author | wenzelm |
Wed, 07 Mar 2018 19:02:22 +0100 | |
changeset 67780 | 7655e6369c9f |
parent 67091 | 1393c2340eec |
child 68450 | 41de07c7a0f3 |
permissions | -rw-r--r-- |
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(* Title: HOL/Map.thy |
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Author: Tobias Nipkow, based on a theory by David von Oheimb |
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Copyright 1997-2003 TU Muenchen |
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The datatype of "maps"; strongly resembles maps in VDM. |
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*) |
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section \<open>Maps\<close> |
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theory Map |
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imports List |
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abbrevs "(=" = "\<subseteq>\<^sub>m" |
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begin |
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type_synonym ('a, 'b) "map" = "'a \<Rightarrow> 'b option" (infixr "\<rightharpoonup>" 0) |
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abbreviation |
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empty :: "'a \<rightharpoonup> 'b" where |
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"empty \<equiv> \<lambda>x. None" |
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definition |
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map_comp :: "('b \<rightharpoonup> 'c) \<Rightarrow> ('a \<rightharpoonup> 'b) \<Rightarrow> ('a \<rightharpoonup> 'c)" (infixl "\<circ>\<^sub>m" 55) where |
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"f \<circ>\<^sub>m g = (\<lambda>k. case g k of None \<Rightarrow> None | Some v \<Rightarrow> f v)" |
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definition |
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map_add :: "('a \<rightharpoonup> 'b) \<Rightarrow> ('a \<rightharpoonup> 'b) \<Rightarrow> ('a \<rightharpoonup> 'b)" (infixl "++" 100) where |
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"m1 ++ m2 = (\<lambda>x. case m2 x of None \<Rightarrow> m1 x | Some y \<Rightarrow> Some y)" |
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definition |
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restrict_map :: "('a \<rightharpoonup> 'b) \<Rightarrow> 'a set \<Rightarrow> ('a \<rightharpoonup> 'b)" (infixl "|`" 110) where |
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"m|`A = (\<lambda>x. if x \<in> A then m x else None)" |
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notation (latex output) |
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restrict_map ("_\<restriction>\<^bsub>_\<^esub>" [111,110] 110) |
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definition |
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dom :: "('a \<rightharpoonup> 'b) \<Rightarrow> 'a set" where |
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"dom m = {a. m a \<noteq> None}" |
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definition |
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ran :: "('a \<rightharpoonup> 'b) \<Rightarrow> 'b set" where |
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"ran m = {b. \<exists>a. m a = Some b}" |
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definition |
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map_le :: "('a \<rightharpoonup> 'b) \<Rightarrow> ('a \<rightharpoonup> 'b) \<Rightarrow> bool" (infix "\<subseteq>\<^sub>m" 50) where |
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"(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)" |
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nonterminal maplets and maplet |
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syntax |
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"_maplet" :: "['a, 'a] \<Rightarrow> maplet" ("_ /\<mapsto>/ _") |
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"_maplets" :: "['a, 'a] \<Rightarrow> maplet" ("_ /[\<mapsto>]/ _") |
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"" :: "maplet \<Rightarrow> maplets" ("_") |
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"_Maplets" :: "[maplet, maplets] \<Rightarrow> maplets" ("_,/ _") |
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"_MapUpd" :: "['a \<rightharpoonup> 'b, maplets] \<Rightarrow> 'a \<rightharpoonup> 'b" ("_/'(_')" [900, 0] 900) |
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"_Map" :: "maplets \<Rightarrow> 'a \<rightharpoonup> 'b" ("(1[_])") |
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syntax (ASCII) |
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"_maplet" :: "['a, 'a] \<Rightarrow> maplet" ("_ /|->/ _") |
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"_maplets" :: "['a, 'a] \<Rightarrow> maplet" ("_ /[|->]/ _") |
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translations |
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"_MapUpd m (_Maplets xy ms)" \<rightleftharpoons> "_MapUpd (_MapUpd m xy) ms" |
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"_MapUpd m (_maplet x y)" \<rightleftharpoons> "m(x := CONST Some y)" |
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"_Map ms" \<rightleftharpoons> "_MapUpd (CONST empty) ms" |
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"_Map (_Maplets ms1 ms2)" \<leftharpoondown> "_MapUpd (_Map ms1) ms2" |
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"_Maplets ms1 (_Maplets ms2 ms3)" \<leftharpoondown> "_Maplets (_Maplets ms1 ms2) ms3" |
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primrec map_of :: "('a \<times> 'b) list \<Rightarrow> 'a \<rightharpoonup> 'b" |
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where |
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"map_of [] = empty" |
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| "map_of (p # ps) = (map_of ps)(fst p \<mapsto> snd p)" |
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definition map_upds :: "('a \<rightharpoonup> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b list \<Rightarrow> 'a \<rightharpoonup> 'b" |
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where "map_upds m xs ys = m ++ map_of (rev (zip xs ys))" |
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translations |
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"_MapUpd m (_maplets x y)" \<rightleftharpoons> "CONST map_upds m x y" |
