author hoelzl Thu Jul 01 11:48:42 2010 +0200 (2010-07-01) changeset 37665 579258a77fec parent 37664 2946b8f057df child 37666 e31fd427c285
Add theory for indicator function.
```     1.1 --- a/src/HOL/IsaMakefile	Thu Jul 01 09:01:09 2010 +0200
1.2 +++ b/src/HOL/IsaMakefile	Thu Jul 01 11:48:42 2010 +0200
1.3 @@ -406,7 +406,8 @@
1.4    Library/Float.thy Library/Formal_Power_Series.thy			\
1.5    Library/Fraction_Field.thy Library/FrechetDeriv.thy Library/Fset.thy	\
1.6    Library/FuncSet.thy Library/Fundamental_Theorem_Algebra.thy		\
1.7 -  Library/Glbs.thy Library/Infinite_Set.thy Library/Inner_Product.thy	\
1.8 +  Library/Glbs.thy Library/Indicator_Function.thy			\
1.9 +  Library/Infinite_Set.thy Library/Inner_Product.thy			\
1.10    Library/HOL_Library_ROOT.ML Library/Kleene_Algebra.thy		\
1.11    Library/LaTeXsugar.thy Library/Lattice_Algebras.thy			\
1.12    Library/Lattice_Syntax.thy Library/Library.thy			\
```
```     2.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
2.2 +++ b/src/HOL/Library/Indicator_Function.thy	Thu Jul 01 11:48:42 2010 +0200
2.3 @@ -0,0 +1,58 @@
2.4 +(*  Title:      HOL/Library/Indicator_Function.thy
2.5 +    Author:     Johannes Hoelzl (TU Muenchen)
2.6 +*)
2.7 +
2.8 +header {* Indicator Function *}
2.9 +
2.10 +theory Indicator_Function
2.11 +imports Main
2.12 +begin
2.13 +
2.14 +definition "indicator S x = (if x \<in> S then 1 else 0)"
2.15 +
2.16 +lemma indicator_simps[simp]:
2.17 +  "x \<in> S \<Longrightarrow> indicator S x = 1"
2.18 +  "x \<notin> S \<Longrightarrow> indicator S x = 0"
2.19 +  unfolding indicator_def by auto
2.20 +
2.21 +lemma
2.22 +  shows indicator_pos_le[intro, simp]: "(0::'a::linordered_semidom) \<le> indicator S x"
2.23 +  and indicator_le_1[intro, simp]: "indicator S x \<le> (1::'a::linordered_semidom)"
2.24 +  and indicator_abs_le_1: "\<bar>indicator S x\<bar> \<le> (1::'a::linordered_idom)"
2.25 +  unfolding indicator_def by auto
2.26 +
2.27 +lemma split_indicator:
2.28 +  "P (indicator S x) \<longleftrightarrow> ((x \<in> S \<longrightarrow> P 1) \<and> (x \<notin> S \<longrightarrow> P 0))"
2.29 +  unfolding indicator_def by auto
2.30 +
2.31 +lemma
2.32 +  shows indicator_inter_arith: "indicator (A \<inter> B) x = indicator A x * (indicator B x::'a::semiring_1)"
2.33 +  and indicator_union_arith: "indicator (A \<union> B) x =  indicator A x + indicator B x - indicator A x * (indicator B x::'a::ring_1)"
2.34 +  and indicator_inter_min: "indicator (A \<inter> B) x = min (indicator A x) (indicator B x::'a::linordered_semidom)"
2.35 +  and indicator_union_max: "indicator (A \<union> B) x = max (indicator A x) (indicator B x::'a::linordered_semidom)"
2.36 +  and indicator_compl: "indicator (- A) x = 1 - (indicator A x::'a::ring_1)"
2.37 +  and indicator_diff: "indicator (A - B) x = indicator A x * (1 - indicator B x::'a::ring_1)"
2.38 +  unfolding indicator_def by (auto simp: min_def max_def)
2.39 +
2.40 +lemma
2.41 +  shows indicator_times: "indicator (A \<times> B) x = indicator A (fst x) * (indicator B (snd x)::'a::semiring_1)"
2.42 +  unfolding indicator_def by (cases x) auto
2.43 +
2.44 +lemma
2.45 +  shows indicator_sum: "indicator (A <+> B) x = (case x of Inl x \<Rightarrow> indicator A x | Inr x \<Rightarrow> indicator B x)"
2.46 +  unfolding indicator_def by (cases x) auto
2.47 +
2.48 +lemma
2.49 +  fixes f :: "'a \<Rightarrow> 'b::semiring_1" assumes "finite A"
2.50 +  shows setsum_mult_indicator[simp]: "(\<Sum>x \<in> A. f x * indicator B x) = (\<Sum>x \<in> A \<inter> B. f x)"
2.51 +  and setsum_indicator_mult[simp]: "(\<Sum>x \<in> A. indicator B x * f x) = (\<Sum>x \<in> A \<inter> B. f x)"
2.52 +  unfolding indicator_def
2.53 +  using assms by (auto intro!: setsum_mono_zero_cong_right split: split_if_asm)
2.54 +
2.55 +lemma setsum_indicator_eq_card:
2.56 +  assumes "finite A"
2.57 +  shows "(SUM x : A. indicator B x) = card (A Int B)"
2.58 +  using setsum_mult_indicator[OF assms, of "%x. 1::nat"]
2.59 +  unfolding card_eq_setsum by simp
2.60 +
2.61 +end
2.62 \ No newline at end of file
```
```     3.1 --- a/src/HOL/Library/Library.thy	Thu Jul 01 09:01:09 2010 +0200
3.2 +++ b/src/HOL/Library/Library.thy	Thu Jul 01 11:48:42 2010 +0200
3.3 @@ -27,6 +27,7 @@
3.4    Fset
3.5    FuncSet
3.6    Fundamental_Theorem_Algebra
3.7 +  Indicator_Function
3.8    Infinite_Set
3.9    Inner_Product
3.10    Lattice_Algebras
```
```     4.1 --- a/src/HOL/Multivariate_Analysis/Cartesian_Euclidean_Space.thy	Thu Jul 01 09:01:09 2010 +0200
4.2 +++ b/src/HOL/Multivariate_Analysis/Cartesian_Euclidean_Space.thy	Thu Jul 01 11:48:42 2010 +0200
4.3 @@ -179,9 +179,6 @@
4.4
4.5  subsection{* Basic componentwise operations on vectors. *}
4.6
4.7 -lemma dimindex_ge_1:"CARD(_::finite) \<ge> 1"
4.8 -  using one_le_card_finite by auto
4.9 -
4.10  instantiation cart :: (times,finite) times
4.11  begin
4.12    definition vector_mult_def : "op * \<equiv> (\<lambda> x y.  (\<chi> i. (x\$i) * (y\$i)))"
```
```     5.1 --- a/src/HOL/Multivariate_Analysis/Integration.thy	Thu Jul 01 09:01:09 2010 +0200
5.2 +++ b/src/HOL/Multivariate_Analysis/Integration.thy	Thu Jul 01 11:48:42 2010 +0200
5.3 @@ -4,7 +4,7 @@
5.4      Translation from HOL light: Robert Himmelmann, TU Muenchen *)
5.5
5.6  theory Integration
5.7 -  imports Derivative "~~/src/HOL/Decision_Procs/Dense_Linear_Order"
5.8 +  imports Derivative "~~/src/HOL/Decision_Procs/Dense_Linear_Order" Indicator_Function
5.9  begin
5.10
5.11  declare [[smt_certificates="~~/src/HOL/Multivariate_Analysis/Integration.certs"]]
5.12 @@ -2468,22 +2468,7 @@
5.13
5.14  subsection {* Negligible sets. *}
5.15
5.16 -definition "indicator s \<equiv> (\<lambda>x. if x \<in> s then 1 else (0::real))"
5.17 -
5.18 -lemma dest_vec1_indicator:
5.19 - "indicator s x = (if x \<in> s then 1 else 0)" unfolding indicator_def by auto
5.20 -
5.21 -lemma indicator_pos_le[intro]: "0 \<le> (indicator s x)" unfolding indicator_def by auto
5.22 -
5.23 -lemma indicator_le_1[intro]: "(indicator s x) \<le> 1" unfolding indicator_def by auto
5.24 -
5.25 -lemma dest_vec1_indicator_abs_le_1: "abs(indicator s x) \<le> 1"
5.26 -  unfolding indicator_def by auto
5.27 -
5.28 -definition "negligible (s::(_::ordered_euclidean_space) set) \<equiv> (\<forall>a b. ((indicator s) has_integral 0) {a..b})"
5.29 -
5.30 -lemma indicator_simps[simp]:"x\<in>s \<Longrightarrow> indicator s x = 1" "x\<notin>s \<Longrightarrow> indicator s x = 0"
5.31 -  unfolding indicator_def by auto
5.32 +definition "negligible (s::('a::ordered_euclidean_space) set) \<equiv> (\<forall>a b. ((indicator s :: 'a\<Rightarrow>real) has_integral 0) {a..b})"
5.33
5.34  subsection {* Negligibility of hyperplane. *}
5.35
5.36 @@ -2543,7 +2528,9 @@
5.37  lemma negligible_standard_hyperplane[intro]: fixes type::"'a::ordered_euclidean_space" assumes k:"k<DIM('a)"
5.38    shows "negligible {x::'a. x\$\$k = (c::real)}"
5.39    unfolding negligible_def has_integral apply(rule,rule,rule,rule)
5.40 -proof- case goal1 from content_doublesplit[OF this k,of a b c] guess d . note d=this let ?i = "indicator {x::'a. x\$\$k = c}"
5.41 +proof-
5.42 +  case goal1 from content_doublesplit[OF this k,of a b c] guess d . note d=this
5.43 +  let ?i = "indicator {x::'a. x\$\$k = c} :: 'a\<Rightarrow>real"
