merged
authornipkow
Mon, 17 Oct 2016 14:37:32 +0200
changeset 64269 c939cc16b605
parent 64266 4699d3b3173e (current diff)
parent 64268 3faa948dc861 (diff)
child 64271 912573107a2e
child 64272 f76b6dda2e56
merged
--- a/src/Doc/Logics/document/HOL.tex	Mon Oct 17 11:07:01 2016 +0200
+++ b/src/Doc/Logics/document/HOL.tex	Mon Oct 17 14:37:32 2016 +0200
@@ -145,7 +145,7 @@
 numbers).  The type class \cldx{plus_ac0} comprises all types for which 0
 and~+ satisfy the laws $x+y=y+x$, $(x+y)+z = x+(y+z)$ and $0+x = x$.  These
 types include the numeric ones \tdx{nat}, \tdx{int} and~\tdx{real} and also
-multisets.  The summation operator \cdx{setsum} is available for all types in
+multisets.  The summation operator \cdx{sum} is available for all types in
 this class. 
 
 Theory \thydx{Ord} defines the syntactic class \cldx{ord} of order
--- a/src/Doc/Main/Main_Doc.thy	Mon Oct 17 11:07:01 2016 +0200
+++ b/src/Doc/Main/Main_Doc.thy	Mon Oct 17 14:37:32 2016 +0200
@@ -408,7 +408,7 @@
 @{const Finite_Set.finite} & @{term_type_only Finite_Set.finite "'a set\<Rightarrow>bool"}\\
 @{const Finite_Set.card} & @{term_type_only Finite_Set.card "'a set \<Rightarrow> nat"}\\
 @{const Finite_Set.fold} & @{term_type_only Finite_Set.fold "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b"}\\
-@{const Groups_Big.setsum} & @{term_type_only Groups_Big.setsum "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'b::comm_monoid_add"}\\
+@{const Groups_Big.sum} & @{term_type_only Groups_Big.sum "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'b::comm_monoid_add"}\\
 @{const Groups_Big.setprod} & @{term_type_only Groups_Big.setprod "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'b::comm_monoid_mult"}\\
 \end{supertabular}
 
@@ -416,8 +416,8 @@
 \subsubsection*{Syntax}
 
 \begin{supertabular}{@ {} l @ {\quad$\equiv$\quad} l l @ {}}
-@{term "setsum (\<lambda>x. x) A"} & @{term[source]"setsum (\<lambda>x. x) A"} & (\<^verbatim>\<open>SUM\<close>)\\
-@{term "setsum (\<lambda>x. t) A"} & @{term[source]"setsum (\<lambda>x. t) A"}\\
+@{term "sum (\<lambda>x. x) A"} & @{term[source]"sum (\<lambda>x. x) A"} & (\<^verbatim>\<open>SUM\<close>)\\
+@{term "sum (\<lambda>x. t) A"} & @{term[source]"sum (\<lambda>x. t) A"}\\
 @{term[source] "\<Sum>x|P. t"} & @{term"\<Sum>x|P. t"}\\
 \multicolumn{2}{@ {}l@ {}}{Similarly for \<open>\<Prod>\<close> instead of \<open>\<Sum>\<close>} & (\<^verbatim>\<open>PROD\<close>)\\
 \end{supertabular}
@@ -463,10 +463,10 @@
 @{term[source] "\<Union>i\<le>n. A"} & @{term[source] "\<Union>i \<in> {..n}. A"}\\
 @{term[source] "\<Union>i<n. A"} & @{term[source] "\<Union>i \<in> {..<n}. A"}\\
 \multicolumn{2}{@ {}l@ {}}{Similarly for \<open>\<Inter>\<close> instead of \<open>\<Union>\<close>}\\
-@{term "setsum (\<lambda>x. t) {a..b}"} & @{term[source] "setsum (\<lambda>x. t) {a..b}"}\\
-@{term "setsum (\<lambda>x. t) {a..<b}"} & @{term[source] "setsum (\<lambda>x. t) {a..<b}"}\\
-@{term "setsum (\<lambda>x. t) {..b}"} & @{term[source] "setsum (\<lambda>x. t) {..b}"}\\
-@{term "setsum (\<lambda>x. t) {..<b}"} & @{term[source] "setsum (\<lambda>x. t) {..<b}"}\\
+@{term "sum (\<lambda>x. t) {a..b}"} & @{term[source] "sum (\<lambda>x. t) {a..b}"}\\
+@{term "sum (\<lambda>x. t) {a..<b}"} & @{term[source] "sum (\<lambda>x. t) {a..<b}"}\\
+@{term "sum (\<lambda>x. t) {..b}"} & @{term[source] "sum (\<lambda>x. t) {..b}"}\\
+@{term "sum (\<lambda>x. t) {..<b}"} & @{term[source] "sum (\<lambda>x. t) {..<b}"}\\
 \multicolumn{2}{@ {}l@ {}}{Similarly for \<open>\<Prod>\<close> instead of \<open>\<Sum>\<close>}\\
 \end{supertabular}
 
--- a/src/Doc/Prog_Prove/Bool_nat_list.thy	Mon Oct 17 11:07:01 2016 +0200
+++ b/src/Doc/Prog_Prove/Bool_nat_list.thy	Mon Oct 17 14:37:32 2016 +0200
@@ -453,9 +453,9 @@
 \end{exercise}
 
 \begin{exercise}
-Define a recursive function @{text "sum ::"} @{typ"nat \<Rightarrow> nat"} such that
-\mbox{@{text"sum n"}} @{text"="} @{text"0 + ... + n"} and prove
-@{prop" sum(n::nat) = n * (n+1) div 2"}.
+Define a recursive function @{text "sum_upto ::"} @{typ"nat \<Rightarrow> nat"} such that
+\mbox{@{text"sum_upto n"}} @{text"="} @{text"0 + ... + n"} and prove
+@{prop" sum_upto (n::nat) = n * (n+1) div 2"}.
 \end{exercise}
 *}
 (*<*)
--- a/src/Doc/Sugar/Sugar.thy	Mon Oct 17 11:07:01 2016 +0200
+++ b/src/Doc/Sugar/Sugar.thy	Mon Oct 17 14:37:32 2016 +0200
@@ -13,9 +13,9 @@
 This document is for those Isabelle users who have mastered
 the art of mixing \LaTeX\ text and Isabelle theories and never want to
 typeset a theorem by hand anymore because they have experienced the
-bliss of writing \verb!@!\verb!{thm[display,mode=latex_sum] setsum_Suc_diff [no_vars]}!
+bliss of writing \verb!@!\verb!{thm[display,mode=latex_sum] sum_Suc_diff [no_vars]}!
 and seeing Isabelle typeset it for them:
-@{thm[display,mode=latex_sum] setsum_Suc_diff[no_vars]}
+@{thm[display,mode=latex_sum] sum_Suc_diff[no_vars]}
 No typos, no omissions, no sweat.
 If you have not experienced that joy, read Chapter 4, \emph{Presenting
 Theories}, @{cite LNCS2283} first.
--- a/src/HOL/Algebra/IntRing.thy	Mon Oct 17 11:07:01 2016 +0200
+++ b/src/HOL/Algebra/IntRing.thy	Mon Oct 17 14:37:32 2016 +0200
@@ -91,7 +91,7 @@
   rewrites int_carrier_eq: "carrier \<Z> = UNIV"
     and int_zero_eq: "zero \<Z> = 0"
     and int_add_eq: "add \<Z> x y = x + y"
-    and int_finsum_eq: "finsum \<Z> f A = setsum f A"
+    and int_finsum_eq: "finsum \<Z> f A = sum f A"
 proof -
   \<comment> "Specification"
   show "abelian_monoid \<Z>" by standard auto
@@ -105,7 +105,7 @@
   note add = this
   show zero: "zero \<Z> = 0"
     by simp
-  show "finsum \<Z> f A = setsum f A"
+  show "finsum \<Z> f A = sum f A"
     by (induct A rule: infinite_finite_induct, auto)
 qed
 
@@ -117,7 +117,7 @@
   rewrites "carrier \<Z> = UNIV"
     and "zero \<Z> = 0"
     and "add \<Z> x y = x + y"
-    and "finsum \<Z> f A = setsum f A"
+    and "finsum \<Z> f A = sum f A"
     and int_a_inv_eq: "a_inv \<Z> x = - x"
     and int_a_minus_eq: "a_minus \<Z> x y = x - y"
 proof -
@@ -150,7 +150,7 @@
   rewrites "carrier \<Z> = UNIV"
     and "zero \<Z> = 0"
     and "add \<Z> x y = x + y"
-    and "finsum \<Z> f A = setsum f A"
+    and "finsum \<Z> f A = sum f A"
     and "a_inv \<Z> x = - x"
     and "a_minus \<Z> x y = x - y"
 proof -
--- a/src/HOL/Analysis/Binary_Product_Measure.thy	Mon Oct 17 11:07:01 2016 +0200
+++ b/src/HOL/Analysis/Binary_Product_Measure.thy	Mon Oct 17 14:37:32 2016 +0200
@@ -767,8 +767,8 @@
     apply (subst B.emeasure_pair_measure_alt[OF X])
     apply (subst emeasure_count_space)
     apply (auto simp add: emeasure_count_space nn_integral_count_space
-                          pos_card of_nat_setsum[symmetric] card_SigmaI[symmetric]
-                simp del: of_nat_setsum card_SigmaI
+                          pos_card of_nat_sum[symmetric] card_SigmaI[symmetric]
+                simp del: of_nat_sum card_SigmaI
                 intro!: arg_cong[where f=card])
     done
 qed
--- a/src/HOL/Analysis/Bochner_Integration.thy	Mon Oct 17 11:07:01 2016 +0200
+++ b/src/HOL/Analysis/Bochner_Integration.thy	Mon Oct 17 14:37:32 2016 +0200
@@ -157,10 +157,10 @@
 
 lemma
   fixes f :: "'a \<Rightarrow> 'b::semiring_1" assumes "finite A"
-  shows setsum_mult_indicator[simp]: "(\<Sum>x \<in> A. f x * indicator (B x) (g x)) = (\<Sum>x\<in>{x\<in>A. g x \<in> B x}. f x)"
-  and setsum_indicator_mult[simp]: "(\<Sum>x \<in> A. indicator (B x) (g x) * f x) = (\<Sum>x\<in>{x\<in>A. g x \<in> B x}. f x)"
+  shows sum_mult_indicator[simp]: "(\<Sum>x \<in> A. f x * indicator (B x) (g x)) = (\<Sum>x\<in>{x\<in>A. g x \<in> B x}. f x)"
+  and sum_indicator_mult[simp]: "(\<Sum>x \<in> A. indicator (B x) (g x) * f x) = (\<Sum>x\<in>{x\<in>A. g x \<in> B x}. f x)"
   unfolding indicator_def
-  using assms by (auto intro!: setsum.mono_neutral_cong_right split: if_split_asm)
+  using assms by (auto intro!: sum.mono_neutral_cong_right split: if_split_asm)
 
 lemma borel_measurable_induct_real[consumes 2, case_names set mult add seq]:
   fixes P :: "('a \<Rightarrow> real) \<Rightarrow> bool"
@@ -220,8 +220,8 @@
     next
       case (insert x F)
       then show ?case
-        by (auto intro!: add mult set setsum_nonneg split: split_indicator split_indicator_asm
-                 simp del: setsum_mult_indicator simp: setsum_nonneg_eq_0_iff)
+        by (auto intro!: add mult set sum_nonneg split: split_indicator split_indicator_asm
+                 simp del: sum_mult_indicator simp: sum_nonneg_eq_0_iff)
     qed
     with U' show "P (U' i)" by simp
   qed
@@ -337,7 +337,7 @@
     (\<Sum>y\<in>f`space M. (\<Sum>z\<in>g`space M.
       if \<exists>x\<in>space M. y = f x \<and> z = g x then measure M {x\<in>space M. g x = z} else 0) *\<^sub>R y)"
     unfolding simple_bochner_integral_def
-  proof (safe intro!: setsum.cong scaleR_cong_right)
+  proof (safe intro!: sum.cong scaleR_cong_right)
     fix y assume y: "y \<in> space M" "f y \<noteq> 0"
     have [simp]: "g ` space M \<inter> {z. \<exists>x\<in>space M. f y = f x \<and> z = g x} =
         {z. \<exists>x\<in>space M. f y = f x \<and> z = g x}"
@@ -359,17 +359,17 @@
     ultimately
     show "measure M {x \<in> space M. f x = f y} =
       (\<Sum>z\<in>g ` space M. if \<exists>x\<in>space M. f y = f x \<and> z = g x then measure M {x \<in> space M. g x = z} else 0)"
-      apply (simp add: setsum.If_cases eq)
+      apply (simp add: sum.If_cases eq)
       apply (subst measure_finite_Union[symmetric])
       apply (auto simp: disjoint_family_on_def less_top)
       done
   qed
   also have "\<dots> = (\<Sum>y\<in>f`space M. (\<Sum>z\<in>g`space M.
       if \<exists>x\<in>space M. y = f x \<and> z = g x then measure M {x\<in>space M. g x = z} *\<^sub>R y else 0))"
-    by (auto intro!: setsum.cong simp: scaleR_setsum_left)
+    by (auto intro!: sum.cong simp: scaleR_sum_left)
   also have "\<dots> = ?r"
-    by (subst setsum.commute)
-       (auto intro!: setsum.cong simp: setsum.If_cases scaleR_setsum_right[symmetric] eq)
+    by (subst sum.commute)
+       (auto intro!: sum.cong simp: sum.If_cases scaleR_sum_right[symmetric] eq)
   finally show "simple_bochner_integral M f = ?r" .
 qed
 
@@ -391,7 +391,7 @@
     by (intro simple_bochner_integral_partition)
        (auto simp: simple_bochner_integrable_compose2 elim: simple_bochner_integrable.cases)
   ultimately show ?thesis
-    by (simp add: setsum.distrib[symmetric] scaleR_add_right)
+    by (simp add: sum.distrib[symmetric] scaleR_add_right)
 qed
 
 lemma (in linear) simple_bochner_integral_linear:
@@ -404,7 +404,7 @@
        (auto simp: simple_bochner_integrable_compose2[where p="\<lambda>x y. f x"] zero
              elim: simple_bochner_integrable.cases)
   also have "\<dots> = f (simple_bochner_integral M g)"
-    by (simp add: simple_bochner_integral_def setsum scaleR)
+    by (simp add: simple_bochner_integral_def sum scaleR)
   finally show ?thesis .
 qed
 
@@ -431,7 +431,7 @@
 proof -
   have "norm (simple_bochner_integral M f) \<le>
     (\<Sum>y\<in>f ` space M. norm (measure M {x \<in> space M. f x = y} *\<^sub>R y))"
-    unfolding simple_bochner_integral_def by (rule norm_setsum)
+    unfolding simple_bochner_integral_def by (rule norm_sum)
   also have "\<dots> = (\<Sum>y\<in>f ` space M. measure M {x \<in> space M. f x = y} *\<^sub>R norm y)"
     by simp
   also have "\<dots> = simple_bochner_integral M (\<lambda>x. norm (f x))"
@@ -444,7 +444,7 @@
 lemma simple_bochner_integral_nonneg[simp]:
   fixes f :: "'a \<Rightarrow> real"
   shows "(\<And>x. 0 \<le> f x) \<Longrightarrow> 0 \<le> simple_bochner_integral M f"
-  by (simp add: setsum_nonneg simple_bochner_integral_def)
+  by (simp add: sum_nonneg simple_bochner_integral_def)
 
 lemma simple_bochner_integral_eq_nn_integral:
   assumes f: "simple_bochner_integrable M f" "\<And>x. 0 \<le> f x"
@@ -473,9 +473,9 @@
     unfolding simple_integral_def
     by (subst simple_bochner_integral_partition[OF f(1), where g="\<lambda>x. ennreal (f x)" and v=enn2real])
        (auto intro: f simple_function_compose1 elim: simple_bochner_integrable.cases
-             intro!: setsum.cong ennreal_cong_mult
-             simp: setsum_ennreal[symmetric] ac_simps ennreal_mult
-             simp del: setsum_ennreal)
+             intro!: sum.cong ennreal_cong_mult
+             simp: sum_ennreal[symmetric] ac_simps ennreal_mult
+             simp del: sum_ennreal)
   also have "\<dots> = (\<integral>\<^sup>+x. f x \<partial>M)"
     using f
     by (intro nn_integral_eq_simple_integral[symmetric])
@@ -714,7 +714,7 @@
   unfolding diff_conv_add_uminus
   by (intro has_bochner_integral_add has_bochner_integral_minus)
 
-lemma has_bochner_integral_setsum:
+lemma has_bochner_integral_sum:
   "(\<And>i. i \<in> I \<Longrightarrow> has_bochner_integral M (f i) (x i)) \<Longrightarrow>
     has_bochner_integral M (\<lambda>x. \<Sum>i\<in>I. f i x) (\<Sum>i\<in>I. x i)"
   by (induct I rule: infinite_finite_induct) auto
@@ -882,7 +882,7 @@
     by (auto simp: space_restrict_space measure_restrict_space[OF \<Omega>(1)] le_infI2
                    simple_bochner_integral_def Collect_restrict
              split: split_indicator split_indicator_asm
-             intro!: setsum.mono_neutral_cong_left arg_cong2[where f=measure])
+             intro!: sum.mono_neutral_cong_left arg_cong2[where f=measure])
 qed
 
 context
@@ -971,8 +971,8 @@
 lemma integrable_zero[simp, intro]: "integrable M (\<lambda>x. 0)"
   by (metis has_bochner_integral_zero integrable.simps)
 
-lemma integrable_setsum[simp, intro]: "(\<And>i. i \<in> I \<Longrightarrow> integrable M (f i)) \<Longrightarrow> integrable M (\<lambda>x. \<Sum>i\<in>I. f i x)"
-  by (metis has_bochner_integral_setsum integrable.simps)
+lemma integrable_sum[simp, intro]: "(\<And>i. i \<in> I \<Longrightarrow> integrable M (f i)) \<Longrightarrow> integrable M (\<lambda>x. \<Sum>i\<in>I. f i x)"
+  by (metis has_bochner_integral_sum integrable.simps)
 
 lemma integrable_indicator[simp, intro]: "A \<in> sets M \<Longrightarrow> emeasure M A < \<infinity> \<Longrightarrow>
   integrable M (\<lambda>x. indicator A x *\<^sub>R c)"
@@ -1045,13 +1045,13 @@
     integral\<^sup>L M (\<lambda>x. f x - g x) = integral\<^sup>L M f - integral\<^sup>L M g"
   by (intro has_bochner_integral_integral_eq has_bochner_integral_diff has_bochner_integral_integrable)
 
-lemma integral_setsum: "(\<And>i. i \<in> I \<Longrightarrow> integrable M (f i)) \<Longrightarrow>
+lemma integral_sum: "(\<And>i. i \<in> I \<Longrightarrow> integrable M (f i)) \<Longrightarrow>
   integral\<^sup>L M (\<lambda>x. \<Sum>i\<in>I. f i x) = (\<Sum>i\<in>I. integral\<^sup>L M (f i))"
-  by (intro has_bochner_integral_integral_eq has_bochner_integral_setsum has_bochner_integral_integrable)
-
-lemma integral_setsum'[simp]: "(\<And>i. i \<in> I =simp=> integrable M (f i)) \<Longrightarrow>
+  by (intro has_bochner_integral_integral_eq has_bochner_integral_sum has_bochner_integral_integrable)
+
+lemma integral_sum'[simp]: "(\<And>i. i \<in> I =simp=> integrable M (f i)) \<Longrightarrow>
   integral\<^sup>L M (\<lambda>x. \<Sum>i\<in>I. f i x) = (\<Sum>i\<in>I. integral\<^sup>L M (f i))"
-  unfolding simp_implies_def by (rule integral_setsum)
+  unfolding simp_implies_def by (rule integral_sum)
 
 lemma integral_bounded_linear: "bounded_linear T \<Longrightarrow> integrable M f \<Longrightarrow>
     integral\<^sup>L M (\<lambda>x. T (f x)) = T (integral\<^sup>L M f)"
@@ -1729,9 +1729,9 @@
     using summable
   proof eventually_elim
     fix j x assume [simp]: "summable (\<lambda>i. norm (f i x))"
-    have "norm (\<Sum>i<j. f i x) \<le> (\<Sum>i<j. norm (f i x))" by (rule norm_setsum)
+    have "norm (\<Sum>i<j. f i x) \<le> (\<Sum>i<j. norm (f i x))" by (rule norm_sum)
     also have "\<dots> \<le> (\<Sum>i. norm (f i x))"
-      using setsum_le_suminf[of "\<lambda>i. norm (f i x)"] unfolding sums_iff by auto
+      using sum_le_suminf[of "\<lambda>i. norm (f i x)"] unfolding sums_iff by auto
     finally show "norm (\<Sum>i<j. f i x) \<le> (\<Sum>i. norm (f i x))" by simp
   qed
 
@@ -1835,7 +1835,7 @@
     have "(\<And>i::'b. i \<in> A \<Longrightarrow> integrable M (f i::'a \<Rightarrow> 'b)) \<Longrightarrow>
     (\<And>i. i \<in> A \<Longrightarrow> P (f i)) \<Longrightarrow> P (\<lambda>x. \<Sum>i\<in>A. f i x)"
     by (induct A rule: infinite_finite_induct) (auto intro!: add) }
-  note setsum = this
+  note sum = this
 
   define s' where [abs_def]: "s' i z = indicator (space M) z *\<^sub>R s i z" for i z
   then have s'_eq_s: "\<And>i x. x \<in> space M \<Longrightarrow> s' i x = s i x"
@@ -1873,7 +1873,7 @@
         using sbi by (auto elim: simple_bochner_integrable.cases simp: less_top)
       finally have "emeasure M {y \<in> space M. s' i y = s' i x} \<noteq> \<infinity>" by simp }
     then show "P (s' i)"
-      by (subst s'_eq) (auto intro!: setsum base simp: less_top)
+      by (subst s'_eq) (auto intro!: sum base simp: less_top)
 
     fix x assume "x \<in> space M" with s show "(\<lambda>i. s' i x) \<longlonglongrightarrow> f x"
       by (simp add: s'_eq_s)
@@ -2191,19 +2191,19 @@
   shows "(\<integral>x. f x \<partial>count_space A) = (\<Sum>a | a \<in> A \<and> f a \<noteq> 0. f a)"
 proof -
   have eq: "\<And>x. x \<in> A \<Longrightarrow> (\<Sum>a | x = a \<and> a \<in> A \<and> f a \<noteq> 0. f a) = (\<Sum>x\<in>{x}. f x)"
-    by (intro setsum.mono_neutral_cong_left) auto
+    by (intro sum.mono_neutral_cong_left) auto
 
   have "(\<integral>x. f x \<partial>count_space A) = (\<integral>x. (\<Sum>a | a \<in> A \<and> f a \<noteq> 0. indicator {a} x *\<^sub>R f a) \<partial>count_space A)"
     by (intro integral_cong refl) (simp add: f eq)
   also have "\<dots> = (\<Sum>a | a \<in> A \<and> f a \<noteq> 0. measure (count_space A) {a} *\<^sub>R f a)"
-    by (subst integral_setsum) (auto intro!: setsum.cong)
+    by (subst integral_sum) (auto intro!: sum.cong)
   finally show ?thesis
     by auto
 qed
 
 lemma lebesgue_integral_count_space_finite: "finite A \<Longrightarrow> (\<integral>x. f x \<partial>count_space A) = (\<Sum>a\<in>A. f a)"
   by (subst lebesgue_integral_count_space_finite_support)
-     (auto intro!: setsum.mono_neutral_cong_left)
+     (auto intro!: sum.mono_neutral_cong_left)
 
 lemma integrable_count_space_nat_iff:
   fixes f :: "nat \<Rightarrow> _::{banach,second_countable_topology}"
@@ -2491,7 +2491,7 @@
       by (auto split: split_indicator simp: eq_commute[of x] cong: conj_cong)
   qed
   also have "\<dots> = (\<Sum>a\<in>A. f a * measure M {a})"
-    using finite by (subst integral_setsum) (auto simp add: integrable_mult_left_iff)
+    using finite by (subst integral_sum) (auto simp add: integrable_mult_left_iff)
   finally show ?thesis .
 qed
 
@@ -2645,7 +2645,7 @@
       (\<Sum>z\<in>s i ` (space N \<times> space M). measure M {y \<in> space M. s i (x, y) = z} *\<^sub>R z)"
       using s(1)[THEN simple_functionD(1)]
       unfolding simple_bochner_integral_def
-      by (intro setsum.mono_neutral_cong_left)
+      by (intro sum.mono_neutral_cong_left)
          (auto simp: eq_commute space_pair_measure image_iff cong: conj_cong) }
   note eq = this
 
--- a/src/HOL/Analysis/Borel_Space.thy	Mon Oct 17 11:07:01 2016 +0200
+++ b/src/HOL/Analysis/Borel_Space.thy	Mon Oct 17 14:37:32 2016 +0200
@@ -969,7 +969,7 @@
   shows "(\<lambda>x. f x + g x) \<in> borel_measurable M"
   using f g by (rule borel_measurable_continuous_Pair) (intro continuous_intros)
 
-lemma borel_measurable_setsum[measurable (raw)]:
+lemma borel_measurable_sum[measurable (raw)]:
   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> 'b::{second_countable_topology, topological_comm_monoid_add}"
   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
   shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> borel_measurable M"
@@ -1646,7 +1646,7 @@
     and borel_measurable_ereal_divide: "(\<lambda>x. f x / g x) \<in> borel_measurable M"
   using assms by (simp_all add: minus_ereal_def divide_ereal_def)
 
-lemma borel_measurable_ereal_setsum[measurable (raw)]:
+lemma borel_measurable_ereal_sum[measurable (raw)]:
   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> ereal"
   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
   shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> borel_measurable M"
--- a/src/HOL/Analysis/Bounded_Continuous_Function.thy	Mon Oct 17 11:07:01 2016 +0200
+++ b/src/HOL/Analysis/Bounded_Continuous_Function.thy	Mon Oct 17 14:37:32 2016 +0200
@@ -426,7 +426,7 @@
     by (simp add: cbox_def)
   then have "(\<Sum>i\<in>Basis. (dist (clamp a b y \<bullet> i) (clamp a b x \<bullet> i))\<^sup>2) \<le>
     (\<Sum>i\<in>Basis. (dist (y \<bullet> i) (x \<bullet> i))\<^sup>2)"
-    by (auto intro!: setsum_mono simp: clamp_def dist_real_def abs_le_square_iff[symmetric])
+    by (auto intro!: sum_mono simp: clamp_def dist_real_def abs_le_square_iff[symmetric])
   then show ?thesis
     by (auto intro: real_sqrt_le_mono
       simp: euclidean_dist_l2[where y=x] euclidean_dist_l2[where y="clamp a b x"] setL2_def)
--- a/src/HOL/Analysis/Bounded_Linear_Function.thy	Mon Oct 17 11:07:01 2016 +0200
+++ b/src/HOL/Analysis/Bounded_Linear_Function.thy	Mon Oct 17 14:37:32 2016 +0200
@@ -43,7 +43,7 @@
   bounded_linear_scaleR_const
   bounded_linear_const_scaleR
   bounded_linear_ident
-  bounded_linear_setsum
+  bounded_linear_sum
   bounded_linear_Pair
   bounded_linear_sub
   bounded_linear_fst_comp
@@ -204,8 +204,8 @@
   scaleR_right
   zero_left
   zero_right
-  setsum_left
-  setsum_right
+  sum_left
+  sum_right
 
 end
 
@@ -340,20 +340,20 @@
 
 subsection \<open>On Euclidean Space\<close>
 
-lemma Zfun_setsum:
+lemma Zfun_sum:
   assumes "finite s"
   assumes f: "\<And>i. i \<in> s \<Longrightarrow> Zfun (f i) F"
-  shows "Zfun (\<lambda>x. setsum (\<lambda>i. f i x) s) F"
+  shows "Zfun (\<lambda>x. sum (\<lambda>i. f i x) s) F"
   using assms by induct (auto intro!: Zfun_zero Zfun_add)
 
 lemma norm_blinfun_euclidean_le:
   fixes a::"'a::euclidean_space \<Rightarrow>\<^sub>L 'b::real_normed_vector"
-  shows "norm a \<le> setsum (\<lambda>x. norm (a x)) Basis"
+  shows "norm a \<le> sum (\<lambda>x. norm (a x)) Basis"
   apply (rule norm_blinfun_bound)
-   apply (simp add: setsum_nonneg)
+   apply (simp add: sum_nonneg)
   apply (subst euclidean_representation[symmetric, where 'a='a])
-  apply (simp only: blinfun.bilinear_simps setsum_distrib_right)
-  apply (rule order.trans[OF norm_setsum setsum_mono])
+  apply (simp only: blinfun.bilinear_simps sum_distrib_right)
+  apply (rule order.trans[OF norm_sum sum_mono])
   apply (simp add: abs_mult mult_right_mono ac_simps Basis_le_norm)
   done
 
@@ -366,7 +366,7 @@
   have "\<And>j. j \<in> Basis \<Longrightarrow> Zfun (\<lambda>x. norm (b x j - a j)) F"
     using assms unfolding tendsto_Zfun_iff Zfun_norm_iff .
   hence "Zfun (\<lambda>x. \<Sum>j\<in>Basis. norm (b x j - a j)) F"
-    by (auto intro!: Zfun_setsum)
+    by (auto intro!: Zfun_sum)
   thus ?thesis
     unfolding tendsto_Zfun_iff
     by (rule Zfun_le)
@@ -387,9 +387,9 @@
   have "(\<Sum>j\<in>Basis. \<Sum>i\<in>Basis. (x \<bullet> i * (f i \<bullet> j)) *\<^sub>R j) \<bullet> b
     = (\<Sum>j\<in>Basis. if j = b then (\<Sum>i\<in>Basis. (x \<bullet> i * (f i \<bullet> j))) else 0)"
     using b
-    by (auto simp add: algebra_simps inner_setsum_left inner_Basis split: if_split intro!: setsum.cong)
+    by (auto simp add: algebra_simps inner_sum_left inner_Basis split: if_split intro!: sum.cong)
   also have "\<dots> = (\<Sum>i\<in>Basis. (x \<bullet> i * (f i \<bullet> b)))"
-    using b by (simp add: setsum.delta)
+    using b by (simp add: sum.delta)
   also have "\<dots> = f x \<bullet> b"
     by (metis (mono_tags, lifting) Linear_Algebra.linear_componentwise linear_axioms)
   finally show "(\<Sum>j\<in>Basis. \<Sum>i\<in>Basis. (x \<bullet> i * (f i \<bullet> j)) *\<^sub>R j) \<bullet> b = f x \<bullet> b" .
@@ -400,15 +400,15 @@
   by transfer simp
 
 lemma blinfun_of_matrix_minus: "blinfun_of_matrix x - blinfun_of_matrix y = blinfun_of_matrix (x - y)"
-  by transfer (auto simp: algebra_simps setsum_subtractf)
+  by transfer (auto simp: algebra_simps sum_subtractf)
 
 lemma norm_blinfun_of_matrix:
   "norm (blinfun_of_matrix a) \<le> (\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. \<bar>a i j\<bar>)"
   apply (rule norm_blinfun_bound)
-   apply (simp add: setsum_nonneg)
-  apply (simp only: blinfun_of_matrix_apply setsum_distrib_right)
-  apply (rule order_trans[OF norm_setsum setsum_mono])
-  apply (rule order_trans[OF norm_setsum setsum_mono])
+   apply (simp add: sum_nonneg)
+  apply (simp only: blinfun_of_matrix_apply sum_distrib_right)
+  apply (rule order_trans[OF norm_sum sum_mono])
+  apply (rule order_trans[OF norm_sum sum_mono])
   apply (simp add: abs_mult mult_right_mono ac_simps Basis_le_norm)
   done
 
@@ -419,7 +419,7 @@
   have "\<And>i j. i \<in> Basis \<Longrightarrow> j \<in> Basis \<Longrightarrow> Zfun (\<lambda>x. norm (b x i j - a i j)) F"
     using assms unfolding tendsto_Zfun_iff Zfun_norm_iff .
   hence "Zfun (\<lambda>x. (\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. \<bar>b x i j - a i j\<bar>)) F"
-    by (auto intro!: Zfun_setsum)
+    by (auto intro!: Zfun_sum)
   thus ?thesis
     unfolding tendsto_Zfun_iff blinfun_of_matrix_minus
     by (rule Zfun_le) (auto intro!: order_trans[OF norm_blinfun_of_matrix])
@@ -480,8 +480,8 @@
     subseq r \<and> (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) i) (l i) < e) sequentially)"
   by (rule compact_lemma_general[where unproj = "\<lambda>e. blinfun_of_matrix (\<lambda>i j. e j \<bullet> i)"])
    (auto intro!: euclidean_eqI[where 'a='b] bounded_linear_image assms
-    simp: blinfun_of_matrix_works blinfun_of_matrix_apply inner_Basis mult_if_delta setsum.delta'
-      scaleR_setsum_left[symmetric])
+    simp: blinfun_of_matrix_works blinfun_of_matrix_apply inner_Basis mult_if_delta sum.delta'
+      scaleR_sum_left[symmetric])
 
 lemma blinfun_euclidean_eqI: "(\<And>i. i \<in> Basis \<Longrightarrow> blinfun_apply x i = blinfun_apply y i) \<Longrightarrow> x = y"
   apply (auto intro!: blinfun_eqI)
@@ -493,7 +493,7 @@
 lemma Blinfun_eq_matrix: "bounded_linear f \<Longrightarrow> Blinfun f = blinfun_of_matrix (\<lambda>i j. f j \<bullet> i)"
   by (intro blinfun_euclidean_eqI)
      (auto simp: blinfun_of_matrix_apply bounded_linear_Blinfun_apply inner_Basis if_distrib
-      cond_application_beta setsum.delta' euclidean_representation
+      cond_application_beta sum.delta' euclidean_representation
       cong: if_cong)
 
 text \<open>TODO: generalize (via @{thm compact_cball})?\<close>
@@ -520,10 +520,10 @@
       also note norm_blinfun_of_matrix
       also have "(\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. \<bar>(f (r n) - l) j \<bullet> i\<bar>) <
         (\<Sum>i\<in>(Basis::'b set). e / real_of_nat DIM('b))"
-      proof (rule setsum_strict_mono)
+      proof (rule sum_strict_mono)
         fix i::'b assume i: "i \<in> Basis"
         have "(\<Sum>j::'a\<in>Basis. \<bar>(f (r n) - l) j \<bullet> i\<bar>) < (\<Sum>j::'a\<in>Basis. e / ?d)"
-        proof (rule setsum_strict_mono)
+        proof (rule sum_strict_mono)
           fix j::'a assume j: "j \<in> Basis"
           have "\<bar>(f (r n) - l) j \<bullet> i\<bar> \<le> norm ((f (r n) - l) j)"
             by (simp add: Basis_le_norm i)
--- a/src/HOL/Analysis/Brouwer_Fixpoint.thy	Mon Oct 17 11:07:01 2016 +0200
+++ b/src/HOL/Analysis/Brouwer_Fixpoint.thy	Mon Oct 17 14:37:32 2016 +0200
@@ -52,13 +52,13 @@
 lemmas Zero_notin_Suc = zero_notin_Suc_image
 lemmas atMost_Suc_eq_insert_0 = Iic_Suc_eq_insert_0
 
-lemma setsum_union_disjoint':
+lemma sum_union_disjoint':
   assumes "finite A"
     and "finite B"
     and "A \<inter> B = {}"
     and "A \<union> B = C"
-  shows "setsum g C = setsum g A + setsum g B"
-  using setsum.union_disjoint[OF assms(1-3)] and assms(4) by auto
+  shows "sum g C = sum g A + sum g B"
+  using sum.union_disjoint[OF assms(1-3)] and assms(4) by auto
 
 lemma pointwise_minimal_pointwise_maximal:
   fixes s :: "(nat \<Rightarrow> nat) set"
@@ -139,19 +139,19 @@
   shows "odd (card {s\<in>S. compo s})"
 proof -
   have "(\<Sum>s | s \<in> S \<and> \<not> compo s. nF s) + (\<Sum>s | s \<in> S \<and> compo s. nF s) = (\<Sum>s\<in>S. nF s)"
-    by (subst setsum.union_disjoint[symmetric]) (auto intro!: setsum.cong)
+    by (subst sum.union_disjoint[symmetric]) (auto intro!: sum.cong)
   also have "\<dots> = (\<Sum>s\<in>S. card {f \<in> {f\<in>F. compo' f \<and> bnd f}. face f s}) +
                   (\<Sum>s\<in>S. card {f \<in> {f\<in>F. compo' f \<and> \<not> bnd f}. face f s})"
-    unfolding setsum.distrib[symmetric]
+    unfolding sum.distrib[symmetric]
     by (subst card_Un_disjoint[symmetric])
-       (auto simp: nF_def intro!: setsum.cong arg_cong[where f=card])
+       (auto simp: nF_def intro!: sum.cong arg_cong[where f=card])
   also have "\<dots> = 1 * card {f\<in>F. compo' f \<and> bnd f} + 2 * card {f\<in>F. compo' f \<and> \<not> bnd f}"
-    using assms(4,5) by (fastforce intro!: arg_cong2[where f="op +"] setsum_multicount)
+    using assms(4,5) by (fastforce intro!: arg_cong2[where f="op +"] sum_multicount)
   finally have "odd ((\<Sum>s | s \<in> S \<and> \<not> compo s. nF s) + card {s\<in>S. compo s})"
     using assms(6,8) by simp
   moreover have "(\<Sum>s | s \<in> S \<and> \<not> compo s. nF s) =
     (\<Sum>s | s \<in> S \<and> \<not> compo s \<and> nF s = 0. nF s) + (\<Sum>s | s \<in> S \<and> \<not> compo s \<and> nF s = 2. nF s)"
-    using assms(7) by (subst setsum.union_disjoint[symmetric]) (fastforce intro!: setsum.cong)+
+    using assms(7) by (subst sum.union_disjoint[symmetric]) (fastforce intro!: sum.cong)+
   ultimately show ?thesis
     by auto
 qed
@@ -1417,9 +1417,9 @@
   fix y :: 'a assume y: "y \<in> unit_cube"
   have "dist 0 y = norm y" by (rule dist_0_norm)
   also have "\<dots> = norm (\<Sum>i\<in>Basis. (y \<bullet> i) *\<^sub>R i)" unfolding euclidean_representation ..
-  also have "\<dots> \<le> (\<Sum>i\<in>Basis. norm ((y \<bullet> i) *\<^sub>R i))" by (rule norm_setsum)
+  also have "\<dots> \<le> (\<Sum>i\<in>Basis. norm ((y \<bullet> i) *\<^sub>R i))" by (rule norm_sum)
   also have "\<dots> \<le> (\<Sum>i::'a\<in>Basis. 1)"
-    by (rule setsum_mono, simp add: y [unfolded mem_unit_cube])
+    by (rule sum_mono, simp add: y [unfolded mem_unit_cube])
   finally show "dist 0 y \<le> (\<Sum>i::'a\<in>Basis. 1)" .
 qed
 
@@ -1679,7 +1679,7 @@
       unfolding inner_diff_left[symmetric]
       by (rule norm_le_l1)
     also have "\<dots> < (\<Sum>(i::'a) \<in> Basis. d / real n)"
-      apply (rule setsum_strict_mono)
+      apply (rule sum_strict_mono)
       using as
       apply auto
       done
@@ -1731,7 +1731,7 @@
     by auto
   {
     have "(\<Sum>i\<in>Basis. \<bar>real (r (b' i)) - real (q (b' i))\<bar>) \<le> (\<Sum>(i::'a)\<in>Basis. 1)"
-      apply (rule setsum_mono)
+      apply (rule sum_mono)
       using rs(1)[OF b'_im]
       apply (auto simp add:* field_simps simp del: of_nat_Suc)
       done
@@ -1743,7 +1743,7 @@
   moreover
   {
     have "(\<Sum>i\<in>Basis. \<bar>real (s (b' i)) - real (q (b' i))\<bar>) \<le> (\<Sum>(i::'a)\<in>Basis. 1)"
-      apply (rule setsum_mono)
+      apply (rule sum_mono)
       using rs(2)[OF b'_im]
       apply (auto simp add:* field_simps simp del: of_nat_Suc)
       done
@@ -1757,7 +1757,7 @@
     unfolding r'_def s'_def z_def
     using \<open>p > 0\<close>
     apply (rule_tac[!] le_less_trans[OF norm_le_l1])
-    apply (auto simp add: field_simps setsum_divide_distrib[symmetric] inner_diff_left)
+    apply (auto simp add: field_simps sum_divide_distrib[symmetric] inner_diff_left)
     done
   then have "\<bar>(f z - z) \<bullet> i\<bar> < d / real n"
     using rs(3) i
--- a/src/HOL/Analysis/Caratheodory.thy	Mon Oct 17 11:07:01 2016 +0200
+++ b/src/HOL/Analysis/Caratheodory.thy	Mon Oct 17 14:37:32 2016 +0200
@@ -20,31 +20,31 @@
 proof -
   have g_def: "g = (\<lambda>m. (\<Sum>n. f (m,n)))"
     using assms by (simp add: fun_eq_iff)
-  have reindex: "\<And>B. (\<Sum>x\<in>B. f (prod_decode x)) = setsum f (prod_decode ` B)"
-    by (simp add: setsum.reindex[OF inj_prod_decode] comp_def)
+  have reindex: "\<And>B. (\<Sum>x\<in>B. f (prod_decode x)) = sum f (prod_decode ` B)"
+    by (simp add: sum.reindex[OF inj_prod_decode] comp_def)
   have "(SUP n. \<Sum>i<n. f (prod_decode i)) = (SUP p : UNIV \<times> UNIV. \<Sum>i<fst p. \<Sum>n<snd p. f (i, n))"
-  proof (intro SUP_eq; clarsimp simp: setsum.cartesian_product reindex)
+  proof (intro SUP_eq; clarsimp simp: sum.cartesian_product reindex)
     fix n
     let ?M = "\<lambda>f. Suc (Max (f ` prod_decode ` {..<n}))"
     { fix a b x assume "x < n" and [symmetric]: "(a, b) = prod_decode x"
       then have "a < ?M fst" "b < ?M snd"
         by (auto intro!: Max_ge le_imp_less_Suc image_eqI) }
-    then have "setsum f (prod_decode ` {..<n}) \<le> setsum f ({..<?M fst} \<times> {..<?M snd})"
-      by (auto intro!: setsum_mono3)
-    then show "\<exists>a b. setsum f (prod_decode ` {..<n}) \<le> setsum f ({..<a} \<times> {..<b})" by auto
+    then have "sum f (prod_decode ` {..<n}) \<le> sum f ({..<?M fst} \<times> {..<?M snd})"
+      by (auto intro!: sum_mono3)
+    then show "\<exists>a b. sum f (prod_decode ` {..<n}) \<le> sum f ({..<a} \<times> {..<b})" by auto
   next
     fix a b
     let ?M = "prod_decode ` {..<Suc (Max (prod_encode ` ({..<a} \<times> {..<b})))}"
     { fix a' b' assume "a' < a" "b' < b" then have "(a', b') \<in> ?M"
         by (auto intro!: Max_ge le_imp_less_Suc image_eqI[where x="prod_encode (a', b')"]) }
-    then have "setsum f ({..<a} \<times> {..<b}) \<le> setsum f ?M"
-      by (auto intro!: setsum_mono3)
-    then show "\<exists>n. setsum f ({..<a} \<times> {..<b}) \<le> setsum f (prod_decode ` {..<n})"
+    then have "sum f ({..<a} \<times> {..<b}) \<le> sum f ?M"
+      by (auto intro!: sum_mono3)
+    then show "\<exists>n. sum f ({..<a} \<times> {..<b}) \<le> sum f (prod_decode ` {..<n})"
       by auto
   qed
   also have "\<dots> = (SUP p. \<Sum>i<p. \<Sum>n. f (i, n))"
-    unfolding suminf_setsum[OF summableI, symmetric]
-    by (simp add: suminf_eq_SUP SUP_pair setsum.commute[of _ "{..< fst _}"])
+    unfolding suminf_sum[OF summableI, symmetric]
+    by (simp add: suminf_eq_SUP SUP_pair sum.commute[of _ "{..< fst _}"])
   finally show ?thesis unfolding g_def
     by (simp add: suminf_eq_SUP)
 qed
@@ -529,7 +529,7 @@
   with \<open>volume M f\<close> have "f (\<Union>(A`I)) = (\<Sum>a\<in>A`I. f a)"
     unfolding volume_def by blast
   also have "\<dots> = (\<Sum>i\<in>I. f (A i))"
-  proof (subst setsum.reindex_nontrivial)
+  proof (subst sum.reindex_nontrivial)
     fix i j assume "i \<in> I" "j \<in> I" "i \<noteq> j" "A i = A j"
     with \<open>disjoint_family_on A I\<close> have "A i = {}"
       by (auto simp: disjoint_family_on_def)
@@ -547,7 +547,7 @@
   shows "volume M \<mu>"
 proof (unfold volume_def, safe)
   fix C assume "finite C" "C \<subseteq> M" "disjoint C"
-  then show "\<mu> (\<Union>C) = setsum \<mu> C"
+  then show "\<mu> (\<Union>C) = sum \<mu> C"
   proof (induct C)
     case (insert c C)
     from insert(1,2,4,5) have "\<mu> (\<Union>insert c C) = \<mu> c + \<mu> (\<Union>C)"
@@ -570,7 +570,7 @@
     fix D assume D: "D \<subseteq> M" "finite D" "disjoint D"
     assume "\<Union>C = \<Union>D"
     have "(\<Sum>d\<in>D. \<mu> d) = (\<Sum>d\<in>D. \<Sum>c\<in>C. \<mu> (c \<inter> d))"
-    proof (intro setsum.cong refl)
+    proof (intro sum.cong refl)
       fix d assume "d \<in> D"
       have Un_eq_d: "(\<Union>c\<in>C. c \<inter> d) = d"
         using \<open>d \<in> D\<close> \<open>\<Union>C = \<Union>D\<close> by auto
@@ -592,7 +592,7 @@
     assume "\<Union>C = \<Union>D"
     with split_sum[OF C D] split_sum[OF D C]
     have "(\<Sum>d\<in>D. \<mu> d) = (\<Sum>c\<in>C. \<mu> c)"
-      by (simp, subst setsum.commute, simp add: ac_simps) }
+      by (simp, subst sum.commute, simp add: ac_simps) }
   note sum_eq = this
 
   { fix C assume C: "C \<subseteq> M" "finite C" "disjoint C"
@@ -612,7 +612,7 @@
     fix a assume "a \<in> ?R" then guess Ca .. note Ca = this
     with \<mu>'[of Ca] \<open>volume M \<mu>\<close>[THEN volume_positive]
     show "0 \<le> \<mu>' a"
-      by (auto intro!: setsum_nonneg)
+      by (auto intro!: sum_nonneg)
   next
     show "\<mu>' {} = 0" using \<mu>'[of "{}"] by auto
   next
@@ -627,7 +627,7 @@
     also have "\<dots> = (\<Sum>c\<in>Ca \<union> Cb. \<mu> c) + (\<Sum>c\<in>Ca \<inter> Cb. \<mu> c)"
       using C_Int_cases volume_empty[OF \<open>volume M \<mu>\<close>] by (elim disjE) simp_all
     also have "\<dots> = (\<Sum>c\<in>Ca. \<mu> c) + (\<Sum>c\<in>Cb. \<mu> c)"
-      using Ca Cb by (simp add: setsum.union_inter)
+      using Ca Cb by (simp add: sum.union_inter)
     also have "\<dots> = \<mu>' a + \<mu>' b"
       using Ca Cb by (simp add: \<mu>')
     finally show "\<mu>' (a \<union> b) = \<mu>' a + \<mu>' b"
@@ -680,7 +680,7 @@
         by (simp add: sums_iff)
     qed
     also have "\<dots> = (\<Sum>c\<in>C. \<mu> c)"
-      using F'(2) by (subst (2) F') (simp add: setsum.reindex)
+      using F'(2) by (subst (2) F') (simp add: sum.reindex)
     finally show "\<mu> (\<Union>C) = (\<Sum>c\<in>C. \<mu> c)" .
   next
     show "\<mu> {} = 0"
@@ -798,9 +798,9 @@
                intro: generated_ringI_Basic)
     also have "\<dots> = (\<Sum>c\<in>C'. \<Sum>n. \<mu>_r (A' n \<inter> c))"
       using C' A'
-      by (intro suminf_setsum G.Int G.finite_Union) (auto intro: generated_ringI_Basic)
+      by (intro suminf_sum G.Int G.finite_Union) (auto intro: generated_ringI_Basic)
     also have "\<dots> = (\<Sum>c\<in>C'. \<mu>_r c)"
-      using eq V C' by (auto intro!: setsum.cong)
+      using eq V C' by (auto intro!: sum.cong)
     also have "\<dots> = \<mu>_r (\<Union>C')"
       using C' Un_A
       by (subst volume_finite_additive[symmetric, OF V(1)])
--- a/src/HOL/Analysis/Cartesian_Euclidean_Space.thy	Mon Oct 17 11:07:01 2016 +0200
+++ b/src/HOL/Analysis/Cartesian_Euclidean_Space.thy	Mon Oct 17 14:37:32 2016 +0200
@@ -12,18 +12,18 @@
   by simp
 
 (*move up?*)
-lemma setsum_UNIV_sum:
+lemma sum_UNIV_sum:
   fixes g :: "'a::finite + 'b::finite \<Rightarrow> _"
   shows "(\<Sum>x\<in>UNIV. g x) = (\<Sum>x\<in>UNIV. g (Inl x)) + (\<Sum>x\<in>UNIV. g (Inr x))"
   apply (subst UNIV_Plus_UNIV [symmetric])
-  apply (subst setsum.Plus)
+  apply (subst sum.Plus)
   apply simp_all
   done
 
-lemma setsum_mult_product:
-  "setsum h {..<A * B :: nat} = (\<Sum>i\<in>{..<A}. \<Sum>j\<in>{..<B}. h (j + i * B))"
-  unfolding setsum_nat_group[of h B A, unfolded atLeast0LessThan, symmetric]
-proof (rule setsum.cong, simp, rule setsum.reindex_cong)
+lemma sum_mult_product:
+  "sum h {..<A * B :: nat} = (\<Sum>i\<in>{..<A}. \<Sum>j\<in>{..<B}. h (j + i * B))"
+  unfolding sum_nat_group[of h B A, unfolded atLeast0LessThan, symmetric]
+proof (rule sum.cong, simp, rule sum.reindex_cong)
   fix i
   show "inj_on (\<lambda>j. j + i * B) {..<B}" by (auto intro!: inj_onI)
   show "{i * B..<i * B + B} = (\<lambda>j. j + i * B) ` {..<B}"
@@ -108,19 +108,19 @@
 
 subsection \<open>A naive proof procedure to lift really trivial arithmetic stuff from the basis of the vector space.\<close>
 
-lemma setsum_cong_aux:
-  "(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> setsum f A = setsum g A"
-  by (auto intro: setsum.cong)
+lemma sum_cong_aux:
+  "(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> sum f A = sum g A"
+  by (auto intro: sum.cong)
 
-hide_fact (open) setsum_cong_aux
+hide_fact (open) sum_cong_aux
 
 method_setup vector = \<open>
 let
   val ss1 =
     simpset_of (put_simpset HOL_basic_ss @{context}
-      addsimps [@{thm setsum.distrib} RS sym,
-      @{thm setsum_subtractf} RS sym, @{thm setsum_distrib_left},
-      @{thm setsum_distrib_right}, @{thm setsum_negf} RS sym])
+      addsimps [@{thm sum.distrib} RS sym,
+      @{thm sum_subtractf} RS sym, @{thm sum_distrib_left},
+      @{thm sum_distrib_right}, @{thm sum_negf} RS sym])
   val ss2 =
     simpset_of (@{context} addsimps
              [@{thm plus_vec_def}, @{thm times_vec_def},
@@ -130,8 +130,8 @@
               @{thm vec_lambda_beta}, @{thm vector_scalar_mult_def}])
   fun vector_arith_tac ctxt ths =
     simp_tac (put_simpset ss1 ctxt)
-    THEN' (fn i => resolve_tac ctxt @{thms Cartesian_Euclidean_Space.setsum_cong_aux} i
-         ORELSE resolve_tac ctxt @{thms setsum.neutral} i
+    THEN' (fn i => resolve_tac ctxt @{thms Cartesian_Euclidean_Space.sum_cong_aux} i
+         ORELSE resolve_tac ctxt @{thms sum.neutral} i
          ORELSE simp_tac (put_simpset HOL_basic_ss ctxt addsimps [@{thm vec_eq_iff}]) i)
     (* THEN' TRY o clarify_tac HOL_cs  THEN' (TRY o rtac @{thm iffI}) *)
     THEN' asm_full_simp_tac (put_simpset ss2 ctxt addsimps ths)
@@ -152,9 +152,9 @@
 lemma vec_cmul: "vec(c * x) = c *s vec x " by vector
 lemma vec_neg: "vec(- x) = - vec x " by vector
 
-lemma vec_setsum:
+lemma vec_sum:
   assumes "finite S"
-  shows "vec(setsum f S) = setsum (vec \<circ> f) S"
+  shows "vec(sum f S) = sum (vec \<circ> f) S"
   using assms
 proof induct
   case empty
@@ -299,8 +299,8 @@
 lemma norm_bound_component_lt_cart: "norm x < e ==> \<bar>x$i\<bar> < e"
   by (metis component_le_norm_cart le_less_trans)
 
-lemma norm_le_l1_cart: "norm x <= setsum(\<lambda>i. \<bar>x$i\<bar>) UNIV"
-  by (simp add: norm_vec_def setL2_le_setsum)
+lemma norm_le_l1_cart: "norm x <= sum(\<lambda>i. \<bar>x$i\<bar>) UNIV"
+  by (simp add: norm_vec_def setL2_le_sum)
 
 lemma scalar_mult_eq_scaleR: "c *s x = c *\<^sub>R x"
   unfolding scaleR_vec_def vector_scalar_mult_def by simp
@@ -309,9 +309,9 @@
   unfolding dist_norm scalar_mult_eq_scaleR
   unfolding scaleR_right_diff_distrib[symmetric] by simp
 
-lemma setsum_component [simp]:
+lemma sum_component [simp]:
   fixes f:: " 'a \<Rightarrow> ('b::comm_monoid_add) ^'n"
-  shows "(setsum f S)$i = setsum (\<lambda>x. (f x)$i) S"
+  shows "(sum f S)$i = sum (\<lambda>x. (f x)$i) S"
 proof (cases "finite S")
   case True
   then show ?thesis by induct simp_all
@@ -320,19 +320,19 @@
   then show ?thesis by simp
 qed
 
-lemma setsum_eq: "setsum f S = (\<chi> i. setsum (\<lambda>x. (f x)$i ) S)"
+lemma sum_eq: "sum f S = (\<chi> i. sum (\<lambda>x. (f x)$i ) S)"
   by (simp add: vec_eq_iff)
 
-lemma setsum_cmul:
+lemma sum_cmul:
   fixes f:: "'c \<Rightarrow> ('a::semiring_1)^'n"
-  shows "setsum (\<lambda>x. c *s f x) S = c *s setsum f S"
-  by (simp add: vec_eq_iff setsum_distrib_left)
+  shows "sum (\<lambda>x. c *s f x) S = c *s sum f S"
+  by (simp add: vec_eq_iff sum_distrib_left)
 
-lemma setsum_norm_allsubsets_bound_cart:
+lemma sum_norm_allsubsets_bound_cart:
   fixes f:: "'a \<Rightarrow> real ^'n"
-  assumes fP: "finite P" and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (setsum f Q) \<le> e"
-  shows "setsum (\<lambda>x. norm (f x)) P \<le> 2 * real CARD('n) *  e"
-  using setsum_norm_allsubsets_bound[OF assms]
+  assumes fP: "finite P" and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (sum f Q) \<le> e"
+  shows "sum (\<lambda>x. norm (f x)) P \<le> 2 * real CARD('n) *  e"
+  using sum_norm_allsubsets_bound[OF assms]
   by simp
 
 subsection\<open>Closures and interiors of halfspaces\<close>
@@ -475,15 +475,15 @@
 
 definition matrix_matrix_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'p^'n \<Rightarrow> 'a ^ 'p ^'m"
     (infixl "**" 70)
-  where "m ** m' == (\<chi> i j. setsum (\<lambda>k. ((m$i)$k) * ((m'$k)$j)) (UNIV :: 'n set)) ::'a ^ 'p ^'m"
+  where "m ** m' == (\<chi> i j. sum (\<lambda>k. ((m$i)$k) * ((m'$k)$j)) (UNIV :: 'n set)) ::'a ^ 'p ^'m"
 
 definition matrix_vector_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n \<Rightarrow> 'a ^ 'm"
     (infixl "*v" 70)
-  where "m *v x \<equiv> (\<chi> i. setsum (\<lambda>j. ((m$i)$j) * (x$j)) (UNIV ::'n set)) :: 'a^'m"
+  where "m *v x \<equiv> (\<chi> i. sum (\<lambda>j. ((m$i)$j) * (x$j)) (UNIV ::'n set)) :: 'a^'m"
 
 definition vector_matrix_mult :: "'a ^ 'm \<Rightarrow> ('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n "
     (infixl "v*" 70)
-  where "v v* m == (\<chi> j. setsum (\<lambda>i. ((m$i)$j) * (v$i)) (UNIV :: 'm set)) :: 'a^'n"
+  where "v v* m == (\<chi> j. sum (\<lambda>i. ((m$i)$j) * (v$i)) (UNIV :: 'm set)) :: 'a^'n"
 
 definition "(mat::'a::zero => 'a ^'n^'n) k = (\<chi> i j. if i = j then k else 0)"
 definition transpose where
@@ -495,14 +495,14 @@
 
 lemma mat_0[simp]: "mat 0 = 0" by (vector mat_def)
 lemma matrix_add_ldistrib: "(A ** (B + C)) = (A ** B) + (A ** C)"
-  by (vector matrix_matrix_mult_def setsum.distrib[symmetric] field_simps)
+  by (vector matrix_matrix_mult_def sum.distrib[symmetric] field_simps)
 
 lemma matrix_mul_lid:
   fixes A :: "'a::semiring_1 ^ 'm ^ 'n"
   shows "mat 1 ** A = A"
   apply (simp add: matrix_matrix_mult_def mat_def)
   apply vector
-  apply (auto simp only: if_distrib cond_application_beta setsum.delta'[OF finite]
+  apply (auto simp only: if_distrib cond_application_beta sum.delta'[OF finite]
     mult_1_left mult_zero_left if_True UNIV_I)
   done
 
@@ -512,26 +512,26 @@
   shows "A ** mat 1 = A"
   apply (simp add: matrix_matrix_mult_def mat_def)
   apply vector
-  apply (auto simp only: if_distrib cond_application_beta setsum.delta[OF finite]
+  apply (auto simp only: if_distrib cond_application_beta sum.delta[OF finite]
     mult_1_right mult_zero_right if_True UNIV_I cong: if_cong)
   done
 
 lemma matrix_mul_assoc: "A ** (B ** C) = (A ** B) ** C"
-  apply (vector matrix_matrix_mult_def setsum_distrib_left setsum_distrib_right mult.assoc)
-  apply (subst setsum.commute)
+  apply (vector matrix_matrix_mult_def sum_distrib_left sum_distrib_right mult.assoc)
+  apply (subst sum.commute)
   apply simp
   done
 
 lemma matrix_vector_mul_assoc: "A *v (B *v x) = (A ** B) *v x"
   apply (vector matrix_matrix_mult_def matrix_vector_mult_def
-    setsum_distrib_left setsum_distrib_right mult.assoc)
-  apply (subst setsum.commute)
+    sum_distrib_left sum_distrib_right mult.assoc)
+  apply (subst sum.commute)
   apply simp
   done
 
 lemma matrix_vector_mul_lid: "mat 1 *v x = (x::'a::semiring_1 ^ 'n)"
   apply (vector matrix_vector_mult_def mat_def)
-  apply (simp add: if_distrib cond_application_beta setsum.delta' cong del: if_weak_cong)
+  apply (simp add: if_distrib cond_application_beta sum.delta' cong del: if_weak_cong)
   done
 
 lemma matrix_transpose_mul:
@@ -548,15 +548,15 @@
   apply (erule_tac x="axis ia 1" in allE)
   apply (erule_tac x="i" in allE)
   apply (auto simp add: if_distrib cond_application_beta axis_def
-    setsum.delta[OF finite] cong del: if_weak_cong)
+    sum.delta[OF finite] cong del: if_weak_cong)
   done
 
 lemma matrix_vector_mul_component: "((A::real^_^_) *v x)$k = (A$k) \<bullet> x"
   by (simp add: matrix_vector_mult_def inner_vec_def)
 
 lemma dot_lmul_matrix: "((x::real ^_) v* A) \<bullet> y = x \<bullet> (A *v y)"
-  apply (simp add: inner_vec_def matrix_vector_mult_def vector_matrix_mult_def setsum_distrib_right setsum_distrib_left ac_simps)
-  apply (subst setsum.commute)
+  apply (simp add: inner_vec_def matrix_vector_mult_def vector_matrix_mult_def sum_distrib_right sum_distrib_left ac_simps)
+  apply (subst sum.commute)
   apply simp
   done
 
@@ -588,27 +588,27 @@
   by (simp add: matrix_vector_mult_def inner_vec_def)
 
 lemma matrix_mult_vsum:
-  "(A::'a::comm_semiring_1^'n^'m) *v x = setsum (\<lambda>i. (x$i) *s column i A) (UNIV:: 'n set)"
+  "(A::'a::comm_semiring_1^'n^'m) *v x = sum (\<lambda>i. (x$i) *s column i A) (UNIV:: 'n set)"
   by (simp add: matrix_vector_mult_def vec_eq_iff column_def mult.commute)
 
 lemma vector_componentwise:
   "(x::'a::ring_1^'n) = (\<chi> j. \<Sum>i\<in>UNIV. (x$i) * (axis i 1 :: 'a^'n) $ j)"
-  by (simp add: axis_def if_distrib setsum.If_cases vec_eq_iff)
+  by (simp add: axis_def if_distrib sum.If_cases vec_eq_iff)
 
-lemma basis_expansion: "setsum (\<lambda>i. (x$i) *s axis i 1) UNIV = (x::('a::ring_1) ^'n)"
-  by (auto simp add: axis_def vec_eq_iff if_distrib setsum.If_cases cong del: if_weak_cong)
+lemma basis_expansion: "sum (\<lambda>i. (x$i) *s axis i 1) UNIV = (x::('a::ring_1) ^'n)"
+  by (auto simp add: axis_def vec_eq_iff if_distrib sum.If_cases cong del: if_weak_cong)
 
 lemma linear_componentwise_expansion:
   fixes f:: "real ^'m \<Rightarrow> real ^ _"
   assumes lf: "linear f"
-  shows "(f x)$j = setsum (\<lambda>i. (x$i) * (f (axis i 1)$j)) (UNIV :: 'm set)" (is "?lhs = ?rhs")
+  shows "(f x)$j = sum (\<lambda>i. (x$i) * (f (axis i 1)$j)) (UNIV :: 'm set)" (is "?lhs = ?rhs")
 proof -
   let ?M = "(UNIV :: 'm set)"
   let ?N = "(UNIV :: 'n set)"
-  have "?rhs = (setsum (\<lambda>i.(x$i) *\<^sub>R f (axis i 1) ) ?M)$j"
-    unfolding setsum_component by simp
+  have "?rhs = (sum (\<lambda>i.(x$i) *\<^sub>R f (axis i 1) ) ?M)$j"
+    unfolding sum_component by simp
   then show ?thesis
-    unfolding linear_setsum_mul[OF lf, symmetric]
+    unfolding linear_sum_mul[OF lf, symmetric]
     unfolding scalar_mult_eq_scaleR[symmetric]
     unfolding basis_expansion
     by simp
@@ -630,7 +630,7 @@
 
 lemma matrix_vector_mul_linear: "linear(\<lambda>x. A *v (x::real ^ _))"
   by (simp add: linear_iff matrix_vector_mult_def vec_eq_iff
-      field_simps setsum_distrib_left setsum.distrib)
+      field_simps sum_distrib_left sum.distrib)
 
 lemma matrix_works:
   assumes lf: "linear f"
@@ -652,14 +652,14 @@
   by (simp add: matrix_eq matrix_works matrix_vector_mul_assoc[symmetric] o_def)
 
 lemma matrix_vector_column:
-  "(A::'a::comm_semiring_1^'n^_) *v x = setsum (\<lambda>i. (x$i) *s ((transpose A)$i)) (UNIV:: 'n set)"
+  "(A::'a::comm_semiring_1^'n^_) *v x = sum (\<lambda>i. (x$i) *s ((transpose A)$i)) (UNIV:: 'n set)"
   by (simp add: matrix_vector_mult_def transpose_def vec_eq_iff mult.commute)
 
 lemma adjoint_matrix: "adjoint(\<lambda>x. (A::real^'n^'m) *v x) = (\<lambda>x. transpose A *v x)"
   apply (rule adjoint_unique)
   apply (simp add: transpose_def inner_vec_def matrix_vector_mult_def
-    setsum_distrib_right setsum_distrib_left)
-  apply (subst setsum.commute)
+    sum_distrib_right sum_distrib_left)
+  apply (subst sum.commute)
   apply (auto simp add: ac_simps)
   done
 
@@ -758,13 +758,13 @@
 lemma matrix_left_invertible_independent_columns:
   fixes A :: "real^'n^'m"
   shows "(\<exists>(B::real ^'m^'n). B ** A = mat 1) \<longleftrightarrow>
-      (\<forall>c. setsum (\<lambda>i. c i *s column i A) (UNIV :: 'n set) = 0 \<longrightarrow> (\<forall>i. c i = 0))"
+      (\<forall>c. sum (\<lambda>i. c i *s column i A) (UNIV :: 'n set) = 0 \<longrightarrow> (\<forall>i. c i = 0))"
     (is "?lhs \<longleftrightarrow> ?rhs")
 proof -
   let ?U = "UNIV :: 'n set"
   { assume k: "\<forall>x. A *v x = 0 \<longrightarrow> x = 0"
     { fix c i
-      assume c: "setsum (\<lambda>i. c i *s column i A) ?U = 0" and i: "i \<in> ?U"
+      assume c: "sum (\<lambda>i. c i *s column i A) ?U = 0" and i: "i \<in> ?U"
       let ?x = "\<chi> i. c i"
       have th0:"A *v ?x = 0"
         using c
@@ -786,7 +786,7 @@
 lemma matrix_right_invertible_independent_rows:
   fixes A :: "real^'n^'m"
   shows "(\<exists>(B::real^'m^'n). A ** B = mat 1) \<longleftrightarrow>
-    (\<forall>c. setsum (\<lambda>i. c i *s row i A) (UNIV :: 'm set) = 0 \<longrightarrow> (\<forall>i. c i = 0))"
+    (\<forall>c. sum (\<lambda>i. c i *s row i A) (UNIV :: 'm set) = 0 \<longrightarrow> (\<forall>i. c i = 0))"
   unfolding left_invertible_transpose[symmetric]
     matrix_left_invertible_independent_columns
   by (simp add: column_transpose)
@@ -797,7 +797,7 @@
 proof -
   let ?U = "UNIV :: 'm set"
   have fU: "finite ?U" by simp
-  have lhseq: "?lhs \<longleftrightarrow> (\<forall>y. \<exists>(x::real^'m). setsum (\<lambda>i. (x$i) *s column i A) ?U = y)"
+  have lhseq: "?lhs \<longleftrightarrow> (\<forall>y. \<exists>(x::real^'m). sum (\<lambda>i. (x$i) *s column i A) ?U = y)"
     unfolding matrix_right_invertible_surjective matrix_mult_vsum surj_def
     apply (subst eq_commute)
     apply rule
@@ -806,10 +806,10 @@
   { assume h: ?lhs
     { fix x:: "real ^'n"
       from h[unfolded lhseq, rule_format, of x] obtain y :: "real ^'m"
-        where y: "setsum (\<lambda>i. (y$i) *s column i A) ?U = x" by blast
+        where y: "sum (\<lambda>i. (y$i) *s column i A) ?U = x" by blast
       have "x \<in> span (columns A)"
         unfolding y[symmetric]
-        apply (rule span_setsum)
+        apply (rule span_sum)
         unfolding scalar_mult_eq_scaleR
         apply (rule span_mul)
         apply (rule span_superset)
@@ -820,11 +820,11 @@
     then have ?rhs unfolding rhseq by blast }
   moreover
   { assume h:?rhs
-    let ?P = "\<lambda>(y::real ^'n). \<exists>(x::real^'m). setsum (\<lambda>i. (x$i) *s column i A) ?U = y"
+    let ?P = "\<lambda>(y::real ^'n). \<exists>(x::real^'m). sum (\<lambda>i. (x$i) *s column i A) ?U = y"
     { fix y
       have "?P y"
       proof (rule span_induct_alt[of ?P "columns A", folded scalar_mult_eq_scaleR])
-        show "\<exists>x::real ^ 'm. setsum (\<lambda>i. (x$i) *s column i A) ?U = 0"
+        show "\<exists>x::real ^ 'm. sum (\<lambda>i. (x$i) *s column i A) ?U = 0"
           by (rule exI[where x=0], simp)
       next
         fix c y1 y2
@@ -832,7 +832,7 @@
         from y1 obtain i where i: "i \<in> ?U" "y1 = column i A"
           unfolding columns_def by blast
         from y2 obtain x:: "real ^'m" where
-          x: "setsum (\<lambda>i. (x$i) *s column i A) ?U = y2" by blast
+          x: "sum (\<lambda>i. (x$i) *s column i A) ?U = y2" by blast
         let ?x = "(\<chi> j. if j = i then c + (x$i) else (x$j))::real^'m"
         show "?P (c*s y1 + y2)"
         proof (rule exI[where x= "?x"], vector, auto simp add: i x[symmetric] if_distrib distrib_left cond_application_beta cong del: if_weak_cong)
@@ -840,18 +840,18 @@
           have th: "\<forall>xa \<in> ?U. (if xa = i then (c + (x$i)) * ((column xa A)$j)
               else (x$xa) * ((column xa A$j))) = (if xa = i then c * ((column i A)$j) else 0) + ((x$xa) * ((column xa A)$j))"
             using i(1) by (simp add: field_simps)
-          have "setsum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j)
-              else (x$xa) * ((column xa A$j))) ?U = setsum (\<lambda>xa. (if xa = i then c * ((column i A)$j) else 0) + ((x$xa) * ((column xa A)$j))) ?U"
-            apply (rule setsum.cong[OF refl])
+          have "sum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j)
+              else (x$xa) * ((column xa A$j))) ?U = sum (\<lambda>xa. (if xa = i then c * ((column i A)$j) else 0) + ((x$xa) * ((column xa A)$j))) ?U"
+            apply (rule sum.cong[OF refl])
             using th apply blast
             done
-          also have "\<dots> = setsum (\<lambda>xa. if xa = i then c * ((column i A)$j) else 0) ?U + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U"
-            by (simp add: setsum.distrib)
-          also have "\<dots> = c * ((column i A)$j) + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U"
-            unfolding setsum.delta[OF fU]
+          also have "\<dots> = sum (\<lambda>xa. if xa = i then c * ((column i A)$j) else 0) ?U + sum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U"
+            by (simp add: sum.distrib)
+          also have "\<dots> = c * ((column i A)$j) + sum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U"
+            unfolding sum.delta[OF fU]
             using i(1) by simp
-          finally show "setsum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j)
-            else (x$xa) * ((column xa A$j))) ?U = c * ((column i A)$j) + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U" .
+          finally show "sum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j)
+            else (x$xa) * ((column xa A$j))) ?U = c * ((column i A)$j) + sum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U" .
         qed
       next
         show "y \<in> span (columns A)"
@@ -984,9 +984,9 @@
     { fix n
       assume n: "\<forall>i. dist (f (r n) $ i) (l $ i) < e / (real_of_nat (card ?d))"
       have "dist (f (r n)) l \<le> (\<Sum>i\<in>?d. dist (f (r n) $ i) (l $ i))"
-        unfolding dist_vec_def using zero_le_dist by (rule setL2_le_setsum)
+        unfolding dist_vec_def using zero_le_dist by (rule setL2_le_sum)
       also have "\<dots> < (\<Sum>i\<in>?d. e / (real_of_nat (card ?d)))"
-        by (rule setsum_strict_mono) (simp_all add: n)
+        by (rule sum_strict_mono) (simp_all add: n)
       finally have "dist (f (r n)) l < e" by simp
     }
     ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"
@@ -1235,7 +1235,7 @@
     where "finite s" "cbox (x - (\<chi> i. d)) (x + (\<chi> i. d)) = convex hull s"
 proof -
   from assms obtain s where "finite s"
-    and "cbox (x - setsum (op *\<^sub>R d) Basis) (x + setsum (op *\<^sub>R d) Basis) = convex hull s"
+    and "cbox (x - sum (op *\<^sub>R d) Basis) (x + sum (op *\<^sub>R d) Basis) = convex hull s"
     by (rule cube_convex_hull)
   with that[of s] show thesis
     by (simp add: const_vector_cart)
@@ -1314,13 +1314,13 @@
 lemma UNIV_3: "UNIV = {1::3, 2::3, 3::3}"
   using exhaust_3 by auto
 
