| import OAI.Dynamics.ConditionalShuffle.OverlayGauge |
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| namespace OAI |
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| noncomputable section |
| open scoped Classical |
| namespace Revealed.Overlay |
| open Thorp Thorp.Conditional Revealed.Split Revealed.Instrument Revealed.Scheduled |
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| 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] |
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| variable {ι α : Type} [Fintype ι] [Fintype α] [DecidableEq α] |
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| 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 |
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| end Revealed.Overlay |
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| end |
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| end OAI |
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