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785c727 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 | 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
end OAI
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