--- a/src/HOL/Map.thy Wed Apr 09 12:52:45 2003 +0200
+++ b/src/HOL/Map.thy Fri Apr 11 23:11:13 2003 +0200
@@ -1,14 +1,14 @@
(* Title: HOL/Map.thy
ID: $Id$
Author: Tobias Nipkow, based on a theory by David von Oheimb
- Copyright 1997 TU Muenchen
+ Copyright 1997-2003 TU Muenchen
The datatype of `maps' (written ~=>); strongly resembles maps in VDM.
*)
-Map = List +
+theory Map = List:
-types ('a,'b) "~=>" = 'a => 'b option (infixr 0)
+types ('a,'b) "~=>" = "'a => 'b option" (infixr 0)
consts
chg_map :: "('b => 'b) => 'a => ('a ~=> 'b) => ('a ~=> 'b)"
@@ -24,11 +24,11 @@
("_/'(_/|->_')" [900,0,0]900)
syntax (xsymbols)
- "~=>" :: [type, type] => type (infixr "\\<leadsto>" 0)
+ "~=>" :: "[type, type] => type" (infixr "\<leadsto>" 0)
map_upd :: "('a ~=> 'b) => 'a => 'b => ('a ~=> 'b)"
- ("_/'(_/\\<mapsto>/_')" [900,0,0]900)
+ ("_/'(_/\<mapsto>/_')" [900,0,0]900)
map_upds :: "('a ~=> 'b) => 'a list => 'b list => ('a ~=> 'b)"
- ("_/'(_/[\\<mapsto>]/_')" [900,0,0]900)
+ ("_/'(_/[\<mapsto>]/_')" [900,0,0]900)
translations
"empty" => "_K None"
@@ -38,12 +38,12 @@
defs
-chg_map_def "chg_map f a m == case m a of None => m | Some b => m(a|->f b)"
+chg_map_def: "chg_map f a m == case m a of None => m | Some b => m(a|->f b)"
-override_def "m1++m2 == %x. case m2 x of None => m1 x | Some y => Some y"
+override_def: "m1++m2 == %x. case m2 x of None => m1 x | Some y => Some y"
-dom_def "dom(m) == {a. m a ~= None}"
-ran_def "ran(m) == {b. ? a. m a = Some b}"
+dom_def: "dom(m) == {a. m a ~= None}"
+ran_def: "ran(m) == {b. ? a. m a = Some b}"
primrec
"map_of [] = empty"
@@ -52,4 +52,279 @@
primrec "t([] [|->]bs) = t"
"t(a#as[|->]bs) = t(a|->hd bs)(as[|->]tl bs)"
+
+section "empty"
+
+lemma empty_upd_none: "empty(x := None) = empty"
+apply (rule ext)
+apply (simp (no_asm))
+done
+declare empty_upd_none [simp]
+
+(* FIXME: what is this sum_case nonsense?? *)
+lemma sum_case_empty_empty: "sum_case empty empty = empty"
+apply (rule ext)
+apply (simp (no_asm) split add: sum.split)
+done
+declare sum_case_empty_empty [simp]
+
+
+section "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: "t(k|->x) ~= empty"
+apply safe
+apply (drule_tac x = "k" in fun_cong)
+apply (simp (no_asm_use))
+done
+declare map_upd_nonempty [simp]
+
+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?? *)
+section "sum_case and empty/map_upd"
+
+lemma sum_case_map_upd_empty: "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
+declare sum_case_map_upd_empty [simp]
+
+lemma sum_case_empty_map_upd: "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
+declare sum_case_empty_map_upd [simp]
+
+lemma sum_case_map_upd_map_upd: "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
+declare sum_case_map_upd_map_upd [simp]
+
+
+section "map_upds"
+
+lemma map_upds_twist [rule_format (no_asm)]: "a ~: set as --> (!m bs. (m(a|->b)(as[|->]bs)) = (m(as[|->]bs)(a|->b)))"
+apply (induct_tac "as")
+apply (auto simp del: fun_upd_apply)
+apply (drule spec)+
+apply (rotate_tac -1)
+apply (erule subst)
+apply (erule fun_upd_twist [THEN subst])
+apply (rule refl)
+done
+declare map_upds_twist [simp]
+
+
+section "chg_map"
+
+lemma chg_map_new: "m a = None ==> chg_map f a m = m"
+apply (unfold chg_map_def)
+apply auto
+done
+
+lemma chg_map_upd: "m a = Some b ==> chg_map f a m = m(a|->f b)"
+apply (unfold chg_map_def)
+apply auto
+done
+
+declare chg_map_new [simp] chg_map_upd [simp]
+
+
+section "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
+
+
+section "option_map related"
+
+lemma option_map_o_empty: "option_map f o empty = empty"
+apply (rule ext)
+apply (simp (no_asm))
+done
+
+lemma option_map_o_map_upd: "option_map f o m(a|->b) = (option_map f o m)(a|->f b)"
+apply (rule ext)
+apply (simp (no_asm))
+done
+
+declare option_map_o_empty [simp] option_map_o_map_upd [simp]
+
+
+section "++"
+
+lemma override_empty: "m ++ empty = m"
+apply (unfold override_def)
+apply (simp (no_asm))
+done
+declare override_empty [simp]
+
+lemma empty_override: "empty ++ m = m"
+apply (unfold override_def)
+apply (rule ext)
+apply (simp split add: option.split)
+done
+declare empty_override [simp]
+
+lemma override_Some_iff [rule_format (no_asm)]:
+ "((m ++ n) k = Some x) = (n k = Some x | n k = None & m k = Some x)"
+apply (unfold override_def)
+apply (simp (no_asm) split add: option.split)
+done
+
+lemmas override_SomeD = override_Some_iff [THEN iffD1, standard]
+declare override_SomeD [dest!]
+
+lemma override_find_right: "!!xx. n k = Some xx ==> (m ++ n) k = Some xx"
+apply (subst override_Some_iff)
+apply fast
+done
+declare override_find_right [simp]
+
+lemma override_None: "((m ++ n) k = None) = (n k = None & m k = None)"
+apply (unfold override_def)
+apply (simp (no_asm) split add: option.split)
+done
+declare override_None [iff]
+
+lemma override_upd: "f ++ g(x|->y) = (f ++ g)(x|->y)"
+apply (unfold override_def)
+apply (rule ext)
+apply auto
+done
+declare override_upd [simp]
+
+lemma map_of_override: "map_of ys ++ map_of xs = map_of (xs@ys)"
+apply (unfold override_def)
+apply (rule sym)
+apply (induct_tac "xs")
+apply (simp (no_asm))
+apply (rule ext)
+apply (simp (no_asm_simp) split add: option.split)
+done
+declare map_of_override [simp]
+
+declare fun_upd_apply [simp del]
+lemma finite_range_map_of_override: "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]
+
+
+section "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: "(a : dom m) = (m a ~= None)"
+apply (unfold dom_def)
+apply auto
+done
+declare domIff [iff]
+declare domIff [simp del]
+
+lemma dom_empty: "dom empty = {}"
+apply (unfold dom_def)
+apply (simp (no_asm))
+done
+declare dom_empty [simp]
+
+lemma dom_map_upd: "dom(m(a|->b)) = insert a (dom m)"
+apply (unfold dom_def)
+apply (simp (no_asm))
+apply blast
+done
+declare dom_map_upd [simp]
+
+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_override: "dom(m++n) = dom n Un dom m"
+apply (unfold dom_def)
+apply auto
+done
+declare dom_override [simp]
+
+section "ran"
+
+lemma ran_empty: "ran empty = {}"
+apply (unfold ran_def)
+apply (simp (no_asm))
+done
+declare ran_empty [simp]
+
+lemma ran_empty': "ran (%u. None) = {}"
+apply (unfold ran_def)
+apply auto
+done
+declare ran_empty' [simp]
+
+lemma ran_map_upd: "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
+declare ran_map_upd [simp]
+
end