awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a p : α
ha0 : a ≠ 0
hp : Irreducible p
hd : p ∣ a
q : α
hq : q ∈ UniqueFactorizationMonoid.factors a
hpq : p ~ᵤ q
⊢ Membership.mem (γ := Multiset { a : Associates α // Irreducible a }) (factors' a)
    (Subtype.mk (p := Irreducible) (Associates.mk p) ⋯)
have apply : ∃ (a_1 : α) (h : a_1 ∈ UniqueFactorizationMonoid.factors a),
  Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) (Associates.mk a_1) sorry =
    Subtype.mk (p := Irreducible) (Associates.mk p) sorry
Tactic: apply Multiset.mem_pmap.mpr
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a p : α
ha0 : a ≠ 0
hp : Irreducible p
hd : p ∣ a
q : α
hq : q ∈ UniqueFactorizationMonoid.factors a
hpq : p ~ᵤ q
⊢ ∃ (a_1 : α) (h : a_1 ∈ UniqueFactorizationMonoid.factors a),
    Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) (Associates.mk a_1) ⋯ =
      Subtype.mk (p := Irreducible) (Associates.mk p) ⋯
have h : ∃ (h : q ∈ UniqueFactorizationMonoid.factors a),
  Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) (Associates.mk q) sorry =
    Subtype.mk (p := Irreducible) (Associates.mk p) sorry
Tactic: use q
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a p : α
ha0 : a ≠ 0
hp : Irreducible p
hd : p ∣ a
q : α
hq : q ∈ UniqueFactorizationMonoid.factors a
hpq : p ~ᵤ q
⊢ ∃ (h : q ∈ UniqueFactorizationMonoid.factors a),
    Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) (Associates.mk q) ⋯ =
      Subtype.mk (p := Irreducible) (Associates.mk p) ⋯
have h : Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) (Associates.mk q) sorry =
  Subtype.mk (p := Irreducible) (Associates.mk p) sorry
Tactic: use hq
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a p : α
ha0 : a ≠ 0
hp : Irreducible p
⊢ Membership.mem (γ := Multiset { a : Associates α // Irreducible a }) (factors' a)
      (Subtype.mk (p := Irreducible) (Associates.mk p) ⋯) ↔
    p ∣ a
have mp : Membership.mem (γ := Multiset { a : Associates α // Irreducible a }) (factors' a)
    (Subtype.mk (p := Irreducible) (Associates.mk p) sorry) →
  p ∣ a
Tactic: constructor
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a p : α
ha0 : a ≠ 0
hp : Irreducible p
mp :
  Membership.mem (γ := Multiset { a : Associates α // Irreducible a }) (factors' a)
      (Subtype.mk (p := Irreducible) (Associates.mk p) ⋯) →
    p ∣ a
⊢ Membership.mem (γ := Multiset { a : Associates α // Irreducible a }) (factors' a)
      (Subtype.mk (p := Irreducible) (Associates.mk p) ⋯) ↔
    p ∣ a
have mpr : p ∣ a →
  Membership.mem (γ := Multiset { a : Associates α // Irreducible a }) (factors' a)
    (Subtype.mk (p := Irreducible) (Associates.mk p) sorry)
Tactic: constructor
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a p : α
ha0 : a ≠ 0
hp : Irreducible p
⊢ Membership.mem (γ := Multiset { a : Associates α // Irreducible a }) (factors' a)
      (Subtype.mk (p := Irreducible) (Associates.mk p) ⋯) →
    p ∣ a
have mp : Membership.mem (γ := Multiset { a : Associates α // Irreducible a }) (factors' a)
    (Subtype.mk (p := Irreducible) (Associates.mk p) sorry) →
  Dvd.dvd (α := Associates α) (Associates.mk p) (Associates.mk a)
Tactic: rw [← mk_dvd_mk]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a p : α
ha0 : a ≠ 0
