awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
x : α
n_1 n : ℕ
h0 : ¬x = 0
⊢ Rel Associated (factors (x * x ^ n)) (factors x + n • factors x)
have neg : Rel Associated (factors x + factors (x ^ n)) (factors x + n • factors x)
Tactic: refine Rel.trans _ (factors_mul h0 (pow_ne_zero n h0)) ?_
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
x : α
n_1 n : ℕ
h0 : ¬x = 0
⊢ Rel Associated (factors x + factors (x ^ n)) (factors x + n • factors x)
have neg : Rel Associated (factors x) (factors x)
Tactic: refine Rel.add ?_ <| factors_pow n
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : DecidableEq α
x : α
hx : x ≠ 0
⊢ prod (∑ a ∈ (factors x).toFinset, HSMul.hSMul (β := Multiset α) (count a (factors x)) {a}) = (factors x).prod
have rw : (factors x).prod = (factors x).prod
Tactic: rw [toFinset_sum_count_nsmul_eq (factors x)]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
h : a ~ᵤ b
⊢ Rel Associated (factors a) (factors b)
have inl : ∀ (h : Associated.{u_1} (M := α) 0 0), Rel.{u_1, u_1} (α := α) Associated (factors 0) (factors 0)
Tactic: rcases iff_iff_and_or_not_and_not.mp h.eq_zero_iff with (⟨rfl, rfl⟩ | ⟨ha, hb⟩)
Cleared hypotheses: a, b, h
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
h : a ~ᵤ b
inl : ∀ (h : Associated.{u_1} (M := α) 0 0), Rel.{u_1, u_1} (α := α) Associated (factors 0) (factors 0)
⊢ Rel Associated (factors a) (factors b)
have inr : ∀ (ha : ¬a = 0) (hb : ¬b = 0), Rel Associated (factors a) (factors b)
Tactic: rcases iff_iff_and_or_not_and_not.mp h.eq_zero_iff with (⟨rfl, rfl⟩ | ⟨ha, hb⟩)
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
h : a ~ᵤ b
ha : ¬a = 0
hb : ¬b = 0
⊢ Rel Associated (factors a) (factors b)
have inr : (factors a).prod ~ᵤ (factors b).prod
Tactic: refine factors_unique irreducible_of_factor irreducible_of_factor ?_
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
x : α
hx : x ≠ 0
h : factors x = 0
⊢ IsUnit x
have contrapose! : ∀ (h : ¬IsUnit x), factors x ≠ 0
Tactic: contrapose! h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
p q : Multiset (Associates α)
⊢ (∀ a ∈ p, Irreducible a) → (∀ a ∈ q, Irreducible a) → p.prod = q.prod → p = q
have apply : ∀ (s : Multiset α),
  (∀ (a : Associates α),
      Membership.mem (γ := Multiset (Associates α)) (map (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) s) a →
        Irreducible a) →
    (∀ a ∈ q, Irreducible a) →
      prod (map (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) s) = q.prod →
        map (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) s = q
Tactic: apply Multiset.induction_on_multiset_quot p
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
p q : Multiset (Associates α)
⊢ ∀ (s : Multiset α),
    (∀ (a : Associates α),
        Membership.mem (γ := Multiset (Associates α)) (map (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) s) a →
          Irreducible a) →
      (∀ a ∈ q, Irreducible a) →
        prod (map (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) s) = q.prod →
          map (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) s = q
have apply : ∀ (s s_1 : Multiset α),
  (∀ (a : Associates α),
      Membership.mem (γ := Multiset (Associates α)) (map (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) s_1) a →
        Irreducible a) →
    (∀ (a : Associates α),
        Membership.mem (γ := Multiset (Associates α)) (map (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) s) a →
          Irreducible a) →
      Eq (α := Associates α) (prod (map (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) s_1))
          (prod (map (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) s)) →
        Eq (α := Multiset (Associates α)) (map (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) s_1)
