author  wenzelm 
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changeset 19947  29b376397cd5 
parent 19656  09be06943252 
child 20800  69c82605efcf 
permissions  rwrr 
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(* Title: HOL/Map.thy 
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ID: $Id$ 

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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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types ('a,'b) "~=>" = "'a => 'b option" (infixr 0) 
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translations (type) "a ~=> b " <= (type) "a => b option" 
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syntax (xsymbols) 
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"~=>" :: "[type, type] => type" (infixr "\<rightharpoonup>" 0) 
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abbreviation 
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empty :: "'a ~=> 'b" 

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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) 
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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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const_syntax (xsymbols) 
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map_comp (infixl "\<circ>\<^sub>m" 55) 
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consts 
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map_add :: "('a ~=> 'b) => ('a ~=> 'b) => ('a ~=> 'b)" (infixl "++" 100) 
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restrict_map :: "('a ~=> 'b) => 'a set => ('a ~=> 'b)" (infixl "`" 110) 
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dom :: "('a ~=> 'b) => 'a set" 
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ran :: "('a ~=> 'b) => 'b set" 

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map_of :: "('a * 'b)list => 'a ~=> 'b" 

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map_upds:: "('a ~=> 'b) => 'a list => 'b list => ('a ~=> 'b)" 
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map_le :: "('a ~=> 'b) => ('a ~=> 'b) => bool" (infix "\<subseteq>\<^sub>m" 50) 
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const_syntax (latex output) 
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restrict_map ("_\<restriction>\<^bsub>_\<^esub>" [111,110] 110) 
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nonterminals 
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maplets 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:=Some y)" 

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"_MapUpd m (_maplets x y)" == "map_upds m x 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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defs 
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map_add_def: "m1++m2 == %x. case m2 x of None => m1 x  Some y => Some y" 
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restrict_map_def: "m`A == %x. if x : A then m x else None" 
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map_upds_def: "m(xs [>] ys) == m ++ map_of (rev(zip xs ys))" 

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dom_def: "dom(m) == {a. m a ~= None}" 
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ran_def: "ran(m) == {b. EX a. m a = Some b}" 
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map_le_def: "m\<^isub>1 \<subseteq>\<^sub>m m\<^isub>2 == ALL a : dom m\<^isub>1. m\<^isub>1 a = m\<^isub>2 a" 
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primrec 
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"map_of [] = empty" 

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"map_of (p#ps) = (map_of ps)(fst p > snd p)" 
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(* special purpose constants that should be defined somewhere else and 
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whose syntax is a bit odd as well: 

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"@chg_map" :: "('a ~=> 'b) => 'a => ('b => 'b) => ('a ~=> 'b)" 

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("_/'(_/\<mapsto>\<lambda>_. _')" [900,0,0,0] 900) 

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"m(x\<mapsto>\<lambda>y. f)" == "chg_map (\<lambda>y. f) x m" 

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map_upd_s::"('a ~=> 'b) => 'a set => 'b => 

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('a ~=> 'b)" ("_/'(_{>}_/')" [900,0,0]900) 

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map_subst::"('a ~=> 'b) => 'b => 'b => 

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

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map_upd_s_def: "m(as{>}b) == %x. if x : as then Some b else m x" 

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map_subst_def: "m(a~>b) == %x. if m x = Some a then Some b else m x" 

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map_upd_s :: "('a ~=> 'b) => 'a set => 'b => ('a ~=> 'b)" 

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("_/'(_/{\<mapsto>}/_')" [900,0,0]900) 

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map_subst :: "('a ~=> 'b) => 'b => 'b => 

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('a ~=> 'b)" ("_/'(_\<leadsto>_/')" [900,0,0]900) 

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subsection {* @{term [source] map_upd_s} *} 

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lemma map_upd_s_apply [simp]: 

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"(m(as{>}b)) x = (if x : as then Some b else m x)" 

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by (simp add: map_upd_s_def) 

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lemma map_subst_apply [simp]: 

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"(m(a~>b)) x = (if m x = Some a then Some b else m x)" 

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by (simp add: map_subst_def) 

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*) 

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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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apply (rule ext) 
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apply (simp (no_asm)) 

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done 

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(* FIXME: what is this sum_case nonsense?? *) 

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lemma sum_case_empty_empty[simp]: "sum_case empty empty = empty" 
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apply (rule ext) 
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apply (simp (no_asm) split add: sum.split) 

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done 

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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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apply (rule ext) 

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apply (simp (no_asm_simp)) 

