| import OAI.Combinatorics.EditDistance.UpperBase |
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| namespace OAI |
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| /-! |
| # Uniform finite-cut embeddings at recursive capacities |
| -/ |
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| namespace EditUpperParameters |
| open scoped BigOperators |
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| lemma log_two_bounds : (1/2 : ℝ) ≤ Real.log 2 ∧ Real.log 2 ≤ 1 := by |
| constructor |
| · have hh := Real.one_sub_inv_le_log_of_pos (by norm_num : (0 : ℝ) < 2) |
| norm_num at hh ⊢ |
| exact hh |
| · have hh := Real.log_le_sub_one_of_pos (by norm_num : (0 : ℝ) < 2) |
| norm_num at hh ⊢ |
| exact hh |
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| lemma scale_count_bound {l : ℕ} (hl : 0 < l) : |
| ((Nat.log 2 l+1 : ℕ) : ℝ) ≤ 2*(1+Real.log (l : ℝ)) := by |
| have hpow := Nat.pow_log_le_self 2 (Nat.ne_of_gt hl) |
| have hpr : (2 : ℝ)^(Nat.log 2 l) ≤ l := by exact_mod_cast hpow |
| have hh := Real.log_le_log (pow_pos (by norm_num : (0 : ℝ)<2) _) hpr |
| rw [Real.log_pow] at hh |
| have hlog : 0 ≤ Real.log (l : ℝ) := Real.log_nonneg (by exact_mod_cast hl) |
| have hm : 0 ≤ (Nat.log 2 l : ℝ) := Nat.cast_nonneg _ |
| have h2 := log_two_bounds |
| push_cast |
| nlinarith |
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| lemma log_square_bound {l : ℕ} (hl : 0 < l) : |
| Real.log (2*(l : ℝ)^2) ≤ 2*(1+Real.log (l : ℝ)) := by |
| have hp : (0 : ℝ)<l := by exact_mod_cast hl |
| rw [Real.log_mul (by norm_num) (pow_ne_zero 2 hp.ne'),Real.log_pow] |
| have h2 := log_two_bounds |
| norm_num only [Nat.cast_ofNat] |
| linarith |
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| lemma step_constant_bound {b l : ℕ} (hl : 0 < l) {D : ℝ} (hD : 0 ≤ D) : |
| 16*((Nat.log 2 l+1 : ℕ) : ℝ)*(2*(b : ℝ)+4*Real.log (2*(l : ℝ)^2)*D) ≤ |
| 512*(1+Real.log (l : ℝ))^2*((b : ℝ)+D) := by |
| have hlog : 0 ≤ Real.log (l : ℝ) := Real.log_nonneg (by exact_mod_cast hl) |
| have hcount := scale_count_bound hl |
| have hsq := log_square_bound hl |
| have hsnn : 0 ≤ Real.log (2*(l : ℝ)^2) := Real.log_nonneg (by |
| have hh : (1 : ℝ) ≤ l := by exact_mod_cast hl |
| nlinarith) |
| have hb := Nat.cast_nonneg (α := ℝ) b |
| have hm := Nat.cast_nonneg (α := ℝ) (Nat.log 2 l+1) |
| have hleft : 2*(b : ℝ)+4*Real.log (2*(l : ℝ)^2)*D ≤ 2*b+8*(1+Real.log (l : ℝ))*D := by nlinarith |
| calc |
| _ ≤ 16*(2*(1+Real.log (l : ℝ)))*(2*b+8*(1+Real.log (l : ℝ))*D) := by |
| apply mul_le_mul |
| · nlinarith |
| · exact hleft |
| · positivity |
| · positivity |
| _ ≤ _ := by nlinarith [mul_nonneg hlog hb,mul_nonneg (sq_nonneg (1+Real.log (l : ℝ))) hD, |
| mul_nonneg (sq_nonneg (Real.log (l : ℝ))) hb] |
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| end EditUpperParameters |
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| namespace EditUpper |
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| universe u |
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| attribute [local instance] Classical.propDecidable |
| open scoped BigOperators |
| open Finset EditDistortion EditWindow EditCutModel |
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| noncomputable def growth (b N : ℕ) : ℝ := 1024*(1+(N : ℝ)*Real.log (b : ℝ))^2 |
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| lemma growth_ge {b N : ℕ} (hb : 1 ≤ b) : 1 ≤ growth b N := by |
| have hlog : 0 ≤ Real.log (b : ℝ) := Real.log_nonneg (by exact_mod_cast hb) |
| have hn := Nat.cast_nonneg (α := ℝ) N |
| dsimp [growth] |
| nlinarith [mul_nonneg hn hlog] |
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| theorem exists_level (b N : ℕ) (hb : 1 ≤ b) (k : ℕ) (hk : k ≤ N) |
| (A : Type u) [Fintype A] : |
| ∃ M : Model (List A), |
| (∀ x y : List A, x.length ≤ b^k → y.length ≤ b^k → (indel x y : ℝ) ≤ M.dist x y) ∧ |
| (∀ x y : List A, M.dist x y ≤ ((b : ℝ)+2)*(growth b N)^k*(indel x y : ℝ)) := by |
| induction k generalizing A with |
| | zero => |
| refine ⟨base,?_,?_⟩ |
| · intro x y hx hy |
| exact base_lower x y (by simpa using hx) (by simpa using hy) |
| · intro x y |
| have hh := base_upper x y |
| have hi := Nat.cast_nonneg (α := ℝ) (indel x y) |
| simp only [pow_zero,mul_one] |
| nlinarith [Nat.cast_nonneg (α := ℝ) b] |
| | succ k ih => |
| obtain ⟨M,hlo,hup⟩ := ih (by omega) (Option A) |
| let D := ((b : ℝ)+2)*(growth b N)^k |
| have hP := growth_ge (N := N) hb |
| have hD : 0 ≤ D := by dsimp [D]; positivity |
| have hpow : 1 ≤ (growth b N)^k := one_le_pow₀ hP |
| have hbD : (b : ℝ) ≤ D := by |
| dsimp [D] |
| have hb0 := Nat.cast_nonneg (α := ℝ) b |
| nlinarith |
| have hl : 0 < b^k := pow_pos (by omega) _ |
| have hkN : (k : ℝ) ≤ N := by exact_mod_cast (show k ≤ N by omega) |
| have hlog : 0 ≤ Real.log (b : ℝ) := Real.log_nonneg (by exact_mod_cast hb) |
| have hT : 1+Real.log ((b^k : ℕ) : ℝ) ≤ 1+(N : ℝ)*Real.log (b : ℝ) := by |
| rw [Nat.cast_pow,Real.log_pow] |
| nlinarith |
| have hTnn : 0 ≤ 1+Real.log ((b^k : ℕ) : ℝ) := by |
| have hh : 0 ≤ Real.log ((b^k : ℕ) : ℝ) := Real.log_nonneg (by exact_mod_cast (Nat.one_le_pow k b (by omega))) |
| linarith |
| have hconst : 512*(1+Real.log ((b^k : ℕ) : ℝ))^2*((b : ℝ)+D) ≤ |
| ((b : ℝ)+2)*(growth b N)^(k+1) := by |
| calc |
| _ ≤ 512*(1+(N : ℝ)*Real.log (b : ℝ))^2*(2*D) := by |
| apply mul_le_mul |
| · gcongr |
| · linarith |
| · positivity |
| · positivity |
| _ = _ := by dsimp [D,growth]; rw [pow_succ]; ring |
| refine ⟨lift M b (b^k),?_,?_⟩ |
| · intro x y hx hy |
| apply lift_lower M b (b^k) hl hlo x y |
| · simpa only [pow_succ,Nat.mul_comm] using hx |
| · simpa only [pow_succ,Nat.mul_comm] using hy |
| · apply model_indel_lipschitz |
| intro p q a |
| exact (lift_insert_raw M hD hup p q a b (b^k) hl).trans |
| ((EditUpperParameters.step_constant_bound hl hD).trans hconst) |
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| end EditUpper |
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| end OAI |
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