          (map (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) s)
Tactic: apply Multiset.induction_on_multiset_quot q
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
p q : Multiset (Associates α)
s t : Multiset α
hs :
  ∀ (a : Associates α),
    Membership.mem (γ := Multiset (Associates α)) (map (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) t) a →
      Irreducible a
ht :
  ∀ (a : Associates α),
    Membership.mem (γ := Multiset (Associates α)) (map (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) s) a →
      Irreducible a
eq :
  Eq (α := Associates α) (prod (map (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) t))
    (prod (map (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) s))
⊢ Eq (α := Multiset (Associates α)) (map (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) t)
    (map (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) s)
have refine_1 : ∀ x ∈ t, Irreducible x
Tactic: refine Multiset.map_mk_eq_map_mk_of_rel (UniqueFactorizationMonoid.factors_unique ?_ ?_ ?_)
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
p q : Multiset (Associates α)
s t : Multiset α
hs :
  ∀ (a : Associates α),
    Membership.mem (γ := Multiset (Associates α)) (map (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) t) a →
      Irreducible a
ht :
  ∀ (a : Associates α),
    Membership.mem (γ := Multiset (Associates α)) (map (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) s) a →
      Irreducible a
eq :
  Eq (α := Associates α) (prod (map (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) t))
    (prod (map (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) s))
refine_1 : ∀ x ∈ t, Irreducible x
⊢ Eq (α := Multiset (Associates α)) (map (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) t)
    (map (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) s)
have refine_2 : ∀ x ∈ s, Irreducible x
Tactic: refine Multiset.map_mk_eq_map_mk_of_rel (UniqueFactorizationMonoid.factors_unique ?_ ?_ ?_)
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
p q : Multiset (Associates α)
s t : Multiset α
hs :
  ∀ (a : Associates α),
    Membership.mem (γ := Multiset (Associates α)) (map (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) t) a →
      Irreducible a
ht :
  ∀ (a : Associates α),
    Membership.mem (γ := Multiset (Associates α)) (map (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) s) a →
      Irreducible a
eq :
  Eq (α := Associates α) (prod (map (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) t))
    (prod (map (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) s))
refine_1 : ∀ x ∈ t, Irreducible x
refine_2 : ∀ x ∈ s, Irreducible x
⊢ Eq (α := Multiset (Associates α)) (map (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) t)
    (map (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) s)
have refine_3 : t.prod ~ᵤ s.prod
Tactic: refine Multiset.map_mk_eq_map_mk_of_rel (UniqueFactorizationMonoid.factors_unique ?_ ?_ ?_)
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (hypothesis)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
p q : Multiset (Associates α)
s t : Multiset α
hs :
  ∀ (a : Associates α),
    Membership.mem (γ := Multiset (Associates α)) (map (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) t) a →
      Irreducible a
ht :
  ∀ (a : Associates α),
    Membership.mem (γ := Multiset (Associates α)) (map (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) s) a →
      Irreducible a
eq :
  Eq (α := Associates α) (prod (map (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) t))
    (prod (map (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) s))
⊢ t.prod ~ᵤ s.prod
