mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.DiscreteValuationRing.Basic

Goal state before:
pp: case neg
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsCancelMulZero R
hR : HasUnitMulPowIrreducibleFactorization R
p : R := Classical.choose hR
spec : Irreducible (Classical.choose hR) ∧ ∀ {x : R}, x ≠ 0 → ∃ n, Associated (Classical.choose hR ^ n) x :=
  Classical.choose_spec hR
x : R
hx : x ≠ 0
q : R
hq : q ∈ Multiset.replicate (Classical.choose ⋯) p
hpq : q = p
b : R
m : ℕ
u : Rˣ
h : p ∣ Classical.choose hR ^ m * b
ha : ¬Classical.choose hR ^ m * ↑u = 0
⊢ p ∣ Classical.choose hR ^ m ∨ p ∣ b

Tactic:
]

Goal state after:
case neg
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsCancelMulZero R
hR : HasUnitMulPowIrreducibleFactorization R
p : R := Classical.choose hR
spec : Irreducible (Classical.choose hR) ∧ ∀ {x : R}, x ≠ 0 → ∃ n, Associated (Classical.choose hR ^ n) x :=
  Classical.choose_spec hR
x : R
hx : x ≠ 0
q : R
hq : q ∈ Multiset.replicate (Classical.choose ⋯) p
hpq : q = p
b : R
m : ℕ
u : Rˣ
h : p ∣ Classical.choose hR ^ m * b
ha : ¬Classical.choose hR ^ m * ↑u = 0
⊢ p ∣ Classical.choose hR ^ m ∨ p ∣ b
Module: Mathlib.RingTheory.AdicCompletion.Basic

Goal state before:
pp: case mp
R : Type u_1
inst✝² : CommRing R
I : Ideal R
M : Type u_4
inst✝¹ : AddCommGroup M
inst✝ : Module R M
⊢ Function.Injective ⇑(of I M) → IsHausdorff I M

Tactic:
refine fun h ↦ ⟨fun x hx ↦ h ?_⟩

Goal state after:
case mp
R : Type u_1
inst✝² : CommRing R
I : Ideal R
M : Type u_4
inst✝¹ : AddCommGroup M
inst✝ : Module R M
h : Function.Injective ⇑(of I M)
x : M
hx : ∀ (n : ℕ), x ≡ 0 [SMOD I ^ n • ⊤]
⊢ (of I M) x = (of I M) 0
Module: Mathlib.RingTheory.AdicCompletion.Basic

Goal state before:
pp: case mpr
R : Type u_1
inst✝² : CommRing R
I : Ideal R
M : Type u_4
inst✝¹ : AddCommGroup M
inst✝ : Module R M
h : IsHausdorff I M
⊢ (of I M).ker = ⊥

Tactic:
]

Goal state after:
case mpr
R : Type u_1
inst✝² : CommRing R
I : Ideal R
M : Type u_4
inst✝¹ : AddCommGroup M
inst✝ : Module R M
h : IsHausdorff I M
⊢ (of I M).ker = ⊥
Module: Mathlib.RingTheory.AdicCompletion.Basic

Goal state before:
pp: case mpr
R : Type u_1
inst✝² : CommRing R
I : Ideal R
M : Type u_4
inst✝¹ : AddCommGroup M
inst✝ : Module R M
h : IsHausdorff I M
x : M
hx : ↑((of I M) x) = ↑0
n : ℕ
⊢ x ≡ 0 [SMOD I ^ n • ⊤]

Tactic:
]

Goal state after:
case mpr
R : Type u_1
inst✝² : CommRing R
I : Ideal R
M : Type u_4
inst✝¹ : AddCommGroup M
inst✝ : Module R M
h : IsHausdorff I M
x : M
hx : ↑((of I M) x) = ↑0
n : ℕ
⊢ x ≡ 0 [SMOD I ^ n • ⊤]
Module: Mathlib.RingTheory.DiscreteValuationRing.Basic

