import Mathlib.Analysis.SpecialFunctions.Log.Base import Mathlib.Tactic namespace OAI /-! Numerical estimates used in the fixed-length binary conversion. -/ open Filter Asymptotics namespace TreeEdit.BinaryLower.Code /-- The segment threshold used in the prescribed-length code construction. -/ noncomputable def segmentThreshold (M : ℕ) : ℕ := Nat.ceil (120 * Real.logb 2 (M : ℝ)) /-- Eventually the error from the short block-pair spans fits within half a block. -/ theorem eventually_short_pair_error : ∀ᶠ M : ℕ in atTop, 0 < M ∧ 2 * (M : ℝ) * segmentThreshold M ≤ (M : ℝ) ^ 2 / 2 := by have hlog : (fun M : ℕ => Real.logb 2 (M : ℝ)) =o[atTop] (fun M : ℕ => (M : ℝ)) := (Real.isLittleO_logb_id_atTop (b := 2)).comp_tendsto tendsto_natCast_atTop_atTop have hb := hlog.bound (show (0 : ℝ) < 1 / 960 by norm_num) filter_upwards [hb, eventually_ge_atTop (8 : ℕ)] with M hM h8 have hpos : 0 < M := by omega have hM0 : (0 : ℝ) ≤ M := Nat.cast_nonneg M have hM8 : (8 : ℝ) ≤ M := by exact_mod_cast h8 have hM1 : (1 : ℝ) ≤ M := by linarith have hlog0 : 0 ≤ Real.logb 2 (M : ℝ) := Real.logb_nonneg (by norm_num) hM1 have hlogbound : Real.logb 2 (M : ℝ) ≤ (1 / 960 : ℝ) * M := by simpa only [Real.norm_eq_abs, abs_of_nonneg hlog0, abs_of_nonneg hM0] using hM have hceil : (segmentThreshold M : ℝ) < 120 * Real.logb 2 (M : ℝ) + 1 := by exact Nat.ceil_lt_add_one (by positivity) have hsmall : (segmentThreshold M : ℝ) ≤ (M : ℝ) / 4 := by linarith refine ⟨hpos, ?_⟩ have := mul_le_mul_of_nonneg_left hsmall (show 0 ≤ 2 * (M : ℝ) by positivity) nlinarith /-- An absolute integer threshold exists, as required in the manuscript. -/ theorem exists_short_pair_threshold : ∃ M₀ : ℕ, ∀ M : ℕ, M₀ ≤ M → 0 < M ∧ 2 * (M : ℝ) * segmentThreshold M ≤ (M : ℝ) ^ 2 / 2 := by exact eventually_atTop.1 eventually_short_pair_error /-- Removing the additive error in the block-matching estimate gives the constant 402. The nonzero original deficit is integral, hence at least one. -/ theorem deficit_from_additive {P r δ e : ℝ} (hP : 0 ≤ P) (hr : 1 ≤ r) (hadd : P * r ≤ 201 * δ + e) (herr : e ≤ P / 2) : P / 402 * r ≤ δ := by have hPr : P ≤ P * r := by nlinarith nlinarith /-- Combining the deficit inequality with the two ordinary-edit comparisons produces exactly the constant 804, without changing the edit metric. -/ theorem edit_lower_from_deficit {P r r' d d' : ℝ} (hP : 0 ≤ P) (hdeficit : P / 402 * r ≤ r') (hleft : r' ≤ d') (hright : d ≤ 2 * r) : P / 804 * d ≤ d' := by have := mul_le_mul_of_nonneg_left hright (show 0 ≤ P / 804 by positivity) nlinarith /-- The equal-label and unequal-label counts combine into the additive estimate. All quantities are natural cardinalities, so no rounding is hidden. -/ theorem block_counts_to_additive {M P a₀ L r matched equal unequal δ : ℕ} (hpartition : matched = equal + unequal) (htotal : matched + δ = M * P) (hdeficit : r + L = M) (hequal : equal ≤ P * L) (hunequal : unequal ≤ 200 * δ + 2 * M * a₀) : P * r ≤ 201 * δ + 2 * M * a₀ := by nlinarith /-- Integral form of the error absorption, including the zero-deficit case. -/ theorem natural_deficit_from_additive {P r δ e : ℕ} (hadd : P * r ≤ 201 * δ + e) (herr : 2 * e ≤ P) : P * r ≤ 402 * δ := by by_cases hr : r = 0 · simp [hr] have hr1 : 1 ≤ r := Nat.one_le_iff_ne_zero.mpr hr have hPr : P ≤ P * r := by nlinarith nlinarith /-- Cast the integral matching conclusion into the exact real-valued lower bound used in the statement of the fixed-length conversion theorem. -/ theorem real_deficit_of_natural {P r δ : ℕ} (h : P * r ≤ 402 * δ) : (P : ℝ) / 402 * r ≤ δ := by have h' : (P : ℝ) * r ≤ 402 * δ := by exact_mod_cast h linarith /-- The two common formulas for the segment threshold agree at positive M. -/ theorem segmentThreshold_eq (M : ℕ) : segmentThreshold M = Nat.ceil (40 * Real.logb 2 ((M : ℝ) ^ 3)) := by unfold segmentThreshold rw [Real.logb_pow] congr 1 ring /-- Elementary failure tail at the source logarithmic segment threshold. -/ theorem code_failure_power_bound {M : ℕ} (hM : 1 ≤ M) : (9 / 10 : ℝ) ^ segmentThreshold M * (M : ℝ) ^ 12 ≤ 1 := by have hM1 : (1 : ℝ) ≤ M := by exact_mod_cast hM have hMpos : (0 : ℝ) < M := by linarith have hlog0 := Real.log_nonneg hM1 have hlog2pos : 0 < Real.log 2 := Real.log_pos (by norm_num) have hlog2le : Real.log 2 ≤ 1 := by have := Real.log_le_sub_one_of_pos (show (0 : ℝ) < 2 by norm_num) norm_num at this ⊢ exact this have hlogbase : Real.log (M : ℝ) ≤ Real.logb 2 (M : ℝ) := by rw [Real.logb, le_div_iff₀ hlog2pos] nlinarith have hthreshold : 120 * Real.log (M : ℝ) ≤ (segmentThreshold M : ℝ) := by have hceil := Nat.le_ceil (120 * Real.logb 2 (M : ℝ)) change 120 * Real.logb 2 (M : ℝ) ≤ (segmentThreshold M : ℝ) at hceil linarith have hbase : (9 / 10 : ℝ) ≤ Real.exp (-(1 / 10 : ℝ)) := by have := Real.add_one_le_exp (-(1 / 10 : ℝ)) linarith have htail : (9 / 10 : ℝ) ^ segmentThreshold M ≤ Real.exp (-(12 * Real.log (M : ℝ))) := by calc (9 / 10 : ℝ) ^ segmentThreshold M ≤ Real.exp (-(1 / 10 : ℝ)) ^ segmentThreshold M := pow_le_pow_left₀ (by norm_num) hbase _ _ = Real.exp ((segmentThreshold M : ℝ) * (-(1 / 10 : ℝ))) := (Real.exp_nat_mul _ _).symm _ ≤ Real.exp (-(12 * Real.log (M : ℝ))) := Real.exp_monotone (by linarith) have hexp : Real.exp (-(12 * Real.log (M : ℝ))) = ((M : ℝ) ^ 12)⁻¹ := by rw [Real.exp_neg] congr 1 have := Real.exp_nat_mul (Real.log (M : ℝ)) 12 norm_num only [Nat.cast_ofNat] at this rw [this, Real.exp_log hMpos] rw [hexp] at htail have hmul := mul_le_mul_of_nonneg_right htail (show 0 ≤ (M : ℝ) ^ 12 by positivity) simpa [ne_of_gt hMpos] using hmul /-- The segment-comparison count times the failure bound is below one. -/ theorem code_union_bound_lt_one {M : ℕ} (hM : 2 ≤ M) : (M : ℝ) ^ 2 * ((M : ℝ) ^ 2 + 1) ^ 3 * (9 / 10 : ℝ) ^ segmentThreshold M < 1 := by have hx : (2 : ℝ) ≤ M := by exact_mod_cast hM have hpos : (0 : ℝ) < M := by linarith have hsq : (1 : ℝ) ≤ (M : ℝ) ^ 2 := by nlinarith have hpoly : (M : ℝ) ^ 2 * ((M : ℝ) ^ 2 + 1) ^ 3 ≤ 8 * (M : ℝ) ^ 8 := by calc (M : ℝ) ^ 2 * ((M : ℝ) ^ 2 + 1) ^ 3 ≤ (M : ℝ) ^ 2 * (2 * (M : ℝ) ^ 2) ^ 3 := by gcongr linarith _ = 8 * (M : ℝ) ^ 8 := by ring have hfour : (8 : ℝ) < (M : ℝ) ^ 4 := by have h := pow_le_pow_left₀ (show (0 : ℝ) ≤ 2 by norm_num) hx 4 norm_num at h linarith have hpoly' : (M : ℝ) ^ 2 * ((M : ℝ) ^ 2 + 1) ^ 3 < (M : ℝ) ^ 12 := by apply hpoly.trans_lt have h := mul_lt_mul_of_pos_right hfour (show (0 : ℝ) < (M : ℝ) ^ 8 by positivity) convert h using 1; ring have hmul := mul_lt_mul_of_pos_right hpoly' (show (0 : ℝ) < (9 / 10 : ℝ) ^ segmentThreshold M by positivity) exact hmul.trans_le (by simpa [mul_comm] using code_failure_power_bound (by omega : 1 ≤ M)) /-- One absolute threshold supplies both probabilistic and deterministic estimates. -/ theorem exists_code_numerical_threshold : ∃ M₀ : ℕ, ∀ M : ℕ, M₀ ≤ M → 2 ≤ M ∧ 2 * (M : ℝ) * segmentThreshold M ≤ (M : ℝ) ^ 2 / 2 ∧ (M : ℝ) ^ 2 * ((M : ℝ) ^ 2 + 1) ^ 3 * (9 / 10 : ℝ) ^ segmentThreshold M < 1 := by rcases exists_short_pair_threshold with ⟨M₀, hM₀⟩ refine ⟨max M₀ 2, fun M hM => ?_⟩ have hMtwo : 2 ≤ M := (le_max_right M₀ 2).trans hM exact ⟨hMtwo, (hM₀ M ((le_max_left M₀ 2).trans hM)).2, code_union_bound_lt_one hMtwo⟩ /-- Alphabet sizes below M only decrease the comparison count. -/ theorem code_union_bound_for_alphabet {K M : ℕ} (hK : K ≤ M) (hM : 2 ≤ M) : (K : ℝ) ^ 2 * ((M : ℝ) ^ 2 + 1) ^ 3 * (9 / 10 : ℝ) ^ segmentThreshold M < 1 := by apply lt_of_le_of_lt _ (code_union_bound_lt_one hM) gcongr /-- The short-pair threshold is no larger than the block width once its error-absorption inequality holds. -/ theorem segmentThreshold_le_square {M : ℕ} (hM : 1 ≤ M) (h : 2 * (M : ℝ) * segmentThreshold M ≤ (M : ℝ) ^ 2 / 2) : segmentThreshold M ≤ M ^ 2 := by have hM' : (1 : ℝ) ≤ M := by exact_mod_cast hM have hnon : (0 : ℝ) ≤ segmentThreshold M := Nat.cast_nonneg _ have h' : (segmentThreshold M : ℝ) ≤ (M : ℝ) ^ 2 := by nlinarith exact_mod_cast h' end TreeEdit.BinaryLower.Code end OAI