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 : R
hx : x ≠ 0
inr :
  LE.le (α := ℤ) 0
    (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x})))) : ℤ)
⊢ LE.le (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) x : WithZero (Multiplicative ℤ))
    (1 : WithZero (Multiplicative ℤ))

Tactic:
rw [v.intValuation_if_neg hx, ← exp_zero, exp_le_exp, Right.neg_nonpos_iff]

Premises used: IsDedekindDomain.HeightOneSpectrum.intValuation_if_neg, Right.neg_nonpos_iff, WithZero.exp_le_exp, WithZero.exp_zero
-- 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
⊢ LE.le (α := ℤ) 0
    (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x})))) : ℤ)

Tactic:
Int.natCast_nonneg _

Premises used: Int.natCast_nonneg
-- 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
⊢ LE.le (α := ℤ) 0
    (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x})))) : ℤ)

Tactic:
exact Int.natCast_nonneg _

Premises used: Int.natCast_nonneg
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
r : R
pos :
  ∀ (hr : r = 0),
    LT.lt (α := WithZero (Multiplicative ℤ))
        ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
        (1 : WithZero (Multiplicative ℤ)) ↔
      v.asIdeal ∣ Ideal.span {r}
neg :
  ∀ (hr : ¬r = 0),
    LT.lt (α := WithZero (Multiplicative ℤ))
        ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
        (1 : WithZero (Multiplicative ℤ)) ↔
      v.asIdeal ∣ Ideal.span {r}
⊢ LT.lt (α := WithZero (Multiplicative ℤ))
      ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
      (1 : WithZero (Multiplicative ℤ)) ↔
    v.asIdeal ∣ Ideal.span {r}

Tactic:
by_cases hr : r = 0
-- 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
⊢ LT.lt (α := WithZero (Multiplicative ℤ))
      ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
      (1 : WithZero (Multiplicative ℤ)) ↔
    v.asIdeal ∣ Ideal.span {r}

Tactic:
simp [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
neg :
  Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r}))) ≠ 0 ↔
    v.asIdeal ∣ Ideal.span {r}
⊢ LT.lt (α := WithZero (Multiplicative ℤ))
      ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
      (1 : WithZero (Multiplicative ℤ)) ↔
    v.asIdeal ∣ Ideal.span {r}

Tactic:
rw [v.intValuation_if_neg hr, ← exp_zero, exp_lt_exp,
      neg_lt_zero, ← Int.ofNat_zero, Int.ofNat_lt, zero_lt_iff]

Premises used: IsDedekindDomain.HeightOneSpectrum.intValuation_if_neg, WithZero.exp_lt_exp, Int.ofNat_lt, neg_lt_zero, Int.ofNat_zero, zero_lt_iff, WithZero.exp_zero
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
r : R
hr rw : ¬r = 0
⊢ Ne (α := Ideal R) (Ideal.span {r}) 0

Tactic:
rw [Ne, Ideal.zero_eq_bot, Ideal.span_singleton_eq_bot]

Premises used: Ideal.span_singleton_eq_bot, Ne, Ideal.zero_eq_bot
-- 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
⊢ ¬r = 0

Tactic:
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
⊢ ¬r = 0

Tactic:
exact 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
⊢ Ne (α := Ideal R) (Ideal.span {r}) 0

Tactic:
have h : (Ideal.span {r} : Ideal R) ≠ 0 := by
      rw [Ne, Ideal.zero_eq_bot, Ideal.span_singleton_eq_bot]
      exact 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
h : Ne (α := Ideal R) (Ideal.span {r}) 0
⊢ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r}))) ≠ 0 ↔
    v.asIdeal ∣ Ideal.span {r}

Tactic:
Associates.count_ne_zero_iff_dvd h v.irreducible

Premises used: Set.Mem, IsDedekindDomain.HeightOneSpectrum.irreducible, Associates.count_ne_zero_iff_dvd
-- 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
h : Ne (α := Ideal R) (Ideal.span {r}) 0
⊢ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r}))) ≠ 0 ↔
    v.asIdeal ∣ Ideal.span {r}

Tactic:
exact Associates.count_ne_zero_iff_dvd h v.irreducible

Premises used: IsDedekindDomain.HeightOneSpectrum.irreducible, Associates.count_ne_zero_iff_dvd
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
r : R
pos :
  ∀ (hr : r = 0),
    LT.lt (α := WithZero (Multiplicative ℤ))
        ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
        (1 : WithZero (Multiplicative ℤ)) ↔
      v.asIdeal ∣ Ideal.span {r}
neg neg₁ :
  ∀ (hr : ¬r = 0),
    LT.lt (α := WithZero (Multiplicative ℤ))
        ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
        (1 : WithZero (Multiplicative ℤ)) ↔
      v.asIdeal ∣ Ideal.span {r}
⊢ LT.lt (α := WithZero (Multiplicative ℤ))
      ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
      (1 : WithZero (Multiplicative ℤ)) ↔
    v.asIdeal ∣ Ideal.span {r}

Tactic:
classical
  by_cases hr : r = 0
  · simp [hr]
  · rw [v.intValuation_if_neg hr, ← exp_zero, exp_lt_exp,
      neg_lt_zero, ← Int.ofNat_zero, Int.ofNat_lt, zero_lt_iff]
    have h : (Ideal.span {r} : Ideal R) ≠ 0 := by
      rw [Ne, Ideal.zero_eq_bot, Ideal.span_singleton_eq_bot]
      exact hr
    exact Associates.count_ne_zero_iff_dvd h v.irreducible

Premises used: WithZero.exp_lt_exp, Int.ofNat_lt, neg_lt_zero, Int.ofNat_zero, zero_lt_iff, WithZero.exp_zero
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
r : R
rw : v.asIdeal ∣ Ideal.span {r} ↔ r ∈ v.asIdeal
rw₁ rw₂ : r ∈ v.asIdeal ↔ r ∈ v.asIdeal
⊢ LT.lt (α := WithZero (Multiplicative ℤ))
      ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
      (1 : WithZero (Multiplicative ℤ)) ↔
    r ∈ v.asIdeal

Tactic:
rw [intValuation_lt_one_iff_dvd, Ideal.dvd_span_singleton]

Premises used: IsDedekindDomain.HeightOneSpectrum.intValuation_lt_one_iff_dvd, Ideal.dvd_span_singleton
-- 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))
      (1 : WithZero (Multiplicative ℤ)) ↔
    r ∈ v.asIdeal.primeCompl

Tactic:
simp [Ideal.primeCompl, ← intValuation_lt_one_iff_mem, LE.le.ge_iff_eq (intValuation_le_one v r)]

Premises used: IsDedekindDomain.HeightOneSpectrum.intValuation_le_one, LE.le.ge_iff_eq, Ideal.primeCompl, IsDedekindDomain.HeightOneSpectrum.intValuation_lt_one_iff_mem
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
r : R
n : ℕ
pos :
  ∀ (hr : r = 0),
    LE.le (α := WithZero (Multiplicative ℤ))
        ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
        (exp (-(↑n : ℤ)) : WithZero (Multiplicative ℤ)) ↔
      v.asIdeal ^ n ∣ Ideal.span {r}
neg :
  ∀ (hr : ¬r = 0),
    LE.le (α := WithZero (Multiplicative ℤ))
        ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
        (exp (-(↑n : ℤ)) : WithZero (Multiplicative ℤ)) ↔
      v.asIdeal ^ n ∣ Ideal.span {r}
⊢ LE.le (α := WithZero (Multiplicative ℤ))
      ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
      (exp (-(↑n : ℤ)) : WithZero (Multiplicative ℤ)) ↔
    v.asIdeal ^ n ∣ Ideal.span {r}

Tactic:
by_cases hr : r = 0
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
r : R
n : ℕ
hr : ¬r = 0
neg :
  LE.le (α := WithZero (Multiplicative ℤ))
      (exp (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r})))) : ℤ)))
      (exp (-(↑n : ℤ))) ↔
    v.asIdeal ^ n ∣ Ideal.span {r}
neg₁ :
  LE.le (α := ℤ)
      (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r})))) : ℤ))
      (-(↑n : ℤ)) ↔
    v.asIdeal ^ n ∣ Ideal.span {r}
neg₂ :
  LE.le (α := ℤ) (↑n : ℤ)
      (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r})))) : ℤ) ↔
    v.asIdeal ^ n ∣ Ideal.span {r}
neg₃ :
  n ≤ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r}))) ↔
    v.asIdeal ^ n ∣ Ideal.span {r}
neg₄ :
  n ≤ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r}))) ↔
    r ∈ v.asIdeal ^ n
neg₅ :
  n ≤ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r}))) ↔
    HPow.hPow (α := Associates (Ideal R)) (Associates.mk v.asIdeal) n ≤ Associates.mk (Ideal.span {r})
neg₆ neg₇ :
  n ≤ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r}))) ↔
    n ≤ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r})))
⊢ LE.le (α := WithZero (Multiplicative ℤ))
      ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
      (exp (-(↑n : ℤ)) : WithZero (Multiplicative ℤ)) ↔
    v.asIdeal ^ n ∣ Ideal.span {r}

Tactic:
rw [v.intValuation_if_neg hr, exp_le_exp, neg_le_neg_iff, Int.ofNat_le,
      Ideal.dvd_span_singleton, ← Associates.le_singleton_iff,
      Associates.prime_pow_dvd_iff_le (Associates.mk_ne_zero'.mpr hr) v.associates_irreducible]

Premises used: Iff.mpr, Associates.prime_pow_dvd_iff_le, IsDedekindDomain.HeightOneSpectrum.intValuation_if_neg, Associates.le_singleton_iff, WithZero.exp_le_exp, Associates.mk_ne_zero', Int.ofNat_le, Ideal.dvd_span_singleton, IsDedekindDomain.HeightOneSpectrum.associates_irreducible, neg_le_neg_iff
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
r : R
n : ℕ
pos :
  ∀ (hr : r = 0),
    LE.le (α := WithZero (Multiplicative ℤ))
        ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
        (exp (-(↑n : ℤ)) : WithZero (Multiplicative ℤ)) ↔
      v.asIdeal ^ n ∣ Ideal.span {r}
neg neg₁ :
  ∀ (hr : ¬r = 0),
    LE.le (α := WithZero (Multiplicative ℤ))
        ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
        (exp (-(↑n : ℤ)) : WithZero (Multiplicative ℤ)) ↔
      v.asIdeal ^ n ∣ Ideal.span {r}
⊢ LE.le (α := WithZero (Multiplicative ℤ))
      ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
      (exp (-(↑n : ℤ)) : WithZero (Multiplicative ℤ)) ↔
    v.asIdeal ^ n ∣ Ideal.span {r}

Tactic:
classical
  by_cases hr : r = 0
  · simp_rw [hr, Valuation.map_zero, Ideal.dvd_span_singleton, zero_le', Submodule.zero_mem]
  · rw [v.intValuation_if_neg hr, exp_le_exp, neg_le_neg_iff, Int.ofNat_le,
      Ideal.dvd_span_singleton, ← Associates.le_singleton_iff,
      Associates.prime_pow_dvd_iff_le (Associates.mk_ne_zero'.mpr hr) v.associates_irreducible]

Premises used: Associates.le_singleton_iff, WithZero.exp_le_exp, Submodule.zero_mem, zero_le', Int.ofNat_le, Ideal.dvd_span_singleton, Valuation.map_zero, neg_le_neg_iff
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
r : R
n : ℕ
rw : v.asIdeal ^ n ∣ Ideal.span {r} ↔ r ∈ v.asIdeal ^ n
rw₁ rw₂ : r ∈ v.asIdeal ^ n ↔ r ∈ v.asIdeal ^ n
⊢ LE.le (α := WithZero (Multiplicative ℤ))
      ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
      (exp (-(↑n : ℤ)) : WithZero (Multiplicative ℤ)) ↔
    r ∈ v.asIdeal ^ n

Tactic:
rw [intValuation_le_pow_iff_dvd, Ideal.dvd_span_singleton]

Premises used: IsDedekindDomain.HeightOneSpectrum.intValuation_le_pow_iff_dvd, Ideal.dvd_span_singleton
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
r : R
n : ℕ
rw : v.asIdeal ^ n ∣ Ideal.span {r} ↔ (↑n : ℕ∞) ≤ emultiplicity v.asIdeal (Ideal.span {r})
rw₁ rw₂ : (↑n : ℕ∞) ≤ emultiplicity v.asIdeal (Ideal.span {r}) ↔ (↑n : ℕ∞) ≤ emultiplicity v.asIdeal (Ideal.span {r})
⊢ LE.le (α := WithZero (Multiplicative ℤ))
      ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
      (exp (-(↑n : ℤ)) : WithZero (Multiplicative ℤ)) ↔
    (↑n : ℕ∞) ≤ emultiplicity v.asIdeal (Ideal.span {r})

Tactic:
rw [intValuation_le_pow_iff_dvd, pow_dvd_iff_le_emultiplicity]

Premises used: IsDedekindDomain.HeightOneSpectrum.intValuation_le_pow_iff_dvd, pow_dvd_iff_le_emultiplicity
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
r : R
n : ℕ
⊢ LT.lt (α := WithZero (Multiplicative ℤ)) 0 (exp 1)

Tactic:
simp
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
r : R
n : ℕ
rw :
  LE.le (α := WithZero (Multiplicative ℤ)) (exp (-(↑n : ℤ)))
      ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r) ↔
    exp (-(↑n : ℤ)) <
      HMul.hMul (α := WithZero (Multiplicative ℤ)) (β := WithZero (Multiplicative ℤ))
        ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ)) (exp 1)
⊢ LE.le (α := WithZero (Multiplicative ℤ)) (exp (-(↑n : ℤ)))
      ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r) ↔
    emultiplicity v.asIdeal (Ideal.span {r}) ≤ (↑n : ℕ∞)

Tactic:
rw [← ENat.lt_coe_add_one_iff, ← ENat.coe_one, ← ENat.coe_add, emultiplicity_lt_iff_not_dvd,
    ← intValuation_le_pow_iff_dvd, not_le, Nat.cast_add, Nat.cast_one, neg_add, exp_add,
    exp_neg 1, mul_inv_lt_iff₀ (by simp)]

Premises used: WithZero.exp_add, not_le, ENat.coe_one, emultiplicity_lt_iff_not_dvd, mul_inv_lt_iff₀, IsDedekindDomain.HeightOneSpectrum.intValuation_le_pow_iff_dvd, neg_add, Nat.cast_add, OfNat.ofNat, Nat.cast_one, ENat.coe_add, ENat.lt_coe_add_one_iff, WithZero.exp_neg
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
r : R
n : ℕ
pos :
  ∀
    (hv :
      Eq (α := WithZero (Multiplicative ℤ))
        ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
        (0 : WithZero (Multiplicative ℤ))),
    LE.le (α := WithZero (Multiplicative ℤ)) (exp (-(↑n : ℤ)))
        ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r) ↔
      exp (-(↑n : ℤ)) <
        HMul.hMul (α := WithZero (Multiplicative ℤ)) (β := WithZero (Multiplicative ℤ))
          ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ)) (exp 1)
neg :
  ∀
    (hv :
      ¬Eq (α := WithZero (Multiplicative ℤ))
          ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
          (0 : WithZero (Multiplicative ℤ))),
    LE.le (α := WithZero (Multiplicative ℤ)) (exp (-(↑n : ℤ)))
        ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r) ↔
      exp (-(↑n : ℤ)) <
        HMul.hMul (α := WithZero (Multiplicative ℤ)) (β := WithZero (Multiplicative ℤ))
          ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ)) (exp 1)
⊢ LE.le (α := WithZero (Multiplicative ℤ)) (exp (-(↑n : ℤ)))
      ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r) ↔
    exp (-(↑n : ℤ)) <
      HMul.hMul (α := WithZero (Multiplicative ℤ)) (β := WithZero (Multiplicative ℤ))
        ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ)) (exp 1)

Tactic:
by_cases hv : v.intValuation r = 0

Premises used: IsDedekindDomain.HeightOneSpectrum.intValuation, DFunLike.coe
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
r : R
n : ℕ
hv :
  Eq (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
    (0 : WithZero (Multiplicative ℤ))
⊢ LE.le (α := WithZero (Multiplicative ℤ)) (exp (-(↑n : ℤ)))
      ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r) ↔
    exp (-(↑n : ℤ)) <
      HMul.hMul (α := WithZero (Multiplicative ℤ)) (β := WithZero (Multiplicative ℤ))
        ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ)) (exp 1)

Tactic:
simp [hv]
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
r : R
n : ℕ
hv :
  ¬Eq (α := WithZero (Multiplicative ℤ))
      ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
      (0 : WithZero (Multiplicative ℤ))
neg neg₁ :
  LE.le (α := WithZero (Multiplicative ℤ)) (exp (-(↑n : ℤ)))
      ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r) ↔
    LE.le (α := WithZero (Multiplicative ℤ)) (exp (-(↑n : ℤ)))
      ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r)
⊢ LE.le (α := WithZero (Multiplicative ℤ)) (exp (-(↑n : ℤ)))
      ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r) ↔
    exp (-(↑n : ℤ)) <
      HMul.hMul (α := WithZero (Multiplicative ℤ)) (β := WithZero (Multiplicative ℤ))
        ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ)) (exp 1)

Tactic:
rw [lt_mul_exp_iff_le hv]

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

Tactic:
have hv : Irreducible (Associates.mk v.asIdeal) := v.associates_irreducible
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
hv : Irreducible (M := Associates (Ideal R)) (Associates.mk v.asIdeal)
rw : DvdNotUnit v.asIdeal (HPow.hPow (β := ℕ) v.asIdeal 2)
⊢ HPow.hPow (β := ℕ) v.asIdeal 2 < v.asIdeal

Tactic:
rw [← Ideal.dvdNotUnit_iff_lt]

Premises used: Ideal.dvdNotUnit_iff_lt
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
hv : Irreducible (M := Associates (Ideal R)) (Associates.mk v.asIdeal)
⊢ DvdNotUnit v.asIdeal (HPow.hPow (β := ℕ) v.asIdeal 2)

Tactic:
⟨v.ne_bot, v.asIdeal, Ideal.isUnit_iff.not.mpr v.isPrime.ne_top, sq v.asIdeal⟩

Premises used: Iff.mpr, IsDedekindDomain.HeightOneSpectrum.asIdeal, Iff.not, sq, DvdNotUnit, And.intro, IsDedekindDomain.HeightOneSpectrum.ne_bot, Exists.intro, Ideal.isUnit_iff, IsDedekindDomain.HeightOneSpectrum.isPrime, Ideal.IsPrime.ne_top
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
hv : Irreducible (M := Associates (Ideal R)) (Associates.mk v.asIdeal)
⊢ DvdNotUnit v.asIdeal (HPow.hPow (β := ℕ) v.asIdeal 2)

