File size: 3,220 Bytes
b317718 | 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 | import OAI.Probability.FreeIsing.Conditioning.CenteredCopyDifference
import OAI.Probability.FreeIsing.Spectral.ColoredKoopman
namespace OAI
noncomputable section
open MeasureTheory
namespace Problem367.CopyDifference
variable {A B : Type*} [MeasurableSpace A] [MeasurableSpace B]
variable {μ : Measure A} {ν : Measure B} {p q : B → A}
variable (hp : MeasurePreserving p ν μ) (hq : MeasurePreserving q ν μ)
variable (τ : A → A) (σ : B → B)
variable (hτ : MeasurePreserving τ μ μ) (hσ : MeasurePreserving σ ν ν)
variable (hpc : p ∘ σ = τ ∘ p) (hqc : q ∘ σ = τ ∘ q)
include hpc hqc in
private theorem symmetry_intertwines (f : Lp ℝ 2 μ) :
koopmanL2 σ hσ (operator hp hq f) = operator hp hq (koopmanL2 τ hτ f) :=
congrArg (fun L : Lp ℝ 2 μ →L[ℝ] Lp ℝ 2 ν => L f)
(intertwines hp hq hτ hσ hpc hqc)
def differenceKoopmanIsometry :
rangeClosure (operator hp hq) →ₗᵢ[ℝ] rangeClosure (operator hp hq) where
toLinearMap := (restrictToRangeClosure (operator hp hq)
(koopmanL2 τ hτ) (koopmanL2 σ hσ)
(symmetry_intertwines hp hq τ σ hτ hσ hpc hqc)).toLinearMap
norm_map' f := koopmanL2_norm σ hσ (f : Lp ℝ 2 ν)
@[simp] theorem differenceKoopmanIsometry_apply_coe
(f : rangeClosure (operator hp hq)) :
(differenceKoopmanIsometry hp hq τ σ hτ hσ hpc hqc f : Lp ℝ 2 ν) =
koopmanL2 σ hσ f := rfl
theorem differenceKoopmanIsometry_involutive (hinv : Function.Involutive σ) :
Function.Involutive (differenceKoopmanIsometry hp hq τ σ hτ hσ hpc hqc) := by
intro f
apply Subtype.ext
exact koopmanL2_involutive σ hσ hinv f
def differenceKoopmanEquiv (hinv : Function.Involutive σ) :
rangeClosure (operator hp hq) ≃ₗᵢ[ℝ] rangeClosure (operator hp hq) :=
LinearIsometryEquiv.ofSurjective
(differenceKoopmanIsometry hp hq τ σ hτ hσ hpc hqc)
(differenceKoopmanIsometry_involutive hp hq τ σ hτ hσ hpc hqc hinv).surjective
@[simp] theorem differenceKoopmanEquiv_apply_coe (hinv : Function.Involutive σ)
(f : rangeClosure (operator hp hq)) :
(differenceKoopmanEquiv hp hq τ σ hτ hσ hpc hqc hinv f : Lp ℝ 2 ν) =
koopmanL2 σ hσ f := rfl
theorem differenceKoopmanEquiv_involutive (hinv : Function.Involutive σ) :
Function.Involutive (differenceKoopmanEquiv hp hq τ σ hτ hσ hpc hqc hinv) :=
differenceKoopmanIsometry_involutive hp hq τ σ hτ hσ hpc hqc hinv
theorem neighborAverage_differenceKoopmanEquiv {d : ℕ}
(τ : Fin d → A → A) (σ : Fin d → 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)
(hinv : ∀ i, Function.Involutive (σ i)) :
ColoredEdgeLifts.neighborAverage
(fun i => differenceKoopmanEquiv hp hq (τ i) (σ i) (hτ i) (hσ i)
(hpc i) (hqc i) (hinv i)) =
averageOnDifferences hp hq τ σ hτ hσ hpc hqc := by
apply ContinuousLinearMap.ext
intro f
apply Subtype.ext
simp [ColoredEdgeLifts.neighborAverage_apply, averageOnDifferences,
restrictToRangeClosure, koopmanAverage]
end Problem367.CopyDifference
end
end OAI
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