agree.v 17 KB
 Robbert Krebbers committed Mar 10, 2016 1 ``````From iris.algebra Require Export cmra. `````` Ralf Jung committed Nov 25, 2016 2 ``````From iris.algebra Require Import list. `````` Robbert Krebbers committed Oct 25, 2016 3 ``````From iris.base_logic Require Import base_logic. `````` Ralf Jung committed Jan 03, 2017 4 5 ``````(* FIXME: This file needs a 'Proof Using' hint. *) `````` Ralf Jung committed Nov 25, 2016 6 7 8 9 ``````Local Arguments validN _ _ _ !_ /. Local Arguments valid _ _ !_ /. Local Arguments op _ _ _ !_ /. Local Arguments pcore _ _ !_ /. `````` Robbert Krebbers committed Nov 11, 2015 10 `````` `````` Robbert Krebbers committed Dec 21, 2015 11 ``````Record agree (A : Type) : Type := Agree { `````` Ralf Jung committed Nov 25, 2016 12 13 `````` agree_car : A; agree_with : list A; `````` Robbert Krebbers committed Nov 11, 2015 14 ``````}. `````` Ralf Jung committed Nov 25, 2016 15 16 17 18 19 20 21 22 23 24 ``````Arguments Agree {_} _ _. Arguments agree_car {_} _. Arguments agree_with {_} _. (* Some theory about set-inclusion on lists and lists of which all elements are equal. TODO: Move this elsewhere. *) Definition list_setincl `(R : relation A) (al bl : list A) := ∀ a, a ∈ al → ∃ b, b ∈ bl ∧ R a b. Definition list_setequiv `(R : relation A) (al bl : list A) := list_setincl R al bl ∧ list_setincl R bl al. `````` Jacques-Henri Jourdan committed Jan 04, 2017 25 26 27 ``````(* list_agrees is carefully written such that, when applied to a singleton, it is convertible to True. This makes working with agreement much more pleasant. *) `````` Ralf Jung committed Nov 25, 2016 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 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 ``````Definition list_agrees `(R : relation A) (al : list A) := match al with | [] => True | [a] => True | a :: al => ∀ b, b ∈ al → R a b end. Lemma list_agrees_alt `(R : relation A) `{Equivalence _ R} al : list_agrees R al ↔ (∀ a b, a ∈ al → b ∈ al → R a b). Proof. destruct al as [|a [|b al]]. - split; last done. intros _ ? ? []%elem_of_nil. - split; last done. intros _ ? ? ->%elem_of_list_singleton ->%elem_of_list_singleton. done. - simpl. split. + intros Hl a' b' [->|Ha']%elem_of_cons. * intros [->|Hb']%elem_of_cons; first done. auto. * intros [->|Hb']%elem_of_cons; first by (symmetry; auto). trans a; last by auto. symmetry. auto. + intros Hl b' Hb'. apply Hl; set_solver. Qed. Section list_theory. Context `(R: relation A) `{Equivalence A R}. Global Instance: PreOrder (list_setincl R). Proof. split. - intros al a Ha. set_solver. - intros al bl cl Hab Hbc a Ha. destruct (Hab _ Ha) as (b & Hb & Rab). destruct (Hbc _ Hb) as (c & Hc & Rbc). exists c. split; first done. by trans b. Qed. Global Instance: Equivalence (list_setequiv R). Proof. split. - by split. - intros ?? [??]. split; auto. - intros ??? [??] [??]. split; etrans; done. Qed. Global Instance list_setincl_subrel `(R' : relation A) : subrelation R R' → subrelation (list_setincl R) (list_setincl R'). Proof. intros HRR' al bl Hab. intros a Ha. destruct (Hab _ Ha) as (b & Hb & HR). exists b. split; first done. exact: HRR'. Qed. Global Instance list_setequiv_subrel `(R' : relation A) : subrelation R R' → subrelation (list_setequiv R) (list_setequiv R'). Proof. intros HRR' ?? [??]