| import Mathlib | |
| namespace OAI | |
| namespace AngularEstimates | |
| theorem squared_secant_identity (l r : ℝ) : | |
| 2 * ((1 + l * r) / 2) - ((l + r) / 2) ^ 2 = | |
| 1 - (r - l) ^ 2 / 4 := by | |
| ring | |
| theorem ordinary_secant_identity (l r : ℝ) : | |
| ((l + r) / 4) ^ 2 - (1 + l * r) / 4 + 1 / 4 = | |
| (r - l) ^ 2 / 16 := by | |
| ring | |
| theorem sector_midpoint_error {l r S Y : ℝ} | |
| (hleft : l * S ≤ Y) (hright : Y ≤ r * S) : | |
| |2 * ((l + r) / 4) * S - Y| ≤ (r - l) * S / 2 := by | |
| rw [abs_le] | |
| constructor <;> nlinarith only [hleft, hright] | |
| theorem cartesian_discriminant_identity | |
| {α β Δ T Y δ S Y₀ : ℝ} | |
| (hsecant : β ^ 2 - α + 1 / 4 = Δ ^ 2 / 16) | |
| (hS : S = T - δ * β) (hY : Y₀ = Y - δ * α) : | |
| T ^ 2 - δ * Y + δ ^ 2 / 4 = | |
| S ^ 2 + δ * (2 * β * S - Y₀) + δ ^ 2 * Δ ^ 2 / 16 := by | |
| rw [hS, hY] | |
| nlinarith only [congrArg (fun x : ℝ => δ ^ 2 * x) hsecant] | |
| theorem cartesian_discriminant_lower | |
| {α β Δ T Y δ S Y₀ : ℝ} | |
| (_hΔ : 0 ≤ Δ) (_hS0 : 0 ≤ S) | |
| (hsecant : β ^ 2 - α + 1 / 4 = Δ ^ 2 / 16) | |
| (hS : S = T - δ * β) (hY : Y₀ = Y - δ * α) | |
| (herror : |2 * β * S - Y₀| ≤ Δ * S / 2) : | |
| (S - |δ| * Δ / 4) ^ 2 ≤ T ^ 2 - δ * Y + δ ^ 2 / 4 := by | |
| have hprod : |δ * (2 * β * S - Y₀)| ≤ |δ| * (Δ * S / 2) := by | |
| rw [abs_mul] | |
| exact mul_le_mul_of_nonneg_left herror (abs_nonneg δ) | |
| have hid := cartesian_discriminant_identity hsecant hS hY | |
| have hsq : |δ| ^ 2 = δ ^ 2 := sq_abs δ | |
| nlinarith only [(abs_le.mp hprod).1, hid, hsq] | |
| theorem cartesian_discriminant_sqrt | |
| {α β Δ T Y δ S Y₀ : ℝ} | |
| (hΔ : 0 ≤ Δ) (hS0 : 0 ≤ S) | |
| (hsecant : β ^ 2 - α + 1 / 4 = Δ ^ 2 / 16) | |
| (hS : S = T - δ * β) (hY : Y₀ = Y - δ * α) | |
| (herror : |2 * β * S - Y₀| ≤ Δ * S / 2) : | |
| S - |δ| * Δ / 4 ≤ Real.sqrt (T ^ 2 - δ * Y + δ ^ 2 / 4) := by | |
| have h := cartesian_discriminant_lower hΔ hS0 hsecant hS hY herror | |
| have hQ : 0 ≤ T ^ 2 - δ * Y + δ ^ 2 / 4 := | |
| (sq_nonneg _).trans h | |
| by_cases hsign : 0 ≤ S - |δ| * Δ / 4 | |
| · exact (Real.le_sqrt hsign hQ).2 h | |
| · exact (le_of_not_ge hsign).trans (Real.sqrt_nonneg _) | |
| theorem selected_sector_sqrt {l r T Y δ S : ℝ} | |
| (hlr : l ≤ r) (hS0 : 0 ≤ S) | |
| (hS : S = T - δ * ((l + r) / 4)) | |
| (hleft : l * S ≤ Y - δ * ((1 + l * r) / 4)) | |
| (hright : Y - δ * ((1 + l * r) / 4) ≤ r * S) : | |
| S - |δ| * (r - l) / 4 ≤ Real.sqrt (T ^ 2 - δ * Y + δ ^ 2 / 4) := by | |
| exact cartesian_discriminant_sqrt (sub_nonneg.mpr hlr) hS0 | |
| (ordinary_secant_identity l r) hS rfl (sector_midpoint_error hleft hright) | |
| theorem cartesian_radial_identity (q : ℝ) : | |
| (q / 2) ^ 2 / 4 + (q / 2) ^ 4 = (q ^ 2 + q ^ 4) / 16 := by | |
| ring | |
| theorem cartesian_obstruction_identity (x y η : ℝ) : | |
| (x ^ 2 / 4 + x ^ 4) + (y ^ 2 / 4 + y ^ 4) + η - | |
| (x ^ 2 + y ^ 2) / 4 - 2 * x ^ 2 * y ^ 2 = | |
| (x ^ 2 - y ^ 2) ^ 2 + η := by | |
| ring | |
| theorem cartesian_obstruction_positive (x y : ℝ) {η : ℝ} (hη : 0 < η) : | |
| 0 < (x ^ 2 / 4 + x ^ 4) + (y ^ 2 / 4 + y ^ 4) + η - | |
| (x ^ 2 + y ^ 2) / 4 - 2 * x ^ 2 * y ^ 2 := by | |
| rw [cartesian_obstruction_identity] | |
| exact add_pos_of_nonneg_of_pos (sq_nonneg _) hη | |
| theorem squared_radius_obstruction (x y : ℝ) : | |
| x ^ 2 + y ^ 2 + 8 * x ^ 2 * y ^ 2 ≤ | |
| x ^ 2 + y ^ 2 + 4 * (x ^ 4 + y ^ 4) := by | |
| nlinarith only [sq_nonneg (x ^ 2 - y ^ 2)] | |
| theorem cartesian_polynomial_bound {Y : ℝ} (hY : |Y| ≤ 3 / 2) : | |
| 0 ≤ Y ^ 2 / 4 + Y ^ 4 ∧ Y ^ 2 / 4 + Y ^ 4 ≤ 45 / 8 := by | |
| have hsq : Y ^ 2 ≤ 9 / 4 := by | |
| nlinarith only [(abs_le.mp hY).1, (abs_le.mp hY).2] | |
| have hfour : Y ^ 4 ≤ 81 / 16 := by | |
| nlinarith only [sq_nonneg (Y ^ 2 - 9 / 4), sq_nonneg Y, hsq, | |
| mul_nonneg (sq_nonneg Y) (sub_nonneg.mpr hsq)] | |
| constructor | |
| · positivity | |
| · linarith | |
| theorem displaced_intercept_bound {Y δ α τ : ℝ} | |
| (hY : |Y| ≤ 1) (hδ : |δ| ≤ τ) (hτ : τ ≤ 1) | |
| (hα0 : 0 ≤ α) (hα1 : α ≤ 1 / 2) : | |
| |Y - δ * α| ≤ 3 / 2 := by | |
| calc | |
| |Y - δ * α| ≤ |Y| + |δ * α| := abs_sub _ _ | |
| _ = |Y| + |δ| * α := by rw [abs_mul, abs_of_nonneg hα0] | |
| _ ≤ 1 + 1 * (1 / 2) := by | |
| gcongr | |
| exact hδ.trans hτ | |
| _ = 3 / 2 := by norm_num | |
| theorem strict_half_of_expansion {ε C : ℝ} | |
| (hε : 0 < ε) (hsmall : ε ≤ 1 / 2000) | |
| (hexp : |C - (1 / 2 - (13 / 6000 : ℝ) * ε ^ 2)| ≤ 2 * ε ^ 4) : | |
| C < 1 / 2 := by | |
| have hsq : ε ^ 2 ≤ (1 / 2000 : ℝ) ^ 2 := | |
| (sq_le_sq₀ hε.le (by norm_num)).2 hsmall | |
| have hcoef : 2 * ε ^ 2 < (13 / 6000 : ℝ) := by | |
| norm_num at hsq | |
| linarith | |
| have hneg := mul_neg_of_pos_of_neg (sq_pos_of_pos hε) (sub_neg.mpr hcoef) | |
| nlinarith only [(abs_le.mp hexp).2, hneg] | |
| end AngularEstimates | |
| end OAI | |