(* Title: HOL/ex/Birthday_Paradox.thy
Author: Lukas Bulwahn, TU Muenchen, 2007
*)
section \<open>A Formulation of the Birthday Paradox\<close>
theory Birthday_Paradox
imports Main "HOL-Library.FuncSet"
begin
section \<open>Cardinality\<close>
lemma card_product_dependent:
assumes "finite S"
assumes "\<forall>x \<in> S. finite (T x)"
shows "card {(x, y). x \<in> S \<and> y \<in> T x} = (\<Sum>x \<in> S. card (T x))"
using card_SigmaI[OF assms, symmetric] by (auto intro!: arg_cong[where f=card] simp add: Sigma_def)
lemma card_extensional_funcset_inj_on:
assumes "finite S" "finite T" "card S \<le> card T"
shows "card {f \<in> extensional_funcset S T. inj_on f S} = fact (card T) div (fact (card T - card S))"
using assms
proof (induct S arbitrary: T rule: finite_induct)
case empty
from this show ?case by (simp add: Collect_conv_if PiE_empty_domain)
next
case (insert x S)
{ fix x
from \<open>finite T\<close> have "finite (T - {x})" by auto
from \<open>finite S\<close> this have "finite (extensional_funcset S (T - {x}))"
by (rule finite_PiE)
moreover
have "{f : extensional_funcset S (T - {x}). inj_on f S} \<subseteq> (extensional_funcset S (T - {x}))" by auto
ultimately have "finite {f : extensional_funcset S (T - {x}). inj_on f S}"
by (auto intro: finite_subset)
} note finite_delete = this
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
from extensional_funcset_extend_domain_inj_on_eq[OF \<open>x \<notin> S\<close>]
have "card {f. f \<in> extensional_funcset (insert x S) T \<and> inj_on f (insert x S)} =
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})"
by metis
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}"
by (simp add: card_image)
also have "card {(y, g). y \<in> T \<and> g \<in> extensional_funcset S (T - {y}) \<and> inj_on g S} =
card {(y, g). y \<in> T \<and> g \<in> {f \<in> extensional_funcset S (T - {y}). inj_on f S}}" by auto
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})"
by (subst card_product_dependent) auto
also from hyps have "... = (card T) * ?k"
by auto
also have "... = card T * fact (card T - 1) div fact (card T - card (insert x S))"
using insert unfolding div_mult1_eq[of "card T" "fact (card T - 1)"]
by (simp add: fact_mod)
also have "... = fact (card T) div fact (card T - card (insert x S))"
using insert by (simp add: fact_reduce[of "card T"])
finally show ?case .
qed
lemma card_extensional_funcset_not_inj_on:
assumes "finite S" "finite T" "card S \<le> card T"
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))"
proof -
have subset: "{f : extensional_funcset S T. inj_on f S} <= extensional_funcset S T" by auto
from finite_subset[OF subset] assms have finite: "finite {f : extensional_funcset S T. inj_on f S}"
by (auto intro!: finite_PiE)
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
from assms this finite subset show ?thesis
by (simp add: card_Diff_subset card_PiE card_extensional_funcset_inj_on prod_constant)
qed
lemma prod_upto_nat_unfold:
"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)}))"
by auto (auto simp add: gr0_conv_Suc atLeastAtMostSuc_conv)
section \<open>Birthday paradox\<close>
lemma birthday_paradox:
assumes "card S = 23" "card T = 365"
shows "2 * card {f \<in> extensional_funcset S T. \<not> inj_on f S} \<ge> card (extensional_funcset S T)"
proof -
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)
from assms show ?thesis
using card_PiE[OF \<open>finite S\<close>, of "\<lambda>i. T"] \<open>finite S\<close>
card_extensional_funcset_not_inj_on[OF \<open>finite S\<close> \<open>finite T\<close> \<open>card S <= card T\<close>]
by (simp add: fact_div_fact prod_upto_nat_unfold prod_constant)
qed
end