l3lab/ntp-mathlib huggingface dataset:
theorem divides_sq_eq_zero {x y} (h : x * x = d * y * y) : x = 0 ∧ y = 0 :=
  let g := x.gcd y
  Or.elim g.eq_zero_or_pos
    (fun H => ⟨Nat.eq_zero_of_gcd_eq_zero_left H, Nat.eq_zero_of_gcd_eq_zero_right H⟩) fun gpos =>
    False.elim <| by
      let ⟨m, n, co, (hx : x = m * g), (hy : y = n * g)⟩ := Nat.exists_coprime gpos
      rw [hx, hy] at h
      have : m * m = d * (n * n) := by
        refine mul_left_cancel₀ (mul_pos gpos gpos).ne' ?_
        -- Porting note: was `simpa [mul_comm, mul_left_comm] using h`
        calc
          g * g * (m * m)
          _ = m * g * (m * g) := by ring
          _ = d * (n * g) * (n * g) := h
          _ = g * g * (d * (n * n)) := by 

Proof state:
d : ℕ
dnsq : Nonsquare d
x y : ℕ
g : ℕ := Nat.gcd x y
gpos : g > 0
m n : ℕ
h : m * g * (m * g) = d * (n * g) * (n * g)
co : Nat.Coprime m n
hx : x = m * g
hy : y = n * g
⊢ d * (n * g) * (n * g) = g * g * (d * (n * n))

Next tactic:
ring
theorem divides_sq_eq_zero {x y} (h : x * x = d * y * y) : x = 0 ∧ y = 0 :=
  let g := x.gcd y
  Or.elim g.eq_zero_or_pos
    (fun H => ⟨Nat.eq_zero_of_gcd_eq_zero_left H, Nat.eq_zero_of_gcd_eq_zero_right H⟩) fun gpos =>
    False.elim <| by
      let ⟨m, n, co, (hx : x = m * g), (hy : y = n * g)⟩ := Nat.exists_coprime gpos
      rw [hx, hy] at h
      have : m * m = d * (n * n) := by
        refine mul_left_cancel₀ (mul_pos gpos gpos).ne' ?_
        -- Porting note: was `simpa [mul_comm, mul_left_comm] using h`
        calc
          g * g * (m * m)
          _ = m * g * (m * g) := by ring
          _ = d * (n * g) * (n * g) := h
          _ = g * g * (d * (n * n)) := by ring
      

Proof state:
d : ℕ
dnsq : Nonsquare d
x y : ℕ
g : ℕ := Nat.gcd x y
gpos : g > 0
m n : ℕ
h : m * g * (m * g) = d * (n * g) * (n * g)
co : Nat.Coprime m n
hx : x = m * g
hy : y = n * g
this : m * m = d * (n * n)
⊢ False

Next tactic:
have co2 :=
        let co1 := co.mul_right co
        co1.mul co1
theorem divides_sq_eq_zero {x y} (h : x * x = d * y * y) : x = 0 ∧ y = 0 :=
  let g := x.gcd y
  Or.elim g.eq_zero_or_pos
    (fun H => ⟨Nat.eq_zero_of_gcd_eq_zero_left H, Nat.eq_zero_of_gcd_eq_zero_right H⟩) fun gpos =>
    False.elim <| by
      let ⟨m, n, co, (hx : x = m * g), (hy : y = n * g)⟩ := Nat.exists_coprime gpos
      rw [hx, hy] at h
      have : m * m = d * (n * n) := by
        refine mul_left_cancel₀ (mul_pos gpos gpos).ne' ?_
        -- Porting note: was `simpa [mul_comm, mul_left_comm] using h`
        calc
          g * g * (m * m)
          _ = m * g * (m * g) := by ring
          _ = d * (n * g) * (n * g) := h
          _ = g * g * (d * (n * n)) := by ring
      have co2 :=
        let co1 := co.mul_right co
        co1.mul co1
      

Proof state:
d : ℕ
dnsq : Nonsquare d
x y : ℕ
g : ℕ := Nat.gcd x y
gpos : g > 0
m n : ℕ
h : m * g * (m * g) = d * (n * g) * (n * g)
co : Nat.Coprime m n
hx : x = m * g
hy : y = n * g
this : m * m = d * (n * n)
co2 : Nat.Coprime (m * m) (n * n)
⊢ False

