| import OAI.Combinatorics.EditDistance.LowerBound |
| import OAI.Combinatorics.EditDistance.DistortionMonotonicity |
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| namespace OAI |
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| /-! |
| # Uniform lower bounds under binary alphabet embeddings |
| -/ |
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| namespace EditDistortion |
| universe u |
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| lemma coordinate_norm {X : Type u} [Fintype X] (x : X) : ‖coordinateEmbedding X x‖ = 1 := by |
| classical |
| change ‖lp.single (E := fun _ : ℕ => ℝ) 1 ((Fintype.equivFin X x).val) (1 : ℝ)‖ = 1 |
| rw [lp.norm_single (by norm_num)] |
| norm_num |
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| lemma coordinate_separation {X : Type u} [Fintype X] (x y : X) (hxy : x ≠ y) : |
| 1 ≤ ‖coordinateEmbedding X x-coordinateEmbedding X y‖ := by |
| classical |
| let e : X ↪ ℕ := (Fintype.equivFin X).toEmbedding.trans ⟨Fin.val,Fin.val_injective⟩ |
| have hne : e x ≠ e y := fun h => hxy (e.injective h) |
| have h := lp.norm_apply_le_norm (by norm_num : (1 : ENNReal) ≠ 0) |
| (coordinateEmbedding X x - coordinateEmbedding X y) (e x) |
| have he : (coordinateEmbedding X x-coordinateEmbedding X y) (e x) = 1 := by |
| change lp.single (E := fun _ : ℕ => ℝ) 1 (e x) (1 : ℝ) (e x) - |
| lp.single (E := fun _ : ℕ => ℝ) 1 (e y) (1 : ℝ) (e x) = 1 |
| rw [lp.single_apply_self,lp.single_apply_ne (E := fun _ : ℕ => ℝ) 1 (e y) (1 : ℝ) hne] |
| norm_num |
| simpa only [he,norm_one] using h |
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| theorem alphabet_coarse_upper (Alpha : Type u) [Fintype Alpha] [Nonempty Alpha] |
| {d : ℕ} (hd : 1 ≤ d) : E Alpha d ≤ 2*(d : ℝ) := by |
| classical |
| let : Finite (Words Alpha d) := finite_words Alpha d |
| let : Fintype (Words Alpha d) := Fintype.ofFinite _ |
| let x : Words Alpha d := ⟨[],by simp⟩ |
| let y : Words Alpha d := ⟨[Classical.arbitrary Alpha],by simpa using hd⟩ |
| let : Nontrivial (Words Alpha d) := ⟨⟨x,y,by simp [x,y]⟩⟩ |
| let f := coordinateEmbedding (Words Alpha d) |
| apply (leastDistortion_le _ (fun v => edit_real_diag v.val) |
| (fun v w h => edit_real_pos v.val w.val (fun hh => h (Subtype.ext hh))) f).trans |
| apply distortion_le_of_bounds _ |
| (fun v w h => edit_real_pos v.val w.val (fun hh => h (Subtype.ext hh))) f (by norm_num) |
| · intro v w |
| by_cases h : v = w |
| · subst w |
| simp [edit_real_diag] |
| · have hp : (1 : ℝ) ≤ edit v.val w.val := by |
| exact_mod_cast Nat.one_le_iff_ne_zero.mpr |
| (mt (cost_eq_zero_iff (s := true)).mp (fun hh => h (Subtype.ext hh))) |
| have hh := norm_sub_le (f v) (f w) |
| have hv : ‖f v‖ = 1 := coordinate_norm v |
| have hw : ‖f w‖ = 1 := coordinate_norm w |
| nlinarith |
| · intro v w |
| by_cases h : v = w |
| · subst w |
| simp [edit_real_diag] |
| · have hs : (1 : ℝ) ≤ ‖f v-f w‖ := coordinate_separation v w h |
| have he : (edit v.val w.val : ℝ) ≤ d := by |
| exact_mod_cast (edit_le_max_length v.val w.val).trans |
| (max_le v.property w.property) |
| nlinarith [Nat.cast_nonneg (α := ℝ) d] |
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| lemma alphabet_values_bddAbove {d : ℕ} (hd : 1 ≤ d) : |
| BddAbove {r : ℝ | ∃ n : ℕ, 2 ≤ n ∧ r = E (Fin n) d} := by |
| refine ⟨2*(d : ℝ),?_⟩ |
| rintro r ⟨n,hn,rfl⟩ |
| let : NeZero n := ⟨by omega⟩ |
| exact alphabet_coarse_upper (Fin n) hd |
|
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| lemma alphabet_le_sup {d n : ℕ} (hd : 1 ≤ d) (hn : 2 ≤ n) : E (Fin n) d ≤ alphabetSup d := |
| le_csSup (alphabet_values_bddAbove hd) ⟨n,hn,rfl⟩ |
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| end EditDistortion |
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| namespace EditLower |
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| universe u |
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| open EditDistortion |
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| theorem all_lower_clauses : ∃ c : ℝ, 0 < c ∧ ∃ d₀ : ℕ, ∀ d : ℕ, d₀ ≤ d → |
| (∀ (Alpha : Type u) [Fintype Alpha], 2 ≤ Fintype.card Alpha → |
| exponentScale c d ≤ E Alpha d) ∧ BinaryWitness c d ∧ |
| exponentScale c d ≤ alphabetSup d := by |
| obtain ⟨D,hD⟩ := Filter.eventually_atTop.mp eventually_length_witness |
| refine ⟨1/1200,by norm_num,max D 1,?_⟩ |
| intro d hd |
| obtain ⟨n,hn,S,hfin,hne,hnt,hlen,hs⟩ := hD d ((le_max_left _ _).trans hd) |
| refine ⟨?_,⟨n,hn,S,hfin,hne,hlen,hs⟩,?_⟩ |
| · intro Alpha inst hAlpha |
| exact hs.trans (binary_subset_lower Alpha hAlpha hn S hfin hnt hlen) |
| · exact (hs.trans (binary_subset_lower (Fin 2) (by simp) hn S hfin hnt hlen)).trans |
| (alphabet_le_sup ((le_max_right _ _).trans hd) (by omega)) |
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| end EditLower |
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| end OAI |
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