paulson@15628: (*  ID:         $Id$
paulson@9944:     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
paulson@9944:     Copyright   1997  University of Cambridge
paulson@9944: *)
paulson@9944: 
wenzelm@11049: header {* The Fibonacci function *}
wenzelm@11049: 
haftmann@16417: theory Fib imports Primes begin
wenzelm@11049: 
wenzelm@11049: text {*
wenzelm@11049:   Fibonacci numbers: proofs of laws taken from:
wenzelm@11049:   R. L. Graham, D. E. Knuth, O. Patashnik.  Concrete Mathematics.
wenzelm@11049:   (Addison-Wesley, 1989)
wenzelm@11049: 
wenzelm@11049:   \bigskip
wenzelm@11049: *}
wenzelm@11049: 
paulson@24549: fun fib :: "nat \<Rightarrow> nat"
paulson@24549: where
paulson@24573:          "fib 0 = 0"
paulson@24573: |        "fib (Suc 0) = 1"
paulson@24549: | fib_2: "fib (Suc (Suc n)) = fib n + fib (Suc n)"
wenzelm@11049: 
wenzelm@11049: text {*
wenzelm@11049:   \medskip The difficulty in these proofs is to ensure that the
wenzelm@11049:   induction hypotheses are applied before the definition of @{term
wenzelm@11049:   fib}.  Towards this end, the @{term fib} equations are not declared
wenzelm@11049:   to the Simplifier and are applied very selectively at first.
wenzelm@11049: *}
wenzelm@11049: 
paulson@24549: text{*We disable @{text fib.fib_2fib_2} for simplification ...*}
paulson@24549: declare fib_2 [simp del]
wenzelm@11049: 
paulson@15628: text{*...then prove a version that has a more restrictive pattern.*}
wenzelm@11049: lemma fib_Suc3: "fib (Suc (Suc (Suc n))) = fib (Suc n) + fib (Suc (Suc n))"
paulson@24549:   by (rule fib_2)
wenzelm@11049: 
wenzelm@11049: text {* \medskip Concrete Mathematics, page 280 *}
wenzelm@11049: 
wenzelm@11049: lemma fib_add: "fib (Suc (n + k)) = fib (Suc k) * fib (Suc n) + fib k * fib n"
paulson@24573: proof (induct n rule: fib.induct)
paulson@24573:   case 1 show ?case by simp
paulson@24573: next
paulson@24573:   case 2 show ?case  by (simp add: fib_2)
paulson@24573: next
paulson@24573:   case 3 thus ?case by (simp add: fib_2 add_mult_distrib2)
paulson@24573: qed
wenzelm@11049: 
paulson@15628: lemma fib_Suc_neq_0: "fib (Suc n) \<noteq> 0"
wenzelm@11049:   apply (induct n rule: fib.induct)
paulson@24549:     apply (simp_all add: fib_2)
wenzelm@11049:   done
wenzelm@11049: 
paulson@15628: lemma fib_Suc_gr_0: "0 < fib (Suc n)"
paulson@15628:   by (insert fib_Suc_neq_0 [of n], simp)  
wenzelm@11049: 
wenzelm@11049: lemma fib_gr_0: "0 < n ==> 0 < fib n"
paulson@15628:   by (case_tac n, auto simp add: fib_Suc_gr_0) 
wenzelm@11049: 
paulson@9944: 
wenzelm@11049: text {*
paulson@15628:   \medskip Concrete Mathematics, page 278: Cassini's identity.  The proof is
paulson@15628:   much easier using integers, not natural numbers!
