src/HOL/Isar_Examples/Fibonacci.thy
author haftmann
Sat Nov 11 18:41:08 2017 +0000 (19 months ago)
changeset 67051 e7e54a0b9197
parent 66453 cc19f7ca2ed6
permissions -rw-r--r--
dedicated definition for coprimality
     1 (*  Title:      HOL/Isar_Examples/Fibonacci.thy
     2     Author:     Gertrud Bauer
     3     Copyright   1999 Technische Universitaet Muenchen
     4 
     5 The Fibonacci function.  Original
     6 tactic script by Lawrence C Paulson.
     7 
     8 Fibonacci numbers: proofs of laws taken from
     9 
    10   R. L. Graham, D. E. Knuth, O. Patashnik.
    11   Concrete Mathematics.
    12   (Addison-Wesley, 1989)
    13 *)
    14 
    15 section \<open>Fib and Gcd commute\<close>
    16 
    17 theory Fibonacci
    18   imports "HOL-Computational_Algebra.Primes"
    19 begin
    20 
    21 text_raw \<open>\<^footnote>\<open>Isar version by Gertrud Bauer. Original tactic script by Larry
    22   Paulson. A few proofs of laws taken from @{cite "Concrete-Math"}.\<close>\<close>
    23 
    24 subsection \<open>Fibonacci numbers\<close>
    25 
    26 fun fib :: "nat \<Rightarrow> nat"
    27   where
    28     "fib 0 = 0"
    29   | "fib (Suc 0) = 1"
    30   | "fib (Suc (Suc x)) = fib x + fib (Suc x)"
    31 
    32 lemma [simp]: "fib (Suc n) > 0"
    33   by (induct n rule: fib.induct) simp_all
    34 
    35 
    36 text \<open>Alternative induction rule.\<close>
    37 
    38 theorem fib_induct: "P 0 \<Longrightarrow> P 1 \<Longrightarrow> (\<And>n. P (n + 1) \<Longrightarrow> P n \<Longrightarrow> P (n + 2)) \<Longrightarrow> P n"
    39   for n :: nat
    40   by (induct rule: fib.induct) simp_all
    41 
    42 
    43 subsection \<open>Fib and gcd commute\<close>
    44 
    45 text \<open>A few laws taken from @{cite "Concrete-Math"}.\<close>
    46 
    47 lemma fib_add: "fib (n + k + 1) = fib (k + 1) * fib (n + 1) + fib k * fib n"
    48   (is "?P n")
    49   \<comment> \<open>see @{cite \<open>page 280\<close> "Concrete-Math"}\<close>
    50 proof (induct n rule: fib_induct)
    51   show "?P 0" by simp
    52   show "?P 1" by simp
    53   fix n
    54   have "fib (n + 2 + k + 1)
    55     = fib (n + k + 1) + fib (n + 1 + k + 1)" by simp
    56   also assume "fib (n + k + 1) = fib (k + 1) * fib (n + 1) + fib k * fib n" (is " _ = ?R1")
    57   also assume "fib (n + 1 + k + 1) = fib (k + 1) * fib (n + 1 + 1) + fib k * fib (n + 1)"
    58     (is " _ = ?R2")
    59   also have "?R1 + ?R2 = fib (k + 1) * fib (n + 2 + 1) + fib k * fib (n + 2)"
    60     by (simp add: add_mult_distrib2)
    61   finally show "?P (n + 2)" .
    62 qed
    63 
    64 lemma coprime_fib_Suc: "coprime (fib n) (fib (n + 1))"
    65   (is "?P n")
    66 proof (induct n rule: fib_induct)
    67   show "?P 0" by simp
    68   show "?P 1" by simp
    69   fix n
    70   assume P: "coprime (fib (n + 1)) (fib (n + 1 + 1))"
    71   have "fib (n + 2 + 1) = fib (n + 1) + fib (n + 2)"
    72     by simp
    73   also have "\<dots> = fib (n + 2) + fib (n + 1)"
    74     by simp
    75   also have "gcd (fib (n + 2)) \<dots> = gcd (fib (n + 2)) (fib (n + 1))"
    76     by (rule gcd_add2)
    77   also have "\<dots> = gcd (fib (n + 1)) (fib (n + 1 + 1))"
    78     by (simp add: gcd.commute)
    79   also have "\<dots> = 1"
    80     using P by simp
    81   finally show "?P (n + 2)"
    82     by (simp add: coprime_iff_gcd_eq_1)
    83 qed
    84 
    85 lemma gcd_mult_add: "(0::nat) < n \<Longrightarrow> gcd (n * k + m) n = gcd m n"
    86 proof -
    87   assume "0 < n"
    88   then have "gcd (n * k + m) n = gcd n (m mod n)"
    89     by (simp add: gcd_non_0_nat add.commute)
    90   also from \<open>0 < n\<close> have "\<dots> = gcd m n"
    91     by (simp add: gcd_non_0_nat)
    92   finally show ?thesis .
    93 qed
    94 
    95 lemma gcd_fib_add: "gcd (fib m) (fib (n + m)) = gcd (fib m) (fib n)"
    96 proof (cases m)
    97   case 0
    98   then show ?thesis by simp
    99 next
   100   case (Suc k)
   101   then have "gcd (fib m) (fib (n + m)) = gcd (fib (n + k + 1)) (fib (k + 1))"
   102     by (simp add: gcd.commute)
   103   also have "fib (n + k + 1) = fib (k + 1) * fib (n + 1) + fib k * fib n"
   104     by (rule fib_add)
   105   also have "gcd \<dots> (fib (k + 1)) = gcd (fib k * fib n) (fib (k + 1))"
   106     by (simp add: gcd_mult_add)
   107   also have "\<dots> = gcd (fib n) (fib (k + 1))"
   108     using coprime_fib_Suc [of k] gcd_mult_left_right_cancel [of "fib (k + 1)" "fib k" "fib n"]
   109     by (simp add: ac_simps)
   110   also have "\<dots> = gcd (fib m) (fib n)"
   111     using Suc by (simp add: gcd.commute)
   112   finally show ?thesis .
   113 qed
   114 
   115 lemma gcd_fib_diff: "gcd (fib m) (fib (n - m)) = gcd (fib m) (fib n)" if "m \<le> n"
   116 proof -
   117   have "gcd (fib m) (fib (n - m)) = gcd (fib m) (fib (n - m + m))"
   118     by (simp add: gcd_fib_add)
   119   also from \<open>m \<le> n\<close> have "n - m + m = n"
   120     by simp
   121   finally show ?thesis .
   122 qed
   123 
   124 lemma gcd_fib_mod: "gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib n)" if "0 < m"
   125 proof (induct n rule: nat_less_induct)
   126   case hyp: (1 n)
   127   show ?case
   128   proof -
   129     have "n mod m = (if n < m then n else (n - m) mod m)"
   130       by (rule mod_if)
   131     also have "gcd (fib m) (fib \<dots>) = gcd (fib m) (fib n)"
   132     proof (cases "n < m")
   133       case True
   134       then show ?thesis by simp
   135     next
   136       case False
   137       then have "m \<le> n" by simp
   138       from \<open>0 < m\<close> and False have "n - m < n"
   139         by simp
   140       with hyp have "gcd (fib m) (fib ((n - m) mod m))
   141           = gcd (fib m) (fib (n - m))" by simp
   142       also have "\<dots> = gcd (fib m) (fib n)"
   143         using \<open>m \<le> n\<close> by (rule gcd_fib_diff)
   144       finally have "gcd (fib m) (fib ((n - m) mod m)) =
   145           gcd (fib m) (fib n)" .
   146       with False show ?thesis by simp
   147     qed
   148     finally show ?thesis .
   149   qed
   150 qed
   151 
   152 theorem fib_gcd: "fib (gcd m n) = gcd (fib m) (fib n)"
   153   (is "?P m n")
   154 proof (induct m n rule: gcd_nat_induct)
   155   fix m n :: nat
   156   show "fib (gcd m 0) = gcd (fib m) (fib 0)"
   157     by simp
   158   assume n: "0 < n"
   159   then have "gcd m n = gcd n (m mod n)"
   160     by (simp add: gcd_non_0_nat)
   161   also assume hyp: "fib \<dots> = gcd (fib n) (fib (m mod n))"
   162   also from n have "\<dots> = gcd (fib n) (fib m)"
   163     by (rule gcd_fib_mod)
   164   also have "\<dots> = gcd (fib m) (fib n)"
   165     by (rule gcd.commute)
   166   finally show "fib (gcd m n) = gcd (fib m) (fib n)" .
   167 qed
   168 
   169 end