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92416a2 | 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 | import OAI.Dynamics.ConditionalShuffle.OverlayGauge
namespace OAI
noncomputable section
open scoped Classical
namespace Revealed.Overlay
open Thorp Thorp.Conditional Revealed.Split Revealed.Instrument Revealed.Scheduled
lemma fairMass_equiv_tv {Ω G : Type} [Fintype Ω] [Fintype G] (f : Ω → G) (e : G ≃ G) :
tv (fairMass (fun ω => e (f ω))) (fun _ => (Fintype.card G : ℝ)⁻¹) =
tv (fairMass f) (fun _ => (Fintype.card G : ℝ)⁻¹) := by
unfold tv
rw [← Equiv.sum_comp e]
congr 1
apply Finset.sum_congr rfl
intro g _
simp only [fairMass, e.injective.eq_iff]
variable {ι α : Type} [Fintype ι] [Fintype α] [DecidableEq α]
lemma expectedTV_instrument (d : ℕ) (L : Sum ι α ≃ Position (d+1)) (σ : ℕ → Bool) (t : ℕ) :
expectedTV d L σ t = Instrument.distance (stepInstrument (d+1) σ) (0,Split.outside L) t := by
unfold expectedTV conditionalLaw
have hh (c : History (d+1) t) :
tv (fairMass (overlayRun d L σ t c)) (fun _ => (Fintype.card (Equiv.Perm α) : ℝ)⁻¹) =
tv (fairMass (resampledRun (stepInstrument (d+1) σ) (xorIncrement d σ) (0,Split.outside L) t c))
(fun _ => (Fintype.card (Equiv.Perm α) : ℝ)⁻¹) := by
have he : overlayRun d L σ t c = fun r =>
((Equiv.mulLeft ((assignment (L.trans (run (d+1) t c)))⁻¹)).trans
(Equiv.mulRight (assignment L)))
(resampledRun (stepInstrument (d+1) σ) (xorIncrement d σ) (0,Split.outside L) t c r) := by
funext r
exact overlayRun_gauge d L σ t c r
rw [he]
exact fairMass_equiv_tv _ _
simp_rw [hh]
exact resampled_distance (stepInstrument (d+1) σ) (xorIncrement d σ) (xorIncrement_law d σ) _ t
end Revealed.Overlay
end
end OAI
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