| import OAI.Combinatorics.BinaryEdit.FiniteTreeInstantiation |
| import OAI.Combinatorics.BinaryEdit.FiniteWordPrimeSeparation |
| import OAI.Combinatorics.BinaryEdit.FiniteTreeTags |
| import OAI.Combinatorics.BinaryEdit.FiniteSeparationArithmetic |
| import OAI.Combinatorics.BinaryEdit.TaggedArrays |
|
|
| namespace OAI |
|
|
| /-! |
| # Geometric separation for the concrete finite tree |
|
|
| The prime construction supplies the half-child condition in the |
| geometric induction. |
| -/ |
|
|
| noncomputable section |
| namespace TreeEdit.BinaryLower.FiniteTree |
|
|
| open scoped BigOperators |
|
|
| / |
| def finiteBlocks {m k : ℕ} (primes : Fin k → Fin m → ℕ) (i₀ : Fin m) |
| (T h : ℕ) (p : List (Fin m)) (x : State (wordPeriods primes)) : |
| Fin T → Fin m → Word (Symbol (wordPeriods primes)) ((m * T) ^ h) := |
| fun a i => ⟨finiteWord primes i₀ T h (p ++ [i]) (x + a.val • wordU primes i₀ p), |
| finiteWord_length primes i₀ T h _ _⟩ |
|
|
| theorem finiteWord_succ_eq_rows {m k : ℕ} (primes : Fin k → Fin m → ℕ) |
| (i₀ : Fin m) (T h : ℕ) (p : List (Fin m)) (x : State (wordPeriods primes)) : |
| finiteWord primes i₀ T (h + 1) p x = |
| TaggedExtraction.rows (finiteBlocks primes i₀ T h p x) := by |
| have hrange : (List.finRange T).map Fin.val = List.range T := by |
| apply List.ext_getElem |
| · simp |
| · intro j hj hj' |
| simp |
| unfold finiteWord nodeWord |
| rw [← hrange, List.flatMap_map] |
| rfl |
|
|
| / |
| def finiteRowShift {m k T : ℕ} (primes : Fin k → Fin m → ℕ) |
| (p : List (Fin m)) (hp : p.length < k) (a b : Fin T) (t : ℤ) : ℤ := |
| (b.val : ℤ) - (a.val : ℤ) + t * |
| (FiniteVectors.levelProduct primes ⟨p.length, hp⟩ : ℤ) |
|
|
| / |
| def separatedChildren {m k T : ℕ} (primes : Fin k → Fin m → ℕ) |
| (i₀ : Fin m) (p : List (Fin m)) (hp : p.length < k) (a b : Fin T) (t : ℤ) : |
| Finset (Fin m) := |
| Finset.univ.filter fun i => ¬ (wordPeriod primes i₀ (p ++ [i]) : ℤ) ∣ |
| finiteRowShift primes p hp a b t |
|
|
| / |
| for arbitrary common-subsequence matchings. -/ |
| theorem finiteWord_separation_of_half_children {m k T : ℕ} |
| (primes : Fin k → Fin m → ℕ) (i₀ : Fin m) (hm : 2 ≤ m) (hT : 0 < T) |
| (hhalf : ∀ p (hp : p.length < k) (a b : Fin T) (t : ℤ), |
| ¬ (wordPeriod primes i₀ p : ℤ) ∣ t → |
| m ≤ 2 * (separatedChildren primes i₀ p hp a b t).card) |
| (h : ℕ) (p : List (Fin m)) (hlen : p.length + h = k) |
| (x : State (wordPeriods primes)) (t : ℤ) |
| (ht : ¬ (wordPeriod primes i₀ p : ℤ) ∣ t) : |
| (((m * T) ^ h : ℕ) : ℝ) / (4 : ℝ) ^ h ≤ |
| (deficit ((m * T) ^ h) (finiteWord primes i₀ T h p x) |