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lemma map_of_Cons_code [code]: |
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"map_of [] k = None" |
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"map_of ((l, v) # ps) k = (if l = k then Some v else map_of ps k)" |
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by simp_all |
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subsection \<open>@{term [source] empty}\<close> |
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lemma empty_upd_none [simp]: "empty(x := None) = empty" |
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by (rule ext) simp |
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subsection \<open>@{term [source] map_upd}\<close> |
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lemma map_upd_triv: "t k = Some x \<Longrightarrow> t(k\<mapsto>x) = t" |
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by (rule ext) simp |
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lemma map_upd_nonempty [simp]: "t(k\<mapsto>x) \<noteq> empty" |
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proof |
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assume "t(k \<mapsto> x) = empty" |
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then have "(t(k \<mapsto> x)) k = None" by simp |
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then show False by simp |
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qed |
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lemma map_upd_eqD1: |
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assumes "m(a\<mapsto>x) = n(a\<mapsto>y)" |
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shows "x = y" |
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proof - |
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from assms have "(m(a\<mapsto>x)) a = (n(a\<mapsto>y)) a" by simp |
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then show ?thesis by simp |
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qed |
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lemma map_upd_Some_unfold: |
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"((m(a\<mapsto>b)) x = Some y) = (x = a \<and> b = y \<or> x \<noteq> a \<and> m x = Some y)" |
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by auto |
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lemma image_map_upd [simp]: "x \<notin> A \<Longrightarrow> m(x \<mapsto> y) ` A = m ` A" |
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by auto |
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lemma finite_range_updI: "finite (range f) \<Longrightarrow> finite (range (f(a\<mapsto>b)))" |
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unfolding image_def |
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apply (simp (no_asm_use) add:full_SetCompr_eq) |
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apply (rule finite_subset) |
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prefer 2 apply assumption |
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apply (auto) |
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done |
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subsection \<open>@{term [source] map_of}\<close> |
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lemma map_of_eq_None_iff: |
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"(map_of xys x = None) = (x \<notin> fst ` (set xys))" |
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by (induct xys) simp_all |
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lemma map_of_eq_Some_iff [simp]: |
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"distinct(map fst xys) \<Longrightarrow> (map_of xys x = Some y) = ((x,y) \<in> set xys)" |
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apply (induct xys) |
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apply simp |
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apply (auto simp: map_of_eq_None_iff [symmetric]) |
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done |
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lemma Some_eq_map_of_iff [simp]: |
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"distinct(map fst xys) \<Longrightarrow> (Some y = map_of xys x) = ((x,y) \<in> set xys)" |
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by (auto simp del: map_of_eq_Some_iff simp: map_of_eq_Some_iff [symmetric]) |
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lemma map_of_is_SomeI [simp]: "\<lbrakk> distinct(map fst xys); (x,y) \<in> set xys \<rbrakk> |
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\<Longrightarrow> map_of xys x = Some y" |
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apply (induct xys) |
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apply simp |
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apply force |
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done |
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lemma map_of_zip_is_None [simp]: |
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"length xs = length ys \<Longrightarrow> (map_of (zip xs ys) x = None) = (x \<notin> set xs)" |
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by (induct rule: list_induct2) simp_all |
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lemma map_of_zip_is_Some: |
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assumes "length xs = length ys" |
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shows "x \<in> set xs \<longleftrightarrow> (\<exists>y. map_of (zip xs ys) x = Some y)" |
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using assms by (induct rule: list_induct2) simp_all |
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lemma map_of_zip_upd: |
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fixes x :: 'a and xs :: "'a list" and ys zs :: "'b list" |
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assumes "length ys = length xs" |
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and "length zs = length xs" |
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and "x \<notin> set xs" |
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and "map_of (zip xs ys)(x \<mapsto> y) = map_of (zip xs zs)(x \<mapsto> z)" |
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shows "map_of (zip xs ys) = map_of (zip xs zs)" |
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proof |
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fix x' :: 'a |
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show "map_of (zip xs ys) x' = map_of (zip xs zs) x'" |