hp : Irreducible p
hd : p ∣ a
⊢ Membership.mem (γ := FactorSet α) (factors (Associates.mk a)) (Associates.mk p)
have rw : Membership.mem (γ := FactorSet α) (↑(factors' a) : WithTop (Multiset { a : Associates α // Irreducible a }))
  (Associates.mk p)
Tactic: rw [factors_mk _ ha0]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a p : α
ha0 : a ≠ 0
hp : Irreducible p
⊢ Membership.mem (γ := FactorSet α) (factors (Associates.mk a)) (Associates.mk p) ↔ p ∣ a
have mp : Membership.mem (γ := FactorSet α) (factors (Associates.mk a)) (Associates.mk p) → p ∣ a
Tactic: constructor
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a p : α
ha0 : a ≠ 0
hp : Irreducible p
mp : Membership.mem (γ := FactorSet α) (factors (Associates.mk a)) (Associates.mk p) → p ∣ a
⊢ Membership.mem (γ := FactorSet α) (factors (Associates.mk a)) (Associates.mk p) ↔ p ∣ a
have mpr : p ∣ a → Membership.mem (γ := FactorSet α) (factors (Associates.mk a)) (Associates.mk p)
Tactic: constructor
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a p : α
ha0 : a ≠ 0
hp : Irreducible p
⊢ Membership.mem (γ := FactorSet α) (factors (Associates.mk a)) (Associates.mk p) → p ∣ a
have mp : Membership.mem (γ := FactorSet α) (factors (Associates.mk a)) (Associates.mk p) →
  Dvd.dvd (α := Associates α) (Associates.mk p) (Associates.mk a)
Tactic: rw [← mk_dvd_mk]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
ha : a ≠ 0
hb : b ≠ 0
h : Ne (α := Associates α) (Associates.mk a ⊓ Associates.mk b) 1
⊢ Ne (α := FactorSet α) (factors (Associates.mk a) ⊓ factors (Associates.mk b)) 0
have contrapose : ∀ (hf : Eq (α := FactorSet α) (factors (Associates.mk a) ⊓ factors (Associates.mk b)) 0),
  Eq (α := Associates α) (Associates.mk a ⊓ Associates.mk b) 1
Tactic: contrapose h with hf
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
ha : a ≠ 0
hb : b ≠ 0
hf : Eq (α := FactorSet α) (factors (Associates.mk a) ⊓ factors (Associates.mk b)) 0
⊢ Eq (α := Associates α) (FactorSet.prod (factors (Associates.mk a) ⊓ factors (Associates.mk b))) 1
have rw : Eq.{u_1 + 1} (α := Associates α) (FactorSet.prod 0) 1
Tactic: rw [hf]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (hypothesis)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
ha : a ≠ 0
hb : b ≠ 0
h : Ne (α := Associates α) (Associates.mk a ⊓ Associates.mk b) 1
⊢ ∃ (p : α), Prime p ∧ p ∣ a ∧ p ∣ b
have hz : Ne (α := FactorSet α) (factors (Associates.mk a) ⊓ factors (Associates.mk b)) 0
Tactic: have hz : factors (Associates.mk a) ⊓ factors (Associates.mk b) ≠ 0 := by
    contrapose h with hf
    change (factors (Associates.mk a) ⊓ factors (Associates.mk b)).prod = 1
    rw [hf]
    exact Multiset.prod_zero
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (hypothesis)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
ha : a ≠ 0
hb : b ≠ 0
h : Ne (α := Associates α) (Associates.mk a ⊓ Associates.mk b) 1
hz : Ne (α := FactorSet α) (factors (Associates.mk a) ⊓ factors (Associates.mk b)) 0
⊢ ∃ (p : α), Prime p ∧ p ∣ a ∧ p ∣ b
have hz₁ : Ne (α := FactorSet α) (↑(factors' a ⊓ factors' b) : WithTop (Multiset { a : Associates α // Irreducible a })) 0
Tactic: rw [factors_mk a ha, factors_mk b hb, ← WithTop.coe_inf] at hz
Cleared hypotheses: hz
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