Goal state before:
pp: case neg
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsCancelMulZero R
hR : HasUnitMulPowIrreducibleFactorization R
p : R := Classical.choose hR
spec : Irreducible (Classical.choose hR) ∧ ∀ {x : R}, x ≠ 0 → ∃ n, Associated (Classical.choose hR ^ n) x :=
  Classical.choose_spec hR
x : R
hx : x ≠ 0
q : R
hq : q ∈ Multiset.replicate (Classical.choose ⋯) p
hpq : q = p
b : R
u : Rˣ
m : ℕ
h : p ∣ Classical.choose hR ^ m.succ * b
ha : ¬Classical.choose hR ^ m.succ * ↑u = 0
hm : ¬m.succ = 0
⊢ p ∣ Classical.choose hR * Classical.choose hR ^ m

Tactic:
]

Goal state after:
case neg
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsCancelMulZero R
hR : HasUnitMulPowIrreducibleFactorization R
p : R := Classical.choose hR
spec : Irreducible (Classical.choose hR) ∧ ∀ {x : R}, x ≠ 0 → ∃ n, Associated (Classical.choose hR ^ n) x :=
  Classical.choose_spec hR
x : R
hx : x ≠ 0
q : R
hq : q ∈ Multiset.replicate (Classical.choose ⋯) p
hpq : q = p
b : R
u : Rˣ
m : ℕ
h : p ∣ Classical.choose hR ^ m.succ * b
ha : ¬Classical.choose hR ^ m.succ * ↑u = 0
hm : ¬m.succ = 0
⊢ p ∣ Classical.choose hR * Classical.choose hR ^ m
Module: Mathlib.RingTheory.AdicCompletion.Basic

Goal state before:
pp: case mp
R : Type u_1
inst✝² : CommRing R
I : Ideal R
M : Type u_4
inst✝¹ : AddCommGroup M
inst✝ : Module R M
h : Function.Surjective ⇑(of I M)
f : ℕ → M
hmn : ∀ {m n : ℕ}, m ≤ n → f m ≡ f n [SMOD I ^ m • ⊤]
u : AdicCompletion I M := ⟨fun n ↦ Submodule.Quotient.mk (f n), ⋯⟩
x : M
hx : (of I M) x = u
n : ℕ
⊢ Submodule.Quotient.mk (f n) = (eval I M n) u

Tactic:
]

Goal state after:
case mp
R : Type u_1
inst✝² : CommRing R
I : Ideal R
M : Type u_4
inst✝¹ : AddCommGroup M
inst✝ : Module R M
h : Function.Surjective ⇑(of I M)
f : ℕ → M
hmn : ∀ {m n : ℕ}, m ≤ n → f m ≡ f n [SMOD I ^ m • ⊤]
u : AdicCompletion I M := ⟨fun n ↦ Submodule.Quotient.mk (f n), ⋯⟩
x : M
hx : (of I M) x = u
n : ℕ
⊢ Submodule.Quotient.mk (f n) = (eval I M n) u
Module: Mathlib.RingTheory.DiscreteValuationRing.Basic

Goal state before:
pp: case h.right
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsCancelMulZero R
hR : HasUnitMulPowIrreducibleFactorization R
p : R := Classical.choose hR
spec : Irreducible (Classical.choose hR) ∧ ∀ {x : R}, x ≠ 0 → ∃ n, Associated (Classical.choose hR ^ n) x :=
  Classical.choose_spec hR
x : R
hx : x ≠ 0
⊢ Associated (p ^ Classical.choose ⋯) x

Tactic:
]

Goal state after:
case h.right
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsCancelMulZero R
hR : HasUnitMulPowIrreducibleFactorization R
p : R := Classical.choose hR
spec : Irreducible (Classical.choose hR) ∧ ∀ {x : R}, x ≠ 0 → ∃ n, Associated (Classical.choose hR ^ n) x :=
  Classical.choose_spec hR
x : R
hx : x ≠ 0
⊢ Associated (p ^ Classical.choose ⋯) x
Module: Mathlib.RingTheory.Valuation.ValuationRing

