src/HOL/Map.thy
author haftmann
Fri Apr 20 11:21:42 2007 +0200 (2007-04-20)
changeset 22744 5cbe966d67a2
parent 22230 bdec4a82f385
child 24331 76f7a8c6e842
permissions -rw-r--r--
Isar definitions are now added explicitly to code theorem table
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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
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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:
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    "map_of xys x = Some y \<Longrightarrow> (x,y) \<in> set xys"
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  apply (induct xys)
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   apply simp
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  apply (clarsimp split: if_splits)
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  done
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lemma map_of_eq_Some_iff [simp]:
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    "distinct(map fst xys) \<Longrightarrow> (map_of xys x = Some y) = ((x,y) \<in> set xys)"
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  apply (induct xys)
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   apply simp
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  apply (auto simp: map_of_eq_None_iff [symmetric])
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  done
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lemma Some_eq_map_of_iff [simp]:
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    "distinct(map fst xys) \<Longrightarrow> (Some y = map_of xys x) = ((x,y) \<in> set xys)"
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  by (auto simp del:map_of_eq_Some_iff simp add: map_of_eq_Some_iff [symmetric])
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lemma map_of_is_SomeI [simp]: "\<lbrakk> distinct(map fst xys); (x,y) \<in> set xys \<rbrakk>
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    \<Longrightarrow> map_of xys x = Some y"
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  apply (induct xys)
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   apply simp
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  apply force
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  done
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lemma map_of_zip_is_None [simp]:
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    "length xs = length ys \<Longrightarrow> (map_of (zip xs ys) x = None) = (x \<notin> set xs)"
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  by (induct rule: list_induct2) simp_all
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lemma finite_range_map_of: "finite (range (map_of xys))"
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  apply (induct xys)
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   apply (simp_all 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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  assumes "inj f"
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  shows "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 f` 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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  assumes 1: "map_of xs k = Some z"
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    and 2: "P k z"
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  shows "map_of (filter (split P) xs) k = Some z"
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  using 1 by (induct xs) (insert 2, 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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  unfolding map_add_def by simp
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lemma empty_map_add[simp]: "empty ++ m = m"
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  unfolding map_add_def by (rule ext) (simp split: option.split)
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lemma map_add_assoc[simp]: "m1 ++ (m2 ++ m3) = (m1 ++ m2) ++ m3"
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  unfolding map_add_def 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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  unfolding map_add_def by (simp 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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  unfolding map_add_def by (simp split: option.split)
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lemma map_add_upd[simp]: "f ++ g(x|->y) = (f ++ g)(x|->y)"
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  unfolding map_add_def by (rule ext) simp
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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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  unfolding map_add_def dom_def inj_on_def
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  by (fastsimp 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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lemma fun_upds_append2_drop [simp]:
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    "size xs = size ys \<Longrightarrow> m(xs[\<mapsto>]ys@zs) = 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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   390
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lemma restrict_map_upds[simp]:
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  "\<lbrakk> length xs = length ys; set xs \<subseteq> D \<rbrakk>
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    \<Longrightarrow> m(xs [\<mapsto>] ys)|`D = (m|`(D - set xs))(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
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  apply (simp add: Diff_insert [symmetric] insert_absorb)
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   399
  apply (simp add: map_upd_upds_conv_if)
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   400
  done
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   401
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   402
wenzelm@17399
   403
subsection {* @{term [source] dom} *}
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   404
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   405
lemma domI: "m a = Some b ==> a : dom m"
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  unfolding dom_def by simp
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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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   411
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   412
lemma domIff [iff, simp del]: "(a : dom m) = (m a ~= None)"
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   413
  unfolding dom_def by simp
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   414
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   415
lemma dom_empty [simp]: "dom empty = {}"
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   416
  unfolding dom_def by simp
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   417
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   418
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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   420
  unfolding dom_def by auto
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   421
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   422
lemma dom_map_of: "dom(map_of xys) = {x. \<exists>y. (x,y) : set xys}"
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   423
  by (induct xys) (auto simp del: fun_upd_apply)
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   424
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   425
lemma dom_map_of_conv_image_fst:
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   426
    "dom(map_of xys) = fst ` (set xys)"
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   427
  unfolding dom_map_of by force
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   428
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   429
lemma dom_map_of_zip [simp]: "[| length xs = length ys; distinct xs |] ==>
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   430
