4 Copyright 1996 University of Cambridge |
4 Copyright 1996 University of Cambridge |
5 |
5 |
6 The "divides" relation, the greatest common divisor and Euclid's algorithm |
6 The "divides" relation, the greatest common divisor and Euclid's algorithm |
7 *) |
7 *) |
8 |
8 |
9 Primes = Main + |
9 theory Primes = Main: |
10 consts |
10 constdefs |
11 dvd :: [i,i]=>o (infixl 50) |
11 divides :: "[i,i]=>o" (infixl "dvd" 50) |
12 is_gcd :: [i,i,i]=>o (* great common divisor *) |
12 "m dvd n == m \<in> nat & n \<in> nat & (\<exists>k \<in> nat. n = m#*k)" |
13 gcd :: [i,i]=>i (* gcd by Euclid's algorithm *) |
13 |
14 |
14 is_gcd :: "[i,i,i]=>o" (* great common divisor *) |
15 defs |
15 "is_gcd(p,m,n) == ((p dvd m) & (p dvd n)) & |
16 dvd_def "m dvd n == m \\<in> nat & n \\<in> nat & (\\<exists>k \\<in> nat. n = m#*k)" |
16 (\<forall>d\<in>nat. (d dvd m) & (d dvd n) --> d dvd p)" |
17 |
17 |
18 is_gcd_def "is_gcd(p,m,n) == ((p dvd m) & (p dvd n)) & |
18 gcd :: "[i,i]=>i" (* gcd by Euclid's algorithm *) |
19 (\\<forall>d\\<in>nat. (d dvd m) & (d dvd n) --> d dvd p)" |
19 "gcd(m,n) == transrec(natify(n), |
20 |
20 %n f. \<lambda>m \<in> nat. |
21 gcd_def "gcd(m,n) == |
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22 transrec(natify(n), |
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23 %n f. \\<lambda>m \\<in> nat. |
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24 if n=0 then m else f`(m mod n)`n) ` natify(m)" |
21 if n=0 then m else f`(m mod n)`n) ` natify(m)" |
25 |
22 |
26 constdefs |
23 coprime :: "[i,i]=>o" (* coprime relation *) |
27 coprime :: [i,i]=>o (* coprime relation *) |
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28 "coprime(m,n) == gcd(m,n) = 1" |
24 "coprime(m,n) == gcd(m,n) = 1" |
29 |
25 |
30 prime :: i (* set of prime numbers *) |
26 prime :: i (* set of prime numbers *) |
31 "prime == {p \\<in> nat. 1<p & (\\<forall>m \\<in> nat. m dvd p --> m=1 | m=p)}" |
27 "prime == {p \<in> nat. 1<p & (\<forall>m \<in> nat. m dvd p --> m=1 | m=p)}" |
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28 |
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29 (************************************************) |
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30 (** Divides Relation **) |
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31 (************************************************) |
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32 |
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33 lemma dvdD: "m dvd n ==> m \<in> nat & n \<in> nat & (\<exists>k \<in> nat. n = m#*k)" |
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34 apply (unfold divides_def) |
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35 apply assumption |
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36 done |
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37 |
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38 lemma dvdE: |
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39 "[|m dvd n; !!k. [|m \<in> nat; n \<in> nat; k \<in> nat; n = m#*k|] ==> P|] ==> P" |
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40 by (blast dest!: dvdD) |
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41 |
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42 lemmas dvd_imp_nat1 = dvdD [THEN conjunct1, standard] |
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43 lemmas dvd_imp_nat2 = dvdD [THEN conjunct2, THEN conjunct1, standard] |
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44 |
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45 |
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46 lemma dvd_0_right [simp]: "m \<in> nat ==> m dvd 0" |
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47 apply (unfold divides_def) |
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48 apply (fast intro: nat_0I mult_0_right [symmetric]) |
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49 done |
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50 |
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51 lemma dvd_0_left: "0 dvd m ==> m = 0" |
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52 by (unfold divides_def, force) |
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53 |
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54 lemma dvd_refl [simp]: "m \<in> nat ==> m dvd m" |
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55 apply (unfold divides_def) |
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56 apply (fast intro: nat_1I mult_1_right [symmetric]) |
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57 done |
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58 |
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59 lemma dvd_trans: "[| m dvd n; n dvd p |] ==> m dvd p" |
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60 apply (unfold divides_def) |
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61 apply (fast intro: mult_assoc mult_type) |
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62 done |
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63 |
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64 lemma dvd_anti_sym: "[| m dvd n; n dvd m |] ==> m=n" |
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65 apply (unfold divides_def) |
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66 apply (force dest: mult_eq_self_implies_10 |
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67 simp add: mult_assoc mult_eq_1_iff) |
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68 done |
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69 |
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70 lemma dvd_mult_left: "[|(i#*j) dvd k; i \<in> nat|] ==> i dvd k" |
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71 apply (unfold divides_def) |
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72 apply (simp add: mult_assoc) |
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73 apply blast |
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74 done |
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75 |
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76 lemma dvd_mult_right: "[|(i#*j) dvd k; j \<in> nat|] ==> j dvd k" |
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77 apply (unfold divides_def) |
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78 apply clarify |
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79 apply (rule_tac x = "i#*k" in bexI) |
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80 apply (simp add: mult_ac) |
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81 apply (rule mult_type) |
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82 done |
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83 |
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84 |
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85 (************************************************) |
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86 (** Greatest Common Divisor **) |
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87 (************************************************) |
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88 |
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89 (* GCD by Euclid's Algorithm *) |
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90 |
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91 lemma gcd_0 [simp]: "gcd(m,0) = natify(m)" |
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92 apply (unfold gcd_def) |
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93 apply (subst transrec) |
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94 apply simp |
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95 done |
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96 |
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97 lemma gcd_natify1 [simp]: "gcd(natify(m),n) = gcd(m,n)" |
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98 by (simp add: gcd_def) |
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99 |
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100 lemma gcd_natify2 [simp]: "gcd(m, natify(n)) = gcd(m,n)" |
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101 by (simp add: gcd_def) |
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102 |
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103 lemma gcd_non_0_raw: |
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104 "[| 0<n; n \<in> nat |] ==> gcd(m,n) = gcd(n, m mod n)" |
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105 apply (unfold gcd_def) |
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106 apply (rule_tac P = "%z. ?left (z) = ?right" in transrec [THEN ssubst]) |
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107 apply (simp add: ltD [THEN mem_imp_not_eq, THEN not_sym] |
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108 mod_less_divisor [THEN ltD]) |
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109 done |
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110 |
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111 lemma gcd_non_0: "0 < natify(n) ==> gcd(m,n) = gcd(n, m mod n)" |
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112 apply (cut_tac m = "m" and n = "natify (n) " in gcd_non_0_raw) |
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113 apply auto |
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114 done |
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115 |
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116 lemma gcd_1 [simp]: "gcd(m,1) = 1" |
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117 by (simp (no_asm_simp) add: gcd_non_0) |
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118 |
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119 lemma dvd_add: "[| k dvd a; k dvd b |] ==> k dvd (a #+ b)" |
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120 apply (unfold divides_def) |
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121 apply (fast intro: add_mult_distrib_left [symmetric] add_type) |
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122 done |
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123 |
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124 lemma dvd_mult: "k dvd n ==> k dvd (m #* n)" |
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125 apply (unfold divides_def) |
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126 apply (fast intro: mult_left_commute mult_type) |
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127 done |
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128 |
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129 lemma dvd_mult2: "k dvd m ==> k dvd (m #* n)" |
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130 apply (subst mult_commute) |
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131 apply (blast intro: dvd_mult) |
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132 done |
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133 |
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134 (* k dvd (m*k) *) |
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135 lemmas dvdI1 [simp] = dvd_refl [THEN dvd_mult, standard] |
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136 lemmas dvdI2 [simp] = dvd_refl [THEN dvd_mult2, standard] |
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137 |
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138 lemma dvd_mod_imp_dvd_raw: |
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139 "[| a \<in> nat; b \<in> nat; k dvd b; k dvd (a mod b) |] ==> k dvd a" |
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140 apply (tactic "div_undefined_case_tac \"b=0\" 1") |
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141 apply (blast intro: mod_div_equality [THEN subst] |
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142 elim: dvdE |
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143 intro!: dvd_add dvd_mult mult_type mod_type div_type) |
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144 done |
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145 |
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146 lemma dvd_mod_imp_dvd: "[| k dvd (a mod b); k dvd b; a \<in> nat |] ==> k dvd a" |
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147 apply (cut_tac b = "natify (b) " in dvd_mod_imp_dvd_raw) |
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148 apply auto |
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149 apply (simp add: divides_def) |
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150 done |
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151 |
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152 (*Imitating TFL*) |
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153 lemma gcd_induct_lemma [rule_format (no_asm)]: "[| n \<in> nat; |
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154 \<forall>m \<in> nat. P(m,0); |
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155 \<forall>m \<in> nat. \<forall>n \<in> nat. 0<n --> P(n, m mod n) --> P(m,n) |] |
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156 ==> \<forall>m \<in> nat. P (m,n)" |
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157 apply (erule_tac i = "n" in complete_induct) |
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158 apply (case_tac "x=0") |
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159 apply (simp (no_asm_simp)) |
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160 apply clarify |
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161 apply (drule_tac x1 = "m" and x = "x" in bspec [THEN bspec]) |
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162 apply (simp_all add: Ord_0_lt_iff) |
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163 apply (blast intro: mod_less_divisor [THEN ltD]) |
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164 done |
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165 |
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166 lemma gcd_induct: "!!P. [| m \<in> nat; n \<in> nat; |
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167 !!m. m \<in> nat ==> P(m,0); |
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168 !!m n. [|m \<in> nat; n \<in> nat; 0<n; P(n, m mod n)|] ==> P(m,n) |] |
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169 ==> P (m,n)" |
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170 by (blast intro: gcd_induct_lemma) |
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171 |
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172 |
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173 |
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174 (* gcd type *) |
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175 |
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176 lemma gcd_type [simp,TC]: "gcd(m, n) \<in> nat" |
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177 apply (subgoal_tac "gcd (natify (m) , natify (n)) \<in> nat") |
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178 apply simp |
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179 apply (rule_tac m = "natify (m) " and n = "natify (n) " in gcd_induct) |
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180 apply auto |
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181 apply (simp add: gcd_non_0) |
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182 done |
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183 |
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184 |
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185 (* Property 1: gcd(a,b) divides a and b *) |
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186 |
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187 lemma gcd_dvd_both: |
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188 "[| m \<in> nat; n \<in> nat |] ==> gcd (m, n) dvd m & gcd (m, n) dvd n" |
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189 apply (rule_tac m = "m" and n = "n" in gcd_induct) |
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190 apply (simp_all add: gcd_non_0) |
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191 apply (blast intro: dvd_mod_imp_dvd_raw nat_into_Ord [THEN Ord_0_lt]) |
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192 done |
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193 |
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194 lemma gcd_dvd1 [simp]: "m \<in> nat ==> gcd(m,n) dvd m" |
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195 apply (cut_tac m = "natify (m) " and n = "natify (n) " in gcd_dvd_both) |
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196 apply auto |
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197 done |
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198 |
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199 lemma gcd_dvd2 [simp]: "n \<in> nat ==> gcd(m,n) dvd n" |
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200 apply (cut_tac m = "natify (m) " and n = "natify (n) " in gcd_dvd_both) |
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201 apply auto |
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202 done |
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203 |
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204 (* if f divides a and b then f divides gcd(a,b) *) |
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205 |
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206 lemma dvd_mod: "[| f dvd a; f dvd b |] ==> f dvd (a mod b)" |
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207 apply (unfold divides_def) |
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208 apply (tactic "div_undefined_case_tac \"b=0\" 1") |
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209 apply auto |
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210 apply (blast intro: mod_mult_distrib2 [symmetric]) |
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211 done |
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212 |
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213 (* Property 2: for all a,b,f naturals, |
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214 if f divides a and f divides b then f divides gcd(a,b)*) |
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215 |
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216 lemma gcd_greatest_raw [rule_format]: |
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217 "[| m \<in> nat; n \<in> nat; f \<in> nat |] |
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218 ==> (f dvd m) --> (f dvd n) --> f dvd gcd(m,n)" |
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219 apply (rule_tac m = "m" and n = "n" in gcd_induct) |
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220 apply (simp_all add: gcd_non_0 dvd_mod) |
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221 done |
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222 |
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223 lemma gcd_greatest: "[| f dvd m; f dvd n; f \<in> nat |] ==> f dvd gcd(m,n)" |
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224 apply (rule gcd_greatest_raw) |
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225 apply (auto simp add: divides_def) |
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226 done |
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227 |
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228 lemma gcd_greatest_iff [simp]: "[| k \<in> nat; m \<in> nat; n \<in> nat |] |
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229 ==> (k dvd gcd (m, n)) <-> (k dvd m & k dvd n)" |
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230 by (blast intro!: gcd_greatest gcd_dvd1 gcd_dvd2 intro: dvd_trans) |
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231 |
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232 |
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233 (* GCD PROOF: GCD exists and gcd fits the definition *) |
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234 |
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235 lemma is_gcd: "[| m \<in> nat; n \<in> nat |] ==> is_gcd(gcd(m,n), m, n)" |
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236 by (simp add: is_gcd_def) |
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237 |
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238 (* GCD is unique *) |
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239 |
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240 lemma is_gcd_unique: "[|is_gcd(m,a,b); is_gcd(n,a,b); m\<in>nat; n\<in>nat|] ==> m=n" |
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241 apply (unfold is_gcd_def) |
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242 apply (blast intro: dvd_anti_sym) |
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243 done |
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244 |
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245 lemma is_gcd_commute: "is_gcd(k,m,n) <-> is_gcd(k,n,m)" |
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246 by (unfold is_gcd_def, blast) |
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247 |
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248 lemma gcd_commute_raw: "[| m \<in> nat; n \<in> nat |] ==> gcd(m,n) = gcd(n,m)" |
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249 apply (rule is_gcd_unique) |
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250 apply (rule is_gcd) |
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251 apply (rule_tac [3] is_gcd_commute [THEN iffD1]) |
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252 apply (rule_tac [3] is_gcd) |
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253 apply auto |
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254 done |
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255 |
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256 lemma gcd_commute: "gcd(m,n) = gcd(n,m)" |
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257 apply (cut_tac m = "natify (m) " and n = "natify (n) " in gcd_commute_raw) |
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258 apply auto |
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259 done |
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260 |
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261 lemma gcd_assoc_raw: "[| k \<in> nat; m \<in> nat; n \<in> nat |] |
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262 ==> gcd (gcd (k, m), n) = gcd (k, gcd (m, n))" |
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263 apply (rule is_gcd_unique) |
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264 apply (rule is_gcd) |
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265 apply (simp_all add: is_gcd_def) |
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266 apply (blast intro: gcd_dvd1 gcd_dvd2 gcd_type intro: dvd_trans) |
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267 done |
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268 |
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269 lemma gcd_assoc: "gcd (gcd (k, m), n) = gcd (k, gcd (m, n))" |
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270 apply (cut_tac k = "natify (k) " and m = "natify (m) " and n = "natify (n) " |
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271 in gcd_assoc_raw) |
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272 apply auto |
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273 done |
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274 |
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275 lemma gcd_0_left [simp]: "gcd (0, m) = natify(m)" |
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276 by (simp add: gcd_commute [of 0]) |
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277 |
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278 lemma gcd_1_left [simp]: "gcd (1, m) = 1" |
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279 by (simp add: gcd_commute [of 1]) |
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280 |
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281 |
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282 (* Multiplication laws *) |
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283 |
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284 lemma gcd_mult_distrib2_raw: |
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285 "[| k \<in> nat; m \<in> nat; n \<in> nat |] |
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286 ==> k #* gcd (m, n) = gcd (k #* m, k #* n)" |
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287 apply (erule_tac m = "m" and n = "n" in gcd_induct) |
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288 apply assumption |
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289 apply (simp) |
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290 apply (case_tac "k = 0") |
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291 apply simp |
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292 apply (simp add: mod_geq gcd_non_0 mod_mult_distrib2 Ord_0_lt_iff) |
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293 done |
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294 |
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295 lemma gcd_mult_distrib2: "k #* gcd (m, n) = gcd (k #* m, k #* n)" |
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296 apply (cut_tac k = "natify (k) " and m = "natify (m) " and n = "natify (n) " |
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297 in gcd_mult_distrib2_raw) |
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298 apply auto |
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299 done |
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300 |
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301 lemma gcd_mult [simp]: "gcd (k, k #* n) = natify(k)" |
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302 apply (cut_tac k = "k" and m = "1" and n = "n" in gcd_mult_distrib2) |
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303 apply auto |
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304 done |
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305 |
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306 lemma gcd_self [simp]: "gcd (k, k) = natify(k)" |
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307 apply (cut_tac k = "k" and n = "1" in gcd_mult) |
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308 apply auto |
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309 done |
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310 |
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311 lemma relprime_dvd_mult: |
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312 "[| gcd (k,n) = 1; k dvd (m #* n); m \<in> nat |] ==> k dvd m" |
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313 apply (cut_tac k = "m" and m = "k" and n = "n" in gcd_mult_distrib2) |
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314 apply auto |
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315 apply (erule_tac b = "m" in ssubst) |
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316 apply (simp add: dvd_imp_nat1) |
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317 done |
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318 |
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319 lemma relprime_dvd_mult_iff: |
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320 "[| gcd (k,n) = 1; m \<in> nat |] ==> k dvd (m #* n) <-> k dvd m" |
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321 by (blast intro: dvdI2 relprime_dvd_mult dvd_trans) |
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322 |
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323 lemma prime_imp_relprime: |
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324 "[| p \<in> prime; ~ (p dvd n); n \<in> nat |] ==> gcd (p, n) = 1" |
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325 apply (unfold prime_def) |
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326 apply clarify |
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327 apply (drule_tac x = "gcd (p,n) " in bspec) |
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328 apply auto |
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329 apply (cut_tac m = "p" and n = "n" in gcd_dvd2) |
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330 apply auto |
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331 done |
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332 |
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333 lemma prime_into_nat: "p \<in> prime ==> p \<in> nat" |
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334 by (unfold prime_def, auto) |
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335 |
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336 lemma prime_nonzero: "p \<in> prime \<Longrightarrow> p\<noteq>0" |
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337 by (unfold prime_def, auto) |
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338 |
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339 |
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340 (*This theorem leads immediately to a proof of the uniqueness of |
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341 factorization. If p divides a product of primes then it is |
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342 one of those primes.*) |
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343 |
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344 lemma prime_dvd_mult: |
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345 "[|p dvd m #* n; p \<in> prime; m \<in> nat; n \<in> nat |] ==> p dvd m \<or> p dvd n" |
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346 by (blast intro: relprime_dvd_mult prime_imp_relprime prime_into_nat) |
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347 |
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348 |
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349 (** Addition laws **) |
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350 |
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351 lemma gcd_add1 [simp]: "gcd (m #+ n, n) = gcd (m, n)" |
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352 apply (subgoal_tac "gcd (m #+ natify (n) , natify (n)) = gcd (m, natify (n))") |
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353 apply simp |
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354 apply (case_tac "natify (n) = 0") |
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355 apply (auto simp add: Ord_0_lt_iff gcd_non_0) |
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356 done |
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357 |
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358 lemma gcd_add2 [simp]: "gcd (m, m #+ n) = gcd (m, n)" |
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359 apply (rule gcd_commute [THEN trans]) |
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360 apply (subst add_commute) |
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361 apply simp |
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362 apply (rule gcd_commute) |
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363 done |
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364 |
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365 lemma gcd_add2' [simp]: "gcd (m, n #+ m) = gcd (m, n)" |
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366 by (subst add_commute, rule gcd_add2) |
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367 |
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368 lemma gcd_add_mult_raw: "k \<in> nat ==> gcd (m, k #* m #+ n) = gcd (m, n)" |
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369 apply (erule nat_induct) |
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370 apply (auto simp add: gcd_add2 add_assoc) |
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371 done |
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372 |
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373 lemma gcd_add_mult: "gcd (m, k #* m #+ n) = gcd (m, n)" |
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374 apply (cut_tac k = "natify (k) " in gcd_add_mult_raw) |
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375 apply auto |
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376 done |
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377 |
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378 |
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379 (* More multiplication laws *) |
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380 |
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381 lemma gcd_mult_cancel_raw: |
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382 "[|gcd (k,n) = 1; m \<in> nat; n \<in> nat|] ==> gcd (k #* m, n) = gcd (m, n)" |
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383 apply (rule dvd_anti_sym) |
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384 apply (rule gcd_greatest) |
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385 apply (rule relprime_dvd_mult [of _ k]) |
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386 apply (simp add: gcd_assoc) |
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387 apply (simp add: gcd_commute) |
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388 apply (simp_all add: mult_commute) |
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389 apply (blast intro: dvdI1 gcd_dvd1 dvd_trans) |
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390 done |
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391 |
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392 lemma gcd_mult_cancel: "gcd (k,n) = 1 ==> gcd (k #* m, n) = gcd (m, n)" |
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393 apply (cut_tac m = "natify (m) " and n = "natify (n) " in gcd_mult_cancel_raw) |
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394 apply auto |
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395 done |
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396 |
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397 |
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398 (*** The square root of a prime is irrational: key lemma ***) |
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399 |
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400 lemma prime_dvd_other_side: |
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401 "\<lbrakk>n#*n = p#*(k#*k); p \<in> prime; n \<in> nat\<rbrakk> \<Longrightarrow> p dvd n" |
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402 apply (subgoal_tac "p dvd n#*n") |
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403 apply (blast dest: prime_dvd_mult) |
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404 apply (rule_tac j = "k#*k" in dvd_mult_left) |
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405 apply (auto simp add: prime_def) |
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406 done |
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407 |
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408 lemma reduction: |
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409 "\<lbrakk>k#*k = p#*(j#*j); p \<in> prime; 0 < k; j \<in> nat; k \<in> nat\<rbrakk> |
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410 \<Longrightarrow> k < p#*j & 0 < j" |
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411 apply (rule ccontr) |
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412 apply (simp add: not_lt_iff_le prime_into_nat) |
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413 apply (erule disjE) |
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414 apply (frule mult_le_mono, assumption+) |
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415 apply (simp add: mult_ac) |
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416 apply (auto dest!: natify_eqE |
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417 simp add: not_lt_iff_le prime_into_nat mult_le_cancel_le1) |
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418 apply (simp add: prime_def) |
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419 apply (blast dest: lt_trans1) |
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420 done |
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421 |
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422 lemma rearrange: "j #* (p#*j) = k#*k \<Longrightarrow> k#*k = p#*(j#*j)" |
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423 by (simp add: mult_ac) |
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424 |
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425 lemma prime_not_square: |
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426 "\<lbrakk>m \<in> nat; p \<in> prime\<rbrakk> \<Longrightarrow> \<forall>k \<in> nat. 0<k \<longrightarrow> m#*m \<noteq> p#*(k#*k)" |
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427 apply (erule complete_induct) |
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428 apply clarify |
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429 apply (frule prime_dvd_other_side) |
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430 apply assumption |
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431 apply assumption |
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432 apply (erule dvdE) |
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433 apply (simp add: mult_assoc mult_cancel1 prime_nonzero prime_into_nat) |
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434 apply (blast dest: rearrange reduction ltD) |
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435 done |
32 |
436 |
33 end |
437 end |