change simp rules for of_nat to work like int did previously (reorient of_nat_Suc, remove of_nat_mult [simp]); preserve original variable names in legacy int theorems
(* 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
*}
consts fib :: "nat => nat"
recdef fib "measure (\<lambda>x. 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) \<noteq> 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)"
proof(induct n rule: fib.induct)
case 1 thus ?case by (simp add: fib.Suc_Suc)
next
case 2 thus ?case by (simp add: fib.Suc_Suc mod_Suc)
next
case (3 x)
have th: "Suc 0 \<noteq> x mod 2 \<longrightarrow> x mod 2 = 0" by presburger
with prems show ?case by (simp add: fib.Suc_Suc add_mult_distrib add_mult_distrib2
mod_Suc zmult_int [symmetric])
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.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 \<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)"
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 = (\<Sum>k \<in> {..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