author | nipkow |
Thu, 11 Sep 2003 22:33:12 +0200 | |
changeset 14186 | 6d2a494e33be |
parent 14180 | d2e550609c40 |
child 14187 | 26dfcd0ac436 |
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 = List: |
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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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map_image::"('b => 'c) => ('a ~=> 'b) => ('a ~=> 'c)" (infixr "`>" 90) |
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restrict_map :: "('a ~=> 'b) => 'a set => ('a ~=> 'b)" ("_|'__" [90, 91] 90) |
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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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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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12114
a8e860c86252
eliminated old "symbols" syntax, use "xsymbols" instead;
wenzelm
parents:
10137
diff
changeset
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syntax (xsymbols) |
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"_maplet" :: "['a, 'a] => maplet" ("_ /\<mapsto>/ _") |
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"_maplets" :: "['a, 'a] => maplet" ("_ /[\<mapsto>]/ _") |
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"~=>" :: "[type, type] => type" (infixr "\<rightharpoonup>" 0) |
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restrict_map :: "('a ~=> 'b) => 'a set => ('a ~=> 'b)" ("_\<lfloor>_" [90, 91] 90) |
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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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map_image_def: "f`>m == option_map f o m" |
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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: "m1 \<subseteq>\<^sub>m m2 == ALL a : dom m1. m1 a = m2 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 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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apply (unfold chg_map_def) |
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apply auto |
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done |
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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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apply (unfold chg_map_def) |
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apply auto |
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done |
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subsection {* @{term map_of} *} |
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lemma map_of_SomeD [rule_format (no_asm)]: "map_of xs k = Some y --> (k,y):set xs" |
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apply (induct_tac "xs") |
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apply auto |
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done |
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lemma map_of_mapk_SomeI [rule_format (no_asm)]: "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_tac "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 (no_asm)]: "(k, x) : set l --> (? x. map_of l k = Some x)" |
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apply (induct_tac "l") |
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apply auto |
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done |
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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_tac "xs") |
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apply auto |
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done |
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lemma finite_range_map_of: "finite (range (map_of l))" |
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apply (induct_tac "l") |
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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_map: "map_of (map (%(a,b). (a,f b)) xs) x = option_map f (map_of xs x)" |
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apply (induct_tac "xs") |
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apply auto |
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done |
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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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apply (subst map_add_Some_iff) |
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apply fast |
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done |
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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) |
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apply 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_tac "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: |
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"finite (range f) ==> finite (range (f ++ map_of l))" |
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apply (induct_tac "l") |
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apply auto |
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apply (erule finite_range_updI) |
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done |
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declare fun_upd_apply [simp] |
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subsection {* @{term map_image} *} |
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lemma map_image_empty [simp]: "f`>empty = empty" |
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by (auto simp: map_image_def empty_def) |
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lemma map_image_upd [simp]: "f`>m(a|->b) = (f`>m)(a|->f b)" |
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apply (auto simp: map_image_def fun_upd_def) |
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by (rule ext, auto) |
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subsection {* @{term restrict_map} *} |
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lemma restrict_map_to_empty[simp]: "m\<lfloor>{} = empty" |
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by(simp add: restrict_map_def) |
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lemma restrict_map_empty[simp]: "empty\<lfloor>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\<lfloor>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\<lfloor>A) x = None" |
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by (auto simp: restrict_map_def) |
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lemma ran_restrictD: "y \<in> ran (m\<lfloor>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\<lfloor>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)\<lfloor>(-{x}) = m\<lfloor>(-{x})" |
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by (rule ext, auto simp: restrict_map_def) |
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lemma restrict_restrict [simp]: "m\<lfloor>A\<lfloor>B = m\<lfloor>(A\<inter>B)" |
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by (rule ext, auto simp: restrict_map_def) |
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lemma restrict_fun_upd[simp]: |
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"m(x := y)\<lfloor>D = (if x \<in> D then (m\<lfloor>(D-{x}))(x := y) else m\<lfloor>D)" |
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by(simp add: restrict_map_def expand_fun_eq) |
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lemma fun_upd_None_restrict[simp]: |
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"(m\<lfloor>D)(x := None) = (if x:D then m\<lfloor>(D - {x}) else m\<lfloor>D)" |
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by(simp add: restrict_map_def expand_fun_eq) |
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lemma fun_upd_restrict: |
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"(m\<lfloor>D)(x := y) = (m\<lfloor>(D-{x}))(x := y)" |
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by(simp add: restrict_map_def expand_fun_eq) |
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lemma fun_upd_restrict_conv[simp]: |
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"x \<in> D \<Longrightarrow> (m\<lfloor>D)(x := y) = (m\<lfloor>(D-{x}))(x := y)" |
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by(simp add: restrict_map_def expand_fun_eq) |
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14100 | 338 |
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subsection {* @{term map_upds} *} |
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lemma map_upds_Nil1[simp]: "m([] [|->] bs) = m" |
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by(simp add:map_upds_def) |
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lemma map_upds_Nil2[simp]: "m(as [|->] []) = m" |
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by(simp add:map_upds_def) |
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lemma map_upds_Cons[simp]: "m(a#as [|->] b#bs) = (m(a|->b))(as[|->]bs)" |
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by(simp add:map_upds_def) |
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lemma map_upd_upds_conv_if: "!!x y ys f. |
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(f(x|->y))(xs [|->] ys) = |
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(if x : set(take (length ys) xs) then f(xs [|->] ys) |
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else (f(xs [|->] ys))(x|->y))" |
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apply(induct xs) |
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apply simp |
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apply(case_tac ys) |
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apply(auto split:split_if simp:fun_upd_twist) |
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done |
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lemma map_upds_twist [simp]: |
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"a ~: set as ==> m(a|->b)(as[|->]bs) = m(as[|->]bs)(a|->b)" |
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apply(insert set_take_subset) |
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apply (fastsimp simp add: map_upd_upds_conv_if) |
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done |
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lemma map_upds_apply_nontin[simp]: |
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"!!ys. x ~: set xs ==> (f(xs[|->]ys)) x = f x" |
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apply(induct xs) |
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apply simp |
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apply(case_tac ys) |
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apply(auto simp: map_upd_upds_conv_if) |
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done |
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lemma restrict_map_upds[simp]: "!!m ys. |
376 |
\<lbrakk> length xs = length ys; set xs \<subseteq> D \<rbrakk> |
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\<Longrightarrow> m(xs [\<mapsto>] ys)\<lfloor>D = (m\<lfloor>(D - set xs))(xs [\<mapsto>] ys)" |
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apply(induct xs) |
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apply simp |
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apply(case_tac ys) |
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apply simp |
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apply(simp add:Diff_insert[symmetric] insert_absorb) |
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apply(simp add: map_upd_upds_conv_if) |
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done |
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subsection {* @{term map_upd_s} *} |
388 |
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lemma map_upd_s_apply [simp]: |
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"(m(as{|->}b)) x = (if x : as then Some b else m x)" |
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by (simp add: map_upd_s_def) |
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||
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lemma map_subst_apply [simp]: |
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"(m(a~>b)) x = (if m x = Some a then Some b else m x)" |
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by (simp add: map_subst_def) |
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||
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subsection {* @{term dom} *} |
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13908 | 398 |
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lemma domI: "m a = Some b ==> a : dom m" |
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apply (unfold dom_def) |
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apply auto |
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done |
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14100 | 403 |
(* declare domI [intro]? *) |
13908 | 404 |
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lemma domD: "a : dom m ==> ? b. m a = Some b" |
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apply (unfold dom_def) |
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apply auto |
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done |
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||
13910 | 410 |
lemma domIff[iff]: "(a : dom m) = (m a ~= None)" |
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apply (unfold dom_def) |
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apply auto |
|
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done |
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declare domIff [simp del] |
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||
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lemma dom_empty[simp]: "dom empty = {}" |
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apply (unfold dom_def) |
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apply (simp (no_asm)) |
|
419 |
done |
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||
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lemma dom_fun_upd[simp]: |
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"dom(f(x := y)) = (if y=None then dom f - {x} else insert x (dom f))" |
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by (simp add:dom_def) blast |
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13908 | 424 |
|
13937 | 425 |
lemma dom_map_of: "dom(map_of xys) = {x. \<exists>y. (x,y) : set xys}" |
426 |
apply(induct xys) |
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apply(auto simp del:fun_upd_apply) |
|
428 |
done |
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||
13908 | 430 |
lemma finite_dom_map_of: "finite (dom (map_of l))" |
431 |
apply (unfold dom_def) |
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apply (induct_tac "l") |
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apply (auto simp add: insert_Collect [symmetric]) |
|
434 |
done |
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435 |
||
14025 | 436 |
lemma dom_map_upds[simp]: |
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"!!m ys. dom(m(xs[|->]ys)) = set(take (length ys) xs) Un dom m" |
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apply(induct xs) |
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apply simp |
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apply(case_tac ys) |
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441 |
apply auto |
|
442 |
done |
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13910 | 443 |
|
14025 | 444 |
lemma dom_map_add[simp]: "dom(m++n) = dom n Un dom m" |
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apply (unfold dom_def) |
446 |
apply auto |
|
447 |
done |
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13910 | 448 |
|
449 |
lemma dom_overwrite[simp]: |
|
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"dom(f(g|A)) = (dom f - {a. a : A - dom g}) Un {a. a : A Int dom g}" |
|
451 |
by(auto simp add: dom_def overwrite_def) |
|
13908 | 452 |
|
14027 | 453 |
lemma map_add_comm: "dom m1 \<inter> dom m2 = {} \<Longrightarrow> m1++m2 = m2++m1" |
454 |
apply(rule ext) |
|
455 |
apply(fastsimp simp:map_add_def split:option.split) |
|
456 |
done |
|
457 |
||
14100 | 458 |
subsection {* @{term ran} *} |
459 |
||
460 |
lemma ranI: "m a = Some b ==> b : ran m" |
|
461 |
by (auto simp add: ran_def) |
|
462 |
(* declare ranI [intro]? *) |
|
13908 | 463 |
|
13910 | 464 |
lemma ran_empty[simp]: "ran empty = {}" |
13908 | 465 |
apply (unfold ran_def) |
466 |
apply (simp (no_asm)) |
|
467 |
done |
|
468 |
||
13910 | 469 |
lemma ran_map_upd[simp]: "m a = None ==> ran(m(a|->b)) = insert b (ran m)" |
13908 | 470 |
apply (unfold ran_def) |
471 |
apply auto |
|
472 |
apply (subgoal_tac "~ (aa = a) ") |
|
473 |
apply auto |
|
474 |
done |
|
13910 | 475 |
|
14100 | 476 |
subsection {* @{text "map_le"} *} |
13910 | 477 |
|
13912 | 478 |
lemma map_le_empty [simp]: "empty \<subseteq>\<^sub>m g" |
13910 | 479 |
by(simp add:map_le_def) |
480 |
||
481 |
lemma map_le_upd[simp]: "f \<subseteq>\<^sub>m g ==> f(a := b) \<subseteq>\<^sub>m g(a := b)" |
|
482 |
by(fastsimp simp add:map_le_def) |
|
483 |
||
484 |
lemma map_le_upds[simp]: |
|
485 |
"!!f g bs. f \<subseteq>\<^sub>m g ==> f(as [|->] bs) \<subseteq>\<^sub>m g(as [|->] bs)" |
|
14025 | 486 |
apply(induct as) |
487 |
apply simp |
|
488 |
apply(case_tac bs) |
|
489 |
apply auto |
|
490 |
done |
|
13908 | 491 |
|
14033 | 492 |
lemma map_le_implies_dom_le: "(f \<subseteq>\<^sub>m g) \<Longrightarrow> (dom f \<subseteq> dom g)" |
493 |
by (fastsimp simp add: map_le_def dom_def) |
|
494 |
||
495 |
lemma map_le_refl [simp]: "f \<subseteq>\<^sub>m f" |
|
496 |
by (simp add: map_le_def) |
|
497 |
||
498 |
lemma map_le_trans: "\<lbrakk> f \<subseteq>\<^sub>m g; g \<subseteq>\<^sub>m h \<rbrakk> \<Longrightarrow> f \<subseteq>\<^sub>m h" |
|
499 |
apply (clarsimp simp add: map_le_def) |
|
500 |
apply (drule_tac x="a" in bspec, fastsimp)+ |
|
501 |
apply assumption |
|
502 |
done |
|
503 |
||
504 |
lemma map_le_antisym: "\<lbrakk> f \<subseteq>\<^sub>m g; g \<subseteq>\<^sub>m f \<rbrakk> \<Longrightarrow> f = g" |
|
505 |
apply (unfold map_le_def) |
|
506 |
apply (rule ext) |
|
507 |
apply (case_tac "x \<in> dom f") |
|
508 |
apply simp |
|
509 |
apply (case_tac "x \<in> dom g") |
|
510 |
apply simp |
|
511 |
apply fastsimp |
|
512 |
done |
|
513 |
||
514 |
lemma map_le_map_add [simp]: "f \<subseteq>\<^sub>m (g ++ f)" |
|
515 |
by (fastsimp simp add: map_le_def) |
|
516 |
||
3981 | 517 |
end |