| import OAI.Geometry.IsometricImmersion.Pulses.ForcingCoefficientSign |
| import OAI.Geometry.IsometricImmersion.Pulses.PulseMoment |
| import Mathlib.MeasureTheory.Integral.Prod |
| import Mathlib.MeasureTheory.Measure.Haar.NormedSpace |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set Filter MeasureTheory |
| open scoped ContDiff Topology Interval |
|
|
| namespace SmoothLocal.Pulse |
| open SmoothLocal.Geometry SmoothLocal.Weighted |
|
|
| def pulseStrip (a delta tau : ℝ) : Set Coord := |
| closedRectangle (-a) a (-(delta/tau)) (delta/tau) |
| def pulsePairRectangle (a delta tau : ℝ) : Set (ℝ × ℝ) := |
| Icc (-(delta/tau)) (delta/tau) ×ˢ Icc (-a) a |
| def pulsePairPoint (q : ℝ × ℝ) : Coord := boxPoint q.2 q.1 |
| def pulseTest (a tau x : ℝ) : ℝ := axisBump a x*Real.cos (tau*x) |
| def pulseMomentWeight (a delta tau : ℝ) (q : ℝ × ℝ) : ℝ := |
| temporalCutoff (tau*q.1/delta)*(axisBump a q.2^2*Real.cos (tau*q.2)^2) |
| def pulseWeightedMoment (a delta tau : ℝ) (f : Coord → ℝ) : ℝ := |
| ∫ q in pulsePairRectangle a delta tau, f (pulsePairPoint q)*pulseTest a tau q.2 |
| ∂volume.prod volume |
|
|
| theorem pulsePairRectangle_isCompact (a delta tau : ℝ) : |
| IsCompact (pulsePairRectangle a delta tau) := isCompact_Icc.prod isCompact_Icc |
|
|
| theorem pulsePairPoint_continuous : Continuous pulsePairPoint := by |
| unfold pulsePairPoint boxPoint |
| fun_prop |
|
|
| theorem pulsePairPoint_mem {a delta tau : ℝ} {q : ℝ × ℝ} |
| (hq : q ∈ pulsePairRectangle a delta tau) : pulsePairPoint q ∈ pulseStrip a delta tau := |
| boxPoint_mem hq.2 hq.1 |
|
|
| theorem pulseStrip_zero_mem {a delta tau : ℝ} (ha : 0 ≤ a) (hd : 0 ≤ delta) (ht : 0 < tau) : |
| boxPoint 0 0 ∈ pulseStrip a delta tau := by |
| apply boxPoint_mem <;> constructor <;> linarith [div_nonneg hd ht.le] |
|
|
| theorem pulsePair_integrableOn {a delta tau : ℝ} {f : Coord → ℝ} |
| (hf : ContinuousOn f (pulseStrip a delta tau)) : |
| IntegrableOn (fun q => f (pulsePairPoint q)) (pulsePairRectangle a delta tau) |
| (volume.prod volume) := |
| (hf.comp pulsePairPoint_continuous.continuousOn (fun _ hq => pulsePairPoint_mem hq)).integrableOn_compact |
| (pulsePairRectangle_isCompact a delta tau) |
|
|
| theorem pulsePair_integral_eq_rectangleIntegral {a delta tau : ℝ} {f : Coord → ℝ} |
| (ha : 0 ≤ a) (hd : 0 ≤ delta) (ht : 0 < tau) |
| (hf : ContinuousOn f (pulseStrip a delta tau)) : |
| (∫ q in pulsePairRectangle a delta tau, f (pulsePairPoint q) ∂volume.prod volume) = |
| rectangleIntegral (-a) a (-(delta/tau)) (delta/tau) f := by |
| have htime : -(delta/tau) ≤ delta/tau := by linarith [div_nonneg hd ht.le] |
| have hspace : -a ≤ a := by linarith |
| rw [show pulsePairRectangle a delta tau = |
| Icc (-(delta/tau)) (delta/tau) ×ˢ Icc (-a) a from rfl, |
| setIntegral_prod _ (pulsePair_integrableOn hf)] |
| unfold rectangleIntegral |
| rw [intervalIntegral.integral_of_le htime,←integral_Icc_eq_integral_Ioc] |
| apply setIntegral_congr_fun measurableSet_Icc |
| intro theta htheta |
| dsimp only |
| rw [intervalIntegral.integral_of_le hspace,←integral_Icc_eq_integral_Ioc] |
| rfl |
|
|
| theorem pulseMomentWeight_continuous (a delta tau : ℝ) : |
| Continuous (pulseMomentWeight a delta tau) := by |
| unfold pulseMomentWeight temporalCutoff |
| exact (((axisBump_contDiff (1/2)).continuous.comp (by fun_prop)).mul |
| ((((axisBump_contDiff a).continuous.comp continuous_snd).pow 2).mul (by fun_prop))) |
|
|
| theorem pulseMomentWeight_nonneg (a delta tau : ℝ) (q : ℝ × ℝ) : |
| 0 ≤ pulseMomentWeight a delta tau q := |
| mul_nonneg (axisBump_nonneg _ _) (mul_nonneg (sq_nonneg _) (sq_nonneg _)) |
|
|
| theorem pulseTest_continuous (a tau : ℝ) : Continuous (pulseTest a tau) := |
| (axisBump_contDiff a).continuous.mul (by fun_prop) |
|
|
| theorem pulseTest_hasCompactSupport {a : ℝ} (ha : 0 < a) (tau : ℝ) : |
| HasCompactSupport (pulseTest a tau) := (axisBump_hasCompactSupport ha).mul_right |
|
|
| theorem spatial_moment_Icc_eq {a : ℝ} (ha : 0 < a) (tau : ℝ) : |
| (∫ x in Icc (-a) a, axisBump a x^2*Real.cos (tau*x)^2) = |
| ∫ x : ℝ, axisBump a x^2*Real.cos (tau*x)^2 := by |
| apply setIntegral_eq_integral_of_forall_compl_eq_zero |
| intro x hx |
| have hz : axisBump a x = 0 := by |
| by_contra hn |
| have h := abs_lt.mp (axisBump_support_bound ha hn) |
| exact hx ⟨h.1.le,h.2.le⟩ |
| simp only [hz,zero_pow (by norm_num : (2 : ℕ) ≠ 0),zero_mul] |
|
|
| theorem temporal_pulse_integral {delta tau : ℝ} (hd : 0 < delta) (ht : 0 < tau) : |
| (∫ theta in Icc (-(delta/tau)) (delta/tau), temporalCutoff (tau*theta/delta)) = |
| (delta/tau)*(∫ theta : ℝ, temporalCutoff theta) := by |
| have hglobal : (∫ theta : ℝ, temporalCutoff (tau*theta/delta)) = |
| (delta/tau)*(∫ theta : ℝ, temporalCutoff theta) := by |
| have heq : (fun theta : ℝ => temporalCutoff (tau*theta/delta)) = |
| fun theta => temporalCutoff ((tau/delta)*theta) := by |
| funext theta |
| congr 1 |
| ring |
| rw [heq,Measure.integral_comp_mul_left] |
| simp only [inv_div,abs_of_pos (div_pos hd ht),smul_eq_mul] |
| rw [←hglobal] |
| apply setIntegral_eq_integral_of_forall_compl_eq_zero |
| intro theta htheta |
| by_contra hn |
| have h := axisBump_support_bound (by norm_num : (0 : ℝ) < 1/2) hn |
| rw [abs_div,abs_mul,abs_of_pos ht,abs_of_pos hd] at h |
| have hmul : tau*|theta| < delta/2 := by |
| have hh := (div_lt_iff₀ hd).mp h |
| linarith |
| have htheta' : |theta| ≤ delta/tau := (le_div_iff₀ ht).mpr (by nlinarith) |
| exact htheta (abs_le.mp htheta') |
|
|
| theorem pulseMomentWeight_integral {a delta tau : ℝ} |
| (ha : 0 < a) (hd : 0 < delta) (ht : 0 < tau) : |
| (∫ q in pulsePairRectangle a delta tau, pulseMomentWeight a delta tau q |
| ∂volume.prod volume) = |
| (delta/tau)*(∫ theta : ℝ, temporalCutoff theta)* |
| (∫ x : ℝ, axisBump a x^2*Real.cos (tau*x)^2) := by |
| rw [show pulsePairRectangle a delta tau = |
| Icc (-(delta/tau)) (delta/tau) ×ˢ Icc (-a) a from rfl] |
| unfold pulseMomentWeight |
| rw [setIntegral_prod_mul (fun theta : ℝ => temporalCutoff (tau*theta/delta)) |
| (fun x : ℝ => axisBump a x^2*Real.cos (tau*x)^2) |
| (Icc (-(delta/tau)) (delta/tau)) (Icc (-a) a), |
| temporal_pulse_integral hd ht,spatial_moment_Icc_eq ha tau] |
|
|
| end SmoothLocal.Pulse |
|
|
| end |
|
|
| end OAI |
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|
