43 by (auto dest: prime_dvd_mult) |
43 by (auto dest: prime_dvd_mult) |
44 |
44 |
45 lemma prime_dvd_power_two: "prime p ==> p dvd m\<twosuperior> ==> p dvd m" |
45 lemma prime_dvd_power_two: "prime p ==> p dvd m\<twosuperior> ==> p dvd m" |
46 by (rule prime_dvd_square) (simp_all add: power2_eq_square) |
46 by (rule prime_dvd_square) (simp_all add: power2_eq_square) |
47 |
47 |
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48 |
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49 lemma exp_eq_1:"(x::nat)^n = 1 \<longleftrightarrow> x = 1 \<or> n = 0" by (induct n, auto) |
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50 lemma exp_mono_lt: "(x::nat) ^ (Suc n) < y ^ (Suc n) \<longleftrightarrow> x < y" |
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51 using power_less_imp_less_base[of x "Suc n" y] power_strict_mono[of x y "Suc n"] |
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52 by auto |
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53 lemma exp_mono_le: "(x::nat) ^ (Suc n) \<le> y ^ (Suc n) \<longleftrightarrow> x \<le> y" |
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54 by (simp only: linorder_not_less[symmetric] exp_mono_lt) |
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55 |
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56 lemma exp_mono_eq: "(x::nat) ^ Suc n = y ^ Suc n \<longleftrightarrow> x = y" |
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57 using power_inject_base[of x n y] by auto |
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58 |
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59 |
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60 lemma even_square: assumes e: "even (n::nat)" shows "\<exists>x. n ^ 2 = 4*x" |
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61 proof- |
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62 from e have "2 dvd n" by presburger |
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63 then obtain k where k: "n = 2*k" using dvd_def by auto |
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64 hence "n^2 = 4* (k^2)" by (simp add: power2_eq_square) |
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65 thus ?thesis by blast |
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66 qed |
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67 |
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68 lemma odd_square: assumes e: "odd (n::nat)" shows "\<exists>x. n ^ 2 = 4*x + 1" |
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69 proof- |
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70 from e have np: "n > 0" by presburger |
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71 from e have "2 dvd (n - 1)" by presburger |
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72 then obtain k where "n - 1 = 2*k" using dvd_def by auto |
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73 hence k: "n = 2*k + 1" using e by presburger |
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74 hence "n^2 = 4* (k^2 + k) + 1" by algebra |
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75 thus ?thesis by blast |
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76 qed |
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77 |
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78 lemma diff_square: "(x::nat)^2 - y^2 = (x+y)*(x - y)" |
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79 proof- |
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80 have "x \<le> y \<or> y \<le> x" by (rule nat_le_linear) |
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81 moreover |
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82 {assume le: "x \<le> y" |
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83 hence "x ^2 \<le> y^2" by (simp only: numeral_2_eq_2 exp_mono_le Let_def) |
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84 with le have ?thesis by simp } |
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85 moreover |
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86 {assume le: "y \<le> x" |
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87 hence le2: "y ^2 \<le> x^2" by (simp only: numeral_2_eq_2 exp_mono_le Let_def) |
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88 from le have "\<exists>z. y + z = x" by presburger |
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89 then obtain z where z: "x = y + z" by blast |
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90 from le2 have "\<exists>z. x^2 = y^2 + z" by presburger |
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91 then obtain z2 where z2: "x^2 = y^2 + z2" by blast |
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92 from z z2 have ?thesis apply simp by algebra } |
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93 ultimately show ?thesis by blast |
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94 qed |
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95 |
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96 (* Elementary theory of divisibility *) |
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97 lemma divides_ge: "(a::nat) dvd b \<Longrightarrow> b = 0 \<or> a \<le> b" unfolding dvd_def by auto |
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98 lemma divides_antisym: "(x::nat) dvd y \<and> y dvd x \<longleftrightarrow> x = y" |
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99 using dvd_anti_sym[of x y] by auto |
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100 |
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101 lemma divides_add_revr: assumes da: "(d::nat) dvd a" and dab:"d dvd (a + b)" |
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102 shows "d dvd b" |
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103 proof- |
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104 from da obtain k where k:"a = d*k" by (auto simp add: dvd_def) |
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105 from dab obtain k' where k': "a + b = d*k'" by (auto simp add: dvd_def) |
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106 from k k' have "b = d *(k' - k)" by (simp add : diff_mult_distrib2) |
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107 thus ?thesis unfolding dvd_def by blast |
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108 qed |
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109 |
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110 declare nat_mult_dvd_cancel_disj[presburger] |
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111 lemma nat_mult_dvd_cancel_disj'[presburger]: |
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112 "(m\<Colon>nat)*k dvd n*k \<longleftrightarrow> k = 0 \<or> m dvd n" unfolding mult_commute[of m k] mult_commute[of n k] by presburger |
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113 |
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114 lemma divides_mul_l: "(a::nat) dvd b ==> (c * a) dvd (c * b)" |
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115 by presburger |
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116 |
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117 lemma divides_mul_r: "(a::nat) dvd b ==> (a * c) dvd (b * c)" by presburger |
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118 lemma divides_cases: "(n::nat) dvd m ==> m = 0 \<or> m = n \<or> 2 * n <= m" |
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119 by (auto simp add: dvd_def) |
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120 lemma divides_le: "m dvd n ==> m <= n \<or> n = (0::nat)" by (auto simp add: dvd_def) |
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121 |
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122 lemma divides_div_not: "(x::nat) = (q * n) + r \<Longrightarrow> 0 < r \<Longrightarrow> r < n ==> ~(n dvd x)" |
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123 proof(auto simp add: dvd_def) |
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124 fix k assume H: "0 < r" "r < n" "q * n + r = n * k" |
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125 from H(3) have r: "r = n* (k -q)" by(simp add: diff_mult_distrib2 mult_commute) |
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126 {assume "k - q = 0" with r H(1) have False by simp} |
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127 moreover |
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128 {assume "k - q \<noteq> 0" with r have "r \<ge> n" by auto |
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129 with H(2) have False by simp} |
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130 ultimately show False by blast |
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131 qed |
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132 lemma divides_exp: "(x::nat) dvd y ==> x ^ n dvd y ^ n" |
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133 by (auto simp add: power_mult_distrib dvd_def) |
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134 |
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135 lemma divides_exp2: "n \<noteq> 0 \<Longrightarrow> (x::nat) ^ n dvd y \<Longrightarrow> x dvd y" |
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136 by (induct n ,auto simp add: dvd_def) |
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137 |
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138 fun fact :: "nat \<Rightarrow> nat" where |
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139 "fact 0 = 1" |
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140 | "fact (Suc n) = Suc n * fact n" |
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141 |
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142 lemma fact_lt: "0 < fact n" by(induct n, simp_all) |
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143 lemma fact_le: "fact n \<ge> 1" using fact_lt[of n] by simp |
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144 lemma fact_mono: assumes le: "m \<le> n" shows "fact m \<le> fact n" |
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145 proof- |
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146 from le have "\<exists>i. n = m+i" by presburger |
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147 then obtain i where i: "n = m+i" by blast |
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148 have "fact m \<le> fact (m + i)" |
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149 proof(induct m) |
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150 case 0 thus ?case using fact_le[of i] by simp |
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151 next |
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152 case (Suc m) |
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153 have "fact (Suc m) = Suc m * fact m" by simp |
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154 have th1: "Suc m \<le> Suc (m + i)" by simp |
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155 from mult_le_mono[of "Suc m" "Suc (m+i)" "fact m" "fact (m+i)", OF th1 Suc.hyps] |
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156 show ?case by simp |
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157 qed |
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158 thus ?thesis using i by simp |
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159 qed |
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160 |
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161 lemma divides_fact: "1 <= p \<Longrightarrow> p <= n ==> p dvd fact n" |
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162 proof(induct n arbitrary: p) |
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163 case 0 thus ?case by simp |
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164 next |
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165 case (Suc n p) |
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166 from Suc.prems have "p = Suc n \<or> p \<le> n" by presburger |
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167 moreover |
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168 {assume "p = Suc n" hence ?case by (simp only: fact.simps dvd_triv_left)} |
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169 moreover |
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170 {assume "p \<le> n" |
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171 with Suc.prems(1) Suc.hyps have th: "p dvd fact n" by simp |
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172 from dvd_mult[OF th] have ?case by (simp only: fact.simps) } |
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173 ultimately show ?case by blast |
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174 qed |
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175 |
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176 declare dvd_triv_left[presburger] |
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177 declare dvd_triv_right[presburger] |
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178 lemma divides_rexp: |
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179 "x dvd y \<Longrightarrow> (x::nat) dvd (y^(Suc n))" by (simp add: dvd_mult2[of x y]) |
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180 |
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181 (* The Bezout theorem is a bit ugly for N; it'd be easier for Z *) |
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182 lemma ind_euclid: |
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183 assumes c: " \<forall>a b. P (a::nat) b \<longleftrightarrow> P b a" and z: "\<forall>a. P a 0" |
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184 and add: "\<forall>a b. P a b \<longrightarrow> P a (a + b)" |
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185 shows "P a b" |
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186 proof(induct n\<equiv>"a+b" arbitrary: a b rule: nat_less_induct) |
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187 fix n a b |
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188 assume H: "\<forall>m < n. \<forall>a b. m = a + b \<longrightarrow> P a b" "n = a + b" |
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189 have "a = b \<or> a < b \<or> b < a" by arith |
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190 moreover {assume eq: "a= b" |
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191 from add[rule_format, OF z[rule_format, of a]] have "P a b" using eq by simp} |
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192 moreover |
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193 {assume lt: "a < b" |
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194 hence "a + b - a < n \<or> a = 0" using H(2) by arith |
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195 moreover |
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196 {assume "a =0" with z c have "P a b" by blast } |
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197 moreover |
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198 {assume ab: "a + b - a < n" |
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199 have th0: "a + b - a = a + (b - a)" using lt by arith |
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200 from add[rule_format, OF H(1)[rule_format, OF ab th0]] |
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201 have "P a b" by (simp add: th0[symmetric])} |
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202 ultimately have "P a b" by blast} |
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203 moreover |
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204 {assume lt: "a > b" |
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205 hence "b + a - b < n \<or> b = 0" using H(2) by arith |
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206 moreover |
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207 {assume "b =0" with z c have "P a b" by blast } |
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208 moreover |
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209 {assume ab: "b + a - b < n" |
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210 have th0: "b + a - b = b + (a - b)" using lt by arith |
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211 from add[rule_format, OF H(1)[rule_format, OF ab th0]] |
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212 have "P b a" by (simp add: th0[symmetric]) |
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213 hence "P a b" using c by blast } |
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214 ultimately have "P a b" by blast} |
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215 ultimately show "P a b" by blast |
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216 qed |
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217 |
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218 lemma bezout_lemma: |
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219 assumes ex: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and> (a * x = b * y + d \<or> b * x = a * y + d)" |
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220 shows "\<exists>d x y. d dvd a \<and> d dvd a + b \<and> (a * x = (a + b) * y + d \<or> (a + b) * x = a * y + d)" |
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221 using ex |
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222 apply clarsimp |
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223 apply (rule_tac x="d" in exI, simp add: dvd_add) |
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224 apply (case_tac "a * x = b * y + d" , simp_all) |
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225 apply (rule_tac x="x + y" in exI) |
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226 apply (rule_tac x="y" in exI) |
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227 apply algebra |
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228 apply (rule_tac x="x" in exI) |
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229 apply (rule_tac x="x + y" in exI) |
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230 apply algebra |
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231 done |
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232 |
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233 lemma bezout_add: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and> (a * x = b * y + d \<or> b * x = a * y + d)" |
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234 apply(induct a b rule: ind_euclid) |
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235 apply blast |
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236 apply clarify |
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237 apply (rule_tac x="a" in exI, simp add: dvd_add) |
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238 apply clarsimp |
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239 apply (rule_tac x="d" in exI) |
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240 apply (case_tac "a * x = b * y + d", simp_all add: dvd_add) |
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241 apply (rule_tac x="x+y" in exI) |
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242 apply (rule_tac x="y" in exI) |
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243 apply algebra |
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244 apply (rule_tac x="x" in exI) |
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245 apply (rule_tac x="x+y" in exI) |
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246 apply algebra |
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247 done |
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248 |
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249 lemma bezout: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and> (a * x - b * y = d \<or> b * x - a * y = d)" |
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250 using bezout_add[of a b] |
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251 apply clarsimp |
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252 apply (rule_tac x="d" in exI, simp) |
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253 apply (rule_tac x="x" in exI) |
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254 apply (rule_tac x="y" in exI) |
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255 apply auto |
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256 done |
