mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Three

Goal state before:
pp: K : Type u_1
inst✝ : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
⊢ ↑(↑⋯.unit ^ 2 + ↑⋯.unit + 1) = ↑0

Tactic:
;

Goal state after:
K : Type u_1
inst✝ : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
⊢ ↑(↑⋯.unit ^ 2 + ↑⋯.unit + 1) = ↑0
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

Tactic:
;

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

Goal state before:
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

Tactic:
;

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
Module: Mathlib.NumberTheory.NumberField.Units.DirichletTheorem

Goal state before:
pp: case refine_2
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
B : Basis { w // w ≠ w₀ } ℝ ({ w // w ≠ w₀ } → ℝ) := Pi.basisFun ℝ { w // w ≠ w₀ }
v : { w // w ≠ w₀ } → logSpace K := fun w ↦ (logEmbedding K) (Additive.ofMul ⋯.choose)
w : { w // w ≠ w₀ }
⊢ -↑(if w₀.IsReal then 1 else 2) < 0

Tactic:
;

Goal state after:
case refine_2
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
B : Basis { w // w ≠ w₀ } ℝ ({ w // w ≠ w₀ } → ℝ) := Pi.basisFun ℝ { w // w ≠ w₀ }
v : { w // w ≠ w₀ } → logSpace K := fun w ↦ (logEmbedding K) (Additive.ofMul ⋯.choose)
w : { w // w ≠ w₀ }
⊢ -↑(if w₀.IsReal then 1 else 2) < 0
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Three

Goal state before:
pp: K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
n : ℤ
x : 𝓞 K
hx : ↑(-⋯.unit ^ 2) - ↑n = 3 * x
⊢ ↑⋯.toInteger = ↑(hζ.toInteger ^ 2)

Tactic:
;

Goal state after:
K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
n : ℤ
x : 𝓞 K
hx : ↑(-⋯.unit ^ 2) - ↑n = 3 * x
⊢ ↑⋯.toInteger = ↑(hζ.toInteger ^ 2)
Module: Mathlib.NumberTheory.NumberField.Units.DirichletTheorem

Goal state before:
pp: case refine_1
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
B : Basis { w // w ≠ w₀ } ℝ ({ w // w ≠ w₀ } → ℝ) := Pi.basisFun ℝ { w // w ≠ w₀ }
v : { w // w ≠ w₀ } → logSpace K := fun w ↦ (logEmbedding K) (Additive.ofMul ⋯.choose)
w x : { w // w ≠ w₀ }
hx : x ∈ univ.erase w
⊢ 0 < ↑(if (↑x).IsReal then 1 else 2)

Tactic:
;

Goal state after:
case refine_1
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
B : Basis { w // w ≠ w₀ } ℝ ({ w // w ≠ w₀ } → ℝ) := Pi.basisFun ℝ { w // w ≠ w₀ }
v : { w // w ≠ w₀ } → logSpace K := fun w ↦ (logEmbedding K) (Additive.ofMul ⋯.choose)
w x : { w // w ≠ w₀ }
hx : x ∈ univ.erase w
⊢ 0 < ↑(if (↑x).IsReal then 1 else 2)
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Three

Goal state before:
pp: case inr.inl
K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
x : 𝓞 K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
this : Finite (𝓞 K ⧸ Ideal.span {λ})
x✝² : Fintype (𝓞 K ⧸ Ideal.span {λ}) := ⋯
x✝¹ : Ring (𝓞 K ⧸ Ideal.span {λ}) := ⋯
x✝ : AddGroup (𝓞 K ⧸ Ideal.span {λ}) := ⋯
h3 : Fintype.card (𝓞 K ⧸ Ideal.span {λ}) = 3
h : (Ideal.Quotient.mk (Ideal.span {λ})) x = 1
⊢ λ ∣ x - 1 ∨ λ ∣ x + 1

Tactic:
;

Goal state after:
case inr.inl
K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
x : 𝓞 K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
this : Finite (𝓞 K ⧸ Ideal.span {λ})
x✝² : Fintype (𝓞 K ⧸ Ideal.span {λ}) := ⋯
x✝¹ : Ring (𝓞 K ⧸ Ideal.span {λ}) := ⋯
x✝ : AddGroup (𝓞 K ⧸ Ideal.span {λ}) := ⋯
h3 : Fintype.card (𝓞 K ⧸ Ideal.span {λ}) = 3
h : (Ideal.Quotient.mk (Ideal.span {λ})) x = 1
⊢ λ ∣ x - 1 ∨ λ ∣ x + 1
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Three

