author | wenzelm |
Thu, 11 Feb 2010 23:00:22 +0100 | |
changeset 35115 | 446c5063e4fd |
parent 34979 | 8cb6e7a42e9c |
child 35159 | df38e92af926 |
permissions | -rw-r--r-- |
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(* Title: HOL/Map.thy |
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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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syntax (xsymbols) |
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"~=>" :: "[type, type] => type" (infixr "\<rightharpoonup>" 0) |
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abbreviation |
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empty :: "'a ~=> 'b" where |
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"empty == %x. None" |
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definition |
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map_comp :: "('b ~=> 'c) => ('a ~=> 'b) => ('a ~=> 'c)" (infixl "o'_m" 55) where |
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"f o_m g = (\<lambda>k. case g k of None \<Rightarrow> None | Some v \<Rightarrow> f v)" |
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notation (xsymbols) |
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map_comp (infixl "\<circ>\<^sub>m" 55) |
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definition |
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map_add :: "('a ~=> 'b) => ('a ~=> 'b) => ('a ~=> 'b)" (infixl "++" 100) where |
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"m1 ++ m2 = (\<lambda>x. case m2 x of None => m1 x | Some y => Some y)" |
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definition |
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restrict_map :: "('a ~=> 'b) => 'a set => ('a ~=> 'b)" (infixl "|`" 110) where |
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"m|`A = (\<lambda>x. if x : A then m x else None)" |
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notation (latex output) |
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restrict_map ("_\<restriction>\<^bsub>_\<^esub>" [111,110] 110) |
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definition |
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dom :: "('a ~=> 'b) => 'a set" where |
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"dom m = {a. m a ~= None}" |
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definition |
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ran :: "('a ~=> 'b) => 'b set" where |
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"ran m = {b. EX a. m a = Some b}" |
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definition |
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map_le :: "('a ~=> 'b) => ('a ~=> 'b) => bool" (infix "\<subseteq>\<^sub>m" 50) where |
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"(m\<^isub>1 \<subseteq>\<^sub>m m\<^isub>2) = (\<forall>a \<in> dom m\<^isub>1. m\<^isub>1 a = m\<^isub>2 a)" |
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nonterminals |
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maplets maplet |
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syntax |
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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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"_maplet" :: "['a, 'a] => maplet" ("_ /\<mapsto>/ _") |
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"_maplets" :: "['a, 'a] => maplet" ("_ /[\<mapsto>]/ _") |
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translations |
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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 := CONST Some y)" |
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"_Map ms" == "_MapUpd (CONST 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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primrec |
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map_of :: "('a \<times> 'b) list \<Rightarrow> 'a \<rightharpoonup> 'b" where |
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"map_of [] = empty" |
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| "map_of (p # ps) = (map_of ps)(fst p \<mapsto> snd p)" |
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definition |
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map_upds :: "('a \<rightharpoonup> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b list \<Rightarrow> 'a \<rightharpoonup> 'b" where |
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"map_upds m xs ys = m ++ map_of (rev (zip xs ys))" |
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translations |
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"_MapUpd m (_maplets x y)" == "CONST map_upds m x y" |
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lemma map_of_Cons_code [code]: |
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"map_of [] k = None" |
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"map_of ((l, v) # ps) k = (if l = k then Some v else map_of ps k)" |
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by simp_all |
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subsection {* @{term [source] empty} *} |
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lemma empty_upd_none [simp]: "empty(x := None) = empty" |
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by (rule ext) simp |
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subsection {* @{term [source] map_upd} *} |
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lemma map_upd_triv: "t k = Some x ==> t(k|->x) = t" |
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by (rule ext) simp |
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lemma map_upd_nonempty [simp]: "t(k|->x) ~= empty" |
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proof |
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assume "t(k \<mapsto> x) = empty" |
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then have "(t(k \<mapsto> x)) k = None" by simp |
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then show False by simp |
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qed |
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lemma map_upd_eqD1: |
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assumes "m(a\<mapsto>x) = n(a\<mapsto>y)" |
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shows "x = y" |
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proof - |
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from prems have "(m(a\<mapsto>x)) a = (n(a\<mapsto>y)) a" by simp |
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then show ?thesis by simp |
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qed |
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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 auto |
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lemma finite_range_updI: "finite (range f) ==> finite (range (f(a|->b)))" |
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unfolding 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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subsection {* @{term [source] 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: "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 map_of_is_SomeI [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 map_of_zip_is_Some: |
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assumes "length xs = length ys" |
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shows "x \<in> set xs \<longleftrightarrow> (\<exists>y. map_of (zip xs ys) x = Some y)" |
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using assms by (induct rule: list_induct2) simp_all |
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lemma map_of_zip_upd: |
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fixes x :: 'a and xs :: "'a list" and ys zs :: "'b list" |
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assumes "length ys = length xs" |
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and "length zs = length xs" |
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and "x \<notin> set xs" |
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and "map_of (zip xs ys)(x \<mapsto> y) = map_of (zip xs zs)(x \<mapsto> z)" |
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shows "map_of (zip xs ys) = map_of (zip xs zs)" |
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proof |
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fix x' :: 'a |
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show "map_of (zip xs ys) x' = map_of (zip xs zs) x'" |
