src/HOL/Map.thy
author nipkow
Wed Aug 04 19:10:45 2004 +0200 (2004-08-04)
changeset 15110 78b5636eabc7
parent 14739 86c6f272ef79
child 15131 c69542757a4d
permissions -rw-r--r--
Added a number of new thms and the new function remove1
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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   1997-2003 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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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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syntax
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  fun_map_comp :: "('b => 'c)  => ('a ~=> 'b) => ('a ~=> 'c)" (infixl "o'_m" 55)
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translations
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  "f o_m m" == "option_map f o m"
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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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syntax (xsymbols)
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  "~=>"     :: "[type, type] => type"    (infixr "\<rightharpoonup>" 0)
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  fun_map_comp :: "('b => 'c)  => ('a ~=> 'b) => ('a ~=> 'c)" (infixl "\<circ>\<^sub>m" 55)
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  "_maplet"  :: "['a, 'a] => maplet"             ("_ /\<mapsto>/ _")
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  "_maplets" :: "['a, 'a] => maplet"             ("_ /[\<mapsto>]/ _")
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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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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: "m\<^isub>1 \<subseteq>\<^sub>m m\<^isub>2  ==  ALL a : dom m\<^isub>1. m\<^isub>1 a = m\<^isub>2 a"
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primrec
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  "map_of [] = empty"
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  "map_of (p#ps) = (map_of ps)(fst p |-> snd p)"
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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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by (unfold chg_map_def, auto)
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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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by (unfold chg_map_def, auto)
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lemma chg_map_other [simp]: "a \<noteq> b \<Longrightarrow> chg_map f a m b = m b"
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by (auto simp: chg_map_def split add: option.split)
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subsection {* @{term map_of} *}
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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_tac xys)
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apply  (simp_all (no_asm) add: image_constant)
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apply (rule finite_subset)
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prefer 2 apply assumption
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apply auto
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done
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lemma map_of_SomeD [rule_format (no_asm)]: "map_of xs k = Some y --> (k,y):set xs"
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by (induct_tac "xs", auto)
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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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by (induct_tac "l", auto)
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lemma map_of_filter_in: 
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"[| map_of xs k = Some z; P k z |] ==> map_of (filter (split P) xs) k = Some z"
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apply (rule mp)
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prefer 2 apply assumption
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apply (erule thin_rl)
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apply (induct_tac "xs", auto)
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done
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lemma map_of_map: "map_of (map (%(a,b). (a,f b)) xs) x = option_map f (map_of xs x)"
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by (induct_tac "xs", auto)
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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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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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apply (unfold map_add_def)
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apply (simp (no_asm) split add: option.split)
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done
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lemma map_add_upd[simp]: "f ++ g(x|->y) = (f ++ g)(x|->y)"
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apply (unfold map_add_def)
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apply (rule ext, auto)
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done
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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", 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 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, simp, 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, simp)
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apply (case_tac ys, 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)
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                                  else (f(xs [|->] ys))(x|->y))"
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apply (induct xs, 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, 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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  "!!m ys. size xs = size ys \<Longrightarrow> m(xs@zs[\<mapsto>]ys) = 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_all
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done
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   383
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lemma fun_upds_append2_drop[simp]:
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  "!!m ys. size xs = size ys \<Longrightarrow> m(xs[\<mapsto>]ys@zs) = 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_all
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done
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   391
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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>
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 \<Longrightarrow> m(xs [\<mapsto>] ys)\<lfloor>D = (m\<lfloor>(D - set xs))(xs [\<mapsto>] ys)"
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apply (induct xs, simp)
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apply (case_tac ys, 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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   401
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   402
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   403
subsection {* @{term map_upd_s} *}
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   405
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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   412
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   413
subsection {* @{term dom} *}
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   415
lemma domI: "m a = Some b ==> a : dom m"
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by (unfold dom_def, auto)
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(* declare domI [intro]? *)
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   418
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   419
lemma domD: "a : dom m ==> ? b. m a = Some b"
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by (unfold dom_def, auto)
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   421
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lemma domIff[iff]: "(a : dom m) = (m a ~= None)"
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   423
by (unfold dom_def, auto)
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   424
