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
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
zero :
  ∀ (hn : Ne (α := ℕ) 0 0) {c : Fin (HAdd.hAdd (α := ℕ) 0 1) → Associates M} (h₁ : StrictMono c)
    (h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (HAdd.hAdd (α := ℕ) 0 1)), r = c i), Irreducible (c 1)
⊢ Irreducible (c 1)
have succ : ∀ (n : ℕ) (hn : n + 1 ≠ 0) {c : Fin (n + 1 + 1) → Associates M} (h₁ : StrictMono c)
  (h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1 + 1)), r = c i), Irreducible (c 1)
Tactic: rcases n with - | n
Cleared hypotheses: n, hn, c, h₁, h₂
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
q : Associates M
hq : q ≠ 0
n : ℕ
hn : n + 1 ≠ 0
c : Fin (n + 1 + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1 + 1)), r = c i
⊢ Irreducible (c 1)
have succ : c 1 ≠ ⊥
Tactic: refine (Associates.isAtom_iff (ne_zero_of_dvd_ne_zero hq (h₂.2 ⟨1, rfl⟩))).mp ⟨?_, fun b hb => ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
q : Associates M
hq : q ≠ 0
n : ℕ
hn : n + 1 ≠ 0
c : Fin (n + 1 + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1 + 1)), r = c i
succ : c 1 ≠ ⊥
⊢ Irreducible (c 1)
have succ₁ : ∀ (b : Associates M) (hb : b < c 1), b = ⊥
Tactic: refine (Associates.isAtom_iff (ne_zero_of_dvd_ne_zero hq (h₂.2 ⟨1, rfl⟩))).mp ⟨?_, fun b hb => ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
q : Associates M
hq : q ≠ 0
n : ℕ
hn : n + 1 ≠ 0
c : Fin (n + 1 + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1 + 1)), r = c i
b : Associates M
hb : b < c 1
⊢ b = ⊥
have succ : ∀ (i : ℕ) (hi : i < n + 1 + 1) (hb : c (⟨i, hi⟩ : Fin (n + 1 + 1)) < c 1), c (⟨i, hi⟩ : Fin (n + 1 + 1)) = ⊥
Tactic: obtain ⟨⟨i, hi⟩, rfl⟩ := h₂.1 (hb.le.trans (h₂.2 ⟨1, rfl⟩))
Cleared hypotheses: b, hb
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
q : Associates M
hq : q ≠ 0
n : ℕ
hn : n + 1 ≠ 0
c : Fin (n + 1 + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1 + 1)), r = c i
i : ℕ
hi : i < n + 1 + 1
hb : c (⟨i, hi⟩ : Fin (n + 1 + 1)) < c 1
⊢ c (⟨i, hi⟩ : Fin (n + 1 + 1)) = ⊥
have succ : ∀ (hi : 0 < n + 1 + 1) (hb : c (⟨0, hi⟩ : Fin (n + 1 + 1)) < c 1), c (⟨0, hi⟩ : Fin (n + 1 + 1)) = ⊥
Tactic: cases i
Cleared hypotheses: i, hi, hb
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
q : Associates M
hq : q ≠ 0
n : ℕ
hn : n + 1 ≠ 0
c : Fin (n + 1 + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1 + 1)), r = c i
i : ℕ
hi : i < n + 1 + 1
hb : c (⟨i, hi⟩ : Fin (n + 1 + 1)) < c 1
succ : ∀ (hi : 0 < n + 1 + 1) (hb : c (⟨0, hi⟩ : Fin (n + 1 + 1)) < c 1), c (⟨0, hi⟩ : Fin (n + 1 + 1)) = ⊥
⊢ c (⟨i, hi⟩ : Fin (n + 1 + 1)) = ⊥
have succ₁ : ∀ (n_1 : ℕ) (hi : n_1 + 1 < n + 1 + 1) (hb : c (⟨n_1 + 1, hi⟩ : Fin (n + 1 + 1)) < c 1),
  c (⟨n_1 + 1, hi⟩ : Fin (n + 1 + 1)) = ⊥
Tactic: cases i
Cleared hypotheses: i, hi, hb
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
p q r : 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
hp : Prime p
hr : r ∣ q
hp' : p ∣ r
⊢ p = c 1
have zero : ∀ (hn : Ne (α := ℕ) 0 0) {c : Fin (HAdd.hAdd (α := ℕ) 0 1) → Associates M} (h₁ : StrictMono c)
  (h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (HAdd.hAdd (α := ℕ) 0 1)), r = c i), p = c 1
Tactic: rcases n with - | n
Cleared hypotheses: n, hn, c, h₁, h₂
