| import OAI.Probability.SATVariance.InsertionVariance | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory | |
| namespace RandomKSAT | |
| open scoped Classical ENNReal | |
| lemma avoidance_probability_lower {n k : ℕ} (hkn : k ≤ n) (hn : 0 < n) | |
| (c : Clause n k) : | |
| 1-(k:ℝ)^2/n ≤ favg (fun d : Clause n k => if Disjoint c.1.1 d.1.1 then (1:ℝ) else 0) := by | |
| let := clause_nonempty n k hkn | |
| have hn0 : (n:ℝ) ≠ 0 := by exact_mod_cast (Nat.ne_of_gt hn) | |
| have hp (v : Fin n) : favg (fun d : Clause n k => if v ∈ d.1.1 then (1:ℝ) else 0) = (k:ℝ)/n := by | |
| apply (eq_div_iff hn0).mpr | |
| nlinarith [support_probability_scaled hkn v] | |
| calc | |
| _ = favg (fun d : Clause n k => 1-∑ v ∈ c.1.1, if v ∈ d.1.1 then (1:ℝ) else 0) := by | |
| rw [favg_sub,favg_const,favg_sum] | |
| simp only [hp,Finset.sum_const,c.1.2,nsmul_eq_mul] | |
| ring | |
| _ ≤ _ := favg_mono (fun d => by | |
| by_cases h : Disjoint c.1.1 d.1.1 | |
| · rw [ite_eq_left h] | |
| have hs : (∑ v ∈ c.1.1, if v ∈ d.1.1 then (1:ℝ) else 0) = 0 := by | |
| apply Finset.sum_eq_zero | |
| intro v hv | |
| simp [Finset.disjoint_left.mp h hv] | |
| rw [hs] | |
| norm_num | |
| · rw [ite_eq_right h] | |
| obtain ⟨v,hv,hd⟩ := Finset.not_disjoint_iff.mp h | |
| have hh := Finset.single_le_sum (f := fun v : Fin n => if v ∈ d.1.1 then (1:ℝ) else 0) | |
| (fun _ _ => by positivity) hv | |
| rw [ite_eq_left hd] at hh | |
| linarith) | |
| lemma log_one_sub_lower {x : ℝ} (hx : 0 ≤ x) (hx' : x ≤ 1/2) : | |
| -2*x ≤ Real.log (1-x) := by | |
| have hp : 0 < 1-x := by linarith | |
| have hi : (1-x)⁻¹ ≤ 1+2*x := by | |
| rw [inv_eq_one_div] | |
| apply (div_le_iff₀ hp).mpr | |
| nlinarith | |
| linarith [Real.one_sub_inv_le_log_of_pos hp] | |
| lemma slow_geometric_lower {x : ℝ} (hx : 0 ≤ x) (hx' : x ≤ 1/2) (J : ℕ) : | |
| Real.exp (-2*x*J) ≤ (1-x)^J := by | |
| have hp : 0 < 1-x := by linarith | |
| rw [←Real.exp_log hp,←Real.exp_nat_mul] | |
| apply Real.exp_le_exp.mpr | |
| have := mul_le_mul_of_nonneg_right (log_one_sub_lower hx hx') (show (0:ℝ) ≤ J by positivity) | |
| nlinarith | |
| def isolated {n k M : ℕ} (w : Fin M → Clause n k) (i : Fin M) : Prop := | |
| ∀ j, j ≠ i → Disjoint (w i).1.1 (w j).1.1 | |
| def isolateSet {n k M : ℕ} (w : Fin M → Clause n k) : Finset (Fin M) := | |
| Finset.univ.filter (isolated w) | |
| lemma isolated_insertNth {n k M : ℕ} (i : Fin (M+1)) (c : Clause n k) (w : Fin M → Clause n k) : | |
| isolated (i.insertNth c w) i ↔ ∀ j : Fin M, Disjoint c.1.1 (w j).1.1 := by | |
| unfold isolated | |
| rw [Fin.forall_iff_succAbove i] | |
| simp only [Fin.insertNth_apply_same, ne_eq, not_true_eq_false, false_implies, true_and, | |
| Fin.succAbove_ne, not_false_eq_true,true_implies,Fin.insertNth_apply_succAbove] | |
| lemma favg_all.{u_1, u_2} {ι : Type u_1} {α : Type u_2} [Fintype ι] [DecidableEq ι] [Fintype α] (p : α → Prop) : | |
