--- a/src/HOL/Big_Operators.thy Mon Feb 27 09:01:49 2012 +0100
+++ b/src/HOL/Big_Operators.thy Mon Feb 27 10:27:21 2012 +0100
@@ -523,6 +523,25 @@
case insert thus ?case by (auto simp: add_strict_mono)
qed
+lemma setsum_strict_mono_ex1:
+fixes f :: "'a \<Rightarrow> 'b::{comm_monoid_add, ordered_cancel_ab_semigroup_add}"
+assumes "finite A" and "ALL x:A. f x \<le> g x" and "EX a:A. f a < g a"
+shows "setsum f A < setsum g A"
+proof-
+ from assms(3) obtain a where a: "a:A" "f a < g a" by blast
+ have "setsum f A = setsum f ((A-{a}) \<union> {a})"
+ by(simp add:insert_absorb[OF `a:A`])
+ also have "\<dots> = setsum f (A-{a}) + setsum f {a}"
+ using `finite A` by(subst setsum_Un_disjoint) auto
+ also have "setsum f (A-{a}) \<le> setsum g (A-{a})"
+ by(rule setsum_mono)(simp add: assms(2))
+ also have "setsum f {a} < setsum g {a}" using a by simp
+ also have "setsum g (A - {a}) + setsum g {a} = setsum g((A-{a}) \<union> {a})"
+ using `finite A` by(subst setsum_Un_disjoint[symmetric]) auto
+ also have "\<dots> = setsum g A" by(simp add:insert_absorb[OF `a:A`])
+ finally show ?thesis by (metis add_right_mono add_strict_left_mono)
+qed
+
lemma setsum_negf:
"setsum (%x. - (f x)::'a::ab_group_add) A = - setsum f A"
proof (cases "finite A")