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
p q r : 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
hp : Prime p
hr : r ∣ q
hp' : p ∣ r
zero :
  ∀ (hn : Ne (α := ℕ) 0 0) {c : Fin (HAdd.hAdd (α := ℕ) 0 1) → Associates M} (h₁ : StrictMono c)
    (h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (HAdd.hAdd (α := ℕ) 0 1)), r = c i), p = c 1
⊢ p = c 1
have succ : ∀ (n : ℕ) (hn : n + 1 ≠ 0) {c : Fin (n + 1 + 1) → Associates M} (h₁ : StrictMono c)
  (h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1 + 1)), r = c i), p = c 1
Tactic: rcases n with - | n
Cleared hypotheses: n, hn, c, h₁, h₂
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
p q r : Associates M
hp : Prime p
hr : r ∣ q
hp' : p ∣ r
n : ℕ
hn : n + 1 ≠ 0
c : Fin (n + 1 + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1 + 1)), r = c i
⊢ p = c 1
have succ : ∀ (i : Fin (n + 1 + 1)) (hp : Prime (c i)) (hp' : c i ∣ r), c i = c 1
Tactic: obtain ⟨i, rfl⟩ := h₂.1 (dvd_trans hp' hr)
Cleared hypotheses: p, hp, hp'
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
q r : Associates M
hr : r ∣ q
n : ℕ
hn : n + 1 ≠ 0
c : Fin (n + 1 + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1 + 1)), r = c i
i : Fin (n + 1 + 1)
hp : Prime (c i)
hp' : c i ∣ r
⊢ c i = c 1
have succ : 1 ≤ i
Tactic: refine congr_arg c (eq_of_le_of_not_lt' ?_ fun hi => ?_)
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
q r : Associates M
hr : r ∣ q
n : ℕ
hn : n + 1 ≠ 0
c : Fin (n + 1 + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1 + 1)), r = c i
i : Fin (n + 1 + 1)
hp : Prime (c i)
hp' : c i ∣ r
succ : 1 ≤ i
⊢ c i = c 1
have succ₁ : ∀ (hi : 1 < i), False
Tactic: refine congr_arg c (eq_of_le_of_not_lt' ?_ fun hi => ?_)
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
q r : Associates M
hr : r ∣ q
n : ℕ
hn : n + 1 ≠ 0
c : Fin (n + 1 + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1 + 1)), r = c i
i : Fin (n + 1 + 1)
hp : Prime (c i)
hp' : c i ∣ r
⊢ 1 ≤ i
have succ : i ≠ 0
Tactic: rw [Fin.le_iff_val_le_val, Fin.val_one, Nat.succ_le_iff, ← Fin.val_zero (n.succ + 1), ←
      Fin.lt_def, Fin.pos_iff_ne_zero]
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
q r : Associates M
hr : r ∣ q
n : ℕ
hn : n + 1 ≠ 0
c : Fin (n + 1 + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1 + 1)), r = c i
i : Fin (n + 1 + 1)
hp : Prime (c i)
hp' : c i ∣ r
hi : 1 < i
⊢ False
have succ : ∀ (hp : Prime (c 0)) (hp' : c 0 ∣ r) (hi : LT.lt (α := Fin (n + 1 + 1)) 1 0), False
Tactic: obtain rfl | ⟨j, rfl⟩ := i.eq_zero_or_eq_succ
Cleared hypotheses: i, hp, hp', hi
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
q r : Associates M
hr : r ∣ q
n : ℕ
hn : n + 1 ≠ 0
c : Fin (n + 1 + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1 + 1)), r = c i
i : Fin (n + 1 + 1)
hp : Prime (c i)
hp' : c i ∣ r
hi : 1 < i
succ : ∀ (hp : Prime (c 0)) (hp' : c 0 ∣ r) (hi : LT.lt (α := Fin (n + 1 + 1)) 1 0), False
⊢ False
have succ₁ : ∀ (j : Fin (n + 1)) (hp : Prime (c j.succ)) (hp' : c j.succ ∣ r) (hi : 1 < j.succ), False
Tactic: obtain rfl | ⟨j, rfl⟩ := i.eq_zero_or_eq_succ
Cleared hypotheses: i, hp, hp', hi
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
q r : Associates M
hr : r ∣ q
n : ℕ
hn : n + 1 ≠ 0
