openAI-math-dataset / lean /OAI /Geometry /IsometricImmersion /Comparison /QApproximationMargins.lean
| import OAI.Geometry.IsometricImmersion.Comparison.QMetricApproximationInputs | |
| import OAI.Geometry.IsometricImmersion.Pulses.PulseEdgeReferenceDistance | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter | |
| open scoped ContDiff Topology | |
| namespace SmoothLocal.Pulse | |
| open SmoothLocal.Geometry | |
| theorem actual_Q_approximation_eventual_margins (Bcompare BQ H : ℝ) | |
| {d dcompare c a : ℝ} (hd : 0 < d) (hdcompare : 0 < dcompare) | |
| (hc : 0 < c) (ha : 0 < a) (N : ℕ) (hN : 1 < N) (delta : ℝ) : | |
| ∀ᶠ tau : ℕ in atTop, | |
| 2 ≤ tau ∧ N ≤ tau ∧ | |
| 0 ≤ metricApproximationAccuracy tau ∧ metricApproximationAccuracy tau ≤ 1 ∧ | |
| (4*max Bcompare 0+2)*metricApproximationAccuracy tau ≤ dcompare/2 ∧ | |
| metricApproximationAccuracy tau ≤ (tau : ℝ)/(tau : ℝ)^N ∧ | |
| 0 ≤ qPulseFirstJetBudget a ha N delta (tau : ℝ) ∧ | |
| qPulseFirstJetBudget a ha N delta (tau : ℝ) ≤ 1 ∧ | |
| (4*max BQ 0+2)*qPulseFirstJetBudget a ha N delta (tau : ℝ) ≤ d/2 ∧ | |
| H*qPulseFirstJetBudget a ha N delta (tau : ℝ) ≤ c/2 := by | |
| let K := 4*max Bcompare 0+2 | |
| have hK : 0 < K := by dsimp only [K]; have hh := le_max_right Bcompare 0; linarith | |
| let epsilon := min 1 (dcompare/(2*K)) | |
| have he : 0 < epsilon := lt_min (by norm_num) (div_pos hdcompare (by positivity)) | |
| obtain ⟨tau0,_,haccurate⟩ := exists_first_error_threshold (by positivity : 0 < 8*epsilon) | |
| obtain ⟨T,_,hQ⟩ := qPulseFirstJetBudget_eventual_margins BQ H hd hc ha N hN delta | |
| have hlarge : ∀ᶠ tau : ℕ in atTop, T ≤ (tau : ℝ) := | |
| (tendsto_natCast_atTop_atTop : Tendsto (fun tau : ℕ => (tau : ℝ)) atTop atTop).eventually | |
| (eventually_ge_atTop T) | |
| filter_upwards [eventually_ge_atTop tau0,eventually_ge_atTop 2,eventually_ge_atTop N,hlarge] | |
| with tau ht0 ht2 htN ht | |
| have ht1 : 1 ≤ tau := by omega | |
| have htpos : (0 : ℝ) < tau := zero_lt_one.trans_le (by exact_mod_cast ht1) | |
| have herr : metricApproximationAccuracy tau ≤ epsilon := by linarith [haccurate tau ht0] | |
| have hsmall : K*metricApproximationAccuracy tau ≤ dcompare/2 := by | |
| have heK : epsilon ≤ dcompare/(2*K) := min_le_right _ _ | |
| have hh := (le_div_iff₀ (show 0 < 2*K by positivity)).mp (herr.trans heK) | |
| nlinarith | |
| exact ⟨ht2,htN,(metricApproximationAccuracy_pos (by omega)).le, | |
| herr.trans (min_le_left _ _),hsmall, | |
| approximation_budget_le_pulse_error (by omega) ht1 (by omega), | |
| qPulseFirstJetBudget_nonneg ha N delta htpos.le,(hQ _ ht).1,(hQ _ ht).2.1,(hQ _ ht).2.2⟩ | |
| end SmoothLocal.Pulse | |
| end | |
| end OAI | |