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 : IsCancelMulZero N
m u : Associates M
n : Associates N
hu' : u ≤ m
d : OrderIso.{u_1, u_2} (↑(Set.Iic m) : Type u_1) (↑(Set.Iic n) : Type u_2)
hu :
  Eq.{u_2 + 1} (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) u hu')) :
      Associates N)
    1
⊢ u = 1
have rw : Eq.{u_1 + 1} (α := Associates M)
  (↑((d.symm : (a : (↑(Set.Iic n) : Type u_2)) → (↑(Set.Iic m) : Type u_1))
        (Subtype.mk (p := fun (x : Associates N) ↦ x ∈ Set.Iic 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) u hu')) :
            Associates N)
          sorry)) :
    Associates M)
  1
Tactic: rw [show u = (d.symm ⟨d ⟨u, hu'⟩, (d ⟨u, hu'⟩).prop⟩) by
        simp only [Subtype.coe_eta, OrderIso.symm_apply_apply, Subtype.coe_mk]]
-- 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 : IsCancelMulZero N
m u : Associates M
n : Associates N
hu' : u ≤ m
d : OrderIso.{u_1, u_2} (↑(Set.Iic m) : Type u_1) (↑(Set.Iic n) : Type u_2)
hu :
  Eq.{u_2 + 1} (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) u hu')) :
      Associates N)
    1
| 1
have rw : (↑((d.symm : (a : { l : Associates N // l ≤ n }) → { l : Associates M // l ≤ m })
      (Subtype.mk (p := fun (l : Associates N) ↦ l ≤ n) 1 sorry)) :
  Associates M)
Tactic: rw [← factor_orderIso_map_one_eq_bot d.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 : IsCancelMulZero N
m u : Associates M
n : Associates N
hu' : u ≤ m
d : OrderIso.{u_1, u_2} (↑(Set.Iic m) : Type u_1) (↑(Set.Iic n) : Type u_2)
hu :
  Eq.{u_2 + 1} (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) u hu')) :
      Associates N)
    1
⊢ Eq.{u_1 + 1} (α := Associates M)
    (↑((d.symm : (a : (↑(Set.Iic n) : Type u_2)) → (↑(Set.Iic m) : Type u_1))
          (Subtype.mk (p := fun (x : Associates N) ↦ x ∈ Set.Iic 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) u hu')) :
              Associates N)
            ⋯)) :
      Associates M)
    1
have conv_rhs : Eq.{u_1 + 1} (α := Associates M)
  (↑((d.symm : (a : (↑(Set.Iic n) : Type u_2)) → (↑(Set.Iic m) : Type u_1))
        (Subtype.mk (p := fun (x : Associates N) ↦ x ∈ Set.Iic 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) u hu')) :
            Associates N)
          sorry)) :
    Associates M)
  (↑((d.symm : (a : { l : Associates N // l ≤ n }) → { l : Associates M // l ≤ m })
        (Subtype.mk (p := fun (l : Associates N) ↦ l ≤ n) 1 sorry)) :
    Associates M)
Tactic: conv_rhs => rw [← factor_orderIso_map_one_eq_bot d.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 : IsCancelMulZero N
m u : Associates M
n : Associates N
hu' : u ≤ m
d : OrderIso.{u_1, u_2} (↑(Set.Iic m) : Type u_1) (↑(Set.Iic n) : Type u_2)
hu : u = 1
⊢ Eq.{u_2 + 1} (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) u hu')) :
      Associates N)
    1
have simp_rw : Eq.{u_2 + 1} (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) 1 sorry)) :
    Associates N)
  1
Tactic: simp_rw [hu]
-- 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 : IsCancelMulZero N
m u : Associates M
n : Associates N
hu' : u ≤ m
d : OrderIso.{u_1, u_2} (↑(Set.Iic m) : Type u_1) (↑(Set.Iic n) : Type u_2)
hu : u = 1
| 1
have rw : (↑((d : (a : { l : Associates M // l ≤ m }) → { l : Associates N // l ≤ n })
      (Subtype.mk (p := fun (l : Associates M) ↦ l ≤ m) 1 sorry)) :
  Associates N)
Tactic: rw [← factor_orderIso_map_one_eq_bot d]
-- 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 : IsCancelMulZero N
m u : Associates M
n : Associates N
hu' : u ≤ m
d : OrderIso.{u_1, u_2} (↑(Set.Iic m) : Type u_1) (↑(Set.Iic n) : Type u_2)
hu : u = 1
⊢ Eq.{u_2 + 1} (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) 1 ⋯)) :
      Associates N)
    1
have conv_rhs : Eq.{u_2 + 1} (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) 1 sorry)) :
    Associates N)
  (↑((d : (a : { l : Associates M // l ≤ m }) → { l : Associates N // l ≤ n })
        (Subtype.mk (p := fun (l : Associates M) ↦ l ≤ m) 1 sorry)) :
    Associates N)
Tactic: conv_rhs => rw [← factor_orderIso_map_one_eq_bot d]
-- 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)
s : ℕ
hs' : p ^ s ≤ m
⊢ 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)
      s ≤
    n
have pos : ∀ (hs : s = 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 sorry)) :
        Associates N)
      s ≤
    n
Tactic: by_cases hs : s = 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
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)
s : ℕ
hs' : p ^ s ≤ m
pos :
  ∀ (hs : s = 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)
        s ≤
      n
⊢ 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)
      s ≤
    n
have neg : ∀ (hs : ¬s = 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 sorry)) :
        Associates N)
      s ≤
    n
Tactic: by_cases hs : s = 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
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)
s : ℕ
hs' : p ^ s ≤ m
hs : ¬s = 0
this :
  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)
      s =
    (↑((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 ^ s) hs')) :
      Associates N)
⊢ 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)
      s ≤
    n
have rw : (↑((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 ^ s) hs')) :
    Associates N) ≤
  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 : 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)
s : ℕ
hs' : p ^ s ≤ m
hs : ¬s = 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)
      s ≤
    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)
    s =
  (↑((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 ^ s) hs')) :
    Associates N)
Tactic: suffices (d ⟨p, dvd_of_mem_normalizedFactors hp⟩ : Associates N) ^ s =
      (d ⟨p ^ s, hs'⟩) by
    rw [this]
    apply Subtype.prop (d ⟨p ^ s, hs'⟩)
-- 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)
s : ℕ
hs' : p ^ s ≤ m
hs : ¬s = 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)
      s =
    (↑((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 ^ s) hs')) :
      Associates N)
have neg : ∀ (c₁ : Fin (s + 1) → Associates M) (hp : c₁ 1 ∈ normalizedFactors m) (hs' : c₁ 1 ^ s ≤ m) (hc₁' : StrictMono c₁)
  (hc₁'' : ∀ {r : Associates M}, r ≤ c₁ 1 ^ s ↔ ∃ (i : Fin (s + 1)), r = c₁ i),
  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) (c₁ 1) sorry)) :
        Associates N)
      s =
    (↑((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) (c₁ 1 ^ s) hs')) :
      Associates N)
Tactic: obtain ⟨c₁, rfl, hc₁', hc₁''⟩ := exists_chain_of_prime_pow hs (prime_of_normalized_factor p hp)
Cleared hypotheses: p, hp, hs'
-- 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 : Associates M
n : Associates N
hn : n ≠ 0
d : OrderIso.{u_1, u_2} (↑(Set.Iic m) : Type u_1) (↑(Set.Iic n) : Type u_2)
s : ℕ
hs : ¬s = 0
c₁ : Fin (s + 1) → Associates M
hp : c₁ 1 ∈ normalizedFactors m
hs' : c₁ 1 ^ s ≤ m
hc₁' : StrictMono c₁
hc₁'' : ∀ {r : Associates M}, r ≤ c₁ 1 ^ s ↔ ∃ (i : Fin (s + 1)), r = c₁ i
c₂ : Fin (s + 1) → Associates N :=
  fun (t : Fin (s + 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) (c₁ t) ⋯)) :
      Associates N)
⊢ 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) (c₁ 1) ⋯)) :
        Associates N)
      s =
    (↑((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) (c₁ 1 ^ s) hs')) :
      Associates N)
have c₂_def : ∀ (t : Fin (s + 1)),
  c₂ t =
    (↑((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) (c₁ t) sorry)) :
      Associates N)
Tactic: have c₂_def : ∀ t, c₂ t = d ⟨c₁ t, _⟩ := fun t => rfl
-- 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 : Associates M
n : Associates N
hn : n ≠ 0
d : OrderIso.{u_1, u_2} (↑(Set.Iic m) : Type u_1) (↑(Set.Iic n) : Type u_2)
s : ℕ
hs : ¬s = 0
c₁ : Fin (s + 1) → Associates M
hp : c₁ 1 ∈ normalizedFactors m
hs' : c₁ 1 ^ s ≤ m
hc₁' : StrictMono c₁
hc₁'' : ∀ {r : Associates M}, r ≤ c₁ 1 ^ s ↔ ∃ (i : Fin (s + 1)), r = c₁ i
c₂ : Fin (s + 1) → Associates N :=
  fun (t : Fin (s + 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) (c₁ t) ⋯)) :
      Associates N)
c₂_def :
  ∀ (t : Fin (s + 1)),
    c₂ t =
      (↑((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) (c₁ t) ⋯)) :
        Associates N)
⊢ c₂ 1 ^ s =
    (↑((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) (c₁ 1 ^ s) hs')) :
      Associates N)
have neg : ∀ (t u : Fin (s + 1)) (h : t < u), c₂ t < c₂ u
Tactic: refine (eq_pow_second_of_chain_of_has_chain hs (fun t u h => ?_)
    (@fun r => ⟨@fun hr => ?_, ?_⟩) ?_).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 : UniqueFactorizationMonoid N
inst_4 : UniqueFactorizationMonoid M
m : Associates M
n : Associates N
hn : n ≠ 0
d : OrderIso.{u_1, u_2} (↑(Set.Iic m) : Type u_1) (↑(Set.Iic n) : Type u_2)
s : ℕ
hs : ¬s = 0
c₁ : Fin (s + 1) → Associates M
hp : c₁ 1 ∈ normalizedFactors m
hs' : c₁ 1 ^ s ≤ m
hc₁' : StrictMono c₁
hc₁'' : ∀ {r : Associates M}, r ≤ c₁ 1 ^ s ↔ ∃ (i : Fin (s + 1)), r = c₁ i
c₂ : Fin (s + 1) → Associates N :=
  fun (t : Fin (s + 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) (c₁ t) ⋯)) :
      Associates N)
c₂_def :
  ∀ (t : Fin (s + 1)),
    c₂ t =
      (↑((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) (c₁ t) ⋯)) :
        Associates N)
neg : ∀ (t u : Fin (s + 1)) (h : t < u), c₂ t < c₂ u
⊢ c₂ 1 ^ s =
    (↑((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) (c₁ 1 ^ s) hs')) :
      Associates N)
have neg₁ : ∀ (r : Associates N)
  (hr :
    r ≤
      (↑((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) (c₁ 1 ^ s) hs')) :
        Associates N)),
  ∃ (i : Fin (s + 1)), r = c₂ i
Tactic: refine (eq_pow_second_of_chain_of_has_chain hs (fun t u h => ?_)
    (@fun r => ⟨@fun hr => ?_, ?_⟩) ?_).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 : UniqueFactorizationMonoid N
inst_4 : UniqueFactorizationMonoid M
m : Associates M
n : Associates N
hn : n ≠ 0
d : OrderIso.{u_1, u_2} (↑(Set.Iic m) : Type u_1) (↑(Set.Iic n) : Type u_2)
s : ℕ
hs : ¬s = 0
c₁ : Fin (s + 1) → Associates M
hp : c₁ 1 ∈ normalizedFactors m
hs' : c₁ 1 ^ s ≤ m
hc₁' : StrictMono c₁
hc₁'' : ∀ {r : Associates M}, r ≤ c₁ 1 ^ s ↔ ∃ (i : Fin (s + 1)), r = c₁ i
c₂ : Fin (s + 1) → Associates N :=
  fun (t : Fin (s + 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) (c₁ t) ⋯)) :
      Associates N)
c₂_def :
  ∀ (t : Fin (s + 1)),
    c₂ t =
      (↑((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) (c₁ t) ⋯)) :
        Associates N)
neg : ∀ (t u : Fin (s + 1)) (h : t < u), c₂ t < c₂ u
neg₁ :
  ∀ (r : Associates N)
    (hr :
      r ≤
        (↑((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) (c₁ 1 ^ s) hs')) :
          Associates N)),
    ∃ (i : Fin (s + 1)), r = c₂ i
⊢ c₂ 1 ^ s =
    (↑((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) (c₁ 1 ^ s) hs')) :
      Associates N)
have neg₂ : ∀ (r : Associates N),
  (∃ (i : Fin (s + 1)), r = c₂ i) →
    r ≤
      (↑((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) (c₁ 1 ^ s) hs')) :
        Associates N)
Tactic: refine (eq_pow_second_of_chain_of_has_chain hs (fun t u h => ?_)
    (@fun r => ⟨@fun hr => ?_, ?_⟩) ?_).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 : UniqueFactorizationMonoid N
inst_4 : UniqueFactorizationMonoid M
m : Associates M
n : Associates N
hn : n ≠ 0
d : OrderIso.{u_1, u_2} (↑(Set.Iic m) : Type u_1) (↑(Set.Iic n) : Type u_2)
s : ℕ
hs : ¬s = 0
c₁ : Fin (s + 1) → Associates M
hp : c₁ 1 ∈ normalizedFactors m
hs' : c₁ 1 ^ s ≤ m
hc₁' : StrictMono c₁
hc₁'' : ∀ {r : Associates M}, r ≤ c₁ 1 ^ s ↔ ∃ (i : Fin (s + 1)), r = c₁ i
c₂ : Fin (s + 1) → Associates N :=
  fun (t : Fin (s + 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) (c₁ t) ⋯)) :
      Associates N)
c₂_def :
  ∀ (t : Fin (s + 1)),
    c₂ t =
      (↑((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) (c₁ t) ⋯)) :
        Associates N)
neg : ∀ (t u : Fin (s + 1)) (h : t < u), c₂ t < c₂ u
neg₁ :
  ∀ (r : Associates N)
    (hr :
      r ≤
        (↑((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) (c₁ 1 ^ s) hs')) :
          Associates N)),
    ∃ (i : Fin (s + 1)), r = c₂ i
neg₂ :
  ∀ (r : Associates N),
    (∃ (i : Fin (s + 1)), r = c₂ i) →
      r ≤
        (↑((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) (c₁ 1 ^ s) hs')) :
          Associates N)
⊢ c₂ 1 ^ s =
    (↑((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) (c₁ 1 ^ s) hs')) :
      Associates N)
have neg₃ : Ne.{u_2 + 1} (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) (c₁ 1 ^ s) hs')) :
    Associates N)
  0
Tactic: refine (eq_pow_second_of_chain_of_has_chain hs (fun t u h => ?_)
    (@fun r => ⟨@fun hr => ?_, ?_⟩) ?_).symm
-- 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 : Associates M
n : Associates N
hn : n ≠ 0
d : OrderIso.{u_1, u_2} (↑(Set.Iic m) : Type u_1) (↑(Set.Iic n) : Type u_2)
s : ℕ
hs : ¬s = 0
c₁ : Fin (s + 1) → Associates M
hp : c₁ 1 ∈ normalizedFactors m
hs' : c₁ 1 ^ s ≤ m
hc₁' : StrictMono c₁
hc₁'' : ∀ {r : Associates M}, r ≤ c₁ 1 ^ s ↔ ∃ (i : Fin (s + 1)), r = c₁ i
c₂ : Fin (s + 1) → Associates N :=
  fun (t : Fin (s + 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) (c₁ t) ⋯)) :
      Associates N)
c₂_def :
  ∀ (t : Fin (s + 1)),
    c₂ t =
      (↑((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) (c₁ t) ⋯)) :
        Associates N)
r : Associates N
hr :
  r ≤
    (↑((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) (c₁ 1 ^ s) hs')) :
      Associates N)
⊢ ∃ (i : Fin (s + 1)), r = c₂ i
have this : r ≤ n
Tactic: have : r ≤ n := hr.trans (d ⟨c₁ 1 ^ s, _⟩).2
-- 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 : Associates M
n : Associates N
hn : n ≠ 0
d : OrderIso.{u_1, u_2} (↑(Set.Iic m) : Type u_1) (↑(Set.Iic n) : Type u_2)
s : ℕ
hs : ¬s = 0
c₁ : Fin (s + 1) → Associates M
hp : c₁ 1 ∈ normalizedFactors m
hs' : c₁ 1 ^ s ≤ m
hc₁' : StrictMono c₁
hc₁'' : ∀ {r : Associates M}, r ≤ c₁ 1 ^ s ↔ ∃ (i : Fin (s + 1)), r = c₁ i
c₂ : Fin (s + 1) → Associates N :=
  fun (t : Fin (s + 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) (c₁ t) ⋯)) :
      Associates N)
c₂_def :
  ∀ (t : Fin (s + 1)),
    c₂ t =
      (↑((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) (c₁ t) ⋯)) :
        Associates N)
r : Associates N
hr :
  r ≤
    (↑((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) (c₁ 1 ^ s) hs')) :
      Associates N)
this_1 : r ≤ n
this :
  LE.le.{u_1} (α := (↑(Set.Iic m) : Type u_1))
    ((d.symm : (a : (↑(Set.Iic n) : Type u_2)) → (↑(Set.Iic m) : Type u_1))
        (Subtype.mk (p := fun (x : Associates N) ↦ x ∈ Set.Iic n) r this_1) :
      (↑(Set.Iic m) : Type u_1))
    (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) (c₁ 1 ^ s) hs' : { x : Associates M // x ∈ Set.Iic m })
⊢ ∃ (i : Fin (s + 1)), r = c₂ i
have obtain : ∀ (i : Fin (s + 1))
  (hi :
    (↑((d.symm : (a : (↑(Set.Iic n) : Type u_2)) → (↑(Set.Iic m) : Type u_1))
            (Subtype.mk (p := fun (x : Associates N) ↦ x ∈ Set.Iic n) r this_1)) :
        Associates M) =
      c₁ i),
  ∃ (i : Fin (s + 1)), r = c₂ i
Tactic: obtain ⟨i, hi⟩ := hc₁''.1 this