(* ID: $Id: Fib.thy,v 1.13 2005/06/17 14:13:09 haftmann 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 *} consts fib :: "nat => nat" recdef fib "measure (λx. x)" 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. *} text{*We disable @{text fib.Suc_Suc} for simplification ...*} declare fib.Suc_Suc [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.Suc_Suc) 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_Suc3) apply (simp_all add: fib.Suc_Suc add_mult_distrib2) done lemma fib_Suc_neq_0: "fib (Suc n) ≠ 0" apply (induct n rule: fib.induct) apply (simp_all add: fib.Suc_Suc) 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)" 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]) apply presburger done 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.Suc_Suc) 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)" apply (induct n rule: nat_less_induct) apply (simp add: mod_if gcd_fib_diff mod_geq) 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_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.Suc_Suc) 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)