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695bd70 | 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 | import Mathlib
namespace OAI
noncomputable section
open Set Filter
open scoped Topology ContDiff
namespace YauCounterexamples
lemma compact_exponential_envelope_log_bound
{E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
{K : Set E} (hK : IsCompact K) {φ : E → ℝ} (hφ : ContDiff ℝ ∞ φ) :
∃ H : ℝ, 0 ≤ H ∧ ∀ n : ℝ, 0 ≤ n → ∀ x ∈ K,
‖fderiv ℝ (fun y => Real.exp (n*φ y)) x‖ ≤ H*n*Real.exp (n*φ x) := by
obtain ⟨C,hC⟩ := hK.exists_bound_of_continuousOn
(hφ.fderiv_right (m:=∞) (by simp)).continuous.continuousOn
refine ⟨max C 0,le_max_right _ _,?_⟩
intro n hn x hx
have hd := (((hφ.differentiable (by simp)) x).hasFDerivAt.const_mul n).exp.fderiv
rw [hd,norm_smul,norm_smul,Real.norm_eq_abs,Real.norm_eq_abs,
abs_of_pos (Real.exp_pos _),abs_of_nonneg hn]
calc
_ ≤ Real.exp (n*φ x)*(n* max C 0) := by gcongr; exact (hC x hx).trans (le_max_left _ _)
_ = _ := by ring
end YauCounterexamples
end
end OAI
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