (Φ: A → PROP) Q: (∃ a, Φ a) ∗ Q ⊣⊢ ∃ a, Φ a ∗ Q. Proof. setoid_rewrite (comm _ _ Q); apply sep_exist_l. Qed. Lemma sep_forall_l {A} P (Ψ : A → PROP) : P ∗ (∀ a, Ψ a) ⊢ ∀ a, P ∗ Ψ a. Proof. by apply forall_intro=> a; rewrite forall_elim. Qed. Lemma sep_forall_r {A} (Φ : A → PROP) Q : (∀ a, Φ a) ∗ Q ⊢ ∀ a, Φ a ∗ Q. Proof. by apply forall_intro=> a; rewrite forall_elim. Qed. Global Instance wand_iff_ne : NonExpansive2 (@bi_wand_iff PROP). Proof. solve_proper. Qed. Global Instance wand_iff_proper : Proper ((⊣⊢) ==> (⊣⊢) ==> (⊣⊢)) (@bi_wand_iff PROP) := ne_proper_2 _. Lemma wand_iff_refl P : emp ⊢ P ∗-∗ P. Proof. apply and_intro; apply wand_intro_l; by rewrite right_id. Qed. Lemma wand_entails P Q : (P -∗ Q)%I → P ⊢ Q. Proof. intros. rewrite -[P]left_id. by apply wand_elim_l'. Qed. Lemma entails_wand P Q : (P ⊢ Q) → (P -∗ Q)%I. Proof. intros ->. apply wand_intro_r. by rewrite left_id. Qed. Lemma equiv_wand_iff P Q : (P ⊣⊢ Q) → (P ∗-∗ Q)%I. Proof. intros ->; apply wand_iff_refl. Qed. Lemma wand_iff_equiv P Q : (P ∗-∗ Q)%I → (P ⊣⊢ Q). Proof. intros HPQ; apply (anti_symm (⊢)); apply wand_entails; rewrite /bi_valid HPQ /bi_wand_iff; auto. Qed. Lemma entails_impl P Q : (P ⊢ Q) → (P → Q)%I. Proof. intros ->. apply impl_intro_l. auto. Qed. Lemma impl_entails P Q `{!Affine P} : (P → Q)%I → P ⊢ Q. Proof. intros HPQ. apply impl_elim with P=>//. by rewrite {1}(affine P). Qed. Lemma equiv_iff P Q : (P ⊣⊢ Q) → (P ↔ Q)%I. Proof. intros ->; apply iff_refl. Qed. Lemma iff_equiv P Q `{!Affine P, !Affine Q} : (P ↔ Q)%I → (P ⊣⊢ Q). Proof. intros HPQ; apply (anti_symm (⊢)); apply: impl_entails; rewrite /bi_valid HPQ /bi_iff; auto. Qed. (* Pure stuff *) Lemma pure_elim φ Q R : (Q ⊢ ⌜φ⌝) → (φ → Q ⊢ R) → Q ⊢ R. Proof. intros HQ HQR. rewrite -(idemp (∧)%I Q) {1}HQ. apply impl_elim_l', pure_elim'=> ?. apply impl_intro_l. rewrite and_elim_l; auto. Qed. Lemma pure_mono φ1 φ2 : (φ1 → φ2) → ⌜φ1⌝ ⊢ ⌜φ2⌝. Proof. auto using pure_elim', pure_intro. Qed. Global Instance pure_mono' : Proper (impl ==> (⊢)) (@bi_pure PROP). Proof. intros φ1 φ2; apply pure_mono. Qed. Global Instance pure_flip_mono : Proper (flip impl ==> flip (⊢)) (@bi_pure PROP). Proof. intros φ1 φ2; apply pure_mono. Qed. Lemma pure_iff φ1 φ2 : (φ1 ↔ φ2) → ⌜φ1⌝ ⊣⊢ ⌜φ2⌝. Proof. intros [??]; apply (anti_symm _); auto using pure_mono. Qed. Lemma pure_elim_l φ Q R : (φ → Q ⊢ R) → ⌜φ⌝ ∧ Q ⊢ R. Proof. intros; apply pure_elim with φ; eauto. Qed. Lemma pure_elim_r φ Q R : (φ → Q ⊢ R) → Q ∧ ⌜φ⌝ ⊢ R. Proof. intros; apply pure_elim with φ; eauto. Qed. Lemma pure_True (φ : Prop) : φ → ⌜φ⌝ ⊣⊢ True. Proof. intros; apply (anti_symm _); auto. Qed. Lemma pure_False (φ : Prop) : ¬φ → ⌜φ⌝ ⊣⊢ False. Proof. intros; apply (anti_symm _); eauto using pure_mono. Qed. Lemma pure_and φ1 φ2 : ⌜φ1 ∧ φ2⌝ ⊣⊢ ⌜φ1⌝ ∧ ⌜φ2⌝. Proof. apply (anti_symm _). - apply and_intro; apply pure_mono; tauto. - eapply (pure_elim φ1); [auto|]=> ?. rewrite and_elim_r. auto using pure_mono. Qed. Lemma pure_or φ1 φ2 : ⌜φ1 ∨ φ2⌝ ⊣⊢ ⌜φ1⌝ ∨ ⌜φ2⌝. Proof. apply (anti_symm _). - eapply pure_elim=> // -[?|?]; auto using pure_mono. - apply or_elim; eauto using pure_mono. Qed. Lemma pure_impl φ1 φ2 : ⌜φ1 → φ2⌝ ⊣⊢ (⌜φ1⌝ → ⌜φ2⌝). Proof. apply (anti_symm _). - apply impl_intro_l. rewrite -pure_and. apply pure_mono. naive_solver. - rewrite -pure_forall_2. apply forall_intro=> ?. by rewrite -(left_id True bi_and (_→_))%I (pure_True φ1) // impl_elim_r. Qed. Lemma pure_forall {A} (φ : A → Prop) : ⌜∀ x, φ x⌝ ⊣⊢ ∀ x, ⌜φ x⌝. Proof. apply (anti_symm _); auto using pure_forall_2. apply forall_intro=> x. eauto using pure_mono. Qed. Lemma pure_exist {A} (φ : A → Prop) : ⌜∃ x, φ x⌝ ⊣⊢ ∃ x, ⌜φ x⌝. Proof. apply (anti_symm _). - eapply pure_elim=> // -[x ?]. rewrite -(exist_intro x); auto using pure_mono. - apply exist_elim=> x. eauto using pure_mono. Qed. Lemma pure_impl_forall φ P : (⌜φ⌝ → P) ⊣⊢ (∀ _ : φ, P). Proof. apply (anti_symm _). - apply forall_intro=> ?. by rewrite pure_True // left_id. - apply impl_intro_l, pure_elim_l=> Hφ. by rewrite (forall_elim Hφ). Qed. Lemma pure_alt φ : ⌜φ⌝ ⊣⊢ ∃ _ : φ, True. Proof. apply (anti_symm _). - eapply pure_elim; eauto=> H. rewrite -(exist_intro H); auto. - by apply exist_elim, pure_intro. Qed. Lemma pure_wand_forall φ P `{!Absorbing P} : (⌜φ⌝ -∗ P) ⊣⊢ (∀ _ : φ, P). Proof. apply (anti_symm _). - apply forall_intro=> Hφ. by rewrite -(left_id emp%I _ (_ -∗ _)%I) (pure_intro emp%I φ) // wand_elim_r. - apply wand_intro_l, wand_elim_l', pure_elim'=> Hφ. `````` Robbert Krebbers committed Oct 30, 2017 577 `````` apply wand_intro_l. rewrite (forall_elim Hφ) comm. by apply absorbing. `````` Robbert Krebbers committed Oct 30, 2017 578 579 580 581 582 583 584 585 586 ``````Qed. Lemma pure_internal_eq {A : ofeT} (x y : A) : ⌜x ≡ y⌝ ⊢ x ≡ y. Proof. apply pure_elim'=> ->. apply internal_eq_refl. Qed. Lemma discrete_eq {A : ofeT} (a b : A) : Discrete a → a ≡ b ⊣⊢ ⌜a ≡ b⌝. Proof. intros. apply (anti_symm _); auto using discrete_eq_1, pure_internal_eq. Qed. `````` Jacques-Henri Jourdan committed Nov 02, 2017 587 588 ``````(* Properties of the affinely modality *) Global Instance affinely_ne : NonExpansive (@bi_affinely PROP). `````` Robbert Krebbers committed Oct 30, 2017 589 ``````Proof. solve_proper. Qed. `````` Jacques-Henri Jourdan committed Nov 02, 2017 590 ``````Global Instance affinely_proper : Proper ((⊣⊢) ==> (⊣⊢)) (@bi_affinely PROP). `````` Robbert Krebbers committed Oct 30, 2017 591 ``````Proof. solve_proper. Qed. `````` Jacques-Henri Jourdan committed Nov 02, 2017 592 ``````Global Instance affinely_mono' : Proper ((⊢) ==> (⊢)) (@bi_affinely PROP). `````` Robbert Krebbers committed Oct 30, 2017 593 ``````Proof. solve_proper. Qed. `````` Jacques-Henri Jourdan committed Nov 02, 2017 594 595 ``````Global Instance affinely_flip_mono' : Proper (flip (⊢) ==> flip (⊢)) (@bi_affinely PROP). `````` Robbert Krebbers committed Oct 30, 2017 596 597 ``````Proof. solve_proper. Qed. `````` Jacques-Henri Jourdan committed Nov 02, 2017 598 599 600 601 602 ``````Lemma affinely_elim_emp P : bi_affinely P ⊢ emp. Proof. rewrite /bi_affinely; auto. Qed. Lemma affinely_elim P : bi_affinely P ⊢ P. Proof. rewrite /bi_affinely; auto. Qed. Lemma affinely_mono P Q : (P ⊢ Q) → bi_affinely P ⊢ bi_affinely Q. `````` Robbert Krebbers committed Oct 30, 2017 603 ``````Proof. by intros ->. Qed. `````` Jacques-Henri Jourdan committed Nov 02, 2017 604 605 ``````Lemma affinely_idemp P : bi_affinely (bi_affinely P) ⊣⊢ bi_affinely P. Proof. by rewrite /bi_affinely assoc idemp. Qed. `````` Robbert Krebbers committed Oct 30, 2017 606 `````` `````` Jacques-Henri Jourdan committed Nov 02, 2017 607 608 ``````Lemma affinely_intro' P Q : (bi_affinely P ⊢ Q) → bi_affinely P ⊢ bi_affinely Q. Proof. intros <-. by rewrite affinely_idemp. Qed. `````` Robbert Krebbers committed Oct 30, 2017 609 `````` `````` Jacques-Henri Jourdan committed Nov 02, 2017 610 611 612 613 614 615 616 ``````Lemma affinely_False : bi_affinely False ⊣⊢ False. Proof. by rewrite /bi_affinely right_absorb. Qed. Lemma affinely_emp : bi_affinely emp ⊣⊢ emp. Proof. by rewrite /bi_affinely (idemp bi_and). Qed. Lemma affinely_or P Q : bi_affinely (P ∨ Q) ⊣⊢ bi_affinely P ∨ bi_affinely Q. Proof. by rewrite /bi_affinely and_or_l. Qed. Lemma affinely_and P Q : bi_affinely (P ∧ Q) ⊣⊢ bi_affinely P ∧ bi_affinely Q. `````` Robbert Krebbers committed Oct 30, 2017 617 ``````Proof. `````` Jacques-Henri Jourdan committed Nov 02, 2017 618 `````` rewrite /bi_affinely -(comm _ P) (assoc _ (_ ∧ _)%I) -!(assoc _ P). `````` Robbert Krebbers committed Oct 30, 2017 619 620 `````` by rewrite idemp !assoc (comm _ P). Qed. `````` Jacques-Henri Jourdan committed Nov 02, 2017 621 ``````Lemma affinely_sep_2 P Q : bi_affinely P ∗ bi_affinely Q ⊢ bi_affinely (P ∗ Q). `````` Robbert Krebbers committed Oct 30, 2017 622 ``````Proof. `````` Jacques-Henri Jourdan committed Nov 02, 2017 623 `````` rewrite /bi_affinely. apply and_intro. `````` Robbert Krebbers committed Oct 30, 2017 624 625 626 `````` - by rewrite !and_elim_l right_id. - by rewrite !and_elim_r. Qed. `````` Jacques-Henri Jourdan committed Dec 04, 2017 627 ``````Lemma affinely_sep `{BiPositive PROP} P Q : `````` Jacques-Henri Jourdan committed Nov 02, 2017 628 `````` bi_affinely (P ∗ Q) ⊣⊢ bi_affinely P ∗ bi_affinely Q. `````` Robbert Krebbers committed Oct 30, 2017 629 ``````Proof. `````` Jacques-Henri Jourdan committed Nov 02, 2017 630 `````` apply (anti_symm _), affinely_sep_2. `````` Jacques-Henri Jourdan committed Dec 04, 2017 631 `````` by rewrite -{1}affinely_idemp bi_positive !