File size: 3,141 Bytes
f61e547 | 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 | import OAI.Probability.FreeIsing.Model
namespace OAI
noncomputable section
open MeasureTheory
namespace Problem367
def fairSpin (u : ℝ) : IsingSpin :=
if u ≤ 1 / 2 then IsingSpin.neg else IsingSpin.pos
lemma measurable_fairSpin : Measurable fairSpin := by
exact measurable_const.ite measurableSet_Iic measurable_const
lemma fairSpin_neg_preimage : fairSpin ⁻¹' {IsingSpin.neg} = Set.Iic (1 / 2 : ℝ) := by
ext u
simp [fairSpin]
lemma fairSpin_pos_preimage : fairSpin ⁻¹' {IsingSpin.pos} = Set.Ioi (1 / 2 : ℝ) := by
ext u
simp [fairSpin, not_le]
lemma fairSpin_neg_mass : uniform01 (fairSpin ⁻¹' {IsingSpin.neg}) = 1 / 2 := by
rw [fairSpin_neg_preimage, uniform01, Measure.restrict_apply measurableSet_Iic]
have h : Set.Iic (1 / 2 : ℝ) ∩ Set.Icc 0 1 = Set.Icc 0 (1 / 2) := by
ext u
simp only [Set.mem_inter_iff, Set.mem_Iic, Set.mem_Icc]
constructor
· rintro ⟨hu, h0, h1⟩
exact ⟨h0, hu⟩
· rintro ⟨h0, hu⟩
exact ⟨hu, h0, by linarith⟩
rw [h, Real.volume_Icc]
norm_num [ENNReal.ofReal_div_of_pos]
lemma fairSpin_pos_mass : uniform01 (fairSpin ⁻¹' {IsingSpin.pos}) = 1 / 2 := by
rw [fairSpin_pos_preimage, uniform01, Measure.restrict_apply measurableSet_Ioi]
have h : Set.Ioi (1 / 2 : ℝ) ∩ Set.Icc 0 1 = Set.Ioc (1 / 2) 1 := by
ext u
simp only [Set.mem_inter_iff, Set.mem_Ioi, Set.mem_Icc, Set.mem_Ioc]
constructor
· rintro ⟨hu, h0, h1⟩
exact ⟨hu, h1⟩
· rintro ⟨hu, h1⟩
exact ⟨hu, by linarith, h1⟩
rw [h, Real.volume_Ioc]
norm_num [ENNReal.ofReal_div_of_pos]
lemma fairSpin_mass (a : IsingSpin) : uniform01 (fairSpin ⁻¹' {a}) = 1 / 2 := by
cases a
· exact fairSpin_neg_mass
· exact fairSpin_pos_mass
def fairSpinFactor (d : ℕ) (U : TreeVertex d → ℝ) : TreeVertex d → IsingSpin :=
fun v => fairSpin (U v)
lemma measurable_fairSpinFactor (d : ℕ) : Measurable (fairSpinFactor d) := by
exact Measurable.of_eval (fun vertex => measurable_fairSpin.comp (measurable_pi_apply vertex))
lemma fairSpinFactor_relabel (d : ℕ) (g : TreeAut d) (U : TreeVertex d → ℝ) :
fairSpinFactor d (relabel g U) = relabel g (fairSpinFactor d U) := rfl
private lemma fairSpin_measurable_singleton (a : IsingSpin) : MeasurableSet ({a} : Set IsingSpin) := by
trivial
lemma fairSpinFactor_cylinder (d : ℕ) (D : Finset (TreeVertex d))
(eta : TreeVertex d → IsingSpin) :
(Measure.map (fairSpinFactor d) (labelMeasure d))
{x | ∀ v ∈ D, x v = eta v} = (1 / 2 : ENNReal) ^ D.card := by
classical
have hmeas : MeasurableSet {x : TreeVertex d → IsingSpin | ∀ v ∈ D, x v = eta v} := by
exact MeasurableSet.pi D.countable_toSet (fun v hv => fairSpin_measurable_singleton (eta v))
rw [Measure.map_apply (measurable_fairSpinFactor d) hmeas]
change (Measure.infinitePi (fun _ : TreeVertex d => uniform01))
(Set.pi (D : Set (TreeVertex d)) (fun v => fairSpin ⁻¹' {eta v})) = _
rw [Measure.infinitePi_pi _ (fun v _ => measurable_fairSpin (fairSpin_measurable_singleton (eta v)))]
simp [fairSpin_mass]
end Problem367
end
end OAI
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