author bulwahn Fri Oct 21 11:17:14 2011 +0200 (2011-10-21) changeset 45231 d85a2fdc586c parent 43238 04c886a1d1a5 child 50123 69b35a75caf3 permissions -rw-r--r--
replacing code_inline by code_unfold, removing obsolete code_unfold, code_inline del now that the ancient code generator is removed
1 (*  Title: HOL/ex/Birthday_Paradox.thy
2     Author: Lukas Bulwahn, TU Muenchen, 2007
3 *)
5 header {* A Formulation of the Birthday Paradox *}
8 imports Main "~~/src/HOL/Fact" "~~/src/HOL/Library/FuncSet"
9 begin
11 section {* Cardinality *}
13 lemma 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 proof -
18   note `finite S`
19   moreover
20   have "{(x, y). x \<in> S \<and> y \<in> T x} = (UN x : S. Pair x ` T x)" by auto
21   moreover
22   from `\<forall>x \<in> S. finite (T x)` have "ALL x:S. finite (Pair x ` T x)" by auto
23   moreover
24   have " ALL i:S. ALL j:S. i ~= j --> Pair i ` T i Int Pair j ` T j = {}" by auto
25   moreover
26   ultimately have "card {(x, y). x \<in> S \<and> y \<in> T x} = (SUM i:S. card (Pair i ` T i))"
27     by (auto, subst card_UN_disjoint) auto
28   also have "... = (SUM x:S. card (T x))"
29     by (subst card_image) (auto intro: inj_onI)
30   finally show ?thesis by auto
31 qed
33 lemma card_extensional_funcset_inj_on:
34   assumes "finite S" "finite T" "card S \<le> card T"
35   shows "card {f \<in> extensional_funcset S T. inj_on f S} = fact (card T) div (fact (card T - card S))"
36 using assms
37 proof (induct S arbitrary: T rule: finite_induct)
38   case empty
39   from this show ?case by (simp add: Collect_conv_if extensional_funcset_empty_domain)
40 next
41   case (insert x S)
42   { fix x
43     from `finite T` have "finite (T - {x})" by auto
44     from `finite S` this have "finite (extensional_funcset S (T - {x}))"
45       by (rule finite_extensional_funcset)
46     moreover
47     have "{f : extensional_funcset S (T - {x}). inj_on f S} \<subseteq> (extensional_funcset S (T - {x}))" by auto
48     ultimately have "finite {f : extensional_funcset S (T - {x}). inj_on f S}"
49       by (auto intro: finite_subset)
50   } note finite_delete = this
51   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
52   from extensional_funcset_extend_domain_inj_on_eq[OF `x \<notin> S`]
53   have "card {f. f : extensional_funcset (insert x S) T & inj_on f (insert x S)} =
54     card ((%(y, g). g(x := y)) ` {(y, g). y : T & g : extensional_funcset S (T - {y}) & inj_on g S})"
55     by metis
56   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}"
57     by (simp add: card_image)
58   also have "card {(y, g). y \<in> T \<and> g \<in> extensional_funcset S (T - {y}) \<and> inj_on g S} =
59     card {(y, g). y \<in> T \<and> g \<in> {f \<in> extensional_funcset S (T - {y}). inj_on f S}}" by auto
60   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})"
61     by (subst card_product_dependent) auto
62   also from hyps have "... = (card T) * ?k"
63     by auto
64   also have "... = card T * fact (card T - 1) div fact (card T - card (insert x S))"
65     using insert unfolding div_mult1_eq[of "card T" "fact (card T - 1)"]
66     by (simp add: fact_mod)
67   also have "... = fact (card T) div fact (card T - card (insert x S))"
68     using insert by (simp add: fact_reduce_nat[of "card T"])
69   finally show ?case .
70 qed
72 lemma card_extensional_funcset_not_inj_on:
73   assumes "finite S" "finite T" "card S \<le> card T"
74   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))"
75 proof -
76   have subset: "{f : extensional_funcset S T. inj_on f S} <= extensional_funcset S T" by auto
77   from finite_subset[OF subset] assms have finite: "finite {f : extensional_funcset S T. inj_on f S}"
78     by (auto intro!: finite_extensional_funcset)
79   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
80   from assms this finite subset show ?thesis
81     by (simp add: card_Diff_subset card_extensional_funcset card_extensional_funcset_inj_on)
82 qed
84 lemma setprod_upto_nat_unfold:
85   "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)}))"
86   by auto (auto simp add: gr0_conv_Suc atLeastAtMostSuc_conv)
88 section {* Birthday paradox *}