File size: 3,283 Bytes
e320e82 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 | import OAI.Combinatorics.EditDistance.RecursiveWords
namespace OAI
/-!
# Length, alphabet, and distortion recurrences for word families
-/
namespace EditFamily
open scoped BigOperators
def familyLength (b P s : ℕ) : ℕ := (2 * (2 * b - 1) * P ^ (b / 2)) ^ s
def familyAlphabet (b P : ℕ) : ℕ → ℕ
| 0 => 2 * P
| s+1 => 1 + (2 * b - 1) * familyAlphabet b P s
noncomputable def familyAlpha (s : ℕ) : ℝ := (24 : ℝ)⁻¹ ^ s
noncomputable def familyBudget (b P h : ℕ) : ℕ → ℝ
| 0 => 2
| s+1 => 2 * (2 * (P : ℝ)) ^ (2 * h) / (P ^ (b / 2) : ℕ) + familyBudget b P h s / h
lemma familyLength_pos {b P : ℕ} (hb : 0 < b) (hP : 0 < P) (s : ℕ) :
0 < familyLength b P s := by
unfold familyLength
exact pow_pos (Nat.mul_pos (Nat.mul_pos (by omega) (by omega)) (pow_pos hP _)) _
lemma familyAlpha_pos (s : ℕ) : 0 < familyAlpha s := by
unfold familyAlpha
positivity
lemma familyAlpha_le_one (s : ℕ) : familyAlpha s ≤ 1 := by
unfold familyAlpha
exact pow_le_one₀ (by norm_num) (by norm_num)
lemma familyBudget_nonneg (b P h s : ℕ) : 0 ≤ familyBudget b P h s := by
induction s with
| zero => norm_num [familyBudget]
| succ s ih =>
exact add_nonneg (by positivity) (div_nonneg ih (Nat.cast_nonneg _))
noncomputable def recursive {J : Type} [Fintype J] [DecidableEq J]
(p : J → ℕ) (hp : ∀ j, (p j).Prime) (hinj : Function.Injective p)
(P : ℕ) (hP : 1 ≤ P) (hpLow : ∀ j, P ≤ p j) (hpHigh : ∀ j, p j ≤ 2 * P)
(h : ℕ) (hh : 0 < h) : (s : ℕ) → (j : J) →
Family (p j) (familyLength (Fintype.card J) P s) (familyAlphabet (Fintype.card J) P s)
(familyAlpha s) (familyBudget (Fintype.card J) P h s)
| 0, j => base (hp j) (hpHigh j)
| s+1, j => by
let F := recursive p hp hinj P hP hpLow hpHigh h hh s
have hn : familyLength (Fintype.card J) P (s+1) =
2 * ((2 * Fintype.card J - 1) * familyLength (Fintype.card J) P s) *
P ^ (Fintype.card J / 2) := by
simp only [familyLength, pow_succ]
ring
have ha : familyAlpha (s+1) = familyAlpha s / 24 := by
simp only [familyAlpha, pow_succ, div_eq_mul_inv]
rw [hn, familyAlphabet, ha, familyBudget]
exact successor j F hp hinj P hP hpLow hpHigh
(familyLength_pos (Fintype.card_pos_iff.mpr ⟨j⟩) (by omega) s)
(familyAlpha_pos s).le (familyAlpha_le_one s)
(familyBudget_nonneg _ _ _ _) h hh
lemma familyAlphabet_pos {b P : ℕ} (hP : 0 < P) (s : ℕ) : 0 < familyAlphabet b P s := by
cases s <;> simp only [familyAlphabet] <;> omega
lemma familyAlphabet_le {b P : ℕ} (hb : 0 < b) (hP : 0 < P) (s : ℕ) :
familyAlphabet b P s ≤ (2 * P + 1) * (2 * b) ^ s := by
induction s with
| zero => simp [familyAlphabet]
| succ s ih =>
have hap := familyAlphabet_pos (b := b) hP s
have hb' : 2 * b - 1 + 1 = 2 * b := by omega
calc
familyAlphabet b P (s+1) = 1 + (2 * b - 1) * familyAlphabet b P s := rfl
_ ≤ 2 * b * familyAlphabet b P s := by
nlinarith [Nat.mul_add (2 * b - 1) 1 (familyAlphabet b P s)]
_ ≤ 2 * b * ((2 * P + 1) * (2 * b) ^ s) := Nat.mul_le_mul_left _ ih
_ = (2 * P + 1) * (2 * b) ^ (s+1) := by rw [pow_succ]; ring
end EditFamily
end OAI
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