src/HOL/Isar_Examples/Fibonacci.thy
author wenzelm
Tue Sep 26 20:54:40 2017 +0200 (23 months ago)
changeset 66695 91500c024c7f
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tuned;
     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 
    25 declare One_nat_def [simp]
    26 
    27 
    28 subsection \<open>Fibonacci numbers\<close>
    29 
    30 fun fib :: "nat \<Rightarrow> nat"
    31   where
    32     "fib 0 = 0"
    33   | "fib (Suc 0) = 1"
    34   | "fib (Suc (Suc x)) = fib x + fib (Suc x)"
    35 
    36 lemma [simp]: "fib (Suc n) > 0"
    37   by (induct n rule: fib.induct) simp_all
    38 
    39 
    40 text \<open>Alternative induction rule.\<close>
    41 
    42 theorem fib_induct: "P 0 \<Longrightarrow> P 1 \<Longrightarrow> (\<And>n. P (n + 1) \<Longrightarrow> P n \<Longrightarrow> P (n + 2)) \<Longrightarrow> P n"
    43   for n :: nat
    44   by (induct rule: fib.induct) simp_all
    45 
    46 
    47 subsection \<open>Fib and gcd commute\<close>
    48 
    49 text \<open>A few laws taken from @{cite "Concrete-Math"}.\<close>
    50 
    51 lemma fib_add: "fib (n + k + 1) = fib (k + 1) * fib (n + 1) + fib k * fib n"
    52   (is "?P n")
    53   \<comment> \<open>see @{cite \<open>page 280\<close> "Concrete-Math"}\<close>
    54 proof (induct n rule: fib_induct)
    55   show "?P 0" by simp
    56   show "?P 1" by simp
    57   fix n
    58   have "fib (n + 2 + k + 1)
    59     = fib (n + k + 1) + fib (n + 1 + k + 1)" by simp
    60   also assume "fib (n + k + 1) = fib (k + 1) * fib (n + 1) + fib k * fib n" (is " _ = ?R1")
    61   also assume "fib (n + 1 + k + 1) = fib (k + 1) * fib (n + 1 + 1) + fib k * fib (n + 1)"
    62     (is " _ = ?R2")
    63   also have "?R1 + ?R2 = fib (k + 1) * fib (n + 2 + 1) + fib k * fib (n + 2)"
    64     by (simp add: add_mult_distrib2)
    65   finally show "?P (n + 2)" .
    66 qed
    67 
    68 lemma gcd_fib_Suc_eq_1: "gcd (fib n) (fib (n + 1)) = 1"
    69   (is "?P n")
    70 proof (induct n rule: fib_induct)
    71   show "?P 0" by simp
    72   show "?P 1" by simp
    73   fix n
    74   have "fib (n + 2 + 1) = fib (n + 1) + fib (n + 2)"
    75     by simp
    76   also have "\<dots> = fib (n + 2) + fib (n + 1)"
    77     by simp
    78   also have "gcd (fib (n + 2)) \<dots> = gcd (fib (n + 2)) (fib (n + 1))"
    79     by (rule gcd_add2)
    80   also have "\<dots> = gcd (fib (n + 1)) (fib (n + 1 + 1))"
    81     by (simp add: gcd.commute)
    82   also assume "\<dots> = 1"
    83   finally show "?P (n + 2)" .
    84 qed
    85 
    86 lemma gcd_mult_add: "(0::nat) < n \<Longrightarrow> gcd (n * k + m) n = gcd m n"
    87 proof -
    88   assume "0 < n"
    89   then have "gcd (n * k + m) n = gcd n (m mod n)"
    90     by (simp add: gcd_non_0_nat add.commute)
    91   also from \<open>0 < n\<close> have "\<dots> = gcd m n"
    92     by (simp add: gcd_non_0_nat)
    93   finally show ?thesis .
    94 qed
    95 
    96 lemma gcd_fib_add: "gcd (fib m) (fib (n + m)) = gcd (fib m) (fib n)"
    97 proof (cases m)
    98   case 0
    99   then show ?thesis by simp
   100 next
   101   case (Suc k)
   102   then have "gcd (fib m) (fib (n + m)) = gcd (fib (n + k + 1)) (fib (k + 1))"
   103     by (simp add: gcd.commute)
   104   also have "fib (n + k + 1) = fib (k + 1) * fib (n + 1) + fib k * fib n"
   105     by (rule fib_add)
   106   also have "gcd \<dots> (fib (k + 1)) = gcd (fib k * fib n) (fib (k + 1))"
   107     by (simp add: gcd_mult_add)
   108   also have "\<dots> = gcd (fib n) (fib (k + 1))"
   109     by (simp only: gcd_fib_Suc_eq_1 gcd_mult_cancel)
   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