src/HOL/Map.thy
changeset 20800 69c82605efcf
parent 19947 29b376397cd5
child 21210 c17fd2df4e9e
--- a/src/HOL/Map.thy	Sat Sep 30 21:39:22 2006 +0200
+++ b/src/HOL/Map.thy	Sat Sep 30 21:39:24 2006 +0200
@@ -12,35 +12,47 @@
 imports List
 begin
 
-types ('a,'b) "~=>" = "'a => 'b option" (infixr 0)
+types ('a,'b) "~=>" = "'a => 'b option"  (infixr 0)
 translations (type) "a ~=> b " <= (type) "a => b option"
 
 syntax (xsymbols)
-  "~=>"     :: "[type, type] => type"    (infixr "\<rightharpoonup>" 0)
+  "~=>" :: "[type, type] => type"  (infixr "\<rightharpoonup>" 0)
 
 abbreviation
-  empty     ::  "'a ~=> 'b"
+  empty :: "'a ~=> 'b"
   "empty == %x. None"
 
 definition
-  map_comp :: "('b ~=> 'c)  => ('a ~=> 'b) => ('a ~=> 'c)" (infixl "o'_m" 55)
-  "f o_m g  == (\<lambda>k. case g k of None \<Rightarrow> None | Some v \<Rightarrow> f v)"
+  map_comp :: "('b ~=> 'c)  => ('a ~=> 'b) => ('a ~=> 'c)"  (infixl "o'_m" 55)
+  "f o_m g = (\<lambda>k. case g k of None \<Rightarrow> None | Some v \<Rightarrow> f v)"
 
 const_syntax (xsymbols)
   map_comp  (infixl "\<circ>\<^sub>m" 55)
 
-consts
-map_add :: "('a ~=> 'b) => ('a ~=> 'b) => ('a ~=> 'b)" (infixl "++" 100)
-restrict_map :: "('a ~=> 'b) => 'a set => ('a ~=> 'b)" (infixl "|`"  110)
-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)"
-map_le  :: "('a ~=> 'b) => ('a ~=> 'b) => bool" (infix "\<subseteq>\<^sub>m" 50)
+definition
+  map_add :: "('a ~=> 'b) => ('a ~=> 'b) => ('a ~=> 'b)"  (infixl "++" 100)
+  "m1 ++ m2 = (\<lambda>x. case m2 x of None => m1 x | Some y => Some y)"
+
+  restrict_map :: "('a ~=> 'b) => 'a set => ('a ~=> 'b)"  (infixl "|`"  110)
+  "m|`A = (\<lambda>x. if x : A then m x else None)"
 
 const_syntax (latex output)
   restrict_map  ("_\<restriction>\<^bsub>_\<^esub>" [111,110] 110)
 
+definition
+  dom :: "('a ~=> 'b) => 'a set"
+  "dom m = {a. m a ~= None}"
+
+  ran :: "('a ~=> 'b) => 'b set"
+  "ran m = {b. EX a. m a = Some b}"
+
+  map_le :: "('a ~=> 'b) => ('a ~=> 'b) => bool"  (infix "\<subseteq>\<^sub>m" 50)
+  "(m\<^isub>1 \<subseteq>\<^sub>m m\<^isub>2) = (\<forall>a \<in> dom m\<^isub>1. m\<^isub>1 a = m\<^isub>2 a)"
+
+consts
+  map_of :: "('a * 'b) list => 'a ~=> 'b"
+  map_upds :: "('a ~=> 'b) => 'a list => 'b list => ('a ~=> 'b)"
+
 nonterminals
   maplets maplet
 
@@ -64,503 +76,472 @@
   "_Map (_Maplets ms1 ms2)"     <= "_MapUpd (_Map ms1) ms2"
   "_Maplets ms1 (_Maplets ms2 ms3)" <= "_Maplets (_Maplets ms1 ms2) ms3"
 
-defs
-map_add_def:   "m1++m2 == %x. case m2 x of None => m1 x | Some y => Some y"
-restrict_map_def: "m|`A == %x. if x : A then m x else None"
-
-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: "m\<^isub>1 \<subseteq>\<^sub>m m\<^isub>2  ==  ALL a : dom m\<^isub>1. m\<^isub>1 a = m\<^isub>2 a"
-
 primrec
   "map_of [] = empty"
   "map_of (p#ps) = (map_of ps)(fst p |-> snd p)"
 
+defs
+  map_upds_def: "m(xs [|->] ys) == m ++ map_of (rev(zip xs ys))"
+
 (* special purpose constants that should be defined somewhere else and
 whose syntax is a bit odd as well:
 
