src/HOL/Map.thy
author haftmann
Thu Aug 27 21:19:48 2015 +0200 (2015-08-27)
changeset 61032 b57df8eecad6
parent 60841 144523e0678e
child 61069 aefe89038dd2
permissions -rw-r--r--
standardized some occurences of ancient "split" alias
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(*  Title:      HOL/Map.thy
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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"; strongly resembles maps in VDM.
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*)
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section \<open>Maps\<close>
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theory Map
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imports List
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begin
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type_synonym ('a, 'b) "map" = "'a \<Rightarrow> 'b option" (infixr "~=>" 0)
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type_notation (xsymbols)
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  "map" (infixr "\<rightharpoonup>" 0)
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abbreviation
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  empty :: "'a \<rightharpoonup> 'b" where
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  "empty \<equiv> \<lambda>x. None"
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definition
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  map_comp :: "('b \<rightharpoonup> 'c) \<Rightarrow> ('a \<rightharpoonup> 'b) \<Rightarrow> ('a \<rightharpoonup> '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 \<rightharpoonup> 'b) \<Rightarrow> ('a \<rightharpoonup> 'b) \<Rightarrow> ('a \<rightharpoonup> 'b)"  (infixl "++" 100) where
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  "m1 ++ m2 = (\<lambda>x. case m2 x of None \<Rightarrow> m1 x | Some y \<Rightarrow> Some y)"
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definition
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  restrict_map :: "('a \<rightharpoonup> 'b) \<Rightarrow> 'a set \<Rightarrow> ('a \<rightharpoonup> 'b)"  (infixl "|`"  110) where
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  "m|`A = (\<lambda>x. if x \<in> 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 \<rightharpoonup> 'b) \<Rightarrow> 'a set" where
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  "dom m = {a. m a \<noteq> None}"
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definition
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  ran :: "('a \<rightharpoonup> 'b) \<Rightarrow> 'b set" where
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  "ran m = {b. \<exists>a. m a = Some b}"
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definition
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  map_le :: "('a \<rightharpoonup> 'b) \<Rightarrow> ('a \<rightharpoonup> 'b) \<Rightarrow> bool"  (infix "\<subseteq>\<^sub>m" 50) where
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  "(m\<^sub>1 \<subseteq>\<^sub>m m\<^sub>2) \<longleftrightarrow> (\<forall>a \<in> dom m\<^sub>1. m\<^sub>1 a = m\<^sub>2 a)"
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nonterminal maplets and maplet
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syntax
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  "_maplet"  :: "['a, 'a] \<Rightarrow> maplet"             ("_ /|->/ _")
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  "_maplets" :: "['a, 'a] \<Rightarrow> maplet"             ("_ /[|->]/ _")
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  ""         :: "maplet \<Rightarrow> maplets"             ("_")
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  "_Maplets" :: "[maplet, maplets] \<Rightarrow> maplets" ("_,/ _")
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  "_MapUpd"  :: "['a \<rightharpoonup> 'b, maplets] \<Rightarrow> 'a \<rightharpoonup> 'b" ("_/'(_')" [900,0]900)
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  "_Map"     :: "maplets \<Rightarrow> 'a \<rightharpoonup> 'b"            ("(1[_])")
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syntax (xsymbols)
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  "_maplet"  :: "['a, 'a] \<Rightarrow> maplet"             ("_ /\<mapsto>/ _")
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  "_maplets" :: "['a, 'a] \<Rightarrow> maplet"             ("_ /[\<mapsto>]/ _")
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translations
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  "_MapUpd m (_Maplets xy ms)"  \<rightleftharpoons> "_MapUpd (_MapUpd m xy) ms"
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  "_MapUpd m (_maplet  x y)"    \<rightleftharpoons> "m(x := CONST Some y)"
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  "_Map ms"                     \<rightleftharpoons> "_MapUpd (CONST empty) ms"
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  "_Map (_Maplets ms1 ms2)"     \<leftharpoondown> "_MapUpd (_Map ms1) ms2"
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  "_Maplets ms1 (_Maplets ms2 ms3)" \<leftharpoondown> "_Maplets (_Maplets ms1 ms2) ms3"
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primrec