(comm _ (bi_affinely P)%I) bi_positive. `````` Robbert Krebbers committed Oct 30, 2017 632 ``````Qed. `````` Jacques-Henri Jourdan committed Nov 02, 2017 633 634 ``````Lemma affinely_forall {A} (Φ : A → PROP) : bi_affinely (∀ a, Φ a) ⊢ ∀ a, bi_affinely (Φ a). `````` Robbert Krebbers committed Oct 30, 2017 635 ``````Proof. apply forall_intro=> a. by rewrite (forall_elim a). Qed. `````` Jacques-Henri Jourdan committed Nov 02, 2017 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 ``````Lemma affinely_exist {A} (Φ : A → PROP) : bi_affinely (∃ a, Φ a) ⊣⊢ ∃ a, bi_affinely (Φ a). Proof. by rewrite /bi_affinely and_exist_l. Qed. Lemma affinely_True_emp : bi_affinely True ⊣⊢ bi_affinely emp. Proof. apply (anti_symm _); rewrite /bi_affinely; auto. Qed. Lemma affinely_and_l P Q : bi_affinely P ∧ Q ⊣⊢ bi_affinely (P ∧ Q). Proof. by rewrite /bi_affinely assoc. Qed. Lemma affinely_and_r P Q : P ∧ bi_affinely Q ⊣⊢ bi_affinely (P ∧ Q). Proof. by rewrite /bi_affinely !assoc (comm _ P). Qed. Lemma affinely_and_lr P Q : bi_affinely P ∧ Q ⊣⊢ P ∧ bi_affinely Q. Proof. by rewrite affinely_and_l affinely_and_r. Qed. (* Properties of the absorbingly modality *) Global Instance absorbingly_ne : NonExpansive (@bi_absorbingly PROP). `````` Robbert Krebbers committed Oct 30, 2017 652 ``````Proof. solve_proper. Qed. `````` Jacques-Henri Jourdan committed Nov 02, 2017 653 ``````Global Instance absorbingly_proper : Proper ((⊣⊢) ==> (⊣⊢)) (@bi_absorbingly PROP). `````` Robbert Krebbers committed Oct 30, 2017 654 ``````Proof. solve_proper. Qed. `````` Jacques-Henri Jourdan committed Nov 02, 2017 655 ``````Global Instance absorbingly_mono' : Proper ((⊢) ==> (⊢)) (@bi_absorbingly PROP). `````` Robbert Krebbers committed Oct 30, 2017 656 ``````Proof. solve_proper. Qed. `````` Jacques-Henri Jourdan committed Nov 02, 2017 657 658 ``````Global Instance absorbingly_flip_mono' : Proper (flip (⊢) ==> flip (⊢)) (@bi_absorbingly PROP). `````` Robbert Krebbers committed Oct 30, 2017 659 660 ``````Proof. solve_proper. Qed. `````` Robbert Krebbers committed Dec 03, 2017 661 ``````Lemma absorbingly_intro P : P ⊢ bi_absorbingly P. `````` Jacques-Henri Jourdan committed Nov 02, 2017 662 ``````Proof. by rewrite /bi_absorbingly -True_sep_2. Qed. `````` Robbert Krebbers committed Dec 03, 2017 663 ``````Lemma absorbingly_mono P Q : (P ⊢ Q) → bi_absorbingly P ⊢ bi_absorbingly Q. `````` Robbert Krebbers committed Oct 30, 2017 664 ``````Proof. by intros ->. Qed. `````` Robbert Krebbers committed Dec 03, 2017 665 ``````Lemma absorbingly_idemp P : bi_absorbingly (bi_absorbingly P) ⊣⊢ bi_absorbingly P. `````` Robbert Krebbers committed Oct 30, 2017 666 ``````Proof. `````` Jacques-Henri Jourdan committed Nov 02, 2017 667 668 `````` apply (anti_symm _), absorbingly_intro. rewrite /bi_absorbingly assoc. apply sep_mono; auto. `````` Robbert Krebbers committed Oct 30, 2017 669 670 ``````Qed. `````` Robbert Krebbers committed Dec 03, 2017 671 ``````Lemma absorbingly_pure φ : bi_absorbingly ⌜ φ ⌝ ⊣⊢ ⌜ φ ⌝. `````` Robbert Krebbers committed Oct 30, 2017 672 ``````Proof. `````` Jacques-Henri Jourdan committed Nov 02, 2017 673 `````` apply (anti_symm _), absorbingly_intro. `````` Robbert Krebbers committed Oct 30, 2017 674 675 `````` apply wand_elim_r', pure_elim'=> ?. apply wand_intro_l; auto. Qed. `````` Robbert Krebbers committed Dec 03, 2017 676 677 ``````Lemma absorbingly_or P Q : bi_absorbingly (P ∨ Q) ⊣⊢ bi_absorbingly P ∨ bi_absorbingly Q. `````` Jacques-Henri Jourdan committed Nov 02, 2017 678 ``````Proof. by rewrite /bi_absorbingly sep_or_l. Qed. `````` Robbert Krebbers committed Dec 03, 2017 679 680 ``````Lemma absorbingly_and P Q : bi_absorbingly (P ∧ Q) ⊢ bi_absorbingly P ∧ bi_absorbingly Q. `````` Jacques-Henri Jourdan committed Nov 02, 2017 681 ``````Proof. apply and_intro; apply absorbingly_mono; auto. Qed. `````` Robbert Krebbers committed Dec 03, 2017 682 683 ``````Lemma absorbingly_forall {A} (Φ : A → PROP) : bi_absorbingly (∀ a, Φ a) ⊢ ∀ a, bi_absorbingly (Φ a). `````` Robbert Krebbers committed Oct 30, 2017 684 ``````Proof. apply forall_intro=> a. by rewrite (forall_elim a). Qed. `````` Robbert Krebbers committed Dec 03, 2017 685 686 ``````Lemma absorbingly_exist {A} (Φ : A → PROP) : bi_absorbingly (∃ a, Φ a) ⊣⊢ ∃ a, bi_absorbingly (Φ a). `````` Jacques-Henri Jourdan committed Nov 02, 2017 687 ``````Proof. by rewrite /bi_absorbingly sep_exist_l. Qed. `````` Robbert Krebbers committed Oct 30, 2017 688 `````` `````` Robbert Krebbers committed Dec 03, 2017 689 ``````Lemma absorbingly_internal_eq {A : ofeT} (x y : A) : bi_absorbingly (x ≡ y) ⊣⊢ x ≡ y. `````` Robbert Krebbers committed Oct 30, 2017 690 ``````Proof. `````` Jacques-Henri Jourdan committed Nov 02, 2017 691 `````` apply (anti_symm _), absorbingly_intro. `````` Robbert Krebbers committed Oct 30, 2017 692 693 694 695 `````` apply wand_elim_r', (internal_eq_rewrite' x y (λ y, True -∗ x ≡ y)%I); auto. apply wand_intro_l, internal_eq_refl. Qed. `````` Robbert Krebbers committed Dec 03, 2017 696 ``````Lemma absorbingly_sep P Q : bi_absorbingly (P ∗ Q) ⊣⊢ bi_absorbingly P ∗ bi_absorbingly Q. `````` Jacques-Henri Jourdan committed Nov 02, 2017 697 ``````Proof. by rewrite -{1}absorbingly_idemp /bi_absorbingly !assoc -!