# HG changeset patch
# User wenzelm
# Date 1393675546 -3600
# Node ID 44055f07cbd8831d4b4d9704209834dacaeac2c5
# Parent  61869776ce1fc5a29c1bce5bea7459e9c6d119b9
more symbols, less parentheses;

diff -r 61869776ce1f -r 44055f07cbd8 src/HOL/Library/BigO.thy
--- a/src/HOL/Library/BigO.thy	Sat Mar 01 12:07:26 2014 +0100
+++ b/src/HOL/Library/BigO.thy	Sat Mar 01 13:05:46 2014 +0100
@@ -19,7 +19,7 @@
 \item We have eliminated the @{text O} operator on sets. (Most uses of this seem
   to be inessential.)
 \item We no longer use @{text "+"} as output syntax for @{text "+o"}
-\item Lemmas involving @{text "sumr"} have been replaced by more general lemmas 
+\item Lemmas involving @{text "sumr"} have been replaced by more general lemmas
   involving `@{text "setsum"}.
 \item The library has been expanded, with e.g.~support for expressions of
   the form @{text "f < g + O(h)"}.
@@ -34,14 +34,12 @@
 
 subsection {* Definitions *}
 
-definition
-  bigo :: "('a => 'b::linordered_idom) => ('a => 'b) set"  ("(1O'(_'))") where
-  "O(f::('a => 'b)) =
-      {h. EX c. ALL x. abs (h x) <= c * abs (f x)}"
+definition bigo :: "('a \<Rightarrow> 'b::linordered_idom) \<Rightarrow> ('a \<Rightarrow> 'b) set"  ("(1O'(_'))")
+  where "O(f:: 'a \<Rightarrow> 'b) = {h. \<exists>c. \<forall>x. abs (h x) \<le> c * abs (f x)}"
 
-lemma bigo_pos_const: "(EX (c::'a::linordered_idom). 
-    ALL x. (abs (h x)) <= (c * (abs (f x))))
-      = (EX c. 0 < c & (ALL x. (abs(h x)) <= (c * (abs (f x)))))"
+lemma bigo_pos_const:
+  "(\<exists>c::'a::linordered_idom. \<forall>x. abs (h x) \<le> c * abs (f x)) \<longleftrightarrow>
+    (\<exists>c. 0 < c \<and> (\<forall>x. abs (h x) \<le> c * abs (f x)))"
   apply auto
   apply (case_tac "c = 0")
   apply simp
@@ -49,7 +47,7 @@
   apply simp
   apply (rule_tac x = "abs c" in exI)
   apply auto
-  apply (subgoal_tac "c * abs(f x) <= abs c * abs (f x)")
+  apply (subgoal_tac "c * abs (f x) \<le> abs c * abs (f x)")
   apply (erule_tac x = x in allE)
   apply force
   apply (rule mult_right_mono)
@@ -57,42 +55,40 @@
   apply (rule abs_ge_zero)
   done
 
-lemma bigo_alt_def: "O(f) = 
-    {h. EX c. (0 < c & (ALL x. abs (h x) <= c * abs (f x)))}"
+lemma bigo_alt_def: "O(f) = {h. \<exists>c. 0 < c \<and> (\<forall>x. abs (h x) \<le> c * abs (f x))}"
   by (auto simp add: bigo_def bigo_pos_const)
 
-lemma bigo_elt_subset [intro]: "f : O(g) ==> O(f) <= O(g)"
+lemma bigo_elt_subset [intro]: "f \<in> O(g) \<Longrightarrow> O(f) \<le> O(g)"
   apply (auto simp add: bigo_alt_def)
   apply (rule_tac x = "ca * c" in exI)
   apply (rule conjI)
   apply (rule mult_pos_pos)
-  apply (assumption)+
+  apply assumption+
   apply (rule allI)
   apply (drule_tac x = "xa" in spec)+
-  apply (subgoal_tac "ca * abs(f xa) <= ca * (c * abs(g xa))")
+  apply (subgoal_tac "ca * abs (f xa) \<le> ca * (c * abs (g xa))")
   apply (erule order_trans)
   apply (simp add: mult_ac)
   apply (rule mult_left_mono, assumption)
   apply (rule order_less_imp_le, assumption)
   done
 
-lemma bigo_refl [intro]: "f : O(f)"
+lemma bigo_refl [intro]: "f \<in> O(f)"
   apply(auto simp add: bigo_def)
   apply(rule_tac x = 1 in exI)
   apply simp
   done
 
-lemma bigo_zero: "0 : O(g)"
+lemma bigo_zero: "0 \<in> O(g)"
   apply (auto simp add: bigo_def func_zero)
   apply (rule_tac x = 0 in exI)
   apply auto
   done
 
-lemma bigo_zero2: "O(%x.0) = {%x.0}"
-  by (auto simp add: bigo_def) 
+lemma bigo_zero2: "O(\<lambda>x. 0) = {\<lambda>x. 0}"
+  by (auto simp add: bigo_def)
 
-lemma bigo_plus_self_subset [intro]: 
-  "O(f) + O(f) <= O(f)"
+lemma bigo_plus_self_subset [intro]: "O(f) + O(f) \<subseteq> O(f)"
   apply (auto simp add: bigo_alt_def set_plus_def)
   apply (rule_tac x = "c + ca" in exI)
   apply auto
@@ -102,30 +98,29 @@
   apply (rule add_mono)
   apply force
   apply force
-done
+  done
 
 lemma bigo_plus_idemp [simp]: "O(f) + O(f) = O(f)"
   apply (rule equalityI)
   apply (rule bigo_plus_self_subset)
-  apply (rule set_zero_plus2) 
+  apply (rule set_zero_plus2)
   apply (rule bigo_zero)
   done
 
-lemma bigo_plus_subset [intro]: "O(f + g) <= O(f) + O(g)"
+lemma bigo_plus_subset [intro]: "O(f + g) \<subseteq> O(f) + O(g)"
   apply (rule subsetI)
   apply (auto simp add: bigo_def bigo_pos_const func_plus set_plus_def)
   apply (subst bigo_pos_const [symmetric])+
-  apply (rule_tac x = 
-    "%n. if abs (g n) <= (abs (f n)) then x n else 0" in exI)
+  apply (rule_tac x = "\<lambda>n. if abs (g n) \<le> (abs (f n)) then x n else 0" in exI)
   apply (rule conjI)
   apply (rule_tac x = "c + c" in exI)
   apply (clarsimp)
   apply (auto)
-  apply (subgoal_tac "c * abs (f xa + g xa) <= (c + c) * abs (f xa)")
+  apply (subgoal_tac "c * abs (f xa + g xa) \<le> (c + c) * abs (f xa)")
   apply (erule_tac x = xa in allE)
   apply (erule order_trans)
   apply (simp)
-  apply (subgoal_tac "c * abs (f xa + g xa) <= c * (abs (f xa) + abs (g xa))")
+  apply (subgoal_tac "c * abs (f xa + g xa) \<le> c * (abs (f xa) + abs (g xa))")
   apply (erule order_trans)
   apply (simp add: ring_distribs)
   apply (rule mult_left_mono)
@@ -133,16 +128,15 @@
   apply (simp add: order_less_le)
   apply (rule mult_nonneg_nonneg)
   apply auto
-  apply (rule_tac x = "%n. if (abs (f n)) <  abs (g n) then x n else 0" 
-     in exI)
+  apply (rule_tac x = "\<lambda>n. if (abs (f n)) < abs (g n) then x n else 0" in exI)
   apply (rule conjI)
   apply (rule_tac x = "c + c" in exI)
   apply auto
-  apply (subgoal_tac "c * abs (f xa + g xa) <= (c + c) * abs (g xa)")
+  apply (subgoal_tac "c * abs (f xa + g xa) \<le> (c + c) * abs (g xa)")
   apply (erule_tac x = xa in allE)
   apply (erule order_trans)
-  apply (simp)
-  apply (subgoal_tac "c * abs (f xa + g xa) <= c * (abs (f xa) + abs (g xa))")
+  apply simp
+  apply (subgoal_tac "c * abs (f xa + g xa) \<le> c * (abs (f xa) + abs (g xa))")
   apply (erule order_trans)
   apply (simp add: ring_distribs)
   apply (rule mult_left_mono)
@@ -153,41 +147,39 @@
   apply simp
   done
 
-lemma bigo_plus_subset2 [intro]: "A <= O(f) ==> B <= O(f) ==> A + B <= O(f)"
-  apply (subgoal_tac "A + B <= O(f) + O(f)")
+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 (auto del: subsetI simp del: bigo_plus_idemp)
   done
 
-lemma bigo_plus_eq: "ALL x. 0 <= f x ==> ALL x. 0 <= g x ==> 
-    O(f + g) = O(f) + O(g)"
+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 (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 <= max c ca")
+  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 <= f xa + g xa")
+  apply (subgoal_tac "0 \<le> f xa + g xa")
   apply (simp add: ring_distribs)
-  apply (subgoal_tac "abs(a xa + b xa) <= abs(a xa) + abs(b xa)")
-  apply (subgoal_tac "abs(a xa) + abs(b xa) <= 
-      max c ca * f xa + max c ca * g xa")
-  apply (force)
+  apply (subgoal_tac "abs (a xa + b xa) \<le> abs (a xa) + abs (b xa)")
+  apply (subgoal_tac "abs (a xa) + abs (b xa) \<le> max c ca * f xa + max c ca * g xa")
+  apply force
   apply (rule add_mono)
-  apply (subgoal_tac "c * f xa <= max c ca * f xa")
-  apply (force)
+  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 <= max c ca * g xa")
-  apply (force)
+  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
@@ -196,8 +188,7 @@
   apply assumption+
   done
 
-lemma bigo_bounded_alt: "ALL x. 0 <= f x ==> ALL x. f x <= c * g x ==> 
-    f : O(g)" 
+lemma bigo_bounded_alt: "\<forall>x. 0 \<le> f x \<Longrightarrow> \<forall>x. f x \<le> c * g x \<Longrightarrow> f \<in> O(g)"
   apply (auto simp add: bigo_def)
   apply (rule_tac x = "abs c" in exI)
   apply auto
@@ -205,14 +196,12 @@
   apply (simp add: abs_mult [symmetric])
   done
 
-lemma bigo_bounded: "ALL x. 0 <= f x ==> ALL x. f x <= g x ==> 
-    f : O(g)" 
+lemma bigo_bounded: "\<forall>x. 0 \<le> f x \<Longrightarrow> \<forall>x. f x \<le> g x \<Longrightarrow> f \<in> O(g)"
   apply (erule bigo_bounded_alt [of f 1 g])
   apply simp
   done
 
-lemma bigo_bounded2: "ALL x. lb x <= f x ==> ALL x. f x <= lb x + g x ==>
-    f : lb +o O(g)"
+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)
@@ -222,21 +211,21 @@
   apply force
   done
 
-lemma bigo_abs: "(%x. abs(f x)) =o O(f)" 
+lemma bigo_abs: "(\<lambda>x. abs (f x)) =o O(f)"
   apply (unfold bigo_def)
   apply auto
   apply (rule_tac x = 1 in exI)
   apply auto
   done
 
-lemma bigo_abs2: "f =o O(%x. abs(f x))"
+lemma bigo_abs2: "f =o O(\<lambda>x. abs (f x))"
   apply (unfold bigo_def)
   apply auto
   apply (rule_tac x = 1 in exI)
   apply auto
   done
 
-lemma bigo_abs3: "O(f) = O(%x. abs(f x))"
+lemma bigo_abs3: "O(f) = O(\<lambda>x. abs (f x))"
   apply (rule equalityI)
   apply (rule bigo_elt_subset)
   apply (rule bigo_abs2)
@@ -244,65 +233,63 @@
   apply (rule bigo_abs)
   done
 
-lemma bigo_abs4: "f =o g +o O(h) ==> 
-    (%x. abs (f x)) =o (%x. abs (g x)) +o O(h)"
+lemma bigo_abs4: "f =o g +o O(h) \<Longrightarrow> (\<lambda>x. abs (f x)) =o (\<lambda>x. abs (g x)) +o O(h)"
   apply (drule set_plus_imp_minus)
   apply (rule set_minus_imp_plus)
   apply (subst fun_diff_def)
 proof -
-  assume a: "f - g : O(h)"
-  have "(%x. abs (f x) - abs (g x)) =o O(%x. abs(abs (f x) - abs (g x)))"
+  assume a: "f - g \<in> O(h)"
+  have "(\<lambda>x. abs (f x) - abs (g x)) =o O(\<lambda>x. abs (abs (f x) - abs (g x)))"
     by (rule bigo_abs2)
-  also have "... <= O(%x. abs (f x - g x))"
+  also have "\<dots> \<subseteq> O(\<lambda>x. abs (f x - g x))"
     apply (rule bigo_elt_subset)
     apply (rule bigo_bounded)
     apply force
     apply (rule allI)
     apply (rule abs_triangle_ineq3)
     done
-  also have "... <= O(f - g)"
+  also have "\<dots> \<subseteq> O(f - g)"
     apply (rule bigo_elt_subset)
     apply (subst fun_diff_def)
     apply (rule bigo_abs)
     done
-  also from a have "... <= O(h)"
+  also from a have "\<dots> \<subseteq> O(h)"
     by (rule bigo_elt_subset)
-  finally show "(%x. abs (f x) - abs (g x)) : O(h)".
+  finally show "(\<lambda>x. abs (f x) - abs (g x)) \<in> O(h)".
 qed
 
-lemma bigo_abs5: "f =o O(g) ==> (%x. abs(f x)) =o O(g)" 
+lemma bigo_abs5: "f =o O(g) \<Longrightarrow> (\<lambda>x. abs (f x)) =o O(g)"
   by (unfold bigo_def, auto)
 
-lemma bigo_elt_subset2 [intro]: "f : g +o O(h) ==> O(f) <= O(g) + O(h)"
+lemma bigo_elt_subset2 [intro]: "f \<in> g +o O(h) \<Longrightarrow> O(f) \<subseteq> O(g) + O(h)"
 proof -
-  assume "f : g +o O(h)"
-  also have "... <= O(g) + O(h)"
+  assume "f \<in> g +o O(h)"
+  also have "\<dots> \<subseteq> O(g) + O(h)"
     by (auto del: subsetI)
-  also have "... = O(%x. abs(g x)) + O(%x. abs(h x))"
+  also have "\<dots> = O(\<lambda>x. abs (g x)) + O(\<lambda>x. abs (h x))"
     apply (subst bigo_abs3 [symmetric])+
     apply (rule refl)
     done
-  also have "... = O((%x. abs(g x)) + (%x. abs(h x)))"
-    by (rule bigo_plus_eq [symmetric], auto)
-  finally have "f : ...".
-  then have "O(f) <= ..."
+  also have "\<dots> = O((\<lambda>x. abs (g x)) + (\<lambda>x. abs (h x)))"
+    by (rule bigo_plus_eq [symmetric]) auto
+  finally have "f \<in> \<dots>" .
+  then have "O(f) \<subseteq> \<dots>"
     by (elim bigo_elt_subset)
-  also have "... = O(%x. abs(g x)) + O(%x. abs(h x))"
+  also have "\<dots> = O(\<lambda>x. abs (g x)) + O(\<lambda>x. abs (h x))"
     by (rule bigo_plus_eq, auto)
   finally show ?thesis
     by (simp add: bigo_abs3 [symmetric])
 qed
 
-lemma bigo_mult [intro]: "O(f)*O(g) <= O(f * g)"
+lemma bigo_mult [intro]: "O(f)*O(g) \<subseteq> O(f * g)"
   apply (rule subsetI)
   apply (subst bigo_def)
   apply (auto simp add: bigo_alt_def set_times_def func_times)
   apply (rule_tac x = "c * ca" in exI)
-  apply(rule allI)
-  apply(erule_tac x = x in allE)+
-  apply(subgoal_tac "c * ca * abs(f x * g x) = 
-      (c * abs(f x)) * (ca * abs(g x))")
-  apply(erule ssubst)
+  apply (rule allI)
+  apply (erule_tac x = x in allE)+
+  apply (subgoal_tac "c * ca * abs (f x * g x) = (c * abs (f x)) * (ca * abs (g x))")
+  apply (erule ssubst)
   apply (subst abs_mult)
   apply (rule mult_mono)
   apply assumption+
@@ -311,24 +298,24 @@
   apply (simp add: mult_ac abs_mult)
   done
 
-lemma bigo_mult2 [intro]: "f *o O(g) <= O(f * g)"
+lemma bigo_mult2 [intro]: "f *o O(g) \<subseteq> O(f * g)"
   apply (auto simp add: bigo_def elt_set_times_def func_times abs_mult)
   apply (rule_tac x = c in exI)
   apply auto
   apply (drule_tac x = x in spec)
-  apply (subgoal_tac "abs(f x) * abs(b x) <= abs(f x) * (c * abs(g x))")
+  apply (subgoal_tac "abs (f x) * abs (b x) \<le> abs (f x) * (c * abs (g x))")
   apply (force simp add: mult_ac)
   apply (rule mult_left_mono, assumption)
   apply (rule abs_ge_zero)
   done
 
-lemma bigo_mult3: "f : O(h) ==> g : O(j) ==> f * g : O(h * j)"
+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 (erule set_times_intro, assumption)
   done
 
-lemma bigo_mult4 [intro]:"f : k +o O(h) ==> g * f : (g * k) +o O(g * h)"
+lemma bigo_mult4 [intro]: "f \<in> k +o O(h) \<Longrightarrow> g * f \<in> (g * k) +o O(g * h)"
   apply (drule set_plus_imp_minus)
   apply (rule set_minus_imp_plus)
   apply (drule bigo_mult3 [where g = g and j = g])
@@ -336,110 +323,117 @@
   done
 
 lemma bigo_mult5:
-  assumes "ALL x. f x ~= 0"
-  shows "O(f * g) <= (f::'a => ('b::linordered_field)) *o O(g)"
+  fixes f :: "'a \<Rightarrow> 'b::linordered_field"
+  assumes "\<forall>x. f x \<noteq> 0"
+  shows "O(f * g) \<subseteq> f *o O(g)"
 proof
   fix h
-  assume "h : O(f * g)"
-  then have "(%x. 1 / (f x)) * h : (%x. 1 / f x) *o O(f * g)"
+  assume "h \<in> O(f * g)"
+  then have "(\<lambda>x. 1 / (f x)) * h \<in> (\<lambda>x. 1 / f x) *o O(f * g)"
     by auto
-  also have "... <= O((%x. 1 / f x) * (f * g))"
+  also have "\<dots> \<subseteq> O((\<lambda>x. 1 / f x) * (f * g))"
     by (rule bigo_mult2)
-  also have "(%x. 1 / f x) * (f * g) = g"
-    apply (simp add: func_times) 
+  also have "(\<lambda>x. 1 / f x) * (f * g) = g"
+    apply (simp add: func_times)
     apply (rule ext)
     apply (simp add: assms nonzero_divide_eq_eq mult_ac)
     done
-  finally have "(%x. (1::'b) / f x) * h : O(g)" .
-  then have "f * ((%x. (1::'b) / f x) * h) : f *o O(g)"
+  finally have "(\<lambda>x. (1::'b) / f x) * h \<in> O(g)" .
+  then have "f * ((\<lambda>x. (1::'b) / f x) * h) \<in> f *o O(g)"
     by auto
-  also have "f * ((%x. (1::'b) / f x) * h) = h"
-    apply (simp add: func_times) 
+  also have "f * ((\<lambda>x. (1::'b) / f x) * h) = h"
+    apply (simp add: func_times)
     apply (rule ext)
     apply (simp add: assms nonzero_divide_eq_eq mult_ac)
     done
-  finally show "h : f *o O(g)" .
+  finally show "h \<in> f *o O(g)" .
 qed
 
-lemma bigo_mult6: "ALL x. f x ~= 0 ==>
-    O(f * g) = (f::'a => ('b::linordered_field)) *o O(g)"
+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)"
   apply (rule equalityI)
   apply (erule bigo_mult5)
   apply (rule bigo_mult2)
   done
 
-lemma bigo_mult7: "ALL x. f x ~= 0 ==>
-    O(f * g) <= O(f::'a => ('b::linordered_field)) * O(g)"
+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)"
   apply (subst bigo_mult6)
   apply assumption
   apply (rule set_times_mono3)
   apply (rule bigo_refl)
   done
 
-lemma bigo_mult8: "ALL x. f x ~= 0 ==>
-    O(f * g) = O(f::'a => ('b::linordered_field)) * O(g)"
+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)"
   apply (rule equalityI)
   apply (erule bigo_mult7)
   apply (rule bigo_mult)
   done
 
-lemma bigo_minus [intro]: "f : O(g) ==> - f : O(g)"
+lemma bigo_minus [intro]: "f \<in> O(g) \<Longrightarrow> - f \<in> O(g)"
   by (auto simp add: bigo_def fun_Compl_def)
 
-lemma bigo_minus2: "f : g +o O(h) ==> -f : -g +o O(h)"
+lemma bigo_minus2: "f \<in> g +o O(h) \<Longrightarrow> - f \<in> -g +o O(h)"
   apply (rule set_minus_imp_plus)
   apply (drule set_plus_imp_minus)
   apply (drule bigo_minus)
   apply simp
   done
 
-lemma bigo_minus3: "O(-f) = O(f)"
+lemma bigo_minus3: "O(- f) = O(f)"
   by (auto simp add: bigo_def fun_Compl_def)
 
-lemma bigo_plus_absorb_lemma1: "f : O(g) ==> f +o O(g) <= O(g)"
+lemma bigo_plus_absorb_lemma1: "f \<in> O(g) \<Longrightarrow> f +o O(g) \<subseteq> O(g)"
 proof -
-  assume a: "f : O(g)"
-  show "f +o O(g) <= O(g)"
+  assume a: "f \<in> O(g)"
+  show "f +o O(g) \<subseteq> O(g)"
   proof -
-    have "f : O(f)" by auto
-    then have "f +o O(g) <= O(f) + 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 "... <= O(g) + O(g)"
+    also have "\<dots> \<subseteq> O(g) + O(g)"
     proof -
-      from a have "O(f) <= O(g)" by (auto del: subsetI)
+      from a have "O(f) \<subseteq> O(g)" by (auto del: subsetI)
       thus ?thesis by (auto del: subsetI)
     qed
-    also have "... <= O(g)" by simp
+    also have "\<dots> \<subseteq> O(g)" by simp
     finally show ?thesis .
   qed
 qed
 
-lemma bigo_plus_absorb_lemma2: "f : O(g) ==> O(g) <= f +o O(g)"
+lemma bigo_plus_absorb_lemma2: "f \<in> O(g) \<Longrightarrow> O(g) \<subseteq> f +o O(g)"
 proof -
-  assume a: "f : O(g)"
-  show "O(g) <= f +o O(g)"
+  assume a: "f \<in> O(g)"
+  show "O(g) \<subseteq> f +o O(g)"
   proof -
-    from a have "-f : O(g)" by auto
-    then have "-f +o O(g) <= O(g)" by (elim bigo_plus_absorb_lemma1)
-    then have "f +o (-f +o O(g)) <= f +o O(g)" by auto
-    also have "f +o (-f +o O(g)) = O(g)"
+    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
 qed
 
-lemma bigo_plus_absorb [simp]: "f : O(g) ==> f +o O(g) = O(g)"
+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_lemma2)
   done
 
