mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.FractionalIdeal.Extended

Goal state before:
pp: A : Type u_1
inst✝⁷ : CommRing A
B : Type u_2
inst✝⁶ : CommRing B
f : A →+* B
K : Type u_3
M : Submonoid A
inst✝⁵ : CommRing K
inst✝⁴ : Algebra A K
inst✝³ : IsLocalization M K
L : Type u_4
N : Submonoid B
inst✝² : CommRing L
inst✝¹ : Algebra B L
inst✝ : IsLocalization N L
hf : M ≤ Submonoid.comap f N
J : Ideal A
hI : ↑J ≤ 1
s : Finset K
hs : ↑s ⊆ ↑↑J
c : ↥s → B
x✝ : ↥s
a : A
ha : a ∈ ↑⊤
hx : (Algebra.linearMap A K) a ∈ s
h : ⟨(Algebra.linearMap A K) a, hx⟩ ∈ Finset.univ
⊢ f a ∈ ↑⊤ ∧ (Algebra.linearMap B L) (f a) = (IsLocalization.map L f hf) ↑⟨(Algebra.linearMap A K) a, hx⟩

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FractionalIdeal.Extended

Goal state before:
pp: A : Type u_1
inst✝⁷ : CommRing A
B : Type u_2
inst✝⁶ : CommRing B
f : A →+* B
K : Type u_3
M : Submonoid A
inst✝⁵ : CommRing K
inst✝⁴ : Algebra A K
inst✝³ : IsLocalization M K
L : Type u_4
N : Submonoid B
inst✝² : CommRing L
inst✝¹ : Algebra B L
inst✝ : IsLocalization N L
hf : M ≤ Submonoid.comap f N
I : FractionalIdeal M K
hI : I ≤ 1
⊢ extended L hf I ≤ 1

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.ClassNumber

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
h : ↑|discr K| < (2 * (π / 4) ^ nrComplexPlaces K * (↑(finrank ℚ K) ^ finrank ℚ K / ↑(finrank ℚ K)!)) ^ 2
⊢ IsPrincipalIdealRing (𝓞 K)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FractionalIdeal.Extended

Goal state before:
pp: A : Type u_1
inst✝⁷ : CommRing A
B : Type u_2
inst✝⁶ : CommRing B
f : A →+* B
K : Type u_3
M : Submonoid A
inst✝⁵ : CommRing K
inst✝⁴ : Algebra A K
inst✝³ : IsLocalization M K
L : Type u_4
N : Submonoid B
inst✝² : CommRing L
inst✝¹ : Algebra B L
inst✝ : IsLocalization N L
hf : M ≤ Submonoid.comap f N
I : FractionalIdeal M K
hI : 1 ∈ I
⊢ (IsLocalization.map L f hf) 1 = 1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FractionalIdeal.Extended

Goal state before:
pp: A : Type u_1
inst✝⁷ : CommRing A
B : Type u_2
inst✝⁶ : CommRing B
f : A →+* B
K : Type u_3
M : Submonoid A
inst✝⁵ : CommRing K
inst✝⁴ : Algebra A K
inst✝³ : IsLocalization M K
L : Type u_4
N : Submonoid B
inst✝² : CommRing L
inst✝¹ : Algebra B L
inst✝ : IsLocalization N L
hf : M ≤ Submonoid.comap f N
I : FractionalIdeal M K
hI : 1 ≤ I
⊢ 1 ≤ extended L hf I

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Trace.Quotient

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : Free R S
inst✝¹ : Module.Finite R S
inst✝ : IsLocalRing R
x : S
⊢ (trace (R ⧸ p) (S ⧸ pS)) ((Ideal.Quotient.mk pS) x) = (Ideal.Quotient.mk p) ((trace R S) x)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FractionalIdeal.Extended

Goal state before:
pp: A : Type u_1
inst✝⁷ : CommRing A
B : Type u_2
inst✝⁶ : CommRing B
f : A →+* B
K : Type u_3
M : Submonoid A
inst✝⁵ : CommRing K
inst✝⁴ : Algebra A K
inst✝³ : IsLocalization M K
L : Type u_4
N : Submonoid B
inst✝² : CommRing L
inst✝¹ : Algebra B L
inst✝ : IsLocalization N L
hf : M ≤ Submonoid.comap f N
I J : FractionalIdeal M K
i : K
hi : i ∈ I
j : K
hj : j ∈ J
hy : i + j ∈ ↑(I + J)
⊢ (IsLocalization.map L f hf) i = (IsLocalization.map L f hf) i