have eq' : Eq.{u_1 + 1} (α := (a : α) → Quot (α := α) (⇑(Associated.setoid α) : α → α → Prop))
  (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) Associates.mk
Tactic: have eq' : (Quot.mk Setoid.r : α → Associates α) = Associates.mk := funext quot_mk_eq_mk
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
p q : Multiset (Associates α)
s t : Multiset α
hs :
  ∀ (a : Associates α),
    Membership.mem (γ := Multiset (Associates α)) (map (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) t) a →
      Irreducible a
ht :
  ∀ (a : Associates α),
    Membership.mem (γ := Multiset (Associates α)) (map (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) s) a →
      Irreducible a
eq :
  Eq (α := Associates α) (prod (map (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) t))
    (prod (map (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) s))
eq' :
  Eq.{u_1 + 1} (α := (a : α) → Quot (α := α) (⇑(Associated.setoid α) : α → α → Prop))
    (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) Associates.mk
⊢ t.prod ~ᵤ s.prod
have refine_3 : ∀ (eq : Eq (α := Associates α) (prod (map Associates.mk t)) (prod (map Associates.mk s)))
  (eq' :
    Eq.{u_1 + 1} (α := (a : α) → Quot (α := α) (⇑(Associated.setoid α) : α → α → Prop))
      (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) Associates.mk),
  t.prod ~ᵤ s.prod
Tactic: rwa [eq', prod_mk, prod_mk, mk_eq_mk_iff_associated] at eq
Cleared hypotheses: eq, eq'
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
p q : Multiset (Associates α)
s t : Multiset α
hs :
  ∀ (a : Associates α),
    Membership.mem (γ := Multiset (Associates α)) (map (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) t) a →
      Irreducible a
ht :
  ∀ (a : Associates α),
    Membership.mem (γ := Multiset (Associates α)) (map (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) s) a →
      Irreducible a
eq :
  Eq (α := Associates α) (prod (map (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) t))
    (prod (map (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) s))
eq' :
  Eq.{u_1 + 1} (α := (a : α) → Quot (α := α) (⇑(Associated.setoid α) : α → α → Prop))
    (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) Associates.mk
refine_3 :
  ∀ (eq : Eq (α := Associates α) (prod (map Associates.mk t)) (prod (map Associates.mk s)))
    (eq' :
      Eq.{u_1 + 1} (α := (a : α) → Quot (α := α) (⇑(Associated.setoid α) : α → α → Prop))
        (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) Associates.mk),
    t.prod ~ᵤ s.prod
⊢ t.prod ~ᵤ s.prod
have refine_3₁ : ∀ (eq : Eq (α := Associates α) (Associates.mk t.prod) (prod (map Associates.mk s)))
  (eq' :
    Eq.{u_1 + 1} (α := (a : α) → Quot (α := α) (⇑(Associated.setoid α) : α → α → Prop))
      (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) Associates.mk),
  t.prod ~ᵤ s.prod
Tactic: rwa [eq', prod_mk, prod_mk, mk_eq_mk_iff_associated] at eq
Cleared hypotheses: eq, eq'
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
p q : Multiset (Associates α)
s t : Multiset α
hs :
  ∀ (a : Associates α),
    Membership.mem (γ := Multiset (Associates α)) (map (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) t) a →
      Irreducible a
ht :
  ∀ (a : Associates α),
    Membership.mem (γ := Multiset (Associates α)) (map (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) s) a →
      Irreducible a
eq :
  Eq (α := Associates α) (prod (map (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) t))
    (prod (map (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) s))
eq' :
  Eq.{u_1 + 1} (α := (a : α) → Quot (α := α) (⇑(Associated.setoid α) : α → α → Prop))
    (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) Associates.mk
