| import OAI.Geometry.IsometricImmersion.Pulses.PulseTemporalMoment | |
| import OAI.Geometry.IsometricImmersion.Comparison.ActualComparisonEquation | |
| import OAI.Geometry.IsometricImmersion.Caps.EllipticAbsorption | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter MeasureTheory | |
| open scoped ContDiff Topology Interval | |
| namespace SmoothLocal.Pulse | |
| open SmoothLocal.Geometry SmoothLocal.Weighted SmoothLocal.Hyperbolic SmoothLocal.Taylor | |
| open SmoothLocal.HighEquation | |
| def testedOperatorSource (R S b c v : Coord → ℝ) (p : Coord) : ℝ := | |
| hyperbolicOperator R S v p-b p*coordPartial 1 v p-c p*coordPartial 0 v p | |
| theorem testedOperatorSource_contDiffOn {R S b c v : Coord → ℝ} {U : Set Coord} | |
| (hR : ContDiffOn ℝ ∞ R U) (hS : ContDiffOn ℝ ∞ S U) | |
| (hb : ContDiffOn ℝ ∞ b U) (hc : ContDiffOn ℝ ∞ c U) | |
| (hv : ContDiffOn ℝ ∞ v U) (hU : IsOpen U) : | |
| ContDiffOn ℝ ∞ (testedOperatorSource R S b c v) U := | |
| ((((partial_contDiffOn (partial_contDiffOn hv hU 1) hU 1).sub | |
| (hR.mul (partial_contDiffOn (partial_contDiffOn hv hU 1) hU 0))).sub | |
| (hS.mul (partial_contDiffOn (partial_contDiffOn hv hU 0) hU 0))).sub | |
| (hb.mul (partial_contDiffOn hv hU 1))).sub (hc.mul (partial_contDiffOn hv hU 0)) | |
| theorem pulseWeightedMoment_eq_tested_rectangle {a delta tau : ℝ} {f : Coord → ℝ} | |
| (ha : 0 < a) (hd : 0 ≤ delta) (ht : 0 < tau) | |
| (hf : ContinuousOn f (pulseStrip a delta tau)) : | |
| pulseWeightedMoment a delta tau f = | |
| rectangleIntegral (-a) a (-(delta/tau)) (delta/tau) (fun p => pulseSpatialTest a tau p*f p) := by | |
| have hh := pulsePair_integral_eq_rectangleIntegral ha.le hd ht | |
| ((pulseSpatialTest_contDiff a tau).continuous.continuousOn.mul hf) | |
| simp only [Pi.mul_def] at hh | |
| have he : (fun q : ℝ × ℝ => pulseSpatialTest a tau (pulsePairPoint q)*f (pulsePairPoint q)) = | |
| (fun q => f (pulsePairPoint q)*pulseTest a tau q.2) := by | |
| funext q | |
| simp only [pulseSpatialTest,pulsePairPoint,pulse_boxPoint_zero,mul_comm] | |
| rw [he] at hh | |
| exact hh | |
| theorem testedOperatorSource_moment_identity {a delta tau : ℝ} | |
| {R S b c v : Coord → ℝ} {U : Set Coord} | |
| (ha : 0 < a) (hd : 0 ≤ delta) (ht : 0 < tau) | |
| (hR : ContDiffOn ℝ ∞ R U) (hS : ContDiffOn ℝ ∞ S U) | |
| (hb : ContDiffOn ℝ ∞ b U) (hc : ContDiffOn ℝ ∞ c U) | |
| (hv : ContDiffOn ℝ ∞ v U) (hU : IsOpen U) (hSU : pulseStrip a delta tau ⊆ U) : | |
| pulseWeightedMoment a delta tau (testedOperatorSource R S b c v) = | |
| rectangleIntegral (-a) a (-(delta/tau)) (delta/tau) | |
| (fun p => pulseSpatialTest a tau p*coordPartial 1 (coordPartial 1 v) p)- | |
| rectangleIntegral (-a) a (-(delta/tau)) (delta/tau) | |
| (fun p => pulseSpatialTest a tau p*R p*coordPartial 0 (coordPartial 1 v) p)- | |
| rectangleIntegral (-a) a (-(delta/tau)) (delta/tau) | |
| (fun p => pulseSpatialTest a tau p*S p*coordPartial 0 (coordPartial 0 v) p)- | |
| rectangleIntegral (-a) a (-(delta/tau)) (delta/tau) | |
| (fun p => pulseSpatialTest a tau p*b p*coordPartial 1 v p)- | |
