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

Premises used: IsDedekindDomain.HeightOneSpectrum.asIdeal, HPow.hPow, Submodule.zero_mem, OfNat.ofNat
-- 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
⊢ Ne (α := Associates (Ideal R)) (Associates.mk (Ideal.span {π})) 0

Tactic:
have hπ : Associates.mk (Ideal.span {π}) ≠ 0 := by
    rw [Associates.mk_ne_zero']
    intro h
    rw [h] at notMem
    exact notMem (Submodule.zero_mem (v.asIdeal ^ 2))
-- 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
hπ : Ne (α := Associates (Ideal R)) (Associates.mk (Ideal.span {π})) 0
h :
  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:
use π
-- 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
hπ : Ne (α := Associates (Ideal R)) (Associates.mk (Ideal.span {π})) 0
h :
  Eq (α := ℤ)
    (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {π})))) : ℤ)) (-1)
⊢ Eq (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) π : WithZero (Multiplicative ℤ))
    (exp (-1) : WithZero (Multiplicative ℤ))

Tactic:
rw [intValuation_if_neg _ (Associates.mk_ne_zero'.mp hπ), exp_inj]

Premises used: WithZero.exp_inj, IsDedekindDomain.HeightOneSpectrum.intValuation_if_neg, Associates.mk_ne_zero', Iff.mp
-- 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
hπ : Ne (α := Associates (Ideal R)) (Associates.mk (Ideal.span {π})) 0
h :
  Eq (α := ℤ) (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {π})))) : ℤ)
    1
⊢ Eq (α := ℤ)
    (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {π})))) : ℤ)) (-1)

Tactic:
apply congr_arg

Premises used: congr_arg
-- 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
hπ : Ne (α := Associates (Ideal R)) (Associates.mk (Ideal.span {π})) 0
h : Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {π}))) = 1
⊢ Eq (α := ℤ) (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {π})))) : ℤ)
    1

Tactic:
rw [← Int.ofNat_one, Int.natCast_inj]

Premises used: Int.ofNat_one, Int.natCast_inj
-- 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
hπ : Ne (α := Associates (Ideal R)) (Associates.mk (Ideal.span {π})) 0
⊢ LE.le (α := Associates (Ideal R)) (Associates.mk v.asIdeal) (Associates.mk (Ideal.span {π}))

Tactic:
rw [← Ideal.dvd_span_singleton, ← Associates.mk_le_mk_iff_dvd] at mem notMem

Premises used: Associates.mk_le_mk_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
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
hπ : Ne (α := Associates (Ideal R)) (Associates.mk (Ideal.span {π})) 0
mem₁ : LE.le (α := Associates (Ideal R)) (Associates.mk v.asIdeal) (Associates.mk (Ideal.span {π}))
⊢ ¬LE.le (α := Associates (Ideal R)) (Associates.mk (HPow.hPow (β := ℕ) v.asIdeal 2)) (Associates.mk (Ideal.span {π}))

Tactic:
rw [← Ideal.dvd_span_singleton, ← Associates.mk_le_mk_iff_dvd] at mem notMem

Premises used: Associates.mk_le_mk_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
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
hπ : Ne (α := Associates (Ideal R)) (Associates.mk (Ideal.span {π})) 0
mem₁ : LE.le (α := Associates (Ideal R)) (Associates.mk v.asIdeal) (Associates.mk (Ideal.span {π}))
notMem₁ :
  ¬LE.le (α := Associates (Ideal R)) (Associates.mk (HPow.hPow (β := ℕ) v.asIdeal 2)) (Associates.mk (Ideal.span {π}))
⊢ Ne (α := Associates (Ideal R)) (Associates.mk (Ideal.span {π})) 0

Tactic:
rw [← Ideal.dvd_span_singleton, ← Associates.mk_le_mk_iff_dvd] at mem notMem

Premises used: Associates.mk_le_mk_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
hv : Irreducible (M := Associates (Ideal R)) (Associates.mk v.asIdeal)
hlt : HPow.hPow (β := ℕ) v.asIdeal 2 < v.asIdeal
π : R
mem : LE.le (α := Associates (Ideal R)) (Associates.mk v.asIdeal) (Associates.mk (Ideal.span {π}))
notMem :
  ¬LE.le (α := Associates (Ideal R)) (Associates.mk (HPow.hPow (β := ℕ) v.asIdeal 2)) (Associates.mk (Ideal.span {π}))
hπ : Ne (α := Associates (Ideal R)) (Associates.mk (Ideal.span {π})) 0
⊢ 1 ≤ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {π})))

Tactic:
rw [← pow_one (Associates.mk v.asIdeal), Associates.prime_pow_dvd_iff_le hπ hv] at mem

Premises used: Associates.mk, Associates.prime_pow_dvd_iff_le, IsDedekindDomain.HeightOneSpectrum.asIdeal, pow_one
-- 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 : LE.le (α := Associates (Ideal R)) (Associates.mk v.asIdeal) (Associates.mk (Ideal.span {π}))
notMem :
  ¬LE.le (α := Associates (Ideal R)) (Associates.mk (HPow.hPow (β := ℕ) v.asIdeal 2)) (Associates.mk (Ideal.span {π}))
hπ : Ne (α := Associates (Ideal R)) (Associates.mk (Ideal.span {π})) 0
mem₁ : 1 ≤ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {π})))
⊢ ¬LE.le (α := Associates (Ideal R)) (Associates.mk (HPow.hPow (β := ℕ) v.asIdeal 2)) (Associates.mk (Ideal.span {π}))

Tactic:
rw [← pow_one (Associates.mk v.asIdeal), Associates.prime_pow_dvd_iff_le hπ hv] at mem

Premises used: Associates.mk, Associates.prime_pow_dvd_iff_le, IsDedekindDomain.HeightOneSpectrum.asIdeal, pow_one
-- 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 : LE.le (α := Associates (Ideal R)) (Associates.mk v.asIdeal) (Associates.mk (Ideal.span {π}))
notMem :
  ¬LE.le (α := Associates (Ideal R)) (Associates.mk (HPow.hPow (β := ℕ) v.asIdeal 2)) (Associates.mk (Ideal.span {π}))
hπ : Ne (α := Associates (Ideal R)) (Associates.mk (Ideal.span {π})) 0
mem₁ : 1 ≤ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {π})))
notMem₁ :
  ¬LE.le (α := Associates (Ideal R)) (Associates.mk (HPow.hPow (β := ℕ) v.asIdeal 2)) (Associates.mk (Ideal.span {π}))
⊢ Ne (α := Associates (Ideal R)) (Associates.mk (Ideal.span {π})) 0

Tactic:
rw [← pow_one (Associates.mk v.asIdeal), Associates.prime_pow_dvd_iff_le hπ hv] at mem

Premises used: Associates.mk, Associates.prime_pow_dvd_iff_le, IsDedekindDomain.HeightOneSpectrum.asIdeal, pow_one
-- 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 : 1 ≤ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {π})))
notMem :
  ¬LE.le (α := Associates (Ideal R)) (Associates.mk (HPow.hPow (β := ℕ) v.asIdeal 2)) (Associates.mk (Ideal.span {π}))
hπ : Ne (α := Associates (Ideal R)) (Associates.mk (Ideal.span {π})) 0
⊢ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {π}))) < 2

Tactic:
rw [Associates.mk_pow, Associates.prime_pow_dvd_iff_le hπ hv, not_le] at notMem

Premises used: not_le, Associates.prime_pow_dvd_iff_le, Associates.mk_pow
-- 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 : 1 ≤ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {π})))
notMem :
  ¬LE.le (α := Associates (Ideal R)) (Associates.mk (HPow.hPow (β := ℕ) v.asIdeal 2)) (Associates.mk (Ideal.span {π}))
hπ : Ne (α := Associates (Ideal R)) (Associates.mk (Ideal.span {π})) 0
notMem₁ : Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {π}))) < 2
⊢ Ne (α := Associates (Ideal R)) (Associates.mk (Ideal.span {π})) 0

Tactic:
rw [Associates.mk_pow, Associates.prime_pow_dvd_iff_le hπ hv, not_le] at notMem

Premises used: not_le, Associates.prime_pow_dvd_iff_le, Associates.mk_pow
-- 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 : 1 ≤ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {π})))
notMem : Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {π}))) < 2
hπ : Ne (α := Associates (Ideal R)) (Associates.mk (Ideal.span {π})) 0
⊢ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {π}))) = 1

Tactic:
Nat.eq_of_le_of_lt_succ mem notMem

Premises used: Nat.eq_of_le_of_lt_succ
-- 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 : 1 ≤ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {π})))
notMem : Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {π}))) < 2
hπ : Ne (α := Associates (Ideal R)) (Associates.mk (Ideal.span {π})) 0
⊢ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {π}))) = 1

Tactic:
exact Nat.eq_of_le_of_lt_succ mem notMem

Premises used: Nat.eq_of_le_of_lt_succ
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
K : Type u_2
S : Type u_3
inst_2 : Field K
inst_3 : CommSemiring S
inst_4 : Algebra R K
inst_5 : IsFractionRing R K
v : HeightOneSpectrum R
π : R
hπ :
  Eq (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) π : WithZero (Multiplicative ℤ))
    (exp (-1) : WithZero (Multiplicative ℤ))
⊢ Ne (α := WithZero (Multiplicative ℤ))
      ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) π : WithZero (Multiplicative ℤ))
      (0 : WithZero (Multiplicative ℤ)) ∧
    Ne (α := WithZero (Multiplicative ℤ))
      ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) π : WithZero (Multiplicative ℤ))
      (1 : WithZero (Multiplicative ℤ))

Tactic:
aesop
-- 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
hv : v.asIdeal = Ideal.span {r}
rw : Irreducible (M := Associates (Ideal R)) (Associates.mk v.asIdeal)
⊢ Eq (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
    (exp (-1) : WithZero (Multiplicative ℤ))

Tactic:
rw [v.intValuation_if_neg hr, ← hv, Associates.count_self, Int.ofNat_one]

Premises used: IsDedekindDomain.HeightOneSpectrum.intValuation_if_neg, Associates.count_self, Int.ofNat_one
-- 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
hv : v.asIdeal = Ideal.span {r}
⊢ 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
r : R
hr : r ≠ 0
hv : v.asIdeal = Ideal.span {r}
⊢ 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
r : R
hr : r ≠ 0
hv : 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 (-1))
rw₁ :
  Eq (α := WithZero (Multiplicative ℤ))
    (exp (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk v.asIdeal))) : ℤ)))
    (exp (-1))
rw₂ : Eq (α := WithZero (Multiplicative ℤ)) (exp (-(↑1 : ℤ))) (exp (-1))
rw₃ : Irreducible (M := Associates (Ideal R)) (Associates.mk v.asIdeal)
rw₄ : Eq (α := WithZero (Multiplicative ℤ)) (exp (-1)) (exp (-1))
rw₅ : Irreducible (M := Associates (Ideal R)) (Associates.mk v.asIdeal)
rw₆ : Eq (α := WithZero (Multiplicative ℤ)) (exp (-1)) (exp (-1))
rw₇ rw₈ : Irreducible (M := Associates (Ideal R)) (Associates.mk v.asIdeal)
⊢ Eq (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
    (exp (-1) : WithZero (Multiplicative ℤ))

Tactic:
classical
  rw [v.intValuation_if_neg hr, ← hv, Associates.count_self, Int.ofNat_one]
  exact v.associates_irreducible

Premises used: IsDedekindDomain.HeightOneSpectrum.intValuation_if_neg, Associates.count_self, Int.ofNat_one, 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 : R
h :
  Eq (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) x : WithZero (Multiplicative ℤ))
    (1 : WithZero (Multiplicative ℤ))
⊢ x ∉ v.asIdeal

Tactic:
simp [← (intValuation_lt_one_iff_mem _ _).not, h]

Premises used: Iff.not, 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
x : R
refine :
  ∀ (h : x ∉ v.asIdeal),
    Eq (α := WithZero (Multiplicative ℤ))
      ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) x : WithZero (Multiplicative ℤ))
      (1 : WithZero (Multiplicative ℤ))
⊢ Eq (α := WithZero (Multiplicative ℤ))
      ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) x : WithZero (Multiplicative ℤ))
      (1 : WithZero (Multiplicative ℤ)) ↔
    x ∉ v.asIdeal

Tactic:
refine ⟨fun h ↦ by simp [← (intValuation_lt_one_iff_mem _ _).not, h], fun h ↦ ?_⟩
-- Module: Mathlib.RingTheory.DedekindDomain.AdicValuation
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
x : R
h : x ∉ v.asIdeal
⊢ LE.le (α := WithZero (Multiplicative ℤ)) (1 : WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) x : WithZero (Multiplicative ℤ))

Tactic:
simp [← not_lt, (v.intValuation_lt_one_iff_mem _).not, h]

Premises used: Iff.not, not_lt, 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
x : R
h : x ∉ v.asIdeal
⊢ Eq (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) x : WithZero (Multiplicative ℤ))
    (1 : WithZero (Multiplicative ℤ))

