mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral

Goal state before:
pp: case refine_1
p : ℕ
hp : Fact (Nat.Prime p)
⊢ ((cyclotomic p ℤ).comp (X + 1)).Monic

Tactic:
show (X + 1 : ℤ[X]) = X + C 1 by simp

Goal state after:
case refine_1
p : ℕ
hp : Fact (Nat.Prime p)
⊢ ((cyclotomic p ℤ).comp (X + C 1)).Monic
Module: Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral

Goal state before:
pp: case refine_1
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
⊢ ((cyclotomic (p ^ (n + 1)) ℤ).comp (X + 1)).Monic

Tactic:
show (X + 1 : ℤ[X]) = X + C 1 by simp

Goal state after:
case refine_1
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
⊢ ((cyclotomic (p ^ (n + 1)) ℤ).comp (X + C 1)).Monic
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: case refine_2
p k : ℕ
K : Type u
inst✝² : Field K
ζ : K
hp : Fact (Nat.Prime p)
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {p ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ (p ^ (k + 1))
hodd : p ≠ 2
this : NumberField K := numberField {p ^ (k + 1)} ℚ K
⊢ Irreducible (Ideal.absNorm (Ideal.span {hζ.toInteger - 1}))

Tactic:
rw [Nat.irreducible_iff_prime, Ideal.absNorm_span_singleton, ← Nat.prime_iff,
    ← Int.prime_iff_natAbs_prime]

Goal state after:
case refine_2
p k : ℕ
K : Type u
inst✝² : Field K
ζ : K
hp : Fact (Nat.Prime p)
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {p ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ (p ^ (k + 1))
hodd : p ≠ 2
this : NumberField K := numberField {p ^ (k + 1)} ℚ K
⊢ Prime ((Algebra.norm ℤ) (hζ.toInteger - 1))
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: case refine_2
k : ℕ
K : Type u
inst✝² : Field K
ζ : K
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {2 ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ (2 ^ (k + 1))
this : NumberField K
⊢ Irreducible (Ideal.absNorm (Ideal.span {hζ.toInteger - 1}))

Tactic:
rw [Nat.irreducible_iff_prime, Ideal.absNorm_span_singleton, ← Nat.prime_iff,
    ← Int.prime_iff_natAbs_prime]

Goal state after:
case refine_2
k : ℕ
K : Type u
inst✝² : Field K
ζ : K
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {2 ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ (2 ^ (k + 1))
this : NumberField K
⊢ Prime ((Algebra.norm ℤ) (hζ.toInteger - 1))
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: case pos
p k : ℕ
K : Type u
inst✝¹ : Field K
ζ : K
hp : Fact (Nat.Prime p)
inst✝ : CharZero K
hcycl : IsCyclotomicExtension {p ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ (p ^ (k + 1))
htwo : p ^ (k + 1) ≠ 2
n : ℤ
x : 𝓞 K
h : hζ.toInteger - ↑n = ↑p * x
pB : PowerBasis ℤ (𝓞 K) := hζ.integralPowerBasisOfPrimePow
hdim : pB.dim = p ^ k * (p - 1)
hk : k = 0
⊢ 1 < p ^ k * (p - 1)

pp: case neg
p k : ℕ
K : Type u
inst✝¹ : Field K
ζ : K
hp : Fact (Nat.Prime p)
inst✝ : CharZero K
hcycl : IsCyclotomicExtension {p ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ (p ^ (k + 1))
htwo : p ^ (k + 1) ≠ 2
n : ℤ
x : 𝓞 K
h : hζ.toInteger - ↑n = ↑p * x
pB : PowerBasis ℤ (𝓞 K) := hζ.integralPowerBasisOfPrimePow
hdim : pB.dim = p ^ k * (p - 1)
hk : ¬k = 0
⊢ 1 < p ^ k * (p - 1)

Tactic:
· simp only [hk, zero_add, pow_one, pow_zero, one_mul, Nat.lt_sub_iff_add_lt,
        Nat.reduceAdd] at htwo ⊢
      exact htwo.symm.lt_of_le hp.1.two_le

