big_op.v 21.6 KB
 Robbert Krebbers committed Mar 24, 2017 1 ``````From iris.algebra Require Export monoid. `````` Robbert Krebbers committed Aug 17, 2017 2 ``````From stdpp Require Export functions gmap gmultiset. `````` Robbert Krebbers committed Mar 24, 2017 3 4 5 ``````Set Default Proof Using "Type*". Local Existing Instances monoid_ne monoid_assoc monoid_comm monoid_left_id monoid_right_id monoid_proper `````` Robbert Krebbers committed Aug 17, 2017 6 7 `````` monoid_homomorphism_rel_po monoid_homomorphism_rel_proper monoid_homomorphism_op_proper `````` Robbert Krebbers committed Mar 24, 2017 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 `````` monoid_homomorphism_ne weak_monoid_homomorphism_proper. (** We define the following big operators with binders build in: - The operator [ [^o list] k ↦ x ∈ l, P ] folds over a list [l]. The binder [x] refers to each element at index [k]. - The operator [ [^o map] k ↦ x ∈ m, P ] folds over a map [m]. The binder [x] refers to each element at index [k]. - The operator [ [^o set] x ∈ X, P ] folds over a set [X]. 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 *) Fixpoint big_opL `{Monoid M o} {A} (f : nat → A → M) (xs : list A) : M := match xs with | [] => monoid_unit | x :: xs => o (f 0 x) (big_opL (λ n, f (S n)) xs) end. Instance: Params (@big_opL) 4. Arguments big_opL {M} o {_ A} _ !_ /. `````` Robbert Krebbers committed Mar 24, 2017 30 ``````Typeclasses Opaque big_opL. `````` Robbert Krebbers committed Mar 24, 2017 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 ``````Notation "'[^' o 'list]' k ↦ x ∈ l , P" := (big_opL o (λ k x, P) l) (at level 200, o at level 1, l at level 10, k, x at level 1, right associativity, format "[^ o list] k ↦ x ∈ l , P") : C_scope. Notation "'[^' o 'list]' x ∈ l , P" := (big_opL o (λ _ x, P) l) (at level 200, o at level 1, l at level 10, x at level 1, right associativity, format "[^ o list] x ∈ l , P") : C_scope. Definition big_opM `{Monoid M o} `{Countable K} {A} (f : K → A → M) (m : gmap K A) : M := big_opL o (λ _, curry f) (map_to_list m). Instance: Params (@big_opM) 7. Arguments big_opM {M} o {_ K _ _ A} _ _ : simpl never. Typeclasses Opaque big_opM. Notation "'[^' o 'map]' k ↦ x ∈ m , P" := (big_opM o (λ k x, P) m) (at level 200, o at level 1, m at level 10, k, x at level 1, right associativity, format "[^ o map] k ↦ x ∈ m , P") : C_scope. Notation "'[^' o 'map]' x ∈ m , P" := (big_opM o (λ _ x, P) m) (at level 200, o at level 1, m at level 10, x at level 1, right associativity, format "[^ o map] x ∈ m , P") : C_scope. Definition big_opS `{Monoid M o} `{Countable A} (f : A → M) (X : gset A) : M := big_opL o (λ _, f) (elements X). Instance: Params (@big_opS) 6. Arguments big_opS {M} o {_ A _ _} _ _ : simpl never. Typeclasses Opaque big_opS. Notation "'[^' o 'set]' x ∈ X , P" := (big_opS o (λ x, P) X) (at level 200, o at level 1, X at level 10, x at level 1, right associativity, format "[^ o set] x ∈ X , P") : C_scope. Definition big_opMS `{Monoid M o} `{Countable A} (f : A → M) (X : gmultiset A) : M := big_opL o (λ _, f) (elements X). Instance: Params (@big_opMS) 7. Arguments big_opMS {M} o {_ A _ _} _ _ : simpl never. Typeclasses