awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
i : ℕ
rw : Eq (α := R) ((f : (a : ℕ) → R) 1 : R) (1 : R)
⊢ Eq (α := R) ((f : (a : ℕ) → R) (HPow.hPow (β := ℕ) i 0) : R) (1 : R)
have rw₁ : Eq.{u_1 + 1} (α := R) 1 1
Tactic: rw [pow_zero, hf.1]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (hypothesis)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
⊢ HMul.hMul (α := R) (β := R) ((f : (a : ℕ) → R) (x.lcm y) : R) ((f : (a : ℕ) → R) (x.gcd y) : R) =
    HMul.hMul (α := R) (β := R) ((f : (a : ℕ) → R) x : R) ((f : (a : ℕ) → R) y : R)
have hfi_zero : ∀ {i : ℕ}, Eq (α := R) ((f : (a : ℕ) → R) (HPow.hPow (β := ℕ) i 0) : R) (1 : R)
Tactic: have hfi_zero : ∀ {i}, f (i ^ 0) = 1 := by
    intro i; rw [pow_zero, hf.1]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, Eq (α := R) ((f : (a : ℕ) → R) (HPow.hPow (β := ℕ) i 0) : R) (1 : R)
⊢ HMul.hMul (α := R) (β := R) ((f : (a : ℕ) → R) (x.lcm y) : R) ((f : (a : ℕ) → R) (x.gcd y) : R) =
    HMul.hMul (α := R) (β := R) ((f : (a : ℕ) → R) x : R) ((f : (a : ℕ) → R) y : R)
have neg : HMul.hMul (α := R) (β := R)
    (∏ x_1 ∈ x.primeFactors ∪ y.primeFactors,
        (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 (((x.lcm y).factorization : (a : ℕ) → ℕ) x_1 : ℕ)) :
      R)
    ((f : (a : ℕ) → R) (x.gcd y) : R) =
  HMul.hMul (α := R) (β := R) ((f : (a : ℕ) → R) x : R) ((f : (a : ℕ) → R) y : R)
Tactic: rw [hf.multiplicative_factorization f (by assumption),
    Finsupp.prod_of_support_subset _ _ _ (fun _ _ => hfi_zero)
      (s := (x.primeFactors ∪ y.primeFactors))]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, Eq (α := R) ((f : (a : ℕ) → R) (HPow.hPow (β := ℕ) i 0) : R) (1 : R)
neg :
  HMul.hMul (α := R) (β := R)
      (∏ x_1 ∈ x.primeFactors ∪ y.primeFactors,
          (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 (((x.lcm y).factorization : (a : ℕ) → ℕ) x_1 : ℕ)) :
        R)
      ((f : (a : ℕ) → R) (x.gcd y) : R) =
    HMul.hMul (α := R) (β := R) ((f : (a : ℕ) → R) x : R) ((f : (a : ℕ) → R) y : R)
⊢ HMul.hMul (α := R) (β := R) ((f : (a : ℕ) → R) (x.lcm y) : R) ((f : (a : ℕ) → R) (x.gcd y) : R) =
    HMul.hMul (α := R) (β := R) ((f : (a : ℕ) → R) x : R) ((f : (a : ℕ) → R) y : R)
have rw : (x.lcm y).factorization.support ⊆ x.primeFactors ∪ y.primeFactors
Tactic: rw [hf.multiplicative_factorization f (by assumption),
    Finsupp.prod_of_support_subset _ _ _ (fun _ _ => hfi_zero)
      (s := (x.primeFactors ∪ y.primeFactors))]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, Eq (α := R) ((f : (a : ℕ) → R) (HPow.hPow (β := ℕ) i 0) : R) (1 : R)
⊢ HMul.hMul (α := R) (β := R)
      (∏ x_1 ∈ x.primeFactors ∪ y.primeFactors,
          (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 (((x.lcm y).factorization : (a : ℕ) → ℕ) x_1 : ℕ)) :
        R)
      ((f : (a : ℕ) → R) (x.gcd y) : R) =
    HMul.hMul (α := R) (β := R) ((f : (a : ℕ) → R) x : R) ((f : (a : ℕ) → R) y : R)
have neg : HMul.hMul (α := R)
    (∏ x_1 ∈ x.primeFactors ∪ y.primeFactors,
      (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 (((x.lcm y).factorization : (a : ℕ) → ℕ) x_1 : ℕ)))
