| import OAI.Probability.GaussianPropeller.Elimination |
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| namespace OAI |
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| open MeasureTheory ProbabilityTheory |
| open scoped ENNReal |
| open scoped RealInnerProductSpace |
| open scoped RealInnerProductSpace |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal RealInnerProductSpace |
| open Filter |
| open scoped Topology |
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped Topology |
| open scoped RealInnerProductSpace |
| open Set Filter |
| open scoped Topology RealInnerProductSpace |
| open scoped NNReal |
| open Set Filter |
| open scoped Topology RealInnerProductSpace NNReal |
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped Topology RealInnerProductSpace |
| open MeasureTheory Set Filter |
| open scoped Topology BigOperators |
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped RealInnerProductSpace Topology |
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped RealInnerProductSpace Topology ENNReal |
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped RealInnerProductSpace Topology ENNReal |
| open Metric |
| open MeasureTheory ProbabilityTheory Set |
| open scoped RealInnerProductSpace ENNReal |
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped ENNReal RealInnerProductSpace |
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| namespace GaussianPropeller.Reduction |
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| theorem all_partition_bound (d:ℕ) (hd:0<d) : |
| ∀ k (A:Fin k→Set (Space d)), IsPartition A → value A≤9/(8*Real.pi) := by |
| classical |
| let : NeZero d := ⟨by omega⟩ |
| intro k |
| induction k using Nat.strong_induction_on with |
| | h k ih => |
| intro A hA |
| by_cases hk:k≤3 |
| · exact partition_bound_small hk hA |
| let : NeZero k := ⟨by omega⟩ |
| obtain ⟨B,hB⟩:=exists_minimal_optimal (d:=d) (k:=k) |
| suffices hb:value B≤9/(8*Real.pi) from (hB.1.2 _ hA).trans hb |
| by_contra hn |
| have hC:9/(8*Real.pi)<value B := lt_of_not_ge hn |
| by_cases hall:∀ j,gaussian d (B j)≠0 |
| · obtain ⟨D⟩:=exists_configuration (by omega) hB hC hall |
| by_cases hk4:k=4 |
| · subst k |
| exact D.impossible_four |
| · exact D.impossible_five_or_more (by omega) |
| · push Not at hall |
| obtain ⟨i,hi⟩:=hall |
| cases k with |
| | zero => omega |
| | succ m => |
| have hp:=remove_null_partition hB.1.1 i hi |
| have hv:=ih m (by omega) (fun j=>B (i.succAbove j)) hp |
| rw [remove_null_value i hi] at hv |
| exact (not_le_of_gt hC) hv |
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| end GaussianPropeller.Reduction |
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| namespace GaussianPropeller |
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| theorem main_bound : MainBound := by |
| intro n hn A hA |
| exact Reduction.all_partition_bound n (by omega) (n+1) A hA |
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| theorem all_partitions : AllPartitions := by |
| constructor |
| · intro d k hd hk A hA |
| exact Reduction.all_partition_bound d hd k A hA |
| · exact Reduction.propeller_attainment |
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| end GaussianPropeller |
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| namespace GaussianPropeller |
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| theorem all_partition_bound_nullMeasurable {d k : ℕ} (hd : 0 < d) |
| (A : Fin k → Set (Space d)) |
| (hA : ∀ i, NullMeasurableSet (A i) (gaussian d)) |
| (hp : ∀ᵐ x ∂gaussian d, ∃! i, x ∈ A i) : |
| value A ≤ 9 / (8 * Real.pi) := by |
| let B : Fin k → Set (Space d) := fun i => toMeasurable (gaussian d) (A i) |
| have he : ∀ i, B i =ᵐ[gaussian d] A i := fun i => (hA i).toMeasurable_ae_eq |
| have hB : IsPartition B := by |
| refine ⟨fun i => measurableSet_toMeasurable _ _, ?_⟩ |
| have hh := ae_all_iff.mpr he |
| filter_upwards [hp,hh] with x hx heq |
| change ∀ i, (x ∈ B i) = (x ∈ A i) at heq |
| obtain ⟨i,hi,hu⟩ := hx |
| exact ⟨i,Eq.mpr (heq i) hi,fun j hj => hu j (Eq.mp (heq j) hj)⟩ |
| rw [←Reduction.value_congr_ae he] |
| exact Reduction.all_partition_bound d hd k B hB |
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| theorem main_bound_nullMeasurable (n : ℕ) (hn : 4 ≤ n) |
| (A : Fin (n+1) → Set (Space n)) |
| (hA : ∀ i, NullMeasurableSet (A i) (gaussian n)) |
| (hp : ∀ᵐ x ∂gaussian n, ∃! i, x ∈ A i) : |
| value A ≤ 9 / (8 * Real.pi) := |
| all_partition_bound_nullMeasurable (by omega) A hA hp |
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| end GaussianPropeller |
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| end OAI |
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