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done 

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lemma map_upd_nonempty[simp]: "t(k>x) ~= empty" 
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apply safe 
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apply (drule_tac x = k in fun_cong) 
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apply (simp (no_asm_use)) 
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done 

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lemma map_upd_eqD1: "m(a\<mapsto>x) = n(a\<mapsto>y) \<Longrightarrow> x = y" 
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by (drule fun_cong [of _ _ a], auto) 

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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 fastsimp 

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lemma finite_range_updI: "finite (range f) ==> finite (range (f(a>b)))" 
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apply (unfold 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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(* FIXME: what is this sum_case nonsense?? *) 

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subsection {* @{term [source] sum_case} and @{term [source] empty}/@{term [source] map_upd} *} 
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lemma sum_case_map_upd_empty[simp]: 
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"sum_case (m(k>y)) empty = (sum_case m empty)(Inl k>y)" 

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apply (rule ext) 
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apply (simp (no_asm) split add: sum.split) 

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done 

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lemma sum_case_empty_map_upd[simp]: 
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"sum_case empty (m(k>y)) = (sum_case empty m)(Inr k>y)" 

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apply (rule ext) 
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apply (simp (no_asm) split add: sum.split) 

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done 

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lemma sum_case_map_upd_map_upd[simp]: 
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"sum_case (m1(k1>y1)) (m2(k2>y2)) = (sum_case (m1(k1>y1)) m2)(Inr k2>y2)" 

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apply (rule ext) 
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apply (simp (no_asm) split add: sum.split) 

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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: 

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"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 finite_range_map_of: "finite (range (map_of xys))" 
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apply (induct xys) 
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apply (simp_all (no_asm) 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 [rule_format]: "map_of xs k = Some y > (k,y):set xs" 
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by (induct "xs", auto) 
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lemma map_of_mapk_SomeI [rule_format]: 
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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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apply (induct "t") 
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apply (auto simp add: inj_eq) 
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done 

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lemma weak_map_of_SomeI [rule_format]: 
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"(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; P k z ] ==> map_of (filter (split P) xs) k = Some z" 

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apply (rule mp) 

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prefer 2 apply assumption 
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apply (erule thin_rl) 
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apply (induct "xs", auto) 
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done 
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lemma map_of_map: "map_of (map (%(a,b). (a,f b)) xs) x = option_map f (map_of xs x)" 

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by (induct "xs", auto) 
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subsection {* @{term [source] option_map} related *} 
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lemma option_map_o_empty[simp]: "option_map f o empty = empty" 
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apply (rule ext) 
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apply (simp (no_asm)) 

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done 

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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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apply (rule ext) 
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apply (simp (no_asm)) 

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done 

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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 intro: ext 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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14025  287 
lemma map_add_empty[simp]: "m ++ empty = m" 
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apply (unfold map_add_def) 

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apply (simp (no_asm)) 
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done 

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lemma empty_map_add[simp]: "empty ++ m = m" 
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apply (unfold map_add_def) 

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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 map_add_assoc[simp]: "m1 ++ (m2 ++ m3) = (m1 ++ m2) ++ m3" 
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apply(rule ext) 

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apply(simp add: map_add_def split:option.split) 

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done 

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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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apply (unfold map_add_def) 
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apply (simp (no_asm) split add: option.split) 
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done 

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lemmas map_add_SomeD = map_add_Some_iff [THEN iffD1, standard] 
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declare map_add_SomeD [dest!] 

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14025  312 
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  315 
lemma map_add_None [iff]: "((m ++ n) k = None) = (n k = None & m k = None)" 
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apply (unfold map_add_def) 

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apply (simp (no_asm) split add: option.split) 
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done 

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lemma map_add_upd[simp]: "f ++ g(x>y) = (f ++ g)(x>y)" 
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apply (unfold map_add_def) 

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apply (rule ext, auto) 
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done 
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14186  325 
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_of_append[simp]: "map_of (xs@ys) = map_of ys ++ map_of xs" 
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apply (unfold map_add_def) 

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apply (induct "xs") 
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apply (simp (no_asm)) 
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apply (rule ext) 

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apply (simp (no_asm_simp) split add: option.split) 

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done 

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declare fun_upd_apply [simp del] 

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lemma finite_range_map_of_map_add: 
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"finite (range f) ==> finite (range (f ++ map_of l))" 

15251  339 
apply (induct "l", auto) 
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apply (erule finite_range_updI) 
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done 

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declare fun_upd_apply [simp] 

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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(fastsimp simp add:map_add_def dom_def inj_on_def split:option.splits) 

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subsection {* @{term [source] restrict_map} *} 
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lemma restrict_map_to_empty[simp]: "m`{} = empty" 
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by(simp add: restrict_map_def) 
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lemma restrict_map_empty[simp]: "empty`D = empty" 
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by(simp add: restrict_map_def) 
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lemma restrict_in [simp]: "x \<in> A \<Longrightarrow> (m`A) x = m x" 
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by (auto simp: restrict_map_def) 
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lemma restrict_out [simp]: "x \<notin> A \<Longrightarrow> (m`A) x = None" 
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by (auto simp: restrict_map_def) 
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lemma ran_restrictD: "y \<in> ran (m`A) \<Longrightarrow> \<exists>x\<in>A. m x = Some y" 
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by (auto simp: restrict_map_def ran_def split: split_if_asm) 
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lemma dom_restrict [simp]: "dom (m`A) = dom m \<inter> A" 
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by (auto simp: restrict_map_def dom_def split: split_if_asm) 
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lemma restrict_upd_same [simp]: "m(x\<mapsto>y)`({x}) = m`({x})" 
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by (rule ext, auto simp: restrict_map_def) 
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lemma restrict_restrict [simp]: "m`A`B = m`(A\<inter>B)" 
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by (rule ext, auto simp: restrict_map_def) 
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lemma restrict_fun_upd[simp]: 
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"m(x := y)`D = (if x \<in> D then (m`(D{x}))(x := y) else m`D)" 
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by(simp add: restrict_map_def expand_fun_eq) 
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lemma fun_upd_None_restrict[simp]: 

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"(m`D)(x := None) = (if x:D then m`(D  {x}) else m`D)" 
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by(simp add: restrict_map_def expand_fun_eq) 
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lemma fun_upd_restrict: 

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"(m`D)(x := y) = (m`(D{x}))(x := y)" 
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by(simp add: restrict_map_def expand_fun_eq) 
385 

386 
lemma fun_upd_restrict_conv[simp]: 

15693  387 
"x \<in> D \<Longrightarrow> (m`D)(x := y) = (m`(D{x}))(x := y)" 
14186  388 
by(simp add: restrict_map_def expand_fun_eq) 
389 

14100  390 

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391 
subsection {* @{term [source] map_upds} *} 
14025  392 

393 
lemma map_upds_Nil1[simp]: "m([] [>] bs) = m" 

394 
by(simp add:map_upds_def) 

395 

396 
lemma map_upds_Nil2[simp]: "m(as [>] []) = m" 

397 
by(simp add:map_upds_def) 

398 

399 
lemma map_upds_Cons[simp]: "m(a#as [>] b#bs) = (m(a>b))(as[>]bs)" 

400 
by(simp add:map_upds_def) 

401 

14187  402 
lemma map_upds_append1[simp]: "\<And>ys m. size xs < size ys \<Longrightarrow> 
403 
m(xs@[x] [\<mapsto>] ys) = m(xs [\<mapsto>] ys)(x \<mapsto> ys!size xs)" 

404 
apply(induct xs) 

405 
apply(clarsimp simp add:neq_Nil_conv) 

14208  406 
apply (case_tac ys, simp, simp) 
14187  407 
done 
408 

409 
lemma map_upds_list_update2_drop[simp]: 

410 
"\<And>m ys i. \<lbrakk>size xs \<le> i; i < size ys\<rbrakk> 

411 
\<Longrightarrow> m(xs[\<mapsto>]ys[i:=y]) = m(xs[\<mapsto>]ys)" 

14208  412 
apply (induct xs, simp) 
413 
apply (case_tac ys, simp) 

14187  414 
apply(simp split:nat.split) 
415 
done 

14025  416 

417 
lemma map_upd_upds_conv_if: "!!x y ys f. 

418 
(f(x>y))(xs [>] ys) = 

419 
(if x : set(take (length ys) xs) then f(xs [>] ys) 

420 
else (f(xs [>] ys))(x>y))" 

14208  421 
apply (induct xs, simp) 
14025  422 
apply(case_tac ys) 
423 
apply(auto split:split_if simp:fun_upd_twist) 

424 
done 

425 

426 
lemma map_upds_twist [simp]: 

427 
"a ~: set as ==> m(a>b)(as[>]bs) = m(as[>]bs)(a>b)" 

428 
apply(insert set_take_subset) 

429 
apply (fastsimp simp add: map_upd_upds_conv_if) 

430 
done 

431 

432 
lemma map_upds_apply_nontin[simp]: 

433 
"!!ys. x ~: set xs ==> (f(xs[>]ys)) x = f x" 

14208  434 
apply (induct xs, simp) 
14025  435 
apply(case_tac ys) 
436 
apply(auto simp: map_upd_upds_conv_if) 

437 
done 

438 

14300  439 
lemma fun_upds_append_drop[simp]: 
440 
"!!m ys. size xs = size ys \<Longrightarrow> m(xs@zs[\<mapsto>]ys) = m(xs[\<mapsto>]ys)" 

441 
apply(induct xs) 

442 
apply (simp) 

443 
apply(case_tac ys) 

444 
apply simp_all 

445 
done 

446 

447 
lemma fun_upds_append2_drop[simp]: 

448 
"!!m ys. size xs = size ys \<Longrightarrow> m(xs[\<mapsto>]ys@zs) = m(xs[\<mapsto>]ys)" 

449 
apply(induct xs) 

450 
apply (simp) 

451 
apply(case_tac ys) 

452 
apply simp_all 

453 
done 

454 

455 

14186  456 
lemma restrict_map_upds[simp]: "!!m ys. 
457 
\<lbrakk> length xs = length ys; set xs \<subseteq> D \<rbrakk> 

15693  458 
\<Longrightarrow> m(xs [\<mapsto>] ys)`D = (m`(D  set xs))(xs [\<mapsto>] ys)" 
14208  459 
apply (induct xs, simp) 
460 
apply (case_tac ys, simp) 

14186  461 
apply(simp add:Diff_insert[symmetric] insert_absorb) 
462 
apply(simp add: map_upd_upds_conv_if) 

463 
done 

464 

465 

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subsection {* @{term [source] dom} *} 
13908  467 

468 
lemma domI: "m a = Some b ==> a : dom m" 

14208  469 
by (unfold dom_def, auto) 
14100  470 
(* declare domI [intro]? *) 
13908  471 

15369  472 
lemma domD: "a : dom m ==> \<exists>b. m a = Some b" 
18447  473 
apply (case_tac "m a") 
474 
apply (auto simp add: dom_def) 

475 
done 

13908  476 

13910  477 
lemma domIff[iff]: "(a : dom m) = (m a ~= None)" 
14208  478 
by (unfold dom_def, auto) 
13908  479 
declare domIff [simp del] 
480 

13910  481 
lemma dom_empty[simp]: "dom empty = {}" 
13908  482 
apply (unfold dom_def) 
483 
apply (simp (no_asm)) 

484 
done 

485 

13910  486 
lemma dom_fun_upd[simp]: 
487 
"dom(f(x := y)) = (if y=None then dom f  {x} else insert x (dom f))" 

488 
by (simp add:dom_def) blast 

13908  489 

13937  490 
lemma dom_map_of: "dom(map_of xys) = {x. \<exists>y. (x,y) : set xys}" 
491 
apply(induct xys) 

492 
apply(auto simp del:fun_upd_apply) 

493 
done 

494 

15304  495 
lemma dom_map_of_conv_image_fst: 
496 
"dom(map_of xys) = fst ` (set xys)" 

497 
by(force simp: dom_map_of) 

498 

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lemma dom_map_of_zip[simp]: "[ length xs = length ys; distinct xs ] ==> 
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500 
dom(map_of(zip xs ys)) = set xs" 
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501 
by(induct rule: list_induct2, simp_all) 
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502 

13908  503 
lemma finite_dom_map_of: "finite (dom (map_of l))" 
504 
apply (unfold dom_def) 

15251  505 
apply (induct "l") 
13908  506 
apply (auto simp add: insert_Collect [symmetric]) 
507 
done 

508 

14025  509 
lemma dom_map_upds[simp]: 
510 
"!!m ys. dom(m(xs[>]ys)) = set(take (length ys) xs) Un dom m" 

14208  511 
apply (induct xs, simp) 
512 
apply (case_tac ys, auto) 

14025  513 
done 
13910  514 

14025  515 
lemma dom_map_add[simp]: "dom(m++n) = dom n Un dom m" 
14208  516 
by (unfold dom_def, auto) 
13910  517 

15691  518 
lemma dom_override_on[simp]: 
519 
"dom(override_on f g A) = 

520 
(dom f  {a. a : A  dom g}) Un {a. a : A Int dom g}" 

521 
by(auto simp add: dom_def override_on_def) 

13908  522 

14027  523 
lemma map_add_comm: "dom m1 \<inter> dom m2 = {} \<Longrightarrow> m1++m2 = m2++m1" 
524 
apply(rule ext) 

18576  525 
apply(force simp: map_add_def dom_def split:option.split) 
14027  526 
done 
527 

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528 
subsection {* @{term [source] ran} *} 
14100  529 

530 
lemma ranI: "m a = Some b ==> b : ran m" 

531 
by (auto simp add: ran_def) 

532 
(* declare ranI [intro]? *) 

13908  533 

13910  534 
lemma ran_empty[simp]: "ran empty = {}" 
13908  535 
apply (unfold ran_def) 
536 
apply (simp (no_asm)) 

537 
done 

538 

13910  539 
lemma ran_map_upd[simp]: "m a = None ==> ran(m(a>b)) = insert b (ran m)" 
14208  540 
apply (unfold ran_def, auto) 
13908  541 
apply (subgoal_tac "~ (aa = a) ") 
542 
apply auto 

543 
done 

13910  544 

14100  545 
subsection {* @{text "map_le"} *} 
13910  546 

13912  547 
lemma map_le_empty [simp]: "empty \<subseteq>\<^sub>m g" 
13910  548 
by(simp add:map_le_def) 
549 

17724  550 
lemma upd_None_map_le [simp]: "f(x := None) \<subseteq>\<^sub>m f" 
14187  551 
by(force simp add:map_le_def) 
552 

13910  553 
lemma map_le_upd[simp]: "f \<subseteq>\<^sub>m g ==> f(a := b) \<subseteq>\<^sub>m g(a := b)" 
554 
by(fastsimp simp add:map_le_def) 

555 

17724  556 
lemma map_le_imp_upd_le [simp]: "m1 \<subseteq>\<^sub>m m2 \<Longrightarrow> m1(x := None) \<subseteq>\<^sub>m m2(x \<mapsto> y)" 
14187  557 
by(force simp add:map_le_def) 
558 

13910  559 
lemma map_le_upds[simp]: 
560 
"!!f g bs. f \<subseteq>\<^sub>m g ==> f(as [>] bs) \<subseteq>\<^sub>m g(as [>] bs)" 

14208  561 
apply (induct as, simp) 
562 
apply (case_tac bs, auto) 

14025  563 
done 
13908  564 

14033  565 
lemma map_le_implies_dom_le: "(f \<subseteq>\<^sub>m g) \<Longrightarrow> (dom f \<subseteq> dom g)" 
566 
by (fastsimp simp add: map_le_def dom_def) 

567 

568 
lemma map_le_refl [simp]: "f \<subseteq>\<^sub>m f" 

569 
by (simp add: map_le_def) 

570 

14187  571 
lemma map_le_trans[trans]: "\<lbrakk> m1 \<subseteq>\<^sub>m m2; m2 \<subseteq>\<^sub>m m3\<rbrakk> \<Longrightarrow> m1 \<subseteq>\<^sub>m m3" 
18447  572 
by (auto simp add: map_le_def dom_def) 
14033  573 

574 
lemma map_le_antisym: "\<lbrakk> f \<subseteq>\<^sub>m g; g \<subseteq>\<^sub>m f \<rbrakk> \<Longrightarrow> f = g" 

575 
apply (unfold map_le_def) 

576 
apply (rule ext) 

14208  577 
apply (case_tac "x \<in> dom f", simp) 
578 
apply (case_tac "x \<in> dom g", simp, fastsimp) 

14033  579 
done 
580 

581 
lemma map_le_map_add [simp]: "f \<subseteq>\<^sub>m (g ++ f)" 

18576  582 
by (fastsimp simp add: map_le_def) 
14033  583 

15304  584 
lemma map_le_iff_map_add_commute: "(f \<subseteq>\<^sub>m f ++ g) = (f++g = g++f)" 
585 
by(fastsimp simp add:map_add_def map_le_def expand_fun_eq split:option.splits) 

586 

15303  587 
lemma map_add_le_mapE: "f++g \<subseteq>\<^sub>m h \<Longrightarrow> g \<subseteq>\<^sub>m h" 
18576  588 
by (fastsimp simp add: map_le_def map_add_def dom_def) 
15303  589 

590 
lemma map_add_le_mapI: "\<lbrakk> f \<subseteq>\<^sub>m h; g \<subseteq>\<^sub>m h; f \<subseteq>\<^sub>m f++g \<rbrakk> \<Longrightarrow> f++g \<subseteq>\<^sub>m h" 

591 
by (clarsimp simp add: map_le_def map_add_def dom_def split:option.splits) 

592 

3981  593 
end 