Magic Square Proof Attempt

proof.lean

-- This module serves as the root of the `MagicSquare` library.
-- Import modules here that should be built as part of the library.
import Mathlib.Data.ZMod.Basic
import Mathlib.Data.Int.Basic
import Mathlib.Tactic.Ring.RingNF
import Mathlib.Algebra.Group.Defs
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Linarith.Frontend
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Complex.Norm
import Mathlib.Analysis.Complex.Arg
import Mathlib.Data.Real.Sqrt
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex

import MagicSquare.Lemmas
open MagicSquare.Lemmas (angle_mod_basic)

-- Gaussian Integer logic
structure GaussianInt where
  re : ℤ -- Real part
  im : ℤ -- Imaginary part
  deriving DecidableEq, Repr

instance : CoeSort GaussianInt (ℤ × ℤ) where
  coe := fun ⟨a, b⟩ => (a, b)

instance : Add GaussianInt where
  add a b := ⟨a.re + b.re, a.im + b.im⟩

instance : Mul GaussianInt where
  mul a b := ⟨a.re * b.re - a.im * b.im, a.re * b.im + a.im * b.re⟩

instance : Neg GaussianInt where
  neg a := ⟨-a.re, -a.im⟩

def GaussianInt.norm (α : GaussianInt) : ℤ :=
  α.re ^ 2 + α.im ^ 2

def GaussianInt.conj (a : GaussianInt) : GaussianInt := ⟨a.re, -a.im⟩

lemma GaussianInt.norm_nonneg (α : GaussianInt) : 0 ≤ α.norm := by
  cases' α with a b
  have h₁ : 0 ≤ a ^ 2 := by apply sq_nonneg
  have h₂ : 0 ≤ b ^ 2 := by apply sq_nonneg
  exact add_nonneg h₁ h₂

instance : HPow GaussianInt ℕ GaussianInt where
  hPow := fun a n =>
    Nat.recOn n
      (⟨1, 0⟩ : GaussianInt)
      (fun _ ih => a * ih)

@[simp]
lemma hPow_zero (a : GaussianInt) : a ^ (0 : ℕ) = ⟨1, 0⟩ := by
  -- `Nat.recOn 0 _ _` reduces directly to the base case
  rfl

@[simp]
lemma hPow_succ (a : GaussianInt) (n : ℕ) :
    a ^ (n + 1) = a * a ^ n := by
  -- `Nat.recOn (n+1) _ _` reduces to the “step” case
  rfl

@[simp] lemma mul_re (a b : GaussianInt) : (a * b).re = a.re * b.re - a.im * b.im := rfl
@[simp] lemma mul_im (a b : GaussianInt) : (a * b).im = a.re * b.im + a.im * b.re := rfl

-- Basic norm properties
theorem gnorm_mul (α β : GaussianInt) :
  (α * β).norm = α.norm * β.norm := by
  -- Expand definitions and simplify using algebraic properties
  simp [GaussianInt.norm, mul_re, mul_im, sq]
  ring  -- Simplify the expression using ring operations

lemma norm_eq_sq_mag (α : GaussianInt) :
  α.norm = (α.re ^ 2 + α.im ^ 2) := rfl

lemma sq_eq_sq_of_nonneg {a b : ℤ} (ha : 0 ≤ a) (hb : 0 ≤ b) (h : a ^ 2 = b ^ 2) : a = b := by
  nlinarith

-- Check if pairs p and q are distinct
-- modulo order and sign
def pairwiseDistinct (p q : ℤ × ℤ) : Prop :=
 let (a, b) := p
 let (c, d) := q
 ¬ ((a = c ∧ b = d) ∨            -- identity
    (a = d ∧ b = c) ∨            -- swap
    (a = -c ∧ b = -d) ∨          -- negate both
    (a = -d ∧ b = -c) ∨          -- swap + negate both
    (a = c ∧ b = -d) ∨           -- negate second
    (a = -c ∧ b = d) ∨           -- negate first
    (a = -d ∧ b = c) ∨           -- swap + negate first
    (a = d ∧ b = -c))            -- swap + negate second

def pairwiseEquivalent (p q : GaussianInt) : Prop :=
  ¬ pairwiseDistinct p q


theorem main_theorem : ¬(∃ (m n p q u v : ℤ),
 (m^2 + n^2)^2 = (p^2 + q^2)^2 ∧ (p^2 + q^2)^2 = (u^2 + v^2)^2 ∧
 4 * m * n * (m^2 - n^2) + 4 * p * q * (p^2 - q^2) + 4 * u * v * (u^2 - v^2) = 0 ∧
 pairwiseDistinct (m, n) (p, q) ∧
 pairwiseDistinct (p, q) (u, v) ∧
 pairwiseDistinct (m, n) (u, v)
):= by
  rintro ⟨m, n, p, q, r, s, h₁, h₂, h₃, h₄, h₅, h₆⟩

