mathlib-initiative/mathlib-tactics huggingface dataset:
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).cramer fun i ↦ (Algebra.trace K L) (x * a * b i)) = (traceMatrix K ⇑b).det • b.equivFun (x * a)
this : Function.Injective (traceMatrix K ⇑b).mulVec
i : ι
a✝ : i ∈ Finset.univ
⊢ IsIntegral A ((traceMatrix K ⇑b).cramer (fun i ↦ (Algebra.trace K L) (x * a * b i)) i)

Tactic:
cramer_apply

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).cramer fun i ↦ (Algebra.trace K L) (x * a * b i)) = (traceMatrix K ⇑b).det • b.equivFun (x * a)
this : Function.Injective (traceMatrix K ⇑b).mulVec
i : ι
a✝ : i ∈ Finset.univ
⊢ IsIntegral A ((traceMatrix K ⇑b).updateCol i fun i ↦ (Algebra.trace K L) (x * a * b i)).det
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
this : Fact (Nat.Prime q)
⊢ ∃ q, Nat.Prime q ∧ ∃ t s, t ≠ 0 ∧ n = q ^ t ∧ ↑p ∣ ↑q ^ s

Tactic:
cases t with
        | zero => simp [← hn'] at hn
        | succ r =>
          rw [← hn'] at hC hμ
          refine ⟨q, hq, r + 1, Module.finrank (ℚ⟮ζ⟯) K, r.add_one_ne_zero, hn'.symm, ?_⟩
          by_cases hq' : q = 2
          · cases r with
            | zero =>
                rw [← hn', hq', zero_add, pow_one] at hn
                exact hn.false.elim
            | succ k =>
                rw [hq'] at hC hμ ⊢
                rwa [hμ.norm_toInteger_sub_one_of_eq_two_pow] at hp
          · rwa [hμ.norm_toInteger_sub_one_of_prime_ne_two hq'] at hp

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

Goal state before:
pp: A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝²¹ : CommRing A
inst✝²⁰ : Field K
inst✝¹⁹ : CommRing B
inst✝¹⁸ : Field L
inst✝¹⁷ : Algebra A K
inst✝¹⁶ : Algebra B L
inst✝¹⁵ : Algebra A B
inst✝¹⁴ : Algebra K L
inst✝¹³ : Algebra A L
inst✝¹² : IsScalarTower A K L
inst✝¹¹ : IsScalarTower A B L
inst✝¹⁰ : IsDomain A
inst✝⁹ : IsFractionRing A K
inst✝⁸ : FiniteDimensional K L
inst✝⁷ : Algebra.IsSeparable K L
inst✝⁶ : IsIntegralClosure B A L
inst✝⁵ : IsFractionRing B L
inst✝⁴ : IsDedekindDomain A
inst✝³ : IsDedekindDomain B
inst✝² : IsTorsionFree A B
inst✝¹ : Module.Finite A B
p : Ideal A
inst✝ : p.IsMaximal
P : Ideal B
e : ℕ
he : e ≠ 0
hp : p ≠ ⊥
hP : P ^ e ∣ Ideal.map (algebraMap A B) p
a : Ideal B
ha : Ideal.map (algebraMap A B) p = P ^ (e - 1) * a
hp' : ¬Ideal.map (algebraMap A B) p = ⊥
ha' : a = ⊥
⊢ Ideal.map (algebraMap A B) p = ⊥

Tactic:
by simpa [ha'] using ha

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

Goal state before:
pp: case a
n : ℕ
K : Type u
inst✝² : Field K
inst✝¹ : NeZero n
inst✝ : NumberField K
hK : IsCyclotomicExtension {n} ℚ K
hζ : IsPrimitiveRoot (zeta n ℚ K) n
μ : ↥(torsion K)
hμ : IsPrimitiveRoot ((algebraMap (𝓞 K) K) ↑↑μ) (torsionOrder K)
h :
  IsPrimitiveRoot
    (zeta n ℚ K ^ (n / n.factorizationLCMLeft (torsionOrder K)) *
      (algebraMap (𝓞 K) K) ↑↑μ ^ (torsionOrder K / n.factorizationLCMRight (torsionOrder K)))
    (n.lcm (torsionOrder K))
this✝ : NeZero (n.lcm (torsionOrder K))
this : IsCyclotomicExtension {n.lcm (torsionOrder K)} ℚ K
h_main : φ n = φ (n.lcm (torsionOrder K))
hn : Even n
⊢ n ∣ torsionOrder K

