lean/OAI/Computability/DirectedFeedback/Games/GenericPolynomial.lean

import Mathlib
import OAI.Computability.DirectedFeedback.Games.NonlinearRecurrence

namespace OAI

namespace DFVSGames.Quadratic

open MvPolynomial
open scoped BigOperators
open scoped nonZeroDivisors

variable {σ : Type*}

theorem zmodTwo_sq (a : ZMod 2) : a ^ 2 = a := by
  classical
  have ha : a = 0 ∨ a = 1 := by
    fin_cases a
    · exact Or.inl rfl
    · exact Or.inr rfl
  rcases ha with rfl | rfl
  · exact zero_pow (by decide)
  · exact one_pow 2

theorem expand_two_eq_sq (p : MvPolynomial σ (ZMod 2)) : expand 2 p = p ^ 2 := by
  induction p using MvPolynomial.induction_on' with
  | monomial m a => simp only [expand_monomial, monomial_pow, zmodTwo_sq]
  | add p q hp hq => rw [map_add, CharTwo.add_sq, hp, hq]

theorem exists_coordinate_separator {X K : Type*} [Field K] [Fintype X]
    (W : Submodule K (X → K)) (v : X → K) (hv : v ∉ W) :
    ∃ a : X → K, (∀ w ∈ W, ∑ x, a x * w x = 0) ∧
      (∑ x, a x * v x) ≠ 0 := by
  classical
  obtain ⟨f, hfv, hfW⟩ := W.exists_dual_map_eq_bot_of_notMem hv inferInstance
  let a : X → K := fun x => f (fun y => if x = y then 1 else 0)
  have hrepr (w : X → K) : f w = ∑ x, a x * w x := by
    rw [LinearMap.pi_apply_eq_sum_univ]
    apply Finset.sum_congr rfl
    intro x _
    exact mul_comm _ _
  refine ⟨a, ?_, ?_⟩
  · intro w hw
    have hmem : f w ∈ W.map f := ⟨w, hw, rfl⟩
    rw [hfW, Submodule.mem_bot, hrepr] at hmem
    exact hmem
  · rwa [← hrepr]

variable {J X : Type*} [Fintype J] [Fintype X]

noncomputable def universalLinearForm (t : J → ZMod 2) (i : Fin 3) :
    MvPolynomial (Fin 3 × J) (ZMod 2) :=
  ∑ j, C (t j) * MvPolynomial.X (i, j)

noncomputable def universalAlignmentColumn (t : X → J → ZMod 2)
    (ij : Fin 3 × J) (x : X) : MvPolynomial (Fin 3 × J) (ZMod 2) :=
  C (t x ij.2) * universalLinearForm (t x) ij.1

theorem universalLinearForm_prod (t : J → ZMod 2) :
    (∏ i, universalLinearForm t i) =
      ∑ v : Fin 3 → J, C (∏ i, t (v i)) * monomial (blockExponent v) 1 := by
  classical
  unfold universalLinearForm
  rw [Fintype.prod_sum]
  apply Finset.sum_congr rfl
  intro v _
  rw [Finset.prod_mul_distrib, ← map_prod, monomial_blockExponent]

noncomputable def cubicPairing (t : X → J → ZMod 2)
    (p : X → MvPolynomial (Fin 3 × J) (ZMod 2)) (v : Fin 3 → J) :
    MvPolynomial (Fin 3 × J) (ZMod 2) :=
  ∑ x, p x * C (∏ i, t x (v i))

noncomputable def alignmentObstruction (t : X → J → ZMod 2)
    (p : X → MvPolynomial (Fin 3 × J) (ZMod 2)) :
    MvPolynomial (Fin 3 × J) (ZMod 2) :=
  ∑ x, p x ^ 2 * ∏ i, universalLinearForm (t x) i

theorem alignmentObstruction_parity_expansion (t : X → J → ZMod 2)
    (p : X → MvPolynomial (Fin 3 × J) (ZMod 2)) :
    alignmentObstruction t p = ∑ v : Fin 3 → J,
      cubicPairing t p v ^ 2 * monomial (blockExponent v) 1 := by
  classical
  unfold alignmentObstruction
  simp_rw [universalLinearForm_prod, Finset.mul_sum]
  rw [Finset.sum_comm]
  apply Finset.sum_congr rfl
  intro v _
  unfold cubicPairing
  rw [CharTwo.sum_sq, Finset.sum_mul]
  apply Finset.sum_congr rfl
  intro x _
  simp only [mul_pow, ← map_pow, zmodTwo_sq, mul_assoc]

theorem alignmentObstruction_ne_zero (t : X → J → ZMod 2)
    (p : X → MvPolynomial (Fin 3 × J) (ZMod 2))
    (v : Fin 3 → J) (hv : cubicPairing t p v ≠ 0) :
    alignmentObstruction t p ≠ 0 := by
  intro h
  rw [alignmentObstruction_parity_expansion] at h
  have hzero : ∀ w, cubicPairing t p w = 0 := by
    apply (sum_expand_mul_blockMonomial_eq_zero_iff _).mp
    simpa only [expand_two_eq_sq] using h
  exact hv (hzero v)

