| import OAI.Dynamics.ConditionalShuffle.ScheduledTransport |
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| namespace OAI |
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| noncomputable section |
| open scoped Classical |
| open Filter Topology |
| namespace Revealed.Overlay |
| open Thorp Thorp.Conditional Revealed.Split Revealed.Instrument Revealed.Scheduled Thorp.Specht |
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| lemma dimension_size_core (d : ℕ) (hd : 16 ≤ d) : 256*(d+7) ≤ 2^(d+6) := by |
| induction d, hd using Nat.le_induction with |
| | base => norm_num |
| | succ d hd ih => |
| rw [show d+1+6 = (d+6)+1 by omega, pow_succ] |
| omega |
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| lemma dimension_size_bound (d : ℕ) (hd : 16 ≤ d) : |
| 256*(d+1+2+4) ≤ Fintype.card (Position (d+1+2+3)) := by |
| rw [card_position] |
| simpa only [Nat.add_assoc] using dimension_size_core d hd |
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| lemma call_bound {ι : Type} [Fintype ι] (d : ℕ) (hd : 16 ≤ d) |
| (hR : (permutationFamily (Position d)).reciprocalSum ≤ 3) |
| (L : Sum ι (Position (d+1+2)) ≃ Position (d+1+2+3)) : |
| expectedTV (d+1+2+2) L (isActive (d+1+2+3)) (65536*21*(d+1+2+3)) ≤ groupRate d := by |
| let g := d+1+2 |
| let e := Split.outside L |
| have h₀ := expectedTV_instrument (d+1+2+2) L (isActive (d+1+2+3)) (65536*21*(d+1+2+3)) |
| have ht : segmentLength g*8 ≤ 65536*21*(g+3) := by |
| unfold segmentLength |
| rw [Hybrid.periodLength_eq] |
| omega |
| have h₁ := Instrument.distance_antitone (stepInstrument (g+3) (isActive (g+3))) (0,e) ht |
| have h₂ := distance_schedule_le (g+3) (segmentLength g*8) (isActive (g+3)) (selectedSchedule g) |
| (fun i hi => selected_active g i.val hi) e |
| have h₃ := distance_periodic (g+3) (segmentLength g) (selectedSchedule g) (selected_periodic g) e 8 |
| rw [interval_congr _ _ (selectedSchedule g) (segmentSchedule g) (selected_segment g)] at h₃ |
| have h₄ := group_distance_bound d (by omega) hR (dimension_size_bound d hd) e |
| have hh := h₁.trans (h₂.trans (h₃.le.trans h₄)) |
| exact h₀.le.trans (Instrument.distance_le_changeFintype _ _ _ _ _ hh) |
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| lemma expectedTV_nonneg {ι α : Type} [Fintype ι] [Fintype α] [DecidableEq α] |
| (d : ℕ) (L : Sum ι α ≃ Position (d+1)) (σ : ℕ → Bool) (t : ℕ) : |
| 0 ≤ expectedTV d L σ t := by |
| unfold expectedTV tv |
| exact mean_nonneg (fun _ => by positivity) |
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| lemma worstCallTV_nonneg (d : ℕ) : 0 ≤ worstCallTV d := by |
| unfold worstCallTV |
| exact (expectedTV_nonneg _ _ _ _).trans |
| (Finset.le_sup' (fun g₀ : State (d+3) => expectedTV (d+2) ((standardLayout d).trans g₀) |
| (isActive (d+3)) (65536*21*(d+3))) (Finset.mem_univ (1 : State (d+3)))) |
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| lemma worstCallTV_bound (d : ℕ) (hd : 16 ≤ d) |
| (hR : (permutationFamily (Position d)).reciprocalSum ≤ 3) : |
| worstCallTV (d+1+2) ≤ groupRate d := by |
| apply Finset.sup'_le |
| intro g₀ _ |
| exact call_bound d hd hR ((standardLayout (d+1+2)).trans g₀) |
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| theorem overlay_call : Tendsto worstCallTV atTop (nhds 0) := by |
| apply (tendsto_add_atTop_iff_nat 3).mp |
| apply squeeze_zero' (Eventually.of_forall (fun d => worstCallTV_nonneg (d+3))) |
| ?_ groupRate_tendsto |
| filter_upwards [Thorp.Block.reciprocal_atMost_three, eventually_ge_atTop 16] with d hR hd |
| exact worstCallTV_bound d hd hR |
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| end Revealed.Overlay |
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| end |
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| end OAI |
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