src/HOL/Map.thy

add lemma of_nat_power

(*  Title:      HOL/Map.thy
    ID:         $Id$
    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)"

defs
  map_upds_def [code func]: "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 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:
  assumes "inj f"
  shows "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 f` 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:
  assumes 1: "map_of xs k = Some z"
    and 2: "P k z"
  shows "map_of (filter (split P) xs) k = Some z"
  using 1 by (induct xs) (insert 2, 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 {* @{term [source] 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"
  unfolding map_add_def by simp

lemma empty_map_add[simp]: "empty ++ m = m"
  unfolding map_add_def by (rule ext) (simp split: option.split)

lemma map_add_assoc[simp]: "m1 ++ (m2 ++ m3) = (m1 ++ m2) ++ m3"
  unfolding map_add_def 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)"
  unfolding map_add_def by (simp 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)"
  unfolding map_add_def by (simp split: option.split)

lemma map_add_upd[simp]: "f ++ g(x|->y) = (f ++ g)(x|->y)"
  unfolding map_add_def by (rule ext) simp

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

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')"
  unfolding map_add_def dom_def inj_on_def
  by (fastsimp 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_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 domI: "m a = Some b ==> a : dom m"
  unfolding dom_def by simp
(* 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)"
  unfolding dom_def by simp

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

lemma dom_fun_upd [simp]:
    "dom(f(x := y)) = (if y=None then dom f - {x} else insert x (dom f))"
  unfolding dom_def by auto

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)"
  unfolding dom_map_of by force

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))"
  unfolding dom_def
  by (induct l) (auto simp add: 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"
  unfolding dom_def by auto

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}"
  unfolding dom_def override_on_def by auto

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)

(* 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)

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

lemma ranI: "m a = Some b ==> b : ran m"
  unfolding ran_def by auto
(* declare ranI [intro]? *)

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

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 add: 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)

end