author  webertj 
Fri, 19 Oct 2012 15:12:52 +0200  
changeset 49962  a8cc904a6820 
parent 46588  4895d7f1be42 
child 53015  a1119cf551e8 
permissions  rwrr 
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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 19972003 TU Muenchen 
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The datatype of `maps' (written ~=>); strongly resembles maps in VDM. 

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*) 

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header {* Maps *} 
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theory Map 
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imports List 
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begin 
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type_synonym ('a,'b) "map" = "'a => 'b option" (infixr "~=>" 0) 
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type_notation (xsymbols) 
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"map" (infixr "\<rightharpoonup>" 0) 
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abbreviation 
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empty :: "'a ~=> 'b" where 
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"empty == %x. None" 
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definition 
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map_comp :: "('b ~=> 'c) => ('a ~=> 'b) => ('a ~=> 'c)" (infixl "o'_m" 55) where 
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"f o_m g = (\<lambda>k. case g k of None \<Rightarrow> None  Some v \<Rightarrow> f v)" 
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notation (xsymbols) 
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map_comp (infixl "\<circ>\<^sub>m" 55) 
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definition 
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map_add :: "('a ~=> 'b) => ('a ~=> 'b) => ('a ~=> 'b)" (infixl "++" 100) where 
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"m1 ++ m2 = (\<lambda>x. case m2 x of None => m1 x  Some y => Some y)" 
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definition 
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restrict_map :: "('a ~=> 'b) => 'a set => ('a ~=> 'b)" (infixl "`" 110) where 
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"m`A = (\<lambda>x. if x : 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 ~=> 'b) => 'a set" where 
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"dom m = {a. m a ~= None}" 
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definition 
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ran :: "('a ~=> 'b) => 'b set" where 
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"ran m = {b. EX a. m a = Some b}" 
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definition 
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map_le :: "('a ~=> 'b) => ('a ~=> 'b) => bool" (infix "\<subseteq>\<^sub>m" 50) where 
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"(m\<^isub>1 \<subseteq>\<^sub>m m\<^isub>2) = (\<forall>a \<in> dom m\<^isub>1. m\<^isub>1 a = m\<^isub>2 a)" 
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nonterminal maplets and maplet 
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syntax 
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"_maplet" :: "['a, 'a] => maplet" ("_ />/ _") 
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"_maplets" :: "['a, 'a] => maplet" ("_ /[>]/ _") 

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"" :: "maplet => maplets" ("_") 

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"_Maplets" :: "[maplet, maplets] => maplets" ("_,/ _") 

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"_MapUpd" :: "['a ~=> 'b, maplets] => 'a ~=> 'b" ("_/'(_')" [900,0]900) 

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"_Map" :: "maplets => 'a ~=> 'b" ("(1[_])") 

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syntax (xsymbols) 
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"_maplet" :: "['a, 'a] => maplet" ("_ /\<mapsto>/ _") 
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"_maplets" :: "['a, 'a] => maplet" ("_ /[\<mapsto>]/ _") 

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translations 
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"_MapUpd m (_Maplets xy ms)" == "_MapUpd (_MapUpd m xy) ms" 
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"_MapUpd m (_maplet x y)" == "m(x := CONST Some y)" 
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"_Map ms" == "_MapUpd (CONST empty) ms" 
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"_Map (_Maplets ms1 ms2)" <= "_MapUpd (_Map ms1) ms2" 
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"_Maplets ms1 (_Maplets ms2 ms3)" <= "_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)" == "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 {* @{term [source] empty} *} 
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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 {* @{term [source] map_upd} *} 
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lemma map_upd_triv: "t k = Some x ==> t(k>x) = t" 

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by (rule ext) simp 
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lemma map_upd_nonempty [simp]: "t(k>x) ~= 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>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) ==> finite (range (f(a>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 {* @{term [source] map_of} *} 
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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_is_SomeD: "map_of xys x = Some y \<Longrightarrow> (x,y) \<in> set xys" 
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apply (induct xys) 

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apply simp 

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apply (clarsimp split: if_splits) 

