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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 |
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8 header{*Binomial Coefficients*} |
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9 |
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10 theory Binomial |
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11 imports SetInterval |
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12 begin |
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13 |
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14 text{*This development is based on the work of Andy Gordon and |
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15 Florian Kammueller*} |
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16 |
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17 consts |
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18 binomial :: "nat \<Rightarrow> nat \<Rightarrow> nat" (infixl "choose" 65) |
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19 |
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20 primrec |
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21 binomial_0: "(0 choose k) = (if k = 0 then 1 else 0)" |
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22 |
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23 binomial_Suc: "(Suc n choose k) = |
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24 (if k = 0 then 1 else (n choose (k - 1)) + (n choose k))" |
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25 |
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26 lemma binomial_n_0 [simp]: "(n choose 0) = 1" |
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27 by (case_tac "n", simp_all) |
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28 |
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29 lemma binomial_0_Suc [simp]: "(0 choose Suc k) = 0" |
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30 by simp |
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31 |
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32 lemma binomial_Suc_Suc [simp]: |
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33 "(Suc n choose Suc k) = (n choose k) + (n choose Suc k)" |
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34 by simp |
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35 |
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36 lemma binomial_eq_0 [rule_format]: "\<forall>k. n < k --> (n choose k) = 0" |
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37 apply (induct "n", auto) |
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38 apply (erule allE) |
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39 apply (erule mp, arith) |
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40 done |
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41 |
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42 declare binomial_0 [simp del] binomial_Suc [simp del] |
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43 |
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44 lemma binomial_n_n [simp]: "(n choose n) = 1" |
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45 apply (induct "n") |
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46 apply (simp_all add: binomial_eq_0) |
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47 done |
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48 |
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49 lemma binomial_Suc_n [simp]: "(Suc n choose n) = Suc n" |
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50 by (induct "n", simp_all) |
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51 |
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52 lemma binomial_1 [simp]: "(n choose Suc 0) = n" |
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53 by (induct "n", simp_all) |
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54 |
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55 lemma zero_less_binomial [rule_format]: "k \<le> n --> 0 < (n choose k)" |
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56 by (rule_tac m = n and n = k in diff_induct, simp_all) |
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57 |
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58 lemma binomial_eq_0_iff: "(n choose k = 0) = (n<k)" |
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59 apply (safe intro!: binomial_eq_0) |
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60 apply (erule contrapos_pp) |
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61 apply (simp add: zero_less_binomial) |
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62 done |
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63 |
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64 lemma zero_less_binomial_iff: "(0 < n choose k) = (k\<le>n)" |
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65 by (simp add: linorder_not_less [symmetric] binomial_eq_0_iff [symmetric]) |
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66 |
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67 (*Might be more useful if re-oriented*) |
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68 lemma Suc_times_binomial_eq [rule_format]: |
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69 "\<forall>k. k \<le> n --> Suc n * (n choose k) = (Suc n choose Suc k) * Suc k" |
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70 apply (induct "n") |
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71 apply (simp add: binomial_0, clarify) |
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72 apply (case_tac "k") |
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73 apply (auto simp add: add_mult_distrib add_mult_distrib2 le_Suc_eq |
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74 binomial_eq_0) |
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75 done |
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76 |
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77 text{*This is the well-known version, but it's harder to use because of the |
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78 need to reason about division.*} |
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79 lemma binomial_Suc_Suc_eq_times: |
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80 "k \<le> n ==> (Suc n choose Suc k) = (Suc n * (n choose k)) div Suc k" |
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81 by (simp add: Suc_times_binomial_eq div_mult_self_is_m zero_less_Suc |
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82 del: mult_Suc mult_Suc_right) |
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83 |
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84 text{*Another version, with -1 instead of Suc.*} |
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85 lemma times_binomial_minus1_eq: |
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86 "[|k \<le> n; 0<k|] ==> (n choose k) * k = n * ((n - 1) choose (k - 1))" |
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87 apply (cut_tac n = "n - 1" and k = "k - 1" in Suc_times_binomial_eq) |
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88 apply (simp split add: nat_diff_split, auto) |
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89 done |
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90 |
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91 subsubsection {* Theorems about @{text "choose"} *} |
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92 |
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93 text {* |
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94 \medskip Basic theorem about @{text "choose"}. By Florian |
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95 Kamm\"uller, tidied by LCP. |
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96 *} |
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97 |
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98 lemma card_s_0_eq_empty: |
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99 "finite A ==> card {B. B \<subseteq> A & card B = 0} = 1" |
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100 apply (simp cong add: conj_cong add: finite_subset [THEN card_0_eq]) |
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101 apply (simp cong add: rev_conj_cong) |
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102 done |
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103 |
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104 lemma choose_deconstruct: "finite M ==> x \<notin> M |
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105 ==> {s. s <= insert x M & card(s) = Suc k} |
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106 = {s. s <= M & card(s) = Suc k} Un |
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107 {s. EX t. t <= M & card(t) = k & s = insert x t}" |
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108 apply safe |
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109 apply (auto intro: finite_subset [THEN card_insert_disjoint]) |
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110 apply (drule_tac x = "xa - {x}" in spec) |
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111 apply (subgoal_tac "x \<notin> xa", auto) |
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112 apply (erule rev_mp, subst card_Diff_singleton) |
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113 apply (auto intro: finite_subset) |
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114 done |
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115 |
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116 text{*There are as many subsets of @{term A} having cardinality @{term k} |
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117 as there are sets obtained from the former by inserting a fixed element |
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118 @{term x} into each.*} |
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119 lemma constr_bij: |
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120 "[|finite A; x \<notin> A|] ==> |
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121 card {B. EX C. C <= A & card(C) = k & B = insert x C} = |
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122 card {B. B <= A & card(B) = k}" |
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123 apply (rule_tac f = "%s. s - {x}" and g = "insert x" in card_bij_eq) |
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124 apply (auto elim!: equalityE simp add: inj_on_def) |
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125 apply (subst Diff_insert0, auto) |
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126 txt {* finiteness of the two sets *} |
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127 apply (rule_tac [2] B = "Pow (A)" in finite_subset) |
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128 apply (rule_tac B = "Pow (insert x A)" in finite_subset) |
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129 apply fast+ |
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130 done |
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131 |
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132 text {* |
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133 Main theorem: combinatorial statement about number of subsets of a set. |
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134 *} |
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135 |
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136 lemma n_sub_lemma: |
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137 "!!A. finite A ==> card {B. B <= A & card B = k} = (card A choose k)" |
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138 apply (induct k) |
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139 apply (simp add: card_s_0_eq_empty, atomize) |
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140 apply (rotate_tac -1, erule finite_induct) |
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141 apply (simp_all (no_asm_simp) cong add: conj_cong |
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142 add: card_s_0_eq_empty choose_deconstruct) |
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143 apply (subst card_Un_disjoint) |
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144 prefer 4 apply (force simp add: constr_bij) |
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145 prefer 3 apply force |
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146 prefer 2 apply (blast intro: finite_Pow_iff [THEN iffD2] |
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147 finite_subset [of _ "Pow (insert x F)", standard]) |
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148 apply (blast intro: finite_Pow_iff [THEN iffD2, THEN [2] finite_subset]) |
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149 done |
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150 |
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151 theorem n_subsets: |
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152 "finite A ==> card {B. B <= A & card B = k} = (card A choose k)" |
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153 by (simp add: n_sub_lemma) |
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154 |
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155 |
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156 text{* The binomial theorem (courtesy of Tobias Nipkow): *} |
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157 |
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158 theorem binomial: "(a+b::nat)^n = (\<Sum>k=0..n. (n choose k) * a^k * b^(n-k))" |
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159 proof (induct n) |
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160 case 0 thus ?case by simp |
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161 next |
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162 case (Suc n) |
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163 have decomp: "{0..n+1} = {0} \<union> {n+1} \<union> {1..n}" |
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164 by (auto simp add:atLeastAtMost_def atLeast_def atMost_def) |
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165 have decomp2: "{0..n} = {0} \<union> {1..n}" |
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166 by (auto simp add:atLeastAtMost_def atLeast_def atMost_def) |
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167 have "(a+b::nat)^(n+1) = (a+b) * (\<Sum>k=0..n. (n choose k) * a^k * b^(n-k))" |
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168 using Suc by simp |
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169 also have "\<dots> = a*(\<Sum>k=0..n. (n choose k) * a^k * b^(n-k)) + |
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170 b*(\<Sum>k=0..n. (n choose k) * a^k * b^(n-k))" |
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171 by(rule nat_distrib) |
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172 also have "\<dots> = (\<Sum>k=0..n. (n choose k) * a^(k+1) * b^(n-k)) + |
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173 (\<Sum>k=0..n. (n choose k) * a^k * b^(n-k+1))" |
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174 by(simp add: setsum_mult mult_ac) |
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175 also have "\<dots> = (\<Sum>k=0..n. (n choose k) * a^k * b^(n+1-k)) + |
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176 (\<Sum>k=1..n+1. (n choose (k - 1)) * a^k * b^(n+1-k))" |
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177 by (simp add:setsum_shift_bounds_cl_Suc_ivl Suc_diff_le |
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178 del:setsum_cl_ivl_Suc) |
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179 also have "\<dots> = a^(n+1) + b^(n+1) + |
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180 (\<Sum>k=1..n. (n choose (k - 1)) * a^k * b^(n+1-k)) + |
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181 (\<Sum>k=1..n. (n choose k) * a^k * b^(n+1-k))" |
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182 by(simp add: decomp2) |
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183 also have |
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184 "\<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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185 by(simp add: nat_distrib setsum_addf binomial.simps) |
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186 also have "\<dots> = (\<Sum>k=0..n+1. (n+1 choose k) * a^k * b^(n+1-k))" |
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187 using decomp by simp |
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188 finally show ?case by simp |
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189 qed |
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190 |
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191 ML |
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192 {* |
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193 val binomial_n_0 = thm"binomial_n_0"; |
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194 val binomial_0_Suc = thm"binomial_0_Suc"; |
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195 val binomial_Suc_Suc = thm"binomial_Suc_Suc"; |
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196 val binomial_eq_0 = thm"binomial_eq_0"; |
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197 val binomial_n_n = thm"binomial_n_n"; |
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198 val binomial_Suc_n = thm"binomial_Suc_n"; |
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199 val binomial_1 = thm"binomial_1"; |
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200 val zero_less_binomial = thm"zero_less_binomial"; |
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201 val binomial_eq_0_iff = thm"binomial_eq_0_iff"; |
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202 val zero_less_binomial_iff = thm"zero_less_binomial_iff"; |
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203 val Suc_times_binomial_eq = thm"Suc_times_binomial_eq"; |
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204 val binomial_Suc_Suc_eq_times = thm"binomial_Suc_Suc_eq_times"; |
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205 val times_binomial_minus1_eq = thm"times_binomial_minus1_eq"; |
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206 *} |
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207 |
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208 end |