mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.NumberTheory.NumberField.Discriminant.Different

Goal state before:
pp: L : Type u_3
inst✝⁶ : Field L
K₁ : Type u_4
K₂ : Type u_5
inst✝⁵ : Field K₁
inst✝⁴ : NumberField K₁
inst✝³ : Field K₂
inst✝² : NumberField K₂
inst✝¹ : Algebra K₁ L
inst✝ : Algebra K₂ L
u v : ℤ
h : u * discr K₁ + v * discr K₂ = 1
⊢ IsCoprime (Ideal.map (algebraMap (𝓞 K₁) (𝓞 L)) (differentIdeal ℤ (𝓞 K₁)))
    (Ideal.map (algebraMap (𝓞 K₂) (𝓞 L)) (differentIdeal ℤ (𝓞 K₂)))

Tactic:
refine Ideal.isCoprime_iff_exists.mpr ⟨u * discr K₁, ?_, v * discr K₂, ?_, ?_⟩

Goal state after:
case refine_1
L : Type u_3
inst✝⁶ : Field L
K₁ : Type u_4
K₂ : Type u_5
inst✝⁵ : Field K₁
inst✝⁴ : NumberField K₁
inst✝³ : Field K₂
inst✝² : NumberField K₂
inst✝¹ : Algebra K₁ L
inst✝ : Algebra K₂ L
u v : ℤ
h : u * discr K₁ + v * discr K₂ = 1
⊢ ↑u * ↑(discr K₁) ∈ Ideal.map (algebraMap (𝓞 K₁) (𝓞 L)) (differentIdeal ℤ (𝓞 K₁))

case refine_2
L : Type u_3
inst✝⁶ : Field L
K₁ : Type u_4
K₂ : Type u_5
inst✝⁵ : Field K₁
inst✝⁴ : NumberField K₁
inst✝³ : Field K₂
inst✝² : NumberField K₂
inst✝¹ : Algebra K₁ L
inst✝ : Algebra K₂ L
u v : ℤ
h : u * discr K₁ + v * discr K₂ = 1
⊢ ↑v * ↑(discr K₂) ∈ Ideal.map (algebraMap (𝓞 K₂) (𝓞 L)) (differentIdeal ℤ (𝓞 K₂))

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

Goal state before:
pp: R : Type u_1
inst✝¹⁴ : CommRing R
inst✝¹³ : IsDomain R
S : Type u_3
inst✝¹² : CommRing S
inst✝¹¹ : IsDomain S
inst✝¹⁰ : IsIntegrallyClosed R
inst✝⁹ : IsIntegrallyClosed S
inst✝⁸ : Algebra R S
inst✝⁷ : Module.Finite R S
inst✝⁶ : IsTorsionFree R S
inst✝⁵ : IsDedekindDomain R
inst✝⁴ : IsDedekindDomain S
R' : Type u_4
inst✝³ : CommRing R'
I : Ideal R'
inst✝² : Algebra R' R
inst✝¹ : Algebra R' S
inst✝ : IsScalarTower R' R S
⊢ (relNorm R) (map (algebraMap R' S) I) = map (algebraMap R' R) I ^ finrank R S

Tactic:
by
  rw [← relNorm_algebraMap, Ideal.map_map, IsScalarTower.algebraMap_eq R' R S]

Goal state after:
Module: Mathlib.RingTheory.Ideal.Norm.RelNorm

Goal state before:
pp: R : Type u_1
inst✝¹³ : CommRing R
inst✝¹² : IsDomain R
S : Type u_3
inst✝¹¹ : CommRing S
inst✝¹⁰ : IsDomain S
inst✝⁹ : IsIntegrallyClosed R
inst✝⁸ : IsIntegrallyClosed S
inst✝⁷ : Algebra R S
inst✝⁶ : Module.Finite R S
inst✝⁵ : IsTorsionFree R S
inst✝⁴ : IsDedekindDomain R
inst✝³ : IsDedekindDomain S
P : Ideal S
p : Ideal R
hPp : P.LiesOver p
inst✝² : p.IsMaximal
inst✝¹ : P.IsPrime
inst✝ : IsGalois (FractionRing R) (FractionRing S)
this✝² : P.IsMaximal
G : Type u_3 := Gal(FractionRing S/FractionRing R)
this✝¹ : MulSemiringAction Gal(FractionRing S/FractionRing R) S :=
  IsIntegralClosure.MulSemiringAction R (FractionRing R) (FractionRing S) S
this✝ : IsGaloisGroup G R S
hp : ¬p = ⊥
s : ℕ
hs : (relNorm R) P = p ^ s
this : s = P.inertiaDeg R
⊢ (relNorm R) P = p ^ P.inertiaDeg R

Tactic:
rwa [this] at hs

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

Goal state before:
pp: 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
⊢ (Algebra.norm ℤ) (hζ.toInteger - 1) = ↑p

Tactic:
by
  simpa only [pow_zero, pow_one] using
    hζ.norm_toInteger_pow_sub_one_of_prime_ne_two (Nat.zero_le _) hodd

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

Goal state before:
pp: case pos
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
h : restrictScalars A (Algebra.trace K L).range ≤ 1
⊢ ⊤ᵛ = ⊤

Tactic:
_root_.eq_top_iff

Goal state after:
case pos
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
h : restrictScalars A (Algebra.trace K L).range ≤ 1
⊢ ⊤ ≤ ⊤ᵛ
Module: Mathlib.RingTheory.DedekindDomain.Different

Goal state before:
pp: case e'_3.h₁
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝¹⁵ : CommRing A
inst✝¹⁴ : Field K
inst✝¹³ : CommRing B
inst✝¹² : Field L
inst✝¹¹ : Algebra A K
inst✝¹⁰ : Algebra B L
inst✝⁹ : Algebra A B
inst✝⁸ : Algebra K L
inst✝⁷ : Algebra A L
inst✝⁶ : IsScalarTower A K L
inst✝⁵ : IsScalarTower A B L
inst✝⁴ : IsDomain A
inst✝³ : IsFractionRing A K
inst✝² : FiniteDimensional K L
inst✝¹ : Algebra.IsSeparable K L
inst✝ : Decidable (IsField A)
⊢ Function.Surjective ⇑(algebraMap A K) ↔ restrictScalars A ⊤ ≤ 1

Tactic:
← RingHom.range_eq_top,

Goal state after:
case e'_3.h₁
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝¹⁵ : CommRing A
inst✝¹⁴ : Field K
inst✝¹³ : CommRing B
inst✝¹² : Field L
inst✝¹¹ : Algebra A K
inst✝¹⁰ : Algebra B L
inst✝⁹ : Algebra A B
inst✝⁸ : Algebra K L
inst✝⁷ : Algebra A L
inst✝⁶ : IsScalarTower A K L
inst✝⁵ : IsScalarTower A B L
inst✝⁴ : IsDomain A
inst✝³ : IsFractionRing A K
inst✝² : FiniteDimensional K L
inst✝¹ : Algebra.IsSeparable K L
inst✝ : Decidable (IsField A)
⊢ (algebraMap A K).range = ⊤ ↔ restrictScalars A ⊤ ≤ 1
Module: Mathlib.RingTheory.DedekindDomain.Different

Goal state before:
pp: case e'_3.h₁
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝¹⁵ : CommRing A
inst✝¹⁴ : Field K
inst✝¹³ : CommRing B
inst✝¹² : Field L
inst✝¹¹ : Algebra A K
inst✝¹⁰ : Algebra B L
inst✝⁹ : Algebra A B
inst✝⁸ : Algebra K L
inst✝⁷ : Algebra A L
inst✝⁶ : IsScalarTower A K L
inst✝⁵ : IsScalarTower A B L
inst✝⁴ : IsDomain A
inst✝³ : IsFractionRing A K
inst✝² : FiniteDimensional K L
inst✝¹ : Algebra.IsSeparable K L
inst✝ : Decidable (IsField A)
⊢ (algebraMap A K).range = ⊤ ↔ restrictScalars A ⊤ ≤ 1

