mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic

Goal state before:
pp: K : Type u_1
inst✝ : Field K
x : K
w : InfinitePlace K
⊢ (normAtPlace w) ((mixedEmbedding K) x) = w x

Tactic:
normAtPlace_apply

Goal state after:
K : Type u_1
inst✝ : Field K
x : K
w : InfinitePlace K
⊢ w x = w x
Module: Mathlib.RingTheory.Invariant.Galois

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝²¹ : CommRing A
inst✝²⁰ : CommRing B
inst✝¹⁹ : Algebra A B
G : Type u_3
inst✝¹⁸ : Finite G
inst✝¹⁷ : Group G
inst✝¹⁶ : MulSemiringAction G B
inst✝¹⁵ : Algebra.IsInvariant A B G
P : Ideal A
Q : Ideal B
inst✝¹⁴ : Q.LiesOver P
inst✝¹³ : P.IsPrime
inst✝¹² : Q.IsPrime
K : Type u_4
L : Type u_5
inst✝¹¹ : Field K
inst✝¹⁰ : Field L
inst✝⁹ : Algebra K L
inst✝⁸ : Algebra (A ⧸ P) K
inst✝⁷ : IsFractionRing (A ⧸ P) K
inst✝⁶ : Algebra (B ⧸ Q) L
inst✝⁵ : IsFractionRing (B ⧸ Q) L
inst✝⁴ : Algebra (A ⧸ P) L
inst✝³ : IsScalarTower (A ⧸ P) (B ⧸ Q) L
inst✝² : IsScalarTower (A ⧸ P) K L
inst✝¹ : SMulCommClass G A B
inst✝ : Algebra.IsSeparable K L
⊢ Module.Finite K L

Tactic:
by
  have : IsGalois K L := { __ := normal G P Q K L }
  have := Finite.of_surjective _ (IsFractionRing.stabilizerHom_surjective G P Q K L)
  apply IsGalois.finiteDimensional_of_finite

Goal state after:
Module: Mathlib.RingTheory.Ideal.Quotient.HasFiniteQuotients

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : HasFiniteQuotients R
B : ℕ
h✝ : Infinite R
s : Finset R
hs : s.card = B + 1
t : Finset R := (s - s) \ {0}
I : Submodule R R
hI : Nat.card (R ⧸ I) ≤ B ∧ I ∉ {⊥}
⊢ (Finset.image (⇑(Ideal.Quotient.mk I)) s).card < s.card

Tactic:
have := finiteQuotient hI.2

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : HasFiniteQuotients R
B : ℕ
h✝ : Infinite R
s : Finset R
hs : s.card = B + 1
t : Finset R := (s - s) \ {0}
I : Submodule R R
hI : Nat.card (R ⧸ I) ≤ B ∧ I ∉ {⊥}
this : Finite (R ⧸ I)
⊢ (Finset.image (⇑(Ideal.Quotient.mk I)) s).card < s.card
Module: Mathlib.NumberTheory.NumberField.ClassNumber

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
h :
  ∀ p ∈ Finset.Icc 1 ⌊(4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|)⌋₊,
    Nat.Prime p →
      ∀ P ∈ (span {↑p}).primesOver (𝓞 K),
        p ^ P.inertiaDeg ℤ ≤
            ⌊(4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|)⌋₊ →
          Submodule.IsPrincipal P
x✝ : ↥(Ideal (𝓞 K))⁰
P : Ideal (𝓞 K)
HP : P ∈ (Ideal (𝓞 K))⁰
hP : (↑⟨P, HP⟩).IsPrime
hPN :
  ↑(absNorm ↑⟨P, HP⟩) ≤ (4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|)
p : ℤ
hp : under ℤ P = ℤ ∙ p
hp0 : p ≠ 0
hpprime : (under ℤ P).IsPrime → _root_.Prime p
⊢ P.LiesOver (span {p})

Tactic:
rcases abs_choice p with h | h

Goal state after:
case inl
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
h✝ :
  ∀ p ∈ Finset.Icc 1 ⌊(4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|)⌋₊,
    Nat.Prime p →
      ∀ P ∈ (span {↑p}).primesOver (𝓞 K),
        p ^ P.inertiaDeg ℤ ≤
            ⌊(4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|)⌋₊ →
          Submodule.IsPrincipal P
x✝ : ↥(Ideal (𝓞 K))⁰
P : Ideal (𝓞 K)
HP : P ∈ (Ideal (𝓞 K))⁰
hP : (↑⟨P, HP⟩).IsPrime
hPN :
  ↑(absNorm ↑⟨P, HP⟩) ≤ (4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|)
p : ℤ
hp : under ℤ P = ℤ ∙ p
hp0 : p ≠ 0
hpprime : (under ℤ P).IsPrime → _root_.Prime p
h : |p| = p
⊢ P.LiesOver (span {p})

case inr
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
h✝ :
  ∀ p ∈ Finset.Icc 1 ⌊(4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|)⌋₊,
    Nat.Prime p →
      ∀ P ∈ (span {↑p}).primesOver (𝓞 K),
        p ^ P.inertiaDeg ℤ ≤
            ⌊(4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|)⌋₊ →
          Submodule.IsPrincipal P
x✝ : ↥(Ideal (𝓞 K))⁰
P : Ideal (𝓞 K)
HP : P ∈ (Ideal (𝓞 K))⁰
hP : (↑⟨P, HP⟩).IsPrime
hPN :
  ↑(absNorm ↑⟨P, HP⟩) ≤ (4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|)
p : ℤ
hp : under ℤ P = ℤ ∙ p
hp0 : p ≠ 0
hpprime : (under ℤ P).IsPrime → _root_.Prime p
h : |p| = -p
⊢ P.LiesOver (span {p})
Module: Mathlib.NumberTheory.NumberField.ClassNumber

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
h :
  ∀ p ∈ Finset.Icc 1 ⌊(4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|)⌋₊,
    Nat.Prime p →
      ∀ P ∈ (span {↑p}).primesOver (𝓞 K),
        p ^ P.inertiaDeg ℤ ≤
            ⌊(4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|)⌋₊ →
          Submodule.IsPrincipal P
x✝ : ↥(Ideal (𝓞 K))⁰
P : Ideal (𝓞 K)
HP : P ∈ (Ideal (𝓞 K))⁰
hP : (↑⟨P, HP⟩).IsPrime
hPN :
  ↑(absNorm ↑⟨P, HP⟩) ≤ (4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|)
p : ℤ
hp : under ℤ P = ℤ ∙ p
hp0 : p ≠ 0
hpprime : (under ℤ P).IsPrime → _root_.Prime p
hlies : P.LiesOver (span {p})
⊢ span {↑p.natAbs} = span {p}

Tactic:
rcases abs_choice p with h | h

Goal state after:
case inl
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
h✝ :
  ∀ p ∈ Finset.Icc 1 ⌊(4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|)⌋₊,
    Nat.Prime p →
      ∀ P ∈ (span {↑p}).primesOver (𝓞 K),
        p ^ P.inertiaDeg ℤ ≤
            ⌊(4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|)⌋₊ →
          Submodule.IsPrincipal P
x✝ : ↥(Ideal (𝓞 K))⁰
P : Ideal (𝓞 K)
HP : P ∈ (Ideal (𝓞 K))⁰
hP : (↑⟨P, HP⟩).IsPrime
hPN :
  ↑(absNorm ↑⟨P, HP⟩) ≤ (4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|)
p : ℤ
hp : under ℤ P = ℤ ∙ p
hp0 : p ≠ 0
hpprime : (under ℤ P).IsPrime → _root_.Prime p
hlies : P.LiesOver (span {p})
h : |p| = p
⊢ span {↑p.natAbs} = span {p}

case inr
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
h✝ :
  ∀ p ∈ Finset.Icc 1 ⌊(4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|)⌋₊,
    Nat.Prime p →
      ∀ P ∈ (span {↑p}).primesOver (𝓞 K),
        p ^ P.inertiaDeg ℤ ≤
            ⌊(4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|)⌋₊ →
          Submodule.IsPrincipal P
x✝ : ↥(Ideal (𝓞 K))⁰
P : Ideal (𝓞 K)
HP : P ∈ (Ideal (𝓞 K))⁰
hP : (↑⟨P, HP⟩).IsPrime
hPN :
  ↑(absNorm ↑⟨P, HP⟩) ≤ (4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|)
p : ℤ
hp : under ℤ P = ℤ ∙ p
hp0 : p ≠ 0
hpprime : (under ℤ P).IsPrime → _root_.Prime p
hlies : P.LiesOver (span {p})
h : |p| = -p
⊢ span {↑p.natAbs} = span {p}
Module: Mathlib.RingTheory.LocalRing.Quotient

