src/HOL/Library/Indicator_Function.thy
 author bulwahn Tue Apr 05 09:38:28 2011 +0200 (2011-04-05) changeset 42231 bc1891226d00 parent 37665 579258a77fec child 45425 7fee7d7abf2f permissions -rw-r--r--
```     1 (*  Title:      HOL/Library/Indicator_Function.thy
```
```     2     Author:     Johannes Hoelzl (TU Muenchen)
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```     3 *)
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```     4
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```     5 header {* Indicator Function *}
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```     6
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```     7 theory Indicator_Function
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```     8 imports Main
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```     9 begin
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```    10
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```    11 definition "indicator S x = (if x \<in> S then 1 else 0)"
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```    12
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```    13 lemma indicator_simps[simp]:
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```    14   "x \<in> S \<Longrightarrow> indicator S x = 1"
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```    15   "x \<notin> S \<Longrightarrow> indicator S x = 0"
```
```    16   unfolding indicator_def by auto
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```    17
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```    18 lemma
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```    19   shows indicator_pos_le[intro, simp]: "(0::'a::linordered_semidom) \<le> indicator S x"
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```    20   and indicator_le_1[intro, simp]: "indicator S x \<le> (1::'a::linordered_semidom)"
```
```    21   and indicator_abs_le_1: "\<bar>indicator S x\<bar> \<le> (1::'a::linordered_idom)"
```
```    22   unfolding indicator_def by auto
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```    23
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```    24 lemma split_indicator:
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```    25   "P (indicator S x) \<longleftrightarrow> ((x \<in> S \<longrightarrow> P 1) \<and> (x \<notin> S \<longrightarrow> P 0))"
```
```    26   unfolding indicator_def by auto
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```    27
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```    28 lemma
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```    29   shows indicator_inter_arith: "indicator (A \<inter> B) x = indicator A x * (indicator B x::'a::semiring_1)"
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```    30   and indicator_union_arith: "indicator (A \<union> B) x =  indicator A x + indicator B x - indicator A x * (indicator B x::'a::ring_1)"
```
```    31   and indicator_inter_min: "indicator (A \<inter> B) x = min (indicator A x) (indicator B x::'a::linordered_semidom)"
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```    32   and indicator_union_max: "indicator (A \<union> B) x = max (indicator A x) (indicator B x::'a::linordered_semidom)"
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```    33   and indicator_compl: "indicator (- A) x = 1 - (indicator A x::'a::ring_1)"
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```    34   and indicator_diff: "indicator (A - B) x = indicator A x * (1 - indicator B x::'a::ring_1)"
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```    35   unfolding indicator_def by (auto simp: min_def max_def)
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```    36
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```    37 lemma
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```    38   shows indicator_times: "indicator (A \<times> B) x = indicator A (fst x) * (indicator B (snd x)::'a::semiring_1)"
```
```    39   unfolding indicator_def by (cases x) auto
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```    40
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```    41 lemma
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```    42   shows 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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```    43   unfolding indicator_def by (cases x) auto
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```    44
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```    45 lemma
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```    46   fixes f :: "'a \<Rightarrow> 'b::semiring_1" assumes "finite A"
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```    47   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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```    48   and setsum_indicator_mult[simp]: "(\<Sum>x \<in> A. indicator B x * f x) = (\<Sum>x \<in> A \<inter> B. f x)"
```
```    49   unfolding indicator_def
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```    50   using assms by (auto intro!: setsum_mono_zero_cong_right split: split_if_asm)
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```    51
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```    52 lemma setsum_indicator_eq_card:
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```    53   assumes "finite A"
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```    54   shows "(SUM x : A. indicator B x) = card (A Int B)"
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```    55   using setsum_mult_indicator[OF assms, of "%x. 1::nat"]
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```    56   unfolding card_eq_setsum by simp
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```    57
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`    58 end`