end DFVSGames.Quadratic

namespace DFVSGames.Quadratic

open scoped BigOperators

theorem eq_top_of_binary_coordinates
    {F X I : Type*} [Field F] [Fintype X]
    (c : I → X → F) (W : Submodule F (X → F))
    (hbinary : ∀ i x, c i x = 0 ∨ c i x = 1)
    (hseparate : ∀ x y, x ≠ y → ∃ i, c i x ≠ c i y)
    (hnonzero : ∀ x, ∃ i, c i x ≠ 0)
    (hcoordinate : ∀ i, c i ∈ W)
    (hstable : ∀ i f, f ∈ W → (fun x => c i x * f x) ∈ W) :
    W = ⊤ := by
  classical
  have hindicator : ∀ x : X, (fun y => if x = y then (1 : F) else 0) ∈ W := by
    intro x
    have hfilter : ∀ s : Finset X,
        ∃ f : X → F, f ∈ W ∧ f x = 1 ∧ ∀ y ∈ s, y ≠ x → f y = 0 := by
      intro s
      induction s using Finset.induction_on with
      | empty =>
          obtain ⟨i, hi⟩ := hnonzero x
          refine ⟨c i, hcoordinate i, (hbinary i x).resolve_left hi, ?_⟩
          simp
      | @insert y s hy ih =>
          obtain ⟨f, hf, hfx, hfzero⟩ := ih
          by_cases hyx : y = x
          · refine ⟨f, hf, hfx, ?_⟩
            intro z hz hzx
            rcases Finset.mem_insert.mp hz with rfl | hz
            · exact (hzx hyx).elim
            · exact hfzero z hz hzx
          · obtain ⟨i, hi⟩ := hseparate x y (Ne.symm hyx)
            rcases hbinary i x with hix | hix <;>
              rcases hbinary i y with hiy | hiy
            · exact (hi (hix.trans hiy.symm)).elim
            · refine ⟨fun z => f z - c i z * f z,
                W.sub_mem hf (hstable i f hf), ?_, ?_⟩
              · simp [hfx, hix]
              · intro z hz hzx
                rcases Finset.mem_insert.mp hz with rfl | hz
                · simp [hiy]
                · simp [hfzero z hz hzx]
            · refine ⟨fun z => c i z * f z, hstable i f hf, ?_, ?_⟩
              · simp [hfx, hix]
              · intro z hz hzx
                rcases Finset.mem_insert.mp hz with rfl | hz
                · simp [hiy]
                · simp [hfzero z hz hzx]
            · exact (hi (hix.trans hiy.symm)).elim
    obtain ⟨f, hf, hfx, hfzero⟩ := hfilter Finset.univ
    have heq : f = fun y => if x = y then (1 : F) else 0 := by
      funext y
      by_cases hxy : x = y
      · subst y
        simp [hfx]
      · simp [hxy, hfzero y (Finset.mem_univ y) (Ne.symm hxy)]
    rwa [← heq]
  apply top_unique
  intro f _
  rw [pi_eq_sum_univ f]
  exact W.sum_mem fun x _ => W.smul_mem (f x) (hindicator x)

def quadraticEvaluationSpan {F X I : Type*} [Field F]
    (c : I → X → F) : Submodule F (X → F) :=
  Submodule.span F (Set.range fun ij : I × I => fun x => c ij.1 x * c ij.2 x)

def cubicEvaluationSpan {F X I : Type*} [Field F]
    (c : I → X → F) : Submodule F (X → F) :=
  Submodule.span F
    (Set.range fun ijk : I × I × I => fun x => c ijk.1 x * c ijk.2.1 x * c ijk.2.2 x)