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proof (cases "x = x'") |
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case True |
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from assms True map_of_zip_is_None [of xs ys x'] |
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have "map_of (zip xs ys) x' = None" by simp |
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moreover from assms True map_of_zip_is_None [of xs zs x'] |
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have "map_of (zip xs zs) x' = None" by simp |
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ultimately show ?thesis by simp |
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next |
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case False from assms |
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have "(map_of (zip xs ys)(x \<mapsto> y)) x' = (map_of (zip xs zs)(x \<mapsto> z)) x'" by auto |
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with False show ?thesis by simp |
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qed |
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qed |
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lemma map_of_zip_inject: |
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assumes "length ys = length xs" |
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and "length zs = length xs" |
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and dist: "distinct xs" |
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and map_of: "map_of (zip xs ys) = map_of (zip xs zs)" |
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shows "ys = zs" |
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using assms(1) assms(2)[symmetric] using dist map_of proof (induct ys xs zs rule: list_induct3) |
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case Nil show ?case by simp |
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next |
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case (Cons y ys x xs z zs) |
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from `map_of (zip (x#xs) (y#ys)) = map_of (zip (x#xs) (z#zs))` |
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have map_of: "map_of (zip xs ys)(x \<mapsto> y) = map_of (zip xs zs)(x \<mapsto> z)" by simp |
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from Cons have "length ys = length xs" and "length zs = length xs" |
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and "x \<notin> set xs" by simp_all |
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then have "map_of (zip xs ys) = map_of (zip xs zs)" using map_of by (rule map_of_zip_upd) |
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with Cons.hyps `distinct (x # xs)` have "ys = zs" by simp |
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moreover from map_of have "y = z" by (rule map_upd_eqD1) |
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ultimately show ?case by simp |
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qed |
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lemma map_of_zip_map: |
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"map_of (zip xs (map f xs)) = (\<lambda>x. if x \<in> set xs then Some (f x) else None)" |
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by (induct xs) (simp_all add: expand_fun_eq) |
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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 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: "map_of xs k = Some y \<Longrightarrow> (k, y) \<in> set xs" |
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by (induct xs) (simp, atomize (full), auto) |
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lemma map_of_mapk_SomeI: |
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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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by (induct t) (auto simp add: inj_eq) |
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lemma weak_map_of_SomeI: "(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 \<Longrightarrow> P k z \<Longrightarrow> map_of (filter (split P) xs) k = Some z" |
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by (induct xs) auto |
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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 {* @{const Option.map} related *} |
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lemma option_map_o_empty [simp]: "Option.map f o empty = empty" |
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by (rule ext) simp |
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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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by (rule ext) simp |
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subsection {* @{term [source] map_comp} related *} |
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lemma map_comp_empty [simp]: |
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"m \<circ>\<^sub>m empty = empty" |
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"empty \<circ>\<^sub>m m = empty" |
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by (auto simp add: map_comp_def intro: ext split: option.splits) |
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lemma map_comp_simps [simp]: |
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"m2 k = None \<Longrightarrow> (m1 \<circ>\<^sub>m m2) k = None" |
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"m2 k = Some k' \<Longrightarrow> (m1 \<circ>\<^sub>m m2) k = m1 k'" |
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by (auto simp add: map_comp_def) |
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lemma map_comp_Some_iff: |
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"((m1 \<circ>\<^sub>m m2) k = Some v) = (\<exists>k'. m2 k = Some k' \<and> m1 k' = Some v)" |
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by (auto simp add: map_comp_def split: option.splits) |
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lemma map_comp_None_iff: |
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"((m1 \<circ>\<^sub>m m2) k = None) = (m2 k = None \<or> (\<exists>k'. m2 k = Some k' \<and> m1 k' = None)) " |
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by (auto simp add: map_comp_def split: option.splits) |
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subsection {* @{text "++"} *} |
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lemma map_add_empty[simp]: "m ++ empty = m" |
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by(simp add: map_add_def) |
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lemma empty_map_add[simp]: "empty ++ m = m" |
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by (rule ext) (simp add: map_add_def split: option.split) |
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lemma map_add_assoc[simp]: "m1 ++ (m2 ++ m3) = (m1 ++ m2) ++ m3" |
24331 | 290 |
by (rule ext) (simp add: map_add_def split: option.split) |
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|
292 |
lemma map_add_Some_iff: |
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24331 | 293 |
"((m ++ n) k = Some x) = (n k = Some x | n k = None & m k = Some x)" |
294 |
by (simp add: map_add_def split: option.split) |
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14025 | 295 |
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lemma map_add_SomeD [dest!]: |
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"(m ++ n) k = Some x \<Longrightarrow> n k = Some x \<or> n k = None \<and> m k = Some x" |
298 |
by (rule map_add_Some_iff [THEN iffD1]) |
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13908 | 299 |
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lemma map_add_find_right [simp]: "!!xx. n k = Some xx ==> (m ++ n) k = Some xx" |
24331 | 301 |
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)" |
24331 | 304 |
by (simp add: map_add_def split: option.split) |
13908 | 305 |
|
14025 | 306 |
lemma map_add_upd[simp]: "f ++ g(x|->y) = (f ++ g)(x|->y)" |
24331 | 307 |
by (rule ext) (simp add: map_add_def) |
13908 | 308 |
|
14186 | 309 |
lemma map_add_upds[simp]: "m1 ++ (m2(xs[\<mapsto>]ys)) = (m1++m2)(xs[\<mapsto>]ys)" |
24331 | 310 |
by (simp add: map_upds_def) |
14186 | 311 |
|
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312 |
lemma map_add_upd_left: "m\<notin>dom e2 \<Longrightarrow> e1(m \<mapsto> u1) ++ e2 = (e1 ++ e2)(m \<mapsto> u1)" |
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313 |
by (rule ext) (auto simp: map_add_def dom_def split: option.split) |
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314 |
|
20800 | 315 |
lemma map_of_append[simp]: "map_of (xs @ ys) = map_of ys ++ map_of xs" |
24331 | 316 |
unfolding map_add_def |
317 |
apply (induct xs) |
|
318 |
apply simp |
|
319 |
apply (rule ext) |
|
320 |
apply (simp split add: option.split) |
|
321 |
done |
|
13908 | 322 |
|
14025 | 323 |
lemma finite_range_map_of_map_add: |
20800 | 324 |
"finite (range f) ==> finite (range (f ++ map_of l))" |
24331 | 325 |
apply (induct l) |
326 |
apply (auto simp del: fun_upd_apply) |
|
327 |
apply (erule finite_range_updI) |
|
328 |
done |
|
13908 | 329 |
|
20800 | 330 |
lemma inj_on_map_add_dom [iff]: |
24331 | 331 |
"inj_on (m ++ m') (dom m') = inj_on m' (dom m')" |
332 |
by (fastsimp simp: map_add_def dom_def inj_on_def split: option.splits) |
|
20800 | 333 |
|
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334 |
lemma map_upds_fold_map_upd: |
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|
335 |
"f(ks[\<mapsto>]vs) = foldl (\<lambda>f (k, v). f(k\<mapsto>v)) f (zip ks vs)" |
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|
336 |
unfolding map_upds_def proof (rule sym, rule zip_obtain_same_length) |
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|
337 |
fix ks :: "'a list" and vs :: "'b list" |
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|
338 |
assume "length ks = length vs" |
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|
339 |
then show "foldl (\<lambda>f (k, v). f(k\<mapsto>v)) f (zip ks vs) = f ++ map_of (rev (zip ks vs))" |
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|
340 |
by (induct arbitrary: f rule: list_induct2) simp_all |
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|
341 |
qed |
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|
342 |
|
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|
343 |
lemma map_add_map_of_foldr: |
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|
344 |
"m ++ map_of ps = foldr (\<lambda>(k, v) m. m(k \<mapsto> v)) ps m" |
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345 |
by (induct ps) (auto simp add: expand_fun_eq map_add_def) |
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346 |
|
15304 | 347 |
|
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348 |
subsection {* @{term [source] restrict_map} *} |
14100 | 349 |
|
20800 | 350 |
lemma restrict_map_to_empty [simp]: "m|`{} = empty" |
24331 | 351 |
by (simp add: restrict_map_def) |
14186 | 352 |
|
31380 | 353 |
lemma restrict_map_insert: "f |` (insert a A) = (f |` A)(a := f a)" |
354 |
by (auto simp add: restrict_map_def intro: ext) |
|
355 |
||
20800 | 356 |
lemma restrict_map_empty [simp]: "empty|`D = empty" |
24331 | 357 |
by (simp add: restrict_map_def) |
14186 | 358 |
|
15693 | 359 |
lemma restrict_in [simp]: "x \<in> A \<Longrightarrow> (m|`A) x = m x" |
24331 | 360 |
by (simp add: restrict_map_def) |
14100 | 361 |
|
15693 | 362 |
lemma restrict_out [simp]: "x \<notin> A \<Longrightarrow> (m|`A) x = None" |
24331 | 363 |
by (simp add: restrict_map_def) |
14100 | 364 |
|
15693 | 365 |
lemma ran_restrictD: "y \<in> ran (m|`A) \<Longrightarrow> \<exists>x\<in>A. m x = Some y" |
24331 | 366 |
by (auto simp: restrict_map_def ran_def split: split_if_asm) |
14100 | 367 |
|
15693 | 368 |
lemma dom_restrict [simp]: "dom (m|`A) = dom m \<inter> A" |
24331 | 369 |
by (auto simp: restrict_map_def dom_def split: split_if_asm) |
14100 | 370 |
|
15693 | 371 |
lemma restrict_upd_same [simp]: "m(x\<mapsto>y)|`(-{x}) = m|`(-{x})" |
24331 | 372 |
by (rule ext) (auto simp: restrict_map_def) |
14100 | 373 |
|
15693 | 374 |
lemma restrict_restrict [simp]: "m|`A|`B = m|`(A\<inter>B)" |
24331 | 375 |
by (rule ext) (auto simp: restrict_map_def) |
14100 | 376 |
|
20800 | 377 |
lemma restrict_fun_upd [simp]: |
24331 | 378 |
"m(x := y)|`D = (if x \<in> D then (m|`(D-{x}))(x := y) else m|`D)" |
379 |
by (simp add: restrict_map_def expand_fun_eq) |
|
14186 | 380 |
|
20800 | 381 |
lemma fun_upd_None_restrict [simp]: |
24331 | 382 |
"(m|`D)(x := None) = (if x:D then m|`(D - {x}) else m|`D)" |
383 |
by (simp add: restrict_map_def expand_fun_eq) |
|
14186 | 384 |
|
20800 | 385 |
lemma fun_upd_restrict: "(m|`D)(x := y) = (m|`(D-{x}))(x := y)" |
24331 | 386 |
by (simp add: restrict_map_def expand_fun_eq) |
14186 | 387 |
|
20800 | 388 |
lemma fun_upd_restrict_conv [simp]: |
24331 | 389 |
"x \<in> D \<Longrightarrow> (m|`D)(x := y) = (m|`(D-{x}))(x := y)" |
390 |
by (simp add: restrict_map_def expand_fun_eq) |
|
14186 | 391 |
|
14100 | 392 |
|
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393 |
subsection {* @{term [source] map_upds} *} |
14025 | 394 |
|
20800 | 395 |
lemma map_upds_Nil1 [simp]: "m([] [|->] bs) = m" |
24331 | 396 |
by (simp add: map_upds_def) |
14025 | 397 |
|
20800 | 398 |
lemma map_upds_Nil2 [simp]: "m(as [|->] []) = m" |
24331 | 399 |
by (simp add:map_upds_def) |
20800 | 400 |
|
401 |
lemma map_upds_Cons [simp]: "m(a#as [|->] b#bs) = (m(a|->b))(as[|->]bs)" |
|
24331 | 402 |
by (simp add:map_upds_def) |
14025 | 403 |
|
20800 | 404 |
lemma map_upds_append1 [simp]: "\<And>ys m. size xs < size ys \<Longrightarrow> |
24331 | 405 |
m(xs@[x] [\<mapsto>] ys) = m(xs [\<mapsto>] ys)(x \<mapsto> ys!size xs)" |
406 |
apply(induct xs) |
|
407 |
apply (clarsimp simp add: neq_Nil_conv) |
|
408 |
apply (case_tac ys) |
|
409 |
apply simp |
|
410 |
apply simp |
|
411 |
done |
|
14187 | 412 |
|
20800 | 413 |
lemma map_upds_list_update2_drop [simp]: |
414 |
"\<lbrakk>size xs \<le> i; i < size ys\<rbrakk> |
|
415 |
\<Longrightarrow> m(xs[\<mapsto>]ys[i:=y]) = m(xs[\<mapsto>]ys)" |
|
24331 | 416 |
apply (induct xs arbitrary: m ys i) |
417 |
apply simp |
|
418 |
apply (case_tac ys) |
|
419 |
apply simp |
|
420 |
apply (simp split: nat.split) |
|
421 |
done |
|
14025 | 422 |
|
20800 | 423 |
lemma map_upd_upds_conv_if: |
424 |
"(f(x|->y))(xs [|->] ys) = |
|
425 |
(if x : set(take (length ys) xs) then f(xs [|->] ys) |
|
426 |
else (f(xs [|->] ys))(x|->y))" |
|
24331 | 427 |
apply (induct xs arbitrary: x y ys f) |
428 |
apply simp |
|
429 |
apply (case_tac ys) |
|
430 |
apply (auto split: split_if simp: fun_upd_twist) |
|
431 |
done |
|
14025 | 432 |
|
433 |
lemma map_upds_twist [simp]: |
|
24331 | 434 |
"a ~: set as ==> m(a|->b)(as[|->]bs) = m(as[|->]bs)(a|->b)" |
435 |
using set_take_subset by (fastsimp simp add: map_upd_upds_conv_if) |
|
14025 | 436 |
|
20800 | 437 |
lemma map_upds_apply_nontin [simp]: |
24331 | 438 |
"x ~: set xs ==> (f(xs[|->]ys)) x = f x" |
439 |
apply (induct xs arbitrary: ys) |
|
440 |
apply simp |
|
441 |
apply (case_tac ys) |
|
442 |
apply (auto simp: map_upd_upds_conv_if) |
|
443 |
done |
|
14025 | 444 |
|
20800 | 445 |
lemma fun_upds_append_drop [simp]: |
24331 | 446 |
"size xs = size ys \<Longrightarrow> m(xs@zs[\<mapsto>]ys) = m(xs[\<mapsto>]ys)" |
447 |
apply (induct xs arbitrary: m ys) |
|
448 |
apply simp |
|
449 |
apply (case_tac ys) |
|
450 |
apply simp_all |
|
451 |
done |
|
14300 | 452 |
|
20800 | 453 |
lemma fun_upds_append2_drop [simp]: |
24331 | 454 |
"size xs = size ys \<Longrightarrow> m(xs[\<mapsto>]ys@zs) = m(xs[\<mapsto>]ys)" |
455 |
apply (induct xs arbitrary: m ys) |
|
456 |
apply simp |
|
457 |
apply (case_tac ys) |
|
458 |
apply simp_all |
|
459 |
done |
|
14300 | 460 |
|
461 |
||
20800 | 462 |
lemma restrict_map_upds[simp]: |
463 |
"\<lbrakk> length xs = length ys; set xs \<subseteq> D \<rbrakk> |
|
464 |
\<Longrightarrow> m(xs [\<mapsto>] ys)|`D = (m|`(D - set xs))(xs [\<mapsto>] ys)" |
|
24331 | 465 |
apply (induct xs arbitrary: m ys) |
466 |
apply simp |
|
467 |
apply (case_tac ys) |
|
468 |
apply simp |
|
469 |
apply (simp add: Diff_insert [symmetric] insert_absorb) |
|
470 |
apply (simp add: map_upd_upds_conv_if) |
|
471 |
done |
|
14186 | 472 |
|
473 |
||
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474 |
subsection {* @{term [source] dom} *} |
13908 | 475 |
|
31080 | 476 |
lemma dom_eq_empty_conv [simp]: "dom f = {} \<longleftrightarrow> f = empty" |
477 |
by(auto intro!:ext simp: dom_def) |
|
478 |
||
13908 | 479 |
lemma domI: "m a = Some b ==> a : dom m" |
24331 | 480 |
by(simp add:dom_def) |
14100 | 481 |
(* declare domI [intro]? *) |
13908 | 482 |
|
15369 | 483 |
lemma domD: "a : dom m ==> \<exists>b. m a = Some b" |
24331 | 484 |
by (cases "m a") (auto simp add: dom_def) |
13908 | 485 |
|
20800 | 486 |
lemma domIff [iff, simp del]: "(a : dom m) = (m a ~= None)" |
24331 | 487 |
by(simp add:dom_def) |
13908 | 488 |
|
20800 | 489 |
lemma dom_empty [simp]: "dom empty = {}" |
24331 | 490 |
by(simp add:dom_def) |
13908 | 491 |
|
20800 | 492 |
lemma dom_fun_upd [simp]: |
24331 | 493 |
"dom(f(x := y)) = (if y=None then dom f - {x} else insert x (dom f))" |
494 |
by(auto simp add:dom_def) |
|
13908 | 495 |
|
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|
496 |
lemma dom_if: |
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|
497 |
"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}" |
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|
498 |
by (auto split: if_splits) |
13937 | 499 |
|
15304 | 500 |
lemma dom_map_of_conv_image_fst: |
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501 |
"dom (map_of xys) = fst ` set xys" |
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|
502 |
by (induct xys) (auto simp add: dom_if) |
15304 | 503 |
|
20800 | 504 |
lemma dom_map_of_zip [simp]: "[| length xs = length ys; distinct xs |] ==> |
24331 | 505 |
dom(map_of(zip xs ys)) = set xs" |
506 |
by (induct rule: list_induct2) simp_all |
|
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|
507 |
|
13908 | 508 |
lemma finite_dom_map_of: "finite (dom (map_of l))" |
24331 | 509 |
by (induct l) (auto simp add: dom_def insert_Collect [symmetric]) |
13908 | 510 |
|
20800 | 511 |
lemma dom_map_upds [simp]: |
24331 | 512 |
"dom(m(xs[|->]ys)) = set(take (length ys) xs) Un dom m" |
513 |
apply (induct xs arbitrary: m ys) |
|
514 |
apply simp |
|
515 |
apply (case_tac ys) |
|
516 |
apply auto |
|
517 |
done |
|
13910 | 518 |
|
20800 | 519 |
lemma dom_map_add [simp]: "dom(m++n) = dom n Un dom m" |
24331 | 520 |
by(auto simp:dom_def) |
13910 | 521 |
|
20800 | 522 |
lemma dom_override_on [simp]: |
523 |
"dom(override_on f g A) = |
|
524 |
(dom f - {a. a : A - dom g}) Un {a. a : A Int dom g}" |
|
24331 | 525 |
by(auto simp: dom_def override_on_def) |
13908 | 526 |
|
14027 | 527 |
lemma map_add_comm: "dom m1 \<inter> dom m2 = {} \<Longrightarrow> m1++m2 = m2++m1" |
24331 | 528 |
by (rule ext) (force simp: map_add_def dom_def split: option.split) |
20800 | 529 |
|
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530 |
lemma map_add_dom_app_simps: |
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|
531 |
"\<lbrakk> m\<in>dom l2 \<rbrakk> \<Longrightarrow> (l1++l2) m = l2 m" |
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|
532 |
"\<lbrakk> m\<notin>dom l1 \<rbrakk> \<Longrightarrow> (l1++l2) m = l2 m" |
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|
533 |
"\<lbrakk> m\<notin>dom l2 \<rbrakk> \<Longrightarrow> (l1++l2) m = l1 m" |
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|
534 |
by (auto simp add: map_add_def split: option.split_asm) |
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|
535 |
|
29622 | 536 |
lemma dom_const [simp]: |
537 |
"dom (\<lambda>x. Some y) = UNIV" |
|
538 |
by auto |
|
539 |
||
22230 | 540 |
(* Due to John Matthews - could be rephrased with dom *) |
541 |
lemma finite_map_freshness: |
|
542 |
"finite (dom (f :: 'a \<rightharpoonup> 'b)) \<Longrightarrow> \<not> finite (UNIV :: 'a set) \<Longrightarrow> |
|
543 |
\<exists>x. f x = None" |
|
544 |
by(bestsimp dest:ex_new_if_finite) |
|
14027 | 545 |
|
28790 | 546 |
lemma dom_minus: |
547 |
"f x = None \<Longrightarrow> dom f - insert x A = dom f - A" |
|
548 |
unfolding dom_def by simp |
|
549 |
||
550 |
lemma insert_dom: |
|
551 |
"f x = Some y \<Longrightarrow> insert x (dom f) = dom f" |
|
552 |
unfolding dom_def by auto |
|
553 |
||
554 |
||
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|
555 |
subsection {* @{term [source] ran} *} |
14100 | 556 |
|
20800 | 557 |
lemma ranI: "m a = Some b ==> b : ran m" |
24331 | 558 |
by(auto simp: ran_def) |
14100 | 559 |
(* declare ranI [intro]? *) |
13908 | 560 |
|
20800 | 561 |
lemma ran_empty [simp]: "ran empty = {}" |
24331 | 562 |
by(auto simp: ran_def) |
13908 | 563 |
|
20800 | 564 |
lemma ran_map_upd [simp]: "m a = None ==> ran(m(a|->b)) = insert b (ran m)" |
24331 | 565 |
unfolding ran_def |
566 |
apply auto |
|
567 |
apply (subgoal_tac "aa ~= a") |
|
568 |
apply auto |
|
569 |
done |
|
20800 | 570 |
|
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|
571 |
lemma ran_distinct: |
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|
572 |
assumes dist: "distinct (map fst al)" |
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|
573 |
shows "ran (map_of al) = snd ` set al" |
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|
574 |
using assms proof (induct al) |
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|
575 |
case Nil then show ?case by simp |
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|
576 |
next |
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|
577 |
case (Cons kv al) |
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|
578 |
then have "ran (map_of al) = snd ` set al" by simp |
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|
579 |
moreover from Cons.prems have "map_of al (fst kv) = None" |
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|
580 |
by (simp add: map_of_eq_None_iff) |
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|
581 |
ultimately show ?case by (simp only: map_of.simps ran_map_upd) simp |
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|
582 |
qed |
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|
583 |
|
13910 | 584 |
|
14100 | 585 |
subsection {* @{text "map_le"} *} |
13910 | 586 |
|
13912 | 587 |
lemma map_le_empty [simp]: "empty \<subseteq>\<^sub>m g" |
24331 | 588 |
by (simp add: map_le_def) |
13910 | 589 |
|
17724 | 590 |
lemma upd_None_map_le [simp]: "f(x := None) \<subseteq>\<^sub>m f" |
24331 | 591 |
by (force simp add: map_le_def) |
14187 | 592 |
|
13910 | 593 |
lemma map_le_upd[simp]: "f \<subseteq>\<^sub>m g ==> f(a := b) \<subseteq>\<^sub>m g(a := b)" |
24331 | 594 |
by (fastsimp simp add: map_le_def) |
13910 | 595 |
|
17724 | 596 |
lemma map_le_imp_upd_le [simp]: "m1 \<subseteq>\<^sub>m m2 \<Longrightarrow> m1(x := None) \<subseteq>\<^sub>m m2(x \<mapsto> y)" |
24331 | 597 |
by (force simp add: map_le_def) |
14187 | 598 |
|
20800 | 599 |
lemma map_le_upds [simp]: |
24331 | 600 |
"f \<subseteq>\<^sub>m g ==> f(as [|->] bs) \<subseteq>\<^sub>m g(as [|->] bs)" |
601 |
apply (induct as arbitrary: f g bs) |
|
602 |
apply simp |
|
603 |
apply (case_tac bs) |
|
604 |
apply auto |
|
605 |
done |
|
13908 | 606 |
|
14033 | 607 |
lemma map_le_implies_dom_le: "(f \<subseteq>\<^sub>m g) \<Longrightarrow> (dom f \<subseteq> dom g)" |
24331 | 608 |
by (fastsimp simp add: map_le_def dom_def) |
14033 | 609 |
|
610 |
lemma map_le_refl [simp]: "f \<subseteq>\<^sub>m f" |
|
24331 | 611 |
by (simp add: map_le_def) |
14033 | 612 |
|
14187 | 613 |
lemma map_le_trans[trans]: "\<lbrakk> m1 \<subseteq>\<^sub>m m2; m2 \<subseteq>\<^sub>m m3\<rbrakk> \<Longrightarrow> m1 \<subseteq>\<^sub>m m3" |
24331 | 614 |
by (auto simp add: map_le_def dom_def) |
14033 | 615 |
|
616 |
lemma map_le_antisym: "\<lbrakk> f \<subseteq>\<^sub>m g; g \<subseteq>\<^sub>m f \<rbrakk> \<Longrightarrow> f = g" |
|
24331 | 617 |
unfolding map_le_def |
618 |
apply (rule ext) |
|
619 |
apply (case_tac "x \<in> dom f", simp) |
|
620 |
apply (case_tac "x \<in> dom g", simp, fastsimp) |
|
621 |
done |
|
14033 | 622 |
|
623 |
lemma map_le_map_add [simp]: "f \<subseteq>\<^sub>m (g ++ f)" |
|
24331 | 624 |
by (fastsimp simp add: map_le_def) |
14033 | 625 |
|
15304 | 626 |
lemma map_le_iff_map_add_commute: "(f \<subseteq>\<^sub>m f ++ g) = (f++g = g++f)" |
24331 | 627 |
by(fastsimp simp: map_add_def map_le_def expand_fun_eq split: option.splits) |
15304 | 628 |
|
15303 | 629 |
lemma map_add_le_mapE: "f++g \<subseteq>\<^sub>m h \<Longrightarrow> g \<subseteq>\<^sub>m h" |
24331 | 630 |
by (fastsimp simp add: map_le_def map_add_def dom_def) |
15303 | 631 |
|
632 |
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" |
|
24331 | 633 |
by (clarsimp simp add: map_le_def map_add_def dom_def split: option.splits) |
15303 | 634 |
|
31080 | 635 |
lemma dom_eq_singleton_conv: "dom f = {x} \<longleftrightarrow> (\<exists>v. f = [x \<mapsto> v])" |
636 |
proof(rule iffI) |
|
637 |
assume "\<exists>v. f = [x \<mapsto> v]" |
|
638 |
thus "dom f = {x}" by(auto split: split_if_asm) |
|
639 |
next |
|
640 |
assume "dom f = {x}" |
|
641 |
then obtain v where "f x = Some v" by auto |
|
642 |
hence "[x \<mapsto> v] \<subseteq>\<^sub>m f" by(auto simp add: map_le_def) |
|
643 |
moreover have "f \<subseteq>\<^sub>m [x \<mapsto> v]" using `dom f = {x}` `f x = Some v` |
|
644 |
by(auto simp add: map_le_def) |
|
645 |
ultimately have "f = [x \<mapsto> v]" by-(rule map_le_antisym) |
|
646 |
thus "\<exists>v. f = [x \<mapsto> v]" by blast |
|
647 |
qed |
|
648 |
||
3981 | 649 |
end |
34979
8cb6e7a42e9c
more correspondence lemmas between related operations
haftmann
parents:
34941
diff
changeset
|
650 |