ha : a ≠ 0
hb : b ≠ 0
h : Ne (α := Associates α) (Associates.mk a ⊓ Associates.mk b) 1
hz : Ne (α := FactorSet α) (↑(factors' a ⊓ factors' b) : WithTop (Multiset { a : Associates α // Irreducible a })) 0
⊢ ∃ (p : α), Prime p ∧ p ∣ a ∧ p ∣ b
have obtain : ∀ (p0 : Associates α) (p0_irr : Irreducible p0)
  (p0_mem :
    Membership.mem (γ := Multiset { a : Associates α // Irreducible a }) (factors' a ⊓ factors' b)
      (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) p0 p0_irr)),
  ∃ (p : α), Prime p ∧ p ∣ a ∧ p ∣ b
Tactic: obtain ⟨⟨p0, p0_irr⟩, p0_mem⟩ := Multiset.exists_mem_of_ne_zero ((mt WithTop.coe_eq_coe.mpr) hz)
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (hypothesis)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
ha : a ≠ 0
hb : b ≠ 0
h : Ne (α := Associates α) (Associates.mk a ⊓ Associates.mk b) 1
hz : Ne (α := FactorSet α) (↑(factors' a ⊓ factors' b) : WithTop (Multiset { a : Associates α // Irreducible a })) 0
p0 : Associates α
p0_irr : Irreducible p0
p0_mem :
  Membership.mem (γ := Multiset { a : Associates α // Irreducible a }) (factors' a ⊓ factors' b)
    (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) p0 p0_irr)
⊢ ∃ (p : α), Prime p ∧ p ∣ a ∧ p ∣ b
have p0_mem₁ : Membership.mem (γ := Multiset { a : Associates α // Irreducible a }) (factors' a ∩ factors' b)
  (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) p0 p0_irr)
Tactic: rw [Multiset.inf_eq_inter] at p0_mem
Cleared hypotheses: p0_mem
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
ha : a ≠ 0
hb : b ≠ 0
h : Ne (α := Associates α) (Associates.mk a ⊓ Associates.mk b) 1
hz : Ne (α := FactorSet α) (↑(factors' a ⊓ factors' b) : WithTop (Multiset { a : Associates α // Irreducible a })) 0
p0 : Associates α
p0_irr : Irreducible p0
p0_mem :
  Membership.mem (γ := Multiset { a : Associates α // Irreducible a }) (factors' a ∩ factors' b)
    (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) p0 p0_irr)
⊢ ∃ (p : α), Prime p ∧ p ∣ a ∧ p ∣ b
have obtain : ∀ (p : α) (p0_irr : Irreducible (M := Associates α) (Associates.mk p))
  (p0_mem :
    Membership.mem (γ := Multiset { a : Associates α // Irreducible a }) (factors' a ∩ factors' b)
      (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) (Associates.mk p) p0_irr)),
  ∃ (p : α), Prime p ∧ p ∣ a ∧ p ∣ b
Tactic: obtain ⟨p, rfl⟩ : ∃ p, Associates.mk p = p0 := Quot.exists_rep p0
Cleared hypotheses: p0, p0_irr, p0_mem
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
ha : a ≠ 0
hb : b ≠ 0
h : Ne (α := Associates α) (Associates.mk a ⊓ Associates.mk b) 1
hz : Ne (α := FactorSet α) (↑(factors' a ⊓ factors' b) : WithTop (Multiset { a : Associates α // Irreducible a })) 0
p : α
p0_irr : Irreducible (M := Associates α) (Associates.mk p)
p0_mem :
  Membership.mem (γ := Multiset { a : Associates α // Irreducible a }) (factors' a ∩ factors' b)
    (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) (Associates.mk p) p0_irr)
⊢ ∃ (p : α), Prime p ∧ p ∣ a ∧ p ∣ b
have refine_1 : Prime p
Tactic: refine ⟨p, ?_, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
ha : a ≠ 0
hb : b ≠ 0
h : Ne (α := Associates α) (Associates.mk a ⊓ Associates.mk b) 1
hz : Ne (α := FactorSet α) (↑(factors' a ⊓ factors' b) : WithTop (Multiset { a : Associates α // Irreducible a })) 0
