awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Moebius (goal)
Goal:
R : Type u_1
inst : AddCommGroup R
f g : ℕ → R
f' : ArithmeticFunction R := { toFun := fun (x : ℕ) ↦ if x = 0 then 0 else f x, map_zero' := ⋯ }
g' : ArithmeticFunction R := { toFun := fun (x : ℕ) ↦ if x = 0 then 0 else g x, map_zero' := ⋯ }
⊢ (∀ n > 0, ∑ i ∈ n.divisors, f i = g n) ↔
    ∀ n > 0, ∑ x ∈ n.divisorsAntidiagonal, HSMul.hSMul (α := ℤ) ((μ : (a : ℕ) → ℤ) x.1 : ℤ) (g x.2) = f n
have trans : (∀ n > 0, ∑ i ∈ n.divisors, f i = g n) ↔ HSMul.hSMul (α := ArithmeticFunction ℤ) (↑ζ : ArithmeticFunction ℤ) f' = g'
Tactic: trans (ζ : ArithmeticFunction ℤ) • f' = g'
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Moebius (goal)
Goal:
R : Type u_1
inst : AddCommGroup R
f g : ℕ → R
f' : ArithmeticFunction R := { toFun := fun (x : ℕ) ↦ if x = 0 then 0 else f x, map_zero' := ⋯ }
g' : ArithmeticFunction R := { toFun := fun (x : ℕ) ↦ if x = 0 then 0 else g x, map_zero' := ⋯ }
trans :
  (∀ n > 0, ∑ i ∈ n.divisors, f i = g n) ↔ HSMul.hSMul (α := ArithmeticFunction ℤ) (↑ζ : ArithmeticFunction ℤ) f' = g'
⊢ (∀ n > 0, ∑ i ∈ n.divisors, f i = g n) ↔
    ∀ n > 0, ∑ x ∈ n.divisorsAntidiagonal, HSMul.hSMul (α := ℤ) ((μ : (a : ℕ) → ℤ) x.1 : ℤ) (g x.2) = f n
have trans₁ : HSMul.hSMul (α := ArithmeticFunction ℤ) (↑ζ : ArithmeticFunction ℤ) f' = g' ↔
  ∀ n > 0, ∑ x ∈ n.divisorsAntidiagonal, HSMul.hSMul (α := ℤ) ((μ : (a : ℕ) → ℤ) x.1 : ℤ) (g x.2) = f n
Tactic: trans (ζ : ArithmeticFunction ℤ) • f' = g'
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Moebius (goal)
Goal:
R : Type u_1
inst : AddCommGroup R
f g : ℕ → R
f' : ArithmeticFunction R := { toFun := fun (x : ℕ) ↦ if x = 0 then 0 else f x, map_zero' := ⋯ }
g' : ArithmeticFunction R := { toFun := fun (x : ℕ) ↦ if x = 0 then 0 else g x, map_zero' := ⋯ }
⊢ (∀ n > 0, ∑ i ∈ n.divisors, f i = g n) ↔ HSMul.hSMul (α := ArithmeticFunction ℤ) (↑ζ : ArithmeticFunction ℤ) f' = g'
have rw : (∀ n > 0, ∑ i ∈ n.divisors, f i = g n) ↔
  ∀ (x : ℕ),
    Eq (α := R) ((HSMul.hSMul (α := ArithmeticFunction ℤ) (↑ζ : ArithmeticFunction ℤ) f' : (a : ℕ) → R) x : R)
      ((g' : (a : ℕ) → R) x : R)
Tactic: rw [ArithmeticFunction.ext_iff]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Moebius (goal)
Goal:
R : Type u_1
inst : AddCommGroup R
f g : ℕ → R
f' : ArithmeticFunction R := { toFun := fun (x : ℕ) ↦ if x = 0 then 0 else f x, map_zero' := ⋯ }
g' : ArithmeticFunction R := { toFun := fun (x : ℕ) ↦ if x = 0 then 0 else g x, map_zero' := ⋯ }
⊢ (∀ n > 0, ∑ i ∈ n.divisors, f i = g n) ↔
    ∀ (x : ℕ),
      Eq (α := R) ((HSMul.hSMul (α := ArithmeticFunction ℤ) (↑ζ : ArithmeticFunction ℤ) f' : (a : ℕ) → R) x : R)
        ((g' : (a : ℕ) → R) x : R)
have h : ∀ (a : ℕ),
  a > 0 → ∑ i ∈ a.divisors, f i = g a ↔
    Eq (α := R) ((HSMul.hSMul (α := ArithmeticFunction ℤ) (↑ζ : ArithmeticFunction ℤ) f' : (a : ℕ) → R) a : R)
      ((g' : (a : ℕ) → R) a : R)
Tactic: apply forall_congr'
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Moebius (goal)