Next tactic:
exact
        Nonsquare.ns d m
          (Nat.dvd_antisymm (by rw [this]; apply dvd_mul_right) <|
            co2.dvd_of_dvd_mul_right <| by simp [this])
theorem divides_sq_eq_zero {x y} (h : x * x = d * y * y) : x = 0 ∧ y = 0 :=
  let g := x.gcd y
  Or.elim g.eq_zero_or_pos
    (fun H => ⟨Nat.eq_zero_of_gcd_eq_zero_left H, Nat.eq_zero_of_gcd_eq_zero_right H⟩) fun gpos =>
    False.elim <| by
      let ⟨m, n, co, (hx : x = m * g), (hy : y = n * g)⟩ := Nat.exists_coprime gpos
      rw [hx, hy] at h
      have : m * m = d * (n * n) := by
        refine mul_left_cancel₀ (mul_pos gpos gpos).ne' ?_
        -- Porting note: was `simpa [mul_comm, mul_left_comm] using h`
        calc
          g * g * (m * m)
          _ = m * g * (m * g) := by ring
          _ = d * (n * g) * (n * g) := h
          _ = g * g * (d * (n * n)) := by ring
      have co2 :=
        let co1 := co.mul_right co
        co1.mul co1
      exact
        Nonsquare.ns d m
          (Nat.dvd_antisymm (by 

Proof state:
d : ℕ
dnsq : Nonsquare d
x y : ℕ
g : ℕ := Nat.gcd x y
gpos : g > 0
m n : ℕ
h : m * g * (m * g) = d * (n * g) * (n * g)
co : Nat.Coprime m n
hx : x = m * g
hy : y = n * g
this : m * m = d * (n * n)
co2 : Nat.Coprime (m * m) (n * n)
⊢ d ∣ m * m

Next tactic:
rw [this]
theorem divides_sq_eq_zero {x y} (h : x * x = d * y * y) : x = 0 ∧ y = 0 :=
  let g := x.gcd y
  Or.elim g.eq_zero_or_pos
    (fun H => ⟨Nat.eq_zero_of_gcd_eq_zero_left H, Nat.eq_zero_of_gcd_eq_zero_right H⟩) fun gpos =>
    False.elim <| by
      let ⟨m, n, co, (hx : x = m * g), (hy : y = n * g)⟩ := Nat.exists_coprime gpos
      rw [hx, hy] at h
      have : m * m = d * (n * n) := by
        refine mul_left_cancel₀ (mul_pos gpos gpos).ne' ?_
        -- Porting note: was `simpa [mul_comm, mul_left_comm] using h`
        calc
          g * g * (m * m)
          _ = m * g * (m * g) := by ring
          _ = d * (n * g) * (n * g) := h
          _ = g * g * (d * (n * n)) := by ring
      have co2 :=
        let co1 := co.mul_right co
        co1.mul co1
      exact
        Nonsquare.ns d m
          (Nat.dvd_antisymm (by rw [this]; 

Proof state:
d : ℕ
dnsq : Nonsquare d
x y : ℕ
g : ℕ := Nat.gcd x y
gpos : g > 0
m n : ℕ
h : m * g * (m * g) = d * (n * g) * (n * g)
co : Nat.Coprime m n
hx : x = m * g
hy : y = n * g
this : m * m = d * (n * n)
co2 : Nat.Coprime (m * m) (n * n)
⊢ d ∣ d * (n * n)

Next tactic:
apply dvd_mul_right
theorem divides_sq_eq_zero {x y} (h : x * x = d * y * y) : x = 0 ∧ y = 0 :=
  let g := x.gcd y
  Or.elim g.eq_zero_or_pos
    (fun H => ⟨Nat.eq_zero_of_gcd_eq_zero_left H, Nat.eq_zero_of_gcd_eq_zero_right H⟩) fun gpos =>
    False.elim <| by
      let ⟨m, n, co, (hx : x = m * g), (hy : y = n * g)⟩ := Nat.exists_coprime gpos
      rw [hx, hy] at h
      have : m * m = d * (n * n) := by
        refine mul_left_cancel₀ (mul_pos gpos gpos).ne' ?_
        -- Porting note: was `simpa [mul_comm, mul_left_comm] using h`
        calc
          g * g * (m * m)
          _ = m * g * (m * g) := by ring
          _ = d * (n * g) * (n * g) := h
          _ = g * g * (d * (n * n)) := by ring
      have co2 :=
        let co1 := co.mul_right co
        co1.mul co1
      exact
        Nonsquare.ns d m
          (Nat.dvd_antisymm (by rw [this]; apply dvd_mul_right) <|
            co2.dvd_of_dvd_mul_right <| by 

Proof state:
d : ℕ
dnsq : Nonsquare d
x y : ℕ
g : ℕ := Nat.gcd x y
gpos : g > 0
m n : ℕ
h : m * g * (m * g) = d * (n * g) * (n * g)
co : Nat.Coprime m n
hx : x = m * g
hy : y = n * g
this : m * m = d * (n * n)
co2 : Nat.Coprime (m * m) (n * n)
⊢ m * m ∣ d * (n * n)

Next tactic:
simp [this]
theorem divides_sq_eq_zero_z {x y : ℤ} (h : x * x = d * y * y) : x = 0 ∧ y = 0 := by
  

Proof state:
d : ℕ
dnsq : Nonsquare d
x y : ℤ
h : x * x = ↑d * y * y
⊢ x = 0 ∧ y = 0

Next tactic:
rw [mul_assoc, ← Int.natAbs_mul_self, ← Int.natAbs_mul_self, ← Int.ofNat_mul, ← mul_assoc] at h
theorem divides_sq_eq_zero_z {x y : ℤ} (h : x * x = d * y * y) : x = 0 ∧ y = 0 := by
  rw [mul_assoc, ← Int.natAbs_mul_self, ← Int.natAbs_mul_self, ← Int.ofNat_mul, ← mul_assoc] at h
  

Proof state:
d : ℕ
dnsq : Nonsquare d
x y : ℤ
h : ↑(Int.natAbs x * Int.natAbs x) = ↑(d * Int.natAbs y * Int.natAbs y)
⊢ x = 0 ∧ y = 0

Next tactic:
exact
    let ⟨h1, h2⟩ := divides_sq_eq_zero (Int.ofNat.inj h)
    ⟨Int.natAbs_eq_zero.mp h1, Int.natAbs_eq_zero.mp h2⟩
theorem not_divides_sq (x y) : (x + 1) * (x + 1) ≠ d * (y + 1) * (y + 1) := fun e => by
  

Proof state:
d : ℕ
dnsq : Nonsquare d
x y : ℕ
e : (x + 1) * (x + 1) = d * (y + 1) * (y + 1)
⊢ False

Next tactic:
have t := (divides_sq_eq_zero e).left
theorem not_divides_sq (x y) : (x + 1) * (x + 1) ≠ d * (y + 1) * (y + 1) := fun e => by
  have t := (divides_sq_eq_zero e).left
  

Proof state:
d : ℕ
dnsq : Nonsquare d
x y : ℕ
e : (x + 1) * (x + 1) = d * (y + 1) * (y + 1)
t : x + 1 = 0
⊢ False

Next tactic:
contradiction
open Int in
theorem nonneg_antisymm : ∀ {a : ℤ√d}, Nonneg a → Nonneg (-a) → a = 0
  | ⟨0, 0⟩, _, _ => rfl
  | ⟨-[x+1], -[y+1]⟩, xy, _ => False.elim xy
  | ⟨(x + 1 : Nat), (y + 1 : Nat)⟩, _, yx => False.elim yx
  | ⟨-[x+1], 0⟩, xy, _ => absurd xy (not_sqLe_succ _ _ _ (by 

Proof state:
d : ℕ
dnsq : Nonsquare d
x : ℕ
xy : Nonneg { re := -[x+1], im := 0 }
x✝ : Nonneg (-{ re := -[x+1], im := 0 })
⊢ 0 < 1

Next tactic:
decide
open Int in
theorem nonneg_antisymm : ∀ {a : ℤ√d}, Nonneg a → Nonneg (-a) → a = 0
  | ⟨0, 0⟩, _, _ => rfl
  | ⟨-[x+1], -[y+1]⟩, xy, _ => False.elim xy
  | ⟨(x + 1 : Nat), (y + 1 : Nat)⟩, _, yx => False.elim yx
  | ⟨-[x+1], 0⟩, xy, _ => absurd xy (not_sqLe_succ _ _ _ (by decide))
  | ⟨(x + 1 : Nat), 0⟩, _, yx => absurd yx (not_sqLe_succ _ _ _ (by 

Proof state:
d : ℕ
dnsq : Nonsquare d
x : ℕ
x✝ : Nonneg { re := ↑(x + 1), im := 0 }
yx : Nonneg (-{ re := ↑(x + 1), im := 0 })
⊢ 0 < 1

Next tactic:
decide
open Int in
theorem nonneg_antisymm : ∀ {a : ℤ√d}, Nonneg a → Nonneg (-a) → a = 0
  | ⟨0, 0⟩, _, _ => rfl
  | ⟨-[x+1], -[y+1]⟩, xy, _ => False.elim xy
  | ⟨(x + 1 : Nat), (y + 1 : Nat)⟩, _, yx => False.elim yx
  | ⟨-[x+1], 0⟩, xy, _ => absurd xy (not_sqLe_succ _ _ _ (by decide))
  | ⟨(x + 1 : Nat), 0⟩, _, yx => absurd yx (not_sqLe_succ _ _ _ (by decide))
  | ⟨0, -[y+1]⟩, xy, _ => absurd xy (not_sqLe_succ _ _ _ d_pos)
  | ⟨0, (y + 1 : Nat)⟩, _, yx => absurd yx (not_sqLe_succ _ _ _ d_pos)
  | ⟨(x + 1 : Nat), -[y+1]⟩, (xy : SqLe _ _ _ _), (yx : SqLe _ _ _ _) => by
    

