Theory Fib

theory Fib
imports Binomial
(*  Title:      HOL/Number_Theory/Fib.thy
Author: Lawrence C. Paulson
Author: 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 => 'a"


(* definition for the natural numbers *)

instantiation nat :: fib
begin

fun fib_nat :: "nat => nat"
where
"fib_nat n =
(if n = 0 then 0 else
(if n = 1 then 1 else
fib (n - 1) + fib (n - 2)))"


instance ..

end

(* definition for the integers *)

instantiation int :: fib
begin

definition fib_int :: "int => int"
where "fib_int n = (if n >= 0 then int (fib (nat n)) else 0)"

instance ..

end


subsection {* Set up Transfer *}

lemma transfer_nat_int_fib:
"(x::int) >= 0 ==> fib (nat x) = nat (fib x)"
unfolding fib_int_def by auto

lemma transfer_nat_int_fib_closure:
"n >= (0::int) ==> fib n >= 0"
by (auto simp add: fib_int_def)

declare transfer_morphism_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 ==> fib n >= 0"
unfolding fib_int_def by auto

declare transfer_morphism_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 ==> fib n = fib (n - 1) + fib (n - 2)"
by simp

declare fib_nat.simps [simp del]

lemma fib_reduce_int: "(n::int) >= 2 ==> 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 ==> 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) ==> P (1::nat) ==>
(!!n. P n ==> P (n + 1) ==> P (n + 2)) ==> 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: field_simps)
apply (subst (1 3 5) fib_reduce_nat)
apply (auto simp add: field_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: "(n::int) >= 0 ==> k >= 0 ==>
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 ==> 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 ==> fib n > 0"
by (frule fib_neq_0_nat, simp)

lemma fib_gr_0_int: "(n::int) > 0 ==> 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: field_simps power2_eq_square fib_reduce_int power_add)
done

lemma fib_Cassini_int: "n >= 0 ==> 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 ==> 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 ==> 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: field_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 ==> 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 ==> n >= 0 ==>
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) ≤ n ==>
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) ==> m ≤ n ==>
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) ==>
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
then have "m ≤ 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 "… = gcd (fib m) (fib (n - m))"
by (simp add: less.hyps diff pos_m)
also have "… = 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 assms 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 ==> n >= 0 ==>
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 = (∑k ∈ {..n}. fib k * fib k)"
apply (induct n)
apply (auto simp add: atMost_plus_one_nat fib_plus_2_nat field_simps)
done

theorem fib_mult_eq_setsum'_nat:
"fib (Suc n) * fib n = (∑k ∈ {..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 ==> fib ((n::int) + 1) * fib n = (∑k ∈ {0..n}. fib k * fib k)"
by (erule fib_mult_eq_setsum_nat [transferred])

end