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  map_of :: "('a \<times> 'b) list \<Rightarrow> 'a \<rightharpoonup> 'b" where
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    "map_of [] = empty"
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  | "map_of (p # ps) = (map_of ps)(fst p \<mapsto> snd p)"
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definition
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  map_upds :: "('a \<rightharpoonup> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b list \<Rightarrow> 'a \<rightharpoonup> 'b" where
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  "map_upds m xs ys = m ++ map_of (rev (zip xs ys))"
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translations
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  "_MapUpd m (_maplets x y)" \<rightleftharpoons> "CONST map_upds m x y"
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lemma map_of_Cons_code [code]:
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  "map_of [] k = None"
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  "map_of ((l, v) # ps) k = (if l = k then Some v else map_of ps k)"
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  by simp_all
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subsection \<open>@{term [source] empty}\<close>
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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 \<open>@{term [source] map_upd}\<close>
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lemma map_upd_triv: "t k = Some x \<Longrightarrow> t(k\<mapsto>x) = t"
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  by (rule ext) simp
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lemma map_upd_nonempty [simp]: "t(k\<mapsto>x) \<noteq> 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 assms 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\<mapsto>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) \<Longrightarrow> finite (range (f(a\<mapsto>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 \<open>@{term [source] map_of}\<close>
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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_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: 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 map_of_zip_is_Some:
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  assumes "length xs = length ys"
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  shows "x \<in> set xs \<longleftrightarrow> (\<exists>y. map_of (zip xs ys) x = Some y)"
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using assms by (induct rule: list_induct2) simp_all
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lemma map_of_zip_upd:
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  fixes x :: 'a and xs :: "'a list" and ys zs :: "'b list"
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  assumes "length ys = length xs"
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    and "length zs = length xs"
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    and "x \<notin> set xs"
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    and "map_of (zip xs ys)(x \<mapsto> y) = map_of (zip xs zs)(x \<mapsto> z)"
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  shows "map_of (zip xs ys) = map_of (zip xs zs)"
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proof
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  fix x' :: 'a
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  show "map_of (zip xs ys) x' = map_of (zip xs zs) x'"
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  proof (cases "x = x'")
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    case True
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    from assms True map_of_zip_is_None [of xs ys x']
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      have "map_of (zip xs ys) x' = None" by simp
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    moreover from assms True map_of_zip_is_None [of xs zs x']
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      have "map_of (zip xs zs) x' = None" by simp
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    ultimately show ?thesis by simp
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  next
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    case False from assms
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      have "(map_of (zip xs ys)(x \<mapsto> y)) x' = (map_of (zip xs zs)(x \<mapsto> z)) x'" by auto
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    with False show ?thesis by simp
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  qed
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qed
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lemma map_of_zip_inject:
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  assumes "length ys = length xs"
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    and "length zs = length xs"
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    and dist: "distinct xs"
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    and map_of: "map_of (zip xs ys) = map_of (zip xs zs)"
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  shows "ys = zs"