Opaque big_opMS. Notation "'[^' o 'mset]' x ∈ X , P" := (big_opMS o (λ x, P) X) (at level 200, o at level 1, X at level 10, x at level 1, right associativity, format "[^ o mset] x ∈ X , P") : C_scope. (** * Properties about big ops *) Section big_op. Context `{Monoid M o}. Implicit Types xs : list M. Infix "`o`" := o (at level 50, left associativity). (** ** Big ops over lists *) Section list. Context {A : Type}. Implicit Types l : list A. Implicit Types f g : nat → A → M. Lemma big_opL_nil f : ([^o list] k↦y ∈ [], f k y) = monoid_unit. Proof. done. Qed. Lemma big_opL_cons f x l : ([^o list] k↦y ∈ x :: l, f k y) = f 0 x `o` [^o list] k↦y ∈ l, f (S k) y. Proof. done. Qed. Lemma big_opL_singleton f x : ([^o list] k↦y ∈ [x], f k y) ≡ f 0 x. Proof. by rewrite /= right_id. Qed. Lemma big_opL_app f l1 l2 : ([^o list] k↦y ∈ l1 ++ l2, f k y) ≡ ([^o list] k↦y ∈ l1, f k y) `o` ([^o list] k↦y ∈ l2, f (length l1 + k) y). Proof. revert f. induction l1 as [|x l1 IH]=> f /=; first by rewrite left_id. by rewrite IH assoc. Qed. `````` Robbert Krebbers committed Jun 12, 2017 95 96 97 `````` Lemma big_opL_unit l : ([^o list] k↦y ∈ l, monoid_unit) ≡ (monoid_unit : M). Proof. induction l; rewrite /= ?left_id //. Qed. `````` Robbert Krebbers committed Mar 24, 2017 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 `````` Lemma big_opL_forall R f g l : Reflexive R → Proper (R ==> R ==> R) o → (∀ k y, l !! k = Some y → R (f k y) (g k y)) → R ([^o list] k ↦ y ∈ l, f k y) ([^o list] k ↦ y ∈ l, g k y). Proof. intros ??. revert f g. induction l as [|x l IH]=> f g ? //=; f_equiv; eauto. Qed. Lemma big_opL_ext f g l : (∀ k y, l !! k = Some y → f k y = g k y) → ([^o list] k ↦ y ∈ l, f k y) = [^o list] k ↦ y ∈ l, g k y. Proof. apply big_opL_forall; apply _. Qed. Lemma big_opL_proper f g l : (∀ k y, l !! k = Some y → f k y ≡ g k y) → ([^o list] k ↦ y ∈ l, f k y) ≡ ([^o list] k ↦ y ∈ l, g k y). Proof. apply big_opL_forall; apply _. Qed. Lemma big_opL_permutation (f : A → M) l1 l2 : l1 ≡ₚ l2 → ([^o list] x ∈ l1, f x) ≡ ([^o list] x ∈ l2, f x). Proof. induction 1 as [|x xs1 xs2 ? IH|x y xs|xs1 xs2 xs3]; simpl; auto. - by rewrite IH. - by rewrite !assoc (comm _ (f x)). - by etrans. Qed. Global Instance big_opL_permutation' (f : A → M) : Proper ((≡ₚ) ==> (≡)) (big_opL o (λ _, f)). Proof. intros xs1 xs2. apply big_opL_permutation. Qed. Global Instance big_opL_ne n : Proper (pointwise_relation _ (pointwise_relation _ (dist n)) ==> eq ==> dist n) (big_opL o (A:=A)). Proof. intros f g Hf m ? <-. apply big_opL_forall; apply _ || intros; apply Hf. Qed. Global Instance big_opL_proper' : Proper (pointwise_relation _ (pointwise_relation _ (≡)) ==> eq ==> (≡)) (big_opL o (A:=A)). Proof. intros f g Hf m ? <-. apply big_opL_forall; apply _ || intros; apply Hf. Qed. Lemma big_opL_consZ_l (f : Z → A → M) x l : ([^o list] k↦y ∈ x :: l, f k y) = f 0 x `o` [^o 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 : ([^o list] k↦y ∈ x :: l, f k y) = f 0 x `o` [^o list] k↦y ∈ l, f (k + 1)%Z y. Proof. rewrite big_opL_cons. auto using big_opL_ext with f_equal lia. Qed. Lemma big_opL_fmap {B} (h : A → B) (f : nat → B → M) l : ([^o list] k↦y ∈ h <\$> l, f k y) ≡ ([^o list] k↦y ∈ l, f