author  nipkow 
Sun, 19 Aug 2007 21:21:37 +0200  
changeset 24331  76f7a8c6e842 
parent 22744  5cbe966d67a2 
child 25483  65de74f62874 
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" 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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consts 

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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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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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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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defs 
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map_upds_def [code func]: "m(xs [>] ys) == m ++ map_of (rev(zip xs ys))" 
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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 prems 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 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: "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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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 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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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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lemma map_add_None [iff]: "((m ++ n) k = None) = (n k = None & m k = None)" 
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by (simp add: map_add_def split: option.split) 
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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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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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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: 
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"finite (range f) ==> finite (range (f ++ map_of l))" 
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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 (fastsimp simp: 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 (simp add: restrict_map_def) 
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lemma restrict_out [simp]: "x \<notin> A \<Longrightarrow> (m`A) x = None" 
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by (simp add: 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: "(m`D)(x := y) = (m`(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`D)(x := y) = (m`(D{x}))(x := y)" 
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by (simp add: restrict_map_def expand_fun_eq) 

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subsection {* @{term [source] 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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"\<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 arbitrary: m ys i) 
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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: 
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"(f(x>y))(xs [>] ys) = 

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

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

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apply (induct xs arbitrary: x y ys f) 
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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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using set_take_subset by (fastsimp simp add: map_upd_upds_conv_if) 

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

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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 fun_upds_append_drop [simp]: 
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"size xs = size ys \<Longrightarrow> m(xs@zs[\<mapsto>]ys) = m(xs[\<mapsto>]ys)" 
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apply (induct xs arbitrary: m ys) 

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apply simp 

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apply (case_tac ys) 

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apply simp_all 

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done 

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20800  377 
lemma fun_upds_append2_drop [simp]: 
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"size xs = size ys \<Longrightarrow> m(xs[\<mapsto>]ys@zs) = m(xs[\<mapsto>]ys)" 
379 
apply (induct xs arbitrary: m ys) 

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apply simp 

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apply (case_tac ys) 

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apply simp_all 

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done 

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385 

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lemma restrict_map_upds[simp]: 
387 
"\<lbrakk> length xs = length ys; set xs \<subseteq> D \<rbrakk> 

388 
\<Longrightarrow> m(xs [\<mapsto>] ys)`D = (m`(D  set xs))(xs [\<mapsto>] ys)" 

24331  389 
apply (induct xs arbitrary: m ys) 
390 
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) 

395 
done 

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397 

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subsection {* @{term [source] dom} *} 
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lemma domI: "m a = Some b ==> a : dom m" 

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by(simp add:dom_def) 
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(* declare domI [intro]? *) 
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lemma domD: "a : dom m ==> \<exists>b. m a = Some b" 
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by (cases "m a") (auto simp add: dom_def) 
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lemma domIff [iff, simp del]: "(a : dom m) = (m a ~= None)" 
24331  408 
by(simp add:dom_def) 
13908  409 

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lemma dom_empty [simp]: "dom empty = {}" 
24331  411 
by(simp add:dom_def) 
13908  412 

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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))" 
415 
by(auto simp add:dom_def) 

13908  416 

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lemma dom_map_of: "dom(map_of xys) = {x. \<exists>y. (x,y) : set xys}" 
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by (induct xys) (auto simp del: fun_upd_apply) 
13937  419 

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lemma dom_map_of_conv_image_fst: 
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"dom(map_of xys) = fst ` (set xys)" 
422 
by(force simp: dom_map_of) 

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20800  424 
lemma dom_map_of_zip [simp]: "[ length xs = length ys; distinct xs ] ==> 
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dom(map_of(zip xs ys)) = set xs" 
426 
by (induct rule: list_induct2) simp_all 

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lemma finite_dom_map_of: "finite (dom (map_of l))" 
24331  429 
by (induct l) (auto simp add: dom_def insert_Collect [symmetric]) 
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lemma dom_map_upds [simp]: 
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"dom(m(xs[>]ys)) = set(take (length ys) xs) Un dom m" 
433 
apply (induct xs arbitrary: m ys) 

434 
apply simp 

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apply (case_tac ys) 

436 
apply auto 

437 
done 

13910  438 

20800  439 
lemma dom_map_add [simp]: "dom(m++n) = dom n Un dom m" 
24331  440 
by(auto simp:dom_def) 
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lemma dom_override_on [simp]: 
443 
"dom(override_on f g A) = 

444 
(dom f  {a. a : A  dom g}) Un {a. a : A Int dom g}" 

24331  445 
by(auto simp: dom_def override_on_def) 
13908  446 

14027  447 
lemma map_add_comm: "dom m1 \<inter> dom m2 = {} \<Longrightarrow> m1++m2 = m2++m1" 
24331  448 
by (rule ext) (force simp: map_add_def dom_def split: option.split) 
20800  449 

22230  450 
(* Due to John Matthews  could be rephrased with dom *) 
451 
lemma finite_map_freshness: 

452 
"finite (dom (f :: 'a \<rightharpoonup> 'b)) \<Longrightarrow> \<not> finite (UNIV :: 'a set) \<Longrightarrow> 

453 
\<exists>x. f x = None" 

454 
by(bestsimp dest:ex_new_if_finite) 

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subsection {* @{term [source] ran} *} 
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20800  458 
lemma ranI: "m a = Some b ==> b : ran m" 
24331  459 
by(auto simp: ran_def) 
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(* declare ranI [intro]? *) 
13908  461 

20800  462 
lemma ran_empty [simp]: "ran empty = {}" 
24331  463 
by(auto simp: ran_def) 
13908  464 

20800  465 
lemma ran_map_upd [simp]: "m a = None ==> ran(m(a>b)) = insert b (ran m)" 
24331  466 
unfolding ran_def 
467 
apply auto 

468 
apply (subgoal_tac "aa ~= a") 

469 
apply auto 

470 
done 

20800  471 

13910  472 

14100  473 
subsection {* @{text "map_le"} *} 
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13912  475 
lemma map_le_empty [simp]: "empty \<subseteq>\<^sub>m g" 
24331  476 
by (simp add: map_le_def) 
13910  477 

17724  478 
lemma upd_None_map_le [simp]: "f(x := None) \<subseteq>\<^sub>m f" 
24331  479 
by (force simp add: map_le_def) 
14187  480 

13910  481 
lemma map_le_upd[simp]: "f \<subseteq>\<^sub>m g ==> f(a := b) \<subseteq>\<^sub>m g(a := b)" 
24331  482 
by (fastsimp simp add: map_le_def) 
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17724  484 
lemma map_le_imp_upd_le [simp]: "m1 \<subseteq>\<^sub>m m2 \<Longrightarrow> m1(x := None) \<subseteq>\<^sub>m m2(x \<mapsto> y)" 
24331  485 
by (force simp add: map_le_def) 
14187  486 

20800  487 
lemma map_le_upds [simp]: 
24331  488 
"f \<subseteq>\<^sub>m g ==> f(as [>] bs) \<subseteq>\<^sub>m g(as [>] bs)" 
489 
apply (induct as arbitrary: f g bs) 

490 
apply simp 

491 
apply (case_tac bs) 

492 
apply auto 

493 
done 

13908  494 

14033  495 
lemma map_le_implies_dom_le: "(f \<subseteq>\<^sub>m g) \<Longrightarrow> (dom f \<subseteq> dom g)" 
24331  496 
by (fastsimp simp add: map_le_def dom_def) 
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498 
lemma map_le_refl [simp]: "f \<subseteq>\<^sub>m f" 

24331  499 
by (simp add: map_le_def) 
14033  500 

14187  501 
lemma map_le_trans[trans]: "\<lbrakk> m1 \<subseteq>\<^sub>m m2; m2 \<subseteq>\<^sub>m m3\<rbrakk> \<Longrightarrow> m1 \<subseteq>\<^sub>m m3" 
24331  502 
by (auto simp add: map_le_def dom_def) 
14033  503 

504 
lemma map_le_antisym: "\<lbrakk> f \<subseteq>\<^sub>m g; g \<subseteq>\<^sub>m f \<rbrakk> \<Longrightarrow> f = g" 

24331  505 
unfolding map_le_def 
506 
apply (rule ext) 

507 
apply (case_tac "x \<in> dom f", simp) 

508 
apply (case_tac "x \<in> dom g", simp, fastsimp) 

509 
done 

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511 
lemma map_le_map_add [simp]: "f \<subseteq>\<^sub>m (g ++ f)" 

24331  512 
by (fastsimp simp add: map_le_def) 
14033  513 

15304  514 
lemma map_le_iff_map_add_commute: "(f \<subseteq>\<^sub>m f ++ g) = (f++g = g++f)" 
24331  515 
by(fastsimp simp: map_add_def map_le_def expand_fun_eq split: option.splits) 
15304  516 

15303  517 
lemma map_add_le_mapE: "f++g \<subseteq>\<^sub>m h \<Longrightarrow> g \<subseteq>\<^sub>m h" 
24331  518 
by (fastsimp simp add: map_le_def map_add_def dom_def) 
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520 
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" 

24331  521 
by (clarsimp simp add: map_le_def map_add_def dom_def split: option.splits) 
15303  522 

3981  523 
end 