twoPowZ

formal source

  34      simpa [B_of_succ, mul_comm] using this
  35
  36/-- Two to an integer power: 2^k for k ∈ ℤ. -/
  37noncomputable def twoPowZ (k : Int) : ℝ :=
  38  if 0 ≤ k then (2 : ℝ) ^ (Int.toNat k)
  39  else 1 / ((2 : ℝ) ^ (Int.toNat (-k)))
  40
  41@[simp] lemma twoPowZ_zero : twoPowZ 0 = 1 := by simp [twoPowZ]
  42@[simp] lemma twoPowZ_ofNat (k : Nat) : twoPowZ (Int.ofNat k) = (2 : ℝ) ^ k := by simp [twoPowZ]
  43@[simp] lemma twoPowZ_negSucc (k : Nat) : twoPowZ (Int.negSucc k) = 1 / ((2 : ℝ) ^ k.succ) := by
  44  simp [twoPowZ]
  45
  46/-- φ-power wrapper. -/
  47noncomputable def PhiPow (x : ℝ) : ℝ := Real.exp (Real.log (Constants.phi) * x)
  48
  49lemma PhiPow_add (x y : ℝ) : PhiPow (x + y) = PhiPow x * PhiPow y := by
  50  unfold PhiPow
  51  have hx : Real.log (Constants.phi) * (x + y)
  52        = Real.log (Constants.phi) * x + Real.log (Constants.phi) * y := by
  53    ring
  54  simp [hx, Real.exp_add]
  55
  56lemma PhiPow_sub (x y : ℝ) : PhiPow (x - y) = PhiPow x / PhiPow y := by
  57  unfold PhiPow
  58  have hx : Real.log (Constants.phi) * (x - y)
  59        = Real.log (Constants.phi) * x + Real.log (Constants.phi) * (-y) := by
  60    ring
  61  calc
  62    Real.exp (Real.log (Constants.phi) * (x - y))
  63        = Real.exp (Real.log (Constants.phi) * x + Real.log (Constants.phi) * (-y)) := by
  64              simp [hx]
  65    _   = Real.exp (Real.log (Constants.phi) * x)
  66            * Real.exp (Real.log (Constants.phi) * (-y)) := by
  67              simp [Real.exp_add]