awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : Semiring R
f : ArithmeticFunction R
x : ℕ
x0 : ¬x = 0
h : {(x, 1)} ⊆ x.divisorsAntidiagonal
neg :
  Eq (α := R) (∑ x ∈ {(x, 1)}, HMul.hMul (α := R) (β := R) ((f : (a : ℕ) → R) x.1 : R) ((1 : (a : ℕ) → R) x.2 : R) : R)
    ((f : (a : ℕ) → R) x : R)
⊢ Eq (α := R)
    (∑ x ∈ x.divisorsAntidiagonal, HMul.hMul (α := R) (β := R) ((f : (a : ℕ) → R) x.1 : R) ((1 : (a : ℕ) → R) x.2 : R) :
      R)
    ((f : (a : ℕ) → R) x : R)
have neg₁ : ∀ x_1 ∈ x.divisorsAntidiagonal,
  x_1 ∉ {(x, 1)} → HMul.hMul (α := R) (β := R) ((f : (a : ℕ) → R) x_1.1 : R) ((1 : (a : ℕ) → R) x_1.2 : R) = 0
Tactic: rw [← sum_subset h]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (hypothesis)
Goal:
R : Type u_1
inst : Semiring R
f : ArithmeticFunction R
x : ℕ
x0 : ¬x = 0
h : {(x, 1)} ⊆ x.divisorsAntidiagonal
y₁ y₂ : ℕ
ymem : (y₁, y₂) ∈ x.divisorsAntidiagonal
ynotMem : (y₁, y₂) ∉ {(x, 1)}
⊢ HMul.hMul (α := R) (β := R) ((f : (a : ℕ) → R) (y₁, y₂).1 : R) ((1 : (a : ℕ) → R) (y₁, y₂).2 : R) = 0
have y2ne : y₂ ≠ 1
Tactic: have y2ne : y₂ ≠ 1 := by
      intro con
      simp_all
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : Semiring R
f : ArithmeticFunction R
⊢ 0 * f = 0
have h : ∀ (x : ℕ), Eq (α := R) ((0 * f : (a : ℕ) → R) x : R) ((0 : (a : ℕ) → R) x : R)
Tactic: ext
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : Semiring R
f : ArithmeticFunction R
⊢ f * 0 = 0
have h : ∀ (x : ℕ), Eq (α := R) ((f * 0 : (a : ℕ) → R) x : R) ((0 : (a : ℕ) → R) x : R)
Tactic: ext
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : Semiring R
a b c : ArithmeticFunction R
⊢ a * (b + c) = a * b + a * c
have h : ∀ (x : ℕ), Eq (α := R) ((a * (b + c) : (a : ℕ) → R) x : R) ((a * b + a * c : (a : ℕ) → R) x : R)
Tactic: ext
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : Semiring R
a b c : ArithmeticFunction R
⊢ (a + b) * c = a * c + b * c
have h : ∀ (x : ℕ), Eq (α := R) (((a + b) * c : (a : ℕ) → R) x : R) ((a * c + b * c : (a : ℕ) → R) x : R)
Tactic: ext
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : CommSemiring R
f g : ArithmeticFunction R
⊢ f * g = g * f
have h : ∀ (x : ℕ), Eq (α := R) ((f * g : (a : ℕ) → R) x : R) ((g * f : (a : ℕ) → R) x : R)
Tactic: ext
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : CommSemiring R
f g : ArithmeticFunction R
x : ℕ
⊢ Eq (α := R) ((f * g : (a : ℕ) → R) x : R) ((g * f : (a : ℕ) → R) x : R)
have h : Eq (α := R)
  (∑ x ∈ x.divisorsAntidiagonal,
      HMul.hMul (α := R) (β := R) ((f : (a : ℕ) → R) (((Equiv.prodComm ℕ ℕ).toEmbedding : (a : ℕ × ℕ) → ℕ × ℕ) x).1 : R)
        ((g : (a : ℕ) → R) (((Equiv.prodComm ℕ ℕ).toEmbedding : (a : ℕ × ℕ) → ℕ × ℕ) x).2 : R) :
    R)
  ((g * f : (a : ℕ) → R) x : R)
Tactic: rw [mul_apply, ← map_swap_divisorsAntidiagonal, sum_map]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
M : Type u_2
inst : Semiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
r : ArithmeticFunction R
⊢ HSMul.hSMul (β := ArithmeticFunction M) r 0 = 0
have h : ∀ (x : ℕ), Eq (α := M) ((HSMul.hSMul (β := ArithmeticFunction M) r 0 : (a : ℕ) → M) x : M) ((0 : (a : ℕ) → M) x : M)
Tactic: ext
