awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.Basic (goal)
Goal:
R_1 : Type u
inst : CommRing R_1
inst_1 : IsDomain R_1
inst_2 : IsDiscreteValuationRing R_1
R_2 : Type u_1
R : Type u_2
inst_3 : CommRing R
inst_4 : IsDomain R
inst_5 : IsDiscreteValuationRing R
x y : R
hy : y ≠ 0
⊢ toWithBotNat (if y ∣ x then 0 else x) < toWithBotNat y
have pos : ∀ (hyx : y ∣ x), toWithBotNat.{u_2} (R := R) 0 < toWithBotNat y
Tactic: split_ifs with hyx
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.Basic (goal)
Goal:
R_1 : Type u
inst : CommRing R_1
inst_1 : IsDomain R_1
inst_2 : IsDiscreteValuationRing R_1
R_2 : Type u_1
R : Type u_2
inst_3 : CommRing R
inst_4 : IsDomain R
inst_5 : IsDiscreteValuationRing R
x y : R
hy : y ≠ 0
pos : ∀ (hyx : y ∣ x), toWithBotNat.{u_2} (R := R) 0 < toWithBotNat y
⊢ toWithBotNat (if y ∣ x then 0 else x) < toWithBotNat y
have neg : ∀ (hyx : ¬y ∣ x), toWithBotNat x < toWithBotNat y
Tactic: split_ifs with hyx
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.Basic (goal)
Goal:
R_1 : Type u
inst : CommRing R_1
inst_1 : IsDomain R_1
inst_2 : IsDiscreteValuationRing R_1
R_2 : Type u_1
R : Type u_2
inst_3 : CommRing R
inst_4 : IsDomain R
inst_5 : IsDiscreteValuationRing R
x y : R
hy : y ≠ 0
hyx : y ∣ x
⊢ toWithBotNat.{u_2} (R := R) 0 < toWithBotNat y
have pos : ⊥ < toWithBotNat y
Tactic: rwa [toWithBotNat_zero, bot_lt_toWithBotNat_iff]
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.Basic (goal)
Goal:
R_1 : Type u
inst : CommRing R_1
inst_1 : IsDomain R_1
inst_2 : IsDiscreteValuationRing R_1
R_2 : Type u_1
R : Type u_2
inst_3 : CommRing R
inst_4 : IsDomain R
inst_5 : IsDiscreteValuationRing R
x y : R
hy : y ≠ 0
⊢ ¬toWithBotNat (x * y) < toWithBotNat x
have pos : ∀ (hx : x = 0), ¬toWithBotNat (x * y) < toWithBotNat x
Tactic: by_cases hx : x = 0
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.Basic (goal)
Goal:
R_1 : Type u
inst : CommRing R_1
inst_1 : IsDomain R_1
inst_2 : IsDiscreteValuationRing R_1
R_2 : Type u_1
R : Type u_2
inst_3 : CommRing R
inst_4 : IsDomain R
inst_5 : IsDiscreteValuationRing R
x y : R
hy : y ≠ 0
pos : ∀ (hx : x = 0), ¬toWithBotNat (x * y) < toWithBotNat x
⊢ ¬toWithBotNat (x * y) < toWithBotNat x
have neg : ∀ (hx : ¬x = 0), ¬toWithBotNat (x * y) < toWithBotNat x
Tactic: by_cases hx : x = 0
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.Basic (goal)
Goal:
R_1 : Type u
inst : CommRing R_1
inst_1 : IsDomain R_1
inst_2 : IsDiscreteValuationRing R_1
R_2 : Type u_1
R : Type u_2
inst_3 : CommRing R
inst_4 : IsDomain R
inst_5 : IsDiscreteValuationRing R
x y : R
hy : y ≠ 0
hx : ¬x = 0
⊢ ¬toWithBotNat (x * y) < toWithBotNat x
