l3lab/ntp-mathlib huggingface dataset:
instance : AddGroupWithOne (X q) :=
  { inferInstanceAs (Monoid (X q)), inferInstanceAs (AddCommGroup (X q)),
      inferInstanceAs (NatCast (X q)) with
    natCast_zero := by ext <;> simp
    natCast_succ := fun _ ↦ by ext <;> simp
    intCast := fun n => ⟨n, 0⟩
    intCast_ofNat := fun n => by ext <;> simp
    intCast_negSucc := fun n => by ext <;> 

Proof state:
case h₁
q : ℕ+
src✝² : Monoid (X q) := inferInstanceAs (Monoid (X q))
src✝¹ : AddCommGroup (X q) := inferInstanceAs (AddCommGroup (X q))
src✝ : NatCast (X q) := inferInstanceAs (NatCast (X q))
n : ℕ
⊢ (IntCast.intCast (Int.negSucc n)).1 = (-↑(n + 1)).1

Next tactic:
simp
instance : AddGroupWithOne (X q) :=
  { inferInstanceAs (Monoid (X q)), inferInstanceAs (AddCommGroup (X q)),
      inferInstanceAs (NatCast (X q)) with
    natCast_zero := by ext <;> simp
    natCast_succ := fun _ ↦ by ext <;> simp
    intCast := fun n => ⟨n, 0⟩
    intCast_ofNat := fun n => by ext <;> simp
    intCast_negSucc := fun n => by ext <;> 

Proof state:
case h₂
q : ℕ+
src✝² : Monoid (X q) := inferInstanceAs (Monoid (X q))
src✝¹ : AddCommGroup (X q) := inferInstanceAs (AddCommGroup (X q))
src✝ : NatCast (X q) := inferInstanceAs (NatCast (X q))
n : ℕ
⊢ (IntCast.intCast (Int.negSucc n)).2 = (-↑(n + 1)).2

Next tactic:
simp
theorem left_distrib (x y z : X q) : x * (y + z) = x * y + x * z := by
  

Proof state:
q : ℕ+
x y z : X q
⊢ x * (y + z) = x * y + x * z

Next tactic:
ext
theorem left_distrib (x y z : X q) : x * (y + z) = x * y + x * z := by
  ext <;> 

Proof state:
case h₁
q : ℕ+
x y z : X q
⊢ (x * (y + z)).1 = (x * y + x * z).1

Next tactic:
dsimp
theorem left_distrib (x y z : X q) : x * (y + z) = x * y + x * z := by
  ext <;> 

Proof state:
case h₂
q : ℕ+
x y z : X q
⊢ (x * (y + z)).2 = (x * y + x * z).2

Next tactic:
dsimp
theorem left_distrib (x y z : X q) : x * (y + z) = x * y + x * z := by
  ext <;> dsimp <;> 

Proof state:
case h₁
q : ℕ+
x y z : X q
⊢ x.1 * (y.1 + z.1) + 3 * x.2 * (y.2 + z.2) = x.1 * y.1 + 3 * x.2 * y.2 + (x.1 * z.1 + 3 * x.2 * z.2)

Next tactic:
ring
theorem left_distrib (x y z : X q) : x * (y + z) = x * y + x * z := by
  ext <;> dsimp <;> 

Proof state:
case h₂
q : ℕ+
x y z : X q
⊢ x.1 * (y.2 + z.2) + x.2 * (y.1 + z.1) = x.1 * y.2 + x.2 * y.1 + (x.1 * z.2 + x.2 * z.1)

Next tactic:
ring
theorem right_distrib (x y z : X q) : (x + y) * z = x * z + y * z := by
  

Proof state:
q : ℕ+
x y z : X q
⊢ (x + y) * z = x * z + y * z

Next tactic:
ext
theorem right_distrib (x y z : X q) : (x + y) * z = x * z + y * z := by
  ext <;> 

Proof state:
case h₁
q : ℕ+
x y z : X q
⊢ ((x + y) * z).1 = (x * z + y * z).1

Next tactic:
dsimp
theorem right_distrib (x y z : X q) : (x + y) * z = x * z + y * z := by
  ext <;> 

Proof state:
case h₂
q : ℕ+
x y z : X q
⊢ ((x + y) * z).2 = (x * z + y * z).2

Next tactic:
dsimp
theorem right_distrib (x y z : X q) : (x + y) * z = x * z + y * z := by
  ext <;> dsimp <;> 

Proof state:
case h₁
q : ℕ+
x y z : X q
⊢ (x.1 + y.1) * z.1 + 3 * (x.2 + y.2) * z.2 = x.1 * z.1 + 3 * x.2 * z.2 + (y.1 * z.1 + 3 * y.2 * z.2)

Next tactic:
ring
theorem right_distrib (x y z : X q) : (x + y) * z = x * z + y * z := by
  ext <;> dsimp <;> 

Proof state:
case h₂
q : ℕ+
x y z : X q
⊢ (x.1 + y.1) * z.2 + (x.2 + y.2) * z.1 = x.1 * z.2 + x.2 * z.1 + (y.1 * z.2 + y.2 * z.1)

