Datasets:

roseking
/

openAI-math-dataset

import Mathlib.Algebra.CharP.Two
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.ZMod.Basic
import Mathlib.LinearAlgebra.Prod
import Mathlib.LinearAlgebra.Span.Basic
import Mathlib.Tactic.FinCases
import Mathlib.Tactic.Ring

namespace OAI

section

namespace UniqueGamesTheorem.Quadratic

abbrev Vec (F : Type*) := Fin 3 → F

variable {F : Type*} [Field F]

/-- The three-coordinate quadratic block from equation (2.4). -/
def Q (x : Vec F) : Vec F := ![x 1 * x 2, x 0 * x 2, x 0 * x 1]

/-- Its polar cross-terms.  In characteristic two these equal
`Q (x + a) + Q x + Q a`. -/
def D (x a : Vec F) : Vec F :=
  ![x 1 * a 2 + a 1 * x 2, x 0 * a 2 + a 0 * x 2,
    x 0 * a 1 + a 0 * x 1]

/-- Field dot product, with all three coordinates displayed. -/
def dot (v y : Vec F) : F := v 0 * y 0 + v 1 * y 1 + v 2 * y 2

@[simp] theorem Q_zero : Q (0 : Vec F) = 0 := by
  ext i
  fin_cases i <;> simp [Q]

theorem Q_add (x a : Vec F) : Q (x + a) = Q x + Q a + D x a := by
  ext i
  fin_cases i <;> simp [Q, D] <;> ring

@[simp] theorem D_zero_left (a : Vec F) : D 0 a = 0 := by
  ext i
  fin_cases i <;> simp [D]

@[simp] theorem D_zero_right (x : Vec F) : D x 0 = 0 := by
  ext i
  fin_cases i <;> simp [D]

theorem D_comm (x a : Vec F) : D x a = D a x := by
  ext i
  fin_cases i <;> simp [D, add_comm]

theorem D_add_left (x y a : Vec F) : D (x + y) a = D x a + D y a := by
  ext i
  fin_cases i <;> simp [D] <;> ring

theorem D_add_right (x a b : Vec F) : D x (a + b) = D x a + D x b := by
  rw [D_comm, D_add_left, D_comm a x, D_comm b x]

theorem D_smul_left (t : F) (x a : Vec F) : D (t • x) a = t • D x a := by
  ext i
  fin_cases i <;> simp [D, smul_eq_mul] <;> ring

theorem D_smul_right (t : F) (x a : Vec F) : D x (t • a) = t • D x a := by
  rw [D_comm, D_smul_left, D_comm a x]

@[simp] theorem dot_zero_right (v : Vec F) : dot v 0 = 0 := by simp [dot]
@[simp] theorem dot_zero_left (y : Vec F) : dot 0 y = 0 := by simp [dot]

theorem dot_add_right (v y z : Vec F) : dot v (y + z) = dot v y + dot v z := by
  simp [dot]
  ring

theorem dot_smul_right (v y : Vec F) (t : F) : dot v (t • y) = t * dot v y := by
  simp [dot, smul_eq_mul]
  ring

theorem dot_smul_left (v y : Vec F) (t : F) : dot (t • v) y = t * dot v y := by
  simp [dot, smul_eq_mul]
  ring

/-- The field line generated by `v`. -/
def line (v : Vec F) : Submodule F (Vec F) := Submodule.span F {v}

/-- The field hyperplane perpendicular to the generator of the line. -/
def hyperplane (v : Vec F) : Submodule F (Vec F) where
  carrier := {y | dot v y = 0}
  zero_mem' := dot_zero_right v
  add_mem' := by
    intro y z hy hz
    rw [Set.mem_ofPred_eq, dot_add_right, hy, hz, add_zero]
  smul_mem' := by
    intro t y hy
    rw [Set.mem_ofPred_eq, dot_smul_right, hy, mul_zero]

@[simp] theorem mem_hyperplane (v y : Vec F) : y ∈ hyperplane v ↔ dot v y = 0 := Iff.rfl

theorem mem_line_iff (v a : Vec F) : a ∈ line v ↔ ∃ t : F, t • v = a :=
  Submodule.mem_span_singleton

section CharacteristicTwo

variable [CharP F 2]

@[simp] theorem vec_add_self (x : Vec F) : x + x = 0 := by
  ext i
  exact CharTwo.add_self_eq_zero (x i)

theorem D_eq_polar (x a : Vec F) : D x a = Q (x + a) + Q x + Q a := by
  rw [Q_add]
  have h : Q x + Q a + D x a + Q x + Q a =
      D x a + (Q x + Q x) + (Q a + Q a) := by ac_rfl
  rw [h, vec_add_self, vec_add_self, add_zero, add_zero]

private theorem cross_self_zero (u w s t : F) :
    (t * u) * (s * w) + (s * u) * (t * w) = 0 := by
  calc
    _ = (t * s * u * w) + (t * s * u * w) := by ring
    _ = 0 := CharTwo.add_self_eq_zero _

/-- The polar expression vanishes on any pair of vectors in the same field line. -/
theorem D_smul_smul_zero (v : Vec F) (t s : F) : D (t • v) (s • v) = 0 := by
  ext i
  fin_cases i <;> simp [D, smul_eq_mul, cross_self_zero]

/-- Every cross-term arising from a shift along `Fv` is perpendicular to `v`. -/
theorem dot_D_smul (v x : Vec F) (t : F) : dot v (D x (t • v)) = 0 := by
  change v 0 * (x 1 * (t * v 2) + (t * v 1) * x 2) +
      v 1 * (x 0 * (t * v 2) + (t * v 0) * x 2) +
      v 2 * (x 0 * (t * v 1) + (t * v 0) * x 1) = 0
  calc
    _ = (t * (v 0 * v 2 * x 1 + v 0 * v 1 * x 2 + v 1 * v 2 * x 0)) +
        (t * (v 0 * v 2 * x 1 + v 0 * v 1 * x 2 + v 1 * v 2 * x 0)) := by ring
    _ = 0 := CharTwo.add_self_eq_zero _

theorem D_mem_hyperplane (v x a : Vec F) (ha : a ∈ line v) :
    D x a ∈ hyperplane v := by
  obtain ⟨t, rfl⟩ := (mem_line_iff v a).mp ha
  exact dot_D_smul v x t

theorem D_eq_zero_of_mem_line (v a b : Vec F)
    (ha : a ∈ line v) (hb : b ∈ line v) : D a b = 0 := by
  obtain ⟨t, rfl⟩ := (mem_line_iff v a).mp ha
  obtain ⟨s, rfl⟩ := (mem_line_iff v b).mp hb
  exact D_smul_smul_zero v t s

/-- The restriction of the quadratic map to one field line is additive. -/
theorem Q_add_of_mem_line (v a b : Vec F)
    (ha : a ∈ line v) (hb : b ∈ line v) : Q (a + b) = Q a + Q b := by
  rw [Q_add, D_eq_zero_of_mem_line v a b ha hb, add_zero]

variable [Algebra (ZMod 2) F]

/-- The block space `U_A`, as an actual binary submodule.  Membership uses
`y + Q a` because subtraction equals addition in characteristic two. -/
def U (v : Vec F) : Submodule (ZMod 2) (Vec F × Vec F) where
  carrier := {p | p.1 ∈ line v ∧ dot v (p.2 + Q p.1) = 0}
  zero_mem' := by simp [Q_zero]
  add_mem' := by
    intro p q hp hq
    refine ⟨(line v).add_mem hp.1 hq.1, ?_⟩
    change dot v ((p.2 + q.2) + Q (p.1 + q.1)) = 0
    rw [Q_add_of_mem_line v p.1 q.1 hp.1 hq.1]
    have h : (p.2 + q.2) + (Q p.1 + Q q.1) =
        (p.2 + Q p.1) + (q.2 + Q q.1) := by ac_rfl
    rw [h, dot_add_right, hp.2, hq.2, add_zero]
  smul_mem' := by
    intro c p hp
    have hc : c = 0 ∨ c = 1 := by
      fin_cases c
      · exact Or.inl rfl
      · exact Or.inr rfl
    rcases hc with rfl | rfl
    · simp [Q_zero]
    · simpa using hp

