awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
a : α
⊢ ∀ (x : α),
    Membership.mem (γ := Multiset α)
        (Multiset.map (α := α) (⇑normalize : (a : α) → α)
          (if h : a = 0 then 0 else Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯))
        x →
      (normalize : (a : α) → α) x = x
have pos : ∀ (h : a = 0) (x : α),
  Membership.mem (γ := Multiset α) (Multiset.map (fun (x : α) ↦ (normalize : (a : α) → α) x) 0) x →
    (normalize : (a : α) → α) x = x
Tactic: split_ifs with h
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
a : α
pos :
  ∀ (h : a = 0) (x : α),
    Membership.mem (γ := Multiset α) (Multiset.map (fun (x : α) ↦ (normalize : (a : α) → α) x) 0) x →
      (normalize : (a : α) → α) x = x
⊢ ∀ (x : α),
    Membership.mem (γ := Multiset α)
        (Multiset.map (α := α) (⇑normalize : (a : α) → α)
          (if h : a = 0 then 0 else Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯))
        x →
      (normalize : (a : α) → α) x = x
have neg : ∀ (h : ¬a = 0) (x : α),
  Membership.mem (γ := Multiset α)
      (Multiset.map (fun (x : α) ↦ (normalize : (a : α) → α) x)
        (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) sorry))
      x →
    (normalize : (a : α) → α) x = x
Tactic: split_ifs with h
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
a : α
h : ¬a = 0
x : α
hx :
  Membership.mem (γ := Multiset α)
    (Multiset.map (fun (x : α) ↦ (normalize : (a : α) → α) x)
      (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯))
    x
⊢ (normalize : (a : α) → α) x = x
have neg : ∀ y ∈ Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) sorry,
  ∀
    (hx :
      Membership.mem (α := α) (γ := Multiset α)
        (Multiset.map (fun (x : α) ↦ (normalize : (a : α) → α) x)
          (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) sorry))
        ((normalize : (a : α) → α) y : α)),
    Eq (α := α) ((normalize : (a : α) → α) ((normalize : (a : α) → α) y : α) : α) ((normalize : (a : α) → α) y : α)
Tactic: obtain ⟨y, _, rfl⟩ := Multiset.mem_map.1 hx
Cleared hypotheses: x, hx
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
a : α
ha : Irreducible a
⊢ normalizedFactors a = singleton (α := α) ((normalize : (a : α) → α) a : α)
have obtain : ∀ (p : α) (a_assoc : a ~ᵤ p) (hp : normalizedFactors a = {p}),
  normalizedFactors a = singleton (α := α) ((normalize : (a : α) → α) a : α)
Tactic: obtain ⟨p, a_assoc, hp⟩ :=
    prime_factors_irreducible ha ⟨prime_of_normalized_factor, prod_normalizedFactors ha.ne_zero⟩
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
a : α
ha : Irreducible a
p : α
a_assoc : a ~ᵤ p
hp : normalizedFactors a = {p}
⊢ p ∈ normalizedFactors a
have rw : Membership.mem (γ := Multiset α) {p} p
Tactic: rw [hp]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (hypothesis)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
a : α
ha : Irreducible a
p : α
a_assoc : a ~ᵤ p
hp : normalizedFactors a = {p}
⊢ normalizedFactors a = singleton (α := α) ((normalize : (a : α) → α) a : α)
have p_mem : p ∈ normalizedFactors a
Tactic: have p_mem : p ∈ normalizedFactors a := by
    rw [hp]
    exact Multiset.mem_singleton_self _
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
a : α
ha : Irreducible a
p : α
a_assoc : a ~ᵤ p
hp : normalizedFactors a = {p}
p_mem : p ∈ normalizedFactors a
⊢ normalizedFactors a = singleton (α := α) ((normalize : (a : α) → α) a : α)
have h : (normalize : (a : α) → α) a = p
Tactic: convert! hp
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
a : α
ha : Irreducible a
p : α
a_assoc : a ~ᵤ p
hp : normalizedFactors a = {p}
p_mem : p ∈ normalizedFactors a
⊢ (normalize : (a : α) → α) a = p
have h : Eq (α := α) ((normalize : (a : α) → α) a : α) ((normalize : (a : α) → α) p : α)
Tactic: rwa [← normalize_normalized_factor p p_mem, normalize_eq_normalize_iff, dvd_dvd_iff_associated]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
a : α
ha : Irreducible a
p : α
a_assoc : a ~ᵤ p
hp : normalizedFactors a = {p}
p_mem : p ∈ normalizedFactors a
h : Eq (α := α) ((normalize : (a : α) → α) a : α) ((normalize : (a : α) → α) p : α)
⊢ (normalize : (a : α) → α) a = p
have h₁ : a ∣ p ∧ p ∣ a
Tactic: rwa [← normalize_normalized_factor p p_mem, normalize_eq_normalize_iff, dvd_dvd_iff_associated]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
a p : α
hp : p ∈ normalizedFactors a
q : α
hq : q ∈ normalizedFactors a
hdvd : p ∣ q
⊢ p = q
have h : p = (normalize : (a : α) → α) p
Tactic: convert!
