(* 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 1' = 1'"
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)) = 1'"
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