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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 46 47 48 49 50 51 52 53 54 55 56 57 58 59 | import OAI.Dynamics.ConditionalShuffle.OverlayLaw
namespace OAI
noncomputable section
open scoped Classical
namespace Revealed.Color
open Thorp Thorp.Conditional Revealed.Overlay
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)))
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)
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
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 _
end Revealed.Color
end
end OAI
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