p : α
p0_irr : Irreducible (M := Associates α) (Associates.mk p)
p0_mem :
  Membership.mem (γ := Multiset { a : Associates α // Irreducible a }) (factors' a ∩ factors' b)
    (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) (Associates.mk p) p0_irr)
refine_1 : Prime p
⊢ ∃ (p : α), Prime p ∧ p ∣ a ∧ p ∣ b
have refine_2 : p ∣ a
Tactic: refine ⟨p, ?_, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
ha : a ≠ 0
hb : b ≠ 0
h : Ne (α := Associates α) (Associates.mk a ⊓ Associates.mk b) 1
hz : Ne (α := FactorSet α) (↑(factors' a ⊓ factors' b) : WithTop (Multiset { a : Associates α // Irreducible a })) 0
p : α
p0_irr : Irreducible (M := Associates α) (Associates.mk p)
p0_mem :
  Membership.mem (γ := Multiset { a : Associates α // Irreducible a }) (factors' a ∩ factors' b)
    (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) (Associates.mk p) p0_irr)
refine_1 : Prime p
refine_2 : p ∣ a
⊢ ∃ (p : α), Prime p ∧ p ∣ a ∧ p ∣ b
have refine_3 : p ∣ b
Tactic: refine ⟨p, ?_, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
ha : a ≠ 0
hb : b ≠ 0
h : Ne (α := Associates α) (Associates.mk a ⊓ Associates.mk b) 1
hz : Ne (α := FactorSet α) (↑(factors' a ⊓ factors' b) : WithTop (Multiset { a : Associates α // Irreducible a })) 0
p : α
p0_irr : Irreducible (M := Associates α) (Associates.mk p)
p0_mem :
  Membership.mem (γ := Multiset { a : Associates α // Irreducible a }) (factors' a ∩ factors' b)
    (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) (Associates.mk p) p0_irr)
⊢ p ∣ a
have refine_2 : LE.le (α := Associates α) (Associates.mk p) (Associates.mk a)
Tactic: apply dvd_of_mk_le_mk
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
ha : a ≠ 0
hb : b ≠ 0
h : Ne (α := Associates α) (Associates.mk a ⊓ Associates.mk b) 1
hz : Ne (α := FactorSet α) (↑(factors' a ⊓ factors' b) : WithTop (Multiset { a : Associates α // Irreducible a })) 0
p : α
p0_irr : Irreducible (M := Associates α) (Associates.mk p)
p0_mem :
  Membership.mem (γ := Multiset { a : Associates α // Irreducible a }) (factors' a ∩ factors' b)
    (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) (Associates.mk p) p0_irr)
⊢ p ∣ b
have refine_3 : LE.le (α := Associates α) (Associates.mk p) (Associates.mk b)
Tactic: apply dvd_of_mk_le_mk
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
ha0 : a ≠ 0
hb0 : b ≠ 0
⊢ Eq (α := Associates α) (Associates.mk a ⊓ Associates.mk b) 1 ↔ ∀ {d : α}, d ∣ a → d ∣ b → ¬Prime d
have mp : Eq (α := Associates α) (Associates.mk a ⊓ Associates.mk b) 1 → ∀ {d : α}, d ∣ a → d ∣ b → ¬Prime d
Tactic: constructor
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
ha0 : a ≠ 0
hb0 : b ≠ 0
mp : Eq (α := Associates α) (Associates.mk a ⊓ Associates.mk b) 1 → ∀ {d : α}, d ∣ a → d ∣ b → ¬Prime d
⊢ Eq (α := Associates α) (Associates.mk a ⊓ Associates.mk b) 1 ↔ ∀ {d : α}, d ∣ a → d ∣ b → ¬Prime d
have mpr : (∀ {d : α}, d ∣ a → d ∣ b → ¬Prime d) → Eq (α := Associates α) (Associates.mk a ⊓ Associates.mk b) 1
Tactic: constructor
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
ha0 : a ≠ 0
hb0 : b ≠ 0
hg : Eq (α := Associates α) (Associates.mk a ⊓ Associates.mk b) 1
p : α
ha : p ∣ a
hb : p ∣ b
hp : Prime p
⊢ False