Goal state before:
pp: case neg
𝒪 : Type u
K : Type v
inst✝² : CommRing 𝒪
inst✝¹ : Field K
inst✝ : Algebra 𝒪 K
h : ∀ (x : K), ∃ a, x = (algebraMap 𝒪 K) a ∨ x⁻¹ = (algebraMap 𝒪 K) a
hinj : Function.Injective ⇑(algebraMap 𝒪 K)
this✝² : IsDomain 𝒪
this✝¹ : FaithfulSMul 𝒪 K
this✝ : IsDomain 𝒪
x : K
a : 𝒪
ha : x = (algebraMap 𝒪 K) a ∨ x * (algebraMap 𝒪 K) a = 1
h0 : ¬a = 0
this : (algebraMap 𝒪 K) a ≠ 0
⊢ ∃ x_1, x * (algebraMap 𝒪 K) ↑x_1.2 = (algebraMap 𝒪 K) x_1.1

Tactic:
]

Goal state after:
case neg
𝒪 : Type u
K : Type v
inst✝² : CommRing 𝒪
inst✝¹ : Field K
inst✝ : Algebra 𝒪 K
h : ∀ (x : K), ∃ a, x = (algebraMap 𝒪 K) a ∨ x⁻¹ = (algebraMap 𝒪 K) a
hinj : Function.Injective ⇑(algebraMap 𝒪 K)
this✝² : IsDomain 𝒪
this✝¹ : FaithfulSMul 𝒪 K
this✝ : IsDomain 𝒪
x : K
a : 𝒪
ha : x = (algebraMap 𝒪 K) a ∨ x * (algebraMap 𝒪 K) a = 1
h0 : ¬a = 0
this : (algebraMap 𝒪 K) a ≠ 0
⊢ ∃ x_1, x * (algebraMap 𝒪 K) ↑x_1.2 = (algebraMap 𝒪 K) x_1.1
Module: Mathlib.RingTheory.AdicCompletion.Basic

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
I : Ideal R
M : Type u_4
inst✝¹ : AddCommGroup M
inst✝ : Module R M
h : IsPrecomplete I M
u : AdicCompletion I M
x : ℕ → M
hx : ∀ (n : ℕ), Submodule.Quotient.mk (x n) = ↑u n
m✝ n✝ : ℕ
hmn : m✝ ≤ n✝
⊢ (transitionMap I M hmn) (Submodule.Quotient.mk (x n✝)) = Submodule.Quotient.mk (x n✝)

Tactic:
]

Goal state after:
R : Type u_1
inst✝² : CommRing R
I : Ideal R
M : Type u_4
inst✝¹ : AddCommGroup M
inst✝ : Module R M
h : IsPrecomplete I M
u : AdicCompletion I M
x : ℕ → M
hx : ∀ (n : ℕ), Submodule.Quotient.mk (x n) = ↑u n
m✝ n✝ : ℕ
hmn : m✝ ≤ n✝
⊢ (transitionMap I M hmn) (Submodule.Quotient.mk (x n✝)) = Submodule.Quotient.mk (x n✝)
Module: Mathlib.RingTheory.DiscreteValuationRing.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : UniqueFactorizationMonoid R
h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q
p : R
hp : Irreducible p
x : R
hx : x ≠ 0
fx : Multiset R
hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x
H : Associates.mk fx.prod = Associates.mk x
⊢ Associates.mk (p ^ fx.card) = Associates.mk x

Tactic:
]

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : UniqueFactorizationMonoid R
h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q
p : R
hp : Irreducible p
x : R
hx : x ≠ 0
fx : Multiset R
hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x
H : Associates.mk fx.prod = Associates.mk x
⊢ Associates.mk (p ^ fx.card) = Associates.mk x
Module: Mathlib.RingTheory.AdicCompletion.Basic

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
I : Ideal R
M : Type u_4
inst✝¹ : AddCommGroup M
inst✝ : Module R M
n : ℕ
r : R
r_in : r ∈ I ^ n
x : AdicCompletion I M
⊢ r • out (↑x n) ∈ I ^ n • ⊤

Tactic:
]

Goal state after:
R : Type u_1
inst✝² : CommRing R
I : Ideal R
M : Type u_4
inst✝¹ : AddCommGroup M
inst✝ : Module R M
n : ℕ
r : R
r_in : r ∈ I ^ n
x : AdicCompletion I M
⊢ r • out (↑x n) ∈ I ^ n • ⊤
Module: Mathlib.RingTheory.DiscreteValuationRing.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : UniqueFactorizationMonoid R
h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q
p : R
hp : Irreducible p
x : R
hx : x ≠ 0
fx : Multiset R
hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x
H : Associates.mk fx.prod = Associates.mk x
⊢ (Multiset.replicate fx.card (Associates.mk p)).prod = (Multiset.map Associates.mk fx).prod

Tactic:
]

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : UniqueFactorizationMonoid R
h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q
p : R
hp : Irreducible p
x : R
hx : x ≠ 0
fx : Multiset R
hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x
H : Associates.mk fx.prod = Associates.mk x
⊢ (Multiset.replicate fx.card (Associates.mk p)).prod = (Multiset.map Associates.mk fx).prod
Module: Mathlib.RingTheory.DiscreteValuationRing.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : UniqueFactorizationMonoid R
h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q
p : R
hp : Irreducible p
x : R
hx : x ≠ 0
fx : Multiset R
hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x
H : Associates.mk fx.prod = Associates.mk x
⊢ (Multiset.map Associates.mk fx).card = fx.card ∧ ∀ b ∈ Multiset.map Associates.mk fx, b = Associates.mk p

Tactic:
]

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : UniqueFactorizationMonoid R
h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q
p : R
hp : Irreducible p
x : R
hx : x ≠ 0
fx : Multiset R
hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x
H : Associates.mk fx.prod = Associates.mk x
⊢ (Multiset.map Associates.mk fx).card = fx.card ∧ ∀ b ∈ Multiset.map Associates.mk fx, b = Associates.mk p
Module: Mathlib.RingTheory.AdicCompletion.Basic

Goal state before:
pp: case refine_1
R : Type u_1
inst✝² : CommRing R
I : Ideal R
M : Type u_4
inst✝¹ : AddCommGroup M
inst✝ : Module R M
m n : ℕ
x : AdicCompletion I M
m_ge : n ≤ m
h : ↑x m ∈ I ^ n • ⊤
⊢ ↑x m ∈ ((I ^ m • ⊤).liftQ ((I ^ n • ⊤).mkQ ∘ₗ LinearMap.id) ⋯).ker

Tactic:
]

Goal state after:
case refine_1
R : Type u_1
inst✝² : CommRing R
I : Ideal R
M : Type u_4
inst✝¹ : AddCommGroup M
inst✝ : Module R M
m n : ℕ
x : AdicCompletion I M
m_ge : n ≤ m
h : ↑x m ∈ I ^ n • ⊤
⊢ ↑x m ∈ ((I ^ m • ⊤).liftQ ((I ^ n • ⊤).mkQ ∘ₗ LinearMap.id) ⋯).ker
Module: Mathlib.RingTheory.DiscreteValuationRing.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : UniqueFactorizationMonoid R
h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q
p : R
hp : Irreducible p
x : R
hx : x ≠ 0
fx : Multiset R
hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x
H : Associates.mk fx.prod = Associates.mk x
q : R
hq : q ∈ fx
⊢ Associated q p

Tactic:
]

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : UniqueFactorizationMonoid R
h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q
p : R
hp : Irreducible p
x : R
hx : x ≠ 0
fx : Multiset R
hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x
H : Associates.mk fx.prod = Associates.mk x
q : R
hq : q ∈ fx
⊢ Associated q p
Module: Mathlib.RingTheory.Jacobson.Polynomial

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
J : Ideal R[X]
j : Ideal R
hj : j ∈ {J | J.IsMaximal} ∧ map C j = J
this : j.IsMaximal
t : ⊥.jacobson = ⊥
⊢ ⊥.jacobson = ⊥

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : CommRing R
J : Ideal R[X]
j : Ideal R
hj : j ∈ {J | J.IsMaximal} ∧ map C j = J
this : j.IsMaximal
t : ⊥.jacobson = ⊥
⊢ ⊥.jacobson = ⊥
Module: Mathlib.RingTheory.Jacobson.Polynomial

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
J : Ideal R[X]
j : Ideal R
hj : j ∈ {J | J.IsMaximal} ∧ map C j = J
this : j.IsMaximal
t : ⊥.jacobson = ⊥
⊢ ⊥.jacobson = ⊥

pp: R : Type u_1
inst✝ : CommRing R
J : Ideal R[X]
j : Ideal R
hj : j ∈ {J | J.IsMaximal} ∧ map C j = J
this : j.IsMaximal
t :
  map j.polynomialQuotientEquivQuotientPolynomial.toRingHom ⊥.jacobson =
    map j.polynomialQuotientEquivQuotientPolynomial.toRingHom ⊥
⊢ Function.Bijective ⇑j.polynomialQuotientEquivQuotientPolynomial.toRingHom

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : CommRing R
J : Ideal R[X]
j : Ideal R
hj : j ∈ {J | J.IsMaximal} ∧ map C j = J
this : j.IsMaximal
t : ⊥.jacobson = ⊥
⊢ ⊥.jacobson = ⊥

R : Type u_1
inst✝ : CommRing R
J : Ideal R[X]
j : Ideal R
hj : j ∈ {J | J.IsMaximal} ∧ map C j = J
this : j.IsMaximal
t :
  map j.polynomialQuotientEquivQuotientPolynomial.toRingHom ⊥.jacobson =
    map j.polynomialQuotientEquivQuotientPolynomial.toRingHom ⊥
⊢ Function.Bijective ⇑j.polynomialQuotientEquivQuotientPolynomial.toRingHom
Module: Mathlib.RingTheory.DiscreteValuationRing.Basic

Goal state before:
pp: case pos
R : Type u
inst✝¹ : CommRing R
inst✝ : UniqueFactorizationMonoid R
h₁ : ∃ p, Irreducible p
h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q
I : Ideal R
I0 : I = ⊥
⊢ Submodule.IsPrincipal ⊥

Tactic:
]

Goal state after:
case pos
R : Type u
inst✝¹ : CommRing R
inst✝ : UniqueFactorizationMonoid R
h₁ : ∃ p, Irreducible p
h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q
I : Ideal R
I0 : I = ⊥
⊢ Submodule.IsPrincipal ⊥
Module: Mathlib.RingTheory.Jacobson.Polynomial

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
h : ⊥.jacobson = ⊥
f : R[X]
hf : f ∈ sInf (map C '' {J | J.IsMaximal})
n : ℕ
this : f.coeff n = 0
⊢ f.coeff n = 0

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : CommRing R
h : ⊥.jacobson = ⊥
f : R[X]
hf : f ∈ sInf (map C '' {J | J.IsMaximal})
n : ℕ
this : f.coeff n = 0
⊢ f.coeff n = 0
Module: Mathlib.RingTheory.Jacobson.Ring

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
h : ∀ (P : Ideal R), P.IsPrime → P.jacobson = P
I : Ideal R
hI : I.IsRadical
x : R
hx : x ∈ I.jacobson
⊢ ∀ ⦃I_1 : Ideal R⦄, I_1 ∈ {J | I ≤ J ∧ J.IsPrime} → x ∈ I_1

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : CommRing R
h : ∀ (P : Ideal R), P.IsPrime → P.jacobson = P
I : Ideal R
hI : I.IsRadical
x : R
hx : x ∈ I.jacobson
⊢ ∀ ⦃I_1 : Ideal R⦄, I_1 ∈ {J | I ≤ J ∧ J.IsPrime} → x ∈ I_1
Module: Mathlib.RingTheory.Jacobson.Ring

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
h : ∀ (P : Ideal R), P.IsPrime → P.jacobson = P
I : Ideal R
hI : I.IsRadical
x : R
hx : x ∈ I.jacobson
P : Ideal R
hP : I ≤ P ∧ P.IsPrime
⊢ x ∈ P