    normalize_eq_normalize hdvd
      ((prime_of_normalized_factor _ hp).irreducible.dvd_symm
        (prime_of_normalized_factor _ hq).irreducible hdvd)
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
a p : α
hp : p ∈ normalizedFactors a
q : α
hq : q ∈ normalizedFactors a
hdvd : p ∣ q
h : p = (normalize : (a : α) → α) p
⊢ p = q
have h₁ : q = (normalize : (a : α) → α) q
Tactic: convert!
    normalize_eq_normalize hdvd
      ((prime_of_normalized_factor _ hp).irreducible.dvd_symm
        (prime_of_normalized_factor _ hq).irreducible hdvd)
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
a p : α
ha0 : a ≠ 0
hp : Irreducible p
h : p ∣ a
b : α
hb : a = p * b
hb0 : b ≠ 0
⊢ p * b ~ᵤ (p ::ₘ normalizedFactors b).prod
have rw : p * b ~ᵤ p * (normalizedFactors b).prod
Tactic: rw [Multiset.prod_cons]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
x : α
hx : x ≠ 0
h : ¬IsUnit x
⊢ ∃ (p : α), p ∈ normalizedFactors x
have obtain : ∀ (p' : α) (hp' : Irreducible p') (hp'x : p' ∣ x), ∃ (p : α), p ∈ normalizedFactors x
Tactic: obtain ⟨p', hp', hp'x⟩ := WfDvdMonoid.exists_irreducible_factor h hx
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
x : α
hx : x ≠ 0
h : ¬IsUnit x
p' : α
hp' : Irreducible p'
hp'x : p' ∣ x
⊢ ∃ (p : α), p ∈ normalizedFactors x
have obtain : ∀ (p : α) (hp : p ∈ normalizedFactors x), p' ~ᵤ p → ∃ (p : α), p ∈ normalizedFactors x
Tactic: obtain ⟨p, hp, _⟩ := exists_mem_normalizedFactors_of_dvd hx hp' hp'x
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
⊢ Eq.{u_1 + 1} (α := Multiset α) (normalizedFactors 1) 0
have inl : ∀ (h : Subsingleton α), Eq.{u_1 + 1} (α := Multiset α) (normalizedFactors 1) 0
Tactic: rcases subsingleton_or_nontrivial α with h | h
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
inl : ∀ (h : Subsingleton α), Eq.{u_1 + 1} (α := Multiset α) (normalizedFactors 1) 0
⊢ Eq.{u_1 + 1} (α := Multiset α) (normalizedFactors 1) 0
have inr : ∀ (h : Nontrivial α), Eq.{u_1 + 1} (α := Multiset α) (normalizedFactors 1) 0
Tactic: rcases subsingleton_or_nontrivial α with h | h
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
h : Nontrivial α
⊢ Eq.{u_1 + 1} (α := Multiset α) (normalizedFactors 1) 0
have inr : Multiset.Rel.{u_1, u_1} (α := α) Associated (normalizedFactors 1) 0
Tactic: rw [← Multiset.rel_zero_right]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
h : Nontrivial α
inr : Multiset.Rel.{u_1, u_1} (α := α) Associated (normalizedFactors 1) 0
rw : Type u_1
⊢ Eq.{u_1 + 1} (α := Multiset α) (normalizedFactors 1) 0
have rw₁ : α → α → Prop
Tactic: rw [← Multiset.rel_zero_right]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
h : Nontrivial α
x : α
hx : Membership.mem (γ := Multiset α) 0 x
⊢ Irreducible x
have inr : False
Tactic: exfalso
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
x y : α
hx : x ≠ 0
hy : y ≠ 0
⊢ Eq.{u_1 + 1} (α := (a : α) → α) (⇑normalize : (a : α) → α)
    (Function.comp (β := Associates α) Associates.out Associates.mk)
have h : ∀ (x : α), (normalize : (a : α) → α) x = Function.comp (β := Associates α) Associates.out Associates.mk x
Tactic: ext
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
x y : α
hx : x ≠ 0
hy : y ≠ 0
x_1 : α
⊢ (normalize : (a : α) → α) x_1 = Function.comp (β := Associates α) Associates.out Associates.mk x_1
have h : (normalize : (a : α) → α) x_1 = Associates.out (Associates.mk x_1)
Tactic: rw [Function.comp_apply, Associates.out_mk]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
x y : α
hx : x ≠ 0
hy : y ≠ 0
x_1 : α
h : (normalize : (a : α) → α) x_1 = Associates.out (Associates.mk x_1)
⊢ (normalize : (a : α) → α) x_1 = Function.comp (β := Associates α) Associates.out Associates.mk x_1
have h₁ : Eq (α := α) ((normalize : (a : α) → α) x_1 : α) ((normalize : (a : α) → α) x_1 : α)
Tactic: rw [Function.comp_apply, Associates.out_mk]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (hypothesis)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
x y : α
hx : x ≠ 0
hy : y ≠ 0
⊢ normalizedFactors (x * y) = normalizedFactors x + normalizedFactors y
have h : Eq.{u_1 + 1} (α := (a : α) → α) (⇑normalize : (a : α) → α)
  (Function.comp (β := Associates α) Associates.out Associates.mk)
Tactic: have h : (normalize : α → α) = Associates.out ∘ Associates.mk := by
    ext
    rw [Function.comp_apply, Associates.out_mk]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
x y : α
hx : x ≠ 0
hy : y ≠ 0
h :
  Eq.{u_1 + 1} (α := (a : α) → α) (⇑normalize : (a : α) → α)
    (Function.comp (β := Associates α) Associates.out Associates.mk)
⊢ normalizedFactors (x * y) = normalizedFactors x + normalizedFactors y
have rw : Multiset.map (α := Associates α) Associates.out
    (Multiset.map Associates.mk (normalizedFactors x + normalizedFactors y)) =
  Multiset.map (α := Associates α) Associates.out (Multiset.map Associates.mk (normalizedFactors (x * y)))
Tactic: rw [← Multiset.map_id' (normalizedFactors (x * y)), ← Multiset.map_id' (normalizedFactors x), ←
    Multiset.map_id' (normalizedFactors y), ← Multiset.map_congr rfl normalize_normalized_factor, ←
    Multiset.map_congr rfl normalize_normalized_factor, ←
    Multiset.map_congr rfl normalize_normalized_factor, ← Multiset.map_add, h, ←
    Multiset.map_map Associates.out, eq_comm, ← Multiset.map_map Associates.out]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
x y : α
hx : x ≠ 0
hy : y ≠ 0
h :
  Eq.{u_1 + 1} (α := (a : α) → α) (⇑normalize : (a : α) → α)
    (Function.comp (β := Associates α) Associates.out Associates.mk)
⊢ Multiset.map (α := Associates α) Associates.out
      (Multiset.map Associates.mk (normalizedFactors x + normalizedFactors y)) =
    Multiset.map (α := Associates α) Associates.out (Multiset.map Associates.mk (normalizedFactors (x * y)))
have refine : Eq (α := Multiset (Associates α)) (Multiset.map Associates.mk (normalizedFactors x + normalizedFactors y))
  (Multiset.map Associates.mk (normalizedFactors (x * y)))
Tactic: refine congr rfl ?_
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
x y : α
hx : x ≠ 0
hy : y ≠ 0
h :
  Eq.{u_1 + 1} (α := (a : α) → α) (⇑normalize : (a : α) → α)
    (Function.comp (β := Associates α) Associates.out Associates.mk)
⊢ Eq (α := Multiset (Associates α)) (Multiset.map Associates.mk (normalizedFactors x + normalizedFactors y))
    (Multiset.map Associates.mk (normalizedFactors (x * y)))
have hst : Multiset.Rel (⇑(Associated.setoid α) : α → α → Prop) (normalizedFactors x + normalizedFactors y)
  (normalizedFactors (x * y))
Tactic: apply Multiset.map_mk_eq_map_mk_of_rel
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
x y : α
hx : x ≠ 0
hy : y ≠ 0
h :
