--- a/src/HOL/IsaMakefile Tue Jun 07 11:10:42 2011 +0200
+++ b/src/HOL/IsaMakefile Tue Jun 07 11:10:57 2011 +0200
@@ -1047,7 +1047,7 @@
$(LOG)/HOL-ex.gz: $(OUT)/HOL Decision_Procs/Commutative_Ring.thy \
Number_Theory/Primes.thy ex/Abstract_NAT.thy ex/Antiquote.thy \
ex/Arith_Examples.thy ex/Arithmetic_Series_Complex.thy ex/BT.thy \
- ex/BinEx.thy ex/Binary.thy ex/Birthday_Paradoxon.thy \
+ ex/BinEx.thy ex/Binary.thy ex/Birthday_Paradox.thy \
ex/CASC_Setup.thy ex/CTL.thy ex/Case_Product.thy \
ex/Chinese.thy ex/Classical.thy ex/CodegenSML_Test.thy \
ex/Coercion_Examples.thy ex/Coherent.thy ex/Dedekind_Real.thy \
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/ex/Birthday_Paradox.thy Tue Jun 07 11:10:57 2011 +0200
@@ -0,0 +1,101 @@
+(* 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
--- a/src/HOL/ex/Birthday_Paradoxon.thy Tue Jun 07 11:10:42 2011 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,101 +0,0 @@
-(* 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
--- a/src/HOL/ex/ROOT.ML Tue Jun 07 11:10:42 2011 +0200
+++ b/src/HOL/ex/ROOT.ML Tue Jun 07 11:10:57 2011 +0200
@@ -73,7 +73,7 @@
"Gauge_Integration",
"Dedekind_Real",
"Quicksort",
- "Birthday_Paradoxon",
+ "Birthday_Paradox",
"List_to_Set_Comprehension_Examples",
"Set_Algebras"
];