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
M : Type u_2
inst : Semiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
r : ArithmeticFunction R
x y : ArithmeticFunction M
⊢ r • (x + y) = r • x + r • y
have h : ∀ (x_1 : ℕ), Eq (α := M) ((r • (x + y) : (a : ℕ) → M) x_1 : M) ((r • x + r • y : (a : ℕ) → M) x_1 : M)
Tactic: ext
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
M : Type u_2
inst : Semiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
r s : ArithmeticFunction R
x : ArithmeticFunction M
⊢ (r + s) • x = r • x + s • x
have h : ∀ (x_1 : ℕ), Eq (α := M) (((r + s) • x : (a : ℕ) → M) x_1 : M) ((r • x + s • x : (a : ℕ) → M) x_1 : M)
Tactic: ext
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
M : Type u_2
inst : Semiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
r : ArithmeticFunction M
⊢ HSMul.hSMul.{u_1, u_2, u_2} (α := ArithmeticFunction R) 0 r = 0
have h : ∀ (x : ℕ),
  Eq (α := M) ((HSMul.hSMul.{u_1, u_2, u_2} (α := ArithmeticFunction R) 0 r : (a : ℕ) → M) x : M)
    ((0 : (a : ℕ) → M) x : M)
Tactic: ext
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : Ring R
f : ℕ → R
hf : Invertible (f 1)
⊢ dirichletInverseFun f hf 0 = 0
have rw : (if Eq (α := ℕ) 0 0 then 0
  else
    if Eq (α := ℕ) 0 1 then @Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne (f 1) _
    else
      -@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne (f 1) _ *
        ∑ d : ↥(properDivisors 0),
          have this := sorry;
          f (0 / (↑d : ℕ)) * dirichletInverseFun f hf (↑d : ℕ)) =
  0
Tactic: rw [dirichletInverseFun, if_pos rfl]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : Ring R
f : ℕ → R
hf : Invertible (f 1)
rw :
  (if Eq (α := ℕ) 0 0 then 0
    else
      if Eq (α := ℕ) 0 1 then @Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne (f 1) _
      else
        -@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne (f 1) _ *
          ∑ d : ↥(properDivisors 0),
            have this := ⋯;
            f (0 / (↑d : ℕ)) * dirichletInverseFun f hf (↑d : ℕ)) =
    0
⊢ dirichletInverseFun f hf 0 = 0
have rw₁ : Eq.{u_1 + 1} (α := R) 0 0
Tactic: rw [dirichletInverseFun, if_pos rfl]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : Ring R
f : ℕ → R
hf : Invertible (f 1)
⊢ dirichletInverseFun f hf 1 = @Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne (f 1) _
have rw : (if Eq (α := ℕ) 1 0 then 0
  else
    if Eq (α := ℕ) 1 1 then @Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne (f 1) _
    else
      -@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne (f 1) _ *
        ∑ d : ↥(properDivisors 1),
          have this := sorry;
          f (1 / (↑d : ℕ)) * dirichletInverseFun f hf (↑d : ℕ)) =
  @Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne (f 1) _
Tactic: rw [dirichletInverseFun, if_neg one_ne_zero, if_pos rfl]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : Ring R
f : ℕ → R
hf : Invertible (f 1)
rw :
  (if Eq (α := ℕ) 1 0 then 0
    else
      if Eq (α := ℕ) 1 1 then @Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne (f 1) _
      else
        -@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne (f 1) _ *
