mathlib-initiative/mathlib-tactics huggingface dataset:
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 : IsIntegrallyClosed A
inst✝¹ : IsDedekindDomain B
inst✝ : IsTorsionFree A B
x : B
hx : K[(algebraMap B L) x] = ⊤
hAx : IsIntegral A x
this : IsFractionRing B L
y : L
hne₁ : (aeval ((algebraMap B L) x)) (derivative (minpoly K ((algebraMap B L) x))) ≠ 0
⊢ (algebraMap B L) ((aeval x) (derivative (minpoly A x))) ≠ 0

Tactic:
by

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

Goal state before:
pp: n₁ n₂ : ℕ
hn₁ : 1 < n₁
hn₂ : 1 < n₂
h : n₁.Coprime n₂
K : Type u
inst✝¹ : Field K
inst✝ : CharZero K
hn : NeZero (n₁ * n₂)
hK : IsCyclotomicExtension {n₁ * n₂} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ (n₁ * n₂)
this✝³ : NumberField K
this✝² : NeZero n₁
this✝¹ : NeZero n₂
this✝ : IsCyclotomicExtension {n₁} ℚ ↥ℚ⟮ζ ^ n₂⟯
hζ₁ : IsPrimitiveRoot (AdjoinSimple.gen ℚ (ζ ^ n₂)) n₁
hK₁ : ℤ[hζ₁.toInteger] = ⊤
this : IsCyclotomicExtension {n₂} ℚ ↥ℚ⟮ζ ^ n₁⟯
hζ₂ : IsPrimitiveRoot (AdjoinSimple.gen ℚ (ζ ^ n₁)) n₂
hK₂ : ℤ[hζ₂.toInteger] = ⊤
⊢ ℚ⟮ζ ^ n₂⟯ ⊔ ℚ⟮ζ ^ n₁⟯ = ⊤

Tactic:
by

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

Goal state before:
pp: K : Type u
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {5} ℚ K
this : Fact (Nat.Prime 5)
⊢ 0 < 5

Tactic:
by

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

Goal state before:
pp: K : Type u
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {5} ℚ K
this✝ : Fact (Nat.Prime 5)
this : (2 * (3 ^ 2 / 4 ^ 2) * (4 ^ 4 / 24)) ^ 2 < (2 * (π ^ 2 / 4 ^ 2) * (4 ^ 4 / 24)) ^ 2
⊢ 125 < (2 * (3 ^ 2 / 4 ^ 2) * (4 ^ 4 / 24)) ^ 2

Tactic:
by

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

Goal state before:
pp: K : Type u
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {5} ℚ K
⊢ IsPrincipalIdealRing (𝓞 K)

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ (fun w ↦ ↑(↑w).mult * Real.log (↑w ((algebraMap (𝓞 K) K) ↑(Additive.toMul 0)))) = 0

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x✝¹ x✝ : Additive (𝓞 K)ˣ
⊢ (fun w ↦ ↑(↑w).mult * Real.log (↑w ((algebraMap (𝓞 K) K) ↑(Additive.toMul (x✝¹ + x✝))))) =
    (fun w ↦ ↑(↑w).mult * Real.log (↑w ((algebraMap (𝓞 K) K) ↑(Additive.toMul x✝¹)))) + fun w ↦
      ↑(↑w).mult * Real.log (↑w ((algebraMap (𝓞 K) K) ↑(Additive.toMul x✝)))

Tactic:
by

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

Goal state before:
pp: n : ℕ
K : Type u
inst✝¹ : Field K
inst✝ : CharZero K
hn : NeZero n
hK : IsCyclotomicExtension {n} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ n
⊢ ℤ[hζ.toInteger] = ⊤

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : (𝓞 K)ˣ
⊢ ∑ w, (logEmbedding K) (Additive.ofMul x) w = -↑w₀.mult * Real.log (w₀ ((algebraMap (𝓞 K) K) ↑x))

Tactic:
by

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

Goal state before:
pp: n : ℕ
K : Type u
inst✝¹ : Field K
inst✝ : CharZero K
hn : NeZero n
ζ : K
hcycl : IsCyclotomicExtension {n} ℚ K
hζ : IsPrimitiveRoot ζ n
⊢ IsIntegralClosure ↥ℤ[ζ] ℤ K