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proof (cases "x = x'") |
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case True |
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from assms True map_of_zip_is_None [of xs ys x'] |
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have "map_of (zip xs ys) x' = None" by simp |
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moreover from assms True map_of_zip_is_None [of xs zs x'] |
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have "map_of (zip xs zs) x' = None" by simp |
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ultimately show ?thesis by simp |
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next |
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case False from assms |
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have "(map_of (zip xs ys)(x \<mapsto> y)) x' = (map_of (zip xs zs)(x \<mapsto> z)) x'" by auto |
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with False show ?thesis by simp |
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qed |
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qed |
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lemma map_of_zip_inject: |
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assumes "length ys = length xs" |
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and "length zs = length xs" |
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and dist: "distinct xs" |
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and map_of: "map_of (zip xs ys) = map_of (zip xs zs)" |
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shows "ys = zs" |
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using assms(1) assms(2)[symmetric] |
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using dist map_of |
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proof (induct ys xs zs rule: list_induct3) |
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case Nil show ?case by simp |
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next |
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case (Cons y ys x xs z zs) |
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from \<open>map_of (zip (x#xs) (y#ys)) = map_of (zip (x#xs) (z#zs))\<close> |
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have map_of: "map_of (zip xs ys)(x \<mapsto> y) = map_of (zip xs zs)(x \<mapsto> z)" by simp |
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from Cons have "length ys = length xs" and "length zs = length xs" |
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and "x \<notin> set xs" by simp_all |
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then have "map_of (zip xs ys) = map_of (zip xs zs)" using map_of by (rule map_of_zip_upd) |
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with Cons.hyps \<open>distinct (x # xs)\<close> have "ys = zs" by simp |
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moreover from map_of have "y = z" by (rule map_upd_eqD1) |
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ultimately show ?case by simp |
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qed |
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lemma map_of_zip_nth: |
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assumes "length xs = length ys" |
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assumes "distinct xs" |
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assumes "i < length ys" |
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shows "map_of (zip xs ys) (xs ! i) = Some (ys ! i)" |
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using assms proof (induct arbitrary: i rule: list_induct2) |
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case Nil |
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then show ?case by simp |
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next |
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case (Cons x xs y ys) |
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then show ?case |
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using less_Suc_eq_0_disj by auto |
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qed |
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||
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lemma map_of_zip_map: |
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"map_of (zip xs (map f xs)) = (\<lambda>x. if x \<in> set xs then Some (f x) else None)" |
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by (induct xs) (simp_all add: fun_eq_iff) |
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lemma finite_range_map_of: "finite (range (map_of xys))" |
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apply (induct xys) |
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apply (simp_all add: image_constant) |
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apply (rule finite_subset) |
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prefer 2 apply assumption |
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apply auto |
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done |
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lemma map_of_SomeD: "map_of xs k = Some y \<Longrightarrow> (k, y) \<in> set xs" |
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by (induct xs) (auto split: if_splits) |
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lemma map_of_mapk_SomeI: |
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"inj f \<Longrightarrow> map_of t k = Some x \<Longrightarrow> |
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map_of (map (case_prod (\<lambda>k. Pair (f k))) t) (f k) = Some x" |
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by (induct t) (auto simp: inj_eq) |
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lemma weak_map_of_SomeI: "(k, x) \<in> set l \<Longrightarrow> \<exists>x. map_of l k = Some x" |
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by (induct l) auto |
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lemma map_of_filter_in: |
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"map_of xs k = Some z \<Longrightarrow> P k z \<Longrightarrow> map_of (filter (case_prod P) xs) k = Some z" |
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by (induct xs) auto |
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lemma map_of_map: |
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"map_of (map (\<lambda>(k, v). (k, f v)) xs) = map_option f \<circ> map_of xs" |
