| import OAI.Geometry.IsometricImmersion.Pulses.PulseFirstJetControl |
| import OAI.Geometry.IsometricImmersion.Pulses.PulseSecondXi |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set Filter |
| open scoped ContDiff Topology Matrix Matrix.Norms.Elementwise |
|
|
| namespace SmoothLocal.Pulse |
| open SmoothLocal.Geometry |
|
|
| def curvatureDensity (g : MetricField) (p : Coord) : ℝ := |
| gaussianCurvature g p*(g p).det |
|
|
| def curvatureDensityJetError (g h : MetricField) (p : Coord) : ℝ := |
| (coordPartial 0 (coordPartial 1 (fun q => g q 0 1)) p- |
| coordPartial 0 (coordPartial 1 (fun q => h q 0 1)) p) - |
| ((coordPartial 0 (coordPartial 0 (fun q => g q 1 1)) p- |
| coordPartial 0 (coordPartial 0 (fun q => h q 1 1)) p) + |
| (coordPartial 1 (coordPartial 1 (fun q => g q 0 0)) p- |
| coordPartial 1 (coordPartial 1 (fun q => h q 0 0)) p))/2 + |
| curvatureLowerTerm g p-curvatureLowerTerm h p |
|
|
| theorem curvatureDensityJetError_eq {g h : MetricField} {U : Set Coord} |
| (hg : SmoothPositiveOn g U) (hh : SmoothPositiveOn h U) (hU : IsOpen U) |
| {p : Coord} (hp : p ∈ U) : |
| curvatureDensityJetError g h p = curvatureDensity g p-curvatureDensity h p := by |
| unfold curvatureDensity curvatureDensityJetError |
| rw [gaussianCurvature_density_principal_part hg hU hp, |
| gaussianCurvature_density_principal_part hh hU hp] |
| ring |
|
|
| theorem curvatureDensityJetError_le (g h : MetricField) (p : Coord) {e : ℝ} |
| (hxy : |coordPartial 0 (coordPartial 1 (fun q => g q 0 1)) p- |
| coordPartial 0 (coordPartial 1 (fun q => h q 0 1)) p| ≤ e) |
| (hxx : |coordPartial 0 (coordPartial 0 (fun q => g q 1 1)) p- |
| coordPartial 0 (coordPartial 0 (fun q => h q 1 1)) p| ≤ e) |
| (hyy : |coordPartial 1 (coordPartial 1 (fun q => g q 0 0)) p- |
| coordPartial 1 (coordPartial 1 (fun q => h q 0 0)) p| ≤ e) : |
| |curvatureDensityJetError g h p| ≤ 2*e+|curvatureLowerTerm g p-curvatureLowerTerm h p| := by |
| let x := coordPartial 0 (coordPartial 1 (fun q => g q 0 1)) p- |
| coordPartial 0 (coordPartial 1 (fun q => h q 0 1)) p |
| let y := coordPartial 0 (coordPartial 0 (fun q => g q 1 1)) p- |
| coordPartial 0 (coordPartial 0 (fun q => h q 1 1)) p |
| let z := coordPartial 1 (coordPartial 1 (fun q => g q 0 0)) p- |
| coordPartial 1 (coordPartial 1 (fun q => h q 0 0)) p |
| have hmain : |x-(y+z)/2| ≤ 2*e := by |
| calc |
| _ ≤ |x|+|(y+z)/2| := abs_sub _ _ |
| _ = |x|+|y+z|/2 := by rw [abs_div]; norm_num |
| _ ≤ e+(e+e)/2 := add_le_add hxy |
| (div_le_div_of_nonneg_right ((abs_add_le y z).trans (add_le_add hxx hyy)) (by norm_num)) |
| _ = 2*e := by ring |
| have heq : curvatureDensityJetError g h p = |
| (x-(y+z)/2)+(curvatureLowerTerm g p-curvatureLowerTerm h p) := by |
| dsimp only [curvatureDensityJetError,x,y,z] |
| ring |
| rw [heq] |
| exact (abs_add_le _ _).trans (add_le_add hmain le_rfl) |
|
|
| def testDensityErrorCoefficient (C a : ℝ) (ha : 0 < a) (delta : ℝ) : ℝ := |
| pulsePrincipalRemainderBound a ha+C*scalarPulseFirstJetBound a ha delta |
|
|
| theorem exists_test_density_slab_bound (B : ℝ) {d : ℝ} (hd : 0 < d) |
| {a : ℝ} (ha : 0 < a) (N : ℕ) (hN : 1 < N) : |
| ∃ C : ℝ, 0 ≤ C ∧ ∀ delta : ℝ, 0 < delta → ∃ T : ℝ, 1 ≤ T ∧ |
| ∀ tau : ℝ, T ≤ tau → ∀ (gStar : MetricField) (q0 : ℝ) (U : Set Coord), |
| SmoothPositiveOn gStar U → SmoothPositiveOn (testMetric gStar q0 a N delta tau) U → |
| IsOpen U → ∀ p ∈ pulseLocalBox a delta tau, |
| inverseShearCoordinates q0 p ∈ U → |
| ‖actualCurvatureFirstInput (metricInShearCoordinates gStar q0) p‖ ≤ B → |
| d ≤ (metricInShearCoordinates gStar q0 p).det → |
| |(curvatureDensity (testMetric gStar q0 a N delta tau) (inverseShearCoordinates q0 p)- |
| curvatureDensity gStar (inverseShearCoordinates q0 p))- |
| pulsePrincipalLeading a N delta tau p| ≤ |
| testDensityErrorCoefficient C a ha delta*tau/tau^N := by |
| obtain ⟨C,hC,hbound⟩ := exists_uniform_actual_pulse_lower_bound B hd ha N hN |
| refine ⟨C,hC,?_⟩ |
| intro delta hdelta |
| obtain ⟨T,hT,hsmall⟩ := hbound delta |
| refine ⟨T,hT,?_⟩ |
| intro tau ht gStar q0 U hg htest hU p hp hpU hjet hdet |
| have ht1 : 1 ≤ tau := hT.trans ht |
| have hL := hsmall tau ht (metricInShearCoordinates gStar q0) p hjet hdet |
| have hR := pulse_principal_remainder_le ha N delta ht1 p |
| have heq := actual_pulse_curvature_density q0 a N delta tau hg htest hU hpU |
| rw [pulse_principal_exact] at heq |
| have hdensity : |
| (curvatureDensity (testMetric gStar q0 a N delta tau) (inverseShearCoordinates q0 p)- |
| curvatureDensity gStar (inverseShearCoordinates q0 p))-pulsePrincipalLeading a N delta tau p = |
| pulsePrincipalRemainder a N delta tau p+ |
| thetaPulseLowerDifference (metricInShearCoordinates gStar q0) (pulseScalar a N delta tau) p := by |
| dsimp only [curvatureDensity] |
| linarith only [heq] |
| rw [hdensity] |
| calc |
| _ ≤ |pulsePrincipalRemainder a N delta tau p|+ |
| |thetaPulseLowerDifference (metricInShearCoordinates gStar q0) (pulseScalar a N delta tau) p| := |
| abs_add_le _ _ |
| _ ≤ pulsePrincipalRemainderBound a ha*tau/tau^N+ |
| C*scalarPulseFirstJetBudget a ha N delta tau := add_le_add hR hL |
| _ = _ := by unfold testDensityErrorCoefficient scalarPulseFirstJetBudget; ring |
|
|
| theorem perturbed_density_slab_with_comparison |
| {gStar gTau : MetricField} {U : Set Coord} {q0 a delta tau A : ℝ} {N : ℕ} {p : Coord} |
| (hgTau : SmoothPositiveOn gTau U) |
| (htest : SmoothPositiveOn (testMetric gStar q0 a N delta tau) U) |
| (hU : IsOpen U) (hp : inverseShearCoordinates q0 p ∈ U) |
| (htestBound : |
| |(curvatureDensity (testMetric gStar q0 a N delta tau) (inverseShearCoordinates q0 p)- |
| curvatureDensity gStar (inverseShearCoordinates q0 p))-pulsePrincipalLeading a N delta tau p| ≤ A) : |
| |(curvatureDensity gTau (inverseShearCoordinates q0 p)- |
| curvatureDensity gStar (inverseShearCoordinates q0 p))-pulsePrincipalLeading a N delta tau p| ≤ |
| |curvatureDensityJetError gTau (testMetric gStar q0 a N delta tau) (inverseShearCoordinates q0 p)|+A := by |
| rw [curvatureDensityJetError_eq hgTau htest hU hp] |
| have heq : (curvatureDensity gTau (inverseShearCoordinates q0 p)- |
| curvatureDensity gStar (inverseShearCoordinates q0 p))-pulsePrincipalLeading a N delta tau p = |
| (curvatureDensity gTau (inverseShearCoordinates q0 p)- |
| curvatureDensity (testMetric gStar q0 a N delta tau) (inverseShearCoordinates q0 p))+ |
| ((curvatureDensity (testMetric gStar q0 a N delta tau) (inverseShearCoordinates q0 p)- |
| curvatureDensity gStar (inverseShearCoordinates q0 p))-pulsePrincipalLeading a N delta tau p) := by ring |
| rw [heq] |
| exact (abs_add_le _ _).trans (add_le_add le_rfl htestBound) |
|
|
| end SmoothLocal.Pulse |
|
|
| end |
|
|
| end OAI |
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