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59d1c80 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 | 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
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