
diff -r f10feaa9b14a -r c1fe30f2bc32 src/HOL/Library/Mapping.thy
--- a/src/HOL/Library/Mapping.thy	Tue Jul 12 14:53:47 2016 +0200
+++ b/src/HOL/Library/Mapping.thy	Tue Jul 12 15:45:32 2016 +0200
@@ -10,15 +10,13 @@
 
 subsection \<open>Parametricity transfer rules\<close>
 
-lemma map_of_foldr: \<comment> \<open>FIXME move\<close>
-  "map_of xs = foldr (\<lambda>(k, v) m. m(k \<mapsto> v)) xs Map.empty"
+lemma map_of_foldr: "map_of xs = foldr (\<lambda>(k, v) m. m(k \<mapsto> v)) xs Map.empty"  (* FIXME move *)
   using map_add_map_of_foldr [of Map.empty] by auto
 
 context includes lifting_syntax
 begin
 
-lemma empty_parametric:
-  "(A ===> rel_option B) Map.empty Map.empty"
+lemma empty_parametric: "(A ===> rel_option B) Map.empty Map.empty"
   by transfer_prover
 
 lemma lookup_parametric: "((A ===> B) ===> A ===> B) (\<lambda>m k. m k) (\<lambda>m k. m k)"
@@ -32,7 +30,7 @@
 
 lemma delete_parametric:
   assumes [transfer_rule]: "bi_unique A"
-  shows "(A ===> (A ===> rel_option B) ===> A ===> rel_option B) 
+  shows "(A ===> (A ===> rel_option B) ===> A ===> rel_option B)
     (\<lambda>k m. m(k := None)) (\<lambda>k m. m(k := None))"
   by transfer_prover
 
@@ -42,7 +40,7 @@
 
 lemma dom_parametric:
   assumes [transfer_rule]: "bi_total A"
-  shows "((A ===> rel_option B) ===> rel_set A) dom dom" 
+  shows "((A ===> rel_option B) ===> rel_set A) dom dom"
   unfolding dom_def [abs_def] Option.is_none_def [symmetric] by transfer_prover
 
 lemma map_of_parametric [transfer_rule]:
@@ -52,51 +50,53 @@
 
 lemma map_entry_parametric [transfer_rule]:
   assumes [transfer_rule]: "bi_unique A"
-  shows "(A ===> (B ===> B) ===> (A ===> rel_option B) ===> A ===> rel_option B) 
+  shows "(A ===> (B ===> B) ===> (A ===> rel_option B) ===> A ===> rel_option B)
     (\<lambda>k f m. (case m k of None \<Rightarrow> m
       | Some v \<Rightarrow> m (k \<mapsto> (f v)))) (\<lambda>k f m. (case m k of None \<Rightarrow> m
       | Some v \<Rightarrow> m (k \<mapsto> (f v))))"
   by transfer_prover
 
-lemma tabulate_parametric: 
+lemma tabulate_parametric:
   assumes [transfer_rule]: "bi_unique A"
-  shows "(list_all2 A ===> (A ===> B) ===> A ===> rel_option B) 
+  shows "(list_all2 A ===> (A ===> B) ===> A ===> rel_option B)
     (\<lambda>ks f. (map_of (map (\<lambda>k. (k, f k)) ks))) (\<lambda>ks f. (map_of (map (\<lambda>k. (k, f k)) ks)))"
   by transfer_prover
 
-lemma bulkload_parametric: 
-  "(list_all2 A ===> HOL.eq ===> rel_option A) 
-    (\<lambda>xs k. if k < length xs then Some (xs ! k) else None) (\<lambda>xs k. if k < length xs then Some (xs ! k) else None)"
+lemma bulkload_parametric:
+  "(list_all2 A ===> HOL.eq ===> rel_option A)
+    (\<lambda>xs k. if k < length xs then Some (xs ! k) else None)
+    (\<lambda>xs k. if k < length xs then Some (xs ! k) else None)"
 proof
   fix xs ys
   assume "list_all2 A xs ys"
-  then show "(HOL.eq ===> rel_option A)
-    (\<lambda>k. if k < length xs then Some (xs ! k) else None)
-    (\<lambda>k. if k < length ys then Some (ys ! k) else None)"
+  then show
+    "(HOL.eq ===> rel_option A)
+      (\<lambda>k. if k < length xs then Some (xs ! k) else None)
+      (\<lambda>k. if k < length ys then Some (ys ! k) else None)"
     apply induct
     apply auto
     unfolding rel_fun_def
-    apply clarsimp 
-    apply (case_tac xa) 
+    apply clarsimp
+    apply (case_tac xa)
     apply (auto dest: list_all2_lengthD list_all2_nthD)
     done
 qed
 