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  using assms(1) assms(2)[symmetric]
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  using dist map_of
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proof (induct ys xs zs rule: list_induct3)
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  case Nil show ?case by simp
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next
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  case (Cons y ys x xs z zs)
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  from \<open>map_of (zip (x#xs) (y#ys)) = map_of (zip (x#xs) (z#zs))\<close>
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    have map_of: "map_of (zip xs ys)(x \<mapsto> y) = map_of (zip xs zs)(x \<mapsto> z)" by simp
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  from Cons have "length ys = length xs" and "length zs = length xs"
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    and "x \<notin> set xs" by simp_all
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  then have "map_of (zip xs ys) = map_of (zip xs zs)" using map_of by (rule map_of_zip_upd)
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  with Cons.hyps \<open>distinct (x # xs)\<close> have "ys = zs" by simp
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  moreover from map_of have "y = z" by (rule map_upd_eqD1)
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  ultimately show ?case by simp
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qed
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lemma map_of_zip_map:
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  "map_of (zip xs (map f xs)) = (\<lambda>x. if x \<in> set xs then Some (f x) else None)"
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  by (induct xs) (simp_all add: fun_eq_iff)
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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) (auto split: if_splits)
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lemma map_of_mapk_SomeI:
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  "inj f \<Longrightarrow> map_of t k = Some x \<Longrightarrow>
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   map_of (map (case_prod (\<lambda>k. Pair (f k))) t) (f k) = Some x"
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by (induct t) (auto simp: inj_eq)
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lemma weak_map_of_SomeI: "(k, x) \<in> set l \<Longrightarrow> \<exists>x. map_of l k = Some x"
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by (induct l) auto
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lemma map_of_filter_in:
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  "map_of xs k = Some z \<Longrightarrow> P k z \<Longrightarrow> map_of (filter (case_prod P) xs) k = Some z"
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by (induct xs) auto
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lemma map_of_map:
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  "map_of (map (\<lambda>(k, v). (k, f v)) xs) = map_option f \<circ> map_of xs"
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  by (induct xs) (auto simp: fun_eq_iff)
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lemma dom_map_option:
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  "dom (\<lambda>k. map_option (f k) (m k)) = dom m"
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  by (simp add: dom_def)
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lemma dom_map_option_comp [simp]:
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  "dom (map_option g \<circ> m) = dom m"
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  using dom_map_option [of "\<lambda>_. g" m] by (simp add: comp_def)
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subsection \<open>@{const map_option} related\<close>
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lemma map_option_o_empty [simp]: "map_option f o empty = empty"
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by (rule ext) simp
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lemma map_option_o_map_upd [simp]:
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  "map_option f o m(a\<mapsto>b) = (map_option f o m)(a\<mapsto>f b)"
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by (rule ext) simp
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subsection \<open>@{term [source] map_comp} related\<close>
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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: map_comp_def 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: 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: 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: map_comp_def split: option.splits)
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subsection \<open>@{text "++"}\<close>
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lemma map_add_empty[simp]: "m ++ empty = m"
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by(simp add: map_add_def)
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lemma empty_map_add[simp]: "empty ++ m = m"
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by (rule ext) (simp add: map_add_def split: option.split)
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lemma map_add_assoc[simp]: "m1 ++ (m2 ++ m3) = (m1 ++ m2) ++ m3"
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by (rule ext) (simp add: map_add_def split: option.split)
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lemma map_add_Some_iff:
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  "((m ++ n) k = Some x) = (n k = Some x | n k = None & m k = Some x)"
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by (simp add: map_add_def split: option.split)
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lemma map_add_SomeD [dest!]:
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  "(m ++ n) k = Some x \<Longrightarrow> n k = Some x \<or> n k = None \<and> m k = Some x"
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by (rule map_add_Some_iff [THEN iffD1])
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wenzelm@60839
   303
lemma map_add_find_right [simp]: "n k = Some xx \<Longrightarrow> (m ++ n) k = Some xx"
nipkow@24331
   304
by (subst map_add_Some_iff) fast
webertj@13908
   305
nipkow@14025
   306
lemma map_add_None [iff]: "((m ++ n) k = None) = (n k = None & m k = None)"
nipkow@24331
   307
by (simp add: map_add_def split: option.split)
webertj@13908
   308
wenzelm@60838
   309
lemma map_add_upd[simp]: "f ++ g(x\<mapsto>y) = (f ++ g)(x\<mapsto>y)"
nipkow@24331
   310
by (rule ext) (simp add: map_add_def)
webertj@13908
   311
nipkow@14186
   312
lemma map_add_upds[simp]: "m1 ++ (m2(xs[\<mapsto>]ys)) = (m1++m2)(xs[\<mapsto>]ys)"
nipkow@24331
   313
by (simp add: map_upds_def)
nipkow@14186
   314
krauss@32236
   315
lemma map_add_upd_left: "m\<notin>dom e2 \<Longrightarrow> e1(m \<mapsto> u1) ++ e2 = (e1 ++ e2)(m \<mapsto> u1)"
krauss@32236
   316
by (rule ext) (auto simp: map_add_def dom_def split: option.split)
krauss@32236
   317
wenzelm@20800
   318
lemma map_of_append[simp]: "map_of (xs @ ys) = map_of ys ++ map_of xs"
nipkow@24331
   319
unfolding map_add_def
nipkow@24331
   320
apply (induct xs)
nipkow@24331
   321
 apply simp
nipkow@24331
   322
apply (rule ext)
nipkow@24331
   323
apply (simp split add: option.split)
nipkow@24331
   324
done
webertj@13908
   325
nipkow@14025
   326
lemma finite_range_map_of_map_add:
wenzelm@60839
   327
  "finite (range f) \<Longrightarrow> finite (range (f ++ map_of l))"
nipkow@24331
   328
apply (induct l)
nipkow@24331
   329
 apply (auto simp del: fun_upd_apply)
nipkow@24331
   330
apply (erule finite_range_updI)
nipkow@24331
   331
done
webertj@13908
   332
wenzelm@20800
   333
lemma inj_on_map_add_dom [iff]:
nipkow@24331
   334
  "inj_on (m ++ m') (dom m') = inj_on m' (dom m')"
nipkow@44890
   335
by (fastforce simp: map_add_def dom_def inj_on_def split: option.splits)
wenzelm@20800
   336
haftmann@34979
   337
lemma map_upds_fold_map_upd:
haftmann@35552
   338
  "m(ks[\<mapsto>]vs) = foldl (\<lambda>m (k, v). m(k \<mapsto> v)) m (zip ks vs)"
haftmann@34979
   339
unfolding map_upds_def proof (rule sym, rule zip_obtain_same_length)
haftmann@34979
   340
  fix ks :: "'a list" and vs :: "'b list"
haftmann@34979
   341
  assume "length ks = length vs"
haftmann@35552
   342
  then show "foldl (\<lambda>m (k, v). m(k\<mapsto>v)) m (zip ks vs) = m ++ map_of (rev (zip ks vs))"
haftmann@35552
   343
    by(induct arbitrary: m rule: list_induct2) simp_all
haftmann@34979
   344
qed
haftmann@34979
   345
haftmann@34979
   346
lemma map_add_map_of_foldr:
haftmann@34979
   347
  "m ++ map_of ps = foldr (\<lambda>(k, v) m. m(k \<mapsto> v)) ps m"
wenzelm@60839
   348
  by (induct ps) (auto simp: fun_eq_iff map_add_def)
haftmann@34979
   349
nipkow@15304
   350
wenzelm@60758
   351
subsection \<open>@{term [source] restrict_map}\<close>
oheimb@14100
   352
wenzelm@20800
   353
lemma restrict_map_to_empty [simp]: "m|`{} = empty"
nipkow@24331
   354
by (simp add: restrict_map_def)
nipkow@14186
   355
haftmann@31380
   356
lemma restrict_map_insert: "f |` (insert a A) = (f |` A)(a := f a)"
wenzelm@60839
   357
by (auto simp: restrict_map_def)
haftmann@31380
   358
wenzelm@20800
   359
lemma restrict_map_empty [simp]: "empty|`D = empty"
nipkow@24331
   360
by (simp add: restrict_map_def)
nipkow@14186
   361
nipkow@15693
   362
lemma restrict_in [simp]: "x \<in> A \<Longrightarrow> (m|`A) x = m x"
nipkow@24331
   363
by (simp add: restrict_map_def)
oheimb@14100
   364
nipkow@15693
   365
lemma restrict_out [simp]: "x \<notin> A \<Longrightarrow> (m|`A) x = None"
nipkow@24331
   366
by (simp add: restrict_map_def)
oheimb@14100
   367
nipkow@15693
   368
lemma ran_restrictD: "y \<in> ran (m|`A) \<Longrightarrow> \<exists>x\<in>A. m x = Some y"
nipkow@24331
   369
by (auto simp: restrict_map_def ran_def split: split_if_asm)
oheimb@14100
   370
nipkow@15693
   371
lemma dom_restrict [simp]: "dom (m|`A) = dom m \<inter> A"
nipkow@24331
   372
by (auto simp: restrict_map_def dom_def split: split_if_asm)
oheimb@14100
   373
nipkow@15693
   374
lemma restrict_upd_same [simp]: "m(x\<mapsto>y)|`(-{x}) = m|`(-{x})"
nipkow@24331
   375
by (rule ext) (auto simp: restrict_map_def)
oheimb@14100
   376
nipkow@15693
   377
lemma restrict_restrict [simp]: "m|`A|`B = m|`(A\<inter>B)"
nipkow@24331
   378
by (rule ext) (auto simp: restrict_map_def)
oheimb@14100
   379
wenzelm@20800
   380
lemma restrict_fun_upd [simp]:
nipkow@24331
   381
  "m(x := y)|`D = (if x \<in> D then (m|`(D-{x}))(x := y) else m|`D)"
nipkow@39302
   382
by (simp add: restrict_map_def fun_eq_iff)
nipkow@14186
   383
wenzelm@20800
   384
lemma fun_upd_None_restrict [simp]:
wenzelm@60839
   385
  "(m|`D)(x := None) = (if x \<in> D then m|`(D - {x}) else m|`D)"
nipkow@39302
   386
by (simp add: restrict_map_def fun_eq_iff)
nipkow@14186
   387
wenzelm@20800
   388
lemma fun_upd_restrict: "(m|`D)(x := y) = (m|`(D-{x}))(x := y)"
nipkow@39302
   389
by (simp add: restrict_map_def fun_eq_iff)
nipkow@14186
   390
wenzelm@20800
   391
lemma fun_upd_restrict_conv [simp]:
nipkow@24331
   392
  "x \<in> D \<Longrightarrow> (m|`D)(x := y) = (m|`(D-{x}))(x := y)"
nipkow@39302
   393
by (simp add: restrict_map_def fun_eq_iff)
nipkow@14186
   394
haftmann@35159
   395
lemma map_of_map_restrict:
haftmann@35159
   396
  "map_of (map (\<lambda>k. (k, f k)) ks) = (Some \<circ> f) |` set ks"
nipkow@39302
   397
  by (induct ks) (simp_all add: fun_eq_iff restrict_map_insert)
haftmann@35159
   398
haftmann@35619
   399
lemma restrict_complement_singleton_eq:
haftmann@35619
   400
  "f |` (- {x}) = f(x := None)"
nipkow@39302
   401
  by (simp add: restrict_map_def fun_eq_iff)
haftmann@35619
   402
oheimb@14100
   403
wenzelm@60758
   404
subsection \<open>@{term [source] map_upds}\<close>
nipkow@14025
   405
wenzelm@60838
   406
lemma map_upds_Nil1 [simp]: "m([] [\<mapsto>] bs) = m"
nipkow@24331
   407
by (simp add: map_upds_def)
nipkow@14025
   408
wenzelm@60838
   409
lemma map_upds_Nil2 [simp]: "m(as [\<mapsto>] []) = m"
nipkow@24331
   410
by (simp add:map_upds_def)
wenzelm@20800
   411
wenzelm@60838
   412
lemma map_upds_Cons [simp]: "m(a#as [\<mapsto>] b#bs) = (m(a\<mapsto>b))(as[\<mapsto>]bs)"
nipkow@24331
   413
by (simp add:map_upds_def)
nipkow@14025
   414
wenzelm@60839
   415
lemma map_upds_append1 [simp]: "size xs < size ys \<Longrightarrow>
nipkow@24331
   416
  m(xs@[x] [\<mapsto>] ys) = m(xs [\<mapsto>] ys)(x \<mapsto> ys!size xs)"
wenzelm@60839
   417
apply(induct xs arbitrary: ys m)
nipkow@24331
   418
 apply (clarsimp simp add: neq_Nil_conv)
nipkow@24331
   419
apply (case_tac ys)
nipkow@24331
   420
 apply simp
nipkow@24331
   421
apply simp
nipkow@24331
   422
done
nipkow@14187
   423
wenzelm@20800
   424
lemma map_upds_list_update2_drop [simp]:
bulwahn@46588
   425
  "size xs \<le> i \<Longrightarrow> m(xs[\<mapsto>]ys[i:=y]) = m(xs[\<mapsto>]ys)"
nipkow@24331
   426
apply (induct xs arbitrary: m ys i)
nipkow@24331
   427
 apply simp
nipkow@24331
   428
apply (case_tac ys)
nipkow@24331
   429
 apply simp
nipkow@24331
   430
apply (simp split: nat.split)
nipkow@24331
   431
done
nipkow@14025
   432
wenzelm@20800
   433
lemma map_upd_upds_conv_if:
wenzelm@60838
   434
  "(f(x\<mapsto>y))(xs [\<mapsto>] ys) =
wenzelm@60839
   435
   (if x \<in> set(take (length ys) xs) then f(xs [\<mapsto>] ys)
wenzelm@60838
   436
                                    else (f(xs [\<mapsto>] ys))(x\<mapsto>y))"
nipkow@24331
   437
apply (induct xs arbitrary: x y ys f)
nipkow@24331
   438
 apply simp
nipkow@24331
   439
apply (case_tac ys)
nipkow@24331
   440
 apply (auto split: split_if simp: fun_upd_twist)
nipkow@24331
   441
done
nipkow@14025
   442
nipkow@14025
   443
lemma map_upds_twist [simp]:
wenzelm@60839
   444
  "a \<notin> set as \<Longrightarrow> m(a\<mapsto>b)(as[\<mapsto>]bs) = m(as[\<mapsto>]bs)(a\<mapsto>b)"
nipkow@44890
   445
using set_take_subset by (fastforce simp add: map_upd_upds_conv_if)
nipkow@14025
   446
wenzelm@20800
   447
lemma map_upds_apply_nontin [simp]:
wenzelm@60839
   448
  "x \<notin> set xs \<Longrightarrow> (f(xs[\<mapsto>]ys)) x = f x"
nipkow@24331
   449
apply (induct xs arbitrary: ys)
nipkow@24331
   450
 apply simp
nipkow@24331
   451
apply (case_tac ys)
nipkow@24331
   452
 apply (auto simp: map_upd_upds_conv_if)
nipkow@24331
   453
done
nipkow@14025
   454
wenzelm@20800
   455
lemma fun_upds_append_drop [simp]:
nipkow@24331
   456
  "size xs = size ys \<Longrightarrow> m(xs@zs[\<mapsto>]ys) = m(xs[\<mapsto>]ys)"
nipkow@24331
   457
apply (induct xs arbitrary: m ys)
nipkow@24331
   458
 apply simp
nipkow@24331
   459
apply (case_tac ys)
nipkow@24331
   460
 apply simp_all
nipkow@24331
   461
done
nipkow@14300
   462
wenzelm@20800
   463
lemma fun_upds_append2_drop [simp]:
nipkow@24331
   464
  "size xs = size ys \<Longrightarrow> m(xs[\<mapsto>]ys@zs) = m(xs[\<mapsto>]ys)"
nipkow@24331
   465
apply (induct xs arbitrary: m ys)
nipkow@24331
   466
 apply simp
nipkow@24331
   467
apply (case_tac ys)
nipkow@24331
   468
 apply simp_all
nipkow@24331
   469
done
nipkow@14300
   470
nipkow@14300
   471
wenzelm@20800
   472
lemma restrict_map_upds[simp]:
wenzelm@20800
   473
  "\<lbrakk> length xs = length ys; set xs \<subseteq> D \<rbrakk>
wenzelm@20800
   474
    \<Longrightarrow> m(xs [\<mapsto>] ys)|`D = (m|`(D - set xs))(xs [\<mapsto>] ys)"
nipkow@24331
   475
apply (induct xs arbitrary: m ys)
nipkow@24331
   476
 apply simp
nipkow@24331
   477
apply (case_tac ys)
nipkow@24331
   478
 apply simp
nipkow@24331
   479
apply (simp add: Diff_insert [symmetric] insert_absorb)
nipkow@24331
   480
apply (simp add: map_upd_upds_conv_if)
nipkow@24331
   481
done
nipkow@14186
   482
nipkow@14186
   483
wenzelm@60758
   484
subsection \<open>@{term [source] dom}\<close>
webertj@13908
   485
nipkow@31080
   486
lemma dom_eq_empty_conv [simp]: "dom f = {} \<longleftrightarrow> f = empty"
huffman@44921
   487
  by (auto simp: dom_def)
nipkow@31080
   488
wenzelm@60839
   489
lemma domI: "m a = Some b \<Longrightarrow> a \<in> dom m"
wenzelm@60839
   490
  by (simp add: dom_def)
oheimb@14100
   491
(* declare domI [intro]? *)
webertj@13908
   492
wenzelm@60839
   493
lemma domD: "a \<in> dom m \<Longrightarrow> \<exists>b. m a = Some b"
wenzelm@60839
   494
  by (cases "m a") (auto simp add: dom_def)
webertj@13908
   495
wenzelm@60839
   496
lemma domIff [iff, simp del]: "a \<in> dom m \<longleftrightarrow> m a \<noteq> None"
wenzelm@60839
   497
  by (simp add: dom_def)
webertj@13908
   498
wenzelm@20800
   499
lemma dom_empty [simp]: "dom empty = {}"
wenzelm@60839
   500
  by (simp add: dom_def)
webertj@13908
   501
wenzelm@20800
   502
lemma dom_fun_upd [simp]:
wenzelm@60839
   503
  "dom(f(x := y)) = (if y = None then dom f - {x} else insert x (dom f))"
wenzelm@60839
   504
  by (auto simp: dom_def)
webertj@13908
   505
haftmann@34979
   506
lemma dom_if:
haftmann@34979
   507
  "dom (\<lambda>x. if P x then f x else g x) = dom f \<inter> {x. P x} \<union> dom g \<inter> {x. \<not> P x}"
haftmann@34979
   508
  by (auto split: if_splits)
nipkow@13937
   509
nipkow@15304
   510
lemma dom_map_of_conv_image_fst:
haftmann@34979
   511
  "dom (map_of xys) = fst ` set xys"
haftmann@34979
   512
  by (induct xys) (auto simp add: dom_if)
nipkow@15304
   513
wenzelm@60839
   514
lemma dom_map_of_zip [simp]: "length xs = length ys \<Longrightarrow> dom (map_of (zip xs ys)) = set xs"
wenzelm@60839
   515
  by (induct rule: list_induct2) (auto simp: dom_if)
nipkow@15110
   516
webertj@13908
   517
lemma finite_dom_map_of: "finite (dom (map_of l))"
wenzelm@60839
   518
  by (induct l) (auto simp: dom_def insert_Collect [symmetric])
webertj@13908
   519
wenzelm@20800
   520
lemma dom_map_upds [simp]:
wenzelm@60839
   521
  "dom(m(xs[\<mapsto>]ys)) = set(take (length ys) xs) \<union> dom m"
nipkow@24331
   522
apply (induct xs arbitrary: m ys)
nipkow@24331
   523
 apply simp
nipkow@24331
   524
apply (case_tac ys)
nipkow@24331
   525
 apply auto
nipkow@24331
   526
done
nipkow@13910
   527
wenzelm@60839
   528
lemma dom_map_add [simp]: "dom (m ++ n) = dom n \<union> dom m"
wenzelm@60839
   529
  by (auto simp: dom_def)
nipkow@13910
   530
wenzelm@20800
   531
lemma dom_override_on [simp]:
wenzelm@60839
   532
  "dom (override_on f g A) =
wenzelm@60839
   533
    (dom f  - {a. a \<in> A - dom g}) \<union> {a. a \<in> A \<inter> dom g}"
wenzelm@60839
   534
  by (auto simp: dom_def override_on_def)
webertj@13908
   535
wenzelm@60839
   536
lemma map_add_comm: "dom m1 \<inter> dom m2 = {} \<Longrightarrow> m1 ++ m2 = m2 ++ m1"
wenzelm@60839
   537
  by (rule ext) (force simp: map_add_def dom_def split: option.split)
wenzelm@20800
   538
krauss@32236
   539
lemma map_add_dom_app_simps:
wenzelm@60839
   540
  "m \<in> dom l2 \<Longrightarrow> (l1 ++ l2) m = l2 m"
wenzelm@60839
   541
  "m \<notin> dom l1 \<Longrightarrow> (l1 ++ l2) m = l2 m"
wenzelm@60839
   542
  "m \<notin> dom l2 \<Longrightarrow> (l1 ++ l2) m = l1 m"
wenzelm@60839
   543
  by (auto simp add: map_add_def split: option.split_asm)
krauss@32236
   544
haftmann@29622
   545
lemma dom_const [simp]:
haftmann@35159
   546
  "dom (\<lambda>x. Some (f x)) = UNIV"
haftmann@29622
   547
  by auto
haftmann@29622
   548
nipkow@22230
   549
(* Due to John Matthews - could be rephrased with dom *)
nipkow@22230
   550
lemma finite_map_freshness:
nipkow@22230
   551
  "finite (dom (f :: 'a \<rightharpoonup> 'b)) \<Longrightarrow> \<not> finite (UNIV :: 'a set) \<Longrightarrow>
nipkow@22230
   552
   \<exists>x. f x = None"
wenzelm@60839
   553
  by (bestsimp dest: ex_new_if_finite)
nipkow@14027
   554
haftmann@28790
   555
lemma dom_minus:
haftmann@28790
   556
  "f x = None \<Longrightarrow> dom f - insert x A = dom f - A"
haftmann@28790
   557
  unfolding dom_def by simp
haftmann@28790
   558
haftmann@28790
   559
lemma insert_dom:
haftmann@28790
   560
  "f x = Some y \<Longrightarrow> insert x (dom f) = dom f"
haftmann@28790
   561
  unfolding dom_def by auto
haftmann@28790
   562
haftmann@35607
   563
lemma map_of_map_keys:
haftmann@35607
   564
  "set xs = dom m \<Longrightarrow> map_of (map (\<lambda>k. (k, the (m k))) xs) = m"
haftmann@35607
   565
  by (rule ext) (auto simp add: map_of_map_restrict restrict_map_def)
haftmann@35607
   566
haftmann@39379
   567
lemma map_of_eqI:
haftmann@39379
   568
  assumes set_eq: "set (map fst xs) = set (map fst ys)"
haftmann@39379
   569
  assumes map_eq: "\<forall>k\<in>set (map fst xs). map_of xs k = map_of ys k"
haftmann@39379
   570
  shows "map_of xs = map_of ys"
haftmann@39379
   571
proof (rule ext)
haftmann@39379
   572
  fix k show "map_of xs k = map_of ys k"
haftmann@39379
   573
  proof (cases "map_of xs k")
wenzelm@60839
   574
    case None
wenzelm@60839
   575
    then have "k \<notin> set (map fst xs)" by (simp add: map_of_eq_None_iff)
