`toZ_fromZ`

proved

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module: IndisputableMonolith.UnitMapping
domain: UnitMapping
line: 16 · github
papers citing: none yet

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IndisputableMonolith.UnitMapping on GitHub at line 16.

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formal source

  13@[simp] def fromZ (δ : ℤ) (n : ℤ) : DeltaSub δ := n
  14@[simp] def toZ   (δ : ℤ) (p : DeltaSub δ) : ℤ := p
  15
  16@[simp] lemma toZ_fromZ (δ : ℤ) (hδ : δ ≠ 0) (n : ℤ) :
  17  toZ δ (fromZ δ n) = n := rfl
  18
  19end LedgerUnits
  20
  21open LedgerUnits
  22
  23/-- Affine map from ℤ to ℝ: n ↦ slope·n + offset. -/
  24structure AffineMapZ where
  25  slope : ℝ
  26  offset : ℝ
  27
  28@[simp] def apply (f : AffineMapZ) (n : ℤ) : ℝ := f.slope * (n : ℝ) + f.offset
  29
  30/-- Map δ-subgroup to ℝ by composing the (stubbed) projection `toZ` with an affine map. -/
  31noncomputable def mapDelta (δ : ℤ) (hδ : δ ≠ 0) (f : AffineMapZ) : DeltaSub δ → ℝ :=
  32  fun p => f.slope * ((LedgerUnits.toZ δ p) : ℝ) + f.offset
  33
  34lemma mapDelta_diff (δ : ℤ) (hδ : δ ≠ 0) (f : AffineMapZ)
  35  (p q : DeltaSub δ) :
  36  mapDelta δ hδ f p - mapDelta δ hδ f q
  37    = f.slope * ((LedgerUnits.toZ δ p - LedgerUnits.toZ δ q : ℤ) : ℝ) := by
  38  classical
  39  calc
  40    mapDelta δ hδ f p - mapDelta δ hδ f q
  41        = (f.slope * (LedgerUnits.toZ δ p : ℝ) + f.offset)
  42            - (f.slope * (LedgerUnits.toZ δ q : ℝ) + f.offset) := by
  43              simp [mapDelta]
  44    _   = f.slope * (LedgerUnits.toZ δ p : ℝ)
  45            - f.slope * (LedgerUnits.toZ δ q : ℝ) := by
  46              ring

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