# HG changeset patch
# User wenzelm
# Date 1468438241 -7200
# Node ID a7c5074a025155b5c2e832038359bdb9341f8d2a
# Parent  ae33d1c2ab26c52c9c93858d309245f319a171f6# Parent  6e29fb72e65942174281178a4ed25684804d7c10
merged

diff -r ae33d1c2ab26 -r a7c5074a0251 NEWS
--- a/NEWS	Wed Jul 13 21:00:03 2016 +0200
+++ b/NEWS	Wed Jul 13 21:30:41 2016 +0200
@@ -79,13 +79,19 @@
 * 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
+* Syntactic 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
 "isabelle.newline" (shortcut ENTER) indents the old and the new line
 according to command keywords only; see also option
 "jedit_indent_newline".
 
+* Semantic indentation for unstructured proof scripts ('apply' etc.) via
+number of subgoals. This requires information of ongoing document
+processing and may thus lag behind, when the user is editing too
+quickly; see also option "jedit_script_indent" and
+"jedit_script_indent_limit".
+
 * Action "isabelle.select-entity" (shortcut CS+ENTER) selects all
 occurences of the formal entity at the caret position. This facilitates
 systematic renaming.
diff -r ae33d1c2ab26 -r a7c5074a0251 src/HOL/Library/AList.thy
--- a/src/HOL/Library/AList.thy	Wed Jul 13 21:00:03 2016 +0200
+++ b/src/HOL/Library/AList.thy	Wed Jul 13 21:30:41 2016 +0200
@@ -73,8 +73,7 @@
         @{term "update k' v' (update k v []) = [(k, v), (k', v')]"}.\<close>
 
 lemma update_swap:
-  "k \<noteq> k' \<Longrightarrow>
-    map_of (update k v (update k' v' al)) = map_of (update k' v' (update k v al))"
+  "k \<noteq> k' \<Longrightarrow> map_of (update k v (update k' v' al)) = map_of (update k' v' (update k v al))"
   by (simp add: update_conv' fun_eq_iff)
 
 lemma update_Some_unfold:
@@ -85,8 +84,8 @@
 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]:
@@ -216,8 +215,8 @@
 
 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"
+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) =
@@ -257,7 +256,7 @@
 
 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 simp_all
   apply clarsimp
   apply (fastforce intro: rev_image_eqI)
   done
@@ -291,7 +290,7 @@
 lemma map_of_delete_aux':
   "distinct (map fst xs) \<Longrightarrow> map_of (delete_aux k xs) = (map_of xs)(k := None)"
   apply (induct xs)
-  apply (fastforce simp add: map_of_eq_None_iff fun_upd_twist)
+   apply (fastforce simp add: map_of_eq_None_iff fun_upd_twist)
   apply (auto intro!: ext)
   apply (simp add: map_of_eq_None_iff)
   done
@@ -318,9 +317,9 @@
 proof
   show "map_of (restrict A al) k = ((map_of al)|` A) k" for k
     apply (induct al)
-    apply simp
+     apply simp
     apply (cases "k \<in> A")
-    apply auto
+     apply auto
     done
 qed
 
diff -r ae33d1c2ab26 -r a7c5074a0251 src/HOL/Library/AList_Mapping.thy
--- a/src/HOL/Library/AList_Mapping.thy	Wed Jul 13 21:00:03 2016 +0200
+++ b/src/HOL/Library/AList_Mapping.thy	Wed Jul 13 21:30:41 2016 +0200
@@ -51,8 +51,11 @@
 proof -
   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: *)
+  show ?thesis
+    apply transfer
+    apply (auto intro!: map_of_eqI)
+     apply (auto dest!: map_of_eq_dom intro: *)
+    done
 qed
 
 lemma map_values_Mapping [code]:
@@ -72,8 +75,8 @@
   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)+
+       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
 
@@ -86,8 +89,8 @@
   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)+
+       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
 
@@ -106,7 +109,7 @@
   apply transfer
   apply (rule ext)
   apply (subst map_of_filter_distinct)
-  apply (simp_all add: map_of_clearjunk split: option.split)
+   apply (simp_all add: map_of_clearjunk split: option.split)
   done
 
 lemma [code nbe]: "HOL.equal (x :: ('a, 'b) mapping) x \<longleftrightarrow> True"
diff -r ae33d1c2ab26 -r a7c5074a0251 src/HOL/Library/BigO.thy
--- a/src/HOL/Library/BigO.thy	Wed Jul 13 21:00:03 2016 +0200
+++ b/src/HOL/Library/BigO.thy	Wed Jul 13 21:30:41 2016 +0200
@@ -5,33 +5,35 @@
 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>
-This library is designed to support asymptotic ``big O'' calculations,
-i.e.~reasoning with expressions of the form $f = O(g)$ and $f = g +
-O(h)$.  An earlier version of this library is described in detail in
-@{cite "Avigad-Donnelly"}.
+  This library is designed to support asymptotic ``big O'' calculations,
+  i.e.~reasoning with expressions of the form \<open>f = O(g)\<close> and \<open>f = g + O(h)\<close>.
+  An earlier version of this library is described in detail in @{cite
+  "Avigad-Donnelly"}.
+
+  The main changes in this version are as follows:
 
-The main changes in this version are as follows:
-\begin{itemize}
-\item We have eliminated the \<open>O\<close> operator on sets. (Most uses of this seem
-  to be inessential.)
-\item We no longer use \<open>+\<close> as output syntax for \<open>+o\<close>
-\item Lemmas involving \<open>sumr\<close> have been replaced by more general lemmas
-  involving `\<open>setsum\<close>.
-\item The library has been expanded, with e.g.~support for expressions of
-  the form \<open>f < g + O(h)\<close>.
-\end{itemize}
+    \<^item> We have eliminated the \<open>O\<close> operator on sets. (Most uses of this seem
+      to be inessential.)
+    \<^item> We no longer use \<open>+\<close> as output syntax for \<open>+o\<close>
+    \<^item> Lemmas involving \<open>sumr\<close> have been replaced by more general lemmas
+      involving `\<open>setsum\<close>.
+    \<^item> The library has been expanded, with e.g.~support for expressions of
+      the form \<open>f < g + O(h)\<close>.
 
-Note also since the Big O library includes rules that demonstrate set
-inclusion, to use the automated reasoners effectively with the library
-one should redeclare the theorem \<open>subsetI\<close> as an intro rule,
-rather than as an \<open>intro!\<close> rule, for example, using
-\<^theory_text>\<open>declare subsetI [del, intro]\<close>.
+  Note also since the Big O library includes rules that demonstrate set
+  inclusion, to use the automated reasoners effectively with the library one
+  should redeclare the theorem \<open>subsetI\<close> as an intro rule, rather than as an
+  \<open>intro!\<close> rule, for example, using \<^theory_text>\<open>declare subsetI [del, intro]\<close>.
 \<close>
 
+
 subsection \<open>Definitions\<close>
 
 definition bigo :: "('a \<Rightarrow> 'b::linordered_idom) \<Rightarrow> ('a \<Rightarrow> 'b) set"  ("(1O'(_'))")
@@ -42,16 +44,16 @@
     (\<exists>c. 0 < c \<and> (\<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>))"
   apply auto
   apply (case_tac "c = 0")
-  apply simp
-  apply (rule_tac x = "1" in exI)
-  apply simp
+   apply simp
+   apply (rule_tac x = "1" in exI)
+   apply simp
   apply (rule_tac x = "\<bar>c\<bar>" in exI)
   apply auto
   apply (subgoal_tac "c * \<bar>f x\<bar> \<le> \<bar>c\<bar> * \<bar>f x\<bar>")
-  apply (erule_tac x = x in allE)
-  apply force
+   apply (erule_tac x = x in allE)
+   apply force
   apply (rule mult_right_mono)
-  apply (rule abs_ge_self)
+   apply (rule abs_ge_self)
   apply (rule abs_ge_zero)
   done
 
@@ -62,19 +64,19 @@
   apply (auto simp add: bigo_alt_def)
   apply (rule_tac x = "ca * c" in exI)
   apply (rule conjI)
-  apply simp
+   apply simp
   apply (rule allI)
   apply (drule_tac x = "xa" in spec)+
   apply (subgoal_tac "ca * \<bar>f xa\<bar> \<le> ca * (c * \<bar>g xa\<bar>)")
-  apply (erule order_trans)
-  apply (simp add: ac_simps)
+   apply (erule order_trans)
+   apply (simp add: ac_simps)
   apply (rule mult_left_mono, assumption)
   apply (rule order_less_imp_le, assumption)
   done
 
 lemma bigo_refl [intro]: "f \<in> O(f)"
-  apply(auto simp add: bigo_def)
-  apply(rule_tac x = 1 in exI)
+  apply (auto simp add: bigo_def)
+  apply (rule_tac x = 1 in exI)
   apply simp
   done
 
@@ -93,15 +95,15 @@
   apply auto
   apply (simp add: ring_distribs func_plus)
   apply (rule order_trans)
-  apply (rule abs_triangle_ineq)
+   apply (rule abs_triangle_ineq)
   apply (rule add_mono)
-  apply force
+   apply force
   apply force
   done
 
 lemma bigo_plus_idemp [simp]: "O(f) + O(f) = O(f)"
   apply (rule equalityI)
-  apply (rule bigo_plus_self_subset)
+   apply (rule bigo_plus_self_subset)
   apply (rule set_zero_plus2)
   apply (rule bigo_zero)
   done
@@ -112,73 +114,73 @@
   apply (subst bigo_pos_const [symmetric])+
   apply (rule_tac x = "\<lambda>n. if \<bar>g n\<bar> \<le> \<bar>f n\<bar> then x n else 0" in exI)
   apply (rule conjI)
-  apply (rule_tac x = "c + c" in exI)
-  apply (clarsimp)
-  apply (subgoal_tac "c * \<bar>f xa + g xa\<bar> \<le> (c + c) * \<bar>f xa\<bar>")
-  apply (erule_tac x = xa in allE)
-  apply (erule order_trans)
-  apply (simp)
-  apply (subgoal_tac "c * \<bar>f xa + g xa\<bar> \<le> c * (\<bar>f xa\<bar> + \<bar>g xa\<bar>)")
-  apply (erule order_trans)
-  apply (simp add: ring_distribs)
-  apply (rule mult_left_mono)
-  apply (simp add: abs_triangle_ineq)
-  apply (simp add: order_less_le)
+   apply (rule_tac x = "c + c" in exI)
+   apply (clarsimp)
+   apply (subgoal_tac "c * \<bar>f xa + g xa\<bar> \<le> (c + c) * \<bar>f xa\<bar>")
+    apply (erule_tac x = xa in allE)
+    apply (erule order_trans)
+    apply (simp)
+   apply (subgoal_tac "c * \<bar>f xa + g xa\<bar> \<le> c * (\<bar>f xa\<bar> + \<bar>g xa\<bar>)")
+    apply (erule order_trans)
+    apply (simp add: ring_distribs)
+   apply (rule mult_left_mono)
+    apply (simp add: abs_triangle_ineq)
+   apply (simp add: order_less_le)
   apply (rule_tac x = "\<lambda>n. if \<bar>f n\<bar> < \<bar>g n\<bar> then x n else 0" in exI)
   apply (rule conjI)
-  apply (rule_tac x = "c + c" in exI)
-  apply auto
+   apply (rule_tac x = "c + c" in exI)
+   apply auto
   apply (subgoal_tac "c * \<bar>f xa + g xa\<bar> \<le> (c + c) * \<bar>g xa\<bar>")
-  apply (erule_tac x = xa in allE)
-  apply (erule order_trans)
-  apply simp
+   apply (erule_tac x = xa in allE)
+   apply (erule order_trans)
+   apply simp
   apply (subgoal_tac "c * \<bar>f xa + g xa\<bar> \<le> c * (\<bar>f xa\<bar> + \<bar>g xa\<bar>)")
-  apply (erule order_trans)
-  apply (simp add: ring_distribs)
+   apply (erule order_trans)
+   apply (simp add: ring_distribs)
   apply (rule mult_left_mono)
-  apply (rule abs_triangle_ineq)
+   apply (rule abs_triangle_ineq)
   apply (simp add: order_less_le)
   done
 
 lemma bigo_plus_subset2 [intro]: "A \<subseteq> O(f) \<Longrightarrow> B \<subseteq> O(f) \<Longrightarrow> A + B \<subseteq> O(f)"
   apply (subgoal_tac "A + B \<subseteq> O(f) + O(f)")
-  apply (erule order_trans)
-  apply simp
+   apply (erule order_trans)
+   apply simp
   apply (auto del: subsetI simp del: bigo_plus_idemp)
   done
 
 lemma bigo_plus_eq: "\<forall>x. 0 \<le> f x \<Longrightarrow> \<forall>x. 0 \<le> g x \<Longrightarrow> O(f + g) = O(f) + O(g)"
   apply (rule equalityI)
-  apply (rule bigo_plus_subset)
+   apply (rule bigo_plus_subset)
   apply (simp add: bigo_alt_def set_plus_def func_plus)
   apply clarify
   apply (rule_tac x = "max c ca" in exI)
   apply (rule conjI)
-  apply (subgoal_tac "c \<le> max c ca")
-  apply (erule order_less_le_trans)
-  apply assumption
-  apply (rule max.cobounded1)
+   apply (subgoal_tac "c \<le> max c ca")
+    apply (erule order_less_le_trans)
+    apply assumption
+   apply (rule max.cobounded1)
   apply clarify
   apply (drule_tac x = "xa" in spec)+
   apply (subgoal_tac "0 \<le> f xa + g xa")
-  apply (simp add: ring_distribs)
-  apply (subgoal_tac "\<bar>a xa + b xa\<bar> \<le> \<bar>a xa\<bar> + \<bar>b xa\<bar>")
-  apply (subgoal_tac "\<bar>a xa\<bar> + \<bar>b xa\<bar> \<le> max c ca * f xa + max c ca * g xa")
-  apply force
-  apply (rule add_mono)
-  apply (subgoal_tac "c * f xa \<le> max c ca * f xa")
-  apply force
-  apply (rule mult_right_mono)
-  apply (rule max.cobounded1)
-  apply assumption
-  apply (subgoal_tac "ca * g xa \<le> max c ca * g xa")
-  apply force
-  apply (rule mult_right_mono)
-  apply (rule max.cobounded2)
-  apply assumption
-  apply (rule abs_triangle_ineq)
+   apply (simp add: ring_distribs)
+   apply (subgoal_tac "\<bar>a xa + b xa\<bar> \<le> \<bar>a xa\<bar> + \<bar>b xa\<bar>")
+    apply (subgoal_tac "\<bar>a xa\<bar> + \<bar>b xa\<bar> \<le> max c ca * f xa + max c ca * g xa")
+     apply force
+    apply (rule add_mono)
+     apply (subgoal_tac "c * f xa \<le> max c ca * f xa")
+      apply force
+     apply (rule mult_right_mono)
+      apply (rule max.cobounded1)
+     apply assumption
+    apply (subgoal_tac "ca * g xa \<le> max c ca * g xa")
+     apply force
+    apply (rule mult_right_mono)
+     apply (rule max.cobounded2)
+    apply assumption
+   apply (rule abs_triangle_ineq)
   apply (rule add_nonneg_nonneg)
-  apply assumption+
+   apply assumption+
   done
 
 lemma bigo_bounded_alt: "\<forall>x. 0 \<le> f x \<Longrightarrow> \<forall>x. f x \<le> c * g x \<Longrightarrow> f \<in> O(g)"
@@ -197,7 +199,7 @@
 lemma bigo_bounded2: "\<forall>x. lb x \<le> f x \<Longrightarrow> \<forall>x. f x \<le> lb x + g x \<Longrightarrow> f \<in> lb +o O(g)"
   apply (rule set_minus_imp_plus)
   apply (rule bigo_bounded)
-  apply (auto simp add: fun_Compl_def func_plus)
+   apply (auto simp add: fun_Compl_def func_plus)
   apply (drule_tac x = x in spec)+
   apply force
   done
@@ -218,8 +220,8 @@
 
 lemma bigo_abs3: "O(f) = O(\<lambda>x. \<bar>f x\<bar>)"
   apply (rule equalityI)
-  apply (rule bigo_elt_subset)
-  apply (rule bigo_abs2)
+   apply (rule bigo_elt_subset)
+   apply (rule bigo_abs2)
   apply (rule bigo_elt_subset)
   apply (rule bigo_abs)
   done
@@ -229,13 +231,13 @@
   apply (rule set_minus_imp_plus)
   apply (subst fun_diff_def)
 proof -
-  assume a: "f - g \<in> O(h)"
+  assume *: "f - g \<in> O(h)"
   have "(\<lambda>x. \<bar>f x\<bar> - \<bar>g x\<bar>) =o O(\<lambda>x. \<bar>\<bar>f x\<bar> - \<bar>g x\<bar>\<bar>)"
     by (rule bigo_abs2)
   also have "\<dots> \<subseteq> O(\<lambda>x. \<bar>f x - g x\<bar>)"
     apply (rule bigo_elt_subset)
     apply (rule bigo_bounded)
-    apply force
+     apply force
     apply (rule allI)
     apply (rule abs_triangle_ineq3)
     done
@@ -244,23 +246,23 @@
     apply (subst fun_diff_def)
     apply (rule bigo_abs)
     done
-  also from a have "\<dots> \<subseteq> O(h)"
+  also from * have "\<dots> \<subseteq> O(h)"
     by (rule bigo_elt_subset)
   finally show "(\<lambda>x. \<bar>f x\<bar> - \<bar>g x\<bar>) \<in> O(h)".
 qed
 
 lemma bigo_abs5: "f =o O(g) \<Longrightarrow> (\<lambda>x. \<bar>f x\<bar>) =o O(g)"
-  by (unfold bigo_def, auto)
+  by (auto simp: bigo_def)
 
-lemma bigo_elt_subset2 [intro]: "f \<in> g +o O(h) \<Longrightarrow> O(f) \<subseteq> O(g) + O(h)"
+lemma bigo_elt_subset2 [intro]:
+  assumes *: "f \<in> g +o O(h)"
+  shows "O(f) \<subseteq> O(g) + O(h)"
 proof -
-  assume "f \<in> g +o O(h)"
-  also have "\<dots> \<subseteq> O(g) + O(h)"
+  note *
+  also have "g +o O(h) \<subseteq> O(g) + O(h)"
     by (auto del: subsetI)
   also have "\<dots> = O(\<lambda>x. \<bar>g x\<bar>) + O(\<lambda>x. \<bar>h x\<bar>)"
-    apply (subst bigo_abs3 [symmetric])+
-    apply (rule refl)
-    done
+    by (subst bigo_abs3 [symmetric])+ (rule refl)
   also have "\<dots> = O((\<lambda>x. \<bar>g x\<bar>) + (\<lambda>x. \<bar>h x\<bar>))"
     by (rule bigo_plus_eq [symmetric]) auto
   finally have "f \<in> \<dots>" .
@@ -280,11 +282,11 @@
   apply (rule allI)
   apply (erule_tac x = x in allE)+
   apply (subgoal_tac "c * ca * \<bar>f x * g x\<bar> = (c * \<bar>f x\<bar>) * (ca * \<bar>g x\<bar>)")
-  apply (erule ssubst)
-  apply (subst abs_mult)
-  apply (rule mult_mono)
-  apply assumption+
-  apply auto
+   apply (erule ssubst)
+   apply (subst abs_mult)
+   apply (rule mult_mono)
+      apply assumption+
+    apply auto
   apply (simp add: ac_simps abs_mult)
   done
 
@@ -294,14 +296,14 @@
   apply auto
   apply (drule_tac x = x in spec)
   apply (subgoal_tac "\<bar>f x\<bar> * \<bar>b x\<bar> \<le> \<bar>f x\<bar> * (c * \<bar>g x\<bar>)")
-  apply (force simp add: ac_simps)
+   apply (force simp add: ac_simps)
   apply (rule mult_left_mono, assumption)
   apply (rule abs_ge_zero)
   done
 
 lemma bigo_mult3: "f \<in> O(h) \<Longrightarrow> g \<in> O(j) \<Longrightarrow> f * g \<in> O(h * j)"
   apply (rule subsetD)
-  apply (rule bigo_mult)
+   apply (rule bigo_mult)
   apply (erule set_times_intro, assumption)
   done
 
@@ -309,7 +311,7 @@
   apply (drule set_plus_imp_minus)
   apply (rule set_minus_imp_plus)
   apply (drule bigo_mult3 [where g = g and j = g])
-  apply (auto simp add: algebra_simps)
+   apply (auto simp add: algebra_simps)
   done
 
 lemma bigo_mult5:
@@ -339,28 +341,25 @@
   finally show "h \<in> f *o O(g)" .
 qed
 
-lemma bigo_mult6:
-  fixes f :: "'a \<Rightarrow> 'b::linordered_field"
-  shows "\<forall>x. f x \<noteq> 0 \<Longrightarrow> O(f * g) = f *o O(g)"
+lemma bigo_mult6: "\<forall>x. f x \<noteq> 0 \<Longrightarrow> O(f * g) = f *o O(g)"
+  for f :: "'a \<Rightarrow> 'b::linordered_field"
   apply (rule equalityI)
-  apply (erule bigo_mult5)
+   apply (erule bigo_mult5)
   apply (rule bigo_mult2)
   done
 
-lemma bigo_mult7:
-  fixes f :: "'a \<Rightarrow> 'b::linordered_field"
-  shows "\<forall>x. f x \<noteq> 0 \<Longrightarrow> O(f * g) \<subseteq> O(f) * O(g)"
+lemma bigo_mult7: "\<forall>x. f x \<noteq> 0 \<Longrightarrow> O(f * g) \<subseteq> O(f) * O(g)"
+  for f :: "'a \<Rightarrow> 'b::linordered_field"
   apply (subst bigo_mult6)
-  apply assumption
+   apply assumption
   apply (rule set_times_mono3)
   apply (rule bigo_refl)
   done
 
-lemma bigo_mult8:
-  fixes f :: "'a \<Rightarrow> 'b::linordered_field"
-  shows "\<forall>x. f x \<noteq> 0 \<Longrightarrow> O(f * g) = O(f) * O(g)"
+lemma bigo_mult8: "\<forall>x. f x \<noteq> 0 \<Longrightarrow> O(f * g) = O(f) * O(g)"
+  for f :: "'a \<Rightarrow> 'b::linordered_field"
   apply (rule equalityI)
-  apply (erule bigo_mult7)
+   apply (erule bigo_mult7)
   apply (rule bigo_mult)
   done
 
@@ -377,65 +376,63 @@
 lemma bigo_minus3: "O(- f) = O(f)"
   by (auto simp add: bigo_def fun_Compl_def)
 
-lemma bigo_plus_absorb_lemma1: "f \<in> O(g) \<Longrightarrow> f +o O(g) \<subseteq> O(g)"
+lemma bigo_plus_absorb_lemma1:
+  assumes *: "f \<in> O(g)"
+  shows "f +o O(g) \<subseteq> O(g)"
 proof -
-  assume a: "f \<in> O(g)"
-  show "f +o O(g) \<subseteq> O(g)"
+  have "f \<in> O(f)" by auto
+  then have "f +o O(g) \<subseteq> O(f) + O(g)"
+    by (auto del: subsetI)
+  also have "\<dots> \<subseteq> O(g) + O(g)"
   proof -
-    have "f \<in> O(f)" by auto
-    then have "f +o O(g) \<subseteq> O(f) + O(g)"
+    from * have "O(f) \<subseteq> O(g)"
       by (auto del: subsetI)
-    also have "\<dots> \<subseteq> O(g) + O(g)"
-    proof -
-      from a have "O(f) \<subseteq> O(g)" by (auto del: subsetI)
-      then show ?thesis by (auto del: subsetI)
-    qed
-    also have "\<dots> \<subseteq> O(g)" by simp
-    finally show ?thesis .
+    then show ?thesis
+      by (auto del: subsetI)
   qed
+  also have "\<dots> \<subseteq> O(g)" by simp
+  finally show ?thesis .
 qed
 
-lemma bigo_plus_absorb_lemma2: "f \<in> O(g) \<Longrightarrow> O(g) \<subseteq> f +o O(g)"
+lemma bigo_plus_absorb_lemma2:
+  assumes *: "f \<in> O(g)"
+  shows "O(g) \<subseteq> f +o O(g)"
 proof -
-  assume a: "f \<in> O(g)"
-  show "O(g) \<subseteq> f +o O(g)"
-  proof -
-    from a have "- f \<in> O(g)"
-      by auto
-    then have "- f +o O(g) \<subseteq> O(g)"
-      by (elim bigo_plus_absorb_lemma1)
-    then have "f +o (- f +o O(g)) \<subseteq> f +o O(g)"
-      by auto
-    also have "f +o (- f +o O(g)) = O(g)"
-      by (simp add: set_plus_rearranges)
-    finally show ?thesis .
-  qed
+  from * have "- f \<in> O(g)"
+    by auto
+  then have "- f +o O(g) \<subseteq> O(g)"
+    by (elim bigo_plus_absorb_lemma1)
+  then have "f +o (- f +o O(g)) \<subseteq> f +o O(g)"
+    by auto
+  also have "f +o (- f +o O(g)) = O(g)"
+    by (simp add: set_plus_rearranges)
+  finally show ?thesis .
 qed
 
 lemma bigo_plus_absorb [simp]: "f \<in> O(g) \<Longrightarrow> f +o O(g) = O(g)"
   apply (rule equalityI)
-  apply (erule bigo_plus_absorb_lemma1)
+   apply (erule bigo_plus_absorb_lemma1)
   apply (erule bigo_plus_absorb_lemma2)
   done
 
 lemma bigo_plus_absorb2 [intro]: "f \<in> O(g) \<Longrightarrow> A \<subseteq> O(g) \<Longrightarrow> f +o A \<subseteq> O(g)"
   apply (subgoal_tac "f +o A \<subseteq> f +o O(g)")
-  apply force+
+   apply force+
   done
 
 lemma bigo_add_commute_imp: "f \<in> g +o O(h) \<Longrightarrow> g \<in> f +o O(h)"
   apply (subst set_minus_plus [symmetric])
   apply (subgoal_tac "g - f = - (f - g)")
-  apply (erule ssubst)
-  apply (rule bigo_minus)
-  apply (subst set_minus_plus)
-  apply assumption
+   apply (erule ssubst)
+   apply (rule bigo_minus)
+   apply (subst set_minus_plus)
+   apply assumption
   apply (simp add: ac_simps)
   done
 
 lemma bigo_add_commute: "f \<in> g +o O(h) \<longleftrightarrow> g \<in> f +o O(h)"
   apply (rule iffI)
-  apply (erule bigo_add_commute_imp)+
+   apply (erule bigo_add_commute_imp)+
   done
 
 lemma bigo_const1: "(\<lambda>x. c) \<in> O(\<lambda>x. 1)"
@@ -446,27 +443,24 @@
   apply (rule bigo_const1)
   done
 
-lemma bigo_const3:
-  fixes c :: "'a::linordered_field"
-  shows "c \<noteq> 0 \<Longrightarrow> (\<lambda>x. 1) \<in> O(\<lambda>x. c)"
+lemma bigo_const3: "c \<noteq> 0 \<Longrightarrow> (\<lambda>x. 1) \<in> O(\<lambda>x. c)"
+  for c :: "'a::linordered_field"
   apply (simp add: bigo_def)
   apply (rule_tac x = "\<bar>inverse c\<bar>" in exI)
   apply (simp add: abs_mult [symmetric])
   done
 
-lemma bigo_const4:
-  fixes c :: "'a::linordered_field"
-  shows "c \<noteq> 0 \<Longrightarrow> O(\<lambda>x. 1) \<subseteq> O(\<lambda>x. c)"
+lemma bigo_const4: "c \<noteq> 0 \<Longrightarrow> O(\<lambda>x. 1) \<subseteq> O(\<lambda>x. c)"
+  for c :: "'a::linordered_field"
   apply (rule bigo_elt_subset)
   apply (rule bigo_const3)
   apply assumption
   done
 
-lemma bigo_const [simp]:
-  fixes c :: "'a::linordered_field"
-  shows "c \<noteq> 0 \<Longrightarrow> O(\<lambda>x. c) = O(\<lambda>x. 1)"
+lemma bigo_const [simp]: "c \<noteq> 0 \<Longrightarrow> O(\<lambda>x. c) = O(\<lambda>x. 1)"
+  for c :: "'a::linordered_field"
   apply (rule equalityI)
-  apply (rule bigo_const2)
+   apply (rule bigo_const2)
   apply (rule bigo_const4)
   apply assumption
   done
@@ -482,37 +476,33 @@
   apply (rule bigo_const_mult1)
   done
 
-lemma bigo_const_mult3:
-  fixes c :: "'a::linordered_field"
-  shows "c \<noteq> 0 \<Longrightarrow> f \<in> O(\<lambda>x. c * f x)"
+lemma bigo_const_mult3: "c \<noteq> 0 \<Longrightarrow> f \<in> O(\<lambda>x. c * f x)"
+  for c :: "'a::linordered_field"
   apply (simp add: bigo_def)
   apply (rule_tac x = "\<bar>inverse c\<bar>" in exI)
   apply (simp add: abs_mult mult.assoc [symmetric])
   done
 
-lemma bigo_const_mult4:
-  fixes c :: "'a::linordered_field"
-  shows "c \<noteq> 0 \<Longrightarrow> O(f) \<subseteq> O(\<lambda>x. c * f x)"
+lemma bigo_const_mult4: "c \<noteq> 0 \<Longrightarrow> O(f) \<subseteq> O(\<lambda>x. c * f x)"
+  for c :: "'a::linordered_field"
   apply (rule bigo_elt_subset)
   apply (rule bigo_const_mult3)
   apply assumption
   done
 
-lemma bigo_const_mult [simp]:
-  fixes c :: "'a::linordered_field"
-  shows "c \<noteq> 0 \<Longrightarrow> O(\<lambda>x. c * f x) = O(f)"
+lemma bigo_const_mult [simp]: "c \<noteq> 0 \<Longrightarrow> O(\<lambda>x. c * f x) = O(f)"
+  for c :: "'a::linordered_field"
   apply (rule equalityI)
-  apply (rule bigo_const_mult2)
+   apply (rule bigo_const_mult2)
   apply (erule bigo_const_mult4)
   done
 
-lemma bigo_const_mult5 [simp]:
-  fixes c :: "'a::linordered_field"
-  shows "c \<noteq> 0 \<Longrightarrow> (\<lambda>x. c) *o O(f) = O(f)"
+lemma bigo_const_mult5 [simp]: "c \<noteq> 0 \<Longrightarrow> (\<lambda>x. c) *o O(f) = O(f)"
+  for c :: "'a::linordered_field"
   apply (auto del: subsetI)
-  apply (rule order_trans)
-  apply (rule bigo_mult2)
-  apply (simp add: func_times)
+   apply (rule order_trans)
+    apply (rule bigo_mult2)
+   apply (simp add: func_times)
   apply (auto intro!: simp add: bigo_def elt_set_times_def func_times)
   apply (rule_tac x = "\<lambda>y. inverse c * x y" in exI)
   apply (simp add: mult.assoc [symmetric] abs_mult)
@@ -525,18 +515,19 @@
   apply (rule_tac x = "ca * \<bar>c\<bar>" in exI)
   apply (rule allI)
   apply (subgoal_tac "ca * \<bar>c\<bar> * \<bar>f x\<bar> = \<bar>c\<bar> * (ca * \<bar>f x\<bar>)")
-  apply (erule ssubst)
-  apply (subst abs_mult)
-  apply (rule mult_left_mono)
-  apply (erule spec)
-  apply simp
-  apply(simp add: ac_simps)
+   apply (erule ssubst)
+   apply (subst abs_mult)
+   apply (rule mult_left_mono)
+    apply (erule spec)
+   apply simp
+  apply (simp add: ac_simps)
   done
 
-lemma bigo_const_mult7 [intro]: "f =o O(g) \<Longrightarrow> (\<lambda>x. c * f x) =o O(g)"
+lemma bigo_const_mult7 [intro]:
+  assumes *: "f =o O(g)"
+  shows "(\<lambda>x. c * f x) =o O(g)"
 proof -
-  assume "f =o O(g)"
-  then have "(\<lambda>x. c) * f =o (\<lambda>x. c) *o O(g)"
+  from * have "(\<lambda>x. c) * f =o (\<lambda>x. c) *o O(g)"
     by auto
   also have "(\<lambda>x. c) * f = (\<lambda>x. c * f x)"
     by (simp add: func_times)
@@ -546,10 +537,9 @@
 qed
 
 lemma bigo_compose1: "f =o O(g) \<Longrightarrow> (\<lambda>x. f (k x)) =o O(\<lambda>x. g (k x))"
-  unfolding bigo_def by auto
+  by (auto simp: bigo_def)
 
-lemma bigo_compose2: "f =o g +o O(h) \<Longrightarrow>
-    (\<lambda>x. f (k x)) =o (\<lambda>x. g (k x)) +o O(\<lambda>x. h(k x))"
+lemma bigo_compose2: "f =o g +o O(h) \<Longrightarrow> (\<lambda>x. f (k x)) =o (\<lambda>x. g (k x)) +o O(\<lambda>x. h(k x))"
   apply (simp only: set_minus_plus [symmetric] fun_Compl_def func_plus)
   apply (drule bigo_compose1)
   apply (simp add: fun_diff_def)
@@ -564,21 +554,21 @@
   apply (auto simp add: bigo_def)
   apply (rule_tac x = "\<bar>c\<bar>" in exI)
   apply (subst abs_of_nonneg) back back
-  apply (rule setsum_nonneg)
-  apply force
+   apply (rule setsum_nonneg)
+   apply force
   apply (subst setsum_right_distrib)
   apply (rule allI)
   apply (rule order_trans)
-  apply (rule setsum_abs)
+   apply (rule setsum_abs)
   apply (rule setsum_mono)
   apply (rule order_trans)
-  apply (drule spec)+
-  apply (drule bspec)+
-  apply assumption+
-  apply (drule bspec)
-  apply assumption+
+   apply (drule spec)+
+   apply (drule bspec)+
+     apply assumption+
+   apply (drule bspec)
+    apply assumption+
   apply (rule mult_right_mono)
-  apply (rule abs_ge_self)
+   apply (rule abs_ge_self)
   apply force
   done
 
@@ -586,7 +576,7 @@
     \<exists>c. \<forall>x y. \<bar>f x y\<bar> \<le> c * h x y \<Longrightarrow>
       (\<lambda>x. \<Sum>y \<in> A x. f x y) =o O(\<lambda>x. \<Sum>y \<in> A x. h x y)"
   apply (rule bigo_setsum_main)
-  apply force
+   apply force
   apply clarsimp
   apply (rule_tac x = c in exI)
   apply force
@@ -600,8 +590,8 @@
 lemma bigo_setsum3: "f =o O(h) \<Longrightarrow>
     (\<lambda>x. \<Sum>y \<in> A x. l x y * f (k x y)) =o O(\<lambda>x. \<Sum>y \<in> A x. \<bar>l x y * h (k x y)\<bar>)"
   apply (rule bigo_setsum1)
-  apply (rule allI)+
-  apply (rule abs_ge_zero)
+   apply (rule allI)+
+   apply (rule abs_ge_zero)
   apply (unfold bigo_def)
   apply auto
   apply (rule_tac x = c in exI)
@@ -609,7 +599,7 @@
   apply (subst abs_mult)+
   apply (subst mult.left_commute)
   apply (rule mult_left_mono)
-  apply (erule spec)
+   apply (erule spec)
   apply (rule abs_ge_zero)
   done
 
@@ -632,13 +622,13 @@
         O(\<lambda>x. \<Sum>y \<in> A x. l x y * h (k x y))"
   apply (subgoal_tac "(\<lambda>x. \<Sum>y \<in> A x. l x y * h (k x y)) =
       (\<lambda>x. \<Sum>y \<in> A x. \<bar>l x y * h (k x y)\<bar>)")
-  apply (erule ssubst)
-  apply (erule bigo_setsum3)
+   apply (erule ssubst)
+   apply (erule bigo_setsum3)
   apply (rule ext)
   apply (rule setsum.cong)
-  apply (rule refl)
+   apply (rule refl)
   apply (subst abs_of_nonneg)
-  apply auto
+   apply auto
   done
 
 lemma bigo_setsum6: "f =o g +o O(h) \<Longrightarrow> \<forall>x y. 0 \<le> l x y \<Longrightarrow>
@@ -651,9 +641,9 @@
   apply (subst setsum_subtractf [symmetric])
   apply (subst right_diff_distrib [symmetric])
   apply (rule bigo_setsum5)
-  apply (subst fun_diff_def [symmetric])
-  apply (drule set_plus_imp_minus)
-  apply auto
+    apply (subst fun_diff_def [symmetric])
+    apply (drule set_plus_imp_minus)
+    apply auto
   done
 
 
@@ -662,25 +652,24 @@
 lemma bigo_useful_intro: "A \<subseteq> O(f) \<Longrightarrow> B \<subseteq> O(f) \<Longrightarrow> A + B \<subseteq> O(f)"
   apply (subst bigo_plus_idemp [symmetric])
   apply (rule set_plus_mono2)
-  apply assumption+
+   apply assumption+
   done
 
 lemma bigo_useful_add: "f =o O(h) \<Longrightarrow> g =o O(h) \<Longrightarrow> f + g =o O(h)"
   apply (subst bigo_plus_idemp [symmetric])
   apply (rule set_plus_intro)
-  apply assumption+
+   apply assumption+
   done
 
-lemma bigo_useful_const_mult:
-  fixes c :: "'a::linordered_field"
-  shows "c \<noteq> 0 \<Longrightarrow> (\<lambda>x. c) * f =o O(h) \<Longrightarrow> f =o O(h)"
+lemma bigo_useful_const_mult: "c \<noteq> 0 \<Longrightarrow> (\<lambda>x. c) * f =o O(h) \<Longrightarrow> f =o O(h)"
+  for c :: "'a::linordered_field"
   apply (rule subsetD)
-  apply (subgoal_tac "(\<lambda>x. 1 / c) *o O(h) \<subseteq> O(h)")
-  apply assumption
-  apply (rule bigo_const_mult6)
+   apply (subgoal_tac "(\<lambda>x. 1 / c) *o O(h) \<subseteq> O(h)")
+    apply assumption
+   apply (rule bigo_const_mult6)
   apply (subgoal_tac "f = (\<lambda>x. 1 / c) * ((\<lambda>x. c) * f)")
-  apply (erule ssubst)
-  apply (erule set_times_intro2)
+   apply (erule ssubst)
+   apply (erule set_times_intro2)
   apply (simp add: func_times)
   done
 
@@ -690,10 +679,10 @@
   apply (rule_tac x = c in exI)
   apply auto
   apply (case_tac "x = 0")
-  apply simp
+   apply simp
   apply (subgoal_tac "x = Suc (x - 1)")
-  apply (erule ssubst) back
-  apply (erule spec)
+   apply (erule ssubst) back
+   apply (erule spec)
   apply simp
   done
 
@@ -702,10 +691,10 @@
        f 0 = g 0 \<Longrightarrow> f =o g +o O(h)"
   apply (rule set_minus_imp_plus)
   apply (rule bigo_fix)
-  apply (subst fun_diff_def)
-  apply (subst fun_diff_def [symmetric])
-  apply (rule set_plus_imp_minus)
-  apply simp
+   apply (subst fun_diff_def)
+   apply (subst fun_diff_def [symmetric])
+   apply (rule set_plus_imp_minus)
+   apply simp
   apply (simp add: fun_diff_def)
   done
 
@@ -721,7 +710,7 @@
   apply (rule_tac x = c in exI)
   apply (rule allI)
   apply (rule order_trans)
-  apply (erule spec)+
+   apply (erule spec)+
   done
 
 lemma bigo_lesseq2: "f =o O(h) \<Longrightarrow> \<forall>x. \<bar>g x\<bar> \<le> f x \<Longrightarrow> g =o O(h)"
@@ -729,7 +718,7 @@
   apply (rule allI)
   apply (drule_tac x = x in spec)
   apply (rule order_trans)
-  apply assumption
+   apply assumption
   apply (rule abs_ge_self)
   done
 
@@ -737,7 +726,7 @@
   apply (erule bigo_lesseq2)
   apply (rule allI)
   apply (subst abs_of_nonneg)
-  apply (erule spec)+
+   apply (erule spec)+
   done
 
 lemma bigo_lesseq4: "f =o O(h) \<Longrightarrow>
@@ -745,75 +734,72 @@
   apply (erule bigo_lesseq1)
   apply (rule allI)
   apply (subst abs_of_nonneg)
-  apply (erule spec)+
+   apply (erule spec)+
   done
 
 lemma bigo_lesso1: "\<forall>x. f x \<le> g x \<Longrightarrow> f <o g =o O(h)"
   apply (unfold lesso_def)
   apply (subgoal_tac "(\<lambda>x. max (f x - g x) 0) = 0")
-  apply (erule ssubst)
-  apply (rule bigo_zero)
+   apply (erule ssubst)
+   apply (rule bigo_zero)
   apply (unfold func_zero)
   apply (rule ext)
   apply (simp split: split_max)
   done
 
-lemma bigo_lesso2: "f =o g +o O(h) \<Longrightarrow>
-    \<forall>x. 0 \<le> k x \<Longrightarrow> \<forall>x. k x \<le> f x \<Longrightarrow> k <o g =o O(h)"
+lemma bigo_lesso2: "f =o g +o O(h) \<Longrightarrow> \<forall>x. 0 \<le> k x \<Longrightarrow> \<forall>x. k x \<le> f x \<Longrightarrow> k <o g =o O(h)"
   apply (unfold lesso_def)
   apply (rule bigo_lesseq4)
-  apply (erule set_plus_imp_minus)
-  apply (rule allI)
-  apply (rule max.cobounded2)
+    apply (erule set_plus_imp_minus)
+   apply (rule allI)
+   apply (rule max.cobounded2)
   apply (rule allI)
   apply (subst fun_diff_def)
   apply (case_tac "0 \<le> k x - g x")
-  apply simp
-  apply (subst abs_of_nonneg)
-  apply (drule_tac x = x in spec) back
-  apply (simp add: algebra_simps)
-  apply (subst diff_conv_add_uminus)+
-  apply (rule add_right_mono)
-  apply (erule spec)
+   apply simp
+   apply (subst abs_of_nonneg)
+    apply (drule_tac x = x in spec) back
+    apply (simp add: algebra_simps)
+   apply (subst diff_conv_add_uminus)+
+   apply (rule add_right_mono)
+   apply (erule spec)
   apply (rule order_trans)
-  prefer 2
-  apply (rule abs_ge_zero)
+   prefer 2
+   apply (rule abs_ge_zero)
   apply (simp add: algebra_simps)
   done
 
-lemma bigo_lesso3: "f =o g +o O(h) \<Longrightarrow>
-    \<forall>x. 0 \<le> k x \<Longrightarrow> \<forall>x. g x \<le> k x \<Longrightarrow> f <o k =o O(h)"
+lemma bigo_lesso3: "f =o g +o O(h) \<Longrightarrow> \<forall>x. 0 \<le> k x \<Longrightarrow> \<forall>x. g x \<le> k x \<Longrightarrow> f <o k =o O(h)"
   apply (unfold lesso_def)
   apply (rule bigo_lesseq4)
-  apply (erule set_plus_imp_minus)
-  apply (rule allI)
-  apply (rule max.cobounded2)
+    apply (erule set_plus_imp_minus)
+   apply (rule allI)
+   apply (rule max.cobounded2)
   apply (rule allI)
   apply (subst fun_diff_def)
   apply (case_tac "0 \<le> f x - k x")
-  apply simp
-  apply (subst abs_of_nonneg)
-  apply (drule_tac x = x in spec) back
-  apply (simp add: algebra_simps)
-  apply (subst diff_conv_add_uminus)+
-  apply (rule add_left_mono)
-  apply (rule le_imp_neg_le)
-  apply (erule spec)
+   apply simp
+   apply (subst abs_of_nonneg)
+    apply (drule_tac x = x in spec) back
+    apply (simp add: algebra_simps)
+   apply (subst diff_conv_add_uminus)+
+   apply (rule add_left_mono)
+   apply (rule le_imp_neg_le)
+   apply (erule spec)
   apply (rule order_trans)
-  prefer 2
-  apply (rule abs_ge_zero)
+   prefer 2
+   apply (rule abs_ge_zero)
   apply (simp add: algebra_simps)
   done
 
-lemma bigo_lesso4:
-  fixes k :: "'a \<Rightarrow> 'b::linordered_field"
-  shows "f <o g =o O(k) \<Longrightarrow> g =o h +o O(k) \<Longrightarrow> f <o h =o O(k)"
+lemma bigo_lesso4: "f <o g =o O(k) \<Longrightarrow> g =o h +o O(k) \<Longrightarrow> f <o h =o O(k)"
+  for k :: "'a \<Rightarrow> 'b::linordered_field"
   apply (unfold lesso_def)
   apply (drule set_plus_imp_minus)
   apply (drule bigo_abs5) back
   apply (simp add: fun_diff_def)
   apply (drule bigo_useful_add)
-  apply assumption
+   apply assumption
   apply (erule bigo_lesseq2) back
   apply (rule allI)
   apply (auto simp add: func_plus fun_diff_def algebra_simps split: split_max abs_split)
@@ -826,7 +812,7 @@
   apply (rule allI)
   apply (drule_tac x = x in spec)
   apply (subgoal_tac "\<bar>max (f x - g x) 0\<bar> = max (f x - g x) 0")
-  apply (clarsimp simp add: algebra_simps)
+   apply (clarsimp simp add: algebra_simps)
   apply (rule abs_of_nonneg)
   apply (rule max.cobounded2)
   done
@@ -834,39 +820,41 @@
 lemma lesso_add: "f <o g =o O(h) \<Longrightarrow> k <o l =o O(h) \<Longrightarrow> (f + k) <o (g + l) =o O(h)"
   apply (unfold lesso_def)
   apply (rule bigo_lesseq3)
-  apply (erule bigo_useful_add)
-  apply assumption
-  apply (force split: split_max)
+    apply (erule bigo_useful_add)
+    apply assumption
+   apply (force split: split_max)
   apply (auto split: split_max simp add: func_plus)
   done
 
-lemma bigo_LIMSEQ1: "f =o O(g) \<Longrightarrow> g \<longlonglongrightarrow> 0 \<Longrightarrow> f \<longlonglongrightarrow> (0::real)"
+lemma bigo_LIMSEQ1: "f =o O(g) \<Longrightarrow> g \<longlonglongrightarrow> 0 \<Longrightarrow> f \<longlonglongrightarrow> 0"
+  for f g :: "nat \<Rightarrow> real"
   apply (simp add: LIMSEQ_iff bigo_alt_def)
   apply clarify
   apply (drule_tac x = "r / c" in spec)
   apply (drule mp)
-  apply simp
+   apply simp
   apply clarify
   apply (rule_tac x = no in exI)
   apply (rule allI)
   apply (drule_tac x = n in spec)+
   apply (rule impI)
   apply (drule mp)
-  apply assumption
+   apply assumption
   apply (rule order_le_less_trans)
-  apply assumption
+   apply assumption
   apply (rule order_less_le_trans)
-  apply (subgoal_tac "c * \<bar>g n\<bar> < c * (r / c)")
-  apply assumption
-  apply (erule mult_strict_left_mono)
-  apply assumption
+   apply (subgoal_tac "c * \<bar>g n\<bar> < c * (r / c)")
+    apply assumption
+   apply (erule mult_strict_left_mono)
+   apply assumption
   apply simp
   done
 
-lemma bigo_LIMSEQ2: "f =o g +o O(h) \<Longrightarrow> h \<longlonglongrightarrow> 0 \<Longrightarrow> f \<longlonglongrightarrow> a \<Longrightarrow> g \<longlonglongrightarrow> (a::real)"
+lemma bigo_LIMSEQ2: "f =o g +o O(h) \<Longrightarrow> h \<longlonglongrightarrow> 0 \<Longrightarrow> f \<longlonglongrightarrow> a \<Longrightarrow> g \<longlonglongrightarrow> a"
+  for f g h :: "nat \<Rightarrow> real"
   apply (drule set_plus_imp_minus)
   apply (drule bigo_LIMSEQ1)
-  apply assumption
+   apply assumption
   apply (simp only: fun_diff_def)
   apply (erule Lim_transform2)
   apply assumption
diff -r ae33d1c2ab26 -r a7c5074a0251 src/HOL/Library/Convex.thy
--- a/src/HOL/Library/Convex.thy	Wed Jul 13 21:00:03 2016 +0200
+++ b/src/HOL/Library/Convex.thy	Wed Jul 13 21:30:41 2016 +0200
@@ -6,7 +6,7 @@
 section \<open>Convexity in real vector spaces\<close>
 
 theory Convex
-imports Product_Vector
+  imports Product_Vector
 begin
 
 subsection \<open>Convexity\<close>
@@ -24,24 +24,27 @@
   shows "u *\<^sub>R x + v *\<^sub>R y \<in> s"
   using assms unfolding convex_def by fast
 
-lemma convex_alt:
-  "convex s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u. 0 \<le> u \<and> u \<le> 1 \<longrightarrow> ((1 - u) *\<^sub>R x + u *\<^sub>R y) \<in> s)"
+lemma convex_alt: "convex s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u. 0 \<le> u \<and> u \<le> 1 \<longrightarrow> ((1 - u) *\<^sub>R x + u *\<^sub>R y) \<in> s)"
   (is "_ \<longleftrightarrow> ?alt")
 proof
-  assume alt[rule_format]: ?alt
-  {
-    fix x y and u v :: real
-    assume mem: "x \<in> s" "y \<in> s"
-    assume "0 \<le> u" "0 \<le> v"
-    moreover
-    assume "u + v = 1"
-    then have "u = 1 - v" by auto
-    ultimately have "u *\<^sub>R x + v *\<^sub>R y \<in> s"
-      using alt[OF mem] by auto
-  }
-  then show "convex s"
-    unfolding convex_def by auto
-qed (auto simp: convex_def)
+  show "convex s" if alt: ?alt
+  proof -
+    {
+      fix x y and u v :: real
+      assume mem: "x \<in> s" "y \<in> s"
+      assume "0 \<le> u" "0 \<le> v"
+      moreover
+      assume "u + v = 1"
+      then have "u = 1 - v" by auto
+      ultimately have "u *\<^sub>R x + v *\<^sub>R y \<in> s"
+        using alt [rule_format, OF mem] by auto
+    }
+    then show ?thesis
+      unfolding convex_def by auto
+  qed
+  show ?alt if "convex s"
+    using that by (auto simp: convex_def)
+qed
 
 lemma convexD_alt:
   assumes "convex s" "a \<in> s" "b \<in> s" "0 \<le> u" "u \<le> 1"
@@ -53,7 +56,7 @@
   shows "((u/(u+v)) *\<^sub>R x + (v/(u+v)) *\<^sub>R y) \<in> S"
   apply (rule convexD)
   using assms
-  apply (simp_all add: zero_le_divide_iff add_divide_distrib [symmetric])
+       apply (simp_all add: zero_le_divide_iff add_divide_distrib [symmetric])
   done
 
 lemma convex_empty[intro,simp]: "convex {}"
@@ -270,12 +273,12 @@
       case False
       then show ?thesis
         using *[rule_format, of "{x, y}" "\<lambda> z. if z = x then 1 - \<mu> else \<mu>"] **
-          by auto
+        by auto
     next
       case True
       then show ?thesis
         using *[rule_format, of "{x, y}" "\<lambda> z. 1"] **
-          by (auto simp: field_simps real_vector.scale_left_diff_distrib)
+        by (auto simp: field_simps real_vector.scale_left_diff_distrib)
     qed
   qed
 qed
@@ -293,8 +296,8 @@
   have if_distrib_arg: "\<And>P f g x. (if P then f else g) x = (if P then f x else g x)"
     by simp
   show "(\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s"
-   using sum[THEN spec[where x="\<lambda>x. if x\<in>t then u x else 0"]] as *
-   by (auto simp: assms setsum.If_cases if_distrib if_distrib_arg)
+    using sum[THEN spec[where x="\<lambda>x. if x\<in>t then u x else 0"]] as *
+    by (auto simp: assms setsum.If_cases if_distrib if_distrib_arg)
 qed (erule_tac x=s in allE, erule_tac x=u in allE, auto)
 
 
@@ -306,39 +309,45 @@
 
 lemma convex_onI [intro?]:
   assumes "\<And>t x y. t > 0 \<Longrightarrow> t < 1 \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow>
-             f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
-  shows   "convex_on A f"
+    f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
+  shows "convex_on A f"
   unfolding convex_on_def
 proof clarify
-  fix x y u v assume A: "x \<in> A" "y \<in> A" "(u::real) \<ge> 0" "v \<ge> 0" "u + v = 1"
-  from A(5) have [simp]: "v = 1 - u" by (simp add: algebra_simps)
-  from A(1-4) show "f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y" using assms[of u y x]
+  fix x y
+  fix u v :: real
+  assume A: "x \<in> A" "y \<in> A" "u \<ge> 0" "v \<ge> 0" "u + v = 1"
+  from A(5) have [simp]: "v = 1 - u"
+    by (simp add: algebra_simps)
+  from A(1-4) show "f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y"
+    using assms[of u y x]
     by (cases "u = 0 \<or> u = 1") (auto simp: algebra_simps)
 qed
 
 lemma convex_on_linorderI [intro?]:
   fixes A :: "('a::{linorder,real_vector}) set"
   assumes "\<And>t x y. t > 0 \<Longrightarrow> t < 1 \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x < y \<Longrightarrow>
-             f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
-  shows   "convex_on A f"
+    f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
+  shows "convex_on A f"
 proof
-  fix t x y assume A: "x \<in> A" "y \<in> A" "(t::real) > 0" "t < 1"
-  with assms[of t x y] assms[of "1 - t" y x]
+  fix x y
+  fix t :: real
+  assume A: "x \<in> A" "y \<in> A" "t > 0" "t < 1"
+  with assms [of t x y] assms [of "1 - t" y x]
   show "f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
     by (cases x y rule: linorder_cases) (auto simp: algebra_simps)
 qed
 
 lemma convex_onD:
   assumes "convex_on A f"
-  shows   "\<And>t x y. t \<ge> 0 \<Longrightarrow> t \<le> 1 \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow>
-             f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
-  using assms unfolding convex_on_def by auto
+  shows "\<And>t x y. t \<ge> 0 \<Longrightarrow> t \<le> 1 \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow>
+    f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
+  using assms by (auto simp: convex_on_def)
 
 lemma convex_onD_Icc:
   assumes "convex_on {x..y} f" "x \<le> (y :: _ :: {real_vector,preorder})"
-  shows   "\<And>t. t \<ge> 0 \<Longrightarrow> t \<le> 1 \<Longrightarrow>
-             f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
-  using assms(2) by (intro convex_onD[OF assms(1)]) simp_all
+  shows "\<And>t. t \<ge> 0 \<Longrightarrow> t \<le> 1 \<Longrightarrow>
+    f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
+  using assms(2) by (intro convex_onD [OF assms(1)]) simp_all
 
 lemma convex_on_subset: "convex_on t f \<Longrightarrow> s \<subseteq> t \<Longrightarrow> convex_on s f"
   unfolding convex_on_def by auto
@@ -370,7 +379,8 @@
     and "convex_on s f"
   shows "convex_on s (\<lambda>x. c * f x)"
 proof -
-  have *: "\<And>u c fx v fy :: real. u * (c * fx) + v * (c * fy) = c * (u * fx + v * fy)"
+  have *: "u * (c * fx) + v * (c * fy) = c * (u * fx + v * fy)"
+    for u c fx v fy :: real
     by (simp add: field_simps)
   show ?thesis using assms(2) and mult_left_mono [OF _ assms(1)]
     unfolding convex_on_def and * by auto
@@ -517,20 +527,24 @@
   assume *: "y > 0" "x > 0" "\<mu> \<ge> 0" "\<mu> \<le> 1"
   {
     assume "\<mu> = 0"
-    then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y = y" by simp
-    then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0" using * by simp
+    then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y = y"
+      by simp
+    then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0"
+      using * by simp
   }
   moreover
   {
     assume "\<mu> = 1"
-    then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0" using * by simp
+    then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0"
+      using * by simp
   }
   moreover
   {
     assume "\<mu> \<noteq> 1" "\<mu> \<noteq> 0"
-    then have "\<mu> > 0" "(1 - \<mu>) > 0" using * by auto
-    then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0" using *
-      by (auto simp: add_pos_pos)
+    then have "\<mu> > 0" "(1 - \<mu>) > 0"
+      using * by auto
+    then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0"
+      using * by (auto simp: add_pos_pos)
   }
   ultimately show "(1 - \<mu>) *\<^sub>R y + \<mu> *\<^sub>R x > 0"
     by fastforce
@@ -550,11 +564,14 @@
   using assms
 proof (induct s arbitrary: a rule: finite_ne_induct)
   case (singleton i)
-  then have ai: "a i = 1" by auto
-  then show ?case by auto
+  then have ai: "a i = 1"
+    by auto
+  then show ?case
+    by auto
 next
   case (insert i s)
-  then have "convex_on C f" by simp
+  then have "convex_on C f"
+    by simp
   from this[unfolded convex_on_def, rule_format]
   have conv: "\<And>x y \<mu>. x \<in> C \<Longrightarrow> y \<in> C \<Longrightarrow> 0 \<le> \<mu> \<Longrightarrow> \<mu> \<le> 1 \<Longrightarrow>
       f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
@@ -593,8 +610,7 @@
       unfolding setsum_divide_distrib by simp
     have "convex C" using insert by auto
     then have asum: "(\<Sum> j \<in> s. ?a j *\<^sub>R y j) \<in> C"
-      using insert convex_setsum[OF \<open>finite s\<close>
-        \<open>convex C\<close> a1 a_nonneg] by auto
+      using insert convex_setsum [OF \<open>finite s\<close> \<open>convex C\<close> a1 a_nonneg] by auto
     have asum_le: "f (\<Sum> j \<in> s. ?a j *\<^sub>R y j) \<le> (\<Sum> j \<in> s. ?a j * f (y j))"
       using a_nonneg a1 insert by blast
     have "f (\<Sum> j \<in> insert i s. a j *\<^sub>R y j) = f ((\<Sum> j \<in> s. a j *\<^sub>R y j) + a i *\<^sub>R y i)"
@@ -611,10 +627,12 @@
       using conv[of "y i" "(\<Sum> j \<in> s. ?a j *\<^sub>R y j)" "a i", OF yai(1) asum yai(2) ai1]
       by (auto simp: add.commute)
     also have "\<dots> \<le> (1 - a i) * (\<Sum> j \<in> s. ?a j * f (y j)) + a i * f (y i)"
-      using add_right_mono[OF mult_left_mono[of _ _ "1 - a i",
-        OF asum_le less_imp_le[OF i0]], of "a i * f (y i)"] by simp
+      using add_right_mono [OF mult_left_mono [of _ _ "1 - a i",
+            OF asum_le less_imp_le[OF i0]], of "a i * f (y i)"]
+      by simp
     also have "\<dots> = (\<Sum> j \<in> s. (1 - a i) * ?a j * f (y j)) + a i * f (y i)"
-      unfolding setsum_right_distrib[of "1 - a i" "\<lambda> j. ?a j * f (y j)"] using i0 by auto
+      unfolding setsum_right_distrib[of "1 - a i" "\<lambda> j. ?a j * f (y j)"]
+      using i0 by auto
     also have "\<dots> = (\<Sum> j \<in> s. a j * f (y j)) + a i * f (y i)"
       using i0 by auto
     also have "\<dots> = (\<Sum> j \<in> insert i s. a j * f (y j))"
@@ -635,9 +653,9 @@
   fix \<mu> :: real
   assume *: "convex_on C f" "x \<in> C" "y \<in> C" "0 \<le> \<mu>" "\<mu> \<le> 1"
   from this[unfolded convex_on_def, rule_format]
-  have "\<And>u v. 0 \<le> u \<Longrightarrow> 0 \<le> v \<Longrightarrow> u + v = 1 \<Longrightarrow> f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y"
+  have "0 \<le> u \<Longrightarrow> 0 \<le> v \<Longrightarrow> u + v = 1 \<Longrightarrow> f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y" for u v
     by auto
-  from this[of "\<mu>" "1 - \<mu>", simplified] *
+  from this [of "\<mu>" "1 - \<mu>", simplified] *
   show "f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
     by auto
 next
@@ -701,8 +719,8 @@
     using * unfolding convex_alt by fastforce
   have geq: "\<mu> * (f x - f ?x) + (1 - \<mu>) * (f y - f ?x) \<ge>
       \<mu> * f' ?x * (x - ?x) + (1 - \<mu>) * f' ?x * (y - ?x)"
-    using add_mono[OF mult_left_mono[OF leq[OF xpos *(2)] \<open>\<mu> \<ge> 0\<close>]
-      mult_left_mono[OF leq[OF xpos *(3)] \<open>1 - \<mu> \<ge> 0\<close>]]
+    using add_mono [OF mult_left_mono [OF leq [OF xpos *(2)] \<open>\<mu> \<ge> 0\<close>]
+        mult_left_mono [OF leq [OF xpos *(3)] \<open>1 - \<mu> \<ge> 0\<close>]]
     by auto
   then have "\<mu> * f x + (1 - \<mu>) * f y - f ?x \<ge> 0"
     by (auto simp: field_simps)
@@ -728,14 +746,14 @@
     have "?\<mu> * x + (1 - ?\<mu>) * y = (y - z) * x / (y - x) + (1 - (y - z) / (y - x)) * y"
       by (auto simp: field_simps)
     also have "\<dots> = ((y - z) * x + (y - x - (y - z)) * y) / (y - x)"
-      using assms unfolding add_divide_distrib by (auto simp: field_simps)
+      using assms by (simp only: add_divide_distrib) (auto simp: field_simps)
     also have "\<dots> = z"
       using assms by (auto simp: field_simps)
     finally show ?thesis
       using comb by auto
   qed
-  show "z \<in> C" using z less assms
-    unfolding atLeastAtMost_iff le_less by auto
+  show "z \<in> C"
+    using z less assms by (auto simp: le_less)
 qed
 
 lemma f''_imp_f':
@@ -744,20 +762,21 @@
     and f': "\<And>x. x \<in> C \<Longrightarrow> DERIV f x :> (f' x)"
     and f'': "\<And>x. x \<in> C \<Longrightarrow> DERIV f' x :> (f'' x)"
     and pos: "\<And>x. x \<in> C \<Longrightarrow> f'' x \<ge> 0"
-    and "x \<in> C" "y \<in> C"
+    and x: "x \<in> C"
+    and y: "y \<in> C"
   shows "f' x * (y - x) \<le> f y - f x"
   using assms
 proof -
-  {
-    fix x y :: real
-    assume *: "x \<in> C" "y \<in> C" "y > x"
-    then have ge: "y - x > 0" "y - x \<ge> 0"
+  have less_imp: "f y - f x \<ge> f' x * (y - x)" "f' y * (x - y) \<le> f x - f y"
+    if *: "x \<in> C" "y \<in> C" "y > x" for x y :: real
+  proof -
+    from * have ge: "y - x > 0" "y - x \<ge> 0"
       by auto
     from * have le: "x - y < 0" "x - y \<le> 0"
       by auto
     then obtain z1 where z1: "z1 > x" "z1 < y" "f y - f x = (y - x) * f' z1"
       using subsetD[OF atMostAtLeast_subset_convex[OF \<open>convex C\<close> \<open>x \<in> C\<close> \<open>y \<in> C\<close> \<open>x < y\<close>],
-        THEN f', THEN MVT2[OF \<open>x < y\<close>, rule_format, unfolded atLeastAtMost_iff[symmetric]]]
+          THEN f', THEN MVT2[OF \<open>x < y\<close>, rule_format, unfolded atLeastAtMost_iff[symmetric]]]
       by auto
     then have "z1 \<in> C"
       using atMostAtLeast_subset_convex \<open>convex C\<close> \<open>x \<in> C\<close> \<open>y \<in> C\<close> \<open>x < y\<close>
@@ -766,11 +785,11 @@
       by (simp add: field_simps)
     obtain z2 where z2: "z2 > x" "z2 < z1" "f' z1 - f' x = (z1 - x) * f'' z2"
       using subsetD[OF atMostAtLeast_subset_convex[OF \<open>convex C\<close> \<open>x \<in> C\<close> \<open>z1 \<in> C\<close> \<open>x < z1\<close>],
-        THEN f'', THEN MVT2[OF \<open>x < z1\<close>, rule_format, unfolded atLeastAtMost_iff[symmetric]]] z1
+          THEN f'', THEN MVT2[OF \<open>x < z1\<close>, rule_format, unfolded atLeastAtMost_iff[symmetric]]] z1
       by auto
     obtain z3 where z3: "z3 > z1" "z3 < y" "f' y - f' z1 = (y - z1) * f'' z3"
       using subsetD[OF atMostAtLeast_subset_convex[OF \<open>convex C\<close> \<open>z1 \<in> C\<close> \<open>y \<in> C\<close> \<open>z1 < y\<close>],
-        THEN f'', THEN MVT2[OF \<open>z1 < y\<close>, rule_format, unfolded atLeastAtMost_iff[symmetric]]] z1
+          THEN f'', THEN MVT2[OF \<open>z1 < y\<close>, rule_format, unfolded atLeastAtMost_iff[symmetric]]] z1
       by auto
     have "f' y - (f x - f y) / (x - y) = f' y - f' z1"
       using * z1' by auto
@@ -818,22 +837,18 @@
       by (simp add: algebra_simps)
     then have "f y - f x - f' x * (y - x) \<ge> 0"
       using ge by auto
-    then have "f y - f x \<ge> f' x * (y - x)" "f' y * (x - y) \<le> f x - f y"
+    then show "f y - f x \<ge> f' x * (y - x)" "f' y * (x - y) \<le> f x - f y"
       using res by auto
-  } note less_imp = this
-  {
-    fix x y :: real
-    assume "x \<in> C" "y \<in> C" "x \<noteq> y"
-    then have"f y - f x \<ge> f' x * (y - x)"
-    unfolding neq_iff using less_imp by auto
-  }
-  moreover
-  {
-    fix x y :: real
-    assume "x \<in> C" "y \<in> C" "x = y"
-    then have "f y - f x \<ge> f' x * (y - x)" by auto
-  }
-  ultimately show ?thesis using assms by blast
+  qed
+  show ?thesis
+  proof (cases "x = y")
+    case True
+    with x y show ?thesis by auto
+  next
+    case False
+    with less_imp x y show ?thesis
+      by (auto simp: neq_iff)
+  qed
 qed
 
 lemma f''_ge0_imp_convex:
@@ -855,10 +870,10 @@
     using DERIV_log by auto
   then have f': "\<And>z. z > 0 \<Longrightarrow> DERIV (\<lambda> z. - log b z) z :> - 1 / (ln b * z)"
     by (auto simp: DERIV_minus)
-  have "\<And>z :: real. z > 0 \<Longrightarrow> DERIV inverse z :> - (inverse z ^ Suc (Suc 0))"
+  have "\<And>z::real. z > 0 \<Longrightarrow> DERIV inverse z :> - (inverse z ^ Suc (Suc 0))"
     using less_imp_neq[THEN not_sym, THEN DERIV_inverse] by auto
   from this[THEN DERIV_cmult, of _ "- 1 / ln b"]
-  have "\<And>z :: real. z > 0 \<Longrightarrow>
+  have "\<And>z::real. z > 0 \<Longrightarrow>
     DERIV (\<lambda> z. (- 1 / ln b) * inverse z) z :> (- 1 / ln b) * (- (inverse z ^ Suc (Suc 0)))"
     by auto
   then have f''0: "\<And>z::real. z > 0 \<Longrightarrow>
@@ -866,9 +881,9 @@
     unfolding inverse_eq_divide by (auto simp: mult.assoc)
   have f''_ge0: "\<And>z::real. z > 0 \<Longrightarrow> 1 / (ln b * z * z) \<ge> 0"
     using \<open>b > 1\<close> by (auto intro!: less_imp_le)
-  from f''_ge0_imp_convex[OF pos_is_convex,
-    unfolded greaterThan_iff, OF f' f''0 f''_ge0]
-  show ?thesis by auto
+  from f''_ge0_imp_convex[OF pos_is_convex, unfolded greaterThan_iff, OF f' f''0 f''_ge0]
+  show ?thesis
+    by auto
 qed
 
 
@@ -876,45 +891,59 @@
 
 lemma convex_on_realI:
   assumes "connected A"
-  assumes "\<And>x. x \<in> A \<Longrightarrow> (f has_real_derivative f' x) (at x)"
-  assumes "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<le> y \<Longrightarrow> f' x \<le> f' y"
-  shows   "convex_on A f"
+    and "\<And>x. x \<in> A \<Longrightarrow> (f has_real_derivative f' x) (at x)"
+    and "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<le> y \<Longrightarrow> f' x \<le> f' y"
+  shows "convex_on A f"
 proof (rule convex_on_linorderI)
   fix t x y :: real
-  assume t: "t > 0" "t < 1" and xy: "x \<in> A" "y \<in> A" "x < y"
+  assume t: "t > 0" "t < 1"
+  assume xy: "x \<in> A" "y \<in> A" "x < y"
   define z where "z = (1 - t) * x + t * y"
-  with \<open>connected A\<close> and xy have ivl: "{x..y} \<subseteq> A" using connected_contains_Icc by blast
+  with \<open>connected A\<close> and xy have ivl: "{x..y} \<subseteq> A"
+    using connected_contains_Icc by blast
 
-  from xy t have xz: "z > x" by (simp add: z_def algebra_simps)
-  have "y - z = (1 - t) * (y - x)" by (simp add: z_def algebra_simps)
-  also from xy t have "... > 0" by (intro mult_pos_pos) simp_all
-  finally have yz: "z < y" by simp
+  from xy t have xz: "z > x"
+    by (simp add: z_def algebra_simps)
+  have "y - z = (1 - t) * (y - x)"
+    by (simp add: z_def algebra_simps)
+  also from xy t have "\<dots> > 0"
+    by (intro mult_pos_pos) simp_all
+  finally have yz: "z < y"
+    by simp
 
   from assms xz yz ivl t have "\<exists>\<xi>. \<xi> > x \<and> \<xi> < z \<and> f z - f x = (z - x) * f' \<xi>"
     by (intro MVT2) (auto intro!: assms(2))
-  then obtain \<xi> where \<xi>: "\<xi> > x" "\<xi> < z" "f' \<xi> = (f z - f x) / (z - x)" by auto
+  then obtain \<xi> where \<xi>: "\<xi> > x" "\<xi> < z" "f' \<xi> = (f z - f x) / (z - x)"
+    by auto
   from assms xz yz ivl t have "\<exists>\<eta>. \<eta> > z \<and> \<eta> < y \<and> f y - f z = (y - z) * f' \<eta>"
     by (intro MVT2) (auto intro!: assms(2))
-  then obtain \<eta> where \<eta>: "\<eta> > z" "\<eta> < y" "f' \<eta> = (f y - f z) / (y - z)" by auto
+  then obtain \<eta> where \<eta>: "\<eta> > z" "\<eta> < y" "f' \<eta> = (f y - f z) / (y - z)"
+    by auto
 
   from \<eta>(3) have "(f y - f z) / (y - z) = f' \<eta>" ..
-  also from \<xi> \<eta> ivl have "\<xi> \<in> A" "\<eta> \<in> A" by auto
-  with \<xi> \<eta> have "f' \<eta> \<ge> f' \<xi>" by (intro assms(3)) auto
+  also from \<xi> \<eta> ivl have "\<xi> \<in> A" "\<eta> \<in> A"
+    by auto
+  with \<xi> \<eta> have "f' \<eta> \<ge> f' \<xi>"
+    by (intro assms(3)) auto
   also from \<xi>(3) have "f' \<xi> = (f z - f x) / (z - x)" .
   finally have "(f y - f z) * (z - x) \<ge> (f z - f x) * (y - z)"
     using xz yz by (simp add: field_simps)
-  also have "z - x = t * (y - x)" by (simp add: z_def algebra_simps)
-  also have "y - z = (1 - t) * (y - x)" by (simp add: z_def algebra_simps)
-  finally have "(f y - f z) * t \<ge> (f z - f x) * (1 - t)" using xy by simp
-  thus "(1 - t) * f x + t * f y \<ge> f ((1 - t) *\<^sub>R x + t *\<^sub>R y)"
+  also have "z - x = t * (y - x)"
+    by (simp add: z_def algebra_simps)
+  also have "y - z = (1 - t) * (y - x)"
+    by (simp add: z_def algebra_simps)
+  finally have "(f y - f z) * t \<ge> (f z - f x) * (1 - t)"
+    using xy by simp
+  then show "(1 - t) * f x + t * f y \<ge> f ((1 - t) *\<^sub>R x + t *\<^sub>R y)"
     by (simp add: z_def algebra_simps)
 qed
 
 lemma convex_on_inverse:
   assumes "A \<subseteq> {0<..}"
-  shows   "convex_on A (inverse :: real \<Rightarrow> real)"
+  shows "convex_on A (inverse :: real \<Rightarrow> real)"
 proof (rule convex_on_subset[OF _ assms], intro convex_on_realI[of _ _ "\<lambda>x. -inverse (x^2)"])
-  fix u v :: real assume "u \<in> {0<..}" "v \<in> {0<..}" "u \<le> v"
+  fix u v :: real
+  assume "u \<in> {0<..}" "v \<in> {0<..}" "u \<le> v"
   with assms show "-inverse (u^2) \<le> -inverse (v^2)"
     by (intro le_imp_neg_le le_imp_inverse_le power_mono) (simp_all)
 qed (insert assms, auto intro!: derivative_eq_intros simp: divide_simps power2_eq_square)
@@ -922,40 +951,47 @@
 lemma convex_onD_Icc':
   assumes "convex_on {x..y} f" "c \<in> {x..y}"
   defines "d \<equiv> y - x"
-  shows   "f c \<le> (f y - f x) / d * (c - x) + f x"
-proof (cases y x rule: linorder_cases)
-  assume less: "x < y"
-  hence d: "d > 0" by (simp add: d_def)
+  shows "f c \<le> (f y - f x) / d * (c - x) + f x"
+proof (cases x y rule: linorder_cases)
+  case less
+  then have d: "d > 0"
+    by (simp add: d_def)
   from assms(2) less have A: "0 \<le> (c - x) / d" "(c - x) / d \<le> 1"
     by (simp_all add: d_def divide_simps)
-  have "f c = f (x + (c - x) * 1)" by simp
-  also from less have "1 = ((y - x) / d)" by (simp add: d_def)
+  have "f c = f (x + (c - x) * 1)"
+    by simp
+  also from less have "1 = ((y - x) / d)"
+    by (simp add: d_def)
   also from d have "x + (c - x) * \<dots> = (1 - (c - x) / d) *\<^sub>R x + ((c - x) / d) *\<^sub>R y"
     by (simp add: field_simps)
-  also have "f \<dots> \<le> (1 - (c - x) / d) * f x + (c - x) / d * f y" using assms less
-    by (intro convex_onD_Icc) simp_all
-  also from d have "\<dots> = (f y - f x) / d * (c - x) + f x" by (simp add: field_simps)
+  also have "f \<dots> \<le> (1 - (c - x) / d) * f x + (c - x) / d * f y"
+    using assms less by (intro convex_onD_Icc) simp_all
+  also from d have "\<dots> = (f y - f x) / d * (c - x) + f x"
+    by (simp add: field_simps)
   finally show ?thesis .
 qed (insert assms(2), simp_all)
 
 lemma convex_onD_Icc'':
   assumes "convex_on {x..y} f" "c \<in> {x..y}"
   defines "d \<equiv> y - x"
-  shows   "f c \<le> (f x - f y) / d * (y - c) + f y"
-proof (cases y x rule: linorder_cases)
-  assume less: "x < y"
-  hence d: "d > 0" by (simp add: d_def)
+  shows "f c \<le> (f x - f y) / d * (y - c) + f y"
+proof (cases x y rule: linorder_cases)
+  case less
+  then have d: "d > 0"
+    by (simp add: d_def)
   from assms(2) less have A: "0 \<le> (y - c) / d" "(y - c) / d \<le> 1"
     by (simp_all add: d_def divide_simps)
-  have "f c = f (y - (y - c) * 1)" by simp
-  also from less have "1 = ((y - x) / d)" by (simp add: d_def)
+  have "f c = f (y - (y - c) * 1)"
+    by simp
+  also from less have "1 = ((y - x) / d)"
+    by (simp add: d_def)
   also from d have "y - (y - c) * \<dots> = (1 - (1 - (y - c) / d)) *\<^sub>R x + (1 - (y - c) / d) *\<^sub>R y"
     by (simp add: field_simps)
-  also have "f \<dots> \<le> (1 - (1 - (y - c) / d)) * f x + (1 - (y - c) / d) * f y" using assms less
-    by (intro convex_onD_Icc) (simp_all add: field_simps)
-  also from d have "\<dots> = (f x - f y) / d * (y - c) + f y" by (simp add: field_simps)
+  also have "f \<dots> \<le> (1 - (1 - (y - c) / d)) * f x + (1 - (y - c) / d) * f y"
+    using assms less by (intro convex_onD_Icc) (simp_all add: field_simps)
+  also from d have "\<dots> = (f x - f y) / d * (y - c) + f y"
+    by (simp add: field_simps)
   finally show ?thesis .
 qed (insert assms(2), simp_all)
 
-
 end
diff -r ae33d1c2ab26 -r a7c5074a0251 src/HOL/Library/Mapping.thy
--- a/src/HOL/Library/Mapping.thy	Wed Jul 13 21:00:03 2016 +0200
+++ b/src/HOL/Library/Mapping.thy	Wed Jul 13 21:30:41 2016 +0200
@@ -74,11 +74,11 @@
       (\<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
+     apply auto
     unfolding rel_fun_def
     apply clarsimp
     apply (case_tac xa)
-    apply (auto dest: list_all2_lengthD list_all2_nthD)
+     apply (auto dest: list_all2_lengthD list_all2_nthD)
     done
 qed
 
diff -r ae33d1c2ab26 -r a7c5074a0251 src/HOL/Library/Set_Algebras.thy
--- a/src/HOL/Library/Set_Algebras.thy	Wed Jul 13 21:00:03 2016 +0200
+++ b/src/HOL/Library/Set_Algebras.thy	Wed Jul 13 21:30:41 2016 +0200
@@ -1,24 +1,26 @@
 (*  Title:      HOL/Library/Set_Algebras.thy
-    Author:     Jeremy Avigad and Kevin Donnelly; Florian Haftmann, TUM
+    Author:     Jeremy Avigad
+    Author:     Kevin Donnelly
+    Author:     Florian Haftmann, TUM
 *)
 
 section \<open>Algebraic operations on sets\<close>
 
 theory Set_Algebras
-imports Main
+  imports Main
 begin
 
 text \<open>
-  This library lifts operations like addition and multiplication to
-  sets.  It was designed to support asymptotic calculations. See the
-  comments at the top of theory \<open>BigO\<close>.
+  This library lifts operations like addition and multiplication to sets. It
+  was designed to support asymptotic calculations. See the comments at the top
+  of @{file "BigO.thy"}.
 \<close>
 
 instantiation set :: (plus) plus
 begin
 
-definition plus_set :: "'a::plus set \<Rightarrow> 'a set \<Rightarrow> 'a set" where
-  set_plus_def: "A + B = {c. \<exists>a\<in>A. \<exists>b\<in>B. c = a + b}"
+definition plus_set :: "'a::plus set \<Rightarrow> 'a set \<Rightarrow> 'a set"
+  where set_plus_def: "A + B = {c. \<exists>a\<in>A. \<exists>b\<in>B. c = a + b}"
 
 instance ..
 
@@ -27,8 +29,8 @@
 instantiation set :: (times) times
 begin
 
-definition times_set :: "'a::times set \<Rightarrow> 'a set \<Rightarrow> 'a set" where
-  set_times_def: "A * B = {c. \<exists>a\<in>A. \<exists>b\<in>B. c = a * b}"
+definition times_set :: "'a::times set \<Rightarrow> 'a set \<Rightarrow> 'a set"
+  where set_times_def: "A * B = {c. \<exists>a\<in>A. \<exists>b\<in>B. c = a * b}"
 
 instance ..
 
@@ -37,8 +39,7 @@
 instantiation set :: (zero) zero
 begin
 
-definition
-  set_zero[simp]: "(0::'a::zero set) = {0}"
+definition set_zero[simp]: "(0::'a::zero set) = {0}"
 
 instance ..
 
@@ -47,21 +48,20 @@
 instantiation set :: (one) one
 begin
 
-definition
-  set_one[simp]: "(1::'a::one set) = {1}"
+definition set_one[simp]: "(1::'a::one set) = {1}"
 
 instance ..
 
 end
 
-definition elt_set_plus :: "'a::plus \<Rightarrow> 'a set \<Rightarrow> 'a set"  (infixl "+o" 70) where
-  "a +o B = {c. \<exists>b\<in>B. c = a + b}"
+definition elt_set_plus :: "'a::plus \<Rightarrow> 'a set \<Rightarrow> 'a set"  (infixl "+o" 70)
+  where "a +o B = {c. \<exists>b\<in>B. c = a + b}"
 
-definition elt_set_times :: "'a::times \<Rightarrow> 'a set \<Rightarrow> 'a set"  (infixl "*o" 80) where
-  "a *o B = {c. \<exists>b\<in>B. c = a * b}"
+definition elt_set_times :: "'a::times \<Rightarrow> 'a set \<Rightarrow> 'a set"  (infixl "*o" 80)
+  where "a *o B = {c. \<exists>b\<in>B. c = a * b}"
 
-abbreviation (input) elt_set_eq :: "'a \<Rightarrow> 'a set \<Rightarrow> bool"  (infix "=o" 50) where
-  "x =o A \<equiv> x \<in> A"
+abbreviation (input) elt_set_eq :: "'a \<Rightarrow> 'a set \<Rightarrow> bool"  (infix "=o" 50)
+  where "x =o A \<equiv> x \<in> A"
 
 instance set :: (semigroup_add) semigroup_add
   by standard (force simp add: set_plus_def add.assoc)
@@ -98,19 +98,21 @@
 lemma set_plus_intro2 [intro]: "b \<in> C \<Longrightarrow> a + b \<in> a +o C"
   by (auto simp add: elt_set_plus_def)
 
-lemma set_plus_rearrange:
-  "((a::'a::comm_monoid_add) +o C) + (b +o D) = (a + b) +o (C + D)"
+lemma set_plus_rearrange: "(a +o C) + (b +o D) = (a + b) +o (C + D)"
+  for a b :: "'a::comm_monoid_add"
   apply (auto simp add: elt_set_plus_def set_plus_def ac_simps)
    apply (rule_tac x = "ba + bb" in exI)
-  apply (auto simp add: ac_simps)
+   apply (auto simp add: ac_simps)
   apply (rule_tac x = "aa + a" in exI)
   apply (auto simp add: ac_simps)
   done
 
-lemma set_plus_rearrange2: "(a::'a::semigroup_add) +o (b +o C) = (a + b) +o C"
+lemma set_plus_rearrange2: "a +o (b +o C) = (a + b) +o C"
+  for a b :: "'a::semigroup_add"
   by (auto simp add: elt_set_plus_def add.assoc)
 
-lemma set_plus_rearrange3: "((a::'a::semigroup_add) +o B) + C = a +o (B + C)"
+lemma set_plus_rearrange3: "(a +o B) + C = a +o (B + C)"
+  for a :: "'a::semigroup_add"
   apply (auto simp add: elt_set_plus_def set_plus_def)
    apply (blast intro: ac_simps)
   apply (rule_tac x = "a + aa" in exI)
@@ -121,7 +123,8 @@
    apply (auto simp add: ac_simps)
   done
 
-theorem set_plus_rearrange4: "C + ((a::'a::comm_monoid_add) +o D) = a +o (C + D)"
+theorem set_plus_rearrange4: "C + (a +o D) = a +o (C + D)"
+  for a :: "'a::comm_monoid_add"
   apply (auto simp add: elt_set_plus_def set_plus_def ac_simps)
    apply (rule_tac x = "aa + ba" in exI)
    apply (auto simp add: ac_simps)
@@ -133,13 +136,15 @@
 lemma set_plus_mono [intro!]: "C \<subseteq> D \<Longrightarrow> a +o C \<subseteq> a +o D"
   by (auto simp add: elt_set_plus_def)
 
-lemma set_plus_mono2 [intro]: "(C::'a::plus set) \<subseteq> D \<Longrightarrow> E \<subseteq> F \<Longrightarrow> C + E \<subseteq> D + F"
+lemma set_plus_mono2 [intro]: "C \<subseteq> D \<Longrightarrow> E \<subseteq> F \<Longrightarrow> C + E \<subseteq> D + F"
+  for C D E F :: "'a::plus set"
   by (auto simp add: set_plus_def)
 
 lemma set_plus_mono3 [intro]: "a \<in> C \<Longrightarrow> a +o D \<subseteq> C + D"
   by (auto simp add: elt_set_plus_def set_plus_def)
 
-lemma set_plus_mono4 [intro]: "(a::'a::comm_monoid_add) \<in> C \<Longrightarrow> a +o D \<subseteq> D + C"
+lemma set_plus_mono4 [intro]: "a \<in> C \<Longrightarrow> a +o D \<subseteq> D + C"
+  for a :: "'a::comm_monoid_add"
   by (auto simp add: elt_set_plus_def set_plus_def ac_simps)
 
 lemma set_plus_mono5: "a \<in> C \<Longrightarrow> B \<subseteq> D \<Longrightarrow> a +o B \<subseteq> C + D"
@@ -166,33 +171,45 @@
   apply auto
   done
 
-lemma set_plus_mono4_b: "(a::'a::comm_monoid_add) : C \<Longrightarrow> x \<in> a +o D \<Longrightarrow> x \<in> D + C"
+lemma set_plus_mono4_b: "a \<in> C \<Longrightarrow> x \<in> a +o D \<Longrightarrow> x \<in> D + C"
+  for a x :: "'a::comm_monoid_add"
   apply (frule set_plus_mono4)
   apply auto
   done
 
-lemma set_zero_plus [simp]: "(0::'a::comm_monoid_add) +o C = C"
+lemma set_zero_plus [simp]: "0 +o C = C"
+  for C :: "'a::comm_monoid_add set"
   by (auto simp add: elt_set_plus_def)
 
-lemma set_zero_plus2: "(0::'a::comm_monoid_add) \<in> A \<Longrightarrow> B \<subseteq> A + B"
+lemma set_zero_plus2: "0 \<in> A \<Longrightarrow> B \<subseteq> A + B"
+  for A B :: "'a::comm_monoid_add set"
   apply (auto simp add: set_plus_def)
   apply (rule_tac x = 0 in bexI)
    apply (rule_tac x = x in bexI)
     apply (auto simp add: ac_simps)
   done
 
-lemma set_plus_imp_minus: "(a::'a::ab_group_add) : b +o C \<Longrightarrow> (a - b) \<in> C"
+lemma set_plus_imp_minus: "a \<in> b +o C \<Longrightarrow> a - b \<in> C"
+  for a b :: "'a::ab_group_add"
   by (auto simp add: elt_set_plus_def ac_simps)
 
-lemma set_minus_imp_plus: "(a::'a::ab_group_add) - b : C \<Longrightarrow> a \<in> b +o C"
+lemma set_minus_imp_plus: "a - b \<in> C \<Longrightarrow> a \<in> b +o C"
+  for a b :: "'a::ab_group_add"
   apply (auto simp add: elt_set_plus_def ac_simps)
   apply (subgoal_tac "a = (a + - b) + b")
-   apply (rule bexI, assumption)
-  apply (auto simp add: ac_simps)
+   apply (rule bexI)
+    apply assumption
+   apply (auto simp add: ac_simps)
   done
 
-lemma set_minus_plus: "(a::'a::ab_group_add) - b \<in> C \<longleftrightarrow> a \<in> b +o C"
-  by (rule iffI, rule set_minus_imp_plus, assumption, rule set_plus_imp_minus)
+lemma set_minus_plus: "a - b \<in> C \<longleftrightarrow> a \<in> b +o C"
+  for a b :: "'a::ab_group_add"
+  apply (rule iffI)
+   apply (rule set_minus_imp_plus)
+   apply assumption
+  apply (rule set_plus_imp_minus)
+  apply assumption
+  done
 
 lemma set_times_intro [intro]: "a \<in> C \<Longrightarrow> b \<in> D \<Longrightarrow> a * b \<in> C * D"
   by (auto simp add: set_times_def)
@@ -205,8 +222,8 @@
 lemma set_times_intro2 [intro!]: "b \<in> C \<Longrightarrow> a * b \<in> a *o C"
   by (auto simp add: elt_set_times_def)
 
-lemma set_times_rearrange:
-  "((a::'a::comm_monoid_mult) *o C) * (b *o D) = (a * b) *o (C * D)"
+lemma set_times_rearrange: "(a *o C) * (b *o D) = (a * b) *o (C * D)"
+  for a b :: "'a::comm_monoid_mult"
   apply (auto simp add: elt_set_times_def set_times_def)
    apply (rule_tac x = "ba * bb" in exI)
    apply (auto simp add: ac_simps)
@@ -214,12 +231,12 @@
   apply (auto simp add: ac_simps)
   done
 
-lemma set_times_rearrange2:
-  "(a::'a::semigroup_mult) *o (b *o C) = (a * b) *o C"
+lemma set_times_rearrange2: "a *o (b *o C) = (a * b) *o C"
+  for a b :: "'a::semigroup_mult"
   by (auto simp add: elt_set_times_def mult.assoc)
 
-lemma set_times_rearrange3:
-  "((a::'a::semigroup_mult) *o B) * C = a *o (B * C)"
+lemma set_times_rearrange3: "(a *o B) * C = a *o (B * C)"
+  for a :: "'a::semigroup_mult"
   apply (auto simp add: elt_set_times_def set_times_def)
    apply (blast intro: ac_simps)
   apply (rule_tac x = "a * aa" in exI)
@@ -230,8 +247,8 @@
    apply (auto simp add: ac_simps)
   done
 
-theorem set_times_rearrange4:
-  "C * ((a::'a::comm_monoid_mult) *o D) = a *o (C * D)"
+theorem set_times_rearrange4: "C * (a *o D) = a *o (C * D)"
+  for a :: "'a::comm_monoid_mult"
   apply (auto simp add: elt_set_times_def set_times_def ac_simps)
    apply (rule_tac x = "aa * ba" in exI)
    apply (auto simp add: ac_simps)
@@ -243,13 +260,15 @@
 lemma set_times_mono [intro]: "C \<subseteq> D \<Longrightarrow> a *o C \<subseteq> a *o D"
   by (auto simp add: elt_set_times_def)
 
-lemma set_times_mono2 [intro]: "(C::'a::times set) \<subseteq> D \<Longrightarrow> E \<subseteq> F \<Longrightarrow> C * E \<subseteq> D * F"
+lemma set_times_mono2 [intro]: "C \<subseteq> D \<Longrightarrow> E \<subseteq> F \<Longrightarrow> C * E \<subseteq> D * F"
+  for C D E F :: "'a::times set"
   by (auto simp add: set_times_def)
 
 lemma set_times_mono3 [intro]: "a \<in> C \<Longrightarrow> a *o D \<subseteq> C * D"
   by (auto simp add: elt_set_times_def set_times_def)
 
-lemma set_times_mono4 [intro]: "(a::'a::comm_monoid_mult) : C \<Longrightarrow> a *o D \<subseteq> D * C"
+lemma set_times_mono4 [intro]: "a \<in> C \<Longrightarrow> a *o D \<subseteq> D * C"
+  for a :: "'a::comm_monoid_mult"
   by (auto simp add: elt_set_times_def set_times_def ac_simps)
 
 lemma set_times_mono5: "a \<in> C \<Longrightarrow> B \<subseteq> D \<Longrightarrow> a *o B \<subseteq> C * D"
@@ -276,30 +295,31 @@
   apply auto
   done
 
-lemma set_times_mono4_b: "(a::'a::comm_monoid_mult) \<in> C \<Longrightarrow> x \<in> a *o D \<Longrightarrow> x \<in> D * C"
+lemma set_times_mono4_b: "a \<in> C \<Longrightarrow> x \<in> a *o D \<Longrightarrow> x \<in> D * C"
+  for a x :: "'a::comm_monoid_mult"
   apply (frule set_times_mono4)
   apply auto
   done
 
-lemma set_one_times [simp]: "(1::'a::comm_monoid_mult) *o C = C"
+lemma set_one_times [simp]: "1 *o C = C"
+  for C :: "'a::comm_monoid_mult set"
   by (auto simp add: elt_set_times_def)
 
-lemma set_times_plus_distrib:
-  "(a::'a::semiring) *o (b +o C) = (a * b) +o (a *o C)"
+lemma set_times_plus_distrib: "a *o (b +o C) = (a * b) +o (a *o C)"
+  for a b :: "'a::semiring"
   by (auto simp add: elt_set_plus_def elt_set_times_def ring_distribs)
 
-lemma set_times_plus_distrib2:
-  "(a::'a::semiring) *o (B + C) = (a *o B) + (a *o C)"
+lemma set_times_plus_distrib2: "a *o (B + C) = (a *o B) + (a *o C)"
+  for a :: "'a::semiring"
   apply (auto simp add: set_plus_def elt_set_times_def ring_distribs)
    apply blast
   apply (rule_tac x = "b + bb" in exI)
   apply (auto simp add: ring_distribs)
   done
 
-lemma set_times_plus_distrib3: "((a::'a::semiring) +o C) * D \<subseteq> a *o D + C * D"
-  apply (auto simp add:
-    elt_set_plus_def elt_set_times_def set_times_def
-    set_plus_def ring_distribs)
+lemma set_times_plus_distrib3: "(a +o C) * D \<subseteq> a *o D + C * D"
+  for a :: "'a::semiring"
+  apply (auto simp: elt_set_plus_def elt_set_times_def set_times_def set_plus_def ring_distribs)
   apply auto
   done
 
@@ -307,23 +327,25 @@
   set_times_plus_distrib
   set_times_plus_distrib2
 
-lemma set_neg_intro: "(a::'a::ring_1) \<in> (- 1) *o C \<Longrightarrow> - a \<in> C"
+lemma set_neg_intro: "a \<in> (- 1) *o C \<Longrightarrow> - a \<in> C"
+  for a :: "'a::ring_1"
   by (auto simp add: elt_set_times_def)
 
-lemma set_neg_intro2: "(a::'a::ring_1) \<in> C \<Longrightarrow> - a \<in> (- 1) *o C"
+lemma set_neg_intro2: "a \<in> C \<Longrightarrow> - a \<in> (- 1) *o C"
+  for a :: "'a::ring_1"
   by (auto simp add: elt_set_times_def)
 
 lemma set_plus_image: "S + T = (\<lambda>(x, y). x + y) ` (S \<times> T)"
-  unfolding set_plus_def by (fastforce simp: image_iff)
+  by (fastforce simp: set_plus_def image_iff)
 
 lemma set_times_image: "S * T = (\<lambda>(x, y). x * y) ` (S \<times> T)"
-  unfolding set_times_def by (fastforce simp: image_iff)
+  by (fastforce simp: set_times_def image_iff)
 
 lemma finite_set_plus: "finite s \<Longrightarrow> finite t \<Longrightarrow> finite (s + t)"
-  unfolding set_plus_image by simp
+  by (simp add: set_plus_image)
 
 lemma finite_set_times: "finite s \<Longrightarrow> finite t \<Longrightarrow> finite (s * t)"
-  unfolding set_times_image by simp
+  by (simp add: set_times_image)
 
 lemma set_setsum_alt:
   assumes fin: "finite I"
diff -r ae33d1c2ab26 -r a7c5074a0251 src/HOL/Library/Stirling.thy
--- a/src/HOL/Library/Stirling.thy	Wed Jul 13 21:00:03 2016 +0200
+++ b/src/HOL/Library/Stirling.thy	Wed Jul 13 21:30:41 2016 +0200
@@ -1,5 +1,9 @@
-(* Authors: Amine Chaieb & Florian Haftmann, TU Muenchen
-            with contributions by Lukas Bulwahn and Manuel Eberl*)
+(*  Title:      HOL/Library/Stirling.thy
+    Author:     Amine Chaieb
+    Author:     Florian Haftmann
+    Author:     Lukas Bulwahn
+    Author:     Manuel Eberl
+*)
 
 section \<open>Stirling numbers of first and second kind\<close>
 
@@ -10,102 +14,105 @@
 subsection \<open>Stirling numbers of the second kind\<close>
 
 fun Stirling :: "nat \<Rightarrow> nat \<Rightarrow> nat"
-where
-  "Stirling 0 0 = 1"
-| "Stirling 0 (Suc k) = 0"
-| "Stirling (Suc n) 0 = 0"
-| "Stirling (Suc n) (Suc k) = Suc k * Stirling n (Suc k) + Stirling n k"
+  where
+    "Stirling 0 0 = 1"
+  | "Stirling 0 (Suc k) = 0"
+  | "Stirling (Suc n) 0 = 0"
+  | "Stirling (Suc n) (Suc k) = Suc k * Stirling n (Suc k) + Stirling n k"
 
-lemma Stirling_1 [simp]:
-  "Stirling (Suc n) (Suc 0) = 1"
+lemma Stirling_1 [simp]: "Stirling (Suc n) (Suc 0) = 1"
   by (induct n) simp_all
 
-lemma Stirling_less [simp]:
-  "n < k \<Longrightarrow> Stirling n k = 0"
+lemma Stirling_less [simp]: "n < k \<Longrightarrow> Stirling n k = 0"
   by (induct n k rule: Stirling.induct) simp_all
 
-lemma Stirling_same [simp]:
-  "Stirling n n = 1"
+lemma Stirling_same [simp]: "Stirling n n = 1"
   by (induct n) simp_all
 
-lemma Stirling_2_2:
-  "Stirling (Suc (Suc n)) (Suc (Suc 0)) = 2 ^ Suc n - 1"
+lemma Stirling_2_2: "Stirling (Suc (Suc n)) (Suc (Suc 0)) = 2 ^ Suc n - 1"
 proof (induct n)
-  case 0 then show ?case by simp
+  case 0
+  then show ?case by simp
 next
   case (Suc n)
   have "Stirling (Suc (Suc (Suc n))) (Suc (Suc 0)) =
-    2 * Stirling (Suc (Suc n)) (Suc (Suc 0)) + Stirling (Suc (Suc n)) (Suc 0)" by simp
+      2 * Stirling (Suc (Suc n)) (Suc (Suc 0)) + Stirling (Suc (Suc n)) (Suc 0)"
+    by simp
   also have "\<dots> = 2 * (2 ^ Suc n - 1) + 1"
     by (simp only: Suc Stirling_1)
   also have "\<dots> = 2 ^ Suc (Suc n) - 1"
   proof -
-    have "(2::nat) ^ Suc n - 1 > 0" by (induct n) simp_all
-    then have "2 * ((2::nat) ^ Suc n - 1) > 0" by simp
-    then have "2 \<le> 2 * ((2::nat) ^ Suc n)" by simp
+    have "(2::nat) ^ Suc n - 1 > 0"
+      by (induct n) simp_all
+    then have "2 * ((2::nat) ^ Suc n - 1) > 0"
+      by simp
+    then have "2 \<le> 2 * ((2::nat) ^ Suc n)"
+      by simp
     with add_diff_assoc2 [of 2 "2 * 2 ^ Suc n" 1]
-      have "2 * 2 ^ Suc n - 2 + (1::nat) = 2 * 2 ^ Suc n + 1 - 2" .
-    then show ?thesis by (simp add: nat_distrib)
+    have "2 * 2 ^ Suc n - 2 + (1::nat) = 2 * 2 ^ Suc n + 1 - 2" .
+    then show ?thesis
+      by (simp add: nat_distrib)
   qed
   finally show ?case by simp
 qed
 
-lemma Stirling_2:
-  "Stirling (Suc n) (Suc (Suc 0)) = 2 ^ n - 1"
+lemma Stirling_2: "Stirling (Suc n) (Suc (Suc 0)) = 2 ^ n - 1"
   using Stirling_2_2 by (cases n) simp_all
 
+
 subsection \<open>Stirling numbers of the first kind\<close>
 
 fun stirling :: "nat \<Rightarrow> nat \<Rightarrow> nat"
-where
-  "stirling 0 0 = 1"
-| "stirling 0 (Suc k) = 0"
-| "stirling (Suc n) 0 = 0"
-| "stirling (Suc n) (Suc k) = n * stirling n (Suc k) + stirling n k"
+  where
+    "stirling 0 0 = 1"
+  | "stirling 0 (Suc k) = 0"
+  | "stirling (Suc n) 0 = 0"
+  | "stirling (Suc n) (Suc k) = n * stirling n (Suc k) + stirling n k"
 
 lemma stirling_0 [simp]: "n > 0 \<Longrightarrow> stirling n 0 = 0"
   by (cases n) simp_all
 
-lemma stirling_less [simp]:
-  "n < k \<Longrightarrow> stirling n k = 0"
+lemma stirling_less [simp]: "n < k \<Longrightarrow> stirling n k = 0"
   by (induct n k rule: stirling.induct) simp_all
 
-lemma stirling_same [simp]:
-  "stirling n n = 1"
+lemma stirling_same [simp]: "stirling n n = 1"
   by (induct n) simp_all
 
-lemma stirling_Suc_n_1:
-  "stirling (Suc n) (Suc 0) = fact n"
+lemma stirling_Suc_n_1: "stirling (Suc n) (Suc 0) = fact n"
   by (induct n) auto
 
-lemma stirling_Suc_n_n:
-  shows "stirling (Suc n) n = Suc n choose 2"
-by (induct n) (auto simp add: numerals(2))
+lemma stirling_Suc_n_n: "stirling (Suc n) n = Suc n choose 2"
+  by (induct n) (auto simp add: numerals(2))
 
 lemma stirling_Suc_n_2:
   assumes "n \<ge> Suc 0"
   shows "stirling (Suc n) 2 = (\<Sum>k=1..n. fact n div k)"
-using assms
+  using assms
 proof (induct n)
-  case 0 from this show ?case by simp
+  case 0
+  then show ?case by simp
 next
   case (Suc n)
   show ?case
   proof (cases n)
-    case 0 from this show ?thesis by (simp add: numerals(2))
+    case 0
+    then show ?thesis
+      by (simp add: numerals(2))
   next
     case Suc
-    from this have geq1: "Suc 0 \<le> n" by simp
+    then have geq1: "Suc 0 \<le> n"
+      by simp
     have "stirling (Suc (Suc n)) 2 = Suc n * stirling (Suc n) 2 + stirling (Suc n) (Suc 0)"
       by (simp only: stirling.simps(4)[of "Suc n"] numerals(2))
-    also have "... = Suc n * (\<Sum>k=1..n. fact n div k) + fact n"
+    also have "\<dots> = Suc n * (\<Sum>k=1..n. fact n div k) + fact n"
       using Suc.hyps[OF geq1]
       by (simp only: stirling_Suc_n_1 of_nat_fact of_nat_add of_nat_mult)
-    also have "... = Suc n * (\<Sum>k=1..n. fact n div k) + Suc n * fact n div Suc n"
+    also have "\<dots> = Suc n * (\<Sum>k=1..n. fact n div k) + Suc n * fact n div Suc n"
       by (metis nat.distinct(1) nonzero_mult_divide_cancel_left)
-    also have "... = (\<Sum>k=1..n. fact (Suc n) div k) + fact (Suc n) div Suc n"
+    also have "\<dots> = (\<Sum>k=1..n. fact (Suc n) div k) + fact (Suc n) div Suc n"
       by (simp add: setsum_right_distrib div_mult_swap dvd_fact)
-    also have "... = (\<Sum>k=1..Suc n. fact (Suc n) div k)" by simp
+    also have "\<dots> = (\<Sum>k=1..Suc n. fact (Suc n) div k)"
+      by simp
     finally show ?thesis .
   qed
 qed
@@ -113,52 +120,60 @@
 lemma of_nat_stirling_Suc_n_2:
   assumes "n \<ge> Suc 0"
   shows "(of_nat (stirling (Suc n) 2)::'a::field_char_0) = fact n * (\<Sum>k=1..n. (1 / of_nat k))"
-using assms
+  using assms
 proof (induct n)
-  case 0 from this show ?case by simp
+  case 0
+  then show ?case by simp
 next
   case (Suc n)
   show ?case
   proof (cases n)
-    case 0 from this show ?thesis by (auto simp add: numerals(2))
+    case 0
+    then show ?thesis
+      by (auto simp add: numerals(2))
   next
     case Suc
-    from this have geq1: "Suc 0 \<le> n" by simp
+    then have geq1: "Suc 0 \<le> n"
+      by simp
     have "(of_nat (stirling (Suc (Suc n)) 2)::'a) =
-      of_nat (Suc n * stirling (Suc n) 2 + stirling (Suc n) (Suc 0))"
+        of_nat (Suc n * stirling (Suc n) 2 + stirling (Suc n) (Suc 0))"
       by (simp only: stirling.simps(4)[of "Suc n"] numerals(2))
-    also have "... = of_nat (Suc n) * (fact n * (\<Sum>k = 1..n. 1 / of_nat k)) + fact n"
+    also have "\<dots> = of_nat (Suc n) * (fact n * (\<Sum>k = 1..n. 1 / of_nat k)) + fact n"
       using Suc.hyps[OF geq1]
       by (simp only: stirling_Suc_n_1 of_nat_fact of_nat_add of_nat_mult)
-    also have "... = fact (Suc n) * (\<Sum>k = 1..n. 1 / of_nat k) + fact (Suc n) * (1 / of_nat (Suc n))"
+    also have "\<dots> = fact (Suc n) * (\<Sum>k = 1..n. 1 / of_nat k) + fact (Suc n) * (1 / of_nat (Suc n))"
       using of_nat_neq_0 by auto
-    also have "... = fact (Suc n) * (\<Sum>k = 1..Suc n. 1 / of_nat k)"
+    also have "\<dots> = fact (Suc n) * (\<Sum>k = 1..Suc n. 1 / of_nat k)"
       by (simp add: distrib_left)
     finally show ?thesis .
   qed
 qed
 
-lemma setsum_stirling:
-  "(\<Sum>k\<le>n. stirling n k) = fact n"
+lemma setsum_stirling: "(\<Sum>k\<le>n. stirling n k) = fact n"
 proof (induct n)
   case 0
-  from this show ?case by simp
+  then show ?case by simp
 next
   case (Suc n)
   have "(\<Sum>k\<le>Suc n. stirling (Suc n) k) = stirling (Suc n) 0 + (\<Sum>k\<le>n. stirling (Suc n) (Suc k))"
     by (simp only: setsum_atMost_Suc_shift)
-  also have "\<dots> = (\<Sum>k\<le>n. stirling (Suc n) (Suc k))" by simp
-  also have "\<dots> = (\<Sum>k\<le>n. n * stirling n (Suc k) + stirling n k)" by simp
+  also have "\<dots> = (\<Sum>k\<le>n. stirling (Suc n) (Suc k))"
+    by simp
+  also have "\<dots> = (\<Sum>k\<le>n. n * stirling n (Suc k) + stirling n k)"
+    by simp
   also have "\<dots> = n * (\<Sum>k\<le>n. stirling n (Suc k)) + (\<Sum>k\<le>n. stirling n k)"
     by (simp add: setsum.distrib setsum_right_distrib)
   also have "\<dots> = n * fact n + fact n"
   proof -
     have "n * (\<Sum>k\<le>n. stirling n (Suc k)) = n * ((\<Sum>k\<le>Suc n. stirling n k) - stirling n 0)"
       by (metis add_diff_cancel_left' setsum_atMost_Suc_shift)
-    also have "\<dots> = n * (\<Sum>k\<le>n. stirling n k)" by (cases n) simp+
-    also have "\<dots> = n * fact n" using Suc.hyps by simp
+    also have "\<dots> = n * (\<Sum>k\<le>n. stirling n k)"
+      by (cases n) simp_all
+    also have "\<dots> = n * fact n"
+      using Suc.hyps by simp
     finally have "n * (\<Sum>k\<le>n. stirling n (Suc k)) = n * fact n" .
-    moreover have "(\<Sum>k\<le>n. stirling n k) = fact n" using Suc.hyps .
+    moreover have "(\<Sum>k\<le>n. stirling n k) = fact n"
+      using Suc.hyps .
     ultimately show ?thesis by simp
   qed
   also have "\<dots> = fact (Suc n)" by simp
@@ -166,26 +181,29 @@
 qed
 
 lemma stirling_pochhammer:
-  "(\<Sum>k\<le>n. of_nat (stirling n k) * x ^ k) = (pochhammer x n :: 'a :: comm_semiring_1)"
-proof (induction n)
+  "(\<Sum>k\<le>n. of_nat (stirling n k) * x ^ k) = (pochhammer x n :: 'a::comm_semiring_1)"
+proof (induct n)
+  case 0
+  then show ?case by simp
+next
   case (Suc n)
   have "of_nat (n * stirling n 0) = (0 :: 'a)" by (cases n) simp_all
-  hence "(\<Sum>k\<le>Suc n. of_nat (stirling (Suc n) k) * x ^ k) =
-            (of_nat (n * stirling n 0) * x ^ 0 +
-            (\<Sum>i\<le>n. of_nat (n * stirling n (Suc i)) * (x ^ Suc i))) +
-            (\<Sum>i\<le>n. of_nat (stirling n i) * (x ^ Suc i))"
+  then have "(\<Sum>k\<le>Suc n. of_nat (stirling (Suc n) k) * x ^ k) =
+      (of_nat (n * stirling n 0) * x ^ 0 +
+      (\<Sum>i\<le>n. of_nat (n * stirling n (Suc i)) * (x ^ Suc i))) +
+      (\<Sum>i\<le>n. of_nat (stirling n i) * (x ^ Suc i))"
     by (subst setsum_atMost_Suc_shift) (simp add: setsum.distrib ring_distribs)
   also have "\<dots> = pochhammer x (Suc n)"
     by (subst setsum_atMost_Suc_shift [symmetric])
-       (simp add: algebra_simps setsum.distrib setsum_right_distrib pochhammer_Suc Suc [symmetric])
+      (simp add: algebra_simps setsum.distrib setsum_right_distrib pochhammer_Suc Suc [symmetric])
   finally show ?case .
-qed simp_all
+qed
 
 
 text \<open>A row of the Stirling number triangle\<close>
 
-definition stirling_row :: "nat \<Rightarrow> nat list" where
-  "stirling_row n = [stirling n k. k \<leftarrow> [0..<Suc n]]"
+definition stirling_row :: "nat \<Rightarrow> nat list"
+  where "stirling_row n = [stirling n k. k \<leftarrow> [0..<Suc n]]"
 
 lemma nth_stirling_row: "k \<le> n \<Longrightarrow> stirling_row n ! k = stirling n k"
   by (simp add: stirling_row_def del: upt_Suc)
@@ -200,82 +218,109 @@
 lemma list_ext:
   assumes "length xs = length ys"
   assumes "\<And>i. i < length xs \<Longrightarrow> xs ! i = ys ! i"
-  shows   "xs = ys"
-using assms
+  shows "xs = ys"
+  using assms
 proof (induction rule: list_induct2)
+  case Nil
+  then show ?case by simp
+next
   case (Cons x xs y ys)
-  from Cons.prems[of 0] have "x = y" by simp
-  moreover from Cons.prems[of "Suc i" for i] have "xs = ys" by (intro Cons.IH) simp
+  from Cons.prems[of 0] have "x = y"
+    by simp
+  moreover from Cons.prems[of "Suc i" for i] have "xs = ys"
+    by (intro Cons.IH) simp
   ultimately show ?case by simp
-qed simp_all
+qed
 
 
 subsubsection \<open>Efficient code\<close>
 
 text \<open>
-  Naively using the defining equations of the Stirling numbers of the first kind to
-  compute them leads to exponential run time due to repeated computations.
-  We can use memoisation to compute them row by row without repeating computations, at
-  the cost of computing a few unneeded values.
+  Naively using the defining equations of the Stirling numbers of the first
+  kind to compute them leads to exponential run time due to repeated
+  computations. We can use memoisation to compute them row by row without
+  repeating computations, at the cost of computing a few unneeded values.
 
   As a bonus, this is very efficient for applications where an entire row of
-  Stirling numbers is needed..
+  Stirling numbers is needed.
 \<close>
 
-definition zip_with_prev :: "('a \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'a list \<Rightarrow> 'b list" where
-  "zip_with_prev f x xs = map (\<lambda>(x,y). f x y) (zip (x # xs) xs)"
+definition zip_with_prev :: "('a \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'a list \<Rightarrow> 'b list"
+  where "zip_with_prev f x xs = map (\<lambda>(x,y). f x y) (zip (x # xs) xs)"
 
 lemma zip_with_prev_altdef:
   "zip_with_prev f x xs =
-     (if xs = [] then [] else f x (hd xs) # [f (xs!i) (xs!(i+1)). i \<leftarrow> [0..<length xs - 1]])"
+    (if xs = [] then [] else f x (hd xs) # [f (xs!i) (xs!(i+1)). i \<leftarrow> [0..<length xs - 1]])"
 proof (cases xs)
+  case Nil
+  then show ?thesis
+    by (simp add: zip_with_prev_def)
+next
   case (Cons y ys)
-  hence "zip_with_prev f x xs = f x (hd xs) # zip_with_prev f y ys"
+  then have "zip_with_prev f x xs = f x (hd xs) # zip_with_prev f y ys"
     by (simp add: zip_with_prev_def)
   also have "zip_with_prev f y ys = map (\<lambda>i. f (xs ! i) (xs ! (i + 1))) [0..<length xs - 1]"
     unfolding Cons
-    by (induction ys arbitrary: y)
-       (simp_all add: zip_with_prev_def upt_conv_Cons map_Suc_upt [symmetric] del: upt_Suc)
-  finally show ?thesis using Cons by simp
-qed (simp add: zip_with_prev_def)
+    by (induct ys arbitrary: y)
+      (simp_all add: zip_with_prev_def upt_conv_Cons map_Suc_upt [symmetric] del: upt_Suc)
+  finally show ?thesis
+    using Cons by simp
+qed
 
 
-primrec stirling_row_aux where
-  "stirling_row_aux n y [] = [1]"
-| "stirling_row_aux n y (x#xs) = (y + n * x) # stirling_row_aux n x xs"
+primrec stirling_row_aux
+  where
+    "stirling_row_aux n y [] = [1]"
+  | "stirling_row_aux n y (x#xs) = (y + n * x) # stirling_row_aux n x xs"
 
 lemma stirling_row_aux_correct:
   "stirling_row_aux n y xs = zip_with_prev (\<lambda>a b. a + n * b) y xs @ [1]"
-  by (induction xs arbitrary: y) (simp_all add: zip_with_prev_def)
+  by (induct xs arbitrary: y) (simp_all add: zip_with_prev_def)
 
 lemma stirling_row_code [code]:
   "stirling_row 0 = [1]"
   "stirling_row (Suc n) = stirling_row_aux n 0 (stirling_row n)"
-proof -
+proof goal_cases
+  case 1
+  show ?case by (simp add: stirling_row_def)
+next
+  case 2
   have "stirling_row (Suc n) =
-          0 # [stirling_row n ! i + stirling_row n ! (i+1) * n. i \<leftarrow> [0..<n]] @ [1]"
+    0 # [stirling_row n ! i + stirling_row n ! (i+1) * n. i \<leftarrow> [0..<n]] @ [1]"
   proof (rule list_ext, goal_cases length nth)
     case (nth i)
-    from nth have "i \<le> Suc n" by simp
-    then consider "i = 0" | j where "i > 0" "i \<le> n" | "i = Suc n" by linarith
-    thus ?case
+    from nth have "i \<le> Suc n"
+      by simp
+    then consider "i = 0 \<or> i = Suc n" | "i > 0" "i \<le> n"
+      by linarith
+    then show ?case
     proof cases
-      assume i: "i > 0" "i \<le> n"
-      from this show ?thesis by (cases i) (simp_all add: nth_append nth_stirling_row)
-    qed (simp_all add: nth_stirling_row nth_append)
-  qed simp
+      case 1
+      then show ?thesis
+        by (auto simp: nth_stirling_row nth_append)
+    next
+      case 2
+      then show ?thesis
+        by (cases i) (simp_all add: nth_append nth_stirling_row)
+    qed
+  next
+    case length
+    then show ?case by simp
+  qed
   also have "0 # [stirling_row n ! i + stirling_row n ! (i+1) * n. i \<leftarrow> [0..<n]] @ [1] =
-               zip_with_prev (\<lambda>a b. a + n * b) 0 (stirling_row n) @ [1]"
+      zip_with_prev (\<lambda>a b. a + n * b) 0 (stirling_row n) @ [1]"
     by (cases n) (auto simp add: zip_with_prev_altdef stirling_row_def hd_map simp del: upt_Suc)
   also have "\<dots> = stirling_row_aux n 0 (stirling_row n)"
     by (simp add: stirling_row_aux_correct)
-  finally show "stirling_row (Suc n) = stirling_row_aux n 0 (stirling_row n)" .
-qed (simp add: stirling_row_def)
+  finally show ?case .
+qed
 
 lemma stirling_code [code]:
-  "stirling n k = (if k = 0 then if n = 0 then 1 else 0
-                   else if k > n then 0 else if k = n then 1
-                   else stirling_row n ! k)"
+  "stirling n k =
+    (if k = 0 then (if n = 0 then 1 else 0)
+     else if k > n then 0
+     else if k = n then 1
+     else stirling_row n ! k)"
   by (simp add: nth_stirling_row)
 
 end
diff -r ae33d1c2ab26 -r a7c5074a0251 src/Pure/Isar/keyword.scala
--- a/src/Pure/Isar/keyword.scala	Wed Jul 13 21:00:03 2016 +0200
+++ b/src/Pure/Isar/keyword.scala	Wed Jul 13 21:30:41 2016 +0200
@@ -66,6 +66,8 @@
 
   val theory_body = Set(THY_LOAD, THY_DECL, THY_DECL_BLOCK, THY_GOAL)
 
+  val prf_script = Set(PRF_SCRIPT)
+
   val proof =
     Set(QED, QED_SCRIPT, QED_BLOCK, QED_GLOBAL, PRF_GOAL, PRF_BLOCK, NEXT_BLOCK, PRF_OPEN,
       PRF_CLOSE, PRF_CHAIN, PRF_DECL, PRF_ASM, PRF_ASM_GOAL, PRF_SCRIPT, PRF_SCRIPT_GOAL,
diff -r ae33d1c2ab26 -r a7c5074a0251 src/Pure/Isar/outer_syntax.scala
--- a/src/Pure/Isar/outer_syntax.scala	Wed Jul 13 21:00:03 2016 +0200
+++ b/src/Pure/Isar/outer_syntax.scala	Wed Jul 13 21:30:41 2016 +0200
@@ -174,9 +174,9 @@
             if (keywords.is_command(tok, Keyword.theory_goal)) (2, 1)
             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.PRF_CLOSE == _)) (y, y - 1)
+            else if (keywords.is_command(tok, Set(Keyword.PRF_BLOCK))) (y + 2, y + 1)
+            else if (keywords.is_command(tok, Set(Keyword.QED_BLOCK))) (y - 1, y - 2)
+            else if (keywords.is_command(tok, Set(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 ae33d1c2ab26 -r a7c5074a0251 src/Pure/Isar/proof.ML
--- a/src/Pure/Isar/proof.ML	Wed Jul 13 21:00:03 2016 +0200
+++ b/src/Pure/Isar/proof.ML	Wed Jul 13 21:30:41 2016 +0200
@@ -42,7 +42,6 @@
   val raw_goal: state -> {context: context, facts: thm list, goal: thm}
   val goal: state -> {context: context, facts: thm list, goal: thm}
   val simple_goal: state -> {context: context, goal: thm}
-  val status_markup: state -> Markup.T
   val let_bind: (term list * term) list -> state -> state
   val let_bind_cmd: (string list * string) list -> state -> state
   val write: Syntax.mode -> (term * mixfix) list -> state -> state
@@ -561,11 +560,6 @@
     val (ctxt, (_, goal)) = get_goal (refine_insert using state);
   in {context = ctxt, goal = goal} end;
 
-fun status_markup state =
-  (case try goal state of
-    SOME {goal, ...} => Markup.proof_state (Thm.nprems_of goal)
-  | NONE => Markup.empty);
-
 fun method_error kind pos state =
   Seq.single (Proof_Display.method_error kind pos (raw_goal state));
 
diff -r ae33d1c2ab26 -r a7c5074a0251 src/Pure/Isar/token.scala
--- a/src/Pure/Isar/token.scala	Wed Jul 13 21:00:03 2016 +0200
+++ b/src/Pure/Isar/token.scala	Wed Jul 13 21:30:41 2016 +0200
@@ -263,6 +263,7 @@
 
   def is_begin: Boolean = is_keyword("begin")
   def is_end: Boolean = is_command("end")
+  def is_begin_or_command: Boolean = is_begin || is_command
 
   def content: String =
     if (kind == Token.Kind.STRING) Scan.Parsers.quoted_content("\"", source)
diff -r ae33d1c2ab26 -r a7c5074a0251 src/Pure/PIDE/command.ML
--- a/src/Pure/PIDE/command.ML	Wed Jul 13 21:00:03 2016 +0200
+++ b/src/Pure/PIDE/command.ML	Wed Jul 13 21:30:41 2016 +0200
@@ -209,17 +209,28 @@
 fun status tr m =
   Toplevel.setmp_thread_position tr (fn () => Output.status (Markup.markup_only m)) ();
 
-fun proof_status tr st =
+fun command_indent tr st =
   (case try Toplevel.proof_of st of
-    SOME prf => status tr (Proof.status_markup prf)
+    SOME prf =>
+      let val keywords = Thy_Header.get_keywords (Proof.theory_of prf) in
+        if Keyword.command_kind keywords (Toplevel.name_of tr) = SOME Keyword.prf_script then
+          (case try Proof.goal prf of
+            SOME {goal, ...} =>
+              let val n = Thm.nprems_of goal
+              in if n > 1 then report tr (Markup.command_indent (n - 1)) else () end
+          | NONE => ())
+        else ()
+      end
   | NONE => ());
 
+
 fun eval_state keywords span tr ({state, ...}: eval_state) =
   let
     val _ = Thread_Attributes.expose_interrupt ();
 
     val st = reset_state keywords tr state;
 
+    val _ = command_indent tr st;
     val _ = status tr Markup.running;
     val (errs1, result) = run keywords true tr st;
     val errs2 = (case result of NONE => [] | SOME st' => check_cmts span tr st');
@@ -235,7 +246,6 @@
         in {failed = true, command = tr, state = st} end
     | SOME st' =>
         let
-          val _ = proof_status tr st';
           val _ = status tr Markup.finished;
         in {failed = false, command = tr, state = st'} end)
   end;
diff -r ae33d1c2ab26 -r a7c5074a0251 src/Pure/PIDE/markup.ML
--- a/src/Pure/PIDE/markup.ML	Wed Jul 13 21:00:03 2016 +0200
+++ b/src/Pure/PIDE/markup.ML	Wed Jul 13 21:30:41 2016 +0200
@@ -155,8 +155,7 @@
   val parse_command_timing_properties:
     Properties.T -> ({file: string, offset: int, name: string} * Time.time) option
   val timingN: string val timing: {elapsed: Time.time, cpu: Time.time, gc: Time.time} -> T
-  val subgoalsN: string
-  val proof_stateN: string val proof_state: int -> T
+  val command_indentN: string val command_indent: int -> T
   val goalN: string val goal: T
   val subgoalN: string val subgoal: string -> T
   val taskN: string
@@ -576,10 +575,12 @@
   | _ => NONE);
 
 
-(* toplevel *)
+(* indentation *)
 
-val subgoalsN = "subgoals";
-val (proof_stateN, proof_state) = markup_int "proof_state" subgoalsN;
+val (command_indentN, command_indent) = markup_int "command_indent" indentN;
+
+
+(* goals *)
 
 val (goalN, goal) = markup_elem "goal";
 val (subgoalN, subgoal) = markup_string "subgoal" nameN;
diff -r ae33d1c2ab26 -r a7c5074a0251 src/Pure/PIDE/markup.scala
--- a/src/Pure/PIDE/markup.scala	Wed Jul 13 21:00:03 2016 +0200
+++ b/src/Pure/PIDE/markup.scala	Wed Jul 13 21:30:41 2016 +0200
@@ -372,10 +372,17 @@
   val COMMAND_TIMING = "command_timing"
 
 
-  /* toplevel */
+  /* command indentation */
 
-  val SUBGOALS = "subgoals"
-  val PROOF_STATE = "proof_state"
+  object Command_Indent
+  {
+    val name = "command_indent"
+    def unapply(markup: Markup): Option[Int] =
+      if (markup.name == name) Indent.unapply(markup.properties) else None
+  }
+
+
+  /* goals */
 
   val GOAL = "goal"
   val SUBGOAL = "subgoal"
diff -r ae33d1c2ab26 -r a7c5074a0251 src/Tools/jEdit/etc/options
--- a/src/Tools/jEdit/etc/options	Wed Jul 13 21:00:03 2016 +0200
+++ b/src/Tools/jEdit/etc/options	Wed Jul 13 21:30:41 2016 +0200
@@ -45,6 +45,12 @@
 public option jedit_indent_newline : bool = true
   -- "indentation of Isabelle keywords on ENTER (action isabelle.newline)"
 
+public option jedit_indent_script : bool = true
+  -- "indent unstructured proof script ('apply' etc.) via number of subgoals"
+
+public option jedit_indent_script_limit : int = 20
+  -- "maximum indentation of unstructured proof script ('apply' etc.)"
+
 
 section "Completion"
 
diff -r ae33d1c2ab26 -r a7c5074a0251 src/Tools/jEdit/src/isabelle.scala
--- a/src/Tools/jEdit/src/isabelle.scala	Wed Jul 13 21:00:03 2016 +0200
+++ b/src/Tools/jEdit/src/isabelle.scala	Wed Jul 13 21:30:41 2016 +0200
@@ -265,6 +265,8 @@
       {
         Isabelle.buffer_syntax(buffer) match {
           case Some(syntax) if buffer.isInstanceOf[Buffer] =>
+            val keywords = syntax.keywords
+
             val caret = text_area.getCaretPosition
             val line = text_area.getCaretLine
             val line_range = JEdit_Lib.line_range(buffer, line)
@@ -282,9 +284,12 @@
             val (tokens1, context1) = line_content(line_range.start, caret, context0)
             val (tokens2, _) = line_content(caret, line_range.stop, context1)
 
-            if (tokens1.nonEmpty && tokens1.head.is_command) buffer.indentLine(line, true)
+            if (tokens1.nonEmpty &&
+                (tokens1.head.is_begin_or_command || keywords.is_quasi_command(tokens1.head)))
+              buffer.indentLine(line, true)
 
-            if (tokens2.isEmpty || tokens2.head.is_command)
+            if (tokens2.isEmpty ||
+                tokens2.head.is_begin_or_command || keywords.is_quasi_command(tokens2.head))
               JEdit_Lib.buffer_edit(buffer) {
                 text_area.setSelectedText("\n")
                 if (!buffer.indentLine(line + 1, true)) text_area.goToStartOfWhiteSpace(false)
diff -r ae33d1c2ab26 -r a7c5074a0251 src/Tools/jEdit/src/rendering.scala
--- a/src/Tools/jEdit/src/rendering.scala	Wed Jul 13 21:00:03 2016 +0200
+++ b/src/Tools/jEdit/src/rendering.scala	Wed Jul 13 21:30:41 2016 +0200
@@ -137,6 +137,9 @@
 
   /* markup elements */
 
+  private val indentation_elements =
+    Markup.Elements(Markup.Command_Indent.name)
+
   private val semantic_completion_elements =
     Markup.Elements(Markup.COMPLETION, Markup.NO_COMPLETION)
 
@@ -295,6 +298,16 @@
   val markdown_item_color4 = color_value("markdown_item_color4")
 
 
+  /* indentation */
+
+  def indentation(range: Text.Range): Int =
+    snapshot.select(range, Rendering.indentation_elements, _ =>
+      {
+        case Text.Info(_, XML.Elem(Markup.Command_Indent(i), _)) => Some(i)
+        case _ => None
+      }).headOption.map(_.info).getOrElse(0)
+
+
   /* completion */
 
   def semantic_completion(completed_range: Option[Text.Range], range: Text.Range)
diff -r ae33d1c2ab26 -r a7c5074a0251 src/Tools/jEdit/src/text_structure.scala
--- a/src/Tools/jEdit/src/text_structure.scala	Wed Jul 13 21:00:03 2016 +0200
+++ b/src/Tools/jEdit/src/text_structure.scala	Wed Jul 13 21:30:41 2016 +0200
@@ -26,13 +26,13 @@
     def iterator(line: Int, lim: Int = limit): Iterator[Text.Info[Token]] =
     {
       val it = Token_Markup.line_token_iterator(syntax, buffer, line, line + lim)
-      if (comments) it.filterNot(_.info.is_space) else it.filter(_.info.is_proper)
+      if (comments) it.filterNot(_.info.is_space) else it.filterNot(_.info.is_improper)
     }
 
     def reverse_iterator(line: Int, lim: Int = limit): Iterator[Text.Info[Token]] =
     {
       val it = Token_Markup.line_token_reverse_iterator(syntax, buffer, line, line - lim)
-      if (comments) it.filterNot(_.info.is_space) else it.filter(_.info.is_proper)
+      if (comments) it.filterNot(_.info.is_space) else it.filterNot(_.info.is_improper)
     }
   }
 
@@ -52,31 +52,52 @@
           val keywords = syntax.keywords
           val nav = new Navigator(syntax, buffer.asInstanceOf[Buffer], true)
 
-          def head_token(line: Int): Option[Token] =
-            nav.iterator(line, 1).toStream.headOption.map(_.info)
-
-          def head_is_quasi_command(line: Int): Boolean =
-            head_token(line) match {
-              case None => false
-              case Some(tok) => keywords.is_quasi_command(tok)
-            }
-
-          def prev_command: Option[Token] =
-            nav.reverse_iterator(prev_line, 1).
-              collectFirst({ case Text.Info(_, tok) if tok.is_command => tok })
-
-          def prev_span: Iterator[Token] =
-            nav.reverse_iterator(prev_line).map(_.info).takeWhile(tok => !tok.is_command)
-
-          def prev_line_span: Iterator[Token] =
-            nav.reverse_iterator(prev_line, 1).map(_.info).takeWhile(tok => !tok.is_command)
+          val indent_size = buffer.getIndentSize
 
 
           def line_indent(line: Int): Int =
             if (line < 0 || line >= buffer.getLineCount) 0
             else buffer.getCurrentIndentForLine(line, null)
 
-          val indent_size = buffer.getIndentSize
+          def line_head(line: Int): Option[Text.Info[Token]] =
+            nav.iterator(line, 1).toStream.headOption
+
+          def head_is_quasi_command(line: Int): Boolean =
+            line_head(line) match {
+              case None => false
+              case Some(Text.Info(_, tok)) => keywords.is_quasi_command(tok)
+            }
+
+          def prev_line_command: Option[Token] =
+            nav.reverse_iterator(prev_line, 1).
+              collectFirst({ case Text.Info(_, tok) if tok.is_begin_or_command => tok })
+
+          def prev_line_span: Iterator[Token] =
+            nav.reverse_iterator(prev_line, 1).map(_.info).takeWhile(tok => !tok.is_begin_or_command)
+
+          def prev_span: Iterator[Token] =
+            nav.reverse_iterator(prev_line).map(_.info).takeWhile(tok => !tok.is_begin_or_command)
+
+
+          val script_indent: Text.Info[Token] => Int =
+          {
+            val opt_rendering: Option[Rendering] =
+              if (PIDE.options.value.bool("jedit_indent_script"))
+                GUI_Thread.now {
+                  (for {
+                    text_area <- JEdit_Lib.jedit_text_areas(buffer)
+                    doc_view <- PIDE.document_view(text_area)
+                  } yield doc_view.get_rendering).toStream.headOption
+                }
+              else None
+            val limit = PIDE.options.value.int("jedit_indent_script_limit")
+            (info: Text.Info[Token]) =>
+              opt_rendering match {
+                case Some(rendering) if keywords.is_command(info.info, Keyword.prf_script) =>
+                  (rendering.indentation(info.range) min limit) max 0
+                case _ => 0
+              }
+          }
 
           def indent_indent(tok: Token): Int =
             if (keywords.is_command(tok, keyword_open)) indent_size
@@ -84,9 +105,24 @@
             else 0
 
           def indent_offset(tok: Token): Int =
-            if (keywords.is_command(tok, Keyword.proof_enclose) || tok.is_begin) indent_size
+            if (keywords.is_command(tok, Keyword.proof_enclose)) indent_size
             else 0
 
+          def indent_structure: Int =
+            nav.reverse_iterator(current_line - 1).scanLeft((0, false))(
+              { case ((ind, _), Text.Info(range, tok)) =>
+                  val ind1 = ind + indent_indent(tok)
+                  if (tok.is_begin_or_command && !keywords.is_command(tok, Keyword.prf_script)) {
+                    val line = buffer.getLineOfOffset(range.start)
+                    line_head(line) match {
+                      case Some(info) if info.info == tok =>
+                        (ind1 + indent_offset(tok) + line_indent(line), true)
+                      case _ => (ind1, false)
+                    }
+                  }
+                  else (ind1, false)
+              }).collectFirst({ case (i, true) => i }).getOrElse(0)
+
           def indent_brackets: Int =
             (0 /: prev_line_span)(
               { case (i, tok) =>
@@ -98,35 +134,20 @@
             if (prev_span.exists(keywords.is_quasi_command(_))) indent_size
             else 0
 
-          def indent_structure: Int =
-            nav.reverse_iterator(current_line - 1).scanLeft((0, false))(
-              { case ((ind, _), Text.Info(range, tok)) =>
-                  val ind1 = ind + indent_indent(tok)
-                  if (tok.is_command) {
-                    val line = buffer.getLineOfOffset(range.start)
-                    if (head_token(line) == Some(tok))
-                      (ind1 + indent_offset(tok) + line_indent(line), true)
-                    else (ind1, false)
-                  }
-                  else (ind1, false)
-              }).collectFirst({ case (i, true) => i }).getOrElse(0)
-
-          def nesting(it: Iterator[Token], open: Token => Boolean, close: Token => Boolean): Int =
-            (0 /: it)({ case (d, tok) => if (open(tok)) d + 1 else if (close(tok)) d - 1 else d })
-
-          def indent_begin: Int =
-            (nesting(nav.iterator(current_line - 1, 1).map(_.info), _.is_begin, _.is_end) max 0) *
-              indent_size
-
           val indent =
-            head_token(current_line) match {
+            line_head(current_line) match {
               case None => indent_structure + indent_brackets + indent_extra
-              case Some(tok) =>
-                if (keywords.is_before_command(tok) ||
-                    keywords.is_command(tok, Keyword.theory)) indent_begin
-                else if (tok.is_command) indent_structure + indent_begin - indent_offset(tok)
+              case Some(info @ Text.Info(range, tok)) =>
+                if (tok.is_begin ||
+                    keywords.is_before_command(tok) ||
+                    keywords.is_command(tok, Keyword.theory)) 0
+                else if (keywords.is_command(tok, Keyword.proof_enclose))
+                  indent_structure + script_indent(info) - indent_offset(tok)
+                else if (keywords.is_command(tok, Keyword.proof))
+                  (indent_structure + script_indent(info) - indent_offset(tok)) max indent_size
+                else if (tok.is_command) indent_structure - indent_offset(tok)
                 else {
-                  prev_command match {
+                  prev_line_command match {
                     case None =>
                       val extra =
                         (keywords.is_quasi_command(tok), head_is_quasi_command(prev_line)) match {
@@ -134,10 +155,10 @@
                           case (true, false) => - indent_extra
                           case (false, true) => indent_extra
                         }
-                      line_indent(prev_line) - indent_offset(tok) + indent_brackets + extra
+                      line_indent(prev_line) + indent_brackets + extra - indent_offset(tok)
                     case Some(prev_tok) =>
-                      indent_structure - indent_offset(tok) - indent_offset(prev_tok) +
-                      indent_brackets - indent_indent(prev_tok) + indent_size
+                      indent_structure + indent_brackets + indent_size - indent_offset(tok) -
+                      indent_offset(prev_tok) - indent_indent(prev_tok)
                   }
                }
             }