refine_3 :
  ∀ (eq : Eq (α := Associates α) (prod (map Associates.mk t)) (prod (map Associates.mk s)))
    (eq' :
      Eq.{u_1 + 1} (α := (a : α) → Quot (α := α) (⇑(Associated.setoid α) : α → α → Prop))
        (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) Associates.mk),
    t.prod ~ᵤ s.prod
refine_3₁ :
  ∀ (eq : Eq (α := Associates α) (Associates.mk t.prod) (prod (map Associates.mk s)))
    (eq' :
      Eq.{u_1 + 1} (α := (a : α) → Quot (α := α) (⇑(Associated.setoid α) : α → α → Prop))
        (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) Associates.mk),
    t.prod ~ᵤ s.prod
⊢ t.prod ~ᵤ s.prod
have refine_3₂ : ∀ (eq : Eq (α := Associates α) (Associates.mk t.prod) (Associates.mk s.prod))
  (eq' :
    Eq.{u_1 + 1} (α := (a : α) → Quot (α := α) (⇑(Associated.setoid α) : α → α → Prop))
      (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) Associates.mk),
  t.prod ~ᵤ s.prod
Tactic: rwa [eq', prod_mk, prod_mk, mk_eq_mk_iff_associated] at eq
Cleared hypotheses: eq, eq'
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
p q : Multiset (Associates α)
s t : Multiset α
hs :
  ∀ (a : Associates α),
    Membership.mem (γ := Multiset (Associates α)) (map (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) t) a →
      Irreducible a
ht :
  ∀ (a : Associates α),
    Membership.mem (γ := Multiset (Associates α)) (map (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) s) a →
      Irreducible a
eq :
  Eq (α := Associates α) (prod (map (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) t))
    (prod (map (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) s))
eq' :
  Eq.{u_1 + 1} (α := (a : α) → Quot (α := α) (⇑(Associated.setoid α) : α → α → Prop))
    (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) Associates.mk
refine_3 :
  ∀ (eq : Eq (α := Associates α) (prod (map Associates.mk t)) (prod (map Associates.mk s)))
    (eq' :
      Eq.{u_1 + 1} (α := (a : α) → Quot (α := α) (⇑(Associated.setoid α) : α → α → Prop))
        (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) Associates.mk),
    t.prod ~ᵤ s.prod
refine_3₁ :
  ∀ (eq : Eq (α := Associates α) (Associates.mk t.prod) (prod (map Associates.mk s)))
    (eq' :
      Eq.{u_1 + 1} (α := (a : α) → Quot (α := α) (⇑(Associated.setoid α) : α → α → Prop))
        (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) Associates.mk),
    t.prod ~ᵤ s.prod
refine_3₂ :
  ∀ (eq : Eq (α := Associates α) (Associates.mk t.prod) (Associates.mk s.prod))
    (eq' :
      Eq.{u_1 + 1} (α := (a : α) → Quot (α := α) (⇑(Associated.setoid α) : α → α → Prop))
        (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) Associates.mk),
    t.prod ~ᵤ s.prod
⊢ t.prod ~ᵤ s.prod
have refine_3₃ : ∀ (eq : t.prod ~ᵤ s.prod)
  (eq' :
    Eq.{u_1 + 1} (α := (a : α) → Quot (α := α) (⇑(Associated.setoid α) : α → α → Prop))
      (Quot.mk (⇑(Associated.setoid α) : α → α → Prop)) Associates.mk),
  t.prod ~ᵤ s.prod
Tactic: rwa [eq', prod_mk, prod_mk, mk_eq_mk_iff_associated] at eq
Cleared hypotheses: eq, eq'
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
⊢ p.prod ≤ q.prod ↔ p ≤ q
have refine : p.prod ≤ q.prod → p ≤ q
Tactic: refine ⟨?_, prod_le_prod⟩
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : q.prod = p.prod * c
⊢ p ≤ q
have refine_1 : ∀ (x : Associates α) (hx : x ∈ p + factors c), Irreducible x
Tactic: refine Multiset.le_iff_exists_add.2 ⟨factors c, unique' hq (fun x hx ↦ ?_) ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : q.prod = p.prod * c
refine_1 : ∀ (x : Associates α) (hx : x ∈ p + factors c), Irreducible x
⊢ p ≤ q
have refine_2 : q.prod = (p + factors c).prod
Tactic: refine Multiset.le_iff_exists_add.2 ⟨factors c, unique' hq (fun x hx ↦ ?_) ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : q.prod = p.prod * c
x : Associates α
hx : x ∈ p + factors c
⊢ Irreducible x
have refine_1 : ∀ (h : x ∈ p), Irreducible x
Tactic: obtain h | h := Multiset.mem_add.1 hx
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : q.prod = p.prod * c
x : Associates α
hx : x ∈ p + factors c
refine_1 : ∀ (h : x ∈ p), Irreducible x
⊢ Irreducible x
have refine_1₁ : ∀ (h : x ∈ factors c), Irreducible x
Tactic: obtain h | h := Multiset.mem_add.1 hx
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : q.prod = p.prod * c
⊢ q.prod = (p + factors c).prod
have refine_2 : p.prod * c = p.prod * (factors c).prod
Tactic: rw [eqc, Multiset.prod_add]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : q.prod = p.prod * c
⊢ p.prod * c = p.prod * (factors c).prod
have refine_2 : c = (factors c).prod
Tactic: congr
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : q.prod = p.prod * c
⊢ c = (factors c).prod
have refine_2 : ∀ (hc : c = 0), False
Tactic: refine associated_iff_eq.mp (factors_prod fun hc => ?_).symm
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : q.prod = p.prod * c
hc : c = 0
⊢ False
have refine_2 : 0 ∈ q
Tactic: refine not_irreducible_zero (hq _ ?_)
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : q.prod = p.prod * c
hc : c = 0
⊢ 0 ∈ q
have refine_2 : q.prod = 0
Tactic: rw [← prod_eq_zero_iff, eqc, hc, mul_zero]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : q.prod = p.prod * c
hc : c = 0
refine_2 : q.prod = 0
⊢ 0 ∈ q
have refine_2₁ : p.prod * c = 0
Tactic: rw [← prod_eq_zero_iff, eqc, hc, mul_zero]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : q.prod = p.prod * c
hc : c = 0
refine_2 : q.prod = 0
refine_2₁ : p.prod * c = 0
⊢ 0 ∈ q
have refine_2₂ : p.prod * 0 = 0
Tactic: rw [← prod_eq_zero_iff, eqc, hc, mul_zero]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : q.prod = p.prod * c
hc : c = 0
refine_2 : q.prod = 0
refine_2₁ : p.prod * c = 0
refine_2₂ : p.prod * 0 = 0
⊢ 0 ∈ q
have refine_2₃ : Eq.{u_1 + 1} (α := Associates α) 0 0
Tactic: rw [← prod_eq_zero_iff, eqc, hc, mul_zero]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
⊢ WellFounded.{u_1 + 1} (α := α) DvdNotUnit
have refine_1 : α → ℕ∞
Tactic: refine RelHomClass.wellFounded
        (RelHom.mk ?_ ?_ : (DvdNotUnit : α → α → Prop) →r ((· < ·) : ℕ∞ → ℕ∞ → Prop)) wellFounded_lt
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
refine_1 : α → ℕ∞
⊢ WellFounded.{u_1 + 1} (α := α) DvdNotUnit
have refine_2 : ∀ {a b : α},
  DvdNotUnit a b →
    LT.lt (α := ℕ∞)
      (if h : a = 0 then ⊤
      else (↑(Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) sorry).card : ℕ∞))
      (if h : b = 0 then ⊤
      else (↑(Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ b) sorry).card : ℕ∞))
Tactic: refine RelHomClass.wellFounded
        (RelHom.mk ?_ ?_ : (DvdNotUnit : α → α → Prop) →r ((· < ·) : ℕ∞ → ℕ∞ → Prop)) wellFounded_lt
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
a : α
⊢ ℕ∞
have pos : (h : a = 0) → ℕ∞
Tactic: by_cases h : a = 0
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
a : α
pos : (h : a = 0) → ℕ∞
⊢ ℕ∞
have neg : (h : ¬a = 0) → ℕ∞
Tactic: by_cases h : a = 0
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
⊢ LT.lt (α := ℕ∞)
    (if h : a = 0 then ⊤
    else (↑(Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯).card : ℕ∞))
    (if h : b = 0 then ⊤
    else (↑(Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ b) ⋯).card : ℕ∞))
have refine_2 : LT.lt (α := ℕ∞) (↑(Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) sorry).card : ℕ∞)
  (if h : b = 0 then ⊤
  else (↑(Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ b) sorry).card : ℕ∞))
Tactic: rw [dif_neg ane0]