| import OAI.Probability.SATVariance.FrozenReplacement | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory | |
| namespace RandomKSAT | |
| open scoped Classical ENNReal | |
| def SetSAT {u s : ℕ} (S : Finset (Assignment u)) (ω : Stream u s) (g : ℕ) : Prop := | |
| ∃ a ∈ S, ∀ i < g, Satisfies (ω i) a | |
| def blockKill (u s : ℕ) (S : Finset (Assignment u)) (g : ℕ) : ℝ := | |
| (streamLaw u s {ω | ¬ SetSAT S ω g}).toReal | |
| lemma setSAT_univ {u s : ℕ} (ω : Stream u s) (g : ℕ) : | |
| SetSAT Finset.univ ω g ↔ PrefixSAT ω g := by | |
| simp [SetSAT, PrefixSAT] | |
| lemma measurableSet_setSAT (u s : ℕ) (S : Finset (Assignment u)) (g : ℕ) : | |
| MeasurableSet {ω : Stream u s | SetSAT S ω g} := by | |
| simp only [SetSAT, Set.ofPred_exists] | |
| apply MeasurableSet.iUnion | |
| intro a | |
| by_cases ha : a ∈ S | |
| · simp only [ha, true_and, Set.ofPred_forall] | |
| apply MeasurableSet.iInter | |
| intro i | |
| apply MeasurableSet.iInter | |
| intro _ | |
| exact (Set.toFinite {c : Clause u s | Satisfies c a}).measurableSet.preimage | |
| (measurable_pi_apply i) | |
| · simp only [ha, false_and, Set.ofPred_false] | |
| exact MeasurableSet.empty | |
| lemma blockKill_nonneg (u s : ℕ) (S : Finset (Assignment u)) (g : ℕ) : | |
| 0 ≤ blockKill u s S g := ENNReal.toReal_nonneg | |
| lemma blockKill_le_one {u s : ℕ} (hsu : s ≤ u) (S : Finset (Assignment u)) (g : ℕ) : | |
| blockKill u s S g ≤ 1 := by | |
| let := streamLaw_probability u s hsu | |
| exact measureReal_le_one (μ := streamLaw u s) | |
| lemma blockKill_empty {u s : ℕ} (hsu : s ≤ u) (g : ℕ) : | |
| blockKill u s ∅ g = 1 := by | |
| let := streamLaw_probability u s hsu | |
| simp [blockKill, SetSAT] | |
| lemma blockKill_compl {u s : ℕ} (hsu : s ≤ u) (S : Finset (Assignment u)) (g : ℕ) : | |
| blockKill u s S g = 1 - (streamLaw u s {ω | SetSAT S ω g}).toReal := by | |
| let := streamLaw_probability u s hsu | |
| have he : {ω : Stream u s | ¬ SetSAT S ω g} = {ω | SetSAT S ω g}ᶜ := rfl | |
| rw [blockKill, he, measure_compl (measurableSet_setSAT u s S g) (measure_ne_top _ _), | |
| measure_univ, ENNReal.toReal_sub_of_le prob_le_one (by simp), ENNReal.toReal_one] | |
| lemma clauseLaw_kills_toReal {u s : ℕ} {S : Finset (Assignment u)} (hS : S.Nonempty) : | |
| (clauseLaw u s {c | Kills S c}).toReal = killRatio u (frozen S).card s := by | |
| rw [clauseLaw_kills hS] | |
| simp [killRatio, ENNReal.toReal_div, ENNReal.toReal_mul, ENNReal.toReal_pow, mul_comm] | |
| lemma blockKill_one {u s : ℕ} (hsu : s ≤ u) {S : Finset (Assignment u)} (hS : S.Nonempty) : | |
| blockKill u s S 1 = killRatio u (frozen S).card s := by | |
| let := clauseLaw_probability u s hsu | |
