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 : 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)
b : Associates N
this_1 : b ≤ n
x : (↑(Set.Iic m) : Type u_1)
hb :
  LT.lt.{u_2} (α := Associates N)
    (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2)) x) : 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)
hx :
  Eq.{u_2 + 1} (α := (↑(Set.Iic n) : Type u_2))
    ((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2)) x : (↑(Set.Iic n) : Type u_2))
    (Subtype.mk (p := fun (x : Associates N) ↦ x ∈ Set.Iic n) b this_1 : { x : Associates N // x ∈ Set.Iic n })
this_2 : OrderBot { l : Associates M // l ≤ m } := Subtype.orderBot (p := fun (l : Associates M) ↦ l ≤ m) ⋯
this : OrderBot { l : Associates N // l ≤ n } := Subtype.orderBot (p := fun (l : Associates N) ↦ l ≤ n) ⋯
⊢ x = ⊥
have refine_2 : ∀ (a : Associates M) (ha : a ∈ Set.Iic m)
  (hb :
    LT.lt.{u_2} (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) a ha)) :
        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 sorry)) :
        Associates N))
  (hx :
    Eq.{u_2 + 1} (α := (↑(Set.Iic n) : Type u_2))
      ((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) a ha) :
        (↑(Set.Iic n) : Type u_2))
      (Subtype.mk (p := fun (x : Associates N) ↦ x ∈ Set.Iic n) b this_1 : { x : Associates N // x ∈ Set.Iic n })),
  Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) a ha = ⊥
Tactic: obtain ⟨a, ha⟩ := x
Cleared hypotheses: x, hb, hx
-- 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 : 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)
b : Associates N
this_1 : b ≤ n
this_2 : OrderBot { l : Associates M // l ≤ m } := Subtype.orderBot (p := fun (l : Associates M) ↦ l ≤ m) ⋯
this : OrderBot { l : Associates N // l ≤ n } := Subtype.orderBot (p := fun (l : Associates N) ↦ l ≤ n) ⋯
a : Associates M
ha : a ∈ Set.Iic m
hb :
  LT.lt.{u_2} (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) a ha)) :
      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)
hx :
  Eq.{u_2 + 1} (α := (↑(Set.Iic n) : Type u_2))
    ((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) a ha) :
      (↑(Set.Iic n) : Type u_2))
    (Subtype.mk (p := fun (x : Associates N) ↦ x ∈ Set.Iic n) b this_1 : { x : Associates N // x ∈ Set.Iic n })
⊢ Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) a ha = ⊥
have refine_2 : a = ⊥
Tactic: rw [Subtype.mk_eq_bot_iff]
-- 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 : 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)
b : Associates N
this_1 : b ≤ n
this_2 : OrderBot { l : Associates M // l ≤ m } := Subtype.orderBot (p := fun (l : Associates M) ↦ l ≤ m) ⋯
this : OrderBot { l : Associates N // l ≤ n } := Subtype.orderBot (p := fun (l : Associates N) ↦ l ≤ n) ⋯
a : Associates M
ha : a ∈ Set.Iic m
hb :
  LT.lt.{u_2} (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) a ha)) :
      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)
hx :
  Eq.{u_2 + 1} (α := (↑(Set.Iic n) : Type u_2))
    ((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) a ha) :
      (↑(Set.Iic n) : Type u_2))
    (Subtype.mk (p := fun (x : Associates N) ↦ x ∈ Set.Iic n) b this_1 : { x : Associates N // x ∈ Set.Iic n })
refine_2 : a = ⊥
⊢ Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) a ha = ⊥
have refine_2₁ : ⊥ ∈ Set.Iic m
Tactic: rw [Subtype.mk_eq_bot_iff]
-- 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 : 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)
⊢ (↑((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
have 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 sorry)) :
        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 sorry)) :
      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
-- 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 : 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) ∈
    normalizedFactors n
have hq'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 sorry)) :
    Associates N) =
  q
Tactic: rw [associated_iff_eq] at hq'
Cleared hypotheses: 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 : 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)
⊢ 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
have 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 sorry)) :
        Associates N)
      n
Tactic: by_cases hn : n = 0
-- 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 : 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
⊢ 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
have 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 sorry)) :
        Associates N)
      n
Tactic: by_cases hn : n = 0
-- 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 : 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
have 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 sorry)) :
        Associates N)
      n
Tactic: by_cases hm : m = 0
-- 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 : 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
⊢ 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
have 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 sorry)) :
        Associates N)
      n
Tactic: by_cases hm : m = 0
-- 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 : 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
have 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 sorry)) :
      Associates N)
    (multiplicity p m) ∣
  n
Tactic: rw [FiniteMultiplicity.of_prime_left (prime_of_normalized_factor p hp) hm
    |>.emultiplicity_eq_multiplicity, ← pow_dvd_iff_le_emultiplicity]
-- 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 : 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
have 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 sorry)) :
      Associates N)
    n ≤
  emultiplicity p m
Tactic: refine le_antisymm (emultiplicity_prime_le_emultiplicity_image_by_factor_orderIso hp d) ?_
-- 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 : 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 p m
have this₁ : 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 sorry)) :
      Associates N)
    n ≤
  emultiplicity (↑(Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) p sorry) : Associates M) m
Tactic: rw [d.symm_apply_apply ⟨p, dvd_of_mem_normalizedFactors hp⟩, Subtype.coe_mk] at this
Cleared hypotheses: 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 : 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
      (↑((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
have 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 sorry)) :
      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 sorry))) :
      Associates M)
    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
-- 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 : 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)
⊢ (associatesEquivOfUniqueUnits : (a : Associates M) → M) (↑l : Associates M) ∣ m
have 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
Tactic: obtain ⟨x, hx⟩ := l
Cleared hypotheses: l
-- 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 : 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
⊢ (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
have rw : x ≤ Associates.mk m
Tactic: rw [Subtype.coe_mk, associatesEquivOfUniqueUnits_apply, out_dvd_iff]
-- 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 : 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)
⊢ (associatesEquivOfUniqueUnits : (a : Associates N) → N) (↑l : Associates N) ∣ n
have 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
Tactic: obtain ⟨x, hx⟩ := l
Cleared hypotheses: l
-- 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 : 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
⊢ (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
have rw : x ≤ Associates.mk n
Tactic: rw [Subtype.coe_mk, associatesEquivOfUniqueUnits_apply, out_dvd_iff]
-- 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)
this :
  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)
⊢ (↑((d : (a : { l : M // l ∣ m }) → { l : N // l ∣ n }) (Subtype.mk (p := fun (l : M) ↦ l ∣ m) p ⋯)) : N) ∈
    normalizedFactors n
have this₁ : Prime.{u_2} (M := N)
  (↑((d : (a : { l : M // l ∣ m }) → { l : N // l ∣ n }) (Subtype.mk (p := fun (l : M) ↦ l ∣ m) p sorry)) : N)
Tactic: simp only [associatesEquivOfUniqueUnits_apply, out_mk, normalize_eq,
      associatesEquivOfUniqueUnits_symm_apply] at this
Cleared hypotheses: 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 :
  Prime.{u_2} (M := N)
    (↑((d : (a : { l : M // l ∣ m }) → { l : N // l ∣ n }) (Subtype.mk (p := fun (l : M) ↦ l ∣ m) p ⋯)) : N)
⊢ (↑((d : (a : { l : M // l ∣ m }) → { l : N // l ∣ n }) (Subtype.mk (p := fun (l : M) ↦ l ∣ m) p ⋯)) : N) ∈
    normalizedFactors n
have obtain : ∀ (q : N) (hq : q ∈ normalizedFactors n)
  (hq' :
    Associated (↑((d : (a : { l : M // l ∣ m }) → { l : N // l ∣ n }) (Subtype.mk (p := fun (l : M) ↦ l ∣ m) p sorry)) : N)
      q),
  (↑((d : (a : { l : M // l ∣ m }) → { l : N // l ∣ n }) (Subtype.mk (p := fun (l : M) ↦ l ∣ m) p sorry)) : N) ∈
    normalizedFactors n
Tactic: obtain ⟨q, hq, hq'⟩ :=
      exists_mem_normalizedFactors_of_dvd hn this.irreducible
        (d ⟨p, by apply dvd_of_mem_normalizedFactors; convert! hp⟩).prop