awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (hypothesis)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : Associates α
⊢ (a * b).factors = a.factors + b.factors
have a₁ : Nontrivial α
Tactic: nontriviality α
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : Associates α
h : Nontrivial α
⊢ (a * b).factors = a.factors + b.factors
have refine : (a * b).factors.prod.factors = (a.factors + b.factors).prod.factors
Tactic: refine FactorSet.unique <| eq_of_factors_eq_factors ?_
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : Associates α
h : Nontrivial α
⊢ (a * b).factors.prod.factors = (a.factors + b.factors).prod.factors
have rw : (a * b).factors.prod.factors = (a.factors.prod * b.factors.prod).factors
Tactic: rw [prod_add, factors_prod, factors_prod, factors_prod]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : Associates α
h : Nontrivial α
rw : (a * b).factors.prod.factors = (a.factors.prod * b.factors.prod).factors
⊢ (a * b).factors.prod.factors = (a.factors + b.factors).prod.factors
have rw₁ : (a * b).factors = (a.factors.prod * b.factors.prod).factors
Tactic: rw [prod_add, factors_prod, factors_prod, factors_prod]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : Associates α
h : Nontrivial α
rw : (a * b).factors.prod.factors = (a.factors.prod * b.factors.prod).factors
rw₁ : (a * b).factors = (a.factors.prod * b.factors.prod).factors
⊢ (a * b).factors.prod.factors = (a.factors + b.factors).prod.factors
have rw₂ : (a * b).factors = (a * b.factors.prod).factors
Tactic: rw [prod_add, factors_prod, factors_prod, factors_prod]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : Associates α
h : Nontrivial α
rw : (a * b).factors.prod.factors = (a.factors.prod * b.factors.prod).factors
rw₁ : (a * b).factors = (a.factors.prod * b.factors.prod).factors
rw₂ : (a * b).factors = (a * b.factors.prod).factors
⊢ (a * b).factors.prod.factors = (a.factors + b.factors).prod.factors
have rw₃ : (a * b).factors = (a * b).factors
Tactic: rw [prod_add, factors_prod, factors_prod, factors_prod]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
s t : Associates α
d : Associates α
eq : t = s * d
⊢ s.factors ≤ t.factors
have rw : s.factors ≤ s.factors + d.factors
Tactic: rw [eq, factors_mul]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : Associates α
⊢ a.factors ≤ b.factors ↔ a ≤ b
have refine : ∀ (h : a.factors ≤ b.factors), a ≤ b
Tactic: refine ⟨fun h ↦ ?_, factors_mono⟩
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (hypothesis)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : Associates α
h : a.factors ≤ b.factors
⊢ a ≤ b
have this : a.factors.prod ≤ b.factors.prod
Tactic: have : a.factors.prod ≤ b.factors.prod := prod_mono 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 : Associates α
h : a.factors ≤ b.factors
this : a.factors.prod ≤ b.factors.prod
⊢ a ≤ b
have rwa : ∀ (this : a ≤ b.factors.prod), a ≤ b
Tactic: rwa [factors_prod, factors_prod] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : Associates α
h : a.factors ≤ b.factors
this : a.factors.prod ≤ b.factors.prod
rwa : ∀ (this : a ≤ b.factors.prod), a ≤ b
⊢ a ≤ b
have rwa₁ : ∀ (this : a ≤ b), a ≤ b
Tactic: rwa [factors_prod, factors_prod] at this
Cleared hypotheses: this
-- 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)
inst_3 : (p : Associates α) → Decidable (Irreducible p)
a b : Associates α
ha : a ≠ 0
hb : b ≠ 0
h : ∀ (p : Associates α), Irreducible p → p.count a.factors = p.count b.factors
⊢ a.factors = b.factors
have obtain : ∀ (sa : Multiset { p : Associates α // Irreducible p })
  (h_sa : a.factors = (↑sa : WithTop (Multiset { p : Associates α // Irreducible p }))), a.factors = b.factors
Tactic: obtain ⟨sa, h_sa⟩ := factors_eq_some_iff_ne_zero.mpr ha
-- 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)
inst_3 : (p : Associates α) → Decidable (Irreducible p)
a b : Associates α
ha : a ≠ 0
hb : b ≠ 0
h : ∀ (p : Associates α), Irreducible p → p.count a.factors = p.count b.factors
sa : Multiset { p : Associates α // Irreducible p }
h_sa : a.factors = (↑sa : WithTop (Multiset { p : Associates α // Irreducible p }))
⊢ a.factors = b.factors
have obtain : ∀ (sb : Multiset { p : Associates α // Irreducible p })
  (h_sb : b.factors = (↑sb : WithTop (Multiset { p : Associates α // Irreducible p }))), a.factors = b.factors
Tactic: obtain ⟨sb, h_sb⟩ := factors_eq_some_iff_ne_zero.mpr hb
-- 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)
inst_3 : (p : Associates α) → Decidable (Irreducible p)
a b : Associates α
ha : a ≠ 0
hb : b ≠ 0
h : ∀ (p : Associates α), Irreducible p → p.count a.factors = p.count b.factors
sa : Multiset { p : Associates α // Irreducible p }
h_sa : a.factors = (↑sa : WithTop (Multiset { p : Associates α // Irreducible p }))
sb : Multiset { p : Associates α // Irreducible p }
h_sb : b.factors = (↑sb : WithTop (Multiset { p : Associates α // Irreducible p }))
⊢ a.factors = b.factors
have simp_all : ∀
  (h :
    ∀ (p : Associates α) (a : Irreducible p),
      Multiset.count (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) p a) sa =
        Multiset.count (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) p a) sb)
  (h_sa : a.factors = (↑sa : WithTop (Multiset { p : Associates α // Irreducible p })))
  (h_sb : b.factors = (↑sb : WithTop (Multiset { p : Associates α // Irreducible p }))), sa = sb
Tactic: simp_all only [count_some, WithTop.coe_eq_coe]
Cleared hypotheses: h, h_sa, h_sb
-- 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)
inst_3 : (p : Associates α) → Decidable (Irreducible p)
a b : Associates α
ha : a ≠ 0
hb : b ≠ 0
sa sb : Multiset { p : Associates α // Irreducible p }
h :
  ∀ (p : Associates α) (a : Irreducible p),
    Multiset.count (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) p a) sa =
      Multiset.count (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) p a) sb
h_sa : a.factors = (↑sa : WithTop (Multiset { p : Associates α // Irreducible p }))
h_sb : b.factors = (↑sb : WithTop (Multiset { p : Associates α // Irreducible p }))
⊢ sa = sb
have a₁ : ∀ (a : { p : Associates α // Irreducible p }), Multiset.count a sa = Multiset.count a sb
Tactic: ext
-- 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)
inst_3 : (p : Associates α) → Decidable (Irreducible p)
a b p : Associates α
hb : b ≠ 0
hp : Irreducible p
h : a.factors ≤ b.factors
⊢ p.count a.factors ≤ p.count b.factors
have pos : ∀ (ha : a = 0), p.count a.factors ≤ p.count b.factors
Tactic: by_cases ha : a = 0
-- 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)
inst_3 : (p : Associates α) → Decidable (Irreducible p)
a b p : Associates α
hb : b ≠ 0
hp : Irreducible p
h : a.factors ≤ b.factors
pos : ∀ (ha : a = 0), p.count a.factors ≤ p.count b.factors
⊢ p.count a.factors ≤ p.count b.factors
have neg : ∀ (ha : ¬a = 0), p.count a.factors ≤ p.count b.factors
Tactic: by_cases ha : a = 0
-- 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)
inst_3 : (p : Associates α) → Decidable (Irreducible p)
a b p : Associates α
hb : b ≠ 0
hp : Irreducible p
h : a.factors ≤ b.factors
ha : ¬a = 0
⊢ p.count a.factors ≤ p.count b.factors
have neg : ∀ (sa : Multiset { p : Associates α // Irreducible p })
  (h_sa : a.factors = (↑sa : WithTop (Multiset { p : Associates α // Irreducible p }))),
  p.count a.factors ≤ p.count b.factors
Tactic: obtain ⟨sa, h_sa⟩ := factors_eq_some_iff_ne_zero.mpr ha