    (∏ x_1 ∈ x.primeFactors ∪ y.primeFactors,
      (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 (((x.gcd y).factorization : (a : ℕ) → ℕ) x_1 : ℕ))) =
  HMul.hMul (α := R) (β := R) ((f : (a : ℕ) → R) x : R) ((f : (a : ℕ) → R) y : R)
Tactic: rw [hf.multiplicative_factorization f (by assumption),
    Finsupp.prod_of_support_subset _ _ _ (fun _ _ => hfi_zero)
      (s := (x.primeFactors ∪ y.primeFactors))]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, Eq (α := R) ((f : (a : ℕ) → R) (HPow.hPow (β := ℕ) i 0) : R) (1 : R)
neg :
  HMul.hMul (α := R)
      (∏ x_1 ∈ x.primeFactors ∪ y.primeFactors,
        (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 (((x.lcm y).factorization : (a : ℕ) → ℕ) x_1 : ℕ)))
      (∏ x_1 ∈ x.primeFactors ∪ y.primeFactors,
        (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 (((x.gcd y).factorization : (a : ℕ) → ℕ) x_1 : ℕ))) =
    HMul.hMul (α := R) (β := R) ((f : (a : ℕ) → R) x : R) ((f : (a : ℕ) → R) y : R)
⊢ HMul.hMul (α := R) (β := R)
      (∏ x_1 ∈ x.primeFactors ∪ y.primeFactors,
          (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 (((x.lcm y).factorization : (a : ℕ) → ℕ) x_1 : ℕ)) :
        R)
      ((f : (a : ℕ) → R) (x.gcd y) : R) =
    HMul.hMul (α := R) (β := R) ((f : (a : ℕ) → R) x : R) ((f : (a : ℕ) → R) y : R)
have rw : (x.gcd y).factorization.support ⊆ x.primeFactors ∪ y.primeFactors
Tactic: rw [hf.multiplicative_factorization f (by assumption),
    Finsupp.prod_of_support_subset _ _ _ (fun _ _ => hfi_zero)
      (s := (x.primeFactors ∪ y.primeFactors))]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, Eq (α := R) ((f : (a : ℕ) → R) (HPow.hPow (β := ℕ) i 0) : R) (1 : R)
⊢ HMul.hMul (α := R)
      (∏ x_1 ∈ x.primeFactors ∪ y.primeFactors,
        (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 (((x.lcm y).factorization : (a : ℕ) → ℕ) x_1 : ℕ)))
      (∏ x_1 ∈ x.primeFactors ∪ y.primeFactors,
        (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 (((x.gcd y).factorization : (a : ℕ) → ℕ) x_1 : ℕ))) =
    HMul.hMul (α := R) (β := R) ((f : (a : ℕ) → R) x : R) ((f : (a : ℕ) → R) y : R)
have neg : HMul.hMul (α := R)
    (∏ x_1 ∈ x.primeFactors ∪ y.primeFactors,
      (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 (((x.lcm y).factorization : (a : ℕ) → ℕ) x_1 : ℕ)))
    (∏ x_1 ∈ x.primeFactors ∪ y.primeFactors,
      (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 (((x.gcd y).factorization : (a : ℕ) → ℕ) x_1 : ℕ))) =
  HMul.hMul (α := R) (β := R)
    (∏ x_1 ∈ x.primeFactors ∪ y.primeFactors,
        (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 ((x.factorization : (a : ℕ) → ℕ) x_1 : ℕ)) :
      R)
    ((f : (a : ℕ) → R) y : R)
Tactic: rw [hf.multiplicative_factorization f (by assumption),
    Finsupp.prod_of_support_subset _ _ _ (fun _ _ => hfi_zero)
      (s := (x.primeFactors ∪ y.primeFactors))]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, Eq (α := R) ((f : (a : ℕ) → R) (HPow.hPow (β := ℕ) i 0) : R) (1 : R)
