src/HOL/Library/Indicator_Function.thy
 author blanchet Wed Sep 24 15:45:55 2014 +0200 (2014-09-24) changeset 58425 246985c6b20b parent 57447 87429bdecad5 child 58729 e8ecc79aee43 permissions -rw-r--r--
simpler proof
```     1 (*  Title:      HOL/Library/Indicator_Function.thy
```
```     2     Author:     Johannes Hoelzl (TU Muenchen)
```
```     3 *)
```
```     4
```
```     5 header {* Indicator Function *}
```
```     6
```
```     7 theory Indicator_Function
```
```     8 imports Complex_Main
```
```     9 begin
```
```    10
```
```    11 definition "indicator S x = (if x \<in> S then 1 else 0)"
```
```    12
```
```    13 lemma indicator_simps[simp]:
```
```    14   "x \<in> S \<Longrightarrow> indicator S x = 1"
```
```    15   "x \<notin> S \<Longrightarrow> indicator S x = 0"
```
```    16   unfolding indicator_def by auto
```
```    17
```
```    18 lemma indicator_pos_le[intro, simp]: "(0::'a::linordered_semidom) \<le> indicator S x"
```
```    19   and indicator_le_1[intro, simp]: "indicator S x \<le> (1::'a::linordered_semidom)"
```
```    20   unfolding indicator_def by auto
```
```    21
```
```    22 lemma indicator_abs_le_1: "\<bar>indicator S x\<bar> \<le> (1::'a::linordered_idom)"
```
```    23   unfolding indicator_def by auto
```
```    24
```
```    25 lemma indicator_eq_0_iff: "indicator A x = (0::_::zero_neq_one) \<longleftrightarrow> x \<notin> A"
```
```    26   by (auto simp: indicator_def)
```
```    27
```
```    28 lemma indicator_eq_1_iff: "indicator A x = (1::_::zero_neq_one) \<longleftrightarrow> x \<in> A"
```
```    29   by (auto simp: indicator_def)
```
```    30
```
```    31 lemma split_indicator: "P (indicator S x) \<longleftrightarrow> ((x \<in> S \<longrightarrow> P 1) \<and> (x \<notin> S \<longrightarrow> P 0))"
```
```    32   unfolding indicator_def by auto
```
```    33
```
```    34 lemma split_indicator_asm: "P (indicator S x) \<longleftrightarrow> (\<not> (x \<in> S \<and> \<not> P 1 \<or> x \<notin> S \<and> \<not> P 0))"
```
```    35   unfolding indicator_def by auto
```
```    36
```
```    37 lemma indicator_inter_arith: "indicator (A \<inter> B) x = indicator A x * (indicator B x::'a::semiring_1)"
```
```    38   unfolding indicator_def by (auto simp: min_def max_def)
```
```    39
```
```    40 lemma indicator_union_arith: "indicator (A \<union> B) x = indicator A x + indicator B x - indicator A x * (indicator B x::'a::ring_1)"
```
```    41   unfolding indicator_def by (auto simp: min_def max_def)
```
```    42
```
```    43 lemma indicator_inter_min: "indicator (A \<inter> B) x = min (indicator A x) (indicator B x::'a::linordered_semidom)"
```
```    44   and indicator_union_max: "indicator (A \<union> B) x = max (indicator A x) (indicator B x::'a::linordered_semidom)"
```
```    45   unfolding indicator_def by (auto simp: min_def max_def)
```
```    46
```
```    47 lemma indicator_disj_union: "A \<inter> B = {} \<Longrightarrow>  indicator (A \<union> B) x = (indicator A x + indicator B x::'a::linordered_semidom)"
```
```    48   by (auto split: split_indicator)
```
```    49
```
```    50 lemma indicator_compl: "indicator (- A) x = 1 - (indicator A x::'a::ring_1)"
```
```    51   and indicator_diff: "indicator (A - B) x = indicator A x * (1 - indicator B x::'a::ring_1)"
```
```    52   unfolding indicator_def by (auto simp: min_def max_def)
```
```    53
```
```    54 lemma indicator_times: "indicator (A \<times> B) x = indicator A (fst x) * (indicator B (snd x)::'a::semiring_1)"
```
```    55   unfolding indicator_def by (cases x) auto
```
```    56
```
```    57 lemma indicator_sum: "indicator (A <+> B) x = (case x of Inl x \<Rightarrow> indicator A x | Inr x \<Rightarrow> indicator B x)"
```
```    58   unfolding indicator_def by (cases x) auto
```
```    59
```
```    60 lemma
```
```    61   fixes f :: "'a \<Rightarrow> 'b::semiring_1" assumes "finite A"
```
```    62   shows setsum_mult_indicator[simp]: "(\<Sum>x \<in> A. f x * indicator B x) = (\<Sum>x \<in> A \<inter> B. f x)"
```
```    63   and setsum_indicator_mult[simp]: "(\<Sum>x \<in> A. indicator B x * f x) = (\<Sum>x \<in> A \<inter> B. f x)"
```
```    64   unfolding indicator_def
```
```    65   using assms by (auto intro!: setsum.mono_neutral_cong_right split: split_if_asm)
```
```    66
```
```    67 lemma setsum_indicator_eq_card:
```
```    68   assumes "finite A"
```
```    69   shows "(SUM x : A. indicator B x) = card (A Int B)"
```
```    70   using setsum_mult_indicator[OF assms, of "%x. 1::nat"]
```
```    71   unfolding card_eq_setsum by simp
```
```    72
```
```    73 lemma setsum_indicator_scaleR[simp]:
```
```    74   "finite A \<Longrightarrow>
```
```    75     (\<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)"
```
```    76   using assms by (auto intro!: setsum.mono_neutral_cong_right split: split_if_asm simp: indicator_def)
```
```    77
```
```    78 lemma LIMSEQ_indicator_incseq:
```
```    79   assumes "incseq A"
```
```    80   shows "(\<lambda>i. indicator (A i) x :: 'a :: {topological_space, one, zero}) ----> indicator (\<Union>i. A i) x"
```
```    81 proof cases
```
```    82   assume "\<exists>i. x \<in> A i"
```
```    83   then obtain i where "x \<in> A i"
```
```    84     by auto
```
```    85   then have
```
```    86     "\<And>n. (indicator (A (n + i)) x :: 'a) = 1"
```
```    87     "(indicator (\<Union> i. A i) x :: 'a) = 1"
```
```    88     using incseqD[OF `incseq A`, of i "n + i" for n] `x \<in> A i` by (auto simp: indicator_def)
```
```    89   then show ?thesis
```
```    90     by (rule_tac LIMSEQ_offset[of _ i]) (simp add: tendsto_const)
```
```    91 qed (auto simp: indicator_def tendsto_const)
```
```    92
```
```    93 lemma LIMSEQ_indicator_UN:
```
```    94   "(\<lambda>k. indicator (\<Union> i<k. A i) x :: 'a :: {topological_space, one, zero}) ----> indicator (\<Union>i. A i) x"
```
```    95 proof -
```
```    96   have "(\<lambda>k. indicator (\<Union> i<k. A i) x::'a) ----> indicator (\<Union>k. \<Union> i<k. A i) x"
```
```    97     by (intro LIMSEQ_indicator_incseq) (auto simp: incseq_def intro: less_le_trans)
```
```    98   also have "(\<Union>k. \<Union> i<k. A i) = (\<Union>i. A i)"
```
```    99     by auto
```
```   100   finally show ?thesis .
```
```   101 qed
```
```   102
```
```   103 lemma LIMSEQ_indicator_decseq:
```
```   104   assumes "decseq A"
```
```   105   shows "(\<lambda>i. indicator (A i) x :: 'a :: {topological_space, one, zero}) ----> indicator (\<Inter>i. A i) x"
```
```   106 proof cases
```
```   107   assume "\<exists>i. x \<notin> A i"
```
```   108   then obtain i where "x \<notin> A i"
```
```   109     by auto
```
```   110   then have
```
```   111     "\<And>n. (indicator (A (n + i)) x :: 'a) = 0"
```
```   112     "(indicator (\<Inter>i. A i) x :: 'a) = 0"
```
```   113     using decseqD[OF `decseq A`, of i "n + i" for n] `x \<notin> A i` by (auto simp: indicator_def)
```
```   114   then show ?thesis
```
```   115     by (rule_tac LIMSEQ_offset[of _ i]) (simp add: tendsto_const)
```
```   116 qed (auto simp: indicator_def tendsto_const)
```
```   117
```
```   118 lemma LIMSEQ_indicator_INT:
```
```   119   "(\<lambda>k. indicator (\<Inter>i<k. A i) x :: 'a :: {topological_space, one, zero}) ----> indicator (\<Inter>i. A i) x"
```
```   120 proof -
```
```   121   have "(\<lambda>k. indicator (\<Inter>i<k. A i) x::'a) ----> indicator (\<Inter>k. \<Inter>i<k. A i) x"
```
```   122     by (intro LIMSEQ_indicator_decseq) (auto simp: decseq_def intro: less_le_trans)
```
```   123   also have "(\<Inter>k. \<Inter> i<k. A i) = (\<Inter>i. A i)"
```
```   124     by auto
```
```   125   finally show ?thesis .
```
```   126 qed
```
```   127
```
```   128 lemma indicator_add:
```
```   129   "A \<inter> B = {} \<Longrightarrow> (indicator A x::_::monoid_add) + indicator B x = indicator (A \<union> B) x"
```
```   130   unfolding indicator_def by auto
```
```   131
```
```   132 lemma of_real_indicator: "of_real (indicator A x) = indicator A x"
```
```   133   by (simp split: split_indicator)
```
```   134
```
```   135 lemma real_of_nat_indicator: "real (indicator A x :: nat) = indicator A x"
```
```   136   by (simp split: split_indicator)
```
```   137
```
```   138 lemma abs_indicator: "\<bar>indicator A x :: 'a::linordered_idom\<bar> = indicator A x"
```
```   139   by (simp split: split_indicator)
```
```   140
```
```   141 lemma mult_indicator_subset:
```
```   142   "A \<subseteq> B \<Longrightarrow> indicator A x * indicator B x = (indicator A x :: 'a::{comm_semiring_1})"
```
```   143   by (auto split: split_indicator simp: fun_eq_iff)
```
```   144
```
```   145 lemma indicator_sums:
```
```   146   assumes "\<And>i j. i \<noteq> j \<Longrightarrow> A i \<inter> A j = {}"
```
```   147   shows "(\<lambda>i. indicator (A i) x::real) sums indicator (\<Union>i. A i) x"
```
```   148 proof cases
```
```   149   assume "\<exists>i. x \<in> A i"
```
```   150   then obtain i where i: "x \<in> A i" ..
```
```   151   with assms have "(\<lambda>i. indicator (A i) x::real) sums (\<Sum>i\<in>{i}. indicator (A i) x)"
```
```   152     by (intro sums_finite) (auto split: split_indicator)
```
```   153   also have "(\<Sum>i\<in>{i}. indicator (A i) x) = indicator (\<Union>i. A i) x"
```
```   154     using i by (auto split: split_indicator)
```
```   155   finally show ?thesis .
```
```   156 qed simp
```
```   157
```
```   158 end
```