awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.TFAE (goal)
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 ^ (n + 1)) (fa ::ₘ fs).prod
have h₁ : Associated (x * x ^ n) (fa * fs.prod)
Tactic: rw [pow_add, Multiset.prod_cons, mul_comm, pow_one]
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.TFAE (hypothesis)
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
⊢ ∃ (n : ℕ), I = HPow.hPow (α := Ideal R) (maximalIdeal R) n
have H' : ∀ (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
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.TFAE (goal)
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 ∈ I, r ≠ 0
have by_contra! : ∀ (h : ∀ (r : R) (_ : r ∈ I), r = 0), False
Tactic: by_contra! h
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.TFAE (goal)
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
⊢ False
have apply : I = ⊥
Tactic: apply hI
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.TFAE (goal)
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 = ⊥
have rw : I ≤ ⊥
Tactic: rw [eq_bot_iff]
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.TFAE (goal)
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
have obtain : ∀ (r : R) (hr₁ : r ∈ I) (hr₂ : r ≠ 0), ∃ (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
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.TFAE (goal)
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
⊢ ∃ (n : ℕ), x ^ n ∈ I
have obtain : ∀ (n : ℕ) (u : Rˣ) (hr₁ : x ^ n * (↑u : R) ∈ I) (hr₂ : x ^ n * (↑u : R) ≠ 0), ∃ (n : ℕ), x ^ n ∈ I
Tactic: obtain ⟨n, u, rfl⟩ := H' r hr₂ (le_maximalIdeal hI' hr₁)
Cleared hypotheses: r, hr₁, hr₂
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.TFAE (goal)
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
⊢ ∃ (n : ℕ), x ^ n ∈ I
have h₁ : x ^ n ∈ I
Tactic: use n
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.TFAE (goal)
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
⊢ x ^ n ∈ I
have h₁ : (↑u : R) * x ^ n ∈ I
Tactic: rwa [← I.unit_mul_mem_iff_mem u.isUnit, mul_comm]
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.TFAE (hypothesis)
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), I = HPow.hPow (α := Ideal R) (maximalIdeal R) n
have this₁ : ∃ (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]
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.TFAE (goal)
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
⊢ ∃ (n : ℕ), I = HPow.hPow (α := Ideal R) (maximalIdeal R) n
have h₁ : I = HPow.hPow (α := Ideal R) (maximalIdeal R) (Nat.find (p := fun (n : ℕ) ↦ x ^ n ∈ I) this)
Tactic: use Nat.find this
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.TFAE (goal)
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
⊢ I = HPow.hPow (α := Ideal R) (maximalIdeal R) (Nat.find (p := fun (n : ℕ) ↦ x ^ n ∈ I) this)
have h₁ : I ≤ HPow.hPow (α := Ideal R) (maximalIdeal R) (Nat.find (p := fun (n : ℕ) ↦ x ^ n ∈ I) this)
Tactic: apply le_antisymm
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.TFAE (goal)
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)
⊢ I = HPow.hPow (α := Ideal R) (maximalIdeal R) (Nat.find (p := fun (n : ℕ) ↦ x ^ n ∈ I) this)
have h₂ : HPow.hPow (α := Ideal R) (maximalIdeal R) (Nat.find (p := fun (n : ℕ) ↦ x ^ n ∈ I) this) ≤ I
Tactic: apply le_antisymm
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.TFAE (goal)
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
⊢ I ≤ HPow.hPow (α := Ideal R) (maximalIdeal R) (Nat.find (p := fun (n : ℕ) ↦ x ^ n ∈ I) this)
have h₁ : ∀ s ∈ I, s ∈ HPow.hPow (α := Ideal R) (maximalIdeal R) (Nat.find (p := fun (n : ℕ) ↦ x ^ n ∈ I) this)
Tactic: change ∀ s ∈ I, s ∈ _
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.TFAE (goal)
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 ∈ I, s ∈ HPow.hPow (α := Ideal R) (maximalIdeal R) (Nat.find (p := fun (n : ℕ) ↦ x ^ n ∈ I) this)
