(*  Title: HOL/ex/Birthday_Paradoxon.thy

     Author: Lukas Bulwahn, TU Muenchen, 2007

 *)

 header {* A Formulation of the Birthday Paradoxon *}

 theory Birthday_Paradoxon

 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 paradoxon *}

 lemma birthday_paradoxon:

   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