mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.NumberTheory.FactorisationProperties

Goal state before:
pp: n m : ℕ
h : n.Abundant
hm : m ≠ 0
⊢ (m * n).Abundant

Tactic:
by

Goal state after:
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 : 𝓞 K
h : ¬λ ∣ x
⊢ λ ^ 4 ∣ x ^ 3 - 1 ∨ λ ^ 4 ∣ x ^ 3 + 1

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FactorisationProperties

Goal state before:
pp: a : ℕ
⊢ Abundant 12

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FactorisationProperties

Goal state before:
pp: ⊢ {n | Even n ∧ n.Abundant}.Infinite

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FactorisationProperties

Goal state before:
pp: a : ℕ
⊢ Abundant 945

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FactorisationProperties

Goal state before:
pp: ⊢ {n | Odd n ∧ n.Abundant}.Infinite

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FactorisationProperties

Goal state before:
pp: ⊢ {n | Odd n ∧ n.Abundant}.Infinite

Tactic:
by
  rw [Set.infinite_iff_exists_gt]
  intro a
  have ha : Abundant 945 := by decide +kernel
  use (2 * a + 1) * 945
  grind [Abundant.mul_left ha (show 2 * a + 1 ≠ 0 by grind)]

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

Goal state before:
pp: a b c : ℤ
ha : a ≠ 0
Hgcd : {a, b, c}.gcd id = 1
h3a : 3 ∣ a
HF : a ^ 3 + b ^ 3 + c ^ 3 = 0
H : ∀ (a b c : ℤ), c ≠ 0 → ¬3 ∣ a → ¬3 ∣ b → 3 ∣ c → IsCoprime a b → a ^ 3 + b ^ 3 ≠ c ^ 3
hbc : IsCoprime (-b) (-c)
h3b : ¬3 ∣ b
h3c : ¬3 ∣ c
⊢ ¬3 ∣ -b

Tactic:
by

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

Goal state before:
pp: a b c : ℤ
ha : a ≠ 0
Hgcd : {a, b, c}.gcd id = 1
h3a : 3 ∣ a
HF : a ^ 3 + b ^ 3 + c ^ 3 = 0
H : ∀ (a b c : ℤ), c ≠ 0 → ¬3 ∣ a → ¬3 ∣ b → 3 ∣ c → IsCoprime a b → a ^ 3 + b ^ 3 ≠ c ^ 3
hbc : IsCoprime (-b) (-c)
h3b : ¬3 ∣ b
h3c : ¬3 ∣ c
⊢ ¬3 ∣ -c

Tactic:
by

Goal state after:
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
⊢ quadraticCharFun F a = 0 ↔ a = 0

Tactic:
by

Goal state after:
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
⊢ quadraticCharFun F 0 = 0

Tactic:
by

Goal state after:
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
⊢ quadraticCharFun F 1 = 1

Tactic:
by

Goal state after:
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
hF : ringChar F = 2
a : F
ha : a ≠ 0
⊢ quadraticCharFun F a = 1

Tactic:
by

Goal state after:
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
hF : ringChar F ≠ 2
a : F
ha : a ≠ 0
⊢ quadraticCharFun F a = if a ^ (Fintype.card F / 2) = 1 then 1 else -1

Tactic:
by

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

Goal state before:
pp: p : ℕ
inst✝ : Fact (Nat.Prime p)
x : (ZMod p)ˣ
hc : ¬p = 2
⊢ ringChar (ZMod p) ≠ 2

Tactic:
by

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

Goal state before:
pp: p : ℕ
inst✝ : Fact (Nat.Prime p)
x : (ZMod p)ˣ
hc : ¬p = 2
h₀ : IsSquare x ↔ x ^ (Fintype.card (ZMod p) / 2) = 1
⊢ (∃ y, y ^ 2 = x) ↔ IsSquare x

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: p : ℕ
inst✝ : Fact (Nat.Prime p)
a : ZMod p
ha : a ≠ 0
⊢ Units.mk0 a ha ^ (p / 2) = 1 ↔ a ^ (p / 2) = 1

Tactic:
by

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

Goal state before:
pp: p : ℕ
inst✝ : Fact (Nat.Prime p)
y : ZMod p
ha : y * y ≠ 0
⊢ y ≠ 0