have neg : x ∣ x * y
Tactic: rw [not_lt, toWithBotNat_le_toWithBotNat_iff hx (mul_ne_zero hx hy)]
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.Basic (goal)
Goal:
K : Type u_1
Γ₀ : Type u_2
O : Type u_3
inst : Field K
inst_1 : LinearOrderedCommGroupWithZero Γ₀
inst_2 : CommRing O
inst_3 : Algebra O K
v : Valuation K Γ₀
hv : v.Integers O
this : IsDomain O := ⋯
inst_4 : IsDiscreteValuationRing O
ϖ : O
h : Irreducible ϖ
⊢ (↑(maximalIdeal O) : Set O) =
    {y : O |
      LE.le.{u_2} (α := Γ₀) ((v : (a : K) → Γ₀) ((algebraMap O K : (a : O) → K) y : K) : Γ₀)
        ((v : (a : K) → Γ₀) ((algebraMap O K : (a : O) → K) ϖ : K) : Γ₀)}
have rw : (↑(maximalIdeal O) : Set O) = (↑(span {ϖ}) : Set O)
Tactic: rw [← hv.coe_span_singleton_eq_setOf_le_v_algebraMap, ← h.maximalIdeal_eq]
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.Basic (goal)
Goal:
K : Type u_1
Γ₀ : Type u_2
O : Type u_3
inst : Field K
inst_1 : LinearOrderedCommGroupWithZero Γ₀
inst_2 : CommRing O
inst_3 : Algebra O K
v : Valuation K Γ₀
hv : v.Integers O
this : IsDomain O := ⋯
inst_4 : IsDiscreteValuationRing O
ϖ : O
h : Irreducible ϖ
rw : (↑(maximalIdeal O) : Set O) = (↑(span {ϖ}) : Set O)
⊢ (↑(maximalIdeal O) : Set O) =
    {y : O |
      LE.le.{u_2} (α := Γ₀) ((v : (a : K) → Γ₀) ((algebraMap O K : (a : O) → K) y : K) : Γ₀)
        ((v : (a : K) → Γ₀) ((algebraMap O K : (a : O) → K) ϖ : K) : Γ₀)}
have rw₁ : (↑(maximalIdeal O) : Set O) = (↑(maximalIdeal O) : Set O)
Tactic: rw [← hv.coe_span_singleton_eq_setOf_le_v_algebraMap, ← h.maximalIdeal_eq]
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.Basic (hypothesis)
Goal:
K : Type u_1
Γ₀ : Type u_2
O : Type u_3
inst : Field K
inst_1 : LinearOrderedCommGroupWithZero Γ₀
inst_2 : CommRing O
inst_3 : Algebra O K
v : Valuation K Γ₀
hv : v.Integers O
this : IsDomain O := ⋯
inst_4 : IsDiscreteValuationRing O
ϖ : O
h : Irreducible ϖ
n : ℕ
⊢ (↑(HPow.hPow (α := Ideal O) (maximalIdeal O) n) : Set O) =
    {y : O |
      (v : (a : K) → Γ₀) ((algebraMap O K : (a : O) → K) y : K) ≤
        HPow.hPow (α := Γ₀) ((v : (a : K) → Γ₀) ((algebraMap O K : (a : O) → K) ϖ : K) : Γ₀) n}
have this₁ : HPow.hPow (α := Γ₀) ((v : (a : K) → Γ₀) ((algebraMap O K : (a : O) → K) ϖ : K) : Γ₀) n =
  (v : (a : K) → Γ₀) ((algebraMap O K : (a : O) → K) (ϖ ^ n) : K)
Tactic: have : (v (algebraMap O K ϖ)) ^ n = v (algebraMap O K (ϖ ^ n)) := by simp
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.Basic (goal)
Goal:
K : Type u_1
Γ₀ : Type u_2
O : Type u_3
inst : Field K
inst_1 : LinearOrderedCommGroupWithZero Γ₀
inst_2 : CommRing O
inst_3 : Algebra O K
v : Valuation K Γ₀
hv : v.Integers O