Tactic:
exact dvd_torsionOrder_of_isPrimitiveRoot hζ

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

Goal state before:
pp: case inr.refine_2
n : ℕ
K : Type u
inst✝² : Field K
inst✝¹ : NeZero n
inst✝ : NumberField K
hK : IsCyclotomicExtension {n} ℚ K
hζ : IsPrimitiveRoot (zeta n ℚ K) n
μ : ↥(torsion K)
hμ : IsPrimitiveRoot ((algebraMap (𝓞 K) K) ↑↑μ) (torsionOrder K)
h :
  IsPrimitiveRoot
    (zeta n ℚ K ^ (n / n.factorizationLCMLeft (torsionOrder K)) *
      (algebraMap (𝓞 K) K) ↑↑μ ^ (torsionOrder K / n.factorizationLCMRight (torsionOrder K)))
    (n.lcm (torsionOrder K))
this✝¹ : NeZero (n.lcm (torsionOrder K))
this✝ : IsCyclotomicExtension {n.lcm (torsionOrder K)} ℚ K
h_main : φ n = φ (n.lcm (torsionOrder K))
hn : Odd n
this : 2 * n = n.lcm (torsionOrder K)
⊢ n ∣ torsionOrder K

Tactic:
exact dvd_torsionOrder_of_isPrimitiveRoot hζ

Goal state after:
Module: Mathlib.RingTheory.Fintype

Goal state before:
pp: case pos
M₀ : Type u_1
inst✝² : MonoidWithZero M₀
inst✝¹ : Nontrivial M₀
inst✝ : Finite M₀
a : M₀
h : IsUnit a
⊢ orderOf ↑h.unit < Nat.card M₀

Tactic:
orderOf_units

Goal state after:
case pos
M₀ : Type u_1
inst✝² : MonoidWithZero M₀
inst✝¹ : Nontrivial M₀
inst✝ : Finite M₀
a : M₀
h : IsUnit a
⊢ orderOf h.unit < Nat.card M₀
Module: Mathlib.RingTheory.DedekindDomain.Different

Goal state before:
pp: A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝²¹ : CommRing A
inst✝²⁰ : Field K
inst✝¹⁹ : CommRing B
inst✝¹⁸ : Field L
inst✝¹⁷ : Algebra A K
inst✝¹⁶ : Algebra B L
inst✝¹⁵ : Algebra A B
inst✝¹⁴ : Algebra K L
inst✝¹³ : Algebra A L
inst✝¹² : IsScalarTower A K L
inst✝¹¹ : IsScalarTower A B L
inst✝¹⁰ : IsDomain A
inst✝⁹ : IsFractionRing A K
inst✝⁸ : FiniteDimensional K L
inst✝⁷ : Algebra.IsSeparable K L
inst✝⁶ : IsIntegralClosure B A L
inst✝⁵ : IsFractionRing B L
inst✝⁴ : IsDedekindDomain A
inst✝³ : IsDedekindDomain B
inst✝² : IsTorsionFree A B
inst✝¹ : Module.Finite A B
p : Ideal A
inst✝ : p.IsMaximal
P : Ideal B
e : ℕ
he : e ≠ 0
hp : p ≠ ⊥
hP✝ : P ^ e ∣ Ideal.map (algebraMap A B) p
a : Ideal B
ha : Ideal.map (algebraMap A B) p = P ^ (e - 1) * a
hp' : ¬Ideal.map (algebraMap A B) p = ⊥
habot : a ≠ ⊥
hPbot : P ≠ ⊥
this✝ : Ideal.map (algebraMap A B) p ∣ a ^ e
this : ∀ x ∈ a, (intTrace A B) x ∈ p
hP : (↑(P ^ (e - 1)))⁻¹ = ↑a / ↑(Ideal.map (algebraMap A B) p)
x : L
hx : x ∈ ↑a / ↑(Ideal.map (algebraMap A B) p)
⊢ ↑(Ideal.map (algebraMap A B) p) ≠ 0

Tactic:
by simpa using hp'

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

Goal state before:
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
n : ℤ
x : 𝓞 K
hx : ↑(-⋯.unit) = 3 * x + ↑n
⊢ hζ.toInteger = ↑3 * -x + ↑(-n)

Tactic:
simp only [Nat.cast_ofNat, mul_neg, Int.cast_neg, ← neg_add, ← hx, Units.val_neg,
      IsUnit.unit_spec, RingOfIntegers.neg_mk, neg_neg]

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

Goal state before:
pp: case 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}) := Fintype.ofFinite (𝓞 K ⧸ Ideal.span {λ})
x✝¹ : Ring (𝓞 K ⧸ Ideal.span {λ}) := (Ideal.Quotient.commRing (Ideal.span {λ})).toRing
x✝ : AddGroup (𝓞 K ⧸ Ideal.span {λ}) := x✝¹.toAddGroupWithOne.toAddGroup
h3 : Fintype.card (𝓞 K ⧸ Ideal.span {λ}) = 3
h : (Ideal.Quotient.mk (Ideal.span {λ})) x = 0
⊢ λ ∣ x ∨ λ ∣ x - 1 ∨ λ ∣ x + 1

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}) := Fintype.ofFinite (𝓞 K ⧸ Ideal.span {λ})
x✝¹ : Ring (𝓞 K ⧸ Ideal.span {λ}) := (Ideal.Quotient.commRing (Ideal.span {λ})).toRing
x✝ : AddGroup (𝓞 K ⧸ Ideal.span {λ}) := x✝¹.toAddGroupWithOne.toAddGroup
h3 : Fintype.card (𝓞 K ⧸ Ideal.span {λ}) = 3
h : (Ideal.Quotient.mk (Ideal.span {λ})) x = 1
⊢ λ ∣ x ∨ λ ∣ x - 1 ∨ λ ∣ x + 1

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}) := Fintype.ofFinite (𝓞 K ⧸ Ideal.span {λ})
x✝¹ : Ring (𝓞 K ⧸ Ideal.span {λ}) := (Ideal.Quotient.commRing (Ideal.span {λ})).toRing
x✝ : AddGroup (𝓞 K ⧸ Ideal.span {λ}) := x✝¹.toAddGroupWithOne.toAddGroup
h3 : Fintype.card (𝓞 K ⧸ Ideal.span {λ}) = 3
h : (Ideal.Quotient.mk (Ideal.span {λ})) x = -1
⊢ λ ∣ x ∨ λ ∣ x - 1 ∨ λ ∣ x + 1

Tactic:
· left
    exact Ideal.mem_span_singleton.1 <| Ideal.Quotient.eq_zero_iff_mem.1 h

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}) := Fintype.ofFinite (𝓞 K ⧸ Ideal.span {λ})
x✝¹ : Ring (𝓞 K ⧸ Ideal.span {λ}) := (Ideal.Quotient.commRing (Ideal.span {λ})).toRing
x✝ : AddGroup (𝓞 K ⧸ Ideal.span {λ}) := x✝¹.toAddGroupWithOne.toAddGroup
h3 : Fintype.card (𝓞 K ⧸ Ideal.span {λ}) = 3
h : (Ideal.Quotient.mk (Ideal.span {λ})) x = 1
⊢ λ ∣ x ∨ λ ∣ x - 1 ∨ λ ∣ x + 1

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}) := Fintype.ofFinite (𝓞 K ⧸ Ideal.span {λ})
x✝¹ : Ring (𝓞 K ⧸ Ideal.span {λ}) := (Ideal.Quotient.commRing (Ideal.span {λ})).toRing
x✝ : AddGroup (𝓞 K ⧸ Ideal.span {λ}) := x✝¹.toAddGroupWithOne.toAddGroup
h3 : Fintype.card (𝓞 K ⧸ Ideal.span {λ}) = 3
h : (Ideal.Quotient.mk (Ideal.span {λ})) x = -1
⊢ λ ∣ x ∨ 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
⊢ Fintype.card (Free.ChooseBasisIndex ℤ (Additive ((𝓞 K)ˣ ⧸ torsion K))) = Fintype.card (Fin (rank K))

