
nipkow@3981: (*  Title:      HOL/Map.thy
nipkow@3981:     Author:     Tobias Nipkow, based on a theory by David von Oheimb
webertj@13908:     Copyright   1997-2003 TU Muenchen
nipkow@3981: 
nipkow@3981: The datatype of `maps' (written ~=>); strongly resembles maps in VDM.
nipkow@3981: *)
nipkow@3981: 
nipkow@13914: header {* Maps *}
nipkow@13914: 
nipkow@15131: theory Map
nipkow@15140: imports List
nipkow@15131: begin
nipkow@3981: 
bulwahn@42163: type_synonym ('a,'b) "map" = "'a => 'b option" (infixr "~=>" 0)
nipkow@3981: 
wenzelm@35427: type_notation (xsymbols)
haftmann@35565:   "map" (infixr "\<rightharpoonup>" 0)
wenzelm@19656: 
nipkow@19378: abbreviation
wenzelm@21404:   empty :: "'a ~=> 'b" where
nipkow@19378:   "empty == %x. None"
nipkow@19378: 
wenzelm@19656: definition
haftmann@25670:   map_comp :: "('b ~=> 'c) => ('a ~=> 'b) => ('a ~=> 'c)"  (infixl "o'_m" 55) where
wenzelm@20800:   "f o_m g = (\<lambda>k. case g k of None \<Rightarrow> None | Some v \<Rightarrow> f v)"
nipkow@19378: 
wenzelm@21210: notation (xsymbols)
wenzelm@19656:   map_comp  (infixl "\<circ>\<^sub>m" 55)
wenzelm@19656: 
wenzelm@20800: definition
wenzelm@21404:   map_add :: "('a ~=> 'b) => ('a ~=> 'b) => ('a ~=> 'b)"  (infixl "++" 100) where
wenzelm@20800:   "m1 ++ m2 = (\<lambda>x. case m2 x of None => m1 x | Some y => Some y)"
wenzelm@20800: 
wenzelm@21404: definition
wenzelm@21404:   restrict_map :: "('a ~=> 'b) => 'a set => ('a ~=> 'b)"  (infixl "|`"  110) where
wenzelm@20800:   "m|`A = (\<lambda>x. if x : A then m x else None)"
nipkow@13910: 
wenzelm@21210: notation (latex output)
wenzelm@19656:   restrict_map  ("_\<restriction>\<^bsub>_\<^esub>" [111,110] 110)
wenzelm@19656: 
wenzelm@20800: definition
wenzelm@21404:   dom :: "('a ~=> 'b) => 'a set" where
wenzelm@20800:   "dom m = {a. m a ~= None}"
wenzelm@20800: 
wenzelm@21404: definition
wenzelm@21404:   ran :: "('a ~=> 'b) => 'b set" where
wenzelm@20800:   "ran m = {b. EX a. m a = Some b}"
wenzelm@20800: 
wenzelm@21404: definition
wenzelm@21404:   map_le :: "('a ~=> 'b) => ('a ~=> 'b) => bool"  (infix "\<subseteq>\<^sub>m" 50) where
wenzelm@53015:   "(m\<^sub>1 \<subseteq>\<^sub>m m\<^sub>2) = (\<forall>a \<in> dom m\<^sub>1. m\<^sub>1 a = m\<^sub>2 a)"
wenzelm@20800: 
wenzelm@41229: nonterminal maplets and maplet
nipkow@14180: 
oheimb@5300: syntax
nipkow@14180:   "_maplet"  :: "['a, 'a] => maplet"             ("_ /|->/ _")
nipkow@14180:   "_maplets" :: "['a, 'a] => maplet"             ("_ /[|->]/ _")
nipkow@14180:   ""         :: "maplet => maplets"             ("_")
nipkow@14180:   "_Maplets" :: "[maplet, maplets] => maplets" ("_,/ _")
nipkow@14180:   "_MapUpd"  :: "['a ~=> 'b, maplets] => 'a ~=> 'b" ("_/'(_')" [900,0]900)
nipkow@14180:   "_Map"     :: "maplets => 'a ~=> 'b"            ("(1[_])")
nipkow@3981: 
wenzelm@12114: syntax (xsymbols)
nipkow@14180:   "_maplet"  :: "['a, 'a] => maplet"             ("_ /\<mapsto>/ _")
nipkow@14180:   "_maplets" :: "['a, 'a] => maplet"             ("_ /[\<mapsto>]/ _")
nipkow@14180: 
oheimb@5300: translations
nipkow@14180:   "_MapUpd m (_Maplets xy ms)"  == "_MapUpd (_MapUpd m xy) ms"
wenzelm@35115:   "_MapUpd m (_maplet  x y)"    == "m(x := CONST Some y)"
wenzelm@19947:   "_Map ms"                     == "_MapUpd (CONST empty) ms"
nipkow@14180:   "_Map (_Maplets ms1 ms2)"     <= "_MapUpd (_Map ms1) ms2"
nipkow@14180:   "_Maplets ms1 (_Maplets ms2 ms3)" <= "_Maplets (_Maplets ms1 ms2) ms3"
nipkow@14180: 
berghofe@5183: primrec
haftmann@34941:   map_of :: "('a \<times> 'b) list \<Rightarrow> 'a \<rightharpoonup> 'b" where
haftmann@34941:     "map_of [] = empty"
haftmann@34941:   | "map_of (p # ps) = (map_of ps)(fst p \<mapsto> snd p)"
oheimb@5300: 
haftmann@34941: definition
haftmann@34941:   map_upds :: "('a \<rightharpoonup> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b list \<Rightarrow> 'a \<rightharpoonup> 'b" where
haftmann@34941:   "map_upds m xs ys = m ++ map_of (rev (zip xs ys))"
haftmann@34941: 
haftmann@34941: translations
haftmann@34941:   "_MapUpd m (_maplets x y)"    == "CONST map_upds m x y"
haftmann@25965: 
haftmann@25965: lemma map_of_Cons_code [code]: 
haftmann@25965:   "map_of [] k = None"
haftmann@25965:   "map_of ((l, v) # ps) k = (if l = k then Some v else map_of ps k)"
haftmann@25965:   by simp_all
haftmann@25965: 
wenzelm@20800: 
wenzelm@17399: subsection {* @{term [source] empty} *}
webertj@13908: 
wenzelm@20800: lemma empty_upd_none [simp]: "empty(x := None) = empty"
nipkow@24331: by (rule ext) simp
webertj@13908: 
webertj@13908: 
wenzelm@17399: subsection {* @{term [source] map_upd} *}
webertj@13908: 
webertj@13908: lemma map_upd_triv: "t k = Some x ==> t(k|->x) = t"
nipkow@24331: by (rule ext) simp
webertj@13908: 
wenzelm@20800: lemma map_upd_nonempty [simp]: "t(k|->x) ~= empty"
wenzelm@20800: proof
wenzelm@20800:   assume "t(k \<mapsto> x) = empty"
wenzelm@20800:   then have "(t(k \<mapsto> x)) k = None" by simp
wenzelm@20800:   then show False by simp
wenzelm@20800: qed
webertj@13908: 
wenzelm@20800: lemma map_upd_eqD1:
wenzelm@20800:   assumes "m(a\<mapsto>x) = n(a\<mapsto>y)"
wenzelm@20800:   shows "x = y"
wenzelm@20800: proof -
wenzelm@41550:   from assms have "(m(a\<mapsto>x)) a = (n(a\<mapsto>y)) a" by simp
wenzelm@20800:   then show ?thesis by simp
wenzelm@20800: qed
oheimb@14100: 
wenzelm@20800: lemma map_upd_Some_unfold:
nipkow@24331:   "((m(a|->b)) x = Some y) = (x = a \<and> b = y \<or> x \<noteq> a \<and> m x = Some y)"
nipkow@24331: by auto
oheimb@14100: 
wenzelm@20800: lemma image_map_upd [simp]: "x \<notin> A \<Longrightarrow> m(x \<mapsto> y) ` A = m ` A"
nipkow@24331: by auto
nipkow@15303: 
webertj@13908: lemma finite_range_updI: "finite (range f) ==> finite (range (f(a|->b)))"
nipkow@24331: unfolding image_def
nipkow@24331: apply (simp (no_asm_use) add:full_SetCompr_eq)
nipkow@24331: apply (rule finite_subset)
nipkow@24331:  prefer 2 apply assumption
nipkow@24331: apply (auto)
nipkow@24331: done
webertj@13908: 
webertj@13908: 
wenzelm@17399: subsection {* @{term [source] map_of} *}
webertj@13908: 
nipkow@15304: lemma map_of_eq_None_iff:
nipkow@24331:   "(map_of xys x = None) = (x \<notin> fst ` (set xys))"
nipkow@24331: by (induct xys) simp_all
nipkow@15304: 
nipkow@24331: lemma map_of_is_SomeD: "map_of xys x = Some y \<Longrightarrow> (x,y) \<in> set xys"
nipkow@24331: apply (induct xys)
nipkow@24331:  apply simp
nipkow@24331: apply (clarsimp split: if_splits)
nipkow@24331: done
nipkow@15304: 
wenzelm@20800: lemma map_of_eq_Some_iff [simp]:
nipkow@24331:   "distinct(map fst xys) \<Longrightarrow> (map_of xys x = Some y) = ((x,y) \<in> set xys)"
nipkow@24331: apply (induct xys)
nipkow@24331:  apply simp
nipkow@24331: apply (auto simp: map_of_eq_None_iff [symmetric])
nipkow@24331: done
nipkow@15304: 
wenzelm@20800: lemma Some_eq_map_of_iff [simp]:
nipkow@24331:   "distinct(map fst xys) \<Longrightarrow> (Some y = map_of xys x) = ((x,y) \<in> set xys)"
nipkow@24331: by (auto simp del:map_of_eq_Some_iff simp add: map_of_eq_Some_iff [symmetric])
nipkow@15304: 
paulson@17724: lemma map_of_is_SomeI [simp]: "\<lbrakk> distinct(map fst xys); (x,y) \<in> set xys \<rbrakk>
wenzelm@20800:     \<Longrightarrow> map_of xys x = Some y"
nipkow@24331: apply (induct xys)
nipkow@24331:  apply simp
nipkow@24331: apply force
nipkow@24331: done
nipkow@15304: 
wenzelm@20800: lemma map_of_zip_is_None [simp]:
nipkow@24331:   "length xs = length ys \<Longrightarrow> (map_of (zip xs ys) x = None) = (x \<notin> set xs)"
nipkow@24331: by (induct rule: list_induct2) simp_all
nipkow@15110: 
haftmann@26443: lemma map_of_zip_is_Some:
haftmann@26443:   assumes "length xs = length ys"
haftmann@26443:   shows "x \<in> set xs \<longleftrightarrow> (\<exists>y. map_of (zip xs ys) x = Some y)"
haftmann@26443: using assms by (induct rule: list_induct2) simp_all
haftmann@26443: 
haftmann@26443: lemma map_of_zip_upd:
haftmann@26443:   fixes x :: 'a and xs :: "'a list" and ys zs :: "'b list"
haftmann@26443:   assumes "length ys = length xs"
haftmann@26443:     and "length zs = length xs"
haftmann@26443:     and "x \<notin> set xs"
haftmann@26443:     and "map_of (zip xs ys)(x \<mapsto> y) = map_of (zip xs zs)(x \<mapsto> z)"
haftmann@26443:   shows "map_of (zip xs ys) = map_of (zip xs zs)"
haftmann@26443: proof
haftmann@26443:   fix x' :: 'a
haftmann@26443:   show "map_of (zip xs ys) x' = map_of (zip xs zs) x'"
haftmann@26443:   proof (cases "x = x'")
haftmann@26443:     case True
haftmann@26443:     from assms True map_of_zip_is_None [of xs ys x']
haftmann@26443:       have "map_of (zip xs ys) x' = None" by simp
haftmann@26443:     moreover from assms True map_of_zip_is_None [of xs zs x']
haftmann@26443:       have "map_of (zip xs zs) x' = None" by simp
haftmann@26443:     ultimately show ?thesis by simp
haftmann@26443:   next
haftmann@26443:     case False from assms
haftmann@26443:       have "(map_of (zip xs ys)(x \<mapsto> y)) x' = (map_of (zip xs zs)(x \<mapsto> z)) x'" by auto
haftmann@26443:     with False show ?thesis by simp
haftmann@26443:   qed
haftmann@26443: qed
haftmann@26443: 
haftmann@26443: lemma map_of_zip_inject:
haftmann@26443:   assumes "length ys = length xs"
haftmann@26443:     and "length zs = length xs"
haftmann@26443:     and dist: "distinct xs"
haftmann@26443:     and map_of: "map_of (zip xs ys) = map_of (zip xs zs)"
haftmann@26443:   shows "ys = zs"
haftmann@26443: using assms(1) assms(2)[symmetric] using dist map_of proof (induct ys xs zs rule: list_induct3)
haftmann@26443:   case Nil show ?case by simp
haftmann@26443: next
haftmann@26443:   case (Cons y ys x xs z zs)
haftmann@26443:   from `map_of (zip (x#xs) (y#ys)) = map_of (zip (x#xs) (z#zs))`
haftmann@26443:     have map_of: "map_of (zip xs ys)(x \<mapsto> y) = map_of (zip xs zs)(x \<mapsto> z)" by simp
haftmann@26443:   from Cons have "length ys = length xs" and "length zs = length xs"
haftmann@26443:     and "x \<notin> set xs" by simp_all
haftmann@26443:   then have "map_of (zip xs ys) = map_of (zip xs zs)" using map_of by (rule map_of_zip_upd)
haftmann@26443:   with Cons.hyps `distinct (x # xs)` have "ys = zs" by simp
haftmann@26443:   moreover from map_of have "y = z" by (rule map_upd_eqD1)
haftmann@26443:   ultimately show ?case by simp
haftmann@26443: qed
haftmann@26443: 
haftmann@33635: lemma map_of_zip_map:
haftmann@33635:   "map_of (zip xs (map f xs)) = (\<lambda>x. if x \<in> set xs then Some (f x) else None)"
nipkow@39302:   by (induct xs) (simp_all add: fun_eq_iff)
haftmann@33635: 
nipkow@15110: lemma finite_range_map_of: "finite (range (map_of xys))"
nipkow@24331: apply (induct xys)
nipkow@24331:  apply (simp_all add: image_constant)
nipkow@24331: apply (rule finite_subset)
nipkow@24331:  prefer 2 apply assumption
nipkow@24331: apply auto
nipkow@24331: done
nipkow@15110: 
wenzelm@20800: lemma map_of_SomeD: "map_of xs k = Some y \<Longrightarrow> (k, y) \<in> set xs"
nipkow@24331: by (induct xs) (simp, atomize (full), auto)
webertj@13908: 
wenzelm@20800: lemma map_of_mapk_SomeI:
nipkow@24331:   "inj f ==> map_of t k = Some x ==>
nipkow@24331:    map_of (map (split (%k. Pair (f k))) t) (f k) = Some x"
nipkow@24331: by (induct t) (auto simp add: inj_eq)
webertj@13908: 
wenzelm@20800: lemma weak_map_of_SomeI: "(k, x) : set l ==> \<exists>x. map_of l k = Some x"
nipkow@24331: by (induct l) auto
webertj@13908: 
wenzelm@20800: lemma map_of_filter_in:
nipkow@24331:   "map_of xs k = Some z \<Longrightarrow> P k z \<Longrightarrow> map_of (filter (split P) xs) k = Some z"
nipkow@24331: by (induct xs) auto
webertj@13908: 
haftmann@35607: lemma map_of_map:
blanchet@55466:   "map_of (map (\<lambda>(k, v). (k, f v)) xs) = map_option f \<circ> map_of xs"
nipkow@39302:   by (induct xs) (auto simp add: fun_eq_iff)
haftmann@35607: 
blanchet@55466: lemma dom_map_option:
blanchet@55466:   "dom (\<lambda>k. map_option (f k) (m k)) = dom m"
haftmann@35607:   by (simp add: dom_def)
webertj@13908: 
haftmann@56545: lemma dom_map_option_comp [simp]:
haftmann@56545:   "dom (map_option g \<circ> m) = dom m"
haftmann@56545:   using dom_map_option [of "\<lambda>_. g" m] by (simp add: comp_def)
haftmann@56545: 
webertj@13908: 
blanchet@55466: subsection {* @{const map_option} related *}
webertj@13908: 
blanchet@55466: lemma map_option_o_empty [simp]: "map_option f o empty = empty"
nipkow@24331: by (rule ext) simp
webertj@13908: 
blanchet@55466: lemma map_option_o_map_upd [simp]:
blanchet@55466:   "map_option f o m(a|->b) = (map_option f o m)(a|->f b)"
nipkow@24331: by (rule ext) simp
wenzelm@20800: 
webertj@13908: 
wenzelm@17399: subsection {* @{term [source] map_comp} related *}
schirmer@17391: 
wenzelm@20800: lemma map_comp_empty [simp]:
nipkow@24331:   "m \<circ>\<^sub>m empty = empty"
nipkow@24331:   "empty \<circ>\<^sub>m m = empty"
huffman@44921: by (auto simp add: map_comp_def split: option.splits)
schirmer@17391: 
wenzelm@20800: lemma map_comp_simps [simp]:
nipkow@24331:   "m2 k = None \<Longrightarrow> (m1 \<circ>\<^sub>m m2) k = None"
nipkow@24331:   "m2 k = Some k' \<Longrightarrow> (m1 \<circ>\<^sub>m m2) k = m1 k'"
nipkow@24331: by (auto simp add: map_comp_def)
schirmer@17391: 
schirmer@17391: lemma map_comp_Some_iff:
nipkow@24331:   "((m1 \<circ>\<^sub>m m2) k = Some v) = (\<exists>k'. m2 k = Some k' \<and> m1 k' = Some v)"
nipkow@24331: by (auto simp add: map_comp_def split: option.splits)
schirmer@17391: 
schirmer@17391: lemma map_comp_None_iff:
nipkow@24331:   "((m1 \<circ>\<^sub>m m2) k = None) = (m2 k = None \<or> (\<exists>k'. m2 k = Some k' \<and> m1 k' = None)) "
nipkow@24331: by (auto simp add: map_comp_def split: option.splits)
webertj@13908: 
wenzelm@20800: 
oheimb@14100: subsection {* @{text "++"} *}
webertj@13908: 
nipkow@14025: lemma map_add_empty[simp]: "m ++ empty = m"
nipkow@24331: by(simp add: map_add_def)
webertj@13908: 
nipkow@14025: lemma empty_map_add[simp]: "empty ++ m = m"
nipkow@24331: by (rule ext) (simp add: map_add_def split: option.split)
webertj@13908: 
nipkow@14025: lemma map_add_assoc[simp]: "m1 ++ (m2 ++ m3) = (m1 ++ m2) ++ m3"
nipkow@24331: by (rule ext) (simp add: map_add_def split: option.split)
wenzelm@20800: 
wenzelm@20800: lemma map_add_Some_iff:
nipkow@24331:   "((m ++ n) k = Some x) = (n k = Some x | n k = None & m k = Some x)"
nipkow@24331: by (simp add: map_add_def split: option.split)
nipkow@14025: 
wenzelm@20800: lemma map_add_SomeD [dest!]:
nipkow@24331:   "(m ++ n) k = Some x \<Longrightarrow> n k = Some x \<or> n k = None \<and> m k = Some x"
nipkow@24331: by (rule map_add_Some_iff [THEN iffD1])
webertj@13908: 
wenzelm@20800: lemma map_add_find_right [simp]: "!!xx. n k = Some xx ==> (m ++ n) k = Some xx"
nipkow@24331: by (subst map_add_Some_iff) fast
webertj@13908: 
nipkow@14025: lemma map_add_None [iff]: "((m ++ n) k = None) = (n k = None & m k = None)"
nipkow@24331: by (simp add: map_add_def split: option.split)
webertj@13908: 
nipkow@14025: lemma map_add_upd[simp]: "f ++ g(x|->y) = (f ++ g)(x|->y)"
nipkow@24331: by (rule ext) (simp add: map_add_def)
webertj@13908: 
nipkow@14186: lemma map_add_upds[simp]: "m1 ++ (m2(xs[\<mapsto>]ys)) = (m1++m2)(xs[\<mapsto>]ys)"
nipkow@24331: by (simp add: map_upds_def)
nipkow@14186: 
krauss@32236: lemma map_add_upd_left: "m\<notin>dom e2 \<Longrightarrow> e1(m \<mapsto> u1) ++ e2 = (e1 ++ e2)(m \<mapsto> u1)"
krauss@32236: by (rule ext) (auto simp: map_add_def dom_def split: option.split)
krauss@32236: 
wenzelm@20800: lemma map_of_append[simp]: "map_of (xs @ ys) = map_of ys ++ map_of xs"
nipkow@24331: unfolding map_add_def
nipkow@24331: apply (induct xs)
nipkow@24331:  apply simp
nipkow@24331: apply (rule ext)
nipkow@24331: apply (simp split add: option.split)
nipkow@24331: done
webertj@13908: 
nipkow@14025: lemma finite_range_map_of_map_add:
wenzelm@20800:   "finite (range f) ==> finite (range (f ++ map_of l))"
nipkow@24331: apply (induct l)
nipkow@24331:  apply (auto simp del: fun_upd_apply)
nipkow@24331: apply (erule finite_range_updI)
nipkow@24331: done
webertj@13908: 
wenzelm@20800: lemma inj_on_map_add_dom [iff]:
nipkow@24331:   "inj_on (m ++ m') (dom m') = inj_on m' (dom m')"
nipkow@44890: by (fastforce simp: map_add_def dom_def inj_on_def split: option.splits)
wenzelm@20800: 
haftmann@34979: lemma map_upds_fold_map_upd:
haftmann@35552:   "m(ks[\<mapsto>]vs) = foldl (\<lambda>m (k, v). m(k \<mapsto> v)) m (zip ks vs)"
haftmann@34979: unfolding map_upds_def proof (rule sym, rule zip_obtain_same_length)
haftmann@34979:   fix ks :: "'a list" and vs :: "'b list"
haftmann@34979:   assume "length ks = length vs"
haftmann@35552:   then show "foldl (\<lambda>m (k, v). m(k\<mapsto>v)) m (zip ks vs) = m ++ map_of (rev (zip ks vs))"
haftmann@35552:     by(induct arbitrary: m rule: list_induct2) simp_all
haftmann@34979: qed
haftmann@34979: 
haftmann@34979: lemma map_add_map_of_foldr:
haftmann@34979:   "m ++ map_of ps = foldr (\<lambda>(k, v) m. m(k \<mapsto> v)) ps m"
nipkow@39302:   by (induct ps) (auto simp add: fun_eq_iff map_add_def)
haftmann@34979: 
nipkow@15304: 
wenzelm@17399: subsection {* @{term [source] restrict_map} *}
oheimb@14100: 
wenzelm@20800: lemma restrict_map_to_empty [simp]: "m|`{} = empty"
nipkow@24331: by (simp add: restrict_map_def)
nipkow@14186: 
haftmann@31380: lemma restrict_map_insert: "f |` (insert a A) = (f |` A)(a := f a)"
huffman@44921: by (auto simp add: restrict_map_def)
haftmann@31380: 
wenzelm@20800: lemma restrict_map_empty [simp]: "empty|`D = empty"
nipkow@24331: by (simp add: restrict_map_def)
nipkow@14186: 
nipkow@15693: lemma restrict_in [simp]: "x \<in> A \<Longrightarrow> (m|`A) x = m x"
nipkow@24331: by (simp add: restrict_map_def)
oheimb@14100: 
nipkow@15693: lemma restrict_out [simp]: "x \<notin> A \<Longrightarrow> (m|`A) x = None"
nipkow@24331: by (simp add: restrict_map_def)
oheimb@14100: 
nipkow@15693: lemma ran_restrictD: "y \<in> ran (m|`A) \<Longrightarrow> \<exists>x\<in>A. m x = Some y"
nipkow@24331: by (auto simp: restrict_map_def ran_def split: split_if_asm)
oheimb@14100: 
nipkow@15693: lemma dom_restrict [simp]: "dom (m|`A) = dom m \<inter> A"
nipkow@24331: by (auto simp: restrict_map_def dom_def split: split_if_asm)
oheimb@14100: 
nipkow@15693: lemma restrict_upd_same [simp]: "m(x\<mapsto>y)|`(-{x}) = m|`(-{x})"
nipkow@24331: by (rule ext) (auto simp: restrict_map_def)
oheimb@14100: 
nipkow@15693: lemma restrict_restrict [simp]: "m|`A|`B = m|`(A\<inter>B)"
nipkow@24331: by (rule ext) (auto simp: restrict_map_def)
oheimb@14100: 
wenzelm@20800: lemma restrict_fun_upd [simp]:
nipkow@24331:   "m(x := y)|`D = (if x \<in> D then (m|`(D-{x}))(x := y) else m|`D)"
nipkow@39302: by (simp add: restrict_map_def fun_eq_iff)
nipkow@14186: 
wenzelm@20800: lemma fun_upd_None_restrict [simp]:
nipkow@24331:   "(m|`D)(x := None) = (if x:D then m|`(D - {x}) else m|`D)"
nipkow@39302: by (simp add: restrict_map_def fun_eq_iff)
nipkow@14186: 
wenzelm@20800: lemma fun_upd_restrict: "(m|`D)(x := y) = (m|`(D-{x}))(x := y)"
nipkow@39302: by (simp add: restrict_map_def fun_eq_iff)
nipkow@14186: 
wenzelm@20800: lemma fun_upd_restrict_conv [simp]:
nipkow@24331:   "x \<in> D \<Longrightarrow> (m|`D)(x := y) = (m|`(D-{x}))(x := y)"
nipkow@39302: by (simp add: restrict_map_def fun_eq_iff)
nipkow@14186: 
haftmann@35159: lemma map_of_map_restrict:
haftmann@35159:   "map_of (map (\<lambda>k. (k, f k)) ks) = (Some \<circ> f) |` set ks"
nipkow@39302:   by (induct ks) (simp_all add: fun_eq_iff restrict_map_insert)
haftmann@35159: 
haftmann@35619: lemma restrict_complement_singleton_eq:
haftmann@35619:   "f |` (- {x}) = f(x := None)"
nipkow@39302:   by (simp add: restrict_map_def fun_eq_iff)
haftmann@35619: 
oheimb@14100: 
wenzelm@17399: subsection {* @{term [source] map_upds} *}
nipkow@14025: 
wenzelm@20800: lemma map_upds_Nil1 [simp]: "m([] [|->] bs) = m"
nipkow@24331: by (simp add: map_upds_def)
nipkow@14025: 
wenzelm@20800: lemma map_upds_Nil2 [simp]: "m(as [|->] []) = m"
nipkow@24331: by (simp add:map_upds_def)
wenzelm@20800: 
wenzelm@20800: lemma map_upds_Cons [simp]: "m(a#as [|->] b#bs) = (m(a|->b))(as[|->]bs)"
nipkow@24331: by (simp add:map_upds_def)
nipkow@14025: 
wenzelm@20800: lemma map_upds_append1 [simp]: "\<And>ys m. size xs < size ys \<Longrightarrow>
nipkow@24331:   m(xs@[x] [\<mapsto>] ys) = m(xs [\<mapsto>] ys)(x \<mapsto> ys!size xs)"
nipkow@24331: apply(induct xs)
nipkow@24331:  apply (clarsimp simp add: neq_Nil_conv)
nipkow@24331: apply (case_tac ys)
nipkow@24331:  apply simp
nipkow@24331: apply simp
nipkow@24331: done
nipkow@14187: 
wenzelm@20800: lemma map_upds_list_update2_drop [simp]:
bulwahn@46588:   "size xs \<le> i \<Longrightarrow> m(xs[\<mapsto>]ys[i:=y]) = m(xs[\<mapsto>]ys)"
nipkow@24331: apply (induct xs arbitrary: m ys i)
nipkow@24331:  apply simp
nipkow@24331: apply (case_tac ys)
nipkow@24331:  apply simp
nipkow@24331: apply (simp split: nat.split)
nipkow@24331: done
nipkow@14025: 
wenzelm@20800: lemma map_upd_upds_conv_if:
wenzelm@20800:   "(f(x|->y))(xs [|->] ys) =
wenzelm@20800:    (if x : set(take (length ys) xs) then f(xs [|->] ys)
wenzelm@20800:                                     else (f(xs [|->] ys))(x|->y))"
nipkow@24331: apply (induct xs arbitrary: x y ys f)
nipkow@24331:  apply simp
nipkow@24331: apply (case_tac ys)
nipkow@24331:  apply (auto split: split_if simp: fun_upd_twist)
nipkow@24331: done
nipkow@14025: 
nipkow@14025: lemma map_upds_twist [simp]:
nipkow@24331:   "a ~: set as ==> m(a|->b)(as[|->]bs) = m(as[|->]bs)(a|->b)"
nipkow@44890: using set_take_subset by (fastforce simp add: map_upd_upds_conv_if)
nipkow@14025: 
wenzelm@20800: lemma map_upds_apply_nontin [simp]:
nipkow@24331:   "x ~: set xs ==> (f(xs[|->]ys)) x = f x"
nipkow@24331: apply (induct xs arbitrary: ys)
nipkow@24331:  apply simp
nipkow@24331: apply (case_tac ys)
nipkow@24331:  apply (auto simp: map_upd_upds_conv_if)
nipkow@24331: done
nipkow@14025: 
wenzelm@20800: lemma fun_upds_append_drop [simp]:
nipkow@24331:   "size xs = size ys \<Longrightarrow> m(xs@zs[\<mapsto>]ys) = m(xs[\<mapsto>]ys)"
nipkow@24331: apply (induct xs arbitrary: m ys)
nipkow@24331:  apply simp
nipkow@24331: apply (case_tac ys)
nipkow@24331:  apply simp_all
nipkow@24331: done
nipkow@14300: 
wenzelm@20800: lemma fun_upds_append2_drop [simp]:
nipkow@24331:   "size xs = size ys \<Longrightarrow> m(xs[\<mapsto>]ys@zs) = m(xs[\<mapsto>]ys)"
nipkow@24331: apply (induct xs arbitrary: m ys)
nipkow@24331:  apply simp
nipkow@24331: apply (case_tac ys)
nipkow@24331:  apply simp_all
nipkow@24331: done
nipkow@14300: 
nipkow@14300: 
wenzelm@20800: lemma restrict_map_upds[simp]:
wenzelm@20800:   "\<lbrakk> length xs = length ys; set xs \<subseteq> D \<rbrakk>
wenzelm@20800:     \<Longrightarrow> m(xs [\<mapsto>] ys)|`D = (m|`(D - set xs))(xs [\<mapsto>] ys)"
nipkow@24331: apply (induct xs arbitrary: m ys)
nipkow@24331:  apply simp
nipkow@24331: apply (case_tac ys)
nipkow@24331:  apply simp
nipkow@24331: apply (simp add: Diff_insert [symmetric] insert_absorb)
nipkow@24331: apply (simp add: map_upd_upds_conv_if)
nipkow@24331: done
nipkow@14186: 
nipkow@14186: 
wenzelm@17399: subsection {* @{term [source] dom} *}
webertj@13908: 
nipkow@31080: lemma dom_eq_empty_conv [simp]: "dom f = {} \<longleftrightarrow> f = empty"
huffman@44921:   by (auto simp: dom_def)
nipkow@31080: 
webertj@13908: lemma domI: "m a = Some b ==> a : dom m"
nipkow@24331: by(simp add:dom_def)
oheimb@14100: (* declare domI [intro]? *)
webertj@13908: 
paulson@15369: lemma domD: "a : dom m ==> \<exists>b. m a = Some b"
nipkow@24331: by (cases "m a") (auto simp add: dom_def)
webertj@13908: 
wenzelm@20800: lemma domIff [iff, simp del]: "(a : dom m) = (m a ~= None)"
nipkow@24331: by(simp add:dom_def)
webertj@13908: 
wenzelm@20800: lemma dom_empty [simp]: "dom empty = {}"
nipkow@24331: by(simp add:dom_def)
webertj@13908: 
wenzelm@20800: lemma dom_fun_upd [simp]:
nipkow@24331:   "dom(f(x := y)) = (if y=None then dom f - {x} else insert x (dom f))"
nipkow@24331: by(auto simp add:dom_def)
webertj@13908: 
haftmann@34979: lemma dom_if:
haftmann@34979:   "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:   by (auto split: if_splits)
nipkow@13937: 
nipkow@15304: lemma dom_map_of_conv_image_fst:
haftmann@34979:   "dom (map_of xys) = fst ` set xys"
haftmann@34979:   by (induct xys) (auto simp add: dom_if)
nipkow@15304: 
bulwahn@46588: lemma dom_map_of_zip [simp]: "length xs = length ys ==> dom (map_of (zip xs ys)) = set xs"
bulwahn@46588: by (induct rule: list_induct2) (auto simp add: dom_if)
nipkow@15110: 
webertj@13908: lemma finite_dom_map_of: "finite (dom (map_of l))"
nipkow@24331: by (induct l) (auto simp add: dom_def insert_Collect [symmetric])
webertj@13908: 
wenzelm@20800: lemma dom_map_upds [simp]:
nipkow@24331:   "dom(m(xs[|->]ys)) = set(take (length ys) xs) Un dom m"
nipkow@24331: apply (induct xs arbitrary: m ys)
nipkow@24331:  apply simp
nipkow@24331: apply (case_tac ys)
nipkow@24331:  apply auto
nipkow@24331: done
nipkow@13910: 
wenzelm@20800: lemma dom_map_add [simp]: "dom(m++n) = dom n Un dom m"
nipkow@24331: by(auto simp:dom_def)
nipkow@13910: 
wenzelm@20800: lemma dom_override_on [simp]:
wenzelm@20800:   "dom(override_on f g A) =
wenzelm@20800:     (dom f  - {a. a : A - dom g}) Un {a. a : A Int dom g}"
nipkow@24331: by(auto simp: dom_def override_on_def)
webertj@13908: 
nipkow@14027: lemma map_add_comm: "dom m1 \<inter> dom m2 = {} \<Longrightarrow> m1++m2 = m2++m1"
nipkow@24331: by (rule ext) (force simp: map_add_def dom_def split: option.split)
wenzelm@20800: 
krauss@32236: lemma map_add_dom_app_simps:
krauss@32236:   "\<lbrakk> m\<in>dom l2 \<rbrakk> \<Longrightarrow> (l1++l2) m = l2 m"
krauss@32236:   "\<lbrakk> m\<notin>dom l1 \<rbrakk> \<Longrightarrow> (l1++l2) m = l2 m"
krauss@32236:   "\<lbrakk> m\<notin>dom l2 \<rbrakk> \<Longrightarrow> (l1++l2) m = l1 m"
krauss@32236: by (auto simp add: map_add_def split: option.split_asm)
krauss@32236: 
haftmann@29622: lemma dom_const [simp]:
haftmann@35159:   "dom (\<lambda>x. Some (f x)) = UNIV"
haftmann@29622:   by auto
haftmann@29622: 
nipkow@22230: (* Due to John Matthews - could be rephrased with dom *)
nipkow@22230: lemma finite_map_freshness:
nipkow@22230:   "finite (dom (f :: 'a \<rightharpoonup> 'b)) \<Longrightarrow> \<not> finite (UNIV :: 'a set) \<Longrightarrow>
nipkow@22230:    \<exists>x. f x = None"
nipkow@22230: by(bestsimp dest:ex_new_if_finite)
nipkow@14027: 
haftmann@28790: lemma dom_minus:
haftmann@28790:   "f x = None \<Longrightarrow> dom f - insert x A = dom f - A"
haftmann@28790:   unfolding dom_def by simp
haftmann@28790: 
haftmann@28790: lemma insert_dom:
haftmann@28790:   "f x = Some y \<Longrightarrow> insert x (dom f) = dom f"
haftmann@28790:   unfolding dom_def by auto
haftmann@28790: 
haftmann@35607: lemma map_of_map_keys:
haftmann@35607:   "set xs = dom m \<Longrightarrow> map_of (map (\<lambda>k. (k, the (m k))) xs) = m"
haftmann@35607:   by (rule ext) (auto simp add: map_of_map_restrict restrict_map_def)
haftmann@35607: 
haftmann@39379: lemma map_of_eqI:
haftmann@39379:   assumes set_eq: "set (map fst xs) = set (map fst ys)"
haftmann@39379:   assumes map_eq: "\<forall>k\<in>set (map fst xs). map_of xs k = map_of ys k"
haftmann@39379:   shows "map_of xs = map_of ys"
haftmann@39379: proof (rule ext)
haftmann@39379:   fix k show "map_of xs k = map_of ys k"
haftmann@39379:   proof (cases "map_of xs k")
haftmann@39379:     case None then have "k \<notin> set (map fst xs)" by (simp add: map_of_eq_None_iff)
haftmann@39379:     with set_eq have "k \<notin> set (map fst ys)" by simp
haftmann@39379:     then have "map_of ys k = None" by (simp add: map_of_eq_None_iff)
haftmann@39379:     with None show ?thesis by simp
haftmann@39379:   next
haftmann@39379:     case (Some v) then have "k \<in> set (map fst xs)" by (auto simp add: dom_map_of_conv_image_fst [symmetric])
haftmann@39379:     with map_eq show ?thesis by auto
haftmann@39379:   qed
haftmann@39379: qed
haftmann@39379: 
haftmann@39379: lemma map_of_eq_dom:
haftmann@39379:   assumes "map_of xs = map_of ys"
haftmann@39379:   shows "fst ` set xs = fst ` set ys"
haftmann@39379: proof -
haftmann@39379:   from assms have "dom (map_of xs) = dom (map_of ys)" by simp
haftmann@39379:   then show ?thesis by (simp add: dom_map_of_conv_image_fst)
haftmann@39379: qed
haftmann@39379: 
nipkow@53820: lemma finite_set_of_finite_maps:
nipkow@53820: assumes "finite A" "finite B"
nipkow@53820: shows  "finite {m. dom m = A \<and> ran m \<subseteq> B}" (is "finite ?S")
nipkow@53820: proof -
nipkow@53820:   let ?S' = "{m. \<forall>x. (x \<in> A \<longrightarrow> m x \<in> Some ` B) \<and> (x \<notin> A \<longrightarrow> m x = None)}"
nipkow@53820:   have "?S = ?S'"
nipkow@53820:   proof
nipkow@53820:     show "?S \<subseteq> ?S'" by(auto simp: dom_def ran_def image_def)
nipkow@53820:     show "?S' \<subseteq> ?S"
nipkow@53820:     proof
nipkow@53820:       fix m assume "m \<in> ?S'"
nipkow@53820:       hence 1: "dom m = A" by force
nipkow@53820:       hence 2: "ran m \<subseteq> B" using `m \<in> ?S'` by(auto simp: dom_def ran_def)
nipkow@53820:       from 1 2 show "m \<in> ?S" by blast
nipkow@53820:     qed
nipkow@53820:   qed
nipkow@53820:   with assms show ?thesis by(simp add: finite_set_of_finite_funs)
nipkow@53820: qed
haftmann@28790: 
wenzelm@17399: subsection {* @{term [source] ran} *}
oheimb@14100: 
wenzelm@20800: lemma ranI: "m a = Some b ==> b : ran m"
nipkow@24331: by(auto simp: ran_def)
oheimb@14100: (* declare ranI [intro]? *)
webertj@13908: 
wenzelm@20800: lemma ran_empty [simp]: "ran empty = {}"
nipkow@24331: by(auto simp: ran_def)
webertj@13908: 
wenzelm@20800: lemma ran_map_upd [simp]: "m a = None ==> ran(m(a|->b)) = insert b (ran m)"
nipkow@24331: unfolding ran_def
nipkow@24331: apply auto
nipkow@24331: apply (subgoal_tac "aa ~= a")
nipkow@24331:  apply auto
nipkow@24331: done
wenzelm@20800: 
haftmann@34979: lemma ran_distinct: 
haftmann@34979:   assumes dist: "distinct (map fst al)" 
haftmann@34979:   shows "ran (map_of al) = snd ` set al"
haftmann@34979: using assms proof (induct al)
haftmann@34979:   case Nil then show ?case by simp
haftmann@34979: next
haftmann@34979:   case (Cons kv al)
haftmann@34979:   then have "ran (map_of al) = snd ` set al" by simp
haftmann@34979:   moreover from Cons.prems have "map_of al (fst kv) = None"
haftmann@34979:     by (simp add: map_of_eq_None_iff)
haftmann@34979:   ultimately show ?case by (simp only: map_of.simps ran_map_upd) simp
haftmann@34979: qed
haftmann@34979: 
nipkow@13910: 
oheimb@14100: subsection {* @{text "map_le"} *}
nipkow@13910: 
kleing@13912: lemma map_le_empty [simp]: "empty \<subseteq>\<^sub>m g"
nipkow@24331: by (simp add: map_le_def)
nipkow@13910: 
paulson@17724: lemma upd_None_map_le [simp]: "f(x := None) \<subseteq>\<^sub>m f"
nipkow@24331: by (force simp add: map_le_def)
nipkow@14187: 
nipkow@13910: lemma map_le_upd[simp]: "f \<subseteq>\<^sub>m g ==> f(a := b) \<subseteq>\<^sub>m g(a := b)"
nipkow@44890: by (fastforce simp add: map_le_def)
nipkow@13910: 
paulson@17724: 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: by (force simp add: map_le_def)
nipkow@14187: 
wenzelm@20800: lemma map_le_upds [simp]:
nipkow@24331:   "f \<subseteq>\<^sub>m g ==> f(as [|->] bs) \<subseteq>\<^sub>m g(as [|->] bs)"
nipkow@24331: apply (induct as arbitrary: f g bs)
nipkow@24331:  apply simp
nipkow@24331: apply (case_tac bs)
nipkow@24331:  apply auto
nipkow@24331: done
webertj@13908: 
webertj@14033: lemma map_le_implies_dom_le: "(f \<subseteq>\<^sub>m g) \<Longrightarrow> (dom f \<subseteq> dom g)"
nipkow@44890: by (fastforce simp add: map_le_def dom_def)
webertj@14033: 
webertj@14033: lemma map_le_refl [simp]: "f \<subseteq>\<^sub>m f"
nipkow@24331: by (simp add: map_le_def)
webertj@14033: 
nipkow@14187: 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: by (auto simp add: map_le_def dom_def)
webertj@14033: 
webertj@14033: lemma map_le_antisym: "\<lbrakk> f \<subseteq>\<^sub>m g; g \<subseteq>\<^sub>m f \<rbrakk> \<Longrightarrow> f = g"
nipkow@24331: unfolding map_le_def
nipkow@24331: apply (rule ext)
nipkow@24331: apply (case_tac "x \<in> dom f", simp)
nipkow@44890: apply (case_tac "x \<in> dom g", simp, fastforce)
nipkow@24331: done
webertj@14033: 
webertj@14033: lemma map_le_map_add [simp]: "f \<subseteq>\<^sub>m (g ++ f)"
nipkow@44890: by (fastforce simp add: map_le_def)
webertj@14033: 
nipkow@15304: lemma map_le_iff_map_add_commute: "(f \<subseteq>\<^sub>m f ++ g) = (f++g = g++f)"
nipkow@44890: by(fastforce simp: map_add_def map_le_def fun_eq_iff split: option.splits)
nipkow@15304: 
nipkow@15303: lemma map_add_le_mapE: "f++g \<subseteq>\<^sub>m h \<Longrightarrow> g \<subseteq>\<^sub>m h"
nipkow@44890: by (fastforce simp add: map_le_def map_add_def dom_def)
nipkow@15303: 
bulwahn@46588: lemma map_add_le_mapI: "\<lbrakk> f \<subseteq>\<^sub>m h; g \<subseteq>\<^sub>m h \<rbrakk> \<Longrightarrow> f++g \<subseteq>\<^sub>m h"
nipkow@24331: by (clarsimp simp add: map_le_def map_add_def dom_def split: option.splits)
nipkow@15303: 
nipkow@31080: lemma dom_eq_singleton_conv: "dom f = {x} \<longleftrightarrow> (\<exists>v. f = [x \<mapsto> v])"
nipkow@31080: proof(rule iffI)
nipkow@31080:   assume "\<exists>v. f = [x \<mapsto> v]"
nipkow@31080:   thus "dom f = {x}" by(auto split: split_if_asm)
nipkow@31080: next
nipkow@31080:   assume "dom f = {x}"
nipkow@31080:   then obtain v where "f x = Some v" by auto
nipkow@31080:   hence "[x \<mapsto> v] \<subseteq>\<^sub>m f" by(auto simp add: map_le_def)
nipkow@31080:   moreover have "f \<subseteq>\<^sub>m [x \<mapsto> v]" using `dom f = {x}` `f x = Some v`
nipkow@31080:     by(auto simp add: map_le_def)
nipkow@31080:   ultimately have "f = [x \<mapsto> v]" by-(rule map_le_antisym)
nipkow@31080:   thus "\<exists>v. f = [x \<mapsto> v]" by blast
nipkow@31080: qed
nipkow@31080: 
haftmann@35565: 
haftmann@35565: subsection {* Various *}
haftmann@35565: 
haftmann@35565: lemma set_map_of_compr:
haftmann@35565:   assumes distinct: "distinct (map fst xs)"
haftmann@35565:   shows "set xs = {(k, v). map_of xs k = Some v}"
haftmann@35565: using assms proof (induct xs)
haftmann@35565:   case Nil then show ?case by simp
haftmann@35565: next
haftmann@35565:   case (Cons x xs)
haftmann@35565:   obtain k v where "x = (k, v)" by (cases x) blast
haftmann@35565:   with Cons.prems have "k \<notin> dom (map_of xs)"
haftmann@35565:     by (simp add: dom_map_of_conv_image_fst)
haftmann@35565:   then have *: "insert (k, v) {(k, v). map_of xs k = Some v} =
haftmann@35565:     {(k', v'). (map_of xs(k \<mapsto> v)) k' = Some v'}"
haftmann@35565:     by (auto split: if_splits)
haftmann@35565:   from Cons have "set xs = {(k, v). map_of xs k = Some v}" by simp
haftmann@35565:   with * `x = (k, v)` show ?case by simp
haftmann@35565: qed
haftmann@35565: 
haftmann@35565: lemma map_of_inject_set:
haftmann@35565:   assumes distinct: "distinct (map fst xs)" "distinct (map fst ys)"
haftmann@35565:   shows "map_of xs = map_of ys \<longleftrightarrow> set xs = set ys" (is "?lhs \<longleftrightarrow> ?rhs")
haftmann@35565: proof
haftmann@35565:   assume ?lhs
haftmann@35565:   moreover from `distinct (map fst xs)` have "set xs = {(k, v). map_of xs k = Some v}"
haftmann@35565:     by (rule set_map_of_compr)
haftmann@35565:   moreover from `distinct (map fst ys)` have "set ys = {(k, v). map_of ys k = Some v}"
haftmann@35565:     by (rule set_map_of_compr)
haftmann@35565:   ultimately show ?rhs by simp
haftmann@35565: next
wenzelm@53374:   assume ?rhs show ?lhs
wenzelm@53374:   proof
haftmann@35565:     fix k
haftmann@35565:     show "map_of xs k = map_of ys k" proof (cases "map_of xs k")
haftmann@35565:       case None
wenzelm@53374:       with `?rhs` have "map_of ys k = None"
haftmann@35565:         by (simp add: map_of_eq_None_iff)
wenzelm@53374:       with None show ?thesis by simp
haftmann@35565:     next
haftmann@35565:       case (Some v)
wenzelm@53374:       with distinct `?rhs` have "map_of ys k = Some v"
haftmann@35565:         by simp
wenzelm@53374:       with Some show ?thesis by simp
haftmann@35565:     qed
haftmann@35565:   qed
haftmann@35565: qed
haftmann@35565: 
nipkow@3981: end