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by (induct xs) (auto simp: fun_eq_iff) |
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lemma dom_map_option: |
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"dom (\<lambda>k. map_option (f k) (m k)) = dom m" |
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by (simp add: dom_def) |
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lemma dom_map_option_comp [simp]: |
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"dom (map_option g \<circ> m) = dom m" |
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using dom_map_option [of "\<lambda>_. g" m] by (simp add: comp_def) |
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subsection \<open>@{const map_option} related\<close> |
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lemma map_option_o_empty [simp]: "map_option f \<circ> empty = empty" |
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by (rule ext) simp |
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lemma map_option_o_map_upd [simp]: |
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"map_option f \<circ> m(a\<mapsto>b) = (map_option f \<circ> m)(a\<mapsto>f b)" |
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by (rule ext) simp |
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subsection \<open>@{term [source] map_comp} related\<close> |
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lemma map_comp_empty [simp]: |
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"m \<circ>\<^sub>m empty = empty" |
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"empty \<circ>\<^sub>m m = empty" |
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by (auto simp: map_comp_def split: option.splits) |
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lemma map_comp_simps [simp]: |
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"m2 k = None \<Longrightarrow> (m1 \<circ>\<^sub>m m2) k = None" |
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"m2 k = Some k' \<Longrightarrow> (m1 \<circ>\<^sub>m m2) k = m1 k'" |
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by (auto simp: map_comp_def) |
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|
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lemma map_comp_Some_iff: |
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"((m1 \<circ>\<^sub>m m2) k = Some v) = (\<exists>k'. m2 k = Some k' \<and> m1 k' = Some v)" |
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by (auto simp: map_comp_def split: option.splits) |
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|
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lemma map_comp_None_iff: |
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"((m1 \<circ>\<^sub>m m2) k = None) = (m2 k = None \<or> (\<exists>k'. m2 k = Some k' \<and> m1 k' = None)) " |
60839 | 288 |
by (auto simp: map_comp_def split: option.splits) |
13908 | 289 |
|
20800 | 290 |
|
61799 | 291 |
subsection \<open>\<open>++\<close>\<close> |
13908 | 292 |
|
14025 | 293 |
lemma map_add_empty[simp]: "m ++ empty = m" |
24331 | 294 |
by(simp add: map_add_def) |
13908 | 295 |
|
14025 | 296 |
lemma empty_map_add[simp]: "empty ++ m = m" |
24331 | 297 |
by (rule ext) (simp add: map_add_def split: option.split) |
13908 | 298 |
|
14025 | 299 |
lemma map_add_assoc[simp]: "m1 ++ (m2 ++ m3) = (m1 ++ m2) ++ m3" |
24331 | 300 |
by (rule ext) (simp add: map_add_def split: option.split) |
20800 | 301 |
|
302 |
lemma map_add_Some_iff: |
|
67091 | 303 |
"((m ++ n) k = Some x) = (n k = Some x \<or> n k = None \<and> m k = Some x)" |
24331 | 304 |
by (simp add: map_add_def split: option.split) |
14025 | 305 |
|
20800 | 306 |
lemma map_add_SomeD [dest!]: |
24331 | 307 |
"(m ++ n) k = Some x \<Longrightarrow> n k = Some x \<or> n k = None \<and> m k = Some x" |
308 |
by (rule map_add_Some_iff [THEN iffD1]) |
|
13908 | 309 |
|
60839 | 310 |
lemma map_add_find_right [simp]: "n k = Some xx \<Longrightarrow> (m ++ n) k = Some xx" |
24331 | 311 |
by (subst map_add_Some_iff) fast |
13908 | 312 |
|
67091 | 313 |
lemma map_add_None [iff]: "((m ++ n) k = None) = (n k = None \<and> m k = None)" |
24331 | 314 |
by (simp add: map_add_def split: option.split) |
13908 | 315 |
|
60838 | 316 |
lemma map_add_upd[simp]: "f ++ g(x\<mapsto>y) = (f ++ g)(x\<mapsto>y)" |
24331 | 317 |
by (rule ext) (simp add: map_add_def) |
13908 | 318 |
|
14186 | 319 |
lemma map_add_upds[simp]: "m1 ++ (m2(xs[\<mapsto>]ys)) = (m1++m2)(xs[\<mapsto>]ys)" |
24331 | 320 |
by (simp add: map_upds_def) |
14186 | 321 |
|
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322 |
lemma map_add_upd_left: "m\<notin>dom e2 \<Longrightarrow> e1(m \<mapsto> u1) ++ e2 = (e1 ++ e2)(m \<mapsto> u1)" |
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323 |
by (rule ext) (auto simp: map_add_def dom_def split: option.split) |
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324 |
|
20800 | 325 |
lemma map_of_append[simp]: "map_of (xs @ ys) = map_of ys ++ map_of xs" |
24331 | 326 |
unfolding map_add_def |
327 |
apply (induct xs) |
|
328 |
apply simp |
|
329 |
apply (rule ext) |
|
63648 | 330 |
apply (simp split: option.split) |
24331 | 331 |
done |
13908 | 332 |
|
14025 | 333 |
lemma finite_range_map_of_map_add: |
60839 | 334 |
"finite (range f) \<Longrightarrow> finite (range (f ++ map_of l))" |
24331 | 335 |
apply (induct l) |
336 |
apply (auto simp del: fun_upd_apply) |
|
337 |
apply (erule finite_range_updI) |
|
338 |
done |
|
13908 | 339 |
|
20800 | 340 |
lemma inj_on_map_add_dom [iff]: |
24331 | 341 |
"inj_on (m ++ m') (dom m') = inj_on m' (dom m')" |
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342 |
by (fastforce simp: map_add_def dom_def inj_on_def split: option.splits) |
20800 | 343 |
|
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344 |
lemma map_upds_fold_map_upd: |
35552 | 345 |
"m(ks[\<mapsto>]vs) = foldl (\<lambda>m (k, v). m(k \<mapsto> v)) m (zip ks vs)" |
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346 |
unfolding map_upds_def proof (rule sym, rule zip_obtain_same_length) |
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347 |
fix ks :: "'a list" and vs :: "'b list" |
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348 |
assume "length ks = length vs" |
35552 | 349 |
then show "foldl (\<lambda>m (k, v). m(k\<mapsto>v)) m (zip ks vs) = m ++ map_of (rev (zip ks vs))" |
350 |
by(induct arbitrary: m rule: list_induct2) simp_all |
|
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351 |
qed |