@[simp] theorem mem_U (v : Vec F) (p : Vec F × Vec F) :
    p ∈ U v ↔ p.1 ∈ line v ∧ dot v (p.2 + Q p.1) = 0 := Iff.rfl

theorem pair_mem_U (v a w : Vec F) (ha : a ∈ line v) (hw : w ∈ hyperplane v) :
    (a, Q a + w) ∈ U v := by
  refine ⟨ha, ?_⟩
  change dot v (Q a + w + Q a) = 0
  have h : Q a + w + Q a = w + (Q a + Q a) := by ac_rfl
  simpa [h] using hw

theorem mem_U_iff_exists (v : Vec F) (p : Vec F × Vec F) :
    p ∈ U v ↔ ∃ a ∈ line v, ∃ w ∈ hyperplane v, p = (a, Q a + w) := by
  constructor
  · intro hp
    refine ⟨p.1, hp.1, p.2 + Q p.1, hp.2, ?_⟩
    apply Prod.ext
    · rfl
    · change p.2 = Q p.1 + (p.2 + Q p.1)
      have h : Q p.1 + (p.2 + Q p.1) = p.2 + (Q p.1 + Q p.1) := by ac_rfl
      rw [h, vec_add_self, add_zero]
  · rintro ⟨a, ha, w, hw, rfl⟩
    exact pair_mem_U v a w ha hw

end CharacteristicTwo
end UniqueGamesTheorem.Quadratic

end

end OAI

	import Mathlib.Algebra.CharP.Two
	import Mathlib.Data.Fin.VecNotation
	import Mathlib.Data.ZMod.Basic
	import Mathlib.LinearAlgebra.Prod
	import Mathlib.LinearAlgebra.Span.Basic
	import Mathlib.Tactic.FinCases
	import Mathlib.Tactic.Ring

	namespace OAI

	section

	namespace UniqueGamesTheorem.Quadratic

	abbrev Vec (F : Type*) := Fin 3 → F

	variable {F : Type*} [Field F]

	/-- The three-coordinate quadratic block from equation (2.4). -/
	def Q (x : Vec F) : Vec F := ![x 1 * x 2, x 0 * x 2, x 0 * x 1]

	/-- Its polar cross-terms. In characteristic two these equal
	`Q (x + a) + Q x + Q a`. -/
	def D (x a : Vec F) : Vec F :=
	![x 1 * a 2 + a 1 * x 2, x 0 * a 2 + a 0 * x 2,
	x 0 * a 1 + a 0 * x 1]

	/-- Field dot product, with all three coordinates displayed. -/
	def dot (v y : Vec F) : F := v 0 * y 0 + v 1 * y 1 + v 2 * y 2

	@[simp] theorem Q_zero : Q (0 : Vec F) = 0 := by
	ext i
	fin_cases i <;> simp [Q]

	theorem Q_add (x a : Vec F) : Q (x + a) = Q x + Q a + D x a := by
	ext i
	fin_cases i <;> simp [Q, D] <;> ring

	@[simp] theorem D_zero_left (a : Vec F) : D 0 a = 0 := by
	ext i
	fin_cases i <;> simp [D]