k (h y)). Proof. revert f. induction l as [|x l IH]=> f; csimpl=> //. by rewrite IH. Qed. Lemma big_opL_opL f g l : ([^o list] k↦x ∈ l, f k x `o` g k x) ≡ ([^o list] k↦x ∈ l, f k x) `o` ([^o list] k↦x ∈ l, g k x). Proof. revert f g; induction l as [|x l IH]=> f g /=; first by rewrite left_id. by rewrite IH -!assoc (assoc _ (g _ _)) [(g _ _ `o` _)]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_forall R f g m : Reflexive R → Proper (R ==> R ==> R) o → (∀ k x, m !! k = Some x → R (f k x) (g k x)) → R ([^o map] k ↦ x ∈ m, f k x) ([^o map] k ↦ x ∈ m, g k x). Proof. intros ?? Hf. apply (big_opL_forall R); auto. intros k [i x] ?%elem_of_list_lookup_2. by apply Hf, elem_of_map_to_list. Qed. Lemma big_opM_ext f g m : (∀ k x, m !! k = Some x → f k x = g k x) → ([^o map] k ↦ x ∈ m, f k x) = ([^o map] k ↦ x ∈ m, g k x). Proof. apply big_opM_forall; apply _. Qed. Lemma big_opM_proper f g m : (∀ k x, m !! k = Some x → f k x ≡ g k x) → ([^o map] k ↦ x ∈ m, f k x) ≡ ([^o map] k ↦ x ∈ m, g k x). Proof. apply big_opM_forall; apply _. Qed. Global Instance big_opM_ne n : Proper (pointwise_relation _ (pointwise_relation _ (dist n)) ==> eq ==> dist n) `````` Robbert Krebbers committed Sep 21, 2017 183 `````` (big_opM o (K:=K) (A:=A)). `````` Robbert Krebbers committed Mar 24, 2017 184 185 186 `````` Proof. intros f g Hf m ? <-. apply big_opM_forall; apply _ || intros; apply Hf. Qed. Global Instance big_opM_proper' : Proper (pointwise_relation _ (pointwise_relation _ (≡)) ==> eq ==> (≡)) `````` Robbert Krebbers committed Sep 21, 2017 187 `````` (big_opM o (K:=K) (A:=A)). `````` Robbert Krebbers committed Mar 24, 2017 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 `````` Proof. intros f g Hf m ? <-. apply big_opM_forall; apply _ || intros; apply Hf. Qed. Lemma big_opM_empty f : ([^o map] k↦x ∈ ∅, f k x) = monoid_unit. Proof. by rewrite /big_opM map_to_list_empty. Qed. Lemma big_opM_insert f m i x : m !! i = None → ([^o map] k↦y ∈ <[i:=x]> m, f k y) ≡ f i x `o` [^o 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 → ([^o map] k↦y ∈ m, f k y) ≡ f i x `o` [^o 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_singleton f i x : ([^o 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. `````` Robbert Krebbers committed Jun 12, 2017 212 213 214 `````` Lemma big_opM_unit m : ([^o map] k↦y ∈ m, monoid_unit) ≡ (monoid_unit : M). Proof. induction m using map_ind; rewrite /= ?big_opM_insert ?left_id //. Qed. `````` Robbert Krebbers committed Mar 24, 2017 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 `````` Lemma big_opM_fmap {B} (h : A → B) (f : K → B → M) m : ([^o map] k↦y ∈ h <\$> m, f k y) ≡ ([^o map] k↦y ∈ m, f k (h y)). Proof. rewrite /big_opM map_to_list_fmap big_opL_fmap. by apply big_opL_proper=> ? [??]. Qed. Lemma big_opM_insert_override (f : K → A → M) m i x x' : m !! i = Some x → f i x ≡ f i x' → ([^o map] k↦y ∈ <[i:=x']> m, f k y) ≡ ([^o map] k↦y ∈ m, f k y). Proof. intros ? Hx. rewrite -insert_delete big_opM_insert ?lookup_delete //. by rewrite -Hx -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 → ([^o map] k↦y ∈ <[i:=x]> m, g k y (<[i:=b]> f k)) ≡ g i x b `o` [^o map] k↦y ∈ m, g k y (f k). Proof. intros. rewrite big_opM_insert // fn_lookup_insert. f_equiv; apply 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 → ([^o map] k↦y ∈ <[i:=x]> m, <[i:=P]> f k) ≡ (P `o` [^o map] k↦y ∈ m, f k). Proof. apply (big_opM_fn_insert (λ _ _, id)). Qed. Lemma big_opM_opM f g m : ([^o map] k↦x ∈ m, f k x `o` g k x) ≡ ([^o map] k↦x ∈ m, f k x) `o` ([^o map] k↦x ∈ m, g k x). Proof. rewrite /big_opM -big_opL_opL. by apply big_opL_proper=> ? [??]. 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_forall R f g X : Reflexive R → Proper (R ==> R ==> R) o → (∀ x, x ∈ X → R (f x) (g x)) → R ([^o set] x ∈ X, f x) ([^o set] x ∈ X, g x). Proof. intros ?? Hf. apply (big_opL_forall R); auto. intros k x ?%elem_of_list_lookup_2. by apply Hf, elem_of_elements. Qed. Lemma big_opS_ext f g X : (∀ x, x ∈ X → f x = g x) → ([^o set] x ∈ X, f x) = ([^o 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) → ([^o set] x ∈ X, f x) ≡ ([^o set] x ∈ X, g x). Proof. apply big_opS_forall; apply _. Qed. Global Instance big_opS_ne n : Proper (pointwise_relation _ (dist n) ==> eq ==> dist n) (big_opS o (A:=A)). Proof. intros f g Hf m ? <-. apply big_opS_forall; apply _ || intros; apply Hf. Qed. Global Instance big_opS_proper' : Proper (pointwise_relation _ (≡) ==> eq ==> (≡)) (big_opS o (A:=A)). Proof. intros f g Hf m ? <-. apply big_opS_forall; apply _ || intros; apply Hf. Qed. Lemma big_opS_empty f : ([^o set] x ∈ ∅, f x) = monoid_unit. Proof. by rewrite /big_opS elements_empty. Qed. Lemma big_opS_insert f X x : x ∉ X → ([^o set] y ∈ {[ x ]} ∪ X, f y) ≡ (f x `o` [^o 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 → ([^o set] y ∈ {[ x ]} ∪ X, f y (<[x:=b]> h y)) ≡ f x b `o` [^o set] y ∈ X, f y (h y). Proof. intros. rewrite big_opS_insert // fn_lookup_insert. f_equiv; apply 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 → ([^o set] y ∈ {[ x ]} ∪ X, <[x:=P]> f y) ≡ (P `o` [^o set] y ∈ X, f y). Proof. apply (big_opS_fn_insert (λ y, id)). Qed. Lemma big_opS_union f X Y : X ⊥ Y → ([^o set] y ∈ X ∪ Y, f y) ≡ ([^o set] y ∈ X, f y) `o` ([^o set] y ∈ Y, f y). Proof. intros. induction X as [|x X ? IH] using collection_ind_L. { by rewrite left_id_L big_opS_empty left_id. } rewrite -assoc_L !big_opS_insert; [|set_solver..]. by rewrite -assoc IH; last set_solver. Qed. Lemma big_opS_delete f X x : x ∈ X → ([^o set] y ∈ X, f y) ≡ f x `o` [^o 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_singleton f x : ([^o set] y ∈ {[ x ]}, f y) ≡ f x. Proof. intros. by rewrite /big_opS elements_singleton /= right_id. Qed. `````` Robbert Krebbers committed Jun 12, 2017 321 322 323 324 325 `````` Lemma big_opS_unit X : ([^o set] y ∈ X, monoid_unit) ≡ (monoid_unit : M). Proof. induction X using collection_ind_L; rewrite /= ?big_opS_insert ?left_id //. Qed. `````` Robbert Krebbers committed Mar 24, 2017 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 `````` Lemma big_opS_opS f g X : ([^o set] y ∈ X, f y `o` g y) ≡ ([^o set] y ∈ X, f y) `o` ([^o set] y ∈ X, g y). Proof. by rewrite /big_opS -big_opL_opL. Qed. End gset. Lemma big_opM_dom `{Countable K} {A} (f : K → M) (m : gmap K A) : ([^o map] k↦_ ∈ m, f k) ≡ ([^o set] k ∈ dom _ m, f k). Proof. induction m as [|i x ?? IH] using map_ind; [by rewrite dom_empty_L|]. by rewrite dom_insert_L big_opM_insert // IH big_opS_insert ?not_elem_of_dom. Qed. (** ** 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) o → (∀ x, x ∈ X → R (f x) (g x)) → R ([^o mset] x ∈ X, f x) ([^o mset] x ∈ X, g x). Proof. intros ?? Hf. apply (big_opL_forall R); auto. intros k x ?