src/HOL/Map.thy
author haftmann
Thu Mar 04 11:22:06 2010 +0100 (2010-03-04)
changeset 35565 56b070cd7ab3
parent 35553 a8c8008a2c9d
child 35607 896f01fe825b
permissions -rw-r--r--
lemmas set_map_of_compr, map_of_inject_set
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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) "map" = "'a => 'b option" (infixr "~=>" 0)
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translations (type) "'a ~=> 'b" <= (type) "'a => 'b option"
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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\<^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 := 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 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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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: expand_fun_eq)
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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: "map_of (map (%(a,b). (a,f b)) xs) x = Option.map f (map_of xs x)"
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by (induct xs) auto
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subsection {* @{const Option.map} related *}
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lemma option_map_o_empty [simp]: "Option.map f o empty = empty"
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by (rule ext) simp
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lemma option_map_o_map_upd [simp]:
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  "Option.map f o m(a|->b) = (Option.map f o m)(a|->f b)"
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by (rule ext) simp
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subsection {* @{term [source] map_comp} related *}
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lemma map_comp_empty [simp]:
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  "m \<circ>\<^sub>m empty = empty"
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  "empty \<circ>\<^sub>m m = empty"
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by (auto simp add: map_comp_def intro: ext split: option.splits)
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lemma map_comp_simps [simp]:
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  "m2 k = None \<Longrightarrow> (m1 \<circ>\<^sub>m m2) k = None"
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  "m2 k = Some k' \<Longrightarrow> (m1 \<circ>\<^sub>m m2) k = m1 k'"
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by (auto simp add: map_comp_def)
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lemma map_comp_Some_iff:
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  "((m1 \<circ>\<^sub>m m2) k = Some v) = (\<exists>k'. m2 k = Some k' \<and> m1 k' = Some v)"
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by (auto simp add: map_comp_def split: option.splits)
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lemma map_comp_None_iff:
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  "((m1 \<circ>\<^sub>m m2) k = None) = (m2 k = None \<or> (\<exists>k'. m2 k = Some k' \<and> m1 k' = None)) "
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by (auto simp add: map_comp_def split: option.splits)
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subsection {* @{text "++"} *}
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lemma map_add_empty[simp]: "m ++ empty = m"
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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)"
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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)
nipkow@14186
   311
krauss@32236
   312
lemma map_add_upd_left: "m\<notin>dom e2 \<Longrightarrow> e1(m \<mapsto> u1) ++ e2 = (e1 ++ e2)(m \<mapsto> u1)"
krauss@32236
   313
by (rule ext) (auto simp: map_add_def dom_def split: option.split)
krauss@32236
   314
wenzelm@20800
   315
lemma map_of_append[simp]: "map_of (xs @ ys) = map_of ys ++ map_of xs"
nipkow@24331
   316
unfolding map_add_def
nipkow@24331
   317
apply (induct xs)
nipkow@24331
   318
 apply simp
nipkow@24331
   319
apply (rule ext)
nipkow@24331
   320
apply (simp split add: option.split)
nipkow@24331
   321
done
webertj@13908
   322
nipkow@14025
   323
lemma finite_range_map_of_map_add:
wenzelm@20800
   324
  "finite (range f) ==> finite (range (f ++ map_of l))"
nipkow@24331
   325
apply (induct l)
nipkow@24331
   326
 apply (auto simp del: fun_upd_apply)
nipkow@24331
   327
apply (erule finite_range_updI)
nipkow@24331
   328
done
webertj@13908
   329
wenzelm@20800
   330
lemma inj_on_map_add_dom [iff]:
nipkow@24331
   331
  "inj_on (m ++ m') (dom m') = inj_on m' (dom m')"
nipkow@24331
   332
by (fastsimp simp: map_add_def dom_def inj_on_def split: option.splits)
wenzelm@20800
   333
haftmann@34979
   334
lemma map_upds_fold_map_upd:
haftmann@35552
   335
  "m(ks[\<mapsto>]vs) = foldl (\<lambda>m (k, v). m(k \<mapsto> v)) m (zip ks vs)"
haftmann@34979
   336
unfolding map_upds_def proof (rule sym, rule zip_obtain_same_length)
haftmann@34979
   337
  fix ks :: "'a list" and vs :: "'b list"
haftmann@34979
   338
  assume "length ks = length vs"
haftmann@35552
   339
  then show "foldl (\<lambda>m (k, v). m(k\<mapsto>v)) m (zip ks vs) = m ++ map_of (rev (zip ks vs))"
haftmann@35552
   340
    by(induct arbitrary: m rule: list_induct2) simp_all
haftmann@34979
   341
qed
haftmann@34979
   342
haftmann@34979
   343
lemma map_add_map_of_foldr:
haftmann@34979
   344
  "m ++ map_of ps = foldr (\<lambda>(k, v) m. m(k \<mapsto> v)) ps m"
haftmann@34979
   345
  by (induct ps) (auto simp add: expand_fun_eq map_add_def)
haftmann@34979
   346
nipkow@15304
   347
wenzelm@17399
   348
subsection {* @{term [source] restrict_map} *}
oheimb@14100
   349
wenzelm@20800
   350
lemma restrict_map_to_empty [simp]: "m|`{} = empty"
nipkow@24331
   351
by (simp add: restrict_map_def)
nipkow@14186
   352
haftmann@31380
   353
lemma restrict_map_insert: "f |` (insert a A) = (f |` A)(a := f a)"
haftmann@31380
   354
by (auto simp add: restrict_map_def intro: ext)
haftmann@31380
   355
wenzelm@20800
   356
lemma restrict_map_empty [simp]: "empty|`D = empty"
nipkow@24331
   357
by (simp add: restrict_map_def)
nipkow@14186
   358
nipkow@15693
   359
lemma restrict_in [simp]: "x \<in> A \<Longrightarrow> (m|`A) x = m x"
nipkow@24331
   360
by (simp add: restrict_map_def)
oheimb@14100
   361
nipkow@15693
   362
lemma restrict_out [simp]: "x \<notin> A \<Longrightarrow> (m|`A) x = None"
nipkow@24331
   363
by (simp add: restrict_map_def)
oheimb@14100
   364
nipkow@15693
   365
lemma ran_restrictD: "y \<in> ran (m|`A) \<Longrightarrow> \<exists>x\<in>A. m x = Some y"
nipkow@24331
   366
by (auto simp: restrict_map_def ran_def split: split_if_asm)
oheimb@14100
   367
nipkow@15693
   368
lemma dom_restrict [simp]: "dom (m|`A) = dom m \<inter> A"
nipkow@24331
   369
by (auto simp: restrict_map_def dom_def split: split_if_asm)
oheimb@14100
   370
nipkow@15693
   371
lemma restrict_upd_same [simp]: "m(x\<mapsto>y)|`(-{x}) = m|`(-{x})"
nipkow@24331
   372
by (rule ext) (auto simp: restrict_map_def)
oheimb@14100
   373
nipkow@15693
   374
lemma restrict_restrict [simp]: "m|`A|`B = m|`(A\<inter>B)"
nipkow@24331
   375
by (rule ext) (auto simp: restrict_map_def)
oheimb@14100
   376
wenzelm@20800
   377
lemma restrict_fun_upd [simp]:
nipkow@24331
   378
  "m(x := y)|`D = (if x \<in> D then (m|`(D-{x}))(x := y) else m|`D)"
nipkow@24331
   379
by (simp add: restrict_map_def expand_fun_eq)
nipkow@14186
   380
wenzelm@20800
   381
lemma fun_upd_None_restrict [simp]:
nipkow@24331
   382
  "(m|`D)(x := None) = (if x:D then m|`(D - {x}) else m|`D)"
nipkow@24331
   383
by (simp add: restrict_map_def expand_fun_eq)
nipkow@14186
   384
wenzelm@20800
   385
lemma fun_upd_restrict: "(m|`D)(x := y) = (m|`(D-{x}))(x := y)"
nipkow@24331
   386
by (simp add: restrict_map_def expand_fun_eq)
nipkow@14186
   387
wenzelm@20800
   388
lemma fun_upd_restrict_conv [simp]:
nipkow@24331
   389
  "x \<in> D \<Longrightarrow> (m|`D)(x := y) = (m|`(D-{x}))(x := y)"
nipkow@24331
   390
by (simp add: restrict_map_def expand_fun_eq)
nipkow@14186
   391
haftmann@35159
   392
lemma map_of_map_restrict:
haftmann@35159
   393
  "map_of (map (\<lambda>k. (k, f k)) ks) = (Some \<circ> f) |` set ks"
haftmann@35159
   394
  by (induct ks) (simp_all add: expand_fun_eq restrict_map_insert)
haftmann@35159
   395
oheimb@14100
   396
wenzelm@17399
   397
subsection {* @{term [source] map_upds} *}
nipkow@14025
   398
wenzelm@20800
   399
lemma map_upds_Nil1 [simp]: "m([] [|->] bs) = m"
nipkow@24331
   400
by (simp add: map_upds_def)
nipkow@14025
   401
wenzelm@20800
   402
lemma map_upds_Nil2 [simp]: "m(as [|->] []) = m"
nipkow@24331
   403
by (simp add:map_upds_def)
wenzelm@20800
   404
wenzelm@20800
   405
lemma map_upds_Cons [simp]: "m(a#as [|->] b#bs) = (m(a|->b))(as[|->]bs)"
nipkow@24331
   406
by (simp add:map_upds_def)
nipkow@14025
   407
wenzelm@20800
   408
lemma map_upds_append1 [simp]: "\<And>ys m. size xs < size ys \<Longrightarrow>
nipkow@24331
   409
  m(xs@[x] [\<mapsto>] ys) = m(xs [\<mapsto>] ys)(x \<mapsto> ys!size xs)"
nipkow@24331
   410
apply(induct xs)
nipkow@24331
   411
 apply (clarsimp simp add: neq_Nil_conv)
nipkow@24331
   412
apply (case_tac ys)
nipkow@24331
   413
 apply simp
nipkow@24331
   414
apply simp
nipkow@24331
   415
done
nipkow@14187
   416
wenzelm@20800
   417
lemma map_upds_list_update2_drop [simp]:
wenzelm@20800
   418
  "\<lbrakk>size xs \<le> i; i < size ys\<rbrakk>
wenzelm@20800
   419
    \<Longrightarrow> m(xs[\<mapsto>]ys[i:=y]) = m(xs[\<mapsto>]ys)"
nipkow@24331
   420
apply (induct xs arbitrary: m ys i)
nipkow@24331
   421
 apply simp
nipkow@24331
   422
apply (case_tac ys)
nipkow@24331
   423
 apply simp
nipkow@24331
   424
apply (simp split: nat.split)
nipkow@24331
   425
done
nipkow@14025
   426
wenzelm@20800
   427
lemma map_upd_upds_conv_if:
wenzelm@20800
   428
  "(f(x|->y))(xs [|->] ys) =
wenzelm@20800
   429
   (if x : set(take (length ys) xs) then f(xs [|->] ys)
wenzelm@20800
   430
                                    else (f(xs [|->] ys))(x|->y))"
nipkow@24331
   431
apply (induct xs arbitrary: x y ys f)
nipkow@24331
   432
 apply simp
nipkow@24331
   433
apply (case_tac ys)
nipkow@24331
   434
 apply (auto split: split_if simp: fun_upd_twist)
nipkow@24331
   435
done
nipkow@14025
   436
nipkow@14025
   437
lemma map_upds_twist [simp]:
nipkow@24331
   438
  "a ~: set as ==> m(a|->b)(as[|->]bs) = m(as[|->]bs)(a|->b)"
nipkow@24331
   439
using set_take_subset by (fastsimp simp add: map_upd_upds_conv_if)
nipkow@14025
   440
wenzelm@20800
   441
lemma map_upds_apply_nontin [simp]:
nipkow@24331
   442
  "x ~: set xs ==> (f(xs[|->]ys)) x = f x"
nipkow@24331
   443
apply (induct xs arbitrary: ys)
nipkow@24331
   444
 apply simp
nipkow@24331
   445
apply (case_tac ys)
nipkow@24331
   446
 apply (auto simp: map_upd_upds_conv_if)
nipkow@24331
   447
done
nipkow@14025
   448
wenzelm@20800
   449
lemma fun_upds_append_drop [simp]:
nipkow@24331
   450
  "size xs = size ys \<Longrightarrow> m(xs@zs[\<mapsto>]ys) = m(xs[\<mapsto>]ys)"
nipkow@24331
   451
apply (induct xs arbitrary: m ys)
nipkow@24331
   452
 apply simp
nipkow@24331
   453
apply (case_tac ys)
nipkow@24331
   454
 apply simp_all
nipkow@24331
   455
done
nipkow@14300
   456
wenzelm@20800
   457
lemma fun_upds_append2_drop [simp]:
nipkow@24331
   458
  "size xs = size ys \<Longrightarrow> m(xs[\<mapsto>]ys@zs) = m(xs[\<mapsto>]ys)"
nipkow@24331
   459
apply (induct xs arbitrary: m ys)
nipkow@24331
   460
 apply simp
nipkow@24331
   461
apply (case_tac ys)
nipkow@24331
   462
 apply simp_all
nipkow@24331
   463
done
nipkow@14300
   464
nipkow@14300
   465
wenzelm@20800
   466
lemma restrict_map_upds[simp]:
wenzelm@20800
   467
  "\<lbrakk> length xs = length ys; set xs \<subseteq> D \<rbrakk>
wenzelm@20800
   468
    \<Longrightarrow> m(xs [\<mapsto>] ys)|`D = (m|`(D - set xs))(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
nipkow@24331
   473
apply (simp add: Diff_insert [symmetric] insert_absorb)
nipkow@24331
   474
apply (simp add: map_upd_upds_conv_if)
nipkow@24331
   475
done
nipkow@14186
   476
nipkow@14186
   477
wenzelm@17399
   478
subsection {* @{term [source] dom} *}
webertj@13908
   479
nipkow@31080
   480
lemma dom_eq_empty_conv [simp]: "dom f = {} \<longleftrightarrow> f = empty"
nipkow@31080
   481
by(auto intro!:ext simp: dom_def)
nipkow@31080
   482
webertj@13908
   483
lemma domI: "m a = Some b ==> a : dom m"
nipkow@24331
   484
by(simp add:dom_def)
oheimb@14100
   485
(* declare domI [intro]? *)
webertj@13908
   486
paulson@15369
   487
lemma domD: "a : dom m ==> \<exists>b. m a = Some b"
nipkow@24331
   488
by (cases "m a") (auto simp add: dom_def)
webertj@13908
   489
wenzelm@20800
   490
lemma domIff [iff, simp del]: "(a : dom m) = (m a ~= None)"
nipkow@24331
   491
by(simp add:dom_def)
webertj@13908
   492
wenzelm@20800
   493
lemma dom_empty [simp]: "dom empty = {}"
nipkow@24331
   494
by(simp add:dom_def)
webertj@13908
   495
wenzelm@20800
   496
lemma dom_fun_upd [simp]:
nipkow@24331
   497
  "dom(f(x := y)) = (if y=None then dom f - {x} else insert x (dom f))"
nipkow@24331
   498
by(auto simp add:dom_def)
webertj@13908
   499
haftmann@34979
   500
lemma dom_if:
haftmann@34979
   501
  "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
   502
  by (auto split: if_splits)
nipkow@13937
   503
nipkow@15304
   504
lemma dom_map_of_conv_image_fst:
haftmann@34979
   505
  "dom (map_of xys) = fst ` set xys"
haftmann@34979
   506
  by (induct xys) (auto simp add: dom_if)
nipkow@15304
   507
wenzelm@20800
   508
lemma dom_map_of_zip [simp]: "[| length xs = length ys; distinct xs |] ==>
nipkow@24331
   509
  dom(map_of(zip xs ys)) = set xs"
nipkow@24331
   510
by (induct rule: list_induct2) simp_all
nipkow@15110
   511
webertj@13908
   512
lemma finite_dom_map_of: "finite (dom (map_of l))"
nipkow@24331
   513
by (induct l) (auto simp add: dom_def insert_Collect [symmetric])
webertj@13908
   514
wenzelm@20800
   515
lemma dom_map_upds [simp]:
nipkow@24331
   516
  "dom(m(xs[|->]ys)) = set(take (length ys) xs) Un dom m"
nipkow@24331
   517
apply (induct xs arbitrary: m ys)
nipkow@24331
   518
 apply simp
nipkow@24331
   519
apply (case_tac ys)
nipkow@24331
   520
 apply auto
nipkow@24331
   521
done
nipkow@13910
   522
wenzelm@20800
   523
lemma dom_map_add [simp]: "dom(m++n) = dom n Un dom m"
nipkow@24331
   524
by(auto simp:dom_def)
nipkow@13910
   525
wenzelm@20800
   526
lemma dom_override_on [simp]:
wenzelm@20800
   527
  "dom(override_on f g A) =
wenzelm@20800
   528
    (dom f  - {a. a : A - dom g}) Un {a. a : A Int dom g}"
nipkow@24331
   529
by(auto simp: dom_def override_on_def)
webertj@13908
   530
nipkow@14027
   531
lemma map_add_comm: "dom m1 \<inter> dom m2 = {} \<Longrightarrow> m1++m2 = m2++m1"
nipkow@24331
   532
by (rule ext) (force simp: map_add_def dom_def split: option.split)
wenzelm@20800
   533
krauss@32236
   534
lemma map_add_dom_app_simps:
krauss@32236
   535
  "\<lbrakk> m\<in>dom l2 \<rbrakk> \<Longrightarrow> (l1++l2) m = l2 m"
krauss@32236
   536
  "\<lbrakk> m\<notin>dom l1 \<rbrakk> \<Longrightarrow> (l1++l2) m = l2 m"
krauss@32236
   537
  "\<lbrakk> m\<notin>dom l2 \<rbrakk> \<Longrightarrow> (l1++l2) m = l1 m"
krauss@32236
   538
by (auto simp add: map_add_def split: option.split_asm)
krauss@32236
   539
haftmann@29622
   540
lemma dom_const [simp]:
haftmann@35159
   541
  "dom (\<lambda>x. Some (f x)) = UNIV"
haftmann@29622
   542
  by auto
haftmann@29622
   543
nipkow@22230
   544
(* Due to John Matthews - could be rephrased with dom *)
nipkow@22230
   545
lemma finite_map_freshness:
nipkow@22230
   546
  "finite (dom (f :: 'a \<rightharpoonup> 'b)) \<Longrightarrow> \<not> finite (UNIV :: 'a set) \<Longrightarrow>
nipkow@22230
   547
   \<exists>x. f x = None"
nipkow@22230
   548
by(bestsimp dest:ex_new_if_finite)
nipkow@14027
   549
haftmann@28790
   550
lemma dom_minus:
haftmann@28790
   551
  "f x = None \<Longrightarrow> dom f - insert x A = dom f - A"
haftmann@28790
   552
  unfolding dom_def by simp
haftmann@28790
   553
haftmann@28790
   554
lemma insert_dom:
haftmann@28790
   555
  "f x = Some y \<Longrightarrow> insert x (dom f) = dom f"
haftmann@28790
   556
  unfolding dom_def by auto
haftmann@28790
   557
haftmann@28790
   558
wenzelm@17399
   559
subsection {* @{term [source] ran} *}
oheimb@14100
   560
wenzelm@20800
   561
lemma ranI: "m a = Some b ==> b : ran m"
nipkow@24331
   562
by(auto simp: ran_def)
oheimb@14100
   563
(* declare ranI [intro]? *)
webertj@13908
   564
wenzelm@20800
   565
lemma ran_empty [simp]: "ran empty = {}"
nipkow@24331
   566
by(auto simp: ran_def)
webertj@13908
   567
wenzelm@20800
   568
lemma ran_map_upd [simp]: "m a = None ==> ran(m(a|->b)) = insert b (ran m)"
nipkow@24331
   569
unfolding ran_def
nipkow@24331
   570
apply auto
nipkow@24331
   571
apply (subgoal_tac "aa ~= a")
nipkow@24331
   572
 apply auto
nipkow@24331
   573
done
wenzelm@20800
   574
haftmann@34979
   575
lemma ran_distinct: 
haftmann@34979
   576
  assumes dist: "distinct (map fst al)" 
haftmann@34979
   577
  shows "ran (map_of al) = snd ` set al"
haftmann@34979
   578
using assms proof (induct al)
haftmann@34979
   579
  case Nil then show ?case by simp
haftmann@34979
   580
next
haftmann@34979
   581
  case (Cons kv al)
haftmann@34979
   582
  then have "ran (map_of al) = snd ` set al" by simp
haftmann@34979
   583
  moreover from Cons.prems have "map_of al (fst kv) = None"
haftmann@34979
   584
    by (simp add: map_of_eq_None_iff)
haftmann@34979
   585
  ultimately show ?case by (simp only: map_of.simps ran_map_upd) simp
haftmann@34979
   586
qed
haftmann@34979
   587
nipkow@13910
   588
oheimb@14100
   589
subsection {* @{text "map_le"} *}
nipkow@13910
   590
kleing@13912
   591
lemma map_le_empty [simp]: "empty \<subseteq>\<^sub>m g"
nipkow@24331
   592
by (simp add: map_le_def)
nipkow@13910
   593
paulson@17724
   594
lemma upd_None_map_le [simp]: "f(x := None) \<subseteq>\<^sub>m f"
nipkow@24331
   595
by (force simp add: map_le_def)
nipkow@14187
   596
nipkow@13910
   597
lemma map_le_upd[simp]: "f \<subseteq>\<^sub>m g ==> f(a := b) \<subseteq>\<^sub>m g(a := b)"
nipkow@24331
   598
by (fastsimp simp add: map_le_def)
nipkow@13910
   599
paulson@17724
   600
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
   601
by (force simp add: map_le_def)
nipkow@14187
   602
wenzelm@20800
   603
lemma map_le_upds [simp]:
nipkow@24331
   604
  "f \<subseteq>\<^sub>m g ==> f(as [|->] bs) \<subseteq>\<^sub>m g(as [|->] bs)"
nipkow@24331
   605
apply (induct as arbitrary: f g bs)
nipkow@24331
   606
 apply simp
nipkow@24331
   607
apply (case_tac bs)
nipkow@24331
   608
 apply auto
nipkow@24331
   609
done
webertj@13908
   610
webertj@14033
   611
lemma map_le_implies_dom_le: "(f \<subseteq>\<^sub>m g) \<Longrightarrow> (dom f \<subseteq> dom g)"
nipkow@24331
   612
by (fastsimp simp add: map_le_def dom_def)
webertj@14033
   613
webertj@14033
   614
lemma map_le_refl [simp]: "f \<subseteq>\<^sub>m f"
nipkow@24331
   615
by (simp add: map_le_def)
webertj@14033
   616
nipkow@14187
   617
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
   618
by (auto simp add: map_le_def dom_def)
webertj@14033
   619
webertj@14033
   620
lemma map_le_antisym: "\<lbrakk> f \<subseteq>\<^sub>m g; g \<subseteq>\<^sub>m f \<rbrakk> \<Longrightarrow> f = g"
nipkow@24331
   621
unfolding map_le_def
nipkow@24331
   622
apply (rule ext)
nipkow@24331
   623
apply (case_tac "x \<in> dom f", simp)
nipkow@24331
   624
apply (case_tac "x \<in> dom g", simp, fastsimp)
nipkow@24331
   625
done
webertj@14033
   626
webertj@14033
   627
lemma map_le_map_add [simp]: "f \<subseteq>\<^sub>m (g ++ f)"
nipkow@24331
   628
by (fastsimp simp add: map_le_def)
webertj@14033
   629
nipkow@15304
   630
lemma map_le_iff_map_add_commute: "(f \<subseteq>\<^sub>m f ++ g) = (f++g = g++f)"
nipkow@24331
   631
by(fastsimp simp: map_add_def map_le_def expand_fun_eq split: option.splits)
nipkow@15304
   632
nipkow@15303
   633
lemma map_add_le_mapE: "f++g \<subseteq>\<^sub>m h \<Longrightarrow> g \<subseteq>\<^sub>m h"
nipkow@24331
   634
by (fastsimp simp add: map_le_def map_add_def dom_def)
nipkow@15303
   635
nipkow@15303
   636
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"
nipkow@24331
   637
by (clarsimp simp add: map_le_def map_add_def dom_def split: option.splits)
nipkow@15303
   638
nipkow@31080
   639
lemma dom_eq_singleton_conv: "dom f = {x} \<longleftrightarrow> (\<exists>v. f = [x \<mapsto> v])"
