| import OAI.Probability.SATVariance.OmissionCompression | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory | |
| namespace RandomKSAT | |
| open scoped Classical ENNReal | |
| lemma deletion_square_density {n r M : ℕ} (hr : 2 ≤ r) (hn : 2*(r+1) ≤ n) | |
| (i : Fin (M+1)) {B : ℝ} (hB : 0 ≤ B) (hM : (M : ℝ) ≤ B*n) : | |
| favg (fun cs : Fin (M+1) → Clause n (r+1) => (omittedTime i cs-finiteTime cs)^2) ≤ | |
| deletionMomentC (r+1) (B*(r+1 : ℕ)^2) (2*B^2*(r+1 : ℕ)^4)*durationLog (r+1) M := by | |
| let := clause_nonempty n (r+1) (by omega) | |
| rw [favg_insertNth i] | |
| apply le_trans (favg_mono fun c : Clause n (r+1) => ?_) (le_of_eq (favg_const _)) | |
| obtain ⟨e,a,hc⟩ := clause_root_presentation c | |
| have he := favg_equiv (Equiv.piCongrRight fun _ : Fin M => splitClauseEquiv e) | |
| (fun w : Fin M → Clause n (r+1) => | |
| (omittedTime i (i.insertNth c w)-finiteTime (i.insertNth c w))^2) | |
| rw [← he] | |
| have hb := favg_mono (fun ps : Fin M → RootData (n-(r+1)) (r+1) => | |
| pow_le_pow_left₀ (omittedTime_nonneg_diff i _) (deletion_difference_le_rewards e i c a hc ps) 2) | |
| exact hb.trans (rootRewards_square_density hr (by omega) e a hB hM) | |
| lemma ell_eq_of_three {n : ℕ} (hn : 1 ≤ n) : ell 3 n = 1+Real.log n := by | |
| have hn0 : (n : ℝ) ≠ 0 := by exact_mod_cast (by omega : n ≠ 0) | |
| simp [ell, Real.log_mul (Real.exp_ne_zero _) hn0] | |
| lemma ell_one_le (k n : ℕ) (hn : 1 ≤ n) : 1 ≤ ell k n := by | |
| by_cases hk : k = 3 | |
| · subst k | |
| rw [ell_eq_of_three hn] | |
| have hh := Real.log_nonneg (show (1 : ℝ) ≤ n by exact_mod_cast hn) | |
| linarith | |
| · simp [ell, hk] | |
| lemma durationLog_density {k n M : ℕ} (hn : 1 ≤ n) {B : ℝ} (hB : 0 ≤ B) | |
| (hM : (M+1 : ℕ) ≤ B*(n : ℝ)) : | |
| durationLog k M ≤ (1+Real.log (B+1))*ell k n := by | |
| have hn0 : (0 : ℝ) < n := by exact_mod_cast (by omega : 0 < n) | |
| have hB0 : 0 < B+1 := by linarith | |
| have hlogB : 0 ≤ Real.log (B+1) := Real.log_nonneg (by linarith) | |
| have hlogn : 0 ≤ Real.log n := Real.log_nonneg (by exact_mod_cast hn) | |
| by_cases hk : k = 3 | |
| · subst k | |
| rw [durationLog, ite_eq_left rfl, ell_eq_of_three hn] | |
| have hm : (M : ℝ)+1 ≤ (B+1)*n := by push_cast at hM; nlinarith | |
| have hl := Real.log_le_log (by positivity : 0 < (M : ℝ)+1) hm | |
| rw [Real.log_mul (ne_of_gt hB0) (ne_of_gt hn0)] at hl | |
| nlinarith [mul_nonneg hlogB hlogn] | |
| · simp only [durationLog, ell, ite_eq_right hk, mul_one] | |
| linarith | |
| lemma extend_upper_bound {f : ℕ → ℝ} (hf : ∀ n, 0 ≤ f n) {k N : ℕ} (hk : 1 ≤ k) | |
| {C : ℝ} (hC : 0 ≤ C) (hlarge : ∀ n, N ≤ n → k ≤ n → f n ≤ C*n*ell k n) : | |
| ∃ D : ℝ, 0 ≤ D ∧ ∀ n, k ≤ n → f n ≤ D*n*ell k n := by | |
| let S := ∑ j ∈ Finset.range N, f j | |
| have hS : 0 ≤ S := Finset.sum_nonneg (fun j _ => hf j) | |
| refine ⟨C+S,add_nonneg hC hS,fun n hkn => ?_⟩ | |
| have hn : 1 ≤ n := hk.trans hkn | |
| have he := ell_one_le k n hn | |
| have hn1 : (1 : ℝ) ≤ n := by exact_mod_cast hn | |
| have hne : 1 ≤ (n : ℝ)*ell k n := by nlinarith | |
| by_cases hN : N ≤ n | |
| · have hh := hlarge n hN hkn | |
| nlinarith [mul_nonneg hS (show 0 ≤ (n : ℝ)*ell k n by positivity)] | |
| · have hs : f n ≤ S := Finset.single_le_sum (fun j _ => hf j) (Finset.mem_range.mpr (by omega)) | |
| nlinarith [mul_nonneg hC (show 0 ≤ (n : ℝ)*ell k n by positivity), | |
| mul_le_mul_of_nonneg_left hne hS] | |
| theorem capped_variance_upper (k : ℕ) (hk : 3 ≤ k) (B : ℝ) (hB : 0 < B) : | |
| ∃ C : ℝ, 0 ≤ C ∧ ∀ n : ℕ, k ≤ n → | |
| variance (T B : Stream n k → ℝ) (streamLaw n k) ≤ C*n*ell k n := by | |
| cases k with | |
| | zero => omega | |
| | succ r => | |
| have hr : 2 ≤ r := by omega | |
| let K := deletionMomentC (r+1) (B*(r+1 : ℕ)^2) (2*B^2*(r+1 : ℕ)^4) | |
| have hK : 0 ≤ K := deletionMomentC_nonneg hr _ _ (by positivity) | |
| let C := B*K*(1+Real.log (B+1)) | |
| have hlogB : 0 ≤ Real.log (B+1) := Real.log_nonneg (by linarith) | |
| have hC : 0 ≤ C := by dsimp [C]; positivity | |
| apply extend_upper_bound (fun n => variance_nonneg _ _) (by omega : 1 ≤ r+1) hC | |
| (N := 2*(r+1)) | |
| intro n hn _ | |
| have hn1 : 1 ≤ n := by omega | |
| have hcap : (cap n B : ℝ) ≤ B*n := Nat.floor_le (by positivity) | |
| have hh := capped_coordinate_omission (by omega : r+1 ≤ n) B | |
| cases he : cap n B with | |
| | zero => | |
| rw [he] at hh | |
| simp only [Finset.univ_eq_empty, Finset.sum_empty] at hh | |
| exact hh.trans (by have hh' := ell_one_le (r+1) n hn1; dsimp [C]; positivity) | |
| | succ M => | |
| have hM : (M : ℝ) ≤ B*n := by rw [he] at hcap; push_cast at hcap; linarith | |
| have hb := Finset.sum_le_sum (fun i (_ : i ∈ (Finset.univ : Finset (Fin (M+1)))) => | |
| deletion_square_density hr hn i hB.le hM) | |
| simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul] at hb | |
| rw [he] at hh hcap | |
| have htime := durationLog_density (k := r+1) hn1 hB.le hcap | |
| have ht := mul_le_mul_of_nonneg_left htime (mul_nonneg (Nat.cast_nonneg (M+1)) hK) | |
| have hf := mul_le_mul_of_nonneg_right hcap | |
| (show 0 ≤ K*((1+Real.log (B+1))*ell (r+1) n) by | |
| have he := ell_one_le (r+1) n hn1 | |
| positivity) | |
| change _ ≤ C*n*ell (r+1) n | |
| change _ ≤ ((M+1 : ℕ) : ℝ)*(K*durationLog (r+1) M) at hb | |
| dsimp only [C] | |
| nlinarith [hh.trans hb] | |
| end RandomKSAT | |
| end | |
| end OAI | |