  "@chg_map" :: "('a ~=> 'b) => 'a => ('b => 'b) => ('a ~=> 'b)"
-					  ("_/'(_/\<mapsto>\<lambda>_. _')"  [900,0,0,0] 900)
+                                          ("_/'(_/\<mapsto>\<lambda>_. _')"  [900,0,0,0] 900)
   "m(x\<mapsto>\<lambda>y. f)" == "chg_map (\<lambda>y. f) x m"
 
-map_upd_s::"('a ~=> 'b) => 'a set => 'b => 
-	    ('a ~=> 'b)"			 ("_/'(_{|->}_/')" [900,0,0]900)
-map_subst::"('a ~=> 'b) => 'b => 'b => 
-	    ('a ~=> 'b)"			 ("_/'(_~>_/')"    [900,0,0]900)
+map_upd_s::"('a ~=> 'b) => 'a set => 'b =>
+            ('a ~=> 'b)"                         ("_/'(_{|->}_/')" [900,0,0]900)
+map_subst::"('a ~=> 'b) => 'b => 'b =>
+            ('a ~=> 'b)"                         ("_/'(_~>_/')"    [900,0,0]900)
 
 map_upd_s_def: "m(as{|->}b) == %x. if x : as then Some b else m x"
 map_subst_def: "m(a~>b)     == %x. if m x = Some a then Some b else m x"
 
   map_upd_s  :: "('a ~=> 'b) => 'a set => 'b => ('a ~=> 'b)"
-				    		 ("_/'(_/{\<mapsto>}/_')" [900,0,0]900)
-  map_subst :: "('a ~=> 'b) => 'b => 'b => 
-	        ('a ~=> 'b)"			 ("_/'(_\<leadsto>_/')"    [900,0,0]900)
+                                                 ("_/'(_/{\<mapsto>}/_')" [900,0,0]900)
+  map_subst :: "('a ~=> 'b) => 'b => 'b =>
+                ('a ~=> 'b)"                     ("_/'(_\<leadsto>_/')"    [900,0,0]900)
 
 
 subsection {* @{term [source] map_upd_s} *}
 
-lemma map_upd_s_apply [simp]: 
+lemma map_upd_s_apply [simp]:
   "(m(as{|->}b)) x = (if x : as then Some b else m x)"
 by (simp add: map_upd_s_def)
 
-lemma map_subst_apply [simp]: 
-  "(m(a~>b)) x = (if m x = Some a then Some b else m x)" 
+lemma map_subst_apply [simp]:
+  "(m(a~>b)) x = (if m x = Some a then Some b else m x)"
 by (simp add: map_subst_def)
 
 *)
 
+
 subsection {* @{term [source] empty} *}
 
-lemma empty_upd_none[simp]: "empty(x := None) = empty"
-apply (rule ext)
-apply (simp (no_asm))
-done
-
+lemma empty_upd_none [simp]: "empty(x := None) = empty"
+  by (rule ext) simp
 
 (* 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
+  by (rule ext) (simp split: sum.split)
+
 
 subsection {* @{term [source] map_upd} *}
 
 lemma map_upd_triv: "t k = Some x ==> t(k|->x) = t"
-apply (rule ext)
-apply (simp (no_asm_simp))
-done
+  by (rule ext) simp
 
-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 map_upd_nonempty [simp]: "t(k|->x) ~= empty"
+proof
+  assume "t(k \<mapsto> x) = empty"
+  then have "(t(k \<mapsto> x)) k = None" by simp
+  then show False by simp
+qed
 
-lemma map_upd_eqD1: "m(a\<mapsto>x) = n(a\<mapsto>y) \<Longrightarrow> x = y"
-by (drule fun_cong [of _ _ a], auto)
+lemma map_upd_eqD1:
+  assumes "m(a\<mapsto>x) = n(a\<mapsto>y)"
+  shows "x = y"
+proof -
+  from prems have "(m(a\<mapsto>x)) a = (n(a\<mapsto>y)) a" by simp
+  then show ?thesis by simp
+qed
 
-lemma map_upd_Some_unfold: 
-  "((m(a|->b)) x = Some y) = (x = a \<and> b = y \<or> x \<noteq> a \<and> m x = Some y)"
-by auto
+lemma map_upd_Some_unfold:
+    "((m(a|->b)) x = Some y) = (x = a \<and> b = y \<or> x \<noteq> a \<and> m x = Some y)"
+  by auto
 