Tactic:
← Module.finrank_eq_card_chooseBasisIndex,

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ finrank ℤ (Additive ((𝓞 K)ˣ ⧸ torsion K)) = Fintype.card (Fin (rank K))
Module: Mathlib.NumberTheory.FactorisationProperties

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

Tactic:
by
  rw [Set.infinite_iff_exists_gt]
  intro a
  have ha : Abundant 12 := by decide
  use (2 * (a + 1)) * 12
  grind [Abundant.mul_left ha (show 2 * (a + 1) ≠ 0 by grind)]

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

Goal state before:
pp: case inr
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
ha : ↑a ≠ 0
hf : ¬(x = 0 ∧ y = 0)
hy : ¬y = 0
⊢ False

Tactic:
rw [eq_one_of_sq_sub_mul_sq_eq_zero ha hy hxy, CharZero.eq_neg_self_iff] at h

Goal state after:
case inr
p : ℕ
inst✝ : Fact (Nat.Prime p)
a : ℤ
h : 1 = 0
x y : ZMod p
hxy : x ^ 2 - ↑a * y ^ 2 = 0
ha : ↑a ≠ 0
hf : ¬(x = 0 ∧ y = 0)
hy : ¬y = 0
⊢ False
Module: Mathlib.RingTheory.DedekindDomain.Different

Goal state before:
pp: case refine_2
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 : B ⧸ P
hx : (Algebra.trace (A ⧸ p) (B ⧸ P)) x ≠ 0
y : B
hy : (Ideal.Quotient.mk (Ideal.map (algebraMap A B) p)) y = e.symm (x, 0)
⊢ (intTrace A B) y ∉ p

Tactic:
rw [← Ideal.Quotient.eq_zero_iff_mem, ← Algebra.trace_quotient_eq_of_isDedekindDomain,
      hy, Algebra.trace_eq_of_algEquiv, Algebra.trace_prod_apply]

Goal state after:
case refine_2
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 : B ⧸ P
hx : (Algebra.trace (A ⧸ p) (B ⧸ P)) x ≠ 0
y : B
hy : (Ideal.Quotient.mk (Ideal.map (algebraMap A B) p)) y = e.symm (x, 0)
⊢ ¬(Algebra.trace (A ⧸ p) (B ⧸ P)) (x, 0).1 + (Algebra.trace (A ⧸ p) (B ⧸ Q)) (x, 0).2 = 0
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
inst✝¹ : Finite (A ⧸ p)
P Q : Ideal B
inst✝ : P.IsMaximal
hPQ : IsCoprime P Q
hP : P * Q = Ideal.map (algebraMap A B) p
⊢ ¬P ∣ differentIdeal A B

Tactic:
have : P.LiesOver p := by
    constructor
    refine ‹p.IsMaximal›.eq_of_le ?_ ?_
    · simpa using ‹P.IsMaximal›.ne_top
    · rw [← Ideal.map_le_iff_le_comap, ← hP]
      exact Ideal.mul_le_left

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
inst✝¹ : Finite (A ⧸ p)
P Q : Ideal B
inst✝ : P.IsMaximal
hPQ : IsCoprime P Q
hP : P * Q = Ideal.map (algebraMap A B) p
this : P.LiesOver p
⊢ ¬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 ≠ ⊥
P : Ideal B
inst✝¹ : P.IsMaximal
inst✝ : P.LiesOver p
H : ¬Algebra.IsSeparable (A ⧸ p) (B ⧸ P)
a : Ideal B
ha : Ideal.map (algebraMap A B) p = P * a
hPa : ¬P ∣ a
K : Type u_1 := FractionRing A
L : Type u_3 := FractionRing B
hp' : ¬Ideal.map (algebraMap A B) p = ⊥
ha' : a = ⊥
⊢ Ideal.map (algebraMap A B) p = ⊥