Tactic:
_root_.eq_top_iff

Goal state after:
case e'_3.h₁
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝¹⁵ : CommRing A
inst✝¹⁴ : Field K
inst✝¹³ : CommRing B
inst✝¹² : Field L
inst✝¹¹ : Algebra A K
inst✝¹⁰ : Algebra B L
inst✝⁹ : Algebra A B
inst✝⁸ : Algebra K L
inst✝⁷ : Algebra A L
inst✝⁶ : IsScalarTower A K L
inst✝⁵ : IsScalarTower A B L
inst✝⁴ : IsDomain A
inst✝³ : IsFractionRing A K
inst✝² : FiniteDimensional K L
inst✝¹ : Algebra.IsSeparable K L
inst✝ : Decidable (IsField A)
⊢ ⊤ ≤ (algebraMap A K).range ↔ restrictScalars A ⊤ ≤ 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✝¹⁷ : CommRing A
inst✝¹⁶ : Field K
inst✝¹⁵ : CommRing B
inst✝¹⁴ : Field L
inst✝¹³ : Algebra A K
inst✝¹² : Algebra B L
inst✝¹¹ : Algebra A B
inst✝¹⁰ : Algebra K L
inst✝⁹ : Algebra A L
inst✝⁸ : IsScalarTower A K L
inst✝⁷ : IsScalarTower A B L
inst✝⁶ : IsFractionRing A K
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : FiniteDimensional K L
inst✝³ : IsIntegralClosure B A L
inst✝² : Algebra.IsSeparable K L
I : Submodule B L
ι : Type u_4
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
b : Basis ι K L
hb : ∀ (i : ι), IsIntegral A (b i)
a x : L
ha : a ∈ I
hx : x ∈ Iᵛ
hinv : IsUnit (traceMatrix K ⇑b).det
H :
  (traceMatrix K ⇑b *ᵥ (traceMatrix K ⇑b).cramer fun i ↦ (Algebra.trace K L) (x * a * b i)) =
    (traceMatrix K ⇑b).det • fun i ↦ (Algebra.trace K L) (x * a * b i)
⊢ IsIntegral A (discr K ⇑b • a * x)

Tactic:
have : Function.Injective (traceMatrix K b).mulVec := by
    rwa [mulVec_injective_iff_isUnit, isUnit_iff_isUnit_det]

Goal state after:
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝¹⁷ : CommRing A
inst✝¹⁶ : Field K
inst✝¹⁵ : CommRing B
inst✝¹⁴ : Field L
inst✝¹³ : Algebra A K
inst✝¹² : Algebra B L
inst✝¹¹ : Algebra A B
inst✝¹⁰ : Algebra K L
inst✝⁹ : Algebra A L
inst✝⁸ : IsScalarTower A K L
inst✝⁷ : IsScalarTower A B L
inst✝⁶ : IsFractionRing A K
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : FiniteDimensional K L
inst✝³ : IsIntegralClosure B A L
inst✝² : Algebra.IsSeparable K L
I : Submodule B L
ι : Type u_4
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
b : Basis ι K L
hb : ∀ (i : ι), IsIntegral A (b i)
a x : L
ha : a ∈ I
hx : x ∈ Iᵛ
hinv : IsUnit (traceMatrix K ⇑b).det
H :
  (traceMatrix K ⇑b *ᵥ (traceMatrix K ⇑b).cramer fun i ↦ (Algebra.trace K L) (x * a * b i)) =
    (traceMatrix K ⇑b).det • fun i ↦ (Algebra.trace K L) (x * a * b i)
this : Function.Injective (traceMatrix K ⇑b).mulVec
⊢ IsIntegral A (discr K ⇑b • a * x)
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 ⊔ F₂ = ⊤
h₃ : IsCoprime (Ideal.map (algebraMap R₁ B) (differentIdeal A R₁)) (Ideal.map (algebraMap R₂ B) (differentIdeal A R₂))
⊢ differentIdeal A B = differentIdeal R₁ B * differentIdeal R₂ B

Tactic:
by
  have := differentIdeal_eq_differentIdeal_mul_differentIdeal A R₂ B
  rwa [← differentIdeal_eq_map_differentIdeal A B R₁ R₂ h₁ h₂ h₃,
    mul_comm] at this

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

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

Tactic:
← PowerBasis.finrank

Goal state after:
p k : ℕ
K : Type u
inst✝² : Field K
hp : Fact (Nat.Prime p)
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {p ^ k} ℚ K
hζ : IsPrimitiveRoot (zeta (p ^ k) ℚ K) (p ^ k)
this : NumberField K
pB₁ : PowerBasis ℤ (𝓞 K) := hζ.integralPowerBasisOfPrimePow
⊢ Module.finrank ℤ (𝓞 K) = Module.finrank ℚ K
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✝¹ : 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
hne₂ : ((aeval ((algebraMap B L) x)) (derivative (minpoly K ((algebraMap B L) x))))⁻¹ ≠ 0
this : IsIntegral A ((algebraMap B L) x)
⊢ (∀ (z : B), y * (algebraMap B L) z ∈ A[(algebraMap B L) x]) ↔
    ∀ a ∈ 1,
      ((traceForm K L) (((algebraMap B L) ((aeval x) (derivative (minpoly A x))))⁻¹ * y)) a ∈ (algebraMap A K).range

Tactic:
← Subalgebra.mem_toSubmodule,

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✝¹ : 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
hne₂ : ((aeval ((algebraMap B L) x)) (derivative (minpoly K ((algebraMap B L) x))))⁻¹ ≠ 0
this : IsIntegral A ((algebraMap B L) x)
⊢ (∀ (z : B), y * (algebraMap B L) z ∈ Subalgebra.toSubmodule A[(algebraMap B L) x]) ↔
    ∀ a ∈ 1,
      ((traceForm K L) (((algebraMap B L) ((aeval x) (derivative (minpoly A x))))⁻¹ * y)) a ∈ (algebraMap A K).range
Module: Mathlib.NumberTheory.NumberField.Units.DirichletTheorem

Goal state before:
pp: case neg
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
tool : ∀ (x : ℝ), 0 ≤ x → x ≤ ↑w.mult * x
hw : ¬w = w₀
hyp : |↑(↑⟨w, hw⟩).mult * Real.log (↑⟨w, hw⟩ ((algebraMap (𝓞 K) K) ↑(Additive.ofMul x)))| ≤ r
⊢ |Real.log (w ((algebraMap (𝓞 K) K) ↑x))| ≤ ↑(Fintype.card (InfinitePlace K)) * r

Tactic:
abs_mul,

Goal state after:
case neg
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
tool : ∀ (x : ℝ), 0 ≤ x → x ≤ ↑w.mult * x
hw : ¬w = w₀
hyp : |↑(↑⟨w, hw⟩).mult| * |Real.log (↑⟨w, hw⟩ ((algebraMap (𝓞 K) K) ↑(Additive.ofMul x)))| ≤ r
⊢ |Real.log (w ((algebraMap (𝓞 K) K) ↑x))| ≤ ↑(Fintype.card (InfinitePlace K)) * r
Module: Mathlib.NumberTheory.NumberField.Units.DirichletTheorem

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w₁ : InfinitePlace K
⊢ ∃ B, minkowskiBound K 1 < ↑(convexBodyLTFactor K) * ↑B

Tactic:
by
    conv => congr; ext; rw [mul_comm]
    exact ENNReal.exists_nat_mul_gt (ENNReal.coe_ne_zero.mpr (convexBodyLTFactor_ne_zero K))
      (ne_of_lt (minkowskiBound_lt_top K 1))

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

Goal state before:
pp: case pos
p : ℕ
inst✝ : Fact (Nat.Prime p)
x : (ZMod p)ˣ
hc : p = 2
⊢ (∃ y, y ^ 2 = x) ↔ x ^ (p / 2) = 1

pp: case neg
p : ℕ
inst✝ : Fact (Nat.Prime p)
x : (ZMod p)ˣ
hc : ¬p = 2
⊢ (∃ y, y ^ 2 = x) ↔ x ^ (p / 2) = 1

Tactic:
· subst hc
    simp only [eq_iff_true_of_subsingleton, exists_const]

Goal state after:
case neg
p : ℕ
inst✝ : Fact (Nat.Prime p)
x : (ZMod p)ˣ
hc : ¬p = 2
⊢ (∃ y, y ^ 2 = x) ↔ x ^ (p / 2) = 1
Module: Mathlib.NumberTheory.FLT.Three

Goal state before:
pp: case inl
a b c : ℤ
ha : a ≠ 0
h3a : 3 ∣ a
Hgcd : {a, b, c}.gcd id = 1
H : ∀ (a b c : ℤ), c ≠ 0 → ¬3 ∣ a → ¬3 ∣ b → 3 ∣ c → IsCoprime a b → a ^ 3 + b ^ 3 ≠ c ^ 3
HF : a ^ 3 + b ^ 3 + c ^ 3 = 0
x : ℤ
h3b : 3 ∣ b
hx : x = a
⊢ 3 ∣ id x

