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

Goal state before:
pp: A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_6
inst✝¹⁹ : CommRing A
inst✝¹⁸ : CommRing B
inst✝¹⁷ : Algebra A B
inst✝¹⁶ : Field K
inst✝¹⁵ : Field L
inst✝¹⁴ : Algebra A K
inst✝¹³ : IsFractionRing A K
inst✝¹² : Algebra K L
inst✝¹¹ : Algebra A L
inst✝¹⁰ : IsScalarTower A K L
inst✝⁹ : Algebra B L
inst✝⁸ : IsScalarTower A B L
inst✝⁷ : IsIntegralClosure B A L
inst✝⁶ : FiniteDimensional K L
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : IsDomain A
inst✝³ : IsDomain B
inst✝² : IsIntegrallyClosed B
inst✝¹ : Algebra.IsIntegral A B
inst✝ : IsTorsionFree A B
x : B
this : IsFractionRing B L
⊢ (algebraMap A K) ((intNorm A B) x) = (norm K) ((algebraMap B L) x)

Tactic:
apply (FractionRing.algEquiv A K).symm.injective

Goal state after:
A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_6
inst✝¹⁹ : CommRing A
inst✝¹⁸ : CommRing B
inst✝¹⁷ : Algebra A B
inst✝¹⁶ : Field K
inst✝¹⁵ : Field L
inst✝¹⁴ : Algebra A K
inst✝¹³ : IsFractionRing A K
inst✝¹² : Algebra K L
inst✝¹¹ : Algebra A L
inst✝¹⁰ : IsScalarTower A K L
inst✝⁹ : Algebra B L
inst✝⁸ : IsScalarTower A B L
inst✝⁷ : IsIntegralClosure B A L
inst✝⁶ : FiniteDimensional K L
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : IsDomain A
inst✝³ : IsDomain B
inst✝² : IsIntegrallyClosed B
inst✝¹ : Algebra.IsIntegral A B
inst✝ : IsTorsionFree A B
x : B
this : IsFractionRing B L
⊢ (FractionRing.algEquiv A K).symm ((algebraMap A K) ((intNorm A B) x)) =
    (FractionRing.algEquiv A K).symm ((norm K) ((algebraMap B L) x))
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
A : Set (mixedSpace K)
hA : ∀ (x : mixedSpace K), x ∈ A ↔ (fun w ↦ ‖x.1 w‖, x.2) ∈ A
inst✝ : NumberField K
hm : MeasurableSet A
⊢ volume A = 2 ^ nrRealPlaces K * volume (plusPart A)

Tactic:
by
  simp only [← measure_congr (iUnion_negAt_plusPart_ae A hA),
    measure_iUnion (disjoint_negAt_plusPart A) (fun _ ↦ measurableSet_negAt_plusPart _ A hm),
    volume_negAt_plusPart hm, tsum_fintype, sum_const, card_univ, Fintype.card_set, nsmul_eq_mul,
    Nat.cast_pow, Nat.cast_ofNat, nrRealPlaces]

Goal state after:
Module: Mathlib.NumberTheory.RamificationInertia.Galois

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝¹⁶ : CommRing A
inst✝¹⁵ : CommRing B
inst✝¹⁴ : Algebra A B
p : Ideal A
G : Type u_3
inst✝¹³ : Group G
inst✝¹² : MulSemiringAction G B
inst✝¹¹ : SMulCommClass G A B
K : Type u_4
L : Type u_5
inst✝¹⁰ : Field K
inst✝⁹ : Field L
inst✝⁸ : Algebra A K
inst✝⁷ : IsFractionRing A K
inst✝⁶ : Algebra B L
inst✝⁵ : Algebra K L
inst✝⁴ : Algebra A L
inst✝³ : IsScalarTower A B L
inst✝² : IsScalarTower A K L
inst✝¹ : IsIntegralClosure B A L
inst✝ : FiniteDimensional K L
σ τ : Gal(L/K)
Q : ↑(p.primesOver B)
⊢ ↑((σ * τ) • Q) = ↑(σ • τ • Q)

Tactic:
change map _ Q.1 = map _ (map _ Q.1)