Goal state before:
pp: case inr.inr
K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
x : 𝓞 K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
this : Finite (𝓞 K ⧸ Ideal.span {λ})
x✝² : Fintype (𝓞 K ⧸ Ideal.span {λ}) := ⋯
x✝¹ : Ring (𝓞 K ⧸ Ideal.span {λ}) := ⋯
x✝ : AddGroup (𝓞 K ⧸ Ideal.span {λ}) := ⋯
h3 : Fintype.card (𝓞 K ⧸ Ideal.span {λ}) = 3
h : (Ideal.Quotient.mk (Ideal.span {λ})) x = -1
⊢ λ ∣ x - 1 ∨ λ ∣ x + 1

Tactic:
;

Goal state after:
case inr.inr
K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
x : 𝓞 K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
this : Finite (𝓞 K ⧸ Ideal.span {λ})
x✝² : Fintype (𝓞 K ⧸ Ideal.span {λ}) := ⋯
x✝¹ : Ring (𝓞 K ⧸ Ideal.span {λ}) := ⋯
x✝ : AddGroup (𝓞 K ⧸ Ideal.span {λ}) := ⋯
h3 : Fintype.card (𝓞 K ⧸ Ideal.span {λ}) = 3
h : (Ideal.Quotient.mk (Ideal.span {λ})) x = -1
⊢ λ ∣ x - 1 ∨ λ ∣ x + 1
Module: Mathlib.NumberTheory.NumberField.Units.DirichletTheorem

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x ζ : (𝓞 K)ˣ
f : Fin (rank K) → ℤ
hζ : ζ ∈ torsion K
h : x = ζ * ∏ i, fundSystem K i ^ f i
⊢ ∑ i, Additive.ofMul ↑(fundSystem K i ^ f i) = ∑ i, f i • Additive.ofMul ↑(fundSystem K i)

Tactic:
;

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x ζ : (𝓞 K)ˣ
f : Fin (rank K) → ℤ
hζ : ζ ∈ torsion K
h : x = ζ * ∏ i, fundSystem K i ^ f i
⊢ ∑ i, Additive.ofMul ↑(fundSystem K i ^ f i) = ∑ i, f i • Additive.ofMul ↑(fundSystem K i)
Module: Mathlib.RingTheory.DedekindDomain.Different

Goal state before:
pp: case pos
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
this✝ : IsLocalization (algebraMapSubmonoid B A⁰) (FractionRing B)
this : FiniteDimensional (FractionRing A) (FractionRing B)
he : e = 0
⊢ 1 ∣ differentIdeal A B

Tactic:
;

Goal state after:
case pos
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
this✝ : IsLocalization (algebraMapSubmonoid B A⁰) (FractionRing B)
this : FiniteDimensional (FractionRing A) (FractionRing B)
he : e = 0
⊢ 1 ∣ differentIdeal A B
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Three

Goal state before:
pp: K : Type u_1
inst✝ : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
x : 𝓞 K
⊢ x ^ 3 - x ^ 2 * 0 + x * 0 - 1 = x ^ 3 - 1

Tactic:
;

Goal state after:
K : Type u_1
inst✝ : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
x : 𝓞 K
⊢ x ^ 3 - x ^ 2 * 0 + x * 0 - 1 = x ^ 3 - 1
Module: Mathlib.NumberTheory.FLT.Three

Goal state before:
pp: ⊢ ∀ {n : ZMod 9}, (castHom ⋯ (ZMod 3)) n ≠ 0 → n ^ 3 = 1 ∨ n ^ 3 = 8

Tactic:
;

Goal state after:
⊢ ∀ {n : ZMod 9}, (castHom ⋯ (ZMod 3)) n ≠ 0 → n ^ 3 = 1 ∨ n ^ 3 = 8
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Three

Goal state before:
pp: K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
x y : 𝓞 K
hy : x - 1 = λ * y
⊢ (x - 1) * (x - hζ.toInteger) * (x - hζ.toInteger ^ 2) = (x - 1) * (x - 1 - λ) * (x - 1 - λ * (hζ.toInteger + 1))

Tactic:
;