5.44    show ?case apply(rule_tac x="\<lambda>x. ball x d" in exI) apply(rule,rule gauge_ball,rule d)
5.45    proof(rule,rule) fix p assume p:"p tagged_division_of {a..b} \<and> (\<lambda>x. ball x d) fine p"
5.46      have *:"(\<Sum>(x, ka)\<in>p. content ka *\<^sub>R ?i x) = (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. abs(x\$\$k - c) \<le> d}) *\<^sub>R ?i x)"
5.47 @@ -2684,13 +2671,13 @@
5.48        have "\<forall>i. \<exists>q. q tagged_division_of {a..b} \<and> (d i) fine q \<and> (\<forall>(x, k)\<in>p. k \<subseteq> (d i) x \<longrightarrow> (x, k) \<in> q)"
5.49          apply(rule,rule tagged_division_finer[OF as(1) d(1)]) by auto
5.50        from choice[OF this] guess q .. note q=conjunctD3[OF this[rule_format]]
5.51 -      have *:"\<And>i. (\<Sum>(x, k)\<in>q i. content k *\<^sub>R indicator s x) \<ge> 0" apply(rule setsum_nonneg,safe)
5.52 +      have *:"\<And>i. (\<Sum>(x, k)\<in>q i. content k *\<^sub>R indicator s x) \<ge> (0::real)" apply(rule setsum_nonneg,safe)
5.53          unfolding real_scaleR_def apply(rule mult_nonneg_nonneg) apply(drule tagged_division_ofD(4)[OF q(1)]) by auto
5.54        have **:"\<And>f g s t. 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"
5.55        proof- case goal1 thus ?case apply-apply(rule setsum_le_included[of s t g snd f]) prefer 4
5.56            apply safe apply(erule_tac x=x in ballE) apply(erule exE) apply(rule_tac x="(xa,x)" in bexI) by auto qed
5.57        have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) \<le> setsum (\<lambda>i. (real i + 1) *
5.58 -                     norm(setsum (\<lambda>(x,k). content k *\<^sub>R indicator s x) (q i))) {0..N+1}"
5.59 +                     norm(setsum (\<lambda>(x,k). content k *\<^sub>R indicator s x :: real) (q i))) {0..N+1}"
5.60          unfolding real_norm_def setsum_right_distrib abs_of_nonneg[OF *] diff_0_right
5.61          apply(rule order_trans,rule setsum_norm) defer apply(subst sum_sum_product) prefer 3
5.62        proof(rule **,safe) show "finite {(i, j) |i j. i \<in> {0..N + 1} \<and> j \<in> q i}" apply(rule finite_product_dependent) using q by auto
5.63 @@ -2780,14 +2767,14 @@
5.64  lemma negligible_unions[intro]: assumes "finite s" "\<forall>t\<in>s. negligible t" shows "negligible(\<Union>s)"
5.65    using assms by(induct,auto)
5.66
5.67 -lemma negligible:  "negligible s \<longleftrightarrow> (\<forall>t::('a::ordered_euclidean_space) set. (indicator s has_integral 0) t)"
5.68 +lemma negligible:  "negligible s \<longleftrightarrow> (\<forall>t::('a::ordered_euclidean_space) set. ((indicator s::'a\<Rightarrow>real) has_integral 0) t)"
5.69    apply safe defer apply(subst negligible_def)
5.70  proof- fix t::"'a set" assume as:"negligible s"
5.71    have *:"(\<lambda>x. if x \<in> s \<inter> t then 1 else 0) = (\<lambda>x. if x\<in>t then if x\<in>s then 1 else 0 else 0)" by(rule ext,auto)
5.72 -  show "(indicator s has_integral 0) t" apply(subst has_integral_alt)
5.73 +  show "((indicator s::'a\<Rightarrow>real) has_integral 0) t" apply(subst has_integral_alt)
5.74      apply(cases,subst if_P,assumption) unfolding if_not_P apply(safe,rule as[unfolded negligible_def,rule_format])
5.75      apply(rule_tac x=1 in exI) apply(safe,rule zero_less_one) apply(rule_tac x=0 in exI)
5.76 -    using negligible_subset[OF as,of "s \<inter> t"] unfolding negligible_def indicator_def unfolding * by auto qed auto
5.77 +    using negligible_subset[OF as,of "s \<inter> t"] unfolding negligible_def indicator_def_raw unfolding * by auto qed auto
5.78
5.79  subsection {* Finite case of the spike theorem is quite commonly needed. *}
5.80
```