awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.RingTheory.ChainOfDivisors
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
N : Type u_2
inst_2 : CommMonoidWithZero N
inst_3 : UniqueFactorizationMonoid N
inst_4 : UniqueFactorizationMonoid M
m p : Associates M
n : Associates N
hn : n ≠ 0
hp : p ∈ normalizedFactors m
d : OrderIso.{u_1, u_2} (↑(Set.Iic m) : Type u_1) (↑(Set.Iic n) : Type u_2)
obtain :
  ∀ (q : Associates N) (hq : q ∈ normalizedFactors n)
    (hq' :
      Associated
        (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
              (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) p ⋯)) :
          Associates N)
        q),
    (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
            (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) p ⋯)) :
        Associates N) ∈
      normalizedFactors n
⊢ (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
          (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) p ⋯)) :
      Associates N) ∈
    normalizedFactors n

Tactic:
obtain ⟨q, hq, hq'⟩ :=
    exists_mem_normalizedFactors_of_dvd hn (map_prime_of_factor_orderIso hn hp d).irreducible
      (d ⟨p, dvd_of_mem_normalizedFactors hp⟩).prop

Premises used: Prime.irreducible, UniqueFactorizationMonoid.dvd_of_mem_normalizedFactors, Subtype.mk, UniqueFactorizationMonoid.exists_mem_normalizedFactors_of_dvd, Subtype.prop, map_prime_of_factor_orderIso, DFunLike.coe
-- Module: Mathlib.RingTheory.ChainOfDivisors
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
N : Type u_2
inst_2 : CommMonoidWithZero N
inst_3 : UniqueFactorizationMonoid N
inst_4 : UniqueFactorizationMonoid M
m p : Associates M
n : Associates N
hn : n ≠ 0
hp : p ∈ normalizedFactors m
d : OrderIso.{u_1, u_2} (↑(Set.Iic m) : Type u_1) (↑(Set.Iic n) : Type u_2)
q : Associates N
hq : q ∈ normalizedFactors n
hq' :
  Associated
    (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
          (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) p ⋯)) :
      Associates N)
    q
⊢ (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
          (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) p ⋯)) :
      Associates N) =
    q

Tactic:
rw [associated_iff_eq] at hq'

Premises used: associated_iff_eq
-- Module: Mathlib.RingTheory.ChainOfDivisors
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
N : Type u_2
inst_2 : CommMonoidWithZero N
inst_3 : UniqueFactorizationMonoid N
inst_4 : UniqueFactorizationMonoid M
m p : Associates M
n : Associates N
hn : n ≠ 0
hp : p ∈ normalizedFactors m
d : OrderIso.{u_1, u_2} (↑(Set.Iic m) : Type u_1) (↑(Set.Iic n) : Type u_2)
q : Associates N
hq : q ∈ normalizedFactors n
hq' :
  (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
          (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) p ⋯)) :
      Associates N) =
    q
rwa rwa₁ rwa₂ rwa₃ : q ∈ normalizedFactors n
⊢ (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
          (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) p ⋯)) :
      Associates N) ∈
    normalizedFactors n

Tactic:
rwa [hq']
-- Module: Mathlib.RingTheory.ChainOfDivisors
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
N : Type u_2
inst_2 : CommMonoidWithZero N
inst_3 : UniqueFactorizationMonoid N
inst_4 : UniqueFactorizationMonoid M
m p : Associates M
n : Associates N
hp : p ∈ normalizedFactors m
d : OrderIso.{u_1, u_2} (↑(Set.Iic m) : Type u_1) (↑(Set.Iic n) : Type u_2)
pos :
  ∀ (hn : n = 0),
    emultiplicity p m ≤
      emultiplicity
        (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
              (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) p ⋯)) :
          Associates N)
        n
neg :
  ∀ (hn : ¬n = 0),
    emultiplicity p m ≤
      emultiplicity
        (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
              (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) p ⋯)) :
          Associates N)
        n
⊢ emultiplicity p m ≤
    emultiplicity
      (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
            (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) p ⋯)) :
        Associates N)
      n