Tactic:
by

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

Goal state before:
pp: p : ℕ
inst✝ : Fact (Nat.Prime p)
a : ZMod p
ha : a ≠ 0
⊢ IsSquare a ↔ a ^ (p / 2) = 1

Tactic:
by

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 : 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✝¹ : Algebra (A ⧸ p) (B ⧸ Q) := Ideal.Quotient.algebraQuotientOfLEComap ⋯
K : Type u_1 := FractionRing A
L : Type u_3 := FractionRing B
this✝ : IsLocalization (algebraMapSubmonoid B A⁰) L
this : FiniteDimensional K L
H : ↑Q ≤ FractionalIdeal.dual A K 1 * ↑(Ideal.map (algebraMap A B) p)
hne : 1 ≠ 0
y✝ : B
x : L
hx : x ∈ FractionalIdeal.dual A K 1
y : A
hy : y ∈ ↑p
⊢ 1 ≠ 0

Tactic:
by

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

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

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: p : ℕ
inst✝ : Fact (Nat.Prime p)
a : ℤ
⊢ ↑(legendreSym p a) = ↑a ^ (p / 2)

Tactic:
by

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

Goal state before:
pp: case neg.inr.inl
F : Type u_1
inst✝² : Field F
inst✝¹ : Fintype F
inst✝ : DecidableEq F
a b : F
ha : ¬a = 0
hb : ¬b = 0
hab : a * b ≠ 0
hF : ¬ringChar F = 2
hb' : b ^ (Fintype.card F / 2) = -1
h : -1 ≠ 1
ha' : a ^ (Fintype.card F / 2) = 1
⊢ (if -a ^ (Fintype.card F / 2) = 1 then 1 else -1) = -if a ^ (Fintype.card F / 2) = 1 then 1 else -1

Tactic:
simp only [ha', h, neg_neg, if_true, if_false]

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

Goal state before:
pp: case neg.inr.inr
F : Type u_1
inst✝² : Field F
inst✝¹ : Fintype F
inst✝ : DecidableEq F
a b : F
ha : ¬a = 0
hb : ¬b = 0
hab : a * b ≠ 0
hF : ¬ringChar F = 2
hb' : b ^ (Fintype.card F / 2) = -1
h : -1 ≠ 1
ha' : a ^ (Fintype.card F / 2) = -1
⊢ (if -a ^ (Fintype.card F / 2) = 1 then 1 else -1) = -if a ^ (Fintype.card F / 2) = 1 then 1 else -1

Tactic:
simp only [ha', h, neg_neg, if_true, if_false]

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

Goal state before:
pp: p : ℕ
inst✝ : Fact (Nat.Prime p)
⊢ legendreSym p 0 = 0

Tactic:
by

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

Goal state before:
pp: p : ℕ
inst✝ : Fact (Nat.Prime p)
⊢ legendreSym p 1 = 1

Tactic:
by

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:
by

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

Goal state before:
pp: p : ℕ
inst✝ : Fact (Nat.Prime p)
a : ℤ
ha : ↑a ≠ 0
⊢ legendreSym p (a ^ 2) = 1

Tactic:
by

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

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

Tactic:
by

Goal state after:
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 b : F
⊢ quadraticCharFun F (a * b) = quadraticCharFun F a * quadraticCharFun F b

Tactic:
by

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

Goal state before:
pp: p : ℕ
inst✝ : Fact (Nat.Prime p)
a : ℕ
ha0 : ↑a ≠ 0
⊢ legendreSym p ↑a = 1 ↔ IsSquare ↑a

Tactic:
by

Goal state after:
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 a = 0

Tactic:
by

Goal state after:
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
⊢ (quadraticChar F) 0 = 0

Tactic:
by

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

Goal state before:
pp: p : ℕ
inst✝ : Fact (Nat.Prime p)
a : ℕ
⊢ legendreSym p ↑a = -1 ↔ ¬IsSquare ↑a

Tactic:
by

Goal state after:
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
⊢ (quadraticChar F) a = 1 ↔ IsSquare a

Tactic:
by

Goal state after:
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
⊢ (quadraticChar F) (a ^ 2) = 1

Tactic:
by

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

Goal state before:
pp: 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
hx : x = a ∨ x = b ∨ x = c
⊢ 3 ∣ b

Tactic:
by

Goal state after:
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
hy : y ≠ 0
hxy : (x ^ 2 - ↑a * y ^ 2) * y⁻¹ ^ 2 = 0
⊢ (x ^ 2 - ↑a * y ^ 2) * y⁻¹ ^ 2 = (x * y⁻¹) ^ 2 - ↑a * (y * y⁻¹) ^ 2

Tactic:
by

Goal state after:
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
⊢ (quadraticChar F) a ^ 2 = 1

Tactic:
by

Goal state after:
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
⊢ ¬(quadraticChar F) a = 1

Tactic:
by

Goal state after:
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
hy : y ≠ 0
hxy : x ^ 2 - ↑a * y ^ 2 = 0
⊢ legendreSym p a = 1

Tactic:
by

Goal state after:
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
⊢ (quadraticChar F) a = -1 ↔ ¬IsSquare a

Tactic:
by

Goal state after:
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
⊢ y ≠ 0

Tactic:
by

Goal state after:
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
hF : ringChar F ≠ 2
⊢ ∃ a, (quadraticChar F) ↑a = -1

Tactic:
by

Goal state after:
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
⊢ legendreSym p a = 1

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: 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
⊢ a ^ 3 + b ^ 3 + c ^ 3 ≠ 0

Tactic:
by

Goal state after:
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
hF : ringChar F ≠ 2
a : F
⊢ ↑((quadraticChar F) a) = a ^ (Fintype.card F / 2)

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: p : ℕ
inst✝ : Fact (Nat.Prime p)
a : ℤ
h : legendreSym p a = -1
x y : ℤ
hxy : ↑p ∣ x ^ 2 - a * y ^ 2
⊢ ↑p ∣ x ∧ ↑p ∣ y

Tactic:
by

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

Goal state before:
pp: p : ℕ
inst✝ : Fact (Nat.Prime p)
hp : p ≠ 2
⊢ legendreSym p (-1) = χ₄ ↑p

Tactic:
by

Goal state after:
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
⊢ (quadraticChar F).IsQuadratic

Tactic:
by

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

Goal state before:
pp: p : ℕ
inst✝ : Fact (Nat.Prime p)
hp : p ≠ 2
a : ℤ
⊢ legendreSym p (-a) = χ₄ ↑p * legendreSym p a

Tactic:
by

Goal state after:
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
hF : ringChar F ≠ 2
⊢ quadraticChar F ≠ 1

Tactic:
by

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

Goal state before:
pp: p : ℕ
inst✝ : Fact (Nat.Prime p)
⊢ IsSquare (-1) ↔ p % 4 ≠ 3

Tactic:
by

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

Goal state before:
pp: p : ℕ
inst✝ : Fact (Nat.Prime p)
x y : ZMod p
hy : y ≠ 0
hxy : x ^ 2 = -y ^ 2
⊢ (x / y) ^ 2 = -1

Tactic:
by

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:
by

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

Goal state before:
pp: F : Type u_1
inst✝² : Field F
inst✝¹ : Fintype F
inst✝ : DecidableEq F
hF : ringChar F ≠ 2
⊢ -2 = -1 * 2

Tactic:
by

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

Goal state before:
pp: H : ∀ (a b c : ℤ), c ≠ 0 → ¬3 ∣ a → ¬3 ∣ b → 3 ∣ c → IsCoprime a b → a ^ 3 + b ^ 3 ≠ c ^ 3
a b c : ℤ
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
Hgcd : {a, b, c}.gcd id = 1
hF : (-c) ^ 3 + a ^ 3 + b ^ 3 = 0
h3c : 3 ∣ c
⊢ 3 ∣ -c

Tactic:
by

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

Goal state before:
pp: F : Type u_1
inst✝² : Field F
inst✝¹ : Fintype F
inst✝ : DecidableEq F
hF : ringChar F ≠ 2
⊢ 4 ∣ 8

Tactic:
by

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

Goal state before:
pp: F : Type u_1
inst✝² : Field F
inst✝¹ : Fintype F
inst✝ : DecidableEq F
hF : ringChar F ≠ 2
⊢ (quadraticChar F) (-2) = χ₈' ↑(Fintype.card F)

Tactic:
by

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 ≠ 5 ∧ Fintype.card F % 8 ≠ 7

Tactic:
by

Goal state after:
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
hF : ringChar F ≠ 2
a : F
s : Finset F := {x | x ^ 2 = a}.toFinset
b : F
h₀ : ¬b = 0
h : a = b * b
⊢ s = [b, -b].toFinset

Tactic:
by

Goal state after:
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
χ : MulChar F F' := (quadraticChar F).ringHomComp (algebraMap ℤ F')
⊢ χ ≠ 1

Tactic:
by

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

Goal state before:
pp: H : ∀ (a b c : ℤ), c ≠ 0 → ¬3 ∣ a → ¬3 ∣ b → 3 ∣ c → IsCoprime a b → a ^ 3 + b ^ 3 ≠ c ^ 3
⊢ FermatLastTheoremFor 3

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝ : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
⊢ 3 ≠ 0

Tactic:
by

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
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
H : FermatLastTheoremForThreeGen hζ
a b c : ℤ
hc : c ≠ 0
ha : ¬3 ∣ a
hb : ¬3 ∣ b
x✝ : 3 ∣ c
hcoprime : IsCoprime a b
h : a ^ 3 + b ^ 3 = c ^ 3
x : ℤ
hx : c = 3 * x
⊢ ↑c ≠ 0

Tactic:
by

Goal state after:
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:
by

Goal state after:
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
hF : ringChar F ≠ 2
a : F
⊢ ↑(#{x | x ^ 2 = a}.toFinset) = (quadraticChar F) a + 1

Tactic:
by

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
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
H : FermatLastTheoremForThreeGen hζ
a b c : ℤ
hc : c ≠ 0
ha : ¬3 ∣ a
hb : ¬3 ∣ b
x✝ : 3 ∣ c
hcoprime : IsCoprime a b
h : a ^ 3 + b ^ 3 = c ^ 3
x : ℤ
hx : c = 3 * x
hdvd : λ ∣ ↑a
⊢ 3 ≠ 2

Tactic:
by

Goal state after:
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
hF : ringChar F ≠ 2
⊢ (quadraticChar F) (-1) = χ₄ ↑(Fintype.card F)

Tactic:
by

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
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
H : FermatLastTheoremForThreeGen hζ
a b c : ℤ
hc : c ≠ 0
ha : ¬3 ∣ a
hb : ¬3 ∣ b
x✝ : 3 ∣ c
hcoprime : IsCoprime a b
h : a ^ 3 + b ^ 3 = c ^ 3
x : ℤ
hx : c = 3 * x
hdvd : (Algebra.norm ℤ) λ ∣ a
⊢ 3 ≠ 2

Tactic:
by

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

Goal state before:
pp: F : Type u_1
inst✝³ : Field F
inst✝² : Fintype F
inst✝¹ : DecidableEq F
hF : ringChar F ≠ 2
p : ℕ
inst✝ : Fact (Nat.Prime p)
hp₁ : p ≠ 2
hp₂ : ringChar F ≠ p
⊢ (quadraticChar F) ↑p = (quadraticChar (ZMod p)) (↑(χ₄ ↑(Fintype.card F)) * ↑(Fintype.card F))

Tactic:
by

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

Goal state before:
pp: F : Type u_1
inst✝¹ : Field F
inst✝ : Fintype F
⊢ IsSquare (-1) ↔ Fintype.card F % 4 ≠ 3

Tactic:
by

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
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
H : FermatLastTheoremForThreeGen hζ
a b c : ℤ
hc : c ≠ 0
ha : ¬3 ∣ a
hb : ¬3 ∣ b
x✝ : 3 ∣ c
hcoprime : IsCoprime a b
h : a ^ 3 + b ^ 3 = c ^ 3
x : ℤ
hx : c = 3 * x
hdvd : λ ∣ ↑b
⊢ 3 ≠ 2

Tactic:
by

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
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
H : FermatLastTheoremForThreeGen hζ
a b c : ℤ
hc : c ≠ 0
ha : ¬3 ∣ a
hb : ¬3 ∣ b
x✝ : 3 ∣ c
hcoprime : IsCoprime a b
h : a ^ 3 + b ^ 3 = c ^ 3
x : ℤ
hx : c = 3 * x
hdvd : (Algebra.norm ℤ) λ ∣ b
⊢ 3 ≠ 2

