src/HOL/Map.thy
author nipkow
Sun Apr 10 17:19:03 2005 +0200 (2005-04-10)
changeset 15691 900cf45ff0a6
parent 15369 090b16d6c6e0
child 15693 3a67e61c6e96
permissions -rw-r--r--
_(_|_) is now override_on
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(*  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:
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 "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)
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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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   324
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   325
subsection {* @{term restrict_map} *}
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   326
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   327
lemma restrict_map_to_empty[simp]: "m|^{} = empty"
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by(simp add: restrict_map_def)
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   329
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   330
lemma restrict_map_empty[simp]: "empty|^D = empty"
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by(simp add: restrict_map_def)
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   332
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   333
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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   335
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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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   338
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   339
lemma ran_restrictD: "y \<in> ran (m|^A) \<Longrightarrow> \<exists>x\<in>A. m x = Some y"
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   340
by (auto simp: restrict_map_def ran_def split: split_if_asm)
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   341
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   342
lemma dom_restrict [simp]: "dom (m|^A) = dom m \<inter> A"
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   343
by (auto simp: restrict_map_def dom_def split: split_if_asm)
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   344
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   345
lemma restrict_upd_same [simp]: "m(x\<mapsto>y)|^(-{x}) = m|^(-{x})"
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   346
by (rule ext, auto simp: restrict_map_def)
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   347
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   348
lemma restrict_restrict [simp]: "m|^A|^B = m|^(A\<inter>B)"
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   349
by (rule ext, auto simp: restrict_map_def)
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   350
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   351
lemma restrict_fun_upd[simp]:
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 "m(x := y)|^D = (if x \<in> D then (m|^(D-{x}))(x := y) else m|^D)"
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by(simp add: restrict_map_def expand_fun_eq)
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   354
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   355
lemma fun_upd_None_restrict[simp]:
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  "(m|^D)(x := None) = (if x:D then m|^(D - {x}) else m|^D)"
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   357
by(simp add: restrict_map_def expand_fun_eq)
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   358
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   359
lemma fun_upd_restrict:
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   360
 "(m|^D)(x := y) = (m|^(D-{x}))(x := y)"
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   361
by(simp add: restrict_map_def expand_fun_eq)
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   362
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   363
lemma fun_upd_restrict_conv[simp]:
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   364
 "x \<in> D \<Longrightarrow> (m|^D)(x := y) = (m|^(D-{x}))(x := y)"
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   365
by(simp add: restrict_map_def expand_fun_eq)
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   366
oheimb@14100
   367
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   368
subsection {* @{term map_upds} *}
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   369
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   370
lemma map_upds_Nil1[simp]: "m([] [|->] bs) = m"
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   371
by(simp add:map_upds_def)
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   372
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   373
lemma map_upds_Nil2[simp]: "m(as [|->] []) = m"
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   374
by(simp add:map_upds_def)
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   375
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   376
lemma map_upds_Cons[simp]: "m(a#as [|->] b#bs) = (m(a|->b))(as[|->]bs)"
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   377
by(simp add:map_upds_def)
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   378
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   379
lemma map_upds_append1[simp]: "\<And>ys m. size xs < size ys \<Longrightarrow>
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   380
  m(xs@[x] [\<mapsto>] ys) = m(xs [\<mapsto>] ys)(x \<mapsto> ys!size xs)"
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   381
apply(induct xs)
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   382
 apply(clarsimp simp add:neq_Nil_conv)
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   383
apply (case_tac ys, simp, simp)
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   384
done
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   385
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   386
lemma map_upds_list_update2_drop[simp]:
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   387
 "\<And>m ys i. \<lbrakk>size xs \<le> i; i < size ys\<rbrakk>
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   388
     \<Longrightarrow> m(xs[\<mapsto>]ys[i:=y]) = m(xs[\<mapsto>]ys)"
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   389
apply (induct xs, simp)
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   390
apply (case_tac ys, simp)
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   391
apply(simp split:nat.split)
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   392
done
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   393
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   394
lemma map_upd_upds_conv_if: "!!x y ys f.
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   395
 (f(x|->y))(xs [|->] ys) =
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   396
 (if x : set(take (length ys) xs) then f(xs [|->] ys)
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   397
                                  else (f(xs [|->] ys))(x|->y))"
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   398
apply (induct xs, simp)
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   399
apply(case_tac ys)
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   400
 apply(auto split:split_if simp:fun_upd_twist)
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   401
done
nipkow@14025
   402
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   403
lemma map_upds_twist [simp]:
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   404
 "a ~: set as ==> m(a|->b)(as[|->]bs) = m(as[|->]bs)(a|->b)"
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   405
apply(insert set_take_subset)
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   406
apply (fastsimp simp add: map_upd_upds_conv_if)
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   407
done
nipkow@14025
   408
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   409
lemma map_upds_apply_nontin[simp]:
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   410
 "!!ys. x ~: set xs ==> (f(xs[|->]ys)) x = f x"
paulson@14208
   411
apply (induct xs, simp)
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   412
apply(case_tac ys)
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   413
 apply(auto simp: map_upd_upds_conv_if)
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   414
done
nipkow@14025
   415
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   416
lemma fun_upds_append_drop[simp]:
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   417
  "!!m ys. size xs = size ys \<Longrightarrow> m(xs@zs[\<mapsto>]ys) = m(xs[\<mapsto>]ys)"
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   418
apply(induct xs)
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   419
 apply (simp)
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   420
apply(case_tac ys)
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   421
apply simp_all
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   422
done
nipkow@14300
   423
nipkow@14300
   424
lemma fun_upds_append2_drop[simp]:
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   425
  "!!m ys. size xs = size ys \<Longrightarrow> m(xs[\<mapsto>]ys@zs) = m(xs[\<mapsto>]ys)"
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   426
apply(induct xs)
nipkow@14300
   427
 apply (simp)
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   428
apply(case_tac ys)
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   429
apply simp_all
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   430
done
nipkow@14300
   431
nipkow@14300
   432
nipkow@14186
   433
lemma restrict_map_upds[simp]: "!!m ys.
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   434
 \<lbrakk> length xs = length ys; set xs \<subseteq> D \<rbrakk>
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   435
 \<Longrightarrow> m(xs [\<mapsto>] ys)|^D = (m|^(D - set xs))(xs [\<mapsto>] ys)"
paulson@14208
   436
apply (induct xs, simp)
paulson@14208
   437
apply (case_tac ys, simp)
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   438
apply(simp add:Diff_insert[symmetric] insert_absorb)
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   439
apply(simp add: map_upd_upds_conv_if)
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   440
done
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   441
nipkow@14186
   442
oheimb@14100
   443
subsection {* @{term map_upd_s} *}
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   444
oheimb@14100
   445
lemma map_upd_s_apply [simp]: 
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   446
  "(m(as{|->}b)) x = (if x : as then Some b else m x)"
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   447
by (simp add: map_upd_s_def)
oheimb@14100
   448
oheimb@14100
   449
lemma map_subst_apply [simp]: 
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   450
  "(m(a~>b)) x = (if m x = Some a then Some b else m x)" 
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   451
by (simp add: map_subst_def)
oheimb@14100
   452
oheimb@14100
   453
subsection {* @{term dom} *}
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   454
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   455
lemma domI: "m a = Some b ==> a : dom m"
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   456
by (unfold dom_def, auto)
oheimb@14100
   457
(* declare domI [intro]? *)
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   458
paulson@15369
   459
lemma domD: "a : dom m ==> \<exists>b. m a = Some b"
paulson@14208
   460
by (unfold dom_def, auto)
webertj@13908
   461
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   462
lemma domIff[iff]: "(a : dom m) = (m a ~= None)"
paulson@14208
   463
by (unfold dom_def, auto)
webertj@13908
   464
declare domIff [simp del]
webertj@13908
   465
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   466
lemma dom_empty[simp]: "dom empty = {}"
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   467
apply (unfold dom_def)
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   468
apply (simp (no_asm))
webertj@13908
   469
done
webertj@13908
   470
nipkow@13910
   471
lemma dom_fun_upd[simp]:
nipkow@13910
   472
 "dom(f(x := y)) = (if y=None then dom f - {x} else insert x (dom f))"
nipkow@13910
   473
by (simp add:dom_def) blast
webertj@13908
   474
nipkow@13937
   475
lemma dom_map_of: "dom(map_of xys) = {x. \<exists>y. (x,y) : set xys}"
nipkow@13937
   476
apply(induct xys)
nipkow@13937
   477
apply(auto simp del:fun_upd_apply)
nipkow@13937
   478
done
nipkow@13937
   479
nipkow@15304
   480
lemma dom_map_of_conv_image_fst:
nipkow@15304
   481
  "dom(map_of xys) = fst ` (set xys)"
nipkow@15304
   482
by(force simp: dom_map_of)
nipkow@15304
   483
nipkow@15110
   484
lemma dom_map_of_zip[simp]: "[| length xs = length ys; distinct xs |] ==>
nipkow@15110
   485
  dom(map_of(zip xs ys)) = set xs"
nipkow@15110
   486
by(induct rule: list_induct2, simp_all)
nipkow@15110
   487
webertj@13908
   488
lemma finite_dom_map_of: "finite (dom (map_of l))"
webertj@13908
   489
apply (unfold dom_def)
paulson@15251
   490
apply (induct "l")
webertj@13908
   491
apply (auto simp add: insert_Collect [symmetric])
webertj@13908
   492
done
webertj@13908
   493
nipkow@14025
   494
lemma dom_map_upds[simp]:
nipkow@14025
   495
 "!!m ys. dom(m(xs[|->]ys)) = set(take (length ys) xs) Un dom m"
paulson@14208
   496
apply (induct xs, simp)
paulson@14208
   497
apply (case_tac ys, auto)
nipkow@14025
   498
done
nipkow@13910
   499
nipkow@14025
   500
lemma dom_map_add[simp]: "dom(m++n) = dom n Un dom m"
paulson@14208
   501
by (unfold dom_def, auto)
nipkow@13910
   502
nipkow@15691
   503
lemma dom_override_on[simp]:
nipkow@15691
   504
 "dom(override_on f g A) =
nipkow@15691
   505
 (dom f  - {a. a : A - dom g}) Un {a. a : A Int dom g}"
nipkow@15691
   506
by(auto simp add: dom_def override_on_def)
webertj@13908
   507
nipkow@14027
   508
lemma map_add_comm: "dom m1 \<inter> dom m2 = {} \<Longrightarrow> m1++m2 = m2++m1"
nipkow@14027
   509
apply(rule ext)
nipkow@14027
   510
apply(fastsimp simp:map_add_def split:option.split)
nipkow@14027
   511
done
nipkow@14027
   512
oheimb@14100
   513
subsection {* @{term ran} *}
oheimb@14100
   514
oheimb@14100
   515
lemma ranI: "m a = Some b ==> b : ran m" 
oheimb@14100
   516
by (auto simp add: ran_def)
oheimb@14100
   517
(* declare ranI [intro]? *)
webertj@13908
   518
nipkow@13910
   519
lemma ran_empty[simp]: "ran empty = {}"
webertj@13908
   520
apply (unfold ran_def)
webertj@13908
   521
apply (simp (no_asm))
webertj@13908
   522
done
webertj@13908
   523
nipkow@13910
   524
lemma ran_map_upd[simp]: "m a = None ==> ran(m(a|->b)) = insert b (ran m)"
paulson@14208
   525
apply (unfold ran_def, auto)
webertj@13908
   526
apply (subgoal_tac "~ (aa = a) ")
webertj@13908
   527
apply auto
webertj@13908
   528
done
nipkow@13910
   529
oheimb@14100
   530
subsection {* @{text "map_le"} *}
nipkow@13910
   531
kleing@13912
   532
lemma map_le_empty [simp]: "empty \<subseteq>\<^sub>m g"
nipkow@13910
   533
by(simp add:map_le_def)
nipkow@13910
   534
nipkow@14187
   535
lemma [simp]: "f(x := None) \<subseteq>\<^sub>m f"
nipkow@14187
   536
by(force simp add:map_le_def)
nipkow@14187
   537
nipkow@13910
   538
lemma map_le_upd[simp]: "f \<subseteq>\<^sub>m g ==> f(a := b) \<subseteq>\<^sub>m g(a := b)"
nipkow@13910
   539
by(fastsimp simp add:map_le_def)
nipkow@13910
   540
nipkow@14187
   541
lemma [simp]: "m1 \<subseteq>\<^sub>m m2 \<Longrightarrow> m1(x := None) \<subseteq>\<^sub>m m2(x \<mapsto> y)"
nipkow@14187
   542
by(force simp add:map_le_def)
nipkow@14187
   543
nipkow@13910
   544
lemma map_le_upds[simp]:
nipkow@13910
   545
 "!!f g bs. f \<subseteq>\<^sub>m g ==> f(as [|->] bs) \<subseteq>\<^sub>m g(as [|->] bs)"
paulson@14208
   546
apply (induct as, simp)
paulson@14208
   547
apply (case_tac bs, auto)
nipkow@14025
   548
done
webertj@13908
   549
webertj@14033
   550
lemma map_le_implies_dom_le: "(f \<subseteq>\<^sub>m g) \<Longrightarrow> (dom f \<subseteq> dom g)"
webertj@14033
   551
  by (fastsimp simp add: map_le_def dom_def)
webertj@14033
   552
webertj@14033
   553
lemma map_le_refl [simp]: "f \<subseteq>\<^sub>m f"
webertj@14033
   554
  by (simp add: map_le_def)
webertj@14033
   555
nipkow@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"
nipkow@14187
   557
by(force simp add:map_le_def)
webertj@14033
   558
webertj@14033
   559
lemma map_le_antisym: "\<lbrakk> f \<subseteq>\<^sub>m g; g \<subseteq>\<^sub>m f \<rbrakk> \<Longrightarrow> f = g"
webertj@14033
   560
  apply (unfold map_le_def)
webertj@14033
   561
  apply (rule ext)
paulson@14208
   562
  apply (case_tac "x \<in> dom f", simp)
paulson@14208
   563
  apply (case_tac "x \<in> dom g", simp, fastsimp)
webertj@14033
   564
done
webertj@14033
   565
webertj@14033
   566
lemma map_le_map_add [simp]: "f \<subseteq>\<^sub>m (g ++ f)"
webertj@14033
   567
  by (fastsimp simp add: map_le_def)
webertj@14033
   568
nipkow@15304
   569
lemma map_le_iff_map_add_commute: "(f \<subseteq>\<^sub>m f ++ g) = (f++g = g++f)"
nipkow@15304
   570
by(fastsimp simp add:map_add_def map_le_def expand_fun_eq split:option.splits)
nipkow@15304
   571
nipkow@15303
   572
lemma map_add_le_mapE: "f++g \<subseteq>\<^sub>m h \<Longrightarrow> g \<subseteq>\<^sub>m h"
nipkow@15303
   573
by (fastsimp simp add: map_le_def map_add_def dom_def)
nipkow@15303
   574
nipkow@15303
   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"
nipkow@15303
   576
by (clarsimp simp add: map_le_def map_add_def dom_def split:option.splits)
nipkow@15303
   577
nipkow@3981
   578
end