File size: 3,246 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 | import OAI.Probability.FreeIsing.Ising.CanonicalArtanhIdentification
import OAI.Probability.FreeIsing.Ising.CommonStopBounds
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter Set
open scoped NNReal
namespace Problem367.IsingObservationFactor
open IsingObservationInnovation IsingCopyEnergy
variable {d : ℕ} {beta : ℝ}
variable {μ : Measure (TreeVertex d → IsingSpin)} [IsProbabilityMeasure μ]
theorem ae_all_copyResponseDriftDifference_primitive
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) (v : TreeVertex d) :
∀ᵐ z ∂copies beta μ, ∀ t : ℝ≥0,
nnrealTimePrimitive (copyResponseDriftDifference beta v) t z =
nnrealTimePrimitive (canonicalResponseDrift beta v) t z.2.1 -
nnrealTimePrimitive (canonicalResponseDrift beta v) t z.2.2 := by
have hi := ae_intervalIntegrable_horizons_of_joint
(μ := observationLaw μ) (a := canonicalResponseDrift beta v)
(fun N => by simpa only [NNReal.coe_natCast] using
canonicalResponseDrift_integrable hμ hb v (N : ℝ≥0))
filter_upwards [(copyProjection_measurePreserving beta μ true).quasiMeasurePreserving.ae hi,
(copyProjection_measurePreserving beta μ false).quasiMeasurePreserving.ae hi]
with z hl hr
intro t
obtain ⟨N, hN⟩ := exists_nat_ge (t : ℝ)
have hrestrict {f : ℝ → ℝ} (hf : IntervalIntegrable f volume 0 (N : ℝ)) :
IntervalIntegrable f volume 0 (t : ℝ) :=
(intervalIntegrable_iff_integrableOn_Ioc_of_le t.coe_nonneg).mpr
(((intervalIntegrable_iff_integrableOn_Ioc_of_le (Nat.cast_nonneg N)).mp hf).mono_set
(Ioc_subset_Ioc le_rfl hN))
change (∫ s in (0 : ℝ)..(t : ℝ),
canonicalResponseDrift beta v s.toNNReal z.2.1 -
canonicalResponseDrift beta v s.toNNReal z.2.2) = _
exact intervalIntegral.integral_sub (hrestrict (hl N)) (hrestrict (hr N))
theorem copied_difference_decomposition
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) (v : TreeVertex d)
{Y : ℝ≥0 → Paths d → ℝ} {Z : Bool → ℝ≥0 → Joint d → ℝ}
(hfield : ∀ᵐ x ∂observationLaw μ, ∀ t,
continuousField beta v t x = Y t x +
nnrealTimePrimitive (canonicalResponseDrift beta v) t x)
(hcopy : ∀ j, ∀ᵐ z ∂copies beta μ, ∀ t,
Y t (projection j z) = Z j t z) :
∀ᵐ z ∂copies beta μ, ∀ t,
continuousDifference beta v t z = (Z true t z - Z false t z) +
nnrealTimePrimitive (copyResponseDriftDifference beta v) t z := by
filter_upwards [(copyProjection_measurePreserving beta μ true).quasiMeasurePreserving.ae hfield,
(copyProjection_measurePreserving beta μ false).quasiMeasurePreserving.ae hfield,
hcopy true, hcopy false, ae_all_copyResponseDriftDifference_primitive hμ hb v]
with z hl hr hzl hzr hprimitive
intro t
simp only [projection, Bool.false_eq_true, ite_true, ite_false] at hl hr hzl hzr
change continuousField beta v t z.2.1 - continuousField beta v t z.2.2 = _
rw [hl t, hr t, hprimitive t]
have hzl' := hzl t
have hzr' := hzr t
change Y t z.2.1 = Z true t z at hzl'
change Y t z.2.2 = Z false t z at hzr'
rw [hzl', hzr']
ring
end Problem367.IsingObservationFactor
end
end OAI
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