openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Spectral /MeasureMagneticLawInvariance.lean
| import OAI.Probability.InvariantIsing.Spectral.MeasureMagneticTransport | |
| import OAI.Probability.InvariantIsing.Core.FiniteFieldLawCoupling | |
| namespace OAI | |
| /-! The finite-field magnetic value depends on the field law, not on its | |
| choice of positive atomic labels. -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Filter Set | |
| open scoped BigOperators | |
| namespace InvariantIsing | |
| theorem measureMagnetic_eq_of_field_law_eq {A B : Type*} [Fintype A] [Fintype 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) | |
| (γ c : A → ℝ) (δ d : B → ℝ) (hγ : ∀ i, 0 < γ i) (hδ : ∀ j, 0 < δ j) | |
| (hγsum : ∑ i, γ i=1) (hδsum : ∑ j, δ j=1) | |
| (he : finiteSpectralMeasure γ c=finiteSpectralMeasure δ d) : | |
| (finiteMagneticFunctional (measureR (ν : Measure ℝ) b) γ c).toReal= | |
| (finiteMagneticFunctional (measureR (ν : Measure ℝ) b) δ d).toReal := by | |
| have hmass := fieldAtomMass_eq_of_law_eq γ c δ d (fun i => (hγ i).le) | |
| (fun j => (hδ j).le) he | |
| have hh := measureMagnetic_transport_le ν a b hcompact hbound ha hb | |
| (sameFieldCoupling γ c δ d) γ c δ d | |
| (sameFieldCoupling_nonneg γ c δ d hγ (fun j => (hδ j).le)) | |
| (fun i => (hγ i).le) hδ hγsum hδsum | |
| (sameFieldCoupling_row γ c δ d hγ hmass) (sameFieldCoupling_col γ c δ d hδ hmass) | |
| have hk := measureMagnetic_transport_le ν a b hcompact hbound ha hb | |
| (sameFieldCoupling δ d γ c) δ d γ c | |
| (sameFieldCoupling_nonneg δ d γ c hδ (fun i => (hγ i).le)) | |
| (fun j => (hδ j).le) hγ hδsum hγsum | |
| (sameFieldCoupling_row δ d γ c hδ (fun x => (hmass x).symm)) | |
| (sameFieldCoupling_col δ d γ c hγ (fun x => (hmass x).symm)) | |
| rw [sameFieldCoupling_cost,add_zero] at hh hk | |
| exact le_antisymm hh hk | |
| end InvariantIsing | |
| end | |
| end OAI | |