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257 |
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258 (* We can get a stronger version with a nonzeroness assumption. *) |
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259 |
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260 lemma bezout_add_strong: assumes nz: "a \<noteq> (0::nat)" |
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261 shows "\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d" |
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262 proof- |
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263 from nz have ap: "a > 0" by simp |
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264 from bezout_add[of a b] |
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265 have "(\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d) \<or> (\<exists>d x y. d dvd a \<and> d dvd b \<and> b * x = a * y + d)" by blast |
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266 moreover |
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267 {fix d x y assume H: "d dvd a" "d dvd b" "a * x = b * y + d" |
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268 from H have ?thesis by blast } |
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269 moreover |
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270 {fix d x y assume H: "d dvd a" "d dvd b" "b * x = a * y + d" |
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271 {assume b0: "b = 0" with H have ?thesis by simp} |
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272 moreover |
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273 {assume b: "b \<noteq> 0" hence bp: "b > 0" by simp |
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274 from divides_le[OF H(2)] b have "d < b \<or> d = b" using le_less by blast |
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275 moreover |
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276 {assume db: "d=b" |
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277 from prems have ?thesis apply simp |
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278 apply (rule exI[where x = b], simp) |
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279 apply (rule exI[where x = b]) |
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280 by (rule exI[where x = "a - 1"], simp add: diff_mult_distrib2)} |
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281 moreover |
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282 {assume db: "d < b" |
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283 {assume "x=0" hence ?thesis using prems by simp } |
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284 moreover |
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285 {assume x0: "x \<noteq> 0" hence xp: "x > 0" by simp |
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286 |
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287 from db have "d \<le> b - 1" by simp |
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288 hence "d*b \<le> b*(b - 1)" by simp |
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289 with xp mult_mono[of "1" "x" "d*b" "b*(b - 1)"] |
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290 have dble: "d*b \<le> x*b*(b - 1)" using bp by simp |
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291 from H (3) have "d + (b - 1) * (b*x) = d + (b - 1) * (a*y + d)" by simp |
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292 hence "d + (b - 1) * a * y + (b - 1) * d = d + (b - 1) * b * x" |
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293 by (simp only: mult_assoc right_distrib) |
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294 hence "a * ((b - 1) * y) + d * (b - 1 + 1) = d + x*b*(b - 1)" by algebra |
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295 hence "a * ((b - 1) * y) = d + x*b*(b - 1) - d*b" using bp by simp |
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296 hence "a * ((b - 1) * y) = d + (x*b*(b - 1) - d*b)" |
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297 by (simp only: diff_add_assoc[OF dble, of d, symmetric]) |
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298 hence "a * ((b - 1) * y) = b*(x*(b - 1) - d) + d" |
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299 by (simp only: diff_mult_distrib2 add_commute mult_ac) |
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300 hence ?thesis using H(1,2) |
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301 apply - |
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302 apply (rule exI[where x=d], simp) |
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303 apply (rule exI[where x="(b - 1) * y"]) |
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304 by (rule exI[where x="x*(b - 1) - d"], simp)} |
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305 ultimately have ?thesis by blast} |
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306 ultimately have ?thesis by blast} |
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307 ultimately have ?thesis by blast} |
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308 ultimately show ?thesis by blast |
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309 qed |
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310 |
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311 (* Greatest common divisor. *) |
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312 lemma gcd_unique: "d dvd a\<and>d dvd b \<and> (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd(a,b)" |
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313 proof(auto) |
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314 assume H: "d dvd a" "d dvd b" "\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d" |
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315 from H(3)[rule_format] gcd_dvd1[of a b] gcd_dvd2[of a b] |
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316 have th: "gcd (a,b) dvd d" by blast |
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317 from dvd_anti_sym[OF th gcd_greatest[OF H(1,2)]] show "d = gcd (a,b)" by blast |
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318 qed |
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319 |
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320 lemma gcd_eq: assumes H: "\<forall>d. d dvd x \<and> d dvd y \<longleftrightarrow> d dvd u \<and> d dvd v" |
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321 shows "gcd (x,y) = gcd(u,v)" |
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322 proof- |
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323 from H have "\<forall>d. d dvd x \<and> d dvd y \<longleftrightarrow> d dvd gcd (u,v)" by simp |
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324 with gcd_unique[of "gcd(u,v)" x y] show ?thesis by auto |
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325 qed |
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326 |
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327 lemma bezout_gcd: "\<exists>x y. a * x - b * y = gcd(a,b) \<or> b * x - a * y = gcd(a,b)" |
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328 proof- |
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329 let ?g = "gcd (a,b)" |
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330 from bezout[of a b] obtain d x y where d: "d dvd a" "d dvd b" "a * x - b * y = d \<or> b * x - a * y = d" by blast |
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331 from d(1,2) have "d dvd ?g" by simp |
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332 then obtain k where k: "?g = d*k" unfolding dvd_def by blast |
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333 from d(3) have "(a * x - b * y)*k = d*k \<or> (b * x - a * y)*k = d*k" by blast |
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334 hence "a * x * k - b * y*k = d*k \<or> b * x * k - a * y*k = d*k" |
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335 by (simp only: diff_mult_distrib) |
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336 hence "a * (x * k) - b * (y*k) = ?g \<or> b * (x * k) - a * (y*k) = ?g" |
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337 by (simp add: k mult_assoc) |
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338 thus ?thesis by blast |
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339 qed |
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340 |
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341 lemma bezout_gcd_strong: assumes a: "a \<noteq> 0" |
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342 shows "\<exists>x y. a * x = b * y + gcd(a,b)" |
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343 proof- |
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344 let ?g = "gcd (a,b)" |
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345 from bezout_add_strong[OF a, of b] |
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346 obtain d x y where d: "d dvd a" "d dvd b" "a * x = b * y + d" by blast |
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347 from d(1,2) have "d dvd ?g" by simp |
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348 then obtain k where k: "?g = d*k" unfolding dvd_def by blast |
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349 from d(3) have "a * x * k = (b * y + d) *k " by auto |
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350 hence "a * (x * k) = b * (y*k) + ?g" by (algebra add: k) |
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351 thus ?thesis by blast |
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352 qed |
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353 |
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354 lemma gcd_mult_distrib: "gcd(a * c, b * c) = c * gcd(a,b)" |
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355 by(simp add: gcd_mult_distrib2 mult_commute) |
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356 |
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357 lemma gcd_bezout: "(\<exists>x y. a * x - b * y = d \<or> b * x - a * y = d) \<longleftrightarrow> gcd(a,b) dvd d" |
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358 (is "?lhs \<longleftrightarrow> ?rhs") |
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359 proof- |
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360 let ?g = "gcd (a,b)" |
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361 {assume H: ?rhs then obtain k where k: "d = ?g*k" unfolding dvd_def by blast |
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362 from bezout_gcd[of a b] obtain x y where xy: "a * x - b * y = ?g \<or> b * x - a * y = ?g" |
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363 by blast |
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364 hence "(a * x - b * y)*k = ?g*k \<or> (b * x - a * y)*k = ?g*k" by auto |
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365 hence "a * x*k - b * y*k = ?g*k \<or> b * x * k - a * y*k = ?g*k" |
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366 by (simp only: diff_mult_distrib) |
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367 hence "a * (x*k) - b * (y*k) = d \<or> b * (x * k) - a * (y*k) = d" |
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368 by (simp add: k[symmetric] mult_assoc) |
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369 hence ?lhs by blast} |
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370 moreover |
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371 {fix x y assume H: "a * x - b * y = d \<or> b * x - a * y = d" |
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372 have dv: "?g dvd a*x" "?g dvd b * y" "?g dvd b*x" "?g dvd a * y" |
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373 using dvd_mult2[OF gcd_dvd1[of a b]] dvd_mult2[OF gcd_dvd2[of a b]] by simp_all |
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374 from dvd_diff[OF dv(1,2)] dvd_diff[OF dv(3,4)] H |
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375 have ?rhs by auto} |
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376 ultimately show ?thesis by blast |
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377 qed |
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378 |
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379 lemma gcd_bezout_sum: assumes H:"a * x + b * y = d" shows "gcd(a,b) dvd d" |
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380 proof- |
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381 let ?g = "gcd (a,b)" |
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382 have dv: "?g dvd a*x" "?g dvd b * y" |
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383 using dvd_mult2[OF gcd_dvd1[of a b]] dvd_mult2[OF gcd_dvd2[of a b]] by simp_all |
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384 from dvd_add[OF dv] H |
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385 show ?thesis by auto |
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386 qed |
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387 |
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388 lemma gcd_mult': "gcd(b,a * b) = b" |
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389 by (simp add: gcd_mult mult_commute[of a b]) |
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390 |
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391 lemma gcd_add: "gcd(a + b,b) = gcd(a,b)" "gcd(b + a,b) = gcd(a,b)" "gcd(a,a + b) = gcd(a,b)" "gcd(a,b + a) = gcd(a,b)" |
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392 apply (simp_all add: gcd_add1) |
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393 by (simp add: gcd_commute gcd_add1) |
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394 |
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395 lemma gcd_sub: "b <= a ==> gcd(a - b,b) = gcd(a,b)" "a <= b ==> gcd(a,b - a) = gcd(a,b)" |
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396 proof- |
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397 {fix a b assume H: "b \<le> (a::nat)" |
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398 hence th: "a - b + b = a" by arith |
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399 from gcd_add(1)[of "a - b" b] th have "gcd(a - b,b) = gcd(a,b)" by simp} |
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400 note th = this |
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401 { |
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402 assume ab: "b \<le> a" |
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403 from th[OF ab] show "gcd (a - b, b) = gcd (a, b)" by blast |
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404 next |
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405 assume ab: "a \<le> b" |
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406 from th[OF ab] show "gcd (a,b - a) = gcd (a, b)" |
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407 by (simp add: gcd_commute)} |
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408 qed |
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409 |
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410 (* Coprimality *) |
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411 |
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412 lemma coprime: "coprime a b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = 1)" |
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413 using gcd_unique[of 1 a b, simplified] by (auto simp add: coprime_def) |
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414 lemma coprime_commute: "coprime a b \<longleftrightarrow> coprime b a" by (simp add: coprime_def gcd_commute) |
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415 |
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416 lemma coprime_bezout: "coprime a b \<longleftrightarrow> (\<exists>x y. a * x - b * y = 1 \<or> b * x - a * y = 1)" |
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417 using coprime_def gcd_bezout by auto |
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418 |
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419 lemma coprime_divprod: "d dvd a * b \<Longrightarrow> coprime d a \<Longrightarrow> d dvd b" |
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420 using relprime_dvd_mult_iff[of d a b] by (auto simp add: coprime_def mult_commute) |
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421 |
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422 lemma coprime_1[simp]: "coprime a 1" by (simp add: coprime_def) |
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423 lemma coprime_1'[simp]: "coprime 1 a" by (simp add: coprime_def) |
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424 lemma coprime_Suc0[simp]: "coprime a (Suc 0)" by (simp add: coprime_def) |
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425 lemma coprime_Suc0'[simp]: "coprime (Suc 0) a" by (simp add: coprime_def) |
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426 |
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427 lemma gcd_coprime: |
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428 assumes z: "gcd(a,b) \<noteq> 0" and a: "a = a' * gcd(a,b)" and b: "b = b' * gcd(a,b)" |
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429 shows "coprime a' b'" |
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430 proof- |
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431 let ?g = "gcd(a,b)" |
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432 {assume bz: "a = 0" from b bz z a have ?thesis by (simp add: gcd_zero coprime_def)} |
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433 moreover |
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434 {assume az: "a\<noteq> 0" |
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435 from z have z': "?g > 0" by simp |
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436 from bezout_gcd_strong[OF az, of b] |
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437 obtain x y where xy: "a*x = b*y + ?g" by blast |
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438 from xy a b have "?g * a'*x = ?g * (b'*y + 1)" by (simp add: ring_simps) |
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439 hence "?g * (a'*x) = ?g * (b'*y + 1)" by (simp add: mult_assoc) |
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440 hence "a'*x = (b'*y + 1)" |
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441 by (simp only: nat_mult_eq_cancel1[OF z']) |
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442 hence "a'*x - b'*y = 1" by simp |
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443 with coprime_bezout[of a' b'] have ?thesis by auto} |
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444 ultimately show ?thesis by blast |
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445 qed |
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446 lemma coprime_0: "coprime d 0 \<longleftrightarrow> d = 1" by (simp add: coprime_def) |
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447 lemma coprime_mul: assumes da: "coprime d a" and db: "coprime d b" |
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448 shows "coprime d (a * b)" |
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449 proof- |
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450 from da have th: "gcd(a, d) = 1" by (simp add: coprime_def gcd_commute) |
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451 from gcd_mult_cancel[of a d b, OF th] db[unfolded coprime_def] have "gcd (d, a*b) = 1" |
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452 by (simp add: gcd_commute) |
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453 thus ?thesis unfolding coprime_def . |
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454 qed |
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455 lemma coprime_lmul2: assumes dab: "coprime d (a * b)" shows "coprime d b" |
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456 using prems unfolding coprime_bezout |
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457 apply clarsimp |
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458 apply (case_tac "d * x - a * b * y = Suc 0 ", simp_all) |
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459 apply (rule_tac x="x" in exI) |
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460 apply (rule_tac x="a*y" in exI) |
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461 apply (simp add: mult_ac) |
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462 apply (rule_tac x="a*x" in exI) |
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463 apply (rule_tac x="y" in exI) |
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464 apply (simp add: mult_ac) |
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465 done |
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466 |
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467 lemma coprime_rmul2: "coprime d (a * b) \<Longrightarrow> coprime d a" |
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468 unfolding coprime_bezout |
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469 apply clarsimp |
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470 apply (case_tac "d * x - a * b * y = Suc 0 ", simp_all) |
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471 apply (rule_tac x="x" in exI) |
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472 apply (rule_tac x="b*y" in exI) |
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473 apply (simp add: mult_ac) |
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474 apply (rule_tac x="b*x" in exI) |
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475 apply (rule_tac x="y" in exI) |
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476 apply (simp add: mult_ac) |
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477 done |
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478 lemma coprime_mul_eq: "coprime d (a * b) \<longleftrightarrow> coprime d a \<and> coprime d b" |
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479 using coprime_rmul2[of d a b] coprime_lmul2[of d a b] coprime_mul[of d a b] |
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480 by blast |
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481 |
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482 lemma gcd_coprime_exists: |
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483 assumes nz: "gcd(a,b) \<noteq> 0" |
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484 shows "\<exists>a' b'. a = a' * gcd(a,b) \<and> b = b' * gcd(a,b) \<and> coprime a' b'" |
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485 proof- |
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486 let ?g = "gcd (a,b)" |
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487 from gcd_dvd1[of a b] gcd_dvd2[of a b] |
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488 obtain a' b' where "a = ?g*a'" "b = ?g*b'" unfolding dvd_def by blast |
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489 hence ab': "a = a'*?g" "b = b'*?g" by algebra+ |
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490 from ab' gcd_coprime[OF nz ab'] show ?thesis by blast |
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491 qed |
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492 |
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493 lemma coprime_exp: "coprime d a ==> coprime d (a^n)" |
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494 by(induct n, simp_all add: coprime_mul) |
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495 |
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496 lemma coprime_exp_imp: "coprime a b ==> coprime (a ^n) (b ^n)" |
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497 by (induct n, simp_all add: coprime_mul_eq coprime_commute coprime_exp) |
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498 lemma coprime_refl[simp]: "coprime n n \<longleftrightarrow> n = 1" by (simp add: coprime_def) |
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499 lemma coprime_plus1[simp]: "coprime (n + 1) n" |
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500 apply (simp add: coprime_bezout) |
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501 apply (rule exI[where x=1]) |
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502 apply (rule exI[where x=1]) |
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503 apply simp |
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504 done |
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505 lemma coprime_minus1: "n \<noteq> 0 ==> coprime (n - 1) n" |
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506 using coprime_plus1[of "n - 1"] coprime_commute[of "n - 1" n] by auto |
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507 |
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508 lemma bezout_gcd_pow: "\<exists>x y. a ^n * x - b ^ n * y = gcd(a,b) ^ n \<or> b ^ n * x - a ^ n * y = gcd(a,b) ^ n" |
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509 proof- |
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510 let ?g = "gcd (a,b)" |
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511 {assume z: "?g = 0" hence ?thesis |
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512 apply (cases n, simp) |
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513 apply arith |
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514 apply (simp only: z power_0_Suc) |
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515 apply (rule exI[where x=0]) |
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516 apply (rule exI[where x=0]) |
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517 by simp} |
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518 moreover |
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519 {assume z: "?g \<noteq> 0" |
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520 from gcd_dvd1[of a b] gcd_dvd2[of a b] obtain a' b' where |
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521 ab': "a = a'*?g" "b = b'*?g" unfolding dvd_def by (auto simp add: mult_ac) |
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522 hence ab'': "?g*a' = a" "?g * b' = b" by algebra+ |
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523 from coprime_exp_imp[OF gcd_coprime[OF z ab'], unfolded coprime_bezout, of n] |
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524 obtain x y where "a'^n * x - b'^n * y = 1 \<or> b'^n * x - a'^n * y = 1" by blast |
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525 hence "?g^n * (a'^n * x - b'^n * y) = ?g^n \<or> ?g^n*(b'^n * x - a'^n * y) = ?g^n" |
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526 using z by auto |
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527 then have "a^n * x - b^n * y = ?g^n \<or> b^n * x - a^n * y = ?g^n" |
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528 using z ab'' by (simp only: power_mult_distrib[symmetric] |
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529 diff_mult_distrib2 mult_assoc[symmetric]) |
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530 hence ?thesis by blast } |
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531 ultimately show ?thesis by blast |
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532 qed |
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533 lemma gcd_exp: "gcd (a^n, b^n) = gcd(a,b)^n" |
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534 proof- |
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535 let ?g = "gcd(a^n,b^n)" |
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536 let ?gn = "gcd(a,b)^n" |
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537 {fix e assume H: "e dvd a^n" "e dvd b^n" |
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538 from bezout_gcd_pow[of a n b] obtain x y |
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539 where xy: "a ^ n * x - b ^ n * y = ?gn \<or> b ^ n * x - a ^ n * y = ?gn" by blast |
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540 from dvd_diff [OF dvd_mult2[OF H(1), of x] dvd_mult2[OF H(2), of y]] |
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541 dvd_diff [OF dvd_mult2[OF H(2), of x] dvd_mult2[OF H(1), of y]] xy |
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542 have "e dvd ?gn" by (cases "a ^ n * x - b ^ n * y = gcd (a, b) ^ n", simp_all)} |
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543 hence th: "\<forall>e. e dvd a^n \<and> e dvd b^n \<longrightarrow> e dvd ?gn" by blast |
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544 from divides_exp[OF gcd_dvd1[of a b], of n] divides_exp[OF gcd_dvd2[of a b], of n] th |
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545 gcd_unique have "?gn = ?g" by blast thus ?thesis by simp |
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546 qed |
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547 |
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548 lemma coprime_exp2: "coprime (a ^ Suc n) (b^ Suc n) \<longleftrightarrow> coprime a b" |
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549 by (simp only: coprime_def gcd_exp exp_eq_1) simp |
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550 |
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551 lemma division_decomp: assumes dc: "(a::nat) dvd b * c" |
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552 shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c" |
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553 proof- |
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554 let ?g = "gcd (a,b)" |
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555 {assume "?g = 0" with dc have ?thesis apply (simp add: gcd_zero) |
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556 apply (rule exI[where x="0"]) |
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557 by (rule exI[where x="c"], simp)} |
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558 moreover |
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559 {assume z: "?g \<noteq> 0" |
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560 from gcd_coprime_exists[OF z] |
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561 obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'" by blast |
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562 from gcd_dvd2[of a b] have thb: "?g dvd b" . |
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563 from ab'(1) have "a' dvd a" unfolding dvd_def by blast |
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564 with dc have th0: "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp |
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565 from dc ab'(1,2) have "a'*?g dvd (b'*?g) *c" by auto |
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566 hence "?g*a' dvd ?g * (b' * c)" by (simp add: mult_assoc) |
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567 with z have th_1: "a' dvd b'*c" by simp |
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568 from coprime_divprod[OF th_1 ab'(3)] have thc: "a' dvd c" . |
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569 from ab' have "a = ?g*a'" by algebra |
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570 with thb thc have ?thesis by blast } |
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571 ultimately show ?thesis by blast |
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572 qed |
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573 |
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574 lemma nat_power_eq_0_iff: "(m::nat) ^ n = 0 \<longleftrightarrow> n \<noteq> 0 \<and> m = 0" by (induct n, auto) |
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575 |
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576 lemma divides_rev: assumes ab: "(a::nat) ^ n dvd b ^n" and n:"n \<noteq> 0" shows "a dvd b" |
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577 proof- |
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578 let ?g = "gcd (a,b)" |
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579 from n obtain m where m: "n = Suc m" by (cases n, simp_all) |
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580 {assume "?g = 0" with ab n have ?thesis by (simp add: gcd_zero)} |
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581 moreover |
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582 {assume z: "?g \<noteq> 0" |
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583 hence zn: "?g ^ n \<noteq> 0" using n by (simp add: neq0_conv) |
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584 from gcd_coprime_exists[OF z] |
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585 obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'" by blast |
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586 from ab have "(a' * ?g) ^ n dvd (b' * ?g)^n" by (simp add: ab'(1,2)[symmetric]) |
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587 hence "?g^n*a'^n dvd ?g^n *b'^n" by (simp only: power_mult_distrib mult_commute) |
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588 with zn z n have th0:"a'^n dvd b'^n" by (auto simp add: nat_power_eq_0_iff) |
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589 have "a' dvd a'^n" by (simp add: m) |
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590 with th0 have "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by simp |
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591 hence th1: "a' dvd b'^m * b'" by (simp add: m mult_commute) |
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592 from coprime_divprod[OF th1 coprime_exp[OF ab'(3), of m]] |
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593 have "a' dvd b'" . |
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594 hence "a'*?g dvd b'*?g" by simp |
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595 with ab'(1,2) have ?thesis by simp } |
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596 ultimately show ?thesis by blast |
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597 qed |
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598 |
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599 lemma divides_mul: assumes mr: "m dvd r" and nr: "n dvd r" and mn:"coprime m n" |
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600 shows "m * n dvd r" |
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601 proof- |
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602 from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'" |
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603 unfolding dvd_def by blast |
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604 from mr n' have "m dvd n'*n" by (simp add: mult_commute) |
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605 hence "m dvd n'" using relprime_dvd_mult_iff[OF mn[unfolded coprime_def]] by simp |
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606 then obtain k where k: "n' = m*k" unfolding dvd_def by blast |
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607 from n' k show ?thesis unfolding dvd_def by auto |
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608 qed |
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609 |
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610 (* A binary form of the Chinese Remainder Theorem. *) |
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611 |
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612 lemma chinese_remainder: assumes ab: "coprime a b" and a:"a \<noteq> 0" and b:"b \<noteq> 0" |
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613 shows "\<exists>x q1 q2. x = u + q1 * a \<and> x = v + q2 * b" |
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614 proof- |
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615 from bezout_add_strong[OF a, of b] bezout_add_strong[OF b, of a] |
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616 obtain d1 x1 y1 d2 x2 y2 where dxy1: "d1 dvd a" "d1 dvd b" "a * x1 = b * y1 + d1" |
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617 and dxy2: "d2 dvd b" "d2 dvd a" "b * x2 = a * y2 + d2" by blast |
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618 from gcd_unique[of 1 a b, simplified ab[unfolded coprime_def], simplified] |
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619 dxy1(1,2) dxy2(1,2) have d12: "d1 = 1" "d2 =1" by auto |
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620 let ?x = "v * a * x1 + u * b * x2" |
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621 let ?q1 = "v * x1 + u * y2" |
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622 let ?q2 = "v * y1 + u * x2" |
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623 from dxy2(3)[simplified d12] dxy1(3)[simplified d12] |
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624 have "?x = u + ?q1 * a" "?x = v + ?q2 * b" by algebra+ |
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625 thus ?thesis by blast |
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626 qed |
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627 |
|
628 (* Primality *) |
|
629 (* A few useful theorems about primes *) |
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630 |
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631 lemma prime_0[simp]: "~prime 0" by (simp add: prime_def) |
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632 lemma prime_1[simp]: "~ prime 1" by (simp add: prime_def) |
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633 lemma prime_Suc0[simp]: "~ prime (Suc 0)" by (simp add: prime_def) |
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634 |
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635 lemma prime_ge_2: "prime p ==> p \<ge> 2" by (simp add: prime_def) |
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636 lemma prime_factor: assumes n: "n \<noteq> 1" shows "\<exists> p. prime p \<and> p dvd n" |
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637 using n |
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638 proof(induct n rule: nat_less_induct) |
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639 fix n |
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640 assume H: "\<forall>m<n. m \<noteq> 1 \<longrightarrow> (\<exists>p. prime p \<and> p dvd m)" "n \<noteq> 1" |
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641 let ?ths = "\<exists>p. prime p \<and> p dvd n" |
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642 {assume "n=0" hence ?ths using two_is_prime by auto} |
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643 moreover |
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644 {assume nz: "n\<noteq>0" |
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645 {assume "prime n" hence ?ths by - (rule exI[where x="n"], simp)} |
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646 moreover |
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647 {assume n: "\<not> prime n" |
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648 with nz H(2) |
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649 obtain k where k:"k dvd n" "k \<noteq> 1" "k \<noteq> n" by (auto simp add: prime_def) |
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650 from dvd_imp_le[OF k(1)] nz k(3) have kn: "k < n" by simp |
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651 from H(1)[rule_format, OF kn k(2)] obtain p where p: "prime p" "p dvd k" by blast |
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652 from dvd_trans[OF p(2) k(1)] p(1) have ?ths by blast} |
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653 ultimately have ?ths by blast} |
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654 ultimately show ?ths by blast |
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655 qed |
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656 |
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657 lemma prime_factor_lt: assumes p: "prime p" and n: "n \<noteq> 0" and npm:"n = p * m" |
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658 shows "m < n" |
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659 proof- |
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660 {assume "m=0" with n have ?thesis by simp} |
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661 moreover |
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662 {assume m: "m \<noteq> 0" |
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663 from npm have mn: "m dvd n" unfolding dvd_def by auto |
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664 from npm m have "n \<noteq> m" using p by auto |
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665 with dvd_imp_le[OF mn] n have ?thesis by simp} |
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666 ultimately show ?thesis by blast |
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667 qed |
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668 |
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669 lemma euclid_bound: "\<exists>p. prime p \<and> n < p \<and> p <= Suc (fact n)" |
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670 proof- |
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671 have f1: "fact n + 1 \<noteq> 1" using fact_le[of n] by arith |
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672 from prime_factor[OF f1] obtain p where p: "prime p" "p dvd fact n + 1" by blast |
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673 from dvd_imp_le[OF p(2)] have pfn: "p \<le> fact n + 1" by simp |
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674 {assume np: "p \<le> n" |
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675 from p(1) have p1: "p \<ge> 1" by (cases p, simp_all) |
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676 from divides_fact[OF p1 np] have pfn': "p dvd fact n" . |
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677 from divides_add_revr[OF pfn' p(2)] p(1) have False by simp} |
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678 hence "n < p" by arith |
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679 with p(1) pfn show ?thesis by auto |
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680 qed |
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681 |
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682 lemma euclid: "\<exists>p. prime p \<and> p > n" using euclid_bound by auto |
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683 lemma primes_infinite: "\<not> (finite {p. prime p})" |
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684 proof (auto simp add: finite_conv_nat_seg_image) |
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685 fix n f |
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686 assume H: "Collect prime = f ` {i. i < (n::nat)}" |
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687 let ?P = "Collect prime" |
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688 let ?m = "Max ?P" |
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689 have P0: "?P \<noteq> {}" using two_is_prime by auto |
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690 from H have fP: "finite ?P" using finite_conv_nat_seg_image by blast |
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691 from Max_in[OF fP P0] have "?m \<in> ?P" . |
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692 from Max_ge[OF fP P0] have contr: "\<forall> p. prime p \<longrightarrow> p \<le> ?m" by blast |
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693 from euclid[of ?m] obtain q where q: "prime q" "q > ?m" by blast |
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694 with contr show False by auto |
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695 qed |
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696 |
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697 lemma coprime_prime: assumes ab: "coprime a b" |
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698 shows "~(prime p \<and> p dvd a \<and> p dvd b)" |
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699 proof |
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700 assume "prime p \<and> p dvd a \<and> p dvd b" |
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701 thus False using ab gcd_greatest[of p a b] by (simp add: coprime_def) |
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702 qed |
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703 lemma coprime_prime_eq: "coprime a b \<longleftrightarrow> (\<forall>p. ~(prime p \<and> p dvd a \<and> p dvd b))" |
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704 (is "?lhs = ?rhs") |
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705 proof- |
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706 {assume "?lhs" with coprime_prime have ?rhs by blast} |
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707 moreover |
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708 {assume r: "?rhs" and c: "\<not> ?lhs" |
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709 then obtain g where g: "g\<noteq>1" "g dvd a" "g dvd b" unfolding coprime_def by blast |
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710 from prime_factor[OF g(1)] obtain p where p: "prime p" "p dvd g" by blast |
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711 from dvd_trans [OF p(2) g(2)] dvd_trans [OF p(2) g(3)] |
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712 have "p dvd a" "p dvd b" . with p(1) r have False by blast} |
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713 ultimately show ?thesis by blast |
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714 qed |
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715 |
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716 lemma prime_coprime: assumes p: "prime p" |
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717 shows "n = 1 \<or> p dvd n \<or> coprime p n" |
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718 using p prime_imp_relprime[of p n] by (auto simp add: coprime_def) |
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719 |
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720 lemma prime_coprime_strong: "prime p \<Longrightarrow> p dvd n \<or> coprime p n" |
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721 using prime_coprime[of p n] by auto |
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722 |
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723 declare coprime_0[simp] |
|
724 |
|
725 lemma coprime_0'[simp]: "coprime 0 d \<longleftrightarrow> d = 1" by (simp add: coprime_commute[of 0 d]) |
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726 lemma coprime_bezout_strong: assumes ab: "coprime a b" and b: "b \<noteq> 1" |
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727 shows "\<exists>x y. a * x = b * y + 1" |
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728 proof- |
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729 from ab b have az: "a \<noteq> 0" by - (rule ccontr, auto) |
|
730 from bezout_gcd_strong[OF az, of b] ab[unfolded coprime_def] |
|
731 show ?thesis by auto |
|
732 qed |
|
733 |
|
734 lemma bezout_prime: assumes p: "prime p" and pa: "\<not> p dvd a" |
|
735 shows "\<exists>x y. a*x = p*y + 1" |
|
736 proof- |
|
737 from p have p1: "p \<noteq> 1" using prime_1 by blast |
|
738 from prime_coprime[OF p, of a] p1 pa have ap: "coprime a p" |
|
739 by (auto simp add: coprime_commute) |
|
740 from coprime_bezout_strong[OF ap p1] show ?thesis . |
|
741 qed |
|
742 lemma prime_divprod: assumes p: "prime p" and pab: "p dvd a*b" |
|
743 shows "p dvd a \<or> p dvd b" |
|
744 proof- |
|
745 {assume "a=1" hence ?thesis using pab by simp } |
|
746 moreover |
|
747 {assume "p dvd a" hence ?thesis by blast} |
|
748 moreover |
|
749 {assume pa: "coprime p a" from coprime_divprod[OF pab pa] have ?thesis .. } |
|
750 ultimately show ?thesis using prime_coprime[OF p, of a] by blast |
|
751 qed |
|
752 |
|
753 lemma prime_divprod_eq: assumes p: "prime p" |
|
754 shows "p dvd a*b \<longleftrightarrow> p dvd a \<or> p dvd b" |
|
755 using p prime_divprod dvd_mult dvd_mult2 by auto |
|
756 |
|
757 lemma prime_divexp: assumes p:"prime p" and px: "p dvd x^n" |
|
758 shows "p dvd x" |
|
759 using px |
|
760 proof(induct n) |
|
761 case 0 thus ?case by simp |
|
762 next |
|
763 case (Suc n) |
|
764 hence th: "p dvd x*x^n" by simp |
|
765 {assume H: "p dvd x^n" |
|
766 from Suc.hyps[OF H] have ?case .} |
|
767 with prime_divprod[OF p th] show ?case by blast |
|
768 qed |
|
769 |
|
770 lemma prime_divexp_n: "prime p \<Longrightarrow> p dvd x^n \<Longrightarrow> p^n dvd x^n" |
|
771 using prime_divexp[of p x n] divides_exp[of p x n] by blast |
|
772 |
|
773 lemma coprime_prime_dvd_ex: assumes xy: "\<not>coprime x y" |
|
774 shows "\<exists>p. prime p \<and> p dvd x \<and> p dvd y" |
|
775 proof- |
|
776 from xy[unfolded coprime_def] obtain g where g: "g \<noteq> 1" "g dvd x" "g dvd y" |
|
777 by blast |
|
778 from prime_factor[OF g(1)] obtain p where p: "prime p" "p dvd g" by blast |
|
779 from g(2,3) dvd_trans[OF p(2)] p(1) show ?thesis by auto |
|
780 qed |
|
781 lemma coprime_sos: assumes xy: "coprime x y" |
|
782 shows "coprime (x * y) (x^2 + y^2)" |
|
783 proof- |
|
784 {assume c: "\<not> coprime (x * y) (x^2 + y^2)" |
|
785 from coprime_prime_dvd_ex[OF c] obtain p |
|
786 where p: "prime p" "p dvd x*y" "p dvd x^2 + y^2" by blast |
|
787 {assume px: "p dvd x" |
|
788 from dvd_mult[OF px, of x] p(3) have "p dvd y^2" |
|
789 unfolding dvd_def |
|
790 apply (auto simp add: power2_eq_square) |
|
791 apply (rule_tac x= "ka - k" in exI) |
|
792 by (simp add: diff_mult_distrib2) |
|
793 with prime_divexp[OF p(1), of y 2] have py: "p dvd y" . |
|
794 from p(1) px py xy[unfolded coprime, rule_format, of p] prime_1 |
|
795 have False by simp } |
|
796 moreover |
|
797 {assume py: "p dvd y" |
|
798 from dvd_mult[OF py, of y] p(3) have "p dvd x^2" |
|
799 unfolding dvd_def |
|
800 apply (auto simp add: power2_eq_square) |
|
801 apply (rule_tac x= "ka - k" in exI) |
|
802 by (simp add: diff_mult_distrib2) |
|
803 with prime_divexp[OF p(1), of x 2] have px: "p dvd x" . |
|
804 from p(1) px py xy[unfolded coprime, rule_format, of p] prime_1 |
|
805 have False by simp } |
|
806 ultimately have False using prime_divprod[OF p(1,2)] by blast} |
|
807 thus ?thesis by blast |
|
808 qed |
|
809 |
|
810 lemma distinct_prime_coprime: "prime p \<Longrightarrow> prime q \<Longrightarrow> p \<noteq> q \<Longrightarrow> coprime p q" |
|
811 unfolding prime_def coprime_prime_eq by blast |
|
812 |
|
813 lemma prime_coprime_lt: assumes p: "prime p" and x: "0 < x" and xp: "x < p" |
|
814 shows "coprime x p" |
|
815 proof- |
|
816 {assume c: "\<not> coprime x p" |
|
817 then obtain g where g: "g \<noteq> 1" "g dvd x" "g dvd p" unfolding coprime_def by blast |
|
818 from dvd_imp_le[OF g(2)] x xp have gp: "g < p" by arith |
|
819 from g(2) x have "g \<noteq> 0" by - (rule ccontr, simp) |
|
820 with g gp p[unfolded prime_def] have False by blast} |
|
821 thus ?thesis by blast |
|
822 qed |
|
823 |
|
824 lemma even_dvd[simp]: "even (n::nat) \<longleftrightarrow> 2 dvd n" by presburger |
|
825 lemma prime_odd: "prime p \<Longrightarrow> p = 2 \<or> odd p" unfolding prime_def by auto |
|
826 |
|
827 (* One property of coprimality is easier to prove via prime factors. *) |
|
828 |
|
829 lemma prime_divprod_pow: |
|
830 assumes p: "prime p" and ab: "coprime a b" and pab: "p^n dvd a * b" |
|
831 shows "p^n dvd a \<or> p^n dvd b" |
|
832 proof- |
|
833 {assume "n = 0 \<or> a = 1 \<or> b = 1" with pab have ?thesis |
|
834 apply (cases "n=0", simp_all) |
|
835 apply (cases "a=1", simp_all) done} |
|
836 moreover |
|
837 {assume n: "n \<noteq> 0" and a: "a\<noteq>1" and b: "b\<noteq>1" |
|
838 then obtain m where m: "n = Suc m" by (cases n, auto) |
|
839 from divides_exp2[OF n pab] have pab': "p dvd a*b" . |
|
840 from prime_divprod[OF p pab'] |
|
841 have "p dvd a \<or> p dvd b" . |
|
842 moreover |
|
843 {assume pa: "p dvd a" |
|
844 have pnba: "p^n dvd b*a" using pab by (simp add: mult_commute) |
|
845 from coprime_prime[OF ab, of p] p pa have "\<not> p dvd b" by blast |
|
846 with prime_coprime[OF p, of b] b |
|
847 have cpb: "coprime b p" using coprime_commute by blast |
|
848 from coprime_exp[OF cpb] have pnb: "coprime (p^n) b" |
|
849 by (simp add: coprime_commute) |
|
850 from coprime_divprod[OF pnba pnb] have ?thesis by blast } |
|
851 moreover |
|
852 {assume pb: "p dvd b" |
|
853 have pnba: "p^n dvd b*a" using pab by (simp add: mult_commute) |
|
854 from coprime_prime[OF ab, of p] p pb have "\<not> p dvd a" by blast |
|
855 with prime_coprime[OF p, of a] a |
|
856 have cpb: "coprime a p" using coprime_commute by blast |
|
857 from coprime_exp[OF cpb] have pnb: "coprime (p^n) a" |
|
858 by (simp add: coprime_commute) |
|
859 from coprime_divprod[OF pab pnb] have ?thesis by blast } |
|
860 ultimately have ?thesis by blast} |
|
861 ultimately show ?thesis by blast |
|
862 qed |
|
863 |
|
864 lemma nat_mult_eq_one: "(n::nat) * m = 1 \<longleftrightarrow> n = 1 \<and> m = 1" (is "?lhs \<longleftrightarrow> ?rhs") |
|
865 proof |
|
866 assume H: "?lhs" |
|
867 hence "n dvd 1" "m dvd 1" unfolding dvd_def by (auto simp add: mult_commute) |
|
868 thus ?rhs by auto |
|
869 next |
|
870 assume ?rhs then show ?lhs by auto |
|
871 qed |
|
872 |
|
873 lemma power_Suc0[simp]: "Suc 0 ^ n = Suc 0" |
|
874 unfolding One_nat_def[symmetric] power_one .. |
|
875 lemma coprime_pow: assumes ab: "coprime a b" and abcn: "a * b = c ^n" |
|
876 shows "\<exists>r s. a = r^n \<and> b = s ^n" |
|
877 using ab abcn |
|
878 proof(induct c arbitrary: a b rule: nat_less_induct) |
|
879 fix c a b |
|
880 assume H: "\<forall>m<c. \<forall>a b. coprime a b \<longrightarrow> a * b = m ^ n \<longrightarrow> (\<exists>r s. a = r ^ n \<and> b = s ^ n)" "coprime a b" "a * b = c ^ n" |
|
881 let ?ths = "\<exists>r s. a = r^n \<and> b = s ^n" |
|
882 {assume n: "n = 0" |
|
883 with H(3) power_one have "a*b = 1" by simp |
|
884 hence "a = 1 \<and> b = 1" by simp |
|
885 hence ?ths |
|
886 apply - |
|
887 apply (rule exI[where x=1]) |
|
888 apply (rule exI[where x=1]) |
|
889 using power_one[of n] |
|
890 by simp} |
|
891 moreover |
|
892 {assume n: "n \<noteq> 0" then obtain m where m: "n = Suc m" by (cases n, auto) |
|
893 {assume c: "c = 0" |
|
894 with H(3) m H(2) have ?ths apply simp |
|
895 apply (cases "a=0", simp_all) |
|
896 apply (rule exI[where x="0"], simp) |
|
897 apply (rule exI[where x="0"], simp) |
|
898 done} |
|
899 moreover |
|
900 {assume "c=1" with H(3) power_one have "a*b = 1" by simp |
|
901 hence "a = 1 \<and> b = 1" by simp |
|
902 hence ?ths |
|
903 apply - |
|
904 apply (rule exI[where x=1]) |
|
905 apply (rule exI[where x=1]) |
|
906 using power_one[of n] |
|
907 by simp} |
|
908 moreover |
|
909 {assume c: "c\<noteq>1" "c \<noteq> 0" |
|
910 from prime_factor[OF c(1)] obtain p where p: "prime p" "p dvd c" by blast |
|
911 from prime_divprod_pow[OF p(1) H(2), unfolded H(3), OF divides_exp[OF p(2), of n]] |
|
912 have pnab: "p ^ n dvd a \<or> p^n dvd b" . |
|
913 from p(2) obtain l where l: "c = p*l" unfolding dvd_def by blast |
|
914 have pn0: "p^n \<noteq> 0" using n prime_ge_2 [OF p(1)] by (simp add: neq0_conv) |
|
915 {assume pa: "p^n dvd a" |
|
916 then obtain k where k: "a = p^n * k" unfolding dvd_def by blast |
|
917 from l have "l dvd c" by auto |
|
918 with dvd_imp_le[of l c] c have "l \<le> c" by auto |
|
919 moreover {assume "l = c" with l c have "p = 1" by simp with p have False by simp} |
|
920 ultimately have lc: "l < c" by arith |
|
921 from coprime_lmul2 [OF H(2)[unfolded k coprime_commute[of "p^n*k" b]]] |
|
922 have kb: "coprime k b" by (simp add: coprime_commute) |
|
923 from H(3) l k pn0 have kbln: "k * b = l ^ n" |
|
924 by (auto simp add: power_mult_distrib) |
|
925 from H(1)[rule_format, OF lc kb kbln] |
|
926 obtain r s where rs: "k = r ^n" "b = s^n" by blast |
|
927 from k rs(1) have "a = (p*r)^n" by (simp add: power_mult_distrib) |
|
928 with rs(2) have ?ths by blast } |
|
929 moreover |
|
930 {assume pb: "p^n dvd b" |
|
931 then obtain k where k: "b = p^n * k" unfolding dvd_def by blast |
|
932 from l have "l dvd c" by auto |
|
933 with dvd_imp_le[of l c] c have "l \<le> c" by auto |
|
934 moreover {assume "l = c" with l c have "p = 1" by simp with p have False by simp} |
|
935 ultimately have lc: "l < c" by arith |
|
936 from coprime_lmul2 [OF H(2)[unfolded k coprime_commute[of "p^n*k" a]]] |
|
937 have kb: "coprime k a" by (simp add: coprime_commute) |
|
938 from H(3) l k pn0 n have kbln: "k * a = l ^ n" |
|
939 by (simp add: power_mult_distrib mult_commute) |
|
940 from H(1)[rule_format, OF lc kb kbln] |
|
941 obtain r s where rs: "k = r ^n" "a = s^n" by blast |
|
942 from k rs(1) have "b = (p*r)^n" by (simp add: power_mult_distrib) |
|
943 with rs(2) have ?ths by blast } |
|
944 ultimately have ?ths using pnab by blast} |
|
945 ultimately have ?ths by blast} |
|
946 ultimately show ?ths by blast |
|
947 qed |
|
948 |
|
949 (* More useful lemmas. *) |
|
950 lemma prime_product: |
|
951 "prime (p*q) \<Longrightarrow> p = 1 \<or> q = 1" unfolding prime_def by auto |
|
952 |
|
953 lemma prime_exp: "prime (p^n) \<longleftrightarrow> prime p \<and> n = 1" |
|
954 proof(induct n) |
|
955 case 0 thus ?case by simp |
|
956 next |
|
957 case (Suc n) |
|
958 {assume "p = 0" hence ?case by simp} |
|
959 moreover |
|
960 {assume "p=1" hence ?case by simp} |
|
961 moreover |
|
962 {assume p: "p \<noteq> 0" "p\<noteq>1" |
|
963 {assume pp: "prime (p^Suc n)" |
|
964 hence "p = 1 \<or> p^n = 1" using prime_product[of p "p^n"] by simp |
|
965 with p have n: "n = 0" |
|
966 by (simp only: exp_eq_1 ) simp |
|
967 with pp have "prime p \<and> Suc n = 1" by simp} |
|
968 moreover |
|
969 {assume n: "prime p \<and> Suc n = 1" hence "prime (p^Suc n)" by simp} |
|
970 ultimately have ?case by blast} |
|
971 ultimately show ?case by blast |
|
972 qed |
|
973 |
|
974 lemma prime_power_mult: |
|
975 assumes p: "prime p" and xy: "x * y = p ^ k" |
|
976 shows "\<exists>i j. x = p ^i \<and> y = p^ j" |
|
977 using xy |
|
978 proof(induct k arbitrary: x y) |
|
979 case 0 thus ?case apply simp by (rule exI[where x="0"], simp) |
|
980 next |
|
981 case (Suc k x y) |
|
982 from Suc.prems have pxy: "p dvd x*y" by auto |
|
983 from prime_divprod[OF p pxy] have pxyc: "p dvd x \<or> p dvd y" . |
|
984 from p have p0: "p \<noteq> 0" by - (rule ccontr, simp) |
|
985 {assume px: "p dvd x" |
|
986 then obtain d where d: "x = p*d" unfolding dvd_def by blast |
|
987 from Suc.prems d have "p*d*y = p^Suc k" by simp |
|
988 hence th: "d*y = p^k" using p0 by simp |
|
989 from Suc.hyps[OF th] obtain i j where ij: "d = p^i" "y = p^j" by blast |
|
990 with d have "x = p^Suc i" by simp |
|
991 with ij(2) have ?case by blast} |
|
992 moreover |
|
993 {assume px: "p dvd y" |
|
994 then obtain d where d: "y = p*d" unfolding dvd_def by blast |
|
995 from Suc.prems d have "p*d*x = p^Suc k" by (simp add: mult_commute) |
|
996 hence th: "d*x = p^k" using p0 by simp |
|
997 from Suc.hyps[OF th] obtain i j where ij: "d = p^i" "x = p^j" by blast |
|
998 with d have "y = p^Suc i" by simp |
|
999 with ij(2) have ?case by blast} |
|
1000 ultimately show ?case using pxyc by blast |
|
1001 qed |
|
1002 |
|
1003 lemma prime_power_exp: assumes p: "prime p" and n:"n \<noteq> 0" |
|
1004 and xn: "x^n = p^k" shows "\<exists>i. x = p^i" |
|
1005 using n xn |
|
1006 proof(induct n arbitrary: k) |
|
1007 case 0 thus ?case by simp |
|
1008 next |
|
1009 case (Suc n k) hence th: "x*x^n = p^k" by simp |
|
1010 {assume "n = 0" with prems have ?case apply simp |
|
1011 by (rule exI[where x="k"],simp)} |
|
1012 moreover |
|
1013 {assume n: "n \<noteq> 0" |
|
1014 from prime_power_mult[OF p th] |
|
1015 obtain i j where ij: "x = p^i" "x^n = p^j"by blast |
|
1016 from Suc.hyps[OF n ij(2)] have ?case .} |
|
1017 ultimately show ?case by blast |
|
1018 qed |
|
1019 |
|
1020 lemma divides_primepow: assumes p: "prime p" |
|
1021 shows "d dvd p^k \<longleftrightarrow> (\<exists> i. i \<le> k \<and> d = p ^i)" |
|
1022 proof |
|
1023 assume H: "d dvd p^k" then obtain e where e: "d*e = p^k" |
|
1024 unfolding dvd_def apply (auto simp add: mult_commute) by blast |
|
1025 from prime_power_mult[OF p e] obtain i j where ij: "d = p^i" "e=p^j" by blast |
|
1026 from prime_ge_2[OF p] have p1: "p > 1" by arith |
|
1027 from e ij have "p^(i + j) = p^k" by (simp add: power_add) |
|
1028 hence "i + j = k" using power_inject_exp[of p "i+j" k, OF p1] by simp |
|
1029 hence "i \<le> k" by arith |
|
1030 with ij(1) show "\<exists>i\<le>k. d = p ^ i" by blast |
|
1031 next |
|
1032 {fix i assume H: "i \<le> k" "d = p^i" |
|
1033 hence "\<exists>j. k = i + j" by arith |
|
1034 then obtain j where j: "k = i + j" by blast |
|
1035 hence "p^k = p^j*d" using H(2) by (simp add: power_add) |
|
1036 hence "d dvd p^k" unfolding dvd_def by auto} |
|
1037 thus "\<exists>i\<le>k. d = p ^ i \<Longrightarrow> d dvd p ^ k" by blast |
|
1038 qed |
|
1039 |
|
1040 lemma coprime_divisors: "d dvd a \<Longrightarrow> e dvd b \<Longrightarrow> coprime a b \<Longrightarrow> coprime d e" |
|
1041 by (auto simp add: dvd_def coprime) |
|
1042 |
48 end |
1043 end |