Goal:
R : Type u_1
inst : AddCommGroup R
f g : ℕ → R
f' : ArithmeticFunction R := { toFun := fun (x : ℕ) ↦ if x = 0 then 0 else f x, map_zero' := ⋯ }
g' : ArithmeticFunction R := { toFun := fun (x : ℕ) ↦ if x = 0 then 0 else g x, map_zero' := ⋯ }
n : ℕ
⊢ n + 1 > 0 → ∑ i ∈ (n + 1).divisors, f i = g (n + 1) ↔
    Eq (α := R) ((HSMul.hSMul (α := ArithmeticFunction ℤ) (↑ζ : ArithmeticFunction ℤ) f' : (a : ℕ) → R) (n + 1) : R)
      ((g' : (a : ℕ) → R) (n + 1) : R)
have h : n + 1 > 0 → ∑ i ∈ (n + 1).divisors, f i = g (n + 1) ↔
  Eq (α := R) (∑ i ∈ (n + 1).divisors, (f' : (a : ℕ) → R) i : R) ((g' : (a : ℕ) → R) (n + 1) : R)
Tactic: rw [coe_zeta_smul_apply]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Moebius (goal)
Goal:
R : Type u_1
inst : AddCommGroup R
f g : ℕ → R
f' : ArithmeticFunction R := { toFun := fun (x : ℕ) ↦ if x = 0 then 0 else f x, map_zero' := ⋯ }
g' : ArithmeticFunction R := { toFun := fun (x : ℕ) ↦ if x = 0 then 0 else g x, map_zero' := ⋯ }
n : ℕ
⊢ n + 1 > 0 → ∑ i ∈ (n + 1).divisors, f i = g (n + 1) ↔
    Eq (α := R) (∑ i ∈ (n + 1).divisors, (f' : (a : ℕ) → R) i : R) ((g' : (a : ℕ) → R) (n + 1) : R)
have h : ∑ i ∈ (n + 1).divisors, f i = g (n + 1) ↔ (∑ x ∈ (n + 1).divisors, if x = 0 then 0 else f x) = g (n + 1)
Tactic: simp only [forall_prop_of_true, succ_pos', f', g', coe_mk, succ_ne_zero, ite_false]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Moebius (goal)
Goal:
R : Type u_1
inst : AddCommGroup R
f g : ℕ → R
f' : ArithmeticFunction R := { toFun := fun (x : ℕ) ↦ if x = 0 then 0 else f x, map_zero' := ⋯ }
g' : ArithmeticFunction R := { toFun := fun (x : ℕ) ↦ if x = 0 then 0 else g x, map_zero' := ⋯ }
n : ℕ
⊢ ∑ i ∈ (n + 1).divisors, f i = g (n + 1) ↔ (∑ x ∈ (n + 1).divisors, if x = 0 then 0 else f x) = g (n + 1)
have h : ∑ i ∈ (n + 1).divisors, f i = g (n + 1) ↔ (n + 1).divisors.sum f = g (n + 1)
Tactic: rw [sum_congr rfl fun x hx => if_neg (pos_of_mem_divisors hx).ne']
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Moebius (goal)
Goal:
R : Type u_1
inst : AddCommGroup R
f g : ℕ → R
f' : ArithmeticFunction R := { toFun := fun (x : ℕ) ↦ if x = 0 then 0 else f x, map_zero' := ⋯ }
g' : ArithmeticFunction R := { toFun := fun (x : ℕ) ↦ if x = 0 then 0 else g x, map_zero' := ⋯ }
n : ℕ
⊢ n > 0 → ∑ i ∈ n.divisors, f i = g n ↔
    Eq (α := R) ((HSMul.hSMul (α := ArithmeticFunction ℤ) (↑ζ : ArithmeticFunction ℤ) f' : (a : ℕ) → R) n : R)
      ((g' : (a : ℕ) → R) n : R)
have h : GT.gt (α := ℕ) 0 0 → ∑ i ∈ divisors 0, f i = g 0 ↔
  Eq (α := R) ((HSMul.hSMul (α := ArithmeticFunction ℤ) (↑ζ : ArithmeticFunction ℤ) f' : (a : ℕ) → R) 0 : R)
    ((g' : (a : ℕ) → R) 0 : R)
Tactic: cases n with
    | zero => simp
    | succ n =>
      rw [coe_zeta_smul_apply]
      simp only [forall_prop_of_true, succ_pos', f', g', coe_mk, succ_ne_zero, ite_false]
      rw [sum_congr rfl fun x hx => if_neg (pos_of_mem_divisors hx).ne']
Cleared hypotheses: n