%elem_of_list_lookup_2. by apply Hf, gmultiset_elem_of_elements. Qed. Lemma big_opMS_ext f g X : (∀ x, x ∈ X → f x = g x) → ([^o mset] x ∈ X, f x) = ([^o 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) → ([^o mset] x ∈ X, f x) ≡ ([^o mset] x ∈ X, g x). Proof. apply big_opMS_forall; apply _. Qed. Global Instance big_opMS_ne n : Proper (pointwise_relation _ (dist n) ==> eq ==> dist n) (big_opMS o (A:=A)). Proof. intros f g Hf m ? <-. apply big_opMS_forall; apply _ || intros; apply Hf. Qed. Global Instance big_opMS_proper' : Proper (pointwise_relation _ (≡) ==> eq ==> (≡)) (big_opMS o (A:=A)). Proof. intros f g Hf m ? <-. apply big_opMS_forall; apply _ || intros; apply Hf. Qed. Lemma big_opMS_empty f : ([^o mset] x ∈ ∅, f x) = monoid_unit. Proof. by rewrite /big_opMS gmultiset_elements_empty. Qed. Lemma big_opMS_union f X Y : ([^o mset] y ∈ X ∪ Y, f y) ≡ ([^o mset] y ∈ X, f y) `o` [^o mset] y ∈ Y, f y. Proof. by rewrite /big_opMS gmultiset_elements_union big_opL_app. Qed. Lemma big_opMS_singleton f x : ([^o mset] y ∈ {[ x ]}, f y) ≡ f x. Proof. intros. by rewrite /big_opMS gmultiset_elements_singleton /= right_id. Qed. Lemma big_opMS_delete f X x : x ∈ X → ([^o mset] y ∈ X, f y) ≡ f x `o` [^o mset] y ∈ X ∖ {[ x ]}, f y. Proof. intros. rewrite -big_opMS_singleton -big_opMS_union. by rewrite -gmultiset_union_difference'. Qed. `````` Robbert Krebbers committed Jun 12, 2017 388 389 390 391 392 393 `````` Lemma big_opMS_unit X : ([^o mset] y ∈ X, monoid_unit) ≡ (monoid_unit : M). Proof. induction X using gmultiset_ind; rewrite /= ?big_opMS_union ?big_opMS_singleton ?left_id //. Qed. `````` Robbert Krebbers committed Mar 24, 2017 394 395 396 397 398 399 400 401 402 403 `````` Lemma big_opMS_opMS f g X : ([^o mset] y ∈ X, f y `o` g y) ≡ ([^o mset] y ∈ X, f y) `o` ([^o mset] y ∈ X, g y). Proof. by rewrite /big_opMS -big_opL_opL. Qed. End gmultiset. End big_op. Section homomorphisms. Context `{Monoid M1 o1, Monoid M2 o2}. Infix "`o1`" := o1 (at level 50, left associativity). Infix "`o2`" := o2 (at level 50, left associativity). `````` Robbert Krebbers committed Aug 17, 2017 404 `````` Instance foo {A} (R : relation A) : RewriteRelation R. `````` Robbert Krebbers committed Mar 24, 2017 405 `````` `````` Robbert Krebbers committed Aug 17, 2017 406 `````` Lemma big_opL_commute {A} (h : M1 → M2) `{!MonoidHomomorphism o1 o2 R h} `````` Robbert Krebbers committed Mar 24, 2017 407 `````` (f : nat → A → M1) l : `````` Robbert Krebbers committed Aug 17, 2017 408 `````` R (h ([^o1 list] k↦x ∈ l, f k x)) ([^o2 list] k↦x ∈ l, h (f k x)). `````` Robbert Krebbers committed Mar 24, 2017 409 410 `````` Proof. revert f. induction l as [|x l IH]=> f /=. `````` Robbert Krebbers committed Aug 17, 2017 411 412 `````` - apply monoid_homomorphism_unit. - by rewrite monoid_homomorphism IH. `````` Robbert Krebbers committed Mar 24, 2017 413 `````` Qed. `````` Robbert Krebbers committed Aug 17, 2017 414 `````` Lemma big_opL_commute1 {A} (h : M1 → M2) `{!WeakMonoidHomomorphism o1 o2 R h} `````` Robbert Krebbers committed Mar 24, 2017 415 `````` (f : nat → A → M1) l : `````` Robbert Krebbers committed Aug 17, 2017 416 `````` l ≠ [] → R (h ([^o1 list] k↦x ∈ l, f k x)) ([^o2 list] k↦x ∈ l, h (f k x)). `````` Robbert Krebbers committed Mar 24, 2017 417 418 419 `````` Proof. intros ?. revert f. induction l as [|x [|x' l'] IH]=> f //. - by rewrite !big_opL_singleton. `````` Robbert Krebbers committed Aug 17, 2017 420 `````` - by rewrite !(big_opL_cons _ x) monoid_homomorphism IH. `````` Robbert Krebbers committed Mar 24, 2017 421 422 423 `````` Qed. Lemma big_opM_commute `{Countable K} {A} (h : M1 → M2) `````` Robbert Krebbers committed Aug 17, 2017 424 425 `````` `{!MonoidHomomorphism o1 o2 R h} (f : K → A → M1) m : R (h ([^o1 map] k↦x ∈ m, f k x)) ([^o2 map] k↦x ∈ m, h (f k x)). `````` Robbert Krebbers committed Mar 24, 2017 426 427 428 429 430 431 `````` Proof. intros. induction m as [|i x m ? IH] using map_ind. - by rewrite !big_opM_empty monoid_homomorphism_unit. - by rewrite !big_opM_insert // monoid_homomorphism -IH. Qed. Lemma big_opM_commute1 `{Countable K} {A} (h : M1 → M2) `````` Robbert Krebbers committed Aug 17, 2017 432 433 `````` `{!WeakMonoidHomomorphism o1 o2 R h} (f : K → A → M1) m : m ≠ ∅ → R (h ([^o1 map] k↦x ∈ m, f k x)) ([^o2 map] k↦x ∈ m, h (f k x)). `````` Robbert Krebbers committed Mar 24, 2017 434 435 436 437 438 439 440 441 `````` Proof. 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 // monoid_homomorphism -IH //. Qed. Lemma big_opS_commute `{Countable A} (h : M1 → M2) `````` Robbert Krebbers committed Aug 17, 2017 442 443 `````` `{!MonoidHomomorphism o1 o2 R h} (f : A → M1) X : R (h ([^o1 set] x ∈ X, f x)) ([^o2 set] x ∈ X, h (f x)). `````` Robbert Krebbers committed Mar 24, 2017 444 445 446 447 448 449 `````` Proof. intros. induction X as [|x X ? IH] using collection_ind_L. - by rewrite !big_opS_empty monoid_homomorphism_unit. - by rewrite !big_opS_insert // monoid_homomorphism -IH. Qed. Lemma big_opS_commute1 `{Countable A} (h : M1 → M2) `````` Robbert Krebbers committed Aug 17, 2017 450 451 `````` `{!WeakMonoidHomomorphism o1 o2 R h} (f : A → M1) X : X ≠ ∅ → R (h ([^o1 set] x ∈ X, f x)) ([^o2 set] x ∈ X, h (f x)). `````` Robbert Krebbers committed Mar 24, 2017 452 453 454 455 456 457 458 459 `````` Proof. 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 // monoid_homomorphism -IH //. Qed. Lemma big_opMS_commute `{Countable A} (h : M1 → M2) `````` Robbert Krebbers committed Aug 17, 2017 460 461 `````` `{!MonoidHomomorphism o1 o2 R h} (f : A → M1) X : R (h ([^o1 mset] x ∈ X, f x)) ([^o2 mset] x ∈ X, h (f x)). `````` Robbert Krebbers committed Mar 24, 2017 462 463 464 465 466 467 `````` Proof. intros. induction X as [|x X IH] using gmultiset_ind. - by rewrite !big_opMS_empty monoid_homomorphism_unit. - by rewrite !big_opMS_union !big_opMS_singleton monoid_homomorphism -IH. Qed. Lemma big_opMS_commute1 `{Countable A} (h : M1 → M2) `````` Robbert Krebbers committed Aug 17, 2017 468 469 `````` `{!WeakMonoidHomomorphism o1 o2 R