/-- The `v`-adic valuation of a sum is bounded above by the maximum of the valuations. -/
theorem IntValuation.map_add_le_max' (x y : R) :
    v.intValuationDef (x + y) ≤ max (v.intValuationDef x) (v.intValuationDef y) := by
  by_cases hx : x = 0
  · rw [hx, zero_add]
    conv_rhs => rw [intValuationDef, if_pos (Eq.refl _)]
    rw [max_eq_right (WithZero.zero_le (v.intValuationDef y))]
  · by_cases hy : y = 0
    · rw [hy, add_zero]
      conv_rhs => rw [max_comm, intValuationDef, if_pos (Eq.refl _)]
      rw [max_eq_right (WithZero.zero_le (v.intValuationDef x))]
    · by_cases hxy : x + y = 0
      · rw [intValuationDef, if_pos hxy]; exact zero_le'
      · rw [v.intValuationDef_if_neg hxy, v.intValuationDef_if_neg hx,
          v.intValuationDef_if_neg hy, WithZero.le_max_iff, IntValuation.le_max_iff_min_le]
        set nmin :=
          min ((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {x})).factors)
            ((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {y})).factors)
        have h_dvd_x : x ∈ v.asIdeal ^ nmin := by
          rw [← Associates.le_singleton_iff x nmin _,
            Associates.prime_pow_dvd_iff_le (Associates.mk_ne_zero'.mpr hx) _]
          exact min_le_left _ _
          apply v.associates_irreducible
        have h_dvd_y : y ∈ v.asIdeal ^ nmin := by
          rw [← Associates.le_singleton_iff y nmin _,
            Associates.prime_pow_dvd_iff_le (Associates.mk_ne_zero'.mpr hy) _]
          exact min_le_right _ _
          apply v.associates_irreducible
        have h_dvd_xy : Associates.mk v.asIdeal ^ nmin ≤ Associates.mk (Ideal.span {x + y}) := by
          rw [Associates.le_singleton_iff]
          exact Ideal.add_mem (v.asIdeal ^ nmin) h_dvd_x h_dvd_y
        rw [Associates.prime_pow_dvd_iff_le (Associates.mk_ne_zero'.mpr hxy) _] at h_dvd_xy
        

Proof state:
case neg
R : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : IsDedekindDomain R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
x y : R
hx : ¬x = 0
hy : ¬y = 0
hxy : ¬x + y = 0
nmin : ℕ :=
  min (Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x}))))
    (Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {y}))))
h_dvd_x : x ∈ v.asIdeal ^ nmin
h_dvd_y : y ∈ v.asIdeal ^ nmin
h_dvd_xy : nmin ≤ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x + y})))
⊢ nmin ≤ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x + y})))
R : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : IsDedekindDomain R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
x y : R
hx : ¬x = 0
hy : ¬y = 0
hxy : ¬x + y = 0
nmin : ℕ :=
  min (Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x}))))
    (Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {y}))))
h_dvd_x : x ∈ v.asIdeal ^ nmin
h_dvd_y : y ∈ v.asIdeal ^ nmin
h_dvd_xy : Associates.mk v.asIdeal ^ nmin ≤ Associates.mk (Ideal.span {x + y})
⊢ Irreducible (Associates.mk v.asIdeal)

Next tactic:
exact h_dvd_xy