| import OAI.Geometry.CylinderCovering.Angular.CapsDefinitions | |
| namespace OAI | |
| open Set | |
| noncomputable section | |
| namespace AngularCaps | |
| theorem sqrt_quarter_bounds {z : ℝ} (hz : 0 ≤ z) : | |
| 1 / 2 ≤ Real.sqrt (1 / 4 + z) ∧ | |
| Real.sqrt (1 / 4 + z) ≤ 1 / 2 + z ∧ | |
| |Real.sqrt (1 / 4 + z) - (1 / 2 + z)| ≤ z ^ 2 := by | |
| let t := Real.sqrt (1 / 4 + z) | |
| have ht0 : 0 ≤ t := Real.sqrt_nonneg _ | |
| have ht2 : t ^ 2 = 1 / 4 + z := Real.sq_sqrt (by linarith) | |
| have ht : 1 / 2 ≤ t := by nlinarith | |
| have hupper : t ≤ 1 / 2 + z := by nlinarith [sq_nonneg (t - 1 / 2)] | |
| refine ⟨ht, hupper, ?_⟩ | |
| change |t - (1 / 2 + z)| ≤ z ^ 2 | |
| rw [abs_of_nonpos (by linarith)] | |
| have hsq : (t - 1 / 2) ^ 2 ≤ z ^ 2 := | |
| (sq_le_sq₀ (by linarith) hz).2 (by linarith) | |
| nlinarith | |
| theorem d_nonneg (q : ℝ) : 0 ≤ d q := by | |
| unfold d | |
| positivity | |
| theorem d_le_one_eighth {q : ℝ} (hq : |q| ≤ 1) : d q ≤ 1 / 8 := by | |
| have hq2 : q ^ 2 ≤ 1 := by nlinarith [abs_le.mp hq, sq_abs q] | |
| have hq4 : q ^ 4 ≤ 1 := by | |
| nlinarith [sq_nonneg q, mul_self_le_mul_self (sq_nonneg q) hq2] | |
| unfold d | |
| linarith | |
| theorem continuous_d : Continuous d := by | |
| unfold d | |
| fun_prop | |
| theorem continuous_squaredRadius (τ η : ℝ) : Continuous (squaredRadius τ η) := by | |
| unfold squaredRadius | |
| exact Real.continuous_sqrt.comp (continuous_const.add | |
| (continuous_const.mul (continuous_d.add continuous_const))) | |
| theorem squaredRadius_pos {τ η q : ℝ} (hη : 0 ≤ η) : | |
| 0 < squaredRadius τ η q := by | |
| have h := sqrt_quarter_bounds (mul_nonneg (sq_nonneg τ) (add_nonneg (d_nonneg q) hη)) | |
| change 0 < Real.sqrt (1 / 4 + τ ^ 2 * (d q + η)) | |
| linarith [h.1] | |
| theorem squaredRadius_expansion {τ η q : ℝ} (hη : 0 ≤ η) (hq : |q| ≤ 1) : | |
| |squaredRadius τ η q - (1 / 2 + τ ^ 2 * (d q + η))| ≤ | |
| (1 / 8 + η) ^ 2 * τ ^ 4 := by | |
| have h := (sqrt_quarter_bounds | |
| (mul_nonneg (sq_nonneg τ) (add_nonneg (d_nonneg q) hη))).2.2 | |
| have hcoeff : (d q + η) ^ 2 ≤ (1 / 8 + η) ^ 2 := | |
| (sq_le_sq₀ (add_nonneg (d_nonneg q) hη) (by linarith)).2 | |
| (by linarith [d_le_one_eighth hq]) | |
| change |Real.sqrt (1 / 4 + τ ^ 2 * (d q + η)) - | |
| (1 / 2 + τ ^ 2 * (d q + η))| ≤ _ | |
| calc | |
| _ ≤ (τ ^ 2 * (d q + η)) ^ 2 := h | |
| _ = (d q + η) ^ 2 * τ ^ 4 := by ring | |
| _ ≤ (1 / 8 + η) ^ 2 * τ ^ 4 := | |
| mul_le_mul_of_nonneg_right hcoeff (by positivity) | |
| theorem radialParameterSet_eq_image {l r : ℝ} {R : ℝ → ℝ} | |
| (hR : ∀ q ∈ Icc l r, 0 ≤ R q) : | |
| radialParameterSet l r R = | |
| (fun p : ℝ × ℝ => (p.1, p.2 * R p.1)) '' (Icc l r ×ˢ Icc 0 1) := by | |
| ext p | |
| constructor | |
| · rintro ⟨hpq, hps⟩ | |
| by_cases hz : R p.1 = 0 | |
| · have hs : p.2 = 0 := by rw [hz] at hps; exact le_antisymm hps.2 hps.1 | |
| refine ⟨(p.1, 0), ⟨hpq, by norm_num⟩, ?_⟩ | |
| apply Prod.ext | |
| · rfl | |
| · simpa using hs.symm | |
| · have hpos : 0 < R p.1 := lt_of_le_of_ne (hR p.1 hpq) (Ne.symm hz) | |
| refine ⟨(p.1, p.2 / R p.1), ⟨hpq, ?_⟩, ?_⟩ | |
| · exact ⟨div_nonneg hps.1 hpos.le, (div_le_one hpos).2 hps.2⟩ | |
| · simp [div_mul_cancel₀ _ hz] | |
| · rintro ⟨p, hp, rfl⟩ | |
| refine ⟨hp.1, mul_nonneg hp.2.1 (hR p.1 hp.1), ?_⟩ | |
| exact mul_le_of_le_one_left (hR p.1 hp.1) hp.2.2 | |
| theorem isCompact_radialParameterSet {l r : ℝ} {R : ℝ → ℝ} | |
| (hRcont : ContinuousOn R (Icc l r)) (hR : ∀ q ∈ Icc l r, 0 ≤ R q) : | |
| IsCompact (radialParameterSet l r R) := by | |
| rw [radialParameterSet_eq_image hR] | |
| apply (isCompact_Icc.prod isCompact_Icc).image_of_continuousOn | |
| exact continuous_fst.continuousOn.prodMk | |
| (continuous_snd.continuousOn.mul | |
| (hRcont.comp continuous_fst.continuousOn (fun _ hp => hp.1))) | |
| theorem isCompact_radialSector_image {E : Type*} [TopologicalSpace E] | |
| {l r : ℝ} {R : ℝ → ℝ} {F : (ℝ × ℝ) → E} | |
| (hRcont : ContinuousOn R (Icc l r)) (hR : ∀ q ∈ Icc l r, 0 ≤ R q) | |
| (hF : ContinuousOn F (radialParameterSet l r R)) : | |
| IsCompact (F '' radialParameterSet l r R) := | |
| (isCompact_radialParameterSet hRcont hR).image_of_continuousOn hF | |
| theorem isCompact_squaredRadius_parameters (l r τ η : ℝ) (hη : 0 ≤ η) : | |
| IsCompact (radialParameterSet l r (squaredRadius τ η)) := | |
| isCompact_radialParameterSet (continuous_squaredRadius τ η).continuousOn | |
| (fun _ _ => (squaredRadius_pos hη).le) | |
| end AngularCaps | |
| end | |
| end OAI | |