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done 

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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 add: 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] using dist map_of 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 `map_of (zip (x#xs) (y#ys)) = map_of (zip (x#xs) (z#zs))` 

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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 `distinct (x # xs)` 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) (simp, atomize (full), auto) 
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lemma map_of_mapk_SomeI: 
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"inj f ==> map_of t k = Some x ==> 
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map_of (map (split (%k. Pair (f k))) t) (f k) = Some x" 

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by (induct t) (auto simp add: inj_eq) 

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lemma weak_map_of_SomeI: "(k, x) : set l ==> \<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 (split 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) = Option.map f \<circ> map_of xs" 

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by (induct xs) (auto simp add: fun_eq_iff) 
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lemma dom_option_map: 

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"dom (\<lambda>k. Option.map (f k) (m k)) = dom m" 

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by (simp add: dom_def) 

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subsection {* @{const Option.map} related *} 
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lemma option_map_o_empty [simp]: "Option.map f o empty = empty" 
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by (rule ext) simp 
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lemma option_map_o_map_upd [simp]: 
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"Option.map f o m(a>b) = (Option.map f o m)(a>f b)" 
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by (rule ext) simp 
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subsection {* @{term [source] map_comp} related *} 
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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 add: 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 add: 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 add: 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 add: map_comp_def split: option.splits) 

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subsection {* @{text "++"} *} 
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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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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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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)" 
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by (simp add: map_add_def split: option.split) 

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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" 
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by (rule map_add_Some_iff [THEN iffD1]) 

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lemma map_add_find_right [simp]: "!!xx. n k = Some xx ==> (m ++ n) k = Some xx" 
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by (subst map_add_Some_iff) fast 
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14025  306 
lemma map_add_None [iff]: "((m ++ n) k = None) = (n k = None & m k = None)" 
24331  307 
by (simp add: map_add_def split: option.split) 
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14025  309 
lemma map_add_upd[simp]: "f ++ g(x>y) = (f ++ g)(x>y)" 
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by (rule ext) (simp add: map_add_def) 
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14186  312 
lemma map_add_upds[simp]: "m1 ++ (m2(xs[\<mapsto>]ys)) = (m1++m2)(xs[\<mapsto>]ys)" 
24331  313 
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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lemma map_of_append[simp]: "map_of (xs @ ys) = map_of ys ++ map_of xs" 
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unfolding map_add_def 
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apply (induct xs) 

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apply simp 

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apply (rule ext) 

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apply (simp split add: option.split) 

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done 

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lemma finite_range_map_of_map_add: 
20800  327 
"finite (range f) ==> finite (range (f ++ map_of l))" 
24331  328 
apply (induct l) 
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apply (auto simp del: fun_upd_apply) 

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apply (erule finite_range_updI) 

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done 

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lemma inj_on_map_add_dom [iff]: 
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"inj_on (m ++ m') (dom m') = inj_on m' (dom m')" 
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by (fastforce simp: map_add_def dom_def inj_on_def split: option.splits) 
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lemma map_upds_fold_map_upd: 
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"m(ks[\<mapsto>]vs) = foldl (\<lambda>m (k, v). m(k \<mapsto> v)) m (zip ks vs)" 
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unfolding map_upds_def proof (rule sym, rule zip_obtain_same_length) 
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fix ks :: "'a list" and vs :: "'b list" 
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assume "length ks = length vs" 
35552  342 
then show "foldl (\<lambda>m (k, v). m(k\<mapsto>v)) m (zip ks vs) = m ++ map_of (rev (zip ks vs))" 
343 
by(induct arbitrary: m rule: list_induct2) simp_all 

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344 
qed 
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345 

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346 
lemma map_add_map_of_foldr: 
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347 
"m ++ map_of ps = foldr (\<lambda>(k, v) m. m(k \<mapsto> v)) ps m" 
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348 
by (induct ps) (auto simp add: fun_eq_iff map_add_def) 
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349 

15304  350 

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351 
subsection {* @{term [source] restrict_map} *} 
14100  352 

