lean/ComparatorChallenges/FiniteCircle.lean

import Mathlib

namespace OAI

namespace FiniteCircle

open scoped BigOperators List

universe u v

inductive EditStep {α : Type u} : List α → List α → Prop
  | insert (u v : List α) (a : α) : EditStep (u ++ v) (u ++ a :: v)
  | delete (u v : List α) (a : α) : EditStep (u ++ a :: v) (u ++ v)
  | substitute (u v : List α) (a b : α) : EditStep (u ++ a :: v) (u ++ b :: v)

inductive EditScript {α : Type u} : ℕ → List α → List α → Prop
  | nil (x : List α) : EditScript 0 x x
  | cons {n : ℕ} {x y z : List α} :
      EditStep x y → EditScript n y z → EditScript (n + 1) x z

noncomputable def editDistance {α : Type u} (x y : List α) : ℕ :=
  sInf {n : ℕ | EditScript n x y}

noncomputable def lcs {α : Type u} (x y : List α) : ℕ :=
  sSup {n : ℕ | ∃ z : List α, z <+ x ∧ z <+ y ∧ z.length = n}

noncomputable def deficit {α : Type u} (x y : List α) : ℕ := x.length - lcs x y

abbrev Words (α : Type u) (d : ℕ) := {x : List α // x.length ≤ d}
abbrev RealL1 := lp (fun _ : ℕ => ℝ) 1

noncomputable def distortion {α : Type u} {X : Type v} (word : X → List α)
    (f : X → RealL1) : ℝ :=
  (sSup {r : ℝ | ∃ x y : X, x ≠ y ∧
    r = ‖f x - f y‖ / (editDistance (word x) (word y) : ℝ)}) *
  (sSup {r : ℝ | ∃ x y : X, x ≠ y ∧
    r = (editDistance (word x) (word y) : ℝ) / ‖f x - f y‖})

noncomputable def optimalDistortion {α : Type u} {X : Type v}
    (word : X → List α) : ℝ :=
  sInf {D | ∃ f : X → RealL1, Function.Injective f ∧ D = distortion word f}

noncomputable def E (α : Type u) (d : ℕ) : ℝ :=
  optimalDistortion (fun x : Words α d => x.val)

noncomputable def setDistortion (S : Set (List Bool)) : ℝ :=
  optimalDistortion (fun x : S => x.val)

noncomputable def alphabetSupremum (d : ℕ) : ℝ :=
  sSup {r | ∃ A : ℕ, 2 ≤ A ∧ r = E (Fin A) d}

noncomputable def exponentScale (d : ℕ) : ℝ :=
  Real.sqrt (Real.log d * Real.log (Real.log d))

def encode {α : Type u} {w : ℕ} (c : α → Fin w → Bool) (x : List α) : List Bool :=
  x.flatMap (fun a => List.ofFn (c a))

def interval {α : Type u} {w : ℕ} (c : α → Fin w → Bool)
    (a : α) (s len : ℕ) : List Bool :=
  ((List.ofFn (c a)).drop s).take len

def OffsetLocal {α : Type u} {w : ℕ} (δ : ℝ) (c : α → Fin w → Bool) : Prop :=
  ∀ a b : α, a ≠ b → ∀ s t : ℕ,
    s + w / 8 ≤ w → t + w / 8 ≤ w →
    (lcs (interval c a s (w / 8)) (interval c b t (w / 8)) : ℝ) ≤
      (1 - δ) * (w / 8 : ℕ)

def indexSpan {w : ℕ} (is : List (Fin w)) : ℕ :=
  match is with
  | [] => 0
  | i :: js => (js.getLastD i).val + 1 - i.val

def AnchorLocal {α : Type u} {w : ℕ} (ε : ℝ) (c : α → Fin w → Bool) : Prop :=
  ∀ a b : α, ∀ is js : List (Fin w),
    is.Pairwise (fun i j => i.val < j.val) →
    js.Pairwise (fun i j => i.val < j.val) →
    is.length = js.length → (w : ℝ) / 4 ≤ is.length →
    (indexSpan is - is.length : ℕ) ≤ ε * w →
    (indexSpan js - js.length : ℕ) ≤ ε * w →
    (∀ ij ∈ is.zip js, c a ij.1 = c b ij.2) →
    a = b ∧ ∃ i : Fin w, (i, i) ∈ is.zip js

def SubstitutionBounds {α : Type u} [Fintype α] {w : ℕ}
    (a C : ℝ) (c : α → Fin w → Bool) : Prop :=
  0 < w ∧ (w : ℝ) ≤ C * Real.log (2 * Fintype.card α) ∧
  (∀ L : ℕ, ∀ x y : List α, x.length = L → y.length = L →
    a * w * deficit x y ≤ (deficit (encode c x) (encode c y) : ℝ) ∧
    deficit (encode c x) (encode c y) ≤ w * deficit x y) ∧
  (∀ x y : List α, editDistance (encode c x) (encode c y) ≤ w * editDistance x y)

