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 : UniqueFactorizationMonoid M
q : Associates M
n : ℕ
hn : n ≠ 0
c : Fin (n + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1)), r = c i
hq : q ≠ 0
i : Fin (n + 1)
hi' : q = c 1 ^ (↑i : ℕ)
j : Fin (n + 1)
hr : c j ∈ Finset.image c Finset.univ
this : c j ≤ c 1 ^ (↑i : ℕ)
h : ∃ i_1 ≤ (↑i : ℕ), Associated (c j) (c 1 ^ i_1)
⊢ Membership.mem (γ := Finset (Associates M)) (Finset.image (fun (m : Fin ((↑i : ℕ) + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ)
    (c j)
have h₃ : ∀ (u : ℕ) (hu : u ≤ (↑i : ℕ)) (hu' : Associated (c j) (c 1 ^ u)),
  Membership.mem (γ := Finset (Associates M)) (Finset.image (fun (m : Fin ((↑i : ℕ) + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ)
    (c j)
Tactic: rcases h with ⟨u, hu, hu'⟩
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
q : Associates M
n : ℕ
hn : n ≠ 0
c : Fin (n + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1)), r = c i
hq : q ≠ 0
i : Fin (n + 1)
hi' : q = c 1 ^ (↑i : ℕ)
j : Fin (n + 1)
hr : c j ∈ Finset.image c Finset.univ
this : c j ≤ c 1 ^ (↑i : ℕ)
u : ℕ
hu : u ≤ (↑i : ℕ)
hu' : Associated (c j) (c 1 ^ u)
⊢ Membership.mem (γ := Finset (Associates M)) (Finset.image (fun (m : Fin ((↑i : ℕ) + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ)
    (c j)
have h : c 1 ^ (↑(⟨u, sorry⟩ : Fin ((↑i : ℕ) + 1)) : ℕ) = c j
Tactic: refine Finset.mem_image.mpr ⟨⟨u, Nat.lt_succ_of_le hu⟩, Finset.mem_univ _, ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
q : Associates M
n : ℕ
hn : n ≠ 0
c : Fin (n + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1)), r = c i
hq : q ≠ 0
i : Fin (n + 1)
hi' : q = c 1 ^ (↑i : ℕ)
j : Fin (n + 1)
hr : c j ∈ Finset.image c Finset.univ
this : c j ≤ c 1 ^ (↑i : ℕ)
u : ℕ
hu : u ≤ (↑i : ℕ)
hu' : Associated (c j) (c 1 ^ u)
⊢ c 1 ^ (↑(⟨u, ⋯⟩ : Fin ((↑i : ℕ) + 1)) : ℕ) = c j
have h : ∀ (hu' : c j = c 1 ^ u), c 1 ^ (↑(⟨u, sorry⟩ : Fin ((↑i : ℕ) + 1)) : ℕ) = c j
Tactic: rwa [associated_iff_eq, eq_comm] at hu'
Cleared hypotheses: hu'
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
q : Associates M
n : ℕ
hn : n ≠ 0
c : Fin (n + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1)), r = c i
hq : q ≠ 0
i : Fin (n + 1)
hi' : q = c 1 ^ (↑i : ℕ)
j : Fin (n + 1)
hr : c j ∈ Finset.image c Finset.univ
this : c j ≤ c 1 ^ (↑i : ℕ)
u : ℕ
hu : u ≤ (↑i : ℕ)
hu' : Associated (c j) (c 1 ^ u)
h : ∀ (hu' : c j = c 1 ^ u), c 1 ^ (↑(⟨u, ⋯⟩ : Fin ((↑i : ℕ) + 1)) : ℕ) = c j
⊢ c 1 ^ (↑(⟨u, ⋯⟩ : Fin ((↑i : ℕ) + 1)) : ℕ) = c j
have h₃ : ∀ (hu' : c 1 ^ u = c j), c 1 ^ (↑(⟨u, sorry⟩ : Fin ((↑i : ℕ) + 1)) : ℕ) = c j
Tactic: rwa [associated_iff_eq, eq_comm] at hu'
Cleared hypotheses: hu'
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
q : Associates M
n : ℕ
hn : n ≠ 0
c : Fin (n + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1)), r = c i
hq : q ≠ 0
i : Fin (n + 1)
hi' : q = c 1 ^ (↑i : ℕ)
j : Fin (n + 1)
hr : c j ∈ Finset.image c Finset.univ
this : c j ≤ c 1 ^ (↑i : ℕ)
⊢ Prime (c 1)
have h : Irreducible (c 1)
Tactic: rw [← irreducible_iff_prime]
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
q : Associates M
n : ℕ
hn : n ≠ 0
c : Fin (n + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1)), r = c i
