| import Mathlib |
| import OAI.Computability.VertexCover.PCP.PoweringLazy |
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| namespace OAI |
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| |
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| namespace UniqueGames.Foundations.PCP.LazySpectral |
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| open PoweringWalks SpectralReturn |
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| noncomputable section |
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| variable {V D : Type*} |
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| theorem half_sum_sq_le (a b : ℝ) : ((a + b) / 2) ^ 2 ≤ (a ^ 2 + b ^ 2) / 2 := by |
| nlinarith [sq_nonneg (a - b)] |
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| theorem lazy_energy_le_average [Fintype V] [Fintype D] [Nonempty D] |
| (G : PortGraph V D) (f : V → ℝ) : |
| energy (averagingOperator (lazyGraph G) f) ≤ |
| (energy f + energy (averagingOperator G f)) / 2 := by |
| calc |
| energy (averagingOperator (lazyGraph G) f) = |
| mean (fun v => ((f v + averagingOperator G f v) / 2) ^ 2) := by |
| unfold energy |
| congr 1 |
| funext v |
| rw [PoweringLazy.averagingOperator_lazy] |
| _ ≤ mean (fun v => (f v ^ 2 + averagingOperator G f v ^ 2) / 2) := |
| mean_mono (fun v => half_sum_sq_le _ _) |
| _ = mean (fun v => (1 / 2 : ℝ) * (f v ^ 2 + averagingOperator G f v ^ 2)) := by |
| congr 1 |
| funext v |
| ring |
| _ = (1 / 2 : ℝ) * (energy f + energy (averagingOperator G f)) := by |
| rw [mean_mul_left, mean_add] |
| rfl |
| _ = (energy f + energy (averagingOperator G f)) / 2 := by ring |
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| theorem lazy_energy_bound [Fintype V] [Fintype D] [Nonempty D] |
| (G : PortGraph V D) (lambda : ℝ) (hG : SpectralCertificate G lambda) |
| (f : V → ℝ) (hf : mean f = 0) : |
| energy (averagingOperator (lazyGraph G) f) ≤ |
| ((1 + lambda ^ 2) / 2) * energy f := by |
| calc |
| energy (averagingOperator (lazyGraph G) f) ≤ |
| (energy f + energy (averagingOperator G f)) / 2 := lazy_energy_le_average G f |
| _ ≤ (energy f + lambda ^ 2 * energy f) / 2 := by |
| have h := hG.contraction f hf |
| linarith |
| _ = ((1 + lambda ^ 2) / 2) * energy f := by ring |
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| theorem lazy_certificate_31_32 [Fintype V] [Fintype D] [Nonempty D] |
| (G : PortGraph V D) (hG : SpectralCertificate G (7 / 8)) : |
| SpectralCertificate (lazyGraph G) (31 / 32) where |
| nonnegative := by norm_num |
| lt_one := by norm_num |
| contraction f hf := by |
| have h := lazy_energy_bound G (7 / 8) hG f hf |
| have hcoef : ((1 + (7 / 8 : ℝ) ^ 2) / 2) ≤ (31 / 32 : ℝ) ^ 2 := by norm_num |
| exact h.trans (mul_le_mul_of_nonneg_right hcoef (energy_nonnegative f)) |
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| end |
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| end UniqueGames.Foundations.PCP.LazySpectral |
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| end OAI |
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