this_1 : IsDomain O := ⋯
inst_4 : IsDiscreteValuationRing O
ϖ : O
h : Irreducible ϖ
n : ℕ
this :
  HPow.hPow (α := Γ₀) ((v : (a : K) → Γ₀) ((algebraMap O K : (a : O) → K) ϖ : K) : Γ₀) n =
    (v : (a : K) → Γ₀) ((algebraMap O K : (a : O) → K) (ϖ ^ n) : K)
⊢ (↑(HPow.hPow (α := Ideal O) (maximalIdeal O) n) : Set O) =
    {y : O |
      (v : (a : K) → Γ₀) ((algebraMap O K : (a : O) → K) y : K) ≤
        HPow.hPow (α := Γ₀) ((v : (a : K) → Γ₀) ((algebraMap O K : (a : O) → K) ϖ : K) : Γ₀) n}
have rw : (↑(HPow.hPow (α := Ideal O) (maximalIdeal O) n) : Set O) =
  {y : O |
    LE.le.{u_2} (α := Γ₀) ((v : (a : K) → Γ₀) ((algebraMap O K : (a : O) → K) y : K) : Γ₀)
      ((v : (a : K) → Γ₀) ((algebraMap O K : (a : O) → K) (ϖ ^ n) : K) : Γ₀)}
Tactic: rw [this, ← hv.coe_span_singleton_eq_setOf_le_v_algebraMap,
      ← Ideal.span_singleton_pow, ← h.maximalIdeal_eq]
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.Basic (goal)
Goal:
K : Type u_1
Γ₀ : Type u_2
O : Type u_3
inst : Field K
inst_1 : LinearOrderedCommGroupWithZero Γ₀
inst_2 : CommRing O
inst_3 : Algebra O K
v : Valuation K Γ₀
hv : v.Integers O
this_1 : IsDomain O := ⋯
inst_4 : IsDiscreteValuationRing O
ϖ : O
h : Irreducible ϖ
n : ℕ
this :
  HPow.hPow (α := Γ₀) ((v : (a : K) → Γ₀) ((algebraMap O K : (a : O) → K) ϖ : K) : Γ₀) n =
    (v : (a : K) → Γ₀) ((algebraMap O K : (a : O) → K) (ϖ ^ n) : K)
rw :
  (↑(HPow.hPow (α := Ideal O) (maximalIdeal O) n) : Set O) =
    {y : O |
      LE.le.{u_2} (α := Γ₀) ((v : (a : K) → Γ₀) ((algebraMap O K : (a : O) → K) y : K) : Γ₀)
        ((v : (a : K) → Γ₀) ((algebraMap O K : (a : O) → K) (ϖ ^ n) : K) : Γ₀)}
⊢ (↑(HPow.hPow (α := Ideal O) (maximalIdeal O) n) : Set O) =
    {y : O |
      (v : (a : K) → Γ₀) ((algebraMap O K : (a : O) → K) y : K) ≤
        HPow.hPow (α := Γ₀) ((v : (a : K) → Γ₀) ((algebraMap O K : (a : O) → K) ϖ : K) : Γ₀) n}
have rw₁ : (↑(HPow.hPow (α := Ideal O) (maximalIdeal O) n) : Set O) = (↑(span {ϖ ^ n}) : Set O)
Tactic: rw [this, ← hv.coe_span_singleton_eq_setOf_le_v_algebraMap,
      ← Ideal.span_singleton_pow, ← h.maximalIdeal_eq]
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.Basic (goal)
Goal:
K : Type u_1
Γ₀ : Type u_2
O : Type u_3
inst : Field K
inst_1 : LinearOrderedCommGroupWithZero Γ₀
inst_2 : CommRing O
inst_3 : Algebra O K
v : Valuation K Γ₀
hv : v.Integers O
this_1 : IsDomain O := ⋯
inst_4 : IsDiscreteValuationRing O
ϖ : O
h : Irreducible ϖ
n : ℕ
this :
  HPow.hPow (α := Γ₀) ((v : (a : K) → Γ₀) ((algebraMap O K : (a : O) → K) ϖ : K) : Γ₀) n =
    (v : (a : K) → Γ₀) ((algebraMap O K : (a : O) → K) (ϖ ^ n) : K)
rw :
  (↑(HPow.hPow (α := Ideal O) (maximalIdeal O) n) : Set O) =
    {y : O |
      LE.le.{u_2} (α := Γ₀) ((v : (a : K) → Γ₀) ((algebraMap O K : (a : O) → K) y : K) : Γ₀)
        ((v : (a : K) → Γ₀) ((algebraMap O K : (a : O) → K) (ϖ ^ n) : K) : Γ₀)}
rw₁ : (↑(HPow.hPow (α := Ideal O) (maximalIdeal O) n) : Set O) = (↑(span {ϖ ^ n}) : Set O)
⊢ (↑(HPow.hPow (α := Ideal O) (maximalIdeal O) n) : Set O) =
    {y : O |
      (v : (a : K) → Γ₀) ((algebraMap O K : (a : O) → K) y : K) ≤
        HPow.hPow (α := Γ₀) ((v : (a : K) → Γ₀) ((algebraMap O K : (a : O) → K) ϖ : K) : Γ₀) n}
have rw₂ : (↑(HPow.hPow (α := Ideal O) (maximalIdeal O) n) : Set O) = (↑(HPow.hPow (α := Ideal O) (span {ϖ}) n) : Set O)
Tactic: rw [this, ← hv.coe_span_singleton_eq_setOf_le_v_algebraMap,
      ← Ideal.span_singleton_pow, ← h.maximalIdeal_eq]
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.Basic (goal)
Goal:
K : Type u_1
Γ₀ : Type u_2
O : Type u_3
inst : Field K
inst_1 : LinearOrderedCommGroupWithZero Γ₀
inst_2 : CommRing O
inst_3 : Algebra O K
v : Valuation K Γ₀
hv : v.Integers O
this_1 : IsDomain O := ⋯
inst_4 : IsDiscreteValuationRing O
ϖ : O
h : Irreducible ϖ
n : ℕ
this :
  HPow.hPow (α := Γ₀) ((v : (a : K) → Γ₀) ((algebraMap O K : (a : O) → K) ϖ : K) : Γ₀) n =
    (v : (a : K) → Γ₀) ((algebraMap O K : (a : O) → K) (ϖ ^ n) : K)
rw :
  (↑(HPow.hPow (α := Ideal O) (maximalIdeal O) n) : Set O) =
    {y : O |
      LE.le.{u_2} (α := Γ₀) ((v : (a : K) → Γ₀) ((algebraMap O K : (a : O) → K) y : K) : Γ₀)
        ((v : (a : K) → Γ₀) ((algebraMap O K : (a : O) → K) (ϖ ^ n) : K) : Γ₀)}
rw₁ : (↑(HPow.hPow (α := Ideal O) (maximalIdeal O) n) : Set O) = (↑(span {ϖ ^ n}) : Set O)
rw₂ : (↑(HPow.hPow (α := Ideal O) (maximalIdeal O) n) : Set O) = (↑(HPow.hPow (α := Ideal O) (span {ϖ}) n) : Set O)
⊢ (↑(HPow.hPow (α := Ideal O) (maximalIdeal O) n) : Set O) =
    {y : O |
      (v : (a : K) → Γ₀) ((algebraMap O K : (a : O) → K) y : K) ≤
        HPow.hPow (α := Γ₀) ((v : (a : K) → Γ₀) ((algebraMap O K : (a : O) → K) ϖ : K) : Γ₀) n}
have rw₃ : (↑(HPow.hPow (α := Ideal O) (maximalIdeal O) n) : Set O) = (↑(HPow.hPow (α := Ideal O) (maximalIdeal O) n) : Set O)
Tactic: rw [this, ← hv.coe_span_singleton_eq_setOf_le_v_algebraMap,
      ← Ideal.span_singleton_pow, ← h.maximalIdeal_eq]
-- 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 ≠ ⊥
⊢ ∃ (n : ℕ), I = HPow.hPow (α := Ideal R) (maximalIdeal R) n
have pos : ∀ (h : IsField R), ∃ (n : ℕ), I = HPow.hPow (α := Ideal R) (maximalIdeal R) n
Tactic: by_cases h : IsField R
-- 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 ≠ ⊥
pos : ∀ (h : IsField R), ∃ (n : ℕ), I = HPow.hPow (α := Ideal R) (maximalIdeal R) n