Tactic:
exact le_antisymm (v.intValuation_le_one x) <| by
    simp [← not_lt, (v.intValuation_lt_one_iff_mem _).not, h]

Premises used: IsDedekindDomain.HeightOneSpectrum.intValuation_le_one, le_antisymm
-- 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
⊢ Eq (α := WithZero (Multiplicative ℤ))
    (((valuation K v : Valuation K (WithZero (Multiplicative ℤ))) : (a : K) → WithZero (Multiplicative ℤ)) x :
      WithZero (Multiplicative ℤ))
    (((v.intValuation.extendToLocalization (S := nonZeroDivisors R) ⋯ K : Valuation K (WithZero (Multiplicative ℤ))) :
          (a : K) → WithZero (Multiplicative ℤ))
        x :
      WithZero (Multiplicative ℤ))

Tactic:
rfl

Premises used: rfl
-- 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
r : R
s : ↥(nonZeroDivisors R)
rw :
  ((v.intValuation.extendToLocalization (S := nonZeroDivisors R) ⋯ K : Valuation K (WithZero (Multiplicative ℤ))) :
        (a : K) → WithZero (Multiplicative ℤ))
      (IsLocalization.mk' K r s) =
    HDiv.hDiv (α := WithZero (Multiplicative ℤ)) (β := WithZero (Multiplicative ℤ))
      ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
      ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) (↑s : R) : WithZero (Multiplicative ℤ))
rw₁ :
  HMul.hMul (α := WithZero (Multiplicative ℤ)) (β := WithZero (Multiplicative ℤ))
      ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
      (((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) (↑s : R))⁻¹ : WithZero (Multiplicative ℤ)) =
    HDiv.hDiv (α := WithZero (Multiplicative ℤ)) (β := WithZero (Multiplicative ℤ))
      ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
      ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) (↑s : R) : WithZero (Multiplicative ℤ))
rw₂ rw₃ :
  HMul.hMul (α := WithZero (Multiplicative ℤ)) (β := WithZero (Multiplicative ℤ))
      ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
      (((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) (↑s : R))⁻¹ : WithZero (Multiplicative ℤ)) =
    HMul.hMul (α := WithZero (Multiplicative ℤ)) (β := WithZero (Multiplicative ℤ))
      ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
      (((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) (↑s : R))⁻¹ : WithZero (Multiplicative ℤ))
⊢ ((valuation K v : Valuation K (WithZero (Multiplicative ℤ))) : (a : K) → WithZero (Multiplicative ℤ))
      (IsLocalization.mk' K r s) =
    HDiv.hDiv (α := WithZero (Multiplicative ℤ)) (β := WithZero (Multiplicative ℤ))
      ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
      ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) (↑s : R) : WithZero (Multiplicative ℤ))

Tactic:
rw [valuation_def, Valuation.extendToLocalization_mk', div_eq_mul_inv]

Premises used: IsDedekindDomain.HeightOneSpectrum.valuation_def, div_eq_mul_inv, Valuation.extendToLocalization_mk'
-- 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
r : R
rw :
  Eq (α := WithZero (Multiplicative ℤ))
    (((v.intValuation.extendToLocalization (S := nonZeroDivisors R) ⋯ K : Valuation K (WithZero (Multiplicative ℤ))) :
          (a : K) → WithZero (Multiplicative ℤ))
        (↑r : K) :
      WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
rw₁ rw₂ :
  Eq (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
⊢ Eq (α := WithZero (Multiplicative ℤ))
    (((valuation K v : Valuation K (WithZero (Multiplicative ℤ))) : (a : K) → WithZero (Multiplicative ℤ)) (↑r : K) :
      WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))

Tactic:
rw [valuation_def, Valuation.extendToLocalization_apply_map_apply]

Premises used: IsDedekindDomain.HeightOneSpectrum.valuation_def, Valuation.extendToLocalization_apply_map_apply
-- 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
r : R
rw :
  LE.le (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
    (1 : WithZero (Multiplicative ℤ))
⊢ LE.le (α := WithZero (Multiplicative ℤ))
    (((valuation K v : Valuation K (WithZero (Multiplicative ℤ))) : (a : K) → WithZero (Multiplicative ℤ)) (↑r : K) :
      WithZero (Multiplicative ℤ))
    (1 : WithZero (Multiplicative ℤ))

Tactic:
rw [valuation_of_algebraMap]

Premises used: IsDedekindDomain.HeightOneSpectrum.valuation_of_algebraMap