Goal state before:
pp: case a
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
this : Module.Finite (R ⧸ p) (S ⧸ pS)
⊢ finrank (R ⧸ p) (S ⧸ pS) ≤ finrank R S

pp: case a
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
this : Module.Finite (R ⧸ p) (S ⧸ pS)
⊢ finrank R S ≤ finrank (R ⧸ p) (S ⧸ pS)

Tactic:
· let b := Module.Free.chooseBasis R S
    conv_rhs => rw [finrank_eq_card_chooseBasisIndex]
    apply finrank_le_of_span_eq_top
    rw [Set.range_comp]
    apply (quotient_span_eq_top_iff_span_eq_top _).mpr b.span_eq

Goal state after:
case a
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
this : Module.Finite (R ⧸ p) (S ⧸ pS)
⊢ finrank R S ≤ finrank (R ⧸ p) (S ⧸ pS)
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
⊢ span B (⇑(IsLocalization.map L (algebraMap A B) ⋯) '' ↑I) = span B (⇑(algebraMap K L) '' ↑I)

Tactic:
IsLocalization.algebraMap_eq_map_map_submonoid A⁰ B K L

Goal state after:
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
⊢ span B (⇑(IsLocalization.map L (algebraMap A B) ⋯) '' ↑I) = span B (⇑(IsLocalization.map L (algebraMap A B) ⋯) '' ↑I)
Module: Mathlib.RingTheory.DedekindDomain.PID

Goal state before:
pp: R : Type u_2
A : Type u_3
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
S : Submonoid R
inst✝ : IsLocalization S A
I : (FractionalIdeal S A)ˣ
v : A
hv : v ∈ ↑I⁻¹
hinv : ↑↑I * ↑↑I⁻¹ = 1
J : Submodule R R := Submodule.comap (Algebra.linearMap R A) (↑↑I * (R ∙ v))
h : J = ⊤
⊢ ↑↑I * (R ∙ v) ≤ ↑↑I * ↑↑I⁻¹

Tactic:
(Submodule.span_singleton_le_iff_mem _ _).2 hv

Goal state after:
R : Type u_2
A : Type u_3
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
S : Submonoid R
inst✝ : IsLocalization S A
I : (FractionalIdeal S A)ˣ
v : A
hv : v ∈ ↑I⁻¹
hinv : ↑↑I * ↑↑I⁻¹ = 1
J : Submodule R R := Submodule.comap (Algebra.linearMap R A) (↑↑I * (R ∙ v))
h : J = ⊤
⊢ ↑↑I * ↑↑I⁻¹ ≤ ↑↑I * ↑↑I⁻¹
Module: Mathlib.NumberTheory.NumberField.Discriminant.Different

Goal state before:
pp: K : Type u_1
𝒪 : Type u_2
inst✝⁷ : Field K
inst✝⁶ : NumberField K
inst✝⁵ : CommRing 𝒪
inst✝⁴ : Algebra 𝒪 K
inst✝³ : IsFractionRing 𝒪 K
inst✝² : IsDedekindDomain 𝒪
inst✝¹ : CharZero 𝒪
inst✝ : Module.Finite ℤ 𝒪
⊢ ↑(differentIdeal ℤ 𝒪) ≤ 1⁻¹

Tactic:
by simpa using FractionalIdeal.coeIdeal_le_one

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

Goal state before:
pp: case refine_3
L : Type u_3
inst✝⁶ : Field L
K₁ : Type u_4
K₂ : Type u_5
inst✝⁵ : Field K₁
inst✝⁴ : NumberField K₁
inst✝³ : Field K₂
inst✝² : NumberField K₂
inst✝¹ : Algebra K₁ L
inst✝ : Algebra K₂ L
u v : ℤ
h : u * discr K₁ + v * discr K₂ = 1
⊢ ↑u * ↑(discr K₁) + ↑v * ↑(discr K₂) = 1

Tactic:
← Int.cast_mul,

Goal state after:
case refine_3
L : Type u_3
inst✝⁶ : Field L
K₁ : Type u_4
K₂ : Type u_5
inst✝⁵ : Field K₁
inst✝⁴ : NumberField K₁
inst✝³ : Field K₂
inst✝² : NumberField K₂
inst✝¹ : Algebra K₁ L
inst✝ : Algebra K₂ L
u v : ℤ
h : u * discr K₁ + v * discr K₂ = 1
⊢ ↑(u * discr K₁) + ↑v * ↑(discr K₂) = 1
Module: Mathlib.NumberTheory.NumberField.Discriminant.Different

