1(*  Title: HOL/ex/Birthday_Paradox.thy
2    Author: Lukas Bulwahn, TU Muenchen, 2007
3*)
4
5section \<open>A Formulation of the Birthday Paradox\<close>
6
7theory Birthday_Paradox
8imports Main "HOL-Library.FuncSet"
9begin
10
11section \<open>Cardinality\<close>
12
13lemma card_product_dependent:
14  assumes "finite S"
15  assumes "\<forall>x \<in> S. finite (T x)"
16  shows "card {(x, y). x \<in> S \<and> y \<in> T x} = (\<Sum>x \<in> S. card (T x))"
17  using card_SigmaI[OF assms, symmetric] by (auto intro!: arg_cong[where f=card] simp add: Sigma_def)
18
19lemma card_extensional_funcset_inj_on:
20  assumes "finite S" "finite T" "card S \<le> card T"
21  shows "card {f \<in> extensional_funcset S T. inj_on f S} = fact (card T) div (fact (card T - card S))"
22using assms
23proof (induct S arbitrary: T rule: finite_induct)
24  case empty
25  from this show ?case by (simp add: Collect_conv_if PiE_empty_domain)
26next
27  case (insert x S)
28  { fix x
29    from \<open>finite T\<close> have "finite (T - {x})" by auto
30    from \<open>finite S\<close> this have "finite (extensional_funcset S (T - {x}))"
31      by (rule finite_PiE)
32    moreover
33    have "{f : extensional_funcset S (T - {x}). inj_on f S} \<subseteq> (extensional_funcset S (T - {x}))" by auto
34    ultimately have "finite {f : extensional_funcset S (T - {x}). inj_on f S}"
35      by (auto intro: finite_subset)
36  } note finite_delete = this
37  from insert have hyps: "\<forall>y \<in> T. card ({g. g \<in> extensional_funcset S (T - {y}) \<and> inj_on g S}) = fact (card T - 1) div fact ((card T - 1) - card S)"(is "\<forall> _ \<in> T. _ = ?k") by auto
38  from extensional_funcset_extend_domain_inj_on_eq[OF \<open>x \<notin> S\<close>]
39  have "card {f. f \<in> extensional_funcset (insert x S) T \<and> inj_on f (insert x S)} =
40    card ((\<lambda>(y, g). g(x := y)) ` {(y, g). y \<in> T \<and> g \<in> extensional_funcset S (T - {y}) \<and> inj_on g S})"
41    by metis
42  also from extensional_funcset_extend_domain_inj_onI[OF \<open>x \<notin> S\<close>, of T] have "\<dots> =  card {(y, g). y \<in> T \<and> g \<in> extensional_funcset S (T - {y}) \<and> inj_on g S}"
43    by (simp add: card_image)
44  also have "card {(y, g). y \<in> T \<and> g \<in> extensional_funcset S (T - {y}) \<and> inj_on g S} =
45    card {(y, g). y \<in> T \<and> g \<in> {f \<in> extensional_funcset S (T - {y}). inj_on f S}}" by auto
46  also from \<open>finite T\<close> finite_delete have "... = (\<Sum>y \<in> T. card {g. g \<in> extensional_funcset S (T - {y}) \<and>  inj_on g S})"
47    by (subst card_product_dependent) auto
48  also from hyps have "... = (card T) * ?k"
49    by auto
50  also have "... = card T * fact (card T - 1) div fact (card T - card (insert x S))"
51    using insert unfolding div_mult1_eq[of "card T" "fact (card T - 1)"]
52    by (simp add: fact_mod)
53  also have "... = fact (card T) div fact (card T - card (insert x S))"
54    using insert by (simp add: fact_reduce[of "card T"])
55  finally show ?case .
56qed
57
58lemma card_extensional_funcset_not_inj_on:
59  assumes "finite S" "finite T" "card S \<le> card T"
60  shows "card {f \<in> extensional_funcset S T. \<not> inj_on f S} = (card T) ^ (card S) - (fact (card T)) div (fact (card T - card S))"
61proof -
62  have subset: "{f : extensional_funcset S T. inj_on f S} <= extensional_funcset S T" by auto
63  from finite_subset[OF subset] assms have finite: "finite {f : extensional_funcset S T. inj_on f S}"
64    by (auto intro!: finite_PiE)
65  have "{f \<in> extensional_funcset S T. \<not> inj_on f S} = extensional_funcset S T - {f \<in> extensional_funcset S T. inj_on f S}" by auto
66  from assms this finite subset show ?thesis
67    by (simp add: card_Diff_subset card_PiE card_extensional_funcset_inj_on prod_constant)
68qed
69
70lemma prod_upto_nat_unfold:
71  "prod f {m..(n::nat)} = (if n < m then 1 else (if n = 0 then f 0 else f n * prod f {m..(n - 1)}))"
72  by auto (auto simp add: gr0_conv_Suc atLeastAtMostSuc_conv)
73
74section \<open>Birthday paradox\<close>
75
76lemma birthday_paradox:
77  assumes "card S = 23" "card T = 365"
78  shows "2 * card {f \<in> extensional_funcset S T. \<not> inj_on f S} \<ge> card (extensional_funcset S T)"
79proof -
80  from \<open>card S = 23\<close> \<open>card T = 365\<close> have "finite S" "finite T" "card S <= card T" by (auto intro: card_ge_0_finite)
81  from assms show ?thesis
82    using card_PiE[OF \<open>finite S\<close>, of "\<lambda>i. T"] \<open>finite S\<close>
83      card_extensional_funcset_not_inj_on[OF \<open>finite S\<close> \<open>finite T\<close> \<open>card S <= card T\<close>]
84    by (simp add: fact_div_fact prod_upto_nat_unfold prod_constant)
85qed
86
87end
88