c : Fin (n + 1 + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1 + 1)), r = c i
j : Fin (n + 1)
hp : Prime (c j.succ)
hp' : c j.succ ∣ r
hi : 1 < j.succ
⊢ False
have succ : 0 < j.castSucc
Tactic: refine
    not_irreducible_of_not_unit_dvdNotUnit
      (DvdNotUnit.not_unit
        (Associates.dvdNotUnit_iff_lt.2 (h₁ (show (0 : Fin (n + 2)) < j.castSucc from ?_))))
      ?_ hp.irreducible
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
q r : Associates M
hr : r ∣ q
n : ℕ
hn : n + 1 ≠ 0
c : Fin (n + 1 + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1 + 1)), r = c i
j : Fin (n + 1)
hp : Prime (c j.succ)
hp' : c j.succ ∣ r
hi : 1 < j.succ
succ : 0 < j.castSucc
⊢ False
have succ₁ : DvdNotUnit (c j.castSucc) (c j.succ)
Tactic: refine
    not_irreducible_of_not_unit_dvdNotUnit
      (DvdNotUnit.not_unit
        (Associates.dvdNotUnit_iff_lt.2 (h₁ (show (0 : Fin (n + 2)) < j.castSucc from ?_))))
      ?_ hp.irreducible
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
q r : Associates M
hr : r ∣ q
n : ℕ
hn : n + 1 ≠ 0
c : Fin (n + 1 + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1 + 1)), r = c i
j : Fin (n + 1)
hp : Prime (c j.succ)
hp' : c j.succ ∣ r
hi : 1 < j.succ
⊢ DvdNotUnit (c j.castSucc) (c j.succ)
have succ : j.castSucc < j.succ
Tactic: refine Associates.dvdNotUnit_iff_lt.2 (h₁ ?_)
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
q : Associates M
n : ℕ
c : Fin (n + 1) → Associates M
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1)), r = c i
m : Finset (Associates M)
hm : ∀ r ∈ m, r ≤ q
r : Associates M
hr : r ≤ q
⊢ r ∈ Finset.image c Finset.univ
have obtain : ∀ (i : Fin (n + 1)) (hi : r = c i), r ∈ Finset.image c Finset.univ
Tactic: obtain ⟨i, hi⟩ := h₂.1 hr
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (hypothesis)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
q : Associates M
n : ℕ
c : Fin (n + 1) → Associates M
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1)), r = c i
m : Finset (Associates M)
hm : ∀ r ∈ m, r ≤ q
⊢ m.card ≤ n + 1
have mem_image : ∀ r ≤ q, r ∈ Finset.image c Finset.univ
Tactic: have mem_image : ∀ r : Associates M, r ≤ q → r ∈ Finset.univ.image c := by
      intro r hr
      obtain ⟨i, hi⟩ := h₂.1 hr
      exact Finset.mem_image.2 ⟨i, Finset.mem_univ _, hi.symm⟩
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
q : Associates M
n : ℕ
c : Fin (n + 1) → Associates M
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1)), r = c i
m : Finset (Associates M)
hm : ∀ r ∈ m, r ≤ q
mem_image : ∀ r ≤ q, r ∈ Finset.image c Finset.univ
⊢ m.card ≤ n + 1
have rw : m.card ≤ Finset.card (α := Fin (n + 1)) Finset.univ
Tactic: rw [← Finset.card_fin (n + 1)]
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
q : Associates M
n : ℕ
c : Fin (n + 1) → Associates M
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1)), r = c i
m : Finset (Associates M)
hm : ∀ r ∈ m, r ≤ q
⊢ m.card ≤ n + 1
have have : ∀ r ≤ q, r ∈ Finset.image c Finset.univ
Tactic: classical
    have mem_image : ∀ r : Associates M, r ≤ q → r ∈ Finset.univ.image c := by
      intro r hr
      obtain ⟨i, hi⟩ := h₂.1 hr
      exact Finset.mem_image.2 ⟨i, Finset.mem_univ _, hi.symm⟩
    rw [← Finset.card_fin (n + 1)]
    exact (Finset.card_le_card fun x hx => mem_image x <| hm x hx).trans Finset.card_image_le