Next tactic:
ring
instance : Ring (X q) :=
  { inferInstanceAs (AddGroupWithOne (X q)), inferInstanceAs (AddCommGroup (X q)),
      inferInstanceAs (Monoid (X q)) with
    left_distrib := left_distrib
    right_distrib := right_distrib
    mul_zero := fun _ ↦ by ext <;> simp
    zero_mul := fun _ ↦ by 

Proof state:
q : ℕ+
src✝² : AddGroupWithOne (X q) := inferInstanceAs (AddGroupWithOne (X q))
src✝¹ : AddCommGroup (X q) := inferInstanceAs (AddCommGroup (X q))
src✝ : Monoid (X q) := inferInstanceAs (Monoid (X q))
x✝ : X q
⊢ 0 * x✝ = 0

Next tactic:
ext
instance : Ring (X q) :=
  { inferInstanceAs (AddGroupWithOne (X q)), inferInstanceAs (AddCommGroup (X q)),
      inferInstanceAs (Monoid (X q)) with
    left_distrib := left_distrib
    right_distrib := right_distrib
    mul_zero := fun _ ↦ by ext <;> simp
    zero_mul := fun _ ↦ by ext <;> 

Proof state:
case h₁
q : ℕ+
src✝² : AddGroupWithOne (X q) := inferInstanceAs (AddGroupWithOne (X q))
src✝¹ : AddCommGroup (X q) := inferInstanceAs (AddCommGroup (X q))
src✝ : Monoid (X q) := inferInstanceAs (Monoid (X q))
x✝ : X q
⊢ (0 * x✝).1 = 0.1

Next tactic:
simp
instance : Ring (X q) :=
  { inferInstanceAs (AddGroupWithOne (X q)), inferInstanceAs (AddCommGroup (X q)),
      inferInstanceAs (Monoid (X q)) with
    left_distrib := left_distrib
    right_distrib := right_distrib
    mul_zero := fun _ ↦ by ext <;> simp
    zero_mul := fun _ ↦ by ext <;> 

Proof state:
case h₂
q : ℕ+
src✝² : AddGroupWithOne (X q) := inferInstanceAs (AddGroupWithOne (X q))
src✝¹ : AddCommGroup (X q) := inferInstanceAs (AddCommGroup (X q))
src✝ : Monoid (X q) := inferInstanceAs (Monoid (X q))
x✝ : X q
⊢ (0 * x✝).2 = 0.2

Next tactic:
simp
instance : Ring (X q) :=
  { inferInstanceAs (AddGroupWithOne (X q)), inferInstanceAs (AddCommGroup (X q)),
      inferInstanceAs (Monoid (X q)) with
    left_distrib := left_distrib
    right_distrib := right_distrib
    mul_zero := fun _ ↦ by 

Proof state:
q : ℕ+
src✝² : AddGroupWithOne (X q) := inferInstanceAs (AddGroupWithOne (X q))
src✝¹ : AddCommGroup (X q) := inferInstanceAs (AddCommGroup (X q))
src✝ : Monoid (X q) := inferInstanceAs (Monoid (X q))
x✝ : X q
⊢ x✝ * 0 = 0

Next tactic:
ext
instance : Ring (X q) :=
  { inferInstanceAs (AddGroupWithOne (X q)), inferInstanceAs (AddCommGroup (X q)),
      inferInstanceAs (Monoid (X q)) with
    left_distrib := left_distrib
    right_distrib := right_distrib
    mul_zero := fun _ ↦ by ext <;> 

Proof state:
case h₁
q : ℕ+
src✝² : AddGroupWithOne (X q) := inferInstanceAs (AddGroupWithOne (X q))
src✝¹ : AddCommGroup (X q) := inferInstanceAs (AddCommGroup (X q))
src✝ : Monoid (X q) := inferInstanceAs (Monoid (X q))
x✝ : X q
⊢ (x✝ * 0).1 = 0.1

Next tactic:
simp
instance : Ring (X q) :=
  { inferInstanceAs (AddGroupWithOne (X q)), inferInstanceAs (AddCommGroup (X q)),
      inferInstanceAs (Monoid (X q)) with
    left_distrib := left_distrib
    right_distrib := right_distrib
    mul_zero := fun _ ↦ by ext <;> 

Proof state:
case h₂
q : ℕ+
src✝² : AddGroupWithOne (X q) := inferInstanceAs (AddGroupWithOne (X q))
src✝¹ : AddCommGroup (X q) := inferInstanceAs (AddCommGroup (X q))
src✝ : Monoid (X q) := inferInstanceAs (Monoid (X q))
x✝ : X q
⊢ (x✝ * 0).2 = 0.2

Next tactic:
simp
instance : CommRing (X q) :=
  { inferInstanceAs (Ring (X q)) with
    mul_comm := fun _ _ ↦ by 

Proof state:
q : ℕ+
src✝ : Ring (X q) := inferInstanceAs (Ring (X q))
x✝¹ x✝ : X q
⊢ x✝¹ * x✝ = x✝ * x✝¹

