awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.LFunction (goal)
Goal:
R : Type u_1
inst : CommSemiring R
q : ℕ
f g : PowerSeries R
hq : 1 < q
k : ℕ
hs :
  Finset.map
      ({ toFun := fun (k : ℕ) ↦ q ^ k, inj' := ⋯ : ℕ ↪ ℕ }.prodMap { toFun := fun (k : ℕ) ↦ q ^ k, inj' := ⋯ : ℕ ↪ ℕ })
      (@Finset.antidiagonal _ Nat.instAddMonoid _ k) ⊆
    (q ^ k).divisorsAntidiagonal
a b : ℕ
hab : (a, b) ∈ (q ^ k).divisorsAntidiagonal
h :
  (a, b) ∉
    Finset.map
      ({ toFun := fun (k : ℕ) ↦ q ^ k, inj' := ⋯ : ℕ ↪ ℕ }.prodMap { toFun := fun (k : ℕ) ↦ q ^ k, inj' := ⋯ : ℕ ↪ ℕ })
      (@Finset.antidiagonal _ Nat.instAddMonoid _ k)
ha : ∃ (i : ℕ), q ^ i = a
hb : ¬∃ (j : ℕ), q ^ j = b
neg :
  HMul.hMul (α := R)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) g) 0
        (a, b).2)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) f) 0
        (a, b).1) =
    0
neg₁ :
  HMul.hMul (α := R) (0 (a, b).2)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) f) 0
        (a, b).1) =
    0
⊢ HMul.hMul (α := R)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) f) 0
        (a, b).1)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) g) 0
        (a, b).2) =
    0
have neg₂ : ¬∃ (a_1 : ℕ), q ^ a_1 = (a, b).2
Tactic: rwa [mul_comm, Function.extend_apply', Pi.zero_apply, zero_mul]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.LFunction (goal)
Goal:
R : Type u_1
inst : CommSemiring R
q : ℕ
f g : PowerSeries R
hq : 1 < q
k : ℕ
hs :
  Finset.map
      ({ toFun := fun (k : ℕ) ↦ q ^ k, inj' := ⋯ : ℕ ↪ ℕ }.prodMap { toFun := fun (k : ℕ) ↦ q ^ k, inj' := ⋯ : ℕ ↪ ℕ })
      (@Finset.antidiagonal _ Nat.instAddMonoid _ k) ⊆
    (q ^ k).divisorsAntidiagonal
a b : ℕ
hab : (a, b) ∈ (q ^ k).divisorsAntidiagonal
h :
  (a, b) ∉
    Finset.map
      ({ toFun := fun (k : ℕ) ↦ q ^ k, inj' := ⋯ : ℕ ↪ ℕ }.prodMap { toFun := fun (k : ℕ) ↦ q ^ k, inj' := ⋯ : ℕ ↪ ℕ })
      (@Finset.antidiagonal _ Nat.instAddMonoid _ k)
ha : ∃ (i : ℕ), q ^ i = a
hb : ¬∃ (j : ℕ), q ^ j = b
neg :
  HMul.hMul (α := R)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) g) 0
        (a, b).2)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) f) 0
        (a, b).1) =
    0
neg₁ :
  HMul.hMul (α := R) (0 (a, b).2)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) f) 0
        (a, b).1) =
    0
neg₂ : ¬∃ (a_1 : ℕ), q ^ a_1 = (a, b).2
⊢ HMul.hMul (α := R)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) f) 0
        (a, b).1)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) g) 0
        (a, b).2) =
    0
have neg₃ : HMul.hMul (α := R) 0
    (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) f) 0
      (a, b).1) =
  0
Tactic: rwa [mul_comm, Function.extend_apply', Pi.zero_apply, zero_mul]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.LFunction (goal)
Goal:
R : Type u_1
inst : CommSemiring R
q : ℕ
f g : PowerSeries R
hq : 1 < q
k : ℕ
hs :
  Finset.map
      ({ toFun := fun (k : ℕ) ↦ q ^ k, inj' := ⋯ : ℕ ↪ ℕ }.prodMap { toFun := fun (k : ℕ) ↦ q ^ k, inj' := ⋯ : ℕ ↪ ℕ })
      (@Finset.antidiagonal _ Nat.instAddMonoid _ k) ⊆
    (q ^ k).divisorsAntidiagonal
a b : ℕ
hab : (a, b) ∈ (q ^ k).divisorsAntidiagonal
h :
  (a, b) ∉
    Finset.map
