| import OAI.Geometry.CylinderCovering.Cartesian.DomainDefinitions |
| import OAI.Geometry.CylinderCovering.Cartesian.Caps |
| import OAI.Geometry.CylinderCovering.Footprints.Footprints |
|
|
| namespace OAI |
|
|
| open Set MeasureTheory intervalIntegral |
|
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| namespace CylinderCovering.CartesianCaps |
|
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| lemma height_lower (τ η y : ℝ) (hη : 0 ≤ η) : 1/2 ≤ height τ η y := by |
| unfold height |
| have h := mul_nonneg (sq_nonneg τ) (add_nonneg (D_nonneg y) hη) |
| linarith |
|
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| lemma height_upper (τ η y : ℝ) (hη : η ≤ 1) (hy : |y| ≤ 1) : |
| height τ η y ≤ 1/2 + (9/4) * τ^2 := by |
| have h := mul_le_mul_of_nonneg_left (show D y + η ≤ 9/4 by linarith [D_le y hy]) (sq_nonneg τ) |
| unfold height |
| linarith |
|
|
| lemma height_le_three_quarters (τ η y : ℝ) (hτ : τ ∈ Icc (0 : ℝ) (1/4)) |
| (hη : η ≤ 1) (hy : |y| ≤ 1) : height τ η y ≤ 3/4 := by |
| have h := height_upper τ η y hη hy |
| have hs := small_square τ hτ |
| linarith |
|
|
| lemma central_subset_domain (τ η b : ℝ) (hτ : τ ∈ Icc (0 : ℝ) (1/4)) |
| (hη : η ≤ 1) (hb : 3/4 ≤ b) : central τ η ⊆ domain τ η b := by |
| rintro ⟨y,s⟩ ⟨hy, hs⟩ |
| have hya : |y| ≤ 1 := abs_le.mpr ⟨by linarith [hy.1], by linarith [hy.2]⟩ |
| refine ⟨(abs_nonneg y).trans hs.1, ?_, hs.1, hs.2⟩ |
| exact hs.2.trans ((height_le_three_quarters τ η y hτ hη hya).trans hb) |
|
|
| lemma domain_enclosure (τ η b : ℝ) (hη : η ≤ 1) (hb : b ≤ 1) : |
| domain τ η b ⊆ central τ η ∪ |
| (Icc (-(1/2 + (9/4)*τ^2)) (-1/2) ×ˢ Icc (1/2) (1/2 + (9/4)*τ^2)) ∪ |
| (Icc (1/2) (1/2 + (9/4)*τ^2) ×ˢ Icc (1/2) (1/2 + (9/4)*τ^2)) := by |
| rintro ⟨y,s⟩ ⟨hs0, hsb, hys, hsh⟩ |
| have hya : |y| ≤ 1 := hys.trans (hsb.trans hb) |
| have hsTop : s ≤ 1/2 + (9/4)*τ^2 := hsh.trans (height_upper τ η y hη hya) |
| have hyTop : |y| ≤ 1/2 + (9/4)*τ^2 := hys.trans hsTop |
| by_cases hy : y ∈ Icc (-1/2 : ℝ) (1/2) |
| · exact Or.inl (Or.inl ⟨hy, hys, hsh⟩) |
| · rw [mem_Icc, not_and_or] at hy |
| rcases hy with hyl | hyr |
| · have hyl' : y < -1/2 := lt_of_not_ge hyl |
| have hsLow : 1/2 ≤ s := by linarith [neg_le_abs y] |
| exact Or.inl (Or.inr ⟨⟨(abs_le.mp hyTop).1, hyl'.le⟩, hsLow, hsTop⟩) |
| · have hyr' : 1/2 < y := lt_of_not_ge hyr |
| have hsLow : 1/2 ≤ s := by linarith [le_abs_self y] |
| exact Or.inr ⟨⟨hyr'.le, (abs_le.mp hyTop).2⟩, hsLow, hsTop⟩ |
|
|
| lemma integral_D : (∫ y in (-1/2 : ℝ)..(1/2), D y) = 1/30 := by |
| unfold D |
| rw [intervalIntegral.integral_add |
| ((show Continuous (fun y : ℝ => y^2/4) by fun_prop).intervalIntegrable _ _) |
| ((show Continuous (fun y : ℝ => y^4) by fun_prop).intervalIntegrable _ _)] |
| rw [intervalIntegral.integral_div, integral_pow, integral_pow] |
| norm_num |
|
|
| lemma integral_abs_half : (∫ y in (-1/2 : ℝ)..(1/2), |y|) = 1/4 := by |
| have hc : Continuous (fun y : ℝ => |y|) := continuous_abs |
| rw [← intervalIntegral.integral_add_adjacent_intervals |
| (hc.intervalIntegrable (-1/2) 0) (hc.intervalIntegrable 0 (1/2))] |
| have hl : (∫ y in (-1/2 : ℝ)..0, |y|) = ∫ y in (-1/2 : ℝ)..0, -y := by |
| apply intervalIntegral.integral_congr |
| intro y hy |
| rw [uIcc_of_le (by norm_num : (-1/2 : ℝ) ≤ 0)] at hy |
| exact abs_of_nonpos hy.2 |
| have hr : (∫ y in (0 : ℝ)..(1/2), |y|) = ∫ y in (0 : ℝ)..(1/2), y := by |
| apply intervalIntegral.integral_congr |
| intro y hy |
| rw [uIcc_of_le (by norm_num : (0 : ℝ) ≤ 1/2)] at hy |
| exact abs_of_nonneg hy.1 |
| rw [hl, hr, intervalIntegral.integral_neg, integral_id, integral_id] |
| norm_num |
|
|
| lemma integral_height (τ η : ℝ) : |
| (∫ y in (-1/2 : ℝ)..(1/2), height τ η y) = 1/2 + τ^2 * (1/30 + η) := by |
| have hD : Continuous D := by unfold D; fun_prop |
| unfold height |
| rw [intervalIntegral.integral_add (continuous_const.intervalIntegrable _ _) |
| ((show Continuous (fun y => τ^2 * (D y + η)) by fun_prop).intervalIntegrable _ _)] |
| rw [intervalIntegral.integral_const, intervalIntegral.integral_const_mul, |
| intervalIntegral.integral_add (hD.intervalIntegrable _ _) (continuous_const.intervalIntegrable _ _), |
| integral_D, intervalIntegral.integral_const] |
| norm_num [smul_eq_mul] |
|
|
| lemma volume_central (τ η : ℝ) (hη : 0 ≤ η) : |
| volume (central τ η) = ENNReal.ofReal (1/4 + τ^2 * (1/30 + η)) := by |
| have hh : Continuous (height τ η) := by unfold height D; fun_prop |
| unfold central |
| rw [Footprints.volume_closedBand_eq_integral continuous_abs hh (by norm_num : (-1/2 : ℝ) ≤ 1/2) |
| (by |
| intro y hy |
| have hya : |y| ≤ 1/2 := abs_le.mpr ⟨by linarith [hy.1], hy.2⟩ |
| exact hya.trans (height_lower τ η y hη))] |
| rw [intervalIntegral.integral_sub (hh.intervalIntegrable _ _) (continuous_abs.intervalIntegrable _ _), |
| integral_height, integral_abs_half] |
| congr 1 |
| ring |
|
|
| lemma volume_rectangle {a b c e : ℝ} (hab : a ≤ b) (hce : c ≤ e) : |
| volume (Icc a b ×ˢ Icc c e) = ENNReal.ofReal ((b-a)*(e-c)) := by |
| change volume {p : ℝ×ℝ | p.1 ∈ Icc a b ∧ p.2 ∈ Icc c e} = _ |
| rw [Footprints.volume_closedBand_eq_integral continuous_const continuous_const hab |
| (by intro x hx; exact hce)] |
| simp only [intervalIntegral.integral_const, smul_eq_mul] |
|
|
| theorem domain_area_bounds (τ η b : ℝ) (hτ : τ ∈ Icc (0 : ℝ) (1/4)) |
| (hη : η ∈ Ioc (0 : ℝ) 1) (hb : b ∈ Icc (3/4 : ℝ) 1) : |
| 1/4 + τ^2 * (1/30 + η) ≤ volume.real (domain τ η b) ∧ |
| volume.real (domain τ η b) ≤ 1/4 + τ^2 * (1/30 + η) + (81/8)*τ^4 := by |
| let A : ℝ := 1/4 + τ^2 * (1/30 + η) |
| let W : ℝ := (81/16)*τ^4 |
| let L : Set (ℝ×ℝ) := Icc (-(1/2 + (9/4)*τ^2)) (-1/2) ×ˢ |
| Icc (1/2) (1/2 + (9/4)*τ^2) |
| let R : Set (ℝ×ℝ) := Icc (1/2) (1/2 + (9/4)*τ^2) ×ˢ |
| Icc (1/2) (1/2 + (9/4)*τ^2) |
| have hη0 : 0 ≤ η := hη.1.le |
| have hA : 0 ≤ A := by dsimp [A]; positivity |
| have hW : 0 ≤ W := by dsimp [W]; positivity |
