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
n : ℕ
hq : 1 < q
pos :
  ∀ (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))
⊢ 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 : ∀ (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))
Tactic: by_cases hn : ∃ a, q ^ a = n
-- 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 pos : ∀ (k : 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
      (q ^ k))
    (∑ x ∈ (q ^ k).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))
Tactic: obtain ⟨k, rfl⟩ := hn
Cleared hypotheses: n, hn
-- 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
⊢ 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
      (q ^ k))
    (∑ x ∈ (q ^ k).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 pos : Eq (α := R) ((PowerSeries.coeff k : (a : PowerSeries R) → R) (f * g) : R)
  (∑ x ∈ (q ^ k).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) :
    R)
Tactic: rw [(Nat.pow_right_injective hq).extend_apply]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.LFunction (hypothesis)
Goal:
R : Type u_1
inst : CommSemiring R
q : ℕ
f g : PowerSeries R
hq : 1 < q
k : ℕ
⊢ Eq (α := R) ((PowerSeries.coeff k : (a : PowerSeries R) → R) (f * g) : R)
    (∑ x ∈ (q ^ k).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) :
      R)
have hs : Finset.map
    ({ toFun := fun (k : ℕ) ↦ q ^ k, inj' := sorry : ℕ ↪ ℕ }.prodMap { toFun := fun (k : ℕ) ↦ q ^ k, inj' := sorry : ℕ ↪ ℕ })
    (@Finset.antidiagonal _ Nat.instAddMonoid _ k) ⊆
  (q ^ k).divisorsAntidiagonal
Tactic: have hs : (Finset.antidiagonal k).map (.prodMap ⟨fun k ↦ q ^ k, Nat.pow_right_injective hq⟩
            ⟨fun k ↦ q ^ k, Nat.pow_right_injective hq⟩) ⊆ (q ^ k).divisorsAntidiagonal :=
          Nat.antidiagonal_map_subset_divisorsAntidiagonal_pow hq k
-- 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
⊢ Eq (α := R) ((PowerSeries.coeff k : (a : PowerSeries R) → R) (f * g) : R)
    (∑ x ∈ (q ^ k).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) :
      R)
have pos : Eq.{u_1 + 1} (α := R)
  (∑ p ∈ @Finset.antidiagonal _ Nat.instAddMonoid _ k,
    HMul.hMul (α := R) (β := R) ((PowerSeries.coeff p.1 : (a : PowerSeries R) → R) f : R)
      ((PowerSeries.coeff p.2 : (a : PowerSeries R) → R) g : R))
  (∑
    x ∈
      Finset.map
        ({ toFun := fun (k : ℕ) ↦ q ^ k, inj' := sorry : ℕ ↪ ℕ }.prodMap
          { toFun := fun (k : ℕ) ↦ q ^ k, inj' := sorry : ℕ ↪ ℕ })
        (@Finset.antidiagonal _ Nat.instAddMonoid _ k),
    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))
Tactic: rw [PowerSeries.coeff_mul k f g, ← Finset.sum_subset hs]
-- 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
pos :
  Eq.{u_1 + 1} (α := R)
    (∑ p ∈ @Finset.antidiagonal _ Nat.instAddMonoid _ k,
      HMul.hMul (α := R) (β := R) ((PowerSeries.coeff p.1 : (a : PowerSeries R) → R) f : R)
        ((PowerSeries.coeff p.2 : (a : PowerSeries R) → R) g : R))
    (∑
      x ∈
        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),
      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))
⊢ Eq (α := R) ((PowerSeries.coeff k : (a : PowerSeries R) → R) (f * g) : R)
    (∑ x ∈ (q ^ k).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) :
      R)
have pos₁ : ∀ x ∈ (q ^ k).divisorsAntidiagonal,
  x ∉
      Finset.map
        ({ toFun := fun (k : ℕ) ↦ q ^ k, inj' := sorry : ℕ ↪ ℕ }.prodMap
          { toFun := fun (k : ℕ) ↦ q ^ k, inj' := sorry : ℕ ↪ ℕ })
        (@Finset.antidiagonal _ Nat.instAddMonoid _ k) →
    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 [PowerSeries.coeff_mul k f g, ← Finset.sum_subset hs]
-- 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)
⊢ 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
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)
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
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 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
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
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
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
⊢ 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) (hab : (q ^ i, b) ∈ (q ^ k).divisorsAntidiagonal)
  (h :
    (q ^ i, b) ∉
      Finset.map
        ({ toFun := fun (k : ℕ) ↦ q ^ k, inj' := sorry : ℕ ↪ ℕ }.prodMap
          { toFun := fun (k : ℕ) ↦ q ^ k, inj' := sorry : ℕ ↪ ℕ })
        (@Finset.antidiagonal _ Nat.instAddMonoid _ k)),
  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, hab, h, 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
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
b : ℕ
hb : ∃ (j : ℕ), q ^ j = b
i : ℕ
hab : (q ^ i, b) ∈ (q ^ k).divisorsAntidiagonal
h :
  (q ^ i, 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)
⊢ 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) (hab : (q ^ i, q ^ j) ∈ (q ^ k).divisorsAntidiagonal)
  (h :
    (q ^ i, q ^ j) ∉
      Finset.map
        ({ toFun := fun (k : ℕ) ↦ q ^ k, inj' := sorry : ℕ ↪ ℕ }.prodMap
          { toFun := fun (k : ℕ) ↦ q ^ k, inj' := sorry : ℕ ↪ ℕ })
        (@Finset.antidiagonal _ Nat.instAddMonoid _ k)),
  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, hab, h
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.LFunction (hypothesis)
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
i j : ℕ
hab : (q ^ i, q ^ j) ∈ (q ^ k).divisorsAntidiagonal
h :
  (q ^ i, q ^ j) ∉
    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)
⊢ 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 : i + j = k ∧ q ^ k ≠ 0
Tactic: rw [Nat.mem_divisorsAntidiagonal, ← pow_add, Nat.pow_right_inj hq] at hab
Cleared hypotheses: hab, h
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.LFunction (hypothesis)
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
i j : ℕ
hab : i + j = k ∧ q ^ k ≠ 0
h :
  (q ^ i, q ^ j) ∉
    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)
⊢ 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 h₁ : ∀ (x : ℕ × ℕ),
  x.1 + x.2 = k →
    ¬({ toFun := fun (k : ℕ) ↦ q ^ k, inj' := sorry : ℕ ↪ ℕ }.prodMap { toFun := fun (k : ℕ) ↦ q ^ k, inj' := sorry : ℕ ↪ ℕ } :
            (a : ℕ × ℕ) → ℕ × ℕ)
          x =
        (q ^ i, q ^ j)
Tactic: simp_rw [Finset.mem_map, not_exists, not_and, Finset.mem_antidiagonal] at h
Cleared hypotheses: h
-- 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
⊢ 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
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
⊢ 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]