hq : q ≠ 0
⊢ q = c 1 ^ n
have obtain : ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)), q = c 1 ^ n
Tactic: classical
    obtain ⟨i, hi'⟩ := element_of_chain_eq_pow_second_of_chain hn h₁ (@fun r => h₂) (dvd_refl q) hq
    convert! hi'
    refine (Nat.lt_succ_iff.1 i.prop).antisymm' (Nat.le_of_succ_le_succ ?_)
    calc
      n + 1 = (Finset.univ : Finset (Fin (n + 1))).card := (Finset.card_fin _).symm
      _ = (Finset.univ.image c).card := (Finset.card_image_iff.mpr h₁.injective.injOn).symm
      _ ≤ (Finset.univ.image fun m : Fin (i + 1) => c 1 ^ (m : ℕ)).card :=
        (Finset.card_le_card ?_)
      _ ≤ (Finset.univ : Finset (Fin (i + 1))).card := Finset.card_image_le
      _ = i + 1 := Finset.card_fin _
    intro r hr
    obtain ⟨j, -, rfl⟩ := Finset.mem_image.1 hr
    have := h₂.2 ⟨j, rfl⟩
    rw [hi'] at this
    have h := (dvd_prime_pow (show Prime (c 1) from ?_) i).1 this
    · rcases h with ⟨u, hu, hu'⟩
      refine Finset.mem_image.mpr ⟨⟨u, Nat.lt_succ_of_le hu⟩, Finset.mem_univ _, ?_⟩
      rwa [associated_iff_eq, eq_comm] at hu'
    · rw [← irreducible_iff_prime]
      exact second_of_chain_is_irreducible hn h₁ (@h₂) hq
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
q : Associates M
n : ℕ
hn : n ≠ 0
c : Fin (n + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1)), r = c i
hq : q ≠ 0
obtain : ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)), q = c 1 ^ n
⊢ q = c 1 ^ n
have h : ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)), n = (↑i : ℕ)
Tactic: classical
    obtain ⟨i, hi'⟩ := element_of_chain_eq_pow_second_of_chain hn h₁ (@fun r => h₂) (dvd_refl q) hq
    convert! hi'
    refine (Nat.lt_succ_iff.1 i.prop).antisymm' (Nat.le_of_succ_le_succ ?_)
    calc
      n + 1 = (Finset.univ : Finset (Fin (n + 1))).card := (Finset.card_fin _).symm
      _ = (Finset.univ.image c).card := (Finset.card_image_iff.mpr h₁.injective.injOn).symm
      _ ≤ (Finset.univ.image fun m : Fin (i + 1) => c 1 ^ (m : ℕ)).card :=
        (Finset.card_le_card ?_)
      _ ≤ (Finset.univ : Finset (Fin (i + 1))).card := Finset.card_image_le
      _ = i + 1 := Finset.card_fin _
    intro r hr
    obtain ⟨j, -, rfl⟩ := Finset.mem_image.1 hr
    have := h₂.2 ⟨j, rfl⟩
    rw [hi'] at this
    have h := (dvd_prime_pow (show Prime (c 1) from ?_) i).1 this
    · rcases h with ⟨u, hu, hu'⟩
      refine Finset.mem_image.mpr ⟨⟨u, Nat.lt_succ_of_le hu⟩, Finset.mem_univ _, ?_⟩
      rwa [associated_iff_eq, eq_comm] at hu'
    · rw [← irreducible_iff_prime]
      exact second_of_chain_is_irreducible hn h₁ (@h₂) hq
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
q : Associates M
n : ℕ
hn : n ≠ 0
c : Fin (n + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1)), r = c i
hq : q ≠ 0
obtain : ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)), q = c 1 ^ n
h : ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)), n = (↑i : ℕ)
⊢ q = c 1 ^ n
have h₃ : ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)), n.succ ≤ (↑i : ℕ).succ
Tactic: classical
    obtain ⟨i, hi'⟩ := element_of_chain_eq_pow_second_of_chain hn h₁ (@fun r => h₂) (dvd_refl q) hq
    convert! hi'
