mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.DedekindDomain.Instances

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
inst✝¹² : CommRing R
inst✝¹¹ : CommRing S
inst✝¹⁰ : CommRing T
inst✝⁹ : IsDomain R
inst✝⁸ : IsDomain S
inst✝⁷ : IsDomain T
inst✝⁶ : Algebra R S
P : Ideal R
inst✝⁵ : P.IsPrime
inst✝⁴ : FaithfulSMul R S
inst✝³ : Algebra S T
inst✝² : Algebra R T
inst✝¹ : IsScalarTower R S T
inst✝ : IsTorsionFree R T
a : Rₚ
b : Sₚ
c : Tₚ
x✝ : R × ↥P.primeCompl
a₁ : R
a₂ : ↥P.primeCompl
this : ↑a₂ ≠ 0
⊢ (Localization.mk (↑a₂ * (a₁, a₂).1) (a₁, a₂).2 • b) • c = Localization.mk (↑a₂ * (a₁, a₂).1) (a₁, a₂).2 • b • c

Tactic:
Localization.mk_eq_mk',

Goal state after:
R : Type u_1
S : Type u_2
T : Type u_3
inst✝¹² : CommRing R
inst✝¹¹ : CommRing S
inst✝¹⁰ : CommRing T
inst✝⁹ : IsDomain R
inst✝⁸ : IsDomain S
inst✝⁷ : IsDomain T
inst✝⁶ : Algebra R S
P : Ideal R
inst✝⁵ : P.IsPrime
inst✝⁴ : FaithfulSMul R S
inst✝³ : Algebra S T
inst✝² : Algebra R T
inst✝¹ : IsScalarTower R S T
inst✝ : IsTorsionFree R T
a : Rₚ
b : Sₚ
c : Tₚ
x✝ : R × ↥P.primeCompl
a₁ : R
a₂ : ↥P.primeCompl
this : ↑a₂ ≠ 0
⊢ (mk' (Localization P.primeCompl) (↑a₂ * (a₁, a₂).1) (a₁, a₂).2 • b) • c =
    mk' (Localization P.primeCompl) (↑a₂ * (a₁, a₂).1) (a₁, a₂).2 • b • c
Module: Mathlib.RingTheory.Ideal.Norm.RelNorm

Goal state before:
pp: R : Type u_1
inst✝²⁴ : CommRing R
inst✝²³ : IsDomain R
S : Type u_3
inst✝²² : CommRing S
inst✝²¹ : IsDomain S
inst✝²⁰ : IsIntegrallyClosed R
inst✝¹⁹ : IsIntegrallyClosed S
inst✝¹⁸ : Algebra R S
inst✝¹⁷ : Module.Finite R S
inst✝¹⁶ : IsTorsionFree R S
I : Ideal S
M : Submonoid R
hM : M ≤ R⁰
Rₘ : Type u_4
Sₘ : Type u_5
inst✝¹⁵ : CommRing Rₘ
inst✝¹⁴ : Algebra R Rₘ
inst✝¹³ : CommRing Sₘ
inst✝¹² : Algebra S Sₘ
inst✝¹¹ : Algebra Rₘ Sₘ
inst✝¹⁰ : Algebra R Sₘ
inst✝⁹ : IsScalarTower R Rₘ Sₘ
inst✝⁸ : IsScalarTower R S Sₘ
inst✝⁷ : IsLocalization M Rₘ
inst✝⁶ : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ
inst✝⁵ : IsIntegrallyClosed Rₘ
inst✝⁴ : IsDomain Rₘ
inst✝³ : IsDomain Sₘ
inst✝² : IsTorsionFree Rₘ Sₘ
inst✝¹ : Module.Finite Rₘ Sₘ
inst✝ : IsIntegrallyClosed Sₘ
K : Type u_1 := FractionRing R
f : Rₘ →+* K := IsLocalization.map K (RingHom.id R) hM
L : Type u_3 := FractionRing S
g : Sₘ →+* L := IsLocalization.map L (RingHom.id S) ⋯
algInst✝² : Algebra Rₘ K := f.toAlgebra
algInst✝¹ : Algebra Sₘ L := g.toAlgebra
algInst✝ : Algebra Rₘ L := ((algebraMap K L).comp f).toAlgebra
scalarTowerInst✝ : IsScalarTower Rₘ K L := IsScalarTower.of_algebraMap_eq' (Eq.refl (algebraMap Rₘ L))
this : IsScalarTower R Rₘ K
x✝ : IsFractionRing Rₘ K := IsFractionRing.isFractionRing_of_isDomain_of_isLocalization M Rₘ K
⊢ spanNorm Rₘ (map (algebraMap S Sₘ) I) = map (algebraMap R Rₘ) (spanNorm R I)

Tactic:
have : IsScalarTower S Sₘ L := IsScalarTower.of_algebraMap_eq'
    (by rw [RingHom.algebraMap_toAlgebra, IsLocalization.map_comp, RingHomCompTriple.comp_eq])

Goal state after:
R : Type u_1
inst✝²⁴ : CommRing R
inst✝²³ : IsDomain R
S : Type u_3
inst✝²² : CommRing S
inst✝²¹ : IsDomain S
inst✝²⁰ : IsIntegrallyClosed R
inst✝¹⁹ : IsIntegrallyClosed S
inst✝¹⁸ : Algebra R S
inst✝¹⁷ : Module.Finite R S
inst✝¹⁶ : IsTorsionFree R S
I : Ideal S
M : Submonoid R
hM : M ≤ R⁰
Rₘ : Type u_4
Sₘ : Type u_5
inst✝¹⁵ : CommRing Rₘ
inst✝¹⁴ : Algebra R Rₘ
inst✝¹³ : CommRing Sₘ
inst✝¹² : Algebra S Sₘ
inst✝¹¹ : Algebra Rₘ Sₘ
inst✝¹⁰ : Algebra R Sₘ
inst✝⁹ : IsScalarTower R Rₘ Sₘ
inst✝⁸ : IsScalarTower R S Sₘ
inst✝⁷ : IsLocalization M Rₘ
inst✝⁶ : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ
inst✝⁵ : IsIntegrallyClosed Rₘ
inst✝⁴ : IsDomain Rₘ
inst✝³ : IsDomain Sₘ
inst✝² : IsTorsionFree Rₘ Sₘ
inst✝¹ : Module.Finite Rₘ Sₘ
inst✝ : IsIntegrallyClosed Sₘ
K : Type u_1 := FractionRing R
f : Rₘ →+* K := IsLocalization.map K (RingHom.id R) hM
L : Type u_3 := FractionRing S
g : Sₘ →+* L := IsLocalization.map L (RingHom.id S) ⋯
algInst✝² : Algebra Rₘ K := f.toAlgebra
algInst✝¹ : Algebra Sₘ L := g.toAlgebra
algInst✝ : Algebra Rₘ L := ((algebraMap K L).comp f).toAlgebra
scalarTowerInst✝ : IsScalarTower Rₘ K L := IsScalarTower.of_algebraMap_eq' (Eq.refl (algebraMap Rₘ L))
this✝ : IsScalarTower R Rₘ K
x✝ : IsFractionRing Rₘ K := IsFractionRing.isFractionRing_of_isDomain_of_isLocalization M Rₘ K
this : IsScalarTower S Sₘ L
⊢ spanNorm Rₘ (map (algebraMap S Sₘ) I) = map (algebraMap R Rₘ) (spanNorm R I)
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 : ¬K₁ ⊓ K₂ = ⊥
⊢ finrank ℚ ↥(K₁ ⊓ K₂) ≠ 1

Tactic:
← IntermediateField.finrank_eq_one_iff

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 : ¬finrank ℚ ↥(K₁ ⊓ K₂) = 1
⊢ finrank ℚ ↥(K₁ ⊓ K₂) ≠ 1
Module: Mathlib.RingTheory.Prime

Goal state before:
pp: R : Type u_1
inst✝² : CommMonoidWithZero R
inst✝¹ : IsCancelMulZero R
α : Type u_2
inst✝ : DecidableEq α
p : α → R
i : α
s : Finset α
his : i ∉ s
ih :
  ∀ {x y a : R},
    (∀ i ∈ s, Prime (p i)) →
      x * y = a * ∏ i ∈ s, p i →
        ∃ t u b c, t ∪ u = s ∧ Disjoint t u ∧ a = b * c ∧ x = b * ∏ i ∈ t, p i ∧ y = c * ∏ i ∈ u, p i
a : R
hp : ∀ i_1 ∈ insert i s, Prime (p i_1)
hpi : Prime (p i)
t u : Finset α
b : R
htus : t ∪ u = s
htu : Disjoint t u
hit : i ∉ t
hiu : i ∉ u
d : R
hbc : a = b * d
hx : (b * ∏ i ∈ t, p i) * (p i * d * ∏ i ∈ u, p i) = a * p i * ∏ x ∈ s, p x
⊢ t ∪ insert i u = insert i s

Tactic:
rw [union_insert, htus]

Goal state after:
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
x : K
h : IsIntegral ℤ x
u : ℤˣ
n n✝ : ℕ
hcycl : IsCyclotomicExtension {p ^ (n✝ + 1)} ℚ K
hζ : IsPrimitiveRoot ζ (p ^ (n✝ + 1))
B : PowerBasis ℚ K := IsPrimitiveRoot.subOnePowerBasis ℚ hζ
hint : IsIntegral ℤ B.gen
this : FiniteDimensional ℚ K := finiteDimensional {p ^ (n✝ + 1)} ℚ K
hun : Algebra.discr ℚ ⇑(IsPrimitiveRoot.subOnePowerBasis ℚ hζ).basis = ↑↑u * ↑p ^ n
H : (algebraMap ℚ K) ↑p ^ n * x ∈ ℤ[B.gen]
h₁ : minpoly ℚ B.gen = map (algebraMap ℤ ℚ) (minpoly ℤ B.gen)
⊢ (minpoly ℤ B.gen).IsEisensteinAt (ℤ ∙ ↑p)

Tactic:
have h₂ := hζ.minpoly_sub_one_eq_cyclotomic_comp (cyclotomic.irreducible_rat (NeZero.pos _))

Goal state after:
p : ℕ
K : Type u
inst✝¹ : Field K
ζ : K
hp : Fact (Nat.Prime p)
inst✝ : CharZero K
x : K
h : IsIntegral ℤ x
u : ℤˣ
n n✝ : ℕ
hcycl : IsCyclotomicExtension {p ^ (n✝ + 1)} ℚ K
hζ : IsPrimitiveRoot ζ (p ^ (n✝ + 1))
B : PowerBasis ℚ K := IsPrimitiveRoot.subOnePowerBasis ℚ hζ
hint : IsIntegral ℤ B.gen
this : FiniteDimensional ℚ K := finiteDimensional {p ^ (n✝ + 1)} ℚ K
hun : Algebra.discr ℚ ⇑(IsPrimitiveRoot.subOnePowerBasis ℚ hζ).basis = ↑↑u * ↑p ^ n
H : (algebraMap ℚ K) ↑p ^ n * x ∈ ℤ[B.gen]
h₁ : minpoly ℚ B.gen = map (algebraMap ℤ ℚ) (minpoly ℤ B.gen)
h₂ : minpoly ℚ (ζ - 1) = (cyclotomic (p ^ (n✝ + 1)) ℚ).comp (X + 1)
⊢ (minpoly ℤ B.gen).IsEisensteinAt (ℤ ∙ ↑p)
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))
htwo : p ^ (k + 1) ≠ 2
⊢ Prime ((Algebra.norm ℤ) (hζ.toInteger - 1))

Tactic:
have := hζ.norm_toInteger_pow_sub_one_of_prime_pow_ne_two zero_le htwo

Goal state after:
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))
htwo : p ^ (k + 1) ≠ 2
this : (Algebra.norm ℤ) (hζ.toInteger ^ p ^ 0 - 1) = ↑p ^ p ^ 0
⊢ Prime ((Algebra.norm ℤ) (hζ.toInteger - 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✝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
I J : Submodule B L
⊢ I ≤ Jᵛ ↔ I * J ≤ 1ᵛ

Tactic:
by
  rw [← le_traceDual_mul_iff, mul_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✝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)
⊢ ⊤ᵛ = if IsField A then ⊤ else ⊥

Tactic:
convert! traceDual_top'
  rw [← IsFractionRing.surjective_iff_isField (R := A) (K := K),
    LinearMap.range_eq_top.mpr (Algebra.trace_surjective K L),
    ← RingHom.range_eq_top, _root_.eq_top_iff]
  simp [SetLike.le_def]

