| import OAI.Geometry.RuledCovering.SingularEnergy |
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| namespace OAI |
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| noncomputable section |
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| open Set MeasureTheory Filter Metric |
| open scoped Topology ContDiff Pointwise |
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| namespace RuledApproximation |
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| lemma ceilingPolynomial_integral (p : ℝ) : |
| (∫ b in Icc (-(1/2:ℝ)) (1/2), ceilingPolynomial p b) = (p+3)/120 := by |
| rw [integral_Icc_eq_integral_Ioc,← intervalIntegral.integral_of_le (by norm_num : -(1/2:ℝ) ≤ 1/2)] |
| have he : ceilingPolynomial p = fun b => (p/4)*b^2+(2-p)*b^4 := by |
| funext b |
| unfold ceilingPolynomial |
| ring |
| rw [he,intervalIntegral.integral_add ((by fun_prop : Continuous (fun b : ℝ => (p/4)*b^2)).intervalIntegrable _ _) |
| ((by fun_prop : Continuous (fun b : ℝ => (2-p)*b^4)).intervalIntegrable _ _), |
| intervalIntegral.integral_const_mul,intervalIntegral.integral_const_mul,integral_pow,integral_pow] |
| norm_num |
| ring |
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| def angularEnergy (p eta : ℝ) : ℝ := |
| (1/4:ℝ)*(p+(2-p)*eta^2)^2+2*eta^2 |
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| lemma angularEnergy_continuous (p : ℝ) : Continuous (angularEnergy p) := by |
| unfold angularEnergy |
| fun_prop |
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| lemma angularEnergy_integral (p : ℝ) : |
| (∫ eta in (-1:ℝ)..1, angularEnergy p eta) = 2*(2*p^2+2*p+13)/15 := by |
| have he : angularEnergy p = fun eta => p^2/4+(p*(2-p)/2+2)*eta^2+((2-p)^2/4)*eta^4 := by |
| funext eta |
| unfold angularEnergy |
| ring |
| have hI₁ : IntervalIntegrable (fun eta : ℝ => p^2/4+(p*(2-p)/2+2)*eta^2) volume (-1) 1 := |
| (by fun_prop : Continuous (fun eta : ℝ => p^2/4+(p*(2-p)/2+2)*eta^2)).intervalIntegrable _ _ |
| have hI₂ : IntervalIntegrable (fun eta : ℝ => ((2-p)^2/4)*eta^4) volume (-1) 1 := |
| (by fun_prop : Continuous (fun eta : ℝ => ((2-p)^2/4)*eta^4)).intervalIntegrable _ _ |
| rw [he,intervalIntegral.integral_add hI₁ hI₂, |
| intervalIntegral.integral_add (intervalIntegrable_const) |
| ((by fun_prop : Continuous (fun eta : ℝ => (p*(2-p)/2+2)*eta^2)).intervalIntegrable _ _), |
| intervalIntegral.integral_const,intervalIntegral.integral_const_mul,intervalIntegral.integral_const_mul, |
| integral_pow,integral_pow] |
| norm_num |
| ring |
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| lemma singularEnergy_polar (p eta : ℝ) {r : ℝ} (hr : 0 < r) : |
| singularEnergy p (vec2 (r*eta) r) = (1/4:ℝ)*(2*r)^(2*p-2)*angularEnergy p eta := by |
| have he : (2*r) • vec2 (eta/2) (1/2) = vec2 (r*eta) r := by |
| ext i |
| fin_cases i |
| · change (2*r)*(eta/2)=r*eta |
| ring |
| · change (2*r)*(1/2)=r |
| ring |
| have hv := singularField_smul p (show 0 < 2*r by positivity) |
| (z := vec2 (eta/2) (1/2)) (by norm_num) |
| rw [he,singularField_level] at hv |
| have hpw : (2*r)^(2*p-2) = ((2*r)^(p-1))^2 := by |
| rw [← Real.rpow_natCast,← Real.rpow_mul (by positivity)] |
| congr 1 |
| ring |
| unfold singularEnergy |
| rw [hv,hpw] |
| simp only [PiLp.smul_apply,smul_eq_mul,vec2_apply_zero,vec2_apply_one] |
| unfold angularEnergy |
| ring |
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| lemma singularEnergy_slice_integral {p : ℝ} (hp : 1 < p) {r : ℝ} (hr : 0 ≤ r) : |
| (∫ b in Icc (-r) r, singularEnergy p (vec2 b r)) = |
| (2*r)^(2*p-1)/8 * (∫ eta in (-1:ℝ)..1, angularEnergy p eta) := by |
| by_cases hr0 : r = 0 |
| · subst r |
| simp [Real.zero_rpow (show 2*p-1 ≠ 0 by linarith)] |
| have hrp : 0 < r := lt_of_le_of_ne hr (Ne.symm hr0) |
| rw [integral_Icc_eq_integral_Ioc,← intervalIntegral.integral_of_le (by linarith : -r ≤ r)] |
| have hs := intervalIntegral.smul_integral_comp_mul_left |
| (a := (-1:ℝ)) (b := 1) (fun b : ℝ => singularEnergy p (vec2 b r)) r |
| simp only [smul_eq_mul,mul_neg_one,mul_one] at hs |
| rw [← hs] |
| have hi : (∫ eta in (-1:ℝ)..1, singularEnergy p (vec2 (r*eta) r)) = |
| (1/4:ℝ)*(2*r)^(2*p-2)*(∫ eta in (-1:ℝ)..1, angularEnergy p eta) := by |
| simp_rw [singularEnergy_polar p _ hrp] |
| exact intervalIntegral.integral_const_mul _ _ |
| rw [hi,show 2*p-1=2*p-2+1 by ring,Real.rpow_add_one (by positivity : 2*r ≠ 0)] |
| ring |
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| lemma singular_radial_integral {p : ℝ} (hp : 0 < p) : |
| (∫ r in (0:ℝ)..(1/2:ℝ), (2*r)^(2*p-1)) = 1/(4*p) := by |
| have he := intervalIntegral.integral_comp_mul_left (a := (0:ℝ)) (b := (1/2:ℝ)) |
| (fun r : ℝ => r^(2*p-1)) (c := (2:ℝ)) (by norm_num) |
| norm_num only [mul_zero,mul_one_div_cancel (by norm_num : (2:ℝ) ≠ 0),smul_eq_mul] at he |
| rw [he,integral_rpow (Or.inl (by linarith : -1 < 2*p-1))] |
| rw [show 2*p-1+1=2*p by ring,Real.one_rpow,Real.zero_rpow (by positivity : 2*p ≠ 0)] |
| field_simp |
| ring |
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| lemma singularEnergy_integral {p : ℝ} (hp : 1 < p) : |
| (∫ z in coneTriangle (1/2), singularEnergy p z) = (2*p^2+2*p+13)/(240*p) := by |
| have hsub : coneTriangle (1/2) ⊆ coneTriangle 2 := fun z hz => |
| ⟨hz.1,hz.2.1.trans (by norm_num),hz.2.2⟩ |
| have heq : ∀ z : Plane, z ∈ coneTriangle (1/2) ↔ |
| z 1 ∈ Icc (0:ℝ) (1/2) ∧ z 0 ∈ Icc (-z 1) (z 1) := by |
| intro z |
| simp only [coneTriangle,mem_ofPred_eq,mem_Icc,← abs_le] |
| tauto |
| rw [plane_setIntegral_horizontal_interval (coneTriangle_isCompact (1/2)).measurableSet |
| ((singularEnergy_integrable hp).mono_set hsub) (s := Icc (0:ℝ) (1/2)) |
| (l := fun r => -r) (u := id) measurableSet_Icc heq] |
| have he : (∫ r in Icc (0:ℝ) (1/2), ∫ b in Icc (-r) r, singularEnergy p (vec2 b r)) = |
| ∫ r in Icc (0:ℝ) (1/2), (2*r)^(2*p-1)/8*(∫ eta in (-1:ℝ)..1, angularEnergy p eta) := by |
| apply setIntegral_congr_fun measurableSet_Icc |
| intro r hr |
| exact singularEnergy_slice_integral hp hr.1 |
| simp only [id_eq] |
| rw [he,angularEnergy_integral p] |
| rw [integral_Icc_eq_integral_Ioc,← intervalIntegral.integral_of_le (by norm_num : (0:ℝ) ≤ 1/2)] |
| have hf : (fun r : ℝ => (2*r)^(2*p-1)/8*(2*(2*p^2+2*p+13)/15)) = |
| fun r => (2*(2*p^2+2*p+13)/15/8)*(2*r)^(2*p-1) := by funext r; ring |
| rw [hf,intervalIntegral.integral_const_mul,singular_radial_integral (by linarith : 0 < p)] |
| field_simp |
| ring |
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| lemma singular_exact_coefficient {p : ℝ} (hp : 1 < p) : |
| (∫ b in Icc (-(1/2:ℝ)) (1/2), ceilingPolynomial p b) - |
| (∫ z in coneTriangle (1/2), singularEnergy p z)/2 = singularCoefficient p := by |
| rw [ceilingPolynomial_integral,singularEnergy_integral hp] |
| unfold singularCoefficient |
| have hp0 : p ≠ 0 := by linarith |
| field_simp |
| ring |
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| lemma energyCorrection_integrable {p : ℝ} (hp : 1 < p) (gamma c : ℝ) : |
| IntegrableOn (energyCorrection p c) (singularDomain p gamma c) := by |
| apply ((singularEnergy_integrable hp).mono_set (singularDomain_subset_cone p gamma c)).div_const 2 |>.mono' |
| (energyCorrection_measurable p c).aestronglyMeasurable |
| exact Filter.Eventually.of_forall (energyCorrection_norm p c) |
| |
| lemma singular_weighted_integral_expansion {p : ℝ} (hp : 1 < p) (gamma c : ℝ) : |
| (∫ z in singularDomain p gamma c, physicalWeight (Real.sqrt 2) (c • singularField p z)) = |
| volume.real (singularDomain p gamma c)+c^2*(∫ z in singularDomain p gamma c, energyCorrection p c z) := by |
| simp_rw [physicalWeight_energy_expansion] |
| rw [integral_add (integrableOn_const (C := (1:ℝ)) (singularDomain_isCompact p gamma c).measure_lt_top.ne) |
| ((energyCorrection_integrable hp gamma c).const_mul (c^2)),integral_const_mul] |
| simp only [integral_const,smul_eq_mul,mul_one,measureReal_def,Measure.restrict_apply_univ] |
| |
| end RuledApproximation |
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| end |
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| end OAI |
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