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a89d282 | 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 | import OAI.Probability.InvariantIsing.Magnetic.MagneticVariationalTransport
import OAI.Probability.InvariantIsing.Magnetic.MagneticPartitionContinuity
namespace OAI
/-! The field transportation bound extends from finite spectral laws to
every compact spectral law by uniform transform approximation. -/
noncomputable section
open MeasureTheory ProbabilityTheory Filter Set
open scoped Topology BigOperators
namespace InvariantIsing
theorem measureMagnetic_transport_le {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)
(w : A → B → ℝ) (γ c : A → ℝ) (δ d : B → ℝ)
(hw : ∀ i j, 0 ≤ w i j) (hγ : ∀ i, 0 ≤ γ i) (hδ : ∀ j, 0 < δ j)
(hγsum : ∑ i, γ i=1) (hδsum : ∑ j, δ j=1)
(hrow : ∀ i, ∑ j, w i j=γ i) (hcol : ∀ j, ∑ i, w i j=δ j) :
(finiteMagneticFunctional (measureR (ν : Measure ℝ) b) γ c).toReal ≤
(finiteMagneticFunctional (measureR (ν : Measure ℝ) b) δ d).toReal+
∑ i, ∑ j, w i j* |c i-d j| := by
let ε := fun k : ℕ => (1 : ℝ)/(k+1)
let D (k : ℕ) : CompactSpectralPartition ν a b (ε k) := Classical.choice
(exists_compact_spectral_partition ν a b (ε k) hcompact hbound (by dsimp [ε]; positivity))
have hε : Tendsto ε atTop (𝓝 0) := tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)
have h1 := compact_partition_magnetic_tendsto ν a b ε D (hbound ha).2 ha hb hbound hε
γ c hγ hγsum
have h2 := compact_partition_magnetic_tendsto ν a b ε D (hbound ha).2 ha hb hbound hε
δ d (fun j => (hδ j).le) hδsum
apply le_of_tendsto_of_tendsto h1 (h2.add_const _)
apply Eventually.of_forall
intro k
let ρ := fun i : (D k).Positive => (D k).weight i
let lam := fun i : (D k).Positive => (D k).value i
have hρ i : 0 < ρ i := i.property
have hs : ∑ i, ρ i=1 := (D k).weight_sum
have heq : measureR ((D k).law : Measure ℝ) (D k).edge=finiteR ρ lam hρ hs := by
funext x
exact measureR_finiteSpectralMeasure ρ lam hρ hs (D k).upper (D k).le_upper x
dsimp only
rw [heq]
exact finiteMagneticFunctional_transport_le ρ lam hρ hs w γ c δ d hw hγ hδ
hγsum hδsum hrow hcol
end InvariantIsing
end
end OAI
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