| import OAI.Analysis.CircleDomains.Modulus.ModelSelectedBarrier |
|
|
| namespace OAI |
|
|
| /-! Choose the finite numerical budgets in the order required by the |
| near-extremal model construction. No analytic conclusion is assumed. -/ |
| open Set Function Filter MeasureTheory Metric |
| open scoped Topology |
| noncomputable section |
| namespace CircleDomainRigidity |
|
|
| theorem exists_barrier_error_budget {η M₁ M₂ N : ℝ} |
| (hη : 0 < η) (hM₁ : 0 ≤ M₁) (hM₂ : 0 ≤ M₂) (hN : 0 ≤ N) : |
| ∃ t : ℝ, 0 < t ∧ M₁*(t+t)+(M₂*(t+t))*N+4*M₁*t < η/3 := by |
| let C := 6*M₁+2*M₂*N+1 |
| have hC : 0 < C := by dsimp only [C]; positivity |
| let t := (η/3)/C |
| have ht : 0 < t := div_pos (by positivity) hC |
| have he : t*C = η/3 := div_mul_cancel₀ _ hC.ne' |
| refine ⟨t,ht,?_⟩ |
| have he' : M₁*(t+t)+(M₂*(t+t))*N+4*M₁*t = t*C-t := by dsimp only [C]; ring |
| rw [he',he] |
| linarith |
|
|
| theorem exists_height_error_scale (C₀ C₁ V : ℝ) {ν b : ℝ} |
| (hν : 0 < ν) (hb : 0 < b) : |
| ∃ t : ℝ, 0 < t ∧ t < b ∧ |
| 2*C₀^2*(2*t+4*Real.pi*t)+4*C₁^2*(t+t)^2*V < ν^2 := by |
| let E := fun t : ℝ => 2*C₀^2*(2*t+4*Real.pi*t)+4*C₁^2*(t+t)^2*V |
| have hc : Continuous E := by dsimp only [E]; fun_prop |
| have hzero : E 0 < ν^2 := by dsimp only [E]; nlinarith [sq_pos_of_pos hν] |
| have ho : IsOpen ({t : ℝ | E t < ν^2} ∩ Iio b) := |
| (isOpen_lt hc continuous_const).inter isOpen_Iio |
| have hn := ho.mem_nhds (show (0 : ℝ) ∈ {t : ℝ | E t < ν^2} ∩ Iio b from ⟨hzero,hb⟩) |
| obtain ⟨r,hr,hsub⟩ := Metric.mem_nhds_iff.mp hn |
| have ht : r/2 ∈ ball (0 : ℝ) r := by |
| simp only [mem_ball, dist_zero_right, Real.norm_eq_abs, abs_of_pos (half_pos hr)] |
| linarith |
| exact ⟨r/2,half_pos hr,(hsub ht).2,(hsub ht).1⟩ |
|
|
| end CircleDomainRigidity |
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|
| end |
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| end OAI |
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|