Tactic:
by simpa [ha'] using ha

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

Goal state before:
pp: A : Type u_1
B : Type u_3
inst✝¹¹ : CommRing A
inst✝¹⁰ : CommRing B
inst✝⁹ : Algebra A B
inst✝⁸ : IsDomain A
inst✝⁷ : IsDedekindDomain A
inst✝⁶ : IsDedekindDomain B
inst✝⁵ : IsTorsionFree A B
inst✝⁴ : Module.Finite A B
inst✝³ : Algebra.IsSeparable (FractionRing A) (FractionRing B)
p : Ideal A
inst✝² : p.IsMaximal
hp : p ≠ ⊥
P : Ideal B
inst✝¹ : P.IsMaximal
inst✝ : P.LiesOver p
H : ¬Algebra.IsSeparable (A ⧸ p) (B ⧸ P)
a : Ideal B
ha : Ideal.map (algebraMap A B) p = P * a
hPa : ¬P ∣ a
K : Type u_1 := FractionRing A
L : Type u_3 := FractionRing B
hp' : ¬Ideal.map (algebraMap A B) p = ⊥
habot : a ≠ ⊥
hPbot : P ≠ ⊥
this : ∀ x ∈ a, (intTrace A B) x ∈ p
hP : (↑P)⁻¹ = ↑a / ↑(Ideal.map (algebraMap A B) p)
x : L
hx : x ∈ ↑a / ↑(Ideal.map (algebraMap A B) p)
⊢ ↑(Ideal.map (algebraMap A B) p) ≠ 0

Tactic:
by simpa using hp'

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
⊢ S.a + ↑η * S.b = S.a + S.b + λ * S.b

Tactic:
by rw [coe_eta]; ring

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
this : λ ^ 2 ∣ λ ^ 4
X : 𝓞 K
hX : ↑S.u₅ * (λ ^ (S.multiplicity - 1) * S.X) ^ 3 = λ ^ 2 * X
Y : 𝓞 K
hY : S.Y ^ 3 - 1 = λ ^ 2 * Y
Z : 𝓞 K
hZ : S.Z ^ 3 - 1 = λ ^ 2 * Z
⊢ ↑S.u₄ - ↑(-1) = ↑S.u₅ * (λ ^ (S.multiplicity - 1) * S.X) ^ 3 - (S.Y ^ 3 - 1) - ↑S.u₄ * (λ ^ 2 * Z)

Tactic:
← hZ,

Goal state after:
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
this : λ ^ 2 ∣ λ ^ 4
X : 𝓞 K
hX : ↑S.u₅ * (λ ^ (S.multiplicity - 1) * S.X) ^ 3 = λ ^ 2 * X
Y : 𝓞 K
hY : S.Y ^ 3 - 1 = λ ^ 2 * Y
Z : 𝓞 K
hZ : S.Z ^ 3 - 1 = λ ^ 2 * Z
⊢ ↑S.u₄ - ↑(-1) = ↑S.u₅ * (λ ^ (S.multiplicity - 1) * S.X) ^ 3 - (S.Y ^ 3 - 1) - ↑S.u₄ * (S.Z ^ 3 - 1)
Module: Mathlib.NumberTheory.FLT.Three

Goal state before:
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
this : λ ^ 2 ∣ λ ^ 4
X : 𝓞 K
hX : ↑S.u₅ * (λ ^ (S.multiplicity - 1) * S.X) ^ 3 = λ ^ 2 * X
Y : 𝓞 K
hY : S.Y ^ 3 - 1 = λ ^ 2 * Y
Z : 𝓞 K
hZ : S.Z ^ 3 + 1 = λ ^ 2 * Z
⊢ ↑S.u₄ - ↑1 = -(↑S.u₅ * (λ ^ (S.multiplicity - 1) * S.X) ^ 3 - (S.Y ^ 3 - 1) - ↑S.u₄ * (λ ^ 2 * Z))