          ∑ d : ↥(properDivisors 1),
            have this := ⋯;
            f (1 / (↑d : ℕ)) * dirichletInverseFun f hf (↑d : ℕ)) =
    @Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne (f 1) _
⊢ dirichletInverseFun f hf 1 = @Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne (f 1) _
have rw₁ : (if Eq (α := ℕ) 1 1 then @Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne (f 1) _
  else
    -@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne (f 1) _ *
      ∑ d : ↥(properDivisors 1),
        have this := sorry;
        f (1 / (↑d : ℕ)) * dirichletInverseFun f hf (↑d : ℕ)) =
  @Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne (f 1) _
Tactic: rw [dirichletInverseFun, if_neg one_ne_zero, if_pos rfl]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : Ring R
f : ℕ → R
hf : Invertible (f 1)
rw :
  (if Eq (α := ℕ) 1 0 then 0
    else
      if Eq (α := ℕ) 1 1 then @Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne (f 1) _
      else
        -@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne (f 1) _ *
          ∑ d : ↥(properDivisors 1),
            have this := ⋯;
            f (1 / (↑d : ℕ)) * dirichletInverseFun f hf (↑d : ℕ)) =
    @Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne (f 1) _
rw₁ :
  (if Eq (α := ℕ) 1 1 then @Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne (f 1) _
    else
      -@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne (f 1) _ *
        ∑ d : ↥(properDivisors 1),
          have this := ⋯;
          f (1 / (↑d : ℕ)) * dirichletInverseFun f hf (↑d : ℕ)) =
    @Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne (f 1) _
⊢ dirichletInverseFun f hf 1 = @Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne (f 1) _
have rw₂ : @Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne (f 1) _ =
  @Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne (f 1) _
Tactic: rw [dirichletInverseFun, if_neg one_ne_zero, if_pos rfl]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : Ring R
f : ℕ → R
hf : Invertible (f 1)
n : ℕ
hn0 : n ≠ 0
hn1 : n ≠ 1
⊢ dirichletInverseFun f hf n =
    -@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne (f 1) _ *
      ∑ d ∈ n.properDivisors, f (n / d) * dirichletInverseFun f hf d
have rw : (-@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne (f 1) _ *
    ∑ d : ↥n.properDivisors,
      have this := sorry;
      f (n / (↑d : ℕ)) * dirichletInverseFun f hf (↑d : ℕ)) =
  -@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne (f 1) _ *
    ∑ d ∈ n.properDivisors, f (n / d) * dirichletInverseFun f hf d
Tactic: rw [dirichletInverseFun, if_neg hn0, if_neg hn1]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : Ring R
f : ℕ → R
hf : Invertible (f 1)
n : ℕ
hn0 : n ≠ 0
hn1 : n ≠ 1
| -@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne (f 1) _ *
    ∑ d ∈ n.properDivisors, f (n / d) * dirichletInverseFun f hf d
have rw : -@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne (f 1) _ *
  ∑ x : ↥n.properDivisors, f (n / (↑x : ℕ)) * dirichletInverseFun f hf (↑x : ℕ)