Goal state after:
case neg
p k : ℕ
K : Type u
inst✝¹ : Field K
ζ : K
hp : Fact (Nat.Prime p)
inst✝ : CharZero K
hcycl : IsCyclotomicExtension {p ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ (p ^ (k + 1))
htwo : p ^ (k + 1) ≠ 2
n : ℤ
x : 𝓞 K
h : hζ.toInteger - ↑n = ↑p * x
pB : PowerBasis ℤ (𝓞 K) := hζ.integralPowerBasisOfPrimePow
hdim : pB.dim = p ^ k * (p - 1)
hk : ¬k = 0
⊢ 1 < p ^ k * (p - 1)
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
inst✝ : Free A R₂
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 K ↥F₂
hb : span A (Set.range ⇑b) = (↑(IsScalarTower.toAlgHom A R₂ ↥F₂)).range
⊢ F₁ ⊔ F₂ = ⊤

Tactic:
rwa [← sup_toSubalgebra_of_isAlgebraic_right, ← top_toSubalgebra, toSubalgebra_inj] at h₂

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

Goal state before:
pp: A : Type u_1
B : Type u_3
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra A B
inst✝⁵ : IsDomain A
inst✝⁴ : IsIntegrallyClosed A
inst✝³ : IsDedekindDomain B
inst✝² : IsTorsionFree A B
inst✝¹ : Module.Finite A B
inst✝ : Algebra.IsSeparable (FractionRing A) (FractionRing B)
⊢ differentIdeal A B ≠ ⊥

Tactic:
by
  let K := FractionRing A
  let L := FractionRing B
  rw [ne_eq, ← FractionalIdeal.coeIdeal_inj (K := L), coeIdeal_differentIdeal (K := K)]
  simp

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

Goal state before:
pp: A : Type u_1
B : Type u_4
inst✝¹⁸ : CommRing A
inst✝¹⁷ : CommRing B
inst✝¹⁶ : Algebra A B
inst✝¹⁵ : IsDomain A
inst✝¹⁴ : IsIntegrallyClosed A
inst✝¹³ : IsDedekindDomain B
inst✝¹² : IsTorsionFree A B
C : Type u_3
inst✝¹¹ : IsDomain B
inst✝¹⁰ : CommRing C
inst✝⁹ : Algebra B C
inst✝⁸ : Algebra A C
inst✝⁷ : IsDedekindDomain C
inst✝⁶ : Module.Finite A B
inst✝⁵ : Module.Finite A C
inst✝⁴ : Module.Finite B C
inst✝³ : IsTorsionFree A C
inst✝² : IsTorsionFree B C
inst✝¹ : IsScalarTower A B C
inst✝ : Algebra.IsSeparable (FractionRing A) (FractionRing C)
this✝³ : Algebra.IsSeparable (FractionRing B) (FractionRing C)
this✝² : Algebra.IsSeparable (FractionRing A) (FractionRing B)
this✝¹ : FiniteDimensional (FractionRing A) (FractionRing B)
this✝ : FiniteDimensional (FractionRing A) (FractionRing C)
this : FiniteDimensional (FractionRing B) (FractionRing C)
⊢ differentIdeal A C = differentIdeal B C * Ideal.map (algebraMap B C) (differentIdeal A B)

Tactic:
← coeIdeal_inj (K := FractionRing C),

Goal state after:
A : Type u_1
B : Type u_4
inst✝¹⁸ : CommRing A
inst✝¹⁷ : CommRing B
inst✝¹⁶ : Algebra A B
inst✝¹⁵ : IsDomain A
inst✝¹⁴ : IsIntegrallyClosed A
inst✝¹³ : IsDedekindDomain B
inst✝¹² : IsTorsionFree A B
C : Type u_3
inst✝¹¹ : IsDomain B
inst✝¹⁰ : CommRing C
inst✝⁹ : Algebra B C
inst✝⁸ : Algebra A C
inst✝⁷ : IsDedekindDomain C
inst✝⁶ : Module.Finite A B
inst✝⁵ : Module.Finite A C
inst✝⁴ : Module.Finite B C
inst✝³ : IsTorsionFree A C
inst✝² : IsTorsionFree B C
inst✝¹ : IsScalarTower A B C
inst✝ : Algebra.IsSeparable (FractionRing A) (FractionRing C)
this✝³ : Algebra.IsSeparable (FractionRing B) (FractionRing C)
this✝² : Algebra.IsSeparable (FractionRing A) (FractionRing B)
this✝¹ : FiniteDimensional (FractionRing A) (FractionRing B)
this✝ : FiniteDimensional (FractionRing A) (FractionRing C)
this : FiniteDimensional (FractionRing B) (FractionRing C)
⊢ ↑(differentIdeal A C) = ↑(differentIdeal B C * Ideal.map (algebraMap B C) (differentIdeal A B))
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
⊢ y ∈ coeSubmodule L (conductor A x) ↔
    y ∈ (algebraMap B L) ((aeval x) (derivative (minpoly A x))) • ↑(FractionalIdeal.dual A K 1)

Tactic:
have hne₁ : aeval (algebraMap B L x) (derivative (minpoly K (algebraMap B L x))) ≠ 0 :=
    (Algebra.IsSeparable.isSeparable _ _).aeval_derivative_ne_zero (minpoly.aeval _ _)

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 : 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
⊢ y ∈ coeSubmodule L (conductor A x) ↔
    y ∈ (algebraMap B L) ((aeval x) (derivative (minpoly A x))) • ↑(FractionalIdeal.dual A 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 : 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
⊢ y ∈ coeSubmodule L (conductor A x) ↔
    ((algebraMap B L) ((aeval x) (derivative (minpoly A x))))⁻¹ * y ∈ ↑(FractionalIdeal.dual A K 1)

Tactic:
FractionalIdeal.mem_coe,

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 : 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
⊢ y ∈ coeSubmodule L (conductor A x) ↔
    ((algebraMap B L) ((aeval x) (derivative (minpoly A x))))⁻¹ * y ∈ FractionalIdeal.dual A K 1
Module: Mathlib.NumberTheory.NumberField.Units.DirichletTheorem

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

Tactic:
← ofMul_toMul x,

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : Additive (𝓞 K)ˣ
⊢ (logEmbedding K) (Additive.ofMul (Additive.toMul x)) = 0 ↔
    Additive.ofMul (Additive.toMul x) ∈ Subgroup.toAddSubgroup (torsion K)
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
hζ : IsPrimitiveRoot ⋯.unit n
⊢ ↑⋯.unit ∈ torsion K

Tactic:
· refine (CommGroup.mem_torsion _).mpr ⟨n, NeZero.pos n, ?_⟩
    rw [isPeriodicPt_mul_iff_pow_eq_one]
    exact hζ.pow_eq_one

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 ≠ ⊥
hPbot : P ≠ ⊥
this✝ : Ideal.map (algebraMap A B) p ∣ a ^ e
this : ∀ x ∈ a, (intTrace A B) x ∈ p
hP : (↑(P ^ (e - 1)))⁻¹ = ↑a / ↑(Ideal.map (algebraMap A B) p)
x : L
hx : x ∈ ↑(↑a / ↑(Ideal.map (algebraMap A B) p))
⊢ x ∈ comap (↑A (Algebra.trace K L)) 1

Tactic:
FractionalIdeal.mem_coe,

Goal state after:
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 ≠ ⊥
hPbot : P ≠ ⊥
this✝ : Ideal.map (algebraMap A B) p ∣ a ^ e
this : ∀ x ∈ a, (intTrace A B) x ∈ p
hP : (↑(P ^ (e - 1)))⁻¹ = ↑a / ↑(Ideal.map (algebraMap A B) p)
x : L
hx : x ∈ ↑a / ↑(Ideal.map (algebraMap A B) p)
⊢ x ∈ comap (↑A (Algebra.trace K L)) 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✝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 ≠ ⊥
hPbot : P ≠ ⊥
this✝ : Ideal.map (algebraMap A B) p ∣ a ^ e
this : ∀ x ∈ a, (intTrace A B) x ∈ p
hP : (↑(P ^ (e - 1)))⁻¹ = ↑a / ↑(Ideal.map (algebraMap A B) p)
x : L
hx : ∀ y ∈ ↑(Ideal.map (algebraMap A B) p), x * y ∈ ↑a
⊢ (Algebra.trace K L) x ∈ ↑(↑p / ↑p)

