File size: 3,099 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 73 74 | import OAI.Dynamics.ConditionalShuffle.ScheduledTransport
namespace OAI
noncomputable section
open scoped Classical
open Filter Topology
namespace Revealed.Overlay
open Thorp Thorp.Conditional Revealed.Split Revealed.Instrument Revealed.Scheduled Thorp.Specht
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
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
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)
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)
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))))
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)
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
end Revealed.Overlay
end
end OAI
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