| import OAI.Geometry.CylinderCovering.Cartesian.CapsDefinitions | |
| namespace OAI | |
| namespace CylinderCovering.CartesianCaps | |
| theorem discriminant_failures_location {τ η x y s : ℝ} | |
| (hτ0 : 0 ≤ τ) (hτ1 : τ ≤ 1) (hη : 0 ≤ η) | |
| (hx : |x| ≤ 1) (hy : |y| ≤ 1) (hs0 : 0 ≤ s) (hs1 : s ≤ 1) | |
| (hfirst : 1 / 4 + τ * x * y + τ ^ 2 * (D y + η / 2 - x ^ 2 / 4) < s ^ 2) | |
| (hsecond : 1 / 4 - τ * x * y + τ ^ 2 * (D x + η / 2 - y ^ 2 / 4) < | |
| (1 - s) ^ 2) : | |
| |s - 1 / 2| ≤ 3 * τ ∧ |s - 1 / 2 - τ * x * y| ≤ 10 * τ ^ 2 := by | |
| have hx2 : x ^ 2 ≤ 1 := by | |
| nlinarith only [(abs_le.mp hx).1, (abs_le.mp hx).2] | |
| have hy2 : y ^ 2 ≤ 1 := by | |
| nlinarith only [(abs_le.mp hy).1, (abs_le.mp hy).2] | |
| have hDy : 0 ≤ D y := by unfold D; positivity | |
| have hDx : 0 ≤ D x := by unfold D; positivity | |
| have hB : -(1 / 4 : ℝ) ≤ D y + η / 2 - x ^ 2 / 4 := by linarith | |
| have hC : -(1 / 4 : ℝ) ≤ D x + η / 2 - y ^ 2 / 4 := by linarith | |
| have hBmul := mul_le_mul_of_nonneg_left hB (sq_nonneg τ) | |
| have hCmul := mul_le_mul_of_nonneg_left hC (sq_nonneg τ) | |
| have hxy : |τ * x * y| ≤ τ := by | |
| rw [abs_mul, abs_mul, abs_of_nonneg hτ0] | |
| calc | |
| τ * |x| * |y| ≤ τ * 1 * 1 := by gcongr | |
| _ = τ := by ring | |
| have hτ2 : τ ^ 2 ≤ τ := by nlinarith | |
| have hcoarse : |s - 1 / 2| ≤ 3 * τ := by | |
| rw [abs_le] | |
| constructor | |
| · by_cases hs : s ≤ 1 / 2 | |
| · have hprod := mul_nonneg hs0 (sub_nonneg.mpr hs) | |
| nlinarith only [hfirst, hBmul, (abs_le.mp hxy).1, hτ2, hprod, hτ0] | |
| · linarith | |
| · by_cases hs : 1 / 2 ≤ s | |
| · have hprod := mul_nonneg (sub_nonneg.mpr hs) (sub_nonneg.mpr hs1) | |
| nlinarith only [hsecond, hCmul, (abs_le.mp hxy).2, hτ2, hprod, hτ0] | |
| · linarith | |
| have hu2 : (s - 1 / 2) ^ 2 ≤ 9 * τ ^ 2 := by | |
| have hh := (sq_le_sq₀ (abs_nonneg (s - 1 / 2)) (by positivity : 0 ≤ 3 * τ)).2 hcoarse | |
| nlinarith only [hh, sq_abs (s - 1 / 2)] | |
| refine ⟨hcoarse, ?_⟩ | |
| rw [abs_le] | |
| constructor <;> nlinarith only [hfirst, hsecond, hBmul, hCmul, hu2, sq_nonneg τ] | |
| theorem discriminant_failures_impossible {τ η x y s : ℝ} | |
| (hτ : 0 < τ) (hτ1 : τ ≤ 1) (hsmall : τ ≤ η / 240) | |
| (hx : |x| ≤ 1) (hy : |y| ≤ 1) (hs0 : 0 ≤ s) (hs1 : s ≤ 1) | |
| (hfirst : 1 / 4 + τ * x * y + τ ^ 2 * (D y + η / 2 - x ^ 2 / 4) < s ^ 2) | |
| (hsecond : 1 / 4 - τ * x * y + τ ^ 2 * (D x + η / 2 - y ^ 2 / 4) < | |
| (1 - s) ^ 2) : False := by | |
| have hη : 0 ≤ η := by linarith | |
| obtain ⟨hcoarse, hfine⟩ := discriminant_failures_location hτ.le hτ1 hη hx hy hs0 hs1 hfirst hsecond | |
| have hxy : |τ * x * y| ≤ τ := by | |
| rw [abs_mul, abs_mul, abs_of_nonneg hτ.le] | |
| calc | |
| τ * |x| * |y| ≤ τ * 1 * 1 := by gcongr | |
| _ = τ := by ring | |
| have hsum : |s - 1 / 2 + τ * x * y| ≤ 4 * τ := | |
| (abs_add_le _ _).trans (by linarith) | |
| have herror : |(s - 1 / 2) ^ 2 - (τ * x * y) ^ 2| ≤ 40 * τ ^ 3 := by | |
| calc | |
| _ = |s - 1 / 2 - τ * x * y| * |s - 1 / 2 + τ * x * y| := by | |
| rw [← abs_mul] | |
| congr 1 | |
| ring | |
| _ ≤ (10 * τ ^ 2) * (4 * τ) := mul_le_mul hfine hsum (abs_nonneg _) (by positivity) | |
| _ = 40 * τ ^ 3 := by ring | |
| have hquartic := mul_nonneg (sq_nonneg τ) (sq_nonneg (x ^ 2 - y ^ 2)) | |
| have hmargin := mul_le_mul_of_nonneg_left hsmall (sq_nonneg τ) | |
| have hτ3 : 0 < τ ^ 3 := pow_pos hτ 3 | |
| unfold D at hfirst hsecond | |
| nlinarith only [hfirst, hsecond, (abs_le.mp herror).2, hquartic, hmargin, hτ3] | |
| theorem discriminant_alternative {τ η x y s : ℝ} | |
| (hτ : 0 < τ) (hτ1 : τ ≤ 1) (hsmall : τ ≤ η / 240) | |
| (hx : |x| ≤ 1) (hy : |y| ≤ 1) (hs0 : 0 ≤ s) (hs1 : s ≤ 1) : | |
| Real.sqrt (s ^ 2 - τ * x * y + τ ^ 2 * x ^ 2 / 4) ≤ | |
| 1 / 2 + τ ^ 2 * (D y + η / 2) ∨ | |
| Real.sqrt ((1 - s) ^ 2 + τ * x * y + τ ^ 2 * y ^ 2 / 4) ≤ | |
| 1 / 2 + τ ^ 2 * (D x + η / 2) := by | |
| by_contra! hfail | |
| have hη : 0 ≤ η := by linarith | |
| have hDx : 0 ≤ D x := by unfold D; positivity | |
| have hDy : 0 ≤ D y := by unfold D; positivity | |
| have hBy : 0 ≤ 1 / 2 + τ ^ 2 * (D y + η / 2) := by positivity | |
| have hBx : 0 ≤ 1 / 2 + τ ^ 2 * (D x + η / 2) := by positivity | |
| have hQy : 0 ≤ s ^ 2 - τ * x * y + τ ^ 2 * x ^ 2 / 4 := | |
| (Real.sqrt_pos.mp (hBy.trans_lt hfail.1)).le | |
| have hQx : 0 ≤ (1 - s) ^ 2 + τ * x * y + τ ^ 2 * y ^ 2 / 4 := | |
| (Real.sqrt_pos.mp (hBx.trans_lt hfail.2)).le | |
| have hy2 := (sq_lt_sq₀ hBy (Real.sqrt_nonneg _)).2 hfail.1 | |
| have hx2 := (sq_lt_sq₀ hBx (Real.sqrt_nonneg _)).2 hfail.2 | |
| rw [Real.sq_sqrt hQy] at hy2 | |
| rw [Real.sq_sqrt hQx] at hx2 | |
| apply discriminant_failures_impossible hτ hτ1 hsmall hx hy hs0 hs1 | |
| · nlinarith only [hy2, sq_nonneg (τ ^ 2 * (D y + η / 2))] | |
| · nlinarith only [hx2, sq_nonneg (τ ^ 2 * (D x + η / 2))] | |
| theorem eventually_discriminant_alternative {η : ℝ} (hη : 0 < η) : | |
| ∃ τ₀ : ℝ, 0 < τ₀ ∧ ∀ τ : ℝ, 0 < τ → τ ≤ τ₀ → | |
| ∀ x y s : ℝ, |x| ≤ 1 → |y| ≤ 1 → 0 ≤ s → s ≤ 1 → | |
| Real.sqrt (s ^ 2 - τ * x * y + τ ^ 2 * x ^ 2 / 4) ≤ | |
| 1 / 2 + τ ^ 2 * (D y + η / 2) ∨ | |
| Real.sqrt ((1 - s) ^ 2 + τ * x * y + τ ^ 2 * y ^ 2 / 4) ≤ | |
| 1 / 2 + τ ^ 2 * (D x + η / 2) := by | |
| refine ⟨min 1 (η / 240), lt_min (by norm_num) (by positivity), ?_⟩ | |
| intro τ hτ hsmall x y s hx hy hs0 hs1 | |
| exact discriminant_alternative hτ (hsmall.trans (min_le_left _ _)) | |
| (hsmall.trans (min_le_right _ _)) hx hy hs0 hs1 | |
| end CylinderCovering.CartesianCaps | |
| end OAI | |