haftmann@39379
   576
    with set_eq have "k \<notin> set (map fst ys)" by simp
haftmann@39379
   577
    then have "map_of ys k = None" by (simp add: map_of_eq_None_iff)
haftmann@39379
   578
    with None show ?thesis by simp
haftmann@39379
   579
  next
wenzelm@60839
   580
    case (Some v)
wenzelm@60839
   581
    then have "k \<in> set (map fst xs)" by (auto simp add: dom_map_of_conv_image_fst [symmetric])
haftmann@39379
   582
    with map_eq show ?thesis by auto
haftmann@39379
   583
  qed
haftmann@39379
   584
qed
haftmann@39379
   585
haftmann@39379
   586
lemma map_of_eq_dom:
haftmann@39379
   587
  assumes "map_of xs = map_of ys"
haftmann@39379
   588
  shows "fst ` set xs = fst ` set ys"
haftmann@39379
   589
proof -
haftmann@39379
   590
  from assms have "dom (map_of xs) = dom (map_of ys)" by simp
haftmann@39379
   591
  then show ?thesis by (simp add: dom_map_of_conv_image_fst)
haftmann@39379
   592
qed
haftmann@39379
   593
nipkow@53820
   594
lemma finite_set_of_finite_maps:
wenzelm@60839
   595
  assumes "finite A" "finite B"
wenzelm@60839
   596
  shows "finite {m. dom m = A \<and> ran m \<subseteq> B}" (is "finite ?S")
nipkow@53820
   597
proof -
nipkow@53820
   598
  let ?S' = "{m. \<forall>x. (x \<in> A \<longrightarrow> m x \<in> Some ` B) \<and> (x \<notin> A \<longrightarrow> m x = None)}"
nipkow@53820
   599
  have "?S = ?S'"
nipkow@53820
   600
  proof
wenzelm@60839
   601
    show "?S \<subseteq> ?S'" by (auto simp: dom_def ran_def image_def)
nipkow@53820
   602
    show "?S' \<subseteq> ?S"
nipkow@53820
   603
    proof
nipkow@53820
   604
      fix m assume "m \<in> ?S'"
nipkow@53820
   605
      hence 1: "dom m = A" by force
wenzelm@60839
   606
      hence 2: "ran m \<subseteq> B" using \<open>m \<in> ?S'\<close> by (auto simp: dom_def ran_def)
nipkow@53820
   607
      from 1 2 show "m \<in> ?S" by blast
nipkow@53820
   608
    qed
nipkow@53820
   609
  qed
nipkow@53820
   610
  with assms show ?thesis by(simp add: finite_set_of_finite_funs)
nipkow@53820
   611
qed
haftmann@28790
   612
wenzelm@60839
   613
wenzelm@60758
   614
subsection \<open>@{term [source] ran}\<close>
oheimb@14100
   615
wenzelm@60839
   616
lemma ranI: "m a = Some b \<Longrightarrow> b \<in> ran m"
wenzelm@60839
   617
  by (auto simp: ran_def)
oheimb@14100
   618
(* declare ranI [intro]? *)
webertj@13908
   619
wenzelm@20800
   620
lemma ran_empty [simp]: "ran empty = {}"
wenzelm@60839
   621
  by (auto simp: ran_def)
webertj@13908
   622
wenzelm@60839
   623
lemma ran_map_upd [simp]: "m a = None \<Longrightarrow> ran(m(a\<mapsto>b)) = insert b (ran m)"
wenzelm@60839
   624
  unfolding ran_def
nipkow@24331
   625
apply auto
wenzelm@60839
   626
apply (subgoal_tac "aa \<noteq> a")
nipkow@24331
   627
 apply auto
nipkow@24331
   628
done
wenzelm@20800
   629
wenzelm@60839
   630
lemma ran_distinct:
wenzelm@60839
   631
  assumes dist: "distinct (map fst al)"
haftmann@34979
   632
  shows "ran (map_of al) = snd ` set al"
wenzelm@60839
   633
  using assms
wenzelm@60839
   634
proof (induct al)
wenzelm@60839
   635
  case Nil
wenzelm@60839
   636
  then show ?case by simp
haftmann@34979
   637
next
haftmann@34979
   638
  case (Cons kv al)
haftmann@34979
   639
  then have "ran (map_of al) = snd ` set al" by simp
haftmann@34979
   640
  moreover from Cons.prems have "map_of al (fst kv) = None"
haftmann@34979
   641
    by (simp add: map_of_eq_None_iff)
haftmann@34979
   642
  ultimately show ?case by (simp only: map_of.simps ran_map_upd) simp
haftmann@34979
   643
qed
haftmann@34979
   644
Andreas@60057
   645
lemma ran_map_option: "ran (\<lambda>x. map_option f (m x)) = f ` ran m"
wenzelm@60839
   646
  by (auto simp add: ran_def)
wenzelm@60839
   647
nipkow@13910
   648
wenzelm@60758
   649
subsection \<open>@{text "map_le"}\<close>
nipkow@13910
   650
kleing@13912
   651
lemma map_le_empty [simp]: "empty \<subseteq>\<^sub>m g"
wenzelm@60839
   652
  by (simp add: map_le_def)
nipkow@13910
   653
paulson@17724
   654
lemma upd_None_map_le [simp]: "f(x := None) \<subseteq>\<^sub>m f"
wenzelm@60839
   655
  by (force simp add: map_le_def)
nipkow@14187
   656
nipkow@13910
   657
lemma map_le_upd[simp]: "f \<subseteq>\<^sub>m g ==> f(a := b) \<subseteq>\<^sub>m g(a := b)"
wenzelm@60839
   658
  by (fastforce simp add: map_le_def)
nipkow@13910
   659
paulson@17724
   660
lemma map_le_imp_upd_le [simp]: "m1 \<subseteq>\<^sub>m m2 \<Longrightarrow> m1(x := None) \<subseteq>\<^sub>m m2(x \<mapsto> y)"
wenzelm@60839
   661
  by (force simp add: map_le_def)
nipkow@14187
   662
wenzelm@20800
   663
lemma map_le_upds [simp]:
wenzelm@60839
   664
  "f \<subseteq>\<^sub>m g \<Longrightarrow> f(as [\<mapsto>] bs) \<subseteq>\<^sub>m g(as [\<mapsto>] bs)"
nipkow@24331
   665
apply (induct as arbitrary: f g bs)
nipkow@24331
   666
 apply simp
nipkow@24331
   667
apply (case_tac bs)
nipkow@24331
   668
 apply auto
nipkow@24331
   669
done
webertj@13908
   670
webertj@14033
   671
lemma map_le_implies_dom_le: "(f \<subseteq>\<^sub>m g) \<Longrightarrow> (dom f \<subseteq> dom g)"
wenzelm@60839
   672
  by (fastforce simp add: map_le_def dom_def)
webertj@14033
   673
webertj@14033
   674
lemma map_le_refl [simp]: "f \<subseteq>\<^sub>m f"
wenzelm@60839
