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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 | import OAI.Dynamics.ConditionalShuffle.PhysicalMix
namespace OAI
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
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 _))
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)
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')
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))
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
end Revealed
end
end OAI
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