author | wenzelm |
Mon, 31 Aug 2015 21:01:21 +0200 | |
changeset 61069 | aefe89038dd2 |
parent 61032 | b57df8eecad6 |
child 61799 | 4cf66f21b764 |
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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begin |
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|
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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" ("_ /|->/ _") |
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"_maplets" :: "['a, 'a] \<Rightarrow> maplet" ("_ /[|->]/ _") |
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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 (xsymbols) |
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"_maplet" :: "['a, 'a] \<Rightarrow> maplet" ("_ /\<mapsto>/ _") |
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"_maplets" :: "['a, 'a] \<Rightarrow> maplet" ("_ /[\<mapsto>]/ _") |
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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 |
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map_of :: "('a \<times> 'b) list \<Rightarrow> 'a \<rightharpoonup> 'b" 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 |
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map_upds :: "('a \<rightharpoonup> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b list \<Rightarrow> 'a \<rightharpoonup> 'b" where |
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"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_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 o 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 o m(a\<mapsto>b) = (map_option f o 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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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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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)) " |
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by (auto simp: map_comp_def split: option.splits) |
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subsection \<open>@{text "++"}\<close> |
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lemma map_add_empty[simp]: "m ++ empty = m" |
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by(simp add: map_add_def) |
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lemma empty_map_add[simp]: "empty ++ m = m" |
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by (rule ext) (simp add: map_add_def split: option.split) |
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|
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lemma map_add_assoc[simp]: "m1 ++ (m2 ++ m3) = (m1 ++ m2) ++ m3" |
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by (rule ext) (simp add: map_add_def split: option.split) |
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|
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lemma map_add_Some_iff: |
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"((m ++ n) k = Some x) = (n k = Some x | n k = None & m k = Some x)" |
291 |
by (simp add: map_add_def split: option.split) |
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|
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lemma map_add_SomeD [dest!]: |
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"(m ++ n) k = Some x \<Longrightarrow> n k = Some x \<or> n k = None \<and> m k = Some x" |
295 |
by (rule map_add_Some_iff [THEN iffD1]) |
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lemma map_add_find_right [simp]: "n k = Some xx \<Longrightarrow> (m ++ n) k = Some xx" |
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by (subst map_add_Some_iff) fast |