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FractionalIdeal.Extended

Goal state before:
pp: A : Type u_1
inst✝⁷ : CommRing A
B : Type u_2
inst✝⁶ : CommRing B
f : A →+* B
K : Type u_3
M : Submonoid A
inst✝⁵ : CommRing K
inst✝⁴ : Algebra A K
inst✝³ : IsLocalization M K
L : Type u_4
N : Submonoid B
inst✝² : CommRing L
inst✝¹ : Algebra B L
inst✝ : IsLocalization N L
hf : M ≤ Submonoid.comap f N
I J : FractionalIdeal M K
i : K
hi : i ∈ I
j : K
hj : j ∈ J
hy : i + j ∈ ↑(I + J)
⊢ (IsLocalization.map L f hf) j = (IsLocalization.map L f hf) j

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Trace.Quotient

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝¹⁴ : CommRing R
inst✝¹³ : CommRing S
inst✝¹² : Algebra R S
p : Ideal R
inst✝¹¹ : p.IsMaximal
Rₚ : Type u_3
Sₚ : Type u_4
inst✝¹⁰ : CommRing Rₚ
inst✝⁹ : CommRing Sₚ
inst✝⁸ : Algebra R Rₚ
inst✝⁷ : IsLocalization.AtPrime Rₚ p
inst✝⁶ : IsLocalRing Rₚ
inst✝⁵ : Algebra S Sₚ
inst✝⁴ : Algebra R Sₚ
inst✝³ : Algebra Rₚ Sₚ
inst✝² : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝¹ : IsScalarTower R S Sₚ
inst✝ : IsScalarTower R Rₚ Sₚ
x : S
⊢ IsScalarTower R (Rₚ ⧸ maximalIdeal Rₚ) (Sₚ ⧸ pSₚ)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalRing.Quotient

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 : IsLocalRing R
inst✝ : Module.Finite R S
s : Set S
H :
  Submodule.restrictScalars R (span (R ⧸ p) (⇑(Ideal.Quotient.mk pS) '' s)) =
    map (↑(IsScalarTower.toAlgHom R S (S ⧸ pS))) (span R s)
hs : span (R ⧸ p) (⇑(Ideal.Quotient.mk pS) '' s) = ⊤
x : S
⊢ (↑(IsScalarTower.toAlgHom R S (S ⧸ pS))).ker = Submodule.restrictScalars R pS

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FractionalIdeal.Extended

Goal state before:
pp: A : Type u_1
inst✝⁷ : CommRing A
B : Type u_2
inst✝⁶ : CommRing B
f : A →+* B
K : Type u_3
M : Submonoid A
inst✝⁵ : CommRing K
inst✝⁴ : Algebra A K
inst✝³ : IsLocalization M K
L : Type u_4
N : Submonoid B
inst✝² : CommRing L
inst✝¹ : Algebra B L
inst✝ : IsLocalization N L
hf : M ≤ Submonoid.comap f N
I J : FractionalIdeal M K
y : K
hy : y ∈ ↑I ∪ ↑J
this : y ∈ I + J
⊢ y ∈ ↑(I + J) ∧ (IsLocalization.map L f hf) y = (IsLocalization.map L f hf) y

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FractionalIdeal.Extended

Goal state before:
pp: A : Type u_1
inst✝⁷ : CommRing A
B : Type u_2
inst✝⁶ : CommRing B
f : A →+* B
K : Type u_3
M : Submonoid A
inst✝⁵ : CommRing K
inst✝⁴ : Algebra A K
inst✝³ : IsLocalization M K
L : Type u_4
N : Submonoid B
inst✝² : CommRing L
inst✝¹ : Algebra B L
inst✝ : IsLocalization N L
hf : M ≤ Submonoid.comap f N
I J : FractionalIdeal M K
⊢ extended L hf (I + J) = extended L hf I + extended L hf J

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FractionalIdeal.Extended

Goal state before:
pp: A : Type u_1
inst✝⁷ : CommRing A
B : Type u_2
inst✝⁶ : CommRing B
f : A →+* B
K : Type u_3
M : Submonoid A
inst✝⁵ : CommRing K
inst✝⁴ : Algebra A K
inst✝³ : IsLocalization M K
L : Type u_4
N : Submonoid B
inst✝² : CommRing L
inst✝¹ : Algebra B L
inst✝ : IsLocalization N L
hf : M ≤ Submonoid.comap f N
I J : FractionalIdeal M K
x : K
hx : x ∈ span A (↑↑I * ↑↑J)
⊢ (IsLocalization.map L f hf) 0 ∈ ↑(span B (⇑(IsLocalization.map L f hf) '' ↑I * ⇑(IsLocalization.map L f hf) '' ↑J))

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Trace.Quotient

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝¹⁴ : CommRing R
inst✝¹³ : CommRing S
inst✝¹² : Algebra R S
p : Ideal R
inst✝¹¹ : p.IsMaximal
Rₚ : Type u_3
Sₚ : Type u_4
inst✝¹⁰ : CommRing Rₚ
inst✝⁹ : CommRing Sₚ
inst✝⁸ : Algebra R Rₚ
inst✝⁷ : IsLocalization.AtPrime Rₚ p
inst✝⁶ : IsLocalRing Rₚ
inst✝⁵ : Algebra S Sₚ
inst✝⁴ : Algebra R Sₚ
inst✝³ : Algebra Rₚ Sₚ
inst✝² : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝¹ : IsScalarTower R S Sₚ
inst✝ : IsScalarTower R Rₚ Sₚ
x : S
⊢ (Algebra.trace (R ⧸ p) (S ⧸ pS)) ((Ideal.Quotient.mk pS) x) =
    (equivQuotMaximalIdeal p Rₚ).symm ((Algebra.trace (Rₚ ⧸ maximalIdeal Rₚ) (Sₚ ⧸ pSₚ)) ((algebraMap S (Sₚ ⧸ pSₚ)) x))

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Trace.Quotient

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
p : Ideal R
inst✝⁵ : p.IsMaximal
x : S
inst✝⁴ : IsDedekindDomain R
inst✝³ : IsDomain S
inst✝² : IsTorsionFree R S
inst✝¹ : Module.Finite R S
inst✝ : IsIntegrallyClosed S
Rₚ : Type u_1 := Localization.AtPrime p
Sₚ : Type u_2 := Localization (algebraMapSubmonoid S p.primeCompl)
this : Algebra Rₚ Sₚ := localizationAlgebra p.primeCompl S
⊢ algebraMap R Sₚ = (algebraMap Rₚ Sₚ).comp (algebraMap R Rₚ)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FractionalIdeal.Extended

Goal state before:
pp: A : Type u_1
inst✝⁷ : CommRing A
B : Type u_2
inst✝⁶ : CommRing B
f : A →+* B
K : Type u_3
M : Submonoid A
inst✝⁵ : CommRing K
inst✝⁴ : Algebra A K
inst✝³ : IsLocalization M K
L : Type u_4
N : Submonoid B
inst✝² : CommRing L
inst✝¹ : Algebra B L
inst✝ : IsLocalization N L
hf : M ≤ Submonoid.comap f N
I J : FractionalIdeal M K
x : K
hx : x ∈ span A (↑↑I * ↑↑J)
i : K
hi : i ∈ ↑↑I
j : K
hj : j ∈ ↑↑J
hy : i * j ∈ ↑↑I * ↑↑J
⊢ (IsLocalization.map L f hf) i = (IsLocalization.map L f hf) i

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FractionalIdeal.Extended

Goal state before:
pp: A : Type u_1
inst✝⁷ : CommRing A
B : Type u_2
inst✝⁶ : CommRing B
f : A →+* B
K : Type u_3
M : Submonoid A
inst✝⁵ : CommRing K
inst✝⁴ : Algebra A K
inst✝³ : IsLocalization M K
L : Type u_4
N : Submonoid B
inst✝² : CommRing L
inst✝¹ : Algebra B L
inst✝ : IsLocalization N L
hf : M ≤ Submonoid.comap f N
I J : FractionalIdeal M K
x : K
hx : x ∈ span A (↑↑I * ↑↑J)
i : K
hi : i ∈ ↑↑I
j : K
hj : j ∈ ↑↑J
hy : i * j ∈ ↑↑I * ↑↑J
⊢ (IsLocalization.map L f hf) j = (IsLocalization.map L f hf) j

