author  nipkow 
Sun, 14 Sep 2003 17:53:27 +0200  
changeset 14187  26dfcd0ac436 
parent 14186  6d2a494e33be 
child 14208  144f45277d5a 
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 = List: 
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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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consts 

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chg_map :: "('b => 'b) => 'a => ('a ~=> 'b) => ('a ~=> 'b)" 
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map_add :: "('a ~=> 'b) => ('a ~=> 'b) => ('a ~=> 'b)" (infixl "++" 100) 
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map_image::"('b => 'c) => ('a ~=> 'b) => ('a ~=> 'c)" (infixr "`>" 90) 

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restrict_map :: "('a ~=> 'b) => 'a set => ('a ~=> 'b)" ("_'__" [90, 91] 90) 

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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 => 

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('a ~=> 'b)" 
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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_le :: "('a ~=> 'b) => ('a ~=> 'b) => bool" (infix "\<subseteq>\<^sub>m" 50) 
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nonterminals 
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maplets maplet 

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syntax 
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empty :: "'a ~=> 'b" 
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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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12114
a8e860c86252
eliminated old "symbols" syntax, use "xsymbols" instead;
wenzelm
parents:
10137
diff
changeset

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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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"~=>" :: "[type, type] => type" (infixr "\<rightharpoonup>" 0) 
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restrict_map :: "('a ~=> 'b) => 'a set => ('a ~=> 'b)" ("_\<lfloor>_" [90, 91] 90) 
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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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"@chg_map" :: "('a ~=> 'b) => 'a => ('b => 'b) => ('a ~=> 'b)" 

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

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translations 

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"empty" => "_K None" 
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"empty" <= "%x. None" 

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"m(x\<mapsto>\<lambda>y. f)" == "chg_map (\<lambda>y. f) x m" 
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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 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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chg_map_def: "chg_map f a m == case m a of None => m  Some b => m(a>f b)" 
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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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map_image_def: "f`>m == option_map f o m" 

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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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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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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: "m1 \<subseteq>\<^sub>m m2 == ALL a : dom m1. m1 a = m2 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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subsection {* @{term 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 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 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 sum_case} and @{term empty}/@{term 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 chg_map} *} 
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lemma chg_map_new[simp]: "m a = None ==> chg_map f a m = m" 
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apply (unfold chg_map_def) 
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apply auto 

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done 

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lemma chg_map_upd[simp]: "m a = Some b ==> chg_map f a m = m(a>f b)" 
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apply (unfold chg_map_def) 
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apply auto 

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done 

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subsection {* @{term map_of} *} 
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lemma map_of_SomeD [rule_format (no_asm)]: "map_of xs k = Some y > (k,y):set xs" 

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apply (induct_tac "xs") 

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apply auto 

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done 

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lemma map_of_mapk_SomeI [rule_format (no_asm)]: "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_tac "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 (no_asm)]: "(k, x) : set l > (? x. map_of l k = Some x)" 

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apply (induct_tac "l") 

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apply auto 

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done 

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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_tac "xs") 

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apply auto 

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done 

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lemma finite_range_map_of: "finite (range (map_of l))" 

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apply (induct_tac "l") 

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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_map: "map_of (map (%(a,b). (a,f b)) xs) x = option_map f (map_of xs x)" 

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apply (induct_tac "xs") 

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apply auto 

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done 

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subsection {* @{term 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 {* @{text "++"} *} 
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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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lemma map_add_find_right[simp]: "!!xx. n k = Some xx ==> (m ++ n) k = Some xx" 
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apply (subst map_add_Some_iff) 

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apply fast 
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done 

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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) 
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apply auto 

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done 

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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_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_tac "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))" 

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apply (induct_tac "l") 
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apply 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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subsection {* @{term map_image} *} 
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lemma map_image_empty [simp]: "f`>empty = empty" 
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by (auto simp: map_image_def empty_def) 

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lemma map_image_upd [simp]: "f`>m(a>b) = (f`>m)(a>f b)" 

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apply (auto simp: map_image_def fun_upd_def) 

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by (rule ext, auto) 

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subsection {* @{term restrict_map} *} 

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lemma restrict_map_to_empty[simp]: "m\<lfloor>{} = empty" 
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by(simp add: restrict_map_def) 

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lemma restrict_map_empty[simp]: "empty\<lfloor>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\<lfloor>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\<lfloor>A) x = None" 

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by (auto simp: restrict_map_def) 

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lemma ran_restrictD: "y \<in> ran (m\<lfloor>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\<lfloor>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)\<lfloor>({x}) = m\<lfloor>({x})" 

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by (rule ext, auto simp: restrict_map_def) 

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lemma restrict_restrict [simp]: "m\<lfloor>A\<lfloor>B = m\<lfloor>(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)\<lfloor>D = (if x \<in> D then (m\<lfloor>(D{x}))(x := y) else m\<lfloor>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\<lfloor>D)(x := None) = (if x:D then m\<lfloor>(D  {x}) else m\<lfloor>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\<lfloor>D)(x := y) = (m\<lfloor>(D{x}))(x := y)" 

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by(simp add: restrict_map_def expand_fun_eq) 

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lemma fun_upd_restrict_conv[simp]: 

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"x \<in> D \<Longrightarrow> (m\<lfloor>D)(x := y) = (m\<lfloor>(D{x}))(x := y)" 

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by(simp add: restrict_map_def expand_fun_eq) 

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subsection {* @{term map_upds} *} 

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

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by(simp add:map_upds_def) 

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

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by(simp add:map_upds_def) 

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

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by(simp add:map_upds_def) 

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

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apply(induct xs) 

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apply(clarsimp simp add:neq_Nil_conv) 

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apply(case_tac ys) 

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apply simp 

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apply simp 

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done 

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lemma map_upds_list_update2_drop[simp]: 

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

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

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apply(induct xs) 

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apply simp 

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apply(case_tac ys) 

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apply simp 

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apply(simp split:nat.split) 

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done 

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lemma map_upd_upds_conv_if: "!!x y ys f. 

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

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

372 
else (f(xs [>] ys))(x>y))" 

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apply(induct xs) 

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apply simp 

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apply(case_tac ys) 

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apply(auto split:split_if simp:fun_upd_twist) 

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done 

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lemma map_upds_twist [simp]: 

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

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apply(insert set_take_subset) 

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apply (fastsimp simp add: map_upd_upds_conv_if) 

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done 

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lemma map_upds_apply_nontin[simp]: 

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

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apply(induct xs) 

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apply simp 

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apply(case_tac ys) 

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apply(auto simp: map_upd_upds_conv_if) 

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done 

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

395 
\<Longrightarrow> m(xs [\<mapsto>] ys)\<lfloor>D = (m\<lfloor>(D  set xs))(xs [\<mapsto>] ys)" 

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apply(induct xs) 

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apply simp 

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apply(case_tac ys) 

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apply simp 

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apply(simp add:Diff_insert[symmetric] insert_absorb) 

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apply(simp add: map_upd_upds_conv_if) 

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done 

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subsection {* @{term 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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subsection {* @{term dom} *} 

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

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apply (unfold dom_def) 

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apply auto 

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done 

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(* declare domI [intro]? *) 
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lemma domD: "a : dom m ==> ? b. m a = Some b" 

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apply (unfold dom_def) 

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apply auto 

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done 

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lemma domIff[iff]: "(a : dom m) = (m a ~= None)" 
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apply (unfold dom_def) 
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apply auto 

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done 

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declare domIff [simp del] 

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lemma dom_empty[simp]: "dom empty = {}" 
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apply (unfold dom_def) 
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apply (simp (no_asm)) 

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done 

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

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by (simp add:dom_def) blast 

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

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apply(auto simp del:fun_upd_apply) 

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done 

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

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apply (induct_tac "l") 

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apply (auto simp add: insert_Collect [symmetric]) 

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done 

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

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apply(induct xs) 

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apply simp 

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apply(case_tac ys) 

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apply auto 

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done 

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lemma dom_map_add[simp]: "dom(m++n) = dom n Un dom m" 
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apply (unfold dom_def) 
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apply auto 

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done 

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lemma dom_overwrite[simp]: 

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

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by(auto simp add: dom_def overwrite_def) 

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lemma map_add_comm: "dom m1 \<inter> dom m2 = {} \<Longrightarrow> m1++m2 = m2++m1" 
472 
apply(rule ext) 

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

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done 

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subsection {* @{term ran} *} 
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lemma ranI: "m a = Some b ==> b : ran m" 

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by (auto simp add: ran_def) 

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(* declare ranI [intro]? *) 

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lemma ran_empty[simp]: "ran empty = {}" 
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apply (unfold ran_def) 
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apply (simp (no_asm)) 

485 
done 

486 

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lemma ran_map_upd[simp]: "m a = None ==> ran(m(a>b)) = insert b (ran m)" 
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apply (unfold ran_def) 
489 
apply auto 

490 
apply (subgoal_tac "~ (aa = a) ") 

491 
apply auto 

492 
done 

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14100  494 
subsection {* @{text "map_le"} *} 
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13912  496 
lemma map_le_empty [simp]: "empty \<subseteq>\<^sub>m g" 
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by(simp add:map_le_def) 
498 

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lemma [simp]: "f(x := None) \<subseteq>\<^sub>m f" 
500 
by(force simp add:map_le_def) 

501 

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lemma map_le_upd[simp]: "f \<subseteq>\<^sub>m g ==> f(a := b) \<subseteq>\<^sub>m g(a := b)" 
503 
by(fastsimp simp add:map_le_def) 

504 

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lemma [simp]: "m1 \<subseteq>\<^sub>m m2 \<Longrightarrow> m1(x := None) \<subseteq>\<^sub>m m2(x \<mapsto> y)" 
506 
by(force simp add:map_le_def) 

507 

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lemma map_le_upds[simp]: 
509 
"!!f g bs. f \<subseteq>\<^sub>m g ==> f(as [>] bs) \<subseteq>\<^sub>m g(as [>] bs)" 

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apply(induct as) 
511 
apply simp 

512 
apply(case_tac bs) 

513 
apply auto 

514 
done 

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lemma map_le_implies_dom_le: "(f \<subseteq>\<^sub>m g) \<Longrightarrow> (dom f \<subseteq> dom g)" 
517 
by (fastsimp simp add: map_le_def dom_def) 

518 

519 
lemma map_le_refl [simp]: "f \<subseteq>\<^sub>m f" 

520 
by (simp add: map_le_def) 

521 

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lemma map_le_trans[trans]: "\<lbrakk> m1 \<subseteq>\<^sub>m m2; m2 \<subseteq>\<^sub>m m3\<rbrakk> \<Longrightarrow> m1 \<subseteq>\<^sub>m m3" 
523 
by(force simp add:map_le_def) 

14033  524 

525 
lemma map_le_antisym: "\<lbrakk> f \<subseteq>\<^sub>m g; g \<subseteq>\<^sub>m f \<rbrakk> \<Longrightarrow> f = g" 

526 
apply (unfold map_le_def) 

527 
apply (rule ext) 

528 
apply (case_tac "x \<in> dom f") 

529 
apply simp 

530 
apply (case_tac "x \<in> dom g") 

531 
apply simp 

532 
apply fastsimp 

533 
done 

534 

535 
lemma map_le_map_add [simp]: "f \<subseteq>\<^sub>m (g ++ f)" 

536 
by (fastsimp simp add: map_le_def) 

537 

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end 