theorem eq_top_of_cubic_le_of_le_quadratic
    {F X I : Type*} [Field F] [Fintype X]
    (c : I → X → F) (W : Submodule F (X → F))
    (hbinary : ∀ i x, c i x = 0 ∨ c i x = 1)
    (hseparate : ∀ x y, x ≠ y → ∃ i, c i x ≠ c i y)
    (hnonzero : ∀ x, ∃ i, c i x ≠ 0)
    (hcubic : cubicEvaluationSpan c ≤ W)
    (hquadratic : W ≤ quadraticEvaluationSpan c) : W = ⊤ := by
  apply eq_top_of_binary_coordinates c W hbinary hseparate hnonzero
  · intro i
    have hi : (fun x => c i x * c i x * c i x) ∈ cubicEvaluationSpan c :=
      Submodule.subset_span ⟨(i, i, i), rfl⟩
    have heq : (fun x => c i x * c i x * c i x) = c i := by
      funext x
      rcases hbinary i x with h | h <;> simp [h]
    rw [heq] at hi
    exact hcubic hi
  · intro k f hf
    have hf' : f ∈ Submodule.span F
        (Set.range fun ij : I × I => fun x => c ij.1 x * c ij.2 x) :=
      hquadratic hf
    clear hf
    induction hf' using Submodule.span_induction with
    | mem f hf =>
        obtain ⟨⟨i, j⟩, rfl⟩ := hf
        have h : (fun x => c k x * c i x * c j x) ∈ cubicEvaluationSpan c :=
          Submodule.subset_span ⟨(k, i, j), rfl⟩
        simpa only [mul_assoc] using hcubic h
    | zero =>
        have heq : (fun x => c k x * (0 : X → F) x) = (0 : X → F) := by
          funext x
          exact mul_zero _
        rw [heq]
        exact W.zero_mem
    | add f g hf hg ihf ihg =>
        have heq : (fun x => c k x * ((f + g) x)) =
            (fun x => c k x * f x) + (fun x => c k x * g x) := by
          funext x
          exact mul_add _ _ _
        rw [heq]
        exact W.add_mem ihf ihg
    | smul a f hf ih =>
        have heq : (fun x => c k x * ((a • f) x)) =
            a • (fun x => c k x * f x) := by
          funext x
          exact mul_left_comm _ _ _
        rw [heq]
        exact W.smul_mem a ih

theorem quadraticEvaluationSpan_eq_top_of_cubic_le
    {F X I : Type*} [Field F] [Fintype X]
    (c : I → X → F)
    (hbinary : ∀ i x, c i x = 0 ∨ c i x = 1)
    (hseparate : ∀ x y, x ≠ y → ∃ i, c i x ≠ c i y)
    (hnonzero : ∀ x, ∃ i, c i x ≠ 0)
    (h : cubicEvaluationSpan c ≤ quadraticEvaluationSpan c) :
    quadraticEvaluationSpan c = ⊤ :=
  eq_top_of_cubic_le_of_le_quadratic c _ hbinary hseparate hnonzero h le_rfl

theorem binary_eq_zero_or_one (x : ZMod 2) : x = 0 ∨ x = 1 := by
  fin_cases x
  · exact Or.inl rfl
  · exact Or.inr rfl

theorem eq_top_of_binary_evaluation_stable
    {F : Type*} [Field F] {r : ℕ}
    (E : Finset (Fin r → ZMod 2))
    (hzero : (0 : Fin r → ZMod 2) ∉ E)
    (φ : ZMod 2 →+* F)
    (W : Submodule F (E → F))
    (hcoordinate : ∀ i : Fin r, (fun t : E => φ (t.val i)) ∈ W)
    (hstable : ∀ (i : Fin r) (f : E → F), f ∈ W →
      (fun t : E => φ (t.val i) * f t) ∈ W) : W = ⊤ := by
  classical
  have hφ : Function.Injective φ := by
    intro a b h
    rcases binary_eq_zero_or_one a with rfl | rfl <;>
      rcases binary_eq_zero_or_one b with rfl | rfl <;> simp_all
  apply eq_top_of_binary_coordinates (fun i (t : E) => φ (t.val i)) W
      _ _ _ hcoordinate hstable
  · intro i t
    rcases binary_eq_zero_or_one (t.val i) with h | h
    · left
      simp [h]
    · right
      simp [h]
  · intro t u htu
    have hne : t.val ≠ u.val := fun h => htu (Subtype.ext h)
    obtain ⟨i, hi⟩ := Function.ne_iff.mp hne
    exact ⟨i, fun h => hi (hφ h)⟩
  · intro t
    have hne : t.val ≠ 0 := by
      intro h
      exact hzero (h ▸ t.property)
    have hi : ∃ i : Fin r, t.val i ≠ 0 := by
      by_contra h
      apply hne
      funext i
      exact Classical.not_not.mp (fun hi => h ⟨i, hi⟩)
    obtain ⟨i, hi⟩ := hi
    exact ⟨i, fun h => hi (hφ (by simpa using h))⟩

end DFVSGames.Quadratic

namespace DFVSGames.Quadratic

theorem exists_cubic_not_mem_of_finrank_lt
    {F X I : Type*} [Field F] [Fintype X]
    (c : I → X → F) (W : Submodule F (X → F))
    (hbinary : ∀ i x, c i x = 0 ∨ c i x = 1)
    (hseparate : ∀ x y, x ≠ y → ∃ i, c i x ≠ c i y)
    (hnonzero : ∀ x, ∃ i, c i x ≠ 0)
    (hquadratic : W ≤ quadraticEvaluationSpan c)
    (hdim : Module.finrank F W < Fintype.card X) :
    ∃ i j k, (fun x => c i x * c j x * c k x) ∉ W := by
  classical
  by_contra h
  have hcubic : cubicEvaluationSpan c ≤ W := by
    apply Submodule.span_le.mpr
    rintro f ⟨⟨i, j, k⟩, rfl⟩
    by_contra hn
    exact h ⟨i, j, k, hn⟩
  have htop := eq_top_of_cubic_le_of_le_quadratic c W hbinary hseparate
    hnonzero hcubic hquadratic
  rw [htop, finrank_top, Module.finrank_fintype_fun_eq_card] at hdim
  exact (lt_irrefl _) hdim

