File size: 3,897 Bytes
b317718 | 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 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 | import OAI.Probability.GaussianPropeller.Elimination
namespace OAI
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
namespace GaussianPropeller.Reduction
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
end GaussianPropeller.Reduction
namespace GaussianPropeller
theorem main_bound : MainBound := by
intro n hn A hA
exact Reduction.all_partition_bound n (by omega) (n+1) A hA
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
end GaussianPropeller
namespace GaussianPropeller
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
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
end GaussianPropeller
end OAI
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