-lemma map_parametric: 
-  "((A ===> B) ===> (C ===> D) ===> (B ===> rel_option C) ===> A ===> rel_option D) 
+lemma map_parametric:
+  "((A ===> B) ===> (C ===> D) ===> (B ===> rel_option C) ===> A ===> rel_option D)
      (\<lambda>f g m. (map_option g \<circ> m \<circ> f)) (\<lambda>f g m. (map_option g \<circ> m \<circ> f))"
   by transfer_prover
-  
-lemma combine_with_key_parametric: 
-  shows "((A ===> B ===> B ===> B) ===> (A ===> rel_option B) ===> (A ===> rel_option B) ===>
-           (A ===> rel_option B)) (\<lambda>f m1 m2 x. combine_options (f x) (m1 x) (m2 x))
-           (\<lambda>f m1 m2 x. combine_options (f x) (m1 x) (m2 x))"
+
+lemma combine_with_key_parametric:
+  "((A ===> B ===> B ===> B) ===> (A ===> rel_option B) ===> (A ===> rel_option B) ===>
+    (A ===> rel_option B)) (\<lambda>f m1 m2 x. combine_options (f x) (m1 x) (m2 x))
+    (\<lambda>f m1 m2 x. combine_options (f x) (m1 x) (m2 x))"
   unfolding combine_options_def by transfer_prover
-  
-lemma combine_parametric: 
-  shows "((B ===> B ===> B) ===> (A ===> rel_option B) ===> (A ===> rel_option B) ===>
-           (A ===> rel_option B)) (\<lambda>f m1 m2 x. combine_options f (m1 x) (m2 x))
-           (\<lambda>f m1 m2 x. combine_options f (m1 x) (m2 x))"
+
+lemma combine_parametric:
+  "((B ===> B ===> B) ===> (A ===> rel_option B) ===> (A ===> rel_option B) ===>
+    (A ===> rel_option B)) (\<lambda>f m1 m2 x. combine_options f (m1 x) (m2 x))
+    (\<lambda>f m1 m2 x. combine_options f (m1 x) (m2 x))"
   unfolding combine_options_def by transfer_prover
 
 end
@@ -105,8 +105,7 @@
 subsection \<open>Type definition and primitive operations\<close>
 
 typedef ('a, 'b) mapping = "UNIV :: ('a \<rightharpoonup> 'b) set"
-  morphisms rep Mapping
-  ..
+  morphisms rep Mapping ..
 
 setup_lifting type_definition_mapping
 
@@ -119,7 +118,7 @@
 definition "lookup_default d m k = (case Mapping.lookup m k of None \<Rightarrow> d | Some v \<Rightarrow> v)"
 
 declare [[code drop: Mapping.lookup]]
-setup \<open>Code.add_eqn (Code.Equation, true) @{thm Mapping.lookup.abs_eq}\<close> \<comment> \<open>FIXME lifting\<close>
+setup \<open>Code.add_eqn (Code.Equation, true) @{thm Mapping.lookup.abs_eq}\<close>  (* FIXME lifting *)
 
 lift_definition update :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping"
   is "\<lambda>k v m. m(k \<mapsto> v)" parametric update_parametric .
@@ -128,7 +127,7 @@
   is "\<lambda>k m. m(k := None)" parametric delete_parametric .
 
 lift_definition filter :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping"
-  is "\<lambda>P m k. case m k of None \<Rightarrow> None | Some v \<Rightarrow> if P k v then Some v else None" . 
+  is "\<lambda>P m k. case m k of None \<Rightarrow> None | Some v \<Rightarrow> if P k v then Some v else None" .
 
 lift_definition keys :: "('a, 'b) mapping \<Rightarrow> 'a set"
   is dom parametric dom_parametric .
@@ -141,20 +140,20 @@
 
 lift_definition map :: "('c \<Rightarrow> 'a) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('c, 'd) mapping"
   is "\<lambda>f g m. (map_option g \<circ> m \<circ> f)" parametric map_parametric .
-  
+
 lift_definition map_values :: "('c \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> ('c, 'a) mapping \<Rightarrow> ('c, 'b) mapping"
-  is "\<lambda>f m x. map_option (f x) (m x)" . 
+  is "\<lambda>f m x. map_option (f x) (m x)" .
 
-lift_definition combine_with_key :: 
+lift_definition combine_with_key ::
   "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a,'b) mapping \<Rightarrow> ('a,'b) mapping \<Rightarrow> ('a,'b) mapping"
   is "\<lambda>f m1 m2 x. combine_options (f x) (m1 x) (m2 x)" parametric combine_with_key_parametric .
 
-lift_definition combine :: 
+lift_definition combine ::
   "('b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a,'b) mapping \<Rightarrow> ('a,'b) mapping \<Rightarrow> ('a,'b) mapping"
   is "\<lambda>f m1 m2 x. combine_options f (m1 x) (m2 x)" parametric combine_parametric .
 
-definition All_mapping where
-  "All_mapping m P \<longleftrightarrow> (\<forall>x. case Mapping.lookup m x of None \<Rightarrow> True | Some y \<Rightarrow> P x y)"
+definition "All_mapping m P \<longleftrightarrow>
+  (\<forall>x. case Mapping.lookup m x of None \<Rightarrow> True | Some y \<Rightarrow> P x y)"
 
 declare [[code drop: map]]
 
@@ -168,52 +167,52 @@
 subsection \<open>Derived operations\<close>
 
 definition ordered_keys :: "('a::linorder, 'b) mapping \<Rightarrow> 'a list"
-where
-  "ordered_keys m = (if finite (keys m) then sorted_list_of_set (keys m) else [])"
+  where "ordered_keys m = (if finite (keys m) then sorted_list_of_set (keys m) else [])"
 
 definition is_empty :: "('a, 'b) mapping \<Rightarrow> bool"
-where
-  "is_empty m \<longleftrightarrow> keys m = {}"
+  where "is_empty m \<longleftrightarrow> keys m = {}"
 
 definition size :: "('a, 'b) mapping \<Rightarrow> nat"
-where
-  "size m = (if finite (keys m) then card (keys m) else 0)"
+  where "size m = (if finite (keys m) then card (keys m) else 0)"
 
 definition replace :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping"
-where
-  "replace k v m = (if k \<in> keys m then update k v m else m)"
+  where "replace k v m = (if k \<in> keys m then update k v m else m)"
 
 definition default :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping"
-where
-  "default k v m = (if k \<in> keys m then m else update k v m)"
+  where "default k v m = (if k \<in> keys m then m else update k v m)"
 
 text \<open>Manual derivation of transfer rule is non-trivial\<close>
 
 lift_definition map_entry :: "'a \<Rightarrow> ('b \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" is
-  "\<lambda>k f m. (case m k of None \<Rightarrow> m
+  "\<lambda>k f m.
+    (case m k of
+      None \<Rightarrow> m
     | Some v \<Rightarrow> m (k \<mapsto> (f v)))" parametric map_entry_parametric .
 
