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f1f7953 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 | import OAI.Analysis.IrrationalRotation.Model.CircleCoupling
import OAI.Analysis.IrrationalRotation.Couplings.CircleCouplingParameters
import OAI.Analysis.IrrationalRotation.Rotation.CircleRawTagEvent
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Set
open scoped ENNReal BigOperators
namespace Invariant233.CircleCoupling
open CircleDigitCoupling CircleParameterData OneCoordinateCoupling
def rawCurrentBlockEvent (p : CircleParameterData) (j : ℕ) : Set Sample :=
{ω | (∃ i ∈ p.block j, label i ω = some (4 : Fin 5)) ∧
∀ i ∈ p.block j, (ω i).2.1 < 1 - 3 / (p.m j : ℝ) ∧
3 / (p.m j : ℝ) < (ω i).2.2}
lemma measurableSet_rawCurrentBlockEvent (p : CircleParameterData) (j : ℕ) :
MeasurableSet (rawCurrentBlockEvent p j) := by
classical
unfold rawCurrentBlockEvent label
measurability
variable {θ : ℝ} {p : CircleParameterData} (hp : CircleParameterLaws θ p)
include hp
private lemma coordinate_probability_for_tag (j i : ℕ) :
IsProbabilityMeasure (coordinateLaw p j i) := by
by_cases hi : i = 0
· simp only [coordinateLaw, ite_eq_left hi, initialLaw]
infer_instance
have hs := p.blockIndex_pos hp.m_pos (show 1 ≤ i by omega)
have hm : (100 : ℝ) ≤ p.m (p.blockIndex i) := by
exact_mod_cast (show 100 ≤ p.m (p.blockIndex i) by have := hp.m_ge_200 _ hs; omega)
have ht : 0 ≤ tag j (p.blockIndex i) ∧ tag j (p.blockIndex i) ≤ 1 := by
unfold tag
split_ifs <;> norm_num
simpa only [coordinateLaw, ite_eq_right hi] using rawLaw_isProbabilityMeasure
(CircleParameterLaws.coupling_kappa_bounds _ hs).1
(CircleParameterLaws.coupling_kappa_bounds _ hs).2 hm
(hp.coupling_a_bounds _ hs).1 (hp.coupling_a_bounds _ hs).2
(hp.coupling_b_bounds _ hs).1 (hp.coupling_b_bounds _ hs).2 ht.1 ht.2
lemma coordinateLaw_currentBlock {j i : ℕ} (hi : i ∈ p.block j) :
coordinateLaw p j i = rawLaw (kappa j) (p.m j) (p.a j) (p.b j) 1 := by
have hi0 : i ≠ 0 := by
have h := (p.mem_block_iff j i).mp hi
omega
simp only [coordinateLaw, ite_eq_right hi0, p.blockIndex_eq_of_mem hp.m_pos hi,
tag, ite_true]
theorem rawCurrentBlockEvent_mass_lower (j : ℕ) (hj : 1 ≤ j) :
Real.exp (-90) ≤ (law p j).real (rawCurrentBlockEvent p j) := by
classical
let := coordinate_probability_for_tag hp j
let μ₀ := rawLaw (kappa j) (p.m j) (p.a j) (p.b j) 1
have : IsProbabilityMeasure μ₀ := rawLaw_isProbabilityMeasure
(CircleParameterLaws.coupling_kappa_bounds j hj).1
(CircleParameterLaws.coupling_kappa_bounds j hj).2
(by exact_mod_cast (show 100 ≤ p.m j by have := hp.m_ge_200 j hj; omega))
(hp.coupling_a_bounds j hj).1 (hp.coupling_a_bounds j hj).2
(hp.coupling_b_bounds j hj).1 (hp.coupling_b_bounds j hj).2
(by norm_num) (by norm_num)
let e : Fin (p.m j) ≃ (p.block j) := Fintype.equivOfCardEq (by simp [p.card_block j hj])
let E := MeasurableEquiv.piCongrLeft (fun _ : Fin (p.m j) => CoordinateSample) e.symm
let f : Sample → Fin (p.m j) → CoordinateSample := fun ω i => ω (e i)
have hf : Measurable f := by unfold f; fun_prop
have hcomp : f = E ∘ (p.block j).restrict := by
funext ω i
simp [E, f, MeasurableEquiv.coe_piCongrLeft, Equiv.piCongrLeft_apply]
have hmap : (law p j).map f = Measure.pi (fun _ : Fin (p.m j) => μ₀) := by
rw [hcomp, ← Measure.map_map E.measurable (Finset.measurable_restrict _)]
change ((Measure.infinitePi (coordinateLaw p j)).map (p.block j).restrict).map E = _
rw [Measure.infinitePi_map_restrict]
have hcoord : (fun i : p.block j => coordinateLaw p j i) = fun _ => μ₀ := by
funext i
exact coordinateLaw_currentBlock hp i.property
rw [hcoord]
exact Measure.pi_map_piCongrLeft e.symm (fun _ : Fin (p.m j) => μ₀)
let A : Set (Fin (p.m j) → CoordinateSample) :=
{z | (∃ i, (z i).1 = some (4 : Fin 5)) ∧
∀ i, (z i).2.1 < 1 - 3 / (p.m j : ℝ) ∧ 3 / (p.m j : ℝ) < (z i).2.2}
have hA : MeasurableSet A := by
dsimp [A]
measurability
have hpre : f ⁻¹' A = rawCurrentBlockEvent p j := by
ext ω
constructor
· rintro ⟨⟨i, hi⟩, hω⟩
refine ⟨⟨e i, (e i).property, hi⟩, ?_⟩
intro k hk
obtain ⟨i, he⟩ := e.surjective ⟨k, hk⟩
simpa only [f, he] using hω i
· rintro ⟨⟨k, hk, htag⟩, hω⟩
obtain ⟨i, he⟩ := e.surjective ⟨k, hk⟩
refine ⟨⟨i, ?_⟩, ?_⟩
· simpa only [f, he, label] using htag
· intro i
exact hω (e i) (e i).property
have hmass : (law p j).real (rawCurrentBlockEvent p j) =
(Measure.pi (fun _ : Fin (p.m j) => μ₀)).real A := by
rw [← hmap]
simp only [measureReal_def, Measure.map_apply hf hA, hpre]
rw [hmass]
exact concrete_circle_currentBlock_mass (p.m j) (hp.m_ge_200 j hj) (kappa j)
(CircleParameterLaws.coupling_kappa_bounds j hj).1
(CircleParameterLaws.coupling_kappa_bounds j hj).2
(fun _ => p.a j) (fun _ => p.b j)
(fun _ => (hp.coupling_a_bounds j hj).1) (fun _ => (hp.coupling_a_bounds j hj).2)
(fun _ => (hp.coupling_b_bounds j hj).1) (fun _ => (hp.coupling_b_bounds j hj).2)
end Invariant233.CircleCoupling
end
end OAI
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