# HG changeset patch
# User wenzelm
# Date 1468332259 -7200
# Node ID b6a1047bc164328b9eeff09bdcbbdeeaa05aadc1
# Parent  3365c8ec67bdc3e2fdf9a94920e755e522f0cf5b# Parent  c1fe30f2bc3276115642ee51799f84d706a7debc
merged

diff -r 3365c8ec67bd -r b6a1047bc164 NEWS
--- a/NEWS	Tue Jul 12 13:55:35 2016 +0200
+++ b/NEWS	Tue Jul 12 16:04:19 2016 +0200
@@ -76,6 +76,9 @@
 
 * Highlighting of entity def/ref positions wrt. cursor.
 
+* Refined folding mode "isabelle" based on Isar syntax: 'next' and 'qed'
+are treated as delimiters for fold structure.
+
 * Improved support for indentation according to Isabelle outer syntax.
 Action "indent-lines" (shortcut C+i) indents the current line according
 to command keywords and some command substructure. Action
diff -r 3365c8ec67bd -r b6a1047bc164 src/HOL/Library/AList.thy
--- a/src/HOL/Library/AList.thy	Tue Jul 12 13:55:35 2016 +0200
+++ b/src/HOL/Library/AList.thy	Tue Jul 12 16:04:19 2016 +0200
@@ -5,7 +5,7 @@
 section \<open>Implementation of Association Lists\<close>
 
 theory AList
-imports Main
+  imports Main
 begin
 
 context
@@ -21,9 +21,9 @@
 subsection \<open>\<open>update\<close> and \<open>updates\<close>\<close>
 
 qualified primrec update :: "'key \<Rightarrow> 'val \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list"
-where
-  "update k v [] = [(k, v)]"
-| "update k v (p # ps) = (if fst p = k then (k, v) # ps else p # update k v ps)"
+  where
+    "update k v [] = [(k, v)]"
+  | "update k v (p # ps) = (if fst p = k then (k, v) # ps else p # update k v ps)"
 
 lemma update_conv': "map_of (update k v al)  = (map_of al)(k\<mapsto>v)"
   by (induct al) (auto simp add: fun_eq_iff)
@@ -82,12 +82,11 @@
     x = k \<and> v = y \<or> x \<noteq> k \<and> map_of al x = Some y"
   by (simp add: update_conv' map_upd_Some_unfold)
 
-lemma image_update [simp]:
-  "x \<notin> A \<Longrightarrow> map_of (update x y al) ` A = map_of al ` A"
+lemma image_update [simp]: "x \<notin> A \<Longrightarrow> map_of (update x y al) ` A = map_of al ` A"
   by (simp add: update_conv')
 
-qualified definition
-    updates :: "'key list \<Rightarrow> 'val list \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list"
+qualified definition updates
+    :: "'key list \<Rightarrow> 'val list \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list"
   where "updates ks vs = fold (case_prod update) (zip ks vs)"
 
 lemma updates_simps [simp]:
@@ -217,66 +216,75 @@
 
 subsection \<open>\<open>update_with_aux\<close> and \<open>delete_aux\<close>\<close>
 
-qualified primrec update_with_aux :: "'val \<Rightarrow> 'key \<Rightarrow> ('val \<Rightarrow> 'val) \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list"
-where
-  "update_with_aux v k f [] = [(k, f v)]"
-| "update_with_aux v k f (p # ps) = (if (fst p = k) then (k, f (snd p)) # ps else p # update_with_aux v k f ps)"
+qualified primrec update_with_aux
+    :: "'val \<Rightarrow> 'key \<Rightarrow> ('val \<Rightarrow> 'val) \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list"
+  where
+    "update_with_aux v k f [] = [(k, f v)]"
+  | "update_with_aux v k f (p # ps) =
+      (if (fst p = k) then (k, f (snd p)) # ps else p # update_with_aux v k f ps)"
 
 text \<open>
   The above @{term "delete"} traverses all the list even if it has found the key.
   This one does not have to keep going because is assumes the invariant that keys are distinct.
 \<close>
 qualified fun delete_aux :: "'key \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list"
-where
-  "delete_aux k [] = []"
-| "delete_aux k ((k', v) # xs) = (if k = k' then xs else (k', v) # delete_aux k xs)"
+  where
+    "delete_aux k [] = []"
+  | "delete_aux k ((k', v) # xs) = (if k = k' then xs else (k', v) # delete_aux k xs)"
 
 lemma map_of_update_with_aux':
-  "map_of (update_with_aux v k f ps) k' = ((map_of ps)(k \<mapsto> (case map_of ps k of None \<Rightarrow> f v | Some v \<Rightarrow> f v))) k'"
-by(induct ps) auto
+  "map_of (update_with_aux v k f ps) k' =
+    ((map_of ps)(k \<mapsto> (case map_of ps k of None \<Rightarrow> f v | Some v \<Rightarrow> f v))) k'"
+  by (induct ps) auto
 
 lemma map_of_update_with_aux:
-  "map_of (update_with_aux v k f ps) = (map_of ps)(k \<mapsto> (case map_of ps k of None \<Rightarrow> f v | Some v \<Rightarrow> f v))"
-by(simp add: fun_eq_iff map_of_update_with_aux')
+  "map_of (update_with_aux v k f ps) =
+    (map_of ps)(k \<mapsto> (case map_of ps k of None \<Rightarrow> f v | Some v \<Rightarrow> f v))"
+  by (simp add: fun_eq_iff map_of_update_with_aux')
 
 lemma dom_update_with_aux: "fst ` set (update_with_aux v k f ps) = {k} \<union> fst ` set ps"
   by (induct ps) auto
 
 lemma distinct_update_with_aux [simp]:
   "distinct (map fst (update_with_aux v k f ps)) = distinct (map fst ps)"
-by(induct ps)(auto simp add: dom_update_with_aux)
+  by (induct ps) (auto simp add: dom_update_with_aux)
 
 lemma set_update_with_aux:
-  "distinct (map fst xs) 
-  \<Longrightarrow> set (update_with_aux v k f xs) = (set xs - {k} \<times> UNIV \<union> {(k, f (case map_of xs k of None \<Rightarrow> v | Some v \<Rightarrow> v))})"
-by(induct xs)(auto intro: rev_image_eqI)
+  "distinct (map fst xs) \<Longrightarrow>
+    set (update_with_aux v k f xs) =
+      (set xs - {k} \<times> UNIV \<union> {(k, f (case map_of xs k of None \<Rightarrow> v | Some v \<Rightarrow> v))})"
+  by (induct xs) (auto intro: rev_image_eqI)
 
 lemma set_delete_aux: "distinct (map fst xs) \<Longrightarrow> set (delete_aux k xs) = set xs - {k} \<times> UNIV"
-apply(induct xs)
-apply simp_all
-apply clarsimp
-apply(fastforce intro: rev_image_eqI)
-done
+  apply (induct xs)
+  apply simp_all
+  apply clarsimp
+  apply (fastforce intro: rev_image_eqI)
+  done
 
 lemma dom_delete_aux: "distinct (map fst ps) \<Longrightarrow> fst ` set (delete_aux k ps) = fst ` set ps - {k}"
-by(auto simp add: set_delete_aux)
+  by (auto simp add: set_delete_aux)
 
-lemma distinct_delete_aux [simp]:
-  "distinct (map fst ps) \<Longrightarrow> distinct (map fst (delete_aux k ps))"
-proof(induct ps)
-  case Nil thus ?case by simp
+lemma distinct_delete_aux [simp]: "distinct (map fst ps) \<Longrightarrow> distinct (map fst (delete_aux k ps))"
+proof (induct ps)
+  case Nil
+  then show ?case by simp
 next
   case (Cons a ps)
-  obtain k' v where a: "a = (k', v)" by(cases a)
+  obtain k' v where a: "a = (k', v)"
+    by (cases a)
   show ?case
-  proof(cases "k' = k")
-    case True with Cons a show ?thesis by simp
+  proof (cases "k' = k")
+    case True
+    with Cons a show ?thesis by simp
   next
     case False
-    with Cons a have "k' \<notin> fst ` set ps" "distinct (map fst ps)" by simp_all
+    with Cons a have "k' \<notin> fst ` set ps" "distinct (map fst ps)"
+      by simp_all
     with False a have "k' \<notin> fst ` set (delete_aux k ps)"
-      by(auto dest!: dom_delete_aux[where k=k])
-    with Cons a show ?thesis by simp
+      by (auto dest!: dom_delete_aux[where k=k])
+    with Cons a show ?thesis
+      by simp
   qed
 qed
 
@@ -290,10 +298,10 @@
 
 lemma map_of_delete_aux:
   "distinct (map fst xs) \<Longrightarrow> map_of (delete_aux k xs) k' = ((map_of xs)(k := None)) k'"
-by(simp add: map_of_delete_aux')
+  by (simp add: map_of_delete_aux')
 
 lemma delete_aux_eq_Nil_conv: "delete_aux k ts = [] \<longleftrightarrow> ts = [] \<or> (\<exists>v. ts = [(k, v)])"
-by(cases ts)(auto split: if_split_asm)
+  by (cases ts) (auto split: if_split_asm)
 
 
 subsection \<open>\<open>restrict\<close>\<close>
@@ -308,16 +316,18 @@
 
 lemma restr_conv': "map_of (restrict A al) = ((map_of al)|` A)"
 proof
-  fix k
-  show "map_of (restrict A al) k = ((map_of al)|` A) k"
-    by (induct al) (simp, cases "k \<in> A", auto)
+  show "map_of (restrict A al) k = ((map_of al)|` A) k" for k
+    apply (induct al)
+    apply simp
+    apply (cases "k \<in> A")
+    apply auto
+    done
 qed
 
 corollary restr_conv: "map_of (restrict A al) k = ((map_of al)|` A) k"
   by (simp add: restr_conv')
 
-lemma distinct_restr:
-  "distinct (map fst al) \<Longrightarrow> distinct (map fst (restrict A al))"
+lemma distinct_restr: "distinct (map fst al) \<Longrightarrow> distinct (map fst (restrict A al))"
   by (induct al) (auto simp add: restrict_eq)
 
 lemma restr_empty [simp]:
@@ -341,8 +351,8 @@
   by (induct al) (auto simp add: restrict_eq)
 
 lemma restr_update[simp]:
- "map_of (restrict D (update x y al)) =
-  map_of ((if x \<in> D then (update x y (restrict (D-{x}) al)) else restrict D al))"
+  "map_of (restrict D (update x y al)) =
+    map_of ((if x \<in> D then (update x y (restrict (D-{x}) al)) else restrict D al))"
   by (simp add: restr_conv' update_conv')
 
 lemma restr_delete [simp]:
@@ -350,12 +360,12 @@
   apply (simp add: delete_eq restrict_eq)
   apply (auto simp add: split_def)
 proof -
-  have "\<And>y. y \<noteq> x \<longleftrightarrow> x \<noteq> y"
+  have "y \<noteq> x \<longleftrightarrow> x \<noteq> y" for y
     by auto
   then show "[p \<leftarrow> al. fst p \<in> D \<and> x \<noteq> fst p] = [p \<leftarrow> al. fst p \<in> D \<and> fst p \<noteq> x]"
     by simp
   assume "x \<notin> D"
-  then have "\<And>y. y \<in> D \<longleftrightarrow> y \<in> D \<and> x \<noteq> y"
+  then have "y \<in> D \<longleftrightarrow> y \<in> D \<and> x \<noteq> y" for y
     by auto
   then show "[p \<leftarrow> al . fst p \<in> D \<and> x \<noteq> fst p] = [p \<leftarrow> al . fst p \<in> D]"
     by simp
@@ -383,9 +393,9 @@
 subsection \<open>\<open>clearjunk\<close>\<close>
 
 qualified function clearjunk  :: "('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list"
-where
-  "clearjunk [] = []"
-| "clearjunk (p#ps) = p # clearjunk (delete (fst p) ps)"
+  where
+    "clearjunk [] = []"
+  | "clearjunk (p#ps) = p # clearjunk (delete (fst p) ps)"
   by pat_completeness auto
 termination
   by (relation "measure length") (simp_all add: less_Suc_eq_le length_delete_le)
@@ -420,7 +430,7 @@
 lemma clearjunk_updates: "clearjunk (updates ks vs al) = updates ks vs (clearjunk al)"
 proof -
   have "clearjunk \<circ> fold (case_prod update) (zip ks vs) =
-    fold (case_prod update) (zip ks vs) \<circ> clearjunk"
+      fold (case_prod update) (zip ks vs) \<circ> clearjunk"
     by (rule fold_commute) (simp add: clearjunk_update case_prod_beta o_def)
   then show ?thesis
     by (simp add: updates_def fun_eq_iff)
@@ -506,10 +516,8 @@
 lemma dom_merge: "fst ` set (merge xs ys) = fst ` set xs \<union> fst ` set ys"
   by (induct ys arbitrary: xs) (auto simp add: dom_update)
 
-lemma distinct_merge:
-  assumes "distinct (map fst xs)"
-  shows "distinct (map fst (merge xs ys))"
-  using assms by (simp add: merge_updates distinct_updates)
+lemma distinct_merge: "distinct (map fst xs) \<Longrightarrow> distinct (map fst (merge xs ys))"
+  by (simp add: merge_updates distinct_updates)
 
 lemma clearjunk_merge: "clearjunk (merge xs ys) = merge (clearjunk xs) ys"
   by (simp add: merge_updates clearjunk_updates)
@@ -542,12 +550,10 @@
 lemma merge_find_right [simp]: "map_of n k = Some v \<Longrightarrow> map_of (merge m n) k = Some v"
   by (simp add: merge_conv')
 
-lemma merge_None [iff]:
-  "(map_of (merge m n) k = None) = (map_of n k = None \<and> map_of m k = None)"
+lemma merge_None [iff]: "(map_of (merge m n) k = None) = (map_of n k = None \<and> map_of m k = None)"
   by (simp add: merge_conv')
 
-lemma merge_upd [simp]:
-  "map_of (merge m (update k v n)) = map_of (update k v (merge m n))"
+lemma merge_upd [simp]: "map_of (merge m (update k v n)) = map_of (update k v (merge m n))"
   by (simp add: update_conv' merge_conv')
 
 lemma merge_updatess [simp]:
@@ -561,20 +567,18 @@
 subsection \<open>\<open>compose\<close>\<close>
 
 qualified function compose :: "('key \<times> 'a) list \<Rightarrow> ('a \<times> 'b) list \<Rightarrow> ('key \<times> 'b) list"
-where
-  "compose [] ys = []"
-| "compose (x # xs) ys =
-    (case map_of ys (snd x) of
-      None \<Rightarrow> compose (delete (fst x) xs) ys
-    | Some v \<Rightarrow> (fst x, v) # compose xs ys)"
+  where
+    "compose [] ys = []"
+  | "compose (x # xs) ys =
+      (case map_of ys (snd x) of
+        None \<Rightarrow> compose (delete (fst x) xs) ys
+      | Some v \<Rightarrow> (fst x, v) # compose xs ys)"
   by pat_completeness auto
 termination
   by (relation "measure (length \<circ> fst)") (simp_all add: less_Suc_eq_le length_delete_le)
 
-lemma compose_first_None [simp]:
-  assumes "map_of xs k = None"
-  shows "map_of (compose xs ys) k = None"
-  using assms by (induct xs ys rule: compose.induct) (auto split: option.splits if_split_asm)
+lemma compose_first_None [simp]: "map_of xs k = None \<Longrightarrow> map_of (compose xs ys) k = None"
+  by (induct xs ys rule: compose.induct) (auto split: option.splits if_split_asm)
 
 lemma compose_conv: "map_of (compose xs ys) k = (map_of ys \<circ>\<^sub>m map_of xs) k"
 proof (induct xs ys rule: compose.induct)
@@ -617,10 +621,8 @@
 lemma compose_conv': "map_of (compose xs ys) = (map_of ys \<circ>\<^sub>m map_of xs)"
   by (rule ext) (rule compose_conv)
 
-lemma compose_first_Some [simp]:
-  assumes "map_of xs k = Some v"
-  shows "map_of (compose xs ys) k = map_of ys v"
-  using assms by (simp add: compose_conv)
+lemma compose_first_Some [simp]: "map_of xs k = Some v \<Longrightarrow> map_of (compose xs ys) k = map_of ys v"
+  by (simp add: compose_conv)
 
 lemma dom_compose: "fst ` set (compose xs ys) \<subseteq> fst ` set xs"
 proof (induct xs ys rule: compose.induct)
@@ -631,19 +633,15 @@
   show ?case
   proof (cases "map_of ys (snd x)")
     case None
-    with "2.hyps"
-    have "fst ` set (compose (delete (fst x) xs) ys) \<subseteq> fst ` set (delete (fst x) xs)"
+    with "2.hyps" have "fst ` set (compose (delete (fst x) xs) ys) \<subseteq> fst ` set (delete (fst x) xs)"
       by simp
-    also
-    have "\<dots> \<subseteq> fst ` set xs"
+    also have "\<dots> \<subseteq> fst ` set xs"
       by (rule dom_delete_subset)
     finally show ?thesis
-      using None
-      by auto
+      using None by auto
   next
     case (Some v)
-    with "2.hyps"
-    have "fst ` set (compose xs ys) \<subseteq> fst ` set xs"
+    with "2.hyps" have "fst ` set (compose xs ys) \<subseteq> fst ` set xs"
       by simp
     with Some show ?thesis
       by auto
@@ -726,10 +724,10 @@
 subsection \<open>\<open>map_entry\<close>\<close>
 
 qualified fun map_entry :: "'key \<Rightarrow> ('val \<Rightarrow> 'val) \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list"
-where
-  "map_entry k f [] = []"
-| "map_entry k f (p # ps) =
-    (if fst p = k then (k, f (snd p)) # ps else p # map_entry k f ps)"
+  where
+    "map_entry k f [] = []"
+  | "map_entry k f (p # ps) =
+      (if fst p = k then (k, f (snd p)) # ps else p # map_entry k f ps)"
 
 lemma map_of_map_entry:
   "map_of (map_entry k f xs) =
@@ -748,10 +746,10 @@
 subsection \<open>\<open>map_default\<close>\<close>
 
 fun map_default :: "'key \<Rightarrow> 'val \<Rightarrow> ('val \<Rightarrow> 'val) \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list"
-where
-  "map_default k v f [] = [(k, v)]"
-| "map_default k v f (p # ps) =
-    (if fst p = k then (k, f (snd p)) # ps else p # map_default k v f ps)"
+  where
+    "map_default k v f [] = [(k, v)]"
+  | "map_default k v f (p # ps) =
+      (if fst p = k then (k, f (snd p)) # ps else p # map_default k v f ps)"
 
 lemma map_of_map_default:
   "map_of (map_default k v f xs) =
diff -r 3365c8ec67bd -r b6a1047bc164 src/HOL/Library/AList_Mapping.thy
--- a/src/HOL/Library/AList_Mapping.thy	Tue Jul 12 13:55:35 2016 +0200
+++ b/src/HOL/Library/AList_Mapping.thy	Tue Jul 12 16:04:19 2016 +0200
@@ -1,51 +1,43 @@
-(* Title: HOL/Library/AList_Mapping.thy
-   Author: Florian Haftmann, TU Muenchen
+(*  Title:      HOL/Library/AList_Mapping.thy
+    Author:     Florian Haftmann, TU Muenchen
 *)
 
 section \<open>Implementation of mappings with Association Lists\<close>
 
 theory AList_Mapping
-imports AList Mapping
+  imports AList Mapping
 begin
 
 lift_definition Mapping :: "('a \<times> 'b) list \<Rightarrow> ('a, 'b) mapping" is map_of .
 
 code_datatype Mapping
 
-lemma lookup_Mapping [simp, code]:
-  "Mapping.lookup (Mapping xs) = map_of xs"
+lemma lookup_Mapping [simp, code]: "Mapping.lookup (Mapping xs) = map_of xs"
   by transfer rule
 
-lemma keys_Mapping [simp, code]:
-  "Mapping.keys (Mapping xs) = set (map fst xs)" 
+lemma keys_Mapping [simp, code]: "Mapping.keys (Mapping xs) = set (map fst xs)"
   by transfer (simp add: dom_map_of_conv_image_fst)
 
-lemma empty_Mapping [code]:
-  "Mapping.empty = Mapping []"
+lemma empty_Mapping [code]: "Mapping.empty = Mapping []"
   by transfer simp
 
-lemma is_empty_Mapping [code]:
-  "Mapping.is_empty (Mapping xs) \<longleftrightarrow> List.null xs"
+lemma is_empty_Mapping [code]: "Mapping.is_empty (Mapping xs) \<longleftrightarrow> List.null xs"
   by (case_tac xs) (simp_all add: is_empty_def null_def)
 
-lemma update_Mapping [code]:
-  "Mapping.update k v (Mapping xs) = Mapping (AList.update k v xs)"
+lemma update_Mapping [code]: "Mapping.update k v (Mapping xs) = Mapping (AList.update k v xs)"
   by transfer (simp add: update_conv')
 
-lemma delete_Mapping [code]:
-  "Mapping.delete k (Mapping xs) = Mapping (AList.delete k xs)"
+lemma delete_Mapping [code]: "Mapping.delete k (Mapping xs) = Mapping (AList.delete k xs)"
   by transfer (simp add: delete_conv')
 
 lemma ordered_keys_Mapping [code]:
   "Mapping.ordered_keys (Mapping xs) = sort (remdups (map fst xs))"
   by (simp only: ordered_keys_def keys_Mapping sorted_list_of_set_sort_remdups) simp
 
-lemma size_Mapping [code]:
-  "Mapping.size (Mapping xs) = length (remdups (map fst xs))"
+lemma size_Mapping [code]: "Mapping.size (Mapping xs) = length (remdups (map fst xs))"
   by (simp add: size_def length_remdups_card_conv dom_map_of_conv_image_fst)
 
-lemma tabulate_Mapping [code]:
-  "Mapping.tabulate ks f = Mapping (map (\<lambda>k. (k, f k)) ks)"
+lemma tabulate_Mapping [code]: "Mapping.tabulate ks f = Mapping (map (\<lambda>k. (k, f k)) ks)"
   by transfer (simp add: map_of_map_restrict)
 
 lemma bulkload_Mapping [code]:
@@ -55,63 +47,69 @@
 lemma equal_Mapping [code]:
   "HOL.equal (Mapping xs) (Mapping ys) \<longleftrightarrow>
     (let ks = map fst xs; ls = map fst ys
-    in (\<forall>l\<in>set ls. l \<in> set ks) \<and> (\<forall>k\<in>set ks. k \<in> set ls \<and> map_of xs k = map_of ys k))"
+     in (\<forall>l\<in>set ls. l \<in> set ks) \<and> (\<forall>k\<in>set ks. k \<in> set ls \<and> map_of xs k = map_of ys k))"
 proof -
-  have aux: "\<And>a b xs. (a, b) \<in> set xs \<Longrightarrow> a \<in> fst ` set xs"
+  have *: "(a, b) \<in> set xs \<Longrightarrow> a \<in> fst ` set xs" for a b xs
     by (auto simp add: image_def intro!: bexI)
   show ?thesis apply transfer
-    by (auto intro!: map_of_eqI) (auto dest!: map_of_eq_dom intro: aux)
+    by (auto intro!: map_of_eqI) (auto dest!: map_of_eq_dom intro: *)
 qed
 
 lemma map_values_Mapping [code]:
-  fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> 'b" and xs :: "('c \<times> 'a) list"
-  shows "Mapping.map_values f (Mapping xs) = Mapping (map (\<lambda>(x,y). (x, f x y)) xs)"
-proof (transfer, rule ext, goal_cases)
-  case (1 f xs x)
-  thus ?case by (induction xs) auto
-qed
+  "Mapping.map_values f (Mapping xs) = Mapping (map (\<lambda>(x,y). (x, f x y)) xs)"
+  for f :: "'c \<Rightarrow> 'a \<Rightarrow> 'b" and xs :: "('c \<times> 'a) list"
+  apply transfer
+  apply (rule ext)
+  subgoal for f xs x by (induct xs) auto
+  done
 
-lemma combine_with_key_code [code]: 
+lemma combine_with_key_code [code]:
   "Mapping.combine_with_key f (Mapping xs) (Mapping ys) =
-     Mapping.tabulate (remdups (map fst xs @ map fst ys)) 
+     Mapping.tabulate (remdups (map fst xs @ map fst ys))
        (\<lambda>x. the (combine_options (f x) (map_of xs x) (map_of ys x)))"
-proof (transfer, rule ext, rule sym, goal_cases)
-  case (1 f xs ys x)
-  show ?case
-  by (cases "map_of xs x"; cases "map_of ys x"; simp)
-     (force simp: map_of_eq_None_iff combine_options_def option.the_def o_def image_iff
-            dest: map_of_SomeD split: option.splits)+
-qed
+  apply transfer
+  apply (rule ext)
+  apply (rule sym)
+  subgoal for f xs ys x
+    apply (cases "map_of xs x"; cases "map_of ys x"; simp)
+    apply (force simp: map_of_eq_None_iff combine_options_def option.the_def o_def image_iff
+      dest: map_of_SomeD split: option.splits)+
+    done
+  done
 
-lemma combine_code [code]: 
+lemma combine_code [code]:
   "Mapping.combine f (Mapping xs) (Mapping ys) =
-     Mapping.tabulate (remdups (map fst xs @ map fst ys)) 
+     Mapping.tabulate (remdups (map fst xs @ map fst ys))
        (\<lambda>x. the (combine_options f (map_of xs x) (map_of ys x)))"
-proof (transfer, rule ext, rule sym, goal_cases)
-  case (1 f xs ys x)
-  show ?case
-  by (cases "map_of xs x"; cases "map_of ys x"; simp)
-     (force simp: map_of_eq_None_iff combine_options_def option.the_def o_def image_iff
-            dest: map_of_SomeD split: option.splits)+
-qed
+  apply transfer
+  apply (rule ext)
+  apply (rule sym)
+  subgoal for f xs ys x
+    apply (cases "map_of xs x"; cases "map_of ys x"; simp)
+    apply (force simp: map_of_eq_None_iff combine_options_def option.the_def o_def image_iff
+      dest: map_of_SomeD split: option.splits)+
+    done
+  done
 
-(* TODO: Move? *)
-lemma map_of_filter_distinct:
+lemma map_of_filter_distinct:  (* TODO: move? *)
   assumes "distinct (map fst xs)"
-  shows   "map_of (filter P xs) x = 
-             (case map_of xs x of None \<Rightarrow> None | Some y \<Rightarrow> if P (x,y) then Some y else None)"
+  shows "map_of (filter P xs) x =
+    (case map_of xs x of
+      None \<Rightarrow> None
+    | Some y \<Rightarrow> if P (x,y) then Some y else None)"
   using assms
   by (auto simp: map_of_eq_None_iff filter_map distinct_map_filter dest: map_of_SomeD
-           simp del: map_of_eq_Some_iff intro!: map_of_is_SomeI split: option.splits)
-(* END TODO *)
-  
+      simp del: map_of_eq_Some_iff intro!: map_of_is_SomeI split: option.splits)
+
 lemma filter_Mapping [code]:
   "Mapping.filter P (Mapping xs) = Mapping (filter (\<lambda>(k,v). P k v) (AList.clearjunk xs))"
- by (transfer, rule ext)
-    (subst map_of_filter_distinct, simp_all add: map_of_clearjunk split: option.split)
+  apply transfer
+  apply (rule ext)
+  apply (subst map_of_filter_distinct)
+  apply (simp_all add: map_of_clearjunk split: option.split)
+  done
 
-lemma [code nbe]:
-  "HOL.equal (x :: ('a, 'b) mapping) x \<longleftrightarrow> True"
+lemma [code nbe]: "HOL.equal (x :: ('a, 'b) mapping) x \<longleftrightarrow> True"
   by (fact equal_refl)
 
 end
diff -r 3365c8ec67bd -r b6a1047bc164 src/HOL/Library/BigO.thy
--- a/src/HOL/Library/BigO.thy	Tue Jul 12 13:55:35 2016 +0200
+++ b/src/HOL/Library/BigO.thy	Tue Jul 12 16:04:19 2016 +0200
@@ -5,7 +5,7 @@
 section \<open>Big O notation\<close>
 
 theory BigO
-imports Complex_Main Function_Algebras Set_Algebras
+  imports Complex_Main Function_Algebras Set_Algebras
 begin
 
 text \<open>
diff -r 3365c8ec67bd -r b6a1047bc164 src/HOL/Library/Bit.thy
--- a/src/HOL/Library/Bit.thy	Tue Jul 12 13:55:35 2016 +0200
+++ b/src/HOL/Library/Bit.thy	Tue Jul 12 16:04:19 2016 +0200
@@ -11,17 +11,14 @@
 subsection \<open>Bits as a datatype\<close>
 
 typedef bit = "UNIV :: bool set"
-  morphisms set Bit
-  ..
+  morphisms set Bit ..
 
 instantiation bit :: "{zero, one}"
 begin
 
-definition zero_bit_def:
-  "0 = Bit False"
+definition zero_bit_def: "0 = Bit False"
 
-definition one_bit_def:
-  "1 = Bit True"
+definition one_bit_def: "1 = Bit True"
 
 instance ..
 
@@ -29,8 +26,9 @@
 
 old_rep_datatype "0::bit" "1::bit"
 proof -
-  fix P and x :: bit
-  assume "P (0::bit)" and "P (1::bit)"
+  fix P :: "bit \<Rightarrow> bool"
+  fix x :: bit
+  assume "P 0" and "P 1"
   then have "\<forall>b. P (Bit b)"
     unfolding zero_bit_def one_bit_def
     by (simp add: all_bool_eq)
@@ -42,21 +40,17 @@
     by (simp add: Bit_inject)
 qed
 
-lemma Bit_set_eq [simp]:
-  "Bit (set b) = b"
+lemma Bit_set_eq [simp]: "Bit (set b) = b"
   by (fact set_inverse)
-  
-lemma set_Bit_eq [simp]:
-  "set (Bit P) = P"
+
+lemma set_Bit_eq [simp]: "set (Bit P) = P"
   by (rule Bit_inverse) rule
 
-lemma bit_eq_iff:
-  "x = y \<longleftrightarrow> (set x \<longleftrightarrow> set y)"
+lemma bit_eq_iff: "x = y \<longleftrightarrow> (set x \<longleftrightarrow> set y)"
   by (auto simp add: set_inject)
 
-lemma Bit_inject [simp]:
-  "Bit P = Bit Q \<longleftrightarrow> (P \<longleftrightarrow> Q)"
-  by (auto simp add: Bit_inject)  
+lemma Bit_inject [simp]: "Bit P = Bit Q \<longleftrightarrow> (P \<longleftrightarrow> Q)"
+  by (auto simp add: Bit_inject)
 
 lemma set [iff]:
   "\<not> set 0"
@@ -68,8 +62,7 @@
   "set 1 \<longleftrightarrow> True"
   by simp_all
 
-lemma set_iff:
-  "set b \<longleftrightarrow> b = 1"
+lemma set_iff: "set b \<longleftrightarrow> b = 1"
   by (cases b) simp_all
 
 lemma bit_eq_iff_set:
@@ -82,42 +75,34 @@
   "Bit True = 1"
   by (simp_all add: zero_bit_def one_bit_def)
 
-lemma bit_not_0_iff [iff]:
-  "(x::bit) \<noteq> 0 \<longleftrightarrow> x = 1"
+lemma bit_not_0_iff [iff]: "x \<noteq> 0 \<longleftrightarrow> x = 1" for x :: bit
   by (simp add: bit_eq_iff)
 
-lemma bit_not_1_iff [iff]:
-  "(x::bit) \<noteq> 1 \<longleftrightarrow> x = 0"
+lemma bit_not_1_iff [iff]: "x \<noteq> 1 \<longleftrightarrow> x = 0" for x :: bit
   by (simp add: bit_eq_iff)
 
 lemma [code]:
   "HOL.equal 0 b \<longleftrightarrow> \<not> set b"
   "HOL.equal 1 b \<longleftrightarrow> set b"
-  by (simp_all add: equal set_iff)  
+  by (simp_all add: equal set_iff)
 
-  
+
 subsection \<open>Type @{typ bit} forms a field\<close>
 
 instantiation bit :: field
 begin
 
-definition plus_bit_def:
-  "x + y = case_bit y (case_bit 1 0 y) x"
+definition plus_bit_def: "x + y = case_bit y (case_bit 1 0 y) x"
 
-definition times_bit_def:
-  "x * y = case_bit 0 y x"
+definition times_bit_def: "x * y = case_bit 0 y x"
 
-definition uminus_bit_def [simp]:
-  "- x = (x :: bit)"
+definition uminus_bit_def [simp]: "- x = x" for x :: bit
 
-definition minus_bit_def [simp]:
-  "x - y = (x + y :: bit)"
+definition minus_bit_def [simp]: "x - y = x + y" for x y :: bit
 
-definition inverse_bit_def [simp]:
-  "inverse x = (x :: bit)"
+definition inverse_bit_def [simp]: "inverse x = x" for x :: bit
 
-definition divide_bit_def [simp]:
-  "x div y = (x * y :: bit)"
+definition divide_bit_def [simp]: "x div y = x * y" for x y :: bit
 
 lemmas field_bit_defs =
   plus_bit_def times_bit_def minus_bit_def uminus_bit_def
@@ -128,18 +113,18 @@
 
 end
 
-lemma bit_add_self: "x + x = (0 :: bit)"
+lemma bit_add_self: "x + x = 0" for x :: bit
   unfolding plus_bit_def by (simp split: bit.split)
 
-lemma bit_mult_eq_1_iff [simp]: "x * y = (1 :: bit) \<longleftrightarrow> x = 1 \<and> y = 1"
+lemma bit_mult_eq_1_iff [simp]: "x * y = 1 \<longleftrightarrow> x = 1 \<and> y = 1" for x y :: bit
   unfolding times_bit_def by (simp split: bit.split)
 
 text \<open>Not sure whether the next two should be simp rules.\<close>
 
-lemma bit_add_eq_0_iff: "x + y = (0 :: bit) \<longleftrightarrow> x = y"
+lemma bit_add_eq_0_iff: "x + y = 0 \<longleftrightarrow> x = y" for x y :: bit
   unfolding plus_bit_def by (simp split: bit.split)
 
-lemma bit_add_eq_1_iff: "x + y = (1 :: bit) \<longleftrightarrow> x \<noteq> y"
+lemma bit_add_eq_1_iff: "x + y = 1 \<longleftrightarrow> x \<noteq> y" for x y :: bit
   unfolding plus_bit_def by (simp split: bit.split)
 
 
@@ -166,37 +151,23 @@
 begin
 
 definition of_bit :: "bit \<Rightarrow> 'a"
-where
-  "of_bit b = case_bit 0 1 b" 
+  where "of_bit b = case_bit 0 1 b"
 
 lemma of_bit_eq [simp, code]:
   "of_bit 0 = 0"
   "of_bit 1 = 1"
   by (simp_all add: of_bit_def)
 
-lemma of_bit_eq_iff:
-  "of_bit x = of_bit y \<longleftrightarrow> x = y"
-  by (cases x) (cases y, simp_all)+
-
-end  
-
-context semiring_1
-begin
-
-lemma of_nat_of_bit_eq:
-  "of_nat (of_bit b) = of_bit b"
-  by (cases b) simp_all
+lemma of_bit_eq_iff: "of_bit x = of_bit y \<longleftrightarrow> x = y"
+  by (cases x) (cases y; simp)+
 
 end
 
-context ring_1
-begin
-
-lemma of_int_of_bit_eq:
-  "of_int (of_bit b) = of_bit b"
+lemma (in semiring_1) of_nat_of_bit_eq: "of_nat (of_bit b) = of_bit b"
   by (cases b) simp_all
 
-end
+lemma (in ring_1) of_int_of_bit_eq: "of_int (of_bit b) = of_bit b"
+  by (cases b) simp_all
 
 hide_const (open) set
 
diff -r 3365c8ec67bd -r b6a1047bc164 src/HOL/Library/Boolean_Algebra.thy
--- a/src/HOL/Library/Boolean_Algebra.thy	Tue Jul 12 13:55:35 2016 +0200
+++ b/src/HOL/Library/Boolean_Algebra.thy	Tue Jul 12 16:04:19 2016 +0200
@@ -5,7 +5,7 @@
 section \<open>Boolean Algebras\<close>
 
 theory Boolean_Algebra
-imports Main
+  imports Main
 begin
 
 locale boolean =
@@ -40,18 +40,18 @@
 lemmas disj_ac = disj.assoc disj.commute disj.left_commute
 
 lemma dual: "boolean disj conj compl one zero"
-apply (rule boolean.intro)
-apply (rule disj_assoc)
-apply (rule conj_assoc)
-apply (rule disj_commute)
-apply (rule conj_commute)
-apply (rule disj_conj_distrib)
-apply (rule conj_disj_distrib)
-apply (rule disj_zero_right)
-apply (rule conj_one_right)
-apply (rule disj_cancel_right)
-apply (rule conj_cancel_right)
-done
+  apply (rule boolean.intro)
+  apply (rule disj_assoc)
+  apply (rule conj_assoc)
+  apply (rule disj_commute)
+  apply (rule conj_commute)
+  apply (rule disj_conj_distrib)
+  apply (rule conj_disj_distrib)
+  apply (rule disj_zero_right)
+  apply (rule conj_one_right)
+  apply (rule disj_cancel_right)
+  apply (rule conj_cancel_right)
+  done
 
 
 subsection \<open>Complement\<close>
@@ -63,99 +63,111 @@
   assumes 4: "a \<squnion> y = \<one>"
   shows "x = y"
 proof -
-  have "(a \<sqinter> x) \<squnion> (x \<sqinter> y) = (a \<sqinter> y) \<squnion> (x \<sqinter> y)" using 1 3 by simp
-  hence "(x \<sqinter> a) \<squnion> (x \<sqinter> y) = (y \<sqinter> a) \<squnion> (y \<sqinter> x)" using conj_commute by simp
-  hence "x \<sqinter> (a \<squnion> y) = y \<sqinter> (a \<squnion> x)" using conj_disj_distrib by simp
-  hence "x \<sqinter> \<one> = y \<sqinter> \<one>" using 2 4 by simp
-  thus "x = y" using conj_one_right by simp
+  have "(a \<sqinter> x) \<squnion> (x \<sqinter> y) = (a \<sqinter> y) \<squnion> (x \<sqinter> y)"
+    using 1 3 by simp
+  then have "(x \<sqinter> a) \<squnion> (x \<sqinter> y) = (y \<sqinter> a) \<squnion> (y \<sqinter> x)"
+    using conj_commute by simp
+  then have "x \<sqinter> (a \<squnion> y) = y \<sqinter> (a \<squnion> x)"
+    using conj_disj_distrib by simp
+  then have "x \<sqinter> \<one> = y \<sqinter> \<one>"
+    using 2 4 by simp
+  then show "x = y"
+    using conj_one_right by simp
 qed
 
-lemma compl_unique: "\<lbrakk>x \<sqinter> y = \<zero>; x \<squnion> y = \<one>\<rbrakk> \<Longrightarrow> \<sim> x = y"
-by (rule complement_unique [OF conj_cancel_right disj_cancel_right])
+lemma compl_unique: "x \<sqinter> y = \<zero> \<Longrightarrow> x \<squnion> y = \<one> \<Longrightarrow> \<sim> x = y"
+  by (rule complement_unique [OF conj_cancel_right disj_cancel_right])
 
 lemma double_compl [simp]: "\<sim> (\<sim> x) = x"
 proof (rule compl_unique)
-  from conj_cancel_right show "\<sim> x \<sqinter> x = \<zero>" by (simp only: conj_commute)
-  from disj_cancel_right show "\<sim> x \<squnion> x = \<one>" by (simp only: disj_commute)
+  from conj_cancel_right show "\<sim> x \<sqinter> x = \<zero>"
+    by (simp only: conj_commute)
+  from disj_cancel_right show "\<sim> x \<squnion> x = \<one>"
+    by (simp only: disj_commute)
 qed
 
-lemma compl_eq_compl_iff [simp]: "(\<sim> x = \<sim> y) = (x = y)"
-by (rule inj_eq [OF inj_on_inverseI], rule double_compl)
+lemma compl_eq_compl_iff [simp]: "\<sim> x = \<sim> y \<longleftrightarrow> x = y"
+  by (rule inj_eq [OF inj_on_inverseI]) (rule double_compl)
 
 
 subsection \<open>Conjunction\<close>
 
 lemma conj_absorb [simp]: "x \<sqinter> x = x"
 proof -
-  have "x \<sqinter> x = (x \<sqinter> x) \<squnion> \<zero>" using disj_zero_right by simp
-  also have "... = (x \<sqinter> x) \<squnion> (x \<sqinter> \<sim> x)" using conj_cancel_right by simp
-  also have "... = x \<sqinter> (x \<squnion> \<sim> x)" using conj_disj_distrib by (simp only:)
-  also have "... = x \<sqinter> \<one>" using disj_cancel_right by simp
-  also have "... = x" using conj_one_right by simp
+  have "x \<sqinter> x = (x \<sqinter> x) \<squnion> \<zero>"
+    using disj_zero_right by simp
+  also have "... = (x \<sqinter> x) \<squnion> (x \<sqinter> \<sim> x)"
+    using conj_cancel_right by simp
+  also have "... = x \<sqinter> (x \<squnion> \<sim> x)"
+    using conj_disj_distrib by (simp only:)
+  also have "... = x \<sqinter> \<one>"
+    using disj_cancel_right by simp
+  also have "... = x"
+    using conj_one_right by simp
   finally show ?thesis .
 qed
 
 lemma conj_zero_right [simp]: "x \<sqinter> \<zero> = \<zero>"
 proof -
-  have "x \<sqinter> \<zero> = x \<sqinter> (x \<sqinter> \<sim> x)" using conj_cancel_right by simp
-  also have "... = (x \<sqinter> x) \<sqinter> \<sim> x" using conj_assoc by (simp only:)
-  also have "... = x \<sqinter> \<sim> x" using conj_absorb by simp
-  also have "... = \<zero>" using conj_cancel_right by simp
+  have "x \<sqinter> \<zero> = x \<sqinter> (x \<sqinter> \<sim> x)"
+    using conj_cancel_right by simp
+  also have "... = (x \<sqinter> x) \<sqinter> \<sim> x"
+    using conj_assoc by (simp only:)
+  also have "... = x \<sqinter> \<sim> x"
+    using conj_absorb by simp
+  also have "... = \<zero>"
+    using conj_cancel_right by simp
   finally show ?thesis .
 qed
 
 lemma compl_one [simp]: "\<sim> \<one> = \<zero>"
-by (rule compl_unique [OF conj_zero_right disj_zero_right])
+  by (rule compl_unique [OF conj_zero_right disj_zero_right])
 
 lemma conj_zero_left [simp]: "\<zero> \<sqinter> x = \<zero>"
-by (subst conj_commute) (rule conj_zero_right)
+  by (subst conj_commute) (rule conj_zero_right)
 
 lemma conj_one_left [simp]: "\<one> \<sqinter> x = x"
-by (subst conj_commute) (rule conj_one_right)
+  by (subst conj_commute) (rule conj_one_right)
 
 lemma conj_cancel_left [simp]: "\<sim> x \<sqinter> x = \<zero>"
-by (subst conj_commute) (rule conj_cancel_right)
+  by (subst conj_commute) (rule conj_cancel_right)
 
 lemma conj_left_absorb [simp]: "x \<sqinter> (x \<sqinter> y) = x \<sqinter> y"
-by (simp only: conj_assoc [symmetric] conj_absorb)
+  by (simp only: conj_assoc [symmetric] conj_absorb)
 
-lemma conj_disj_distrib2:
-  "(y \<squnion> z) \<sqinter> x = (y \<sqinter> x) \<squnion> (z \<sqinter> x)"
-by (simp only: conj_commute conj_disj_distrib)
+lemma conj_disj_distrib2: "(y \<squnion> z) \<sqinter> x = (y \<sqinter> x) \<squnion> (z \<sqinter> x)"
+  by (simp only: conj_commute conj_disj_distrib)
 
-lemmas conj_disj_distribs =
-   conj_disj_distrib conj_disj_distrib2
+lemmas conj_disj_distribs = conj_disj_distrib conj_disj_distrib2
 
 
 subsection \<open>Disjunction\<close>
 
 lemma disj_absorb [simp]: "x \<squnion> x = x"
-by (rule boolean.conj_absorb [OF dual])
+  by (rule boolean.conj_absorb [OF dual])
 
 lemma disj_one_right [simp]: "x \<squnion> \<one> = \<one>"
-by (rule boolean.conj_zero_right [OF dual])
+  by (rule boolean.conj_zero_right [OF dual])
 
 lemma compl_zero [simp]: "\<sim> \<zero> = \<one>"
-by (rule boolean.compl_one [OF dual])
+  by (rule boolean.compl_one [OF dual])
 
 lemma disj_zero_left [simp]: "\<zero> \<squnion> x = x"
-by (rule boolean.conj_one_left [OF dual])
+  by (rule boolean.conj_one_left [OF dual])
 
 lemma disj_one_left [simp]: "\<one> \<squnion> x = \<one>"
-by (rule boolean.conj_zero_left [OF dual])
+  by (rule boolean.conj_zero_left [OF dual])
 
 lemma disj_cancel_left [simp]: "\<sim> x \<squnion> x = \<one>"
-by (rule boolean.conj_cancel_left [OF dual])
+  by (rule boolean.conj_cancel_left [OF dual])
 
 lemma disj_left_absorb [simp]: "x \<squnion> (x \<squnion> y) = x \<squnion> y"
-by (rule boolean.conj_left_absorb [OF dual])
+  by (rule boolean.conj_left_absorb [OF dual])
 
-lemma disj_conj_distrib2:
-  "(y \<sqinter> z) \<squnion> x = (y \<squnion> x) \<sqinter> (z \<squnion> x)"
-by (rule boolean.conj_disj_distrib2 [OF dual])
+lemma disj_conj_distrib2: "(y \<sqinter> z) \<squnion> x = (y \<squnion> x) \<sqinter> (z \<squnion> x)"
+  by (rule boolean.conj_disj_distrib2 [OF dual])
 
-lemmas disj_conj_distribs =
-   disj_conj_distrib disj_conj_distrib2
+lemmas disj_conj_distribs = disj_conj_distrib disj_conj_distrib2
 
 
 subsection \<open>De Morgan's Laws\<close>
@@ -178,7 +190,7 @@
 qed
 
 lemma de_Morgan_disj [simp]: "\<sim> (x \<squnion> y) = \<sim> x \<sqinter> \<sim> y"
-by (rule boolean.de_Morgan_conj [OF dual])
+  by (rule boolean.de_Morgan_conj [OF dual])
 
 end
 
@@ -198,7 +210,7 @@
   have "?t \<squnion> (z \<sqinter> x \<sqinter> \<sim> x) \<squnion> (z \<sqinter> y \<sqinter> \<sim> y) =
         ?t \<squnion> (x \<sqinter> y \<sqinter> \<sim> y) \<squnion> (x \<sqinter> z \<sqinter> \<sim> z)"
     by (simp only: conj_cancel_right conj_zero_right)
-  thus "(x \<oplus> y) \<oplus> z = x \<oplus> (y \<oplus> z)"
+  then show "(x \<oplus> y) \<oplus> z = x \<oplus> (y \<oplus> z)"
     apply (simp only: xor_def de_Morgan_disj de_Morgan_conj double_compl)
     apply (simp only: conj_disj_distribs conj_ac disj_ac)
     done
@@ -212,57 +224,55 @@
 
 lemmas xor_ac = xor.assoc xor.commute xor.left_commute
 
-lemma xor_def2:
-  "x \<oplus> y = (x \<squnion> y) \<sqinter> (\<sim> x \<squnion> \<sim> y)"
-by (simp only: xor_def conj_disj_distribs
-               disj_ac conj_ac conj_cancel_right disj_zero_left)
+lemma xor_def2: "x \<oplus> y = (x \<squnion> y) \<sqinter> (\<sim> x \<squnion> \<sim> y)"
+  by (simp only: xor_def conj_disj_distribs disj_ac conj_ac conj_cancel_right disj_zero_left)
 
 lemma xor_zero_right [simp]: "x \<oplus> \<zero> = x"
-by (simp only: xor_def compl_zero conj_one_right conj_zero_right disj_zero_right)
+  by (simp only: xor_def compl_zero conj_one_right conj_zero_right disj_zero_right)
 
 lemma xor_zero_left [simp]: "\<zero> \<oplus> x = x"
-by (subst xor_commute) (rule xor_zero_right)
+  by (subst xor_commute) (rule xor_zero_right)
 
 lemma xor_one_right [simp]: "x \<oplus> \<one> = \<sim> x"
-by (simp only: xor_def compl_one conj_zero_right conj_one_right disj_zero_left)
+  by (simp only: xor_def compl_one conj_zero_right conj_one_right disj_zero_left)
 
 lemma xor_one_left [simp]: "\<one> \<oplus> x = \<sim> x"
-by (subst xor_commute) (rule xor_one_right)
+  by (subst xor_commute) (rule xor_one_right)
 
 lemma xor_self [simp]: "x \<oplus> x = \<zero>"
-by (simp only: xor_def conj_cancel_right conj_cancel_left disj_zero_right)
+  by (simp only: xor_def conj_cancel_right conj_cancel_left disj_zero_right)
 
 lemma xor_left_self [simp]: "x \<oplus> (x \<oplus> y) = y"
-by (simp only: xor_assoc [symmetric] xor_self xor_zero_left)
+  by (simp only: xor_assoc [symmetric] xor_self xor_zero_left)
 
 lemma xor_compl_left [simp]: "\<sim> x \<oplus> y = \<sim> (x \<oplus> y)"
-apply (simp only: xor_def de_Morgan_disj de_Morgan_conj double_compl)
-apply (simp only: conj_disj_distribs)
-apply (simp only: conj_cancel_right conj_cancel_left)
-apply (simp only: disj_zero_left disj_zero_right)
-apply (simp only: disj_ac conj_ac)
-done
+  apply (simp only: xor_def de_Morgan_disj de_Morgan_conj double_compl)
+  apply (simp only: conj_disj_distribs)
+  apply (simp only: conj_cancel_right conj_cancel_left)
+  apply (simp only: disj_zero_left disj_zero_right)
+  apply (simp only: disj_ac conj_ac)
+  done
 
 lemma xor_compl_right [simp]: "x \<oplus> \<sim> y = \<sim> (x \<oplus> y)"
-apply (simp only: xor_def de_Morgan_disj de_Morgan_conj double_compl)
-apply (simp only: conj_disj_distribs)
-apply (simp only: conj_cancel_right conj_cancel_left)
-apply (simp only: disj_zero_left disj_zero_right)
-apply (simp only: disj_ac conj_ac)
-done
+  apply (simp only: xor_def de_Morgan_disj de_Morgan_conj double_compl)
+  apply (simp only: conj_disj_distribs)
+  apply (simp only: conj_cancel_right conj_cancel_left)
+  apply (simp only: disj_zero_left disj_zero_right)
+  apply (simp only: disj_ac conj_ac)
+  done
 
 lemma xor_cancel_right: "x \<oplus> \<sim> x = \<one>"
-by (simp only: xor_compl_right xor_self compl_zero)
+  by (simp only: xor_compl_right xor_self compl_zero)
 
 lemma xor_cancel_left: "\<sim> x \<oplus> x = \<one>"
-by (simp only: xor_compl_left xor_self compl_zero)
+  by (simp only: xor_compl_left xor_self compl_zero)
 
 lemma conj_xor_distrib: "x \<sqinter> (y \<oplus> z) = (x \<sqinter> y) \<oplus> (x \<sqinter> z)"
 proof -
-  have "(x \<sqinter> y \<sqinter> \<sim> z) \<squnion> (x \<sqinter> \<sim> y \<sqinter> z) =
+  have *: "(x \<sqinter> y \<sqinter> \<sim> z) \<squnion> (x \<sqinter> \<sim> y \<sqinter> z) =
         (y \<sqinter> x \<sqinter> \<sim> x) \<squnion> (z \<sqinter> x \<sqinter> \<sim> x) \<squnion> (x \<sqinter> y \<sqinter> \<sim> z) \<squnion> (x \<sqinter> \<sim> y \<sqinter> z)"
     by (simp only: conj_cancel_right conj_zero_right disj_zero_left)
-  thus "x \<sqinter> (y \<oplus> z) = (x \<sqinter> y) \<oplus> (x \<sqinter> z)"
+  then show "x \<sqinter> (y \<oplus> z) = (x \<sqinter> y) \<oplus> (x \<sqinter> z)"
     by (simp (no_asm_use) only:
         xor_def de_Morgan_disj de_Morgan_conj double_compl
         conj_disj_distribs conj_ac disj_ac)
@@ -272,7 +282,7 @@
 proof -
   have "x \<sqinter> (y \<oplus> z) = (x \<sqinter> y) \<oplus> (x \<sqinter> z)"
     by (rule conj_xor_distrib)
-  thus "(y \<oplus> z) \<sqinter> x = (y \<sqinter> x) \<oplus> (z \<sqinter> x)"
+  then show "(y \<oplus> z) \<sqinter> x = (y \<sqinter> x) \<oplus> (z \<sqinter> x)"
     by (simp only: conj_commute)
 qed
 
diff -r 3365c8ec67bd -r b6a1047bc164 src/HOL/Library/Char_ord.thy
--- a/src/HOL/Library/Char_ord.thy	Tue Jul 12 13:55:35 2016 +0200
+++ b/src/HOL/Library/Char_ord.thy	Tue Jul 12 16:04:19 2016 +0200
@@ -5,7 +5,7 @@
 section \<open>Order on characters\<close>
 
 theory Char_ord
-imports Main
+  imports Main
 begin
 
 instantiation char :: linorder
@@ -20,17 +20,19 @@
 end
 
 lemma less_eq_char_simps:
-  "(0 :: char) \<le> c"
+  "0 \<le> c"
   "Char k \<le> 0 \<longleftrightarrow> numeral k mod 256 = (0 :: nat)"
   "Char k \<le> Char l \<longleftrightarrow> numeral k mod 256 \<le> (numeral l mod 256 :: nat)"
+  for c :: char
   by (simp_all add: Char_def less_eq_char_def)
 
 lemma less_char_simps:
-  "\<not> c < (0 :: char)"
+  "\<not> c < 0"
   "0 < Char k \<longleftrightarrow> (0 :: nat) < numeral k mod 256"
   "Char k < Char l \<longleftrightarrow> numeral k mod 256 < (numeral l mod 256 :: nat)"
+  for c :: char
   by (simp_all add: Char_def less_char_def)
-  
+
 instantiation char :: distrib_lattice
 begin
 
@@ -45,28 +47,34 @@
 instantiation String.literal :: linorder
 begin
 
-context includes literal.lifting begin
-lift_definition less_literal :: "String.literal \<Rightarrow> String.literal \<Rightarrow> bool" is "ord.lexordp op <" .
-lift_definition less_eq_literal :: "String.literal \<Rightarrow> String.literal \<Rightarrow> bool" is "ord.lexordp_eq op <" .
+context includes literal.lifting
+begin
+
+lift_definition less_literal :: "String.literal \<Rightarrow> String.literal \<Rightarrow> bool"
+  is "ord.lexordp op <" .
+
+lift_definition less_eq_literal :: "String.literal \<Rightarrow> String.literal \<Rightarrow> bool"
+  is "ord.lexordp_eq op <" .
 
 instance
 proof -
   interpret linorder "ord.lexordp_eq op <" "ord.lexordp op < :: string \<Rightarrow> string \<Rightarrow> bool"
-    by(rule linorder.lexordp_linorder[where less_eq="op \<le>"])(unfold_locales)
+    by (rule linorder.lexordp_linorder[where less_eq="op \<le>"]) unfold_locales
   show "PROP ?thesis"
-    by(intro_classes)(transfer, simp add: less_le_not_le linear)+
+    by intro_classes (transfer, simp add: less_le_not_le linear)+
 qed
 
 end
+
 end
 
-lemma less_literal_code [code]: 
+lemma less_literal_code [code]:
   "op < = (\<lambda>xs ys. ord.lexordp op < (String.explode xs) (String.explode ys))"
-by(simp add: less_literal.rep_eq fun_eq_iff)
+  by (simp add: less_literal.rep_eq fun_eq_iff)
 
 lemma less_eq_literal_code [code]:
   "op \<le> = (\<lambda>xs ys. ord.lexordp_eq op < (String.explode xs) (String.explode ys))"
-by(simp add: less_eq_literal.rep_eq fun_eq_iff)
+  by (simp add: less_eq_literal.rep_eq fun_eq_iff)
 
 lifting_update literal.lifting
 lifting_forget literal.lifting
diff -r 3365c8ec67bd -r b6a1047bc164 src/HOL/Library/Combine_PER.thy
--- a/src/HOL/Library/Combine_PER.thy	Tue Jul 12 13:55:35 2016 +0200
+++ b/src/HOL/Library/Combine_PER.thy	Tue Jul 12 16:04:19 2016 +0200
@@ -1,47 +1,37 @@
-(* Author: Florian Haftmann, TU Muenchen *)
+(*  Author:     Florian Haftmann, TU Muenchen *)
 
 section \<open>A combinator to build partial equivalence relations from a predicate and an equivalence relation\<close>
 
 theory Combine_PER
-imports Main "~~/src/HOL/Library/Lattice_Syntax"
+  imports Main "~~/src/HOL/Library/Lattice_Syntax"
 begin
 
 definition combine_per :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"
-where
-  "combine_per P R = (\<lambda>x y. P x \<and> P y) \<sqinter> R"
+  where "combine_per P R = (\<lambda>x y. P x \<and> P y) \<sqinter> R"
 
 lemma combine_per_simp [simp]:
-  fixes R (infixl "\<approx>" 50)
-  shows "combine_per P R x y \<longleftrightarrow> P x \<and> P y \<and> x \<approx> y"
+  "combine_per P R x y \<longleftrightarrow> P x \<and> P y \<and> x \<approx> y" for R (infixl "\<approx>" 50)
   by (simp add: combine_per_def)
 
-lemma combine_per_top [simp]:
-  "combine_per \<top> R = R"
+lemma combine_per_top [simp]: "combine_per \<top> R = R"
   by (simp add: fun_eq_iff)
 
-lemma combine_per_eq [simp]:
-  "combine_per P HOL.eq = HOL.eq \<sqinter> (\<lambda>x y. P x)"
+lemma combine_per_eq [simp]: "combine_per P HOL.eq = HOL.eq \<sqinter> (\<lambda>x y. P x)"
   by (auto simp add: fun_eq_iff)
 
-lemma symp_combine_per:
-  "symp R \<Longrightarrow> symp (combine_per P R)"
+lemma symp_combine_per: "symp R \<Longrightarrow> symp (combine_per P R)"
   by (auto simp add: symp_def sym_def combine_per_def)
 
-lemma transp_combine_per:
-  "transp R \<Longrightarrow> transp (combine_per P R)"
+lemma transp_combine_per: "transp R \<Longrightarrow> transp (combine_per P R)"
   by (auto simp add: transp_def trans_def combine_per_def)
 
-lemma combine_perI:
-  fixes R (infixl "\<approx>" 50)
-  shows "P x \<Longrightarrow> P y \<Longrightarrow> x \<approx> y \<Longrightarrow> combine_per P R x y"
+lemma combine_perI: "P x \<Longrightarrow> P y \<Longrightarrow> x \<approx> y \<Longrightarrow> combine_per P R x y" for R (infixl "\<approx>" 50)
   by (simp add: combine_per_def)
 
-lemma symp_combine_per_symp:
-  "symp R \<Longrightarrow> symp (combine_per P R)"
+lemma symp_combine_per_symp: "symp R \<Longrightarrow> symp (combine_per P R)"
   by (auto intro!: sympI elim: sympE)
 
-lemma transp_combine_per_transp:
-  "transp R \<Longrightarrow> transp (combine_per P R)"
+lemma transp_combine_per_transp: "transp R \<Longrightarrow> transp (combine_per P R)"
   by (auto intro!: transpI elim: transpE)
 
 lemma equivp_combine_per_part_equivp [intro?]:
@@ -50,8 +40,10 @@
   shows "part_equivp (combine_per P R)"
 proof -
   from \<open>\<exists>x. P x\<close> obtain x where "P x" ..
-  moreover from \<open>equivp R\<close> have "x \<approx> x" by (rule equivp_reflp)
-  ultimately have "\<exists>x. P x \<and> x \<approx> x" by blast
+  moreover from \<open>equivp R\<close> have "x \<approx> x"
+    by (rule equivp_reflp)
+  ultimately have "\<exists>x. P x \<and> x \<approx> x"
+    by blast
   with \<open>equivp R\<close> show ?thesis
     by (auto intro!: part_equivpI symp_combine_per_symp transp_combine_per_transp
       elim: equivpE)
diff -r 3365c8ec67bd -r b6a1047bc164 src/HOL/Library/Mapping.thy
--- a/src/HOL/Library/Mapping.thy	Tue Jul 12 13:55:35 2016 +0200
+++ b/src/HOL/Library/Mapping.thy	Tue Jul 12 16:04:19 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
-
diff -r 3365c8ec67bd -r b6a1047bc164 src/Pure/Isar/outer_syntax.scala
--- a/src/Pure/Isar/outer_syntax.scala	Tue Jul 12 13:55:35 2016 +0200
+++ b/src/Pure/Isar/outer_syntax.scala	Tue Jul 12 16:04:19 2016 +0200
@@ -142,14 +142,28 @@
 
   /* line-oriented structure */
 
+  private val close_structure =
+    Set(Keyword.NEXT_BLOCK, Keyword.QED_BLOCK, Keyword.PRF_CLOSE)
+
   def line_structure(tokens: List[Token], structure: Outer_Syntax.Line_Structure)
     : Outer_Syntax.Line_Structure =
   {
     val improper1 = tokens.forall(_.is_improper)
     val command1 = tokens.exists(_.is_command)
 
+    val command_depth =
+      tokens.iterator.filter(_.is_proper).toStream.headOption match {
+        case Some(tok) =>
+          if (keywords.is_command(tok, close_structure))
+            Some(structure.after_span_depth - 1)
+          else None
+        case None => None
+      }
+
     val depth1 =
-      if (tokens.exists(keywords.is_command(_, Keyword.theory))) 0
+      if (tokens.exists(tok =>
+            keywords.is_before_command(tok) || keywords.is_command(tok, Keyword.theory))) 0
+      else if (command_depth.isDefined) command_depth.get
       else if (command1) structure.after_span_depth
       else structure.span_depth
 
@@ -161,7 +175,8 @@
             else if (keywords.is_command(tok, Keyword.theory)) (1, 0)
             else if (keywords.is_command(tok, Keyword.proof_open)) (y + 2, y + 1)
             else if (keywords.is_command(tok, Keyword.PRF_BLOCK == _)) (y + 2, y + 1)
-            else if (keywords.is_command(tok, Keyword.QED_BLOCK == _)) (y + 1, y - 2)
+            else if (keywords.is_command(tok, Keyword.QED_BLOCK == _)) (y - 1, y - 2)
+            else if (keywords.is_command(tok, Keyword.PRF_CLOSE == _)) (y, y - 1)
             else if (keywords.is_command(tok, Keyword.proof_close)) (y + 1, y - 1)
             else if (keywords.is_command(tok, Keyword.qed_global)) (1, 0)
             else (x, y)
diff -r 3365c8ec67bd -r b6a1047bc164 src/Tools/jEdit/src/fold_handling.scala
--- a/src/Tools/jEdit/src/fold_handling.scala	Tue Jul 12 13:55:35 2016 +0200
+++ b/src/Tools/jEdit/src/fold_handling.scala	Tue Jul 12 16:04:19 2016 +0200
@@ -34,12 +34,12 @@
     override def getPrecedingFoldLevels(
       buffer: JEditBuffer, line: Int, seg: Segment, level: Int): java.util.List[Integer] =
     {
-      val struct = Token_Markup.Line_Context.next(buffer, line).structure
+      val structure = Token_Markup.Line_Context.next(buffer, line).structure
       val result =
-        if (line > 0 && struct.command)
+        if (line > 0 && structure.command)
           Range.inclusive(line - 1, 0, -1).iterator.
             map(i => Token_Markup.Line_Context.next(buffer, i).structure).
-            takeWhile(_.improper).map(_ => struct.depth max 0).toList
+            takeWhile(_.improper).map(_ => structure.depth max 0).toList
         else Nil
 
       if (result.isEmpty) null