Goal state after:
A : Type u_1
B : Type u_2
inst✝¹⁶ : CommRing A
inst✝¹⁵ : CommRing B
inst✝¹⁴ : Algebra A B
p : Ideal A
G : Type u_3
inst✝¹³ : Group G
inst✝¹² : MulSemiringAction G B
inst✝¹¹ : SMulCommClass G A B
K : Type u_4
L : Type u_5
inst✝¹⁰ : Field K
inst✝⁹ : Field L
inst✝⁸ : Algebra A K
inst✝⁷ : IsFractionRing A K
inst✝⁶ : Algebra B L
inst✝⁵ : Algebra K L
inst✝⁴ : Algebra A L
inst✝³ : IsScalarTower A B L
inst✝² : IsScalarTower A K L
inst✝¹ : IsIntegralClosure B A L
inst✝ : FiniteDimensional K L
σ τ : Gal(L/K)
Q : ↑(p.primesOver B)
⊢ map ((galRestrict A K L B) (σ * τ)) ↑Q = map ((galRestrict A K L B) σ) (map ((galRestrict A K L B) τ) ↑Q)
Module: Mathlib.RingTheory.Ideal.Quotient.HasFiniteQuotients

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : HasFiniteQuotients R
B : ℕ
⊢ {I | Submodule.cardQuot I ≤ B}.Finite

Tactic:
classical
  rcases finite_or_infinite R
  · apply Set.toFinite
  -- if `R` is infinite, then we can pick a finite set `s` of cardinality `B + 1`
  obtain ⟨s, hs⟩ := Infinite.exists_subset_card_eq R (B + 1)
  -- and consider the finite set `t` of nonzero differences
  let t := (s - s) \ {0}
  refine Set.Finite.of_sdiff ?_ (Set.finite_singleton ⊥)
  -- in a ring with finite quotients, each nonzero element is contained in only finitely many ideals
  -- so it is enough to show that each ideal `I` of norm at most `B` contains some element of `t`
  suffices {I | Submodule.cardQuot I ≤ B} \ {⊥} ⊆ ⋃ x ∈ t, {I | x ∈ I} from
    (t.finite_toSet.biUnion fun x hx ↦ finite_setOfPred_mem x (by grind)).subset this
  intro I hI
  rw [Set.mem_sdiff, Set.mem_ofPred, Submodule.cardQuot_apply] at hI
  simp_rw [Set.mem_iUnion, exists_prop, Set.mem_ofPred_eq]
  -- `s` has cardinality `B + 1`, but the quotient `R ⧸ I` has cardinality at most `B`
  replace hs : (s.image (Ideal.Quotient.mk I)).card < s.card := by
    have := finiteQuotient hI.2
    have := Fintype.ofFinite (R ⧸ I)
    grw [Finset.card_le_univ, Fintype.card_eq_nat_card, hI.1, hs, Nat.lt_add_one_iff]
  -- so we can find distinct `x, y ∈ s` with the desired collision `x - y ∈ I`
  obtain ⟨x, hx, y, hy, hxy, h⟩ := Finset.exists_ne_map_eq_of_card_image_lt hs
  refine ⟨x - y, ?_, (Submodule.Quotient.eq I).mp h⟩
  refine Finset.mem_sdiff.mpr ⟨Finset.mem_sub.mpr ⟨x, hx, y, hy, rfl⟩, ?_⟩
  rwa [Finset.notMem_singleton, sub_ne_zero]

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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
I : Ideal ℤ
hI : I ≠ ⊥
⊢ Finite (ℤ ⧸ I)

Tactic:
obtain ⟨n, rfl⟩ := Submodule.IsPrincipal.principal I

Goal state after:
R : Type u_1
inst✝ : CommRing R
n : ℤ
hI : ℤ ∙ n ≠ ⊥
⊢ Finite (ℤ ⧸ ℤ ∙ n)
Module: Mathlib.NumberTheory.NumberField.Discriminant.Basic