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FractionalIdeal.Extended

Goal state before:
pp: A : Type u_1
inst✝⁷ : CommRing A
B : Type u_2
inst✝⁶ : CommRing B
f : A →+* B
K : Type u_3
M : Submonoid A
inst✝⁵ : CommRing K
inst✝⁴ : Algebra A K
inst✝³ : IsLocalization M K
L : Type u_4
N : Submonoid B
inst✝² : CommRing L
inst✝¹ : Algebra B L
inst✝ : IsLocalization N L
hf : M ≤ Submonoid.comap f N
I J : FractionalIdeal M K
x : K
hx : x ∈ span A (↑↑I * ↑↑J)
i : K
hi : i ∈ ↑↑I
j : K
hj : j ∈ ↑↑J
hy : i * j ∈ ↑↑I * ↑↑J
⊢ (IsLocalization.map L f hf) i * (IsLocalization.map L f hf) j = (IsLocalization.map L f hf) (i * j)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Trace.Quotient

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
p : Ideal R
inst✝⁵ : p.IsMaximal
x : S
inst✝⁴ : IsDedekindDomain R
inst✝³ : IsDomain S
inst✝² : IsTorsionFree R S
inst✝¹ : Module.Finite R S
inst✝ : IsIntegrallyClosed S
Rₚ : Type u_1 := Localization.AtPrime p
Sₚ : Type u_2 := Localization (algebraMapSubmonoid S p.primeCompl)
this✝² : Algebra Rₚ Sₚ := localizationAlgebra p.primeCompl S
this✝¹ : IsScalarTower R Rₚ Sₚ
this✝ : IsLocalization (Submonoid.map (algebraMap R S) p.primeCompl) Sₚ
e : algebraMapSubmonoid S p.primeCompl ≤ S⁰
this : IsDomain Sₚ
⊢ IsTorsionFree Rₚ Sₚ

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalRing.Quotient

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 : IsLocalRing R
inst✝ : Module.Finite R S
s : Set S
⊢ span (R ⧸ p) (⇑(Ideal.Quotient.mk pS) '' s) = ⊤ ↔ span R s = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Trace.Quotient

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
p : Ideal R
inst✝⁵ : p.IsMaximal
x : S
inst✝⁴ : IsDedekindDomain R
inst✝³ : IsDomain S
inst✝² : IsTorsionFree R S
inst✝¹ : Module.Finite R S
inst✝ : IsIntegrallyClosed S
Rₚ : Type u_1 := Localization.AtPrime p
Sₚ : Type u_2 := Localization (algebraMapSubmonoid S p.primeCompl)
this✝⁵ : Algebra Rₚ Sₚ := localizationAlgebra p.primeCompl S
this✝⁴ : IsScalarTower R Rₚ Sₚ
this✝³ : IsLocalization (Submonoid.map (algebraMap R S) p.primeCompl) Sₚ
e : algebraMapSubmonoid S p.primeCompl ≤ S⁰
this✝² : IsDomain Sₚ
this✝¹ : IsTorsionFree Rₚ Sₚ
this✝ : Module.Finite Rₚ Sₚ
this : IsIntegrallyClosed Sₚ
⊢ IsPrincipalIdealRing Rₚ

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FractionalIdeal.Extended

Goal state before:
pp: case refine_2
A : Type u_1
inst✝⁷ : CommRing A
B : Type u_2
inst✝⁶ : CommRing B
f : A →+* B
K : Type u_3
M : Submonoid A
inst✝⁵ : CommRing K
inst✝⁴ : Algebra A K
inst✝³ : IsLocalization M K
L : Type u_4
N : Submonoid B
inst✝² : CommRing L
inst✝¹ : Algebra B L
inst✝ : IsLocalization N L
hf : M ≤ Submonoid.comap f N
I J : FractionalIdeal M K
x✝ : L
h : x✝ ∈ ⇑(IsLocalization.map L f hf) '' ↑I * ⇑(IsLocalization.map L f hf) '' ↑J
⊢ x✝ ∈ ↑(span B (⇑(IsLocalization.map L f hf) '' ↑(I * J)))

