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cfa2fd2 | 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 | import Mathlib
namespace OAI
namespace TreeEdit
universe u
/-- A single unit-cost insertion, deletion or substitution, at any position. -/
inductive EditStep {α : Type u} : List α → List α → Prop
| insert (p q : List α) (a : α) : EditStep (p ++ q) (p ++ a :: q)
| delete (p q : List α) (a : α) : EditStep (p ++ a :: q) (p ++ q)
| substitute (p q : List α) (a b : α) : EditStep (p ++ a :: q) (p ++ b :: q)
/-- A script of precisely `n` edits. There is no restriction on its intermediate lengths. -/
inductive EditScript {α : Type u} : List α → List α → ℕ → Prop
| nil (x : List α) : EditScript x x 0
| cons {x y z : List α} {n : ℕ} :
EditStep x y → EditScript y z n → EditScript x z (n + 1)
/-- Ordinary unit-cost edit distance. The set is nonempty (delete then insert). -/
noncomputable def ed {α : Type u} (x y : List α) : ℕ :=
sInf {n | EditScript x y n}
/-- The full infinite-dimensional real ℓ₁ space. -/
abbrev RealL1 := lp (fun _ : ℕ => ℝ) 1
/-- Product of the two maximal pairwise distance ratios. -/
noncomputable def distortion (W : Finset (List Bool)) (f : W → RealL1) : ℝ :=
sSup {r : ℝ | ∃ x y : W, x ≠ y ∧
r = dist (f x) (f y) / (ed x.val y.val : ℝ)} *
sSup {r : ℝ | ∃ x y : W, x ≠ y ∧
r = (ed x.val y.val : ℝ) / dist (f x) (f y)}
/-- Infimum over all injective ℓ₁ maps, used on finite sets with at least two points. -/
noncomputable def c1 (W : Finset (List Bool)) : ℝ :=
sInf {D : ℝ | ∃ f : W → RealL1, Function.Injective f ∧ distortion W f = D}
/-- The binary main endpoint, for every sufficiently large cap (natural logarithms). -/
def BinaryLowerBound : Prop :=
∃ c : ℝ, 0 < c ∧ ∃ d₀ : ℕ, ∀ d : ℕ, d₀ ≤ d →
∃ n : ℕ, n ≤ d ∧ ∃ W : Finset (List Bool),
2 ≤ W.card ∧ (∀ w ∈ W, w.length = n) ∧
Real.exp (c * Real.sqrt (Real.log (d : ℝ) * Real.log (Real.log (d : ℝ)))) ≤ c1 W
theorem binary_torus : BinaryLowerBound := by
sorry
end TreeEdit
end OAI
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