| import OAI.Geometry.RCD.Metric.EvolutionInitialLimit |
|
|
| namespace OAI |
|
|
| /-! # A short-time sufficient form of the dimensional evolution bound |
|
|
| Bounding the exponential integral by its endpoint retains the sharp |
| coefficient at time zero, as required for the integrated Bochner limit. |
| -/ |
|
|
| namespace RegularPointCoordinates |
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|
| theorem gronwallBound_one_le {d k e : ℝ} (hk : 0 ≤ k) (he : 0 ≤ e) : |
| gronwallBound d k e 1 ≤ (d+e)*Real.exp k := by |
| by_cases hk0 : k=0 |
| · simp only [hk0,gronwallBound_K0,mul_one,Real.exp_zero,le_refl] |
| · have hkp : 0 < k := lt_of_le_of_ne hk (Ne.symm hk0) |
| have h := Real.add_one_le_exp (-k) |
| have hh := mul_le_mul_of_nonneg_right h (Real.exp_pos k).le |
| rw [Real.exp_neg,inv_mul_cancel₀ (Real.exp_ne_zero k)] at hh |
| have hr : (Real.exp k-1)/k ≤ Real.exp k := by |
| apply (div_le_iff₀ hkp).mpr |
| nlinarith only [hh] |
| have hr' := mul_le_mul_of_nonneg_left hr he |
| simp only [gronwallBound_of_K_ne_0 hk0,mul_one] |
| calc d*Real.exp k+e/k*(Real.exp k-1) |
| = d*Real.exp k+e*((Real.exp k-1)/k) := by ring |
| _ ≤ d*Real.exp k+e*Real.exp k := add_le_add le_rfl hr' |
| _ = (d+e)*Real.exp k := by ring |
|
|
| end RegularPointCoordinates |
|
|
| end OAI |
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|
