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c270adf | 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 61 | 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; exact hq.1 j; rfl
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 <;> simp
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 <;> simp [hgood]
end LargeIndependentSets
end OAI
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