import OAI.Geometry.IsometricImmersion.Comparison.ActualComparisonSource import OAI.Geometry.IsometricImmersion.Estimates.MovingSourceBounds namespace OAI noncomputable section open Set Filter MeasureTheory open scoped ContDiff Topology namespace SmoothLocal.Pulse open SmoothLocal.Geometry SmoothLocal.HighEquation SmoothLocal.Taylor SmoothLocal.Hyperbolic def pulseLeadingBound (a : ℝ) (ha : 0 < a) : ℝ := axisBumpDerivativeBound a ha 0*axisBumpDerivativeBound (1/2) (by norm_num) 0/2 theorem pulseLeadingBound_nonneg {a : ℝ} (ha : 0 < a) : 0 ≤ pulseLeadingBound a ha := by exact div_nonneg (mul_nonneg (axisBumpDerivativeBound_pos ha 0).le (axisBumpDerivativeBound_pos (by norm_num : (0 : ℝ)<1/2) 0).le) (by norm_num) theorem pulsePrincipalLeading_abs_bound {a tau : ℝ} (ha : 0 < a) (ht : 0 < tau) (N : ℕ) (delta : ℝ) (p : Coord) : |pulsePrincipalLeading a N delta tau p| ≤ pulseLeadingBound a ha*tau^2/tau^N := by let A0 := axisBumpDerivativeBound a ha 0 let P0 := axisBumpDerivativeBound (1/2) (by norm_num) 0 have hA0 : 0 ≤ A0 := (axisBumpDerivativeBound_pos ha 0).le have hP0 : 0 ≤ P0 := (axisBumpDerivativeBound_pos (by norm_num : (0 : ℝ)<1/2) 0).le have haxis : |axisBump a (p 0)| ≤ A0 := by simpa only [norm_iteratedFDeriv_zero,Real.norm_eq_abs] using axisBump_derivative_le ha 0 (p 0) have htemp : |temporalCutoff (tau*p 1/delta)| ≤ P0 := by simpa only [temporalCutoff,norm_iteratedFDeriv_zero,Real.norm_eq_abs] using axisBump_derivative_le (by norm_num : (0 : ℝ)<1/2) 0 (tau*p 1/delta) have hq : 0 ≤ tau^2/(2*tau^N) := by positivity calc _ = (tau^2/(2*tau^N))*|axisBump a (p 0)| * |temporalCutoff (tau*p 1/delta)| *|Real.cos (tau*p 0)| := by rw [pulsePrincipalLeading,abs_mul,abs_mul,abs_mul,abs_of_nonneg hq] _ ≤ (tau^2/(2*tau^N))*A0*P0*1 := mul_le_mul (mul_le_mul (mul_le_mul_of_nonneg_left haxis hq) htemp (abs_nonneg _) (mul_nonneg hq hA0)) (Real.abs_cos_le_one _) (abs_nonneg _) (mul_nonneg (mul_nonneg hq hA0) hP0) _ = _ := by dsimp [pulseLeadingBound,A0,P0]; field_simp [ht.ne'] def comparisonSourceBudget (A Cfirst Ctest Ccompare Cres a : ℝ) (ha : 0 < a) (N : ℕ) (delta tau epsilon epsilonQ : ℝ) : ℝ := 2*A*(pulseLeadingBound a ha*tau^2/tau^N)+ qForcingRemainderBudget A Cfirst Ctest Ccompare a ha N delta tau epsilon epsilonQ+Cres/tau^N theorem comparisonSourceBudget_nonneg {A Cfirst Ctest Ccompare Cres a : ℝ} (ha : 0 < a) (hA : 0 ≤ A) (hCf : 0 ≤ Cfirst) (hCt : 0 ≤ Ctest) (hCc : 0 ≤ Ccompare) (hCr : 0 ≤ Cres) (N : ℕ) (delta : ℝ) {tau epsilon epsilonQ : ℝ} (ht : 0 ≤ tau) (he : 0 ≤ epsilon) (heQ : 0 ≤ epsilonQ) : 0 ≤ comparisonSourceBudget A Cfirst Ctest Ccompare Cres a ha N delta tau epsilon epsilonQ := by have hrem := qForcingRemainderBudget_nonneg ha hA hCf hCt hCc N delta ht he heQ have hlead := pulseLeadingBound_nonneg ha unfold comparisonSourceBudget positivity theorem exists_actual_comparison_source_pointwise_bound (B Bcompare BQ D A Cres : ℝ) {d dcompare c : ℝ} (hd : 0 < d) (hdcompare : 0 < dcompare) (hc : 0 < c) (hD : 0 ≤ D) (hA : 0 ≤ A) {a : ℝ} (ha : 0 < a) (N : ℕ) (hN : 1 < N) : ∃ H Cfirst Ctest Ccompare : ℝ, 0 ≤ H ∧ 0 ≤ Cfirst ∧ 0 ≤ Ctest ∧ 0 ≤ Ccompare ∧ ∀ delta : ℝ, 0 < delta → ∃ T : ℝ, 1 ≤ T ∧ ∀ tau : ℝ, T ≤ tau → ∀ (gStar gTau : MetricField) (q0 : ℝ) (U : Set Coord), SmoothPositiveOn gStar U → SmoothPositiveOn (testMetric gStar q0 a N delta tau) U → SmoothPositiveOn gTau U → IsOpen U → ∀ (z P : Coord → ℝ), ContDiffOn ℝ ∞ z U → ∀ epsilon epsilonQ : ℝ, epsilon ≤ 1 → (4*max Bcompare 0+2)*epsilon ≤ dcompare/2 → epsilonQ ≤ 1 → (4*max BQ 0+2)*epsilonQ ≤ d/2 → H*epsilonQ ≤ c/2 → ∀ p : Coord, QPulsePointBounds gStar gTau z U q0 a delta tau N B Bcompare BQ D A d dcompare c epsilon epsilonQ p → |qResidual (metricInShearCoordinates gStar q0) P p| ≤ Cres/tau^N → |actualComparisonSource gStar gTau z q0 P p| ≤ comparisonSourceBudget A Cfirst Ctest Ccompare Cres a ha N delta tau epsilon epsilonQ := by obtain ⟨H,Cfirst,Ctest,Ccompare,hH,hCf,hCt,hCc,hglobal⟩ := exists_actual_Q_pulse_forcing_bound_global B Bcompare BQ D A hd hdcompare hc hD hA ha N hN refine ⟨H,Cfirst,Ctest,Ccompare,hH,hCf,hCt,hCc,?_⟩ intro delta hdelt obtain ⟨T,hT,hglobalT⟩ := hglobal delta hdelt refine ⟨T,hT,?_⟩ intro tau ht gStar gTau q0 U hgStar hgTest hgTau hU z P hz epsilon epsilonQ he1 hesmall heQ1 heQsmall hxxsmall p hpoint hres have htpos : 0 < tau := zero_lt_one.trans_le (hT.trans ht) have herr := hglobalT tau ht gStar gTau q0 U hgStar hgTest hgTau hU p hpoint.point_mem z hz epsilon epsilonQ hpoint.referenceFirst hpoint.referenceDet hpoint.testFirst hpoint.testDet hpoint.approximationFirst he1 hesmall hpoint.approximationSecond hpoint.referenceState hpoint.referenceXX hpoint.baseFirstDistance heQ1 heQsmall hxxsmall hpoint.actualDensity hpoint.referenceCoefficient have hlead : |actualShearedQLeading gStar z q0 a N delta tau p| ≤ 2*A*(pulseLeadingBound a ha*tau^2/tau^N) := by rw [actualShearedQLeading,abs_mul,abs_mul] rw [abs_of_nonneg (by norm_num : (0 : ℝ) ≤ 2)] exact mul_le_mul (mul_le_mul_of_nonneg_left hpoint.referenceCoefficient (by norm_num)) (pulsePrincipalLeading_abs_bound ha htpos N delta p) (abs_nonneg _) (by positivity) have hforce : |actualShearedQForcing gStar gTau z q0 p| ≤ 2*A*(pulseLeadingBound a ha*tau^2/tau^N)+ qForcingRemainderBudget A Cfirst Ctest Ccompare a ha N delta tau epsilon epsilonQ := by have htri : |actualShearedQForcing gStar gTau z q0 p| ≤ |actualShearedQLeading gStar z q0 a N delta tau p|+ |actualShearedQForcing gStar gTau z q0 p-actualShearedQLeading gStar z q0 a N delta tau p| := by have he : actualShearedQLeading gStar z q0 a N delta tau p+ (actualShearedQForcing gStar gTau z q0 p-actualShearedQLeading gStar z q0 a N delta tau p) = actualShearedQForcing gStar gTau z q0 p := by ring simpa only [he] using abs_add_le (actualShearedQLeading gStar z q0 a N delta tau p) (actualShearedQForcing gStar gTau z q0 p-actualShearedQLeading gStar z q0 a N delta tau p) exact htri.trans (add_le_add hlead herr) exact (abs_sub _ _).trans (add_le_add hforce hres) end SmoothLocal.Pulse end end OAI