openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Algebra /FixedRankContradiction.lean
| 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 | |