Tactic:
by_cases hn : n = 0
-- Module: Mathlib.RingTheory.ChainOfDivisors
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
N : Type u_2
inst_2 : CommMonoidWithZero N
inst_3 : UniqueFactorizationMonoid N
inst_4 : UniqueFactorizationMonoid M
m p : Associates M
n : Associates N
hp : p ∈ normalizedFactors m
d : OrderIso.{u_1, u_2} (↑(Set.Iic m) : Type u_1) (↑(Set.Iic n) : Type u_2)
hn : n = 0
⊢ emultiplicity p m ≤
    emultiplicity
      (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
            (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) p ⋯)) :
        Associates N)
      n

Tactic:
simp [hn]
-- Module: Mathlib.RingTheory.ChainOfDivisors
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
N : Type u_2
inst_2 : CommMonoidWithZero N
inst_3 : UniqueFactorizationMonoid N
inst_4 : UniqueFactorizationMonoid M
m p : Associates M
n : Associates N
hp : p ∈ normalizedFactors m
d : OrderIso.{u_1, u_2} (↑(Set.Iic m) : Type u_1) (↑(Set.Iic n) : Type u_2)
hn : ¬n = 0
pos :
  ∀ (hm : m = 0),
    emultiplicity p m ≤
      emultiplicity
        (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
              (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) p ⋯)) :
          Associates N)
        n
neg :
  ∀ (hm : ¬m = 0),
    emultiplicity p m ≤
      emultiplicity
        (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
              (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) p ⋯)) :
          Associates N)
        n
⊢ emultiplicity p m ≤
    emultiplicity
      (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
            (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) p ⋯)) :
        Associates N)
      n

Tactic:
by_cases hm : m = 0
-- Module: Mathlib.RingTheory.ChainOfDivisors
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
N : Type u_2
inst_2 : CommMonoidWithZero N
inst_3 : UniqueFactorizationMonoid N
inst_4 : UniqueFactorizationMonoid M
m p : Associates M
n : Associates N
hp : p ∈ normalizedFactors m
d : OrderIso.{u_1, u_2} (↑(Set.Iic m) : Type u_1) (↑(Set.Iic n) : Type u_2)
hn : ¬n = 0
hm : m = 0
⊢ emultiplicity p m ≤
    emultiplicity
      (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
            (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) p ⋯)) :
        Associates N)
      n

Tactic:
simp [hm] at hp
-- Module: Mathlib.RingTheory.ChainOfDivisors
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
N : Type u_2
inst_2 : CommMonoidWithZero N
inst_3 : UniqueFactorizationMonoid N
inst_4 : UniqueFactorizationMonoid M
m p : Associates M
n : Associates N
hp : p ∈ normalizedFactors m
d : OrderIso.{u_1, u_2} (↑(Set.Iic m) : Type u_1) (↑(Set.Iic n) : Type u_2)
hn : ¬n = 0
hm : ¬m = 0
neg :
  HPow.hPow (α := Associates N)
      (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
            (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) p ⋯)) :
        Associates N)
      (multiplicity p m) ∣
    n
⊢ emultiplicity p m ≤
    emultiplicity
      (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
            (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) p ⋯)) :
        Associates N)
      n

Tactic:
rw [FiniteMultiplicity.of_prime_left (prime_of_normalized_factor p hp) hm
    |>.emultiplicity_eq_multiplicity, ← pow_dvd_iff_le_emultiplicity]

Premises used: FiniteMultiplicity.emultiplicity_eq_multiplicity, pow_dvd_iff_le_emultiplicity, UniqueFactorizationMonoid.prime_of_normalized_factor, FiniteMultiplicity.of_prime_left
-- Module: Mathlib.RingTheory.ChainOfDivisors
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
N : Type u_2
inst_2 : CommMonoidWithZero N
inst_3 : UniqueFactorizationMonoid N
inst_4 : UniqueFactorizationMonoid M
m p : Associates M
n : Associates N
hp : p ∈ normalizedFactors m
d : OrderIso.{u_1, u_2} (↑(Set.Iic m) : Type u_1) (↑(Set.Iic n) : Type u_2)
hn : ¬n = 0
hm : ¬m = 0
⊢ HPow.hPow (α := Associates N)
      (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
            (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) p ⋯)) :
        Associates N)
      (multiplicity p m) ∣
    n

Tactic:
apply pow_image_of_prime_by_factor_orderIso_dvd hn hp d (pow_multiplicity_dvd ..)