have 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
Tactic: by_contra! ⟨s, hs₁, hs₂⟩
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.TFAE (goal)
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)
⊢ False
have h₁ : s ∈ HPow.hPow (α := Ideal R) (maximalIdeal R) (Nat.find (p := fun (n : ℕ) ↦ x ^ n ∈ I) this)
Tactic: apply hs₂
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.TFAE (goal)
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)
⊢ s ∈ HPow.hPow (α := Ideal R) (maximalIdeal R) (Nat.find (p := fun (n : ℕ) ↦ x ^ n ∈ I) this)
have pos : ∀ (hs₃ : s = 0), s ∈ HPow.hPow (α := Ideal R) (maximalIdeal R) (Nat.find (p := fun (n : ℕ) ↦ x ^ n ∈ I) this)
Tactic: by_cases hs₃ : s = 0
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.TFAE (goal)
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)
⊢ s ∈ HPow.hPow (α := Ideal R) (maximalIdeal R) (Nat.find (p := fun (n : ℕ) ↦ x ^ n ∈ I) this)
have neg : ∀ (hs₃ : ¬s = 0), s ∈ HPow.hPow (α := Ideal R) (maximalIdeal R) (Nat.find (p := fun (n : ℕ) ↦ x ^ n ∈ I) this)
Tactic: by_cases hs₃ : s = 0
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.TFAE (goal)
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)
hs₃ : s = 0
⊢ s ∈ HPow.hPow (α := Ideal R) (maximalIdeal R) (Nat.find (p := fun (n : ℕ) ↦ x ^ n ∈ I) this)
have pos : 0 ∈ HPow.hPow (α := Ideal R) (maximalIdeal R) (Nat.find (p := fun (n : ℕ) ↦ x ^ n ∈ I) this)
Tactic: rw [hs₃]
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.TFAE (goal)
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)
hs₃ : ¬s = 0
⊢ s ∈ HPow.hPow (α := Ideal R) (maximalIdeal R) (Nat.find (p := fun (n : ℕ) ↦ x ^ n ∈ I) this)
have neg : ∀ (n : ℕ) (u : Rˣ) (hs₁ : x ^ n * (↑u : R) ∈ I)
  (hs₂ : x ^ n * (↑u : R) ∉ HPow.hPow (α := Ideal R) (maximalIdeal R) (Nat.find (p := fun (n : ℕ) ↦ x ^ n ∈ I) this))
  (hs₃ : ¬x ^ n * (↑u : R) = 0),
  x ^ n * (↑u : R) ∈ HPow.hPow (α := Ideal R) (maximalIdeal R) (Nat.find (p := fun (n : ℕ) ↦ x ^ n ∈ I) this)
Tactic: obtain ⟨n, u, rfl⟩ := H' s hs₃ (le_maximalIdeal hI' hs₁)
Cleared hypotheses: s, hs₁, hs₂, hs₃
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.TFAE (goal)
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
n : ℕ
u : Rˣ
hs₁ : x ^ n * (↑u : R) ∈ I
hs₂ : x ^ n * (↑u : R) ∉ HPow.hPow (α := Ideal R) (maximalIdeal R) (Nat.find (p := fun (n : ℕ) ↦ x ^ n ∈ I) this)
hs₃ : ¬x ^ n * (↑u : R) = 0
⊢ x ^ n * (↑u : R) ∈ HPow.hPow (α := Ideal R) (maximalIdeal R) (Nat.find (p := fun (n : ℕ) ↦ x ^ n ∈ I) this)
have neg : ∀ (hs₁ : x ^ n ∈ I)
  (hs₂ : x ^ n * (↑u : R) ∉ HPow.hPow (α := Ideal R) (maximalIdeal R) (Nat.find (p := fun (n : ℕ) ↦ x ^ n ∈ I) this))
  (hs₃ : ¬x ^ n * (↑u : R) = 0),
  x ^ n ∈ HPow.hPow (α := Ideal R) (maximalIdeal R) (Nat.find (p := fun (n : ℕ) ↦ x ^ n ∈ I) this)
Tactic: rw [mul_comm, Ideal.unit_mul_mem_iff_mem _ u.isUnit] at hs₁ ⊢
Cleared hypotheses: hs₁, hs₂, hs₃
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.TFAE (goal)
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
n : ℕ
u : Rˣ
hs₁ : x ^ n ∈ I
hs₂ : x ^ n * (↑u : R) ∉ HPow.hPow (α := Ideal R) (maximalIdeal R) (Nat.find (p := fun (n : ℕ) ↦ x ^ n ∈ I) this)
hs₃ : ¬x ^ n * (↑u : R) = 0
⊢ x ^ n ∈ HPow.hPow (α := Ideal R) (maximalIdeal R) (Nat.find (p := fun (n : ℕ) ↦ x ^ n ∈ I) this)
have neg : x ^ n ∈ HPow.hPow (α := Ideal R) (maximalIdeal R) n
Tactic: apply Ideal.pow_le_pow_right (Nat.find_min' this hs₁)
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.TFAE (goal)
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
n : ℕ
u : Rˣ
hs₁ : x ^ n ∈ I
hs₂ : x ^ n * (↑u : R) ∉ HPow.hPow (α := Ideal R) (maximalIdeal R) (Nat.find (p := fun (n : ℕ) ↦ x ^ n ∈ I) this)
hs₃ : ¬x ^ n * (↑u : R) = 0
⊢ x ^ n ∈ HPow.hPow (α := Ideal R) (maximalIdeal R) n
