(*  Title:      HOL/Map.thy

     Author:     Tobias Nipkow, based on a theory by David von Oheimb

     Copyright   1997-2003 TU Muenchen

 The datatype of `maps' (written ~=>); strongly resembles maps in VDM.

 *)

 header {* Maps *}

 theory Map

 imports List

 begin

 types ('a,'b) "~=>" = "'a => 'b option"  (infixr "~=>" 0)

 translations (type) "a ~=> b " <= (type) "a => b option"

 syntax (xsymbols)

   "~=>" :: "[type, type] => type"  (infixr "\<rightharpoonup>" 0)

 abbreviation

   empty :: "'a ~=> 'b" where

   "empty == %x. None"

 definition

   map_comp :: "('b ~=> 'c) => ('a ~=> 'b) => ('a ~=> 'c)"  (infixl "o'_m" 55) where

   "f o_m g = (\<lambda>k. case g k of None \<Rightarrow> None | Some v \<Rightarrow> f v)"

 notation (xsymbols)

   map_comp  (infixl "\<circ>\<^sub>m" 55)

 definition

   map_add :: "('a ~=> 'b) => ('a ~=> 'b) => ('a ~=> 'b)"  (infixl "++" 100) where

   "m1 ++ m2 = (\<lambda>x. case m2 x of None => m1 x | Some y => Some y)"

 definition

   restrict_map :: "('a ~=> 'b) => 'a set => ('a ~=> 'b)"  (infixl "|`"  110) where

   "m|`A = (\<lambda>x. if x : A then m x else None)"

 notation (latex output)

   restrict_map  ("_\<restriction>\<^bsub>_\<^esub>" [111,110] 110)

 definition

   dom :: "('a ~=> 'b) => 'a set" where

   "dom m = {a. m a ~= None}"

 definition

   ran :: "('a ~=> 'b) => 'b set" where

   "ran m = {b. EX a. m a = Some b}"

 definition

   map_le :: "('a ~=> 'b) => ('a ~=> 'b) => bool"  (infix "\<subseteq>\<^sub>m" 50) where

   "(m\<^isub>1 \<subseteq>\<^sub>m m\<^isub>2) = (\<forall>a \<in> dom m\<^isub>1. m\<^isub>1 a = m\<^isub>2 a)"

 consts

   map_of :: "('a * 'b) list => 'a ~=> 'b"

   map_upds :: "('a ~=> 'b) => 'a list => 'b list => ('a ~=> 'b)"

 nonterminals

   maplets maplet

 syntax

   "_maplet"  :: "['a, 'a] => maplet"             ("_ /|->/ _")

   "_maplets" :: "['a, 'a] => maplet"             ("_ /[|->]/ _")

   ""         :: "maplet => maplets"             ("_")

   "_Maplets" :: "[maplet, maplets] => maplets" ("_,/ _")

   "_MapUpd"  :: "['a ~=> 'b, maplets] => 'a ~=> 'b" ("_/'(_')" [900,0]900)

   "_Map"     :: "maplets => 'a ~=> 'b"            ("(1[_])")

 syntax (xsymbols)

   "_maplet"  :: "['a, 'a] => maplet"             ("_ /\<mapsto>/ _")

   "_maplets" :: "['a, 'a] => maplet"             ("_ /[\<mapsto>]/ _")

 translations

   "_MapUpd m (_Maplets xy ms)"  == "_MapUpd (_MapUpd m xy) ms"

   "_MapUpd m (_maplet  x y)"    == "m(x:=Some y)"

   "_MapUpd m (_maplets x y)"    == "map_upds m x y"

   "_Map ms"                     == "_MapUpd (CONST empty) ms"

   "_Map (_Maplets ms1 ms2)"     <= "_MapUpd (_Map ms1) ms2"

   "_Maplets ms1 (_Maplets ms2 ms3)" <= "_Maplets (_Maplets ms1 ms2) ms3"

 primrec

   "map_of [] = empty"

   "map_of (p#ps) = (map_of ps)(fst p |-> snd p)"

 declare map_of.simps [code del]

 lemma map_of_Cons_code [code]: 

   "map_of [] k = None"

   "map_of ((l, v) # ps) k = (if l = k then Some v else map_of ps k)"

   by simp_all

 defs

   map_upds_def [code]: "m(xs [|->] ys) == m ++ map_of (rev(zip xs ys))"

 subsection {* @{term [source] empty} *}

 lemma empty_upd_none [simp]: "empty(x := None) = empty"

 by (rule ext) simp

 subsection {* @{term [source] map_upd} *}

 lemma map_upd_triv: "t k = Some x ==> t(k|->x) = t"

 by (rule ext) simp

 lemma map_upd_nonempty [simp]: "t(k|->x) ~= empty"

 proof

   assume "t(k \<mapsto> x) = empty"

   then have "(t(k \<mapsto> x)) k = None" by simp

   then show False by simp

 qed

 lemma map_upd_eqD1:

   assumes "m(a\<mapsto>x) = n(a\<mapsto>y)"

   shows "x = y"

 proof -

   from prems have "(m(a\<mapsto>x)) a = (n(a\<mapsto>y)) a" by simp

   then show ?thesis by simp

 qed

 lemma map_upd_Some_unfold:

   "((m(a|->b)) x = Some y) = (x = a \<and> b = y \<or> x \<noteq> a \<and> m x = Some y)"

 by auto

 lemma image_map_upd [simp]: "x \<notin> A \<Longrightarrow> m(x \<mapsto> y) ` A = m ` A"

 by auto

 lemma finite_range_updI: "finite (range f) ==> finite (range (f(a|->b)))"

 unfolding image_def

 apply (simp (no_asm_use) add:full_SetCompr_eq)

 apply (rule finite_subset)

  prefer 2 apply assumption

 apply (auto)

 done

 subsection {* @{term [source] map_of} *}

 lemma map_of_eq_None_iff:

   "(map_of xys x = None) = (x \<notin> fst ` (set xys))"

 by (induct xys) simp_all

 lemma map_of_is_SomeD: "map_of xys x = Some y \<Longrightarrow> (x,y) \<in> set xys"

 apply (induct xys)

  apply simp

 apply (clarsimp split: if_splits)

 done

 lemma map_of_eq_Some_iff [simp]:

   "distinct(map fst xys) \<Longrightarrow> (map_of xys x = Some y) = ((x,y) \<in> set xys)"

 apply (induct xys)

  apply simp

 apply (auto simp: map_of_eq_None_iff [symmetric])

 done

 lemma Some_eq_map_of_iff [simp]:

   "distinct(map fst xys) \<Longrightarrow> (Some y = map_of xys x) = ((x,y) \<in> set xys)"

 by (auto simp del:map_of_eq_Some_iff simp add: map_of_eq_Some_iff [symmetric])

 lemma map_of_is_SomeI [simp]: "\<lbrakk> distinct(map fst xys); (x,y) \<in> set xys \<rbrakk>

     \<Longrightarrow> map_of xys x = Some y"

 apply (induct xys)

  apply simp

 apply force

 done

 lemma map_of_zip_is_None [simp]:

   "length xs = length ys \<Longrightarrow> (map_of (zip xs ys) x = None) = (x \<notin> set xs)"

 by (induct rule: list_induct2) simp_all

 lemma map_of_zip_is_Some:

   assumes "length xs = length ys"

   shows "x \<in> set xs \<longleftrightarrow> (\<exists>y. map_of (zip xs ys) x = Some y)"

 using assms by (induct rule: list_induct2) simp_all

 lemma map_of_zip_upd:

   fixes x :: 'a and xs :: "'a list" and ys zs :: "'b list"

   assumes "length ys = length xs"

     and "length zs = length xs"

     and "x \<notin> set xs"

     and "map_of (zip xs ys)(x \<mapsto> y) = map_of (zip xs zs)(x \<mapsto> z)"

   shows "map_of (zip xs ys) = map_of (zip xs zs)"

 proof

   fix x' :: 'a

   show "map_of (zip xs ys) x' = map_of (zip xs zs) x'"

   proof (cases "x = x'")

     case True

     from assms True map_of_zip_is_None [of xs ys x']

       have "map_of (zip xs ys) x' = None" by simp

     moreover from assms True map_of_zip_is_None [of xs zs x']

       have "map_of (zip xs zs) x' = None" by simp

     ultimately show ?thesis by simp

   next

     case False from assms

       have "(map_of (zip xs ys)(x \<mapsto> y)) x' = (map_of (zip xs zs)(x \<mapsto> z)) x'" by auto

     with False show ?thesis by simp

   qed

 qed

 lemma map_of_zip_inject:

   assumes "length ys = length xs"

     and "length zs = length xs"

     and dist: "distinct xs"

     and map_of: "map_of (zip xs ys) = map_of (zip xs zs)"

   shows "ys = zs"

 using assms(1) assms(2)[symmetric] using dist map_of proof (induct ys xs zs rule: list_induct3)

   case Nil show ?case by simp

 next

   case (Cons y ys x xs z zs)

   from `map_of (zip (x#xs) (y#ys)) = map_of (zip (x#xs) (z#zs))`

     have map_of: "map_of (zip xs ys)(x \<mapsto> y) = map_of (zip xs zs)(x \<mapsto> z)" by simp

   from Cons have "length ys = length xs" and "length zs = length xs"

     and "x \<notin> set xs" by simp_all

   then have "map_of (zip xs ys) = map_of (zip xs zs)" using map_of by (rule map_of_zip_upd)

   with Cons.hyps `distinct (x # xs)` have "ys = zs" by simp

   moreover from map_of have "y = z" by (rule map_upd_eqD1)

   ultimately show ?case by simp

 qed

 lemma map_of_zip_map:

   "map_of (zip xs (map f xs)) = (\<lambda>x. if x \<in> set xs then Some (f x) else None)"

   by (induct xs) (simp_all add: expand_fun_eq)

 lemma finite_range_map_of: "finite (range (map_of xys))"

 apply (induct xys)

  apply (simp_all add: image_constant)

 apply (rule finite_subset)

  prefer 2 apply assumption

 apply auto

 done

 lemma map_of_SomeD: "map_of xs k = Some y \<Longrightarrow> (k, y) \<in> set xs"

 by (induct xs) (simp, atomize (full), auto)

 lemma map_of_mapk_SomeI:

   "inj f ==> map_of t k = Some x ==>

    map_of (map (split (%k. Pair (f k))) t) (f k) = Some x"

 by (induct t) (auto simp add: inj_eq)

 lemma weak_map_of_SomeI: "(k, x) : set l ==> \<exists>x. map_of l k = Some x"

 by (induct l) auto

 lemma map_of_filter_in:

   "map_of xs k = Some z \<Longrightarrow> P k z \<Longrightarrow> map_of (filter (split P) xs) k = Some z"

 by (induct xs) auto

 lemma map_of_map: "map_of (map (%(a,b). (a,f b)) xs) x = Option.map f (map_of xs x)"

 by (induct xs) auto

 subsection {* @{const Option.map} related *}

 lemma option_map_o_empty [simp]: "Option.map f o empty = empty"

 by (rule ext) simp

 lemma option_map_o_map_upd [simp]:

   "Option.map f o m(a|->b) = (Option.map f o m)(a|->f b)"

 by (rule ext) simp

 subsection {* @{term [source] map_comp} related *}

 lemma map_comp_empty [simp]:

   "m \<circ>\<^sub>m empty = empty"

   "empty \<circ>\<^sub>m m = empty"

 by (auto simp add: map_comp_def intro: ext split: option.splits)

 lemma map_comp_simps [simp]:

   "m2 k = None \<Longrightarrow> (m1 \<circ>\<^sub>m m2) k = None"

   "m2 k = Some k' \<Longrightarrow> (m1 \<circ>\<^sub>m m2) k = m1 k'"

 by (auto simp add: map_comp_def)

 lemma map_comp_Some_iff:

   "((m1 \<circ>\<^sub>m m2) k = Some v) = (\<exists>k'. m2 k = Some k' \<and> m1 k' = Some v)"

 by (auto simp add: map_comp_def split: option.splits)

 lemma map_comp_None_iff:

   "((m1 \<circ>\<^sub>m m2) k = None) = (m2 k = None \<or> (\<exists>k'. m2 k = Some k' \<and> m1 k' = None)) "

 by (auto simp add: map_comp_def split: option.splits)

 subsection {* @{text "++"} *}

 lemma map_add_empty[simp]: "m ++ empty = m"

 by(simp add: map_add_def)

 lemma empty_map_add[simp]: "empty ++ m = m"

 by (rule ext) (simp add: map_add_def split: option.split)

 lemma map_add_assoc[simp]: "m1 ++ (m2 ++ m3) = (m1 ++ m2) ++ m3"

 by (rule ext) (simp add: map_add_def split: option.split)

 lemma map_add_Some_iff:

   "((m ++ n) k = Some x) = (n k = Some x | n k = None & m k = Some x)"

 by (simp add: map_add_def split: option.split)

 lemma map_add_SomeD [dest!]:

   "(m ++ n) k = Some x \<Longrightarrow> n k = Some x \<or> n k = None \<and> m k = Some x"

 by (rule map_add_Some_iff [THEN iffD1])

 lemma map_add_find_right [simp]: "!!xx. n k = Some xx ==> (m ++ n) k = Some xx"

 by (subst map_add_Some_iff) fast

 lemma map_add_None [iff]: "((m ++ n) k = None) = (n k = None & m k = None)"

 by (simp add: map_add_def split: option.split)

 lemma map_add_upd[simp]: "f ++ g(x|->y) = (f ++ g)(x|->y)"

 by (rule ext) (simp add: map_add_def)

 lemma map_add_upds[simp]: "m1 ++ (m2(xs[\<mapsto>]ys)) = (m1++m2)(xs[\<mapsto>]ys)"

 by (simp add: map_upds_def)

 lemma map_add_upd_left: "m\<notin>dom e2 \<Longrightarrow> e1(m \<mapsto> u1) ++ e2 = (e1 ++ e2)(m \<mapsto> u1)"

 by (rule ext) (auto simp: map_add_def dom_def split: option.split)

 lemma map_of_append[simp]: "map_of (xs @ ys) = map_of ys ++ map_of xs"

 unfolding map_add_def

 apply (induct xs)

  apply simp

 apply (rule ext)

 apply (simp split add: option.split)

 done

 lemma finite_range_map_of_map_add:

   "finite (range f) ==> finite (range (f ++ map_of l))"

 apply (induct l)

  apply (auto simp del: fun_upd_apply)

 apply (erule finite_range_updI)

 done

 lemma inj_on_map_add_dom [iff]:

   "inj_on (m ++ m') (dom m') = inj_on m' (dom m')"

 by (fastsimp simp: map_add_def dom_def inj_on_def split: option.splits)

 subsection {* @{term [source] restrict_map} *}

 lemma restrict_map_to_empty [simp]: "m|`{} = empty"

 by (simp add: restrict_map_def)

 lemma restrict_map_insert: "f |` (insert a A) = (f |` A)(a := f a)"

 by (auto simp add: restrict_map_def intro: ext)

 lemma restrict_map_empty [simp]: "empty|`D = empty"

 by (simp add: restrict_map_def)

 lemma restrict_in [simp]: "x \<in> A \<Longrightarrow> (m|`A) x = m x"

 by (simp add: restrict_map_def)

 lemma restrict_out [simp]: "x \<notin> A \<Longrightarrow> (m|`A) x = None"

 by (simp add: restrict_map_def)

 lemma ran_restrictD: "y \<in> ran (m|`A) \<Longrightarrow> \<exists>x\<in>A. m x = Some y"

 by (auto simp: restrict_map_def ran_def split: split_if_asm)

 lemma dom_restrict [simp]: "dom (m|`A) = dom m \<inter> A"

 by (auto simp: restrict_map_def dom_def split: split_if_asm)

 lemma restrict_upd_same [simp]: "m(x\<mapsto>y)|`(-{x}) = m|`(-{x})"

 by (rule ext) (auto simp: restrict_map_def)

 lemma restrict_restrict [simp]: "m|`A|`B = m|`(A\<inter>B)"

 by (rule ext) (auto simp: restrict_map_def)

 lemma restrict_fun_upd [simp]:

   "m(x := y)|`D = (if x \<in> D then (m|`(D-{x}))(x := y) else m|`D)"

 by (simp add: restrict_map_def expand_fun_eq)

 lemma fun_upd_None_restrict [simp]:

   "(m|`D)(x := None) = (if x:D then m|`(D - {x}) else m|`D)"

 by (simp add: restrict_map_def expand_fun_eq)

 lemma fun_upd_restrict: "(m|`D)(x := y) = (m|`(D-{x}))(x := y)"

 by (simp add: restrict_map_def expand_fun_eq)

 lemma fun_upd_restrict_conv [simp]:

   "x \<in> D \<Longrightarrow> (m|`D)(x := y) = (m|`(D-{x}))(x := y)"

 by (simp add: restrict_map_def expand_fun_eq)

 subsection {* @{term [source] map_upds} *}

 lemma map_upds_Nil1 [simp]: "m([] [|->] bs) = m"

 by (simp add: map_upds_def)

 lemma map_upds_Nil2 [simp]: "m(as [|->] []) = m"

 by (simp add:map_upds_def)

 lemma map_upds_Cons [simp]: "m(a#as [|->] b#bs) = (m(a|->b))(as[|->]bs)"

 by (simp add:map_upds_def)

 lemma map_upds_append1 [simp]: "\<And>ys m. size xs < size ys \<Longrightarrow>

   m(xs@[x] [\<mapsto>] ys) = m(xs [\<mapsto>] ys)(x \<mapsto> ys!size xs)"

 apply(induct xs)

  apply (clarsimp simp add: neq_Nil_conv)

 apply (case_tac ys)

  apply simp

 apply simp

 done

 lemma map_upds_list_update2_drop [simp]:

   "\<lbrakk>size xs \<le> i; i < size ys\<rbrakk>

     \<Longrightarrow> m(xs[\<mapsto>]ys[i:=y]) = m(xs[\<mapsto>]ys)"

 apply (induct xs arbitrary: m ys i)

  apply simp

 apply (case_tac ys)

  apply simp

 apply (simp split: nat.split)

 done

 lemma map_upd_upds_conv_if:

   "(f(x|->y))(xs [|->] ys) =

    (if x : set(take (length ys) xs) then f(xs [|->] ys)

                                     else (f(xs [|->] ys))(x|->y))"

 apply (induct xs arbitrary: x y ys f)

  apply simp

 apply (case_tac ys)

  apply (auto split: split_if simp: fun_upd_twist)

 done

 lemma map_upds_twist [simp]:

   "a ~: set as ==> m(a|->b)(as[|->]bs) = m(as[|->]bs)(a|->b)"

 using set_take_subset by (fastsimp simp add: map_upd_upds_conv_if)

 lemma map_upds_apply_nontin [simp]:

   "x ~: set xs ==> (f(xs[|->]ys)) x = f x"

 apply (induct xs arbitrary: ys)

  apply simp

 apply (case_tac ys)

  apply (auto simp: map_upd_upds_conv_if)

 done

 lemma fun_upds_append_drop [simp]:

   "size xs = size ys \<Longrightarrow> m(xs@zs[\<mapsto>]ys) = m(xs[\<mapsto>]ys)"

 apply (induct xs arbitrary: m ys)

  apply simp

 apply (case_tac ys)

  apply simp_all

 done

 lemma fun_upds_append2_drop [simp]:

   "size xs = size ys \<Longrightarrow> m(xs[\<mapsto>]ys@zs) = m(xs[\<mapsto>]ys)"

 apply (induct xs arbitrary: m ys)

  apply simp

 apply (case_tac ys)

  apply simp_all

 done

 lemma restrict_map_upds[simp]:

   "\<lbrakk> length xs = length ys; set xs \<subseteq> D \<rbrakk>

     \<Longrightarrow> m(xs [\<mapsto>] ys)|`D = (m|`(D - set xs))(xs [\<mapsto>] ys)"

 apply (induct xs arbitrary: m ys)

  apply simp

 apply (case_tac ys)

  apply simp

 apply (simp add: Diff_insert [symmetric] insert_absorb)

 apply (simp add: map_upd_upds_conv_if)

 done

 subsection {* @{term [source] dom} *}

 lemma dom_eq_empty_conv [simp]: "dom f = {} \<longleftrightarrow> f = empty"

 by(auto intro!:ext simp: dom_def)

 lemma domI: "m a = Some b ==> a : dom m"

 by(simp add:dom_def)

 (* declare domI [intro]? *)

 lemma domD: "a : dom m ==> \<exists>b. m a = Some b"

 by (cases "m a") (auto simp add: dom_def)

 lemma domIff [iff, simp del]: "(a : dom m) = (m a ~= None)"

 by(simp add:dom_def)

 lemma dom_empty [simp]: "dom empty = {}"

 by(simp add:dom_def)

 lemma dom_fun_upd [simp]:

   "dom(f(x := y)) = (if y=None then dom f - {x} else insert x (dom f))"

 by(auto simp add:dom_def)

 lemma dom_map_of: "dom(map_of xys) = {x. \<exists>y. (x,y) : set xys}"

 by (induct xys) (auto simp del: fun_upd_apply)

 lemma dom_map_of_conv_image_fst:

   "dom(map_of xys) = fst ` (set xys)"

 by(force simp: dom_map_of)

 lemma dom_map_of_zip [simp]: "[| length xs = length ys; distinct xs |] ==>

   dom(map_of(zip xs ys)) = set xs"

 by (induct rule: list_induct2) simp_all

 lemma finite_dom_map_of: "finite (dom (map_of l))"

 by (induct l) (auto simp add: dom_def insert_Collect [symmetric])

 lemma dom_map_upds [simp]:

   "dom(m(xs[|->]ys)) = set(take (length ys) xs) Un dom m"

 apply (induct xs arbitrary: m ys)

  apply simp

 apply (case_tac ys)

  apply auto

 done

 lemma dom_map_add [simp]: "dom(m++n) = dom n Un dom m"

 by(auto simp:dom_def)

 lemma dom_override_on [simp]:

   "dom(override_on f g A) =

     (dom f  - {a. a : A - dom g}) Un {a. a : A Int dom g}"

 by(auto simp: dom_def override_on_def)

 lemma map_add_comm: "dom m1 \<inter> dom m2 = {} \<Longrightarrow> m1++m2 = m2++m1"

 by (rule ext) (force simp: map_add_def dom_def split: option.split)

 lemma map_add_dom_app_simps:

   "\<lbrakk> m\<in>dom l2 \<rbrakk> \<Longrightarrow> (l1++l2) m = l2 m"

   "\<lbrakk> m\<notin>dom l1 \<rbrakk> \<Longrightarrow> (l1++l2) m = l2 m"

   "\<lbrakk> m\<notin>dom l2 \<rbrakk> \<Longrightarrow> (l1++l2) m = l1 m"

 by (auto simp add: map_add_def split: option.split_asm)

 lemma dom_const [simp]:

   "dom (\<lambda>x. Some y) = UNIV"

   by auto

 lemma dom_if:

   "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}"

   by (auto split: if_splits)

 (* Due to John Matthews - could be rephrased with dom *)

 lemma finite_map_freshness:

   "finite (dom (f :: 'a \<rightharpoonup> 'b)) \<Longrightarrow> \<not> finite (UNIV :: 'a set) \<Longrightarrow>

    \<exists>x. f x = None"

 by(bestsimp dest:ex_new_if_finite)

 lemma dom_minus:

   "f x = None \<Longrightarrow> dom f - insert x A = dom f - A"

   unfolding dom_def by simp

 lemma insert_dom:

   "f x = Some y \<Longrightarrow> insert x (dom f) = dom f"

   unfolding dom_def by auto

 subsection {* @{term [source] ran} *}

 lemma ranI: "m a = Some b ==> b : ran m"

 by(auto simp: ran_def)

 (* declare ranI [intro]? *)

 lemma ran_empty [simp]: "ran empty = {}"

 by(auto simp: ran_def)

 lemma ran_map_upd [simp]: "m a = None ==> ran(m(a|->b)) = insert b (ran m)"

 unfolding ran_def

 apply auto

 apply (subgoal_tac "aa ~= a")

  apply auto

 done

 subsection {* @{text "map_le"} *}

 lemma map_le_empty [simp]: "empty \<subseteq>\<^sub>m g"

 by (simp add: map_le_def)

 lemma upd_None_map_le [simp]: "f(x := None) \<subseteq>\<^sub>m f"

 by (force simp add: map_le_def)

 lemma map_le_upd[simp]: "f \<subseteq>\<^sub>m g ==> f(a := b) \<subseteq>\<^sub>m g(a := b)"

 by (fastsimp simp add: map_le_def)

 lemma map_le_imp_upd_le [simp]: "m1 \<subseteq>\<^sub>m m2 \<Longrightarrow> m1(x := None) \<subseteq>\<^sub>m m2(x \<mapsto> y)"

 by (force simp add: map_le_def)

 lemma map_le_upds [simp]:

   "f \<subseteq>\<^sub>m g ==> f(as [|->] bs) \<subseteq>\<^sub>m g(as [|->] bs)"

 apply (induct as arbitrary: f g bs)

  apply simp

 apply (case_tac bs)

  apply auto

 done

 lemma map_le_implies_dom_le: "(f \<subseteq>\<^sub>m g) \<Longrightarrow> (dom f \<subseteq> dom g)"

 by (fastsimp simp add: map_le_def dom_def)

 lemma map_le_refl [simp]: "f \<subseteq>\<^sub>m f"

 by (simp add: map_le_def)

 lemma map_le_trans[trans]: "\<lbrakk> m1 \<subseteq>\<^sub>m m2; m2 \<subseteq>\<^sub>m m3\<rbrakk> \<Longrightarrow> m1 \<subseteq>\<^sub>m m3"

 by (auto simp add: map_le_def dom_def)

 lemma map_le_antisym: "\<lbrakk> f \<subseteq>\<^sub>m g; g \<subseteq>\<^sub>m f \<rbrakk> \<Longrightarrow> f = g"

 unfolding map_le_def

 apply (rule ext)

 apply (case_tac "x \<in> dom f", simp)

 apply (case_tac "x \<in> dom g", simp, fastsimp)

 done

 lemma map_le_map_add [simp]: "f \<subseteq>\<^sub>m (g ++ f)"

 by (fastsimp simp add: map_le_def)

 lemma map_le_iff_map_add_commute: "(f \<subseteq>\<^sub>m f ++ g) = (f++g = g++f)"

 by(fastsimp simp: map_add_def map_le_def expand_fun_eq split: option.splits)

 lemma map_add_le_mapE: "f++g \<subseteq>\<^sub>m h \<Longrightarrow> g \<subseteq>\<^sub>m h"

 by (fastsimp simp add: map_le_def map_add_def dom_def)

 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"

 by (clarsimp simp add: map_le_def map_add_def dom_def split: option.splits)

 lemma dom_eq_singleton_conv: "dom f = {x} \<longleftrightarrow> (\<exists>v. f = [x \<mapsto> v])"

 proof(rule iffI)

   assume "\<exists>v. f = [x \<mapsto> v]"

   thus "dom f = {x}" by(auto split: split_if_asm)

 next

   assume "dom f = {x}"

   then obtain v where "f x = Some v" by auto

   hence "[x \<mapsto> v] \<subseteq>\<^sub>m f" by(auto simp add: map_le_def)

   moreover have "f \<subseteq>\<^sub>m [x \<mapsto> v]" using `dom f = {x}` `f x = Some v`

     by(auto simp add: map_le_def)

   ultimately have "f = [x \<mapsto> v]" by-(rule map_le_antisym)

   thus "\<exists>v. f = [x \<mapsto> v]" by blast

 qed

 end