| import OAI.Geometry.CylinderCovering.Radial.OverlapDefinitions |
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| namespace OAI |
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| noncomputable section |
| namespace CylinderCovering.RadialOverlap |
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| theorem overlap_identities : IdentitiesStatement := by |
| intro ε x y t p q h |
| obtain ⟨hq, hp⟩ := h |
| dsimp [rho, sigma, T, S, RadialCaps.G] |
| refine ⟨?_, ?_, ?_, ?_, ?_⟩ |
| · linear_combination ε * q * hp - ε * p * hq |
| · linear_combination ε * x * hq |
| · linear_combination -ε * y * hp |
| · linear_combination -2 * ε * x * hq |
| · linear_combination 2 * ε * y * hp |
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| private theorem product_abs_le {x q : ℝ} (hx : |x| ≤ 2) (hq : |q| ≤ 1) : |
| |x * q| ≤ 2 := by |
| rw [abs_mul] |
| nlinarith [abs_nonneg x, abs_nonneg q] |
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| private theorem weighted_square_le {x q : ℝ} (hx : |x| ≤ 2) (hq : |q| ≤ 1) : |
| x ^ 2 * RadialCaps.G q ≤ 4 := by |
| have hx2 : x ^ 2 ≤ 4 := by nlinarith [sq_abs x, abs_nonneg x] |
| have hq2 : q ^ 2 ≤ 1 := by nlinarith [sq_abs q, abs_nonneg q] |
| have hG0 : 0 ≤ RadialCaps.G q := by unfold RadialCaps.G; positivity |
| have hG1 : RadialCaps.G q ≤ 1 := by unfold RadialCaps.G; linarith |
| nlinarith [mul_nonneg (sub_nonneg.mpr hx2) hG0] |
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| theorem complementary_bound : ComplementaryBoundStatement := by |
| intro ε x y t p q hε hsmall hx hy hp hq hroot hT hS |
| obtain ⟨_, hTi, hSi, _, _⟩ := overlap_identities ε x y t p q hroot |
| have hxq := product_abs_le hx hq |
| have hyp := product_abs_le hy hp |
| have hcT : 1 / 2 ≤ 1 + ε * x * q := by |
| have hh := mul_le_mul_of_nonneg_left (abs_le.mp hxq).1 hε.le |
| nlinarith |
| have hcS : 1 / 2 ≤ 1 - ε * y * p := by |
| have hh := mul_le_mul_of_nonneg_left (abs_le.mp hyp).2 hε.le |
| nlinarith |
| have hw : x ^ 2 * RadialCaps.G q + y ^ 2 * RadialCaps.G p ≤ 8 := by |
| linarith [weighted_square_le hx hq, weighted_square_le hy hp] |
| have hsum : -8 * ε ^ 2 ≤ |
| (1 + ε * x * q) * T ε x t q + (1 - ε * y * p) * S ε y t p := by |
| rw [hTi, hSi] |
| have hh := mul_le_mul_of_nonneg_left hw (sq_nonneg ε) |
| nlinarith only [hh] |
| have hT0 : (1 + ε * x * q) * T ε x t q ≤ 0 := |
| mul_nonpos_of_nonneg_of_nonpos (by linarith) hT |
| have hS0 : (1 - ε * y * p) * S ε y t p ≤ 0 := |
| mul_nonpos_of_nonneg_of_nonpos (by linarith) hS |
| have hTa := mul_le_mul_of_nonpos_right hcT hT |
| have hSa := mul_le_mul_of_nonpos_right hcS hS |
| rw [abs_of_nonpos hT, abs_of_nonpos hS] |
| constructor <;> nlinarith only [hsum, hT0, hS0, hTa, hSa] |
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| theorem monotonicity_defect (p q : ℝ) : |
| (p ^ 2 - q ^ 2) * (RadialCaps.G p - RadialCaps.G q) = |
| (p ^ 2 - q ^ 2) ^ 2 / 2 := by |
| unfold RadialCaps.G |
| ring |
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| theorem physical_complementary_bound {k τ A B : ℝ} (hk : 0 < k) |
| (hA : |A| ≤ 16 * (k * τ) ^ 2) (hB : |B| ≤ 16 * (k * τ) ^ 2) : |
| |A / k| ≤ 16 * k * τ ^ 2 ∧ |B / k| ≤ 16 * k * τ ^ 2 := by |
| rw [abs_div, abs_div, abs_of_pos hk] |
| constructor |
| · apply (div_le_iff₀ hk).2 |
| nlinarith only [hA] |
| · apply (div_le_iff₀ hk).2 |
| nlinarith only [hB] |
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| end CylinderCovering.RadialOverlap |
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| end |
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| end OAI |
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