-lemma image_map_upd[simp]: "x \<notin> A \<Longrightarrow> m(x \<mapsto> y) ` A = m ` A"
-by fastsimp
+lemma image_map_upd [simp]: "x \<notin> A \<Longrightarrow> m(x \<mapsto> y) ` A = m ` A"
+  by auto
 
 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
+  unfolding 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 {* @{term [source] sum_case} and @{term [source] empty}/@{term [source] 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_map_upd_empty [simp]:
+    "sum_case (m(k|->y)) empty = (sum_case m empty)(Inl k|->y)"
+  by (rule ext) (simp split: sum.split)
 
-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_empty_map_upd [simp]:
+    "sum_case empty (m(k|->y)) = (sum_case empty m)(Inr k|->y)"
+  by (rule ext) (simp split: sum.split)
 
-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
+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)"
+  by (rule ext) (simp split: sum.split)
 
 
 subsection {* @{term [source] map_of} *}
 
 lemma map_of_eq_None_iff:
- "(map_of xys x = None) = (x \<notin> fst ` (set xys))"
-by (induct xys) simp_all
+    "(map_of xys x = None) = (x \<notin> fst ` (set xys))"
+  by (induct xys) simp_all
 
 lemma map_of_is_SomeD:
- "map_of xys x = Some y \<Longrightarrow> (x,y) \<in> set xys"
-apply(induct xys)
- apply simp
-apply(clarsimp split:if_splits)
-done
+    "map_of xys x = Some y \<Longrightarrow> (x,y) \<in> set xys"
+  apply (induct xys)
+   apply simp
+  apply (clarsimp split: if_splits)
+  done
 
-lemma map_of_eq_Some_iff[simp]:
- "distinct(map fst xys) \<Longrightarrow> (map_of xys x = Some y) = ((x,y) \<in> set xys)"
-apply(induct xys)
- apply(simp)
-apply(auto simp:map_of_eq_None_iff[symmetric])
-done
+lemma map_of_eq_Some_iff [simp]:
+    "distinct(map fst xys) \<Longrightarrow> (map_of xys x = Some y) = ((x,y) \<in> set xys)"
+  apply (induct xys)
+   apply simp
+  apply (auto simp: map_of_eq_None_iff [symmetric])
+  done
 
-lemma Some_eq_map_of_iff[simp]:
- "distinct(map fst xys) \<Longrightarrow> (Some y = map_of xys x) = ((x,y) \<in> set xys)"
-by(auto simp del:map_of_eq_Some_iff simp add:map_of_eq_Some_iff[symmetric])
+lemma Some_eq_map_of_iff [simp]:
+    "distinct(map fst xys) \<Longrightarrow> (Some y = map_of xys x) = ((x,y) \<in> set xys)"
+  by (auto simp del:map_of_eq_Some_iff simp add: map_of_eq_Some_iff [symmetric])
 
 lemma map_of_is_SomeI [simp]: "\<lbrakk> distinct(map fst xys); (x,y) \<in> set xys \<rbrakk>
-  \<Longrightarrow> map_of xys x = Some y"
-apply (induct xys)
- apply simp
-apply force
-done
+    \<Longrightarrow> map_of xys x = Some y"
+  apply (induct xys)
+   apply simp
+  apply force
+  done
 
-lemma map_of_zip_is_None[simp]:
-  "length xs = length ys \<Longrightarrow> (map_of (zip xs ys) x = None) = (x \<notin> set xs)"
-by (induct rule:list_induct2, simp_all)
+lemma map_of_zip_is_None [simp]:
+    "length xs = length ys \<Longrightarrow> (map_of (zip xs ys) x = None) = (x \<notin> set xs)"
+  by (induct rule: list_induct2) simp_all
 
 lemma finite_range_map_of: "finite (range (map_of xys))"
-apply (induct xys)
-apply  (simp_all (no_asm) add: image_constant)
-apply (rule finite_subset)
-prefer 2 apply assumption
-apply auto
-done
+  apply (induct xys)
+   apply (simp_all add: image_constant)
+  apply (rule finite_subset)
+   prefer 2 apply assumption
+  apply auto
+  done
 
-lemma map_of_SomeD [rule_format]: "map_of xs k = Some y --> (k,y):set xs"
-by (induct "xs", auto)
+lemma map_of_SomeD: "map_of xs k = Some y \<Longrightarrow> (k, y) \<in> set xs"
+  by (induct xs) (simp, atomize (full), auto)
 