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|
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lemma map_add_None [iff]: "((m ++ n) k = None) = (n k = None & m k = None)" |
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by (simp add: map_add_def split: option.split) |
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|
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lemma map_add_upd[simp]: "f ++ g(x\<mapsto>y) = (f ++ g)(x\<mapsto>y)" |
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by (rule ext) (simp add: map_add_def) |
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|
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lemma map_add_upds[simp]: "m1 ++ (m2(xs[\<mapsto>]ys)) = (m1++m2)(xs[\<mapsto>]ys)" |
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by (simp add: map_upds_def) |
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lemma map_add_upd_left: "m\<notin>dom e2 \<Longrightarrow> e1(m \<mapsto> u1) ++ e2 = (e1 ++ e2)(m \<mapsto> u1)" |
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by (rule ext) (auto simp: map_add_def dom_def split: option.split) |
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311 |
|
20800 | 312 |
lemma map_of_append[simp]: "map_of (xs @ ys) = map_of ys ++ map_of xs" |
24331 | 313 |
unfolding map_add_def |
314 |
apply (induct xs) |
|
315 |
apply simp |
|
316 |
apply (rule ext) |
|
317 |
apply (simp split add: option.split) |
|
318 |
done |
|
13908 | 319 |
|
14025 | 320 |
lemma finite_range_map_of_map_add: |
60839 | 321 |
"finite (range f) \<Longrightarrow> finite (range (f ++ map_of l))" |
24331 | 322 |
apply (induct l) |
323 |
apply (auto simp del: fun_upd_apply) |
|
324 |
apply (erule finite_range_updI) |
|
325 |
done |
|
13908 | 326 |
|
20800 | 327 |
lemma inj_on_map_add_dom [iff]: |
24331 | 328 |
"inj_on (m ++ m') (dom m') = inj_on m' (dom m')" |
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329 |
by (fastforce simp: map_add_def dom_def inj_on_def split: option.splits) |
20800 | 330 |
|
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|
331 |
lemma map_upds_fold_map_upd: |
35552 | 332 |
"m(ks[\<mapsto>]vs) = foldl (\<lambda>m (k, v). m(k \<mapsto> v)) m (zip ks vs)" |
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|
333 |
unfolding map_upds_def proof (rule sym, rule zip_obtain_same_length) |
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|
334 |
fix ks :: "'a list" and vs :: "'b list" |
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335 |
assume "length ks = length vs" |
35552 | 336 |
then show "foldl (\<lambda>m (k, v). m(k\<mapsto>v)) m (zip ks vs) = m ++ map_of (rev (zip ks vs))" |
337 |
by(induct arbitrary: m rule: list_induct2) simp_all |
|
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338 |
qed |
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|
339 |
|
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|
340 |
lemma map_add_map_of_foldr: |
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|
341 |
"m ++ map_of ps = foldr (\<lambda>(k, v) m. m(k \<mapsto> v)) ps m" |
60839 | 342 |
by (induct ps) (auto simp: fun_eq_iff map_add_def) |
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|
343 |
|
15304 | 344 |
|
60758 | 345 |
subsection \<open>@{term [source] restrict_map}\<close> |
14100 | 346 |
|
20800 | 347 |
lemma restrict_map_to_empty [simp]: "m|`{} = empty" |
24331 | 348 |
by (simp add: restrict_map_def) |
14186 | 349 |
|
31380 | 350 |
lemma restrict_map_insert: "f |` (insert a A) = (f |` A)(a := f a)" |
60839 | 351 |
by (auto simp: restrict_map_def) |
31380 | 352 |
|
20800 | 353 |
lemma restrict_map_empty [simp]: "empty|`D = empty" |
24331 | 354 |
by (simp add: restrict_map_def) |
