| import OAI.Probability.FreeIsing.Conditioning.CopyDifference |
| import OAI.Probability.FreeIsing.Spectral.KoopmanInvolutions |
| import OAI.Probability.FreeIsing.Spectral.SpectralClosure |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory |
|
|
| namespace Problem367.CopyDifference |
|
|
| variable {A B : Type*} [MeasurableSpace A] [MeasurableSpace B] |
| variable {μ : Measure A} {ν : Measure B} [IsProbabilityMeasure μ] |
| variable {p q : B → A} |
|
|
| lemma inner_const_one_self : |
| inner ℝ (Lp.const 2 μ (1 : ℝ)) (Lp.const 2 μ (1 : ℝ)) = 1 := by |
| rw [inner_const_one_eq_integral] |
| calc |
| (∫ x, Lp.const 2 μ (1 : ℝ) x ∂μ) = ∫ _ : A, (1 : ℝ) ∂μ := |
| integral_congr_ae (Lp.coeFn_const 2 μ (1 : ℝ)) |
| _ = 1 := by simp |
|
|
| lemma centered_source_cover |
| (hp : MeasurePreserving p ν μ) (hq : MeasurePreserving q ν μ) : |
| ∀ f : Lp ℝ 2 μ, ∃ z ∈ centeredL2 μ, operator hp hq z = operator hp hq f := by |
| apply SpectralClosure.source_cover_of_kills_direction (operator hp hq) |
| (innerSL ℝ (Lp.const 2 μ (1 : ℝ))) (Lp.const 2 μ (1 : ℝ)) |
| · exact inner_const_one_self |
| · exact operator_eq_zero_of_ae_const hp hq _ 1 (Lp.coeFn_const 2 μ (1 : ℝ)) |
|
|
| lemma rangeClosure_centered_eq |
| (hp : MeasurePreserving p ν μ) (hq : MeasurePreserving q ν μ) : |
| rangeClosure ((operator hp hq).domRestrict (centeredL2 μ)) = |
| rangeClosure (operator hp hq) := by |
| exact SpectralClosure.rangeClosure_domRestrict_eq (operator hp hq) (centeredL2 μ) |
| (centered_source_cover hp hq) |
|
|
| lemma norm_restrict_le_of_centered_source |
| (hp : MeasurePreserving p ν μ) (hq : MeasurePreserving q ν μ) |
| (S : Lp ℝ 2 μ →L[ℝ] Lp ℝ 2 μ) (T : Lp ℝ 2 ν →L[ℝ] Lp ℝ 2 ν) |
| (hS : ∀ f ∈ centeredL2 μ, S f ∈ centeredL2 μ) |
| (hT : IsSelfAdjoint T) |
| (hcomm : ∀ f, T (operator hp hq f) = operator hp hq (S f)) |
| {ρ : ℝ} (hρ : 0 ≤ ρ) (hbound : ‖S.restrict hS‖ ≤ ρ) : |
| ‖restrictToRangeClosure (operator hp hq) S T hcomm‖ ≤ ρ := by |
| exact SpectralClosure.norm_restrict_le_of_source_subspace S T (operator hp hq) |
| (centeredL2 μ) hS hT hcomm (centered_source_cover hp hq) hρ hbound |
|
|
| section Averaging |
|
|
| variable {ι : Type*} [Fintype ι] |
|
|
| omit [IsProbabilityMeasure μ] in |
|
|
| lemma average_intertwines_apply |
| (hp : MeasurePreserving p ν μ) (hq : MeasurePreserving q ν μ) |
| (τ : ι → A → A) (σ : ι → B → B) |
| (hτ : ∀ i, MeasurePreserving (τ i) μ μ) |
| (hσ : ∀ i, MeasurePreserving (σ i) ν ν) |
| (hpc : ∀ i, p ∘ σ i = τ i ∘ p) (hqc : ∀ i, q ∘ σ i = τ i ∘ q) |
| (f : Lp ℝ 2 μ) : |
| koopmanAverage σ hσ (operator hp hq f) = operator hp hq (koopmanAverage τ hτ f) := by |
| simp only [koopmanAverage, smul_apply, sum_apply, map_smul, map_sum] |
| congr 1 |
| apply Finset.sum_congr rfl |
| intro i hi |
| exact congrArg (fun L : Lp ℝ 2 μ →L[ℝ] Lp ℝ 2 ν => L f) |
| (intertwines hp hq (hτ i) (hσ i) (hpc i) (hqc i)) |
|
|
| def averageOnDifferences |
| (hp : MeasurePreserving p ν μ) (hq : MeasurePreserving q ν μ) |
| (τ : ι → A → A) (σ : ι → B → B) |
| (hτ : ∀ i, MeasurePreserving (τ i) μ μ) |
| (hσ : ∀ i, MeasurePreserving (σ i) ν ν) |
| (hpc : ∀ i, p ∘ σ i = τ i ∘ p) (hqc : ∀ i, q ∘ σ i = τ i ∘ q) : |
| rangeClosure (operator hp hq) →L[ℝ] rangeClosure (operator hp hq) := |
| restrictToRangeClosure (operator hp hq) (koopmanAverage τ hτ) (koopmanAverage σ hσ) |
| (average_intertwines_apply hp hq τ σ hτ hσ hpc hqc) |
|
|
| omit [IsProbabilityMeasure μ] in |
| lemma averageOnDifferences_isSelfAdjoint |
| (hp : MeasurePreserving p ν μ) (hq : MeasurePreserving q ν μ) |
| (τ : ι → A → A) (σ : ι → B → B) |
| (hτ : ∀ i, MeasurePreserving (τ i) μ μ) |
| (hσ : ∀ i, MeasurePreserving (σ i) ν ν) |
| (hσinv : ∀ i, Function.Involutive (σ i)) |
| (hpc : ∀ i, p ∘ σ i = τ i ∘ p) (hqc : ∀ i, q ∘ σ i = τ i ∘ q) : |
| IsSelfAdjoint (averageOnDifferences hp hq τ σ hτ hσ hpc hqc) := |
| isSelfAdjoint_restrictToRangeClosure _ _ _ |
| (average_intertwines_apply hp hq τ σ hτ hσ hpc hqc) |
| (koopmanAverage_selfAdjoint σ hσ hσinv) |
|
|
| lemma norm_averageOnDifferences_le |
| (hp : MeasurePreserving p ν μ) (hq : MeasurePreserving q ν μ) |
| (τ : ι → A → A) (σ : ι → B → B) |
| (hτ : ∀ i, MeasurePreserving (τ i) μ μ) |
| (hσ : ∀ i, MeasurePreserving (σ i) ν ν) |
| (hτinv : ∀ i, Function.Involutive (τ i)) |
| (hσinv : ∀ i, Function.Involutive (σ i)) |
| (hpc : ∀ i, p ∘ σ i = τ i ∘ p) (hqc : ∀ i, q ∘ σ i = τ i ∘ q) |
| {ρ : ℝ} (hρ : 0 ≤ ρ) (hbound : ‖centeredKoopmanAverage τ hτ hτinv‖ ≤ ρ) : |
| ‖averageOnDifferences hp hq τ σ hτ hσ hpc hqc‖ ≤ ρ := by |
| exact norm_restrict_le_of_centered_source hp hq (koopmanAverage τ hτ) |
| (koopmanAverage σ hσ) (koopmanAverage_preserves_centered τ hτ hτinv) |
| (koopmanAverage_selfAdjoint σ hσ hσinv) |
| (average_intertwines_apply hp hq τ σ hτ hσ hpc hqc) hρ hbound |
|
|
| end Averaging |
|
|
| end Problem367.CopyDifference |
|
|
| end |
|
|
| end OAI |
|
|