-lemma map_of_mapk_SomeI [rule_format]:
-     "inj f ==> map_of t k = Some x -->  
-        map_of (map (split (%k. Pair (f k))) t) (f k) = Some x"
-apply (induct "t")
-apply  (auto simp add: inj_eq)
-done
+lemma map_of_mapk_SomeI:
+  assumes "inj f"
+  shows "map_of t k = Some x ==>
+    map_of (map (split (%k. Pair (f k))) t) (f k) = Some x"
+  by (induct t) (auto simp add: `inj f` inj_eq)
 
-lemma weak_map_of_SomeI [rule_format]:
-     "(k, x) : set l --> (\<exists>x. map_of l k = Some x)"
-by (induct "l", auto)
+lemma weak_map_of_SomeI: "(k, x) : set l ==> \<exists>x. map_of l k = Some x"
+  by (induct l) auto
 
-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 "xs", auto)
-done
+lemma map_of_filter_in:
+  assumes 1: "map_of xs k = Some z"
+    and 2: "P k z"
+  shows "map_of (filter (split P) xs) k = Some z"
+  using 1 by (induct xs) (insert 2, auto)
 
 lemma map_of_map: "map_of (map (%(a,b). (a,f b)) xs) x = option_map f (map_of xs x)"
-by (induct "xs", auto)
+  by (induct xs) auto
 
 
 subsection {* @{term [source] 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_empty [simp]: "option_map f o empty = empty"
+  by (rule ext) simp
 
-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
+lemma option_map_o_map_upd [simp]:
+    "option_map f o m(a|->b) = (option_map f o m)(a|->f b)"
+  by (rule ext) simp
+
 
 subsection {* @{term [source] map_comp} related *}
 
-lemma map_comp_empty [simp]: 
-  "m \<circ>\<^sub>m empty = empty"
-  "empty \<circ>\<^sub>m m = empty"
+lemma map_comp_empty [simp]:
+    "m \<circ>\<^sub>m empty = empty"
+    "empty \<circ>\<^sub>m m = empty"
   by (auto simp add: map_comp_def intro: ext split: option.splits)
 
-lemma map_comp_simps [simp]: 
-  "m2 k = None \<Longrightarrow> (m1 \<circ>\<^sub>m m2) k = None"
-  "m2 k = Some k' \<Longrightarrow> (m1 \<circ>\<^sub>m m2) k = m1 k'" 
+lemma map_comp_simps [simp]:
+    "m2 k = None \<Longrightarrow> (m1 \<circ>\<^sub>m m2) k = None"
+    "m2 k = Some k' \<Longrightarrow> (m1 \<circ>\<^sub>m m2) k = m1 k'"
   by (auto simp add: map_comp_def)
 
 lemma map_comp_Some_iff:
-  "((m1 \<circ>\<^sub>m m2) k = Some v) = (\<exists>k'. m2 k = Some k' \<and> m1 k' = Some v)" 
+    "((m1 \<circ>\<^sub>m m2) k = Some v) = (\<exists>k'. m2 k = Some k' \<and> m1 k' = Some v)"
   by (auto simp add: map_comp_def split: option.splits)
 
 lemma map_comp_None_iff:
-  "((m1 \<circ>\<^sub>m m2) k = None) = (m2 k = None \<or> (\<exists>k'. m2 k = Some k' \<and> m1 k' = None)) " 
+    "((m1 \<circ>\<^sub>m m2) k = None) = (m2 k = None \<or> (\<exists>k'. m2 k = Some k' \<and> m1 k' = None)) "
   by (auto simp add: map_comp_def split: option.splits)
 
+
 subsection {* @{text "++"} *}
 
 lemma map_add_empty[simp]: "m ++ empty = m"
-apply (unfold map_add_def)
-apply (simp (no_asm))
-done
+  unfolding map_add_def by simp
 
 lemma empty_map_add[simp]: "empty ++ m = m"
-apply (unfold map_add_def)
-apply (rule ext)
-apply (simp split add: option.split)
-done
+  unfolding map_add_def by (rule ext) (simp split: option.split)
 
 lemma map_add_assoc[simp]: "m1 ++ (m2 ++ m3) = (m1 ++ m2) ++ m3"
-apply(rule ext)
-apply(simp add: map_add_def split:option.split)
-done
+  unfolding map_add_def by (rule ext) (simp add: map_add_def split: option.split)
+
+lemma map_add_Some_iff:
+    "((m ++ n) k = Some x) = (n k = Some x | n k = None & m k = Some x)"
+  unfolding map_add_def by (simp split: option.split)
 
-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
+lemma map_add_SomeD [dest!]:
+    "(m ++ n) k = Some x \<Longrightarrow> n k = Some x \<or> n k = None \<and> m k = Some x"
+  by (rule map_add_Some_iff [THEN iffD1])
 
