| import OAI.Geometry.CylinderCovering.Cartesian.CapsDefinitions | |
| namespace OAI | |
| namespace CylinderCovering.CartesianCaps | |
| open Set | |
| lemma D_nonneg (a : ℝ) : 0 ≤ D a := by | |
| unfold D | |
| positivity | |
| lemma D_le (a : ℝ) (ha : |a| ≤ 1) : D a ≤ 5 / 4 := by | |
| have h2 : a ^ 2 ≤ 1 := (sq_le_one_iff_abs_le_one a).2 ha | |
| have h4 : a ^ 4 ≤ 1 := by nlinarith [mul_nonneg (sub_nonneg.mpr h2) (sq_nonneg a)] | |
| unfold D | |
| linarith | |
| lemma D_sub_factor (a b : ℝ) : | |
| D a - D b = (a - b) * ((a + b) * (1 / 4 + a ^ 2 + b ^ 2)) := by | |
| unfold D | |
| ring | |
| lemma D_lipschitz (a b : ℝ) (ha : |a| ≤ 1) (hb : |b| ≤ 1) : | |
| |D a - D b| ≤ (9 / 2) * |a - b| := by | |
| have h2a : a ^ 2 ≤ 1 := (sq_le_one_iff_abs_le_one a).2 ha | |
| have h2b : b ^ 2 ≤ 1 := (sq_le_one_iff_abs_le_one b).2 hb | |
| have hab : |a + b| ≤ 2 := (abs_add_le a b).trans (by linarith) | |
| have hfac : |(a + b) * (1 / 4 + a ^ 2 + b ^ 2)| ≤ 9 / 2 := by | |
| rw [abs_mul, abs_of_nonneg (by positivity : 0 ≤ (1 / 4 : ℝ) + a ^ 2 + b ^ 2)] | |
| calc | |
| |a + b| * (1 / 4 + a ^ 2 + b ^ 2) ≤ 2 * (1 / 4 + a ^ 2 + b ^ 2) := | |
| mul_le_mul_of_nonneg_right hab (by positivity) | |
| _ ≤ 9 / 2 := by linarith | |
| rw [D_sub_factor, abs_mul] | |
| nlinarith [abs_nonneg (a-b)] | |
| lemma abs_mul_le_one (a b : ℝ) (ha : |a| ≤ 1) (hb : |b| ≤ 1) : | |
| |a * b| ≤ 1 := by | |
| rw [abs_mul] | |
| calc | |
| |a| * |b| ≤ 1 * 1 := mul_le_mul ha hb (abs_nonneg _) (by norm_num) | |
| _ = 1 := by norm_num | |
| lemma D_ray_lipschitz (q s t : ℝ) (hq : |q| ≤ 1) | |
| (hs : s ∈ Icc (0 : ℝ) 1) (ht : t ∈ Icc (0 : ℝ) 1) : | |
| |D (q * s) - D (q * t)| ≤ (9 / 2) * |s - t| := by | |
| have hs' : |s| ≤ 1 := by rw [abs_of_nonneg hs.1]; exact hs.2 | |
| have ht' : |t| ≤ 1 := by rw [abs_of_nonneg ht.1]; exact ht.2 | |
| have h := D_lipschitz (q*s) (q*t) (abs_mul_le_one q s hq hs') | |
| (abs_mul_le_one q t hq ht') | |
| have hmul : |q*s-q*t| ≤ |s-t| := by | |
| rw [← mul_sub, abs_mul] | |
| nlinarith [abs_nonneg (s-t)] | |
| linarith | |
| lemma small_square (τ : ℝ) (hτ : τ ∈ Icc (0 : ℝ) (1/4)) : τ ^ 2 ≤ 1/16 := by | |
| nlinarith [hτ.1, hτ.2] | |
| lemma residual_strictMono (τ η q : ℝ) (hτ : τ ∈ Icc (0 : ℝ) (1/4)) | |
| (hq : |q| ≤ 1) : StrictMonoOn (residual τ η q) (Icc (0 : ℝ) 1) := by | |
| intro s hs t ht hst | |
| have hl := D_ray_lipschitz q t s hq ht hs | |
| rw [abs_of_pos (sub_pos.mpr hst)] at hl | |
| have hd : D (q*t) - D (q*s) ≤ (9/2)*(t-s) := (le_abs_self _).trans hl | |
| have hsq := small_square τ hτ | |
| have hmult := mul_le_mul_of_nonneg_left hd (sq_nonneg τ) | |
| have hb := mul_nonneg (sub_nonneg.mpr hsq) (sub_nonneg.mpr hst.le) | |