. split; exact: list_setincl_subrel. Qed. Global Instance list_setincl_perm : subrelation (≡ₚ) (list_setincl R). Proof. intros al bl Hab a Ha. exists a. split; last done. by rewrite -Hab. Qed. Global Instance list_setincl_app l : Proper (list_setincl R ==> list_setincl R) (app l). Proof. intros al bl Hab a [Ha|Ha]%elem_of_app. - exists a. split; last done. apply elem_of_app. by left. - destruct (Hab _ Ha) as (b & Hb & HR). exists b. split; last done. apply elem_of_app. by right. Qed. Global Instance list_setequiv_app l : Proper (list_setequiv R ==> list_setequiv R) (app l). Proof. intros al bl [??]. split; apply list_setincl_app; done. Qed. Global Instance: subrelation (≡ₚ) (flip (list_setincl R)). Proof. intros ???. apply list_setincl_perm. done. Qed. Global Instance list_agrees_setincl : Proper (flip (list_setincl R) ==> impl) (list_agrees R). Proof. move=> al bl /= Hab /list_agrees_alt Hal. apply (list_agrees_alt _) => a b Ha Hb. destruct (Hab _ Ha) as (a' & Ha' & HRa). destruct (Hab _ Hb) as (b' & Hb' & HRb). trans a'; first done. etrans; last done. eapply Hal; done. Qed. Global Instance list_agrees_setequiv : Proper (list_setequiv R ==> iff) (list_agrees R). Proof. intros ?? [??]. split; by apply: list_agrees_setincl. Qed. Lemma list_setincl_contains al bl : (∀ x, x ∈ al → x ∈ bl) → list_setincl R al bl. Proof. intros Hin a Ha. exists a. split; last done. naive_solver. Qed. Lemma list_setequiv_equiv al bl : (∀ x, x ∈ al ↔ x ∈ bl) → list_setequiv R al bl. Proof. intros Hin. split; apply list_setincl_contains; naive_solver. Qed. Lemma list_agrees_contains al bl : (∀ x, x ∈ bl → x ∈ al) → list_agrees R al → list_agrees R bl. Proof. intros ?. by eapply (list_agrees_setincl _),list_setincl_contains. Qed. Lemma list_agrees_equiv al bl : (∀ x, x ∈ bl ↔ x ∈ al) → list_agrees R al ↔ list_agrees R bl. Proof. intros ?. by eapply (list_agrees_setequiv _), list_setequiv_equiv. Qed. Lemma list_setincl_singleton a b : R a b → list_setincl R [a] [b]. Proof. intros HR c ->%elem_of_list_singleton. exists b. split; last done. apply elem_of_list_singleton. done. Qed. Lemma list_setincl_singleton_rev a b : list_setincl R [a] [b] → R a b. Proof. intros Hl. destruct (Hl a) as (? & ->%elem_of_list_singleton & HR); last done. by apply elem_of_list_singleton. Qed. Lemma list_setequiv_singleton a b : R a b → list_setequiv R [a] [b]. Proof. intros ?. split; by apply list_setincl_singleton. Qed. Lemma list_agrees_iff_setincl al a : a ∈ al → list_agrees R al ↔ list_setincl R al [a]. Proof. intros Hin. split. - move=>/list_agrees_alt Hl b Hb. exists a. split; first set_solver+. exact: Hl. - intros Hl. apply (list_agrees_alt _)=> b c Hb Hc. destruct (Hl _ Hb) as (? & ->%elem_of_list_singleton & ?). destruct (Hl _ Hc) as (? & ->%elem_of_list_singleton & ?). by trans a. Qed. Lemma list_setincl_singleton_in al a : a ∈ al → list_setincl R [a] al. Proof. intros Hin b ->%elem_of_list_singleton. exists a. split; done. Qed. Global Instance list_setincl_ext : subrelation (Forall2 R) (list_setincl R). Proof. move=>al bl. induction 1. - intros ? []%elem_of_nil. - intros a [->|Ha]%elem_of_cons. + eexists. split; first constructor. done. + destruct (IHForall2 _ Ha) as (b & ? & ?). exists b. split; first by constructor. done. Qed. Global Instance list_setequiv_ext : subrelation (Forall2 R) (list_setequiv R). Proof. move=>al bl ?. split; apply list_setincl_ext; done. Qed. Lemma list_agrees_subrel `(R' : relation A) `{Equivalence _ R'} : subrelation R R' → ∀ l, list_agrees R l → list_agrees