| import Mathlib |
| import OAI.Combinatorics.IndependentSets.Geometry.NoUniformlyBadFinitePatterns |
| import OAI.Combinatorics.IndependentSets.Repetition.FiniteGame |
|
|
| namespace OAI |
|
|
| namespace LargeIndependentSets |
| open MeasureTheory |
| open scoped BigOperators |
| open scoped Classical |
|
|
| noncomputable def simplexLaw {J : Type*} [Fintype J] [MeasurableSpace J] |
| (q : J → ℝ) (hq : q ∈ finiteProbabilitySimplex J) : ProbabilityMeasure J := by |
| let p : PMF J := PMF.ofFintype (fun j => ENNReal.ofReal (q j)) (by |
| rw [← ENNReal.ofReal_sum_of_nonneg (fun j _ => hq.1 j), hq.2, ENNReal.ofReal_one]) |
| exact ⟨p.toMeasure, inferInstance⟩ |
|
|
| lemma simplexLaw_apply_real {J : Type*} [Fintype J] [MeasurableSpace J] |
| [MeasurableSingletonClass J] (q : J → ℝ) (hq : q ∈ finiteProbabilitySimplex J) |
| (E : Set J) : |
| (simplexLaw q hq E : ℝ) = ∑ j, if j ∈ E then q j else 0 := by |
| classical |
| have hn : ∀ j, 0 ≤ if j ∈ E then q j else 0 := by |
| intro j; split_ifs |
| have he : (simplexLaw q hq : Measure J) E = |
| ENNReal.ofReal (∑ j, if j ∈ E then q j else 0) := by |
| change (PMF.ofFintype (fun j => ENNReal.ofReal (q j)) _).toMeasure E = _ |
| rw [PMF.toMeasure_apply_fintype, ENNReal.ofReal_sum_of_nonneg (fun j _ => hn j)] |
| apply Finset.sum_congr rfl |
| intro j _ |
| simp only [Set.indicator_apply, PMF.ofFintype_apply] |
| split_ifs < |
| rw [← ProbabilityMeasure.measureReal_eq_coe_coeFn] |
| change ENNReal.toReal ((simplexLaw q hq : Measure J) E) = _ |
| rw [he, ENNReal.toReal_ofReal (Finset.sum_nonneg (fun j _ => hn j))] |
| |
| theorem real_pattern_alignment (s d : ℕ) (ξ : ℝ) (hξ : 0 < ξ) : |
| ∃ r ≥ s, ∃ p : (Fin s ↪o Fin r) → ℝ, p ∈ finiteProbabilitySimplex (Fin s ↪o Fin r) ∧ |
| ∀ C : ListPattern s d (Fin r), |
| ∑ e, p e * (if ¬GoodPatternOn (C.restrict e) Finset.univ then 1 else 0) ≤ ξ := by |
| classical |
| by_contra h |
| push Not at h |
| apply no_uniformly_bad_finite_patterns hξ |
| intro r hr |
| have : Nonempty (Fin s ↪o Fin r) := ⟨Fin.castLEOrderEmb hr⟩ |
| obtain ⟨q, hq, hbad⟩ := finite_minimax_counterstatement |
| (fun (e : Fin s ↪o Fin r) (C : ListPattern s d (Fin r)) => |
| if ¬GoodPatternOn (C.restrict e) Finset.univ then (1 : ℝ) else 0) ξ (h r hr) |
| refine ⟨simplexLaw q hq, ?_⟩ |
| intro e |
| rw [simplexLaw_apply_real] |
| have hb := hbad e |
| simp only [mul_ite, mul_one, mul_zero] at hb |
| refine hb.le.trans (Finset.sum_le_sum fun pattern _ => ?_) |
| by_cases hgood : GoodPatternOn (pattern.restrict e) Finset.univ < |
|
|
| end LargeIndependentSets |
|
|
| end OAI |
|
|