end DFVSGames.Quadratic

namespace DFVSGames.Quadratic

open MvPolynomial
open scoped nonZeroDivisors

theorem exists_polynomial_alignment_annihilator
    {J X : Type*} [Fintype J] [Fintype X]
    (t : X → J → ZMod 2) (ht : Function.Injective t)
    (hnonzero : ∀ x, t x ≠ 0) (hcard : 3 * Fintype.card J < Fintype.card X) :
    ∃ p : X → MvPolynomial (Fin 3 × J) (ZMod 2),
      (∀ ij, ∑ x, p x * universalAlignmentColumn t ij x = 0) ∧
      ∃ v : Fin 3 → J, cubicPairing t p v ≠ 0 := by
  classical
  let P := MvPolynomial (Fin 3 × J) (ZMod 2)
  let K := FractionRing P
  let f : P →+* K := algebraMap P K
  let φ : ZMod 2 →+* K := f.comp C
  let c : J → X → K := fun j x => φ (t x j)
  let M : Matrix X (Fin 3 × J) K := fun x ij => f (universalAlignmentColumn t ij x)
  let W := LinearMap.range M.mulVecLin
  have hquadratic : W ≤ quadraticEvaluationSpan c := by
    change LinearMap.range M.mulVecLin ≤ _
    rw [Matrix.range_mulVecLin]
    apply Submodule.span_le.mpr
    rintro _ ⟨ij, rfl⟩
    have heq : M.col ij = ∑ j, f (MvPolynomial.X (ij.1, j)) •
        (fun x => c ij.2 x * c j x) := by
      funext x
      simp only [Finset.sum_apply, Pi.smul_apply, smul_eq_mul]
      change f (C (t x ij.2) * ∑ j, C (t x j) * MvPolynomial.X (ij.1, j)) =
        ∑ j, f (MvPolynomial.X (ij.1, j)) * (f (C (t x ij.2)) * f (C (t x j)))
      simp only [map_mul, map_sum, Finset.mul_sum]
      apply Finset.sum_congr rfl
      intro j _
      ac_rfl
    rw [heq]
    apply Submodule.sum_mem
    intro j _
    apply Submodule.smul_mem
    exact Submodule.subset_span ⟨(ij.2, j), rfl⟩
  have hbinary : ∀ j x, c j x = 0 ∨ c j x = 1 := by
    intro j x
    rcases binary_eq_zero_or_one (t x j) with h | h
    · left
      simp [c, h]
    · right
      simp [c, h]
  have hseparate : ∀ x y, x ≠ y → ∃ j, c j x ≠ c j y := by
    intro x y hxy
    have hfun : t x ≠ t y := fun h => hxy (ht h)
    obtain ⟨j, hj⟩ := Function.ne_iff.mp hfun
    exact ⟨j, fun h => hj (φ.injective h)⟩
  have hnz : ∀ x, ∃ j, c j x ≠ 0 := by
    intro x
    obtain ⟨j, hj⟩ := Function.ne_iff.mp (hnonzero x)
    refine ⟨j, ?_⟩
    intro h
    exact hj (φ.injective (by simpa [c] using h))
  have hdim : Module.finrank K W < Fintype.card X := by
    have hle : Module.finrank K W ≤ 3 * Fintype.card J := by
      calc
        Module.finrank K W ≤ Module.finrank K ((Fin 3 × J) → K) :=
          LinearMap.finrank_range_le M.mulVecLin
        _ = 3 * Fintype.card J := by
          rw [Module.finrank_fintype_fun_eq_card]
          simp
    exact hle.trans_lt hcard
  obtain ⟨i, j, k, hijk⟩ := exists_cubic_not_mem_of_finrank_lt
    c W hbinary hseparate hnz hquadratic hdim
  let v : Fin 3 → J := ![i, j, k]
  have hcubic : (fun x => ∏ b : Fin 3, c (v b) x) ∉ W := by
    simpa [v, Fin.prod_univ_succ, mul_assoc] using hijk
  obtain ⟨a, ha, hav⟩ := exists_coordinate_separator W _ hcubic
  obtain ⟨b, hb⟩ := IsLocalization.exist_integer_multiples_of_finite P⁰ a
  choose p hp using hb
  have hp' (x : X) : f (p x) = f (b : P) * a x := by
    simpa only [Algebra.smul_def] using hp x
  have hbne : f (b : P) ≠ 0 :=
    IsFractionRing.to_map_ne_zero_of_mem_nonZeroDivisors b.property
  refine ⟨p, ?_, v, ?_⟩
  · intro ij
    apply IsFractionRing.injective P K
    change f (∑ x, p x * universalAlignmentColumn t ij x) = f 0
    rw [map_sum, map_zero]
    simp only [map_mul, hp', mul_assoc]
    rw [← Finset.mul_sum]
    have hcol : M.col ij ∈ W := by
      change M.col ij ∈ LinearMap.range M.mulVecLin
      rw [Matrix.range_mulVecLin]
      exact Submodule.subset_span ⟨ij, rfl⟩
    have hsum := ha (M.col ij) hcol
    change (∑ x, a x * f (universalAlignmentColumn t ij x)) = 0 at hsum
    rw [hsum, mul_zero]
  · have heq : f (cubicPairing t p v) =
        f (b : P) * ∑ x, a x * ∏ i : Fin 3, c (v i) x := by
      unfold cubicPairing
      rw [map_sum, Finset.mul_sum]
      apply Finset.sum_congr rfl
      intro x _
      simp [hp', c, φ, map_prod, mul_assoc]
    intro hz
    have : f (b : P) * (∑ x, a x * ∏ i : Fin 3, c (v i) x) = 0 := by
      rw [← heq, hz, map_zero]
    exact (mul_ne_zero hbne hav) this

