src/HOL/Map.thy
author haftmann
Fri Jun 19 07:53:35 2015 +0200 (2015-06-19)
changeset 60517 f16e4fb20652
parent 60057 86fa63ce8156
child 60758 d8d85a8172b5
permissions -rw-r--r--
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
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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' (written ~=>); strongly resembles maps in VDM.
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*)
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section {* Maps *}
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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 => '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 ~=> '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\<^sub>1 \<subseteq>\<^sub>m m\<^sub>2) = (\<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] => 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 := CONST Some 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 :: "('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)"    == "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 {* @{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 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|->b)) x = Some y) = (x = a \<and> b = y \<or> x \<noteq> a \<and> m x = Some y)"
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by auto
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lemma image_map_upd [simp]: "x \<notin> A \<Longrightarrow> m(x \<mapsto> y) ` A = m ` A"
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by auto
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lemma finite_range_updI: "finite (range f) ==> finite (range (f(a|->b)))"
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unfolding image_def
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apply (simp (no_asm_use) add:full_SetCompr_eq)
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apply (rule finite_subset)
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 prefer 2 apply assumption
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apply (auto)
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done
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subsection {* @{term [source] map_of} *}
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lemma map_of_eq_None_iff:
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  "(map_of xys x = None) = (x \<notin> fst ` (set xys))"
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by (induct xys) simp_all
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lemma map_of_is_SomeD: "map_of xys x = Some y \<Longrightarrow> (x,y) \<in> set xys"
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apply (induct xys)
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 apply simp
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apply (clarsimp split: if_splits)
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done
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lemma map_of_eq_Some_iff [simp]:
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  "distinct(map fst xys) \<Longrightarrow> (map_of xys x = Some y) = ((x,y) \<in> set xys)"
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apply (induct xys)
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 apply simp
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apply (auto simp: map_of_eq_None_iff [symmetric])
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done
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lemma Some_eq_map_of_iff [simp]:
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  "distinct(map fst xys) \<Longrightarrow> (Some y = map_of xys x) = ((x,y) \<in> set xys)"
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by (auto simp del:map_of_eq_Some_iff simp add: map_of_eq_Some_iff [symmetric])
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lemma map_of_is_SomeI [simp]: "\<lbrakk> distinct(map fst xys); (x,y) \<in> set xys \<rbrakk>
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    \<Longrightarrow> map_of xys x = Some y"
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apply (induct xys)
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 apply simp
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apply force
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done
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lemma map_of_zip_is_None [simp]:
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  "length xs = length ys \<Longrightarrow> (map_of (zip xs ys) x = None) = (x \<notin> set xs)"
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by (induct rule: list_induct2) simp_all
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lemma 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] using dist map_of 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 `map_of (zip (x#xs) (y#ys)) = map_of (zip (x#xs) (z#zs))`
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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 `distinct (x # xs)` 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) (simp, atomize (full), auto)
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lemma map_of_mapk_SomeI:
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  "inj f ==> map_of t k = Some x ==>
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   map_of (map (split (%k. Pair (f k))) t) (f k) = Some x"
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by (induct t) (auto simp add: inj_eq)
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lemma weak_map_of_SomeI: "(k, x) : set l ==> \<exists>x. map_of l k = Some x"
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by (induct l) auto
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lemma map_of_filter_in:
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  "map_of xs k = Some z \<Longrightarrow> P k z \<Longrightarrow> map_of (filter (split P) xs) k = Some z"
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by (induct xs) auto
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lemma map_of_map:
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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 add: 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 {* @{const map_option} related *}
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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|->b) = (map_option 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 split: option.splits)
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lemma map_comp_simps [simp]:
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  "m2 k = None \<Longrightarrow> (m1 \<circ>\<^sub>m m2) k = None"
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  "m2 k = Some k' \<Longrightarrow> (m1 \<circ>\<^sub>m m2) k = m1 k'"
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by (auto simp add: map_comp_def)
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lemma map_comp_Some_iff:
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  "((m1 \<circ>\<^sub>m m2) k = Some v) = (\<exists>k'. m2 k = Some k' \<and> m1 k' = Some v)"
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by (auto simp add: map_comp_def split: option.splits)
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lemma map_comp_None_iff:
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  "((m1 \<circ>\<^sub>m m2) k = None) = (m2 k = None \<or> (\<exists>k'. m2 k = Some k' \<and> m1 k' = None)) "
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by (auto simp add: map_comp_def split: option.splits)
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subsection {* @{text "++"} *}
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lemma map_add_empty[simp]: "m ++ empty = m"
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by(simp add: map_add_def)
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lemma empty_map_add[simp]: "empty ++ m = m"
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by (rule ext) (simp add: map_add_def split: option.split)
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lemma map_add_assoc[simp]: "m1 ++ (m2 ++ m3) = (m1 ++ m2) ++ m3"
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by (rule ext) (simp add: map_add_def split: option.split)
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lemma map_add_Some_iff:
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  "((m ++ n) k = Some x) = (n k = Some x | n k = None & m k = Some x)"
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by (simp add: map_add_def split: option.split)
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lemma map_add_SomeD [dest!]:
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  "(m ++ n) k = Some x \<Longrightarrow> n k = Some x \<or> n k = None \<and> m k = Some x"
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by (rule map_add_Some_iff [THEN iffD1])
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lemma map_add_find_right [simp]: "!!xx. n k = Some xx ==> (m ++ n) k = Some xx"
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by (subst map_add_Some_iff) fast
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lemma map_add_None [iff]: "((m ++ n) k = None) = (n k = None & m k = None)"
nipkow@24331
   311
by (simp add: map_add_def split: option.split)
webertj@13908
   312
nipkow@14025
   313
lemma map_add_upd[simp]: "f ++ g(x|->y) = (f ++ g)(x|->y)"
nipkow@24331
   314
by (rule ext) (simp add: map_add_def)
webertj@13908
   315
nipkow@14186
   316
lemma map_add_upds[simp]: "m1 ++ (m2(xs[\<mapsto>]ys)) = (m1++m2)(xs[\<mapsto>]ys)"
nipkow@24331
   317
by (simp add: map_upds_def)
nipkow@14186
   318
krauss@32236
   319
lemma map_add_upd_left: "m\<notin>dom e2 \<Longrightarrow> e1(m \<mapsto> u1) ++ e2 = (e1 ++ e2)(m \<mapsto> u1)"
krauss@32236
   320
by (rule ext) (auto simp: map_add_def dom_def split: option.split)
krauss@32236
   321
wenzelm@20800
   322
lemma map_of_append[simp]: "map_of (xs @ ys) = map_of ys ++ map_of xs"
nipkow@24331
   323
unfolding map_add_def
nipkow@24331
   324
apply (induct xs)
nipkow@24331
   325
 apply simp
nipkow@24331
   326
apply (rule ext)
nipkow@24331
   327
apply (simp split add: option.split)
nipkow@24331
   328
done
webertj@13908
   329
nipkow@14025
   330
lemma finite_range_map_of_map_add:
wenzelm@20800
   331
  "finite (range f) ==> finite (range (f ++ map_of l))"
nipkow@24331
   332
apply (induct l)
nipkow@24331
   333
 apply (auto simp del: fun_upd_apply)
nipkow@24331
   334
apply (erule finite_range_updI)
nipkow@24331
   335
done
webertj@13908
   336
wenzelm@20800
   337
lemma inj_on_map_add_dom [iff]:
nipkow@24331
   338
  "inj_on (m ++ m') (dom m') = inj_on m' (dom m')"
nipkow@44890
   339
by (fastforce simp: map_add_def dom_def inj_on_def split: option.splits)
wenzelm@20800
   340
haftmann@34979
   341
lemma map_upds_fold_map_upd:
haftmann@35552
   342
  "m(ks[\<mapsto>]vs) = foldl (\<lambda>m (k, v). m(k \<mapsto> v)) m (zip ks vs)"
haftmann@34979
   343
unfolding map_upds_def proof (rule sym, rule zip_obtain_same_length)
haftmann@34979
   344
  fix ks :: "'a list" and vs :: "'b list"
haftmann@34979
   345
  assume "length ks = length vs"
haftmann@35552
   346
  then show "foldl (\<lambda>m (k, v). m(k\<mapsto>v)) m (zip ks vs) = m ++ map_of (rev (zip ks vs))"
haftmann@35552
   347
    by(induct arbitrary: m rule: list_induct2) simp_all
haftmann@34979
   348
qed
haftmann@34979
   349
haftmann@34979
   350
lemma map_add_map_of_foldr:
haftmann@34979
   351
  "m ++ map_of ps = foldr (\<lambda>(k, v) m. m(k \<mapsto> v)) ps m"
nipkow@39302
   352
  by (induct ps) (auto simp add: fun_eq_iff map_add_def)
haftmann@34979
   353
nipkow@15304
   354
wenzelm@17399
   355
subsection {* @{term [source] restrict_map} *}
oheimb@14100
   356
wenzelm@20800
   357
lemma restrict_map_to_empty [simp]: "m|`{} = empty"
nipkow@24331
   358
by (simp add: restrict_map_def)
nipkow@14186
   359
haftmann@31380
   360
lemma restrict_map_insert: "f |` (insert a A) = (f |` A)(a := f a)"
huffman@44921
   361
by (auto simp add: restrict_map_def)
haftmann@31380
   362
wenzelm@20800
   363
lemma restrict_map_empty [simp]: "empty|`D = empty"
nipkow@24331
   364
by (simp add: restrict_map_def)
nipkow@14186
   365
nipkow@15693
   366
lemma restrict_in [simp]: "x \<in> A \<Longrightarrow> (m|`A) x = m x"
nipkow@24331
   367
by (simp add: restrict_map_def)
oheimb@14100
   368
nipkow@15693
   369
lemma restrict_out [simp]: "x \<notin> A \<Longrightarrow> (m|`A) x = None"
nipkow@24331
   370
by (simp add: restrict_map_def)
oheimb@14100
   371
nipkow@15693
   372
lemma ran_restrictD: "y \<in> ran (m|`A) \<Longrightarrow> \<exists>x\<in>A. m x = Some y"
nipkow@24331
   373
by (auto simp: restrict_map_def ran_def split: split_if_asm)
oheimb@14100
   374
nipkow@15693
   375
lemma dom_restrict [simp]: "dom (m|`A) = dom m \<inter> A"
nipkow@24331
   376
by (auto simp: restrict_map_def dom_def split: split_if_asm)
oheimb@14100
   377
nipkow@15693
   378
lemma restrict_upd_same [simp]: "m(x\<mapsto>y)|`(-{x}) = m|`(-{x})"
nipkow@24331
   379
by (rule ext) (auto simp: restrict_map_def)
oheimb@14100
   380
nipkow@15693
   381
lemma restrict_restrict [simp]: "m|`A|`B = m|`(A\<inter>B)"
nipkow@24331
   382
by (rule ext) (auto simp: restrict_map_def)
oheimb@14100
   383
wenzelm@20800
   384
lemma restrict_fun_upd [simp]:
nipkow@24331
   385
  "m(x := y)|`D = (if x \<in> D then (m|`(D-{x}))(x := y) else m|`D)"
nipkow@39302
   386
by (simp add: restrict_map_def fun_eq_iff)
nipkow@14186
   387
wenzelm@20800
   388
lemma fun_upd_None_restrict [simp]:
nipkow@24331
   389
  "(m|`D)(x := None) = (if x:D then m|`(D - {x}) else m|`D)"
nipkow@39302
   390
by (simp add: restrict_map_def fun_eq_iff)
nipkow@14186
   391
wenzelm@20800
   392
lemma fun_upd_restrict: "(m|`D)(x := y) = (m|`(D-{x}))(x := y)"
nipkow@39302
   393
by (simp add: restrict_map_def fun_eq_iff)
nipkow@14186
   394
wenzelm@20800
   395
lemma fun_upd_restrict_conv [simp]:
nipkow@24331
   396
  "x \<in> D \<Longrightarrow> (m|`D)(x := y) = (m|`(D-{x}))(x := y)"
nipkow@39302
   397
by (simp add: restrict_map_def fun_eq_iff)
nipkow@14186
   398
haftmann@35159
   399
lemma map_of_map_restrict:
haftmann@35159
   400
  "map_of (map (\<lambda>k. (k, f k)) ks) = (Some \<circ> f) |` set ks"
nipkow@39302
   401
  by (induct ks) (simp_all add: fun_eq_iff restrict_map_insert)
haftmann@35159
   402
haftmann@35619
   403
lemma restrict_complement_singleton_eq:
haftmann@35619
   404
  "f |` (- {x}) = f(x := None)"
nipkow@39302
   405
  by (simp add: restrict_map_def fun_eq_iff)
haftmann@35619
   406
oheimb@14100
   407
wenzelm@17399
   408
subsection {* @{term [source] map_upds} *}
nipkow@14025
   409
wenzelm@20800
   410
lemma map_upds_Nil1 [simp]: "m([] [|->] bs) = m"
nipkow@24331
   411
by (simp add: map_upds_def)
nipkow@14025
   412
wenzelm@20800
   413
lemma map_upds_Nil2 [simp]: "m(as [|->] []) = m"
nipkow@24331
   414
by (simp add:map_upds_def)
wenzelm@20800
   415
wenzelm@20800
   416
lemma map_upds_Cons [simp]: "m(a#as [|->] b#bs) = (m(a|->b))(as[|->]bs)"
nipkow@24331
   417
by (simp add:map_upds_def)
nipkow@14025
   418
wenzelm@20800
   419
lemma map_upds_append1 [simp]: "\<And>ys m. size xs < size ys \<Longrightarrow>
nipkow@24331
   420
  m(xs@[x] [\<mapsto>] ys) = m(xs [\<mapsto>] ys)(x \<mapsto> ys!size xs)"
nipkow@24331
   421
apply(induct xs)
nipkow@24331
   422
 apply (clarsimp simp add: neq_Nil_conv)
nipkow@24331
   423
apply (case_tac ys)
nipkow@24331
   424
 apply simp
nipkow@24331
   425
apply simp
nipkow@24331
   426
done
nipkow@14187
   427
wenzelm@20800
   428
lemma map_upds_list_update2_drop [simp]:
bulwahn@46588
   429
  "size xs \<le> i \<Longrightarrow> m(xs[\<mapsto>]ys[i:=y]) = m(xs[\<mapsto>]ys)"
nipkow@24331
   430
apply (induct xs arbitrary: m ys i)
nipkow@24331
   431
 apply simp
nipkow@24331
   432
apply (case_tac ys)
nipkow@24331
   433
 apply simp
nipkow@24331
   434
apply (simp split: nat.split)
nipkow@24331
   435
done
nipkow@14025
   436
wenzelm@20800
   437
lemma map_upd_upds_conv_if:
wenzelm@20800
   438
  "(f(x|->y))(xs [|->] ys) =
wenzelm@20800
   439
   (if x : set(take (length ys) xs) then f(xs [|->] ys)
wenzelm@20800
   440
                                    else (f(xs [|->] ys))(x|->y))"
nipkow@24331
   441
apply (induct xs arbitrary: x y ys f)
nipkow@24331
   442
 apply simp
nipkow@24331
   443
apply (case_tac ys)
nipkow@24331
   444
 apply (auto split: split_if simp: fun_upd_twist)
nipkow@24331
   445
done
nipkow@14025
   446
nipkow@14025
   447
lemma map_upds_twist [simp]:
nipkow@24331
   448
  "a ~: set as ==> m(a|->b)(as[|->]bs) = m(as[|->]bs)(a|->b)"
nipkow@44890
   449
using set_take_subset by (fastforce simp add: map_upd_upds_conv_if)
nipkow@14025
   450
wenzelm@20800
   451
lemma map_upds_apply_nontin [simp]:
nipkow@24331
   452
  "x ~: set xs ==> (f(xs[|->]ys)) x = f x"
nipkow@24331
   453
apply (induct xs arbitrary: ys)
nipkow@24331
   454
 apply simp
nipkow@24331
   455
apply (case_tac ys)
nipkow@24331
   456
 apply (auto simp: map_upd_upds_conv_if)
nipkow@24331
   457
done
nipkow@14025
   458
wenzelm@20800
   459
lemma fun_upds_append_drop [simp]:
nipkow@24331
   460
  "size xs = size ys \<Longrightarrow> m(xs@zs[\<mapsto>]ys) = m(xs[\<mapsto>]ys)"
nipkow@24331
   461
apply (induct xs arbitrary: m ys)
nipkow@24331
   462
 apply simp
nipkow@24331
   463
apply (case_tac ys)
nipkow@24331
   464
 apply simp_all
nipkow@24331
   465
done
nipkow@14300
   466
wenzelm@20800
   467
lemma fun_upds_append2_drop [simp]:
nipkow@24331
   468
  "size xs = size ys \<Longrightarrow> m(xs[\<mapsto>]ys@zs) = m(xs[\<mapsto>]ys)"
nipkow@24331
   469
apply (induct xs arbitrary: m ys)
nipkow@24331
   470
 apply simp
nipkow@24331
   471
apply (case_tac ys)
nipkow@24331
   472
 apply simp_all
nipkow@24331
   473
done
nipkow@14300
   474
nipkow@14300
   475
wenzelm@20800
   476
lemma restrict_map_upds[simp]:
wenzelm@20800
   477
  "\<lbrakk> length xs = length ys; set xs \<subseteq> D \<rbrakk>
wenzelm@20800
   478
    \<Longrightarrow> m(xs [\<mapsto>] ys)|`D = (m|`(D - set xs))(xs [\<mapsto>] ys)"
nipkow@24331
   479
apply (induct xs arbitrary: m ys)
nipkow@24331
   480
 apply simp
nipkow@24331
   481
apply (case_tac ys)
nipkow@24331
   482
 apply simp
nipkow@24331
   483
apply (simp add: Diff_insert [symmetric] insert_absorb)
nipkow@24331
   484
apply (simp add: map_upd_upds_conv_if)
nipkow@24331
   485
done
nipkow@14186
   486
nipkow@14186
   487
wenzelm@17399
   488
subsection {* @{term [source] dom} *}
webertj@13908
   489
nipkow@31080
   490
lemma dom_eq_empty_conv [simp]: "dom f = {} \<longleftrightarrow> f = empty"
huffman@44921
   491
  by (auto simp: dom_def)
nipkow@31080
   492
webertj@13908
   493
lemma domI: "m a = Some b ==> a : dom m"
nipkow@24331
   494
by(simp add:dom_def)
oheimb@14100
   495
(* declare domI [intro]? *)
webertj@13908
   496
paulson@15369
   497
lemma domD: "a : dom m ==> \<exists>b. m a = Some b"
nipkow@24331
   498
by (cases "m a") (auto simp add: dom_def)
webertj@13908
   499
wenzelm@20800
   500
lemma domIff [iff, simp del]: "(a : dom m) = (m a ~= None)"
nipkow@24331
   501
by(simp add:dom_def)
webertj@13908
   502
wenzelm@20800
   503
lemma dom_empty [simp]: "dom empty = {}"
nipkow@24331
   504
by(simp add:dom_def)
webertj@13908
   505
wenzelm@20800
   506
lemma dom_fun_upd [simp]:
nipkow@24331
   507
  "dom(f(x := y)) = (if y=None then dom f - {x} else insert x (dom f))"
nipkow@24331
   508
by(auto simp add:dom_def)
webertj@13908
   509
haftmann@34979
   510
lemma dom_if:
haftmann@34979
   511
  "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
   512
  by (auto split: if_splits)
nipkow@13937
   513
nipkow@15304
   514
lemma dom_map_of_conv_image_fst:
haftmann@34979
   515
  "dom (map_of xys) = fst ` set xys"
haftmann@34979
   516
  by (induct xys) (auto simp add: dom_if)
nipkow@15304
   517
bulwahn@46588
   518
lemma dom_map_of_zip [simp]: "length xs = length ys ==> dom (map_of (zip xs ys)) = set xs"
bulwahn@46588
   519
by (induct rule: list_induct2) (auto simp add: dom_if)
nipkow@15110
   520
webertj@13908
   521
lemma finite_dom_map_of: "finite (dom (map_of l))"
nipkow@24331
   522
by (induct l) (auto simp add: dom_def insert_Collect [symmetric])
webertj@13908
   523
wenzelm@20800
   524
lemma dom_map_upds [simp]:
nipkow@24331
   525
  "dom(m(xs[|->]ys)) = set(take (length ys) xs) Un dom m"
nipkow@24331
   526
apply (induct xs arbitrary: m ys)
nipkow@24331
   527
 apply simp
nipkow@24331
   528
apply (case_tac ys)
nipkow@24331
   529
 apply auto
nipkow@24331
   530
done
nipkow@13910
   531
wenzelm@20800
   532
lemma dom_map_add [simp]: "dom(m++n) = dom n Un dom m"
nipkow@24331
   533
by(auto simp:dom_def)
nipkow@13910
   534
wenzelm@20800
   535
lemma dom_override_on [simp]:
wenzelm@20800
   536
  "dom(override_on f g A) =
wenzelm@20800
   537
    (dom f  - {a. a : A - dom g}) Un {a. a : A Int dom g}"
nipkow@24331
   538
by(auto simp: dom_def override_on_def)
webertj@13908
   539
nipkow@14027
   540
lemma map_add_comm: "dom m1 \<inter> dom m2 = {} \<Longrightarrow> m1++m2 = m2++m1"
nipkow@24331
   541
by (rule ext) (force simp: map_add_def dom_def split: option.split)
wenzelm@20800
   542
krauss@32236
   543
lemma map_add_dom_app_simps:
krauss@32236
   544
  "\<lbrakk> m\<in>dom l2 \<rbrakk> \<Longrightarrow> (l1++l2) m = l2 m"
krauss@32236
   545
  "\<lbrakk> m\<notin>dom l1 \<rbrakk> \<Longrightarrow> (l1++l2) m = l2 m"
krauss@32236
   546
  "\<lbrakk> m\<notin>dom l2 \<rbrakk> \<Longrightarrow> (l1++l2) m = l1 m"
krauss@32236
   547
by (auto simp add: map_add_def split: option.split_asm)
krauss@32236
   548
haftmann@29622
   549
lemma dom_const [simp]:
haftmann@35159
   550
  "dom (\<lambda>x. Some (f x)) = UNIV"
haftmann@29622
   551
  by auto
haftmann@29622
   552
nipkow@22230
   553
(* Due to John Matthews - could be rephrased with dom *)
nipkow@22230
   554
lemma finite_map_freshness:
nipkow@22230
   555
  "finite (dom (f :: 'a \<rightharpoonup> 'b)) \<Longrightarrow> \<not> finite (UNIV :: 'a set) \<Longrightarrow>
nipkow@22230
   556
   \<exists>x. f x = None"
nipkow@22230
   557
by(bestsimp dest:ex_new_if_finite)
nipkow@14027
   558
haftmann@28790
   559
lemma dom_minus:
haftmann@28790
   560
  "f x = None \<Longrightarrow> dom f - insert x A = dom f - A"
haftmann@28790
   561
  unfolding dom_def by simp
haftmann@28790
   562
haftmann@28790
   563
lemma insert_dom:
haftmann@28790
   564
  "f x = Some y \<Longrightarrow> insert x (dom f) = dom f"
haftmann@28790
   565
  unfolding dom_def by auto
haftmann@28790
   566
haftmann@35607
   567
lemma map_of_map_keys:
haftmann@35607
   568
  "set xs = dom m \<Longrightarrow> map_of (map (\<lambda>k. (k, the (m k))) xs) = m"
haftmann@35607
   569
  by (rule ext) (auto simp add: map_of_map_restrict restrict_map_def)
haftmann@35607
   570
haftmann@39379
   571
lemma map_of_eqI:
haftmann@39379
   572
  assumes set_eq: "set (map fst xs) = set (map fst ys)"
haftmann@39379
   573
  assumes map_eq: "\<forall>k\<in>set (map fst xs). map_of xs k = map_of ys k"
haftmann@39379
   574
  shows "map_of xs = map_of ys"
haftmann@39379
   575
proof (rule ext)
haftmann@39379
   576
  fix k show "map_of xs k = map_of ys k"
haftmann@39379
   577
  proof (cases "map_of xs k")
haftmann@39379
   578
    case None then have "k \<notin> set (map fst xs)" by (simp add: map_of_eq_None_iff)
haftmann@39379
   579
    with set_eq have "k \<notin> set (map fst ys)" by simp
haftmann@39379
   580
    then have "map_of ys k = None" by (simp add: map_of_eq_None_iff)
haftmann@39379
   581
    with None show ?thesis by simp
haftmann@39379
   582
  next
haftmann@39379
   583
    case (Some v) then have "k \<in> set (map fst xs)" by (auto simp add: dom_map_of_conv_image_fst [symmetric])
haftmann@39379
   584
    with map_eq show ?thesis by auto
haftmann@39379
   585
  qed
haftmann@39379
   586
qed
haftmann@39379
   587
haftmann@39379
   588
lemma map_of_eq_dom:
haftmann@39379
   589
  assumes "map_of xs = map_of ys"
haftmann@39379
   590
  shows "fst ` set xs = fst ` set ys"
haftmann@39379
   591
proof -
haftmann@39379
   592
  from assms have "dom (map_of xs) = dom (map_of ys)" by simp
haftmann@39379
   593
  then show ?thesis by (simp add: dom_map_of_conv_image_fst)
haftmann@39379
   594
qed
haftmann@39379
   595
nipkow@53820
   596
lemma finite_set_of_finite_maps:
nipkow@53820
   597
assumes "finite A" "finite B"
nipkow@53820
   598
shows  "finite {m. dom m = A \<and> ran m \<subseteq> B}" (is "finite ?S")
nipkow@53820
   599
proof -
nipkow@53820
   600
  let ?S' = "{m. \<forall>x. (x \<in> A \<longrightarrow> m x \<in> Some ` B) \<and> (x \<notin> A \<longrightarrow> m x = None)}"
nipkow@53820
   601
  have "?S = ?S'"
nipkow@53820
   602
  proof
nipkow@53820
   603
    show "?S \<subseteq> ?S'" by(auto simp: dom_def ran_def image_def)
nipkow@53820
   604
    show "?S' \<subseteq> ?S"
nipkow@53820
   605
    proof
nipkow@53820
   606
      fix m assume "m \<in> ?S'"
nipkow@53820
   607
      hence 1: "dom m = A" by force
nipkow@53820
   608
      hence 2: "ran m \<subseteq> B" using `m \<in> ?S'` by(auto simp: dom_def ran_def)
nipkow@53820
   609
      from 1 2 show "m \<in> ?S" by blast
nipkow@53820
   610
    qed
nipkow@53820
   611
  qed
nipkow@53820
   612
  with assms show ?thesis by(simp add: finite_set_of_finite_funs)
nipkow@53820
   613
qed
haftmann@28790
   614
wenzelm@17399
   615
subsection {* @{term [source] ran} *}
oheimb@14100
   616
wenzelm@20800
   617
lemma ranI: "m a = Some b ==> b : ran m"
nipkow@24331
   618
by(auto simp: ran_def)
oheimb@14100
   619
(* declare ranI [intro]? *)
webertj@13908
   620
wenzelm@20800
   621
lemma ran_empty [simp]: "ran empty = {}"
nipkow@24331
   622
by(auto simp: ran_def)
webertj@13908
   623
wenzelm@20800
   624
lemma ran_map_upd [simp]: "m a = None ==> ran(m(a|->b)) = insert b (ran m)"
nipkow@24331
   625
unfolding ran_def
nipkow@24331
   626
apply auto
nipkow@24331
   627
apply (subgoal_tac "aa ~= a")
nipkow@24331
   628
 apply auto
nipkow@24331
   629
done
wenzelm@20800
   630
haftmann@34979
   631
lemma ran_distinct: 
haftmann@34979
   632
  assumes dist: "distinct (map fst al)" 
haftmann@34979
   633
  shows "ran (map_of al) = snd ` set al"
haftmann@34979
   634
using assms proof (induct al)
haftmann@34979
   635
  case Nil then show ?case by simp
haftmann@34979
   636
next
haftmann@34979
   637
  case (Cons kv al)
haftmann@34979
   638
  then have "ran (map_of al) = snd ` set al" by simp
haftmann@34979
   639
  moreover from Cons.prems have "map_of al (fst kv) = None"
haftmann@34979
   640
    by (simp add: map_of_eq_None_iff)
haftmann@34979
   641
  ultimately show ?case by (simp only: map_of.simps ran_map_upd) simp
haftmann@34979
   642
qed
haftmann@34979
   643
Andreas@60057
   644
lemma ran_map_option: "ran (\<lambda>x. map_option f (m x)) = f ` ran m"
Andreas@60057
   645
by(auto simp add: ran_def)
nipkow@13910
   646
oheimb@14100
   647
subsection {* @{text "map_le"} *}
nipkow@13910
   648
kleing@13912
   649
lemma map_le_empty [simp]: "empty \<subseteq>\<^sub>m g"
nipkow@24331
   650
by (simp add: map_le_def)
nipkow@13910
   651
paulson@17724
   652
lemma upd_None_map_le [simp]: "f(x := None) \<subseteq>\<^sub>m f"
nipkow@24331
   653
by (force simp add: map_le_def)
nipkow@14187
   654
nipkow@13910
   655
lemma map_le_upd[simp]: "f \<subseteq>\<^sub>m g ==> f(a := b) \<subseteq>\<^sub>m g(a := b)"
nipkow@44890
   656
by (fastforce simp add: map_le_def)
nipkow@13910
   657
paulson@17724
   658
lemma map_le_imp_upd_le [simp]: "m1 \<subseteq>\<^sub>m m2 \<Longrightarrow> m1(x := None) \<subseteq>\<^sub>m m2(x \<mapsto> y)"
nipkow@24331
   659
by (force simp add: map_le_def)
nipkow@14187
   660
wenzelm@20800
   661
lemma map_le_upds [simp]:
nipkow@24331
   662
  "f \<subseteq>\<^sub>m g ==> f(as [|->] bs) \<subseteq>\<^sub>m g(as [|->] bs)"
nipkow@24331
   663
apply (induct as arbitrary: f g bs)
nipkow@24331
   664
 apply simp
nipkow@24331
   665
apply (case_tac bs)
nipkow@24331
   666
 apply auto
nipkow@24331
   667
done
webertj@13908
   668
webertj@14033
   669
lemma map_le_implies_dom_le: "(f \<subseteq>\<^sub>m g) \<Longrightarrow> (dom f \<subseteq> dom g)"
nipkow@44890
   670
by (fastforce simp add: map_le_def dom_def)
webertj@14033
   671
webertj@14033
   672
lemma map_le_refl [simp]: "f \<subseteq>\<^sub>m f"
nipkow@24331
   673
by (simp add: map_le_def)
webertj@14033
   674
nipkow@14187
   675
lemma map_le_trans[trans]: "\<lbrakk> m1 \<subseteq>\<^sub>m m2; m2 \<subseteq>\<^sub>m m3\<rbrakk> \<Longrightarrow> m1 \<subseteq>\<^sub>m m3"
nipkow@24331
   676
by (auto simp add: map_le_def dom_def)
webertj@14033
   677
webertj@14033
   678
lemma map_le_antisym: "\<lbrakk> f \<subseteq>\<^sub>m g; g \<subseteq>\<^sub>m f \<rbrakk> \<Longrightarrow> f = g"
nipkow@24331
   679
unfolding map_le_def
nipkow@24331
   680
apply (rule ext)
nipkow@24331
   681
apply (case_tac "x \<in> dom f", simp)
nipkow@44890
   682
apply (case_tac "x \<in> dom g", simp, fastforce)
nipkow@24331
   683
done
webertj@14033
   684
webertj@14033
   685
lemma map_le_map_add [simp]: "f \<subseteq>\<^sub>m (g ++ f)"
nipkow@44890
   686
by (fastforce simp add: map_le_def)
webertj@14033
   687
nipkow@15304
   688
lemma map_le_iff_map_add_commute: "(f \<subseteq>\<^sub>m f ++ g) = (f++g = g++f)"
nipkow@44890
   689
by(fastforce simp: map_add_def map_le_def fun_eq_iff split: option.splits)
nipkow@15304
   690
nipkow@15303
   691
lemma map_add_le_mapE: "f++g \<subseteq>\<^sub>m h \<Longrightarrow> g \<subseteq>\<^sub>m h"
nipkow@44890
   692
by (fastforce simp add: map_le_def map_add_def dom_def)
nipkow@15303
   693
bulwahn@46588
   694
lemma map_add_le_mapI: "\<lbrakk> f \<subseteq>\<^sub>m h; g \<subseteq>\<^sub>m h \<rbrakk> \<Longrightarrow> f++g \<subseteq>\<^sub>m h"
nipkow@24331
   695
by (clarsimp simp add: map_le_def map_add_def dom_def split: option.splits)
nipkow@15303
   696
nipkow@31080
   697
lemma dom_eq_singleton_conv: "dom f = {x} \<longleftrightarrow> (\<exists>v. f = [x \<mapsto> v])"
nipkow@31080
   698
proof(rule iffI)
nipkow@31080
   699
  assume "\<exists>v. f = [x \<mapsto> v]"
nipkow@31080
   700
  thus "dom f = {x}" by(auto split: split_if_asm)
nipkow@31080
   701
next
nipkow@31080
   702
  assume "dom f = {x}"
nipkow@31080
   703
  then obtain v where "f x = Some v" by auto
nipkow@31080
   704
  hence "[x \<mapsto> v] \<subseteq>\<^sub>m f" by(auto simp add: map_le_def)
nipkow@31080
   705
  moreover have "f \<subseteq>\<^sub>m [x \<mapsto> v]" using `dom f = {x}` `f x = Some v`
nipkow@31080
   706
    by(auto simp add: map_le_def)
nipkow@31080
   707
  ultimately have "f = [x \<mapsto> v]" by-(rule map_le_antisym)
nipkow@31080
   708
  thus "\<exists>v. f = [x \<mapsto> v]" by blast
nipkow@31080
   709
qed
nipkow@31080
   710
haftmann@35565
   711
haftmann@35565
   712
subsection {* Various *}
haftmann@35565
   713
haftmann@35565
   714
lemma set_map_of_compr:
haftmann@35565
   715
  assumes distinct: "distinct (map fst xs)"
haftmann@35565
   716
  shows "set xs = {(k, v). map_of xs k = Some v}"
haftmann@35565
   717
using assms proof (induct xs)
haftmann@35565
   718
  case Nil then show ?case by simp
haftmann@35565
   719
next
haftmann@35565
   720
  case (Cons x xs)
haftmann@35565
   721
  obtain k v where "x = (k, v)" by (cases x) blast
haftmann@35565
   722
  with Cons.prems have "k \<notin> dom (map_of xs)"
haftmann@35565
   723
    by (simp add: dom_map_of_conv_image_fst)
haftmann@35565
   724
  then have *: "insert (k, v) {(k, v). map_of xs k = Some v} =
haftmann@35565
   725
    {(k', v'). (map_of xs(k \<mapsto> v)) k' = Some v'}"
haftmann@35565
   726
    by (auto split: if_splits)
haftmann@35565
   727
  from Cons have "set xs = {(k, v). map_of xs k = Some v}" by simp
haftmann@35565
   728
  with * `x = (k, v)` show ?case by simp
haftmann@35565
   729
qed
haftmann@35565
   730
haftmann@35565
   731
lemma map_of_inject_set:
haftmann@35565
   732
  assumes distinct: "distinct (map fst xs)" "distinct (map fst ys)"
haftmann@35565
   733
  shows "map_of xs = map_of ys \<longleftrightarrow> set xs = set ys" (is "?lhs \<longleftrightarrow> ?rhs")
haftmann@35565
   734
proof
haftmann@35565
   735
  assume ?lhs
haftmann@35565
   736
  moreover from `distinct (map fst xs)` have "set xs = {(k, v). map_of xs k = Some v}"
haftmann@35565
   737
    by (rule set_map_of_compr)
haftmann@35565
   738
  moreover from `distinct (map fst ys)` have "set ys = {(k, v). map_of ys k = Some v}"
haftmann@35565
   739
    by (rule set_map_of_compr)
haftmann@35565
   740
  ultimately show ?rhs by simp
haftmann@35565
   741
next
wenzelm@53374
   742
  assume ?rhs show ?lhs
wenzelm@53374
   743
  proof
haftmann@35565
   744
    fix k
haftmann@35565
   745
    show "map_of xs k = map_of ys k" proof (cases "map_of xs k")
haftmann@35565
   746
      case None
wenzelm@53374
   747
      with `?rhs` have "map_of ys k = None"
haftmann@35565
   748
        by (simp add: map_of_eq_None_iff)
wenzelm@53374
   749
      with None show ?thesis by simp
haftmann@35565
   750
    next
haftmann@35565
   751
      case (Some v)
wenzelm@53374
   752
      with distinct `?rhs` have "map_of ys k = Some v"
haftmann@35565
   753
        by simp
wenzelm@53374
   754
      with Some show ?thesis by simp
haftmann@35565
   755
    qed
haftmann@35565
   756
  qed
haftmann@35565
   757
qed
haftmann@35565
   758
nipkow@3981
   759
end