-lemma setsum_1: "setsum f (UNIV::1 set) = f 1"
+lemma sum_1: "sum f (UNIV::1 set) = f 1"
   unfolding UNIV_1 by simp
 
-lemma setsum_2: "setsum f (UNIV::2 set) = f 1 + f 2"
+lemma sum_2: "sum f (UNIV::2 set) = f 1 + f 2"
   unfolding UNIV_2 by simp
 
-lemma setsum_3: "setsum f (UNIV::3 set) = f 1 + f 2 + f 3"
+lemma sum_3: "sum f (UNIV::3 set) = f 1 + f 2 + f 3"
   unfolding UNIV_3 by (simp add: ac_simps)
 
 instantiation num1 :: cart_one
--- a/src/HOL/Analysis/Cauchy_Integral_Theorem.thy	Mon Oct 17 11:07:01 2016 +0200
+++ b/src/HOL/Analysis/Cauchy_Integral_Theorem.thy	Mon Oct 17 14:37:32 2016 +0200
@@ -1551,9 +1551,9 @@
     "(\<And>z. z \<in> path_image g \<Longrightarrow> f z = 0) \<Longrightarrow> (f has_contour_integral 0) g"
   by (rule has_contour_integral_eq [OF has_contour_integral_0]) auto
 
-lemma has_contour_integral_setsum:
+lemma has_contour_integral_sum:
     "\<lbrakk>finite s; \<And>a. a \<in> s \<Longrightarrow> (f a has_contour_integral i a) p\<rbrakk>
-     \<Longrightarrow> ((\<lambda>x. setsum (\<lambda>a. f a x) s) has_contour_integral setsum i s) p"
+     \<Longrightarrow> ((\<lambda>x. sum (\<lambda>a. f a x) s) has_contour_integral sum i s) p"
   by (induction s rule: finite_induct) (auto simp: has_contour_integral_0 has_contour_integral_add)
 
 
@@ -1613,10 +1613,10 @@
 lemma contour_integral_0 [simp]: "contour_integral g (\<lambda>x. 0) = 0"
   by (simp add: contour_integral_unique has_contour_integral_0)
 
-lemma contour_integral_setsum:
+lemma contour_integral_sum:
     "\<lbrakk>finite s; \<And>a. a \<in> s \<Longrightarrow> (f a) contour_integrable_on p\<rbrakk>
-     \<Longrightarrow> contour_integral p (\<lambda>x. setsum (\<lambda>a. f a x) s) = setsum (\<lambda>a. contour_integral p (f a)) s"
-  by (auto simp: contour_integral_unique has_contour_integral_setsum has_contour_integral_integral)
+     \<Longrightarrow> contour_integral p (\<lambda>x. sum (\<lambda>a. f a x) s) = sum (\<lambda>a. contour_integral p (f a)) s"
+  by (auto simp: contour_integral_unique has_contour_integral_sum has_contour_integral_integral)
 
 lemma contour_integrable_eq:
     "\<lbrakk>f contour_integrable_on p; \<And>x. x \<in> path_image p \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> g contour_integrable_on p"
@@ -1655,11 +1655,11 @@
   using has_contour_integral_div contour_integrable_on_def
   by fastforce
 
-lemma contour_integrable_setsum:
+lemma contour_integrable_sum:
     "\<lbrakk>finite s; \<And>a. a \<in> s \<Longrightarrow> (f a) contour_integrable_on p\<rbrakk>
-     \<Longrightarrow> (\<lambda>x. setsum (\<lambda>a. f a x) s) contour_integrable_on p"
+     \<Longrightarrow> (\<lambda>x. sum (\<lambda>a. f a x) s) contour_integrable_on p"
    unfolding contour_integrable_on_def
-   by (metis has_contour_integral_setsum)
+   by (metis has_contour_integral_sum)
 
 
 subsection\<open>Reversing a path integral\<close>
@@ -5919,9 +5919,9 @@
   have sumeq: "(\<Sum>i = 0..n.
                of_nat (n choose i) * (deriv ((deriv ^^ i) f) z * (deriv ^^ (n - i)) g z + deriv ((deriv ^^ (n - i)) g) z * (deriv ^^ i) f z)) =
             g z * deriv ((deriv ^^ n) f) z + (\<Sum>i = 0..n. (deriv ^^ i) f z * (of_nat (Suc n choose i) * (deriv ^^ (Suc n - i)) g z))"
-    apply (simp add: bb algebra_simps setsum.distrib)
-    apply (subst (4) setsum_Suc_reindex)
-    apply (auto simp: algebra_simps Suc_diff_le intro: setsum.cong)
+    apply (simp add: bb algebra_simps sum.distrib)
+    apply (subst (4) sum_Suc_reindex)
+    apply (auto simp: algebra_simps Suc_diff_le intro: sum.cong)
     done
   show ?case
     apply (simp only: funpow.simps o_apply)
@@ -5929,7 +5929,7 @@
     apply (rule DERIV_transform_within_open
              [of "\<lambda>w. (\<Sum>i = 0..n. of_nat (n choose i) * (deriv ^^ i) f w * (deriv ^^ (n - i)) g w)"])
     apply (simp add: algebra_simps)
-    apply (rule DERIV_cong [OF DERIV_setsum])
+    apply (rule DERIV_cong [OF DERIV_sum])
     apply (rule DERIV_cmult)
     apply (auto simp: intro: DERIV_mult * sumeq \<open>open s\<close> Suc.prems Suc.IH [symmetric])
     done
@@ -6210,7 +6210,7 @@
         by (metis (no_types) dist_norm divide_eq_1_iff less_irrefl mem_ball norm_minus_commute r w)
       have "norm ((\<Sum>k<N. (w - z) ^ k * f u / (u - z) ^ Suc k) * (u - w) - f u)
             = norm ((\<Sum>k<N. (((w - z) / (u - z)) ^ k)) * f u * (u - w) / (u - z) - f u)"
-        unfolding setsum_distrib_right setsum_divide_distrib power_divide by (simp add: algebra_simps)
+        unfolding sum_distrib_right sum_divide_distrib power_divide by (simp add: algebra_simps)
       also have "... = norm ((((w - z) / (u - z)) ^ N - 1) * (u - w) / (((w - z) / (u - z) - 1) * (u - z)) - 1) * norm (f u)"
         using \<open>0 < B\<close>
         apply (auto simp: geometric_sum [OF wzu_not1])
@@ -6242,7 +6242,7 @@
              contour_integral (circlepath z r) (\<lambda>u. \<Sum>k<x. (w - z) ^ k * (f u / (u - z) ^ Suc k)) =
              (\<Sum>k<x. contour_integral (circlepath z r) (\<lambda>u. f u / (u - z) ^ Suc k) * (w - z) ^ k)"
     apply (rule eventuallyI)
-    apply (subst contour_integral_setsum, simp)
+    apply (subst contour_integral_sum, simp)
     using contour_integrable_lmul [OF cint, of "(w - z) ^ a" for a] apply (simp add: field_simps)
     apply (simp only: contour_integral_lmul cint algebra_simps)
     done
@@ -6251,7 +6251,7 @@
     unfolding sums_def
     apply (rule Lim_transform_eventually [OF eq])
     apply (rule contour_integral_uniform_limit_circlepath [OF eventuallyI *])
-    apply (rule contour_integrable_setsum, simp)
+    apply (rule contour_integrable_sum, simp)
     apply (rule contour_integrable_lmul)
     apply (rule Cauchy_higher_derivative_integral_circlepath [OF contf holf])
     using \<open>0 < r\<close>
@@ -6589,7 +6589,7 @@
   have "(\<forall>x\<in>s. (\<lambda>n. \<Sum>i<n. f i x) \<longlonglongrightarrow> g x)"
     using g by (force simp add: lim_sequentially)
   moreover have "\<exists>g'. \<forall>x\<in>s. (g has_field_derivative g' x) (at x) \<and> (\<lambda>n. \<Sum>i<n. f' i x) \<longlonglongrightarrow> g' x"
-    by (rule has_complex_derivative_uniform_sequence [OF s]) (auto intro: * hfd DERIV_setsum)+
+    by (rule has_complex_derivative_uniform_sequence [OF s]) (auto intro: * hfd DERIV_sum)+
   ultimately show ?thesis
     by (force simp add: sums_def  conj_commute intro: that)
 qed
--- a/src/HOL/Analysis/Complex_Analysis_Basics.thy	Mon Oct 17 11:07:01 2016 +0200
+++ b/src/HOL/Analysis/Complex_Analysis_Basics.thy	Mon Oct 17 14:37:32 2016 +0200
@@ -269,7 +269,7 @@
   fixes l::complex
   shows "f sums l \<Longrightarrow> (\<And>n. f n \<in> \<real>) \<Longrightarrow> l \<in> \<real>"
 unfolding sums_def
-by (metis real_lim_sequentially setsum_in_Reals)
+by (metis real_lim_sequentially sum_in_Reals)
 
 lemma Lim_null_comparison_Re:
   assumes "eventually (\<lambda>x. norm(f x) \<le> Re(g x)) F" "(g \<longlongrightarrow> 0) F" shows "(f \<longlongrightarrow> 0) F"
@@ -335,7 +335,7 @@
      "op + c field_differentiable F"
   by (simp add: field_differentiable_add)
 
-lemma field_differentiable_setsum [derivative_intros]:
+lemma field_differentiable_sum [derivative_intros]:
   "(\<And>i. i \<in> I \<Longrightarrow> (f i) field_differentiable F) \<Longrightarrow> (\<lambda>z. \<Sum>i\<in>I. f i z) field_differentiable F"
   by (induct I rule: infinite_finite_induct)
      (auto intro: field_differentiable_add field_differentiable_const)
@@ -502,9 +502,9 @@
   "f holomorphic_on s \<Longrightarrow> (\<lambda>z. (f z)^n) holomorphic_on s"
   unfolding holomorphic_on_def by (metis field_differentiable_power)
 
-lemma holomorphic_on_setsum [holomorphic_intros]:
-  "(\<And>i. i \<in> I \<Longrightarrow> (f i) holomorphic_on s) \<Longrightarrow> (\<lambda>x. setsum (\<lambda>i. f i x) I) holomorphic_on s"
-  unfolding holomorphic_on_def by (metis field_differentiable_setsum)
+lemma holomorphic_on_sum [holomorphic_intros]:
+  "(\<And>i. i \<in> I \<Longrightarrow> (f i) holomorphic_on s) \<Longrightarrow> (\<lambda>x. sum (\<lambda>i. f i x) I) holomorphic_on s"
+  unfolding holomorphic_on_def by (metis field_differentiable_sum)
 
 lemma DERIV_deriv_iff_field_differentiable:
   "DERIV f x :> deriv f x \<longleftrightarrow> f field_differentiable at x"
@@ -788,8 +788,8 @@
   "f analytic_on s \<Longrightarrow> (\<lambda>z. (f z) ^ n) analytic_on s"
 by (induct n) (auto simp: analytic_on_const analytic_on_mult)
 
-lemma analytic_on_setsum:
-  "(\<And>i. i \<in> I \<Longrightarrow> (f i) analytic_on s) \<Longrightarrow> (\<lambda>x. setsum (\<lambda>i. f i x) I) analytic_on s"
+lemma analytic_on_sum:
+  "(\<And>i. i \<in> I \<Longrightarrow> (f i) analytic_on s) \<Longrightarrow> (\<lambda>x. sum (\<lambda>i. f i x) I) analytic_on s"
   by (induct I rule: infinite_finite_induct) (auto simp: analytic_on_const analytic_on_add)
 
 lemma deriv_left_inverse:
@@ -951,7 +951,7 @@
       then have "cmod h * cmod ((\<Sum>i<n. f' i y) - g' y) \<le> cmod h * e"
         by (auto simp: antisym_conv2 mult_le_cancel_left norm_triangle_ineq2)
       then show "cmod ((\<Sum>i<n. h * f' i y) - g' y * h) \<le> e * cmod h"
-        by (simp add: norm_mult [symmetric] field_simps setsum_distrib_left)
+        by (simp add: norm_mult [symmetric] field_simps sum_distrib_left)
     qed
   } note ** = this
   show ?thesis
@@ -1064,9 +1064,9 @@
 
 subsection \<open>Taylor on Complex Numbers\<close>
 
-lemma setsum_Suc_reindex:
+lemma sum_Suc_reindex:
   fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
-    shows  "setsum f {0..n} = f 0 - f (Suc n) + setsum (\<lambda>i. f (Suc i)) {0..n}"
+    shows  "sum f {0..n} = f 0 - f (Suc n) + sum (\<lambda>i. f (Suc i)) {0..n}"
 by (induct n) auto
 
 lemma complex_taylor:
@@ -1146,7 +1146,7 @@
   have "(\<Sum>i\<le>n. f i z * (z - z) ^ i / (fact i)) = (\<Sum>i\<le>n. (f i z / (fact i)) * 0 ^ i)"
     by simp
   also have "\<dots> = f 0 z / (fact 0)"
-    by (subst setsum_zero_power) simp
+    by (subst sum_zero_power) simp
   finally have "cmod (f 0 z - (\<Sum>i\<le>n. f i w * (z - w) ^ i / (fact i)))
                 \<le> cmod ((\<Sum>i\<le>n. f i w * (z - w) ^ i / (fact i)) -
                         (\<Sum>i\<le>n. f i z * (z - z) ^ i / (fact i)))"
@@ -1203,7 +1203,7 @@
              (\<Sum>i = 0..n.
                  (f (Suc (Suc i)) u * ((z-u) ^ Suc i) - of_nat (Suc i) * (f (Suc i) u * (z-u) ^ i)) /
                  (fact (Suc i)))"
-       by (subst setsum_Suc_reindex) simp
+       by (subst sum_Suc_reindex) simp
     also have "... = f (Suc 0) u -
              (f (Suc (Suc n)) u * ((z-u) ^ Suc n) - (of_nat (Suc n)) * (z-u) ^ n * f (Suc n) u) /
              (fact (Suc n)) +
@@ -1215,7 +1215,7 @@
              (f (Suc (Suc n)) u * (z-u) ^ Suc n - of_nat (Suc n) * (z-u) ^ n * f (Suc n) u) /
              (fact (Suc n)) +
              f (Suc (Suc n)) u * (z-u) ^ Suc n / (fact (Suc n)) - f (Suc 0) u"
-      by (subst setsum_Suc_diff) auto
+      by (subst sum_Suc_diff) auto
     also have "... = f (Suc n) u * (z-u) ^ n / (fact n)"
       by (simp only: algebra_simps diff_divide_distrib fact_cancel)
     finally have "(\<Sum>i = 0..n. (f (Suc i) u * (z - u) ^ i
--- a/src/HOL/Analysis/Complex_Transcendental.thy	Mon Oct 17 11:07:01 2016 +0200
+++ b/src/HOL/Analysis/Complex_Transcendental.thy	Mon Oct 17 14:37:32 2016 +0200
@@ -597,7 +597,7 @@
 text\<open>32-bit Approximation to e\<close>
 lemma e_approx_32: "\<bar>exp(1) - 5837465777 / 2147483648\<bar> \<le> (inverse(2 ^ 32)::real)"
   using Taylor_exp [of 1 14] exp_le
-  apply (simp add: setsum_distrib_right in_Reals_norm Re_exp atMost_nat_numeral fact_numeral)
+  apply (simp add: sum_distrib_right in_Reals_norm Re_exp atMost_nat_numeral fact_numeral)
   apply (simp only: pos_le_divide_eq [symmetric], linarith)
   done
 
--- a/src/HOL/Analysis/Conformal_Mappings.thy	Mon Oct 17 11:07:01 2016 +0200
+++ b/src/HOL/Analysis/Conformal_Mappings.thy	Mon Oct 17 14:37:32 2016 +0200
@@ -612,7 +612,7 @@
     have "powf sums f w"
       unfolding powf_def by (rule holomorphic_power_series [OF holfb w])
     moreover have "(\<Sum>i<n. powf i) = f \<xi>"
-      apply (subst Groups_Big.comm_monoid_add_class.setsum.setdiff_irrelevant [symmetric])
+      apply (subst Groups_Big.comm_monoid_add_class.sum.setdiff_irrelevant [symmetric])
       apply (simp add:)
       apply (simp only: dfz sing)
       apply (simp add: powf_def)
@@ -1052,7 +1052,7 @@
   proof (cases "\<forall>i\<le>n. 0<i \<longrightarrow> a i = 0")
     case True
     then have [simp]: "\<And>z. f z = a 0"
-      by (simp add: that setsum_atMost_shift)
+      by (simp add: that sum_atMost_shift)
     have False using compf [of "{a 0}"] by simp
     then show ?thesis ..
   next
@@ -1073,7 +1073,7 @@
       using k apply auto
       done
     with k m show ?thesis
-      by (rule_tac x=m in exI) (auto simp: that comm_monoid_add_class.setsum.mono_neutral_right)
+      by (rule_tac x=m in exI) (auto simp: that comm_monoid_add_class.sum.mono_neutral_right)
   qed
   have "((inverse \<circ> f) \<longlongrightarrow> 0) at_infinity"
   proof (rule Lim_at_infinityI)
@@ -2758,9 +2758,9 @@
       finally show ?thesis .
     qed
   ultimately show ?case unfolding p_circ_def
-    apply (subst (asm) setsum.cong[OF refl,
+    apply (subst (asm) sum.cong[OF refl,
         of pts _ "\<lambda>p. winding_number g p * contour_integral (circlepath p (h p)) f"])
-    by (auto simp add:setsum.insert[OF \<open>finite pts\<close> \<open>p\<notin>pts\<close>] algebra_simps)
+    by (auto simp add:sum.insert[OF \<open>finite pts\<close> \<open>p\<notin>pts\<close>] algebra_simps)
 qed
 
 lemma Cauchy_theorem_singularities:
@@ -2791,17 +2791,17 @@
       show "\<forall>p\<in>s. 0 < h p \<and> (\<forall>w\<in>cball p (h p). w \<in> s \<and> (w \<noteq> p \<longrightarrow> w \<notin> pts1))"
         by (simp add: avoid pts1_def)
     qed
-  moreover have "setsum circ pts2=0"
+  moreover have "sum circ pts2=0"
     proof -
       have "winding_number g p=0" when "p\<in>pts2" for p
         using  \<open>pts2 \<inter> s={}\<close> that homo[rule_format,of p] by auto
       thus ?thesis unfolding circ_def
-        apply (intro setsum.neutral)
+        apply (intro sum.neutral)
         by auto
     qed
-  moreover have "?R=setsum circ pts1 + setsum circ pts2"
+  moreover have "?R=sum circ pts1 + sum circ pts2"
     unfolding circ_def
-    using setsum.union_disjoint[OF _ _ \<open>pts1 \<inter> pts2 = {}\<close>] \<open>finite pts\<close> \<open>pts=pts1 \<union> pts2\<close>
+    using sum.union_disjoint[OF _ _ \<open>pts1 \<inter> pts2 = {}\<close>] \<open>finite pts\<close> \<open>pts=pts1 \<union> pts2\<close>
     by blast
   ultimately show ?thesis
     apply (fold circ_def)
@@ -2826,7 +2826,7 @@
       = (\<Sum>p\<in>pts. winding_number g p * contour_integral (circlepath p (h p)) f)"
     using Cauchy_theorem_singularities[OF assms avoid] .
   also have "... = (\<Sum>p\<in>pts.  c * winding_number g p * residue f p)"
-    proof (intro setsum.cong)
+    proof (intro sum.cong)
       show "pts = pts" by simp
     next
       fix x assume "x \<in> pts"
@@ -2846,7 +2846,7 @@
         qed
     qed
   also have "... = c * (\<Sum>p\<in>pts. winding_number g p * residue f p)"
-    by (simp add: setsum_distrib_left algebra_simps)
+    by (simp add: sum_distrib_left algebra_simps)
   finally show ?thesis unfolding c_def .
 qed
 
@@ -3437,12 +3437,12 @@
     qed
   also have "... = (\<Sum>p\<in>zeros. w p * cont p) + (\<Sum>p\<in>poles. w p * cont p)"
     using finite
-    apply (subst setsum.union_disjoint)
+    apply (subst sum.union_disjoint)
     by (auto simp add:zeros_def)
   also have "... = c * ((\<Sum>p\<in>zeros. w p *  h p * zorder f p) - (\<Sum>p\<in>poles. w p *  h p * porder f p))"
     proof -
       have "(\<Sum>p\<in>zeros. w p * cont p) = (\<Sum>p\<in>zeros. c * w p *  h p * zorder f p)"
-        proof (rule setsum.cong[of zeros zeros,simplified])
+        proof (rule sum.cong[of zeros zeros,simplified])
           fix p assume "p \<in> zeros"
           show "w p * cont p = c * w p * h p * (zorder f p)"
             proof (cases "p\<in>s")
@@ -3459,10 +3459,10 @@
             qed
         qed
       then have "(\<Sum>p\<in>zeros. w p * cont p) = c * (\<Sum>p\<in>zeros.  w p *  h p * zorder f p)"
-        apply (subst setsum_distrib_left)
+        apply (subst sum_distrib_left)
         by (simp add:algebra_simps)
       moreover have "(\<Sum>p\<in>poles. w p * cont p) = (\<Sum>p\<in>poles.  - c * w p *  h p * porder f p)"
-        proof (rule setsum.cong[of poles poles,simplified])
+        proof (rule sum.cong[of poles poles,simplified])
           fix p assume "p \<in> poles"
           show "w p * cont p = - c * w p * h p * (porder f p)"
             proof (cases "p\<in>s")
@@ -3479,7 +3479,7 @@
             qed
         qed
       then have "(\<Sum>p\<in>poles. w p * cont p) = - c * (\<Sum>p\<in>poles. w p *  h p * porder f p)"
-        apply (subst setsum_distrib_left)
+        apply (subst sum_distrib_left)
         by (simp add:algebra_simps)
       ultimately show ?thesis by (simp add: right_diff_distrib)
     qed
--- a/src/HOL/Analysis/Continuous_Extension.thy	Mon Oct 17 11:07:01 2016 +0200
+++ b/src/HOL/Analysis/Continuous_Extension.thy	Mon Oct 17 14:37:32 2016 +0200
@@ -20,19 +20,19 @@
   obtains F :: "['a set, 'a] \<Rightarrow> real"
     where "\<And>U. U \<in> \<C> \<Longrightarrow> continuous_on S (F U) \<and> (\<forall>x \<in> S. 0 \<le> F U x)"
       and "\<And>x U. \<lbrakk>U \<in> \<C>; x \<in> S; x \<notin> U\<rbrakk> \<Longrightarrow> F U x = 0"
-      and "\<And>x. x \<in> S \<Longrightarrow> supp_setsum (\<lambda>W. F W x) \<C> = 1"
+      and "\<And>x. x \<in> S \<Longrightarrow> supp_sum (\<lambda>W. F W x) \<C> = 1"
       and "\<And>x. x \<in> S \<Longrightarrow> \<exists>V. open V \<and> x \<in> V \<and> finite {U \<in> \<C>. \<exists>x\<in>V. F U x \<noteq> 0}"
 proof (cases "\<exists>W. W \<in> \<C> \<and> S \<subseteq> W")
   case True
     then obtain W where "W \<in> \<C>" "S \<subseteq> W" by metis
     then show ?thesis
       apply (rule_tac F = "\<lambda>V x. if V = W then 1 else 0" in that)
-      apply (auto simp: continuous_on_const supp_setsum_def support_on_def)
+      apply (auto simp: continuous_on_const supp_sum_def support_on_def)
       done
 next
   case False
-    have nonneg: "0 \<le> supp_setsum (\<lambda>V. setdist {x} (S - V)) \<C>" for x
-      by (simp add: supp_setsum_def setsum_nonneg)
+    have nonneg: "0 \<le> supp_sum (\<lambda>V. setdist {x} (S - V)) \<C>" for x
+      by (simp add: supp_sum_def sum_nonneg)
     have sd_pos: "0 < setdist {x} (S - V)" if "V \<in> \<C>" "x \<in> S" "x \<in> V" for V x
     proof -
       have "closedin (subtopology euclidean S) (S - V)"
@@ -41,7 +41,7 @@
         show ?thesis
           by (simp add: order_class.order.order_iff_strict)
     qed
-    have ss_pos: "0 < supp_setsum (\<lambda>V. setdist {x} (S - V)) \<C>" if "x \<in> S" for x
+    have ss_pos: "0 < supp_sum (\<lambda>V. setdist {x} (S - V)) \<C>" if "x \<in> S" for x
     proof -
       obtain U where "U \<in> \<C>" "x \<in> U" using \<open>x \<in> S\<close> \<open>S \<subseteq> \<Union>\<C>\<close>
         by blast
@@ -52,20 +52,20 @@
       have "x \<notin> closure (S - U)"
         by (metis \<open>U \<in> \<C>\<close> \<open>x \<in> U\<close> less_irrefl sd_pos setdist_eq_0_sing_1 that)
       then show ?thesis
-        apply (simp add: setdist_eq_0_sing_1 supp_setsum_def support_on_def)
-        apply (rule ordered_comm_monoid_add_class.setsum_pos2 [OF *, of U])
+        apply (simp add: setdist_eq_0_sing_1 supp_sum_def support_on_def)
+        apply (rule ordered_comm_monoid_add_class.sum_pos2 [OF *, of U])
         using \<open>U \<in> \<C>\<close> \<open>x \<in> U\<close> False
         apply (auto simp: setdist_pos_le sd_pos that)
         done
     qed
     define F where
-      "F \<equiv> \<lambda>W x. if x \<in> S then setdist {x} (S - W) / supp_setsum (\<lambda>V. setdist {x} (S - V)) \<C>
+      "F \<equiv> \<lambda>W x. if x \<in> S then setdist {x} (S - W) / supp_sum (\<lambda>V. setdist {x} (S - V)) \<C>
                  else 0"
     show ?thesis
     proof (rule_tac F = F in that)
       have "continuous_on S (F U)" if "U \<in> \<C>" for U
       proof -
-        have *: "continuous_on S (\<lambda>x. supp_setsum (\<lambda>V. setdist {x} (S - V)) \<C>)"
+        have *: "continuous_on S (\<lambda>x. supp_sum (\<lambda>V. setdist {x} (S - V)) \<C>)"
         proof (clarsimp simp add: continuous_on_eq_continuous_within)
           fix x assume "x \<in> S"
           then obtain X where "open X" and x: "x \<in> S \<inter> X" and finX: "finite {U \<in> \<C>. U \<inter> X \<noteq> {}}"
@@ -73,15 +73,15 @@
           then have OSX: "openin (subtopology euclidean S) (S \<inter> X)" by blast
           have sumeq: "\<And>x. x \<in> S \<inter> X \<Longrightarrow>
                      (\<Sum>V | V \<in> \<C> \<and> V \<inter> X \<noteq> {}. setdist {x} (S - V))
-                     = supp_setsum (\<lambda>V. setdist {x} (S - V)) \<C>"
-            apply (simp add: supp_setsum_def)
-            apply (rule setsum.mono_neutral_right [OF finX])
+                     = supp_sum (\<lambda>V. setdist {x} (S - V)) \<C>"
+            apply (simp add: supp_sum_def)
+            apply (rule sum.mono_neutral_right [OF finX])
             apply (auto simp: setdist_eq_0_sing_1 support_on_def subset_iff)
             apply (meson DiffI closure_subset disjoint_iff_not_equal subsetCE)
             done
-          show "continuous (at x within S) (\<lambda>x. supp_setsum (\<lambda>V. setdist {x} (S - V)) \<C>)"
+          show "continuous (at x within S) (\<lambda>x. supp_sum (\<lambda>V. setdist {x} (S - V)) \<C>)"
             apply (rule continuous_transform_within_openin
-                     [where f = "\<lambda>x. (setsum (\<lambda>V. setdist {x} (S - V)) {V \<in> \<C>. V \<inter> X \<noteq> {}})"
+                     [where f = "\<lambda>x. (sum (\<lambda>V. setdist {x} (S - V)) {V \<in> \<C>. V \<inter> X \<noteq> {}})"
                         and S ="S \<inter> X"])
             apply (rule continuous_intros continuous_at_setdist continuous_at_imp_continuous_at_within OSX x)+
             apply (simp add: sumeq)
@@ -101,9 +101,9 @@
       show "\<And>x U. \<lbrakk>U \<in> \<C>; x \<in> S; x \<notin> U\<rbrakk> \<Longrightarrow> F U x = 0"
         by (simp add: setdist_eq_0_sing_1 closure_def F_def)
     next
-      show "supp_setsum (\<lambda>W. F W x) \<C> = 1" if "x \<in> S" for x
+      show "supp_sum (\<lambda>W. F W x) \<C> = 1" if "x \<in> S" for x
         using that ss_pos [OF that]
-        by (simp add: F_def divide_simps supp_setsum_divide_distrib [symmetric])
+        by (simp add: F_def divide_simps supp_sum_divide_distrib [symmetric])
     next
       show "\<exists>V. open V \<and> x \<in> V \<and> finite {U \<in> \<C>. \<exists>x\<in>V. F U x \<noteq> 0}" if "x \<in> S" for x
         using fin [OF that] that
@@ -378,11 +378,11 @@
     where Hcont: "\<And>Z. Z \<in> \<C> \<Longrightarrow> continuous_on (U-S) (H Z)"
       and Hge0: "\<And>Z x. \<lbrakk>Z \<in> \<C>; x \<in> U-S\<rbrakk> \<Longrightarrow> 0 \<le> H Z x"
       and Heq0: "\<And>x Z. \<lbrakk>Z \<in> \<C>; x \<in> U-S; x \<notin> Z\<rbrakk> \<Longrightarrow> H Z x = 0"
-      and H1: "\<And>x. x \<in> U-S \<Longrightarrow> supp_setsum (\<lambda>W. H W x) \<C> = 1"
+      and H1: "\<And>x. x \<in> U-S \<Longrightarrow> supp_sum (\<lambda>W. H W x) \<C> = 1"
       and Hfin: "\<And>x. x \<in> U-S \<Longrightarrow> \<exists>V. open V \<and> x \<in> V \<and> finite {U \<in> \<C>. \<exists>x\<in>V. H U x \<noteq> 0}"
     apply (rule subordinate_partition_of_unity [OF USsub _ fin])
     using nbrhd by auto
-  define g where "g \<equiv> \<lambda>x. if x \<in> S then f x else supp_setsum (\<lambda>T. H T x *\<^sub>R f(\<A> T)) \<C>"
+  define g where "g \<equiv> \<lambda>x. if x \<in> S then f x else supp_sum (\<lambda>T. H T x *\<^sub>R f(\<A> T)) \<C>"
   show ?thesis
   proof (rule that)
     show "continuous_on U g"
@@ -420,8 +420,8 @@
               then show ?thesis
                 using VA [OF \<open>T \<in> \<C>\<close>] by (auto simp: d)
             qed
-            have "supp_setsum (\<lambda>T. H T y *\<^sub>R f (\<A> T)) \<C> \<in> ball (f a) e"
-              apply (rule convex_supp_setsum [OF convex_ball])
+            have "supp_sum (\<lambda>T. H T y *\<^sub>R f (\<A> T)) \<C> \<in> ball (f a) e"
+              apply (rule convex_supp_sum [OF convex_ball])
               apply (simp_all add: False H1 Hge0 \<open>y \<in> U\<close>)
               by (metis dist_commute *)
             then show ?thesis
@@ -447,7 +447,7 @@
           done
         show ?thesis
         proof (rule continuous_transform_within_openin [OF _ oUS])
-          show "continuous (at a within U) (\<lambda>x. supp_setsum (\<lambda>T. H T x *\<^sub>R f (\<A> T)) \<C>)"
+          show "continuous (at a within U) (\<lambda>x. supp_sum (\<lambda>T. H T x *\<^sub>R f (\<A> T)) \<C>)"
           proof (rule continuous_transform_within_openin)
             show "continuous (at a within U)
                     (\<lambda>x. \<Sum>T\<in>{U \<in> \<C>. \<exists>x\<in>N. H U x \<noteq> 0}. H T x *\<^sub>R f (\<A> T))"
@@ -460,21 +460,21 @@
           next
             show "\<And>x. x \<in> (U - S) \<inter> N \<Longrightarrow>
                          (\<Sum>T \<in> {U \<in> \<C>. \<exists>x\<in>N. H U x \<noteq> 0}. H T x *\<^sub>R f (\<A> T))
-                         = supp_setsum (\<lambda>T. H T x *\<^sub>R f (\<A> T)) \<C>"
-              by (auto simp: supp_setsum_def support_on_def
-                       intro: setsum.mono_neutral_right [OF finN])
+                         = supp_sum (\<lambda>T. H T x *\<^sub>R f (\<A> T)) \<C>"
+              by (auto simp: supp_sum_def support_on_def
+                       intro: sum.mono_neutral_right [OF finN])
           qed
         next
           show "a \<in> U - S" using False \<open>a \<in> U\<close> by blast
         next
-          show "\<And>x. x \<in> U - S \<Longrightarrow> supp_setsum (\<lambda>T. H T x *\<^sub>R f (\<A> T)) \<C> = g x"
+          show "\<And>x. x \<in> U - S \<Longrightarrow> supp_sum (\<lambda>T. H T x *\<^sub>R f (\<A> T)) \<C> = g x"
             by (simp add: g_def)
         qed
       qed
     qed
     show "g ` U \<subseteq> C"
       using \<open>f ` S \<subseteq> C\<close> VA
-      by (fastforce simp: g_def Hge0 intro!: convex_supp_setsum [OF \<open>convex C\<close>] H1)
+      by (fastforce simp: g_def Hge0 intro!: convex_supp_sum [OF \<open>convex C\<close>] H1)
     show "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
       by (simp add: g_def)
   qed
--- a/src/HOL/Analysis/Convex_Euclidean_Space.thy	Mon Oct 17 11:07:01 2016 +0200
+++ b/src/HOL/Analysis/Convex_Euclidean_Space.thy	Mon Oct 17 14:37:32 2016 +0200
@@ -69,9 +69,9 @@
     by (auto intro: finite_subset[OF assms])
   have ***: "\<And>i. i \<in> Basis \<Longrightarrow> (\<Sum>i\<in>d. f i *\<^sub>R i) \<bullet> i = (\<Sum>x\<in>d. if x = i then f x else 0)"
     using d
-    by (auto intro!: setsum.cong simp: inner_Basis inner_setsum_left)
-  show ?thesis
-    unfolding euclidean_eq_iff[where 'a='a] by (auto simp: setsum.delta[OF **] ***)
+    by (auto intro!: sum.cong simp: inner_Basis inner_sum_left)
+  show ?thesis
+    unfolding euclidean_eq_iff[where 'a='a] by (auto simp: sum.delta[OF **] ***)
 qed
 
 lemma independent_substdbasis: "d \<subseteq> Basis \<Longrightarrow> independent d"
@@ -106,7 +106,7 @@
   shows "finite B \<and> card B = dim (span B)"
   using assms basis_card_eq_dim[of B "span B"] span_inc by auto
 
-lemma setsum_not_0: "setsum f A \<noteq> 0 \<Longrightarrow> \<exists>a \<in> A. f a \<noteq> 0"
+lemma sum_not_0: "sum f A \<noteq> 0 \<Longrightarrow> \<exists>a \<in> A. f a \<noteq> 0"
   by (rule ccontr) auto
 
 lemma subset_translation_eq [simp]:
@@ -306,7 +306,7 @@
 
 subsection \<open>Explicit expressions for convexity in terms of arbitrary sums\<close>
 
-lemma convex_setsum:
+lemma convex_sum:
   fixes C :: "'a::real_vector set"
   assumes "finite s"
     and "convex C"
@@ -320,55 +320,55 @@
   then show ?case by simp
 next
   case (insert i s) note IH = this(3)
-  have "a i + setsum a s = 1"
+  have "a i + sum a s = 1"
     and "0 \<le> a i"
     and "\<forall>j\<in>s. 0 \<le> a j"
     and "y i \<in> C"
     and "\<forall>j\<in>s. y j \<in> C"
     using insert.hyps(1,2) insert.prems by simp_all
-  then have "0 \<le> setsum a s"
-    by (simp add: setsum_nonneg)
+  then have "0 \<le> sum a s"
+    by (simp add: sum_nonneg)
   have "a i *\<^sub>R y i + (\<Sum>j\<in>s. a j *\<^sub>R y j) \<in> C"
-  proof (cases "setsum a s = 0")
+  proof (cases "sum a s = 0")
     case True
-    with \<open>a i + setsum a s = 1\<close> have "a i = 1"
+    with \<open>a i + sum a s = 1\<close> have "a i = 1"
       by simp
-    from setsum_nonneg_0 [OF \<open>finite s\<close> _ True] \<open>\<forall>j\<in>s. 0 \<le> a j\<close> have "\<forall>j\<in>s. a j = 0"
+    from sum_nonneg_0 [OF \<open>finite s\<close> _ True] \<open>\<forall>j\<in>s. 0 \<le> a j\<close> have "\<forall>j\<in>s. a j = 0"
       by simp
     show ?thesis using \<open>a i = 1\<close> and \<open>\<forall>j\<in>s. a j = 0\<close> and \<open>y i \<in> C\<close>
       by simp
   next
     case False
-    with \<open>0 \<le> setsum a s\<close> have "0 < setsum a s"
+    with \<open>0 \<le> sum a s\<close> have "0 < sum a s"
       by simp
-    then have "(\<Sum>j\<in>s. (a j / setsum a s) *\<^sub>R y j) \<in> C"
+    then have "(\<Sum>j\<in>s. (a j / sum a s) *\<^sub>R y j) \<in> C"
       using \<open>\<forall>j\<in>s. 0 \<le> a j\<close> and \<open>\<forall>j\<in>s. y j \<in> C\<close>
-      by (simp add: IH setsum_divide_distrib [symmetric])
+      by (simp add: IH sum_divide_distrib [symmetric])
     from \<open>convex C\<close> and \<open>y i \<in> C\<close> and this and \<open>0 \<le> a i\<close>
-      and \<open>0 \<le> setsum a s\<close> and \<open>a i + setsum a s = 1\<close>
-    have "a i *\<^sub>R y i + setsum a s *\<^sub>R (\<Sum>j\<in>s. (a j / setsum a s) *\<^sub>R y j) \<in> C"
+      and \<open>0 \<le> sum a s\<close> and \<open>a i + sum a s = 1\<close>
+    have "a i *\<^sub>R y i + sum a s *\<^sub>R (\<Sum>j\<in>s. (a j / sum a s) *\<^sub>R y j) \<in> C"
       by (rule convexD)
     then show ?thesis
-      by (simp add: scaleR_setsum_right False)
+      by (simp add: scaleR_sum_right False)
   qed
   then show ?case using \<open>finite s\<close> and \<open>i \<notin> s\<close>
     by simp
 qed
 
 lemma convex:
-  "convex s \<longleftrightarrow> (\<forall>(k::nat) u x. (\<forall>i. 1\<le>i \<and> i\<le>k \<longrightarrow> 0 \<le> u i \<and> x i \<in>s) \<and> (setsum u {1..k} = 1)
-      \<longrightarrow> setsum (\<lambda>i. u i *\<^sub>R x i) {1..k} \<in> s)"
+  "convex s \<longleftrightarrow> (\<forall>(k::nat) u x. (\<forall>i. 1\<le>i \<and> i\<le>k \<longrightarrow> 0 \<le> u i \<and> x i \<in>s) \<and> (sum u {1..k} = 1)
+      \<longrightarrow> sum (\<lambda>i. u i *\<^sub>R x i) {1..k} \<in> s)"
 proof safe
   fix k :: nat
   fix u :: "nat \<Rightarrow> real"
   fix x
   assume "convex s"
     "\<forall>i. 1 \<le> i \<and> i \<le> k \<longrightarrow> 0 \<le> u i \<and> x i \<in> s"
-    "setsum u {1..k} = 1"
-  with convex_setsum[of "{1 .. k}" s] show "(\<Sum>j\<in>{1 .. k}. u j *\<^sub>R x j) \<in> s"
-    by auto
-next
-  assume *: "\<forall>k u x. (\<forall> i :: nat. 1 \<le> i \<and> i \<le> k \<longrightarrow> 0 \<le> u i \<and> x i \<in> s) \<and> setsum u {1..k} = 1
+    "sum u {1..k} = 1"
+  with convex_sum[of "{1 .. k}" s] show "(\<Sum>j\<in>{1 .. k}. u j *\<^sub>R x j) \<in> s"
+    by auto
+next
+  assume *: "\<forall>k u x. (\<forall> i :: nat. 1 \<le> i \<and> i \<le> k \<longrightarrow> 0 \<le> u i \<and> x i \<in> s) \<and> sum u {1..k} = 1
     \<longrightarrow> (\<Sum>i = 1..k. u i *\<^sub>R (x i :: 'a)) \<in> s"
   {
     fix \<mu> :: real
@@ -381,14 +381,14 @@
       by auto
     then have card: "card ({1 :: nat .. 2} \<inter> - {x. x = 1}) = 1"
       by simp
-    then have "setsum ?u {1 .. 2} = 1"
-      using setsum.If_cases[of "{(1 :: nat) .. 2}" "\<lambda> x. x = 1" "\<lambda> x. \<mu>" "\<lambda> x. 1 - \<mu>"]
+    then have "sum ?u {1 .. 2} = 1"
+      using sum.If_cases[of "{(1 :: nat) .. 2}" "\<lambda> x. x = 1" "\<lambda> x. \<mu>" "\<lambda> x. 1 - \<mu>"]
       by auto
     with *[rule_format, of "2" ?u ?x] have s: "(\<Sum>j \<in> {1..2}. ?u j *\<^sub>R ?x j) \<in> s"
       using mu xy by auto
     have grarr: "(\<Sum>j \<in> {Suc (Suc 0)..2}. ?u j *\<^sub>R ?x j) = (1 - \<mu>) *\<^sub>R y"
-      using setsum_head_Suc[of "Suc (Suc 0)" 2 "\<lambda> j. (1 - \<mu>) *\<^sub>R y"] by auto
-    from setsum_head_Suc[of "Suc 0" 2 "\<lambda> j. ?u j *\<^sub>R ?x j", simplified this]
+      using sum_head_Suc[of "Suc (Suc 0)" 2 "\<lambda> j. (1 - \<mu>) *\<^sub>R y"] by auto
+    from sum_head_Suc[of "Suc 0" 2 "\<lambda> j. ?u j *\<^sub>R ?x j", simplified this]
     have "(\<Sum>j \<in> {1..2}. ?u j *\<^sub>R ?x j) = \<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y"
       by auto
     then have "(1 - \<mu>) *\<^sub>R y + \<mu> *\<^sub>R x \<in> s"
@@ -402,18 +402,18 @@
 lemma convex_explicit:
   fixes s :: "'a::real_vector set"
   shows "convex s \<longleftrightarrow>
-    (\<forall>t u. finite t \<and> t \<subseteq> s \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and> setsum u t = 1 \<longrightarrow> setsum (\<lambda>x. u x *\<^sub>R x) t \<in> s)"
+    (\<forall>t u. finite t \<and> t \<subseteq> s \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and> sum u t = 1 \<longrightarrow> sum (\<lambda>x. u x *\<^sub>R x) t \<in> s)"
 proof safe
   fix t
   fix u :: "'a \<Rightarrow> real"
   assume "convex s"
     and "finite t"
-    and "t \<subseteq> s" "\<forall>x\<in>t. 0 \<le> u x" "setsum u t = 1"
+    and "t \<subseteq> s" "\<forall>x\<in>t. 0 \<le> u x" "sum u t = 1"
   then show "(\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s"
-    using convex_setsum[of t s u "\<lambda> x. x"] by auto
+    using convex_sum[of t s u "\<lambda> x. x"] by auto
 next
   assume *: "\<forall>t. \<forall> u. finite t \<and> t \<subseteq> s \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and>
-    setsum u t = 1 \<longrightarrow> (\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s"
+    sum u t = 1 \<longrightarrow> (\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s"
   show "convex s"
     unfolding convex_alt
   proof safe
@@ -437,7 +437,7 @@
 
 lemma convex_finite:
   assumes "finite s"
-  shows "convex s \<longleftrightarrow> (\<forall>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<longrightarrow> setsum (\<lambda>x. u x *\<^sub>R x) s \<in> s)"
+  shows "convex s \<longleftrightarrow> (\<forall>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = 1 \<longrightarrow> sum (\<lambda>x. u x *\<^sub>R x) s \<in> s)"
   unfolding convex_explicit
   apply safe
   subgoal for u by (erule allE [where x=s], erule allE [where x=u]) auto
@@ -445,12 +445,12 @@
   proof -
     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
-    assume sum: "\<forall>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<longrightarrow> (\<Sum>x\<in>s. u x *\<^sub>R x) \<in> s"
-    assume *: "\<forall>x\<in>t. 0 \<le> u x" "setsum u t = 1"
+    assume sum: "\<forall>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = 1 \<longrightarrow> (\<Sum>x\<in>s. u x *\<^sub>R x) \<in> s"
+    assume *: "\<forall>x\<in>t. 0 \<le> u x" "sum u t = 1"
     assume "t \<subseteq> s"
     then have "s \<inter> t = t" by auto
     with sum[THEN spec[where x="\<lambda>x. if x\<in>t then u x else 0"]] * show "(\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s"
-      by (auto simp: assms setsum.If_cases if_distrib if_distrib_arg)
+      by (auto simp: assms sum.If_cases if_distrib if_distrib_arg)
   qed
   done
 
@@ -704,7 +704,7 @@
     by fastforce
 qed
 
-lemma convex_on_setsum:
+lemma convex_on_sum:
   fixes a :: "'a \<Rightarrow> real"
     and y :: "'a \<Rightarrow> 'b::real_vector"
     and f :: "'b \<Rightarrow> real"
@@ -736,7 +736,7 @@
     then have "(\<Sum> j \<in> s. a j) = 0"
       using insert by auto
     then have "\<And>j. j \<in> s \<Longrightarrow> a j = 0"
-      using insert by (fastforce simp: setsum_nonneg_eq_0_iff)
+      using insert by (fastforce simp: sum_nonneg_eq_0_iff)
     then show ?thesis
       using insert by auto
   next
@@ -746,7 +746,7 @@
     have fis: "finite (insert i s)"
       using insert by auto
     then have ai1: "a i \<le> 1"
-      using setsum_nonneg_leq_bound[of "insert i s" a] insert by simp
+      using sum_nonneg_leq_bound[of "insert i s" a] insert by simp
     then have "a i < 1"
       using False by auto
     then have i0: "1 - a i > 0"
@@ -757,23 +757,23 @@
     have "(\<Sum> j \<in> insert i s. a j) = 1"
       using insert by auto
     then have "(\<Sum> j \<in> s. a j) = 1 - a i"
-      using setsum.insert insert by fastforce
+      using sum.insert insert by fastforce
     then have "(\<Sum> j \<in> s. a j) / (1 - a i) = 1"
       using i0 by auto
     then have a1: "(\<Sum> j \<in> s. ?a j) = 1"
-      unfolding setsum_divide_distrib by simp
+      unfolding sum_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_sum [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)"
-      using setsum.insert[of s i "\<lambda> j. a j *\<^sub>R y j", OF \<open>finite s\<close> \<open>i \<notin> s\<close>] insert
+      using sum.insert[of s i "\<lambda> j. a j *\<^sub>R y j", OF \<open>finite s\<close> \<open>i \<notin> s\<close>] insert
       by (auto simp only: add.commute)
     also have "\<dots> = f (((1 - a i) * inverse (1 - a i)) *\<^sub>R (\<Sum> j \<in> s. a j *\<^sub>R y j) + a i *\<^sub>R y i)"
       using i0 by auto
     also have "\<dots> = f ((1 - a i) *\<^sub>R (\<Sum> j \<in> s. (a j * inverse (1 - a i)) *\<^sub>R y j) + a i *\<^sub>R y i)"
-      using scaleR_right.setsum[of "inverse (1 - a i)" "\<lambda> j. a j *\<^sub>R y j" s, symmetric]
+      using scaleR_right.sum[of "inverse (1 - a i)" "\<lambda> j. a j *\<^sub>R y j" s, symmetric]
       by (auto simp: algebra_simps)
     also have "\<dots> = f ((1 - a i) *\<^sub>R (\<Sum> j \<in> s. ?a j *\<^sub>R y j) + a i *\<^sub>R y i)"
       by (auto simp: divide_inverse)
@@ -785,7 +785,7 @@
             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_distrib_left[of "1 - a i" "\<lambda> j. ?a j * f (y j)"]
+      unfolding sum_distrib_left[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
@@ -1148,19 +1148,19 @@
   finally show ?thesis .
 qed (insert assms(2), simp_all)
 
-lemma convex_supp_setsum:
-  assumes "convex S" and 1: "supp_setsum u I = 1"
+lemma convex_supp_sum:
+  assumes "convex S" and 1: "supp_sum u I = 1"
       and "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> u i \<and> (u i = 0 \<or> f i \<in> S)"
-    shows "supp_setsum (\<lambda>i. u i *\<^sub>R f i) I \<in> S"
+    shows "supp_sum (\<lambda>i. u i *\<^sub>R f i) I \<in> S"
 proof -
   have fin: "finite {i \<in> I. u i \<noteq> 0}"
-    using 1 setsum.infinite by (force simp: supp_setsum_def support_on_def)
-  then have eq: "supp_setsum (\<lambda>i. u i *\<^sub>R f i) I = setsum (\<lambda>i. u i *\<^sub>R f i) {i \<in> I. u i \<noteq> 0}"
-    by (force intro: setsum.mono_neutral_left simp: supp_setsum_def support_on_def)
+    using 1 sum.infinite by (force simp: supp_sum_def support_on_def)
+  then have eq: "supp_sum (\<lambda>i. u i *\<^sub>R f i) I = sum (\<lambda>i. u i *\<^sub>R f i) {i \<in> I. u i \<noteq> 0}"
+    by (force intro: sum.mono_neutral_left simp: supp_sum_def support_on_def)
   show ?thesis
     apply (simp add: eq)
-    apply (rule convex_setsum [OF fin \<open>convex S\<close>])
-    using 1 assms apply (auto simp: supp_setsum_def support_on_def)
+    apply (rule convex_sum [OF fin \<open>convex S\<close>])
+    using 1 assms apply (auto simp: supp_sum_def support_on_def)
     done
 qed
 
@@ -1335,18 +1335,18 @@
   shows "sphere a r = {} \<longleftrightarrow> r < 0"
 by (auto simp: sphere_def dist_norm) (metis dist_norm le_less_linear vector_choose_dist)
 
-lemma setsum_delta_notmem:
+lemma sum_delta_notmem:
   assumes "x \<notin> s"
-  shows "setsum (\<lambda>y. if (y = x) then P x else Q y) s = setsum Q s"
-    and "setsum (\<lambda>y. if (x = y) then P x else Q y) s = setsum Q s"
-    and "setsum (\<lambda>y. if (y = x) then P y else Q y) s = setsum Q s"
-    and "setsum (\<lambda>y. if (x = y) then P y else Q y) s = setsum Q s"
-  apply (rule_tac [!] setsum.cong)
+  shows "sum (\<lambda>y. if (y = x) then P x else Q y) s = sum Q s"
+    and "sum (\<lambda>y. if (x = y) then P x else Q y) s = sum Q s"
+    and "sum (\<lambda>y. if (y = x) then P y else Q y) s = sum Q s"
+    and "sum (\<lambda>y. if (x = y) then P y else Q y) s = sum Q s"
+  apply (rule_tac [!] sum.cong)
   using assms
   apply auto
   done
 
-lemma setsum_delta'':
+lemma sum_delta'':
   fixes s::"'a::real_vector set"
   assumes "finite s"
   shows "(\<Sum>x\<in>s. (if y = x then f x else 0) *\<^sub>R x) = (if y\<in>s then (f y) *\<^sub>R y else 0)"
@@ -1354,7 +1354,7 @@
   have *: "\<And>x y. (if y = x then f x else (0::real)) *\<^sub>R x = (if x=y then (f x) *\<^sub>R x else 0)"
     by auto
   show ?thesis
-    unfolding * using setsum.delta[OF assms, of y "\<lambda>x. f x *\<^sub>R x"] by auto
+    unfolding * using sum.delta[OF assms, of y "\<lambda>x. f x *\<^sub>R x"] by auto
 qed
 
 lemma if_smult: "(if P then x else (y::real)) *\<^sub>R v = (if P then x *\<^sub>R v else y *\<^sub>R v)"
@@ -1429,7 +1429,7 @@
 lemma affine:
   fixes V::"'a::real_vector set"
   shows "affine V \<longleftrightarrow>
-    (\<forall>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> V \<and> setsum u s = 1 \<longrightarrow> (setsum (\<lambda>x. (u x) *\<^sub>R x)) s \<in> V)"
+    (\<forall>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> V \<and> sum u s = 1 \<longrightarrow> (sum (\<lambda>x. (u x) *\<^sub>R x)) s \<in> V)"
   unfolding affine_def
   apply rule
   apply(rule, rule, rule)
@@ -1439,7 +1439,7 @@
 proof -
   fix x y u v
   assume as: "x \<in> V" "y \<in> V" "u + v = (1::real)"
-    "\<forall>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> V \<and> setsum u s = 1 \<longrightarrow> (\<Sum>x\<in>s. u x *\<^sub>R x) \<in> V"
+    "\<forall>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> V \<and> sum u s = 1 \<longrightarrow> (\<Sum>x\<in>s. u x *\<^sub>R x) \<in> V"
   then show "u *\<^sub>R x + v *\<^sub>R y \<in> V"
     apply (cases "x = y")
     using as(4)[THEN spec[where x="{x,y}"], THEN spec[where x="\<lambda>w. if w = x then u else v"]]
@@ -1449,7 +1449,7 @@
 next
   fix s u
   assume as: "\<forall>x\<in>V. \<forall>y\<in>V. \<forall>u v. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> V"
-    "finite s" "s \<noteq> {}" "s \<subseteq> V" "setsum u s = (1::real)"
+    "finite s" "s \<noteq> {}" "s \<subseteq> V" "sum u s = (1::real)"
   define n where "n = card s"
   have "card s = 0 \<or> card s = 1 \<or> card s = 2 \<or> card s > 2" by auto
   then show "(\<Sum>x\<in>s. u x *\<^sub>R x) \<in> V"
@@ -1461,7 +1461,7 @@
       unfolding card_Suc_eq by auto
     then show ?thesis
       using as(1)[THEN bspec[where x=a], THEN bspec[where x=b]] using as(4,5)
-      by (auto simp add: setsum_clauses(2))
+      by (auto simp add: sum_clauses(2))
   next
     assume "card s > 2"
     then show ?thesis using as and n_def
@@ -1473,15 +1473,15 @@
       fix s :: "'a set" and u :: "'a \<Rightarrow> real"
       assume IA:
         "\<And>u s.  \<lbrakk>2 < card s; \<forall>x\<in>V. \<forall>y\<in>V. \<forall>u v. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> V; finite s;
-          s \<noteq> {}; s \<subseteq> V; setsum u s = 1; n = card s \<rbrakk> \<Longrightarrow> (\<Sum>x\<in>s. u x *\<^sub>R x) \<in> V"
+          s \<noteq> {}; s \<subseteq> V; sum u s = 1; n = card s \<rbrakk> \<Longrightarrow> (\<Sum>x\<in>s. u x *\<^sub>R x) \<in> V"
         and as:
           "Suc n = card s" "2 < card s" "\<forall>x\<in>V. \<forall>y\<in>V. \<forall>u v. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> V"
-           "finite s" "s \<noteq> {}" "s \<subseteq> V" "setsum u s = 1"
+           "finite s" "s \<noteq> {}" "s \<subseteq> V" "sum u s = 1"
       have "\<exists>x\<in>s. u x \<noteq> 1"
       proof (rule ccontr)
         assume "\<not> ?thesis"
-        then have "setsum u s = real_of_nat (card s)"
-          unfolding card_eq_setsum by auto
+        then have "sum u s = real_of_nat (card s)"
+          unfolding card_eq_sum by auto
         then show False
           using as(7) and \<open>card s > 2\<close>
           by (metis One_nat_def less_Suc0 Zero_not_Suc of_nat_1 of_nat_eq_iff numeral_2_eq_2)
@@ -1495,9 +1495,9 @@
         done
       have *: "s = insert x (s - {x})" "finite (s - {x})"
         using \<open>x\<in>s\<close> and as(4) by auto
-      have **: "setsum u (s - {x}) = 1 - u x"
-        using setsum_clauses(2)[OF *(2), of u x, unfolded *(1)[symmetric] as(7)] by auto
-      have ***: "inverse (1 - u x) * setsum u (s - {x}) = 1"
+      have **: "sum u (s - {x}) = 1 - u x"
+        using sum_clauses(2)[OF *(2), of u x, unfolded *(1)[symmetric] as(7)] by auto
+      have ***: "inverse (1 - u x) * sum u (s - {x}) = 1"
         unfolding ** using \<open>u x \<noteq> 1\<close> by auto
       have "(\<Sum>xa\<in>s - {x}. (inverse (1 - u x) * u xa) *\<^sub>R xa) \<in> V"
       proof (cases "card (s - {x}) > 2")
@@ -1511,7 +1511,7 @@
         qed auto
         then show ?thesis
           apply (rule_tac IA[of "s - {x}" "\<lambda>y. (inverse (1 - u x) * u y)"])
-          unfolding setsum_distrib_left[symmetric]
+          unfolding sum_distrib_left[symmetric]
           using as and *** and True
           apply auto
           done
@@ -1524,21 +1524,21 @@
         then show ?thesis
           using as(3)[THEN bspec[where x=a], THEN bspec[where x=b]]
           using *** *(2) and \<open>s \<subseteq> V\<close>
-          unfolding setsum_distrib_left
-          by (auto simp add: setsum_clauses(2))
+          unfolding sum_distrib_left
+          by (auto simp add: sum_clauses(2))
       qed
       then have "u x + (1 - u x) = 1 \<Longrightarrow>
           u x *\<^sub>R x + (1 - u x) *\<^sub>R ((\<Sum>xa\<in>s - {x}. u xa *\<^sub>R xa) /\<^sub>R (1 - u x)) \<in> V"
         apply -
         apply (rule as(3)[rule_format])
-        unfolding  Real_Vector_Spaces.scaleR_right.setsum
+        unfolding  Real_Vector_Spaces.scaleR_right.sum
         using x(1) as(6)
         apply auto
         done
       then show "(\<Sum>x\<in>s. u x *\<^sub>R x) \<in> V"
-        unfolding scaleR_scaleR[symmetric] and scaleR_right.setsum [symmetric]
+        unfolding scaleR_scaleR[symmetric] and scaleR_right.sum [symmetric]
         apply (subst *)
-        unfolding setsum_clauses(2)[OF *(2)]
+        unfolding sum_clauses(2)[OF *(2)]
         using \<open>u x \<noteq> 1\<close>
         apply auto
         done
@@ -1554,7 +1554,7 @@
 
 lemma affine_hull_explicit:
   "affine hull p =
-    {y. \<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum u s = 1 \<and> setsum (\<lambda>v. (u v) *\<^sub>R v) s = y}"
+    {y. \<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> sum u s = 1 \<and> sum (\<lambda>v. (u v) *\<^sub>R v) s = y}"
   apply (rule hull_unique)
   apply (subst subset_eq)
   prefer 3
@@ -1567,7 +1567,7 @@
 proof -
   fix x
   assume "x\<in>p"
-  then show "\<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = x"
+  then show "\<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> sum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = x"
     apply (rule_tac x="{x}" in exI)
     apply (rule_tac x="\<lambda>x. 1" in exI)
     apply auto
@@ -1575,12 +1575,12 @@
 next
   fix t x s u
   assume as: "p \<subseteq> t" "affine t" "finite s" "s \<noteq> {}"
-    "s \<subseteq> p" "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x"
+    "s \<subseteq> p" "sum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x"
   then show "x \<in> t"
     using as(2)[unfolded affine, THEN spec[where x=s], THEN spec[where x=u]]
     by auto
 next
-  show "affine {y. \<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y}"
+  show "affine {y. \<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> sum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y}"
     unfolding affine_def
     apply (rule, rule, rule, rule, rule)
     unfolding mem_Collect_eq
@@ -1588,26 +1588,26 @@
     fix u v :: real
     assume uv: "u + v = 1"
     fix x
-    assume "\<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = x"
+    assume "\<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> sum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = x"
     then obtain sx ux where
-      x: "finite sx" "sx \<noteq> {}" "sx \<subseteq> p" "setsum ux sx = 1" "(\<Sum>v\<in>sx. ux v *\<^sub>R v) = x"
+      x: "finite sx" "sx \<noteq> {}" "sx \<subseteq> p" "sum ux sx = 1" "(\<Sum>v\<in>sx. ux v *\<^sub>R v) = x"
       by auto
     fix y
-    assume "\<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y"
+    assume "\<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> sum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y"
     then obtain sy uy where
-      y: "finite sy" "sy \<noteq> {}" "sy \<subseteq> p" "setsum uy sy = 1" "(\<Sum>v\<in>sy. uy v *\<^sub>R v) = y" by auto
+      y: "finite sy" "sy \<noteq> {}" "sy \<subseteq> p" "sum uy sy = 1" "(\<Sum>v\<in>sy. uy v *\<^sub>R v) = y" by auto
     have xy: "finite (sx \<union> sy)"
       using x(1) y(1) by auto
     have **: "(sx \<union> sy) \<inter> sx = sx" "(sx \<union> sy) \<inter> sy = sy"
       by auto
     show "\<exists>s ua. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and>
-        setsum ua s = 1 \<and> (\<Sum>v\<in>s. ua v *\<^sub>R v) = u *\<^sub>R x + v *\<^sub>R y"
+        sum ua s = 1 \<and> (\<Sum>v\<in>s. ua v *\<^sub>R v) = u *\<^sub>R x + v *\<^sub>R y"
       apply (rule_tac x="sx \<union> sy" in exI)
       apply (rule_tac x="\<lambda>a. (if a\<in>sx then u * ux a else 0) + (if a\<in>sy then v * uy a else 0)" in exI)
-      unfolding scaleR_left_distrib setsum.distrib if_smult scaleR_zero_left
-        ** setsum.inter_restrict[OF xy, symmetric]
-      unfolding scaleR_scaleR[symmetric] Real_Vector_Spaces.scaleR_right.setsum [symmetric]
-        and setsum_distrib_left[symmetric]
+      unfolding scaleR_left_distrib sum.distrib if_smult scaleR_zero_left
+        ** sum.inter_restrict[OF xy, symmetric]
+      unfolding scaleR_scaleR[symmetric] Real_Vector_Spaces.scaleR_right.sum [symmetric]
+        and sum_distrib_left[symmetric]
       unfolding x y
       using x(1-3) y(1-3) uv
       apply simp
@@ -1617,7 +1617,7 @@
 
 lemma affine_hull_finite:
   assumes "finite s"
-  shows "affine hull s = {y. \<exists>u. setsum u s = 1 \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = y}"
+  shows "affine hull s = {y. \<exists>u. sum u s = 1 \<and> sum (\<lambda>v. u v *\<^sub>R v) s = y}"
   unfolding affine_hull_explicit and set_eq_iff and mem_Collect_eq
   apply (rule, rule)
   apply (erule exE)+
@@ -1627,9 +1627,9 @@
   apply (erule conjE)
 proof -
   fix x u
-  assume "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x"
+  assume "sum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x"
   then show "\<exists>sa u. finite sa \<and>
-      \<not> (\<forall>x. (x \<in> sa) = (x \<in> {})) \<and> sa \<subseteq> s \<and> setsum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *\<^sub>R v) = x"
+      \<not> (\<forall>x. (x \<in> sa) = (x \<in> {})) \<and> sa \<subseteq> s \<and> sum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *\<^sub>R v) = x"
     apply (rule_tac x=s in exI, rule_tac x=u in exI)
     using assms
     apply auto
@@ -1639,10 +1639,10 @@
   assume "t \<subseteq> s"
   then have *: "s \<inter> t = t"
     by auto
-  assume "finite t" "\<not> (\<forall>x. (x \<in> t) = (x \<in> {}))" "setsum u t = 1" "(\<Sum>v\<in>t. u v *\<^sub>R v) = x"
-  then show "\<exists>u. setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = x"
+  assume "finite t" "\<not> (\<forall>x. (x \<in> t) = (x \<in> {}))" "sum u t = 1" "(\<Sum>v\<in>t. u v *\<^sub>R v) = x"
+  then show "\<exists>u. sum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = x"
     apply (rule_tac x="\<lambda>x. if x\<in>t then u x else 0" in exI)
-    unfolding if_smult scaleR_zero_left and setsum.inter_restrict[OF assms, symmetric] and *
+    unfolding if_smult scaleR_zero_left and sum.inter_restrict[OF assms, symmetric] and *
     apply auto
     done
 qed
@@ -1653,22 +1653,22 @@
 lemma affine_hull_empty[simp]: "affine hull {} = {}"
   by (rule hull_unique) auto
 
-(*could delete: it simply rewrites setsum expressions, but it's used twice*)
+(*could delete: it simply rewrites sum expressions, but it's used twice*)
 lemma affine_hull_finite_step:
   fixes y :: "'a::real_vector"
   shows
-    "(\<exists>u. setsum u {} = w \<and> setsum (\<lambda>x. u x *\<^sub>R x) {} = y) \<longleftrightarrow> w = 0 \<and> y = 0" (is ?th1)
+    "(\<exists>u. sum u {} = w \<and> sum (\<lambda>x. u x *\<^sub>R x) {} = y) \<longleftrightarrow> w = 0 \<and> y = 0" (is ?th1)
     and
     "finite s \<Longrightarrow>
-      (\<exists>u. setsum u (insert a s) = w \<and> setsum (\<lambda>x. u x *\<^sub>R x) (insert a s) = y) \<longleftrightarrow>
-      (\<exists>v u. setsum u s = w - v \<and> setsum (\<lambda>x. u x *\<^sub>R x) s = y - v *\<^sub>R a)" (is "_ \<Longrightarrow> ?lhs = ?rhs")
+      (\<exists>u. sum u (insert a s) = w \<and> sum (\<lambda>x. u x *\<^sub>R x) (insert a s) = y) \<longleftrightarrow>
+      (\<exists>v u. sum u s = w - v \<and> sum (\<lambda>x. u x *\<^sub>R x) s = y - v *\<^sub>R a)" (is "_ \<Longrightarrow> ?lhs = ?rhs")
 proof -
   show ?th1 by simp
   assume fin: "finite s"
   show "?lhs = ?rhs"
   proof
     assume ?lhs
-    then obtain u where u: "setsum u (insert a s) = w \<and> (\<Sum>x\<in>insert a s. u x *\<^sub>R x) = y"
+    then obtain u where u: "sum u (insert a s) = w \<and> (\<Sum>x\<in>insert a s. u x *\<^sub>R x) = y"
       by auto
     show ?rhs
     proof (cases "a \<in> s")
@@ -1689,7 +1689,7 @@
     qed
   next
     assume ?rhs
-    then obtain v u where vu: "setsum u s = w - v"  "(\<Sum>x\<in>s. u x *\<^sub>R x) = y - v *\<^sub>R a"
+    then obtain v u where vu: "sum u s = w - v"  "(\<Sum>x\<in>s. u x *\<^sub>R x) = y - v *\<^sub>R a"
       by auto
     have *: "\<And>x M. (if x = a then v else M) *\<^sub>R x = (if x = a then v *\<^sub>R x else M *\<^sub>R x)"
       by auto
@@ -1698,11 +1698,11 @@
       case True
       then show ?thesis
         apply (rule_tac x="\<lambda>x. (if x=a then v else 0) + u x" in exI)
-        unfolding setsum_clauses(2)[OF fin]
+        unfolding sum_clauses(2)[OF fin]
         apply simp
-        unfolding scaleR_left_distrib and setsum.distrib
+        unfolding scaleR_left_distrib and sum.distrib
         unfolding vu and * and scaleR_zero_left
-        apply (auto simp add: setsum.delta[OF fin])
+        apply (auto simp add: sum.delta[OF fin])
         done
     next
       case False
@@ -1711,9 +1711,9 @@
         "\<And>x. x \<in> s \<Longrightarrow> u x *\<^sub>R x = (if x = a then v *\<^sub>R x else u x *\<^sub>R x)" by auto
       from False show ?thesis
         apply (rule_tac x="\<lambda>x. if x=a then v else u x" in exI)
-        unfolding setsum_clauses(2)[OF fin] and * using vu
-        using setsum.cong [of s _ "\<lambda>x. u x *\<^sub>R x" "\<lambda>x. if x = a then v *\<^sub>R x else u x *\<^sub>R x", OF _ **(2)]
-        using setsum.cong [of s _ u "\<lambda>x. if x = a then v else u x", OF _ **(1)]
+        unfolding sum_clauses(2)[OF fin] and * using vu
+        using sum.cong [of s _ "\<lambda>x. u x *\<^sub>R x" "\<lambda>x. if x = a then v *\<^sub>R x else u x *\<^sub>R x", OF _ **(2)]
+        using sum.cong [of s _ u "\<lambda>x. if x = a then v else u x", OF _ **(1)]
         apply auto
         done
     qed
@@ -1728,7 +1728,7 @@
   have *:
     "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real)"
     "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::'a)" by auto
-  have "?lhs = {y. \<exists>u. setsum u {a, b} = 1 \<and> (\<Sum>v\<in>{a, b}. u v *\<^sub>R v) = y}"
+  have "?lhs = {y. \<exists>u. sum u {a, b} = 1 \<and> (\<Sum>v\<in>{a, b}. u v *\<^sub>R v) = y}"
     using affine_hull_finite[of "{a,b}"] by auto
   also have "\<dots> = {y. \<exists>v u. u b = 1 - v \<and> u b *\<^sub>R b = y - v *\<^sub>R a}"
     by (simp add: affine_hull_finite_step(2)[of "{b}" a])
@@ -1797,7 +1797,7 @@
   apply (erule conjE)+
 proof -
   fix x t u
-  assume as: "finite t" "t \<noteq> {}" "t \<subseteq> insert a s" "setsum u t = 1" "(\<Sum>v\<in>t. u v *\<^sub>R v) = x"
+  assume as: "finite t" "t \<noteq> {}" "t \<subseteq> insert a s" "sum u t = 1" "(\<Sum>v\<in>t. u v *\<^sub>R v) = x"
   have "(\<lambda>x. x - a) ` (t - {a}) \<subseteq> {x - a |x. x \<in> s}"
     using as(3) by auto
   then show "\<exists>v. x = a + v \<and> (\<exists>S u. finite S \<and> S \<subseteq> {x - a |x. x \<in> s} \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = v)"
@@ -1808,8 +1808,8 @@
     apply (rule conjI) using as(1) apply simp
     apply (erule conjI)
     using as(1)
-    apply (simp add: setsum.reindex[unfolded inj_on_def] scaleR_right_diff_distrib
-      setsum_subtractf scaleR_left.setsum[symmetric] setsum_diff1 scaleR_left_diff_distrib)
+    apply (simp add: sum.reindex[unfolded inj_on_def] scaleR_right_diff_distrib
+      sum_subtractf scaleR_left.sum[symmetric] sum_diff1 scaleR_left_diff_distrib)
     unfolding as
     apply simp
     done
@@ -1828,16 +1828,16 @@
     unfolding span_explicit by auto
   define f where "f = (\<lambda>x. x + a) ` t"
   have f: "finite f" "f \<subseteq> s" "(\<Sum>v\<in>f. u (v - a) *\<^sub>R (v - a)) = y - a"
-    unfolding f_def using obt by (auto simp add: setsum.reindex[unfolded inj_on_def])
+    unfolding f_def using obt by (auto simp add: sum.reindex[unfolded inj_on_def])
   have *: "f \<inter> {a} = {}" "f \<inter> - {a} = f"
     using f(2) assms by auto
-  show "\<exists>sa u. finite sa \<and> sa \<noteq> {} \<and> sa \<subseteq> insert a s \<and> setsum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *\<^sub>R v) = y"
+  show "\<exists>sa u. finite sa \<and> sa \<noteq> {} \<and> sa \<subseteq> insert a s \<and> sum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *\<^sub>R v) = y"
     apply (rule_tac x = "insert a f" in exI)
-    apply (rule_tac x = "\<lambda>x. if x=a then 1 - setsum (\<lambda>x. u (x - a)) f else u (x - a)" in exI)
+    apply (rule_tac x = "\<lambda>x. if x=a then 1 - sum (\<lambda>x. u (x - a)) f else u (x - a)" in exI)
     using assms and f
-    unfolding setsum_clauses(2)[OF f(1)] and if_smult
-    unfolding setsum.If_cases[OF f(1), of "\<lambda>x. x = a"]
-    apply (auto simp add: setsum_subtractf scaleR_left.setsum algebra_simps *)
+    unfolding sum_clauses(2)[OF f(1)] and if_smult
+    unfolding sum.If_cases[OF f(1), of "\<lambda>x. x = a"]
+    apply (auto simp add: sum_subtractf scaleR_left.sum algebra_simps *)
     done
 qed
 
@@ -2281,8 +2281,8 @@
 
 lemma affine_dependent_explicit:
   "affine_dependent p \<longleftrightarrow>
-    (\<exists>s u. finite s \<and> s \<subseteq> p \<and> setsum u s = 0 \<and>
-      (\<exists>v\<in>s. u v \<noteq> 0) \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = 0)"
+    (\<exists>s u. finite s \<and> s \<subseteq> p \<and> sum u s = 0 \<and>
+      (\<exists>v\<in>s. u v \<noteq> 0) \<and> sum (\<lambda>v. u v *\<^sub>R v) s = 0)"
   unfolding affine_dependent_def affine_hull_explicit mem_Collect_eq
   apply rule
   apply (erule bexE, erule exE, erule exE)
@@ -2293,25 +2293,25 @@
   apply (erule bexE)
 proof -
   fix x s u
-  assume as: "x \<in> p" "finite s" "s \<noteq> {}" "s \<subseteq> p - {x}" "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x"
+  assume as: "x \<in> p" "finite s" "s \<noteq> {}" "s \<subseteq> p - {x}" "sum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x"
   have "x \<notin> s" using as(1,4) by auto
-  show "\<exists>s u. finite s \<and> s \<subseteq> p \<and> setsum u s = 0 \<and> (\<exists>v\<in>s. u v \<noteq> 0) \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = 0"
+  show "\<exists>s u. finite s \<and> s \<subseteq> p \<and> sum u s = 0 \<and> (\<exists>v\<in>s. u v \<noteq> 0) \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = 0"
     apply (rule_tac x="insert x s" in exI, rule_tac x="\<lambda>v. if v = x then - 1 else u v" in exI)
-    unfolding if_smult and setsum_clauses(2)[OF as(2)] and setsum_delta_notmem[OF \<open>x\<notin>s\<close>] and as
+    unfolding if_smult and sum_clauses(2)[OF as(2)] and sum_delta_notmem[OF \<open>x\<notin>s\<close>] and as
     using as
     apply auto
     done
 next
   fix s u v
-  assume as: "finite s" "s \<subseteq> p" "setsum u s = 0" "(\<Sum>v\<in>s. u v *\<^sub>R v) = 0" "v \<in> s" "u v \<noteq> 0"
+  assume as: "finite s" "s \<subseteq> p" "sum u s = 0" "(\<Sum>v\<in>s. u v *\<^sub>R v) = 0" "v \<in> s" "u v \<noteq> 0"
   have "s \<noteq> {v}"
     using as(3,6) by auto
-  then show "\<exists>x\<in>p. \<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p - {x} \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = x"
+  then show "\<exists>x\<in>p. \<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p - {x} \<and> sum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = x"
     apply (rule_tac x=v in bexI)
     apply (rule_tac x="s - {v}" in exI)
     apply (rule_tac x="\<lambda>x. - (1 / u v) * u x" in exI)
-    unfolding scaleR_scaleR[symmetric] and scaleR_right.setsum [symmetric]
-    unfolding setsum_distrib_left[symmetric] and setsum_diff1[OF as(1)]
+    unfolding scaleR_scaleR[symmetric] and scaleR_right.sum [symmetric]
+    unfolding sum_distrib_left[symmetric] and sum_diff1[OF as(1)]
     using as
     apply auto
     done
@@ -2321,24 +2321,24 @@
   fixes s :: "'a::real_vector set"
   assumes "finite s"
   shows "affine_dependent s \<longleftrightarrow>
-    (\<exists>u. setsum u s = 0 \<and> (\<exists>v\<in>s. u v \<noteq> 0) \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = 0)"
+    (\<exists>u. sum u s = 0 \<and> (\<exists>v\<in>s. u v \<noteq> 0) \<and> sum (\<lambda>v. u v *\<^sub>R v) s = 0)"
   (is "?lhs = ?rhs")
 proof
   have *: "\<And>vt u v. (if vt then u v else 0) *\<^sub>R v = (if vt then (u v) *\<^sub>R v else 0::'a)"
     by auto
   assume ?lhs
   then obtain t u v where
-    "finite t" "t \<subseteq> s" "setsum u t = 0" "v\<in>t" "u v \<noteq> 0"  "(\<Sum>v\<in>t. u v *\<^sub>R v) = 0"
+    "finite t" "t \<subseteq> s" "sum u t = 0" "v\<in>t" "u v \<noteq> 0"  "(\<Sum>v\<in>t. u v *\<^sub>R v) = 0"
     unfolding affine_dependent_explicit by auto
   then show ?rhs
     apply (rule_tac x="\<lambda>x. if x\<in>t then u x else 0" in exI)
-    apply auto unfolding * and setsum.inter_restrict[OF assms, symmetric]
+    apply auto unfolding * and sum.inter_restrict[OF assms, symmetric]
     unfolding Int_absorb1[OF \<open>t\<subseteq>s\<close>]
     apply auto
     done
 next
   assume ?rhs
-  then obtain u v where "setsum u s = 0"  "v\<in>s" "u v \<noteq> 0" "(\<Sum>v\<in>s. u v *\<^sub>R v) = 0"
+  then obtain u v where "sum u s = 0"  "v\<in>s" "u v \<noteq> 0" "(\<Sum>v\<in>s. u v *\<^sub>R v) = 0"
     by auto
   then show ?lhs unfolding affine_dependent_explicit
     using assms by auto
@@ -2718,7 +2718,7 @@
   shows "convex hull s =
     {y. \<exists>k u x.
       (\<forall>i\<in>{1::nat .. k}. 0 \<le> u i \<and> x i \<in> s) \<and>
-      (setsum u {1..k} = 1) \<and> (setsum (\<lambda>i. u i *\<^sub>R x i) {1..k} = y)}"
+      (sum u {1..k} = 1) \<and> (sum (\<lambda>i. u i *\<^sub>R x i) {1..k} = y)}"
   (is "?xyz = ?hull")
   apply (rule hull_unique)
   apply rule
@@ -2743,7 +2743,7 @@
     fix x k u y
     assume assm:
       "\<forall>i\<in>{1::nat..k}. 0 \<le> u i \<and> y i \<in> s"
-      "setsum u {1..k} = 1" "(\<Sum>i = 1..k. u i *\<^sub>R y i) = x"
+      "sum u {1..k} = 1" "(\<Sum>i = 1..k. u i *\<^sub>R y i) = x"
     show "x\<in>t"
       unfolding assm(3) [symmetric]
       apply (rule as(2)[unfolded convex, rule_format])
@@ -2755,10 +2755,10 @@
   assume uv: "0 \<le> u" "0 \<le> v" "u + v = (1::real)"
   assume xy: "x \<in> ?hull" "y \<in> ?hull"
   from xy obtain k1 u1 x1 where
-    x: "\<forall>i\<in>{1::nat..k1}. 0\<le>u1 i \<and> x1 i \<in> s" "setsum u1 {Suc 0..k1} = 1" "(\<Sum>i = Suc 0..k1. u1 i *\<^sub>R x1 i) = x"
+    x: "\<forall>i\<in>{1::nat..k1}. 0\<le>u1 i \<and> x1 i \<in> s" "sum u1 {Suc 0..k1} = 1" "(\<Sum>i = Suc 0..k1. u1 i *\<^sub>R x1 i) = x"
     by auto
   from xy obtain k2 u2 x2 where
-    y: "\<forall>i\<in>{1::nat..k2}. 0\<le>u2 i \<and> x2 i \<in> s" "setsum u2 {Suc 0..k2} = 1" "(\<Sum>i = Suc 0..k2. u2 i *\<^sub>R x2 i) = y"
+    y: "\<forall>i\<in>{1::nat..k2}. 0\<le>u2 i \<and> x2 i \<in> s" "sum u2 {Suc 0..k2} = 1" "(\<Sum>i = Suc 0..k2. u2 i *\<^sub>R x2 i) = y"
     by auto
   have *: "\<And>P (x1::'a) x2 s1 s2 i.
     (if P i then s1 else s2) *\<^sub>R (if P i then x1 else x2) = (if P i then s1 *\<^sub>R x1 else s2 *\<^sub>R x2)"
@@ -2779,9 +2779,9 @@
     apply (rule, rule)
     defer
     apply rule
-    unfolding * and setsum.If_cases[OF finite_atLeastAtMost[of 1 "k1 + k2"]] and
-      setsum.reindex[OF inj] and o_def Collect_mem_eq
-    unfolding scaleR_scaleR[symmetric] scaleR_right.setsum [symmetric] setsum_distrib_left[symmetric]
+    unfolding * and sum.If_cases[OF finite_atLeastAtMost[of 1 "k1 + k2"]] and
+      sum.reindex[OF inj] and o_def Collect_mem_eq
+    unfolding scaleR_scaleR[symmetric] scaleR_right.sum [symmetric] sum_distrib_left[symmetric]
   proof -
     fix i
     assume i: "i \<in> {1..k1+k2}"
@@ -2807,22 +2807,22 @@
   fixes s :: "'a::real_vector set"
   assumes "finite s"
   shows "convex hull s = {y. \<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and>
-    setsum u s = 1 \<and> setsum (\<lambda>x. u x *\<^sub>R x) s = y}"
+    sum u s = 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) s = y}"
   (is "?HULL = ?set")
 proof (rule hull_unique, auto simp add: convex_def[of ?set])
   fix x
   assume "x \<in> s"
-  then show "\<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> (\<Sum>x\<in>s. u x *\<^sub>R x) = x"
+  then show "\<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = 1 \<and> (\<Sum>x\<in>s. u x *\<^sub>R x) = x"
     apply (rule_tac x="\<lambda>y. if x=y then 1 else 0" in exI)
     apply auto
-    unfolding setsum.delta'[OF assms] and setsum_delta''[OF assms]
+    unfolding sum.delta'[OF assms] and sum_delta''[OF assms]
     apply auto
     done
 next
   fix u v :: real
   assume uv: "0 \<le> u" "0 \<le> v" "u + v = 1"
-  fix ux assume ux: "\<forall>x\<in>s. 0 \<le> ux x" "setsum ux s = (1::real)"
-  fix uy assume uy: "\<forall>x\<in>s. 0 \<le> uy x" "setsum uy s = (1::real)"
+  fix ux assume ux: "\<forall>x\<in>s. 0 \<le> ux x" "sum ux s = (1::real)"
+  fix uy assume uy: "\<forall>x\<in>s. 0 \<le> uy x" "sum uy s = (1::real)"
   {
     fix x
     assume "x\<in>s"
@@ -2832,15 +2832,15 @@
   }
   moreover
   have "(\<Sum>x\<in>s. u * ux x + v * uy x) = 1"
-    unfolding setsum.distrib and setsum_distrib_left[symmetric] and ux(2) uy(2)
+    unfolding sum.distrib and sum_distrib_left[symmetric] and ux(2) uy(2)
     using uv(3) by auto
   moreover
   have "(\<Sum>x\<in>s. (u * ux x + v * uy x) *\<^sub>R x) = u *\<^sub>R (\<Sum>x\<in>s. ux x *\<^sub>R x) + v *\<^sub>R (\<Sum>x\<in>s. uy x *\<^sub>R x)"
-    unfolding scaleR_left_distrib and setsum.distrib and scaleR_scaleR[symmetric]
-      and scaleR_right.setsum [symmetric]
+    unfolding scaleR_left_distrib and sum.distrib and scaleR_scaleR[symmetric]
+      and scaleR_right.sum [symmetric]
     by auto
   ultimately
-  show "\<exists>uc. (\<forall>x\<in>s. 0 \<le> uc x) \<and> setsum uc s = 1 \<and>
+  show "\<exists>uc. (\<forall>x\<in>s. 0 \<le> uc x) \<and> sum uc s = 1 \<and>
       (\<Sum>x\<in>s. uc x *\<^sub>R x) = u *\<^sub>R (\<Sum>x\<in>s. ux x *\<^sub>R x) + v *\<^sub>R (\<Sum>x\<in>s. uy x *\<^sub>R x)"
     apply (rule_tac x="\<lambda>x. u * ux x + v * uy x" in exI)
     apply auto
@@ -2849,7 +2849,7 @@
   fix t
   assume t: "s \<subseteq> t" "convex t"
   fix u
-  assume u: "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = (1::real)"
+  assume u: "\<forall>x\<in>s. 0 \<le> u x" "sum u s = (1::real)"
   then show "(\<Sum>x\<in>s. u x *\<^sub>R x) \<in> t"
     using t(2)[unfolded convex_explicit, THEN spec[where x=s], THEN spec[where x=u]]
     using assms and t(1) by auto
@@ -2861,14 +2861,14 @@
 lemma convex_hull_explicit:
   fixes p :: "'a::real_vector set"
   shows "convex hull p =
-    {y. \<exists>s u. finite s \<and> s \<subseteq> p \<and> (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = y}"
+    {y. \<exists>s u. finite s \<and> s \<subseteq> p \<and> (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = 1 \<and> sum (\<lambda>v. u v *\<^sub>R v) s = y}"
   (is "?lhs = ?rhs")
 proof -
   {
     fix x
     assume "x\<in>?lhs"
     then obtain k u y where
-        obt: "\<forall>i\<in>{1::nat..k}. 0 \<le> u i \<and> y i \<in> p" "setsum u {1..k} = 1" "(\<Sum>i = 1..k. u i *\<^sub>R y i) = x"
+        obt: "\<forall>i\<in>{1::nat..k}. 0 \<le> u i \<and> y i \<in> p" "sum u {1..k} = 1" "(\<Sum>i = 1..k. u i *\<^sub>R y i) = x"
       unfolding convex_hull_indexed by auto
 
     have fin: "finite {1..k}" by auto
@@ -2876,24 +2876,24 @@
     {
       fix j
       assume "j\<in>{1..k}"
-      then have "y j \<in> p" "0 \<le> setsum u {i. Suc 0 \<le> i \<and> i \<le> k \<and> y i = y j}"
+      then have "y j \<in> p" "0 \<le> sum u {i. Suc 0 \<le> i \<and> i \<le> k \<and> y i = y j}"
         using obt(1)[THEN bspec[where x=j]] and obt(2)
         apply simp
-        apply (rule setsum_nonneg)
+        apply (rule sum_nonneg)
         using obt(1)
         apply auto
         done
     }
     moreover
-    have "(\<Sum>v\<in>y ` {1..k}. setsum u {i \<in> {1..k}. y i = v}) = 1"
-      unfolding setsum_image_gen[OF fin, symmetric] using obt(2) by auto
-    moreover have "(\<Sum>v\<in>y ` {1..k}. setsum u {i \<in> {1..k}. y i = v} *\<^sub>R v) = x"
-      using setsum_image_gen[OF fin, of "\<lambda>i. u i *\<^sub>R y i" y, symmetric]
-      unfolding scaleR_left.setsum using obt(3) by auto
+    have "(\<Sum>v\<in>y ` {1..k}. sum u {i \<in> {1..k}. y i = v}) = 1"
+      unfolding sum_image_gen[OF fin, symmetric] using obt(2) by auto
+    moreover have "(\<Sum>v\<in>y ` {1..k}. sum u {i \<in> {1..k}. y i = v} *\<^sub>R v) = x"
+      using sum_image_gen[OF fin, of "\<lambda>i. u i *\<^sub>R y i" y, symmetric]
+      unfolding scaleR_left.sum using obt(3) by auto
     ultimately
-    have "\<exists>s u. finite s \<and> s \<subseteq> p \<and> (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = x"
+    have "\<exists>s u. finite s \<and> s \<subseteq> p \<and> (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = x"
       apply (rule_tac x="y ` {1..k}" in exI)
-      apply (rule_tac x="\<lambda>v. setsum u {i\<in>{1..k}. y i = v}" in exI)
+      apply (rule_tac x="\<lambda>v. sum u {i\<in>{1..k}. y i = v}" in exI)
       apply auto
       done
     then have "x\<in>?rhs" by auto
@@ -2903,7 +2903,7 @@
     fix y
     assume "y\<in>?rhs"
     then obtain s u where
-      obt: "finite s" "s \<subseteq> p" "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = y"
+      obt: "finite s" "s \<subseteq> p" "\<forall>x\<in>s. 0 \<le> u x" "sum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = y"
       by auto
 
     obtain f where f: "inj_on f {1..card s}" "f ` {1..card s} = s"
@@ -2934,18 +2934,18 @@
       then have "card {x. Suc 0 \<le> x \<and> x \<le> card s \<and> f x = y} = 1" by auto
       then have "(\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x)) = u y"
           "(\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x) *\<^sub>R f x) = u y *\<^sub>R y"
-        by (auto simp add: setsum_constant_scaleR)
+        by (auto simp add: sum_constant_scaleR)
     }
     then have "(\<Sum>x = 1..card s. u (f x)) = 1" "(\<Sum>i = 1..card s. u (f i) *\<^sub>R f i) = y"
-      unfolding setsum_image_gen[OF *(1), of "\<lambda>x. u (f x) *\<^sub>R f x" f]
-        and setsum_image_gen[OF *(1), of "\<lambda>x. u (f x)" f]
+      unfolding sum_image_gen[OF *(1), of "\<lambda>x. u (f x) *\<^sub>R f x" f]
+        and sum_image_gen[OF *(1), of "\<lambda>x. u (f x)" f]
       unfolding f
-      using setsum.cong [of s s "\<lambda>y. (\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x) *\<^sub>R f x)" "\<lambda>v. u v *\<^sub>R v"]
-      using setsum.cong [of s s "\<lambda>y. (\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x))" u]
+      using sum.cong [of s s "\<lambda>y. (\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x) *\<^sub>R f x)" "\<lambda>v. u v *\<^sub>R v"]
+      using sum.cong [of s s "\<lambda>y. (\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x))" u]
       unfolding obt(4,5)
       by auto
     ultimately
-    have "\<exists>k u x. (\<forall>i\<in>{1..k}. 0 \<le> u i \<and> x i \<in> p) \<and> setsum u {1..k} = 1 \<and>
+    have "\<exists>k u x. (\<forall>i\<in>{1..k}. 0 \<le> u i \<and> x i \<in> p) \<and> sum u {1..k} = 1 \<and>
         (\<Sum>i::nat = 1..k. u i *\<^sub>R x i) = y"
       apply (rule_tac x="card s" in exI)
       apply (rule_tac x="u \<circ> f" in exI)
@@ -2966,8 +2966,8 @@
   fixes s :: "'a::real_vector set"
   assumes "finite s"
   shows
-    "(\<exists>u. (\<forall>x\<in>insert a s. 0 \<le> u x) \<and> setsum u (insert a s) = w \<and> setsum (\<lambda>x. u x *\<^sub>R x) (insert a s) = y)
-      \<longleftrightarrow> (\<exists>v\<ge>0. \<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = w - v \<and> setsum (\<lambda>x. u x *\<^sub>R x) s = y - v *\<^sub>R a)"
+    "(\<exists>u. (\<forall>x\<in>insert a s. 0 \<le> u x) \<and> sum u (insert a s) = w \<and> sum (\<lambda>x. u x *\<^sub>R x) (insert a s) = y)
+      \<longleftrightarrow> (\<exists>v\<ge>0. \<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = w - v \<and> sum (\<lambda>x. u x *\<^sub>R x) s = y - v *\<^sub>R a)"
   (is "?lhs = ?rhs")
 proof (rule, case_tac[!] "a\<in>s")
   assume "a \<in> s"
@@ -2981,7 +2981,7 @@
 next
   assume ?lhs
   then obtain u where
-      u: "\<forall>x\<in>insert a s. 0 \<le> u x" "setsum u (insert a s) = w" "(\<Sum>x\<in>insert a s. u x *\<^sub>R x) = y"
+      u: "\<forall>x\<in>insert a s. 0 \<le> u x" "sum u (insert a s) = w" "(\<Sum>x\<in>insert a s. u x *\<^sub>R x) = y"
     by auto
   assume "a \<notin> s"
   then show ?rhs
@@ -2989,7 +2989,7 @@
     using u(1)[THEN bspec[where x=a]]
     apply simp
     apply (rule_tac x=u in exI)
-    using u[unfolded setsum_clauses(2)[OF assms]] and \<open>a\<notin>s\<close>
+    using u[unfolded sum_clauses(2)[OF assms]] and \<open>a\<notin>s\<close>
     apply auto
     done
 next
@@ -2997,34 +2997,34 @@
   then have *: "insert a s = s" by auto
   have fin: "finite (insert a s)" using assms by auto
   assume ?rhs
-  then obtain v u where uv: "v\<ge>0" "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = w - v" "(\<Sum>x\<in>s. u x *\<^sub>R x) = y - v *\<^sub>R a"
+  then obtain v u where uv: "v\<ge>0" "\<forall>x\<in>s. 0 \<le> u x" "sum u s = w - v" "(\<Sum>x\<in>s. u x *\<^sub>R x) = y - v *\<^sub>R a"
     by auto
   show ?lhs
     apply (rule_tac x = "\<lambda>x. (if a = x then v else 0) + u x" in exI)
-    unfolding scaleR_left_distrib and setsum.distrib and setsum_delta''[OF fin] and setsum.delta'[OF fin]
-    unfolding setsum_clauses(2)[OF assms]
+    unfolding scaleR_left_distrib and sum.distrib and sum_delta''[OF fin] and sum.delta'[OF fin]
+    unfolding sum_clauses(2)[OF assms]
     using uv and uv(2)[THEN bspec[where x=a]] and \<open>a\<in>s\<close>
     apply auto
     done
 next
   assume ?rhs
   then obtain v u where
-    uv: "v\<ge>0" "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = w - v" "(\<Sum>x\<in>s. u x *\<^sub>R x) = y - v *\<^sub>R a"
+    uv: "v\<ge>0" "\<forall>x\<in>s. 0 \<le> u x" "sum u s = w - v" "(\<Sum>x\<in>s. u x *\<^sub>R x) = y - v *\<^sub>R a"
     by auto
   moreover
   assume "a \<notin> s"
   moreover
-  have "(\<Sum>x\<in>s. if a = x then v else u x) = setsum u s"
+  have "(\<Sum>x\<in>s. if a = x then v else u x) = sum u s"
     and "(\<Sum>x\<in>s. (if a = x then v else u x) *\<^sub>R x) = (\<Sum>x\<in>s. u x *\<^sub>R x)"
-    apply (rule_tac setsum.cong) apply rule
+    apply (rule_tac sum.cong) apply rule
     defer
-    apply (rule_tac setsum.cong) apply rule
+    apply (rule_tac sum.cong) apply rule
     using \<open>a \<notin> s\<close>
     apply auto
     done
   ultimately show ?lhs
     apply (rule_tac x="\<lambda>x. if a = x then v else u x" in exI)
-    unfolding setsum_clauses(2)[OF assms]
+    unfolding sum_clauses(2)[OF assms]
     apply auto
     done
 qed
@@ -3150,12 +3150,12 @@
   then have "finite (insert a t)" and "insert a t \<subseteq> insert a s"
     by auto
   moreover have *: "\<And>P Q. (\<Sum>x\<in>t. (if x = a then P x else Q x)) = (\<Sum>x\<in>t. Q x)"
-    apply (rule setsum.cong)
+    apply (rule sum.cong)
     using \<open>a\<notin>s\<close> \<open>t\<subseteq>s\<close>
     apply auto
     done
   have "(\<Sum>x\<in>insert a t. if x = a then - (\<Sum>x\<in>t. u (x - a)) else u (x - a)) = 0"
-    unfolding setsum_clauses(2)[OF fin]
+    unfolding sum_clauses(2)[OF fin]
     using \<open>a\<notin>s\<close> \<open>t\<subseteq>s\<close>
     apply auto
     unfolding *
@@ -3168,17 +3168,17 @@
     apply auto
     done
   moreover have *: "\<And>P Q. (\<Sum>x\<in>t. (if x = a then P x else Q x) *\<^sub>R x) = (\<Sum>x\<in>t. Q x *\<^sub>R x)"
-    apply (rule setsum.cong)
+    apply (rule sum.cong)
     using \<open>a\<notin>s\<close> \<open>t\<subseteq>s\<close>
     apply auto
     done
   have "(\<Sum>x\<in>t. u (x - a)) *\<^sub>R a = (\<Sum>v\<in>t. u (v - a) *\<^sub>R v)"
-    unfolding scaleR_left.setsum
-    unfolding t_def and setsum.reindex[OF inj] and o_def
+    unfolding scaleR_left.sum
+    unfolding t_def and sum.reindex[OF inj] and o_def
     using obt(5)
-    by (auto simp add: setsum.distrib scaleR_right_distrib)
+    by (auto simp add: sum.distrib scaleR_right_distrib)
   then have "(\<Sum>v\<in>insert a t. (if v = a then - (\<Sum>x\<in>t. u (x - a)) else u (v - a)) *\<^sub>R v) = 0"
-    unfolding setsum_clauses(2)[OF fin]
+    unfolding sum_clauses(2)[OF fin]
     using \<open>a\<notin>s\<close> \<open>t\<subseteq>s\<close>
     by (auto simp add: *)
   ultimately show ?thesis
@@ -3337,10 +3337,10 @@
   assumes "0 \<in> affine hull S"
   shows "dependent S"
 proof -
-  obtain s u where s_u: "finite s \<and> s \<noteq> {} \<and> s \<subseteq> S \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = 0"
+  obtain s u where s_u: "finite s \<and> s \<noteq> {} \<and> s \<subseteq> S \<and> sum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = 0"
     using assms affine_hull_explicit[of S] by auto
   then have "\<exists>v\<in>s. u v \<noteq> 0"
-    using setsum_not_0[of "u" "s"] by auto
+    using sum_not_0[of "u" "s"] by auto
   then have "finite s \<and> s \<subseteq> S \<and> (\<exists>v\<in>s. u v \<noteq> 0 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = 0)"
     using s_u by auto
   then show ?thesis
@@ -4131,18 +4131,18 @@
   { fix y
     assume yt: "y \<in> affine hull t" and yu: "y \<in> affine hull u"
     then obtain a b
-           where a1 [simp]: "setsum a t = 1" and [simp]: "setsum (\<lambda>v. a v *\<^sub>R v) t = y"
-             and [simp]: "setsum b u = 1" "setsum (\<lambda>v. b v *\<^sub>R v) u = y"
+           where a1 [simp]: "sum a t = 1" and [simp]: "sum (\<lambda>v. a v *\<^sub>R v) t = y"
+             and [simp]: "sum b u = 1" "sum (\<lambda>v. b v *\<^sub>R v) u = y"
       by (auto simp: affine_hull_finite \<open>finite t\<close> \<open>finite u\<close>)
     define c where "c x = (if x \<in> t then a x else if x \<in> u then -(b x) else 0)" for x
     have [simp]: "s \<inter> t = t" "s \<inter> - t \<inter> u = u" using assms by auto
-    have "setsum c s = 0"
-      by (simp add: c_def comm_monoid_add_class.setsum.If_cases \<open>finite s\<close> setsum_negf)
+    have "sum c s = 0"
+      by (simp add: c_def comm_monoid_add_class.sum.If_cases \<open>finite s\<close> sum_negf)
     moreover have "~ (\<forall>v\<in>s. c v = 0)"
-      by (metis (no_types) IntD1 \<open>s \<inter> t = t\<close> a1 c_def setsum_not_0 zero_neq_one)
+      by (metis (no_types) IntD1 \<open>s \<inter> t = t\<close> a1 c_def sum_not_0 zero_neq_one)
     moreover have "(\<Sum>v\<in>s. c v *\<^sub>R v) = 0"
-      by (simp add: c_def if_smult setsum_negf
-             comm_monoid_add_class.setsum.If_cases \<open>finite s\<close>)
+      by (simp add: c_def if_smult sum_negf
+             comm_monoid_add_class.sum.If_cases \<open>finite s\<close>)
     ultimately have False
       using assms \<open>finite s\<close> by (auto simp: affine_dependent_explicit)
   }
@@ -4220,13 +4220,13 @@
   fixes p :: "('a::euclidean_space) set"
   shows "convex hull p =
     {y. \<exists>s u. finite s \<and> s \<subseteq> p \<and> card s \<le> aff_dim p + 1 \<and>
-      (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = y}"
+      (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = 1 \<and> sum (\<lambda>v. u v *\<^sub>R v) s = y}"
   unfolding convex_hull_explicit set_eq_iff mem_Collect_eq
 proof (intro allI iffI)
   fix y
   let ?P = "\<lambda>n. \<exists>s u. finite s \<and> card s = n \<and> s \<subseteq> p \<and> (\<forall>x\<in>s. 0 \<le> u x) \<and>
-    setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y"
-  assume "\<exists>s u. finite s \<and> s \<subseteq> p \<and> (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y"
+    sum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y"
+  assume "\<exists>s u. finite s \<and> s \<subseteq> p \<and> (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y"
   then obtain N where "?P N" by auto
   then have "\<exists>n\<le>N. (\<forall>k<n. \<not> ?P k) \<and> ?P n"
     apply (rule_tac ex_least_nat_le)
@@ -4235,7 +4235,7 @@
   then obtain n where "?P n" and smallest: "\<forall>k<n. \<not> ?P k"
     by blast
   then obtain s u where obt: "finite s" "card s = n" "s\<subseteq>p" "\<forall>x\<in>s. 0 \<le> u x"
-    "setsum u s = 1"  "(\<Sum>v\<in>s. u v *\<^sub>R v) = y" by auto
+    "sum u s = 1"  "(\<Sum>v\<in>s. u v *\<^sub>R v) = y" by auto
 
   have "card s \<le> aff_dim p + 1"
   proof (rule ccontr, simp only: not_le)
@@ -4243,19 +4243,19 @@
     then have "affine_dependent s"
       using affine_dependent_biggerset[OF obt(1)] independent_card_le_aff_dim not_less obt(3)
       by blast
-    then obtain w v where wv: "setsum w s = 0" "v\<in>s" "w v \<noteq> 0" "(\<Sum>v\<in>s. w v *\<^sub>R v) = 0"
+    then obtain w v where wv: "sum w s = 0" "v\<in>s" "w v \<noteq> 0" "(\<Sum>v\<in>s. w v *\<^sub>R v) = 0"
       using affine_dependent_explicit_finite[OF obt(1)] by auto
     define i where "i = (\<lambda>v. (u v) / (- w v)) ` {v\<in>s. w v < 0}"
     define t where "t = Min i"
     have "\<exists>x\<in>s. w x < 0"
     proof (rule ccontr, simp add: not_less)
       assume as:"\<forall>x\<in>s. 0 \<le> w x"
-      then have "setsum w (s - {v}) \<ge> 0"
-        apply (rule_tac setsum_nonneg)
+      then have "sum w (s - {v}) \<ge> 0"
+        apply (rule_tac sum_nonneg)
         apply auto
         done
-      then have "setsum w s > 0"
-        unfolding setsum.remove[OF obt(1) \<open>v\<in>s\<close>]
+      then have "sum w s > 0"
+        unfolding sum.remove[OF obt(1) \<open>v\<in>s\<close>]
         using as[THEN bspec[where x=v]]  \<open>v\<in>s\<close>  \<open>w v \<noteq> 0\<close> by auto
       then show False using wv(1) by auto
     qed
@@ -4291,12 +4291,12 @@
     obtain a where "a \<in> s" and "t = (\<lambda>v. (u v) / (- w v)) a" and "w a < 0"
       using Min_in[OF _ \<open>i\<noteq>{}\<close>] and obt(1) unfolding i_def t_def by auto
     then have a: "a \<in> s" "u a + t * w a = 0" by auto
-    have *: "\<And>f. setsum f (s - {a}) = setsum f s - ((f a)::'b::ab_group_add)"
-      unfolding setsum.remove[OF obt(1) \<open>a\<in>s\<close>] by auto
+    have *: "\<And>f. sum f (s - {a}) = sum f s - ((f a)::'b::ab_group_add)"
+      unfolding sum.remove[OF obt(1) \<open>a\<in>s\<close>] by auto
     have "(\<Sum>v\<in>s. u v + t * w v) = 1"
-      unfolding setsum.distrib wv(1) setsum_distrib_left[symmetric] obt(5) by auto
+      unfolding sum.distrib wv(1) sum_distrib_left[symmetric] obt(5) by auto
     moreover have "(\<Sum>v\<in>s. u v *\<^sub>R v + (t * w v) *\<^sub>R v) - (u a *\<^sub>R a + (t * w a) *\<^sub>R a) = y"
-      unfolding setsum.distrib obt(6) scaleR_scaleR[symmetric] scaleR_right.setsum [symmetric] wv(4)
+      unfolding sum.distrib obt(6) scaleR_scaleR[symmetric] scaleR_right.sum [symmetric] wv(4)
       using a(2) [THEN eq_neg_iff_add_eq_0 [THEN iffD2]] by simp
     ultimately have "?P (n - 1)"
       apply (rule_tac x="(s - {a})" in exI)
@@ -4308,7 +4308,7 @@
       using smallest[THEN spec[where x="n - 1"]] by auto
   qed
   then show "\<exists>s u. finite s \<and> s \<subseteq> p \<and> card s \<le> aff_dim p + 1 \<and>
-      (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y"
+      (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y"
     using obt by auto
 qed auto
 
@@ -4332,7 +4332,7 @@
   fixes p :: "('a::euclidean_space) set"
   shows "convex hull p =
             {y. \<exists>s u. finite s \<and> s \<subseteq> p \<and> card s \<le> DIM('a) + 1 \<and>
-              (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = y}"
+              (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = 1 \<and> sum (\<lambda>v. u v *\<^sub>R v) s = y}"
         (is "?lhs = ?rhs")
 proof (intro set_eqI iffI)
   fix x
@@ -4356,7 +4356,7 @@
   fix x
   assume "x \<in> convex hull p"
   then obtain s u where "finite s" "s \<subseteq> p" "card s \<le> DIM('a) + 1"
-    "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x"
+    "\<forall>x\<in>s. 0 \<le> u x" "sum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x"
     unfolding convex_hull_caratheodory by auto
   then show "\<exists>s. finite s \<and> s \<subseteq> p \<and> card s \<le> DIM('a) + 1 \<and> x \<in> convex hull s"
     apply (rule_tac x=s in exI)
@@ -5071,8 +5071,8 @@
   unfolding mem_Collect_eq ball_simps(8)
 proof (rule, rule)
   fix a
-  assume "\<exists>sa u. finite sa \<and> sa \<subseteq> s \<and> (\<forall>x\<in>sa. 0 \<le> u x) \<and> setsum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *\<^sub>R v) = a"
-  then obtain t u where obt: "finite t" "t\<subseteq>s" "\<forall>x\<in>t. 0 \<le> u x" "setsum u t = 1" "(\<Sum>v\<in>t. u v *\<^sub>R v) = a"
+  assume "\<exists>sa u. finite sa \<and> sa \<subseteq> s \<and> (\<forall>x\<in>sa. 0 \<le> u x) \<and> sum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *\<^sub>R v) = a"
+  then obtain t u where obt: "finite t" "t\<subseteq>s" "\<forall>x\<in>t. 0 \<le> u x" "sum u t = 1" "(\<Sum>v\<in>t. u v *\<^sub>R v) = a"
     by auto
 
   from assms[unfolded open_contains_cball] obtain b
@@ -5083,7 +5083,7 @@
   define i where "i = b ` t"
 
   show "\<exists>e > 0.
-    cball a e \<subseteq> {y. \<exists>sa u. finite sa \<and> sa \<subseteq> s \<and> (\<forall>x\<in>sa. 0 \<le> u x) \<and> setsum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *\<^sub>R v) = y}"
+    cball a e \<subseteq> {y. \<exists>sa u. finite sa \<and> sa \<subseteq> s \<and> (\<forall>x\<in>sa. 0 \<le> u x) \<and> sum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *\<^sub>R v) = y}"
     apply (rule_tac x = "Min i" in exI)
     unfolding subset_eq
     apply rule
@@ -5125,12 +5125,12 @@
     have *: "inj_on (\<lambda>v. v + (y - a)) t"
       unfolding inj_on_def by auto
     have "(\<Sum>v\<in>(\<lambda>v. v + (y - a)) ` t. u (v - (y - a))) = 1"
-      unfolding setsum.reindex[OF *] o_def using obt(4) by auto
+      unfolding sum.reindex[OF *] o_def using obt(4) by auto
     moreover have "(\<Sum>v\<in>(\<lambda>v. v + (y - a)) ` t. u (v - (y - a)) *\<^sub>R v) = y"
-      unfolding setsum.reindex[OF *] o_def using obt(4,5)
-      by (simp add: setsum.distrib setsum_subtractf scaleR_left.setsum[symmetric] scaleR_right_distrib)
+      unfolding sum.reindex[OF *] o_def using obt(4,5)
+      by (simp add: sum.distrib sum_subtractf scaleR_left.sum[symmetric] scaleR_right_distrib)
     ultimately
-    show "\<exists>sa u. finite sa \<and> (\<forall>x\<in>sa. x \<in> s) \<and> (\<forall>x\<in>sa. 0 \<le> u x) \<and> setsum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *\<^sub>R v) = y"
+    show "\<exists>sa u. finite sa \<and> (\<forall>x\<in>sa. x \<in> s) \<and> (\<forall>x\<in>sa. 0 \<le> u x) \<and> sum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *\<^sub>R v) = y"
       apply (rule_tac x="(\<lambda>v. v + (y - a)) ` t" in exI)
       apply (rule_tac x="\<lambda>v. u (v - (y - a))" in exI)
       using obt(1, 3)
@@ -6167,44 +6167,44 @@
 
 lemma radon_ex_lemma:
   assumes "finite c" "affine_dependent c"
-  shows "\<exists>u. setsum u c = 0 \<and> (\<exists>v\<in>c. u v \<noteq> 0) \<and> setsum (\<lambda>v. u v *\<^sub>R v) c = 0"
+  shows "\<exists>u. sum u c = 0 \<and> (\<exists>v\<in>c. u v \<noteq> 0) \<and> sum (\<lambda>v. u v *\<^sub>R v) c = 0"
 proof -
   from assms(2)[unfolded affine_dependent_explicit]
   obtain s u where
-      "finite s" "s \<subseteq> c" "setsum u s = 0" "\<exists>v\<in>s. u v \<noteq> 0" "(\<Sum>v\<in>s. u v *\<^sub>R v) = 0"
+      "finite s" "s \<subseteq> c" "sum u s = 0" "\<exists>v\<in>s. u v \<noteq> 0" "(\<Sum>v\<in>s. u v *\<^sub>R v) = 0"
     by blast
   then show ?thesis
     apply (rule_tac x="\<lambda>v. if v\<in>s then u v else 0" in exI)
-    unfolding if_smult scaleR_zero_left and setsum.inter_restrict[OF assms(1), symmetric]
+    unfolding if_smult scaleR_zero_left and sum.inter_restrict[OF assms(1), symmetric]
     apply (auto simp add: Int_absorb1)
     done
 qed
 
 lemma radon_s_lemma:
   assumes "finite s"
-    and "setsum f s = (0::real)"
-  shows "setsum f {x\<in>s. 0 < f x} = - setsum f {x\<in>s. f x < 0}"
+    and "sum f s = (0::real)"
+  shows "sum f {x\<in>s. 0 < f x} = - sum f {x\<in>s. f x < 0}"
 proof -
   have *: "\<And>x. (if f x < 0 then f x else 0) + (if 0 < f x then f x else 0) = f x"
     by auto
   show ?thesis
-    unfolding add_eq_0_iff[symmetric] and setsum.inter_filter[OF assms(1)]
-      and setsum.distrib[symmetric] and *
+    unfolding add_eq_0_iff[symmetric] and sum.inter_filter[OF assms(1)]
+      and sum.distrib[symmetric] and *
     using assms(2)
     by assumption
 qed
 
 lemma radon_v_lemma:
   assumes "finite s"
-    and "setsum f s = 0"
+    and "sum f s = 0"
     and "\<forall>x. g x = (0::real) \<longrightarrow> f x = (0::'a::euclidean_space)"
-  shows "(setsum f {x\<in>s. 0 < g x}) = - setsum f {x\<in>s. g x < 0}"
+  shows "(sum f {x\<in>s. 0 < g x}) = - sum f {x\<in>s. g x < 0}"
 proof -
   have *: "\<And>x. (if 0 < g x then f x else 0) + (if g x < 0 then f x else 0) = f x"
     using assms(3) by auto
   show ?thesis
-    unfolding eq_neg_iff_add_eq_0 and setsum.inter_filter[OF assms(1)]
-      and setsum.distrib[symmetric] and *
+    unfolding eq_neg_iff_add_eq_0 and sum.inter_filter[OF assms(1)]
+      and sum.distrib[symmetric] and *
     using assms(2)
     apply assumption
     done
@@ -6214,12 +6214,12 @@
   assumes "finite c" "affine_dependent c"
   shows "\<exists>m p. m \<inter> p = {} \<and> m \<union> p = c \<and> (convex hull m) \<inter> (convex hull p) \<noteq> {}"
 proof -
-  obtain u v where uv: "setsum u c = 0" "v\<in>c" "u v \<noteq> 0"  "(\<Sum>v\<in>c. u v *\<^sub>R v) = 0"
+  obtain u v where uv: "sum u c = 0" "v\<in>c" "u v \<noteq> 0"  "(\<Sum>v\<in>c. u v *\<^sub>R v) = 0"
     using radon_ex_lemma[OF assms] by auto
   have fin: "finite {x \<in> c. 0 < u x}" "finite {x \<in> c. 0 > u x}"
     using assms(1) by auto
-  define z  where "z = inverse (setsum u {x\<in>c. u x > 0}) *\<^sub>R setsum (\<lambda>x. u x *\<^sub>R x) {x\<in>c. u x > 0}"
-  have "setsum u {x \<in> c. 0 < u x} \<noteq> 0"
+  define z  where "z = inverse (sum u {x\<in>c. u x > 0}) *\<^sub>R sum (\<lambda>x. u x *\<^sub>R x) {x\<in>c. u x > 0}"
+  have "sum u {x \<in> c. 0 < u x} \<noteq> 0"
   proof (cases "u v \<ge> 0")
     case False
     then have "u v < 0" by auto
@@ -6227,36 +6227,36 @@
     proof (cases "\<exists>w\<in>{x \<in> c. 0 < u x}. u w > 0")
       case True
       then show ?thesis
-        using setsum_nonneg_eq_0_iff[of _ u, OF fin(1)] by auto
+        using sum_nonneg_eq_0_iff[of _ u, OF fin(1)] by auto
     next
       case False
-      then have "setsum u c \<le> setsum (\<lambda>x. if x=v then u v else 0) c"
-        apply (rule_tac setsum_mono)
+      then have "sum u c \<le> sum (\<lambda>x. if x=v then u v else 0) c"
+        apply (rule_tac sum_mono)
         apply auto
         done
       then show ?thesis
-        unfolding setsum.delta[OF assms(1)] using uv(2) and \<open>u v < 0\<close> and uv(1) by auto
+        unfolding sum.delta[OF assms(1)] using uv(2) and \<open>u v < 0\<close> and uv(1) by auto
     qed
-  qed (insert setsum_nonneg_eq_0_iff[of _ u, OF fin(1)] uv(2-3), auto)
-
-  then have *: "setsum u {x\<in>c. u x > 0} > 0"
+  qed (insert sum_nonneg_eq_0_iff[of _ u, OF fin(1)] uv(2-3), auto)
+
+  then have *: "sum u {x\<in>c. u x > 0} > 0"
     unfolding less_le
     apply (rule_tac conjI)
-    apply (rule_tac setsum_nonneg)
-    apply auto
-    done
-  moreover have "setsum u ({x \<in> c. 0 < u x} \<union> {x \<in> c. u x < 0}) = setsum u c"
+    apply (rule_tac sum_nonneg)
+    apply auto
+    done
+  moreover have "sum u ({x \<in> c. 0 < u x} \<union> {x \<in> c. u x < 0}) = sum u c"
     "(\<Sum>x\<in>{x \<in> c. 0 < u x} \<union> {x \<in> c. u x < 0}. u x *\<^sub>R x) = (\<Sum>x\<in>c. u x *\<^sub>R x)"
     using assms(1)
-    apply (rule_tac[!] setsum.mono_neutral_left)
-    apply auto
-    done
-  then have "setsum u {x \<in> c. 0 < u x} = - setsum u {x \<in> c. 0 > u x}"
+    apply (rule_tac[!] sum.mono_neutral_left)
+    apply auto
+    done
+  then have "sum u {x \<in> c. 0 < u x} = - sum u {x \<in> c. 0 > u x}"
     "(\<Sum>x\<in>{x \<in> c. 0 < u x}. u x *\<^sub>R x) = - (\<Sum>x\<in>{x \<in> c. 0 > u x}. u x *\<^sub>R x)"
     unfolding eq_neg_iff_add_eq_0
     using uv(1,4)
-    by (auto simp add: setsum.union_inter_neutral[OF fin, symmetric])
-  moreover have "\<forall>x\<in>{v \<in> c. u v < 0}. 0 \<le> inverse (setsum u {x \<in> c. 0 < u x}) * - u x"
+    by (auto simp add: sum.union_inter_neutral[OF fin, symmetric])
+  moreover have "\<forall>x\<in>{v \<in> c. u v < 0}. 0 \<le> inverse (sum u {x \<in> c. 0 < u x}) * - u x"
     apply rule
     apply (rule mult_nonneg_nonneg)
     using *
@@ -6265,11 +6265,11 @@
   ultimately have "z \<in> convex hull {v \<in> c. u v \<le> 0}"
     unfolding convex_hull_explicit mem_Collect_eq
     apply (rule_tac x="{v \<in> c. u v < 0}" in exI)
-    apply (rule_tac x="\<lambda>y. inverse (setsum u {x\<in>c. u x > 0}) * - u y" in exI)
-    using assms(1) unfolding scaleR_scaleR[symmetric] scaleR_right.setsum [symmetric] and z_def
-    apply (auto simp add: setsum_negf setsum_distrib_left[symmetric])
-    done
-  moreover have "\<forall>x\<in>{v \<in> c. 0 < u v}. 0 \<le> inverse (setsum u {x \<in> c. 0 < u x}) * u x"
+    apply (rule_tac x="\<lambda>y. inverse (sum u {x\<in>c. u x > 0}) * - u y" in exI)
+    using assms(1) unfolding scaleR_scaleR[symmetric] scaleR_right.sum [symmetric] and z_def
+    apply (auto simp add: sum_negf sum_distrib_left[symmetric])
+    done
+  moreover have "\<forall>x\<in>{v \<in> c. 0 < u v}. 0 \<le> inverse (sum u {x \<in> c. 0 < u x}) * u x"
     apply rule
     apply (rule mult_nonneg_nonneg)
     using *
@@ -6278,11 +6278,11 @@
   then have "z \<in> convex hull {v \<in> c. u v > 0}"
     unfolding convex_hull_explicit mem_Collect_eq
     apply (rule_tac x="{v \<in> c. 0 < u v}" in exI)
-    apply (rule_tac x="\<lambda>y. inverse (setsum u {x\<in>c. u x > 0}) * u y" in exI)
+    apply (rule_tac x="\<lambda>y. inverse (sum u {x\<in>c. u x > 0}) * u y" in exI)
     using assms(1)
-    unfolding scaleR_scaleR[symmetric] scaleR_right.setsum [symmetric] and z_def
+    unfolding scaleR_scaleR[symmetric] scaleR_right.sum [symmetric] and z_def
     using *
-    apply (auto simp add: setsum_negf setsum_distrib_left[symmetric])
+    apply (auto simp add: sum_negf sum_distrib_left[symmetric])
     done
   ultimately show ?thesis
     apply (rule_tac x="{v\<in>c. u v \<le> 0}" in exI)
@@ -6297,7 +6297,7 @@
 proof -
   from assms[unfolded affine_dependent_explicit]
   obtain s u where
-      "finite s" "s \<subseteq> c" "setsum u s = 0" "\<exists>v\<in>s. u v \<noteq> 0" "(\<Sum>v\<in>s. u v *\<^sub>R v) = 0"
+      "finite s" "s \<subseteq> c" "sum u s = 0" "\<exists>v\<in>s. u v \<noteq> 0" "(\<Sum>v\<in>s. u v *\<^sub>R v) = 0"
     by blast
   then have *: "finite s" "affine_dependent s" and s: "s \<subseteq> c"
     unfolding affine_dependent_explicit by auto
@@ -6460,10 +6460,10 @@
 lemma convex_on:
   assumes "convex s"
   shows "convex_on s f \<longleftrightarrow>
-    (\<forall>k u x. (\<forall>i\<in>{1..k::nat}. 0 \<le> u i \<and> x i \<in> s) \<and> setsum u {1..k} = 1 \<longrightarrow>
-      f (setsum (\<lambda>i. u i *\<^sub>R x i) {1..k} ) \<le> setsum (\<lambda>i. u i * f(x i)) {1..k})"
+    (\<forall>k u x. (\<forall>i\<in>{1..k::nat}. 0 \<le> u i \<and> x i \<in> s) \<and> sum u {1..k} = 1 \<longrightarrow>
+      f (sum (\<lambda>i. u i *\<^sub>R x i) {1..k} ) \<le> sum (\<lambda>i. u i * f(x i)) {1..k})"
   unfolding convex_epigraph_convex[OF assms] convex epigraph_def Ball_def mem_Collect_eq
-  unfolding fst_setsum snd_setsum fst_scaleR snd_scaleR
+  unfolding fst_sum snd_sum fst_scaleR snd_scaleR
   apply safe
   apply (drule_tac x=k in spec)
   apply (drule_tac x=u in spec)
@@ -6473,7 +6473,7 @@
   apply simp
   apply (rule_tac y="\<Sum>i = 1..k. u i * f (fst (x i))" in order_trans)
   defer
-  apply (rule setsum_mono)
+  apply (rule sum_mono)
   apply (erule_tac x=i in allE)
   unfolding real_scaleR_def
   apply (rule mult_left_mono)
@@ -6692,11 +6692,11 @@
   fix x
   assume "x \<in> convex hull s"
   then obtain k u v where
-    obt: "\<forall>i\<in>{1..k::nat}. 0 \<le> u i \<and> v i \<in> s" "setsum u {1..k} = 1" "(\<Sum>i = 1..k. u i *\<^sub>R v i) = x"
+    obt: "\<forall>i\<in>{1..k::nat}. 0 \<le> u i \<and> v i \<in> s" "sum u {1..k} = 1" "(\<Sum>i = 1..k. u i *\<^sub>R v i) = x"
     unfolding convex_hull_indexed mem_Collect_eq by auto
   have "(\<Sum>i = 1..k. u i * f (v i)) \<le> b"
-    using setsum_mono[of "{1..k}" "\<lambda>i. u i * f (v i)" "\<lambda>i. u i * b"]
-    unfolding setsum_distrib_right[symmetric] obt(2) mult_1
+    using sum_mono[of "{1..k}" "\<lambda>i. u i * f (v i)" "\<lambda>i. u i * b"]
+    unfolding sum_distrib_right[symmetric] obt(2) mult_1
     apply (drule_tac meta_mp)
     apply (rule mult_left_mono)
     using assms(2) obt(1)
@@ -6709,8 +6709,8 @@
     by auto
 qed
 
-lemma inner_setsum_Basis[simp]: "i \<in> Basis \<Longrightarrow> (\<Sum>Basis) \<bullet> i = 1"
-  by (simp add: inner_setsum_left setsum.If_cases inner_Basis)
+lemma inner_sum_Basis[simp]: "i \<in> Basis \<Longrightarrow> (\<Sum>Basis) \<bullet> i = 1"
+  by (simp add: inner_sum_left sum.If_cases inner_Basis)
 
 lemma convex_set_plus:
   assumes "convex s" and "convex t" shows "convex (s + t)"
@@ -6722,7 +6722,7 @@
   finally show "convex (s + t)" .
 qed
 
-lemma convex_set_setsum:
+lemma convex_set_sum:
   assumes "\<And>i. i \<in> A \<Longrightarrow> convex (B i)"
   shows "convex (\<Sum>i\<in>A. B i)"
 proof (cases "finite A")
@@ -6730,11 +6730,11 @@
     by induct (auto simp: convex_set_plus)
 qed auto
 
-lemma finite_set_setsum:
+lemma finite_set_sum:
   assumes "finite A" and "\<forall>i\<in>A. finite (B i)" shows "finite (\<Sum>i\<in>A. B i)"
   using assms by (induct set: finite, simp, simp add: finite_set_plus)
 
-lemma set_setsum_eq:
+lemma set_sum_eq:
   "finite A \<Longrightarrow> (\<Sum>i\<in>A. B i) = {\<Sum>i\<in>A. f i |f. \<forall>i\<in>A. f i \<in> B i}"
   apply (induct set: finite)
   apply simp
@@ -6743,24 +6743,24 @@
   apply (rule_tac x="fun_upd f x a" in exI)
   apply simp
   apply (rule_tac f="\<lambda>x. a + x" in arg_cong)
-  apply (rule setsum.cong [OF refl])
+  apply (rule sum.cong [OF refl])
   apply clarsimp
   apply fast
   done
 
-lemma box_eq_set_setsum_Basis:
+lemma box_eq_set_sum_Basis:
   shows "{x. \<forall>i\<in>Basis. x\<bullet>i \<in> B i} = (\<Sum>i\<in>Basis. image (\<lambda>x. x *\<^sub>R i) (B i))"
-  apply (subst set_setsum_eq [OF finite_Basis])
+  apply (subst set_sum_eq [OF finite_Basis])
   apply safe
   apply (fast intro: euclidean_representation [symmetric])
-  apply (subst inner_setsum_left)
+  apply (subst inner_sum_left)
   apply (subgoal_tac "(\<Sum>x\<in>Basis. f x \<bullet> i) = f i \<bullet> i")
   apply (drule (1) bspec)
   apply clarsimp
-  apply (frule setsum.remove [OF finite_Basis])
+  apply (frule sum.remove [OF finite_Basis])
   apply (erule trans)
   apply simp
-  apply (rule setsum.neutral)
+  apply (rule sum.neutral)
   apply clarsimp
   apply (frule_tac x=i in bspec, assumption)
   apply (drule_tac x=x in bspec, assumption)
@@ -6770,7 +6770,7 @@
   apply simp
   done
 
-lemma convex_hull_set_setsum:
+lemma convex_hull_set_sum:
   "convex hull (\<Sum>i\<in>A. B i) = (\<Sum>i\<in>A. convex hull (B i))"
 proof (cases "finite A")
   assume "finite A" then show ?thesis
@@ -6796,19 +6796,19 @@
   (is "?int = convex hull ?points")
 proof -
   have One[simp]: "\<And>i. i \<in> Basis \<Longrightarrow> One \<bullet> i = 1"
-    by (simp add: inner_setsum_left setsum.If_cases inner_Basis)
+    by (simp add: inner_sum_left sum.If_cases inner_Basis)
   have "?int = {x. \<forall>i\<in>Basis. x \<bullet> i \<in> cbox 0 1}"
     by (auto simp: cbox_def)
   also have "\<dots> = (\<Sum>i\<in>Basis. (\<lambda>x. x *\<^sub>R i) ` cbox 0 1)"
-    by (simp only: box_eq_set_setsum_Basis)
+    by (simp only: box_eq_set_sum_Basis)
   also have "\<dots> = (\<Sum>i\<in>Basis. (\<lambda>x. x *\<^sub>R i) ` (convex hull {0, 1}))"
     by (simp only: convex_hull_eq_real_cbox zero_le_one)
   also have "\<dots> = (\<Sum>i\<in>Basis. convex hull ((\<lambda>x. x *\<^sub>R i) ` {0, 1}))"
     by (simp only: convex_hull_linear_image linear_scaleR_left)
   also have "\<dots> = convex hull (\<Sum>i\<in>Basis. (\<lambda>x. x *\<^sub>R i) ` {0, 1})"
-    by (simp only: convex_hull_set_setsum)
+    by (simp only: convex_hull_set_sum)
   also have "\<dots> = convex hull {x. \<forall>i\<in>Basis. x\<bullet>i \<in> {0, 1}}"
-    by (simp only: box_eq_set_setsum_Basis)
+    by (simp only: box_eq_set_sum_Basis)
   also have "convex hull {x. \<forall>i\<in>Basis. x\<bullet>i \<in> {0, 1}} = convex hull ?points"
     by simp
   finally show ?thesis .
@@ -6939,7 +6939,7 @@
            op + a ` (\<lambda>x. \<Sum>k\<in>Basis. ((b \<bullet> k - a \<bullet> k) * (x \<bullet> k)) *\<^sub>R k) ` cbox 0 One"
 using assms
 apply (simp add: box_ne_empty image_stretch_interval cbox_translation [symmetric])
-apply (simp add: min_def max_def algebra_simps setsum_subtractf euclidean_representation)
+apply (simp add: min_def max_def algebra_simps sum_subtractf euclidean_representation)
 done
 
 lemma closed_interval_as_convex_hull:
@@ -6953,7 +6953,7 @@
   obtain s::"'a set" where "finite s" and eq: "cbox 0 One = convex hull s"
     by (blast intro: unit_cube_convex_hull)
   have lin: "linear (\<lambda>x. \<Sum>k\<in>Basis. ((b \<bullet> k - a \<bullet> k) * (x \<bullet> k)) *\<^sub>R k)"
-    by (rule linear_compose_setsum) (auto simp: algebra_simps linearI)
+    by (rule linear_compose_sum) (auto simp: algebra_simps linearI)
   have "finite (op + a ` (\<lambda>x. \<Sum>k\<in>Basis. ((b \<bullet> k - a \<bullet> k) * (x \<bullet> k)) *\<^sub>R k) ` s)"
     by (rule finite_imageI \<open>finite s\<close>)+
   then show ?thesis
@@ -7141,13 +7141,13 @@
   proof
     define c where "c = (\<Sum>i\<in>Basis. (\<lambda>a. a *\<^sub>R i) ` {x\<bullet>i - d, x\<bullet>i + d})"
     show "finite c"
-      unfolding c_def by (simp add: finite_set_setsum)
+      unfolding c_def by (simp add: finite_set_sum)
     have 1: "convex hull c = {a. \<forall>i\<in>Basis. a \<bullet> i \<in> cbox (x \<bullet> i - d) (x \<bullet> i + d)}"
-      unfolding box_eq_set_setsum_Basis
-      unfolding c_def convex_hull_set_setsum
+      unfolding box_eq_set_sum_Basis
+      unfolding c_def convex_hull_set_sum
       apply (subst convex_hull_linear_image [symmetric])
       apply (simp add: linear_iff scaleR_add_left)
-      apply (rule setsum.cong [OF refl])
+      apply (rule sum.cong [OF refl])
       apply (rule image_cong [OF _ refl])
       apply (rule convex_hull_eq_real_cbox)
       apply (cut_tac \<open>0 < d\<close>, simp)
@@ -7168,10 +7168,10 @@
       unfolding 2
       apply clarsimp
       apply (subst euclidean_dist_l2)
-      apply (rule order_trans [OF setL2_le_setsum])
+      apply (rule order_trans [OF setL2_le_sum])
       apply (rule zero_le_dist)
       unfolding e'
-      apply (rule setsum_mono)
+      apply (rule sum_mono)
       apply simp
       done
   qed
@@ -8232,18 +8232,18 @@
   assumes "finite s"
     and "0 \<notin> s"
   shows "convex hull (insert 0 s) =
-    {y. (\<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s \<le> 1 \<and> setsum (\<lambda>x. u x *\<^sub>R x) s = y)}"
+    {y. (\<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s \<le> 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) s = y)}"
   unfolding convex_hull_finite[OF finite.insertI[OF assms(1)]]
   apply (rule set_eqI, rule)
   unfolding mem_Collect_eq
   apply (erule_tac[!] exE)
   apply (erule_tac[!] conjE)+
-  unfolding setsum_clauses(2)[OF \<open>finite s\<close>]
+  unfolding sum_clauses(2)[OF \<open>finite s\<close>]
   apply (rule_tac x=u in exI)
   defer
-  apply (rule_tac x="\<lambda>x. if x = 0 then 1 - setsum u s else u x" in exI)
+  apply (rule_tac x="\<lambda>x. if x = 0 then 1 - sum u s else u x" in exI)
   using assms(2)
-  unfolding if_smult and setsum_delta_notmem[OF assms(2)]
+  unfolding if_smult and sum_delta_notmem[OF assms(2)]
   apply auto
   done
 
