-- Mathlib/RingTheory/UniqueFactorizationDomain/Multiplicity.lean
theorem count_normalizedFactors_eq [DecidableEq R] {p x : R} (hp : Irreducible p)
    (hnorm : normalize p = p) {n : ℕ} (hle : p ^ n ∣ x) (hlt : ¬p ^ (n + 1) ∣ x) :
    (normalizedFactors x).count p = n

Proof state:
case neg.a
R : Type u_2
inst✝³ : CancelCommMonoidWithZero R
inst✝² : UniqueFactorizationMonoid R
inst✝¹ : NormalizationMonoid R
inst✝ : DecidableEq R
p x : R
hp : Irreducible p
hnorm : normalize p = p
n : ℕ
hle : p ^ n ∣ x
hlt : ¬p ^ (n + 1) ∣ x
hx0 : ¬x = 0
⊢ ↑(count p (normalizedFactors x)) = ↑n

Tactic:
convert (emultiplicity_eq_count_normalizedFactors hp hx0).symm

Result:
case h.e'_2.h.e'_3.h.e'_3
R : Type u_2
inst✝³ : CancelCommMonoidWithZero R
inst✝² : UniqueFactorizationMonoid R
inst✝¹ : NormalizationMonoid R
inst✝ : DecidableEq R
p x : R
hp : Irreducible p
hnorm : normalize p = p
n : ℕ
hle : p ^ n ∣ x
hlt : ¬p ^ (n + 1) ∣ x
hx0 : ¬x = 0
⊢ p = normalize p

case h.e'_3
R : Type u_2
inst✝³ : CancelCommMonoidWithZero R
inst✝² : UniqueFactorizationMonoid R
inst✝¹ : NormalizationMonoid R
inst✝ : DecidableEq R
p x : R
hp : Irreducible p
hnorm : normalize p = p
n : ℕ
hle : p ^ n ∣ x
hlt : ¬p ^ (n + 1) ∣ x
hx0 : ¬x = 0
⊢ ↑n = emultiplicity p x

Proof state:
case h.e'_3
R : Type u_2
inst✝³ : CancelCommMonoidWithZero R
inst✝² : UniqueFactorizationMonoid R
inst✝¹ : NormalizationMonoid R
inst✝ : DecidableEq R
p x : R
hp : Irreducible p
hnorm : normalize p = p
n : ℕ
hle : p ^ n ∣ x
hlt : ¬p ^ (n + 1) ∣ x
hx0 : ¬x = 0
⊢ ↑n = emultiplicity p x

Tactic:
exact (emultiplicity_eq_coe.mpr ⟨hle, hlt⟩).symm

Result:
no goals

Proof state:
case h.e'_2.h.e'_3.h.e'_3
R : Type u_2
inst✝³ : CancelCommMonoidWithZero R
inst✝² : UniqueFactorizationMonoid R
inst✝¹ : NormalizationMonoid R
inst✝ : DecidableEq R
p x : R
hp : Irreducible p
hnorm : normalize p = p
n : ℕ
hle : p ^ n ∣ x
hlt : ¬p ^ (n + 1) ∣ x
hx0 : ¬x = 0
⊢ p = normalize p

Tactic:
exact hnorm.symm

Result:
no goals

Proof state:
case pos
R : Type u_2
inst✝³ : CancelCommMonoidWithZero R
inst✝² : UniqueFactorizationMonoid R
inst✝¹ : NormalizationMonoid R
inst✝ : DecidableEq R
p x : R
hp : Irreducible p
hnorm : normalize p = p
n : ℕ
hle : p ^ n ∣ x
hlt : ¬p ^ (n + 1) ∣ x
hx0 : x = 0
⊢ count p (normalizedFactors x) = n

Tactic:
simp [hx0] at hlt

Result:
no goals

Proof state:
case neg
R : Type u_2
inst✝³ : CancelCommMonoidWithZero R
inst✝² : UniqueFactorizationMonoid R
inst✝¹ : NormalizationMonoid R
inst✝ : DecidableEq R
p x : R
hp : Irreducible p
hnorm : normalize p = p
n : ℕ
hle : p ^ n ∣ x
hlt : ¬p ^ (n + 1) ∣ x
hx0 : ¬x = 0
⊢ count p (normalizedFactors x) = n

Tactic:
apply Nat.cast_injective (R := ℕ∞)

Result:
case neg.a
R : Type u_2
inst✝³ : CancelCommMonoidWithZero R
inst✝² : UniqueFactorizationMonoid R
inst✝¹ : NormalizationMonoid R
inst✝ : DecidableEq R
p x : R
hp : Irreducible p
hnorm : normalize p = p
n : ℕ
hle : p ^ n ∣ x
hlt : ¬p ^ (n + 1) ∣ x
hx0 : ¬x = 0
⊢ ↑(count p (normalizedFactors x)) = ↑n

Proof state:
R : Type u_2
inst✝³ : CancelCommMonoidWithZero R
inst✝² : UniqueFactorizationMonoid R
inst✝¹ : NormalizationMonoid R
inst✝ : DecidableEq R
p x : R
hp : Irreducible p
hnorm : normalize p = p
n : ℕ
hle : p ^ n ∣ x
hlt : ¬p ^ (n + 1) ∣ x
⊢ count p (normalizedFactors x) = n

Tactic:
by_cases hx0 : x = 0

Result:
case pos
R : Type u_2
inst✝³ : CancelCommMonoidWithZero R
inst✝² : UniqueFactorizationMonoid R
inst✝¹ : NormalizationMonoid R
inst✝ : DecidableEq R
p x : R
hp : Irreducible p
hnorm : normalize p = p
n : ℕ
hle : p ^ n ∣ x
hlt : ¬p ^ (n + 1) ∣ x
hx0 : x = 0
⊢ count p (normalizedFactors x) = n

case neg
R : Type u_2
inst✝³ : CancelCommMonoidWithZero R
inst✝² : UniqueFactorizationMonoid R
inst✝¹ : NormalizationMonoid R
inst✝ : DecidableEq R
p x : R
hp : Irreducible p
hnorm : normalize p = p
n : ℕ
hle : p ^ n ∣ x
hlt : ¬p ^ (n + 1) ∣ x
hx0 : ¬x = 0
⊢ count p (normalizedFactors x) = n