cmra_big_op.v 25.7 KB
 Robbert Krebbers committed Mar 21, 2016 1 ``````From iris.algebra Require Export cmra list. `````` Robbert Krebbers committed Nov 15, 2016 2 ``````From iris.prelude Require Import functions gmap gmultiset. `````` 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 `````` `````` Robbert Krebbers committed Nov 15, 2016 59 60 61 62 63 64 65 66 ``````Definition big_opMS {M : ucmraT} `{Countable A} (X : gmultiset A) (f : A → M) : M := [⋅] (f <\$> elements X). Instance: Params (@big_opMS) 5. Typeclasses Opaque big_opMS. Notation "'[⋅' 'mset' ] x ∈ X , P" := (big_opMS X (λ x, P)) (at level 200, X at level 10, x at level 1, right associativity, format "[⋅ 'mset' ] x ∈ X , P") : C_scope. `````` Robbert Krebbers committed Feb 01, 2016 67 68 ``````(** * Properties about big ops *) Section big_op. `````` Robbert Krebbers committed Sep 28, 2016 69 70 ``````Context {M : ucmraT}. Implicit Types xs : list M. `````` Robbert Krebbers committed Feb 01, 2016 71 72 `````` (** * Big ops *) `````` Robbert Krebbers committed Sep 28, 2016 73 74 75 76 77 ``````Lemma big_op_Forall2 R : Reflexive R → Proper (R ==> R ==> R) (@op M _) → Proper (Forall2 R ==> R) (@big_op M). Proof. rewrite /Proper /respectful. induction 3; eauto. Qed. `````` Robbert Krebbers committed Sep 28, 2016 78 ``````Global Instance big_op_ne n : Proper (dist n ==> dist n) (@big_op M). `````` Robbert Krebbers committed Sep 28, 2016 79 ``````Proof. apply big_op_Forall2; apply _. Qed. `````` Robbert Krebbers committed Sep 28, 2016 80 81 82 ``````Global Instance big_op_proper : Proper ((≡) ==> (≡)) (@big_op M) := ne_proper _. Lemma big_op_nil : [⋅] (@nil M) = ∅. `````` Robbert Krebbers committed Feb 01, 2016 83 ``````Proof. done. Qed. `````` Robbert Krebbers committed Sep 28, 2016 84 ``````Lemma big_op_cons x xs : [⋅] (x :: xs) = x ⋅ [⋅] xs. `````` Robbert Krebbers committed Feb 01, 2016 85 ``````Proof. done. Qed. `````` Robbert Krebbers committed Sep 28, 2016 86 87 88 89 90 91 92 93 94 95 ``````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 96 97 ``````Proof. induction 1 as [|x xs1 xs2 ? IH|x y xs|xs1 xs2 xs3]; simpl; auto. `````` Robbert Krebbers committed Feb 17, 2016 98 99 `````` - by rewrite IH. - by rewrite !assoc (comm _ x). `````` Ralf Jung committed Feb 20, 2016 100 `````` - by trans (big_op xs2). `````` Robbert Krebbers committed Feb 01, 2016 101 ``````Qed. `````` Robbert Krebbers committed Sep 28, 2016 102 103 `````` Lemma big_op_contains xs ys : xs `contains` ys → [⋅] xs ≼ [⋅] ys. `````` Robbert Krebbers committed Feb 01, 2016 104 ``````Proof. `````` Robbert Krebbers committed Feb 17, 2016 105 106 `````` intros [xs' ->]%contains_Permutation. rewrite big_op_app; apply cmra_included_l. `````` Robbert Krebbers committed Feb 01, 2016 107 ``````Qed. `````` Robbert Krebbers committed Sep 28, 2016 108 109 `````` Lemma big_op_delete xs i x : xs !! i = Some x → x ⋅ [⋅] delete i xs ≡ [⋅] xs. `````` Robbert Krebbers committed Feb 01, 2016 110 111 ``````Proof. by intros; rewrite {2}(delete_Permutation xs i x). Qed. `````` Robbert Krebbers committed Sep 28, 2016 112 ``````Lemma big_sep_elem_of xs x : x ∈ xs → x ≼ [⋅] xs. `````` Robbert Krebbers committed Feb 01, 2016 113 ``````Proof. `````` Robbert Krebbers committed Sep 28, 2016 114 115 `````` intros [i ?]