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Moebius (goal)
Goal:
R : Type u_1
inst : AddCommGroup R
f g : ℕ → R
f' : ArithmeticFunction R := { toFun := fun (x : ℕ) ↦ if x = 0 then 0 else f x, map_zero' := ⋯ }
g' : ArithmeticFunction R := { toFun := fun (x : ℕ) ↦ if x = 0 then 0 else g x, map_zero' := ⋯ }
n : ℕ
h :
  GT.gt (α := ℕ) 0 0 → ∑ i ∈ divisors 0, f i = g 0 ↔
    Eq (α := R) ((HSMul.hSMul (α := ArithmeticFunction ℤ) (↑ζ : ArithmeticFunction ℤ) f' : (a : ℕ) → R) 0 : R)
      ((g' : (a : ℕ) → R) 0 : R)
⊢ n > 0 → ∑ i ∈ n.divisors, f i = g n ↔
    Eq (α := R) ((HSMul.hSMul (α := ArithmeticFunction ℤ) (↑ζ : ArithmeticFunction ℤ) f' : (a : ℕ) → R) n : R)
      ((g' : (a : ℕ) → R) n : R)
have h₁ : ∀ (n : ℕ),
  n + 1 > 0 → ∑ i ∈ (n + 1).divisors, f i = g (n + 1) ↔
    Eq (α := R) ((HSMul.hSMul (α := ArithmeticFunction ℤ) (↑ζ : ArithmeticFunction ℤ) f' : (a : ℕ) → R) (n + 1) : R)
      ((g' : (a : ℕ) → R) (n + 1) : R)
Tactic: cases n with
    | zero => simp
    | succ n =>
      rw [coe_zeta_smul_apply]
      simp only [forall_prop_of_true, succ_pos', f', g', coe_mk, succ_ne_zero, ite_false]
      rw [sum_congr rfl fun x hx => if_neg (pos_of_mem_divisors hx).ne']
Cleared hypotheses: n
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Moebius (goal)
Goal:
R : Type u_1
inst : AddCommGroup R
f g : ℕ → R
f' : ArithmeticFunction R := { toFun := fun (x : ℕ) ↦ if x = 0 then 0 else f x, map_zero' := ⋯ }
g' : ArithmeticFunction R := { toFun := fun (x : ℕ) ↦ if x = 0 then 0 else g x, map_zero' := ⋯ }
⊢ HSMul.hSMul (α := ArithmeticFunction ℤ) (↑ζ : ArithmeticFunction ℤ) f' = g' ↔
    ∀ n > 0, ∑ x ∈ n.divisorsAntidiagonal, HSMul.hSMul (α := ℤ) ((μ : (a : ℕ) → ℤ) x.1 : ℤ) (g x.2) = f n
have trans : HSMul.hSMul (α := ArithmeticFunction ℤ) (↑ζ : ArithmeticFunction ℤ) f' = g' ↔ μ • g' = f'
Tactic: trans μ • g' = f'
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Moebius (goal)
Goal:
R : Type u_1
inst : AddCommGroup R
f g : ℕ → R
f' : ArithmeticFunction R := { toFun := fun (x : ℕ) ↦ if x = 0 then 0 else f x, map_zero' := ⋯ }
g' : ArithmeticFunction R := { toFun := fun (x : ℕ) ↦ if x = 0 then 0 else g x, map_zero' := ⋯ }
trans : HSMul.hSMul (α := ArithmeticFunction ℤ) (↑ζ : ArithmeticFunction ℤ) f' = g' ↔ μ • g' = f'
⊢ HSMul.hSMul (α := ArithmeticFunction ℤ) (↑ζ : ArithmeticFunction ℤ) f' = g' ↔
    ∀ n > 0, ∑ x ∈ n.divisorsAntidiagonal, HSMul.hSMul (α := ℤ) ((μ : (a : ℕ) → ℤ) x.1 : ℤ) (g x.2) = f n
have trans₁ : μ • g' = f' ↔ ∀ n > 0, ∑ x ∈ n.divisorsAntidiagonal, HSMul.hSMul (α := ℤ) ((μ : (a : ℕ) → ℤ) x.1 : ℤ) (g x.2) = f n
Tactic: trans μ • g' = f'
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Moebius (goal)
Goal:
R : Type u_1
inst : AddCommGroup R
f g : ℕ → R
f' : ArithmeticFunction R := { toFun := fun (x : ℕ) ↦ if x = 0 then 0 else f x, map_zero' := ⋯ }
g' : ArithmeticFunction R := { toFun := fun (x : ℕ) ↦ if x = 0 then 0 else g x, map_zero' := ⋯ }
⊢ HSMul.hSMul (α := ArithmeticFunction ℤ) (↑ζ : ArithmeticFunction ℤ) f' = g' ↔ μ • g' = f'
have mp : HSMul.hSMul (α := ArithmeticFunction ℤ) (↑ζ : ArithmeticFunction ℤ) f' = g' → μ • g' = f'
Tactic: constructor
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Moebius (goal)
Goal:
R : Type u_1
inst : AddCommGroup R
f g : ℕ → R
f' : ArithmeticFunction R := { toFun := fun (x : ℕ) ↦ if x = 0 then 0 else f x, map_zero' := ⋯ }
g' : ArithmeticFunction R := { toFun := fun (x : ℕ) ↦ if x = 0 then 0 else g x, map_zero' := ⋯ }
mp : HSMul.hSMul (α := ArithmeticFunction ℤ) (↑ζ : ArithmeticFunction ℤ) f' = g' → μ • g' = f'
⊢ HSMul.hSMul (α := ArithmeticFunction ℤ) (↑ζ : ArithmeticFunction ℤ) f' = g' ↔ μ • g' = f'
have mpr : μ • g' = f' → HSMul.hSMul (α := ArithmeticFunction ℤ) (↑ζ : ArithmeticFunction ℤ) f' = g'
Tactic: constructor
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Moebius (goal)
Goal:
R : Type u_1
inst : AddCommGroup R
f g : ℕ → R
f' : ArithmeticFunction R := { toFun := fun (x : ℕ) ↦ if x = 0 then 0 else f x, map_zero' := ⋯ }
g' : ArithmeticFunction R := { toFun := fun (x : ℕ) ↦ if x = 0 then 0 else g x, map_zero' := ⋯ }
⊢ μ • g' = f' ↔ ∀ n > 0, ∑ x ∈ n.divisorsAntidiagonal, HSMul.hSMul (α := ℤ) ((μ : (a : ℕ) → ℤ) x.1 : ℤ) (g x.2) = f n
have rw : (∀ (x : ℕ), Eq (α := R) ((μ • g' : (a : ℕ) → R) x : R) ((f' : (a : ℕ) → R) x : R)) ↔
  ∀ n > 0, ∑ x ∈ n.divisorsAntidiagonal, HSMul.hSMul (α := ℤ) ((μ : (a : ℕ) → ℤ) x.1 : ℤ) (g x.2) = f n
Tactic: rw [ArithmeticFunction.ext_iff]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Moebius (goal)
Goal:
R : Type u_1
inst : AddCommGroup R
f g : ℕ → R
f' : ArithmeticFunction R := { toFun := fun (x : ℕ) ↦ if x = 0 then 0 else f x, map_zero' := ⋯ }
g' : ArithmeticFunction R := { toFun := fun (x : ℕ) ↦ if x = 0 then 0 else g x, map_zero' := ⋯ }
⊢ (∀ (x : ℕ), Eq (α := R) ((μ • g' : (a : ℕ) → R) x : R) ((f' : (a : ℕ) → R) x : R)) ↔
    ∀ n > 0, ∑ x ∈ n.divisorsAntidiagonal, HSMul.hSMul (α := ℤ) ((μ : (a : ℕ) → ℤ) x.1 : ℤ) (g x.2) = f n
have h : ∀ (a : ℕ),
  Eq (α := R) ((μ • g' : (a : ℕ) → R) a : R) ((f' : (a : ℕ) → R) a : R) ↔
    a > 0 → ∑ x ∈ a.divisorsAntidiagonal, HSMul.hSMul (α := ℤ) ((μ : (a : ℕ) → ℤ) x.1 : ℤ) (g x.2) = f a
Tactic: apply forall_congr'
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Moebius (goal)
Goal:
R : Type u_1
inst : AddCommGroup R
f g : ℕ → R
f' : ArithmeticFunction R := { toFun := fun (x : ℕ) ↦ if x = 0 then 0 else f x, map_zero' := ⋯ }
g' : ArithmeticFunction R := { toFun := fun (x : ℕ) ↦ if x = 0 then 0 else g x, map_zero' := ⋯ }
n : ℕ
⊢ Eq (α := R) ((μ • g' : (a : ℕ) → R) (n + 1) : R) ((f' : (a : ℕ) → R) (n + 1) : R) ↔
    n + 1 > 0 → ∑ x ∈ (n + 1).divisorsAntidiagonal, HSMul.hSMul (α := ℤ) ((μ : (a : ℕ) → ℤ) x.1 : ℤ) (g x.2) = f (n + 1)
have h : ∑ x ∈ (n + 1).divisorsAntidiagonal, HSMul.hSMul (α := ℤ) ((μ : (a : ℕ) → ℤ) x.1 : ℤ) (if x.2 = 0 then 0 else g x.2) =
    f (n + 1) ↔
  ∑ x ∈ (n + 1).divisorsAntidiagonal, HSMul.hSMul (α := ℤ) ((μ : (a : ℕ) → ℤ) x.1 : ℤ) (g x.2) = f (n + 1)
Tactic: simp only [forall_prop_of_true, succ_pos', smul_apply, f', g', coe_mk, succ_ne_zero,
        ite_false]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Moebius (goal)
Goal:
R : Type u_1
inst : AddCommGroup R
f g : ℕ → R
f' : ArithmeticFunction R := { toFun := fun (x : ℕ) ↦ if x = 0 then 0 else f x, map_zero' := ⋯ }
g' : ArithmeticFunction R := { toFun := fun (x : ℕ) ↦ if x = 0 then 0 else g x, map_zero' := ⋯ }
n : ℕ
⊢ ∑ x ∈ (n + 1).divisorsAntidiagonal, HSMul.hSMul (α := ℤ) ((μ : (a : ℕ) → ℤ) x.1 : ℤ) (if x.2 = 0 then 0 else g x.2) =
      f (n + 1) ↔
    ∑ x ∈ (n + 1).divisorsAntidiagonal, HSMul.hSMul (α := ℤ) ((μ : (a : ℕ) → ℤ) x.1 : ℤ) (g x.2) = f (n + 1)
have rw : ∀ (x : ℕ × ℕ) (hx : x ∈ (n + 1).divisorsAntidiagonal),
  HSMul.hSMul (α := ℤ) ((μ : (a : ℕ) → ℤ) x.1 : ℤ) (if x.2 = 0 then 0 else g x.2) =
    HSMul.hSMul (α := ℤ) ((μ : (a : ℕ) → ℤ) x.1 : ℤ) (g x.2)
Tactic: rw [sum_congr rfl fun x hx => ?_]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Moebius (goal)
Goal:
R : Type u_1
inst : AddCommGroup R
f g : ℕ → R
f' : ArithmeticFunction R := { toFun := fun (x : ℕ) ↦ if x = 0 then 0 else f x, map_zero' := ⋯ }
g' : ArithmeticFunction R := { toFun := fun (x : ℕ) ↦ if x = 0 then 0 else g x, map_zero' := ⋯ }
n : ℕ
x : ℕ × ℕ
hx : x ∈ (n + 1).divisorsAntidiagonal
⊢ HSMul.hSMul (α := ℤ) ((μ : (a : ℕ) → ℤ) x.1 : ℤ) (if x.2 = 0 then 0 else g x.2) =
    HSMul.hSMul (α := ℤ) ((μ : (a : ℕ) → ℤ) x.1 : ℤ) (g x.2)
have rw : HSMul.hSMul (α := ℤ) ((μ : (a : ℕ) → ℤ) x.1 : ℤ) (g x.2) = HSMul.hSMul (α := ℤ) ((μ : (a : ℕ) → ℤ) x.1 : ℤ) (g x.2)
Tactic: rw [if_neg (pos_of_mem_divisors (snd_mem_divisors_of_mem_antidiagonal hx)).ne']
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Moebius (goal)
Goal:
R : Type u_1
inst : AddCommGroup R
f g : ℕ → R
f' : ArithmeticFunction R := { toFun := fun (x : ℕ) ↦ if x = 0 then 0 else f x, map_zero' := ⋯ }
g' : ArithmeticFunction R := { toFun := fun (x : ℕ) ↦ if x = 0 then 0 else g x, map_zero' := ⋯ }
n : ℕ
⊢ Eq (α := R) ((μ • g' : (a : ℕ) → R) n : R) ((f' : (a : ℕ) → R) n : R) ↔
    n > 0 → ∑ x ∈ n.divisorsAntidiagonal, HSMul.hSMul (α := ℤ) ((μ : (a : ℕ) → ℤ) x.1 : ℤ) (g x.2) = f n
have h : Eq (α := R) ((μ • g' : (a : ℕ) → R) 0 : R) ((f' : (a : ℕ) → R) 0 : R) ↔
  GT.gt (α := ℕ) 0 0 → ∑ x ∈ divisorsAntidiagonal 0, HSMul.hSMul (α := ℤ) ((μ : (a : ℕ) → ℤ) x.1 : ℤ) (g x.2) = f 0
Tactic: cases n with
    | zero => simp
    | succ n =>
      simp only [forall_prop_of_true, succ_pos', smul_apply, f', g', coe_mk, succ_ne_zero,
        ite_false]
      rw [sum_congr rfl fun x hx => ?_]
      rw [if_neg (pos_of_mem_divisors (snd_mem_divisors_of_mem_antidiagonal hx)).ne']
Cleared hypotheses: n
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Moebius (goal)
Goal:
R : Type u_1
inst : AddCommGroup R
f g : ℕ → R
f' : ArithmeticFunction R := { toFun := fun (x : ℕ) ↦ if x = 0 then 0 else f x, map_zero' := ⋯ }
g' : ArithmeticFunction R := { toFun := fun (x : ℕ) ↦ if x = 0 then 0 else g x, map_zero' := ⋯ }
n : ℕ
h :
  Eq (α := R) ((μ • g' : (a : ℕ) → R) 0 : R) ((f' : (a : ℕ) → R) 0 : R) ↔
    GT.gt (α := ℕ) 0 0 → ∑ x ∈ divisorsAntidiagonal 0, HSMul.hSMul (α := ℤ) ((μ : (a : ℕ) → ℤ) x.1 : ℤ) (g x.2) = f 0
⊢ Eq (α := R) ((μ • g' : (a : ℕ) → R) n : R) ((f' : (a : ℕ) → R) n : R) ↔
    n > 0 → ∑ x ∈ n.divisorsAntidiagonal, HSMul.hSMul (α := ℤ) ((μ : (a : ℕ) → ℤ) x.1 : ℤ) (g x.2) = f n
have h₁ : ∀ (n : ℕ),
  Eq (α := R) ((μ • g' : (a : ℕ) → R) (n + 1) : R) ((f' : (a : ℕ) → R) (n + 1) : R) ↔
    n + 1 > 0 → ∑ x ∈ (n + 1).divisorsAntidiagonal, HSMul.hSMul (α := ℤ) ((μ : (a : ℕ) → ℤ) x.1 : ℤ) (g x.2) = f (n + 1)
Tactic: cases n with
    | zero => simp
    | succ n =>
      simp only [forall_prop_of_true, succ_pos', smul_apply, f', g', coe_mk, succ_ne_zero,
        ite_false]
      rw [sum_congr rfl fun x hx => ?_]
      rw [if_neg (pos_of_mem_divisors (snd_mem_divisors_of_mem_antidiagonal hx)).ne']
Cleared hypotheses: n
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Moebius (goal)
Goal:
R : Type u_1
inst : NonAssocRing R
f g : ℕ → R
⊢ (∀ n > 0, ∑ i ∈ n.divisors, f i = g n) ↔
    ∀ n > 0, ∑ x ∈ n.divisorsAntidiagonal, (↑((μ : (a : ℕ) → ℤ) x.1) : R) * g x.2 = f n
have rw : (∀ n > 0, ∑ x ∈ n.divisorsAntidiagonal, HSMul.hSMul (α := ℤ) ((μ : (a : ℕ) → ℤ) x.1 : ℤ) (g x.2) = f n) ↔
  ∀ n > 0, ∑ x ∈ n.divisorsAntidiagonal, (↑((μ : (a : ℕ) → ℤ) x.1) : R) * g x.2 = f n
Tactic: rw [sum_eq_iff_sum_smul_moebius_eq]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Moebius (goal)
Goal:
R : Type u_1
inst : NonAssocRing R
f g : ℕ → R
⊢ (∀ n > 0, ∑ x ∈ n.divisorsAntidiagonal, HSMul.hSMul (α := ℤ) ((μ : (a : ℕ) → ℤ) x.1 : ℤ) (g x.2) = f n) ↔
    ∀ n > 0, ∑ x ∈ n.divisorsAntidiagonal, (↑((μ : (a : ℕ) → ℤ) x.1) : R) * g x.2 = f n
have h : ∀ (a : ℕ),
  a > 0 → ∑ x ∈ a.divisorsAntidiagonal, HSMul.hSMul (α := ℤ) ((μ : (a : ℕ) → ℤ) x.1 : ℤ) (g x.2) = f a ↔
    a > 0 → ∑ x ∈ a.divisorsAntidiagonal, (↑((μ : (a : ℕ) → ℤ) x.1) : R) * g x.2 = f a
Tactic: apply forall_congr'
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Moebius (goal)
Goal:
R : Type u_1
inst : NonAssocRing R
f g : ℕ → R
⊢ ∀ (a : ℕ),
    a > 0 → ∑ x ∈ a.divisorsAntidiagonal, HSMul.hSMul (α := ℤ) ((μ : (a : ℕ) → ℤ) x.1 : ℤ) (g x.2) = f a ↔
      a > 0 → ∑ x ∈ a.divisorsAntidiagonal, (↑((μ : (a : ℕ) → ℤ) x.1) : R) * g x.2 = f a
have h : ∀ a > 0,
  ∀ (x : ℕ × ℕ) (_hx : x ∈ a.divisorsAntidiagonal),
    HSMul.hSMul (α := ℤ) ((μ : (a : ℕ) → ℤ) x.1 : ℤ) (g x.2) = (↑((μ : (a : ℕ) → ℤ) x.1) : R) * g x.2
Tactic: refine fun a => imp_congr_right fun _ => (sum_congr rfl fun x _hx => ?_).congr_left
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Moebius (goal)
Goal:
R : Type u_1
inst : NonAssocRing R
f g : ℕ → R
a : ℕ
h : a > 0
x : ℕ × ℕ
_hx : x ∈ a.divisorsAntidiagonal
⊢ HSMul.hSMul (α := ℤ) ((μ : (a : ℕ) → ℤ) x.1 : ℤ) (g x.2) = (↑((μ : (a : ℕ) → ℤ) x.1) : R) * g x.2
have h : (↑((μ : (a : ℕ) → ℤ) x.1) : R) * g x.2 = (↑((μ : (a : ℕ) → ℤ) x.1) : R) * g x.2
Tactic: rw [zsmul_eq_mul]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Moebius (goal)
Goal:
R : Type u_1
inst : CommGroupWithZero R
f g : ℕ → R
hf : ∀ (n : ℕ), 0 < n → f n ≠ 0
hg : ∀ (n : ℕ), 0 < n → g n ≠ 0
⊢ (∀ n > 0, ∏ i ∈ n.divisors, f i = g n) ↔
    ∀ n > 0, ∏ x ∈ n.divisorsAntidiagonal, HPow.hPow (β := ℤ) (g x.2) ((μ : (a : ℕ) → ℤ) x.1 : ℤ) = f n
have refine_1 : ∀ (n : ℕ),
  n > 0 → ∏ i ∈ n.divisors, f i = g n ↔
    n > 0 →
      Eq.{u_1 + 1} (α := Rˣ) (∏ i ∈ n.divisors, if h : 0 < i then Units.mk0 (f i) sorry else 1)
        (if h : 0 < n then Units.mk0 (g n) sorry else 1)
Tactic: refine
      Iff.trans
        (Iff.trans (forall_congr' fun n => ?_)
          (@prod_eq_iff_prod_pow_moebius_eq Rˣ _
            (fun n => if h : 0 < n then Units.mk0 (f n) (hf n h) else 1) fun n =>
            if h : 0 < n then Units.mk0 (g n) (hg n h) else 1))
        (forall_congr' fun n => ?_)
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Moebius (goal)
Goal:
R : Type u_1
inst : CommGroupWithZero R
f g : ℕ → R
hf : ∀ (n : ℕ), 0 < n → f n ≠ 0
hg : ∀ (n : ℕ), 0 < n → g n ≠ 0
refine_1 :
  ∀ (n : ℕ),
    n > 0 → ∏ i ∈ n.divisors, f i = g n ↔
      n > 0 →
        Eq.{u_1 + 1} (α := Rˣ) (∏ i ∈ n.divisors, if h : 0 < i then Units.mk0 (f i) ⋯ else 1)
          (if h : 0 < n then Units.mk0 (g n) ⋯ else 1)
⊢ (∀ n > 0, ∏ i ∈ n.divisors, f i = g n) ↔
    ∀ n > 0, ∏ x ∈ n.divisorsAntidiagonal, HPow.hPow (β := ℤ) (g x.2) ((μ : (a : ℕ) → ℤ) x.1 : ℤ) = f n
have refine_2 : ∀ (n : ℕ),
  n > 0 →
      Eq.{u_1 + 1} (α := Rˣ)
        (∏ x ∈ n.divisorsAntidiagonal,
          HPow.hPow (α := Rˣ) (β := ℤ) (if h : 0 < x.2 then Units.mk0 (g x.2) sorry else 1) ((μ : (a : ℕ) → ℤ) x.1 : ℤ))
        (if h : 0 < n then Units.mk0 (f n) sorry else 1) ↔
    n > 0 → ∏ x ∈ n.divisorsAntidiagonal, HPow.hPow (β := ℤ) (g x.2) ((μ : (a : ℕ) → ℤ) x.1 : ℤ) = f n
Tactic: refine
      Iff.trans
        (Iff.trans (forall_congr' fun n => ?_)
          (@prod_eq_iff_prod_pow_moebius_eq Rˣ _
            (fun n => if h : 0 < n then Units.mk0 (f n) (hf n h) else 1) fun n =>
            if h : 0 < n then Units.mk0 (g n) (hg n h) else 1))
        (forall_congr' fun n => ?_)
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Moebius (goal)
Goal:
R : Type u_1
inst : CommGroupWithZero R
f g : ℕ → R
hf : ∀ (n : ℕ), 0 < n → f n ≠ 0
hg : ∀ (n : ℕ), 0 < n → g n ≠ 0
n : ℕ
⊢ n > 0 → ∏ i ∈ n.divisors, f i = g n ↔
    n > 0 →
      Eq.{u_1 + 1} (α := Rˣ) (∏ i ∈ n.divisors, if h : 0 < i then Units.mk0 (f i) ⋯ else 1)
        (if h : 0 < n then Units.mk0 (g n) ⋯ else 1)
have refine_1 : ∀ (hn : n > 0),
  ∏ i ∈ n.divisors, f i = g n ↔
    Eq.{u_1 + 1} (α := Rˣ) (∏ i ∈ n.divisors, if h : 0 < i then Units.mk0 (f i) sorry else 1)
      (if h : 0 < n then Units.mk0 (g n) sorry else 1)
Tactic: refine imp_congr_right fun hn => ?_
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Moebius (goal)
Goal:
R : Type u_1
inst : CommGroupWithZero R
f g : ℕ → R
hf : ∀ (n : ℕ), 0 < n → f n ≠ 0
hg : ∀ (n : ℕ), 0 < n → g n ≠ 0
n : ℕ
⊢ n > 0 →
      Eq.{u_1 + 1} (α := Rˣ)
        (∏ x ∈ n.divisorsAntidiagonal,
          HPow.hPow (α := Rˣ) (β := ℤ) (if h : 0 < x.2 then Units.mk0 (g x.2) ⋯ else 1) ((μ : (a : ℕ) → ℤ) x.1 : ℤ))
        (if h : 0 < n then Units.mk0 (f n) ⋯ else 1) ↔
    n > 0 → ∏ x ∈ n.divisorsAntidiagonal, HPow.hPow (β := ℤ) (g x.2) ((μ : (a : ℕ) → ℤ) x.1 : ℤ) = f n
have refine_2 : ∀ (hn : n > 0),
  Eq.{u_1 + 1} (α := Rˣ)
      (∏ x ∈ n.divisorsAntidiagonal,
        HPow.hPow (α := Rˣ) (β := ℤ) (if h : 0 < x.2 then Units.mk0 (g x.2) sorry else 1) ((μ : (a : ℕ) → ℤ) x.1 : ℤ))
      (if h : 0 < n then Units.mk0 (f n) sorry else 1) ↔
    ∏ x ∈ n.divisorsAntidiagonal, HPow.hPow (β := ℤ) (g x.2) ((μ : (a : ℕ) → ℤ) x.1 : ℤ) = f n
Tactic: refine imp_congr_right fun hn => ?_
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Moebius (goal)
Goal:
R : Type u_1
inst : CommGroupWithZero R
f g : ℕ → R
hf : ∀ (n : ℕ), 0 < n → f n ≠ 0
hg : ∀ (n : ℕ), 0 < n → g n ≠ 0
n : ℕ
hn : n > 0
⊢ ∏ i ∈ n.divisors, f i = g n ↔
    Eq.{u_1 + 1} (α := Rˣ) (∏ i ∈ n.divisors, if h : 0 < i then Units.mk0 (f i) ⋯ else 1)
      (if h : 0 < n then Units.mk0 (g n) ⋯ else 1)
have rw : ∀ x ∈ n.divisors, f x = (Units.coeHom R : (a : Rˣ) → R) (if h : 0 < x then Units.mk0 (f x) sorry else 1)
Tactic: rw [dif_pos hn, ← Units.val_inj, ← Units.coeHom_apply, map_prod, Units.val_mk0,
      prod_congr rfl _]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Moebius (goal)
Goal:
R : Type u_1
inst : CommGroupWithZero R
f g : ℕ → R
hf : ∀ (n : ℕ), 0 < n → f n ≠ 0
hg : ∀ (n : ℕ), 0 < n → g n ≠ 0
n : ℕ
hn : n > 0
x : ℕ
hx : x ∈ n.divisors
⊢ f x = (Units.coeHom R : (a : Rˣ) → R) (if h : 0 < x then Units.mk0 (f x) ⋯ else 1)
have rw : f x = (Units.coeHom R : (a : Rˣ) → R) (Units.mk0 (f x) sorry)
Tactic: rw [dif_pos (pos_of_mem_divisors hx), Units.coeHom_apply, Units.val_mk0]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Moebius (goal)
Goal:
R : Type u_1
inst : CommGroupWithZero R
f g : ℕ → R
hf : ∀ (n : ℕ), 0 < n → f n ≠ 0
hg : ∀ (n : ℕ), 0 < n → g n ≠ 0
n : ℕ
hn : n > 0
x : ℕ
hx : x ∈ n.divisors
rw : f x = (Units.coeHom R : (a : Rˣ) → R) (Units.mk0 (f x) ⋯)
⊢ f x = (Units.coeHom R : (a : Rˣ) → R) (if h : 0 < x then Units.mk0 (f x) ⋯ else 1)
have rw₁ : f x = (↑(Units.mk0 (f x) sorry) : R)
Tactic: rw [dif_pos (pos_of_mem_divisors hx), Units.coeHom_apply, Units.val_mk0]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Moebius (goal)
Goal:
R : Type u_1
inst : CommGroupWithZero R
f g : ℕ → R
hf : ∀ (n : ℕ), 0 < n → f n ≠ 0
hg : ∀ (n : ℕ), 0 < n → g n ≠ 0
n : ℕ
hn : n > 0
x : ℕ
hx : x ∈ n.divisors
rw : f x = (Units.coeHom R : (a : Rˣ) → R) (Units.mk0 (f x) ⋯)
rw₁ : f x = (↑(Units.mk0 (f x) ⋯) : R)
⊢ f x = (Units.coeHom R : (a : Rˣ) → R) (if h : 0 < x then Units.mk0 (f x) ⋯ else 1)
have rw₂ : f x = f x
Tactic: rw [dif_pos (pos_of_mem_divisors hx), Units.coeHom_apply, Units.val_mk0]