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2f887db | 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 | import OAI.Computability.PerfectCompleteness.Decoding.TwoResponseCollisionLemmas
import OAI.Computability.PerfectCompleteness.Reduction.FixedCallBudgetLemmas
namespace OAI
section
namespace PerfectCompleteness.FixedRankContradiction
open scoped BigOperators Classical
open UniqueGamesTheorem.Foundations.Games
noncomputable section
theorem probability_split {Ω : Type*} [Fintype Ω]
(μ : FiniteDistribution Ω) (event low : Ω → Bool) :
μ.probability event = μ.probability (fun x => event x && low x) +
μ.probability (fun x => event x && !(low x)) := by
unfold FiniteDistribution.probability
rw [← Finset.sum_add_distrib]
apply Finset.sum_congr rfl
intro x _
cases he : event x <;> cases hl : low x <;> simp [he, hl]
variable {δ : ℚ} {hδ : 0 < δ} (p : FixedParameters.Parameters δ hδ)
def gamma (height : Nat) : ℝ :=
UpperParameterScalars.gamma (InitialParameters.useful δ) p.plan.density
(InitialParameters.inverse δ) p.plan.order (FixedRows.rows p.plan height)
theorem impossible {Ω : Type*} [Fintype Ω] (μ : FiniteDistribution Ω)
(event low : Ω → Bool) (height : Nat) (hheight : height ≤ p.plan.depth)
(hupper : gamma p height ≤ μ.probability event)
(hhigh : μ.probability (fun x => event x && !(low x)) ^ 2 ≤
gamma p height ^ 2 / 8 + 2 * p.accuracy)
(hlow : μ.probability (fun x => event x && low x) < p.accuracy) : False := by
have hgamma : 0 < gamma p height :=
UpperParameterScalars.gamma_pos (InitialParameters.useful_pos hδ)
p.plan.density_pos (InitialParameters.inverse_pos hδ)
p.plan.order (FixedRows.rows p.plan height)
have hsmall := p.accuracy_small height hheight
simp only [CommonAccuracy.tolerance, lt_min_iff] at hsmall
have hsquare : p.accuracy < gamma p height ^ 2 / 640 := hsmall.2.2.2.1
have hlinear : p.accuracy < gamma p height / 40 := hsmall.2.2.2.2
have hhalf : μ.probability (fun x => event x && !(low x)) < gamma p height / 2 := by
by_contra hn
have hge := le_of_not_gt hn
have hnonneg := μ.probability_nonnegative (fun x => event x && !(low x))
have hsq := (sq_le_sq₀ (by positivity : 0 ≤ gamma p height / 2) hnonneg).mpr hge
nlinarith
have hsplit := probability_split μ event low
linarith
end
end PerfectCompleteness.FixedRankContradiction
end
end OAI
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