big_op.v 66.8 KB
 Robbert Krebbers committed Oct 30, 2017 1 ``````From iris.algebra Require Export big_op. `````` Ralf Jung committed Mar 21, 2018 2 ``````From iris.bi Require Import derived_laws_sbi plainly. `````` Robbert Krebbers committed Feb 20, 2019 3 ``````From stdpp Require Import countable fin_sets functions. `````` Ralf Jung committed Jan 05, 2017 4 ``````Set Default Proof Using "Type". `````` Robbert Krebbers committed Jun 15, 2018 5 ``````Import interface.bi derived_laws_bi.bi derived_laws_sbi.bi. `````` Robbert Krebbers committed Feb 14, 2016 6 `````` `````` Dan Frumin committed Apr 07, 2019 7 ``````(** Notations for unary variants *) `````` Ralf Jung committed Jun 05, 2018 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 ``````Notation "'[∗' 'list]' k ↦ x ∈ l , P" := (big_opL bi_sep (λ k x, P) l) : bi_scope. Notation "'[∗' 'list]' x ∈ l , P" := (big_opL bi_sep (λ _ x, P) l) : bi_scope. Notation "'[∗]' Ps" := (big_opL bi_sep (λ _ x, x) Ps) : bi_scope. Notation "'[∧' 'list]' k ↦ x ∈ l , P" := (big_opL bi_and (λ k x, P) l) : bi_scope. Notation "'[∧' 'list]' x ∈ l , P" := (big_opL bi_and (λ _ x, P) l) : bi_scope. Notation "'[∧]' Ps" := (big_opL bi_and (λ _ x, x) Ps) : bi_scope. Notation "'[∗' 'map]' k ↦ x ∈ m , P" := (big_opM bi_sep (λ k x, P) m) : bi_scope. Notation "'[∗' 'map]' x ∈ m , P" := (big_opM bi_sep (λ _ x, P) m) : bi_scope. Notation "'[∗' 'set]' x ∈ X , P" := (big_opS bi_sep (λ x, P) X) : bi_scope. Notation "'[∗' 'mset]' x ∈ X , P" := (big_opMS bi_sep (λ x, P) X) : bi_scope. `````` Robbert Krebbers committed Aug 24, 2016 28 `````` `````` Dan Frumin committed Apr 07, 2019 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 ``````(** Definitions and notations for binary variants *) (** A version of the separating big operator that ranges over two lists. This version also ensures that both lists have the same length. Although this version can be defined in terms of the unary using a [zip] (see [big_sepL2_alt]), we do not define it that way to get better computational behavior (for [simpl]). *) Fixpoint big_sepL2 {PROP : bi} {A B} (Φ : nat → A → B → PROP) (l1 : list A) (l2 : list B) : PROP := match l1, l2 with | [], [] => emp | x1 :: l1, x2 :: l2 => Φ 0 x1 x2 ∗ big_sepL2 (λ n, Φ (S n)) l1 l2 | _, _ => False end%I. Instance: Params (@big_sepL2) 3 := {}. Arguments big_sepL2 {PROP A B} _ !_ !_ /. Typeclasses Opaque big_sepL2. Notation "'[∗' 'list]' k ↦ x1 ; x2 ∈ l1 ; l2 , P" := (big_sepL2 (λ k x1 x2, P) l1 l2) : bi_scope. Notation "'[∗' 'list]' x1 ; x2 ∈ l1 ; l2 , P" := (big_sepL2 (λ _ x1 x2, P) l1 l2) : bi_scope. Definition big_sepM2 {PROP : bi} `{Countable K} {A B} (Φ : K → A → B → PROP) (m1 : gmap K A) (m2 : gmap K B) : PROP := (⌜ ∀ k, is_Some (m1 !! k) ↔ is_Some (m2 !! k) ⌝ ∧ [∗ map] k ↦ xy ∈ map_zip m1 m2, Φ k xy.1 xy.2)%I. Instance: Params (@big_sepM2) 6 := {}. Typeclasses Opaque big_sepM2. Notation "'[∗' 'map]' k ↦ x1 ; x2 ∈ m1 ; m2 , P" := (big_sepM2 (λ k x1 x2, P) m1 m2) : bi_scope. Notation "'[∗' 'map]' x1 ; x2 ∈ m1 ; m2 , P" := (big_sepM2 (λ _ x1 x2, P) m1 m2) : bi_scope. `````` Robbert Krebbers committed Apr 08, 2016 60 ``````(** * Properties *) `````` Robbert Krebbers committed Oct 30, 2017 61 62 ``````Section bi_big_op. Context {PROP : bi}. `````` Robbert Krebbers committed Oct 31, 2018 63 ``````Implicit Types P Q : PROP. `````` Robbert Krebbers committed Oct 30, 2017 64 ``````Implicit Types Ps Qs : list PROP. `````` Robbert Krebbers committed Feb 14, 2016 65 66 ``````Implicit Types A : Type. `````` Robbert Krebbers committed Aug 24, 2016 67 ``````(** ** Big ops over lists *) `````` Robbert Krebbers committed Oct 30, 2017 68 ``````Section sep_list. `````` Robbert Krebbers committed Aug 24, 2016 69 70 `````` Context {A : Type}. Implicit Types l : list A. `````` Robbert Krebbers committed Oct 30, 2017 71 `````` Implicit Types Φ Ψ : nat → A → PROP. `````` Robbert Krebbers committed Aug 24, 2016 72 `````` `````` Robbert Krebbers committed Oct 30, 2017 73 `````` Lemma big_sepL_nil Φ : ([∗ list] k↦y ∈ nil, Φ k y) ⊣⊢ emp. `````` Robbert Krebbers committed Sep 28, 2016 74 `````` Proof. done. Qed. `````` Jacques-Henri Jourdan committed Dec 04, 2017 75 `````` Lemma big_sepL_nil' `{BiAffine PROP} P Φ : P ⊢ [∗ list] k↦y ∈ nil, Φ k y. `````` Robbert Krebbers committed Oct 30, 2017 76 `````` Proof. apply (affine _). Qed. `````` Robbert Krebbers committed Sep 28, 2016 77 `````` Lemma big_sepL_cons Φ x l : `````` Robbert Krebbers committed Nov 03, 2016 78 `````` ([∗ list] k↦y ∈ x :: l, Φ k y) ⊣⊢ Φ 0 x ∗ [∗ list] k↦y ∈ l, Φ (S k) y. `````` Robbert Krebbers committed Sep 28, 2016 79 `````` Proof. by rewrite big_opL_cons. Qed. `````` Robbert Krebbers committed Nov 03, 2016 80 `````` Lemma big_sepL_singleton Φ x : ([∗ list] k↦y ∈ [x], Φ k y) ⊣⊢ Φ 0 x. `````` Robbert Krebbers committed Sep 28, 2016 81 82 `````` Proof. by rewrite big_opL_singleton. Qed. Lemma big_sepL_app Φ l1 l2 : `````` Robbert Krebbers committed Nov 03, 2016 83 84 `````` ([∗ list] k↦y ∈ l1 ++ l2, Φ k y) ⊣⊢ ([∗ list] k↦y ∈ l1, Φ k y) ∗ ([∗ list] k↦y ∈ l2, Φ (length l1 + k) y). `````` Robbert Krebbers committed Sep 28, 2016 85 86 `````` Proof. by rewrite big_opL_app. Qed. `````` Robbert Krebbers committed Aug 24, 2016 87 88 `````` Lemma big_sepL_mono Φ Ψ l : (∀ k y, l !! k = Some y → Φ k y ⊢ Ψ k y) → `````` Robbert Krebbers committed Nov 03, 2016 89 `````` ([∗ list] k ↦ y ∈ l, Φ k y) ⊢ [∗ list] k ↦ y ∈ l, Ψ k y. `````` Robbert Krebbers committed Sep 28, 2016 90 `````` Proof. apply big_opL_forall; apply _. Qed. `````` Robbert Krebbers committed Aug 24, 2016 91 92 `````` Lemma big_sepL_proper Φ Ψ l : (∀ k y, l !! k = Some y → Φ k y ⊣⊢ Ψ k y) → `````` Robbert Krebbers committed Nov 03, 2016 93 `````` ([∗ list] k ↦ y ∈ l, Φ k y) ⊣⊢ ([∗ list] k ↦ y ∈ l, Ψ k y). `````` Robbert Krebbers committed Sep 28, 2016 94 `````` Proof. apply big_opL_proper. Qed. `````` Jacques-Henri Jourdan committed Dec 04, 2017 95 `````` Lemma big_sepL_submseteq `{BiAffine PROP} (Φ : A → PROP) l1 l2 : `````` Robbert Krebbers committed Jan 