openAI-math-dataset / lean /OAI /Combinatorics /IndependentSets /Geometry /NoUniformlyBadFinitePatterns.lean
| import Mathlib | |
| import OAI.Combinatorics.IndependentSets.Reduction.RestrictionLawBad | |
| namespace OAI | |
| namespace LargeIndependentSets | |
| open MeasureTheory Filter | |
| open scoped Topology | |
| theorem no_uniformly_bad_finite_patterns {s d : ℕ} {ξ : ℝ} (hξ : 0 < ξ) | |
| (hcounter : ∀ r, s ≤ r → ∃ μ : ProbabilityMeasure (ListPattern s d (Fin r)), | |
| ∀ e : Fin s ↪o Fin r, | |
| ξ ≤ (μ {C | ¬GoodPatternOn (C.restrict e) Finset.univ} : ℝ)) : False := by | |
| classical | |
| have hex (k : ℕ) := homogeneous_bad_finite_law hcounter k | |
| (1 / ((k : ℝ) + 1)) (by positivity) | |
| choose μ hhom hbad using hex | |
| let seq (k : ℕ) := extensionLaw (rationalExhaustion k) (rationalExhaustion_card k) (μ k) | |
| let natRat : ℕ ↪o ℚ := ⟨⟨Nat.cast, Nat.cast_injective⟩, by simp⟩ | |
| let e₀ : Fin s ↪o ℚ := (Fin.valOrderEmb s).trans natRat | |
| apply no_asymptotically_invariant_bad_sequence hξ seq _ e₀ | |
| · filter_upwards [eventually_range_rationalExhaustion e₀] with k hk | |
| rw [← restrictionLaw_bad] | |
| change ξ ≤ ((restrictionLaw e₀ (extensionLaw (rationalExhaustion k) | |
| (rationalExhaustion_card k) (μ k))) {D | ¬GoodPatternOn D Finset.univ} : ℝ) | |
| rw [restriction_extensionLaw_inside _ _ _ hk, restrictionLaw_bad] | |
| exact hbad k _ | |
| · intro n e f | |
| filter_upwards [eventually_ge_atTop n, | |
| eventually_range_rationalExhaustion e, eventually_range_rationalExhaustion f] | |
| with k hnk he hf | |
| change dist (restrictionLaw e (extensionLaw (rationalExhaustion k) _ (μ k))) | |
| (restrictionLaw f (extensionLaw (rationalExhaustion k) _ (μ k))) ≤ _ | |
| rw [restriction_extensionLaw_inside _ _ _ he, restriction_extensionLaw_inside _ _ _ hf] | |
| exact (hhom k n hnk _ _).le | |
| end LargeIndependentSets | |
| end OAI | |