neg :
  HMul.hMul (α := R)
      (∏ x_1 ∈ x.primeFactors ∪ y.primeFactors,
        (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 (((x.lcm y).factorization : (a : ℕ) → ℕ) x_1 : ℕ)))
      (∏ x_1 ∈ x.primeFactors ∪ y.primeFactors,
        (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 (((x.gcd y).factorization : (a : ℕ) → ℕ) x_1 : ℕ))) =
    HMul.hMul (α := R) (β := R)
      (∏ x_1 ∈ x.primeFactors ∪ y.primeFactors,
          (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 ((x.factorization : (a : ℕ) → ℕ) x_1 : ℕ)) :
        R)
      ((f : (a : ℕ) → R) y : R)
⊢ HMul.hMul (α := R)
      (∏ x_1 ∈ x.primeFactors ∪ y.primeFactors,
        (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 (((x.lcm y).factorization : (a : ℕ) → ℕ) x_1 : ℕ)))
      (∏ x_1 ∈ x.primeFactors ∪ y.primeFactors,
        (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 (((x.gcd y).factorization : (a : ℕ) → ℕ) x_1 : ℕ))) =
    HMul.hMul (α := R) (β := R) ((f : (a : ℕ) → R) x : R) ((f : (a : ℕ) → R) y : R)
have rw : x.factorization.support ⊆ x.primeFactors ∪ y.primeFactors
Tactic: rw [hf.multiplicative_factorization f (by assumption),
    Finsupp.prod_of_support_subset _ _ _ (fun _ _ => hfi_zero)
      (s := (x.primeFactors ∪ y.primeFactors))]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, Eq (α := R) ((f : (a : ℕ) → R) (HPow.hPow (β := ℕ) i 0) : R) (1 : R)
⊢ HMul.hMul (α := R)
      (∏ x_1 ∈ x.primeFactors ∪ y.primeFactors,
        (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 (((x.lcm y).factorization : (a : ℕ) → ℕ) x_1 : ℕ)))
      (∏ x_1 ∈ x.primeFactors ∪ y.primeFactors,
        (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 (((x.gcd y).factorization : (a : ℕ) → ℕ) x_1 : ℕ))) =
    HMul.hMul (α := R) (β := R)
      (∏ x_1 ∈ x.primeFactors ∪ y.primeFactors,
          (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 ((x.factorization : (a : ℕ) → ℕ) x_1 : ℕ)) :
        R)
      ((f : (a : ℕ) → R) y : R)
have neg : HMul.hMul (α := R)
    (∏ x_1 ∈ x.primeFactors ∪ y.primeFactors,
      (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 (((x.lcm y).factorization : (a : ℕ) → ℕ) x_1 : ℕ)))
    (∏ x_1 ∈ x.primeFactors ∪ y.primeFactors,
      (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 (((x.gcd y).factorization : (a : ℕ) → ℕ) x_1 : ℕ))) =
  HMul.hMul (α := R)
    (∏ x_1 ∈ x.primeFactors ∪ y.primeFactors,
      (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 ((x.factorization : (a : ℕ) → ℕ) x_1 : ℕ)))
    (∏ x ∈ x.primeFactors ∪ y.primeFactors,
      (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x ((y.factorization : (a : ℕ) → ℕ) x : ℕ)))
Tactic: rw [hf.multiplicative_factorization f (by assumption),
    Finsupp.prod_of_support_subset _ _ _ (fun _ _ => hfi_zero)
      (s := (x.primeFactors ∪ y.primeFactors))]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, Eq (α := R) ((f : (a : ℕ) → R) (HPow.hPow (β := ℕ) i 0) : R) (1 : R)
neg :
  HMul.hMul (α := R)
      (∏ x_1 ∈ x.primeFactors ∪ y.primeFactors,
        (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 (((x.lcm y).factorization : (a : ℕ) → ℕ) x_1 : ℕ)))
      (∏ x_1 ∈ x.primeFactors ∪ y.primeFactors,
        (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 (((x.gcd y).factorization : (a : ℕ) → ℕ) x_1 : ℕ))) =
    HMul.hMul (α := R)
      (∏ x_1 ∈ x.primeFactors ∪ y.primeFactors,
        (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 ((x.factorization : (a : ℕ) → ℕ) x_1 : ℕ)))
      (∏ x ∈ x.primeFactors ∪ y.primeFactors,
        (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x ((y.factorization : (a : ℕ) → ℕ) x : ℕ)))
⊢ HMul.hMul (α := R)
      (∏ x_1 ∈ x.primeFactors ∪ y.primeFactors,
        (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 (((x.lcm y).factorization : (a : ℕ) → ℕ) x_1 : ℕ)))
      (∏ x_1 ∈ x.primeFactors ∪ y.primeFactors,
        (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 (((x.gcd y).factorization : (a : ℕ) → ℕ) x_1 : ℕ))) =
    HMul.hMul (α := R) (β := R)
      (∏ x_1 ∈ x.primeFactors ∪ y.primeFactors,
          (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 ((x.factorization : (a : ℕ) → ℕ) x_1 : ℕ)) :
        R)
      ((f : (a : ℕ) → R) y : R)
have rw : y.factorization.support ⊆ x.primeFactors ∪ y.primeFactors
Tactic: rw [hf.multiplicative_factorization f (by assumption),
    Finsupp.prod_of_support_subset _ _ _ (fun _ _ => hfi_zero)
      (s := (x.primeFactors ∪ y.primeFactors))]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, Eq (α := R) ((f : (a : ℕ) → R) (HPow.hPow (β := ℕ) i 0) : R) (1 : R)
⊢ HMul.hMul (α := R) (β := R) ((f : (a : ℕ) → R) (x.lcm y) : R) ((f : (a : ℕ) → R) (x.gcd y) : R) =
    HMul.hMul (α := R) (β := R) ((f : (a : ℕ) → R) x : R) ((f : (a : ℕ) → R) y : R)
have neg : HMul.hMul (α := R)
    (∏ x_1 ∈ x.primeFactors ∪ y.primeFactors,
      (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 (((x.lcm y).factorization : (a : ℕ) → ℕ) x_1 : ℕ)))
    (∏ x_1 ∈ x.primeFactors ∪ y.primeFactors,
      (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 (((x.gcd y).factorization : (a : ℕ) → ℕ) x_1 : ℕ))) =
  HMul.hMul (α := R)
    (∏ x_1 ∈ x.primeFactors ∪ y.primeFactors,
      (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 ((x.factorization : (a : ℕ) → ℕ) x_1 : ℕ)))
    (∏ x ∈ x.primeFactors ∪ y.primeFactors,
      (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x ((y.factorization : (a : ℕ) → ℕ) x : ℕ)))
Tactic: iterate 4 rw [hf.multiplicative_factorization f (by assumption),
    Finsupp.prod_of_support_subset _ _ _ (fun _ _ => hfi_zero)
      (s := (x.primeFactors ∪ y.primeFactors))]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, Eq (α := R) ((f : (a : ℕ) → R) (HPow.hPow (β := ℕ) i 0) : R) (1 : R)
neg :
  HMul.hMul (α := R)
      (∏ x_1 ∈ x.primeFactors ∪ y.primeFactors,
        (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 (((x.lcm y).factorization : (a : ℕ) → ℕ) x_1 : ℕ)))
      (∏ x_1 ∈ x.primeFactors ∪ y.primeFactors,
        (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 (((x.gcd y).factorization : (a : ℕ) → ℕ) x_1 : ℕ))) =
    HMul.hMul (α := R)
      (∏ x_1 ∈ x.primeFactors ∪ y.primeFactors,
        (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 ((x.factorization : (a : ℕ) → ℕ) x_1 : ℕ)))
      (∏ x ∈ x.primeFactors ∪ y.primeFactors,
        (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x ((y.factorization : (a : ℕ) → ℕ) x : ℕ)))
⊢ HMul.hMul (α := R) (β := R) ((f : (a : ℕ) → R) (x.lcm y) : R) ((f : (a : ℕ) → R) (x.gcd y) : R) =
    HMul.hMul (α := R) (β := R) ((f : (a : ℕ) → R) x : R) ((f : (a : ℕ) → R) y : R)
have iterate : y.factorization.support ⊆ x.primeFactors ∪ y.primeFactors
Tactic: iterate 4 rw [hf.multiplicative_factorization f (by assumption),
    Finsupp.prod_of_support_subset _ _ _ (fun _ _ => hfi_zero)
      (s := (x.primeFactors ∪ y.primeFactors))]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, Eq (α := R) ((f : (a : ℕ) → R) (HPow.hPow (β := ℕ) i 0) : R) (1 : R)
neg :
  HMul.hMul (α := R)
      (∏ x_1 ∈ x.primeFactors ∪ y.primeFactors,
        (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 (((x.lcm y).factorization : (a : ℕ) → ℕ) x_1 : ℕ)))
      (∏ x_1 ∈ x.primeFactors ∪ y.primeFactors,
        (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 (((x.gcd y).factorization : (a : ℕ) → ℕ) x_1 : ℕ))) =
    HMul.hMul (α := R)
      (∏ x_1 ∈ x.primeFactors ∪ y.primeFactors,
        (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 ((x.factorization : (a : ℕ) → ℕ) x_1 : ℕ)))
      (∏ x ∈ x.primeFactors ∪ y.primeFactors,
        (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x ((y.factorization : (a : ℕ) → ℕ) x : ℕ)))
iterate : y.factorization.support ⊆ x.primeFactors ∪ y.primeFactors
⊢ HMul.hMul (α := R) (β := R) ((f : (a : ℕ) → R) (x.lcm y) : R) ((f : (a : ℕ) → R) (x.gcd y) : R) =
    HMul.hMul (α := R) (β := R) ((f : (a : ℕ) → R) x : R) ((f : (a : ℕ) → R) y : R)
have iterate₁ : x.factorization.support ⊆ x.primeFactors ∪ y.primeFactors
Tactic: iterate 4 rw [hf.multiplicative_factorization f (by assumption),
    Finsupp.prod_of_support_subset _ _ _ (fun _ _ => hfi_zero)
      (s := (x.primeFactors ∪ y.primeFactors))]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, Eq (α := R) ((f : (a : ℕ) → R) (HPow.hPow (β := ℕ) i 0) : R) (1 : R)
neg :
  HMul.hMul (α := R)
      (∏ x_1 ∈ x.primeFactors ∪ y.primeFactors,
        (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 (((x.lcm y).factorization : (a : ℕ) → ℕ) x_1 : ℕ)))
      (∏ x_1 ∈ x.primeFactors ∪ y.primeFactors,
        (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 (((x.gcd y).factorization : (a : ℕ) → ℕ) x_1 : ℕ))) =
    HMul.hMul (α := R)
      (∏ x_1 ∈ x.primeFactors ∪ y.primeFactors,
        (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 ((x.factorization : (a : ℕ) → ℕ) x_1 : ℕ)))
      (∏ x ∈ x.primeFactors ∪ y.primeFactors,
        (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x ((y.factorization : (a : ℕ) → ℕ) x : ℕ)))
iterate : y.factorization.support ⊆ x.primeFactors ∪ y.primeFactors
iterate₁ : x.factorization.support ⊆ x.primeFactors ∪ y.primeFactors