have neg : Membership.mem (γ := Ideal R) (maximalIdeal R) x
Tactic: apply Ideal.pow_mem_pow
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.TFAE (goal)
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
⊢ HPow.hPow (α := Ideal R) (maximalIdeal R) (Nat.find (p := fun (n : ℕ) ↦ x ^ n ∈ I) this) ≤ I
have h₁ : x ^ Nat.find (p := fun (n : ℕ) ↦ x ^ n ∈ I) this ∈ (↑I : Set R)
Tactic: rw [hx, Ideal.span_singleton_pow, Ideal.span_le, Set.singleton_subset_iff]
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.TFAE (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsNoetherianRing R
inst_2 : IsLocalRing R
inst_3 : IsDomain R
h' : Submodule.IsPrincipal (maximalIdeal R)
I : Ideal R
hI : I ≠ ⊥
h : ¬IsField R
⊢ ∃ (n : ℕ), I = HPow.hPow (α := Ideal R) (maximalIdeal R) n
have neg : ∀ (x : R) (hx : Eq (α := Ideal R) (maximalIdeal R) (Ideal.span {x})),
  ∃ (n : ℕ), I = HPow.hPow (α := Ideal R) (maximalIdeal R) n
Tactic: classical
  obtain ⟨x, hx : _ = Ideal.span _⟩ := h'
  by_cases hI' : I = ⊤
  · use 0; rw [pow_zero, hI', Ideal.one_eq_top]
  have H : ∀ r : R, ¬IsUnit r ↔ x ∣ r := fun r =>
    (SetLike.ext_iff.mp hx r).trans Ideal.mem_span_singleton
  have : x ≠ 0 := by
    rintro rfl
    apply Ring.ne_bot_of_isMaximal_of_not_isField (maximalIdeal.isMaximal R) h
    simp [hx]
  have hx' := IsDiscreteValuationRing.irreducible_of_span_eq_maximalIdeal x this hx
  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
  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]
  use Nat.find this
  apply le_antisymm
  · change ∀ s ∈ I, s ∈ _
    by_contra! ⟨s, hs₁, hs₂⟩
    apply hs₂
    by_cases hs₃ : s = 0; · rw [hs₃]; exact zero_mem _
    obtain ⟨n, u, rfl⟩ := H' s hs₃ (le_maximalIdeal hI' hs₁)
    rw [mul_comm, Ideal.unit_mul_mem_iff_mem _ u.isUnit] at hs₁ ⊢
    apply Ideal.pow_le_pow_right (Nat.find_min' this hs₁)
    apply Ideal.pow_mem_pow
    exact (H _).mpr (dvd_refl _)
  · rw [hx, Ideal.span_singleton_pow, Ideal.span_le, Set.singleton_subset_iff]
    exact Nat.find_spec this
Cleared hypotheses: h'
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.TFAE (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsNoetherianRing R
inst_2 : IsLocalRing R
inst_3 : IsDomain R
h' : Submodule.IsPrincipal (maximalIdeal R)
I : Ideal R
hI : I ≠ ⊥
h : ¬IsField R
neg :
  ∀ (x : R) (hx : Eq (α := Ideal R) (maximalIdeal R) (Ideal.span {x})),
    ∃ (n : ℕ), I = HPow.hPow (α := Ideal R) (maximalIdeal R) n
⊢ ∃ (n : ℕ), I = HPow.hPow (α := Ideal R) (maximalIdeal R) n
have pos : ∀ (x : R) (hx : Eq (α := Ideal R) (maximalIdeal R) (Ideal.span {x})) (hI' : I = ⊤),
  ∃ (n : ℕ), I = HPow.hPow (α := Ideal R) (maximalIdeal R) n
Tactic: classical
  obtain ⟨x, hx : _ = Ideal.span _⟩ := h'
  by_cases hI' : I = ⊤
  · use 0; rw [pow_zero, hI', Ideal.one_eq_top]
  have H : ∀ r : R, ¬IsUnit r ↔ x ∣ r := fun r =>
    (SetLike.ext_iff.mp hx r).trans Ideal.mem_span_singleton
  have : x ≠ 0 := by
    rintro rfl
    apply Ring.ne_bot_of_isMaximal_of_not_isField (maximalIdeal.isMaximal R) h
    simp [hx]
  have hx' := IsDiscreteValuationRing.irreducible_of_span_eq_maximalIdeal x this hx
  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
  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]
  use Nat.find this
  apply le_antisymm
  · change ∀ s ∈ I, s ∈ _
    by_contra! ⟨s, hs₁, hs₂⟩
    apply hs₂
    by_cases hs₃ : s = 0; · rw [hs₃]; exact zero_mem _
    obtain ⟨n, u, rfl⟩ := H' s hs₃ (le_maximalIdeal hI' hs₁)
    rw [mul_comm, Ideal.unit_mul_mem_iff_mem _ u.isUnit] at hs₁ ⊢
    apply Ideal.pow_le_pow_right (Nat.find_min' this hs₁)
    apply Ideal.pow_mem_pow
    exact (H _).mpr (dvd_refl _)
  · rw [hx, Ideal.span_singleton_pow, Ideal.span_le, Set.singleton_subset_iff]
    exact Nat.find_spec this
Cleared hypotheses: h'