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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 | import Mathlib
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
namespace RandomKSAT
abbrev Assignment (n : ℕ) := Fin n → Bool
def Clause (n k : ℕ) :=
(s : {s : Finset (Fin n) // s.card = k}) × (s.1 → Bool)
instance clauseFintype (n k : ℕ) : Fintype (Clause n k) := by
classical
unfold Clause
infer_instance
instance clauseMeasurableSpace (n k : ℕ) : MeasurableSpace (Clause n k) := ⊤
instance clauseMeasurableSingletonClass (n k : ℕ) :
MeasurableSingletonClass (Clause n k) := ⟨fun _ => trivial⟩
def Satisfies {n k : ℕ} (c : Clause n k) (a : Assignment n) : Prop :=
∃ v : c.1.1, a v = c.2 v
abbrev Stream (n k : ℕ) := ℕ → Clause n k
def clauseLaw (n k : ℕ) : Measure (Clause n k) := uniformOn Set.univ
def streamLaw (n k : ℕ) : Measure (Stream n k) :=
Measure.infinitePi (fun _ : ℕ => clauseLaw n k)
def PrefixSAT {n k : ℕ} (ω : Stream n k) (m : ℕ) : Prop :=
∃ a : Assignment n, ∀ i < m, Satisfies (ω i) a
def firstFailure {n k : ℕ} (ω : Stream n k) : ℕ∞ :=
sInf ((fun m : ℕ => (m : ℕ∞)) '' {m | ¬ PrefixSAT ω m})
def H {n k : ℕ} (ω : Stream n k) : ℝ := (firstFailure ω).toNat
def cap (n : ℕ) (B : ℝ) : ℕ := ⌊B * n⌋₊
def T {n k : ℕ} (B : ℝ) (ω : Stream n k) : ℝ :=
(min (firstFailure ω) (cap n B : ℕ∞)).toNat
def ell (k n : ℕ) : ℝ :=
if k = 3 then Real.log (Real.exp 1 * n) else 1
def U (k : ℕ) : ℝ := Real.log 2 / (-Real.log (1 - ((2 : ℝ) ^ k)⁻¹))
end RandomKSAT
end
end OAI
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