14186 | 355 |
|
15693 | 356 |
lemma restrict_in [simp]: "x \<in> A \<Longrightarrow> (m|`A) x = m x" |
24331 | 357 |
by (simp add: restrict_map_def) |
14100 | 358 |
|
15693 | 359 |
lemma restrict_out [simp]: "x \<notin> A \<Longrightarrow> (m|`A) x = None" |
24331 | 360 |
by (simp add: restrict_map_def) |
14100 | 361 |
|
15693 | 362 |
lemma ran_restrictD: "y \<in> ran (m|`A) \<Longrightarrow> \<exists>x\<in>A. m x = Some y" |
24331 | 363 |
by (auto simp: restrict_map_def ran_def split: split_if_asm) |
14100 | 364 |
|
15693 | 365 |
lemma dom_restrict [simp]: "dom (m|`A) = dom m \<inter> A" |
24331 | 366 |
by (auto simp: restrict_map_def dom_def split: split_if_asm) |
14100 | 367 |
|
15693 | 368 |
lemma restrict_upd_same [simp]: "m(x\<mapsto>y)|`(-{x}) = m|`(-{x})" |
24331 | 369 |
by (rule ext) (auto simp: restrict_map_def) |
14100 | 370 |
|
15693 | 371 |
lemma restrict_restrict [simp]: "m|`A|`B = m|`(A\<inter>B)" |
24331 | 372 |
by (rule ext) (auto simp: restrict_map_def) |
14100 | 373 |
|
20800 | 374 |
lemma restrict_fun_upd [simp]: |
24331 | 375 |
"m(x := y)|`D = (if x \<in> D then (m|`(D-{x}))(x := y) else m|`D)" |
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|
376 |
by (simp add: restrict_map_def fun_eq_iff) |
14186 | 377 |
|
20800 | 378 |
lemma fun_upd_None_restrict [simp]: |
60839 | 379 |
"(m|`D)(x := None) = (if x \<in> D then m|`(D - {x}) else m|`D)" |
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changeset
|
380 |
by (simp add: restrict_map_def fun_eq_iff) |
14186 | 381 |
|
20800 | 382 |
lemma fun_upd_restrict: "(m|`D)(x := y) = (m|`(D-{x}))(x := y)" |
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|
383 |
by (simp add: restrict_map_def fun_eq_iff) |
14186 | 384 |
|
20800 | 385 |
lemma fun_upd_restrict_conv [simp]: |
24331 | 386 |
"x \<in> D \<Longrightarrow> (m|`D)(x := y) = (m|`(D-{x}))(x := y)" |
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|
387 |
by (simp add: restrict_map_def fun_eq_iff) |
14186 | 388 |
|
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|
389 |
lemma map_of_map_restrict: |
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|
390 |
"map_of (map (\<lambda>k. (k, f k)) ks) = (Some \<circ> f) |` set ks" |
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changeset
|
391 |
by (induct ks) (simp_all add: fun_eq_iff restrict_map_insert) |
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|
392 |
|
35619 | 393 |
lemma restrict_complement_singleton_eq: |
394 |
"f |` (- {x}) = f(x := None)" |
|
39302
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|
395 |
by (simp add: restrict_map_def fun_eq_iff) |
35619 | 396 |
|
14100 | 397 |
|
60758 | 398 |
subsection \<open>@{term [source] map_upds}\<close> |
14025 | 399 |
|
60838 | 400 |
lemma map_upds_Nil1 [simp]: "m([] [\<mapsto>] bs) = m" |
24331 | 401 |
by (simp add: map_upds_def) |
14025 | 402 |
|
60838 | 403 |
lemma map_upds_Nil2 [simp]: "m(as [\<mapsto>] []) = m" |
24331 | 404 |
by (simp add:map_upds_def) |
20800 | 405 |
|
60838 | 406 |
lemma map_upds_Cons [simp]: "m(a#as [\<mapsto>] b#bs) = (m(a\<mapsto>b))(as[\<mapsto>]bs)" |
24331 | 407 |
by (simp add:map_upds_def) |
14025 | 408 |
|
60839 | 409 |
lemma map_upds_append1 [simp]: "size xs < size ys \<Longrightarrow> |
24331 | 410 |
m(xs@[x] [\<mapsto>] ys) = m(xs [\<mapsto>] ys)(x \<mapsto> ys!size xs)" |
60839 | 411 |
apply(induct xs arbitrary: ys m) |
24331 | 412 |
apply (clarsimp simp add: neq_Nil_conv) |
413 |
apply (case_tac ys) |
|
414 |
apply simp |
|
415 |
apply simp |
|
416 |
done |
|
14187 | 417 |
|
20800 | 418 |
lemma map_upds_list_update2_drop [simp]: |
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|
419 |
