openAI-math-dataset / lean /OAI /Geometry /IsometricImmersion /Comparison /ActualComparisonSource.lean
| import OAI.Geometry.IsometricImmersion.Estimates.ActualQMomentPositive | |
| import OAI.Geometry.IsometricImmersion.Taylor.TaylorResidual | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter MeasureTheory | |
| open scoped ContDiff Topology Matrix Matrix.Norms.Elementwise | |
| namespace SmoothLocal.Pulse | |
| open SmoothLocal.Geometry SmoothLocal.HighEquation SmoothLocal.Taylor | |
| def actualComparisonSource (gStar gTau : MetricField) (z : Coord → ℝ) | |
| (q0 : ℝ) (P : Coord → ℝ) (p : Coord) : ℝ := | |
| actualShearedQForcing gStar gTau z q0 p - | |
| qResidual (metricInShearCoordinates gStar q0) P p | |
| theorem actualComparisonSource_moment {gStar gTau : MetricField} {z P : Coord → ℝ} | |
| {q0 a delta tau : ℝ} | |
| (hQ : ContinuousOn (actualShearedQForcing gStar gTau z q0) (pulseStrip a delta tau)) | |
| (hP : ContinuousOn (qResidual (metricInShearCoordinates gStar q0) P) | |
| (pulseStrip a delta tau)) : | |
| pulseWeightedMoment a delta tau (actualComparisonSource gStar gTau z q0 P) = | |
| pulseWeightedMoment a delta tau (actualShearedQForcing gStar gTau z q0) - | |
| pulseWeightedMoment a delta tau (qResidual (metricInShearCoordinates gStar q0) P) := | |
| pulseWeightedMoment_sub hQ hP | |
| theorem actualComparisonSource_moment_triangle {gStar gTau : MetricField} {z P : Coord → ℝ} | |
| {q0 a delta tau : ℝ} | |
| (hQ : ContinuousOn (actualShearedQForcing gStar gTau z q0) (pulseStrip a delta tau)) | |
| (hP : ContinuousOn (qResidual (metricInShearCoordinates gStar q0) P) | |
| (pulseStrip a delta tau)) : | |
| |pulseWeightedMoment a delta tau (actualShearedQForcing gStar gTau z q0)| - | |
| |pulseWeightedMoment a delta tau (qResidual (metricInShearCoordinates gStar q0) P)| ≤ | |
| |pulseWeightedMoment a delta tau (actualComparisonSource gStar gTau z q0 P)| := by | |
| rw [actualComparisonSource_moment hQ hP] | |
| have ht : |pulseWeightedMoment a delta tau (actualShearedQForcing gStar gTau z q0)| ≤ | |
| |pulseWeightedMoment a delta tau (actualShearedQForcing gStar gTau z q0) - | |
| pulseWeightedMoment a delta tau (qResidual (metricInShearCoordinates gStar q0) P)| + | |
| |pulseWeightedMoment a delta tau (qResidual (metricInShearCoordinates gStar q0) P)| := by | |
| calc | |
| _ = |(pulseWeightedMoment a delta tau (actualShearedQForcing gStar gTau z q0) - | |
| pulseWeightedMoment a delta tau (qResidual (metricInShearCoordinates gStar q0) P)) + | |
| pulseWeightedMoment a delta tau (qResidual (metricInShearCoordinates gStar q0) P)| := by | |
| congr 1 | |
| ring | |
| _ ≤ _ := abs_add_le _ _ | |
| linarith | |
| theorem taylor_moment_absorption_threshold {R c3 : ℝ} | |
| (hR : 0 ≤ R) (hc3 : 0 < c3) (N : ℕ) : | |
| ∃ T : ℝ, 1 ≤ T ∧ ∀ tau : ℝ, T ≤ tau → ∀ delta : ℝ, 0 ≤ delta → | |
| R*delta/tau^(N+1) ≤ (c3/2)*delta*tau/tau^N := by | |
| refine ⟨max 1 (2*R/c3),le_max_left _ _,?_⟩ | |
| intro tau ht delta hdelta | |
| have ht1 : 1 ≤ tau := (le_max_left _ _).trans ht | |
| have htpos : 0 < tau := zero_lt_one.trans_le ht1 | |
| have hthreshold : 2*R/c3 ≤ tau := (le_max_right _ _).trans ht | |
| have hh := (div_le_iff₀ hc3).mp hthreshold | |
| have hcoeff : R/tau ≤ (c3/2)*tau := by | |
| calc | |
| R/tau ≤ R := div_le_self hR ht1 | |
| _ ≤ (c3/2)*tau := by linarith only [hh] | |
| have hscale : 0 ≤ delta/tau^N := div_nonneg hdelta (pow_nonneg htpos.le N) | |
| calc | |
| _ = (R/tau)*(delta/tau^N) := by rw [pow_succ]; field_simp [htpos.ne'] | |
| _ ≤ ((c3/2)*tau)*(delta/tau^N) := mul_le_mul_of_nonneg_right hcoeff hscale | |
| _ = _ := by ring | |
| end SmoothLocal.Pulse | |
| end | |
| end OAI | |