(* 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"
"empty == %x. None"
definition
map_comp :: "('b ~=> 'c) => ('a ~=> 'b) => ('a ~=> 'c)" (infixl "o'_m" 55)
"f o_m g = (\<lambda>k. case g k of None \<Rightarrow> None | Some v \<Rightarrow> f v)"
const_syntax (xsymbols)
map_comp (infixl "\<circ>\<^sub>m" 55)
definition
map_add :: "('a ~=> 'b) => ('a ~=> 'b) => ('a ~=> 'b)" (infixl "++" 100)
"m1 ++ m2 = (\<lambda>x. case m2 x of None => m1 x | Some y => Some y)"
restrict_map :: "('a ~=> 'b) => 'a set => ('a ~=> 'b)" (infixl "|`" 110)
"m|`A = (\<lambda>x. if x : A then m x else None)"
const_syntax (latex output)
restrict_map ("_\<restriction>\<^bsub>_\<^esub>" [111,110] 110)
definition
dom :: "('a ~=> 'b) => 'a set"
"dom m = {a. m a ~= None}"
ran :: "('a ~=> 'b) => 'b set"
"ran m = {b. EX a. m a = Some b}"
map_le :: "('a ~=> 'b) => ('a ~=> 'b) => bool" (infix "\<subseteq>\<^sub>m" 50)
"(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: "m(xs [|->] ys) == m ++ map_of (rev(zip xs ys))"
(* special purpose constants that should be defined somewhere else and
whose syntax is a bit odd as well:
"@chg_map" :: "('a ~=> 'b) => 'a => ('b => 'b) => ('a ~=> 'b)"
("_/'(_/\<mapsto>\<lambda>_. _')" [900,0,0,0] 900)
"m(x\<mapsto>\<lambda>y. f)" == "chg_map (\<lambda>y. f) x m"
map_upd_s::"('a ~=> 'b) => 'a set => 'b =>
('a ~=> 'b)" ("_/'(_{|->}_/')" [900,0,0]900)
map_subst::"('a ~=> 'b) => 'b => 'b =>
('a ~=> 'b)" ("_/'(_~>_/')" [900,0,0]900)
map_upd_s_def: "m(as{|->}b) == %x. if x : as then Some b else m x"
map_subst_def: "m(a~>b) == %x. if m x = Some a then Some b else m x"
map_upd_s :: "('a ~=> 'b) => 'a set => 'b => ('a ~=> 'b)"
("_/'(_/{\<mapsto>}/_')" [900,0,0]900)
map_subst :: "('a ~=> 'b) => 'b => 'b =>
('a ~=> 'b)" ("_/'(_\<leadsto>_/')" [900,0,0]900)
subsection {* @{term [source] map_upd_s} *}
lemma map_upd_s_apply [simp]:
"(m(as{|->}b)) x = (if x : as then Some b else m x)"
by (simp add: map_upd_s_def)
lemma map_subst_apply [simp]:
"(m(a~>b)) x = (if m x = Some a then Some b else m x)"
by (simp add: map_subst_def)
*)
subsection {* @{term [source] empty} *}
lemma empty_upd_none [simp]: "empty(x := None) = empty"
by (rule ext) simp
(* FIXME: what is this sum_case nonsense?? *)
lemma sum_case_empty_empty[simp]: "sum_case empty empty = empty"
by (rule ext) (simp split: sum.split)
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
(* FIXME: what is this sum_case nonsense?? *)
subsection {* @{term [source] sum_case} and @{term [source] empty}/@{term [source] map_upd} *}
lemma sum_case_map_upd_empty [simp]:
"sum_case (m(k|->y)) empty = (sum_case m empty)(Inl k|->y)"
by (rule ext) (simp split: sum.split)
lemma sum_case_empty_map_upd [simp]:
"sum_case empty (m(k|->y)) = (sum_case empty m)(Inr k|->y)"
by (rule ext) (simp split: sum.split)
lemma sum_case_map_upd_map_upd [simp]:
"sum_case (m1(k1|->y1)) (m2(k2|->y2)) = (sum_case (m1(k1|->y1)) m2)(Inr k2|->y2)"
by (rule ext) (simp split: sum.split)
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)
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