-- 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)
inst_3 : (p : Associates α) → Decidable (Irreducible p)
a b p : Associates α
hb : b ≠ 0
hp : Irreducible p
h : a.factors ≤ b.factors
ha : ¬a = 0
sa : Multiset { p : Associates α // Irreducible p }
h_sa : a.factors = (↑sa : WithTop (Multiset { p : Associates α // Irreducible p }))
⊢ p.count a.factors ≤ p.count b.factors
have neg : ∀ (sb : Multiset { p : Associates α // Irreducible p })
  (h_sb : b.factors = (↑sb : WithTop (Multiset { p : Associates α // Irreducible p }))),
  p.count a.factors ≤ p.count b.factors
Tactic: obtain ⟨sb, h_sb⟩ := factors_eq_some_iff_ne_zero.mpr hb
-- 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)
inst_3 : (p : Associates α) → Decidable (Irreducible p)
a b p : Associates α
hb : b ≠ 0
hp : Irreducible p
h : a.factors ≤ b.factors
ha : ¬a = 0
sa : Multiset { p : Associates α // Irreducible p }
h_sa : a.factors = (↑sa : WithTop (Multiset { p : Associates α // Irreducible p }))
sb : Multiset { p : Associates α // Irreducible p }
h_sb : b.factors = (↑sb : WithTop (Multiset { p : Associates α // Irreducible p }))
⊢ p.count a.factors ≤ p.count b.factors
have neg : ∀ (h_sa : a.factors = (↑sa : WithTop (Multiset { p : Associates α // Irreducible p })))
  (sb : Multiset { p : Associates α // Irreducible p })
  (h :
    (↑sa : WithTop (Multiset { p : Associates α // Irreducible p })) ≤
      (↑sb : WithTop (Multiset { p : Associates α // Irreducible p })))
  (h_sb : b.factors = (↑sb : WithTop (Multiset { p : Associates α // Irreducible p }))),
  p.count (↑sa : WithTop (Multiset { p : Associates α // Irreducible p })) ≤
    p.count (↑sb : WithTop (Multiset { p : Associates α // Irreducible p }))
Tactic: rw [h_sa, h_sb] at h ⊢
Cleared hypotheses: h, h_sa, sb, h_sb
-- 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)
inst_3 : (p : Associates α) → Decidable (Irreducible p)
a b p : Associates α
hb : b ≠ 0
hp : Irreducible p
ha : ¬a = 0
sa : Multiset { p : Associates α // Irreducible p }
h_sa : a.factors = (↑sa : WithTop (Multiset { p : Associates α // Irreducible p }))
sb : Multiset { p : Associates α // Irreducible p }
h :
  (↑sa : WithTop (Multiset { p : Associates α // Irreducible p })) ≤
    (↑sb : WithTop (Multiset { p : Associates α // Irreducible p }))
h_sb : b.factors = (↑sb : WithTop (Multiset { p : Associates α // Irreducible p }))
⊢ p.count (↑sa : WithTop (Multiset { p : Associates α // Irreducible p })) ≤
    p.count (↑sb : WithTop (Multiset { p : Associates α // Irreducible p }))
have neg : Multiset.count (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) p hp) sa ≤
  Multiset.count (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) p hp) sb
Tactic: rw [count_some hp, count_some hp]
-- 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)
inst_3 : (p : Associates α) → Decidable (Irreducible p)
a b p : Associates α
hb : b ≠ 0
hp : Irreducible p
ha : ¬a = 0
sa : Multiset { p : Associates α // Irreducible p }
h_sa : a.factors = (↑sa : WithTop (Multiset { p : Associates α // Irreducible p }))
sb : Multiset { p : Associates α // Irreducible p }
h :
  (↑sa : WithTop (Multiset { p : Associates α // Irreducible p })) ≤
    (↑sb : WithTop (Multiset { p : Associates α // Irreducible p }))
h_sb : b.factors = (↑sb : WithTop (Multiset { p : Associates α // Irreducible p }))
⊢ Multiset.count (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) p hp) sa ≤
    Multiset.count (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) p hp) sb
have h₁ : sa ≤ sb
Tactic: rw [WithTop.coe_le_coe] at h
Cleared hypotheses: h, h_sb
-- 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 b : FactorSet α
⊢ a.prod ≤ b.prod ↔ a ≤ b
have refine : ∀ (h : a.prod ≤ b.prod), a ≤ b
Tactic: refine ⟨fun h ↦ ?_, prod_mono⟩
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (hypothesis)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : Nontrivial α
a b : FactorSet α
h : a.prod ≤ b.prod
⊢ a ≤ b
have this : a.prod.factors ≤ b.prod.factors
Tactic: have : a.prod.factors ≤ b.prod.factors := factors_mono h
-- 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 b : FactorSet α
h : a.prod ≤ b.prod
this : a.prod.factors ≤ b.prod.factors
⊢ a ≤ b
have rwa : ∀ (this : a ≤ b.prod.factors), a ≤ b
Tactic: rwa [prod_factors, prod_factors] at this
Cleared hypotheses: this
-- 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 b : FactorSet α
h : a.prod ≤ b.prod
this : a.prod.factors ≤ b.prod.factors
rwa : ∀ (this : a ≤ b.prod.factors), a ≤ b
⊢ a ≤ b
have rwa₁ : ∀ (this : a ≤ b), a ≤ b
Tactic: rwa [prod_factors, prod_factors] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (hypothesis)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : Associates α
⊢ (a.factors ⊔ b.factors).prod * (a.factors ⊓ b.factors).prod = a * b
have a₁ : Nontrivial α
Tactic: nontriviality α
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : Associates α
h : Nontrivial α
⊢ (a.factors ⊔ b.factors).prod * (a.factors ⊓ b.factors).prod = a * b
have refine : ((a.factors ⊔ b.factors).prod * (a.factors ⊓ b.factors).prod).factors = (a * b).factors
Tactic: refine eq_of_factors_eq_factors ?_
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : Associates α
h : Nontrivial α
⊢ ((a.factors ⊔ b.factors).prod * (a.factors ⊓ b.factors).prod).factors = (a * b).factors
have rw : (a.factors ⊔ b.factors + a.factors ⊓ b.factors).prod.factors = (a * b).factors
Tactic: rw [← prod_add, prod_factors, factors_mul, FactorSet.sup_add_inf_eq_add]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : Associates α
h : Nontrivial α
rw : (a.factors ⊔ b.factors + a.factors ⊓ b.factors).prod.factors = (a * b).factors
⊢ ((a.factors ⊔ b.factors).prod * (a.factors ⊓ b.factors).prod).factors = (a * b).factors
have rw₁ : a.factors ⊔ b.factors + a.factors ⊓ b.factors = (a * b).factors
Tactic: rw [← prod_add, prod_factors, factors_mul, FactorSet.sup_add_inf_eq_add]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : Associates α
h : Nontrivial α
rw : (a.factors ⊔ b.factors + a.factors ⊓ b.factors).prod.factors = (a * b).factors
rw₁ : a.factors ⊔ b.factors + a.factors ⊓ b.factors = (a * b).factors
⊢ ((a.factors ⊔ b.factors).prod * (a.factors ⊓ b.factors).prod).factors = (a * b).factors
have rw₂ : a.factors ⊔ b.factors + a.factors ⊓ b.factors = a.factors + b.factors
Tactic: rw [← prod_add, prod_factors, factors_mul, FactorSet.sup_add_inf_eq_add]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : Associates α
h : Nontrivial α
rw : (a.factors ⊔ b.factors + a.factors ⊓ b.factors).prod.factors = (a * b).factors
rw₁ : a.factors ⊔ b.factors + a.factors ⊓ b.factors = (a * b).factors
rw₂ : a.factors ⊔ b.factors + a.factors ⊓ b.factors = a.factors + b.factors
⊢ ((a.factors ⊔ b.factors).prod * (a.factors ⊓ b.factors).prod).factors = (a * b).factors
have rw₃ : a.factors + b.factors = a.factors + b.factors
Tactic: rw [← prod_add, prod_factors, factors_mul, FactorSet.sup_add_inf_eq_add]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a p : Associates α
hm : p ∈ a.factors
⊢ p ∣ a
have inl : ∀ (hm : Membership.mem (γ := FactorSet α) (factors 0) p), p ∣ 0
Tactic: rcases eq_or_ne a 0 with rfl | ha0
Cleared hypotheses: a, hm
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a p : Associates α
hm : p ∈ a.factors
inl : ∀ (hm : Membership.mem (γ := FactorSet α) (factors 0) p), p ∣ 0
⊢ p ∣ a
have inr : ∀ (ha0 : a ≠ 0), p ∣ a
Tactic: rcases eq_or_ne a 0 with rfl | 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 : Associates α
hm : p ∈ a.factors
ha0 : a ≠ 0
⊢ p ∣ a
have inr : ∀ (a0 : α) (nza : a0 ≠ 0) (ha' : Associates.mk a0 = a), p ∣ a
Tactic: obtain ⟨a0, nza, ha'⟩ := exists_non_zero_rep 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 : Associates α
hm : p ∈ a.factors
ha0 : a ≠ 0
a0 : α
nza : a0 ≠ 0
ha' : Associates.mk a0 = a
⊢ p ∣ a
have inr : p ∣ a.factors.prod
Tactic: rw [← Associates.factors_prod a]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a p : Associates α
hm : p ∈ a.factors
ha0 : a ≠ 0
a0 : α
nza : a0 ≠ 0
ha' : Associates.mk a0 = a
⊢ p ∣ a.factors.prod
have inr : ∀ (hm : Membership.mem (γ := FactorSet α) (↑(factors' a0) : WithTop (Multiset { a : Associates α // Irreducible a })) p)
  (nza : a0 ≠ 0) (ha' : Associates.mk a0 = a),
  p ∣ FactorSet.prod (↑(factors' a0) : WithTop (Multiset { a : Associates α // Irreducible a }))
Tactic: rw [← ha', factors_mk a0 nza] at hm ⊢
Cleared hypotheses: hm, nza, ha'
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a p : Associates α
ha0 : a ≠ 0
a0 : α
hm : Membership.mem (γ := FactorSet α) (↑(factors' a0) : WithTop (Multiset { a : Associates α // Irreducible a })) p
nza : a0 ≠ 0
ha' : Associates.mk a0 = a
⊢ p ∣ Multiset.prod (Multiset.map (Subtype.val (p := fun (a : Associates α) ↦ Irreducible a)) (factors' a0))
have inr : Membership.mem (γ := Multiset (Associates α))
  (Multiset.map (Subtype.val (p := fun (a : Associates α) ↦ Irreducible a)) (factors' a0)) p
Tactic: apply Multiset.dvd_prod
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a p : Associates α
ha0 : a ≠ 0
a0 : α
hm : Membership.mem (γ := FactorSet α) (↑(factors' a0) : WithTop (Multiset { a : Associates α // Irreducible a })) p
nza : a0 ≠ 0
ha' : Associates.mk a0 = a
⊢ Membership.mem (γ := Multiset (Associates α))
    (Multiset.map (Subtype.val (p := fun (a : Associates α) ↦ Irreducible a)) (factors' a0)) p
have inr : ∃ (a : { a : Associates α // Irreducible a }),
  Membership.mem (γ := Multiset { a : Associates α // Irreducible a }) (factors' a0) a ∧ (↑a : Associates α) = p
Tactic: apply Multiset.mem_map.mpr
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a : α
p : Associates α
hp : Irreducible p
hz : a ≠ 0
h_mem :
  Membership.mem (γ := Multiset { a : Associates α // Irreducible a }) (factors' a) (Subtype.mk (p := Irreducible) p hp)
⊢ p ∣ Associates.mk a
have haveI : ∀ (this : DecidableEq (Associates α)), p ∣ Associates.mk a
Tactic: haveI := Classical.decEq (Associates α)
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a : α
p : Associates α
hp : Irreducible p
hz : a ≠ 0
h_mem :
  Membership.mem (γ := Multiset { a : Associates α // Irreducible a }) (factors' a) (Subtype.mk (p := Irreducible) p hp)
this : DecidableEq (Associates α)
⊢ p ∣ Associates.mk a
have hm : Membership.mem (γ := FactorSet α) (factors (Associates.mk a)) p
Tactic: apply dvd_of_mem_factors
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a : α
p : Associates α
hp : Irreducible p
hz : a ≠ 0
h_mem :
  Membership.mem (γ := Multiset { a : Associates α // Irreducible a }) (factors' a) (Subtype.mk (p := Irreducible) p hp)
this : DecidableEq (Associates α)
⊢ Membership.mem (γ := FactorSet α) (factors (Associates.mk a)) p
have hm : Membership.mem (γ := FactorSet α) (↑(factors' a) : WithTop (Multiset { a : Associates α // Irreducible a })) p
Tactic: rw [factors_mk _ hz]
-- 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 := Multiset { a : Associates α // Irreducible a }) (factors' a)
    (Subtype.mk (p := Irreducible) (Associates.mk p) ⋯)
have obtain : ∀ (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) sorry)
Tactic: obtain ⟨q, hq, hpq⟩ := exists_mem_factors_of_dvd ha0 hp hd