Goal state before:
pp: case ha
A : Type u_2
inst✝¹ : Field A
inst✝ : CharZero A
N : ℕ
D : ℕ := ⋯
B : ℝ≥0 := ⋯
C : ℕ := ⋯
x✝¹ : { F // FiniteDimensional ℚ ↥F }
K : IntermediateField ℚ A
hK₀ : FiniteDimensional ℚ ↥K
x✝ : ⟨K, hK₀⟩ ∈ {K | {w | w.IsReal}.Nonempty ∧ |discr ↥↑K| ≤ ↑N}
hK₂ : |discr ↥↑⟨K, hK₀⟩| ≤ ↑N
this✝¹ : CharZero ↥K
this✝ : NumberField ↥K
w₀ : InfinitePlace ↥↑⟨K, hK₀⟩
hw₀ : w₀ ∈ {w | w.IsReal}
this : minkowskiBound (↥K) 1 < ↑(convexBodyLTFactor ↥K) * ↑B
x : 𝓞 ↥K
hx₁ : ℚ⟮↑x⟯ = ⊤
hx₂ : ∀ (w : InfinitePlace ↥K), w ↑x < ↑(max B 1)
hx : IsIntegral ℤ ((algebraMap (𝓞 ↥K) ↥K) x)
i : ℕ
⊢ 1 ≤ max (↑B) 1

Tactic:
exact le_max_right _ 1

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

Goal state before:
pp: case ha
A : Type u_2
inst✝¹ : Field A
inst✝ : CharZero A
N : ℕ
D : ℕ := ⋯
B : ℝ≥0 := ⋯
C : ℕ := ⋯
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 : 𝓞 ↥K
hx₁ : ℚ⟮↑x⟯ = ⊤
hx₂ : ∀ (w : InfinitePlace ↥K), w ↑x < √(1 + ↑(boundOfDiscBdd N) ^ 2)
hx : IsIntegral ℤ ((algebraMap (𝓞 ↥K) ↥K) x)
i : ℕ
⊢ 1 ≤ max (√(1 + ↑(boundOfDiscBdd N) ^ 2)) 1

Tactic:
exact le_max_right _ 1

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✝¹⁸ : CommRing A
inst✝¹⁷ : IsDomain A
inst✝¹⁶ : CommRing B
inst✝¹⁵ : IsDomain B
inst✝¹⁴ : Algebra A B
inst✝¹³ : IsTorsionFree A B
inst✝¹² : Field K
inst✝¹¹ : Field L
inst✝¹⁰ : Algebra A K
inst✝⁹ : Algebra B L
inst✝⁸ : IsFractionRing A K
inst✝⁷ : IsFractionRing B L
I : FractionalIdeal A⁰ K
inst✝⁶ : Algebra K L
inst✝⁵ : Algebra A L
inst✝⁴ : IsScalarTower A B L
inst✝³ : IsScalarTower A K L
inst✝² : Algebra.IsIntegral A B
inst✝¹ : IsIntegrallyClosed A
inst✝ : IsIntegrallyClosed B
hI : ∃ x ∈ I, x ∉ 1
⊢ ∃ x ∈ (extendedHom L B) I, x ∉ 1

Tactic:
obtain ⟨x, hx₁, hx₂⟩ := hI

Goal state after:
A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝¹⁸ : CommRing A
inst✝¹⁷ : IsDomain A
inst✝¹⁶ : CommRing B
inst✝¹⁵ : IsDomain B
inst✝¹⁴ : Algebra A B
inst✝¹³ : IsTorsionFree A B
inst✝¹² : Field K
inst✝¹¹ : Field L
inst✝¹⁰ : Algebra A K
inst✝⁹ : Algebra B L
inst✝⁸ : IsFractionRing A K
inst✝⁷ : IsFractionRing B L
I : FractionalIdeal A⁰ K
inst✝⁶ : Algebra K L
inst✝⁵ : Algebra A L
inst✝⁴ : IsScalarTower A B L
inst✝³ : IsScalarTower A K L
inst✝² : Algebra.IsIntegral A B
inst✝¹ : IsIntegrallyClosed A
inst✝ : IsIntegrallyClosed B
x : K
hx₁ : x ∈ I
hx₂ : x ∉ 1
⊢ ∃ x ∈ (extendedHom L B) I, x ∉ 1
Module: Mathlib.RingTheory.DedekindDomain.PID

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
A : Type u_2
inst✝² : CommRing A
inst✝¹ : Algebra R A
S : Submonoid R
inst✝ : IsLocalization S A
hS : S ≤ R⁰
hf : {I | I.IsMaximal}.Finite
I I' : FractionalIdeal S A
hinv : ↑I * ↑I' = ↑1
hinv' : I * I' = 1
s : Finset (Ideal R) := hf.toFinset
this : DecidableEq (Ideal R)
coprime : ∀ M ∈ s, ∀ M' ∈ s.erase M, M ⊔ M' = ⊤
⊢ (↑I).IsPrincipal

Tactic:
have nle : ∀ M ∈ s, ¬⨅ M' ∈ s.erase M, M' ≤ M := fun M hM =>
    left_lt_sup.1
      ((hf.mem_toFinset.1 hM).lt_top.trans_eq (Ideal.sup_iInf_eq_top <| coprime M hM).symm)

Goal state after:
R : Type u_1
inst✝³ : CommRing R
A : Type u_2
inst✝² : CommRing A
inst✝¹ : Algebra R A
S : Submonoid R
inst✝ : IsLocalization S A
hS : S ≤ R⁰
hf : {I | I.IsMaximal}.Finite
I I' : FractionalIdeal S A
hinv : ↑I * ↑I' = ↑1
hinv' : I * I' = 1
s : Finset (Ideal R) := hf.toFinset
this : DecidableEq (Ideal R)
coprime : ∀ M ∈ s, ∀ M' ∈ s.erase M, M ⊔ M' = ⊤
nle : ∀ M ∈ s, ¬⨅ M' ∈ s.erase M, M' ≤ M
⊢ (↑I).IsPrincipal
Module: Mathlib.NumberTheory.NumberField.Discriminant.Different

Goal state before:
pp: case inl
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
this : ↑(Ideal.absNorm (differentIdeal ℤ 𝒪)) ∈ differentIdeal ℤ 𝒪
h : discr K = ↑(discr K).natAbs
⊢ ↑(discr K) ∈ differentIdeal ℤ 𝒪

Tactic:
| inl h =>

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

Goal state before:
pp: L : Type u_3
inst✝² : Field L
inst✝¹ : NumberField L
K₁ K₂ : IntermediateField ℚ L
inst✝ : IsGalois ℚ ↥K₁
h : IsCoprime (discr ↥K₁) (discr ↥K₂)
this : IsUnit (discr ↥(K₁ ⊓ K₂))
⊢ K₁ ⊓ K₂ = ⊥

Tactic:
contrapose! this

Goal state after:
L : Type u_3
inst✝² : Field L
inst✝¹ : NumberField L
K₁ K₂ : IntermediateField ℚ L
inst✝ : IsGalois ℚ ↥K₁
h : IsCoprime (discr ↥K₁) (discr ↥K₂)
this : K₁ ⊓ K₂ ≠ ⊥
⊢ ¬IsUnit (discr ↥(K₁ ⊓ K₂))
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.inv • ↑↑u * ↑p ^ n) • x ∈ ℤ[B.gen]
⊢ x ∈ ℤ[ζ]

Tactic:
Units.inv_eq_val_inv,

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.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: K : Type u
inst✝² : Field K
ζ : K
inst✝¹ : CharZero K
this : NumberField K
n✝ : ℕ
inst✝ : IsCyclotomicExtension {2 ^ (n✝ + 1 + 1)} ℚ K
hζ : IsPrimitiveRoot ζ (2 ^ (n✝ + 1 + 1))
⊢ (Algebra.norm ℚ) ((algebraMap (𝓞 K) K) (hζ.toInteger - 1)) = (algebraMap ℤ ℚ) 2

Tactic:
algebraMap_int_eq

Goal state after:
K : Type u
inst✝² : Field K
ζ : K
inst✝¹ : CharZero K
this : NumberField K
n✝ : ℕ
inst✝ : IsCyclotomicExtension {2 ^ (n✝ + 1 + 1)} ℚ K
hζ : IsPrimitiveRoot ζ (2 ^ (n✝ + 1 + 1))
⊢ (Algebra.norm ℚ) ((algebraMap (𝓞 K) K) (hζ.toInteger - 1)) = (Int.castRingHom ℚ) 2
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: p : ℕ
K : Type u
inst✝¹ : Field K
ζ : K
hp : Fact (Nat.Prime p)
inst✝ : CharZero K
hcycl : IsCyclotomicExtension {p} ℚ K
hζ : IsPrimitiveRoot ζ p
hodd : p ≠ 2
⊢ ¬∃ n, ↑p ∣ hζ.toInteger - ↑n

Tactic:
have : IsCyclotomicExtension {p ^ (0 + 1)} ℚ K := by simpa using hcycl
  replace hζ : IsPrimitiveRoot ζ (p ^ (0 + 1)) := by simpa using hζ
  exact not_exists_int_prime_dvd_sub_of_prime_ne_two hζ hodd

