src/HOL/Map.thy
author haftmann
Sun, 31 Jan 2010 14:51:32 +0100
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permissions -rw-r--r--
more correspondence lemmas between related operations
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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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header {* Maps *}
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theory Map
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imports List
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begin
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types ('a,'b) "~=>" = "'a => 'b option"  (infixr "~=>" 0)
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translations (type) "a ~=> b " <= (type) "a => b option"
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syntax (xsymbols)
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  "~=>" :: "[type, type] => type"  (infixr "\<rightharpoonup>" 0)
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abbreviation
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  empty :: "'a ~=> 'b" where
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  "empty == %x. None"
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definition
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  map_comp :: "('b ~=> 'c) => ('a ~=> 'b) => ('a ~=> 'c)"  (infixl "o'_m" 55) where
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  "f o_m g = (\<lambda>k. case g k of None \<Rightarrow> None | Some v \<Rightarrow> f v)"
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notation (xsymbols)
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  map_comp  (infixl "\<circ>\<^sub>m" 55)
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definition
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  map_add :: "('a ~=> 'b) => ('a ~=> 'b) => ('a ~=> 'b)"  (infixl "++" 100) where
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  "m1 ++ m2 = (\<lambda>x. case m2 x of None => m1 x | Some y => Some y)"
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definition
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  restrict_map :: "('a ~=> 'b) => 'a set => ('a ~=> 'b)"  (infixl "|`"  110) where
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  "m|`A = (\<lambda>x. if x : A then m x else None)"
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notation (latex output)
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  restrict_map  ("_\<restriction>\<^bsub>_\<^esub>" [111,110] 110)
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definition
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  dom :: "('a ~=> 'b) => 'a set" where
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  "dom m = {a. m a ~= None}"
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definition
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  ran :: "('a ~=> 'b) => 'b set" where
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  "ran m = {b. EX a. m a = Some b}"
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definition
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  map_le :: "('a ~=> 'b) => ('a ~=> 'b) => bool"  (infix "\<subseteq>\<^sub>m" 50) where
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  "(m\<^isub>1 \<subseteq>\<^sub>m m\<^isub>2) = (\<forall>a \<in> dom m\<^isub>1. m\<^isub>1 a = m\<^isub>2 a)"
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nonterminals
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  maplets maplet
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syntax
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  "_maplet"  :: "['a, 'a] => maplet"             ("_ /|->/ _")
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  "_maplets" :: "['a, 'a] => maplet"             ("_ /[|->]/ _")
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  ""         :: "maplet => maplets"             ("_")
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  "_Maplets" :: "[maplet, maplets] => maplets" ("_,/ _")
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  "_MapUpd"  :: "['a ~=> 'b, maplets] => 'a ~=> 'b" ("_/'(_')" [900,0]900)
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  "_Map"     :: "maplets => 'a ~=> 'b"            ("(1[_])")
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syntax (xsymbols)
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  "_maplet"  :: "['a, 'a] => maplet"             ("_ /\<mapsto>/ _")
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  "_maplets" :: "['a, 'a] => maplet"             ("_ /[\<mapsto>]/ _")
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translations
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  "_MapUpd m (_Maplets xy ms)"  == "_MapUpd (_MapUpd m xy) ms"
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  "_MapUpd m (_maplet  x y)"    == "m(x:=Some y)"
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  "_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 prems have "(m(a\<mapsto>x)) a = (n(a\<mapsto>y)) a" by simp
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  then show ?thesis by simp
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qed
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lemma map_upd_Some_unfold:
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  "((m(a|->b)) x = Some y) = (x = a \<and> b = y \<or> x \<noteq> a \<and> m x = Some y)"
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by auto
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lemma image_map_upd [simp]: "x \<notin> A \<Longrightarrow> m(x \<mapsto> y) ` A = m ` A"
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by auto
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lemma finite_range_updI: "finite (range f) ==> finite (range (f(a|->b)))"
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unfolding image_def
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apply (simp (no_asm_use) add:full_SetCompr_eq)
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apply (rule finite_subset)
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 prefer 2 apply assumption
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apply (auto)
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done
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subsection {* @{term [source] map_of} *}
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lemma map_of_eq_None_iff:
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  "(map_of xys x = None) = (x \<notin> fst ` (set xys))"
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by (induct xys) simp_all
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lemma map_of_is_SomeD: "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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   157
by (auto simp del:map_of_eq_Some_iff simp add: map_of_eq_Some_iff [symmetric])
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   158
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   159
lemma map_of_is_SomeI [simp]: "\<lbrakk> distinct(map fst xys); (x,y) \<in> set xys \<rbrakk>
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   160
    \<Longrightarrow> map_of xys x = Some y"
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   161
apply (induct xys)
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   162
 apply simp
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   163
apply force
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   164
done
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   165
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   166
lemma map_of_zip_is_None [simp]:
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   167
  "length xs = length ys \<Longrightarrow> (map_of (zip xs ys) x = None) = (x \<notin> set xs)"
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   168
by (induct rule: list_induct2) simp_all
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   169
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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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   173
using assms by (induct rule: list_induct2) simp_all
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   174
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   175
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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   179
    and "x \<notin> set xs"
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   180
    and "map_of (zip xs ys)(x \<mapsto> y) = map_of (zip xs zs)(x \<mapsto> z)"
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   181
  shows "map_of (zip xs ys) = map_of (zip xs zs)"
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   182
proof
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   183
  fix x' :: 'a
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   184
  show "map_of (zip xs ys) x' = map_of (zip xs zs) x'"
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   185
  proof (cases "x = x'")
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   186
    case True
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   187
    from assms True map_of_zip_is_None [of xs ys x']
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   188
      have "map_of (zip xs ys) x' = None" by simp
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   189
    moreover from assms True map_of_zip_is_None [of xs zs x']
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   190
      have "map_of (zip xs zs) x' = None" by simp
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   191
    ultimately show ?thesis by simp
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   192
  next
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   193
    case False from assms
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   194
      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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   195
    with False show ?thesis by simp
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   196
  qed
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   197
qed
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   198
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   199
lemma map_of_zip_inject:
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   200
  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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   203
    and map_of: "map_of (zip xs ys) = map_of (zip xs zs)"
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   204
  shows "ys = zs"
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   205
using assms(1) assms(2)[symmetric] using dist map_of proof (induct ys xs zs rule: list_induct3)
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   206
  case Nil show ?case by simp
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   207
next
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   208
  case (Cons y ys x xs z zs)
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   209
  from `map_of (zip (x#xs) (y#ys)) = map_of (zip (x#xs) (z#zs))`
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   210
    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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   211
  from Cons have "length ys = length xs" and "length zs = length xs"
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   212
    and "x \<notin> set xs" by simp_all
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   213
  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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   214
  with Cons.hyps `distinct (x # xs)` have "ys = zs" by simp
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   215
  moreover from map_of have "y = z" by (rule map_upd_eqD1)
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   216
  ultimately show ?case by simp
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   217
qed
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   218
33635
dcaada178c6f moved lemma map_of_zip_map to Map.thy
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   219
lemma map_of_zip_map:
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   220
  "map_of (zip xs (map f xs)) = (\<lambda>x. if x \<in> set xs then Some (f x) else None)"
dcaada178c6f moved lemma map_of_zip_map to Map.thy
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   221
  by (induct xs) (simp_all add: expand_fun_eq)
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   222
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   223
lemma finite_range_map_of: "finite (range (map_of xys))"
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   224
apply (induct xys)
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   225
 apply (simp_all add: image_constant)
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   226
apply (rule finite_subset)
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   227
 prefer 2 apply assumption
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   228
apply auto
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   229
done
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   230
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   231
lemma map_of_SomeD: "map_of xs k = Some y \<Longrightarrow> (k, y) \<in> set xs"
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   232
by (induct xs) (simp, atomize (full), auto)
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   233
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   234
lemma map_of_mapk_SomeI:
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   235
  "inj f ==> map_of t k = Some x ==>
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   236
   map_of (map (split (%k. Pair (f k))) t) (f k) = Some x"
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   237
by (induct t) (auto simp add: inj_eq)
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   238
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   239
lemma weak_map_of_SomeI: "(k, x) : set l ==> \<exists>x. map_of l k = Some x"
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   240
by (induct l) auto
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   241
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   242
lemma map_of_filter_in:
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   243
  "map_of xs k = Some z \<Longrightarrow> P k z \<Longrightarrow> map_of (filter (split P) xs) k = Some z"
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   244
by (induct xs) auto
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   245
30235
58d147683393 Made Option a separate theory and renamed option_map to Option.map
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   246
lemma map_of_map: "map_of (map (%(a,b). (a,f b)) xs) x = Option.map f (map_of xs x)"
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   247
by (induct xs) auto
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   248
4bdfa9f77254 Map.ML integrated into Map.thy
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   249
30235
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   250
subsection {* @{const Option.map} related *}
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   251
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58d147683393 Made Option a separate theory and renamed option_map to Option.map
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   252
lemma option_map_o_empty [simp]: "Option.map f o empty = empty"
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   253
by (rule ext) simp
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   254
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   255
lemma option_map_o_map_upd [simp]:
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58d147683393 Made Option a separate theory and renamed option_map to Option.map
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   256
  "Option.map f o m(a|->b) = (Option.map f o m)(a|->f b)"
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   257
by (rule ext) simp
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   258
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   259
17399
56a3a4affedc @{term [source] ...} in subsections probably more robust;
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   260
subsection {* @{term [source] map_comp} related *}
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   261
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   262
lemma map_comp_empty [simp]:
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   263
  "m \<circ>\<^sub>m empty = empty"
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   264
  "empty \<circ>\<^sub>m m = empty"
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diff changeset
   265
by (auto simp add: map_comp_def intro: ext split: option.splits)
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diff changeset
   266
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   267
lemma map_comp_simps [simp]:
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   268
  "m2 k = None \<Longrightarrow> (m1 \<circ>\<^sub>m m2) k = None"
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   269
  "m2 k = Some k' \<Longrightarrow> (m1 \<circ>\<^sub>m m2) k = m1 k'"
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diff changeset
   270
by (auto simp add: map_comp_def)
17391
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   271
c6338ed6caf8 removed syntax fun_map_comp;
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   272
lemma map_comp_Some_iff:
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   273
  "((m1 \<circ>\<^sub>m m2) k = Some v) = (\<exists>k'. m2 k = Some k' \<and> m1 k' = Some v)"
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   274
by (auto simp add: map_comp_def split: option.splits)
17391
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diff changeset
   275
c6338ed6caf8 removed syntax fun_map_comp;
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   276
lemma map_comp_None_iff:
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   277
  "((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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   278
by (auto simp add: map_comp_def split: option.splits)
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   279
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   280
14100
804be4c4b642 added map_image, restrict_map, some thms
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   281
subsection {* @{text "++"} *}
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   282
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   283
lemma map_add_empty[simp]: "m ++ empty = m"
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   284
by(simp add: map_add_def)
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   285
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   286
lemma empty_map_add[simp]: "empty ++ m = m"
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   287
by (rule ext) (simp add: map_add_def split: option.split)
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diff changeset
   288
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   289
lemma map_add_assoc[simp]: "m1 ++ (m2 ++ m3) = (m1 ++ m2) ++ m3"
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diff changeset
   290
by (rule ext) (simp add: map_add_def split: option.split)
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diff changeset
   291
69c82605efcf tuned specifications and proofs;
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   292
lemma map_add_Some_iff:
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diff changeset
   293
  "((m ++ n) k = Some x) = (n k = Some x | n k = None & m k = Some x)"
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diff changeset
   294
by (simp add: map_add_def split: option.split)
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diff changeset
   295
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   296
lemma map_add_SomeD [dest!]:
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   297
  "(m ++ n) k = Some x \<Longrightarrow> n k = Some x \<or> n k = None \<and> m k = Some x"
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diff changeset
   298
by (rule map_add_Some_iff [THEN iffD1])
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   299
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   300
lemma map_add_find_right [simp]: "!!xx. n k = Some xx ==> (m ++ n) k = Some xx"
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diff changeset
   301
by (subst map_add_Some_iff) fast
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diff changeset
   302
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   303
lemma map_add_None [iff]: "((m ++ n) k = None) = (n k = None & m k = None)"
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   304
by (simp add: map_add_def split: option.split)
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lemma map_add_upd[simp]: "f ++ g(x|->y) = (f ++ g)(x|->y)"
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by (rule ext) (simp add: map_add_def)
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lemma map_add_upds[simp]: "m1 ++ (m2(xs[\<mapsto>]ys)) = (m1++m2)(xs[\<mapsto>]ys)"
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by (simp add: map_upds_def)
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   311
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lemma map_add_upd_left: "m\<notin>dom e2 \<Longrightarrow> e1(m \<mapsto> u1) ++ e2 = (e1 ++ e2)(m \<mapsto> u1)"
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by (rule ext) (auto simp: map_add_def dom_def split: option.split)
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lemma map_of_append[simp]: "map_of (xs @ ys) = map_of ys ++ map_of xs"
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unfolding map_add_def
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apply (induct xs)
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 apply simp
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apply (rule ext)
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apply (simp split add: option.split)
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done
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lemma finite_range_map_of_map_add:
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  "finite (range f) ==> finite (range (f ++ map_of l))"
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apply (induct l)
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 apply (auto simp del: fun_upd_apply)
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apply (erule finite_range_updI)
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done
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lemma inj_on_map_add_dom [iff]:
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  "inj_on (m ++ m') (dom m') = inj_on m' (dom m')"
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by (fastsimp simp: map_add_def dom_def inj_on_def split: option.splits)
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lemma map_upds_fold_map_upd:
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  "f(ks[\<mapsto>]vs) = foldl (\<lambda>f (k, v). f(k\<mapsto>v)) f (zip ks vs)"
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unfolding map_upds_def proof (rule sym, rule zip_obtain_same_length)
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  fix ks :: "'a list" and vs :: "'b list"
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  assume "length ks = length vs"
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  then show "foldl (\<lambda>f (k, v). f(k\<mapsto>v)) f (zip ks vs) = f ++ map_of (rev (zip ks vs))"
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    by (induct arbitrary: f rule: list_induct2) simp_all
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qed
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lemma map_add_map_of_foldr:
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  "m ++ map_of ps = foldr (\<lambda>(k, v) m. m(k \<mapsto> v)) ps m"
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  by (induct ps) (auto simp add: expand_fun_eq map_add_def)
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   346
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   347
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subsection {* @{term [source] restrict_map} *}
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   349
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lemma restrict_map_to_empty [simp]: "m|`{} = empty"
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by (simp add: restrict_map_def)
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   352
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   353
lemma restrict_map_insert: "f |` (insert a A) = (f |` A)(a := f a)"
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   354
by (auto simp add: restrict_map_def intro: ext)
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   355
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   356
lemma restrict_map_empty [simp]: "empty|`D = empty"
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   357
by (simp add: restrict_map_def)
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   358
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lemma restrict_in [simp]: "x \<in> A \<Longrightarrow> (m|`A) x = m x"
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by (simp add: restrict_map_def)
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   361
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   362
lemma restrict_out [simp]: "x \<notin> A \<Longrightarrow> (m|`A) x = None"
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   363
by (simp add: restrict_map_def)
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   364
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   365
lemma ran_restrictD: "y \<in> ran (m|`A) \<Longrightarrow> \<exists>x\<in>A. m x = Some y"
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   366
by (auto simp: restrict_map_def ran_def split: split_if_asm)
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   367
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   368
lemma dom_restrict [simp]: "dom (m|`A) = dom m \<inter> A"
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   369
by (auto simp: restrict_map_def dom_def split: split_if_asm)
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   370
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   371
lemma restrict_upd_same [simp]: "m(x\<mapsto>y)|`(-{x}) = m|`(-{x})"
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   372
by (rule ext) (auto simp: restrict_map_def)
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   373
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   374
lemma restrict_restrict [simp]: "m|`A|`B = m|`(A\<inter>B)"
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   375
by (rule ext) (auto simp: restrict_map_def)
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   376
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   377
lemma restrict_fun_upd [simp]:
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   378
  "m(x := y)|`D = (if x \<in> D then (m|`(D-{x}))(x := y) else m|`D)"
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   379
by (simp add: restrict_map_def expand_fun_eq)
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   380
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   381
lemma fun_upd_None_restrict [simp]:
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   382
  "(m|`D)(x := None) = (if x:D then m|`(D - {x}) else m|`D)"
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   383
by (simp add: restrict_map_def expand_fun_eq)
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diff changeset
   384
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   385
lemma fun_upd_restrict: "(m|`D)(x := y) = (m|`(D-{x}))(x := y)"
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   386
by (simp add: restrict_map_def expand_fun_eq)
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   387
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   388
lemma fun_upd_restrict_conv [simp]:
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   389
  "x \<in> D \<Longrightarrow> (m|`D)(x := y) = (m|`(D-{x}))(x := y)"
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   390
by (simp add: restrict_map_def expand_fun_eq)
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   391
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diff changeset
   392
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   393
subsection {* @{term [source] map_upds} *}
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   394
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   395
lemma map_upds_Nil1 [simp]: "m([] [|->] bs) = m"
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   396
by (simp add: map_upds_def)
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   397
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   398
lemma map_upds_Nil2 [simp]: "m(as [|->] []) = m"
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   399
by (simp add:map_upds_def)
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   400
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   401
lemma map_upds_Cons [simp]: "m(a#as [|->] b#bs) = (m(a|->b))(as[|->]bs)"
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   402
by (simp add:map_upds_def)
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diff changeset
   403
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   404
lemma map_upds_append1 [simp]: "\<And>ys m. size xs < size ys \<Longrightarrow>
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diff changeset
   405
  m(xs@[x] [\<mapsto>] ys) = m(xs [\<mapsto>] ys)(x \<mapsto> ys!size xs)"
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diff changeset
   406
apply(induct xs)
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diff changeset
   407
 apply (clarsimp simp add: neq_Nil_conv)
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   408
apply (case_tac ys)
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diff changeset
   409
 apply simp
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diff changeset
   410
apply simp
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diff changeset
   411
done
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   412
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   413
lemma map_upds_list_update2_drop [simp]:
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   414
  "\<lbrakk>size xs \<le> i; i < size ys\<rbrakk>
69c82605efcf tuned specifications and proofs;
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   415
    \<Longrightarrow> m(xs[\<mapsto>]ys[i:=y]) = m(xs[\<mapsto>]ys)"
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diff changeset
   416
apply (induct xs arbitrary: m ys i)
76f7a8c6e842 Made UN_Un simp
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diff changeset
   417
 apply simp
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   418
apply (case_tac ys)
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diff changeset
   419
 apply simp
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   420
apply (simp split: nat.split)
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diff changeset
   421
done
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   422
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   423
lemma map_upd_upds_conv_if:
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   424
  "(f(x|->y))(xs [|->] ys) =
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   425
   (if x : set(take (length ys) xs) then f(xs [|->] ys)
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   426
                                    else (f(xs [|->] ys))(x|->y))"
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diff changeset
   427
apply (induct xs arbitrary: x y ys f)
76f7a8c6e842 Made UN_Un simp
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diff changeset
   428
 apply simp
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diff changeset
   429
apply (case_tac ys)
76f7a8c6e842 Made UN_Un simp
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diff changeset
   430
 apply (auto split: split_if simp: fun_upd_twist)
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diff changeset
   431
done
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   432
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diff changeset
   433
lemma map_upds_twist [simp]:
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diff changeset
   434
  "a ~: set as ==> m(a|->b)(as[|->]bs) = m(as[|->]bs)(a|->b)"
76f7a8c6e842 Made UN_Un simp
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diff changeset
   435
using set_take_subset by (fastsimp simp add: map_upd_upds_conv_if)
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diff changeset
   436
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   437
lemma map_upds_apply_nontin [simp]:
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diff changeset
   438
  "x ~: set xs ==> (f(xs[|->]ys)) x = f x"
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   439
apply (induct xs arbitrary: ys)
76f7a8c6e842 Made UN_Un simp
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diff changeset
   440
 apply simp
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diff changeset
   441
apply (case_tac ys)
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diff changeset
   442
 apply (auto simp: map_upd_upds_conv_if)
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diff changeset
   443
done
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   444
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   445
lemma fun_upds_append_drop [simp]:
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diff changeset
   446
  "size xs = size ys \<Longrightarrow> m(xs@zs[\<mapsto>]ys) = m(xs[\<mapsto>]ys)"
76f7a8c6e842 Made UN_Un simp
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diff changeset
   447
apply (induct xs arbitrary: m ys)
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diff changeset
   448
 apply simp
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   449
apply (case_tac ys)
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diff changeset
   450
 apply simp_all
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diff changeset
   451
done
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diff changeset
   452
20800
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diff changeset
   453
lemma fun_upds_append2_drop [simp]:
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diff changeset
   454
  "size xs = size ys \<Longrightarrow> m(xs[\<mapsto>]ys@zs) = m(xs[\<mapsto>]ys)"
76f7a8c6e842 Made UN_Un simp
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diff changeset
   455
apply (induct xs arbitrary: m ys)
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parents: 22744
diff changeset
   456
 apply simp
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   457
apply (case_tac ys)
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   458
 apply simp_all
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   459
done
14300
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14208
diff changeset
   460
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14208
diff changeset
   461
20800
69c82605efcf tuned specifications and proofs;
wenzelm
parents: 19947
diff changeset
   462
lemma restrict_map_upds[simp]:
69c82605efcf tuned specifications and proofs;
wenzelm
parents: 19947
diff changeset
   463
  "\<lbrakk> length xs = length ys; set xs \<subseteq> D \<rbrakk>
69c82605efcf tuned specifications and proofs;
wenzelm
parents: 19947
diff changeset
   464
    \<Longrightarrow> m(xs [\<mapsto>] ys)|`D = (m|`(D - set xs))(xs [\<mapsto>] ys)"
24331
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   465
apply (induct xs arbitrary: m ys)
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   466
 apply simp
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   467
apply (case_tac ys)
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   468
 apply simp
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   469
apply (simp add: Diff_insert [symmetric] insert_absorb)
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   470
apply (simp add: map_upd_upds_conv_if)
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   471
done
14186
6d2a494e33be Added a number of thms about map restriction.
nipkow
parents: 14180
diff changeset
   472
6d2a494e33be Added a number of thms about map restriction.
nipkow
parents: 14180
diff changeset
   473
17399
56a3a4affedc @{term [source] ...} in subsections probably more robust;
wenzelm
parents: 17391
diff changeset
   474
subsection {* @{term [source] dom} *}
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   475
31080
21ffc770ebc0 lemmas by Andreas Lochbihler
nipkow
parents: 30935
diff changeset
   476
lemma dom_eq_empty_conv [simp]: "dom f = {} \<longleftrightarrow> f = empty"
21ffc770ebc0 lemmas by Andreas Lochbihler
nipkow
parents: 30935
diff changeset
   477
by(auto intro!:ext simp: dom_def)
21ffc770ebc0 lemmas by Andreas Lochbihler
nipkow
parents: 30935
diff changeset
   478
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   479
lemma domI: "m a = Some b ==> a : dom m"
24331
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   480
by(simp add:dom_def)
14100
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   481
(* declare domI [intro]? *)
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   482
15369
paulson
parents: 15304
diff changeset
   483
lemma domD: "a : dom m ==> \<exists>b. m a = Some b"
24331
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   484
by (cases "m a") (auto simp add: dom_def)
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   485
20800
69c82605efcf tuned specifications and proofs;
wenzelm
parents: 19947
diff changeset
   486
lemma domIff [iff, simp del]: "(a : dom m) = (m a ~= None)"
24331
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   487
by(simp add:dom_def)
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   488
20800
69c82605efcf tuned specifications and proofs;
wenzelm
parents: 19947
diff changeset
   489
lemma dom_empty [simp]: "dom empty = {}"
24331
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   490
by(simp add:dom_def)
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   491
20800
69c82605efcf tuned specifications and proofs;
wenzelm
parents: 19947
diff changeset
   492
lemma dom_fun_upd [simp]:
24331
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   493
  "dom(f(x := y)) = (if y=None then dom f - {x} else insert x (dom f))"
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   494
by(auto simp add:dom_def)
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   495
34979
8cb6e7a42e9c more correspondence lemmas between related operations
haftmann
parents: 34941
diff changeset
   496
lemma dom_if:
8cb6e7a42e9c more correspondence lemmas between related operations
haftmann
parents: 34941
diff changeset
   497
  "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}"
8cb6e7a42e9c more correspondence lemmas between related operations
haftmann
parents: 34941
diff changeset
   498
  by (auto split: if_splits)
13937
e9d57517c9b1 added a thm
nipkow
parents: 13914
diff changeset
   499
15304
3514ca74ac54 Added more lemmas
nipkow
parents: 15303
diff changeset
   500
lemma dom_map_of_conv_image_fst:
34979
8cb6e7a42e9c more correspondence lemmas between related operations
haftmann
parents: 34941
diff changeset
   501
  "dom (map_of xys) = fst ` set xys"
8cb6e7a42e9c more correspondence lemmas between related operations
haftmann
parents: 34941
diff changeset
   502
  by (induct xys) (auto simp add: dom_if)
15304
3514ca74ac54 Added more lemmas
nipkow
parents: 15303
diff changeset
   503
20800
69c82605efcf tuned specifications and proofs;
wenzelm
parents: 19947
diff changeset
   504
lemma dom_map_of_zip [simp]: "[| length xs = length ys; distinct xs |] ==>
24331
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   505
  dom(map_of(zip xs ys)) = set xs"
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   506
by (induct rule: list_induct2) simp_all
15110
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 14739
diff changeset
   507
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   508
lemma finite_dom_map_of: "finite (dom (map_of l))"
24331
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   509
by (induct l) (auto simp add: dom_def insert_Collect [symmetric])
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   510
20800
69c82605efcf tuned specifications and proofs;
wenzelm
parents: 19947
diff changeset
   511
lemma dom_map_upds [simp]:
24331
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   512
  "dom(m(xs[|->]ys)) = set(take (length ys) xs) Un dom m"
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   513
apply (induct xs arbitrary: m ys)
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   514
 apply simp
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   515
apply (case_tac ys)
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   516
 apply auto
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   517
done
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   518
20800
69c82605efcf tuned specifications and proofs;
wenzelm
parents: 19947
diff changeset
   519
lemma dom_map_add [simp]: "dom(m++n) = dom n Un dom m"
24331
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   520
by(auto simp:dom_def)
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   521
20800
69c82605efcf tuned specifications and proofs;
wenzelm
parents: 19947
diff changeset
   522
lemma dom_override_on [simp]:
69c82605efcf tuned specifications and proofs;
wenzelm
parents: 19947
diff changeset
   523
  "dom(override_on f g A) =
69c82605efcf tuned specifications and proofs;
wenzelm
parents: 19947
diff changeset
   524
    (dom f  - {a. a : A - dom g}) Un {a. a : A Int dom g}"
24331
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   525
by(auto simp: dom_def override_on_def)
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   526
14027
68d247b7b14b *** empty log message ***
nipkow
parents: 14026
diff changeset
   527
lemma map_add_comm: "dom m1 \<inter> dom m2 = {} \<Longrightarrow> m1++m2 = m2++m1"
24331
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   528
by (rule ext) (force simp: map_add_def dom_def split: option.split)
20800
69c82605efcf tuned specifications and proofs;
wenzelm
parents: 19947
diff changeset
   529
32236
0203e1006f1b some lemmas about maps (contributed by Peter Lammich)
krauss
parents: 31380
diff changeset
   530
lemma map_add_dom_app_simps:
0203e1006f1b some lemmas about maps (contributed by Peter Lammich)
krauss
parents: 31380
diff changeset
   531
  "\<lbrakk> m\<in>dom l2 \<rbrakk> \<Longrightarrow> (l1++l2) m = l2 m"
0203e1006f1b some lemmas about maps (contributed by Peter Lammich)
krauss
parents: 31380
diff changeset
   532
  "\<lbrakk> m\<notin>dom l1 \<rbrakk> \<Longrightarrow> (l1++l2) m = l2 m"
0203e1006f1b some lemmas about maps (contributed by Peter Lammich)
krauss
parents: 31380
diff changeset
   533
  "\<lbrakk> m\<notin>dom l2 \<rbrakk> \<Longrightarrow> (l1++l2) m = l1 m"
0203e1006f1b some lemmas about maps (contributed by Peter Lammich)
krauss
parents: 31380
diff changeset
   534
by (auto simp add: map_add_def split: option.split_asm)
0203e1006f1b some lemmas about maps (contributed by Peter Lammich)
krauss
parents: 31380
diff changeset
   535
29622
2eeb09477ed3 lemmas dom_const, dom_if
haftmann
parents: 28790
diff changeset
   536
lemma dom_const [simp]:
2eeb09477ed3 lemmas dom_const, dom_if
haftmann
parents: 28790
diff changeset
   537
  "dom (\<lambda>x. Some y) = UNIV"
2eeb09477ed3 lemmas dom_const, dom_if
haftmann
parents: 28790
diff changeset
   538
  by auto
2eeb09477ed3 lemmas dom_const, dom_if
haftmann
parents: 28790
diff changeset
   539
22230
bdec4a82f385 a few additions and deletions
nipkow
parents: 21404
diff changeset
   540
(* Due to John Matthews - could be rephrased with dom *)
bdec4a82f385 a few additions and deletions
nipkow
parents: 21404
diff changeset
   541
lemma finite_map_freshness:
bdec4a82f385 a few additions and deletions
nipkow
parents: 21404
diff changeset
   542
  "finite (dom (f :: 'a \<rightharpoonup> 'b)) \<Longrightarrow> \<not> finite (UNIV :: 'a set) \<Longrightarrow>
bdec4a82f385 a few additions and deletions
nipkow
parents: 21404
diff changeset
   543
   \<exists>x. f x = None"
bdec4a82f385 a few additions and deletions
nipkow
parents: 21404
diff changeset
   544
by(bestsimp dest:ex_new_if_finite)
14027
68d247b7b14b *** empty log message ***
nipkow
parents: 14026
diff changeset
   545
28790
2efba7b18c5b lemmas about dom and minus / insert
haftmann
parents: 28562
diff changeset
   546
lemma dom_minus:
2efba7b18c5b lemmas about dom and minus / insert
haftmann
parents: 28562
diff changeset
   547
  "f x = None \<Longrightarrow> dom f - insert x A = dom f - A"
2efba7b18c5b lemmas about dom and minus / insert
haftmann
parents: 28562
diff changeset
   548
  unfolding dom_def by simp
2efba7b18c5b lemmas about dom and minus / insert
haftmann
parents: 28562
diff changeset
   549
2efba7b18c5b lemmas about dom and minus / insert
haftmann
parents: 28562
diff changeset
   550
lemma insert_dom:
2efba7b18c5b lemmas about dom and minus / insert
haftmann
parents: 28562
diff changeset
   551
  "f x = Some y \<Longrightarrow> insert x (dom f) = dom f"
2efba7b18c5b lemmas about dom and minus / insert
haftmann
parents: 28562
diff changeset
   552
  unfolding dom_def by auto
2efba7b18c5b lemmas about dom and minus / insert
haftmann
parents: 28562
diff changeset
   553
2efba7b18c5b lemmas about dom and minus / insert
haftmann
parents: 28562
diff changeset
   554
17399
56a3a4affedc @{term [source] ...} in subsections probably more robust;
wenzelm
parents: 17391
diff changeset
   555
subsection {* @{term [source] ran} *}
14100
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   556
20800
69c82605efcf tuned specifications and proofs;
wenzelm
parents: 19947
diff changeset
   557
lemma ranI: "m a = Some b ==> b : ran m"
24331
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   558
by(auto simp: ran_def)
14100
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   559
(* declare ranI [intro]? *)
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   560
20800
69c82605efcf tuned specifications and proofs;
wenzelm
parents: 19947
diff changeset
   561
lemma ran_empty [simp]: "ran empty = {}"
24331
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   562
by(auto simp: ran_def)
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   563
20800
69c82605efcf tuned specifications and proofs;
wenzelm
parents: 19947
diff changeset
   564
lemma ran_map_upd [simp]: "m a = None ==> ran(m(a|->b)) = insert b (ran m)"
24331
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   565
unfolding ran_def
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   566
apply auto
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   567
apply (subgoal_tac "aa ~= a")
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   568
 apply auto
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   569
done
20800
69c82605efcf tuned specifications and proofs;
wenzelm
parents: 19947
diff changeset
   570
34979
8cb6e7a42e9c more correspondence lemmas between related operations
haftmann
parents: 34941
diff changeset
   571
lemma ran_distinct: 
8cb6e7a42e9c more correspondence lemmas between related operations
haftmann
parents: 34941
diff changeset
   572
  assumes dist: "distinct (map fst al)" 
8cb6e7a42e9c more correspondence lemmas between related operations
haftmann
parents: 34941
diff changeset
   573
  shows "ran (map_of al) = snd ` set al"
8cb6e7a42e9c more correspondence lemmas between related operations
haftmann
parents: 34941
diff changeset
   574
using assms proof (induct al)
8cb6e7a42e9c more correspondence lemmas between related operations
haftmann
parents: 34941
diff changeset
   575
  case Nil then show ?case by simp
8cb6e7a42e9c more correspondence lemmas between related operations
haftmann
parents: 34941
diff changeset
   576
next
8cb6e7a42e9c more correspondence lemmas between related operations
haftmann
parents: 34941
diff changeset
   577
  case (Cons kv al)
8cb6e7a42e9c more correspondence lemmas between related operations
haftmann
parents: 34941
diff changeset
   578
  then have "ran (map_of al) = snd ` set al" by simp
8cb6e7a42e9c more correspondence lemmas between related operations
haftmann
parents: 34941
diff changeset
   579
  moreover from Cons.prems have "map_of al (fst kv) = None"
8cb6e7a42e9c more correspondence lemmas between related operations
haftmann
parents: 34941
diff changeset
   580
    by (simp add: map_of_eq_None_iff)
8cb6e7a42e9c more correspondence lemmas between related operations
haftmann
parents: 34941
diff changeset
   581
  ultimately show ?case by (simp only: map_of.simps ran_map_upd) simp
8cb6e7a42e9c more correspondence lemmas between related operations
haftmann
parents: 34941
diff changeset
   582
qed
8cb6e7a42e9c more correspondence lemmas between related operations
haftmann
parents: 34941
diff changeset
   583
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   584
14100
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   585
subsection {* @{text "map_le"} *}
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   586
13912
3c0a340be514 fixed document
kleing
parents: 13910
diff changeset
   587
lemma map_le_empty [simp]: "empty \<subseteq>\<^sub>m g"
24331
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   588
by (simp add: map_le_def)
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   589
17724
e969fc0a4925 simprules need names
paulson
parents: 17399
diff changeset
   590
lemma upd_None_map_le [simp]: "f(x := None) \<subseteq>\<^sub>m f"
24331
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   591
by (force simp add: map_le_def)
14187
26dfcd0ac436 Added new theorems
nipkow
parents: 14186
diff changeset
   592
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   593
lemma map_le_upd[simp]: "f \<subseteq>\<^sub>m g ==> f(a := b) \<subseteq>\<^sub>m g(a := b)"
24331
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   594
by (fastsimp simp add: map_le_def)
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   595
17724
e969fc0a4925 simprules need names
paulson
parents: 17399
diff changeset
   596
lemma map_le_imp_upd_le [simp]: "m1 \<subseteq>\<^sub>m m2 \<Longrightarrow> m1(x := None) \<subseteq>\<^sub>m m2(x \<mapsto> y)"
24331
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   597
by (force simp add: map_le_def)
14187
26dfcd0ac436 Added new theorems
nipkow
parents: 14186
diff changeset
   598
20800
69c82605efcf tuned specifications and proofs;
wenzelm
parents: 19947
diff changeset
   599
lemma map_le_upds [simp]:
24331
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   600
  "f \<subseteq>\<^sub>m g ==> f(as [|->] bs) \<subseteq>\<^sub>m g(as [|->] bs)"
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   601
apply (induct as arbitrary: f g bs)
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   602
 apply simp
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   603
apply (case_tac bs)
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   604
 apply auto
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   605
done
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   606
14033
bc723de8ec95 Added a few lemmas about map_le
webertj
parents: 14027
diff changeset
   607
lemma map_le_implies_dom_le: "(f \<subseteq>\<^sub>m g) \<Longrightarrow> (dom f \<subseteq> dom g)"
24331
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   608
by (fastsimp simp add: map_le_def dom_def)
14033
bc723de8ec95 Added a few lemmas about map_le
webertj
parents: 14027
diff changeset
   609
bc723de8ec95 Added a few lemmas about map_le
webertj
parents: 14027
diff changeset
   610
lemma map_le_refl [simp]: "f \<subseteq>\<^sub>m f"
24331
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   611
by (simp add: map_le_def)
14033
bc723de8ec95 Added a few lemmas about map_le
webertj
parents: 14027
diff changeset
   612
14187
26dfcd0ac436 Added new theorems
nipkow
parents: 14186
diff changeset
   613
lemma map_le_trans[trans]: "\<lbrakk> m1 \<subseteq>\<^sub>m m2; m2 \<subseteq>\<^sub>m m3\<rbrakk> \<Longrightarrow> m1 \<subseteq>\<^sub>m m3"
24331
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   614
by (auto simp add: map_le_def dom_def)
14033
bc723de8ec95 Added a few lemmas about map_le
webertj
parents: 14027
diff changeset
   615
bc723de8ec95 Added a few lemmas about map_le
webertj
parents: 14027
diff changeset
   616
lemma map_le_antisym: "\<lbrakk> f \<subseteq>\<^sub>m g; g \<subseteq>\<^sub>m f \<rbrakk> \<Longrightarrow> f = g"
24331
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   617
unfolding map_le_def
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   618
apply (rule ext)
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   619
apply (case_tac "x \<in> dom f", simp)
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   620
apply (case_tac "x \<in> dom g", simp, fastsimp)
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   621
done
14033
bc723de8ec95 Added a few lemmas about map_le
webertj
parents: 14027
diff changeset
   622
bc723de8ec95 Added a few lemmas about map_le
webertj
parents: 14027
diff changeset
   623
lemma map_le_map_add [simp]: "f \<subseteq>\<^sub>m (g ++ f)"
24331
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   624
by (fastsimp simp add: map_le_def)
14033
bc723de8ec95 Added a few lemmas about map_le
webertj
parents: 14027
diff changeset
   625
15304
3514ca74ac54 Added more lemmas
nipkow
parents: 15303
diff changeset
   626
lemma map_le_iff_map_add_commute: "(f \<subseteq>\<^sub>m f ++ g) = (f++g = g++f)"
24331
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   627
by(fastsimp simp: map_add_def map_le_def expand_fun_eq split: option.splits)
15304
3514ca74ac54 Added more lemmas
nipkow
parents: 15303
diff changeset
   628
15303
eedbb8d22ca2 added lemmas
nipkow
parents: 15251
diff changeset
   629
lemma map_add_le_mapE: "f++g \<subseteq>\<^sub>m h \<Longrightarrow> g \<subseteq>\<^sub>m h"
24331
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   630
by (fastsimp simp add: map_le_def map_add_def dom_def)
15303
eedbb8d22ca2 added lemmas
nipkow
parents: 15251
diff changeset
   631
eedbb8d22ca2 added lemmas
nipkow
parents: 15251
diff changeset
   632
lemma map_add_le_mapI: "\<lbrakk> f \<subseteq>\<^sub>m h; g \<subseteq>\<^sub>m h; f \<subseteq>\<^sub>m f++g \<rbrakk> \<Longrightarrow> f++g \<subseteq>\<^sub>m h"
24331
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 22744
diff changeset
   633
by (clarsimp simp add: map_le_def map_add_def dom_def split: option.splits)
15303
eedbb8d22ca2 added lemmas
nipkow
parents: 15251
diff changeset
   634
31080
21ffc770ebc0 lemmas by Andreas Lochbihler
nipkow
parents: 30935
diff changeset
   635
lemma dom_eq_singleton_conv: "dom f = {x} \<longleftrightarrow> (\<exists>v. f = [x \<mapsto> v])"
21ffc770ebc0 lemmas by Andreas Lochbihler
nipkow
parents: 30935
diff changeset
   636
proof(rule iffI)
21ffc770ebc0 lemmas by Andreas Lochbihler
nipkow
parents: 30935
diff changeset
   637
  assume "\<exists>v. f = [x \<mapsto> v]"
21ffc770ebc0 lemmas by Andreas Lochbihler
nipkow
parents: 30935
diff changeset
   638
  thus "dom f = {x}" by(auto split: split_if_asm)
21ffc770ebc0 lemmas by Andreas Lochbihler
nipkow
parents: 30935
diff changeset
   639
next
21ffc770ebc0 lemmas by Andreas Lochbihler
nipkow
parents: 30935
diff changeset
   640
  assume "dom f = {x}"
21ffc770ebc0 lemmas by Andreas Lochbihler
nipkow
parents: 30935
diff changeset
   641
  then obtain v where "f x = Some v" by auto
21ffc770ebc0 lemmas by Andreas Lochbihler
nipkow
parents: 30935
diff changeset
   642
  hence "[x \<mapsto> v] \<subseteq>\<^sub>m f" by(auto simp add: map_le_def)
21ffc770ebc0 lemmas by Andreas Lochbihler
nipkow
parents: 30935
diff changeset
   643
  moreover have "f \<subseteq>\<^sub>m [x \<mapsto> v]" using `dom f = {x}` `f x = Some v`
21ffc770ebc0 lemmas by Andreas Lochbihler
nipkow
parents: 30935
diff changeset
   644
    by(auto simp add: map_le_def)
21ffc770ebc0 lemmas by Andreas Lochbihler
nipkow
parents: 30935
diff changeset
   645
  ultimately have "f = [x \<mapsto> v]" by-(rule map_le_antisym)
21ffc770ebc0 lemmas by Andreas Lochbihler
nipkow
parents: 30935
diff changeset
   646
  thus "\<exists>v. f = [x \<mapsto> v]" by blast
21ffc770ebc0 lemmas by Andreas Lochbihler
nipkow
parents: 30935
diff changeset
   647
qed
21ffc770ebc0 lemmas by Andreas Lochbihler
nipkow
parents: 30935
diff changeset
   648
3981
b4f93a8da835 Added the new theory Map.
nipkow
parents:
diff changeset
   649
end
34979
8cb6e7a42e9c more correspondence lemmas between related operations
haftmann
parents: 34941
diff changeset
   650