%elem_of_list_lookup. rewrite -big_op_delete //. apply cmra_included_l. `````` Robbert Krebbers committed Feb 01, 2016 116 ``````Qed. `````` Robbert Krebbers committed Sep 28, 2016 117 118 119 120 121 122 123 `````` (** ** Big ops over lists *) Section list. Context {A : Type}. Implicit Types l : list A. Implicit Types f g : nat → A → M. `````` Robbert Krebbers committed Sep 28, 2016 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 `````` 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_forall R f g l : Reflexive R → Proper (R ==> R ==> R) (@op M _) → (∀ k y, l !! k = Some y → R (f k y) (g k y)) → R ([⋅ list] k ↦ y ∈ l, f k y) ([⋅ list] k ↦ y ∈ l, g k y). Proof. intros ? Hop. revert f g. induction l as [|x l IH]=> f g Hf; [done|]. rewrite !big_opL_cons. apply Hop; eauto. Qed. `````` Robbert Krebbers committed Sep 28, 2016 145 146 147 `````` 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. `````` Robbert Krebbers committed Sep 28, 2016 148 `````` Proof. apply big_opL_forall; apply _. Qed. `````` Robbert Krebbers committed Oct 03, 2016 149 150 151 152 `````` Lemma big_opL_ext 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. apply big_opL_forall; apply _. Qed. `````` Robbert Krebbers committed Sep 28, 2016 153 154 155 `````` 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). `````` Robbert Krebbers committed Sep 28, 2016 156 `````` Proof. apply big_opL_forall; apply _. Qed. `````` Robbert Krebbers committed Sep 28, 2016 157 158 159 160 `````` Global Instance big_opL_ne l n : Proper (pointwise_relation _ (pointwise_relation _ (dist n)) ==> (dist n)) (big_opL (M:=M) l). `````` Robbert Krebbers committed Sep 28, 2016 161 `````` Proof. intros f g Hf. apply big_opL_forall; apply _ || intros; apply Hf. Qed. `````` Robbert Krebbers committed Sep 28, 2016 162 163 164 `````` Global Instance big_opL_proper' l : Proper (pointwise_relation _ (pointwise_relation _ (≡)) ==> (≡)) (big_opL (M:=M) l). `````` Robbert Krebbers committed Sep 28, 2016 165 `````` Proof. intros f g Hf. apply big_opL_forall; apply _ || intros; apply Hf. Qed. `````` Robbert Krebbers committed Sep 28, 2016 166 167 168 `````` Global Instance big_opL_mono' l : Proper (pointwise_relation _ (pointwise_relation _ (≼)) ==> (≼)) (big_opL (M:=M) l). `````` Robbert Krebbers committed Sep 28, 2016 169 `````` Proof. intros f g Hf. apply big_opL_forall; apply _ || intros; apply Hf. Qed. `````` Robbert Krebbers committed Sep 28, 2016 170 `````` `````` Robbert