| import OAI.Combinatorics.BinaryEdit.FiniteSeparation |
| import OAI.Combinatorics.BinaryEdit.FiniteWordL1Mean |
| import OAI.Combinatorics.BinaryEdit.ParameterImage |
| import OAI.Combinatorics.BinaryEdit.FiniteAlphabet |
| import OAI.Combinatorics.BinaryEdit.FiniteFamilyTransfer |
| import OAI.Combinatorics.BinaryEdit.PrimeSupply |
| import OAI.Combinatorics.BinaryEdit.EditDeficit |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace TreeEdit.BinaryLower |
|
|
| open FiniteTree |
| open scoped BigOperators |
|
|
| /-- A finite-alphabet family with distortion constant `1/34`, constructed |
| from finite coordinate groups and their word images. -/ |
| theorem exists_finite_source_family : |
| ∃ c₀ : ℝ, 0 < c₀ ∧ ∃ K : ℕ, ∀ k : ℕ, K ≤ k → |
| ∃ Γ : Type, ∃ inst : Fintype Γ, |
| 1 ≤ @Fintype.card Γ inst ∧ @Fintype.card Γ inst ≤ finiteSourceLength k ∧ |
| ∃ W : Finset (Word Γ (finiteSourceLength k)), 2 ≤ W.card ∧ |
| c₀ * ((k : ℝ) / 4) ^ k ≤ |
| c1 {x : Word Γ (finiteSourceLength k) |
| (fun x y => (ed x.val.val y.val.val : ℝ)) := by |
| classical |
| obtain ⟨K, hK⟩ := PrimeSupply.finite_tree_odd_prime_family |
| refine ⟨1 / 34, by norm_num, K, ?_⟩ |
| intro k hk |
| obtain ⟨hk2, primes, hprimes⟩ := hK k hk |
| let p : Fin k → Fin (20 * k) → ℕ := fun q i => primes (q, i) |
| have hpprime : ∀ q i, Nat.Prime (p q i) := fun q i => (hprimes (q, i)).1 |
| have hppos : ∀ q i, 0 < p q i := fun q i => (hpprime q i).pos |
| have hplarge : ∀ q i, k ^ 4 ≤ p q i := fun q i => (hprimes (q, i)).2.2.1 |
| have hpupper : ∀ q i, p q i ≤ k ^ 5 := fun q i => (hprimes (q, i)).2.2.2 |
| have hpinj : Function.Injective (fun z : Fin k × Fin (20 * k) => p z.1 z.2) := |
| primes.injective |
| let i₀ : Fin (20 * k) := ⟨0, by omega⟩ |
| let Γ := Symbol (wordPeriods p) |
| let A := State (wordPeriods p) |
| let : Finite Γ := finite_wordSymbol p hppos |
| let : Fintype Γ := Fintype.ofFinite Γ |
| let : Finite A := finite_wordState p hppos |
| let : Fintype A := Fintype.ofFinite A |
| have hΓ := FiniteAlphabet.prime_product_alphabet_bounds hk2 p hppos hpupper |
| change Finite Γ ∧ 1 ≤ Nat.card Γ ∧ Nat.card Γ ≤ finiteSourceLength k at hΓ |
| have hΓ1 : 1 ≤ Fintype.card Γ := by simpa only [Nat.card_eq_fintype_card] using hΓ.2.1 |
| have hΓM : Fintype.card Γ ≤ finiteSourceLength k := by |
| simpa only [Nat.card_eq_fintype_card] using hΓ.2.2 |
| let G : A → Word Γ (finiteSourceLength k) := rootWord p i₀ (k ^ (20 * k)) |
| let z : A := wordW p i₀ [] |
| have hMnat : 0 < finiteSourceLength k := by |
| unfold finiteSourceLength |
| positivity |
| have hM : (0 : ℝ) < finiteSourceLength k := by exact_mod_cast hMnat |
| have hkpos : (0 : ℝ) < k := by exact_mod_cast (show 0 < k by omega) |
| have hsep : ∀ x : A, (finiteSourceLength k : ℝ) / (4 : ℝ) ^ k ≤ |
| (ed (G x).val (G (x + z)).val : ℝ) := by |
| intro x |
| have hd := finiteWord_root_deficit hk2 p i₀ hpinj hpprime hplarge x |
| have he := deficit_le_editDistance |
| (finiteWord_length p i₀ (k ^ (20 * k)) k [] x) |
| (finiteWord_length p i₀ (k ^ (20 * k)) k [] (x + z)) |
| exact hd.trans (Nat.cast_le.mpr he) |
| have hmean : ∀ F : A → RealL1, |
| (∀ x y, ‖F x - F y‖ ≤ (ed (G x).val (G y).val : ℝ)) → |
| (∑ x : A, ‖F (x + z) - F x‖) / Fintype.card A ≤ |
| 34 * (finiteSourceLength k : ℝ) / (k : ℝ) ^ k := by |
| intro F hF |
| have h := finiteWord_root_mean_bound34 p i₀ hk2 hppos hpupper F hF |
| simpa only [Fintype.expect_eq_sum_div_card, finiteSourceLength, Nat.cast_pow] using h |
| obtain ⟨W, _, hW, hdist⟩ := ParameterImage.exists_word_image_lower_of_root_bound |
| G z ((finiteSourceLength k : ℝ) / (4 : ℝ) ^ k) |
| (34 * (finiteSourceLength k : ℝ) / (k : ℝ) ^ k) |
| (by positivity) (by positivity) hsep hmean |
| refine ⟨Γ, inferInstance, hΓ1, hΓM, W, hW, ?_⟩ |
| have hratio : (finiteSourceLength k : ℝ) / (4 : ℝ) ^ k / |
| (34 * (finiteSourceLength k : ℝ) / (k : ℝ) ^ k) = |
| (1 / 34 : ℝ) * ((k : ℝ) / 4) ^ k := by |
| rw [div_pow] |
| field_simp |
| rw [hratio] at hdist |
| exact hdist |
|
|
| end TreeEdit.BinaryLower |
|
|
| end |
|
|
| end OAI |
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|
