-- Mathlib/RingTheory/DedekindDomain/Factorization.lean
theorem count_mul' (I I' : FractionalIdeal R⁰ K) [Decidable (I ≠ 0 ∧ I' ≠ 0)] :
    count K v (I * I') = if I ≠ 0 ∧ I' ≠ 0 then count K v I + count K v I' else 0

Proof state:
case pos
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDedekindDomain R
v : HeightOneSpectrum R
I I' : FractionalIdeal R⁰ K
inst✝ : Decidable (I ≠ 0 ∧ I' ≠ 0)
h : I ≠ 0 ∧ I' ≠ 0
⊢ count K v (I * I') = count K v I + count K v I'

Tactic:
exact count_mul K v h.1 h.2

Result:
no goals

Proof state:
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDedekindDomain R
v : HeightOneSpectrum R
I I' : FractionalIdeal R⁰ K
inst✝ : Decidable (I ≠ 0 ∧ I' ≠ 0)
⊢ count K v (I * I') = if I ≠ 0 ∧ I' ≠ 0 then count K v I + count K v I' else 0

Tactic:
split_ifs with h

Result:
case pos
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDedekindDomain R
v : HeightOneSpectrum R
I I' : FractionalIdeal R⁰ K
inst✝ : Decidable (I ≠ 0 ∧ I' ≠ 0)
h : I ≠ 0 ∧ I' ≠ 0
⊢ count K v (I * I') = count K v I + count K v I'

case neg
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDedekindDomain R
v : HeightOneSpectrum R
I I' : FractionalIdeal R⁰ K
inst✝ : Decidable (I ≠ 0 ∧ I' ≠ 0)
h : ¬(I ≠ 0 ∧ I' ≠ 0)
⊢ count K v (I * I') = 0

Proof state:
case pos
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDedekindDomain R
v : HeightOneSpectrum R
I I' : FractionalIdeal R⁰ K
inst✝ : Decidable (I ≠ 0 ∧ I' ≠ 0)
h : I ≠ 0 → I' = 0
hI : I = 0
⊢ count K v (I * I') = 0

Tactic:
rw [hI, MulZeroClass.zero_mul, count, dif_pos (Eq.refl _)]

Result:
no goals

Proof state:
case neg
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDedekindDomain R
v : HeightOneSpectrum R
I I' : FractionalIdeal R⁰ K
inst✝ : Decidable (I ≠ 0 ∧ I' ≠ 0)
h : I ≠ 0 → I' = 0
⊢ count K v (I * I') = 0

Tactic:
by_cases hI : I = 0

Result:
case pos
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDedekindDomain R
v : HeightOneSpectrum R
I I' : FractionalIdeal R⁰ K
inst✝ : Decidable (I ≠ 0 ∧ I' ≠ 0)
h : I ≠ 0 → I' = 0
hI : I = 0
⊢ count K v (I * I') = 0

case neg
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDedekindDomain R
v : HeightOneSpectrum R
I I' : FractionalIdeal R⁰ K
inst✝ : Decidable (I ≠ 0 ∧ I' ≠ 0)
h : I ≠ 0 → I' = 0
hI : ¬I = 0
⊢ count K v (I * I') = 0

Proof state:
case neg
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDedekindDomain R
v : HeightOneSpectrum R
I I' : FractionalIdeal R⁰ K
inst✝ : Decidable (I ≠ 0 ∧ I' ≠ 0)
h : ¬(I ≠ 0 ∧ I' ≠ 0)
⊢ count K v (I * I') = 0

Tactic:
push_neg at h

Result:
case neg
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDedekindDomain R
v : HeightOneSpectrum R
I I' : FractionalIdeal R⁰ K
inst✝ : Decidable (I ≠ 0 ∧ I' ≠ 0)
h : I ≠ 0 → I' = 0
⊢ count K v (I * I') = 0

Proof state:
case neg
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDedekindDomain R
v : HeightOneSpectrum R
I I' : FractionalIdeal R⁰ K
inst✝ : Decidable (I ≠ 0 ∧ I' ≠ 0)
h : I ≠ 0 → I' = 0
hI : ¬I = 0
⊢ count K v (I * I') = 0

Tactic:
rw [h hI, MulZeroClass.mul_zero, count, dif_pos (Eq.refl _)]

Result:
no goals