⊢ HMul.hMul (α := R) (β := R) ((f : (a : ℕ) → R) (x.lcm y) : R) ((f : (a : ℕ) → R) (x.gcd y) : R) =
    HMul.hMul (α := R) (β := R) ((f : (a : ℕ) → R) x : R) ((f : (a : ℕ) → R) y : R)
have iterate₂ : (x.gcd y).factorization.support ⊆ x.primeFactors ∪ y.primeFactors
Tactic: iterate 4 rw [hf.multiplicative_factorization f (by assumption),
    Finsupp.prod_of_support_subset _ _ _ (fun _ _ => hfi_zero)
      (s := (x.primeFactors ∪ y.primeFactors))]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, Eq (α := R) ((f : (a : ℕ) → R) (HPow.hPow (β := ℕ) i 0) : R) (1 : R)
neg :
  HMul.hMul (α := R)
      (∏ x_1 ∈ x.primeFactors ∪ y.primeFactors,
        (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 (((x.lcm y).factorization : (a : ℕ) → ℕ) x_1 : ℕ)))
      (∏ x_1 ∈ x.primeFactors ∪ y.primeFactors,
        (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 (((x.gcd y).factorization : (a : ℕ) → ℕ) x_1 : ℕ))) =
    HMul.hMul (α := R)
      (∏ x_1 ∈ x.primeFactors ∪ y.primeFactors,
        (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 ((x.factorization : (a : ℕ) → ℕ) x_1 : ℕ)))
      (∏ x ∈ x.primeFactors ∪ y.primeFactors,
        (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x ((y.factorization : (a : ℕ) → ℕ) x : ℕ)))
iterate : y.factorization.support ⊆ x.primeFactors ∪ y.primeFactors
iterate₁ : x.factorization.support ⊆ x.primeFactors ∪ y.primeFactors
iterate₂ : (x.gcd y).factorization.support ⊆ x.primeFactors ∪ y.primeFactors
⊢ HMul.hMul (α := R) (β := R) ((f : (a : ℕ) → R) (x.lcm y) : R) ((f : (a : ℕ) → R) (x.gcd y) : R) =
    HMul.hMul (α := R) (β := R) ((f : (a : ℕ) → R) x : R) ((f : (a : ℕ) → R) y : R)
have iterate₃ : (x.lcm y).factorization.support ⊆ x.primeFactors ∪ y.primeFactors
Tactic: iterate 4 rw [hf.multiplicative_factorization f (by assumption),
    Finsupp.prod_of_support_subset _ _ _ (fun _ _ => hfi_zero)
      (s := (x.primeFactors ∪ y.primeFactors))]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, Eq (α := R) ((f : (a : ℕ) → R) (HPow.hPow (β := ℕ) i 0) : R) (1 : R)
⊢ HMul.hMul (α := R)
      (∏ x_1 ∈ x.primeFactors ∪ y.primeFactors,
        (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 (((x.lcm y).factorization : (a : ℕ) → ℕ) x_1 : ℕ)))
      (∏ x_1 ∈ x.primeFactors ∪ y.primeFactors,
        (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 (((x.gcd y).factorization : (a : ℕ) → ℕ) x_1 : ℕ))) =
    HMul.hMul (α := R)
      (∏ x_1 ∈ x.primeFactors ∪ y.primeFactors,
        (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 ((x.factorization : (a : ℕ) → ℕ) x_1 : ℕ)))
      (∏ x ∈ x.primeFactors ∪ y.primeFactors,
        (f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x ((y.factorization : (a : ℕ) → ℕ) x : ℕ)))
have neg : Eq.{u_1 + 1} (α := R)
  (∏ x_1 ∈ x.primeFactors ∪ y.primeFactors,
    HMul.hMul (α := R) (β := R)
      ((f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 (((x.lcm y).factorization : (a : ℕ) → ℕ) x_1 : ℕ)) : R)
      ((f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 (((x.gcd y).factorization : (a : ℕ) → ℕ) x_1 : ℕ)) : R))