Goal state after:
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 J : FractionalIdeal B⁰ L
⊢ dual A K 1 * J⁻¹ / (dual A K 1 * I⁻¹) = I / J

Tactic:
mul_div_mul_comm,

Goal state after:
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 J : FractionalIdeal B⁰ L
⊢ dual A K 1 / dual A K 1 * (J⁻¹ / I⁻¹) = I / J
Module: Mathlib.RingTheory.DedekindDomain.Different

Goal state before:
pp: case refine_1
A : Type u_1
K : Type u_2
L : Type u
B : Type u_5
inst✝⁴¹ : CommRing A
inst✝⁴⁰ : Field K
inst✝³⁹ : CommRing B
inst✝³⁸ : Field L
inst✝³⁷ : Algebra A K
inst✝³⁶ : Algebra B L
inst✝³⁵ : Algebra A B
inst✝³⁴ : Algebra K L
inst✝³³ : Algebra A L
inst✝³² : IsScalarTower A K L
inst✝³¹ : IsScalarTower A B L
inst✝³⁰ : IsDomain A
inst✝²⁹ : IsFractionRing A K
inst✝²⁸ : FiniteDimensional K L
inst✝²⁷ : Algebra.IsSeparable K L
inst✝²⁶ : IsIntegralClosure B A L
inst✝²⁵ : IsFractionRing B L
inst✝²⁴ : IsIntegrallyClosed A
inst✝²³ : IsDedekindDomain B
C : Type u_3
M : Type u_4
inst✝²² : CommRing C
inst✝²¹ : IsDedekindDomain C
inst✝²⁰ : Field M
inst✝¹⁹ : Algebra C M
inst✝¹⁸ : IsFractionRing C M
inst✝¹⁷ : Algebra A C
inst✝¹⁶ : Algebra B C
inst✝¹⁵ : Algebra A M
inst✝¹⁴ : Algebra B M
inst✝¹³ : Algebra K M
inst✝¹² : Algebra L M
inst✝¹¹ : IsScalarTower A C M
inst✝¹⁰ : IsScalarTower A K M
inst✝⁹ : IsScalarTower B C M
inst✝⁸ : IsScalarTower B L M
inst✝⁷ : IsScalarTower K L M
inst✝⁶ : IsIntegralClosure C A M
inst✝⁵ : FiniteDimensional K M
inst✝⁴ : FiniteDimensional L M
inst✝³ : Algebra.IsSeparable K M
inst✝² : IsIntegralClosure C B M
inst✝¹ : Algebra.IsSeparable L M
inst✝ : IsTorsionFree B C
this : IsLocalization (algebraMapSubmonoid C B⁰) M
h : B⁰ ≤ Submonoid.comap (algebraMap B C) C⁰
h_alg : ∀ {x : L}, (algebraMap L M) x = (IsLocalization.map M (algebraMap B C) h) x
x : M
hx : x ∈ dual A K 1
⊢ (C ∙ x) * span C (⇑(IsLocalization.map M (algebraMap B C) ⋯) '' ↑(dual A K 1)⁻¹) ≤ traceDual B L 1

Tactic:
span_mul_span,

Goal state after:
case refine_1
A : Type u_1
K : Type u_2
L : Type u
B : Type u_5
inst✝⁴¹ : CommRing A
inst✝⁴⁰ : Field K
inst✝³⁹ : CommRing B
inst✝³⁸ : Field L
inst✝³⁷ : Algebra A K
inst✝³⁶ : Algebra B L
inst✝³⁵ : Algebra A B
inst✝³⁴ : Algebra K L
inst✝³³ : Algebra A L
inst✝³² : IsScalarTower A K L
inst✝³¹ : IsScalarTower A B L
inst✝³⁰ : IsDomain A
inst✝²⁹ : IsFractionRing A K
inst✝²⁸ : FiniteDimensional K L
inst✝²⁷ : Algebra.IsSeparable K L
inst✝²⁶ : IsIntegralClosure B A L
inst✝²⁵ : IsFractionRing B L
inst✝²⁴ : IsIntegrallyClosed A
inst✝²³ : IsDedekindDomain B
C : Type u_3
M : Type u_4
inst✝²² : CommRing C
inst✝²¹ : IsDedekindDomain C
inst✝²⁰ : Field M
inst✝¹⁹ : Algebra C M
inst✝¹⁸ : IsFractionRing C M
inst✝¹⁷ : Algebra A C
inst✝¹⁶ : Algebra B C
inst✝¹⁵ : Algebra A M
inst✝¹⁴ : Algebra B M
inst✝¹³ : Algebra K M
inst✝¹² : Algebra L M
inst✝¹¹ : IsScalarTower A C M
inst✝¹⁰ : IsScalarTower A K M
inst✝⁹ : IsScalarTower B C M
inst✝⁸ : IsScalarTower B L M
inst✝⁷ : IsScalarTower K L M
inst✝⁶ : IsIntegralClosure C A M
inst✝⁵ : FiniteDimensional K M
inst✝⁴ : FiniteDimensional L M
inst✝³ : Algebra.IsSeparable K M
inst✝² : IsIntegralClosure C B M
inst✝¹ : Algebra.IsSeparable L M
inst✝ : IsTorsionFree B C
this : IsLocalization (algebraMapSubmonoid C B⁰) M
h : B⁰ ≤ Submonoid.comap (algebraMap B C) C⁰
h_alg : ∀ {x : L}, (algebraMap L M) x = (IsLocalization.map M (algebraMap B C) h) x
x : M
hx : x ∈ dual A K 1
⊢ span C ({x} * ⇑(IsLocalization.map M (algebraMap B C) ⋯) '' ↑(dual A K 1)⁻¹) ≤ traceDual B L 1
Module: Mathlib.RingTheory.DedekindDomain.Different

Goal state before:
pp: case refine_1
A : Type u_1
K : Type u_2
L : Type u
B : Type u_5
inst✝⁴¹ : CommRing A
inst✝⁴⁰ : Field K
inst✝³⁹ : CommRing B
inst✝³⁸ : Field L
inst✝³⁷ : Algebra A K
inst✝³⁶ : Algebra B L
inst✝³⁵ : Algebra A B
inst✝³⁴ : Algebra K L
inst✝³³ : Algebra A L
inst✝³² : IsScalarTower A K L
inst✝³¹ : IsScalarTower A B L
inst✝³⁰ : IsDomain A
inst✝²⁹ : IsFractionRing A K
inst✝²⁸ : FiniteDimensional K L
inst✝²⁷ : Algebra.IsSeparable K L
inst✝²⁶ : IsIntegralClosure B A L
inst✝²⁵ : IsFractionRing B L
inst✝²⁴ : IsIntegrallyClosed A
inst✝²³ : IsDedekindDomain B
C : Type u_3
M : Type u_4
inst✝²² : CommRing C
inst✝²¹ : IsDedekindDomain C
inst✝²⁰ : Field M
inst✝¹⁹ : Algebra C M
inst✝¹⁸ : IsFractionRing C M
inst✝¹⁷ : Algebra A C
inst✝¹⁶ : Algebra B C
inst✝¹⁵ : Algebra A M
inst✝¹⁴ : Algebra B M
inst✝¹³ : Algebra K M
inst✝¹² : Algebra L M
inst✝¹¹ : IsScalarTower A C M
inst✝¹⁰ : IsScalarTower A K M
inst✝⁹ : IsScalarTower B C M
inst✝⁸ : IsScalarTower B L M
inst✝⁷ : IsScalarTower K L M
inst✝⁶ : IsIntegralClosure C A M
inst✝⁵ : FiniteDimensional K M
inst✝⁴ : FiniteDimensional L M
inst✝³ : Algebra.IsSeparable K M
inst✝² : IsIntegralClosure C B M
inst✝¹ : Algebra.IsSeparable L M
inst✝ : IsTorsionFree B C
this : IsLocalization (algebraMapSubmonoid C B⁰) M
h : B⁰ ≤ Submonoid.comap (algebraMap B C) C⁰
h_alg : ∀ {x : L}, (algebraMap L M) x = (IsLocalization.map M (algebraMap B C) h) x
x : M
hx : x ∈ dual A K 1
a : L
ha : a ∈ ↑(dual A K 1)⁻¹
m : C
⊢ a * (Algebra.trace L M) ((algebraMap C M) m * x) ∈ ↑1

Tactic:
refine (mem_inv_iff (by simp)).mp ha _ (trace_mem_dual_one A K L B C M _ ?_)