nipkow@31080
   640
proof(rule iffI)
nipkow@31080
   641
  assume "\<exists>v. f = [x \<mapsto> v]"
nipkow@31080
   642
  thus "dom f = {x}" by(auto split: split_if_asm)
nipkow@31080
   643
next
nipkow@31080
   644
  assume "dom f = {x}"
nipkow@31080
   645
  then obtain v where "f x = Some v" by auto
nipkow@31080
   646
  hence "[x \<mapsto> v] \<subseteq>\<^sub>m f" by(auto simp add: map_le_def)
nipkow@31080
   647
  moreover have "f \<subseteq>\<^sub>m [x \<mapsto> v]" using `dom f = {x}` `f x = Some v`
nipkow@31080
   648
    by(auto simp add: map_le_def)
nipkow@31080
   649
  ultimately have "f = [x \<mapsto> v]" by-(rule map_le_antisym)
nipkow@31080
   650
  thus "\<exists>v. f = [x \<mapsto> v]" by blast
nipkow@31080
   651
qed
nipkow@31080
   652
haftmann@35565
   653
haftmann@35565
   654
subsection {* Various *}
haftmann@35565
   655
haftmann@35565
   656
lemma set_map_of_compr:
haftmann@35565
   657
  assumes distinct: "distinct (map fst xs)"
haftmann@35565
   658
  shows "set xs = {(k, v). map_of xs k = Some v}"
haftmann@35565
   659
using assms proof (induct xs)
haftmann@35565
   660
  case Nil then show ?case by simp
haftmann@35565
   661
next
haftmann@35565
   662
  case (Cons x xs)
haftmann@35565
   663
  obtain k v where "x = (k, v)" by (cases x) blast
haftmann@35565
   664
  with Cons.prems have "k \<notin> dom (map_of xs)"
haftmann@35565
   665
    by (simp add: dom_map_of_conv_image_fst)
haftmann@35565
   666
  then have *: "insert (k, v) {(k, v). map_of xs k = Some v} =
haftmann@35565
   667
    {(k', v'). (map_of xs(k \<mapsto> v)) k' = Some v'}"
haftmann@35565
   668
    by (auto split: if_splits)
haftmann@35565
   669
  from Cons have "set xs = {(k, v). map_of xs k = Some v}" by simp
haftmann@35565
   670
  with * `x = (k, v)` show ?case by simp
haftmann@35565
   671
qed
haftmann@35565
   672
haftmann@35565
   673
lemma map_of_inject_set:
haftmann@35565
   674
  assumes distinct: "distinct (map fst xs)" "distinct (map fst ys)"
haftmann@35565
   675
  shows "map_of xs = map_of ys \<longleftrightarrow> set xs = set ys" (is "?lhs \<longleftrightarrow> ?rhs")
haftmann@35565
   676
proof
haftmann@35565
   677
  assume ?lhs
haftmann@35565
   678
  moreover from `distinct (map fst xs)` have "set xs = {(k, v). map_of xs k = Some v}"
haftmann@35565
   679
    by (rule set_map_of_compr)
haftmann@35565
   680
  moreover from `distinct (map fst ys)` have "set ys = {(k, v). map_of ys k = Some v}"
haftmann@35565
   681
    by (rule set_map_of_compr)
haftmann@35565
   682
  ultimately show ?rhs by simp
haftmann@35565
   683
next
haftmann@35565
   684
  assume ?rhs show ?lhs proof
haftmann@35565
   685
    fix k
haftmann@35565
   686
    show "map_of xs k = map_of ys k" proof (cases "map_of xs k")
haftmann@35565
   687
      case None
haftmann@35565
   688
      moreover with `?rhs` have "map_of ys k = None"
haftmann@35565
   689
        by (simp add: map_of_eq_None_iff)
haftmann@35565
   690
      ultimately show ?thesis by simp
haftmann@35565
   691
    next
haftmann@35565
   692
      case (Some v)
haftmann@35565
   693
      moreover with distinct `?rhs` have "map_of ys k = Some v"
haftmann@35565
   694
        by simp
haftmann@35565
   695
      ultimately show ?thesis by simp
haftmann@35565
   696
    qed
haftmann@35565
   697
  qed
haftmann@35565
   698
qed
haftmann@35565
   699
nipkow@3981
   700
end
haftmann@34979
   701