File size: 2,914 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 | import OAI.Dynamics.ConditionalShuffle.OverlayMain
namespace OAI
noncomputable section
open scoped BigOperators Classical
open Filter Topology
namespace Revealed.Overlay
open Thorp Thorp.Conditional
def prefixMean (D p t : ℕ) (w : History D p) (f : History D p → History D t → ℝ) : ℝ :=
mean (fun ω : History D (p+t) =>
if (splitHistory D p t ω).1 = w then
f (splitHistory D p t ω).1 (splitHistory D p t ω).2 else 0) /
fairMass (fun ω : History D (p+t) => (splitHistory D p t ω).1) w
lemma prefixMean_eq (D p t : ℕ) (w : History D p) (f : History D p → History D t → ℝ) :
prefixMean D p t w f = mean (f w) := by
unfold prefixMean
rw [← mean_equiv (splitHistory D p t).symm,
fairMass_eq_mean, ← mean_equiv (splitHistory D p t).symm]
simp only [Function.comp_def, Equiv.apply_symm_apply]
rw [mean_prod, mean_prod]
simp only [mean_indicator]
have hconst (a : History D p) :
mean (fun _ : History D t => if a = w then (1 : ℝ) else 0) =
if a = w then (1 : ℝ) else 0 := mean_const _
simp only [hconst]
simp only [mean, Finset.sum_ite_eq', Finset.mem_univ, ite_true]
have hc : (Fintype.card (History D p) : ℝ) ≠ 0 := by
exact_mod_cast (Fintype.card_ne_zero : Fintype.card (History D p) ≠ 0)
field_simp
def preCallLayout (d p : ℕ) (g₀ : State (d+3)) (w : History (d+3) p) (phase : ℕ) :
Sum (OutsideLabels d) (Position d) ≃ Position (d+3) :=
((standardLayout d).trans (run (d+3) p w * g₀)).trans (rotate (d+3)^phase)
def preCallRisk (d p : ℕ) (g₀ : State (d+3)) (w : History (d+3) p) (phase : ℕ) : ℝ :=
prefixMean (d+3) p (65536*21*(d+3)) w (fun a c =>
tv (conditionalLaw (d+2) (preCallLayout d p g₀ a phase) (isActive (d+3))
(65536*21*(d+3)) c)
(fun _ => (Fintype.card (Equiv.Perm (Position d)) : ℝ)⁻¹))
lemma preCallRisk_eq (d p : ℕ) (g₀ : State (d+3)) (w : History (d+3) p) (phase : ℕ) :
preCallRisk d p g₀ w phase = expectedTV (d+2) (preCallLayout d p g₀ w phase)
(isActive (d+3)) (65536*21*(d+3)) := by
exact prefixMean_eq _ _ _ _ _
lemma preCallRisk_le (d p : ℕ) (g₀ : State (d+3)) (w : History (d+3) p) (phase : ℕ) :
preCallRisk d p g₀ w phase ≤ worstCallTV d := by
rw [preCallRisk_eq]
exact Finset.le_sup' (fun g : State (d+3) => expectedTV (d+2) ((standardLayout d).trans g)
(isActive (d+3)) (65536*21*(d+3)))
(Finset.mem_univ ((run (d+3) p w * g₀).trans (rotate (d+3)^phase)))
theorem overlay_call_uniform_prehistory :
∀ ε : ℝ, 0 < ε → ∀ᶠ d : ℕ in atTop,
∀ (p : ℕ) (g₀ : State (d+3)) (w : History (d+3) p) (phase : ℕ),
preCallRisk d p g₀ w phase < ε := by
intro ε hε
filter_upwards [(tendsto_order.mp overlay_call).2 ε hε] with d hd
intro p g₀ w phase
exact (preCallRisk_le d p g₀ w phase).trans_lt hd
end Revealed.Overlay
end
end OAI
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