Tactic: rw [← Finset.sum_attach, Finset.attach_eq_univ]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : Ring R
f : ℕ → R
hf : Invertible (f 1)
n : ℕ
hn0 : n ≠ 0
hn1 : n ≠ 1
⊢ (-@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne (f 1) _ *
      ∑ d : ↥n.properDivisors,
        have this := ⋯;
        f (n / (↑d : ℕ)) * dirichletInverseFun f hf (↑d : ℕ)) =
    -@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne (f 1) _ *
      ∑ d ∈ n.properDivisors, f (n / d) * dirichletInverseFun f hf d
have conv_rhs : (-@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne (f 1) _ *
    ∑ d : ↥n.properDivisors,
      have this := sorry;
      f (n / (↑d : ℕ)) * dirichletInverseFun f hf (↑d : ℕ)) =
  -@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne (f 1) _ *
    ∑ d ∈ n.properDivisors, f (n / d) * dirichletInverseFun f hf d
Tactic: conv_rhs => rw [← Finset.sum_attach, Finset.attach_eq_univ]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : Ring R
f : ℕ → R
hf : Invertible (f 1)
n : ℕ
hn0 : n ≠ 0
hn1 : n ≠ 1
conv_rhs :
  (-@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne (f 1) _ *
      ∑ d : ↥n.properDivisors,
        have this := ⋯;
        f (n / (↑d : ℕ)) * dirichletInverseFun f hf (↑d : ℕ)) =
    -@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne (f 1) _ *
      ∑ d ∈ n.properDivisors, f (n / d) * dirichletInverseFun f hf d
⊢ (-@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne (f 1) _ *
      ∑ d : ↥n.properDivisors,
        have this := ⋯;
        f (n / (↑d : ℕ)) * dirichletInverseFun f hf (↑d : ℕ)) =
    -@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne (f 1) _ *
      ∑ d ∈ n.properDivisors, f (n / d) * dirichletInverseFun f hf d
have conv_rhs₁ : (-@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne (f 1) _ *
    ∑ d : ↥n.properDivisors,
      have this := sorry;
      f (n / (↑d : ℕ)) * dirichletInverseFun f hf (↑d : ℕ)) =
  -@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne (f 1) _ *
    ∑ x : ↥n.properDivisors, f (n / (↑x : ℕ)) * dirichletInverseFun f hf (↑x : ℕ)
Tactic: conv_rhs => rw [← Finset.sum_attach, Finset.attach_eq_univ]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : Ring R
f : ArithmeticFunction R
hf : Invertible (α := R) ((f : (a : ℕ) → R) 1 : R)
⊢ f * dirichletInverse (⇑f : (a : ℕ) → R) hf = 1
have h : ∀ (n : ℕ), Eq (α := R) ((f * dirichletInverse (⇑f : (a : ℕ) → R) hf : (a : ℕ) → R) n : R) ((1 : (a : ℕ) → R) n : R)
Tactic: ext n
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : Ring R
f : ArithmeticFunction R
hf : Invertible (α := R) ((f : (a : ℕ) → R) 1 : R)
n : ℕ
⊢ Eq (α := R) ((f * dirichletInverse (⇑f : (a : ℕ) → R) hf : (a : ℕ) → R) n : R) ((1 : (a : ℕ) → R) n : R)
have pos : ∀ (hn0 : n = 0),
  Eq (α := R) ((f * dirichletInverse (⇑f : (a : ℕ) → R) hf : (a : ℕ) → R) n : R) ((1 : (a : ℕ) → R) n : R)
Tactic: by_cases hn0 : n = 0
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : Ring R
f : ArithmeticFunction R
hf : Invertible (α := R) ((f : (a : ℕ) → R) 1 : R)
n : ℕ
pos :
  ∀ (hn0 : n = 0),
    Eq (α := R) ((f * dirichletInverse (⇑f : (a : ℕ) → R) hf : (a : ℕ) → R) n : R) ((1 : (a : ℕ) → R) n : R)