-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"
-by (subst map_add_Some_iff, fast)
+lemma map_add_find_right [simp]: "!!xx. n k = Some xx ==> (m ++ n) k = Some xx"
+  by (subst map_add_Some_iff) fast
 
 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
+  unfolding map_add_def by (simp split: option.split)
 
 lemma map_add_upd[simp]: "f ++ g(x|->y) = (f ++ g)(x|->y)"
-apply (unfold map_add_def)
-apply (rule ext, auto)
-done
+  unfolding map_add_def by (rule ext) simp
 
 lemma map_add_upds[simp]: "m1 ++ (m2(xs[\<mapsto>]ys)) = (m1++m2)(xs[\<mapsto>]ys)"
-by(simp add:map_upds_def)
+  by (simp add: map_upds_def)
 
-lemma map_of_append[simp]: "map_of (xs@ys) = map_of ys ++ map_of xs"
-apply (unfold map_add_def)
-apply (induct "xs")
-apply (simp (no_asm))
-apply (rule ext)
-apply (simp (no_asm_simp) split add: option.split)
-done
+lemma map_of_append[simp]: "map_of (xs @ ys) = map_of ys ++ map_of xs"
+  unfolding map_add_def
+  apply (induct xs)
+   apply simp
+  apply (rule ext)
+  apply (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 "l", auto)
-apply (erule finite_range_updI)
-done
-declare fun_upd_apply [simp]
+  "finite (range f) ==> finite (range (f ++ map_of l))"
+  apply (induct l)
+   apply (auto simp del: fun_upd_apply)
+  apply (erule finite_range_updI)
+  done
 
-lemma inj_on_map_add_dom[iff]:
- "inj_on (m ++ m') (dom m') = inj_on m' (dom m')"
-by(fastsimp simp add:map_add_def dom_def inj_on_def split:option.splits)
+lemma inj_on_map_add_dom [iff]:
+    "inj_on (m ++ m') (dom m') = inj_on m' (dom m')"
+  unfolding map_add_def dom_def inj_on_def
+  by (fastsimp split: option.splits)
+
 
 subsection {* @{term [source] restrict_map} *}
 
-lemma restrict_map_to_empty[simp]: "m|`{} = empty"
-by(simp add: restrict_map_def)
+lemma restrict_map_to_empty [simp]: "m|`{} = empty"
+  by (simp add: restrict_map_def)
 
-lemma restrict_map_empty[simp]: "empty|`D = empty"
-by(simp add: restrict_map_def)
+lemma restrict_map_empty [simp]: "empty|`D = empty"
+  by (simp add: restrict_map_def)
 
 lemma restrict_in [simp]: "x \<in> A \<Longrightarrow> (m|`A) x = m x"
-by (auto simp: restrict_map_def)
+  by (simp add: restrict_map_def)
 
 lemma restrict_out [simp]: "x \<notin> A \<Longrightarrow> (m|`A) x = None"
-by (auto simp: restrict_map_def)
+  by (simp add: restrict_map_def)
 
 lemma ran_restrictD: "y \<in> ran (m|`A) \<Longrightarrow> \<exists>x\<in>A. m x = Some y"
-by (auto simp: restrict_map_def ran_def split: split_if_asm)
+  by (auto simp: restrict_map_def ran_def split: split_if_asm)
 
 lemma dom_restrict [simp]: "dom (m|`A) = dom m \<inter> A"
-by (auto simp: restrict_map_def dom_def split: split_if_asm)
+  by (auto simp: restrict_map_def dom_def split: split_if_asm)
 
 lemma restrict_upd_same [simp]: "m(x\<mapsto>y)|`(-{x}) = m|`(-{x})"
-by (rule ext, auto simp: restrict_map_def)
+  by (rule ext) (auto simp: restrict_map_def)
 
 lemma restrict_restrict [simp]: "m|`A|`B = m|`(A\<inter>B)"
-by (rule ext, auto simp: restrict_map_def)
+  by (rule ext) (auto simp: restrict_map_def)
 
-lemma restrict_fun_upd[simp]:
- "m(x := y)|`D = (if x \<in> D then (m|`(D-{x}))(x := y) else m|`D)"
-by(simp add: restrict_map_def expand_fun_eq)
+lemma restrict_fun_upd [simp]:
+    "m(x := y)|`D = (if x \<in> D then (m|`(D-{x}))(x := y) else m|`D)"
+  by (simp add: restrict_map_def expand_fun_eq)
 
