| import OAI.Geometry.CylinderCovering.Angular.Estimates |
| import OAI.Geometry.CylinderCovering.Cartesian.Caps |
| import OAI.Geometry.CylinderCovering.Cartesian.DomainDefinitions |
|
|
| namespace OAI |
|
|
| namespace AngularCartesianDirect |
|
|
| open CylinderCovering.CartesianCaps |
|
|
| theorem D_lipschitz_displaced (x y : ℝ) |
| (hx : |x| ≤ 3 / 2) (hy : |y| ≤ 3 / 2) : |
| |D x - D y| ≤ (57 / 4) * |x - y| := by |
| have hx2 : x ^ 2 ≤ 9 / 4 := by |
| nlinarith only [(abs_le.mp hx).1, (abs_le.mp hx).2] |
| have hy2 : y ^ 2 ≤ 9 / 4 := by |
| nlinarith only [(abs_le.mp hy).1, (abs_le.mp hy).2] |
| have hsum : |x + y| ≤ 3 := (abs_add_le x y).trans (by linarith) |
| have hfactor : |(x + y) * (1 / 4 + x ^ 2 + y ^ 2)| ≤ 57 / 4 := by |
| rw [abs_mul, abs_of_nonneg (by positivity : 0 ≤ (1 / 4 : ℝ) + x ^ 2 + y ^ 2)] |
| calc |
| |x + y| * (1 / 4 + x ^ 2 + y ^ 2) ≤ |
| 3 * (1 / 4 + x ^ 2 + y ^ 2) := |
| mul_le_mul_of_nonneg_right hsum (by positivity) |
| _ ≤ 57 / 4 := by linarith |
| rw [D_sub_factor, abs_mul] |
| nlinarith only [mul_le_mul_of_nonneg_left hfactor (abs_nonneg (x-y))] |
|
|
| theorem displaced_polynomial_error {Y δ α τ : ℝ} |
| (hY : |Y| ≤ 1) (hδ : |δ| ≤ τ) (hτ : τ ≤ 1) |
| (hα0 : 0 ≤ α) (hα1 : α ≤ 1 / 2) : |
| |D (Y - δ * α) - D Y| ≤ 8 * τ := by |
| have hτ0 : 0 ≤ τ := (abs_nonneg δ).trans hδ |
| have hdisp : |Y - δ * α - Y| ≤ τ / 2 := by |
| have hmul := mul_le_mul hδ hα1 hα0 hτ0 |
| simpa only [sub_sub_cancel_left, abs_neg, abs_mul, abs_of_nonneg hα0, |
| mul_one_div] using hmul |
| have hlip := D_lipschitz_displaced (Y-δ*α) Y |
| (AngularEstimates.displaced_intercept_bound hY hδ hτ hα0 hα1) |
| (hY.trans (by norm_num)) |
| nlinarith only [hlip, hdisp, hτ0] |
|
|
| theorem failed_cap_gt {S H : ℝ} (hH : H < 1) |
| (hfail : ¬ (S ≤ 1 ∧ S ≤ H)) : H < S := by |
| by_contra hn |
| have hSH : S ≤ H := le_of_not_gt hn |
| exact hfail ⟨hSH.trans hH.le, hSH⟩ |
|
|
| theorem discriminant_gain {α β Δ T Y δ S τ η M : ℝ} |
| (hY : |Y| ≤ 1) (hδ : |δ| ≤ τ) (hτ0 : 0 ≤ τ) (hτ1 : τ ≤ 1) |
| (hα0 : 0 ≤ α) (hα1 : α ≤ 1 / 2) |
| (hΔ : 0 ≤ Δ) (_hM : 0 ≤ M) (hmesh : Δ ≤ M * τ ^ 2) |
| (hsmall : (32 + M) * τ ≤ 2 * η) |
| (houter : τ ^ 2 * (45 / 8 + η) < 1 / 2) |
| (hS0 : 0 ≤ S) (hS : S = T - δ * β) |
| (hsecant : β ^ 2 - α + 1 / 4 = Δ ^ 2 / 16) |
| (hsector : |2 * β * S - (Y - δ * α)| ≤ Δ * S / 2) |
| (hfail : ¬ (S ≤ 1 ∧ S ≤ 1 / 2 + τ ^ 2 * (D (Y - δ * α) + η))) : |
| 1 / 2 + τ ^ 2 * (D Y + η / 2) < |
| Real.sqrt (T ^ 2 - δ * Y + δ ^ 2 / 4) := by |
| have hDbound : D (Y - δ * α) ≤ 45 / 8 := |
| (AngularEstimates.cartesian_polynomial_bound |
| (AngularEstimates.displaced_intercept_bound hY hδ hτ1 hα0 hα1)).2 |
| have hH : 1 / 2 + τ ^ 2 * (D (Y - δ * α) + η) < 1 := by |
| have hm := mul_le_mul_of_nonneg_left hDbound (sq_nonneg τ) |
| nlinarith only [hm, houter] |
| have hcap := failed_cap_gt hH hfail |
| have herr := displaced_polynomial_error hY hδ hτ1 hα0 hα1 |
| have hD := mul_le_mul_of_nonneg_left (abs_le.mp herr).1 (sq_nonneg τ) |
| have hmesh' : |δ| * Δ ≤ M * τ ^ 3 := by |
| have hm := mul_le_mul hδ hmesh hΔ hτ0 |
| nlinarith only [hm] |
| have hmargin := mul_le_mul_of_nonneg_left hsmall (sq_nonneg τ) |
| have hsqrt := AngularEstimates.cartesian_discriminant_sqrt hΔ hS0 |
| hsecant hS rfl hsector |
| nlinarith only [hcap, hD, hmesh', hmargin, hsqrt] |
|
|
| theorem smallness_radius {η M : ℝ} (hη : 0 < η) (hM : 0 ≤ M) : |
| ∃ τ₀ : ℝ, 0 < τ₀ ∧ ∀ τ : ℝ, 0 ≤ τ → τ ≤ τ₀ → |
| τ ≤ 1 ∧ (32 + M) * τ ≤ 2 * η ∧ |
| τ ^ 2 * (45 / 8 + η) < 1 / 2 := by |
| have hd : 0 < (32 : ℝ) + M := by linarith |
| have he : 0 < 4 * (6 + η) := by positivity |
| refine ⟨min 1 (min (1 / (4 * (6 + η))) (η / (32 + M))), |
| lt_min (by norm_num) (lt_min (div_pos (by norm_num) he) (div_pos hη hd)), ?_⟩ |
| intro τ hτ hbound |
| have hτ1 : τ ≤ 1 := hbound.trans (min_le_left _ _) |
| have hτrest := hbound.trans (min_le_right _ _) |
| have hτa : τ ≤ 1 / (4 * (6 + η)) := hτrest.trans (min_le_left _ _) |
| have hτb : τ ≤ η / (32 + M) := hτrest.trans (min_le_right _ _) |
| have ha := (le_div_iff₀ he).mp hτa |
| have hb := (le_div_iff₀ hd).mp hτb |
| have hsq : τ ^ 2 ≤ τ := by nlinarith only [hτ, hτ1] |
| have hc := mul_le_mul_of_nonneg_right hsq |
| (show 0 ≤ (45 / 8 : ℝ) + η by linarith) |
| refine ⟨hτ1, ?_, ?_⟩ |
| · nlinarith only [hb, hη] |
| · nlinarith only [hc, ha, hτ] |
|
|
| theorem selected_domain_gain {l r T Y δ S τ η M : ℝ} |
| (hl : -1 ≤ l) (hlr : l ≤ r) (hr : r ≤ 1) |
| (hY : |Y| ≤ 1) (hδ : |δ| ≤ τ) (hτ0 : 0 ≤ τ) (hτ1 : τ ≤ 1) |
| (hM : 0 ≤ M) (hmesh : r - l ≤ M * τ ^ 2) |
| (hsmall : (32 + M) * τ ≤ 2 * η) |
| (houter : τ ^ 2 * (45 / 8 + η) < 1 / 2) |
| (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) |
| (hfail : (Y - δ * ((1+l*r)/4), S) ∉ domain τ η 1) : |
| 1 / 2 + τ ^ 2 * (D Y + η / 2) < |
| Real.sqrt (T ^ 2 - δ * Y + δ ^ 2 / 4) := by |
| have hal : |l| ≤ 1 := abs_le.mpr ⟨hl, hlr.trans hr⟩ |
| have har : |r| ≤ 1 := abs_le.mpr ⟨hl.trans hlr, hr⟩ |
| have hprod := CylinderCovering.CartesianCaps.abs_mul_le_one l r hal har |
| have hα0 : 0 ≤ (1+l*r)/4 := by linarith [(abs_le.mp hprod).1] |
| have hα1 : (1+l*r)/4 ≤ 1/2 := by linarith [(abs_le.mp hprod).2] |
| have hangle : |Y - δ * ((1+l*r)/4)| ≤ S := by |
| rw [abs_le] |
| constructor |
| · nlinarith only [hleft, mul_le_mul_of_nonneg_right hl hS0] |
| · nlinarith only [hright, mul_le_mul_of_nonneg_right hr hS0] |
| apply discriminant_gain hY hδ hτ0 hτ1 hα0 hα1 (sub_nonneg.mpr hlr) |
| hM hmesh hsmall houter hS0 hS (AngularEstimates.ordinary_secant_identity l r) |
| (AngularEstimates.sector_midpoint_error hleft hright) |
| intro hf |
| exact hfail ⟨hS0, hf.1, hangle, hf.2⟩ |
|
|
| theorem uniform_selected_domain_gain {η M : ℝ} (hη : 0 < η) (hM : 0 ≤ M) : |
| ∃ τ₀ : ℝ, 0 < τ₀ ∧ ∀ τ : ℝ, 0 ≤ τ → τ ≤ τ₀ → |
| ∀ l r T Y δ S : ℝ, |
| -1 ≤ l → l ≤ r → r ≤ 1 → |Y| ≤ 1 → |δ| ≤ τ → |
| r-l ≤ M*τ^2 → 0 ≤ S → S = T-δ*((l+r)/4) → |
| l*S ≤ Y-δ*((1+l*r)/4) → Y-δ*((1+l*r)/4) ≤ r*S → |
| (Y-δ*((1+l*r)/4), S) ∉ domain τ η 1 → |
| 1/2+τ^2*(D Y+η/2) < Real.sqrt (T^2-δ*Y+δ^2/4) := by |
| obtain ⟨τ₀, hτ₀, hsmall⟩ := smallness_radius hη hM |
| refine ⟨τ₀, hτ₀, ?_⟩ |
| intro τ hτ hτle l r T Y δ S hl hlr hr hY hδ hmesh hS0 hS hleft hright hfail |
| obtain ⟨hτ1, hmarg, houter⟩ := hsmall τ hτ hτle |
| exact selected_domain_gain hl hlr hr hY hδ hτ hτ1 hM hmesh hmarg houter |
| hS0 hS hleft hright hfail |
|
|
| end AngularCartesianDirect |
|
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| end OAI |
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