declare domIff [simp del]
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   425
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   426
lemma dom_empty[simp]: "dom empty = {}"
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   427
apply (unfold dom_def)
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   428
apply (simp (no_asm))
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   429
done
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   430
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   431
lemma dom_fun_upd[simp]:
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   432
 "dom(f(x := y)) = (if y=None then dom f - {x} else insert x (dom f))"
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   433
by (simp add:dom_def) blast
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   434
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   435
lemma dom_map_of: "dom(map_of xys) = {x. \<exists>y. (x,y) : set xys}"
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   436
apply(induct xys)
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   437
apply(auto simp del:fun_upd_apply)
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   438
done
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   439
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   440
lemma dom_map_of_zip[simp]: "[| length xs = length ys; distinct xs |] ==>
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   441
  dom(map_of(zip xs ys)) = set xs"
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   442
by(induct rule: list_induct2, simp_all)
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   443
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   444
lemma finite_dom_map_of: "finite (dom (map_of l))"
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   445
apply (unfold dom_def)
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   446
apply (induct_tac "l")
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   447
apply (auto simp add: insert_Collect [symmetric])
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   448
done
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   449
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   450
lemma dom_map_upds[simp]:
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   451
 "!!m ys. dom(m(xs[|->]ys)) = set(take (length ys) xs) Un dom m"
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   452
apply (induct xs, simp)
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   453
apply (case_tac ys, auto)
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   454
done
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   455
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   456
lemma dom_map_add[simp]: "dom(m++n) = dom n Un dom m"
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   457
by (unfold dom_def, auto)
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   458
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   459
lemma dom_overwrite[simp]:
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   460
 "dom(f(g|A)) = (dom f  - {a. a : A - dom g}) Un {a. a : A Int dom g}"
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   461
by(auto simp add: dom_def overwrite_def)
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   462
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   463
lemma map_add_comm: "dom m1 \<inter> dom m2 = {} \<Longrightarrow> m1++m2 = m2++m1"
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   464
apply(rule ext)
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   465
apply(fastsimp simp:map_add_def split:option.split)
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   466
done
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   467
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   468
subsection {* @{term ran} *}
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   469
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   470
lemma ranI: "m a = Some b ==> b : ran m" 
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   471
by (auto simp add: ran_def)
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   472
(* declare ranI [intro]? *)
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   473
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   474
lemma ran_empty[simp]: "ran empty = {}"
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   475
apply (unfold ran_def)
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   476
apply (simp (no_asm))
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   477
done
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   478
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   479
lemma ran_map_upd[simp]: "m a = None ==> ran(m(a|->b)) = insert b (ran m)"
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   480
apply (unfold ran_def, auto)
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   481
apply (subgoal_tac "~ (aa = a) ")
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   482
apply auto
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   483
done
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   484
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   485
subsection {* @{text "map_le"} *}
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   486
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   487
lemma map_le_empty [simp]: "empty \<subseteq>\<^sub>m g"
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   488
by(simp add:map_le_def)
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   489
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   490
lemma [simp]: "f(x := None) \<subseteq>\<^sub>m f"
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   491
by(force simp add:map_le_def)
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   492
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   493
lemma map_le_upd[simp]: "f \<subseteq>\<^sub>m g ==> f(a := b) \<subseteq>\<^sub>m g(a := b)"
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   494
by(fastsimp simp add:map_le_def)
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   495
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   496
lemma [simp]: "m1 \<subseteq>\<^sub>m m2 \<Longrightarrow> m1(x := None) \<subseteq>\<^sub>m m2(x \<mapsto> y)"
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   497
by(force simp add:map_le_def)
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   498
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   499
lemma map_le_upds[simp]:
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   500
 "!!f g bs. f \<subseteq>\<^sub>m g ==> f(as [|->] bs) \<subseteq>\<^sub>m g(as [|->] bs)"
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   501
apply (induct as, simp)
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   502
apply (case_tac bs, auto)
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   503
done
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   504
webertj@14033
   505
lemma map_le_implies_dom_le: "(f \<subseteq>\<^sub>m g) \<Longrightarrow> (dom f \<subseteq> dom g)"
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   506
  by (fastsimp simp add: map_le_def dom_def)
webertj@14033
   507
webertj@14033
   508
lemma map_le_refl [simp]: "f \<subseteq>\<^sub>m f"
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   509
  by (simp add: map_le_def)
webertj@14033
   510
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   511
lemma map_le_trans[trans]: "\<lbrakk> m1 \<subseteq>\<^sub>m m2; m2 \<subseteq>\<^sub>m m3\<rbrakk> \<Longrightarrow> m1 \<subseteq>\<^sub>m m3"
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   512
by(force simp add:map_le_def)
webertj@14033
   513
webertj@14033
   514
lemma map_le_antisym: "\<lbrakk> f \<subseteq>\<^sub>m g; g \<subseteq>\<^sub>m f \<rbrakk> \<Longrightarrow> f = g"
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   515
  apply (unfold map_le_def)
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   516
  apply (rule ext)
paulson@14208
   517
  apply (case_tac "x \<in> dom f", simp)
paulson@14208
   518
  apply (case_tac "x \<in> dom g", simp, fastsimp)
webertj@14033
   519
done
webertj@14033
   520
webertj@14033
   521
lemma map_le_map_add [simp]: "f \<subseteq>\<^sub>m (g ++ f)"
webertj@14033
   522
  by (fastsimp simp add: map_le_def)
webertj@14033
   523
nipkow@3981
   524
end