20800  353 
lemma restrict_map_to_empty [simp]: "m`{} = empty" 
24331  354 
by (simp add: restrict_map_def) 
14186  355 

31380  356 
lemma restrict_map_insert: "f ` (insert a A) = (f ` A)(a := f a)" 
44921  357 
by (auto simp add: restrict_map_def) 
31380  358 

20800  359 
lemma restrict_map_empty [simp]: "empty`D = empty" 
24331  360 
by (simp add: restrict_map_def) 
14186  361 

15693  362 
lemma restrict_in [simp]: "x \<in> A \<Longrightarrow> (m`A) x = m x" 
24331  363 
by (simp add: restrict_map_def) 
14100  364 

15693  365 
lemma restrict_out [simp]: "x \<notin> A \<Longrightarrow> (m`A) x = None" 
24331  366 
by (simp add: restrict_map_def) 
14100  367 

15693  368 
lemma ran_restrictD: "y \<in> ran (m`A) \<Longrightarrow> \<exists>x\<in>A. m x = Some y" 
24331  369 
by (auto simp: restrict_map_def ran_def split: split_if_asm) 
14100  370 

15693  371 
lemma dom_restrict [simp]: "dom (m`A) = dom m \<inter> A" 
24331  372 
by (auto simp: restrict_map_def dom_def split: split_if_asm) 
14100  373 

15693  374 
lemma restrict_upd_same [simp]: "m(x\<mapsto>y)`({x}) = m`({x})" 
24331  375 
by (rule ext) (auto simp: restrict_map_def) 
14100  376 

15693  377 
lemma restrict_restrict [simp]: "m`A`B = m`(A\<inter>B)" 
24331  378 
by (rule ext) (auto simp: restrict_map_def) 
14100  379 

20800  380 
lemma restrict_fun_upd [simp]: 
24331  381 
"m(x := y)`D = (if x \<in> D then (m`(D{x}))(x := y) else m`D)" 
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382 
by (simp add: restrict_map_def fun_eq_iff) 
14186  383 

20800  384 
lemma fun_upd_None_restrict [simp]: 
24331  385 
"(m`D)(x := None) = (if x:D then m`(D  {x}) else m`D)" 
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386 
by (simp add: restrict_map_def fun_eq_iff) 
14186  387 

20800  388 
lemma fun_upd_restrict: "(m`D)(x := y) = (m`(D{x}))(x := y)" 
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389 
by (simp add: restrict_map_def fun_eq_iff) 
14186  390 

20800  391 
lemma fun_upd_restrict_conv [simp]: 
24331  392 
"x \<in> D \<Longrightarrow> (m`D)(x := y) = (m`(D{x}))(x := y)" 
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393 
by (simp add: restrict_map_def fun_eq_iff) 
14186  394 

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395 
lemma map_of_map_restrict: 
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396 
"map_of (map (\<lambda>k. (k, f k)) ks) = (Some \<circ> f) ` set ks" 
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changeset

397 
by (induct ks) (simp_all add: fun_eq_iff restrict_map_insert) 
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398 

35619  399 
lemma restrict_complement_singleton_eq: 
400 
"f ` ( {x}) = f(x := None)" 

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changeset

401 
by (simp add: restrict_map_def fun_eq_iff) 
35619  402 

14100  403 

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404 
subsection {* @{term [source] map_upds} *} 
14025  405 

20800  406 
lemma map_upds_Nil1 [simp]: "m([] [>] bs) = m" 
24331  407 
by (simp add: map_upds_def) 
14025  408 

20800  409 
lemma map_upds_Nil2 [simp]: "m(as [>] []) = m" 
24331  410 
by (simp add:map_upds_def) 
20800  411 

412 
lemma map_upds_Cons [simp]: "m(a#as [>] b#bs) = (m(a>b))(as[>]bs)" 

24331  413 
by (simp add:map_upds_def) 
14025  414 

20800  415 
lemma map_upds_append1 [simp]: "\<And>ys m. size xs < size ys \<Longrightarrow> 
24331  416 
m(xs@[x] [\<mapsto>] ys) = m(xs [\<mapsto>] ys)(x \<mapsto> ys!size xs)" 
417 
apply(induct xs) 