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : CommRing R
h : ∀ (P : Ideal R), P.IsPrime → P.jacobson = P
I : Ideal R
hI : I.IsRadical
x : R
hx : x ∈ I.jacobson
P : Ideal R
hP : I ≤ P ∧ P.IsPrime
⊢ x ∈ P
Module: Mathlib.RingTheory.Jacobson.Ring

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
h : ∀ (P : Ideal R), P.IsPrime → P.jacobson = P
I : Ideal R
hI : I.IsRadical
x : R
hx : ∀ ⦃I_1 : Ideal R⦄, I_1 ∈ {J | I ≤ J ∧ J.IsMaximal} → x ∈ I_1
P : Ideal R
hP : I ≤ P ∧ P.IsPrime
⊢ x ∈ P

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : CommRing R
h : ∀ (P : Ideal R), P.IsPrime → P.jacobson = P
I : Ideal R
hI : I.IsRadical
x : R
hx : ∀ ⦃I_1 : Ideal R⦄, I_1 ∈ {J | I ≤ J ∧ J.IsMaximal} → x ∈ I_1
P : Ideal R
hP : I ≤ P ∧ P.IsPrime
⊢ x ∈ P
Module: Mathlib.RingTheory.Jacobson.Ring

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
h : ∀ (P : Ideal R), P.IsPrime → P.jacobson = P
I : Ideal R
hI : I.IsRadical
x : R
hx : ∀ ⦃I_1 : Ideal R⦄, I_1 ∈ {J | I ≤ J ∧ J.IsMaximal} → x ∈ I_1
P : Ideal R
hP : I ≤ P ∧ P.IsPrime
⊢ ∀ ⦃I : Ideal R⦄, I ∈ {J | P ≤ J ∧ J.IsMaximal} → x ∈ I

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : CommRing R
h : ∀ (P : Ideal R), P.IsPrime → P.jacobson = P
I : Ideal R
hI : I.IsRadical
x : R
hx : ∀ ⦃I_1 : Ideal R⦄, I_1 ∈ {J | I ≤ J ∧ J.IsMaximal} → x ∈ I_1
P : Ideal R
hP : I ≤ P ∧ P.IsPrime
⊢ ∀ ⦃I : Ideal R⦄, I ∈ {J | P ≤ J ∧ J.IsMaximal} → x ∈ I
Module: Mathlib.RingTheory.Jacobson.Ring

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
H : IsJacobsonRing R
f : R →+* S
hf : Function.Surjective ⇑f
⊢ ∀ {I : Ideal S}, I.IsPrime → ∃ M, (∀ J ∈ M, J.IsMaximal ∨ J = ⊤) ∧ I = sInf M

Tactic:
]

Goal state after:
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
H : IsJacobsonRing R
f : R →+* S
hf : Function.Surjective ⇑f
⊢ ∀ {I : Ideal S}, I.IsPrime → ∃ M, (∀ J ∈ M, J.IsMaximal ∨ J = ⊤) ∧ I = sInf M
Module: Mathlib.RingTheory.Jacobson.Ring

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : Algebra.IsIntegral R S
inst✝ : IsJacobsonRing R
⊢ ∀ (P : Ideal S), P.IsPrime → P.jacobson = P

Tactic:
]

Goal state after:
R : Type u_1
S : Type u_2
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : Algebra.IsIntegral R S
inst✝ : IsJacobsonRing R
⊢ ∀ (P : Ideal S), P.IsPrime → P.jacobson = P
Module: Mathlib.RingTheory.AdicCompletion.Basic

Goal state before:
pp: R : Type u_1
inst✝⁵ : CommRing R
I : Ideal R
M : Type u_4
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
N : Type u_5
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : IsAdicComplete I N
f : (n : ℕ) → M →ₗ[R] N ⧸ I ^ n • ⊤
h : ∀ {m n : ℕ} (hle : m ≤ n), Submodule.factorPow I N hle ∘ₗ f n = f m
n : ℕ
x : M
⊢ (AdicCompletion.eval I N n) ((of I N) ((ofLinearEquiv I N).symm ((AdicCompletion.lift I f ⋯) x))) = (f n) x

Tactic:
]

Goal state after:
R : Type u_1
inst✝⁵ : CommRing R
I : Ideal R
M : Type u_4
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
N : Type u_5
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : IsAdicComplete I N
f : (n : ℕ) → M →ₗ[R] N ⧸ I ^ n • ⊤
h : ∀ {m n : ℕ} (hle : m ≤ n), Submodule.factorPow I N hle ∘ₗ f n = f m
n : ℕ
x : M
⊢ (AdicCompletion.eval I N n) ((of I N) ((ofLinearEquiv I N).symm ((AdicCompletion.lift I f ⋯) x))) = (f n) x
Module: Mathlib.RingTheory.Jacobson.Ring

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : Algebra.IsIntegral R S
inst✝ : IsJacobsonRing R
P : Ideal S
hP : P.IsPrime
hP_top : ¬comap (algebraMap R S) P = ⊤
⊢ comap (algebraMap R S) P ≠ ⊤

Tactic:
]

Goal state after:
R : Type u_1
S : Type u_2
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : Algebra.IsIntegral R S
inst✝ : IsJacobsonRing R
P : Ideal S
hP : P.IsPrime
hP_top : ¬comap (algebraMap R S) P = ⊤
⊢ comap (algebraMap R S) P ≠ ⊤
Module: Mathlib.RingTheory.Jacobson.Ring

Goal state before:
pp: case neg
R : Type u_1
S : Type u_2
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : Algebra.IsIntegral R S
inst✝ : IsJacobsonRing R
P : Ideal S
hP : P.IsPrime
hP_top : ¬comap (algebraMap R S) P = ⊤
this : Nontrivial (R ⧸ comap (algebraMap R S) P)
⊢ ⊥.jacobson = ⊥

Tactic:
]

Goal state after:
case neg
R : Type u_1
S : Type u_2
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : Algebra.IsIntegral R S
inst✝ : IsJacobsonRing R
P : Ideal S
hP : P.IsPrime
hP_top : ¬comap (algebraMap R S) P = ⊤
this : Nontrivial (R ⧸ comap (algebraMap R S) P)
⊢ ⊥.jacobson = ⊥
Module: Mathlib.RingTheory.Jacobson.Ring

Goal state before:
pp: case neg
R : Type u_1
S : Type u_2
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : Algebra.IsIntegral R S
inst✝ : IsJacobsonRing R
P : Ideal S
hP : P.IsPrime
hP_top : ¬comap (algebraMap R S) P = ⊤
this : Nontrivial (R ⧸ comap (algebraMap R S) P)
⊢ sInf (comap (algebraMap (R ⧸ comap (algebraMap R S) P) (S ⧸ P)) '' {J | ⊥ ≤ J ∧ J.IsMaximal}) ≤ ⊥.jacobson

Tactic:
]

Goal state after:
case neg
R : Type u_1
S : Type u_2
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : Algebra.IsIntegral R S
inst✝ : IsJacobsonRing R
P : Ideal S
hP : P.IsPrime
hP_top : ¬comap (algebraMap R S) P = ⊤
this : Nontrivial (R ⧸ comap (algebraMap R S) P)
⊢ sInf (comap (algebraMap (R ⧸ comap (algebraMap R S) P) (S ⧸ P)) '' {J | ⊥ ≤ J ∧ J.IsMaximal}) ≤ ⊥.jacobson
Module: Mathlib.RingTheory.Jacobson.Ring

Goal state before:
pp: case mp
R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
y : R
inst✝¹ : Algebra R S
inst✝ : Away y S
H : IsJacobsonRing R
J : Ideal S
h : J.IsMaximal
hJ : (under R J).IsPrime ∧ Disjoint ↑(powers y) ↑(under R J)
⊢ (under R J).IsMaximal

Tactic:
]

Goal state after:
case mp
R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
y : R
inst✝¹ : Algebra R S
inst✝ : Away y S
H : IsJacobsonRing R
J : Ideal S
h : J.IsMaximal
hJ : (under R J).IsPrime ∧ Disjoint ↑(powers y) ↑(under R J)
⊢ (under R J).IsMaximal
Module: Mathlib.RingTheory.DiscreteValuationRing.Basic

Goal state before:
pp: case a
R : Type u
inst✝¹ : CommRing R
inst✝ : UniqueFactorizationMonoid R
h₁ : ∃ p, Irreducible p
h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q
I : Ideal R
I0 : ¬I = ⊥
x : R
hxI : x ∈ I
hx0 : x ≠ 0
p : R
left✝ : Irreducible p
H : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (p ^ n) x
ex : ∃ n, p ^ n ∈ I
⊢ p ^ Nat.find ex ∈ I

Tactic:
]

Goal state after:
case a
R : Type u
inst✝¹ : CommRing R
inst✝ : UniqueFactorizationMonoid R
h₁ : ∃ p, Irreducible p
h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q
I : Ideal R
I0 : ¬I = ⊥
x : R
hxI : x ∈ I
hx0 : x ≠ 0
p : R
left✝ : Irreducible p
H : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (p ^ n) x
ex : ∃ n, p ^ n ∈ I
⊢ p ^ Nat.find ex ∈ I
Module: Mathlib.RingTheory.Jacobson.Ring

Goal state before:
pp: case mp
R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
y : R
inst✝¹ : Algebra R S
inst✝ : Away y S
H : IsJacobsonRing R
J : Ideal S
h : J.IsMaximal
hJ : (under R J).IsPrime ∧ Disjoint ↑(powers y) ↑(under R J)
this : ¬∀ ⦃I : Ideal R⦄, I ∈ {J_1 | under R J ≤ J_1 ∧ J_1.IsMaximal} → y ∈ I
⊢ (under R J).IsMaximal

Tactic:
]

Goal state after:
case mp
R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
y : R
inst✝¹ : Algebra R S
inst✝ : Away y S
H : IsJacobsonRing R
J : Ideal S
h : J.IsMaximal
hJ : (under R J).IsPrime ∧ Disjoint ↑(powers y) ↑(under R J)
this : ¬∀ ⦃I : Ideal R⦄, I ∈ {J_1 | under R J ≤ J_1 ∧ J_1.IsMaximal} → y ∈ I
⊢ (under R J).IsMaximal
Module: Mathlib.RingTheory.DiscreteValuationRing.Basic

Goal state before:
pp: R : Type u
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : UniqueFactorizationMonoid R
h₁ : ∃ p, Irreducible p
h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q
⊢ IsPrincipalIdealRing R ∧ ∃! P, P ≠ ⊥ ∧ P.IsPrime

Tactic:
]

Goal state after:
R : Type u
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : UniqueFactorizationMonoid R
h₁ : ∃ p, Irreducible p
h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q
⊢ IsPrincipalIdealRing R ∧ ∃! P, P ≠ ⊥ ∧ P.IsPrime
Module: Mathlib.RingTheory.Valuation.Basic

Goal state before:
pp: R : Type u_3
Γ₀ : Type u_4
inst✝¹ : Ring R
inst✝ : LinearOrderedCommMonoidWithZero Γ₀
v : Valuation R Γ₀
x y : R
⊢ v (x + y) ≤ max (v x) (v y)

Tactic:
]

Goal state after:
R : Type u_3
Γ₀ : Type u_4
inst✝¹ : Ring R
inst✝ : LinearOrderedCommMonoidWithZero Γ₀
v : Valuation R Γ₀
x y : R
⊢ v (x + y) ≤ max (v x) (v y)
Module: Mathlib.RingTheory.DiscreteValuationRing.Basic

Goal state before:
pp: case refine_1
R : Type u
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : UniqueFactorizationMonoid R
h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q
PID : IsPrincipalIdealRing R
p : R
hp : Irreducible p
⊢ ∃ x ∈ span {p}, x ≠ 0

Tactic:
]

Goal state after:
case refine_1
R : Type u
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : UniqueFactorizationMonoid R
h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q
PID : IsPrincipalIdealRing R
p : R
hp : Irreducible p
⊢ ∃ x ∈ span {p}, x ≠ 0