1(* Title: HOL/Map.thy 2 Author: Tobias Nipkow, based on a theory by David von Oheimb 3 Copyright 1997-2003 TU Muenchen 4 5The datatype of "maps"; strongly resembles maps in VDM. 6*) 7 8section \<open>Maps\<close> 9 10theory Map 11 imports List 12 abbrevs "(=" = "\<subseteq>\<^sub>m" 13begin 14 15type_synonym ('a, 'b) "map" = "'a \<Rightarrow> 'b option" (infixr "\<rightharpoonup>" 0) 16 17abbreviation 18 empty :: "'a \<rightharpoonup> 'b" where 19 "empty \<equiv> \<lambda>x. None" 20 21definition 22 map_comp :: "('b \<rightharpoonup> 'c) \<Rightarrow> ('a \<rightharpoonup> 'b) \<Rightarrow> ('a \<rightharpoonup> 'c)" (infixl "\<circ>\<^sub>m" 55) where 23 "f \<circ>\<^sub>m g = (\<lambda>k. case g k of None \<Rightarrow> None | Some v \<Rightarrow> f v)" 24 25definition 26 map_add :: "('a \<rightharpoonup> 'b) \<Rightarrow> ('a \<rightharpoonup> 'b) \<Rightarrow> ('a \<rightharpoonup> 'b)" (infixl "++" 100) where 27 "m1 ++ m2 = (\<lambda>x. case m2 x of None \<Rightarrow> m1 x | Some y \<Rightarrow> Some y)" 28 29definition 30 restrict_map :: "('a \<rightharpoonup> 'b) \<Rightarrow> 'a set \<Rightarrow> ('a \<rightharpoonup> 'b)" (infixl "|`" 110) where 31 "m|`A = (\<lambda>x. if x \<in> A then m x else None)" 32 33notation (latex output) 34 restrict_map ("_\<restriction>\<^bsub>_\<^esub>" [111,110] 110) 35 36definition 37 dom :: "('a \<rightharpoonup> 'b) \<Rightarrow> 'a set" where 38 "dom m = {a. m a \<noteq> None}" 39 40definition 41 ran :: "('a \<rightharpoonup> 'b) \<Rightarrow> 'b set" where 42 "ran m = {b. \<exists>a. m a = Some b}" 43 44definition 45 map_le :: "('a \<rightharpoonup> 'b) \<Rightarrow> ('a \<rightharpoonup> 'b) \<Rightarrow> bool" (infix "\<subseteq>\<^sub>m" 50) where 46 "(m\<^sub>1 \<subseteq>\<^sub>m m\<^sub>2) \<longleftrightarrow> (\<forall>a \<in> dom m\<^sub>1. m\<^sub>1 a = m\<^sub>2 a)" 47 48nonterminal maplets and maplet 49 50syntax 51 "_maplet" :: "['a, 'a] \<Rightarrow> maplet" ("_ /\<mapsto>/ _") 52 "_maplets" :: "['a, 'a] \<Rightarrow> maplet" ("_ /[\<mapsto>]/ _") 53 "" :: "maplet \<Rightarrow> maplets" ("_") 54 "_Maplets" :: "[maplet, maplets] \<Rightarrow> maplets" ("_,/ _") 55 "_MapUpd" :: "['a \<rightharpoonup> 'b, maplets] \<Rightarrow> 'a \<rightharpoonup> 'b" ("_/'(_')" [900, 0] 900) 56 "_Map" :: "maplets \<Rightarrow> 'a \<rightharpoonup> 'b" ("(1[_])") 57 58syntax (ASCII) 59 "_maplet" :: "['a, 'a] \<Rightarrow> maplet" ("_ /|->/ _") 60 "_maplets" :: "['a, 'a] \<Rightarrow> maplet" ("_ /[|->]/ _") 61 62translations 63 "_MapUpd m (_Maplets xy ms)" \<rightleftharpoons> "_MapUpd (_MapUpd m xy) ms" 64 "_MapUpd m (_maplet x y)" \<rightleftharpoons> "m(x := CONST Some y)" 65 "_Map ms" \<rightleftharpoons> "_MapUpd (CONST empty) ms" 66 "_Map (_Maplets ms1 ms2)" \<leftharpoondown> "_MapUpd (_Map ms1) ms2" 67 "_Maplets ms1 (_Maplets ms2 ms3)" \<leftharpoondown> "_Maplets (_Maplets ms1 ms2) ms3" 68 69primrec map_of :: "('a \<times> 'b) list \<Rightarrow> 'a \<rightharpoonup> 'b" 70where 71 "map_of [] = empty" 72| "map_of (p # ps) = (map_of ps)(fst p \<mapsto> snd p)" 73 74definition map_upds :: "('a \<rightharpoonup> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b list \<Rightarrow> 'a \<rightharpoonup> 'b" 75 where "map_upds m xs ys = m ++ map_of (rev (zip xs ys))" 76translations 77 "_MapUpd m (_maplets x y)" \<rightleftharpoons> "CONST map_upds m x y" 78 79lemma map_of_Cons_code [code]: 80 "map_of [] k = None" 81 "map_of ((l, v) # ps) k = (if l = k then Some v else map_of ps k)" 82 by simp_all 83 84 85subsection \<open>@{term [source] empty}\<close> 86 87lemma empty_upd_none [simp]: "empty(x := None) = empty" 88 by (rule ext) simp 89 90 91subsection \<open>@{term [source] map_upd}\<close> 92 93lemma map_upd_triv: "t k = Some x \<Longrightarrow> t(k\<mapsto>x) = t" 94 by (rule ext) simp 95 96lemma map_upd_nonempty [simp]: "t(k\<mapsto>x) \<noteq> empty" 97proof 98 assume "t(k \<mapsto> x) = empty" 99 then have "(t(k \<mapsto> x)) k = None" by simp 100 then show False by simp 101qed 102 103lemma map_upd_eqD1: 104 assumes "m(a\<mapsto>x) = n(a\<mapsto>y)" 105 shows "x = y" 106proof - 107 from assms have "(m(a\<mapsto>x)) a = (n(a\<mapsto>y)) a" by simp 108 then show ?thesis by simp 109qed 110 111lemma map_upd_Some_unfold: 112 "((m(a\<mapsto>b)) x = Some y) = (x = a \<and> b = y \<or> x \<noteq> a \<and> m x = Some y)" 113by auto 114 115lemma image_map_upd [simp]: "x \<notin> A \<Longrightarrow> m(x \<mapsto> y) ` A = m ` A" 116by auto 117 118lemma finite_range_updI: "finite (range f) \<Longrightarrow> finite (range (f(a\<mapsto>b)))" 119unfolding image_def 120apply (simp (no_asm_use) add:full_SetCompr_eq) 121apply (rule finite_subset) 122 prefer 2 apply assumption 123apply (auto) 124done 125 126 127subsection \<open>@{term [source] map_of}\<close> 128 129lemma map_of_eq_empty_iff [simp]: 130 "map_of xys = empty \<longleftrightarrow> xys = []" 131proof 132 show "map_of xys = empty \<Longrightarrow> xys = []" 133 by (induction xys) simp_all 134qed simp 135 136lemma empty_eq_map_of_iff [simp]: 137 "empty = map_of xys \<longleftrightarrow> xys = []" 138by(subst eq_commute) simp 139 140lemma map_of_eq_None_iff: 141 "(map_of xys x = None) = (x \<notin> fst ` (set xys))" 142by (induct xys) simp_all 143 144lemma map_of_eq_Some_iff [simp]: 145 "distinct(map fst xys) \<Longrightarrow> (map_of xys x = Some y) = ((x,y) \<in> set xys)" 146apply (induct xys) 147 apply simp 148apply (auto simp: map_of_eq_None_iff [symmetric]) 149done 150 151lemma Some_eq_map_of_iff [simp]: 152 "distinct(map fst xys) \<Longrightarrow> (Some y = map_of xys x) = ((x,y) \<in> set xys)" 153by (auto simp del: map_of_eq_Some_iff simp: map_of_eq_Some_iff [symmetric]) 154 155lemma map_of_is_SomeI [simp]: "\<lbrakk> distinct(map fst xys); (x,y) \<in> set xys \<rbrakk> 156 \<Longrightarrow> map_of xys x = Some y" 157apply (induct xys) 158 apply simp 159apply force 160done 161 162lemma map_of_zip_is_None [simp]: 163 "length xs = length ys \<Longrightarrow> (map_of (zip xs ys) x = None) = (x \<notin> set xs)" 164by (induct rule: list_induct2) simp_all 165 166lemma map_of_zip_is_Some: 167 assumes "length xs = length ys" 168 shows "x \<in> set xs \<longleftrightarrow> (\<exists>y. map_of (zip xs ys) x = Some y)" 169using assms by (induct rule: list_induct2) simp_all 170 171lemma map_of_zip_upd: 172 fixes x :: 'a and xs :: "'a list" and ys zs :: "'b list" 173 assumes "length ys = length xs" 174 and "length zs = length xs" 175 and "x \<notin> set xs" 176 and "map_of (zip xs ys)(x \<mapsto> y) = map_of (zip xs zs)(x \<mapsto> z)" 177 shows "map_of (zip xs ys) = map_of (zip xs zs)" 178proof 179 fix x' :: 'a 180 show "map_of (zip xs ys) x' = map_of (zip xs zs) x'" 181 proof (cases "x = x'") 182 case True 183 from assms True map_of_zip_is_None [of xs ys x'] 184 have "map_of (zip xs ys) x' = None" by simp 185 moreover from assms True map_of_zip_is_None [of xs zs x'] 186 have "map_of (zip xs zs) x' = None" by simp 187 ultimately show ?thesis by simp 188 next 189 case False from assms 190 have "(map_of (zip xs ys)(x \<mapsto> y)) x' = (map_of (zip xs zs)(x \<mapsto> z)) x'" by auto 191 with False show ?thesis by simp 192 qed 193qed 194 195lemma map_of_zip_inject: 196 assumes "length ys = length xs" 197 and "length zs = length xs" 198 and dist: "distinct xs" 199 and map_of: "map_of (zip xs ys) = map_of (zip xs zs)" 200 shows "ys = zs" 201 using assms(1) assms(2)[symmetric] 202 using dist map_of 203proof (induct ys xs zs rule: list_induct3) 204 case Nil show ?case by simp 205next 206 case (Cons y ys x xs z zs) 207 from \<open>map_of (zip (x#xs) (y#ys)) = map_of (zip (x#xs) (z#zs))\<close> 208 have map_of: "map_of (zip xs ys)(x \<mapsto> y) = map_of (zip xs zs)(x \<mapsto> z)" by simp 209 from Cons have "length ys = length xs" and "length zs = length xs" 210 and "x \<notin> set xs" by simp_all 211 then have "map_of (zip xs ys) = map_of (zip xs zs)" using map_of by (rule map_of_zip_upd) 212 with Cons.hyps \<open>distinct (x # xs)\<close> have "ys = zs" by simp 213 moreover from map_of have "y = z" by (rule map_upd_eqD1) 214 ultimately show ?case by simp 215qed 216 217lemma map_of_zip_nth: 218 assumes "length xs = length ys" 219 assumes "distinct xs" 220 assumes "i < length ys" 221 shows "map_of (zip xs ys) (xs ! i) = Some (ys ! i)" 222using assms proof (induct arbitrary: i rule: list_induct2) 223 case Nil 224 then show ?case by simp 225next 226 case (Cons x xs y ys) 227 then show ?case 228 using less_Suc_eq_0_disj by auto 229qed 230 231lemma map_of_zip_map: 232 "map_of (zip xs (map f xs)) = (\<lambda>x. if x \<in> set xs then Some (f x) else None)" 233 by (induct xs) (simp_all add: fun_eq_iff) 234 235lemma finite_range_map_of: "finite (range (map_of xys))" 236apply (induct xys) 237 apply (simp_all add: image_constant) 238apply (rule finite_subset) 239 prefer 2 apply assumption 240apply auto 241done 242 243lemma map_of_SomeD: "map_of xs k = Some y \<Longrightarrow> (k, y) \<in> set xs" 244 by (induct xs) (auto split: if_splits) 245 246lemma map_of_mapk_SomeI: 247 "inj f \<Longrightarrow> map_of t k = Some x \<Longrightarrow> 248 map_of (map (case_prod (\<lambda>k. Pair (f k))) t) (f k) = Some x" 249by (induct t) (auto simp: inj_eq) 250 251lemma weak_map_of_SomeI: "(k, x) \<in> set l \<Longrightarrow> \<exists>x. map_of l k = Some x" 252by (induct l) auto 253 254lemma map_of_filter_in: 255 "map_of xs k = Some z \<Longrightarrow> P k z \<Longrightarrow> map_of (filter (case_prod P) xs) k = Some z" 256by (induct xs) auto 257 258lemma map_of_map: 259 "map_of (map (\<lambda>(k, v). (k, f v)) xs) = map_option f \<circ> map_of xs" 260 by (induct xs) (auto simp: fun_eq_iff) 261 262lemma dom_map_option: 263 "dom (\<lambda>k. map_option (f k) (m k)) = dom m" 264 by (simp add: dom_def) 265 266lemma dom_map_option_comp [simp]: 267 "dom (map_option g \<circ> m) = dom m" 268 using dom_map_option [of "\<lambda>_. g" m] by (simp add: comp_def) 269 270 271subsection \<open>\<^const>\<open>map_option\<close> related\<close> 272 273lemma map_option_o_empty [simp]: "map_option f \<circ> empty = empty" 274by (rule ext) simp 275 276lemma map_option_o_map_upd [simp]: 277 "map_option f \<circ> m(a\<mapsto>b) = (map_option f \<circ> m)(a\<mapsto>f b)" 278by (rule ext) simp 279 280 281subsection \<open>@{term [source] map_comp} related\<close> 282 283lemma map_comp_empty [simp]: 284 "m \<circ>\<^sub>m empty = empty" 285 "empty \<circ>\<^sub>m m = empty" 286by (auto simp: map_comp_def split: option.splits) 287 288lemma map_comp_simps [simp]: 289 "m2 k = None \<Longrightarrow> (m1 \<circ>\<^sub>m m2) k = None" 290 "m2 k = Some k' \<Longrightarrow> (m1 \<circ>\<^sub>m m2) k = m1 k'" 291by (auto simp: map_comp_def) 292 293lemma map_comp_Some_iff: 294 "((m1 \<circ>\<^sub>m m2) k = Some v) = (\<exists>k'. m2 k = Some k' \<and> m1 k' = Some v)" 295by (auto simp: map_comp_def split: option.splits) 296 297lemma map_comp_None_iff: 298 "((m1 \<circ>\<^sub>m m2) k = None) = (m2 k = None \<or> (\<exists>k'. m2 k = Some k' \<and> m1 k' = None)) " 299by (auto simp: map_comp_def split: option.splits) 300 301 302subsection \<open>\<open>++\<close>\<close> 303 304lemma map_add_empty[simp]: "m ++ empty = m" 305by(simp add: map_add_def) 306 307lemma empty_map_add[simp]: "empty ++ m = m" 308by (rule ext) (simp add: map_add_def split: option.split) 309 310lemma map_add_assoc[simp]: "m1 ++ (m2 ++ m3) = (m1 ++ m2) ++ m3" 311by (rule ext) (simp add: map_add_def split: option.split) 312 313lemma map_add_Some_iff: 314 "((m ++ n) k = Some