mathlib-initiative/mathlib-tactics huggingface dataset:
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
f : ℕ → M
h : ∀ (n : ℕ), f n ≡ f (n + 1) [SMOD I ^ n • ⊤]
m n✝ n : ℕ
hmn : m ≤ n
ih : f m ≡ f n [SMOD I ^ m • ⊤]
⊢ f n ≡ f (n + 1) [SMOD I ^ m • ⊤]

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
f : ℕ → M
h : ∀ (n : ℕ), f n ≡ f (n + 1) [SMOD I ^ n • ⊤]
m n✝ n : ℕ
hmn : m ≤ n
ih : f m ≡ f n [SMOD I ^ m • ⊤]
⊢ f n ≡ f (n + 1) [SMOD I ^ m • ⊤]
Module: Mathlib.RingTheory.DiscreteValuationRing.Basic

Goal state before:
pp: case inl
R : Type u_1
inst✝ : CommRing R
ϖ : R
hϖ : Irreducible ϖ
hR : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (ϖ ^ n) x
p : R
hp : Irreducible p
n : ℕ
hn : Associated (ϖ ^ n) p
this : Irreducible (ϖ ^ n)
H : n < 1
⊢ Associated p ϖ

pp: case inr.inl
R : Type u_1
inst✝ : CommRing R
ϖ : R
hϖ : Irreducible ϖ
hR : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (ϖ ^ n) x
p : R
hp : Irreducible p
hn : Associated (ϖ ^ 1) p
this : Irreducible (ϖ ^ 1)
⊢ Associated p ϖ

pp: case inr.inr
R : Type u_1
inst✝ : CommRing R
ϖ : R
hϖ : Irreducible ϖ
hR : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (ϖ ^ n) x
p : R
hp : Irreducible p
n : ℕ
hn : Associated (ϖ ^ n) p
this : Irreducible (ϖ ^ n)
H : 1 < n
⊢ Associated p ϖ

Tactic:
·

Goal state after:
case inl
R : Type u_1
inst✝ : CommRing R
ϖ : R
hϖ : Irreducible ϖ
hR : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (ϖ ^ n) x
p : R
hp : Irreducible p
n : ℕ
hn : Associated (ϖ ^ n) p
this : Irreducible (ϖ ^ n)
H : n < 1
⊢ Associated p ϖ
Module: Mathlib.RingTheory.DiscreteValuationRing.Basic

Goal state before:
pp: case inr.inl
R : Type u_1
inst✝ : CommRing R
ϖ : R
hϖ : Irreducible ϖ
hR : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (ϖ ^ n) x
p : R
hp : Irreducible p
hn : Associated (ϖ ^ 1) p
this : Irreducible (ϖ ^ 1)
⊢ Associated p ϖ

pp: case inr.inr
R : Type u_1
inst✝ : CommRing R
ϖ : R
hϖ : Irreducible ϖ
hR : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (ϖ ^ n) x
p : R
hp : Irreducible p
n : ℕ
hn : Associated (ϖ ^ n) p
this : Irreducible (ϖ ^ n)
H : 1 < n
⊢ Associated p ϖ

Tactic:
·

Goal state after:
case inr.inl
R : Type u_1
inst✝ : CommRing R
ϖ : R
hϖ : Irreducible ϖ
hR : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (ϖ ^ n) x
p : R
hp : Irreducible p
hn : Associated (ϖ ^ 1) p
this : Irreducible (ϖ ^ 1)
⊢ Associated p ϖ
Module: Mathlib.RingTheory.Valuation.ValuationRing

Goal state before:
pp: case inl
R : Type u_1
inst✝⁶ : CommRing R
inst✝⁵ : IsDomain R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsLocalRing R
inst✝ : IsBezout R
g a b : R
e : Ideal.span {a * g, b * g} = Ideal.span {g}
x y : R
e' : x * (a * g) + y * (b * g) = g
h : g = 0
⊢ a * g ∣ b * g ∨ b * g ∣ a * g

pp: case inr
R : Type u_1
inst✝⁶ : CommRing R
inst✝⁵ : IsDomain R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsLocalRing R
inst✝ : IsBezout R
g a b : R
e : Ideal.span {a * g, b * g} = Ideal.span {g}
x y : R
e' : x * (a * g) + y * (b * g) = g
h : g ≠ 0
⊢ a * g ∣ b * g ∨ b * g ∣ a * g

Tactic:
·

Goal state after:
case inl
R : Type u_1
inst✝⁶ : CommRing R
inst✝⁵ : IsDomain R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsLocalRing R
inst✝ : IsBezout R
g a b : R
e : Ideal.span {a * g, b * g} = Ideal.span {g}
x y : R
e' : x * (a * g) + y * (b * g) = g
h : g = 0
⊢ a * g ∣ b * g ∨ b * g ∣ a * g
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
f : AdicCauchySequence I M
k : ℕ
h : ∀ n ≥ k, ∃ m ≥ n, ∃ l ≥ n, ↑f m ∈ I ^ l • ⊤
n : ℕ
⊢ n + k ≥ k

Tactic:
by lia

Goal state after:
Module: Mathlib.RingTheory.DiscreteValuationRing.Basic

Goal state before:
pp: case inr.inr
R : Type u_1
inst✝ : CommRing R
ϖ : R
hϖ : Irreducible ϖ
hR : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (ϖ ^ n) x
p : R
hp : Irreducible p
n : ℕ
hn : Associated (ϖ ^ n) p
this : Irreducible (ϖ ^ n)
H : 1 < n
⊢ Associated p ϖ

Tactic:
·

Goal state after:
case inr.inr
R : Type u_1
inst✝ : CommRing R
ϖ : R
hϖ : Irreducible ϖ
hR : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (ϖ ^ n) x
p : R
hp : Irreducible p
n : ℕ
hn : Associated (ϖ ^ n) p
this : Irreducible (ϖ ^ n)
H : 1 < n
⊢ Associated p ϖ
Module: Mathlib.RingTheory.DiscreteValuationRing.Basic

Goal state before:
pp: case inr.inr.inl
R : Type u_1
inst✝ : CommRing R
ϖ : R
hϖ : Irreducible ϖ
hR : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (ϖ ^ n) x
p : R
hp : Irreducible p
n : ℕ
hn : Associated (ϖ ^ (1 + n + 1)) p
this : Irreducible (ϖ * ϖ ^ (1 + n))
H : 1 < 1 + n + 1
H0 : IsUnit ϖ
⊢ Associated p ϖ

pp: case inr.inr.inr
R : Type u_1
inst✝ : CommRing R
ϖ : R
hϖ : Irreducible ϖ
hR : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (ϖ ^ n) x
p : R
hp : Irreducible p
n : ℕ
hn : Associated (ϖ ^ (1 + n + 1)) p
this : Irreducible (ϖ * ϖ ^ (1 + n))
H : 1 < 1 + n + 1
H0 : IsUnit (ϖ ^ (1 + n))
⊢ Associated p ϖ

Tactic:
·

Goal state after:
case inr.inr.inl
R : Type u_1
inst✝ : CommRing R
ϖ : R
hϖ : Irreducible ϖ
hR : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (ϖ ^ n) x
p : R
hp : Irreducible p
n : ℕ
hn : Associated (ϖ ^ (1 + n + 1)) p
this : Irreducible (ϖ * ϖ ^ (1 + n))
H : 1 < 1 + n + 1
H0 : IsUnit ϖ
⊢ Associated p ϖ
Module: Mathlib.RingTheory.DiscreteValuationRing.Basic

Goal state before:
pp: case inr.inr.inr
R : Type u_1
inst✝ : CommRing R
ϖ : R
hϖ : Irreducible ϖ
hR : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (ϖ ^ n) x
p : R
hp : Irreducible p
n : ℕ
hn : Associated (ϖ ^ (1 + n + 1)) p
this : Irreducible (ϖ * ϖ ^ (1 + n))
H : 1 < 1 + n + 1
H0 : IsUnit (ϖ ^ (1 + n))
⊢ Associated p ϖ

Tactic:
·

Goal state after:
case inr.inr.inr
R : Type u_1
inst✝ : CommRing R
ϖ : R
hϖ : Irreducible ϖ
hR : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (ϖ ^ n) x
p : R
hp : Irreducible p
n : ℕ
hn : Associated (ϖ ^ (1 + n + 1)) p
this : Irreducible (ϖ * ϖ ^ (1 + n))
H : 1 < 1 + n + 1
H0 : IsUnit (ϖ ^ (1 + n))
⊢ Associated p ϖ
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
f : AdicCauchySequence I M
k : ℕ
h : ∀ n ≥ k, ∃ m ≥ n, ∃ l ≥ n, ↑f m ∈ I ^ l • ⊤
n m : ℕ
hnm : m ≥ n + k
l : ℕ
hnl : l ≥ n + k
hl : ↑f m ∈ I ^ l • ⊤
⊢ n ≤ l

Tactic:
by lia

Goal state after:
Module: Mathlib.RingTheory.DiscreteValuationRing.Basic

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

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

Tactic:
·

Goal state after:
case h.left
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsCancelMulZero R
hR : HasUnitMulPowIrreducibleFactorization R
p : R := ⋯
spec : Irreducible (Classical.choose hR) ∧ ∀ {x : R}, x ≠ 0 → ∃ n, Associated (Classical.choose hR ^ n) x := ⋯
x : R
hx : x ≠ 0
⊢ ∀ b ∈ Multiset.replicate (Classical.choose ⋯) p, Prime b
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
x : AdicCompletion I M
a : ℕ → M
ha : ∀ (n : ℕ), Submodule.Quotient.mk (a n) = ↑x n
⊢ IsAdicCauchy I M a

pp: case refine_2
R : Type u_1
inst✝² : CommRing R
I : Ideal R
M : Type u_4
inst✝¹ : AddCommGroup M
inst✝ : Module R M
x : AdicCompletion I M
a : ℕ → M
ha : ∀ (n : ℕ), Submodule.Quotient.mk (a n) = ↑x n
⊢ (mk I M) ⟨a, ?refine_1⟩ = x

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
x : AdicCompletion I M
a : ℕ → M
ha : ∀ (n : ℕ), Submodule.Quotient.mk (a n) = ↑x n
⊢ IsAdicCauchy I M a
Module: Mathlib.RingTheory.DiscreteValuationRing.Basic

Goal state before:
pp: case pos
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
a b : R
h : p ∣ a * b
ha : a = 0
⊢ p ∣ a ∨ p ∣ b

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
a b : R
h : p ∣ a * b
ha : ¬a = 0
⊢ p ∣ a ∨ p ∣ b

Tactic:
·

Goal state after:
case pos
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
a b : R
h : p ∣ a * b
ha : a = 0
⊢ p ∣ a ∨ p ∣ b
Module: Mathlib.RingTheory.AdicCompletion.Basic

Goal state before:
pp: case refine_2
R : Type u_1
inst✝² : CommRing R
I : Ideal R
M : Type u_4
inst✝¹ : AddCommGroup M
inst✝ : Module R M
x : AdicCompletion I M
a : ℕ → M
ha : ∀ (n : ℕ), Submodule.Quotient.mk (a n) = ↑x n
⊢ (mk I M) ⟨a, ⋯⟩ = x

Tactic:
·

Goal state after:
case refine_2
R : Type u_1
inst✝² : CommRing R
I : Ideal R
M : Type u_4
inst✝¹ : AddCommGroup M
inst✝ : Module R M
x : AdicCompletion I M
a : ℕ → M
ha : ∀ (n : ℕ), Submodule.Quotient.mk (a n) = ↑x n
⊢ (mk I M) ⟨a, ⋯⟩ = x
Module: Mathlib.RingTheory.Valuation.ValuationRing

Goal state before:
pp: R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
tfae_1_iff_2 : ValuationRing R ↔ ∀ (x : FractionRing R), IsLocalization.IsInteger R x ∨ IsLocalization.IsInteger R x⁻¹
tfae_1_iff_3 : ValuationRing R ↔ Std.Total fun x1 x2 ↦ x1 ∣ x2
tfae_1_iff_4 : ValuationRing R ↔ Std.Total fun x1 x2 ↦ x1 ≤ x2
tfae_1_iff_5 : ValuationRing R ↔ IsLocalRing R ∧ IsBezout R
⊢ [ValuationRing R, ∀ (x : FractionRing R), IsLocalization.IsInteger R x ∨ IsLocalization.IsInteger R x⁻¹,
      Std.Total fun x1 x2 ↦ x1 ∣ x2, Std.Total fun x1 x2 ↦ x1 ≤ x2, IsLocalRing R ∧ IsBezout R].TFAE

Tactic:
tfae_finish

Goal state after:
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

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
⊢ IsHausdorff I M → Function.Injective ⇑(of I M)

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
⊢ Function.Injective ⇑(of I M) → IsHausdorff I M
Module: Mathlib.RingTheory.DiscreteValuationRing.Basic

Goal state before:
pp: case pos
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
hm : m = 0
⊢ p ∣ Classical.choose hR ^ m ∨ p ∣ b

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
hm : ¬m = 0
⊢ p ∣ Classical.choose hR ^ m ∨ p ∣ b

Tactic:
·

Goal state after:
case pos
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
hm : m = 0
⊢ p ∣ Classical.choose hR ^ m ∨ p ∣ b
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
⊢ IsHausdorff I M → Function.Injective ⇑(of I M)

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
⊢ IsHausdorff I M → Function.Injective ⇑(of I M)
Module: Mathlib.RingTheory.Valuation.ValuationRing

Goal state before:
pp: case pos
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✝¹ : IsDomain 𝒪
this✝ : FaithfulSMul 𝒪 K
this : IsDomain 𝒪
x : K
a : 𝒪
ha : x = (algebraMap 𝒪 K) a ∨ x⁻¹ = (algebraMap 𝒪 K) a
h0 : a = 0
⊢ ∃ x_1, x * (algebraMap 𝒪 K) ↑x_1.2 = (algebraMap 𝒪 K) x_1.1

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✝¹ : IsDomain 𝒪
this✝ : FaithfulSMul 𝒪 K
this : IsDomain 𝒪
x : K
a : 𝒪
ha : x = (algebraMap 𝒪 K) a ∨ x⁻¹ = (algebraMap 𝒪 K) a
h0 : ¬a = 0
⊢ ∃ x_1, x * (algebraMap 𝒪 K) ↑x_1.2 = (algebraMap 𝒪 K) x_1.1

Tactic:
·

Goal state after:
case pos
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✝¹ : IsDomain 𝒪
this✝ : FaithfulSMul 𝒪 K
this : IsDomain 𝒪
x : K
a : 𝒪
ha : x = (algebraMap 𝒪 K) a ∨ x⁻¹ = (algebraMap 𝒪 K) a
h0 : 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: 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
⊢ (of I M) x = 0 ↔ x = 0

Tactic:
refine ⟨fun hx ↦ h.haus x fun n ↦ ?_, fun hx ↦ by simp [hx]⟩

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.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.Surjective ⇑(of I M) → IsPrecomplete I M

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
⊢ IsPrecomplete I M → Function.Surjective ⇑(of I M)

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
⊢ Function.Surjective ⇑(of I M) → IsPrecomplete I M
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✝¹ : IsDomain 𝒪
this✝ : FaithfulSMul 𝒪 K
this : IsDomain 𝒪
x : K
a : 𝒪
ha : x = (algebraMap 𝒪 K) a ∨ x⁻¹ = (algebraMap 𝒪 K) a
h0 : ¬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✝¹ : IsDomain 𝒪
this✝ : FaithfulSMul 𝒪 K
this : IsDomain 𝒪
x : K
a : 𝒪
ha : x = (algebraMap 𝒪 K) a ∨ x⁻¹ = (algebraMap 𝒪 K) a
h0 : ¬a = 0
⊢ ∃ x_1, x * (algebraMap 𝒪 K) ↑x_1.2 = (algebraMap 𝒪 K) x_1.1
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 := ⋯
spec : Irreducible (Classical.choose hR) ∧ ∀ {x : R}, x ≠ 0 → ∃ n, Associated (Classical.choose hR ^ n) x := ⋯
x : R
hx : x ≠ 0
⊢ Associated (Multiset.replicate (Classical.choose ⋯) p).prod x

Tactic:
·

Goal state after:
case h.right
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsCancelMulZero R
hR : HasUnitMulPowIrreducibleFactorization R
p : R := ⋯
spec : Irreducible (Classical.choose hR) ∧ ∀ {x : R}, x ≠ 0 → ∃ n, Associated (Classical.choose hR ^ n) x := ⋯
x : R
hx : x ≠ 0
⊢ Associated (Multiset.replicate (Classical.choose ⋯) p).prod x
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
⊢ IsPrecomplete I M → Function.Surjective ⇑(of I M)

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
⊢ IsPrecomplete I M → Function.Surjective ⇑(of I M)
Module: Mathlib.RingTheory.Valuation.ValuationRing

Goal state before:
pp: case inl
𝒪 : Type u
K : Type v
Γ : Type w
inst✝³ : CommRing 𝒪
inst✝² : Field K
inst✝¹ : Algebra 𝒪 K
inst✝ : LinearOrderedCommGroupWithZero Γ
v : Valuation K Γ
hh : v.Integers 𝒪
this : IsDomain 𝒪
a b : 𝒪
h : v ((algebraMap 𝒪 K) a) ≤ v ((algebraMap 𝒪 K) b)
⊢ ∃ c, a * c = b ∨ b * c = a

pp: case inr
𝒪 : Type u
K : Type v
Γ : Type w
inst✝³ : CommRing 𝒪
inst✝² : Field K
inst✝¹ : Algebra 𝒪 K
inst✝ : LinearOrderedCommGroupWithZero Γ
v : Valuation K Γ
hh : v.Integers 𝒪
this : IsDomain 𝒪
a b : 𝒪
h : v ((algebraMap 𝒪 K) b) ≤ v ((algebraMap 𝒪 K) a)
⊢ ∃ c, a * c = b ∨ b * c = a

Tactic:
·

Goal state after:
case inl
𝒪 : Type u
K : Type v
Γ : Type w
inst✝³ : CommRing 𝒪
inst✝² : Field K
inst✝¹ : Algebra 𝒪 K
inst✝ : LinearOrderedCommGroupWithZero Γ
v : Valuation K Γ
hh : v.Integers 𝒪
this : IsDomain 𝒪
a b : 𝒪
h : v ((algebraMap 𝒪 K) a) ≤ v ((algebraMap 𝒪 K) b)
⊢ ∃ c, a * c = b ∨ b * c = a
Module: Mathlib.RingTheory.Valuation.ValuationRing

Goal state before:
pp: case inr
𝒪 : Type u
K : Type v
Γ : Type w
inst✝³ : CommRing 𝒪
inst✝² : Field K
inst✝¹ : Algebra 𝒪 K
inst✝ : LinearOrderedCommGroupWithZero Γ
v : Valuation K Γ
hh : v.Integers 𝒪
this : IsDomain 𝒪
a b : 𝒪
h : v ((algebraMap 𝒪 K) b) ≤ v ((algebraMap 𝒪 K) a)
⊢ ∃ c, a * c = b ∨ b * c = a

Tactic:
·

Goal state after:
case inr
𝒪 : Type u
K : Type v
Γ : Type w
inst✝³ : CommRing 𝒪
inst✝² : Field K
inst✝¹ : Algebra 𝒪 K
inst✝ : LinearOrderedCommGroupWithZero Γ
v : Valuation K Γ
hh : v.Integers 𝒪
this : IsDomain 𝒪
a b : 𝒪
h : v ((algebraMap 𝒪 K) b) ≤ v ((algebraMap 𝒪 K) a)
⊢ ∃ c, a * c = b ∨ b * c = a
Module: Mathlib.RingTheory.Valuation.ValuationRing

Goal state before:
pp: case refine_1
𝒪 : Type u
K : Type v
inst✝⁴ : CommRing 𝒪
inst✝³ : Field K
inst✝² : Algebra 𝒪 K
inst✝¹ : IsDomain 𝒪
inst✝ : ValuationRing 𝒪
h : IsFractionRing 𝒪 K
x : K
⊢ ∃ a, x = (algebraMap 𝒪 K) a ∨ x⁻¹ = (algebraMap 𝒪 K) a

pp: case refine_2
𝒪 : Type u
K : Type v
inst✝⁴ : CommRing 𝒪
inst✝³ : Field K
inst✝² : Algebra 𝒪 K
inst✝¹ : IsDomain 𝒪
inst✝ : ValuationRing 𝒪
h : (∀ (x : K), ∃ a, x = (algebraMap 𝒪 K) a ∨ x⁻¹ = (algebraMap 𝒪 K) a) ∧ Function.Injective ⇑(algebraMap 𝒪 K)
⊢ IsFractionRing 𝒪 K

Tactic:
·

Goal state after:
case refine_1
𝒪 : Type u
K : Type v
inst✝⁴ : CommRing 𝒪
inst✝³ : Field K
inst✝² : Algebra 𝒪 K
inst✝¹ : IsDomain 𝒪
inst✝ : ValuationRing 𝒪
h : IsFractionRing 𝒪 K
x : K
⊢ ∃ a, x = (algebraMap 𝒪 K) a ∨ x⁻¹ = (algebraMap 𝒪 K) a
Module: Mathlib.RingTheory.Valuation.ValuationRing

Goal state before:
pp: case refine_2
𝒪 : Type u
K : Type v
inst✝⁴ : CommRing 𝒪
inst✝³ : Field K
inst✝² : Algebra 𝒪 K
inst✝¹ : IsDomain 𝒪
inst✝ : ValuationRing 𝒪
h : (∀ (x : K), ∃ a, x = (algebraMap 𝒪 K) a ∨ x⁻¹ = (algebraMap 𝒪 K) a) ∧ Function.Injective ⇑(algebraMap 𝒪 K)
⊢ IsFractionRing 𝒪 K

Tactic:
·

Goal state after:
case refine_2
𝒪 : Type u
K : Type v
inst✝⁴ : CommRing 𝒪
inst✝³ : Field K
inst✝² : Algebra 𝒪 K
inst✝¹ : IsDomain 𝒪
inst✝ : ValuationRing 𝒪
h : (∀ (x : K), ∃ a, x = (algebraMap 𝒪 K) a ∨ x⁻¹ = (algebraMap 𝒪 K) a) ∧ Function.Injective ⇑(algebraMap 𝒪 K)
⊢ IsFractionRing 𝒪 K
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 n = 0

pp: case refine_2
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 n = 0
⊢ ↑x m ∈ I ^ n • ⊤

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 n = 0
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 I

pp: case neg
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 I

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 I
Module: Mathlib.RingTheory.Jacobson.Ring

Goal state before:
pp: case pos
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 = ⊤
⊢ P.jacobson = P

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 = ⊤
⊢ P.jacobson = P

Tactic:
·

Goal state after:
case pos
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 = ⊤
⊢ P.jacobson = P
Module: Mathlib.RingTheory.DiscreteValuationRing.Basic

Goal state before:
pp: case h.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
⊢ I ≤ span {p ^ Nat.find ex}

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
⊢ span {p ^ Nat.find ex} ≤ I

Tactic:
·

Goal state after:
case h.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
⊢ I ≤ span {p ^ Nat.find ex}