| have he : {ω : Stream u s | ¬ SetSAT S ω 1} = | |
| Set.pi ({0} : Finset ℕ) (fun _ => {c : Clause u s | Kills S c}) := by | |
| ext ω | |
| simp [SetSAT, Kills] | |
| rw [blockKill, he, streamLaw, Measure.infinitePi_pi] | |
| · simpa using clauseLaw_kills_toReal hS | |
| · intro _ _ | |
| exact (Set.toFinite _).measurableSet | |
| lemma blockKill_individual {u s : ℕ} (hsu : s ≤ u) {S : Finset (Assignment u)} | |
| (hS : S.Nonempty) (g : ℕ) : | |
| 1 - (1 - killRatio u (frozen S).card s)^g ≤ blockKill u s S g := by | |
| let := clauseLaw_probability u s hsu | |
| let := streamLaw_probability u s hsu | |
| have hsub : {ω : Stream u s | SetSAT S ω g} ⊆ | |
| Set.pi (Finset.range g) (fun _ => {c : Clause u s | ¬ Kills S c}) := by | |
| rintro ω ⟨a, ha, hω⟩ i hi hk | |
| exact hk a ha (hω i (Finset.mem_range.mp hi)) | |
| have ht := measure_mono (μ := streamLaw u s) hsub | |
| rw [streamLaw, Measure.infinitePi_pi _ (fun _ _ => (Set.toFinite _).measurableSet)] at ht | |
| have he : clauseLaw u s {c | ¬ Kills S c} = 1-clauseLaw u s {c | Kills S c} := by | |
| exact (measure_compl (Set.toFinite _).measurableSet (measure_ne_top _ _)).trans | |
| (by rw [measure_univ]; rfl) | |
| simp only [he, Finset.prod_const, Finset.card_range] at ht | |
| have ht' := ENNReal.toReal_mono (by finiteness) ht | |
| rw [ENNReal.toReal_pow, ENNReal.toReal_sub_of_le prob_le_one (by simp), | |
| ENNReal.toReal_one, clauseLaw_kills_toReal hS] at ht' | |
| rw [blockKill_compl hsu] | |
| change ((streamLaw u s) {ω | SetSAT S ω g}).toReal ≤ _ at ht' | |
| linarith | |
| lemma set_survival_le {u s : ℕ} (hsu : s ≤ u) (S : Finset (Assignment u)) (g : ℕ) : | |
| streamLaw u s {ω | SetSAT S ω g} ≤ | |
| (S.card : ℝ≥0∞) * (1-((2 : ℝ≥0∞)^s)⁻¹)^g := by | |
| have he : {ω : Stream u s | SetSAT S ω g} = | |
| ⋃ a : S, {ω | ∀ i < g, Satisfies (ω i) a} := by | |
| ext ω | |
| simp [SetSAT] | |
| rw [he] | |
| apply (measure_iUnion_fintype_le _ _).trans_eq | |
| simp [assignment_survival hsu] | |
| lemma frozen_card_le {u : ℕ} (S : Finset (Assignment u)) : (frozen S).card ≤ u := by | |
| simpa using Finset.card_le_univ (frozen S) | |
| lemma rate_toReal (s : ℕ) : | |
| (1-((2 : ℝ≥0∞)^s)⁻¹).toReal = 1-((2 : ℝ)^s)⁻¹ := by | |
| rw [ENNReal.toReal_sub_of_le (ENNReal.inv_le_one.mpr (one_le_pow₀ (by norm_num))) | |
| (by simp)] | |
| simp | |
| lemma real_rate_nonneg (s : ℕ) : 0 ≤ 1-((2 : ℝ)^s)⁻¹ := by | |
| exact sub_nonneg.mpr ((inv_le_one₀ (by positivity)).mpr (one_le_pow₀ (by norm_num))) | |
| lemma survival_exp_bound {u s : ℕ} (hsu : s ≤ u) (S : Finset (Assignment u)) (g : ℕ) : | |
| (streamLaw u s {ω | SetSAT S ω g}).toReal ≤ | |
| (S.card : ℝ) * Real.exp (-((2 : ℝ)^s)⁻¹ * g) := by | |
| have ht := ENNReal.toReal_mono (by finiteness) (set_survival_le hsu S g) | |
| rw [ENNReal.toReal_mul, ENNReal.toReal_natCast, ENNReal.toReal_pow, rate_toReal] at ht | |
| apply ht.trans | |
| apply mul_le_mul_of_nonneg_left _ (by positivity) | |
| have he : 1-((2 : ℝ)^s)⁻¹ ≤ Real.exp (-((2 : ℝ)^s)⁻¹) := by | |
| linarith [Real.add_one_le_exp (-((2 : ℝ)^s)⁻¹)] | |
| calc | |
| _ ≤ (Real.exp (-((2 : ℝ)^s)⁻¹))^g := pow_le_pow_left₀ (real_rate_nonneg s) he g | |
| _ = _ := by rw [← Real.exp_nat_mul]; congr 1; ring | |
| lemma blockKill_exp_full {u s : ℕ} (hsu : s ≤ u) (S : Finset (Assignment u)) (g : ℕ) : | |
| 1 - (2 : ℝ)^u * Real.exp (-((2 : ℝ)^s)⁻¹ * g) ≤ blockKill u s S g := by | |
| have hScard : (S.card : ℝ) ≤ (2 : ℝ)^u := by | |
| exact_mod_cast (by simpa using Finset.card_le_univ S : S.card ≤ 2^u) | |
| have ht := (survival_exp_bound hsu S g).trans | |
| (mul_le_mul_of_nonneg_right hScard (Real.exp_nonneg _)) | |
| rw [blockKill_compl hsu] | |
| linarith | |
| lemma blockKill_exp_individual {u s : ℕ} (hu : 0 < u) (hsu : s ≤ u) | |
| {S : Finset (Assignment u)} (hS : S.Nonempty) (g : ℕ) : | |
| 1 - Real.exp (-(g : ℝ)*killRatio u (frozen S).card s) ≤ blockKill u s S g := by | |
| have hk : killRatio u (frozen S).card s ≤ 1 := | |
| (killRatio_le_half_pow hu (frozen_card_le S) hsu).trans | |
| (pow_le_one₀ (by norm_num) (by norm_num)) | |
| have he : 1-killRatio u (frozen S).card s ≤ | |
| Real.exp (-killRatio u (frozen S).card s) := by | |
| linarith [Real.add_one_le_exp (-killRatio u (frozen S).card s)] | |
| have ht := pow_le_pow_left₀ (sub_nonneg.mpr hk) he g | |
| rw [← Real.exp_nat_mul] at ht | |
| have hm := blockKill_individual hsu hS g | |
| have hx : (g : ℝ) * (-killRatio u (frozen S).card s) = | |
| -(g : ℝ) * killRatio u (frozen S).card s := by ring | |
| rw [hx] at ht | |
| exact le_trans (sub_le_sub_left ht 1) hm | |
| lemma one_sub_exp_neg_two_mul_ge {h : ℝ} (h₀ : 0 ≤ h) (h₁ : h ≤ 1/2) : | |
| h ≤ 1-Real.exp (-2*h) := by | |
| have hre : Real.exp (-2*h) ≤ (1+2*h)⁻¹ := by | |
| have he := Real.add_one_le_exp (2*h) | |
| rw [show -2*h = -(2*h) by ring, Real.exp_neg] | |
| exact inv_anti₀ (by positivity) (by linarith) | |
| have hi : (1+2*h)⁻¹ ≤ 1-h := by | |
| rw [inv_eq_one_div] | |
| apply (div_le_iff₀ (by positivity : (0 : ℝ) < 1+2*h)).2 | |
| nlinarith | |
| linarith | |
| lemma blockKill_long {u s g : ℕ} (hu : 0 < u) (hsu : s ≤ u) | |
| (hg : (2 : ℝ)^(s+1)*u ≤ g) (S : Finset (Assignment u)) : | |
| (1 : ℝ)/3 ≤ blockKill u s S g := by | |
| have he : -((2 : ℝ)^s)⁻¹ * g ≤ -2*u := by | |
| have h2 : (0 : ℝ) < (2 : ℝ)^s := by positivity | |
| rw [pow_succ] at hg | |
| have h := mul_le_mul_of_nonneg_left hg (le_of_lt (inv_pos.mpr h2)) | |
| have hc : ((2 : ℝ)^s)⁻¹ * ((2:ℝ)^s*2*u) = 2*u := by field_simp | |
| rw [hc] at h | |
| linarith | |
| have hexp : (2 : ℝ)*Real.exp (-2) ≤ 2/3 := by | |
| have ht : (3 : ℝ) ≤ Real.exp 2 := by linarith [Real.add_one_le_exp 2] | |
| have hi := inv_anti₀ (by norm_num : (0 : ℝ)<3) ht | |
| rw [Real.exp_neg] | |
| linarith | |
| have hh : (2 : ℝ)^u * Real.exp (-2*u) ≤ 2/3 := by | |
| have hi : (1 : ℕ) ≤ u := hu | |
| calc | |
| _ = (2*Real.exp (-2))^u := by rw [mul_pow, ← Real.exp_nat_mul]; congr 2; ring | |
| _ ≤ ((2 : ℝ)/3)^u := pow_le_pow_left₀ (by positivity) hexp u | |
| _ ≤ ((2 : ℝ)/3)^1 := pow_le_pow_of_le_one (by norm_num) (by norm_num) hi | |
| _ = _ := by norm_num | |
| have hb := blockKill_exp_full hsu S g | |
| have ht := mul_le_mul_of_nonneg_left (Real.exp_le_exp.mpr he) | |
| (by positivity : (0 : ℝ) ≤ (2:ℝ)^u) | |
| linarith | |
| def replacementK (k : ℕ) : ℝ := ((2 : ℝ)^k * ((k-1 : ℕ) : ℝ))^(k-1) | |
| lemma replacementK_succ (r : ℕ) : replacementK (r+1) = ((2 : ℝ)^(r+1)*r)^r := by | |
| simp [replacementK] | |
| lemma replacement_numeric {u b r g : ℕ} (hr : 2 ≤ r) (hbu : b ≤ u) | |
| (hru : r+1 ≤ u) (hg : 1 ≤ g) {R : ℝ} (hR : 0 ≤ R) | |
| (hind : 1-Real.exp (-(g : ℝ)*killRatio u b (r+1)) ≤ R) | |
| (hlong : (2 : ℝ)^(r+2)*u ≤ g → (1 : ℝ)/3 ≤ R) : | |
| killRatio u b r - replacementK (r+1)/(g : ℝ)^r ≤ R := by | |
| have hr₀ : (0 : ℝ) < r := by exact_mod_cast (by omega : 0 < r) | |
| have hg₀ : (0 : ℝ) < g := by exact_mod_cast hg | |
| have hu₀ : (0 : ℝ) < u := by exact_mod_cast (by omega : 0 < u) | |
| have hquarter : killRatio u b r ≤ 1/4 := by | |
| apply (killRatio_le_half_pow (by omega) hbu (by omega)).trans | |
| have h := pow_le_pow_of_le_one (by norm_num : (0 : ℝ) ≤ 1/2) (by norm_num) hr | |
| norm_num at h ⊢ | |
| exact h | |
| have hK : (2 : ℝ)*(r+1) ≤ (2 : ℝ)^(r+1)*r := by | |
| have h2 : (4 : ℝ) ≤ (2 : ℝ)^(r+1) := by | |
| calc | |
| _ = (2 : ℝ)^2 := by norm_num | |
| _ ≤ _ := pow_le_pow_right₀ (by norm_num) (by omega) | |
| have hr₁ : (1 : ℝ) ≤ r := by exact_mod_cast (by omega : 1 ≤ r) | |
| nlinarith | |
| rw [replacementK_succ, ← div_pow] | |
| by_cases hrb : r+1 ≤ b | |
| · by_cases hlarge : 2 ≤ (g : ℝ)/(r+1) * (killRatio u b r)^((r : ℝ)⁻¹) | |
| · have hp := frozen_power (by omega : 1 ≤ r) hbu hrb | |
| have hex : ((r : ℝ)+1)/r = 1+(r : ℝ)⁻¹ := by field_simp | |
| rw [hex, Real.rpow_add_of_nonneg (killRatio_nonneg u b r) zero_le_one | |
| (by positivity), Real.rpow_one] at hp | |
| have hs : 2*killRatio u b r ≤ (g : ℝ)*killRatio u b (r+1) := by | |
| have h₁ := mul_le_mul_of_nonneg_left hlarge (killRatio_nonneg u b r) | |
| have h₂ := mul_le_mul_of_nonneg_left hp hg₀.le | |
| calc | |
| 2*killRatio u b r = killRatio u b r*2 := by ring | |
| _ ≤ killRatio u b r * ((g : ℝ)/(r+1) * (killRatio u b r)^((r : ℝ)⁻¹)) := h₁ | |
| _ = (g : ℝ) * (killRatio u b r * (killRatio u b r)^((r : ℝ)⁻¹)/(r+1)) := by ring | |
| _ ≤ _ := h₂ | |
| have ht := Real.exp_le_exp.mpr (neg_le_neg hs) | |
| simp only [← neg_mul] at ht | |
| have hh := one_sub_exp_neg_two_mul_ge (killRatio_nonneg u b r) (by linarith : | |
| killRatio u b r ≤ 1/2) | |
| have herr : 0 ≤ ((2:ℝ)^(r+1)*r/g)^r := by positivity | |
| linarith | |
| · have hroot : (killRatio u b r)^((r : ℝ)⁻¹) ≤ 2*(r+1)/(g : ℝ) := by | |
| apply (le_div_iff₀ hg₀).2 | |
| have hh := (not_le.mp hlarge).le | |
| rw [div_mul_eq_mul_div] at hh | |
| have hh' := (div_le_iff₀ (by positivity : (0:ℝ)<r+1)).mp hh | |
| linarith | |
| have hbound : killRatio u b r ≤ ((2:ℝ)^(r+1)*r/g)^r := by | |
| calc | |
| killRatio u b r = ((killRatio u b r)^((r : ℝ)⁻¹))^r := | |
| (Real.rpow_inv_natCast_pow (killRatio_nonneg u b r) (by omega : r ≠ 0)).symm | |
| _ ≤ _ := pow_le_pow_left₀ (Real.rpow_nonneg (killRatio_nonneg u b r) _) | |
| (hroot.trans (div_le_div_of_nonneg_right hK hg₀.le)) r | |
| linarith | |
| · by_cases hbr : b < r | |
| · rw [killRatio_eq_zero hbr] | |
| have herr : 0 ≤ ((2:ℝ)^(r+1)*r/g)^r := by positivity | |
| linarith | |
| · have hbr : b = r := by omega | |
| subst b | |
| by_cases hlong' : (2 : ℝ)^(r+2)*u ≤ g | |
| · have hh := hlong hlong' | |
| have herr : 0 ≤ ((2:ℝ)^(r+1)*r/g)^r := by positivity | |
| linarith | |
| · have hg' : (g : ℝ) ≤ (2 : ℝ)^(r+2)*u := (not_le.mp hlong').le | |
| have hd : (r : ℝ)/(2*u) ≤ (2 : ℝ)^(r+1)*r/g := by | |
| apply (div_le_div_iff₀ (by positivity) hg₀).2 | |
| rw [show r+2 = (r+1)+1 by omega, pow_succ] at hg' | |
| nlinarith [mul_le_mul_of_nonneg_left hg' hr₀.le] | |
| have hh := (killRatio_le_density_pow hbu (by omega : r ≤ u)).trans | |
| (pow_le_pow_left₀ (by positivity) hd r) | |
| linarith | |
| lemma one_replacement {u r g : ℕ} (hr : 2 ≤ r) (hru : r+1 ≤ u) (hg : 1 ≤ g) | |
| (S : Finset (Assignment u)) : | |
| blockKill u r S 1 - replacementK (r+1)/(g : ℝ)^r ≤ blockKill u (r+1) S g := by | |
| by_cases hS : S.Nonempty | |
| · rw [blockKill_one (by omega) hS] | |
| apply replacement_numeric hr (frozen_card_le S) hru hg (blockKill_nonneg _ _ _ _) | |
| · exact blockKill_exp_individual (by omega) hru hS g | |
| · intro hlong | |
| apply blockKill_long (by omega) hru _ S | |
| simpa [Nat.add_assoc] using hlong | |
| · have he : S = ∅ := Finset.not_nonempty_iff_eq_empty.mp hS | |
| subst S | |
| rw [blockKill_empty (by omega), blockKill_empty hru] | |
| have hK : 0 ≤ replacementK (r+1)/(g : ℝ)^r := by | |
| rw [replacementK_succ] | |
| positivity | |
| linarith | |
| end RandomKSAT | |
| end | |
| end OAI | |