	@[simp] theorem D_zero_right (x : Vec F) : D x 0 = 0 := by
	ext i
	fin_cases i <;> simp [D]

	theorem D_comm (x a : Vec F) : D x a = D a x := by
	ext i
	fin_cases i <;> simp [D, add_comm]

	theorem D_add_left (x y a : Vec F) : D (x + y) a = D x a + D y a := by
	ext i
	fin_cases i <;> simp [D] <;> ring

	theorem D_add_right (x a b : Vec F) : D x (a + b) = D x a + D x b := by
	rw [D_comm, D_add_left, D_comm a x, D_comm b x]

	theorem D_smul_left (t : F) (x a : Vec F) : D (t • x) a = t • D x a := by
	ext i
	fin_cases i <;> simp [D, smul_eq_mul] <;> ring

	theorem D_smul_right (t : F) (x a : Vec F) : D x (t • a) = t • D x a := by
	rw [D_comm, D_smul_left, D_comm a x]

	@[simp] theorem dot_zero_right (v : Vec F) : dot v 0 = 0 := by simp [dot]
	@[simp] theorem dot_zero_left (y : Vec F) : dot 0 y = 0 := by simp [dot]

	theorem dot_add_right (v y z : Vec F) : dot v (y + z) = dot v y + dot v z := by
	simp [dot]
	ring

	theorem dot_smul_right (v y : Vec F) (t : F) : dot v (t • y) = t * dot v y := by
	simp [dot, smul_eq_mul]
	ring

	theorem dot_smul_left (v y : Vec F) (t : F) : dot (t • v) y = t * dot v y := by
	simp [dot, smul_eq_mul]
	ring

	/-- The field line generated by `v`. -/
	def line (v : Vec F) : Submodule F (Vec F) := Submodule.span F {v}

	/-- The field hyperplane perpendicular to the generator of the line. -/
	def hyperplane (v : Vec F) : Submodule F (Vec F) where
	carrier := {y \| dot v y = 0}
	zero_mem' := dot_zero_right v
	add_mem' := by
	intro y z hy hz
	rw [Set.mem_ofPred_eq, dot_add_right, hy, hz, add_zero]
	smul_mem' := by
	intro t y hy
	rw [Set.mem_ofPred_eq, dot_smul_right, hy, mul_zero]

	@[simp] theorem mem_hyperplane (v y : Vec F) : y ∈ hyperplane v ↔ dot v y = 0 := Iff.rfl

	theorem mem_line_iff (v a : Vec F) : a ∈ line v ↔ ∃ t : F, t • v = a :=
	Submodule.mem_span_singleton

	section CharacteristicTwo

	variable [CharP F 2]

	@[simp] theorem vec_add_self (x : Vec F) : x + x = 0 := by
	ext i
	exact CharTwo.add_self_eq_zero (x i)

	theorem D_eq_polar (x a : Vec F) : D x a = Q (x + a) + Q x + Q a := by
	rw [Q_add]
	have h : Q x + Q a + D x a + Q x + Q a =
	D x a + (Q x + Q x) + (Q a + Q a) := by ac_rfl
	rw [h, vec_add_self, vec_add_self, add_zero, add_zero]

	private theorem cross_self_zero (u w s t : F) :
	(t * u) * (s * w) + (s * u) * (t * w) = 0 := by
	calc
	_ = (t * s * u * w) + (t * s * u * w) := by ring
	_ = 0 := CharTwo.add_self_eq_zero _

	/-- The polar expression vanishes on any pair of vectors in the same field line. -/
	theorem D_smul_smul_zero (v : Vec F) (t s : F) : D (t • v) (s • v) = 0 := by
	ext i
	fin_cases i <;> simp [D, smul_eq_mul, cross_self_zero]

