| import OAI.Geometry.IsometricImmersion.Comparison.ShearFirstJetComparison |
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| namespace OAI |
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| noncomputable section |
| open Set Filter |
| open scoped ContDiff Topology Matrix Matrix.Norms.Elementwise |
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| namespace SmoothLocal.Pulse |
| open SmoothLocal.Geometry SmoothLocal.HighEquation |
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| def qPulseFirstJetBudget (a : ℝ) (ha : 0 < a) (N : ℕ) (delta tau : ℝ) : ℝ := |
| (8+scalarPulseFirstJetBound a ha delta)*tau/tau^N |
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| theorem qPulseFirstJetBudget_nonneg {a : ℝ} (ha : 0 < a) (N : ℕ) (delta : ℝ) |
| {tau : ℝ} (ht : 0 ≤ tau) : 0 ≤ qPulseFirstJetBudget a ha N delta tau := |
| div_nonneg (mul_nonneg (by linarith [scalarPulseFirstJetBound_nonneg ha delta]) ht) |
| (pow_nonneg ht N) |
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| theorem actual_sheared_first_jet_error_nat |
| {gStar gTau : MetricField} {U : Set Coord} {q0 a delta : ℝ} {N n : ℕ} |
| (hgStar : SmoothPositiveOn gStar U) |
| (hgTest : SmoothPositiveOn (testMetric gStar q0 a N delta n) U) |
| (hgTau : SmoothPositiveOn gTau U) (hU : IsOpen U) |
| (hq : |q0| ≤ 1) (ha : 0 < a) (hN : 0 < N) (hn : 1 ≤ n) (hlarge : N-1 ≤ n) |
| {p : Coord} (hp : inverseShearCoordinates q0 p ∈ U) |
| (hjet : ‖actualCurvatureFirstInput gTau (inverseShearCoordinates q0 p)- |
| actualCurvatureFirstInput (testMetric gStar q0 a N delta n) |
| (inverseShearCoordinates q0 p)‖ ≤ 1/(n : ℝ)^n) : |
| ‖actualCurvatureFirstInput (metricInShearCoordinates gTau q0) p- |
| actualCurvatureFirstInput (metricInShearCoordinates gStar q0) p‖ ≤ |
| qPulseFirstJetBudget a ha N delta n := by |
| have ht : (1 : ℝ) ≤ (n : ℝ) := by exact_mod_cast hn |
| have hfirst := actual_sheared_solution_first_jet_error hgStar hgTest hgTau hU hq ha ht hp hjet |
| calc |
| _ ≤ 8*(1/(n : ℝ)^n)+scalarPulseFirstJetBudget a ha N delta n := hfirst |
| _ ≤ 8*((n : ℝ)/(n : ℝ)^N)+scalarPulseFirstJetBudget a ha N delta n := |
| add_le_add (mul_le_mul_of_nonneg_left |
| (approximation_budget_le_pulse_error hN hn hlarge) (by norm_num : (0 : ℝ) ≤ 8)) le_rfl |
| _ = _ := by unfold qPulseFirstJetBudget scalarPulseFirstJetBudget; ring |
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| theorem qPulseFirstJetBudget_tendsto_zero {a : ℝ} (ha : 0 < a) (N : ℕ) |
| (hN : 1 < N) (delta : ℝ) : |
| Tendsto (qPulseFirstJetBudget a ha N delta) atTop (𝓝 0) := by |
| have hn : N-1 ≠ 0 := by omega |
| have hi := (tendsto_inv_atTop_zero : Tendsto (fun t : ℝ => t⁻¹) atTop (𝓝 0)).pow (N-1) |
| have h : Tendsto (fun t : ℝ => (8+scalarPulseFirstJetBound a ha delta)*(t⁻¹)^(N-1)) |
| atTop (𝓝 0) := by |
| simpa only [zero_pow hn,mul_zero] using hi.const_mul (8+scalarPulseFirstJetBound a ha delta) |
| apply h.congr' |
| filter_upwards [eventually_gt_atTop (0 : ℝ)] with t ht |
| have hN' : N = 1+(N-1) := by omega |
| have hr : t/t^N = (t⁻¹)^(N-1) := by |
| conv_lhs => rw [hN',pow_add,pow_one] |
| rw [inv_pow] |
| field_simp [ht.ne'] |
| simp only [qPulseFirstJetBudget,mul_div_assoc,hr] |
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| theorem qPulseFirstJetBudget_eventual_margins (B H : ℝ) {d c a : ℝ} |
| (hd : 0 < d) (hc : 0 < c) (ha : 0 < a) (N : ℕ) (hN : 1 < N) |
| (delta : ℝ) : |
| ∃ T : ℝ, 1 ≤ T ∧ ∀ tau : ℝ, T ≤ tau → |
| qPulseFirstJetBudget a ha N delta tau ≤ 1 ∧ |
| (4*max B 0+2)*qPulseFirstJetBudget a ha N delta tau ≤ d/2 ∧ |
| H*qPulseFirstJetBudget a ha N delta tau ≤ c/2 := by |
| have htend := qPulseFirstJetBudget_tendsto_zero ha N hN delta |
| have h1 : ∀ᶠ t : ℝ in atTop, qPulseFirstJetBudget a ha N delta t < 1 := |
| htend.eventually (eventually_lt_nhds zero_lt_one) |
| have hd' : Tendsto (fun t : ℝ => (4*max B 0+2)*qPulseFirstJetBudget a ha N delta t) |
| atTop (𝓝 0) := by simpa only [mul_zero] using htend.const_mul (4*max B 0+2) |
| have hc' : Tendsto (fun t : ℝ => H*qPulseFirstJetBudget a ha N delta t) |
| atTop (𝓝 0) := by simpa only [mul_zero] using htend.const_mul H |
| have hev : ∀ᶠ t : ℝ in atTop, |
| qPulseFirstJetBudget a ha N delta t ≤ 1 ∧ |
| (4*max B 0+2)*qPulseFirstJetBudget a ha N delta t ≤ d/2 ∧ |
| H*qPulseFirstJetBudget a ha N delta t ≤ c/2 := by |
| filter_upwards [h1,hd'.eventually (eventually_lt_nhds (half_pos hd)), |
| hc'.eventually (eventually_lt_nhds (half_pos hc))] with t ht1 htd htc |
| exact ⟨ht1.le,htd.le,htc.le⟩ |
| obtain ⟨T,hT⟩ := eventually_atTop.mp hev |
| exact ⟨max T 1,le_max_right _ _,fun t ht => hT t ((le_max_left _ _).trans ht)⟩ |
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| end SmoothLocal.Pulse |
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| end |
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| end OAI |
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