def TwoCodes (a C δ ε : ℝ) : Prop :=
  ∀ (α : Type) [Fintype α], 2 ≤ Fintype.card α →
    (∃ (w : ℕ) (c : α → Fin w → Bool),
      SubstitutionBounds a C c ∧ OffsetLocal δ c) ∧
    (∃ (w : ℕ) (c : α → Fin w → Bool),
      SubstitutionBounds a C c ∧ AnchorLocal ε c)

def period (h : ℕ) (p : ℕ → ℕ → ℕ) (l : ℕ) : ℕ := ∏ k : Fin h, p l k

def circleSize (h : ℕ) (p : ℕ → ℕ → ℕ) : ℕ :=
  2 * ∏ l : Fin h, period h p (l.val + 1)

def MaskPrimes (h : ℕ) (p : ℕ → ℕ → ℕ) : Prop :=
  (∀ l k : Fin h, Nat.Prime (p (l.val + 1) k)) ∧
  Function.Injective (fun lk : Fin h × Fin h => p (lk.1.val + 1) lk.2) ∧
  ∀ l k : Fin h, h ^ 3 ≤ p (l.val + 1) k ∧ p (l.val + 1) k ≤ 2 * h ^ 3

def ExcludedPrimes (m : ℕ) (p : ℕ → ℕ → ℕ) : Prop :=
  (∀ l k : Fin m, Nat.Prime (p (l.val + 1) k)) ∧
  Function.Injective (fun lk : Fin m × Fin m => p (lk.1.val + 1) lk.2) ∧
  ∀ l k : Fin m, m ^ 3 < p (l.val + 1) k ∧ p (l.val + 1) k ≤ m ^ 5

abbrev MaskTag (h : ℕ) (p : ℕ → ℕ → ℕ) (l : ℕ) := Σ k : Fin h, Fin (p l k)

def maskTags (h : ℕ) (p : ℕ → ℕ → ℕ) (l : ℕ) : List (MaskTag h p l) :=
  (List.ofFn fun k : Fin h => List.ofFn fun b : Fin (p l k) => Sigma.mk k b).flatten

def MaskAlphabet (h : ℕ) (p : ℕ → ℕ → ℕ) : ℕ → Type
  | 0 => Bool
  | l + 1 => MaskTag h p (l + 1) × MaskAlphabet h p l

instance maskAlphabetFintype (h : ℕ) (p : ℕ → ℕ → ℕ) (l : ℕ) :
    Fintype (MaskAlphabet h p l) := by
  induction l with
  | zero => exact inferInstanceAs (Fintype Bool)
  | succ l ih => exact @instFintypeProd _ _ inferInstance ih

def MaskState (h : ℕ) (p : ℕ → ℕ → ℕ) (M : ℕ) : ℕ → Type
  | 0 => ZMod M
  | l + 1 => MaskTag h p (l + 1) → MaskState h p M l

def shiftMaskState (h : ℕ) (p : ℕ → ℕ → ℕ) (M : ℕ) :
    (l : ℕ) → MaskState h p M l → ZMod M → MaskState h p M l
  | 0, v, θ => (show ZMod M from v) + θ
  | l + 1, v, θ => fun j => shiftMaskState h p M l (v j) θ

def maskGamma (h : ℕ) (p : ℕ → ℕ → ℕ) (M l a : ℕ)
    (j : MaskTag h p l) : ZMod M :=
  (∑ k : Fin h, ((M / p l k * (a % p l k) : ℕ) : ZMod M)) +
    ((j.2.val * (M / p l j.1) * (a % p l j.1) : ℕ) : ZMod M)

def maskWord (h : ℕ) (p : ℕ → ℕ → ℕ) (R M : ℕ) :
    (l : ℕ) → MaskState h p M l → List (MaskAlphabet h p l)
  | 0, v => (List.range M).map (fun b => decide (((b : ZMod M) - (show ZMod M from v)).val < M / 2))
  | l + 1, v => (List.range (R * period h p (l + 1))).flatMap fun a =>
      (maskTags h p (l + 1)).flatMap fun j =>
        (maskWord h p R M l
          (shiftMaskState h p M l (v j) (maskGamma h p M (l + 1) a j))).map
          (fun x => (j, x))

def ExcludedAlphabet (m L : ℕ) : ℕ → Type
  | 0 => Fin L × Bool
  | l + 1 => Fin m × ExcludedAlphabet m L l

instance excludedAlphabetFintype (m L l : ℕ) : Fintype (ExcludedAlphabet m L l) := by
  induction l with
  | zero => exact inferInstanceAs (Fintype (Fin L × Bool))
  | succ l ih => exact @instFintypeProd _ _ inferInstance ih