Tactic:
exact hx ▸ h3a

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

Goal state before:
pp: case inl.inl
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
z : 𝓞 K
hz : S'.c = λ * z
x : 𝓞 K
hx : S'.a ^ 3 - 1 = λ ^ 4 * x
y : 𝓞 K
hy : S'.b ^ 3 - 1 = λ ^ 4 * y
⊢ λ ^ 4 ∣ S'.a ^ 3 - 1 ∧ λ ^ 4 ∣ S'.b ^ 3 + 1 ∨ λ ^ 4 ∣ S'.a ^ 3 + 1 ∧ λ ^ 4 ∣ S'.b ^ 3 - 1

pp: case inl.inr
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
z : 𝓞 K
hz : S'.c = λ * z
x : 𝓞 K
hx : S'.a ^ 3 - 1 = λ ^ 4 * x
y : 𝓞 K
hy : S'.b ^ 3 + 1 = λ ^ 4 * y
⊢ λ ^ 4 ∣ S'.a ^ 3 - 1 ∧ λ ^ 4 ∣ S'.b ^ 3 + 1 ∨ λ ^ 4 ∣ S'.a ^ 3 + 1 ∧ λ ^ 4 ∣ S'.b ^ 3 - 1

pp: case inr.inl
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
z : 𝓞 K
hz : S'.c = λ * z
x : 𝓞 K
hx : S'.a ^ 3 + 1 = λ ^ 4 * x
y : 𝓞 K
hy : S'.b ^ 3 - 1 = λ ^ 4 * y
⊢ λ ^ 4 ∣ S'.a ^ 3 - 1 ∧ λ ^ 4 ∣ S'.b ^ 3 + 1 ∨ λ ^ 4 ∣ S'.a ^ 3 + 1 ∧ λ ^ 4 ∣ S'.b ^ 3 - 1

pp: case inr.inr
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
z : 𝓞 K
hz : S'.c = λ * z
x : 𝓞 K
hx : S'.a ^ 3 + 1 = λ ^ 4 * x
y : 𝓞 K
hy : S'.b ^ 3 + 1 = λ ^ 4 * y
⊢ λ ^ 4 ∣ S'.a ^ 3 - 1 ∧ λ ^ 4 ∣ S'.b ^ 3 + 1 ∨ λ ^ 4 ∣ S'.a ^ 3 + 1 ∧ λ ^ 4 ∣ S'.b ^ 3 - 1

Tactic:
· exfalso
    replace hζ : IsPrimitiveRoot ζ (3 ^ 1) := by rwa [pow_one]
    refine hζ.toInteger_sub_one_not_dvd_two (by decide) ⟨S'.u * λ ^ 2 * z ^ 3 - λ ^ 3 * (x + y), ?_⟩
    symm
    calc _ = S'.u * (λ * z) ^ 3 - λ ^ 4 * x - λ ^ 4 * y := by ring
    _ = (S'.a ^ 3 + S'.b ^ 3) - (S'.a ^ 3 - 1) - (S'.b ^ 3 - 1) := by rw [← hx, ← hy, ← hz, ← S'.H]
    _ = 2 := by ring

Goal state after:
case inl.inr
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
z : 𝓞 K
hz : S'.c = λ * z
x : 𝓞 K
hx : S'.a ^ 3 - 1 = λ ^ 4 * x
y : 𝓞 K
hy : S'.b ^ 3 + 1 = λ ^ 4 * y
⊢ λ ^ 4 ∣ S'.a ^ 3 - 1 ∧ λ ^ 4 ∣ S'.b ^ 3 + 1 ∨ λ ^ 4 ∣ S'.a ^ 3 + 1 ∧ λ ^ 4 ∣ S'.b ^ 3 - 1

case inr.inl
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
z : 𝓞 K
hz : S'.c = λ * z
x : 𝓞 K
hx : S'.a ^ 3 + 1 = λ ^ 4 * x
y : 𝓞 K
hy : S'.b ^ 3 - 1 = λ ^ 4 * y
⊢ λ ^ 4 ∣ S'.a ^ 3 - 1 ∧ λ ^ 4 ∣ S'.b ^ 3 + 1 ∨ λ ^ 4 ∣ S'.a ^ 3 + 1 ∧ λ ^ 4 ∣ S'.b ^ 3 - 1

case inr.inr
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
z : 𝓞 K
hz : S'.c = λ * z
x : 𝓞 K
hx : S'.a ^ 3 + 1 = λ ^ 4 * x
y : 𝓞 K
hy : S'.b ^ 3 + 1 = λ ^ 4 * y
⊢ λ ^ 4 ∣ S'.a ^ 3 - 1 ∧ λ ^ 4 ∣ S'.b ^ 3 + 1 ∨ λ ^ 4 ∣ S'.a ^ 3 + 1 ∧ λ ^ 4 ∣ S'.b ^ 3 - 1
Module: Mathlib.NumberTheory.Fermat

Goal state before:
pp: case hd
n : ℕ
h : 3 ^ 2 ^ (2 ^ n - 1) = -1
this : Fact (2 < 2 ^ 2 ^ n + 1)
key : 2 ^ n = 2 ^ n - 1 + 1
p : ℕ
hp1 : Prime p
hp2 : p ∣ 2 ^ 2 ^ n + 1 - 1
⊢ 3 ^ ((2 ^ 2 ^ n + 1 - 1) / p) ≠ 1

Tactic:
rw [Nat.add_sub_cancel, (Nat.prime_dvd_prime_iff_eq hp1 prime_two).mp (hp1.dvd_of_dvd_pow hp2),
        key, pow_succ, Nat.mul_div_cancel _ two_pos, ← pow_succ, ← key, h]

Goal state after:
case hd
n : ℕ
h : 3 ^ 2 ^ (2 ^ n - 1) = -1
this : Fact (2 < 2 ^ 2 ^ n + 1)
key : 2 ^ n = 2 ^ n - 1 + 1
p : ℕ
hp1 : Prime p
hp2 : p ∣ 2 ^ 2 ^ n + 1 - 1
⊢ -1 ≠ 1
Module: Mathlib.NumberTheory.FrobeniusNumber

Goal state before:
pp: case neg
s : Set ℕ
h0 : ¬setGcd s = 0
t : Finset ℕ
hts : ↑t ⊆ s
a : ↥t → ℤ
eq : ∑ i, a i • ↑↑i = ↑(setGcd s)
x : ℕ
hxs : x ∈ s
hx : x ≠ 0
n : ℕ := x / setGcd s * ∑ i, (-a i).toNat * ↑i
c : ℕ
ge : n + c ≥ n
dvd : setGcd s ∣ n + c
q : ℕ := c / x
r : ℕ := c % x
rx : ℕ
hrx : r = setGcd s * rx
i : ↥t
a✝ : i ∈ Finset.univ
hai : ¬0 ≤ a i
⊢ ↑rx * a i + ↑(x / setGcd s) * ↑(-a i).toNat = ↑(x / setGcd s - rx) * ↑(a i).natAbs

Tactic:
rw [cast_sub, Int.natCast_natAbs, abs_eq_neg_self.mpr (by lia), sub_mul,
      ← Int.eq_natCast_toNat.mpr (by lia), mul_neg (rx : ℤ), sub_neg_eq_add, add_comm]

Goal state after:
case neg
s : Set ℕ
h0 : ¬setGcd s = 0
t : Finset ℕ
hts : ↑t ⊆ s
a : ↥t → ℤ
eq : ∑ i, a i • ↑↑i = ↑(setGcd s)
x : ℕ
hxs : x ∈ s
hx : x ≠ 0
n : ℕ := x / setGcd s * ∑ i, (-a i).toNat * ↑i
c : ℕ
ge : n + c ≥ n
dvd : setGcd s ∣ n + c
q : ℕ := c / x
r : ℕ := c % x
rx : ℕ
hrx : r = setGcd s * rx
i : ↥t
a✝ : i ∈ Finset.univ
hai : ¬0 ≤ a i
⊢ rx ≤ x / setGcd s
Module: Mathlib.NumberTheory.FermatPsp

Goal state before:
pp: b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_prime : Prime p
p_gt_two : 2 < p
not_dvd : ¬p ∣ b * (b ^ 2 - 1)
A : ℕ := (b ^ p - 1) / (b - 1)
B : ℕ := (b ^ p + 1) / (b + 1)
hA : p < A
hi_A : 1 < A
hi_B : 1 < B
hi_b : 0 < b
hi_bsquared : 0 < b ^ 2 - 1
hi_bpowtwop : 1 ≤ b ^ (2 * p)
hi_bpowpsubone : 1 ≤ b ^ (p - 1)
p_odd : Odd p
AB_not_prime : ¬Prime (A * B)
AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)
hd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1
ha₁ : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)
ha₂ : 2 ∣ b ^ p + b
ha₃ : p ∣ b ^ (p - 1) - 1
ha₄ : b ^ 2 - 1 ∣ b ^ (p - 1) - 1
ha₅ : (b ^ 2 - 1) * (2 * p) ∣ (b ^ 2 - 1) * (A * B - 1)
⊢ 2 * p ∣ A * B - 1

