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135052a | 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 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 | import OAI.Probability.SATVariance.LifetimeSharpness
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter
open scoped Classical ENNReal Topology
namespace RandomKSAT
lemma sharp_parameters {k g : ℕ} (hg : 2 ≤ g) :
g^(k+1) ≤ g^(k+2) ∧ k ≤ g^(k+1) ∧ 0 < g^(k+2) := by
refine ⟨?_, ?_, by positivity⟩
· exact Nat.pow_le_pow_right (by omega) (by omega)
· exact (Nat.le_of_lt Nat.lt_two_pow_self).trans
((Nat.pow_le_pow_left hg k).trans (Nat.pow_le_pow_right (by omega) (by omega)))
lemma replacement_scaled_limit (k : ℕ) (hk : 3 ≤ k) :
Tendsto (fun g : ℕ => (g : ℝ)^(k-1) * blockKill (g^(k+2)) k
(subcube (g^(k+2)) (g^(k+1))) g) atTop (𝓝 (((2 : ℝ)^k)⁻¹)) := by
let h := fun g : ℕ => killRatio (g^(k+2)) (g^(k+1)) k
have hlim : Tendsto (fun g : ℕ => (g : ℝ)^k * h g) atTop (𝓝 (((2 : ℝ)^k)⁻¹)) :=
killRatio_limit (by omega : 0 < k+1) k
have elo : Tendsto (fun g : ℕ => ((g : ℝ)^k * h g)^2 / (g : ℝ)^(k-1))
atTop (𝓝 0) := (hlim.pow 2).div_atTop (tendsto_nat_pow_real (k-1) (by omega))
have elo' : Tendsto (fun g : ℕ => (g : ℝ)^k * h g - ((g : ℝ)^k * h g)^2 / (g : ℝ)^(k-1))
atTop (𝓝 (((2 : ℝ)^k)⁻¹)) := by simpa using hlim.sub elo
have eup : Tendsto (fun g : ℕ => (g : ℝ)^k * h g + (k : ℝ)^2 / g)
atTop (𝓝 (((2 : ℝ)^k)⁻¹)) := by
simpa using hlim.add (tendsto_const_div_atTop_nhds_zero_nat ((k : ℝ)^2))
apply tendsto_of_tendsto_of_tendsto_of_le_of_le' elo' eup
· filter_upwards [eventually_ge_atTop 2] with g hg
obtain ⟨hbu, hkb, hu⟩ := sharp_parameters (k := k) hg
have hl := mul_le_mul_of_nonneg_left (subcube_block_lower hbu (hkb.trans hbu) g)
(pow_nonneg (Nat.cast_nonneg g) (k-1))
have he : (g : ℝ)^(k-1) * ((g : ℝ) * h g - (g : ℝ)^2 * (h g)^2) =
(g : ℝ)^k * h g - ((g : ℝ)^k * h g)^2 / (g : ℝ)^(k-1) := by
have hg0 : (g : ℝ) ≠ 0 := by positivity
have hk' : k = (k-1)+1 := by omega
have hpow : (g : ℝ)^k = (g : ℝ)^(k-1) * g := by
conv_lhs => rw [hk']
rw [pow_succ]
rw [hpow]
simp only [pow_succ]
field_simp
rwa [show killRatio (g^(k+2)) (g^(k+1)) k = h g from rfl, he] at hl
· filter_upwards [eventually_ge_atTop 2] with g hg
obtain ⟨hbu, hkb, hu⟩ := sharp_parameters (k := k) hg
have hl := mul_le_mul_of_nonneg_left (subcube_block_upper hu hbu (hkb.trans hbu) g)
(pow_nonneg (Nat.cast_nonneg g) (k-1))
have he : (g : ℝ)^(k-1) * ((g : ℝ) * h g + (g : ℝ)^2 * (k : ℝ)^2 / (g^(k+2) : ℕ)) =
(g : ℝ)^k * h g + (k : ℝ)^2 / g := by
have hg0 : (g : ℝ) ≠ 0 := by positivity
have hk' : k = (k-1)+1 := by omega
have hk2 : k+2 = (k-1)+3 := by omega
rw [hk2]
have hpow : (g : ℝ)^k = (g : ℝ)^(k-1) * g := by
conv_lhs => rw [hk']
rw [pow_succ]
rw [hpow]
push_cast
simp only [pow_succ]
field_simp
rwa [show killRatio (g^(k+2)) (g^(k+1)) k = h g from rfl, he] at hl
lemma replacement_gap_scaled_limit (k : ℕ) (hk : 3 ≤ k) :
Tendsto (fun g : ℕ => (g : ℝ)^(k-1) *
(blockKill (g^(k+2)) (k-1) (subcube (g^(k+2)) (g^(k+1))) 1 -
blockKill (g^(k+2)) k (subcube (g^(k+2)) (g^(k+1))) g))
atTop (𝓝 (((2 : ℝ)^k)⁻¹)) := by
have hq := killRatio_limit (by omega : 0 < k+1) (k-1)
have hr := replacement_scaled_limit k hk
have hc : ((2 : ℝ)^(k-1))⁻¹ - ((2 : ℝ)^k)⁻¹ = ((2 : ℝ)^k)⁻¹ := by
have he : k = (k-1)+1 := by omega
conv_lhs => rhs; rw [he, pow_succ]
conv_rhs => rw [he, pow_succ]
field_simp
ring
have hh := hq.sub hr
rw [hc] at hh
apply hh.congr'
filter_upwards [eventually_ge_atTop 2] with g hg
obtain ⟨hbu, hkb, hu⟩ := sharp_parameters (k := k) hg
rw [subcube_one hbu (by omega : k-1 ≤ g^(k+2))]
ring
theorem replacement_sharp (k : ℕ) (hk : 3 ≤ k) :
∃ u : ℕ → ℕ, ∃ S : ∀ g, Finset (Assignment (u g)),
(∀ g, 2 ≤ g → k ≤ u g ∧ (S g).Nonempty) ∧
Asymptotics.IsEquivalent Filter.atTop
(fun g => blockKill (u g) (k-1) (S g) 1 - blockKill (u g) k (S g) g)
(fun g => ((2 : ℝ)^k)⁻¹ * (g : ℝ)^(-((k-1 : ℕ) : ℝ))) := by
refine ⟨fun g => g^(k+2), fun g => subcube (g^(k+2)) (g^(k+1)), ?_, ?_⟩
· intro g hg
obtain ⟨hbu, hkb, hu⟩ := sharp_parameters (k := k) hg
exact ⟨hkb.trans hbu, subcube_nonempty _ _⟩
· have hc : ((2 : ℝ)^k)⁻¹ ≠ 0 := by positivity
apply (Asymptotics.isEquivalent_iff_tendsto_one (by
filter_upwards [eventually_ge_atTop 2] with g hg
exact mul_ne_zero hc (ne_of_gt (Real.rpow_pos_of_pos (by positivity) _)))).mpr
have hh := (replacement_gap_scaled_limit k hk).div_const (((2 : ℝ)^k)⁻¹)
rw [div_self hc] at hh
apply hh.congr'
filter_upwards [eventually_ge_atTop 2] with g hg
simp only [Pi.div_apply]
rw [Real.rpow_neg (by positivity), Real.rpow_natCast]
field_simp
theorem no_uniform_littleO_error (k : ℕ) (hk : 3 ≤ k) :
¬ ∃ e : ℕ → ℝ,
Asymptotics.IsLittleO Filter.atTop e (fun g => (g : ℝ)^(-((k-1 : ℕ) : ℝ))) ∧
∀ u, k ≤ u → ∀ S : Finset (Assignment u), S.Nonempty → ∀ g, 2 ≤ g →
blockKill u (k-1) S 1 - e g ≤ blockKill u k S g := by
rintro ⟨e, he, hbound⟩
have hc : 0 < ((2 : ℝ)^k)⁻¹ := by positivity
have hp := (replacement_gap_scaled_limit k hk).eventually_const_lt (show
((2 : ℝ)^k)⁻¹ / 2 < ((2 : ℝ)^k)⁻¹ by linarith)
have hsmall := he.tendsto_div_nhds_zero.eventually_lt_const (show
(0 : ℝ) < ((2 : ℝ)^k)⁻¹ / 2 by positivity)
obtain ⟨g, hg, hp, he'⟩ := ((eventually_ge_atTop 2).and (hp.and hsmall)).exists
obtain ⟨hbu, hkb, hu⟩ := sharp_parameters (k := k) hg
have hb := hbound (g^(k+2)) (hkb.trans hbu)
(subcube (g^(k+2)) (g^(k+1))) (subcube_nonempty _ _) g hg
have hb' : blockKill (g^(k+2)) (k-1) (subcube (g^(k+2)) (g^(k+1))) 1 -
blockKill (g^(k+2)) k (subcube (g^(k+2)) (g^(k+1))) g ≤ e g := by linarith
have hm := mul_le_mul_of_nonneg_left hb' (pow_nonneg (Nat.cast_nonneg (α := ℝ) g) (k-1))
rw [Real.rpow_neg (by positivity), Real.rpow_natCast, div_inv_eq_mul] at he'
nlinarith
end RandomKSAT
end
end OAI
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