  -- Translate into ℤ[i]
  let α : GaussianInt := ⟨m, n⟩
  let β : GaussianInt := ⟨p, q⟩
  let γ : GaussianInt := ⟨r, s⟩
  have h₁' : α.norm = β.norm := by
    apply sq_eq_sq_of_nonneg (GaussianInt.norm_nonneg α) (GaussianInt.norm_nonneg β)
    exact h₁
  have h₂' : β.norm = γ.norm := by
    apply sq_eq_sq_of_nonneg (GaussianInt.norm_nonneg β) (GaussianInt.norm_nonneg γ)
    exact h₂
  have h_norm : α.norm = β.norm ∧ β.norm = γ.norm := ⟨h₁', h₂'⟩
  have h_imag : (α^4).im + (β^4).im + (γ^4).im = 0 := by
    simp [hPow_succ, mul_im, mul_re, h₃]
    linear_combination h₃

  have h_norm_val : α.norm = β.norm := h_norm.1
  have h_norm_val' : β.norm = γ.norm := h_norm.2
  let t := α.norm

  -- Prove that α β and γ are on the unit circle
  let α_unit : ℂ := ⟨α.re / Real.sqrt t, α.im / Real.sqrt t⟩
  let β_unit : ℂ := ⟨β.re / Real.sqrt t, β.im / Real.sqrt t⟩
  let γ_unit : ℂ := ⟨γ.re / Real.sqrt t, γ.im / Real.sqrt t⟩

  have h_unit_norm : ‖α_unit‖ = 1 ∧ ‖β_unit‖ = 1 ∧ ‖γ_unit‖ = 1 := by
    have ht : (0 : ℝ) < t := by
      sorry

    apply And.intro
    . -- proof of α
      sorry
    . apply And.intro
      . -- proof of β
        sorry
      . -- proof of γ
        sorry

  -- Define angles from normalized complex numbers
  let θ_α := Complex.arg α_unit
  let θ_β := Complex.arg β_unit
  let θ_γ := Complex.arg γ_unit

  -- Express imaginary parts using sine
  have h_sin_sum : Real.sin (4 * θ_α) + Real.sin (4 * θ_β) + Real.sin (4 * θ_γ) = 0 := by
    exact sorry

  have h_angle_symm : ∃ (φ ψ : ℝ),
      Real.sin φ + Real.sin ψ + Real.sin (-(φ + ψ)) = 0 := by
    sorry
  obtain ⟨φ, ψ, hφψ⟩ := h_angle_symm

  -- Conclude two angles must be negatives modulo 2π
  -- have h_angle_mod :
  --   (∃ k : ℤ, φ + ψ = (k : ℝ) * Real.pi) ∨
  --   (∃ k : ℤ, φ = -ψ + (k : ℝ) * 2 * Real.pi) := by sorry

  have h_angle_mod :
      (∃ k : ℤ, φ + ψ = (k : ℝ) * Real.pi) ∨
      (∃ k : ℤ, φ = (k : ℝ) * 2 * Real.pi) ∨
      (∃ k : ℤ, ψ = (k : ℝ) * 2 * Real.pi) := by
      sorry

  -- Connect angles to Gaussian integer equivalence
  have h_equiv : α = β.conj ∨ β = γ.conj ∨ γ = α.conj := by
    sorry

  rcases h_equiv with hαβ | hβγ | hγα
  · -- α equals the conjugate of β  →  contradicts h₄
    have h_re : m = p := by
      simpa [α, β, GaussianInt.conj] using congrArg GaussianInt.re hαβ
    have h_im : n = -q := by
      simpa [α, β, GaussianInt.conj] using congrArg GaussianInt.im hαβ
    have : False := by
      -- after substituting `h_re`, `h_im`, the disjunction in `pairwiseDistinct`
      -- is true, so `h₄ : ¬ True`, hence `h₄ trivial` is a contradiction
      have hcontra : True → False := by
        simp [pairwiseDistinct, h_re, h_im] at h₄
      exact hcontra trivial
    exact this
  · -- β equals the conjugate of γ  →  contradicts h₅
    have h_re : p = r := by
      simpa [β, γ, GaussianInt.conj] using congrArg GaussianInt.re hβγ
    have h_im : q = -s := by
      simpa [β, γ, GaussianInt.conj] using congrArg GaussianInt.im hβγ
    have : False := by
      have hcontra : True → False := by
        simp [pairwiseDistinct, h_re, h_im] at h₅
      exact hcontra trivial
    exact this
  · -- γ equals the conjugate of α  →  contradicts h₆
    have h_re : r = m := by
      simpa [γ, α, GaussianInt.conj] using congrArg GaussianInt.re hγα
    have h_im : s = -n := by
      simpa [γ, α, GaussianInt.conj] using congrArg GaussianInt.im hγα
    have : False := by
      have hcontra : True → False := by
        simp [pairwiseDistinct, h_re, h_im] at h₆
      exact hcontra trivial
    exact this