cmra_big_op.v 14.4 KB
 Robbert Krebbers committed Mar 21, 2016 1 ``````From iris.algebra Require Export cmra list. `````` Robbert Krebbers committed Sep 28, 2016 2 ``````From iris.prelude Require Import functions gmap. `````` Robbert Krebbers committed Feb 01, 2016 3 `````` `````` Robbert Krebbers committed Sep 28, 2016 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 ``````(** The operator [ [⋅] Ps ] folds [⋅] over the list [Ps]. This operator is not a quantifier, so it binds strongly. Apart from that, we define the following big operators with binders build in: - The operator [ [⋅ list] k ↦ x ∈ l, P ] folds over a list [l]. The binder [x] refers to each element at index [k]. - The operator [ [⋅ map] k ↦ x ∈ m, P ] folds over a map [m]. The binder [x] refers to each element at index [k]. - The operator [ [⋅ set] x ∈ X, P ] folds over a set [m]. The binder [x] refers to each element. Since these big operators are like quantifiers, they have the same precedence as [∀] and [∃]. *) (** * Big ops over lists *) (* This is the basic building block for other big ops *) Fixpoint big_op {M : ucmraT} (xs : list M) : M := `````` Robbert Krebbers committed Feb 01, 2016 22 `````` match xs with [] => ∅ | x :: xs => x ⋅ big_op xs end. `````` Robbert Krebbers committed May 27, 2016 23 24 ``````Arguments big_op _ !_ /. Instance: Params (@big_op) 1. `````` Robbert Krebbers committed Sep 28, 2016 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 ``````Notation "'[⋅]' xs" := (big_op xs) (at level 20) : C_scope. (** * Other big ops *) Definition big_opL {M : ucmraT} {A} (l : list A) (f : nat → A → M) : M := [⋅] (imap f l). Instance: Params (@big_opL) 2. Typeclasses Opaque big_opL. Notation "'[⋅' 'list' ] k ↦ x ∈ l , P" := (big_opL l (λ k x, P)) (at level 200, l at level 10, k, x at level 1, right associativity, format "[⋅ list ] k ↦ x ∈ l , P") : C_scope. Notation "'[⋅' 'list' ] x ∈ l , P" := (big_opL l (λ _ x, P)) (at level 200, l at level 10, x at level 1, right associativity, format "[⋅ list ] x ∈ l , P") : C_scope. Definition big_opM {M : ucmraT} `{Countable K} {A} (m : gmap K A) (f : K → A → M) : M := [⋅] (curry f <\$> map_to_list m). Instance: Params (@big_opM) 6. Typeclasses Opaque big_opM. Notation "'[⋅' 'map' ] k ↦ x ∈ m , P" := (big_opM m (λ k x, P)) (at level 200, m at level 10, k, x at level 1, right associativity, format "[⋅ map ] k ↦ x ∈ m , P") : C_scope. `````` Robbert Krebbers committed Sep 28, 2016 47 48 49 ``````Notation "'[⋅' 'map' ] x ∈ m , P" := (big_opM m (λ _ x, P)) (at level 200, m at level 10, x at level 1, right associativity, format "[⋅ map ] x ∈ m , P") : C_scope. `````` Robbert Krebbers committed Sep 28, 2016 50 51 52 53 54 55 56 57 `````` Definition big_opS {M : ucmraT} `{Countable A} (X : gset A) (f : A → M) : M := [⋅] (f <\$> elements X). Instance: Params (@big_opS) 5. Typeclasses Opaque big_opS. Notation "'[⋅' 'set' ] x ∈ X , P" := (big_opS X (λ x, P)) (at level 200, X at level 10, x at level 1, right associativity, format "[⋅ set ] x ∈ X , P") : C_scope. `````` Robbert Krebbers committed Feb 01, 2016 58 59 60 `````` (** * Properties about big ops *) Section big_op. `````` Robbert Krebbers committed Sep 28, 2016 61 62 ``````Context {M : ucmraT}. Implicit Types xs : list M. `````` Robbert Krebbers committed Feb 01, 2016 63 64 `````` (** * Big ops *) `````` Robbert Krebbers committed Sep 28, 2016 65 66 67 68 69 ``````Global Instance big_op_ne n : Proper (dist n ==> dist n) (@big_op M). Proof. by induction 1; simpl; repeat apply (_ : Proper (_ ==> _ ==> _) op). Qed. Global Instance big_op_proper : Proper ((≡) ==> (≡)) (@big_op M) := ne_proper _. Lemma big_op_nil : [⋅] (@nil M) = ∅. `````` Robbert Krebbers committed Feb 01, 2016 70 ``````Proof. done. Qed. `````` Robbert Krebbers committed Sep 28, 2016 71 ``````Lemma big_op_cons x xs : [⋅] (x :: xs) = x ⋅ [⋅] xs. `````` Robbert Krebbers committed Feb 01, 2016 72 ``````Proof. done. Qed. `````` Robbert Krebbers committed Sep 28, 2016 73 74 75 76 77 78 79 80 81 82 ``````Lemma big_op_app xs ys : [⋅] (xs ++ ys) ≡ [⋅] xs ⋅ [⋅] ys. Proof. induction xs as [|x xs IH]; simpl; first by rewrite ?left_id. by rewrite IH assoc. Qed. Lemma big_op_mono xs ys : Forall2 (≼) xs ys → [⋅] xs ≼ [⋅] ys. Proof. induction 1 as [|x y xs ys Hxy ? IH]; simpl; eauto using cmra_mono. Qed. Global Instance big_op_permutation : Proper ((≡ₚ) ==> (≡)) (@big_op M). `````` Robbert Krebbers committed Feb 01, 2016 83 84 ``````Proof. induction 1 as [|x xs1 xs2 ? IH|x y xs|xs1 xs2 xs3]; simpl; auto. `````` Robbert Krebbers committed Feb 17, 2016 85 86 `````` - by rewrite IH. - by rewrite !assoc (comm _ x). `````` Ralf Jung committed Feb 20, 2016 87 `````` - by trans (big_op xs2). `````` Robbert Krebbers committed Feb 01, 2016 88 ``````Qed. `````` Robbert Krebbers committed Sep 28, 2016 89 90 `````` Lemma big_op_contains xs ys : xs `contains` ys → [⋅] xs ≼ [⋅] ys. `````` Robbert Krebbers committed Feb 01, 2016 91 ``````Proof. `````` Robbert Krebbers committed Feb 17, 2016 92 93 `````` intros [xs' ->]%contains_Permutation. rewrite big_op_app; apply cmra_included_l. `````` Robbert Krebbers committed Feb 01, 2016 94 ``````Qed. `````` Robbert Krebbers committed Sep 28, 2016 95 96 `````` Lemma big_op_delete xs i x : xs !! i = Some x → x ⋅ [⋅] delete i xs ≡ [⋅] xs. `````` Robbert Krebbers committed Feb 01, 2016 97 98 ``````Proof. by intros; rewrite {2}(delete_Permutation xs i x). Qed. `````` Robbert Krebbers committed Sep 28, 2016 99 ``````Lemma big_sep_elem_of xs x : x ∈ xs → x ≼ [⋅] xs. `````` Robbert Krebbers committed Feb 01, 2016 100 ``````Proof. `````` Robbert Krebbers committed Sep 28, 2016 101 102 `````` intros [i ?]%elem_of_list_lookup. rewrite -big_op_delete //. apply cmra_included_l. `````` Robbert Krebbers committed Feb 01, 2016 103 ``````Qed. `````` Robbert Krebbers committed Sep 28, 2016 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 `````` (** ** Big ops over lists *) Section list. Context {A : Type}. Implicit Types l : list A. Implicit Types f g : nat → A → M. Lemma big_opL_mono f g l : (∀ k y, l !! k = Some y → f k y ≼ g k y) → ([⋅ list] k ↦ y ∈ l, f k y) ≼ [⋅ list] k ↦ y ∈ l, g k y. Proof. intros Hf. apply big_op_mono. revert f g Hf. induction l as [|x l IH]=> f g Hf; first constructor. rewrite !imap_cons; constructor; eauto. Qed. Lemma big_opL_proper f g l : (∀ k y, l !! k = Some y → f k y ≡ g k y) → ([⋅ list] k ↦ y ∈ l, f k y) ≡ ([⋅ list] k ↦ y ∈ l, g k y). Proof. intros Hf; apply big_op_proper. revert f g Hf. induction l as [|x l IH]=> f g Hf; first constructor. rewrite !imap_cons; constructor; eauto. Qed. Global Instance big_opL_ne l n : Proper (pointwise_relation _ (pointwise_relation _ (dist n)) ==> (dist n)) (big_opL (M:=M) l). Proof. intros f g Hf. apply big_op_ne. revert f g Hf. induction l as [|x l IH]=> f g Hf; first constructor. rewrite !imap_cons; constructor. by apply Hf. apply IH=> n'; apply Hf. Qed. Global Instance big_opL_proper' l : Proper (pointwise_relation _ (pointwise_relation _ (≡)) ==> (≡)) (big_opL (M:=M) l). Proof. intros f1 f2 Hf. by apply big_opL_proper; intros; last apply Hf. Qed. Global Instance big_opL_mono' l : Proper (pointwise_relation _ (pointwise_relation _ (≼)) ==> (≼)) (big_opL (M:=M) l). Proof. intros f1 f2 Hf. by apply big_opL_mono; intros; last apply Hf. Qed. Lemma big_opL_nil f : ([⋅ list] k↦y ∈ nil, f k y) = ∅. Proof. done. Qed. Lemma big_opL_cons f x l : ([⋅ list] k↦y ∈ x :: l, f k y) = f 0 x ⋅ [⋅ list] k↦y ∈ l, f (S k) y. Proof. by rewrite /big_opL imap_cons. Qed. Lemma big_opL_singleton f x : ([⋅ list] k↦y ∈ [x], f k y) ≡ f 0 x. Proof. by rewrite big_opL_cons big_opL_nil right_id. Qed. Lemma big_opL_app f l1 l2 : ([⋅ list] k↦y ∈ l1 ++ l2, f k y) ≡ ([⋅ list] k↦y ∈ l1, f k y) ⋅ ([⋅ list] k↦y ∈ l2, f (length l1 + k) y). Proof. by rewrite /big_opL imap_app big_op_app. Qed. Lemma big_opL_lookup f l i x : l !! i = Some x → f i x ≼ [⋅ list] k↦y ∈ l, f k y. Proof. intros. rewrite -(take_drop_middle l i x) // big_opL_app big_opL_cons. rewrite Nat.add_0_r take_length_le; eauto using lookup_lt_Some, Nat.lt_le_incl. eapply transitivity, cmra_included_r; eauto using cmra_included_l. Qed. Lemma big_opL_elem_of (f : A → M) l x : x ∈ l → f x ≼ [⋅ list] y ∈ l, f y. Proof. intros [i ?]%elem_of_list_lookup; eauto using (big_opL_lookup (λ _, f)). Qed. Lemma big_opL_fmap {B} (h : A → B) (f : nat → B → M) l : ([⋅ list] k↦y ∈ h <\$> l, f k y) ≡ ([⋅ list] k↦y ∈ l, f k (h y)). Proof. by rewrite /big_opL imap_fmap. Qed. Lemma big_opL_opL f g l : ([⋅ list] k↦x ∈ l, f k x ⋅ g k x) ≡ ([⋅ list] k↦x ∈ l, f k x) ⋅ ([⋅ list] k↦x ∈ l, g k x). Proof. revert f g; induction l as [|x l IH]=> f g. { by rewrite !big_opL_nil left_id. } rewrite !big_opL_cons IH. by rewrite -!assoc (assoc _ (g _ _)) [(g _ _ ⋅ _)]comm -!assoc. Qed. End list. (** ** Big ops over finite maps *) Section gmap. Context `{Countable K} {A : Type}. Implicit Types m : gmap K A. Implicit Types f g : K → A → M. Lemma big_opM_mono f g m1 m2 : m1 ⊆ m2 → (∀ k x, m2 !! k = Some x → f k x ≼ g k x) → ([⋅ map] k ↦ x ∈ m1, f k x) ≼ [⋅ map] k ↦ x ∈ m2, g k x. Proof. intros HX Hf. trans ([⋅ map] k↦x ∈ m2, f k x). - by apply big_op_contains, fmap_contains, map_to_list_contains. - apply big_op_mono, Forall2_fmap, Forall_Forall2. apply Forall_forall=> -[i x] ? /=. by apply Hf, elem_of_map_to_list. Qed. Lemma big_opM_proper f g m : (∀ k x, m !! k = Some x → f k x ≡ g k x) → ([⋅ map] k ↦ x ∈ m, f k x) ≡ ([⋅ map] k ↦ x ∈ m, g k x). Proof. intros Hf. apply big_op_proper, equiv_Forall2, Forall2_fmap, Forall_Forall2. apply Forall_forall=> -[i x] ? /=. by apply Hf, elem_of_map_to_list. Qed. Global Instance big_opM_ne m n : Proper (pointwise_relation _ (pointwise_relation _ (dist n)) ==> (dist n)) (big_opM (M:=M) m). Proof. intros f1 f2 Hf. apply big_op_ne, Forall2_fmap. apply Forall_Forall2, Forall_true=> -[i x]; apply Hf. Qed. Global Instance big_opM_proper' m : Proper (pointwise_relation _ (pointwise_relation _ (≡)) ==> (≡)) (big_opM (M:=M) m). Proof. intros f1 f2 Hf. by apply big_opM_proper; intros; last apply Hf. Qed. Global Instance big_opM_mono' m : Proper (pointwise_relation _ (pointwise_relation _ (≼)) ==> (≼)) (big_opM (M:=M) m). Proof. intros f1 f2 Hf. by apply big_opM_mono; intros; last apply Hf. Qed. Lemma big_opM_empty f : ([⋅ map] k↦x ∈ ∅, f k x) = ∅. Proof. by rewrite /big_opM map_to_list_empty. Qed. Lemma big_opM_insert f m i x : m !! i = None → ([⋅ map] k↦y ∈ <[i:=x]> m, f k y) ≡ f i x ⋅ [⋅ map] k↦y ∈ m, f k y. Proof. intros ?. by rewrite /big_opM map_to_list_insert. Qed. Lemma big_opM_delete f m i x : m !! i = Some x → ([⋅ map] k↦y ∈ m, f k y) ≡ f i x ⋅ [⋅ map] k↦y ∈ delete i m, f k y. Proof. intros. rewrite -big_opM_insert ?lookup_delete //. by rewrite insert_delete insert_id. Qed. Lemma big_opM_lookup f m i x : m !! i = Some x → f i x ≼ [⋅ map] k↦y ∈ m, f k y. Proof. intros. rewrite big_opM_delete //. apply cmra_included_l. Qed. Lemma big_opM_singleton f i x : ([⋅ map] k↦y ∈ {[i:=x]}, f k y) ≡ f i x. Proof. rewrite -insert_empty big_opM_insert/=; last auto using lookup_empty. by rewrite big_opM_empty right_id. Qed. Lemma big_opM_fmap {B} (h : A → B) (f : K → B → M) m : ([⋅ map] k↦y ∈ h <\$> m, f k y) ≡ ([⋅ map] k↦y ∈ m, f k (h y)). Proof. rewrite /big_opM map_to_list_fmap -list_fmap_compose. f_equiv; apply reflexive_eq, list_fmap_ext. by intros []. done. Qed. Lemma big_opM_insert_override (f : K → M) m i x y : m !! i = Some x → ([⋅ map] k↦_ ∈ <[i:=y]> m, f k) ≡ ([⋅ map] k↦_ ∈ m, f k). Proof. intros. rewrite -insert_delete big_opM_insert ?lookup_delete //. by rewrite -big_opM_delete. Qed. Lemma big_opM_fn_insert {B} (g : K → A → B → M) (f : K → B) m i (x : A) b : m !! i = None → ([⋅ map] k↦y ∈ <[i:=x]> m, g k y (<[i:=b]> f k)) ≡ (g i x b ⋅ [⋅ map] k↦y ∈ m, g k y (f k)). Proof. intros. rewrite big_opM_insert // fn_lookup_insert. apply cmra_op_proper', big_opM_proper; auto=> k y ?. by rewrite fn_lookup_insert_ne; last set_solver. Qed. Lemma big_opM_fn_insert' (f : K → M) m i x P : m !! i = None → ([⋅ map] k↦y ∈ <[i:=x]> m, <[i:=P]> f k) ≡ (P ⋅ [⋅ map] k↦y ∈ m, f k). Proof. apply (big_opM_fn_insert (λ _ _, id)). Qed. Lemma big_opM_opM f g m : ([⋅ map] k↦x ∈ m, f k x ⋅ g k x) ≡ ([⋅ map] k↦x ∈ m, f k x) ⋅ ([⋅ map] k↦x ∈ m, g k x). Proof. rewrite /big_opM. induction (map_to_list m) as [|[i x] l IH]; csimpl; rewrite ?right_id //. by rewrite IH -!assoc (assoc _ (g _ _)) [(g _ _ ⋅ _)]comm -!assoc. Qed. End gmap. (** ** Big ops over finite sets *) Section gset. Context `{Countable A}. Implicit Types X : gset A. Implicit Types f : A → M. Lemma big_opS_mono f g X Y : X ⊆ Y → (∀ x, x ∈ Y → f x ≼ g x) → ([⋅ set] x ∈ X, f x) ≼ [⋅ set] x ∈ Y, g x. Proof. intros HX Hf. trans ([⋅ set] x ∈ Y, f x). - by apply big_op_contains, fmap_contains, elements_contains. - apply big_op_mono, Forall2_fmap, Forall_Forall2. apply Forall_forall=> x ? /=. by apply Hf, elem_of_elements. Qed. Lemma big_opS_proper f g X Y : X ≡ Y → (∀ x, x ∈ X → x ∈ Y → f x ≡ g x) → ([⋅ set] x ∈ X, f x) ≡ ([⋅ set] x ∈ Y, g x). Proof. intros HX Hf. trans ([⋅ set] x ∈ Y, f x). - apply big_op_permutation. by rewrite HX. - apply big_op_proper, equiv_Forall2, Forall2_fmap, Forall_Forall2. apply Forall_forall=> x ? /=. apply Hf; first rewrite HX; by apply elem_of_elements. Qed. Lemma big_opS_ne X n : Proper (pointwise_relation _ (dist n) ==> dist n) (big_opS (M:=M) X). Proof. intros f1 f2 Hf. apply big_op_ne, Forall2_fmap. apply Forall_Forall2, Forall_true=> x; apply Hf. Qed. Lemma big_opS_proper' X : Proper (pointwise_relation _ (≡) ==> (≡)) (big_opS (M:=M) X). Proof. intros f1 f2 Hf. apply big_opS_proper; naive_solver. Qed. Lemma big_opS_mono' X : Proper (pointwise_relation _ (≼) ==> (≼)) (big_opS (M:=M) X). Proof. intros f1 f2 Hf. apply big_opS_mono; naive_solver. Qed. Lemma big_opS_empty f : ([⋅ set] x ∈ ∅, f x) = ∅. Proof. by rewrite /big_opS elements_empty. Qed. Lemma big_opS_insert f X x : x ∉ X → ([⋅ set] y ∈ {[ x ]} ∪ X, f y) ≡ (f x ⋅ [⋅ set] y ∈ X, f y). Proof. intros. by rewrite /big_opS elements_union_singleton. Qed. Lemma big_opS_fn_insert {B} (f : A → B → M) h X x b : x ∉ X → ([⋅ set] y ∈ {[ x ]} ∪ X, f y (<[x:=b]> h y)) ≡ (f x b ⋅ [⋅ set] y ∈ X, f y (h y)). Proof. intros. rewrite big_opS_insert // fn_lookup_insert. apply cmra_op_proper', big_opS_proper; auto=> y ??. by rewrite fn_lookup_insert_ne; last set_solver. Qed. Lemma big_opS_fn_insert' f X x P : x ∉ X → ([⋅ set] y ∈ {[ x ]} ∪ X, <[x:=P]> f y) ≡ (P ⋅ [⋅ set] y ∈ X, f y). Proof. apply (big_opS_fn_insert (λ y, id)). Qed. Lemma big_opS_delete f X x : x ∈ X → ([⋅ set] y ∈ X, f y) ≡ f x ⋅ [⋅ set] y ∈ X ∖ {[ x ]}, f y. Proof. intros. rewrite -big_opS_insert; last set_solver. by rewrite -union_difference_L; last set_solver. Qed. Lemma big_opS_elem_of f X x : x ∈ X → f x ≼ [⋅ set] y ∈ X, f y. Proof. intros. rewrite big_opS_delete //. apply cmra_included_l. Qed. Lemma big_opS_singleton f x : ([⋅ set] y ∈ {[ x ]}, f y) ≡ f x. Proof. intros. by rewrite /big_opS elements_singleton /= right_id. Qed. Lemma big_opS_opS f g X : ([⋅ set] y ∈ X, f y ⋅ g y) ≡ ([⋅ set] y ∈ X, f y) ⋅ ([⋅ set] y ∈ X, g y). Proof. rewrite /big_opS. induction (elements X) as [|x l IH]; csimpl; first by rewrite ?right_id. by rewrite IH -!assoc (assoc _ (g _)) [(g _ ⋅ _)]comm -!assoc. Qed. End gset. `````` Robbert Krebbers committed Feb 01, 2016 373 ``End big_op.``