⊢ Eq (α := R) ((f * dirichletInverse (⇑f : (a : ℕ) → R) hf : (a : ℕ) → R) n : R) ((1 : (a : ℕ) → R) n : R)
have neg : ∀ (hn0 : ¬n = 0),
  Eq (α := R) ((f * dirichletInverse (⇑f : (a : ℕ) → R) hf : (a : ℕ) → R) n : R) ((1 : (a : ℕ) → R) n : R)
Tactic: by_cases hn0 : n = 0
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : Ring R
f : ArithmeticFunction R
hf : Invertible (α := R) ((f : (a : ℕ) → R) 1 : R)
n : ℕ
hn0 : ¬n = 0
⊢ Eq (α := R) ((f * dirichletInverse (⇑f : (a : ℕ) → R) hf : (a : ℕ) → R) n : R) ((1 : (a : ℕ) → R) n : R)
have pos : ∀ (hn1 : n = 1),
  Eq (α := R) ((f * dirichletInverse (⇑f : (a : ℕ) → R) hf : (a : ℕ) → R) n : R) ((1 : (a : ℕ) → R) n : R)
Tactic: by_cases hn1 : n = 1
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : Ring R
f : ArithmeticFunction R
hf : Invertible (α := R) ((f : (a : ℕ) → R) 1 : R)
n : ℕ
hn0 : ¬n = 0
pos :
  ∀ (hn1 : n = 1),
    Eq (α := R) ((f * dirichletInverse (⇑f : (a : ℕ) → R) hf : (a : ℕ) → R) n : R) ((1 : (a : ℕ) → R) n : R)
⊢ Eq (α := R) ((f * dirichletInverse (⇑f : (a : ℕ) → R) hf : (a : ℕ) → R) n : R) ((1 : (a : ℕ) → R) n : R)
have neg : ∀ (hn1 : ¬n = 1),
  Eq (α := R) ((f * dirichletInverse (⇑f : (a : ℕ) → R) hf : (a : ℕ) → R) n : R) ((1 : (a : ℕ) → R) n : R)
Tactic: by_cases hn1 : n = 1
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : Ring R
f : ArithmeticFunction R
hf : Invertible (α := R) ((f : (a : ℕ) → R) 1 : R)
n : ℕ
hn0 : ¬n = 0
hn1 : ¬n = 1
⊢ Eq (α := R) ((f * dirichletInverse (⇑f : (a : ℕ) → R) hf : (a : ℕ) → R) n : R) ((1 : (a : ℕ) → R) n : R)
have neg : Eq.{u_1 + 1} (α := R)
  (∑ i ∈ cons n n.properDivisors sorry,
      HMul.hMul (α := R) ((f : (a : ℕ) → R) (n / i) : R) (dirichletInverseFun (⇑f : (a : ℕ) → R) hf i) :
    R)
  ((1 : (a : ℕ) → R) n : R)
Tactic: rw [dirichletInverse, mul_apply, coe_mk,
    Nat.sum_divisorsAntidiagonal' fun x y ↦ f x * dirichletInverseFun f hf y,
    ← Nat.cons_self_properDivisors hn0]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : CommRing R
f : ArithmeticFunction R
hf : Invertible (α := R) ((f : (a : ℕ) → R) 1 : R)
⊢ dirichletInverse (⇑f : (a : ℕ) → R) hf * f = 1
have rw : f * dirichletInverse (⇑f : (a : ℕ) → R) hf = 1
Tactic: rw [mul_comm, self_mul_dirichletInverse]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : CommRing R
f : ArithmeticFunction R
hf : Invertible (α := R) ((f : (a : ℕ) → R) 1 : R)
rw : f * dirichletInverse (⇑f : (a : ℕ) → R) hf = 1
⊢ dirichletInverse (⇑f : (a : ℕ) → R) hf * f = 1
have rw₁ : Eq.{u_1 + 1} (α := ArithmeticFunction R) 1 1
Tactic: rw [mul_comm, self_mul_dirichletInverse]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : CommRing R
f : ArithmeticFunction R
⊢ IsUnit f ↔ IsUnit (M := R) ((f : (a : ℕ) → R) 1 : R)
have mp : IsUnit f → IsUnit (M := R) ((f : (a : ℕ) → R) 1 : R)
Tactic: constructor
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : CommRing R
f : ArithmeticFunction R
mp : IsUnit f → IsUnit (M := R) ((f : (a : ℕ) → R) 1 : R)
⊢ IsUnit f ↔ IsUnit (M := R) ((f : (a : ℕ) → R) 1 : R)
have mpr : IsUnit (M := R) ((f : (a : ℕ) → R) 1 : R) → IsUnit f
Tactic: constructor
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : CommRing R
f : (ArithmeticFunction R)ˣ
⊢ IsUnit (M := R) (((↑f : ArithmeticFunction R) : (a : ℕ) → R) 1 : R)
have mp : HMul.hMul (α := R) (β := R) (((↑f : ArithmeticFunction R) : (a : ℕ) → R) 1 : R)
    (((↑f⁻¹ : ArithmeticFunction R) : (a : ℕ) → R) 1 : R) =
  1
Tactic: refine ⟨⟨f.val 1, f⁻¹.val 1, ?_, ?_⟩, rfl⟩
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : CommRing R
f : (ArithmeticFunction R)ˣ
mp :
  HMul.hMul (α := R) (β := R) (((↑f : ArithmeticFunction R) : (a : ℕ) → R) 1 : R)
      (((↑f⁻¹ : ArithmeticFunction R) : (a : ℕ) → R) 1 : R) =
    1
⊢ IsUnit (M := R) (((↑f : ArithmeticFunction R) : (a : ℕ) → R) 1 : R)
have mp₁ : HMul.hMul (α := R) (β := R) (((↑f⁻¹ : ArithmeticFunction R) : (a : ℕ) → R) 1 : R)
    (((↑f : ArithmeticFunction R) : (a : ℕ) → R) 1 : R) =
  1
Tactic: refine ⟨⟨f.val 1, f⁻¹.val 1, ?_, ?_⟩, rfl⟩
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : CommRing R
f : (ArithmeticFunction R)ˣ
⊢ HMul.hMul (α := R) (β := R) (((↑f : ArithmeticFunction R) : (a : ℕ) → R) 1 : R)
      (((↑f⁻¹ : ArithmeticFunction R) : (a : ℕ) → R) 1 : R) =
    1
have mp : Eq (α := R) (((↑f : ArithmeticFunction R) * (↑f⁻¹ : ArithmeticFunction R) : (a : ℕ) → R) 1 : R) (1 : R)
Tactic: rw [← ArithmeticFunction.mul_apply_one, Units.mul_inv, one_one]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : CommRing R
f : (ArithmeticFunction R)ˣ
mp : Eq (α := R) (((↑f : ArithmeticFunction R) * (↑f⁻¹ : ArithmeticFunction R) : (a : ℕ) → R) 1 : R) (1 : R)
⊢ HMul.hMul (α := R) (β := R) (((↑f : ArithmeticFunction R) : (a : ℕ) → R) 1 : R)
      (((↑f⁻¹ : ArithmeticFunction R) : (a : ℕ) → R) 1 : R) =
    1
have mp₁ : Eq.{u_1 + 1} (α := R) ((1 : (a : ℕ) → R) 1 : R) (1 : R)
Tactic: rw [← ArithmeticFunction.mul_apply_one, Units.mul_inv, one_one]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : CommRing R
f : (ArithmeticFunction R)ˣ
mp : Eq (α := R) (((↑f : ArithmeticFunction R) * (↑f⁻¹ : ArithmeticFunction R) : (a : ℕ) → R) 1 : R) (1 : R)
mp₁ : Eq.{u_1 + 1} (α := R) ((1 : (a : ℕ) → R) 1 : R) (1 : R)
⊢ HMul.hMul (α := R) (β := R) (((↑f : ArithmeticFunction R) : (a : ℕ) → R) 1 : R)
      (((↑f⁻¹ : ArithmeticFunction R) : (a : ℕ) → R) 1 : R) =
    1
have mp₂ : Eq.{u_1 + 1} (α := R) 1 1
Tactic: rw [← ArithmeticFunction.mul_apply_one, Units.mul_inv, one_one]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : CommRing R
f : (ArithmeticFunction R)ˣ
⊢ HMul.hMul (α := R) (β := R) (((↑f⁻¹ : ArithmeticFunction R) : (a : ℕ) → R) 1 : R)
      (((↑f : ArithmeticFunction R) : (a : ℕ) → R) 1 : R) =
    1
have mp : Eq (α := R) (((↑f⁻¹ : ArithmeticFunction R) * (↑f : ArithmeticFunction R) : (a : ℕ) → R) 1 : R) (1 : R)
Tactic: rw [← ArithmeticFunction.mul_apply_one, Units.inv_mul, one_one]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : CommRing R
f : (ArithmeticFunction R)ˣ
mp : Eq (α := R) (((↑f⁻¹ : ArithmeticFunction R) * (↑f : ArithmeticFunction R) : (a : ℕ) → R) 1 : R) (1 : R)
⊢ HMul.hMul (α := R) (β := R) (((↑f⁻¹ : ArithmeticFunction R) : (a : ℕ) → R) 1 : R)
      (((↑f : ArithmeticFunction R) : (a : ℕ) → R) 1 : R) =
    1
have mp₁ : Eq.{u_1 + 1} (α := R) ((1 : (a : ℕ) → R) 1 : R) (1 : R)