(comm _ P) !assoc. Qed. `````` Robbert Krebbers committed Dec 03, 2017 698 ``````Lemma absorbingly_True_emp : bi_absorbingly True ⊣⊢ bi_absorbingly emp. `````` Jacques-Henri Jourdan committed Nov 02, 2017 699 ``````Proof. by rewrite absorbingly_pure /bi_absorbingly right_id. Qed. `````` Robbert Krebbers committed Dec 03, 2017 700 ``````Lemma absorbingly_wand P Q : bi_absorbingly (P -∗ Q) ⊢ bi_absorbingly P -∗ bi_absorbingly Q. `````` Jacques-Henri Jourdan committed Nov 02, 2017 701 ``````Proof. apply wand_intro_l. by rewrite -absorbingly_sep wand_elim_r. Qed. `````` Robbert Krebbers committed Oct 30, 2017 702 `````` `````` Robbert Krebbers committed Dec 03, 2017 703 ``````Lemma absorbingly_sep_l P Q : bi_absorbingly P ∗ Q ⊣⊢ bi_absorbingly (P ∗ Q). `````` Jacques-Henri Jourdan committed Nov 02, 2017 704 ``````Proof. by rewrite /bi_absorbingly assoc. Qed. `````` Robbert Krebbers committed Dec 03, 2017 705 ``````Lemma absorbingly_sep_r P Q : P ∗ bi_absorbingly Q ⊣⊢ bi_absorbingly (P ∗ Q). `````` Jacques-Henri Jourdan committed Nov 02, 2017 706 ``````Proof. by rewrite /bi_absorbingly !assoc (comm _ P). Qed. `````` Robbert Krebbers committed Dec 03, 2017 707 ``````Lemma absorbingly_sep_lr P Q : bi_absorbingly P ∗ Q ⊣⊢ P ∗ bi_absorbingly Q. `````` Jacques-Henri Jourdan committed Nov 02, 2017 708 ``````Proof. by rewrite absorbingly_sep_l absorbingly_sep_r. Qed. `````` Robbert Krebbers committed Oct 30, 2017 709 `````` `````` Jacques-Henri Jourdan committed Dec 04, 2017 710 ``````Lemma affinely_absorbingly `{!BiPositive PROP} P : `````` Robbert Krebbers committed Dec 03, 2017 711 `````` bi_affinely (bi_absorbingly P) ⊣⊢ bi_affinely P. `````` Robbert Krebbers committed Oct 30, 2017 712 ``````Proof. `````` Jacques-Henri Jourdan committed Nov 02, 2017 713 714 `````` apply (anti_symm _), affinely_mono, absorbingly_intro. by rewrite /bi_absorbingly affinely_sep affinely_True_emp affinely_emp left_id. `````` Robbert Krebbers committed Oct 30, 2017 715 716 ``````Qed. `````` 717 ``````(* Affine and absorbing propositions *) `````` Robbert Krebbers committed Oct 30, 2017 718 ``````Global Instance Affine_proper : Proper ((⊣⊢) ==> iff) (@Affine PROP). `````` Robbert Krebbers committed Oct 30, 2017 719 ``````Proof. solve_proper. Qed. `````` Robbert Krebbers committed Oct 30, 2017 720 721 ``````Global Instance Absorbing_proper : Proper ((⊣⊢) ==> iff) (@Absorbing PROP). Proof. solve_proper. Qed. `````` Robbert Krebbers committed Oct 30, 2017 722 `````` `````` Jacques-Henri Jourdan committed Nov 02, 2017 723 724 ``````Lemma affine_affinely P `{!Affine P} : bi_affinely P ⊣⊢ P. Proof. rewrite /bi_affinely. apply (anti_symm _); auto. Qed. `````` Robbert Krebbers committed Dec 03, 2017 725 ``````Lemma absorbing_absorbingly P `{!Absorbing P} : bi_absorbingly P ⊣⊢ P. `````` Jacques-Henri Jourdan committed Nov 02, 2017 726 ``````Proof. by apply (anti_symm _), absorbingly_intro. Qed. `````` Robbert Krebbers committed Oct 30, 2017 727 `````` `````` Robbert Krebbers committed Oct 30, 2017 728 729 730 ``````Lemma True_affine_all_affine P : Affine (True%I : PROP) → Affine P. Proof. rewrite /Affine=> <-; auto. Qed. Lemma emp_absorbing_all_absorbing P : Absorbing (emp%I : PROP) → Absorbing P. `````` Robbert Krebbers committed Oct 30, 2017 731 732 ``````Proof. intros. rewrite /Absorbing -{2}(left_id emp%I _ P). `````` Jacques-Henri Jourdan committed Nov 02, 2017 733 `````` by rewrite -(absorbing emp) absorbingly_sep_l left_id. `````` Robbert Krebbers committed Oct 30, 2017 734 ``````Qed. `````` Robbert Krebbers committed Oct 30, 2017 735 736 `````` Lemma sep_elim_l P Q `{H : TCOr (Affine Q) (Absorbing P)} : P ∗ Q ⊢ P. `````` Robbert Krebbers committed Oct 30, 2017 737 738 739 740 741 ``````Proof. destruct H. - by rewrite (affine Q) right_id. - by rewrite (True_intro Q) comm. Qed. `````` Robbert Krebbers committed Oct 30, 2017 742 743 744 ``````Lemma sep_elim_r P Q `{H : TCOr (Affine P) (Absorbing Q)} : P ∗ Q ⊢ Q. Proof. by rewrite comm sep_elim_l. Qed. `````` Robbert Krebbers committed Oct 30, 2017 745 746 ``````Lemma sep_and P Q `{HPQ : TCOr (TCAnd (Affine P) (Affine Q)) (TCAnd (Absorbing P) (Absorbing Q))} : `````` Robbert Krebbers committed Oct 30, 2017 747 `````` P ∗ Q ⊢ P ∧ Q. `````` Robbert Krebbers committed Oct 30, 2017 748 749 750 751 ``````Proof. destruct HPQ as [[??]|[??]]; apply and_intro; apply: sep_elim_l || apply: sep_elim_r. Qed. `````` Robbert Krebbers committed Oct 30, 2017 752 `````` `````` Jacques-Henri Jourdan committed Nov 02, 2017 753 754 ``````Lemma affinely_intro P Q `{!Affine P} : (P ⊢ Q) → P ⊢ bi_affinely Q. Proof. intros <-. by rewrite affine_affinely. Qed. `````` Robbert Krebbers committed Oct 30, 2017 755 756 757 758 759 760 761 762 763 764 765 `````` Lemma emp_and P `{!Affine P} : emp ∧ P ⊣⊢ P. Proof. apply (anti_symm _); auto. Qed. Lemma and_emp P `{!Affine P} : P ∧ emp ⊣⊢ P. Proof. apply (anti_symm _); auto. Qed. Lemma emp_or P `{!Affine P} : emp ∨ P ⊣⊢ emp. Proof. apply (anti_symm _); auto. Qed. Lemma or_emp P `{!Affine P} : P ∨ emp ⊣⊢ emp. Proof. apply (anti_symm _); auto. Qed. Lemma True_sep P `{!Absorbing P} : True ∗ P ⊣⊢ P. `````` Robbert Krebbers committed Oct 30, 2017 766 ``````Proof. apply (anti_symm _); auto using True_sep_2. Qed. `````` Robbert Krebbers committed Oct 30, 2017 767 ``````Lemma sep_True P `{!Absorbing P} : P ∗ True ⊣⊢ P. `````` Robbert Krebbers committed Oct 30, 2017 768 ``````Proof. by rewrite comm True_sep. Qed. `````` Robbert Krebbers committed Oct 30, 2017 769 `````` `````` Jacques-Henri Jourdan committed Dec 04, 2017 770 771 ``````Section bi_affine. Context `{BiAffine PROP}. `````` Robbert Krebbers committed Oct 30, 2017 772 `````` `````` Jacques-Henri Jourdan committed Dec 04, 2017 773 `````` Global Instance bi_affine_absorbing P : Absorbing P | 0. `````` Jacques-Henri Jourdan committed Nov 02, 2017 774 `````` Proof. by rewrite /Absorbing /bi_absorbingly (affine True%I) left_id. Qed. `````` Jacques-Henri Jourdan committed Dec 04, 2017 775 `````` Global Instance bi_affine_positive : BiPositive PROP. `````` Jacques-Henri Jourdan committed Nov 02, 2017 776 `````` Proof. intros P Q. by rewrite !affine_affinely. Qed. `````` Robbert Krebbers committed Oct 30, 2017 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 `````` Lemma True_emp : True ⊣⊢ emp. Proof. apply (anti_symm _); auto using affine. Qed. Global Instance emp_and' : LeftId (⊣⊢) emp%I (@bi_and PROP). Proof. intros P. by rewrite -True_emp left_id. Qed. Global Instance and_emp' : RightId (⊣⊢) emp%I (@bi_and PROP). Proof. intros P. by rewrite -True_emp right_id. Qed. Global Instance True_sep' : LeftId (⊣⊢) True%I (@bi_sep PROP). Proof. intros P. by rewrite True_emp left_id. Qed. Global Instance sep_True' : RightId (⊣⊢) True%I (@bi_sep PROP). Proof. intros P. by rewrite True_emp right_id. Qed. Lemma impl_wand_1 P Q : (P → Q) ⊢ P -∗ Q. Proof. apply wand_intro_l. by rewrite sep_and impl_elim_r. Qed. Lemma decide_emp φ `{!Decision φ} (P : PROP) : (if decide φ then P else emp) ⊣⊢ (⌜φ⌝ → P). Proof. destruct (decide _). - by rewrite pure_True // True_impl. - by rewrite pure_False // False_impl True_emp. Qed. `````` Jacques-Henri Jourdan committed Dec 04, 2017 801 ``````End bi_affine. `````` Robbert Krebbers committed Oct 30, 2017 802 `````` `````` Jacques-Henri Jourdan committed Nov 03, 2017 803 ``````(* Properties of the persistence modality *) `````` Robbert Krebbers committed Oct 30, 2017 804 805 806 807 808 809 ``````Hint Resolve persistently_mono. Global Instance persistently_mono' : Proper ((⊢) ==> (⊢)) (@bi_persistently PROP). Proof. intros P Q; apply persistently_mono. Qed. Global Instance persistently_flip_mono' : Proper (flip (⊢) ==> flip (⊢)) (@bi_persistently PROP). Proof. intros P Q; apply persistently_mono. Qed. `````` Robbert Krebbers committed Oct 30, 2017 810 `````` `````` Robbert Krebbers committed Dec 03, 2017 811 812 ``````Lemma absorbingly_persistently P : bi_absorbingly (bi_persistently P) ⊣⊢ bi_persistently P. `````` Robbert Krebbers committed Oct 30, 2017 813 ``````Proof. `````` Jacques-Henri Jourdan committed Nov 02, 2017 814 815 `````` apply (anti_symm _), absorbingly_intro. by rewrite /bi_absorbingly comm persistently_absorbing. `````` Robbert Krebbers committed Oct 30, 2017 816 ``````Qed. `````` Robbert Krebbers committed Oct 30, 2017 817 `````` `````` Jacques-Henri Jourdan committed Nov 02, 2017 818 819 ``````Lemma persistently_and_sep_assoc P Q R : bi_persistently P ∧ (Q ∗ R) ⊣⊢ (bi_persistently P ∧ Q) ∗ R. `````` Robbert Krebbers committed Oct 30, 2017 820 ``````Proof. `````` Robbert Krebbers committed Oct 30, 2017 821 822 823 `````` apply (anti_symm (⊢)). - rewrite {1}persistently_idemp_2 persistently_and_sep_elim assoc. apply sep_mono_l, and_intro. `````` 824 `````` + by rewrite and_elim_r persistently_absorbing. `````` Robbert Krebbers committed Oct 30, 2017 825 826 `````` + by rewrite and_elim_l left_id. - apply and_intro. `````` 827 `````` + by rewrite and_elim_l persistently_absorbing. `````` Robbert Krebbers committed Oct 30, 2017 828 `````` + by rewrite and_elim_r. `````` Robbert Krebbers committed Oct 30, 2017 829 ``````Qed. `````` Jacques-Henri Jourdan committed Nov 02, 2017 830 ``````Lemma persistently_and_emp_elim P : emp ∧ bi_persistently P ⊢ P. `````` Robbert Krebbers committed Oct 30, 2017 831 ``````Proof. by rewrite comm persistently_and_sep_elim right_id and_elim_r. Qed. `````` Robbert Krebbers committed Dec 03, 2017 832 ``````Lemma persistently_elim_absorbingly P : bi_persistently P ⊢ bi_absorbingly P. `````` Robbert Krebbers committed Oct 30, 2017 833 ``````Proof. `````` Jacques-Henri Jourdan committed Nov 02, 2017 834 `````` rewrite -(right_id True%I _ (bi_persistently _)%I) -{1}(left_id emp%I _ True%I). `````` Robbert Krebbers committed Oct 30, 2017 835 `````` by rewrite persistently_and_sep_assoc (comm bi_and) persistently_and_emp_elim comm. `````` Robbert Krebbers committed Oct 30, 2017 836 ``````Qed. `````` Jacques-Henri Jourdan committed Nov 02, 2017 837 ``````Lemma persistently_elim P `{!Absorbing P} : bi_persistently P ⊢ P. `````` Jacques-Henri Jourdan committed Nov 02, 2017 838 ``````Proof. by rewrite persistently_elim_absorbingly absorbing_absorbingly. Qed. `````` Robbert Krebbers committed Oct 30, 2017 839 `````` `````` Jacques-Henri Jourdan committed Nov 02, 2017 840 841 ``````Lemma persistently_idemp_1 P : bi_persistently (bi_persistently P) ⊢ bi_persistently P. `````` Jacques-Henri Jourdan committed Nov 02, 2017 842 ``````Proof. by rewrite persistently_elim_absorbingly absorbingly_persistently. Qed. `````` Jacques-Henri Jourdan committed Nov 02, 2017 843 844 ``````Lemma persistently_idemp P : bi_persistently (bi_persistently P) ⊣⊢ bi_persistently P. `````` Robbert Krebbers committed Oct 30, 2017 845 ``````Proof. apply (anti_symm _); auto using persistently_idemp_1, persistently_idemp_2. Qed. `````` Robbert Krebbers committed Oct 30, 2017 846 `````` `````` Jacques-Henri Jourdan committed Nov 02, 2017 847 848 ``````Lemma persistently_intro' P Q : (bi_persistently P ⊢ Q) → bi_persistently P ⊢ bi_persistently Q. `````` Robbert Krebbers committed Oct 30, 2017 849 850 ``````Proof. intros <-. apply persistently_idemp_2. Qed. `````` Jacques-Henri Jourdan committed Nov 02, 2017 851 ``````Lemma persistently_pure φ : bi_persistently ⌜φ⌝ ⊣⊢ ⌜φ⌝. `````` Robbert Krebbers committed Oct 30, 2017 852 ``````Proof. `````` 853 854 `````` apply (anti_symm _). { by rewrite persistently_elim_absorbingly absorbingly_pure. } `````` Robbert Krebbers committed Oct 30, 2017 855 `````` apply pure_elim'=> Hφ. `````` Jacques-Henri Jourdan committed Nov 02, 2017 856 `````` trans (∀ x : False, bi_persistently True : PROP)%I; [by apply forall_intro|]. `````` Robbert Krebbers committed Oct 30, 2017 857 `````` rewrite persistently_forall_2. auto using persistently_mono, pure_intro. `````` Robbert Krebbers committed Oct 30, 2017 858 ``````Qed. `````` Jacques-Henri Jourdan committed Nov 02, 2017 859 860 ``````Lemma persistently_forall {A} (Ψ : A → PROP) : bi_persistently (∀ a, Ψ a) ⊣⊢ ∀ a, bi_persistently (Ψ a). `````` Robbert Krebbers committed Oct 30, 2017 861 862 863 864 ``````Proof. apply (anti_symm _); auto using persistently_forall_2. apply forall_intro=> x. by rewrite (forall_elim x). Qed. `````` Jacques-Henri Jourdan committed Nov 02, 2017 865 866 ``````Lemma persistently_exist {A} (Ψ : A → PROP) : bi_persistently (∃ a, Ψ a) ⊣⊢ ∃ a, bi_persistently (Ψ a). `````` Robbert Krebbers committed Oct 30, 2017 867 868 869 870 ``````Proof. apply (anti_symm _); auto using persistently_exist_1. apply exist_elim=> x. by rewrite (exist_intro x). Qed. `````` Jacques-Henri Jourdan committed Nov 02, 2017 871 872 ``````Lemma persistently_and P Q : bi_persistently (P ∧ Q) ⊣⊢ bi_persistently P ∧ bi_persistently Q. `````` Robbert Krebbers committed Oct 30, 2017 873 ``````Proof. rewrite !and_alt persistently_forall. by apply forall_proper=> -[]. Qed. `````` Jacques-Henri Jourdan committed Nov 02, 2017 874 875 ``````Lemma persistently_or P Q : bi_persistently (P ∨ Q) ⊣⊢ bi_persistently P ∨ bi_persistently Q. `````` Robbert Krebbers committed Oct 30, 2017 876 ``````Proof. rewrite !or_alt persistently_exist. by apply exist_proper=> -[]. Qed. `````` Jacques-Henri Jourdan committed Nov 02, 2017 877 878 ``````Lemma persistently_impl P Q : bi_persistently (P → Q) ⊢ bi_persistently P → bi_persistently Q. `````` Robbert Krebbers committed Oct 30, 2017 879 880 881 882 883 ``````Proof. apply impl_intro_l; rewrite -persistently_and. apply persistently_mono, impl_elim with P; auto. Qed. `````` Jacques-Henri Jourdan committed Nov 02, 2017 884 885 ``````Lemma persistently_sep_dup P : bi_persistently P ⊣⊢ bi_persistently P ∗ bi_persistently P. `````` Robbert Krebbers committed Oct 30, 2017 886 ``````Proof. `````` 887 888 889 890 891 `````` apply (anti_symm _). - rewrite -{1}(idemp bi_and (bi_persistently _)). by rewrite -{2}(left_id emp%I _ (bi_persistently _)) persistently_and_sep_assoc and_elim_l. - by rewrite persistently_absorbing. `````` Robbert Krebbers committed Oct 30, 2017 892 893 ``````Qed. `````` Jacques-Henri Jourdan committed Nov 02, 2017 894 ``````Lemma persistently_and_sep_l_1 P Q : bi_persistently P ∧ Q ⊢ bi_persistently P ∗ Q. `````` Robbert Krebbers committed Oct 30, 2017 895 896 897 ``````Proof. by rewrite -{1}(left_id emp%I _ Q%I) persistently_and_sep_assoc and_elim_l. Qed. `````` Jacques-Henri Jourdan committed Nov 02, 2017 898 ``````Lemma persistently_and_sep_r_1 P Q : P ∧ bi_persistently Q ⊢ P ∗ bi_persistently Q. `````` Robbert Krebbers committed Oct 30, 2017 899 900 ``````Proof. by rewrite !(comm _ P) persistently_and_sep_l_1. Qed. `````` Jacques-Henri Jourdan committed Nov 03, 2017 901 902 903 904 905 906 ``````Lemma persistently_emp_intro P : P ⊢ bi_persistently emp. Proof. by rewrite -plainly_elim_persistently -plainly_emp_intro. Qed. Lemma persistently_internal_eq {A : ofeT} (a b : A) : bi_persistently (a ≡ b) ⊣⊢ a ≡ b. Proof. `````` 907 908 `````` apply (anti_symm (⊢)). { by rewrite persistently_elim_absorbingly absorbingly_internal_eq. } `````` Jacques-Henri Jourdan committed Nov 03, 2017 909 910 911 912 `````` apply (internal_eq_rewrite' a b (λ b, bi_persistently (a ≡ b))%I); auto. rewrite -(internal_eq_refl emp%I a). apply persistently_emp_intro. Qed. `````` Jacques-Henri Jourdan committed Nov 02, 2017 913 ``````Lemma persistently_True_emp : bi_persistently True ⊣⊢ bi_persistently emp. `````` Robbert Krebbers committed Oct 30, 2017 914 ``````Proof. apply (anti_symm _); auto using persistently_emp_intro. Qed. `````` Jacques-Henri Jourdan committed Nov 02, 2017 915 ``````Lemma persistently_and_sep P Q : bi_persistently (P ∧ Q) ⊢ bi_persistently (P ∗ Q). `````` Robbert Krebbers committed Oct 30, 2017 916 ``````Proof. `````` Robbert Krebbers committed Oct 30, 2017 917 918 919 920 921 `````` rewrite persistently_and. rewrite -{1}persistently_idemp -persistently_and -{1}(left_id emp%I _ Q%I). by rewrite persistently_and_sep_assoc (comm bi_and) persistently_and_emp_elim. Qed. `````` Jacques-Henri Jourdan committed Nov 02, 2017 922 ``````Lemma persistently_affinely P : bi_persistently (bi_affinely P) ⊣⊢ bi_persistently P. `````` Robbert Krebbers committed Oct 30, 2017 923 ``````Proof. `````` Jacques-Henri Jourdan committed Nov 02, 2017 924 `````` by rewrite /bi_affinely persistently_and -persistently_True_emp `````` Robbert Krebbers committed Oct 30, 2017 925 `````` persistently_pure left_id. `````` Robbert Krebbers committed Oct 30, 2017 926 927 ``````Qed. `````` Jacques-Henri Jourdan committed Nov 02, 2017 928 929 ``````Lemma and_sep_persistently P Q : bi_persistently P ∧ bi_persistently Q ⊣⊢ bi_persistently P ∗ bi_persistently Q. `````` Robbert Krebbers committed Oct 30, 2017 930 ``````Proof. `````` 931 932 933 934 `````` apply (anti_symm _); auto using persistently_and_sep_l_1. apply and_intro. - by rewrite persistently_absorbing. - by rewrite comm persistently_absorbing. `````` Robbert Krebbers committed Oct 30, 2017 935 ``````Qed. `````` Jacques-Henri Jourdan committed Nov 02, 2017 936 937 ``````Lemma persistently_sep_2 P Q : bi_persistently P ∗ bi_persistently Q ⊢ bi_persistently (P ∗ Q). `````` Robbert Krebbers committed Oct 30, 2017 938 ``````Proof. by rewrite -persistently_and_sep persistently_and -and_sep_persistently. Qed. `````` Jacques-Henri Jourdan committed Dec 04, 2017 939 ``````Lemma persistently_sep `{BiPositive PROP} P Q : `````` Jacques-Henri Jourdan committed Nov 02, 2017 940 `````` bi_persistently (P ∗ Q) ⊣⊢ bi_persistently P ∗ bi_persistently Q. `````` Robbert Krebbers committed Oct 30, 2017 941 942 ``````Proof. apply (anti_symm _); auto using persistently_sep_2. `````` 943 944 945 `````` rewrite -persistently_affinely affinely_sep -and_sep_persistently. apply and_intro. - by rewrite (affinely_elim_emp Q) right_id affinely_elim. - by rewrite (affinely_elim_emp P) left_id affinely_elim. `````` Robbert Krebbers committed Oct 30, 2017 946 ``````Qed. `````` Robbert Krebbers committed Oct 30, 2017 947 `````` `````` Jacques-Henri Jourdan committed Nov 02, 2017 948 949 ``````Lemma persistently_wand P Q : bi_persistently (P -∗ Q) ⊢ bi_persistently P -∗ bi_persistently Q. `````` Robbert Krebbers committed Oct 30, 2017 950 ``````Proof. apply wand_intro_r. by rewrite persistently_sep_2 wand_elim_l. Qed. `````` Robbert Krebbers committed Oct 30, 2017 951 `````` `````` Jacques-Henri Jourdan committed Nov 02, 2017 952 953 ``````Lemma persistently_entails_l P Q : (P ⊢ bi_persistently Q) → P ⊢ bi_persistently Q ∗ P. `````` Robbert Krebbers committed Oct 30, 2017 954 ``````Proof. intros; rewrite -persistently_and_sep_l_1; auto. Qed. `````` Jacques-Henri Jourdan committed Nov 02, 2017 955 956 ``````Lemma persistently_entails_r P Q : (P ⊢ bi_persistently Q) → P ⊢ P ∗ bi_persistently Q. `````` Robbert Krebbers committed Oct 30, 2017 957 958 ``````Proof. intros; rewrite -persistently_and_sep_r_1; auto. Qed. `````` Jacques-Henri Jourdan committed Nov 02, 2017 959 960 ``````Lemma persistently_impl_wand_2 P Q : bi_persistently (P -∗ Q) ⊢ bi_persistently (P → Q). `````` Robbert Krebbers committed Oct 30, 2017 961 962 963 964 965 966 ``````Proof. apply persistently_intro', impl_intro_r. rewrite -{2}(left_id emp%I _ P%I) persistently_and_sep_assoc. by rewrite (comm bi_and) persistently_and_emp_elim wand_elim_l. Qed. `````` Robbert Krebbers committed Dec 03, 2017 967 ``````Lemma impl_wand_persistently_2 P Q : (bi_persistently P -∗ Q) ⊢ (bi_persistently P → Q). `````` Robbert Krebbers committed Dec 03, 2017 968 969 ``````Proof. apply impl_intro_l. by rewrite persistently_and_sep_l_1 wand_elim_r. Qed. `````` Jacques-Henri Jourdan committed Nov 02, 2017 970 ``````Section persistently_affinely_bi. `````` Jacques-Henri Jourdan committed Dec 04, 2017 971 `````` Context `{BiAffine PROP}. `````` Robbert Krebbers committed Oct 30, 2017 972 `````` `````` Jacques-Henri Jourdan committed Nov 02, 2017 973 `````` Lemma persistently_emp : bi_persistently emp ⊣⊢ emp. `````` Robbert Krebbers committed Oct 30, 2017 974 975 `````` Proof. by rewrite -!True_emp persistently_pure. Qed. `````` Jacques-Henri Jourdan committed Nov 02, 2017 976 977 `````` Lemma persistently_and_sep_l P Q : bi_persistently P ∧ Q ⊣⊢ bi_persistently P ∗ Q. `````` Robbert Krebbers committed Oct 30, 2017 978 979 980 981 `````` Proof. apply (anti_symm (⊢)); eauto using persistently_and_sep_l_1, sep_and with typeclass_instances. Qed. `````` Jacques-Henri Jourdan committed Nov 02, 2017 982 `````` Lemma persistently_and_sep_r P Q : P ∧ bi_persistently Q ⊣⊢ P ∗ bi_persistently Q. `````` Robbert Krebbers committed Oct 30, 2017 983 984 `````` Proof. by rewrite !(comm _ P) persistently_and_sep_l. Qed. `````` Jacques-Henri Jourdan committed Nov 02, 2017 985 986 `````` Lemma persistently_impl_wand P Q : bi_persistently (P → Q) ⊣⊢ bi_persistently (P -∗ Q). `````` Robbert Krebbers committed Oct 30, 2017 987 988 989 `````` Proof. apply (anti_symm (⊢)); auto using persistently_impl_wand_2. apply persistently_intro', wand_intro_l. `````` Robbert Krebbers committed Oct 30, 2017 990 `````` by rewrite -persistently_and_sep_r persistently_elim impl_elim_r. `````` Robbert Krebbers committed Oct 30, 2017 991 992 `````` Qed. `````` Robbert Krebbers committed Dec 03, 2017 993 `````` Lemma impl_wand_persistently P Q : (bi_persistently P → Q) ⊣⊢ (bi_persistently P -∗ Q). `````` Robbert Krebbers committed Dec 03, 2017 994 `````` Proof. `````` Robbert Krebbers committed Dec 03, 2017