Goal state after:
K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
x y : 𝓞 K
hy : x - 1 = λ * y
⊢ (x - 1) * (x - hζ.toInteger) * (x - hζ.toInteger ^ 2) = (x - 1) * (x - 1 - λ) * (x - 1 - λ * (hζ.toInteger + 1))
Module: Mathlib.RingTheory.DedekindDomain.Different

Goal state before:
pp: case neg
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 : IsDedekindDomain A
inst✝³ : IsDedekindDomain B
inst✝² : IsTorsionFree A B
inst✝¹ : Module.Finite A B
inst✝ : Algebra.IsSeparable (FractionRing A) (FractionRing B)
p : Ideal A
P Q : Ideal B
hP : P * Q = Ideal.map (algebraMap A B) p
x : B
hxQ : x ∈ Q
hx : (intTrace A B) x ∉ p
hp : ¬p = ⊥
this : Algebra (A ⧸ p) (B ⧸ Q) := Ideal.Quotient.algebraQuotientOfLEComap ⋯
⊢ ¬P ∣ differentIdeal A B

Tactic:
let K := FractionRing A

Goal state after:
case neg
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 : IsDedekindDomain A
inst✝³ : IsDedekindDomain B
inst✝² : IsTorsionFree A B
inst✝¹ : Module.Finite A B
inst✝ : Algebra.IsSeparable (FractionRing A) (FractionRing B)
p : Ideal A
P Q : Ideal B
hP : P * Q = Ideal.map (algebraMap A B) p
x : B
hxQ : x ∈ Q
hx : (intTrace A B) x ∉ p
hp : ¬p = ⊥
this : Algebra (A ⧸ p) (B ⧸ Q) := Ideal.Quotient.algebraQuotientOfLEComap ⋯
K : Type u_1 := FractionRing A
⊢ ¬P ∣ differentIdeal A B
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Three

Goal state before:
pp: K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
x y : 𝓞 K
hy : x - 1 = λ * y
⊢ λ * y * (λ * y - λ) * (λ * y - λ * (↑η + 1)) = λ ^ 3 * (y * (y - 1) * (y - (↑η + 1)))

Tactic:
;

Goal state after:
K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
x y : 𝓞 K
hy : x - 1 = λ * y
⊢ λ * y * (λ * y - λ) * (λ * y - λ * (↑η + 1)) = λ ^ 3 * (y * (y - 1) * (y - (↑η + 1)))
Module: Mathlib.NumberTheory.LegendreSymbol.Basic

Goal state before:
pp: case mp
p : ℕ
inst✝ : Fact (Nat.Prime p)
a : ZMod p
ha : a ≠ 0
y : (ZMod p)ˣ
hy : ↑y * ↑y = a
⊢ IsSquare a

Tactic:
;

Goal state after:
case mp
p : ℕ
inst✝ : Fact (Nat.Prime p)
a : ZMod p
ha : a ≠ 0
y : (ZMod p)ˣ
hy : ↑y * ↑y = a
⊢ IsSquare a
Module: Mathlib.NumberTheory.LegendreSymbol.Basic

Goal state before:
pp: case mpr
p : ℕ
inst✝ : Fact (Nat.Prime p)
y : ZMod p
ha : y * y ≠ 0
hy : y ≠ 0
⊢ ↑(Units.mk0 y hy) * ↑(Units.mk0 y hy) = y * y

Tactic:
;

Goal state after:
case mpr
p : ℕ
inst✝ : Fact (Nat.Prime p)
y : ZMod p
ha : y * y ≠ 0
hy : y ≠ 0
⊢ ↑(Units.mk0 y hy) * ↑(Units.mk0 y hy) = y * y
Module: Mathlib.NumberTheory.LegendreSymbol.Basic

Goal state before:
pp: case inl
inst✝ : Fact (Nat.Prime 2)
⊢ ∀ {a : ZMod 2}, a ≠ 0 → a ^ (2 / 2) = 1 ∨ a ^ (2 / 2) = -1

Tactic:
;

Goal state after:
case inl
inst✝ : Fact (Nat.Prime 2)
⊢ ∀ {a : ZMod 2}, a ≠ 0 → a ^ (2 / 2) = 1 ∨ a ^ (2 / 2) = -1
Module: Mathlib.NumberTheory.LegendreSymbol.Basic

Goal state before:
pp: case inl
inst✝ : Fact (Nat.Prime 2)
a : ZMod 2
⊢ a ≠ 0 → a ^ (2 / 2) = 1 ∨ a ^ (2 / 2) = -1

Tactic:
;