⊢ ∃ (n : ℕ), I = HPow.hPow (α := Ideal R) (maximalIdeal R) n
have neg : ∀ (h : ¬IsField R), ∃ (n : ℕ), I = HPow.hPow (α := Ideal R) (maximalIdeal R) n
Tactic: by_cases h : IsField R
-- 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: obtain ⟨x, hx : _ = Ideal.span _⟩ := h'
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
I : Ideal R
hI : I ≠ ⊥
h : ¬IsField R
x : R
hx : Eq (α := Ideal R) (maximalIdeal R) (Ideal.span {x})
⊢ ∃ (n : ℕ), I = HPow.hPow (α := Ideal R) (maximalIdeal R) n
have pos : ∀ (hI' : I = ⊤), ∃ (n : ℕ), I = HPow.hPow (α := Ideal R) (maximalIdeal R) n
Tactic: by_cases hI' : I = ⊤
-- 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})
pos : ∀ (hI' : I = ⊤), ∃ (n : ℕ), I = HPow.hPow (α := Ideal R) (maximalIdeal R) n
⊢ ∃ (n : ℕ), I = HPow.hPow (α := Ideal R) (maximalIdeal R) n
have neg : ∀ (hI' : ¬I = ⊤), ∃ (n : ℕ), I = HPow.hPow (α := Ideal R) (maximalIdeal R) n
Tactic: by_cases hI' : I = ⊤
-- 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 = ⊤
⊢ ∃ (n : ℕ), I = HPow.hPow (α := Ideal R) (maximalIdeal R) n
have h₁ : I = HPow.hPow (α := Ideal R) (β := ℕ) (maximalIdeal R) 0
Tactic: use 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 = ⊤
⊢ I = HPow.hPow (α := Ideal R) (β := ℕ) (maximalIdeal R) 0
have h₁ : I = 1
Tactic: rw [pow_zero, hI', Ideal.one_eq_top]
-- 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 = ⊤
h₁ : I = 1
⊢ I = HPow.hPow (α := Ideal R) (β := ℕ) (maximalIdeal R) 0
have h₂ : Eq.{u_1 + 1} (α := Ideal R) ⊤ 1
Tactic: rw [pow_zero, hI', Ideal.one_eq_top]
-- 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 = ⊤
h₁ : I = 1
h₂ : Eq.{u_1 + 1} (α := Ideal R) ⊤ 1
⊢ I = HPow.hPow (α := Ideal R) (β := ℕ) (maximalIdeal R) 0
have h₃ : Eq.{u_1 + 1} (α := Ideal R) ⊤ ⊤
Tactic: rw [pow_zero, hI', Ideal.one_eq_top]
-- 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 = ⊤
⊢ ∃ (n : ℕ), I = HPow.hPow (α := Ideal R) (maximalIdeal R) n
have H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
Tactic: have H : ∀ r : R, ¬IsUnit r ↔ x ∣ r := fun r =>
    (SetLike.ext_iff.mp hx r).trans Ideal.mem_span_singleton
-- 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
hI' : ¬I = ⊤
hx : Eq (α := Ideal R) (maximalIdeal R) (Ideal.span {0})
H : ∀ (r : R), ¬IsUnit r ↔ 0 ∣ r
⊢ False
have apply : Eq (α := Ideal R) (maximalIdeal R) ⊥
Tactic: apply Ring.ne_bot_of_isMaximal_of_not_isField (maximalIdeal.isMaximal R) h
-- 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
⊢ ∃ (n : ℕ), I = HPow.hPow (α := Ideal R) (maximalIdeal R) n
have this : x ≠ 0
Tactic: have : x ≠ 0 := by
    rintro rfl
    apply Ring.ne_bot_of_isMaximal_of_not_isField (maximalIdeal.isMaximal R) h
    simp [hx]