Tactic:
rcases Set.mem_mul.mp h with ⟨y, ⟨i, hi, rfl⟩, z, ⟨j, hj, rfl⟩, rfl⟩

Goal state after:
case refine_2
A : Type u_1
inst✝⁷ : CommRing A
B : Type u_2
inst✝⁶ : CommRing B
f : A →+* B
K : Type u_3
M : Submonoid A
inst✝⁵ : CommRing K
inst✝⁴ : Algebra A K
inst✝³ : IsLocalization M K
L : Type u_4
N : Submonoid B
inst✝² : CommRing L
inst✝¹ : Algebra B L
inst✝ : IsLocalization N L
hf : M ≤ Submonoid.comap f N
I J : FractionalIdeal M K
i : K
hi : i ∈ ↑I
j : K
hj : j ∈ ↑J
h :
  (IsLocalization.map L f hf) i * (IsLocalization.map L f hf) j ∈
    ⇑(IsLocalization.map L f hf) '' ↑I * ⇑(IsLocalization.map L f hf) '' ↑J
⊢ (IsLocalization.map L f hf) i * (IsLocalization.map L f hf) j ∈ ↑(span B (⇑(IsLocalization.map L f hf) '' ↑(I * J)))
Module: Mathlib.RingTheory.FractionalIdeal.Extended

Goal state before:
pp: A : Type u_1
inst✝⁷ : CommRing A
B : Type u_2
inst✝⁶ : CommRing B
f : A →+* B
K : Type u_3
M : Submonoid A
inst✝⁵ : CommRing K
inst✝⁴ : Algebra A K
inst✝³ : IsLocalization M K
L : Type u_4
N : Submonoid B
inst✝² : CommRing L
inst✝¹ : Algebra B L
inst✝ : IsLocalization N L
hf : M ≤ Submonoid.comap f N
I J : FractionalIdeal M K
i : K
hi : i ∈ ↑I
j : K
hj : j ∈ ↑J
h :
  (IsLocalization.map L f hf) i * (IsLocalization.map L f hf) j ∈
    ⇑(IsLocalization.map L f hf) '' ↑I * ⇑(IsLocalization.map L f hf) '' ↑J
⊢ (IsLocalization.map L f hf) (i * j) = (IsLocalization.map L f hf) i * (IsLocalization.map L f hf) j

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FractionalIdeal.Extended

Goal state before:
pp: A : Type u_1
inst✝⁷ : CommRing A
B : Type u_2
inst✝⁶ : CommRing B
f : A →+* B
K : Type u_3
M : Submonoid A
inst✝⁵ : CommRing K
inst✝⁴ : Algebra A K
inst✝³ : IsLocalization M K
L : Type u_4
N : Submonoid B
inst✝² : CommRing L
inst✝¹ : Algebra B L
inst✝ : IsLocalization N L
hf : M ≤ Submonoid.comap f N
I J : FractionalIdeal M K
⊢ extended L hf (I * J) = extended L hf I * extended L hf J

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FractionalIdeal.Extended

Goal state before:
pp: A : Type u_1
inst✝¹¹ : CommRing A
B : Type u_2
inst✝¹⁰ : CommRing B
f : A →+* B
K : Type u_3
M : Submonoid A
inst✝⁹ : CommRing K
inst✝⁸ : Algebra A K
inst✝⁷ : IsLocalization M K
L : Type u_4
N : Submonoid B
inst✝⁶ : CommRing L
inst✝⁵ : Algebra B L
inst✝⁴ : IsLocalization N L
hf : M ≤ Submonoid.comap f N
I : FractionalIdeal M K
C : Type u_5
W : Type u_6
inst✝³ : CommRing C
inst✝² : CommRing W
inst✝¹ : Algebra C W
P : Submonoid C
inst✝ : IsLocalization P W
g : B →+* C
hg : N ≤ Submonoid.comap g P
x : L
hx : x ∈ ↑(extended L hf I)
hx' : x ∈ span B (⇑(IsLocalization.map L f hf) '' ↑I)
⊢ (IsLocalization.map W g hg) 0 ∈ ↑(span C (⇑(IsLocalization.map W (g.comp f) ⋯) '' ↑I))

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Trace.Quotient

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
p : Ideal R
inst✝⁵ : p.IsMaximal
x : S
inst✝⁴ : IsDedekindDomain R
inst✝³ : IsDomain S
inst✝² : IsTorsionFree R S
inst✝¹ : Module.Finite R S
inst✝ : IsIntegrallyClosed S
⊢ (trace (R ⧸ p) (S ⧸ pS)) ((Ideal.Quotient.mk pS) x) = (Ideal.Quotient.mk p) ((intTrace R S) x)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FractionalIdeal.Extended

Goal state before:
pp: A : Type u_1
inst✝¹¹ : CommRing A
B : Type u_2
inst✝¹⁰ : CommRing B
f : A →+* B
K : Type u_3
M : Submonoid A
inst✝⁹ : CommRing K
inst✝⁸ : Algebra A K
inst✝⁷ : IsLocalization M K
L : Type u_4
N : Submonoid B
inst✝⁶ : CommRing L
inst✝⁵ : Algebra B L
inst✝⁴ : IsLocalization N L
hf : M ≤ Submonoid.comap f N
I : FractionalIdeal M K
C : Type u_5
W : Type u_6
inst✝³ : CommRing C
inst✝² : CommRing W
inst✝¹ : Algebra C W
P : Submonoid C
inst✝ : IsLocalization P W
g : B →+* C
hg : N ≤ Submonoid.comap g P
x : L
hx : x ∈ ↑(extended L hf I)
hx' : x ∈ span B (⇑(IsLocalization.map L f hf) '' ↑I)
z : K
hz : z ∈ ↑I
⊢ (IsLocalization.map W (g.comp f) ⋯) z = (IsLocalization.map W g hg) ((IsLocalization.map L f hf) z)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalRing.Quotient

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : IsLocalRing R
inst✝¹ : Module.Finite R S
inst✝ : Free R S
⊢ finrank (R ⧸ p) (S ⧸ pS) = finrank R S

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalRing.Quotient

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
inst✝³ : IsLocalRing R
inst✝² : Module.Finite R S
inst✝¹ : Free R S
ι : Type u_3
inst✝ : Fintype ι
b : Basis ι R S
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalRing.Quotient

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
inst✝³ : IsLocalRing R
inst✝² : Module.Finite R S
inst✝¹ : Free R S
ι : Type u_3
inst✝ : Fintype ι
b : Basis ι R S
i : ι
⊢ (basisQuotient b) i = (Ideal.Quotient.mk pS) (b i)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalRing.Quotient

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
inst✝³ : IsLocalRing R
inst✝² : Module.Finite R S
inst✝¹ : Free R S
ι : Type u_4
inst✝ : Fintype ι
b : Basis ι R S
x : S
i : ι
⊢ ((basisQuotient b).repr ((Ideal.Quotient.mk pS) x)) i = (Ideal.Quotient.mk p) ((b.repr x) i)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalRing.Quotient

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsLocalRing R
I : Ideal R
inst✝ : IsArtinianRing (R ⧸ I)
hI : ¬I = ⊤
this✝¹ : Nontrivial (R ⧸ I)
this✝ : IsLocalRing (R ⧸ I)
this : IsLocalHom (Ideal.Quotient.mk I)
n : ℕ
hn : ⊥.jacobson ^ n = 0
⊢ Ideal.map (Ideal.Quotient.mk I) p = maximalIdeal (R ⧸ I)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalRing.Quotient

Goal state before:
pp: case neg
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsLocalRing R
I : Ideal R
inst✝ : IsArtinianRing (R ⧸ I)
hI : ¬I = ⊤
this✝² : Nontrivial (R ⧸ I)
this✝¹ : IsLocalRing (R ⧸ I)
this✝ : IsLocalHom (Ideal.Quotient.mk I)
n : ℕ
hn : ⊥.jacobson ^ n = 0
this : Ideal.map (Ideal.Quotient.mk I) p = maximalIdeal (R ⧸ I)
⊢ ∃ n, p ^ n ≤ I