418 
apply (clarsimp simp add: neq_Nil_conv) 

419 
apply (case_tac ys) 

420 
apply simp 

421 
apply simp 

422 
done 

14187  423 

20800  424 
lemma map_upds_list_update2_drop [simp]: 
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425 
"size xs \<le> i \<Longrightarrow> m(xs[\<mapsto>]ys[i:=y]) = m(xs[\<mapsto>]ys)" 
24331  426 
apply (induct xs arbitrary: m ys i) 
427 
apply simp 

428 
apply (case_tac ys) 

429 
apply simp 

430 
apply (simp split: nat.split) 

431 
done 

14025  432 

20800  433 
lemma map_upd_upds_conv_if: 
434 
"(f(x>y))(xs [>] ys) = 

435 
(if x : set(take (length ys) xs) then f(xs [>] ys) 

436 
else (f(xs [>] ys))(x>y))" 

24331  437 
apply (induct xs arbitrary: x y ys f) 
438 
apply simp 

439 
apply (case_tac ys) 

440 
apply (auto split: split_if simp: fun_upd_twist) 

441 
done 

14025  442 

443 
lemma map_upds_twist [simp]: 

24331  444 
"a ~: set as ==> m(a>b)(as[>]bs) = m(as[>]bs)(a>b)" 
44890
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changeset

445 
using set_take_subset by (fastforce simp add: map_upd_upds_conv_if) 
14025  446 

20800  447 
lemma map_upds_apply_nontin [simp]: 
24331  448 
"x ~: set xs ==> (f(xs[>]ys)) x = f x" 
449 
apply (induct xs arbitrary: ys) 

450 
apply simp 

451 
apply (case_tac ys) 

452 
apply (auto simp: map_upd_upds_conv_if) 

453 
done 

14025  454 

20800  455 
lemma fun_upds_append_drop [simp]: 
24331  456 
"size xs = size ys \<Longrightarrow> m(xs@zs[\<mapsto>]ys) = m(xs[\<mapsto>]ys)" 
457 
apply (induct xs arbitrary: m ys) 

458 
apply simp 

459 
apply (case_tac ys) 

460 
apply simp_all 

461 
done 

14300  462 

20800  463 
lemma fun_upds_append2_drop [simp]: 
24331  464 
"size xs = size ys \<Longrightarrow> m(xs[\<mapsto>]ys@zs) = m(xs[\<mapsto>]ys)" 
465 
apply (induct xs arbitrary: m ys) 

466 
apply simp 

467 
apply (case_tac ys) 

468 
apply simp_all 

469 
done 

14300  470 

471 

20800  472 
lemma restrict_map_upds[simp]: 
473 
"\<lbrakk> length xs = length ys; set xs \<subseteq> D \<rbrakk> 

474 
\<Longrightarrow> m(xs [\<mapsto>] ys)`D = (m`(D  set xs))(xs [\<mapsto>] ys)" 

24331  475 
apply (induct xs arbitrary: m ys) 
476 
apply simp 

477 
apply (case_tac ys) 

478 
apply simp 

479 
apply (simp add: Diff_insert [symmetric] insert_absorb) 

480 
apply (simp add: map_upd_upds_conv_if) 

481 
done 

14186  482 

483 

17399
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484 
subsection {* @{term [source] dom} *} 
13908  485 

31080  486 
lemma dom_eq_empty_conv [simp]: "dom f = {} \<longleftrightarrow> f = empty" 
44921  487 
by (auto simp: dom_def) 
31080  488 

13908  489 
lemma domI: "m a = Some b ==> a : dom m" 
24331  490 
by(simp add:dom_def) 
14100  491 
(* declare domI [intro]? *) 
13908  492 

15369  493 
lemma domD: "a : dom m ==> \<exists>b. m a = Some b" 
24331  494 
by (cases "m a") (auto simp add: dom_def) 
13908  495 

20800  496 
lemma domIff [iff, simp del]: "(a : dom m) = (m a ~= None)" 
24331  497 
by(simp add:dom_def) 
13908  498 

20800  499 
lemma dom_empty [simp]: "dom empty = {}" 
24331  500 
by(simp add:dom_def) 
13908  501 

20800  502 
lemma dom_fun_upd [simp]: 
24331  503 
"dom(f(x := y)) = (if y=None then dom f  {x} else insert x (dom f))" 
504 
by(auto simp add:dom_def) 

13908  505 

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506 
lemma dom_if: 
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507 
"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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508 
by (auto split: if_splits) 
13937  509 

15304  510 
lemma dom_map_of_conv_image_fst: 
34979
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511 
"dom (map_of xys) = fst ` set xys" 
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changeset

512 
by (induct xys) (auto simp add: dom_if) 
15304  513 

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514 
lemma dom_map_of_zip [simp]: "length xs = length ys ==> dom (map_of (zip xs ys)) = set xs" 
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changeset

515 
by (induct rule: list_induct2) (auto simp add: dom_if) 
15110
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Added a number of new thms and the new function remove1
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changeset

516 

13908  517 
lemma finite_dom_map_of: "finite (dom (map_of l))" 
24331  518 
by (induct l) (auto simp add: dom_def insert_Collect [symmetric]) 
13908  519 

20800  520 
lemma dom_map_upds [simp]: 
24331  521 
"dom(m(xs[>]ys)) = set(take (length ys) xs) Un dom m" 
522 
apply (induct xs arbitrary: m ys) 

523 
apply simp 

524 
apply (case_tac ys) 

525 
apply auto 

526 
done 

13910  527 

20800  528 
lemma dom_map_add [simp]: "dom(m++n) = dom n Un dom m" 
24331  529 
by(auto simp:dom_def) 
13910  530 

20800  531 
lemma dom_override_on [simp]: 
532 
"dom(override_on f g A) = 

533 
(dom f  {a. a : A  dom g}) Un {a. a : A Int dom g}" 

24331  534 
by(auto simp: dom_def override_on_def) 
13908  535 

14027  536 
lemma map_add_comm: "dom m1 \<inter> dom m2 = {} \<Longrightarrow> m1++m2 = m2++m1" 
24331  537 
by (rule ext) (force simp: map_add_def dom_def split: option.split) 
20800  538 

32236
0203e1006f1b
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krauss
parents:
31380
diff
changeset

539 
lemma map_add_dom_app_simps: 
0203e1006f1b
some lemmas about maps (contributed by Peter Lammich)
krauss
parents:
31380
diff
changeset

540 
"\<lbrakk> m\<in>dom l2 \<rbrakk> \<Longrightarrow> (l1++l2) m = l2 m" 
0203e1006f1b
some lemmas about maps (contributed by Peter Lammich)
krauss
parents:
31380
diff
changeset

541 
"\<lbrakk> m\<notin>dom l1 \<rbrakk> \<Longrightarrow> (l1++l2) m = l2 m" 
0203e1006f1b
some lemmas about maps (contributed by Peter Lammich)
krauss
parents:
31380
diff
changeset

542 
"\<lbrakk> m\<notin>dom l2 \<rbrakk> \<Longrightarrow> (l1++l2) m = l1 m" 
0203e1006f1b
some lemmas about maps (contributed by Peter Lammich)
krauss
parents:
31380
diff
changeset

543 
by (auto simp add: map_add_def split: option.split_asm) 
0203e1006f1b
some lemmas about maps (contributed by Peter Lammich)
krauss
parents:
31380
diff
changeset

544 

29622  545 
lemma dom_const [simp]: 
35159
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changeset

546 
"dom (\<lambda>x. Some (f x)) = UNIV" 
29622  547 
by auto 
548 

22230  549 
(* Due to John Matthews  could be rephrased with dom *) 
550 
lemma finite_map_freshness: 

551 
"finite (dom (f :: 'a \<rightharpoonup> 'b)) \<Longrightarrow> \<not> finite (UNIV :: 'a set) \<Longrightarrow> 