Krebbers committed Oct 03, 2016 171 172 173 174 175 176 177 `````` Lemma big_opL_consZ_l (f : Z → A → M) x l : ([⋅ list] k↦y ∈ x :: l, f k y) = f 0 x ⋅ [⋅ list] k↦y ∈ l, f (1 + k)%Z y. Proof. rewrite big_opL_cons. auto using big_opL_ext with f_equal lia. Qed. Lemma big_opL_consZ_r (f : Z → A → M) x l : ([⋅ list] k↦y ∈ x :: l, f k y) = f 0 x ⋅ [⋅ list] k↦y ∈ l, f (k + 1)%Z y. Proof. rewrite big_opL_cons. auto using big_opL_ext with f_equal lia. Qed. `````` Robbert Krebbers committed Sep 28, 2016 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 `````` 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. `````` Robbert Krebbers committed Sep 28, 2016 212 213 214 215 216 217 218 219 220 `````` Lemma big_opM_forall R f g m : Reflexive R → Proper (R ==> R ==> R) (@op M _) → (∀ k x, m !! k = Some x → R (f k x) (g k x)) → R ([⋅ map] k ↦ x ∈ m, f k x) ([⋅ map] k ↦ x ∈ m, g k x). Proof. intros ?? Hf. apply (big_op_Forall2 R _ _), Forall2_fmap, Forall_Forall2. apply Forall_forall=> -[i x] ? /=. by apply Hf, elem_of_map_to_list. Qed. `````` Robbert Krebbers committed Sep 28, 2016 221 222 223 224 `````` 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. `````` Robbert Krebbers committed Sep 28, 2016 225 `````` intros Hm Hf. trans ([⋅ map] k↦x ∈ m2, f k x). `````` Robbert Krebbers committed Sep 28, 2016 226 `````` - by apply big_op_contains, fmap_contains, map_to_list_contains. `````` Robbert Krebbers committed Sep 28, 2016 227 `````` - apply big_opM_forall; apply _ || auto. `````` Robbert Krebbers committed Sep 28, 2016 228 `````` Qed. `````` Robbert Krebbers committed Oct 03, 2016 229 230 231 232 `````` Lemma big_opM_ext 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. apply big_opM_forall; apply _. Qed. `````` Robbert Krebbers committed Sep 28, 2016 233 234 235 `````` 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). `````` Robbert Krebbers committed Sep 28, 2016 236 `````` Proof. apply big_opM_forall; apply _. Qed. `````` Robbert Krebbers committed Sep 28, 2016 237 238 239 240 `````` Global Instance big_opM_ne m n : Proper (pointwise_relation _ (pointwise_relation _ (dist n)) ==> (dist n)) (big_opM (M:=M) m). `````` Robbert Krebbers committed Sep 28, 2016 241 `````` Proof. intros f g Hf. apply big_opM_forall; apply _ || intros; apply Hf. Qed. `````` Robbert Krebbers committed Sep 28, 2016 242 243 244 `````` Global Instance big_opM_proper' m : Proper (pointwise_relation _ (pointwise_relation _ (≡)) ==> (≡)) (big_opM (M:=M) m). `````` Robbert Krebbers committed Sep 28, 2016 245 `````` Proof. intros f g Hf. apply big_opM_forall; apply _ || intros; apply Hf. Qed. `````` Robbert Krebbers committed Sep 28, 2016 246 247 248 `````` Global Instance big_opM_mono' m : Proper (pointwise_relation _ (pointwise_relation _ (≼)) ==> (≼)) (big_opM (M:=M) m). `````` Robbert Krebbers committed Sep 28, 2016 249 `````` Proof. intros f g Hf. apply big_opM_forall; apply _ || intros; apply Hf. Qed. `````` Robbert Krebbers committed Sep 28, 2016 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 `````` 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. `````` Robbert Krebbers committed Sep 28, 2016 323 324 325 326 327 328 329 330 331 `````` Lemma big_opS_forall R f g X : Reflexive R → Proper (R ==> R ==> R) (@op M _) → (∀ x, x ∈ X → R (f x) (g x)) → R ([⋅ set] x ∈ X, f x) ([⋅ set] x ∈ X, g x). Proof. intros ?? Hf. apply (big_op_Forall2 R _ _), Forall2_fmap, Forall_Forall2. apply Forall_forall=> x ? /=. by apply Hf, elem_of_elements. Qed. `````` Robbert Krebbers committed Sep 28, 2016 332 333 334 335 336 337 `````` 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. `````` Robbert Krebbers committed Sep 28, 2016 338 `````` - apply big_opS_forall; apply _ || auto. `````` Robbert Krebbers committed Sep 28, 2016 339 `````` Qed. `````` Robbert Krebbers committed Oct 03, 2016 340 341 342 343 344 345 346 347 `````` Lemma big_opS_ext f g X : (∀ x, x ∈ X → f x = g x) → ([⋅ set] x ∈ X, f x) = ([⋅ set] x ∈ X, g x). Proof. apply big_opS_forall; apply _. Qed. Lemma big_opS_proper f g X : (∀ x, x ∈ X → f x ≡ g x) → ([⋅ set] x ∈ X, f x) ≡ ([⋅ set] x ∈ X, g x). Proof. apply big_opS_forall; apply _. Qed. `````` Robbert Krebbers committed Sep 28, 2016 348 349 350 `````` Lemma big_opS_ne X n : Proper (pointwise_relation _ (dist n) ==> dist n) (big_opS (M:=M) X). `````` Robbert Krebbers committed Sep 28, 2016 351 `````` Proof. intros f g Hf. apply big_opS_forall; apply _ || intros; apply Hf. Qed. `````` Robbert Krebbers committed Sep 28, 2016 352 353 `````` Lemma big_opS_proper' X : Proper (pointwise_relation _ (≡) ==> (≡)) (big_opS (M:=M) X). `````` Robbert Krebbers committed Sep 28, 2016 354 `````` Proof. intros f g Hf. apply big_opS_forall; apply _ || intros; apply Hf. Qed. `````` Robbert Krebbers committed Sep 28, 2016 355 356 `````` Lemma big_opS_mono' X : Proper (pointwise_relation _ (≼) ==> (≼)) (big_opS (M:=M) X). `````` Robbert Krebbers committed Sep 28, 2016 357 `````` Proof. intros f g Hf. apply big_opS_forall; apply _ || intros; apply Hf. Qed. `````` Robbert Krebbers committed Sep 28, 2016 358 359 360 361 362 363 364 365 366 367 368 369 370 `````` 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. `````` Robbert Krebbers committed Oct 03, 2016 371 `````` apply cmra_op_proper', big_opS_proper; auto=> y ?. `````` Robbert Krebbers committed Sep 28, 2016 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 `````` 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 Nov 15, 2016 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 `````` (** ** Big ops over finite msets *) Section gmultiset. Context `{Countable A}. Implicit Types X : gmultiset A. Implicit Types f : A → M. Lemma big_opMS_forall R f g X : Reflexive R → Proper (R ==> R ==> R) (@op M _) → (∀ x, x ∈ X → R (f x) (g x)) → R ([⋅ mset] x ∈ X, f x) ([⋅ mset] x ∈ X, g x). Proof. intros ?? Hf. apply (big_op_Forall2 R _ _), Forall2_fmap, Forall_Forall2. apply Forall_forall=> x ? /=. by apply Hf, gmultiset_elem_of_elements. Qed. Lemma big_opMS_mono f g X Y : X ⊆ Y → (∀ x, x ∈ Y → f x ≼ g x) → ([⋅ mset] x ∈ X, f x) ≼ [⋅ mset] x ∈ Y, g x. Proof. intros HX Hf. trans ([⋅ mset] x ∈ Y, f x). - by apply big_op_contains, fmap_contains, gmultiset_elements_contains. - apply big_opMS_forall; apply _ || auto. Qed. Lemma big_opMS_ext f g X : (∀ x, x ∈ X → f x = g x) → ([⋅ mset] x ∈ X, f x) = ([⋅ mset] x ∈ X, g x). Proof. apply big_opMS_forall; apply _. Qed. Lemma big_opMS_proper f g X : (∀ x, x ∈ X → f x ≡ g x) → ([⋅ mset] x ∈ X, f x) ≡ ([⋅ mset] x ∈ X, g x). Proof. apply big_opMS_forall; apply _. Qed. Lemma big_opMS_ne X n : Proper (pointwise_relation _ (dist n) ==> dist n) (big_opMS (M:=M) X). Proof. intros f g Hf. apply big_opMS_forall; apply _ || intros; apply Hf. Qed. Lemma big_opMS_proper' X : Proper (pointwise_relation _ (≡) ==> (≡)) (big_opMS (M:=M) X). Proof. intros f g Hf. apply big_opMS_forall; apply _ || intros; apply Hf. Qed. Lemma big_opMS_mono' X : Proper (pointwise_relation _ (≼) ==> (≼)) (big_opMS (M:=M) X). Proof. intros f g Hf. apply big_opMS_forall; apply _ || intros; apply Hf. Qed. Lemma big_opMS_empty f : ([⋅ mset] x ∈ ∅, f x) = ∅. Proof. by rewrite /big_opMS gmultiset_elements_empty. Qed. Lemma big_opMS_union f X Y : ([⋅ mset] y ∈ X ∪ Y, f y) ≡ ([⋅ mset] y ∈ X, f y) ⋅ [⋅ mset] y ∈ Y, f y. Proof. by rewrite /big_opMS gmultiset_elements_union fmap_app big_op_app. Qed. Lemma big_opMS_singleton f x : ([⋅ mset] y ∈ {[ x ]}, f y) ≡ f x. Proof. intros. by rewrite /big_opMS gmultiset_elements_singleton /= right_id. Qed. `````` Robbert Krebbers committed Nov 18, 2016 455 456 457 `````` Lemma big_opMS_delete f X x : x ∈ X → ([⋅ mset] y ∈ X, f y) ≡ f x ⋅ [⋅ mset] y ∈ X ∖ {[ x ]}, f y. Proof. `````` Robbert Krebbers committed Nov 19, 2016 458 459 `````` intros. rewrite -big_opMS_singleton -big_opMS_union. by rewrite -gmultiset_union_difference'. `````` Robbert Krebbers committed Nov 18, 2016 460 461 462 463 464 `````` Qed. Lemma big_opMS_elem_of f X x : x ∈ X → f x ≼ [⋅ mset] y ∈ X, f y. Proof. intros. rewrite big_opMS_delete //. apply cmra_included_l. Qed. `````` Robbert Krebbers committed Nov 19, 2016 465 `````` Lemma big_opMS_opMS f g X : `````` Robbert Krebbers committed Nov 15, 2016 466 467 468 469 470 471 472 `````` ([⋅ mset] y ∈ X, f y ⋅ g y) ≡ ([⋅ mset] y ∈ X, f y) ⋅ ([⋅ mset] y ∈ X, g y). Proof. rewrite /big_opMS. induction (elements X) as [|x l IH]; csimpl; first by rewrite ?right_id. by rewrite IH -!assoc (assoc _ (g _)) [(g _ ⋅ _)]comm -!assoc. Qed. End gmultiset. `````` Robbert Krebbers committed Feb 01, 2016 473 ``````End big_op. `````` Robbert Krebbers committed Sep 28, 2016 474 `````` `````` Robbert