Next tactic:
ext
instance : CommRing (X q) :=
  { inferInstanceAs (Ring (X q)) with
    mul_comm := fun _ _ ↦ by ext <;> 

Proof state:
case h₁
q : ℕ+
src✝ : Ring (X q) := inferInstanceAs (Ring (X q))
x✝¹ x✝ : X q
⊢ (x✝¹ * x✝).1 = (x✝ * x✝¹).1

Next tactic:
dsimp
instance : CommRing (X q) :=
  { inferInstanceAs (Ring (X q)) with
    mul_comm := fun _ _ ↦ by ext <;> 

Proof state:
case h₂
q : ℕ+
src✝ : Ring (X q) := inferInstanceAs (Ring (X q))
x✝¹ x✝ : X q
⊢ (x✝¹ * x✝).2 = (x✝ * x✝¹).2

Next tactic:
dsimp
instance : CommRing (X q) :=
  { inferInstanceAs (Ring (X q)) with
    mul_comm := fun _ _ ↦ by ext <;> dsimp <;> 

Proof state:
case h₁
q : ℕ+
src✝ : Ring (X q) := inferInstanceAs (Ring (X q))
x✝¹ x✝ : X q
⊢ x✝¹.1 * x✝.1 + 3 * x✝¹.2 * x✝.2 = x✝.1 * x✝¹.1 + 3 * x✝.2 * x✝¹.2

Next tactic:
ring
instance : CommRing (X q) :=
  { inferInstanceAs (Ring (X q)) with
    mul_comm := fun _ _ ↦ by ext <;> dsimp <;> 

Proof state:
case h₂
q : ℕ+
src✝ : Ring (X q) := inferInstanceAs (Ring (X q))
x✝¹ x✝ : X q
⊢ x✝¹.1 * x✝.2 + x✝¹.2 * x✝.1 = x✝.1 * x✝¹.2 + x✝.2 * x✝¹.1

Next tactic:
ring
@[norm_cast]
theorem coe_mul (n m : ℤ) : ((n * m : ℤ) : X q) = (n : X q) * (m : X q) := by 

Proof state:
q : ℕ+
n m : ℤ
⊢ ↑(n * m) = ↑n * ↑m

Next tactic:
ext
@[norm_cast]
theorem coe_mul (n m : ℤ) : ((n * m : ℤ) : X q) = (n : X q) * (m : X q) := by ext <;> 

Proof state:
case h₁
q : ℕ+
n m : ℤ
⊢ (↑(n * m)).1 = (↑n * ↑m).1

Next tactic:
simp
@[norm_cast]
theorem coe_mul (n m : ℤ) : ((n * m : ℤ) : X q) = (n : X q) * (m : X q) := by ext <;> 

Proof state:
case h₂
q : ℕ+
n m : ℤ
⊢ (↑(n * m)).2 = (↑n * ↑m).2

Next tactic:
simp
@[norm_cast]
theorem coe_nat (n : ℕ) : ((n : ℤ) : X q) = (n : X q) := by 

Proof state:
q : ℕ+
n : ℕ
⊢ ↑↑n = ↑n

Next tactic:
ext
@[norm_cast]
theorem coe_nat (n : ℕ) : ((n : ℤ) : X q) = (n : X q) := by ext <;> 

Proof state:
case h₁
q : ℕ+
n : ℕ
⊢ (↑↑n).1 = (↑n).1

Next tactic:
simp
@[norm_cast]
theorem coe_nat (n : ℕ) : ((n : ℤ) : X q) = (n : X q) := by ext <;> 

Proof state:
case h₂
q : ℕ+
n : ℕ
⊢ (↑↑n).2 = (↑n).2

Next tactic:
simp
/-- The cardinality of `X` is `q^2`. -/
theorem card_eq : Fintype.card (X q) = q ^ 2 := by
  

Proof state:
q : ℕ+
⊢ Fintype.card (X q) = ↑q ^ 2

Next tactic:
dsimp [X]
/-- The cardinality of `X` is `q^2`. -/
theorem card_eq : Fintype.card (X q) = q ^ 2 := by
  dsimp [X]
  

Proof state:
q : ℕ+
⊢ Fintype.card (ZMod ↑q × ZMod ↑q) = ↑q ^ 2

Next tactic:
rw [Fintype.card_prod, ZMod.card q, sq]
/-- There are strictly fewer than `q^2` units, since `0` is not a unit. -/
nonrec theorem card_units_lt (w : 1 < q) : Fintype.card (X q)ˣ < q ^ 2 := by
  

Proof state:
q : ℕ+
w : 1 < q
⊢ Fintype.card (X q)ˣ < ↑q ^ 2

Next tactic:
have : Fact (1 < (q : ℕ)) := ⟨w⟩
/-- There are strictly fewer than `q^2` units, since `0` is not a unit. -/
nonrec theorem card_units_lt (w : 1 < q) : Fintype.card (X q)ˣ < q ^ 2 := by
  have : Fact (1 < (q : ℕ)) := ⟨w⟩
  

Proof state:
q : ℕ+
w : 1 < q
this : Fact (1 < ↑q)
⊢ Fintype.card (X q)ˣ < ↑q ^ 2