552 
\<exists>x. f x = None" 

553 
by(bestsimp dest:ex_new_if_finite) 

14027  554 

28790  555 
lemma dom_minus: 
556 
"f x = None \<Longrightarrow> dom f  insert x A = dom f  A" 

557 
unfolding dom_def by simp 

558 

559 
lemma insert_dom: 

560 
"f x = Some y \<Longrightarrow> insert x (dom f) = dom f" 

561 
unfolding dom_def by auto 

562 

35607  563 
lemma map_of_map_keys: 
564 
"set xs = dom m \<Longrightarrow> map_of (map (\<lambda>k. (k, the (m k))) xs) = m" 

565 
by (rule ext) (auto simp add: map_of_map_restrict restrict_map_def) 

566 

39379
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parents:
39302
diff
changeset

567 
lemma map_of_eqI: 
ab1b070aa412
moved lemmas map_of_eqI and map_of_eq_dom to Map.thy
haftmann
parents:
39302
diff
changeset

568 
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

569 
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

570 
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

571 
proof (rule ext) 
ab1b070aa412
moved lemmas map_of_eqI and map_of_eq_dom to Map.thy
haftmann
parents:
39302
diff
changeset

572 
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:
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diff
changeset

573 
proof (cases "map_of xs k") 
ab1b070aa412
moved lemmas map_of_eqI and map_of_eq_dom to Map.thy
haftmann
parents:
39302
diff
changeset

574 
case None then have "k \<notin> set (map fst xs)" by (simp add: map_of_eq_None_iff) 
ab1b070aa412
moved lemmas map_of_eqI and map_of_eq_dom to Map.thy
haftmann
parents:
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diff
changeset

575 
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

576 
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:
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diff
changeset

577 
with None show ?thesis by simp 
ab1b070aa412
moved lemmas map_of_eqI and map_of_eq_dom to Map.thy
haftmann
parents:
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diff
changeset

578 
next 
ab1b070aa412
moved lemmas map_of_eqI and map_of_eq_dom to Map.thy
haftmann
parents:
39302
diff
changeset

579 
case (Some v) then have "k \<in> set (map fst xs)" by (auto simp add: dom_map_of_conv_image_fst [symmetric]) 
ab1b070aa412
moved lemmas map_of_eqI and map_of_eq_dom to Map.thy
haftmann
parents:
39302
diff
changeset

580 
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

581 
qed 
ab1b070aa412
moved lemmas map_of_eqI and map_of_eq_dom to Map.thy
haftmann
parents:
39302
diff
changeset

582 
qed 
ab1b070aa412
moved lemmas map_of_eqI and map_of_eq_dom to Map.thy
haftmann
parents:
39302
diff
changeset

583 

ab1b070aa412
moved lemmas map_of_eqI and map_of_eq_dom to Map.thy
haftmann
parents:
39302
diff
changeset

584 
lemma map_of_eq_dom: 
ab1b070aa412
moved lemmas map_of_eqI and map_of_eq_dom to Map.thy
haftmann
parents:
39302
diff
changeset

585 
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

586 
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

587 
proof  
ab1b070aa412
moved lemmas map_of_eqI and map_of_eq_dom to Map.thy
haftmann
parents:
39302
diff
changeset

588 
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

589 
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

590 
qed 
ab1b070aa412
moved lemmas map_of_eqI and map_of_eq_dom to Map.thy
haftmann
parents:
39302
diff
changeset

591 

28790  592 

17399
56a3a4affedc
@{term [source] ...} in subsections probably more robust;
wenzelm
parents:
17391
diff
changeset

593 
subsection {* @{term [source] ran} *} 
14100  594 

20800  595 
lemma ranI: "m a = Some b ==> b : ran m" 
24331  596 
by(auto simp: ran_def) 
14100  597 
(* declare ranI [intro]? *) 
13908  598 

20800  599 
lemma ran_empty [simp]: "ran empty = {}" 
24331  600 
by(auto simp: ran_def) 
13908  601 

