(*  Title:      HOL/NumberTheory/Fib.thy

    ID:         $Id$

    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

    Copyright   1997  University of Cambridge

*)

header {* The Fibonacci function *}

theory Fib = Primes:

text {*

  Fibonacci numbers: proofs of laws taken from:

  R. L. Graham, D. E. Knuth, O. Patashnik.  Concrete Mathematics.

  (Addison-Wesley, 1989)

  \bigskip

*}

consts fib :: "nat => nat"

recdef fib  less_than

  zero: "fib 0  = 0"

  one:  "fib (Suc 0) = Suc 0"

  Suc_Suc: "fib (Suc (Suc x)) = fib x + fib (Suc x)"

text {*

  \medskip The difficulty in these proofs is to ensure that the

  induction hypotheses are applied before the definition of @{term

  fib}.  Towards this end, the @{term fib} equations are not declared

  to the Simplifier and are applied very selectively at first.

*}

declare fib.Suc_Suc [simp del]

lemma fib_Suc3: "fib (Suc (Suc (Suc n))) = fib (Suc n) + fib (Suc (Suc n))"

  apply (rule fib.Suc_Suc)

  done

text {* \medskip Concrete Mathematics, page 280 *}

lemma fib_add: "fib (Suc (n + k)) = fib (Suc k) * fib (Suc n) + fib k * fib n"

  apply (induct n rule: fib.induct)

    prefer 3

    txt {* simplify the LHS just enough to apply the induction hypotheses *}

    apply (simp add: fib.Suc_Suc [of "Suc (m + n)", standard])

    apply (simp_all (no_asm_simp) add: fib.Suc_Suc add_mult_distrib add_mult_distrib2)

    done

lemma fib_Suc_neq_0 [simp]: "fib (Suc n) \<noteq> 0"

  apply (induct n rule: fib.induct)

    apply (simp_all add: fib.Suc_Suc)

  done

lemma [simp]: "0 < fib (Suc n)"

  apply (simp add: neq0_conv [symmetric])

  done

lemma fib_gr_0: "0 < n ==> 0 < fib n"

  apply (rule not0_implies_Suc [THEN exE])

   apply auto

  done

text {*

  \medskip Concrete Mathematics, page 278: Cassini's identity.  It is

  much easier to prove using integers!

*}

lemma fib_Cassini: "int (fib (Suc (Suc n)) * fib n) =

  (if n mod 2 = 0 then int (fib (Suc n) * fib (Suc n)) - Numeral1

   else int (fib (Suc n) * fib (Suc n)) + Numeral1)"

  apply (induct n rule: fib.induct)

    apply (simp add: fib.Suc_Suc)

   apply (simp add: fib.Suc_Suc mod_Suc)

  apply (simp add: fib.Suc_Suc

    add_mult_distrib add_mult_distrib2 mod_Suc zmult_int [symmetric] zmult_ac)

  apply (subgoal_tac "x mod 2 < 2", arith)

  apply simp

  done

text {* \medskip Towards Law 6.111 of Concrete Mathematics *}

lemma gcd_fib_Suc_eq_1: "gcd (fib n, fib (Suc n)) = Suc 0"

  apply (induct n rule: fib.induct)

    prefer 3

    apply (simp add: gcd_commute fib_Suc3)

   apply (simp_all add: fib.Suc_Suc)

  done

lemma gcd_fib_add: "gcd (fib m, fib (n + m)) = gcd (fib m, fib n)"

  apply (simp (no_asm) add: gcd_commute [of "fib m"])

  apply (case_tac "m = 0")

   apply simp

  apply (clarify dest!: not0_implies_Suc)

  apply (simp add: fib_add)

  apply (simp add: add_commute gcd_non_0)

  apply (simp add: gcd_non_0 [symmetric])

  apply (simp add: gcd_fib_Suc_eq_1 gcd_mult_cancel)

  done

lemma gcd_fib_diff: "m \<le> n ==> gcd (fib m, fib (n - m)) = gcd (fib m, fib n)"

  apply (rule gcd_fib_add [symmetric, THEN trans])

  apply simp

  done

lemma gcd_fib_mod: "0 < m ==> gcd (fib m, fib (n mod m)) = gcd (fib m, fib n)"

  apply (induct n rule: nat_less_induct)

  apply (subst mod_if)

  apply (simp add: gcd_fib_diff mod_geq not_less_iff_le diff_less)

  done

lemma fib_gcd: "fib (gcd (m, n)) = gcd (fib m, fib n)"  -- {* Law 6.111 *}

  apply (induct m n rule: gcd_induct)

   apply simp

  apply (simp add: gcd_non_0)

  apply (simp add: gcd_commute gcd_fib_mod)

  done

lemma fib_mult_eq_setsum:

    "fib (Suc n) * fib n = setsum (\<lambda>k. fib k * fib k) (atMost n)"

  apply (induct n rule: fib.induct)

    apply (auto simp add: atMost_Suc fib.Suc_Suc)

  apply (simp add: add_mult_distrib add_mult_distrib2)

  done

end