Next tactic:
convert card_units_lt (X q)
/-- There are strictly fewer than `q^2` units, since `0` is not a unit. -/
nonrec theorem card_units_lt (w : 1 < q) : Fintype.card (X q)ˣ < q ^ 2 := by
  have : Fact (1 < (q : ℕ)) := ⟨w⟩
  convert card_units_lt (X q)
  

Proof state:
case h.e'_4
q : ℕ+
w : 1 < q
this : Fact (1 < ↑q)
⊢ ↑q ^ 2 = Fintype.card (X q)

Next tactic:
rw [card_eq]
theorem ω_mul_ωb (q : ℕ+) : (ω : X q) * ωb = 1 := by
  

Proof state:
q✝ q : ℕ+
⊢ ω * ωb = 1

Next tactic:
dsimp [ω, ωb]
theorem ω_mul_ωb (q : ℕ+) : (ω : X q) * ωb = 1 := by
  dsimp [ω, ωb]
  

Proof state:
q✝ q : ℕ+
⊢ (2, 1) * (2, -1) = 1

Next tactic:
ext
theorem ω_mul_ωb (q : ℕ+) : (ω : X q) * ωb = 1 := by
  dsimp [ω, ωb]
  ext <;> 

Proof state:
case h₁
q✝ q : ℕ+
⊢ ((2, 1) * (2, -1)).1 = 1.1

Next tactic:
simp
theorem ω_mul_ωb (q : ℕ+) : (ω : X q) * ωb = 1 := by
  dsimp [ω, ωb]
  ext <;> 

Proof state:
case h₂
q✝ q : ℕ+
⊢ ((2, 1) * (2, -1)).2 = 1.2

Next tactic:
simp
theorem ω_mul_ωb (q : ℕ+) : (ω : X q) * ωb = 1 := by
  dsimp [ω, ωb]
  ext <;> simp; 

Proof state:
case h₁
q✝ q : ℕ+
⊢ 2 * 2 + -3 = 1

Next tactic:
ring
theorem ωb_mul_ω (q : ℕ+) : (ωb : X q) * ω = 1 := by
  

Proof state:
q✝ q : ℕ+
⊢ ωb * ω = 1

Next tactic:
rw [mul_comm, ω_mul_ωb]
/-- A closed form for the recurrence relation. -/
theorem closed_form (i : ℕ) : (s i : X q) = (ω : X q) ^ 2 ^ i + (ωb : X q) ^ 2 ^ i := by
  

Proof state:
q : ℕ+
i : ℕ
⊢ ↑(s i) = ω ^ 2 ^ i + ωb ^ 2 ^ i

Next tactic:
induction' i with i ih
/-- A closed form for the recurrence relation. -/
theorem closed_form (i : ℕ) : (s i : X q) = (ω : X q) ^ 2 ^ i + (ωb : X q) ^ 2 ^ i := by
  induction' i with i ih
  · 

Proof state:
case zero
q : ℕ+
⊢ ↑(s zero) = ω ^ 2 ^ zero + ωb ^ 2 ^ zero

Next tactic:
dsimp [s, ω, ωb]
/-- A closed form for the recurrence relation. -/
theorem closed_form (i : ℕ) : (s i : X q) = (ω : X q) ^ 2 ^ i + (ωb : X q) ^ 2 ^ i := by
  induction' i with i ih
  · dsimp [s, ω, ωb]
    

Proof state:
case zero
q : ℕ+
⊢ ↑4 = (2, 1) ^ 1 + (2, -1) ^ 1

Next tactic:
ext
/-- A closed form for the recurrence relation. -/
theorem closed_form (i : ℕ) : (s i : X q) = (ω : X q) ^ 2 ^ i + (ωb : X q) ^ 2 ^ i := by
  induction' i with i ih
  · dsimp [s, ω, ωb]
    ext <;> 

Proof state:
case zero.h₁
q : ℕ+
⊢ (↑4).1 = ((2, 1) ^ 1 + (2, -1) ^ 1).1

Next tactic:
norm_num
/-- A closed form for the recurrence relation. -/
theorem closed_form (i : ℕ) : (s i : X q) = (ω : X q) ^ 2 ^ i + (ωb : X q) ^ 2 ^ i := by
  induction' i with i ih
  · dsimp [s, ω, ωb]
    ext <;> 

Proof state:
case zero.h₂
q : ℕ+
⊢ (↑4).2 = ((2, 1) ^ 1 + (2, -1) ^ 1).2

Next tactic:
norm_num
/-- A closed form for the recurrence relation. -/
theorem closed_form (i : ℕ) : (s i : X q) = (ω : X q) ^ 2 ^ i + (ωb : X q) ^ 2 ^ i := by
  induction' i with i ih
  · dsimp [s, ω, ωb]
    ext <;> norm_num
  · 

Proof state:
case succ
q : ℕ+
i : ℕ
ih : ↑(s i) = ω ^ 2 ^ i + ωb ^ 2 ^ i
⊢ ↑(s (succ i)) = ω ^ 2 ^ succ i + ωb ^ 2 ^ succ i

