lean/OAI/Algebra/DepthFive/Basic.lean

import Mathlib

namespace OAI

noncomputable section

open scoped BigOperators
universe u

namespace Problem335

inductive D5Leaf (K : Type u) (n : ℕ) where
  | scalar : K → D5Leaf K n
  | variable : (Fin n × Fin n × Fin n) → D5Leaf K n

def d5LeafDegree {K : Type u} {n : ℕ} : D5Leaf K n → ℕ
  | D5Leaf.scalar _ => 0
  | D5Leaf.variable _ => 1

structure Depth5Circuit (K : Type u) [CommSemiring K] (n : ℕ) where
  leafCount : ℕ
  leaves : Fin leafCount → D5Leaf K n
  bottomCount : ℕ
  bottomInputs : Fin bottomCount → List (K × Fin leafCount)
  bottomDegree : Fin bottomCount → ℕ
  bottomHomogeneous :
    ∀ i entry, entry ∈ bottomInputs i → d5LeafDegree (leaves entry.2) = bottomDegree i
  bottomEmpty :
    ∀ i, bottomInputs i = [] → bottomDegree i = 0
  lowerCount : ℕ
  lowerInputs : Fin lowerCount → List (Fin bottomCount)
  middleCount : ℕ
  middleInputs : Fin middleCount → List (K × Fin lowerCount)
  middleDegree : Fin middleCount → ℕ
  middleHomogeneous :
    ∀ i entry, entry ∈ middleInputs i →
      ((lowerInputs entry.2).map bottomDegree).sum = middleDegree i
  middleEmpty :
    ∀ i, middleInputs i = [] → middleDegree i = 0
  upperCount : ℕ
  upperInputs : Fin upperCount → List (Fin middleCount)
  outputInputs : List (K × Fin upperCount)
  outputDegree : ℕ
  outputHomogeneous :
    ∀ entry, entry ∈ outputInputs →
      ((upperInputs entry.2).map middleDegree).sum = outputDegree
  outputEmpty :
    outputInputs = [] → outputDegree = 0

def d5LeafValue {K : Type u} [CommSemiring K] {n : ℕ}
    (x : D5Leaf K n) : MvPolynomial (Fin n × Fin n × Fin n) K :=
  match x with
  | D5Leaf.scalar a => MvPolynomial.C a
  | D5Leaf.variable i => MvPolynomial.X i

def bottomValue {K : Type u} [CommSemiring K] {n : ℕ}
    (c : Depth5Circuit K n) (i : Fin c.bottomCount) :
    MvPolynomial (Fin n × Fin n × Fin n) K := by
  classical
  exact ((c.bottomInputs i).map (fun entry =>
    MvPolynomial.C entry.1 * d5LeafValue (c.leaves entry.2))).sum

def lowerValue {K : Type u} [CommSemiring K] {n : ℕ}
    (c : Depth5Circuit K n) (i : Fin c.lowerCount) :
    MvPolynomial (Fin n × Fin n × Fin n) K := by
  classical
  exact ((c.lowerInputs i).map (fun j => bottomValue c j)).prod

def middleValue {K : Type u} [CommSemiring K] {n : ℕ}
    (c : Depth5Circuit K n) (i : Fin c.middleCount) :
    MvPolynomial (Fin n × Fin n × Fin n) K := by
  classical
  exact ((c.middleInputs i).map (fun entry =>
    MvPolynomial.C entry.1 * lowerValue c entry.2)).sum

def upperValue {K : Type u} [CommSemiring K] {n : ℕ}
    (c : Depth5Circuit K n) (i : Fin c.upperCount) :
    MvPolynomial (Fin n × Fin n × Fin n) K := by
  classical
  exact ((c.upperInputs i).map (fun j => middleValue c j)).prod

def circuitValue {K : Type u} [CommSemiring K] {n : ℕ}
    (c : Depth5Circuit K n) : MvPolynomial (Fin n × Fin n × Fin n) K := by
  classical
  exact (c.outputInputs.map (fun entry =>
    MvPolynomial.C entry.1 * upperValue c entry.2)).sum

def circuitSize {K : Type u} [CommSemiring K] {n : ℕ}
    (c : Depth5Circuit K n) : ℕ :=
  c.leafCount + c.bottomCount + c.lowerCount + c.middleCount + c.upperCount + 1

def immLayer (K : Type u) [CommSemiring K] (n : ℕ) (t : Fin n) :
    Matrix (Fin n) (Fin n) (MvPolynomial (Fin n × Fin n × Fin n) K) :=
  fun i j => MvPolynomial.X (t, i, j)

def imm (K : Type u) [CommSemiring K] (n : ℕ) :
    MvPolynomial (Fin n × Fin n × Fin n) K := by
  classical
  by_cases h : 0 < n
  · let matrices :
        List (Matrix (Fin n) (Fin n)
          (MvPolynomial (Fin n × Fin n × Fin n) K)) :=
      List.ofFn (fun t : Fin n => immLayer K n t)
    exact matrices.prod ⟨0, h⟩ ⟨0, h⟩
  · exact 0

def ceilSqrt (n : ℕ) : ℕ :=
  Nat.sqrt n + if (Nat.sqrt n) ^ 2 = n then 0 else 1

def ceilDiv (n t : ℕ) : ℕ :=
  (n + t - 1) / t

def upperGateBound (n : ℕ) : ℕ :=
  let t := ceilSqrt n
  let r := ceilDiv n t
  2 * n ^ 3 + r * n ^ 2 + r * n ^ (t + 1) + n ^ (r - 1) + 1

end Problem335

end

end OAI