Tactic:
exact Nat.dvd_of_mul_dvd_mul_left hi_bsquared ha₅

Goal state after:
Module: Mathlib.NumberTheory.FrobeniusNumber

Goal state before:
pp: s : Set ℕ
h : setGcd s = 1 ∧ 1 ∉ s
n : ℕ
hn : ∀ m ≥ n, setGcd s ∣ m → m ∈ AddSubmonoid.closure s
P : ℕ → Prop := fun n ↦ n ∉ AddSubmonoid.closure s
this : P 1
⊢ ∃ n, FrobeniusNumber n s

Tactic:
classical
    refine ⟨findGreatest P n, frobeniusNumber_iff.mpr ⟨findGreatest_spec (P := P) (m := 1)
      (le_of_not_gt fun lt ↦ this (hn _ lt.le h.1.dvd)) this, fun k gt ↦ ?_⟩⟩
    obtain le | le := le_total k n
    · exact of_not_not (findGreatest_is_greatest gt le)
    · exact hn k le (h.1.dvd.trans <| one_dvd k)

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

Goal state before:
pp: K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
⊢ ¬λ ^ 2 ∣ S.a + ↑η ^ 2 * S.b

Tactic:
intro ⟨k, hk⟩
  rcases S.hab with ⟨k', hk'⟩
  refine S.hb ⟨(k - k') * (-η), ?_⟩
  rw [show S.a + η ^ 2 * S.b = S.a + S.b - S.b + η ^ 2 * S.b by ring, hk',
    show λ ^ 2 * k' - S.b + η ^ 2 * S.b = λ * (S.b * (η + 1) + λ * k') by rw [coe_eta]; ring,
    pow_two, mul_assoc] at hk
  simp only [mul_eq_mul_left_iff, hζ.zeta_sub_one_prime'.ne_zero, or_false] at hk
  apply_fun (· * -↑η) at hk
  rw [show (S.b * (η + 1) + λ * k') * -η = (-S.b) * (η ^ 2 + η + 1 - 1) - η * λ * k' by ring,
    eta_sq, show -S.b * (-↑η - 1 + ↑η + 1 - 1) = S.b by ring, sub_eq_iff_eq_add] at hk
  rw [hk]
  ring

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

Goal state before:
pp: K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
p : 𝓞 K
hp : Prime p
hpab : p ∣ S.a + S.b
hpaηb : p ∣ S.a + ↑η * S.b
⊢ Associated p λ

Tactic:
suffices p_lam : p ∣ λ from hp.associated_of_dvd hζ.zeta_sub_one_prime' p_lam
  rw [← one_mul S.a, ← one_mul S.b] at hpab
  rw [← one_mul S.a] at hpaηb
  have := dvd_mul_sub_mul_mul_gcd_of_dvd hpab hpaηb
  rwa [one_mul, one_mul, coe_eta, IsUnit.dvd_mul_right <| (gcd_isUnit_iff _ _).2 S.coprime] at this

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

Goal state before:
pp: K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
⊢ (∃ X, Associated (X ^ 3) S.x) ∧ (∃ Y, Associated (Y ^ 3) S.y) ∧ ∃ Z, Associated (Z ^ 3) S.z

Tactic:
by
  have h₁ := S.isCoprime_x_z.mul_left S.isCoprime_y_z
  have h₂ : Associated (S.w ^ 3) (S.x * S.y * S.z) :=
    ⟨S.u, by rw [x_mul_y_mul_z_eq_u_mul_w_cube S, mul_comm]⟩
  obtain ⟨T, h₃⟩ := exists_associated_pow_of_associated_pow_mul h₁ h₂
  exact ⟨exists_associated_pow_of_associated_pow_mul S.isCoprime_x_y h₃,
    exists_associated_pow_of_associated_pow_mul S.isCoprime_x_y.symm (mul_comm _ S.x ▸ h₃),
    exists_associated_pow_of_associated_pow_mul h₁.symm (mul_comm _ S.z ▸ h₂)⟩

Goal state after: