diff -r 7247cb62406c -r 1d43f86f48be src/HOL/Metis_Examples/Big_O.thy
--- a/src/HOL/Metis_Examples/Big_O.thy	Mon Dec 28 18:03:26 2015 +0100
+++ b/src/HOL/Metis_Examples/Big_O.thy	Mon Dec 28 19:23:15 2015 +0100
@@ -493,7 +493,7 @@
 
 lemma bigo_setsum_main: "\<forall>x. \<forall>y \<in> A x. 0 <= h x y \<Longrightarrow>
     \<exists>c. \<forall>x. \<forall>y \<in> A x. \<bar>f x y\<bar> <= c * (h x y) \<Longrightarrow>
-      (\<lambda>x. SUM y : A x. f x y) =o O(\<lambda>x. SUM y : A x. h x y)"
+      (\<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 = "\<bar>c\<bar>" in exI)
 apply (subst abs_of_nonneg) back back
@@ -508,18 +508,18 @@
 
 lemma bigo_setsum1: "\<forall>x y. 0 <= h x y \<Longrightarrow>
     \<exists>c. \<forall>x y. \<bar>f x y\<bar> <= c * (h x y) \<Longrightarrow>
-      (\<lambda>x. SUM y : A x. f x y) =o O(\<lambda>x. SUM y : A x. h x y)"
+      (\<lambda>x. \<Sum>y \<in> A x. f x y) =o O(\<lambda>x. \<Sum>y \<in> A x. h x y)"
 by (metis (no_types) bigo_setsum_main)
 
 lemma bigo_setsum2: "\<forall>y. 0 <= h y \<Longrightarrow>
     \<exists>c. \<forall>y. \<bar>f y\<bar> <= c * (h y) \<Longrightarrow>
-      (\<lambda>x. SUM y : A x. f y) =o O(\<lambda>x. SUM y : A x. h y)"
+      (\<lambda>x. \<Sum>y \<in> A x. f y) =o O(\<lambda>x. \<Sum>y \<in> A x. h y)"
 apply (rule bigo_setsum1)
 by metis+
 
 lemma bigo_setsum3: "f =o O(h) \<Longrightarrow>
-    (\<lambda>x. SUM y : A x. (l x y) * f(k x y)) =o
-      O(\<lambda>x. SUM y : A x. \<bar>l x y * h(k x y)\<bar>)"
+    (\<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)
@@ -528,9 +528,9 @@
 by (metis abs_ge_zero mult.left_commute mult_left_mono)
 
 lemma bigo_setsum4: "f =o g +o O(h) \<Longrightarrow>
-    (\<lambda>x. SUM y : A x. l x y * f(k x y)) =o
-      (\<lambda>x. SUM y : A x. l x y * g(k x y)) +o
-        O(\<lambda>x. SUM y : A x. \<bar>l x y * h(k x y)\<bar>)"
+    (\<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. \<bar>l x y * h(k x y)\<bar>)"
 apply (rule set_minus_imp_plus)
 apply (subst fun_diff_def)
 apply (subst setsum_subtractf [symmetric])
@@ -540,10 +540,10 @@
 
 lemma bigo_setsum5: "f =o O(h) \<Longrightarrow> \<forall>x y. 0 <= l x y \<Longrightarrow>
     \<forall>x. 0 <= h x \<Longrightarrow>
-      (\<lambda>x. SUM y : A x. (l x y) * f(k x y)) =o
-        O(\<lambda>x. SUM y : A x. (l x y) * h(k x y))"
-apply (subgoal_tac "(\<lambda>x. SUM y : A x. (l x y) * h(k x y)) =
-      (\<lambda>x. SUM y : A x. \<bar>(l x y) * h(k x y)\<bar>)")
+      (\<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. \<bar>(l x y) * h(k x y)\<bar>)")
  apply (erule ssubst)
  apply (erule bigo_setsum3)
 apply (rule ext)
@@ -553,9 +553,9 @@
 
 lemma bigo_setsum6: "f =o g +o O(h) \<Longrightarrow> \<forall>x y. 0 <= l x y \<Longrightarrow>
     \<forall>x. 0 <= h x \<Longrightarrow>
-      (\<lambda>x. SUM y : A x. (l x y) * f(k x y)) =o
-        (\<lambda>x. SUM y : A x. (l x y) * g(k x y)) +o
-          O(\<lambda>x. SUM y : A x. (l x y) * h(k x y))"
+      (\<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])