      ({ toFun := fun (k : ℕ) ↦ q ^ k, inj' := ⋯ : ℕ ↪ ℕ }.prodMap { toFun := fun (k : ℕ) ↦ q ^ k, inj' := ⋯ : ℕ ↪ ℕ })
      (@Finset.antidiagonal _ Nat.instAddMonoid _ k)
ha : ∃ (i : ℕ), q ^ i = a
hb : ¬∃ (j : ℕ), q ^ j = b
neg :
  HMul.hMul (α := R)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) g) 0
        (a, b).2)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) f) 0
        (a, b).1) =
    0
neg₁ :
  HMul.hMul (α := R) (0 (a, b).2)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) f) 0
        (a, b).1) =
    0
neg₂ : ¬∃ (a_1 : ℕ), q ^ a_1 = (a, b).2
neg₃ :
  HMul.hMul (α := R) 0
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) f) 0
        (a, b).1) =
    0
neg₄ : ¬∃ (a_1 : ℕ), q ^ a_1 = (a, b).2
⊢ HMul.hMul (α := R)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) f) 0
        (a, b).1)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) g) 0
        (a, b).2) =
    0
have neg₅ : Eq.{u_1 + 1} (α := R) 0 0
Tactic: rwa [mul_comm, Function.extend_apply', Pi.zero_apply, zero_mul]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.LFunction (goal)
Goal:
R : Type u_1
inst : CommSemiring R
q : ℕ
f g : PowerSeries R
hq : 1 < q
k : ℕ
hs :
  Finset.map
      ({ toFun := fun (k : ℕ) ↦ q ^ k, inj' := ⋯ : ℕ ↪ ℕ }.prodMap { toFun := fun (k : ℕ) ↦ q ^ k, inj' := ⋯ : ℕ ↪ ℕ })
      (@Finset.antidiagonal _ Nat.instAddMonoid _ k) ⊆
    (q ^ k).divisorsAntidiagonal
a b : ℕ
hab : (a, b) ∈ (q ^ k).divisorsAntidiagonal
h :
  (a, b) ∉
    Finset.map
      ({ toFun := fun (k : ℕ) ↦ q ^ k, inj' := ⋯ : ℕ ↪ ℕ }.prodMap { toFun := fun (k : ℕ) ↦ q ^ k, inj' := ⋯ : ℕ ↪ ℕ })
      (@Finset.antidiagonal _ Nat.instAddMonoid _ k)
ha : ¬∃ (i : ℕ), q ^ i = a
⊢ HMul.hMul (α := R)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) f) 0
        (a, b).1)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) g) 0
        (a, b).2) =
    0
have neg : HMul.hMul (α := R) (0 (a, b).1)
    (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) g) 0
      (a, b).2) =
  0
Tactic: rwa [Function.extend_apply', Pi.zero_apply, zero_mul]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.LFunction (goal)
Goal:
R : Type u_1
inst : CommSemiring R
q : ℕ
f g : PowerSeries R
hq : 1 < q
k : ℕ
hs :
  Finset.map
      ({ toFun := fun (k : ℕ) ↦ q ^ k, inj' := ⋯ : ℕ ↪ ℕ }.prodMap { toFun := fun (k : ℕ) ↦ q ^ k, inj' := ⋯ : ℕ ↪ ℕ })
      (@Finset.antidiagonal _ Nat.instAddMonoid _ k) ⊆
    (q ^ k).divisorsAntidiagonal
a b : ℕ
hab : (a, b) ∈ (q ^ k).divisorsAntidiagonal
h :
  (a, b) ∉
    Finset.map
      ({ toFun := fun (k : ℕ) ↦ q ^ k, inj' := ⋯ : ℕ ↪ ℕ }.prodMap { toFun := fun (k : ℕ) ↦ q ^ k, inj' := ⋯ : ℕ ↪ ℕ })
      (@Finset.antidiagonal _ Nat.instAddMonoid _ k)
ha : ¬∃ (i : ℕ), q ^ i = a
neg :
  HMul.hMul (α := R) (0 (a, b).1)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) g) 0
        (a, b).2) =
    0
⊢ HMul.hMul (α := R)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) f) 0
        (a, b).1)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) g) 0
        (a, b).2) =
    0
have neg₁ : ¬∃ (a_1 : ℕ), q ^ a_1 = (a, b).1
Tactic: rwa [Function.extend_apply', Pi.zero_apply, zero_mul]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.LFunction (goal)
Goal:
R : Type u_1
inst : CommSemiring R
q : ℕ
f g : PowerSeries R
hq : 1 < q
k : ℕ
hs :
  Finset.map
      ({ toFun := fun (k : ℕ) ↦ q ^ k, inj' := ⋯ : ℕ ↪ ℕ }.prodMap { toFun := fun (k : ℕ) ↦ q ^ k, inj' := ⋯ : ℕ ↪ ℕ })
      (@Finset.antidiagonal _ Nat.instAddMonoid _ k) ⊆
    (q ^ k).divisorsAntidiagonal
a b : ℕ
hab : (a, b) ∈ (q ^ k).divisorsAntidiagonal
h :
  (a, b) ∉
    Finset.map
      ({ toFun := fun (k : ℕ) ↦ q ^ k, inj' := ⋯ : ℕ ↪ ℕ }.prodMap { toFun := fun (k : ℕ) ↦ q ^ k, inj' := ⋯ : ℕ ↪ ℕ })
      (@Finset.antidiagonal _ Nat.instAddMonoid _ k)
ha : ¬∃ (i : ℕ), q ^ i = a
neg :
  HMul.hMul (α := R) (0 (a, b).1)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) g) 0
        (a, b).2) =
    0
neg₁ : ¬∃ (a_1 : ℕ), q ^ a_1 = (a, b).1
⊢ HMul.hMul (α := R)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) f) 0
        (a, b).1)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) g) 0
        (a, b).2) =
    0
have neg₂ : HMul.hMul (α := R) 0
    (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) g) 0
      (a, b).2) =
  0
Tactic: rwa [Function.extend_apply', Pi.zero_apply, zero_mul]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.LFunction (goal)
Goal:
R : Type u_1
inst : CommSemiring R
q : ℕ
f g : PowerSeries R
hq : 1 < q
k : ℕ
hs :
  Finset.map
      ({ toFun := fun (k : ℕ) ↦ q ^ k, inj' := ⋯ : ℕ ↪ ℕ }.prodMap { toFun := fun (k : ℕ) ↦ q ^ k, inj' := ⋯ : ℕ ↪ ℕ })
      (@Finset.antidiagonal _ Nat.instAddMonoid _ k) ⊆
    (q ^ k).divisorsAntidiagonal
a b : ℕ
hab : (a, b) ∈ (q ^ k).divisorsAntidiagonal
h :
  (a, b) ∉
    Finset.map
      ({ toFun := fun (k : ℕ) ↦ q ^ k, inj' := ⋯ : ℕ ↪ ℕ }.prodMap { toFun := fun (k : ℕ) ↦ q ^ k, inj' := ⋯ : ℕ ↪ ℕ })
      (@Finset.antidiagonal _ Nat.instAddMonoid _ k)
ha : ¬∃ (i : ℕ), q ^ i = a
neg :
  HMul.hMul (α := R) (0 (a, b).1)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) g) 0
        (a, b).2) =
    0
neg₁ : ¬∃ (a_1 : ℕ), q ^ a_1 = (a, b).1
neg₂ :
  HMul.hMul (α := R) 0
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) g) 0
        (a, b).2) =
    0
neg₃ : ¬∃ (a_1 : ℕ), q ^ a_1 = (a, b).1
⊢ HMul.hMul (α := R)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) f) 0
        (a, b).1)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) g) 0
        (a, b).2) =
    0
have neg₄ : Eq.{u_1 + 1} (α := R) 0 0
Tactic: rwa [Function.extend_apply', Pi.zero_apply, zero_mul]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.LFunction (goal)
Goal:
R : Type u_1
inst : CommSemiring R
q : ℕ
f g : PowerSeries R
n : ℕ
hq : 1 < q
hn : ¬∃ (a : ℕ), q ^ a = n
⊢ Eq.{u_1 + 1} (α := R)
    (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) (f * g)) 0 n)
    (∑ x ∈ n.divisorsAntidiagonal,
      HMul.hMul (α := R)
        (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) f) 0 x.1)
        (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) g) 0 x.2))
have neg : ∀ x ∈ n.divisorsAntidiagonal,
  HMul.hMul (α := R)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) f) 0 x.1)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) g) 0 x.2) =
    0
Tactic: rw [Function.extend_apply' _ _ _ hn, Pi.zero_apply, Finset.sum_eq_zero]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.LFunction (hypothesis)
Goal:
R : Type u_1
inst : CommSemiring R
q : ℕ
f g : PowerSeries R