Tactic: rw [← ArithmeticFunction.mul_apply_one, Units.inv_mul, one_one]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : CommRing R
f : (ArithmeticFunction R)ˣ
mp : Eq (α := R) (((↑f⁻¹ : ArithmeticFunction R) * (↑f : ArithmeticFunction R) : (a : ℕ) → R) 1 : R) (1 : R)
mp₁ : Eq.{u_1 + 1} (α := R) ((1 : (a : ℕ) → R) 1 : R) (1 : R)
⊢ HMul.hMul (α := R) (β := R) (((↑f⁻¹ : ArithmeticFunction R) : (a : ℕ) → R) 1 : R)
      (((↑f : ArithmeticFunction R) : (a : ℕ) → R) 1 : R) =
    1
have mp₂ : Eq.{u_1 + 1} (α := R) 1 1
Tactic: rw [← ArithmeticFunction.mul_apply_one, Units.inv_mul, one_one]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : CommRing R
f : ArithmeticFunction R
⊢ IsUnit (M := R) ((f : (a : ℕ) → R) 1 : R) → IsUnit f
have mpr : Invertible (α := R) ((f : (a : ℕ) → R) 1 : R) → Invertible f
Tactic: suffices Invertible (f 1) → Invertible f by simpa using Nonempty.map this
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (hypothesis)
Goal:
R : Type u_1
ι : Type u_2
inst : CommMonoidWithZero R
g : ι → ℕ
f : ArithmeticFunction R
hf : f.IsMultiplicative
a : ι
s : Finset ι
has : a ∉ s
ih :
  ∀ (hs : (↑s : Set ι).Pairwise (Coprime on g)),
    Eq (α := R) ((f : (a : ℕ) → R) (∏ i ∈ s, g i) : R) (∏ i ∈ s, (f : (a : ℕ) → R) (g i) : R)
hs : (↑(insert a s) : Set ι).Pairwise (Coprime on g)
⊢ Eq (α := R) ((f : (a : ℕ) → R) (∏ i ∈ insert a s, g i) : R) (∏ i ∈ insert a s, (f : (a : ℕ) → R) (g i) : R)
have hs₁ : (↑s : Set ι).Pairwise (Coprime on g) ∧ ∀ b ∈ (↑s : Set ι), a ≠ b → (Coprime on g) a b
Tactic: rw [coe_insert, Set.pairwise_insert_of_symmetric (Coprime.symmetric.comap g)] at hs
Cleared hypotheses: hs
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
ι : Type u_2
inst : CommMonoidWithZero R
g : ι → ℕ
f : ArithmeticFunction R
hf : f.IsMultiplicative
a : ι
s : Finset ι
has : a ∉ s
ih :
  ∀ (hs : (↑s : Set ι).Pairwise (Coprime on g)),
    Eq (α := R) ((f : (a : ℕ) → R) (∏ i ∈ s, g i) : R) (∏ i ∈ s, (f : (a : ℕ) → R) (g i) : R)
hs : (↑s : Set ι).Pairwise (Coprime on g) ∧ ∀ b ∈ (↑s : Set ι), a ≠ b → (Coprime on g) a b
⊢ Eq (α := R) ((f : (a : ℕ) → R) (∏ i ∈ insert a s, g i) : R) (∏ i ∈ insert a s, (f : (a : ℕ) → R) (g i) : R)
have insert : (g a).gcd (∏ x ∈ s, g x) = 1
Tactic: rw [prod_insert has, prod_insert has, hf.map_mul_of_coprime, ih hs.1]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
ι : Type u_2
inst : CommMonoidWithZero R
g : ι → ℕ
f : ArithmeticFunction R
hf : f.IsMultiplicative
s : Finset ι
hs : (↑s : Set ι).Pairwise (Coprime on g)
⊢ Eq (α := R) ((f : (a : ℕ) → R) (∏ i ∈ s, g i) : R) (∏ i ∈ s, (f : (a : ℕ) → R) (g i) : R)
have empty : ∀ (hs : (↑∅ : Set ι).Pairwise (Coprime on g)),
  Eq (α := R) ((f : (a : ℕ) → R) (∏ i ∈ ∅, g i) : R) (∏ i ∈ ∅, (f : (a : ℕ) → R) (g i) : R)
Tactic: induction s using Finset.induction_on with
    | empty => simp [hf]
    | insert _ _ has ih =>
      rw [coe_insert, Set.pairwise_insert_of_symmetric (Coprime.symmetric.comap g)] at hs
      rw [prod_insert has, prod_insert has, hf.map_mul_of_coprime, ih hs.1]
      exact Coprime.prod_right fun i hi => hs.2 _ hi (hi.ne_of_notMem has).symm
Cleared hypotheses: s, hs