"size xs \<le> i \<Longrightarrow> m(xs[\<mapsto>]ys[i:=y]) = m(xs[\<mapsto>]ys)" |
24331 | 420 |
apply (induct xs arbitrary: m ys i) |
421 |
apply simp |
|
422 |
apply (case_tac ys) |
|
423 |
apply simp |
|
424 |
apply (simp split: nat.split) |
|
425 |
done |
|
14025 | 426 |
|
20800 | 427 |
lemma map_upd_upds_conv_if: |
60838 | 428 |
"(f(x\<mapsto>y))(xs [\<mapsto>] ys) = |
60839 | 429 |
(if x \<in> set(take (length ys) xs) then f(xs [\<mapsto>] ys) |
60838 | 430 |
else (f(xs [\<mapsto>] ys))(x\<mapsto>y))" |
24331 | 431 |
apply (induct xs arbitrary: x y ys f) |
432 |
apply simp |
|
433 |
apply (case_tac ys) |
|
434 |
apply (auto split: split_if simp: fun_upd_twist) |
|
435 |
done |
|
14025 | 436 |
|
437 |
lemma map_upds_twist [simp]: |
|
60839 | 438 |
"a \<notin> set as \<Longrightarrow> m(a\<mapsto>b)(as[\<mapsto>]bs) = m(as[\<mapsto>]bs)(a\<mapsto>b)" |
44890
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42163
diff
changeset
|
439 |
using set_take_subset by (fastforce simp add: map_upd_upds_conv_if) |
14025 | 440 |
|
20800 | 441 |
lemma map_upds_apply_nontin [simp]: |
60839 | 442 |
"x \<notin> set xs \<Longrightarrow> (f(xs[\<mapsto>]ys)) x = f x" |
24331 | 443 |
apply (induct xs arbitrary: ys) |
444 |
apply simp |
|
445 |
apply (case_tac ys) |
|
446 |
apply (auto simp: map_upd_upds_conv_if) |
|
447 |
done |
|
14025 | 448 |
|
20800 | 449 |
lemma fun_upds_append_drop [simp]: |
24331 | 450 |
"size xs = size ys \<Longrightarrow> m(xs@zs[\<mapsto>]ys) = m(xs[\<mapsto>]ys)" |
451 |
apply (induct xs arbitrary: m ys) |
|
452 |
apply simp |
|
453 |
apply (case_tac ys) |
|
454 |
apply simp_all |
|
455 |
done |
|
14300 | 456 |
|
20800 | 457 |
lemma fun_upds_append2_drop [simp]: |
24331 | 458 |
"size xs = size ys \<Longrightarrow> m(xs[\<mapsto>]ys@zs) = m(xs[\<mapsto>]ys)" |
459 |
apply (induct xs arbitrary: m ys) |
|
460 |
apply simp |
|
461 |
apply (case_tac ys) |
|
462 |
apply simp_all |
|
463 |
done |
|
14300 | 464 |
|
465 |
||
20800 | 466 |
lemma restrict_map_upds[simp]: |
467 |
"\<lbrakk> length xs = length ys; set xs \<subseteq> D \<rbrakk> |
|
468 |
\<Longrightarrow> m(xs [\<mapsto>] ys)|`D = (m|`(D - set xs))(xs [\<mapsto>] ys)" |
|
24331 | 469 |
apply (induct xs arbitrary: m ys) |
470 |
apply simp |
|
471 |
apply (case_tac ys) |
|
472 |
apply simp |
|
473 |
apply (simp add: Diff_insert [symmetric] insert_absorb) |
|
474 |
apply (simp add: map_upd_upds_conv_if) |
|
475 |
done |
|
14186 | 476 |
|
477 |
||
60758 | 478 |
subsection \<open>@{term [source] dom}\<close> |
13908 | 479 |
|
31080 | 480 |
lemma dom_eq_empty_conv [simp]: "dom f = {} \<longleftrightarrow> f = empty" |
44921 | 481 |
by (auto simp: dom_def) |
31080 | 482 |
|
60839 | 483 |
lemma domI: "m a = Some b \<Longrightarrow> a \<in> dom m" |
484 |
by (simp add: dom_def) |
|
14100 | 485 |
(* declare domI [intro]? *) |
13908 | 486 |
|
60839 | 487 |
lemma domD: "a \<in> dom m \<Longrightarrow> \<exists>b. m a = Some b" |
488 |
by (cases "m a") (auto simp add: dom_def) |
|
13908 | 489 |
|
60839 | 490 |
lemma domIff [iff, simp del]: "a \<in> dom m \<longleftrightarrow> m a \<noteq> None" |
491 |
by (simp add: dom_def) |
|
13908 | 492 |
|
20800 | 493 |
lemma dom_empty [simp]: "dom empty = {}" |
60839 | 494 |
by (simp add: dom_def) |
13908 | 495 |
|
20800 | 496 |
lemma dom_fun_upd [simp]: |
60839 | 497 |
"dom(f(x := y)) = (if y = None then dom f - {x} else insert x (dom f))" |
498 |
by (auto simp: dom_def) |
|
13908 | 499 |
|
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diff
changeset
|
500 |
lemma dom_if: |
8cb6e7a42e9c
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changeset
|
501 |
"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}" |
8cb6e7a42e9c
more correspondence lemmas between related operations