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|
352 |
|
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|
353 |
lemma map_add_map_of_foldr: |
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|
354 |
"m ++ map_of ps = foldr (\<lambda>(k, v) m. m(k \<mapsto> v)) ps m" |
60839 | 355 |
by (induct ps) (auto simp: fun_eq_iff map_add_def) |
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356 |
|
15304 | 357 |
|
60758 | 358 |
subsection \<open>@{term [source] restrict_map}\<close> |
14100 | 359 |
|
20800 | 360 |
lemma restrict_map_to_empty [simp]: "m|`{} = empty" |
24331 | 361 |
by (simp add: restrict_map_def) |
14186 | 362 |
|
31380 | 363 |
lemma restrict_map_insert: "f |` (insert a A) = (f |` A)(a := f a)" |
60839 | 364 |
by (auto simp: restrict_map_def) |
31380 | 365 |
|
20800 | 366 |
lemma restrict_map_empty [simp]: "empty|`D = empty" |
24331 | 367 |
by (simp add: restrict_map_def) |
14186 | 368 |
|
15693 | 369 |
lemma restrict_in [simp]: "x \<in> A \<Longrightarrow> (m|`A) x = m x" |
24331 | 370 |
by (simp add: restrict_map_def) |
14100 | 371 |
|
15693 | 372 |
lemma restrict_out [simp]: "x \<notin> A \<Longrightarrow> (m|`A) x = None" |
24331 | 373 |
by (simp add: restrict_map_def) |
14100 | 374 |
|
15693 | 375 |
lemma ran_restrictD: "y \<in> ran (m|`A) \<Longrightarrow> \<exists>x\<in>A. m x = Some y" |
62390 | 376 |
by (auto simp: restrict_map_def ran_def split: if_split_asm) |
14100 | 377 |
|
15693 | 378 |
lemma dom_restrict [simp]: "dom (m|`A) = dom m \<inter> A" |
62390 | 379 |
by (auto simp: restrict_map_def dom_def split: if_split_asm) |
14100 | 380 |
|
15693 | 381 |
lemma restrict_upd_same [simp]: "m(x\<mapsto>y)|`(-{x}) = m|`(-{x})" |
24331 | 382 |
by (rule ext) (auto simp: restrict_map_def) |
14100 | 383 |
|
15693 | 384 |
lemma restrict_restrict [simp]: "m|`A|`B = m|`(A\<inter>B)" |
24331 | 385 |
by (rule ext) (auto simp: restrict_map_def) |
14100 | 386 |
|
20800 | 387 |
lemma restrict_fun_upd [simp]: |
24331 | 388 |
"m(x := y)|`D = (if x \<in> D then (m|`(D-{x}))(x := y) else m|`D)" |
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|
389 |
by (simp add: restrict_map_def fun_eq_iff) |
14186 | 390 |
|
20800 | 391 |
lemma fun_upd_None_restrict [simp]: |
60839 | 392 |
"(m|`D)(x := None) = (if x \<in> D then m|`(D - {x}) else m|`D)" |
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|
393 |
by (simp add: restrict_map_def fun_eq_iff) |
14186 | 394 |
|
20800 | 395 |
lemma fun_upd_restrict: "(m|`D)(x := y) = (m|`(D-{x}))(x := y)" |
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|
396 |
by (simp add: restrict_map_def fun_eq_iff) |
14186 | 397 |
|
20800 | 398 |
lemma fun_upd_restrict_conv [simp]: |
24331 | 399 |
"x \<in> D \<Longrightarrow> (m|`D)(x := y) = (m|`(D-{x}))(x := y)" |
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|
400 |
by (simp add: restrict_map_def fun_eq_iff) |
14186 | 401 |
|
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|
402 |
lemma map_of_map_restrict: |
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|
403 |
"map_of (map (\<lambda>k. (k, f k)) ks) = (Some \<circ> f) |` set ks" |
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|
404 |
by (induct ks) (simp_all add: fun_eq_iff restrict_map_insert) |
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|
405 |
|
35619 | 406 |
lemma restrict_complement_singleton_eq: |
407 |
"f |` (- {x}) = f(x := None)" |
|
39302
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|
408 |
by (simp add: restrict_map_def fun_eq_iff) |
35619 | 409 |
|
14100 | 410 |
|
60758 | 411 |
subsection \<open>@{term [source] map_upds}\<close> |
14025 | 412 |
|
60838 | 413 |
lemma map_upds_Nil1 [simp]: "m([] [\<mapsto>] bs) = m" |
24331 | 414 |
by (simp add: map_upds_def) |
14025 | 415 |
|
60838 | 416 |
lemma map_upds_Nil2 [simp]: "m(as [\<mapsto>] []) = m" |
24331 | 417 |
by (simp add:map_upds_def) |
20800 | 418 |
|
60838 | 419 |
lemma map_upds_Cons [simp]: "m(a#as [\<mapsto>] b#bs) = (m(a\<mapsto>b))(as[\<mapsto>]bs)" |
24331 | 420 |
by (simp add:map_upds_def) |
14025 | 421 |
|
60839 | 422 |
lemma map_upds_append1 [simp]: "size xs < size ys \<Longrightarrow> |
24331 | 423 |
m(xs@[x] [\<mapsto>] ys) = m(xs [\<mapsto>] ys)(x \<mapsto> ys!size xs)" |
60839 | 424 |
apply(induct xs arbitrary: ys m) |
24331 | 425 |
apply (clarsimp simp add: neq_Nil_conv) |
426 |
apply (case_tac ys) |
|
427 |
apply simp |
|
428 |
apply simp |
|
429 |
done |
|
14187 | 430 |
|
20800 | 431 |
lemma map_upds_list_update2_drop [simp]: |
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|
432 |
"size xs \<le> i \<Longrightarrow> m(xs[\<mapsto>]ys[i:=y]) = m(xs[\<mapsto>]ys)" |
24331 | 433 |
apply (induct xs arbitrary: m ys i) |
434 |
apply simp |
|
435 |
apply (case_tac ys) |
|
436 |
apply simp |
|
437 |
apply (simp split: nat.split) |
|
438 |
done |
|
14025 | 439 |
|
20800 | 440 |
lemma map_upd_upds_conv_if: |
60838 | 441 |
"(f(x\<mapsto>y))(xs [\<mapsto>] ys) = |
60839 | 442 |
(if x \<in> set(take (length ys) xs) then f(xs [\<mapsto>] ys) |
60838 | 443 |
else (f(xs [\<mapsto>] ys))(x\<mapsto>y))" |
24331 | 444 |
apply (induct xs arbitrary: x y ys f) |
445 |
apply simp |
|
446 |
apply (case_tac ys) |
|
62390 | 447 |
apply (auto split: if_split simp: fun_upd_twist) |
24331 | 448 |
done |
14025 | 449 |
|
450 |
lemma map_upds_twist [simp]: |
|
60839 | 451 |
"a \<notin> set as \<Longrightarrow> m(a\<mapsto>b)(as[\<mapsto>]bs) = m(as[\<mapsto>]bs)(a\<mapsto>b)" |
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|
452 |
using set_take_subset by (fastforce simp add: map_upd_upds_conv_if) |
14025 | 453 |
|
20800 | 454 |
lemma map_upds_apply_nontin [simp]: |
60839 | 455 |
"x \<notin> set xs \<Longrightarrow> (f(xs[\<mapsto>]ys)) x = f x" |
24331 | 456 |
apply (induct xs arbitrary: ys) |
457 |
apply simp |
|
458 |
apply (case_tac ys) |
|
459 |
apply (auto simp: map_upd_upds_conv_if) |
|
460 |
done |
|
14025 | 461 |
|
20800 | 462 |
lemma fun_upds_append_drop [simp]: |
24331 | 463 |
"size xs = size ys \<Longrightarrow> m(xs@zs[\<mapsto>]ys) = m(xs[\<mapsto>]ys)" |
464 |
apply (induct xs arbitrary: m ys) |
|
465 |
apply simp |
|
466 |
apply (case_tac ys) |
|
467 |
apply simp_all |
|
468 |
done |
|
14300 | 469 |
|
20800 | 470 |
lemma fun_upds_append2_drop [simp]: |
24331 | 471 |
"size xs = size ys \<Longrightarrow> m(xs[\<mapsto>]ys@zs) = m(xs[\<mapsto>]ys)" |
472 |
apply (induct xs arbitrary: m ys) |
|
473 |
apply simp |
|
474 |
apply (case_tac ys) |
|
475 |
apply simp_all |
|
476 |
done |
|
14300 | 477 |
|
478 |
||
20800 | 479 |
lemma restrict_map_upds[simp]: |
480 |
"\<lbrakk> length xs = length ys; set xs \<subseteq> D \<rbrakk> |
|
481 |
\<Longrightarrow> m(xs [\<mapsto>] ys)|`D = (m|`(D - set xs))(xs [\<mapsto>] ys)" |
|
24331 | 482 |
apply (induct xs arbitrary: m ys) |
483 |
apply simp |
|
484 |
apply (case_tac ys) |
|
485 |
apply simp |
|
486 |
apply (simp add: Diff_insert [symmetric] insert_absorb) |
|
487 |
apply (simp add: map_upd_upds_conv_if) |
|
488 |
done |
|
14186 | 489 |
|
490 |
||
60758 | 491 |
subsection \<open>@{term [source] dom}\<close> |
13908 | 492 |
|
31080 | 493 |
lemma dom_eq_empty_conv [simp]: "dom f = {} \<longleftrightarrow> f = empty" |
44921 | 494 |
by (auto simp: dom_def) |
31080 | 495 |
|
60839 | 496 |
lemma domI: "m a = Some b \<Longrightarrow> a \<in> dom m" |
497 |
by (simp add: dom_def) |
|
14100 | 498 |
(* declare domI [intro]? *) |
13908 | 499 |
|
60839 | 500 |
lemma domD: "a \<in> dom m \<Longrightarrow> \<exists>b. m a = Some b" |
501 |
by (cases "m a") (auto simp add: dom_def) |
|
13908 | 502 |
|
66010 | 503 |
lemma domIff [iff, simp del, code_unfold]: "a \<in> dom m \<longleftrightarrow> m a \<noteq> None" |
60839 | 504 |
by (simp add: dom_def) |
13908 | 505 |
|
20800 | 506 |
lemma dom_empty [simp]: "dom empty = {}" |
60839 | 507 |
by (simp add: dom_def) |
13908 | 508 |
|
20800 | 509 |
lemma dom_fun_upd [simp]: |
60839 | 510 |
"dom(f(x := y)) = (if y = None then dom f - {x} else insert x (dom f))" |
511 |
by (auto simp: dom_def) |
|
13908 | 512 |
|
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513 |
lemma dom_if: |
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514 |
"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}" |
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|
515 |
by (auto split: if_splits) |
13937 | 516 |
|
15304 | 517 |
lemma dom_map_of_conv_image_fst: |
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|
518 |
"dom (map_of xys) = fst ` set xys" |
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|
519 |
by (induct xys) (auto simp add: dom_if) |
15304 | 520 |
|
60839 | 521 |
lemma dom_map_of_zip [simp]: "length xs = length ys \<Longrightarrow> dom (map_of (zip xs ys)) = set xs" |
522 |
by (induct rule: list_induct2) (auto simp: dom_if) |
|
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|
523 |
|
13908 | 524 |
lemma finite_dom_map_of: "finite (dom (map_of l))" |
60839 | 525 |
by (induct l) (auto simp: dom_def insert_Collect [symmetric]) |
13908 | 526 |
|
20800 | 527 |
lemma dom_map_upds [simp]: |
60839 | 528 |
"dom(m(xs[\<mapsto>]ys)) = set(take (length ys) xs) \<union> dom m" |
24331 | 529 |
apply (induct xs arbitrary: m ys) |
530 |
apply simp |
|
531 |
apply (case_tac ys) |
|
532 |
apply auto |
|
533 |
done |
|
13910 | 534 |
|
60839 | 535 |
lemma dom_map_add [simp]: "dom (m ++ n) = dom n \<union> dom m" |
536 |
by (auto simp: dom_def) |
|
13910 | 537 |
|
20800 | 538 |
lemma dom_override_on [simp]: |
60839 | 539 |
"dom (override_on f g A) = |
540 |
(dom f - {a. a \<in> A - dom g}) \<union> {a. a \<in> A \<inter> dom g}" |
|
541 |
by (auto simp: dom_def override_on_def) |
|
13908 | 542 |
|
60839 | 543 |
lemma map_add_comm: "dom m1 \<inter> dom m2 = {} \<Longrightarrow> m1 ++ m2 = m2 ++ m1" |
544 |
by (rule ext) (force simp: map_add_def dom_def split: option.split) |
|
20800 | 545 |
|
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|
546 |
lemma map_add_dom_app_simps: |
60839 | 547 |
"m \<in> dom l2 \<Longrightarrow> (l1 ++ l2) m = l2 m" |
548 |
"m \<notin> dom l1 \<Longrightarrow> (l1 ++ l2) m = l2 m" |
|
549 |
"m \<notin> dom l2 \<Longrightarrow> (l1 ++ l2) m = l1 m" |
|
550 |
by (auto simp add: map_add_def split: option.split_asm) |
|
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|
551 |
|
29622 | 552 |
lemma dom_const [simp]: |
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|
553 |
"dom (\<lambda>x. Some (f x)) = UNIV" |
29622 | 554 |
by auto |
555 |
||
22230 | 556 |
(* Due to John Matthews - could be rephrased with dom *) |
557 |
lemma finite_map_freshness: |
|
558 |
"finite (dom (f :: 'a \<rightharpoonup> 'b)) \<Longrightarrow> \<not> finite (UNIV :: 'a set) \<Longrightarrow> |
|
559 |
\<exists>x. f x = None" |
|
60839 | 560 |
by (bestsimp dest: ex_new_if_finite) |
14027 | 561 |
|
28790 | 562 |
lemma dom_minus: |
563 |
"f x = None \<Longrightarrow> dom f - insert x A = dom f - A" |
|
564 |
unfolding dom_def by simp |
|
565 |
||
566 |
lemma insert_dom: |
|
567 |
"f x = Some y \<Longrightarrow> insert x (dom f) = dom f" |
|
568 |
unfolding dom_def by auto |
|
569 |
||
35607 | 570 |
lemma map_of_map_keys: |
571 |
"set xs = dom m \<Longrightarrow> map_of (map (\<lambda>k. (k, the (m k))) xs) = m" |
|
572 |
by (rule ext) (auto simp add: map_of_map_restrict restrict_map_def) |
|
573 |
||
39379
ab1b070aa412
moved lemmas map_of_eqI and map_of_eq_dom to Map.thy
haftmann
parents:
39302
diff
changeset
|
574 |
lemma map_of_eqI: |
ab1b070aa412
moved lemmas map_of_eqI and map_of_eq_dom to Map.thy
haftmann
parents:
39302
diff
changeset
|
575 |
assumes set_eq: "set (map fst xs) = set (map fst ys)" |
ab1b070aa412
moved lemmas map_of_eqI and map_of_eq_dom to Map.thy
haftmann
parents:
39302
diff
changeset
|
576 |
assumes map_eq: "\<forall>k\<in>set (map fst xs). map_of xs k = map_of ys k" |
ab1b070aa412
moved lemmas map_of_eqI and map_of_eq_dom to Map.thy
haftmann
parents:
39302
diff
changeset
|
577 |
shows "map_of xs = map_of ys" |
ab1b070aa412
moved lemmas map_of_eqI and map_of_eq_dom to Map.thy
haftmann
parents:
39302
diff
changeset
|
578 |
proof (rule ext) |
ab1b070aa412
moved lemmas map_of_eqI and map_of_eq_dom to Map.thy
haftmann
parents:
39302
diff
changeset
|
579 |
fix k show "map_of xs k = map_of ys k" |
ab1b070aa412
moved lemmas map_of_eqI and map_of_eq_dom to Map.thy
haftmann
parents:
39302
diff
changeset
|
580 |
proof (cases "map_of xs k") |
60839 | 581 |
case None |
582 |
then have "k \<notin> set (map fst xs)" by (simp add: map_of_eq_None_iff) |
|
39379
ab1b070aa412
moved lemmas map_of_eqI and map_of_eq_dom to Map.thy
haftmann
parents:
39302
diff
changeset
|
583 |
with set_eq have "k \<notin> set (map fst ys)" by simp |
ab1b070aa412
moved lemmas map_of_eqI and map_of_eq_dom to Map.thy
haftmann
parents:
39302
diff
changeset
|
584 |
then have "map_of ys k = None" by (simp add: map_of_eq_None_iff) |
ab1b070aa412
moved lemmas map_of_eqI and map_of_eq_dom to Map.thy
haftmann
parents:
39302
diff
changeset
|
585 |
with None show ?thesis by simp |
ab1b070aa412
moved lemmas map_of_eqI and map_of_eq_dom to Map.thy
haftmann
parents:
39302
diff
changeset
|
586 |
next |
60839 | 587 |
case (Some v) |
588 |
then have "k \<in> set (map fst xs)" by (auto simp add: dom_map_of_conv_image_fst [symmetric]) |
|
39379
ab1b070aa412
moved lemmas map_of_eqI and map_of_eq_dom to Map.thy
haftmann
parents:
39302
diff
changeset
|
589 |
with map_eq show ?thesis by auto |
ab1b070aa412
moved lemmas map_of_eqI and map_of_eq_dom to Map.thy
haftmann
parents:
39302
diff
changeset
|
590 |
qed |
ab1b070aa412
moved lemmas map_of_eqI and map_of_eq_dom to Map.thy
haftmann
parents:
39302
diff
changeset
|
591 |
qed |
ab1b070aa412
moved lemmas map_of_eqI and map_of_eq_dom to Map.thy
haftmann
parents:
39302
diff
changeset
|
592 |
|
ab1b070aa412
moved lemmas map_of_eqI and map_of_eq_dom to Map.thy
haftmann
parents:
39302
diff
changeset
|
593 |
lemma map_of_eq_dom: |
ab1b070aa412
moved lemmas map_of_eqI and map_of_eq_dom to Map.thy
haftmann
parents:
39302
diff
changeset
|
594 |
assumes "map_of xs = map_of ys" |
ab1b070aa412
moved lemmas map_of_eqI and map_of_eq_dom to Map.thy
haftmann
parents:
39302
diff
changeset
|
595 |
shows "fst ` set xs = fst ` set ys" |
ab1b070aa412
moved lemmas map_of_eqI and map_of_eq_dom to Map.thy
haftmann
parents:
39302
diff
changeset
|
596 |
proof - |
ab1b070aa412
moved lemmas map_of_eqI and map_of_eq_dom to Map.thy
haftmann
parents:
39302
diff
changeset
|
597 |
from assms have "dom (map_of xs) = dom (map_of ys)" by simp |
ab1b070aa412
moved lemmas map_of_eqI and map_of_eq_dom to Map.thy
haftmann
parents:
39302
diff
changeset
|
598 |
then show ?thesis by (simp add: dom_map_of_conv_image_fst) |
ab1b070aa412
moved lemmas map_of_eqI and map_of_eq_dom to Map.thy
haftmann
parents:
39302
diff
changeset
|
599 |
qed |
ab1b070aa412
moved lemmas map_of_eqI and map_of_eq_dom to Map.thy
haftmann
parents:
39302
diff
changeset
|
600 |
|
53820 | 601 |
lemma finite_set_of_finite_maps: |
60839 | 602 |
assumes "finite A" "finite B" |
603 |
shows "finite {m. dom m = A \<and> ran m \<subseteq> B}" (is "finite ?S") |
|
53820 | 604 |
proof - |
605 |
let ?S' = "{m. \<forall>x. (x \<in> A \<longrightarrow> m x \<in> Some ` B) \<and> (x \<notin> A \<longrightarrow> m x = None)}" |
|
606 |
have "?S = ?S'" |
|
607 |
proof |
|
60839 | 608 |
show "?S \<subseteq> ?S'" by (auto simp: dom_def ran_def image_def) |
53820 | 609 |
show "?S' \<subseteq> ?S" |
610 |
proof |
|
611 |
fix m assume "m \<in> ?S'" |
|
612 |
hence 1: "dom m = A" by force |
|
60839 | 613 |
hence 2: "ran m \<subseteq> B" using \<open>m \<in> ?S'\<close> by (auto simp: dom_def ran_def) |
53820 | 614 |
from 1 2 show "m \<in> ?S" by blast |
615 |
qed |
|
616 |
qed |
|
617 |
with assms show ?thesis by(simp add: finite_set_of_finite_funs) |
|
618 |
qed |
|
28790 | 619 |
|
60839 | 620 |
|
60758 | 621 |
subsection \<open>@{term [source] ran}\<close> |
14100 | 622 |
|
60839 | 623 |
lemma ranI: "m a = Some b \<Longrightarrow> b \<in> ran m" |
624 |
by (auto simp: ran_def) |
|
14100 | 625 |
(* declare ranI [intro]? *) |
13908 | 626 |
|
20800 | 627 |
lemma ran_empty [simp]: "ran empty = {}" |
60839 | 628 |
by (auto simp: ran_def) |
13908 | 629 |
|
60839 | 630 |
lemma ran_map_upd [simp]: "m a = None \<Longrightarrow> ran(m(a\<mapsto>b)) = insert b (ran m)" |
631 |
unfolding ran_def |
|
24331 | 632 |
apply auto |
60839 | 633 |
apply (subgoal_tac "aa \<noteq> a") |
24331 | 634 |
apply auto |
635 |
done |
|
20800 | 636 |
|
66583 | 637 |
lemma ran_map_add: |
638 |
assumes "dom m1 \<inter> dom m2 = {}" |
|
639 |
shows "ran (m1 ++ m2) = ran m1 \<union> ran m2" |
|
640 |
proof |
|
641 |
show "ran (m1 ++ m2) \<subseteq> ran m1 \<union> ran m2" |
|
642 |
unfolding ran_def by auto |
|
643 |
next |
|
644 |
show "ran m1 \<union> ran m2 \<subseteq> ran (m1 ++ m2)" |
|
645 |
proof - |
|
646 |
have "(m1 ++ m2) x = Some y" if "m1 x = Some y" for x y |
|
647 |
using assms map_add_comm that by fastforce |
|
648 |
moreover have "(m1 ++ m2) x = Some y" if "m2 x = Some y" for x y |
|
649 |
using assms that by auto |
|
650 |
ultimately show ?thesis |
|
651 |
unfolding ran_def by blast |
|
652 |
qed |
|
653 |
qed |
|
654 |
||
655 |
lemma finite_ran: |
|
656 |
assumes "finite (dom p)" |
|
657 |
shows "finite (ran p)" |
|
658 |
proof - |
|
659 |
have "ran p = (\<lambda>x. the (p x)) ` dom p" |
|
660 |
unfolding ran_def by force |
|
661 |
from this \<open>finite (dom p)\<close> show ?thesis by auto |
|
662 |
qed |
|
663 |
||
60839 | 664 |
lemma ran_distinct: |
665 |
assumes dist: "distinct (map fst al)" |
|
34979
8cb6e7a42e9c
more correspondence lemmas between related operations
haftmann
parents:
34941
diff
changeset
|
666 |
shows "ran (map_of al) = snd ` set al" |
60839 | 667 |
using assms |
668 |
proof (induct al) |
|
669 |
case Nil |
|
670 |
then show ?case by simp |
|
34979
8cb6e7a42e9c
more correspondence lemmas between related operations
haftmann
parents:
34941
diff
changeset
|
671 |
next |
8cb6e7a42e9c
more correspondence lemmas between related operations
haftmann
parents:
34941
diff
changeset
|
672 |
case (Cons kv al) |
8cb6e7a42e9c
more correspondence lemmas between related operations
haftmann
parents:
34941
diff
changeset
|
673 |
then have "ran (map_of al) = snd ` set al" by simp |
8cb6e7a42e9c
more correspondence lemmas between related operations
haftmann
parents:
34941
diff
changeset
|
674 |
moreover from Cons.prems have "map_of al (fst kv) = None" |
8cb6e7a42e9c
more correspondence lemmas between related operations
haftmann
parents:
34941
diff
changeset
|
675 |
by (simp add: map_of_eq_None_iff) |
8cb6e7a42e9c
more correspondence lemmas between related operations
haftmann
parents:
34941
diff
changeset
|
676 |
ultimately show ?case by (simp only: map_of.simps ran_map_upd) simp |
8cb6e7a42e9c
more correspondence lemmas between related operations
haftmann
parents:
34941
diff
changeset
|
677 |
qed |
8cb6e7a42e9c
more correspondence lemmas between related operations
haftmann
parents:
34941
diff
changeset
|
678 |
|
66584 | 679 |
lemma ran_map_of_zip: |
680 |
assumes "length xs = length ys" "distinct xs" |
|
681 |
shows "ran (map_of (zip xs ys)) = set ys" |
|
682 |
using assms by (simp add: ran_distinct set_map[symmetric]) |
|
683 |
||
60057 | 684 |
lemma ran_map_option: "ran (\<lambda>x. map_option f (m x)) = f ` ran m" |
60839 | 685 |
by (auto simp add: ran_def) |
686 |
||
13910 | 687 |
|
61799 | 688 |
subsection \<open>\<open>map_le\<close>\<close> |
13910 | 689 |
|
13912 | 690 |
lemma map_le_empty [simp]: "empty \<subseteq>\<^sub>m g" |
60839 | 691 |
by (simp add: map_le_def) |
13910 | 692 |
|
17724 | 693 |
lemma upd_None_map_le [simp]: "f(x := None) \<subseteq>\<^sub>m f" |
60839 | 694 |
by (force simp add: map_le_def) |
14187 | 695 |
|
13910 | 696 |
lemma map_le_upd[simp]: "f \<subseteq>\<^sub>m g ==> f(a := b) \<subseteq>\<^sub>m g(a := b)" |
60839 | 697 |
by (fastforce simp add: map_le_def) |
13910 | 698 |
|
17724 | 699 |
lemma map_le_imp_upd_le [simp]: "m1 \<subseteq>\<^sub>m m2 \<Longrightarrow> m1(x := None) \<subseteq>\<^sub>m m2(x \<mapsto> y)" |
60839 | 700 |
by (force simp add: map_le_def) |
14187 | 701 |
|
20800 | 702 |
lemma map_le_upds [simp]: |
60839 | 703 |
"f \<subseteq>\<^sub>m g \<Longrightarrow> f(as [\<mapsto>] bs) \<subseteq>\<^sub>m g(as [\<mapsto>] bs)" |
24331 | 704 |
apply (induct as arbitrary: f g bs) |
705 |
apply simp |
|
706 |
apply (case_tac bs) |
|
707 |
apply auto |
|
708 |
done |
|
13908 | 709 |
|
14033 | 710 |
lemma map_le_implies_dom_le: "(f \<subseteq>\<^sub>m g) \<Longrightarrow> (dom f \<subseteq> dom g)" |
60839 | 711 |
by (fastforce simp add: map_le_def dom_def) |
14033 | 712 |
|
713 |
lemma map_le_refl [simp]: "f \<subseteq>\<^sub>m f" |
|
60839 | 714 |
by (simp add: map_le_def) |
14033 | 715 |
|
14187 | 716 |
lemma map_le_trans[trans]: "\<lbrakk> m1 \<subseteq>\<^sub>m m2; m2 \<subseteq>\<^sub>m m3\<rbrakk> \<Longrightarrow> m1 \<subseteq>\<^sub>m m3" |
60839 | 717 |
by (auto simp add: map_le_def dom_def) |
14033 | 718 |
|
719 |
lemma map_le_antisym: "\<lbrakk> f \<subseteq>\<^sub>m g; g \<subseteq>\<^sub>m f \<rbrakk> \<Longrightarrow> f = g" |
|
24331 | 720 |
unfolding map_le_def |
721 |
apply (rule ext) |
|
722 |
apply (case_tac "x \<in> dom f", simp) |
|
44890
22f665a2e91c
new fastforce replacing fastsimp - less confusing name
nipkow
parents:
42163
diff
changeset
|
723 |
apply (case_tac "x \<in> dom g", simp, fastforce) |
24331 | 724 |
done |
14033 | 725 |
|
60839 | 726 |
lemma map_le_map_add [simp]: "f \<subseteq>\<^sub>m g ++ f" |
727 |
by (fastforce simp: map_le_def) |
|
14033 | 728 |
|
60839 | 729 |
lemma map_le_iff_map_add_commute: "f \<subseteq>\<^sub>m f ++ g \<longleftrightarrow> f ++ g = g ++ f" |
730 |
by (fastforce simp: map_add_def map_le_def fun_eq_iff split: option.splits) |
|
15304 | 731 |
|
60839 | 732 |
lemma map_add_le_mapE: "f ++ g \<subseteq>\<^sub>m h \<Longrightarrow> g \<subseteq>\<^sub>m h" |
733 |
by (fastforce simp: map_le_def map_add_def dom_def) |
|
15303 | 734 |
|
60839 | 735 |
lemma map_add_le_mapI: "\<lbrakk> f \<subseteq>\<^sub>m h; g \<subseteq>\<^sub>m h \<rbrakk> \<Longrightarrow> f ++ g \<subseteq>\<^sub>m h" |
736 |
by (auto simp: map_le_def map_add_def dom_def split: option.splits) |
|
15303 | 737 |
|
63828 | 738 |
lemma map_add_subsumed1: "f \<subseteq>\<^sub>m g \<Longrightarrow> f++g = g" |
739 |
by (simp add: map_add_le_mapI map_le_antisym) |
|
740 |
||
741 |
lemma map_add_subsumed2: "f \<subseteq>\<^sub>m g \<Longrightarrow> g++f = g" |
|
742 |
by (metis map_add_subsumed1 map_le_iff_map_add_commute) |
|
743 |
||
31080 | 744 |
lemma dom_eq_singleton_conv: "dom f = {x} \<longleftrightarrow> (\<exists>v. f = [x \<mapsto> v])" |
63834 | 745 |
(is "?lhs \<longleftrightarrow> ?rhs") |
746 |
proof |
|
747 |
assume ?rhs |
|
748 |
then show ?lhs by (auto split: if_split_asm) |
|
31080 | 749 |
next |
63834 | 750 |
assume ?lhs |
751 |
then obtain v where v: "f x = Some v" by auto |
|
752 |
show ?rhs |
|
753 |
proof |
|
754 |
show "f = [x \<mapsto> v]" |
|
755 |
proof (rule map_le_antisym) |
|
756 |
show "[x \<mapsto> v] \<subseteq>\<^sub>m f" |
|
757 |
using v by (auto simp add: map_le_def) |
|
758 |
show "f \<subseteq>\<^sub>m [x \<mapsto> v]" |
|
759 |
using \<open>dom f = {x}\<close> \<open>f x = Some v\<close> by (auto simp add: map_le_def) |
|
760 |
qed |
|
761 |
qed |
|
31080 | 762 |
qed |
763 |
||
35565 | 764 |
|
60758 | 765 |
subsection \<open>Various\<close> |
35565 | 766 |
|
767 |
lemma set_map_of_compr: |
|
768 |
assumes distinct: "distinct (map fst xs)" |
|
769 |
shows "set xs = {(k, v). map_of xs k = Some v}" |
|
60839 | 770 |
using assms |
771 |
proof (induct xs) |
|
772 |
case Nil |
|
773 |
then show ?case by simp |
|
35565 | 774 |
next |
775 |
case (Cons x xs) |
|
776 |
obtain k v where "x = (k, v)" by (cases x) blast |
|
777 |
with Cons.prems have "k \<notin> dom (map_of xs)" |
|
778 |
by (simp add: dom_map_of_conv_image_fst) |
|
779 |
then have *: "insert (k, v) {(k, v). map_of xs k = Some v} = |
|
780 |
{(k', v'). (map_of xs(k \<mapsto> v)) k' = Some v'}" |
|
781 |
by (auto split: if_splits) |
|
782 |
from Cons have "set xs = {(k, v). map_of xs k = Some v}" by simp |
|
60758 | 783 |
with * \<open>x = (k, v)\<close> show ?case by simp |
35565 | 784 |
qed |
785 |
||
67051 | 786 |
lemma eq_key_imp_eq_value: |
787 |
"v1 = v2" |
|
788 |
if "distinct (map fst xs)" "(k, v1) \<in> set xs" "(k, v2) \<in> set xs" |
|
789 |
proof - |
|
790 |
from that have "inj_on fst (set xs)" |
|
791 |
by (simp add: distinct_map) |
|
792 |
moreover have "fst (k, v1) = fst (k, v2)" |
|
793 |
by simp |
|
794 |
ultimately have "(k, v1) = (k, v2)" |
|
795 |
by (rule inj_onD) (fact that)+ |
|
796 |
then show ?thesis |
|
797 |
by simp |
|
798 |
qed |
|
799 |
||
35565 | 800 |
lemma map_of_inject_set: |
801 |
assumes distinct: "distinct (map fst xs)" "distinct (map fst ys)" |
|
802 |
shows "map_of xs = map_of ys \<longleftrightarrow> set xs = set ys" (is "?lhs \<longleftrightarrow> ?rhs") |
|
803 |
proof |
|
804 |
assume ?lhs |
|
60758 | 805 |
moreover from \<open>distinct (map fst xs)\<close> have "set xs = {(k, v). map_of xs k = Some v}" |
35565 | 806 |
by (rule set_map_of_compr) |
60758 | 807 |
moreover from \<open>distinct (map fst ys)\<close> have "set ys = {(k, v). map_of ys k = Some v}" |
35565 | 808 |
by (rule set_map_of_compr) |
809 |
ultimately show ?rhs by simp |
|
810 |
next |
|
53374
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset
|
811 |
assume ?rhs show ?lhs |
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset
|
812 |
proof |
35565 | 813 |
fix k |
60839 | 814 |
show "map_of xs k = map_of ys k" |
815 |
proof (cases "map_of xs k") |
|
35565 | 816 |
case None |
60758 | 817 |
with \<open>?rhs\<close> have "map_of ys k = None" |
35565 | 818 |
by (simp add: map_of_eq_None_iff) |
53374
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset
|
819 |
with None show ?thesis by simp |
35565 | 820 |
next |
821 |
case (Some v) |
|
60758 | 822 |
with distinct \<open>?rhs\<close> have "map_of ys k = Some v" |
35565 | 823 |
by simp |
53374
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset
|
824 |
with Some show ?thesis by simp |
35565 | 825 |
qed |
826 |
qed |
|
827 |
qed |
|
828 |
||
3981 | 829 |
end |