-lemma fun_upd_None_restrict[simp]:
-  "(m|`D)(x := None) = (if x:D then m|`(D - {x}) else m|`D)"
-by(simp add: restrict_map_def expand_fun_eq)
+lemma fun_upd_None_restrict [simp]:
+    "(m|`D)(x := None) = (if x:D then m|`(D - {x}) else m|`D)"
+  by (simp add: restrict_map_def expand_fun_eq)
 
-lemma fun_upd_restrict:
- "(m|`D)(x := y) = (m|`(D-{x}))(x := y)"
-by(simp add: restrict_map_def expand_fun_eq)
+lemma fun_upd_restrict: "(m|`D)(x := y) = (m|`(D-{x}))(x := y)"
+  by (simp add: restrict_map_def expand_fun_eq)
 
-lemma fun_upd_restrict_conv[simp]:
- "x \<in> D \<Longrightarrow> (m|`D)(x := y) = (m|`(D-{x}))(x := y)"
-by(simp add: restrict_map_def expand_fun_eq)
+lemma fun_upd_restrict_conv [simp]:
+    "x \<in> D \<Longrightarrow> (m|`D)(x := y) = (m|`(D-{x}))(x := y)"
+  by (simp add: restrict_map_def expand_fun_eq)
 
 
 subsection {* @{term [source] map_upds} *}
 
-lemma map_upds_Nil1[simp]: "m([] [|->] bs) = m"
-by(simp add:map_upds_def)
+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_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_upds_Cons[simp]: "m(a#as [|->] b#bs) = (m(a|->b))(as[|->]bs)"
-by(simp add:map_upds_def)
-
-lemma map_upds_append1[simp]: "\<And>ys m. size xs < size ys \<Longrightarrow>
-  m(xs@[x] [\<mapsto>] ys) = m(xs [\<mapsto>] ys)(x \<mapsto> ys!size xs)"
-apply(induct xs)
- apply(clarsimp simp add:neq_Nil_conv)
-apply (case_tac ys, simp, simp)
-done
+lemma map_upds_append1 [simp]: "\<And>ys m. size xs < size ys \<Longrightarrow>
+    m(xs@[x] [\<mapsto>] ys) = m(xs [\<mapsto>] ys)(x \<mapsto> ys!size xs)"
+  apply(induct xs)
+   apply (clarsimp simp add: neq_Nil_conv)
+  apply (case_tac ys)
+   apply simp
+  apply simp
+  done
 
-lemma map_upds_list_update2_drop[simp]:
- "\<And>m ys i. \<lbrakk>size xs \<le> i; i < size ys\<rbrakk>
-     \<Longrightarrow> m(xs[\<mapsto>]ys[i:=y]) = m(xs[\<mapsto>]ys)"
-apply (induct xs, simp)
-apply (case_tac ys, simp)
-apply(simp split:nat.split)
-done
+lemma map_upds_list_update2_drop [simp]:
+  "\<lbrakk>size xs \<le> i; i < size ys\<rbrakk>
+    \<Longrightarrow> m(xs[\<mapsto>]ys[i:=y]) = m(xs[\<mapsto>]ys)"
+  apply (induct xs arbitrary: m ys i)
+   apply simp
+  apply (case_tac ys)
+   apply simp
+  apply (simp split: nat.split)
+  done
 
-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, simp)
-apply(case_tac ys)
- apply(auto split:split_if simp:fun_upd_twist)
-done
+lemma map_upd_upds_conv_if:
+  "(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 arbitrary: x y ys f)
+   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
+    "a ~: set as ==> m(a|->b)(as[|->]bs) = m(as[|->]bs)(a|->b)"
+  using set_take_subset by (fastsimp simp add: map_upd_upds_conv_if)
 
-lemma map_upds_apply_nontin[simp]:
- "!!ys. x ~: set xs ==> (f(xs[|->]ys)) x = f x"
-apply (induct xs, simp)
-apply(case_tac ys)
- apply(auto simp: map_upd_upds_conv_if)
-done
+lemma map_upds_apply_nontin [simp]:
+    "x ~: set xs ==> (f(xs[|->]ys)) x = f x"
+  apply (induct xs arbitrary: ys)
+   apply simp
+  apply (case_tac ys)
+   apply (auto simp: map_upd_upds_conv_if)
+  done
 
-lemma fun_upds_append_drop[simp]:
-  "!!m ys. size xs = size ys \<Longrightarrow> m(xs@zs[\<mapsto>]ys) = m(xs[\<mapsto>]ys)"
-apply(induct xs)
- apply (simp)
-apply(case_tac ys)
-apply simp_all
-done
+lemma fun_upds_append_drop [simp]:
+    "size xs = size ys \<Longrightarrow> m(xs@zs[\<mapsto>]ys) = m(xs[\<mapsto>]ys)"
+  apply (induct xs arbitrary: m ys)
+   apply simp
+  apply (case_tac ys)
+   apply simp_all
+  done
 
-lemma fun_upds_append2_drop[simp]:
-  "!!m ys. size xs = size ys \<Longrightarrow> m(xs[\<mapsto>]ys@zs) = m(xs[\<mapsto>]ys)"
-apply(induct xs)
- apply (simp)
-apply(case_tac ys)
-apply simp_all
-done
+lemma fun_upds_append2_drop [simp]:
+    "size xs = size ys \<Longrightarrow> m(xs[\<mapsto>]ys@zs) = m(xs[\<mapsto>]ys)"
+  apply (induct xs arbitrary: m ys)
+   apply simp
+  apply (case_tac ys)
+   apply simp_all
+  done
 
 
-lemma restrict_map_upds[simp]: "!!m ys.
- \<lbrakk> length xs = length ys; set xs \<subseteq> D \<rbrakk>
- \<Longrightarrow> m(xs [\<mapsto>] ys)|`D = (m|`(D - set xs))(xs [\<mapsto>] ys)"
-apply (induct xs, simp)
-apply (case_tac ys, simp)
-apply(simp add:Diff_insert[symmetric] insert_absorb)
-apply(simp add: map_upd_upds_conv_if)
-done
+lemma restrict_map_upds[simp]:
+  "\<lbrakk> length xs = length ys; set xs \<subseteq> D \<rbrakk>
+    \<Longrightarrow> m(xs [\<mapsto>] ys)|`D = (m|`(D - set xs))(xs [\<mapsto>] ys)"
+  apply (induct xs arbitrary: m ys)
+   apply simp
+  apply (case_tac ys)
+   apply simp
+  apply (simp add: Diff_insert [symmetric] insert_absorb)
+  apply (simp add: map_upd_upds_conv_if)
+  done
 
 
 subsection {* @{term [source] dom} *}
 
 lemma domI: "m a = Some b ==> a : dom m"
-by (unfold dom_def, auto)
+  unfolding dom_def by simp
 (* declare domI [intro]? *)
 
 lemma domD: "a : dom m ==> \<exists>b. m a = Some b"
-apply (case_tac "m a") 
-apply (auto simp add: dom_def) 
-done
+  by (cases "m a") (auto simp add: dom_def)
 
-lemma domIff[iff]: "(a : dom m) = (m a ~= None)"
-by (unfold dom_def, auto)
-declare domIff [simp del]
+lemma domIff [iff, simp del]: "(a : dom m) = (m a ~= None)"
+  unfolding dom_def by simp
 
-lemma dom_empty[simp]: "dom empty = {}"
-apply (unfold dom_def)
-apply (simp (no_asm))
-done
+lemma dom_empty [simp]: "dom empty = {}"
+  unfolding dom_def by simp
 
-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_fun_upd [simp]:
+    "dom(f(x := y)) = (if y=None then dom f - {x} else insert x (dom f))"
+  unfolding dom_def by auto
 
 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
+  by (induct xys) (auto simp del: fun_upd_apply)
 
 lemma dom_map_of_conv_image_fst:
-  "dom(map_of xys) = fst ` (set xys)"
-by(force simp: dom_map_of)
+    "dom(map_of xys) = fst ` (set xys)"
+  unfolding dom_map_of by force
 
-lemma dom_map_of_zip[simp]: "[| length xs = length ys; distinct xs |] ==>
-  dom(map_of(zip xs ys)) = set xs"
-by(induct rule: list_induct2, simp_all)
+lemma dom_map_of_zip [simp]: "[| length xs = length ys; distinct xs |] ==>
+    dom(map_of(zip xs ys)) = set xs"
+  by (induct rule: list_induct2) simp_all
 
 lemma finite_dom_map_of: "finite (dom (map_of l))"
-apply (unfold dom_def)
-apply (induct "l")
-apply (auto simp add: insert_Collect [symmetric])
-done
+  unfolding dom_def
+  by (induct l) (auto simp add: insert_Collect [symmetric])
 
-lemma dom_map_upds[simp]:
- "!!m ys. dom(m(xs[|->]ys)) = set(take (length ys) xs) Un dom m"
-apply (induct xs, simp)
-apply (case_tac ys, auto)
-done
+lemma dom_map_upds [simp]:
+    "dom(m(xs[|->]ys)) = set(take (length ys) xs) Un dom m"
+  apply (induct xs arbitrary: m ys)
+   apply simp
+  apply (case_tac ys)
+   apply auto
+  done
 
-lemma dom_map_add[simp]: "dom(m++n) = dom n Un dom m"
-by (unfold dom_def, auto)
+lemma dom_map_add [simp]: "dom(m++n) = dom n Un dom m"
+  unfolding dom_def by auto
 
-lemma dom_override_on[simp]:
- "dom(override_on f g A) =
- (dom f  - {a. a : A - dom g}) Un {a. a : A Int dom g}"
-by(auto simp add: dom_def override_on_def)
+lemma dom_override_on [simp]:
+  "dom(override_on f g A) =
+    (dom f  - {a. a : A - dom g}) Un {a. a : A Int dom g}"
+  unfolding dom_def override_on_def by auto
 
 lemma map_add_comm: "dom m1 \<inter> dom m2 = {} \<Longrightarrow> m1++m2 = m2++m1"
-apply(rule ext)
-apply(force simp: map_add_def dom_def split:option.split) 
-done
+  by (rule ext) (force simp: map_add_def dom_def split: option.split)
+
 
 subsection {* @{term [source] ran} *}
 
-lemma ranI: "m a = Some b ==> b : ran m" 
-by (auto simp add: ran_def)
+lemma ranI: "m a = Some b ==> b : ran m"
+  unfolding ran_def by auto
 (* declare ranI [intro]? *)
 
-lemma ran_empty[simp]: "ran empty = {}"
-apply (unfold ran_def)
-apply (simp (no_asm))
-done
+lemma ran_empty [simp]: "ran empty = {}"
+  unfolding ran_def by simp
 
-lemma ran_map_upd[simp]: "m a = None ==> ran(m(a|->b)) = insert b (ran m)"
-apply (unfold ran_def, auto)
-apply (subgoal_tac "~ (aa = a) ")
-apply auto
-done
+lemma ran_map_upd [simp]: "m a = None ==> ran(m(a|->b)) = insert b (ran m)"
+  unfolding ran_def
+  apply auto
+  apply (subgoal_tac "aa ~= a")
+   apply auto
+  done
+
 
 subsection {* @{text "map_le"} *}
 
 lemma map_le_empty [simp]: "empty \<subseteq>\<^sub>m g"
-by(simp add:map_le_def)
+  by (simp add: map_le_def)
 
 lemma upd_None_map_le [simp]: "f(x := None) \<subseteq>\<^sub>m f"
-by(force simp add:map_le_def)
+  by (force 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)
+  by (fastsimp simp add: map_le_def)
 
 lemma map_le_imp_upd_le [simp]: "m1 \<subseteq>\<^sub>m m2 \<Longrightarrow> m1(x := None) \<subseteq>\<^sub>m m2(x \<mapsto> y)"
-by(force simp add:map_le_def)
+  by (force 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, simp)
-apply (case_tac bs, auto)
-done
+lemma map_le_upds [simp]:
+    "f \<subseteq>\<^sub>m g ==> f(as [|->] bs) \<subseteq>\<^sub>m g(as [|->] bs)"
+  apply (induct as arbitrary: f g bs)
+   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)
@@ -572,22 +553,22 @@
   by (auto simp add: map_le_def dom_def)
 
 lemma map_le_antisym: "\<lbrakk> f \<subseteq>\<^sub>m g; g \<subseteq>\<^sub>m f \<rbrakk> \<Longrightarrow> f = g"
-  apply (unfold map_le_def)
+  unfolding map_le_def
   apply (rule ext)
   apply (case_tac "x \<in> dom f", simp)
   apply (case_tac "x \<in> dom g", simp, fastsimp)
-done
+  done
 
 lemma map_le_map_add [simp]: "f \<subseteq>\<^sub>m (g ++ f)"
   by (fastsimp simp add: map_le_def)
 
 lemma map_le_iff_map_add_commute: "(f \<subseteq>\<^sub>m f ++ g) = (f++g = g++f)"
-by(fastsimp simp add:map_add_def map_le_def expand_fun_eq split:option.splits)
+  by (fastsimp simp add: map_add_def map_le_def expand_fun_eq split: option.splits)
 
 lemma map_add_le_mapE: "f++g \<subseteq>\<^sub>m h \<Longrightarrow> g \<subseteq>\<^sub>m h"
-by (fastsimp simp add: map_le_def map_add_def dom_def)
+  by (fastsimp simp add: map_le_def map_add_def dom_def)
 
 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"
-by (clarsimp simp add: map_le_def map_add_def dom_def split:option.splits)
+  by (clarsimp simp add: map_le_def map_add_def dom_def split: option.splits)
 
 end