| have hCvol : volume (central τ η) = ENNReal.ofReal A := volume_central τ η hη.1.le |
| have hLvol : volume L = ENNReal.ofReal W := by |
| dsimp [L] |
| rw [volume_rectangle (by nlinarith [sq_nonneg τ]) (by nlinarith [sq_nonneg τ])] |
| congr 1 |
| dsimp [W] |
| ring |
| have hRvol : volume R = ENNReal.ofReal W := by |
| dsimp [R] |
| rw [volume_rectangle (by nlinarith [sq_nonneg τ]) (by nlinarith [sq_nonneg τ])] |
| congr 1 |
| dsimp [W] |
| ring |
| have hlow : ENNReal.ofReal A ≤ volume (domain τ η b) := by |
| rw [← hCvol] |
| exact measure_mono (central_subset_domain τ η b hτ hη.2 hb.1) |
| have hup : volume (domain τ η b) ≤ ENNReal.ofReal (A + (81/8)*τ^4) := by |
| calc |
| volume (domain τ η b) ≤ volume ((central τ η ∪ L) ∪ R) := |
| measure_mono (domain_enclosure τ η b hη.2 hb.2) |
| _ ≤ volume (central τ η ∪ L) + volume R := measure_union_le _ _ |
| _ ≤ (volume (central τ η) + volume L) + volume R := |
| add_le_add (measure_union_le _ _) le_rfl |
| _ = (ENNReal.ofReal A + ENNReal.ofReal W) + ENNReal.ofReal W := by rw [hCvol, hLvol, hRvol] |
| _ = ENNReal.ofReal ((A+W)+W) := by rw [ENNReal.ofReal_add (add_nonneg hA hW) hW, ENNReal.ofReal_add hA hW] |
| _ = ENNReal.ofReal (A + (81/8)*τ^4) := by congr 1; dsimp [W]; ring |
| have hfinite : volume (domain τ η b) ≠ ⊤ := |
| (lt_of_le_of_lt hup ENNReal.ofReal_lt_top).ne |
| have hlowR := ENNReal.toReal_mono hfinite hlow |
| have hupR := ENNReal.toReal_mono ENNReal.ofReal_ne_top hup |
| rw [ENNReal.toReal_ofReal hA] at hlowR |
| rw [ENNReal.toReal_ofReal (by positivity : 0 ≤ A + (81/8)*τ^4)] at hupR |
| exact ⟨hlowR, hupR⟩ |
|
|
| lemma domain_isClosed (τ η b : ℝ) : IsClosed (domain τ η b) := by |
| unfold domain |
| change IsClosed ({p : ℝ×ℝ | 0 ≤ p.2} ∩ ({p : ℝ×ℝ | p.2 ≤ b} ∩ |
| ({p : ℝ×ℝ | |p.1| ≤ p.2} ∩ {p : ℝ×ℝ | p.2 ≤ height τ η p.1}))) |
| refine (isClosed_le continuous_const continuous_snd).inter |
| ((isClosed_le continuous_snd continuous_const).inter |
| ((isClosed_le (by fun_prop) continuous_snd).inter (isClosed_le continuous_snd ?_))) |
| unfold height D |
| fun_prop |
|
|
| lemma domain_isCompact (τ η b : ℝ) (hb : b ≤ 1) : IsCompact (domain τ η b) := by |
| apply (isCompact_Icc.prod isCompact_Icc).of_isClosed_subset (domain_isClosed τ η b) |
| (show domain τ η b ⊆ Icc (-1 : ℝ) 1 ×ˢ Icc (0 : ℝ) 1 from ?_) |
| rintro ⟨y,s⟩ ⟨hs0, hsb, hys, hsh⟩ |
| exact ⟨abs_le.mp (hys.trans (hsb.trans hb)), hs0, hsb.trans hb⟩ |
|
|
| lemma domain_radial (τ η b : ℝ) (hτ : τ ∈ Icc (0 : ℝ) (1/4)) |
| (hη : η ∈ Ioc (0 : ℝ) 1) (hb : b ∈ Icc (3/4 : ℝ) 1) : |
| domain τ η b = {p : ℝ×ℝ | ∃ q ∈ Icc (-1 : ℝ) 1, |
| ∃ s ∈ Icc (0 : ℝ) (cap τ η q), p = (q*s,s)} := by |
| ext p |
| rcases p with ⟨y,s⟩ |
| constructor |
| · rintro ⟨hs0, hsb, hys, hsh⟩ |
| by_cases hs : s = 0 |
| · have hy : y = 0 := abs_nonpos_iff.mp (by simpa [hs] using hys) |
| refine ⟨0, by norm_num, 0, ?_, ?_⟩ |
| · exact ⟨le_rfl, (cap_mem_unit τ η 0 hτ hη (by norm_num)).1⟩ |
| · simp [hy, hs] |
| · have hspos : 0 < s := lt_of_le_of_ne hs0 (Ne.symm hs) |
| have hq : |y/s| ≤ 1 := by |
| rw [abs_div, abs_of_pos hspos] |
| exact (div_le_one hspos).2 hys |
| have he : (y/s)*s = y := div_mul_cancel₀ y hs |
| refine ⟨y/s, abs_le.mp hq, s, ?_, ?_⟩ |
| · rw [← aperture_eq_Icc τ η b (y/s) hτ hη hb hq] |
| exact ⟨hs0, hsb, by simpa [height, he] using hsh⟩ |
| · simp [he] |
| · rintro ⟨q, hq, t, ht, hp⟩ |
| have haq : |q| ≤ 1 := abs_le.mpr hq |
| have ha : t ∈ aperture τ η b q := by |
| rw [aperture_eq_Icc τ η b q hτ hη hb haq] |
| exact ht |
| have hym : |q*t| ≤ t := by |
| rw [abs_mul, abs_of_nonneg ht.1] |
| simpa using mul_le_mul_of_nonneg_right haq ht.1 |
| have hcoords : y = q*t ∧ s = t := Prod.mk.inj hp |
| rcases hcoords with ⟨rfl,rfl⟩ |
| exact ⟨ha.1, ha.2.1, hym, ha.2.2⟩ |
|
|
| theorem domain_area_expansion (τ η b : ℝ) (hτ : τ ∈ Icc (0 : ℝ) (1/4)) |
| (hη : η ∈ Ioc (0 : ℝ) 1) (hb : b ∈ Icc (3/4 : ℝ) 1) : |
| |volume.real (domain τ η b) - (1/4 + τ^2 * (1/30 + η))| ≤ (81/8)*τ^4 := by |
| have h := domain_area_bounds τ η b hτ hη hb |
| rw [abs_of_nonneg (sub_nonneg.mpr h.1)] |
| linarith [h.2] |
|
|
| lemma untruncated_contains_ray (τ η q : ℝ) (hτ : 0 < τ) (hη : 0 ≤ η) |
| (hq : q ≠ 0) : ∃ B : ℝ, 1 ≤ B ∧ ∀ s : ℝ, B ≤ s → s ≤ height τ η (q*s) := by |
| let K : ℝ := τ^2 * q^4 |
| have hK : 0 < K := by dsimp [K]; positivity |
| refine ⟨max 1 (1/K), le_max_left _ _, ?_⟩ |
| intro s hs |
| have hs1 : 1 ≤ s := (le_max_left _ _).trans hs |
| have hs0 : 0 ≤ s := by linarith |
| have hdiv : 1/K ≤ s := (le_max_right _ _).trans hs |
| have hKs : 1 ≤ K*s := by |
| have h := (div_le_iff₀ hK).1 hdiv |
| nlinarith |
| have hs2 : 1 ≤ s^2 := by nlinarith |
| have hs3 : s ≤ s^3 := by nlinarith [mul_le_mul_of_nonneg_right hs2 hs0] |
| have hleading : s ≤ K*s^4 := calc |
| s ≤ s^3 := hs3 |
| _ ≤ (K*s)*s^3 := by simpa using mul_le_mul_of_nonneg_right hKs (show 0 ≤ s^3 by positivity) |
| _ = K*s^4 := by ring |
| have he : height τ η (q*s) = 1/2 + τ^2*((q*s)^2/4+η) + K*s^4 := by |
| unfold height D |
| dsimp [K] |
| ring |
| rw [he] |
| have hp : 0 ≤ τ^2*((q*s)^2/4+η) := by positivity |
| linarith |
|
|
| lemma untruncated_not_bddAbove (τ η q : ℝ) (hτ : 0 < τ) (hη : 0 ≤ η) |
| (hq : q ≠ 0) : ¬ BddAbove {s : ℝ | 0 ≤ s ∧ s ≤ height τ η (q*s)} := by |
| obtain ⟨B, hB, htail⟩ := untruncated_contains_ray τ η q hτ hη hq |
| rintro ⟨M,hM⟩ |
| have hBs : B ≤ max B (M+1) := le_max_left _ _ |
| have h0 : 0 ≤ max B (M+1) := by linarith |
| have hle := hM ⟨h0, htail _ hBs⟩ |
| have hgt : M+1 ≤ max B (M+1) := le_max_right _ _ |
| linarith |
|
|
| lemma untruncated_gap (τ η q : ℝ) (hτ : τ ∈ Icc (0 : ℝ) (1/4)) |
| (hη : η ≤ 1) (hq : |q| ≤ 1) : ¬ 1 ≤ height τ η (q*1) := by |
| have h := height_le_three_quarters τ η q hτ hη hq |
| simpa only [mul_one] using (show ¬ 1 ≤ height τ η q by linarith) |
|
|
| end CylinderCovering.CartesianCaps |
|
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| end OAI |
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|
