(*  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)) - 1

    else int (fib (Suc n) * fib (Suc n)) + 1)"

   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 = (\<Sum>k \<in> {..n}. fib k * fib k)"

   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