   675
  by (simp add: map_le_def)
webertj@14033
   676
nipkow@14187
   677
lemma map_le_trans[trans]: "\<lbrakk> m1 \<subseteq>\<^sub>m m2; m2 \<subseteq>\<^sub>m m3\<rbrakk> \<Longrightarrow> m1 \<subseteq>\<^sub>m m3"
wenzelm@60839
   678
  by (auto simp add: map_le_def dom_def)
webertj@14033
   679
webertj@14033
   680
lemma map_le_antisym: "\<lbrakk> f \<subseteq>\<^sub>m g; g \<subseteq>\<^sub>m f \<rbrakk> \<Longrightarrow> f = g"
nipkow@24331
   681
unfolding map_le_def
nipkow@24331
   682
apply (rule ext)
nipkow@24331
   683
apply (case_tac "x \<in> dom f", simp)
nipkow@44890
   684
apply (case_tac "x \<in> dom g", simp, fastforce)
nipkow@24331
   685
done
webertj@14033
   686
wenzelm@60839
   687
lemma map_le_map_add [simp]: "f \<subseteq>\<^sub>m g ++ f"
wenzelm@60839
   688
  by (fastforce simp: map_le_def)
webertj@14033
   689
wenzelm@60839
   690
lemma map_le_iff_map_add_commute: "f \<subseteq>\<^sub>m f ++ g \<longleftrightarrow> f ++ g = g ++ f"
wenzelm@60839
   691
  by (fastforce simp: map_add_def map_le_def fun_eq_iff split: option.splits)
nipkow@15304
   692
wenzelm@60839
   693
lemma map_add_le_mapE: "f ++ g \<subseteq>\<^sub>m h \<Longrightarrow> g \<subseteq>\<^sub>m h"
wenzelm@60839
   694
  by (fastforce simp: map_le_def map_add_def dom_def)
nipkow@15303
   695
wenzelm@60839
   696
lemma map_add_le_mapI: "\<lbrakk> f \<subseteq>\<^sub>m h; g \<subseteq>\<^sub>m h \<rbrakk> \<Longrightarrow> f ++ g \<subseteq>\<^sub>m h"
wenzelm@60839
   697
  by (auto simp: map_le_def map_add_def dom_def split: option.splits)
nipkow@15303
   698
nipkow@31080
   699
lemma dom_eq_singleton_conv: "dom f = {x} \<longleftrightarrow> (\<exists>v. f = [x \<mapsto> v])"
nipkow@31080
   700
proof(rule iffI)
nipkow@31080
   701
  assume "\<exists>v. f = [x \<mapsto> v]"
nipkow@31080
   702
  thus "dom f = {x}" by(auto split: split_if_asm)
nipkow@31080
   703
next
nipkow@31080
   704
  assume "dom f = {x}"
nipkow@31080
   705
  then obtain v where "f x = Some v" by auto
nipkow@31080
   706
  hence "[x \<mapsto> v] \<subseteq>\<^sub>m f" by(auto simp add: map_le_def)
wenzelm@60758
   707
  moreover have "f \<subseteq>\<^sub>m [x \<mapsto> v]" using \<open>dom f = {x}\<close> \<open>f x = Some v\<close>
nipkow@31080
   708
    by(auto simp add: map_le_def)
nipkow@31080
   709
  ultimately have "f = [x \<mapsto> v]" by-(rule map_le_antisym)
nipkow@31080
   710
  thus "\<exists>v. f = [x \<mapsto> v]" by blast
nipkow@31080
   711
qed
nipkow@31080
   712
haftmann@35565
   713
wenzelm@60758
   714
subsection \<open>Various\<close>
haftmann@35565
   715
haftmann@35565
   716
lemma set_map_of_compr:
haftmann@35565
   717
  assumes distinct: "distinct (map fst xs)"
haftmann@35565
   718
  shows "set xs = {(k, v). map_of xs k = Some v}"
wenzelm@60839
   719
  using assms
wenzelm@60839
   720
proof (induct xs)
wenzelm@60839
   721
  case Nil
wenzelm@60839
   722
  then show ?case by simp
haftmann@35565
   723
next
haftmann@35565
   724
  case (Cons x xs)
haftmann@35565
   725
  obtain k v where "x = (k, v)" by (cases x) blast
haftmann@35565
   726
  with Cons.prems have "k \<notin> dom (map_of xs)"
haftmann@35565
   727
    by (simp add: dom_map_of_conv_image_fst)
haftmann@35565
   728
  then have *: "insert (k, v) {(k, v). map_of xs k = Some v} =
haftmann@35565
   729
    {(k', v'). (map_of xs(k \<mapsto> v)) k' = Some v'}"
haftmann@35565
   730
    by (auto split: if_splits)
haftmann@35565
   731
  from Cons have "set xs = {(k, v). map_of xs k = Some v}" by simp
wenzelm@60758
   732
  with * \<open>x = (k, v)\<close> show ?case by simp
haftmann@35565
   733
qed
haftmann@35565
   734
haftmann@35565
   735
lemma map_of_inject_set:
haftmann@35565
   736
  assumes distinct: "distinct (map fst xs)" "distinct (map fst ys)"
haftmann@35565
   737
  shows "map_of xs = map_of ys \<longleftrightarrow> set xs = set ys" (is "?lhs \<longleftrightarrow> ?rhs")
haftmann@35565
   738
proof
haftmann@35565
   739
  assume ?lhs
wenzelm@60758
   740
  moreover from \<open>distinct (map fst xs)\<close> have "set xs = {(k, v). map_of xs k = Some v}"
haftmann@35565
   741
    by (rule set_map_of_compr)
wenzelm@60758
   742
  moreover from \<open>distinct (map fst ys)\<close> have "set ys = {(k, v). map_of ys k = Some v}"
haftmann@35565
   743
    by (rule set_map_of_compr)
haftmann@35565
   744
  ultimately show ?rhs by simp
haftmann@35565
   745
next
wenzelm@53374
   746
  assume ?rhs show ?lhs
wenzelm@53374
   747
  proof
haftmann@35565
   748
    fix k
wenzelm@60839
   749
    show "map_of xs k = map_of ys k"
wenzelm@60839
   750
    proof (cases "map_of xs k")
haftmann@35565
   751
      case None
wenzelm@60758
   752
      with \<open>?rhs\<close> have "map_of ys k = None"
haftmann@35565
   753
        by (simp add: map_of_eq_None_iff)
wenzelm@53374
   754
      with None show ?thesis by simp
haftmann@35565
   755
    next
haftmann@35565
   756
      case (Some v)
wenzelm@60758
   757
      with distinct \<open>?rhs\<close> have "map_of ys k = Some v"
haftmann@35565
   758
        by simp
wenzelm@53374
   759
      with Some show ?thesis by simp
haftmann@35565
   760
    qed
haftmann@35565
   761
  qed
haftmann@35565
   762
qed
haftmann@35565
   763
nipkow@3981
   764
end