06, 2017 96 `````` l1 ⊆+ l2 → ([∗ list] y ∈ l2, Φ y) ⊢ [∗ list] y ∈ l1, Φ y. `````` Robbert Krebbers committed Oct 30, 2017 97 98 99 `````` Proof. intros [l ->]%submseteq_Permutation. by rewrite big_sepL_app sep_elim_l. Qed. `````` Robbert Krebbers committed Aug 24, 2016 100 `````` `````` Robbert Krebbers committed Mar 24, 2017 101 102 `````` Global Instance big_sepL_mono' : Proper (pointwise_relation _ (pointwise_relation _ (⊢)) ==> (=) ==> (⊢)) `````` Robbert Krebbers committed Oct 30, 2017 103 `````` (big_opL (@bi_sep PROP) (A:=A)). `````` Robbert Krebbers committed Mar 24, 2017 104 `````` Proof. intros f g Hf m ? <-. apply big_opL_forall; apply _ || intros; apply Hf. Qed. `````` Robbert Krebbers committed Jun 14, 2018 105 `````` Global Instance big_sepL_id_mono' : `````` Robbert Krebbers committed Oct 31, 2018 106 `````` Proper (Forall2 (⊢) ==> (⊢)) (big_opL (@bi_sep PROP) (λ _ P, P)). `````` Robbert Krebbers committed Mar 24, 2017 107 `````` Proof. by induction 1 as [|P Q Ps Qs HPQ ? IH]; rewrite /= ?HPQ ?IH. Qed. `````` Robbert Krebbers committed Aug 24, 2016 108 `````` `````` Ralf Jung committed Apr 05, 2018 109 `````` Lemma big_sepL_emp l : ([∗ list] k↦y ∈ l, emp) ⊣⊢@{PROP} emp. `````` Robbert Krebbers committed Oct 30, 2017 110 111 `````` Proof. by rewrite big_opL_unit. Qed. `````` Jacques-Henri Jourdan committed Dec 05, 2016 112 113 114 115 `````` Lemma big_sepL_lookup_acc Φ l i x : l !! i = Some x → ([∗ list] k↦y ∈ l, Φ k y) ⊢ Φ i x ∗ (Φ i x -∗ ([∗ list] k↦y ∈ l, Φ k y)). Proof. `````` Robbert Krebbers committed Mar 24, 2017 116 117 118 `````` intros Hli. rewrite -(take_drop_middle l i x) // big_sepL_app /=. rewrite Nat.add_0_r take_length_le; eauto using lookup_lt_Some, Nat.lt_le_incl. rewrite assoc -!(comm _ (Φ _ _)) -assoc. by apply sep_mono_r, wand_intro_l. `````` Jacques-Henri Jourdan committed Dec 05, 2016 119 120 `````` Qed. `````` Robbert Krebbers committed Oct 30, 2017 121 `````` Lemma big_sepL_lookup Φ l i x `{!Absorbing (Φ i x)} : `````` Robbert Krebbers committed Nov 03, 2016 122 `````` l !! i = Some x → ([∗ list] k↦y ∈ l, Φ k y) ⊢ Φ i x. `````` Robbert Krebbers committed Oct 30, 2017 123 `````` Proof. intros. rewrite big_sepL_lookup_acc //. by rewrite sep_elim_l. Qed. `````` Robbert Krebbers committed Aug 24, 2016 124 `````` `````` Robbert Krebbers committed Oct 30, 2017 125 `````` Lemma big_sepL_elem_of (Φ : A → PROP) l x `{!Absorbing (Φ x)} : `````` Robbert Krebbers committed Nov 03, 2016 126 `````` x ∈ l → ([∗ list] y ∈ l, Φ y) ⊢ Φ x. `````` Robbert Krebbers committed Mar 24, 2017 127 128 129 `````` Proof. intros [i ?]%elem_of_list_lookup; eauto using (big_sepL_lookup (λ _, Φ)). Qed. `````` Robbert Krebbers committed Aug 28, 2016 130 `````` `````` Robbert Krebbers committed Oct 30, 2017 131 `````` Lemma big_sepL_fmap {B} (f : A → B) (Φ : nat → B → PROP) l : `````` Robbert Krebbers committed Nov 03, 2016 132 `````` ([∗ list] k↦y ∈ f <\$> l, Φ k y) ⊣⊢ ([∗ list] k↦y ∈ l, Φ k (f y)). `````` Robbert Krebbers committed Sep 28, 2016 133 `````` Proof. by rewrite big_opL_fmap. Qed. `````` Robbert Krebbers committed Aug 24, 2016 134 135 `````` Lemma big_sepL_sepL Φ Ψ l : `````` Robbert Krebbers committed Nov 03, 2016 136 137 `````` ([∗ list] k↦x ∈ l, Φ k x ∗ Ψ k x) ⊣⊢ ([∗ list] k↦x ∈ l, Φ k x) ∗ ([∗ list] k↦x ∈ l, Ψ k x). `````` Robbert Krebbers committed Sep 28, 2016 138 `````` Proof. by rewrite big_opL_opL. Qed. `````` Robbert Krebbers committed Sep 28, 2016 139 `````` `````` Robbert Krebbers committed Nov 27, 2016 140 141 142 `````` Lemma big_sepL_and 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). `````` Robbert Krebbers committed Oct 30, 2017 143 `````` Proof. auto using and_intro, big_sepL_mono, and_elim_l, and_elim_r. Qed. `````` Robbert Krebbers committed Nov 27, 2016 144 `````` `````` Jacques-Henri Jourdan committed Dec 04, 2017 145 `````` Lemma big_sepL_persistently `{BiAffine PROP} Φ l : `````` Robbert Krebbers committed Mar 04, 2018 146 `````` ([∗ list] k↦x ∈ l, Φ k x) ⊣⊢ [∗ list] k↦x ∈ l, (Φ k x). `````` Robbert Krebbers committed Sep 28, 2016 147 `````` Proof. apply (big_opL_commute _). Qed. `````` Robbert Krebbers committed Aug 24, 2016 148 `````` `````` Jacques-Henri Jourdan committed Dec 04, 2017 149 `````` Lemma big_sepL_forall `{BiAffine PROP} Φ l : `````` Robbert Krebbers committed Oct 25, 2017 150 `````` (∀ k x, Persistent (Φ k x)) → `````` Ralf Jung committed Nov 22, 2016 151 `````` ([∗ list] k↦x ∈ l, Φ k x) ⊣⊢ (∀ k x, ⌜l !! k = Some x⌝ → Φ k x). `````` Robbert Krebbers committed Aug 24, 2016 152 153 154 `````` Proof. intros HΦ. apply (anti_symm _). { apply forall_intro=> k; apply forall_intro=> x. `````` Robbert Krebbers committed Oct 30, 2017 155 156 `````` apply impl_intro_l, pure_elim_l=> ?; by apply: big_sepL_lookup. } revert Φ HΦ. induction l as [|x l IH]=> Φ HΦ; [by auto using big_sepL_nil'|]. `````` Robbert Krebbers committed Oct 30, 2017 157 `````` rewrite big_sepL_cons. rewrite -persistent_and_sep; apply and_intro. `````` Robbert Krebbers committed Nov 21, 2016 158 `````` - by rewrite (forall_elim 0) (forall_elim x) pure_True // True_impl. `````` Robbert Krebbers committed Aug 24, 2016 159 160 161 162 `````` - rewrite -IH. apply forall_intro=> k; by rewrite (forall_elim (S k)). Qed. Lemma big_sepL_impl Φ Ψ l : `````` Robbert Krebbers committed Oct 30, 2017 163 `````` ([∗ list] k↦x ∈ l, Φ k x) -∗ `````` Jacques-Henri Jourdan committed Nov 02, 2017 164 `````` □ (∀ k x, ⌜l !! k = Some x⌝ → Φ k x -∗ Ψ k x) -∗ `````` Robbert Krebbers committed Oct 30, 2017 165 `````` [∗ list] k↦x ∈ l, Ψ k x. `````` Robbert Krebbers committed Aug 24, 2016 166 `````` Proof. `````` Robbert Krebbers committed Oct 30, 2017 167 168 `````` apply wand_intro_l. revert Φ Ψ. induction l as [|x l IH]=> Φ Ψ /=. { by rewrite sep_elim_r. } `````` 169 `````` rewrite intuitionistically_sep_dup -assoc [(□ _ ∗ _)%I]comm -!assoc assoc. `````` Robbert Krebbers committed Oct 30, 2017 170 171 `````` apply sep_mono. - rewrite (forall_elim 0) (forall_elim x) pure_True // True_impl. `````` 172 `````` by rewrite intuitionistically_elim wand_elim_l. `````` Robbert Krebbers committed Oct 30, 2017 173 `````` - rewrite comm -(IH (Φ ∘ S) (Ψ ∘ S)) /=. `````` Jacques-Henri Jourdan committed Nov 02, 2017 174 `````` apply sep_mono_l, affinely_mono, persistently_mono. `````` Robbert Krebbers committed Oct 30, 2017 175 `````` apply forall_intro=> k. by rewrite (forall_elim (S k)). `````` Robbert Krebbers committed Aug 24, 2016 176 177 `````` Qed. `````` Robbert Krebbers committed Apr 04, 2018 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 `````` Lemma big_sepL_delete Φ l i x : l !! i = Some x → ([∗ list] k↦y ∈ l, Φ k y) ⊣⊢ Φ i x ∗ [∗ list] k↦y ∈ l, if decide (k = i) then emp else Φ k y. Proof. intros. rewrite -(take_drop_middle l i x) // !big_sepL_app /= Nat.add_0_r. rewrite take_length_le; last eauto using lookup_lt_Some, Nat.lt_le_incl. rewrite decide_True // left_id. rewrite assoc -!(comm _ (Φ _ _)) -assoc. do 2 f_equiv. - apply big_sepL_proper=> k y Hk. apply lookup_lt_Some in Hk. rewrite take_length in Hk. by rewrite decide_False; last lia. - apply big_sepL_proper=> k y _. by rewrite decide_False; last lia. Qed. Lemma big_sepL_delete' `{!BiAffine PROP} Φ l i x : l !! i = Some x → ([∗ list] k↦y ∈ l, Φ k y) ⊣⊢ Φ i x ∗ [∗ list] k↦y ∈ l, ⌜ k ≠ i ⌝ → Φ k y. Proof. intros. rewrite big_sepL_delete //. (do 2 f_equiv)=> k y. rewrite -decide_emp. by repeat case_decide. Qed. `````` Robbert Krebbers committed Oct 31, 2018 200 201 202 203 `````` Lemma big_sepL_replicate l P : [∗] replicate (length l) P ⊣⊢ [∗ list] y ∈ l, P. Proof. induction l as [|x l]=> //=; by f_equiv. Qed. `````` Robbert Krebbers committed Oct 30, 2017 204 `````` Global Instance big_sepL_nil_persistent Φ : `````` Robbert Krebbers committed Oct 25, 2017 205 `````` Persistent ([∗ list] k↦x ∈ [], Φ k x). `````` Robbert Krebbers committed Mar 24, 2017 206 `````` Proof. simpl; apply _. Qed. `````` Robbert Krebbers committed Oct 30, 2017 207 `````` Global Instance big_sepL_persistent Φ l : `````` Robbert Krebbers committed Oct 25, 2017 208 `````` (∀ k x, Persistent (Φ k x)) → Persistent ([∗ list] k↦x ∈ l, Φ k x). `````` Robbert Krebbers committed Mar 24, 2017 209 `````` Proof. revert Φ. induction l as [|x l IH]=> Φ ? /=; apply _. Qed. `````` Robbert Krebbers committed Oct 30, 2017 210 `````` Global Instance big_sepL_persistent_id Ps : `````` Robbert Krebbers committed Oct 25, 2017 211 `````` TCForall Persistent Ps → Persistent ([∗] Ps). `````` Robbert Krebbers committed Mar 24, 2017 212 `````` Proof. induction 1; simpl; apply _. Qed. `````` Aleš Bizjak committed Oct 30, 2017 213 `````` `````` Robbert Krebbers committed Oct 30, 2017 214 215 216 `````` Global Instance big_sepL_nil_affine Φ : Affine ([∗ list] k↦x ∈ [], Φ k x). Proof. simpl; apply _. Qed. `````` Aleš Bizjak committed Oct 30, 2017 217 218 219 `````` Global Instance big_sepL_affine Φ l : (∀ k x, Affine (Φ k x)) → Affine ([∗ list] k↦x ∈ l, Φ k x). Proof. revert Φ. induction l as [|x l IH]=> Φ ? /=; apply _. Qed. `````` Robbert Krebbers committed Oct 30, 2017 220 221 `````` Global Instance big_sepL_affine_id Ps : TCForall Affine Ps → Affine ([∗] Ps). Proof. induction 1; simpl; apply _. Qed. `````` Robbert Krebbers committed Oct 30, 2017 222 ``````End sep_list. `````` Robbert Krebbers committed Aug 24, 2016 223 `````` `````` Robbert Krebbers committed Jun 15, 2018 224 ``````Section sep_list_more. `````` Ralf Jung committed Dec 20, 2016 225 226 `````` Context {A : Type}. Implicit Types l : list A. `````` Robbert Krebbers committed Oct 30, 2017 227 `````` Implicit Types Φ Ψ : nat → A → PROP. `````` Ralf Jung committed Dec 20, 2016 228 229 230 `````` (* Some lemmas depend on the generalized versions of the above ones. *) Lemma big_sepL_zip_with {B C} Φ f (l1 : list B) (l2 : list C) : `````` Robbert Krebbers committed Mar 14, 2017 231 `````` ([∗ list] k↦x ∈ zip_with f l1 l2, Φ k x) `````` Robbert Krebbers committed Oct 30, 2017 232 `````` ⊣⊢ ([∗ list] k↦x ∈ l1, if l2 !! k is Some y then Φ k (f x y) else emp). `````` Ralf Jung committed Dec 20, 2016 233 `````` Proof. `````` Robbert Krebbers committed Oct 30, 2017 234 235 236 `````` revert Φ l2; induction l1 as [|x l1 IH]=> Φ [|y l2] //=. - by rewrite big_sepL_emp left_id. - by rewrite IH. `````` Ralf Jung committed Dec 20, 2016 237 `````` Qed. `````` Robbert Krebbers committed Jun 15, 2018 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 ``````End sep_list_more. Lemma big_sepL2_alt {A B} (Φ : nat → A → B → PROP) l1 l2 : ([∗ list] k↦y1;y2 ∈ l1; l2, Φ k y1 y2) ⊣⊢ ⌜ length l1 = length l2 ⌝ ∧ [∗ list] k ↦ y ∈ zip l1 l2, Φ k (y.1) (y.2). Proof. apply (anti_symm _). - apply and_intro. + revert Φ l2. induction l1 as [|x1 l1 IH]=> Φ -[|x2 l2] /=; auto using pure_intro, False_elim. rewrite IH sep_elim_r. apply pure_mono; auto. + revert Φ l2. induction l1 as [|x1 l1 IH]=> Φ -[|x2 l2] /=; auto using pure_intro, False_elim. by rewrite IH. - apply pure_elim_l=> /Forall2_same_length Hl. revert Φ. induction Hl as [|x1 l1 x2 l2 _ _ IH]=> Φ //=. by rewrite -IH. Qed. (** ** Big ops over two lists *) Section sep_list2. Context {A B : Type}. Implicit Types Φ Ψ : nat → A → B → PROP. Lemma big_sepL2_nil Φ : ([∗ list] k↦y1;y2 ∈ []; [], Φ k y1 y2) ⊣⊢ emp. Proof. done. Qed. Lemma big_sepL2_nil' `{BiAffine PROP} P Φ : P ⊢ [∗ list] k↦y1;y2 ∈ [];[], Φ k y1 y2. Proof. apply (affine _). Qed. Lemma big_sepL2_cons Φ x1 x2 l1 l2 : ([∗ list] k↦y1;y2 ∈ x1 :: l1; x2 :: l2, Φ k y1 y2) ⊣⊢ Φ 0 x1 x2 ∗ [∗ list] k↦y1;y2 ∈ l1;l2, Φ (S k) y1 y2. Proof. done. Qed. Lemma big_sepL2_cons_inv_l Φ x1 l1 l2 : ([∗ list] k↦y1;y2 ∈ x1 :: l1; l2, Φ k y1 y2) -∗ ∃ x2 l2', ⌜ l2 = x2 :: l2' ⌝ ∧ Φ 0 x1 x2 ∗ [∗ list] k↦y1;y2 ∈ l1;l2', Φ (S k) y1 y2. Proof. destruct l2 as [|x2 l2]; simpl; auto using False_elim. by rewrite -(exist_intro x2) -(exist_intro l2) pure_True // left_id. Qed. Lemma big_sepL2_cons_inv_r Φ x2 l1 l2 : ([∗ list] k↦y1;y2 ∈ l1; x2 :: l2, Φ k y1 y2) -∗ ∃ x1 l1', ⌜ l1 = x1 :: l1' ⌝ ∧ Φ 0 x1 x2 ∗ [∗ list] k↦y1;y2 ∈ l1';l2, Φ (S k) y1 y2. Proof. destruct l1 as [|x1 l1]; simpl; auto using False_elim. by rewrite -(exist_intro x1) -(exist_intro l1) pure_True // left_id. Qed. Lemma big_sepL2_singleton Φ x1 x2 : ([∗ list] k↦y1;y2 ∈ [x1];[x2], Φ k y1 y2) ⊣⊢ Φ 0 x1 x2. Proof. by rewrite /= right_id. Qed. Lemma big_sepL2_length Φ l1 l2 : ([∗ list] k↦y1;y2 ∈ l1; l2, Φ k y1 y2) -∗ ⌜ length l1 = length l2 ⌝. Proof. by rewrite big_sepL2_alt and_elim_l. Qed. Lemma big_sepL2_app Φ l1 l2 l1' l2' : ([∗ list] k↦y1;y2 ∈ l1; l1', Φ k y1 y2) -∗ ([∗ list] k↦y1;y2 ∈ l2; l2', Φ (length l1 + k) y1 y2) -∗ ([∗ list] k↦y1;y2 ∈ l1 ++ l2; l1' ++ l2', Φ k y1 y2). Proof. apply wand_intro_r. revert Φ l1'. induction l1 as [|x1 l1 IH]=> Φ -[|x1' l1'] /=. - by rewrite left_id. - rewrite left_absorb. apply False_elim. - rewrite left_absorb. apply False_elim. - by rewrite -assoc IH. Qed. Lemma big_sepL2_app_inv_l Φ l1' l1'' l2 : ([∗ list] k↦y1;y2 ∈ l1' ++ l1''; l2, Φ k y1 y2) -∗ ∃ l2' l2'', ⌜ l2 = l2' ++ l2'' ⌝ ∧ ([∗ list] k↦y1;y2 ∈ l1';l2', Φ k y1 y2) ∗ ([∗ list] k↦y1;y2 ∈ l1'';l2'', Φ (length l1' + k) y1 y2). Proof. rewrite -(exist_intro (take (length l1') l2)) -(exist_intro (drop (length l1') l2)) take_drop pure_True // left_id. revert Φ l2. induction l1' as [|x1 l1' IH]=> Φ -[|x2 l2] /=; [by rewrite left_id|by rewrite left_id|apply False_elim|]. by rewrite IH -assoc. Qed. Lemma big_sepL2_app_inv_r Φ l1 l2' l2'' : ([∗ list] k↦y1;y2 ∈ l1; l2' ++ l2'', Φ k y1 y2) -∗ ∃ l1' l1'', ⌜ l1 = l1' ++ l1'' ⌝ ∧ ([∗ list] k↦y1;y2 ∈ l1';l2', Φ k y1 y2) ∗ ([∗ list] k↦y1;y2 ∈ l1'';l2'', Φ (length l2' + k) y1 y2). Proof. rewrite -(exist_intro (take (length l2') l1)) -(exist_intro (drop (length l2') l1)) take_drop pure_True // left_id. revert Φ l1. induction l2' as [|x2 l2' IH]=> Φ -[|x1 l1] /=; [by rewrite left_id|by rewrite left_id|apply False_elim|]. by rewrite IH -assoc. Qed. Lemma big_sepL2_mono Φ Ψ l1 l2 : (∀ k y1 y2, l1 !! k = Some y1 → l2 !! k = Some y2 → Φ k y1 y2 ⊢ Ψ k y1 y2) → ([∗ list] k ↦ y1;y2 ∈ l1;l2, Φ k y1 y2) ⊢ [∗ list] k ↦ y1;y2 ∈ l1;l2, Ψ k y1 y2. Proof. intros H. rewrite !big_sepL2_alt. f_equiv. apply big_sepL_mono=> k [y1 y2]. rewrite lookup_zip_with=> ?; simplify_option_eq; auto. Qed. Lemma big_sepL2_proper Φ Ψ l1 l2 : (∀ k y1 y2, l1 !! k = Some y1 → l2 !! k = Some y2 → Φ k y1 y2 ⊣⊢ Ψ k y1 y2) → ([∗ list] k ↦ y1;y2 ∈ l1;l2, Φ k y1 y2) ⊣⊢ [∗ list] k ↦ y1;y2 ∈ l1;l2, Ψ k y1 y2. Proof. intros; apply (anti_symm _); apply big_sepL2_mono; auto using equiv_entails, equiv_entails_sym. Qed. Global Instance big_sepL2_ne n : Proper (pointwise_relation _ (pointwise_relation _ (pointwise_relation _ (dist n))) ==> (=) ==> (=) ==> (dist n)) (big_sepL2 (PROP:=PROP) (A:=A) (B:=B)). Proof. intros Φ1 Φ2 HΦ x1 ? <- x2 ? <-. rewrite !big_sepL2_alt. f_equiv. f_equiv=> k [y1 y2]. apply HΦ. Qed. Global Instance big_sepL2_mono' : Proper (pointwise_relation _ (pointwise_relation _ (pointwise_relation _ (⊢))) ==> (=) ==> (=) ==> (⊢)) (big_sepL2 (PROP:=PROP) (A:=A) (B:=B)). Proof. intros f g Hf l1 ? <- l2 ? <-. apply big_sepL2_mono; intros; apply Hf. Qed. Global Instance big_sepL2_proper' : Proper (pointwise_relation _ (pointwise_relation _ (pointwise_relation _ (⊣⊢))) ==> (=) ==> (=) ==> (⊣⊢)) (big_sepL2 (PROP:=PROP) (A:=A) (B:=B)). Proof. intros f g Hf l1 ? <- l2 ? <-. apply big_sepL2_proper; intros; apply Hf. Qed. Lemma big_sepL2_lookup_acc Φ l1 l2 i x1 x2 : l1 !! i = Some x1 → l2 !! i = Some x2 → ([∗ list] k↦y1;y2 ∈ l1;l2, Φ k y1 y2) ⊢ Φ i x1 x2 ∗ (Φ i x1 x2 -∗ ([∗ list] k↦y1;y2 ∈ l1;l2, Φ k y1 y2)). Proof. intros Hl1 Hl2. rewrite big_sepL2_alt. apply pure_elim_l=> Hl. rewrite {1}big_sepL_lookup_acc; last by rewrite lookup_zip_with; simplify_option_eq. by rewrite pure_True // left_id. Qed. Lemma big_sepL2_lookup Φ l1 l2 i x1 x2 `{!Absorbing (Φ i x1 x2)} : l1 !! i = Some x1 → l2 !! i = Some x2 → ([∗ list] k↦y1;y2 ∈ l1;l2, Φ k y1 y2) ⊢ Φ i x1 x2. Proof. intros. rewrite big_sepL2_lookup_acc //. by rewrite sep_elim_l. Qed. Lemma big_sepL2_fmap_l {A'} (f : A → A') (Φ : nat → A' → B → PROP) l1 l2 : ([∗ list] k↦y1;y2 ∈ f <\$> l1; l2, Φ k y1 y2) ⊣⊢ ([∗ list] k↦y1;y2 ∈ l1;l2, Φ k (f y1) y2). Proof. rewrite !big_sepL2_alt fmap_length zip_with_fmap_l zip_with_zip big_sepL_fmap. by f_equiv; f_equiv=> k [??]. Qed. Lemma big_sepL2_fmap_r {B'} (g : B → B') (Φ : nat → A → B' → PROP) l1 l2 : ([∗ list] k↦y1;y2 ∈ l1; g <\$> l2, Φ k y1 y2) ⊣⊢ ([∗ list] k↦y1;y2 ∈ l1;l2, Φ k y1 (g y2)). Proof. rewrite !big_sepL2_alt fmap_length zip_with_fmap_r zip_with_zip big_sepL_fmap. by f_equiv; f_equiv=> k [??]. Qed. Lemma big_sepL2_sepL2 Φ Ψ l1 l2 : ([∗ list] k↦y1;y2 ∈ l1;l2, Φ k y1 y2 ∗ Ψ k y1 y2) ⊣⊢ ([∗ list] k↦y1;y2 ∈ l1;l2, Φ k y1 y2) ∗ ([∗ list] k↦y1;y2 ∈ l1;l2, Ψ k y1 y2). Proof. rewrite !big_sepL2_alt big_sepL_sepL !persistent_and_affinely_sep_l. rewrite -assoc (assoc _ _ ( _)%I). rewrite -(comm bi_sep ( _)%I). rewrite -assoc (assoc _ _ ( _)%I) -!persistent_and_affinely_sep_l. by rewrite affinely_and_r persistent_and_affinely_sep_l idemp. Qed. Lemma big_sepL2_and Φ Ψ l1 l2 : ([∗ list] k↦y1;y2 ∈ l1;l2, Φ k y1 y2 ∧ Ψ k y1 y2) ⊢ ([∗ list] k↦y1;y2 ∈ l1;l2, Φ k y1 y2) ∧ ([∗ list] k↦y1;y2 ∈ l1;l2, Ψ k y1 y2). Proof. auto using and_intro, big_sepL2_mono, and_elim_l, and_elim_r. Qed. Lemma big_sepL2_persistently `{BiAffine PROP} Φ l1 l2 : ([∗ list] k↦y1;y2 ∈ l1;l2, Φ k y1 y2) ⊣⊢ [∗ list] k↦y1;y2 ∈ l1;l2, (Φ k y1 y2). Proof. by rewrite !big_sepL2_alt persistently_and persistently_pure big_sepL_persistently. Qed. Lemma big_sepL2_impl Φ Ψ l1 l2 : ([∗ list] k↦y1;y2 ∈ l1;l2, Φ k y1 y2) -∗ □ (∀ k x1 x2, ⌜l1 !! k = Some x1⌝ → ⌜l2 !! k = Some x2⌝ → Φ k x1 x2 -∗ Ψ k x1 x2) -∗ [∗ list] k↦y1;y2 ∈ l1;l2, Ψ k y1 y2. Proof. apply wand_intro_l. revert Φ Ψ l2. induction l1 as [|x1 l1 IH]=> Φ Ψ [|x2 l2] /=; [by rewrite sep_elim_r..|]. rewrite intuitionistically_sep_dup -assoc [(□ _ ∗ _)%I]comm -!assoc assoc. apply sep_mono. - rewrite (forall_elim 0) (forall_elim x1) (forall_elim x2) !pure_True // !True_impl. by rewrite intuitionistically_elim wand_elim_l. - rewrite comm -(IH (Φ ∘ S) (Ψ ∘ S)) /=. apply sep_mono_l, affinely_mono, persistently_mono. apply forall_intro=> k. by rewrite (forall_elim (S k)). Qed. Global Instance big_sepL2_nil_persistent Φ : Persistent ([∗ list] k↦y1;y2 ∈ []; [], Φ k y1 y2). Proof. simpl; apply _. Qed. Global Instance big_sepL2_persistent Φ l1 l2 : (∀ k x1 x2, Persistent (Φ k x1 x2)) → Persistent ([∗ list] k↦y1;y2 ∈ l1;l2, Φ k y1 y2). Proof. rewrite big_sepL2_alt. apply _. Qed. Global Instance big_sepL2_nil_affine Φ : Affine ([∗ list] k↦y1;y2 ∈ []; [], Φ k y1 y2). Proof. simpl; apply _. Qed. Global Instance big_sepL2_affine Φ l1 l2 : (∀ k x1 x2, Affine (Φ k x1 x2)) → Affine ([∗ list] k↦y1;y2 ∈ l1;l2, Φ k y1 y2). Proof. rewrite big_sepL2_alt. apply _. Qed. `````` Robbert Krebbers committed Oct 30, 2017 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 ``````End sep_list2. Section and_list. Context {A : Type}. Implicit Types l : list A. Implicit Types Φ Ψ : nat → A → PROP. Lemma big_andL_nil Φ : ([∧ list] k↦y ∈ nil, Φ k y) ⊣⊢ True. Proof. done. Qed. Lemma big_andL_nil' P Φ : P ⊢ [∧ list] k↦y ∈ nil, Φ k y. Proof. by apply pure_intro. Qed. Lemma big_andL_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 big_opL_cons. Qed. Lemma big_andL_singleton Φ x : ([∧ list] k↦y ∈ [x], Φ k y) ⊣⊢ Φ 0 x. Proof. by rewrite big_opL_singleton. Qed. Lemma big_andL_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 big_opL_app. Qed. Lemma big_andL_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. apply big_opL_forall; apply _. Qed. Lemma big_andL_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. apply big_opL_proper. Qed. Lemma big_andL_submseteq (Φ : A → PROP) l1 l2 : l1 ⊆+ l2 → ([∧ list] y ∈ l2, Φ y) ⊢ [∧ list] y ∈ l1, Φ y. Proof. intros [l ->]%submseteq_Permutation. by rewrite big_andL_app and_elim_l. Qed. Global Instance big_andL_mono' : Proper (pointwise_relation _ (pointwise_relation _ (⊢)) ==> (=) ==> (⊢)) (big_opL (@bi_and PROP) (A:=A)). Proof. intros f g Hf m ? <-. apply big_opL_forall; apply _ || intros; apply Hf. Qed. `````` Robbert Krebbers committed Jun 14, 2018 488 `````` Global Instance big_andL_id_mono' : `````` Robbert Krebbers committed Oct 31, 2018 489 `````` Proper (Forall2 (⊢) ==> (⊢)) (big_opL (@bi_and PROP) (λ _ P, P)). `````` Robbert Krebbers committed Oct 30, 2017 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 `````` Proof. by induction 1 as [|P Q Ps Qs HPQ ? IH]; rewrite /= ?HPQ ?IH. Qed. Lemma big_andL_lookup Φ l i x `{!Absorbing (Φ 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_andL_app /=. rewrite Nat.add_0_r take_length_le; eauto using lookup_lt_Some, Nat.lt_le_incl, and_elim_l', and_elim_r'. Qed. Lemma big_andL_elem_of (Φ : A → PROP) l x `{!Absorbing (Φ x)} : x ∈ l → ([∧ list] y ∈ l, Φ y) ⊢ Φ x. Proof. intros [i ?]%elem_of_list_lookup; eauto using (big_andL_lookup (λ _, Φ)). Qed. Lemma big_andL_fmap {B} (f : A → B) (Φ : nat → B → PROP) l : ([∧ list] k↦y ∈ f <\$> l, Φ k y) ⊣⊢ ([∧ list] k↦y ∈ l, Φ k (f y)). Proof. by rewrite big_opL_fmap. Qed. Lemma big_andL_andL 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. by rewrite big_opL_opL. Qed. Lemma big_andL_and 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. auto using and_intro, big_andL_mono, and_elim_l, and_elim_r. Qed. Lemma big_andL_persistently Φ l : `````` Robbert Krebbers committed Mar 04, 2018 521 `````` ([∧ list] k↦x ∈ l, Φ k x) ⊣⊢ [∧ list] k↦x ∈ l, (Φ k x). `````` Robbert Krebbers committed Oct 30, 2017 522 523 `````` Proof. apply (big_opL_commute _). Qed. `````` Jacques-Henri Jourdan committed Dec 04, 2017 524 `````` Lemma big_andL_forall `{BiAffine PROP} Φ l : `````` Robbert Krebbers committed Oct 30, 2017 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 `````` ([∧ list] k↦x ∈ l, Φ k x) ⊣⊢ (∀ k x, ⌜l !! k = Some x⌝ → Φ k x). Proof. apply (anti_symm _). { apply forall_intro=> k; apply forall_intro=> x. apply impl_intro_l, pure_elim_l=> ?; by apply: big_andL_lookup. } revert Φ. induction l as [|x l IH]=> Φ; [by auto using big_andL_nil'|]. rewrite big_andL_cons. apply and_intro. - by rewrite (forall_elim 0) (forall_elim x) pure_True // True_impl. - rewrite -IH. apply forall_intro=> k; by rewrite (forall_elim (S k)). Qed. Global Instance big_andL_nil_persistent Φ : Persistent ([∧ list] k↦x ∈ [], Φ k x). Proof. simpl; apply _. Qed. Global Instance big_andL_persistent Φ l : (∀ k x, Persistent (Φ k x)) → Persistent ([∧ list] k↦x ∈ l, Φ k x). Proof. revert Φ. induction l as [|x l IH]=> Φ ? /=; apply _. Qed. End and_list. `````` Robbert Krebbers committed Aug 24, 2016 543 `````` `````` Robbert Krebbers committed Apr 08, 2016 544 ``````(** ** Big ops over finite maps *) `````` Dan Frumin committed Apr 07, 2019 545 ``````Section map. `````` Robbert Krebbers committed Feb 17, 2016 546 547 `````` Context `{Countable K} {A : Type}. Implicit Types m : gmap K A. `````` Robbert Krebbers committed Oct 30, 2017 548 `````` Implicit Types Φ Ψ : K → A → PROP. `````` Robbert Krebbers committed Feb 14, 2016 549 `````` `````` Robbert Krebbers committed Oct 30, 2017 550 551 552 553 `````` Lemma big_sepM_mono 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. Proof. apply big_opM_forall; apply _ || auto. Qed. `````` Robbert Krebbers committed Jul 22, 2016 554 555 `````` Lemma big_sepM_proper Φ Ψ m : (∀ k x, m !! k = Some x → Φ k x ⊣⊢ Ψ k x) → `````` Robbert Krebbers committed Nov 03, 2016 556 `````` ([∗ map] k ↦ x ∈ m, Φ k x) ⊣⊢ ([∗ map] k ↦ x ∈ m, Ψ k x). `````` Robbert Krebbers committed Sep 28, 2016 557 `````` Proof. apply big_opM_proper. Qed. `````` Jacques-Henri Jourdan committed Dec 04, 2017 558 `````` Lemma big_sepM_subseteq `{BiAffine PROP} Φ m1 m2 : `````` Robbert Krebbers committed Oct 30, 2017 559 560 `````` m2 ⊆ m1 → ([∗ map] k ↦ x ∈ m1, Φ k x) ⊢ [∗ map] k ↦ x ∈ m2, Φ k x. Proof. intros. by apply big_sepL_submseteq, map_to_list_submseteq. Qed. `````` Robbert Krebbers committed Feb 17, 2016 561 `````` `````` Robbert Krebbers committed Mar 24, 2017 562 563 `````` Global Instance big_sepM_mono' : Proper (pointwise_relation _ (pointwise_relation _ (⊢)) ==> (=) ==> (⊢)) `````` Robbert Krebbers committed Oct 30, 2017 564 565 `````` (big_opM (@bi_sep PROP) (K:=K) (A:=A)). Proof. intros f g Hf m ? <-. apply big_sepM_mono=> ???; apply Hf. Qed. `````` Robbert Krebbers committed Feb 17, 2016 566 `````` `````` Robbert Krebbers committed Oct 30, 2017 567 `````` Lemma big_sepM_empty Φ : ([∗ map] k↦x ∈ ∅, Φ k x) ⊣⊢ emp. `````` Robbert Krebbers committed Sep 28, 2016 568 `````` Proof. by rewrite big_opM_empty. Qed. `````` Jacques-Henri Jourdan committed Dec 04, 2017 569 `````` Lemma big_sepM_empty' `{BiAffine PROP} P Φ : P ⊢ [∗ map] k↦x ∈ ∅, Φ k x. `````` Robbert Krebbers committed Oct 30, 2017 570 `````` Proof. rewrite big_sepM_empty. apply: affine. Qed. `````` Robbert Krebbers committed May 30, 2016 571 `````` `````` Robbert Krebbers committed May 31, 2016 572 `````` Lemma big_sepM_insert Φ m i x : `````` Robbert Krebbers committed May 24, 2016 573 `````` m !! i = None → `````` Robbert Krebbers committed Nov 03, 2016 574 `````` ([∗ map] k↦y ∈ <[i:=x]> m, Φ k y) ⊣⊢ Φ i x ∗ [∗ map] k↦y ∈ m, Φ k y. `````` Robbert Krebbers committed Mar 24, 2017 575 `````` Proof. apply big_opM_insert. Qed. `````` Robbert Krebbers committed May 30, 2016 576 `````` `````` Robbert Krebbers committed May 31, 2016 577 `````` Lemma big_sepM_delete Φ m i x : `````` Robbert Krebbers committed May 24, 2016 578 `````` m !! i = Some x → `````` Robbert Krebbers committed Nov 03, 2016 579 `````` ([∗ map] k↦y ∈ m, Φ k y) ⊣⊢ Φ i x ∗ [∗ map] k↦y ∈ delete i m, Φ k y. `````` Robbert Krebbers committed Mar 24, 2017 580 `````` Proof. apply big_opM_delete. Qed. `````` Robbert Krebbers committed May 30, 2016 581 `````` `````` Robbert Krebbers committed Dec 12, 2018 582 583 584 585 586 587 588 589 590 591 592 593 594 `````` Lemma big_sepM_insert_2 Φ m i x : TCOr (∀ x, Affine (Φ i x)) (Absorbing (Φ i x)) → Φ i x -∗ ([∗ map] k↦y ∈ m, Φ k y) -∗ [∗ map] k↦y ∈ <[i:=x]> m, Φ k y. Proof. intros Ha. apply wand_intro_r. destruct (m !! i) as [y|] eqn:Hi; last first. { by rewrite -big_sepM_insert. } assert (TCOr (Affine (Φ i y)) (Absorbing (Φ i x))). { destruct Ha; try apply _. } rewrite big_sepM_delete // assoc. rewrite (sep_elim_l (Φ i x)) -big_sepM_insert ?lookup_delete //. by rewrite insert_delete. Qed. `````` Robbert Krebbers committed Nov 24, 2016 595 596 597 598 599 600 601 `````` Lemma big_sepM_lookup_acc Φ m i x : m !! i = Some x → ([∗ map] k↦y ∈ m, Φ k y) ⊢ Φ i x ∗ (Φ i x -∗ ([∗ map] k↦y ∈ m, Φ k y)). Proof. intros. rewrite big_sepM_delete //. by apply sep_mono_r, wand_intro_l. Qed. `````` Robbert Krebbers committed Oct 30, 2017 602 `````` Lemma big_sepM_lookup Φ m i x `{!Absorbing (Φ i x)} : `````` Robbert Krebbers committed Nov 03, 2016 603 `````` m !! i = Some x → ([∗ map] k↦y ∈ m, Φ k y) ⊢ Φ i x. `````` Robbert Krebbers committed Oct 30, 2017 604 `````` Proof. intros. rewrite big_sepM_lookup_acc //. by rewrite sep_elim_l. Qed. `````` Robbert Krebbers committed Nov 24, 2016 605 `````` `````` Robbert Krebbers committed Oct 30, 2017 606 `````` Lemma big_sepM_lookup_dom (Φ : K → PROP) m i `{!Absorbing (Φ i)} : `````` Robbert Krebbers committed Nov 20, 2016 607 608 `````` is_Some (m !! i) → ([∗ map] k↦_ ∈ m, Φ k) ⊢ Φ i. Proof. intros [x ?]. by eapply (big_sepM_lookup (λ i x, Φ i)). Qed. `````` Robbert Krebbers committed May 31, 2016 609 `````` `````` Robbert Krebbers committed Nov 03, 2016 610 `````` Lemma big_sepM_singleton Φ i x : ([∗ map] k↦y ∈ {[i:=x]}, Φ k y) ⊣⊢ Φ i x. `````` Robbert Krebbers committed Sep 28, 2016 611 `````` Proof. by rewrite big_opM_singleton. Qed. `````` Ralf Jung committed Feb 17, 2016 612 `````` `````` Robbert Krebbers committed Oct 30, 2017 613 `````` Lemma big_sepM_fmap {B} (f : A → B) (Φ : K → B → PROP) m : `````` Robbert Krebbers committed Nov 03, 2016 614 `````` ([∗ map] k↦y ∈ f <\$> m, Φ k y) ⊣⊢ ([∗ map] k↦y ∈ m, Φ k (f y)). `````` Robbert Krebbers committed Sep 28, 2016 615 `````` Proof. by rewrite big_opM_fmap. Qed. `````` Robbert Krebbers committed May 31, 2016 616 `````` `````` Robbert Krebbers committed Dec 02, 2016 617 618 619 `````` Lemma big_sepM_insert_override Φ m i x x' : m !! i = Some x → (Φ i x ⊣⊢ Φ i x') → ([∗ map] k↦y ∈ <[i:=x']> m, Φ k y) ⊣⊢ ([∗ map] k↦y ∈ m, Φ k y). `````` Robbert Krebbers committed Mar 24, 2017 620 `````` Proof. apply big_opM_insert_override. Qed. `````` Robbert Krebbers committed May 31, 2016 621 `````` `````` Robbert Krebbers committed Dec 02, 2016 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 `````` Lemma big_sepM_insert_override_1 Φ m i x x' : m !! i = Some x → ([∗ map] k↦y ∈ <[i:=x']> m, Φ k y) ⊢ (Φ i x' -∗ Φ i x) -∗ ([∗ map] k↦y ∈ m, Φ k y). Proof. intros ?. apply wand_intro_l. rewrite -insert_delete big_sepM_insert ?lookup_delete //. by rewrite assoc wand_elim_l -big_sepM_delete. Qed. Lemma big_sepM_insert_override_2 Φ m i x x' : m !! i = Some x → ([∗ map] k↦y ∈ m, Φ k y) ⊢ (Φ i x -∗ Φ i x') -∗ ([∗ map] k↦y ∈ <[i:=x']> m, Φ k y). Proof. intros ?. apply wand_intro_l. rewrite {1}big_sepM_delete //; rewrite assoc wand_elim_l. rewrite -insert_delete big_sepM_insert ?lookup_delete //. Qed. `````` Dan Frumin committed Feb 03, 2019 642 643 644 645 646 647 648 649 650 651 652 `````` Lemma big_sepM_insert_acc Φ m i x : m !! i = Some x → ([∗ map] k↦y ∈ m, Φ k y) ⊢ Φ i x ∗ (∀ x', Φ i x' -∗ ([∗ map] k↦y ∈ <[i:=x']> m, Φ k y)). Proof. intros ?. rewrite {1}big_sepM_delete //. apply sep_mono; [done|]. apply forall_intro=> x'. rewrite -insert_delete big_sepM_insert ?lookup_delete //. by apply wand_intro_l. Qed. `````` Robbert Krebbers committed Oct 30, 2017 653 `````` Lemma big_sepM_fn_insert {B} (Ψ : K → A → B → PROP) (f : K → B) m i x b : `````` Robbert Krebbers committed May 31, 2016 654 `````` m !! i = None → `````` Robbert Krebbers committed Nov 03, 2016 655 656 `````` ([∗ 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 Mar 24, 2017 657 `````` Proof. apply big_opM_fn_insert. Qed. `````` Robbert Krebbers committed Sep 28, 2016 658 `````` `````` Robbert Krebbers committed Oct 30, 2017 659 `````` Lemma big_sepM_fn_insert' (Φ : K → PROP) m i x P : `````` Robbert Krebbers committed May 31, 2016 660 `````` m !! i = None → `````` Robbert Krebbers committed Nov 03, 2016 661 `````` ([∗ map] k↦y ∈ <[i:=x]> m, <[i:=P]> Φ k) ⊣⊢ (P ∗ [∗ map] k↦y ∈ m, Φ k). `````` Robbert Krebbers committed Mar 24, 2017 662 `````` Proof. apply big_opM_fn_insert'. Qed. `````` Robbert Krebbers committed May 31, 2016 663 `````` `````` Dan Frumin committed Nov 01, 2018 664 665 666 667 668 669 `````` Lemma big_sepM_union Φ m1 m2 : m1 ##ₘ m2 → ([∗ map] k↦y ∈ m1 ∪ m2, Φ k y) ⊣⊢ ([∗ map] k↦y ∈ m1, Φ k y) ∗ ([∗ map] k↦y ∈ m2, Φ k y). Proof. apply big_opM_union. Qed. `````` Robbert Krebbers committed Feb 18, 2016 670 `````` Lemma big_sepM_sepM Φ Ψ m : `````` Robbert Krebbers committed Nov 27, 2016 671 `````` ([∗ map] k↦x ∈ m, Φ k x ∗ Ψ k x) `````` Robbert Krebbers committed Nov 03, 2016 672 `````` ⊣⊢ ([∗ map] k↦x ∈ m, Φ k x) ∗ ([∗ map] k↦x ∈ m, Ψ k x). `````` Robbert Krebbers committed Mar 24, 2017 673 `````` Proof. apply big_opM_opM. Qed. `````` Robbert Krebbers committed May 31, 2016 674 `````` `````` Robbert Krebbers committed Nov 27, 2016 675 676 677 `````` Lemma big_sepM_and Φ Ψ m : ([∗ map] k↦x ∈ m, Φ k x ∧ Ψ k x) ⊢ ([∗ map] k↦x ∈ m, Φ k x) ∧ ([∗ map] k↦x ∈ m, Ψ k x). `````` Robbert Krebbers committed Oct 30, 2017 678 `````` Proof. auto using and_intro, big_sepM_mono, and_elim_l, and_elim_r. Qed. `````` Robbert Krebbers committed Sep 28, 2016 679 `````` `````` Jacques-Henri Jourdan committed Dec 04, 2017 680 `````` Lemma big_sepM_persistently `{BiAffine PROP} Φ m : `````` Robbert Krebbers committed Mar 04, 2018 681 `````` ( ([∗ map] k↦x ∈ m, Φ k x)) ⊣⊢ ([∗ map] k↦x ∈ m, (Φ k x)). `````` Robbert Krebbers committed Sep 28, 2016 682 `````` Proof. apply (big_opM_commute _). Qed. `````` Robbert Krebbers committed May 31, 2016 683 `````` `````` Jacques-Henri Jourdan committed Dec 04, 2017 684 `````` Lemma big_sepM_forall `{BiAffine PROP} Φ m : `````` Robbert Krebbers committed Oct 25, 2017 685 `````` (∀ k x, Persistent (Φ k x)) → `````` Ralf Jung committed Nov 22, 2016 686 `````` ([∗ map] k↦x ∈ m, Φ k x) ⊣⊢ (∀ k x, ⌜m !! k = Some x⌝ → Φ k x). `````` Robbert Krebbers committed May 31, 2016 687 688 689 `````` Proof. intros. apply (anti_symm _). { apply forall_intro=> k; apply forall_intro=> x. `````` Robbert Krebbers committed Oct 30, 2017 690 691 `````` apply impl_intro_l, pure_elim_l=> ?; by apply: big_sepM_lookup. } induction m as [|i x m ? IH] using map_ind; auto using big_sepM_empty'. `````` Robbert Krebbers committed Oct 30, 2017 692 `````` rewrite big_sepM_insert // -persistent_and_sep. apply and_intro. `````` Robbert Krebbers committed Sep 28, 2016 693 `````` - rewrite (forall_elim i) (forall_elim x) lookup_insert. `````` Robbert Krebbers committed Nov 21, 2016 694 `````` by rewrite pure_True // True_impl. `````` Robbert Krebbers committed May 31, 2016 695 `````` - rewrite -IH. apply forall_mono=> k; apply forall_mono=> y. `````` Robbert Krebbers committed Sep 28, 2016 696 697 `````` apply impl_intro_l, pure_elim_l=> ?. rewrite lookup_insert_ne; last by intros ?; simplify_map_eq. `````` Robbert Krebbers committed Nov 21, 2016 698 `````` by rewrite pure_True // True_impl. `````` Robbert Krebbers committed May 31, 2016 699 700 701 `````` Qed. Lemma big_sepM_impl Φ Ψ m : `````` Robbert Krebbers committed Oct 30, 2017 702 `````` ([∗ map] k↦x ∈ m, Φ k x) -∗ `````` Jacques-Henri Jourdan committed Nov 02, 2017 703 `````` □ (∀ k x, ⌜m !! k = Some x⌝ → Φ k x -∗ Ψ k x) -∗ `````` Robbert Krebbers committed Oct 30, 2017 704 `````` [∗ map] k↦x ∈ m, Ψ k x. `````` Robbert Krebbers committed May 31, 2016 705 `````` Proof. `````` Robbert Krebbers committed Oct 30, 2017 706 707 `````` apply wand_intro_l. induction m as [|i x m ? IH] using map_ind. { by rewrite sep_elim_r. } `````` 708 `````` rewrite !big_sepM_insert // intuitionistically_sep_dup. `````` Jacques-Henri Jourdan committed Nov 02, 2017 709 `````` rewrite -assoc [(□ _ ∗ _)%I]comm -!assoc assoc. apply sep_mono. `````` Robbert Krebbers committed Oct 30, 2017 710 `````` - rewrite (forall_elim i) (forall_elim x) pure_True ?lookup_insert //. `````` 711 `````` by rewrite True_impl intuitionistically_elim wand_elim_l. `````` Robbert Krebbers committed Oct 30, 2017 712 `````` - rewrite comm -IH /=. `````` Jacques-Henri Jourdan committed Nov 02, 2017 713 `````` apply sep_mono_l, affinely_mono, persistently_mono, forall_mono=> k. `````` Robbert Krebbers committed Oct 30, 2017 714 715 716 `````` apply forall_mono=> y. apply impl_intro_l, pure_elim_l=> ?. rewrite lookup_insert_ne; last by intros ?; simplify_map_eq. by rewrite pure_True // True_impl. `````` Robbert Krebbers committed May 31, 2016 717 `````` Qed. `````` Robbert Krebbers committed Aug 24, 2016 718 `````` `````` Robbert Krebbers committed Oct 30, 2017 719 `````` Global Instance big_sepM_empty_persistent Φ : `````` Robbert Krebbers committed Oct 25, 2017 720 `````` Persistent ([∗ map] k↦x ∈ ∅, Φ k x). `````` Robbert Krebbers committed Sep 28, 2016 721 `````` Proof. rewrite /big_opM map_to_list_empty. apply _. Qed. `````` Robbert Krebbers committed Oct 30, 2017 722 `````` Global Instance big_sepM_persistent Φ m : `````` Robbert Krebbers committed Oct 25, 2017 723 `````` (∀ k x, Persistent (Φ k x)) → Persistent ([∗ map] k↦x ∈ m, Φ k x). `````` Robbert Krebbers committed Mar 24, 2017 724 `````` Proof. intros. apply big_sepL_persistent=> _ [??]; apply _. Qed. `````` Aleš Bizjak committed Oct 30, 2017 725 `````` `````` Robbert Krebbers committed Oct 30, 2017 726 727 728 `````` Global Instance big_sepM_empty_affine Φ : Affine ([∗ map] k↦x ∈ ∅, Φ k x). Proof. rewrite /big_opM map_to_list_empty. apply _. Qed. `````` Aleš Bizjak committed Oct 30, 2017 729 730 `````` Global Instance big_sepM_affine Φ m : (∀ k x, Affine (Φ k x)) → Affine ([∗ map] k↦x ∈ m, Φ k x). `````` Robbert Krebbers committed Oct 30, 2017 731 `````` Proof. intros. apply big_sepL_affine=> _ [??]; apply _. Qed. `````` Dan Frumin committed Apr 07, 2019 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 ``````End map. (** ** Big ops over two maps *) Section map2. Context `{Countable K} {A B : Type}. Implicit Types Φ Ψ : K → A → B → PROP. Lemma big_sepM2_dom Φ m1 m2 : ([∗ map] k↦y1;y2 ∈ m1; m2, Φ k y1 y2) -∗ ⌜ dom (gset K) m1 = dom (gset K) m2 ⌝. Proof. rewrite /big_sepM2 and_elim_l. apply pure_mono=>Hm. set_unfold=>k. by rewrite !elem_of_dom. Qed. Lemma big_sepM2_mono Φ Ψ m1 m2 : (∀ k y1 y2, m1 !! k = Some y1 → m2 !! k = Some y2 → Φ k y1 y2 ⊢ Ψ k y1 y2) → ([∗ map] k ↦ y1;y2 ∈ m1;m2, Φ k y1 y2) ⊢ [∗ map] k ↦ y1;y2 ∈ m1;m2, Ψ k y1 y2. Proof. intros Hm1m2. rewrite /big_sepM2. apply and_mono_r, big_sepM_mono. intros k [x1 x2]. rewrite map_lookup_zip_with. specialize (Hm1m2 k x1 x2). destruct (m1 !! k) as [y1|]; last done. destruct (m2 !! k) as [y2|]; simpl; last done. intros ?; simplify_eq. by apply Hm1m2. Qed. Lemma big_sepM2_proper Φ Ψ m1 m2 : (∀ k y1 y2, m1 !! k = Some y1 → m2 !! k = Some y2 → Φ k y1 y2 ⊣⊢ Ψ k y1 y2) → ([∗ map] k ↦ y1;y2 ∈ m1;m2, Φ k y1 y2) ⊣⊢ [∗ map] k ↦ y1;y2 ∈ m1;m2, Ψ k y1 y2. Proof. intros; apply (anti_symm _); apply big_sepM2_mono; auto using equiv_entails, equiv_entails_sym. Qed. Global Instance big_sepM2_ne n : Proper (pointwise_relation _ (pointwise_relation _ (pointwise_relation _ (dist n))) ==> (=) ==> (=) ==> (dist n)) (big_sepM2 (PROP:=PROP) (K:=K) (A:=A) (B:=B)). Proof. intros Φ1 Φ2 HΦ x1 ? <- x2 ? <-. rewrite /big_sepM2. f_equiv. f_equiv=> k [y1 y2]. apply HΦ. Qed. Global Instance big_sepM2_mono' : Proper (pointwise_relation _ (pointwise_relation _ (pointwise_relation _ (⊢))) ==> (=) ==> (=) ==> (⊢)) (big_sepM2 (PROP:=PROP) (K:=K) (A:=A) (B:=B)). Proof. intros f g Hf m1 ? <- m2 ? <-. apply big_sepM2_mono; intros; apply Hf. Qed. Global Instance big_sepM2_proper' : Proper (pointwise_relation _ (pointwise_relation _ (pointwise_relation _ (⊣⊢))) ==> (=) ==> (=) ==> (⊣⊢)) (big_sepM2 (PROP:=PROP) (K:=K) (A:=A) (B:=B)). Proof. intros f g Hf m1 ? <- m2 ? <-. apply big_sepM2_proper; intros; apply Hf. Qed. Lemma big_sepM2_empty Φ : ([∗ map] k↦y1;y2 ∈ ∅; ∅, Φ k y1 y2) ⊣⊢ emp. Proof. rewrite /big_sepM2 pure_True ?left_id //. intros k. rewrite !lookup_empty; split; by inversion 1. Qed. Lemma big_sepM2_empty' `{BiAffine PROP} P Φ : P ⊢ [∗ map] k↦y1;y2 ∈ ∅;∅, Φ k y1 y2. Proof. rewrite big_sepM2_empty. apply (affine _). Qed. Lemma big_sepM2_empty_l m1 Φ : ([∗ map] k↦y1;y2 ∈ m1; ∅, Φ k y1 y2) ⊢ ⌜m1 = ∅⌝. Proof. rewrite big_sepM2_dom dom_empty_L. apply pure_mono, dom_empty_inv_L. Qed. Lemma big_sepM2_empty_r m2 Φ : ([∗ map] k↦y1;y2 ∈ ∅; m2, Φ k y1 y2) ⊢ ⌜m2 = ∅⌝. Proof. rewrite big_sepM2_dom dom_empty_L. apply pure_mono=>?. eapply (dom_empty_inv_L (D:=gset K)). eauto. Qed. Lemma big_sepM2_insert Φ m1 m2 i x1 x2 : m1 !! i = None → m2 !! i = None → ([∗ map] k↦y1;y2 ∈ <[i:=x1]>m1; <[i:=x2]>m2, Φ k y1 y2) ⊣⊢ Φ i x1 x2 ∗ [∗ map] k↦y1;y2 ∈ m1;m2, Φ k y1 y2. Proof. intros Hm1 Hm2. rewrite /big_sepM2 -map_insert_zip_with. `````` Robbert Krebbers committed Apr 07, 2019 812 813 `````` rewrite big_sepM_insert; last by rewrite map_lookup_zip_with Hm1. `````` Dan Frumin committed Apr 07, 2019 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 `````` rewrite !persistent_and_affinely_sep_l /=. rewrite sep_assoc (sep_comm _ (Φ _ _ _)) -sep_assoc. repeat apply sep_proper=>//. apply affinely_proper, pure_proper. split. - intros H1 k. destruct (decide (i = k)) as [->|?]. + rewrite Hm1 Hm2 //. by split; intros ?; exfalso; eapply is_Some_None. + specialize (H1 k). revert H1. rewrite !lookup_insert_ne //. - intros H1 k. destruct (decide (i = k)) as [->|?]. + rewrite !lookup_insert. split; by econstructor. + rewrite !lookup_insert_ne //. Qed. Lemma big_sepM2_delete Φ m1 m2 i x1 x2 : m1 !! i = Some x1 → m2 !! i = Some x2 → ([∗ map] k↦x;y ∈ m1;m2, Φ k x y) ⊣⊢ Φ i x1 x2 ∗ [∗ map] k↦x;y ∈ delete i m1;delete i m2, Φ k x y. Proof. rewrite /big_sepM2 => Hx1 Hx2. rewrite !persistent_and_affinely_sep_l /=. rewrite sep_assoc (sep_comm (Φ _ _ _)) -sep_assoc. apply sep_proper. - apply affinely_proper, pure_proper. split. + intros Hm k. destruct (decide (i = k)) as [->|?]. { rewrite !lookup_delete. split; intros []%is_Some_None. } rewrite !lookup_delete_ne //. + intros Hm k. specialize (Hm k). revert Hm. destruct (decide (i = k)) as [->|?]. { intros _. rewrite Hx1 Hx2. split; eauto. } rewrite !lookup_delete_ne //. - rewrite -map_delete_zip_with. apply (big_sepM_delete (λ i xx, Φ i xx.1 xx.2) (map_zip m1 m2) i (x1,x2)). by rewrite map_lookup_zip_with Hx1 Hx2. Qed. Lemma big_sepM2_insert_acc Φ m1 m2 i x1 x2 : m1 !! i = Some x1 → m2 !! i = Some x2 → ([∗ map] k↦y1;y2 ∈ m1;m2, Φ k y1 y2) ⊢ Φ i x1 x2 ∗ (∀ x1' x2', Φ i x1' x2' -∗ ([∗ map] k↦y1;y2 ∈ <[i:=x1']>m1;<[i:=x2']>m2, Φ k y1 y2)). Proof. intros ??. rewrite {1}big_sepM2_delete //. apply sep_mono; [done|]. apply forall_intro=> x1'. apply forall_intro=> x2'. rewrite -(insert_delete m1) -(insert_delete m2) big_sepM2_insert ?lookup_delete //. by apply wand_intro_l. Qed. Lemma big_sepM2_insert_2 Φ m1 m2 i x1 x2 : TCOr (∀ x y, Affine (Φ i x y)) (Absorbing (Φ i x1 x2)) → Φ i x1 x2 -∗ ([∗ map] k↦y1;y2 ∈ m1;m2, Φ k y1 y2) -∗ ([∗ map] k↦y1;y2 ∈ <[i:=x1]>m1; <[i:=x2]>m2, Φ k y1 y2). Proof. intros Ha. rewrite /big_sepM2. assert (TCOr (∀ x, Affine (Φ i x.1 x.2)) (Absorbing (Φ i x1 x2))). { destruct Ha; try apply _. } apply wand_intro_r. rewrite !persistent_and_affinely_sep_l /=. rewrite (sep_comm (Φ _ _ _)) -sep_assoc. apply sep_mono. { apply affinely_mono, pure_mono. intros Hm k. destruct (decide (i = k)) as [->|]. - rewrite !lookup_insert. split; eauto. - rewrite !lookup_insert_ne //. } rewrite (big_sepM_insert_2 (λ k xx, Φ k xx.1 xx.2) (map_zip m1 m2) i (x1, x2)). rewrite map_insert_zip_with. apply wand_elim_r. Qed. Lemma big_sepM2_lookup_acc Φ m1 m2 i x1 x2 : m1 !! i = Some x1 → m2 !! i = Some x2 → ([∗ map] k↦y1;y2 ∈ m1;m2, Φ k y1 y2) ⊢ Φ i x1 x2 ∗ (Φ i x1 x2 -∗ ([∗ map] k↦y1;y2 ∈ m1;m2, Φ k y1 y2)). Proof. intros Hm1 Hm2. etrans; first apply big_sepM2_insert_acc=>//. apply sep_mono_r. rewrite (forall_elim x1) (forall_elim x2). rewrite !insert_id //. Qed. Lemma big_sepM2_lookup Φ m1 ``````