  85
  86/-- The gradient maximum is non-increasing under one NS time step when
  87the sub-Kolmogorov condition holds. -/
  88theorem gradient_nonincreasing (data : OneStepData)
  89    (hsubK : data.subKolmogorov)
  90    (newGradMax : ℝ)
  91    (h_update : newGradMax ≤
  92      data.gradMax * (1 + data.dt * (data.advectionBound - data.viscousRate))) :
  93    newGradMax ≤ data.gradMax := by
  94  have hfactor := one_step_factor_le_one data hsubK
  95  exact le_trans h_update (mul_le_of_le_one_right data.gradMax_nonneg hfactor)
  96
  97/-- The sub-Kolmogorov condition is preserved by one step. -/
  98theorem subK_preserved (data : OneStepData)
  99    (hsubK : data.subKolmogorov)
 100    (newGradMax : ℝ)
 101    (h_update : newGradMax ≤
 102      data.gradMax * (1 + data.dt * (data.advectionBound - data.viscousRate)))
 103    (newData : OneStepData)
 104    (h_same_params : newData.viscousRate = data.viscousRate)
 105    (h_new_grad : newData.gradMax = newGradMax) :
 106    newData.subKolmogorov := by
 107  unfold OneStepData.subKolmogorov
 108  rw [h_new_grad, h_same_params]
 109  exact le_trans (gradient_nonincreasing data hsubK newGradMax h_update) hsubK
 110
 111/-! ## Inductive Propagation -/
 112
 113/-- The sub-Kolmogorov condition propagates through arbitrarily many
 114time steps by induction. -/
 115theorem subK_propagation
 116    (gradients : ℕ → ℝ) (bound : ℝ)
 117    (h_init : gradients 0 ≤ bound)
 118    (h_step : ∀ n, gradients (n + 1) ≤ gradients n) :

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