n : ℕ
hq : 1 < q
hn : ¬∃ (a : ℕ), q ^ a = n
a b : ℕ
hk : (a, b) ∈ n.divisorsAntidiagonal
⊢ HMul.hMul (α := R)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) f) 0
        (a, b).1)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) g) 0
        (a, b).2) =
    0
have hab : (a, b).1 * (a, b).2 = n
Tactic: obtain ⟨hab, -⟩ := Nat.mem_divisorsAntidiagonal.mp hk
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.LFunction (goal)
Goal:
R : Type u_1
inst : CommSemiring R
q : ℕ
f g : PowerSeries R
n : ℕ
hq : 1 < q
hn : ¬∃ (a : ℕ), q ^ a = n
a b : ℕ
hk : (a, b) ∈ n.divisorsAntidiagonal
hab : (a, b).1 * (a, b).2 = n
⊢ HMul.hMul (α := R)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) f) 0
        (a, b).1)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) g) 0
        (a, b).2) =
    0
have pos : ∀ (ha : ∃ (i : ℕ), q ^ i = a),
  HMul.hMul (α := R)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) f) 0
        (a, b).1)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) g) 0
        (a, b).2) =
    0
Tactic: by_cases ha : ∃ i, q ^ i = a
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.LFunction (goal)
Goal:
R : Type u_1
inst : CommSemiring R
q : ℕ
f g : PowerSeries R
n : ℕ
hq : 1 < q
hn : ¬∃ (a : ℕ), q ^ a = n
a b : ℕ
hk : (a, b) ∈ n.divisorsAntidiagonal
hab : (a, b).1 * (a, b).2 = n
pos :
  ∀ (ha : ∃ (i : ℕ), q ^ i = a),
    HMul.hMul (α := R)
        (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) f) 0
          (a, b).1)
        (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) g) 0
          (a, b).2) =
      0
⊢ HMul.hMul (α := R)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) f) 0
        (a, b).1)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) g) 0
        (a, b).2) =
    0
have neg : ∀ (ha : ¬∃ (i : ℕ), q ^ i = a),
  HMul.hMul (α := R)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) f) 0
        (a, b).1)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) g) 0
        (a, b).2) =
    0
Tactic: by_cases ha : ∃ i, q ^ i = a
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.LFunction (goal)
Goal:
R : Type u_1
inst : CommSemiring R
q : ℕ
f g : PowerSeries R
n : ℕ
hq : 1 < q
hn : ¬∃ (a : ℕ), q ^ a = n
a b : ℕ
hk : (a, b) ∈ n.divisorsAntidiagonal
hab : (a, b).1 * (a, b).2 = n
ha : ∃ (i : ℕ), q ^ i = a
⊢ HMul.hMul (α := R)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) f) 0
        (a, b).1)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) g) 0
        (a, b).2) =
    0
have pos : ∀ (hb : ∃ (j : ℕ), q ^ j = b),
  HMul.hMul (α := R)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) f) 0
        (a, b).1)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) g) 0
        (a, b).2) =
    0
Tactic: by_cases hb : ∃ j, q ^ j = b
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.LFunction (goal)
Goal:
R : Type u_1
inst : CommSemiring R
q : ℕ
f g : PowerSeries R
n : ℕ
hq : 1 < q
hn : ¬∃ (a : ℕ), q ^ a = n
a b : ℕ
hk : (a, b) ∈ n.divisorsAntidiagonal
hab : (a, b).1 * (a, b).2 = n
ha : ∃ (i : ℕ), q ^ i = a
pos :
  ∀ (hb : ∃ (j : ℕ), q ^ j = b),
    HMul.hMul (α := R)
        (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) f) 0
          (a, b).1)
        (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) g) 0
          (a, b).2) =
      0
⊢ HMul.hMul (α := R)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) f) 0
        (a, b).1)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) g) 0
        (a, b).2) =
    0
have neg : ∀ (hb : ¬∃ (j : ℕ), q ^ j = b),
  HMul.hMul (α := R)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) f) 0
        (a, b).1)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) g) 0
        (a, b).2) =
    0
Tactic: by_cases hb : ∃ j, q ^ j = b
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.LFunction (goal)
Goal:
R : Type u_1
inst : CommSemiring R
q : ℕ
f g : PowerSeries R
n : ℕ
hq : 1 < q
hn : ¬∃ (a : ℕ), q ^ a = n
a b : ℕ
hk : (a, b) ∈ n.divisorsAntidiagonal
hab : (a, b).1 * (a, b).2 = n
ha : ∃ (i : ℕ), q ^ i = a
hb : ∃ (j : ℕ), q ^ j = b
⊢ HMul.hMul (α := R)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) f) 0
        (a, b).1)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) g) 0
        (a, b).2) =
    0
have pos : ∀ (i : ℕ) (hk : (q ^ i, b) ∈ n.divisorsAntidiagonal) (hab : (q ^ i, b).1 * (q ^ i, b).2 = n),
  HMul.hMul (α := R)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) f) 0
        (q ^ i, b).1)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) g) 0
        (q ^ i, b).2) =
    0
Tactic: obtain ⟨i, rfl⟩ := ha
Cleared hypotheses: a, hk, hab, ha
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.LFunction (goal)
Goal:
R : Type u_1
inst : CommSemiring R
q : ℕ
f g : PowerSeries R
n : ℕ
hq : 1 < q
hn : ¬∃ (a : ℕ), q ^ a = n
b : ℕ
hb : ∃ (j : ℕ), q ^ j = b
i : ℕ
hk : (q ^ i, b) ∈ n.divisorsAntidiagonal
hab : (q ^ i, b).1 * (q ^ i, b).2 = n
⊢ HMul.hMul (α := R)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) f) 0
        (q ^ i, b).1)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) g) 0
        (q ^ i, b).2) =
    0
have pos : ∀ (j : ℕ) (hk : (q ^ i, q ^ j) ∈ n.divisorsAntidiagonal) (hab : (q ^ i, q ^ j).1 * (q ^ i, q ^ j).2 = n),
  HMul.hMul (α := R)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) f) 0
        (q ^ i, q ^ j).1)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) g) 0
        (q ^ i, q ^ j).2) =
    0
Tactic: obtain ⟨j, rfl⟩ := hb
Cleared hypotheses: b, hb, hk, hab
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.LFunction (hypothesis)
Goal:
R : Type u_1
inst : CommSemiring R
q : ℕ
f g : PowerSeries R
n : ℕ
hq : 1 < q
hn : ¬∃ (a : ℕ), q ^ a = n
i j : ℕ
hk : (q ^ i, q ^ j) ∈ n.divisorsAntidiagonal
hab : (q ^ i, q ^ j).1 * (q ^ i, q ^ j).2 = n
⊢ HMul.hMul (α := R)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) f) 0
        (q ^ i, q ^ j).1)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) g) 0
        (q ^ i, q ^ j).2) =
    0
have hab₁ : q ^ (i + j) = n
Tactic: rw [← pow_add] at hab
Cleared hypotheses: hab
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.LFunction (goal)
Goal:
R : Type u_1
inst : CommSemiring R
q : ℕ
f g : PowerSeries R
n : ℕ
hq : 1 < q
hn : ¬∃ (a : ℕ), q ^ a = n
a b : ℕ
hk : (a, b) ∈ n.divisorsAntidiagonal
hab : (a, b).1 * (a, b).2 = n
ha : ∃ (i : ℕ), q ^ i = a
hb : ¬∃ (j : ℕ), q ^ j = b
⊢ HMul.hMul (α := R)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) f) 0
        (a, b).1)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) g) 0
        (a, b).2) =
    0
have neg : HMul.hMul (α := R)
    (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) g) 0 (a, b).2)
    (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) f) 0
      (a, b).1) =
  0
Tactic: rwa [mul_comm, Function.extend_apply', Pi.zero_apply, zero_mul]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.LFunction (goal)
Goal:
R : Type u_1
inst : CommSemiring R
q : ℕ
f g : PowerSeries R
n : ℕ
hq : 1 < q
hn : ¬∃ (a : ℕ), q ^ a = n
a b : ℕ
hk : (a, b) ∈ n.divisorsAntidiagonal
hab : (a, b).1 * (a, b).2 = n
ha : ∃ (i : ℕ), q ^ i = a
hb : ¬∃ (j : ℕ), q ^ j = b
neg :
  HMul.hMul (α := R)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) g) 0
        (a, b).2)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) f) 0
        (a, b).1) =
    0
⊢ HMul.hMul (α := R)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) f) 0
        (a, b).1)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) g) 0
        (a, b).2) =
    0
have neg₁ : HMul.hMul (α := R) (0 (a, b).2)
    (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) f) 0
      (a, b).1) =
  0
Tactic: rwa [mul_comm, Function.extend_apply', Pi.zero_apply, zero_mul]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.LFunction (goal)
Goal:
R : Type u_1
inst : CommSemiring R
q : ℕ
f g : PowerSeries R
n : ℕ
hq : 1 < q
hn : ¬∃ (a : ℕ), q ^ a = n
a b : ℕ
hk : (a, b) ∈ n.divisorsAntidiagonal
hab : (a, b).1 * (a, b).2 = n
ha : ∃ (i : ℕ), q ^ i = a
hb : ¬∃ (j : ℕ), q ^ j = b
neg :
  HMul.hMul (α := R)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) g) 0
        (a, b).2)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) f) 0
        (a, b).1) =
    0
neg₁ :
  HMul.hMul (α := R) (0 (a, b).2)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) f) 0
        (a, b).1) =
    0
⊢ HMul.hMul (α := R)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) f) 0
        (a, b).1)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) g) 0
        (a, b).2) =
    0
have neg₂ : ¬∃ (a_1 : ℕ), q ^ a_1 = (a, b).2
Tactic: rwa [mul_comm, Function.extend_apply', Pi.zero_apply, zero_mul]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.LFunction (goal)
Goal:
R : Type u_1
inst : CommSemiring R
q : ℕ
f g : PowerSeries R
n : ℕ
hq : 1 < q
hn : ¬∃ (a : ℕ), q ^ a = n
a b : ℕ
hk : (a, b) ∈ n.divisorsAntidiagonal
hab : (a, b).1 * (a, b).2 = n
ha : ∃ (i : ℕ), q ^ i = a
hb : ¬∃ (j : ℕ), q ^ j = b
neg :
  HMul.hMul (α := R)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) g) 0
        (a, b).2)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) f) 0
        (a, b).1) =
    0
neg₁ :
  HMul.hMul (α := R) (0 (a, b).2)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) f) 0
        (a, b).1) =
    0
neg₂ : ¬∃ (a_1 : ℕ), q ^ a_1 = (a, b).2
⊢ HMul.hMul (α := R)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) f) 0
        (a, b).1)
      (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) g) 0
        (a, b).2) =
    0
have neg₃ : HMul.hMul (α := R) 0
    (Function.extend (fun (x : ℕ) ↦ q ^ x) (fun (x : ℕ) ↦ (PowerSeries.coeff x : (a : PowerSeries R) → R) f) 0
      (a, b).1) =
  0
Tactic: rwa [mul_comm, Function.extend_apply', Pi.zero_apply, zero_mul]