src/HOL/Map.thy
author nipkow
Wed Sep 03 18:20:57 2003 +0200 (2003-09-03)
changeset 14180 d2e550609c40
parent 14134 0fdf5708c7a8
child 14186 6d2a494e33be
permissions -rw-r--r--
Introduced new syntax for maplets x |-> y
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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 = 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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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_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_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_valF_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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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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subsection {* @{term map_upd_s} *}
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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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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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subsection {* @{term dom} *}
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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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(* declare domI [intro]? *)
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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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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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lemma dom_empty[simp]: "dom empty = {}"
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apply (unfold dom_def)
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apply (simp (no_asm))
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done
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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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lemma dom_map_of: "dom(map_of xys) = {x. \<exists>y. (x,y) : set xys}"
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apply(induct xys)
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apply(auto simp del:fun_upd_apply)
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done
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lemma finite_dom_map_of: "finite (dom (map_of l))"
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apply (unfold dom_def)
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apply (induct_tac "l")
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apply (auto simp add: insert_Collect [symmetric])
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done
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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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 apply auto
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done
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lemma dom_map_add[simp]: "dom(m++n) = dom n Un 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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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}"
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by(auto simp add: dom_def overwrite_def)
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lemma map_add_comm: "dom m1 \<inter> dom m2 = {} \<Longrightarrow> m1++m2 = m2++m1"
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apply(rule ext)
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apply(fastsimp simp:map_add_def split:option.split)
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done
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subsection {* @{term ran} *}
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lemma ranI: "m a = Some b ==> b : ran m" 
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by (auto simp add: ran_def)
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(* declare ranI [intro]? *)
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lemma ran_empty[simp]: "ran empty = {}"
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apply (unfold ran_def)
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apply (simp (no_asm))
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done
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lemma ran_map_upd[simp]: "m a = None ==> ran(m(a|->b)) = insert b (ran m)"
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apply (unfold ran_def)
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apply auto
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apply (subgoal_tac "~ (aa = a) ")
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apply auto
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done
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subsection {* @{text "map_le"} *}
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lemma map_le_empty [simp]: "empty \<subseteq>\<^sub>m g"
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by(simp add:map_le_def)
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lemma map_le_upd[simp]: "f \<subseteq>\<^sub>m g ==> f(a := b) \<subseteq>\<^sub>m g(a := b)"
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by(fastsimp simp add:map_le_def)
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lemma map_le_upds[simp]:
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 "!!f g bs. f \<subseteq>\<^sub>m g ==> f(as [|->] bs) \<subseteq>\<^sub>m g(as [|->] bs)"
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apply(induct as)
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 apply simp
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apply(case_tac bs)
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 apply auto
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done
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lemma map_le_implies_dom_le: "(f \<subseteq>\<^sub>m g) \<Longrightarrow> (dom f \<subseteq> dom g)"
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  by (fastsimp simp add: map_le_def dom_def)
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lemma map_le_refl [simp]: "f \<subseteq>\<^sub>m f"
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  by (simp add: map_le_def)
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lemma map_le_trans: "\<lbrakk> f \<subseteq>\<^sub>m g; g \<subseteq>\<^sub>m h \<rbrakk> \<Longrightarrow> f \<subseteq>\<^sub>m h"
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  apply (clarsimp simp add: map_le_def)
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  apply (drule_tac x="a" in bspec, fastsimp)+
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  apply assumption
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done
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lemma map_le_antisym: "\<lbrakk> f \<subseteq>\<^sub>m g; g \<subseteq>\<^sub>m f \<rbrakk> \<Longrightarrow> f = g"
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  apply (unfold map_le_def)
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  apply (rule ext)
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  apply (case_tac "x \<in> dom f")
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    apply simp
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  apply (case_tac "x \<in> dom g")
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    apply simp
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  apply fastsimp
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done
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lemma map_le_map_add [simp]: "f \<subseteq>\<^sub>m (g ++ f)"
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  by (fastsimp simp add: map_le_def)
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end