Proof state:
d : ℕ
dnsq : Nonsquare d
x y : ℕ
xy : SqLe (y + 1) d (x + 1) 1
yx : SqLe (Nat.add x 0 + 1) 1 (Nat.succ y) d
⊢ { re := ↑(x + 1), im := -[y+1] } = 0

Next tactic:
let t := le_antisymm yx xy
open Int in
theorem nonneg_antisymm : ∀ {a : ℤ√d}, Nonneg a → Nonneg (-a) → a = 0
  | ⟨0, 0⟩, _, _ => rfl
  | ⟨-[x+1], -[y+1]⟩, xy, _ => False.elim xy
  | ⟨(x + 1 : Nat), (y + 1 : Nat)⟩, _, yx => False.elim yx
  | ⟨-[x+1], 0⟩, xy, _ => absurd xy (not_sqLe_succ _ _ _ (by decide))
  | ⟨(x + 1 : Nat), 0⟩, _, yx => absurd yx (not_sqLe_succ _ _ _ (by decide))
  | ⟨0, -[y+1]⟩, xy, _ => absurd xy (not_sqLe_succ _ _ _ d_pos)
  | ⟨0, (y + 1 : Nat)⟩, _, yx => absurd yx (not_sqLe_succ _ _ _ d_pos)
  | ⟨(x + 1 : Nat), -[y+1]⟩, (xy : SqLe _ _ _ _), (yx : SqLe _ _ _ _) => by
    let t := le_antisymm yx xy
    

Proof state:
d : ℕ
dnsq : Nonsquare d
x y : ℕ
xy : SqLe (y + 1) d (x + 1) 1
yx : SqLe (Nat.add x 0 + 1) 1 (Nat.succ y) d
t : 1 * (Nat.add x 0 + 1) * (Nat.add x 0 + 1) = d * Nat.succ y * Nat.succ y := le_antisymm yx xy
⊢ { re := ↑(x + 1), im := -[y+1] } = 0

Next tactic:
rw [one_mul] at t
open Int in
theorem nonneg_antisymm : ∀ {a : ℤ√d}, Nonneg a → Nonneg (-a) → a = 0
  | ⟨0, 0⟩, _, _ => rfl
  | ⟨-[x+1], -[y+1]⟩, xy, _ => False.elim xy
  | ⟨(x + 1 : Nat), (y + 1 : Nat)⟩, _, yx => False.elim yx
  | ⟨-[x+1], 0⟩, xy, _ => absurd xy (not_sqLe_succ _ _ _ (by decide))
  | ⟨(x + 1 : Nat), 0⟩, _, yx => absurd yx (not_sqLe_succ _ _ _ (by decide))
  | ⟨0, -[y+1]⟩, xy, _ => absurd xy (not_sqLe_succ _ _ _ d_pos)
  | ⟨0, (y + 1 : Nat)⟩, _, yx => absurd yx (not_sqLe_succ _ _ _ d_pos)
  | ⟨(x + 1 : Nat), -[y+1]⟩, (xy : SqLe _ _ _ _), (yx : SqLe _ _ _ _) => by
    let t := le_antisymm yx xy
    rw [one_mul] at t
    

Proof state:
d : ℕ
dnsq : Nonsquare d
x y : ℕ
xy : SqLe (y + 1) d (x + 1) 1
yx : SqLe (Nat.add x 0 + 1) 1 (Nat.succ y) d
t : (Nat.add x 0 + 1) * (Nat.add x 0 + 1) = d * Nat.succ y * Nat.succ y
⊢ { re := ↑(x + 1), im := -[y+1] } = 0

Next tactic:
exact absurd t (not_divides_sq _ _)
open Int in
theorem nonneg_antisymm : ∀ {a : ℤ√d}, Nonneg a → Nonneg (-a) → a = 0
  | ⟨0, 0⟩, _, _ => rfl
  | ⟨-[x+1], -[y+1]⟩, xy, _ => False.elim xy
  | ⟨(x + 1 : Nat), (y + 1 : Nat)⟩, _, yx => False.elim yx
  | ⟨-[x+1], 0⟩, xy, _ => absurd xy (not_sqLe_succ _ _ _ (by decide))
  | ⟨(x + 1 : Nat), 0⟩, _, yx => absurd yx (not_sqLe_succ _ _ _ (by decide))
  | ⟨0, -[y+1]⟩, xy, _ => absurd xy (not_sqLe_succ _ _ _ d_pos)
  | ⟨0, (y + 1 : Nat)⟩, _, yx => absurd yx (not_sqLe_succ _ _ _ d_pos)
  | ⟨(x + 1 : Nat), -[y+1]⟩, (xy : SqLe _ _ _ _), (yx : SqLe _ _ _ _) => by
    let t := le_antisymm yx xy
    rw [one_mul] at t
    exact absurd t (not_divides_sq _ _)
  | ⟨-[x+1], (y + 1 : Nat)⟩, (xy : SqLe _ _ _ _), (yx : SqLe _ _ _ _) => by
    