 lemma map_entry_code [code]:
-  "map_entry k f m = (case lookup m k of None \<Rightarrow> m
+  "map_entry k f m =
+    (case lookup m k of
+      None \<Rightarrow> m
     | Some v \<Rightarrow> update k (f v) m)"
   by transfer rule
 
 definition map_default :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping"
-where
-  "map_default k v f m = map_entry k f (default k v m)" 
+  where "map_default k v f m = map_entry k f (default k v m)"
 
 definition of_alist :: "('k \<times> 'v) list \<Rightarrow> ('k, 'v) mapping"
-where
-  "of_alist xs = foldr (\<lambda>(k, v) m. update k v m) xs empty"
+  where "of_alist xs = foldr (\<lambda>(k, v) m. update k v m) xs empty"
 
 instantiation mapping :: (type, type) equal
 begin
 
-definition
-  "HOL.equal m1 m2 \<longleftrightarrow> (\<forall>k. lookup m1 k = lookup m2 k)"
+definition "HOL.equal m1 m2 \<longleftrightarrow> (\<forall>k. lookup m1 k = lookup m2 k)"
 
 instance
-  by standard (unfold equal_mapping_def, transfer, auto)
+  apply standard
+  unfolding equal_mapping_def
+  apply transfer
+  apply auto
+  done
 
 end
 
@@ -222,9 +221,9 @@
 
 lemma [transfer_rule]:
   assumes [transfer_rule]: "bi_total A"
-  assumes [transfer_rule]: "bi_unique B"
+    and [transfer_rule]: "bi_unique B"
   shows "(pcr_mapping A B ===> pcr_mapping A B ===> op=) HOL.eq HOL.equal"
-  by (unfold equal) transfer_prover
+  unfolding equal by transfer_prover
 
 lemma of_alist_transfer [transfer_rule]:
   assumes [transfer_rule]: "bi_unique R1"
@@ -236,195 +235,186 @@
 
 subsection \<open>Properties\<close>
 
-lemma mapping_eqI:
-  "(\<And>x. lookup m x = lookup m' x) \<Longrightarrow> m = m'"
+lemma mapping_eqI: "(\<And>x. lookup m x = lookup m' x) \<Longrightarrow> m = m'"
   by transfer (simp add: fun_eq_iff)
 
-lemma mapping_eqI': 
-  assumes "\<And>x. x \<in> Mapping.keys m \<Longrightarrow> Mapping.lookup_default d m x = Mapping.lookup_default d m' x" 
-      and "Mapping.keys m = Mapping.keys m'"
-  shows   "m = m'"
+lemma mapping_eqI':
+  assumes "\<And>x. x \<in> Mapping.keys m \<Longrightarrow> Mapping.lookup_default d m x = Mapping.lookup_default d m' x"
+    and "Mapping.keys m = Mapping.keys m'"
+  shows "m = m'"
 proof (intro mapping_eqI)
-  fix x
-  show "Mapping.lookup m x = Mapping.lookup m' x"
+  show "Mapping.lookup m x = Mapping.lookup m' x" for x
   proof (cases "Mapping.lookup m x")
     case None
-    hence "x \<notin> Mapping.keys m" by transfer (simp add: dom_def)
-    hence "x \<notin> Mapping.keys m'" by (simp add: assms)
-    hence "Mapping.lookup m' x = None" by transfer (simp add: dom_def)
-    with None show ?thesis by simp
+    then have "x \<notin> Mapping.keys m"
+      by transfer (simp add: dom_def)
+    then have "x \<notin> Mapping.keys m'"
+      by (simp add: assms)
+    then have "Mapping.lookup m' x = None"
+      by transfer (simp add: dom_def)
+    with None show ?thesis
+      by simp
   next
     case (Some y)
-    hence A: "x \<in> Mapping.keys m" by transfer (simp add: dom_def)
-    hence "x \<in> Mapping.keys m'" by (simp add: assms)
-    hence "\<exists>y'. Mapping.lookup m' x = Some y'" by transfer (simp add: dom_def)
-    with Some assms(1)[OF A] show ?thesis by (auto simp add: lookup_default_def)
+    then have A: "x \<in> Mapping.keys m"
+      by transfer (simp add: dom_def)
+    then have "x \<in> Mapping.keys m'"
+      by (simp add: assms)
+    then have "\<exists>y'. Mapping.lookup m' x = Some y'"
+      by transfer (simp add: dom_def)
+    with Some assms(1)[OF A] show ?thesis
+      by (auto simp add: lookup_default_def)
   qed
 qed
 
-lemma lookup_update:
-  "lookup (update k v m) k = Some v" 
+lemma lookup_update: "lookup (update k v m) k = Some v"
   by transfer simp
 
-lemma lookup_update_neq:
-  "k \<noteq> k' \<Longrightarrow> lookup (update k v m) k' = lookup m k'" 
+lemma lookup_update_neq: "k \<noteq> k' \<Longrightarrow> lookup (update k v m) k' = lookup m k'"
   by transfer simp
 
-lemma lookup_update': 
-  "Mapping.lookup (update k v m) k' = (if k = k' then Some v else lookup m k')"
+lemma lookup_update': "Mapping.lookup (update k v m) k' = (if k = k' then Some v else lookup m k')"
   by (auto simp: lookup_update lookup_update_neq)
 
-lemma lookup_empty:
-  "lookup empty k = None" 
+lemma lookup_empty: "lookup empty k = None"
   by transfer simp
 
 lemma lookup_filter:
-  "lookup (filter P m) k = 
-     (case lookup m k of None \<Rightarrow> None | Some v \<Rightarrow> if P k v then Some v else None)"
+  "lookup (filter P m) k =
+    (case lookup m k of
+      None \<Rightarrow> None
+    | Some v \<Rightarrow> if P k v then Some v else None)"
   by transfer simp_all
 
-lemma lookup_map_values:
-  "lookup (map_values f m) k = map_option (f k) (lookup m k)"
+lemma lookup_map_values: "lookup (map_values f m) k = map_option (f k) (lookup m k)"
   by transfer simp_all
 
 lemma lookup_default_empty: "lookup_default d empty k = d"
   by (simp add: lookup_default_def lookup_empty)
 
-lemma lookup_default_update:
-  "lookup_default d (update k v m) k = v" 
+lemma lookup_default_update: "lookup_default d (update k v m) k = v"
   by (simp add: lookup_default_def lookup_update)
 
 lemma lookup_default_update_neq:
-  "k \<noteq> k' \<Longrightarrow> lookup_default d (update k v m) k' = lookup_default d m k'" 
+  "k \<noteq> k' \<Longrightarrow> lookup_default d (update k v m) k' = lookup_default d m k'"
   by (simp add: lookup_default_def lookup_update_neq)
 
-lemma lookup_default_update': 
+lemma lookup_default_update':
   "lookup_default d (update k v m) k' = (if k = k' then v else lookup_default d m k')"
   by (auto simp: lookup_default_update lookup_default_update_neq)
 
 lemma lookup_default_filter:
-  "lookup_default d (filter P m) k =  
+  "lookup_default d (filter P m) k =
      (if P k (lookup_default d m k) then lookup_default d m k else d)"
   by (simp add: lookup_default_def lookup_filter split: option.splits)
 
 lemma lookup_default_map_values:
   "lookup_default (f k d) (map_values f m) k = f k (lookup_default d m k)"
-  by (simp add: lookup_default_def lookup_map_values split: option.splits)  
+  by (simp add: lookup_default_def lookup_map_values split: option.splits)
 
 lemma lookup_combine_with_key:
-  "Mapping.lookup (combine_with_key f m1 m2) x = 
-     combine_options (f x) (Mapping.lookup m1 x) (Mapping.lookup m2 x)"
+  "Mapping.lookup (combine_with_key f m1 m2) x =
+    combine_options (f x) (Mapping.lookup m1 x) (Mapping.lookup m2 x)"
   by transfer (auto split: option.splits)
-  
+
 lemma combine_altdef: "combine f m1 m2 = combine_with_key (\<lambda>_. f) m1 m2"
   by transfer' (rule refl)
 
 lemma lookup_combine:
-  "Mapping.lookup (combine f m1 m2) x = 
+  "Mapping.lookup (combine f m1 m2) x =
      combine_options f (Mapping.lookup m1 x) (Mapping.lookup m2 x)"
   by transfer (auto split: option.splits)
-  
-lemma lookup_default_neutral_combine_with_key: 
+
+lemma lookup_default_neutral_combine_with_key:
   assumes "\<And>x. f k d x = x" "\<And>x. f k x d = x"
-  shows   "Mapping.lookup_default d (combine_with_key f m1 m2) k = 
-             f k (Mapping.lookup_default d m1 k) (Mapping.lookup_default d m2 k)"
+  shows "Mapping.lookup_default d (combine_with_key f m1 m2) k =
+    f k (Mapping.lookup_default d m1 k) (Mapping.lookup_default d m2 k)"
   by (auto simp: lookup_default_def lookup_combine_with_key assms split: option.splits)
-  
-lemma lookup_default_neutral_combine: 
+
+lemma lookup_default_neutral_combine:
   assumes "\<And>x. f d x = x" "\<And>x. f x d = x"
-  shows   "Mapping.lookup_default d (combine f m1 m2) x = 
-             f (Mapping.lookup_default d m1 x) (Mapping.lookup_default d m2 x)"
+  shows "Mapping.lookup_default d (combine f m1 m2) x =
+    f (Mapping.lookup_default d m1 x) (Mapping.lookup_default d m2 x)"
   by (auto simp: lookup_default_def lookup_combine assms split: option.splits)
 
-lemma lookup_map_entry:
-  "lookup (map_entry x f m) x = map_option f (lookup m x)"
+lemma lookup_map_entry: "lookup (map_entry x f m) x = map_option f (lookup m x)"
   by transfer (auto split: option.splits)
 
-lemma lookup_map_entry_neq:
-  "x \<noteq> y \<Longrightarrow> lookup (map_entry x f m) y = lookup m y"
+lemma lookup_map_entry_neq: "x \<noteq> y \<Longrightarrow> lookup (map_entry x f m) y = lookup m y"
   by transfer (auto split: option.splits)
 
 lemma lookup_map_entry':
-  "lookup (map_entry x f m) y = 
+  "lookup (map_entry x f m) y =
      (if x = y then map_option f (lookup m y) else lookup m y)"
   by transfer (auto split: option.splits)
-  
-lemma lookup_default:
-  "lookup (default x d m) x = Some (lookup_default d m x)"
-    unfolding lookup_default_def default_def
-    by transfer (auto split: option.splits)
 
-lemma lookup_default_neq:
-  "x \<noteq> y \<Longrightarrow> lookup (default x d m) y = lookup m y"
-    unfolding lookup_default_def default_def
-    by transfer (auto split: option.splits)
+lemma lookup_default: "lookup (default x d m) x = Some (lookup_default d m x)"
+  unfolding lookup_default_def default_def
+  by transfer (auto split: option.splits)
+
+lemma lookup_default_neq: "x \<noteq> y \<Longrightarrow> lookup (default x d m) y = lookup m y"
+  unfolding lookup_default_def default_def
+  by transfer (auto split: option.splits)
 
 lemma lookup_default':
-  "lookup (default x d m) y = 
-     (if x = y then Some (lookup_default d m x) else lookup m y)"
+  "lookup (default x d m) y =
+    (if x = y then Some (lookup_default d m x) else lookup m y)"
   unfolding lookup_default_def default_def
   by transfer (auto split: option.splits)
-  
-lemma lookup_map_default:
-  "lookup (map_default x d f m) x = Some (f (lookup_default d m x))"
-    unfolding lookup_default_def default_def
-    by (simp add: map_default_def lookup_map_entry lookup_default lookup_default_def)
 
-lemma lookup_map_default_neq:
-  "x \<noteq> y \<Longrightarrow> lookup (map_default x d f m) y = lookup m y"
-    unfolding lookup_default_def default_def
-    by (simp add: map_default_def lookup_map_entry_neq lookup_default_neq) 
+lemma lookup_map_default: "lookup (map_default x d f m) x = Some (f (lookup_default d m x))"
+  unfolding lookup_default_def default_def
+  by (simp add: map_default_def lookup_map_entry lookup_default lookup_default_def)
+
+lemma lookup_map_default_neq: "x \<noteq> y \<Longrightarrow> lookup (map_default x d f m) y = lookup m y"
+  unfolding lookup_default_def default_def
+  by (simp add: map_default_def lookup_map_entry_neq lookup_default_neq)
 
 lemma lookup_map_default':
-  "lookup (map_default x d f m) y = 
-     (if x = y then Some (f (lookup_default d m x)) else lookup m y)"
-    unfolding lookup_default_def default_def
-    by (simp add: map_default_def lookup_map_entry' lookup_default' lookup_default_def)  
+  "lookup (map_default x d f m) y =
+    (if x = y then Some (f (lookup_default d m x)) else lookup m y)"
+  unfolding lookup_default_def default_def
+  by (simp add: map_default_def lookup_map_entry' lookup_default' lookup_default_def)
 
-lemma lookup_tabulate: 
+lemma lookup_tabulate:
   assumes "distinct xs"
-  shows   "Mapping.lookup (Mapping.tabulate xs f) x = (if x \<in> set xs then Some (f x) else None)"
+  shows "Mapping.lookup (Mapping.tabulate xs f) x = (if x \<in> set xs then Some (f x) else None)"
   using assms by transfer (auto simp: map_of_eq_None_iff o_def dest!: map_of_SomeD)
 
 lemma lookup_of_alist: "Mapping.lookup (Mapping.of_alist xs) k = map_of xs k"
   by transfer simp_all
 
-lemma keys_is_none_rep [code_unfold]:
-  "k \<in> keys m \<longleftrightarrow> \<not> (Option.is_none (lookup m k))"
+lemma keys_is_none_rep [code_unfold]: "k \<in> keys m \<longleftrightarrow> \<not> (Option.is_none (lookup m k))"
   by transfer (auto simp add: Option.is_none_def)
 
 lemma update_update:
   "update k v (update k w m) = update k v m"
   "k \<noteq> l \<Longrightarrow> update k v (update l w m) = update l w (update k v m)"
-  by (transfer, simp add: fun_upd_twist)+
+  by (transfer; simp add: fun_upd_twist)+
 
-lemma update_delete [simp]:
-  "update k v (delete k m) = update k v m"
+lemma update_delete [simp]: "update k v (delete k m) = update k v m"
   by transfer simp
 
 lemma delete_update:
   "delete k (update k v m) = delete k m"
   "k \<noteq> l \<Longrightarrow> delete k (update l v m) = update l v (delete k m)"
-  by (transfer, simp add: fun_upd_twist)+
+  by (transfer; simp add: fun_upd_twist)+
 
-lemma delete_empty [simp]:
-  "delete k empty = empty"
+lemma delete_empty [simp]: "delete k empty = empty"
   by transfer simp
 
 lemma replace_update:
   "k \<notin> keys m \<Longrightarrow> replace k v m = m"
   "k \<in> keys m \<Longrightarrow> replace k v m = update k v m"
-  by (transfer, auto simp add: replace_def fun_upd_twist)+
-  
+  by (transfer; auto simp add: replace_def fun_upd_twist)+
+
 lemma map_values_update: "map_values f (update k v m) = update k (f k v) (map_values f m)"
   by transfer (simp_all add: fun_eq_iff)
-  
-lemma size_mono:
-  "finite (keys m') \<Longrightarrow> keys m \<subseteq> keys m' \<Longrightarrow> size m \<le> size m'"
+
+lemma size_mono: "finite (keys m') \<Longrightarrow> keys m \<subseteq> keys m' \<Longrightarrow> size m \<le> size m'"
   unfolding size_def by (auto intro: card_mono)
 
-lemma size_empty [simp]:
-  "size empty = 0"
+lemma size_empty [simp]: "size empty = 0"
   unfolding size_def by transfer simp
 
 lemma size_update:
@@ -432,13 +422,11 @@
     (if k \<in> keys m then size m else Suc (size m))"
   unfolding size_def by transfer (auto simp add: insert_dom)
 
-lemma size_delete:
-  "size (delete k m) = (if k \<in> keys m then size m - 1 else size m)"
+lemma size_delete: "size (delete k m) = (if k \<in> keys m then size m - 1 else size m)"
   unfolding size_def by transfer simp
 
-lemma size_tabulate [simp]:
-  "size (tabulate ks f) = length (remdups ks)"
-  unfolding size_def by transfer (auto simp add: map_of_map_restrict  card_set comp_def)
+lemma size_tabulate [simp]: "size (tabulate ks f) = length (remdups ks)"
+  unfolding size_def by transfer (auto simp add: map_of_map_restrict card_set comp_def)
 
 lemma keys_filter: "keys (filter P m) \<subseteq> keys m"
   by transfer (auto split: option.splits)
@@ -446,131 +434,113 @@
 lemma size_filter: "finite (keys m) \<Longrightarrow> size (filter P m) \<le> size m"
   by (intro size_mono keys_filter)
 
-
-lemma bulkload_tabulate:
-  "bulkload xs = tabulate [0..<length xs] (nth xs)"
+lemma bulkload_tabulate: "bulkload xs = tabulate [0..<length xs] (nth xs)"
   by transfer (auto simp add: map_of_map_restrict)
 
-lemma is_empty_empty [simp]:
-  "is_empty empty"
-  unfolding is_empty_def by transfer simp 
-
-lemma is_empty_update [simp]:
-  "\<not> is_empty (update k v m)"
+lemma is_empty_empty [simp]: "is_empty empty"
   unfolding is_empty_def by transfer simp
 
-lemma is_empty_delete:
-  "is_empty (delete k m) \<longleftrightarrow> is_empty m \<or> keys m = {k}"
+lemma is_empty_update [simp]: "\<not> is_empty (update k v m)"
+  unfolding is_empty_def by transfer simp
+
+lemma is_empty_delete: "is_empty (delete k m) \<longleftrightarrow> is_empty m \<or> keys m = {k}"
   unfolding is_empty_def by transfer (auto simp del: dom_eq_empty_conv)
 
-lemma is_empty_replace [simp]:
-  "is_empty (replace k v m) \<longleftrightarrow> is_empty m"
+lemma is_empty_replace [simp]: "is_empty (replace k v m) \<longleftrightarrow> is_empty m"
   unfolding is_empty_def replace_def by transfer auto
 
-lemma is_empty_default [simp]:
-  "\<not> is_empty (default k v m)"
+lemma is_empty_default [simp]: "\<not> is_empty (default k v m)"
   unfolding is_empty_def default_def by transfer auto
 
-lemma is_empty_map_entry [simp]:
-  "is_empty (map_entry k f m) \<longleftrightarrow> is_empty m"
+lemma is_empty_map_entry [simp]: "is_empty (map_entry k f m) \<longleftrightarrow> is_empty m"
   unfolding is_empty_def by transfer (auto split: option.split)
 
-lemma is_empty_map_values [simp]:
-  "is_empty (map_values f m) \<longleftrightarrow> is_empty m"
+lemma is_empty_map_values [simp]: "is_empty (map_values f m) \<longleftrightarrow> is_empty m"
   unfolding is_empty_def by transfer (auto simp: fun_eq_iff)
 
-lemma is_empty_map_default [simp]:
-  "\<not> is_empty (map_default k v f m)"
+lemma is_empty_map_default [simp]: "\<not> is_empty (map_default k v f m)"
   by (simp add: map_default_def)
 
-lemma keys_dom_lookup:
-  "keys m = dom (Mapping.lookup m)"
+lemma keys_dom_lookup: "keys m = dom (Mapping.lookup m)"
   by transfer rule
 
-lemma keys_empty [simp]:
-  "keys empty = {}"
+lemma keys_empty [simp]: "keys empty = {}"
   by transfer simp
 
-lemma keys_update [simp]:
-  "keys (update k v m) = insert k (keys m)"
+lemma keys_update [simp]: "keys (update k v m) = insert k (keys m)"
   by transfer simp
 
-lemma keys_delete [simp]:
-  "keys (delete k m) = keys m - {k}"
+lemma keys_delete [simp]: "keys (delete k m) = keys m - {k}"
   by transfer simp
 
-lemma keys_replace [simp]:
-  "keys (replace k v m) = keys m"
+lemma keys_replace [simp]: "keys (replace k v m) = keys m"
   unfolding replace_def by transfer (simp add: insert_absorb)
 
-lemma keys_default [simp]:
-  "keys (default k v m) = insert k (keys m)"
+lemma keys_default [simp]: "keys (default k v m) = insert k (keys m)"
   unfolding default_def by transfer (simp add: insert_absorb)
 
-lemma keys_map_entry [simp]:
-  "keys (map_entry k f m) = keys m"
+lemma keys_map_entry [simp]: "keys (map_entry k f m) = keys m"
   by transfer (auto split: option.split)
 
-lemma keys_map_default [simp]:
-  "keys (map_default k v f m) = insert k (keys m)"
+lemma keys_map_default [simp]: "keys (map_default k v f m) = insert k (keys m)"
   by (simp add: map_default_def)
 
-lemma keys_map_values [simp]:
-  "keys (map_values f m) = keys m"
+lemma keys_map_values [simp]: "keys (map_values f m) = keys m"
   by transfer (simp_all add: dom_def)
 
-lemma keys_combine_with_key [simp]: 
+lemma keys_combine_with_key [simp]:
   "Mapping.keys (combine_with_key f m1 m2) = Mapping.keys m1 \<union> Mapping.keys m2"
-  by transfer (auto simp: dom_def combine_options_def split: option.splits)  
+  by transfer (auto simp: dom_def combine_options_def split: option.splits)
 
 lemma keys_combine [simp]: "Mapping.keys (combine f m1 m2) = Mapping.keys m1 \<union> Mapping.keys m2"
   by (simp add: combine_altdef)
 
-lemma keys_tabulate [simp]:
-  "keys (tabulate ks f) = set ks"
+lemma keys_tabulate [simp]: "keys (tabulate ks f) = set ks"
   by transfer (simp add: map_of_map_restrict o_def)
 
 lemma keys_of_alist [simp]: "keys (of_alist xs) = set (List.map fst xs)"
   by transfer (simp_all add: dom_map_of_conv_image_fst)
 
-lemma keys_bulkload [simp]:
-  "keys (bulkload xs) = {0..<length xs}"
+lemma keys_bulkload [simp]: "keys (bulkload xs) = {0..<length xs}"
   by (simp add: bulkload_tabulate)
 
-lemma distinct_ordered_keys [simp]:
-  "distinct (ordered_keys m)"
+lemma distinct_ordered_keys [simp]: "distinct (ordered_keys m)"
   by (simp add: ordered_keys_def)
 
-lemma ordered_keys_infinite [simp]:
-  "\<not> finite (keys m) \<Longrightarrow> ordered_keys m = []"
+lemma ordered_keys_infinite [simp]: "\<not> finite (keys m) \<Longrightarrow> ordered_keys m = []"
   by (simp add: ordered_keys_def)
 
-lemma ordered_keys_empty [simp]:
-  "ordered_keys empty = []"
+lemma ordered_keys_empty [simp]: "ordered_keys empty = []"
   by (simp add: ordered_keys_def)
 
 lemma ordered_keys_update [simp]:
   "k \<in> keys m \<Longrightarrow> ordered_keys (update k v m) = ordered_keys m"
-  "finite (keys m) \<Longrightarrow> k \<notin> keys m \<Longrightarrow> ordered_keys (update k v m) = insort k (ordered_keys m)"
-  by (simp_all add: ordered_keys_def) (auto simp only: sorted_list_of_set_insert [symmetric] insert_absorb)
+  "finite (keys m) \<Longrightarrow> k \<notin> keys m \<Longrightarrow>
+    ordered_keys (update k v m) = insort k (ordered_keys m)"
+  by (simp_all add: ordered_keys_def)
+    (auto simp only: sorted_list_of_set_insert [symmetric] insert_absorb)
 
-lemma ordered_keys_delete [simp]:
-  "ordered_keys (delete k m) = remove1 k (ordered_keys m)"
+lemma ordered_keys_delete [simp]: "ordered_keys (delete k m) = remove1 k (ordered_keys m)"
 proof (cases "finite (keys m)")
-  case False then show ?thesis by simp
+  case False
+  then show ?thesis by simp
 next
-  case True note fin = True
+  case fin: True
   show ?thesis
   proof (cases "k \<in> keys m")
-    case False with fin have "k \<notin> set (sorted_list_of_set (keys m))" by simp
-    with False show ?thesis by (simp add: ordered_keys_def remove1_idem)
+    case False
+    with fin have "k \<notin> set (sorted_list_of_set (keys m))"
+      by simp
+    with False show ?thesis
+      by (simp add: ordered_keys_def remove1_idem)
   next
-    case True with fin show ?thesis by (simp add: ordered_keys_def sorted_list_of_set_remove)
+    case True
+    with fin show ?thesis
+      by (simp add: ordered_keys_def sorted_list_of_set_remove)
   qed
 qed
 
-lemma ordered_keys_replace [simp]:
-  "ordered_keys (replace k v m) = ordered_keys m"
+lemma ordered_keys_replace [simp]: "ordered_keys (replace k v m) = ordered_keys m"
   by (simp add: replace_def)
 
 lemma ordered_keys_default [simp]:
@@ -578,8 +548,7 @@
   "finite (keys m) \<Longrightarrow> k \<notin> keys m \<Longrightarrow> ordered_keys (default k v m) = insort k (ordered_keys m)"
   by (simp_all add: default_def)
 
-lemma ordered_keys_map_entry [simp]:
-  "ordered_keys (map_entry k f m) = ordered_keys m"
+lemma ordered_keys_map_entry [simp]: "ordered_keys (map_entry k f m) = ordered_keys m"
   by (simp add: ordered_keys_def)
 
 lemma ordered_keys_map_default [simp]:
@@ -587,16 +556,13 @@
   "finite (keys m) \<Longrightarrow> k \<notin> keys m \<Longrightarrow> ordered_keys (map_default k v f m) = insort k (ordered_keys m)"
   by (simp_all add: map_default_def)
 
-lemma ordered_keys_tabulate [simp]:
-  "ordered_keys (tabulate ks f) = sort (remdups ks)"
+lemma ordered_keys_tabulate [simp]: "ordered_keys (tabulate ks f) = sort (remdups ks)"
   by (simp add: ordered_keys_def sorted_list_of_set_sort_remdups)
 
-lemma ordered_keys_bulkload [simp]:
-  "ordered_keys (bulkload ks) = [0..<length ks]"
+lemma ordered_keys_bulkload [simp]: "ordered_keys (bulkload ks) = [0..<length ks]"
   by (simp add: ordered_keys_def)
 
-lemma tabulate_fold:
-  "tabulate xs f = fold (\<lambda>k m. update k (f k) m) xs empty"
+lemma tabulate_fold: "tabulate xs f = fold (\<lambda>k m. update k (f k) m) xs empty"
 proof transfer
   fix f :: "'a \<Rightarrow> 'b" and xs
   have "map_of (List.map (\<lambda>k. (k, f k)) xs) = foldr (\<lambda>k m. m(k \<mapsto> f k)) xs Map.empty"
@@ -613,21 +579,23 @@
 
 lemma All_mapping_empty [simp]: "All_mapping Mapping.empty P"
   by (auto simp: All_mapping_def lookup_empty)
-  
-lemma All_mapping_update_iff: 
+
+lemma All_mapping_update_iff:
   "All_mapping (Mapping.update k v m) P \<longleftrightarrow> P k v \<and> All_mapping m (\<lambda>k' v'. k = k' \<or> P k' v')"
-  unfolding All_mapping_def 
+  unfolding All_mapping_def
 proof safe
   assume "\<forall>x. case Mapping.lookup (Mapping.update k v m) x of None \<Rightarrow> True | Some y \<Rightarrow> P x y"
-  hence A: "case Mapping.lookup (Mapping.update k v m) x of None \<Rightarrow> True | Some y \<Rightarrow> P x y" for x
+  then have *: "case Mapping.lookup (Mapping.update k v m) x of None \<Rightarrow> True | Some y \<Rightarrow> P x y" for x
     by blast
-  from A[of k] show "P k v" by (simp add: lookup_update)
+  from *[of k] show "P k v"
+    by (simp add: lookup_update)
   show "case Mapping.lookup m x of None \<Rightarrow> True | Some v' \<Rightarrow> k = x \<or> P x v'" for x
-    using A[of x] by (auto simp add: lookup_update' split: if_splits option.splits)
+    using *[of x] by (auto simp add: lookup_update' split: if_splits option.splits)
 next
   assume "P k v"
   assume "\<forall>x. case Mapping.lookup m x of None \<Rightarrow> True | Some v' \<Rightarrow> k = x \<or> P x v'"
-  hence A: "case Mapping.lookup m x of None \<Rightarrow> True | Some v' \<Rightarrow> k = x \<or> P x v'" for x by blast
+  then have A: "case Mapping.lookup m x of None \<Rightarrow> True | Some v' \<Rightarrow> k = x \<or> P x v'" for x
+    by blast
   show "case Mapping.lookup (Mapping.update k v m) x of None \<Rightarrow> True | Some xa \<Rightarrow> P x xa" for x
     using \<open>P k v\<close> A[of x] by (auto simp: lookup_update' split: option.splits)
 qed
@@ -636,31 +604,28 @@
   "P k v \<Longrightarrow> All_mapping m (\<lambda>k' v'. k = k' \<or> P k' v') \<Longrightarrow> All_mapping (Mapping.update k v m) P"
   by (simp add: All_mapping_update_iff)
 
-lemma All_mapping_filter_iff:
-  "All_mapping (filter P m) Q \<longleftrightarrow> All_mapping m (\<lambda>k v. P k v \<longrightarrow> Q k v)"
+lemma All_mapping_filter_iff: "All_mapping (filter P m) Q \<longleftrightarrow> All_mapping m (\<lambda>k v. P k v \<longrightarrow> Q k v)"
   by (auto simp: All_mapping_def lookup_filter split: option.splits)
 
-lemma All_mapping_filter:
-  "All_mapping m Q \<Longrightarrow> All_mapping (filter P m) Q"
+lemma All_mapping_filter: "All_mapping m Q \<Longrightarrow> All_mapping (filter P m) Q"
   by (auto simp: All_mapping_filter_iff intro: All_mapping_mono)
 
-lemma All_mapping_map_values:
-  "All_mapping (map_values f m) P \<longleftrightarrow> All_mapping m (\<lambda>k v. P k (f k v))"
+lemma All_mapping_map_values: "All_mapping (map_values f m) P \<longleftrightarrow> All_mapping m (\<lambda>k v. P k (f k v))"
   by (auto simp: All_mapping_def lookup_map_values split: option.splits)
 
-lemma All_mapping_tabulate: 
-  "(\<forall>x\<in>set xs. P x (f x)) \<Longrightarrow> All_mapping (Mapping.tabulate xs f) P"
-  unfolding All_mapping_def 
-  by (intro allI,  transfer) (auto split: option.split dest!: map_of_SomeD)
+lemma All_mapping_tabulate: "(\<forall>x\<in>set xs. P x (f x)) \<Longrightarrow> All_mapping (Mapping.tabulate xs f) P"
+  unfolding All_mapping_def
+  apply (intro allI)
+  apply transfer
+  apply (auto split: option.split dest!: map_of_SomeD)
+  done
 
 lemma All_mapping_alist:
   "(\<And>k v. (k, v) \<in> set xs \<Longrightarrow> P k v) \<Longrightarrow> All_mapping (Mapping.of_alist xs) P"
   by (auto simp: All_mapping_def lookup_of_alist dest!: map_of_SomeD split: option.splits)
 
-
-lemma combine_empty [simp]:
-  "combine f Mapping.empty y = y" "combine f y Mapping.empty = y"
-  by (transfer, force)+
+lemma combine_empty [simp]: "combine f Mapping.empty y = y" "combine f y Mapping.empty = y"
+  by (transfer; force)+
 
 lemma (in abel_semigroup) comm_monoid_set_combine: "comm_monoid_set (combine f) Mapping.empty"
   by standard (transfer fixing: f, simp add: combine_options_ac[of f] ac_simps)+
@@ -679,19 +644,16 @@
     using that by (induction xs) simp_all
   from this[of "remdups xs"] show ?thesis by simp
 qed
-  
-lemma keys_fold_combine:
-  assumes "finite A"
-  shows   "Mapping.keys (combine.F g A) = (\<Union>x\<in>A. Mapping.keys (g x))"
-  using assms by (induction A rule: finite_induct) simp_all
+
+lemma keys_fold_combine: "finite A \<Longrightarrow> Mapping.keys (combine.F g A) = (\<Union>x\<in>A. Mapping.keys (g x))"
+  by (induct A rule: finite_induct) simp_all
 
 end
 
-  
+
 subsection \<open>Code generator setup\<close>
 
 hide_const (open) empty is_empty rep lookup lookup_default filter update delete ordered_keys
   keys size replace default map_entry map_default tabulate bulkload map map_values combine of_alist
 
 end
-