Goal state before:
pp: case refine_3
L : Type u_3
inst✝⁶ : Field L
K₁ : Type u_4
K₂ : Type u_5
inst✝⁵ : Field K₁
inst✝⁴ : NumberField K₁
inst✝³ : Field K₂
inst✝² : NumberField K₂
inst✝¹ : Algebra K₁ L
inst✝ : Algebra K₂ L
u v : ℤ
h : u * discr K₁ + v * discr K₂ = 1
⊢ ↑(u * discr K₁) + ↑v * ↑(discr K₂) = 1

Tactic:
← Int.cast_mul,

Goal state after:
case refine_3
L : Type u_3
inst✝⁶ : Field L
K₁ : Type u_4
K₂ : Type u_5
inst✝⁵ : Field K₁
inst✝⁴ : NumberField K₁
inst✝³ : Field K₂
inst✝² : NumberField K₂
inst✝¹ : Algebra K₁ L
inst✝ : Algebra K₂ L
u v : ℤ
h : u * discr K₁ + v * discr K₂ = 1
⊢ ↑(u * discr K₁) + ↑(v * discr K₂) = 1
Module: Mathlib.RingTheory.Ideal.Norm.RelNorm

Goal state before:
pp: R : Type u_1
inst✝⁸ : CommRing R
inst✝⁷ : IsDomain R
S : Type u_3
inst✝⁶ : CommRing S
inst✝⁵ : IsDomain S
inst✝⁴ : IsIntegrallyClosed R
inst✝³ : IsIntegrallyClosed S
inst✝² : Algebra R S
inst✝¹ : Module.Finite R S
inst✝ : IsTorsionFree R S
eq_bot_or_top : ∀ (I : Ideal R), I = ⊥ ∨ I = ⊤
I J : Ideal S
⊢ spanNorm R (I * J) ≤ spanNorm R I * spanNorm R J

Tactic:
rcases eq_bot_or_top (spanNorm R I) with hI | hI

Goal state after:
case inl
R : Type u_1
inst✝⁸ : CommRing R
inst✝⁷ : IsDomain R
S : Type u_3
inst✝⁶ : CommRing S
inst✝⁵ : IsDomain S
inst✝⁴ : IsIntegrallyClosed R
inst✝³ : IsIntegrallyClosed S
inst✝² : Algebra R S
inst✝¹ : Module.Finite R S
inst✝ : IsTorsionFree R S
eq_bot_or_top : ∀ (I : Ideal R), I = ⊥ ∨ I = ⊤
I J : Ideal S
hI : spanNorm R I = ⊥
⊢ spanNorm R (I * J) ≤ spanNorm R I * spanNorm R J

case inr
R : Type u_1
inst✝⁸ : CommRing R
inst✝⁷ : IsDomain R
S : Type u_3
inst✝⁶ : CommRing S
inst✝⁵ : IsDomain S
inst✝⁴ : IsIntegrallyClosed R
inst✝³ : IsIntegrallyClosed S
inst✝² : Algebra R S
inst✝¹ : Module.Finite R S
inst✝ : IsTorsionFree R S
eq_bot_or_top : ∀ (I : Ideal R), I = ⊥ ∨ I = ⊤
I J : Ideal S
hI : spanNorm R I = ⊤
⊢ spanNorm R (I * J) ≤ spanNorm R I * spanNorm R J
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: case refine_1
p k : ℕ
K : Type u
inst✝¹ : Field K
ζ : K
hp : Fact (Nat.Prime p)
inst✝ : CharZero K
hcycl : IsCyclotomicExtension {p ^ k} ℚ K
hζ : IsPrimitiveRoot ζ (p ^ k)
x : K
h : IsIntegral ℤ x
B : PowerBasis ℚ K := IsPrimitiveRoot.subOnePowerBasis ℚ hζ
hint : IsIntegral ℤ B.gen
this : FiniteDimensional ℚ K := finiteDimensional {p ^ k} ℚ K
u : ℤˣ
n : ℕ
hun : Algebra.discr ℚ ⇑(IsPrimitiveRoot.subOnePowerBasis ℚ hζ).basis = ↑↑u * ↑p ^ n
H : (↑↑u⁻¹ * ↑↑u * ↑p ^ n) • x ∈ ℤ[B.gen]
⊢ x ∈ ℤ[ζ]

