| import OAI.Geometry.IsometricImmersion.Pulses.PulseMomentError |
| import OAI.Geometry.IsometricImmersion.Caps.EnergyCauchySchwarz |
| import OAI.Geometry.IsometricImmersion.Pulses.PulseDerivativeBounds |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set Filter MeasureTheory |
| open scoped ContDiff Topology |
|
|
| namespace SmoothLocal.Pulse |
| open SmoothLocal.Geometry SmoothLocal.Weighted SmoothLocal.Hyperbolic |
|
|
| theorem pulseTest_contDiff (a tau : ℝ) : ContDiff ℝ ∞ (pulseTest a tau) := |
| (axisBump_contDiff a).mul (Real.contDiff_cos.comp (contDiff_const.mul contDiff_id)) |
|
|
| theorem pulseTest_hasDerivAt (a tau x : ℝ) : |
| HasDerivAt (pulseTest a tau) |
| (deriv (axisBump a) x*Real.cos (tau*x)-tau*axisBump a x*Real.sin (tau*x)) x := by |
| have hχ := ((axisBump_contDiff a).differentiable (by simp) x).hasDerivAt |
| have hc := (Real.hasDerivAt_cos (tau*x)).comp x ((hasDerivAt_id x).const_mul tau) |
| have h := hχ.mul hc |
| simp only [Function.comp_def,mul_one] at h |
| convert h using 1 <;> (first | rfl | ring) |
|
|
| theorem pulseTest_deriv (a tau x : ℝ) : |
| deriv (pulseTest a tau) x = |
| deriv (axisBump a) x*Real.cos (tau*x)-tau*axisBump a x*Real.sin (tau*x) := |
| (pulseTest_hasDerivAt a tau x).deriv |
|
|
| def pulseTestDerivativeBound (a : ℝ) (ha : 0 < a) : ℝ := |
| axisBumpDerivativeBound a ha 1+axisBumpDerivativeBound a ha 0 |
|
|
| theorem pulseTestDerivativeBound_nonneg {a : ℝ} (ha : 0 < a) : |
| 0 ≤ pulseTestDerivativeBound a ha := |
| add_nonneg (axisBumpDerivativeBound_pos ha 1).le (axisBumpDerivativeBound_pos ha 0).le |
|
|
| theorem pulseTest_deriv_bound {a tau : ℝ} (ha : 0 < a) (ht : 1 ≤ tau) (x : ℝ) : |
| |deriv (pulseTest a tau) x| ≤ pulseTestDerivativeBound a ha*tau := by |
| have hA0 := (axisBumpDerivativeBound_pos ha 0).le |
| have hA1 := (axisBumpDerivativeBound_pos ha 1).le |
| have hχ : |axisBump a x| ≤ axisBumpDerivativeBound a ha 0 := by |
| simpa only [norm_iteratedFDeriv_zero,Real.norm_eq_abs] using axisBump_derivative_le ha 0 x |
| have hχ' : |deriv (axisBump a) x| ≤ axisBumpDerivativeBound a ha 1 := by |
| simpa only [norm_iteratedFDeriv_eq_norm_iteratedDeriv,iteratedDeriv_one,Real.norm_eq_abs] |
| using axisBump_derivative_le ha 1 x |
| have ht0 : 0 ≤ tau := zero_le_one.trans ht |
| rw [pulseTest_deriv] |
| calc |
| _ ≤ |deriv (axisBump a) x*Real.cos (tau*x)|+ |
| |tau*axisBump a x*Real.sin (tau*x)| := abs_sub _ _ |
| _ ≤ axisBumpDerivativeBound a ha 1+tau*axisBumpDerivativeBound a ha 0 := by |
| apply add_le_add |
| · rw [abs_mul] |
| simpa only [mul_one] using mul_le_mul hχ' (Real.abs_cos_le_one _) (abs_nonneg _) hA1 |
| · rw [abs_mul,abs_mul,abs_of_nonneg ht0] |
| simpa only [mul_one] using mul_le_mul |
| (mul_le_mul_of_nonneg_left hχ ht0) (Real.abs_sin_le_one _) (abs_nonneg _) (mul_nonneg ht0 hA0) |
| _ ≤ _ := by |
| unfold pulseTestDerivativeBound |
| nlinarith [mul_le_mul_of_nonneg_left ht hA1] |
|
|
| theorem real_Icc_memLp_two {f : ℝ → ℝ} {l r : ℝ} (hf : ContinuousOn f (Icc l r)) : |
| MemLp f 2 (volume.restrict (Icc l r)) := |
| (memLp_two_iff_integrable_sq (hf.aestronglyMeasurable measurableSet_Icc)).mpr |
| ((hf.pow 2).integrableOn_Icc) |
|
|
| theorem sqrt_Icc_square_le_of_abs {f : ℝ → ℝ} {l r F : ℝ} |
| (hlr : l ≤ r) (hF : 0 ≤ F) (hf : ContinuousOn f (Icc l r)) |
| (hbound : ∀ x ∈ Icc l r, |f x| ≤ F) : |
| Real.sqrt (∫ x in Icc l r, f x^2) ≤ F*Real.sqrt (r-l) := by |
| have hsq : (∫ x in Icc l r, f x^2) ≤ F^2*(r-l) := by |
| have hm := integral_mono_ae (μ := volume.restrict (Icc l r)) ((hf.pow 2).integrableOn_Icc) |
| ((continuous_const : Continuous (fun _ : ℝ => F^2)).continuousOn.integrableOn_Icc) (by |
| filter_upwards [ae_restrict_mem measurableSet_Icc] with x hx |
| have hh := (sq_le_sq₀ (abs_nonneg (f x)) hF).mpr (hbound x hx) |
| simpa only [Pi.pow_apply,sq_abs] using hh) |
| rw [setIntegral_const,Real.volume_real_Icc_of_le hlr,smul_eq_mul] at hm |
| simpa only [Pi.pow_apply,mul_comm] using hm |
| apply (Real.sqrt_le_sqrt hsq).trans_eq |
| rw [Real.sqrt_mul (sq_nonneg F),Real.sqrt_sq_eq_abs,abs_of_nonneg hF] |
|
|
| theorem real_Icc_pairing_bound {f h : ℝ → ℝ} {l r F G : ℝ} |
| (hlr : l ≤ r) (hF : 0 ≤ F) (hf : ContinuousOn f (Icc l r)) |
| (hh : ContinuousOn h (Icc l r)) (hbound : ∀ x ∈ Icc l r, |f x| ≤ F) |
| (hL2 : Real.sqrt (∫ x in Icc l r, h x^2) ≤ G) : |
| |∫ x in Icc l r, f x*h x| ≤ F*Real.sqrt (r-l)*G := by |
| have hcs := integral_product_abs_le_sqrt_squares (real_Icc_memLp_two hf) (real_Icc_memLp_two hh) |
| exact hcs.trans (mul_le_mul (sqrt_Icc_square_le_of_abs hlr hF hf hbound) hL2 |
| (Real.sqrt_nonneg _) (mul_nonneg hF (Real.sqrt_nonneg _))) |
|
|
| theorem pulse_strip_pairing_bound (i : Fin 2) {a delta tau F S0 G : ℝ} |
| {test : ℝ → ℝ} {S v : Coord → ℝ} {U : Set Coord} |
| (ha : 0 < a) (hd : 0 ≤ delta) (ht : 0 < tau) |
| (hF : 0 ≤ F) (hS0 : 0 ≤ S0) (_ : 0 ≤ G) |
| (hU : IsOpen U) (hSU : pulseStrip a delta tau ⊆ U) |
| (hS : ContDiffOn ℝ ∞ S U) (hv : ContDiffOn ℝ ∞ v U) |
| (htest : Continuous test) (htestB : ∀ x, |test x| ≤ F) |
| (hSB : ∀ p ∈ pulseStrip a delta tau, |S p| ≤ S0) |
| (hL2 : ∀ theta ∈ Icc (-(delta/tau)) (delta/tau), |
| Real.sqrt (∫ x in Icc (-a) a, (coordPartial i v (boxPoint x theta))^2) ≤ G) : |
| |∫ theta in Icc (-(delta/tau)) (delta/tau), ∫ x in Icc (-a) a, |
| test x*S (boxPoint x theta)*coordPartial i v (boxPoint x theta)| ≤ |
| (F*S0*Real.sqrt (2*a)*G)*(2*delta/tau) := by |
| have hslice (theta : ℝ) (htheta : theta ∈ Icc (-(delta/tau)) (delta/tau)) : |
| |∫ x in Icc (-a) a, test x*S (boxPoint x theta)*coordPartial i v (boxPoint x theta)| ≤ |
| F*S0*Real.sqrt (2*a)*G := by |
| have hmap : ∀ x ∈ Icc (-a) a, boxPoint x theta ∈ U := |
| fun x hx => hSU (boxPoint_mem hx htheta) |
| have hSc : ContinuousOn (fun x => S (boxPoint x theta)) (Icc (-a) a) := |
| hS.continuousOn.comp (by unfold boxPoint; fun_prop) hmap |
| have hvc : ContinuousOn (fun x => coordPartial i v (boxPoint x theta)) (Icc (-a) a) := |
| (partial_contDiffOn hv hU i).continuousOn.comp (by unfold boxPoint; fun_prop) hmap |
| have hproduct : ∀ x ∈ Icc (-a) a, |test x*S (boxPoint x theta)| ≤ F*S0 := by |
| intro x hx |
| rw [abs_mul] |
| exact mul_le_mul (htestB x) (hSB _ (boxPoint_mem hx htheta)) (abs_nonneg _) hF |
| have hh := real_Icc_pairing_bound (by linarith : -a ≤ a) (mul_nonneg hF hS0) |
| (htest.continuousOn.mul hSc) hvc hproduct (hL2 theta htheta) |
| simpa only [Pi.mul_apply,sub_neg_eq_add,←two_mul] using hh |
| have htime : -(delta/tau) ≤ delta/tau := by linarith [div_nonneg hd ht.le] |
| have hmajor : Integrable (fun _ : ℝ => F*S0*Real.sqrt (2*a)*G) |
| (volume.restrict (Icc (-(delta/tau)) (delta/tau))) := |
| (continuous_const : Continuous |
| (fun _ : ℝ => F*S0*Real.sqrt (2*a)*G)).continuousOn.integrableOn_Icc |
| have hpoint : ∀ᵐ theta ∂volume.restrict (Icc (-(delta/tau)) (delta/tau)), |
| ‖∫ x in Icc (-a) a, test x*S (boxPoint x theta)*coordPartial i v (boxPoint x theta)‖ ≤ |
| F*S0*Real.sqrt (2*a)*G := by |
| filter_upwards [ae_restrict_mem measurableSet_Icc] with theta htheta |
| simpa only [Real.norm_eq_abs] using hslice theta htheta |
| have hi := norm_integral_le_of_norm_le hmajor hpoint |
| rw [setIntegral_const,Real.volume_real_Icc_of_le htime,smul_eq_mul] at hi |
| simp only [Real.norm_eq_abs] at hi |
| apply hi.trans_eq |
| ring |
|
|
| theorem pulse_derivative_pairing_delta_sq (i : Fin 2) {a delta tau S0 M : ℝ} {N : ℕ} |
| {S v : Coord → ℝ} {U : Set Coord} |
| (ha : 0 < a) (hd : 0 ≤ delta) (ht : 1 ≤ tau) |
| (hS0 : 0 ≤ S0) (hM : 0 ≤ M) (hU : IsOpen U) (hSU : pulseStrip a delta tau ⊆ U) |
| (hS : ContDiffOn ℝ ∞ S U) (hv : ContDiffOn ℝ ∞ v U) |
| (hSB : ∀ p ∈ pulseStrip a delta tau, |S p| ≤ S0) |
| (hL2 : ∀ theta ∈ Icc (-(delta/tau)) (delta/tau), |
| Real.sqrt (∫ x in Icc (-a) a, (coordPartial i v (boxPoint x theta))^2) ≤ M*delta*tau/tau^N) : |
| |∫ theta in Icc (-(delta/tau)) (delta/tau), ∫ x in Icc (-a) a, |
| deriv (pulseTest a tau) x*S (boxPoint x theta)*coordPartial i v (boxPoint x theta)| ≤ |
| (2*pulseTestDerivativeBound a ha*S0*Real.sqrt (2*a)*M)*delta^2*tau/tau^N := by |
| have htpos : 0 < tau := zero_lt_one.trans_le ht |
| have hF : 0 ≤ pulseTestDerivativeBound a ha*tau := |
| mul_nonneg (pulseTestDerivativeBound_nonneg ha) htpos.le |
| have hG : 0 ≤ M*delta*tau/tau^N := by positivity |
| have htest : Continuous (deriv (pulseTest a tau)) := |
| (contDiff_infty_iff_deriv.mp (pulseTest_contDiff a tau)).2.continuous |
| have hh := pulse_strip_pairing_bound i ha hd htpos hF hS0 hG hU hSU hS hv htest |
| (pulseTest_deriv_bound ha ht) hSB hL2 |
| apply hh.trans_eq |
| field_simp [htpos.ne'] |
| |
| theorem pulse_test_pairing_extra_inv_tau (i : Fin 2) {a delta tau S0 M : ℝ} {N : ℕ} |
| {S v : Coord → ℝ} {U : Set Coord} |
| (ha : 0 < a) (hd : 0 ≤ delta) (ht : 1 ≤ tau) |
| (hS0 : 0 ≤ S0) (hM : 0 ≤ M) (hU : IsOpen U) (hSU : pulseStrip a delta tau ⊆ U) |
| (hS : ContDiffOn ℝ ∞ S U) (hv : ContDiffOn ℝ ∞ v U) |
| (hSB : ∀ p ∈ pulseStrip a delta tau, |S p| ≤ S0) |
| (hL2 : ∀ theta ∈ Icc (-(delta/tau)) (delta/tau), |
| Real.sqrt (∫ x in Icc (-a) a, (coordPartial i v (boxPoint x theta))^2) ≤ M*delta*tau/tau^N) : |
| |∫ theta in Icc (-(delta/tau)) (delta/tau), ∫ x in Icc (-a) a, |
| pulseTest a tau x*S (boxPoint x theta)*coordPartial i v (boxPoint x theta)| ≤ |
| (2*axisBumpDerivativeBound a ha 0*S0*Real.sqrt (2*a)*M)*delta^2/tau^N := by |
| have htpos : 0 < tau := zero_lt_one.trans_le ht |
| have hF := (axisBumpDerivativeBound_pos ha 0).le |
| have hG : 0 ≤ M*delta*tau/tau^N := by positivity |
| have htest (x : ℝ) : |pulseTest a tau x| ≤ axisBumpDerivativeBound a ha 0 := by |
| have hχ : |axisBump a x| ≤ axisBumpDerivativeBound a ha 0 := by |
| simpa only [norm_iteratedFDeriv_zero,Real.norm_eq_abs] using axisBump_derivative_le ha 0 x |
| exact (pulseTest_abs_le a tau x).trans ((le_abs_self _).trans hχ) |
| have hh := pulse_strip_pairing_bound i ha hd htpos hF hS0 hG hU hSU hS hv |
| (pulseTest_continuous a tau) htest hSB hL2 |
| apply hh.trans_eq |
| field_simp [htpos.ne'] |
|
|
| end SmoothLocal.Pulse |
|
|
| end |
|
|
| end OAI |
|
|