  Eq.{u_1 + 1} (α := (a : α) → α) (⇑normalize : (a : α) → α)
    (Function.comp (β := Associates α) Associates.out Associates.mk)
⊢ Multiset.Rel (⇑(Associated.setoid α) : α → α → Prop) (normalizedFactors x + normalizedFactors y)
    (normalizedFactors (x * y))
have hst : ∀ x_1 ∈ normalizedFactors x + normalizedFactors y, Irreducible x_1
Tactic: apply factors_unique
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
x y : α
hx : x ≠ 0
hy : y ≠ 0
h :
  Eq.{u_1 + 1} (α := (a : α) → α) (⇑normalize : (a : α) → α)
    (Function.comp (β := Associates α) Associates.out Associates.mk)
hst : ∀ x_1 ∈ normalizedFactors x + normalizedFactors y, Irreducible x_1
⊢ Multiset.Rel (⇑(Associated.setoid α) : α → α → Prop) (normalizedFactors x + normalizedFactors y)
    (normalizedFactors (x * y))
have hst₁ : ∀ x_1 ∈ normalizedFactors (x * y), Irreducible x_1
Tactic: apply factors_unique
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
x y : α
hx : x ≠ 0
hy : y ≠ 0
h :
  Eq.{u_1 + 1} (α := (a : α) → α) (⇑normalize : (a : α) → α)
    (Function.comp (β := Associates α) Associates.out Associates.mk)
hst : ∀ x_1 ∈ normalizedFactors x + normalizedFactors y, Irreducible x_1
hst₁ : ∀ x_1 ∈ normalizedFactors (x * y), Irreducible x_1
⊢ Multiset.Rel (⇑(Associated.setoid α) : α → α → Prop) (normalizedFactors x + normalizedFactors y)
    (normalizedFactors (x * y))
have hst₂ : (normalizedFactors x + normalizedFactors y).prod ~ᵤ (normalizedFactors (x * y)).prod
Tactic: apply factors_unique
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
x_1 y : α
hx_1 : x_1 ≠ 0
hy : y ≠ 0
h :
  Eq.{u_1 + 1} (α := (a : α) → α) (⇑normalize : (a : α) → α)
    (Function.comp (β := Associates α) Associates.out Associates.mk)
x : α
hx : x ∈ normalizedFactors x_1 + normalizedFactors y
⊢ Irreducible x
have hst : ∀ (hx : x ∈ normalizedFactors x_1), Irreducible x
Tactic: rcases Multiset.mem_add.1 hx with (hx | hx)
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
x_1 y : α
hx_1 : x_1 ≠ 0
hy : y ≠ 0
h :
  Eq.{u_1 + 1} (α := (a : α) → α) (⇑normalize : (a : α) → α)
    (Function.comp (β := Associates α) Associates.out Associates.mk)
x : α
hx : x ∈ normalizedFactors x_1 + normalizedFactors y
hst : ∀ (hx : x ∈ normalizedFactors x_1), Irreducible x
⊢ Irreducible x
have hst₁ : ∀ (hx : x ∈ normalizedFactors y), Irreducible x
Tactic: rcases Multiset.mem_add.1 hx with (hx | hx)
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
x y : α
hx : x ≠ 0
hy : y ≠ 0
h :
  Eq.{u_1 + 1} (α := (a : α) → α) (⇑normalize : (a : α) → α)
    (Function.comp (β := Associates α) Associates.out Associates.mk)
⊢ (normalizedFactors x + normalizedFactors y).prod ~ᵤ (normalizedFactors (x * y)).prod
have hst : (normalizedFactors x).prod * (normalizedFactors y).prod ~ᵤ (normalizedFactors (x * y)).prod
Tactic: rw [Multiset.prod_add]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
x : α
n : ℕ
ih : normalizedFactors (x ^ n) = n • normalizedFactors x
⊢ normalizedFactors (x ^ (n + 1)) = (n + 1) • normalizedFactors x
have pos : ∀ (h0 : x = 0), normalizedFactors (x ^ (n + 1)) = (n + 1) • normalizedFactors x
Tactic: by_cases h0 : x = 0
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
x : α
n : ℕ
ih : normalizedFactors (x ^ n) = n • normalizedFactors x
pos : ∀ (h0 : x = 0), normalizedFactors (x ^ (n + 1)) = (n + 1) • normalizedFactors x
⊢ normalizedFactors (x ^ (n + 1)) = (n + 1) • normalizedFactors x
have neg : ∀ (h0 : ¬x = 0), normalizedFactors (x ^ (n + 1)) = (n + 1) • normalizedFactors x
Tactic: by_cases h0 : x = 0
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
x : α
n : ℕ
ih : normalizedFactors (x ^ n) = n • normalizedFactors x
h0 : ¬x = 0
⊢ normalizedFactors (x ^ (n + 1)) = (n + 1) • normalizedFactors x
have neg : normalizedFactors (x * x ^ n) = (n + 1) • normalizedFactors x
Tactic: rw [pow_succ', succ_nsmul', normalizedFactors_mul h0 (pow_ne_zero _ h0), ih]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
x : α
n : ℕ
ih : normalizedFactors (x ^ n) = n • normalizedFactors x
h0 : ¬x = 0
neg : normalizedFactors (x * x ^ n) = (n + 1) • normalizedFactors x
⊢ normalizedFactors (x ^ (n + 1)) = (n + 1) • normalizedFactors x
have neg₁ : normalizedFactors (x * x ^ n) = normalizedFactors x + n • normalizedFactors x
Tactic: rw [pow_succ', succ_nsmul', normalizedFactors_mul h0 (pow_ne_zero _ h0), ih]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
x : α
n : ℕ
ih : normalizedFactors (x ^ n) = n • normalizedFactors x
h0 : ¬x = 0
neg : normalizedFactors (x * x ^ n) = (n + 1) • normalizedFactors x
neg₁ : normalizedFactors (x * x ^ n) = normalizedFactors x + n • normalizedFactors x
⊢ normalizedFactors (x ^ (n + 1)) = (n + 1) • normalizedFactors x
have neg₂ : normalizedFactors x + normalizedFactors (x ^ n) = normalizedFactors x + n • normalizedFactors x
Tactic: rw [pow_succ', succ_nsmul', normalizedFactors_mul h0 (pow_ne_zero _ h0), ih]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
x : α
n : ℕ
ih : normalizedFactors (x ^ n) = n • normalizedFactors x
h0 : ¬x = 0
neg : normalizedFactors (x * x ^ n) = (n + 1) • normalizedFactors x
neg₁ : normalizedFactors (x * x ^ n) = normalizedFactors x + n • normalizedFactors x
neg₂ : normalizedFactors x + normalizedFactors (x ^ n) = normalizedFactors x + n • normalizedFactors x
⊢ normalizedFactors (x ^ (n + 1)) = (n + 1) • normalizedFactors x
have neg₃ : normalizedFactors x + n • normalizedFactors x = normalizedFactors x + n • normalizedFactors x
Tactic: rw [pow_succ', succ_nsmul', normalizedFactors_mul h0 (pow_ne_zero _ h0), ih]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
x : α
n : ℕ
⊢ normalizedFactors (x ^ n) = n • normalizedFactors x
have zero : normalizedFactors (HPow.hPow (β := ℕ) x 0) = HSMul.hSMul (α := ℕ) 0 (normalizedFactors x)
Tactic: induction n with
  | zero => simp [zero_nsmul]
  | succ n ih =>
    by_cases h0 : x = 0
    · simp [h0, zero_pow n.succ_ne_zero, nsmul_zero]
    rw [pow_succ', succ_nsmul', normalizedFactors_mul h0 (pow_ne_zero _ h0), ih]
Cleared hypotheses: n
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
x : α
n : ℕ
zero : normalizedFactors (HPow.hPow (β := ℕ) x 0) = HSMul.hSMul (α := ℕ) 0 (normalizedFactors x)
⊢ normalizedFactors (x ^ n) = n • normalizedFactors x
have succ : ∀ (n : ℕ) (ih : normalizedFactors (x ^ n) = n • normalizedFactors x),
  normalizedFactors (x ^ (n + 1)) = (n + 1) • normalizedFactors x
Tactic: induction n with
  | zero => simp [zero_nsmul]
  | succ n ih =>
    by_cases h0 : x = 0
    · simp [h0, zero_pow n.succ_ne_zero, nsmul_zero]
    rw [pow_succ', succ_nsmul', normalizedFactors_mul h0 (pow_ne_zero _ h0), ih]
Cleared hypotheses: n