-- Mathlib/RingTheory/DiscreteValuationRing/TFAE.lean
theorem maximalIdeal_isPrincipal_of_isDedekindDomain [IsLocalRing R] [IsDomain R]
    [IsDedekindDomain R] : (maximalIdeal R).IsPrincipal

Proof state:
R : Type u_1
inst✝³ : CommRing R
inst✝² : IsLocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝¹ : (Ideal.span {a}).radical = maximalIdeal R
this✝ : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this✝ = n + 1
b : R
hb₁ : b ∈ maximalIdeal R ^ n
hb₂ : b ∉ Ideal.span {a}
hb₃ : ∀ m ∈ maximalIdeal R, ∃ k, k * a = b * m
hb₄ : b ≠ 0
K : Type u_1 := FractionRing R
x : K := (algebraMap R K) b / (algebraMap R K) a
M : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)
ha₃ : (algebraMap R K) a ≠ 0
hx : ¬∀ y ∈ M, x * y ∈ M
this : (Algebra.linearMap R K) 1 ∈ Submodule.map (DistribMulAction.toLinearMap R K x) M
⊢ ∃ y ∈ maximalIdeal R, b * y = a

Tactic:
obtain ⟨_, ⟨y, hy, rfl⟩, hy' : x * algebraMap R K y = algebraMap R K 1⟩ := this

Result:
case intro.intro.intro.intro
R : Type u_1
inst✝³ : CommRing R
inst✝² : IsLocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝ : (Ideal.span {a}).radical = maximalIdeal R
this : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this = n + 1
b : R
hb₁ : b ∈ maximalIdeal R ^ n
hb₂ : b ∉ Ideal.span {a}
hb₃ : ∀ m ∈ maximalIdeal R, ∃ k, k * a = b * m
hb₄ : b ≠ 0
K : Type u_1 := FractionRing R
x : K := (algebraMap R K) b / (algebraMap R K) a
M : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)
ha₃ : (algebraMap R K) a ≠ 0
hx : ¬∀ y ∈ M, x * y ∈ M
y : R
hy : y ∈ ↑(maximalIdeal R)
hy' : x * (algebraMap R K) y = (algebraMap R K) 1
⊢ ∃ y ∈ maximalIdeal R, b * y = a

Proof state:
case pos.refine_1
R : Type u_1
inst✝³ : CommRing R
inst✝² : IsLocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝ : (Ideal.span {a}).radical = maximalIdeal R
this : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this = n + 1
b : R
hb₁ : b ∈ maximalIdeal R ^ n
hb₂ : b ∉ Ideal.span {a}
hb₃ : ∀ m ∈ maximalIdeal R, ∃ k, k * a = b * m
hb₄ : b ≠ 0
K : Type u_1 := FractionRing R
x : K := (algebraMap R K) b / (algebraMap R K) a
M : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)
ha₃ : (algebraMap R K) a ≠ 0
hx : ∀ y ∈ M, x * y ∈ M
⊢ (Algebra.linearMap R K) a ≠ 0

Tactic:
exact (IsFractionRing.to_map_eq_zero_iff (K := K)).not.mpr ha₂

Result:
no goals

Proof state:
R : Type u_1
inst✝³ : CommRing R
inst✝² : IsLocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝ : (Ideal.span {a}).radical = maximalIdeal R
this : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this = n + 1
⊢ ∃ b ∈ maximalIdeal R ^ n, b ∉ Ideal.span {a}

Tactic:
by_contra! h'

Result:
R : Type u_1
inst✝³ : CommRing R
inst✝² : IsLocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝ : (Ideal.span {a}).radical = maximalIdeal R
this : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this = n + 1
h' : ∀ b ∈ maximalIdeal R ^ n, b ∈ Ideal.span {a}
⊢ False

Proof state:
R : Type u_1
inst✝³ : CommRing R
inst✝² : IsLocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
⊢ Ideal.span {a} ≤ maximalIdeal R

Tactic:
rwa [Ideal.span_le, Set.singleton_subset_iff]

Result:
no goals

Proof state:
case pos.refine_3.intro.a
R : Type u_1
inst✝³ : CommRing R
inst✝² : IsLocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝¹ : (Ideal.span {a}).radical = maximalIdeal R
this✝ : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this✝ = n + 1
b : R
hb₁ : b ∈ maximalIdeal R ^ n
hb₂ : b ∉ Ideal.span {a}
hb₃ : ∀ m ∈ maximalIdeal R, ∃ k, k * a = b * m
hb₄ : b ≠ 0
K : Type u_1 := FractionRing R
x : K := (algebraMap R K) b / (algebraMap R K) a
M : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)
ha₃ : (algebraMap R K) a ≠ 0
hx : ∀ y ∈ M, x * y ∈ M
this : IsIntegral R x
y : R
e : (algebraMap R K) y = x
⊢ (algebraMap R K) (y * a) = (algebraMap R K) b

Tactic:
rw [map_mul, e, div_mul_cancel₀ _ ha₃]

Result:
no goals

Proof state:
case pos.refine_1
R : Type u_1
inst✝³ : CommRing R
inst✝² : IsLocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝ : (Ideal.span {a}).radical = maximalIdeal R
this : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this = n + 1
b : R
hb₁ : b ∈ maximalIdeal R ^ n
hb₂ : b ∉ Ideal.span {a}
hb₃ : ∀ m ∈ maximalIdeal R, ∃ k, k * a = b * m
hb₄ : b ≠ 0
K : Type u_1 := FractionRing R
x : K := (algebraMap R K) b / (algebraMap R K) a
M : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)
ha₃ : (algebraMap R K) a ≠ 0
hx : ∀ y ∈ M, x * y ∈ M
⊢ ∃ x ∈ M, x ≠ 0

Tactic:
refine ⟨_, ⟨a, ha₁, rfl⟩, ?_⟩

Result:
case pos.refine_1
R : Type u_1
inst✝³ : CommRing R
inst✝² : IsLocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝ : (Ideal.span {a}).radical = maximalIdeal R
this : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn : Nat.find this = n + 1
b : R
hb₁ : b ∈ maximalIdeal R ^ n
hb₂ : b ∉ Ideal.span {a}
hb₃ : ∀ m ∈ maximalIdeal R, ∃ k, k * a = b * m
hb₄ : b ≠ 0
K : Type u_1 := FractionRing R
x : K := (algebraMap R K) b / (algebraMap R K) a
M : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)
ha₃ : (algebraMap R K) a ≠ 0
hx : ∀ y ∈ M, x * y ∈ M
⊢ (Algebra.linearMap R K) a ≠ 0

Proof state:
case a
R : Type u_1
inst✝³ : CommRing R
inst✝² : IsLocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
⊢ sInf {J | Ideal.span {a} ≤ J ∧ J.IsPrime} ≤ maximalIdeal R

Tactic:
exact sInf_le ⟨hle, inferInstance⟩

Result:
no goals

Proof state:
case h
R : Type u_1
inst✝³ : CommRing R
inst✝² : IsLocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this : (Ideal.span {a}).radical = maximalIdeal R
⊢ (Ideal.span {a}).radical.FG

Tactic:
exact IsNoetherian.noetherian _

Result:
no goals