(* ID: $Id: Fib.thy,v 1.24 2007/11/09 17:59:56 krauss Exp $ Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 1997 University of Cambridge *) header {* The Fibonacci function *} theory Fib imports Primes begin text {* Fibonacci numbers: proofs of laws taken from: R. L. Graham, D. E. Knuth, O. Patashnik. Concrete Mathematics. (Addison-Wesley, 1989) \bigskip *} fun fib :: "nat => nat" where "fib 0 = 0" | "fib (Suc 0) = 1" | fib_2: "fib (Suc (Suc n)) = fib n + fib (Suc n)" 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. *} text{*We disable @{text fib.fib_2fib_2} for simplification ...*} declare fib_2 [simp del] text{*...then prove a version that has a more restrictive pattern.*} lemma fib_Suc3: "fib (Suc (Suc (Suc n))) = fib (Suc n) + fib (Suc (Suc n))" by (rule fib_2) text {* \medskip Concrete Mathematics, page 280 *} lemma fib_add: "fib (Suc (n + k)) = fib (Suc k) * fib (Suc n) + fib k * fib n" proof (induct n rule: fib.induct) case 1 show ?case by simp next case 2 show ?case by (simp add: fib_2) next case 3 thus ?case by (simp add: fib_2 add_mult_distrib2) qed lemma fib_Suc_neq_0: "fib (Suc n) ≠ 0" apply (induct n rule: fib.induct) apply (simp_all add: fib_2) done lemma fib_Suc_gr_0: "0 < fib (Suc n)" by (insert fib_Suc_neq_0 [of n], simp) lemma fib_gr_0: "0 < n ==> 0 < fib n" by (case_tac n, auto simp add: fib_Suc_gr_0) text {* \medskip Concrete Mathematics, page 278: Cassini's identity. The proof is much easier using integers, not natural numbers! *} lemma fib_Cassini_int: "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)" proof(induct n rule: fib.induct) case 1 thus ?case by (simp add: fib_2) next case 2 thus ?case by (simp add: fib_2 mod_Suc) next case (3 x) have "Suc 0 ≠ x mod 2 --> x mod 2 = 0" by presburger with "3.hyps" show ?case by (simp add: fib.simps add_mult_distrib add_mult_distrib2) qed text{*We now obtain a version for the natural numbers via the coercion function @{term int}.*} theorem fib_Cassini: "fib (Suc (Suc n)) * fib n = (if n mod 2 = 0 then fib (Suc n) * fib (Suc n) - 1 else fib (Suc n) * fib (Suc n) + 1)" apply (rule int_int_eq [THEN iffD1]) apply (simp add: fib_Cassini_int) apply (subst zdiff_int [symmetric]) apply (insert fib_Suc_gr_0 [of n], simp_all) done text {* \medskip Toward 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_2) done lemma gcd_fib_add: "gcd (fib m, fib (n + m)) = gcd (fib m, fib n)" apply (simp add: gcd_commute [of "fib m"]) apply (case_tac m) apply simp apply (simp add: fib_add) apply (simp add: add_commute gcd_non_0 [OF fib_Suc_gr_0]) apply (simp add: gcd_non_0 [OF fib_Suc_gr_0, symmetric]) apply (simp add: gcd_fib_Suc_eq_1 gcd_mult_cancel) done lemma gcd_fib_diff: "m ≤ n ==> gcd (fib m, fib (n - m)) = gcd (fib m, fib n)" by (simp add: gcd_fib_add [symmetric, of _ "n-m"]) lemma gcd_fib_mod: "0 < m ==> gcd (fib m, fib (n mod m)) = gcd (fib m, fib n)" proof (induct n rule: less_induct) case (less n) from less.prems have pos_m: "0 < m" . show "gcd (fib m, fib (n mod m)) = gcd (fib m, fib n)" proof (cases "m < n") case True note m_n = True then have m_n': "m ≤ n" by auto with pos_m have pos_n: "0 < n" by auto with pos_m m_n have diff: "n - m < n" by auto have "gcd (fib m, fib (n mod m)) = gcd (fib m, fib ((n - m) mod m))" by (simp add: mod_if [of n]) (insert m_n, auto) also have "… = gcd (fib m, fib (n - m))" by (simp add: less.hyps diff pos_m) also have "… = gcd (fib m, fib n)" by (simp add: gcd_fib_diff m_n') finally show "gcd (fib m, fib (n mod m)) = gcd (fib m, fib n)" . next case False then show "gcd (fib m, fib (n mod m)) = gcd (fib m, fib n)" by (cases "m = n") auto qed qed lemma fib_gcd: "fib (gcd (m, n)) = gcd (fib m, fib n)" -- {* Law 6.111 *} apply (induct m n rule: gcd_induct) apply (simp_all add: gcd_non_0 gcd_commute gcd_fib_mod) done theorem fib_mult_eq_setsum: "fib (Suc n) * fib n = (∑k ∈ {..n}. fib k * fib k)" apply (induct n rule: fib.induct) apply (auto simp add: atMost_Suc fib_2) apply (simp add: add_mult_distrib add_mult_distrib2) done end
lemma fib_Suc3:
fib (Suc (Suc (Suc n))) = fib (Suc n) + fib (Suc (Suc n))
lemma fib_add:
fib (Suc (n + k)) = fib (Suc k) * fib (Suc n) + fib k * fib n
lemma fib_Suc_neq_0:
fib (Suc n) ≠ 0
lemma fib_Suc_gr_0:
0 < fib (Suc n)
lemma fib_gr_0:
0 < n ==> 0 < fib n
lemma fib_Cassini_int:
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)
theorem fib_Cassini:
fib (Suc (Suc n)) * fib n =
(if n mod 2 = 0 then fib (Suc n) * fib (Suc n) - 1
else fib (Suc n) * fib (Suc n) + 1)
lemma gcd_fib_Suc_eq_1:
gcd (fib n, fib (Suc n)) = Suc 0
lemma gcd_fib_add:
gcd (fib m, fib (n + m)) = gcd (fib m, fib n)
lemma gcd_fib_diff:
m ≤ n ==> gcd (fib m, fib (n - m)) = gcd (fib m, fib n)
lemma gcd_fib_mod:
0 < m ==> gcd (fib m, fib (n mod m)) = gcd (fib m, fib n)
lemma fib_gcd:
fib (gcd (m, n)) = gcd (fib m, fib n)
theorem fib_mult_eq_setsum:
fib (Suc n) * fib n = (∑k≤n. fib k * fib k)