Tactic:
← Int.cast_mul,

Goal state after:
case refine_1
p k : ℕ
K : Type u
inst✝¹ : Field K
ζ : K
hp : Fact (Nat.Prime p)
inst✝ : CharZero K
hcycl : IsCyclotomicExtension {p ^ k} ℚ K
hζ : IsPrimitiveRoot ζ (p ^ k)
x : K
h : IsIntegral ℤ x
B : PowerBasis ℚ K := IsPrimitiveRoot.subOnePowerBasis ℚ hζ
hint : IsIntegral ℤ B.gen
this : FiniteDimensional ℚ K := finiteDimensional {p ^ k} ℚ K
u : ℤˣ
n : ℕ
hun : Algebra.discr ℚ ⇑(IsPrimitiveRoot.subOnePowerBasis ℚ hζ).basis = ↑↑u * ↑p ^ n
H : (↑(↑u⁻¹ * ↑u) * ↑p ^ n) • x ∈ ℤ[B.gen]
⊢ x ∈ ℤ[ζ]
Module: Mathlib.RingTheory.DedekindDomain.LinearDisjoint

Goal state before:
pp: A : Type u_1
B : Type u_2
K : Type u_3
L : Type u_4
inst✝⁴⁶ : CommRing A
inst✝⁴⁵ : Field K
inst✝⁴⁴ : Algebra A K
inst✝⁴³ : IsFractionRing A K
inst✝⁴² : CommRing B
inst✝⁴¹ : Field L
inst✝⁴⁰ : Algebra B L
inst✝³⁹ : Algebra A L
inst✝³⁸ : Algebra K L
inst✝³⁷ : FiniteDimensional K L
inst✝³⁶ : IsScalarTower A K L
R₁ : Type u_5
R₂ : Type u_6
inst✝³⁵ : CommRing R₁
inst✝³⁴ : CommRing R₂
inst✝³³ : IsDomain R₁
inst✝³² : Algebra A R₁
inst✝³¹ : Algebra A R₂
inst✝³⁰ : Algebra R₁ B
inst✝²⁹ : Algebra R₂ B
inst✝²⁸ : Algebra R₁ L
inst✝²⁷ : Algebra R₂ L
inst✝²⁶ : IsScalarTower A R₁ L
inst✝²⁵ : IsScalarTower R₁ B L
inst✝²⁴ : IsScalarTower R₂ B L
inst✝²³ : Module.Finite A R₂
F₁ F₂ : IntermediateField K L
inst✝²² : Algebra R₁ ↥F₁
inst✝²¹ : Algebra R₂ ↥F₂
inst✝²⁰ : IsTorsionFree R₁ ↥F₁
inst✝¹⁹ : IsScalarTower A (↥F₂) L
inst✝¹⁸ : IsScalarTower A R₂ ↥F₂
inst✝¹⁷ : IsScalarTower R₁ (↥F₁) L
inst✝¹⁶ : IsScalarTower R₂ (↥F₂) L
inst✝¹⁵ : Algebra.IsSeparable K ↥F₂
inst✝¹⁴ : Algebra.IsSeparable (↥F₁) L
inst✝¹³ : IsDomain A
inst✝¹² : IsDedekindDomain B
inst✝¹¹ : IsDedekindDomain R₁
inst✝¹⁰ : IsDedekindDomain R₂
inst✝⁹ : IsFractionRing B L
inst✝⁸ : IsFractionRing R₁ ↥F₁
inst✝⁷ : IsFractionRing R₂ ↥F₂
inst✝⁶ : IsIntegrallyClosed A
inst✝⁵ : IsIntegralClosure B R₁ L
inst✝⁴ : IsTorsionFree R₁ B
inst✝³ : IsTorsionFree R₂ B
inst✝² : Algebra.IsIntegral R₂ B
inst✝¹ : Free A R₂
inst✝ : IsLocalization (algebraMapSubmonoid R₂ A⁰) ↥F₂
h₁ : F₁.LinearDisjoint ↥F₂
h₂ : F₁ ⊔ F₂ = ⊤
this✝ : Algebra.IsSeparable (FractionRing A) (FractionRing R₂)
this :
  span B ↑(Submodule.restrictScalars R₁ (traceDual R₁ (↥F₁) 1)) ≤
    span B ↑(span R₁ (⇑(algebraMap (↥F₂) L) '' ↑(traceDual A K 1)))
⊢ traceDual R₁ (↥F₁) 1 ≤ span B (⇑(algebraMap (↥F₂) L) '' ↑(traceDual A K 1))

Tactic:
rwa [← span_coe_eq_restrictScalars, span_span_of_tower, span_span_of_tower, span_eq] at this

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

Goal state before:
pp: p k : ℕ
K : Type u
inst✝² : Field K
hp : Fact (Nat.Prime p)
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {p ^ k} ℚ K
hζ : IsPrimitiveRoot (zeta (p ^ k) ℚ K) (p ^ k)
this : NumberField K
pB₁ : PowerBasis ℤ (𝓞 K) := hζ.integralPowerBasisOfPrimePow
⊢ NumberField.discr K = (-1) ^ (φ (p ^ k) / 2) * ↑p ^ (p ^ (k - 1) * ((p - 1) * k - 1))

Tactic:
apply (algebraMap ℤ ℚ).injective_int

Goal state after:
p k : ℕ
K : Type u
inst✝² : Field K
hp : Fact (Nat.Prime p)
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {p ^ k} ℚ K
hζ : IsPrimitiveRoot (zeta (p ^ k) ℚ K) (p ^ k)
this : NumberField K
pB₁ : PowerBasis ℤ (𝓞 K) := ⋯
⊢ (algebraMap ℤ ℚ) (NumberField.discr K) =
    (algebraMap ℤ ℚ) ((-1) ^ (φ (p ^ k) / 2) * ↑p ^ (p ^ (k - 1) * ((p - 1) * k - 1)))
Module: Mathlib.RingTheory.DedekindDomain.Different

Goal state before:
pp: A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝¹⁸ : CommRing A
inst✝¹⁷ : Field K
inst✝¹⁶ : CommRing B
inst✝¹⁵ : Field L
inst✝¹⁴ : Algebra A K
inst✝¹³ : Algebra B L
inst✝¹² : Algebra A B
inst✝¹¹ : Algebra K L
inst✝¹⁰ : Algebra A L
inst✝⁹ : IsScalarTower A K L
inst✝⁸ : IsScalarTower A B L
inst✝⁷ : IsDomain A
inst✝⁶ : IsFractionRing A K
inst✝⁵ : FiniteDimensional K L
inst✝⁴ : Algebra.IsSeparable K L
inst✝³ : IsIntegralClosure B A L
inst✝² : IsFractionRing B L
inst✝¹ : IsIntegrallyClosed A
inst✝ : IsDedekindDomain B
I : FractionalIdeal B⁰ L
⊢ dual A K (dual A K I) = I

Tactic:
rw [dual_eq_mul_inv, dual_eq_mul_inv A K (I := I), mul_inv, inv_inv, ← mul_assoc, mul_inv_cancel₀,
    one_mul]
  rw [dual_ne_zero_iff]
  exact one_ne_zero

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.LinearDisjoint

Goal state before:
pp: A : Type u_1
B : Type u_2
K : Type u_3
L : Type u_4
inst✝⁵⁴ : CommRing A
inst✝⁵³ : Field K
inst✝⁵² : Algebra A K
inst✝⁵¹ : IsFractionRing A K
inst✝⁵⁰ : CommRing B
inst✝⁴⁹ : Field L
inst✝⁴⁸ : Algebra B L
inst✝⁴⁷ : Algebra A L
inst✝⁴⁶ : Algebra K L
inst✝⁴⁵ : FiniteDimensional K L
inst✝⁴⁴ : IsScalarTower A K L
R₁ : Type u_5
R₂ : Type u_6
inst✝⁴³ : CommRing R₁
inst✝⁴² : CommRing R₂
inst✝⁴¹ : IsDomain R₁
inst✝⁴⁰ : Algebra A R₁
inst✝³⁹ : Algebra A R₂
inst✝³⁸ : Algebra R₁ B
inst✝³⁷ : Algebra R₂ B
inst✝³⁶ : Algebra R₁ L
inst✝³⁵ : Algebra R₂ L
inst✝³⁴ : IsScalarTower A R₁ L
inst✝³³ : IsScalarTower R₁ B L
inst✝³² : IsScalarTower R₂ B L
inst✝³¹ : Module.Finite A R₂
F₁ F₂ : IntermediateField K L
inst✝³⁰ : Algebra R₁ ↥F₁
inst✝²⁹ : Algebra R₂ ↥F₂
inst✝²⁸ : IsTorsionFree R₁ ↥F₁
inst✝²⁷ : IsScalarTower A (↥F₂) L
inst✝²⁶ : IsScalarTower A R₂ ↥F₂
inst✝²⁵ : IsScalarTower R₁ (↥F₁) L
inst✝²⁴ : IsScalarTower R₂ (↥F₂) L
inst✝²³ : Algebra.IsSeparable K ↥F₂
inst✝²² : Algebra.IsSeparable (↥F₁) L
inst✝²¹ : IsDomain A
inst✝²⁰ : IsDedekindDomain B
inst✝¹⁹ : IsDedekindDomain R₁
inst✝¹⁸ : IsDedekindDomain R₂
inst✝¹⁷ : IsFractionRing B L
inst✝¹⁶ : IsFractionRing R₁ ↥F₁
inst✝¹⁵ : IsFractionRing R₂ ↥F₂
inst✝¹⁴ : IsIntegrallyClosed A
inst✝¹³ : IsIntegralClosure B R₁ L
inst✝¹² : IsTorsionFree R₁ B
inst✝¹¹ : IsTorsionFree R₂ B
inst✝¹⁰ : Algebra A B
inst✝⁹ : Module.Finite A B
inst✝⁸ : IsTorsionFree A B
inst✝⁷ : IsTorsionFree A R₁
inst✝⁶ : IsTorsionFree A R₂
inst✝⁵ : Module.Finite A R₁
inst✝⁴ : Module.Finite R₂ B
inst✝³ : IsScalarTower A R₂ B
inst✝² : Module.Finite R₁ B
inst✝¹ : Algebra.IsSeparable (FractionRing A) (FractionRing B)
inst✝ : IsScalarTower A R₁ B
h₁ : F₁.LinearDisjoint ↥F₂
h₂ : F₁.toSubalgebra ⊔ F₂.toSubalgebra = ⊤
h₃ : IsCoprime (Ideal.map (algebraMap R₁ B) (differentIdeal A R₁)) (Ideal.map (algebraMap R₂ B) (differentIdeal A R₂))
ι : Type u_7
b : Basis ι A R₂
this : Free A R₂
v : ι → B := fun i ↦ (algebraMap R₂ B) (b i)
b₂ : Basis ι K ↥F₂ := localizationLocalization K A⁰ (↥F₂) b
⊢ LinearIndependent ↥F₁ fun x ↦ (algebraMap (↥F₂) L) ((localizationLocalization K A⁰ (↥F₂) b) x)

Tactic:
exact h₁.linearIndependent_right b₂.linearIndependent

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Units.DirichletTheorem

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
s : AddSubgroup (Additive (𝓞 K)ˣ)
⊢ s ⊔ (logEmbedding K).ker ⊔ (logEmbedding K).ker = s ⊔ (logEmbedding K).ker

Tactic:
sup_right_idem

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
s : AddSubgroup (Additive (𝓞 K)ˣ)
⊢ s ⊔ (logEmbedding K).ker = s ⊔ (logEmbedding K).ker
Module: Mathlib.NumberTheory.NumberField.Units.DirichletTheorem

Goal state before:
pp: case succ
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w₁ : InfinitePlace K
B : ℕ
hB : minkowskiBound K 1 < ↑(convexBodyLTFactor K) * ↑B
n : ℕ
⊢ ↑((Algebra.norm ℤ) ↑(seq K w₁ hB (n + 1))).natAbs ≤ ↑B

Tactic:
Nat.cast_natAbs,

Goal state after:
case succ
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w₁ : InfinitePlace K
B : ℕ
hB : minkowskiBound K 1 < ↑(convexBodyLTFactor K) * ↑B
n : ℕ
⊢ ↑|(Algebra.norm ℤ) ↑(seq K w₁ hB (n + 1))| ≤ ↑B
Module: Mathlib.NumberTheory.LegendreSymbol.Basic

Goal state before:
pp: p : ℕ
inst✝ : Fact (Nat.Prime p)
a : ZMod p
ha : a ≠ 0
⊢ Units.mk0 a ha ^ (p / 2) = 1 ↔ a ^ (p / 2) = 1

Tactic:
by simp [Units.ext_iff]

Goal state after:
Module: Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic

Goal state before:
pp: case pos
F : Type u_1
inst✝² : Field F
inst✝¹ : Fintype F
inst✝ : DecidableEq F
a : F
ha : a = 0
⊢ (quadraticChar F) a = 0 ∨ (quadraticChar F) a = 1 ∨ (quadraticChar F) a = -1

Tactic:
left; rw [ha]; exact quadraticChar_zero

Goal state after:
Module: Mathlib.NumberTheory.FLT.Three

Goal state before:
pp: K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
H : FermatLastTheoremForThreeGen hζ
a b c : ℤ
hc : c ≠ 0
ha : ¬3 ∣ a
hb : ¬3 ∣ b
x✝ : 3 ∣ c
hcoprime : IsCoprime a b
h : a ^ 3 + b ^ 3 = c ^ 3
x : ℤ
hx : c = 3 * x
⊢ False

Tactic:
refine H a b c 1 (by simp [hc]) (fun hdvd ↦ ha ?_) (fun hdvd ↦ hb ?_) ?_ ?_ ?_

Goal state after:
case refine_1
K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
H : FermatLastTheoremForThreeGen hζ
a b c : ℤ
hc : c ≠ 0
ha : ¬3 ∣ a
hb : ¬3 ∣ b
x✝ : 3 ∣ c
hcoprime : IsCoprime a b
h : a ^ 3 + b ^ 3 = c ^ 3
x : ℤ
hx : c = 3 * x
hdvd : λ ∣ ↑a
⊢ 3 ∣ a

case refine_2
K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
H : FermatLastTheoremForThreeGen hζ
a b c : ℤ
hc : c ≠ 0
ha : ¬3 ∣ a
hb : ¬3 ∣ b
x✝ : 3 ∣ c
hcoprime : IsCoprime a b
h : a ^ 3 + b ^ 3 = c ^ 3
x : ℤ
hx : c = 3 * x
hdvd : λ ∣ ↑b
⊢ 3 ∣ b

case refine_3
K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
H : FermatLastTheoremForThreeGen hζ
a b c : ℤ
hc : c ≠ 0
ha : ¬3 ∣ a
hb : ¬3 ∣ b
x✝ : 3 ∣ c
hcoprime : IsCoprime a b
h : a ^ 3 + b ^ 3 = c ^ 3
x : ℤ
hx : c = 3 * x
⊢ λ ∣ ↑c

case refine_4
K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
H : FermatLastTheoremForThreeGen hζ
a b c : ℤ
hc : c ≠ 0
ha : ¬3 ∣ a
hb : ¬3 ∣ b
x✝ : 3 ∣ c
hcoprime : IsCoprime a b
h : a ^ 3 + b ^ 3 = c ^ 3
x : ℤ
hx : c = 3 * x
⊢ IsCoprime ↑a ↑b

case refine_5
K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
H : FermatLastTheoremForThreeGen hζ
a b c : ℤ
hc : c ≠ 0
ha : ¬3 ∣ a
hb : ¬3 ∣ b
x✝ : 3 ∣ c
hcoprime : IsCoprime a b
h : a ^ 3 + b ^ 3 = c ^ 3
x : ℤ
hx : c = 3 * x
⊢ ↑a ^ 3 + ↑b ^ 3 = ↑1 * ↑c ^ 3
Module: Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic

Goal state before:
pp: case pos
F : Type u_1
inst✝¹ : Field F
inst✝ : Fintype F
hF : ringChar F = 2
⊢ IsSquare (-1) ↔ Fintype.card F % 4 ≠ 3

Tactic:
simp only [FiniteField.isSquare_of_char_two hF, Ne, true_iff]
    exact fun hf ↦
      one_ne_zero <|
        (Nat.odd_of_mod_four_eq_three hf).symm.trans <| FiniteField.even_card_of_char_two hF

Goal state after:
Module: Mathlib.NumberTheory.Fermat

Goal state before:
pp: n : ℕ
⊢ ↑((n + 1).fermatNumber ^ 2 - 2 * (n.fermatNumber - 1) ^ 2) = ↑(n + 1).fermatNumber ^ 2 - 2 * (↑n.fermatNumber - 1) ^ 2

Tactic:
Nat.cast_sub <| two_mul_fermatNumber_sub_one_sq_le_fermatNumber_sq n

Goal state after:
n : ℕ
⊢ ↑((n + 1).fermatNumber ^ 2) - ↑(2 * (n.fermatNumber - 1) ^ 2) =
    ↑(n + 1).fermatNumber ^ 2 - 2 * (↑n.fermatNumber - 1) ^ 2