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parents:
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diff
changeset
|
502 |
by (auto split: if_splits) |
13937 | 503 |
|
15304 | 504 |
lemma dom_map_of_conv_image_fst: |
34979
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34941
diff
changeset
|
505 |
"dom (map_of xys) = fst ` set xys" |
8cb6e7a42e9c
more correspondence lemmas between related operations
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diff
changeset
|
506 |
by (induct xys) (auto simp add: dom_if) |
15304 | 507 |
|
60839 | 508 |
lemma dom_map_of_zip [simp]: "length xs = length ys \<Longrightarrow> dom (map_of (zip xs ys)) = set xs" |
509 |
by (induct rule: list_induct2) (auto simp: dom_if) |
|
15110
78b5636eabc7
Added a number of new thms and the new function remove1
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14739
diff
changeset
|
510 |
|
13908 | 511 |
lemma finite_dom_map_of: "finite (dom (map_of l))" |
60839 | 512 |
by (induct l) (auto simp: dom_def insert_Collect [symmetric]) |
13908 | 513 |
|
20800 | 514 |
lemma dom_map_upds [simp]: |
60839 | 515 |
"dom(m(xs[\<mapsto>]ys)) = set(take (length ys) xs) \<union> dom m" |
24331 | 516 |
apply (induct xs arbitrary: m ys) |
517 |
apply simp |
|
518 |
apply (case_tac ys) |
|
519 |
apply auto |
|
520 |
done |
|
13910 | 521 |
|
60839 | 522 |
lemma dom_map_add [simp]: "dom (m ++ n) = dom n \<union> dom m" |
523 |
by (auto simp: dom_def) |
|
13910 | 524 |
|
20800 | 525 |
lemma dom_override_on [simp]: |
60839 | 526 |
"dom (override_on f g A) = |
527 |
(dom f - {a. a \<in> A - dom g}) \<union> {a. a \<in> A \<inter> dom g}" |
|
528 |
by (auto simp: dom_def override_on_def) |
|
13908 | 529 |
|
60839 | 530 |
lemma map_add_comm: "dom m1 \<inter> dom m2 = {} \<Longrightarrow> m1 ++ m2 = m2 ++ m1" |
531 |
by (rule ext) (force simp: map_add_def dom_def split: option.split) |
|
20800 | 532 |
|
32236
0203e1006f1b
some lemmas about maps (contributed by Peter Lammich)
krauss
parents:
31380
diff
changeset
|
533 |
lemma map_add_dom_app_simps: |
60839 | 534 |
"m \<in> dom l2 \<Longrightarrow> (l1 ++ l2) m = l2 m" |
535 |
"m \<notin> dom l1 \<Longrightarrow> (l1 ++ l2) m = l2 m" |
|
536 |
"m \<notin> dom l2 \<Longrightarrow> (l1 ++ l2) m = l1 m" |
|
537 |
by (auto simp add: map_add_def split: option.split_asm) |
|
32236
0203e1006f1b
some lemmas about maps (contributed by Peter Lammich)
krauss
parents:
31380
diff
changeset
|
538 |
|
29622 | 539 |
lemma dom_const [simp]: |
35159
df38e92af926
added lemma map_of_map_restrict; generalized lemma dom_const
haftmann
parents:
35115
diff
changeset
|
540 |
"dom (\<lambda>x. Some (f x)) = UNIV" |
29622 | 541 |
by auto |
542 |
||
22230 | 543 |
(* Due to John Matthews - could be rephrased with dom *) |
544 |
lemma finite_map_freshness: |
|
545 |
"finite (dom (f :: 'a \<rightharpoonup> 'b)) \<Longrightarrow> \<not> finite (UNIV :: 'a set) \<Longrightarrow> |
|
546 |
\<exists>x. f x = None" |
|
60839 | 547 |
by (bestsimp dest: ex_new_if_finite) |
14027 | 548 |
|
28790 | 549 |
lemma dom_minus: |
550 |
"f x = None \<Longrightarrow> dom f - insert x A = dom f - A" |
|
551 |
unfolding dom_def by simp |
|
552 |
||
553 |
lemma insert_dom: |
|
554 |
"f x = Some y \<Longrightarrow> insert x (dom f) = dom f" |
|
555 |
unfolding dom_def by auto |
|
556 |
||
35607 | 557 |
lemma map_of_map_keys: |
558 |
"set xs = dom m \<Longrightarrow> map_of (map (\<lambda>k. (k, the (m k))) xs) = m" |
|
559 |
by (rule ext) (auto simp add: map_of_map_restrict restrict_map_def) |
|
560 |
||
39379
ab1b070aa412
moved lemmas map_of_eqI and map_of_eq_dom to Map.thy
haftmann
parents:
39302
diff
changeset
|
561 |
lemma map_of_eqI: |