Goal state after:
case inl
inst✝ : Fact (Nat.Prime 2)
a : ZMod 2
⊢ a ≠ 0 → a ^ (2 / 2) = 1 ∨ a ^ (2 / 2) = -1
Module: Mathlib.NumberTheory.LegendreSymbol.Basic

Goal state before:
pp: case neg
a : ℤ
inst✝ : Fact (Nat.Prime 2)
hc : ringChar (ZMod 2) = 2
b : ZMod 2
⊢ ¬b = 0 → 1 = b ^ 1

Tactic:
;

Goal state after:
case neg
a : ℤ
inst✝ : Fact (Nat.Prime 2)
hc : ringChar (ZMod 2) = 2
b : ZMod 2
⊢ ¬b = 0 → 1 = b ^ 1
Module: Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic

Goal state before:
pp: F : Type u_1
inst✝² : Field F
inst✝¹ : Fintype F
inst✝ : DecidableEq F
a : F
ha : ¬IsUnit a
⊢ quadraticCharFun F 0 = 0

Tactic:
;

Goal state after:
F : Type u_1
inst✝² : Field F
inst✝¹ : Fintype F
inst✝ : DecidableEq F
a : F
ha : ¬IsUnit a
⊢ quadraticCharFun F 0 = 0
Module: Mathlib.NumberTheory.LegendreSymbol.Basic

Goal state before:
pp: p : ℕ
inst✝ : Fact (Nat.Prime p)
a : ℕ
⊢ ¬IsSquare ↑↑a ↔ ¬IsSquare ↑a

Tactic:
;

Goal state after:
p : ℕ
inst✝ : Fact (Nat.Prime p)
a : ℕ
⊢ ¬IsSquare ↑↑a ↔ ¬IsSquare ↑a
Module: Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic

Goal state before:
pp: F : Type u_1
inst✝² : Field F
inst✝¹ : Fintype F
inst✝ : DecidableEq F
a : F
ha : a ≠ 0
h : (quadraticChar F) a = -1
⊢ ¬-1 = 1

Tactic:
;

Goal state after:
F : Type u_1
inst✝² : Field F
inst✝¹ : Fintype F
inst✝ : DecidableEq F
a : F
ha : a ≠ 0
h : (quadraticChar F) a = -1
⊢ ¬-1 = 1
Module: Mathlib.NumberTheory.LegendreSymbol.Basic

Goal state before:
pp: p : ℕ
inst✝ : Fact (Nat.Prime p)
a : ℤ
ha : ↑a ≠ 0
x y : ZMod p
hx : x ≠ 0
hxy : x ^ 2 - ↑a * y ^ 2 = 0
hy : y ≠ 0
⊢ legendreSym p a = 1

Tactic:
exact eq_one_of_sq_sub_mul_sq_eq_zero ha hy hxy

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

Goal state before:
pp: case pos
F : Type u_1
inst✝² : Field F
inst✝¹ : Fintype F
inst✝ : DecidableEq F
a : F
ha : a = 0
⊢ (quadraticChar F) a = 0

Tactic:
;

Goal state after:
case pos
F : Type u_1
inst✝² : Field F
inst✝¹ : Fintype F
inst✝ : DecidableEq F
a : F
ha : a = 0
⊢ (quadraticChar F) a = 0
Module: Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic

Goal state before:
pp: case pos
F : Type u_1
inst✝² : Field F
inst✝¹ : Fintype F
inst✝ : DecidableEq F
a : F
ha : a = 0
⊢ (quadraticChar F) 0 = 0

Tactic:
;

Goal state after:
case pos
F : Type u_1
inst✝² : Field F
inst✝¹ : Fintype F
inst✝ : DecidableEq F
a : F
ha : a = 0
⊢ (quadraticChar F) 0 = 0
Module: Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic

Goal state before:
pp: case neg
F : Type u_1
inst✝² : Field F
inst✝¹ : Fintype F
inst✝ : DecidableEq F
a : F
ha : ¬a = 0
⊢ (quadraticChar F) a = 1 ∨ (quadraticChar F) a = -1

Tactic:
;

Goal state after:
case neg
F : Type u_1
inst✝² : Field F
inst✝¹ : Fintype F
inst✝ : DecidableEq F
a : F
ha : ¬a = 0
⊢ (quadraticChar F) a = 1 ∨ (quadraticChar F) a = -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
⊢ P * P ∣ P * a

Tactic:
;

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
⊢ P * P ∣ P * a
Module: Mathlib.RingTheory.DedekindDomain.Different

Goal state before:
pp: case neg
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
⊢ P ∣ differentIdeal A B

Tactic:
let K := FractionRing A

Goal state after:
case neg
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
⊢ P ∣ differentIdeal A B
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 ≠ ⊥
a : Ideal B
K : Type u_1 := FractionRing A
L : Type u_3 := FractionRing B
hp' : ¬Ideal.map (algebraMap A B) p = ⊥
habot : a ≠ ⊥
inst✝¹ : ⊥.IsMaximal
inst✝ : ⊥.LiesOver p
H : ¬Algebra.IsSeparable (A ⧸ p) (B ⧸ ⊥)
ha : Ideal.map (algebraMap A B) p = ⊥ * a
hPa : ¬⊥ ∣ a
⊢ False

Tactic:
;

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 ≠ ⊥
a : Ideal B
K : Type u_1 := FractionRing A
L : Type u_3 := FractionRing B
hp' : ¬Ideal.map (algebraMap A B) p = ⊥
habot : a ≠ ⊥
inst✝¹ : ⊥.IsMaximal
inst✝ : ⊥.LiesOver p
H : ¬Algebra.IsSeparable (A ⧸ p) (B ⧸ ⊥)
ha : Ideal.map (algebraMap A B) p = ⊥ * a
hPa : ¬⊥ ∣ a
⊢ False
Module: Mathlib.NumberTheory.Fermat

Goal state before:
pp: n : ℕ
this : 3 ≤ n.fermatNumber
⊢ n.fermatNumber ≠ 1

Tactic:
;

Goal state after:
n : ℕ
this : 3 ≤ n.fermatNumber
⊢ n.fermatNumber ≠ 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
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 - 1 + (S'.b ^ 3 + 1)) = ↑S'.u⁻¹ * (S'.a ^ 3 + S'.b ^ 3)

Tactic:
;

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
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 - 1 + (S'.b ^ 3 + 1)) = ↑S'.u⁻¹ * (S'.a ^ 3 + S'.b ^ 3)
Module: Mathlib.NumberTheory.FermatPsp

Goal state before:
pp: b p : ℕ
x✝ : 2 ≤ b
hp : Odd p
⊢ (b ^ p - 1) / (b - 1) * ((b ^ p + 1) / (b + 1)) = (b ^ (2 * p) - 1) / (b ^ 2 - 1)

Tactic:
have q₁ : b - 1 ∣ b ^ p - 1 := by simpa only [one_pow] using Nat.sub_dvd_pow_sub_pow b 1 p
  have q₂ : b + 1 ∣ b ^ p + 1 := by simpa only [one_pow] using hp.nat_add_dvd_pow_add_pow b 1
  convert! Nat.div_mul_div_comm q₁ q₂ using 2 <;> rw [mul_comm (_ - 1), ← Nat.sq_sub_sq]
  ring_nf

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 + 1 + (S'.b ^ 3 - 1)) = ↑S'.u⁻¹ * (S'.a ^ 3 + S'.b ^ 3)

Tactic:
;

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
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 + 1 + (S'.b ^ 3 - 1)) = ↑S'.u⁻¹ * (S'.a ^ 3 + S'.b ^ 3)
Module: Mathlib.RingTheory.Ideal.NatInt

Goal state before:
pp: x✝ : ℕ
⊢ x✝ ∈ ↑(maximalIdeal ℕ) ↔ x✝ ∈ {1}ᶜ

Tactic:
;

Goal state after:
x✝ : ℕ
⊢ x✝ ∈ ↑(maximalIdeal ℕ) ↔ x✝ ∈ {1}ᶜ
Module: Mathlib.RingTheory.Ideal.NatInt

Goal state before:
pp: case inl
h : 0 ∈ maximalIdeal ℕ
lt : 0 < 1
⊢ 0 ∈ span {2, 3}

Tactic:
;

Goal state after:
case inl
h : 0 ∈ maximalIdeal ℕ
lt : 0 < 1
⊢ 0 ∈ span {2, 3}
Module: Mathlib.NumberTheory.FLT.Three

Goal state before:
pp: K : Type u_1
inst✝ : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S' : Solution' hζ
⊢ S'.a ^ 3 + S'.a ^ 2 * S'.b * (-↑⋯.unit - 1 + ↑η + 1) + S'.a * S'.b ^ 2 * (-↑⋯.unit - 1 + ↑η + 1) + S'.b ^ 3 =
    S'.a ^ 3 + S'.b ^ 3