x) = (n k = Some x \<or> n k = None \<and> m k = Some x)" 315by (simp add: map_add_def split: option.split) 316 317lemma map_add_SomeD [dest!]: 318 "(m ++ n) k = Some x \<Longrightarrow> n k = Some x \<or> n k = None \<and> m k = Some x" 319by (rule map_add_Some_iff [THEN iffD1]) 320 321lemma map_add_find_right [simp]: "n k = Some xx \<Longrightarrow> (m ++ n) k = Some xx" 322by (subst map_add_Some_iff) fast 323 324lemma map_add_None [iff]: "((m ++ n) k = None) = (n k = None \<and> m k = None)" 325by (simp add: map_add_def split: option.split) 326 327lemma map_add_upd[simp]: "f ++ g(x\<mapsto>y) = (f ++ g)(x\<mapsto>y)" 328by (rule ext) (simp add: map_add_def) 329 330lemma map_add_upds[simp]: "m1 ++ (m2(xs[\<mapsto>]ys)) = (m1++m2)(xs[\<mapsto>]ys)" 331by (simp add: map_upds_def) 332 333lemma map_add_upd_left: "m\<notin>dom e2 \<Longrightarrow> e1(m \<mapsto> u1) ++ e2 = (e1 ++ e2)(m \<mapsto> u1)" 334by (rule ext) (auto simp: map_add_def dom_def split: option.split) 335 336lemma map_of_append[simp]: "map_of (xs @ ys) = map_of ys ++ map_of xs" 337unfolding map_add_def 338apply (induct xs) 339 apply simp 340apply (rule ext) 341apply (simp split: option.split) 342done 343 344lemma finite_range_map_of_map_add: 345 "finite (range f) \<Longrightarrow> finite (range (f ++ map_of l))" 346apply (induct l) 347 apply (auto simp del: fun_upd_apply) 348apply (erule finite_range_updI) 349done 350 351lemma inj_on_map_add_dom [iff]: 352 "inj_on (m ++ m') (dom m') = inj_on m' (dom m')" 353by (fastforce simp: map_add_def dom_def inj_on_def split: option.splits) 354 355lemma map_upds_fold_map_upd: 356 "m(ks[\<mapsto>]vs) = foldl (\<lambda>m (k, v). m(k \<mapsto> v)) m (zip ks vs)" 357unfolding map_upds_def proof (rule sym, rule zip_obtain_same_length) 358 fix ks :: "'a list" and vs :: "'b list" 359 assume "length ks = length vs" 360 then show "foldl (\<lambda>m (k, v). m(k\<mapsto>v)) m (zip ks vs) = m ++ map_of (rev (zip ks vs))" 361 by(induct arbitrary: m rule: list_induct2) simp_all 362qed 363 364lemma map_add_map_of_foldr: 365 "m ++ map_of ps = foldr (\<lambda>(k, v) m. m(k \<mapsto> v)) ps m" 366 by (induct ps) (auto simp: fun_eq_iff map_add_def) 367 368 369subsection \<open>@{term [source] restrict_map}\<close> 370 371lemma restrict_map_to_empty [simp]: "m|`{} = empty" 372by (simp add: restrict_map_def) 373 374lemma restrict_map_insert: "f |` (insert a A) = (f |` A)(a := f a)" 375by (auto simp: restrict_map_def) 376 377lemma restrict_map_empty [simp]: "empty|`D = empty" 378by (simp add: restrict_map_def) 379 380lemma restrict_in [simp]: "x \<in> A \<Longrightarrow> (m|`A) x = m x" 381by (simp add: restrict_map_def) 382 383lemma restrict_out [simp]: "x \<notin> A \<Longrightarrow> (m|`A) x = None" 384by (simp add: restrict_map_def) 385 386lemma ran_restrictD: "y \<in> ran (m|`A) \<Longrightarrow> \<exists>x\<in>A. m x = Some y" 387by (auto simp: restrict_map_def ran_def split: if_split_asm) 388 389lemma dom_restrict [simp]: "dom (m|`A) = dom m \<inter> A" 390by (auto simp: restrict_map_def dom_def split: if_split_asm) 391 392lemma restrict_upd_same [simp]: "m(x\<mapsto>y)|`(-{x}) = m|`(-{x})" 393by (rule ext) (auto simp: restrict_map_def) 394 395lemma restrict_restrict [simp]: "m|`A|`B = m|`(A\<inter>B)" 396by (rule ext) (auto simp: restrict_map_def) 397 398lemma restrict_fun_upd [simp]: 399 "m(x := y)|`D = (if x \<in> D then (m|`(D-{x}))(x := y) else m|`D)" 400by (simp add: restrict_map_def fun_eq_iff) 401 402lemma fun_upd_None_restrict [simp]: 403 "(m|`D)(x := None) = (if x \<in> D then m|`(D - {x}) else m|`D)" 404by (simp add: restrict_map_def fun_eq_iff) 405 406lemma fun_upd_restrict: "(m|`D)(x := y) = (m|`(D-{x}))(x := y)" 407by (simp add: restrict_map_def fun_eq_iff) 408 409lemma fun_upd_restrict_conv [simp]: 410 "x \<in> D \<Longrightarrow> (m|`D)(x := y) = (m|`(D-{x}))(x := y)" 411by (simp add: restrict_map_def fun_eq_iff) 412 413lemma map_of_map_restrict: 414 "map_of (map (\<lambda>k. (k, f k)) ks) = (Some \<circ> f) |` set ks" 415 by (induct ks) (simp_all add: fun_eq_iff restrict_map_insert) 416 417lemma restrict_complement_singleton_eq: 418 "f |` (- {x}) = f(x := None)" 419 by (simp add: restrict_map_def fun_eq_iff) 420 421 422subsection \<open>@{term [source] map_upds}\<close> 423 424lemma map_upds_Nil1 [simp]: "m([] [\<mapsto>] bs) = m" 425by (simp add: map_upds_def) 426 427lemma map_upds_Nil2 [simp]: "m(as [\<mapsto>] []) = m" 428by (simp add:map_upds_def) 429 430lemma map_upds_Cons [simp]: "m(a#as [\<mapsto>] b#bs) = (m(a\<mapsto>b))(as[\<mapsto>]bs)" 431by (simp add:map_upds_def) 432 433lemma map_upds_append1 [simp]: "size xs < size ys \<Longrightarrow> 434 m(xs@[x] [\<mapsto>] ys) = m(xs [\<mapsto>] ys)(x \<mapsto> ys!size xs)" 435apply(induct xs arbitrary: ys m) 436 apply (clarsimp simp add: neq_Nil_conv) 437apply (case_tac ys) 438 apply simp 439apply simp 440done 441 442lemma map_upds_list_update2_drop [simp]: 443 "size xs \<le> i \<Longrightarrow> m(xs[\<mapsto>]ys[i:=y]) = m(xs[\<mapsto>]ys)" 444apply (induct xs arbitrary: m ys i) 445 apply simp 446apply (case_tac ys) 447 apply simp 448apply (simp split: nat.split) 449done 450 451lemma map_upd_upds_conv_if: 452 "(f(x\<mapsto>y))(xs [\<mapsto>] ys) = 453 (if x \<in> set(take (length ys) xs) then f(xs [\<mapsto>] ys) 454 else (f(xs [\<mapsto>] ys))(x\<mapsto>y))" 455apply (induct xs arbitrary: x y ys f) 456 apply simp 457apply (case_tac ys) 458 apply (auto split: if_split simp: fun_upd_twist) 459done 460 461lemma map_upds_twist [simp]: 462 "a \<notin> set as \<Longrightarrow> m(a\<mapsto>b)(as[\<mapsto>]bs) = m(as[\<mapsto>]bs)(a\<mapsto>b)" 463using set_take_subset by (fastforce simp add: map_upd_upds_conv_if) 464 465lemma map_upds_apply_nontin [simp]: 466 "x \<notin> set xs \<Longrightarrow> (f(xs[\<mapsto>]ys)) x = f x" 467apply (induct xs arbitrary: ys) 468 apply simp 469apply (case_tac ys) 470 apply (auto simp: map_upd_upds_conv_if) 471done 472 473lemma fun_upds_append_drop [simp]: 474 "size xs = size ys \<Longrightarrow> m(xs@zs[\<mapsto>]ys) = m(xs[\<mapsto>]ys)" 475apply (induct xs arbitrary: m ys) 476 apply simp 477apply (case_tac ys) 478 apply simp_all 479done 480 481lemma fun_upds_append2_drop [simp]: 482 "size xs = size ys \<Longrightarrow> m(xs[\<mapsto>]ys@zs) = m(xs[\<mapsto>]ys)" 483apply (induct xs arbitrary: m ys) 484 apply simp 485apply (case_tac ys) 486 apply simp_all 487done 488 489 490lemma restrict_map_upds[simp]: 491 "\<lbrakk> length xs = length ys; set xs \<subseteq> D \<rbrakk> 492 \<Longrightarrow> m(xs [\<mapsto>] ys)|`D = (m|`(D - set xs))(xs [\<mapsto>] ys)" 493apply (induct xs arbitrary: m ys) 494 apply simp 495apply (case_tac ys) 496 apply simp 497apply (simp add: Diff_insert [symmetric] insert_absorb) 498apply (simp add: map_upd_upds_conv_if) 499done 500 501 502subsection \<open>@{term [source] dom}\<close> 503 504lemma dom_eq_empty_conv [simp]: "dom f = {} \<longleftrightarrow> f = empty" 505 by (auto simp: dom_def) 506 507lemma domI: "m a = Some b \<Longrightarrow> a \<in> dom m" 508 by (simp add: dom_def) 509(* declare domI [intro]? *) 510 511lemma domD: "a \<in> dom m \<Longrightarrow> \<exists>b. m a = Some b" 512 by (cases "m a") (auto simp add: dom_def) 513 514lemma domIff [iff, simp del, code_unfold]: "a \<in> dom m \<longleftrightarrow> m a \<noteq> None" 515 by (simp add: dom_def) 516 517lemma dom_empty [simp]: "dom empty = {}" 518 by (simp add: dom_def) 519 520lemma dom_fun_upd [simp]: 521 "dom(f(x := y)) = (if y = None then dom f - {x} else insert x (dom f))" 522 by (auto simp: dom_def) 523 524lemma dom_if: 525 "dom (\<lambda>x. if P x then f x else g x) = dom f \<inter> {x. P x} \<union> dom g \<inter> {x. \<not> P