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a2ee19a | 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 | import OAI.Probability.RandomSAT.Convergence
namespace OAI
/-!
For every fixed clause size at least three, the proper-clause satisfiability
probability has a positive finite limiting threshold. No conclusion is asserted
at the critical density.
-/
namespace FixedClauseThreshold
open Finset
noncomputable section
attribute [local instance] Classical.propDecidable
def limitingCenter (k : ℕ) : ℝ := Filter.limsup (fun n : ℕ => center n k) Filter.atTop
theorem center_tendsto (k : ℕ) (hk : 3 ≤ k) :
Filter.Tendsto (fun n : ℕ => center n k) Filter.atTop (nhds (limitingCenter k)) := by
have hb := center_bounds_eventually k hk
have hUpper : Filter.IsBoundedUnder (· ≤ ·) Filter.atTop (fun n : ℕ => center n k) :=
Filter.isBoundedUnder_of_eventually_le (hb.mono (fun _ h => h.2))
have hLower : Filter.IsBoundedUnder (· ≥ ·) Filter.atTop (fun n : ℕ => center n k) :=
Filter.isBoundedUnder_of_eventually_ge (hb.mono (fun _ h => h.1))
apply summable_min_convergence (fun n => center n k) (by norm_num : (0 : ℝ) ≤ 4)
(delta_pos hk) hUpper hLower
obtain ⟨N, hadd, hnext⟩ := center_comparisons k hk
refine ⟨N, ?_, ?_⟩
· intro r hr s hs
simpa only [densityWindow, Nat.cast_min] using hadd r hr s hs
· intro s hs
exact hnext s hs
theorem limitingCenter_bounds (k : ℕ) (hk : 3 ≤ k) :
lowerCenterBound ≤ limitingCenter k ∧ limitingCenter k ≤ (2 : ℝ)^k + 1 := by
have ht := center_tendsto k hk
have hb := center_bounds_eventually k hk
exact ⟨ge_of_tendsto ht (hb.mono (fun _ h => h.1)), le_of_tendsto ht (hb.mono (fun _ h => h.2))⟩
theorem limitingCenter_pos (k : ℕ) (hk : 3 ≤ k) : 0 < limitingCenter k :=
lowerCenterBound_pos.trans_le (limitingCenter_bounds k hk).1
theorem limitingCenter_lt_cap (k : ℕ) (hk : 3 ≤ k) : limitingCenter k < capMultiplier k := by
have hp : (1 : ℝ) < 2^k := one_lt_pow₀ (by norm_num) (by omega : k ≠ 0)
have he : (capMultiplier k : ℝ) = 2^k * 2 := by simp [capMultiplier, pow_succ]
rw [he]
have h := (limitingCenter_bounds k hk).2
linarith
theorem floor_density_le_cap {n k : ℕ} {c : ℝ} (hc0 : 0 ≤ c) (hcL : c ≤ capMultiplier k) :
⌊c * (n : ℝ)⌋₊ ≤ mainCap n k := by
have h := (Nat.floor_le (mul_nonneg hc0 (Nat.cast_nonneg n : (0 : ℝ) ≤ n))).trans
(mul_le_mul_of_nonneg_right hcL (Nat.cast_nonneg n : (0 : ℝ) ≤ n))
exact_mod_cast (show (⌊c * (n : ℝ)⌋₊ : ℝ) ≤ (mainCap n k : ℝ) by
simpa only [mainCap, Nat.cast_mul] using h)
theorem subcritical_limit (k : ℕ) (hk : 3 ≤ k) (c : ℝ) (hc0 : 0 ≤ c)
(hc : c < limitingCenter k) :
Filter.Tendsto (fun n : ℕ => properSATProbability n k ⌊c * (n : ℝ)⌋₊)
Filter.atTop (nhds 1) := by
have hcL : c ≤ capMultiplier k := (hc.trans (limitingCenter_lt_cap k hk)).le
have hcenter : Filter.Tendsto (fun n : ℕ => center n k - densityWindow n k)
Filter.atTop (nhds (limitingCenter k)) := by
simpa only [sub_zero] using (center_tendsto k hk).sub (densityWindow_tendsto k hk)
have hmargin := hcenter.eventually (Ioi_mem_nhds hc)
have hlower : ∀ᶠ n : ℕ in Filter.atTop,
1 - concentrationError n k ≤ properSATProbability n k ⌊c * (n : ℝ)⌋₊ := by
filter_upwards [hmargin, Filter.eventually_ge_atTop (k + 1)] with n hmargin hn
apply properSATProbability_center_lower hk hn (floor_density_le_cap hc0 hcL)
have hn0 : 0 < n := by omega
have hn' : (0 : ℝ) < n := by exact_mod_cast hn0
rw [fluctuationWindow_eq_mul hn0]
have hmul := mul_le_mul_of_nonneg_left hmargin.le hn'.le
have hf := Nat.floor_le (mul_nonneg hc0 hn'.le)
nlinarith
have hlowT : Filter.Tendsto (fun n : ℕ => 1 - concentrationError n k) Filter.atTop (nhds 1) := by
simpa only [sub_zero] using (concentrationError_tendsto k hk).const_sub 1
exact tendsto_of_tendsto_of_tendsto_of_le_of_le' hlowT tendsto_const_nhds hlower
(Filter.Eventually.of_forall fun n => properSATProbability_le_one n k _)
theorem supercritical_limit_below_cap (k : ℕ) (hk : 3 ≤ k) (c : ℝ)
(hc : limitingCenter k < c) (hcL : c ≤ capMultiplier k) :
Filter.Tendsto (fun n : ℕ => properSATProbability n k ⌊c * (n : ℝ)⌋₊)
Filter.atTop (nhds 0) := by
have hc0 : 0 ≤ c := ((limitingCenter_pos k hk).trans hc).le
have hd : Filter.Tendsto (fun n : ℕ => 2 * densityWindow n k) Filter.atTop (nhds 0) := by
simpa only [mul_zero] using (densityWindow_tendsto k hk).const_mul (2 : ℝ)
have hcenter : Filter.Tendsto (fun n : ℕ => center n k + 2 * densityWindow n k)
Filter.atTop (nhds (limitingCenter k)) := by
simpa only [add_zero] using (center_tendsto k hk).add hd
have hmargin := hcenter.eventually_lt_const hc
have hupper : ∀ᶠ n : ℕ in Filter.atTop,
properSATProbability n k ⌊c * (n : ℝ)⌋₊ ≤ concentrationError n k := by
filter_upwards [hmargin, Filter.eventually_ge_atTop (k + 1)] with n hmargin hn
apply properSATProbability_center_upper hk hn (floor_density_le_cap hc0 hcL)
have hn0 : 0 < n := by omega
have hn' : (0 : ℝ) < n := by exact_mod_cast hn0
have hw := fluctuationWindow_one_le (n := n) hk hn0
have he := fluctuationWindow_eq_mul (k := k) hn0
have hmul := mul_le_mul_of_nonneg_left hmargin.le hn'.le
have hf := Nat.lt_floor_add_one (c * (n : ℝ))
nlinarith
exact squeeze_zero' (Filter.Eventually.of_forall fun n => properSATProbability_nonneg n k _)
hupper (concentrationError_tendsto k hk)
theorem supercritical_limit (k : ℕ) (hk : 3 ≤ k) (c : ℝ)
(hc : limitingCenter k < c) :
Filter.Tendsto (fun n : ℕ => properSATProbability n k ⌊c * (n : ℝ)⌋₊)
Filter.atTop (nhds 0) := by
let d := min c (capMultiplier k : ℝ)
have hd : limitingCenter k < d := lt_min hc (limitingCenter_lt_cap k hk)
have hdL : d ≤ capMultiplier k := min_le_right _ _
have ht := supercritical_limit_below_cap k hk d hd hdL
apply squeeze_zero' (Filter.Eventually.of_forall fun n => properSATProbability_nonneg n k _) _ ht
filter_upwards [Filter.eventually_ge_atTop k] with n hn
apply properSATProbability_antitone hn
exact Nat.floor_mono (mul_le_mul_of_nonneg_right (min_le_left c (capMultiplier k : ℝ)) (Nat.cast_nonneg n))
theorem main (k : ℕ) (hk : 3 ≤ k) : HasLimitingThreshold k := by
refine ⟨limitingCenter k, limitingCenter_pos k hk, ?_, ?_⟩
· intro c hc0 hc
exact subcritical_limit k hk c hc0 hc
· intro c hc
exact supercritical_limit k hk c hc
end
end FixedClauseThreshold
end OAI
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