Tactic:
;

Goal state after:
K : Type u_1
inst✝ : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S' : Solution' hζ
⊢ S'.a ^ 3 + S'.a ^ 2 * S'.b * (-↑⋯.unit - 1 + ↑η + 1) + S'.a * S'.b ^ 2 * (-↑⋯.unit - 1 + ↑η + 1) + S'.b ^ 3 =
    S'.a ^ 3 + S'.b ^ 3
Module: Mathlib.NumberTheory.FrobeniusNumber

Goal state before:
pp: case inl
s : Set ℕ
m : ℕ
h : setGcd s ∣ m
⊢ setGcd s ∣ m

pp: case inr
s : Set ℕ
n : ℕ
h : setGcd s ∣ n
m : ℕ
hm : m ∈ s
⊢ setGcd s ∣ m

Tactic:
;

Goal state after:
case inl
s : Set ℕ
m : ℕ
h : setGcd s ∣ m
⊢ setGcd s ∣ m

case inr
s : Set ℕ
n : ℕ
h : setGcd s ∣ n
m : ℕ
hm : m ∈ s
⊢ setGcd s ∣ m
Module: Mathlib.NumberTheory.FrobeniusNumber

Goal state before:
pp: case refl
s : Set ℕ
h0 : setGcd s = 0
x✝ : 0 ≥ 0
dvd : setGcd s ∣ 0
⊢ 0 ∈ span ↑∅

Tactic:
;

Goal state after:
case refl
s : Set ℕ
h0 : setGcd s = 0
x✝ : 0 ≥ 0
dvd : setGcd s ∣ 0
⊢ 0 ∈ span ↑∅
Module: Mathlib.RingTheory.DedekindDomain.Different

Goal state before:
pp: case inl
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)
inst✝ : ⊥.IsPrime
⊢ IsUnramifiedAt A ⊥

Tactic:
let K := FractionRing A

Goal state after:
case inl
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)
inst✝ : ⊥.IsPrime
K : Type u_1 := FractionRing A
⊢ IsUnramifiedAt A ⊥
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)
inst✝ : ⊥.IsPrime
K : Type u_1 := FractionRing A
L : Type u_3 := FractionRing B
x✝ : B
⊢ x✝ ∈ B⁰ ↔ x✝ ∈ ⊥.primeCompl

Tactic:
;

Goal state after:
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)
inst✝ : ⊥.IsPrime
K : Type u_1 := FractionRing A
L : Type u_3 := FractionRing B
x✝ : B
⊢ x✝ ∈ B⁰ ↔ x✝ ∈ ⊥.primeCompl
Module: Mathlib.NumberTheory.FLT.Three

Goal state before:
pp: K : Type u_1
inst✝ : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
⊢ S.a + hζ.toInteger * S.b = S.a + S.b + λ * S.b

Tactic:
;

Goal state after:
K : Type u_1
inst✝ : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
⊢ S.a + hζ.toInteger * S.b = S.a + S.b + λ * S.b
Module: Mathlib.NumberTheory.FLT.Three

Goal state before:
pp: K : Type u_1
inst✝ : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
⊢ S.a + hζ.toInteger ^ 2 * S.b = S.a + S.b + λ ^ 2 * S.b + 2 * λ * S.b

Tactic:
;

Goal state after:
K : Type u_1
inst✝ : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
⊢ S.a + hζ.toInteger ^ 2 * S.b = S.a + S.b + λ ^ 2 * S.b + 2 * λ * S.b
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
k k' : 𝓞 K
hk : λ ^ 2 * k' - S.b + ↑η ^ 2 * S.b = λ ^ 2 * k
hk' : S.a + S.b = λ ^ 2 * k'
⊢ λ ^ 2 * k' - S.b + hζ.toInteger ^ 2 * S.b = λ * (S.b * (hζ.toInteger + 1) + λ * k')

Tactic:
;

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
k k' : 𝓞 K
hk : λ ^ 2 * k' - S.b + ↑η ^ 2 * S.b = λ ^ 2 * k
hk' : S.a + S.b = λ ^ 2 * k'
⊢ λ ^ 2 * k' - S.b + hζ.toInteger ^ 2 * S.b = λ * (S.b * (hζ.toInteger + 1) + λ * k')