def ExcludedState (m L : ℕ) : ℕ → Type
  | 0 => ZMod L
  | l + 1 => Fin m → ExcludedState m L l

def shiftExcludedState (m L : ℕ) :
    (l : ℕ) → ExcludedState m L l → ZMod L → ExcludedState m L l
  | 0, v, θ => (show ZMod L from v) + θ
  | l + 1, v, θ => fun i => shiftExcludedState m L l (v i) θ

def excludedTranslation (m : ℕ) (p : ℕ → ℕ → ℕ) (L l s : ℕ) (i : Fin m) :
    ZMod L := ∑ j ∈ Finset.univ.erase i, ((L / p l j * (s % p l j) : ℕ) : ZMod L)

def excludedWord (m : ℕ) (p : ℕ → ℕ → ℕ) (R L : ℕ) :
    (l : ℕ) → ExcludedState m L l → List (ExcludedAlphabet m L l)
  | 0, v => List.ofFn fun b : Fin L =>
      (b, decide (((b.val : ZMod L) - (show ZMod L from v)).val < L / 2))
  | l + 1, v => (List.range (R * period m p (l + 1))).flatMap fun s =>
      (List.ofFn fun i : Fin m =>
        (excludedWord m p R L l
          (shiftExcludedState m L l (v i) (excludedTranslation m p L (l + 1) s i))).map
          (fun x => (i, x))).flatten

def MaskWitness (a C δ lower : ℝ) (d : ℕ) : Prop :=
  ∃ h : ℕ, 2 ≤ h ∧ ∃ p : ℕ → ℕ → ℕ, MaskPrimes h p ∧
    ∃ (w : ℕ) (c : MaskAlphabet h p h → Fin w → Bool),
      SubstitutionBounds a C c ∧ OffsetLocal δ c ∧
      ∃ n : ℕ, 0 < n ∧ n ≤ d ∧
        (∀ v : MaskState h p (circleSize h p) h,
          (encode c (maskWord h p (h ^ (2 * h)) (circleSize h p) h v)).length = n) ∧
        Real.exp (lower * exponentScale d) ≤ setDistortion
          (Set.range fun v : MaskState h p (circleSize h p) h =>
            encode c (maskWord h p (h ^ (2 * h)) (circleSize h p) h v))

def ExcludedWitness (a C ε lower : ℝ) (d : ℕ) : Prop :=
  ∃ m : ℕ, 2 ≤ m ∧ ∃ p : ℕ → ℕ → ℕ, ExcludedPrimes m p ∧
    ∃ (w : ℕ) (c : ExcludedAlphabet m (circleSize m p) m → Fin w → Bool),
      SubstitutionBounds a C c ∧ AnchorLocal ε c ∧
      ∃ n : ℕ, 0 < n ∧ n ≤ d ∧
        (∀ v : ExcludedState m (circleSize m p) m,
          (encode c (excludedWord m p (m ^ (2 * m)) (circleSize m p) m v)).length = n) ∧
        Real.exp (lower * exponentScale d) ≤ setDistortion
          (Set.range fun v : ExcludedState m (circleSize m p) m =>
            encode c (excludedWord m p (m ^ (2 * m)) (circleSize m p) m v))

def UniformUpper (C : ℝ) (d₀ : ℕ) : Prop :=
  ∀ (α : Type) [Fintype α], 2 ≤ Fintype.card α → ∀ d : ℕ, d₀ ≤ d →
    ∃ (M : ℕ) (F : Words α d → Fin M → ℝ), ∀ x y : Words α d,
      (editDistance x.val y.val : ℝ) ≤ ∑ i, |F x i - F y i| ∧
      (∑ i, |F x i - F y i|) ≤
        Real.exp (C * exponentScale d) * editDistance x.val y.val

def FullMain : Prop :=
  ∃ c C a Ccode δ ε : ℝ, ∃ d₀ : ℕ,
    0 < c ∧ 0 < C ∧ 0 < a ∧ 0 < Ccode ∧ 0 < δ ∧ 0 < ε ∧
    TwoCodes a Ccode δ ε ∧ UniformUpper C d₀ ∧
    (∀ (α : Type) [Fintype α], 2 ≤ Fintype.card α → ∀ d : ℕ, d₀ ≤ d →
      Real.exp (c * exponentScale d) ≤ E α d ∧
      E α d ≤ Real.exp (C * exponentScale d)) ∧
    (∀ d : ℕ, d₀ ≤ d →
      MaskWitness a Ccode δ c d ∧ ExcludedWitness a Ccode ε c d ∧
      Real.exp (c * exponentScale d) ≤ alphabetSupremum d ∧
      alphabetSupremum d ≤ Real.exp (C * exponentScale d))

end FiniteCircle

theorem finiteCircle_fullMain : FiniteCircle.FullMain := by
  sorry

end OAI