theorem Nat.nextPowerOfTwo_dec {n power : Nat} (h₁ : power > 0) (h₂ : power < n) : n - power * 2 < n - power

def Nat.nextPowerOfTwo (n : Nat) : Nat

Equations

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

@[irreducible]

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

Equations

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

def Nat.isPowerOfTwo (n : Nat) : Prop

Equations

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

theorem Nat.one_isPowerOfTwo : Nat.isPowerOfTwo 1

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) : (Nat.nextPowerOfTwo.go n power h₁).isPowerOfTwo