Tactic:
exact ⟨v.ne_bot, v.asIdeal, Ideal.isUnit_iff.not.mpr v.isPrime.ne_top, sq v.asIdeal⟩
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
hv : Irreducible (M := Associates (Ideal R)) (Associates.mk v.asIdeal)
⊢ HPow.hPow (β := ℕ) v.asIdeal 2 < v.asIdeal

Tactic:
have hlt : v.asIdeal ^ 2 < v.asIdeal := by
    rw [← Ideal.dvdNotUnit_iff_lt]
    exact ⟨v.ne_bot, v.asIdeal, Ideal.isUnit_iff.not.mpr v.isPrime.ne_top, sq v.asIdeal⟩
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
hv : Irreducible (M := Associates (Ideal R)) (Associates.mk v.asIdeal)
hlt : HPow.hPow (β := ℕ) v.asIdeal 2 < v.asIdeal
obtain :
  ∀ (π : R) (mem : π ∈ v.asIdeal) (notMem : π ∉ HPow.hPow (β := ℕ) v.asIdeal 2),
    ∃ (π : R),
      Eq (α := WithZero (Multiplicative ℤ))
        ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) π : WithZero (Multiplicative ℤ))
        (exp (-1) : WithZero (Multiplicative ℤ))
⊢ ∃ (π : R),
    Eq (α := WithZero (Multiplicative ℤ))
      ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) π : WithZero (Multiplicative ℤ))
      (exp (-1) : WithZero (Multiplicative ℤ))

Tactic:
obtain ⟨π, mem, notMem⟩ := SetLike.exists_of_lt hlt

Premises used: SetLike.exists_of_lt
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
hv : Irreducible (M := Associates (Ideal R)) (Associates.mk v.asIdeal)
hlt : HPow.hPow (β := ℕ) v.asIdeal 2 < v.asIdeal
π : R
mem : π ∈ v.asIdeal
notMem : π ∉ HPow.hPow (β := ℕ) v.asIdeal 2
rw : π ≠ 0
⊢ Ne (α := Associates (Ideal R)) (Associates.mk (Ideal.span {π})) 0

Tactic:
rw [Associates.mk_ne_zero']

Premises used: Associates.mk_ne_zero'
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
hv : Irreducible (M := Associates (Ideal R)) (Associates.mk v.asIdeal)
hlt : HPow.hPow (β := ℕ) v.asIdeal 2 < v.asIdeal
π : R
mem : π ∈ v.asIdeal
notMem : π ∉ HPow.hPow (β := ℕ) v.asIdeal 2
h : π = 0
⊢ 0 ∉ HPow.hPow (β := ℕ) v.asIdeal 2

Tactic:
rw [h] at notMem
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
hv : Irreducible (M := Associates (Ideal R)) (Associates.mk v.asIdeal)
hlt : HPow.hPow (β := ℕ) v.asIdeal 2 < v.asIdeal
π : R
mem : π ∈ v.asIdeal
notMem : π ∉ HPow.hPow (β := ℕ) v.asIdeal 2
h : π = 0
notMem₁ : 0 ∉ HPow.hPow (β := ℕ) v.asIdeal 2
⊢ π = 0

Tactic:
rw [h] at notMem
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
hv : Irreducible (M := Associates (Ideal R)) (Associates.mk v.asIdeal)
hlt : HPow.hPow (β := ℕ) v.asIdeal 2 < v.asIdeal
π : R
mem : π ∈ v.asIdeal
notMem : 0 ∉ HPow.hPow (β := ℕ) v.asIdeal 2
h : π = 0
⊢ False

Tactic:
notMem (Submodule.zero_mem (v.asIdeal ^ 2))

Premises used: IsDedekindDomain.HeightOneSpectrum.asIdeal, HPow.hPow, Submodule.zero_mem, OfNat.ofNat