awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
N : Type u_2
inst_2 : CommMonoidWithZero N
inst_3 : Subsingleton.{u_1 + 1} Mˣ
inst_4 : Subsingleton.{u_2 + 1} Nˣ
inst_5 : UniqueFactorizationMonoid M
inst_6 : UniqueFactorizationMonoid N
m p : M
n : N
hm : m ≠ 0
hn : n ≠ 0
hp : p ∈ normalizedFactors m
d : Equiv.{max 1 (u_1 + 1), max 1 (u_2 + 1)} { l : M // l ∣ m } { l : N // l ∣ n }
hd :
  ∀ (l l' : { l : M // l ∣ m }),
    Dvd.dvd.{u_2} (α := N) (↑((d : (a : { l : M // l ∣ m }) → { l : N // l ∣ n }) l) : N)
        (↑((d : (a : { l : M // l ∣ m }) → { l : N // l ∣ n }) l') : N) ↔
      (↑l : M) ∣ (↑l' : M)
⊢ (↑((d : (a : { l : M // l ∣ m }) → { l : N // l ∣ n }) (Subtype.mk (p := fun (l : M) ↦ l ∣ m) p ⋯)) : N) ∈
    normalizedFactors n
have Prime : Prime.{u_2} (M := N)
  (↑((d : (a : { l : M // l ∣ m }) → { l : N // l ∣ n })
        (Subtype.mk (p := fun (l : M) ↦ l ∣ m)
          ((associatesEquivOfUniqueUnits : (a : Associates M) → M)
            ((MulEquiv.symm associatesEquivOfUniqueUnits : (a : M) → Associates M) p : Associates M))
          sorry)) :
    N)
Tactic: suffices
    Prime (d ⟨associatesEquivOfUniqueUnits (associatesEquivOfUniqueUnits.symm p), by
            simp [dvd_of_mem_normalizedFactors hp]⟩ : N) by
    simp only [associatesEquivOfUniqueUnits_apply, out_mk, normalize_eq,
      associatesEquivOfUniqueUnits_symm_apply] at this
    obtain ⟨q, hq, hq'⟩ :=
      exists_mem_normalizedFactors_of_dvd hn this.irreducible
        (d ⟨p, by apply dvd_of_mem_normalizedFactors; convert! hp⟩).prop
    rwa [associated_iff_eq.mp hq']
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
N : Type u_2
inst_2 : CommMonoidWithZero N
inst_3 : Subsingleton.{u_1 + 1} Mˣ
inst_4 : Subsingleton.{u_2 + 1} Nˣ
inst_5 : UniqueFactorizationMonoid M
inst_6 : UniqueFactorizationMonoid N
m p : M
n : N
hm : m ≠ 0
hn : n ≠ 0
hp : p ∈ normalizedFactors m
d : Equiv.{max 1 (u_1 + 1), max 1 (u_2 + 1)} { l : M // l ∣ m } { l : N // l ∣ n }
hd :
  ∀ (l l' : { l : M // l ∣ m }),
    Dvd.dvd.{u_2} (α := N) (↑((d : (a : { l : M // l ∣ m }) → { l : N // l ∣ n }) l) : N)
        (↑((d : (a : { l : M // l ∣ m }) → { l : N // l ∣ n }) l') : N) ↔
      (↑l : M) ∣ (↑l' : M)
⊢ Eq.{u_2 + 1} (α := Associates N)
    (Associates.mk
      (↑((d : (a : { l : M // l ∣ m }) → { l : N // l ∣ n })
            (Subtype.mk (p := fun (l : M) ↦ l ∣ m)
              ((associatesEquivOfUniqueUnits : (a : Associates M) → M)
                ((MulEquiv.symm associatesEquivOfUniqueUnits : (a : M) → Associates M) p : Associates M))
              ⋯)) :
        N))
    (↑((mkFactorOrderIsoOfFactorDvdEquiv hd :
            (a : (↑(Set.Iic (Associates.mk m)) : Type u_1)) → (↑(Set.Iic (Associates.mk n)) : Type u_2))
          (Subtype.mk (p := fun (x : Associates M) ↦
            Membership.mem (γ := Set (Associates M)) (Set.Iic (Associates.mk m)) x)
            ((MulEquiv.symm associatesEquivOfUniqueUnits : (a : M) → Associates M) p) ⋯)) :
      Associates N)
have rw : Eq.{u_2 + 1} (α := Associates N)
  (Associates.mk
    (↑((d : (a : { l : M // l ∣ m }) → { l : N // l ∣ n })
          (Subtype.mk (p := fun (l : M) ↦ l ∣ m)
            ((associatesEquivOfUniqueUnits : (a : Associates M) → M)
              ((MulEquiv.symm associatesEquivOfUniqueUnits : (a : M) → Associates M) p : Associates M))
            sorry)) :
      N))
  (Associates.mk
    (↑((d : (a : { l : M // l ∣ m }) → { l : N // l ∣ n })
          (Subtype.mk (p := fun (l : M) ↦ l ∣ m)
            ((associatesEquivOfUniqueUnits : (a : Associates M) → M)
              ((↑(Subtype.mk (p := fun (x : Associates M) ↦
                      Membership.mem (γ := Set (Associates M)) (Set.Iic (Associates.mk m)) x)
                      ((MulEquiv.symm associatesEquivOfUniqueUnits : (a : M) → Associates M) p) sorry) :
                  Associates M) :
                Associates M))
            sorry)) :
      N))
Tactic: rw [mkFactorOrderIsoOfFactorDvdEquiv_apply_coe]
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (hypothesis)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
N : Type u_2
inst_2 : CommMonoidWithZero N
inst_3 : Subsingleton.{u_1 + 1} Mˣ
inst_4 : Subsingleton.{u_2 + 1} Nˣ
inst_5 : UniqueFactorizationMonoid M
inst_6 : UniqueFactorizationMonoid N
m p : M
n : N
hm : m ≠ 0
hn : n ≠ 0
hp : p ∈ normalizedFactors m
d : Equiv.{max 1 (u_1 + 1), max 1 (u_2 + 1)} { l : M // l ∣ m } { l : N // l ∣ n }
hd :
  ∀ (l l' : { l : M // l ∣ m }),
    Dvd.dvd.{u_2} (α := N) (↑((d : (a : { l : M // l ∣ m }) → { l : N // l ∣ n }) l) : N)
        (↑((d : (a : { l : M // l ∣ m }) → { l : N // l ∣ n }) l') : N) ↔
      (↑l : M) ∣ (↑l' : M)
⊢ Prime.{u_2} (M := N)
    (↑((d : (a : { l : M // l ∣ m }) → { l : N // l ∣ n })
          (Subtype.mk (p := fun (l : M) ↦ l ∣ m)
            ((associatesEquivOfUniqueUnits : (a : Associates M) → M)
              ((MulEquiv.symm associatesEquivOfUniqueUnits : (a : M) → Associates M) p : Associates M))
            ⋯)) :
      N)
have this : Eq.{u_2 + 1} (α := Associates N)
  (Associates.mk
    (↑((d : (a : { l : M // l ∣ m }) → { l : N // l ∣ n })
          (Subtype.mk (p := fun (l : M) ↦ l ∣ m)
            ((associatesEquivOfUniqueUnits : (a : Associates M) → M)
              ((MulEquiv.symm associatesEquivOfUniqueUnits : (a : M) → Associates M) p : Associates M))
            sorry)) :
      N))
  (↑((mkFactorOrderIsoOfFactorDvdEquiv hd :
          (a : (↑(Set.Iic (Associates.mk m)) : Type u_1)) → (↑(Set.Iic (Associates.mk n)) : Type u_2))
        (Subtype.mk (p := fun (x : Associates M) ↦
          Membership.mem (γ := Set (Associates M)) (Set.Iic (Associates.mk m)) x)
          ((MulEquiv.symm associatesEquivOfUniqueUnits : (a : M) → Associates M) p) sorry)) :
    Associates N)
Tactic: have :
    Associates.mk
        (d ⟨associatesEquivOfUniqueUnits (associatesEquivOfUniqueUnits.symm p), by
              simp only [dvd_of_mem_normalizedFactors hp, associatesEquivOfUniqueUnits_apply,
                out_mk, normalize_eq, associatesEquivOfUniqueUnits_symm_apply]⟩ : N) =
      ↑(mkFactorOrderIsoOfFactorDvdEquiv hd
          ⟨associatesEquivOfUniqueUnits.symm p, by
            simp only [associatesEquivOfUniqueUnits_symm_apply]
            exact mk_dvd_mk.mpr (dvd_of_mem_normalizedFactors hp)⟩) := by
    rw [mkFactorOrderIsoOfFactorDvdEquiv_apply_coe]
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
N : Type u_2
inst_2 : CommMonoidWithZero N
inst_3 : Subsingleton.{u_1 + 1} Mˣ
inst_4 : Subsingleton.{u_2 + 1} Nˣ
inst_5 : UniqueFactorizationMonoid M
inst_6 : UniqueFactorizationMonoid N
m p : M
n : N
hm : m ≠ 0
hn : n ≠ 0
hp : p ∈ normalizedFactors m
d : Equiv.{max 1 (u_1 + 1), max 1 (u_2 + 1)} { l : M // l ∣ m } { l : N // l ∣ n }
hd :
  ∀ (l l' : { l : M // l ∣ m }),
    Dvd.dvd.{u_2} (α := N) (↑((d : (a : { l : M // l ∣ m }) → { l : N // l ∣ n }) l) : N)
        (↑((d : (a : { l : M // l ∣ m }) → { l : N // l ∣ n }) l') : N) ↔
      (↑l : M) ∣ (↑l' : M)
this :
  Eq.{u_2 + 1} (α := Associates N)
    (Associates.mk
      (↑((d : (a : { l : M // l ∣ m }) → { l : N // l ∣ n })
            (Subtype.mk (p := fun (l : M) ↦ l ∣ m)
              ((associatesEquivOfUniqueUnits : (a : Associates M) → M)
                ((MulEquiv.symm associatesEquivOfUniqueUnits : (a : M) → Associates M) p : Associates M))
              ⋯)) :
        N))
    (↑((mkFactorOrderIsoOfFactorDvdEquiv hd :
            (a : (↑(Set.Iic (Associates.mk m)) : Type u_1)) → (↑(Set.Iic (Associates.mk n)) : Type u_2))
          (Subtype.mk (p := fun (x : Associates M) ↦
            Membership.mem (γ := Set (Associates M)) (Set.Iic (Associates.mk m)) x)
            ((MulEquiv.symm associatesEquivOfUniqueUnits : (a : M) → Associates M) p) ⋯)) :
      Associates N)
⊢ Prime.{u_2} (M := N)
    (↑((d : (a : { l : M // l ∣ m }) → { l : N // l ∣ n })
          (Subtype.mk (p := fun (l : M) ↦ l ∣ m)
            ((associatesEquivOfUniqueUnits : (a : Associates M) → M)
              ((MulEquiv.symm associatesEquivOfUniqueUnits : (a : M) → Associates M) p : Associates M))
            ⋯)) :
      N)
have rw : Prime.{u_2} (M := Associates N)
  (↑((mkFactorOrderIsoOfFactorDvdEquiv hd :
          (a : (↑(Set.Iic (Associates.mk m)) : Type u_1)) → (↑(Set.Iic (Associates.mk n)) : Type u_2))
        (Subtype.mk (p := fun (x : Associates M) ↦
          Membership.mem (γ := Set (Associates M)) (Set.Iic (Associates.mk m)) x)
          ((MulEquiv.symm associatesEquivOfUniqueUnits : (a : M) → Associates M) p) sorry)) :
    Associates N)
Tactic: rw [← Associates.prime_mk, this]
