| import OAI.Dynamics.ConditionalShuffle.OverlayLaw |
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|
| namespace OAI |
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| noncomputable section |
| open scoped Classical |
| namespace Revealed.Color |
| open Thorp Thorp.Conditional Revealed.Overlay |
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| def predictResample (d : ℕ) (M : (t : ℕ) → History (d+1) t → Position d → Bool) : |
| (t : ℕ) → History (d+1) t → History (d+1) t → History (d+1) t |
| | 0, _, _ => Fin.elim0 |
| | t+1, c, r => Fin.snoc (predictResample d M t (Fin.init c) (Fin.init r)) |
| (maskedCoin (M t (Fin.init c)) (c (Fin.last t)) (r (Fin.last t))) |
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| def historyXor (d t : ℕ) (c : History (d+1) t) : Equiv.Perm (History (d+1) t) := |
| Function.Involutive.toPerm (fun r i x => c i x ^^ r i x) (by |
| intro r; funext i x |
| change (c i x ^^ (c i x ^^ r i x)) = r i x |
| cases c i x <;> cases r i x <;> rfl) |
|
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| lemma resample_xor (d : ℕ) (M : (t : ℕ) → History (d+1) t → Position d → Bool) |
| (t : ℕ) (c r : History (d+1) t) : |
| predictResample d M t c (historyXor d t c r) = |
| predictableHistory d (fun k a _ => M k a) t c r := by |
| induction t with |
| | zero => exact Subsingleton.elim _ _ |
| | succ t ih => |
| rw [predictResample, predictableHistory] |
| apply congrArg₂ Fin.snoc |
| · exact ih (Fin.init c) (Fin.init r) |
| · funext x |
| change (if M t (Fin.init c) x then c (Fin.last t) x ^^ r (Fin.last t) x else c (Fin.last t) x) = |
| (c (Fin.last t) x ^^ (if M t (Fin.init c) x then r (Fin.last t) x else false)) |
| cases M t (Fin.init c) x <;> simp |
|
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| lemma predictResample_stationary (d : ℕ) |
| (M : (t : ℕ) → History (d+1) t → Position d → Bool) (t : ℕ) |
| (f : History (d+1) t → ℝ) : |
| mean (fun c => mean (fun r => f (predictResample d M t c r))) = mean f := by |
| calc |
| _ = mean (fun c => mean (fun r => f (predictableHistory d (fun k a _ => M k a) t c r))) := by |
| apply mean_congr; intro c |
| have he := mean_equiv (historyXor d t c) (fun r => f (predictResample d M t c r)) |
| change mean (fun r => f (predictResample d M t c (historyXor d t c r))) = _ at he |
| simp_rw [resample_xor] at he |
| exact he.symm |
| _ = mean (fun r => mean (fun c => f (predictableHistory d (fun k a _ => M k a) t c r))) := mean_comm _ |
| _ = mean (fun _ : History (d+1) t => mean f) := by |
| apply mean_congr; intro r |
| exact mean_equiv (predictableEquiv d (fun k a _ => M k a) t r) f |
| _ = mean f := mean_const _ |
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| end Revealed.Color |
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| end |
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| end OAI |
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