x}" 526 by (auto split: if_splits) 527 528lemma dom_map_of_conv_image_fst: 529 "dom (map_of xys) = fst ` set xys" 530 by (induct xys) (auto simp add: dom_if) 531 532lemma dom_map_of_zip [simp]: "length xs = length ys \<Longrightarrow> dom (map_of (zip xs ys)) = set xs" 533 by (induct rule: list_induct2) (auto simp: dom_if) 534 535lemma finite_dom_map_of: "finite (dom (map_of l))" 536 by (induct l) (auto simp: dom_def insert_Collect [symmetric]) 537 538lemma dom_map_upds [simp]: 539 "dom(m(xs[\<mapsto>]ys)) = set(take (length ys) xs) \<union> dom m" 540apply (induct xs arbitrary: m ys) 541 apply simp 542apply (case_tac ys) 543 apply auto 544done 545 546lemma dom_map_add [simp]: "dom (m ++ n) = dom n \<union> dom m" 547 by (auto simp: dom_def) 548 549lemma dom_override_on [simp]: 550 "dom (override_on f g A) = 551 (dom f - {a. a \<in> A - dom g}) \<union> {a. a \<in> A \<inter> dom g}" 552 by (auto simp: dom_def override_on_def) 553 554lemma map_add_comm: "dom m1 \<inter> dom m2 = {} \<Longrightarrow> m1 ++ m2 = m2 ++ m1" 555 by (rule ext) (force simp: map_add_def dom_def split: option.split) 556 557lemma map_add_dom_app_simps: 558 "m \<in> dom l2 \<Longrightarrow> (l1 ++ l2) m = l2 m" 559 "m \<notin> dom l1 \<Longrightarrow> (l1 ++ l2) m = l2 m" 560 "m \<notin> dom l2 \<Longrightarrow> (l1 ++ l2) m = l1 m" 561 by (auto simp add: map_add_def split: option.split_asm) 562 563lemma dom_const [simp]: 564 "dom (\<lambda>x. Some (f x)) = UNIV" 565 by auto 566 567(* Due to John Matthews - could be rephrased with dom *) 568lemma finite_map_freshness: 569 "finite (dom (f :: 'a \<rightharpoonup> 'b)) \<Longrightarrow> \<not> finite (UNIV :: 'a set) \<Longrightarrow> 570 \<exists>x. f x = None" 571 by (bestsimp dest: ex_new_if_finite) 572 573lemma dom_minus: 574 "f x = None \<Longrightarrow> dom f - insert x A = dom f - A" 575 unfolding dom_def by simp 576 577lemma insert_dom: 578 "f x = Some y \<Longrightarrow> insert x (dom f) = dom f" 579 unfolding dom_def by auto 580 581lemma map_of_map_keys: 582 "set xs = dom m \<Longrightarrow> map_of (map (\<lambda>k. (k, the (m k))) xs) = m" 583 by (rule ext) (auto simp add: map_of_map_restrict restrict_map_def) 584 585lemma map_of_eqI: 586 assumes set_eq: "set (map fst xs) = set (map fst ys)" 587 assumes map_eq: "\<forall>k\<in>set (map fst xs). map_of xs k = map_of ys k" 588 shows "map_of xs = map_of ys" 589proof (rule ext) 590 fix k show "map_of xs k = map_of ys k" 591 proof (cases "map_of xs k") 592 case None 593 then have "k \<notin> set (map fst xs)" by (simp add: map_of_eq_None_iff) 594 with set_eq have "k \<notin> set (map fst ys)" by simp 595 then have "map_of ys k = None" by (simp add: map_of_eq_None_iff) 596 with None show ?thesis by simp 597 next 598 case (Some v) 599 then have "k \<in> set (map fst xs)" by (auto simp add: dom_map_of_conv_image_fst [symmetric]) 600 with map_eq show ?thesis by auto 601 qed 602qed 603 604lemma map_of_eq_dom: 605 assumes "map_of xs = map_of ys" 606 shows "fst ` set xs = fst ` set ys" 607proof - 608 from assms have "dom (map_of xs) = dom (map_of ys)" by simp 609 then show ?thesis by (simp add: dom_map_of_conv_image_fst) 610qed 611 612lemma finite_set_of_finite_maps: 613 assumes "finite A" "finite B" 614 shows "finite {m. dom m = A \<and> ran m \<subseteq> B}" (is "finite ?S") 615proof - 616 let ?S' = "{m. \<forall>x. (x \<in> A \<longrightarrow> m x \<in> Some ` B) \<and> (x \<notin> A \<longrightarrow> m x = None)}" 617 have "?S = ?S'" 618 proof 619 show "?S \<subseteq> ?S'" by (auto simp: dom_def ran_def image_def) 620 show "?S' \<subseteq> ?S" 621 proof 622 fix m assume "m \<in> ?S'" 623 hence 1: "dom m = A" by force 624 hence 2: "ran m \<subseteq> B" using \<open>m \<in> ?S'\<close> by (auto simp: dom_def ran_def) 625 from 1 2 show "m \<in> ?S" by blast 626 qed 627 qed 628 with assms show ?thesis by(simp add: finite_set_of_finite_funs) 629qed 630 631 632subsection \<open>@{term [source] ran}\<close> 633 634lemma ranI: "m a = Some b \<Longrightarrow> b \<in> ran m" 635 by (auto simp: ran_def) 636(* declare ranI [intro]? *) 637 638lemma ran_empty [simp]: "ran empty = {}" 639 by (auto simp: ran_def) 640 641lemma ran_map_upd [simp]: "m a = None \<Longrightarrow> ran(m(a\<mapsto>b)) = insert b (ran m)" 642 unfolding ran_def 643apply auto 644apply (subgoal_tac "aa \<noteq> a") 645 apply auto 646done 647 648lemma ran_map_add: 649 assumes "dom m1 \<inter> dom m2 = {}" 650 shows "ran (m1 ++ m2) = ran m1 \<union> ran m2" 651proof 652 show "ran (m1 ++ m2) \<subseteq> ran m1 \<union> ran m2" 653 unfolding ran_def by auto 654next 655 show "ran m1 \<union> ran m2 \<subseteq> ran (m1 ++ m2)" 656 proof - 657 have "(m1 ++ m2) x = Some y" if "m1 x = Some y" for x y 658 using assms map_add_comm that by fastforce 659 moreover have "(m1 ++ m2) x = Some y" if "m2 x = Some y" for x y 660 using assms that by auto 661 ultimately show ?thesis 662 unfolding ran_def by blast 663 qed 664qed 665 666lemma finite_ran: 667 assumes "finite (dom p)" 668 shows "finite (ran p)" 669proof - 670 have "ran p = (\<lambda>x. the (p x)) ` dom p" 671 unfolding ran_def by force 672 from this \<open>finite (dom p)\<close> show ?thesis by auto 673qed 674 675lemma ran_distinct: 676 assumes dist: "distinct (map fst al)" 677 shows "ran (map_of al) = snd ` set al" 678 using assms 679proof (induct al) 680 case Nil 681 then show ?case by simp 682next 683 case (Cons kv al) 684 then have "ran (map_of al) = snd ` set al" by simp 685 moreover from Cons.prems have "map_of al (fst kv) = None" 686 by (simp add: map_of_eq_None_iff) 687 ultimately show ?case by (simp only: map_of.simps ran_map_upd) simp 688qed 689 690lemma ran_map_of_zip: 691 assumes "length xs = length ys" "distinct xs" 692 shows "ran (map_of (zip xs ys)) = set ys" 693using assms by (simp add: ran_distinct set_map[symmetric]) 694 695lemma ran_map_option: "ran (\<lambda>x. map_option f (m x)) = f ` ran m" 696 by (auto simp add: ran_def) 697 698 699subsection \<open>\<open>map_le\<close>\<close> 700 701lemma map_le_empty [simp]: "empty \<subseteq>\<^sub>m g" 702 by (simp add: map_le_def) 703 704lemma upd_None_map_le [simp]: "f(x := None) \<subseteq>\<^sub>m f" 705 by (force simp add: map_le_def) 706 707lemma map_le_upd[simp]: "f \<subseteq>\<^sub>m g ==> f(a := b) \<subseteq>\<^sub>m g(a := b)" 708 by (fastforce simp add: map_le_def) 709 710lemma map_le_imp_upd_le [simp]: "m1 \<subseteq>\<^sub>m m2 \<Longrightarrow> m1(x := None) \<subseteq>\<^sub>m m2(x \<mapsto> y)" 711 by (force simp add: map_le_def) 712 713lemma map_le_upds [simp]: 714 "f \<subseteq>\<^sub>m g \<Longrightarrow> f(as [\<mapsto>] bs) \<subseteq>\<^sub>m g(as [\<mapsto>] bs)" 715apply (induct as arbitrary: f g bs) 716 apply simp 717apply (case_tac bs) 718 apply auto 719done 720 721lemma map_le_implies_dom_le: "(f \<subseteq>\<^sub>m g) \<Longrightarrow> (dom f \<subseteq> dom g)" 722 by (fastforce simp add: map_le_def dom_def) 723 724lemma map_le_refl [simp]: "f \<subseteq>\<^sub>m f" 725 by (simp add: map_le_def) 726 727lemma map_le_trans[trans]: "\<lbrakk> m1 \<subseteq>\<^sub>m m2; m2 \<subseteq>\<^sub>m m3\<rbrakk> \<Longrightarrow> m1 \<subseteq>\<^sub>m m3" 728 by (auto simp add: map_le_def dom_def) 729 730lemma map_le_antisym: "\<lbrakk> f \<subseteq>\<^sub>m g; g \<subseteq>\<^sub>m f \<rbrakk> \<Longrightarrow> f = g" 731unfolding map_le_def 732apply (rule ext) 733apply (case_tac "x \<in> dom f", simp) 734apply (case_tac "x \<in> dom g", simp, fastforce) 735done 736 737lemma map_le_map_add [simp]: "f \<subseteq>\<^sub>m g ++ f" 738 by (fastforce simp: map_le_def) 739 740lemma map_le_iff_map_add_commute: "f \<subseteq>\<^sub>m f ++ g \<longleftrightarrow> f ++ g = g ++ f" 741 by (fastforce simp: map_add_def map_le_def fun_eq_iff split: option.splits) 742 743lemma map_add_le_mapE: "f ++ g \<subseteq>\<^sub>m h \<Longrightarrow> g \<subseteq>\<^sub>m h" 744 by (fastforce simp: map_le_def map_add_def dom_def) 745 746lemma map_add_le_mapI: "\<lbrakk> f \<subseteq>\<^sub>m h; g \<subseteq>\<^sub>m h \<rbrakk> \<Longrightarrow> f ++ g \<subseteq>\<^sub>m h" 747 by (auto simp: map_le_def map_add_def dom_def split: option.splits) 748 749lemma map_add_subsumed1: "f \<subseteq>\<^sub>m g \<Longrightarrow> f++g = g" 750by (simp add: map_add_le_mapI map_le_antisym) 751 752lemma map_add_subsumed2: "f \<subseteq>\<^sub>m g \<Longrightarrow> g++f = g" 753by (metis map_add_subsumed1 map_le_iff_map_add_commute) 754 755lemma dom_eq_singleton_conv: "dom f = {x} \<longleftrightarrow> (\<exists>v. f = [x \<mapsto> v])" 756 (is "?lhs \<longleftrightarrow> ?rhs") 757proof 758 assume ?rhs 759 then show ?lhs by (auto split: if_split_asm) 760next 761 assume ?lhs 762 then obtain v where v: "f x = Some v" by auto 763 show ?rhs 764 proof 765 show "f = [x \<mapsto> v]" 766 proof (rule map_le_antisym) 767 show "[x \<mapsto> v] \<subseteq>\<^sub>m f" 768 using v by (auto simp add: map_le_def) 769 show "f \<subseteq>\<^sub>m [x \<mapsto> v]" 770 using \<open>dom f = {x}\<close> \<open>f x = Some v\<close> by (auto simp add: map_le_def) 771 qed 772 qed 773qed 774 775lemma map_add_eq_empty_iff[simp]: 776 "(f++g = empty) \<longleftrightarrow> f = empty \<and> g = empty" 777by (metis map_add_None) 778 779lemma empty_eq_map_add_iff[simp]: 780 "(empty = f++g) \<longleftrightarrow> f = empty \<and> g = empty" 781by(subst map_add_eq_empty_iff[symmetric])(rule eq_commute) 782 783 784subsection \<open>Various\<close> 785 786lemma set_map_of_compr: 787 assumes distinct: "distinct (map fst xs)" 788 shows "set xs = {(k, v). map_of xs k = Some v}" 789 using assms 790proof (induct xs) 791 case Nil 792 then show ?case by simp 793next 794 case (Cons x xs) 795 obtain k v where "x = (k, v)" by (cases x) blast 796 with Cons.prems have "k \<notin> dom (map_of xs)" 797 by (simp add: dom_map_of_conv_image_fst) 798 then have *: "insert (k, v) {(k, v). map_of xs k = Some v} = 799 {(k', v'). (map_of xs(k \<mapsto> v)) k' = Some v'}" 800 by (auto split: if_splits) 801 from Cons have "set xs = {(k, v). map_of xs k = Some v}" by simp 802 with * \<open>x = (k, v)\<close> show ?case by simp 803qed 804 805lemma eq_key_imp_eq_value: 806 "v1 = v2" 807 if "distinct (map fst xs)" "(k, v1) \<in> set xs" "(k, v2) \<in> set xs" 808proof - 809 from that have "inj_on fst (set xs)" 810 by (simp add: distinct_map) 811 moreover have "fst (k, v1) = fst (k, v2)" 812 by simp 813 ultimately have "(k, v1) = (k, v2)" 814 by (rule inj_onD) (fact that)+ 815 then show ?thesis 816 by simp 817qed 818 819lemma map_of_inject_set: 820 assumes distinct: "distinct (map fst xs)" "distinct (map fst ys)" 821 shows "map_of xs = map_of ys \<longleftrightarrow> set xs = set ys" (is "?lhs \<longleftrightarrow> ?rhs") 822proof 823 assume ?lhs 824 moreover from \<open>distinct (map fst xs)\<close> have "set xs = {(k, v). map_of xs k = Some v}" 825 by (rule set_map_of_compr) 826 moreover from \<open>distinct (map fst ys)\<close> have "set ys = {(k, v). map_of ys k = Some v}" 827 by (rule set_map_of_compr) 828 ultimately show ?rhs by simp 829next 830 assume ?rhs show ?lhs 831 proof 832 fix k 833 show "map_of xs k = map_of ys k" 834 proof (cases "map_of xs k") 835 case None 836 with \<open>?rhs\<close> have "map_of ys k = None" 837 by (simp add: map_of_eq_None_iff) 838 with None show ?thesis by simp 839 next 840 case (Some v) 841 with distinct \<open>?rhs\<close> have "map_of ys k = Some v" 842 by simp 843 with Some show ?thesis by simp 844 qed 845 qed 846qed 847 848hide_const (open) Map.empty 849 850end 851