# HG changeset patch
# User wenzelm
# Date 1468412895 -7200
# Node ID 151bb79536a75eafe5c00dbb6c5246b03a888cd6
# Parent  f3781c5fb03f445926406000d61ccca7cdb51d49
misc tuning and modernization;

diff -r f3781c5fb03f -r 151bb79536a7 src/HOL/Library/BigO.thy
--- a/src/HOL/Library/BigO.thy	Tue Jul 12 22:54:37 2016 +0200
+++ b/src/HOL/Library/BigO.thy	Wed Jul 13 14:28:15 2016 +0200
@@ -9,29 +9,28 @@
 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 have been replaced by more general lemmas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 have been replaced by more general lemmas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 +41,16 @@
     (\<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 +61,19 @@
   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 +92,15 @@
   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 +111,73 @@
   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 +196,7 @@
 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 +217,8 @@
 
 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 +228,13 @@
   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 +243,23 @@
     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 +279,11 @@
   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 +293,14 @@
   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 +308,7 @@
   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 +338,25 @@
   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 +373,63 @@
 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 +440,24 @@
   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 +473,33 @@
   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 +512,19 @@
   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 +534,9 @@
 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 +551,21 @@
   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 +573,7 @@
     \<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 +587,8 @@
 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 +596,7 @@
   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 +619,13 @@
         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 +638,9 @@
   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 +649,24 @@
 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 +676,10 @@
   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 +688,10 @@
        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 +707,7 @@
   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 +715,7 @@
   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 +723,7 @@
   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 +731,72 @@
   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 +809,7 @@
   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 +817,41 @@
 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 f3781c5fb03f -r 151bb79536a7 src/HOL/Library/Set_Algebras.thy
--- a/src/HOL/Library/Set_Algebras.thy	Tue Jul 12 22:54:37 2016 +0200
+++ b/src/HOL/Library/Set_Algebras.thy	Wed Jul 13 14:28:15 2016 +0200
@@ -1,5 +1,7 @@
 (*  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>
@@ -11,14 +13,14 @@
 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>.
+  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"