    dom(map_of(zip xs ys)) = set xs"
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   431
  by (induct rule: list_induct2) simp_all
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   432
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   433
lemma finite_dom_map_of: "finite (dom (map_of l))"
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   434
  unfolding dom_def
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   435
  by (induct l) (auto simp add: insert_Collect [symmetric])
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   436
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   437
lemma dom_map_upds [simp]:
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   438
    "dom(m(xs[|->]ys)) = set(take (length ys) xs) Un dom m"
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   439
  apply (induct xs arbitrary: m ys)
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   440
   apply simp
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   441
  apply (case_tac ys)
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   442
   apply auto
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   443
  done
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   444
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   445
lemma dom_map_add [simp]: "dom(m++n) = dom n Un dom m"
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   446
  unfolding dom_def by auto
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   447
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   448
lemma dom_override_on [simp]:
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   449
  "dom(override_on f g A) =
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   450
    (dom f  - {a. a : A - dom g}) Un {a. a : A Int dom g}"
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   451
  unfolding dom_def override_on_def by auto
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   452
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   453
lemma map_add_comm: "dom m1 \<inter> dom m2 = {} \<Longrightarrow> m1++m2 = m2++m1"
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   454
  by (rule ext) (force simp: map_add_def dom_def split: option.split)
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   455
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   456
(* Due to John Matthews - could be rephrased with dom *)
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   457
lemma finite_map_freshness:
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   458
  "finite (dom (f :: 'a \<rightharpoonup> 'b)) \<Longrightarrow> \<not> finite (UNIV :: 'a set) \<Longrightarrow>
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   459
   \<exists>x. f x = None"
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   460
by(bestsimp dest:ex_new_if_finite)
nipkow@14027
   461
wenzelm@17399
   462
subsection {* @{term [source] ran} *}
oheimb@14100
   463
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   464
lemma ranI: "m a = Some b ==> b : ran m"
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   465
  unfolding ran_def by auto
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(* declare ranI [intro]? *)
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   467
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   468
lemma ran_empty [simp]: "ran empty = {}"
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   469
  unfolding ran_def by simp
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   470
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   471
lemma ran_map_upd [simp]: "m a = None ==> ran(m(a|->b)) = insert b (ran m)"
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   472
  unfolding ran_def
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   473
  apply auto
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   474
  apply (subgoal_tac "aa ~= a")
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   475
   apply auto
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   476
  done
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   477
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   478
oheimb@14100
   479
subsection {* @{text "map_le"} *}
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   480
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   481
lemma map_le_empty [simp]: "empty \<subseteq>\<^sub>m g"
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   482
  by (simp add: map_le_def)
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   483
paulson@17724
   484
lemma upd_None_map_le [simp]: "f(x := None) \<subseteq>\<^sub>m f"
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   485
  by (force simp add: map_le_def)
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   486
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   487
lemma map_le_upd[simp]: "f \<subseteq>\<^sub>m g ==> f(a := b) \<subseteq>\<^sub>m g(a := b)"
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   488
  by (fastsimp simp add: map_le_def)
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   489
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   490
lemma map_le_imp_upd_le [simp]: "m1 \<subseteq>\<^sub>m m2 \<Longrightarrow> m1(x := None) \<subseteq>\<^sub>m m2(x \<mapsto> y)"
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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_upds [simp]:
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   494
    "f \<subseteq>\<^sub>m g ==> f(as [|->] bs) \<subseteq>\<^sub>m g(as [|->] bs)"
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   495
  apply (induct as arbitrary: f g bs)
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   496
   apply simp
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   497
  apply (case_tac bs)
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   498
   apply auto
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   499
  done
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   500
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   501
lemma map_le_implies_dom_le: "(f \<subseteq>\<^sub>m g) \<Longrightarrow> (dom f \<subseteq> dom g)"
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   502
  by (fastsimp simp add: map_le_def dom_def)
webertj@14033
   503
webertj@14033
   504
lemma map_le_refl [simp]: "f \<subseteq>\<^sub>m f"
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   505
  by (simp add: map_le_def)
webertj@14033
   506
nipkow@14187
   507
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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   508
  by (auto simp add: map_le_def dom_def)
webertj@14033
   509
webertj@14033
   510
lemma map_le_antisym: "\<lbrakk> f \<subseteq>\<^sub>m g; g \<subseteq>\<^sub>m f \<rbrakk> \<Longrightarrow> f = g"
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   511
  unfolding map_le_def
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   512
  apply (rule ext)
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   513
  apply (case_tac "x \<in> dom f", simp)
paulson@14208
   514
  apply (case_tac "x \<in> dom g", simp, fastsimp)
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   515
  done
webertj@14033
   516
webertj@14033
   517
lemma map_le_map_add [simp]: "f \<subseteq>\<^sub>m (g ++ f)"
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   518
  by (fastsimp simp add: map_le_def)
webertj@14033
   519
nipkow@15304
   520
lemma map_le_iff_map_add_commute: "(f \<subseteq>\<^sub>m f ++ g) = (f++g = g++f)"
wenzelm@20800
   521
  by (fastsimp simp add: map_add_def map_le_def expand_fun_eq split: option.splits)
nipkow@15304
   522
nipkow@15303
   523
lemma map_add_le_mapE: "f++g \<subseteq>\<^sub>m h \<Longrightarrow> g \<subseteq>\<^sub>m h"
wenzelm@20800
   524
  by (fastsimp simp add: map_le_def map_add_def dom_def)
nipkow@15303
   525
nipkow@15303
   526
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"
wenzelm@20800
   527
  by (clarsimp simp add: map_le_def map_add_def dom_def split: option.splits)
nipkow@15303
   528
nipkow@3981
   529
end