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
N : Type u_2
inst_2 : CommMonoidWithZero N
inst_3 : Subsingleton.{u_1 + 1} Mˣ
inst_4 : Subsingleton.{u_2 + 1} Nˣ
inst_5 : UniqueFactorizationMonoid M
inst_6 : UniqueFactorizationMonoid N
m p : M
n : N
hm : m ≠ 0
hn : n ≠ 0
hp : p ∈ normalizedFactors m
d : Equiv.{max 1 (u_1 + 1), max 1 (u_2 + 1)} { l : M // l ∣ m } { l : N // l ∣ n }
hd :
  ∀ (l l' : { l : M // l ∣ m }),
    Dvd.dvd.{u_2} (α := N) (↑((d : (a : { l : M // l ∣ m }) → { l : N // l ∣ n }) l) : N)
        (↑((d : (a : { l : M // l ∣ m }) → { l : N // l ∣ n }) l') : N) ↔
      (↑l : M) ∣ (↑l' : M)
this_1 :
  Eq.{u_2 + 1} (α := Associates N)
    (Associates.mk
      (↑((d : (a : { l : M // l ∣ m }) → { l : N // l ∣ n })
            (Subtype.mk (p := fun (l : M) ↦ l ∣ m)
              ((associatesEquivOfUniqueUnits : (a : Associates M) → M)
                ((MulEquiv.symm associatesEquivOfUniqueUnits : (a : M) → Associates M) p : Associates M))
              ⋯)) :
        N))
    (↑((mkFactorOrderIsoOfFactorDvdEquiv hd :
            (a : (↑(Set.Iic (Associates.mk m)) : Type u_1)) → (↑(Set.Iic (Associates.mk n)) : Type u_2))
          (Subtype.mk (p := fun (x : Associates M) ↦
            Membership.mem (γ := Set (Associates M)) (Set.Iic (Associates.mk m)) x)
            ((MulEquiv.symm associatesEquivOfUniqueUnits : (a : M) → Associates M) p) ⋯)) :
      Associates N)
this : DecidableEq (Associates M) := Classical.decEq (Associates M)
⊢ Prime.{u_2} (M := Associates N)
    (↑((mkFactorOrderIsoOfFactorDvdEquiv hd :
            (a : (↑(Set.Iic (Associates.mk m)) : Type u_1)) → (↑(Set.Iic (Associates.mk n)) : Type u_2))
          (Subtype.mk (p := fun (x : Associates M) ↦
            Membership.mem (γ := Set (Associates M)) (Set.Iic (Associates.mk m)) x)
            ((MulEquiv.symm associatesEquivOfUniqueUnits : (a : M) → Associates M) p) ⋯)) :
      Associates N)
have refine : Membership.mem (α := Associates M) (γ := Multiset (Associates M)) (normalizedFactors (Associates.mk m))
  ((MulEquiv.symm associatesEquivOfUniqueUnits : (a : M) → Associates M) p : Associates M)
Tactic: refine map_prime_of_factor_orderIso (mk_ne_zero.mpr hn) ?_ _
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
N : Type u_2
inst_2 : CommMonoidWithZero N
inst_3 : Subsingleton.{u_1 + 1} Mˣ
inst_4 : Subsingleton.{u_2 + 1} Nˣ
inst_5 : UniqueFactorizationMonoid M
inst_6 : UniqueFactorizationMonoid N
m p : M
n : N
hm : m ≠ 0
hn : n ≠ 0
hp : p ∈ normalizedFactors m
d : Equiv.{max 1 (u_1 + 1), max 1 (u_2 + 1)} { l : M // l ∣ m } { l : N // l ∣ n }
hd :
  ∀ (l l' : { l : M // l ∣ m }),
    Dvd.dvd.{u_2} (α := N) (↑((d : (a : { l : M // l ∣ m }) → { l : N // l ∣ n }) l) : N)
        (↑((d : (a : { l : M // l ∣ m }) → { l : N // l ∣ n }) l') : N) ↔
      (↑l : M) ∣ (↑l' : M)
this_1 :
  Eq.{u_2 + 1} (α := Associates N)
