awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
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}))))
⊢ x ∈ v.asIdeal ^ nmin

Tactic:
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 _ _
          exact v.associates_irreducible
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
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
rw : nmin ≤ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {y})))
rw₁ : Irreducible (M := Associates (Ideal R)) (Associates.mk v.asIdeal)
⊢ y ∈ v.asIdeal ^ nmin

Tactic:
rw [← Associates.le_singleton_iff y nmin _,
            Associates.prime_pow_dvd_iff_le (Associates.mk_ne_zero'.mpr hy) _]

Premises used: Iff.mpr, Associates.prime_pow_dvd_iff_le, Associates.le_singleton_iff, Associates.mk_ne_zero'
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
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
⊢ nmin ≤ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {y})))

Tactic:
min_le_right _ _

Premises used: min_le_right
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
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
⊢ nmin ≤ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {y})))

Tactic:
exact min_le_right _ _

Premises used: min_le_right
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
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
⊢ Irreducible (M := Associates (Ideal R)) (Associates.mk v.asIdeal)

Tactic:
v.associates_irreducible

Premises used: IsDedekindDomain.HeightOneSpectrum.associates_irreducible
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
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
⊢ Irreducible (M := Associates (Ideal R)) (Associates.mk v.asIdeal)

Tactic:
exact v.associates_irreducible

Premises used: IsDedekindDomain.HeightOneSpectrum.associates_irreducible
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
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
⊢ y ∈ v.asIdeal ^ nmin

Tactic:
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 _ _
          exact v.associates_irreducible
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
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
rw : x + y ∈ v.asIdeal ^ nmin
⊢ HPow.hPow (α := Associates (Ideal R)) (Associates.mk v.asIdeal) nmin ≤ Associates.mk (Ideal.span {x + y})

Tactic:
rw [Associates.le_singleton_iff]

Premises used: Associates.le_singleton_iff
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
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
⊢ x + y ∈ v.asIdeal ^ nmin

Tactic:
Ideal.add_mem (v.asIdeal ^ nmin) h_dvd_x h_dvd_y

Premises used: IsDedekindDomain.HeightOneSpectrum.asIdeal, Ideal.add_mem, HPow.hPow
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
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
⊢ x + y ∈ v.asIdeal ^ nmin

Tactic:
exact Ideal.add_mem (v.asIdeal ^ nmin) h_dvd_x h_dvd_y

Premises used: IsDedekindDomain.HeightOneSpectrum.asIdeal, Ideal.add_mem, HPow.hPow
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
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
⊢ HPow.hPow (α := Associates (Ideal R)) (Associates.mk v.asIdeal) nmin ≤ Associates.mk (Ideal.span {x + y})

Tactic:
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
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
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 : HPow.hPow (α := Associates (Ideal R)) (Associates.mk v.asIdeal) nmin ≤ Associates.mk (Ideal.span {x + y})
neg :
  ∀
    (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})))
rw : Irreducible (M := Associates (Ideal R)) (Associates.mk v.asIdeal)
⊢ nmin ≤ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x + y})))

Tactic:
rw [Associates.prime_pow_dvd_iff_le (Associates.mk_ne_zero'.mpr hxy) _] at h_dvd_xy

Premises used: Iff.mpr, Associates.prime_pow_dvd_iff_le, Associates.mk_ne_zero'
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
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})))

Tactic:
h_dvd_xy
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
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})))

Tactic:
exact h_dvd_xy
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
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 : HPow.hPow (α := Associates (Ideal R)) (Associates.mk v.asIdeal) nmin ≤ Associates.mk (Ideal.span {x + y})
⊢ Irreducible (M := Associates (Ideal R)) (Associates.mk v.asIdeal)

Tactic:
v.associates_irreducible

Premises used: IsDedekindDomain.HeightOneSpectrum.associates_irreducible
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
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 : HPow.hPow (α := Associates (Ideal R)) (Associates.mk v.asIdeal) nmin ≤ Associates.mk (Ideal.span {x + y})
⊢ Irreducible (M := Associates (Ideal R)) (Associates.mk v.asIdeal)

Tactic:
exact v.associates_irreducible

Premises used: IsDedekindDomain.HeightOneSpectrum.associates_irreducible
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
x y : R
pos : ∀ (hx : x = 0), v.intValuationDef (x + y) ≤ max (v.intValuationDef x) (v.intValuationDef y)
neg neg₁ : ∀ (hx : ¬x = 0), v.intValuationDef (x + y) ≤ max (v.intValuationDef x) (v.intValuationDef y)
⊢ v.intValuationDef (x + y) ≤ max (v.intValuationDef x) (v.intValuationDef y)

Tactic:
classical
  by_cases hx : x = 0
  · rw [hx, zero_add]
    order
  · by_cases hy : y = 0
    · rw [hy, add_zero]
      order
    · 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,
          le_max_iff]
        simp only [exp_le_exp, neg_le_neg_iff, Nat.cast_le, ← min_le_iff]
        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 _ _
          exact 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 _ _
          exact 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
        · exact h_dvd_xy
        exact v.associates_irreducible

Premises used: Associates.mk, Associates.count, IsDedekindDomain.HeightOneSpectrum.intValuationDef, min_le_iff, IsDedekindDomain.HeightOneSpectrum.asIdeal, WithZero.exp_le_exp, zero_add, zero_le', Ideal.span, Nat.cast_le, add_zero, Associates.factors, Singleton.singleton, le_max_iff, IsDedekindDomain.HeightOneSpectrum.associates_irreducible, Min.min, neg_le_neg_iff
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
r : R
v : HeightOneSpectrum R
⊢ (v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r = v.intValuationDef r

Tactic:
rfl

Premises used: IsDedekindDomain.HeightOneSpectrum.intValuationDef, rfl, ite
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
r : R
⊢ Eq (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
    (if r = 0 then 0
      else
        exp
          (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r})))) : ℤ)) :
      WithZero (Multiplicative ℤ))

Tactic:
rfl

Premises used: IsDedekindDomain.HeightOneSpectrum.intValuationDef, rfl, ite
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
r : R
hr : r ≠ 0
⊢ Eq (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
    (exp (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r})))) : ℤ)) :
      WithZero (Multiplicative ℤ))

Tactic:
intValuationDef_if_neg _ hr

Premises used: IsDedekindDomain.HeightOneSpectrum.intValuationDef_if_neg, IsDedekindDomain.HeightOneSpectrum.intValuationDef, ite
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
r : R
hr : r ≠ 0
⊢ Ne (α := Ideal R) (Ideal.span {r}) 0

Tactic:
have hsr : Ideal.span {r} ≠ 0 := Submodule.span_singleton_eq_bot.mp.mt hr
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
r : R
hr : r ≠ 0
hsr : Ne (α := Ideal R) (Ideal.span {r}) 0
⊢ FiniteMultiplicity v.asIdeal (Ideal.span {r})

Tactic:
have hfm : FiniteMultiplicity v.asIdeal (Ideal.span {r}) :=
    FiniteMultiplicity.of_prime_left v.prime hsr
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
r : R
hr : r ≠ 0
hsr : Ne (α := Ideal R) (Ideal.span {r}) 0
hfm : FiniteMultiplicity v.asIdeal (Ideal.span {r})
rw :
  Eq (α := WithZero (Multiplicative ℤ))
    (exp (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r})))) : ℤ)))
    (exp (-(↑(multiplicity v.asIdeal (Ideal.span {r})) : ℤ)))
rw₁ :
  Eq (α := ℤ)
    (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r})))) : ℤ))
    (-(↑(multiplicity v.asIdeal (Ideal.span {r})) : ℤ))
rw₂ :
  Eq (α := ℤ) (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r})))) : ℤ)
    (↑(multiplicity v.asIdeal (Ideal.span {r})) : ℤ)
rw₃ :
  Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r}))) =
    multiplicity v.asIdeal (Ideal.span {r})
rw₄ :
  Eq (α := ℕ∞)
    (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r})))) : ℕ∞)
    (↑(multiplicity v.asIdeal (Ideal.span {r})) : ℕ∞)
rw₅ :
  (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r})))) : ℕ∞) =
    emultiplicity v.asIdeal (Ideal.span {r})
rw₆ :
  Eq (α := ℕ∞)
    (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r})))) : ℕ∞)
    (↑(Multiset.count (α := Ideal R) ((normalize : (a : Ideal R) → Ideal R) v.asIdeal : Ideal R)
          (UniqueFactorizationMonoid.normalizedFactors (Ideal.span {r}) : Multiset (Ideal R))) :
      ℕ∞)
rw₇ :
  Eq (α := ℕ∞)
    (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r})))) : ℕ∞)
    (↑(Multiset.count v.asIdeal (UniqueFactorizationMonoid.normalizedFactors (Ideal.span {r}))) : ℕ∞)
rw₈ rw₉ :
  Eq (α := ℕ∞) (↑(Multiset.count v.asIdeal (UniqueFactorizationMonoid.normalizedFactors (Ideal.span {r}))) : ℕ∞)
    (↑(Multiset.count v.asIdeal (UniqueFactorizationMonoid.normalizedFactors (Ideal.span {r}))) : ℕ∞)
⊢ Eq (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
    (exp (-(↑(multiplicity v.asIdeal (Ideal.span {r})) : ℤ)) : WithZero (Multiplicative ℤ))

Tactic:
rw [v.intValuation_if_neg hr, exp_inj, neg_inj, Int.natCast_inj, ← ENat.coe_inj,
    ← FiniteMultiplicity.emultiplicity_eq_multiplicity hfm,
    UniqueFactorizationMonoid.emultiplicity_eq_count_normalizedFactors (irreducible v) hsr,
    normalize_eq, Ideal.count_associates_factors_eq hsr v.isPrime v.ne_bot]

Premises used: WithZero.exp_inj, IsDedekindDomain.HeightOneSpectrum.intValuation_if_neg, FiniteMultiplicity.emultiplicity_eq_multiplicity, IsDedekindDomain.HeightOneSpectrum.irreducible, ENat.coe_inj, Ideal.count_associates_factors_eq, neg_inj, UniqueFactorizationMonoid.emultiplicity_eq_count_normalizedFactors, IsDedekindDomain.HeightOneSpectrum.ne_bot, IsDedekindDomain.HeightOneSpectrum.isPrime, normalize_eq, Int.natCast_inj
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
x : R
hx : x ≠ 0
rw :
  Ne (α := WithZero (Multiplicative ℤ))
    (exp (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x})))) : ℤ))) 0
⊢ Ne (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) x : WithZero (Multiplicative ℤ))
    (0 : WithZero (Multiplicative ℤ))

Tactic:
rw [v.intValuation_if_neg hx]

Premises used: IsDedekindDomain.HeightOneSpectrum.intValuation_if_neg
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
x : R
hx : x ≠ 0
⊢ Ne (α := WithZero (Multiplicative ℤ))
    (exp (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x})))) : ℤ))) 0

Tactic:
WithZero.coe_ne_zero

Premises used: WithZero.coe_ne_zero, WithZero.exp
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
x : R
hx : x ≠ 0
⊢ Ne (α := WithZero (Multiplicative ℤ))
    (exp (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x})))) : ℤ))) 0

Tactic:
exact WithZero.coe_ne_zero

Premises used: WithZero.coe_ne_zero
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
x : ↥(nonZeroDivisors R)
⊢ Ne (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) (↑x : R) : WithZero (Multiplicative ℤ))
    (0 : WithZero (Multiplicative ℤ))

Tactic:
v.intValuation_ne_zero x (nonZeroDivisors.coe_ne_zero x)

Premises used: IsDedekindDomain.HeightOneSpectrum.intValuation_ne_zero, nonZeroDivisors.coe_ne_zero, Subtype.val
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
x : ↥(nonZeroDivisors R)
rw :
  LT.lt (α := WithZero (Multiplicative ℤ)) 0
    (exp
      (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {(↑x : R)})))) :
          ℤ)))
⊢ LT.lt (α := WithZero (Multiplicative ℤ)) (0 : WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) (↑x : R) : WithZero (Multiplicative ℤ))

Tactic:
rw [v.intValuation_if_neg (nonZeroDivisors.coe_ne_zero x)]

Premises used: IsDedekindDomain.HeightOneSpectrum.intValuation_if_neg, nonZeroDivisors.coe_ne_zero
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
x : ↥(nonZeroDivisors R)
⊢ LT.lt (α := WithZero (Multiplicative ℤ)) 0
    (exp
      (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {(↑x : R)})))) :
          ℤ)))

Tactic:
WithZero.zero_lt_coe _

Premises used: WithZero.exp, WithZero.zero_lt_coe
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
x : ↥(nonZeroDivisors R)
⊢ LT.lt (α := WithZero (Multiplicative ℤ)) 0
    (exp
      (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {(↑x : R)})))) :
          ℤ)))

Tactic:
exact WithZero.zero_lt_coe _

Premises used: WithZero.zero_lt_coe
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
x : R
inl :
  LE.le (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) 0 : WithZero (Multiplicative ℤ))
    (1 : WithZero (Multiplicative ℤ))
inr :
  ∀ (hx : x ≠ 0),
    LE.le (α := WithZero (Multiplicative ℤ))
      ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) x : WithZero (Multiplicative ℤ))
      (1 : WithZero (Multiplicative ℤ))
⊢ LE.le (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) x : WithZero (Multiplicative ℤ))
    (1 : WithZero (Multiplicative ℤ))

Tactic:
obtain rfl | hx := eq_or_ne x 0

Premises used: eq_or_ne, OfNat.ofNat, rfl
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
⊢ LE.le (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) 0 : WithZero (Multiplicative ℤ))
    (1 : WithZero (Multiplicative ℤ))

Tactic:
simp