Tactic:
← hZ,

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
this : λ ^ 2 ∣ λ ^ 4
X : 𝓞 K
hX : ↑S.u₅ * (λ ^ (S.multiplicity - 1) * S.X) ^ 3 = λ ^ 2 * X
Y : 𝓞 K
hY : S.Y ^ 3 - 1 = λ ^ 2 * Y
Z : 𝓞 K
hZ : S.Z ^ 3 + 1 = λ ^ 2 * Z
⊢ ↑S.u₄ - ↑1 = -(↑S.u₅ * (λ ^ (S.multiplicity - 1) * S.X) ^ 3 - (S.Y ^ 3 - 1) - ↑S.u₄ * (S.Z ^ 3 + 1))
Module: Mathlib.NumberTheory.Height.Basic

Goal state before:
pp: case inr
K : Type u_1
inst✝² : Field K
inst✝¹ : AdmissibleAbsValues K
ι : Type u_2
inst✝ : Finite ι
x : ι → K
e : ↑(support x) ⊕ ↑(support x)ᶜ ≃ ι := Equiv.Set.sumCompl (support x)
val : ↑(support x)ᶜ
⊢ x (e (Sum.inr val)) = Sum.elim (fun i ↦ x ↑i) 0 (Sum.inr val)

Tactic:
| inr val =>

Goal state after:
Module: Mathlib.NumberTheory.Height.MvPolynomial

Goal state before:
pp: case inr
K : Type u_1
inst✝⁴ : Field K
ι : Type u_2
ι' : Type u_3
inst✝³ : Fintype ι
inst✝² : Finite ι'
inst✝¹ : AdmissibleAbsValues K
inst✝ : Nonempty ι
A : ι' × ι → K
x : ι → K
H₀ : 1 ≤ ↑(Nat.card ι) ^ totalWeight K * mulHeight A * mulHeight x
hι' : Nonempty ι'
⊢ (mulHeight fun j ↦ ∑ i, A (j, i) * x i) ≤ ↑(Nat.card ι) ^ totalWeight K * mulHeight A * mulHeight x

Tactic:
rcases eq_or_ne (fun j ↦ ∑ i, A (j, i) * x i) 0 with h | h

Goal state after:
case inr.inl
K : Type u_1
inst✝⁴ : Field K
ι : Type u_2
ι' : Type u_3
inst✝³ : Fintype ι
inst✝² : Finite ι'
inst✝¹ : AdmissibleAbsValues K
inst✝ : Nonempty ι
A : ι' × ι → K
x : ι → K
H₀ : 1 ≤ ↑(Nat.card ι) ^ totalWeight K * mulHeight A * mulHeight x
hι' : Nonempty ι'
h : (fun j ↦ ∑ i, A (j, i) * x i) = 0
⊢ (mulHeight fun j ↦ ∑ i, A (j, i) * x i) ≤ ↑(Nat.card ι) ^ totalWeight K * mulHeight A * mulHeight x

case inr.inr
K : Type u_1
inst✝⁴ : Field K
ι : Type u_2
ι' : Type u_3
inst✝³ : Fintype ι
inst✝² : Finite ι'
inst✝¹ : AdmissibleAbsValues K
inst✝ : Nonempty ι
A : ι' × ι → K
x : ι → K
H₀ : 1 ≤ ↑(Nat.card ι) ^ totalWeight K * mulHeight A * mulHeight x
hι' : Nonempty ι'
h : (fun j ↦ ∑ i, A (j, i) * x i) ≠ 0
⊢ (mulHeight fun j ↦ ∑ i, A (j, i) * x i) ≤ ↑(Nat.card ι) ^ totalWeight K * mulHeight A * mulHeight x
Module: Mathlib.NumberTheory.FLT.Three

Goal state before:
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
this : λ ^ 2 ∣ λ ^ 4
X : 𝓞 K
hX : ↑S.u₅ * (λ ^ (S.multiplicity - 1) * S.X) ^ 3 = λ ^ 2 * X
Y : 𝓞 K
hY : S.Y ^ 3 + 1 = λ ^ 2 * Y
Z : 𝓞 K
hZ : S.Z ^ 3 - 1 = λ ^ 2 * Z
⊢ ↑S.u₄ - ↑1 = ↑S.u₅ * (λ ^ (S.multiplicity - 1) * S.X) ^ 3 - (S.Y ^ 3 + 1) - ↑S.u₄ * (λ ^ 2 * Z)

Tactic:
← hZ,

Goal state after:
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
this : λ ^ 2 ∣ λ ^ 4
X : 𝓞 K
hX : ↑S.u₅ * (λ ^ (S.multiplicity - 1) * S.X) ^ 3 = λ ^ 2 * X
Y : 𝓞 K
hY : S.Y ^ 3 + 1 = λ ^ 2 * Y
Z : 𝓞 K
hZ : S.Z ^ 3 - 1 = λ ^ 2 * Z
⊢ ↑S.u₄ - ↑1 = ↑S.u₅ * (λ ^ (S.multiplicity - 1) * S.X) ^ 3 - (S.Y ^ 3 + 1) - ↑S.u₄ * (S.Z ^ 3 - 1)
Module: Mathlib.NumberTheory.Harmonic.ZetaAsymp

Goal state before:
pp: case hf.refine_1
n : ℕ
hn : 0 < n
s : ℝ
hs : 1 < s
hv : ∀ x ∈ uIcc (↑n) (↑n + 1), 0 < x
⊢ 0 < ↑n

pp: case hf.refine_2
n : ℕ
hn : 0 < n
s : ℝ
hs : 1 < s
hv : ∀ x ∈ uIcc (↑n) (↑n + 1), 0 < x
⊢ 0 < ↑n + 1

Tactic:
· exact_mod_cast hn

Goal state after:
case hf.refine_2
n : ℕ
hn : 0 < n
s : ℝ
hs : 1 < s
hv : ∀ x ∈ uIcc (↑n) (↑n + 1), 0 < x
⊢ 0 < ↑n + 1
Module: Mathlib.NumberTheory.Height.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : AdmissibleAbsValues K
x : K
⊢ mulHeight₁ x = mulHeight ![x, 1]

Tactic:
have H (v : AbsoluteValue K ℝ) (x : K) : v x ⊔ 1 = ⨆ i, v (![x, 1] i) := by
    have (i : Fin 2) : v (![x, 1] i) = ![v x, 1] i := by fin_cases i <;> simp
    simpa [this] using max_eq_iSup (v x) 1
  have hx : ![x, 1] ≠ 0 := by simp
  simp only [mulHeight₁_eq, mulHeight_eq hx, H]

Goal state after:
Module: Mathlib.NumberTheory.Harmonic.ZetaAsymp

Goal state before:
pp: case hg.refine_1
n : ℕ
hn : 0 < n
s : ℝ
hs : 1 < s
hv : ∀ x ∈ uIcc (↑n) (↑n + 1), 0 < x
⊢ 0 < ↑n

pp: case hg.refine_2
n : ℕ
hn : 0 < n
s : ℝ
hs : 1 < s
hv : ∀ x ∈ uIcc (↑n) (↑n + 1), 0 < x
⊢ 0 < ↑n + 1

Tactic:
· exact_mod_cast hn

Goal state after:
case hg.refine_2
n : ℕ
hn : 0 < n
s : ℝ
hs : 1 < s
hv : ∀ x ∈ uIcc (↑n) (↑n + 1), 0 < x
⊢ 0 < ↑n + 1
Module: Mathlib.NumberTheory.Height.Basic

Goal state before:
pp: case inr
K : Type u_1
inst✝¹ : Field K
inst✝ : AdmissibleAbsValues K
x y : K
hy : y ≠ 0
⊢ mulHeight₁ (x / y) = mulHeight ![x, y]

Tactic:
rw [mulHeight₁_eq_mulHeight, ← mulHeight_smul_eq_mulHeight _ hy]

Goal state after:
case inr
K : Type u_1
inst✝¹ : Field K
inst✝ : AdmissibleAbsValues K
x y : K
hy : y ≠ 0
⊢ mulHeight (y • ![x / y, 1]) = mulHeight ![x, y]