author  nipkow 
Mon, 14 Apr 2003 18:52:13 +0200  
changeset 13910  f9a9ef16466f 
parent 13909  a5247a49c85e 
child 13912  3c0a340be514 
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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theory Map = List: 
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types ('a,'b) "~=>" = "'a => 'b option" (infixr 0) 
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consts 

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chg_map :: "('b => 'b) => 'a => ('a ~=> 'b) => ('a ~=> 'b)" 
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override:: "('a ~=> 'b) => ('a ~=> 'b) => ('a ~=> 'b)" (infixl "++" 100) 
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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)" ("_/'(_[>]_/')" [900,0,0]900) 
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map_le :: "('a ~=> 'b) => ('a ~=> 'b) => bool" (infix "\<subseteq>\<^sub>m" 50) 

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syntax 
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empty :: "'a ~=> 'b" 
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map_upd :: "('a ~=> 'b) => 'a => 'b => ('a ~=> 'b)" 
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("_/'(_/>_')" [900,0,0]900) 
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syntax (xsymbols) 
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"~=>" :: "[type, type] => type" (infixr "\<leadsto>" 0) 
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map_upd :: "('a ~=> 'b) => 'a => 'b => ('a ~=> 'b)" 
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("_/'(_/\<mapsto>/_')" [900,0,0]900) 
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map_upds :: "('a ~=> 'b) => 'a list => 'b list => ('a ~=> 'b)" 
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("_/'(_/[\<mapsto>]/_')" [900,0,0]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(a>b)" == "m(a:=Some b)" 

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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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override_def: "m1++m2 == %x. case m2 x of None => m1 x  Some y => Some y" 
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dom_def: "dom(m) == {a. m a ~= None}" 
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ran_def: "ran(m) == {b. ? 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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primrec "t([] [>]bs) = t" 

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"t(a#as[>]bs) = t(a>hd bs)(as[>]tl bs)" 

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section {* 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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section {* 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 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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section {* sum\_case and empty/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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section {* map\_upds *} 
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lemma map_upd_upds_conv_if: 
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"!!x y ys f. (f(x>y))(xs [>] ys) = 

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(if x : set xs then f(xs [>] ys) else (f(xs [>] ys))(x>y))" 

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apply(induct xs) 

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apply simp 

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apply(simp split:split_if add:fun_upd_twist eq_sym_conv) 

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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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by (simp add: map_upd_upds_conv_if) 

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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(simp add: fun_upd_apply map_upd_upds_conv_if split:split_if) 

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done 

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section {* 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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section {* 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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section {* 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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section {* ++ *} 
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lemma override_empty[simp]: "m ++ empty = m" 
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apply (unfold override_def) 
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apply (simp (no_asm)) 

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done 

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lemma empty_override[simp]: "empty ++ m = m" 
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apply (unfold override_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 override_Some_iff [rule_format (no_asm)]: 

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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 override_def) 

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

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done 

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lemmas override_SomeD = override_Some_iff [THEN iffD1, standard] 

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declare override_SomeD [dest!] 

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lemma override_find_right[simp]: "!!xx. n k = Some xx ==> (m ++ n) k = Some xx" 
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apply (subst override_Some_iff) 
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apply fast 

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done 

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lemma override_None [iff]: "((m ++ n) k = None) = (n k = None & m k = None)" 
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apply (unfold override_def) 
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apply (simp (no_asm) split add: option.split) 

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done 

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lemma override_upd[simp]: "f ++ g(x>y) = (f ++ g)(x>y)" 
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apply (unfold override_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_of_override[simp]: "map_of ys ++ map_of xs = map_of (xs@ys)" 
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apply (unfold override_def) 
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apply (rule sym) 

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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_override: "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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section {* 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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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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(* 

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lemma dom_map_upd[simp]: "dom(m(a>b)) = insert a (dom m)" 

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apply (unfold dom_def) 
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apply (simp (no_asm)) 

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apply blast 

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done 

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*) 
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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]: "!!m vs. dom(m(xs[>]vs)) = set xs Un dom m" 
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by(induct xs, simp_all) 

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lemma dom_override[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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section {* ran *} 
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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)) 

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done 

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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) 
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apply auto 

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

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apply auto 

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done 

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section{* @{text"\<subseteq>\<^sub>m"} *} 

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lemma [simp]: "empty \<subseteq>\<^sub>m g" 

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by(simp add:map_le_def) 

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

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by(fastsimp simp add:map_le_def) 

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lemma map_le_upds[simp]: 

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"!!f g bs. f \<subseteq>\<^sub>m g ==> f(as [>] bs) \<subseteq>\<^sub>m g(as [>] bs)" 

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by(induct as, auto) 

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end 