    refine (Nat.lt_succ_iff.1 i.prop).antisymm' (Nat.le_of_succ_le_succ ?_)
    calc
      n + 1 = (Finset.univ : Finset (Fin (n + 1))).card := (Finset.card_fin _).symm
      _ = (Finset.univ.image c).card := (Finset.card_image_iff.mpr h₁.injective.injOn).symm
      _ ≤ (Finset.univ.image fun m : Fin (i + 1) => c 1 ^ (m : ℕ)).card :=
        (Finset.card_le_card ?_)
      _ ≤ (Finset.univ : Finset (Fin (i + 1))).card := Finset.card_image_le
      _ = i + 1 := Finset.card_fin _
    intro r hr
    obtain ⟨j, -, rfl⟩ := Finset.mem_image.1 hr
    have := h₂.2 ⟨j, rfl⟩
    rw [hi'] at this
    have h := (dvd_prime_pow (show Prime (c 1) from ?_) i).1 this
    · rcases h with ⟨u, hu, hu'⟩
      refine Finset.mem_image.mpr ⟨⟨u, Nat.lt_succ_of_le hu⟩, Finset.mem_univ _, ?_⟩
      rwa [associated_iff_eq, eq_comm] at hu'
    · rw [← irreducible_iff_prime]
      exact second_of_chain_is_irreducible hn h₁ (@h₂) hq
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
q : Associates M
n : ℕ
hn : n ≠ 0
c : Fin (n + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1)), r = c i
hq : q ≠ 0
obtain : ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)), q = c 1 ^ n
h : ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)), n = (↑i : ℕ)
h₃ : ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)), n.succ ≤ (↑i : ℕ).succ
⊢ q = c 1 ^ n
have h₄ : ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)),
  Finset.image c Finset.univ ⊆ Finset.image (fun (m : Fin ((↑i : ℕ) + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ
Tactic: classical
    obtain ⟨i, hi'⟩ := element_of_chain_eq_pow_second_of_chain hn h₁ (@fun r => h₂) (dvd_refl q) hq
    convert! hi'
    refine (Nat.lt_succ_iff.1 i.prop).antisymm' (Nat.le_of_succ_le_succ ?_)
    calc
      n + 1 = (Finset.univ : Finset (Fin (n + 1))).card := (Finset.card_fin _).symm
      _ = (Finset.univ.image c).card := (Finset.card_image_iff.mpr h₁.injective.injOn).symm
      _ ≤ (Finset.univ.image fun m : Fin (i + 1) => c 1 ^ (m : ℕ)).card :=
        (Finset.card_le_card ?_)
      _ ≤ (Finset.univ : Finset (Fin (i + 1))).card := Finset.card_image_le
      _ = i + 1 := Finset.card_fin _
    intro r hr
    obtain ⟨j, -, rfl⟩ := Finset.mem_image.1 hr
    have := h₂.2 ⟨j, rfl⟩
    rw [hi'] at this
    have h := (dvd_prime_pow (show Prime (c 1) from ?_) i).1 this
    · rcases h with ⟨u, hu, hu'⟩
      refine Finset.mem_image.mpr ⟨⟨u, Nat.lt_succ_of_le hu⟩, Finset.mem_univ _, ?_⟩
      rwa [associated_iff_eq, eq_comm] at hu'
    · rw [← irreducible_iff_prime]
      exact second_of_chain_is_irreducible hn h₁ (@h₂) hq
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
q : Associates M
n : ℕ
hn : n ≠ 0
c : Fin (n + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1)), r = c i
hq : q ≠ 0
obtain : ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)), q = c 1 ^ n
h : ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)), n = (↑i : ℕ)
h₃ : ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)), n.succ ≤ (↑i : ℕ).succ
h₄ :
  ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)),
    Finset.image c Finset.univ ⊆ Finset.image (fun (m : Fin ((↑i : ℕ) + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ
⊢ q = c 1 ^ n
