| import OAI.Geometry.CylinderCovering.Angular.CartesianDirect |
| import OAI.Geometry.CylinderCovering.Angular.CartesianObstruction |
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| namespace OAI |
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| namespace AngularCartesianDirect |
| open CylinderCovering.CartesianCaps |
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| 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 |
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| 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 |
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| end AngularCartesianDirect |
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| end OAI |
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