awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
K : Type u_2
inst_2 : Field K
inst_3 : Algebra R K
inst_4 : IsFractionRing R K
v : HeightOneSpectrum R
x : K
hv :
  LE.le (α := WithZero (Multiplicative ℤ))
    (((valuation K v : Valuation K (WithZero (Multiplicative ℤ))) : (a : K) → WithZero (Multiplicative ℤ)) x :
      WithZero (Multiplicative ℤ))
    (1 : WithZero (Multiplicative ℤ))
n : R
d : ↥v.asIdeal.primeCompl
hnd : HMul.hMul (β := K) x ((algebraMap R K : (a : R) → K) n : K) = (algebraMap R K : (a : R) → K) (↑d : R)
inr :
  1 ≤
    HMul.hMul (α := WithZero (Multiplicative ℤ)) (β := WithZero (Multiplicative ℤ))
      (((valuation K v : Valuation K (WithZero (Multiplicative ℤ))) : (a : K) → WithZero (Multiplicative ℤ)) x :
        WithZero (Multiplicative ℤ))
      ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) n : WithZero (Multiplicative ℤ))
⊢ Eq (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) n : WithZero (Multiplicative ℤ))
    (1 : WithZero (Multiplicative ℤ))

Tactic:
apply eq_one_of_one_le_mul_right hv (intValuation_le_one v n)

Premises used: IsDedekindDomain.HeightOneSpectrum.intValuation_le_one, eq_one_of_one_le_mul_right
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
a b : R
hv :
  LE.le (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) b : WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) a : WithZero (Multiplicative ℤ))
γ : Multiplicative ℤ
pos :
  ∀ (ha : a = 0),
    ∃ (y : R), (v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) (b - y * a) < (↑γ : WithZero (Multiplicative ℤ))
neg :
  ∀ (ha : ¬a = 0),
    ∃ (y : R), (v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) (b - y * a) < (↑γ : WithZero (Multiplicative ℤ))
⊢ ∃ (y : R), (v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) (b - y * a) < (↑γ : WithZero (Multiplicative ℤ))

Tactic:
by_cases ha: a = 0
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
a b : R
hv :
  LE.le (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) b : WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) a : WithZero (Multiplicative ℤ))
γ : Multiplicative ℤ
ha : a = 0
pos :
  ∀
    (hv :
      LE.le (α := WithZero (Multiplicative ℤ))
        ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) b : WithZero (Multiplicative ℤ))
        ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) 0 : WithZero (Multiplicative ℤ))),
    ∃ (y : R), (v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) (b - y * 0) < (↑γ : WithZero (Multiplicative ℤ))
⊢ ∃ (y : R), (v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) (b - y * a) < (↑γ : WithZero (Multiplicative ℤ))

Tactic:
subst ha
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
b : R
γ : Multiplicative ℤ
hv :
  LE.le (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) b : WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) 0 : WithZero (Multiplicative ℤ))
⊢ Eq (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) b : WithZero (Multiplicative ℤ))
    (0 : WithZero (Multiplicative ℤ))

Tactic:
rw [map_zero, le_zero_iff] at hv

Premises used: le_zero_iff, map_zero
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
b : R
γ : Multiplicative ℤ
hv :
  Eq (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) b : WithZero (Multiplicative ℤ))
    (0 : WithZero (Multiplicative ℤ))
⊢ (v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) (b - HMul.hMul (α := R) 0 0) <
    (↑γ : WithZero (Multiplicative ℤ))

Tactic:
simp [hv]
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
b : R
γ : Multiplicative ℤ
hv :
  Eq (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) b : WithZero (Multiplicative ℤ))
    (0 : WithZero (Multiplicative ℤ))
⊢ ∃ (y : R), (v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) (b - y * 0) < (↑γ : WithZero (Multiplicative ℤ))

Tactic:
exact ⟨0, by simp [hv]⟩
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
a b : R
hv :
  LE.le (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) b : WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) a : WithZero (Multiplicative ℤ))
γ : Multiplicative ℤ
ha : ¬a = 0
⊢ Ne (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) a : WithZero (Multiplicative ℤ))
    (0 : WithZero (Multiplicative ℤ))

Tactic:
have hvaz := intValuation_ne_zero v a ha
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
a b : R
hv :
  LE.le (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) b : WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) a : WithZero (Multiplicative ℤ))
γ : Multiplicative ℤ
ha : ¬a = 0
hvaz :
  Ne (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) a : WithZero (Multiplicative ℤ))
    (0 : WithZero (Multiplicative ℤ))
⊢ (↑γ : WithZero (Multiplicative ℤ)) ≠ 0

Tactic:
have hγz : WithZero.coe γ ≠ 0 := WithZero.coe_ne_zero
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
a b : R
hv :
  LE.le (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) b : WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) a : WithZero (Multiplicative ℤ))
γ : Multiplicative ℤ
ha : ¬a = 0
hvaz :
  Ne (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) a : WithZero (Multiplicative ℤ))
    (0 : WithZero (Multiplicative ℤ))
hγz : (↑γ : WithZero (Multiplicative ℤ)) ≠ 0
neg :
  ∀ (n : ℕ)
    (hna :
      LT.lt (α := WithZero (Multiplicative ℤ)) (exp (-(↑n : ℤ)))
        ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) a))
    (hnγ : exp (-(↑n : ℤ)) < (↑γ : WithZero (Multiplicative ℤ))),
    ∃ (y : R), (v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) (b - y * a) < (↑γ : WithZero (Multiplicative ℤ))
⊢ ∃ (y : R), (v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) (b - y * a) < (↑γ : WithZero (Multiplicative ℤ))

Tactic:
obtain ⟨n, hna, hnγ⟩ := exists_exp_neg_natCast_lt_and_lt hvaz hγz

Premises used: WithZero.exists_exp_neg_natCast_lt_and_lt
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
a b : R
hv :
  LE.le (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) b : WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) a : WithZero (Multiplicative ℤ))
γ : Multiplicative ℤ
ha : ¬a = 0
hvaz :
  Ne (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) a : WithZero (Multiplicative ℤ))
    (0 : WithZero (Multiplicative ℤ))
hγz : (↑γ : WithZero (Multiplicative ℤ)) ≠ 0
n : ℕ
hna :
  LT.lt (α := WithZero (Multiplicative ℤ)) (exp (-(↑n : ℤ)))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) a)
hnγ : exp (-(↑n : ℤ)) < (↑γ : WithZero (Multiplicative ℤ))
neg :
  ∃ (a_1 : R),
    LE.le (α := WithZero (Multiplicative ℤ))
      ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) (b - a_1 * a) : WithZero (Multiplicative ℤ))
      (exp (-(↑n : ℤ)) : WithZero (Multiplicative ℤ))
⊢ ∃ (y : R), (v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) (b - y * a) < (↑γ : WithZero (Multiplicative ℤ))

Tactic:
apply Exists.imp (fun _ h ↦ lt_of_le_of_lt h hnγ)

Premises used: lt_of_le_of_lt, Exists.imp
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
a b : R
hv :
  LE.le (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) b : WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) a : WithZero (Multiplicative ℤ))
γ : Multiplicative ℤ
ha : ¬a = 0
hvaz :
  Ne (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) a : WithZero (Multiplicative ℤ))
    (0 : WithZero (Multiplicative ℤ))
hγz : (↑γ : WithZero (Multiplicative ℤ)) ≠ 0
n : ℕ
hna :
  LT.lt (α := WithZero (Multiplicative ℤ)) (exp (-(↑n : ℤ)))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) a)
hnγ : exp (-(↑n : ℤ)) < (↑γ : WithZero (Multiplicative ℤ))
grw :
  LE.le (α := WithZero (Multiplicative ℤ)) (exp (-(↑n : ℤ)))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) a)
grw₁ grw₂ :
  LE.le (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) a : WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) a : WithZero (Multiplicative ℤ))
⊢ emultiplicity v.asIdeal (Ideal.span {a}) ≤ (↑n : ℕ∞)

Tactic:
grw [← exp_le_intValuation_iff_emultiplicity_le, hna]

Premises used: IsDedekindDomain.HeightOneSpectrum.exp_le_intValuation_iff_emultiplicity_le
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
a b : R
hv :
  LE.le (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) b : WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) a : WithZero (Multiplicative ℤ))
γ : Multiplicative ℤ
ha : ¬a = 0
hvaz :
  Ne (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) a : WithZero (Multiplicative ℤ))
    (0 : WithZero (Multiplicative ℤ))
hγz : (↑γ : WithZero (Multiplicative ℤ)) ≠ 0
n : ℕ
hna :
  LT.lt (α := WithZero (Multiplicative ℤ)) (exp (-(↑n : ℤ)))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) a)
hnγ : exp (-(↑n : ℤ)) < (↑γ : WithZero (Multiplicative ℤ))
⊢ emultiplicity v.asIdeal (Ideal.span {a}) ≤ (↑n : ℕ∞)

Tactic:
have hvn : emultiplicity v.asIdeal (Ideal.span {a}) ≤ n := by
      grw [← exp_le_intValuation_iff_emultiplicity_le, hna]
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
a b : R
hv :
  LE.le (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) b : WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) a : WithZero (Multiplicative ℤ))
γ : Multiplicative ℤ
ha : ¬a = 0
hvaz :
  Ne (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) a : WithZero (Multiplicative ℤ))
    (0 : WithZero (Multiplicative ℤ))
hγz : (↑γ : WithZero (Multiplicative ℤ)) ≠ 0
n : ℕ
hna :
  LT.lt (α := WithZero (Multiplicative ℤ)) (exp (-(↑n : ℤ)))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) a)
hnγ : exp (-(↑n : ℤ)) < (↑γ : WithZero (Multiplicative ℤ))
hvn : emultiplicity v.asIdeal (Ideal.span {a}) ≤ (↑n : ℕ∞)
rwa :
  LE.le (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) b : WithZero (Multiplicative ℤ))
    (exp (-(↑(multiplicity v.asIdeal (Ideal.span {a})) : ℤ)) : WithZero (Multiplicative ℤ))
rwa₁ rwa₂ rwa₃ rwa₄ :
  LE.le (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) b : WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) a : WithZero (Multiplicative ℤ))
⊢ b ∈ v.asIdeal ^ multiplicity v.asIdeal (Ideal.span {a})

Tactic:
rwa [← intValuation_le_pow_iff_mem, ← v.intValuation_eq_exp_neg_multiplicity ha]

Premises used: IsDedekindDomain.HeightOneSpectrum.intValuation_eq_exp_neg_multiplicity, IsDedekindDomain.HeightOneSpectrum.intValuation_le_pow_iff_mem
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
a b : R
hv :
  LE.le (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) b : WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) a : WithZero (Multiplicative ℤ))
γ : Multiplicative ℤ
ha : ¬a = 0
hvaz :
  Ne (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) a : WithZero (Multiplicative ℤ))
    (0 : WithZero (Multiplicative ℤ))
hγz : (↑γ : WithZero (Multiplicative ℤ)) ≠ 0
n : ℕ
hna :
  LT.lt (α := WithZero (Multiplicative ℤ)) (exp (-(↑n : ℤ)))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) a)
hnγ : exp (-(↑n : ℤ)) < (↑γ : WithZero (Multiplicative ℤ))
hvn : emultiplicity v.asIdeal (Ideal.span {a}) ≤ (↑n : ℕ∞)
⊢ b ∈ v.asIdeal ^ multiplicity v.asIdeal (Ideal.span {a})

Tactic:
have hb : b ∈ v.asIdeal ^ multiplicity v.asIdeal (Ideal.span {a}) := by
      rwa [← intValuation_le_pow_iff_mem, ← v.intValuation_eq_exp_neg_multiplicity ha]
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
a b : R
hv :
  LE.le (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) b : WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) a : WithZero (Multiplicative ℤ))
γ : Multiplicative ℤ
ha : ¬a = 0
hvaz :
  Ne (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) a : WithZero (Multiplicative ℤ))
    (0 : WithZero (Multiplicative ℤ))
hγz : (↑γ : WithZero (Multiplicative ℤ)) ≠ 0
n : ℕ
hna :
  LT.lt (α := WithZero (Multiplicative ℤ)) (exp (-(↑n : ℤ)))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) a)
hnγ : exp (-(↑n : ℤ)) < (↑γ : WithZero (Multiplicative ℤ))
hvn : emultiplicity v.asIdeal (Ideal.span {a}) ≤ (↑n : ℕ∞)
hb : b ∈ v.asIdeal ^ multiplicity v.asIdeal (Ideal.span {a})
rwa : ¬Eq (α := Ideal R) (Ideal.span {a}) ⊥
rwa₁ rwa₂ rwa₃ rwa₄ : ¬a = 0
⊢ Ne (α := Ideal R) (Ideal.span {a}) ⊥

Tactic:
rwa [ne_eq, Ideal.span_singleton_eq_bot]

Premises used: Ideal.span_singleton_eq_bot, ne_eq
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
a b : R
hv :
  LE.le (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) b : WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) a : WithZero (Multiplicative ℤ))
γ : Multiplicative ℤ
ha : ¬a = 0
hvaz :
  Ne (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) a : WithZero (Multiplicative ℤ))
    (0 : WithZero (Multiplicative ℤ))
hγz : (↑γ : WithZero (Multiplicative ℤ)) ≠ 0
n : ℕ
hna :
  LT.lt (α := WithZero (Multiplicative ℤ)) (exp (-(↑n : ℤ)))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) a)
hnγ : exp (-(↑n : ℤ)) < (↑γ : WithZero (Multiplicative ℤ))
hvn : emultiplicity v.asIdeal (Ideal.span {a}) ≤ (↑n : ℕ∞)
hb : b ∈ v.asIdeal ^ multiplicity v.asIdeal (Ideal.span {a})
⊢ Ne (α := Ideal R) (Ideal.span {a}) ⊥

Tactic:
have hnz : Ideal.span {a} ≠ ⊥ := by rwa [ne_eq, Ideal.span_singleton_eq_bot]
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
a b : R
hv :
  LE.le (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) b : WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) a : WithZero (Multiplicative ℤ))
γ : Multiplicative ℤ
ha : ¬a = 0
hvaz :
  Ne (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) a : WithZero (Multiplicative ℤ))
    (0 : WithZero (Multiplicative ℤ))
hγz : (↑γ : WithZero (Multiplicative ℤ)) ≠ 0
n : ℕ
hna :
  LT.lt (α := WithZero (Multiplicative ℤ)) (exp (-(↑n : ℤ)))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) a)
hnγ : exp (-(↑n : ℤ)) < (↑γ : WithZero (Multiplicative ℤ))
hvn : emultiplicity v.asIdeal (Ideal.span {a}) ≤ (↑n : ℕ∞)
hb : b ∈ v.asIdeal ^ multiplicity v.asIdeal (Ideal.span {a})
hnz : Ne (α := Ideal R) (Ideal.span {a}) ⊥
⊢ ∃ (a_1 : R),
    LE.le (α := WithZero (Multiplicative ℤ))
      ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) (b - a_1 * a) : WithZero (Multiplicative ℤ))
      (exp (-(↑n : ℤ)) : WithZero (Multiplicative ℤ))

Tactic:
simpa [← Ideal.irreducible_pow_sup_of_ge hnz v.irreducible n hvn, Submodule.mem_sup,
      ← eq_sub_iff_add_eq, ← intValuation_le_pow_iff_mem, Ideal.mem_span_singleton'] using hb

Premises used: Ideal.mem_span_singleton', IsDedekindDomain.HeightOneSpectrum.intValuation_le_pow_iff_mem, eq_sub_iff_add_eq, IsDedekindDomain.HeightOneSpectrum.irreducible, Submodule.mem_sup, Ideal.irreducible_pow_sup_of_ge
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
K : Type u_2
inst_2 : Field K
inst_3 : Algebra R K
inst_4 : IsFractionRing R K
v : HeightOneSpectrum R
x : K
hv :
  LE.le (α := WithZero (Multiplicative ℤ))
    (((valuation K v : Valuation K (WithZero (Multiplicative ℤ))) : (a : K) → WithZero (Multiplicative ℤ)) x :
      WithZero (Multiplicative ℤ))
    (1 : WithZero (Multiplicative ℤ))
γ : (WithZero (Multiplicative ℤ))ˣ
obtain :
  ∀ (n d : R) (hd : d ∈ v.asIdeal.primeCompl)
    (hnd :
      HMul.hMul (β := K) x
          ((algebraMap R K : (a : R) → K) (↑(Subtype.mk (p := fun (x : R) ↦ x ∈ v.asIdeal.primeCompl) d hd) : R) : K) =
        (algebraMap R K : (a : R) → K) n),
    ∃ (a : R),
      ((valuation K v : Valuation K (WithZero (Multiplicative ℤ))) : (a : K) → WithZero (Multiplicative ℤ))
          (HSub.hSub (α := K) ((algebraMap R K : (a : R) → K) a : K) x) <
        (↑γ : WithZero (Multiplicative ℤ))
⊢ ∃ (a : R),
    ((valuation K v : Valuation K (WithZero (Multiplicative ℤ))) : (a : K) → WithZero (Multiplicative ℤ))
        (HSub.hSub (α := K) ((algebraMap R K : (a : R) → K) a : K) x) <
      (↑γ : WithZero (Multiplicative ℤ))

Tactic:
obtain ⟨n, ⟨d, hd⟩, hnd⟩ := exists_primeCompl_mul_eq_of_integer v x hv

Premises used: IsDedekindDomain.HeightOneSpectrum.exists_primeCompl_mul_eq_of_integer
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
K : Type u_2
inst_2 : Field K
inst_3 : Algebra R K
inst_4 : IsFractionRing R K
v : HeightOneSpectrum R
x : K
hv :
  LE.le (α := WithZero (Multiplicative ℤ))
    (((valuation K v : Valuation K (WithZero (Multiplicative ℤ))) : (a : K) → WithZero (Multiplicative ℤ)) x :
      WithZero (Multiplicative ℤ))
    (1 : WithZero (Multiplicative ℤ))
γ : (WithZero (Multiplicative ℤ))ˣ
n d : R
hd : d ∈ v.asIdeal.primeCompl
hnd :
  HMul.hMul (β := K) x
      ((algebraMap R K : (a : R) → K) (↑(Subtype.mk (p := fun (x : R) ↦ x ∈ v.asIdeal.primeCompl) d hd) : R) : K) =
    (algebraMap R K : (a : R) → K) n
⊢ Eq (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) d : WithZero (Multiplicative ℤ))
    (1 : WithZero (Multiplicative ℤ))

Tactic:
rw [← intValuation_eq_one_iff_mem_primeCompl] at hd

Premises used: IsDedekindDomain.HeightOneSpectrum.intValuation_eq_one_iff_mem_primeCompl
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
K : Type u_2
inst_2 : Field K
inst_3 : Algebra R K
inst_4 : IsFractionRing R K
v : HeightOneSpectrum R
x : K
hv :
  LE.le (α := WithZero (Multiplicative ℤ))
    (((valuation K v : Valuation K (WithZero (Multiplicative ℤ))) : (a : K) → WithZero (Multiplicative ℤ)) x :
      WithZero (Multiplicative ℤ))
    (1 : WithZero (Multiplicative ℤ))
γ : (WithZero (Multiplicative ℤ))ˣ
n d : R
hd : d ∈ v.asIdeal.primeCompl
hnd :
  HMul.hMul (β := K) x
      ((algebraMap R K : (a : R) → K) (↑(Subtype.mk (p := fun (x : R) ↦ x ∈ v.asIdeal.primeCompl) d hd) : R) : K) =
    (algebraMap R K : (a : R) → K) n
hd₁ :
  Eq (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) d : WithZero (Multiplicative ℤ))
    (1 : WithZero (Multiplicative ℤ))
⊢ HMul.hMul (β := K) x
      ((algebraMap R K : (a : R) → K) (↑(Subtype.mk (p := fun (x : R) ↦ x ∈ v.asIdeal.primeCompl) d hd) : R) : K) =
    (algebraMap R K : (a : R) → K) n

Tactic:
rw [← intValuation_eq_one_iff_mem_primeCompl] at hd

Premises used: IsDedekindDomain.HeightOneSpectrum.intValuation_eq_one_iff_mem_primeCompl
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
K : Type u_2
inst_2 : Field K
inst_3 : Algebra R K
inst_4 : IsFractionRing R K
v : HeightOneSpectrum R
x : K
hv :
  LE.le (α := WithZero (Multiplicative ℤ))
    (((valuation K v : Valuation K (WithZero (Multiplicative ℤ))) : (a : K) → WithZero (Multiplicative ℤ)) x :
      WithZero (Multiplicative ℤ))
    (1 : WithZero (Multiplicative ℤ))
γ : (WithZero (Multiplicative ℤ))ˣ
n d : R
hd_1 : d ∈ v.asIdeal.primeCompl
hd :
  Eq (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) d : WithZero (Multiplicative ℤ))
    (1 : WithZero (Multiplicative ℤ))
hnd :
  HMul.hMul (β := K) x
      ((algebraMap R K : (a : R) → K) (↑(Subtype.mk (p := fun (x : R) ↦ x ∈ v.asIdeal.primeCompl) d hd_1) : R) : K) =
    (algebraMap R K : (a : R) → K) n
grw :
  LE.le (α := WithZero (Multiplicative ℤ)) (1 : WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) d : WithZero (Multiplicative ℤ))
grw₁ grw₂ : LE.le (α := WithZero (Multiplicative ℤ)) 1 1
⊢ LE.le (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) n : WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) d : WithZero (Multiplicative ℤ))

Tactic:
grw [v.intValuation_le_one n, hd]

Premises used: IsDedekindDomain.HeightOneSpectrum.intValuation_le_one
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
K : Type u_2
inst_2 : Field K
inst_3 : Algebra R K
inst_4 : IsFractionRing R K
v : HeightOneSpectrum R
x : K
hv :
  LE.le (α := WithZero (Multiplicative ℤ))
    (((valuation K v : Valuation K (WithZero (Multiplicative ℤ))) : (a : K) → WithZero (Multiplicative ℤ)) x :
      WithZero (Multiplicative ℤ))
    (1 : WithZero (Multiplicative ℤ))
γ : (WithZero (Multiplicative ℤ))ˣ
n d : R
hd_1 : d ∈ v.asIdeal.primeCompl
hd :
  Eq (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) d : WithZero (Multiplicative ℤ))
    (1 : WithZero (Multiplicative ℤ))
hnd :
  HMul.hMul (β := K) x
      ((algebraMap R K : (a : R) → K) (↑(Subtype.mk (p := fun (x : R) ↦ x ∈ v.asIdeal.primeCompl) d hd_1) : R) : K) =
    (algebraMap R K : (a : R) → K) n
⊢ LE.le (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) n : WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) d : WithZero (Multiplicative ℤ))

Tactic:
have hd' : v.intValuation n ≤ v.intValuation d := by grw [v.intValuation_le_one n, hd]