Proof state:
d : ℕ
dnsq : Nonsquare d
x y : ℕ
xy : SqLe (x + 1) 1 (y + 1) d
yx : SqLe (Nat.add y 0 + 1) d (Nat.succ x) 1
⊢ { re := -[x+1], im := ↑(y + 1) } = 0

Next tactic:
let t := le_antisymm xy yx
open Int in
theorem nonneg_antisymm : ∀ {a : ℤ√d}, Nonneg a → Nonneg (-a) → a = 0
  | ⟨0, 0⟩, _, _ => rfl
  | ⟨-[x+1], -[y+1]⟩, xy, _ => False.elim xy
  | ⟨(x + 1 : Nat), (y + 1 : Nat)⟩, _, yx => False.elim yx
  | ⟨-[x+1], 0⟩, xy, _ => absurd xy (not_sqLe_succ _ _ _ (by decide))
  | ⟨(x + 1 : Nat), 0⟩, _, yx => absurd yx (not_sqLe_succ _ _ _ (by decide))
  | ⟨0, -[y+1]⟩, xy, _ => absurd xy (not_sqLe_succ _ _ _ d_pos)
  | ⟨0, (y + 1 : Nat)⟩, _, yx => absurd yx (not_sqLe_succ _ _ _ d_pos)
  | ⟨(x + 1 : Nat), -[y+1]⟩, (xy : SqLe _ _ _ _), (yx : SqLe _ _ _ _) => by
    let t := le_antisymm yx xy
    rw [one_mul] at t
    exact absurd t (not_divides_sq _ _)
  | ⟨-[x+1], (y + 1 : Nat)⟩, (xy : SqLe _ _ _ _), (yx : SqLe _ _ _ _) => by
    let t := le_antisymm xy yx
    

Proof state:
d : ℕ
dnsq : Nonsquare d
x y : ℕ
xy : SqLe (x + 1) 1 (y + 1) d
yx : SqLe (Nat.add y 0 + 1) d (Nat.succ x) 1
t : 1 * (x + 1) * (x + 1) = d * (y + 1) * (y + 1) := le_antisymm xy yx
⊢ { re := -[x+1], im := ↑(y + 1) } = 0

Next tactic:
rw [one_mul] at t
open Int in
theorem nonneg_antisymm : ∀ {a : ℤ√d}, Nonneg a → Nonneg (-a) → a = 0
  | ⟨0, 0⟩, _, _ => rfl
  | ⟨-[x+1], -[y+1]⟩, xy, _ => False.elim xy
  | ⟨(x + 1 : Nat), (y + 1 : Nat)⟩, _, yx => False.elim yx
  | ⟨-[x+1], 0⟩, xy, _ => absurd xy (not_sqLe_succ _ _ _ (by decide))
  | ⟨(x + 1 : Nat), 0⟩, _, yx => absurd yx (not_sqLe_succ _ _ _ (by decide))
  | ⟨0, -[y+1]⟩, xy, _ => absurd xy (not_sqLe_succ _ _ _ d_pos)
  | ⟨0, (y + 1 : Nat)⟩, _, yx => absurd yx (not_sqLe_succ _ _ _ d_pos)
  | ⟨(x + 1 : Nat), -[y+1]⟩, (xy : SqLe _ _ _ _), (yx : SqLe _ _ _ _) => by
    let t := le_antisymm yx xy
    rw [one_mul] at t
    exact absurd t (not_divides_sq _ _)
  | ⟨-[x+1], (y + 1 : Nat)⟩, (xy : SqLe _ _ _ _), (yx : SqLe _ _ _ _) => by
    let t := le_antisymm xy yx
    rw [one_mul] at t
    

Proof state:
d : ℕ
dnsq : Nonsquare d
x y : ℕ
xy : SqLe (x + 1) 1 (y + 1) d
yx : SqLe (Nat.add y 0 + 1) d (Nat.succ x) 1
t : (x + 1) * (x + 1) = d * (y + 1) * (y + 1)
⊢ { re := -[x+1], im := ↑(y + 1) } = 0

Next tactic:
exact absurd t (not_divides_sq _ _)
theorem le_antisymm {a b : ℤ√d} (ab : a ≤ b) (ba : b ≤ a) : a = b :=
  eq_of_sub_eq_zero <| nonneg_antisymm ba (by 

Proof state:
d : ℕ
dnsq : Nonsquare d
a b : ℤ√↑d
ab : a ≤ b
ba : b ≤ a
⊢ Nonneg (-(a - b))

