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1 (* Title: HOL/Binomial.thy |
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2 ID: $Id$ |
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3 Author: Lawrence C Paulson |
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4 Copyright 1997 University of Cambridge |
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5 *) |
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6 |
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7 header{*Binomial Coefficients*} |
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8 |
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9 theory Binomial |
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10 imports Main |
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11 begin |
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12 |
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13 text{*This development is based on the work of Andy Gordon and |
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14 Florian Kammueller*} |
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15 |
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16 consts |
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17 binomial :: "nat \<Rightarrow> nat \<Rightarrow> nat" (infixl "choose" 65) |
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18 |
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19 primrec |
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20 binomial_0: "(0 choose k) = (if k = 0 then 1 else 0)" |
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21 |
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22 binomial_Suc: "(Suc n choose k) = |
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23 (if k = 0 then 1 else (n choose (k - 1)) + (n choose k))" |
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24 |
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25 lemma binomial_n_0 [simp]: "(n choose 0) = 1" |
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26 by (cases n) simp_all |
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27 |
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28 lemma binomial_0_Suc [simp]: "(0 choose Suc k) = 0" |
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29 by simp |
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30 |
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31 lemma binomial_Suc_Suc [simp]: |
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32 "(Suc n choose Suc k) = (n choose k) + (n choose Suc k)" |
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33 by simp |
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34 |
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35 lemma binomial_eq_0 [rule_format]: "\<forall>k. n < k --> (n choose k) = 0" |
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36 apply (induct "n") |
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37 apply auto |
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38 done |
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39 |
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40 declare binomial_0 [simp del] binomial_Suc [simp del] |
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41 |
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42 lemma binomial_n_n [simp]: "(n choose n) = 1" |
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43 apply (induct "n") |
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44 apply (simp_all add: binomial_eq_0) |
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45 done |
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46 |
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47 lemma binomial_Suc_n [simp]: "(Suc n choose n) = Suc n" |
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48 by (induct "n", simp_all) |
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49 |
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50 lemma binomial_1 [simp]: "(n choose Suc 0) = n" |
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51 by (induct "n", simp_all) |
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52 |
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53 lemma zero_less_binomial [rule_format]: "k \<le> n --> 0 < (n choose k)" |
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54 by (rule_tac m = n and n = k in diff_induct, simp_all) |
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55 |
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56 lemma binomial_eq_0_iff: "(n choose k = 0) = (n<k)" |
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57 apply (safe intro!: binomial_eq_0) |
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58 apply (erule contrapos_pp) |
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59 apply (simp add: zero_less_binomial) |
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60 done |
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61 |
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62 lemma zero_less_binomial_iff: "(0 < n choose k) = (k\<le>n)" |
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63 by (simp add: linorder_not_less [symmetric] binomial_eq_0_iff [symmetric]) |
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64 |
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65 (*Might be more useful if re-oriented*) |
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66 lemma Suc_times_binomial_eq [rule_format]: |
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67 "\<forall>k. k \<le> n --> Suc n * (n choose k) = (Suc n choose Suc k) * Suc k" |
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68 apply (induct "n") |
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69 apply (simp add: binomial_0, clarify) |
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70 apply (case_tac "k") |
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71 apply (auto simp add: add_mult_distrib add_mult_distrib2 le_Suc_eq |
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72 binomial_eq_0) |
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73 done |
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74 |
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75 text{*This is the well-known version, but it's harder to use because of the |
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76 need to reason about division.*} |
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77 lemma binomial_Suc_Suc_eq_times: |
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78 "k \<le> n ==> (Suc n choose Suc k) = (Suc n * (n choose k)) div Suc k" |
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79 by (simp add: Suc_times_binomial_eq div_mult_self_is_m zero_less_Suc |
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80 del: mult_Suc mult_Suc_right) |
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81 |
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82 text{*Another version, with -1 instead of Suc.*} |
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83 lemma times_binomial_minus1_eq: |
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84 "[|k \<le> n; 0<k|] ==> (n choose k) * k = n * ((n - 1) choose (k - 1))" |
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85 apply (cut_tac n = "n - 1" and k = "k - 1" in Suc_times_binomial_eq) |
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86 apply (simp split add: nat_diff_split, auto) |
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87 done |
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88 |
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89 subsubsection {* Theorems about @{text "choose"} *} |
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90 |
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91 text {* |
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92 \medskip Basic theorem about @{text "choose"}. By Florian |
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93 Kamm\"uller, tidied by LCP. |
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94 *} |
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95 |
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96 lemma card_s_0_eq_empty: |
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97 "finite A ==> card {B. B \<subseteq> A & card B = 0} = 1" |
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98 apply (simp cong add: conj_cong add: finite_subset [THEN card_0_eq]) |
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99 apply (simp cong add: rev_conj_cong) |
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100 done |
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101 |
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102 lemma choose_deconstruct: "finite M ==> x \<notin> M |
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103 ==> {s. s <= insert x M & card(s) = Suc k} |
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104 = {s. s <= M & card(s) = Suc k} Un |
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105 {s. EX t. t <= M & card(t) = k & s = insert x t}" |
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106 apply safe |
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107 apply (auto intro: finite_subset [THEN card_insert_disjoint]) |
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108 apply (drule_tac x = "xa - {x}" in spec) |
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109 apply (subgoal_tac "x \<notin> xa", auto) |
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110 apply (erule rev_mp, subst card_Diff_singleton) |
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111 apply (auto intro: finite_subset) |
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112 done |
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113 |
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114 text{*There are as many subsets of @{term A} having cardinality @{term k} |
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115 as there are sets obtained from the former by inserting a fixed element |
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116 @{term x} into each.*} |
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117 lemma constr_bij: |
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118 "[|finite A; x \<notin> A|] ==> |
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119 card {B. EX C. C <= A & card(C) = k & B = insert x C} = |
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120 card {B. B <= A & card(B) = k}" |
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121 apply (rule_tac f = "%s. s - {x}" and g = "insert x" in card_bij_eq) |
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122 apply (auto elim!: equalityE simp add: inj_on_def) |
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123 apply (subst Diff_insert0, auto) |
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124 txt {* finiteness of the two sets *} |
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125 apply (rule_tac [2] B = "Pow (A)" in finite_subset) |
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126 apply (rule_tac B = "Pow (insert x A)" in finite_subset) |
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127 apply fast+ |
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128 done |
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129 |
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130 text {* |
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131 Main theorem: combinatorial statement about number of subsets of a set. |
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132 *} |
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133 |
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134 lemma n_sub_lemma: |
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135 "!!A. finite A ==> card {B. B <= A & card B = k} = (card A choose k)" |
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136 apply (induct k) |
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137 apply (simp add: card_s_0_eq_empty, atomize) |
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138 apply (rotate_tac -1, erule finite_induct) |
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139 apply (simp_all (no_asm_simp) cong add: conj_cong |
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140 add: card_s_0_eq_empty choose_deconstruct) |
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141 apply (subst card_Un_disjoint) |
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142 prefer 4 apply (force simp add: constr_bij) |
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143 prefer 3 apply force |
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144 prefer 2 apply (blast intro: finite_Pow_iff [THEN iffD2] |
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145 finite_subset [of _ "Pow (insert x F)", standard]) |
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146 apply (blast intro: finite_Pow_iff [THEN iffD2, THEN [2] finite_subset]) |
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147 done |
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148 |
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149 theorem n_subsets: |
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150 "finite A ==> card {B. B <= A & card B = k} = (card A choose k)" |
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151 by (simp add: n_sub_lemma) |
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152 |
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153 |
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154 text{* The binomial theorem (courtesy of Tobias Nipkow): *} |
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155 |
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156 theorem binomial: "(a+b::nat)^n = (\<Sum>k=0..n. (n choose k) * a^k * b^(n-k))" |
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157 proof (induct n) |
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158 case 0 thus ?case by simp |
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159 next |
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160 case (Suc n) |
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161 have decomp: "{0..n+1} = {0} \<union> {n+1} \<union> {1..n}" |
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162 by (auto simp add:atLeastAtMost_def atLeast_def atMost_def) |
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163 have decomp2: "{0..n} = {0} \<union> {1..n}" |
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164 by (auto simp add:atLeastAtMost_def atLeast_def atMost_def) |
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165 have "(a+b::nat)^(n+1) = (a+b) * (\<Sum>k=0..n. (n choose k) * a^k * b^(n-k))" |
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166 using Suc by simp |
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167 also have "\<dots> = a*(\<Sum>k=0..n. (n choose k) * a^k * b^(n-k)) + |
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168 b*(\<Sum>k=0..n. (n choose k) * a^k * b^(n-k))" |
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169 by(rule nat_distrib) |
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170 also have "\<dots> = (\<Sum>k=0..n. (n choose k) * a^(k+1) * b^(n-k)) + |
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171 (\<Sum>k=0..n. (n choose k) * a^k * b^(n-k+1))" |
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172 by(simp add: setsum_right_distrib mult_ac) |
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173 also have "\<dots> = (\<Sum>k=0..n. (n choose k) * a^k * b^(n+1-k)) + |
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174 (\<Sum>k=1..n+1. (n choose (k - 1)) * a^k * b^(n+1-k))" |
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175 by (simp add:setsum_shift_bounds_cl_Suc_ivl Suc_diff_le |
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176 del:setsum_cl_ivl_Suc) |
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177 also have "\<dots> = a^(n+1) + b^(n+1) + |
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178 (\<Sum>k=1..n. (n choose (k - 1)) * a^k * b^(n+1-k)) + |
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179 (\<Sum>k=1..n. (n choose k) * a^k * b^(n+1-k))" |
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180 by(simp add: decomp2) |
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181 also have |
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182 "\<dots> = a^(n+1) + b^(n+1) + (\<Sum>k=1..n. (n+1 choose k) * a^k * b^(n+1-k))" |
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183 by(simp add: nat_distrib setsum_addf binomial.simps) |
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184 also have "\<dots> = (\<Sum>k=0..n+1. (n+1 choose k) * a^k * b^(n+1-k))" |
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185 using decomp by simp |
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186 finally show ?case by simp |
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187 qed |
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188 |
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189 end |