(* Title: HOL/ex/Birthday_Paradox.thy
Author: Lukas Bulwahn, TU Muenchen, 2007
*)
header {* A Formulation of the Birthday Paradox *}
theory Birthday_Paradox
imports Main "~~/src/HOL/Fact" "~~/src/HOL/Library/FuncSet"
begin
section {* Cardinality *}
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))"
proof -
note `finite S`
moreover
have "{(x, y). x \<in> S \<and> y \<in> T x} = (UN x : S. Pair x ` T x)" by auto
moreover
from `\<forall>x \<in> S. finite (T x)` have "ALL x:S. finite (Pair x ` T x)" by auto
moreover
have " ALL i:S. ALL j:S. i ~= j --> Pair i ` T i Int Pair j ` T j = {}" by auto
moreover
ultimately have "card {(x, y). x \<in> S \<and> y \<in> T x} = (SUM i:S. card (Pair i ` T i))"
by (auto, subst card_UN_disjoint) auto
also have "... = (SUM x:S. card (T x))"
by (subst card_image) (auto intro: inj_onI)
finally show ?thesis by auto
qed
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 extensional_funcset_empty_domain)
next
case (insert x S)
{ fix x
from `finite T` have "finite (T - {x})" by auto
from `finite S` this have "finite (extensional_funcset S (T - {x}))"
by (rule finite_extensional_funcset)
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 `x \<notin> S`]
have "card {f. f : extensional_funcset (insert x S) T & inj_on f (insert x S)} =
card ((%(y, g). g(x := y)) ` {(y, g). y : T & g : extensional_funcset S (T - {y}) & inj_on g S})"
by metis
also from extensional_funcset_extend_domain_inj_onI[OF `x \<notin> S`, of T] have "... = card {(y, g). y : T & g : extensional_funcset S (T - {y}) & 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 `finite T` 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_nat[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_extensional_funcset)
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_extensional_funcset card_extensional_funcset_inj_on)
qed
lemma setprod_upto_nat_unfold:
"setprod f {m..(n::nat)} = (if n < m then 1 else (if n = 0 then f 0 else f n * setprod f {m..(n - 1)}))"
by auto (auto simp add: gr0_conv_Suc atLeastAtMostSuc_conv)
section {* Birthday paradox *}
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 `card S = 23` `card T = 365` have "finite S" "finite T" "card S <= card T" by (auto intro: card_ge_0_finite)
from assms show ?thesis
using card_extensional_funcset[OF `finite S`, of T]
card_extensional_funcset_not_inj_on[OF `finite S` `finite T` `card S <= card T`]
by (simp add: fact_div_fact setprod_upto_nat_unfold)
qed
end