-- 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
⊢ ∃ (n : ℕ), I = HPow.hPow (α := Ideal R) (maximalIdeal R) n
have hx' : Irreducible x
Tactic: have hx' := IsDiscreteValuationRing.irreducible_of_span_eq_maximalIdeal x this hx
-- 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
r : R
hr₁ : r ≠ 0
hr₂ : r ∈ nonunits R
⊢ ∃ (n : ℕ), Associated (x ^ n) r
have obtain : ∀ (f : Multiset R) (hf₁ : ∀ b ∈ f, Irreducible b) (hf₂ : f ≠ ∅) (hr₁ : f.prod ≠ 0) (hr₂ : f.prod ∈ nonunits R),
  ∃ (n : ℕ), Associated (x ^ n) f.prod
Tactic: obtain ⟨f, hf₁, rfl, hf₂⟩ := (WfDvdMonoid.not_unit_iff_exists_factors_eq r hr₁).mp hr₂
Cleared hypotheses: r, hr₁, hr₂
-- 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
f : Multiset R
hf₁ : ∀ b ∈ f, Irreducible b
hf₂ : f ≠ ∅
hr₁ : f.prod ≠ 0
hr₂ : f.prod ∈ nonunits R
⊢ ∃ (n : ℕ), Associated (x ^ n) f.prod
have this₁ : ∀ 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)
-- 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
f : Multiset R
hf₂ : f ≠ ∅
this : ∀ b ∈ f, Associated x b
⊢ ∃ (n : ℕ), Associated (x ^ n) f.prod
have 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
Tactic: induction f using Multiset.induction with
    | empty => exact (hf₂ rfl).elim
    | cons fa fs fh => ?_
Cleared hypotheses: f, hf₂, 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
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
have 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
Tactic: rcases eq_or_ne fs ∅ with (rfl | hf')
Cleared hypotheses: fs, fh, hf₂, 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
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
⊢ ∃ (n : ℕ), Associated (x ^ n) (fa ::ₘ fs).prod
have cons₁ : ∀ (hf' : fs ≠ ∅), ∃ (n : ℕ), Associated (x ^ n) (fa ::ₘ fs).prod
Tactic: rcases eq_or_ne fs ∅ with (rfl | hf')
-- 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
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
have h₁ : Associated (HPow.hPow (β := ℕ) x 1) (fa ::ₘ ∅).prod
Tactic: use 1
-- 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
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 (HPow.hPow (β := ℕ) x 1) (fa ::ₘ ∅).prod
have h₁ : Associated x fa
Tactic: rw [pow_one, Multiset.prod_cons, Multiset.empty_eq_zero, Multiset.prod_zero, mul_one]
-- 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), Associated (x ^ n) (fa ::ₘ fs).prod
have cons : ∀ (n : ℕ) (hn : Associated (x ^ n) 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)
-- 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
⊢ ∃ (n : ℕ), Associated (x ^ n) (fa ::ₘ fs).prod
have h₁ : Associated (x ^ (n + 1)) (fa ::ₘ fs).prod
Tactic: use n + 1