-- Mathlib/RingTheory/ChainOfDivisors.lean
theorem element_of_chain_eq_pow_second_of_chain {q r : Associates M} {n : ℕ} (hn : n ≠ 0)
    {c : Fin (n + 1) → Associates M} (h₁ : StrictMono c) (h₂ : ∀ {r}, r ≤ q ↔ ∃ i, r = c i)
    (hr : r ∣ q) (hq : q ≠ 0) : ∃ i : Fin (n + 1), r = c 1 ^ (i : ℕ)

Proof state:
case a
M : Type u_1
inst✝¹ : CancelCommMonoidWithZero M
inst✝ : 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, r = c i
hr : r ∣ q
hq : q ≠ 0
i : ℕ := (normalizedFactors r).card
⊢ ∀ b ∈ normalizedFactors r, b = c 1

Tactic:
intro b hb

Result:
case a
M : Type u_1
inst✝¹ : CancelCommMonoidWithZero M
inst✝ : 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, r = c i
hr : r ∣ q
hq : q ≠ 0
i : ℕ := (normalizedFactors r).card
b : Associates M
hb : b ∈ normalizedFactors r
⊢ b = c 1

Proof state:
M : Type u_1
inst✝¹ : CancelCommMonoidWithZero M
inst✝ : 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, r = c i
hr : r ∣ q
hq : q ≠ 0
i : ℕ := (normalizedFactors r).card
⊢ normalizedFactors r = Multiset.replicate i (c 1)

Tactic:
apply Multiset.eq_replicate_of_mem

Result:
case a
M : Type u_1
inst✝¹ : CancelCommMonoidWithZero M
inst✝ : 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, r = c i
hr : r ∣ q
hq : q ≠ 0
i : ℕ := (normalizedFactors r).card
⊢ ∀ b ∈ normalizedFactors r, b = c 1

Proof state:
M : Type u_1
inst✝¹ : CancelCommMonoidWithZero M
inst✝ : 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, 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.image (fun m => c 1 ^ ↑m) Finset.univ).card = i + 1

Tactic:
conv_rhs => rw [← Finset.card_fin (i + 1)]

Result:
M : Type u_1
inst✝¹ : CancelCommMonoidWithZero M
inst✝ : 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, 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.image (fun m => c 1 ^ ↑m) Finset.univ).card = Finset.univ.card

Proof state:
M : Type u_1
inst✝¹ : CancelCommMonoidWithZero M
inst✝ : 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, 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
this : (Finset.image (fun m => c 1 ^ ↑m) Finset.univ).card = i + 1
⊢ ∀ r ∈ Finset.image (fun m => c 1 ^ ↑m) Finset.univ, r ≤ q

Tactic:
simp only [Finset.mem_image]

Result:
M : Type u_1
inst✝¹ : CancelCommMonoidWithZero M
inst✝ : 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, 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
this : (Finset.image (fun m => c 1 ^ ↑m) Finset.univ).card = i + 1
⊢ ∀ (r : Associates M), (∃ a ∈ Finset.univ, c 1 ^ ↑a = r) → r ≤ q

Proof state:
case zero
M : Type u_1
inst✝¹ : CancelCommMonoidWithZero M
inst✝ : UniqueFactorizationMonoid M
q r : Associates M
hr : r ∣ q
hq : q ≠ 0
i : ℕ := (normalizedFactors r).card
hn : 0 ≠ 0
c : Fin (0 + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i
hi : normalizedFactors r = Multiset.replicate i (c 1)
H : r = c 1 ^ i
⊢ (Finset.image (fun m => c 1 ^ ↑m) Finset.univ).card = Finset.univ.card

Tactic:
contradiction

Result:
no goals

Proof state:
M : Type u_1
inst✝¹ : CancelCommMonoidWithZero M
inst✝ : 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, 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 = c 1 ^ ↑m'
⊢ 1 ≠ 0

Tactic:
simp

Result:
no goals

Proof state:
M : Type u_1
inst✝¹ : CancelCommMonoidWithZero M
inst✝ : 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, 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
this : (Finset.image (fun m => c 1 ^ ↑m) Finset.univ).card = i + 1
⊢ ∀ (r : Associates M), (∃ a ∈ Finset.univ, c 1 ^ ↑a = r) → r ≤ q

Tactic:
rintro r ⟨a, _, rfl⟩

Result:
case intro.intro
M : Type u_1
inst✝¹ : CancelCommMonoidWithZero M
inst✝ : 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, 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
this : (Finset.image (fun m => c 1 ^ ↑m) Finset.univ).card = i + 1
a : Fin (i + 1)
left✝ : a ∈ Finset.univ
⊢ c 1 ^ ↑a ≤ q

Proof state:
M : Type u_1
inst✝¹ : CancelCommMonoidWithZero M
inst✝ : 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, 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

Tactic:
rw [associated_iff_eq, hi, Multiset.prod_replicate] at this

Result:
M : Type u_1
inst✝¹ : CancelCommMonoidWithZero M
inst✝ : 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, 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

Proof state:
M : Type u_1
inst✝¹ : CancelCommMonoidWithZero M
inst✝ : 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, r = c i
hr : r ∣ q
hq : q ≠ 0
⊢ ∃ i, r = c 1 ^ ↑i

Tactic:
let i := Multiset.card (normalizedFactors r)

Result:
M : Type u_1
inst✝¹ : CancelCommMonoidWithZero M
inst✝ : 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, r = c i
hr : r ∣ q
hq : q ≠ 0
i : ℕ := (normalizedFactors r).card
⊢ ∃ i, r = c 1 ^ ↑i

Proof state:
M : Type u_1
inst✝¹ : CancelCommMonoidWithZero M
inst✝ : 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, 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

Tactic:
have := UniqueFactorizationMonoid.prod_normalizedFactors (ne_zero_of_dvd_ne_zero hq hr)

Result:
M : Type u_1
inst✝¹ : CancelCommMonoidWithZero M
inst✝ : 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, 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