-lemma bigo_plus_absorb2 [intro]: "f : O(g) ==> A <= O(g) ==> f +o A <= O(g)"
-  apply (subgoal_tac "f +o A <= f +o O(g)")
+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+
   done
 
-lemma bigo_add_commute_imp: "f : g +o O(h) ==> g : f +o O(h)"
+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)
@@ -449,63 +443,88 @@
   apply (simp add: add_ac)
   done
 
-lemma bigo_add_commute: "(f : g +o O(h)) = (g : f +o O(h))"
+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)+
   done
 
-lemma bigo_const1: "(%x. c) : O(%x. 1)"
+lemma bigo_const1: "(\<lambda>x. c) \<in> O(\<lambda>x. 1)"
   by (auto simp add: bigo_def mult_ac)
 
-lemma bigo_const2 [intro]: "O(%x. c) <= O(%x. 1)"
+lemma bigo_const2 [intro]: "O(\<lambda>x. c) \<subseteq> O(\<lambda>x. 1)"
   apply (rule bigo_elt_subset)
   apply (rule bigo_const1)
   done
 
-lemma bigo_const3: "(c::'a::linordered_field) ~= 0 ==> (%x. 1) : O(%x. c)"
+lemma bigo_const3:
+  fixes c :: "'a::linordered_field"
+  shows "c \<noteq> 0 \<Longrightarrow> (\<lambda>x. 1) \<in> O(\<lambda>x. c)"
   apply (simp add: bigo_def)
-  apply (rule_tac x = "abs(inverse c)" in exI)
+  apply (rule_tac x = "abs (inverse c)" in exI)
   apply (simp add: abs_mult [symmetric])
   done
 
-lemma bigo_const4: "(c::'a::linordered_field) ~= 0 ==> O(%x. 1) <= O(%x. c)"
-  by (rule bigo_elt_subset, rule bigo_const3, assumption)
+lemma bigo_const4:
+  fixes c :: "'a::linordered_field"
+  shows "c \<noteq> 0 \<Longrightarrow> O(\<lambda>x. 1) \<subseteq> O(\<lambda>x. c)"
+  apply (rule bigo_elt_subset)
+  apply (rule bigo_const3)
+  apply assumption
+  done
 
-lemma bigo_const [simp]: "(c::'a::linordered_field) ~= 0 ==> 
-    O(%x. c) = O(%x. 1)"
-  by (rule equalityI, rule bigo_const2, rule bigo_const4, assumption)
+lemma bigo_const [simp]:
+  fixes c :: "'a::linordered_field"
+  shows "c \<noteq> 0 \<Longrightarrow> O(\<lambda>x. c) = O(\<lambda>x. 1)"
+  apply (rule equalityI)
+  apply (rule bigo_const2)
+  apply (rule bigo_const4)
+  apply assumption
+  done
 
-lemma bigo_const_mult1: "(%x. c * f x) : O(f)"
+lemma bigo_const_mult1: "(\<lambda>x. c * f x) \<in> O(f)"
   apply (simp add: bigo_def)
-  apply (rule_tac x = "abs(c)" in exI)
+  apply (rule_tac x = "abs c" in exI)
   apply (auto simp add: abs_mult [symmetric])
   done
 
-lemma bigo_const_mult2: "O(%x. c * f x) <= O(f)"
-  by (rule bigo_elt_subset, rule bigo_const_mult1)
+lemma bigo_const_mult2: "O(\<lambda>x. c * f x) \<subseteq> O(f)"
+  apply (rule bigo_elt_subset)
+  apply (rule bigo_const_mult1)
+  done
 
-lemma bigo_const_mult3: "(c::'a::linordered_field) ~= 0 ==> f : O(%x. c * f x)"
+lemma bigo_const_mult3:
+  fixes c :: "'a::linordered_field"
+  shows "c \<noteq> 0 \<Longrightarrow> f \<in> O(\<lambda>x. c * f x)"
   apply (simp add: bigo_def)
-  apply (rule_tac x = "abs(inverse c)" in exI)
+  apply (rule_tac x = "abs (inverse c)" in exI)
   apply (simp add: abs_mult [symmetric] mult_assoc [symmetric])
   done
 
-lemma bigo_const_mult4: "(c::'a::linordered_field) ~= 0 ==> 
-    O(f) <= O(%x. c * f x)"
-  by (rule bigo_elt_subset, rule bigo_const_mult3, assumption)
+lemma bigo_const_mult4:
+  fixes c :: "'a::linordered_field"
+  shows "c \<noteq> 0 \<Longrightarrow> O(f) \<subseteq> O(\<lambda>x. c * f x)"
+  apply (rule bigo_elt_subset)
+  apply (rule bigo_const_mult3)
+  apply assumption
+  done
 
-lemma bigo_const_mult [simp]: "(c::'a::linordered_field) ~= 0 ==> 
-    O(%x. c * f x) = O(f)"
-  by (rule equalityI, rule bigo_const_mult2, erule bigo_const_mult4)
+lemma bigo_const_mult [simp]:
+  fixes c :: "'a::linordered_field"
+  shows "c \<noteq> 0 \<Longrightarrow> O(\<lambda>x. c * f x) = O(f)"
+  apply (rule equalityI)
+  apply (rule bigo_const_mult2)
+  apply (erule bigo_const_mult4)
+  done
 
-lemma bigo_const_mult5 [simp]: "(c::'a::linordered_field) ~= 0 ==> 
-    (%x. c) *o O(f) = O(f)"
+lemma bigo_const_mult5 [simp]:
+  fixes c :: "'a::linordered_field"
+  shows "c \<noteq> 0 \<Longrightarrow> (\<lambda>x. c) *o O(f) = O(f)"
   apply (auto del: subsetI)
   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 = "%y. inverse c * x y" in exI)
+  apply (rule_tac x = "\<lambda>y. inverse c * x y" in exI)
   apply (simp add: mult_assoc [symmetric] abs_mult)
   apply (rule_tac x = "abs (inverse c) * ca" in exI)
   apply (rule allI)
@@ -515,11 +534,11 @@
   apply force
   done
 
-lemma bigo_const_mult6 [intro]: "(%x. c) *o O(f) <= O(f)"
+lemma bigo_const_mult6 [intro]: "(\<lambda>x. c) *o O(f) \<subseteq> O(f)"
   apply (auto intro!: simp add: bigo_def elt_set_times_def func_times)
-  apply (rule_tac x = "ca * (abs c)" in exI)
+  apply (rule_tac x = "ca * abs c" in exI)
   apply (rule allI)
-  apply (subgoal_tac "ca * abs(c) * abs(f x) = abs(c) * (ca * abs(f x))")
+  apply (subgoal_tac "ca * abs c * abs (f x) = abs c * (ca * abs (f x))")
   apply (erule ssubst)
   apply (subst abs_mult)
   apply (rule mult_left_mono)
@@ -528,33 +547,34 @@
   apply(simp add: mult_ac)
   done
 
-lemma bigo_const_mult7 [intro]: "f =o O(g) ==> (%x. c * f x) =o O(g)"
+lemma bigo_const_mult7 [intro]: "f =o O(g) \<Longrightarrow> (\<lambda>x. c * f x) =o O(g)"
 proof -
   assume "f =o O(g)"
-  then have "(%x. c) * f =o (%x. c) *o O(g)"
+  then have "(\<lambda>x. c) * f =o (\<lambda>x. c) *o O(g)"
     by auto
-  also have "(%x. c) * f = (%x. c * f x)"
+  also have "(\<lambda>x. c) * f = (\<lambda>x. c * f x)"
     by (simp add: func_times)
-  also have "(%x. c) *o O(g) <= O(g)"
+  also have "(\<lambda>x. c) *o O(g) \<subseteq> O(g)"
     by (auto del: subsetI)
   finally show ?thesis .
 qed
 
-lemma bigo_compose1: "f =o O(g) ==> (%x. f(k x)) =o O(%x. g(k x))"
-by (unfold bigo_def, auto)
+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
 
-lemma bigo_compose2: "f =o g +o O(h) ==> (%x. f(k x)) =o (%x. g(k x)) +o 
-    O(%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)
+  apply (drule bigo_compose1)
+  apply (simp add: fun_diff_def)
   done
 
 
 subsection {* Setsum *}
 
-lemma bigo_setsum_main: "ALL x. ALL y : A x. 0 <= h x y ==> 
-    EX c. ALL x. ALL y : A x. abs(f x y) <= c * (h x y) ==>
-      (%x. SUM y : A x. f x y) =o O(%x. SUM y : A x. h x y)"  
+lemma bigo_setsum_main: "\<forall>x. \<forall>y \<in> A x. 0 \<le> h x y \<Longrightarrow>
+    \<exists>c. \<forall>x. \<forall>y \<in> A x. abs (f x y) \<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 (auto simp add: bigo_def)
   apply (rule_tac x = "abs c" in exI)
   apply (subst abs_of_nonneg) back back
@@ -571,14 +591,14 @@
   apply assumption+
   apply (drule bspec)
   apply assumption+
-  apply (rule mult_right_mono) 
+  apply (rule mult_right_mono)
   apply (rule abs_ge_self)
   apply force
   done
 
-lemma bigo_setsum1: "ALL x y. 0 <= h x y ==> 
-    EX c. ALL x y. abs(f x y) <= c * (h x y) ==>
-      (%x. SUM y : A x. f x y) =o O(%x. SUM y : A x. h x y)"
+lemma bigo_setsum1: "\<forall>x y. 0 \<le> h x y \<Longrightarrow>
+    \<exists>c. \<forall>x y. abs (f x y) \<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 clarsimp
@@ -586,14 +606,13 @@
   apply force
   done
 
-lemma bigo_setsum2: "ALL y. 0 <= h y ==> 
-    EX c. ALL y. abs(f y) <= c * (h y) ==>
-      (%x. SUM y : A x. f y) =o O(%x. SUM y : A x. h y)"
-  by (rule bigo_setsum1, auto)  
+lemma bigo_setsum2: "\<forall>y. 0 \<le> h y \<Longrightarrow>
+    \<exists>c. \<forall>y. abs (f y) \<le> c * (h y) \<Longrightarrow>
+      (\<lambda>x. \<Sum>y \<in> A x. f y) =o O(\<lambda>x. \<Sum>y \<in> A x. h y)"
+  by (rule bigo_setsum1) auto
 
-lemma bigo_setsum3: "f =o O(h) ==>
-    (%x. SUM y : A x. (l x y) * f(k x y)) =o
-      O(%x. SUM y : A x. abs(l x y * h(k x y)))"
+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. abs (l x y * h (k x y)))"
   apply (rule bigo_setsum1)
   apply (rule allI)+
   apply (rule abs_ge_zero)
@@ -608,10 +627,10 @@
   apply (rule abs_ge_zero)
   done
 
-lemma bigo_setsum4: "f =o g +o O(h) ==>
-    (%x. SUM y : A x. l x y * f(k x y)) =o
-      (%x. SUM y : A x. l x y * g(k x y)) +o
-        O(%x. SUM y : A x. abs(l x y * h(k x y)))"
+lemma bigo_setsum4: "f =o g +o O(h) \<Longrightarrow>
+    (\<lambda>x. \<Sum>y \<in> A x. l x y * f (k x y)) =o
+      (\<lambda>x. \<Sum>y \<in> A x. l x y * g (k x y)) +o
+        O(\<lambda>x. \<Sum>y \<in> A x. abs (l x y * h (k x y)))"
   apply (rule set_minus_imp_plus)
   apply (subst fun_diff_def)
   apply (subst setsum_subtractf [symmetric])
@@ -621,12 +640,12 @@
   apply (erule set_plus_imp_minus)
   done
 
-lemma bigo_setsum5: "f =o O(h) ==> ALL x y. 0 <= l x y ==> 
-    ALL x. 0 <= h x ==>
-      (%x. SUM y : A x. (l x y) * f(k x y)) =o
-        O(%x. SUM y : A x. (l x y) * h(k x y))" 
-  apply (subgoal_tac "(%x. SUM y : A x. (l x y) * h(k x y)) = 
-      (%x. SUM y : A x. abs((l x y) * h(k x y)))")
+lemma bigo_setsum5: "f =o O(h) \<Longrightarrow> \<forall>x y. 0 \<le> l x y \<Longrightarrow>
+    \<forall>x. 0 \<le> h x \<Longrightarrow>
+      (\<lambda>x. \<Sum>y \<in> A x. l x y * f (k x y)) =o
+        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. abs (l x y * h (k x y)))")
   apply (erule ssubst)
   apply (erule bigo_setsum3)
   apply (rule ext)
@@ -636,11 +655,11 @@
   apply auto
   done
 
-lemma bigo_setsum6: "f =o g +o O(h) ==> ALL x y. 0 <= l x y ==>
-    ALL x. 0 <= h x ==>
-      (%x. SUM y : A x. (l x y) * f(k x y)) =o
-        (%x. SUM y : A x. (l x y) * g(k x y)) +o
-          O(%x. SUM y : A x. (l x y) * h(k x y))" 
+lemma bigo_setsum6: "f =o g +o O(h) \<Longrightarrow> \<forall>x y. 0 \<le> l x y \<Longrightarrow>
+    \<forall>x. 0 \<le> h x \<Longrightarrow>
+      (\<lambda>x. \<Sum>y \<in> A x. l x y * f (k x y)) =o
+        (\<lambda>x. \<Sum>y \<in> A x. l x y * g (k x y)) +o
+          O(\<lambda>x. \<Sum>y \<in> A x. l x y * h (k x y))"
   apply (rule set_minus_imp_plus)
   apply (subst fun_diff_def)
   apply (subst setsum_subtractf [symmetric])
@@ -654,33 +673,32 @@
 
 subsection {* Misc useful stuff *}
 
-lemma bigo_useful_intro: "A <= O(f) ==> B <= O(f) ==>
-  A + B <= O(f)"
+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+
   done
 
-lemma bigo_useful_add: "f =o O(h) ==> g =o O(h) ==> f + g =o O(h)"
+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+
   done
-  
-lemma bigo_useful_const_mult: "(c::'a::linordered_field) ~= 0 ==> 
-    (%x. c) * f =o O(h) ==> f =o O(h)"
+
+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)"
   apply (rule subsetD)
-  apply (subgoal_tac "(%x. 1 / c) *o O(h) <= O(h)")
+  apply (subgoal_tac "(\<lambda>x. 1 / c) *o O(h) \<subseteq> O(h)")
   apply assumption
   apply (rule bigo_const_mult6)
-  apply (subgoal_tac "f = (%x. 1 / c) * ((%x. c) * f)")
+  apply (subgoal_tac "f = (\<lambda>x. 1 / c) * ((\<lambda>x. c) * f)")
   apply (erule ssubst)
   apply (erule set_times_intro2)
   apply (simp add: func_times)
   done
 
-lemma bigo_fix: "(%x. f ((x::nat) + 1)) =o O(%x. h(x + 1)) ==> f 0 = 0 ==>
-    f =o O(h)"
+lemma bigo_fix: "(\<lambda>x::nat. f (x + 1)) =o O(\<lambda>x. h (x + 1)) \<Longrightarrow> f 0 = 0 \<Longrightarrow> f =o O(h)"
   apply (simp add: bigo_alt_def)
   apply auto
   apply (rule_tac x = c in exI)
@@ -696,9 +714,9 @@
   apply simp
   done
 
-lemma bigo_fix2: 
-    "(%x. f ((x::nat) + 1)) =o (%x. g(x + 1)) +o O(%x. h(x + 1)) ==> 
-       f 0 = g 0 ==> f =o g +o O(h)"
+lemma bigo_fix2:
+    "(\<lambda>x. f ((x::nat) + 1)) =o (\<lambda>x. g(x + 1)) +o O(\<lambda>x. h(x + 1)) \<Longrightarrow>
+       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)
@@ -711,13 +729,10 @@
 
 subsection {* Less than or equal to *}
 
-definition
-  lesso :: "('a => 'b::linordered_idom) => ('a => 'b) => ('a => 'b)"
-    (infixl "<o" 70) where
-  "f <o g = (%x. max (f x - g x) 0)"
+definition lesso :: "('a \<Rightarrow> 'b::linordered_idom) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"  (infixl "<o" 70)
+  where "f <o g = (\<lambda>x. max (f x - g x) 0)"
 
-lemma bigo_lesseq1: "f =o O(h) ==> ALL x. abs (g x) <= abs (f x) ==>
-    g =o O(h)"
+lemma bigo_lesseq1: "f =o O(h) \<Longrightarrow> \<forall>x. abs (g x) \<le> abs (f x) \<Longrightarrow> g =o O(h)"
   apply (unfold bigo_def)
   apply clarsimp
   apply (rule_tac x = c in exI)
@@ -726,8 +741,7 @@
   apply (erule spec)+
   done
 
-lemma bigo_lesseq2: "f =o O(h) ==> ALL x. abs (g x) <= f x ==>
-      g =o O(h)"
+lemma bigo_lesseq2: "f =o O(h) \<Longrightarrow> \<forall>x. abs (g x) \<le> f x \<Longrightarrow> g =o O(h)"
   apply (erule bigo_lesseq1)
   apply (rule allI)
   apply (drule_tac x = x in spec)
@@ -736,26 +750,24 @@
   apply (rule abs_ge_self)
   done
 
-lemma bigo_lesseq3: "f =o O(h) ==> ALL x. 0 <= g x ==> ALL x. g x <= f x ==>
-    g =o O(h)"
+lemma bigo_lesseq3: "f =o O(h) \<Longrightarrow> \<forall>x. 0 \<le> g x \<Longrightarrow> \<forall>x. g x \<le> f x \<Longrightarrow> g =o O(h)"
   apply (erule bigo_lesseq2)
   apply (rule allI)
   apply (subst abs_of_nonneg)
   apply (erule spec)+
   done
 
-lemma bigo_lesseq4: "f =o O(h) ==>
-    ALL x. 0 <= g x ==> ALL x. g x <= abs (f x) ==>
-      g =o O(h)"
+lemma bigo_lesseq4: "f =o O(h) \<Longrightarrow>
+    \<forall>x. 0 \<le> g x \<Longrightarrow> \<forall>x. g x \<le> abs (f x) \<Longrightarrow> g =o O(h)"
   apply (erule bigo_lesseq1)
   apply (rule allI)
   apply (subst abs_of_nonneg)
   apply (erule spec)+
   done
 
-lemma bigo_lesso1: "ALL x. f x <= g x ==> f <o g =o O(h)"
+lemma bigo_lesso1: "\<forall>x. f x \<le> g x \<Longrightarrow> f <o g =o O(h)"
   apply (unfold lesso_def)
-  apply (subgoal_tac "(%x. max (f x - g x) 0) = 0")
+  apply (subgoal_tac "(\<lambda>x. max (f x - g x) 0) = 0")
   apply (erule ssubst)
   apply (rule bigo_zero)
   apply (unfold func_zero)
@@ -763,9 +775,8 @@
   apply (simp split: split_max)
   done
 
-lemma bigo_lesso2: "f =o g +o O(h) ==>
-    ALL x. 0 <= k x ==> ALL x. k x <= f x ==>
-      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)
@@ -773,7 +784,7 @@
   apply (rule max.cobounded2)
   apply (rule allI)
   apply (subst fun_diff_def)
-  apply (case_tac "0 <= k x - g x")
+  apply (case_tac "0 \<le> k x - g x")
   apply simp
   apply (subst abs_of_nonneg)
   apply (drule_tac x = x in spec) back
@@ -781,15 +792,14 @@
   apply (subst diff_conv_add_uminus)+
   apply (rule add_right_mono)
   apply (erule spec)
-  apply (rule order_trans) 
+  apply (rule order_trans)
   prefer 2
   apply (rule abs_ge_zero)
   apply (simp add: algebra_simps)
   done
 
-lemma bigo_lesso3: "f =o g +o O(h) ==>
-    ALL x. 0 <= k x ==> ALL x. g x <= k x ==>
-      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)
@@ -797,7 +807,7 @@
   apply (rule max.cobounded2)
   apply (rule allI)
   apply (subst fun_diff_def)
-  apply (case_tac "0 <= f x - k x")
+  apply (case_tac "0 \<le> f x - k x")
   apply simp
   apply (subst abs_of_nonneg)
   apply (drule_tac x = x in spec) back
@@ -806,14 +816,15 @@
   apply (rule add_left_mono)
   apply (rule le_imp_neg_le)
   apply (erule spec)
-  apply (rule order_trans) 
+  apply (rule order_trans)
   prefer 2
   apply (rule abs_ge_zero)
   apply (simp add: algebra_simps)
   done
 
-lemma bigo_lesso4: "f <o g =o O(k::'a=>'b::linordered_field) ==>
-    g =o h +o O(k) ==> f <o h =o O(k)"
+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)"
   apply (unfold lesso_def)
   apply (drule set_plus_imp_minus)
   apply (drule bigo_abs5) back
@@ -822,25 +833,22 @@
   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)
+  apply (auto simp add: func_plus fun_diff_def algebra_simps split: split_max abs_split)
   done
 
-lemma bigo_lesso5: "f <o g =o O(h) ==>
-    EX C. ALL x. f x <= g x + C * abs(h x)"
+lemma bigo_lesso5: "f <o g =o O(h) \<Longrightarrow> \<exists>C. \<forall>x. f x \<le> g x + C * abs (h x)"
   apply (simp only: lesso_def bigo_alt_def)
   apply clarsimp
   apply (rule_tac x = c in exI)
   apply (rule allI)
   apply (drule_tac x = x in spec)
-  apply (subgoal_tac "abs(max (f x - g x) 0) = max (f x - g x) 0")
-  apply (clarsimp simp add: algebra_simps) 
+  apply (subgoal_tac "abs (max (f x - g x) 0) = max (f x - g x) 0")
+  apply (clarsimp simp add: algebra_simps)
   apply (rule abs_of_nonneg)
   apply (rule max.cobounded2)
   done
 
-lemma lesso_add: "f <o g =o O(h) ==>
-      k <o l =o O(h) ==> (f + k) <o (g + l) =o O(h)"
+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)
@@ -849,7 +857,7 @@
   apply (auto split: split_max simp add: func_plus)
   done
 
-lemma bigo_LIMSEQ1: "f =o O(g) ==> g ----> 0 ==> f ----> (0::real)"
+lemma bigo_LIMSEQ1: "f =o O(g) \<Longrightarrow> g ----> 0 \<Longrightarrow> f ----> (0::real)"
   apply (simp add: LIMSEQ_iff bigo_alt_def)
   apply clarify
   apply (drule_tac x = "r / c" in spec)
@@ -866,21 +874,20 @@
   apply (rule order_le_less_trans)
   apply assumption
   apply (rule order_less_le_trans)
-  apply (subgoal_tac "c * abs(g n) < c * (r / c)")
+  apply (subgoal_tac "c * abs (g n) < c * (r / c)")
   apply assumption
   apply (erule mult_strict_left_mono)
   apply assumption
   apply simp
-done
+  done
 
-lemma bigo_LIMSEQ2: "f =o g +o O(h) ==> h ----> 0 ==> f ----> a 
-    ==> g ----> (a::real)"
+lemma bigo_LIMSEQ2: "f =o g +o O(h) \<Longrightarrow> h ----> 0 \<Longrightarrow> f ----> a \<Longrightarrow> g ----> (a::real)"
   apply (drule set_plus_imp_minus)
   apply (drule bigo_LIMSEQ1)
   apply assumption
   apply (simp only: fun_diff_def)
   apply (erule LIMSEQ_diff_approach_zero2)
   apply assumption
-done
+  done
 
 end