Next tactic:
calc
      (s (i + 1) : X q) = (s i ^ 2 - 2 : ℤ) := rfl
      _ = (s i : X q) ^ 2 - 2 := by push_cast; rfl
      _ = (ω ^ 2 ^ i + ωb ^ 2 ^ i) ^ 2 - 2 := by rw [ih]
      _ = (ω ^ 2 ^ i) ^ 2 + (ωb ^ 2 ^ i) ^ 2 + 2 * (ωb ^ 2 ^ i * ω ^ 2 ^ i) - 2 := by ring
      _ = (ω ^ 2 ^ i) ^ 2 + (ωb ^ 2 ^ i) ^ 2 := by
        rw [← mul_pow ωb ω, ωb_mul_ω, one_pow, mul_one, add_sub_cancel]
      _ = ω ^ 2 ^ (i + 1) + ωb ^ 2 ^ (i + 1) := by rw [← pow_mul, ← pow_mul, _root_.pow_succ']
/-- A closed form for the recurrence relation. -/
theorem closed_form (i : ℕ) : (s i : X q) = (ω : X q) ^ 2 ^ i + (ωb : X q) ^ 2 ^ i := by
  induction' i with i ih
  · dsimp [s, ω, ωb]
    ext <;> norm_num
  · calc
      (s (i + 1) : X q) = (s i ^ 2 - 2 : ℤ) := rfl
      _ = (s i : X q) ^ 2 - 2 := by 

Proof state:
q : ℕ+
i : ℕ
ih : ↑(s i) = ω ^ 2 ^ i + ωb ^ 2 ^ i
⊢ ↑(s i ^ 2 - 2) = ↑(s i) ^ 2 - 2

Next tactic:
push_cast
/-- A closed form for the recurrence relation. -/
theorem closed_form (i : ℕ) : (s i : X q) = (ω : X q) ^ 2 ^ i + (ωb : X q) ^ 2 ^ i := by
  induction' i with i ih
  · dsimp [s, ω, ωb]
    ext <;> norm_num
  · calc
      (s (i + 1) : X q) = (s i ^ 2 - 2 : ℤ) := rfl
      _ = (s i : X q) ^ 2 - 2 := by push_cast; 

Proof state:
q : ℕ+
i : ℕ
ih : ↑(s i) = ω ^ 2 ^ i + ωb ^ 2 ^ i
⊢ ↑(s i) ^ 2 - 2 = ↑(s i) ^ 2 - 2

Next tactic:
rfl
/-- A closed form for the recurrence relation. -/
theorem closed_form (i : ℕ) : (s i : X q) = (ω : X q) ^ 2 ^ i + (ωb : X q) ^ 2 ^ i := by
  induction' i with i ih
  · dsimp [s, ω, ωb]
    ext <;> norm_num
  · calc
      (s (i + 1) : X q) = (s i ^ 2 - 2 : ℤ) := rfl
      _ = (s i : X q) ^ 2 - 2 := by push_cast; rfl
      _ = (ω ^ 2 ^ i + ωb ^ 2 ^ i) ^ 2 - 2 := by 

Proof state:
q : ℕ+
i : ℕ
ih : ↑(s i) = ω ^ 2 ^ i + ωb ^ 2 ^ i
⊢ ↑(s i) ^ 2 - 2 = (ω ^ 2 ^ i + ωb ^ 2 ^ i) ^ 2 - 2

Next tactic:
rw [ih]
/-- A closed form for the recurrence relation. -/
theorem closed_form (i : ℕ) : (s i : X q) = (ω : X q) ^ 2 ^ i + (ωb : X q) ^ 2 ^ i := by
  induction' i with i ih
  · dsimp [s, ω, ωb]
    ext <;> norm_num
  · calc
      (s (i + 1) : X q) = (s i ^ 2 - 2 : ℤ) := rfl
      _ = (s i : X q) ^ 2 - 2 := by push_cast; rfl
      _ = (ω ^ 2 ^ i + ωb ^ 2 ^ i) ^ 2 - 2 := by rw [ih]
      _ = (ω ^ 2 ^ i) ^ 2 + (ωb ^ 2 ^ i) ^ 2 + 2 * (ωb ^ 2 ^ i * ω ^ 2 ^ i) - 2 := by 

Proof state:
q : ℕ+
i : ℕ
ih : ↑(s i) = ω ^ 2 ^ i + ωb ^ 2 ^ i
⊢ (ω ^ 2 ^ i + ωb ^ 2 ^ i) ^ 2 - 2 = (ω ^ 2 ^ i) ^ 2 + (ωb ^ 2 ^ i) ^ 2 + 2 * (ωb ^ 2 ^ i * ω ^ 2 ^ i) - 2

Next tactic:
ring
/-- A closed form for the recurrence relation. -/
theorem closed_form (i : ℕ) : (s i : X q) = (ω : X q) ^ 2 ^ i + (ωb : X q) ^ 2 ^ i := by
  induction' i with i ih
  · dsimp [s, ω, ωb]
    ext <;> norm_num
  · calc
      (s (i + 1) : X q) = (s i ^ 2 - 2 : ℤ) := rfl
      _ = (s i : X q) ^ 2 - 2 := by push_cast; rfl
      _ = (ω ^ 2 ^ i + ωb ^ 2 ^ i) ^ 2 - 2 := by rw [ih]
      _ = (ω ^ 2 ^ i) ^ 2 + (ωb ^ 2 ^ i) ^ 2 + 2 * (ωb ^ 2 ^ i * ω ^ 2 ^ i) - 2 := by ring
      _ = (ω ^ 2 ^ i) ^ 2 + (ωb ^ 2 ^ i) ^ 2 := by
        

Proof state:
q : ℕ+
i : ℕ
ih : ↑(s i) = ω ^ 2 ^ i + ωb ^ 2 ^ i
⊢ (ω ^ 2 ^ i) ^ 2 + (ωb ^ 2 ^ i) ^ 2 + 2 * (ωb ^ 2 ^ i * ω ^ 2 ^ i) - 2 = (ω ^ 2 ^ i) ^ 2 + (ωb ^ 2 ^ i) ^ 2

Next tactic:
rw [← mul_pow ωb ω, ωb_mul_ω, one_pow, mul_one, add_sub_cancel]
/-- A closed form for the recurrence relation. -/
theorem closed_form (i : ℕ) : (s i : X q) = (ω : X q) ^ 2 ^ i + (ωb : X q) ^ 2 ^ i := by
  induction' i with i ih
  · dsimp [s, ω, ωb]
    ext <;> norm_num
  · calc
      (s (i + 1) : X q) = (s i ^ 2 - 2 : ℤ) := rfl
      _ = (s i : X q) ^ 2 - 2 := by push_cast; rfl
      _ = (ω ^ 2 ^ i + ωb ^ 2 ^ i) ^ 2 - 2 := by rw [ih]
      _ = (ω ^ 2 ^ i) ^ 2 + (ωb ^ 2 ^ i) ^ 2 + 2 * (ωb ^ 2 ^ i * ω ^ 2 ^ i) - 2 := by ring
      _ = (ω ^ 2 ^ i) ^ 2 + (ωb ^ 2 ^ i) ^ 2 := by
        rw [← mul_pow ωb ω, ωb_mul_ω, one_pow, mul_one, add_sub_cancel]
      _ = ω ^ 2 ^ (i + 1) + ωb ^ 2 ^ (i + 1) := by 

Proof state:
q : ℕ+
i : ℕ
ih : ↑(s i) = ω ^ 2 ^ i + ωb ^ 2 ^ i
⊢ (ω ^ 2 ^ i) ^ 2 + (ωb ^ 2 ^ i) ^ 2 = ω ^ 2 ^ (i + 1) + ωb ^ 2 ^ (i + 1)

Next tactic:
rw [← pow_mul, ← pow_mul, _root_.pow_succ']
/-- If `1 < p`, then `q p`, the smallest prime factor of `mersenne p`, is more than 2. -/
theorem two_lt_q (p' : ℕ) : 2 < q (p' + 2) := by
  

Proof state:
p' : ℕ
⊢ 2 < q (p' + 2)

Next tactic:
refine (minFac_prime (one_lt_mersenne ?_).ne').two_le.lt_of_ne' ?_
/-- If `1 < p`, then `q p`, the smallest prime factor of `mersenne p`, is more than 2. -/
theorem two_lt_q (p' : ℕ) : 2 < q (p' + 2) := by
  refine (minFac_prime (one_lt_mersenne ?_).ne').two_le.lt_of_ne' ?_
  · 

Proof state:
case refine_1
p' : ℕ
⊢ 1 < p' + 2

Next tactic:
exact le_add_left _ _
/-- If `1 < p`, then `q p`, the smallest prime factor of `mersenne p`, is more than 2. -/
theorem two_lt_q (p' : ℕ) : 2 < q (p' + 2) := by
  refine (minFac_prime (one_lt_mersenne ?_).ne').two_le.lt_of_ne' ?_
  · exact le_add_left _ _
  · 

Proof state:
case refine_2
p' : ℕ
⊢ minFac (mersenne (p' + 2)) ≠ 2

Next tactic:
rw [Ne.def, minFac_eq_two_iff, mersenne, Nat.pow_succ']
/-- If `1 < p`, then `q p`, the smallest prime factor of `mersenne p`, is more than 2. -/
theorem two_lt_q (p' : ℕ) : 2 < q (p' + 2) := by
  refine (minFac_prime (one_lt_mersenne ?_).ne').two_le.lt_of_ne' ?_
  · exact le_add_left _ _
  · rw [Ne.def, minFac_eq_two_iff, mersenne, Nat.pow_succ']
    

Proof state:
case refine_2
p' : ℕ
⊢ ¬2 ∣ 2 * 2 ^ (p' + 1) - 1

Next tactic:
exact Nat.two_not_dvd_two_mul_sub_one (Nat.one_le_two_pow _)
theorem ω_pow_formula (p' : ℕ) (h : lucasLehmerResidue (p' + 2) = 0) :
    ∃ k : ℤ,
      (ω : X (q (p' + 2))) ^ 2 ^ (p' + 1) =
        k * mersenne (p' + 2) * (ω : X (q (p' + 2))) ^ 2 ^ p' - 1 := by
  

Proof state:
p' : ℕ
h : lucasLehmerResidue (p' + 2) = 0
⊢ ∃ k, ω ^ 2 ^ (p' + 1) = ↑k * ↑(mersenne (p' + 2)) * ω ^ 2 ^ p' - 1

Next tactic:
dsimp [lucasLehmerResidue] at h
theorem ω_pow_formula (p' : ℕ) (h : lucasLehmerResidue (p' + 2) = 0) :
    ∃ k : ℤ,
      (ω : X (q (p' + 2))) ^ 2 ^ (p' + 1) =
        k * mersenne (p' + 2) * (ω : X (q (p' + 2))) ^ 2 ^ p' - 1 := by
  dsimp [lucasLehmerResidue] at h
  

Proof state:
p' : ℕ
h : sZMod (p' + 2) (p' + 2 - 2) = 0
⊢ ∃ k, ω ^ 2 ^ (p' + 1) = ↑k * ↑(mersenne (p' + 2)) * ω ^ 2 ^ p' - 1

Next tactic:
rw [sZMod_eq_s p'] at h
theorem ω_pow_formula (p' : ℕ) (h : lucasLehmerResidue (p' + 2) = 0) :
    ∃ k : ℤ,
      (ω : X (q (p' + 2))) ^ 2 ^ (p' + 1) =
        k * mersenne (p' + 2) * (ω : X (q (p' + 2))) ^ 2 ^ p' - 1 := by
  dsimp [lucasLehmerResidue] at h
  rw [sZMod_eq_s p'] at h
  

Proof state:
p' : ℕ
h : ↑(s (p' + 2 - 2)) = 0
⊢ ∃ k, ω ^ 2 ^ (p' + 1) = ↑k * ↑(mersenne (p' + 2)) * ω ^ 2 ^ p' - 1

Next tactic:
simp? [ZMod.int_cast_zmod_eq_zero_iff_dvd] at h says
    simp only [ge_iff_le, add_le_iff_nonpos_left, nonpos_iff_eq_zero, add_tsub_cancel_right,
      ZMod.int_cast_zmod_eq_zero_iff_dvd, gt_iff_lt, zero_lt_two, pow_pos, cast_pred, cast_pow,
      cast_ofNat] at h
theorem ω_pow_formula (p' : ℕ) (h : lucasLehmerResidue (p' + 2) = 0) :
    ∃ k : ℤ,
      (ω : X (q (p' + 2))) ^ 2 ^ (p' + 1) =
        k * mersenne (p' + 2) * (ω : X (q (p' + 2))) ^ 2 ^ p' - 1 := by
  dsimp [lucasLehmerResidue] at h
  rw [sZMod_eq_s p'] at h
  simp? [ZMod.int_cast_zmod_eq_zero_iff_dvd] at h says
    

Proof state:
p' : ℕ
h : ↑(s (p' + 2 - 2)) = 0
⊢ ∃ k, ω ^ 2 ^ (p' + 1) = ↑k * ↑(mersenne (p' + 2)) * ω ^ 2 ^ p' - 1

Next tactic:
simp only [ge_iff_le, add_le_iff_nonpos_left, nonpos_iff_eq_zero, add_tsub_cancel_right,
      ZMod.int_cast_zmod_eq_zero_iff_dvd, gt_iff_lt, zero_lt_two, pow_pos, cast_pred, cast_pow,
      cast_ofNat] at h
theorem ω_pow_formula (p' : ℕ) (h : lucasLehmerResidue (p' + 2) = 0) :
    ∃ k : ℤ,
      (ω : X (q (p' + 2))) ^ 2 ^ (p' + 1) =
        k * mersenne (p' + 2) * (ω : X (q (p' + 2))) ^ 2 ^ p' - 1 := by
  dsimp [lucasLehmerResidue] at h
  rw [sZMod_eq_s p'] at h
  simp? [ZMod.int_cast_zmod_eq_zero_iff_dvd] at h says
    simp only [ge_iff_le, add_le_iff_nonpos_left, nonpos_iff_eq_zero, add_tsub_cancel_right,
      ZMod.int_cast_zmod_eq_zero_iff_dvd, gt_iff_lt, zero_lt_two, pow_pos, cast_pred, cast_pow,
      cast_ofNat] at h
  

Proof state:
p' : ℕ
h : 2 ^ (p' + 2) - 1 ∣ s p'
⊢ ∃ k, ω ^ 2 ^ (p' + 1) = ↑k * ↑(mersenne (p' + 2)) * ω ^ 2 ^ p' - 1

Next tactic:
cases' h with k h
theorem ω_pow_formula (p' : ℕ) (h : lucasLehmerResidue (p' + 2) = 0) :
    ∃ k : ℤ,
      (ω : X (q (p' + 2))) ^ 2 ^ (p' + 1) =
        k * mersenne (p' + 2) * (ω : X (q (p' + 2))) ^ 2 ^ p' - 1 := by
  dsimp [lucasLehmerResidue] at h
  rw [sZMod_eq_s p'] at h
  simp? [ZMod.int_cast_zmod_eq_zero_iff_dvd] at h says
    simp only [ge_iff_le, add_le_iff_nonpos_left, nonpos_iff_eq_zero, add_tsub_cancel_right,
      ZMod.int_cast_zmod_eq_zero_iff_dvd, gt_iff_lt, zero_lt_two, pow_pos, cast_pred, cast_pow,
      cast_ofNat] at h
  cases' h with k h
  

Proof state:
case intro
p' : ℕ
k : ℤ
h : s p' = (2 ^ (p' + 2) - 1) * k
⊢ ∃ k, ω ^ 2 ^ (p' + 1) = ↑k * ↑(mersenne (p' + 2)) * ω ^ 2 ^ p' - 1

Next tactic:
use k
theorem ω_pow_formula (p' : ℕ) (h : lucasLehmerResidue (p' + 2) = 0) :
    ∃ k : ℤ,
      (ω : X (q (p' + 2))) ^ 2 ^ (p' + 1) =
        k * mersenne (p' + 2) * (ω : X (q (p' + 2))) ^ 2 ^ p' - 1 := by
  dsimp [lucasLehmerResidue] at h
  rw [sZMod_eq_s p'] at h
  simp? [ZMod.int_cast_zmod_eq_zero_iff_dvd] at h says
    simp only [ge_iff_le, add_le_iff_nonpos_left, nonpos_iff_eq_zero, add_tsub_cancel_right,
      ZMod.int_cast_zmod_eq_zero_iff_dvd, gt_iff_lt, zero_lt_two, pow_pos, cast_pred, cast_pow,
      cast_ofNat] at h
  cases' h with k h
  use k
  

Proof state:
case h
p' : ℕ
k : ℤ
h : s p' = (2 ^ (p' + 2) - 1) * k
⊢ ω ^ 2 ^ (p' + 1) = ↑k * ↑(mersenne (p' + 2)) * ω ^ 2 ^ p' - 1

Next tactic:
replace h := congr_arg (fun n : ℤ => (n : X (q (p' + 2)))) h
theorem ω_pow_formula (p' : ℕ) (h : lucasLehmerResidue (p' + 2) = 0) :
    ∃ k : ℤ,
      (ω : X (q (p' + 2))) ^ 2 ^ (p' + 1) =
        k * mersenne (p' + 2) * (ω : X (q (p' + 2))) ^ 2 ^ p' - 1 := by
  dsimp [lucasLehmerResidue] at h
  rw [sZMod_eq_s p'] at h
  simp? [ZMod.int_cast_zmod_eq_zero_iff_dvd] at h says
    simp only [ge_iff_le, add_le_iff_nonpos_left, nonpos_iff_eq_zero, add_tsub_cancel_right,
      ZMod.int_cast_zmod_eq_zero_iff_dvd, gt_iff_lt, zero_lt_two, pow_pos, cast_pred, cast_pow,
      cast_ofNat] at h
  cases' h with k h
  use k
  replace h := congr_arg (fun n : ℤ => (n : X (q (p' + 2)))) h
  -- coercion from ℤ to X q
  

Proof state:
case h
p' : ℕ
k : ℤ
h : (fun n => ↑n) (s p') = (fun n => ↑n) ((2 ^ (p' + 2) - 1) * k)
⊢ ω ^ 2 ^ (p' + 1) = ↑k * ↑(mersenne (p' + 2)) * ω ^ 2 ^ p' - 1

Next tactic:
dsimp at h
theorem ω_pow_formula (p' : ℕ) (h : lucasLehmerResidue (p' + 2) = 0) :
    ∃ k : ℤ,
      (ω : X (q (p' + 2))) ^ 2 ^ (p' + 1) =
        k * mersenne (p' + 2) * (ω : X (q (p' + 2))) ^ 2 ^ p' - 1 := by
  dsimp [lucasLehmerResidue] at h
  rw [sZMod_eq_s p'] at h
  simp? [ZMod.int_cast_zmod_eq_zero_iff_dvd] at h says
    simp only [ge_iff_le, add_le_iff_nonpos_left, nonpos_iff_eq_zero, add_tsub_cancel_right,
      ZMod.int_cast_zmod_eq_zero_iff_dvd, gt_iff_lt, zero_lt_two, pow_pos, cast_pred, cast_pow,
      cast_ofNat] at h
  cases' h with k h
  use k
  replace h := congr_arg (fun n : ℤ => (n : X (q (p' + 2)))) h
  -- coercion from ℤ to X q
  dsimp at h
  

Proof state:
case h
p' : ℕ
k : ℤ
h : ↑(s p') = ↑((2 ^ (p' + 2) - 1) * k)
⊢ ω ^ 2 ^ (p' + 1) = ↑k * ↑(mersenne (p' + 2)) * ω ^ 2 ^ p' - 1

Next tactic:
rw [closed_form] at h