mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.IntegralClosure.IntegralRestrict

Goal state before:
pp: A : Type u_1
B : Type u_6
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra A B
inst✝⁵ : IsDomain A
inst✝⁴ : IsIntegrallyClosed A
inst✝³ : IsDomain B
inst✝² : IsIntegrallyClosed B
inst✝¹ : Module.Finite A B
inst✝ : IsTorsionFree A B
x : B
⊢ x ∣ (algebraMap A B) ((intNorm A B) x)

Tactic:
by
  classical
  have : FiniteDimensional (FractionRing A) (FractionRing B) := .of_isLocalization A B A⁰
  by_cases hx : x = 0
  · exact ⟨1, by simp [hx]⟩
  let K := FractionRing A
  let L := FractionRing B
  let E := AlgebraicClosure L
  suffices IsIntegral A ((algebraMap B L x)⁻¹ * (algebraMap A L (intNorm A B x))) by
    obtain ⟨y, hy⟩ := IsIntegrallyClosed.isIntegral_iff.mp <|
      _root_.IsIntegral.tower_top (A := B) this
    refine ⟨y, ?_⟩
    apply FaithfulSMul.algebraMap_injective B L
    rw [← algebraMap_apply, map_mul, hy, mul_inv_cancel_left₀]
    exact (map_ne_zero_iff _ (FaithfulSMul.algebraMap_injective B L)).mpr hx
  rw [← isIntegral_algHom_iff (toAlgHom A L E)
    (FaithfulSMul.algebraMap_injective L E), coe_toAlgHom', map_mul, map_inv₀,
    algebraMap_apply A K L, algebraMap_intNorm (L := L), ← algebraMap_apply, ← algebraMap_apply,
    norm_eq_prod_roots _ (IsAlgClosed.splits _), ← Multiset.prod_erase
    (a := algebraMap B E x)]
  · have := IsTorsionFree.trans_faithfulSMul B L E
    rw [mul_pow, ← mul_pow_sub_one (Nat.pos_iff_ne_zero.1 Module.finrank_pos) (algebraMap B E x),
      mul_assoc, inv_mul_cancel_left₀]
    · refine IsIntegral.mul (IsIntegral.pow ?_ _)
        (IsIntegral.pow (IsIntegral.multiset_prod (fun a ha ↦ ⟨minpoly A x, minpoly.monic
          (IsIntegral.isIntegral x), ?_⟩)) _)
      · exact (isIntegral_algebraMap_iff (isTorsionFree_iff_algebraMap_injective.1 this)).mpr
          (IsIntegral.isIntegral x)
      · replace ha := Multiset.erase_subset _ _ ha
        suffices (aeval a) ((minpoly A x).map (algebraMap A K)) = 0 by simpa
        rw [← minpoly.isIntegrallyClosed_eq_field_fractions K L (IsIntegral.isIntegral x)]
        simp only [mem_roots', ne_eq, Polynomial.map_eq_zero, IsRoot.def, eval_map_algebraMap] at ha
        exact ha.2
    · exact (map_ne_zero_iff _ (FaithfulSMul.algebraMap_injective B E)).mpr hx
  · simp only [mem_roots', ne_eq, Polynomial.map_eq_zero, IsRoot.def, eval_map_algebraMap]
    refine ⟨minpoly.ne_zero (IsIntegral.isIntegral _), ?_⟩
    simp [algebraMap_apply B L E, aeval_algebraMap_apply]

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

Goal state before:
pp: case neg
K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsGalois ℚ 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),
        ⌊(4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|)⌋₊ <
            p ^ P.inertiaDeg ℤ ∨
          Submodule.IsPrincipal P
p : ℕ
hpmem :
  p ∈ Finset.Icc 1 ⌊(4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|)⌋₊
hp : Nat.Prime p
P : Ideal (𝓞 K)
x✝ : P ∈ (span {↑p}).primesOver (𝓞 K)
hple :
  p ^ P.inertiaDeg ℤ ≤ ⌊(4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|)⌋₊
hP1 : P.IsPrime
hP2 : P.LiesOver (span {↑p})
Q : Ideal (𝓞 K)
H :
  ⌊(4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|)⌋₊ <
      p ^ Q.inertiaDeg ℤ ∨
    Submodule.IsPrincipal Q
hQ1 : Q.IsPrime
hQ2 : Q.LiesOver (span {↑p})
this : (span {↑p}).IsMaximal
h :
  ¬⌊(4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|)⌋₊ <
      p ^ P.inertiaDeg ℤ
⊢ Submodule.IsPrincipal P

Tactic:
inertiaDeg_eq_of_isGaloisGroup (span {↑p}) Q P (K ≃ₐ[ℚ] K)

Goal state after:
case neg
K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsGalois ℚ 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),
        ⌊(4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|)⌋₊ <
            p ^ P.inertiaDeg ℤ ∨
          Submodule.IsPrincipal P
p : ℕ
hpmem :
  p ∈ Finset.Icc 1 ⌊(4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|)⌋₊
hp : Nat.Prime p
P : Ideal (𝓞 K)
x✝ : P ∈ (span {↑p}).primesOver (𝓞 K)
hple :
  p ^ P.inertiaDeg ℤ ≤ ⌊(4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|)⌋₊
hP1 : P.IsPrime
hP2 : P.LiesOver (span {↑p})
Q : Ideal (𝓞 K)
H :
  ⌊(4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|)⌋₊ <
      p ^ P.inertiaDeg ℤ ∨
    Submodule.IsPrincipal Q
hQ1 : Q.IsPrime
hQ2 : Q.LiesOver (span {↑p})
this : (span {↑p}).IsMaximal
h :
  ¬⌊(4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|)⌋₊ <
      p ^ P.inertiaDeg ℤ
⊢ Submodule.IsPrincipal P
Module: Mathlib.RingTheory.Ideal.Quotient.Index

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
ι : Type u_3
I : ι → Ideal R
s : Finset ι
hI : ∀ i ∈ s, (I i).FG
hI' : ∀ i ∈ s, Finite (R ⧸ I i)
⊢ Finite (R ⧸ ∏ i ∈ s, I i)

Tactic:
by
  classical
  induction s using Finset.induction_on with
  | empty => simp only [Finset.prod_empty, one_eq_top]; infer_instance
  | insert a s has IH =>
    rw [Finset.prod_insert has, mul_comm]
    have := hI' a (by simp)
    have := IH (fun i hi ↦ hI _ (by simp [hi])) (fun i hi ↦ hI' _ (by simp [hi]))
    exact Submodule.finite_quotient_smul _ (hI a (by simp))

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

Goal state before:
pp: case ht
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
⊢ ⇑(IsLocalization.map L f hf) '' (↑I ∪ ↑J) ⊆ ↑(span B (⇑(IsLocalization.map L f hf) '' ↑(I + J)))

Tactic:
rintro _ ⟨y, hy, rfl⟩
    suffices y ∈ I + J from SetLike.mem_coe.mpr <| Submodule.subset_span ⟨y, by simp [this]⟩
    exact hy.elim (fun h ↦ (mem_add I J y).mpr ⟨y, h, 0, zero_mem J, add_zero y⟩)
      (fun h ↦ (mem_add I J y).mpr ⟨0, zero_mem I, y, h, zero_add y⟩)

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

Goal state before:
pp: case he
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
this : IsScalarTower R (Rₚ ⧸ maximalIdeal Rₚ) (Sₚ ⧸ pSₚ)
x : R
⊢ (algebraMap R (Sₚ ⧸ pSₚ)) x = (algebraMap S (Sₚ ⧸ pSₚ)) ((algebraMap R S) x)

Tactic:
← IsScalarTower.algebraMap_apply

Goal state after:
case he
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
this : IsScalarTower R (Rₚ ⧸ maximalIdeal Rₚ) (Sₚ ⧸ pSₚ)
x : R
⊢ (algebraMap R (Sₚ ⧸ pSₚ)) x = (algebraMap R (Sₚ ⧸ pSₚ)) x
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)
b : Basis (Free.ChooseBasisIndex (R ⧸ p) (S ⧸ pS)) (R ⧸ p) (S ⧸ pS) := Free.chooseBasis (R ⧸ p) (S ⧸ pS)
⊢ finrank R S ≤ finrank (R ⧸ p) (S ⧸ pS)

Tactic:
choose b' hb' using fun i ↦ Ideal.Quotient.mk_surjective (b i)

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)
b : Basis (Free.ChooseBasisIndex (R ⧸ p) (S ⧸ pS)) (R ⧸ p) (S ⧸ pS) := Free.chooseBasis (R ⧸ p) (S ⧸ pS)
b' : Free.ChooseBasisIndex (R ⧸ p) (S ⧸ pS) → S
hb' : ∀ (i : Free.ChooseBasisIndex (R ⧸ p) (S ⧸ pS)), (Ideal.Quotient.mk pS) (b' i) = b i
⊢ finrank R S ≤ finrank (R ⧸ p) (S ⧸ pS)
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✝⁷ : Algebra Rₚ Sₚ := localizationAlgebra p.primeCompl S
this✝⁶ : IsScalarTower R Rₚ Sₚ
this✝⁵ : IsLocalization (Submonoid.map (algebraMap R S) p.primeCompl) Sₚ
e : algebraMapSubmonoid S p.primeCompl ≤ S⁰
this✝⁴ : IsDomain Sₚ
this✝³ : IsTorsionFree Rₚ Sₚ
this✝² : Module.Finite Rₚ Sₚ
this✝¹ : IsIntegrallyClosed Sₚ
this✝ : IsPrincipalIdealRing Rₚ
this : Free Rₚ Sₚ
⊢ (algebraMap Rₚ (Rₚ ⧸ maximalIdeal Rₚ)) ((algebraMap R Rₚ) ((intTrace R S) x)) =
    (equivQuotMaximalIdeal p Rₚ) ((Ideal.Quotient.mk p) ((intTrace R S) x))

Tactic:
← IsScalarTower.algebraMap_apply

Goal state after:
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✝⁷ : Algebra Rₚ Sₚ := localizationAlgebra p.primeCompl S
this✝⁶ : IsScalarTower R Rₚ Sₚ
this✝⁵ : IsLocalization (Submonoid.map (algebraMap R S) p.primeCompl) Sₚ
e : algebraMapSubmonoid S p.primeCompl ≤ S⁰
this✝⁴ : IsDomain Sₚ
this✝³ : IsTorsionFree Rₚ Sₚ
this✝² : Module.Finite Rₚ Sₚ
this✝¹ : IsIntegrallyClosed Sₚ
this✝ : IsPrincipalIdealRing Rₚ
this : Free Rₚ Sₚ
⊢ (algebraMap R (Rₚ ⧸ maximalIdeal Rₚ)) ((intTrace R S) x) =
    (equivQuotMaximalIdeal p Rₚ) ((Ideal.Quotient.mk p) ((intTrace R S) x))
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 : Ideal.map (Ideal.Quotient.mk I) p ^ n = 0
this : Ideal.map (Ideal.Quotient.mk I) p = maximalIdeal (R ⧸ I)
⊢ ∃ n, p ^ n ≤ I

Tactic:
← Ideal.map_pow,

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 : Ideal.map (Ideal.Quotient.mk I) (p ^ n) = 0
this : Ideal.map (Ideal.Quotient.mk I) p = maximalIdeal (R ⧸ I)
⊢ ∃ n, p ^ n ≤ I
Module: Mathlib.RingTheory.DedekindDomain.PID

Goal state before:
pp: R : Type u_1
inst✝¹³ : CommRing R
inst✝¹² : IsDedekindDomain R
S : Type u_2
inst✝¹¹ : CommRing S
inst✝¹⁰ : Algebra R S
inst✝⁹ : Module.IsTorsionFree R S
inst✝⁸ : Module.Finite R S
p : Ideal R
hp0 : p ≠ ⊥
inst✝⁷ : p.IsPrime
Sₚ : Type u_3
inst✝⁶ : CommRing Sₚ
inst✝⁵ : Algebra S Sₚ
inst✝⁴ : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝³ : Algebra R Sₚ
inst✝² : IsScalarTower R S Sₚ
inst✝¹ : IsDedekindDomain Sₚ
inst✝ : IsDomain S
P : Ideal Sₚ
hP : P.IsPrime
hP0 : P ≠ ⊥
non_zero_div : Algebra.algebraMapSubmonoid S p.primeCompl ≤ S⁰
this✝¹ : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra p.primeCompl S
this✝ : IsScalarTower R (Localization.AtPrime p) Sₚ
pid : IsPrincipalIdealRing (Localization.AtPrime p)
p' : Ideal (Localization.AtPrime p)
hpu : ∀ (y : Ideal (Localization.AtPrime p)), (fun P ↦ P ≠ ⊥ ∧ P.IsPrime) y → y = p'
hp'0 : p' ≠ ⊥
hp'p : p'.IsPrime
this : IsLocalRing.maximalIdeal (Localization.AtPrime p) ≠ ⊥
⊢ Ideal.map (algebraMap R Sₚ) p ≠ 0

Tactic:
rw [Ne, zero_eq_bot, Ideal.map_eq_bot_iff_of_injective]

Goal state after:
R : Type u_1
inst✝¹³ : CommRing R
inst✝¹² : IsDedekindDomain R
S : Type u_2
inst✝¹¹ : CommRing S
inst✝¹⁰ : Algebra R S
inst✝⁹ : Module.IsTorsionFree R S
inst✝⁸ : Module.Finite R S
p : Ideal R
hp0 : p ≠ ⊥
inst✝⁷ : p.IsPrime
Sₚ : Type u_3
inst✝⁶ : CommRing Sₚ
inst✝⁵ : Algebra S Sₚ
inst✝⁴ : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝³ : Algebra R Sₚ
inst✝² : IsScalarTower R S Sₚ
inst✝¹ : IsDedekindDomain Sₚ
inst✝ : IsDomain S
P : Ideal Sₚ
hP : P.IsPrime
hP0 : P ≠ ⊥
non_zero_div : Algebra.algebraMapSubmonoid S p.primeCompl ≤ S⁰
this✝¹ : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra p.primeCompl S
this✝ : IsScalarTower R (Localization.AtPrime p) Sₚ
pid : IsPrincipalIdealRing (Localization.AtPrime p)
p' : Ideal (Localization.AtPrime p)
hpu : ∀ (y : Ideal (Localization.AtPrime p)), (fun P ↦ P ≠ ⊥ ∧ P.IsPrime) y → y = p'
hp'0 : p' ≠ ⊥
hp'p : p'.IsPrime
this : IsLocalRing.maximalIdeal (Localization.AtPrime p) ≠ ⊥
⊢ ¬p = ⊥

R : Type u_1
inst✝¹³ : CommRing R
inst✝¹² : IsDedekindDomain R
S : Type u_2
inst✝¹¹ : CommRing S
inst✝¹⁰ : Algebra R S
inst✝⁹ : Module.IsTorsionFree R S
inst✝⁸ : Module.Finite R S
p : Ideal R
hp0 : p ≠ ⊥
inst✝⁷ : p.IsPrime
Sₚ : Type u_3
inst✝⁶ : CommRing Sₚ
inst✝⁵ : Algebra S Sₚ
inst✝⁴ : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝³ : Algebra R Sₚ
inst✝² : IsScalarTower R S Sₚ
inst✝¹ : IsDedekindDomain Sₚ
inst✝ : IsDomain S
P : Ideal Sₚ
hP : P.IsPrime
hP0 : P ≠ ⊥
non_zero_div : Algebra.algebraMapSubmonoid S p.primeCompl ≤ S⁰
this✝¹ : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra p.primeCompl S
this✝ : IsScalarTower R (Localization.AtPrime p) Sₚ
pid : IsPrincipalIdealRing (Localization.AtPrime p)
p' : Ideal (Localization.AtPrime p)
hpu : ∀ (y : Ideal (Localization.AtPrime p)), (fun P ↦ P ≠ ⊥ ∧ P.IsPrime) y → y = p'
hp'0 : p' ≠ ⊥
hp'p : p'.IsPrime
this : IsLocalRing.maximalIdeal (Localization.AtPrime p) ≠ ⊥
⊢ Function.Injective ⇑(algebraMap R Sₚ)
Module: Mathlib.RingTheory.Ideal.Norm.RelNorm

Goal state before:
pp: R : Type u_1
inst✝²⁴ : CommRing R
inst✝²³ : IsDomain R
S : Type u_3
inst✝²² : CommRing S
inst✝²¹ : IsDomain S
inst✝²⁰ : IsIntegrallyClosed R
inst✝¹⁹ : IsIntegrallyClosed S
inst✝¹⁸ : Algebra R S
inst✝¹⁷ : Module.Finite R S
inst✝¹⁶ : IsTorsionFree R S
I : Ideal S
M : Submonoid R
hM : M ≤ R⁰
Rₘ : Type u_4
Sₘ : Type u_5
inst✝¹⁵ : CommRing Rₘ
inst✝¹⁴ : Algebra R Rₘ
inst✝¹³ : CommRing Sₘ
inst✝¹² : Algebra S Sₘ
inst✝¹¹ : Algebra Rₘ Sₘ
inst✝¹⁰ : Algebra R Sₘ
inst✝⁹ : IsScalarTower R Rₘ Sₘ
inst✝⁸ : IsScalarTower R S Sₘ
inst✝⁷ : IsLocalization M Rₘ
inst✝⁶ : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ
inst✝⁵ : IsIntegrallyClosed Rₘ
inst✝⁴ : IsDomain Rₘ
inst✝³ : IsDomain Sₘ
inst✝² : IsTorsionFree Rₘ Sₘ
inst✝¹ : Module.Finite Rₘ Sₘ
inst✝ : IsIntegrallyClosed Sₘ
⊢ spanNorm Rₘ (map (algebraMap S Sₘ) I) = map (algebraMap R Rₘ) (spanNorm R I)

Tactic:
let K := FractionRing R

Goal state after:
R : Type u_1
inst✝²⁴ : CommRing R
inst✝²³ : IsDomain R
S : Type u_3
inst✝²² : CommRing S
inst✝²¹ : IsDomain S
inst✝²⁰ : IsIntegrallyClosed R
inst✝¹⁹ : IsIntegrallyClosed S
inst✝¹⁸ : Algebra R S
inst✝¹⁷ : Module.Finite R S
inst✝¹⁶ : IsTorsionFree R S
I : Ideal S
M : Submonoid R
hM : M ≤ R⁰
Rₘ : Type u_4
Sₘ : Type u_5
inst✝¹⁵ : CommRing Rₘ
inst✝¹⁴ : Algebra R Rₘ
inst✝¹³ : CommRing Sₘ
inst✝¹² : Algebra S Sₘ
inst✝¹¹ : Algebra Rₘ Sₘ
inst✝¹⁰ : Algebra R Sₘ
inst✝⁹ : IsScalarTower R Rₘ Sₘ
inst✝⁸ : IsScalarTower R S Sₘ
inst✝⁷ : IsLocalization M Rₘ
inst✝⁶ : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ
inst✝⁵ : IsIntegrallyClosed Rₘ
inst✝⁴ : IsDomain Rₘ
inst✝³ : IsDomain Sₘ
inst✝² : IsTorsionFree Rₘ Sₘ
inst✝¹ : Module.Finite Rₘ Sₘ
inst✝ : IsIntegrallyClosed Sₘ
K : Type u_1 := FractionRing R
⊢ spanNorm Rₘ (map (algebraMap S Sₘ) I) = map (algebraMap R Rₘ) (spanNorm R I)
Module: Mathlib.NumberTheory.NumberField.Discriminant.Different

Goal state before:
pp: K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
L : Type u_3
inst✝² : Field L
inst✝¹ : NumberField L
inst✝ : Algebra K L
⊢ discr K ^ finrank K L ∣ ↑(Ideal.absNorm (differentIdeal (𝓞 K) (𝓞 L))) * ↑((discr K).natAbs ^ finrank K L)

Tactic:
Nat.cast_pow,

Goal state after:
K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
L : Type u_3
inst✝² : Field L
inst✝¹ : NumberField L
inst✝ : Algebra K L
⊢ discr K ^ finrank K L ∣ ↑(Ideal.absNorm (differentIdeal (𝓞 K) (𝓞 L))) * ↑(discr K).natAbs ^ finrank K L
Module: Mathlib.RingTheory.Ideal.Norm.RelNorm

Goal state before:
pp: R : Type u_1
inst✝¹⁰ : CommRing R
inst✝⁹ : IsDomain R
S : Type u_3
inst✝⁸ : CommRing S
inst✝⁷ : IsDomain S
inst✝⁶ : IsIntegrallyClosed R
inst✝⁵ : IsIntegrallyClosed S
inst✝⁴ : Algebra R S
inst✝³ : Module.Finite R S
inst✝² : IsTorsionFree R S
inst✝¹ : IsDedekindDomain R
inst✝ : IsDedekindDomain S
I P : Ideal R
hPd : P.IsMaximal
P' : Submonoid S := Algebra.algebraMapSubmonoid S P.primeCompl
Rₚ : Type u_1 := Localization.AtPrime P
⊢ map (algebraMap R (Localization.AtPrime P)) (spanNorm R (map (algebraMap R S) I)) =
    map (algebraMap R (Localization.AtPrime P)) (I ^ finrank R S)

Tactic:
let K := FractionRing R

Goal state after:
R : Type u_1
inst✝¹⁰ : CommRing R
inst✝⁹ : IsDomain R
S : Type u_3
inst✝⁸ : CommRing S
inst✝⁷ : IsDomain S
inst✝⁶ : IsIntegrallyClosed R
inst✝⁵ : IsIntegrallyClosed S
inst✝⁴ : Algebra R S
inst✝³ : Module.Finite R S
inst✝² : IsTorsionFree R S
inst✝¹ : IsDedekindDomain R
inst✝ : IsDedekindDomain S
I P : Ideal R
hPd : P.IsMaximal
P' : Submonoid S := Algebra.algebraMapSubmonoid S P.primeCompl
Rₚ : Type u_1 := Localization.AtPrime P
K : Type u_1 := FractionRing R
⊢ map (algebraMap R (Localization.AtPrime P)) (spanNorm R (map (algebraMap R S) I)) =
    map (algebraMap R (Localization.AtPrime P)) (I ^ finrank R S)
Module: Mathlib.RingTheory.Prime

Goal state before:
pp: R : Type u_1
inst✝² : CommMonoidWithZero R
inst✝¹ : IsCancelMulZero R
α : Type u_2
inst✝ : DecidableEq α
p : α → R
x y a : R
hp : ∀ i ∈ ∅, Prime (p i)
hx : x * y = a * ∏ i ∈ ∅, p i
⊢ ∅ ∪ ∅ = ∅ ∧ Disjoint ∅ ∅ ∧ a = x * y ∧ x = x * ∏ i ∈ ∅, p i ∧ y = y * ∏ i ∈ ∅, p i

Tactic:
simp [hx]

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

Goal state before:
pp: case refine_1.zero
p : ℕ
K : Type u
inst✝¹ : Field K
ζ : K
hp : Fact (Nat.Prime p)
inst✝ : CharZero K
u : ℤˣ
n : ℕ
hcycl : IsCyclotomicExtension {p ^ 0} ℚ K
hζ : IsPrimitiveRoot ζ (p ^ 0)
B : PowerBasis ℚ K := IsPrimitiveRoot.subOnePowerBasis ℚ hζ
hint : IsIntegral ℤ B.gen
this✝¹ : FiniteDimensional ℚ K := finiteDimensional {p ^ 0} ℚ K
hun : Algebra.discr ℚ ⇑(IsPrimitiveRoot.subOnePowerBasis ℚ hζ).basis = ↑↑u * ↑p ^ n
this✝ : IsCyclotomicExtension {1} ℚ K
y : ℚ
H : (algebraMap ℚ K) ↑p ^ n * (algebraMap ℚ K) y ∈ ℤ[B.gen]
this : (algebraMap ℚ K) y ∈ ⊥
h : IsIntegral ℤ y
z : ℤ
hz : (algebraMap ℤ ℚ) z = y
⊢ (algebraMap ℚ K) ((algebraMap ℤ ℚ) z) ∈ ℤ[ζ]

Tactic:
← IsScalarTower.algebraMap_apply

Goal state after:
case refine_1.zero
p : ℕ
K : Type u
inst✝¹ : Field K
ζ : K
hp : Fact (Nat.Prime p)
inst✝ : CharZero K
u : ℤˣ
n : ℕ
hcycl : IsCyclotomicExtension {p ^ 0} ℚ K
hζ : IsPrimitiveRoot ζ (p ^ 0)
B : PowerBasis ℚ K := IsPrimitiveRoot.subOnePowerBasis ℚ hζ
hint : IsIntegral ℤ B.gen
this✝¹ : FiniteDimensional ℚ K := finiteDimensional {p ^ 0} ℚ K
hun : Algebra.discr ℚ ⇑(IsPrimitiveRoot.subOnePowerBasis ℚ hζ).basis = ↑↑u * ↑p ^ n
this✝ : IsCyclotomicExtension {1} ℚ K
y : ℚ
H : (algebraMap ℚ K) ↑p ^ n * (algebraMap ℚ K) y ∈ ℤ[B.gen]
this : (algebraMap ℚ K) y ∈ ⊥
h : IsIntegral ℤ y
z : ℤ
hz : (algebraMap ℤ ℚ) z = y
⊢ (algebraMap ℤ K) z ∈ ℤ[ζ]
Module: Mathlib.RingTheory.DedekindDomain.Different

Goal state before:
pp: case neg
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝¹⁰ : CommRing A
inst✝⁹ : Field K
inst✝⁸ : CommRing B
inst✝⁷ : Field L
inst✝⁶ : Algebra A K
inst✝⁵ : Algebra B L
inst✝⁴ : Algebra A B
inst✝³ : Algebra K L
inst✝² : Algebra A L
inst✝¹ : IsScalarTower A K L
inst✝ : IsScalarTower A B L
b c : L
hc : ∀ (a : L), ∃ x, (algebraMap A K) x = (Algebra.trace K L) (c * a)
hc0 : c ≠ 0
⊢ ∃ x, (algebraMap A K) x = (Algebra.trace K L) b

Tactic:
simpa [hc0] using hc (c⁻¹ * b)

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

Goal state before:
pp: case e'_2
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
this : pB₁.dim = (IsPrimitiveRoot.powerBasis ℚ hζ).dim
i : Fin (IsPrimitiveRoot.powerBasis ℚ hζ).dim
⊢ (Module.Basis.localizationLocalization ℚ ℤ⁰ K pB₁.basis) ((finCongr this).symm i) =
    (IsPrimitiveRoot.powerBasis ℚ hζ).basis i

Tactic:
Module.Basis.localizationLocalization_apply,

Goal state after:
case e'_2
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
this : pB₁.dim = (IsPrimitiveRoot.powerBasis ℚ hζ).dim
i : Fin (IsPrimitiveRoot.powerBasis ℚ hζ).dim
⊢ (algebraMap (𝓞 K) K) (pB₁.basis ((finCongr this).symm i)) = (IsPrimitiveRoot.powerBasis ℚ hζ).basis i
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✝¹⁹ : CommRing A
inst✝¹⁸ : Field K
inst✝¹⁷ : CommRing B
inst✝¹⁶ : Field L
inst✝¹⁵ : Algebra A K
inst✝¹⁴ : Algebra B L
inst✝¹³ : Algebra A B
inst✝¹² : Algebra K L
inst✝¹¹ : Algebra A L
inst✝¹⁰ : IsScalarTower A K L
inst✝⁹ : IsScalarTower A B L
inst✝⁸ : IsDomain A
inst✝⁷ : IsFractionRing A K
inst✝⁶ : FiniteDimensional K L
inst✝⁵ : Algebra.IsSeparable K L
inst✝⁴ : IsIntegralClosure B A L
inst✝³ : IsIntegrallyClosed A
inst✝² : IsDedekindDomain B
inst✝¹ : IsTorsionFree A B
inst✝ : IsFractionRing B L
this : ↑↑(FractionRing.algEquiv B L) ∘ₗ Algebra.linearMap B (FractionRing B) = Algebra.linearMap B L
x✝ : A
⊢ (algebraMap A (FractionRing B)) x✝ = (algebraMap B (FractionRing B)) ((algebraMap A B) x✝)

Tactic:
← IsScalarTower.algebraMap_apply

Goal state after:
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝¹⁹ : CommRing A
inst✝¹⁸ : Field K
inst✝¹⁷ : CommRing B
inst✝¹⁶ : Field L
inst✝¹⁵ : Algebra A K
inst✝¹⁴ : Algebra B L
inst✝¹³ : Algebra A B
inst✝¹² : Algebra K L
inst✝¹¹ : Algebra A L
inst✝¹⁰ : IsScalarTower A K L
inst✝⁹ : IsScalarTower A B L
inst✝⁸ : IsDomain A
inst✝⁷ : IsFractionRing A K
inst✝⁶ : FiniteDimensional K L
inst✝⁵ : Algebra.IsSeparable K L
inst✝⁴ : IsIntegralClosure B A L
inst✝³ : IsIntegrallyClosed A
inst✝² : IsDedekindDomain B
inst✝¹ : IsTorsionFree A B
inst✝ : IsFractionRing B L
this : ↑↑(FractionRing.algEquiv B L) ∘ₗ Algebra.linearMap B (FractionRing B) = Algebra.linearMap B L
x✝ : A
⊢ (algebraMap A (FractionRing B)) x✝ = (algebraMap A (FractionRing B)) x✝