theorem alignmentObstruction_eval_eq_zero
    {J X F : Type*} [Fintype J] [Fintype X] [Field F] [CharP F 2]
    (t : X → J → ZMod 2)
    (p : X → MvPolynomial (Fin 3 × J) (ZMod 2))
    (hann : ∀ ij, ∑ x, p x * universalAlignmentColumn t ij x = 0)
    (ρ : MvPolynomial (Fin 3 × J) (ZMod 2) →+* F)
    (c : X → F) (a : (Fin 3 × J) → F)
    (hc : ∀ x, c x ^ 2 = ρ (∏ i, universalLinearForm (t x) i))
    (ha : ∀ x, c x = ∑ ij, ρ (universalAlignmentColumn t ij x) * a ij) :
    ρ (alignmentObstruction t p) = 0 := by
  classical
  have hann' (ij : Fin 3 × J) :
      (∑ x, ρ (p x) * ρ (universalAlignmentColumn t ij x)) = 0 := by
    have h := congrArg ρ (hann ij)
    simpa only [map_sum, map_mul, map_zero] using h
  have hpair : (∑ x, ρ (p x) * c x) = 0 := by
    simp_rw [ha, Finset.mul_sum]
    rw [Finset.sum_comm]
    apply Finset.sum_eq_zero
    intro ij _
    simp only [← mul_assoc]
    rw [← Finset.sum_mul, hann', zero_mul]
  calc
    ρ (alignmentObstruction t p) = ∑ x, ρ (p x) ^ 2 *
        ρ (∏ i, universalLinearForm (t x) i) := by
          simp only [alignmentObstruction, map_sum, map_mul, map_pow]
    _ = (∑ x, ρ (p x) * c x) ^ 2 := by
      rw [CharTwo.sum_sq]
      apply Finset.sum_congr rfl
      intro x _
      rw [mul_pow, hc]
    _ = 0 := by rw [hpair, zero_pow (by decide)]

theorem exists_nonzero_alignment_obstruction
    {J X : Type*} [Fintype J] [Fintype X]
    (t : X → J → ZMod 2) (ht : Function.Injective t)
    (hnonzero : ∀ x, t x ≠ 0) (hcard : 3 * Fintype.card J < Fintype.card X) :
    ∃ P : MvPolynomial (Fin 3 × J) (ZMod 2), P ≠ 0 ∧
      ∀ (F : Type*) [Field F] [CharP F 2]
        (ρ : MvPolynomial (Fin 3 × J) (ZMod 2) →+* F)
        (c : X → F) (a : (Fin 3 × J) → F),
        (∀ x, c x ^ 2 = ρ (∏ i, universalLinearForm (t x) i)) →
        (∀ x, c x = ∑ ij, ρ (universalAlignmentColumn t ij x) * a ij) →
        ρ P = 0 := by
  obtain ⟨p, hann, v, hv⟩ := exists_polynomial_alignment_annihilator t ht hnonzero hcard
  refine ⟨alignmentObstruction t p, alignmentObstruction_ne_zero t p v hv, ?_⟩
  intro F _ _ ρ c a hc ha
  exact alignmentObstruction_eval_eq_zero t p hann ρ c a hc ha

end DFVSGames.Quadratic

namespace DFVSGames.Quadratic

open Module
open scoped BigOperators

noncomputable section

variable {F J : Type*} [Field F] [CharP F 2] [Algebra (ZMod 2) F] [Fintype J]

def parameterMap (x : (Fin 3 × J) → F) : (J → ZMod 2) →ₗ[ZMod 2] (Fin 3 → F) :=
  (Pi.basisFun (ZMod 2) J).constr (ZMod 2) (fun j i => x (i, j))

omit [CharP F 2] in
theorem parameterMap_apply (x : (Fin 3 × J) → F) (t : J → ZMod 2) (i : Fin 3) :
    parameterMap x t i = ∑ j, algebraMap (ZMod 2) F (t j) * x (i, j) := by
  classical
  simp [parameterMap, Basis.constr_apply_fintype, Algebra.smul_def]

omit [CharP F 2] in
theorem parameterMap_basis (x : (Fin 3 × J) → F) (j : J) (i : Fin 3) :
    parameterMap x ((Pi.basisFun (ZMod 2) J) j) i = x (i, j) := by
  exact congrFun ((Pi.basisFun (ZMod 2) J).constr_basis (ZMod 2)
    (fun j i => x (i, j)) j) i

omit [CharP F 2] in

theorem eval_universalLinearForm (x : (Fin 3 × J) → F) (t : J → ZMod 2) (i : Fin 3) :
    MvPolynomial.eval₂ (algebraMap (ZMod 2) F) x (universalLinearForm t i) =
      parameterMap x t i := by
  rw [parameterMap_apply]
  simp [universalLinearForm]

omit [CharP F 2] in
theorem dot_parameter_lift (x : (Fin 3 × J) → F)
    (g : (J → ZMod 2) →ₗ[ZMod 2] (Fin 3 → F)) (t : J → ZMod 2) :
    (∑ i, parameterMap x t i * g t i) =
      ∑ ij : Fin 3 × J,
        MvPolynomial.eval₂ (algebraMap (ZMod 2) F) x
          (universalAlignmentColumn (fun t : J → ZMod 2 => t) ij t) *
          g ((Pi.basisFun (ZMod 2) J) ij.2) ij.1 := by
  have h := dot_lift_eq_alignment_mulVec (Pi.basisFun (ZMod 2) J)
    (fun t : J → ZMod 2 => t) (parameterMap x) g t
  simpa [Matrix.mulVec, dotProduct, alignmentMatrix, Pi.basisFun_repr,
    universalAlignmentColumn, eval_universalLinearForm] using h

variable [Fintype F]

theorem alignmentRightHandSide_sq (x : (Fin 3 × J) → F) (t : J → ZMod 2) :
    alignmentRightHandSide (parameterMap x t) ^ 2 =
      MvPolynomial.eval₂ (algebraMap (ZMod 2) F) x
        (∏ i : Fin 3, universalLinearForm t i) := by
  simp [alignmentRightHandSide, Fin.prod_univ_succ, eval_universalLinearForm, mul_assoc]

end

end DFVSGames.Quadratic

namespace DFVSGames.Quadratic

open scoped BigOperators

noncomputable section

theorem exists_all_lifts_alignment_polynomial (J : Type*) [Fintype J] [DecidableEq J] :
    ∃ P : MvPolynomial (Fin 3 × J) (ZMod 2), P ≠ 0 ∧
      ∀ (F : Type*) [Field F] [Fintype F] [CharP F 2] [Algebra (ZMod 2) F]
        (x : (Fin 3 × J) → F),
        MvPolynomial.eval₂ (algebraMap (ZMod 2) F) x P ≠ 0 →
        ∀ g : (J → ZMod 2) →ₗ[ZMod 2] (Fin 3 → F),
          (alignmentEvent (R := ZMod 2) (parameterMap x) g
            (fun t => alignmentRightHandSide (parameterMap x t))).card ≤
              3 * Fintype.card J := by
  classical
  let V := J → ZMod 2
  let Esets := {E : Finset V // (0 : V) ∉ E ∧ E.card = 3 * Fintype.card J + 1}
  have hex (E : Esets) := exists_nonzero_alignment_obstruction
    (fun t : E.val => (t : V)) Subtype.val_injective
    (fun t : E.val => by
      intro hz
      apply E.property.1
      simpa only [hz] using t.property)
    (show 3 * Fintype.card J < Fintype.card E.val by
      simpa only [Fintype.card_coe, E.property.2] using
        Nat.lt_succ_self (3 * Fintype.card J))
  choose p hp hvanish using hex
  let P : MvPolynomial (Fin 3 × J) (ZMod 2) := ∏ E : Esets, p E
  refine ⟨P, Finset.prod_ne_zero_iff.mpr (fun E _ => hp E), ?_⟩
  intro F _ _ _ _ x hP g
  by_contra hbad
  have hlarge : 3 * Fintype.card J + 1 ≤
      (alignmentEvent (R := ZMod 2) (parameterMap x) g
        (fun t => alignmentRightHandSide (parameterMap x t))).card := by omega
  obtain ⟨E, hsub, hcard⟩ := Finset.exists_subset_card_eq hlarge
  have hzero : (0 : V) ∉ E := by
    intro hz
    have hmem := hsub hz
    simp only [alignmentEvent, Finset.mem_filter, Finset.mem_univ, true_and] at hmem
    exact hmem.1 rfl
  let E' : Esets := ⟨E, hzero, hcard⟩
  let ρ : MvPolynomial (Fin 3 × J) (ZMod 2) →+* F :=
    MvPolynomial.eval₂Hom (algebraMap (ZMod 2) F) x
  let c : E → F := fun t => alignmentRightHandSide (parameterMap x (t : V))
  let a : (Fin 3 × J) → F :=
    fun ij => g ((Pi.basisFun (ZMod 2) J) ij.2) ij.1
  have hc (t : E) :
      c t ^ 2 = ρ (∏ i : Fin 3, universalLinearForm (t : V) i) := by
    exact alignmentRightHandSide_sq x t
  have ha (t : E) : c t = ∑ ij, ρ
      (universalAlignmentColumn (fun t : E => (t : V)) ij t) * a ij := by
    have hmem := hsub t.property
    simp only [alignmentEvent, Finset.mem_filter, Finset.mem_univ, true_and] at hmem
    have ht := hmem.2
    calc
      c t = ∑ i, parameterMap x (t : V) i * g (t : V) i := ht.symm
      _ = _ := dot_parameter_lift x g t
  have hz : ρ (p E') = 0 := hvanish E' F ρ c a hc ha
  apply hP
  change ρ P = 0
  simp only [P, map_prod]
  exact Finset.prod_eq_zero (Finset.mem_univ E') hz

end

end DFVSGames.Quadratic

namespace DFVSGames.Quadratic

open MvPolynomial

theorem exists_binary_coordinate_one {J : Type*} (t : J → ZMod 2) (ht : t ≠ 0) :
    ∃ j, t j = 1 := by
  obtain ⟨j, hj⟩ := Function.ne_iff.mp ht
  exact ⟨j, (binary_eq_zero_or_one (t j)).resolve_left hj⟩

theorem exists_binary_minor_one {J : Type*} (t u : J → ZMod 2)
    (ht : t ≠ 0) (hu : u ≠ 0) (htu : t ≠ u) :
    ∃ j k, t j * u k + t k * u j = 1 := by
  obtain ⟨j, hj⟩ := exists_binary_coordinate_one t ht
  by_cases huj : u j = 0
  · obtain ⟨k, hk⟩ := exists_binary_coordinate_one u hu
    exact ⟨j, k, by simp [hj, huj, hk]⟩
  · have huj' : u j = 1 := (binary_eq_zero_or_one (u j)).resolve_left huj
    obtain ⟨k, hk⟩ := Function.ne_iff.mp htu
    refine ⟨j, k, ?_⟩
    rcases binary_eq_zero_or_one (t k) with htk | htk <;>
      rcases binary_eq_zero_or_one (u k) with huk | huk
    · exact (hk (htk.trans huk.symm)).elim
    · simp [hj, huj', htk, huk]
    · simp [hj, huj', htk, huk]
    · exact (hk (htk.trans huk.symm)).elim

variable {J : Type*} [Fintype J]

noncomputable def selectorAssignment (j k : J) : (Fin 3 × J) → ZMod 2 := by
  classical
  exact fun x => if x.1 = 0 then (if x.2 = j then 1 else 0)
    else if x.1 = 1 then (if x.2 = k then 1 else 0) else 0

theorem eval_universalLinearForm_selector_zero (t : J → ZMod 2) (j k : J) :
    eval (selectorAssignment j k) (universalLinearForm t 0) = t j := by
  classical
  simp [universalLinearForm, selectorAssignment]

theorem eval_universalLinearForm_selector_one (t : J → ZMod 2) (j k : J) :
    eval (selectorAssignment j k) (universalLinearForm t 1) = t k := by
  classical
  simp [universalLinearForm, selectorAssignment]

theorem universalLinearForm_zero_ne_zero (t : J → ZMod 2) (ht : t ≠ 0) :
    universalLinearForm t 0 ≠ 0 := by
  obtain ⟨j, hj⟩ := exists_binary_coordinate_one t ht
  intro h
  have hc := congrArg (eval (selectorAssignment j j)) h
  rw [eval_universalLinearForm_selector_zero, map_zero, hj] at hc
  exact one_ne_zero hc

noncomputable def universalPairMinor (t u : J → ZMod 2) :
    MvPolynomial (Fin 3 × J) (ZMod 2) :=
  universalLinearForm t 0 * universalLinearForm u 1 +
    universalLinearForm t 1 * universalLinearForm u 0

theorem universalPairMinor_ne_zero (t u : J → ZMod 2)
    (ht : t ≠ 0) (hu : u ≠ 0) (htu : t ≠ u) :
    universalPairMinor t u ≠ 0 := by
  obtain ⟨j, k, hjk⟩ := exists_binary_minor_one t u ht hu htu
  intro h
  have hc := congrArg (eval (selectorAssignment j k)) h
  simp only [universalPairMinor, map_add, map_mul,
    eval_universalLinearForm_selector_zero, eval_universalLinearForm_selector_one,
    map_zero] at hc
  rw [hjk] at hc
  exact one_ne_zero hc

theorem universalPairMinor_eq_zero_of_proportional
    {F : Type*} [Field F] [CharP F 2]
    (t u : J → ZMod 2)
    (ρ : MvPolynomial (Fin 3 × J) (ZMod 2) →+* F) (a : F)
    (h : ∀ i, ρ (universalLinearForm t i) = a * ρ (universalLinearForm u i)) :
    ρ (universalPairMinor t u) = 0 := by
  simp only [universalPairMinor, map_add, map_mul, h]
  simpa only [mul_assoc, mul_left_comm, mul_comm] using
    (CharTwo.add_self_eq_zero
      (a * ρ (universalLinearForm u 0) * ρ (universalLinearForm u 1)))

end DFVSGames.Quadratic

namespace DFVSGames.Quadratic

open MvPolynomial
open scoped BigOperators

noncomputable section

variable (J : Type*) [Fintype J]

def nonzeroBinaryInputs : Finset (J → ZMod 2) := by
  classical
  exact Finset.univ.erase 0

def distinctNonzeroBinaryPairs : Finset ((J → ZMod 2) × (J → ZMod 2)) := by
  classical
  exact Finset.univ.filter (fun tu => tu.1 ≠ 0 ∧ tu.2 ≠ 0 ∧ tu.1 ≠ tu.2)

def separationPolynomial : MvPolynomial (Fin 3 × J) (ZMod 2) :=
  (∏ t ∈ nonzeroBinaryInputs J, universalLinearForm t 0) *
    ∏ tu ∈ distinctNonzeroBinaryPairs J, universalPairMinor tu.1 tu.2

theorem separationPolynomial_ne_zero : separationPolynomial J ≠ 0 := by
  classical
  apply mul_ne_zero
  · apply Finset.prod_ne_zero_iff.mpr
    intro t ht
    exact universalLinearForm_zero_ne_zero t
      (Finset.mem_erase.mp ht).1
  · apply Finset.prod_ne_zero_iff.mpr
    intro tu htu
    have h := (Finset.mem_filter.mp htu).2
    exact universalPairMinor_ne_zero tu.1 tu.2 h.1 h.2.1 h.2.2

variable {J}

theorem parameterMap_separated_of_eval_ne_zero
    {F : Type*} [Field F] [CharP F 2] [Algebra (ZMod 2) F]
    (x : (Fin 3 × J) → F)
    (hx : eval₂ (algebraMap (ZMod 2) F) x (separationPolynomial J) ≠ 0) :
    Function.Injective (parameterMap x) ∧
      ∀ t u : J → ZMod 2, t ≠ 0 → u ≠ 0 →
        (∃ a : F, parameterMap x t = a • parameterMap x u) → t = u := by
  classical
  let ρ : MvPolynomial (Fin 3 × J) (ZMod 2) →+* F :=
    eval₂Hom (algebraMap (ZMod 2) F) x
  have hprod :
      (∏ t ∈ nonzeroBinaryInputs J, ρ (universalLinearForm t 0)) *
        (∏ tu ∈ distinctNonzeroBinaryPairs J, ρ (universalPairMinor tu.1 tu.2)) ≠ 0 := by
    change ρ (separationPolynomial J) ≠ 0 at hx
    simpa only [separationPolynomial, map_mul, map_prod] using hx
  have hlinear := (mul_ne_zero_iff.mp hprod).1
  have hminor := (mul_ne_zero_iff.mp hprod).2
  have hnonzero (t : J → ZMod 2) (ht : t ≠ 0) : parameterMap x t 0 ≠ 0 := by
    have hmem : t ∈ nonzeroBinaryInputs J := by
      simp [nonzeroBinaryInputs, ht]
    have h := Finset.prod_ne_zero_iff.mp hlinear t hmem
    change eval₂ (algebraMap (ZMod 2) F) x (universalLinearForm t 0) ≠ 0 at h
    rwa [eval_universalLinearForm] at h
  constructor
  · intro t u htu
    by_contra hne
    have hz : parameterMap x (t - u) = 0 := by
      rw [map_sub, htu, sub_self]
    exact hnonzero (t - u) (sub_ne_zero.mpr hne) (congrFun hz 0)
  · intro t u ht hu hprop
    by_contra hne
    have hmem : (t, u) ∈ distinctNonzeroBinaryPairs J := by
      simp [distinctNonzeroBinaryPairs, ht, hu, hne]
    have hnz := Finset.prod_ne_zero_iff.mp hminor (t, u) hmem
    obtain ⟨a, ha⟩ := hprop
    apply hnz
    apply universalPairMinor_eq_zero_of_proportional t u ρ a
    intro i
    change eval₂ (algebraMap (ZMod 2) F) x (universalLinearForm t i) =
      a * eval₂ (algebraMap (ZMod 2) F) x (universalLinearForm u i)
    rw [eval_universalLinearForm, eval_universalLinearForm]
    simpa only [Pi.smul_apply, smul_eq_mul] using congrFun ha i

end

end DFVSGames.Quadratic

end OAI