Tactic:
by

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

Goal state before:
pp: F : Type u_1
inst✝² : Field F
inst✝¹ : Fintype F
hF : ringChar F ≠ 2
p : ℕ
inst✝ : Fact (Nat.Prime p)
hp : p ≠ 2
⊢ IsSquare ↑p ↔ (quadraticChar (ZMod p)) (↑(χ₄ ↑(Fintype.card F)) * ↑(Fintype.card F)) ≠ -1

Tactic:
by

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 : 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
⊢ ¬P ∣ differentIdeal A B

Tactic:
by

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
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
H : FermatLastTheoremForThreeGen hζ
a b c : ℤ
hc : c ≠ 0
ha : ¬3 ∣ a
hb : ¬3 ∣ b
x✝ : 3 ∣ c
hcoprime : IsCoprime a b
h : a ^ 3 + b ^ 3 = c ^ 3
x : ℤ
hx : c = 3 * x
⊢ ↑c = ↑3 * ↑x

Tactic:
by

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

Goal state before:
pp: A : Type u_1
B : Type u_3
inst✝¹² : CommRing A
inst✝¹¹ : CommRing B
inst✝¹⁰ : Algebra A B
inst✝⁹ : IsDomain A
inst✝⁸ : IsDedekindDomain A
inst✝⁷ : IsDedekindDomain B
inst✝⁶ : IsTorsionFree A B
inst✝⁵ : Module.Finite A B
inst✝⁴ : Algebra.IsSeparable (FractionRing A) (FractionRing B)
p : Ideal A
inst✝³ : p.IsMaximal
P Q : Ideal B
inst✝² : P.IsMaximal
inst✝¹ : P.LiesOver p
hPQ : IsCoprime P Q
hP : P * Q = Ideal.map (algebraMap A B) p
inst✝ : Algebra.IsSeparable (A ⧸ p) (B ⧸ P)
⊢ p ≤ Ideal.comap (algebraMap A B) Q

Tactic:
by

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
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
⊢ FermatLastTheoremForThreeGen hζ → FermatLastTheoremFor 3

Tactic:
by

Goal state after:
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
⊢ ∃ S₁, S₁.isMinimal

Tactic:
by

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

Goal state before:
pp: A : Type u_1
B : Type u_3
inst✝¹² : CommRing A
inst✝¹¹ : CommRing B
inst✝¹⁰ : Algebra A B
inst✝⁹ : IsDomain A
inst✝⁸ : IsDedekindDomain A
inst✝⁷ : IsDedekindDomain B
inst✝⁶ : IsTorsionFree A B
inst✝⁵ : Module.Finite A B
inst✝⁴ : Algebra.IsSeparable (FractionRing A) (FractionRing B)
p : Ideal A
inst✝³ : p.IsMaximal
P Q : Ideal B
inst✝² : P.IsMaximal
inst✝¹ : P.LiesOver p
hPQ : IsCoprime P Q
hP : P * Q = Ideal.map (algebraMap A B) p
inst✝ : Algebra.IsSeparable (A ⧸ p) (B ⧸ P)
this✝¹ : Algebra (A ⧸ p) (B ⧸ Q) := Ideal.Quotient.algebraQuotientOfLEComap ⋯
this✝ : IsScalarTower A (A ⧸ p) (B ⧸ Q)
this : Module.Finite (A ⧸ p) (B ⧸ Q)
e : (B ⧸ Ideal.map (algebraMap A B) p) ≃ₐ[A ⧸ p] (B ⧸ P) × B ⧸ Q :=
  let __spread.0 := (Ideal.quotEquivOfEq ⋯).trans (P.quotientMulEquivQuotientProd Q hPQ);
  { toEquiv := __spread.0.toEquiv, map_mul' := ⋯, map_add' := ⋯, commutes' := ⋯ }
⊢ ∃ x, (Algebra.trace (A ⧸ p) (B ⧸ P)) x ≠ 0

Tactic:
by

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
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
⊢ IsPrimitiveRoot ζ (3 ^ 1)

Tactic:
by

Goal state after: