src/HOL/Map.thy
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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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theory Map = List:
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types ('a,'b) "~=>" = "'a => 'b option" (infixr 0)
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consts
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chg_map	:: "('b => 'b) => 'a => ('a ~=> 'b) => ('a ~=> 'b)"
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override:: "('a ~=> 'b) => ('a ~=> 'b) => ('a ~=> 'b)" (infixl "++" 100)
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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)"		 ("_/'(_[|->]_/')" [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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empty	::  "'a ~=> 'b"
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map_upd	:: "('a ~=> 'b) => 'a => 'b => ('a ~=> 'b)"
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					 ("_/'(_/|->_')"   [900,0,0]900)
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syntax (xsymbols)
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  "~=>"     :: "[type, type] => type"    (infixr "\<leadsto>" 0)
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  map_upd   :: "('a ~=> 'b) => 'a      => 'b      => ('a ~=> 'b)"
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					  ("_/'(_/\<mapsto>/_')"  [900,0,0]900)
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  map_upds  :: "('a ~=> 'b) => 'a list => 'b list => ('a ~=> 'b)"
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				         ("_/'(_/[\<mapsto>]/_')" [900,0,0]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(a|->b)" == "m(a:=Some b)"
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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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override_def: "m1++m2 == %x. case m2 x of None => m1 x | Some y => Some y"
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dom_def: "dom(m) == {a. m a ~= None}"
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ran_def: "ran(m) == {b. ? 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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primrec "t([]  [|->]bs) = t"
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        "t(a#as[|->]bs) = t(a|->hd bs)(as[|->]tl bs)"
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section {* 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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section {* 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 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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section {* sum\_case and empty/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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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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section {* map\_upds *}
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lemma map_upd_upds_conv_if:
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 "!!x y ys f. (f(x|->y))(xs [|->] ys) =
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              (if x : set xs then f(xs [|->] ys) else (f(xs [|->] ys))(x|->y))"
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apply(induct xs)
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 apply simp
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apply(simp split:split_if add:fun_upd_twist eq_sym_conv)
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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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by (simp add: map_upd_upds_conv_if)
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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(simp add: fun_upd_apply map_upd_upds_conv_if split:split_if)
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done
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section {* 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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section {* 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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section {* 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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section {* ++ *}
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lemma override_empty[simp]: "m ++ empty = m"
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apply (unfold override_def)
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apply (simp (no_asm))
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done
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lemma empty_override[simp]: "empty ++ m = m"
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apply (unfold override_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 override_Some_iff [rule_format (no_asm)]: 
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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 override_def)
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apply (simp (no_asm) split add: option.split)
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done
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lemmas override_SomeD = override_Some_iff [THEN iffD1, standard]
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declare override_SomeD [dest!]
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lemma override_find_right[simp]: "!!xx. n k = Some xx ==> (m ++ n) k = Some xx"
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apply (subst override_Some_iff)
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apply fast
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done
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lemma override_None [iff]: "((m ++ n) k = None) = (n k = None & m k = None)"
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apply (unfold override_def)
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apply (simp (no_asm) split add: option.split)
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done
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lemma override_upd[simp]: "f ++ g(x|->y) = (f ++ g)(x|->y)"
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apply (unfold override_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_override[simp]: "map_of ys ++ map_of xs = map_of (xs@ys)"
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apply (unfold override_def)
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apply (rule sym)
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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_override: "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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   261
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section {* 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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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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   283
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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(*
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lemma dom_map_upd[simp]: "dom(m(a|->b)) = insert a (dom m)"
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apply (unfold dom_def)
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apply (simp (no_asm))
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apply blast
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   293
done
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*)
13908
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   296
lemma finite_dom_map_of: "finite (dom (map_of l))"
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   297
apply (unfold dom_def)
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   298
apply (induct_tac "l")
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apply (auto simp add: insert_Collect [symmetric])
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   300
done
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lemma dom_map_upds[simp]: "!!m vs. dom(m(xs[|->]vs)) = set xs Un dom m"
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by(induct xs, simp_all)
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   304
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lemma dom_override[simp]: "dom(m++n) = dom n Un dom m"
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   306
apply (unfold dom_def)
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apply auto
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done
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   309
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lemma dom_overwrite[simp]:
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   311
 "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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   313
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section {* ran *}
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   315
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lemma ran_empty[simp]: "ran empty = {}"
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apply (unfold ran_def)
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   318
apply (simp (no_asm))
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   319
done
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   320
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   321
lemma ran_map_upd[simp]: "m a = None ==> ran(m(a|->b)) = insert b (ran m)"
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   322
apply (unfold ran_def)
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   323
apply auto
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   324
apply (subgoal_tac "~ (aa = a) ")
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   325
apply auto
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   326
done
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section {* 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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by(induct as, auto)
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end