Tactic:
FractionalIdeal.mem_coe,

Goal state after:
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 ≠ ⊥
hPbot : P ≠ ⊥
this✝ : Ideal.map (algebraMap A B) p ∣ a ^ e
this : ∀ x ∈ a, (intTrace A B) x ∈ p
hP : (↑(P ^ (e - 1)))⁻¹ = ↑a / ↑(Ideal.map (algebraMap A B) p)
x : L
hx : ∀ y ∈ ↑(Ideal.map (algebraMap A B) p), x * y ∈ ↑a
⊢ (Algebra.trace K L) x ∈ ↑p / ↑p
Module: Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum

Goal state before:
pp: F : Type u_1
inst✝⁵ : Field F
inst✝⁴ : Fintype F
inst✝³ : DecidableEq F
hF : ringChar F ≠ 2
F' : Type u_2
inst✝² : Field F'
inst✝¹ : Fintype F'
inst✝ : DecidableEq F'
hF' : ringChar F' ≠ 2
h : ringChar F' ≠ ringChar F
⊢ (quadraticChar F) ↑(Fintype.card F') = (quadraticChar F') (↑((quadraticChar F) (-1)) * ↑(Fintype.card F))

Tactic:
let χ := (quadraticChar F).ringHomComp (algebraMap ℤ F')

Goal state after:
F : Type u_1
inst✝⁵ : Field F
inst✝⁴ : Fintype F
inst✝³ : DecidableEq F
hF : ringChar F ≠ 2
F' : Type u_2
inst✝² : Field F'
inst✝¹ : Fintype F'
inst✝ : DecidableEq F'
hF' : ringChar F' ≠ 2
h : ringChar F' ≠ ringChar F
χ : MulChar F F' := (quadraticChar F).ringHomComp (algebraMap ℤ F')
⊢ (quadraticChar F) ↑(Fintype.card F') = (quadraticChar F') (↑((quadraticChar F) (-1)) * ↑(Fintype.card F))
Module: Mathlib.NumberTheory.LucasPrimality

Goal state before:
pp: p : ℕ
hP : Nat.Prime p
this : Fact (Nat.Prime p)
g : (ZMod p)ˣ
hg : ∀ (x : (ZMod p)ˣ), x ∈ Subgroup.zpowers g
h1 : (Units.coeHom (ZMod p)) g ^ (p - 1) = 1 ∧ ∀ m < p - 1, 0 < m → (Units.coeHom (ZMod p)) g ^ m ≠ 1
h2 : 0 < p - 1
⊢ ∃ a, a ^ (p - 1) = 1 ∧ ∀ (q : ℕ), Nat.Prime q → q ∣ p - 1 → a ^ ((p - 1) / q) ≠ 1

Tactic:
refine ⟨g, h1.1, fun q hq hqd ↦ ?_⟩

Goal state after:
p : ℕ
hP : Nat.Prime p
this : Fact (Nat.Prime p)
g : (ZMod p)ˣ
hg : ∀ (x : (ZMod p)ˣ), x ∈ Subgroup.zpowers g
h1 : (Units.coeHom (ZMod p)) g ^ (p - 1) = 1 ∧ ∀ m < p - 1, 0 < m → (Units.coeHom (ZMod p)) g ^ m ≠ 1
h2 : 0 < p - 1
q : ℕ
hq : Nat.Prime q
hqd : q ∣ p - 1
⊢ ↑g ^ ((p - 1) / q) ≠ 1
Module: Mathlib.NumberTheory.Fermat

Goal state before:
pp: m n : ℕ
⊢ m < n → m.fermatNumber < n.fermatNumber

Tactic:
simp only [fermatNumber, add_lt_add_iff_right, Nat.pow_lt_pow_iff_right (one_lt_two : 1 < 2),
    imp_self]

