| import OAI.Dynamics.ConditionalShuffle.PhysicalMix |
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| namespace OAI |
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| noncomputable section |
| open Filter Topology |
| namespace Revealed.Instrument |
| lemma distance_le_changeFintype {E S G Ω : Type} (s₁ s₂ : Fintype S) |
| [Fintype G] [Group G] [Fintype Ω] (I : Data E S G Ω) (e : E) (t : ℕ) {r : ℝ} |
| (h : @distance E S G Ω s₁ _ _ _ I e t ≤ r) : |
| @distance E S G Ω s₂ _ _ _ I e t ≤ r := by |
| cases Subsingleton.elim s₁ s₂ |
| exact h |
| end Revealed.Instrument |
| namespace Revealed |
| open scoped Classical |
| open Thorp Thorp.Conditional Split Physical |
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| lemma worstConditionalTV_nonneg (d t : ℕ) : 0 ≤ worstConditionalTV d t := by |
| exact (expectedTV_nonneg d t (Equiv.refl _)).trans |
| (Finset.le_sup' (fun g₀ => expectedTV d t g₀) (Finset.mem_univ _)) |
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| lemma worstConditionalTV_le {d t : ℕ} {r : ℝ} |
| (h : ∀ g₀, expectedTV d t g₀ ≤ r) : worstConditionalTV d t ≤ r := by |
| exact Finset.sup'_le Finset.univ_nonempty _ (fun g₀ _ => h g₀) |
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| lemma expectedTV_late_bound (d t : ℕ) (hd : 1 ≤ d) |
| (hR : (Thorp.Specht.permutationFamily (Position d)).reciprocalSum ≤ 3) |
| (ht : 3200*(d+2+2) ≤ t) (g₀ : State (d+2+2)) : |
| expectedTV (d+1+2) t g₀ ≤ fullConditionalRate d := by |
| let : Fintype (Outside (Position (d+1+2)) (Position (d+1+2)) (Position (d+2+2))) := inferInstance |
| let e := Split.outside (splitLayout (d+1+2) g₀) |
| have h₁ := expectedTV_le_instrument (d+1+2) t g₀ |
| have h₂ : Instrument.distance (instrument (d+2+2)) e t ≤ |
| Instrument.distance (instrument (d+2+2)) e (3200*(d+2+2)) := |
| Instrument.distance_antitone (instrument (d+2+2)) e ht |
| have h₃ := physical_distance_bound d hd hR e |
| have h₃' : Instrument.distance (instrument (d+2+2)) e (3200*(d+2+2)) ≤ fullConditionalRate d := |
| Instrument.distance_le_changeFintype _ _ _ _ _ h₃ |
| exact h₁.trans (h₂.trans h₃') |
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| theorem conditional_mixing_shift : |
| Tendsto (fun d : ℕ => worstConditionalTV (d+1+2) |
| (40000*(400+1)*(d+2+2))) atTop (nhds 0) := by |
| apply squeeze_zero' (Eventually.of_forall (fun d => worstConditionalTV_nonneg _ _)) |
| _ fullConditionalRate_tendsto |
| filter_upwards [Thorp.Block.reciprocal_atMost_three, eventually_ge_atTop 1] with d hR hd |
| exact worstConditionalTV_le (expectedTV_late_bound d _ hd hR (by omega)) |
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| theorem conditional_mixing : |
| Tendsto (fun d : ℕ => worstConditionalTV d (40000*(400+1)*(d+1))) |
| atTop (nhds 0) := by |
| apply (tendsto_add_atTop_iff_nat 3).mp |
| simpa only [Nat.add_assoc, Nat.add_comm, Nat.add_left_comm] using conditional_mixing_shift |
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| end Revealed |
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| end |
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| end OAI |
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