wenzelm@11049: *}
wenzelm@11049: 
paulson@15628: lemma fib_Cassini_int:
paulson@15003:  "int (fib (Suc (Suc n)) * fib n) =
paulson@11868:   (if n mod 2 = 0 then int (fib (Suc n) * fib (Suc n)) - 1
paulson@11868:    else int (fib (Suc n) * fib (Suc n)) + 1)"
chaieb@23315: proof(induct n rule: fib.induct)
paulson@24549:   case 1 thus ?case by (simp add: fib_2)
chaieb@23315: next
paulson@24549:   case 2 thus ?case by (simp add: fib_2 mod_Suc)
chaieb@23315: next 
chaieb@23315:   case (3 x) 
paulson@24573:   have "Suc 0 \<noteq> x mod 2 \<longrightarrow> x mod 2 = 0" by presburger
paulson@24573:   with "3.hyps" show ?case by (simp add: fib_2 add_mult_distrib add_mult_distrib2)
chaieb@23315: qed
paulson@15628: 
paulson@15628: text{*We now obtain a version for the natural numbers via the coercion 
paulson@15628:    function @{term int}.*}
paulson@15628: theorem fib_Cassini:
paulson@15628:  "fib (Suc (Suc n)) * fib n =
paulson@15628:   (if n mod 2 = 0 then fib (Suc n) * fib (Suc n) - 1
paulson@15628:    else fib (Suc n) * fib (Suc n) + 1)"
paulson@15628:   apply (rule int_int_eq [THEN iffD1]) 
huffman@23431:   apply (simp add: fib_Cassini_int)
paulson@15628:   apply (subst zdiff_int [symmetric]) 
paulson@15628:    apply (insert fib_Suc_gr_0 [of n], simp_all)
wenzelm@11049:   done
wenzelm@11049: 
wenzelm@11049: 
paulson@15628: text {* \medskip Toward Law 6.111 of Concrete Mathematics *}
wenzelm@11049: 
wenzelm@11701: lemma gcd_fib_Suc_eq_1: "gcd (fib n, fib (Suc n)) = Suc 0"
wenzelm@11049:   apply (induct n rule: fib.induct)
wenzelm@11049:     prefer 3
wenzelm@11049:     apply (simp add: gcd_commute fib_Suc3)
paulson@24549:    apply (simp_all add: fib_2)
wenzelm@11049:   done
wenzelm@11049: 
wenzelm@11049: lemma gcd_fib_add: "gcd (fib m, fib (n + m)) = gcd (fib m, fib n)"
paulson@15628:   apply (simp add: gcd_commute [of "fib m"])
paulson@15628:   apply (case_tac m)
paulson@15628:    apply simp 
wenzelm@11049:   apply (simp add: fib_add)
paulson@15628:   apply (simp add: add_commute gcd_non_0 [OF fib_Suc_gr_0])
paulson@15628:   apply (simp add: gcd_non_0 [OF fib_Suc_gr_0, symmetric])
wenzelm@11049:   apply (simp add: gcd_fib_Suc_eq_1 gcd_mult_cancel)
wenzelm@11049:   done
wenzelm@11049: 
wenzelm@11049: lemma gcd_fib_diff: "m \<le> n ==> gcd (fib m, fib (n - m)) = gcd (fib m, fib n)"
paulson@15628:   by (simp add: gcd_fib_add [symmetric, of _ "n-m"]) 
wenzelm@11049: 
wenzelm@11049: lemma gcd_fib_mod: "0 < m ==> gcd (fib m, fib (n mod m)) = gcd (fib m, fib n)"
haftmann@23688: proof (induct n rule: less_induct)
haftmann@23688:   case (less n)
haftmann@23688:   from less.prems have pos_m: "0 < m" .
haftmann@23688:   show "gcd (fib m, fib (n mod m)) = gcd (fib m, fib n)"
haftmann@23688:   proof (cases "m < n")
haftmann@23688:     case True note m_n = True
haftmann@23688:     then have m_n': "m \<le> n" by auto
haftmann@23688:     with pos_m have pos_n: "0 < n" by auto
haftmann@23688:     with pos_m m_n have diff: "n - m < n" by auto
haftmann@23688:     have "gcd (fib m, fib (n mod m)) = gcd (fib m, fib ((n - m) mod m))"
haftmann@23688:     by (simp add: mod_if [of n]) (insert m_n, auto)
haftmann@23688:     also have "\<dots> = gcd (fib m, fib (n - m))" by (simp add: less.hyps diff pos_m)
haftmann@23688:     also have "\<dots> = gcd (fib m, fib n)" by (simp add: gcd_fib_diff m_n')
haftmann@23688:     finally show "gcd (fib m, fib (n mod m)) = gcd (fib m, fib n)" .
haftmann@23688:   next
haftmann@23688:     case False then show "gcd (fib m, fib (n mod m)) = gcd (fib m, fib n)"
haftmann@23688:     by (cases "m = n") auto
haftmann@23688:   qed
haftmann@23688: qed
wenzelm@11049: 
wenzelm@11049: lemma fib_gcd: "fib (gcd (m, n)) = gcd (fib m, fib n)"  -- {* Law 6.111 *}
wenzelm@11049:   apply (induct m n rule: gcd_induct)
paulson@15628:   apply (simp_all add: gcd_non_0 gcd_commute gcd_fib_mod)
wenzelm@11049:   done
wenzelm@11049: 
paulson@15628: theorem fib_mult_eq_setsum:
wenzelm@11786:     "fib (Suc n) * fib n = (\<Sum>k \<in> {..n}. fib k * fib k)"
wenzelm@11049:   apply (induct n rule: fib.induct)
paulson@24549:     apply (auto simp add: atMost_Suc fib_2)
wenzelm@11049:   apply (simp add: add_mult_distrib add_mult_distrib2)
wenzelm@11049:   done
paulson@9944: 
paulson@9944: end