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝ : Field K
x : (𝓞 K)ˣ
w : InfinitePlace K
⊢ ↑w.mult * Real.log (w ((algebraMap (𝓞 K) K) ↑x)) = 0 ↔ w ((algebraMap (𝓞 K) K) ↑x) = 1

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : (𝓞 K)ˣ
h : (logEmbedding K) (Additive.ofMul x) = 0
w : InfinitePlace K
hw : w = w₀
this : -↑w₀.mult * Real.log (w₀ ((algebraMap (𝓞 K) K) ↑x)) = 0
⊢ w ((algebraMap (𝓞 K) K) ↑x) = 1

Tactic:
by

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 : IsIntegrallyClosed A
inst✝¹ : IsDedekindDomain B
inst✝ : IsTorsionFree A B
x : B
hx : K[(algebraMap B L) x] = ⊤
hAx : IsIntegral A x
this✝ : IsFractionRing B L
y : L
hne₁ : (aeval ((algebraMap B L) x)) (derivative (minpoly K ((algebraMap B L) x))) ≠ 0
this : (algebraMap B L) ((aeval x) (derivative (minpoly A x))) ≠ 0
⊢ ((aeval ((algebraMap B L) x)) (derivative (minpoly K ((algebraMap B L) x))))⁻¹ ≠ 0

Tactic:
by

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

Goal state before:
pp: n : ℕ
hn : NeZero n
⊢ IsIntegralClosure (CyclotomicRing n ℤ ℚ) ℤ (CyclotomicField n ℚ)

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : (𝓞 K)ˣ
⊢ (logEmbedding K) (Additive.ofMul x) = 0 ↔ x ∈ torsion K

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ (logEmbedding K).ker = Subgroup.toAddSubgroup (torsion K)

Tactic:
by

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
P₁ : LinearIndependent R₁ v
⊢ ⊤ ≤ span R₁ (Set.range v)

Tactic:
by

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

Goal state before:
pp: n : ℕ
K : Type u
inst✝³ : Field K
ζ : K
inst✝² : NeZero n
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {n} ℚ K
hζ : IsPrimitiveRoot ζ n
⊢ hζ.integralPowerBasis.dim = φ n

Tactic:
by

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)
⊢ AddSubgroup.map (logEmbedding K) (s ⊔ Subgroup.toAddSubgroup (torsion K)) = AddSubgroup.map (logEmbedding K) s

Tactic:
by

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

Goal state before:
pp: p k n : ℕ
K : Type u
inst✝³ : Field K
ζ : K
hp : Fact (Nat.Prime p)
inst✝² : NeZero n
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {n} ℚ K
hζ : IsPrimitiveRoot ζ n
⊢ ℤ[⟨ζ - 1, ⋯⟩] = ⊤

Tactic:
by

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

Goal state before:
pp: n : ℕ
K : Type u
inst✝³ : Field K
ζ : K
inst✝² : NeZero n
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {n} ℚ K
hζ : IsPrimitiveRoot ζ n
⊢ hζ.subOneIntegralPowerBasis.gen = ⟨ζ - 1, ⋯⟩

Tactic:
by

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
i : ι
⊢ (ofIsCoprimeDifferentIdeal A B R₁ R₂ h₁ h₂ h₃ b) i = (algebraMap R₂ B) (b i)

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
r : ℝ
x : (𝓞 K)ˣ
hr : 0 ≤ r
h : ‖(logEmbedding K) x‖ ≤ r
w : { w // w ≠ w₀ }
⊢ |(logEmbedding K) (Additive.ofMul x) w| ≤ r

Tactic:
by

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

Goal state before:
pp: n : ℕ
K : Type u
inst✝¹ : Field K
inst✝ : NeZero n
ζ : K
hζ : IsPrimitiveRoot ζ n
⊢ n ∣ torsionOrder K

Tactic:
by

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

Goal state before:
pp: n : ℕ
K : Type u
inst✝² : Field K
inst✝¹ : NeZero n
inst✝ : NumberField K
hK : IsCyclotomicExtension {n} ℚ K
hζ : IsPrimitiveRoot (zeta n ℚ K) n
⊢ ∃ μ, orderOf μ = torsionOrder K

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
r : ℝ
x✝ : (𝓞 K)ˣ
hr : 0 ≤ r
h : ‖(logEmbedding K) (Additive.ofMul x✝)‖ ≤ r
w : InfinitePlace K
x : ℝ
hx : 0 ≤ x
⊢ x ≤ ↑w.mult * x

Tactic:
by

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 : IsIntegrallyClosed A
inst✝¹ : IsDedekindDomain B
inst✝ : IsTorsionFree A B
x : B
hx : K[(algebraMap B L) x] = ⊤
⊢ conductor A x * differentIdeal A B = Ideal.span {(aeval x) (derivative (minpoly A x))}

Tactic:
by

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

Goal state before:
pp: n : ℕ
K : Type u
inst✝² : Field K
inst✝¹ : NeZero n
inst✝ : NumberField K
hK : IsCyclotomicExtension {n} ℚ K
hζ : IsPrimitiveRoot (zeta n ℚ K) n
μ : ↥(torsion K)
hμ : IsPrimitiveRoot ((algebraMap (𝓞 K) K) ↑↑μ) (torsionOrder K)
h :
  IsPrimitiveRoot
    (zeta n ℚ K ^ (n / n.factorizationLCMLeft (torsionOrder K)) *
      (algebraMap (𝓞 K) K) ↑↑μ ^ (torsionOrder K / n.factorizationLCMRight (torsionOrder K)))
    (n.lcm (torsionOrder K))
this✝ : NeZero (n.lcm (torsionOrder K))
this : IsCyclotomicExtension ({n.lcm (torsionOrder K)} ∪ {n}) ℚ K
⊢ n.lcm (torsionOrder K) ∈ {n.lcm (torsionOrder K)}

Tactic:
by

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

Goal state before:
pp: n : ℕ
K : Type u
inst✝² : Field K
inst✝¹ : NeZero n
inst✝ : NumberField K
hK : IsCyclotomicExtension {n} ℚ K
hζ : IsPrimitiveRoot (zeta n ℚ K) n
μ : ↥(torsion K)
hμ : IsPrimitiveRoot ((algebraMap (𝓞 K) K) ↑↑μ) (torsionOrder K)
h :
  IsPrimitiveRoot
    (zeta n ℚ K ^ (n / n.factorizationLCMLeft (torsionOrder K)) *
      (algebraMap (𝓞 K) K) ↑↑μ ^ (torsionOrder K / n.factorizationLCMRight (torsionOrder K)))
    (n.lcm (torsionOrder K))
this : NeZero (n.lcm (torsionOrder K))
⊢ IsCyclotomicExtension {n.lcm (torsionOrder K)} ℚ K

Tactic:
by

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 : IsIntegrallyClosed A
inst✝¹ : IsDedekindDomain B
inst✝ : IsTorsionFree A B
x : B
hx : K[(algebraMap B L) x] = ⊤
⊢ (aeval x) (derivative (minpoly A x)) ∈ differentIdeal A B

Tactic:
by

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✝⁵⁵ : CommRing A
inst✝⁵⁴ : Field K
inst✝⁵³ : Algebra A K
inst✝⁵² : IsFractionRing A K
inst✝⁵¹ : CommRing B
inst✝⁵⁰ : Field L
inst✝⁴⁹ : Algebra B L
inst✝⁴⁸ : Algebra A L
inst✝⁴⁷ : Algebra K L
inst✝⁴⁶ : FiniteDimensional K L
inst✝⁴⁵ : IsScalarTower A K L
R₁ : Type u_5
R₂ : Type u_6
inst✝⁴⁴ : CommRing R₁
inst✝⁴³ : CommRing R₂
inst✝⁴² : IsDomain R₁
inst✝⁴¹ : Algebra A R₁
inst✝⁴⁰ : Algebra A R₂
inst✝³⁹ : Algebra R₁ B
inst✝³⁸ : Algebra R₂ B
inst✝³⁷ : Algebra R₁ L
inst✝³⁶ : Algebra R₂ L
inst✝³⁵ : IsScalarTower A R₁ L
inst✝³⁴ : IsScalarTower R₁ B L
inst✝³³ : IsScalarTower R₂ B L
inst✝³² : Module.Finite A R₂
F₁ F₂ : IntermediateField K L
inst✝³¹ : Algebra R₁ ↥F₁
inst✝³⁰ : Algebra R₂ ↥F₂
inst✝²⁹ : IsTorsionFree R₁ ↥F₁
inst✝²⁸ : IsScalarTower A (↥F₂) L
inst✝²⁷ : IsScalarTower A R₂ ↥F₂
inst✝²⁶ : IsScalarTower R₁ (↥F₁) L
inst✝²⁵ : IsScalarTower R₂ (↥F₂) L
inst✝²⁴ : Algebra.IsSeparable K ↥F₂
inst✝²³ : Algebra.IsSeparable (↥F₁) L
inst✝²² : IsDomain A
inst✝²¹ : IsDedekindDomain B
inst✝²⁰ : IsDedekindDomain R₁
inst✝¹⁹ : IsDedekindDomain R₂
inst✝¹⁸ : IsFractionRing B L
inst✝¹⁷ : IsFractionRing R₁ ↥F₁
inst✝¹⁶ : IsFractionRing R₂ ↥F₂
inst✝¹⁵ : IsIntegrallyClosed A
inst✝¹⁴ : IsIntegralClosure B R₁ L
inst✝¹³ : IsTorsionFree R₁ B
inst✝¹² : IsTorsionFree R₂ B
inst✝¹¹ : Algebra A B
inst✝¹⁰ : Module.Finite A B
inst✝⁹ : IsTorsionFree A B
inst✝⁸ : IsTorsionFree A R₁
inst✝⁷ : IsTorsionFree A R₂
inst✝⁶ : Module.Finite A R₁
inst✝⁵ : Module.Finite R₂ B
inst✝⁴ : IsScalarTower A R₂ B
inst✝³ : Module.Finite R₁ B
inst✝² : Algebra.IsSeparable (FractionRing A) (FractionRing B)
inst✝¹ : 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))
inst✝ : Free A R₂
B₁ : Basis (Free.ChooseBasisIndex A R₂) R₁ B :=
  Basis.ofIsCoprimeDifferentIdeal A B R₁ R₂ h₁ h₂ h₃ (Free.chooseBasis A R₂)
x : B
i : Free.ChooseBasisIndex A R₂
x✝ : i ∈ Finset.univ
⊢ (algebraMap R₁ B) ((B₁.repr x) i) ∈ (IsScalarTower.toAlgHom A R₁ B).range

Tactic:
by

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✝²¹ : CommRing A
inst✝²⁰ : Field K
inst✝¹⁹ : CommRing B
inst✝¹⁸ : Field L
inst✝¹⁷ : Algebra A K
inst✝¹⁶ : Algebra B L
inst✝¹⁵ : Algebra A B
inst✝¹⁴ : Algebra K L
inst✝¹³ : Algebra A L
inst✝¹² : IsScalarTower A K L
inst✝¹¹ : IsScalarTower A B L
inst✝¹⁰ : IsDomain A
inst✝⁹ : IsFractionRing A K
inst✝⁸ : FiniteDimensional K L
inst✝⁷ : Algebra.IsSeparable K L
inst✝⁶ : IsIntegralClosure B A L
inst✝⁵ : IsFractionRing B L
inst✝⁴ : IsDedekindDomain A
inst✝³ : IsDedekindDomain B
inst✝² : IsTorsionFree A B
inst✝¹ : Module.Finite A B
p : Ideal A
inst✝ : p.IsMaximal
P : Ideal B
e : ℕ
he : e ≠ 0
hp : p ≠ ⊥
hP : P ^ e ∣ Ideal.map (algebraMap A B) p
a : Ideal B
ha : Ideal.map (algebraMap A B) p = P ^ (e - 1) * a
hp' : ¬Ideal.map (algebraMap A B) p = ⊥
ha' : a = ⊥
⊢ Ideal.map (algebraMap A B) p = ⊥

Tactic:
by

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✝⁵⁵ : CommRing A
inst✝⁵⁴ : Field K
inst✝⁵³ : Algebra A K
inst✝⁵² : IsFractionRing A K
inst✝⁵¹ : CommRing B
inst✝⁵⁰ : Field L
inst✝⁴⁹ : Algebra B L
inst✝⁴⁸ : Algebra A L
inst✝⁴⁷ : Algebra K L
inst✝⁴⁶ : FiniteDimensional K L
inst✝⁴⁵ : IsScalarTower A K L
R₁ : Type u_5
R₂ : Type u_6
inst✝⁴⁴ : CommRing R₁
inst✝⁴³ : CommRing R₂
inst✝⁴² : IsDomain R₁
inst✝⁴¹ : Algebra A R₁
inst✝⁴⁰ : Algebra A R₂
inst✝³⁹ : Algebra R₁ B
inst✝³⁸ : Algebra R₂ B
inst✝³⁷ : Algebra R₁ L
inst✝³⁶ : Algebra R₂ L
inst✝³⁵ : IsScalarTower A R₁ L
inst✝³⁴ : IsScalarTower R₁ B L
inst✝³³ : IsScalarTower R₂ B L
inst✝³² : Module.Finite A R₂
F₁ F₂ : IntermediateField K L
inst✝³¹ : Algebra R₁ ↥F₁
inst✝³⁰ : Algebra R₂ ↥F₂
inst✝²⁹ : IsTorsionFree R₁ ↥F₁
inst✝²⁸ : IsScalarTower A (↥F₂) L
inst✝²⁷ : IsScalarTower A R₂ ↥F₂
inst✝²⁶ : IsScalarTower R₁ (↥F₁) L
inst✝²⁵ : IsScalarTower R₂ (↥F₂) L
inst✝²⁴ : Algebra.IsSeparable K ↥F₂
inst✝²³ : Algebra.IsSeparable (↥F₁) L
inst✝²² : IsDomain A
inst✝²¹ : IsDedekindDomain B
inst✝²⁰ : IsDedekindDomain R₁
inst✝¹⁹ : IsDedekindDomain R₂
inst✝¹⁸ : IsFractionRing B L
inst✝¹⁷ : IsFractionRing R₁ ↥F₁
inst✝¹⁶ : IsFractionRing R₂ ↥F₂
inst✝¹⁵ : IsIntegrallyClosed A
inst✝¹⁴ : IsIntegralClosure B R₁ L
inst✝¹³ : IsTorsionFree R₁ B
inst✝¹² : IsTorsionFree R₂ B
inst✝¹¹ : Algebra A B
inst✝¹⁰ : Module.Finite A B
inst✝⁹ : IsTorsionFree A B
inst✝⁸ : IsTorsionFree A R₁
inst✝⁷ : IsTorsionFree A R₂
inst✝⁶ : Module.Finite A R₁
inst✝⁵ : Module.Finite R₂ B
inst✝⁴ : IsScalarTower A R₂ B
inst✝³ : Module.Finite R₁ B
inst✝² : Algebra.IsSeparable (FractionRing A) (FractionRing B)
inst✝¹ : 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))
inst✝ : Free A R₂
B₁ : Basis (Free.ChooseBasisIndex A R₂) R₁ B :=
  Basis.ofIsCoprimeDifferentIdeal A B R₁ R₂ h₁ h₂ h₃ (Free.chooseBasis A R₂)
x : B
i : Free.ChooseBasisIndex A R₂
x✝ : i ∈ Finset.univ
⊢ B₁ i ∈ (IsScalarTower.toAlgHom A R₂ B).range

Tactic:
by

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✝⁵⁵ : CommRing A
inst✝⁵⁴ : Field K
inst✝⁵³ : Algebra A K
inst✝⁵² : IsFractionRing A K
inst✝⁵¹ : CommRing B
inst✝⁵⁰ : Field L
inst✝⁴⁹ : Algebra B L
inst✝⁴⁸ : Algebra A L
inst✝⁴⁷ : Algebra K L
inst✝⁴⁶ : FiniteDimensional K L
inst✝⁴⁵ : IsScalarTower A K L
R₁ : Type u_5
R₂ : Type u_6
inst✝⁴⁴ : CommRing R₁
inst✝⁴³ : CommRing R₂
inst✝⁴² : IsDomain R₁
inst✝⁴¹ : Algebra A R₁
inst✝⁴⁰ : Algebra A R₂
inst✝³⁹ : Algebra R₁ B
inst✝³⁸ : Algebra R₂ B
inst✝³⁷ : Algebra R₁ L
inst✝³⁶ : Algebra R₂ L
inst✝³⁵ : IsScalarTower A R₁ L
inst✝³⁴ : IsScalarTower R₁ B L
inst✝³³ : IsScalarTower R₂ B L
inst✝³² : Module.Finite A R₂
F₁ F₂ : IntermediateField K L
inst✝³¹ : Algebra R₁ ↥F₁
inst✝³⁰ : Algebra R₂ ↥F₂
inst✝²⁹ : IsTorsionFree R₁ ↥F₁
inst✝²⁸ : IsScalarTower A (↥F₂) L
inst✝²⁷ : IsScalarTower A R₂ ↥F₂
inst✝²⁶ : IsScalarTower R₁ (↥F₁) L
inst✝²⁵ : IsScalarTower R₂ (↥F₂) L
inst✝²⁴ : Algebra.IsSeparable K ↥F₂
inst✝²³ : Algebra.IsSeparable (↥F₁) L
inst✝²² : IsDomain A
inst✝²¹ : IsDedekindDomain B
inst✝²⁰ : IsDedekindDomain R₁
inst✝¹⁹ : IsDedekindDomain R₂
inst✝¹⁸ : IsFractionRing B L
inst✝¹⁷ : IsFractionRing R₁ ↥F₁
inst✝¹⁶ : IsFractionRing R₂ ↥F₂
inst✝¹⁵ : IsIntegrallyClosed A
inst✝¹⁴ : IsIntegralClosure B R₁ L
inst✝¹³ : IsTorsionFree R₁ B
inst✝¹² : IsTorsionFree R₂ B
inst✝¹¹ : Algebra A B
inst✝¹⁰ : Module.Finite A B
inst✝⁹ : IsTorsionFree A B
inst✝⁸ : IsTorsionFree A R₁
inst✝⁷ : IsTorsionFree A R₂
inst✝⁶ : Module.Finite A R₁
inst✝⁵ : Module.Finite R₂ B
inst✝⁴ : IsScalarTower A R₂ B
inst✝³ : Module.Finite R₁ B
inst✝² : Algebra.IsSeparable (FractionRing A) (FractionRing B)
inst✝¹ : 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))
inst✝ : Free A R₂
⊢ (IsScalarTower.toAlgHom A R₁ B).range ⊔ (IsScalarTower.toAlgHom A R₂ B).range = ⊤

Tactic:
by

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✝²¹ : CommRing A
inst✝²⁰ : Field K
inst✝¹⁹ : CommRing B
inst✝¹⁸ : Field L
inst✝¹⁷ : Algebra A K
inst✝¹⁶ : Algebra B L
inst✝¹⁵ : Algebra A B
inst✝¹⁴ : Algebra K L
inst✝¹³ : Algebra A L
inst✝¹² : IsScalarTower A K L
inst✝¹¹ : IsScalarTower A B L
inst✝¹⁰ : IsDomain A
inst✝⁹ : IsFractionRing A K
inst✝⁸ : FiniteDimensional K L
inst✝⁷ : Algebra.IsSeparable K L
inst✝⁶ : IsIntegralClosure B A L
inst✝⁵ : IsFractionRing B L
inst✝⁴ : IsDedekindDomain A
inst✝³ : IsDedekindDomain B
inst✝² : IsTorsionFree A B
inst✝¹ : Module.Finite A B
p : Ideal A
inst✝ : p.IsMaximal
P : Ideal B
e : ℕ
he : e ≠ 0
hp : p ≠ ⊥
hP : P ^ e ∣ Ideal.map (algebraMap A B) p
a : Ideal B
ha : Ideal.map (algebraMap A B) p = P ^ (e - 1) * a
hp' : ¬Ideal.map (algebraMap A B) p = ⊥
habot : a ≠ ⊥
⊢ P ≠ ⊥

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
r : ℝ
x : (𝓞 K)ˣ
hr : 0 ≤ r
h : ‖(logEmbedding K) (Additive.ofMul x)‖ ≤ r
w : InfinitePlace K
⊢ |Real.log (w ((algebraMap (𝓞 K) K) ↑x))| ≤ ↑(Fintype.card (InfinitePlace K)) * r

Tactic:
by

Goal state after: