author | haftmann |
Sat, 28 Jun 2014 21:09:17 +0200 | |
changeset 57427 | 91f9e4148460 |
parent 57418 | 6ab1c7cb0b8d |
child 57446 | 06e195515deb |
permissions | -rw-r--r-- |
37665 | 1 |
(* Title: HOL/Library/Indicator_Function.thy |
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Author: Johannes Hoelzl (TU Muenchen) |
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*) |
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header {* Indicator Function *} |
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theory Indicator_Function |
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introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
54408
diff
changeset
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imports Complex_Main |
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begin |
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definition "indicator S x = (if x \<in> S then 1 else 0)" |
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lemma indicator_simps[simp]: |
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"x \<in> S \<Longrightarrow> indicator S x = 1" |
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"x \<notin> S \<Longrightarrow> indicator S x = 0" |
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unfolding indicator_def by auto |
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lemma indicator_pos_le[intro, simp]: "(0::'a::linordered_semidom) \<le> indicator S x" |
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and indicator_le_1[intro, simp]: "indicator S x \<le> (1::'a::linordered_semidom)" |
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unfolding indicator_def by auto |
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lemma indicator_abs_le_1: "\<bar>indicator S x\<bar> \<le> (1::'a::linordered_idom)" |
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unfolding indicator_def by auto |
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lemma indicator_eq_0_iff: "indicator A x = (0::_::zero_neq_one) \<longleftrightarrow> x \<notin> A" |
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by (auto simp: indicator_def) |
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lemma indicator_eq_1_iff: "indicator A x = (1::_::zero_neq_one) \<longleftrightarrow> x \<in> A" |
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by (auto simp: indicator_def) |
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lemma split_indicator: |
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"P (indicator S x) \<longleftrightarrow> ((x \<in> S \<longrightarrow> P 1) \<and> (x \<notin> S \<longrightarrow> P 0))" |
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unfolding indicator_def by auto |
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lemma indicator_inter_arith: "indicator (A \<inter> B) x = indicator A x * (indicator B x::'a::semiring_1)" |
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unfolding indicator_def by (auto simp: min_def max_def) |
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lemma indicator_union_arith: "indicator (A \<union> B) x = indicator A x + indicator B x - indicator A x * (indicator B x::'a::ring_1)" |
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unfolding indicator_def by (auto simp: min_def max_def) |
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lemma indicator_inter_min: "indicator (A \<inter> B) x = min (indicator A x) (indicator B x::'a::linordered_semidom)" |
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and indicator_union_max: "indicator (A \<union> B) x = max (indicator A x) (indicator B x::'a::linordered_semidom)" |
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unfolding indicator_def by (auto simp: min_def max_def) |
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lemma indicator_compl: "indicator (- A) x = 1 - (indicator A x::'a::ring_1)" |
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and indicator_diff: "indicator (A - B) x = indicator A x * (1 - indicator B x::'a::ring_1)" |
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unfolding indicator_def by (auto simp: min_def max_def) |
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lemma indicator_times: "indicator (A \<times> B) x = indicator A (fst x) * (indicator B (snd x)::'a::semiring_1)" |
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unfolding indicator_def by (cases x) auto |
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lemma indicator_sum: "indicator (A <+> B) x = (case x of Inl x \<Rightarrow> indicator A x | Inr x \<Rightarrow> indicator B x)" |
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unfolding indicator_def by (cases x) auto |
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lemma |
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fixes f :: "'a \<Rightarrow> 'b::semiring_1" assumes "finite A" |
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shows setsum_mult_indicator[simp]: "(\<Sum>x \<in> A. f x * indicator B x) = (\<Sum>x \<in> A \<inter> B. f x)" |
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and setsum_indicator_mult[simp]: "(\<Sum>x \<in> A. indicator B x * f x) = (\<Sum>x \<in> A \<inter> B. f x)" |
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unfolding indicator_def |
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using assms by (auto intro!: setsum.mono_neutral_cong_right split: split_if_asm) |
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lemma setsum_indicator_eq_card: |
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assumes "finite A" |
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shows "(SUM x : A. indicator B x) = card (A Int B)" |
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using setsum_mult_indicator[OF assms, of "%x. 1::nat"] |
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unfolding card_eq_setsum by simp |
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||
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
54408
diff
changeset
|
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lemma setsum_indicator_scaleR[simp]: |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
54408
diff
changeset
|
69 |
"finite A \<Longrightarrow> |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
54408
diff
changeset
|
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(\<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)" |
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using assms by (auto intro!: setsum.mono_neutral_cong_right split: split_if_asm simp: indicator_def) |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
54408
diff
changeset
|
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end |