Module: Mathlib.NumberTheory.SumFourSquares

Goal state before:
pp: case neg
p : ℕ
hp : Prime p
this✝ : Fact (Prime p)
natAbs_iff :
  ∀ {a b c d : ℤ} {k : ℕ},
    a.natAbs ^ 2 + b.natAbs ^ 2 + c.natAbs ^ 2 + d.natAbs ^ 2 = k ↔ a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = ↑k
hm✝ : ∃ m < p, 0 < m ∧ ∃ a b c d, a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = m * p
m : ℕ
hmin : ∀ m_1 < m, ¬(m_1 < p ∧ 0 < m_1 ∧ ∃ a b c d, a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = m_1 * p)
hmp : m < p
hm₀ : 0 < m
a b c d : ℕ
habcd : a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = m * p
hm₁ : 1 < m
this : NeZero m
hm : ¬2 ∣ m
⊢ False

Tactic:
· -- For each `x` in `a`, `b`, `c`, `d`, take a number `f x ≡ x [ZMOD m]` with least possible
    -- absolute value
    obtain ⟨f, hf_lt, hf_mod⟩ :
        ∃ f : ℕ → ℤ, (∀ x, 2 * (f x).natAbs < m) ∧ ∀ x, (f x : ZMod m) = x := by
      refine ⟨fun x ↦ (x : ZMod m).valMinAbs, fun x ↦ ?_, fun x ↦ (x : ZMod m).coe_valMinAbs⟩
      exact (mul_le_mul' le_rfl (x : ZMod m).natAbs_valMinAbs_le).trans_lt
        (Nat.mul_div_lt_iff_not_dvd.2 hm)
    -- Since `|f x| ^ 2 = (f x) ^ 2 ≡ x ^ 2 [ZMOD m]`, we have
    -- `m ∣ |f a| ^ 2 + |f b| ^ 2 + |f c| ^ 2 + |f d| ^ 2`
    obtain ⟨r, hr⟩ :
        m ∣ (f a).natAbs ^ 2 + (f b).natAbs ^ 2 + (f c).natAbs ^ 2 + (f d).natAbs ^ 2 := by
      simp only [← Int.natCast_dvd_natCast, ← ZMod.intCast_zmod_eq_zero_iff_dvd]
      push_cast [hf_mod, sq_abs]
      norm_cast
      simp [habcd]
    -- The quotient `r` is not zero, because otherwise `f a = f b = f c = f d = 0`, hence
    -- `m` divides each `a`, `b`, `c`, `d`, thus `m ∣ p` which is impossible.
    rcases (zero_le r).eq_or_lt with rfl | hr₀
    · replace hr : f a = 0 ∧ f b = 0 ∧ f c = 0 ∧ f d = 0 := by simpa [and_assoc] using hr
      obtain ⟨⟨a, rfl⟩, ⟨b, rfl⟩, ⟨c, rfl⟩, ⟨d, rfl⟩⟩ : m ∣ a ∧ m ∣ b ∧ m ∣ c ∧ m ∣ d := by
        simp only [← ZMod.natCast_eq_zero_iff, ← hf_mod, hr, Int.cast_zero, and_self]
      have : m * m ∣ m * p := habcd ▸ ⟨a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2, by ring⟩
      rw [mul_dvd_mul_iff_left hm₀.ne'] at this
      exact (hp.eq_one_or_self_of_dvd _ this).elim hm₁.ne' hmp.ne
    -- Since `2 * |f x| < m` for each `x ∈ {a, b, c, d}`, we have `r < m`
    have hrm : r < m := by
      rw [mul_comm] at hr
      apply lt_of_sum_four_squares_eq_mul hr <;> apply hf_lt
    -- Now it suffices to represent `r * p` as a sum of four squares
    -- More precisely, we will represent `(m * r) * (m * p)` as a sum of squares of four numbers,
    -- each of them is divisible by `m`
    rsuffices ⟨w, x, y, z, hw, hx, hy, hz, h⟩ : ∃ w x y z : ℤ, ↑m ∣ w ∧ ↑m ∣ x ∧ ↑m ∣ y ∧ ↑m ∣ z ∧
      w ^ 2 + x ^ 2 + y ^ 2 + z ^ 2 = ↑(m * r) * ↑(m * p)
    · have : (w / m) ^ 2 + (x / m) ^ 2 + (y / m) ^ 2 + (z / m) ^ 2 = ↑(r * p) := by
        refine mul_left_cancel₀ (pow_ne_zero 2 (Nat.cast_ne_zero.2 hm₀.ne')) ?_
        conv_rhs => rw [← Nat.cast_pow, ← Nat.cast_mul, sq m, mul_mul_mul_comm, Nat.cast_mul, ← h]
        simp only [mul_add, ← mul_pow, Int.mul_ediv_cancel', *]
      rw [← natAbs_iff] at this
      exact hmin r hrm ⟨hrm.trans hmp, hr₀, _, _, _, _, this⟩
    -- To do the last step, we apply the Euler's four square identity once more
    replace hr : (f b) ^ 2 + (f a) ^ 2 + (f d) ^ 2 + (-f c) ^ 2 = ↑(m * r) := by
      rw [← natAbs_iff, natAbs_neg, ← hr]
      ac_rfl
    have := congr_arg₂ (· * Nat.cast ·) hr habcd
    simp only [← _root_.euler_four_squares, Nat.cast_add, Nat.cast_pow] at this
    refine ⟨_, _, _, _, ?_, ?_, ?_, ?_, this⟩
    · simp [← ZMod.intCast_zmod_eq_zero_iff_dvd, hf_mod, mul_comm]
    · suffices ((a : ZMod m) ^ 2 + (b : ZMod m) ^ 2 + (c : ZMod m) ^ 2 + (d : ZMod m) ^ 2) = 0 by
        simpa [← ZMod.intCast_zmod_eq_zero_iff_dvd, hf_mod, sq, add_comm, add_assoc,
          add_left_comm] using this
      norm_cast
      simp [habcd]
    · simp [← ZMod.intCast_zmod_eq_zero_iff_dvd, hf_mod, mul_comm]
    · simp [← ZMod.intCast_zmod_eq_zero_iff_dvd, hf_mod, mul_comm]

Goal state after: