src/HOL/NumberTheory/Fib.thy
author obua
Mon Apr 10 16:00:34 2006 +0200 (2006-04-10)
changeset 19404 9bf2cdc9e8e8
parent 18156 5a971b272f78
child 20272 0ca998e83447
permissions -rw-r--r--
Moved stuff from Ring_and_Field to Matrix
     1 (*  ID:         $Id$
     2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     3     Copyright   1997  University of Cambridge
     4 *)
     5 
     6 header {* The Fibonacci function *}
     7 
     8 theory Fib imports Primes begin
     9 
    10 text {*
    11   Fibonacci numbers: proofs of laws taken from:
    12   R. L. Graham, D. E. Knuth, O. Patashnik.  Concrete Mathematics.
    13   (Addison-Wesley, 1989)
    14 
    15   \bigskip
    16 *}
    17 
    18 consts fib :: "nat => nat"
    19 recdef fib  "measure (\<lambda>x. x)"
    20    zero:    "fib 0  = 0"
    21    one:     "fib (Suc 0) = Suc 0"
    22    Suc_Suc: "fib (Suc (Suc x)) = fib x + fib (Suc x)"
    23 
    24 text {*
    25   \medskip The difficulty in these proofs is to ensure that the
    26   induction hypotheses are applied before the definition of @{term
    27   fib}.  Towards this end, the @{term fib} equations are not declared
    28   to the Simplifier and are applied very selectively at first.
    29 *}
    30 
    31 text{*We disable @{text fib.Suc_Suc} for simplification ...*}
    32 declare fib.Suc_Suc [simp del]
    33 
    34 text{*...then prove a version that has a more restrictive pattern.*}
    35 lemma fib_Suc3: "fib (Suc (Suc (Suc n))) = fib (Suc n) + fib (Suc (Suc n))"
    36   by (rule fib.Suc_Suc)
    37 
    38 
    39 text {* \medskip Concrete Mathematics, page 280 *}
    40 
    41 lemma fib_add: "fib (Suc (n + k)) = fib (Suc k) * fib (Suc n) + fib k * fib n"
    42   apply (induct n rule: fib.induct)
    43     prefer 3
    44     txt {* simplify the LHS just enough to apply the induction hypotheses *}
    45     apply (simp add: fib_Suc3)
    46     apply (simp_all add: fib.Suc_Suc add_mult_distrib2)
    47     done
    48 
    49 lemma fib_Suc_neq_0: "fib (Suc n) \<noteq> 0"
    50   apply (induct n rule: fib.induct)
    51     apply (simp_all add: fib.Suc_Suc)
    52   done
    53 
    54 lemma fib_Suc_gr_0: "0 < fib (Suc n)"
    55   by (insert fib_Suc_neq_0 [of n], simp)  
    56 
    57 lemma fib_gr_0: "0 < n ==> 0 < fib n"
    58   by (case_tac n, auto simp add: fib_Suc_gr_0) 
    59 
    60 
    61 text {*
    62   \medskip Concrete Mathematics, page 278: Cassini's identity.  The proof is
    63   much easier using integers, not natural numbers!
    64 *}
    65 
    66 lemma fib_Cassini_int:
    67  "int (fib (Suc (Suc n)) * fib n) =
    68   (if n mod 2 = 0 then int (fib (Suc n) * fib (Suc n)) - 1
    69    else int (fib (Suc n) * fib (Suc n)) + 1)"
    70   apply (induct n rule: fib.induct)
    71     apply (simp add: fib.Suc_Suc)
    72    apply (simp add: fib.Suc_Suc mod_Suc)
    73   apply (simp add: fib.Suc_Suc add_mult_distrib add_mult_distrib2
    74                    mod_Suc zmult_int [symmetric])
    75   apply (presburger (no_abs))
    76   done
    77 
    78 text{*We now obtain a version for the natural numbers via the coercion 
    79    function @{term int}.*}
    80 theorem fib_Cassini:
    81  "fib (Suc (Suc n)) * fib n =
    82   (if n mod 2 = 0 then fib (Suc n) * fib (Suc n) - 1
    83    else fib (Suc n) * fib (Suc n) + 1)"
    84   apply (rule int_int_eq [THEN iffD1]) 
    85   apply (simp add: fib_Cassini_int) 
    86   apply (subst zdiff_int [symmetric]) 
    87    apply (insert fib_Suc_gr_0 [of n], simp_all)
    88   done
    89 
    90 
    91 text {* \medskip Toward Law 6.111 of Concrete Mathematics *}
    92 
    93 lemma gcd_fib_Suc_eq_1: "gcd (fib n, fib (Suc n)) = Suc 0"
    94   apply (induct n rule: fib.induct)
    95     prefer 3
    96     apply (simp add: gcd_commute fib_Suc3)
    97    apply (simp_all add: fib.Suc_Suc)
    98   done
    99 
   100 lemma gcd_fib_add: "gcd (fib m, fib (n + m)) = gcd (fib m, fib n)"
   101   apply (simp add: gcd_commute [of "fib m"])
   102   apply (case_tac m)
   103    apply simp 
   104   apply (simp add: fib_add)
   105   apply (simp add: add_commute gcd_non_0 [OF fib_Suc_gr_0])
   106   apply (simp add: gcd_non_0 [OF fib_Suc_gr_0, symmetric])
   107   apply (simp add: gcd_fib_Suc_eq_1 gcd_mult_cancel)
   108   done
   109 
   110 lemma gcd_fib_diff: "m \<le> n ==> gcd (fib m, fib (n - m)) = gcd (fib m, fib n)"
   111   by (simp add: gcd_fib_add [symmetric, of _ "n-m"]) 
   112 
   113 lemma gcd_fib_mod: "0 < m ==> gcd (fib m, fib (n mod m)) = gcd (fib m, fib n)"
   114   apply (induct n rule: nat_less_induct)
   115   apply (simp add: mod_if gcd_fib_diff mod_geq)
   116   done
   117 
   118 lemma fib_gcd: "fib (gcd (m, n)) = gcd (fib m, fib n)"  -- {* Law 6.111 *}
   119   apply (induct m n rule: gcd_induct)
   120   apply (simp_all add: gcd_non_0 gcd_commute gcd_fib_mod)
   121   done
   122 
   123 theorem fib_mult_eq_setsum:
   124     "fib (Suc n) * fib n = (\<Sum>k \<in> {..n}. fib k * fib k)"
   125   apply (induct n rule: fib.induct)
   126     apply (auto simp add: atMost_Suc fib.Suc_Suc)
   127   apply (simp add: add_mult_distrib add_mult_distrib2)
   128   done
   129 
   130 end