src/HOL/NumberTheory/Fib.thy
```     1.1 --- a/src/HOL/NumberTheory/Fib.thy	Sat Feb 03 17:43:34 2001 +0100
1.2 +++ b/src/HOL/NumberTheory/Fib.thy	Sun Feb 04 19:31:13 2001 +0100
1.3 @@ -1,17 +1,125 @@
1.4 -(*  Title:      ex/Fib
1.5 +(*  Title:      HOL/NumberTheory/Fib.thy
1.6      ID:         \$Id\$
1.7      Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
1.8      Copyright   1997  University of Cambridge
1.9 -
1.10 -The Fibonacci function.  Demonstrates the use of recdef.
1.11  *)
1.12
1.13 -Fib = Primes +
1.14 +header {* The Fibonacci function *}
1.15 +
1.16 +theory Fib = Primes:
1.17 +
1.18 +text {*
1.19 +  Fibonacci numbers: proofs of laws taken from:
1.20 +  R. L. Graham, D. E. Knuth, O. Patashnik.  Concrete Mathematics.
1.22 +
1.23 +  \bigskip
1.24 +*}
1.25 +
1.26 +consts fib :: "nat => nat"
1.27 +recdef fib  less_than
1.28 +  zero: "fib 0 = 0"
1.29 +  one:  "fib 1 = 1"
1.30 +  Suc_Suc: "fib (Suc (Suc x)) = fib x + fib (Suc x)"
1.31 +
1.32 +text {*
1.33 +  \medskip The difficulty in these proofs is to ensure that the
1.34 +  induction hypotheses are applied before the definition of @{term
1.35 +  fib}.  Towards this end, the @{term fib} equations are not declared
1.36 +  to the Simplifier and are applied very selectively at first.
1.37 +*}
1.38 +
1.39 +declare fib.Suc_Suc [simp del]
1.40 +
1.41 +lemma fib_Suc3: "fib (Suc (Suc (Suc n))) = fib (Suc n) + fib (Suc (Suc n))"
1.42 +  apply (rule fib.Suc_Suc)
1.43 +  done
1.44 +
1.45 +
1.46 +text {* \medskip Concrete Mathematics, page 280 *}
1.47 +
1.48 +lemma fib_add: "fib (Suc (n + k)) = fib (Suc k) * fib (Suc n) + fib k * fib n"
1.49 +  apply (induct n rule: fib.induct)
1.50 +    prefer 3
1.51 +    txt {* simplify the LHS just enough to apply the induction hypotheses *}
1.52 +    apply (simp add: fib.Suc_Suc [of "Suc (m + n)", standard])
1.54 +    done
1.55 +
1.56 +lemma fib_Suc_neq_0 [simp]: "fib (Suc n) \<noteq> 0"
1.57 +  apply (induct n rule: fib.induct)
1.58 +    apply (simp_all add: fib.Suc_Suc)
1.59 +  done
1.60 +
1.61 +lemma [simp]: "0 < fib (Suc n)"
1.62 +  apply (simp add: neq0_conv [symmetric])
1.63 +  done
1.64 +
1.65 +lemma fib_gr_0: "0 < n ==> 0 < fib n"
1.66 +  apply (rule not0_implies_Suc [THEN exE])
1.67 +   apply auto
1.68 +  done
1.69 +
1.70
1.71 -consts fib  :: "nat => nat"
1.72 -recdef fib "less_than"
1.73 -  zero    "fib 0 = 0"
1.74 -  one     "fib 1 = 1"
1.75 -  Suc_Suc "fib (Suc (Suc x)) = fib x + fib (Suc x)"
1.76 +text {*
1.77 +  \medskip Concrete Mathematics, page 278: Cassini's identity.  It is
1.78 +  much easier to prove using integers!
1.79 +*}
1.80 +
1.81 +lemma fib_Cassini: "int (fib (Suc (Suc n)) * fib n) =
1.82 +  (if n mod #2 = 0 then int (fib (Suc n) * fib (Suc n)) - #1
1.83 +   else int (fib (Suc n) * fib (Suc n)) + #1)"
1.84 +  apply (induct n rule: fib.induct)
1.85 +    apply (simp add: fib.Suc_Suc)
1.86 +   apply (simp add: fib.Suc_Suc mod_Suc)
1.87 +  apply (simp add: fib.Suc_Suc
1.89 +  done
1.90 +
1.91 +
1.93 +
1.94 +lemma gcd_fib_Suc_eq_1: "gcd (fib n, fib (Suc n)) = 1"
1.95 +  apply (induct n rule: fib.induct)
1.96 +    prefer 3
1.97 +    apply (simp add: gcd_commute fib_Suc3)
1.98 +   apply (simp_all add: fib.Suc_Suc)
1.99 +  done
1.100 +
1.101 +lemma gcd_fib_add: "gcd (fib m, fib (n + m)) = gcd (fib m, fib n)"
1.102 +  apply (simp (no_asm) add: gcd_commute [of "fib m"])
1.103 +  apply (case_tac "m = 0")
1.104 +   apply simp
1.105 +  apply (clarify dest!: not0_implies_Suc)
1.108 +  apply (simp add: gcd_non_0 [symmetric])
1.109 +  apply (simp add: gcd_fib_Suc_eq_1 gcd_mult_cancel)
1.110 +  done
1.111 +
1.112 +lemma gcd_fib_diff: "m \<le> n ==> gcd (fib m, fib (n - m)) = gcd (fib m, fib n)"
1.113 +  apply (rule gcd_fib_add [symmetric, THEN trans])
1.114 +  apply simp
1.115 +  done
1.116 +
1.117 +lemma gcd_fib_mod: "0 < m ==> gcd (fib m, fib (n mod m)) = gcd (fib m, fib n)"
1.118 +  apply (induct n rule: nat_less_induct)
1.119 +  apply (subst mod_if)
1.120 +  apply (simp add: gcd_fib_diff mod_geq not_less_iff_le diff_less)
1.121 +  done
1.122 +
1.123 +lemma fib_gcd: "fib (gcd (m, n)) = gcd (fib m, fib n)"  -- {* Law 6.111 *}
1.124 +  apply (induct m n rule: gcd_induct)
1.125 +   apply simp
1.126 +  apply (simp add: gcd_non_0)
1.127 +  apply (simp add: gcd_commute gcd_fib_mod)
1.128 +  done
1.129 +
1.130 +lemma fib_mult_eq_setsum:
1.131 +    "fib (Suc n) * fib n = setsum (\<lambda>k. fib k * fib k) (atMost n)"
1.132 +  apply (induct n rule: fib.induct)
1.133 +    apply (auto simp add: atMost_Suc fib.Suc_Suc)