(* Title: HOL/Library/Indicator_Function.thy
Author: Johannes Hoelzl (TU Muenchen)
*)
header {* Indicator Function *}
theory Indicator_Function
imports Complex_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 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)"
unfolding indicator_def by auto
lemma indicator_abs_le_1: "\<bar>indicator S x\<bar> \<le> (1::'a::linordered_idom)"
unfolding indicator_def by auto
lemma indicator_eq_0_iff: "indicator A x = (0::_::zero_neq_one) \<longleftrightarrow> x \<notin> A"
by (auto simp: indicator_def)
lemma indicator_eq_1_iff: "indicator A x = (1::_::zero_neq_one) \<longleftrightarrow> x \<in> A"
by (auto simp: indicator_def)
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 indicator_inter_arith: "indicator (A \<inter> B) x = indicator A x * (indicator B x::'a::semiring_1)"
unfolding indicator_def by (auto simp: min_def max_def)
lemma indicator_union_arith: "indicator (A \<union> B) x = indicator A x + indicator B x - indicator A x * (indicator B x::'a::ring_1)"
unfolding indicator_def by (auto simp: min_def max_def)
lemma 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)"
unfolding indicator_def by (auto simp: min_def max_def)
lemma 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 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 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_neutral_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
lemma setsum_indicator_scaleR[simp]:
"finite A \<Longrightarrow>
(\<Sum>x \<in> A. indicator (B x) (g x) *\<^sub>R f x) = (\<Sum>x \<in> {x\<in>A. g x \<in> B x}. f x::'a::real_vector)"
using assms by (auto intro!: setsum.mono_neutral_cong_right split: split_if_asm simp: indicator_def)
end