author | nipkow |
Sun, 10 Apr 2005 17:19:03 +0200 | |
changeset 15691 | 900cf45ff0a6 |
parent 15369 | 090b16d6c6e0 |
child 15693 | 3a67e61c6e96 |
permissions | -rw-r--r-- |
3981 | 1 |
(* Title: HOL/Map.thy |
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ID: $Id$ |
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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) "~=>" = "'a => 'b option" (infixr 0) |
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translations (type) "a ~=> b " <= (type) "a => b option" |
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consts |
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chg_map :: "('b => 'b) => 'a => ('a ~=> 'b) => ('a ~=> 'b)" |
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map_add :: "('a ~=> 'b) => ('a ~=> 'b) => ('a ~=> 'b)" (infixl "++" 100) |
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restrict_map :: "('a ~=> 'b) => 'a set => ('a ~=> 'b)" (infixl "|^" 110) |
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dom :: "('a ~=> 'b) => 'a set" |
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ran :: "('a ~=> 'b) => 'b set" |
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map_of :: "('a * 'b)list => 'a ~=> 'b" |
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map_upds:: "('a ~=> 'b) => 'a list => 'b list => |
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('a ~=> 'b)" |
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map_upd_s::"('a ~=> 'b) => 'a set => 'b => |
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('a ~=> 'b)" ("_/'(_{|->}_/')" [900,0,0]900) |
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map_subst::"('a ~=> 'b) => 'b => 'b => |
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('a ~=> 'b)" ("_/'(_~>_/')" [900,0,0]900) |
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map_le :: "('a ~=> 'b) => ('a ~=> 'b) => bool" (infix "\<subseteq>\<^sub>m" 50) |
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syntax |
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fun_map_comp :: "('b => 'c) => ('a ~=> 'b) => ('a ~=> 'c)" (infixl "o'_m" 55) |
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translations |
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"f o_m m" == "option_map f o m" |
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nonterminals |
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maplets maplet |
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syntax |
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empty :: "'a ~=> 'b" |
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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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"~=>" :: "[type, type] => type" (infixr "\<rightharpoonup>" 0) |
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fun_map_comp :: "('b => 'c) => ('a ~=> 'b) => ('a ~=> 'c)" (infixl "\<circ>\<^sub>m" 55) |
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"_maplet" :: "['a, 'a] => maplet" ("_ /\<mapsto>/ _") |
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"_maplets" :: "['a, 'a] => maplet" ("_ /[\<mapsto>]/ _") |
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restrict_map :: "('a ~=> 'b) => 'a set => ('a ~=> 'b)" (infixl "\<upharpoonright>" 110) --"requires amssymb!" |
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map_upd_s :: "('a ~=> 'b) => 'a set => 'b => ('a ~=> 'b)" |
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("_/'(_/{\<mapsto>}/_')" [900,0,0]900) |
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map_subst :: "('a ~=> 'b) => 'b => 'b => |
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('a ~=> 'b)" ("_/'(_\<leadsto>_/')" [900,0,0]900) |
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"@chg_map" :: "('a ~=> 'b) => 'a => ('b => 'b) => ('a ~=> 'b)" |
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("_/'(_/\<mapsto>\<lambda>_. _')" [900,0,0,0] 900) |
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translations |
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"empty" => "_K None" |
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"empty" <= "%x. None" |
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"m(x\<mapsto>\<lambda>y. f)" == "chg_map (\<lambda>y. f) x m" |
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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:=Some y)" |
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"_MapUpd m (_maplets x y)" == "map_upds m x y" |
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"_Map ms" == "_MapUpd 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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defs |
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chg_map_def: "chg_map f a m == case m a of None => m | Some b => m(a|->f b)" |
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map_add_def: "m1++m2 == %x. case m2 x of None => m1 x | Some y => Some y" |
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restrict_map_def: "m|^A == %x. if x : A then m x else None" |
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map_upds_def: "m(xs [|->] ys) == m ++ map_of (rev(zip xs ys))" |
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map_upd_s_def: "m(as{|->}b) == %x. if x : as then Some b else m x" |
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map_subst_def: "m(a~>b) == %x. if m x = Some a then Some b else m x" |
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dom_def: "dom(m) == {a. m a ~= None}" |
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ran_def: "ran(m) == {b. EX a. m a = Some b}" |
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map_le_def: "m\<^isub>1 \<subseteq>\<^sub>m m\<^isub>2 == ALL a : dom m\<^isub>1. m\<^isub>1 a = m\<^isub>2 a" |
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primrec |
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"map_of [] = empty" |
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"map_of (p#ps) = (map_of ps)(fst p |-> snd p)" |
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subsection {* @{term empty} *} |
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lemma empty_upd_none[simp]: "empty(x := None) = empty" |
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apply (rule ext) |
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apply (simp (no_asm)) |
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done |
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(* FIXME: what is this sum_case nonsense?? *) |
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lemma sum_case_empty_empty[simp]: "sum_case empty empty = empty" |
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apply (rule ext) |
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apply (simp (no_asm) split add: sum.split) |
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done |
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subsection {* @{term map_upd} *} |
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lemma map_upd_triv: "t k = Some x ==> t(k|->x) = t" |
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apply (rule ext) |
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apply (simp (no_asm_simp)) |
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done |
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lemma map_upd_nonempty[simp]: "t(k|->x) ~= empty" |
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apply safe |
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apply (drule_tac x = k in fun_cong) |
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apply (simp (no_asm_use)) |
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done |
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lemma map_upd_eqD1: "m(a\<mapsto>x) = n(a\<mapsto>y) \<Longrightarrow> x = y" |
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by (drule fun_cong [of _ _ a], auto) |
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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 fastsimp |
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lemma finite_range_updI: "finite (range f) ==> finite (range (f(a|->b)))" |
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apply (unfold 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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(* FIXME: what is this sum_case nonsense?? *) |
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subsection {* @{term sum_case} and @{term empty}/@{term map_upd} *} |
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lemma sum_case_map_upd_empty[simp]: |
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"sum_case (m(k|->y)) empty = (sum_case m empty)(Inl k|->y)" |
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apply (rule ext) |
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apply (simp (no_asm) split add: sum.split) |
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done |
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lemma sum_case_empty_map_upd[simp]: |
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"sum_case empty (m(k|->y)) = (sum_case empty m)(Inr k|->y)" |
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apply (rule ext) |
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apply (simp (no_asm) split add: sum.split) |
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done |
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lemma sum_case_map_upd_map_upd[simp]: |
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"sum_case (m1(k1|->y1)) (m2(k2|->y2)) = (sum_case (m1(k1|->y1)) m2)(Inr k2|->y2)" |
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apply (rule ext) |
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apply (simp (no_asm) split add: sum.split) |
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done |
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subsection {* @{term chg_map} *} |
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lemma chg_map_new[simp]: "m a = None ==> chg_map f a m = m" |
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by (unfold chg_map_def, auto) |
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lemma chg_map_upd[simp]: "m a = Some b ==> chg_map f a m = m(a|->f b)" |
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by (unfold chg_map_def, auto) |
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lemma chg_map_other [simp]: "a \<noteq> b \<Longrightarrow> chg_map f a m b = m b" |
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by (auto simp: chg_map_def split add: option.split) |
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subsection {* @{term 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: |
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"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 [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 finite_range_map_of: "finite (range (map_of xys))" |
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apply (induct xys) |
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apply (simp_all (no_asm) 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 [rule_format]: "map_of xs k = Some y --> (k,y):set xs" |
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by (induct "xs", auto) |
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lemma map_of_mapk_SomeI [rule_format]: |
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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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apply (induct "t") |
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apply (auto simp add: inj_eq) |
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done |
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lemma weak_map_of_SomeI [rule_format]: |
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"(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; P k z |] ==> map_of (filter (split P) xs) k = Some z" |
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apply (rule mp) |
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prefer 2 apply assumption |
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apply (erule thin_rl) |
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apply (induct "xs", auto) |
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done |
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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 {* @{term option_map} related *} |
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lemma option_map_o_empty[simp]: "option_map f o empty = empty" |
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apply (rule ext) |
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apply (simp (no_asm)) |
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done |
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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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apply (rule ext) |
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apply (simp (no_asm)) |
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done |
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subsection {* @{text "++"} *} |
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lemma map_add_empty[simp]: "m ++ empty = m" |
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apply (unfold map_add_def) |
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apply (simp (no_asm)) |
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done |
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lemma empty_map_add[simp]: "empty ++ m = m" |
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apply (unfold map_add_def) |
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apply (rule ext) |
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apply (simp split add: option.split) |
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done |
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lemma map_add_assoc[simp]: "m1 ++ (m2 ++ m3) = (m1 ++ m2) ++ m3" |
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apply(rule ext) |
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apply(simp add: map_add_def split:option.split) |
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done |
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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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apply (unfold map_add_def) |
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apply (simp (no_asm) split add: option.split) |
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done |
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lemmas map_add_SomeD = map_add_Some_iff [THEN iffD1, standard] |
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declare map_add_SomeD [dest!] |
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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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apply (unfold map_add_def) |
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apply (simp (no_asm) split add: option.split) |
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done |
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lemma map_add_upd[simp]: "f ++ g(x|->y) = (f ++ g)(x|->y)" |
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apply (unfold map_add_def) |
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apply (rule ext, auto) |
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done |
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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) |
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lemma map_of_append[simp]: "map_of (xs@ys) = map_of ys ++ map_of xs" |
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apply (unfold map_add_def) |
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apply (induct "xs") |
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apply (simp (no_asm)) |
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apply (rule ext) |
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apply (simp (no_asm_simp) split add: option.split) |
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done |
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declare fun_upd_apply [simp del] |
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lemma finite_range_map_of_map_add: |
315 |
"finite (range f) ==> finite (range (f ++ map_of l))" |
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apply (induct "l", auto) |
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apply (erule finite_range_updI) |
318 |
done |
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declare fun_upd_apply [simp] |
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lemma inj_on_map_add_dom[iff]: |
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"inj_on (m ++ m') (dom m') = inj_on m' (dom m')" |
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by(fastsimp simp add:map_add_def dom_def inj_on_def split:option.splits) |
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subsection {* @{term restrict_map} *} |
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lemma restrict_map_to_empty[simp]: "m|^{} = empty" |
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by(simp add: restrict_map_def) |
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lemma restrict_map_empty[simp]: "empty|^D = empty" |
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by(simp add: restrict_map_def) |
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lemma restrict_in [simp]: "x \<in> A \<Longrightarrow> (m|^A) x = m x" |
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by (auto simp: restrict_map_def) |
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lemma restrict_out [simp]: "x \<notin> A \<Longrightarrow> (m|^A) x = None" |
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by (auto simp: restrict_map_def) |
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lemma ran_restrictD: "y \<in> ran (m|^A) \<Longrightarrow> \<exists>x\<in>A. m x = Some y" |
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by (auto simp: restrict_map_def ran_def split: split_if_asm) |
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lemma dom_restrict [simp]: "dom (m|^A) = dom m \<inter> A" |
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by (auto simp: restrict_map_def dom_def split: split_if_asm) |
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lemma restrict_upd_same [simp]: "m(x\<mapsto>y)|^(-{x}) = m|^(-{x})" |
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by (rule ext, auto simp: restrict_map_def) |
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lemma restrict_restrict [simp]: "m|^A|^B = m|^(A\<inter>B)" |
14100 | 349 |
by (rule ext, auto simp: restrict_map_def) |
350 |
||
14186 | 351 |
lemma restrict_fun_upd[simp]: |
15691 | 352 |
"m(x := y)|^D = (if x \<in> D then (m|^(D-{x}))(x := y) else m|^D)" |
14186 | 353 |
by(simp add: restrict_map_def expand_fun_eq) |
354 |
||
355 |
lemma fun_upd_None_restrict[simp]: |
|
15691 | 356 |
"(m|^D)(x := None) = (if x:D then m|^(D - {x}) else m|^D)" |
14186 | 357 |
by(simp add: restrict_map_def expand_fun_eq) |
358 |
||
359 |
lemma fun_upd_restrict: |
|
15691 | 360 |
"(m|^D)(x := y) = (m|^(D-{x}))(x := y)" |
14186 | 361 |
by(simp add: restrict_map_def expand_fun_eq) |
362 |
||
363 |
lemma fun_upd_restrict_conv[simp]: |
|
15691 | 364 |
"x \<in> D \<Longrightarrow> (m|^D)(x := y) = (m|^(D-{x}))(x := y)" |
14186 | 365 |
by(simp add: restrict_map_def expand_fun_eq) |
366 |
||
14100 | 367 |
|
368 |
subsection {* @{term map_upds} *} |
|
14025 | 369 |
|
370 |
lemma map_upds_Nil1[simp]: "m([] [|->] bs) = m" |
|
371 |
by(simp add:map_upds_def) |
|
372 |
||
373 |
lemma map_upds_Nil2[simp]: "m(as [|->] []) = m" |
|
374 |
by(simp add:map_upds_def) |
|
375 |
||
376 |
lemma map_upds_Cons[simp]: "m(a#as [|->] b#bs) = (m(a|->b))(as[|->]bs)" |
|
377 |
by(simp add:map_upds_def) |
|
378 |
||
14187 | 379 |
lemma map_upds_append1[simp]: "\<And>ys m. size xs < size ys \<Longrightarrow> |
380 |
m(xs@[x] [\<mapsto>] ys) = m(xs [\<mapsto>] ys)(x \<mapsto> ys!size xs)" |
|
381 |
apply(induct xs) |
|
382 |
apply(clarsimp simp add:neq_Nil_conv) |
|
14208 | 383 |
apply (case_tac ys, simp, simp) |
14187 | 384 |
done |
385 |
||
386 |
lemma map_upds_list_update2_drop[simp]: |
|
387 |
"\<And>m ys i. \<lbrakk>size xs \<le> i; i < size ys\<rbrakk> |
|
388 |
\<Longrightarrow> m(xs[\<mapsto>]ys[i:=y]) = m(xs[\<mapsto>]ys)" |
|
14208 | 389 |
apply (induct xs, simp) |
390 |
apply (case_tac ys, simp) |
|
14187 | 391 |
apply(simp split:nat.split) |
392 |
done |
|
14025 | 393 |
|
394 |
lemma map_upd_upds_conv_if: "!!x y ys f. |
|
395 |
(f(x|->y))(xs [|->] ys) = |
|
396 |
(if x : set(take (length ys) xs) then f(xs [|->] ys) |
|
397 |
else (f(xs [|->] ys))(x|->y))" |
|
14208 | 398 |
apply (induct xs, simp) |
14025 | 399 |
apply(case_tac ys) |
400 |
apply(auto split:split_if simp:fun_upd_twist) |
|
401 |
done |
|
402 |
||
403 |
lemma map_upds_twist [simp]: |
|
404 |
"a ~: set as ==> m(a|->b)(as[|->]bs) = m(as[|->]bs)(a|->b)" |
|
405 |
apply(insert set_take_subset) |
|
406 |
apply (fastsimp simp add: map_upd_upds_conv_if) |
|
407 |
done |
|
408 |
||
409 |
lemma map_upds_apply_nontin[simp]: |
|
410 |
"!!ys. x ~: set xs ==> (f(xs[|->]ys)) x = f x" |
|
14208 | 411 |
apply (induct xs, simp) |
14025 | 412 |
apply(case_tac ys) |
413 |
apply(auto simp: map_upd_upds_conv_if) |
|
414 |
done |
|
415 |
||
14300 | 416 |
lemma fun_upds_append_drop[simp]: |
417 |
"!!m ys. size xs = size ys \<Longrightarrow> m(xs@zs[\<mapsto>]ys) = m(xs[\<mapsto>]ys)" |
|
418 |
apply(induct xs) |
|
419 |
apply (simp) |
|
420 |
apply(case_tac ys) |
|
421 |
apply simp_all |
|
422 |
done |
|
423 |
||
424 |
lemma fun_upds_append2_drop[simp]: |
|
425 |
"!!m ys. size xs = size ys \<Longrightarrow> m(xs[\<mapsto>]ys@zs) = m(xs[\<mapsto>]ys)" |
|
426 |
apply(induct xs) |
|
427 |
apply (simp) |
|
428 |
apply(case_tac ys) |
|
429 |
apply simp_all |
|
430 |
done |
|
431 |
||
432 |
||
14186 | 433 |
lemma restrict_map_upds[simp]: "!!m ys. |
434 |
\<lbrakk> length xs = length ys; set xs \<subseteq> D \<rbrakk> |
|
15691 | 435 |
\<Longrightarrow> m(xs [\<mapsto>] ys)|^D = (m|^(D - set xs))(xs [\<mapsto>] ys)" |
14208 | 436 |
apply (induct xs, simp) |
437 |
apply (case_tac ys, simp) |
|
14186 | 438 |
apply(simp add:Diff_insert[symmetric] insert_absorb) |
439 |
apply(simp add: map_upd_upds_conv_if) |
|
440 |
done |
|
441 |
||
442 |
||
14100 | 443 |
subsection {* @{term map_upd_s} *} |
444 |
||
445 |
lemma map_upd_s_apply [simp]: |
|
446 |
"(m(as{|->}b)) x = (if x : as then Some b else m x)" |
|
447 |
by (simp add: map_upd_s_def) |
|
448 |
||
449 |
lemma map_subst_apply [simp]: |
|
450 |
"(m(a~>b)) x = (if m x = Some a then Some b else m x)" |
|
451 |
by (simp add: map_subst_def) |
|
452 |
||
453 |
subsection {* @{term dom} *} |
|
13908 | 454 |
|
455 |
lemma domI: "m a = Some b ==> a : dom m" |
|
14208 | 456 |
by (unfold dom_def, auto) |
14100 | 457 |
(* declare domI [intro]? *) |
13908 | 458 |
|
15369 | 459 |
lemma domD: "a : dom m ==> \<exists>b. m a = Some b" |
14208 | 460 |
by (unfold dom_def, auto) |
13908 | 461 |
|
13910 | 462 |
lemma domIff[iff]: "(a : dom m) = (m a ~= None)" |
14208 | 463 |
by (unfold dom_def, auto) |
13908 | 464 |
declare domIff [simp del] |
465 |
||
13910 | 466 |
lemma dom_empty[simp]: "dom empty = {}" |
13908 | 467 |
apply (unfold dom_def) |
468 |
apply (simp (no_asm)) |
|
469 |
done |
|
470 |
||
13910 | 471 |
lemma dom_fun_upd[simp]: |
472 |
"dom(f(x := y)) = (if y=None then dom f - {x} else insert x (dom f))" |
|
473 |
by (simp add:dom_def) blast |
|
13908 | 474 |
|
13937 | 475 |
lemma dom_map_of: "dom(map_of xys) = {x. \<exists>y. (x,y) : set xys}" |
476 |
apply(induct xys) |
|
477 |
apply(auto simp del:fun_upd_apply) |
|
478 |
done |
|
479 |
||
15304 | 480 |
lemma dom_map_of_conv_image_fst: |
481 |
"dom(map_of xys) = fst ` (set xys)" |
|
482 |
by(force simp: dom_map_of) |
|
483 |
||
15110
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
14739
diff
changeset
|
484 |
lemma dom_map_of_zip[simp]: "[| length xs = length ys; distinct xs |] ==> |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
14739
diff
changeset
|
485 |
dom(map_of(zip xs ys)) = set xs" |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
14739
diff
changeset
|
486 |
by(induct rule: list_induct2, simp_all) |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
14739
diff
changeset
|
487 |
|
13908 | 488 |
lemma finite_dom_map_of: "finite (dom (map_of l))" |
489 |
apply (unfold dom_def) |
|
15251 | 490 |
apply (induct "l") |
13908 | 491 |
apply (auto simp add: insert_Collect [symmetric]) |
492 |
done |
|
493 |
||
14025 | 494 |
lemma dom_map_upds[simp]: |
495 |
"!!m ys. dom(m(xs[|->]ys)) = set(take (length ys) xs) Un dom m" |
|
14208 | 496 |
apply (induct xs, simp) |
497 |
apply (case_tac ys, auto) |
|
14025 | 498 |
done |
13910 | 499 |
|
14025 | 500 |
lemma dom_map_add[simp]: "dom(m++n) = dom n Un dom m" |
14208 | 501 |
by (unfold dom_def, auto) |
13910 | 502 |
|
15691 | 503 |
lemma dom_override_on[simp]: |
504 |
"dom(override_on f g A) = |
|
505 |
(dom f - {a. a : A - dom g}) Un {a. a : A Int dom g}" |
|
506 |
by(auto simp add: dom_def override_on_def) |
|
13908 | 507 |
|
14027 | 508 |
lemma map_add_comm: "dom m1 \<inter> dom m2 = {} \<Longrightarrow> m1++m2 = m2++m1" |
509 |
apply(rule ext) |
|
510 |
apply(fastsimp simp:map_add_def split:option.split) |
|
511 |
done |
|
512 |
||
14100 | 513 |
subsection {* @{term ran} *} |
514 |
||
515 |
lemma ranI: "m a = Some b ==> b : ran m" |
|
516 |
by (auto simp add: ran_def) |
|
517 |
(* declare ranI [intro]? *) |
|
13908 | 518 |
|
13910 | 519 |
lemma ran_empty[simp]: "ran empty = {}" |
13908 | 520 |
apply (unfold ran_def) |
521 |
apply (simp (no_asm)) |
|
522 |
done |
|
523 |
||
13910 | 524 |
lemma ran_map_upd[simp]: "m a = None ==> ran(m(a|->b)) = insert b (ran m)" |
14208 | 525 |
apply (unfold ran_def, auto) |
13908 | 526 |
apply (subgoal_tac "~ (aa = a) ") |
527 |
apply auto |
|
528 |
done |
|
13910 | 529 |
|
14100 | 530 |
subsection {* @{text "map_le"} *} |
13910 | 531 |
|
13912 | 532 |
lemma map_le_empty [simp]: "empty \<subseteq>\<^sub>m g" |
13910 | 533 |
by(simp add:map_le_def) |
534 |
||
14187 | 535 |
lemma [simp]: "f(x := None) \<subseteq>\<^sub>m f" |
536 |
by(force simp add:map_le_def) |
|
537 |
||
13910 | 538 |
lemma map_le_upd[simp]: "f \<subseteq>\<^sub>m g ==> f(a := b) \<subseteq>\<^sub>m g(a := b)" |
539 |
by(fastsimp simp add:map_le_def) |
|
540 |
||
14187 | 541 |
lemma [simp]: "m1 \<subseteq>\<^sub>m m2 \<Longrightarrow> m1(x := None) \<subseteq>\<^sub>m m2(x \<mapsto> y)" |
542 |
by(force simp add:map_le_def) |
|
543 |
||
13910 | 544 |
lemma map_le_upds[simp]: |
545 |
"!!f g bs. f \<subseteq>\<^sub>m g ==> f(as [|->] bs) \<subseteq>\<^sub>m g(as [|->] bs)" |
|
14208 | 546 |
apply (induct as, simp) |
547 |
apply (case_tac bs, auto) |
|
14025 | 548 |
done |
13908 | 549 |
|
14033 | 550 |
lemma map_le_implies_dom_le: "(f \<subseteq>\<^sub>m g) \<Longrightarrow> (dom f \<subseteq> dom g)" |
551 |
by (fastsimp simp add: map_le_def dom_def) |
|
552 |
||
553 |
lemma map_le_refl [simp]: "f \<subseteq>\<^sub>m f" |
|
554 |
by (simp add: map_le_def) |
|
555 |
||
14187 | 556 |
lemma map_le_trans[trans]: "\<lbrakk> m1 \<subseteq>\<^sub>m m2; m2 \<subseteq>\<^sub>m m3\<rbrakk> \<Longrightarrow> m1 \<subseteq>\<^sub>m m3" |
557 |
by(force simp add:map_le_def) |
|
14033 | 558 |
|
559 |
lemma map_le_antisym: "\<lbrakk> f \<subseteq>\<^sub>m g; g \<subseteq>\<^sub>m f \<rbrakk> \<Longrightarrow> f = g" |
|
560 |
apply (unfold map_le_def) |
|
561 |
apply (rule ext) |
|
14208 | 562 |
apply (case_tac "x \<in> dom f", simp) |
563 |
apply (case_tac "x \<in> dom g", simp, fastsimp) |
|
14033 | 564 |
done |
565 |
||
566 |
lemma map_le_map_add [simp]: "f \<subseteq>\<^sub>m (g ++ f)" |
|
567 |
by (fastsimp simp add: map_le_def) |
|
568 |
||
15304 | 569 |
lemma map_le_iff_map_add_commute: "(f \<subseteq>\<^sub>m f ++ g) = (f++g = g++f)" |
570 |
by(fastsimp simp add:map_add_def map_le_def expand_fun_eq split:option.splits) |
|
571 |
||
15303 | 572 |
lemma map_add_le_mapE: "f++g \<subseteq>\<^sub>m h \<Longrightarrow> g \<subseteq>\<^sub>m h" |
573 |
by (fastsimp simp add: map_le_def map_add_def dom_def) |
|
574 |
||
575 |
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" |
|
576 |
by (clarsimp simp add: map_le_def map_add_def dom_def split:option.splits) |
|
577 |
||
3981 | 578 |
end |