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b37e936 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 | 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
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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