awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.RingTheory.DiscreteValuationRing.TFAE
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsNoetherianRing R
inst_2 : IsLocalRing R
inst_3 : IsDomain R
I : Ideal R
hI : I ≠ ⊥
h : ¬IsField R
x : R
hx : Eq (α := Ideal R) (maximalIdeal R) (Ideal.span {x})
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this : x ≠ 0
hx' : Irreducible x
f : Multiset R
hf₁ : ∀ b ∈ f, Irreducible b
hf₂ : f ≠ ∅
hr₁ : f.prod ≠ 0
hr₂ : f.prod ∈ nonunits R
b : R
hb : b ∈ f
⊢ Associated x b

Tactic:
Irreducible.associated_of_dvd hx' (hf₁ b hb) ((H b).mp (hf₁ b hb).1)

Premises used: Irreducible.associated_of_dvd, Irreducible.not_isUnit, Iff.mp
-- Module: Mathlib.RingTheory.DiscreteValuationRing.TFAE
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsNoetherianRing R
inst_2 : IsLocalRing R
inst_3 : IsDomain R
I : Ideal R
hI : I ≠ ⊥
h : ¬IsField R
x : R
hx : Eq (α := Ideal R) (maximalIdeal R) (Ideal.span {x})
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this : x ≠ 0
hx' : Irreducible x
f : Multiset R
hf₁ : ∀ b ∈ f, Irreducible b
hf₂ : f ≠ ∅
hr₁ : f.prod ≠ 0
hr₂ : f.prod ∈ nonunits R
b : R
hb : b ∈ f
⊢ Associated x b

Tactic:
exact Irreducible.associated_of_dvd hx' (hf₁ b hb) ((H b).mp (hf₁ b hb).1)

Premises used: Irreducible.associated_of_dvd, Irreducible.not_isUnit, Iff.mp
-- Module: Mathlib.RingTheory.DiscreteValuationRing.TFAE
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsNoetherianRing R
inst_2 : IsLocalRing R
inst_3 : IsDomain R
I : Ideal R
hI : I ≠ ⊥
h : ¬IsField R
x : R
hx : Eq (α := Ideal R) (maximalIdeal R) (Ideal.span {x})
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this : x ≠ 0
hx' : Irreducible x
f : Multiset R
hf₁ : ∀ b ∈ f, Irreducible b
hf₂ : f ≠ ∅
hr₁ : f.prod ≠ 0
hr₂ : f.prod ∈ nonunits R
⊢ ∀ b ∈ f, Associated x b

Tactic:
have : ∀ b ∈ f, Associated x b := by
      intro b hb
      exact Irreducible.associated_of_dvd hx' (hf₁ b hb) ((H b).mp (hf₁ b hb).1)
-- Module: Mathlib.RingTheory.DiscreteValuationRing.TFAE
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsNoetherianRing R
inst_2 : IsLocalRing R
inst_3 : IsDomain R
I : Ideal R
hI : I ≠ ⊥
h : ¬IsField R
x : R
hx : Eq (α := Ideal R) (maximalIdeal R) (Ideal.span {x})
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this_1 : x ≠ 0
hx' : Irreducible x
hf₂ : Ne.{u_1 + 1} (α := Multiset R) 0 ∅
this : ∀ (b : R), Membership.mem (γ := Multiset R) 0 b → Associated x b
⊢ ∃ (n : ℕ), Associated (x ^ n) (Multiset.prod 0)

Tactic:
(hf₂ rfl).elim

Premises used: False.elim, rfl
-- Module: Mathlib.RingTheory.DiscreteValuationRing.TFAE
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsNoetherianRing R
inst_2 : IsLocalRing R
inst_3 : IsDomain R
I : Ideal R
hI : I ≠ ⊥
h : ¬IsField R
x : R
hx : Eq (α := Ideal R) (maximalIdeal R) (Ideal.span {x})
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this_1 : x ≠ 0
hx' : Irreducible x
hf₂ : Ne.{u_1 + 1} (α := Multiset R) 0 ∅
this : ∀ (b : R), Membership.mem (γ := Multiset R) 0 b → Associated x b
⊢ ∃ (n : ℕ), Associated (x ^ n) (Multiset.prod 0)

Tactic:
exact (hf₂ rfl).elim

Premises used: False.elim, rfl
-- Module: Mathlib.RingTheory.DiscreteValuationRing.TFAE
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsNoetherianRing R
inst_2 : IsLocalRing R
inst_3 : IsDomain R
I : Ideal R
hI : I ≠ ⊥
h : ¬IsField R
x : R
hx : Eq (α := Ideal R) (maximalIdeal R) (Ideal.span {x})
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this_1 : x ≠ 0
hx' : Irreducible x
fa : R
fs : Multiset R
fh : ∀ (hf₂ : fs ≠ ∅) (this : ∀ b ∈ fs, Associated x b), ∃ (n : ℕ), Associated (x ^ n) fs.prod
hf₂ : fa ::ₘ fs ≠ ∅
this : ∀ b ∈ fa ::ₘ fs, Associated x b
⊢ ∃ (n : ℕ), Associated (x ^ n) (fa ::ₘ fs).prod

Tactic:
?_
-- Module: Mathlib.RingTheory.DiscreteValuationRing.TFAE
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsNoetherianRing R
inst_2 : IsLocalRing R
inst_3 : IsDomain R
I : Ideal R
hI : I ≠ ⊥
h : ¬IsField R
x : R
hx : Eq (α := Ideal R) (maximalIdeal R) (Ideal.span {x})
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this_1 : x ≠ 0
hx' : Irreducible x
f : Multiset R
hf₂ : f ≠ ∅
this : ∀ b ∈ f, Associated x b
cons :
  ∀ (fa : R) (fs : Multiset R)
    (fh : ∀ (hf₂ : fs ≠ ∅) (this : ∀ b ∈ fs, Associated x b), ∃ (n : ℕ), Associated (x ^ n) fs.prod)
    (hf₂ : fa ::ₘ fs ≠ ∅) (this : ∀ b ∈ fa ::ₘ fs, Associated x b), ∃ (n : ℕ), Associated (x ^ n) (fa ::ₘ fs).prod
⊢ ∃ (n : ℕ), Associated (x ^ n) f.prod

Tactic:
induction f using Multiset.induction with
    | empty => exact (hf₂ rfl).elim
    | cons fa fs fh => ?_

Premises used: False.elim
-- Module: Mathlib.RingTheory.DiscreteValuationRing.TFAE
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsNoetherianRing R
inst_2 : IsLocalRing R
inst_3 : IsDomain R
I : Ideal R
hI : I ≠ ⊥
h : ¬IsField R
x : R
hx : Eq (α := Ideal R) (maximalIdeal R) (Ideal.span {x})
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this_1 : x ≠ 0
hx' : Irreducible x
fa : R
fs : Multiset R
fh : ∀ (hf₂ : fs ≠ ∅) (this : ∀ b ∈ fs, Associated x b), ∃ (n : ℕ), Associated (x ^ n) fs.prod
hf₂ : fa ::ₘ fs ≠ ∅
this : ∀ b ∈ fa ::ₘ fs, Associated x b
cons :
  ∀
    (fh :
      ∀ (hf₂ : Ne.{u_1 + 1} (α := Multiset R) ∅ ∅)
        (this : ∀ (b : R), Membership.mem (γ := Multiset R) ∅ b → Associated x b),
        ∃ (n : ℕ), Associated (x ^ n) (Multiset.prod ∅))
    (hf₂ : fa ::ₘ ∅ ≠ ∅) (this : ∀ b ∈ fa ::ₘ ∅, Associated x b), ∃ (n : ℕ), Associated (x ^ n) (fa ::ₘ ∅).prod
cons₁ : ∀ (hf' : fs ≠ ∅), ∃ (n : ℕ), Associated (x ^ n) (fa ::ₘ fs).prod
⊢ ∃ (n : ℕ), Associated (x ^ n) (fa ::ₘ fs).prod

Tactic:
rcases eq_or_ne fs ∅ with (rfl | hf')

Premises used: eq_or_ne, EmptyCollection.emptyCollection, rfl
-- Module: Mathlib.RingTheory.DiscreteValuationRing.TFAE
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsNoetherianRing R
inst_2 : IsLocalRing R
inst_3 : IsDomain R
I : Ideal R
hI : I ≠ ⊥
h : ¬IsField R
x : R
hx : Eq (α := Ideal R) (maximalIdeal R) (Ideal.span {x})
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this_1 : x ≠ 0
hx' : Irreducible x
fa : R
fh :
  ∀ (hf₂ : Ne.{u_1 + 1} (α := Multiset R) ∅ ∅)
    (this : ∀ (b : R), Membership.mem (γ := Multiset R) ∅ b → Associated x b),
    ∃ (n : ℕ), Associated (x ^ n) (Multiset.prod ∅)
hf₂ : fa ::ₘ ∅ ≠ ∅
this : ∀ b ∈ fa ::ₘ ∅, Associated x b
h₁ : Associated (HPow.hPow (β := ℕ) x 1) (fa ::ₘ ∅).prod
⊢ ∃ (n : ℕ), Associated (x ^ n) (fa ::ₘ ∅).prod

