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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 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 | 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
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