@@ -8268,19 +8268,19 @@
   proof -
     fix x :: "'a::euclidean_space"
     fix u
-    assume as: "\<forall>x\<in>?D. 0 \<le> u x" "setsum u ?D \<le> 1" "(\<Sum>x\<in>?D. u x *\<^sub>R x) = x"
+    assume as: "\<forall>x\<in>?D. 0 \<le> u x" "sum u ?D \<le> 1" "(\<Sum>x\<in>?D. u x *\<^sub>R x) = x"
     have *: "\<forall>i\<in>Basis. i:d \<longrightarrow> u i = x\<bullet>i"
       and "(\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0)"
       using as(3)
       unfolding substdbasis_expansion_unique[OF assms]
       by auto
-    then have **: "setsum u ?D = setsum (op \<bullet> x) ?D"
+    then have **: "sum u ?D = sum (op \<bullet> x) ?D"
       apply -
-      apply (rule setsum.cong)
+      apply (rule sum.cong)
       using assms
       apply auto
       done
-    have "(\<forall>i\<in>Basis. 0 \<le> x\<bullet>i) \<and> setsum (op \<bullet> x) ?D \<le> 1"
+    have "(\<forall>i\<in>Basis. 0 \<le> x\<bullet>i) \<and> sum (op \<bullet> x) ?D \<le> 1"
     proof (rule,rule)
       fix i :: 'a
       assume i: "i \<in> Basis"
@@ -8293,12 +8293,12 @@
         using \<open>(\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0)\<close>[rule_format, OF i] by auto
       ultimately show "0 \<le> x\<bullet>i" by auto
     qed (insert as(2)[unfolded **], auto)
-    then show "(\<forall>i\<in>Basis. 0 \<le> x\<bullet>i) \<and> setsum (op \<bullet> x) ?D \<le> 1 \<and> (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0)"
+    then show "(\<forall>i\<in>Basis. 0 \<le> x\<bullet>i) \<and> sum (op \<bullet> x) ?D \<le> 1 \<and> (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0)"
       using \<open>(\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0)\<close> by auto
   next
     fix x :: "'a::euclidean_space"
-    assume as: "\<forall>i\<in>Basis. 0 \<le> x \<bullet> i" "setsum (op \<bullet> x) ?D \<le> 1" "(\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0)"
-    show "\<exists>u. (\<forall>x\<in>?D. 0 \<le> u x) \<and> setsum u ?D \<le> 1 \<and> (\<Sum>x\<in>?D. u x *\<^sub>R x) = x"
+    assume as: "\<forall>i\<in>Basis. 0 \<le> x \<bullet> i" "sum (op \<bullet> x) ?D \<le> 1" "(\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0)"
+    show "\<exists>u. (\<forall>x\<in>?D. 0 \<le> u x) \<and> sum u ?D \<le> 1 \<and> (\<Sum>x\<in>?D. u x *\<^sub>R x) = x"
       using as d
       unfolding substdbasis_expansion_unique[OF assms]
       apply (rule_tac x="inner x" in exI)
@@ -8309,12 +8309,12 @@
 
 lemma std_simplex:
   "convex hull (insert 0 Basis) =
-    {x::'a::euclidean_space. (\<forall>i\<in>Basis. 0 \<le> x\<bullet>i) \<and> setsum (\<lambda>i. x\<bullet>i) Basis \<le> 1}"
+    {x::'a::euclidean_space. (\<forall>i\<in>Basis. 0 \<le> x\<bullet>i) \<and> sum (\<lambda>i. x\<bullet>i) Basis \<le> 1}"
   using substd_simplex[of Basis] by auto
 
 lemma interior_std_simplex:
   "interior (convex hull (insert 0 Basis)) =
-    {x::'a::euclidean_space. (\<forall>i\<in>Basis. 0 < x\<bullet>i) \<and> setsum (\<lambda>i. x\<bullet>i) Basis < 1}"
+    {x::'a::euclidean_space. (\<forall>i\<in>Basis. 0 < x\<bullet>i) \<and> sum (\<lambda>i. x\<bullet>i) Basis < 1}"
   apply (rule set_eqI)
   unfolding mem_interior std_simplex
   unfolding subset_eq mem_Collect_eq Ball_def mem_ball
@@ -8326,8 +8326,8 @@
 proof -
   fix x :: 'a
   fix e
-  assume "e > 0" and as: "\<forall>xa. dist x xa < e \<longrightarrow> (\<forall>x\<in>Basis. 0 \<le> xa \<bullet> x) \<and> setsum (op \<bullet> xa) Basis \<le> 1"
-  show "(\<forall>xa\<in>Basis. 0 < x \<bullet> xa) \<and> setsum (op \<bullet> x) Basis < 1"
+  assume "e > 0" and as: "\<forall>xa. dist x xa < e \<longrightarrow> (\<forall>x\<in>Basis. 0 \<le> xa \<bullet> x) \<and> sum (op \<bullet> xa) Basis \<le> 1"
+  show "(\<forall>xa\<in>Basis. 0 < x \<bullet> xa) \<and> sum (op \<bullet> x) Basis < 1"
     apply safe
   proof -
     fix i :: 'a
@@ -8343,15 +8343,15 @@
     have "\<And>i. i \<in> Basis \<Longrightarrow> (x + (e / 2) *\<^sub>R (SOME i. i\<in>Basis)) \<bullet> i =
       x\<bullet>i + (if i = (SOME i. i\<in>Basis) then e/2 else 0)"
       by (auto simp: SOME_Basis inner_Basis inner_simps)
-    then have *: "setsum (op \<bullet> (x + (e / 2) *\<^sub>R (SOME i. i\<in>Basis))) Basis =
-      setsum (\<lambda>i. x\<bullet>i + (if (SOME i. i\<in>Basis) = i then e/2 else 0)) Basis"
-      apply (rule_tac setsum.cong)
+    then have *: "sum (op \<bullet> (x + (e / 2) *\<^sub>R (SOME i. i\<in>Basis))) Basis =
+      sum (\<lambda>i. x\<bullet>i + (if (SOME i. i\<in>Basis) = i then e/2 else 0)) Basis"
+      apply (rule_tac sum.cong)
       apply auto
       done
-    have "setsum (op \<bullet> x) Basis < setsum (op \<bullet> (x + (e / 2) *\<^sub>R (SOME i. i\<in>Basis))) Basis"
-      unfolding * setsum.distrib
+    have "sum (op \<bullet> x) Basis < sum (op \<bullet> (x + (e / 2) *\<^sub>R (SOME i. i\<in>Basis))) Basis"
+      unfolding * sum.distrib
       using \<open>e > 0\<close> DIM_positive[where 'a='a]
-      apply (subst setsum.delta')
+      apply (subst sum.delta')
       apply (auto simp: SOME_Basis)
       done
     also have "\<dots> \<le> 1"
@@ -8359,13 +8359,13 @@
       apply (drule_tac as[rule_format])
       apply auto
       done
-    finally show "setsum (op \<bullet> x) Basis < 1" by auto
+    finally show "sum (op \<bullet> x) Basis < 1" by auto
   qed
 next
   fix x :: 'a
-  assume as: "\<forall>i\<in>Basis. 0 < x \<bullet> i" "setsum (op \<bullet> x) Basis < 1"
+  assume as: "\<forall>i\<in>Basis. 0 < x \<bullet> i" "sum (op \<bullet> x) Basis < 1"
   obtain a :: 'b where "a \<in> UNIV" using UNIV_witness ..
-  let ?d = "(1 - setsum (op \<bullet> x) Basis) / real (DIM('a))"
+  let ?d = "(1 - sum (op \<bullet> x) Basis) / real (DIM('a))"
   have "Min ((op \<bullet> x) ` Basis) > 0"
     apply (rule Min_grI)
     using as(1)
@@ -8373,7 +8373,7 @@
     done
   moreover have "?d > 0"
     using as(2) by (auto simp: Suc_le_eq DIM_positive)
-  ultimately show "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> (\<forall>i\<in>Basis. 0 \<le> y \<bullet> i) \<and> setsum (op \<bullet> y) Basis \<le> 1"
+  ultimately show "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> (\<forall>i\<in>Basis. 0 \<le> y \<bullet> i) \<and> sum (op \<bullet> y) Basis \<le> 1"
     apply (rule_tac x="min (Min ((op \<bullet> x) ` Basis)) D" for D in exI)
     apply rule
     defer
@@ -8381,8 +8381,8 @@
   proof -
     fix y
     assume y: "dist x y < min (Min (op \<bullet> x ` Basis)) ?d"
-    have "setsum (op \<bullet> y) Basis \<le> setsum (\<lambda>i. x\<bullet>i + ?d) Basis"
-    proof (rule setsum_mono)
+    have "sum (op \<bullet> y) Basis \<le> sum (\<lambda>i. x\<bullet>i + ?d) Basis"
+    proof (rule sum_mono)
       fix i :: 'a
       assume i: "i \<in> Basis"
       then have "\<bar>y\<bullet>i - x\<bullet>i\<bar> < ?d"
@@ -8395,9 +8395,9 @@
       then show "y \<bullet> i \<le> x \<bullet> i + ?d" by auto
     qed
     also have "\<dots> \<le> 1"
-      unfolding setsum.distrib setsum_constant
+      unfolding sum.distrib sum_constant
       by (auto simp add: Suc_le_eq)
-    finally show "(\<forall>i\<in>Basis. 0 \<le> y \<bullet> i) \<and> setsum (op \<bullet> y) Basis \<le> 1"
+    finally show "(\<forall>i\<in>Basis. 0 \<le> y \<bullet> i) \<and> sum (op \<bullet> y) Basis \<le> 1"
     proof safe
       fix i :: 'a
       assume i: "i \<in> Basis"
@@ -8419,13 +8419,13 @@
     "a \<in> interior(convex hull (insert 0 Basis))"
 proof -
   let ?D = "Basis :: 'a set"
-  let ?a = "setsum (\<lambda>b::'a. inverse (2 * real DIM('a)) *\<^sub>R b) Basis"
+  let ?a = "sum (\<lambda>b::'a. inverse (2 * real DIM('a)) *\<^sub>R b) Basis"
   {
     fix i :: 'a
     assume i: "i \<in> Basis"
     have "?a \<bullet> i = inverse (2 * real DIM('a))"
-      by (rule trans[of _ "setsum (\<lambda>j. if i = j then inverse (2 * real DIM('a)) else 0) ?D"])
-         (simp_all add: setsum.If_cases i) }
+      by (rule trans[of _ "sum (\<lambda>j. if i = j then inverse (2 * real DIM('a)) else 0) ?D"])
+         (simp_all add: sum.If_cases i) }
   note ** = this
   show ?thesis
     apply (rule that[of ?a])
@@ -8436,15 +8436,15 @@
     show "0 < ?a \<bullet> i"
       unfolding **[OF i] by (auto simp add: Suc_le_eq DIM_positive)
   next
-    have "setsum (op \<bullet> ?a) ?D = setsum (\<lambda>i. inverse (2 * real DIM('a))) ?D"
-      apply (rule setsum.cong)
+    have "sum (op \<bullet> ?a) ?D = sum (\<lambda>i. inverse (2 * real DIM('a))) ?D"
+      apply (rule sum.cong)
       apply rule
       apply auto
       done
     also have "\<dots> < 1"
-      unfolding setsum_constant divide_inverse[symmetric]
+      unfolding sum_constant divide_inverse[symmetric]
       by (auto simp add: field_simps)
-    finally show "setsum (op \<bullet> ?a) ?D < 1" by auto
+    finally show "sum (op \<bullet> ?a) ?D < 1" by auto
   qed
 qed
 
@@ -8478,11 +8478,11 @@
         "ball x e \<inter> {xa. (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> xa\<bullet>i = 0)} \<subseteq> convex hull (insert 0 ?p)"
         using mem_rel_interior_ball[of x "convex hull (insert 0 ?p)"] h0 by auto
       then have as: "\<forall>xa. dist x xa < e \<and> (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> xa\<bullet>i = 0) \<longrightarrow>
-        (\<forall>i\<in>d. 0 \<le> xa \<bullet> i) \<and> setsum (op \<bullet> xa) d \<le> 1"
+        (\<forall>i\<in>d. 0 \<le> xa \<bullet> i) \<and> sum (op \<bullet> xa) d \<le> 1"
         unfolding ball_def unfolding substd_simplex[OF assms] using assms by auto
       have x0: "(\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0)"
         using x rel_interior_subset  substd_simplex[OF assms] by auto
-      have "(\<forall>i\<in>d. 0 < x \<bullet> i) \<and> setsum (op \<bullet> x) d < 1 \<and> (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0)"
+      have "(\<forall>i\<in>d. 0 < x \<bullet> i) \<and> sum (op \<bullet> x) d < 1 \<and> (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0)"
         apply rule
         apply rule
       proof -
@@ -8509,22 +8509,22 @@
           by auto
         have "\<And>i. i \<in> Basis \<Longrightarrow> (x + (e / 2) *\<^sub>R a) \<bullet> i = x\<bullet>i + (if i = a then e/2 else 0)"
           using a d by (auto simp: inner_simps inner_Basis)
-        then have *: "setsum (op \<bullet> (x + (e / 2) *\<^sub>R a)) d =
-          setsum (\<lambda>i. x\<bullet>i + (if a = i then e/2 else 0)) d"
-          using d by (intro setsum.cong) auto
+        then have *: "sum (op \<bullet> (x + (e / 2) *\<^sub>R a)) d =
+          sum (\<lambda>i. x\<bullet>i + (if a = i then e/2 else 0)) d"
+          using d by (intro sum.cong) auto
         have "a \<in> Basis"
           using \<open>a \<in> d\<close> d by auto
         then have h1: "(\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> (x + (e / 2) *\<^sub>R a) \<bullet> i = 0)"
           using x0 d \<open>a\<in>d\<close> by (auto simp add: inner_add_left inner_Basis)
-        have "setsum (op \<bullet> x) d < setsum (op \<bullet> (x + (e / 2) *\<^sub>R a)) d"
-          unfolding * setsum.distrib
+        have "sum (op \<bullet> x) d < sum (op \<bullet> (x + (e / 2) *\<^sub>R a)) d"
+          unfolding * sum.distrib
           using \<open>e > 0\<close> \<open>a \<in> d\<close>
           using \<open>finite d\<close>
-          by (auto simp add: setsum.delta')
+          by (auto simp add: sum.delta')
         also have "\<dots> \<le> 1"
           using ** h1 as[rule_format, of "x + (e / 2) *\<^sub>R a"]
           by auto
-        finally show "setsum (op \<bullet> x) d < 1 \<and> (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0)"
+        finally show "sum (op \<bullet> x) d < 1 \<and> (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0)"
           using x0 by auto
       qed
     }
@@ -8543,7 +8543,7 @@
         unfolding substd_simplex[OF assms] by fastforce
       obtain a where a: "a \<in> d"
         using \<open>d \<noteq> {}\<close> by auto
-      let ?d = "(1 - setsum (op \<bullet> x) d) / real (card d)"
+      let ?d = "(1 - sum (op \<bullet> x) d) / real (card d)"
       have "0 < card d" using \<open>d \<noteq> {}\<close> \<open>finite d\<close>
         by (simp add: card_gt_0_iff)
       have "Min ((op \<bullet> x) ` d) > 0"
@@ -8564,8 +8564,8 @@
         fix y :: 'a
         assume y: "dist x y < min (Min (op \<bullet> x ` d)) ?d"
         assume y2: "\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> y\<bullet>i = 0"
-        have "setsum (op \<bullet> y) d \<le> setsum (\<lambda>i. x\<bullet>i + ?d) d"
-        proof (rule setsum_mono)
+        have "sum (op \<bullet> y) d \<le> sum (\<lambda>i. x\<bullet>i + ?d) d"
+        proof (rule sum_mono)
           fix i
           assume "i \<in> d"
           with d have i: "i \<in> Basis"
@@ -8579,9 +8579,9 @@
           then show "y \<bullet> i \<le> x \<bullet> i + ?d" by auto
         qed
         also have "\<dots> \<le> 1"
-          unfolding setsum.distrib setsum_constant  using \<open>0 < card d\<close>
+          unfolding sum.distrib sum_constant  using \<open>0 < card d\<close>
           by auto
-        finally show "setsum (op \<bullet> y) d \<le> 1" .
+        finally show "sum (op \<bullet> y) d \<le> 1" .
 
         fix i :: 'a
         assume i: "i \<in> Basis"
@@ -8600,7 +8600,7 @@
     }
     ultimately have
       "\<And>x. x \<in> rel_interior (convex hull insert 0 d) \<longleftrightarrow>
-        x \<in> {x. (\<forall>i\<in>d. 0 < x \<bullet> i) \<and> setsum (op \<bullet> x) d < 1 \<and> (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0)}"
+        x \<in> {x. (\<forall>i\<in>d. 0 < x \<bullet> i) \<and> sum (op \<bullet> x) d < 1 \<and> (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0)}"
       by blast
     then show ?thesis by (rule set_eqI)
   qed
@@ -8613,7 +8613,7 @@
     where "a \<in> rel_interior (convex hull (insert 0 d))"
 proof -
   let ?D = d
-  let ?a = "setsum (\<lambda>b::'a::euclidean_space. inverse (2 * real (card d)) *\<^sub>R b) ?D"
+  let ?a = "sum (\<lambda>b::'a::euclidean_space. inverse (2 * real (card d)) *\<^sub>R b) ?D"
   have "finite d"
     apply (rule finite_subset)
     using assms(2)
@@ -8625,10 +8625,10 @@
     fix i
     assume "i \<in> d"
     have "?a \<bullet> i = inverse (2 * real (card d))"
-      apply (rule trans[of _ "setsum (\<lambda>j. if i = j then inverse (2 * real (card d)) else 0) ?D"])
-      unfolding inner_setsum_left
-      apply (rule setsum.cong)
-      using \<open>i \<in> d\<close> \<open>finite d\<close> setsum.delta'[of d i "(\<lambda>k. inverse (2 * real (card d)))"]
+      apply (rule trans[of _ "sum (\<lambda>j. if i = j then inverse (2 * real (card d)) else 0) ?D"])
+      unfolding inner_sum_left
+      apply (rule sum.cong)
+      using \<open>i \<in> d\<close> \<open>finite d\<close> sum.delta'[of d i "(\<lambda>k. inverse (2 * real (card d)))"]
         d1 assms(2)
       by (auto simp: inner_Basis set_rev_mp[OF _ assms(2)])
   }
@@ -8645,17 +8645,17 @@
       by auto
     finally show "0 < ?a \<bullet> i" by auto
   next
-    have "setsum (op \<bullet> ?a) ?D = setsum (\<lambda>i. inverse (2 * real (card d))) ?D"
-      by (rule setsum.cong) (rule refl, rule **)
+    have "sum (op \<bullet> ?a) ?D = sum (\<lambda>i. inverse (2 * real (card d))) ?D"
+      by (rule sum.cong) (rule refl, rule **)
     also have "\<dots> < 1"
-      unfolding setsum_constant divide_real_def[symmetric]
+      unfolding sum_constant divide_real_def[symmetric]
       by (auto simp add: field_simps)
-    finally show "setsum (op \<bullet> ?a) ?D < 1" by auto
+    finally show "sum (op \<bullet> ?a) ?D < 1" by auto
   next
     fix i
     assume "i \<in> Basis" and "i \<notin> d"
     have "?a \<in> span d"
-    proof (rule span_setsum[of d "(\<lambda>b. b /\<^sub>R (2 * real (card d)))" d])
+    proof (rule span_sum[of d "(\<lambda>b. b /\<^sub>R (2 * real (card d)))" d])
       {
         fix x :: "'a::euclidean_space"
         assume "x \<in> d"
@@ -10232,20 +10232,20 @@
   assumes "finite I"
   assumes "\<forall>i\<in>I. convex (S i) \<and> (S i) \<noteq> {}"
   shows "convex hull (\<Union>(S ` I)) =
-    {setsum (\<lambda>i. c i *\<^sub>R s i) I | c s. (\<forall>i\<in>I. c i \<ge> 0) \<and> setsum c I = 1 \<and> (\<forall>i\<in>I. s i \<in> S i)}"
+    {sum (\<lambda>i. c i *\<^sub>R s i) I | c s. (\<forall>i\<in>I. c i \<ge> 0) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. s i \<in> S i)}"
   (is "?lhs = ?rhs")
 proof -
   have "?lhs \<supseteq> ?rhs"
   proof
     fix x
     assume "x : ?rhs"
-    then obtain c s where *: "setsum (\<lambda>i. c i *\<^sub>R s i) I = x" "setsum c I = 1"
+    then obtain c s where *: "sum (\<lambda>i. c i *\<^sub>R s i) I = x" "sum c I = 1"
       "(\<forall>i\<in>I. c i \<ge> 0) \<and> (\<forall>i\<in>I. s i \<in> S i)" by auto
     then have "\<forall>i\<in>I. s i \<in> convex hull (\<Union>(S ` I))"
       using hull_subset[of "\<Union>(S ` I)" convex] by auto
     then show "x \<in> ?lhs"
       unfolding *(1)[symmetric]
-      apply (subst convex_setsum[of I "convex hull \<Union>(S ` I)" c s])
+      apply (subst convex_sum[of I "convex hull \<Union>(S ` I)" c s])
       using * assms convex_convex_hull
       apply auto
       done
@@ -10265,14 +10265,14 @@
       fix x
       assume "x \<in> S i"
       define c where "c j = (if j = i then 1::real else 0)" for j
-      then have *: "setsum c I = 1"
-        using \<open>finite I\<close> \<open>i \<in> I\<close> setsum.delta[of I i "\<lambda>j::'a. 1::real"]
+      then have *: "sum c I = 1"
+        using \<open>finite I\<close> \<open>i \<in> I\<close> sum.delta[of I i "\<lambda>j::'a. 1::real"]
         by auto
       define s where "s j = (if j = i then x else p j)" for j
       then have "\<forall>j. c j *\<^sub>R s j = (if j = i then x else 0)"
         using c_def by (auto simp add: algebra_simps)
-      then have "x = setsum (\<lambda>i. c i *\<^sub>R s i) I"
-        using s_def c_def \<open>finite I\<close> \<open>i \<in> I\<close> setsum.delta[of I i "\<lambda>j::'a. x"]
+      then have "x = sum (\<lambda>i. c i *\<^sub>R s i) I"
+        using s_def c_def \<open>finite I\<close> \<open>i \<in> I\<close> sum.delta[of I i "\<lambda>j::'a. x"]
         by auto
       then have "x \<in> ?rhs"
         apply auto
@@ -10292,20 +10292,20 @@
     fix x y
     assume xy: "x \<in> ?rhs \<and> y \<in> ?rhs"
     from xy obtain c s where
-      xc: "x = setsum (\<lambda>i. c i *\<^sub>R s i) I \<and> (\<forall>i\<in>I. c i \<ge> 0) \<and> setsum c I = 1 \<and> (\<forall>i\<in>I. s i \<in> S i)"
+      xc: "x = sum (\<lambda>i. c i *\<^sub>R s i) I \<and> (\<forall>i\<in>I. c i \<ge> 0) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. s i \<in> S i)"
       by auto
     from xy obtain d t where
-      yc: "y = setsum (\<lambda>i. d i *\<^sub>R t i) I \<and> (\<forall>i\<in>I. d i \<ge> 0) \<and> setsum d I = 1 \<and> (\<forall>i\<in>I. t i \<in> S i)"
+      yc: "y = sum (\<lambda>i. d i *\<^sub>R t i) I \<and> (\<forall>i\<in>I. d i \<ge> 0) \<and> sum d I = 1 \<and> (\<forall>i\<in>I. t i \<in> S i)"
       by auto
     define e where "e i = u * c i + v * d i" for i
     have ge0: "\<forall>i\<in>I. e i \<ge> 0"
       using e_def xc yc uv by simp
-    have "setsum (\<lambda>i. u * c i) I = u * setsum c I"
-      by (simp add: setsum_distrib_left)
-    moreover have "setsum (\<lambda>i. v * d i) I = v * setsum d I"
-      by (simp add: setsum_distrib_left)
-    ultimately have sum1: "setsum e I = 1"
-      using e_def xc yc uv by (simp add: setsum.distrib)
+    have "sum (\<lambda>i. u * c i) I = u * sum c I"
+      by (simp add: sum_distrib_left)
+    moreover have "sum (\<lambda>i. v * d i) I = v * sum d I"
+      by (simp add: sum_distrib_left)
+    ultimately have sum1: "sum e I = 1"
+      using e_def xc yc uv by (simp add: sum.distrib)
     define q where "q i = (if e i = 0 then p i else (u * c i / e i) *\<^sub>R s i + (v * d i / e i) *\<^sub>R t i)"
       for i
     {
@@ -10348,11 +10348,11 @@
     }
     then have *: "\<forall>i\<in>I. (u * c i) *\<^sub>R s i + (v * d i) *\<^sub>R t i = e i *\<^sub>R q i"
       by auto
-    have "u *\<^sub>R x + v *\<^sub>R y = setsum (\<lambda>i. (u * c i) *\<^sub>R s i + (v * d i) *\<^sub>R t i) I"
-      using xc yc by (simp add: algebra_simps scaleR_right.setsum setsum.distrib)
-    also have "\<dots> = setsum (\<lambda>i. e i *\<^sub>R q i) I"
+    have "u *\<^sub>R x + v *\<^sub>R y = sum (\<lambda>i. (u * c i) *\<^sub>R s i + (v * d i) *\<^sub>R t i) I"
+      using xc yc by (simp add: algebra_simps scaleR_right.sum sum.distrib)
+    also have "\<dots> = sum (\<lambda>i. e i *\<^sub>R q i) I"
       using * by auto
-    finally have "u *\<^sub>R x + v *\<^sub>R y = setsum (\<lambda>i. (e i) *\<^sub>R (q i)) I"
+    finally have "u *\<^sub>R x + v *\<^sub>R y = sum (\<lambda>i. (e i) *\<^sub>R (q i)) I"
       by auto
     then have "u *\<^sub>R x + v *\<^sub>R y \<in> ?rhs"
       using ge0 sum1 qs by auto
@@ -10380,13 +10380,13 @@
   then have "convex hull (\<Union>(s ` I)) = convex hull (S \<union> T)"
     by auto
   moreover have "convex hull \<Union>(s ` I) =
-    {\<Sum> i\<in>I. c i *\<^sub>R sa i | c sa. (\<forall>i\<in>I. 0 \<le> c i) \<and> setsum c I = 1 \<and> (\<forall>i\<in>I. sa i \<in> s i)}"
+    {\<Sum> i\<in>I. c i *\<^sub>R sa i | c sa. (\<forall>i\<in>I. 0 \<le> c i) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. sa i \<in> s i)}"
       apply (subst convex_hull_finite_union[of I s])
       using assms s_def I_def
       apply auto
       done
   moreover have
-    "{\<Sum>i\<in>I. c i *\<^sub>R sa i | c sa. (\<forall>i\<in>I. 0 \<le> c i) \<and> setsum c I = 1 \<and> (\<forall>i\<in>I. sa i \<in> s i)} \<le> ?rhs"
+    "{\<Sum>i\<in>I. c i *\<^sub>R sa i | c sa. (\<forall>i\<in>I. 0 \<le> c i) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. sa i \<in> s i)} \<le> ?rhs"
     using s_def I_def by auto
   ultimately show "?lhs \<subseteq> ?rhs" by auto
   {
@@ -10432,8 +10432,8 @@
 lemma rel_interior_sum_gen:
   fixes S :: "'a \<Rightarrow> 'n::euclidean_space set"
   assumes "\<forall>i\<in>I. convex (S i)"
-  shows "rel_interior (setsum S I) = setsum (\<lambda>i. rel_interior (S i)) I"
-  apply (subst setsum_set_cond_linear[of convex])
+  shows "rel_interior (sum S I) = sum (\<lambda>i. rel_interior (S i)) I"
+  apply (subst sum_set_cond_linear[of convex])
   using rel_interior_sum rel_interior_sing[of "0"] assms
   apply (auto simp add: convex_set_plus)
   done
@@ -10460,14 +10460,14 @@
   assumes "finite I"
     and "I \<noteq> {}"
   assumes "\<forall>i\<in>I. convex (S i) \<and> cone (S i) \<and> S i \<noteq> {}"
-  shows "convex hull (\<Union>(S ` I)) = setsum S I"
+  shows "convex hull (\<Union>(S ` I)) = sum S I"
   (is "?lhs = ?rhs")
 proof -
   {
     fix x
     assume "x \<in> ?lhs"
     then obtain c xs where
-      x: "x = setsum (\<lambda>i. c i *\<^sub>R xs i) I \<and> (\<forall>i\<in>I. c i \<ge> 0) \<and> setsum c I = 1 \<and> (\<forall>i\<in>I. xs i \<in> S i)"
+      x: "x = sum (\<lambda>i. c i *\<^sub>R xs i) I \<and> (\<forall>i\<in>I. c i \<ge> 0) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. xs i \<in> S i)"
       using convex_hull_finite_union[of I S] assms by auto
     define s where "s i = c i *\<^sub>R xs i" for i
     {
@@ -10477,24 +10477,24 @@
         using s_def x assms mem_cone[of "S i" "xs i" "c i"] by auto
     }
     then have "\<forall>i\<in>I. s i \<in> S i" by auto
-    moreover have "x = setsum s I" using x s_def by auto
+    moreover have "x = sum s I" using x s_def by auto
     ultimately have "x \<in> ?rhs"
-      using set_setsum_alt[of I S] assms by auto
+      using set_sum_alt[of I S] assms by auto
   }
   moreover
   {
     fix x
     assume "x \<in> ?rhs"
-    then obtain s where x: "x = setsum s I \<and> (\<forall>i\<in>I. s i \<in> S i)"
-      using set_setsum_alt[of I S] assms by auto
+    then obtain s where x: "x = sum s I \<and> (\<forall>i\<in>I. s i \<in> S i)"
+      using set_sum_alt[of I S] assms by auto
     define xs where "xs i = of_nat(card I) *\<^sub>R s i" for i
-    then have "x = setsum (\<lambda>i. ((1 :: real) / of_nat(card I)) *\<^sub>R xs i) I"
+    then have "x = sum (\<lambda>i. ((1 :: real) / of_nat(card I)) *\<^sub>R xs i) I"
       using x assms by auto
     moreover have "\<forall>i\<in>I. xs i \<in> S i"
       using x xs_def assms by (simp add: cone_def)
     moreover have "\<forall>i\<in>I. (1 :: real) / of_nat (card I) \<ge> 0"
       by auto
-    moreover have "setsum (\<lambda>i. (1 :: real) / of_nat (card I)) I = 1"
+    moreover have "sum (\<lambda>i. (1 :: real) / of_nat (card I)) I = 1"
       using assms by auto
     ultimately have "x \<in> ?lhs"
       apply (subst convex_hull_finite_union[of I S])
@@ -10526,12 +10526,12 @@
     using A_def I_def by auto
   then have "convex hull (\<Union>(A ` I)) = convex hull (S \<union> T)"
     by auto
-  moreover have "convex hull \<Union>(A ` I) = setsum A I"
+  moreover have "convex hull \<Union>(A ` I) = sum A I"
     apply (subst convex_hull_finite_union_cones[of I A])
     using assms A_def I_def
     apply auto
     done
-  moreover have "setsum A I = S + T"
+  moreover have "sum A I = S + T"
     using A_def I_def
     unfolding set_plus_def
     apply auto
@@ -10546,7 +10546,7 @@
   assumes "finite I"
     and "\<forall>i\<in>I. convex (S i) \<and> S i \<noteq> {}"
   shows "rel_interior (convex hull (\<Union>(S ` I))) =
-    {setsum (\<lambda>i. c i *\<^sub>R s i) I | c s. (\<forall>i\<in>I. c i > 0) \<and> setsum c I = 1 \<and>
+    {sum (\<lambda>i. c i *\<^sub>R s i) I | c s. (\<forall>i\<in>I. c i > 0) \<and> sum c I = 1 \<and>
       (\<forall>i\<in>I. s i \<in> rel_interior(S i))}"
   (is "?lhs = ?rhs")
 proof (cases "I = {}")
@@ -10621,7 +10621,7 @@
     by blast
   then have "K0 = convex hull (\<Union>(K ` I))"
     using geq by auto
-  also have "\<dots> = setsum K I"
+  also have "\<dots> = sum K I"
     apply (subst convex_hull_finite_union_cones[of I K])
     using assms
     apply blast
@@ -10634,8 +10634,8 @@
     using assms cone_cone_hull \<open>\<forall>i\<in>I. K i \<noteq> {}\<close> K_def
     apply auto
     done
-  finally have "K0 = setsum K I" by auto
-  then have *: "rel_interior K0 = setsum (\<lambda>i. (rel_interior (K i))) I"
+  finally have "K0 = sum K I" by auto
+  then have *: "rel_interior K0 = sum (\<lambda>i. (rel_interior (K i))) I"
     using rel_interior_sum_gen[of I K] convK by auto
   {
     fix x
@@ -10643,8 +10643,8 @@
     then have "(1::real, x) \<in> rel_interior K0"
       using K0_def C0_def rel_interior_convex_cone_aux[of C0 "1::real" x] convex_convex_hull
       by auto
-    then obtain k where k: "(1::real, x) = setsum k I \<and> (\<forall>i\<in>I. k i \<in> rel_interior (K i))"
-      using \<open>finite I\<close> * set_setsum_alt[of I "\<lambda>i. rel_interior (K i)"] by auto
+    then obtain k where k: "(1::real, x) = sum k I \<and> (\<forall>i\<in>I. k i \<in> rel_interior (K i))"
+      using \<open>finite I\<close> * set_sum_alt[of I "\<lambda>i. rel_interior (K i)"] by auto
     {
       fix i
       assume "i \<in> I"
@@ -10656,8 +10656,8 @@
     then obtain c s where
       cs: "\<forall>i\<in>I. k i = (c i, c i *\<^sub>R s i) \<and> 0 < c i \<and> s i \<in> rel_interior (S i)"
       by metis
-    then have "x = (\<Sum>i\<in>I. c i *\<^sub>R s i) \<and> setsum c I = 1"
-      using k by (simp add: setsum_prod)
+    then have "x = (\<Sum>i\<in>I. c i *\<^sub>R s i) \<and> sum c I = 1"
+      using k by (simp add: sum_prod)
     then have "x \<in> ?rhs"
       using k
       apply auto
@@ -10671,8 +10671,8 @@
   {
     fix x
     assume "x \<in> ?rhs"
-    then obtain c s where cs: "x = setsum (\<lambda>i. c i *\<^sub>R s i) I \<and>
-        (\<forall>i\<in>I. c i > 0) \<and> setsum c I = 1 \<and> (\<forall>i\<in>I. s i \<in> rel_interior (S i))"
+    then obtain c s where cs: "x = sum (\<lambda>i. c i *\<^sub>R s i) I \<and>
+        (\<forall>i\<in>I. c i > 0) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. s i \<in> rel_interior (S i))"
       by auto
     define k where "k i = (c i, c i *\<^sub>R s i)" for i
     {
@@ -10682,10 +10682,10 @@
         by auto
     }
     then have "(1::real, x) \<in> rel_interior K0"
-      using K0_def * set_setsum_alt[of I "(\<lambda>i. rel_interior (K i))"] assms k_def cs
+      using K0_def * set_sum_alt[of I "(\<lambda>i. rel_interior (K i))"] assms k_def cs
       apply auto
       apply (rule_tac x = k in exI)
-      apply (simp add: setsum_prod)
+      apply (simp add: sum_prod)
       done
     then have "x \<in> ?lhs"
       using K0_def C0_def rel_interior_convex_cone_aux[of C0 1 x]
@@ -10801,18 +10801,18 @@
 proof -
   have fin: "finite s" "finite t" using assms aff_independent_finite finite_subset by auto
     { fix u v x
-      assume uv: "setsum u t = 1" "\<forall>x\<in>s. 0 \<le> v x" "setsum v s = 1"
+      assume uv: "sum u t = 1" "\<forall>x\<in>s. 0 \<le> v x" "sum v s = 1"
                  "(\<Sum>x\<in>s. v x *\<^sub>R x) = (\<Sum>v\<in>t. u v *\<^sub>R v)" "x \<in> t"
       then have s: "s = (s - t) \<union> t" \<comment>\<open>split into separate cases\<close>
         using assms by auto
       have [simp]: "(\<Sum>x\<in>t. v x *\<^sub>R x) + (\<Sum>x\<in>s - t. v x *\<^sub>R x) = (\<Sum>x\<in>t. u x *\<^sub>R x)"
-                   "setsum v t + setsum v (s - t) = 1"
+                   "sum v t + sum v (s - t) = 1"
         using uv fin s
-        by (auto simp: setsum.union_disjoint [symmetric] Un_commute)
+        by (auto simp: sum.union_disjoint [symmetric] Un_commute)
       have "(\<Sum>x\<in>s. if x \<in> t then v x - u x else v x) = 0"
            "(\<Sum>x\<in>s. (if x \<in> t then v x - u x else v x) *\<^sub>R x) = 0"
         using uv fin
-        by (subst s, subst setsum.union_disjoint, auto simp: algebra_simps setsum_subtractf)+
+        by (subst s, subst sum.union_disjoint, auto simp: algebra_simps sum_subtractf)+
     } note [simp] = this
   have "convex hull t \<subseteq> affine hull t"
     using convex_hull_subset_affine_hull by blast
@@ -10890,14 +10890,14 @@
 lemma explicit_subset_rel_interior_convex_hull:
   fixes s :: "'a::euclidean_space set"
   shows "finite s
-         \<Longrightarrow> {y. \<exists>u. (\<forall>x \<in> s. 0 < u x \<and> u x < 1) \<and> setsum u s = 1 \<and> setsum (\<lambda>x. u x *\<^sub>R x) s = y}
+         \<Longrightarrow> {y. \<exists>u. (\<forall>x \<in> s. 0 < u x \<and> u x < 1) \<and> sum u s = 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) s = y}
              \<subseteq> rel_interior (convex hull s)"
   by (force simp add:  rel_interior_convex_hull_union [where S="\<lambda>x. {x}" and I=s, simplified])
 
 lemma explicit_subset_rel_interior_convex_hull_minimal:
   fixes s :: "'a::euclidean_space set"
   shows "finite s
-         \<Longrightarrow> {y. \<exists>u. (\<forall>x \<in> s. 0 < u x) \<and> setsum u s = 1 \<and> setsum (\<lambda>x. u x *\<^sub>R x) s = y}
+         \<Longrightarrow> {y. \<exists>u. (\<forall>x \<in> s. 0 < u x) \<and> sum u s = 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) s = y}
              \<subseteq> rel_interior (convex hull s)"
   by (force simp add:  rel_interior_convex_hull_union [where S="\<lambda>x. {x}" and I=s, simplified])
 
@@ -10905,7 +10905,7 @@
   fixes s :: "'a::euclidean_space set"
   assumes "~ affine_dependent s"
   shows "rel_interior(convex hull s) =
-         {y. \<exists>u. (\<forall>x \<in> s. 0 < u x) \<and> setsum u s = 1 \<and> setsum (\<lambda>x. u x *\<^sub>R x) s = y}"
+         {y. \<exists>u. (\<forall>x \<in> s. 0 < u x) \<and> sum u s = 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) s = y}"
          (is "?lhs = ?rhs")
 proof
   show "?rhs \<le> ?lhs"
@@ -10926,14 +10926,14 @@
       have "(\<Sum>x\<in>s. if x = a then - d else if x = b then d else 0) = 0"
            "(\<Sum>x\<in>s. (if x = a then - d else if x = b then d else 0) *\<^sub>R x) = d *\<^sub>R b - d *\<^sub>R a"
         using ab fs
-        by (subst s, subst setsum.union_disjoint, auto)+
+        by (subst s, subst sum.union_disjoint, auto)+
     } note [simp] = this
     { fix y
       assume y: "y \<in> convex hull s" "y \<notin> ?rhs"
       { fix u T a
-        assume ua: "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = 1" "\<not> 0 < u a" "a \<in> s"
+        assume ua: "\<forall>x\<in>s. 0 \<le> u x" "sum u s = 1" "\<not> 0 < u a" "a \<in> s"
            and yT: "y = (\<Sum>x\<in>s. u x *\<^sub>R x)" "y \<in> T" "open T"
-           and sb: "T \<inter> affine hull s \<subseteq> {w. \<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> (\<Sum>x\<in>s. u x *\<^sub>R x) = w}"
+           and sb: "T \<inter> affine hull s \<subseteq> {w. \<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = 1 \<and> (\<Sum>x\<in>s. u x *\<^sub>R x) = w}"
         have ua0: "u a = 0"
           using ua by auto
         obtain b where b: "b\<in>s" "a \<noteq> b"
@@ -10949,7 +10949,7 @@
         then have "y - d *\<^sub>R (a - b) \<in> T \<inter> affine hull s"
           using d e yT by auto
         then obtain v where "\<forall>x\<in>s. 0 \<le> v x"
-                            "setsum v s = 1"
+                            "sum v s = 1"
                             "(\<Sum>x\<in>s. v x *\<^sub>R x) = (\<Sum>x\<in>s. u x *\<^sub>R x) - d *\<^sub>R (a - b)"
           using subsetD [OF sb] yT
           by auto
@@ -10958,7 +10958,7 @@
           apply (simp add: affine_dependent_explicit_finite fs)
           apply (drule_tac x="\<lambda>x. (v x - u x) - (if x = a then -d else if x = b then d else 0)" in spec)
           using ua b d
-          apply (auto simp: algebra_simps setsum_subtractf setsum.distrib)
+          apply (auto simp: algebra_simps sum_subtractf sum.distrib)
           done
       } note * = this
       have "y \<notin> rel_interior (convex hull s)"
@@ -10979,7 +10979,7 @@
    "~ affine_dependent s
         ==> interior(convex hull s) =
              (if card(s) \<le> DIM('a) then {}
-              else {y. \<exists>u. (\<forall>x \<in> s. 0 < u x) \<and> setsum u s = 1 \<and> (\<Sum>x\<in>s. u x *\<^sub>R x) = y})"
+              else {y. \<exists>u. (\<forall>x \<in> s. 0 < u x) \<and> sum u s = 1 \<and> (\<Sum>x\<in>s. u x *\<^sub>R x) = y})"
   apply (simp add: aff_independent_finite empty_interior_convex_hull, clarify)
   apply (rule trans [of _ "rel_interior(convex hull s)"])
   apply (simp add: affine_hull_convex_hull affine_independent_span_gt rel_interior_interior)
@@ -10991,10 +10991,10 @@
   shows
    "interior(convex hull s) =
              (if card(s) \<le> DIM('a) then {}
-              else {y. \<exists>u. (\<forall>x \<in> s. 0 < u x \<and> u x < 1) \<and> setsum u s = 1 \<and> (\<Sum>x\<in>s. u x *\<^sub>R x) = y})"
+              else {y. \<exists>u. (\<forall>x \<in> s. 0 < u x \<and> u x < 1) \<and> sum u s = 1 \<and> (\<Sum>x\<in>s. u x *\<^sub>R x) = y})"
 proof -
   { fix u :: "'a \<Rightarrow> real" and a
-    assume "card Basis < card s" and u: "\<And>x. x\<in>s \<Longrightarrow> 0 < u x" "setsum u s = 1" and a: "a \<in> s"
+    assume "card Basis < card s" and u: "\<And>x. x\<in>s \<Longrightarrow> 0 < u x" "sum u s = 1" and a: "a \<in> s"
     then have cs: "Suc 0 < card s"
       by (metis DIM_positive less_trans_Suc)
     obtain b where b: "b \<in> s" "a \<noteq> b"
@@ -11007,10 +11007,10 @@
       then show thesis
       by (blast intro: that)
     qed
-    have "u a + u b \<le> setsum u {a,b}"
+    have "u a + u b \<le> sum u {a,b}"
       using a b by simp
-    also have "... \<le> setsum u s"
-      apply (rule Groups_Big.setsum_mono2)
+    also have "... \<le> sum u s"
+      apply (rule Groups_Big.sum_mono2)
       using a b u
       apply (auto simp: less_imp_le aff_independent_finite assms)
       done
@@ -11156,7 +11156,7 @@
   fixes s :: "'a::euclidean_space set"
   assumes "~ affine_dependent s"
   shows "rel_frontier(convex hull s) =
-         {y. \<exists>u. (\<forall>x \<in> s. 0 \<le> u x) \<and> (\<exists>x \<in> s. u x = 0) \<and> setsum u s = 1 \<and> setsum (\<lambda>x. u x *\<^sub>R x) s = y}"
+         {y. \<exists>u. (\<forall>x \<in> s. 0 \<le> u x) \<and> (\<exists>x \<in> s. u x = 0) \<and> sum u s = 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) s = y}"
 proof -
   have fs: "finite s"
     using assms by (simp add: aff_independent_finite)
@@ -11172,7 +11172,7 @@
     apply (rule_tac x=s in exI)
     apply (auto simp: fs)
     apply (rule_tac x = "\<lambda>x. u x - v x" in exI)
-    apply (force simp: setsum_subtractf scaleR_diff_left)
+    apply (force simp: sum_subtractf scaleR_diff_left)
     done
 qed
 
@@ -11181,7 +11181,7 @@
   assumes "~ affine_dependent s"
   shows "frontier(convex hull s) =
          {y. \<exists>u. (\<forall>x \<in> s. 0 \<le> u x) \<and> (DIM ('a) < card s \<longrightarrow> (\<exists>x \<in> s. u x = 0)) \<and>
-             setsum u s = 1 \<and> setsum (\<lambda>x. u x *\<^sub>R x) s = y}"
+             sum u s = 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) s = y}"
 proof -
   have fs: "finite s"
     using assms by (simp add: aff_independent_finite)
@@ -11211,21 +11211,21 @@
   have fs: "finite s"
     using assms by (simp add: aff_independent_finite)
   { fix u a
-  have "\<forall>x\<in>s. 0 \<le> u x \<Longrightarrow> a \<in> s \<Longrightarrow> u a = 0 \<Longrightarrow> setsum u s = 1 \<Longrightarrow>
+  have "\<forall>x\<in>s. 0 \<le> u x \<Longrightarrow> a \<in> s \<Longrightarrow> u a = 0 \<Longrightarrow> sum u s = 1 \<Longrightarrow>
             \<exists>x v. x \<in> s \<and>
                   (\<forall>x\<in>s - {x}. 0 \<le> v x) \<and>
-                      setsum v (s - {x}) = 1 \<and> (\<Sum>x\<in>s - {x}. v x *\<^sub>R x) = (\<Sum>x\<in>s. u x *\<^sub>R x)"
+                      sum v (s - {x}) = 1 \<and> (\<Sum>x\<in>s - {x}. v x *\<^sub>R x) = (\<Sum>x\<in>s. u x *\<^sub>R x)"
     apply (rule_tac x=a in exI)
     apply (rule_tac x=u in exI)
-    apply (simp add: Groups_Big.setsum_diff1 fs)
+    apply (simp add: Groups_Big.sum_diff1 fs)
     done }
   moreover
   { fix a u
-    have "a \<in> s \<Longrightarrow> \<forall>x\<in>s - {a}. 0 \<le> u x \<Longrightarrow> setsum u (s - {a}) = 1 \<Longrightarrow>
+    have "a \<in> s \<Longrightarrow> \<forall>x\<in>s - {a}. 0 \<le> u x \<Longrightarrow> sum u (s - {a}) = 1 \<Longrightarrow>
             \<exists>v. (\<forall>x\<in>s. 0 \<le> v x) \<and>
-                 (\<exists>x\<in>s. v x = 0) \<and> setsum v s = 1 \<and> (\<Sum>x\<in>s. v x *\<^sub>R x) = (\<Sum>x\<in>s - {a}. u x *\<^sub>R x)"
+                 (\<exists>x\<in>s. v x = 0) \<and> sum v s = 1 \<and> (\<Sum>x\<in>s. v x *\<^sub>R x) = (\<Sum>x\<in>s - {a}. u x *\<^sub>R x)"
     apply (rule_tac x="\<lambda>x. if x = a then 0 else u x" in exI)
-    apply (auto simp: setsum.If_cases Diff_eq if_smult fs)
+    apply (auto simp: sum.If_cases Diff_eq if_smult fs)
     done }
   ultimately show ?thesis
     using assms
@@ -12496,7 +12496,7 @@
     have "x = (\<Sum>b\<in>Basis. (x \<bullet> b) *\<^sub>R b)"
       by (simp add: euclidean_representation)
     also have "... = (\<Sum>b \<in> Basis - {k}. (x \<bullet> b) *\<^sub>R b)"
-      by (auto simp: setsum.remove [of _ k] inner_commute assms that)
+      by (auto simp: sum.remove [of _ k] inner_commute assms that)
     finally have "x = (\<Sum>b\<in>Basis - {k}. (x \<bullet> b) *\<^sub>R b)" .
     then show ?thesis
       by (simp add: Linear_Algebra.span_finite) metis
@@ -12960,20 +12960,20 @@
 proof -
   have [simp]: "finite s" "finite t"
     using aff_independent_finite assms by blast+
-    have "setsum u (s \<inter> t) = 1 \<and>
+    have "sum u (s \<inter> t) = 1 \<and>
           (\<Sum>v\<in>s \<inter> t. u v *\<^sub>R v) = (\<Sum>v\<in>s. u v *\<^sub>R v)"
-      if [simp]:  "setsum u s = 1"
-                 "setsum v t = 1"
+      if [simp]:  "sum u s = 1"
+                 "sum v t = 1"
          and eq: "(\<Sum>x\<in>t. v x *\<^sub>R x) = (\<Sum>x\<in>s. u x *\<^sub>R x)" for u v
     proof -
     define f where "f x = (if x \<in> s then u x else 0) - (if x \<in> t then v x else 0)" for x
-    have "setsum f (s \<union> t) = 0"
-      apply (simp add: f_def setsum_Un setsum_subtractf)
-      apply (simp add: setsum.inter_restrict [symmetric] Int_commute)
+    have "sum f (s \<union> t) = 0"
+      apply (simp add: f_def sum_Un sum_subtractf)
+      apply (simp add: sum.inter_restrict [symmetric] Int_commute)
       done
     moreover have "(\<Sum>x\<in>(s \<union> t). f x *\<^sub>R x) = 0"
-      apply (simp add: f_def setsum_Un scaleR_left_diff_distrib setsum_subtractf)
-      apply (simp add: if_smult setsum.inter_restrict [symmetric] Int_commute eq
+      apply (simp add: f_def sum_Un scaleR_left_diff_distrib sum_subtractf)
+      apply (simp add: if_smult sum.inter_restrict [symmetric] Int_commute eq
           cong del: if_weak_cong)
       done
     ultimately have "\<And>v. v \<in> s \<union> t \<Longrightarrow> f v = 0"
@@ -12981,12 +12981,12 @@
       by blast
     then have u [simp]: "\<And>x. x \<in> s \<Longrightarrow> u x = (if x \<in> t then v x else 0)"
       by (simp add: f_def) presburger
-    have "setsum u (s \<inter> t) = setsum u s"
-      by (simp add: setsum.inter_restrict)
-    then have "setsum u (s \<inter> t) = 1"
+    have "sum u (s \<inter> t) = sum u s"
+      by (simp add: sum.inter_restrict)
+    then have "sum u (s \<inter> t) = 1"
       using that by linarith
     moreover have "(\<Sum>v\<in>s \<inter> t. u v *\<^sub>R v) = (\<Sum>v\<in>s. u v *\<^sub>R v)"
-      by (auto simp: if_smult setsum.inter_restrict intro: setsum.cong)
+      by (auto simp: if_smult sum.inter_restrict intro: sum.cong)
     ultimately show ?thesis
       by force
     qed
@@ -13063,19 +13063,19 @@
   have [simp]: "finite S"
     by (simp add: aff_independent_finite assms)
   then obtain b where b0: "\<And>s. s \<in> S \<Longrightarrow> 0 \<le> b s"
-                  and b1: "setsum b S = 1" and aeq: "a = (\<Sum>s\<in>S. b s *\<^sub>R s)"
+                  and b1: "sum b S = 1" and aeq: "a = (\<Sum>s\<in>S. b s *\<^sub>R s)"
     using a by (auto simp: convex_hull_finite)
   have fin [simp]: "finite T" "finite T'"
     using assms infinite_super \<open>finite S\<close> by blast+
   then obtain c c' where c0: "\<And>t. t \<in> insert a T \<Longrightarrow> 0 \<le> c t"
-                     and c1: "setsum c (insert a T) = 1"
+                     and c1: "sum c (insert a T) = 1"
                      and xeq: "x = (\<Sum>t \<in> insert a T. c t *\<^sub>R t)"
                      and c'0: "\<And>t. t \<in> insert a T' \<Longrightarrow> 0 \<le> c' t"
-                     and c'1: "setsum c' (insert a T') = 1"
+                     and c'1: "sum c' (insert a T') = 1"
                      and x'eq: "x = (\<Sum>t \<in> insert a T'. c' t *\<^sub>R t)"
     using x x' by (auto simp: convex_hull_finite)
   with fin anot
-  have sumTT': "setsum c T = 1 - c a" "setsum c' T' = 1 - c' a"
+  have sumTT': "sum c T = 1 - c a" "sum c' T' = 1 - c' a"
    and wsumT: "(\<Sum>t \<in> T. c t *\<^sub>R t) = x - c a *\<^sub>R a"
     by simp_all
   have wsumT': "(\<Sum>t \<in> T'. c' t *\<^sub>R t) = x - c' a *\<^sub>R a"
@@ -13083,23 +13083,23 @@
   define cc  where "cc \<equiv> \<lambda>x. if x \<in> T then c x else 0"
   define cc' where "cc' \<equiv> \<lambda>x. if x \<in> T' then c' x else 0"
   define dd  where "dd \<equiv> \<lambda>x. cc x - cc' x + (c a - c' a) * b x"
-  have sumSS': "setsum cc S = 1 - c a" "setsum cc' S = 1 - c' a"
+  have sumSS': "sum cc S = 1 - c a" "sum cc' S = 1 - c' a"
     unfolding cc_def cc'_def  using S
-    by (simp_all add: Int_absorb1 Int_absorb2 setsum_subtractf setsum.inter_restrict [symmetric] sumTT')
+    by (simp_all add: Int_absorb1 Int_absorb2 sum_subtractf sum.inter_restrict [symmetric] sumTT')
   have wsumSS: "(\<Sum>t \<in> S. cc t *\<^sub>R t) = x - c a *\<^sub>R a" "(\<Sum>t \<in> S. cc' t *\<^sub>R t) = x - c' a *\<^sub>R a"
     unfolding cc_def cc'_def  using S
-    by (simp_all add: Int_absorb1 Int_absorb2 if_smult setsum.inter_restrict [symmetric] wsumT wsumT' cong: if_cong)
-  have sum_dd0: "setsum dd S = 0"
+    by (simp_all add: Int_absorb1 Int_absorb2 if_smult sum.inter_restrict [symmetric] wsumT wsumT' cong: if_cong)
+  have sum_dd0: "sum dd S = 0"
     unfolding dd_def  using S
-    by (simp add: sumSS' comm_monoid_add_class.setsum.distrib setsum_subtractf
-                  algebra_simps setsum_distrib_right [symmetric] b1)
+    by (simp add: sumSS' comm_monoid_add_class.sum.distrib sum_subtractf
+                  algebra_simps sum_distrib_right [symmetric] b1)
   have "(\<Sum>v\<in>S. (b v * x) *\<^sub>R v) = x *\<^sub>R (\<Sum>v\<in>S. b v *\<^sub>R v)" for x
-    by (simp add: pth_5 real_vector.scale_setsum_right mult.commute)
+    by (simp add: pth_5 real_vector.scale_sum_right mult.commute)
   then have *: "(\<Sum>v\<in>S. (b v * x) *\<^sub>R v) = x *\<^sub>R a" for x
     using aeq by blast
   have "(\<Sum>v \<in> S. dd v *\<^sub>R v) = 0"
     unfolding dd_def using S
-    by (simp add: * wsumSS setsum.distrib setsum_subtractf algebra_simps)
+    by (simp add: * wsumSS sum.distrib sum_subtractf algebra_simps)
   then have dd0: "dd v = 0" if "v \<in> S" for v
     using naff that \<open>finite S\<close> sum_dd0 unfolding affine_dependent_explicit
     apply (simp only: not_ex)
@@ -13110,7 +13110,7 @@
   then show ?thesis
   proof cases
     case 1
-    then have "setsum cc S \<le> setsum cc' S"
+    then have "sum cc S \<le> sum cc' S"
       by (simp add: sumSS')
     then have le: "cc x \<le> cc' x" if "x \<in> S" for x
       using dd0 [OF that] 1 b0 mult_left_mono that
@@ -13124,32 +13124,32 @@
         by (simp add: c0 cc_def)
       show "0 \<le> (cc(a := c a)) a"
         by (simp add: c0)
-      have "setsum (cc(a := c a)) (insert a (T \<inter> T')) = c a + setsum (cc(a := c a)) (T \<inter> T')"
+      have "sum (cc(a := c a)) (insert a (T \<inter> T')) = c a + sum (cc(a := c a)) (T \<inter> T')"
         by (simp add: anot)
-      also have "... = c a + setsum (cc(a := c a)) S"
+      also have "... = c a + sum (cc(a := c a)) S"
         apply simp
-        apply (rule setsum.mono_neutral_left)
+        apply (rule sum.mono_neutral_left)
         using \<open>T \<subseteq> S\<close> apply (auto simp: \<open>a \<notin> S\<close> cc0)
         done
       also have "... = c a + (1 - c a)"
-        by (metis \<open>a \<notin> S\<close> fun_upd_other setsum.cong sumSS')
-      finally show "setsum (cc(a := c a)) (insert a (T \<inter> T')) = 1"
+        by (metis \<open>a \<notin> S\<close> fun_upd_other sum.cong sumSS')
+      finally show "sum (cc(a := c a)) (insert a (T \<inter> T')) = 1"
         by simp
       have "(\<Sum>x\<in>insert a (T \<inter> T'). (cc(a := c a)) x *\<^sub>R x) = c a *\<^sub>R a + (\<Sum>x \<in> T \<inter> T'. (cc(a := c a)) x *\<^sub>R x)"
         by (simp add: anot)
       also have "... = c a *\<^sub>R a + (\<Sum>x \<in> S. (cc(a := c a)) x *\<^sub>R x)"
         apply simp
-        apply (rule setsum.mono_neutral_left)
+        apply (rule sum.mono_neutral_left)
         using \<open>T \<subseteq> S\<close> apply (auto simp: \<open>a \<notin> S\<close> cc0)
         done
       also have "... = c a *\<^sub>R a + x - c a *\<^sub>R a"
-        by (simp add: wsumSS \<open>a \<notin> S\<close> if_smult setsum_delta_notmem)
+        by (simp add: wsumSS \<open>a \<notin> S\<close> if_smult sum_delta_notmem)
       finally show "(\<Sum>x\<in>insert a (T \<inter> T'). (cc(a := c a)) x *\<^sub>R x) = x"
         by simp
     qed
   next
     case 2
-    then have "setsum cc' S \<le> setsum cc S"
+    then have "sum cc' S \<le> sum cc S"
       by (simp add: sumSS')
     then have le: "cc' x \<le> cc x" if "x \<in> S" for x
       using dd0 [OF that] 2 b0 mult_left_mono that
@@ -13163,26 +13163,26 @@
         by (simp add: c'0 cc'_def)
       show "0 \<le> (cc'(a := c' a)) a"
         by (simp add: c'0)
-      have "setsum (cc'(a := c' a)) (insert a (T \<inter> T')) = c' a + setsum (cc'(a := c' a)) (T \<inter> T')"
+      have "sum (cc'(a := c' a)) (insert a (T \<inter> T')) = c' a + sum (cc'(a := c' a)) (T \<inter> T')"
         by (simp add: anot)
-      also have "... = c' a + setsum (cc'(a := c' a)) S"
+      also have "... = c' a + sum (cc'(a := c' a)) S"
         apply simp
-        apply (rule setsum.mono_neutral_left)
+        apply (rule sum.mono_neutral_left)
         using \<open>T \<subseteq> S\<close> apply (auto simp: \<open>a \<notin> S\<close> cc0)
         done
       also have "... = c' a + (1 - c' a)"
-        by (metis \<open>a \<notin> S\<close> fun_upd_other setsum.cong sumSS')
-      finally show "setsum (cc'(a := c' a)) (insert a (T \<inter> T')) = 1"
+        by (metis \<open>a \<notin> S\<close> fun_upd_other sum.cong sumSS')
+      finally show "sum (cc'(a := c' a)) (insert a (T \<inter> T')) = 1"
         by simp
       have "(\<Sum>x\<in>insert a (T \<inter> T'). (cc'(a := c' a)) x *\<^sub>R x) = c' a *\<^sub>R a + (\<Sum>x \<in> T \<inter> T'. (cc'(a := c' a)) x *\<^sub>R x)"
         by (simp add: anot)
       also have "... = c' a *\<^sub>R a + (\<Sum>x \<in> S. (cc'(a := c' a)) x *\<^sub>R x)"
         apply simp
-        apply (rule setsum.mono_neutral_left)
+        apply (rule sum.mono_neutral_left)
         using \<open>T \<subseteq> S\<close> apply (auto simp: \<open>a \<notin> S\<close> cc0)
         done
       also have "... = c a *\<^sub>R a + x - c a *\<^sub>R a"
-        by (simp add: wsumSS \<open>a \<notin> S\<close> if_smult setsum_delta_notmem)
+        by (simp add: wsumSS \<open>a \<notin> S\<close> if_smult sum_delta_notmem)
       finally show "(\<Sum>x\<in>insert a (T \<inter> T'). (cc'(a := c' a)) x *\<^sub>R x) = x"
         by simp
     qed
@@ -13458,10 +13458,10 @@
     have "(\<Sum>v\<in>D - C. a v *\<^sub>R v) \<in> S"
       apply (subst eq)
       apply (rule subspace_neg [OF \<open>subspace S\<close>])
-      apply (rule subspace_setsum [OF \<open>subspace S\<close>])
+      apply (rule subspace_sum [OF \<open>subspace S\<close>])
       by (meson subsetCE subspace_mul \<open>C \<subseteq> S\<close> \<open>subspace S\<close>)
     moreover have "(\<Sum>v\<in>D - C. a v *\<^sub>R v) \<in> T"
-      apply (rule subspace_setsum [OF \<open>subspace T\<close>])
+      apply (rule subspace_sum [OF \<open>subspace T\<close>])
       by (meson DiffD1 \<open>D \<subseteq> T\<close> \<open>subspace T\<close> subset_eq subspace_def)
     ultimately have "(\<Sum>v \<in> D-C. a v *\<^sub>R v) \<in> span B"
       using B by blast
@@ -13475,12 +13475,12 @@
     have f2: "(\<Sum>v\<in>C \<inter> D. e v *\<^sub>R v) = (\<Sum>v\<in>D - C. a v *\<^sub>R v)"
       using Beq e by presburger
     have f3: "(\<Sum>v\<in>C \<union> D. a v *\<^sub>R v) = (\<Sum>v\<in>C - D. a v *\<^sub>R v) + (\<Sum>v\<in>D - C. a v *\<^sub>R v) + (\<Sum>v\<in>C \<inter> D. a v *\<^sub>R v)"
-      using \<open>finite C\<close> \<open>finite D\<close> setsum.union_diff2 by blast
+      using \<open>finite C\<close> \<open>finite D\<close> sum.union_diff2 by blast
     have f4: "(\<Sum>v\<in>C \<union> (D - C). a v *\<^sub>R v) = (\<Sum>v\<in>C. a v *\<^sub>R v) + (\<Sum>v\<in>D - C. a v *\<^sub>R v)"
-      by (meson Diff_disjoint \<open>finite C\<close> \<open>finite D\<close> finite_Diff setsum.union_disjoint)
+      by (meson Diff_disjoint \<open>finite C\<close> \<open>finite D\<close> finite_Diff sum.union_disjoint)
     have "(\<Sum>v\<in>C. cc v *\<^sub>R v) = 0"
       using 0 f2 f3 f4
-      apply (simp add: cc_def Beq if_smult \<open>finite C\<close> setsum.If_cases algebra_simps setsum.distrib)
+      apply (simp add: cc_def Beq if_smult \<open>finite C\<close> sum.If_cases algebra_simps sum.distrib)
       apply (simp add: add.commute add.left_commute diff_eq)
       done
     then have "\<And>v. v \<in> C \<Longrightarrow> cc v = 0"
@@ -13498,14 +13498,14 @@
     qed
     with 0 have Dcc0: "(\<Sum>v\<in>D. a v *\<^sub>R v) = 0"
       apply (subst Deq)
-      by (simp add: \<open>finite B\<close> \<open>finite D\<close> setsum_Un)
+      by (simp add: \<open>finite B\<close> \<open>finite D\<close> sum_Un)
     then have D0: "\<And>v. v \<in> D \<Longrightarrow> a v = 0"
       using independent_explicit \<open>independent D\<close> by blast
     show ?thesis
       using v C0 D0 Beq by blast
   qed
   then have "independent (C \<union> (D - C))"
-    by (simp add: independent_explicit \<open>finite C\<close> \<open>finite D\<close> setsum_Un del: Un_Diff_cancel)
+    by (simp add: independent_explicit \<open>finite C\<close> \<open>finite D\<close> sum_Un del: Un_Diff_cancel)
   then have indCUD: "independent (C \<union> D)" by simp
   have "dim (S \<inter> T) = card B"
     by (rule dim_unique [OF B indB refl])
@@ -13660,8 +13660,8 @@
     then have "a \<bullet> a = a \<bullet> (\<Sum>v\<in>T. U v *\<^sub>R v)"
       by simp
     also have "... = 0"
-      apply (simp add: inner_setsum_right)
-      apply (rule comm_monoid_add_class.setsum.neutral)
+      apply (simp add: inner_sum_right)
+      apply (rule comm_monoid_add_class.sum.neutral)
       by (metis "0" DiffE \<open>T \<subseteq> S - {a}\<close> mult_not_zero singletonI subsetCE \<open>a \<in> S\<close>)
     finally show ?thesis
       using \<open>0 \<notin> S\<close> \<open>a \<in> S\<close> by auto
@@ -13707,12 +13707,12 @@
        if "x \<in> S" for x
   proof -
     have "a \<bullet> x = (\<Sum>y\<in>S. if y = x then y \<bullet> a else 0)"
-      by (simp add: \<open>finite S\<close> inner_commute setsum.delta that)
+      by (simp add: \<open>finite S\<close> inner_commute sum.delta that)
     also have "... =  (\<Sum>b\<in>S. b \<bullet> a * (b \<bullet> x) / (b \<bullet> b))"
-      apply (rule setsum.cong [OF refl], simp)
+      apply (rule sum.cong [OF refl], simp)
       by (meson S orthogonal_def pairwise_def that)
    finally show ?thesis
-     by (simp add: orthogonal_def algebra_simps inner_setsum_left)
+     by (simp add: orthogonal_def algebra_simps inner_sum_left)
   qed
   then show ?thesis
     using orthogonal_to_span orthogonal_commute x by blast
@@ -13746,7 +13746,7 @@
     using spanU by simp
   also have "... = span (insert a (S \<union> T))"
     apply (rule eq_span_insert_eq)
-    apply (simp add: a'_def span_neg span_setsum span_clauses(1) span_mul)
+    apply (simp add: a'_def span_neg span_sum span_clauses(1) span_mul)
     done
   also have "... = span (S \<union> insert a T)"
     by simp
@@ -13866,7 +13866,7 @@
     by (simp add: Gram_Schmidt_step \<open>pairwise orthogonal T\<close> \<open>span T = span S\<close> orthogonal_commute that)
   show ?thesis
     apply (rule_tac y = "?a" and z = "x - ?a" in that)
-      apply (meson \<open>T \<subseteq> span S\<close> span_mul span_setsum subsetCE)
+      apply (meson \<open>T \<subseteq> span S\<close> span_mul span_sum subsetCE)
      apply (fact orth, simp)
     done
 qed
--- a/src/HOL/Analysis/Derivative.thy	Mon Oct 17 11:07:01 2016 +0200
+++ b/src/HOL/Analysis/Derivative.thy	Mon Oct 17 14:37:32 2016 +0200
@@ -2195,13 +2195,13 @@
   fixes f :: "nat \<Rightarrow> 'a::real_normed_vector \<Rightarrow> 'b::banach"
   assumes "convex s"
     and "\<And>n x. x \<in> s \<Longrightarrow> ((f n) has_derivative (f' n x)) (at x within s)"
-    and "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm (setsum (\<lambda>i. f' i x h) {..<n} - g' x h) \<le> e * norm h"
+    and "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm (sum (\<lambda>i. f' i x h) {..<n} - g' x h) \<le> e * norm h"
     and "x \<in> s"
     and "(\<lambda>n. f n x) sums l"
   shows "\<exists>g. \<forall>x\<in>s. (\<lambda>n. f n x) sums (g x) \<and> (g has_derivative g' x) (at x within s)"
   unfolding sums_def
   apply (rule has_derivative_sequence[OF assms(1) _ assms(3)])
-  apply (metis assms(2) has_derivative_setsum)
+  apply (metis assms(2) has_derivative_sum)
   using assms(4-5)
   unfolding sums_def
   apply auto
@@ -2224,7 +2224,7 @@
     {
       fix n :: nat and x h :: 'a assume nx: "n \<ge> N" "x \<in> s"
       have "norm ((\<Sum>i<n. f' i x * h) - g' x * h) = norm ((\<Sum>i<n. f' i x) - g' x) * norm h"
-        by (simp add: norm_mult [symmetric] ring_distribs setsum_distrib_right)
+        by (simp add: norm_mult [symmetric] ring_distribs sum_distrib_right)
       also from N[OF nx] have "norm ((\<Sum>i<n. f' i x) - g' x) \<le> e" by simp
       hence "norm ((\<Sum>i<n. f' i x) - g' x) * norm h \<le> e * norm h"
         by (intro mult_right_mono) simp_all
@@ -2591,7 +2591,7 @@
     "\<forall>\<^sub>F n in sequentially. ((\<lambda>x. \<Sum>i<n. ?e i x) \<longlongrightarrow> A) ?F"
     by eventually_elim
       (auto intro!: tendsto_eq_intros
-        simp: power_0_left if_distrib cond_application_beta setsum.delta
+        simp: power_0_left if_distrib cond_application_beta sum.delta
         cong: if_cong)
   ultimately
   have [tendsto_intros]: "((\<lambda>x. \<Sum>i. ?e i x) \<longlongrightarrow> A) ?F"
--- a/src/HOL/Analysis/Determinants.thy	Mon Oct 17 11:07:01 2016 +0200
+++ b/src/HOL/Analysis/Determinants.thy	Mon Oct 17 14:37:32 2016 +0200
@@ -13,7 +13,7 @@
 subsection \<open>Trace\<close>
 
 definition trace :: "'a::semiring_1^'n^'n \<Rightarrow> 'a"
-  where "trace A = setsum (\<lambda>i. ((A$i)$i)) (UNIV::'n set)"
+  where "trace A = sum (\<lambda>i. ((A$i)$i)) (UNIV::'n set)"
 
 lemma trace_0: "trace (mat 0) = 0"
   by (simp add: trace_def mat_def)
@@ -22,14 +22,14 @@
   by (simp add: trace_def mat_def)
 
 lemma trace_add: "trace ((A::'a::comm_semiring_1^'n^'n) + B) = trace A + trace B"
-  by (simp add: trace_def setsum.distrib)
+  by (simp add: trace_def sum.distrib)
 
 lemma trace_sub: "trace ((A::'a::comm_ring_1^'n^'n) - B) = trace A - trace B"
-  by (simp add: trace_def setsum_subtractf)
+  by (simp add: trace_def sum_subtractf)
 
 lemma trace_mul_sym: "trace ((A::'a::comm_semiring_1^'n^'m) ** B) = trace (B**A)"
   apply (simp add: trace_def matrix_matrix_mult_def)
-  apply (subst setsum.commute)
+  apply (subst sum.commute)
   apply (simp add: mult.commute)
   done
 
@@ -37,7 +37,7 @@
 
 definition det:: "'a::comm_ring_1^'n^'n \<Rightarrow> 'a" where
   "det A =
-    setsum (\<lambda>p. of_int (sign p) * setprod (\<lambda>i. A$i$p i) (UNIV :: 'n set))
+    sum (\<lambda>p. of_int (sign p) * setprod (\<lambda>i. A$i$p i) (UNIV :: 'n set))
       {p. p permutes (UNIV :: 'n set)}"
 
 text \<open>A few general lemmas we need below.\<close>
@@ -94,8 +94,8 @@
   }
   then show ?thesis
     unfolding det_def
-    apply (subst setsum_permutations_inverse)
-    apply (rule setsum.cong)
+    apply (subst sum_permutations_inverse)
+    apply (rule sum.cong)
     apply (rule refl)
     apply blast
     done
@@ -128,7 +128,7 @@
   }
   then have p0: "\<forall>p \<in> ?PU - {id}. ?pp p = 0"
     by blast
-  from setsum.mono_neutral_cong_left[OF fPU id0 p0] show ?thesis
+  from sum.mono_neutral_cong_left[OF fPU id0 p0] show ?thesis
     unfolding det_def by (simp add: sign_id)
 qed
 
@@ -159,7 +159,7 @@
   }
   then have p0: "\<forall>p \<in> ?PU -{id}. ?pp p = 0"
     by blast
-  from setsum.mono_neutral_cong_left[OF fPU id0 p0] show ?thesis
+  from sum.mono_neutral_cong_left[OF fPU id0 p0] show ?thesis
     unfolding det_def by (simp add: sign_id)
 qed
 
@@ -189,7 +189,7 @@
   }
   then have p0: "\<forall>p \<in> ?PU - {id}. ?pp p = 0"
     by blast
-  from setsum.mono_neutral_cong_left[OF fPU id0 p0] show ?thesis
+  from sum.mono_neutral_cong_left[OF fPU id0 p0] show ?thesis
     unfolding det_def by (simp add: sign_id)
 qed
 
@@ -226,9 +226,9 @@
   fixes A :: "'a::comm_ring_1^'n^'n"
   assumes p: "p permutes (UNIV :: 'n::finite set)"
   shows "det (\<chi> i. A$p i :: 'a^'n^'n) = of_int (sign p) * det A"
-  apply (simp add: det_def setsum_distrib_left mult.assoc[symmetric])
+  apply (simp add: det_def sum_distrib_left mult.assoc[symmetric])
   apply (subst sum_permutations_compose_right[OF p])
-proof (rule setsum.cong)
+proof (rule sum.cong)
   let ?U = "UNIV :: 'n set"
   let ?PU = "{p. p permutes ?U}"
   fix q
@@ -303,7 +303,7 @@
   shows "det A = 0"
   using r
   apply (simp add: row_def det_def vec_eq_iff)
-  apply (rule setsum.neutral)
+  apply (rule sum.neutral)
   apply (auto simp: sign_nz)
   done
 
@@ -321,8 +321,8 @@
   shows "det((\<chi> i. if i = k then a i + b i else c i)::'a::comm_ring_1^'n^'n) =
     det((\<chi> i. if i = k then a i else c i)::'a::comm_ring_1^'n^'n) +
     det((\<chi> i. if i = k then b i else c i)::'a::comm_ring_1^'n^'n)"
-  unfolding det_def vec_lambda_beta setsum.distrib[symmetric]
-proof (rule setsum.cong)
+  unfolding det_def vec_lambda_beta sum.distrib[symmetric]
+proof (rule sum.cong)
   let ?U = "UNIV :: 'n set"
   let ?pU = "{p. p permutes ?U}"
   let ?f = "(\<lambda>i. if i = k then a i + b i else c i)::'n \<Rightarrow> 'a::comm_ring_1^'n"
@@ -372,8 +372,8 @@
   fixes a b :: "'n::finite \<Rightarrow> _ ^ 'n"
   shows "det((\<chi> i. if i = k then c *s a i else b i)::'a::comm_ring_1^'n^'n) =
     c * det((\<chi> i. if i = k then a i else b i)::'a::comm_ring_1^'n^'n)"
-  unfolding det_def vec_lambda_beta setsum_distrib_left
-proof (rule setsum.cong)
+  unfolding det_def vec_lambda_beta sum_distrib_left
+proof (rule sum.cong)
   let ?U = "UNIV :: 'n set"
   let ?pU = "{p. p permutes ?U}"
   let ?f = "(\<lambda>i. if i = k then c*s a i else b i)::'n \<Rightarrow> 'a::comm_ring_1^'n"
@@ -535,15 +535,15 @@
 lemma Cart_lambda_cong: "(\<And>x. f x = g x) \<Longrightarrow> (vec_lambda f::'a^'n) = (vec_lambda g :: 'a^'n)"
   by (rule iffD1[OF vec_lambda_unique]) vector
 
-lemma det_linear_row_setsum:
+lemma det_linear_row_sum:
   assumes fS: "finite S"
-  shows "det ((\<chi> i. if i = k then setsum (a i) S else c i)::'a::comm_ring_1^'n^'n) =
-    setsum (\<lambda>j. det ((\<chi> i. if i = k then a  i j else c i)::'a^'n^'n)) S"
+  shows "det ((\<chi> i. if i = k then sum (a i) S else c i)::'a::comm_ring_1^'n^'n) =
+    sum (\<lambda>j. det ((\<chi> i. if i = k then a  i j else c i)::'a^'n^'n)) S"
 proof (induct rule: finite_induct[OF fS])
   case 1
   then show ?case
     apply simp
-    unfolding setsum.empty det_row_0[of k]
+    unfolding sum.empty det_row_0[of k]
     apply rule
     done
 next
@@ -577,11 +577,11 @@
 qed
 
 
-lemma det_linear_rows_setsum_lemma:
+lemma det_linear_rows_sum_lemma:
   assumes fS: "finite S"
     and fT: "finite T"
-  shows "det ((\<chi> i. if i \<in> T then setsum (a i) S else c i):: 'a::comm_ring_1^'n^'n) =
-    setsum (\<lambda>f. det((\<chi> i. if i \<in> T then a i (f i) else c i)::'a^'n^'n))
+  shows "det ((\<chi> i. if i \<in> T then sum (a i) S else c i):: 'a::comm_ring_1^'n^'n) =
+    sum (\<lambda>f. det((\<chi> i. if i \<in> T then a i (f i) else c i)::'a^'n^'n))
       {f. (\<forall>i \<in> T. f i \<in> S) \<and> (\<forall>i. i \<notin> T \<longrightarrow> f i = i)}"
   using fT
 proof (induct T arbitrary: a c set: finite)
@@ -604,48 +604,48 @@
     by simp
   from \<open>z \<notin> T\<close> have nz: "\<And>i. i \<in> T \<Longrightarrow> i = z \<longleftrightarrow> False"
     by auto
-  have "det (\<chi> i. if i \<in> insert z T then setsum (a i) S else c i) =
-    det (\<chi> i. if i = z then setsum (a i) S else if i \<in> T then setsum (a i) S else c i)"
+  have "det (\<chi> i. if i \<in> insert z T then sum (a i) S else c i) =
+    det (\<chi> i. if i = z then sum (a i) S else if i \<in> T then sum (a i) S else c i)"
     unfolding insert_iff thif ..
-  also have "\<dots> = (\<Sum>j\<in>S. det (\<chi> i. if i \<in> T then setsum (a i) S else if i = z then a i j else c i))"
-    unfolding det_linear_row_setsum[OF fS]
+  also have "\<dots> = (\<Sum>j\<in>S. det (\<chi> i. if i \<in> T then sum (a i) S else if i = z then a i j else c i))"
+    unfolding det_linear_row_sum[OF fS]
     apply (subst thif2)
     using nz
     apply (simp cong del: if_weak_cong cong add: if_cong)
     done
   finally have tha:
-    "det (\<chi> i. if i \<in> insert z T then setsum (a i) S else c i) =
+    "det (\<chi> i. if i \<in> insert z T then sum (a i) S else c i) =
      (\<Sum>(j, f)\<in>S \<times> ?F T. det (\<chi> i. if i \<in> T then a i (f i)
                                 else if i = z then a i j
                                 else c i))"
-    unfolding insert.hyps unfolding setsum.cartesian_product by blast
+    unfolding insert.hyps unfolding sum.cartesian_product by blast
   show ?case unfolding tha
     using \<open>z \<notin> T\<close>
-    by (intro setsum.reindex_bij_witness[where i="?k" and j="?h"])
+    by (intro sum.reindex_bij_witness[where i="?k" and j="?h"])
        (auto intro!: cong[OF refl[of det]] simp: vec_eq_iff)
 qed
 
-lemma det_linear_rows_setsum:
+lemma det_linear_rows_sum:
   fixes S :: "'n::finite set"
   assumes fS: "finite S"
-  shows "det (\<chi> i. setsum (a i) S) =
-    setsum (\<lambda>f. det (\<chi> i. a i (f i) :: 'a::comm_ring_1 ^ 'n^'n)) {f. \<forall>i. f i \<in> S}"
+  shows "det (\<chi> i. sum (a i) S) =
+    sum (\<lambda>f. det (\<chi> i. a i (f i) :: 'a::comm_ring_1 ^ 'n^'n)) {f. \<forall>i. f i \<in> S}"
 proof -
   have th0: "\<And>x y. ((\<chi> i. if i \<in> (UNIV:: 'n set) then x i else y i) :: 'a^'n^'n) = (\<chi> i. x i)"
     by vector
-  from det_linear_rows_setsum_lemma[OF fS, of "UNIV :: 'n set" a, unfolded th0, OF finite]
+  from det_linear_rows_sum_lemma[OF fS, of "UNIV :: 'n set" a, unfolded th0, OF finite]
   show ?thesis by simp
 qed
 
-lemma matrix_mul_setsum_alt:
+lemma matrix_mul_sum_alt:
   fixes A B :: "'a::comm_ring_1^'n^'n"
-  shows "A ** B = (\<chi> i. setsum (\<lambda>k. A$i$k *s B $ k) (UNIV :: 'n set))"
-  by (vector matrix_matrix_mult_def setsum_component)
+  shows "A ** B = (\<chi> i. sum (\<lambda>k. A$i$k *s B $ k) (UNIV :: 'n set))"
+  by (vector matrix_matrix_mult_def sum_component)
 
 lemma det_rows_mul:
   "det((\<chi> i. c i *s a i)::'a::comm_ring_1^'n^'n) =
     setprod (\<lambda>i. c i) (UNIV:: 'n set) * det((\<chi> i. a i)::'a^'n^'n)"
-proof (simp add: det_def setsum_distrib_left cong add: setprod.cong, rule setsum.cong)
+proof (simp add: det_def sum_distrib_left cong add: setprod.cong, rule sum.cong)
   let ?U = "UNIV :: 'n set"
   let ?PU = "{p. p permutes ?U}"
   fix p
@@ -735,11 +735,11 @@
       by blast
     let ?s = "\<lambda>p. of_int (sign p)"
     let ?f = "\<lambda>q. ?s p * (\<Prod>i\<in> ?U. A $ i $ p i) * (?s q * (\<Prod>i\<in> ?U. B $ i $ q i))"
-    have "(setsum (\<lambda>q. ?s q *
+    have "(sum (\<lambda>q. ?s q *
         (\<Prod>i\<in> ?U. (\<chi> i. A $ i $ p i *s B $ p i :: 'a^'n^'n) $ i $ q i)) ?PU) =
-      (setsum (\<lambda>q. ?s p * (\<Prod>i\<in> ?U. A $ i $ p i) * (?s q * (\<Prod>i\<in> ?U. B $ i $ q i))) ?PU)"
+      (sum (\<lambda>q. ?s p * (\<Prod>i\<in> ?U. A $ i $ p i) * (?s q * (\<Prod>i\<in> ?U. B $ i $ q i))) ?PU)"
       unfolding sum_permutations_compose_right[OF permutes_inv[OF p], of ?f]
-    proof (rule setsum.cong)
+    proof (rule sum.cong)
       fix q
       assume qU: "q \<in> ?PU"
       then have q: "q permutes ?U"
@@ -769,14 +769,14 @@
         by (simp add: sign_nz th00 field_simps sign_idempotent sign_compose)
     qed rule
   }
-  then have th2: "setsum (\<lambda>f. det (\<chi> i. A$i$f i *s B$f i)) ?PU = det A * det B"
-    unfolding det_def setsum_product
-    by (rule setsum.cong [OF refl])
-  have "det (A**B) = setsum (\<lambda>f.  det (\<chi> i. A $ i $ f i *s B $ f i)) ?F"
-    unfolding matrix_mul_setsum_alt det_linear_rows_setsum[OF fU]
+  then have th2: "sum (\<lambda>f. det (\<chi> i. A$i$f i *s B$f i)) ?PU = det A * det B"
+    unfolding det_def sum_product
+    by (rule sum.cong [OF refl])
+  have "det (A**B) = sum (\<lambda>f.  det (\<chi> i. A $ i $ f i *s B $ f i)) ?F"
+    unfolding matrix_mul_sum_alt det_linear_rows_sum[OF fU]
     by simp
-  also have "\<dots> = setsum (\<lambda>f. det (\<chi> i. A$i$f i *s B$f i)) ?PU"
-    using setsum.mono_neutral_cong_left[OF fF PUF zth, symmetric]
+  also have "\<dots> = sum (\<lambda>f. det (\<chi> i. A$i$f i *s B$f i)) ?PU"
+    using sum.mono_neutral_cong_left[OF fF PUF zth, symmetric]
     unfolding det_rows_mul by auto
   finally show ?thesis unfolding th2 .
 qed
@@ -812,7 +812,7 @@
     let ?U = "UNIV :: 'n set"
     have fU: "finite ?U"
       by simp
-    from H obtain c i where c: "setsum (\<lambda>i. c i *s row i A) ?U = 0"
+    from H obtain c i where c: "sum (\<lambda>i. c i *s row i A) ?U = 0"
       and iU: "i \<in> ?U"
       and ci: "c i \<noteq> 0"
       unfolding invertible_righ_inverse
@@ -824,8 +824,8 @@
       apply simp
       done
     from c ci
-    have thr0: "- row i A = setsum (\<lambda>j. (1/ c i) *s (c j *s row j A)) (?U - {i})"
-      unfolding setsum.remove[OF fU iU] setsum_cmul
+    have thr0: "- row i A = sum (\<lambda>j. (1/ c i) *s (c j *s row j A)) (?U - {i})"
+      unfolding sum.remove[OF fU iU] sum_cmul
       apply -
       apply (rule vector_mul_lcancel_imp[OF ci])
       apply (auto simp add: field_simps)
@@ -834,7 +834,7 @@
       done
     have thr: "- row i A \<in> span {row j A| j. j \<noteq> i}"
       unfolding thr0
-      apply (rule span_setsum)
+      apply (rule span_sum)
       apply simp
       apply (rule span_mul [where 'a="real^'n", folded scalar_mult_eq_scaleR])+
       apply (rule span_superset)
@@ -855,7 +855,7 @@
 lemma cramer_lemma_transpose:
   fixes A:: "real^'n^'n"
     and x :: "real^'n"
-  shows "det ((\<chi> i. if i = k then setsum (\<lambda>i. x$i *s row i A) (UNIV::'n set)
+  shows "det ((\<chi> i. if i = k then sum (\<lambda>i. x$i *s row i A) (UNIV::'n set)
                              else row i A)::real^'n^'n) = x$k * det A"
   (is "?lhs = ?rhs")
 proof -
@@ -876,14 +876,14 @@
     by simp
   have thd0: "det (\<chi> i. if i = k then row k A + (\<Sum>i \<in> ?Uk. x $ i *s row i A) else row i A) = det A"
     apply (rule det_row_span)
-    apply (rule span_setsum)
+    apply (rule span_sum)
     apply (rule span_mul [where 'a="real^'n", folded scalar_mult_eq_scaleR])+
     apply (rule span_superset)
     apply auto
     done
   show "?lhs = x$k * det A"
     apply (subst U)
-    unfolding setsum.insert[OF fUk kUk]
+    unfolding sum.insert[OF fUk kUk]
     apply (subst th00)
     unfolding add.assoc
     apply (subst det_row_add)
@@ -900,8 +900,8 @@
   shows "det((\<chi> i j. if j = k then (A *v x)$i else A$i$j):: real^'n^'n) = x$k * det A"
 proof -
   let ?U = "UNIV :: 'n set"
-  have *: "\<And>c. setsum (\<lambda>i. c i *s row i (transpose A)) ?U = setsum (\<lambda>i. c i *s column i A) ?U"
-    by (auto simp add: row_transpose intro: setsum.cong)
+  have *: "\<And>c. sum (\<lambda>i. c i *s row i (transpose A)) ?U = sum (\<lambda>i. c i *s column i A) ?U"
+    by (auto simp add: row_transpose intro: sum.cong)
   show ?thesis
     unfolding matrix_mult_vsum
     unfolding cramer_lemma_transpose[of k x "transpose A", unfolded det_transpose, symmetric]
@@ -995,7 +995,7 @@
         simplified matrix_works[OF lf, symmetric] dot_matrix_vector_mul]
       have "?A$i$j = ?m1 $ i $ j"
         by (simp add: inner_vec_def matrix_matrix_mult_def columnvector_def rowvector_def
-            th0 setsum.delta[OF fU] mat_def axis_def)
+            th0 sum.delta[OF fU] mat_def axis_def)
     }
     then have "orthogonal_matrix ?mf"
       unfolding orthogonal_matrix
@@ -1219,7 +1219,7 @@
   have f12: "finite {2::2}" "1 \<notin> {2::2}" by auto
   show ?thesis
     unfolding det_def UNIV_2
-    unfolding setsum_over_permutations_insert[OF f12]
+    unfolding sum_over_permutations_insert[OF f12]
     unfolding permutes_sing
     by (simp add: sign_swap_id sign_id swap_id_eq)
 qed
@@ -1240,8 +1240,8 @@
 
   show ?thesis
     unfolding det_def UNIV_3
-    unfolding setsum_over_permutations_insert[OF f123]
-    unfolding setsum_over_permutations_insert[OF f23]
+    unfolding sum_over_permutations_insert[OF f123]
+    unfolding sum_over_permutations_insert[OF f23]
     unfolding permutes_sing
     by (simp add: sign_swap_id permutation_swap_id sign_compose sign_id swap_id_eq)
 qed
--- a/src/HOL/Analysis/Equivalence_Lebesgue_Henstock_Integration.thy	Mon Oct 17 11:07:01 2016 +0200
+++ b/src/HOL/Analysis/Equivalence_Lebesgue_Henstock_Integration.thy	Mon Oct 17 14:37:32 2016 +0200
@@ -182,15 +182,15 @@
       have "(\<Sum>(x, k)\<in>?p. content k *\<^sub>R f x) =
         (\<Sum>(x, k)\<in>?T ` (p \<inter> s). content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>p - s. content k *\<^sub>R f x)"
         using p(1)[THEN tagged_division_ofD(1)]
-        by (safe intro!: setsum.union_inter_neutral) (auto simp: s_def T_def)
+        by (safe intro!: sum.union_inter_neutral) (auto simp: s_def T_def)
       also have "(\<Sum>(x, k)\<in>?T ` (p \<inter> s). content k *\<^sub>R f x) = (\<Sum>(x, k)\<in>p \<inter> s. content k *\<^sub>R f (T X k))"
-      proof (subst setsum.reindex_nontrivial, safe)
+      proof (subst sum.reindex_nontrivial, safe)
         fix x1 x2 k assume 1: "(x1, k) \<in> p" "(x1, k) \<in> s" and 2: "(x2, k) \<in> p" "(x2, k) \<in> s"
           and eq: "content k *\<^sub>R f (T X k) \<noteq> 0"
         with tagged_division_ofD(5)[OF p(1), of x1 k x2 k] tagged_division_ofD(4)[OF p(1), of x1 k]
         show "x1 = x2"
           by (auto simp: content_eq_0_interior)
-      qed (use p in \<open>auto intro!: setsum.cong\<close>)
+      qed (use p in \<open>auto intro!: sum.cong\<close>)
       finally have eq: "(\<Sum>(x, k)\<in>?p. content k *\<^sub>R f x) =
         (\<Sum>(x, k)\<in>p \<inter> s. content k *\<^sub>R f (T X k)) + (\<Sum>(x, k)\<in>p - s. content k *\<^sub>R f x)" .
 
@@ -270,11 +270,11 @@
         using \<open>0 < e\<close> by (simp add: split_beta)
     qed (use \<open>a < b\<close> in auto)
     also have "\<dots> = (\<Sum>(x, k)\<in>p \<inter> s. content k * (b - a))"
-      by (simp add: setsum_distrib_right split_beta')
+      by (simp add: sum_distrib_right split_beta')
     also have "\<dots> \<le> (\<Sum>(x, k)\<in>p \<inter> s. content k * (f (T ?F k) - f (T ?E k)))"
-      using parts(3) by (auto intro!: setsum_mono mult_left_mono diff_mono)
+      using parts(3) by (auto intro!: sum_mono mult_left_mono diff_mono)
     also have "\<dots> = (\<Sum>(x, k)\<in>p \<inter> s. content k * f (T ?F k)) - (\<Sum>(x, k)\<in>p \<inter> s. content k * f (T ?E k))"
-      by (auto intro!: setsum.cong simp: field_simps setsum_subtractf[symmetric])
+      by (auto intro!: sum.cong simp: field_simps sum_subtractf[symmetric])
     also have "\<dots> = (\<Sum>(x, k)\<in>?B. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>?A. content k *\<^sub>R f x)"
       by (subst (1 2) parts) auto
     also have "\<dots> \<le> norm ((\<Sum>(x, k)\<in>?B. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>?A. content k *\<^sub>R f x))"
@@ -401,13 +401,13 @@
       proof (rule has_integral_monotone_convergence_increasing)
         let ?f = "\<lambda>k x. \<Sum>i<k. if x \<in> F i \<inter> box a b then 1 else 0 :: real"
         show "\<And>k. (?f k has_integral (\<Sum>i<k. ?M (F i))) (box a b)"
-          using union.IH by (auto intro!: has_integral_setsum simp del: Int_iff)
+          using union.IH by (auto intro!: has_integral_sum simp del: Int_iff)
         show "\<And>k x. ?f k x \<le> ?f (Suc k) x"
-          by (intro setsum_mono2) auto
+          by (intro sum_mono2) auto
         from union(1) have *: "\<And>x i j. x \<in> F i \<Longrightarrow> x \<in> F j \<longleftrightarrow> j = i"
           by (auto simp add: disjoint_family_on_def)
         show "\<And>x. (\<lambda>k. ?f k x) \<longlonglongrightarrow> (if x \<in> UNION UNIV F \<inter> box a b then 1 else 0)"
-          apply (auto simp: * setsum.If_cases Iio_Int_singleton)
+          apply (auto simp: * sum.If_cases Iio_Int_singleton)
           apply (rule_tac k="Suc xa" in LIMSEQ_offset)
           apply simp
           done
@@ -757,7 +757,7 @@
   shows "(f has_integral (integral\<^sup>L lborel f)) UNIV"
 proof -
   have "((\<lambda>x. \<Sum>b\<in>Basis. (f x \<bullet> b) *\<^sub>R b) has_integral (\<Sum>b\<in>Basis. integral\<^sup>L lborel (\<lambda>x. f x \<bullet> b) *\<^sub>R b)) UNIV"
-    using f by (intro has_integral_setsum finite_Basis ballI has_integral_scaleR_left has_integral_integral_real) auto
+    using f by (intro has_integral_sum finite_Basis ballI has_integral_scaleR_left has_integral_integral_real) auto
   also have eq_f: "(\<lambda>x. \<Sum>b\<in>Basis. (f x \<bullet> b) *\<^sub>R b) = f"
     by (simp add: fun_eq_iff euclidean_representation)
   also have "(\<Sum>b\<in>Basis. integral\<^sup>L lborel (\<lambda>x. f x \<bullet> b) *\<^sub>R b) = integral\<^sup>L lborel f"
@@ -804,7 +804,7 @@
   shows "(f has_integral (integral\<^sup>L lebesgue f)) UNIV"
 proof -
   have "((\<lambda>x. \<Sum>b\<in>Basis. (f x \<bullet> b) *\<^sub>R b) has_integral (\<Sum>b\<in>Basis. integral\<^sup>L lebesgue (\<lambda>x. f x \<bullet> b) *\<^sub>R b)) UNIV"
-    using f by (intro has_integral_setsum finite_Basis ballI has_integral_scaleR_left has_integral_integral_lebesgue_real) auto
+    using f by (intro has_integral_sum finite_Basis ballI has_integral_scaleR_left has_integral_integral_lebesgue_real) auto
   also have eq_f: "(\<lambda>x. \<Sum>b\<in>Basis. (f x \<bullet> b) *\<^sub>R b) = f"
     by (simp add: fun_eq_iff euclidean_representation)
   also have "(\<Sum>b\<in>Basis. integral\<^sup>L lebesgue (\<lambda>x. f x \<bullet> b) *\<^sub>R b) = integral\<^sup>L lebesgue f"
@@ -1245,10 +1245,10 @@
       using \<open>0 < B\<close> \<open>\<D>' \<subseteq> \<D>\<close> that vu_pos by fastforce
     have "{} \<notin> \<D>'"
       using cbox \<open>\<D>' \<subseteq> \<D>\<close> interior_empty by blast
-    have "(\<Sum>k\<in>fbx`\<D>'. measure lebesgue k) \<le> setsum (measure lebesgue o fbx) \<D>'"
-      by (rule setsum_image_le [OF \<open>finite \<D>'\<close>]) (force simp: fbx_def)
+    have "(\<Sum>k\<in>fbx`\<D>'. measure lebesgue k) \<le> sum (measure lebesgue o fbx) \<D>'"
+      by (rule sum_image_le [OF \<open>finite \<D>'\<close>]) (force simp: fbx_def)
     also have "\<dots> \<le> (\<Sum>X\<in>\<D>'. (2 * B * DIM('M)) ^ DIM('N) * measure lebesgue X)"
-    proof (rule setsum_mono)
+    proof (rule sum_mono)
       fix X assume "X \<in> \<D>'"
       then have "X \<in> \<D>" using \<open>\<D>' \<subseteq> \<D>\<close> by blast
       then have ufvf: "cbox (uf X) (vf X) = X"
@@ -1281,8 +1281,8 @@
         by (subst (3) ufvf[symmetric]) simp
       finally show "(measure lebesgue \<circ> fbx) X \<le> (2 * B * DIM('M)) ^ DIM('N) * measure lebesgue X" .
     qed
-    also have "\<dots> = (2 * B * DIM('M)) ^ DIM('N) * setsum (measure lebesgue) \<D>'"
-      by (simp add: setsum_distrib_left)
+    also have "\<dots> = (2 * B * DIM('M)) ^ DIM('N) * sum (measure lebesgue) \<D>'"
+      by (simp add: sum_distrib_left)
     also have "\<dots> \<le> e / 2"
     proof -
       have div: "\<D>' division_of \<Union>\<D>'"
@@ -1306,9 +1306,9 @@
         qed
         finally show "\<Union>\<D>' \<subseteq> T" .
       qed
-      have "setsum (measure lebesgue) \<D>' = setsum content \<D>'"
-        using  \<open>\<D>' \<subseteq> \<D>\<close> cbox by (force intro: setsum.cong)
-      then have "(2 * B * DIM('M)) ^ DIM('N) * setsum (measure lebesgue) \<D>' =
+      have "sum (measure lebesgue) \<D>' = sum content \<D>'"
+        using  \<open>\<D>' \<subseteq> \<D>\<close> cbox by (force intro: sum.cong)
+      then have "(2 * B * DIM('M)) ^ DIM('N) * sum (measure lebesgue) \<D>' =
                  (2 * B * real DIM('M)) ^ DIM('N) * measure lebesgue (\<Union>\<D>')"
         using content_division [OF div] by auto
       also have "\<dots> \<le> (2 * B * real DIM('M)) ^ DIM('N) * measure lebesgue T"
@@ -1322,7 +1322,7 @@
     qed
     finally show ?thesis .
   qed
-  then have e2: "setsum (measure lebesgue) \<G> \<le> e / 2" if "\<G> \<subseteq> fbx ` \<D>" "finite \<G>" for \<G>
+  then have e2: "sum (measure lebesgue) \<G> \<le> e / 2" if "\<G> \<subseteq> fbx ` \<D>" "finite \<G>" for \<G>
     by (metis finite_subset_image that)
   show "\<exists>W\<in>lmeasurable. f ` S \<subseteq> W \<and> measure lebesgue W < e"
   proof (intro bexI conjI)
@@ -1339,7 +1339,7 @@
       have "norm (y - zf X) \<le> (\<Sum>i\<in>Basis. \<bar>(y - zf X) \<bullet> i\<bar>)"
         by (rule norm_le_l1)
       also have "\<dots> \<le> real DIM('M) * prj1 (vf X - uf X)"
-      proof (rule setsum_bounded_above)
+      proof (rule sum_bounded_above)
         fix j::'M assume j: "j \<in> Basis"
         show "\<bar>(y - zf X) \<bullet> j\<bar> \<le> prj1 (vf X - uf X)"
           using yin zin j
@@ -1533,7 +1533,7 @@
 lemma absolutely_integrable_bounded_variation:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes f: "f absolutely_integrable_on UNIV"
-  obtains B where "\<forall>d. d division_of (\<Union>d) \<longrightarrow> setsum (\<lambda>k. norm (integral k f)) d \<le> B"
+  obtains B where "\<forall>d. d division_of (\<Union>d) \<longrightarrow> sum (\<lambda>k. norm (integral k f)) d \<le> B"
 proof (rule that[of "integral UNIV (\<lambda>x. norm (f x))"]; safe)
   fix d :: "'a set set" assume d: "d division_of \<Union>d"
   have *: "k \<in> d \<Longrightarrow> f absolutely_integrable_on k" for k
@@ -1541,11 +1541,11 @@
   note d' = division_ofD[OF d]
 
   have "(\<Sum>k\<in>d. norm (integral k f)) = (\<Sum>k\<in>d. norm (LINT x:k|lebesgue. f x))"
-    by (intro setsum.cong refl arg_cong[where f=norm] set_lebesgue_integral_eq_integral(2)[symmetric] *)
+    by (intro sum.cong refl arg_cong[where f=norm] set_lebesgue_integral_eq_integral(2)[symmetric] *)
   also have "\<dots> \<le> (\<Sum>k\<in>d. LINT x:k|lebesgue. norm (f x))"
-    by (intro setsum_mono set_integral_norm_bound *)
+    by (intro sum_mono set_integral_norm_bound *)
   also have "\<dots> = (\<Sum>k\<in>d. integral k (\<lambda>x. norm (f x)))"
-    by (intro setsum.cong refl set_lebesgue_integral_eq_integral(2) set_integrable_norm *)
+    by (intro sum.cong refl set_lebesgue_integral_eq_integral(2) set_integrable_norm *)
   also have "\<dots> \<le> integral (\<Union>d) (\<lambda>x. norm (f x))"
     using integrable_on_subdivision[OF d] assms f unfolding absolutely_integrable_on_def
     by (subst integral_combine_division_topdown[OF _ d]) auto
@@ -1556,20 +1556,20 @@
 qed
 
 lemma helplemma:
-  assumes "setsum (\<lambda>x. norm (f x - g x)) s < e"
+  assumes "sum (\<lambda>x. norm (f x - g x)) s < e"
     and "finite s"
-  shows "\<bar>setsum (\<lambda>x. norm(f x)) s - setsum (\<lambda>x. norm(g x)) s\<bar> < e"
-  unfolding setsum_subtractf[symmetric]
-  apply (rule le_less_trans[OF setsum_abs])
+  shows "\<bar>sum (\<lambda>x. norm(f x)) s - sum (\<lambda>x. norm(g x)) s\<bar> < e"
+  unfolding sum_subtractf[symmetric]
+  apply (rule le_less_trans[OF sum_abs])
   apply (rule le_less_trans[OF _ assms(1)])
-  apply (rule setsum_mono)
+  apply (rule sum_mono)
   apply (rule norm_triangle_ineq3)
   done
 
 lemma bounded_variation_absolutely_integrable_interval:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
   assumes f: "f integrable_on cbox a b"
-    and *: "\<forall>d. d division_of (cbox a b) \<longrightarrow> setsum (\<lambda>k. norm(integral k f)) d \<le> B"
+    and *: "\<forall>d. d division_of (cbox a b) \<longrightarrow> sum (\<lambda>k. norm(integral k f)) d \<le> B"
   shows "f absolutely_integrable_on cbox a b"
 proof -
   let ?f = "\<lambda>d. \<Sum>k\<in>d. norm (integral k f)" and ?D = "{d. d division_of (cbox a b)}"
@@ -1749,7 +1749,7 @@
           done
       qed
       have sum_p': "(\<Sum>(x, k)\<in>p'. norm (integral k f)) = (\<Sum>k\<in>snd ` p'. norm (integral k f))"
-        apply (subst setsum.over_tagged_division_lemma[OF p'',of "\<lambda>k. norm (integral k f)"])
+        apply (subst sum.over_tagged_division_lemma[OF p'',of "\<lambda>k. norm (integral k f)"])
         unfolding norm_eq_zero
          apply (rule integral_null)
         apply (simp add: content_eq_0_interior)
@@ -1775,7 +1775,7 @@
           (\<lambda>(k,l). k \<inter> l) ` {(k,l)|k l. k \<in> d \<and> l \<in> snd ` p}"
           by auto
         have "(\<Sum>k\<in>d. norm (integral k f)) \<le> (\<Sum>i\<in>d. \<Sum>l\<in>snd ` p. norm (integral (i \<inter> l) f))"
-        proof (rule setsum_mono, goal_cases)
+        proof (rule sum_mono, goal_cases)
           case k: (1 k)
           from d'(4)[OF this] guess u v by (elim exE) note uv=this
           define d' where "d' = {cbox u v \<inter> l |l. l \<in> snd ` p \<and>  cbox u v \<inter> l \<noteq> {}}"
@@ -1786,16 +1786,16 @@
             apply (rule division_of_tagged_division[OF p(1)])
             apply (rule uvab)
             done
-          then have "norm (integral k f) \<le> setsum (\<lambda>k. norm (integral k f)) d'"
+          then have "norm (integral k f) \<le> sum (\<lambda>k. norm (integral k f)) d'"
             unfolding uv
             apply (subst integral_combine_division_topdown[of _ _ d'])
             apply (rule integrable_on_subcbox[OF assms(1) uvab])
             apply assumption
-            apply (rule setsum_norm_le)
+            apply (rule sum_norm_le)
             apply auto
             done
           also have "\<dots> = (\<Sum>k\<in>{k \<inter> l |l. l \<in> snd ` p}. norm (integral k f))"
-            apply (rule setsum.mono_neutral_left)
+            apply (rule sum.mono_neutral_left)
             apply (subst Setcompr_eq_image)
             apply (rule finite_imageI)+
             apply fact
@@ -1813,7 +1813,7 @@
           qed
           also have "\<dots> = (\<Sum>l\<in>snd ` p. norm (integral (k \<inter> l) f))"
             unfolding Setcompr_eq_image
-            apply (rule setsum.reindex_nontrivial [unfolded o_def])
+            apply (rule sum.reindex_nontrivial [unfolded o_def])
             apply (rule finite_imageI)
             apply (rule p')
           proof goal_cases
@@ -1845,7 +1845,7 @@
           unfolding split_def ..
         also have "\<dots> = (\<Sum>k\<in>{i \<inter> l |i l. i \<in> d \<and> l \<in> snd ` p}. norm (integral k f))"
           unfolding *
-          apply (rule setsum.reindex_nontrivial [symmetric, unfolded o_def])
+          apply (rule sum.reindex_nontrivial [symmetric, unfolded o_def])
           apply (rule finite_product_dependent)
           apply fact
           apply (rule finite_imageI)
@@ -1886,7 +1886,7 @@
         qed
         also have "\<dots> = (\<Sum>(x, k)\<in>p'. norm (integral k f))"
           unfolding sum_p'
-          apply (rule setsum.mono_neutral_right)
+          apply (rule sum.mono_neutral_right)
           apply (subst *)
           apply (rule finite_imageI[OF finite_product_dependent])
           apply fact
@@ -1929,7 +1929,7 @@
         note pdfin = finite_cartesian_product[OF p'(1) d'(1)]
         have "(\<Sum>(x, k)\<in>p'. norm (content k *\<^sub>R f x)) = (\<Sum>(x, k)\<in>?S. \<bar>content k\<bar> * norm (f x))"
           unfolding norm_scaleR
-          apply (rule setsum.mono_neutral_left)
+          apply (rule sum.mono_neutral_left)
           apply (subst *)
           apply (rule finite_imageI)
           apply fact
@@ -1943,7 +1943,7 @@
           done
         also have "\<dots> = (\<Sum>((x,l),i)\<in>p \<times> d. \<bar>content (l \<inter> i)\<bar> * norm (f x))"
           unfolding *
-          apply (subst setsum.reindex_nontrivial)
+          apply (subst sum.reindex_nontrivial)
           apply fact
           unfolding split_paired_all
           unfolding o_def split_def snd_conv fst_conv mem_Sigma_iff prod.inject
@@ -1981,7 +1981,7 @@
           apply (rule p')
           apply rule
           apply (rule d')
-          apply (rule setsum.cong)
+          apply (rule sum.cong)
           apply (rule refl)
           unfolding split_paired_all split_conv
         proof -
@@ -1990,7 +1990,7 @@
           note xl = p'(2-4)[OF this]
           from this(3) guess u v by (elim exE) note uv=this
           have "(\<Sum>i\<in>d. \<bar>content (l \<inter> i)\<bar>) = (\<Sum>k\<in>d. content (k \<inter> cbox u v))"
-            apply (rule setsum.cong)
+            apply (rule sum.cong)
             apply (rule refl)
             apply (drule d'(4))
             apply safe
@@ -1999,9 +1999,9 @@
             apply (subst abs_of_nonneg)
             apply auto
             done
-          also have "\<dots> = setsum content {k \<inter> cbox u v| k. k \<in> d}"
+          also have "\<dots> = sum content {k \<inter> cbox u v| k. k \<in> d}"
             unfolding Setcompr_eq_image
-            apply (rule setsum.reindex_nontrivial [unfolded o_def, symmetric])
+            apply (rule sum.reindex_nontrivial [unfolded o_def, symmetric])
             apply (rule d')
           proof goal_cases
             case prems: (1 k y)
@@ -2019,8 +2019,8 @@
             finally show ?case
               unfolding uv Int_interval content_eq_0_interior ..
           qed
-          also have "\<dots> = setsum content {cbox u v \<inter> k |k. k \<in> d \<and> cbox u v \<inter> k \<noteq> {}}"
-            apply (rule setsum.mono_neutral_right)
+          also have "\<dots> = sum content {cbox u v \<inter> k |k. k \<in> d \<and> cbox u v \<inter> k \<noteq> {}}"
+            apply (rule sum.mono_neutral_right)
             unfolding Setcompr_eq_image
             apply (rule finite_imageI)
             apply (rule d')
@@ -2040,7 +2040,7 @@
               by auto
           qed
           finally show "(\<Sum>i\<in>d. \<bar>content (l \<inter> i)\<bar> * norm (f x)) = content l *\<^sub>R norm (f x)"
-            unfolding setsum_distrib_right[symmetric] real_scaleR_def
+            unfolding sum_distrib_right[symmetric] real_scaleR_def
             apply (subst(asm) additive_content_division[OF division_inter_1[OF d(1)]])
             using xl(2)[unfolded uv]
             unfolding uv
@@ -2056,7 +2056,7 @@
 lemma bounded_variation_absolutely_integrable:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
   assumes "f integrable_on UNIV"
-    and "\<forall>d. d division_of (\<Union>d) \<longrightarrow> setsum (\<lambda>k. norm (integral k f)) d \<le> B"
+    and "\<forall>d. d division_of (\<Union>d) \<longrightarrow> sum (\<lambda>k. norm (integral k f)) d \<le> B"
   shows "f absolutely_integrable_on UNIV"
 proof (rule absolutely_integrable_onI, fact, rule)
   let ?f = "\<lambda>d. \<Sum>k\<in>d. norm (integral k f)" and ?D = "{d. d division_of  (\<Union>d)}"
@@ -2083,7 +2083,7 @@
       using f_int[of a b] unfolding absolutely_integrable_on_def by auto
   next
     case prems: (2 e)
-    have "\<exists>y\<in>setsum (\<lambda>k. norm (integral k f)) ` {d. d division_of \<Union>d}. \<not> y \<le> ?S - e"
+    have "\<exists>y\<in>sum (\<lambda>k. norm (integral k f)) ` {d. d division_of \<Union>d}. \<not> y \<le> ?S - e"
     proof (rule ccontr)
       assume "\<not> ?thesis"
       then have "?S \<le> ?S - e"
@@ -2092,7 +2092,7 @@
         using prems by auto
     qed
     then obtain K where *: "\<exists>x\<in>{d. d division_of \<Union>d}. K = (\<Sum>k\<in>x. norm (integral k f))"
-      "SUPREMUM {d. d division_of \<Union>d} (setsum (\<lambda>k. norm (integral k f))) - e < K"
+      "SUPREMUM {d. d division_of \<Union>d} (sum (\<lambda>k. norm (integral k f))) - e < K"
       by (auto simp add: image_iff not_le)
     from this(1) obtain d where "d division_of \<Union>d"
       and "K = (\<Sum>k\<in>d. norm (integral k f))"
@@ -2118,8 +2118,8 @@
         apply (rule d(2))
       proof goal_cases
         case 1
-        have "(\<Sum>k\<in>d. norm (integral k f)) \<le> setsum (\<lambda>k. integral k (\<lambda>x. norm (f x))) d"
-          apply (intro setsum_mono)
+        have "(\<Sum>k\<in>d. norm (integral k f)) \<le> sum (\<lambda>k. integral k (\<lambda>x. norm (f x))) d"
+          apply (intro sum_mono)
           subgoal for k
             using d'(4)[of k] f_int
             by (auto simp: absolutely_integrable_on_def integral_norm_bound_integral)
@@ -2188,7 +2188,7 @@
             using d1(2)[rule_format,OF conjI[OF p(1,2)]]
             by (simp only: real_norm_def)
           show "(\<Sum>(x, k)\<in>p. content k *\<^sub>R norm (f x)) = (\<Sum>(x, k)\<in>p. norm (content k *\<^sub>R f x))"
-            apply (rule setsum.cong)
+            apply (rule sum.cong)
             apply (rule refl)
             unfolding split_paired_all split_conv
             apply (drule tagged_division_ofD(4)[OF p(1)])
@@ -2198,7 +2198,7 @@
             done
           show "(\<Sum>(x, k)\<in>p. norm (integral k f)) \<le> ?S"
             using partial_division_of_tagged_division[of p "cbox a b"] p(1)
-            apply (subst setsum.over_tagged_division_lemma[OF p(1)])
+            apply (subst sum.over_tagged_division_lemma[OF p(1)])
             apply (simp add: content_eq_0_interior)
             apply (intro cSUP_upper2[OF D(2), of "snd ` p"])
             apply (auto simp: tagged_partial_division_of_def)
@@ -2230,10 +2230,10 @@
   using integrable_bounded_linear[of h lebesgue "\<lambda>x. indicator s x *\<^sub>R f x"]
   by (simp add: linear_simps[of h])
 
-lemma absolutely_integrable_setsum:
+lemma absolutely_integrable_sum:
   fixes f :: "'a \<Rightarrow> 'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
   assumes "finite t" and "\<And>a. a \<in> t \<Longrightarrow> (f a) absolutely_integrable_on s"
-  shows "(\<lambda>x. setsum (\<lambda>a. f a x) t) absolutely_integrable_on s"
+  shows "(\<lambda>x. sum (\<lambda>a. f a x) t) absolutely_integrable_on s"
   using assms(1,2) by induct auto
 
 lemma absolutely_integrable_integrable_bound:
--- a/src/HOL/Analysis/Euclidean_Space.thy	Mon Oct 17 11:07:01 2016 +0200
+++ b/src/HOL/Analysis/Euclidean_Space.thy	Mon Oct 17 14:37:32 2016 +0200
@@ -53,9 +53,9 @@
 lemma (in euclidean_space) SOME_Basis: "(SOME i. i \<in> Basis) \<in> Basis"
   by (metis ex_in_conv nonempty_Basis someI_ex)
 
-lemma (in euclidean_space) inner_setsum_left_Basis[simp]:
+lemma (in euclidean_space) inner_sum_left_Basis[simp]:
     "b \<in> Basis \<Longrightarrow> inner (\<Sum>i\<in>Basis. f i *\<^sub>R i) b = f b"
-  by (simp add: inner_setsum_left inner_Basis if_distrib comm_monoid_add_class.setsum.If_cases)
+  by (simp add: inner_sum_left inner_Basis if_distrib comm_monoid_add_class.sum.If_cases)
 
 lemma (in euclidean_space) euclidean_eqI:
   assumes b: "\<And>b. b \<in> Basis \<Longrightarrow> inner x b = inner y b" shows "x = y"
@@ -70,16 +70,16 @@
   "x = y \<longleftrightarrow> (\<forall>b\<in>Basis. inner x b = inner y b)"
   by (auto intro: euclidean_eqI)
 
-lemma (in euclidean_space) euclidean_representation_setsum:
+lemma (in euclidean_space) euclidean_representation_sum:
   "(\<Sum>i\<in>Basis. f i *\<^sub>R i) = b \<longleftrightarrow> (\<forall>i\<in>Basis. f i = inner b i)"
   by (subst euclidean_eq_iff) simp
 
-lemma (in euclidean_space) euclidean_representation_setsum':
+lemma (in euclidean_space) euclidean_representation_sum':
   "b = (\<Sum>i\<in>Basis. f i *\<^sub>R i) \<longleftrightarrow> (\<forall>i\<in>Basis. f i = inner b i)"
-  by (auto simp add: euclidean_representation_setsum[symmetric])
+  by (auto simp add: euclidean_representation_sum[symmetric])
 
 lemma (in euclidean_space) euclidean_representation: "(\<Sum>b\<in>Basis. inner x b *\<^sub>R b) = x"
-  unfolding euclidean_representation_setsum by simp
+  unfolding euclidean_representation_sum by simp
 
 lemma (in euclidean_space) choice_Basis_iff:
   fixes P :: "'a \<Rightarrow> real \<Rightarrow> bool"
@@ -96,15 +96,15 @@
   shows "(\<forall>i\<in>Basis. \<exists>x\<in>A. P i x) \<longleftrightarrow> (\<exists>x. \<forall>i\<in>Basis. inner x i \<in> A \<and> P i (inner x i))"
 by (simp add: choice_Basis_iff Bex_def)
 
-lemma (in euclidean_space) euclidean_representation_setsum_fun:
+lemma (in euclidean_space) euclidean_representation_sum_fun:
     "(\<lambda>x. \<Sum>b\<in>Basis. inner (f x) b *\<^sub>R b) = f"
-  by (rule ext) (simp add: euclidean_representation_setsum)
+  by (rule ext) (simp add: euclidean_representation_sum)
 
 lemma euclidean_isCont:
   assumes "\<And>b. b \<in> Basis \<Longrightarrow> isCont (\<lambda>x. (inner (f x) b) *\<^sub>R b) x"
     shows "isCont f x"
-  apply (subst euclidean_representation_setsum_fun [symmetric])
-  apply (rule isCont_setsum)
+  apply (subst euclidean_representation_sum_fun [symmetric])
+  apply (rule isCont_sum)
   apply (blast intro: assms)
   done
 
@@ -115,16 +115,16 @@
   by (meson DIM_positive Suc_leI)
 
 
-lemma setsum_inner_Basis_scaleR [simp]:
+lemma sum_inner_Basis_scaleR [simp]:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_vector"
   assumes "b \<in> Basis" shows "(\<Sum>i\<in>Basis. (inner i b) *\<^sub>R f i) = f b"
-  by (simp add: comm_monoid_add_class.setsum.remove [OF finite_Basis assms]
-         assms inner_not_same_Basis comm_monoid_add_class.setsum.neutral)
+  by (simp add: comm_monoid_add_class.sum.remove [OF finite_Basis assms]
+         assms inner_not_same_Basis comm_monoid_add_class.sum.neutral)
 
-lemma setsum_inner_Basis_eq [simp]:
+lemma sum_inner_Basis_eq [simp]:
   assumes "b \<in> Basis" shows "(\<Sum>i\<in>Basis. (inner i b) * f i) = f b"
-  by (simp add: comm_monoid_add_class.setsum.remove [OF finite_Basis assms]
-         assms inner_not_same_Basis comm_monoid_add_class.setsum.neutral)
+  by (simp add: comm_monoid_add_class.sum.remove [OF finite_Basis assms]
+         assms inner_not_same_Basis comm_monoid_add_class.sum.neutral)
 
 subsection \<open>Subclass relationships\<close>
 
@@ -188,15 +188,15 @@
 definition
   "Basis = (\<lambda>u. (u, 0)) ` Basis \<union> (\<lambda>v. (0, v)) ` Basis"
 
-lemma setsum_Basis_prod_eq:
+lemma sum_Basis_prod_eq:
   fixes f::"('a*'b)\<Rightarrow>('a*'b)"
-  shows "setsum f Basis = setsum (\<lambda>i. f (i, 0)) Basis + setsum (\<lambda>i. f (0, i)) Basis"
+  shows "sum f Basis = sum (\<lambda>i. f (i, 0)) Basis + sum (\<lambda>i. f (0, i)) Basis"
 proof -
   have "inj_on (\<lambda>u. (u::'a, 0::'b)) Basis" "inj_on (\<lambda>u. (0::'a, u::'b)) Basis"
     by (auto intro!: inj_onI Pair_inject)
   thus ?thesis
     unfolding Basis_prod_def
-    by (subst setsum.union_disjoint) (auto simp: Basis_prod_def setsum.reindex)
+    by (subst sum.union_disjoint) (auto simp: Basis_prod_def sum.reindex)
 qed
 
 instance proof
--- a/src/HOL/Analysis/Fashoda_Theorem.thy	Mon Oct 17 11:07:01 2016 +0200
+++ b/src/HOL/Analysis/Fashoda_Theorem.thy	Mon Oct 17 14:37:32 2016 +0200
@@ -17,13 +17,13 @@
 lemma interval_bij_affine:
   "interval_bij (a,b) (u,v) = (\<lambda>x. (\<Sum>i\<in>Basis. ((v\<bullet>i - u\<bullet>i) / (b\<bullet>i - a\<bullet>i) * (x\<bullet>i)) *\<^sub>R i) +
     (\<Sum>i\<in>Basis. (u\<bullet>i - (v\<bullet>i - u\<bullet>i) / (b\<bullet>i - a\<bullet>i) * (a\<bullet>i)) *\<^sub>R i))"
-  by (auto simp: setsum.distrib[symmetric] scaleR_add_left[symmetric] interval_bij_def fun_eq_iff
-    field_simps inner_simps add_divide_distrib[symmetric] intro!: setsum.cong)
+  by (auto simp: sum.distrib[symmetric] scaleR_add_left[symmetric] interval_bij_def fun_eq_iff
+    field_simps inner_simps add_divide_distrib[symmetric] intro!: sum.cong)
 
 lemma continuous_interval_bij:
   fixes a b :: "'a::euclidean_space"
   shows "continuous (at x) (interval_bij (a, b) (u, v))"
-  by (auto simp add: divide_inverse interval_bij_def intro!: continuous_setsum continuous_intros)
+  by (auto simp add: divide_inverse interval_bij_def intro!: continuous_sum continuous_intros)
 
 lemma continuous_on_interval_bij: "continuous_on s (interval_bij (a, b) (u, v))"
   apply(rule continuous_at_imp_continuous_on)
@@ -35,7 +35,7 @@
   assumes "x \<in> cbox a b"
     and "cbox u v \<noteq> {}"
   shows "interval_bij (a, b) (u, v) x \<in> cbox u v"
-  apply (simp only: interval_bij_def split_conv mem_box inner_setsum_left_Basis cong: ball_cong)
+  apply (simp only: interval_bij_def split_conv mem_box inner_sum_left_Basis cong: ball_cong)
   apply safe
 proof -
   fix i :: 'a
--- a/src/HOL/Analysis/Finite_Cartesian_Product.thy	Mon Oct 17 11:07:01 2016 +0200
+++ b/src/HOL/Analysis/Finite_Cartesian_Product.thy	Mon Oct 17 14:37:32 2016 +0200
@@ -373,10 +373,10 @@
     assume "M \<le> m" "M \<le> n"
     have "dist (X m) (X n) = setL2 (\<lambda>i. dist (X m $ i) (X n $ i)) UNIV"
       unfolding dist_vec_def ..
-    also have "\<dots> \<le> setsum (\<lambda>i. dist (X m $ i) (X n $ i)) UNIV"
-      by (rule setL2_le_setsum [OF zero_le_dist])
-    also have "\<dots> < setsum (\<lambda>i::'n. ?s) UNIV"
-      by (rule setsum_strict_mono, simp_all add: M \<open>M \<le> m\<close> \<open>M \<le> n\<close>)
+    also have "\<dots> \<le> sum (\<lambda>i. dist (X m $ i) (X n $ i)) UNIV"
+      by (rule setL2_le_sum [OF zero_le_dist])
+    also have "\<dots> < sum (\<lambda>i::'n. ?s) UNIV"
+      by (rule sum_strict_mono, simp_all add: M \<open>M \<le> m\<close> \<open>M \<le> n\<close>)
     also have "\<dots> = r"
       by simp
     finally have "dist (X m) (X n) < r" .
@@ -449,7 +449,7 @@
 instantiation vec :: (real_inner, finite) real_inner
 begin
 
-definition "inner x y = setsum (\<lambda>i. inner (x$i) (y$i)) UNIV"
+definition "inner x y = sum (\<lambda>i. inner (x$i) (y$i)) UNIV"
 
 instance proof
   fix r :: real and x y z :: "'a ^ 'b"
@@ -458,16 +458,16 @@
     by (simp add: inner_commute)
   show "inner (x + y) z = inner x z + inner y z"
     unfolding inner_vec_def
-    by (simp add: inner_add_left setsum.distrib)
+    by (simp add: inner_add_left sum.distrib)
   show "inner (scaleR r x) y = r * inner x y"
     unfolding inner_vec_def
-    by (simp add: setsum_distrib_left)
+    by (simp add: sum_distrib_left)
   show "0 \<le> inner x x"
     unfolding inner_vec_def
-    by (simp add: setsum_nonneg)
+    by (simp add: sum_nonneg)
   show "inner x x = 0 \<longleftrightarrow> x = 0"
     unfolding inner_vec_def
-    by (simp add: vec_eq_iff setsum_nonneg_eq_0_iff)
+    by (simp add: vec_eq_iff sum_nonneg_eq_0_iff)
   show "norm x = sqrt (inner x x)"
     unfolding inner_vec_def norm_vec_def setL2_def
     by (simp add: power2_norm_eq_inner)
@@ -493,22 +493,22 @@
   unfolding inner_vec_def
   apply (cases "i = j")
   apply clarsimp
-  apply (subst setsum.remove [of _ j], simp_all)
-  apply (rule setsum.neutral, simp add: axis_def)
-  apply (rule setsum.neutral, simp add: axis_def)
+  apply (subst sum.remove [of _ j], simp_all)
+  apply (rule sum.neutral, simp add: axis_def)
+  apply (rule sum.neutral, simp add: axis_def)
   done
 
-lemma setsum_single:
+lemma sum_single:
   assumes "finite A" and "k \<in> A" and "f k = y"
   assumes "\<And>i. i \<in> A \<Longrightarrow> i \<noteq> k \<Longrightarrow> f i = 0"
   shows "(\<Sum>i\<in>A. f i) = y"
-  apply (subst setsum.remove [OF assms(1,2)])
-  apply (simp add: setsum.neutral assms(3,4))
+  apply (subst sum.remove [OF assms(1,2)])
+  apply (simp add: sum.neutral assms(3,4))
   done
 
 lemma inner_axis: "inner x (axis i y) = inner (x $ i) y"
   unfolding inner_vec_def
-  apply (rule_tac k=i in setsum_single)
+  apply (rule_tac k=i in sum_single)
   apply simp_all
   apply (simp add: axis_def)
   done
--- a/src/HOL/Analysis/Gamma_Function.thy	Mon Oct 17 11:07:01 2016 +0200
+++ b/src/HOL/Analysis/Gamma_Function.thy	Mon Oct 17 14:37:32 2016 +0200
@@ -407,7 +407,7 @@
     moreover have "N \<ge> 2" "N \<ge> M" unfolding N_def by simp_all
     moreover have "(\<Sum>k=m..n. 1/(of_nat k)\<^sup>2) < e'" if "m \<ge> N" for m n
       using M[OF order.trans[OF \<open>N \<ge> M\<close> that]] unfolding real_norm_def
-      by (subst (asm) abs_of_nonneg) (auto intro: setsum_nonneg simp: divide_simps)
+      by (subst (asm) abs_of_nonneg) (auto intro: sum_nonneg simp: divide_simps)
     moreover note calculation
   } note N = this
 
@@ -432,25 +432,25 @@
       by (simp add: ln_Gamma_series_def algebra_simps)
     also have "(\<Sum>k=1..n. Ln (t / of_nat k + 1)) - (\<Sum>k=1..m. Ln (t / of_nat k + 1)) =
                  (\<Sum>k\<in>{1..n}-{1..m}. Ln (t / of_nat k + 1))" using mn
-      by (simp add: setsum_diff)
+      by (simp add: sum_diff)
     also from mn have "{1..n}-{1..m} = {Suc m..n}" by fastforce
     also have "-(t * Ln (of_nat n)) - (-(t * Ln (of_nat m))) =
                    (\<Sum>k = Suc m..n. t * Ln (of_nat (k - 1)) - t * Ln (of_nat k))" using mn
-      by (subst setsum_telescope'' [symmetric]) simp_all
+      by (subst sum_telescope'' [symmetric]) simp_all
     also have "... = (\<Sum>k = Suc m..n. t * Ln (of_nat (k - 1) / of_nat k))" using mn N
-      by (intro setsum_cong_Suc)
+      by (intro sum_cong_Suc)
          (simp_all del: of_nat_Suc add: field_simps Ln_of_nat Ln_of_nat_over_of_nat)
     also have "of_nat (k - 1) / of_nat k = 1 - 1 / (of_nat k :: complex)" if "k \<in> {Suc m..n}" for k
       using that of_nat_eq_0_iff[of "Suc i" for i] by (cases k) (simp_all add: divide_simps)
     hence "(\<Sum>k = Suc m..n. t * Ln (of_nat (k - 1) / of_nat k)) =
              (\<Sum>k = Suc m..n. t * Ln (1 - 1 / of_nat k))" using mn N
-      by (intro setsum.cong) simp_all
-    also note setsum.distrib [symmetric]
+      by (intro sum.cong) simp_all
+    also note sum.distrib [symmetric]
     also have "norm (\<Sum>k=Suc m..n. t * Ln (1 - 1/of_nat k) + Ln (t/of_nat k + 1)) \<le>
       (\<Sum>k=Suc m..n. 2 * (norm t + (norm t)\<^sup>2) / (real_of_nat k)\<^sup>2)" using t_nz N(2) mn norm_t
-      by (intro order.trans[OF norm_setsum setsum_mono[OF ln_Gamma_series_complex_converges_aux]]) simp_all
+      by (intro order.trans[OF norm_sum sum_mono[OF ln_Gamma_series_complex_converges_aux]]) simp_all
     also have "... \<le> 2 * (norm t + norm t^2) * (\<Sum>k=Suc m..n. 1 / (of_nat k)\<^sup>2)"
-      by (simp add: setsum_distrib_left)
+      by (simp add: sum_distrib_left)
     also have "... < 2 * (norm t + norm t^2) * e'" using mn z t_nz
       by (intro mult_strict_left_mono N mult_pos_pos add_pos_pos) simp_all
     also from e''_pos have "... = e * ((cmod t + (cmod t)\<^sup>2) / e'')"
@@ -508,7 +508,7 @@
   from assms have "z \<noteq> 0" by (intro notI) auto
   with assms have "exp (ln_Gamma_series z n) =
           (of_nat n) powr z / (z * (\<Prod>k=1..n. exp (Ln (z / of_nat k + 1))))"
-    unfolding ln_Gamma_series_def powr_def by (simp add: exp_diff exp_setsum)
+    unfolding ln_Gamma_series_def powr_def by (simp add: exp_diff exp_sum)
   also from assms have "(\<Prod>k=1..n. exp (Ln (z / of_nat k + 1))) = (\<Prod>k=1..n. z / of_nat k + 1)"
     by (intro setprod.cong[OF refl], subst exp_Ln) (auto simp: field_simps plus_of_nat_eq_0_imp)
   also have "... = (\<Prod>k=1..n. z + k) / fact n"
@@ -540,12 +540,12 @@
   proof eventually_elim
     fix n :: nat assume n: "n > 0"
     have "(\<Sum>k<n. ?f k) = (\<Sum>k=1..n. z / of_nat k - ln (1 + z / of_nat k))"
-      by (subst atLeast0LessThan [symmetric], subst setsum_shift_bounds_Suc_ivl [symmetric],
+      by (subst atLeast0LessThan [symmetric], subst sum_shift_bounds_Suc_ivl [symmetric],
           subst atLeastLessThanSuc_atLeastAtMost) simp_all
     also have "\<dots> = z * of_real (harm n) - (\<Sum>k=1..n. ln (1 + z / of_nat k))"
-      by (simp add: harm_def setsum_subtractf setsum_distrib_left divide_inverse)
+      by (simp add: harm_def sum_subtractf sum_distrib_left divide_inverse)
     also from n have "\<dots> - ?g n = 0"
-      by (simp add: ln_Gamma_series_def setsum_subtractf algebra_simps Ln_of_nat)
+      by (simp add: ln_Gamma_series_def sum_subtractf algebra_simps Ln_of_nat)
     finally show "(\<Sum>k<n. ?f k) - ?g n = 0" .
   qed
   show "(\<lambda>n. (\<Sum>k<n. ?f k) - ?g n) \<longlonglongrightarrow> 0" by (subst tendsto_cong[OF A]) simp_all
@@ -654,13 +654,13 @@
     fix m :: nat
     have "{..<m+k} = {..<k} \<union> {k..<m+k}" by auto
     also have "(\<Sum>n\<in>\<dots>. f n) = (\<Sum>n<k. f n) + (\<Sum>n=k..<m+k. f n)"
-      by (rule setsum.union_disjoint) auto
+      by (rule sum.union_disjoint) auto
     also have "(\<Sum>n=k..<m+k. f n) = (\<Sum>n=0..<m+k-k. f (n + k))"
-      by (intro setsum.reindex_cong[of "\<lambda>n. n + k"])
+      by (intro sum.reindex_cong[of "\<lambda>n. n + k"])
          (simp, subst image_add_atLeastLessThan, auto)
     finally show "(\<Sum>n<k. f n) + (\<Sum>n<m. f (n + k)) = (\<Sum>n<m+k. f n)" by (simp add: atLeast0LessThan)
   qed
-  finally have "(\<lambda>a. setsum f {..<a}) \<longlonglongrightarrow> lim (\<lambda>m. setsum f {..<m + k})"
+  finally have "(\<lambda>a. sum f {..<a}) \<longlonglongrightarrow> lim (\<lambda>m. sum f {..<m + k})"
     by (auto simp: convergent_LIMSEQ_iff dest: LIMSEQ_offset)
   thus ?thesis by (auto simp: summable_iff_convergent convergent_def)
 qed
@@ -733,10 +733,10 @@
             sums Digamma z" by (simp add: add_ac)
   hence "(\<lambda>m. (\<Sum>n<m. of_real (ln (real (Suc n) + 1)) - of_real (ln (real n + 1))) - 
               (\<Sum>n<m. inverse (z + of_nat n))) \<longlonglongrightarrow> Digamma z"
-    by (simp add: sums_def setsum_subtractf)
+    by (simp add: sums_def sum_subtractf)
   also have "(\<lambda>m. (\<Sum>n<m. of_real (ln (real (Suc n) + 1)) - of_real (ln (real n + 1)))) = 
                  (\<lambda>m. of_real (ln (m + 1)) :: 'a)"
-    by (subst setsum_lessThan_telescope) simp_all
+    by (subst sum_lessThan_telescope) simp_all
   finally show ?thesis by (rule Lim_transform) (insert lim, simp)
 qed
 
@@ -944,7 +944,7 @@
       using z' n by (intro uniformly_convergent_mult Polygamma_converges) (simp_all add: n'_def)
     thus "uniformly_convergent_on (ball z d)
               (\<lambda>k z. \<Sum>i<k. - of_nat n' * inverse ((z + of_nat i :: 'a) ^ (n'+1)))"
-      by (subst (asm) setsum_distrib_left) simp
+      by (subst (asm) sum_distrib_left) simp
   qed (insert Polygamma_converges'[OF z' n'] d, simp_all)
   also have "(\<Sum>k. - of_nat n' * inverse ((z + of_nat k) ^ (n' + 1))) =
                (- of_nat n') * (\<Sum>k. inverse ((z + of_nat k) ^ (n' + 1)))"
@@ -2573,7 +2573,7 @@
 proof -
   have "(\<Prod>k=1..n. exp (z * of_real (ln (1 + 1 / of_nat k)))) =
           exp (z * of_real (\<Sum>k = 1..n. ln (1 + 1 / real_of_nat k)))"
-    by (subst exp_setsum [symmetric]) (simp_all add: setsum_distrib_left)
+    by (subst exp_sum [symmetric]) (simp_all add: sum_distrib_left)
   also have "(\<Sum>k=1..n. ln (1 + 1 / of_nat k) :: real) = ln (\<Prod>k=1..n. 1 + 1 / real_of_nat k)"
     by (subst ln_setprod [symmetric]) (auto intro!: add_pos_nonneg)
   also have "(\<Prod>k=1..n. 1 + 1 / of_nat k :: real) = (\<Prod>k=1..n. (of_nat k + 1) / of_nat k)"
@@ -2654,9 +2654,9 @@
   have "(\<lambda>n. \<Sum>k=1..n. z / of_nat k - ln (1 + z / of_nat k)) \<longlonglongrightarrow> ln_Gamma z + euler_mascheroni * z + ln z"
     using ln_Gamma_series'_aux[OF False]
     by (simp only: atLeastLessThanSuc_atLeastAtMost [symmetric] One_nat_def
-                   setsum_shift_bounds_Suc_ivl sums_def atLeast0LessThan)
+                   sum_shift_bounds_Suc_ivl sums_def atLeast0LessThan)
   from tendsto_exp[OF this] False z have "?f \<longlonglongrightarrow> z * exp (euler_mascheroni * z) * Gamma z"
-    by (simp add: exp_add exp_setsum exp_diff mult_ac Gamma_complex_altdef A)
+    by (simp add: exp_add exp_sum exp_diff mult_ac Gamma_complex_altdef A)
   from tendsto_mult[OF tendsto_const[of "exp (-euler_mascheroni * z) / z"] this] z
     show "Gamma_series_weierstrass z \<longlonglongrightarrow> Gamma z"
     by (simp add: exp_minus divide_simps Gamma_series_weierstrass_def [abs_def])
--- a/src/HOL/Analysis/Harmonic_Numbers.thy	Mon Oct 17 11:07:01 2016 +0200
+++ b/src/HOL/Analysis/Harmonic_Numbers.thy	Mon Oct 17 14:37:32 2016 +0200
@@ -25,7 +25,7 @@
   thus "ln 2 < (1::real)" by (subst (asm) exp_less_cancel_iff) simp
 qed
 
-lemma setsum_Suc_diff':
+lemma sum_Suc_diff':
   fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
   assumes "m \<le> n"
   shows "(\<Sum>i = m..<n. f (Suc i) - f i) = f n - f m"
@@ -44,10 +44,10 @@
   by (simp add: harm_def)
 
 lemma harm_nonneg: "harm n \<ge> (0 :: 'a :: {real_normed_field,linordered_field})"
-  unfolding harm_def by (intro setsum_nonneg) simp_all
+  unfolding harm_def by (intro sum_nonneg) simp_all
 
 lemma harm_pos: "n > 0 \<Longrightarrow> harm n > (0 :: 'a :: {real_normed_field,linordered_field})"
-  unfolding harm_def by (intro setsum_pos) simp_all
+  unfolding harm_def by (intro sum_pos) simp_all
 
 lemma of_real_harm: "of_real (harm n) = harm n"
   unfolding harm_def by simp
@@ -73,7 +73,7 @@
   also have "\<dots> \<longleftrightarrow> convergent (\<lambda>n. \<Sum>k=Suc 0..Suc n. inverse (of_nat k) :: real)"
     unfolding harm_def[abs_def] by (subst convergent_Suc_iff) simp_all
   also have "... \<longleftrightarrow> convergent (\<lambda>n. \<Sum>k\<le>n. inverse (of_nat (Suc k)) :: real)"
-    by (subst setsum_shift_bounds_cl_Suc_ivl) (simp add: atLeast0AtMost)
+    by (subst sum_shift_bounds_cl_Suc_ivl) (simp add: atLeast0AtMost)
   also have "... \<longleftrightarrow> summable (\<lambda>n. inverse (of_nat n) :: real)"
     by (subst summable_Suc_iff [symmetric]) (simp add: summable_iff_convergent')
   also have "\<not>..." by (rule not_summable_harmonic)
@@ -126,7 +126,7 @@
   "euler_mascheroni.sum_integral_diff_series n = harm (Suc n) - ln (of_nat (Suc n))"
 proof -
   have "harm (Suc n) = (\<Sum>k=0..n. inverse (of_nat k + 1) :: real)" unfolding harm_def
-    unfolding One_nat_def by (subst setsum_shift_bounds_cl_Suc_ivl) (simp add: add_ac)
+    unfolding One_nat_def by (subst sum_shift_bounds_cl_Suc_ivl) (simp add: add_ac)
   moreover have "((\<lambda>x. inverse (x + 1) :: real) has_integral ln (of_nat n + 1) - ln (0 + 1))
                    {0..of_nat n}"
     by (intro fundamental_theorem_of_calculus)
@@ -201,16 +201,16 @@
     fix n :: nat assume n: "n > 0"
     have "(\<Sum>k<2*n. (-1)^k / real_of_nat (Suc k)) =
               (\<Sum>k<2*n. ((-1)^k + ?g k) / of_nat (Suc k)) - (\<Sum>k<2*n. ?g k / of_nat (Suc k))"
-      by (simp add: setsum.distrib algebra_simps divide_inverse)
+      by (simp add: sum.distrib algebra_simps divide_inverse)
     also have "(\<Sum>k<2*n. ((-1)^k + ?g k) / real_of_nat (Suc k)) = harm (2*n)"
-      unfolding harm_altdef by (intro setsum.cong) (auto simp: field_simps)
+      unfolding harm_altdef by (intro sum.cong) (auto simp: field_simps)
     also have "(\<Sum>k<2*n. ?g k / real_of_nat (Suc k)) = (\<Sum>k|k<2*n \<and> odd k. ?g k / of_nat (Suc k))"
-      by (intro setsum.mono_neutral_right) auto
+      by (intro sum.mono_neutral_right) auto
     also have "\<dots> = (\<Sum>k|k<2*n \<and> odd k. 2 / (real_of_nat (Suc k)))"
-      by (intro setsum.cong) auto
+      by (intro sum.cong) auto
     also have "(\<Sum>k|k<2*n \<and> odd k. 2 / (real_of_nat (Suc k))) = harm n"
       unfolding harm_altdef
-      by (intro setsum.reindex_cong[of "\<lambda>n. 2*n+1"]) (auto simp: inj_on_def field_simps elim!: oddE)
+      by (intro sum.reindex_cong[of "\<lambda>n. 2*n+1"]) (auto simp: inj_on_def field_simps elim!: oddE)
     also have "harm (2*n) - harm n = ?em (2*n) - ?em n + ln 2" using n
       by (simp_all add: algebra_simps ln_mult)
     finally show "?em (2*n) - ?em n + ln 2 = (\<Sum>k<2*n. (-1)^k / real_of_nat (Suc k))" ..
@@ -388,9 +388,9 @@
   finally have "euler_mascheroni \<ge> (\<Sum>k\<le>n. D k) + 1 / (of_nat (2 * (n+2)))"
     by (simp add: inv_def field_simps)
   also have "(\<Sum>k\<le>n. D k) = harm (Suc n) - (\<Sum>k\<le>n. ln (real_of_nat (Suc k+1)) - ln (of_nat (k+1)))"
-    unfolding harm_altdef D_def by (subst lessThan_Suc_atMost) (simp add:  setsum.distrib setsum_subtractf)
+    unfolding harm_altdef D_def by (subst lessThan_Suc_atMost) (simp add:  sum.distrib sum_subtractf)
   also have "(\<Sum>k\<le>n. ln (real_of_nat (Suc k+1)) - ln (of_nat (k+1))) = ln (of_nat (n+2))"
-    by (subst atLeast0AtMost [symmetric], subst setsum_Suc_diff) simp_all
+    by (subst atLeast0AtMost [symmetric], subst sum_Suc_diff) simp_all
   finally show "euler_mascheroni \<ge> harm (Suc n) - ln (real_of_nat (n + 2)) + 1/real_of_nat (2 * (n + 2))"
     by simp
 
@@ -422,9 +422,9 @@
   finally have "euler_mascheroni \<le> (\<Sum>k\<le>n. D k) + 1 / of_nat (2 * (n+1))"
     by (simp add: inv_def field_simps)
   also have "(\<Sum>k\<le>n. D k) = harm (Suc n) - (\<Sum>k\<le>n. ln (real_of_nat (Suc k+1)) - ln (of_nat (k+1)))"
-    unfolding harm_altdef D_def by (subst lessThan_Suc_atMost) (simp add:  setsum.distrib setsum_subtractf)
+    unfolding harm_altdef D_def by (subst lessThan_Suc_atMost) (simp add:  sum.distrib sum_subtractf)
   also have "(\<Sum>k\<le>n. ln (real_of_nat (Suc k+1)) - ln (of_nat (k+1))) = ln (of_nat (n+2))"
-    by (subst atLeast0AtMost [symmetric], subst setsum_Suc_diff) simp_all
+    by (subst atLeast0AtMost [symmetric], subst sum_Suc_diff) simp_all
   finally show "euler_mascheroni \<le> harm (Suc n) - ln (real_of_nat (n + 2)) + 1/real_of_nat (2 * (n + 1))"
     by simp
 qed
--- a/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy	Mon Oct 17 11:07:01 2016 +0200
+++ b/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy	Mon Oct 17 14:37:32 2016 +0200
@@ -176,11 +176,11 @@
   unfolding forall_in_division[OF assms(2)]
   by (metis antisym_conv assms content_pos_le content_subset division_ofD(2))
 
-lemma setsum_content_null:
+lemma sum_content_null:
   assumes "content (cbox a b) = 0"
     and "p tagged_division_of (cbox a b)"
-  shows "setsum (\<lambda>(x,k). content k *\<^sub>R f x) p = (0::'a::real_normed_vector)"
-proof (rule setsum.neutral, rule)
+  shows "sum (\<lambda>(x,k). content k *\<^sub>R f x) p = (0::'a::real_normed_vector)"
+proof (rule sum.neutral, rule)
   fix y
   assume y: "y \<in> p"
   obtain x k where xk: "y = (x, k)"
@@ -199,12 +199,12 @@
 lemma operative_content[intro]: "add.operative content"
   by (force simp add: add.operative_def content_split[symmetric] content_eq_0_interior)
 
-lemma additive_content_division: "d division_of (cbox a b) \<Longrightarrow> setsum content d = content (cbox a b)"
-  by (metis operative_content setsum.operative_division)
+lemma additive_content_division: "d division_of (cbox a b) \<Longrightarrow> sum content d = content (cbox a b)"
+  by (metis operative_content sum.operative_division)
 
 lemma additive_content_tagged_division:
-  "d tagged_division_of (cbox a b) \<Longrightarrow> setsum (\<lambda>(x,l). content l) d = content (cbox a b)"
-  unfolding setsum.operative_tagged_division[OF operative_content, symmetric] by blast
+  "d tagged_division_of (cbox a b) \<Longrightarrow> sum (\<lambda>(x,l). content l) d = content (cbox a b)"
+  unfolding sum.operative_tagged_division[OF operative_content, symmetric] by blast
 
 lemma content_real_eq_0: "content {a .. b::real} = 0 \<longleftrightarrow> a \<ge> b"
   by (metis atLeastatMost_empty_iff2 content_empty content_real diff_self eq_iff le_cases le_iff_diff_le_0)
@@ -234,14 +234,14 @@
   "(f has_integral y) (cbox a b) \<longleftrightarrow>
     (\<forall>e>0. \<exists>d. gauge d \<and>
       (\<forall>p. p tagged_division_of (cbox a b) \<and> d fine p \<longrightarrow>
-        norm (setsum (\<lambda>(x,k). content(k) *\<^sub>R f x) p - y) < e))"
+        norm (sum (\<lambda>(x,k). content(k) *\<^sub>R f x) p - y) < e))"
   by (auto simp: dist_norm eventually_division_filter has_integral_def tendsto_iff)
 
 lemma has_integral_real:
   "(f has_integral y) {a .. b::real} \<longleftrightarrow>
     (\<forall>e>0. \<exists>d. gauge d \<and>
       (\<forall>p. p tagged_division_of {a .. b} \<and> d fine p \<longrightarrow>
-        norm (setsum (\<lambda>(x,k). content(k) *\<^sub>R f x) p - y) < e))"
+        norm (sum (\<lambda>(x,k). content(k) *\<^sub>R f x) p - y) < e))"
   unfolding box_real[symmetric]
   by (rule has_integral)
 
@@ -364,7 +364,7 @@
   shows "((\<lambda>x. c) has_integral (content (cbox a b) *\<^sub>R c)) (cbox a b)"
   using eventually_division_filter_tagged_division[of "cbox a b"]
      additive_content_tagged_division[of _ a b]
-  by (auto simp: has_integral_cbox split_beta' scaleR_setsum_left[symmetric]
+  by (auto simp: has_integral_cbox split_beta' scaleR_sum_left[symmetric]
            elim!: eventually_mono intro!: tendsto_cong[THEN iffD1, OF _ tendsto_const])
 
 lemma has_integral_const_real [intro]:
@@ -391,7 +391,7 @@
     unfolding has_integral_cbox
     using eventually_division_filter_tagged_division[of "cbox a b"]
     by (subst tendsto_cong[where g="\<lambda>_. 0"])
-       (auto elim!: eventually_mono intro!: setsum.neutral simp: tag_in_interval)
+       (auto elim!: eventually_mono intro!: sum.neutral simp: tag_in_interval)
   {
     presume "\<not> (\<exists>a b. s = cbox a b) \<Longrightarrow> ?thesis"
     with assms lem show ?thesis
@@ -425,7 +425,7 @@
   from pos_bounded obtain B where B: "0 < B" "\<And>x. norm (h x) \<le> norm x * B"
     by blast
   have lem: "\<And>a b y f::'n\<Rightarrow>'a. (f has_integral y) (cbox a b) \<Longrightarrow> ((h \<circ> f) has_integral h y) (cbox a b)"
-    unfolding has_integral_cbox by (drule tendsto) (simp add: setsum scaleR split_beta')
+    unfolding has_integral_cbox by (drule tendsto) (simp add: sum scaleR split_beta')
   {
     presume "\<not> (\<exists>a b. s = cbox a b) \<Longrightarrow> ?thesis"
     then show ?thesis
@@ -531,7 +531,7 @@
     ((\<lambda>x. f x + g x) has_integral (k + l)) (cbox a b)"
     for f :: "'n \<Rightarrow> 'a" and g a b k l
     unfolding has_integral_cbox
-    by (simp add: split_beta' scaleR_add_right setsum.distrib[abs_def] tendsto_add)
+    by (simp add: split_beta' scaleR_add_right sum.distrib[abs_def] tendsto_add)
   {
     presume "\<not> (\<exists>a b. s = cbox a b) \<Longrightarrow> ?thesis"
     then show ?thesis
@@ -666,10 +666,10 @@
   shows "integral s (\<lambda>x. f x \<bullet> k) = integral s f \<bullet> k"
   unfolding integral_linear[OF assms(1) bounded_linear_inner_left,unfolded o_def] ..
 
-lemma has_integral_setsum:
+lemma has_integral_sum:
   assumes "finite t"
     and "\<forall>a\<in>t. ((f a) has_integral (i a)) s"
-  shows "((\<lambda>x. setsum (\<lambda>a. f a x) t) has_integral (setsum i t)) s"
+  shows "((\<lambda>x. sum (\<lambda>a. f a x) t) has_integral (sum i t)) s"
   using assms(1) subset_refl[of t]
 proof (induct rule: finite_subset_induct)
   case empty
@@ -680,16 +680,16 @@
     by (simp add: has_integral_add)
 qed
 
-lemma integral_setsum:
+lemma integral_sum:
   "\<lbrakk>finite t;  \<forall>a\<in>t. (f a) integrable_on s\<rbrakk> \<Longrightarrow>
-   integral s (\<lambda>x. setsum (\<lambda>a. f a x) t) = setsum (\<lambda>a. integral s (f a)) t"
-  by (auto intro: has_integral_setsum integrable_integral)
-
-lemma integrable_setsum:
-  "\<lbrakk>finite t;  \<forall>a\<in>t. (f a) integrable_on s\<rbrakk> \<Longrightarrow> (\<lambda>x. setsum (\<lambda>a. f a x) t) integrable_on s"
+   integral s (\<lambda>x. sum (\<lambda>a. f a x) t) = sum (\<lambda>a. integral s (f a)) t"
+  by (auto intro: has_integral_sum integrable_integral)
+
+lemma integrable_sum:
+  "\<lbrakk>finite t;  \<forall>a\<in>t. (f a) integrable_on s\<rbrakk> \<Longrightarrow> (\<lambda>x. sum (\<lambda>a. f a x) t) integrable_on s"
   unfolding integrable_on_def
   apply (drule bchoice)
-  using has_integral_setsum[of t]
+  using has_integral_sum[of t]
   apply auto
   done
 
@@ -760,7 +760,7 @@
 lemma has_integral_null [intro]: "content(cbox a b) = 0 \<Longrightarrow> (f has_integral 0) (cbox a b)"
   unfolding has_integral_cbox
   using eventually_division_filter_tagged_division[of "cbox a b"]
-  by (subst tendsto_cong[where g="\<lambda>_. 0"]) (auto elim: eventually_mono intro: setsum_content_null)
+  by (subst tendsto_cong[where g="\<lambda>_. 0"]) (auto elim: eventually_mono intro: sum_content_null)
 
 lemma has_integral_null_real [intro]: "content {a .. b::real} = 0 \<Longrightarrow> (f has_integral 0) {a .. b}"
   by (metis box_real(2) has_integral_null)
@@ -840,7 +840,7 @@
   hence "\<forall>b\<in>Basis. (((\<lambda>x. x *\<^sub>R b) \<circ> (\<lambda>x. f x \<bullet> b)) has_integral ((y \<bullet> b) *\<^sub>R b)) A"
     by (intro ballI has_integral_linear) (simp_all add: bounded_linear_scaleR_left)
   hence "((\<lambda>x. \<Sum>b\<in>Basis. (f x \<bullet> b) *\<^sub>R b) has_integral (\<Sum>b\<in>Basis. (y \<bullet> b) *\<^sub>R b)) A"
-    by (intro has_integral_setsum) (simp_all add: o_def)
+    by (intro has_integral_sum) (simp_all add: o_def)
   thus "(f has_integral y) A" by (simp add: euclidean_representation)
 qed
 
@@ -865,7 +865,7 @@
   hence "\<forall>b\<in>Basis. (((\<lambda>x. x *\<^sub>R b) \<circ> (\<lambda>x. f x \<bullet> b)) has_integral (y b *\<^sub>R b)) A"
     by (intro ballI has_integral_linear) (simp_all add: bounded_linear_scaleR_left)
   hence "((\<lambda>x. \<Sum>b\<in>Basis. (f x \<bullet> b) *\<^sub>R b) has_integral (\<Sum>b\<in>Basis. y b *\<^sub>R b)) A"
-    by (intro has_integral_setsum) (simp_all add: o_def)
+    by (intro has_integral_sum) (simp_all add: o_def)
   thus "f integrable_on A" by (auto simp: integrable_on_def o_def euclidean_representation)
 qed
 
@@ -885,7 +885,7 @@
   have "integral A f = integral A (\<lambda>x. \<Sum>b\<in>Basis. (f x \<bullet> b) *\<^sub>R b)"
     by (simp add: euclidean_representation)
   also from integrable have "\<dots> = (\<Sum>a\<in>Basis. integral A (\<lambda>x. (f x \<bullet> a) *\<^sub>R a))"
-    by (subst integral_setsum) (simp_all add: o_def)
+    by (subst integral_sum) (simp_all add: o_def)
   finally show ?thesis .
 qed
 
@@ -944,7 +944,7 @@
   from choice[OF this] guess p .. note p = conjunctD2[OF this[rule_format]]
   have dp: "\<And>i n. i\<le>n \<Longrightarrow> d i fine p n"
     using p(2) unfolding fine_inters by auto
-  have "Cauchy (\<lambda>n. setsum (\<lambda>(x,k). content k *\<^sub>R (f x)) (p n))"
+  have "Cauchy (\<lambda>n. sum (\<lambda>(x,k). content k *\<^sub>R (f x)) (p n))"
   proof (rule CauchyI, goal_cases)
     case (1 e)
     then guess N unfolding real_arch_inverse[of e] .. note N=this
@@ -1124,9 +1124,9 @@
         unfolding xk split_conv by auto
     } note [simp] = this
     have lem3: "\<And>g :: 'a set \<Rightarrow> 'a set. finite p \<Longrightarrow>
-                  setsum (\<lambda>(x, k). content k *\<^sub>R f x) {(x,g k) |x k. (x,k) \<in> p \<and> g k \<noteq> {}} =
-                  setsum (\<lambda>(x, k). content k *\<^sub>R f x) ((\<lambda>(x, k). (x, g k)) ` p)"
-      by (rule setsum.mono_neutral_left) auto
+                  sum (\<lambda>(x, k). content k *\<^sub>R f x) {(x,g k) |x k. (x,k) \<in> p \<and> g k \<noteq> {}} =
+                  sum (\<lambda>(x, k). content k *\<^sub>R f x) ((\<lambda>(x, k). (x, g k)) ` p)"
+      by (rule sum.mono_neutral_left) auto
     let ?M1 = "{(x, kk \<inter> {x. x\<bullet>k \<le> c}) |x kk. (x, kk) \<in> p \<and> kk \<inter> {x. x\<bullet>k \<le> c} \<noteq> {}}"
     have d1_fine: "d1 fine ?M1"
       by (force intro: fineI dest: fineD[OF p(2)] simp add: split: if_split_asm)
@@ -1218,10 +1218,10 @@
       also have "\<dots> = (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. x \<bullet> k \<le> c}) *\<^sub>R f x) +
         (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. c \<le> x \<bullet> k}) *\<^sub>R f x) - (i + j)"
         unfolding lem3[OF p(3)]
-        by (subst (1 2) setsum.reindex_nontrivial[OF p(3)])
+        by (subst (1 2) sum.reindex_nontrivial[OF p(3)])
            (auto intro!: k eq0 tagged_division_split_left_inj_content[OF p(1)] tagged_division_split_right_inj_content[OF p(1)]
                  simp: cont_eq)+
-      also note setsum.distrib[symmetric]
+      also note sum.distrib[symmetric]
       also have "\<And>x. x \<in> p \<Longrightarrow>
                     (\<lambda>(x,ka). content (ka \<inter> {x. x \<bullet> k \<le> c}) *\<^sub>R f x) x +
                     (\<lambda>(x,ka). content (ka \<inter> {x. c \<le> x \<bullet> k}) *\<^sub>R f x) x =
@@ -1236,7 +1236,7 @@
           unfolding uv content_split[OF k,of u v c]
           by auto
       qed
-      note setsum.cong [OF _ this]
+      note sum.cong [OF _ this]
       finally have "(\<Sum>(x, k)\<in>{(x, kk \<inter> {x. x \<bullet> k \<le> c}) |x kk. (x, kk) \<in> p \<and> kk \<inter> {x. x \<bullet> k \<le> c} \<noteq> {}}. content k *\<^sub>R f x) - i +
         ((\<Sum>(x, k)\<in>{(x, kk \<inter> {x. c \<le> x \<bullet> k}) |x kk. (x, kk) \<in> p \<and> kk \<inter> {x. c \<le> x \<bullet> k} \<noteq> {}}. content k *\<^sub>R f x) - j) =
         (\<Sum>(x, ka)\<in>p. content ka *\<^sub>R f x) - (i + j)"
@@ -1258,7 +1258,7 @@
   obtains d where "gauge d"
     "\<forall>p1 p2. p1 tagged_division_of (cbox a b \<inter> {x. x\<bullet>k \<le> c}) \<and> d fine p1 \<and>
         p2 tagged_division_of (cbox a b \<inter> {x. x\<bullet>k \<ge> c}) \<and> d fine p2 \<longrightarrow>
-        norm ((setsum (\<lambda>(x,k). content k *\<^sub>R f x) p1 + setsum (\<lambda>(x,k). content k *\<^sub>R f x) p2) - i) < e"
+        norm ((sum (\<lambda>(x,k). content k *\<^sub>R f x) p1 + sum (\<lambda>(x,k). content k *\<^sub>R f x) p2) - i) < e"
 proof -
   guess d using has_integralD[OF assms(1-2)] . note d=this
   { fix p1 p2
@@ -1287,9 +1287,9 @@
           using e k by (auto simp: inner_simps inner_not_same_Basis)
         have "(\<Sum>i\<in>Basis. \<bar>(x - (x + (e / 2 ) *\<^sub>R k)) \<bullet> i\<bar>) =
               (\<Sum>i\<in>Basis. (if i = k then e / 2 else 0))"
-          using "*" by (blast intro: setsum.cong)
+          using "*" by (blast intro: sum.cong)
         also have "\<dots> < e"
-          apply (subst setsum.delta)
+          apply (subst sum.delta)
           using e
           apply auto
           done
@@ -1308,7 +1308,7 @@
     }
     then have "norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x) - i) =
                norm ((\<Sum>(x, k)\<in>p1 \<union> p2. content k *\<^sub>R f x) - i)"
-      by (subst setsum.union_inter_neutral) (auto simp: p1 p2)
+      by (subst sum.union_inter_neutral) (auto simp: p1 p2)
     also have "\<dots> < e"
       by (rule k d(2) p12 fine_union p1 p2)+
     finally have "norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x) - i) < e" .
@@ -1426,20 +1426,20 @@
 lemma dsum_bound:
   assumes "p division_of (cbox a b)"
     and "norm c \<le> e"
-  shows "norm (setsum (\<lambda>l. content l *\<^sub>R c) p) \<le> e * content(cbox a b)"
+  shows "norm (sum (\<lambda>l. content l *\<^sub>R c) p) \<le> e * content(cbox a b)"
 proof -
-  have sumeq: "(\<Sum>i\<in>p. \<bar>content i\<bar>) = setsum content p"
-    apply (rule setsum.cong)
+  have sumeq: "(\<Sum>i\<in>p. \<bar>content i\<bar>) = sum content p"
+    apply (rule sum.cong)
     using assms
     apply simp
     apply (metis abs_of_nonneg assms(1) content_pos_le division_ofD(4))
     done
   have e: "0 \<le> e"
     using assms(2) norm_ge_zero order_trans by blast
-  have "norm (setsum (\<lambda>l. content l *\<^sub>R c) p) \<le> (\<Sum>i\<in>p. norm (content i *\<^sub>R c))"
-    using norm_setsum by blast
+  have "norm (sum (\<lambda>l. content l *\<^sub>R c) p) \<le> (\<Sum>i\<in>p. norm (content i *\<^sub>R c))"
+    using norm_sum by blast
   also have "...  \<le> e * (\<Sum>i\<in>p. \<bar>content i\<bar>)"
-    by (simp add: setsum_distrib_left[symmetric] mult.commute assms(2) mult_right_mono setsum_nonneg)
+    by (simp add: sum_distrib_left[symmetric] mult.commute assms(2) mult_right_mono sum_nonneg)
   also have "... \<le> e * content (cbox a b)"
     apply (rule mult_left_mono [OF _ e])
     apply (simp add: sumeq)
@@ -1451,7 +1451,7 @@
 lemma rsum_bound:
   assumes p: "p tagged_division_of (cbox a b)"
       and "\<forall>x\<in>cbox a b. norm (f x) \<le> e"
-    shows "norm (setsum (\<lambda>(x,k). content k *\<^sub>R f x) p) \<le> e * content (cbox a b)"
+    shows "norm (sum (\<lambda>(x,k). content k *\<^sub>R f x) p) \<le> e * content (cbox a b)"
 proof (cases "cbox a b = {}")
   case True show ?thesis
     using p unfolding True tagged_division_of_trivial by auto
@@ -1459,7 +1459,7 @@
   case False
   then have e: "e \<ge> 0"
     by (meson ex_in_conv assms(2) norm_ge_zero order_trans)
-  have setsum_le: "setsum (content \<circ> snd) p \<le> content (cbox a b)"
+  have sum_le: "sum (content \<circ> snd) p \<le> content (cbox a b)"
     unfolding additive_content_tagged_division[OF p, symmetric] split_def
     by (auto intro: eq_refl)
   have con: "\<And>xk. xk \<in> p \<Longrightarrow> 0 \<le> content (snd xk)"
@@ -1468,15 +1468,15 @@
   have norm: "\<And>xk. xk \<in> p \<Longrightarrow> norm (f (fst xk)) \<le> e"
     unfolding fst_conv using tagged_division_ofD(2,3)[OF p] assms
     by (metis prod.collapse subset_eq)
-  have "norm (setsum (\<lambda>(x,k). content k *\<^sub>R f x) p) \<le> (\<Sum>i\<in>p. norm (case i of (x, k) \<Rightarrow> content k *\<^sub>R f x))"
-    by (rule norm_setsum)
+  have "norm (sum (\<lambda>(x,k). content k *\<^sub>R f x) p) \<le> (\<Sum>i\<in>p. norm (case i of (x, k) \<Rightarrow> content k *\<^sub>R f x))"
+    by (rule norm_sum)
   also have "...  \<le> e * content (cbox a b)"
     unfolding split_def norm_scaleR
-    apply (rule order_trans[OF setsum_mono])
+    apply (rule order_trans[OF sum_mono])
     apply (rule mult_left_mono[OF _ abs_ge_zero, of _ e])
     apply (metis norm)
-    unfolding setsum_distrib_right[symmetric]
-    using con setsum_le
+    unfolding sum_distrib_right[symmetric]
+    using con sum_le
     apply (auto simp: mult.commute intro: mult_left_mono [OF _ e])
     done
   finally show ?thesis .
@@ -1485,10 +1485,10 @@
 lemma rsum_diff_bound:
   assumes "p tagged_division_of (cbox a b)"
     and "\<forall>x\<in>cbox a b. norm (f x - g x) \<le> e"
-  shows "norm (setsum (\<lambda>(x,k). content k *\<^sub>R f x) p - setsum (\<lambda>(x,k). content k *\<^sub>R g x) p) \<le>
+  shows "norm (sum (\<lambda>(x,k). content k *\<^sub>R f x) p - sum (\<lambda>(x,k). content k *\<^sub>R g x) p) \<le>
          e * content (cbox a b)"
   apply (rule order_trans[OF _ rsum_bound[OF assms]])
-  apply (simp add: split_def scaleR_diff_right setsum_subtractf eq_refl)
+  apply (simp add: split_def scaleR_diff_right sum_subtractf eq_refl)
   done
 
 lemma has_integral_bound:
@@ -1534,9 +1534,9 @@
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "p tagged_division_of (cbox a b)"
       and "\<forall>x\<in>cbox a b. (f x)\<bullet>i \<le> (g x)\<bullet>i"
-    shows "(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p)\<bullet>i \<le> (setsum (\<lambda>(x,k). content k *\<^sub>R g x) p)\<bullet>i"
-unfolding inner_setsum_left
-proof (rule setsum_mono, clarify)
+    shows "(sum (\<lambda>(x,k). content k *\<^sub>R f x) p)\<bullet>i \<le> (sum (\<lambda>(x,k). content k *\<^sub>R g x) p)\<bullet>i"
+unfolding inner_sum_left
+proof (rule sum_mono, clarify)
   fix a b
   assume ab: "(a, b) \<in> p"
   note tagged = tagged_division_ofD(2-4)[OF assms(1) ab]
@@ -1877,7 +1877,7 @@
   also have "(\<Prod>j\<in>Basis. (b' 0 - a' 0) \<bullet> j) = 0"
     using k *
     by (intro setprod_zero bexI[OF _ k])
-       (auto simp: b'_def a'_def inner_diff inner_setsum_left inner_not_same_Basis intro!: setsum.cong)
+       (auto simp: b'_def a'_def inner_diff inner_sum_left inner_not_same_Basis intro!: sum.cong)
   also have "((\<lambda>d. \<Prod>j\<in>Basis. (b' d - a' d) \<bullet> j) \<longlongrightarrow> 0) (at_right 0) =
     ((\<lambda>d. content (cbox a b \<inter> {x. \<bar>x\<bullet>k - c\<bar> \<le> d})) \<longlongrightarrow> 0) (at_right 0)"
   proof (intro tendsto_cong eventually_at_rightI)
@@ -1952,7 +1952,7 @@
     assume p: "p tagged_division_of (cbox a b) \<and> (\<lambda>x. ball x d) fine p"
     have *: "(\<Sum>(x, ka)\<in>p. content ka *\<^sub>R ?i x) =
       (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. \<bar>x\<bullet>k - c\<bar> \<le> d}) *\<^sub>R ?i x)"
-      apply (rule setsum.cong)
+      apply (rule sum.cong)
       apply (rule refl)
       unfolding split_paired_all real_scaleR_def mult_cancel_right split_conv
       apply cases
@@ -1989,7 +1989,7 @@
       unfolding diff_0_right *
       unfolding real_scaleR_def real_norm_def
       apply (subst abs_of_nonneg)
-      apply (rule setsum_nonneg)
+      apply (rule sum_nonneg)
       apply rule
       unfolding split_paired_all split_conv
       apply (rule mult_nonneg_nonneg)
@@ -2005,14 +2005,14 @@
     proof -
       have "(\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) * ?i x) \<le>
         (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}))"
-        apply (rule setsum_mono)
+        apply (rule sum_mono)
         unfolding split_paired_all split_conv
         apply (rule mult_right_le_one_le)
         apply (drule p'(4))
         apply (auto simp add:interval_doublesplit[OF k])
         done
       also have "\<dots> < e"
-      proof (subst setsum.over_tagged_division_lemma[OF p[THEN conjunct1]], goal_cases)
+      proof (subst sum.over_tagged_division_lemma[OF p[THEN conjunct1]], goal_cases)
         case prems: (1 u v)
         then have *: "content (cbox u v) = 0"
           unfolding content_eq_0_interior by simp
@@ -2027,8 +2027,8 @@
           by (blast intro: antisym)
       next
         have "(\<Sum>l\<in>snd ` p. content (l \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d})) =
-          setsum content ((\<lambda>l. l \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d})`{l\<in>snd ` p. l \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d} \<noteq> {}})"
-        proof (subst (2) setsum.reindex_nontrivial)
+          sum content ((\<lambda>l. l \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d})`{l\<in>snd ` p. l \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d} \<noteq> {}})"
+        proof (subst (2) sum.reindex_nontrivial)
           fix x y assume "x \<in> {l \<in> snd ` p. l \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d} \<noteq> {}}" "y \<in> {l \<in> snd ` p. l \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d} \<noteq> {}}"
             "x \<noteq> y" and eq: "x \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d} = y \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}"
           then obtain x' y' where "(x', x) \<in> p" "x \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d} \<noteq> {}" "(y', y) \<in> p" "y \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d} \<noteq> {}"
@@ -2041,7 +2041,7 @@
             by (auto simp: eq)
           then show "content (x \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) = 0"
             using p'(4)[OF \<open>(x', x) \<in> p\<close>] by (auto simp: interval_doublesplit[OF k] content_eq_0_interior simp del: interior_Int)
-        qed (insert p'(1), auto intro!: setsum.mono_neutral_right)
+        qed (insert p'(1), auto intro!: sum.mono_neutral_right)
         also have "\<dots> \<le> norm (\<Sum>l\<in>(\<lambda>l. l \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d})`{l\<in>snd ` p. l \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d} \<noteq> {}}. content l *\<^sub>R 1::real)"
           by simp
         also have "\<dots> \<le> 1 * content (cbox a b \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d})"
@@ -2144,15 +2144,15 @@
         by (auto intro: tagged_division_finer[OF as(1) d(1)])
       from choice[OF this] guess q .. note q=conjunctD3[OF this[rule_format]]
       have *: "\<And>i. (\<Sum>(x, k)\<in>q i. content k *\<^sub>R indicator s x) \<ge> (0::real)"
-        apply (rule setsum_nonneg)
+        apply (rule sum_nonneg)
         apply safe
         unfolding real_scaleR_def
         apply (drule tagged_division_ofD(4)[OF q(1)])
         apply (auto intro: mult_nonneg_nonneg)
         done
       have **: "finite s \<Longrightarrow> finite t \<Longrightarrow> (\<forall>(x,y) \<in> t. (0::real) \<le> g(x,y)) \<Longrightarrow>
-        (\<forall>y\<in>s. \<exists>x. (x,y) \<in> t \<and> f(y) \<le> g(x,y)) \<Longrightarrow> setsum f s \<le> setsum g t" for f g s t
-        apply (rule setsum_le_included[of s t g snd f])
+        (\<forall>y\<in>s. \<exists>x. (x,y) \<in> t \<and> f(y) \<le> g(x,y)) \<Longrightarrow> sum f s \<le> sum g t" for f g s t
+        apply (rule sum_le_included[of s t g snd f])
         prefer 4
         apply safe
         apply (erule_tac x=x in ballE)
@@ -2160,11 +2160,11 @@
         apply (rule_tac x="(xa,x)" in bexI)
         apply auto
         done
-      have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) \<le> setsum (\<lambda>i. (real i + 1) *
-        norm (setsum (\<lambda>(x,k). content k *\<^sub>R indicator s x :: real) (q i))) {..N+1}"
-        unfolding real_norm_def setsum_distrib_left abs_of_nonneg[OF *] diff_0_right
+      have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) \<le> sum (\<lambda>i. (real i + 1) *
+        norm (sum (\<lambda>(x,k). content k *\<^sub>R indicator s x :: real) (q i))) {..N+1}"
+        unfolding real_norm_def sum_distrib_left abs_of_nonneg[OF *] diff_0_right
         apply (rule order_trans)
-        apply (rule norm_setsum)
+        apply (rule norm_sum)
         apply (subst sum_sum_product)
         prefer 3
       proof (rule **, safe)
@@ -2229,8 +2229,8 @@
           apply auto
           done
       qed (insert as, auto)
-      also have "\<dots> \<le> setsum (\<lambda>i. e / 2 / 2 ^ i) {..N+1}"
-      proof (rule setsum_mono, goal_cases)
+      also have "\<dots> \<le> sum (\<lambda>i. e / 2 / 2 ^ i) {..N+1}"
+      proof (rule sum_mono, goal_cases)
         case (1 i)
         then show ?case
           apply (subst mult.commute, subst pos_le_divide_eq[symmetric])
@@ -2240,7 +2240,7 @@
           done
       qed
       also have "\<dots> < e * inverse 2 * 2"
-        unfolding divide_inverse setsum_distrib_left[symmetric]
+        unfolding divide_inverse sum_distrib_left[symmetric]
         apply (rule mult_strict_left_mono)
         unfolding power_inverse [symmetric] lessThan_Suc_atMost[symmetric]
         apply (subst geometric_sum)
@@ -2640,14 +2640,14 @@
 lemma has_integral_factor_content:
   "(f has_integral i) (cbox a b) \<longleftrightarrow>
     (\<forall>e>0. \<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of (cbox a b) \<and> d fine p \<longrightarrow>
-      norm (setsum (\<lambda>(x,k). content k *\<^sub>R f x) p - i) \<le> e * content (cbox a b)))"
+      norm (sum (\<lambda>(x,k). content k *\<^sub>R f x) p - i) \<le> e * content (cbox a b)))"
 proof (cases "content (cbox a b) = 0")
   case True
   show ?thesis
     unfolding has_integral_null_eq[OF True]
     apply safe
     apply (rule, rule, rule gauge_trivial, safe)
-    unfolding setsum_content_null[OF True] True
+    unfolding sum_content_null[OF True] True
     defer
     apply (erule_tac x=1 in allE)
     apply safe
@@ -2655,7 +2655,7 @@
     apply (rule fine_division_exists[of _ a b])
     apply assumption
     apply (erule_tac x=p in allE)
-    unfolding setsum_content_null[OF True]
+    unfolding sum_content_null[OF True]
     apply auto
     done
 next
@@ -2696,7 +2696,7 @@
 lemma has_integral_factor_content_real:
   "(f has_integral i) {a .. b::real} \<longleftrightarrow>
     (\<forall>e>0. \<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a .. b}  \<and> d fine p \<longrightarrow>
-      norm (setsum (\<lambda>(x,k). content k *\<^sub>R f x) p - i) \<le> e * content {a .. b} ))"
+      norm (sum (\<lambda>(x,k). content k *\<^sub>R f x) p - i) \<le> e * content {a .. b} ))"
   unfolding box_real[symmetric]
   by (rule has_integral_factor_content)
 
@@ -2738,10 +2738,10 @@
     show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f' x) - (f b - f a)) \<le> e * content (cbox a b)"
       unfolding content_real[OF assms(1), simplified box_real[symmetric]] additive_tagged_division_1[OF assms(1) as(1)[simplified box_real],of f,symmetric]
       unfolding additive_tagged_division_1[OF assms(1) as(1)[simplified box_real],of "\<lambda>x. x",symmetric]
-      unfolding setsum_distrib_left
+      unfolding sum_distrib_left
       defer
-      unfolding setsum_subtractf[symmetric]
-    proof (rule setsum_norm_le,safe)
+      unfolding sum_subtractf[symmetric]
+    proof (rule sum_norm_le,safe)
       fix x k
       assume "(x, k) \<in> p"
       note xk = tagged_division_ofD(2-4)[OF as(1) this]
@@ -2809,7 +2809,7 @@
 
 subsection \<open>Taylor series expansion\<close>
 
-lemma (in bounded_bilinear) setsum_prod_derivatives_has_vector_derivative:
+lemma (in bounded_bilinear) sum_prod_derivatives_has_vector_derivative:
   assumes "p>0"
   and f0: "Df 0 = f"
   and Df: "\<And>m t. m < p \<Longrightarrow> a \<le> t \<Longrightarrow> t \<le> b \<Longrightarrow>
@@ -2835,7 +2835,7 @@
     prod (Df (Suc i) t) (Dg (p - Suc i) t))) =
     (\<Sum>i\<le>(Suc p'). ?f i - ?f (Suc i))"
     by (auto simp: algebra_simps p'(2) numeral_2_eq_2 * lessThan_Suc_atMost)
-  also note setsum_telescope
+  also note sum_telescope
   finally
   have "(\<Sum>i<p. (-1) ^ i *\<^sub>R (prod (Df i t) (Dg (Suc (p - Suc i)) t) +
     prod (Df (Suc i) t) (Dg (p - Suc i) t)))
@@ -2884,7 +2884,7 @@
     unfolding Dg_def
     by (auto intro!: derivative_eq_intros simp: has_vector_derivative_def fact_eq divide_simps)
   let ?sum = "\<lambda>t. \<Sum>i<p. (- 1) ^ i *\<^sub>R Dg i t *\<^sub>R Df (p - Suc i) t"
-  from setsum_prod_derivatives_has_vector_derivative[of _ Dg _ _ _ Df,
+  from sum_prod_derivatives_has_vector_derivative[of _ Dg _ _ _ Df,
       OF \<open>p > 0\<close> g0 Dg f0 Df]
   have deriv: "\<And>t. a \<le> t \<Longrightarrow> t \<le> b \<Longrightarrow>
     (?sum has_vector_derivative
@@ -2903,7 +2903,7 @@
   then have "?sum b = f b"
     using Suc_pred'[OF \<open>p > 0\<close>]
     by (simp add: diff_eq_eq Dg_def power_0_left le_Suc_eq if_distrib
-        cond_application_beta setsum.If_cases f0)
+        cond_application_beta sum.If_cases f0)
   also
   have "{..<p} = (\<lambda>x. p - x - 1) ` {..<p}"
   proof safe
@@ -2914,7 +2914,7 @@
   qed simp
   from _ this
   have "?sum a = (\<Sum>i<p. ((b - a) ^ i / fact i) *\<^sub>R Df i a)"
-    by (rule setsum.reindex_cong) (auto simp add: inj_on_def Dg_def one)
+    by (rule sum.reindex_cong) (auto simp add: inj_on_def Dg_def one)
   finally show c: ?case .
   case 2 show ?case using c integral_unique by force
   case 3 show ?case using c by force
@@ -3003,10 +3003,10 @@
         unfolding euclidean_eq_iff[where 'a='a] using i by auto
       have *: "Basis = insert i (Basis - {i})"
         using i by auto
-      have "norm (y - x) < e + setsum (\<lambda>i. 0) Basis"
+      have "norm (y - x) < e + sum (\<lambda>i. 0) Basis"
         apply (rule le_less_trans[OF norm_le_l1])
         apply (subst *)
-        apply (subst setsum.insert)
+        apply (subst sum.insert)
         prefer 3
         apply (rule add_less_le_mono)
       proof -
@@ -3436,9 +3436,9 @@
           using inj(1) unfolding inj_on_def by fastforce
         have "(\<Sum>(x, k)\<in>(\<lambda>(x, k). (g x, g ` k)) ` p. content k *\<^sub>R f x) - i = r *\<^sub>R (\<Sum>(x, k)\<in>p. content k *\<^sub>R f (g x)) - i" (is "?l = _")
           using assms(7)
-          apply (simp only: algebra_simps add_left_cancel scaleR_right.setsum)
-          apply (subst setsum.reindex_bij_betw[symmetric, where h="\<lambda>(x, k). (g x, g ` k)" and S=p])
-          apply (auto intro!: * setsum.cong simp: bij_betw_def dest!: p(4))
+          apply (simp only: algebra_simps add_left_cancel scaleR_right.sum)
+          apply (subst sum.reindex_bij_betw[symmetric, where h="\<lambda>(x, k). (g x, g ` k)" and S=p])
+          apply (auto intro!: * sum.cong simp: bij_betw_def dest!: p(4))
           done
       also have "\<dots> = r *\<^sub>R ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f (g x)) - (1 / r) *\<^sub>R i)" (is "_ = ?r")
         unfolding scaleR_diff_right scaleR_scaleR
@@ -3865,18 +3865,18 @@
     have **: "\<And>n1 s1 n2 s2::real. n2 \<le> s2 / 2 \<Longrightarrow> n1 - s1 \<le> s2 / 2 \<Longrightarrow> n1 + n2 \<le> s1 + s2"
       by arith
     show ?case
-      unfolding content_real[OF assms(1)] and *[of "\<lambda>x. x"] *[of f] setsum_subtractf[symmetric] split_minus
-      unfolding setsum_distrib_left
+      unfolding content_real[OF assms(1)] and *[of "\<lambda>x. x"] *[of f] sum_subtractf[symmetric] split_minus
+      unfolding sum_distrib_left
       apply (subst(2) pA)
       apply (subst pA)
-      unfolding setsum.union_disjoint[OF pA(2-)]
+      unfolding sum.union_disjoint[OF pA(2-)]
     proof (rule norm_triangle_le, rule **, goal_cases)
       case 1
       show ?case
         apply (rule order_trans)
-        apply (rule setsum_norm_le)
+        apply (rule sum_norm_le)
         defer
-        apply (subst setsum_divide_distrib)
+        apply (subst sum_divide_distrib)
         apply (rule order_refl)
         apply safe
         apply (unfold not_le o_def split_conv fst_conv)
@@ -3930,12 +3930,12 @@
       case 2
       show ?case
         apply (rule *)
-        apply (rule setsum_nonneg)
+        apply (rule sum_nonneg)
         apply rule
         apply (unfold split_paired_all split_conv)
         defer
-        unfolding setsum.union_disjoint[OF pA(2-),symmetric] pA(1)[symmetric]
-        unfolding setsum_distrib_left[symmetric]
+        unfolding sum.union_disjoint[OF pA(2-),symmetric] pA(1)[symmetric]
+        unfolding sum_distrib_left[symmetric]
         apply (subst additive_tagged_division_1[OF _ as(1)])
         apply (rule assms)
       proof -
@@ -3948,12 +3948,12 @@
         then show "0 \<le> e * ((Sup k) - (Inf k))"
           unfolding uv using e by (auto simp add: field_simps)
       next
-        have *: "\<And>s f t e. setsum f s = setsum f t \<Longrightarrow> norm (setsum f t) \<le> e \<Longrightarrow> norm (setsum f s) \<le> e"
+        have *: "\<And>s f t e. sum f s = sum f t \<Longrightarrow> norm (sum f t) \<le> e \<Longrightarrow> norm (sum f s) \<le> e"
           by auto
         show "norm (\<Sum>(x, k)\<in>p \<inter> ?A. content k *\<^sub>R f' x -
           (f ((Sup k)) - f ((Inf k)))) \<le> e * (b - a) / 2"
           apply (rule *[where t1="p \<inter> {t. fst t \<in> {a, b} \<and> content(snd t) \<noteq> 0}"])
-          apply (rule setsum.mono_neutral_right[OF pA(2)])
+          apply (rule sum.mono_neutral_right[OF pA(2)])
           defer
           apply rule
           unfolding split_paired_all split_conv o_def
@@ -3975,7 +3975,7 @@
           have *: "p \<inter> {t. fst t \<in> {a, b} \<and> content(snd t) \<noteq> 0} =
             {t. t\<in>p \<and> fst t = a \<and> content(snd t) \<noteq> 0} \<union> {t. t\<in>p \<and> fst t = b \<and> content(snd t) \<noteq> 0}"
             by blast
-          have **: "norm (setsum f s) \<le> e"
+          have **: "norm (sum f s) \<le> e"
             if "\<forall>x y. x \<in> s \<and> y \<in> s \<longrightarrow> x = y"
             and "\<forall>x. x \<in> s \<longrightarrow> norm (f x) \<le> e"
             and "e > 0"
@@ -3995,7 +3995,7 @@
           case 2
           show ?case
             apply (subst *)
-            apply (subst setsum.union_disjoint)
+            apply (subst sum.union_disjoint)
             prefer 4
             apply (rule order_trans[of _ "e * (b - a)/4 + e * (b - a)/4"])
             apply (rule norm_triangle_le,rule add_mono)
@@ -4425,7 +4425,7 @@
       then show ?thesis
         unfolding ** box_real
         apply -
-        apply (subst setsum.insert)
+        apply (subst sum.insert)
         apply (rule p')
         unfolding split_conv
         defer
@@ -5003,7 +5003,7 @@
       show "c \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> d \<bullet> i" if "norm x \<le> B" and "i \<in> Basis"
         using that and Basis_le_norm[OF \<open>i\<in>Basis\<close>, of x]
         unfolding c_def d_def
-        by (auto simp add: field_simps setsum_negf)
+        by (auto simp add: field_simps sum_negf)
     qed
     have "ball 0 C \<subseteq> cbox c d"
       apply (rule subsetI)
@@ -5014,7 +5014,7 @@
       show "c \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> d \<bullet> i"
         using Basis_le_norm[OF i, of x] and x i
         unfolding c_def d_def
-        by (auto simp: setsum_negf)
+        by (auto simp: sum_negf)
     qed
     from C(2)[OF this] have "\<exists>y. (f has_integral y) (cbox a b)"
       unfolding has_integral_restrict_closed_subintervals_eq[OF c_d,symmetric]
@@ -5040,7 +5040,7 @@
         then show "c \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> d \<bullet> i"
           using Basis_le_norm[of i x]
           unfolding c_def d_def
-          by (auto simp add: field_simps setsum_negf)
+          by (auto simp add: field_simps sum_negf)
       qed
       have "ball 0 C \<subseteq> cbox c d"
         apply (rule subsetI)
@@ -5051,7 +5051,7 @@
         then show "c \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> d \<bullet> i"
           using Basis_le_norm[of i x]
           unfolding c_def d_def
-          by (auto simp: setsum_negf)
+          by (auto simp: sum_negf)
       qed
       note C(2)[OF this] then guess z .. note z = conjunctD2[OF this, unfolded s]
       note this[unfolded has_integral_restrict_closed_subintervals_eq[OF c_d]]
@@ -5425,7 +5425,7 @@
         then show ?case
           using Basis_le_norm[of i x] \<open>i\<in>Basis\<close>
           using n N
-          by (auto simp add: field_simps setsum_negf)
+          by (auto simp add: field_simps sum_negf)
       qed
     }
     then show ?case
@@ -5482,7 +5482,7 @@
           then show ?case
             using Basis_le_norm[of i x] \<open>i \<in> Basis\<close>
             using n
-            by (auto simp add: field_simps setsum_negf)
+            by (auto simp add: field_simps sum_negf)
         qed
       qed
     qed
@@ -5541,9 +5541,9 @@
       and "0 \<le> (\<Sum>(x, k)\<in>p2. content k *\<^sub>R h x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)"
       and "(\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R g x) \<ge> 0"
       and "0 \<le> (\<Sum>(x, k)\<in>p1. content k *\<^sub>R h x) - (\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x)"
-      unfolding setsum_subtractf[symmetric]
+      unfolding sum_subtractf[symmetric]
       apply -
-      apply (rule_tac[!] setsum_nonneg)
+      apply (rule_tac[!] sum_nonneg)
       apply safe
       unfolding real_scaleR_def right_diff_distrib[symmetric]
       apply (rule_tac[!] mult_nonneg_nonneg)
@@ -5774,7 +5774,7 @@
   assumes "finite t"
     and "\<forall>s\<in>t. (f has_integral (i s)) s"
     and "\<forall>s\<in>t. \<forall>s'\<in>t. s \<noteq> s' \<longrightarrow> negligible (s \<inter> s')"
-  shows "(f has_integral (setsum i t)) (\<Union>t)"
+  shows "(f has_integral (sum i t)) (\<Union>t)"
 proof -
   note * = has_integral_restrict_univ[symmetric, of f]
   have **: "negligible (\<Union>((\<lambda>(a,b). a \<inter> b) ` {(a,b). a \<in> t \<and> b \<in> {y. y \<in> t \<and> a \<noteq> y}}))"
@@ -5787,7 +5787,7 @@
     apply auto
     done
   note assms(2)[unfolded *]
-  note has_integral_setsum[OF assms(1) this]
+  note has_integral_sum[OF assms(1) this]
   then show ?thesis
     unfolding *
     apply -
@@ -5807,12 +5807,12 @@
         unfolding if_P[OF True]
         apply (rule trans)
         defer
-        apply (rule setsum.cong)
+        apply (rule sum.cong)
         apply (rule refl)
         apply (subst *)
         apply assumption
         apply (rule refl)
-        unfolding setsum.delta[OF assms(1)]
+        unfolding sum.delta[OF assms(1)]
         using s
         apply auto
         done
@@ -5827,7 +5827,7 @@
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   assumes "d division_of s"
     and "\<forall>k\<in>d. (f has_integral (i k)) k"
-  shows "(f has_integral (setsum i d)) s"
+  shows "(f has_integral (sum i d)) s"
 proof -
   note d = division_ofD[OF assms(1)]
   show ?thesis
@@ -5854,7 +5854,7 @@
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   assumes "d division_of s"
     and "\<forall>k\<in>d. f integrable_on k"
-  shows "integral s f = setsum (\<lambda>i. integral i f) d"
+  shows "integral s f = sum (\<lambda>i. integral i f) d"
   apply (rule integral_unique)
   apply (rule has_integral_combine_division[OF assms(1)])
   using assms(2)
@@ -5867,7 +5867,7 @@
   assumes "f integrable_on s"
     and "d division_of k"
     and "k \<subseteq> s"
-  shows "(f has_integral (setsum (\<lambda>i. integral i f) d)) k"
+  shows "(f has_integral (sum (\<lambda>i. integral i f) d)) k"
   apply (rule has_integral_combine_division[OF assms(2)])
   apply safe
   unfolding has_integral_integral[symmetric]
@@ -5887,7 +5887,7 @@
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   assumes "f integrable_on s"
     and "d division_of s"
-  shows "integral s f = setsum (\<lambda>i. integral i f) d"
+  shows "integral s f = sum (\<lambda>i. integral i f) d"
   apply (rule integral_unique)
   apply (rule has_integral_combine_division_topdown)
   using assms
@@ -5941,17 +5941,17 @@
     apply auto
     done
   also have "(\<Sum>k\<in>snd`p. integral k f) = (\<Sum>(x, k)\<in>p. integral k f)"
-    by (intro setsum.over_tagged_division_lemma[OF assms(1), symmetric] integral_null)
+    by (intro sum.over_tagged_division_lemma[OF assms(1), symmetric] integral_null)
        (simp add: content_eq_0_interior)
   finally show ?thesis
-    using assms by (auto simp add: has_integral_iff intro!: setsum.cong)
+    using assms by (auto simp add: has_integral_iff intro!: sum.cong)
 qed
 
 lemma integral_combine_tagged_division_bottomup:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   assumes "p tagged_division_of (cbox a b)"
     and "\<forall>(x,k)\<in>p. f integrable_on k"
-  shows "integral (cbox a b) f = setsum (\<lambda>(x,k). integral k f) p"
+  shows "integral (cbox a b) f = sum (\<lambda>(x,k). integral k f) p"
   apply (rule integral_unique)
   apply (rule has_integral_combine_tagged_division[OF assms(1)])
   using assms(2)
@@ -5962,7 +5962,7 @@
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   assumes "f integrable_on cbox a b"
     and "p tagged_division_of (cbox a b)"
-  shows "(f has_integral (setsum (\<lambda>(x,k). integral k f) p)) (cbox a b)"
+  shows "(f has_integral (sum (\<lambda>(x,k). integral k f) p)) (cbox a b)"
   apply (rule has_integral_combine_tagged_division[OF assms(2)])
   apply safe
 proof goal_cases
@@ -5976,7 +5976,7 @@
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   assumes "f integrable_on cbox a b"
     and "p tagged_division_of (cbox a b)"
-  shows "integral (cbox a b) f = setsum (\<lambda>(x,k). integral k f) p"
+  shows "integral (cbox a b) f = sum (\<lambda>(x,k). integral k f) p"
   apply (rule integral_unique)
   apply (rule has_integral_combine_tagged_division_topdown)
   using assms
@@ -5992,9 +5992,9 @@
     and "e > 0"
     and "gauge d"
     and "(\<forall>p. p tagged_division_of (cbox a b) \<and> d fine p \<longrightarrow>
-      norm (setsum (\<lambda>(x,k). content k *\<^sub>R f x) p - integral(cbox a b) f) < e)"
+      norm (sum (\<lambda>(x,k). content k *\<^sub>R f x) p - integral(cbox a b) f) < e)"
     and p: "p tagged_partial_division_of (cbox a b)" "d fine p"
-  shows "norm (setsum (\<lambda>(x,k). content k *\<^sub>R f x - integral k f) p) \<le> e"
+  shows "norm (sum (\<lambda>(x,k). content k *\<^sub>R f x - integral k f) p) \<le> e"
   (is "?x \<le> e")
 proof -
   { presume "\<And>k. 0<k \<Longrightarrow> ?x \<le> e + k" then show ?thesis by (blast intro: field_le_epsilon) }
@@ -6012,7 +6012,7 @@
     using q' unfolding r_def by auto
 
   have "\<forall>i\<in>r. \<exists>p. p tagged_division_of i \<and> d fine p \<and>
-    norm (setsum (\<lambda>(x,j). content j *\<^sub>R f x) p - integral i f) < k / (real (card r) + 1)"
+    norm (sum (\<lambda>(x,j). content j *\<^sub>R f x) p - integral i f) < k / (real (card r) + 1)"
     apply safe
   proof goal_cases
     case (1 i)
@@ -6100,7 +6100,7 @@
 
   then have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>\<Union>(qq ` r). content k *\<^sub>R f x) -
     integral (cbox a b) f) < e"
-    apply (subst setsum.union_inter_neutral[symmetric])
+    apply (subst sum.union_inter_neutral[symmetric])
     apply (rule p')
     prefer 3
     apply assumption
@@ -6124,9 +6124,9 @@
       unfolding uv content_eq_0_interior[symmetric] by auto
   qed auto
 
-  then have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) + setsum (setsum (\<lambda>(x, k). content k *\<^sub>R f x))
+  then have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) + sum (sum (\<lambda>(x, k). content k *\<^sub>R f x))
     (qq ` r) - integral (cbox a b) f) < e"
-    apply (subst (asm) setsum.Union_comp)
+    apply (subst (asm) sum.Union_comp)
     prefer 2
     unfolding split_paired_all split_conv image_iff
     apply (erule bexE)+
@@ -6150,11 +6150,11 @@
       by auto
   qed (insert qq, auto)
 
-  then have **: "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) + setsum (setsum (\<lambda>(x, k). content k *\<^sub>R f x) \<circ> qq) r -
+  then have **: "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) + sum (sum (\<lambda>(x, k). content k *\<^sub>R f x) \<circ> qq) r -
     integral (cbox a b) f) < e"
-    apply (subst (asm) setsum.reindex_nontrivial)
+    apply (subst (asm) sum.reindex_nontrivial)
     apply fact
-    apply (rule setsum.neutral)
+    apply (rule sum.neutral)
     apply rule
     unfolding split_paired_all split_conv
     defer
@@ -6180,28 +6180,28 @@
   qed
 
   have "?x =  norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p. integral k f))"
-    unfolding split_def setsum_subtractf ..
+    unfolding split_def sum_subtractf ..
   also have "\<dots> \<le> e + k"
-    apply (rule *[OF **, where ir1="setsum (\<lambda>k. integral k f) r"])
+    apply (rule *[OF **, where ir1="sum (\<lambda>k. integral k f) r"])
   proof goal_cases
     case 1
     have *: "k * real (card r) / (1 + real (card r)) < k"
       using k by (auto simp add: field_simps)
     show ?case
-      apply (rule le_less_trans[of _ "setsum (\<lambda>x. k / (real (card r) + 1)) r"])
-      unfolding setsum_subtractf[symmetric]
-      apply (rule setsum_norm_le)
+      apply (rule le_less_trans[of _ "sum (\<lambda>x. k / (real (card r) + 1)) r"])
+      unfolding sum_subtractf[symmetric]
+      apply (rule sum_norm_le)
       apply rule
       apply (drule qq)
       defer
-      unfolding divide_inverse setsum_distrib_right[symmetric]
+      unfolding divide_inverse sum_distrib_right[symmetric]
       unfolding divide_inverse[symmetric]
       using * apply (auto simp add: field_simps)
       done
   next
     case 2
     have *: "(\<Sum>(x, k)\<in>p. integral k f) = (\<Sum>k\<in>snd ` p. integral k f)"
-      apply (subst setsum.reindex_nontrivial)
+      apply (subst sum.reindex_nontrivial)
       apply fact
       unfolding split_paired_all snd_conv split_def o_def
     proof -
@@ -6221,7 +6221,7 @@
     from q(1) have **: "snd ` p \<union> q = q" by auto
     show ?case
       unfolding integral_combine_division_topdown[OF assms(1) q(2)] * r_def
-      using ** q'(1) p'(1) setsum.union_disjoint [of "snd ` p" "q - snd ` p" "\<lambda>k. integral k f", symmetric]
+      using ** q'(1) p'(1) sum.union_disjoint [of "snd ` p" "q - snd ` p" "\<lambda>k. integral k f", symmetric]
         by simp
   qed
   finally show "?x \<le> e + k" .
@@ -6233,12 +6233,12 @@
     and "e > 0"
     and "gauge d"
     and "\<forall>p. p tagged_division_of (cbox a b) \<and> d fine p \<longrightarrow>
-      norm (setsum (\<lambda>(x,k). content k *\<^sub>R f x) p - integral (cbox a b) f) < e"
+      norm (sum (\<lambda>(x,k). content k *\<^sub>R f x) p - integral (cbox a b) f) < e"
     and "p tagged_partial_division_of (cbox a b)"
     and "d fine p"
-  shows "setsum (\<lambda>(x,k). norm (content k *\<^sub>R f x - integral k f)) p \<le> 2 * real (DIM('n)) * e"
+  shows "sum (\<lambda>(x,k). norm (content k *\<^sub>R f x - integral k f)) p \<le> 2 * real (DIM('n)) * e"
   unfolding split_def
-  apply (rule setsum_norm_allsubsets_bound)
+  apply (rule sum_norm_allsubsets_bound)
   defer
   apply (rule henstock_lemma_part1[unfolded split_def,OF assms(1-3)])
   apply safe
@@ -6260,7 +6260,7 @@
     and "e > 0"
   obtains d where "gauge d"
     and "\<forall>p. p tagged_partial_division_of (cbox a b) \<and> d fine p \<longrightarrow>
-      setsum (\<lambda>(x,k). norm(content k *\<^sub>R f x - integral k f)) p < e"
+      sum (\<lambda>(x,k). norm(content k *\<^sub>R f x - integral k f)) p < e"
 proof -
   have *: "e / (2 * (real DIM('n) + 1)) > 0" using assms(2) by simp
   from integrable_integral[OF assms(1),unfolded has_integral[of f],rule_format,OF this]
@@ -6287,7 +6287,7 @@
 
 lemma sum_gp_basic:
   fixes x :: "'a::ring_1"
-  shows "(1 - x) * setsum (\<lambda>i. x^i) {0 .. n} = (1 - x^(Suc n))"
+  shows "(1 - x) * sum (\<lambda>i. x^i) {0 .. n} = (1 - x^(Suc n))"
 proof -
   define y where "y = 1 - x"
   have "y * (\<Sum>i=0..n. (1 - y) ^ i) = 1 - (1 - y) ^ Suc n"
@@ -6298,7 +6298,7 @@
 
 lemma sum_gp_multiplied:
   assumes mn: "m \<le> n"
-  shows "((1::'a::{field}) - x) * setsum (op ^ x) {m..n} = x^m - x^ Suc n"
+  shows "((1::'a::{field}) - x) * sum (op ^ x) {m..n} = x^m - x^ Suc n"
   (is "?lhs = ?rhs")
 proof -
   let ?S = "{0..(n - m)}"
@@ -6314,8 +6314,8 @@
     done
   have th: "op ^ x \<circ> op + m = (\<lambda>i. x^m * x^i)"
     by (rule ext) (simp add: power_add power_mult)
-  from setsum.reindex[OF i, of "op ^ x", unfolded f th setsum_distrib_left[symmetric]]
-  have "?lhs = x^m * ((1 - x) * setsum (op ^ x) {0..n - m})"
+  from sum.reindex[OF i, of "op ^ x", unfolded f th sum_distrib_left[symmetric]]
+  have "?lhs = x^m * ((1 - x) * sum (op ^ x) {0..n - m})"
     by simp
   then show ?thesis
     unfolding sum_gp_basic
@@ -6324,7 +6324,7 @@
 qed
 
 lemma sum_gp:
-  "setsum (op ^ (x::'a::{field})) {m .. n} =
+  "sum (op ^ (x::'a::{field})) {m .. n} =
     (if n < m then 0
      else if x = 1 then of_nat ((n + 1) - m)
      else (x^ m - x^ (Suc n)) / (1 - x))"
@@ -6357,7 +6357,7 @@
 qed
 
 lemma sum_gp_offset:
-  "setsum (op ^ (x::'a::{field})) {m .. m+n} =
+  "sum (op ^ (x::'a::{field})) {m .. m+n} =
     (if x = 1 then of_nat n + 1 else x^m * (1 - x^Suc n) / (1 - x))"
   unfolding sum_gp[of x m "m + n"] power_Suc
   by (simp add: field_simps power_add)
@@ -6497,12 +6497,12 @@
         case 1
         show ?case
           apply (rule order_trans[of _ "\<Sum>(x, k)\<in>p. content k * (e / (4 * content (cbox a b)))"])
-          unfolding setsum_subtractf[symmetric]
+          unfolding sum_subtractf[symmetric]
           apply (rule order_trans)
-          apply (rule norm_setsum)
-          apply (rule setsum_mono)
+          apply (rule norm_sum)
+          apply (rule sum_mono)
           unfolding split_paired_all split_conv
-          unfolding split_def setsum_distrib_right[symmetric] scaleR_diff_right[symmetric]
+          unfolding split_def sum_distrib_right[symmetric] scaleR_diff_right[symmetric]
           unfolding additive_content_tagged_division[OF p(1), unfolded split_def]
         proof -
           fix x k
@@ -6524,22 +6524,22 @@
         show ?case
           apply (rule le_less_trans[of _ "norm (\<Sum>j = 0..s.
             \<Sum>(x, k)\<in>{xk\<in>p. m (fst xk) = j}. content k *\<^sub>R f (m x) x - integral k (f (m x)))"])
-          apply (subst setsum_group)
+          apply (subst sum_group)
           apply fact
           apply (rule finite_atLeastAtMost)
           defer
           apply (subst split_def)+
-          unfolding setsum_subtractf
+          unfolding sum_subtractf
           apply rule
         proof -
           show "norm (\<Sum>j = 0..s. \<Sum>(x, k)\<in>{xk \<in> p.
             m (fst xk) = j}. content k *\<^sub>R f (m x) x - integral k (f (m x))) < e / 2"
-            apply (rule le_less_trans[of _ "setsum (\<lambda>i. e / 2^(i+2)) {0..s}"])
-            apply (rule setsum_norm_le)
+            apply (rule le_less_trans[of _ "sum (\<lambda>i. e / 2^(i+2)) {0..s}"])
+            apply (rule sum_norm_le)
           proof
             show "(\<Sum>i = 0..s. e / 2 ^ (i + 2)) < e / 2"
               unfolding power_add divide_inverse inverse_mult_distrib
-              unfolding setsum_distrib_left[symmetric] setsum_distrib_right[symmetric]
+              unfolding sum_distrib_left[symmetric] sum_distrib_right[symmetric]
               unfolding power_inverse [symmetric] sum_gp
               apply(rule mult_strict_left_mono[OF _ e])
               unfolding power2_eq_square
@@ -6550,10 +6550,10 @@
             show "norm (\<Sum>(x, k)\<in>{xk \<in> p. m (fst xk) = t}. content k *\<^sub>R f (m x) x -
               integral k (f (m x))) \<le> e / 2 ^ (t + 2)"
               apply (rule order_trans
-                [of _ "norm (setsum (\<lambda>(x,k). content k *\<^sub>R f t x - integral k (f t)) {xk \<in> p. m (fst xk) = t})"])
+                [of _ "norm (sum (\<lambda>(x,k). content k *\<^sub>R f t x - integral k (f t)) {xk \<in> p. m (fst xk) = t})"])
               apply (rule eq_refl)
               apply (rule arg_cong[where f=norm])
-              apply (rule setsum.cong)
+              apply (rule sum.cong)
               apply (rule refl)
               defer
               apply (rule henstock_lemma_part1)
@@ -6587,7 +6587,7 @@
           apply (rule comb[of r])
           apply (rule comb[of s])
           apply (rule i'[unfolded real_inner_1_right])
-          apply (rule_tac[1-2] setsum_mono)
+          apply (rule_tac[1-2] sum_mono)
           unfolding split_paired_all split_conv
           apply (rule_tac[1-2] integral_le[OF ])
         proof safe
@@ -7013,7 +7013,7 @@
       defer
       apply (rule d1(2)[OF conjI[OF p(1)]])
       defer
-      apply (rule setsum_norm_le)
+      apply (rule sum_norm_le)
     proof safe
       fix x k
       assume "(x, k) \<in> p"
--- a/src/HOL/Analysis/Homeomorphism.thy	Mon Oct 17 11:07:01 2016 +0200
+++ b/src/HOL/Analysis/Homeomorphism.thy	Mon Oct 17 14:37:32 2016 +0200
@@ -999,22 +999,22 @@
   proof
     show "linear f"
       unfolding f_def
-      by (intro linear_compose_setsum linearI ballI) (auto simp: algebra_simps)
+      by (intro linear_compose_sum linearI ballI) (auto simp: algebra_simps)
     show "linear g"
       unfolding g_def
-      by (intro linear_compose_setsum linearI ballI) (auto simp: algebra_simps)
+      by (intro linear_compose_sum linearI ballI) (auto simp: algebra_simps)
     have *: "(\<Sum>a \<in> Basis. a \<bullet> basf b * (x \<bullet> basg a)) = x \<bullet> b" if "b \<in> Basis" for x b
       using sbf that by auto
     show gf: "g (f x) = x" for x
       apply (rule euclidean_eqI)
-      apply (simp add: f_def g_def inner_setsum_left scaleR_setsum_left algebra_simps)
-      apply (simp add: Groups_Big.setsum_distrib_left [symmetric] *)
+      apply (simp add: f_def g_def inner_sum_left scaleR_sum_left algebra_simps)
+      apply (simp add: Groups_Big.sum_distrib_left [symmetric] *)
       done
     show "basf(0,1) \<in> Basis"
       using b01 sbf by auto
     then show "f(x,0) \<bullet> basf(0,1) = 0" for x
-      apply (simp add: f_def inner_setsum_left)
-      apply (rule comm_monoid_add_class.setsum.neutral)
+      apply (simp add: f_def inner_sum_left)
+      apply (rule comm_monoid_add_class.sum.neutral)
       using b01 inner_not_same_Basis by fastforce
   qed
 qed
--- a/src/HOL/Analysis/Inner_Product.thy	Mon Oct 17 11:07:01 2016 +0200
+++ b/src/HOL/Analysis/Inner_Product.thy	Mon Oct 17 14:37:32 2016 +0200
@@ -46,7 +46,7 @@
 lemma inner_diff_left: "inner (x - y) z = inner x z - inner y z"
   using inner_add_left [of x "- y" z] by simp
 
-lemma inner_setsum_left: "inner (\<Sum>x\<in>A. f x) y = (\<Sum>x\<in>A. inner (f x) y)"
+lemma inner_sum_left: "inner (\<Sum>x\<in>A. f x) y = (\<Sum>x\<in>A. inner (f x) y)"
   by (cases "finite A", induct set: finite, simp_all add: inner_add_left)
 
 text \<open>Transfer distributivity rules to right argument.\<close>
@@ -66,8 +66,8 @@
 lemma inner_diff_right: "inner x (y - z) = inner x y - inner x z"
   using inner_diff_left [of y z x] by (simp only: inner_commute)
 
-lemma inner_setsum_right: "inner x (\<Sum>y\<in>A. f y) = (\<Sum>y\<in>A. inner x (f y))"
-  using inner_setsum_left [of f A x] by (simp only: inner_commute)
+lemma inner_sum_right: "inner x (\<Sum>y\<in>A. f y) = (\<Sum>y\<in>A. inner x (f y))"
+  using inner_sum_left [of f A x] by (simp only: inner_commute)
 
 lemmas inner_add [algebra_simps] = inner_add_left inner_add_right
 lemmas inner_diff [algebra_simps]  = inner_diff_left inner_diff_right
--- a/src/HOL/Analysis/Integral_Test.thy	Mon Oct 17 11:07:01 2016 +0200
+++ b/src/HOL/Analysis/Integral_Test.thy	Mon Oct 17 14:37:32 2016 +0200
@@ -43,11 +43,11 @@
     with Suc show ?case by simp
   qed simp_all
   also have "... \<le> (\<Sum>k<n. integral {of_nat k..of_nat (Suc k)} (\<lambda>_::real. f (of_nat k)))"
-    by (intro setsum_mono integral_le int) (auto intro: dec)
+    by (intro sum_mono integral_le int) (auto intro: dec)
   also have "... = (\<Sum>k<n. f (of_nat k))" by simp
   also have "\<dots> - (\<Sum>k\<le>n. f (of_nat k)) = -(\<Sum>k\<in>{..n} - {..<n}. f (of_nat k))"
-    by (subst setsum_diff) auto
-  also have "\<dots> \<le> 0" by (auto intro!: setsum_nonneg nonneg)
+    by (subst sum_diff) auto
+  also have "\<dots> \<le> 0" by (auto intro!: sum_nonneg nonneg)
   finally show "sum_integral_diff_series n \<ge> 0" by simp
 qed
 
--- a/src/HOL/Analysis/L2_Norm.thy	Mon Oct 17 11:07:01 2016 +0200
+++ b/src/HOL/Analysis/L2_Norm.thy	Mon Oct 17 14:37:32 2016 +0200
@@ -28,10 +28,10 @@
 lemma setL2_insert [simp]:
   "\<lbrakk>finite F; a \<notin> F\<rbrakk> \<Longrightarrow>
     setL2 f (insert a F) = sqrt ((f a)\<^sup>2 + (setL2 f F)\<^sup>2)"
-  unfolding setL2_def by (simp add: setsum_nonneg)
+  unfolding setL2_def by (simp add: sum_nonneg)
 
 lemma setL2_nonneg [simp]: "0 \<le> setL2 f A"
-  unfolding setL2_def by (simp add: setsum_nonneg)
+  unfolding setL2_def by (simp add: sum_nonneg)
 
 lemma setL2_0': "\<forall>a\<in>A. f a = 0 \<Longrightarrow> setL2 f A = 0"
   unfolding setL2_def by simp
@@ -44,7 +44,7 @@
   assumes "\<And>i. i \<in> K \<Longrightarrow> 0 \<le> f i"
   shows "setL2 f K \<le> setL2 g K"
   unfolding setL2_def
-  by (simp add: setsum_nonneg setsum_mono power_mono assms)
+  by (simp add: sum_nonneg sum_mono power_mono assms)
 
 lemma setL2_strict_mono:
   assumes "finite K" and "K \<noteq> {}"
@@ -52,27 +52,27 @@
   assumes "\<And>i. i \<in> K \<Longrightarrow> 0 \<le> f i"
   shows "setL2 f K < setL2 g K"
   unfolding setL2_def
-  by (simp add: setsum_strict_mono power_strict_mono assms)
+  by (simp add: sum_strict_mono power_strict_mono assms)
 
 lemma setL2_right_distrib:
   "0 \<le> r \<Longrightarrow> r * setL2 f A = setL2 (\<lambda>x. r * f x) A"
   unfolding setL2_def
   apply (simp add: power_mult_distrib)
-  apply (simp add: setsum_distrib_left [symmetric])
-  apply (simp add: real_sqrt_mult setsum_nonneg)
+  apply (simp add: sum_distrib_left [symmetric])
+  apply (simp add: real_sqrt_mult sum_nonneg)
   done
 
 lemma setL2_left_distrib:
   "0 \<le> r \<Longrightarrow> setL2 f A * r = setL2 (\<lambda>x. f x * r) A"
   unfolding setL2_def
   apply (simp add: power_mult_distrib)
-  apply (simp add: setsum_distrib_right [symmetric])
-  apply (simp add: real_sqrt_mult setsum_nonneg)
+  apply (simp add: sum_distrib_right [symmetric])
+  apply (simp add: real_sqrt_mult sum_nonneg)
   done
 
 lemma setL2_eq_0_iff: "finite A \<Longrightarrow> setL2 f A = 0 \<longleftrightarrow> (\<forall>x\<in>A. f x = 0)"
   unfolding setL2_def
-  by (simp add: setsum_nonneg setsum_nonneg_eq_0_iff)
+  by (simp add: sum_nonneg sum_nonneg_eq_0_iff)
 
 lemma setL2_triangle_ineq:
   shows "setL2 (\<lambda>i. f i + g i) A \<le> setL2 f A + setL2 g A"
@@ -106,8 +106,8 @@
   apply simp
   done
 
-lemma setL2_le_setsum [rule_format]:
-  "(\<forall>i\<in>A. 0 \<le> f i) \<longrightarrow> setL2 f A \<le> setsum f A"
+lemma setL2_le_sum [rule_format]:
+  "(\<forall>i\<in>A. 0 \<le> f i) \<longrightarrow> setL2 f A \<le> sum f A"
   apply (cases "finite A")
   apply (induct set: finite)
   apply simp
@@ -124,7 +124,7 @@
   apply simp
   done
 
-lemma setL2_le_setsum_abs: "setL2 f A \<le> (\<Sum>i\<in>A. \<bar>f i\<bar>)"
+lemma setL2_le_sum_abs: "setL2 f A \<le> (\<Sum>i\<in>A. \<bar>f i\<bar>)"
   apply (cases "finite A")
   apply (induct set: finite)
   apply simp
@@ -155,7 +155,7 @@
   apply (rule order_trans)
   apply (rule power_mono)
   apply (erule add_left_mono)
-  apply (simp add: setsum_nonneg)
+  apply (simp add: sum_nonneg)
   apply (simp add: power2_sum)
   apply (simp add: power_mult_distrib)
   apply (simp add: distrib_left distrib_right)
@@ -166,6 +166,6 @@
 
 lemma member_le_setL2: "\<lbrakk>finite A; i \<in> A\<rbrakk> \<Longrightarrow> f i \<le> setL2 f A"
   unfolding setL2_def
-  by (auto intro!: member_le_setsum real_le_rsqrt)
+  by (auto intro!: member_le_sum real_le_rsqrt)
 
 end
--- a/src/HOL/Analysis/Lebesgue_Measure.thy	Mon Oct 17 11:07:01 2016 +0200
+++ b/src/HOL/Analysis/Lebesgue_Measure.thy	Mon Oct 17 14:37:32 2016 +0200
@@ -82,7 +82,7 @@
           by fastforce
 
         have "(\<Sum>i\<in>S. F (r i) - F (l i)) = (F (r m) - F (l m)) + (\<Sum>i\<in>S - {m}. F (r i) - F (l i))"
-          using m psubset by (intro setsum.remove) auto
+          using m psubset by (intro sum.remove) auto
         also have "(\<Sum>i\<in>S - {m}. F (r i) - F (l i)) \<le> F b - F (r m)"
         proof (intro psubset.IH)
           show "S - {m} \<subset> S"
@@ -144,7 +144,7 @@
             by (intro psubset) auto
           also have "\<dots> \<le> (\<Sum>i\<in>S. F (r i) - F (l i))"
             using psubset.prems
-            by (intro setsum_mono2 psubset) (auto intro: less_imp_le)
+            by (intro sum_mono2 psubset) (auto intro: less_imp_le)
           finally show ?thesis .
         next
           assume "\<not> ?R"
@@ -155,13 +155,13 @@
 
           have "(\<Sum>i\<in>S. F (r i) - F (l i)) \<ge> (\<Sum>i\<in>?S1 \<union> ?S2 \<union> {j}. F (r i) - F (l i))"
             using \<open>j \<in> S\<close> \<open>finite S\<close> psubset.prems j
-            by (intro setsum_mono2) (auto intro: less_imp_le)
+            by (intro sum_mono2) (auto intro: less_imp_le)
           also have "(\<Sum>i\<in>?S1 \<union> ?S2 \<union> {j}. F (r i) - F (l i)) =
             (\<Sum>i\<in>?S1. F (r i) - F (l i)) + (\<Sum>i\<in>?S2 . F (r i) - F (l i)) + (F (r j) - F (l j))"
             using psubset(1) psubset.prems(1) j
-            apply (subst setsum.union_disjoint)
+            apply (subst sum.union_disjoint)
             apply simp_all
-            apply (subst setsum.union_disjoint)
+            apply (subst sum.union_disjoint)
             apply auto
             apply (metis less_le_not_le)
             done
@@ -245,7 +245,7 @@
       apply arith
       by (rule deltai_gt0)
     also have "... \<le> (\<Sum>i \<in> S. F(r i) - F(l i) + epsilon / 2^(i+2))"
-      apply (rule setsum_mono)
+      apply (rule sum_mono)
       apply simp
       apply (rule order_trans)
       apply (rule less_imp_le)
@@ -253,11 +253,11 @@
       by auto
     also have "... = (\<Sum>i \<in> S. F(r i) - F(l i)) +
         (epsilon / 4) * (\<Sum>i \<in> S. (1 / 2)^i)" (is "_ = ?t + _")
-      by (subst setsum.distrib) (simp add: field_simps setsum_distrib_left)
+      by (subst sum.distrib) (simp add: field_simps sum_distrib_left)
     also have "... \<le> ?t + (epsilon / 4) * (\<Sum> i < Suc n. (1 / 2)^i)"
       apply (rule add_left_mono)
       apply (rule mult_left_mono)
-      apply (rule setsum_mono2)
+      apply (rule sum_mono2)
       using egt0 apply auto
       by (frule Sbound, auto)
     also have "... \<le> ?t + (epsilon / 2)"
@@ -281,11 +281,11 @@
     also have "... = (\<Sum>i\<in>S. F (r i) - F (l i)) + epsilon"
       by auto
     also have "... \<le> (\<Sum>i\<le>n. F (r i) - F (l i)) + epsilon"
-      using finS Sbound Sprop by (auto intro!: add_right_mono setsum_mono3)
+      using finS Sbound Sprop by (auto intro!: add_right_mono sum_mono3)
     finally have "ennreal (F b - F a) \<le> (\<Sum>i\<le>n. ennreal (F (r i) - F (l i))) + epsilon"
-      using egt0 by (simp add: ennreal_plus[symmetric] setsum_nonneg del: ennreal_plus)
+      using egt0 by (simp add: ennreal_plus[symmetric] sum_nonneg del: ennreal_plus)
     then show "ennreal (F b - F a) \<le> (\<Sum>i. ennreal (F (r i) - F (l i))) + (epsilon :: real)"
-      by (rule order_trans) (auto intro!: add_mono setsum_le_suminf simp del: setsum_ennreal)
+      by (rule order_trans) (auto intro!: add_mono sum_le_suminf simp del: sum_ennreal)
   qed
   moreover have "(\<Sum>i. ennreal (F (r i) - F (l i))) \<le> ennreal (F b - F a)"
     using \<open>a \<le> b\<close> by (auto intro!: suminf_le_const ennreal_le_iff[THEN iffD2] claim1)
@@ -637,7 +637,7 @@
   fixes t :: "'a::euclidean_space"
   shows "c \<noteq> 0 \<Longrightarrow> lborel = density (distr lborel borel (\<lambda>x. t + c *\<^sub>R x)) (\<lambda>_. \<bar>c\<bar>^DIM('a))"
   using lborel_affine_euclidean[where c="\<lambda>_::'a. c" and t=t]
-  unfolding scaleR_scaleR[symmetric] scaleR_setsum_right[symmetric] euclidean_representation setprod_constant by simp
+  unfolding scaleR_scaleR[symmetric] scaleR_sum_right[symmetric] euclidean_representation setprod_constant by simp
 
 lemma lborel_real_affine:
   "c \<noteq> 0 \<Longrightarrow> lborel = density (distr lborel borel (\<lambda>x. t + c * x)) (\<lambda>_. ennreal (abs c))"
@@ -722,7 +722,7 @@
 next
   assume "x \<noteq> 0" then show ?thesis
     using lebesgue_affine_measurable[of "\<lambda>_. x" 0]
-    unfolding scaleR_scaleR[symmetric] scaleR_setsum_right[symmetric] euclidean_representation
+    unfolding scaleR_scaleR[symmetric] scaleR_sum_right[symmetric] euclidean_representation
     by (auto simp add: ac_simps)
 qed
 
@@ -747,7 +747,7 @@
       using bij_vimage_eq_inv_image[OF \<open>bij ?T'\<close>, of S] by auto
 
     have trans_eq_T: "(\<lambda>x. \<delta> + (\<Sum>j\<in>Basis. (m * (x \<bullet> j)) *\<^sub>R j)) = T m \<delta>" for m \<delta>
-      unfolding T_def[abs_def] scaleR_scaleR[symmetric] scaleR_setsum_right[symmetric]
+      unfolding T_def[abs_def] scaleR_scaleR[symmetric] scaleR_sum_right[symmetric]
       by (auto simp add: euclidean_representation ac_simps)
 
     have T[measurable]: "T r d \<in> lebesgue \<rightarrow>\<^sub>M lebesgue" for r d
--- a/src/HOL/Analysis/Linear_Algebra.thy	Mon Oct 17 11:07:01 2016 +0200
+++ b/src/HOL/Analysis/Linear_Algebra.thy	Mon Oct 17 14:37:32 2016 +0200
@@ -141,9 +141,9 @@
 lemma linear_uminus: "linear uminus"
 by (simp add: linear_iff)
 
-lemma linear_compose_setsum:
+lemma linear_compose_sum:
   assumes lS: "\<forall>a \<in> S. linear (f a)"
-  shows "linear (\<lambda>x. setsum (\<lambda>a. f a x) S)"
+  shows "linear (\<lambda>x. sum (\<lambda>a. f a x) S)"
 proof (cases "finite S")
   case True
   then show ?thesis
@@ -173,9 +173,9 @@
 lemma linear_diff: "linear f \<Longrightarrow> f (x - y) = f x - f y"
   using linear_add [of f x "- y"] by (simp add: linear_neg)
 
-lemma linear_setsum:
+lemma linear_sum:
   assumes f: "linear f"
-  shows "f (setsum g S) = setsum (f \<circ> g) S"
+  shows "f (sum g S) = sum (f \<circ> g) S"
 proof (cases "finite S")
   case True
   then show ?thesis
@@ -186,10 +186,10 @@
     by (simp add: linear_0 [OF f])
 qed
 
-lemma linear_setsum_mul:
+lemma linear_sum_mul:
   assumes lin: "linear f"
-  shows "f (setsum (\<lambda>i. c i *\<^sub>R v i) S) = setsum (\<lambda>i. c i *\<^sub>R f (v i)) S"
-  using linear_setsum[OF lin, of "\<lambda>i. c i *\<^sub>R v i" , unfolded o_def] linear_cmul[OF lin]
+  shows "f (sum (\<lambda>i. c i *\<^sub>R v i) S) = sum (\<lambda>i. c i *\<^sub>R f (v i)) S"
+  using linear_sum[OF lin, of "\<lambda>i. c i *\<^sub>R v i" , unfolded o_def] linear_cmul[OF lin]
   by simp
 
 lemma linear_injective_0:
@@ -250,10 +250,10 @@
 lemma subspace_diff: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x - y \<in> S"
   using subspace_add [of S x "- y"] by (simp add: subspace_neg)
 
-lemma (in real_vector) subspace_setsum:
+lemma (in real_vector) subspace_sum:
   assumes sA: "subspace A"
     and f: "\<And>x. x \<in> B \<Longrightarrow> f x \<in> A"
-  shows "setsum f B \<in> A"
+  shows "sum f B \<in> A"
 proof (cases "finite B")
   case True
   then show ?thesis
@@ -460,8 +460,8 @@
 lemma span_diff: "x \<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x - y \<in> span S"
   by (metis subspace_span subspace_diff)
 
-lemma (in real_vector) span_setsum: "(\<And>x. x \<in> A \<Longrightarrow> f x \<in> span S) \<Longrightarrow> setsum f A \<in> span S"
-  by (rule subspace_setsum [OF subspace_span])
+lemma (in real_vector) span_sum: "(\<And>x. x \<in> A \<Longrightarrow> f x \<in> span S) \<Longrightarrow> sum f A \<in> span S"
+  by (rule subspace_sum [OF subspace_span])
 
 lemma span_add_eq: "x \<in> span S \<Longrightarrow> x + y \<in> span S \<longleftrightarrow> y \<in> span S"
   by (metis add_minus_cancel scaleR_minus1_left subspace_def subspace_span)
@@ -619,16 +619,16 @@
 text \<open>An explicit expansion is sometimes needed.\<close>
 
 lemma span_explicit:
-  "span P = {y. \<exists>S u. finite S \<and> S \<subseteq> P \<and> setsum (\<lambda>v. u v *\<^sub>R v) S = y}"
+  "span P = {y. \<exists>S u. finite S \<and> S \<subseteq> P \<and> sum (\<lambda>v. u v *\<^sub>R v) S = y}"
   (is "_ = ?E" is "_ = {y. ?h y}" is "_ = {y. \<exists>S u. ?Q S u y}")
 proof -
   {
     fix x
     assume "?h x"
-    then obtain S u where "finite S" and "S \<subseteq> P" and "setsum (\<lambda>v. u v *\<^sub>R v) S = x"
+    then obtain S u where "finite S" and "S \<subseteq> P" and "sum (\<lambda>v. u v *\<^sub>R v) S = x"
       by blast
     then have "x \<in> span P"
-      by (auto intro: span_setsum span_mul span_superset)
+      by (auto intro: span_sum span_mul span_superset)
   }
   moreover
   have "\<forall>x \<in> span P. ?h x"
@@ -640,7 +640,7 @@
     assume x: "x \<in> P"
     assume hy: "?h y"
     from hy obtain S u where fS: "finite S" and SP: "S\<subseteq>P"
-      and u: "setsum (\<lambda>v. u v *\<^sub>R v) S = y" by blast
+      and u: "sum (\<lambda>v. u v *\<^sub>R v) S = y" by blast
     let ?S = "insert x S"
     let ?u = "\<lambda>y. if y = x then (if x \<in> S then u y + c else c) else u y"
     from fS SP x have th0: "finite (insert x S)" "insert x S \<subseteq> P"
@@ -648,19 +648,19 @@
     have "?Q ?S ?u (c*\<^sub>R x + y)"
     proof cases
       assume xS: "x \<in> S"
-      have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = (\<Sum>v\<in>S - {x}. u v *\<^sub>R v) + (u x + c) *\<^sub>R x"
-        using xS by (simp add: setsum.remove [OF fS xS] insert_absorb)
+      have "sum (\<lambda>v. ?u v *\<^sub>R v) ?S = (\<Sum>v\<in>S - {x}. u v *\<^sub>R v) + (u x + c) *\<^sub>R x"
+        using xS by (simp add: sum.remove [OF fS xS] insert_absorb)
       also have "\<dots> = (\<Sum>v\<in>S. u v *\<^sub>R v) + c *\<^sub>R x"
-        by (simp add: setsum.remove [OF fS xS] algebra_simps)
+        by (simp add: sum.remove [OF fS xS] algebra_simps)
       also have "\<dots> = c*\<^sub>R x + y"
         by (simp add: add.commute u)
-      finally have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = c*\<^sub>R x + y" .
+      finally have "sum (\<lambda>v. ?u v *\<^sub>R v) ?S = c*\<^sub>R x + y" .
       then show ?thesis using th0 by blast
     next
       assume xS: "x \<notin> S"
       have th00: "(\<Sum>v\<in>S. (if v = x then c else u v) *\<^sub>R v) = y"
         unfolding u[symmetric]
-        apply (rule setsum.cong)
+        apply (rule sum.cong)
         using xS
         apply auto
         done
@@ -674,13 +674,13 @@
 qed
 
 lemma dependent_explicit:
-  "dependent P \<longleftrightarrow> (\<exists>S u. finite S \<and> S \<subseteq> P \<and> (\<exists>v\<in>S. u v \<noteq> 0 \<and> setsum (\<lambda>v. u v *\<^sub>R v) S = 0))"
+  "dependent P \<longleftrightarrow> (\<exists>S u. finite S \<and> S \<subseteq> P \<and> (\<exists>v\<in>S. u v \<noteq> 0 \<and> sum (\<lambda>v. u v *\<^sub>R v) S = 0))"
   (is "?lhs = ?rhs")
 proof -
   {
     assume dP: "dependent P"
     then obtain a S u where aP: "a \<in> P" and fS: "finite S"
-      and SP: "S \<subseteq> P - {a}" and ua: "setsum (\<lambda>v. u v *\<^sub>R v) S = a"
+      and SP: "S \<subseteq> P - {a}" and ua: "sum (\<lambda>v. u v *\<^sub>R v) S = a"
       unfolding dependent_def span_explicit by blast
     let ?S = "insert a S"
     let ?u = "\<lambda>y. if y = a then - 1 else u y"
@@ -689,11 +689,11 @@
       by blast
     from fS SP aP have th0: "finite ?S" "?S \<subseteq> P" "?v \<in> ?S" "?u ?v \<noteq> 0"
       by auto
-    have s0: "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = 0"
+    have s0: "sum (\<lambda>v. ?u v *\<^sub>R v) ?S = 0"
       using fS aS
       apply simp
       apply (subst (2) ua[symmetric])
-      apply (rule setsum.cong)
+      apply (rule sum.cong)
       apply auto
       done
     with th0 have ?rhs by fast
@@ -705,18 +705,18 @@
       and SP: "S \<subseteq> P"
       and vS: "v \<in> S"
       and uv: "u v \<noteq> 0"
-      and u: "setsum (\<lambda>v. u v *\<^sub>R v) S = 0"
+      and u: "sum (\<lambda>v. u v *\<^sub>R v) S = 0"
     let ?a = v
     let ?S = "S - {v}"
     let ?u = "\<lambda>i. (- u i) / u v"
     have th0: "?a \<in> P" "finite ?S" "?S \<subseteq> P"
       using fS SP vS by auto
-    have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S =
-      setsum (\<lambda>v. (- (inverse (u ?a))) *\<^sub>R (u v *\<^sub>R v)) S - ?u v *\<^sub>R v"
-      using fS vS uv by (simp add: setsum_diff1 field_simps)
+    have "sum (\<lambda>v. ?u v *\<^sub>R v) ?S =
+      sum (\<lambda>v. (- (inverse (u ?a))) *\<^sub>R (u v *\<^sub>R v)) S - ?u v *\<^sub>R v"
+      using fS vS uv by (simp add: sum_diff1 field_simps)
     also have "\<dots> = ?a"
-      unfolding scaleR_right.setsum [symmetric] u using uv by simp
-    finally have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = ?a" .
+      unfolding scaleR_right.sum [symmetric] u using uv by simp
+    finally have "sum (\<lambda>v. ?u v *\<^sub>R v) ?S = ?a" .
     with th0 have ?lhs
       unfolding dependent_def span_explicit
       apply -
@@ -740,7 +740,7 @@
     by (force simp: dependent_explicit)
   with assms show ?rhs
     apply (rule_tac x="\<lambda>v. if v \<in> T then u v else 0" in exI)
-    apply (auto simp: setsum.mono_neutral_right)
+    apply (auto simp: sum.mono_neutral_right)
     done
 next
   assume ?rhs  with assms show ?lhs
@@ -753,7 +753,7 @@
   apply safe
   subgoal for x S u
     by (intro exI[of _ "\<lambda>x. if x \<in> S then u x else 0"])
-        (auto intro!: setsum.mono_neutral_cong_right)
+        (auto intro!: sum.mono_neutral_cong_right)
   apply auto
   done
 
@@ -764,8 +764,8 @@
   apply safe
   subgoal for S u v
     apply (intro exI[of _ "\<lambda>x. if x \<in> S then u x else 0"])
-    apply (subst setsum.mono_neutral_cong_left[where T=S])
-    apply (auto intro!: setsum.mono_neutral_cong_right cong: rev_conj_cong)
+    apply (subst sum.mono_neutral_cong_left[where T=S])
+    apply (auto intro!: sum.mono_neutral_cong_right cong: rev_conj_cong)
     done
   apply auto
   done
@@ -794,13 +794,13 @@
     then show "finite {x. X x - Y x \<noteq> 0}" "{x. X x - Y x \<noteq> 0} \<subseteq> B"
       using X Y by (auto dest: finite_subset)
     then have "(\<Sum>x | X x - Y x \<noteq> 0. (X x - Y x) *\<^sub>R x) = (\<Sum>v\<in>{S. X S \<noteq> 0} \<union> {S. Y S \<noteq> 0}. (X v - Y v) *\<^sub>R v)"
-      using X Y by (intro setsum.mono_neutral_cong_left) auto
+      using X Y by (intro sum.mono_neutral_cong_left) auto
     also have "\<dots> = (\<Sum>v\<in>{S. X S \<noteq> 0} \<union> {S. Y S \<noteq> 0}. X v *\<^sub>R v) - (\<Sum>v\<in>{S. X S \<noteq> 0} \<union> {S. Y S \<noteq> 0}. Y v *\<^sub>R v)"
-      by (simp add: scaleR_diff_left setsum_subtractf assms)
+      by (simp add: scaleR_diff_left sum_subtractf assms)
     also have "(\<Sum>v\<in>{S. X S \<noteq> 0} \<union> {S. Y S \<noteq> 0}. X v *\<^sub>R v) = (\<Sum>v\<in>{S. X S \<noteq> 0}. X v *\<^sub>R v)"
-      using X Y by (intro setsum.mono_neutral_cong_right) auto
+      using X Y by (intro sum.mono_neutral_cong_right) auto
     also have "(\<Sum>v\<in>{S. X S \<noteq> 0} \<union> {S. Y S \<noteq> 0}. Y v *\<^sub>R v) = (\<Sum>v\<in>{S. Y S \<noteq> 0}. Y v *\<^sub>R v)"
-      using X Y by (intro setsum.mono_neutral_cong_right) auto
+      using X Y by (intro sum.mono_neutral_cong_right) auto
     finally show "(\<Sum>x | X x - Y x \<noteq> 0. (X x - Y x) *\<^sub>R x) = 0"
       using assms by simp
   qed
@@ -927,7 +927,7 @@
 
 lemma span_finite:
   assumes fS: "finite S"
-  shows "span S = {y. \<exists>u. setsum (\<lambda>v. u v *\<^sub>R v) S = y}"
+  shows "span S = {y. \<exists>u. sum (\<lambda>v. u v *\<^sub>R v) S = y}"
   (is "_ = ?rhs")
 proof -
   {
@@ -935,18 +935,18 @@
     assume y: "y \<in> span S"
     from y obtain S' u where fS': "finite S'"
       and SS': "S' \<subseteq> S"
-      and u: "setsum (\<lambda>v. u v *\<^sub>R v) S' = y"
+      and u: "sum (\<lambda>v. u v *\<^sub>R v) S' = y"
       unfolding span_explicit by blast
     let ?u = "\<lambda>x. if x \<in> S' then u x else 0"
-    have "setsum (\<lambda>v. ?u v *\<^sub>R v) S = setsum (\<lambda>v. u v *\<^sub>R v) S'"
-      using SS' fS by (auto intro!: setsum.mono_neutral_cong_right)
-    then have "setsum (\<lambda>v. ?u v *\<^sub>R v) S = y" by (metis u)
+    have "sum (\<lambda>v. ?u v *\<^sub>R v) S = sum (\<lambda>v. u v *\<^sub>R v) S'"
+      using SS' fS by (auto intro!: sum.mono_neutral_cong_right)
+    then have "sum (\<lambda>v. ?u v *\<^sub>R v) S = y" by (metis u)
     then have "y \<in> ?rhs" by auto
   }
   moreover
   {
     fix y u
-    assume u: "setsum (\<lambda>v. u v *\<^sub>R v) S = y"
+    assume u: "sum (\<lambda>v. u v *\<^sub>R v) S = y"
     then have "y \<in> span S" using fS unfolding span_explicit by auto
   }
   ultimately show ?thesis by blast
@@ -970,10 +970,10 @@
   proof (rule independentD_unique)
     have "(\<Sum>z | X x z + X y z \<noteq> 0. (X x z + X y z) *\<^sub>R z)
       = (\<Sum>z\<in>{z. X x z \<noteq> 0} \<union> {z. X y z \<noteq> 0}. (X x z + X y z) *\<^sub>R z)"
-      by (intro setsum.mono_neutral_cong_left) (auto intro: X)
+      by (intro sum.mono_neutral_cong_left) (auto intro: X)
     also have "\<dots> = (\<Sum>z\<in>{z. X x z \<noteq> 0}. X x z *\<^sub>R z) + (\<Sum>z\<in>{z. X y z \<noteq> 0}. X y z *\<^sub>R z)"
-      by (auto simp add: scaleR_add_left setsum.distrib
-               intro!: arg_cong2[where f="op +"]  setsum.mono_neutral_cong_right X)
+      by (auto simp add: scaleR_add_left sum.distrib
+               intro!: arg_cong2[where f="op +"]  sum.mono_neutral_cong_right X)
     also have "\<dots> = x + y"
       by (simp add: X(3)[symmetric])
     also have "\<dots> = (\<Sum>z | X (x + y) z \<noteq> 0. X (x + y) z *\<^sub>R z)"
@@ -994,7 +994,7 @@
       "finite {z. c * X x z \<noteq> 0}" "{z. c * X x z \<noteq> 0} \<subseteq> B' "
       using X(1,2) by auto
     show "(\<Sum>z | X (c *\<^sub>R x) z \<noteq> 0. X (c *\<^sub>R x) z *\<^sub>R z) = (\<Sum>z | c * X x z \<noteq> 0. (c * X x z) *\<^sub>R z)"
-      unfolding scaleR_scaleR[symmetric] scaleR_setsum_right[symmetric]
+      unfolding scaleR_scaleR[symmetric] scaleR_sum_right[symmetric]
       by (cases "c = 0") (auto simp: X(3)[symmetric])
   qed
 
@@ -1013,14 +1013,14 @@
     fix x y
     have *: "(\<Sum>z | X x z + X y z \<noteq> 0. (X x z + X y z) *\<^sub>R f' z)
       = (\<Sum>z\<in>{z. X x z \<noteq> 0} \<union> {z. X y z \<noteq> 0}. (X x z + X y z) *\<^sub>R f' z)"
-      by (intro setsum.mono_neutral_cong_left) (auto intro: X)
+      by (intro sum.mono_neutral_cong_left) (auto intro: X)
     show "g (x + y) = g x + g y"
       unfolding g_def X_add *
-      by (auto simp add: scaleR_add_left setsum.distrib
-               intro!: arg_cong2[where f="op +"]  setsum.mono_neutral_cong_right X)
+      by (auto simp add: scaleR_add_left sum.distrib
+               intro!: arg_cong2[where f="op +"]  sum.mono_neutral_cong_right X)
   next
     show "g (r *\<^sub>R x) = r *\<^sub>R g x" for r x
-      by (auto simp add: g_def X_cmult scaleR_setsum_right intro!: setsum.mono_neutral_cong_left X)
+      by (auto simp add: g_def X_cmult scaleR_sum_right intro!: sum.mono_neutral_cong_left X)
   qed
   moreover have "\<forall>x\<in>B. g x = f x"
     using \<open>B \<subseteq> B'\<close> by (auto simp: g_f' f'_def)
@@ -1436,9 +1436,9 @@
   shows "(f x) \<bullet> j = (\<Sum>i\<in>Basis. (x\<bullet>i) * (f i\<bullet>j))" (is "?lhs = ?rhs")
 proof -
   have "?rhs = (\<Sum>i\<in>Basis. (x\<bullet>i) *\<^sub>R (f i))\<bullet>j"
-    by (simp add: inner_setsum_left)
+    by (simp add: inner_sum_left)
   then show ?thesis
-    unfolding linear_setsum_mul[OF lf, symmetric]
+    unfolding linear_sum_mul[OF lf, symmetric]
     unfolding euclidean_representation ..
 qed
 
@@ -1475,30 +1475,30 @@
 lemma norm_triangle_lt: "norm x + norm y < e \<Longrightarrow> norm (x + y) < e"
   by (rule norm_triangle_ineq [THEN le_less_trans])
 
-lemma setsum_clauses:
-  shows "setsum f {} = 0"
-    and "finite S \<Longrightarrow> setsum f (insert x S) = (if x \<in> S then setsum f S else f x + setsum f S)"
+lemma sum_clauses:
+  shows "sum f {} = 0"
+    and "finite S \<Longrightarrow> sum f (insert x S) = (if x \<in> S then sum f S else f x + sum f S)"
   by (auto simp add: insert_absorb)
 
-lemma setsum_norm_le:
+lemma sum_norm_le:
   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
   assumes fg: "\<forall>x \<in> S. norm (f x) \<le> g x"
-  shows "norm (setsum f S) \<le> setsum g S"
-  by (rule order_trans [OF norm_setsum setsum_mono]) (simp add: fg)
-
-lemma setsum_norm_bound:
+  shows "norm (sum f S) \<le> sum g S"
+  by (rule order_trans [OF norm_sum sum_mono]) (simp add: fg)
+
+lemma sum_norm_bound:
   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
   assumes K: "\<forall>x \<in> S. norm (f x) \<le> K"
-  shows "norm (setsum f S) \<le> of_nat (card S) * K"
-  using setsum_norm_le[OF K] setsum_constant[symmetric]
+  shows "norm (sum f S) \<le> of_nat (card S) * K"
+  using sum_norm_le[OF K] sum_constant[symmetric]
   by simp
 
-lemma setsum_group:
+lemma sum_group:
   assumes fS: "finite S" and fT: "finite T" and fST: "f ` S \<subseteq> T"
-  shows "setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) T = setsum g S"
-  apply (subst setsum_image_gen[OF fS, of g f])
-  apply (rule setsum.mono_neutral_right[OF fT fST])
-  apply (auto intro: setsum.neutral)
+  shows "sum (\<lambda>y. sum g {x. x \<in> S \<and> f x = y}) T = sum g S"
+  apply (subst sum_image_gen[OF fS, of g f])
+  apply (rule sum.mono_neutral_right[OF fT fST])
+  apply (auto intro: sum.neutral)
   done
 
 lemma vector_eq_ldot: "(\<forall>x. x \<bullet> y = x \<bullet> z) \<longleftrightarrow> y = z"
@@ -1557,11 +1557,11 @@
   by (auto simp: pairwise_def orthogonal_clauses)
 
 lemma orthogonal_rvsum:
-    "\<lbrakk>finite s; \<And>y. y \<in> s \<Longrightarrow> orthogonal x (f y)\<rbrakk> \<Longrightarrow> orthogonal x (setsum f s)"
+    "\<lbrakk>finite s; \<And>y. y \<in> s \<Longrightarrow> orthogonal x (f y)\<rbrakk> \<Longrightarrow> orthogonal x (sum f s)"
   by (induction s rule: finite_induct) (auto simp: orthogonal_clauses)
 
 lemma orthogonal_lvsum:
-    "\<lbrakk>finite s; \<And>x. x \<in> s \<Longrightarrow> orthogonal (f x) y\<rbrakk> \<Longrightarrow> orthogonal (setsum f s) y"
+    "\<lbrakk>finite s; \<And>x. x \<in> s \<Longrightarrow> orthogonal (f x) y\<rbrakk> \<Longrightarrow> orthogonal (sum f s) y"
   by (induction s rule: finite_induct) (auto simp: orthogonal_clauses)
 
 lemma norm_add_Pythagorean:
@@ -1574,15 +1574,15 @@
     by (simp add: power2_norm_eq_inner)
 qed
 
-lemma norm_setsum_Pythagorean:
+lemma norm_sum_Pythagorean:
   assumes "finite I" "pairwise (\<lambda>i j. orthogonal (f i) (f j)) I"
-    shows "(norm (setsum f I))\<^sup>2 = (\<Sum>i\<in>I. (norm (f i))\<^sup>2)"
+    shows "(norm (sum f I))\<^sup>2 = (\<Sum>i\<in>I. (norm (f i))\<^sup>2)"
 using assms
 proof (induction I rule: finite_induct)
   case empty then show ?case by simp
 next
   case (insert x I)
-  then have "orthogonal (f x) (setsum f I)"
+  then have "orthogonal (f x) (sum f I)"
     by (metis pairwise_insert orthogonal_rvsum)
   with insert show ?case
     by (simp add: pairwise_insert norm_add_Pythagorean)
@@ -1630,25 +1630,25 @@
 lemma bilinear_rsub: "bilinear h \<Longrightarrow> h z (x - y) = h z x - h z y"
   using bilinear_radd [of h _ x "- y"] by (simp add: bilinear_rneg)
 
-lemma bilinear_setsum:
+lemma bilinear_sum:
   assumes bh: "bilinear h"
     and fS: "finite S"
     and fT: "finite T"
-  shows "h (setsum f S) (setsum g T) = setsum (\<lambda>(i,j). h (f i) (g j)) (S \<times> T) "
+  shows "h (sum f S) (sum g T) = sum (\<lambda>(i,j). h (f i) (g j)) (S \<times> T) "
 proof -
-  have "h (setsum f S) (setsum g T) = setsum (\<lambda>x. h (f x) (setsum g T)) S"
-    apply (rule linear_setsum[unfolded o_def])
+  have "h (sum f S) (sum g T) = sum (\<lambda>x. h (f x) (sum g T)) S"
+    apply (rule linear_sum[unfolded o_def])
     using bh fS
     apply (auto simp add: bilinear_def)
     done
-  also have "\<dots> = setsum (\<lambda>x. setsum (\<lambda>y. h (f x) (g y)) T) S"
-    apply (rule setsum.cong, simp)
-    apply (rule linear_setsum[unfolded o_def])
+  also have "\<dots> = sum (\<lambda>x. sum (\<lambda>y. h (f x) (g y)) T) S"
+    apply (rule sum.cong, simp)
+    apply (rule linear_sum[unfolded o_def])
     using bh fT
     apply (auto simp add: bilinear_def)
     done
   finally show ?thesis
-    unfolding setsum.cartesian_product .
+    unfolding sum.cartesian_product .
 qed
 
 
@@ -1694,10 +1694,10 @@
     have "f x \<bullet> y = f (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R i) \<bullet> y"
       by (simp add: euclidean_representation)
     also have "\<dots> = (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R f i) \<bullet> y"
-      unfolding linear_setsum[OF lf]
+      unfolding linear_sum[OF lf]
       by (simp add: linear_cmul[OF lf])
     finally show "f x \<bullet> y = x \<bullet> ?w"
-      by (simp add: inner_setsum_left inner_setsum_right mult.commute)
+      by (simp add: inner_sum_left inner_sum_right mult.commute)
   qed
   then show ?thesis
     unfolding adjoint_def choice_iff
@@ -1876,7 +1876,7 @@
   apply simp
   apply clarify
   apply (drule_tac f="inner a" in arg_cong)
-  apply (simp add: inner_Basis inner_setsum_right eq_commute)
+  apply (simp add: inner_Basis inner_sum_right eq_commute)
   done
 
 lemma span_Basis [simp]: "span Basis = UNIV"
@@ -1897,27 +1897,27 @@
 
 lemma norm_le_l1: "norm x \<le> (\<Sum>b\<in>Basis. \<bar>x \<bullet> b\<bar>)"
   apply (subst euclidean_representation[of x, symmetric])
-  apply (rule order_trans[OF norm_setsum])
-  apply (auto intro!: setsum_mono)
+  apply (rule order_trans[OF norm_sum])
+  apply (auto intro!: sum_mono)
   done
 
-lemma setsum_norm_allsubsets_bound:
+lemma sum_norm_allsubsets_bound:
   fixes f :: "'a \<Rightarrow> 'n::euclidean_space"
   assumes fP: "finite P"
-    and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (setsum f Q) \<le> e"
+    and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (sum f Q) \<le> e"
   shows "(\<Sum>x\<in>P. norm (f x)) \<le> 2 * real DIM('n) * e"
 proof -
   have "(\<Sum>x\<in>P. norm (f x)) \<le> (\<Sum>x\<in>P. \<Sum>b\<in>Basis. \<bar>f x \<bullet> b\<bar>)"
-    by (rule setsum_mono) (rule norm_le_l1)
+    by (rule sum_mono) (rule norm_le_l1)
   also have "(\<Sum>x\<in>P. \<Sum>b\<in>Basis. \<bar>f x \<bullet> b\<bar>) = (\<Sum>b\<in>Basis. \<Sum>x\<in>P. \<bar>f x \<bullet> b\<bar>)"
-    by (rule setsum.commute)
+    by (rule sum.commute)
   also have "\<dots> \<le> of_nat (card (Basis :: 'n set)) * (2 * e)"
-  proof (rule setsum_bounded_above)
+  proof (rule sum_bounded_above)
     fix i :: 'n
     assume i: "i \<in> Basis"
     have "norm (\<Sum>x\<in>P. \<bar>f x \<bullet> i\<bar>) \<le>
       norm ((\<Sum>x\<in>P \<inter> - {x. f x \<bullet> i < 0}. f x) \<bullet> i) + norm ((\<Sum>x\<in>P \<inter> {x. f x \<bullet> i < 0}. f x) \<bullet> i)"
-      by (simp add: abs_real_def setsum.If_cases[OF fP] setsum_negf norm_triangle_ineq4 inner_setsum_left
+      by (simp add: abs_real_def sum.If_cases[OF fP] sum_negf norm_triangle_ineq4 inner_sum_left
         del: real_norm_def)
     also have "\<dots> \<le> e + e"
       unfolding real_norm_def
@@ -1943,9 +1943,9 @@
     let ?g = "\<lambda>b. (x \<bullet> b) *\<^sub>R f b"
     have "norm (f x) = norm (f (\<Sum>b\<in>Basis. (x \<bullet> b) *\<^sub>R b))"
       unfolding euclidean_representation ..
-    also have "\<dots> = norm (setsum ?g Basis)"
-      by (simp add: linear_setsum [OF lf] linear_cmul [OF lf])
-    finally have th0: "norm (f x) = norm (setsum ?g Basis)" .
+    also have "\<dots> = norm (sum ?g Basis)"
+      by (simp add: linear_sum [OF lf] linear_cmul [OF lf])
+    finally have th0: "norm (f x) = norm (sum ?g Basis)" .
     have th: "\<forall>b\<in>Basis. norm (?g b) \<le> norm (f b) * norm x"
     proof
       fix i :: 'a
@@ -1958,9 +1958,9 @@
         apply (auto simp add: field_simps)
         done
     qed
-    from setsum_norm_le[of _ ?g, OF th]
+    from sum_norm_le[of _ ?g, OF th]
     show "norm (f x) \<le> ?B * norm x"
-      unfolding th0 setsum_distrib_right by metis
+      unfolding th0 sum_distrib_right by metis
   qed
 qed
 
@@ -2012,17 +2012,17 @@
 proof (clarify intro!: exI[of _ "\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)"])
   fix x :: 'm
   fix y :: 'n
-  have "norm (h x y) = norm (h (setsum (\<lambda>i. (x \<bullet> i) *\<^sub>R i) Basis) (setsum (\<lambda>i. (y \<bullet> i) *\<^sub>R i) Basis))"
+  have "norm (h x y) = norm (h (sum (\<lambda>i. (x \<bullet> i) *\<^sub>R i) Basis) (sum (\<lambda>i. (y \<bullet> i) *\<^sub>R i) Basis))"
     apply (subst euclidean_representation[where 'a='m])
     apply (subst euclidean_representation[where 'a='n])
     apply rule
     done
-  also have "\<dots> = norm (setsum (\<lambda> (i,j). h ((x \<bullet> i) *\<^sub>R i) ((y \<bullet> j) *\<^sub>R j)) (Basis \<times> Basis))"
-    unfolding bilinear_setsum[OF bh finite_Basis finite_Basis] ..
+  also have "\<dots> = norm (sum (\<lambda> (i,j). h ((x \<bullet> i) *\<^sub>R i) ((y \<bullet> j) *\<^sub>R j)) (Basis \<times> Basis))"
+    unfolding bilinear_sum[OF bh finite_Basis finite_Basis] ..
   finally have th: "norm (h x y) = \<dots>" .
   show "norm (h x y) \<le> (\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)) * norm x * norm y"
-    apply (auto simp add: setsum_distrib_right th setsum.cartesian_product)
-    apply (rule setsum_norm_le)
+    apply (auto simp add: sum_distrib_right th sum.cartesian_product)
+    apply (rule sum_norm_le)
     apply simp
     apply (auto simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh]
       field_simps simp del: scaleR_scaleR)
@@ -2353,7 +2353,7 @@
   from \<open>\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C\<close>
   obtain C where C: "finite C" "card C \<le> card B"
     "span C = span B" "pairwise orthogonal C" by blast
-  let ?a = "a - setsum (\<lambda>x. (x \<bullet> a / (x \<bullet> x)) *\<^sub>R x) C"
+  let ?a = "a - sum (\<lambda>x. (x \<bullet> a / (x \<bullet> x)) *\<^sub>R x) C"
   let ?C = "insert ?a C"
   from C(1) have fC: "finite ?C"
     by simp
@@ -2367,7 +2367,7 @@
       apply (simp only: scaleR_right_diff_distrib th0)
       apply (rule span_add_eq)
       apply (rule span_mul)
-      apply (rule span_setsum)
+      apply (rule span_sum)
       apply (rule span_mul)
       apply (rule span_superset)
       apply assumption
@@ -2384,10 +2384,10 @@
       using C by simp
     have "orthogonal ?a y"
       unfolding orthogonal_def
-      unfolding inner_diff inner_setsum_left right_minus_eq
-      unfolding setsum.remove [OF \<open>finite C\<close> \<open>y \<in> C\<close>]
+      unfolding inner_diff inner_sum_left right_minus_eq
+      unfolding sum.remove [OF \<open>finite C\<close> \<open>y \<in> C\<close>]
       apply (clarsimp simp add: inner_commute[of y a])
-      apply (rule setsum.neutral)
+      apply (rule sum.neutral)
       apply clarsimp
       apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format])
       using \<open>y \<in> C\<close> by auto
@@ -2446,10 +2446,10 @@
   from span_mono[OF B(2)] span_mono[OF B(3)]
   have sSB: "span S = span B"
     by (simp add: span_span)
-  let ?a = "a - setsum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B"
-  have "setsum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B \<in> span S"
+  let ?a = "a - sum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B"
+  have "sum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B \<in> span S"
     unfolding sSB
-    apply (rule span_setsum)
+    apply (rule span_sum)
     apply (rule span_mul)
     apply (rule span_superset)
     apply assumption
@@ -2471,10 +2471,10 @@
       have "?a \<bullet> x = 0"
         apply (subst B')
         using fB fth
-        unfolding setsum_clauses(2)[OF fth]
+        unfolding sum_clauses(2)[OF fth]
         apply simp unfolding inner_simps
-        apply (clarsimp simp add: inner_add inner_setsum_left)
-        apply (rule setsum.neutral, rule ballI)
+        apply (clarsimp simp add: inner_add inner_sum_left)
+        apply (rule sum.neutral, rule ballI)
         apply (simp only: inner_commute)
         apply (auto simp add: x field_simps
           intro: B(5)[unfolded pairwise_def orthogonal_def, rule_format])
@@ -3016,7 +3016,7 @@
 
 lemma (in euclidean_space) euclidean_inner: "inner x y = (\<Sum>b\<in>Basis. (x \<bullet> b) * (y \<bullet> b))"
   by (subst (1 2) euclidean_representation [symmetric])
-    (simp add: inner_setsum_right inner_Basis ac_simps)
+    (simp add: inner_sum_right inner_Basis ac_simps)
 
 lemma norm_le_infnorm:
   fixes x :: "'a::euclidean_space"
@@ -3033,7 +3033,7 @@
     unfolding power_mult_distrib d2
     apply (subst euclidean_inner)
     apply (subst power2_abs[symmetric])
-    apply (rule order_trans[OF setsum_bounded_above[where K="\<bar>infnorm x\<bar>\<^sup>2"]])
+    apply (rule order_trans[OF sum_bounded_above[where K="\<bar>infnorm x\<bar>\<^sup>2"]])
     apply (auto simp add: power2_eq_square[symmetric])
     apply (subst power2_abs[symmetric])
     apply (rule power_mono)
--- a/src/HOL/Analysis/Measure_Space.thy	Mon Oct 17 11:07:01 2016 +0200
+++ b/src/HOL/Analysis/Measure_Space.thy	Mon Oct 17 14:37:32 2016 +0200
@@ -21,7 +21,7 @@
   have **: "\<And>n. f n * indicator (A n) x = (if n = i then f n else 0 :: ennreal)"
     using \<open>x \<in> A i\<close> assms unfolding disjoint_family_on_def indicator_def by auto
   then have "\<And>n. (\<Sum>j<n. f j * indicator (A j) x) = (if i < n then f i else 0 :: ennreal)"
-    by (auto simp: setsum.If_cases)
+    by (auto simp: sum.If_cases)
   moreover have "(SUP n. if i < n then f i else 0) = (f i :: ennreal)"
   proof (rule SUP_eqI)
     fix y :: ennreal assume "\<And>n. n \<in> UNIV \<Longrightarrow> (if i < n then f i else 0) \<le> y"
@@ -41,7 +41,7 @@
   show ?thesis using * by simp
 qed simp
 
-lemma setsum_indicator_disjoint_family:
+lemma sum_indicator_disjoint_family:
   fixes f :: "'d \<Rightarrow> 'e::semiring_1"
   assumes d: "disjoint_family_on A P" and "x \<in> A j" and "finite P" and "j \<in> P"
   shows "(\<Sum>i\<in>P. f i * indicator (A i) x) = f j"
@@ -51,7 +51,7 @@
     by auto
   thus ?thesis
     unfolding indicator_def
-    by (simp add: if_distrib setsum.If_cases[OF \<open>finite P\<close>])
+    by (simp add: if_distrib sum.If_cases[OF \<open>finite P\<close>])
 qed
 
 text \<open>
@@ -302,7 +302,7 @@
   from fin_not_0 have "(\<Sum>i. \<mu> (F i)) = (\<Sum>i | \<mu> (F i) \<noteq> 0. \<mu> (F i))"
     by (rule suminf_finite) auto
   also have "\<dots> = (\<Sum>i | F i \<noteq> {}. \<mu> (F i))"
-    using fin_not_empty F_subset by (rule setsum.mono_neutral_left) auto
+    using fin_not_empty F_subset by (rule sum.mono_neutral_left) auto
   also have "\<dots> = \<mu> (\<Union>i\<in>{i. F i \<noteq> {}}. F i)"
     using \<open>positive M \<mu>\<close> \<open>additive M \<mu>\<close> fin_not_empty disj_not_empty F by (intro additive_sum) auto
   also have "\<dots> = \<mu> (\<Union>i. F i)"
@@ -476,7 +476,7 @@
   "A \<in> sets M \<Longrightarrow> B \<in> sets M \<Longrightarrow> emeasure M (A \<union> B) = emeasure M A + emeasure M (B - A)"
   using plus_emeasure[of A M "B - A"] by auto
 
-lemma setsum_emeasure:
+lemma sum_emeasure:
   "F`I \<subseteq> sets M \<Longrightarrow> disjoint_family_on F I \<Longrightarrow> finite I \<Longrightarrow>
     (\<Sum>i\<in>I. emeasure M (F i)) = emeasure M (\<Union>i\<in>I. F i)"
   by (metis sets.additive_sum emeasure_positive emeasure_additive)
@@ -704,20 +704,20 @@
   "A \<noteq> {} \<Longrightarrow> {x} \<in> sets M \<Longrightarrow> A \<in> sets M \<Longrightarrow> x \<notin> A \<Longrightarrow> emeasure M (insert x A) = emeasure M {x} + emeasure M A"
   by (rule emeasure_insert)
 
-lemma emeasure_eq_setsum_singleton:
+lemma emeasure_eq_sum_singleton:
   assumes "finite S" "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M"
   shows "emeasure M S = (\<Sum>x\<in>S. emeasure M {x})"
-  using setsum_emeasure[of "\<lambda>x. {x}" S M] assms
+  using sum_emeasure[of "\<lambda>x. {x}" S M] assms
   by (auto simp: disjoint_family_on_def subset_eq)
 
-lemma setsum_emeasure_cover:
+lemma sum_emeasure_cover:
   assumes "finite S" and "A \<in> sets M" and br_in_M: "B ` S \<subseteq> sets M"
   assumes A: "A \<subseteq> (\<Union>i\<in>S. B i)"
   assumes disj: "disjoint_family_on B S"
   shows "emeasure M A = (\<Sum>i\<in>S. emeasure M (A \<inter> (B i)))"
 proof -
   have "(\<Sum>i\<in>S. emeasure M (A \<inter> (B i))) = emeasure M (\<Union>i\<in>S. A \<inter> (B i))"
-  proof (rule setsum_emeasure)
+  proof (rule sum_emeasure)
     show "disjoint_family_on (\<lambda>i. A \<inter> B i) S"
       using \<open>disjoint_family_on B S\<close>
       unfolding disjoint_family_on_def by auto
@@ -749,11 +749,11 @@
   fix X assume "X \<in> sets M"
   then have X: "X \<subseteq> A" by auto
   then have "emeasure M X = (\<Sum>a\<in>X. emeasure M {a})"
-    using \<open>finite A\<close> by (subst emeasure_eq_setsum_singleton) (auto dest: finite_subset)
+    using \<open>finite A\<close> by (subst emeasure_eq_sum_singleton) (auto dest: finite_subset)
   also have "\<dots> = (\<Sum>a\<in>X. emeasure N {a})"
-    using X eq by (auto intro!: setsum.cong)
+    using X eq by (auto intro!: sum.cong)
   also have "\<dots> = emeasure N X"
-    using X \<open>finite A\<close> by (subst emeasure_eq_setsum_singleton) (auto dest: finite_subset)
+    using X \<open>finite A\<close> by (subst emeasure_eq_sum_singleton) (auto dest: finite_subset)
   finally show "emeasure M X = emeasure N X" .
 qed simp
 
@@ -1320,7 +1320,7 @@
       have "emeasure M (\<Union>i\<le>n. F i) \<le> (\<Sum>i\<le>n. emeasure M (F i))"
         using F by (auto intro!: emeasure_subadditive_finite)
       also have "\<dots> < \<infinity>"
-        using F by (auto simp: setsum_Pinfty less_top)
+        using F by (auto simp: sum_Pinfty less_top)
       finally show ?thesis by simp
     qed
     show "incseq (\<lambda>n. \<Union>i\<le>n. F i)"
@@ -1517,7 +1517,7 @@
   "finite S \<Longrightarrow> A`S \<subseteq> sets M \<Longrightarrow> disjoint_family_on A S \<Longrightarrow> (\<And>i. i \<in> S \<Longrightarrow> emeasure M (A i) \<noteq> \<infinity>) \<Longrightarrow>
     measure M (\<Union>i\<in>S. A i) = (\<Sum>i\<in>S. measure M (A i))"
   by (induction S rule: finite_induct)
-     (auto simp: disjoint_family_on_insert measure_Union setsum_emeasure[symmetric] sets.countable_UN'[OF countable_finite])
+     (auto simp: disjoint_family_on_insert measure_Union sum_emeasure[symmetric] sets.countable_UN'[OF countable_finite])
 
 lemma measure_Diff:
   assumes finite: "emeasure M A \<noteq> \<infinity>"
@@ -1576,7 +1576,7 @@
   show ?thesis
     using emeasure_subadditive_finite[OF A] fin
     unfolding emeasure_eq_ennreal_measure[OF *]
-    by (simp_all add: setsum_nonneg emeasure_eq_ennreal_measure)
+    by (simp_all add: sum_nonneg emeasure_eq_ennreal_measure)
 qed
 
 lemma measure_subadditive_countably:
@@ -1611,11 +1611,11 @@
 lemma measure_Diff_null_set: "A \<in> sets M \<Longrightarrow> B \<in> null_sets M \<Longrightarrow> measure M (A - B) = measure M A"
   by (simp add: measure_def emeasure_Diff_null_set)
 
-lemma measure_eq_setsum_singleton:
+lemma measure_eq_sum_singleton:
   "finite S \<Longrightarrow> (\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M) \<Longrightarrow> (\<And>x. x \<in> S \<Longrightarrow> emeasure M {x} \<noteq> \<infinity>) \<Longrightarrow>
     measure M S = (\<Sum>x\<in>S. measure M {x})"
-  using emeasure_eq_setsum_singleton[of S M]
-  by (intro measure_eq_emeasure_eq_ennreal) (auto simp: setsum_nonneg emeasure_eq_ennreal_measure)
+  using emeasure_eq_sum_singleton[of S M]
+  by (intro measure_eq_emeasure_eq_ennreal) (auto simp: sum_nonneg emeasure_eq_ennreal_measure)
 
 lemma Lim_measure_incseq:
   assumes A: "range A \<subseteq> sets M" "incseq A" and fin: "emeasure M (\<Union>i. A i) \<noteq> \<infinity>"
@@ -1886,10 +1886,10 @@
   by (rule measure_subadditive_countably)
      (simp_all add: ennreal_suminf_neq_top emeasure_eq_measure)
 
-lemma (in finite_measure) finite_measure_eq_setsum_singleton:
+lemma (in finite_measure) finite_measure_eq_sum_singleton:
   assumes "finite S" and *: "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M"
   shows "measure M S = (\<Sum>x\<in>S. measure M {x})"
-  using measure_eq_setsum_singleton[OF assms] by simp
+  using measure_eq_sum_singleton[OF assms] by simp
 
 lemma (in finite_measure) finite_Lim_measure_incseq:
   assumes A: "range A \<subseteq> sets M" "incseq A"
@@ -1985,10 +1985,10 @@
   from someI_ex[OF this] assms
   have prob_some: "\<And> x. x \<in> s \<Longrightarrow> measure M {x} = measure M {SOME y. y \<in> s}" by blast
   have "measure M s = (\<Sum> x \<in> s. measure M {x})"
-    using finite_measure_eq_setsum_singleton[OF s] by simp
+    using finite_measure_eq_sum_singleton[OF s] by simp
   also have "\<dots> = (\<Sum> x \<in> s. measure M {SOME y. y \<in> s})" using prob_some by auto
   also have "\<dots> = real (card s) * measure M {(SOME x. x \<in> s)}"
-    using setsum_constant assms by simp
+    using sum_constant assms by simp
   finally show ?thesis by simp
 qed simp
 
@@ -3581,7 +3581,7 @@
       using \<open>?M \<noteq> 0\<close>
       by (simp add: \<open>card X = Suc (Suc n)\<close> field_simps less_le)
     also have "\<dots> \<le> (\<Sum>x\<in>X. ?m x)"
-      by (rule setsum_mono) fact
+      by (rule sum_mono) fact
     also have "\<dots> = measure M (\<Union>x\<in>X. {x})"
       using singleton_sets \<open>finite X\<close>
       by (intro finite_measure_finite_Union[symmetric]) (auto simp: disjoint_family_on_def)
--- a/src/HOL/Analysis/Nonnegative_Lebesgue_Integration.thy	Mon Oct 17 11:07:01 2016 +0200
+++ b/src/HOL/Analysis/Nonnegative_Lebesgue_Integration.thy	Mon Oct 17 14:37:32 2016 +0200
@@ -68,9 +68,9 @@
 proof -
   have "?r = (\<Sum>y \<in> f ` space M.
     (if y = f x then y * indicator (f -` {y} \<inter> space M) x else 0))"
-    by (auto intro!: setsum.cong)
+    by (auto intro!: sum.cong)
   also have "... =  f x *  indicator (f -` {f x} \<inter> space M) x"
-    using assms by (auto dest: simple_functionD simp: setsum.delta)
+    using assms by (auto dest: simple_functionD simp: sum.delta)
   also have "... = f x" using x by (auto simp: indicator_def)
   finally show ?thesis by auto
 qed
@@ -188,7 +188,7 @@
   and simple_function_inverse[intro, simp] = simple_function_compose[where g="inverse"]
   and simple_function_max[intro, simp] = simple_function_compose2[where h=max]
 
-lemma simple_function_setsum[intro, simp]:
+lemma simple_function_sum[intro, simp]:
   assumes "\<And>i. i \<in> P \<Longrightarrow> simple_function M (f i)"
   shows "simple_function M (\<lambda>x. \<Sum>i\<in>P. f i x)"
 proof cases
@@ -362,10 +362,10 @@
       case (insert x S)
       { fix z have "(\<Sum>y\<in>S. y * indicator (u -` {y} \<inter> space M) z) = 0 \<or>
           x * indicator (u -` {x} \<inter> space M) z = 0"
-          using insert by (subst setsum_eq_0_iff) (auto simp: indicator_def) }
+          using insert by (subst sum_eq_0_iff) (auto simp: indicator_def) }
       note disj = this
       from insert show ?case
-        by (auto intro!: add mult set simple_functionD u simple_function_setsum disj)
+        by (auto intro!: add mult set simple_functionD u simple_function_sum disj)
     qed
   qed fact
 qed
@@ -530,7 +530,7 @@
     (\<Sum>y\<in>f`space M. y * (\<Sum>z\<in>g`space M.
       if \<exists>x\<in>space M. y = f x \<and> z = g x then emeasure M {x\<in>space M. g x = z} else 0))"
     unfolding simple_integral_def
-  proof (safe intro!: setsum.cong ennreal_mult_left_cong)
+  proof (safe intro!: sum.cong ennreal_mult_left_cong)
     fix y assume y: "y \<in> space M" "f y \<noteq> 0"
     have [simp]: "g ` space M \<inter> {z. \<exists>x\<in>space M. f y = f x \<and> z = g x} =
         {z. \<exists>x\<in>space M. f y = f x \<and> z = g x}"
@@ -543,17 +543,17 @@
       by (rule rev_finite_subset) auto
     then show "emeasure M (f -` {f y} \<inter> space M) =
       (\<Sum>z\<in>g ` space M. if \<exists>x\<in>space M. f y = f x \<and> z = g x then emeasure M {x \<in> space M. g x = z} else 0)"
-      apply (simp add: setsum.If_cases)
-      apply (subst setsum_emeasure)
+      apply (simp add: sum.If_cases)
+      apply (subst sum_emeasure)
       apply (auto simp: disjoint_family_on_def eq)
       done
   qed
   also have "\<dots> = (\<Sum>y\<in>f`space M. (\<Sum>z\<in>g`space M.
       if \<exists>x\<in>space M. y = f x \<and> z = g x then y * emeasure M {x\<in>space M. g x = z} else 0))"
-    by (auto intro!: setsum.cong simp: setsum_distrib_left)
+    by (auto intro!: sum.cong simp: sum_distrib_left)
   also have "\<dots> = ?r"
-    by (subst setsum.commute)
-       (auto intro!: setsum.cong simp: setsum.If_cases scaleR_setsum_right[symmetric] eq)
+    by (subst sum.commute)
+       (auto intro!: sum.cong simp: sum.If_cases scaleR_sum_right[symmetric] eq)
   finally show "integral\<^sup>S M f = ?r" .
 qed
 
@@ -566,7 +566,7 @@
     by (intro simple_function_partition) (auto intro: f g)
   also have "\<dots> = (\<Sum>y\<in>(\<lambda>x. (f x, g x))`space M. fst y * emeasure M {x\<in>space M. (f x, g x) = y}) +
     (\<Sum>y\<in>(\<lambda>x. (f x, g x))`space M. snd y * emeasure M {x\<in>space M. (f x, g x) = y})"
-    using assms(2,4) by (auto intro!: setsum.cong distrib_right simp: setsum.distrib[symmetric])
+    using assms(2,4) by (auto intro!: sum.cong distrib_right simp: sum.distrib[symmetric])
   also have "(\<Sum>y\<in>(\<lambda>x. (f x, g x))`space M. fst y * emeasure M {x\<in>space M. (f x, g x) = y}) = (\<integral>\<^sup>Sx. f x \<partial>M)"
     by (intro simple_function_partition[symmetric]) (auto intro: f g)
   also have "(\<Sum>y\<in>(\<lambda>x. (f x, g x))`space M. snd y * emeasure M {x\<in>space M. (f x, g x) = y}) = (\<integral>\<^sup>Sx. g x \<partial>M)"
@@ -574,14 +574,14 @@
   finally show ?thesis .
 qed
 
-lemma simple_integral_setsum[simp]:
+lemma simple_integral_sum[simp]:
   assumes "\<And>i x. i \<in> P \<Longrightarrow> 0 \<le> f i x"
   assumes "\<And>i. i \<in> P \<Longrightarrow> simple_function M (f i)"
   shows "(\<integral>\<^sup>Sx. (\<Sum>i\<in>P. f i x) \<partial>M) = (\<Sum>i\<in>P. integral\<^sup>S M (f i))"
 proof cases
   assume "finite P"
   from this assms show ?thesis
-    by induct (auto simp: simple_function_setsum simple_integral_add setsum_nonneg)
+    by induct (auto simp: simple_function_sum simple_integral_add sum_nonneg)
 qed auto
 
 lemma simple_integral_mult[simp]:
@@ -592,7 +592,7 @@
     using f by (intro simple_function_partition) auto
   also have "\<dots> = c * integral\<^sup>S M f"
     using f unfolding simple_integral_def
-    by (subst setsum_distrib_left) (auto simp: mult.assoc Int_def conj_commute)
+    by (subst sum_distrib_left) (auto simp: mult.assoc Int_def conj_commute)
   finally show ?thesis .
 qed
 
@@ -605,7 +605,7 @@
   have "integral\<^sup>S M f = (\<Sum>y\<in>(\<lambda>x. (f x, g x))`space M. fst y * ?\<mu> (\<lambda>x. (f x, g x) = y))"
     using f g by (intro simple_function_partition) auto
   also have "\<dots> \<le> (\<Sum>y\<in>(\<lambda>x. (f x, g x))`space M. snd y * ?\<mu> (\<lambda>x. (f x, g x) = y))"
-  proof (clarsimp intro!: setsum_mono)
+  proof (clarsimp intro!: sum_mono)
     fix x assume "x \<in> space M"
     let ?M = "?\<mu> (\<lambda>y. f y = f x \<and> g y = g x)"
     show "f x * ?M \<le> g x * ?M"
@@ -664,14 +664,14 @@
     using assms by (intro simple_function_partition) auto
   also have "\<dots> = (\<Sum>y\<in>(\<lambda>x. (f x, indicator A x::ennreal))`space M.
     if snd y = 1 then fst y * emeasure M (f -` {fst y} \<inter> space M \<inter> A) else 0)"
-    by (auto simp: indicator_def split: if_split_asm intro!: arg_cong2[where f="op *"] arg_cong2[where f=emeasure] setsum.cong)
+    by (auto simp: indicator_def split: if_split_asm intro!: arg_cong2[where f="op *"] arg_cong2[where f=emeasure] sum.cong)
   also have "\<dots> = (\<Sum>y\<in>(\<lambda>x. (f x, 1::ennreal))`A. fst y * emeasure M (f -` {fst y} \<inter> space M \<inter> A))"
-    using assms by (subst setsum.If_cases) (auto intro!: simple_functionD(1) simp: eq)
+    using assms by (subst sum.If_cases) (auto intro!: simple_functionD(1) simp: eq)
   also have "\<dots> = (\<Sum>y\<in>fst`(\<lambda>x. (f x, 1::ennreal))`A. y * emeasure M (f -` {y} \<inter> space M \<inter> A))"
-    by (subst setsum.reindex [of fst]) (auto simp: inj_on_def)
+    by (subst sum.reindex [of fst]) (auto simp: inj_on_def)
   also have "\<dots> = (\<Sum>x \<in> f ` space M. x * emeasure M (f -` {x} \<inter> space M \<inter> A))"
     using A[THEN sets.sets_into_space]
-    by (intro setsum.mono_neutral_cong_left simple_functionD f) (auto simp: image_comp comp_def eq2)
+    by (intro sum.mono_neutral_cong_left simple_functionD f) (auto simp: image_comp comp_def eq2)
   finally show ?thesis .
 qed
 
@@ -848,7 +848,7 @@
       let ?B = "\<lambda>c i. {x\<in>space M. ?au x = c \<and> c \<le> f i x}"
       have "integral\<^sup>S M ?au = (\<Sum>c\<in>?au`space M. c * (SUP i. emeasure M (?B c i)))"
         unfolding simple_integral_def
-      proof (intro setsum.cong ennreal_mult_left_cong refl)
+      proof (intro sum.cong ennreal_mult_left_cong refl)
         fix c assume "c \<in> ?au ` space M" "c \<noteq> 0"
         { fix x' assume x': "x' \<in> space M" "?au x' = c"
           with \<open>c \<noteq> 0\<close> u_range have "?au x' < 1 * u x'"
@@ -866,7 +866,7 @@
       qed
       also have "\<dots> = (SUP i. \<Sum>c\<in>?au`space M. c * emeasure M (?B c i))"
         unfolding SUP_mult_left_ennreal using f
-        by (intro ennreal_SUP_setsum[symmetric])
+        by (intro ennreal_SUP_sum[symmetric])
            (auto intro!: mult_mono emeasure_mono simp: incseq_def le_fun_def intro: order_trans)
       also have "\<dots> \<le> (SUP i. integral\<^sup>N M (f i))"
       proof (intro SUP_subset_mono order_refl)
@@ -874,7 +874,7 @@
         have "(\<Sum>c\<in>?au`space M. c * emeasure M (?B c i)) =
           (\<integral>\<^sup>Sx. (a * u x) * indicator {x\<in>space M. a * u x \<le> f i x} x \<partial>M)"
           by (subst simple_integral_indicator)
-             (auto intro!: setsum.cong ennreal_mult_left_cong arg_cong2[where f=emeasure])
+             (auto intro!: sum.cong ennreal_mult_left_cong arg_cong2[where f=emeasure])
         also have "\<dots> = (\<integral>\<^sup>+x. (a * u x) * indicator {x\<in>space M. a * u x \<le> f i x} x \<partial>M)"
           by (rule nn_integral_eq_simple_integral[symmetric]) simp
         also have "\<dots> \<le> (\<integral>\<^sup>+x. f i x \<partial>M)"
@@ -1031,7 +1031,7 @@
   "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (\<integral>\<^sup>+ x. f x + g x \<partial>M) = integral\<^sup>N M f + integral\<^sup>N M g"
   using nn_integral_linear[of f M g 1] by simp
 
-lemma nn_integral_setsum:
+lemma nn_integral_sum:
   "(\<And>i. i \<in> P \<Longrightarrow> f i \<in> borel_measurable M) \<Longrightarrow> (\<integral>\<^sup>+ x. (\<Sum>i\<in>P. f i x) \<partial>M) = (\<Sum>i\<in>P. integral\<^sup>N M (f i))"
   by (induction P rule: infinite_finite_induct) (auto simp: nn_integral_add)
 
@@ -1044,10 +1044,10 @@
   have "(\<Sum>i. integral\<^sup>N M (f i)) = (SUP n. \<Sum>i<n. integral\<^sup>N M (f i))"
     by (rule suminf_eq_SUP)
   also have "\<dots> = (SUP n. \<integral>\<^sup>+x. (\<Sum>i<n. f i x) \<partial>M)"
-    unfolding nn_integral_setsum[OF f] ..
+    unfolding nn_integral_sum[OF f] ..
   also have "\<dots> = \<integral>\<^sup>+x. (SUP n. \<Sum>i<n. f i x) \<partial>M" using f all_pos
     by (intro nn_integral_monotone_convergence_SUP_AE[symmetric])
-       (elim AE_mp, auto simp: setsum_nonneg simp del: setsum_lessThan_Suc intro!: AE_I2 setsum_mono3)
+       (elim AE_mp, auto simp: sum_nonneg simp del: sum_lessThan_Suc intro!: AE_I2 sum_mono3)
   also have "\<dots> = \<integral>\<^sup>+x. (\<Sum>i. f i x) \<partial>M" using all_pos
     by (intro nn_integral_cong_AE) (auto simp: suminf_eq_SUP)
   finally show ?thesis by simp
@@ -1534,7 +1534,7 @@
     by (simp add: simple_function_def)
   show "integral\<^sup>S M x \<le> integral\<^sup>S N x" for x
     using le_measureD3[OF \<open>M \<le> N\<close>]
-    by (auto simp add: simple_integral_def intro!: setsum_mono mult_mono)
+    by (auto simp add: simple_integral_def intro!: sum_mono mult_mono)
 qed
 
 lemma nn_integral_empty:
@@ -1587,17 +1587,17 @@
   have *: "(\<integral>\<^sup>+x. max 0 (f x) \<partial>count_space A) =
     (\<integral>\<^sup>+ x. (\<Sum>a|a\<in>A \<and> 0 < f a. f a * indicator {a} x) \<partial>count_space A)"
     by (auto intro!: nn_integral_cong
-             simp add: indicator_def if_distrib setsum.If_cases[OF A] max_def le_less)
+             simp add: indicator_def if_distrib sum.If_cases[OF A] max_def le_less)
   also have "\<dots> = (\<Sum>a|a\<in>A \<and> 0 < f a. \<integral>\<^sup>+ x. f a * indicator {a} x \<partial>count_space A)"
-    by (subst nn_integral_setsum) (simp_all add: AE_count_space  less_imp_le)
+    by (subst nn_integral_sum) (simp_all add: AE_count_space  less_imp_le)
   also have "\<dots> = (\<Sum>a|a\<in>A \<and> 0 < f a. f a)"
-    by (auto intro!: setsum.cong simp: one_ennreal_def[symmetric] max_def)
+    by (auto intro!: sum.cong simp: one_ennreal_def[symmetric] max_def)
   finally show ?thesis by (simp add: max.absorb2)
 qed
 
 lemma nn_integral_count_space_finite:
     "finite A \<Longrightarrow> (\<integral>\<^sup>+x. f x \<partial>count_space A) = (\<Sum>a\<in>A. f a)"
-  by (auto intro!: setsum.mono_neutral_left simp: nn_integral_count_space less_le)
+  by (auto intro!: sum.mono_neutral_left simp: nn_integral_count_space less_le)
 
 lemma nn_integral_count_space':
   assumes "finite A" "\<And>x. x \<in> B \<Longrightarrow> x \<notin> A \<Longrightarrow> f x = 0" "A \<subseteq> B"
@@ -1607,7 +1607,7 @@
     using assms(2,3)
     by (intro nn_integral_count_space finite_subset[OF _ \<open>finite A\<close>]) (auto simp: less_le)
   also have "\<dots> = (\<Sum>a\<in>A. f a)"
-    using assms by (intro setsum.mono_neutral_cong_left) (auto simp: less_le)
+    using assms by (intro sum.mono_neutral_cong_left) (auto simp: less_le)
   finally show ?thesis .
 qed
 
@@ -1624,9 +1624,9 @@
   shows "(\<integral>\<^sup>+x. f x * indicator A x \<partial>M) = (\<Sum>x\<in>A. f x * emeasure M {x})"
 proof -
   from f have "(\<integral>\<^sup>+x. f x * indicator A x \<partial>M) = (\<integral>\<^sup>+x. (\<Sum>a\<in>A. f a * indicator {a} x) \<partial>M)"
-    by (intro nn_integral_cong) (auto simp: indicator_def if_distrib[where f="\<lambda>a. x * a" for x] setsum.If_cases)
+    by (intro nn_integral_cong) (auto simp: indicator_def if_distrib[where f="\<lambda>a. x * a" for x] sum.If_cases)
   also have "\<dots> = (\<Sum>a\<in>A. f a * emeasure M {a})"
-    by (subst nn_integral_setsum) auto
+    by (subst nn_integral_sum) auto
   finally show ?thesis .
 qed
 
@@ -1680,7 +1680,7 @@
   shows "(\<integral>\<^sup>+x. \<integral>\<^sup>+i. f i x \<partial>count_space I \<partial>M) = (\<integral>\<^sup>+i. \<integral>\<^sup>+x. f i x \<partial>M \<partial>count_space I)"
 proof cases
   assume "finite I" then show ?thesis
-    by (simp add: nn_integral_count_space_finite nn_integral_setsum)
+    by (simp add: nn_integral_count_space_finite nn_integral_sum)
 next
   assume "infinite I"
   then have [simp]: "I \<noteq> {}"
@@ -1823,7 +1823,7 @@
         then obtain m' where "m' < m" and m': "(SUP i:?Y. f i) = f (I m')" by auto
         have "I m' \<in> Y" using I by blast
         have "(\<Sum>x<m. g x n) \<le> (\<Sum>x<m. f (I m') x)"
-        proof(rule setsum_mono)
+        proof(rule sum_mono)
           fix x
           assume "x \<in> {..<m}"
           hence "x < m" by simp
@@ -1920,7 +1920,7 @@
   using simple_function_restrict_space_ennreal[THEN iffD1, OF \<Omega>, THEN simple_functionD(1)]
   by (auto simp add: space_restrict_space emeasure_restrict_space[OF \<Omega>(1)] le_infI2 simple_integral_def
            split: split_indicator split_indicator_asm
-           intro!: setsum.mono_neutral_cong_left ennreal_mult_left_cong arg_cong2[where f=emeasure])
+           intro!: sum.mono_neutral_cong_left ennreal_mult_left_cong arg_cong2[where f=emeasure])
 
 lemma nn_integral_restrict_space:
   assumes \<Omega>[simp]: "\<Omega> \<inter> space M \<in> sets M"
@@ -2234,12 +2234,12 @@
 
 lemma emeasure_point_measure_finite:
   "finite A \<Longrightarrow> X \<subseteq> A \<Longrightarrow> emeasure (point_measure A f) X = (\<Sum>a\<in>X. f a)"
-  by (subst emeasure_point_measure) (auto dest: finite_subset intro!: setsum.mono_neutral_left simp: less_le)
+  by (subst emeasure_point_measure) (auto dest: finite_subset intro!: sum.mono_neutral_left simp: less_le)
 
 lemma emeasure_point_measure_finite2:
   "X \<subseteq> A \<Longrightarrow> finite X \<Longrightarrow> emeasure (point_measure A f) X = (\<Sum>a\<in>X. f a)"
   by (subst emeasure_point_measure)
-     (auto dest: finite_subset intro!: setsum.mono_neutral_left simp: less_le)
+     (auto dest: finite_subset intro!: sum.mono_neutral_left simp: less_le)
 
 lemma null_sets_point_measure_iff:
   "X \<in> null_sets (point_measure A f) \<longleftrightarrow> X \<subseteq> A \<and> (\<forall>x\<in>X. f x = 0)"
@@ -2259,7 +2259,7 @@
 
 lemma nn_integral_point_measure_finite:
   "finite A \<Longrightarrow> integral\<^sup>N (point_measure A f) g = (\<Sum>a\<in>A. f a * g a)"
-  by (subst nn_integral_point_measure) (auto intro!: setsum.mono_neutral_left simp: less_le)
+  by (subst nn_integral_point_measure) (auto intro!: sum.mono_neutral_left simp: less_le)
 
 subsubsection \<open>Uniform measure\<close>
 
--- a/src/HOL/Analysis/Ordered_Euclidean_Space.thy	Mon Oct 17 11:07:01 2016 +0200
+++ b/src/HOL/Analysis/Ordered_Euclidean_Space.thy	Mon Oct 17 14:37:32 2016 +0200
@@ -46,7 +46,7 @@
 
 lemma inner_Basis_inf_left: "i \<in> Basis \<Longrightarrow> inf x y \<bullet> i = inf (x \<bullet> i) (y \<bullet> i)"
   and inner_Basis_sup_left: "i \<in> Basis \<Longrightarrow> sup x y \<bullet> i = sup (x \<bullet> i) (y \<bullet> i)"
-  by (simp_all add: eucl_inf eucl_sup inner_setsum_left inner_Basis if_distrib comm_monoid_add_class.setsum.delta
+  by (simp_all add: eucl_inf eucl_sup inner_sum_left inner_Basis if_distrib comm_monoid_add_class.sum.delta
       cong: if_cong)
 
 lemma inner_Basis_INF_left: "i \<in> Basis \<Longrightarrow> (INF x:X. f x) \<bullet> i = (INF x:X. f x \<bullet> i)"
@@ -65,7 +65,7 @@
   shows "a\<bullet>i \<le> x\<bullet>i" "x\<bullet>i \<le> b\<bullet>i"
   using assms
   unfolding euclidean_inner[of a i] euclidean_inner[of x i] euclidean_inner[of b i]
-  by (auto intro!: ordered_comm_monoid_add_class.setsum_mono mult_right_mono simp: eucl_le)
+  by (auto intro!: ordered_comm_monoid_add_class.sum_mono mult_right_mono simp: eucl_le)
 
 lemma inner_nonneg_nonneg:
   shows "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a \<bullet> b"
@@ -86,7 +86,7 @@
   assumes i_s: "\<And>b. b \<in> Basis \<Longrightarrow> (i b \<bullet> b) \<in> (\<lambda>x. x \<bullet> b) ` s"
   shows "Sup s = X"
   using assms
-  unfolding eucl_Sup euclidean_representation_setsum
+  unfolding eucl_Sup euclidean_representation_sum
   by (auto intro!: conditionally_complete_lattice_class.cSup_eq_maximum)
 
 lemma Inf_eq_minimum_componentwise:
@@ -95,7 +95,7 @@
   assumes i_s: "\<And>b. b \<in> Basis \<Longrightarrow> (i b \<bullet> b) \<in> (\<lambda>x. x \<bullet> b) ` s"
   shows "Inf s = X"
   using assms
-  unfolding eucl_Inf euclidean_representation_setsum
+  unfolding eucl_Inf euclidean_representation_sum
   by (auto intro!: conditionally_complete_lattice_class.cInf_eq_minimum)
 
 end
@@ -117,7 +117,7 @@
   hence "Inf ?proj = x \<bullet> b"
     by (auto intro!: conditionally_complete_lattice_class.cInf_eq_minimum)
   hence "x \<bullet> b = Inf X \<bullet> b"
-    by (auto simp: eucl_Inf inner_setsum_left inner_Basis if_distrib \<open>b \<in> Basis\<close> setsum.delta
+    by (auto simp: eucl_Inf inner_sum_left inner_Basis if_distrib \<open>b \<in> Basis\<close> sum.delta
       cong: if_cong)
   with x show "\<exists>x. x \<in> X \<and> x \<bullet> b = Inf X \<bullet> b \<and> (\<forall>y. y \<in> X \<longrightarrow> x \<bullet> b \<le> y \<bullet> b)" by blast
 qed
@@ -139,7 +139,7 @@
   hence "Sup ?proj = x \<bullet> b"
     by (auto intro!: cSup_eq_maximum)
   hence "x \<bullet> b = Sup X \<bullet> b"
-    by (auto simp: eucl_Sup[where 'a='a] inner_setsum_left inner_Basis if_distrib \<open>b \<in> Basis\<close> setsum.delta
+    by (auto simp: eucl_Sup[where 'a='a] inner_sum_left inner_Basis if_distrib \<open>b \<in> Basis\<close> sum.delta
       cong: if_cong)
   with x show "\<exists>x. x \<in> X \<and> x \<bullet> b = Sup X \<bullet> b \<and> (\<forall>y. y \<in> X \<longrightarrow> y \<bullet> b \<le> x \<bullet> b)" by blast
 qed
@@ -167,7 +167,7 @@
 
 instance prod :: (ordered_euclidean_space, ordered_euclidean_space) ordered_euclidean_space
   by standard
-    (auto intro!: add_mono simp add: euclidean_representation_setsum'  Ball_def inner_prod_def
+    (auto intro!: add_mono simp add: euclidean_representation_sum'  Ball_def inner_prod_def
       in_Basis_prod_iff inner_Basis_inf_left inner_Basis_sup_left inner_Basis_INF_left Inf_prod_def
       inner_Basis_SUP_left Sup_prod_def less_prod_def less_eq_prod_def eucl_le[where 'a='a]
       eucl_le[where 'a='b] abs_prod_def abs_inner)
@@ -241,7 +241,7 @@
   eucl_less (infix "<e" 50)
 
 lemma One_nonneg: "0 \<le> (\<Sum>Basis::'a::ordered_euclidean_space)"
-  by (auto intro: setsum_nonneg)
+  by (auto intro: sum_nonneg)
 
 lemma
   fixes a b::"'a::ordered_euclidean_space"
@@ -264,7 +264,7 @@
 
 instance
   apply standard
-  unfolding euclidean_representation_setsum'
+  unfolding euclidean_representation_sum'
   apply (auto simp: less_eq_vec_def inf_vec_def sup_vec_def Inf_vec_def Sup_vec_def inner_axis
     Basis_vec_def inner_Basis_inf_left inner_Basis_sup_left inner_Basis_INF_left
     inner_Basis_SUP_left eucl_le[where 'a='a] less_le_not_le abs_vec_def abs_inner)
--- a/src/HOL/Analysis/Path_Connected.thy	Mon Oct 17 11:07:01 2016 +0200
+++ b/src/HOL/Analysis/Path_Connected.thy	Mon Oct 17 14:37:32 2016 +0200
@@ -5760,18 +5760,18 @@
     obtain a where x: "x = (\<Sum>v \<in> B. a v *\<^sub>R v)"
       using \<open>finite B\<close> \<open>span B = S\<close> \<open>x \<in> S\<close> span_finite by fastforce
     have "f x = (\<Sum>v \<in> B. f (a v *\<^sub>R v))"
-      using linear_setsum [OF \<open>linear f\<close>] x by auto
+      using linear_sum [OF \<open>linear f\<close>] x by auto
     also have "... = (\<Sum>v \<in> B. a v *\<^sub>R f v)"
-      using \<open>linear f\<close> by (simp add: linear_setsum linear.scaleR)
+      using \<open>linear f\<close> by (simp add: linear_sum linear.scaleR)
     also have "... = (\<Sum>v \<in> B. a v *\<^sub>R fb v)"
-      by (simp add: ffb cong: setsum.cong)
+      by (simp add: ffb cong: sum.cong)
     finally have "norm (f x)^2 = norm (\<Sum>v\<in>B. a v *\<^sub>R fb v)^2" by simp
     also have "... = (\<Sum>v\<in>B. norm ((a v *\<^sub>R fb v))^2)"
-      apply (rule norm_setsum_Pythagorean [OF \<open>finite B\<close>])
+      apply (rule norm_sum_Pythagorean [OF \<open>finite B\<close>])
       apply (rule pairwise_ortho_scaleR [OF pairwise_orth_fb])
       done
     also have "... = norm x ^2"
-      by (simp add: x pairwise_ortho_scaleR Borth norm_setsum_Pythagorean [OF \<open>finite B\<close>])
+      by (simp add: x pairwise_ortho_scaleR Borth norm_sum_Pythagorean [OF \<open>finite B\<close>])
     finally show ?thesis
       by (simp add: norm_eq_sqrt_inner)
   qed
@@ -5823,28 +5823,28 @@
     obtain a where x: "x = (\<Sum>v \<in> B. a v *\<^sub>R v)"
       using \<open>finite B\<close> \<open>span B = S\<close> \<open>x \<in> S\<close> span_finite by fastforce
     have "f x = (\<Sum>v \<in> B. f (a v *\<^sub>R v))"
-      using linear_setsum [OF \<open>linear f\<close>] x by auto
+      using linear_sum [OF \<open>linear f\<close>] x by auto
     also have "... = (\<Sum>v \<in> B. a v *\<^sub>R f v)"
-      using \<open>linear f\<close> by (simp add: linear_setsum linear.scaleR)
+      using \<open>linear f\<close> by (simp add: linear_sum linear.scaleR)
     also have "... = (\<Sum>v \<in> B. a v *\<^sub>R fb v)"
-      by (simp add: ffb cong: setsum.cong)
+      by (simp add: ffb cong: sum.cong)
     finally have *: "f x = (\<Sum>v\<in>B. a v *\<^sub>R fb v)" .
     then have "(norm (f x))\<^sup>2 = (norm (\<Sum>v\<in>B. a v *\<^sub>R fb v))\<^sup>2" by simp
     also have "... = (\<Sum>v\<in>B. norm ((a v *\<^sub>R fb v))^2)"
-      apply (rule norm_setsum_Pythagorean [OF \<open>finite B\<close>])
+      apply (rule norm_sum_Pythagorean [OF \<open>finite B\<close>])
       apply (rule pairwise_ortho_scaleR [OF pairwise_orth_fb])
       done
     also have "... = (norm x)\<^sup>2"
-      by (simp add: x pairwise_ortho_scaleR Borth norm_setsum_Pythagorean [OF \<open>finite B\<close>])
+      by (simp add: x pairwise_ortho_scaleR Borth norm_sum_Pythagorean [OF \<open>finite B\<close>])
     finally show "norm (f x) = norm x"
       by (simp add: norm_eq_sqrt_inner)
     have "g (f x) = g (\<Sum>v\<in>B. a v *\<^sub>R fb v)" by (simp add: *)
     also have "... = (\<Sum>v\<in>B. g (a v *\<^sub>R fb v))"
-      using \<open>linear g\<close> by (simp add: linear_setsum linear.scaleR)
+      using \<open>linear g\<close> by (simp add: linear_sum linear.scaleR)
     also have "... = (\<Sum>v\<in>B. a v *\<^sub>R g (fb v))"
       by (simp add: \<open>linear g\<close> linear.scaleR)
     also have "... = (\<Sum>v\<in>B. a v *\<^sub>R v)"
-      apply (rule setsum.cong [OF refl])
+      apply (rule sum.cong [OF refl])
       using \<open>bij_betw fb B C\<close> gb_def bij_betwE bij_betw_inv_into_left gb_def ggb by fastforce
     also have "... = x"
       using x by blast
@@ -5857,14 +5857,14 @@
     obtain a where x: "x = (\<Sum>v \<in> C. a v *\<^sub>R v)"
       using \<open>finite C\<close> \<open>span C = T\<close> \<open>x \<in> T\<close> span_finite by fastforce
     have "g x = (\<Sum>v \<in> C. g (a v *\<^sub>R v))"
-      using linear_setsum [OF \<open>linear g\<close>] x by auto
+      using linear_sum [OF \<open>linear g\<close>] x by auto
     also have "... = (\<Sum>v \<in> C. a v *\<^sub>R g v)"
-      using \<open>linear g\<close> by (simp add: linear_setsum linear.scaleR)
+      using \<open>linear g\<close> by (simp add: linear_sum linear.scaleR)
     also have "... = (\<Sum>v \<in> C. a v *\<^sub>R gb v)"
-      by (simp add: ggb cong: setsum.cong)
+      by (simp add: ggb cong: sum.cong)
     finally have "f (g x) = f (\<Sum>v\<in>C. a v *\<^sub>R gb v)" by simp
     also have "... = (\<Sum>v\<in>C. f (a v *\<^sub>R gb v))"
-      using \<open>linear f\<close> by (simp add: linear_setsum linear.scaleR)
+      using \<open>linear f\<close> by (simp add: linear_sum linear.scaleR)
     also have "... = (\<Sum>v\<in>C. a v *\<^sub>R f (gb v))"
       by (simp add: \<open>linear f\<close> linear.scaleR)
     also have "... = (\<Sum>v\<in>C. a v *\<^sub>R v)"
--- a/src/HOL/Analysis/Poly_Roots.thy	Mon Oct 17 11:07:01 2016 +0200
+++ b/src/HOL/Analysis/Poly_Roots.thy	Mon Oct 17 14:37:32 2016 +0200
@@ -10,67 +10,67 @@
 
 subsection\<open>Geometric progressions\<close>
 
-lemma setsum_gp_basic:
+lemma sum_gp_basic:
   fixes x :: "'a::{comm_ring,monoid_mult}"
   shows "(1 - x) * (\<Sum>i\<le>n. x^i) = 1 - x^Suc n"
   by (simp only: one_diff_power_eq [of "Suc n" x] lessThan_Suc_atMost)
 
-lemma setsum_gp0:
+lemma sum_gp0:
   fixes x :: "'a::{comm_ring,division_ring}"
   shows   "(\<Sum>i\<le>n. x^i) = (if x = 1 then of_nat(n + 1) else (1 - x^Suc n) / (1 - x))"
-  using setsum_gp_basic[of x n]
+  using sum_gp_basic[of x n]
   by (simp add: mult.commute divide_simps)
 
-lemma setsum_power_add:
+lemma sum_power_add:
   fixes x :: "'a::{comm_ring,monoid_mult}"
   shows "(\<Sum>i\<in>I. x^(m+i)) = x^m * (\<Sum>i\<in>I. x^i)"
-  by (simp add: setsum_distrib_left power_add)
+  by (simp add: sum_distrib_left power_add)
 
-lemma setsum_power_shift:
+lemma sum_power_shift:
   fixes x :: "'a::{comm_ring,monoid_mult}"
   assumes "m \<le> n"
   shows "(\<Sum>i=m..n. x^i) = x^m * (\<Sum>i\<le>n-m. x^i)"
 proof -
   have "(\<Sum>i=m..n. x^i) = x^m * (\<Sum>i=m..n. x^(i-m))"
-    by (simp add: setsum_distrib_left power_add [symmetric])
+    by (simp add: sum_distrib_left power_add [symmetric])
   also have "(\<Sum>i=m..n. x^(i-m)) = (\<Sum>i\<le>n-m. x^i)"
-    using \<open>m \<le> n\<close> by (intro setsum.reindex_bij_witness[where j="\<lambda>i. i - m" and i="\<lambda>i. i + m"]) auto
+    using \<open>m \<le> n\<close> by (intro sum.reindex_bij_witness[where j="\<lambda>i. i - m" and i="\<lambda>i. i + m"]) auto
   finally show ?thesis .
 qed
 
-lemma setsum_gp_multiplied:
+lemma sum_gp_multiplied:
   fixes x :: "'a::{comm_ring,monoid_mult}"
   assumes "m \<le> n"
   shows "(1 - x) * (\<Sum>i=m..n. x^i) = x^m - x^Suc n"
 proof -
   have  "(1 - x) * (\<Sum>i=m..n. x^i) = x^m * (1 - x) * (\<Sum>i\<le>n-m. x^i)"
-    by (metis mult.assoc mult.commute assms setsum_power_shift)
+    by (metis mult.assoc mult.commute assms sum_power_shift)
   also have "... =x^m * (1 - x^Suc(n-m))"
-    by (metis mult.assoc setsum_gp_basic)
+    by (metis mult.assoc sum_gp_basic)
   also have "... = x^m - x^Suc n"
     using assms
     by (simp add: algebra_simps) (metis le_add_diff_inverse power_add)
   finally show ?thesis .
 qed
 
-lemma setsum_gp:
+lemma sum_gp:
   fixes x :: "'a::{comm_ring,division_ring}"
   shows   "(\<Sum>i=m..n. x^i) =
                (if n < m then 0
                 else if x = 1 then of_nat((n + 1) - m)
                 else (x^m - x^Suc n) / (1 - x))"
-using setsum_gp_multiplied [of m n x]
+using sum_gp_multiplied [of m n x]
 apply auto
 by (metis eq_iff_diff_eq_0 mult.commute nonzero_divide_eq_eq)
 
-lemma setsum_gp_offset:
+lemma sum_gp_offset:
   fixes x :: "'a::{comm_ring,division_ring}"
   shows   "(\<Sum>i=m..m+n. x^i) =
        (if x = 1 then of_nat n + 1 else x^m * (1 - x^Suc n) / (1 - x))"
-  using setsum_gp [of x m "m+n"]
+  using sum_gp [of x m "m+n"]
   by (auto simp: power_add algebra_simps)
 
-lemma setsum_gp_strict:
+lemma sum_gp_strict:
   fixes x :: "'a::{comm_ring,division_ring}"
   shows "(\<Sum>i<n. x^i) = (if x = 1 then of_nat n else (1 - x^n) / (1 - x))"
   by (induct n) (auto simp: algebra_simps divide_simps)
@@ -84,13 +84,13 @@
 proof -
   have "(\<Sum>i\<le>n. a i * x^i) - (\<Sum>i\<le>n. a i * y^i) =
         (\<Sum>i\<le>n. a i * (x^i - y^i))"
-    by (simp add: algebra_simps setsum_subtractf [symmetric])
+    by (simp add: algebra_simps sum_subtractf [symmetric])
   also have "... = (\<Sum>i\<le>n. a i * (x - y) * (\<Sum>j<i. y^(i - Suc j) * x^j))"
     by (simp add: power_diff_sumr2 ac_simps)
   also have "... = (x - y) * (\<Sum>i\<le>n. (\<Sum>j<i. a i * y^(i - Suc j) * x^j))"
-    by (simp add: setsum_distrib_left ac_simps)
+    by (simp add: sum_distrib_left ac_simps)
   also have "... = (x - y) * (\<Sum>j<n. (\<Sum>i=Suc j..n. a i * y^(i - Suc j) * x^j))"
-    by (simp add: nested_setsum_swap')
+    by (simp add: nested_sum_swap')
   finally show ?thesis .
 qed
 
@@ -102,7 +102,7 @@
   { fix j
     have "(\<Sum>k = Suc j..n. a k * y^(k - Suc j) * x^j) =
           (\<Sum>k <n - j. a (Suc (j + k)) * y^k * x^j)"
-      by (rule setsum.reindex_bij_witness[where i="\<lambda>i. i + Suc j" and j="\<lambda>i. i - Suc j"]) auto }
+      by (rule sum.reindex_bij_witness[where i="\<lambda>i. i + Suc j" and j="\<lambda>i. i - Suc j"]) auto }
   then show ?thesis
     by (simp add: sub_polyfun)
 qed
@@ -115,7 +115,7 @@
   { fix z
     have "(\<Sum>i\<le>n. c i * z^i) - (\<Sum>i\<le>n. c i * a^i) =
           (z - a) * (\<Sum>j<n. (\<Sum>k = Suc j..n. c k * a^(k - Suc j)) * z^j)"
-      by (simp add: sub_polyfun setsum_distrib_right)
+      by (simp add: sub_polyfun sum_distrib_right)
     then have "(\<Sum>i\<le>n. c i * z^i) =
           (z - a) * (\<Sum>j<n. (\<Sum>k = Suc j..n. c k * a^(k - Suc j)) * z^j)
           + (\<Sum>i\<le>n. c i * a^i)"
@@ -241,7 +241,7 @@
    have "\<exists>k\<le>n. b k \<noteq> 0"
      apply (rule ccontr)
      using polyfun_extremal [OF extr_prem, of 1]
-     apply (auto simp: eventually_at_infinity b simp del: setsum_atMost_Suc)
+     apply (auto simp: eventually_at_infinity b simp del: sum_atMost_Suc)
      apply (drule_tac x="of_real ba" in spec, simp)
      done
    then show ?thesis using Suc.IH [of b] ins_ab
--- a/src/HOL/Analysis/Polytope.thy	Mon Oct 17 11:07:01 2016 +0200
+++ b/src/HOL/Analysis/Polytope.thy	Mon Oct 17 14:37:32 2016 +0200
@@ -293,8 +293,8 @@
   proof -
     have waff: "w \<in> affine hull T"
       using convex_hull_subset_affine_hull w by blast
-    obtain a b where a: "\<And>i. i \<in> S \<Longrightarrow> 0 \<le> a i" and asum: "setsum a S = 1" and aeqx: "(\<Sum>i\<in>S. a i *\<^sub>R i) = x"
-                 and b: "\<And>i. i \<in> S \<Longrightarrow> 0 \<le> b i" and bsum: "setsum b S = 1" and beqy: "(\<Sum>i\<in>S. b i *\<^sub>R i) = y"
+    obtain a b where a: "\<And>i. i \<in> S \<Longrightarrow> 0 \<le> a i" and asum: "sum a S = 1" and aeqx: "(\<Sum>i\<in>S. a i *\<^sub>R i) = x"
+                 and b: "\<And>i. i \<in> S \<Longrightarrow> 0 \<le> b i" and bsum: "sum b S = 1" and beqy: "(\<Sum>i\<in>S. b i *\<^sub>R i) = y"
       using x y by (auto simp: assms convex_hull_finite)
     obtain u where "(1 - u) *\<^sub>R x + u *\<^sub>R y \<in> convex hull T" "x \<noteq> y" and weq: "w = (1 - u) *\<^sub>R x + u *\<^sub>R y"
                and u01: "0 < u" "u < 1"
@@ -302,43 +302,43 @@
     define c where "c i = (1 - u) * a i + u * b i" for i
     have cge0: "\<And>i. i \<in> S \<Longrightarrow> 0 \<le> c i"
       using a b u01 by (simp add: c_def)
-    have sumc1: "setsum c S = 1"
-      by (simp add: c_def setsum.distrib setsum_distrib_left [symmetric] asum bsum)
+    have sumc1: "sum c S = 1"
+      by (simp add: c_def sum.distrib sum_distrib_left [symmetric] asum bsum)
     have sumci_xy: "(\<Sum>i\<in>S. c i *\<^sub>R i) = (1 - u) *\<^sub>R x + u *\<^sub>R y"
-      apply (simp add: c_def setsum.distrib scaleR_left_distrib)
-      by (simp only: scaleR_scaleR [symmetric] Real_Vector_Spaces.scaleR_right.setsum [symmetric] aeqx beqy)
+      apply (simp add: c_def sum.distrib scaleR_left_distrib)
+      by (simp only: scaleR_scaleR [symmetric] Real_Vector_Spaces.scaleR_right.sum [symmetric] aeqx beqy)
     show ?thesis
-    proof (cases "setsum c (S - T) = 0")
+    proof (cases "sum c (S - T) = 0")
       case True
       have ci0: "\<And>i. i \<in> (S - T) \<Longrightarrow> c i = 0"
-        using True cge0 by (simp add: \<open>finite S\<close> setsum_nonneg_eq_0_iff)
+        using True cge0 by (simp add: \<open>finite S\<close> sum_nonneg_eq_0_iff)
       have a0: "a i = 0" if "i \<in> (S - T)" for i
         using ci0 [OF that] u01 a [of i] b [of i] that
         by (simp add: c_def Groups.ordered_comm_monoid_add_class.add_nonneg_eq_0_iff)
-      have [simp]: "setsum a T = 1"
-        using assms by (metis setsum.mono_neutral_cong_right a0 asum)
+      have [simp]: "sum a T = 1"
+        using assms by (metis sum.mono_neutral_cong_right a0 asum)
       show ?thesis
         apply (simp add: convex_hull_finite \<open>finite T\<close>)
         apply (rule_tac x=a in exI)
         using a0 assms
-        apply (auto simp: cge0 a aeqx [symmetric] setsum.mono_neutral_right)
+        apply (auto simp: cge0 a aeqx [symmetric] sum.mono_neutral_right)
         done
     next
       case False
-      define k where "k = setsum c (S - T)"
+      define k where "k = sum c (S - T)"
       have "k > 0" using False
-        unfolding k_def by (metis DiffD1 antisym_conv cge0 setsum_nonneg not_less)
-      have weq_sumsum: "w = setsum (\<lambda>x. c x *\<^sub>R x) T + setsum (\<lambda>x. c x *\<^sub>R x) (S - T)"
-        by (metis (no_types) add.commute S(1) S(2) setsum.subset_diff sumci_xy weq)
+        unfolding k_def by (metis DiffD1 antisym_conv cge0 sum_nonneg not_less)
+      have weq_sumsum: "w = sum (\<lambda>x. c x *\<^sub>R x) T + sum (\<lambda>x. c x *\<^sub>R x) (S - T)"
+        by (metis (no_types) add.commute S(1) S(2) sum.subset_diff sumci_xy weq)
       show ?thesis
       proof (cases "k = 1")
         case True
-        then have "setsum c T = 0"
-          by (simp add: S k_def setsum_diff sumc1)
-        then have [simp]: "setsum c (S - T) = 1"
-          by (simp add: S setsum_diff sumc1)
+        then have "sum c T = 0"
+          by (simp add: S k_def sum_diff sumc1)
+        then have [simp]: "sum c (S - T) = 1"
+          by (simp add: S sum_diff sumc1)
         have ci0: "\<And>i. i \<in> T \<Longrightarrow> c i = 0"
-          by (meson \<open>finite T\<close> \<open>setsum c T = 0\<close> \<open>T \<subseteq> S\<close> cge0 setsum_nonneg_eq_0_iff subsetCE)
+          by (meson \<open>finite T\<close> \<open>sum c T = 0\<close> \<open>T \<subseteq> S\<close> cge0 sum_nonneg_eq_0_iff subsetCE)
         then have [simp]: "(\<Sum>i\<in>S-T. c i *\<^sub>R i) = w"
           by (simp add: weq_sumsum)
         have "w \<in> convex hull (S - T)"
@@ -350,22 +350,22 @@
           using disj waff by blast
       next
         case False
-        then have sumcf: "setsum c T = 1 - k"
-          by (simp add: S k_def setsum_diff sumc1)
+        then have sumcf: "sum c T = 1 - k"
+          by (simp add: S k_def sum_diff sumc1)
         have "(\<Sum>i\<in>T. c i *\<^sub>R i) /\<^sub>R (1 - k) \<in> convex hull T"
           apply (simp add: convex_hull_finite fin)
           apply (rule_tac x="\<lambda>i. inverse (1-k) * c i" in exI)
           apply auto
-          apply (metis sumcf cge0 inverse_nonnegative_iff_nonnegative mult_nonneg_nonneg S(2) setsum_nonneg subsetCE)
-          apply (metis False mult.commute right_inverse right_minus_eq setsum_distrib_left sumcf)
-          by (metis (mono_tags, lifting) scaleR_right.setsum scaleR_scaleR setsum.cong)
-        with \<open>0 < k\<close>  have "inverse(k) *\<^sub>R (w - setsum (\<lambda>i. c i *\<^sub>R i) T) \<in> affine hull T"
+          apply (metis sumcf cge0 inverse_nonnegative_iff_nonnegative mult_nonneg_nonneg S(2) sum_nonneg subsetCE)
+          apply (metis False mult.commute right_inverse right_minus_eq sum_distrib_left sumcf)
+          by (metis (mono_tags, lifting) scaleR_right.sum scaleR_scaleR sum.cong)
+        with \<open>0 < k\<close>  have "inverse(k) *\<^sub>R (w - sum (\<lambda>i. c i *\<^sub>R i) T) \<in> affine hull T"
           by (simp add: affine_diff_divide [OF affine_affine_hull] False waff convex_hull_subset_affine_hull [THEN subsetD])
-        moreover have "inverse(k) *\<^sub>R (w - setsum (\<lambda>x. c x *\<^sub>R x) T) \<in> convex hull (S - T)"
+        moreover have "inverse(k) *\<^sub>R (w - sum (\<lambda>x. c x *\<^sub>R x) T) \<in> convex hull (S - T)"
           apply (simp add: weq_sumsum convex_hull_finite fin)
           apply (rule_tac x="\<lambda>i. inverse k * c i" in exI)
           using \<open>k > 0\<close> cge0
-          apply (auto simp: scaleR_right.setsum setsum_distrib_left [symmetric] k_def [symmetric])
+          apply (auto simp: scaleR_right.sum sum_distrib_left [symmetric] k_def [symmetric])
           done
         ultimately show ?thesis
           using disj by blast
@@ -2727,7 +2727,7 @@
           have "norm (x - y) \<le> (\<Sum>b\<in>Basis. \<bar>(x-y) \<bullet> b\<bar>)"
             by (rule norm_le_l1)
           also have "... \<le> of_nat (DIM('a)) * (e / 2 / DIM('a))"
-          proof (rule setsum_bounded_above)
+          proof (rule sum_bounded_above)
             fix i::'a
             assume "i \<in> Basis"
             then have I': "\<And>z b. \<lbrakk>z \<in> C; b = z * e / (2 * real DIM('a))\<rbrakk> \<Longrightarrow> i \<bullet> x \<le> b \<and> i \<bullet> y \<le> b \<or> i \<bullet> x \<ge> b \<and> i \<bullet> y \<ge> b"
--- a/src/HOL/Analysis/Regularity.thy	Mon Oct 17 11:07:01 2016 +0200
+++ b/src/HOL/Analysis/Regularity.thy	Mon Oct 17 14:37:32 2016 +0200
@@ -260,7 +260,7 @@
     also have "(\<lambda>n. \<Sum>i<n. M (D i)) \<longlonglongrightarrow> (\<Sum>i. M (D i))"
       by (intro summable_LIMSEQ) auto
     finally have measure_LIMSEQ: "(\<lambda>n. \<Sum>i<n. measure M (D i)) \<longlonglongrightarrow> measure M (\<Union>i. D i)"
-      by (simp add: emeasure_eq_measure measure_nonneg setsum_nonneg)
+      by (simp add: emeasure_eq_measure measure_nonneg sum_nonneg)
     have "(\<Union>i. D i) \<in> sets M" using \<open>range D \<subseteq> sets M\<close> by auto
 
     case 1
@@ -274,12 +274,12 @@
       then obtain n0 where n0: "\<bar>(\<Sum>i<n0. measure M (D i)) - measure M (\<Union>i. D i)\<bar> < e/2"
         unfolding choice_iff by blast
       have "ennreal (\<Sum>i<n0. measure M (D i)) = (\<Sum>i<n0. M (D i))"
-        by (auto simp add: emeasure_eq_measure setsum_nonneg measure_nonneg)
-      also have "\<dots> \<le> (\<Sum>i. M (D i))" by (rule setsum_le_suminf) auto
+        by (auto simp add: emeasure_eq_measure sum_nonneg measure_nonneg)
+      also have "\<dots> \<le> (\<Sum>i. M (D i))" by (rule sum_le_suminf) auto
       also have "\<dots> = M (\<Union>i. D i)" by (simp add: M)
       also have "\<dots> = measure M (\<Union>i. D i)" by (simp add: emeasure_eq_measure)
       finally have n0: "measure M (\<Union>i. D i) - (\<Sum>i<n0. measure M (D i)) < e/2"
-        using n0 by (auto simp: measure_nonneg setsum_nonneg)
+        using n0 by (auto simp: measure_nonneg sum_nonneg)
       have "\<forall>i. \<exists>K. K \<subseteq> D i \<and> compact K \<and> emeasure M (D i) \<le> emeasure M K + e/(2*Suc n0)"
       proof
         fix i
@@ -301,16 +301,16 @@
       have "measure M (\<Union>i. D i) < (\<Sum>i<n0. measure M (D i)) + e/2" using n0 by simp
       also have "(\<Sum>i<n0. measure M (D i)) \<le> (\<Sum>i<n0. measure M (K i) + e/(2*Suc n0))"
         using K \<open>0 < e\<close>
-        by (auto intro: setsum_mono simp: emeasure_eq_measure measure_nonneg ennreal_plus[symmetric] simp del: ennreal_plus)
+        by (auto intro: sum_mono simp: emeasure_eq_measure measure_nonneg ennreal_plus[symmetric] simp del: ennreal_plus)
       also have "\<dots> = (\<Sum>i<n0. measure M (K i)) + (\<Sum>i<n0. e/(2*Suc n0))"
-        by (simp add: setsum.distrib)
+        by (simp add: sum.distrib)
       also have "\<dots> \<le> (\<Sum>i<n0. measure M (K i)) +  e / 2" using \<open>0 < e\<close>
         by (auto simp: field_simps intro!: mult_left_mono)
       finally
       have "measure M (\<Union>i. D i) < (\<Sum>i<n0. measure M (K i)) + e / 2 + e / 2"
         by auto
       hence "M (\<Union>i. D i) < M ?K + e"
-        using \<open>0<e\<close> by (auto simp: mK emeasure_eq_measure measure_nonneg setsum_nonneg ennreal_less_iff ennreal_plus[symmetric] simp del: ennreal_plus)
+        using \<open>0<e\<close> by (auto simp: mK emeasure_eq_measure measure_nonneg sum_nonneg ennreal_less_iff ennreal_plus[symmetric] simp del: ennreal_plus)
       moreover
       have "?K \<subseteq> (\<Union>i. D i)" using K by auto
       moreover
@@ -332,7 +332,7 @@
         from INF_approx_ennreal[OF \<open>0 < e/(2 powr Suc i)\<close> this]
         show "\<exists>U. D i \<subseteq> U \<and> open U \<and> e/(2 powr Suc i) > emeasure M U - emeasure M (D i)"
           using \<open>0<e\<close>
-          by (auto simp: emeasure_eq_measure measure_nonneg setsum_nonneg ennreal_less_iff ennreal_plus[symmetric] ennreal_minus
+          by (auto simp: emeasure_eq_measure measure_nonneg sum_nonneg ennreal_less_iff ennreal_plus[symmetric] ennreal_minus
                          finite_measure_mono sb
                    simp del: ennreal_plus)
       qed
--- a/src/HOL/Analysis/Summation_Tests.thy	Mon Oct 17 11:07:01 2016 +0200
+++ b/src/HOL/Analysis/Summation_Tests.thy	Mon Oct 17 14:37:32 2016 +0200
@@ -218,7 +218,7 @@
   assumes mono: "\<And>m n. 0 < m \<Longrightarrow> m \<le> n \<Longrightarrow> f n \<le> f m"
   shows   "(\<Sum>k=1..<n. f k) \<ge> (\<Sum>k=1..<n. f (2 * 2 ^ natlog2 k))" (is "?thesis1")
           "(\<Sum>k=1..<n. f k) \<le> (\<Sum>k=1..<n. f (2 ^ natlog2 k))" (is "?thesis2")
-  by (intro setsum_mono mono pow_natlog2_ge pow_natlog2_le, simp, simp)+
+  by (intro sum_mono mono pow_natlog2_ge pow_natlog2_le, simp, simp)+
 
 private lemma condensation_condense1: "(\<Sum>k=1..<2^n. f (2 ^ natlog2 k)) = (\<Sum>k<n. 2^k * f (2 ^ k))"
 proof (induction n)
@@ -226,10 +226,10 @@
   have "{1..<2^Suc n} = {1..<2^n} \<union> {2^n..<(2^Suc n :: nat)}" by auto
   also have "(\<Sum>k\<in>\<dots>. f (2 ^ natlog2 k)) =
                  (\<Sum>k<n. 2^k * f (2^k)) + (\<Sum>k = 2^n..<2^Suc n. f (2^natlog2 k))"
-    by (subst setsum.union_disjoint) (insert Suc, auto)
+    by (subst sum.union_disjoint) (insert Suc, auto)
   also have "natlog2 k = n" if "k \<in> {2^n..<2^Suc n}" for k using that by (intro natlog2_eqI) simp_all
   hence "(\<Sum>k = 2^n..<2^Suc n. f (2^natlog2 k)) = (\<Sum>(_::nat) = 2^n..<2^Suc n. f (2^n))"
-    by (intro setsum.cong) simp_all
+    by (intro sum.cong) simp_all
   also have "\<dots> = 2^n * f (2^n)" by (simp add: of_nat_power)
   finally show ?case by simp
 qed simp
@@ -240,10 +240,10 @@
   have "{1..<2^Suc n} = {1..<2^n} \<union> {2^n..<(2^Suc n :: nat)}" by auto
   also have "(\<Sum>k\<in>\<dots>. f (2 * 2 ^ natlog2 k)) =
                  (\<Sum>k<n. 2^k * f (2^Suc k)) + (\<Sum>k = 2^n..<2^Suc n. f (2 * 2^natlog2 k))"
-    by (subst setsum.union_disjoint) (insert Suc, auto)
+    by (subst sum.union_disjoint) (insert Suc, auto)
   also have "natlog2 k = n" if "k \<in> {2^n..<2^Suc n}" for k using that by (intro natlog2_eqI) simp_all
   hence "(\<Sum>k = 2^n..<2^Suc n. f (2*2^natlog2 k)) = (\<Sum>(_::nat) = 2^n..<2^Suc n. f (2^Suc n))"
-    by (intro setsum.cong) simp_all
+    by (intro sum.cong) simp_all
   also have "\<dots> = 2^n * f (2^Suc n)" by (simp add: of_nat_power)
   finally show ?case by simp
 qed simp
@@ -267,11 +267,11 @@
   hence "convergent (\<lambda>n. \<Sum>k<n. f' k) \<longleftrightarrow> Bseq (\<lambda>n. \<Sum>k<n. f' k)"
     by (rule monoseq_imp_convergent_iff_Bseq)
   also have "\<dots> \<longleftrightarrow> Bseq (\<lambda>n. \<Sum>k=1..<n. f' k)" unfolding One_nat_def
-    by (subst setsum_shift_lb_Suc0_0_upt) (simp_all add: f'_def atLeast0LessThan)
+    by (subst sum_shift_lb_Suc0_0_upt) (simp_all add: f'_def atLeast0LessThan)
   also have "\<dots> \<longleftrightarrow> Bseq (\<lambda>n. \<Sum>k=1..<n. f k)" unfolding f'_def by simp
   also have "\<dots> \<longleftrightarrow> Bseq (\<lambda>n. \<Sum>k=1..<2^n. f k)"
     by (rule nonneg_incseq_Bseq_subseq_iff[symmetric])
-       (auto intro!: setsum_nonneg incseq_SucI nonneg simp: subseq_def)
+       (auto intro!: sum_nonneg incseq_SucI nonneg simp: subseq_def)
   also have "\<dots> \<longleftrightarrow> Bseq (\<lambda>n. \<Sum>k<n. 2^k * f (2^k))"
   proof (intro iffI)
     assume A: "Bseq (\<lambda>n. \<Sum>k=1..<2^n. f k)"
@@ -279,20 +279,20 @@
     proof (intro always_eventually allI)
       fix n :: nat
       have "norm (\<Sum>k<n. 2^k * f (2^Suc k)) = (\<Sum>k<n. 2^k * f (2^Suc k))" unfolding real_norm_def
-        by (intro abs_of_nonneg setsum_nonneg ballI mult_nonneg_nonneg nonneg) simp_all
+        by (intro abs_of_nonneg sum_nonneg ballI mult_nonneg_nonneg nonneg) simp_all
       also have "\<dots> \<le> (\<Sum>k=1..<2^n. f k)"
         by (subst condensation_condense2 [symmetric]) (intro condensation_inequality mono')
       also have "\<dots> = norm \<dots>" unfolding real_norm_def
-        by (intro abs_of_nonneg[symmetric] setsum_nonneg ballI mult_nonneg_nonneg nonneg)
+        by (intro abs_of_nonneg[symmetric] sum_nonneg ballI mult_nonneg_nonneg nonneg)
       finally show "norm (\<Sum>k<n. 2 ^ k * f (2 ^ Suc k)) \<le> norm (\<Sum>k=1..<2^n. f k)" .
     qed
     from this and A have "Bseq (\<lambda>n. \<Sum>k<n. 2^k * f (2^Suc k))" by (rule Bseq_eventually_mono)
     from Bseq_mult[OF Bfun_const[of 2] this] have "Bseq (\<lambda>n. \<Sum>k<n. 2^Suc k * f (2^Suc k))"
-      by (simp add: setsum_distrib_left setsum_distrib_right mult_ac)
+      by (simp add: sum_distrib_left sum_distrib_right mult_ac)
     hence "Bseq (\<lambda>n. (\<Sum>k=Suc 0..<Suc n. 2^k * f (2^k)) + f 1)"
-      by (intro Bseq_add, subst setsum_shift_bounds_Suc_ivl) (simp add: atLeast0LessThan)
+      by (intro Bseq_add, subst sum_shift_bounds_Suc_ivl) (simp add: atLeast0LessThan)
     hence "Bseq (\<lambda>n. (\<Sum>k=0..<Suc n. 2^k * f (2^k)))"
-      by (subst setsum_head_upt_Suc) (simp_all add: add_ac)
+      by (subst sum_head_upt_Suc) (simp_all add: add_ac)
     thus "Bseq (\<lambda>n. (\<Sum>k<n. 2^k * f (2^k)))"
       by (subst (asm) Bseq_Suc_iff) (simp add: atLeast0LessThan)
   next
@@ -301,11 +301,11 @@
     proof (intro always_eventually allI)
       fix n :: nat
       have "norm (\<Sum>k=1..<2^n. f k) = (\<Sum>k=1..<2^n. f k)" unfolding real_norm_def
-        by (intro abs_of_nonneg setsum_nonneg ballI mult_nonneg_nonneg nonneg)
+        by (intro abs_of_nonneg sum_nonneg ballI mult_nonneg_nonneg nonneg)
       also have "\<dots> \<le> (\<Sum>k<n. 2^k * f (2^k))"
         by (subst condensation_condense1 [symmetric]) (intro condensation_inequality mono')
       also have "\<dots> = norm \<dots>" unfolding real_norm_def
-        by (intro abs_of_nonneg [symmetric] setsum_nonneg ballI mult_nonneg_nonneg nonneg) simp_all
+        by (intro abs_of_nonneg [symmetric] sum_nonneg ballI mult_nonneg_nonneg nonneg) simp_all
       finally show "norm (\<Sum>k=1..<2^n. f k) \<le> norm (\<Sum>k<n. 2^k * f (2^k))" .
     qed
     from this and A show "Bseq (\<lambda>n. \<Sum>k=1..<2^n. f k)" by (rule Bseq_eventually_mono)
@@ -384,7 +384,7 @@
     by (simp add: summable_iff_convergent convergent_norm)
   hence "convergent (\<lambda>n. abs (\<Sum>k<n. inverse (of_nat k)) :: real)" by (simp only: norm_of_real)
   also have "(\<lambda>n. abs (\<Sum>k<n. inverse (of_nat k)) :: real) = (\<lambda>n. \<Sum>k<n. inverse (of_nat k))"
-    by (intro ext abs_of_nonneg setsum_nonneg) auto
+    by (intro ext abs_of_nonneg sum_nonneg) auto
   also have "convergent \<dots> \<longleftrightarrow> summable (\<lambda>k. inverse (of_nat k) :: real)"
     by (simp add: summable_iff_convergent)
   finally show False using summable_real_powr_iff[of "-1"] by (simp add: powr_minus)
@@ -424,31 +424,31 @@
     have n: "n > m" by (simp add: n_def)
 
     from r have "r * norm (\<Sum>k\<le>n. f k) = norm (\<Sum>k\<le>n. r * f k)"
-      by (simp add: setsum_distrib_left[symmetric] abs_mult)
+      by (simp add: sum_distrib_left[symmetric] abs_mult)
     also from n have "{..n} = {..m} \<union> {Suc m..n}" by auto
     hence "(\<Sum>k\<le>n. r * f k) = (\<Sum>k\<in>{..m} \<union> {Suc m..n}. r * f k)" by (simp only:)
     also have "\<dots> = (\<Sum>k\<le>m. r * f k) + (\<Sum>k=Suc m..n. r * f k)"
-      by (subst setsum.union_disjoint) auto
+      by (subst sum.union_disjoint) auto
     also have "norm \<dots> \<le> norm (\<Sum>k\<le>m. r * f k) + norm (\<Sum>k=Suc m..n. r * f k)"
       by (rule norm_triangle_ineq)
     also from r less_imp_le[OF m(1)] have "(\<Sum>k=Suc m..n. r * f k) \<ge> 0"
-      by (intro setsum_nonneg) auto
+      by (intro sum_nonneg) auto
     hence "norm (\<Sum>k=Suc m..n. r * f k) = (\<Sum>k=Suc m..n. r * f k)" by simp
     also have "(\<Sum>k=Suc m..n. r * f k) = (\<Sum>k=m..<n. r * f (Suc k))"
-     by (subst setsum_shift_bounds_Suc_ivl [symmetric])
+     by (subst sum_shift_bounds_Suc_ivl [symmetric])
           (simp only: atLeastLessThanSuc_atLeastAtMost)
     also from m have "\<dots> \<le> (\<Sum>k=m..<n. p k * f k - p (Suc k) * f (Suc k))"
-      by (intro setsum_mono[OF less_imp_le]) simp_all
+      by (intro sum_mono[OF less_imp_le]) simp_all
     also have "\<dots> = -(\<Sum>k=m..<n. p (Suc k) * f (Suc k) - p k * f k)"
-      by (simp add: setsum_negf [symmetric] algebra_simps)
+      by (simp add: sum_negf [symmetric] algebra_simps)
     also from n have "\<dots> = p m * f m - p n * f n"
-      by (cases n, simp, simp only: atLeastLessThanSuc_atLeastAtMost, subst setsum_Suc_diff) simp_all
+      by (cases n, simp, simp only: atLeastLessThanSuc_atLeastAtMost, subst sum_Suc_diff) simp_all
     also from less_imp_le[OF m(1)] m(2) n have "\<dots> \<le> p m * f m" by simp
     finally show "norm (\<Sum>k\<le>n. f k) \<le> (norm (\<Sum>k\<le>m. r * f k) + p m * f m) / r" using r
       by (subst pos_le_divide_eq[OF r(1)]) (simp only: mult_ac)
   qed
   moreover have "(\<Sum>k\<le>n. f k) \<le> (\<Sum>k\<le>n'. f k)" if "Suc m \<le> n" "n \<le> n'" for n n'
-    using less_imp_le[OF m(1)] that by (intro setsum_mono2) auto
+    using less_imp_le[OF m(1)] that by (intro sum_mono2) auto
   ultimately show "convergent (\<lambda>n. \<Sum>k\<le>n. f k)" by (rule Bseq_monoseq_convergent'_inc)
 qed
 
@@ -905,7 +905,7 @@
   from r have "summable (\<lambda>n. (\<Sum>i\<le>n. (f i * of_real r^i) * (g (n - i) * of_real r^(n - i))))"
     by (intro summable_Cauchy_product abs_summable_in_conv_radius) simp_all
   thus "summable (\<lambda>n. (\<Sum>i\<le>n. f i * g (n - i)) * of_real r ^ n)"
-    by (simp add: algebra_simps of_real_def power_add [symmetric] scaleR_setsum_right)
+    by (simp add: algebra_simps of_real_def power_add [symmetric] scaleR_sum_right)
 qed
 
 lemma le_conv_radius_iff:
--- a/src/HOL/Analysis/Tagged_Division.thy	Mon Oct 17 11:07:01 2016 +0200
+++ b/src/HOL/Analysis/Tagged_Division.thy	Mon Oct 17 14:37:32 2016 +0200
@@ -30,8 +30,8 @@
 lemma sum_sum_product:
   assumes "finite s"
     and "\<forall>i\<in>s. finite (t i)"
-  shows "setsum (\<lambda>i. setsum (x i) (t i)::real) s =
-    setsum (\<lambda>(i,j). x i j) {(i,j) | i j. i \<in> s \<and> j \<in> t i}"
+  shows "sum (\<lambda>i. sum (x i) (t i)::real) s =
+    sum (\<lambda>(i,j). x i j) {(i,j) | i j. i \<in> s \<and> j \<in> t i}"
   using assms
 proof induct
   case (insert a s)
@@ -39,14 +39,14 @@
     (\<lambda>y. (a,y)) ` (t a) \<union> {(i, j) |i j. i \<in> s \<and> j \<in> t i}" by auto
   show ?case
     unfolding *
-    apply (subst setsum.union_disjoint)
-    unfolding setsum.insert[OF insert(1-2)]
+    apply (subst sum.union_disjoint)
+    unfolding sum.insert[OF insert(1-2)]
     prefer 4
     apply (subst insert(3))
     unfolding add_right_cancel
   proof -
-    show "setsum (x a) (t a) = (\<Sum>(xa, y)\<in> Pair a ` t a. x xa y)"
-      apply (subst setsum.reindex)
+    show "sum (x a) (t a) = (\<Sum>(xa, y)\<in> Pair a ` t a. x xa y)"
+      apply (subst sum.reindex)
       unfolding inj_on_def
       apply auto
       done
@@ -196,13 +196,13 @@
 lemma interval_upperbound[simp]:
   "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i \<Longrightarrow>
     interval_upperbound (cbox a b) = (b::'a::euclidean_space)"
-  unfolding interval_upperbound_def euclidean_representation_setsum cbox_def
+  unfolding interval_upperbound_def euclidean_representation_sum cbox_def
   by (safe intro!: cSup_eq) auto
 
 lemma interval_lowerbound[simp]:
   "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i \<Longrightarrow>
     interval_lowerbound (cbox a b) = (a::'a::euclidean_space)"
-  unfolding interval_lowerbound_def euclidean_representation_setsum cbox_def
+  unfolding interval_lowerbound_def euclidean_representation_sum cbox_def
   by (safe intro!: cInf_eq) auto
 
 lemmas interval_bounds = interval_upperbound interval_lowerbound
@@ -230,7 +230,7 @@
   have "(\<Sum>i\<in>Basis. (SUP x:A \<times> B. x \<bullet> (0, i)) *\<^sub>R i) = (\<Sum>i\<in>Basis. (SUP x:B. x \<bullet> i) *\<^sub>R i)"
       by (subst (2) snd_image_times') (simp del: snd_image_times add: o_def inner_Pair_0)
   ultimately show ?thesis unfolding interval_upperbound_def
-      by (subst setsum_Basis_prod_eq) (auto simp add: setsum_prod)
+      by (subst sum_Basis_prod_eq) (auto simp add: sum_prod)
 qed
 
 lemma interval_lowerbound_Times:
@@ -244,7 +244,7 @@
   have "(\<Sum>i\<in>Basis. (INF x:A \<times> B. x \<bullet> (0, i)) *\<^sub>R i) = (\<Sum>i\<in>Basis. (INF x:B. x \<bullet> i) *\<^sub>R i)"
       by (subst (2) snd_image_times') (simp del: snd_image_times add: o_def inner_Pair_0)
   ultimately show ?thesis unfolding interval_lowerbound_def
-      by (subst setsum_Basis_prod_eq) (auto simp add: setsum_prod)
+      by (subst sum_Basis_prod_eq) (auto simp add: sum_prod)
 qed
 
 subsection \<open>The notion of a gauge --- simply an open set containing the point.\<close>
@@ -1359,7 +1359,7 @@
   assumes g: "operative g" shows "g {} = \<^bold>1"
 proof -
   have *: "cbox One (-One) = ({}::'b set)"
-    by (auto simp: box_eq_empty inner_setsum_left inner_Basis setsum.If_cases ex_in_conv)
+    by (auto simp: box_eq_empty inner_sum_left inner_Basis sum.If_cases ex_in_conv)
   moreover have "box One (-One) = ({}::'b set)"
     using box_subset_cbox[of One "-One"] by (auto simp: *)
   ultimately show ?thesis
@@ -1438,7 +1438,7 @@
     apply rule
     unfolding mem_Collect_eq split_beta
     apply (erule bexE conjE)+
-    apply (simp only: mem_Collect_eq inner_setsum_left_Basis simp_thms)
+    apply (simp only: mem_Collect_eq inner_sum_left_Basis simp_thms)
     apply (erule exE conjE)+
   proof
     fix i l x
@@ -1931,7 +1931,7 @@
   fixes f :: "real \<Rightarrow> 'a::real_normed_vector"
   assumes "a \<le> b"
     and "p tagged_division_of {a..b}"
-  shows "setsum (\<lambda>(x,k). f(Sup k) - f(Inf k)) p = f b - f a"
+  shows "sum (\<lambda>(x,k). f(Sup k) - f(Inf k)) p = f b - f a"
 proof -
   let ?f = "(\<lambda>k::(real) set. if k = {} then 0 else f(interval_upperbound k) - f(interval_lowerbound k))"
   have ***: "\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i"
@@ -1941,11 +1941,11 @@
     by auto
   have **: "cbox a b \<noteq> {}"
     using assms(1) by auto
-  note setsum.operative_tagged_division[OF * assms(2)[simplified box_real[symmetric]]]
+  note sum.operative_tagged_division[OF * assms(2)[simplified box_real[symmetric]]]
   note * = this[unfolded if_not_P[OF **] interval_bounds[OF ***],symmetric]
   show ?thesis
     unfolding *
-    apply (rule setsum.cong)
+    apply (rule sum.cong)
     unfolding split_paired_all split_conv
     using assms(2)
     apply auto
@@ -1959,7 +1959,7 @@
   fixes f :: "real \<Rightarrow> 'a::real_normed_vector"
   assumes "a \<le> b"
     and "p tagged_division_of {a..b}"
-  shows "setsum (\<lambda>(x,k). f (Sup k) - f(Inf k)) p = f b - f a"
+  shows "sum (\<lambda>(x,k). f (Sup k) - f(Inf k)) p = f b - f a"
   using additive_tagged_division_1[OF _ assms(2), of f]
   using assms(1)
   by auto
@@ -2264,24 +2264,24 @@
     if e: "0 < e" for e
   proof -
     obtain n where n: "(\<Sum>i\<in>Basis. b \<bullet> i - a \<bullet> i) / e < 2 ^ n"
-      using real_arch_pow[of 2 "(setsum (\<lambda>i. b\<bullet>i - a\<bullet>i) Basis) / e"] by auto
+      using real_arch_pow[of 2 "(sum (\<lambda>i. b\<bullet>i - a\<bullet>i) Basis) / e"] by auto
     show ?thesis
     proof (rule exI [where x=n], clarify)
       fix x y
       assume xy: "x\<in>cbox (A n) (B n)" "y\<in>cbox (A n) (B n)"
-      have "dist x y \<le> setsum (\<lambda>i. \<bar>(x - y)\<bullet>i\<bar>) Basis"
+      have "dist x y \<le> sum (\<lambda>i. \<bar>(x - y)\<bullet>i\<bar>) Basis"
         unfolding dist_norm by(rule norm_le_l1)
-      also have "\<dots> \<le> setsum (\<lambda>i. B n\<bullet>i - A n\<bullet>i) Basis"
-      proof (rule setsum_mono)
+      also have "\<dots> \<le> sum (\<lambda>i. B n\<bullet>i - A n\<bullet>i) Basis"
+      proof (rule sum_mono)
         fix i :: 'a
         assume i: "i \<in> Basis"
         show "\<bar>(x - y) \<bullet> i\<bar> \<le> B n \<bullet> i - A n \<bullet> i"
           using xy[unfolded mem_box,THEN bspec, OF i]
           by (auto simp: inner_diff_left)
       qed
-      also have "\<dots> \<le> setsum (\<lambda>i. b\<bullet>i - a\<bullet>i) Basis / 2^n"
-        unfolding setsum_divide_distrib
-      proof (rule setsum_mono)
+      also have "\<dots> \<le> sum (\<lambda>i. b\<bullet>i - a\<bullet>i) Basis / 2^n"
+        unfolding sum_divide_distrib
+      proof (rule sum_mono)
         show "B n \<bullet> i - A n \<bullet> i \<le> (b \<bullet> i - a \<bullet> i) / 2 ^ n" if i: "i \<in> Basis" for i
         proof (induct n)
           case 0
@@ -2600,9 +2600,9 @@
                if "(\<And>i. i \<in> Basis \<Longrightarrow> \<bar>x \<bullet> i - y \<bullet> i\<bar> \<le> \<epsilon> / (2 * real DIM('a)))" for y
           proof -
             have "norm (x - y) \<le> (\<Sum>i\<in>Basis. \<bar>x \<bullet> i - y \<bullet> i\<bar>)"
-              by (metis (no_types, lifting) inner_diff_left norm_le_l1 setsum.cong)
+              by (metis (no_types, lifting) inner_diff_left norm_le_l1 sum.cong)
             also have "... \<le> DIM('a) * (\<epsilon> / (2 * real DIM('a)))"
-              by (meson setsum_bounded_above that)
+              by (meson sum_bounded_above that)
             also have "... = \<epsilon> / 2"
               by (simp add: divide_simps)
             also have "... < \<epsilon>"
@@ -2814,4 +2814,4 @@
   "eventually (\<lambda>p. p tagged_division_of s) (division_filter s)"
   unfolding eventually_division_filter by (intro exI[of _ "\<lambda>x. ball x 1"]) auto
 
-end
\ No newline at end of file
+end
--- a/src/HOL/Analysis/Topology_Euclidean_Space.thy	Mon Oct 17 11:07:01 2016 +0200
+++ b/src/HOL/Analysis/Topology_Euclidean_Space.thy	Mon Oct 17 14:37:32 2016 +0200
@@ -58,22 +58,22 @@
 lemma finite_support[intro]: "finite s \<Longrightarrow> finite (support_on s f)"
   unfolding support_on_def by auto
 
-(* TODO: is supp_setsum really needed? TODO: Generalize to Finite_Set.fold *)
-definition (in comm_monoid_add) supp_setsum :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
+(* TODO: is supp_sum really needed? TODO: Generalize to Finite_Set.fold *)
+definition (in comm_monoid_add) supp_sum :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
 where
-  "supp_setsum f s = (\<Sum>x\<in>support_on s f. f x)"
-
-lemma supp_setsum_empty[simp]: "supp_setsum f {} = 0"
-  unfolding supp_setsum_def by auto
-
-lemma supp_setsum_insert[simp]:
+  "supp_sum f s = (\<Sum>x\<in>support_on s f. f x)"
+
+lemma supp_sum_empty[simp]: "supp_sum f {} = 0"
+  unfolding supp_sum_def by auto
+
+lemma supp_sum_insert[simp]:
   "finite (support_on s f) \<Longrightarrow>
-    supp_setsum f (insert x s) = (if x \<in> s then supp_setsum f s else f x + supp_setsum f s)"
-  by (simp add: supp_setsum_def in_support_on insert_absorb)
-
-lemma supp_setsum_divide_distrib: "supp_setsum f A / (r::'a::field) = supp_setsum (\<lambda>n. f n / r) A"
+    supp_sum f (insert x s) = (if x \<in> s then supp_sum f s else f x + supp_sum f s)"
+  by (simp add: supp_sum_def in_support_on insert_absorb)
+
+lemma supp_sum_divide_distrib: "supp_sum f A / (r::'a::field) = supp_sum (\<lambda>n. f n / r) A"
   by (cases "r = 0")
-     (auto simp: supp_setsum_def setsum_divide_distrib intro!: setsum.cong support_on_cong)
+     (auto simp: supp_sum_def sum_divide_distrib intro!: sum.cong support_on_cong)
 
 (*END OF SUPPORT, ETC.*)
 
@@ -1164,7 +1164,7 @@
     have "dist x y = sqrt (\<Sum>i\<in>Basis. (dist (x \<bullet> i) (y \<bullet> i))\<^sup>2)"
       unfolding setL2_def[symmetric] by (rule euclidean_dist_l2)
     also have "\<dots> < sqrt (\<Sum>(i::'a)\<in>Basis. e^2 / real (DIM('a)))"
-    proof (rule real_sqrt_less_mono, rule setsum_strict_mono)
+    proof (rule real_sqrt_less_mono, rule sum_strict_mono)
       fix i :: "'a"
       assume i: "i \<in> Basis"
       have "a i < y\<bullet>i \<and> y\<bullet>i < b i"
@@ -1497,7 +1497,7 @@
   moreover have "\<And>x b::'a. \<And>n::nat.  \<bar>x \<bullet> b\<bar> < real n \<longleftrightarrow> - real n < x \<bullet> b \<and> x \<bullet> b < real n"
     by auto
   ultimately show ?thesis
-    by (auto simp: box_def inner_setsum_left inner_Basis setsum.If_cases)
+    by (auto simp: box_def inner_sum_left inner_Basis sum.If_cases)
 qed
 
 text \<open>Intervals in general, including infinite and mixtures of open and closed.\<close>
@@ -1562,10 +1562,10 @@
 qed
 
 lemma cbox01_nonempty [simp]: "cbox 0 One \<noteq> {}"
-  by (simp add: box_ne_empty inner_Basis inner_setsum_left setsum_nonneg)
+  by (simp add: box_ne_empty inner_Basis inner_sum_left sum_nonneg)
 
 lemma box01_nonempty [simp]: "box 0 One \<noteq> {}"
-  by (simp add: box_ne_empty inner_Basis inner_setsum_left) (simp add: setsum.remove)
+  by (simp add: box_ne_empty inner_Basis inner_sum_left) (simp add: sum.remove)
 
 
 subsection\<open>Connectedness\<close>
@@ -5046,10 +5046,10 @@
       have "dist (f (r n)) l \<le> (\<Sum>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i))"
         apply (subst euclidean_dist_l2)
         using zero_le_dist
-        apply (rule setL2_le_setsum)
+        apply (rule setL2_le_sum)
         done
       also have "\<dots> < (\<Sum>i\<in>(Basis::'a set). e / (real_of_nat DIM('a)))"
-        apply (rule setsum_strict_mono)
+        apply (rule sum_strict_mono)
         using n
         apply auto
         done
@@ -5448,7 +5448,7 @@
 by (frule summable_LIMSEQ_zero) (simp add: convergent_imp_bounded)
 
 lemma summable_imp_sums_bounded:
-   "summable f \<Longrightarrow> bounded (range (\<lambda>n. setsum f {..<n}))"
+   "summable f \<Longrightarrow> bounded (range (\<lambda>n. sum f {..<n}))"
 by (auto simp: summable_def sums_def dest: convergent_imp_bounded)
 
 lemma power_series_conv_imp_absconv_weak:
@@ -7064,17 +7064,17 @@
       show ?thesis
       proof (rule image_eqI)
         show "y = f (x + (\<Sum>i\<in>Basis. (u \<bullet> i) *\<^sub>R inv f i))"
-          apply (simp add: linear_add linear_setsum linear.scaleR \<open>linear f\<close> surj_f_inv_f \<open>surj f\<close>)
+          apply (simp add: linear_add linear_sum linear.scaleR \<open>linear f\<close> surj_f_inv_f \<open>surj f\<close>)
           apply (simp add: euclidean_representation u_def)
           done
         have "dist (x + (\<Sum>i\<in>Basis. (u \<bullet> i) *\<^sub>R inv f i)) x \<le> (\<Sum>i\<in>Basis. norm ((u \<bullet> i) *\<^sub>R inv f i))"
-          by (simp add: dist_norm setsum_norm_le)
+          by (simp add: dist_norm sum_norm_le)
         also have "... = (\<Sum>i\<in>Basis. \<bar>u \<bullet> i\<bar> * norm (inv f i))"
           by (simp add: )
         also have "... \<le> (\<Sum>i\<in>Basis. \<bar>u \<bullet> i\<bar>) * B"
-          by (simp add: B setsum_distrib_right setsum_mono mult_left_mono)
+          by (simp add: B sum_distrib_right sum_mono mult_left_mono)
         also have "... \<le> DIM('b) * dist y (f x) * B"
-          apply (rule mult_right_mono [OF setsum_bounded_above])
+          apply (rule mult_right_mono [OF sum_bounded_above])
           using \<open>0 < B\<close> by (auto simp add: Basis_le_norm dist_norm u_def)
         also have "... < e"
           by (metis mult.commute mult.left_commute that)
@@ -8200,7 +8200,7 @@
     }
     then have "(\<Sum>i\<in>Basis. \<bar>x \<bullet> i\<bar>) \<le> ?b"
       apply -
-      apply (rule setsum_mono)
+      apply (rule sum_mono)
       apply auto
       done
     then have "norm x \<le> ?b"
@@ -9164,9 +9164,9 @@
     have "finite d"
       using finite_subset [OF d finite_Basis] .
     then have "(\<Sum>i\<in>d. (x \<bullet> i) *\<^sub>R i) \<in> span d"
-      by (simp add: span_setsum span_clauses)
+      by (simp add: span_sum span_clauses)
     also have "(\<Sum>i\<in>d. (x \<bullet> i) *\<^sub>R i) = (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R i)"
-      by (rule setsum.mono_neutral_cong_left [OF finite_Basis d]) (auto simp add: x)
+      by (rule sum.mono_neutral_cong_left [OF finite_Basis d]) (auto simp add: x)
     finally show "x \<in> span d"
       unfolding euclidean_representation .
   qed
@@ -9950,10 +9950,10 @@
     have *: "setL2 (\<lambda>i. dist (f x \<bullet> i) (l \<bullet> i)) Basis < e"
              if "\<forall>i\<in>Basis. dist (f x \<bullet> i) (l \<bullet> i) < e / real DIM('b)" for x
     proof -
-      have "setL2 (\<lambda>i. dist (f x \<bullet> i) (l \<bullet> i)) Basis \<le> setsum (\<lambda>i. dist (f x \<bullet> i) (l \<bullet> i)) Basis"
-        by (simp add: setL2_le_setsum)
+      have "setL2 (\<lambda>i. dist (f x \<bullet> i) (l \<bullet> i)) Basis \<le> sum (\<lambda>i. dist (f x \<bullet> i) (l \<bullet> i)) Basis"
+        by (simp add: setL2_le_sum)
       also have "... < DIM('b) * (e / real DIM('b))"
-        apply (rule setsum_bounded_above_strict)
+        apply (rule sum_bounded_above_strict)
         using that by auto
       also have "... = e"
         by (simp add: field_simps)
@@ -9998,14 +9998,14 @@
     by (auto simp: bounded_linear_def bounded_linear_axioms_def linear_componentwise_iff [symmetric] ball_conj_distrib)
   then obtain F where F: "\<And>i x. i \<in> Basis \<Longrightarrow> \<bar>f' x \<bullet> i\<bar> \<le> norm x * F i"
     by metis
-  have "norm (f' x) \<le> norm x * setsum F Basis" for x
+  have "norm (f' x) \<le> norm x * sum F Basis" for x
   proof -
     have "norm (f' x) \<le> (\<Sum>i\<in>Basis. \<bar>f' x \<bullet> i\<bar>)"
       by (rule norm_le_l1)
     also have "... \<le> (\<Sum>i\<in>Basis. norm x * F i)"