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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 | 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
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