Tactic:
use 1
-- Module: Mathlib.RingTheory.DiscreteValuationRing.TFAE
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsNoetherianRing R
inst_2 : IsLocalRing R
inst_3 : IsDomain R
I : Ideal R
hI : I ≠ ⊥
h : ¬IsField R
x : R
hx : Eq (α := Ideal R) (maximalIdeal R) (Ideal.span {x})
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this_1 : x ≠ 0
hx' : Irreducible x
fa : R
fh :
  ∀ (hf₂ : Ne.{u_1 + 1} (α := Multiset R) ∅ ∅)
    (this : ∀ (b : R), Membership.mem (γ := Multiset R) ∅ b → Associated x b),
    ∃ (n : ℕ), Associated (x ^ n) (Multiset.prod ∅)
hf₂ : fa ::ₘ ∅ ≠ ∅
this : ∀ b ∈ fa ::ₘ ∅, Associated x b
h₁ : Associated x fa
⊢ Associated (HPow.hPow (β := ℕ) x 1) (fa ::ₘ ∅).prod

Tactic:
rw [pow_one, Multiset.prod_cons, Multiset.empty_eq_zero, Multiset.prod_zero, mul_one]

Premises used: Multiset.prod_zero, Multiset.prod_cons, Multiset.empty_eq_zero, pow_one, mul_one
-- Module: Mathlib.RingTheory.DiscreteValuationRing.TFAE
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsNoetherianRing R
inst_2 : IsLocalRing R
inst_3 : IsDomain R
I : Ideal R
hI : I ≠ ⊥
h : ¬IsField R
x : R
hx : Eq (α := Ideal R) (maximalIdeal R) (Ideal.span {x})
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this_1 : x ≠ 0
hx' : Irreducible x
fa : R
fh :
  ∀ (hf₂ : Ne.{u_1 + 1} (α := Multiset R) ∅ ∅)
    (this : ∀ (b : R), Membership.mem (γ := Multiset R) ∅ b → Associated x b),
    ∃ (n : ℕ), Associated (x ^ n) (Multiset.prod ∅)
hf₂ : fa ::ₘ ∅ ≠ ∅
this : ∀ b ∈ fa ::ₘ ∅, Associated x b
⊢ Associated x fa

Tactic:
this _ (Multiset.mem_cons_self _ _)

Premises used: Multiset.mem_cons_self
-- Module: Mathlib.RingTheory.DiscreteValuationRing.TFAE
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsNoetherianRing R
inst_2 : IsLocalRing R
inst_3 : IsDomain R
I : Ideal R
hI : I ≠ ⊥
h : ¬IsField R
x : R
hx : Eq (α := Ideal R) (maximalIdeal R) (Ideal.span {x})
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this_1 : x ≠ 0
hx' : Irreducible x
fa : R
fh :
  ∀ (hf₂ : Ne.{u_1 + 1} (α := Multiset R) ∅ ∅)
    (this : ∀ (b : R), Membership.mem (γ := Multiset R) ∅ b → Associated x b),
    ∃ (n : ℕ), Associated (x ^ n) (Multiset.prod ∅)
hf₂ : fa ::ₘ ∅ ≠ ∅
this : ∀ b ∈ fa ::ₘ ∅, Associated x b
⊢ Associated x fa

Tactic:
exact this _ (Multiset.mem_cons_self _ _)

Premises used: Multiset.mem_cons_self
-- Module: Mathlib.RingTheory.DiscreteValuationRing.TFAE
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsNoetherianRing R
inst_2 : IsLocalRing R
inst_3 : IsDomain R
I : Ideal R
hI : I ≠ ⊥
h : ¬IsField R
x : R
hx : Eq (α := Ideal R) (maximalIdeal R) (Ideal.span {x})
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this_1 : x ≠ 0
hx' : Irreducible x
fa : R
fs : Multiset R
fh : ∀ (hf₂ : fs ≠ ∅) (this : ∀ b ∈ fs, Associated x b), ∃ (n : ℕ), Associated (x ^ n) fs.prod
hf₂ : fa ::ₘ fs ≠ ∅
this : ∀ b ∈ fa ::ₘ fs, Associated x b
hf' : fs ≠ ∅
cons : ∀ (n : ℕ) (hn : Associated (x ^ n) fs.prod), ∃ (n : ℕ), Associated (x ^ n) (fa ::ₘ fs).prod
⊢ ∃ (n : ℕ), Associated (x ^ n) (fa ::ₘ fs).prod

