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+++ b/src/HOL/Old_Number_Theory/Fib.thy Tue Sep 01 15:39:33 2009 +0200
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+(* ID: $Id$
+ 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 \<Rightarrow> 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) \<noteq> 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 \<noteq> x mod 2 \<longrightarrow> 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 \<le> 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 \<le> 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 "\<dots> = gcd (fib m) (fib (n - m))" by (simp add: less.hyps diff pos_m)
+ also have "\<dots> = 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 = (\<Sum>k \<in> {..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