| import OAI.Probability.InvariantIsing.Core.FiniteIndexCoupling |
| import OAI.Probability.InvariantIsing.Spectral.MeasureMagneticTransport |
|
|
| namespace OAI |
|
|
| /-! The magnetic transportation estimate on a common probability space. -/ |
| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped BigOperators |
| namespace InvariantIsing |
|
|
| lemma finiteIndex_magnetic_sub_le {Ω A B : Type*} [MeasurableSpace Ω] |
| [Fintype A] [MeasurableSpace A] [MeasurableSingletonClass A] |
| [Fintype B] [MeasurableSpace B] [MeasurableSingletonClass B] |
| (P : Measure Ω) [IsProbabilityMeasure P] |
| (g : Ω → A) (h : Ω → B) (hg : Measurable g) (hh : Measurable h) |
| (c : A → ℝ) (d : B → ℝ) (hpos : ∀ b, 0 < finiteIndexWeight P h b) |
| (ν : ProbabilityMeasure ℝ) (a b : ℝ) |
| (hcompact : IsCompact (ν : Measure ℝ).support) |
| (hbound : (ν : Measure ℝ).support ⊆ Icc a b) |
| (ha : a∈(ν : Measure ℝ).support) (hb : b∈(ν : Measure ℝ).support) : |
| (finiteMagneticFunctional (measureR (ν : Measure ℝ) b) (finiteIndexWeight P g) c).toReal- |
| (finiteMagneticFunctional (measureR (ν : Measure ℝ) b) (finiteIndexWeight P h) d).toReal ≤ |
| ∫ x, |c (g x)-d (h x)| ∂P := by |
| have ht := measureMagnetic_transport_le ν a b hcompact hbound ha hb |
| (finiteIndexCoupling P g h) (finiteIndexWeight P g) c (finiteIndexWeight P h) d |
| (fun _ _ => measureReal_nonneg) (fun _ => measureReal_nonneg) hpos |
| (finiteIndexWeight_sum P g hg) (finiteIndexWeight_sum P h hh) |
| (finiteIndexCoupling_row P g h hh) (finiteIndexCoupling_col P g h hg) |
| rw [← finiteIndexCoupling_integral P g h hg hh] at ht |
| linarith |
|
|
| end InvariantIsing |
|
|
| end |
|
|
| end OAI |
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|
