# Fibonacci Numbers #

This file defines the fibonacci series, proves results about it and introduces methods to compute it quickly.

# The Fibonacci Sequence #

## Summary #

Definition of the Fibonacci sequence F₀ = 0, F₁ = 1, Fₙ₊₂ = Fₙ + Fₙ₊₁.

## Main Definitions #

• Nat.fib returns the stream of Fibonacci numbers.

## Main Statements #

• Nat.fib_add_two: shows that fib indeed satisfies the Fibonacci recurrence Fₙ₊₂ = Fₙ + Fₙ₊₁..
• Nat.fib_gcd: fib n is a strong divisibility sequence.
• Nat.fib_succ_eq_sum_choose: fib is given by the sum of Nat.choose along an antidiagonal.
• Nat.fib_succ_eq_succ_sum: shows that F₀ + F₁ + ⋯ + Fₙ = Fₙ₊₂ - 1.
• Nat.fib_two_mul and Nat.fib_two_mul_add_one are the basis for an efficient algorithm to compute fib (see Nat.fastFib). There are bit0/bit1 variants of these can be used to simplify fib expressions: simp only [Nat.fib_bit0, Nat.fib_bit1, Nat.fib_bit0_succ, Nat.fib_bit1_succ, Nat.fib_one, Nat.fib_two].

## Implementation Notes #

For efficiency purposes, the sequence is defined using Stream.iterate.

## Tags #

fib, fibonacci

def Nat.fib (n : ) :

Implementation of the fibonacci sequence satisfying fib 0 = 0, fib 1 = 1, fib (n + 2) = fib n + fib (n + 1).

Note: We use a stream iterator for better performance when compared to the naive recursive implementation.

Equations
• = ((fun (p : ) => (p.2, p.1 + p.2))^[n] (0, 1)).1
Instances For
@[simp]
theorem Nat.fib_zero :
= 0
@[simp]
theorem Nat.fib_one :
= 1
@[simp]
theorem Nat.fib_two :
= 1
theorem Nat.fib_add_two {n : } :
Nat.fib (n + 2) = + Nat.fib (n + 1)

Shows that fib indeed satisfies the Fibonacci recurrence Fₙ₊₂ = Fₙ + Fₙ₊₁.

theorem Nat.fib_add_one {n : } :
n 0Nat.fib (n + 1) = Nat.fib (n - 1) +
theorem Nat.fib_le_fib_succ {n : } :
Nat.fib (n + 1)
@[simp]
theorem Nat.fib_eq_zero {n : } :
= 0 n = 0
@[simp]
theorem Nat.fib_pos {n : } :
0 < 0 < n
theorem Nat.fib_lt_fib_succ {n : } (hn : 2 n) :
< Nat.fib (n + 1)
StrictMono fun (n : ) => Nat.fib (n + 2)

fib (n + 2) is strictly monotone.

theorem Nat.fib_lt_fib {m : } (hm : 2 m) {n : } :
< m < n
theorem Nat.le_fib_self {n : } (five_le_n : 5 n) :
n
theorem Nat.le_fib_add_one (n : ) :
n + 1
theorem Nat.fib_coprime_fib_succ (n : ) :
().Coprime (Nat.fib (n + 1))

Subsequent Fibonacci numbers are coprime, see https://proofwiki.org/wiki/Consecutive_Fibonacci_Numbers_are_Coprime

theorem Nat.fib_add (m : ) (n : ) :
Nat.fib (m + n + 1) = * + Nat.fib (m + 1) * Nat.fib (n + 1)
theorem Nat.fib_two_mul (n : ) :
Nat.fib (2 * n) = * (2 * Nat.fib (n + 1) - )
theorem Nat.fib_two_mul_add_one (n : ) :
Nat.fib (2 * n + 1) = Nat.fib (n + 1) ^ 2 + ^ 2
theorem Nat.fib_two_mul_add_two (n : ) :
Nat.fib (2 * n + 2) = Nat.fib (n + 1) * (2 * + Nat.fib (n + 1))

Computes (Nat.fib n, Nat.fib (n + 1)) using the binary representation of n. Supports Nat.fastFib.

Equations
• One or more equations did not get rendered due to their size.
Instances For
def Nat.fastFib (n : ) :

Computes Nat.fib n using the binary representation of n. Proved to be equal to Nat.fib in Nat.fast_fib_eq.

Equations
• n.fastFib = n.fastFibAux.1
Instances For
theorem Nat.fast_fib_aux_bit_ff (n : ) :
().fastFibAux = let p := n.fastFibAux; (p.1 * (2 * p.2 - p.1), p.2 ^ 2 + p.1 ^ 2)
theorem Nat.fast_fib_aux_bit_tt (n : ) :
().fastFibAux = let p := n.fastFibAux; (p.2 ^ 2 + p.1 ^ 2, p.2 * (2 * p.1 + p.2))
theorem Nat.fast_fib_aux_eq (n : ) :
n.fastFibAux = (, Nat.fib (n + 1))
theorem Nat.fast_fib_eq (n : ) :
n.fastFib =
theorem Nat.gcd_fib_add_self (m : ) (n : ) :
().gcd (Nat.fib (n + m)) = ().gcd ()
theorem Nat.gcd_fib_add_mul_self (m : ) (n : ) (k : ) :
().gcd (Nat.fib (n + k * m)) = ().gcd ()
theorem Nat.fib_gcd (m : ) (n : ) :
Nat.fib (m.gcd n) = ().gcd ()

fib n is a strong divisibility sequence, see https://proofwiki.org/wiki/GCD_of_Fibonacci_Numbers

theorem Nat.fib_dvd (m : ) (n : ) (h : m n) :
theorem Nat.fib_succ_eq_sum_choose (n : ) :
Nat.fib (n + 1) = p, p.1.choose p.2
theorem Nat.fib_succ_eq_succ_sum (n : ) :
Nat.fib (n + 1) = k, + 1