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

Goal state before:
pp: A : Type u_1
B : Type u_3
inst✝¹¹ : CommRing A
inst✝¹⁰ : CommRing B
inst✝⁹ : Algebra A B
inst✝⁸ : IsDomain A
inst✝⁷ : IsDedekindDomain A
inst✝⁶ : IsDedekindDomain B
inst✝⁵ : IsTorsionFree A B
inst✝⁴ : Module.Finite A B
inst✝³ : Algebra.IsSeparable (FractionRing A) (FractionRing B)
p : Ideal A
inst✝² : p.IsMaximal
hp : p ≠ ⊥
P : Ideal B
inst✝¹ : P.IsMaximal
inst✝ : P.LiesOver p
H : ¬Algebra.IsSeparable (A ⧸ p) (B ⧸ P)
a : Ideal B
ha : Ideal.map (algebraMap A B) p = P * a
hPa : ¬P ∣ a
K : Type u_1 := FractionRing A
L : Type u_3 := FractionRing B
hp' : ¬Ideal.map (algebraMap A B) p = ⊥
habot : a ≠ ⊥
hPbot : P ≠ ⊥
this : ∀ x ∈ a, (intTrace A B) x ∈ p
hP : (↑P)⁻¹ = ↑a / ↑(Ideal.map (algebraMap A B) p)
x : L
hx : x ∈ ↑(↑a / ↑(Ideal.map (algebraMap A B) p))
⊢ x ∈ comap (↑A (Algebra.trace K L)) 1

Tactic:
FractionalIdeal.mem_coe,

Goal state after:
A : Type u_1
B : Type u_3
inst✝¹¹ : CommRing A
inst✝¹⁰ : CommRing B
inst✝⁹ : Algebra A B
inst✝⁸ : IsDomain A
inst✝⁷ : IsDedekindDomain A
inst✝⁶ : IsDedekindDomain B
inst✝⁵ : IsTorsionFree A B
inst✝⁴ : Module.Finite A B
inst✝³ : Algebra.IsSeparable (FractionRing A) (FractionRing B)
p : Ideal A
inst✝² : p.IsMaximal
hp : p ≠ ⊥
P : Ideal B
inst✝¹ : P.IsMaximal
inst✝ : P.LiesOver p
H : ¬Algebra.IsSeparable (A ⧸ p) (B ⧸ P)
a : Ideal B
ha : Ideal.map (algebraMap A B) p = P * a
hPa : ¬P ∣ a
K : Type u_1 := FractionRing A
L : Type u_3 := FractionRing B
hp' : ¬Ideal.map (algebraMap A B) p = ⊥
habot : a ≠ ⊥
hPbot : P ≠ ⊥
this : ∀ x ∈ a, (intTrace A B) x ∈ p
hP : (↑P)⁻¹ = ↑a / ↑(Ideal.map (algebraMap A B) p)
x : L
hx : x ∈ ↑a / ↑(Ideal.map (algebraMap A B) p)
⊢ x ∈ comap (↑A (Algebra.trace K L)) 1
Module: Mathlib.RingTheory.DedekindDomain.Different

Goal state before:
pp: A : Type u_1
B : Type u_3
inst✝¹¹ : CommRing A
inst✝¹⁰ : CommRing B
inst✝⁹ : Algebra A B
inst✝⁸ : IsDomain A
inst✝⁷ : IsDedekindDomain A
inst✝⁶ : IsDedekindDomain B
inst✝⁵ : IsTorsionFree A B
inst✝⁴ : Module.Finite A B
inst✝³ : Algebra.IsSeparable (FractionRing A) (FractionRing B)
p : Ideal A
inst✝² : p.IsMaximal
hp : p ≠ ⊥
P : Ideal B
inst✝¹ : P.IsMaximal
inst✝ : P.LiesOver p
H : ¬Algebra.IsSeparable (A ⧸ p) (B ⧸ P)
a : Ideal B
ha : Ideal.map (algebraMap A B) p = P * a
hPa : ¬P ∣ a
K : Type u_1 := FractionRing A
L : Type u_3 := FractionRing B
hp' : ¬Ideal.map (algebraMap A B) p = ⊥
habot : a ≠ ⊥
hPbot : P ≠ ⊥
this : ∀ x ∈ a, (intTrace A B) x ∈ p
hP : (↑P)⁻¹ = ↑a / ↑(Ideal.map (algebraMap A B) p)
x : L
hx : ∀ y ∈ ↑(Ideal.map (algebraMap A B) p), x * y ∈ ↑a
⊢ (Algebra.trace K L) x ∈ ↑(↑p / ↑p)

Tactic:
FractionalIdeal.mem_coe,

Goal state after:
A : Type u_1
B : Type u_3
inst✝¹¹ : CommRing A
inst✝¹⁰ : CommRing B
inst✝⁹ : Algebra A B
inst✝⁸ : IsDomain A
inst✝⁷ : IsDedekindDomain A
inst✝⁶ : IsDedekindDomain B
inst✝⁵ : IsTorsionFree A B
inst✝⁴ : Module.Finite A B
inst✝³ : Algebra.IsSeparable (FractionRing A) (FractionRing B)
p : Ideal A
inst✝² : p.IsMaximal
hp : p ≠ ⊥
P : Ideal B
inst✝¹ : P.IsMaximal
inst✝ : P.LiesOver p
H : ¬Algebra.IsSeparable (A ⧸ p) (B ⧸ P)
a : Ideal B
ha : Ideal.map (algebraMap A B) p = P * a
hPa : ¬P ∣ a
K : Type u_1 := FractionRing A
L : Type u_3 := FractionRing B
hp' : ¬Ideal.map (algebraMap A B) p = ⊥
habot : a ≠ ⊥
hPbot : P ≠ ⊥
this : ∀ x ∈ a, (intTrace A B) x ∈ p
hP : (↑P)⁻¹ = ↑a / ↑(Ideal.map (algebraMap A B) p)
x : L
hx : ∀ y ∈ ↑(Ideal.map (algebraMap A B) p), x * y ∈ ↑a
⊢ (Algebra.trace K L) x ∈ ↑p / ↑p
Module: Mathlib.NumberTheory.Fermat

Goal state before:
pp: n p : ℕ
hn : 1 < n
hp : Prime p
hpdvd : p ∣ n.fermatNumber
this : Fact (Prime p)
hp2 : p ≠ 2
hp8 : p % 8 = 1
⊢ ∃ k, p = k * 2 ^ (n + 2) + 1

Tactic:
obtain ⟨a, ha⟩ := (exists_sq_eq_two_iff hp2).mpr (Or.inl hp8)

Goal state after:
n p : ℕ
hn : 1 < n
hp : Prime p
hpdvd : p ∣ n.fermatNumber
this : Fact (Prime p)
hp2 : p ≠ 2
hp8 : p % 8 = 1
a : ZMod p
ha : 2 = a * a
⊢ ∃ k, p = k * 2 ^ (n + 2) + 1
Module: Mathlib.NumberTheory.FLT.Three

Goal state before:
pp: K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S' : Solution' hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
hm : FiniteMultiplicity λ S'.c
this : λ ^ 4 ∣ S'.c ^ 3
⊢ λ ^ 2 ∣ S'.c

Tactic:
rw [pow_dvd_iff_le_emultiplicity, emultiplicity_pow hζ.zeta_sub_one_prime',
    hm.emultiplicity_eq_multiplicity] at this

Goal state after:
K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S' : Solution' hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
hm : FiniteMultiplicity λ S'.c
this : ↑4 ≤ ↑3 * ↑(multiplicity λ S'.c)
⊢ λ ^ 2 ∣ S'.c
Module: Mathlib.RingTheory.Ideal.NatInt

Goal state before:
pp: ⊢ maximalIdeal ℕ = span {2, 3}

Tactic:
refine le_antisymm (fun n h ↦ ?_) (span_le.mpr <| Set.pair_subset (by simp) (by simp))
  obtain lt | lt := (mem_maximalIdeal_iff.mp h).lt_or_gt
  · obtain rfl := lt_one_iff.mp lt; exact zero_mem _
  exact mem_span_pair.mpr <|
    exists_add_mul_eq_of_gcd_dvd_of_mul_pred_le 2 3 n (by simp) (show 2 ≤ n by lia)

Goal state after: