| 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 | |