| rectangleIntegral (-a) a (-(delta/tau)) (delta/tau) | |
| (fun p => pulseSpatialTest a tau p*c p*coordPartial 0 v p) := by | |
| have hlr : -a ≤ a := by linarith | |
| have hbt : -(delta/tau) ≤ delta/tau := by linarith [div_nonneg hd ht.le] | |
| have hψ : ContinuousOn (pulseSpatialTest a tau) (pulseStrip a delta tau) := | |
| (pulseSpatialTest_contDiff a tau).continuous.continuousOn | |
| have hvi (i : Fin 2) := (partial_contDiffOn hv hU i).continuousOn.mono hSU | |
| have hvij (i j : Fin 2) := (partial_contDiffOn (partial_contDiffOn hv hU j) hU i).continuousOn.mono hSU | |
| have h0 := hψ.mul (hvij 1 1) | |
| have h1 := (hψ.mul (hR.continuousOn.mono hSU)).mul (hvij 0 1) | |
| have h2 := (hψ.mul (hS.continuousOn.mono hSU)).mul (hvij 0 0) | |
| have h3 := (hψ.mul (hb.continuousOn.mono hSU)).mul (hvi 1) | |
| have h4 := (hψ.mul (hc.continuousOn.mono hSU)).mul (hvi 0) | |
| rw [pulseWeightedMoment_eq_tested_rectangle ha hd ht | |
| ((testedOperatorSource_contDiffOn hR hS hb hc hv hU).continuousOn.mono hSU)] | |
| have heq : (fun p => pulseSpatialTest a tau p*testedOperatorSource R S b c v p) = | |
| fun p => (((pulseSpatialTest a tau p*coordPartial 1 (coordPartial 1 v) p- | |
| pulseSpatialTest a tau p*R p*coordPartial 0 (coordPartial 1 v) p)- | |
| pulseSpatialTest a tau p*S p*coordPartial 0 (coordPartial 0 v) p)- | |
| pulseSpatialTest a tau p*b p*coordPartial 1 v p)- | |
| pulseSpatialTest a tau p*c p*coordPartial 0 v p := by | |
| funext p | |
| unfold testedOperatorSource hyperbolicOperator | |
| ring | |
| have hs4 := rectangleIntegral_sub hlr hbt (((h0.sub h1).sub h2).sub h3) h4 | |
| have hs3 := rectangleIntegral_sub hlr hbt ((h0.sub h1).sub h2) h3 | |
| have hs2 := rectangleIntegral_sub hlr hbt (h0.sub h1) h2 | |
| have hs1 := rectangleIntegral_sub hlr hbt h0 h1 | |
| simp only [Pi.mul_def,Pi.sub_def] at hs4 hs3 hs2 hs1 | |
| rw [heq,hs4,hs3,hs2,hs1] | |
| theorem testedOperatorSource_moment_cost {a delta tau K K1 M B : ℝ} {N : ℕ} | |
| {R S b c v : Coord → ℝ} {U : Set Coord} {I : Set ℝ} | |
| (ha : 0 < a) (hd : 0 ≤ delta) (ht : 1 ≤ tau) | |
| (hK : 0 ≤ K) (hK1 : 0 ≤ K1) (hM : 0 ≤ M) (hB : 0 ≤ B) | |
| (hR : ContDiffOn ℝ ∞ R U) (hS : ContDiffOn ℝ ∞ S U) | |
| (hb : ContDiffOn ℝ ∞ b U) (hc : ContDiffOn ℝ ∞ c U) | |
| (hv : ContDiffOn ℝ ∞ v U) (hU : IsOpen U) (hSU : pulseStrip a delta tau ⊆ U) | |
| (hcoeff : ∀ p ∈ pulseStrip a delta tau, |R p| ≤ K ∧ |S p| ≤ K ∧ |b p| ≤ K ∧ |c p| ≤ K) | |
| (hderiv : ∀ p ∈ pulseStrip a delta tau, |coordPartial 0 R p| ≤ K1 ∧ |coordPartial 0 S p| ≤ K1) | |
| (hL2 : ∀ theta ∈ Icc (-(delta/tau)) (delta/tau), ∀ i : Fin 2, | |
| Real.sqrt (∫ x in Icc (-a) a, (coordPartial i v (boxPoint x theta))^2) ≤ M*delta*tau/tau^N) | |
| (hI : IsOpen I) (hsub : Icc (-a) a ⊆ I) | |
| (hupper : ∀ x ∈ I, (![x,delta/tau] : Coord) ∈ U) | |
| (hinitial : ∀ x ∈ Icc (-a) a, coordPartial 1 v (boxPoint x (-(delta/tau))) = 0) | |
| (hjet : ∀ k ≤ N, ∀ x ∈ Icc (-a) a, | |
| ‖iteratedFDeriv ℝ k (heightCauchyVelocity v (delta/tau)) x‖ ≤ B) : | |
| |pulseWeightedMoment a delta tau (testedOperatorSource R S b c v)| ≤ | |
| ((2*a)*cutoffEdgeDerivativeBound a ha N B)/tau^N+ | |
| (4*pulseTestDerivativeBound a ha*K*Real.sqrt (2*a)*M)*delta^2*tau/tau^N+ | |
| (4*axisBumpDerivativeBound a ha 0*(K1+K)*Real.sqrt (2*a)*M)*delta^2/tau^N := by | |
| have htpos := zero_lt_one.trans_le ht | |
| have hlr : -a ≤ a := by linarith | |
| have hbt : -(delta/tau) ≤ delta/tau := by linarith [div_nonneg hd htpos.le] | |
| have htime := pulse_temporal_moment_finite_bound ha hB hd htpos hv hU hSU hI hsub hupper hinitial N hjet | |
| have hRcost := pulse_spatial_second_cost 1 ha hd ht hK hK1 hM hR hv hU hSU | |
| (fun p hp => (hcoeff p hp).1) (fun p hp => (hderiv p hp).1) (fun theta ht => hL2 theta ht 1) | |
| have hScost := pulse_spatial_second_cost 0 ha hd ht hK hK1 hM hS hv hU hSU | |
| (fun p hp => (hcoeff p hp).2.1) (fun p hp => (hderiv p hp).2) (fun theta ht => hL2 theta ht 0) | |
| have hbcost := pulse_test_pairing_extra_inv_tau 1 ha hd ht hK hM hU hSU hb hv | |
| (fun p hp => (hcoeff p hp).2.2.1) (fun theta ht => hL2 theta ht 1) | |
| have hccost := pulse_test_pairing_extra_inv_tau 0 ha hd ht hK hM hU hSU hc hv | |
| (fun p hp => (hcoeff p hp).2.2.2) (fun theta ht => hL2 theta ht 0) | |
| have hfirst (F : Coord → ℝ) (i : Fin 2) : | |
| rectangleIntegral (-a) a (-(delta/tau)) (delta/tau) | |
| (fun p => pulseSpatialTest a tau p*F p*coordPartial i v p) = | |
| ∫ theta in Icc (-(delta/tau)) (delta/tau), ∫ x in Icc (-a) a, | |
| pulseTest a tau x*F (boxPoint x theta)*coordPartial i v (boxPoint x theta) := by | |
| rw [rectangleIntegral_eq_Icc_slices _ hlr hbt] | |
| simp only [pulseSpatialTest,pulse_boxPoint_zero] | |
| rw [testedOperatorSource_moment_identity ha hd htpos hR hS hb hc hv hU hSU] | |
| rw [← hfirst b 1] at hbcost | |
| rw [← hfirst c 0] at hccost | |
| have htriangle (a0 a1 a2 a3 a4 : ℝ) : | |
| |(((a0-a1)-a2)-a3)-a4| ≤ (((|a0|+|a1|)+|a2|)+|a3|)+|a4| := by | |
| have h01 := abs_sub a0 a1 | |
| have h02 := (abs_sub (a0-a1) a2).trans (add_le_add h01 le_rfl) | |
| have h03 := (abs_sub ((a0-a1)-a2) a3).trans (add_le_add h02 le_rfl) | |
| exact (abs_sub (((a0-a1)-a2)-a3) a4).trans (add_le_add h03 le_rfl) | |
| calc | |
| _ ≤ (((|rectangleIntegral (-a) a (-(delta/tau)) (delta/tau) | |
| (fun p => pulseSpatialTest a tau p*coordPartial 1 (coordPartial 1 v) p)|+ | |
| |rectangleIntegral (-a) a (-(delta/tau)) (delta/tau) | |
| (fun p => pulseSpatialTest a tau p*R p*coordPartial 0 (coordPartial 1 v) p)|)+ | |
| |rectangleIntegral (-a) a (-(delta/tau)) (delta/tau) | |
| (fun p => pulseSpatialTest a tau p*S p*coordPartial 0 (coordPartial 0 v) p)|)+ | |
| |rectangleIntegral (-a) a (-(delta/tau)) (delta/tau) | |
| (fun p => pulseSpatialTest a tau p*b p*coordPartial 1 v p)|)+ | |
| |rectangleIntegral (-a) a (-(delta/tau)) (delta/tau) | |
| (fun p => pulseSpatialTest a tau p*c p*coordPartial 0 v p)| := by | |
| exact htriangle _ _ _ _ _ | |
| _ ≤ _ := by | |
| have hsum := add_le_add (add_le_add (add_le_add (add_le_add htime hRcost) hScost) hbcost) hccost | |
| apply hsum.trans_eq | |
| ring | |
| end SmoothLocal.Pulse | |
| end | |
| end OAI | |