Goal state after:
case refine_1
A : Type u_1
K : Type u_2
L : Type u
B : Type u_5
inst✝⁴¹ : CommRing A
inst✝⁴⁰ : Field K
inst✝³⁹ : CommRing B
inst✝³⁸ : Field L
inst✝³⁷ : Algebra A K
inst✝³⁶ : Algebra B L
inst✝³⁵ : Algebra A B
inst✝³⁴ : Algebra K L
inst✝³³ : Algebra A L
inst✝³² : IsScalarTower A K L
inst✝³¹ : IsScalarTower A B L
inst✝³⁰ : IsDomain A
inst✝²⁹ : IsFractionRing A K
inst✝²⁸ : FiniteDimensional K L
inst✝²⁷ : Algebra.IsSeparable K L
inst✝²⁶ : IsIntegralClosure B A L
inst✝²⁵ : IsFractionRing B L
inst✝²⁴ : IsIntegrallyClosed A
inst✝²³ : IsDedekindDomain B
C : Type u_3
M : Type u_4
inst✝²² : CommRing C
inst✝²¹ : IsDedekindDomain C
inst✝²⁰ : Field M
inst✝¹⁹ : Algebra C M
inst✝¹⁸ : IsFractionRing C M
inst✝¹⁷ : Algebra A C
inst✝¹⁶ : Algebra B C
inst✝¹⁵ : Algebra A M
inst✝¹⁴ : Algebra B M
inst✝¹³ : Algebra K M
inst✝¹² : Algebra L M
inst✝¹¹ : IsScalarTower A C M
inst✝¹⁰ : IsScalarTower A K M
inst✝⁹ : IsScalarTower B C M
inst✝⁸ : IsScalarTower B L M
inst✝⁷ : IsScalarTower K L M
inst✝⁶ : IsIntegralClosure C A M
inst✝⁵ : FiniteDimensional K M
inst✝⁴ : FiniteDimensional L M
inst✝³ : Algebra.IsSeparable K M
inst✝² : IsIntegralClosure C B M
inst✝¹ : Algebra.IsSeparable L M
inst✝ : IsTorsionFree B C
this : IsLocalization (algebraMapSubmonoid C B⁰) M
h : B⁰ ≤ Submonoid.comap (algebraMap B C) C⁰
h_alg : ∀ {x : L}, (algebraMap L M) x = (IsLocalization.map M (algebraMap B C) h) x
x : M
hx : x ∈ dual A K 1
a : L
ha : a ∈ ↑(dual A K 1)⁻¹
m : C
⊢ (algebraMap C M) m * x ∈ dual A K 1
Module: Mathlib.RingTheory.DedekindDomain.Different

Goal state before:
pp: case refine_2
A : Type u_1
K : Type u_2
L : Type u
B : Type u_5
inst✝⁴¹ : CommRing A
inst✝⁴⁰ : Field K
inst✝³⁹ : CommRing B
inst✝³⁸ : Field L
inst✝³⁷ : Algebra A K
inst✝³⁶ : Algebra B L
inst✝³⁵ : Algebra A B
inst✝³⁴ : Algebra K L
inst✝³³ : Algebra A L
inst✝³² : IsScalarTower A K L
inst✝³¹ : IsScalarTower A B L
inst✝³⁰ : IsDomain A
inst✝²⁹ : IsFractionRing A K
inst✝²⁸ : FiniteDimensional K L
inst✝²⁷ : Algebra.IsSeparable K L
inst✝²⁶ : IsIntegralClosure B A L
inst✝²⁵ : IsFractionRing B L
inst✝²⁴ : IsIntegrallyClosed A
inst✝²³ : IsDedekindDomain B
C : Type u_3
M : Type u_4
inst✝²² : CommRing C
inst✝²¹ : IsDedekindDomain C
inst✝²⁰ : Field M
inst✝¹⁹ : Algebra C M
inst✝¹⁸ : IsFractionRing C M
inst✝¹⁷ : Algebra A C
inst✝¹⁶ : Algebra B C
inst✝¹⁵ : Algebra A M
inst✝¹⁴ : Algebra B M
inst✝¹³ : Algebra K M
inst✝¹² : Algebra L M
inst✝¹¹ : IsScalarTower A C M
inst✝¹⁰ : IsScalarTower A K M
inst✝⁹ : IsScalarTower B C M
inst✝⁸ : IsScalarTower B L M
inst✝⁷ : IsScalarTower K L M
inst✝⁶ : IsIntegralClosure C A M
inst✝⁵ : FiniteDimensional K M
inst✝⁴ : FiniteDimensional L M
inst✝³ : Algebra.IsSeparable K M
inst✝² : IsIntegralClosure C B M
inst✝¹ : Algebra.IsSeparable L M
inst✝ : IsTorsionFree B C
this : IsLocalization (algebraMapSubmonoid C B⁰) M
h : B⁰ ≤ Submonoid.comap (algebraMap B C) C⁰
h_alg : ∀ {x : L}, (algebraMap L M) x = (IsLocalization.map M (algebraMap B C) h) x
⊢ span C ↑↑(dual B L 1) * span C (⇑(IsLocalization.map M (algebraMap B C) ⋯) '' ↑(dual A K 1)) ≤ ↑(dual A K 1)

Tactic:
span_mul_span,

Goal state after:
case refine_2
A : Type u_1
K : Type u_2
L : Type u
B : Type u_5
inst✝⁴¹ : CommRing A
inst✝⁴⁰ : Field K
inst✝³⁹ : CommRing B
inst✝³⁸ : Field L
inst✝³⁷ : Algebra A K
inst✝³⁶ : Algebra B L
inst✝³⁵ : Algebra A B
inst✝³⁴ : Algebra K L
inst✝³³ : Algebra A L
inst✝³² : IsScalarTower A K L
inst✝³¹ : IsScalarTower A B L
inst✝³⁰ : IsDomain A
inst✝²⁹ : IsFractionRing A K
inst✝²⁸ : FiniteDimensional K L
inst✝²⁷ : Algebra.IsSeparable K L
inst✝²⁶ : IsIntegralClosure B A L
inst✝²⁵ : IsFractionRing B L
inst✝²⁴ : IsIntegrallyClosed A
inst✝²³ : IsDedekindDomain B
C : Type u_3
M : Type u_4
inst✝²² : CommRing C
inst✝²¹ : IsDedekindDomain C
inst✝²⁰ : Field M
inst✝¹⁹ : Algebra C M
inst✝¹⁸ : IsFractionRing C M
inst✝¹⁷ : Algebra A C
inst✝¹⁶ : Algebra B C
inst✝¹⁵ : Algebra A M
inst✝¹⁴ : Algebra B M
inst✝¹³ : Algebra K M
inst✝¹² : Algebra L M
inst✝¹¹ : IsScalarTower A C M
inst✝¹⁰ : IsScalarTower A K M
inst✝⁹ : IsScalarTower B C M
inst✝⁸ : IsScalarTower B L M
inst✝⁷ : IsScalarTower K L M
inst✝⁶ : IsIntegralClosure C A M
inst✝⁵ : FiniteDimensional K M
inst✝⁴ : FiniteDimensional L M
inst✝³ : Algebra.IsSeparable K M
inst✝² : IsIntegralClosure C B M
inst✝¹ : Algebra.IsSeparable L M
inst✝ : IsTorsionFree B C
this : IsLocalization (algebraMapSubmonoid C B⁰) M
h : B⁰ ≤ Submonoid.comap (algebraMap B C) C⁰
h_alg : ∀ {x : L}, (algebraMap L M) x = (IsLocalization.map M (algebraMap B C) h) x
⊢ span C (↑↑(dual B L 1) * ⇑(IsLocalization.map M (algebraMap B C) ⋯) '' ↑(dual A K 1)) ≤ ↑(dual A K 1)
Module: Mathlib.NumberTheory.NumberField.Units.DirichletTheorem

Goal state before:
pp: case inl
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
r : ℝ
hr : r < 0
⊢ (↑(unitLattice K) ∩ Metric.closedBall 0 r).Finite

pp: case inr
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
r : ℝ
hr : 0 ≤ r
⊢ (↑(unitLattice K) ∩ Metric.closedBall 0 r).Finite

Tactic:
· convert! Set.finite_empty
    rw [Metric.closedBall_eq_empty.mpr hr]
    exact Set.inter_empty _

Goal state after:
case inr
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
r : ℝ
hr : 0 ≤ r
⊢ (↑(unitLattice K) ∩ Metric.closedBall 0 r).Finite