1 (* Title: HOL/Old_Number_Theory/Fib.thy |
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2 Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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3 Copyright 1997 University of Cambridge |
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4 *) |
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5 |
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6 section \<open>The Fibonacci function\<close> |
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7 |
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8 theory Fib |
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9 imports Primes |
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10 begin |
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11 |
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12 text \<open> |
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13 Fibonacci numbers: proofs of laws taken from: |
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14 R. L. Graham, D. E. Knuth, O. Patashnik. Concrete Mathematics. |
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15 (Addison-Wesley, 1989) |
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16 |
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17 \bigskip |
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18 \<close> |
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19 |
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20 fun fib :: "nat \<Rightarrow> nat" |
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21 where |
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22 "fib 0 = 0" |
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23 | "fib (Suc 0) = 1" |
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24 | fib_2: "fib (Suc (Suc n)) = fib n + fib (Suc n)" |
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25 |
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26 text \<open> |
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27 \medskip The difficulty in these proofs is to ensure that the |
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28 induction hypotheses are applied before the definition of @{term |
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29 fib}. Towards this end, the @{term fib} equations are not declared |
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30 to the Simplifier and are applied very selectively at first. |
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31 \<close> |
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32 |
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33 text\<open>We disable \<open>fib.fib_2fib_2\<close> for simplification ...\<close> |
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34 declare fib_2 [simp del] |
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35 |
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36 text\<open>...then prove a version that has a more restrictive pattern.\<close> |
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37 lemma fib_Suc3: "fib (Suc (Suc (Suc n))) = fib (Suc n) + fib (Suc (Suc n))" |
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38 by (rule fib_2) |
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39 |
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40 text \<open>\medskip Concrete Mathematics, page 280\<close> |
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41 |
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42 lemma fib_add: "fib (Suc (n + k)) = fib (Suc k) * fib (Suc n) + fib k * fib n" |
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43 proof (induct n rule: fib.induct) |
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44 case 1 show ?case by simp |
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45 next |
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46 case 2 show ?case by (simp add: fib_2) |
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47 next |
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48 case 3 thus ?case by (simp add: fib_2 add_mult_distrib2) |
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49 qed |
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50 |
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51 lemma fib_Suc_neq_0: "fib (Suc n) \<noteq> 0" |
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52 apply (induct n rule: fib.induct) |
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53 apply (simp_all add: fib_2) |
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54 done |
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55 |
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56 lemma fib_Suc_gr_0: "0 < fib (Suc n)" |
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57 by (insert fib_Suc_neq_0 [of n], simp) |
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58 |
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59 lemma fib_gr_0: "0 < n ==> 0 < fib n" |
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60 by (case_tac n, auto simp add: fib_Suc_gr_0) |
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61 |
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62 |
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63 text \<open> |
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64 \medskip Concrete Mathematics, page 278: Cassini's identity. The proof is |
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65 much easier using integers, not natural numbers! |
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66 \<close> |
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67 |
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68 lemma fib_Cassini_int: |
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69 "int (fib (Suc (Suc n)) * fib n) = |
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70 (if n mod 2 = 0 then int (fib (Suc n) * fib (Suc n)) - 1 |
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71 else int (fib (Suc n) * fib (Suc n)) + 1)" |
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72 proof(induct n rule: fib.induct) |
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73 case 1 thus ?case by (simp add: fib_2) |
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74 next |
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75 case 2 thus ?case by (simp add: fib_2 mod_Suc) |
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76 next |
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77 case (3 x) |
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78 have "Suc 0 \<noteq> x mod 2 \<longrightarrow> x mod 2 = 0" by presburger |
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79 with "3.hyps" show ?case by (simp add: fib.simps add_mult_distrib add_mult_distrib2) |
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80 qed |
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81 |
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82 text\<open>We now obtain a version for the natural numbers via the coercion |
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83 function @{term int}.\<close> |
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84 theorem fib_Cassini: |
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85 "fib (Suc (Suc n)) * fib n = |
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86 (if n mod 2 = 0 then fib (Suc n) * fib (Suc n) - 1 |
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87 else fib (Suc n) * fib (Suc n) + 1)" |
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88 apply (rule of_nat_eq_iff [where 'a = int, THEN iffD1]) |
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89 using fib_Cassini_int apply (auto simp add: Suc_leI fib_Suc_gr_0 of_nat_diff) |
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90 done |
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91 |
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92 |
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93 text \<open>\medskip Toward Law 6.111 of Concrete Mathematics\<close> |
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94 |
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95 lemma gcd_fib_Suc_eq_1: "gcd (fib n) (fib (Suc n)) = Suc 0" |
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96 apply (induct n rule: fib.induct) |
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97 prefer 3 |
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98 apply (simp add: gcd_commute fib_Suc3) |
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99 apply (simp_all add: fib_2) |
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100 done |
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101 |
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102 lemma gcd_fib_add: "gcd (fib m) (fib (n + m)) = gcd (fib m) (fib n)" |
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103 apply (simp add: gcd_commute [of "fib m"]) |
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104 apply (case_tac m) |
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105 apply simp |
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106 apply (simp add: fib_add) |
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107 apply (simp add: add.commute gcd_non_0 [OF fib_Suc_gr_0]) |
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108 apply (simp add: gcd_non_0 [OF fib_Suc_gr_0, symmetric]) |
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109 apply (simp add: gcd_fib_Suc_eq_1 gcd_mult_cancel) |
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110 done |
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111 |
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112 lemma gcd_fib_diff: "m \<le> n ==> gcd (fib m) (fib (n - m)) = gcd (fib m) (fib n)" |
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113 by (simp add: gcd_fib_add [symmetric, of _ "n-m"]) |
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114 |
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115 lemma gcd_fib_mod: "0 < m ==> gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib n)" |
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116 proof (induct n rule: less_induct) |
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117 case (less n) |
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118 from less.prems have pos_m: "0 < m" . |
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119 show "gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib n)" |
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120 proof (cases "m < n") |
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121 case True note m_n = True |
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122 then have m_n': "m \<le> n" by auto |
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123 with pos_m have pos_n: "0 < n" by auto |
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124 with pos_m m_n have diff: "n - m < n" by auto |
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125 have "gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib ((n - m) mod m))" |
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126 by (simp add: mod_if [of n]) (insert m_n, auto) |
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127 also have "\<dots> = gcd (fib m) (fib (n - m))" by (simp add: less.hyps diff pos_m) |
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128 also have "\<dots> = gcd (fib m) (fib n)" by (simp add: gcd_fib_diff m_n') |
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129 finally show "gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib n)" . |
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130 next |
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131 case False then show "gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib n)" |
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132 by (cases "m = n") auto |
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133 qed |
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134 qed |
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135 |
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136 lemma fib_gcd: "fib (gcd m n) = gcd (fib m) (fib n)" \<comment> \<open>Law 6.111\<close> |
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137 apply (induct m n rule: gcd_induct) |
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138 apply (simp_all add: gcd_non_0 gcd_commute gcd_fib_mod) |
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139 done |
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140 |
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141 theorem fib_mult_eq_sum: |
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142 "fib (Suc n) * fib n = (\<Sum>k \<in> {..n}. fib k * fib k)" |
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143 apply (induct n rule: fib.induct) |
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144 apply (auto simp add: atMost_Suc fib_2) |
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145 apply (simp add: add_mult_distrib add_mult_distrib2) |
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146 done |
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147 |
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148 end |
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