have mp : Dvd.dvd (α := Associates α) (Associates.mk p) 1
Tactic: refine (Associates.prime_mk.mpr hp).not_unit (isUnit_of_dvd_one ?_)
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
ha0 : a ≠ 0
hb0 : b ≠ 0
hg : Eq (α := Associates α) (Associates.mk a ⊓ Associates.mk b) 1
p : α
ha : p ∣ a
hb : p ∣ b
hp : Prime p
⊢ Dvd.dvd (α := Associates α) (Associates.mk p) 1
have mp : Dvd.dvd (α := Associates α) (Associates.mk p) (Associates.mk a ⊓ Associates.mk b)
Tactic: rw [← hg]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
ha0 : a ≠ 0
hb0 : b ≠ 0
⊢ (∀ {d : α}, d ∣ a → d ∣ b → ¬Prime d) → Eq (α := Associates α) (Associates.mk a ⊓ Associates.mk b) 1
have mpr : ¬Eq (α := Associates α) (Associates.mk a ⊓ Associates.mk b) 1 → ¬∀ {d : α}, d ∣ a → d ∣ b → ¬Prime d
Tactic: contrapose
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
ha0 : a ≠ 0
hb0 : b ≠ 0
hg : ¬Eq (α := Associates α) (Associates.mk a ⊓ Associates.mk b) 1
hc : ∀ {d : α}, d ∣ a → d ∣ b → ¬Prime d
⊢ False
have mpr : ∀ (p : α) (hp : Prime p) (hpa : p ∣ a) (hpb : p ∣ b), False
Tactic: obtain ⟨p, hp, hpa, hpb⟩ := exists_prime_dvd_of_not_inf_one ha0 hb0 hg
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : Nontrivial α
p : Associates α
hp : Irreducible p
⊢ p.factors.prod =
    FactorSet.prod
      (↑{Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) p hp} :
        WithTop (Multiset { a : Associates α // Irreducible a }))
have rw : p =
  match (motive := FactorSet α → Associates α)
    (↑{Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) p hp} :
      WithTop (Multiset { a : Associates α // Irreducible a })) with
  | none => 0
  | some s => (Multiset.map (Subtype.val (p := fun (a : Associates α) ↦ Irreducible a)) s).prod
Tactic: rw [factors_prod, FactorSet.prod.eq_def]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : Nontrivial α
p : Associates α
hp : Irreducible p
⊢ p =
    match (motive := FactorSet α → Associates α)
      (↑{Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) p hp} :
        WithTop (Multiset { a : Associates α // Irreducible a })) with
    | none => 0
    | some s => (Multiset.map (Subtype.val (p := fun (a : Associates α) ↦ Irreducible a)) s).prod
have dsimp : p = Multiset.prod {p}
Tactic: dsimp
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : Nontrivial α
p : Associates α
hp : Irreducible p
⊢ p = Multiset.prod {p}
have rw : p = p
Tactic: rw [prod_singleton]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : Nontrivial α
p : Associates α
hp : Irreducible p
k : ℕ
⊢ (p ^ k).factors.prod =
    FactorSet.prod
      (↑(replicate k (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) p hp)) :
        WithTop (Multiset { a : Associates α // Irreducible a }))
have rw : p ^ k =
  match (motive := FactorSet α → Associates α)
    (↑(replicate k (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) p hp)) :
      WithTop (Multiset { a : Associates α // Irreducible a })) with
  | none => 0
  | some s => (Multiset.map (Subtype.val (p := fun (a : Associates α) ↦ Irreducible a)) s).prod
Tactic: rw [Associates.factors_prod, FactorSet.prod.eq_def]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : Nontrivial α
p : Associates α
hp : Irreducible p
k : ℕ
⊢ p ^ k =
    (Multiset.map (Subtype.val (p := fun (a : Associates α) ↦ Irreducible a))
        (replicate k (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) p hp))).prod
have rw : p ^ k = (replicate k (↑(Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) p hp) : Associates α)).prod
Tactic: rw [Multiset.map_replicate, Multiset.prod_replicate, Subtype.coe_mk]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : Nontrivial α
p : Associates α
hp : Irreducible p
k : ℕ
rw : p ^ k = (replicate k (↑(Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) p hp) : Associates α)).prod
⊢ p ^ k =
    (Multiset.map (Subtype.val (p := fun (a : Associates α) ↦ Irreducible a))
        (replicate k (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) p hp))).prod
have rw₁ : p ^ k = (↑(Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) p hp) : Associates α) ^ k
Tactic: rw [Multiset.map_replicate, Multiset.prod_replicate, Subtype.coe_mk]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : Nontrivial α
p : Associates α
hp : Irreducible p
k : ℕ
rw : p ^ k = (replicate k (↑(Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) p hp) : Associates α)).prod
rw₁ : p ^ k = (↑(Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) p hp) : Associates α) ^ k
⊢ p ^ k =
    (Multiset.map (Subtype.val (p := fun (a : Associates α) ↦ Irreducible a))
        (replicate k (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) p hp))).prod
have rw₂ : p ^ k = p ^ k
Tactic: rw [Multiset.map_replicate, Multiset.prod_replicate, Subtype.coe_mk]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : DecidableEq (Associates α)
m p : Associates α
h₁ : m ≠ 0
h₂ : Irreducible p
k : ℕ
⊢ p ^ k ≤ m ↔ k ≤ bcount (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) p h₂) m.factors
have rcases : ∀ (m : α) (hm : m ≠ 0) (h₁ : Ne (α := Associates α) (Associates.mk m) 0),
  p ^ k ≤ Associates.mk m ↔
    k ≤ bcount (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) p h₂) (factors (Associates.mk m))
Tactic: rcases Associates.exists_non_zero_rep h₁ with ⟨m, hm, rfl⟩
Cleared hypotheses: m, h₁
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (hypothesis)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : DecidableEq (Associates α)
p : Associates α
h₂ : Irreducible p
k : ℕ
m : α
hm : m ≠ 0
h₁ : Ne (α := Associates α) (Associates.mk m) 0
⊢ p ^ k ≤ Associates.mk m ↔
    k ≤ bcount (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) p h₂) (factors (Associates.mk m))
have this : Nontrivial α
Tactic: have := nontrivial_of_ne _ _ hm
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : Nontrivial α
⊢ Eq.{u_1 + 1} (α := FactorSet α) (factors 1) 0
have h : Eq.{u_1 + 1} (α := Associates α) (FactorSet.prod (factors 1)) (FactorSet.prod 0)
Tactic: apply FactorSet.unique
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : Nontrivial α
⊢ Eq.{u_1 + 1} (α := Associates α) (FactorSet.prod (factors 1)) (FactorSet.prod 0)
have h : Eq.{u_1 + 1} (α := Associates α) 1 (FactorSet.prod 0)
Tactic: rw [Associates.factors_prod]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : Nontrivial α
a : Associates α
⊢ (HPow.hPow (β := ℕ) a 0).factors = HSMul.hSMul (α := ℕ) 0 a.factors
have zero : Eq.{u_1 + 1} (α := FactorSet α) (factors 1) 0
Tactic: rw [zero_nsmul, pow_zero]