mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity

Goal state before:
pp: case succ.mpr
R : Type u_2
inst✝² : CommMonoidWithZero R
inst✝¹ : UniqueFactorizationMonoid R
inst✝ : NormalizationMonoid R
a : R
ha : Irreducible a
n : ℕ
ih : ∀ {b : R}, b ≠ 0 → (a ^ n ∣ b ↔ replicate n (normalize a) ≤ normalizedFactors b)
b : R
hb : b ≠ 0
⊢ (∃ u, normalizedFactors b = replicate (n + 1) (normalize a) + u) → a ^ (n + 1) ∣ b

Tactic:
]

Goal state after:
case succ.mpr
R : Type u_2
inst✝² : CommMonoidWithZero R
inst✝¹ : UniqueFactorizationMonoid R
inst✝ : NormalizationMonoid R
a : R
ha : Irreducible a
n : ℕ
ih : ∀ {b : R}, b ≠ 0 → (a ^ n ∣ b ↔ replicate n (normalize a) ≤ normalizedFactors b)
b : R
hb : b ≠ 0
⊢ (∃ u, normalizedFactors b = replicate (n + 1) (normalize a) + u) → a ^ (n + 1) ∣ b
Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity

Goal state before:
pp: case succ.mpr
R : Type u_2
inst✝² : CommMonoidWithZero R
inst✝¹ : UniqueFactorizationMonoid R
inst✝ : NormalizationMonoid R
a : R
ha : Irreducible a
n : ℕ
ih : ∀ {b : R}, b ≠ 0 → (a ^ n ∣ b ↔ replicate n (normalize a) ≤ normalizedFactors b)
b : R
hb : b ≠ 0
u : Multiset R
hu : normalizedFactors b = replicate (n + 1) (normalize a) + u
⊢ a ^ (n + 1) ∣ normalize a ^ (n + 1) * u.prod

Tactic:
]

Goal state after:
case succ.mpr
R : Type u_2
inst✝² : CommMonoidWithZero R
inst✝¹ : UniqueFactorizationMonoid R
inst✝ : NormalizationMonoid R
a : R
ha : Irreducible a
n : ℕ
ih : ∀ {b : R}, b ≠ 0 → (a ^ n ∣ b ↔ replicate n (normalize a) ≤ normalizedFactors b)
b : R
hb : b ≠ 0
u : Multiset R
hu : normalizedFactors b = replicate (n + 1) (normalize a) + u
⊢ a ^ (n + 1) ∣ normalize a ^ (n + 1) * u.prod
Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity

Goal state before:
pp: case succ.mpr
R : Type u_2
inst✝² : CommMonoidWithZero R
inst✝¹ : UniqueFactorizationMonoid R
inst✝ : NormalizationMonoid R
a : R
ha : Irreducible a
n : ℕ
ih : ∀ {b : R}, b ≠ 0 → (a ^ n ∣ b ↔ replicate n (normalize a) ≤ normalizedFactors b)
b : R
hb : b ≠ 0
u : Multiset R
hu : normalizedFactors b = replicate (n + 1) (normalize a) + u
⊢ a ^ (n + 1) ∣ normalize a ^ (n + 1) * u.prod

Tactic:
exact (Associated.pow_pow <| associated_normalize a).dvd.trans (Dvd.intro u.prod rfl)

Goal state after:
Module: Mathlib.Algebra.Category.CoalgCat.ComonEquivalence

Goal state before:
pp: R : Type u
inst✝ : CommRing R
K L : CoalgCat R
⊢ ModuleCat.Hom.hom Δ =
    ↑(TensorProduct.tensorTensorTensorComm R ↑K.toModuleCat ↑K.toModuleCat ↑L.toModuleCat ↑L.toModuleCat) ∘ₗ
      TensorProduct.map CoalgebraStruct.comul CoalgebraStruct.comul

Tactic:
]

Goal state after:
R : Type u
inst✝ : CommRing R
K L : CoalgCat R
⊢ ModuleCat.Hom.hom Δ =
    ↑(TensorProduct.tensorTensorTensorComm R ↑K.toModuleCat ↑K.toModuleCat ↑L.toModuleCat ↑L.toModuleCat) ∘ₗ
      TensorProduct.map CoalgebraStruct.comul CoalgebraStruct.comul
Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity

Goal state before:
pp: case a
R : Type u_2
inst✝³ : CommMonoidWithZero R
inst✝² : UniqueFactorizationMonoid R
inst✝¹ : NormalizationMonoid R
inst✝ : DecidableEq R
a b : R
ha : Irreducible a
hb : b ≠ 0
⊢ ¬replicate (count (normalize a) (normalizedFactors b) + 1) (normalize a) ≤ normalizedFactors b

Tactic:
← le_count_iff_replicate_le

Goal state after:
case a
R : Type u_2
inst✝³ : CommMonoidWithZero R
inst✝² : UniqueFactorizationMonoid R
inst✝¹ : NormalizationMonoid R
inst✝ : DecidableEq R
a b : R
ha : Irreducible a
hb : b ≠ 0
⊢ ¬count (normalize a) (normalizedFactors b) + 1 ≤ count (normalize a) (normalizedFactors b)
Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity

Goal state before:
pp: case a
R : Type u_2
inst✝³ : CommMonoidWithZero R
inst✝² : UniqueFactorizationMonoid R
inst✝¹ : NormalizationMonoid R
inst✝ : DecidableEq R
a b : R
ha : Irreducible a
hb : b ≠ 0
⊢ ¬count (normalize a) (normalizedFactors b) + 1 ≤ count (normalize a) (normalizedFactors b)

Tactic:
]

Goal state after:
case a
R : Type u_2
inst✝³ : CommMonoidWithZero R
inst✝² : UniqueFactorizationMonoid R
inst✝¹ : NormalizationMonoid R
inst✝ : DecidableEq R
a b : R
ha : Irreducible a
hb : b ≠ 0
⊢ ¬count (normalize a) (normalizedFactors b) + 1 ≤ count (normalize a) (normalizedFactors b)
Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity

Goal state before:
pp: case a
R : Type u_2
inst✝³ : CommMonoidWithZero R
inst✝² : UniqueFactorizationMonoid R
inst✝¹ : NormalizationMonoid R
inst✝ : DecidableEq R
a b : R
ha : Irreducible a
hb : b ≠ 0
⊢ replicate (count (normalize a) (normalizedFactors b)) (normalize a) ≤ normalizedFactors b

Tactic:
← le_count_iff_replicate_le

Goal state after:
case a
R : Type u_2
inst✝³ : CommMonoidWithZero R
inst✝² : UniqueFactorizationMonoid R
inst✝¹ : NormalizationMonoid R
inst✝ : DecidableEq R
a b : R
ha : Irreducible a
hb : b ≠ 0
⊢ count (normalize a) (normalizedFactors b) ≤ count (normalize a) (normalizedFactors b)
Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity

Goal state before:
pp: case a
R : Type u_2
inst✝³ : CommMonoidWithZero R
inst✝² : UniqueFactorizationMonoid R
inst✝¹ : NormalizationMonoid R
inst✝ : DecidableEq R
a b : R
ha : Irreducible a
hb : b ≠ 0
⊢ count (normalize a) (normalizedFactors b) ≤ count (normalize a) (normalizedFactors b)

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity

Goal state before:
pp: R : Type u_2
inst✝³ : CommMonoidWithZero R
inst✝² : UniqueFactorizationMonoid R
inst✝¹ : NormalizationMonoid R
inst✝ : DecidableEq R
a b : R
ha : Irreducible a
hb : b ≠ 0
this : multiplicity a b = count (normalize a) (normalizedFactors b)
⊢ multiplicity a b = count (normalize a) (normalizedFactors b)

Tactic:
]

Goal state after:
R : Type u_2
inst✝³ : CommMonoidWithZero R
inst✝² : UniqueFactorizationMonoid R
inst✝¹ : NormalizationMonoid R
inst✝ : DecidableEq R
a b : R
ha : Irreducible a
hb : b ≠ 0
this : multiplicity a b = count (normalize a) (normalizedFactors b)
⊢ multiplicity a b = count (normalize a) (normalizedFactors b)
Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity

Goal state before:
pp: case inl.succ
R : Type u_2
inst✝³ : CommMonoidWithZero R
inst✝² : UniqueFactorizationMonoid R
inst✝¹ : NormalizationMonoid R
inst✝ : DecidableEq R
x : R
hnorm : normalize 0 = 0
n✝ : ℕ
hle : 0 ∣ x
hlt : ¬0 ∣ x
⊢ count 0 (normalizedFactors x) = n✝ + 1

Tactic:
]

Goal state after:
case inl.succ
R : Type u_2
inst✝³ : CommMonoidWithZero R
inst✝² : UniqueFactorizationMonoid R
inst✝¹ : NormalizationMonoid R
inst✝ : DecidableEq R
x : R
hnorm : normalize 0 = 0
n✝ : ℕ
hle : 0 ∣ x
hlt : ¬0 ∣ x
⊢ count 0 (normalizedFactors x) = n✝ + 1
Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity

Goal state before:
pp: R : Type u_2
inst✝³ : CommMonoidWithZero R
inst✝² : UniqueFactorizationMonoid R
inst✝¹ : NormalizationMonoid R
inst✝ : DecidableEq R
x : R
hx : x ≠ 0
⊢ (normalizedFactors x).prod ~ᵤ x

Tactic:
]

Goal state after:
R : Type u_2
inst✝³ : CommMonoidWithZero R
inst✝² : UniqueFactorizationMonoid R
inst✝¹ : NormalizationMonoid R
inst✝ : DecidableEq R
x : R
hx : x ≠ 0
⊢ (normalizedFactors x).prod ~ᵤ x
Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity

Goal state before:
pp: case neg
R : Type u_2
inst✝¹ : CommMonoidWithZero R
inst✝ : UniqueFactorizationMonoid R
a b : R
ha : a ≠ 0
h : ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b
hb : ¬b = 0
this : StrongNormalizationMonoid R := UniqueFactorizationMonoid.strongNormalizationMonoid
⊢ ∀ (a_1 : R), Multiset.count a_1 (normalizedFactors a) ≤ Multiset.count a_1 (normalizedFactors b)

Tactic:
]

Goal state after:
case neg
R : Type u_2
inst✝¹ : CommMonoidWithZero R
inst✝ : UniqueFactorizationMonoid R
a b : R
ha : a ≠ 0
h : ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b
hb : ¬b = 0
this : StrongNormalizationMonoid R := UniqueFactorizationMonoid.strongNormalizationMonoid
⊢ ∀ (a_1 : R), Multiset.count a_1 (normalizedFactors a) ≤ Multiset.count a_1 (normalizedFactors b)
Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity

Goal state before:
pp: case pos
R : Type u_2
inst✝¹ : CommMonoidWithZero R
inst✝ : UniqueFactorizationMonoid R
a b : R
ha : a ≠ 0
h : ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b
hb : ¬b = 0
this : StrongNormalizationMonoid R := UniqueFactorizationMonoid.strongNormalizationMonoid
q : R
hq : q ∈ normalizedFactors a
hqprime : Prime q
h1 : emultiplicity q a = ↑(Multiset.count q (normalizedFactors a))
h2 : emultiplicity q b = ↑(Multiset.count q (normalizedFactors b))
⊢ Multiset.count q (normalizedFactors a) ≤ Multiset.count q (normalizedFactors b)

Tactic:
]

Goal state after:
case pos
R : Type u_2
inst✝¹ : CommMonoidWithZero R
inst✝ : UniqueFactorizationMonoid R
a b : R
ha : a ≠ 0
h : ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b
hb : ¬b = 0
this : StrongNormalizationMonoid R := UniqueFactorizationMonoid.strongNormalizationMonoid
q : R
hq : q ∈ normalizedFactors a
hqprime : Prime q
h1 : emultiplicity q a = ↑(Multiset.count q (normalizedFactors a))
h2 : emultiplicity q b = ↑(Multiset.count q (normalizedFactors b))
⊢ Multiset.count q (normalizedFactors a) ≤ Multiset.count q (normalizedFactors b)
Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity

Goal state before:
pp: case neg
R : Type u_2
inst✝¹ : CommMonoidWithZero R
inst✝ : UniqueFactorizationMonoid R
a b : R
n : ℕ
hn : n ≠ 0
ha : ¬a = 0
⊢ (∀ (p : R), Prime p → emultiplicity p (a ^ n) ≤ emultiplicity p (b ^ n)) →
    ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b

Tactic:
]

Goal state after:
case neg
R : Type u_2
inst✝¹ : CommMonoidWithZero R
inst✝ : UniqueFactorizationMonoid R
a b : R
n : ℕ
hn : n ≠ 0
ha : ¬a = 0
⊢ (∀ (p : R), Prime p → emultiplicity p (a ^ n) ≤ emultiplicity p (b ^ n)) →
    ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b
Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity

Goal state before:
pp: case neg
R : Type u_2
inst✝¹ : CommMonoidWithZero R
inst✝ : UniqueFactorizationMonoid R
a b : R
n : ℕ
hn : n ≠ 0
ha : ¬a = 0
H : ∀ (p : R), Prime p → emultiplicity p (a ^ n) ≤ emultiplicity p (b ^ n)
p : R
hp : Prime p
this : emultiplicity p a ≤ emultiplicity p b
⊢ emultiplicity p a ≤ emultiplicity p b

Tactic:
]

Goal state after:
case neg
R : Type u_2
inst✝¹ : CommMonoidWithZero R
inst✝ : UniqueFactorizationMonoid R
a b : R
n : ℕ
hn : n ≠ 0
ha : ¬a = 0
H : ∀ (p : R), Prime p → emultiplicity p (a ^ n) ≤ emultiplicity p (b ^ n)
p : R
hp : Prime p
this : emultiplicity p a ≤ emultiplicity p b
⊢ emultiplicity p a ≤ emultiplicity p b
Module: Mathlib.Algebra.Category.CoalgCat.ComonEquivalence

Goal state before:
pp: R : Type u
inst✝⁶ : CommRing R
M N : Type u
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup N
inst✝³ : Module R M
inst✝² : Module R N
inst✝¹ : Coalgebra R M
inst✝ : Coalgebra R N
⊢ ModuleCat.Hom.hom Δ = CoalgebraStruct.comul

Tactic:
]

Goal state after:
R : Type u
inst✝⁶ : CommRing R
M N : Type u
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup N
inst✝³ : Module R M
inst✝² : Module R N
inst✝¹ : Coalgebra R M
inst✝ : Coalgebra R N
⊢ ModuleCat.Hom.hom Δ = CoalgebraStruct.comul
Module: Mathlib.Algebra.Category.CoalgCat.ComonEquivalence

Goal state before:
pp: R : Type u
inst✝⁹ : CommRing R
M N P : Type u
inst✝⁸ : AddCommGroup M
inst✝⁷ : AddCommGroup N
inst✝⁶ : AddCommGroup P
inst✝⁵ : Module R M
inst✝⁴ : Module R N
inst✝³ : Module R P
inst✝² : Coalgebra R M
inst✝¹ : Coalgebra R N
inst✝ : Coalgebra R P
⊢ ModuleCat.Hom.hom Δ = CoalgebraStruct.comul

Tactic:
]

Goal state after:
R : Type u
inst✝⁹ : CommRing R
M N P : Type u
inst✝⁸ : AddCommGroup M
inst✝⁷ : AddCommGroup N
inst✝⁶ : AddCommGroup P
inst✝⁵ : Module R M
inst✝⁴ : Module R N
inst✝³ : Module R P
inst✝² : Coalgebra R M
inst✝¹ : Coalgebra R N
inst✝ : Coalgebra R P
⊢ ModuleCat.Hom.hom Δ = CoalgebraStruct.comul
Module: Mathlib.Algebra.Category.CoalgCat.ComonEquivalence

Goal state before:
pp: R : Type u
inst✝⁶ : CommRing R
M N : Type u
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup N
inst✝³ : Module R M
inst✝² : Module R N
inst✝¹ : Coalgebra R M
inst✝ : Coalgebra R N
⊢ ModuleCat.Hom.hom ε = CoalgebraStruct.counit

Tactic:
]

Goal state after:
R : Type u
inst✝⁶ : CommRing R
M N : Type u
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup N
inst✝³ : Module R M
inst✝² : Module R N
inst✝¹ : Coalgebra R M
inst✝ : Coalgebra R N
⊢ ModuleCat.Hom.hom ε = CoalgebraStruct.counit
Module: Mathlib.Algebra.Category.CoalgCat.ComonEquivalence

Goal state before:
pp: R : Type u
inst✝⁹ : CommRing R
M N P : Type u
inst✝⁸ : AddCommGroup M
inst✝⁷ : AddCommGroup N
inst✝⁶ : AddCommGroup P
inst✝⁵ : Module R M
inst✝⁴ : Module R N
inst✝³ : Module R P
inst✝² : Coalgebra R M
inst✝¹ : Coalgebra R N
inst✝ : Coalgebra R P
⊢ ModuleCat.Hom.hom ε = CoalgebraStruct.counit

Tactic:
]

Goal state after:
R : Type u
inst✝⁹ : CommRing R
M N P : Type u
inst✝⁸ : AddCommGroup M
inst✝⁷ : AddCommGroup N
inst✝⁶ : AddCommGroup P
inst✝⁵ : Module R M
inst✝⁴ : Module R N
inst✝³ : Module R P
inst✝² : Coalgebra R M
inst✝¹ : Coalgebra R N
inst✝ : Coalgebra R P
⊢ ModuleCat.Hom.hom ε = CoalgebraStruct.counit
Module: Mathlib.RingTheory.Localization.Away.Basic

Goal state before:
pp: R : Type u_1
inst✝³ : CommSemiring R
S : Type u_2
inst✝² : CommSemiring S
inst✝¹ : Algebra R S
x : R
inst✝ : Away x S
s : S
⊢ Associated (s * (algebraMap R S) x ^ (sec x s).2) s

Tactic:
]

Goal state after:
R : Type u_1
inst✝³ : CommSemiring R
S : Type u_2
inst✝² : CommSemiring S
inst✝¹ : Algebra R S
x : R
inst✝ : Away x S
s : S
⊢ Associated (s * (algebraMap R S) x ^ (sec x s).2) s
Module: Mathlib.RingTheory.Localization.Away.Basic

Goal state before:
pp: case h
R : Type u_1
inst✝³ : CommSemiring R
S : Type u_2
inst✝² : CommSemiring S
inst✝¹ : Algebra R S
r : R
inst✝ : Away r S
u : Rˣ
a b : R
hab : (algebraMap R S) a = (algebraMap R S) b
n : ℕ
hn : r ^ n * a = r ^ n * b
⊢ ↑u ^ n * (r ^ n * b) = ↑u ^ n * (r ^ n * b)

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Localization.Away.Basic

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
P : Type u_3
inst✝¹ : CommSemiring P
x : R
inst✝ : Away x S
g : R →+* P
hg : IsUnit (g x)
y : ↥(Submonoid.powers x)
n : ℕ
hn : (fun x_1 ↦ x ^ x_1) n = ↑y
⊢ IsUnit (g x ^ n)

Tactic:
]

Goal state after:
R : Type u_1
inst✝⁴ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
P : Type u_3
inst✝¹ : CommSemiring P
x : R
inst✝ : Away x S
g : R →+* P
hg : IsUnit (g x)
y : ↥(Submonoid.powers x)
n : ℕ
hn : (fun x_1 ↦ x ^ x_1) n = ↑y
⊢ IsUnit (g x ^ n)
Module: Mathlib.RingTheory.Localization.Away.Basic

Goal state before:
pp: R : Type u_1
inst✝⁷ : CommSemiring R
P : Type u_3
inst✝⁶ : CommSemiring P
Q : Type u_4
inst✝⁵ : CommSemiring Q
inst✝⁴ : Algebra P Q
S : Type u_5
inst✝³ : CommRing S
inst✝² : Algebra R S
f : R →+* P
r : R
inst✝¹ : Away r S
inst✝ : Away (f r) Q
⊢ (∀ (a : S), (map S Q f r) a = 0 → a = 0) ↔ ∀ (a : R), f a = 0 → ∃ n, r ^ n * a = 0

Tactic:
]

Goal state after:
R : Type u_1
inst✝⁷ : CommSemiring R
P : Type u_3
inst✝⁶ : CommSemiring P
Q : Type u_4
inst✝⁵ : CommSemiring Q
inst✝⁴ : Algebra P Q
S : Type u_5
inst✝³ : CommRing S
inst✝² : Algebra R S
f : R →+* P
r : R
inst✝¹ : Away r S
inst✝ : Away (f r) Q
⊢ (∀ (a : S), (map S Q f r) a = 0 → a = 0) ↔ ∀ (a : R), f a = 0 → ∃ n, r ^ n * a = 0
Module: Mathlib.RingTheory.Localization.Away.Basic

Goal state before:
pp: R : Type u_1
inst✝²⁰ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝¹⁹ : CommSemiring S
inst✝¹⁸ : Algebra R S
P : Type u_3
inst✝¹⁷ : CommSemiring P
x : R
inst✝¹⁶ : Away x S
g : R →+* P
Q : Type u_4
inst✝¹⁵ : CommSemiring Q
inst✝¹⁴ : Algebra P Q
A : Type u_5
inst✝¹³ : CommSemiring A
inst✝¹² : Algebra R A
B : Type u_6
inst✝¹¹ : CommSemiring B
inst✝¹⁰ : Algebra R B
Aₚ : Type u_7
inst✝⁹ : CommSemiring Aₚ
inst✝⁸ : Algebra A Aₚ
inst✝⁷ : Algebra R Aₚ
inst✝⁶ : IsScalarTower R A Aₚ
Bₚ : Type u_8
inst✝⁵ : CommSemiring Bₚ
inst✝⁴ : Algebra B Bₚ
inst✝³ : Algebra R Bₚ
inst✝² : IsScalarTower R B Bₚ
f : A →ₐ[R] B
a : A
inst✝¹ : Away a Aₚ
inst✝ : Away (f a) Bₚ
r : R
⊢ ↑↑(IsLocalization.map Bₚ ↑f ⋯) ((algebraMap A Aₚ) ((algebraMap R A) r)) = ((algebraMap B Bₚ).comp (algebraMap R B)) r

Tactic:
]

Goal state after:
R : Type u_1
inst✝²⁰ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝¹⁹ : CommSemiring S
inst✝¹⁸ : Algebra R S
P : Type u_3
inst✝¹⁷ : CommSemiring P
x : R
inst✝¹⁶ : Away x S
g : R →+* P
Q : Type u_4
inst✝¹⁵ : CommSemiring Q
inst✝¹⁴ : Algebra P Q
A : Type u_5
inst✝¹³ : CommSemiring A
inst✝¹² : Algebra R A
B : Type u_6
inst✝¹¹ : CommSemiring B
inst✝¹⁰ : Algebra R B
Aₚ : Type u_7
inst✝⁹ : CommSemiring Aₚ
inst✝⁸ : Algebra A Aₚ
inst✝⁷ : Algebra R Aₚ
inst✝⁶ : IsScalarTower R A Aₚ
Bₚ : Type u_8
inst✝⁵ : CommSemiring Bₚ
inst✝⁴ : Algebra B Bₚ
inst✝³ : Algebra R Bₚ
inst✝² : IsScalarTower R B Bₚ
f : A →ₐ[R] B
a : A
inst✝¹ : Away a Aₚ
inst✝ : Away (f a) Bₚ
r : R
⊢ ↑↑(IsLocalization.map Bₚ ↑f ⋯) ((algebraMap A Aₚ) ((algebraMap R A) r)) = ((algebraMap B Bₚ).comp (algebraMap R B)) r
Module: Mathlib.RingTheory.Localization.Away.Basic

Goal state before:
pp: case refine_3
R : Type u_1
inst✝⁸ : CommSemiring R
S : Type u_2
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
T : Type u_5
inst✝⁵ : CommSemiring T
inst✝⁴ : Algebra S T
inst✝³ : Algebra R T
inst✝² : IsScalarTower R S T
x y : R
inst✝¹ : Away x S
inst✝ : Away ((algebraMap R S) y) T
a b : R
h : (algebraMap S T) ((algebraMap R S) a) = (algebraMap R T) b
⊢ ∃ n, (y * x) ^ n * a = (y * x) ^ n * b

Tactic:
]

Goal state after:
case refine_3
R : Type u_1
inst✝⁸ : CommSemiring R
S : Type u_2
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
T : Type u_5
inst✝⁵ : CommSemiring T
inst✝⁴ : Algebra S T
inst✝³ : Algebra R T
inst✝² : IsScalarTower R S T
x y : R
inst✝¹ : Away x S
inst✝ : Away ((algebraMap R S) y) T
a b : R
h : (algebraMap S T) ((algebraMap R S) a) = (algebraMap R T) b
⊢ ∃ n, (y * x) ^ n * a = (y * x) ^ n * b
Module: Mathlib.RingTheory.Localization.Away.Basic

Goal state before:
pp: case refine_3
R : Type u_1
inst✝⁸ : CommSemiring R
S : Type u_2
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
T : Type u_5
inst✝⁵ : CommSemiring T
inst✝⁴ : Algebra S T
inst✝³ : Algebra R T
inst✝² : IsScalarTower R S T
x y : R
inst✝¹ : Away x S
inst✝ : Away ((algebraMap R S) y) T
a b : R
h : (algebraMap S T) ((algebraMap R S) a) = (algebraMap S T) ((algebraMap R S) b)
⊢ ∃ n, (y * x) ^ n * a = (y * x) ^ n * b

Tactic:
]

Goal state after:
case refine_3
R : Type u_1
inst✝⁸ : CommSemiring R
S : Type u_2
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
T : Type u_5
inst✝⁵ : CommSemiring T
inst✝⁴ : Algebra S T
inst✝³ : Algebra R T
inst✝² : IsScalarTower R S T
x y : R
inst✝¹ : Away x S
inst✝ : Away ((algebraMap R S) y) T
a b : R
h : (algebraMap S T) ((algebraMap R S) a) = (algebraMap S T) ((algebraMap R S) b)
⊢ ∃ n, (y * x) ^ n * a = (y * x) ^ n * b
Module: Mathlib.RingTheory.TwoSidedIdeal.Operations

Goal state before:
pp: R : Type u_1
inst✝ : NonUnitalNonAssocRing R
s : Set R
x : R
hx : x ∈ s
⊢ (span s).ringCon x 0

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : NonUnitalNonAssocRing R
s : Set R
x : R
hx : x ∈ s
⊢ (span s).ringCon x 0
Module: Mathlib.RingTheory.TwoSidedIdeal.Operations

Goal state before:
pp: R : Type u_1
inst✝ : NonUnitalNonAssocRing R
s : Set R
x : R
⊢ x ∈ { ringCon := sInf {s_1 | ∀ (x y : R), x - y ∈ s → s_1 x y} } → ∀ (I : TwoSidedIdeal R), s ⊆ ↑I → x ∈ I

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : NonUnitalNonAssocRing R
s : Set R
x : R
⊢ x ∈ { ringCon := sInf {s_1 | ∀ (x y : R), x - y ∈ s → s_1 x y} } → ∀ (I : TwoSidedIdeal R), s ⊆ ↑I → x ∈ I
Module: Mathlib.RingTheory.TwoSidedIdeal.Operations

Goal state before:
pp: R : Type u_1
inst✝ : NonUnitalNonAssocRing R
s : Set R
x : R
h : x ∈ { ringCon := sInf {s_1 | ∀ (x y : R), x - y ∈ s → s_1 x y} }
I : TwoSidedIdeal R
⊢ s ⊆ ↑I → x ∈ I

Tactic:
hI

Goal state after:
R : Type u_1
inst✝ : NonUnitalNonAssocRing R
s : Set R
x : R
h : x ∈ { ringCon := sInf {s_1 | ∀ (x y : R), x - y ∈ s → s_1 x y} }
I : TwoSidedIdeal R
hI : s ⊆ ↑I
⊢ x ∈ I
Module: Mathlib.RingTheory.TwoSidedIdeal.Operations

Goal state before:
pp: R : Type u_1
inst✝ : NonUnitalNonAssocRing R
s : Set R
x✝ : R
h : x✝ ∈ { ringCon := sInf {s_1 | ∀ (x y : R), x - y ∈ s → s_1 x y} }
I : TwoSidedIdeal R
x y : R
hI : I.ringCon x y
hxy : x - y ∈ s
⊢ I.ringCon x y

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : NonUnitalNonAssocRing R
s : Set R
x✝ : R
h : x✝ ∈ { ringCon := sInf {s_1 | ∀ (x y : R), x - y ∈ s → s_1 x y} }
I : TwoSidedIdeal R
x y : R
hI : I.ringCon x y
hxy : x - y ∈ s
⊢ I.ringCon x y
Module: Mathlib.RingTheory.TwoSidedIdeal.Operations

Goal state before:
pp: R : Type u_1
inst✝ : NonUnitalNonAssocRing R
s t : Set R
h : s ⊆ t
x : R
hx : ∀ (I : TwoSidedIdeal R), s ⊆ ↑I → x ∈ I
⊢ ∀ (I : TwoSidedIdeal R), t ⊆ ↑I → x ∈ I

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : NonUnitalNonAssocRing R
s t : Set R
h : s ⊆ t
x : R
hx : ∀ (I : TwoSidedIdeal R), s ⊆ ↑I → x ∈ I
⊢ ∀ (I : TwoSidedIdeal R), t ⊆ ↑I → x ∈ I
Module: Mathlib.RingTheory.TwoSidedIdeal.Operations

Goal state before:
pp: R : Type u_1
inst✝ : NonUnitalNonAssocRing R
s : Set R
I : TwoSidedIdeal R
⊢ (fun a b ↦ a - b ∈ s) ≤ ⇑I.ringCon ↔ s ⊆ ↑I

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : NonUnitalNonAssocRing R
s : Set R
I : TwoSidedIdeal R
⊢ (fun a b ↦ a - b ∈ s) ≤ ⇑I.ringCon ↔ s ⊆ ↑I
Module: Mathlib.RingTheory.TwoSidedIdeal.Operations

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝³ : NonUnitalNonAssocRing R
inst✝² : NonUnitalNonAssocRing S
F : Type u_3
inst✝¹ : FunLike F R S
f : F
inst✝ : NonUnitalRingHomClass F R S
I J : TwoSidedIdeal S
h : ↑I ⊆ ↑J
⊢ (fun I ↦ { ringCon := I.ringCon.comap f }) I ≤ (fun I ↦ { ringCon := I.ringCon.comap f }) J

Tactic:
]

Goal state after:
R : Type u_1
S : Type u_2
inst✝³ : NonUnitalNonAssocRing R
inst✝² : NonUnitalNonAssocRing S
F : Type u_3
inst✝¹ : FunLike F R S
f : F
inst✝ : NonUnitalRingHomClass F R S
I J : TwoSidedIdeal S
h : ↑I ⊆ ↑J
⊢ (fun I ↦ { ringCon := I.ringCon.comap f }) I ≤ (fun I ↦ { ringCon := I.ringCon.comap f }) J
Module: Mathlib.RingTheory.Localization.Away.Basic

Goal state before:
pp: case h
R : Type u_1
inst✝⁸ : CommSemiring R
S : Type u_2
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
T : Type u_5
inst✝⁵ : CommSemiring T
inst✝⁴ : Algebra S T
inst✝³ : Algebra R T
inst✝² : IsScalarTower R S T
x y : R
inst✝¹ : Away x S
inst✝ : Away ((algebraMap R S) y) T
a b : R
h : (algebraMap S T) ((algebraMap R S) a) = (algebraMap S T) ((algebraMap R S) b)
n : ℕ
hn : (algebraMap R S) (y ^ n * a) = (algebraMap R S) (y ^ n * b)
m : ℕ
hm : x ^ m * (y ^ n * a) = x ^ m * (y ^ n * b)
⊢ y ^ m * x ^ n * (x ^ m * (y ^ n * b)) = y ^ m * x ^ n * (x ^ m * (y ^ n * b))

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.TwoSidedIdeal.Operations

Goal state before:
pp: R : Type u_1
inst✝ : NonUnitalRing R
s : Set R
h_left : ∀ (x y : R), y ∈ s → x * y ∈ s
h_right : ∀ (y x : R), y ∈ s → y * x ∈ s
z : R
h_left' : ∀ {x y : R}, y ∈ closure s → x * y ∈ closure s
h_right' : ∀ {y x : R}, y ∈ closure s → y * x ∈ closure s
I : TwoSidedIdeal R := mk' ↑(closure s) ⋯ ⋯ ⋯ ⋯ ⋯
⊢ (∀ (I : TwoSidedIdeal R), s ⊆ ↑I → z ∈ I) ↔ z ∈ I

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : NonUnitalRing R
s : Set R
h_left : ∀ (x y : R), y ∈ s → x * y ∈ s
h_right : ∀ (y x : R), y ∈ s → y * x ∈ s
z : R
h_left' : ∀ {x y : R}, y ∈ closure s → x * y ∈ closure s
h_right' : ∀ {y x : R}, y ∈ closure s → y * x ∈ closure s
I : TwoSidedIdeal R := mk' ↑(closure s) ⋯ ⋯ ⋯ ⋯ ⋯
⊢ (∀ (I : TwoSidedIdeal R), s ⊆ ↑I → z ∈ I) ↔ z ∈ I
Module: Mathlib.RingTheory.Localization.Away.Basic

Goal state before:
pp: R : Type u_1
inst✝⁵ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁴ : CommSemiring S
inst✝³ : Algebra R S
P : Type u_3
inst✝² : CommSemiring P
inst✝¹ : IsLocalization M S
x : R
e : IsUnit x
inst✝ : Away x S
⊢ x ∈ ↑(IsUnit.submonoid R)

Tactic:
]

Goal state after:
R : Type u_1
inst✝⁵ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁴ : CommSemiring S
inst✝³ : Algebra R S
P : Type u_3
inst✝² : CommSemiring P
inst✝¹ : IsLocalization M S
x : R
e : IsUnit x
inst✝ : Away x S
⊢ x ∈ ↑(IsUnit.submonoid R)