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
q : Associates M
n : ℕ
c : Fin (n + 1) → Associates M
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1)), r = c i
m : Finset (Associates M)
hm : ∀ r ∈ m, r ≤ q
have : ∀ r ≤ q, r ∈ Finset.image c Finset.univ
⊢ m.card ≤ n + 1
have have₁ : ∀ (mem_image : ∀ r ≤ q, r ∈ Finset.image c Finset.univ), m.card ≤ n + 1
Tactic: classical
    have mem_image : ∀ r : Associates M, r ≤ q → r ∈ Finset.univ.image c := by
      intro r hr
      obtain ⟨i, hi⟩ := h₂.1 hr
      exact Finset.mem_image.2 ⟨i, Finset.mem_univ _, hi.symm⟩
    rw [← Finset.card_fin (n + 1)]
    exact (Finset.card_le_card fun x hx => mem_image x <| hm x hx).trans Finset.card_image_le
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
q : Associates M
n : ℕ
c : Fin (n + 1) → Associates M
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1)), r = c i
m : Finset (Associates M)
hm : ∀ r ∈ m, r ≤ q
have : ∀ r ≤ q, r ∈ Finset.image c Finset.univ
have₁ : ∀ (mem_image : ∀ r ≤ q, r ∈ Finset.image c Finset.univ), m.card ≤ n + 1
have₂ : ∀ r ≤ q, r ∈ Finset.image c Finset.univ
have₃ have₄ : ∀ (mem_image : ∀ r ≤ q, r ∈ Finset.image c Finset.univ), m.card ≤ n + 1
⊢ m.card ≤ n + 1
have have₅ : ∀ (mem_image : ∀ r ≤ q, r ∈ Finset.image c Finset.univ), m.card ≤ Finset.card (α := Fin (n + 1)) Finset.univ
Tactic: classical
    have mem_image : ∀ r : Associates M, r ≤ q → r ∈ Finset.univ.image c := by
      intro r hr
      obtain ⟨i, hi⟩ := h₂.1 hr
      exact Finset.mem_image.2 ⟨i, Finset.mem_univ _, hi.symm⟩
    rw [← Finset.card_fin (n + 1)]
    exact (Finset.card_le_card fun x hx => mem_image x <| hm x hx).trans Finset.card_image_le
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
q r : 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
hr : r ∣ q
hq : q ≠ 0
i : ℕ := (normalizedFactors r).card
⊢ normalizedFactors r = Multiset.replicate i (c 1)
have a : ∀ b ∈ normalizedFactors r, b = c 1
Tactic: apply Multiset.eq_replicate_of_mem
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (hypothesis)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
q r : 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
hr : r ∣ q
hq : q ≠ 0
i : ℕ := (normalizedFactors r).card
⊢ ∃ (i : Fin (n + 1)), r = c 1 ^ (↑i : ℕ)
have hi : normalizedFactors r = Multiset.replicate i (c 1)
Tactic: have hi : normalizedFactors r = Multiset.replicate i (c 1) := by
      apply Multiset.eq_replicate_of_mem
      intro b hb
      refine
        eq_second_of_chain_of_prime_dvd hn h₁ (@fun r' => h₂) (prime_of_normalized_factor b hb) hr
          (dvd_of_mem_normalizedFactors hb)
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (hypothesis)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
q r : 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
hr : r ∣ q
hq : q ≠ 0
i : ℕ := (normalizedFactors r).card
hi : normalizedFactors r = Multiset.replicate i (c 1)
⊢ r = c 1 ^ i
have this : Associated (normalizedFactors r).prod r
Tactic: have := UniqueFactorizationMonoid.prod_normalizedFactors (ne_zero_of_dvd_ne_zero hq hr)
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (hypothesis)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
q r : 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
hr : r ∣ q
hq : q ≠ 0
i : ℕ := (normalizedFactors r).card
hi : normalizedFactors r = Multiset.replicate i (c 1)
this : Associated (normalizedFactors r).prod r
⊢ r = c 1 ^ i
have this₁ : c 1 ^ i = r
Tactic: rw [associated_iff_eq, hi, Multiset.prod_replicate] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
q r : 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
hr : r ∣ q
hq : q ≠ 0
i : ℕ := (normalizedFactors r).card
hi : normalizedFactors r = Multiset.replicate i (c 1)
this : c 1 ^ i = r
⊢ r = c 1 ^ i
have rw : r = r
Tactic: rw [this]
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (hypothesis)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
q r : 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
hr : r ∣ q
hq : q ≠ 0
i : ℕ := (normalizedFactors r).card
hi : normalizedFactors r = Multiset.replicate i (c 1)
⊢ ∃ (i : Fin (n + 1)), r = c 1 ^ (↑i : ℕ)
have H : r = c 1 ^ i
Tactic: have H : r = c 1 ^ i := by
      have := UniqueFactorizationMonoid.prod_normalizedFactors (ne_zero_of_dvd_ne_zero hq hr)
      rw [associated_iff_eq, hi, Multiset.prod_replicate] at this
      rw [this]
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
q r : 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
hr : r ∣ q
hq : q ≠ 0
i : ℕ := (normalizedFactors r).card
hi : normalizedFactors r = Multiset.replicate i (c 1)
H : r = c 1 ^ i
⊢ ∃ (i : Fin (n + 1)), r = c 1 ^ (↑i : ℕ)
have refine : i < n + 1
Tactic: refine ⟨⟨i, ?_⟩, H⟩
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
q r : 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
hr : r ∣ q
hq : q ≠ 0
i : ℕ := (normalizedFactors r).card
hi : normalizedFactors r = Multiset.replicate i (c 1)
H : r = c 1 ^ i
| i + 1
have rw : Finset.card (α := Fin (i + 1)) Finset.univ
Tactic: rw [← Finset.card_fin (i + 1)]
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
q r : 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
hr : r ∣ q
hq : q ≠ 0
i : ℕ := (normalizedFactors r).card
hi : normalizedFactors r = Multiset.replicate i (c 1)
H : r = c 1 ^ i
⊢ Finset.card.{u_1} (α := Associates M) (Finset.image (fun (m : Fin (i + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ) = i + 1
have conv_rhs : Finset.card.{u_1} (α := Associates M) (Finset.image (fun (m : Fin (i + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ) =
  Finset.card (α := Fin (i + 1)) Finset.univ
Tactic: conv_rhs => rw [← Finset.card_fin (i + 1)]
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
q r : 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
hr : r ∣ q
hq : q ≠ 0
i : ℕ := (normalizedFactors r).card
hi : normalizedFactors r = Multiset.replicate i (c 1)
H : r = c 1 ^ i
⊢ Finset.card.{u_1} (α := Associates M) (Finset.image (fun (m : Fin (i + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ) =
    Finset.card (α := Fin (i + 1)) Finset.univ
have zero : ∀ (hn : Ne (α := ℕ) 0 0) {c : Fin (HAdd.hAdd (α := ℕ) 0 1) → Associates M} (h₁ : StrictMono c)
  (h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (HAdd.hAdd (α := ℕ) 0 1)), r = c i)
  (hi : normalizedFactors r = Multiset.replicate i (c 1)) (H : r = c 1 ^ i),
  Finset.card.{u_1} (α := Associates M) (Finset.image (fun (m : Fin (i + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ) =
    Finset.card (α := Fin (i + 1)) Finset.univ
Tactic: cases n
Cleared hypotheses: n, hn, c, h₁, h₂, hi, H
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
q r : 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
hr : r ∣ q
hq : q ≠ 0
i : ℕ := (normalizedFactors r).card
hi : normalizedFactors r = Multiset.replicate i (c 1)
H : r = c 1 ^ i
zero :
  ∀ (hn : Ne (α := ℕ) 0 0) {c : Fin (HAdd.hAdd (α := ℕ) 0 1) → Associates M} (h₁ : StrictMono c)
    (h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (HAdd.hAdd (α := ℕ) 0 1)), r = c i)
    (hi : normalizedFactors r = Multiset.replicate i (c 1)) (H : r = c 1 ^ i),
    Finset.card.{u_1} (α := Associates M) (Finset.image (fun (m : Fin (i + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ) =
      Finset.card (α := Fin (i + 1)) Finset.univ
⊢ Finset.card.{u_1} (α := Associates M) (Finset.image (fun (m : Fin (i + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ) =
    Finset.card (α := Fin (i + 1)) Finset.univ
have succ : ∀ (n : ℕ) (hn : n + 1 ≠ 0) {c : Fin (n + 1 + 1) → Associates M} (h₁ : StrictMono c)
  (h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1 + 1)), r = c i)
  (hi : normalizedFactors r = Multiset.replicate i (c 1)) (H : r = c 1 ^ i),
  Finset.card.{u_1} (α := Associates M) (Finset.image (fun (m : Fin (i + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ) =
    Finset.card (α := Fin (i + 1)) Finset.univ
Tactic: cases n
Cleared hypotheses: n, hn, c, h₁, h₂, hi, H
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
q r : Associates M
hr : r ∣ q
hq : q ≠ 0
i : ℕ := (normalizedFactors r).card
n : ℕ
hn : n + 1 ≠ 0
c : Fin (n + 1 + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1 + 1)), r = c i
hi : normalizedFactors r = Multiset.replicate i (c 1)
H : r = c 1 ^ i
⊢ Finset.card.{u_1} (α := Associates M) (Finset.image (fun (m : Fin (i + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ) =
    Finset.card (α := Fin (i + 1)) Finset.univ
have succ : Set.InjOn.{0, u_1} (β := Associates M) (fun (m : Fin (i + 1)) ↦ c 1 ^ (↑m : ℕ)) (↑Finset.univ : Set (Fin (i + 1)))
Tactic: rw [Finset.card_image_iff]
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
q r : Associates M
hr : r ∣ q
hq : q ≠ 0
i : ℕ := (normalizedFactors r).card
n : ℕ
hn : n + 1 ≠ 0
c : Fin (n + 1 + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1 + 1)), r = c i
hi : normalizedFactors r = Multiset.replicate i (c 1)
H : r = c 1 ^ i
⊢ Set.InjOn.{0, u_1} (β := Associates M) (fun (m : Fin (i + 1)) ↦ c 1 ^ (↑m : ℕ)) (↑Finset.univ : Set (Fin (i + 1)))
have succ : ∀ (m m' : Fin (i + 1)) (h : c 1 ^ (↑m : ℕ) = c 1 ^ (↑m' : ℕ)), (↑m : ℕ) = (↑m' : ℕ)
Tactic: refine Set.injOn_of_injective (fun m m' h => Fin.ext ?_)
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
q r : Associates M
hr : r ∣ q
hq : q ≠ 0
i : ℕ := (normalizedFactors r).card
n : ℕ
hn : n + 1 ≠ 0
c : Fin (n + 1 + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1 + 1)), r = c i
hi : normalizedFactors r = Multiset.replicate i (c 1)
H : r = c 1 ^ i
m m' : Fin (i + 1)
h : c 1 ^ (↑m : ℕ) = c 1 ^ (↑m' : ℕ)
⊢ (↑m : ℕ) = (↑m' : ℕ)
have succ : c 1 ≠ 0
Tactic: refine
        pow_injective_of_not_isUnit (element_of_chain_not_isUnit_of_index_ne_zero (by simp) h₁) ?_ h