Premises used: pow_multiplicity_dvd, pow_image_of_prime_by_factor_orderIso_dvd
-- Module: Mathlib.RingTheory.ChainOfDivisors
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
N : Type u_2
inst_2 : CommMonoidWithZero N
inst_3 : UniqueFactorizationMonoid N
inst_4 : UniqueFactorizationMonoid M
m p : Associates M
n : Associates N
hn : n ≠ 0
hp : p ∈ normalizedFactors m
d : OrderIso.{u_1, u_2} (↑(Set.Iic m) : Type u_1) (↑(Set.Iic n) : Type u_2)
⊢ emultiplicity p m =
    emultiplicity
      (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
            (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) p ⋯)) :
        Associates N)
      n

Tactic:
le_antisymm (emultiplicity_prime_le_emultiplicity_image_by_factor_orderIso hp d) ?_

Premises used: le_antisymm, emultiplicity_prime_le_emultiplicity_image_by_factor_orderIso
-- Module: Mathlib.RingTheory.ChainOfDivisors
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
N : Type u_2
inst_2 : CommMonoidWithZero N
inst_3 : UniqueFactorizationMonoid N
inst_4 : UniqueFactorizationMonoid M
m p : Associates M
n : Associates N
hn : n ≠ 0
hp : p ∈ normalizedFactors m
d : OrderIso.{u_1, u_2} (↑(Set.Iic m) : Type u_1) (↑(Set.Iic n) : Type u_2)
refine :
  emultiplicity
      (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
            (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) p ⋯)) :
        Associates N)
      n ≤
    emultiplicity p m
⊢ emultiplicity p m =
    emultiplicity
      (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
            (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) p ⋯)) :
        Associates N)
      n

Tactic:
refine le_antisymm (emultiplicity_prime_le_emultiplicity_image_by_factor_orderIso hp d) ?_

Premises used: le_antisymm, emultiplicity_prime_le_emultiplicity_image_by_factor_orderIso
-- Module: Mathlib.RingTheory.ChainOfDivisors
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
N : Type u_2
inst_2 : CommMonoidWithZero N
inst_3 : UniqueFactorizationMonoid N
inst_4 : UniqueFactorizationMonoid M
m p : Associates M
n : Associates N
hn : n ≠ 0
hp : p ∈ normalizedFactors m
d : OrderIso.{u_1, u_2} (↑(Set.Iic m) : Type u_1) (↑(Set.Iic n) : Type u_2)
this :
  emultiplicity
      (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
            (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) p ⋯)) :
        Associates N)
      n ≤
    emultiplicity
      (↑((d.symm : (a : (↑(Set.Iic n) : Type u_2)) → (↑(Set.Iic m) : Type u_1))
            ((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
              (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) p ⋯))) :
        Associates M)
      m
⊢ emultiplicity
      (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
            (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) p ⋯)) :
        Associates N)
      n ≤
    emultiplicity (↑(Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) p ⋯) : Associates M) m

Tactic:
rw [d.symm_apply_apply ⟨p, dvd_of_mem_normalizedFactors hp⟩, Subtype.coe_mk] at this

Premises used: Subtype.coe_mk, UniqueFactorizationMonoid.dvd_of_mem_normalizedFactors, Subtype.mk, OrderIso.symm_apply_apply
-- Module: Mathlib.RingTheory.ChainOfDivisors
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
N : Type u_2
inst_2 : CommMonoidWithZero N
inst_3 : UniqueFactorizationMonoid N
inst_4 : UniqueFactorizationMonoid M
m p : Associates M
n : Associates N
hn : n ≠ 0
hp : p ∈ normalizedFactors m
d : OrderIso.{u_1, u_2} (↑(Set.Iic m) : Type u_1) (↑(Set.Iic n) : Type u_2)
this :
  emultiplicity
      (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
            (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) p ⋯)) :
        Associates N)
      n ≤
    emultiplicity (↑(Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) p ⋯) : Associates M) m
⊢ emultiplicity
      (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
            (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) p ⋯)) :
        Associates N)
      n ≤
    emultiplicity p m

Tactic:
this
-- Module: Mathlib.RingTheory.ChainOfDivisors
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
N : Type u_2
inst_2 : CommMonoidWithZero N
inst_3 : UniqueFactorizationMonoid N
inst_4 : UniqueFactorizationMonoid M
m p : Associates M
n : Associates N
hn : n ≠ 0
hp : p ∈ normalizedFactors m
d : OrderIso.{u_1, u_2} (↑(Set.Iic m) : Type u_1) (↑(Set.Iic n) : Type u_2)
this :
  emultiplicity
      (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
            (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) p ⋯)) :
        Associates N)
      n ≤
    emultiplicity (↑(Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) p ⋯) : Associates M) m
⊢ emultiplicity
      (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
            (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) p ⋯)) :
        Associates N)
      n ≤
    emultiplicity p m

Tactic:
exact this
-- Module: Mathlib.RingTheory.ChainOfDivisors
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
N : Type u_2
inst_2 : CommMonoidWithZero N
inst_3 : UniqueFactorizationMonoid N
inst_4 : UniqueFactorizationMonoid M
m p : Associates M
n : Associates N
hn : n ≠ 0
hp : p ∈ normalizedFactors m
d : OrderIso.{u_1, u_2} (↑(Set.Iic m) : Type u_1) (↑(Set.Iic n) : Type u_2)
suffices :
  emultiplicity
      (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
            (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) p ⋯)) :
        Associates N)
      n ≤
    emultiplicity
      (↑((d.symm : (a : (↑(Set.Iic n) : Type u_2)) → (↑(Set.Iic m) : Type u_1))
            ((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
              (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) p ⋯))) :
        Associates M)
      m
⊢ emultiplicity
      (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
            (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) p ⋯)) :
        Associates N)
      n ≤
    emultiplicity p m

Tactic:
suffices emultiplicity (↑(d ⟨p, dvd_of_mem_normalizedFactors hp⟩)) n ≤
      emultiplicity (↑(d.symm (d ⟨p, dvd_of_mem_normalizedFactors hp⟩))) m by
    rw [d.symm_apply_apply ⟨p, dvd_of_mem_normalizedFactors hp⟩, Subtype.coe_mk] at this
    exact this
-- Module: Mathlib.RingTheory.ChainOfDivisors
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
N : Type u_2
inst_2 : CommMonoidWithZero N
inst_3 : UniqueFactorizationMonoid N
inst_4 : UniqueFactorizationMonoid M
m p : Associates M
n : Associates N
hn : n ≠ 0
hp : p ∈ normalizedFactors m
d : OrderIso.{u_1, u_2} (↑(Set.Iic m) : Type u_1) (↑(Set.Iic n) : Type u_2)
this : DecidableEq (Associates N) := Classical.decEq (Associates N)
⊢ emultiplicity
      (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
            (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) p ⋯)) :
        Associates N)
      n ≤
    emultiplicity
      (↑((d.symm : (a : (↑(Set.Iic n) : Type u_2)) → (↑(Set.Iic m) : Type u_1))
            ((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
              (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) p ⋯))) :
        Associates M)
      m

Tactic:
simpa only [Subtype.coe_eta] using
    emultiplicity_prime_le_emultiplicity_image_by_factor_orderIso
      (mem_normalizedFactors_factor_orderIso_of_mem_normalizedFactors hn hp d) d.symm

Premises used: OrderIso.symm, emultiplicity_prime_le_emultiplicity_image_by_factor_orderIso, Subtype.coe_eta, mem_normalizedFactors_factor_orderIso_of_mem_normalizedFactors
-- Module: Mathlib.RingTheory.ChainOfDivisors
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 : IsCancelMulZero N
m : M
n : N
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)
l : (↑(Set.Iic (Associates.mk m)) : Type u_1)
obtain :
  ∀ (x : Associates M) (hx : Membership.mem (γ := Set (Associates M)) (Set.Iic (Associates.mk m)) x),
    (associatesEquivOfUniqueUnits : (a : Associates M) → M)
        (↑(Subtype.mk (p := fun (x : Associates M) ↦
              Membership.mem (γ := Set (Associates M)) (Set.Iic (Associates.mk m)) x) x hx) :
          Associates M) ∣
      m
⊢ (associatesEquivOfUniqueUnits : (a : Associates M) → M) (↑l : Associates M) ∣ m

Tactic:
obtain ⟨x, hx⟩ := l
-- Module: Mathlib.RingTheory.ChainOfDivisors
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 : IsCancelMulZero N
m : M
n : N
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)
x : Associates M
hx : Membership.mem (γ := Set (Associates M)) (Set.Iic (Associates.mk m)) x
rw : x ≤ Associates.mk 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) x hx) :
        Associates M) ∣
    m

Tactic:
rw [Subtype.coe_mk, associatesEquivOfUniqueUnits_apply, out_dvd_iff]

Premises used: Subtype.coe_mk, Associates.out_dvd_iff, associatesEquivOfUniqueUnits_apply
-- Module: Mathlib.RingTheory.ChainOfDivisors
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 : IsCancelMulZero N
m : M
n : N
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)
x : Associates M
hx : Membership.mem (γ := Set (Associates M)) (Set.Iic (Associates.mk m)) x
⊢ x ≤ Associates.mk m

Tactic:
hx

Premises used: Set.Mem, setOf, Set.Iic
-- Module: Mathlib.RingTheory.ChainOfDivisors
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 : IsCancelMulZero N
m : M
n : N
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)
x : Associates M
hx : Membership.mem (γ := Set (Associates M)) (Set.Iic (Associates.mk m)) x
⊢ x ≤ Associates.mk m

Tactic:
exact hx
-- Module: Mathlib.RingTheory.ChainOfDivisors
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 : IsCancelMulZero N
m : M
n : N
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)
l : (↑(Set.Iic (Associates.mk n)) : Type u_2)
obtain :
  ∀ (x : Associates N) (hx : Membership.mem (γ := Set (Associates N)) (Set.Iic (Associates.mk n)) x),
    (associatesEquivOfUniqueUnits : (a : Associates N) → N)
        (↑(Subtype.mk (p := fun (x : Associates N) ↦
              Membership.mem (γ := Set (Associates N)) (Set.Iic (Associates.mk n)) x) x hx) :
          Associates N) ∣
      n
⊢ (associatesEquivOfUniqueUnits : (a : Associates N) → N) (↑l : Associates N) ∣ n

Tactic:
obtain ⟨x, hx⟩ := l
-- Module: Mathlib.RingTheory.ChainOfDivisors
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 : IsCancelMulZero N
m : M
n : N
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)
x : Associates N
hx : Membership.mem (γ := Set (Associates N)) (Set.Iic (Associates.mk n)) x
rw : x ≤ Associates.mk n
⊢ (associatesEquivOfUniqueUnits : (a : Associates N) → N)
      (↑(Subtype.mk (p := fun (x : Associates N) ↦
            Membership.mem (γ := Set (Associates N)) (Set.Iic (Associates.mk n)) x) x hx) :
        Associates N) ∣
    n

Tactic:
rw [Subtype.coe_mk, associatesEquivOfUniqueUnits_apply, out_dvd_iff]

Premises used: Subtype.coe_mk, Associates.out_dvd_iff, associatesEquivOfUniqueUnits_apply
-- Module: Mathlib.RingTheory.ChainOfDivisors
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 : IsCancelMulZero N
m : M
n : N
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)
x : Associates N
hx : Membership.mem (γ := Set (Associates N)) (Set.Iic (Associates.mk n)) x
⊢ x ≤ Associates.mk n

Tactic:
hx

Premises used: Set.Mem, setOf, Set.Iic
-- Module: Mathlib.RingTheory.ChainOfDivisors
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 : IsCancelMulZero N
m : M
n : N
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)
x : Associates N
hx : Membership.mem (γ := Set (Associates N)) (Set.Iic (Associates.mk n)) x
⊢ x ≤ Associates.mk n

Tactic:
exact hx