| import OAI.Dynamics.ConditionalShuffle.OverlayMain |
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| namespace OAI |
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| noncomputable section |
| open scoped BigOperators Classical |
| open Filter Topology |
| namespace Revealed.Overlay |
| open Thorp Thorp.Conditional |
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| 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 |
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| 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 |
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| 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) |
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| 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)) |
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| 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 _ _ _ _ _ |
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| 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))) |
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| 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 |
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| end Revealed.Overlay |
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| end |
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| end OAI |
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