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✝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 : IsFractionRing A K
inst✝³ : IsDomain A
inst✝² : IsFractionRing B L
inst✝¹ : IsDomain B
inst✝ : FaithfulSMul A B
I : Submodule B (FractionRing B)
x : L
y : FractionRing B
hy : y ∈ I
⊢ (∃ y_1,
      (algebraMap A (FractionRing A)) y_1 =
        (Algebra.trace (FractionRing A) (FractionRing B)) ((FractionRing.algEquiv B L).symm x * y)) ↔
    ∃ y_1, (algebraMap A K) y_1 = (Algebra.trace K L) (x * (FractionRing.algEquiv B L) y)

Tactic:
rw [Algebra.trace_eq_of_equiv_equiv (FractionRing.algEquiv A K).toRingEquiv
    (FractionRing.algEquiv B L).toRingEquiv]

Goal state after:
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 : IsFractionRing A K
inst✝³ : IsDomain A
inst✝² : IsFractionRing B L
inst✝¹ : IsDomain B
inst✝ : FaithfulSMul A B
I : Submodule B (FractionRing B)
x : L
y : FractionRing B
hy : y ∈ I
⊢ (∃ y_1,
      (algebraMap A (FractionRing A)) y_1 =
        (FractionRing.algEquiv A K).toRingEquiv.symm
          ((Algebra.trace K L) ((FractionRing.algEquiv B L).toRingEquiv ((FractionRing.algEquiv B L).symm x * y)))) ↔
    ∃ y_1, (algebraMap A K) y_1 = (Algebra.trace K L) (x * (FractionRing.algEquiv B L) y)

case he
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 : IsFractionRing A K
inst✝³ : IsDomain A
inst✝² : IsFractionRing B L
inst✝¹ : IsDomain B
inst✝ : FaithfulSMul A B
I : Submodule B (FractionRing B)
x : L
y : FractionRing B
hy : y ∈ I
⊢ (algebraMap K L).comp ↑(FractionRing.algEquiv A K).toRingEquiv =
    (↑(FractionRing.algEquiv B L).toRingEquiv).comp (algebraMap (FractionRing A) (FractionRing B))
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: case pos
n : ℕ
hn✝ : 2 ≤ n
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
ζ : K
p : ℕ
hF : Fact (Nat.Prime p)
hζ : IsPrimitiveRoot ζ n
this : NeZero n
μ : ↥ℚ⟮ζ⟯
hC : IsCyclotomicExtension {n} ℚ ↥ℚ⟮ζ⟯
hμ : IsPrimitiveRoot μ n
hp : ↑p ∣ (Algebra.norm ℤ) (hμ.toInteger - 1) ^ Module.finrank (↥ℚ⟮ζ⟯) K
h : (Algebra.norm ℤ) (hζ.toInteger - 1) = (Algebra.norm ℤ) (hμ.toInteger - 1) ^ Module.finrank (↥ℚ⟮ζ⟯) K
hn : 2 < n
q : ℕ
hq : Nat.Prime q
t : ℕ
hn' : q ^ t = n
⊢ ∃ q, Nat.Prime q ∧ ∃ t s, t ≠ 0 ∧ n = q ^ t ∧ ↑p ∣ ↑q ^ s

Tactic:
have : Fact (Nat.Prime q) := ⟨hq⟩

Goal state after:
case pos
n : ℕ
hn✝ : 2 ≤ n
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
ζ : K
p : ℕ
hF : Fact (Nat.Prime p)
hζ : IsPrimitiveRoot ζ n
this✝ : NeZero n
μ : ↥ℚ⟮ζ⟯
hC : IsCyclotomicExtension {n} ℚ ↥ℚ⟮ζ⟯
hμ : IsPrimitiveRoot μ n
hp : ↑p ∣ (Algebra.norm ℤ) (hμ.toInteger - 1) ^ Module.finrank (↥ℚ⟮ζ⟯) K
h : (Algebra.norm ℤ) (hζ.toInteger - 1) = (Algebra.norm ℤ) (hμ.toInteger - 1) ^ Module.finrank (↥ℚ⟮ζ⟯) K
hn : 2 < n
q : ℕ
hq : Nat.Prime q
t : ℕ
hn' : q ^ t = n
this : Fact (Nat.Prime q)
⊢ ∃ q, Nat.Prime q ∧ ∃ t s, t ≠ 0 ∧ n = q ^ t ∧ ↑p ∣ ↑q ^ s
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 (Polynomial.map (algebraMap A K) (minpoly A x))) ≠ 0
⊢ (algebraMap B L) ((aeval x) (derivative (minpoly A x))) ≠ 0

Tactic:
derivative_map,

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)) (Polynomial.map (algebraMap A K) (derivative (minpoly A x))) ≠ 0
⊢ (algebraMap B L) ((aeval x) (derivative (minpoly A x))) ≠ 0
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 : ℕ
hB : minkowskiBound K 1 < ↑(convexBodyLTFactor K) * ↑B
x : 𝓞 K
hx : x ≠ 0
hx' : (algebraMap (𝓞 K) K) x ≠ 0
f : InfinitePlace K → ℝ≥0 := fun w ↦ ⟨w ↑x / 2, ⋯⟩
this : ∀ (w : InfinitePlace K), w ≠ w₁ → f w ≠ 0
g : InfinitePlace K → ℝ≥0
h_geqf : ∀ (w : InfinitePlace K), w ≠ w₁ → g w = f w
h_gprod : ∏ w, g w ^ w.mult = ↑B
y : 𝓞 K
h_ynz : y ≠ 0
h_yle : ∀ (w : InfinitePlace K), w ↑y < ↑(g w)
⊢ ∏ w, ↑(g w) ^ w.mult ≤ ↑B

Tactic:
simp_rw [← NNReal.coe_pow, ← NNReal.coe_prod]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w₁ : InfinitePlace K
B : ℕ
hB : minkowskiBound K 1 < ↑(convexBodyLTFactor K) * ↑B
x : 𝓞 K
hx : x ≠ 0
hx' : (algebraMap (𝓞 K) K) x ≠ 0
f : InfinitePlace K → ℝ≥0 := fun w ↦ ⟨w ↑x / 2, ⋯⟩
this : ∀ (w : InfinitePlace K), w ≠ w₁ → f w ≠ 0
g : InfinitePlace K → ℝ≥0
h_geqf : ∀ (w : InfinitePlace K), w ≠ w₁ → g w = f w
h_gprod : ∏ w, g w ^ w.mult = ↑B
y : 𝓞 K
h_ynz : y ≠ 0
h_yle : ∀ (w : InfinitePlace K), w ↑y < ↑(g w)
⊢ ↑(∏ a, g a ^ a.mult) ≤ ↑B
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 : ℕ
hB : minkowskiBound K 1 < ↑(convexBodyLTFactor K) * ↑B
n : ℕ
⊢ ((Algebra.norm ℤ) ↑(seq K w₁ hB n)).natAbs ≤ B

Tactic:
by
  cases n with
  | zero =>
      have : 1 ≤ B := by
        contrapose! hB
        simp only [Nat.lt_one_iff.mp hB, CharP.cast_eq_zero, mul_zero, zero_le]
      simp only [ne_eq, seq, map_one, Int.natAbs_one, this]
  | succ n =>
      rw [← Nat.cast_le (α := ℚ), Nat.cast_natAbs, Int.cast_abs, Algebra.coe_norm_int]
      exact (seq_next K w₁ hB (seq K w₁ hB n).prop).choose_spec.2.2

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

Goal state before:
pp: case «0»
K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
hcong : ∃ n, 3 ∣ ↑1 - ↑n
⊢ 1 = 1 ∨ 1 = -1

pp: case «1»
K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
hcong : ∃ n, 3 ∣ ↑(-1) - ↑n
⊢ -1 = 1 ∨ -1 = -1

pp: case «2»
K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
hcong : ∃ n, 3 ∣ ↑⋯.unit - ↑n
⊢ ⋯.unit = 1 ∨ ⋯.unit = -1

pp: case «3»
K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
hcong : ∃ n, 3 ∣ ↑(-⋯.unit) - ↑n
⊢ -⋯.unit = 1 ∨ -⋯.unit = -1

pp: case «4»
K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
hcong : ∃ n, 3 ∣ ↑(⋯.unit ^ 2) - ↑n
⊢ ⋯.unit ^ 2 = 1 ∨ ⋯.unit ^ 2 = -1

pp: case «5»
K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
hcong : ∃ n, 3 ∣ ↑(-⋯.unit ^ 2) - ↑n
⊢ -⋯.unit ^ 2 = 1 ∨ -⋯.unit ^ 2 = -1

Tactic:
· left; rfl

Goal state after:
case «1»
K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
hcong : ∃ n, 3 ∣ ↑(-1) - ↑n
⊢ -1 = 1 ∨ -1 = -1

case «2»
K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
hcong : ∃ n, 3 ∣ ↑⋯.unit - ↑n
⊢ ⋯.unit = 1 ∨ ⋯.unit = -1

case «3»
K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
hcong : ∃ n, 3 ∣ ↑(-⋯.unit) - ↑n
⊢ -⋯.unit = 1 ∨ -⋯.unit = -1

case «4»
K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
hcong : ∃ n, 3 ∣ ↑(⋯.unit ^ 2) - ↑n
⊢ ⋯.unit ^ 2 = 1 ∨ ⋯.unit ^ 2 = -1

case «5»
K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
hcong : ∃ n, 3 ∣ ↑(-⋯.unit ^ 2) - ↑n
⊢ -⋯.unit ^ 2 = 1 ∨ -⋯.unit ^ 2 = -1
Module: Mathlib.NumberTheory.NumberField.Units.DirichletTheorem

Goal state before:
pp: case refine_1
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ (Submodule.map (MonoidHom.toAdditive (QuotientGroup.mk' (torsion K))).toIntLinearMap ⊤).FG

Tactic:
Submodule.map_top,

Goal state after:
case refine_1
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ (MonoidHom.toAdditive (QuotientGroup.mk' (torsion K))).toIntLinearMap.range.FG
Module: Mathlib.RingTheory.DedekindDomain.Different

Goal state before:
pp: A : Type u_1
B : Type u_3
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : IsDomain A
inst✝⁵ : IsDedekindDomain A
inst✝⁴ : IsDedekindDomain B
inst✝³ : IsTorsionFree A B
inst✝² : Module.Finite A B
inst✝¹ : Algebra.IsSeparable (FractionRing A) (FractionRing B)
p : Ideal A
inst✝ : p.IsMaximal
P : Ideal B
e : ℕ
hp : p ≠ ⊥
hP : P ^ e ∣ Ideal.map (algebraMap A B) p
⊢ P ^ (e - 1) ∣ differentIdeal A B

Tactic:
by
  have : IsLocalization (algebraMapSubmonoid B A⁰) (FractionRing B) :=
    IsIntegralClosure.isLocalization _ (FractionRing A) _ _
  have : FiniteDimensional (FractionRing A) (FractionRing B) := .of_isLocalization A B A⁰
  by_cases he : e = 0
  · rw [he, pow_zero]; exact one_dvd _
  exact pow_sub_one_dvd_differentIdeal_aux A (FractionRing A) (FractionRing B) _ he hp hP

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

Goal state before:
pp: p : ℕ
inst✝ : Fact (Nat.Prime p)
a b : ℤ
⊢ legendreSym p (a * b) = legendreSym p a * legendreSym p b

Tactic:
simp [legendreSym, Int.cast_mul, map_mul]

Goal state after:
Module: Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum

Goal state before:
pp: F : Type u_1
inst✝¹ : Field F
inst✝ : Fintype F
⊢ IsSquare 2 ↔ Fintype.card F % 8 ≠ 3 ∧ Fintype.card F % 8 ≠ 5

Tactic:
classical
  by_cases hF : ringChar F = 2
  · have h := FiniteField.even_card_of_char_two hF
    simp only [FiniteField.isSquare_of_char_two hF, true_iff]
    lia
  · have h := FiniteField.odd_card_of_char_ne_two hF
    rw [← quadraticChar_one_iff_isSquare (Ring.two_ne_zero hF), quadraticChar_two hF,
      χ₈_nat_eq_if_mod_eight]
    lia

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 : 𝓞 K
hx : S'.a ^ 3 - 1 = λ ^ 4 * x
y : 𝓞 K
hy : S'.b ^ 3 + 1 = λ ^ 4 * y
⊢ ↑S'.u⁻¹ * (S'.a ^ 3 + S'.b ^ 3) = ↑S'.u⁻¹ * (↑S'.u * S'.c ^ 3)

Tactic:
by rw [S'.H]

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 : 𝓞 K
hx : S'.a ^ 3 + 1 = λ ^ 4 * x
y : 𝓞 K
hy : S'.b ^ 3 - 1 = λ ^ 4 * y
⊢ ↑S'.u⁻¹ * (S'.a ^ 3 + S'.b ^ 3) = ↑S'.u⁻¹ * (↑S'.u * S'.c ^ 3)

Tactic:
by rw [S'.H]

Goal state after:
Module: Mathlib.RingTheory.Ideal.NatInt

Goal state before:
pp: s : Set ℕ
⊢ 1 ∈ span s ↔ 1 ∈ s

Tactic:
rw [← SetLike.mem_coe, ← not_iff_not]

Goal state after:
s : Set ℕ
⊢ 1 ∉ ↑(span s) ↔ 1 ∉ s
Module: Mathlib.NumberTheory.FrobeniusNumber

Goal state before:
pp: m n✝ : ℕ
s✝ t : Set ℕ
n : ℕ
s : Submodule ℕ ℕ
⊢ s.FG

Tactic:
by
    have ⟨t, n, hts, hn⟩ := exists_mem_span_nat_finset_of_ge s
    classical
    refine ⟨t ∪ {m ∈ Finset.range n | m ∈ s}, (Submodule.span_le.mpr ?_).antisymm fun m hm ↦ ?_⟩
    · simpa using ⟨hts, fun _ ↦ And.right⟩
    obtain le | gt := le_or_gt n m
    · exact Submodule.span_mono (by simp) (hn m le (setGcd_dvd_of_mem hm))
    · exact Submodule.subset_span (by simpa using .inr ⟨gt, hm⟩)

Goal state after:
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 : 2 * p * (b ^ 2 - 1) ∣ (b ^ 2 - 1) * (A * B - 1)
ha₇ : A * B ∣ b ^ (2 * p) - 1
q : ℕ
hq : A * B - 1 = 2 * p * q
ha₈ : b ^ (2 * p) - 1 ∣ b ^ (A * B - 1) - 1
⊢ (A * B).ProbablePrime b

Tactic:
exact dvd_trans ha₇ ha₈

Goal state after:
Module: Mathlib.NumberTheory.HeckeRing.Defs

Goal state before:
pp: G : Type u_1
inst✝ : Group G
Δ : Submonoid G
H₁ H₂ : Subgroup G
h : IsHeckeTriple Δ H₁ H₂
⊢ Δ ≤ (commensurator H₁).toSubmonoid

Tactic:
h.commensurable.eq

Goal state after:
G : Type u_1
inst✝ : Group G
Δ : Submonoid G
H₁ H₂ : Subgroup G
h : IsHeckeTriple Δ H₁ H₂
⊢ Δ ≤ (commensurator H₂).toSubmonoid
Module: Mathlib.NumberTheory.Height.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : AdmissibleAbsValues K
ι : Type u_2
ι' : Type u_3
e : ι ≃ ι'
x : ι' → K
H : ∀ (v : AbsoluteValue K ℝ), ⨆ i, v (x (e i)) = ⨆ i, v (x i)
⊢ mulHeight (x ∘ ⇑e) = mulHeight x

Tactic:
rcases eq_or_ne x 0 with rfl | hx

Goal state after:
case inl
K : Type u_1
inst✝¹ : Field K
inst✝ : AdmissibleAbsValues K
ι : Type u_2
ι' : Type u_3
e : ι ≃ ι'
H : ∀ (v : AbsoluteValue K ℝ), ⨆ i, v (0 (e i)) = ⨆ i, v (0 i)
⊢ mulHeight (0 ∘ ⇑e) = mulHeight 0

case inr
K : Type u_1
inst✝¹ : Field K
inst✝ : AdmissibleAbsValues K
ι : Type u_2
ι' : Type u_3
e : ι ≃ ι'
x : ι' → K
H : ∀ (v : AbsoluteValue K ℝ), ⨆ i, v (x (e i)) = ⨆ i, v (x i)
hx : x ≠ 0
⊢ mulHeight (x ∘ ⇑e) = mulHeight x