Next tactic:
rwa [neg_sub]
protected theorem eq_zero_or_eq_zero_of_mul_eq_zero : ∀ {a b : ℤ√d}, a * b = 0 → a = 0 ∨ b = 0
  | ⟨x, y⟩, ⟨z, w⟩, h => by
    

Proof state:
d : ℕ
dnsq : Nonsquare d
x y z w : ℤ
h : { re := x, im := y } * { re := z, im := w } = 0
⊢ { re := x, im := y } = 0 ∨ { re := z, im := w } = 0

Next tactic:
injection h with h1 h2
protected theorem eq_zero_or_eq_zero_of_mul_eq_zero : ∀ {a b : ℤ√d}, a * b = 0 → a = 0 ∨ b = 0
  | ⟨x, y⟩, ⟨z, w⟩, h => by
    injection h with h1 h2
    

Proof state:
d : ℕ
dnsq : Nonsquare d
x y z w : ℤ
h1 : { re := x, im := y }.re * { re := z, im := w }.re + ↑d * { re := x, im := y }.im * { re := z, im := w }.im = 0
h2 : { re := x, im := y }.re * { re := z, im := w }.im + { re := x, im := y }.im * { re := z, im := w }.re = 0
⊢ { re := x, im := y } = 0 ∨ { re := z, im := w } = 0

Next tactic:
have h1 : x * z = -(d * y * w) := eq_neg_of_add_eq_zero_left h1
protected theorem eq_zero_or_eq_zero_of_mul_eq_zero : ∀ {a b : ℤ√d}, a * b = 0 → a = 0 ∨ b = 0
  | ⟨x, y⟩, ⟨z, w⟩, h => by
    injection h with h1 h2
    have h1 : x * z = -(d * y * w) := eq_neg_of_add_eq_zero_left h1
    

Proof state:
d : ℕ
dnsq : Nonsquare d
x y z w : ℤ
h1✝ : { re := x, im := y }.re * { re := z, im := w }.re + ↑d * { re := x, im := y }.im * { re := z, im := w }.im = 0
h2 : { re := x, im := y }.re * { re := z, im := w }.im + { re := x, im := y }.im * { re := z, im := w }.re = 0
h1 : x * z = -(↑d * y * w)
⊢ { re := x, im := y } = 0 ∨ { re := z, im := w } = 0

Next tactic:
have h2 : x * w = -(y * z) := eq_neg_of_add_eq_zero_left h2
protected theorem eq_zero_or_eq_zero_of_mul_eq_zero : ∀ {a b : ℤ√d}, a * b = 0 → a = 0 ∨ b = 0
  | ⟨x, y⟩, ⟨z, w⟩, h => by
    injection h with h1 h2
    have h1 : x * z = -(d * y * w) := eq_neg_of_add_eq_zero_left h1
    have h2 : x * w = -(y * z) := eq_neg_of_add_eq_zero_left h2
    

Proof state:
d : ℕ
dnsq : Nonsquare d
x y z w : ℤ
h1✝ : { re := x, im := y }.re * { re := z, im := w }.re + ↑d * { re := x, im := y }.im * { re := z, im := w }.im = 0
h2✝ : { re := x, im := y }.re * { re := z, im := w }.im + { re := x, im := y }.im * { re := z, im := w }.re = 0
h1 : x * z = -(↑d * y * w)
h2 : x * w = -(y * z)
⊢ { re := x, im := y } = 0 ∨ { re := z, im := w } = 0

Next tactic:
have fin : x * x = d * y * y → (⟨x, y⟩ : ℤ√d) = 0 := fun e =>
      match x, y, divides_sq_eq_zero_z e with
      | _, _, ⟨rfl, rfl⟩ => rfl
protected theorem eq_zero_or_eq_zero_of_mul_eq_zero : ∀ {a b : ℤ√d}, a * b = 0 → a = 0 ∨ b = 0
  | ⟨x, y⟩, ⟨z, w⟩, h => by
    injection h with h1 h2
    have h1 : x * z = -(d * y * w) := eq_neg_of_add_eq_zero_left h1
    have h2 : x * w = -(y * z) := eq_neg_of_add_eq_zero_left h2
    have fin : x * x = d * y * y → (⟨x, y⟩ : ℤ√d) = 0 := fun e =>
      match x, y, divides_sq_eq_zero_z e with
      | _, _, ⟨rfl, rfl⟩ => rfl
    

Proof state:
d : ℕ
dnsq : Nonsquare d
x y z w : ℤ
h1✝ : { re := x, im := y }.re * { re := z, im := w }.re + ↑d * { re := x, im := y }.im * { re := z, im := w }.im = 0
h2✝ : { re := x, im := y }.re * { re := z, im := w }.im + { re := x, im := y }.im * { re := z, im := w }.re = 0
h1 : x * z = -(↑d * y * w)
h2 : x * w = -(y * z)
fin : x * x = ↑d * y * y → { re := x, im := y } = 0
⊢ { re := x, im := y } = 0 ∨ { re := z, im := w } = 0

Next tactic:
exact
      if z0 : z = 0 then
        if w0 : w = 0 then
          Or.inr
            (match z, w, z0, w0 with
            | _, _, rfl, rfl => rfl)
        else
          Or.inl <|
            fin <|
              mul_right_cancel₀ w0 <|
                calc
                  x * x * w = -y * (x * z) := by simp [h2, mul_assoc, mul_left_comm]
                  _ = d * y * y * w := by simp [h1, mul_assoc, mul_left_comm]
      else
        Or.inl <|
          fin <|
            mul_right_cancel₀ z0 <|
              calc
                x * x * z = d * -y * (x * w) := by simp [h1, mul_assoc, mul_left_comm]
                _ = d * y * y * z := by simp [h2, mul_assoc, mul_left_comm]
protected theorem eq_zero_or_eq_zero_of_mul_eq_zero : ∀ {a b : ℤ√d}, a * b = 0 → a = 0 ∨ b = 0
  | ⟨x, y⟩, ⟨z, w⟩, h => by
    injection h with h1 h2
    have h1 : x * z = -(d * y * w) := eq_neg_of_add_eq_zero_left h1
    have h2 : x * w = -(y * z) := eq_neg_of_add_eq_zero_left h2
    have fin : x * x = d * y * y → (⟨x, y⟩ : ℤ√d) = 0 := fun e =>
      match x, y, divides_sq_eq_zero_z e with
      | _, _, ⟨rfl, rfl⟩ => rfl
    exact
      if z0 : z = 0 then
        if w0 : w = 0 then
          Or.inr
            (match z, w, z0, w0 with
            | _, _, rfl, rfl => rfl)
        else
          Or.inl <|
            fin <|
              mul_right_cancel₀ w0 <|
                calc
                  x * x * w = -y * (x * z) := by 

Proof state:
d : ℕ
dnsq : Nonsquare d
x y z w : ℤ
h1✝ : { re := x, im := y }.re * { re := z, im := w }.re + ↑d * { re := x, im := y }.im * { re := z, im := w }.im = 0
h2✝ : { re := x, im := y }.re * { re := z, im := w }.im + { re := x, im := y }.im * { re := z, im := w }.re = 0
h1 : x * z = -(↑d * y * w)
h2 : x * w = -(y * z)
fin : x * x = ↑d * y * y → { re := x, im := y } = 0
z0 : z = 0
w0 : ¬w = 0
⊢ x * x * w = -y * (x * z)

Next tactic:
simp [h2, mul_assoc, mul_left_comm]
protected theorem eq_zero_or_eq_zero_of_mul_eq_zero : ∀ {a b : ℤ√d}, a * b = 0 → a = 0 ∨ b = 0
  | ⟨x, y⟩, ⟨z, w⟩, h => by
    injection h with h1 h2
    have h1 : x * z = -(d * y * w) := eq_neg_of_add_eq_zero_left h1
    have h2 : x * w = -(y * z) := eq_neg_of_add_eq_zero_left h2
    have fin : x * x = d * y * y → (⟨x, y⟩ : ℤ√d) = 0 := fun e =>
      match x, y, divides_sq_eq_zero_z e with
      | _, _, ⟨rfl, rfl⟩ => rfl
    exact
      if z0 : z = 0 then
        if w0 : w = 0 then
          Or.inr
            (match z, w, z0, w0 with
            | _, _, rfl, rfl => rfl)
        else
          Or.inl <|
            fin <|
              mul_right_cancel₀ w0 <|
                calc
                  x * x * w = -y * (x * z) := by simp [h2, mul_assoc, mul_left_comm]
                  _ = d * y * y * w := by 

Proof state:
d : ℕ
dnsq : Nonsquare d
x y z w : ℤ
h1✝ : { re := x, im := y }.re * { re := z, im := w }.re + ↑d * { re := x, im := y }.im * { re := z, im := w }.im = 0
h2✝ : { re := x, im := y }.re * { re := z, im := w }.im + { re := x, im := y }.im * { re := z, im := w }.re = 0
h1 : x * z = -(↑d * y * w)
h2 : x * w = -(y * z)
fin : x * x = ↑d * y * y → { re := x, im := y } = 0
z0 : z = 0
w0 : ¬w = 0
⊢ -y * (x * z) = ↑d * y * y * w

Next tactic:
simp [h1, mul_assoc, mul_left_comm]
protected theorem eq_zero_or_eq_zero_of_mul_eq_zero : ∀ {a b : ℤ√d}, a * b = 0 → a = 0 ∨ b = 0
  | ⟨x, y⟩, ⟨z, w⟩, h => by
    injection h with h1 h2
    have h1 : x * z = -(d * y * w) := eq_neg_of_add_eq_zero_left h1
    have h2 : x * w = -(y * z) := eq_neg_of_add_eq_zero_left h2
    have fin : x * x = d * y * y → (⟨x, y⟩ : ℤ√d) = 0 := fun e =>
      match x, y, divides_sq_eq_zero_z e with
      | _, _, ⟨rfl, rfl⟩ => rfl
    exact
      if z0 : z = 0 then
        if w0 : w = 0 then
          Or.inr
            (match z, w, z0, w0 with
            | _, _, rfl, rfl => rfl)
        else
          Or.inl <|
            fin <|
              mul_right_cancel₀ w0 <|
                calc
                  x * x * w = -y * (x * z) := by simp [h2, mul_assoc, mul_left_comm]
                  _ = d * y * y * w := by simp [h1, mul_assoc, mul_left_comm]
      else
        Or.inl <|
          fin <|
            mul_right_cancel₀ z0 <|
              calc
                x * x * z = d * -y * (x * w) := by 

Proof state:
d : ℕ
dnsq : Nonsquare d
x y z w : ℤ
h1✝ : { re := x, im := y }.re * { re := z, im := w }.re + ↑d * { re := x, im := y }.im * { re := z, im := w }.im = 0
h2✝ : { re := x, im := y }.re * { re := z, im := w }.im + { re := x, im := y }.im * { re := z, im := w }.re = 0
h1 : x * z = -(↑d * y * w)
h2 : x * w = -(y * z)
fin : x * x = ↑d * y * y → { re := x, im := y } = 0
z0 : ¬z = 0
⊢ x * x * z = ↑d * -y * (x * w)

Next tactic:
simp [h1, mul_assoc, mul_left_comm]
protected theorem eq_zero_or_eq_zero_of_mul_eq_zero : ∀ {a b : ℤ√d}, a * b = 0 → a = 0 ∨ b = 0
  | ⟨x, y⟩, ⟨z, w⟩, h => by
    injection h with h1 h2
    have h1 : x * z = -(d * y * w) := eq_neg_of_add_eq_zero_left h1
    have h2 : x * w = -(y * z) := eq_neg_of_add_eq_zero_left h2
    have fin : x * x = d * y * y → (⟨x, y⟩ : ℤ√d) = 0 := fun e =>
      match x, y, divides_sq_eq_zero_z e with
      | _, _, ⟨rfl, rfl⟩ => rfl
    exact
      if z0 : z = 0 then
        if w0 : w = 0 then
          Or.inr
            (match z, w, z0, w0 with
            | _, _, rfl, rfl => rfl)
        else
          Or.inl <|
            fin <|
              mul_right_cancel₀ w0 <|
                calc
                  x * x * w = -y * (x * z) := by simp [h2, mul_assoc, mul_left_comm]
                  _ = d * y * y * w := by simp [h1, mul_assoc, mul_left_comm]
      else
        Or.inl <|
          fin <|
            mul_right_cancel₀ z0 <|
              calc
                x * x * z = d * -y * (x * w) := by simp [h1, mul_assoc, mul_left_comm]
                _ = d * y * y * z := by 

Proof state:
d : ℕ
dnsq : Nonsquare d
x y z w : ℤ
h1✝ : { re := x, im := y }.re * { re := z, im := w }.re + ↑d * { re := x, im := y }.im * { re := z, im := w }.im = 0
h2✝ : { re := x, im := y }.re * { re := z, im := w }.im + { re := x, im := y }.im * { re := z, im := w }.re = 0
h1 : x * z = -(↑d * y * w)
h2 : x * w = -(y * z)
fin : x * x = ↑d * y * y → { re := x, im := y } = 0
z0 : ¬z = 0
⊢ ↑d * -y * (x * w) = ↑d * y * y * z

Next tactic:
simp [h2, mul_assoc, mul_left_comm]
instance : LinearOrderedCommRing (ℤ√d) :=
  { Zsqrtd.commRing, Zsqrtd.linearOrder, Zsqrtd.nontrivial with
    add_le_add_left := Zsqrtd.add_le_add_left
    mul_pos := Zsqrtd.mul_pos
    zero_le_one := by 

Proof state:
d : ℕ
dnsq : Nonsquare d
src✝² : CommRing (ℤ√↑d) := commRing
src✝¹ : LinearOrder (ℤ√↑d) := linearOrder
src✝ : Nontrivial (ℤ√↑d) := nontrivial
⊢ 0 ≤ 1

Next tactic:
trivial