src/HOL/Map.thy

Added a few lemmas about map_le

(*  Title:      HOL/Map.thy
    ID:         $Id$
    Author:     Tobias Nipkow, based on a theory by David von Oheimb
    Copyright   1997-2003 TU Muenchen

The datatype of `maps' (written ~=>); strongly resembles maps in VDM.
*)

header {* Maps *}

theory Map = List:

types ('a,'b) "~=>" = "'a => 'b option" (infixr 0)

consts
chg_map	:: "('b => 'b) => 'a => ('a ~=> 'b) => ('a ~=> 'b)"
map_add:: "('a ~=> 'b) => ('a ~=> 'b) => ('a ~=> 'b)" (infixl "++" 100)
dom	:: "('a ~=> 'b) => 'a set"
ran	:: "('a ~=> 'b) => 'b set"
map_of	:: "('a * 'b)list => 'a ~=> 'b"
map_upds:: "('a ~=> 'b) => 'a list => 'b list => 
	    ('a ~=> 'b)"		 ("_/'(_[|->]_/')" [900,0,0]900)
map_le  :: "('a ~=> 'b) => ('a ~=> 'b) => bool" (infix "\<subseteq>\<^sub>m" 50)

syntax
empty	::  "'a ~=> 'b"
map_upd	:: "('a ~=> 'b) => 'a => 'b => ('a ~=> 'b)"
					 ("_/'(_/|->_')"   [900,0,0]900)

syntax (xsymbols)
  "~=>"     :: "[type, type] => type"    (infixr "\<leadsto>" 0)
  map_upd   :: "('a ~=> 'b) => 'a      => 'b      => ('a ~=> 'b)"
					  ("_/'(_/\<mapsto>/_')"  [900,0,0]900)
  map_upds  :: "('a ~=> 'b) => 'a list => 'b list => ('a ~=> 'b)"
				         ("_/'(_/[\<mapsto>]/_')" [900,0,0]900)

translations
  "empty"    => "_K None"
  "empty"    <= "%x. None"

  "m(a|->b)" == "m(a:=Some b)"

defs
chg_map_def:  "chg_map f a m == case m a of None => m | Some b => m(a|->f b)"

map_add_def: "m1++m2 == %x. case m2 x of None => m1 x | Some y => Some y"

map_upds_def: "m(xs [|->] ys) == m ++ map_of (rev(zip xs ys))"

dom_def: "dom(m) == {a. m a ~= None}"
ran_def: "ran(m) == {b. EX a. m a = Some b}"

map_le_def: "m1 \<subseteq>\<^sub>m m2  ==  ALL a : dom m1. m1 a = m2 a"

primrec
  "map_of [] = empty"
  "map_of (p#ps) = (map_of ps)(fst p |-> snd p)"


subsection {* empty *}

lemma empty_upd_none[simp]: "empty(x := None) = empty"
apply (rule ext)
apply (simp (no_asm))
done


(* FIXME: what is this sum_case nonsense?? *)
lemma sum_case_empty_empty[simp]: "sum_case empty empty = empty"
apply (rule ext)
apply (simp (no_asm) split add: sum.split)
done

subsection {* map\_upd *}

lemma map_upd_triv: "t k = Some x ==> t(k|->x) = t"
apply (rule ext)
apply (simp (no_asm_simp))
done

lemma map_upd_nonempty[simp]: "t(k|->x) ~= empty"
apply safe
apply (drule_tac x = "k" in fun_cong)
apply (simp (no_asm_use))
done

lemma finite_range_updI: "finite (range f) ==> finite (range (f(a|->b)))"
apply (unfold image_def)
apply (simp (no_asm_use) add: full_SetCompr_eq)
apply (rule finite_subset)
prefer 2 apply (assumption)
apply auto
done


(* FIXME: what is this sum_case nonsense?? *)
subsection {* sum\_case and empty/map\_upd *}

lemma sum_case_map_upd_empty[simp]:
 "sum_case (m(k|->y)) empty =  (sum_case m empty)(Inl k|->y)"
apply (rule ext)
apply (simp (no_asm) split add: sum.split)
done

lemma sum_case_empty_map_upd[simp]:
 "sum_case empty (m(k|->y)) =  (sum_case empty m)(Inr k|->y)"
apply (rule ext)
apply (simp (no_asm) split add: sum.split)
done

lemma sum_case_map_upd_map_upd[simp]:
 "sum_case (m1(k1|->y1)) (m2(k2|->y2)) = (sum_case (m1(k1|->y1)) m2)(Inr k2|->y2)"
apply (rule ext)
apply (simp (no_asm) split add: sum.split)
done


subsection {* chg\_map *}

lemma chg_map_new[simp]: "m a = None   ==> chg_map f a m = m"
apply (unfold chg_map_def)
apply auto
done

lemma chg_map_upd[simp]: "m a = Some b ==> chg_map f a m = m(a|->f b)"
apply (unfold chg_map_def)
apply auto
done


subsection {* map\_of *}

lemma map_of_SomeD [rule_format (no_asm)]: "map_of xs k = Some y --> (k,y):set xs"
apply (induct_tac "xs")
apply  auto
done

lemma map_of_mapk_SomeI [rule_format (no_asm)]: "inj f ==> map_of t k = Some x -->  
   map_of (map (split (%k. Pair (f k))) t) (f k) = Some x"
apply (induct_tac "t")
apply  (auto simp add: inj_eq)
done

lemma weak_map_of_SomeI [rule_format (no_asm)]: "(k, x) : set l --> (? x. map_of l k = Some x)"
apply (induct_tac "l")
apply  auto
done

lemma map_of_filter_in: 
"[| map_of xs k = Some z; P k z |] ==> map_of (filter (split P) xs) k = Some z"
apply (rule mp)
prefer 2 apply (assumption)
apply (erule thin_rl)
apply (induct_tac "xs")
apply  auto
done

lemma finite_range_map_of: "finite (range (map_of l))"
apply (induct_tac "l")
apply  (simp_all (no_asm) add: image_constant)
apply (rule finite_subset)
prefer 2 apply (assumption)
apply auto
done

lemma map_of_map: "map_of (map (%(a,b). (a,f b)) xs) x = option_map f (map_of xs x)"
apply (induct_tac "xs")
apply auto
done


subsection {* option\_map related *}

lemma option_map_o_empty[simp]: "option_map f o empty = empty"
apply (rule ext)
apply (simp (no_asm))
done

lemma option_map_o_map_upd[simp]:
 "option_map f o m(a|->b) = (option_map f o m)(a|->f b)"
apply (rule ext)
apply (simp (no_asm))
done


subsection {* ++ *}

lemma map_add_empty[simp]: "m ++ empty = m"
apply (unfold map_add_def)
apply (simp (no_asm))
done

lemma empty_map_add[simp]: "empty ++ m = m"
apply (unfold map_add_def)
apply (rule ext)
apply (simp split add: option.split)
done

lemma map_add_assoc[simp]: "m1 ++ (m2 ++ m3) = (m1 ++ m2) ++ m3"
apply(rule ext)
apply(simp add: map_add_def split:option.split)
done

lemma map_add_Some_iff: 
 "((m ++ n) k = Some x) = (n k = Some x | n k = None & m k = Some x)"
apply (unfold map_add_def)
apply (simp (no_asm) split add: option.split)
done

lemmas map_add_SomeD = map_add_Some_iff [THEN iffD1, standard]
declare map_add_SomeD [dest!]

lemma map_add_find_right[simp]: "!!xx. n k = Some xx ==> (m ++ n) k = Some xx"
apply (subst map_add_Some_iff)
apply fast
done

lemma map_add_None [iff]: "((m ++ n) k = None) = (n k = None & m k = None)"
apply (unfold map_add_def)
apply (simp (no_asm) split add: option.split)
done

lemma map_add_upd[simp]: "f ++ g(x|->y) = (f ++ g)(x|->y)"
apply (unfold map_add_def)
apply (rule ext)
apply auto
done

lemma map_of_append[simp]: "map_of (xs@ys) = map_of ys ++ map_of xs"
apply (unfold map_add_def)
apply (induct_tac "xs")
apply (simp (no_asm))
apply (rule ext)
apply (simp (no_asm_simp) split add: option.split)
done

declare fun_upd_apply [simp del]
lemma finite_range_map_of_map_add:
 "finite (range f) ==> finite (range (f ++ map_of l))"
apply (induct_tac "l")
apply  auto
apply (erule finite_range_updI)
done
declare fun_upd_apply [simp]


subsection {* map\_upds *}

lemma map_upds_Nil1[simp]: "m([] [|->] bs) = m"
by(simp add:map_upds_def)

lemma map_upds_Nil2[simp]: "m(as [|->] []) = m"
by(simp add:map_upds_def)

lemma map_upds_Cons[simp]: "m(a#as [|->] b#bs) = (m(a|->b))(as[|->]bs)"
by(simp add:map_upds_def)


lemma map_upd_upds_conv_if: "!!x y ys f.
 (f(x|->y))(xs [|->] ys) =
 (if x : set(take (length ys) xs) then f(xs [|->] ys)
                                  else (f(xs [|->] ys))(x|->y))"
apply(induct xs)
 apply simp
apply(case_tac ys)
 apply(auto split:split_if simp:fun_upd_twist)
done

lemma map_upds_twist [simp]:
 "a ~: set as ==> m(a|->b)(as[|->]bs) = m(as[|->]bs)(a|->b)"
apply(insert set_take_subset)
apply (fastsimp simp add: map_upd_upds_conv_if)
done

lemma map_upds_apply_nontin[simp]:
 "!!ys. x ~: set xs ==> (f(xs[|->]ys)) x = f x"
apply(induct xs)
 apply simp
apply(case_tac ys)
 apply(auto simp: map_upd_upds_conv_if)
done

subsection {* dom *}

lemma domI: "m a = Some b ==> a : dom m"
apply (unfold dom_def)
apply auto
done

lemma domD: "a : dom m ==> ? b. m a = Some b"
apply (unfold dom_def)
apply auto
done

lemma domIff[iff]: "(a : dom m) = (m a ~= None)"
apply (unfold dom_def)
apply auto
done
declare domIff [simp del]

lemma dom_empty[simp]: "dom empty = {}"
apply (unfold dom_def)
apply (simp (no_asm))
done

lemma dom_fun_upd[simp]:
 "dom(f(x := y)) = (if y=None then dom f - {x} else insert x (dom f))"
by (simp add:dom_def) blast

lemma dom_map_of: "dom(map_of xys) = {x. \<exists>y. (x,y) : set xys}"
apply(induct xys)
apply(auto simp del:fun_upd_apply)
done

lemma finite_dom_map_of: "finite (dom (map_of l))"
apply (unfold dom_def)
apply (induct_tac "l")
apply (auto simp add: insert_Collect [symmetric])
done

lemma dom_map_upds[simp]:
 "!!m ys. dom(m(xs[|->]ys)) = set(take (length ys) xs) Un dom m"
apply(induct xs)
 apply simp
apply(case_tac ys)
 apply auto
done

lemma dom_map_add[simp]: "dom(m++n) = dom n Un dom m"
apply (unfold dom_def)
apply auto
done

lemma dom_overwrite[simp]:
 "dom(f(g|A)) = (dom f  - {a. a : A - dom g}) Un {a. a : A Int dom g}"
by(auto simp add: dom_def overwrite_def)

lemma map_add_comm: "dom m1 \<inter> dom m2 = {} \<Longrightarrow> m1++m2 = m2++m1"
apply(rule ext)
apply(fastsimp simp:map_add_def split:option.split)
done

subsection {* ran *}

lemma ran_empty[simp]: "ran empty = {}"
apply (unfold ran_def)
apply (simp (no_asm))
done

lemma ran_map_upd[simp]: "m a = None ==> ran(m(a|->b)) = insert b (ran m)"
apply (unfold ran_def)
apply auto
apply (subgoal_tac "~ (aa = a) ")
apply auto
done

subsection {* map\_le *}

lemma map_le_empty [simp]: "empty \<subseteq>\<^sub>m g"
by(simp add:map_le_def)

lemma map_le_upd[simp]: "f \<subseteq>\<^sub>m g ==> f(a := b) \<subseteq>\<^sub>m g(a := b)"
by(fastsimp simp add:map_le_def)

lemma map_le_upds[simp]:
 "!!f g bs. f \<subseteq>\<^sub>m g ==> f(as [|->] bs) \<subseteq>\<^sub>m g(as [|->] bs)"
apply(induct as)
 apply simp
apply(case_tac bs)
 apply auto
done

lemma map_le_implies_dom_le: "(f \<subseteq>\<^sub>m g) \<Longrightarrow> (dom f \<subseteq> dom g)"
  by (fastsimp simp add: map_le_def dom_def)

lemma map_le_refl [simp]: "f \<subseteq>\<^sub>m f"
  by (simp add: map_le_def)

lemma map_le_trans: "\<lbrakk> f \<subseteq>\<^sub>m g; g \<subseteq>\<^sub>m h \<rbrakk> \<Longrightarrow> f \<subseteq>\<^sub>m h"
  apply (clarsimp simp add: map_le_def)
  apply (drule_tac x="a" in bspec, fastsimp)+
  apply assumption
done

lemma map_le_antisym: "\<lbrakk> f \<subseteq>\<^sub>m g; g \<subseteq>\<^sub>m f \<rbrakk> \<Longrightarrow> f = g"
  apply (unfold map_le_def)
  apply (rule ext)
  apply (case_tac "x \<in> dom f")
    apply simp
  apply (case_tac "x \<in> dom g")
    apply simp
  apply fastsimp
done

lemma map_le_map_add [simp]: "f \<subseteq>\<^sub>m (g ++ f)"
  by (fastsimp simp add: map_le_def)

end