ab1b070aa412
moved lemmas map_of_eqI and map_of_eq_dom to Map.thy
haftmann
parents:
39302
diff
changeset
|
562 |
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
|
563 |
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
|
564 |
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
|
565 |
proof (rule ext) |
ab1b070aa412
moved lemmas map_of_eqI and map_of_eq_dom to Map.thy
haftmann
parents:
39302
diff
changeset
|
566 |
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
|
567 |
proof (cases "map_of xs k") |
60839 | 568 |
case None |
569 |
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
|
570 |
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
|
571 |
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
|
572 |
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
|
573 |
next |
60839 | 574 |
case (Some v) |
575 |
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
|
576 |
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
|
577 |
qed |
ab1b070aa412
moved lemmas map_of_eqI and map_of_eq_dom to Map.thy
haftmann
parents:
39302
diff
changeset
|
578 |
qed |
ab1b070aa412
moved lemmas map_of_eqI and map_of_eq_dom to Map.thy
haftmann
parents:
39302
diff
changeset
|
579 |
|
ab1b070aa412
moved lemmas map_of_eqI and map_of_eq_dom to Map.thy
haftmann
parents:
39302
diff
changeset
|
580 |
lemma map_of_eq_dom: |
ab1b070aa412
moved lemmas map_of_eqI and map_of_eq_dom to Map.thy
haftmann
parents:
39302
diff
changeset
|
581 |
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
|
582 |
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
|
583 |
proof - |
ab1b070aa412
moved lemmas map_of_eqI and map_of_eq_dom to Map.thy
haftmann
parents:
39302
diff
changeset
|
584 |
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
|
585 |
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
|
586 |
qed |
ab1b070aa412
moved lemmas map_of_eqI and map_of_eq_dom to Map.thy
haftmann
parents:
39302
diff
changeset
|
587 |
|
53820 | 588 |
lemma finite_set_of_finite_maps: |
60839 | 589 |
assumes "finite A" "finite B" |
590 |
shows "finite {m. dom m = A \<and> ran m \<subseteq> B}" (is "finite ?S") |
|
53820 | 591 |
proof - |
592 |
let ?S' = "{m. \<forall>x. (x \<in> A \<longrightarrow> m x \<in> Some ` B) \<and> (x \<notin> A \<longrightarrow> m x = None)}" |
|
593 |
have "?S = ?S'" |
|
594 |
proof |
|
60839 | 595 |
show "?S \<subseteq> ?S'" by (auto simp: dom_def ran_def image_def) |
53820 | 596 |
show "?S' \<subseteq> ?S" |
597 |
proof |
|
598 |
fix m assume "m \<in> ?S'" |
|
599 |
hence 1: "dom m = A" by force |
|
60839 | 600 |
hence 2: "ran m \<subseteq> B" using \<open>m \<in> ?S'\<close> by (auto simp: dom_def ran_def) |
53820 | 601 |
from 1 2 show "m \<in> ?S" by blast |
602 |
qed |
|
603 |
qed |
|
604 |
with assms show ?thesis by(simp add: finite_set_of_finite_funs) |
|
605 |
qed |
|
28790 | 606 |
|
60839 | 607 |
|
60758 | 608 |
subsection \<open>@{term [source] ran}\<close> |
14100 | 609 |
|
60839 | 610 |
lemma ranI: "m a = Some b \<Longrightarrow> b \<in> ran m" |
611 |
by (auto simp: ran_def) |
|
14100 | 612 |
(* declare ranI [intro]? *) |
13908 | 613 |
|
20800 | 614 |
lemma ran_empty [simp]: "ran empty = {}" |
60839 | 615 |
by (auto simp: ran_def) |
13908 | 616 |
|
60839 | 617 |
lemma ran_map_upd [simp]: "m a = None \<Longrightarrow> ran(m(a\<mapsto>b)) = insert b (ran m)" |
618 |
unfolding ran_def |
|
24331 | 619 |
apply auto |
60839 | 620 |
apply (subgoal_tac "aa \<noteq> a") |
24331 | 621 |
apply auto |
622 |
done |
|
20800 | 623 |
|
60839 | 624 |
lemma ran_distinct: |
625 |
assumes dist: "distinct (map fst al)" |
|
34979
8cb6e7a42e9c
more correspondence lemmas between related operations
haftmann
parents:
34941
diff
changeset
|
626 |
shows "ran (map_of al) = snd ` set al" |
60839 | 627 |
using assms |
628 |
proof (induct al) |
|
629 |
case Nil |
|
630 |
then show ?case by simp |
|
34979
8cb6e7a42e9c
more correspondence lemmas between related operations
haftmann
parents:
34941
diff
changeset
|
631 |
next |
8cb6e7a42e9c
more correspondence lemmas between related operations
haftmann
parents:
34941
diff
changeset
|
632 |
case (Cons kv al) |
8cb6e7a42e9c
more correspondence lemmas between related operations
haftmann
parents:
34941
diff
changeset
|
633 |
then have "ran (map_of al) = snd ` set al" by simp |
8cb6e7a42e9c
more correspondence lemmas between related operations
haftmann
parents:
34941
diff
changeset
|
634 |
moreover from Cons.prems have "map_of al (fst kv) = None" |
8cb6e7a42e9c
more correspondence lemmas between related operations
haftmann
parents:
34941
diff
changeset
|
635 |
by (simp add: map_of_eq_None_iff) |
8cb6e7a42e9c
more correspondence lemmas between related operations
haftmann
parents:
34941
diff
changeset
|
636 |
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
|
637 |
qed |
8cb6e7a42e9c
more correspondence lemmas between related operations
haftmann
parents:
34941
diff
changeset
|
638 |
|
60057 | 639 |
lemma ran_map_option: "ran (\<lambda>x. map_option f (m x)) = f ` ran m" |
60839 | 640 |
by (auto simp add: ran_def) |
641 |
||
13910 | 642 |
|
60758 | 643 |
subsection \<open>@{text "map_le"}\<close> |
13910 | 644 |
|
13912 | 645 |
lemma map_le_empty [simp]: "empty \<subseteq>\<^sub>m g" |
60839 | 646 |
by (simp add: map_le_def) |
13910 | 647 |
|
17724 | 648 |
lemma upd_None_map_le [simp]: "f(x := None) \<subseteq>\<^sub>m f" |
60839 | 649 |
by (force simp add: map_le_def) |
14187 | 650 |
|
13910 | 651 |
lemma map_le_upd[simp]: "f \<subseteq>\<^sub>m g ==> f(a := b) \<subseteq>\<^sub>m g(a := b)" |
60839 | 652 |
by (fastforce simp add: map_le_def) |
13910 | 653 |
|
17724 | 654 |
lemma map_le_imp_upd_le [simp]: "m1 \<subseteq>\<^sub>m m2 \<Longrightarrow> m1(x := None) \<subseteq>\<^sub>m m2(x \<mapsto> y)" |
60839 | 655 |
by (force simp add: map_le_def) |
14187 | 656 |
|
20800 | 657 |
lemma map_le_upds [simp]: |
60839 | 658 |
"f \<subseteq>\<^sub>m g \<Longrightarrow> f(as [\<mapsto>] bs) \<subseteq>\<^sub>m g(as [\<mapsto>] bs)" |
24331 | 659 |
apply (induct as arbitrary: f g bs) |
660 |
apply simp |
|
661 |
apply (case_tac bs) |
|
662 |
apply auto |
|
663 |
done |
|
13908 | 664 |
|
14033 | 665 |
lemma map_le_implies_dom_le: "(f \<subseteq>\<^sub>m g) \<Longrightarrow> (dom f \<subseteq> dom g)" |
60839 | 666 |
by (fastforce simp add: map_le_def dom_def) |
14033 | 667 |
|
668 |
lemma map_le_refl [simp]: "f \<subseteq>\<^sub>m f" |
|
60839 | 669 |
by (simp add: map_le_def) |
14033 | 670 |
|
14187 | 671 |
lemma map_le_trans[trans]: "\<lbrakk> m1 \<subseteq>\<^sub>m m2; m2 \<subseteq>\<^sub>m m3\<rbrakk> \<Longrightarrow> m1 \<subseteq>\<^sub>m m3" |
60839 | 672 |
by (auto simp add: map_le_def dom_def) |
14033 | 673 |
|
674 |
lemma map_le_antisym: "\<lbrakk> f \<subseteq>\<^sub>m g; g \<subseteq>\<^sub>m f \<rbrakk> \<Longrightarrow> f = g" |
|
24331 | 675 |
unfolding map_le_def |
676 |
apply (rule ext) |
|
677 |
apply (case_tac "x \<in> dom f", simp) |
|
44890
22f665a2e91c
new fastforce replacing fastsimp - less confusing name
nipkow
parents:
42163
diff
changeset
|
678 |
apply (case_tac "x \<in> dom g", simp, fastforce) |
24331 | 679 |
done |
14033 | 680 |
|
60839 | 681 |
lemma map_le_map_add [simp]: "f \<subseteq>\<^sub>m g ++ f" |
682 |
by (fastforce simp: map_le_def) |
|
14033 | 683 |
|
60839 | 684 |
lemma map_le_iff_map_add_commute: "f \<subseteq>\<^sub>m f ++ g \<longleftrightarrow> f ++ g = g ++ f" |
685 |
by (fastforce simp: map_add_def map_le_def fun_eq_iff split: option.splits) |
|
15304 | 686 |
|
60839 | 687 |
lemma map_add_le_mapE: "f ++ g \<subseteq>\<^sub>m h \<Longrightarrow> g \<subseteq>\<^sub>m h" |
688 |
by (fastforce simp: map_le_def map_add_def dom_def) |
|
15303 | 689 |
|
60839 | 690 |
lemma map_add_le_mapI: "\<lbrakk> f \<subseteq>\<^sub>m h; g \<subseteq>\<^sub>m h \<rbrakk> \<Longrightarrow> f ++ g \<subseteq>\<^sub>m h" |
691 |
by (auto simp: map_le_def map_add_def dom_def split: option.splits) |
|
15303 | 692 |
|
31080 | 693 |
lemma dom_eq_singleton_conv: "dom f = {x} \<longleftrightarrow> (\<exists>v. f = [x \<mapsto> v])" |
694 |
proof(rule iffI) |
|
695 |
assume "\<exists>v. f = [x \<mapsto> v]" |
|
696 |
thus "dom f = {x}" by(auto split: split_if_asm) |
|
697 |
next |
|
698 |
assume "dom f = {x}" |
|
699 |
then obtain v where "f x = Some v" by auto |
|
700 |
hence "[x \<mapsto> v] \<subseteq>\<^sub>m f" by(auto simp add: map_le_def) |
|
60758 | 701 |
moreover have "f \<subseteq>\<^sub>m [x \<mapsto> v]" using \<open>dom f = {x}\<close> \<open>f x = Some v\<close> |
31080 | 702 |
by(auto simp add: map_le_def) |
703 |
ultimately have "f = [x \<mapsto> v]" by-(rule map_le_antisym) |
|
704 |
thus "\<exists>v. f = [x \<mapsto> v]" by blast |
|
705 |
qed |
|
706 |
||
35565 | 707 |
|
60758 | 708 |
subsection \<open>Various\<close> |
35565 | 709 |
|
710 |
lemma set_map_of_compr: |
|
711 |
assumes distinct: "distinct (map fst xs)" |
|
712 |
shows "set xs = {(k, v). map_of xs k = Some v}" |
|
60839 | 713 |
using assms |
714 |
proof (induct xs) |
|
715 |
case Nil |
|
716 |
then show ?case by simp |
|
35565 | 717 |
next |
718 |
case (Cons x xs) |
|
719 |
obtain k v where "x = (k, v)" by (cases x) blast |
|
720 |
with Cons.prems have "k \<notin> dom (map_of xs)" |
|
721 |
by (simp add: dom_map_of_conv_image_fst) |
|
722 |
then have *: "insert (k, v) {(k, v). map_of xs k = Some v} = |
|
723 |
{(k', v'). (map_of xs(k \<mapsto> v)) k' = Some v'}" |
|
724 |
by (auto split: if_splits) |
|
725 |
from Cons have "set xs = {(k, v). map_of xs k = Some v}" by simp |
|
60758 | 726 |
with * \<open>x = (k, v)\<close> show ?case by simp |
35565 | 727 |
qed |
728 |
||
729 |
lemma map_of_inject_set: |
|
730 |
assumes distinct: "distinct (map fst xs)" "distinct (map fst ys)" |
|
731 |
shows "map_of xs = map_of ys \<longleftrightarrow> set xs = set ys" (is "?lhs \<longleftrightarrow> ?rhs") |
|
732 |
proof |
|
733 |
assume ?lhs |
|
60758 | 734 |
moreover from \<open>distinct (map fst xs)\<close> have "set xs = {(k, v). map_of xs k = Some v}" |
35565 | 735 |
by (rule set_map_of_compr) |
60758 | 736 |
moreover from \<open>distinct (map fst ys)\<close> have "set ys = {(k, v). map_of ys k = Some v}" |
35565 | 737 |
by (rule set_map_of_compr) |
738 |
ultimately show ?rhs by simp |
|
739 |
next |
|
53374
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset
|
740 |
assume ?rhs show ?lhs |
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset
|
741 |
proof |
35565 | 742 |
fix k |
60839 | 743 |
show "map_of xs k = map_of ys k" |
744 |
proof (cases "map_of xs k") |
|
35565 | 745 |
case None |
60758 | 746 |
with \<open>?rhs\<close> have "map_of ys k = None" |
35565 | 747 |
by (simp add: map_of_eq_None_iff) |
53374
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset
|
748 |
with None show ?thesis by simp |
35565 | 749 |
next |
750 |
case (Some v) |
|
60758 | 751 |
with distinct \<open>?rhs\<close> have "map_of ys k = Some v" |
35565 | 752 |
by simp |
53374
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset
|
753 |
with Some show ?thesis by simp |
35565 | 754 |
qed |
755 |
qed |
|
756 |
qed |
|
757 |
||
3981 | 758 |
end |