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1 (* Title: HOL/GroupTheory/Exponent |
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2 ID: $Id$ |
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3 Author: Florian Kammueller, with new proofs by L C Paulson |
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4 |
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5 exponent p s yields the greatest power of p that divides s. |
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6 *) |
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7 |
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8 header{*The Combinatorial Argument Underlying the First Sylow Theorem*} |
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9 |
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10 theory Exponent = Main + Primes: |
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11 |
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12 constdefs |
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13 exponent :: "[nat, nat] => nat" |
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14 "exponent p s == if p \<in> prime then (GREATEST r. p^r dvd s) else 0" |
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15 |
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16 subsection{*Prime Theorems*} |
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17 |
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18 lemma prime_imp_one_less: "p \<in> prime ==> Suc 0 < p" |
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19 by (unfold prime_def, force) |
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20 |
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21 lemma prime_iff: |
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22 "(p \<in> prime) = (Suc 0 < p & (\<forall>a b. p dvd a*b --> (p dvd a) | (p dvd b)))" |
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23 apply (auto simp add: prime_imp_one_less) |
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24 apply (blast dest!: prime_dvd_mult) |
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25 apply (auto simp add: prime_def) |
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26 apply (erule dvdE) |
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27 apply (case_tac "k=0", simp) |
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28 apply (drule_tac x = m in spec) |
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29 apply (drule_tac x = k in spec) |
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30 apply (simp add: dvd_mult_cancel1 dvd_mult_cancel2, auto) |
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31 done |
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32 |
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33 lemma zero_less_prime_power: "p \<in> prime ==> 0 < p^a" |
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34 by (force simp add: prime_iff) |
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35 |
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36 |
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37 lemma le_extend_mult: "[| 0 < c; a <= b |] ==> a <= b * (c::nat)" |
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38 apply (rule_tac P = "%x. x <= b * c" in subst) |
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39 apply (rule mult_1_right) |
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40 apply (rule mult_le_mono, auto) |
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41 done |
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42 |
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43 lemma insert_partition: |
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44 "[| x \<notin> F; \<forall>c1\<in>insert x F. \<forall>c2 \<in> insert x F. c1 \<noteq> c2 --> c1 \<inter> c2 = {}|] |
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45 ==> x \<inter> \<Union> F = {}" |
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46 by auto |
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47 |
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48 (* main cardinality theorem *) |
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49 lemma card_partition [rule_format]: |
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50 "finite C ==> |
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51 finite (\<Union> C) --> |
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52 (\<forall>c\<in>C. card c = k) --> |
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53 (\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 --> c1 \<inter> c2 = {}) --> |
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54 k * card(C) = card (\<Union> C)" |
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55 apply (erule finite_induct, simp) |
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56 apply (simp add: card_insert_disjoint card_Un_disjoint insert_partition |
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57 finite_subset [of _ "\<Union> (insert x F)"]) |
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58 done |
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59 |
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60 lemma zero_less_card_empty: "[| finite S; S \<noteq> {} |] ==> 0 < card(S)" |
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61 by (rule ccontr, simp) |
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62 |
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63 |
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64 lemma prime_dvd_cases: |
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65 "[| p*k dvd m*n; p \<in> prime |] |
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66 ==> (\<exists>x. k dvd x*n & m = p*x) | (\<exists>y. k dvd m*y & n = p*y)" |
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67 apply (simp add: prime_iff) |
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68 apply (frule dvd_mult_left) |
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69 apply (subgoal_tac "p dvd m | p dvd n") |
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70 prefer 2 apply blast |
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71 apply (erule disjE) |
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72 apply (rule disjI1) |
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73 apply (rule_tac [2] disjI2) |
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74 apply (erule_tac n = m in dvdE) |
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75 apply (erule_tac [2] n = n in dvdE, auto) |
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76 apply (rule_tac [2] k = p in dvd_mult_cancel) |
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77 apply (rule_tac k = p in dvd_mult_cancel) |
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78 apply (simp_all add: mult_ac) |
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79 done |
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80 |
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81 |
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82 lemma prime_power_dvd_cases [rule_format (no_asm)]: "p \<in> prime |
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83 ==> \<forall>m n. p^c dvd m*n --> |
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84 (\<forall>a b. a+b = Suc c --> p^a dvd m | p^b dvd n)" |
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85 apply (induct_tac "c") |
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86 apply clarify |
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87 apply (case_tac "a") |
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88 apply simp |
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89 apply simp |
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90 (*inductive step*) |
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91 apply simp |
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92 apply clarify |