Krebbers committed Oct 02, 2016 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 ``````(** Option *) Lemma big_opL_None {M : cmraT} {A} (f : nat → A → option M) l : ([⋅ list] k↦x ∈ l, f k x) = None ↔ ∀ k x, l !! k = Some x → f k x = None. Proof. revert f. induction l as [|x l IH]=> f //=. rewrite big_opL_cons op_None IH. split. - intros [??] [|k] y ?; naive_solver. - intros Hl. split. by apply (Hl 0). intros k. apply (Hl (S k)). Qed. Lemma big_opM_None {M : cmraT} `{Countable K} {A} (f : K → A → option M) m : ([⋅ map] k↦x ∈ m, f k x) = None ↔ ∀ k x, m !! k = Some x → f k x = None. Proof. induction m as [|i x m ? IH] using map_ind=> //=. rewrite -equiv_None big_opM_insert // equiv_None op_None IH. split. { intros [??] k y. rewrite lookup_insert_Some; naive_solver. } intros Hm; split. - apply (Hm i). by simplify_map_eq. - intros k y ?. apply (Hm k). by simplify_map_eq. Qed. Lemma big_opS_None {M : cmraT} `{Countable A} (f : A → option M) X : ([⋅ set] x ∈ X, f x) = None ↔ ∀ x, x ∈ X → f x = None. Proof. induction X as [|x X ? IH] using collection_ind_L; [done|]. rewrite -equiv_None big_opS_insert // equiv_None op_None IH. set_solver. Qed. `````` Robbert Krebbers committed Nov 19, 2016 500 501 502 503 504 505 506 507 ``````Lemma big_opMS_None {M : cmraT} `{Countable A} (f : A → option M) X : ([⋅ mset] x ∈ X, f x) = None ↔ ∀ x, x ∈ X → f x = None. Proof. induction X as [|x X IH] using gmultiset_ind. { rewrite big_opMS_empty. set_solver. } rewrite -equiv_None big_opMS_union big_opMS_singleton equiv_None op_None IH. set_solver. Qed. `````` Robbert Krebbers committed Oct 02, 2016 508 509 `````` (** Commuting with respect to homomorphisms *) `````` Robbert Krebbers committed Sep 28, 2016 510 ``````Lemma big_opL_commute {M1 M2 : ucmraT} {A} (h : M1 → M2) `````` Robbert Krebbers committed Sep 28, 2016 511 `````` `{!UCMRAHomomorphism h} (f : nat → A → M1) l : `````` Robbert Krebbers committed Sep 28, 2016 512 513 `````` h ([⋅ list] k↦x ∈ l, f k x) ≡ ([⋅ list] k↦x ∈ l, h (f k x)). Proof. `````` Robbert Krebbers committed Sep 28, 2016 514 515 516 `````` revert f. induction l as [|x l IH]=> f. - by rewrite !big_opL_nil ucmra_homomorphism_unit. - by rewrite !big_opL_cons cmra_homomorphism -IH. `````` Robbert Krebbers committed Sep 28, 2016 517 518 ``````Qed. Lemma big_opL_commute1 {M1 M2 : ucmraT} {A} (h : M1 → M2) `````` Robbert Krebbers committed Sep 28, 2016 519 520 `````` `{!CMRAHomomorphism h} (f : nat → A → M1) l : l ≠ [] → h ([⋅ list] k↦x ∈ l, f k x) ≡ ([⋅ list] k↦x ∈ l, h (f k x)). `````` Robbert Krebbers committed Sep 28, 2016 521 ``````Proof. `````` Robbert Krebbers committed Sep 28, 2016 522 `````` intros ?. revert f. induction l as [|x [|x' l'] IH]=> f //. `````` Robbert Krebbers committed Sep 28, 2016 523 `````` - by rewrite !big_opL_singleton. `````` Robbert Krebbers committed Sep 28, 2016 524 `````` - by rewrite !