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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 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 | import OAI.Dynamics.ConditionalShuffle.ConditionalMixing
namespace OAI
noncomputable section
open Filter Topology
namespace Thorp
lemma distance_antitone (d : ℕ) : Antitone (distance d) := by
apply antitone_nat_of_succ_le
intro t
unfold distance uniform
rw [law_add]
have hh := FiniteLaw.tv_conv_contract (law d 1) (law d t)
(law_sum d 1) (law_sum d t) 0 (law_nonneg d 1)
simpa only [mul_zero, sub_zero, one_mul] using hh
lemma full_mixing_time {T : ℕ → ℕ} (hT : ∀ᶠ d in atTop, 1600*d ≤ T d) :
Tendsto (fun d : ℕ => distance d (T d)) atTop (nhds 0) := by
apply squeeze_zero' (Eventually.of_forall (fun d => by unfold distance tv; positivity))
_ full_mixing
filter_upwards [hT] with d hd
exact distance_antitone d hd
lemma full_mixing_constant (C : ℕ) (hC : 1600 ≤ C) :
Tendsto (fun d : ℕ => distance d (C*d)) atTop (nhds 0) :=
full_mixing_time (Eventually.of_forall (fun d => Nat.mul_le_mul_right d hC))
lemma support_lower_every_time (d t : ℕ) (hd : 1 ≤ d) :
1 - (2 : ℝ)^(t*2^(d-1)) / ((2^d).factorial : ℝ) ≤ distance d t := by
obtain ⟨k, rfl⟩ := Nat.exists_eq_succ_of_ne_zero (by omega : d ≠ 0)
simpa only [Nat.succ_eq_add_one, Nat.add_sub_cancel, mul_comm] using distance_support_lower k t
end Thorp
namespace Revealed
open Thorp
lemma twoHalfC_pos : 0 < twoHalfC := by
have hx : 0 < (3 + twoHalfP) / 4 := by norm_num [twoHalfP, twoHalfL]
have hy : (3 + twoHalfP) / 4 < 1 := by norm_num [twoHalfP, twoHalfL]
have hh : Real.logb 2 ((3 + twoHalfP) / 4) < 0 :=
Real.logb_neg (by norm_num) hx hy
unfold twoHalfC
linarith
lemma twoHalfC_lt_one : twoHalfC < 1 := by
have hx : (1 : ℝ)/2 < (3 + twoHalfP) / 4 := by norm_num [twoHalfP, twoHalfL]
have hh := Real.logb_lt_logb (by norm_num : (1:ℝ)<2) (by norm_num : (0:ℝ)<1/2) hx
have hl : Real.logb (2:ℝ) (1/2) = -1 := by simp
rw [hl] at hh
unfold twoHalfC
linarith
lemma twoHalfB_ge_four : 4 ≤ twoHalfB := by
have h : (4:ℝ) ≤ 10 / twoHalfC := by
rw [le_div_iff₀ twoHalfC_pos]
linarith [twoHalfC_lt_one]
exact (by exact_mod_cast h.trans (Nat.le_ceil _))
theorem main :
Tendsto (fun d : ℕ => worstConditionalTV d (40000 * (400 + 1) * (d + 1)))
atTop (nhds 0) ∧
Tendsto (fun d : ℕ => distance d (16040400 * d)) atTop (nhds 0) ∧
(∀ (d t : ℕ) (g₀ : State d),
tv (lawFrom d t g₀) (uniform d) = distance d t) ∧
(∀ (d t : ℕ), 1 ≤ d →
1 - (2 : ℝ) ^ (t * 2 ^ (d - 1)) / ((2 ^ d).factorial : ℝ) ≤ distance d t) ∧
Asymptotics.IsTheta atTop (fun d : ℕ => (mixingTime d : ℝ)) orderFunction ∧
Tendsto (fun d : ℕ => distance d (171798691968 * d)) atTop (nhds 0) ∧
Tendsto (fun d : ℕ => distance d (33030144 * d)) atTop (nhds 0) ∧
Tendsto (fun d : ℕ => distance d (400 * twoHalfB * d)) atTop (nhds 0) := by
refine ⟨conditional_mixing, full_mixing_constant _ (by norm_num), worst_start_eq,
support_lower_every_time, mixingTime_isTheta_orderFunction,
full_mixing_constant _ (by norm_num), full_mixing_constant _ (by norm_num), ?_⟩
exact full_mixing_constant _ (by have h := twoHalfB_ge_four; omega)
end Revealed
end
end OAI
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