h} (f : A → M1) X : X ≠ ∅ → R (h ([^o1 mset] x ∈ X, f x)) ([^o2 mset] x ∈ X, h (f x)). `````` Robbert Krebbers committed Mar 24, 2017 470 471 472 473 474 475 476 477 478 479 `````` 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 monoid_homomorphism -IH //. Qed. Context `{!LeibnizEquiv M2}. Lemma big_opL_commute_L {A} (h : M1 → M2) `````` Robbert Krebbers committed Aug 17, 2017 480 `````` `{!MonoidHomomorphism o1 o2 (≡) h} (f : nat → A → M1) l : `````` Robbert Krebbers committed Mar 24, 2017 481 482 483 `````` h ([^o1 list] k↦x ∈ l, f k x) = ([^o2 list] k↦x ∈ l, h (f k x)). Proof. unfold_leibniz. by apply big_opL_commute. Qed. Lemma big_opL_commute1_L {A} (h : M1 → M2) `````` Robbert Krebbers committed Aug 17, 2017 484 `````` `{!WeakMonoidHomomorphism o1 o2 (≡) h} (f : nat → A → M1) l : `````` Robbert Krebbers committed Mar 24, 2017 485 486 487 488 `````` l ≠ [] → h ([^o1 list] k↦x ∈ l, f k x) = ([^o2 list] k↦x ∈ l, h (f k x)). Proof. unfold_leibniz. by apply big_opL_commute1. Qed. Lemma big_opM_commute_L `{Countable K} {A} (h : M1 → M2) `````` Robbert Krebbers committed Aug 17, 2017 489 `````` `{!MonoidHomomorphism o1 o2 (≡) h} (f : K → A → M1) m : `````` Robbert Krebbers committed Mar 24, 2017 490 491 492 `````` h ([^o1 map] k↦x ∈ m, f k x) = ([^o2 map] k↦x ∈ m, h (f k x)). Proof. unfold_leibniz. by apply big_opM_commute. Qed. Lemma big_opM_commute1_L `{Countable K} {A} (h : M1 → M2) `````` Robbert Krebbers committed Aug 17, 2017 493 `````` `{!WeakMonoidHomomorphism o1 o2 (≡) h} (f : K → A → M1) m : `````` Robbert Krebbers committed Mar 24, 2017 494 495 496 497 `````` m ≠ ∅ → h ([^o1 map] k↦x ∈ m, f k x) = ([^o2 map] k↦x ∈ m, h (f k x)). Proof. unfold_leibniz. by apply big_opM_commute1. Qed. Lemma big_opS_commute_L `{Countable A} (h : M1 → M2) `````` Robbert Krebbers committed Aug 17, 2017 498 `````` `{!MonoidHomomorphism o1 o2 (≡) h} (f : A → M1) X : `````` Robbert Krebbers committed Mar 24, 2017 499 500 501 `````` h ([^o1 set] x ∈ X, f x) = ([^o2 set] x ∈ X, h (f x)). Proof. unfold_leibniz. by apply big_opS_commute. Qed. Lemma big_opS_commute1_L `{ Countable A} (h : M1 → M2) `````` Robbert Krebbers committed Aug 17, 2017 502 `````` `{!WeakMonoidHomomorphism o1 o2 (≡) h} (f : A → M1) X : `````` Robbert Krebbers committed Mar 24, 2017 503 504 505 506 `````` X ≠ ∅ → h ([^o1 set] x ∈ X, f x) = ([^o2 set] x ∈ X, h (f x)). Proof. intros. rewrite <-leibniz_equiv_iff. by apply big_opS_commute1. Qed. Lemma big_opMS_commute_L `{Countable A} (h : M1 → M2) `````` Robbert Krebbers committed Aug 17, 2017 507 `````` `{!MonoidHomomorphism o1 o2 (≡) h} (f : A → M1) X : `````` Robbert Krebbers committed Mar 24, 2017 508 509 510 `````` h ([^o1 mset] x ∈ X, f x) = ([^o2 mset] x ∈ X, h (f x)). Proof. unfold_leibniz. by apply big_opMS_commute. Qed. Lemma big_opMS_commute1_L `{Countable A} (h : M1 → M2) `````` Robbert Krebbers committed Aug 17, 2017 511 `````` `{!WeakMonoidHomomorphism o1 o2 (≡) h} (f : A → M1) X : `````` Robbert Krebbers committed Mar 24, 2017 512 513 514 `````` X ≠ ∅ → h ([^o1 mset] x ∈ X, f x) = ([^o2 mset] x ∈ X, h (f x)). Proof. intros. rewrite <-leibniz_equiv_iff. by apply big_opMS_commute1. Qed. End homomorphisms.``````