(big_opL_cons _ x) cmra_homomorphism -IH. `````` Robbert Krebbers committed Sep 28, 2016 525 526 527 ``````Qed. Lemma big_opM_commute {M1 M2 : ucmraT} `{Countable K} {A} (h : M1 → M2) `````` Robbert Krebbers committed Sep 28, 2016 528 `````` `{!UCMRAHomomorphism h} (f : K → A → M1) m : `````` Robbert Krebbers committed Sep 28, 2016 529 530 `````` h ([⋅ map] k↦x ∈ m, f k x) ≡ ([⋅ map] k↦x ∈ m, h (f k x)). Proof. `````` Robbert Krebbers committed Sep 28, 2016 531 532 533 `````` intros. induction m as [|i x m ? IH] using map_ind. - by rewrite !big_opM_empty ucmra_homomorphism_unit. - by rewrite !big_opM_insert // cmra_homomorphism -IH. `````` Robbert Krebbers committed Sep 28, 2016 534 535 ``````Qed. Lemma big_opM_commute1 {M1 M2 : ucmraT} `{Countable K} {A} (h : M1 → M2) `````` Robbert Krebbers committed Sep 28, 2016 536 537 `````` `{!CMRAHomomorphism h} (f : K → A → M1) m : m ≠ ∅ → h ([⋅ map] k↦x ∈ m, f k x) ≡ ([⋅ map] k↦x ∈ m, h (f k x)). `````` Robbert Krebbers committed Sep 28, 2016 538 ``````Proof. `````` Robbert Krebbers committed Sep 28, 2016 539 540 541 542 `````` intros. induction m as [|i x m ? IH] using map_ind; [done|]. destruct (decide (m = ∅)) as [->|]. - by rewrite !big_opM_insert // !big_opM_empty !right_id. - by rewrite !big_opM_insert // cmra_homomorphism -IH //. `````` Robbert Krebbers committed Sep 28, 2016 543 544 ``````Qed. `````` Robbert Krebbers committed Sep 28, 2016 545 546 ``````Lemma big_opS_commute {M1 M2 : ucmraT} `{Countable A} (h : M1 → M2) `{!UCMRAHomomorphism h} (f : A → M1) X : `````` Robbert Krebbers committed Sep 28, 2016 547 548 `````` h ([⋅ set] x ∈ X, f x) ≡ ([⋅ set] x ∈ X, h (f x)). Proof. `````` Robbert Krebbers committed Sep 28, 2016 549 550 551 `````` intros. induction X as [|x X ? IH] using collection_ind_L. - by rewrite !big_opS_empty ucmra_homomorphism_unit. - by rewrite !big_opS_insert // cmra_homomorphism -IH. `````` Robbert Krebbers committed Sep 28, 2016 552 ``````Qed. `````` Robbert Krebbers committed Sep 28, 2016 553 554 555 ``````Lemma big_opS_commute1 {M1 M2 : ucmraT} `{Countable A} (h : M1 → M2) `{!CMRAHomomorphism h} (f : A → M1) X : X ≠ ∅ → h ([⋅ set] x ∈ X, f x) ≡ ([⋅ set] x ∈ X, h (f x)). `````` Robbert Krebbers committed Sep 28, 2016 556 ``````Proof. `````` Robbert Krebbers committed Sep 28, 2016 557 558 559 560 `````` intros. induction X as [|x X ? IH] using collection_ind_L; [done|]. destruct (decide (X = ∅)) as [->|]. - by rewrite !big_opS_insert // !big_opS_empty !right_id. - by rewrite !big_opS_insert // cmra_homomorphism -IH //. `````` Robbert Krebbers committed Sep 28, 2016 561 ``````Qed. `````` Robbert Krebbers committed Sep 28, 2016 562 `````` `````` Robbert Krebbers committed Nov 19, 2016 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 ``````Lemma big_opMS_commute {M1 M2 : ucmraT} `{Countable A} (h : M1 → M2) `{!UCMRAHomomorphism h} (f : A → M1) X : h ([⋅ mset] x ∈ X, f x) ≡ ([⋅ mset] x ∈ X, h (f x)). Proof. intros. induction X as [|x X IH] using gmultiset_ind. - by rewrite !big_opMS_empty ucmra_homomorphism_unit. - by rewrite !big_opMS_union !big_opMS_singleton cmra_homomorphism -IH. Qed. Lemma big_opMS_commute1 {M1 M2 : ucmraT} `{Countable A} (h : M1 → M2) `{!CMRAHomomorphism h} (f : A → M1) X : X ≠ ∅ → h ([⋅ mset] x ∈ X, f x) ≡ ([⋅ mset] x ∈ X, h (f x)). Proof. intros. induction X as [|x X IH] using gmultiset_ind; [done|]. destruct (decide (X = ∅)) as [->|]. - by rewrite !big_opMS_union !big_opMS_singleton !big_opMS_empty !right_id. - by rewrite !big_opMS_union !big_opMS_singleton cmra_homomorphism -IH //. Qed. `````` Robbert Krebbers committed Sep 28, 2016 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 ``````Lemma big_opL_commute_L {M1 M2 : ucmraT} `{!LeibnizEquiv M2} {A} (h : M1 → M2) `{!UCMRAHomomorphism h} (f : nat → A → M1) l : h ([⋅ list] k↦x ∈ l, f k x) = ([⋅ list] k↦x ∈ l, h (f k x)). Proof. unfold_leibniz. by apply big_opL_commute. Qed. Lemma big_opL_commute1_L {M1 M2 : ucmraT} `{!LeibnizEquiv M2} {A} (h : M1 → M2) `{!CMRAHomomorphism h} (f : nat → A → M1) l : l ≠ [] → h ([⋅ list] k↦x ∈ l, f k x) = ([⋅ list] k↦x ∈ l, h (f k x)). Proof. unfold_leibniz. by apply big_opL_commute1. Qed. Lemma big_opM_commute_L {M1 M2 : ucmraT} `{!LeibnizEquiv M2, Countable K} {A} (h : M1 → M2) `{!UCMRAHomomorphism h} (f : K → A → M1) m : h ([⋅ map] k↦x ∈ m, f k x) = ([⋅ map] k↦x ∈ m, h (f k x)). Proof. unfold_leibniz. by apply big_opM_commute. Qed. Lemma big_opM_commute1_L {M1 M2 : ucmraT} `{!LeibnizEquiv M2, Countable K} {A} (h : M1 → M2) `{!CMRAHomomorphism h} (f : K → A → M1) m : m ≠ ∅ → h ([⋅ map] k↦x ∈ m, f k x) = ([⋅ map] k↦x ∈ m, h (f k x)). Proof. unfold_leibniz. by apply big_opM_commute1. Qed. Lemma big_opS_commute_L {M1 M2 : ucmraT} `{!LeibnizEquiv M2, Countable A} (h : M1 → M2) `{!UCMRAHomomorphism h} (f : A → M1) X : h ([⋅ set] x ∈ X, f x) = ([⋅ set] x ∈ X, h (f x)). Proof. unfold_leibniz. by apply big_opS_commute. Qed. Lemma big_opS_commute1_L {M1 M2 : ucmraT} `{!LeibnizEquiv M2, Countable A} (h : M1 → M2) `{!CMRAHomomorphism h} (f : A → M1) X : X ≠ ∅ → h ([⋅ set] x ∈ X, f x) = ([⋅ set] x ∈ X, h (f x)). Proof. intros. rewrite <-leibniz_equiv_iff. by apply big_opS_commute1. Qed. `````` Robbert Krebbers committed Nov 19, 2016 607 608 609 610 611 612 613 614 615 `````` Lemma big_opMS_commute_L {M1 M2 : ucmraT} `{!LeibnizEquiv M2, Countable A} (h : M1 → M2) `{!UCMRAHomomorphism h} (f : A → M1) X : h ([⋅ mset] x ∈ X, f x) = ([⋅ mset] x ∈ X, h (f x)). Proof. unfold_leibniz. by apply big_opMS_commute. Qed. Lemma big_opMS_commute1_L {M1 M2 : ucmraT} `{!LeibnizEquiv M2, Countable A} (h : M1 → M2) `{!CMRAHomomorphism h} (f : A → M1) X : X ≠ ∅ → h ([⋅ mset] x ∈ X, f x) = ([⋅ mset] x ∈ X, h (f x)). Proof. intros. rewrite <-leibniz_equiv_iff. by apply big_opMS_commute1. Qed.``````