File size: 2,912 Bytes
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 60 61 62 63 64 65 66 67 68 69 70 71 72 | import OAI.Dynamics.ConditionalShuffle.Main
namespace OAI
noncomputable section
open scoped Classical
namespace Revealed.Overlay
open Thorp Thorp.Conditional Split
variable {ι α : Type} [fintype_ι : Fintype ι] [fintype_α : Fintype α] [decidableEq_α : DecidableEq α]
def isActive (D s : ℕ) : Bool := decide (s % (21 * D) ≥ 20 * D)
def xorCoins (d : ℕ) (L : Sum ι α ≃ Position (d+1)) (active : Bool)
(c r : Coins (d+1)) : Coins (d+1) := fun x =>
c x ^^ (if active && freePairMask d (avoid (fun i => L (.inl i))) x then r x else false)
lemma xorCoins_occupied (d : ℕ) (L : Sum ι α ≃ Position (d+1)) (active : Bool)
(c r : Coins (d+1)) (i : ι) :
step (d+1) (xorCoins d L active c r) (L (.inl i)) = step (d+1) c (L (.inl i)) := by
let retained_fintype_ι := fintype_ι
let retained_fintype_α := fintype_α
let retained_decidableEq_α := decidableEq_α
have hf : avoid (fun j => L (.inl j)) (L (.inl i)) = false := by
simp [avoid]
obtain ⟨⟨b,x⟩, hx⟩ := (splitPosition d).symm.surjective (L (.inl i))
change Fin.cons b x = L (.inl i) at hx
rw [← hx] at hf ⊢
calc
_ = scatterPosition d (b ^^ xorCoins d L active c r x, x) := step_cons d _ b x
_ = scatterPosition d (b ^^ c x, x) := by
cases b <;> simp [xorCoins, freePairMask, hf]
_ = _ := (step_cons d c b x).symm
def multiplier (d : ℕ) (L : Sum ι α ≃ Position (d+1)) (active : Bool)
(c r : Coins (d+1)) : Equiv.Perm α :=
relative (L.trans (step (d+1) c)) (L.trans (step (d+1) (xorCoins d L active c r)))
(fun i => (xorCoins_occupied d L active c r i).symm)
def overlayRun (d : ℕ) (L : Sum ι α ≃ Position (d+1)) (schedule : ℕ → Bool) :
(t : ℕ) → History (d+1) t → History (d+1) t → Equiv.Perm α
| 0, _, _ => 1
| t+1, c, r =>
multiplier d (L.trans (run (d+1) t (Fin.init c))) (schedule t)
(c (Fin.last t)) (r (Fin.last t)) *
overlayRun d L schedule t (Fin.init c) (Fin.init r)
def conditionalLaw (d : ℕ) (L : Sum ι α ≃ Position (d+1)) (schedule : ℕ → Bool)
(t : ℕ) (c : History (d+1) t) : Equiv.Perm α → ℝ :=
fairMass (overlayRun d L schedule t c)
def expectedTV (d : ℕ) (L : Sum ι α ≃ Position (d+1)) (schedule : ℕ → Bool)
(t : ℕ) : ℝ := mean (fun c : History (d+1) t =>
tv (conditionalLaw d L schedule t c)
(fun _ => (Fintype.card (Equiv.Perm α) : ℝ)⁻¹))
abbrev OutsideLabels (d : ℕ) := Fin 7 × Position d
def standardLayout (d : ℕ) : Sum (OutsideLabels d) (Position d) ≃ Position (d+3) :=
Fintype.equivOfCardEq (by simp [OutsideLabels, Position, pow_add]; omega)
def worstCallTV (d : ℕ) : ℝ :=
(Finset.univ : Finset (State (d+3))).sup' ⟨1, Finset.mem_univ _⟩ (fun g₀ : State (d+3) =>
expectedTV (d+2) ((standardLayout d).trans g₀) (isActive (d+3))
(65536 * 21 * (d+3)))
end Revealed.Overlay
end
end OAI
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