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93 apply (erule prime_dvd_cases [THEN disjE], assumption, auto) |
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94 (*case 1: p dvd m*) |
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95 apply (case_tac "a") |
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96 apply simp |
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97 apply clarify |
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98 apply (drule spec, drule spec, erule (1) notE impE) |
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99 apply (drule_tac x = nat in spec) |
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100 apply (drule_tac x = b in spec) |
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101 apply simp |
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102 apply (blast intro: dvd_refl mult_dvd_mono) |
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103 (*case 2: p dvd n*) |
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104 apply (case_tac "b") |
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105 apply simp |
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106 apply clarify |
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107 apply (drule spec, drule spec, erule (1) notE impE) |
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108 apply (drule_tac x = a in spec) |
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109 apply (drule_tac x = nat in spec, simp) |
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110 apply (blast intro: dvd_refl mult_dvd_mono) |
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111 done |
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112 |
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113 (*needed in this form in Sylow.ML*) |
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114 lemma div_combine: |
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115 "[| p \<in> prime; ~ (p ^ (Suc r) dvd n); p^(a+r) dvd n*k |] |
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116 ==> p ^ a dvd k" |
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117 by (drule_tac a = "Suc r" and b = a in prime_power_dvd_cases, assumption, auto) |
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118 |
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119 (*Lemma for power_dvd_bound*) |
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120 lemma Suc_le_power: "Suc 0 < p ==> Suc n <= p^n" |
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121 apply (induct_tac "n") |
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122 apply (simp (no_asm_simp)) |
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123 apply simp |
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124 apply (subgoal_tac "2 * n + 2 <= p * p^n", simp) |
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125 apply (subgoal_tac "2 * p^n <= p * p^n") |
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126 (*?arith_tac should handle all of this!*) |
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127 apply (rule order_trans) |
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128 prefer 2 apply assumption |
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129 apply (drule_tac k = 2 in mult_le_mono2, simp) |
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130 apply (rule mult_le_mono1, simp) |
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131 done |
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132 |
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133 (*An upper bound for the n such that p^n dvd a: needed for GREATEST to exist*) |
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134 lemma power_dvd_bound: "[|p^n dvd a; Suc 0 < p; 0 < a|] ==> n < a" |
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135 apply (drule dvd_imp_le) |
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136 apply (drule_tac [2] n = n in Suc_le_power, auto) |
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137 done |
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138 |
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139 |
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140 subsection{*Exponent Theorems*} |
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141 |
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142 lemma exponent_ge [rule_format]: |
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143 "[|p^k dvd n; p \<in> prime; 0<n|] ==> k <= exponent p n" |
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144 apply (simp add: exponent_def) |
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145 apply (erule Greatest_le) |
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146 apply (blast dest: prime_imp_one_less power_dvd_bound) |
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147 done |
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148 |
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149 lemma power_exponent_dvd: "0<s ==> (p ^ exponent p s) dvd s" |
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150 apply (simp add: exponent_def) |
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151 apply clarify |
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152 apply (rule_tac k = 0 in GreatestI) |
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153 prefer 2 apply (blast dest: prime_imp_one_less power_dvd_bound, simp) |
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154 done |
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155 |
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156 lemma power_Suc_exponent_Not_dvd: |
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157 "[|(p * p ^ exponent p s) dvd s; p \<in> prime |] ==> s=0" |
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158 apply (subgoal_tac "p ^ Suc (exponent p s) dvd s") |
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159 prefer 2 apply simp |
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160 apply (rule ccontr) |
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161 apply (drule exponent_ge, auto) |
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162 done |
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163 |
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164 lemma exponent_power_eq [simp]: "p \<in> prime ==> exponent p (p^a) = a" |
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165 apply (simp (no_asm_simp) add: exponent_def) |
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166 apply (rule Greatest_equality, simp) |
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167 apply (simp (no_asm_simp) add: prime_imp_one_less power_dvd_imp_le) |
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168 done |
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169 |
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170 lemma exponent_equalityI: |
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171 "!r::nat. (p^r dvd a) = (p^r dvd b) ==> exponent p a = exponent p b" |
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172 by (simp (no_asm_simp) add: exponent_def) |
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173 |
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174 lemma exponent_eq_0 [simp]: "p \<notin> prime ==> exponent p s = 0" |
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175 by (simp (no_asm_simp) add: exponent_def) |
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176 |
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177 |
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178 (* exponent_mult_add, easy inclusion. Could weaken p \<in> prime to Suc 0 < p *) |
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179 lemma exponent_mult_add1: |
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180 "[| 0 < a; 0 < b |] |
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181 ==> (exponent p a) + (exponent p b) <= exponent p (a * b)" |
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182 apply (case_tac "p \<in> prime") |
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183 apply (rule exponent_ge) |
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184 apply (auto simp add: power_add) |
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185 apply (blast intro: prime_imp_one_less power_exponent_dvd mult_dvd_mono) |
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186 done |
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187 |
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188 (* exponent_mult_add, opposite inclusion *) |
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189 lemma exponent_mult_add2: "[| 0 < a; 0 < b |] |
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190 ==> exponent p (a * b) <= (exponent p a) + (exponent p b)" |
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191 apply (case_tac "p \<in> prime") |
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192 apply (rule leI, clarify) |
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193 apply (cut_tac p = p and s = "a*b" in power_exponent_dvd, auto) |
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194 apply (subgoal_tac "p ^ (Suc (exponent p a + exponent p b)) dvd a * b") |
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195 apply (rule_tac [2] le_imp_power_dvd [THEN dvd_trans]) |
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196 prefer 3 apply assumption |
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197 prefer 2 apply simp |
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198 apply (frule_tac a = "Suc (exponent p a) " and b = "Suc (exponent p b) " in prime_power_dvd_cases) |
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199 apply (assumption, force, simp) |
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200 apply (blast dest: power_Suc_exponent_Not_dvd) |
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201 done |
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202 |
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203 lemma exponent_mult_add: |
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204 "[| 0 < a; 0 < b |] |
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205 ==> exponent p (a * b) = (exponent p a) + (exponent p b)" |
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206 by (blast intro: exponent_mult_add1 exponent_mult_add2 order_antisym) |
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207 |
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208 |
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209 lemma not_divides_exponent_0: "~ (p dvd n) ==> exponent p n = 0" |
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210 apply (case_tac "exponent p n", simp) |
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211 apply (case_tac "n", simp) |
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212 apply (cut_tac s = n and p = p in power_exponent_dvd) |
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213 apply (auto dest: dvd_mult_left) |
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214 done |
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215 |
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216 lemma exponent_1_eq_0 [simp]: "exponent p (Suc 0) = 0" |
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217 apply (case_tac "p \<in> prime") |
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218 apply (auto simp add: prime_iff not_divides_exponent_0) |
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219 done |
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220 |
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221 |
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222 subsection{*Lemmas for the Main Combinatorial Argument*} |
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223 |
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224 lemma p_fac_forw_lemma: |
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225 "[| 0 < (m::nat); 0<k; k < p^a; (p^r) dvd (p^a)* m - k |] ==> r <= a" |
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226 apply (rule notnotD) |
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227 apply (rule notI) |
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228 apply (drule contrapos_nn [OF _ leI, THEN notnotD], assumption) |
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229 apply (drule_tac m = a in less_imp_le) |
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230 apply (drule le_imp_power_dvd) |
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231 apply (drule_tac n = "p ^ r" in dvd_trans, assumption) |
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232 apply (frule_tac m = k in less_imp_le) |
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233 apply (drule_tac c = m in le_extend_mult, assumption) |
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234 apply (drule_tac k = "p ^ a" and m = " (p ^ a) * m" in dvd_diffD1) |
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235 prefer 2 apply assumption |
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236 apply (rule dvd_refl [THEN dvd_mult2]) |
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237 apply (drule_tac n = k in dvd_imp_le, auto) |
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238 done |
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239 |
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240 lemma p_fac_forw: "[| 0 < (m::nat); 0<k; k < p^a; (p^r) dvd (p^a)* m - k |] |
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241 ==> (p^r) dvd (p^a) - k" |
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242 apply (frule_tac k1 = k and i = p in p_fac_forw_lemma [THEN le_imp_power_dvd], auto) |
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243 apply (subgoal_tac "p^r dvd p^a*m") |
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244 prefer 2 apply (blast intro: dvd_mult2) |
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245 apply (drule dvd_diffD1) |
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246 apply assumption |
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247 prefer 2 apply (blast intro: dvd_diff) |
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248 apply (drule less_imp_Suc_add, auto) |
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249 done |
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250 |
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251 |
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252 lemma r_le_a_forw: "[| 0 < (k::nat); k < p^a; 0 < p; (p^r) dvd (p^a) - k |] ==> r <= a" |
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253 by (rule_tac m = "Suc 0" in p_fac_forw_lemma, auto) |
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254 |
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255 lemma p_fac_backw: "[| 0<m; 0<k; 0 < (p::nat); k < p^a; (p^r) dvd p^a - k |] |
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256 ==> (p^r) dvd (p^a)*m - k" |
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257 apply (frule_tac k1 = k and i = p in r_le_a_forw [THEN le_imp_power_dvd], auto) |
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258 apply (subgoal_tac "p^r dvd p^a*m") |
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259 prefer 2 apply (blast intro: dvd_mult2) |
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260 apply (drule dvd_diffD1) |
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261 apply assumption |
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262 prefer 2 apply (blast intro: dvd_diff) |
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263 apply (drule less_imp_Suc_add, auto) |
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264 done |
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265 |
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266 lemma exponent_p_a_m_k_equation: "[| 0<m; 0<k; 0 < (p::nat); k < p^a |] |
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267 ==> exponent p (p^a * m - k) = exponent p (p^a - k)" |
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268 apply (blast intro: exponent_equalityI p_fac_forw p_fac_backw) |
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269 done |
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270 |
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271 text{*Suc rules that we have to delete from the simpset*} |
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272 lemmas bad_Sucs = binomial_Suc_Suc mult_Suc mult_Suc_right |
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273 |
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274 (*The bound K is needed; otherwise it's too weak to be used.*) |
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275 lemma p_not_div_choose_lemma [rule_format]: |
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276 "[| \<forall>i. Suc i < K --> exponent p (Suc i) = exponent p (Suc(j+i))|] |
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277 ==> k<K --> exponent p ((j+k) choose k) = 0" |
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278 apply (case_tac "p \<in> prime") |
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279 prefer 2 apply simp |
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280 apply (induct_tac "k") |
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281 apply (simp (no_asm)) |
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282 (*induction step*) |
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283 apply (subgoal_tac "0 < (Suc (j+n) choose Suc n) ") |
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284 prefer 2 apply (simp add: zero_less_binomial_iff, clarify) |
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285 apply (subgoal_tac "exponent p ((Suc (j+n) choose Suc n) * Suc n) = |
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286 exponent p (Suc n)") |
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287 txt{*First, use the assumed equation. We simplify the LHS to |
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288 @{term "exponent p (Suc (j + n) choose Suc n) + exponent p (Suc n)"} |
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289 the common terms cancel, proving the conclusion.*} |
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290 apply (simp del: bad_Sucs add: exponent_mult_add) |
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291 txt{*Establishing the equation requires first applying |
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292 @{text Suc_times_binomial_eq} ...*} |
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293 apply (simp del: bad_Sucs add: Suc_times_binomial_eq [symmetric]) |
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294 txt{*...then @{text exponent_mult_add} and the quantified premise.*} |
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295 apply (simp del: bad_Sucs add: zero_less_binomial_iff exponent_mult_add) |
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296 done |
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297 |
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298 (*The lemma above, with two changes of variables*) |
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299 lemma p_not_div_choose: |
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300 "[| k<K; k<=n; |
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301 \<forall>j. 0<j & j<K --> exponent p (n - k + (K - j)) = exponent p (K - j)|] |
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302 ==> exponent p (n choose k) = 0" |
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303 apply (cut_tac j = "n-k" and k = k and p = p in p_not_div_choose_lemma) |
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304 prefer 3 apply simp |
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305 prefer 2 apply assumption |
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306 apply (drule_tac x = "K - Suc i" in spec) |
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307 apply (simp add: Suc_diff_le) |
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308 done |
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309 |
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310 |
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311 lemma const_p_fac_right: |
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312 "0 < m ==> exponent p ((p^a * m - Suc 0) choose (p^a - Suc 0)) = 0" |
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313 apply (case_tac "p \<in> prime") |
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314 prefer 2 apply simp |
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315 apply (frule_tac a = a in zero_less_prime_power) |
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316 apply (rule_tac K = "p^a" in p_not_div_choose) |
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317 apply simp |
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318 apply simp |
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319 apply (case_tac "m") |
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320 apply (case_tac [2] "p^a") |
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321 apply auto |
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322 (*now the hard case, simplified to |
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323 exponent p (Suc (p ^ a * m + i - p ^ a)) = exponent p (Suc i) *) |
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324 apply (subgoal_tac "0<p") |
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325 prefer 2 apply (force dest!: prime_imp_one_less) |
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326 apply (subst exponent_p_a_m_k_equation, auto) |
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327 done |
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328 |
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329 lemma const_p_fac: |
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330 "0 < m ==> exponent p (((p^a) * m) choose p^a) = exponent p m" |
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331 apply (case_tac "p \<in> prime") |
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332 prefer 2 apply simp |
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333 apply (subgoal_tac "0 < p^a * m & p^a <= p^a * m") |
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334 prefer 2 apply (force simp add: prime_iff) |
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335 txt{*A similar trick to the one used in @{text p_not_div_choose_lemma}: |
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336 insert an equation; use @{text exponent_mult_add} on the LHS; on the RHS, |
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337 first |
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338 transform the binomial coefficient, then use @{text exponent_mult_add}.*} |
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339 apply (subgoal_tac "exponent p ((( (p^a) * m) choose p^a) * p^a) = |
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340 a + exponent p m") |
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341 apply (simp del: bad_Sucs add: zero_less_binomial_iff exponent_mult_add prime_iff) |
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342 txt{*one subgoal left!*} |
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343 apply (subst times_binomial_minus1_eq, simp, simp) |
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344 apply (subst exponent_mult_add, simp) |
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345 apply (simp (no_asm_simp) add: zero_less_binomial_iff) |
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346 apply arith |
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347 apply (simp del: bad_Sucs add: exponent_mult_add const_p_fac_right) |
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348 done |
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349 |
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350 |
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351 end |