| unfold residual | |
| nlinarith | |
| lemma root_exists (τ η q : ℝ) (hτ : τ ∈ Icc (0 : ℝ) (1/4)) | |
| (hη : η ∈ Ioc (0 : ℝ) 1) (hq : |q| ≤ 1) : | |
| ∃ s : ℝ, s ∈ Icc (1/2 : ℝ) (3/4) ∧ residual τ η q s = 0 := by | |
| have hc : ContinuousOn (residual τ η q) (Icc (1/2 : ℝ) (3/4)) := by | |
| unfold residual D | |
| fun_prop | |
| have hleft : residual τ η q (1/2) ≤ 0 := by | |
| have hp : 0 ≤ τ ^ 2 * (D (q * (1/2)) + η) := | |
| mul_nonneg (sq_nonneg τ) (by linarith [D_nonneg (q*(1/2)), hη.1]) | |
| unfold residual | |
| linarith | |
| have habs : |q * (3/4)| ≤ 1 := abs_mul_le_one q (3/4) hq (by norm_num) | |
| have hd := D_le (q*(3/4)) habs | |
| have hsq := small_square τ hτ | |
| have hright : 0 ≤ residual τ η q (3/4) := by | |
| have h := mul_le_mul_of_nonneg_left (show D (q*(3/4))+η ≤ 9/4 by linarith [hη.2]) (sq_nonneg τ) | |
| unfold residual | |
| nlinarith | |
| obtain ⟨s, hs, he⟩ := intermediate_value_Icc (by norm_num : (1/2 : ℝ) ≤ 3/4) hc ⟨hleft, hright⟩ | |
| exact ⟨s, hs, he⟩ | |
| lemma cap_spec (τ η q : ℝ) (hτ : τ ∈ Icc (0 : ℝ) (1/4)) | |
| (hη : η ∈ Ioc (0 : ℝ) 1) (hq : |q| ≤ 1) : | |
| cap τ η q ∈ Icc (1/2 : ℝ) (3/4) ∧ residual τ η q (cap τ η q) = 0 := by | |
| have h := root_exists τ η q hτ hη hq | |
| unfold cap | |
| rw [dite_eq_left h] | |
| exact Classical.choose_spec h | |
| lemma cap_equation (τ η q : ℝ) (hτ : τ ∈ Icc (0 : ℝ) (1/4)) | |
| (hη : η ∈ Ioc (0 : ℝ) 1) (hq : |q| ≤ 1) : | |
| cap τ η q = 1/2 + τ^2 * (D (q * cap τ η q) + η) := by | |
| have h := (cap_spec τ η q hτ hη hq).2 | |
| unfold residual at h | |
| linarith | |
| lemma cap_mem_unit (τ η q : ℝ) (hτ : τ ∈ Icc (0 : ℝ) (1/4)) | |
| (hη : η ∈ Ioc (0 : ℝ) 1) (hq : |q| ≤ 1) : | |
| cap τ η q ∈ Icc (0 : ℝ) 1 := by | |
| have h := (cap_spec τ η q hτ hη hq).1 | |
| constructor <;> linarith [h.1, h.2] | |
| lemma cap_unique (τ η q s : ℝ) (hτ : τ ∈ Icc (0 : ℝ) (1/4)) | |
| (hη : η ∈ Ioc (0 : ℝ) 1) (hq : |q| ≤ 1) | |
| (hs : s ∈ Icc (0 : ℝ) 1) (he : residual τ η q s = 0) : | |
| s = cap τ η q := by | |
| apply (residual_strictMono τ η q hτ hq).injOn hs (cap_mem_unit τ η q hτ hη hq) | |
| rw [he, (cap_spec τ η q hτ hη hq).2] | |
| lemma residual_nonpos_iff (τ η q s : ℝ) (hτ : τ ∈ Icc (0 : ℝ) (1/4)) | |
| (hη : η ∈ Ioc (0 : ℝ) 1) (hq : |q| ≤ 1) | |
| (hs : s ∈ Icc (0 : ℝ) 1) : | |
| residual τ η q s ≤ 0 ↔ s ≤ cap τ η q := by | |
| have hc := cap_mem_unit τ η q hτ hη hq | |
| have he := (cap_spec τ η q hτ hη hq).2 | |
| have hm := residual_strictMono τ η q hτ hq | |
| constructor | |
| · intro h | |
| by_contra hn | |
| have hlt := hm hc hs (lt_of_not_ge hn) | |
| rw [he] at hlt | |
| linarith | |
| · intro h | |
| have hle := hm.monotoneOn hs hc h | |
| rwa [he] at hle | |
| lemma aperture_eq_Icc (τ η b q : ℝ) (hτ : τ ∈ Icc (0 : ℝ) (1/4)) | |
| (hη : η ∈ Ioc (0 : ℝ) 1) (hb : b ∈ Icc (3/4 : ℝ) 1) | |
| (hq : |q| ≤ 1) : aperture τ η b q = Icc 0 (cap τ η q) := by | |
| ext s | |
| constructor | |
| · rintro ⟨hs0, hsb, hs⟩ | |
| refine ⟨hs0, (residual_nonpos_iff τ η q s hτ hη hq ⟨hs0, hsb.trans hb.2⟩).1 ?_⟩ | |
| unfold residual | |
| linarith | |
| · rintro ⟨hs0, hsR⟩ | |
| have hR := (cap_spec τ η q hτ hη hq).1 | |
| have hsb : s ≤ b := by linarith [hR.2, hb.1] | |
| refine ⟨hs0, hsb, ?_⟩ | |
| have h := (residual_nonpos_iff τ η q s hτ hη hq ⟨hs0, hsb.trans hb.2⟩).2 hsR | |
| unfold residual at h | |
| linarith | |
| lemma cap_lipschitz (τ η : ℝ) (hτ : τ ∈ Icc (0 : ℝ) (1/4)) | |
| (hη : η ∈ Ioc (0 : ℝ) 1) (p q : ℝ) (hp : |p| ≤ 1) (hq : |q| ≤ 1) : | |
| |cap τ η p - cap τ η q| ≤ |p - q| := by | |
| have hRp := cap_mem_unit τ η p hτ hη hp | |
| have hRq := cap_mem_unit τ η q hτ hη hq | |
| have hRap : |cap τ η p| ≤ 1 := by rw [abs_of_nonneg hRp.1]; exact hRp.2 | |
| have hRaq : |cap τ η q| ≤ 1 := by rw [abs_of_nonneg hRq.1]; exact hRq.2 | |
| have hd := D_lipschitz (p * cap τ η p) (q * cap τ η q) | |
| (abs_mul_le_one p _ hp hRap) (abs_mul_le_one q _ hq hRaq) | |
| have hm : |p * cap τ η p - q * cap τ η q| ≤ | |
| |cap τ η p - cap τ η q| + |p-q| := by | |
| calc | |
| |p * cap τ η p - q * cap τ η q| = | |
| |p * (cap τ η p - cap τ η q) + (p-q) * cap τ η q| := by congr 1; ring | |
| _ ≤ |p * (cap τ η p - cap τ η q)| + |(p-q) * cap τ η q| := abs_add_le _ _ | |
| _ ≤ |cap τ η p - cap τ η q| + |p-q| := by | |
| rw [abs_mul, abs_mul] | |
| have h1 := mul_le_mul_of_nonneg_right hp (abs_nonneg (cap τ η p - cap τ η q)) | |
| have h2 := mul_le_mul_of_nonneg_left hRaq (abs_nonneg (p-q)) | |
| nlinarith | |
| have he : cap τ η p - cap τ η q = | |
| τ^2 * (D (p * cap τ η p) - D (q * cap τ η q)) := by | |
| linarith [cap_equation τ η p hτ hη hp, cap_equation τ η q hτ hη hq] | |
| have ha : |cap τ η p - cap τ η q| = | |
| τ^2 * |D (p * cap τ η p) - D (q * cap τ η q)| := by | |
| rw [he, abs_mul, abs_of_nonneg (sq_nonneg τ)] | |
| have hu : |cap τ η p - cap τ η q| ≤ | |
| τ^2 * ((9/2) * (|cap τ η p - cap τ η q| + |p-q|)) := by | |
| rw [ha] | |
| apply mul_le_mul_of_nonneg_left _ (sq_nonneg τ) | |
| linarith | |
| have ht := mul_le_mul_of_nonneg_right (small_square τ hτ) | |
| (show 0 ≤ (9/2 : ℝ) * (|cap τ η p - cap τ η q| + |p-q|) by positivity) | |
| nlinarith [abs_nonneg (cap τ η p - cap τ η q), abs_nonneg (p-q)] | |
| lemma cap_continuousOn (τ η : ℝ) (hτ : τ ∈ Icc (0 : ℝ) (1/4)) | |
| (hη : η ∈ Ioc (0 : ℝ) 1) : ContinuousOn (cap τ η) (Icc (-1 : ℝ) 1) := by | |
| have h : LipschitzOnWith 1 (cap τ η) (Icc (-1 : ℝ) 1) := by | |
| apply LipschitzOnWith.of_dist_le_mul | |
| intro p hp q hq | |
| simpa [Real.dist_eq] using cap_lipschitz τ η hτ hη p q | |
| (abs_le.mpr hp) (abs_le.mpr hq) | |
| exact h.continuousOn | |
| lemma D_half (q : ℝ) : D (q * (1/2)) = d q := by | |
| unfold D d | |
| ring | |
| lemma cap_quadratic_bound (τ η q : ℝ) (hτ : τ ∈ Icc (0 : ℝ) (1/4)) | |
| (hη : η ∈ Ioc (0 : ℝ) 1) (hq : |q| ≤ 1) : | |
| |cap τ η q - 1/2| ≤ (9/4) * τ^2 := by | |
| have hc := (cap_spec τ η q hτ hη hq).1 | |
| have hcu := cap_mem_unit τ η q hτ hη hq | |
| have hca : |cap τ η q| ≤ 1 := by rw [abs_of_nonneg hcu.1]; exact hcu.2 | |
| have hd := D_le (q * cap τ η q) (abs_mul_le_one q _ hq hca) | |
| rw [abs_of_nonneg (by linarith [hc.1] : 0 ≤ cap τ η q - 1/2)] | |
| have hm := mul_le_mul_of_nonneg_left (show D (q * cap τ η q) + η ≤ 9/4 by linarith [hη.2]) (sq_nonneg τ) | |
| linarith [cap_equation τ η q hτ hη hq] | |
| lemma cap_fourth_order (τ η q : ℝ) (hτ : τ ∈ Icc (0 : ℝ) (1/4)) | |
| (hη : η ∈ Ioc (0 : ℝ) 1) (hq : |q| ≤ 1) : | |
| |cap τ η q - (1/2 + τ^2 * (d q + η))| ≤ 12 * τ^4 := by | |
| have hc := cap_mem_unit τ η q hτ hη hq | |
| have hl := D_ray_lipschitz q (cap τ η q) (1/2) hq hc (by norm_num) | |
| have hb := cap_quadratic_bound τ η q hτ hη hq | |
| have he : cap τ η q - (1/2 + τ^2 * (d q + η)) = | |
| τ^2 * (D (q * cap τ η q) - D (q * (1/2))) := by | |
| rw [D_half] | |
| linarith [cap_equation τ η q hτ hη hq] | |
| rw [he, abs_mul, abs_of_nonneg (sq_nonneg τ)] | |
| calc | |
| τ^2 * |D (q * cap τ η q) - D (q * (1/2))| ≤ τ^2 * ((9/2) * ((9/4) * τ^2)) := by | |
| apply mul_le_mul_of_nonneg_left _ (sq_nonneg τ) | |
| linarith | |
| _ ≤ 12 * τ^4 := by nlinarith [sq_nonneg (τ^2)] | |
| theorem cap_properties (τ η : ℝ) (hτ : τ ∈ Icc (0 : ℝ) (1/4)) | |
| (hη : η ∈ Ioc (0 : ℝ) 1) : | |
| ContinuousOn (cap τ η) (Icc (-1 : ℝ) 1) ∧ | |
| ∀ q ∈ Icc (-1 : ℝ) 1, | |
| cap τ η q ∈ Icc (1/2 : ℝ) (3/4) ∧ | |
| residual τ η q (cap τ η q) = 0 ∧ | |
| (∀ s ∈ Icc (0 : ℝ) 1, residual τ η q s = 0 → s = cap τ η q) ∧ | |
| |cap τ η q - (1/2 + τ^2 * (d q + η))| ≤ 12 * τ^4 ∧ | |
| ∀ b ∈ Icc (3/4 : ℝ) 1, aperture τ η b q = Icc 0 (cap τ η q) := by | |
| refine ⟨cap_continuousOn τ η hτ hη, ?_⟩ | |
| intro q hq | |
| have haq : |q| ≤ 1 := abs_le.mpr hq | |
| refine ⟨(cap_spec τ η q hτ hη haq).1, (cap_spec τ η q hτ hη haq).2, ?_, | |
| cap_fourth_order τ η q hτ hη haq, ?_⟩ | |
| · intro s hs he | |
| exact cap_unique τ η q s hτ hη haq hs he | |
| · intro b hb | |
| exact aperture_eq_Icc τ η b q hτ hη hb haq | |
| end CylinderCovering.CartesianCaps | |
| end OAI | |