Tactic:
jacobson_eq_maximalIdeal _ bot_ne_top,

Goal state after:
case neg
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsLocalRing R
I : Ideal R
inst✝ : IsArtinianRing (R ⧸ I)
hI : ¬I = ⊤
this✝² : Nontrivial (R ⧸ I)
this✝¹ : IsLocalRing (R ⧸ I)
this✝ : IsLocalHom (Ideal.Quotient.mk I)
n : ℕ
hn : maximalIdeal (R ⧸ I) ^ n = 0
this : Ideal.map (Ideal.Quotient.mk I) p = maximalIdeal (R ⧸ I)
⊢ ∃ n, p ^ n ≤ I
Module: Mathlib.RingTheory.LocalRing.Quotient

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsLocalRing R
I : Ideal R
inst✝ : IsArtinianRing (R ⧸ I)
⊢ ∃ n, p ^ n ≤ I

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalRing.Quotient

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
inst✝² : IsLocalRing R
inst✝¹ : IsNoetherianRing R
inst✝ : Finite (ResidueField R)
I : Ideal R
⊢ Finite (R ⧸ I) ↔ ∃ n, p ^ n ≤ I

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FractionalIdeal.Extended

Goal state before:
pp: A : Type u_1
inst✝¹¹ : CommRing A
B : Type u_2
inst✝¹⁰ : CommRing B
f : A →+* B
K : Type u_3
M : Submonoid A
inst✝⁹ : CommRing K
inst✝⁸ : Algebra A K
inst✝⁷ : IsLocalization M K
L : Type u_4
N : Submonoid B
inst✝⁶ : CommRing L
inst✝⁵ : Algebra B L
inst✝⁴ : IsLocalization N L
hf : M ≤ Submonoid.comap f N
I : FractionalIdeal M K
C : Type u_5
W : Type u_6
inst✝³ : CommRing C
inst✝² : CommRing W
inst✝¹ : Algebra C W
P : Submonoid C
inst✝ : IsLocalization P W
g : B →+* C
hg : N ≤ Submonoid.comap g P
⊢ extended W hg (extended L hf I) = extended W ⋯ I

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FractionalIdeal.Extended

Goal state before:
pp: A : Type u_1
inst✝⁷ : CommRing A
B : Type u_2
inst✝⁶ : CommRing B
f : A →+* B
K : Type u_3
M : Submonoid A
inst✝⁵ : CommRing K
inst✝⁴ : Algebra A K
inst✝³ : IsLocalization M K
L : Type u_4
N : Submonoid B
inst✝² : CommRing L
inst✝¹ : Algebra B L
inst✝ : IsLocalization N L
hf : M ≤ Submonoid.comap f N
I₀ : Ideal A
a : A
ha : a ∈ ↑I₀
⊢ (algebraMap B L) (f a) = (IsLocalization.map L f hf) ((Algebra.linearMap A K) a)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FractionalIdeal.Extended

Goal state before:
pp: A : Type u_1
inst✝⁷ : CommRing A
B : Type u_2
inst✝⁶ : CommRing B
f : A →+* B
K : Type u_3
M : Submonoid A
inst✝⁵ : CommRing K
inst✝⁴ : Algebra A K
inst✝³ : IsLocalization M K
L : Type u_4
N : Submonoid B
inst✝² : CommRing L
inst✝¹ : Algebra B L
inst✝ : IsLocalization N L
hf : M ≤ Submonoid.comap f N
I₀ : Ideal A
⊢ extended L hf ↑I₀ = ↑(Ideal.map f I₀)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Discriminant.Basic

Goal state before:
pp: A : Type u_2
inst✝¹ : Field A
inst✝ : CharZero A
N : ℕ
D : ℕ := rankOfDiscrBdd N
B : ℝ≥0 := boundOfDiscBdd N
C : ℕ := ⌈max (sqrt (1 + B ^ 2)) 1 ^ D * ↑(D.choose (D / 2))⌉₊
x✝¹ : { F // FiniteDimensional ℚ ↥F }
K : IntermediateField ℚ A
hK₀ : FiniteDimensional ℚ ↥K
x✝ : ⟨K, hK₀⟩ ∈ {K | {w | w.IsComplex}.Nonempty ∧ |discr ↥↑K| ≤ ↑N}
hK₂ : |discr ↥↑⟨K, hK₀⟩| ≤ ↑N
this✝¹ : CharZero ↥K
this✝ : NumberField ↥K
w₀ : InfinitePlace ↥↑⟨K, hK₀⟩
hw₀ : w₀ ∈ {w | w.IsComplex}
this : minkowskiBound (↥K) 1 < ↑(convexBodyLT'Factor ↥K) * ↑(boundOfDiscBdd N)
⊢ ∃ x,
    (∃ i,
        ∃ (_ : i.natDegree ≤ D ∧ ∀ (i_1 : ℕ), i.coeff i_1 ∈ Set.Icc (-↑C) ↑C),
          x ∈ ↑(Polynomial.map (algebraMap ℤ A) i).roots.toFinset) ∧
      K = ℚ⟮x⟯

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FractionalIdeal.Extended

Goal state before:
pp: A : Type u_1
inst✝⁷ : CommRing A
B : Type u_2
inst✝⁶ : CommRing B
f : A →+* B
K : Type u_3
M : Submonoid A
inst✝⁵ : CommRing K
inst✝⁴ : Algebra A K
inst✝³ : IsLocalization M K
L : Type u_4
N : Submonoid B
inst✝² : CommRing L
inst✝¹ : Algebra B L
inst✝ : IsLocalization N L
hf : M ≤ Submonoid.comap f N
x : K
⊢ extended L hf (spanSingleton M x) = spanSingleton N ((IsLocalization.map L f hf) x)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FractionalIdeal.Extended

Goal state before:
pp: A : Type u_1
inst✝¹¹ : CommRing A
B : Type u_2
inst✝¹⁰ : CommRing B
f : A →+* B
K : Type u_3
M : Submonoid A
inst✝⁹ : CommRing K
inst✝⁸ : Algebra A K
inst✝⁷ : IsLocalization M K
L : Type u_4
N : Submonoid B
inst✝⁶ : CommRing L
inst✝⁵ : Algebra B L
inst✝⁴ : IsLocalization N L
hf : M ≤ Submonoid.comap f N
C : Type u_5
W : Type u_6
inst✝³ : CommRing C
inst✝² : CommRing W
inst✝¹ : Algebra C W
P : Submonoid C
inst✝ : IsLocalization P W
g : B →+* C
hg : N ≤ Submonoid.comap g P
⊢ (extendedHom' W hg).comp (extendedHom' L hf) = extendedHom' W ⋯

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Discriminant.Basic

Goal state before:
pp: A : Type u_2
inst✝¹ : Field A
inst✝ : CharZero A
N : ℕ
D : ℕ := rankOfDiscrBdd N
B : ℝ≥0 := boundOfDiscBdd N
C : ℕ := ⌈max (sqrt (1 + B ^ 2)) 1 ^ D * ↑(D.choose (D / 2))⌉₊
x✝¹ : { F // FiniteDimensional ℚ ↥F }
K : IntermediateField ℚ A
hK₀ : FiniteDimensional ℚ ↥K
x✝ : ⟨K, hK₀⟩ ∈ {K | {w | w.IsComplex}.Nonempty ∧ |discr ↥↑K| ≤ ↑N}
hK₂ : |discr ↥↑⟨K, hK₀⟩| ≤ ↑N
this✝ : CharZero ↥K
this : NumberField ↥K
w₀ : InfinitePlace ↥↑⟨K, hK₀⟩
hw₀ : w₀ ∈ {w | w.IsComplex}
⊢ ↑B = 1 * ↑B

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FractionalIdeal.Extended

Goal state before:
pp: A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝¹⁶ : CommRing A
inst✝¹⁵ : IsDomain A
inst✝¹⁴ : CommRing B
inst✝¹³ : IsDomain B
inst✝¹² : Algebra A B
inst✝¹¹ : IsTorsionFree A B
inst✝¹⁰ : Field K
inst✝⁹ : Field L
inst✝⁸ : Algebra A K
inst✝⁷ : Algebra B L
inst✝⁶ : IsFractionRing A K
inst✝⁵ : IsFractionRing B L
inst✝⁴ : Algebra K L
inst✝³ : Algebra A L
inst✝² : IsScalarTower A B L
inst✝¹ : IsScalarTower A K L
inst✝ : Algebra.IsIntegral A B
I : FractionalIdeal A⁰ K
⊢ ↑((extendedHom L B) I) = span B (⇑(algebraMap K L) '' ↑I)

Tactic:
by

Goal state after: