| import OAI.Combinatorics.EditDistance.UpperBound |
| import OAI.Combinatorics.EditDistance.AlphabetLower |
|
|
| namespace OAI |
|
|
| /-! |
| # Simultaneous whole-domain maps and fixed-length binary upper bounds |
| -/ |
|
|
| namespace EditUpper |
| open EditDistortion EditCutModel Filter |
| universe u |
|
|
| lemma coarse_whole_map (A : Type u) [Fintype A] (d : ℕ) : |
| ∃ f : Words A d → L1, ∀ x y, |
| (edit x.val y.val : ℝ) ≤ ‖f x-f y‖ ∧ |
| ‖f x-f y‖ ≤ (2*(d : ℝ))*(edit x.val y.val : ℝ) := by |
| classical |
| let : Finite (Words A d) := finite_words A d |
| let : Fintype (Words A d) := Fintype.ofFinite _ |
| let f := fun x : Words A d => (d : ℝ) • coordinateEmbedding (Words A d) x |
| have hn (x y : Words A d) : ‖f x-f y‖ = (d : ℝ)* |
| ‖coordinateEmbedding (Words A d) x-coordinateEmbedding (Words A d) y‖ := by |
| dsimp [f] |
| rw [← smul_sub,norm_smul,Real.norm_eq_abs,abs_of_nonneg (Nat.cast_nonneg d)] |
| refine ⟨f,?_⟩ |
| intro x y |
| by_cases hxy : x = y |
| · subst y |
| simp only [sub_self,norm_zero,edit_real_diag,mul_zero,le_refl,and_self] |
| · have hlo := coordinate_separation x y hxy |
| have hu := norm_sub_le (coordinateEmbedding (Words A d) x) (coordinateEmbedding (Words A d) y) |
| rw [coordinate_norm,coordinate_norm] at hu |
| have he : (edit x.val y.val : ℝ) ≤ d := by |
| exact_mod_cast (edit_le_max_length x.val y.val).trans (max_le x.property y.property) |
| have hep : (1 : ℝ) ≤ edit x.val y.val := by |
| exact_mod_cast Nat.one_le_iff_ne_zero.mpr |
| (mt (cost_eq_zero_iff (s := true)).mp (fun hh => hxy (Subtype.ext hh))) |
| rw [hn] |
| have hd := Nat.cast_nonneg (α := ℝ) d |
| constructor <;> nlinarith |
|
|
| theorem eventual_whole_maps : ∀ᶠ d : ℕ in atTop, |
| ∀ (A : Type u) [Fintype A], ∃ f : Words A d → L1, ∀ x y, |
| (edit x.val y.val : ℝ) ≤ ‖f x-f y‖ ∧ |
| ‖f x-f y‖ ≤ exponentScale 32 d*(edit x.val y.val : ℝ) := by |
| filter_upwards [EditUpperParameters.eventual_choice] with d hc |
| rcases hc with ⟨hN,hb,hcap,hbound⟩ |
| intro A inst |
| let N := EditUpperParameters.level (Real.log (d : ℝ)) |
| let b := EditUpperParameters.branching (Real.log (d : ℝ)) |
| obtain ⟨M,hlo,hup⟩ := exists_level b N (by omega) N le_rfl A |
| let D := ((b : ℝ)+2)*(growth b N)^N |
| have hD : 0 ≤ D := by |
| have hh := growth_ge (N := N) (show 1 ≤ b by omega) |
| dsimp [D] |
| positivity |
| have hnum : 2*D ≤ exponentScale 32 d := by |
| simpa [D,growth,exponentScale,b,N,mul_assoc] using hbound |
| refine ⟨fun x => M.vector x.val,?_⟩ |
| intro x y |
| rw [M.vector_distance] |
| constructor |
| · exact (show (edit x.val y.val : ℝ) ≤ indel x.val y.val by |
| exact_mod_cast edit_le_indel x.val y.val).trans |
| (hlo _ _ (x.property.trans hcap) (y.property.trans hcap)) |
| · have hi : (indel x.val y.val : ℝ) ≤ 2*(edit x.val y.val : ℝ) := by |
| exact_mod_cast indel_le_two_edit x.val y.val |
| calc |
| _ ≤ D*(indel x.val y.val : ℝ) := hup _ _ |
| _ ≤ D*(2*(edit x.val y.val : ℝ)) := mul_le_mul_of_nonneg_left hi hD |
| _ = (2*D)*(edit x.val y.val : ℝ) := by ring |
| _ ≤ _ := mul_le_mul_of_nonneg_right hnum (Nat.cast_nonneg _) |
|
|
| theorem whole_maps_all_lengths : ∃ C : ℝ, 0 < C ∧ ∀ d : ℕ, 3 ≤ d → |
| ∀ (A : Type u) [Fintype A], ∃ f : Words A d → L1, ∀ x y, |
| (edit x.val y.val : ℝ) ≤ ‖f x-f y‖ ∧ |
| ‖f x-f y‖ ≤ exponentScale C d*(edit x.val y.val : ℝ) := by |
| obtain ⟨D,hD⟩ := eventually_atTop.mp eventual_whole_maps.{u} |
| let S := Real.sqrt (Real.log 3*Real.log (Real.log 3)) |
| have hl3 : 1 < Real.log 3 := by linarith [Real.log_three_gt_d9] |
| have hll3 : 0 < Real.log (Real.log 3) := Real.log_pos hl3 |
| have hS : 0 < S := Real.sqrt_pos.mpr (mul_pos (by linarith) hll3) |
| let C := 32+2*(D : ℝ)/S |
| have hDC : 0 ≤ 2*(D : ℝ)/S := by positivity |
| have hC : 0 < C := by dsimp [C]; linarith |
| have h32 : 32 ≤ C := by dsimp [C]; linarith |
| refine ⟨C,hC,?_⟩ |
| intro d hd A inst |
| have hm : 0 ≤ Real.sqrt (Real.log (d : ℝ)*Real.log (Real.log (d : ℝ))) := Real.sqrt_nonneg _ |
| by_cases hDd : D ≤ d |
| · obtain ⟨f,hf⟩ := hD d hDd A |
| refine ⟨f,fun x y => ⟨(hf x y).1,?_⟩⟩ |
| apply (hf x y).2.trans |
| apply mul_le_mul_of_nonneg_right _ (Nat.cast_nonneg _) |
| apply Real.exp_le_exp.mpr |
| exact mul_le_mul_of_nonneg_right h32 hm |
| · obtain ⟨f,hf⟩ := coarse_whole_map A d |
| have hl : Real.log 3 ≤ Real.log (d : ℝ) := |
| Real.log_le_log (by norm_num) (by exact_mod_cast hd) |
| have hll : Real.log (Real.log 3) ≤ Real.log (Real.log (d : ℝ)) := |
| Real.log_le_log (by linarith) hl |
| have hST : S ≤ Real.sqrt (Real.log (d : ℝ)*Real.log (Real.log (d : ℝ))) := by |
| apply Real.sqrt_le_sqrt |
| exact mul_le_mul hl hll hll3.le (by linarith) |
| have hCS : 2*(D : ℝ) ≤ C*S := by |
| have he : (2*(D : ℝ)/S)*S = 2*(D : ℝ) := div_mul_cancel₀ _ hS.ne' |
| dsimp [C] |
| nlinarith |
| have h2 : 2*(d : ℝ) ≤ exponentScale C d := by |
| have hdD : (d : ℝ) ≤ D := by exact_mod_cast (show d ≤ D by omega) |
| have hmC := mul_le_mul_of_nonneg_left hST hC.le |
| have hexp := Real.add_one_le_exp (C*Real.sqrt (Real.log (d : ℝ)*Real.log (Real.log (d : ℝ)))) |
| unfold exponentScale |
| linarith |
| exact ⟨f,fun x y => ⟨(hf x y).1,(hf x y).2.trans |
| (mul_le_mul_of_nonneg_right h2 (Nat.cast_nonneg _))⟩⟩ |
|
|
| theorem fixed_length_binary_upper : ∃ C₀ : ℝ, 0 < C₀ ∧ ∀ m : ℕ, 3 ≤ m → |
| ∃ g : (Fin m → Bool) → L1, ∀ x y : Fin m → Bool, |
| (edit (List.ofFn x) (List.ofFn y) : ℝ) ≤ ‖g x-g y‖ ∧ |
| ‖g x-g y‖ ≤ exponentScale C₀ m*(edit (List.ofFn x) (List.ofFn y) : ℝ) := by |
| obtain ⟨C,hC,h⟩ := whole_maps_all_lengths.{0} |
| refine ⟨C,hC,?_⟩ |
| intro m hm |
| obtain ⟨f,hf⟩ := h m hm Bool |
| let word : (Fin m → Bool) → Words Bool m := fun x => ⟨List.ofFn x,by simp⟩ |
| exact ⟨fun x => f (word x),fun x y => hf (word x) (word y)⟩ |
|
|
| end EditUpper |
|
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| end OAI |
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