| (finiteWord primes i₀ T h p (x + t • wordW primes i₀ p)) : ℝ) := by |
| classical |
| induction h generalizing p x t with |
| | zero => |
| have hp : p.length = k := by simpa using hlen |
| simpa using (finiteWord_leaf_deficit primes i₀ T p hp x t ht).symm.le |
| | succ h ih => |
| have hp : p.length < k := by omega |
| let B : ℕ := (m * T) ^ h |
| let q : ℝ := (B : ℝ) / (4 : ℝ) ^ h |
| let d : ℝ := (m : ℝ) * q / 2 |
| let X := finiteBlocks primes i₀ T h p x |
| let Y := finiteBlocks primes i₀ T h p (x + t • wordW primes i₀ p) |
| have hB : 0 < B := by dsimp [B]; exact pow_pos (Nat.mul_pos (by omega) hT) _ |
| have hq : 0 ≤ q := by dsimp [q]; positivity |
| have hd : 0 ≤ d := by dsimp [d]; positivity |
| have hlocal (a b : Fin T) : d ≤ ∑ i, (deficit B (X a i).val (Y b i).val : ℝ) := by |
| apply row_deficit_charge (separatedChildren primes i₀ p hp a b t) _ q hq |
| (fun i => Nat.cast_nonneg _) _ (hhalf p hp a b t ht) |
| intro i hi |
| have hnot : ¬ (wordPeriod primes i₀ (p ++ [i]) : ℤ) ∣ |
| finiteRowShift primes p hp a b t := (Finset.mem_filter.mp hi).2 |
| change q ≤ (deficit B |
| (finiteWord primes i₀ T h (p ++ [i]) (x + a.val • wordU primes i₀ p)) |
| (finiteWord primes i₀ T h (p ++ [i]) |
| (x + t • wordW primes i₀ p + b.val • wordU primes i₀ p)) : ℝ) |
| rw [finiteWord_child_relative primes i₀ i T h p hlen x t a.val b.val] |
| exact ih (p ++ [i]) (by simp; omega) _ _ hnot |
| have hX : ∀ (a : Fin T) (i : Fin m) (s : Symbol (wordPeriods primes)), |
| s ∈ (X a i).val → s ∈ subtreeAlphabet (wordPeriods primes) (p ++ [i]) := by |
| intro a i s hs |
| exact mem_subtreeAlphabet_of_mem (wordPeriods primes) |
| ⟨List.replicate k i₀, by simp⟩ (wordU primes i₀) h (p ++ [i]) |
| (by simp; omega) _ s hs |
| have hY : ∀ (a : Fin T) (i : Fin m) (s : Symbol (wordPeriods primes)), |
| s ∈ (Y a i).val → s ∈ subtreeAlphabet (wordPeriods primes) (p ++ [i]) := by |
| intro a i s hs |
| exact mem_subtreeAlphabet_of_mem (wordPeriods primes) |
| ⟨List.replicate k i₀, by simp⟩ (wordU primes i₀) h (p ++ [i]) |
| (by simp; omega) _ s hs |
| have hdg : d ≤ (((m - 1) * B : ℕ) : ℝ) := by |
| have hh := finite_charge_le_gap m h hm (B : ℝ) (Nat.cast_nonneg _) |
| simpa [d, q, Nat.cast_mul, Nat.cast_sub (show 1 ≤ m by omega)] using hh |
| have hsep := (Tagged.rows_with_ordered_tags hT hm hB |
| (fun i => subtreeAlphabet (wordPeriods primes) (p ++ [i])) |
| (subtreeAlphabet_children_pairwise_disjoint (wordPeriods primes) p) |
| X Y hX hY d hd hlocal).2.2 hdg |
| have hN : T * m * B = (m * T) ^ (h + 1) := by dsimp [B]; rw [pow_succ]; ring |
| have hcharge : (T : ℝ) * d / 2 = (((m * T) ^ (h + 1) : ℕ) : ℝ) / (4 : ℝ) ^ (h + 1) := by |
| dsimp [d, q, B] |
| push_cast |
| rw [pow_succ, pow_succ] |
| ring |
| rw [hcharge, hN] at hsep |
| simpa only [X, Y, ← finiteWord_succ_eq_rows] using hsep |
|
|
| / |
| theorem finiteWord_root_deficit_of_half_children {m k T : ℕ} |
| (primes : Fin k → Fin m → ℕ) (i₀ : Fin m) (hm : 2 ≤ m) (hT : 0 < T) |
| (hhalf : ∀ p (hp : p.length < k) (a b : Fin T) (t : ℤ), |
| ¬ (wordPeriod primes i₀ p : ℤ) ∣ t → |
| m ≤ 2 * (separatedChildren primes i₀ p hp a b t).card) |
| (x : State (wordPeriods primes)) : |
| (((m * T) ^ k : ℕ) : ℝ) / (4 : ℝ) ^ k ≤ |
| (deficit ((m * T) ^ k) (finiteWord primes i₀ T k [] x) |
| (finiteWord primes i₀ T k [] (x + wordW primes i₀ [])) : ℝ) := by |
| have h := finiteWord_separation_of_half_children primes i₀ hm hT hhalf |
| k [] (by simp) x 1 (by simp) |
| simpa using h |
|
|
| / |
| theorem finiteWord_separation {k : ℕ} (hk : 2 ≤ k) |
| (primes : Fin k → Fin (20 * k) → ℕ) (i₀ : Fin (20 * k)) |
| (hinj : Function.Injective |
| (fun z : Fin k × Fin (20 * k) => primes z.1 z.2)) |
| (hprime : ∀ q i, Nat.Prime (primes q i)) |
| (hlarge : ∀ q i, k ^ 4 ≤ primes q i) |
| (h : ℕ) (p : List (Fin (20 * k))) (hlen : p.length + h = k) |
| (x : State (wordPeriods primes)) (t : ℤ) |
| (ht : ¬ (wordPeriod primes i₀ p : ℤ) ∣ t) : |
| (((20 * k * k ^ (20 * k)) ^ h : ℕ) : ℝ) / (4 : ℝ) ^ h ≤ |
| (deficit ((20 * k * k ^ (20 * k)) ^ h) |
| (finiteWord primes i₀ (k ^ (20 * k)) h p x) |
| (finiteWord primes i₀ (k ^ (20 * k)) h p |
| (x + t • wordW primes i₀ p)) : ℝ) := by |
| apply finiteWord_separation_of_half_children primes i₀ (by omega) |
| (pow_pos (by omega) _) _ h p hlen x t ht |
| intro p hp a b t ht |
| exact finiteWord_half_children hk primes i₀ p hp hinj hprime hlarge a b t ht |
|
|
| / |
| prescribed half-period shift. -/ |
| theorem finiteWord_root_deficit {k : ℕ} (hk : 2 ≤ k) |
| (primes : Fin k → Fin (20 * k) → ℕ) (i₀ : Fin (20 * k)) |
| (hinj : Function.Injective |
| (fun z : Fin k × Fin (20 * k) => primes z.1 z.2)) |
| (hprime : ∀ q i, Nat.Prime (primes q i)) |
| (hlarge : ∀ q i, k ^ 4 ≤ primes q i) |
| (x : State (wordPeriods primes)) : |
| (((20 * k * k ^ (20 * k)) ^ k : ℕ) : ℝ) / (4 : ℝ) ^ k ≤ |
| (deficit ((20 * k * k ^ (20 * k)) ^ k) |
| (finiteWord primes i₀ (k ^ (20 * k)) k [] x) |
| (finiteWord primes i₀ (k ^ (20 * k)) k [] |
| (x + wordW primes i₀ [])) : ℝ) := by |
| have h := finiteWord_separation hk primes i₀ hinj hprime hlarge k [] |
| (by simp) x 1 (by simp) |
| simpa using h |
|
|
| end TreeEdit.BinaryLower.FiniteTree |
|
|
| end |
|
|
| end OAI |
|
|