| favg (fun w : ι → α => if ∀ i, p (w i) then (1:ℝ) else 0) = | |
| (favg (fun a => if p a then (1:ℝ) else 0))^(Fintype.card ι) := by | |
| have he (w : ι → α) : (if ∀ i, p (w i) then (1:ℝ) else 0) = | |
| ∏ i, if p (w i) then (1:ℝ) else 0 := by simp only [Finset.prod_boole,Finset.mem_univ,true_implies] | |
| simp only [he] | |
| rw [favg_pi_product (fun _ a => if p a then (1:ℝ) else 0)] | |
| simp | |
| lemma isolate_probability_lower {n k M R : ℕ} (hk : 1 ≤ k) (hkn : k ≤ n) | |
| (hn : 2*k^2 ≤ n) (hM : M ≤ 2*R*n) (i : Fin (M+1)) : | |
| Real.exp (-4*(R:ℝ)*k^2) ≤ favg (fun w : Fin (M+1) → Clause n k => if isolated w i then (1:ℝ) else 0) := by | |
| let := clause_nonempty n k hkn | |
| have hn' : 0 < n := by omega | |
| have hnR : (0:ℝ) < n := by exact_mod_cast hn' | |
| have hx : 0 ≤ (k:ℝ)^2/n := by positivity | |
| have hx' : (k:ℝ)^2/n ≤ 1/2 := by | |
| apply (div_le_iff₀ hnR).mpr | |
| have hnR' : 2*(k:ℝ)^2 ≤ n := by exact_mod_cast hn | |
| linarith | |
| have he : Real.exp (-4*(R:ℝ)*k^2) ≤ (1-(k:ℝ)^2/n)^M := by | |
| apply le_trans ?_ (slow_geometric_lower hx hx' M) | |
| apply Real.exp_le_exp.mpr | |
| have hMR : (M:ℝ) ≤ 2*R*n := by exact_mod_cast hM | |
| have hh := mul_le_mul_of_nonneg_right hMR hx | |
| have hd : (2*(R:ℝ)*n)*((k:ℝ)^2/n) = 2*R*k^2 := by field_simp | |
| rw [hd] at hh | |
| nlinarith | |
| rw [favg_insertNth i] | |
| simp only [isolated_insertNth] | |
| have hm (c : Clause n k) : | |
| favg (fun w : Fin M → Clause n k => if ∀ j, Disjoint c.1.1 (w j).1.1 then (1:ℝ) else 0) = | |
| (favg (fun d : Clause n k => if Disjoint c.1.1 d.1.1 then (1:ℝ) else 0))^M := by | |
| have hh := favg_all (ι := Fin M) (fun d : Clause n k => Disjoint c.1.1 d.1.1) | |
| simp only [Fintype.card_fin] at hh | |
| convert hh using 1 | |
| · congr 1 | |
| funext w | |
| by_cases h : ∀ j : Fin M, Disjoint c.1.1 (w j).1.1 <;> simp [h] | |
| · congr 2 | |
| funext d | |
| by_cases h : Disjoint c.1.1 d.1.1 <;> simp [h] | |
| simp only [hm] | |
| rw [←favg_const (α := Clause n k) (Real.exp (-4*(R:ℝ)*k^2))] | |
| exact favg_mono (fun c => he.trans (pow_le_pow_left₀ (by linarith) | |
| (avoidance_probability_lower hkn hn' c) M)) | |
| lemma isolate_count_lower {n k M R : ℕ} (hk : 1 ≤ k) (hkn : k ≤ n) | |
| (hn : 2*k^2 ≤ n) (hM : M ≤ 2*R*n) : | |
| (M:ℝ)*Real.exp (-4*(R:ℝ)*k^2) ≤ | |
| favg (fun w : Fin M → Clause n k => ((isolateSet w).card:ℝ)) := by | |
| have he (w : Fin M → Clause n k) : ((isolateSet w).card:ℝ) = | |
| ∑ i, if isolated w i then (1:ℝ) else 0 := by simp [isolateSet] | |
| simp only [he] | |
| rw [favg_sum] | |
| calc | |
| _ = ∑ _ : Fin M, Real.exp (-4*(R:ℝ)*k^2) := by simp | |
| _ ≤ _ := Finset.sum_le_sum (fun i _ => by | |
| cases M with | |
| | zero => exact i.elim0 | |
| | succ M => exact isolate_probability_lower hk hkn hn (by omega) i) | |
| end RandomKSAT | |
| end | |
| end OAI | |