Tactic:
obtain ⟨n, hn⟩ := fh hf' fun b hb => this _ (Multiset.mem_cons_of_mem hb)

Premises used: Multiset.mem_cons_of_mem
-- Module: Mathlib.RingTheory.DiscreteValuationRing.TFAE
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsNoetherianRing R
inst_2 : IsLocalRing R
inst_3 : IsDomain R
I : Ideal R
hI : I ≠ ⊥
h : ¬IsField R
x : R
hx : Eq (α := Ideal R) (maximalIdeal R) (Ideal.span {x})
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this_1 : x ≠ 0
hx' : Irreducible x
fa : R
fs : Multiset R
fh : ∀ (hf₂ : fs ≠ ∅) (this : ∀ b ∈ fs, Associated x b), ∃ (n : ℕ), Associated (x ^ n) fs.prod
hf₂ : fa ::ₘ fs ≠ ∅
this : ∀ b ∈ fa ::ₘ fs, Associated x b
hf' : fs ≠ ∅
n : ℕ
hn : Associated (x ^ n) fs.prod
h₁ : Associated (x ^ (n + 1)) (fa ::ₘ fs).prod
⊢ ∃ (n : ℕ), Associated (x ^ n) (fa ::ₘ fs).prod

Tactic:
use n + 1
-- Module: Mathlib.RingTheory.DiscreteValuationRing.TFAE
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsNoetherianRing R
inst_2 : IsLocalRing R
inst_3 : IsDomain R
I : Ideal R
hI : I ≠ ⊥
h : ¬IsField R
x : R
hx : Eq (α := Ideal R) (maximalIdeal R) (Ideal.span {x})
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this_1 : x ≠ 0
hx' : Irreducible x
fa : R
fs : Multiset R
fh : ∀ (hf₂ : fs ≠ ∅) (this : ∀ b ∈ fs, Associated x b), ∃ (n : ℕ), Associated (x ^ n) fs.prod
hf₂ : fa ::ₘ fs ≠ ∅
this : ∀ b ∈ fa ::ₘ fs, Associated x b
hf' : fs ≠ ∅
n : ℕ
hn : Associated (x ^ n) fs.prod
h₁ : Associated (x * x ^ n) (fa * fs.prod)
⊢ Associated (x ^ (n + 1)) (fa ::ₘ fs).prod

Tactic:
rw [pow_add, Multiset.prod_cons, mul_comm, pow_one]

Premises used: Multiset.prod_cons, pow_add, pow_one, mul_comm
-- Module: Mathlib.RingTheory.DiscreteValuationRing.TFAE
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsNoetherianRing R
inst_2 : IsLocalRing R
inst_3 : IsDomain R
I : Ideal R
hI : I ≠ ⊥
h : ¬IsField R
x : R
hx : Eq (α := Ideal R) (maximalIdeal R) (Ideal.span {x})
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this_1 : x ≠ 0
hx' : Irreducible x
fa : R
fs : Multiset R
fh : ∀ (hf₂ : fs ≠ ∅) (this : ∀ b ∈ fs, Associated x b), ∃ (n : ℕ), Associated (x ^ n) fs.prod
hf₂ : fa ::ₘ fs ≠ ∅
this : ∀ b ∈ fa ::ₘ fs, Associated x b
hf' : fs ≠ ∅
n : ℕ
hn : Associated (x ^ n) fs.prod
⊢ Associated (x * x ^ n) (fa * fs.prod)

Tactic:
Associated.mul_mul (this _ (Multiset.mem_cons_self _ _)) hn

Premises used: Multiset.mem_cons_self, Associated.mul_mul
-- Module: Mathlib.RingTheory.DiscreteValuationRing.TFAE
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsNoetherianRing R
inst_2 : IsLocalRing R
inst_3 : IsDomain R
I : Ideal R
hI : I ≠ ⊥
h : ¬IsField R
x : R
hx : Eq (α := Ideal R) (maximalIdeal R) (Ideal.span {x})
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this_1 : x ≠ 0
hx' : Irreducible x
fa : R
fs : Multiset R
fh : ∀ (hf₂ : fs ≠ ∅) (this : ∀ b ∈ fs, Associated x b), ∃ (n : ℕ), Associated (x ^ n) fs.prod
hf₂ : fa ::ₘ fs ≠ ∅
this : ∀ b ∈ fa ::ₘ fs, Associated x b
hf' : fs ≠ ∅
n : ℕ
hn : Associated (x ^ n) fs.prod
⊢ Associated (x * x ^ n) (fa * fs.prod)

Tactic:
exact Associated.mul_mul (this _ (Multiset.mem_cons_self _ _)) hn

Premises used: Multiset.mem_cons_self, Associated.mul_mul
-- Module: Mathlib.RingTheory.DiscreteValuationRing.TFAE
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsNoetherianRing R
inst_2 : IsLocalRing R
inst_3 : IsDomain R
I : Ideal R
hI : I ≠ ⊥
h : ¬IsField R
x : R
hx : Eq (α := Ideal R) (maximalIdeal R) (Ideal.span {x})
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this : x ≠ 0
hx' : Irreducible x
⊢ ∀ (r : R), r ≠ 0 → r ∈ nonunits R → ∃ (n : ℕ), Associated (x ^ n) r

Tactic:
have H' : ∀ r : R, r ≠ 0 → r ∈ nonunits R → ∃ n : ℕ, Associated (x ^ n) r := by
    intro r hr₁ hr₂
    obtain ⟨f, hf₁, rfl, hf₂⟩ := (WfDvdMonoid.not_unit_iff_exists_factors_eq r hr₁).mp hr₂
    have : ∀ b ∈ f, Associated x b := by
      intro b hb
      exact Irreducible.associated_of_dvd hx' (hf₁ b hb) ((H b).mp (hf₁ b hb).1)
    clear hr₁ hr₂ hf₁
    induction f using Multiset.induction with
    | empty => exact (hf₂ rfl).elim
    | cons fa fs fh => ?_
    rcases eq_or_ne fs ∅ with (rfl | hf')
    · use 1
      rw [pow_one, Multiset.prod_cons, Multiset.empty_eq_zero, Multiset.prod_zero, mul_one]
      exact this _ (Multiset.mem_cons_self _ _)
    · obtain ⟨n, hn⟩ := fh hf' fun b hb => this _ (Multiset.mem_cons_of_mem hb)
      use n + 1
      rw [pow_add, Multiset.prod_cons, mul_comm, pow_one]
      exact Associated.mul_mul (this _ (Multiset.mem_cons_self _ _)) hn
-- Module: Mathlib.RingTheory.DiscreteValuationRing.TFAE
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsNoetherianRing R
inst_2 : IsLocalRing R
inst_3 : IsDomain R
I : Ideal R
hI : I ≠ ⊥
h : ¬IsField R
x : R
hx : Eq (α := Ideal R) (maximalIdeal R) (Ideal.span {x})
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this : x ≠ 0
hx' : Irreducible x
H' : ∀ (r : R), r ≠ 0 → r ∈ nonunits R → ∃ (n : ℕ), Associated (x ^ n) r
by_contra! : ∀ (h : ∀ (r : R) (_ : r ∈ I), r = 0), False
⊢ ∃ r ∈ I, r ≠ 0

Tactic:
by_contra! h
-- Module: Mathlib.RingTheory.DiscreteValuationRing.TFAE
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsNoetherianRing R
inst_2 : IsLocalRing R
inst_3 : IsDomain R
I : Ideal R
hI : I ≠ ⊥
h_1 : ¬IsField R
x : R
hx : Eq (α := Ideal R) (maximalIdeal R) (Ideal.span {x})
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this : x ≠ 0
hx' : Irreducible x
H' : ∀ (r : R), r ≠ 0 → r ∈ nonunits R → ∃ (n : ℕ), Associated (x ^ n) r
h : ∀ (r : R) (_ : r ∈ I), r = 0
apply : I = ⊥
⊢ False

Tactic:
apply hI
-- Module: Mathlib.RingTheory.DiscreteValuationRing.TFAE
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsNoetherianRing R
inst_2 : IsLocalRing R
inst_3 : IsDomain R
I : Ideal R
hI : I ≠ ⊥
h_1 : ¬IsField R
x : R
hx : Eq (α := Ideal R) (maximalIdeal R) (Ideal.span {x})
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this : x ≠ 0
hx' : Irreducible x
H' : ∀ (r : R), r ≠ 0 → r ∈ nonunits R → ∃ (n : ℕ), Associated (x ^ n) r
h : ∀ (r : R) (_ : r ∈ I), r = 0
rw : I ≤ ⊥
⊢ I = ⊥

Tactic:
rw [eq_bot_iff]

Premises used: eq_bot_iff
-- Module: Mathlib.RingTheory.DiscreteValuationRing.TFAE
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsNoetherianRing R
inst_2 : IsLocalRing R
inst_3 : IsDomain R
I : Ideal R
hI : I ≠ ⊥
h_1 : ¬IsField R
x : R
hx : Eq (α := Ideal R) (maximalIdeal R) (Ideal.span {x})
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this : x ≠ 0
hx' : Irreducible x
H' : ∀ (r : R), r ≠ 0 → r ∈ nonunits R → ∃ (n : ℕ), Associated (x ^ n) r
h : ∀ (r : R) (_ : r ∈ I), r = 0
⊢ I ≤ ⊥

Tactic:
h

Premises used: Set.Mem, setOf, Set.singleton
-- Module: Mathlib.RingTheory.DiscreteValuationRing.TFAE
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsNoetherianRing R
inst_2 : IsLocalRing R
inst_3 : IsDomain R
I : Ideal R
hI : I ≠ ⊥
h_1 : ¬IsField R
x : R
hx : Eq (α := Ideal R) (maximalIdeal R) (Ideal.span {x})
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this : x ≠ 0
hx' : Irreducible x
H' : ∀ (r : R), r ≠ 0 → r ∈ nonunits R → ∃ (n : ℕ), Associated (x ^ n) r
h : ∀ (r : R) (_ : r ∈ I), r = 0
⊢ I ≤ ⊥

Tactic:
exact h
-- Module: Mathlib.RingTheory.DiscreteValuationRing.TFAE
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsNoetherianRing R
inst_2 : IsLocalRing R
inst_3 : IsDomain R
I : Ideal R
hI : I ≠ ⊥
h : ¬IsField R
x : R
hx : Eq (α := Ideal R) (maximalIdeal R) (Ideal.span {x})
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this : x ≠ 0
hx' : Irreducible x
H' : ∀ (r : R), r ≠ 0 → r ∈ nonunits R → ∃ (n : ℕ), Associated (x ^ n) r
obtain : ∀ (r : R) (hr₁ : r ∈ I) (hr₂ : r ≠ 0), ∃ (n : ℕ), x ^ n ∈ I
⊢ ∃ (n : ℕ), x ^ n ∈ I

Tactic:
obtain ⟨r, hr₁, hr₂⟩ : ∃ r : R, r ∈ I ∧ r ≠ 0 := by
      by_contra! h; apply hI; rw [eq_bot_iff]; exact h
-- Module: Mathlib.RingTheory.DiscreteValuationRing.TFAE
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsNoetherianRing R
inst_2 : IsLocalRing R
inst_3 : IsDomain R
I : Ideal R
hI : I ≠ ⊥
h : ¬IsField R
x : R
hx : Eq (α := Ideal R) (maximalIdeal R) (Ideal.span {x})
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this : x ≠ 0
hx' : Irreducible x
H' : ∀ (r : R), r ≠ 0 → r ∈ nonunits R → ∃ (n : ℕ), Associated (x ^ n) r
r : R
hr₁ : r ∈ I
hr₂ : r ≠ 0
obtain : ∀ (n : ℕ) (u : Rˣ) (hr₁ : x ^ n * (↑u : R) ∈ I) (hr₂ : x ^ n * (↑u : R) ≠ 0), ∃ (n : ℕ), x ^ n ∈ I
⊢ ∃ (n : ℕ), x ^ n ∈ I

Tactic:
obtain ⟨n, u, rfl⟩ := H' r hr₂ (le_maximalIdeal hI' hr₁)

Premises used: IsLocalRing.le_maximalIdeal, rfl
-- Module: Mathlib.RingTheory.DiscreteValuationRing.TFAE
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsNoetherianRing R
inst_2 : IsLocalRing R
inst_3 : IsDomain R
I : Ideal R
hI : I ≠ ⊥
h : ¬IsField R
x : R
hx : Eq (α := Ideal R) (maximalIdeal R) (Ideal.span {x})
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this : x ≠ 0
hx' : Irreducible x
H' : ∀ (r : R), r ≠ 0 → r ∈ nonunits R → ∃ (n : ℕ), Associated (x ^ n) r
n : ℕ
u : Rˣ
hr₁ : x ^ n * (↑u : R) ∈ I
hr₂ : x ^ n * (↑u : R) ≠ 0
h₁ : x ^ n ∈ I
⊢ ∃ (n : ℕ), x ^ n ∈ I

Tactic:
use n
-- Module: Mathlib.RingTheory.DiscreteValuationRing.TFAE
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsNoetherianRing R
inst_2 : IsLocalRing R
inst_3 : IsDomain R
I : Ideal R
hI : I ≠ ⊥
h : ¬IsField R
x : R
hx : Eq (α := Ideal R) (maximalIdeal R) (Ideal.span {x})
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this : x ≠ 0
hx' : Irreducible x
H' : ∀ (r : R), r ≠ 0 → r ∈ nonunits R → ∃ (n : ℕ), Associated (x ^ n) r
n : ℕ
u : Rˣ
hr₁ : x ^ n * (↑u : R) ∈ I
hr₂ : x ^ n * (↑u : R) ≠ 0
h₁ : (↑u : R) * x ^ n ∈ I
h₂ h₃ h₄ h₅ : x ^ n * (↑u : R) ∈ I
⊢ x ^ n ∈ I

Tactic:
rwa [← I.unit_mul_mem_iff_mem u.isUnit, mul_comm]

Premises used: Ideal.unit_mul_mem_iff_mem, mul_comm, Units.isUnit
-- Module: Mathlib.RingTheory.DiscreteValuationRing.TFAE
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsNoetherianRing R
inst_2 : IsLocalRing R
inst_3 : IsDomain R
I : Ideal R
hI : I ≠ ⊥
h : ¬IsField R
x : R
hx : Eq (α := Ideal R) (maximalIdeal R) (Ideal.span {x})
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this : x ≠ 0
hx' : Irreducible x
H' : ∀ (r : R), r ≠ 0 → r ∈ nonunits R → ∃ (n : ℕ), Associated (x ^ n) r
⊢ ∃ (n : ℕ), x ^ n ∈ I

Tactic:
have : ∃ n : ℕ, x ^ n ∈ I := by
    obtain ⟨r, hr₁, hr₂⟩ : ∃ r : R, r ∈ I ∧ r ≠ 0 := by
      by_contra! h; apply hI; rw [eq_bot_iff]; exact h
    obtain ⟨n, u, rfl⟩ := H' r hr₂ (le_maximalIdeal hI' hr₁)
    use n
    rwa [← I.unit_mul_mem_iff_mem u.isUnit, mul_comm]
-- Module: Mathlib.RingTheory.DiscreteValuationRing.TFAE
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsNoetherianRing R
inst_2 : IsLocalRing R
inst_3 : IsDomain R
I : Ideal R
hI : I ≠ ⊥
h : ¬IsField R
x : R
hx : Eq (α := Ideal R) (maximalIdeal R) (Ideal.span {x})
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this_1 : x ≠ 0
hx' : Irreducible x
H' : ∀ (r : R), r ≠ 0 → r ∈ nonunits R → ∃ (n : ℕ), Associated (x ^ n) r
this : ∃ (n : ℕ), x ^ n ∈ I
h₁ : I = HPow.hPow (α := Ideal R) (maximalIdeal R) (Nat.find (p := fun (n : ℕ) ↦ x ^ n ∈ I) this)
⊢ ∃ (n : ℕ), I = HPow.hPow (α := Ideal R) (maximalIdeal R) n

Tactic:
use Nat.find this

Premises used: Nat.find
-- Module: Mathlib.RingTheory.DiscreteValuationRing.TFAE
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsNoetherianRing R
inst_2 : IsLocalRing R
inst_3 : IsDomain R
I : Ideal R
hI : I ≠ ⊥
h : ¬IsField R
x : R
hx : Eq (α := Ideal R) (maximalIdeal R) (Ideal.span {x})
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this_1 : x ≠ 0
hx' : Irreducible x
H' : ∀ (r : R), r ≠ 0 → r ∈ nonunits R → ∃ (n : ℕ), Associated (x ^ n) r
this : ∃ (n : ℕ), x ^ n ∈ I
h₁ : I ≤ HPow.hPow (α := Ideal R) (maximalIdeal R) (Nat.find (p := fun (n : ℕ) ↦ x ^ n ∈ I) this)
h₂ : HPow.hPow (α := Ideal R) (maximalIdeal R) (Nat.find (p := fun (n : ℕ) ↦ x ^ n ∈ I) this) ≤ I
⊢ I = HPow.hPow (α := Ideal R) (maximalIdeal R) (Nat.find (p := fun (n : ℕ) ↦ x ^ n ∈ I) this)

Tactic:
apply le_antisymm

Premises used: le_antisymm
-- Module: Mathlib.RingTheory.DiscreteValuationRing.TFAE
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsNoetherianRing R
inst_2 : IsLocalRing R
inst_3 : IsDomain R
I : Ideal R
hI : I ≠ ⊥
h : ¬IsField R
x : R
hx : Eq (α := Ideal R) (maximalIdeal R) (Ideal.span {x})
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this_1 : x ≠ 0
hx' : Irreducible x
H' : ∀ (r : R), r ≠ 0 → r ∈ nonunits R → ∃ (n : ℕ), Associated (x ^ n) r
this : ∃ (n : ℕ), x ^ n ∈ I
h₁ : ∀ s ∈ I, s ∈ HPow.hPow (α := Ideal R) (maximalIdeal R) (Nat.find (p := fun (n : ℕ) ↦ x ^ n ∈ I) this)
⊢ I ≤ HPow.hPow (α := Ideal R) (maximalIdeal R) (Nat.find (p := fun (n : ℕ) ↦ x ^ n ∈ I) this)

Tactic:
change ∀ s ∈ I, s ∈ _
-- Module: Mathlib.RingTheory.DiscreteValuationRing.TFAE
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsNoetherianRing R
inst_2 : IsLocalRing R
inst_3 : IsDomain R
I : Ideal R
hI : I ≠ ⊥
h : ¬IsField R
x : R
hx : Eq (α := Ideal R) (maximalIdeal R) (Ideal.span {x})
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this_1 : x ≠ 0
hx' : Irreducible x
H' : ∀ (r : R), r ≠ 0 → r ∈ nonunits R → ∃ (n : ℕ), Associated (x ^ n) r
this : ∃ (n : ℕ), x ^ n ∈ I
h₁ :
  ∀ (s : R) (hs₁ : s ∈ I)
    (hs₂ : s ∉ HPow.hPow (α := Ideal R) (maximalIdeal R) (Nat.find (p := fun (n : ℕ) ↦ x ^ n ∈ I) this)), False
⊢ ∀ s ∈ I, s ∈ HPow.hPow (α := Ideal R) (maximalIdeal R) (Nat.find (p := fun (n : ℕ) ↦ x ^ n ∈ I) this)

Tactic:
by_contra! ⟨s, hs₁, hs₂⟩
-- Module: Mathlib.RingTheory.DiscreteValuationRing.TFAE
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsNoetherianRing R
inst_2 : IsLocalRing R
inst_3 : IsDomain R
I : Ideal R
hI : I ≠ ⊥
h : ¬IsField R
x : R
hx : Eq (α := Ideal R) (maximalIdeal R) (Ideal.span {x})
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this_1 : x ≠ 0
hx' : Irreducible x
H' : ∀ (r : R), r ≠ 0 → r ∈ nonunits R → ∃ (n : ℕ), Associated (x ^ n) r
this : ∃ (n : ℕ), x ^ n ∈ I
s : R
hs₁ : s ∈ I
hs₂ : s ∉ HPow.hPow (α := Ideal R) (maximalIdeal R) (Nat.find (p := fun (n : ℕ) ↦ x ^ n ∈ I) this)
h₁ : s ∈ HPow.hPow (α := Ideal R) (maximalIdeal R) (Nat.find (p := fun (n : ℕ) ↦ x ^ n ∈ I) this)
⊢ False

Tactic:
apply hs₂
-- Module: Mathlib.RingTheory.DiscreteValuationRing.TFAE
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsNoetherianRing R
inst_2 : IsLocalRing R
inst_3 : IsDomain R
I : Ideal R
hI : I ≠ ⊥
h : ¬IsField R
x : R
hx : Eq (α := Ideal R) (maximalIdeal R) (Ideal.span {x})
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this_1 : x ≠ 0
hx' : Irreducible x
H' : ∀ (r : R), r ≠ 0 → r ∈ nonunits R → ∃ (n : ℕ), Associated (x ^ n) r
this : ∃ (n : ℕ), x ^ n ∈ I
s : R
hs₁ : s ∈ I
hs₂ : s ∉ HPow.hPow (α := Ideal R) (maximalIdeal R) (Nat.find (p := fun (n : ℕ) ↦ x ^ n ∈ I) this)
pos : ∀ (hs₃ : s = 0), s ∈ HPow.hPow (α := Ideal R) (maximalIdeal R) (Nat.find (p := fun (n : ℕ) ↦ x ^ n ∈ I) this)
neg : ∀ (hs₃ : ¬s = 0), s ∈ HPow.hPow (α := Ideal R) (maximalIdeal R) (Nat.find (p := fun (n : ℕ) ↦ x ^ n ∈ I) this)
⊢ s ∈ HPow.hPow (α := Ideal R) (maximalIdeal R) (Nat.find (p := fun (n : ℕ) ↦ x ^ n ∈ I) this)

Tactic:
by_cases hs₃ : s = 0