File size: 4,172 Bytes
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 | import OAI.Geometry.CylinderCovering.Angular.CartesianDirect
import OAI.Geometry.CylinderCovering.Angular.CartesianObstruction
namespace OAI
namespace AngularCartesianDirect
open CylinderCovering.CartesianCaps
theorem selected_pair_domains {τ η M x y s l₁ r₁ l₂ r₂ S T : ℝ}
(hτ : 0 < τ) (hτ1 : τ ≤ 1) (hsmall : τ ≤ η / 240)
(hM : 0 ≤ M) (hgain : (32 + M) * τ ≤ 2 * η)
(houter : τ ^ 2 * (45 / 8 + η) < 1 / 2)
(hx : |x| ≤ 1) (hy : |y| ≤ 1) (hs0 : 0 ≤ s) (hs1 : s ≤ 1)
(hl₁ : -1 ≤ l₁) (hlr₁ : l₁ ≤ r₁) (hr₁ : r₁ ≤ 1)
(hl₂ : -1 ≤ l₂) (hlr₂ : l₂ ≤ r₂) (hr₂ : r₂ ≤ 1)
(hmesh₁ : r₁ - l₁ ≤ M * τ ^ 2) (hmesh₂ : r₂ - l₂ ≤ M * τ ^ 2)
(hS0 : 0 ≤ S) (hT0 : 0 ≤ T)
(hS : S = s - (τ * x) * ((l₁ + r₁) / 4))
(hT : T = 1 - s + (τ * y) * ((l₂ + r₂) / 4))
(hleft₁ : l₁ * S ≤ y - (τ * x) * ((1 + l₁ * r₁) / 4))
(hright₁ : y - (τ * x) * ((1 + l₁ * r₁) / 4) ≤ r₁ * S)
(hleft₂ : l₂ * T ≤ x + (τ * y) * ((1 + l₂ * r₂) / 4))
(hright₂ : x + (τ * y) * ((1 + l₂ * r₂) / 4) ≤ r₂ * T) :
(y - (τ * x) * ((1 + l₁ * r₁) / 4), S) ∈ domain τ η 1 ∨
(x + (τ * y) * ((1 + l₂ * r₂) / 4), T) ∈ domain τ η 1 := by
by_contra! hfail
have hδ₁ : |τ * x| ≤ τ := by
rw [abs_mul, abs_of_nonneg hτ.le]
simpa using mul_le_mul_of_nonneg_left hx hτ.le
have hδ₂ : |-(τ * y)| ≤ τ := by
rw [abs_neg, abs_mul, abs_of_nonneg hτ.le]
simpa using mul_le_mul_of_nonneg_left hy hτ.le
have hfirst := selected_domain_gain hl₁ hlr₁ hr₁ hy hδ₁ hτ.le hτ1 hM hmesh₁
hgain houter hS0 hS hleft₁ hright₁ hfail.1
have hsecond := selected_domain_gain hl₂ hlr₂ hr₂ hx hδ₂ hτ.le hτ1 hM hmesh₂
hgain houter hT0
(show T = (1-s) - (-(τ*y)) * ((l₂+r₂)/4) by nlinarith only [hT])
(by simpa only [neg_mul, sub_neg_eq_add] using hleft₂)
(by simpa only [neg_mul, sub_neg_eq_add] using hright₂)
(by simpa only [neg_mul, sub_neg_eq_add] using hfail.2)
have hQ₁ : s ^ 2 - (τ * x) * y + (τ * x) ^ 2 / 4 =
s ^ 2 - τ * x * y + τ ^ 2 * x ^ 2 / 4 := by ring
have hQ₂ : (1 - s) ^ 2 - (-(τ * y)) * x + (-(τ * y)) ^ 2 / 4 =
(1 - s) ^ 2 + τ * x * y + τ ^ 2 * y ^ 2 / 4 := by ring
rw [hQ₁] at hfirst
rw [hQ₂] at hsecond
rcases discriminant_alternative hτ hτ1 hsmall hx hy hs0 hs1 with h | h
· exact (not_lt_of_ge h) hfirst
· exact (not_lt_of_ge h) hsecond
theorem uniform_selected_pair_domains {η M : ℝ} (hη : 0 < η) (hM : 0 ≤ M) :
∃ τ₀ : ℝ, 0 < τ₀ ∧ ∀ τ : ℝ, 0 < τ → τ ≤ τ₀ →
∀ x y s l₁ r₁ l₂ r₂ S T : ℝ,
|x| ≤ 1 → |y| ≤ 1 → 0 ≤ s → s ≤ 1 →
-1 ≤ l₁ → l₁ ≤ r₁ → r₁ ≤ 1 → -1 ≤ l₂ → l₂ ≤ r₂ → r₂ ≤ 1 →
r₁ - l₁ ≤ M * τ ^ 2 → r₂ - l₂ ≤ M * τ ^ 2 → 0 ≤ S → 0 ≤ T →
S = s - (τ * x) * ((l₁ + r₁) / 4) →
T = 1 - s + (τ * y) * ((l₂ + r₂) / 4) →
l₁ * S ≤ y - (τ * x) * ((1 + l₁ * r₁) / 4) →
y - (τ * x) * ((1 + l₁ * r₁) / 4) ≤ r₁ * S →
l₂ * T ≤ x + (τ * y) * ((1 + l₂ * r₂) / 4) →
x + (τ * y) * ((1 + l₂ * r₂) / 4) ≤ r₂ * T →
(y - (τ * x) * ((1 + l₁ * r₁) / 4), S) ∈ domain τ η 1 ∨
(x + (τ * y) * ((1 + l₂ * r₂) / 4), T) ∈ domain τ η 1 := by
obtain ⟨δ, hδ, hδspec⟩ := smallness_radius hη hM
refine ⟨min δ (η / 240), lt_min hδ (by positivity), ?_⟩
intro τ hτ hτsmall x y s l₁ r₁ l₂ r₂ S T hx hy hs0 hs1 hl₁ hlr₁ hr₁
hl₂ hlr₂ hr₂ hmesh₁ hmesh₂ hS0 hT0 hS hT hleft₁ hright₁ hleft₂ hright₂
obtain ⟨hτ1, hgain, houter⟩ := hδspec τ hτ.le (hτsmall.trans (min_le_left _ _))
exact selected_pair_domains hτ hτ1 (hτsmall.trans (min_le_right _ _)) hM hgain houter
hx hy hs0 hs1 hl₁ hlr₁ hr₁ hl₂ hlr₂ hr₂ hmesh₁ hmesh₂ hS0 hT0 hS hT
hleft₁ hright₁ hleft₂ hright₂
end AngularCartesianDirect
end OAI
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