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).

Implementation Notes

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

Tags

fib, fibonacci

def Nat.fib (n : ℕ) : 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

• Nat.fib n = ((fun (p : ℕ × ℕ) => (p.2, p.1 + p.2))^[n] (0, 1)).1

@[simp]

theorem Nat.fib_zero : fib 0 = 0

@[simp]

theorem Nat.fib_one : fib 1 = 1

@[simp]

theorem Nat.fib_two : fib 2 = 1

theorem Nat.fib_add_two {n : ℕ} : fib (n + 2) = fib n + fib (n + 1)

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

theorem Nat.fib_add_one {n : ℕ} : n ≠ 0 → fib (n + 1) = fib (n - 1) + fib n

theorem Nat.fib_le_fib_succ {n : ℕ} : fib n ≤ fib (n + 1)

theorem Nat.fib_mono : Monotone fib

@[simp]

theorem Nat.fib_eq_zero {n : ℕ} : fib n = 0 ↔ n = 0

@[simp]

theorem Nat.fib_pos {n : ℕ} : 0 < fib n ↔ 0 < n

theorem Nat.fib_add_two_sub_fib_add_one {n : ℕ} : fib (n + 2) - fib (n + 1) = fib n

theorem Nat.fib_lt_fib_succ {n : ℕ} (hn : 2 ≤ n) : fib n < fib (n + 1)

theorem Nat.fib_add_two_strictMono : StrictMono fun (n : ℕ) => fib (n + 2)

fib (n + 2) is strictly monotone.

theorem Nat.fib_strictMonoOn : StrictMonoOn fib (Set.Ici 2)

theorem Nat.fib_lt_fib {m : ℕ} (hm : 2 ≤ m) {n : ℕ} : fib m < fib n ↔ m < n

theorem Nat.le_fib_self {n : ℕ} (five_le_n : 5 ≤ n) : n ≤ fib n

theorem Nat.le_fib_add_one (n : ℕ) : n ≤ fib n + 1

theorem Nat.fib_coprime_fib_succ (n : ℕ) : (fib n).Coprime (fib (n + 1))

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

theorem Nat.fib_add (m n : ℕ) : fib (m + n + 1) = fib m * fib n + fib (m + 1) * fib (n + 1)

See https://proofwiki.org/wiki/Fibonacci_Number_in_terms_of_Smaller_Fibonacci_Numbers

theorem Nat.fib_two_mul (n : ℕ) : fib (2 * n) = fib n * (2 * fib (n + 1) - fib n)

theorem Nat.fib_two_mul_add_one (n : ℕ) : fib (2 * n + 1) = fib (n + 1) ^ 2 + fib n ^ 2

theorem Nat.fib_two_mul_add_two (n : ℕ) : fib (2 * n + 2) = fib (n + 1) * (2 * fib n + fib (n + 1))

def Nat.fastFibAux : ℕ → ℕ × ℕ

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.

def Nat.fastFib (n : ℕ) : 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

theorem Nat.fast_fib_aux_bit_ff (n : ℕ) : (bit false 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 : ℕ) : (bit true 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) : n.fastFibAux = (fib n, fib (n + 1))

theorem Nat.fast_fib_eq (n : ℕ) : n.fastFib = fib n

theorem Nat.gcd_fib_add_self (m n : ℕ) : (fib m).gcd (fib (n + m)) = (fib m).gcd (fib n)

theorem Nat.gcd_fib_add_mul_self (m n k : ℕ) : (fib m).gcd (fib (n + k * m)) = (fib m).gcd (fib n)

theorem Nat.fib_gcd (m n : ℕ) : fib (m.gcd n) = (fib m).gcd (fib n)

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

theorem Nat.fib_dvd (m n : ℕ) (h : m ∣ n) : fib m ∣ fib n

theorem Nat.fib_succ_eq_sum_choose (n : ℕ) : fib (n + 1) = ∑ p ∈ Finset.antidiagonal n, p.1.choose p.2

theorem Nat.fib_succ_eq_succ_sum (n : ℕ) : fib (n + 1) = ∑ k ∈ Finset.range n, fib k + 1