20800  602 
lemma ran_map_upd [simp]: "m a = None ==> ran(m(a>b)) = insert b (ran m)" 
24331  603 
unfolding ran_def 
604 
apply auto 

605 
apply (subgoal_tac "aa ~= a") 

606 
apply auto 

607 
done 

20800  608 

34979
8cb6e7a42e9c
more correspondence lemmas between related operations
haftmann
parents:
34941
diff
changeset

609 
lemma ran_distinct: 
8cb6e7a42e9c
more correspondence lemmas between related operations
haftmann
parents:
34941
diff
changeset

610 
assumes dist: "distinct (map fst al)" 
8cb6e7a42e9c
more correspondence lemmas between related operations
haftmann
parents:
34941
diff
changeset

611 
shows "ran (map_of al) = snd ` set al" 
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more correspondence lemmas between related operations
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using assms proof (induct al) 
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case Nil then show ?case by simp 
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next 
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case (Cons kv al) 
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616 
then have "ran (map_of al) = snd ` set al" by simp 
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617 
moreover from Cons.prems have "map_of al (fst kv) = None" 
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618 
by (simp add: map_of_eq_None_iff) 
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619 
ultimately show ?case by (simp only: map_of.simps ran_map_upd) simp 
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620 
qed 
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621 

13910  622 

14100  623 
subsection {* @{text "map_le"} *} 
13910  624 

13912  625 
lemma map_le_empty [simp]: "empty \<subseteq>\<^sub>m g" 
24331  626 
by (simp add: map_le_def) 
13910  627 

17724  628 
lemma upd_None_map_le [simp]: "f(x := None) \<subseteq>\<^sub>m f" 
24331  629 
by (force simp add: map_le_def) 
14187  630 

13910  631 
lemma map_le_upd[simp]: "f \<subseteq>\<^sub>m g ==> f(a := b) \<subseteq>\<^sub>m g(a := b)" 
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by (fastforce simp add: map_le_def) 
13910  633 

17724  634 
lemma map_le_imp_upd_le [simp]: "m1 \<subseteq>\<^sub>m m2 \<Longrightarrow> m1(x := None) \<subseteq>\<^sub>m m2(x \<mapsto> y)" 
24331  635 
by (force simp add: map_le_def) 
14187  636 

20800  637 
lemma map_le_upds [simp]: 
24331  638 
"f \<subseteq>\<^sub>m g ==> f(as [>] bs) \<subseteq>\<^sub>m g(as [>] bs)" 
639 
apply (induct as arbitrary: f g bs) 

640 
apply simp 

641 
apply (case_tac bs) 

642 
apply auto 

643 
done 

13908  644 

14033  645 
lemma map_le_implies_dom_le: "(f \<subseteq>\<^sub>m g) \<Longrightarrow> (dom f \<subseteq> dom g)" 
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by (fastforce simp add: map_le_def dom_def) 
14033  647 

648 
lemma map_le_refl [simp]: "f \<subseteq>\<^sub>m f" 

24331  649 
by (simp add: map_le_def) 
14033  650 

14187  651 
lemma map_le_trans[trans]: "\<lbrakk> m1 \<subseteq>\<^sub>m m2; m2 \<subseteq>\<^sub>m m3\<rbrakk> \<Longrightarrow> m1 \<subseteq>\<^sub>m m3" 
24331  652 
by (auto simp add: map_le_def dom_def) 
14033  653 

654 
lemma map_le_antisym: "\<lbrakk> f \<subseteq>\<^sub>m g; g \<subseteq>\<^sub>m f \<rbrakk> \<Longrightarrow> f = g" 

24331  655 
unfolding map_le_def 
656 
apply (rule ext) 

657 
apply (case_tac "x \<in> dom f", simp) 

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apply (case_tac "x \<in> dom g", simp, fastforce) 
24331  659 
done 
14033  660 

661 
lemma map_le_map_add [simp]: "f \<subseteq>\<^sub>m (g ++ f)" 

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662 
by (fastforce simp add: map_le_def) 
14033  663 

15304  664 
lemma map_le_iff_map_add_commute: "(f \<subseteq>\<^sub>m f ++ g) = (f++g = g++f)" 
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665 
by(fastforce simp: map_add_def map_le_def fun_eq_iff split: option.splits) 
15304  666 

15303  667 
lemma map_add_le_mapE: "f++g \<subseteq>\<^sub>m h \<Longrightarrow> g \<subseteq>\<^sub>m h" 
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668 
by (fastforce simp add: map_le_def map_add_def dom_def) 
15303  669 

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lemma map_add_le_mapI: "\<lbrakk> f \<subseteq>\<^sub>m h; g \<subseteq>\<^sub>m h \<rbrakk> \<Longrightarrow> f++g \<subseteq>\<^sub>m h" 
24331  671 
by (clarsimp simp add: map_le_def map_add_def dom_def split: option.splits) 
15303  672 

31080  673 
lemma dom_eq_singleton_conv: "dom f = {x} \<longleftrightarrow> (\<exists>v. f = [x \<mapsto> v])" 
674 
proof(rule iffI) 

675 
assume "\<exists>v. f = [x \<mapsto> v]" 

676 
thus "dom f = {x}" by(auto split: split_if_asm) 

677 
next 

678 
assume "dom f = {x}" 

679 
then obtain v where "f x = Some v" by auto 

680 
hence "[x \<mapsto> v] \<subseteq>\<^sub>m f" by(auto simp add: map_le_def) 

681 
moreover have "f \<subseteq>\<^sub>m [x \<mapsto> v]" using `dom f = {x}` `f x = Some v` 

682 
by(auto simp add: map_le_def) 

683 
ultimately have "f = [x \<mapsto> v]" by(rule map_le_antisym) 

684 
thus "\<exists>v. f = [x \<mapsto> v]" by blast 

685 
qed 

686 

35565  687 

688 
subsection {* Various *} 

689 

690 
lemma set_map_of_compr: 

691 
assumes distinct: "distinct (map fst xs)" 

692 
shows "set xs = {(k, v). map_of xs k = Some v}" 

693 
using assms proof (induct xs) 

694 
case Nil then show ?case by simp 

695 
next 

696 
case (Cons x xs) 

697 
obtain k v where "x = (k, v)" by (cases x) blast 

698 
with Cons.prems have "k \<notin> dom (map_of xs)" 

699 
by (simp add: dom_map_of_conv_image_fst) 

700 
then have *: "insert (k, v) {(k, v). map_of xs k = Some v} = 

701 
{(k', v'). (map_of xs(k \<mapsto> v)) k' = Some v'}" 

702 
by (auto split: if_splits) 

703 
from Cons have "set xs = {(k, v). map_of xs k = Some v}" by simp 

704 
with * `x = (k, v)` show ?case by simp 

705 
qed 

706 

707 
lemma map_of_inject_set: 

708 
assumes distinct: "distinct (map fst xs)" "distinct (map fst ys)" 

709 
shows "map_of xs = map_of ys \<longleftrightarrow> set xs = set ys" (is "?lhs \<longleftrightarrow> ?rhs") 

710 
proof 

711 
assume ?lhs 

712 
moreover from `distinct (map fst xs)` have "set xs = {(k, v). map_of xs k = Some v}" 

713 
by (rule set_map_of_compr) 

714 
moreover from `distinct (map fst ys)` have "set ys = {(k, v). map_of ys k = Some v}" 

715 
by (rule set_map_of_compr) 

716 
ultimately show ?rhs by simp 

717 
next 

718 
assume ?rhs show ?lhs proof 

719 
fix k 

720 
show "map_of xs k = map_of ys k" proof (cases "map_of xs k") 

721 
case None 

722 
moreover with `?rhs` have "map_of ys k = None" 

723 
by (simp add: map_of_eq_None_iff) 

724 
ultimately show ?thesis by simp 

725 
next 

726 
case (Some v) 

727 
moreover with distinct `?rhs` have "map_of ys k = Some v" 

728 
by simp 

729 
ultimately show ?thesis by simp 

730 
qed 

731 
qed 

732 
qed 

733 

3981  734 
end 