Documentation

Init.Data.Nat.Power2

theorem Nat.nextPowerOfTwo_dec {n power : Nat} (h₁ : power > 0) (h₂ : power < n) :

n - power * 2 < n - power

def Nat.nextPowerOfTwo (n : Nat) :

Returns the least power of two that's greater than or equal to n.

Examples:

Equations

n.nextPowerOfTwo = Nat.nextPowerOfTwo.go n 1 Nat.nextPowerOfTwo.proof_1

@[irreducible]

def Nat.nextPowerOfTwo.go (n power : Nat) (h : power > 0) :

Equations

Nat.nextPowerOfTwo.go n power h = if power < n then Nat.nextPowerOfTwo.go n (power * 2) ⋯ else power

def Nat.isPowerOfTwo (n : Nat) :

A natural number n is a power of two if there exists some k : Nat such that n = 2 ^ k.

Equations

n.isPowerOfTwo = ∃ (k : Nat ), n = 2 ^ k

theorem Nat.isPowerOfTwo_one :

@[reducible, inline, deprecated Nat.isPowerOfTwo_one (since := "2025-03-18")]

abbrev Nat.one_isPowerOfTwo :

Equations

Nat.one_isPowerOfTwo = Nat.isPowerOfTwo_one

theorem Nat.mul2_isPowerOfTwo_of_isPowerOfTwo {n : Nat} (h : n.isPowerOfTwo) :

(n * 2).isPowerOfTwo

theorem Nat.pos_of_isPowerOfTwo {n : Nat} (h : n.isPowerOfTwo) :

n > 0

theorem Nat.isPowerOfTwo_nextPowerOfTwo (n : Nat) :

n.nextPowerOfTwo.isPowerOfTwo

@[irreducible]

theorem Nat.isPowerOfTwo_nextPowerOfTwo.isPowerOfTwo_go (n power : Nat) (h₁ : power > 0) (h₂ : power.isPowerOfTwo) :

(nextPowerOfTwo.go n power h₁).isPowerOfTwo