have h₅ : ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)) ⦃r : Associates M⦄ (hr : r ∈ Finset.image c Finset.univ),
  Membership.mem (γ := Finset (Associates M)) (Finset.image (fun (m : Fin ((↑i : ℕ) + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ)
    r
Tactic: classical
    obtain ⟨i, hi'⟩ := element_of_chain_eq_pow_second_of_chain hn h₁ (@fun r => h₂) (dvd_refl q) hq
    convert! hi'
    refine (Nat.lt_succ_iff.1 i.prop).antisymm' (Nat.le_of_succ_le_succ ?_)
    calc
      n + 1 = (Finset.univ : Finset (Fin (n + 1))).card := (Finset.card_fin _).symm
      _ = (Finset.univ.image c).card := (Finset.card_image_iff.mpr h₁.injective.injOn).symm
      _ ≤ (Finset.univ.image fun m : Fin (i + 1) => c 1 ^ (m : ℕ)).card :=
        (Finset.card_le_card ?_)
      _ ≤ (Finset.univ : Finset (Fin (i + 1))).card := Finset.card_image_le
      _ = i + 1 := Finset.card_fin _
    intro r hr
    obtain ⟨j, -, rfl⟩ := Finset.mem_image.1 hr
    have := h₂.2 ⟨j, rfl⟩
    rw [hi'] at this
    have h := (dvd_prime_pow (show Prime (c 1) from ?_) i).1 this
    · rcases h with ⟨u, hu, hu'⟩
      refine Finset.mem_image.mpr ⟨⟨u, Nat.lt_succ_of_le hu⟩, Finset.mem_univ _, ?_⟩
      rwa [associated_iff_eq, eq_comm] at hu'
    · rw [← irreducible_iff_prime]
      exact second_of_chain_is_irreducible hn h₁ (@h₂) hq
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
q : Associates M
n : ℕ
hn : n ≠ 0
c : Fin (n + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1)), r = c i
hq : q ≠ 0
obtain : ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)), q = c 1 ^ n
h : ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)), n = (↑i : ℕ)
h₃ : ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)), n.succ ≤ (↑i : ℕ).succ
h₄ :
  ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)),
    Finset.image c Finset.univ ⊆ Finset.image (fun (m : Fin ((↑i : ℕ) + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ
h₅ :
  ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)) ⦃r : Associates M⦄ (hr : r ∈ Finset.image c Finset.univ),
    Membership.mem (γ := Finset (Associates M))
      (Finset.image (fun (m : Fin ((↑i : ℕ) + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ) r
⊢ q = c 1 ^ n
have h₆ : ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)) (j : Fin (n + 1)) (hr : c j ∈ Finset.image c Finset.univ),
  Membership.mem (γ := Finset (Associates M)) (Finset.image (fun (m : Fin ((↑i : ℕ) + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ)
    (c j)
Tactic: classical
    obtain ⟨i, hi'⟩ := element_of_chain_eq_pow_second_of_chain hn h₁ (@fun r => h₂) (dvd_refl q) hq
    convert! hi'
    refine (Nat.lt_succ_iff.1 i.prop).antisymm' (Nat.le_of_succ_le_succ ?_)
    calc
      n + 1 = (Finset.univ : Finset (Fin (n + 1))).card := (Finset.card_fin _).symm
      _ = (Finset.univ.image c).card := (Finset.card_image_iff.mpr h₁.injective.injOn).symm
      _ ≤ (Finset.univ.image fun m : Fin (i + 1) => c 1 ^ (m : ℕ)).card :=
        (Finset.card_le_card ?_)
      _ ≤ (Finset.univ : Finset (Fin (i + 1))).card := Finset.card_image_le
      _ = i + 1 := Finset.card_fin _
    intro r hr
    obtain ⟨j, -, rfl⟩ := Finset.mem_image.1 hr
    have := h₂.2 ⟨j, rfl⟩
    rw [hi'] at this
    have h := (dvd_prime_pow (show Prime (c 1) from ?_) i).1 this
    · rcases h with ⟨u, hu, hu'⟩
      refine Finset.mem_image.mpr ⟨⟨u, Nat.lt_succ_of_le hu⟩, Finset.mem_univ _, ?_⟩
      rwa [associated_iff_eq, eq_comm] at hu'
    · rw [← irreducible_iff_prime]
      exact second_of_chain_is_irreducible hn h₁ (@h₂) hq
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
q : Associates M
n : ℕ
hn : n ≠ 0
c : Fin (n + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1)), r = c i
hq : q ≠ 0
obtain : ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)), q = c 1 ^ n
h : ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)), n = (↑i : ℕ)
h₃ : ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)), n.succ ≤ (↑i : ℕ).succ
h₄ :
  ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)),
    Finset.image c Finset.univ ⊆ Finset.image (fun (m : Fin ((↑i : ℕ) + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ
h₅ :
  ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)) ⦃r : Associates M⦄ (hr : r ∈ Finset.image c Finset.univ),
    Membership.mem (γ := Finset (Associates M))
      (Finset.image (fun (m : Fin ((↑i : ℕ) + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ) r
h₆ :
  ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)) (j : Fin (n + 1)) (hr : c j ∈ Finset.image c Finset.univ),
    Membership.mem (γ := Finset (Associates M))
      (Finset.image (fun (m : Fin ((↑i : ℕ) + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ) (c j)
⊢ q = c 1 ^ n
have h₇ : ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)) (j : Fin (n + 1)) (hr : c j ∈ Finset.image c Finset.univ)
  (this : c j ≤ q),
  Membership.mem (γ := Finset (Associates M)) (Finset.image (fun (m : Fin ((↑i : ℕ) + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ)
    (c j)
Tactic: classical
    obtain ⟨i, hi'⟩ := element_of_chain_eq_pow_second_of_chain hn h₁ (@fun r => h₂) (dvd_refl q) hq
    convert! hi'
    refine (Nat.lt_succ_iff.1 i.prop).antisymm' (Nat.le_of_succ_le_succ ?_)
    calc
      n + 1 = (Finset.univ : Finset (Fin (n + 1))).card := (Finset.card_fin _).symm
      _ = (Finset.univ.image c).card := (Finset.card_image_iff.mpr h₁.injective.injOn).symm
      _ ≤ (Finset.univ.image fun m : Fin (i + 1) => c 1 ^ (m : ℕ)).card :=
        (Finset.card_le_card ?_)
      _ ≤ (Finset.univ : Finset (Fin (i + 1))).card := Finset.card_image_le
      _ = i + 1 := Finset.card_fin _
    intro r hr
    obtain ⟨j, -, rfl⟩ := Finset.mem_image.1 hr
    have := h₂.2 ⟨j, rfl⟩
    rw [hi'] at this
    have h := (dvd_prime_pow (show Prime (c 1) from ?_) i).1 this
    · rcases h with ⟨u, hu, hu'⟩
      refine Finset.mem_image.mpr ⟨⟨u, Nat.lt_succ_of_le hu⟩, Finset.mem_univ _, ?_⟩
      rwa [associated_iff_eq, eq_comm] at hu'
    · rw [← irreducible_iff_prime]
      exact second_of_chain_is_irreducible hn h₁ (@h₂) hq
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
q : Associates M
n : ℕ
hn : n ≠ 0
c : Fin (n + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1)), r = c i
hq : q ≠ 0
obtain : ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)), q = c 1 ^ n
h : ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)), n = (↑i : ℕ)
h₃ : ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)), n.succ ≤ (↑i : ℕ).succ
h₄ :
  ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)),
    Finset.image c Finset.univ ⊆ Finset.image (fun (m : Fin ((↑i : ℕ) + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ
h₅ :
  ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)) ⦃r : Associates M⦄ (hr : r ∈ Finset.image c Finset.univ),
    Membership.mem (γ := Finset (Associates M))
      (Finset.image (fun (m : Fin ((↑i : ℕ) + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ) r
h₆ :
  ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)) (j : Fin (n + 1)) (hr : c j ∈ Finset.image c Finset.univ),
    Membership.mem (γ := Finset (Associates M))
      (Finset.image (fun (m : Fin ((↑i : ℕ) + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ) (c j)
h₇ :
  ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)) (j : Fin (n + 1)) (hr : c j ∈ Finset.image c Finset.univ)
    (this : c j ≤ q),
    Membership.mem (γ := Finset (Associates M))
      (Finset.image (fun (m : Fin ((↑i : ℕ) + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ) (c j)
⊢ q = c 1 ^ n
have h₈ : ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)) (j : Fin (n + 1)) (hr : c j ∈ Finset.image c Finset.univ)
  (this : c j ≤ c 1 ^ (↑i : ℕ)),
  Membership.mem (γ := Finset (Associates M)) (Finset.image (fun (m : Fin ((↑i : ℕ) + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ)
    (c j)
Tactic: classical
    obtain ⟨i, hi'⟩ := element_of_chain_eq_pow_second_of_chain hn h₁ (@fun r => h₂) (dvd_refl q) hq
    convert! hi'
    refine (Nat.lt_succ_iff.1 i.prop).antisymm' (Nat.le_of_succ_le_succ ?_)
    calc
      n + 1 = (Finset.univ : Finset (Fin (n + 1))).card := (Finset.card_fin _).symm
      _ = (Finset.univ.image c).card := (Finset.card_image_iff.mpr h₁.injective.injOn).symm
      _ ≤ (Finset.univ.image fun m : Fin (i + 1) => c 1 ^ (m : ℕ)).card :=
        (Finset.card_le_card ?_)
      _ ≤ (Finset.univ : Finset (Fin (i + 1))).card := Finset.card_image_le
      _ = i + 1 := Finset.card_fin _
    intro r hr
    obtain ⟨j, -, rfl⟩ := Finset.mem_image.1 hr
    have := h₂.2 ⟨j, rfl⟩
    rw [hi'] at this
    have h := (dvd_prime_pow (show Prime (c 1) from ?_) i).1 this
    · rcases h with ⟨u, hu, hu'⟩
      refine Finset.mem_image.mpr ⟨⟨u, Nat.lt_succ_of_le hu⟩, Finset.mem_univ _, ?_⟩
      rwa [associated_iff_eq, eq_comm] at hu'
    · rw [← irreducible_iff_prime]
      exact second_of_chain_is_irreducible hn h₁ (@h₂) hq
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
q : Associates M
n : ℕ
hn : n ≠ 0
c : Fin (n + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1)), r = c i
hq : q ≠ 0
obtain : ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)), q = c 1 ^ n
h : ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)), n = (↑i : ℕ)
h₃ : ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)), n.succ ≤ (↑i : ℕ).succ
h₄ :
  ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)),
    Finset.image c Finset.univ ⊆ Finset.image (fun (m : Fin ((↑i : ℕ) + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ
h₅ :
  ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)) ⦃r : Associates M⦄ (hr : r ∈ Finset.image c Finset.univ),
    Membership.mem (γ := Finset (Associates M))
      (Finset.image (fun (m : Fin ((↑i : ℕ) + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ) r
h₆ :
  ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)) (j : Fin (n + 1)) (hr : c j ∈ Finset.image c Finset.univ),
    Membership.mem (γ := Finset (Associates M))
      (Finset.image (fun (m : Fin ((↑i : ℕ) + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ) (c j)
h₇ :
  ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)) (j : Fin (n + 1)) (hr : c j ∈ Finset.image c Finset.univ)
    (this : c j ≤ q),
    Membership.mem (γ := Finset (Associates M))
      (Finset.image (fun (m : Fin ((↑i : ℕ) + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ) (c j)
h₈ :
  ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)) (j : Fin (n + 1)) (hr : c j ∈ Finset.image c Finset.univ)
    (this : c j ≤ c 1 ^ (↑i : ℕ)),
    Membership.mem (γ := Finset (Associates M))
      (Finset.image (fun (m : Fin ((↑i : ℕ) + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ) (c j)
⊢ q = c 1 ^ n
have h₉ : ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)) (j : Fin (n + 1)) (hr : c j ∈ Finset.image c Finset.univ)
  (this : c j ≤ c 1 ^ (↑i : ℕ)) (h : ∃ i_1 ≤ (↑i : ℕ), Associated (c j) (c 1 ^ i_1)),
  Membership.mem (γ := Finset (Associates M)) (Finset.image (fun (m : Fin ((↑i : ℕ) + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ)
    (c j)
Tactic: classical
    obtain ⟨i, hi'⟩ := element_of_chain_eq_pow_second_of_chain hn h₁ (@fun r => h₂) (dvd_refl q) hq
    convert! hi'
    refine (Nat.lt_succ_iff.1 i.prop).antisymm' (Nat.le_of_succ_le_succ ?_)
    calc
      n + 1 = (Finset.univ : Finset (Fin (n + 1))).card := (Finset.card_fin _).symm
      _ = (Finset.univ.image c).card := (Finset.card_image_iff.mpr h₁.injective.injOn).symm
      _ ≤ (Finset.univ.image fun m : Fin (i + 1) => c 1 ^ (m : ℕ)).card :=
        (Finset.card_le_card ?_)
      _ ≤ (Finset.univ : Finset (Fin (i + 1))).card := Finset.card_image_le
      _ = i + 1 := Finset.card_fin _
    intro r hr
    obtain ⟨j, -, rfl⟩ := Finset.mem_image.1 hr
    have := h₂.2 ⟨j, rfl⟩
    rw [hi'] at this
    have h := (dvd_prime_pow (show Prime (c 1) from ?_) i).1 this
    · rcases h with ⟨u, hu, hu'⟩
      refine Finset.mem_image.mpr ⟨⟨u, Nat.lt_succ_of_le hu⟩, Finset.mem_univ _, ?_⟩
      rwa [associated_iff_eq, eq_comm] at hu'
    · rw [← irreducible_iff_prime]
      exact second_of_chain_is_irreducible hn h₁ (@h₂) 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 : IsCancelMulZero N
m : Associates M
n : Associates N
d : OrderIso.{u_1, u_2} { l : Associates M // l ≤ m } { l : Associates N // l ≤ n }
this_1 : 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) ⋯
⊢ Eq.{u_2 + 1} (α := Associates N)
    (↑((d : (a : { l : Associates M // l ≤ m }) → { l : Associates N // l ≤ n })
          (Subtype.mk (p := fun (l : Associates M) ↦ l ≤ m) 1 ⋯)) :
      Associates N)
    1
have simp : Eq.{u_2 + 1} (α := { x : Associates N // x ≤ n })
  ((d : (a : { l : Associates M // l ≤ m }) → { l : Associates N // l ≤ n }) ⊥) ⊥
Tactic: simp only [← Associates.bot_eq_one, Subtype.mk_bot, bot_le, Subtype.coe_eq_bot_iff]