R' l. Proof. move=> HR l /list_agrees_alt Hl. apply (list_agrees_alt _)=> a b Ha Hb. by apply HR, Hl. Qed. Section fmap. Context `(R' : relation B) (f : A → B) {Hf: Proper (R ==> R') f}. Global Instance list_setincl_fmap : Proper (list_setincl R ==> list_setincl R') (fmap f). Proof. intros al bl Hab a' (a & -> & Ha)%elem_of_list_fmap. destruct (Hab _ Ha) as (b & Hb & HR). exists (f b). split; first eapply elem_of_list_fmap; eauto. Qed. Global Instance list_setequiv_fmap : Proper (list_setequiv R ==> list_setequiv R') (fmap f). Proof. intros ?? [??]. split; apply list_setincl_fmap; done. Qed. Lemma list_agrees_fmap `{Equivalence _ R'} al : list_agrees R al → list_agrees R' (f <\$> al). Proof. move=> /list_agrees_alt Hl. apply <-(list_agrees_alt R')=> a' b'. intros (a & -> & Ha)%elem_of_list_fmap (b & -> & Hb)%elem_of_list_fmap. apply Hf. exact: Hl. Qed. End fmap. End list_theory. `````` Robbert Krebbers committed Nov 11, 2015 218 219 `````` Section agree. `````` Ralf Jung committed Nov 22, 2016 220 ``````Context {A : ofeT}. `````` Robbert Krebbers committed Nov 11, 2015 221 `````` `````` Ralf Jung committed Nov 25, 2016 222 ``````Definition agree_list (x : agree A) := agree_car x :: agree_with x. `````` Robbert Krebbers committed Feb 24, 2016 223 `````` `````` Ralf Jung committed Nov 25, 2016 224 225 226 227 ``````Instance agree_validN : ValidN (agree A) := λ n x, list_agrees (dist n) (agree_list x). Instance agree_valid : Valid (agree A) := λ x, list_agrees (equiv) (agree_list x). `````` Robbert Krebbers committed Feb 24, 2016 228 `````` `````` Robbert Krebbers committed Jan 14, 2016 229 ``````Instance agree_dist : Dist (agree A) := λ n x y, `````` Ralf Jung committed Nov 25, 2016 230 231 232 233 234 235 236 `````` list_setequiv (dist n) (agree_list x) (agree_list y). Instance agree_equiv : Equiv (agree A) := λ x y, ∀ n, list_setequiv (dist n) (agree_list x) (agree_list y). Definition agree_dist_incl n (x y : agree A) := list_setincl (dist n) (agree_list x) (agree_list y). `````` Ralf Jung committed Nov 22, 2016 237 ``````Definition agree_ofe_mixin : OfeMixin (agree A). `````` Robbert Krebbers committed Nov 11, 2015 238 239 ``````Proof. split. `````` Ralf Jung committed Nov 25, 2016 240 241 242 243 244 245 `````` - intros x y; split; intros Hxy; done. - split; rewrite /dist /agree_dist; intros ? *. + reflexivity. + by symmetry. + intros. etrans; eassumption. - intros ???. apply list_setequiv_subrel=>??. apply dist_S. `````` Robbert Krebbers committed Nov 11, 2015 246 ``````Qed. `````` Ralf Jung committed Nov 22, 2016 247 248 ``````Canonical Structure agreeC := OfeT (agree A) agree_ofe_mixin. `````` Robbert Krebbers committed Jan 14, 2016 249 ``````Program Instance agree_op : Op (agree A) := λ x y, `````` Robbert Krebbers committed Jul 21, 2016 250 `````` {| agree_car := agree_car x; `````` Ralf Jung committed Nov 25, 2016 251 `````` agree_with := agree_with x ++ agree_car y :: agree_with y |}. `````` Robbert Krebbers committed May 28, 2016 252 ``````Instance agree_pcore : PCore (agree A) := Some. `````` Robbert Krebbers committed Feb 24, 2016 253 `````` `````` Robbert Krebbers committed Feb 11, 2016 254 ``````Instance: Comm (≡) (@op (agree A) _). `````` Ralf Jung committed Nov 25, 2016 255 256 ``````Proof. intros x y n. apply: list_setequiv_equiv. set_solver. Qed. `````` Ralf Jung committed Feb 23, 2016 257 ``````Lemma agree_idemp (x : agree A) : x ⋅ x ≡ x. `````` Ralf Jung committed Nov 25, 2016 258 259 ``````Proof. intros n. apply: list_setequiv_equiv. set_solver. Qed. `````` Robbert Krebbers committed Jan 16, 2016 260 261 ``````Instance: ∀ n : nat, Proper (dist n ==> impl) (@validN (agree A) _ n). Proof. `````` Ralf Jung committed Nov 25, 2016 262 263 `````` intros n x y. rewrite /dist /validN /agree_dist /agree_validN. by intros ->. `````` Robbert Krebbers committed Jan 16, 2016 264 ``````Qed. `````` Ralf Jung committed Nov 25, 2016 265 266 267 268 269 270 ``````Instance: ∀ n : nat, Proper (equiv ==> iff) (@validN (agree A) _ n). Proof. intros n ???. assert (x ≡{n}≡ y) as Hxy by by apply equiv_dist. split; rewrite Hxy; done. Qed. `````` Robbert Krebbers committed Nov 11, 2015 271 272 ``````Instance: ∀ x : agree A, Proper (dist n ==> dist n) (op x). Proof. `````` Ralf Jung committed Nov 25, 2016 273 274 `````` intros n x y1 y2. rewrite /dist /agree_dist /agree_list /=. rewrite !app_comm_cons. apply: list_setequiv_app. `````` Robbert Krebbers committed Nov 11, 2015 275 ``````Qed. `````` Robbert Krebbers committed Jan 14, 2016 276 ``````Instance: Proper (dist n ==> dist n ==> dist n) (@op (agree A) _). `````` Robbert Krebbers committed Feb 11, 2016 277 ``````Proof. by intros n x1 x2 Hx y1 y2 Hy; rewrite Hy !(comm _ _ y2) Hx. Qed. `````` Robbert Krebbers committed Nov 12, 2015 278 ``````Instance: Proper ((≡) ==> (≡) ==> (≡)) op := ne_proper_2 _. `````` Robbert Krebbers committed Feb 11, 2016 279 ``````Instance: Assoc (≡) (@op (agree A) _). `````` Ralf Jung committed Nov 25, 2016 280 ``````Proof. intros x y z n. apply: list_setequiv_equiv. set_solver. Qed. `````` Robbert Krebbers committed Feb 24, 2016 281 `````` `````` Robbert Krebbers committed Feb 26, 2016 282 283 284 285 286 ``````Lemma agree_included (x y : agree A) : x ≼ y ↔ y ≡ x ⋅ y. Proof. split; [|by intros ?; exists y]. by intros [z Hz]; rewrite Hz assoc agree_idemp. Qed. `````` Ralf Jung committed Nov 25, 2016 287 288 289 290 291 292 293 294 295 296 297 298 ``````Lemma agree_op_inv_inclN n x1 x2 : ✓{n} (x1 ⋅ x2) → agree_dist_incl n x1 x2. Proof. rewrite /validN /= => /list_agrees_alt Hv a /elem_of_cons Ha. exists (agree_car x2). split; first by constructor. eapply Hv. - simpl. destruct Ha as [->|Ha]; set_solver. - simpl. set_solver+. Qed. Lemma agree_op_invN n (x1 x2 : agree A) : ✓{n} (x1 ⋅ x2) → x1 ≡{n}≡ x2. Proof. intros Hxy. split; apply agree_op_inv_inclN; first done. by rewrite comm. Qed. `````` Robbert Krebbers committed Feb 24, 2016 299 300 301 ``````Lemma agree_valid_includedN n (x y : agree A) : ✓{n} y → x ≼{n} y → x ≡{n}≡ y. Proof. move=> Hval [z Hy]; move: Hval; rewrite Hy. `````` Ralf Jung committed Nov 25, 2016 302 `````` by move=> /agree_op_invN->; rewrite agree_idemp. `````` Robbert Krebbers committed Feb 24, 2016 303 304 ``````Qed. `````` Robbert Krebbers committed Jan 14, 2016 305 ``````Definition agree_cmra_mixin : CMRAMixin (agree A). `````` Robbert Krebbers committed Nov 11, 2015 306 ``````Proof. `````` Robbert Krebbers committed May 28, 2016 307 `````` apply cmra_total_mixin; try apply _ || by eauto. `````` Ralf Jung committed Nov 25, 2016 308 309 310 311 312 313 `````` - move=>x. split. + move=>/list_agrees_alt Hx n. apply (list_agrees_alt _)=> a b Ha Hb. apply equiv_dist, Hx; done. + intros Hx. apply (list_agrees_alt _)=> a b Ha Hb. apply equiv_dist=>n. eapply (list_agrees_alt _); first (by apply Hx); done. - intros n x. apply (list_agrees_subrel _ _)=>??. apply dist_S. `````` Robbert Krebbers committed May 28, 2016 314 `````` - intros x. apply agree_idemp. `````` Ralf Jung committed Nov 25, 2016 315 `````` - intros ??? Hl. apply: list_agrees_contains Hl. set_solver. `````` Robbert Krebbers committed Aug 14, 2016 316 `````` - intros n x y1 y2 Hval Hx; exists x, x; simpl; split. `````` Robbert Krebbers committed Feb 24, 2016 317 `````` + by rewrite agree_idemp. `````` Ralf Jung committed Nov 25, 2016 318 `````` + by move: Hval; rewrite Hx; move=> /agree_op_invN->; rewrite agree_idemp. `````` Robbert Krebbers committed Nov 11, 2015 319 ``````Qed. `````` Robbert Krebbers committed May 25, 2016 320 ``````Canonical Structure agreeR : cmraT := `````` Ralf Jung committed Nov 22, 2016 321 `````` CMRAT (agree A) agree_ofe_mixin agree_cmra_mixin. `````` Robbert Krebbers committed Jan 14, 2016 322 `````` `````` Robbert Krebbers committed May 28, 2016 323 324 ``````Global Instance agree_total : CMRATotal agreeR. Proof. rewrite /CMRATotal; eauto. Qed. `````` Robbert Krebbers committed Mar 15, 2016 325 ``````Global Instance agree_persistent (x : agree A) : Persistent x. `````` Robbert Krebbers committed May 28, 2016 326 ``````Proof. by constructor. Qed. `````` Robbert Krebbers committed Mar 15, 2016 327 `````` `````` Ralf Jung committed Nov 25, 2016 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 ``````Lemma agree_op_inv (x1 x2 : agree A) : ✓ (x1 ⋅ x2) → x1 ≡ x2. Proof. intros ?. apply equiv_dist=>n. by apply agree_op_invN, cmra_valid_validN. Qed. Global Instance agree_discrete : Discrete A → CMRADiscrete agreeR. Proof. intros HD. split. - intros x y Hxy n. eapply list_setequiv_subrel; last exact Hxy. clear -HD. intros x y ?. apply equiv_dist, HD. done. - rewrite /valid /cmra_valid /agree_valid /validN /cmra_validN /agree_validN /=. move=> x. apply (list_agrees_subrel _ _). clear -HD. intros x y. apply HD. Qed. Definition to_agree (x : A) : agree A := {| agree_car := x; agree_with := [] |}. `````` Robbert Krebbers committed Jul 21, 2016 346 `````` `````` Robbert Krebbers committed Nov 16, 2015 347 ``````Global Instance to_agree_ne n : Proper (dist n ==> dist n) to_agree. `````` Ralf Jung committed Nov 25, 2016 348 349 350 351 ``````Proof. intros x1 x2 Hx; rewrite /= /dist /agree_dist /=. exact: list_setequiv_singleton. Qed. `````` Robbert Krebbers committed Jan 16, 2016 352 ``````Global Instance to_agree_proper : Proper ((≡) ==> (≡)) to_agree := ne_proper _. `````` Robbert Krebbers committed Jul 21, 2016 353 `````` `````` Ralf Jung committed Nov 25, 2016 354 355 356 357 358 359 ``````Global Instance to_agree_injN n : Inj (dist n) (dist n) (to_agree). Proof. intros a b [Hxy%list_setincl_singleton_rev _]. done. Qed. Global Instance to_agree_inj : Inj (≡) (≡) (to_agree). Proof. intros a b ?. apply equiv_dist=>n. apply to_agree_injN. by apply equiv_dist. Qed. `````` Robbert Krebbers committed Jul 21, 2016 360 `````` `````` Ralf Jung committed Dec 13, 2016 361 ``````Lemma to_agree_uninjN n (x : agree A) : ✓{n} x → ∃ y : A, to_agree y ≡{n}≡ x. `````` Robbert Krebbers committed Jul 21, 2016 362 ``````Proof. `````` Ralf Jung committed Nov 25, 2016 363 364 365 366 367 368 `````` intros Hl. exists (agree_car x). rewrite /dist /agree_dist /=. split. - apply: list_setincl_singleton_in. set_solver+. - apply (list_agrees_iff_setincl _); first set_solver+. done. Qed. `````` Ralf Jung committed Dec 13, 2016 369 370 371 372 373 374 375 376 377 378 ``````Lemma to_agree_uninj (x : agree A) : ✓ x → ∃ y : A, to_agree y ≡ x. Proof. intros Hl. exists (agree_car x). rewrite /dist /agree_dist /=. split. - apply: list_setincl_singleton_in. set_solver+. - apply (list_agrees_iff_setincl _); first set_solver+. eapply list_agrees_subrel; last exact: Hl; [apply _..|]. intros ???. by apply equiv_dist. Qed. `````` Ralf Jung committed Nov 25, 2016 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 ``````Lemma to_agree_included (a b : A) : to_agree a ≼ to_agree b ↔ a ≡ b. Proof. split. - intros (x & Heq). apply equiv_dist=>n. destruct (Heq n) as [_ Hincl]. (* TODO: This could become a generic lemma about list_setincl. *) destruct (Hincl a) as (? & ->%elem_of_list_singleton & ?); first set_solver+. done. - intros Hab. rewrite Hab. eexists. symmetry. eapply agree_idemp. Qed. Lemma to_agree_comp_valid (a b : A) : ✓ (to_agree a ⋅ to_agree b) ↔ a ≡ b. Proof. split. - (* TODO: can this be derived from other stuff? Otherwise, should probably become sth. generic about list_agrees. *) intros Hv. apply Hv; simpl; set_solver. - intros ->. rewrite agree_idemp. done. `````` Robbert Krebbers committed Jul 21, 2016 395 ``````Qed. `````` Robbert Krebbers committed Feb 13, 2016 396 397 `````` (** Internalized properties *) `````` Robbert Krebbers committed May 31, 2016 398 ``````Lemma agree_equivI {M} a b : to_agree a ≡ to_agree b ⊣⊢ (a ≡ b : uPred M). `````` Robbert Krebbers committed Feb 25, 2016 399 ``````Proof. `````` Ralf Jung committed Nov 25, 2016 400 401 402 `````` uPred.unseal. do 2 split. - intros Hx. exact: to_agree_injN. - intros Hx. exact: to_agree_ne. `````` Robbert Krebbers committed Feb 25, 2016 403 ``````Qed. `````` Ralf Jung committed Mar 10, 2016 404 ``````Lemma agree_validI {M} x y : ✓ (x ⋅ y) ⊢ (x ≡ y : uPred M). `````` Ralf Jung committed Nov 25, 2016 405 ``````Proof. uPred.unseal; split=> r n _ ?; by apply: agree_op_invN. Qed. `````` Robbert Krebbers committed Nov 11, 2015 406 407 ``````End agree. `````` Robbert Krebbers committed Jan 14, 2016 408 ``````Arguments agreeC : clear implicits. `````` Robbert Krebbers committed Mar 01, 2016 409 ``````Arguments agreeR : clear implicits. `````` Robbert Krebbers committed Jan 14, 2016 410 `````` `````` Robbert Krebbers committed Dec 21, 2015 411 ``````Program Definition agree_map {A B} (f : A → B) (x : agree A) : agree B := `````` Ralf Jung committed Nov 25, 2016 412 `````` {| agree_car := f (agree_car x); agree_with := f <\$> (agree_with x) |}. `````` Robbert Krebbers committed Jan 14, 2016 413 ``````Lemma agree_map_id {A} (x : agree A) : agree_map id x = x. `````` Ralf Jung committed Nov 25, 2016 414 ``````Proof. rewrite /agree_map /= list_fmap_id. by destruct x. Qed. `````` Robbert Krebbers committed Jan 16, 2016 415 416 ``````Lemma agree_map_compose {A B C} (f : A → B) (g : B → C) (x : agree A) : agree_map (g ∘ f) x = agree_map g (agree_map f x). `````` Ralf Jung committed Nov 25, 2016 417 ``````Proof. rewrite /agree_map /= list_fmap_compose. done. Qed. `````` Robbert Krebbers committed Dec 15, 2015 418 `````` `````` Robbert Krebbers committed Nov 11, 2015 419 ``````Section agree_map. `````` Ralf Jung committed Nov 22, 2016 420 `````` Context {A B : ofeT} (f : A → B) `{Hf: ∀ n, Proper (dist n ==> dist n) f}. `````` Robbert Krebbers committed Feb 26, 2016 421 `````` Instance agree_map_ne n : Proper (dist n ==> dist n) (agree_map f). `````` Ralf Jung committed Nov 25, 2016 422 423 424 425 426 `````` Proof. intros x y Hxy. change (list_setequiv (dist n)(f <\$> (agree_list x))(f <\$> (agree_list y))). eapply list_setequiv_fmap; last exact Hxy. apply _. Qed. `````` Robbert Krebbers committed Feb 26, 2016 427 `````` Instance agree_map_proper : Proper ((≡) ==> (≡)) (agree_map f) := ne_proper _. `````` Ralf Jung committed Nov 25, 2016 428 `````` `````` Robbert Krebbers committed Dec 15, 2015 429 430 `````` Lemma agree_map_ext (g : A → B) x : (∀ x, f x ≡ g x) → agree_map f x ≡ agree_map g x. `````` Ralf Jung committed Nov 25, 2016 431 432 433 434 435 436 `````` Proof. intros Hfg n. apply: list_setequiv_ext. change (f <\$> (agree_list x) ≡{n}≡ g <\$> (agree_list x)). apply list_fmap_ext_ne=>y. by apply equiv_dist. Qed. `````` Robbert Krebbers committed Dec 15, 2015 437 `````` Global Instance agree_map_monotone : CMRAMonotone (agree_map f). `````` Robbert Krebbers committed Nov 11, 2015 438 `````` Proof. `````` Robbert Krebbers committed Feb 26, 2016 439 `````` split; first apply _. `````` Ralf Jung committed Nov 25, 2016 440 441 442 `````` - intros n x. rewrite /cmra_validN /validN /= /agree_validN /= => ?. change (list_agrees (dist n) (f <\$> agree_list x)). eapply (list_agrees_fmap _ _ _); done. `````` Robbert Krebbers committed Feb 26, 2016 443 `````` - intros x y; rewrite !agree_included=> ->. `````` Ralf Jung committed Nov 25, 2016 444 445 `````` rewrite /equiv /agree_equiv /agree_map /agree_op /agree_list /=. rewrite !fmap_app=>n. apply: list_setequiv_equiv. set_solver+. `````` Robbert Krebbers committed Nov 11, 2015 446 447 `````` Qed. End agree_map. `````` Robbert Krebbers committed Nov 16, 2015 448 `````` `````` Robbert Krebbers committed Feb 04, 2016 449 450 451 ``````Definition agreeC_map {A B} (f : A -n> B) : agreeC A -n> agreeC B := CofeMor (agree_map f : agreeC A → agreeC B). Instance agreeC_map_ne A B n : Proper (dist n ==> dist n) (@agreeC_map A B). `````` Robbert Krebbers committed Nov 16, 2015 452 ``````Proof. `````` Ralf Jung committed Nov 25, 2016 453 454 455 `````` intros f g Hfg x. apply: list_setequiv_ext. change (f <\$> (agree_list x) ≡{n}≡ g <\$> (agree_list x)). apply list_fmap_ext_ne. done. `````` Robbert Krebbers committed Nov 16, 2015 456 ``````Qed. `````` Ralf Jung committed Feb 05, 2016 457 `````` `````` Robbert Krebbers committed Mar 02, 2016 458 459 460 461 ``````Program Definition agreeRF (F : cFunctor) : rFunctor := {| rFunctor_car A B := agreeR (cFunctor_car F A B); rFunctor_map A1 A2 B1 B2 fg := agreeC_map (cFunctor_map F fg) |}. `````` Robbert Krebbers committed Mar 07, 2016 462 463 464 ``````Next Obligation. intros ? A1 A2 B1 B2 n ???; simpl. by apply agreeC_map_ne, cFunctor_ne. Qed. `````` Robbert Krebbers committed Mar 02, 2016 465 466 467 468 469 470 471 472 ``````Next Obligation. intros F A B x; simpl. rewrite -{2}(agree_map_id x). apply agree_map_ext=>y. by rewrite cFunctor_id. Qed. Next Obligation. intros F A1 A2 A3 B1 B2 B3 f g f' g' x; simpl. rewrite -agree_map_compose. apply agree_map_ext=>y; apply cFunctor_compose. Qed. `````` Ralf Jung committed Mar 07, 2016 473 474 475 476 477 478 479 `````` Instance agreeRF_contractive F : cFunctorContractive F → rFunctorContractive (agreeRF F). Proof. intros ? A1 A2 B1 B2 n ???; simpl. by apply agreeC_map_ne, cFunctor_contractive. Qed.``````