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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 | import OAI.Geometry.IsometricImmersion.Pulses.ActualForcingMoment
namespace OAI
noncomputable section
open Set Filter MeasureTheory
open scoped ContDiff Topology
namespace SmoothLocal.Pulse
open SmoothLocal.Geometry SmoothLocal.Weighted
theorem pulseTest_abs_le (a tau x : ℝ) : |pulseTest a tau x| ≤ axisBump a x := by
unfold pulseTest
rw [abs_mul,abs_of_nonneg (axisBump_nonneg a x)]
simpa only [mul_one] using mul_le_mul_of_nonneg_left (Real.abs_cos_le_one _) (axisBump_nonneg a x)
theorem axisBump_Icc_integral {a : ℝ} (ha : 0 < a) :
(∫ x in Icc (-a) a, axisBump a x) = ∫ x : ℝ, axisBump a x := by
apply setIntegral_eq_integral_of_forall_compl_eq_zero
intro x hx
by_contra hn
have h := abs_lt.mp (axisBump_support_bound ha hn)
exact hx ⟨h.1.le,h.2.le⟩
theorem pulse_spatial_mass_integral {a delta tau : ℝ}
(ha : 0 < a) (hd : 0 ≤ delta) (ht : 0 < tau) :
(∫ q in pulsePairRectangle a delta tau, axisBump a q.2 ∂volume.prod volume) =
(2*delta/tau)*(∫ x : ℝ, axisBump a x) := by
have hw : -(delta/tau) ≤ delta/tau := by linarith [div_nonneg hd ht.le]
have hh := setIntegral_prod_mul (μ := (volume : Measure ℝ)) (ν := (volume : Measure ℝ))
(fun _ : ℝ => (1 : ℝ)) (axisBump a) (Icc (-(delta/tau)) (delta/tau)) (Icc (-a) a)
simp only [one_mul] at hh
rw [show pulsePairRectangle a delta tau =
Icc (-(delta/tau)) (delta/tau) ×ˢ Icc (-a) a from rfl,hh,
setIntegral_const,Real.volume_real_Icc_of_le hw,smul_eq_mul,mul_one,axisBump_Icc_integral ha]
ring
theorem pulseWeightedMoment_bound {a delta tau B : ℝ} {f : Coord → ℝ}
(ha : 0 < a) (hd : 0 ≤ delta) (ht : 0 < tau) (hB : 0 ≤ B)
(hf : ∀ p ∈ pulseStrip a delta tau, |f p| ≤ B) :
|pulseWeightedMoment a delta tau f| ≤
B*(2*delta/tau)*(∫ x : ℝ, axisBump a x) := by
have hmajor : IntegrableOn (fun q : ℝ × ℝ => B*axisBump a q.2)
(pulsePairRectangle a delta tau) (volume.prod volume) :=
((continuous_const.mul ((axisBump_contDiff a).continuous.comp continuous_snd)).continuousOn).integrableOn_compact
(pulsePairRectangle_isCompact a delta tau)
have hpoint : ∀ᵐ q ∂(volume.prod volume).restrict (pulsePairRectangle a delta tau),
‖f (pulsePairPoint q)*pulseTest a tau q.2‖ ≤ B*axisBump a q.2 := by
filter_upwards [ae_restrict_mem (pulsePairRectangle_isCompact a delta tau).measurableSet] with q hq
rw [Real.norm_eq_abs,abs_mul]
exact mul_le_mul (hf _ (pulsePairPoint_mem hq)) (pulseTest_abs_le a tau q.2)
(abs_nonneg _) hB
have h := norm_integral_le_of_norm_le hmajor hpoint
simp only [Real.norm_eq_abs,integral_const_mul] at h
rw [pulse_spatial_mass_integral ha hd ht] at h
simpa only [pulseWeightedMoment,mul_assoc] using h
theorem pulseWeightedMoment_integrable {a delta tau : ℝ} {f : Coord → ℝ}
(hf : ContinuousOn f (pulseStrip a delta tau)) :
IntegrableOn (fun q => f (pulsePairPoint q)*pulseTest a tau q.2)
(pulsePairRectangle a delta tau) (volume.prod volume) :=
((hf.comp pulsePairPoint_continuous.continuousOn (fun _ hq => pulsePairPoint_mem hq)).mul
((pulseTest_continuous a tau).comp continuous_snd).continuousOn).integrableOn_compact
(pulsePairRectangle_isCompact a delta tau)
theorem pulseWeightedMoment_sub {a delta tau : ℝ} {f h : Coord → ℝ}
(hf : ContinuousOn f (pulseStrip a delta tau)) (hh : ContinuousOn h (pulseStrip a delta tau)) :
pulseWeightedMoment a delta tau (fun p => f p-h p) =
pulseWeightedMoment a delta tau f-pulseWeightedMoment a delta tau h := by
unfold pulseWeightedMoment
simp_rw [sub_mul]
exact integral_sub (pulseWeightedMoment_integrable hf) (pulseWeightedMoment_integrable hh)
theorem pulseWeightedMoment_lower_of_remainder {a delta tau B L : ℝ} {f h : Coord → ℝ}
(ha : 0 < a) (hd : 0 ≤ delta) (ht : 0 < tau) (hB : 0 ≤ B)
(hf : ContinuousOn f (pulseStrip a delta tau)) (hh : ContinuousOn h (pulseStrip a delta tau))
(hrem : ∀ p ∈ pulseStrip a delta tau, |f p-h p| ≤ B)
(hlead : L ≤ |pulseWeightedMoment a delta tau h|) :
L-B*(2*delta/tau)*(∫ x : ℝ, axisBump a x) ≤ |pulseWeightedMoment a delta tau f| := by
have herr := pulseWeightedMoment_bound ha hd ht hB hrem
rw [pulseWeightedMoment_sub hf hh] at herr
have htriangle : |pulseWeightedMoment a delta tau h| ≤
|pulseWeightedMoment a delta tau f|+
|pulseWeightedMoment a delta tau f-pulseWeightedMoment a delta tau h| := by
calc
_ = |pulseWeightedMoment a delta tau f-
(pulseWeightedMoment a delta tau f-pulseWeightedMoment a delta tau h)| := by congr 1; ring
_ ≤ _ := abs_sub _ _
linarith
theorem pulseWeightedMoment_lower_order_error {a delta tau C : ℝ} {N : ℕ} {f : Coord → ℝ}
(ha : 0 < a) (hd : 0 ≤ delta) (ht : 0 < tau) (hC : 0 ≤ C)
(hf : ∀ p ∈ pulseStrip a delta tau, |f p| ≤ C*tau/tau^N) :
|pulseWeightedMoment a delta tau f| ≤
(2*C*(∫ x : ℝ, axisBump a x))*delta/tau^N := by
have hB : 0 ≤ C*tau/tau^N := div_nonneg (mul_nonneg hC ht.le) (pow_nonneg ht.le N)
apply (pulseWeightedMoment_bound ha hd ht hB hf).trans_eq
field_simp [ht.ne']
end SmoothLocal.Pulse
end
end OAI
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