--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Indicator_Function.thy Thu Jul 01 11:48:42 2010 +0200
@@ -0,0 +1,58 @@
+(* Title: HOL/Library/Indicator_Function.thy
+ Author: Johannes Hoelzl (TU Muenchen)
+*)
+
+header {* Indicator Function *}
+
+theory Indicator_Function
+imports Main
+begin
+
+definition "indicator S x = (if x \<in> S then 1 else 0)"
+
+lemma indicator_simps[simp]:
+ "x \<in> S \<Longrightarrow> indicator S x = 1"
+ "x \<notin> S \<Longrightarrow> indicator S x = 0"
+ unfolding indicator_def by auto
+
+lemma
+ shows indicator_pos_le[intro, simp]: "(0::'a::linordered_semidom) \<le> indicator S x"
+ and indicator_le_1[intro, simp]: "indicator S x \<le> (1::'a::linordered_semidom)"
+ and indicator_abs_le_1: "\<bar>indicator S x\<bar> \<le> (1::'a::linordered_idom)"
+ unfolding indicator_def by auto
+
+lemma split_indicator:
+ "P (indicator S x) \<longleftrightarrow> ((x \<in> S \<longrightarrow> P 1) \<and> (x \<notin> S \<longrightarrow> P 0))"
+ unfolding indicator_def by auto
+
+lemma
+ shows indicator_inter_arith: "indicator (A \<inter> B) x = indicator A x * (indicator B x::'a::semiring_1)"
+ and indicator_union_arith: "indicator (A \<union> B) x = indicator A x + indicator B x - indicator A x * (indicator B x::'a::ring_1)"
+ and indicator_inter_min: "indicator (A \<inter> B) x = min (indicator A x) (indicator B x::'a::linordered_semidom)"
+ and indicator_union_max: "indicator (A \<union> B) x = max (indicator A x) (indicator B x::'a::linordered_semidom)"
+ and indicator_compl: "indicator (- A) x = 1 - (indicator A x::'a::ring_1)"
+ and indicator_diff: "indicator (A - B) x = indicator A x * (1 - indicator B x::'a::ring_1)"
+ unfolding indicator_def by (auto simp: min_def max_def)
+
+lemma
+ shows indicator_times: "indicator (A \<times> B) x = indicator A (fst x) * (indicator B (snd x)::'a::semiring_1)"
+ unfolding indicator_def by (cases x) auto
+
+lemma
+ shows indicator_sum: "indicator (A <+> B) x = (case x of Inl x \<Rightarrow> indicator A x | Inr x \<Rightarrow> indicator B x)"
+ unfolding indicator_def by (cases x) auto
+
+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) = (\<Sum>x \<in> A \<inter> B. f x)"
+ and setsum_indicator_mult[simp]: "(\<Sum>x \<in> A. indicator B x * f x) = (\<Sum>x \<in> A \<inter> B. f x)"
+ unfolding indicator_def
+ using assms by (auto intro!: setsum_mono_zero_cong_right split: split_if_asm)
+
+lemma setsum_indicator_eq_card:
+ assumes "finite A"
+ shows "(SUM x : A. indicator B x) = card (A Int B)"
+ using setsum_mult_indicator[OF assms, of "%x. 1::nat"]
+ unfolding card_eq_setsum by simp
+
+end
\ No newline at end of file