--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Number_Theory/Fib.thy Tue Sep 01 15:39:33 2009 +0200
@@ -0,0 +1,319 @@
+(* Title: Fib.thy
+ Authors: Lawrence C. Paulson, Jeremy Avigad
+
+
+Defines the fibonacci function.
+
+The original "Fib" is due to Lawrence C. Paulson, and was adapted by
+Jeremy Avigad.
+*)
+
+
+header {* Fib *}
+
+theory Fib
+imports Binomial
+begin
+
+
+subsection {* Main definitions *}
+
+class fib =
+
+fixes
+ fib :: "'a \<Rightarrow> 'a"
+
+
+(* definition for the natural numbers *)
+
+instantiation nat :: fib
+
+begin
+
+fun
+ fib_nat :: "nat \<Rightarrow> nat"
+where
+ "fib_nat n =
+ (if n = 0 then 0 else
+ (if n = 1 then 1 else
+ fib (n - 1) + fib (n - 2)))"
+
+instance proof qed
+
+end
+
+(* definition for the integers *)
+
+instantiation int :: fib
+
+begin
+
+definition
+ fib_int :: "int \<Rightarrow> int"
+where
+ "fib_int n = (if n >= 0 then int (fib (nat n)) else 0)"
+
+instance proof qed
+
+end
+
+
+subsection {* Set up Transfer *}
+
+
+lemma transfer_nat_int_fib:
+ "(x::int) >= 0 \<Longrightarrow> fib (nat x) = nat (fib x)"
+ unfolding fib_int_def by auto
+
+lemma transfer_nat_int_fib_closure:
+ "n >= (0::int) \<Longrightarrow> fib n >= 0"
+ by (auto simp add: fib_int_def)
+
+declare TransferMorphism_nat_int[transfer add return:
+ transfer_nat_int_fib transfer_nat_int_fib_closure]
+
+lemma transfer_int_nat_fib:
+ "fib (int n) = int (fib n)"
+ unfolding fib_int_def by auto
+
+lemma transfer_int_nat_fib_closure:
+ "is_nat n \<Longrightarrow> fib n >= 0"
+ unfolding fib_int_def by auto
+
+declare TransferMorphism_int_nat[transfer add return:
+ transfer_int_nat_fib transfer_int_nat_fib_closure]
+
+
+subsection {* Fibonacci numbers *}
+
+lemma fib_0_nat [simp]: "fib (0::nat) = 0"
+ by simp
+
+lemma fib_0_int [simp]: "fib (0::int) = 0"
+ unfolding fib_int_def by simp
+
+lemma fib_1_nat [simp]: "fib (1::nat) = 1"
+ by simp
+
+lemma fib_Suc_0_nat [simp]: "fib (Suc 0) = Suc 0"
+ by simp
+
+lemma fib_1_int [simp]: "fib (1::int) = 1"
+ unfolding fib_int_def by simp
+
+lemma fib_reduce_nat: "(n::nat) >= 2 \<Longrightarrow> fib n = fib (n - 1) + fib (n - 2)"
+ by simp
+
+declare fib_nat.simps [simp del]
+
+lemma fib_reduce_int: "(n::int) >= 2 \<Longrightarrow> fib n = fib (n - 1) + fib (n - 2)"
+ unfolding fib_int_def
+ by (auto simp add: fib_reduce_nat nat_diff_distrib)
+
+lemma fib_neg_int [simp]: "(n::int) < 0 \<Longrightarrow> fib n = 0"
+ unfolding fib_int_def by auto
+
+lemma fib_2_nat [simp]: "fib (2::nat) = 1"
+ by (subst fib_reduce_nat, auto)
+
+lemma fib_2_int [simp]: "fib (2::int) = 1"
+ by (subst fib_reduce_int, auto)
+
+lemma fib_plus_2_nat: "fib ((n::nat) + 2) = fib (n + 1) + fib n"
+ by (subst fib_reduce_nat, auto simp add: One_nat_def)
+(* the need for One_nat_def is due to the natdiff_cancel_numerals
+ procedure *)
+
+lemma fib_induct_nat: "P (0::nat) \<Longrightarrow> P (1::nat) \<Longrightarrow>
+ (!!n. P n \<Longrightarrow> P (n + 1) \<Longrightarrow> P (n + 2)) \<Longrightarrow> P n"
+ apply (atomize, induct n rule: nat_less_induct)
+ apply auto
+ apply (case_tac "n = 0", force)
+ apply (case_tac "n = 1", force)
+ apply (subgoal_tac "n >= 2")
+ apply (frule_tac x = "n - 1" in spec)
+ apply (drule_tac x = "n - 2" in spec)
+ apply (drule_tac x = "n - 2" in spec)
+ apply auto
+ apply (auto simp add: One_nat_def) (* again, natdiff_cancel *)
+done
+
+lemma fib_add_nat: "fib ((n::nat) + k + 1) = fib (k + 1) * fib (n + 1) +
+ fib k * fib n"
+ apply (induct n rule: fib_induct_nat)
+ apply auto
+ apply (subst fib_reduce_nat)
+ apply (auto simp add: ring_simps)
+ apply (subst (1 3 5) fib_reduce_nat)
+ apply (auto simp add: ring_simps Suc_eq_plus1)
+(* hmmm. Why doesn't "n + (1 + (1 + k))" simplify to "n + k + 2"? *)
+ apply (subgoal_tac "n + (k + 2) = n + (1 + (1 + k))")
+ apply (erule ssubst) back back
+ apply (erule ssubst) back
+ apply auto
+done
+
+lemma fib_add'_nat: "fib (n + Suc k) = fib (Suc k) * fib (Suc n) +
+ fib k * fib n"
+ using fib_add_nat by (auto simp add: One_nat_def)
+
+
+(* transfer from nats to ints *)
+lemma fib_add_int [rule_format]: "(n::int) >= 0 \<Longrightarrow> k >= 0 \<Longrightarrow>
+ fib (n + k + 1) = fib (k + 1) * fib (n + 1) +
+ fib k * fib n "
+
+ by (rule fib_add_nat [transferred])
+
+lemma fib_neq_0_nat: "(n::nat) > 0 \<Longrightarrow> fib n ~= 0"
+ apply (induct n rule: fib_induct_nat)
+ apply (auto simp add: fib_plus_2_nat)
+done
+
+lemma fib_gr_0_nat: "(n::nat) > 0 \<Longrightarrow> fib n > 0"
+ by (frule fib_neq_0_nat, simp)
+
+lemma fib_gr_0_int: "(n::int) > 0 \<Longrightarrow> fib n > 0"
+ unfolding fib_int_def by (simp add: fib_gr_0_nat)
+
+text {*
+ \medskip Concrete Mathematics, page 278: Cassini's identity. The proof is
+ much easier using integers, not natural numbers!
+*}
+
+lemma fib_Cassini_aux_int: "fib (int n + 2) * fib (int n) -
+ (fib (int n + 1))^2 = (-1)^(n + 1)"
+ apply (induct n)
+ apply (auto simp add: ring_simps power2_eq_square fib_reduce_int
+ power_add)
+done
+
+lemma fib_Cassini_int: "n >= 0 \<Longrightarrow> fib (n + 2) * fib n -
+ (fib (n + 1))^2 = (-1)^(nat n + 1)"
+ by (insert fib_Cassini_aux_int [of "nat n"], auto)
+
+(*
+lemma fib_Cassini'_int: "n >= 0 \<Longrightarrow> fib (n + 2) * fib n =
+ (fib (n + 1))^2 + (-1)^(nat n + 1)"
+ by (frule fib_Cassini_int, simp)
+*)
+
+lemma fib_Cassini'_int: "n >= 0 \<Longrightarrow> fib ((n::int) + 2) * fib n =
+ (if even n then tsub ((fib (n + 1))^2) 1
+ else (fib (n + 1))^2 + 1)"
+ apply (frule fib_Cassini_int, auto simp add: pos_int_even_equiv_nat_even)
+ apply (subst tsub_eq)
+ apply (insert fib_gr_0_int [of "n + 1"], force)
+ apply auto
+done
+
+lemma fib_Cassini_nat: "fib ((n::nat) + 2) * fib n =
+ (if even n then (fib (n + 1))^2 - 1
+ else (fib (n + 1))^2 + 1)"
+
+ by (rule fib_Cassini'_int [transferred, of n], auto)
+
+
+text {* \medskip Toward Law 6.111 of Concrete Mathematics *}
+
+lemma coprime_fib_plus_1_nat: "coprime (fib (n::nat)) (fib (n + 1))"
+ apply (induct n rule: fib_induct_nat)
+ apply auto
+ apply (subst (2) fib_reduce_nat)
+ apply (auto simp add: Suc_eq_plus1) (* again, natdiff_cancel *)
+ apply (subst add_commute, auto)
+ apply (subst gcd_commute_nat, auto simp add: ring_simps)
+done
+
+lemma coprime_fib_Suc_nat: "coprime (fib n) (fib (Suc n))"
+ using coprime_fib_plus_1_nat by (simp add: One_nat_def)
+
+lemma coprime_fib_plus_1_int:
+ "n >= 0 \<Longrightarrow> coprime (fib (n::int)) (fib (n + 1))"
+ by (erule coprime_fib_plus_1_nat [transferred])
+
+lemma gcd_fib_add_nat: "gcd (fib (m::nat)) (fib (n + m)) = gcd (fib m) (fib n)"
+ apply (simp add: gcd_commute_nat [of "fib m"])
+ apply (rule cases_nat [of _ m])
+ apply simp
+ apply (subst add_assoc [symmetric])
+ apply (simp add: fib_add_nat)
+ apply (subst gcd_commute_nat)
+ apply (subst mult_commute)
+ apply (subst gcd_add_mult_nat)
+ apply (subst gcd_commute_nat)
+ apply (rule gcd_mult_cancel_nat)
+ apply (rule coprime_fib_plus_1_nat)
+done
+
+lemma gcd_fib_add_int [rule_format]: "m >= 0 \<Longrightarrow> n >= 0 \<Longrightarrow>
+ gcd (fib (m::int)) (fib (n + m)) = gcd (fib m) (fib n)"
+ by (erule gcd_fib_add_nat [transferred])
+
+lemma gcd_fib_diff_nat: "(m::nat) \<le> n \<Longrightarrow>
+ gcd (fib m) (fib (n - m)) = gcd (fib m) (fib n)"
+ by (simp add: gcd_fib_add_nat [symmetric, of _ "n-m"])
+
+lemma gcd_fib_diff_int: "0 <= (m::int) \<Longrightarrow> m \<le> n \<Longrightarrow>
+ gcd (fib m) (fib (n - m)) = gcd (fib m) (fib n)"
+ by (simp add: gcd_fib_add_int [symmetric, of _ "n-m"])
+
+lemma gcd_fib_mod_nat: "0 < (m::nat) \<Longrightarrow>
+ 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_nat 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 gcd_fib_mod_int:
+ assumes "0 < (m::int)" and "0 <= n"
+ shows "gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib n)"
+
+ apply (rule gcd_fib_mod_nat [transferred])
+ using prems apply auto
+done
+
+lemma fib_gcd_nat: "fib (gcd (m::nat) n) = gcd (fib m) (fib n)"
+ -- {* Law 6.111 *}
+ apply (induct m n rule: gcd_nat_induct)
+ apply (simp_all add: gcd_non_0_nat gcd_commute_nat gcd_fib_mod_nat)
+done
+
+lemma fib_gcd_int: "m >= 0 \<Longrightarrow> n >= 0 \<Longrightarrow>
+ fib (gcd (m::int) n) = gcd (fib m) (fib n)"
+ by (erule fib_gcd_nat [transferred])
+
+lemma atMost_plus_one_nat: "{..(k::nat) + 1} = insert (k + 1) {..k}"
+ by auto
+
+theorem fib_mult_eq_setsum_nat:
+ "fib ((n::nat) + 1) * fib n = (\<Sum>k \<in> {..n}. fib k * fib k)"
+ apply (induct n)
+ apply (auto simp add: atMost_plus_one_nat fib_plus_2_nat ring_simps)
+done
+
+theorem fib_mult_eq_setsum'_nat:
+ "fib (Suc n) * fib n = (\<Sum>k \<in> {..n}. fib k * fib k)"
+ using fib_mult_eq_setsum_nat by (simp add: One_nat_def)
+
+theorem fib_mult_eq_setsum_int [rule_format]:
+ "n >= 0 \<Longrightarrow> fib ((n::int) + 1) * fib n = (\<Sum>k \<in> {0..n}. fib k * fib k)"
+ by (erule fib_mult_eq_setsum_nat [transferred])
+
+end