	/-- Every cross-term arising from a shift along `Fv` is perpendicular to `v`. -/
	theorem dot_D_smul (v x : Vec F) (t : F) : dot v (D x (t • v)) = 0 := by
	change v 0 * (x 1 * (t * v 2) + (t * v 1) * x 2) +
	v 1 * (x 0 * (t * v 2) + (t * v 0) * x 2) +
	v 2 * (x 0 * (t * v 1) + (t * v 0) * x 1) = 0
	calc
	_ = (t * (v 0 * v 2 * x 1 + v 0 * v 1 * x 2 + v 1 * v 2 * x 0)) +
	(t * (v 0 * v 2 * x 1 + v 0 * v 1 * x 2 + v 1 * v 2 * x 0)) := by ring
	_ = 0 := CharTwo.add_self_eq_zero _

	theorem D_mem_hyperplane (v x a : Vec F) (ha : a ∈ line v) :
	D x a ∈ hyperplane v := by
	obtain ⟨t, rfl⟩ := (mem_line_iff v a).mp ha
	exact dot_D_smul v x t

	theorem D_eq_zero_of_mem_line (v a b : Vec F)
	(ha : a ∈ line v) (hb : b ∈ line v) : D a b = 0 := by
	obtain ⟨t, rfl⟩ := (mem_line_iff v a).mp ha
	obtain ⟨s, rfl⟩ := (mem_line_iff v b).mp hb
	exact D_smul_smul_zero v t s

	/-- The restriction of the quadratic map to one field line is additive. -/
	theorem Q_add_of_mem_line (v a b : Vec F)
	(ha : a ∈ line v) (hb : b ∈ line v) : Q (a + b) = Q a + Q b := by
	rw [Q_add, D_eq_zero_of_mem_line v a b ha hb, add_zero]

	variable [Algebra (ZMod 2) F]

	/-- The block space `U_A`, as an actual binary submodule. Membership uses
	`y + Q a` because subtraction equals addition in characteristic two. -/
	def U (v : Vec F) : Submodule (ZMod 2) (Vec F × Vec F) where
	carrier := {p \| p.1 ∈ line v ∧ dot v (p.2 + Q p.1) = 0}
	zero_mem' := by simp [Q_zero]
	add_mem' := by
	intro p q hp hq
	refine ⟨(line v).add_mem hp.1 hq.1, ?_⟩
	change dot v ((p.2 + q.2) + Q (p.1 + q.1)) = 0
	rw [Q_add_of_mem_line v p.1 q.1 hp.1 hq.1]
	have h : (p.2 + q.2) + (Q p.1 + Q q.1) =
	(p.2 + Q p.1) + (q.2 + Q q.1) := by ac_rfl
	rw [h, dot_add_right, hp.2, hq.2, add_zero]
	smul_mem' := by
	intro c p hp
	have hc : c = 0 ∨ c = 1 := by
	fin_cases c
	· exact Or.inl rfl
	· exact Or.inr rfl
	rcases hc with rfl \| rfl
	· simp [Q_zero]
	· simpa using hp

	@[simp] theorem mem_U (v : Vec F) (p : Vec F × Vec F) :
	p ∈ U v ↔ p.1 ∈ line v ∧ dot v (p.2 + Q p.1) = 0 := Iff.rfl

	theorem pair_mem_U (v a w : Vec F) (ha : a ∈ line v) (hw : w ∈ hyperplane v) :
	(a, Q a + w) ∈ U v := by
	refine ⟨ha, ?_⟩
	change dot v (Q a + w + Q a) = 0
	have h : Q a + w + Q a = w + (Q a + Q a) := by ac_rfl
	simpa [h] using hw

	theorem mem_U_iff_exists (v : Vec F) (p : Vec F × Vec F) :
	p ∈ U v ↔ ∃ a ∈ line v, ∃ w ∈ hyperplane v, p = (a, Q a + w) := by
	constructor
	· intro hp
	refine ⟨p.1, hp.1, p.2 + Q p.1, hp.2, ?_⟩
	apply Prod.ext
	· rfl
	· change p.2 = Q p.1 + (p.2 + Q p.1)
	have h : Q p.1 + (p.2 + Q p.1) = p.2 + (Q p.1 + Q p.1) := by ac_rfl
	rw [h, vec_add_self, add_zero]
	· rintro ⟨a, ha, w, hw, rfl⟩
	exact pair_mem_U v a w ha hw

	end CharacteristicTwo
	end UniqueGamesTheorem.Quadratic

	end

	end OAI