    (Associates.mk
      (↑((d : (a : { l : M // l ∣ m }) → { l : N // l ∣ n })
            (Subtype.mk (p := fun (l : M) ↦ l ∣ m)
              ((associatesEquivOfUniqueUnits : (a : Associates M) → M)
                ((MulEquiv.symm associatesEquivOfUniqueUnits : (a : M) → Associates M) p : Associates M))
              ⋯)) :
        N))
    (↑((mkFactorOrderIsoOfFactorDvdEquiv hd :
            (a : (↑(Set.Iic (Associates.mk m)) : Type u_1)) → (↑(Set.Iic (Associates.mk n)) : Type u_2))
          (Subtype.mk (p := fun (x : Associates M) ↦
            Membership.mem (γ := Set (Associates M)) (Set.Iic (Associates.mk m)) x)
            ((MulEquiv.symm associatesEquivOfUniqueUnits : (a : M) → Associates M) p) ⋯)) :
      Associates N)
this : DecidableEq (Associates M) := Classical.decEq (Associates M)
⊢ Membership.mem (α := Associates M) (γ := Multiset (Associates M)) (normalizedFactors (Associates.mk m))
    ((MulEquiv.symm associatesEquivOfUniqueUnits : (a : M) → Associates M) p : Associates M)
have obtain : ∀ (q : Associates M) (hq : Membership.mem (γ := Multiset (Associates M)) (normalizedFactors (Associates.mk m)) q)
  (hq' : Associated (Associates.mk p) q),
  Membership.mem (α := Associates M) (γ := Multiset (Associates M)) (normalizedFactors (Associates.mk m))
    ((MulEquiv.symm associatesEquivOfUniqueUnits : (a : M) → Associates M) p : Associates M)
Tactic: obtain ⟨q, hq, hq'⟩ :=
    exists_mem_normalizedFactors_of_dvd (mk_ne_zero.mpr hm)
      (prime_mk.mpr (prime_of_normalized_factor p (by convert! hp))).irreducible
      (mk_le_mk_of_dvd (dvd_of_mem_normalizedFactors hp))
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
N : Type u_2
inst_2 : CommMonoidWithZero N
inst_3 : Subsingleton.{u_1 + 1} Mˣ
inst_4 : Subsingleton.{u_2 + 1} Nˣ
inst_5 : UniqueFactorizationMonoid M
inst_6 : UniqueFactorizationMonoid N
m p : M
n : N
hm : m ≠ 0
hn : n ≠ 0
hp : p ∈ normalizedFactors m
d : Equiv.{max 1 (u_1 + 1), max 1 (u_2 + 1)} { l : M // l ∣ m } { l : N // l ∣ n }
hd :
  ∀ (l l' : { l : M // l ∣ m }),
    Dvd.dvd.{u_2} (α := N) (↑((d : (a : { l : M // l ∣ m }) → { l : N // l ∣ n }) l) : N)
        (↑((d : (a : { l : M // l ∣ m }) → { l : N // l ∣ n }) l') : N) ↔
      (↑l : M) ∣ (↑l' : M)
⊢ emultiplicity (↑((d : (a : { l : M // l ∣ m }) → { l : N // l ∣ n }) (Subtype.mk (p := fun (l : M) ↦ l ∣ m) p ⋯)) : N)
      n =
    emultiplicity p m
have h : emultiplicity p m =
  emultiplicity (↑((d : (a : { l : M // l ∣ m }) → { l : N // l ∣ n }) (Subtype.mk (p := fun (l : M) ↦ l ∣ m) p sorry)) : N)
    n
Tactic: apply Eq.symm
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
N : Type u_2
inst_2 : CommMonoidWithZero N
inst_3 : Subsingleton.{u_1 + 1} Mˣ
inst_4 : Subsingleton.{u_2 + 1} Nˣ
inst_5 : UniqueFactorizationMonoid M
inst_6 : UniqueFactorizationMonoid N
m p : M
n : N
hm : m ≠ 0
hn : n ≠ 0
hp : p ∈ normalizedFactors m
d : Equiv.{max 1 (u_1 + 1), max 1 (u_2 + 1)} { l : M // l ∣ m } { l : N // l ∣ n }
hd :
  ∀ (l l' : { l : M // l ∣ m }),
    Dvd.dvd.{u_2} (α := N) (↑((d : (a : { l : M // l ∣ m }) → { l : N // l ∣ n }) l) : N)
        (↑((d : (a : { l : M // l ∣ m }) → { l : N // l ∣ n }) l') : N) ↔
      (↑l : M) ∣ (↑l' : M)
⊢ emultiplicity p m =
    emultiplicity
      (↑((d : (a : { l : M // l ∣ m }) → { l : N // l ∣ n }) (Subtype.mk (p := fun (l : M) ↦ l ∣ m) p ⋯)) : N) n
have h : emultiplicity (α := Associates M) (Associates.mk p) (Associates.mk m) =
  emultiplicity (α := Associates N)
    (Associates.mk
      (↑((d : (a : { l : M // l ∣ m }) → { l : N // l ∣ n })
            (Subtype.mk (p := fun (l : M) ↦ l ∣ m)
              ((associatesEquivOfUniqueUnits : (a : Associates M) → M)
                ((MulEquiv.symm associatesEquivOfUniqueUnits : (a : M) → Associates M) p : Associates M))
              sorry)) :
        N))
    (Associates.mk n)
Tactic: suffices emultiplicity (Associates.mk p) (Associates.mk m) = emultiplicity (Associates.mk
    ↑(d ⟨associatesEquivOfUniqueUnits (associatesEquivOfUniqueUnits.symm p), by
      simp [dvd_of_mem_normalizedFactors hp]⟩)) (Associates.mk n) by
    simpa only [emultiplicity_mk_eq_emultiplicity, associatesEquivOfUniqueUnits_symm_apply,
      associatesEquivOfUniqueUnits_apply, out_mk, normalize_eq] using this
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (hypothesis)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
N : Type u_2
inst_2 : CommMonoidWithZero N
inst_3 : Subsingleton.{u_1 + 1} Mˣ
inst_4 : Subsingleton.{u_2 + 1} Nˣ
inst_5 : UniqueFactorizationMonoid M
inst_6 : UniqueFactorizationMonoid N
m p : M
n : N
hm : m ≠ 0
hn : n ≠ 0
hp : p ∈ normalizedFactors m
d : Equiv.{max 1 (u_1 + 1), max 1 (u_2 + 1)} { l : M // l ∣ m } { l : N // l ∣ n }
hd :
  ∀ (l l' : { l : M // l ∣ m }),
    Dvd.dvd.{u_2} (α := N) (↑((d : (a : { l : M // l ∣ m }) → { l : N // l ∣ n }) l) : N)
        (↑((d : (a : { l : M // l ∣ m }) → { l : N // l ∣ n }) l') : N) ↔
      (↑l : M) ∣ (↑l' : M)
⊢ emultiplicity (α := Associates M) (Associates.mk p) (Associates.mk m) =
    emultiplicity (α := Associates N)
      (Associates.mk
        (↑((d : (a : { l : M // l ∣ m }) → { l : N // l ∣ n })
              (Subtype.mk (p := fun (l : M) ↦ l ∣ m)
                ((associatesEquivOfUniqueUnits : (a : Associates M) → M)
                  ((MulEquiv.symm associatesEquivOfUniqueUnits : (a : M) → Associates M) p : Associates M))
                ⋯)) :
          N))
      (Associates.mk n)
have this : Eq.{u_2 + 1} (α := Associates N)
  (Associates.mk
    (↑((d : (a : { l : M // l ∣ m }) → { l : N // l ∣ n })
          (Subtype.mk (p := fun (l : M) ↦ l ∣ m)
            ((associatesEquivOfUniqueUnits : (a : Associates M) → M)
              ((MulEquiv.symm associatesEquivOfUniqueUnits : (a : M) → Associates M) p : Associates M))
            sorry)) :
      N))
  (↑((mkFactorOrderIsoOfFactorDvdEquiv hd :
          (a : (↑(Set.Iic (Associates.mk m)) : Type u_1)) → (↑(Set.Iic (Associates.mk n)) : Type u_2))
        (Subtype.mk (p := fun (x : Associates M) ↦
          Membership.mem (γ := Set (Associates M)) (Set.Iic (Associates.mk m)) x)
          ((MulEquiv.symm associatesEquivOfUniqueUnits : (a : M) → Associates M) p) sorry)) :
    Associates N)
Tactic: have : Associates.mk (d ⟨associatesEquivOfUniqueUnits (associatesEquivOfUniqueUnits.symm p), by
    simp only [dvd_of_mem_normalizedFactors hp, associatesEquivOfUniqueUnits_symm_apply,
      associatesEquivOfUniqueUnits_apply, out_mk, normalize_eq]⟩ : N) =
    ↑(mkFactorOrderIsoOfFactorDvdEquiv hd ⟨associatesEquivOfUniqueUnits.symm p, by
      rw [associatesEquivOfUniqueUnits_symm_apply]
      exact mk_le_mk_of_dvd (dvd_of_mem_normalizedFactors hp)⟩) := by
    rw [mkFactorOrderIsoOfFactorDvdEquiv_apply_coe]
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
N : Type u_2
inst_2 : CommMonoidWithZero N
inst_3 : Subsingleton.{u_1 + 1} Mˣ
inst_4 : Subsingleton.{u_2 + 1} Nˣ
inst_5 : UniqueFactorizationMonoid M
inst_6 : UniqueFactorizationMonoid N
m p : M
n : N
hm : m ≠ 0
hn : n ≠ 0
hp : p ∈ normalizedFactors m
d : Equiv.{max 1 (u_1 + 1), max 1 (u_2 + 1)} { l : M // l ∣ m } { l : N // l ∣ n }
hd :
  ∀ (l l' : { l : M // l ∣ m }),
    Dvd.dvd.{u_2} (α := N) (↑((d : (a : { l : M // l ∣ m }) → { l : N // l ∣ n }) l) : N)
        (↑((d : (a : { l : M // l ∣ m }) → { l : N // l ∣ n }) l') : N) ↔
      (↑l : M) ∣ (↑l' : M)
this :
  Eq.{u_2 + 1} (α := Associates N)
    (Associates.mk
      (↑((d : (a : { l : M // l ∣ m }) → { l : N // l ∣ n })
            (Subtype.mk (p := fun (l : M) ↦ l ∣ m)
              ((associatesEquivOfUniqueUnits : (a : Associates M) → M)
                ((MulEquiv.symm associatesEquivOfUniqueUnits : (a : M) → Associates M) p : Associates M))
              ⋯)) :
        N))
    (↑((mkFactorOrderIsoOfFactorDvdEquiv hd :
            (a : (↑(Set.Iic (Associates.mk m)) : Type u_1)) → (↑(Set.Iic (Associates.mk n)) : Type u_2))
          (Subtype.mk (p := fun (x : Associates M) ↦
            Membership.mem (γ := Set (Associates M)) (Set.Iic (Associates.mk m)) x)
            ((MulEquiv.symm associatesEquivOfUniqueUnits : (a : M) → Associates M) p) ⋯)) :
      Associates N)
⊢ emultiplicity (α := Associates M) (Associates.mk p) (Associates.mk m) =
    emultiplicity (α := Associates N)
      (Associates.mk
        (↑((d : (a : { l : M // l ∣ m }) → { l : N // l ∣ n })
              (Subtype.mk (p := fun (l : M) ↦ l ∣ m)
                ((associatesEquivOfUniqueUnits : (a : Associates M) → M)
                  ((MulEquiv.symm associatesEquivOfUniqueUnits : (a : M) → Associates M) p : Associates M))
                ⋯)) :
          N))
      (Associates.mk n)
have h : emultiplicity (α := Associates M) (Associates.mk p) (Associates.mk m) =
  emultiplicity (α := Associates N)
    (↑((mkFactorOrderIsoOfFactorDvdEquiv hd :
            (a : (↑(Set.Iic (Associates.mk m)) : Type u_1)) → (↑(Set.Iic (Associates.mk n)) : Type u_2))
          (Subtype.mk (p := fun (x : Associates M) ↦
            Membership.mem (γ := Set (Associates M)) (Set.Iic (Associates.mk m)) x)
            ((MulEquiv.symm associatesEquivOfUniqueUnits : (a : M) → Associates M) p) sorry)) :
      Associates N)
    (Associates.mk n)
Tactic: rw [this]
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
N : Type u_2
inst_2 : CommMonoidWithZero N
inst_3 : Subsingleton.{u_1 + 1} Mˣ
inst_4 : Subsingleton.{u_2 + 1} Nˣ
inst_5 : UniqueFactorizationMonoid M
inst_6 : UniqueFactorizationMonoid N
m p : M
n : N
hm : m ≠ 0
hn : n ≠ 0
hp : p ∈ normalizedFactors m
d : Equiv.{max 1 (u_1 + 1), max 1 (u_2 + 1)} { l : M // l ∣ m } { l : N // l ∣ n }
hd :
  ∀ (l l' : { l : M // l ∣ m }),
    Dvd.dvd.{u_2} (α := N) (↑((d : (a : { l : M // l ∣ m }) → { l : N // l ∣ n }) l) : N)
        (↑((d : (a : { l : M // l ∣ m }) → { l : N // l ∣ n }) l') : N) ↔
      (↑l : M) ∣ (↑l' : M)
this :
  Eq.{u_2 + 1} (α := Associates N)
    (Associates.mk
      (↑((d : (a : { l : M // l ∣ m }) → { l : N // l ∣ n })
            (Subtype.mk (p := fun (l : M) ↦ l ∣ m)
              ((associatesEquivOfUniqueUnits : (a : Associates M) → M)
                ((MulEquiv.symm associatesEquivOfUniqueUnits : (a : M) → Associates M) p : Associates M))
              ⋯)) :
        N))
    (↑((mkFactorOrderIsoOfFactorDvdEquiv hd :
            (a : (↑(Set.Iic (Associates.mk m)) : Type u_1)) → (↑(Set.Iic (Associates.mk n)) : Type u_2))
          (Subtype.mk (p := fun (x : Associates M) ↦
            Membership.mem (γ := Set (Associates M)) (Set.Iic (Associates.mk m)) x)
            ((MulEquiv.symm associatesEquivOfUniqueUnits : (a : M) → Associates M) p) ⋯)) :
      Associates N)
⊢ emultiplicity (α := Associates M) (Associates.mk p) (Associates.mk m) =
    emultiplicity (α := Associates N)
      (↑((mkFactorOrderIsoOfFactorDvdEquiv hd :
              (a : (↑(Set.Iic (Associates.mk m)) : Type u_1)) → (↑(Set.Iic (Associates.mk n)) : Type u_2))
            (Subtype.mk (p := fun (x : Associates M) ↦
              Membership.mem (γ := Set (Associates M)) (Set.Iic (Associates.mk m)) x)
              ((MulEquiv.symm associatesEquivOfUniqueUnits : (a : M) → Associates M) p) ⋯)) :
        Associates N)
      (Associates.mk n)
have h : Membership.mem (γ := Multiset (Associates M)) (normalizedFactors (Associates.mk m)) (Associates.mk p)
Tactic: refine
    emultiplicity_prime_eq_emultiplicity_image_by_factor_orderIso (mk_ne_zero.mpr hn) ?_
      (mkFactorOrderIsoOfFactorDvdEquiv hd)