  (∏ x_1 ∈ x.primeFactors ∪ y.primeFactors,
    HMul.hMul (α := R) (β := R)
      ((f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 ((x.factorization : (a : ℕ) → ℕ) x_1 : ℕ)) : R)
      ((f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 ((y.factorization : (a : ℕ) → ℕ) x_1 : ℕ)) : R))
Tactic: rw [← prod_mul_distrib, ← prod_mul_distrib]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, Eq (α := R) ((f : (a : ℕ) → R) (HPow.hPow (β := ℕ) i 0) : R) (1 : R)
⊢ Eq.{u_1 + 1} (α := R)
    (∏ x_1 ∈ x.primeFactors ∪ y.primeFactors,
      HMul.hMul (α := R) (β := R)
        ((f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 (((x.lcm y).factorization : (a : ℕ) → ℕ) x_1 : ℕ)) : R)
        ((f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 (((x.gcd y).factorization : (a : ℕ) → ℕ) x_1 : ℕ)) : R))
    (∏ x_1 ∈ x.primeFactors ∪ y.primeFactors,
      HMul.hMul (α := R) (β := R)
        ((f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 ((x.factorization : (a : ℕ) → ℕ) x_1 : ℕ)) : R)
        ((f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 ((y.factorization : (a : ℕ) → ℕ) x_1 : ℕ)) : R))
have neg : ∀ x_1 ∈ x.primeFactors ∪ y.primeFactors,
  HMul.hMul (α := R) (β := R)
      ((f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 (((x.lcm y).factorization : (a : ℕ) → ℕ) x_1 : ℕ)) : R)
      ((f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 (((x.gcd y).factorization : (a : ℕ) → ℕ) x_1 : ℕ)) : R) =
    HMul.hMul (α := R) (β := R)
      ((f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 ((x.factorization : (a : ℕ) → ℕ) x_1 : ℕ)) : R)
      ((f : (a : ℕ) → R) (HPow.hPow (β := ℕ) x_1 ((y.factorization : (a : ℕ) → ℕ) x_1 : ℕ)) : R)
Tactic: apply prod_congr rfl
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, Eq (α := R) ((f : (a : ℕ) → R) (HPow.hPow (β := ℕ) i 0) : R) (1 : R)
p : ℕ
h : p ∈ x.primeFactors ∪ y.primeFactors
⊢ HMul.hMul (α := R) (β := R)
      ((f : (a : ℕ) → R) (HPow.hPow (β := ℕ) p (((x.lcm y).factorization : (a : ℕ) → ℕ) p : ℕ)) : R)
      ((f : (a : ℕ) → R) (HPow.hPow (β := ℕ) p (((x.gcd y).factorization : (a : ℕ) → ℕ) p : ℕ)) : R) =
    HMul.hMul (α := R) (β := R) ((f : (a : ℕ) → R) (HPow.hPow (β := ℕ) p ((x.factorization : (a : ℕ) → ℕ) p : ℕ)) : R)
      ((f : (a : ℕ) → R) (HPow.hPow (β := ℕ) p ((y.factorization : (a : ℕ) → ℕ) p : ℕ)) : R)
have neg : ∀ (h : LE.le (α := ℕ) ((x.factorization : (a : ℕ) → ℕ) p : ℕ) ((y.factorization : (a : ℕ) → ℕ) p : ℕ)),
  HMul.hMul (α := R) (β := R)
      ((f : (a : ℕ) → R) (HPow.hPow (β := ℕ) p (((x.lcm y).factorization : (a : ℕ) → ℕ) p : ℕ)) : R)
      ((f : (a : ℕ) → R) (HPow.hPow (β := ℕ) p (((x.gcd y).factorization : (a : ℕ) → ℕ) p : ℕ)) : R) =
    HMul.hMul (α := R) (β := R) ((f : (a : ℕ) → R) (HPow.hPow (β := ℕ) p ((x.factorization : (a : ℕ) → ℕ) p : ℕ)) : R)
      ((f : (a : ℕ) → R) (HPow.hPow (β := ℕ) p ((y.factorization : (a : ℕ) → ℕ) p : ℕ)) : R)
Tactic: rcases Nat.le_or_le (x.factorization p) (y.factorization p) with h | h