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

Goal state before:
pp: case inl
k : Type u_1
inst✝ : Field k
p q r : ℕ
hp : p ≠ 0
hq : q ≠ 0
hr : r ≠ 0
hineq : q * r + r * p + p * q ≤ p * q * r
chp : ↑p ≠ 0
chq : ↑q ≠ 0
chr : ↑r ≠ 0
a b c : k[X]
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hab : IsCoprime a b
u v w : k
hu : u ≠ 0
hv : v ≠ 0
hw : w ≠ 0
heq : C u * a ^ p + C v * b ^ q + C w * c ^ r = 0
hbc : IsCoprime b c
hca : IsCoprime c a
hCu : C u ≠ 0
hCv : C v ≠ 0
hCw : C w ≠ 0
hap : a ^ p ≠ 0
hbq : b ^ q ≠ 0
hcr : c ^ r ≠ 0
habp : IsCoprime (C u * a ^ p) (C v * b ^ q)
hbcp : IsCoprime (C v * b ^ q) (C w * c ^ r)
hcap : IsCoprime (C w * c ^ r) (C u * a ^ p)
habcp : IsCoprime (C u * a ^ p * (C v * b ^ q)) (C w * c ^ r)
nd_lt :
  (C u).natDegree + (a ^ p).natDegree + 1 ≤ (radical a * radical b * radical c).natDegree ∧
    (C v).natDegree + (b ^ q).natDegree + 1 ≤ (radical a * radical b * radical c).natDegree ∧
      (C w * c ^ r).natDegree + 1 ≤ (radical a * radical b * radical c).natDegree
⊢ derivative a = 0 ∧ derivative b = 0 ∧ derivative c = 0

Tactic:
natDegree_mul hCw hcr,

Goal state after:
case inl
k : Type u_1
inst✝ : Field k
p q r : ℕ
hp : p ≠ 0
hq : q ≠ 0
hr : r ≠ 0
hineq : q * r + r * p + p * q ≤ p * q * r
chp : ↑p ≠ 0
chq : ↑q ≠ 0
chr : ↑r ≠ 0
a b c : k[X]
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hab : IsCoprime a b
u v w : k
hu : u ≠ 0
hv : v ≠ 0
hw : w ≠ 0
heq : C u * a ^ p + C v * b ^ q + C w * c ^ r = 0
hbc : IsCoprime b c
hca : IsCoprime c a
hCu : C u ≠ 0
hCv : C v ≠ 0
hCw : C w ≠ 0
hap : a ^ p ≠ 0
hbq : b ^ q ≠ 0
hcr : c ^ r ≠ 0
habp : IsCoprime (C u * a ^ p) (C v * b ^ q)
hbcp : IsCoprime (C v * b ^ q) (C w * c ^ r)
hcap : IsCoprime (C w * c ^ r) (C u * a ^ p)
habcp : IsCoprime (C u * a ^ p * (C v * b ^ q)) (C w * c ^ r)
nd_lt :
  (C u).natDegree + (a ^ p).natDegree + 1 ≤ (radical a * radical b * radical c).natDegree ∧
    (C v).natDegree + (b ^ q).natDegree + 1 ≤ (radical a * radical b * radical c).natDegree ∧
      (C w).natDegree + (c ^ r).natDegree + 1 ≤ (radical a * radical b * radical c).natDegree
⊢ derivative a = 0 ∧ derivative b = 0 ∧ derivative c = 0
Module: Mathlib.NumberTheory.FLT.Polynomial

Goal state before:
pp: case inl
k : Type u_1
inst✝ : Field k
p q r : ℕ
hp : p ≠ 0
hq : q ≠ 0
hr : r ≠ 0
hineq : q * r + r * p + p * q ≤ p * q * r
chp : ↑p ≠ 0
chq : ↑q ≠ 0
chr : ↑r ≠ 0
a b c : k[X]
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hab : IsCoprime a b
u v w : k
hu : u ≠ 0
hv : v ≠ 0
hw : w ≠ 0
heq : C u * a ^ p + C v * b ^ q + C w * c ^ r = 0
hbc : IsCoprime b c
hca : IsCoprime c a
hCu : C u ≠ 0
hCv : C v ≠ 0
hCw : C w ≠ 0
hap : a ^ p ≠ 0
hbq : b ^ q ≠ 0
hcr : c ^ r ≠ 0
habp : IsCoprime (C u * a ^ p) (C v * b ^ q)
hbcp : IsCoprime (C v * b ^ q) (C w * c ^ r)
hcap : IsCoprime (C w * c ^ r) (C u * a ^ p)
habcp : IsCoprime (C u * a ^ p * (C v * b ^ q)) (C w * c ^ r)
nd_lt :
  (C u * a ^ p).natDegree + 1 ≤ (radical (C u * a ^ p * (C v * b ^ q) * (C w * c ^ r))).natDegree ∧
    (C v * b ^ q).natDegree + 1 ≤ (radical (C u * a ^ p * (C v * b ^ q) * (C w * c ^ r))).natDegree ∧
      (C w * c ^ r).natDegree + 1 ≤ (radical (C u * a ^ p * (C v * b ^ q) * (C w * c ^ r))).natDegree
⊢ derivative a = 0 ∧ derivative b = 0 ∧ derivative c = 0

Tactic:
simp_rw [radical_mul habcp.isRelPrime, radical_mul habp.isRelPrime,
      radical_mul_of_isUnit_left hu.isUnit_C,
      radical_mul_of_isUnit_left hv.isUnit_C,
      radical_mul_of_isUnit_left hw.isUnit_C,
      radical_pow a hp, radical_pow b hq, radical_pow c hr,
      natDegree_mul hCu hap,
      natDegree_mul hCv hbq,
      natDegree_mul hCw hcr,
      natDegree_C, natDegree_pow, zero_add,
      ← radical_mul hab.isRelPrime,
      ← radical_mul (hca.symm.mul_left hbc).isRelPrime] at nd_lt

Goal state after:
case inl
k : Type u_1
inst✝ : Field k
p q r : ℕ
hp : p ≠ 0
hq : q ≠ 0
hr : r ≠ 0
hineq : q * r + r * p + p * q ≤ p * q * r
chp : ↑p ≠ 0
chq : ↑q ≠ 0
chr : ↑r ≠ 0
a b c : k[X]
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hab : IsCoprime a b
u v w : k
hu : u ≠ 0
hv : v ≠ 0
hw : w ≠ 0
heq : C u * a ^ p + C v * b ^ q + C w * c ^ r = 0
hbc : IsCoprime b c
hca : IsCoprime c a
hCu : C u ≠ 0
hCv : C v ≠ 0
hCw : C w ≠ 0
hap : a ^ p ≠ 0
hbq : b ^ q ≠ 0
hcr : c ^ r ≠ 0
habp : IsCoprime (C u * a ^ p) (C v * b ^ q)
hbcp : IsCoprime (C v * b ^ q) (C w * c ^ r)
hcap : IsCoprime (C w * c ^ r) (C u * a ^ p)
habcp : IsCoprime (C u * a ^ p * (C v * b ^ q)) (C w * c ^ r)
nd_lt :
  p * a.natDegree + 1 ≤ (radical (a * b * c)).natDegree ∧
    q * b.natDegree + 1 ≤ (radical (a * b * c)).natDegree ∧ r * c.natDegree + 1 ≤ (radical (a * b * c)).natDegree
⊢ derivative a = 0 ∧ derivative b = 0 ∧ derivative c = 0
Module: Mathlib.RingTheory.FractionalIdeal.Norm

Goal state before:
pp: R : Type u_1
inst✝⁶ : CommRing R
inst✝⁵ : IsDedekindDomain R
inst✝⁴ : Free ℤ R
inst✝³ : Module.Finite ℤ R
K : Type u_2
inst✝² : CommRing K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
I₀ : Ideal R
⊢ 1 • ↑↑I₀ = Submodule.map (Algebra.linearMap R K) I₀

Tactic:
by rw [one_smul]; rfl

Goal state after:
Module: Mathlib.NumberTheory.PythagoreanTriples

Goal state before:
pp: case h.inr
z m n : ℤ
h : PythagoreanTriple (2 * m * n) (m ^ 2 - n ^ 2) z ∧ (2 * m * n).gcd (m ^ 2 - n ^ 2) = 1
co : m.gcd n = 1
pp : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
⊢ (2 * m * n = m ^ 2 - n ^ 2 ∧ m ^ 2 - n ^ 2 = 2 * m * n ∨ 2 * m * n = 2 * m * n ∧ m ^ 2 - n ^ 2 = m ^ 2 - n ^ 2) ∧
    (z = m ^ 2 + n ^ 2 ∨ z = -(m ^ 2 + n ^ 2)) ∧ m.gcd n = 1 ∧ (m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0)

Tactic:
refine ⟨Or.inr ⟨rfl, rfl⟩, ?_, co, pp⟩

Goal state after:
case h.inr
z m n : ℤ
h : PythagoreanTriple (2 * m * n) (m ^ 2 - n ^ 2) z ∧ (2 * m * n).gcd (m ^ 2 - n ^ 2) = 1
co : m.gcd n = 1
pp : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
⊢ z = m ^ 2 + n ^ 2 ∨ z = -(m ^ 2 + n ^ 2)
Module: Mathlib.Topology.Instances.Complex

Goal state before:
pp: case e'_4
K : Subfield ℂ
ψ : ↥K →+* ℂ
hc : UniformContinuous ⇑ψ
this✝ : IsTopologicalDivisionRing ℂ :=
  { toIsTopologicalRing := NormedDivisionRing.to_isTopologicalDivisionRing.toIsTopologicalRing,
    toContinuousInv₀ := NormedDivisionRing.to_isTopologicalDivisionRing.toContinuousInv₀ }
this : IsTopologicalRing ↥K.topologicalClosure := Subring.instIsTopologicalRing K.topologicalClosure.toSubring
ι : ↥K → ↥K.topologicalClosure := ⇑(Subfield.inclusion ⋯)
ui : IsUniformInducing ι
x : { x // x ∈ closure (id '' ?m.826) }
⊢ ↑K.topologicalClosure = closure (id '' ?m.829 ↑x)

Tactic:
simp only [id, Set.image_id']

Goal state after:
case e'_4
K : Subfield ℂ
ψ : ↥K →+* ℂ
hc : UniformContinuous ⇑ψ
this✝ : IsTopologicalDivisionRing ℂ :=
  { toIsTopologicalRing := NormedDivisionRing.to_isTopologicalDivisionRing.toIsTopologicalRing,
    toContinuousInv₀ := NormedDivisionRing.to_isTopologicalDivisionRing.toContinuousInv₀ }
this : IsTopologicalRing ↥K.topologicalClosure := Subring.instIsTopologicalRing K.topologicalClosure.toSubring
ι : ↥K → ↥K.topologicalClosure := ⇑(Subfield.inclusion ⋯)
ui : IsUniformInducing ι
x : { x // x ∈ closure (id '' ?m.826) }
⊢ ↑K.topologicalClosure = closure (?m.829 ↑x)
Module: Mathlib.Topology.Instances.Complex

Goal state before:
pp: K : Subfield ℂ
ψ : ↥K →+* ℂ
hc : UniformContinuous ⇑ψ
this✝ : IsTopologicalDivisionRing ℂ :=
  { toIsTopologicalRing := NormedDivisionRing.to_isTopologicalDivisionRing.toIsTopologicalRing,
    toContinuousInv₀ := NormedDivisionRing.to_isTopologicalDivisionRing.toContinuousInv₀ }
this : IsTopologicalRing ↥K.topologicalClosure := Subring.instIsTopologicalRing K.topologicalClosure.toSubring
ι : ↥K → ↥K.topologicalClosure := ⇑(Subfield.inclusion ⋯)
ui : IsUniformInducing ι
j : { x // x ∈ closure (id '' ↑K) } → ↑↑K.topologicalClosure := fun x ↦ ⟨↑x, ⋯⟩
y : ℂ
hy : y ∈ ↑K.topologicalClosure
⊢ closure (id '' {x | x ∈ K}) = ↑K.topologicalClosure

Tactic:
simp only [id, Set.image_id']

Goal state after:
K : Subfield ℂ
ψ : ↥K →+* ℂ
hc : UniformContinuous ⇑ψ
this✝ : IsTopologicalDivisionRing ℂ :=
  { toIsTopologicalRing := NormedDivisionRing.to_isTopologicalDivisionRing.toIsTopologicalRing,
    toContinuousInv₀ := NormedDivisionRing.to_isTopologicalDivisionRing.toContinuousInv₀ }
this : IsTopologicalRing ↥K.topologicalClosure := Subring.instIsTopologicalRing K.topologicalClosure.toSubring
ι : ↥K → ↥K.topologicalClosure := ⇑(Subfield.inclusion ⋯)
ui : IsUniformInducing ι
j : { x // x ∈ closure (id '' ↑K) } → ↑↑K.topologicalClosure := fun x ↦ ⟨↑x, ⋯⟩
y : ℂ
hy : y ∈ ↑K.topologicalClosure
⊢ closure {x | x ∈ K} = ↑K.topologicalClosure
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x✝¹ : InfinitePlace K
x✝ : x✝¹ ∈ Finset.univ
⊢ x✝¹.mult = ∑ i with mk i = x✝¹, 1

Tactic:
Finset.sum_const,

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x✝¹ : InfinitePlace K
x✝ : x✝¹ ∈ Finset.univ
⊢ x✝¹.mult = #{i | mk i = x✝¹} • 1
Module: Mathlib.NumberTheory.NumberField.Units.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ Finite ↥(torsion K)

Tactic:
refine Set.Finite.of_finite_image ?_ (coe_injective K).injOn
  refine (Embeddings.finite_of_norm_le K ℂ 1).subset
    (fun a ⟨u, ⟨h_tors, h_ua⟩⟩ => ⟨?_, fun φ => ?_⟩)
  · rw [← h_ua]
    exact u.val.prop
  · rw [← h_ua]
    exact le_of_eq ((eq_iff_eq _ 1).mp ((mem_torsion K).mp h_tors) φ)

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
this : ∀ (w : { w // w.IsComplex }), #{φ | mkComplex φ = w} = 2
⊢ ∑ j, ∑ i with mkComplex i = j, 1 = 2 * nrComplexPlaces K

Tactic:
Finset.sum_const,

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
this : ∀ (w : { w // w.IsComplex }), #{φ | mkComplex φ = w} = 2
⊢ ∑ x, #{i | mkComplex i = x} • 1 = 2 * nrComplexPlaces K
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
this : ∀ (w : { w // w.IsComplex }), #{φ | mkComplex φ = w} = 2
⊢ ∑ x, 2 = ∑ i, 2 * 1

Tactic:
Finset.sum_const,

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
this : ∀ (w : { w // w.IsComplex }), #{φ | mkComplex φ = w} = 2
⊢ #Finset.univ • 2 = #Finset.univ • (2 * 1)
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody

Goal state before:
pp: case refine_2
K : Type u_1
inst✝¹ : Field K
f : InfinitePlace K → ℝ≥0
inst✝ : NumberField K
w₁ : InfinitePlace K
B : ℝ≥0
hf : ∀ (w : InfinitePlace K), w ≠ w₁ → f w ≠ 0
S : ℝ≥0 := ∏ w ∈ Finset.univ.erase w₁, f w ^ w.mult
⊢ S ≠ 0

Tactic:
rw [Finset.prod_ne_zero_iff]
      exact fun w hw => pow_ne_zero _ (hf w (Finset.ne_of_mem_erase hw))

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

Goal state before:
pp: K : Type u_1
inst✝ : Field K
x : (K →+* ℂ) → ℂ
φ : K →+* ℂ
hx : x ∈ Submodule.span ℝ (Set.range ⇑(canonicalEmbedding K))
⊢ (starRingEnd ℂ) (x φ) = x (ComplexEmbedding.conjugate φ)

Tactic:
refine Submodule.span_induction ?_ ?_ (fun _ _ _ _ hx hy => ?_) (fun a _ _ hx => ?_) hx
  · rintro _ ⟨x, rfl⟩
    rw [apply_at, apply_at, ComplexEmbedding.conjugate_coe_eq]
  · rw [Pi.zero_apply, Pi.zero_apply, map_zero]
  · rw [Pi.add_apply, Pi.add_apply, map_add, hx, hy]
  · rw [Pi.smul_apply, Complex.real_smul, map_mul, Complex.conj_ofReal]
    exact congrArg ((a : ℂ) * ·) hx

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
r : ℝ
⊢ (↑(integerLattice K) ∩ Metric.closedBall 0 r).Finite

Tactic:
obtain hr | _ := lt_or_ge r 0
  · simp [Metric.closedBall_eq_empty.2 hr]
  · have heq : ∀ x, canonicalEmbedding K x ∈ Metric.closedBall 0 r ↔
        ∀ φ : K →+* ℂ, ‖φ x‖ ≤ r := by
      intro x; rw [← norm_le_iff, mem_closedBall_zero_iff]
    convert! (Embeddings.finite_of_norm_le K ℂ r).image (canonicalEmbedding K)
    ext; constructor
    · rintro ⟨⟨_, ⟨x, rfl⟩, rfl⟩, hx⟩
      exact ⟨x, ⟨SetLike.coe_mem x, fun φ => (heq _).mp hx φ⟩, rfl⟩
    · rintro ⟨x, ⟨hx1, hx2⟩, rfl⟩
      exact ⟨⟨x, ⟨⟨x, hx1⟩, rfl⟩, rfl⟩, (heq x).mpr hx2⟩

Goal state after:
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : mixedSpace K
⊢ ∑ w, ↑w.mult * (normAtPlace w) x = 0 ↔ ∀ (w : InfinitePlace K), (normAtPlace w) x = 0

Tactic:
Finset.sum_eq_zero_iff_of_nonneg
    fun _ _ ↦ mul_nonneg (Nat.cast_pos.mpr mult_pos).le (normAtPlace_nonneg _ _)

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : mixedSpace K
⊢ (∀ i ∈ Finset.univ, ↑i.mult * (normAtPlace i) x = 0) ↔ ∀ (w : InfinitePlace K), (normAtPlace w) x = 0
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody

Goal state before:
pp: case refine_2
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : mixedSpace K
w : InfinitePlace K
x✝ : w ∈ Finset.univ
⊢ 0 ≤ ∑ x_1 ∈ Finset.univ.erase w, ↑x_1.mult * (normAtPlace x_1) x

Tactic:
· exact Finset.sum_nonneg (fun _ _ => mul_nonneg (Nat.cast_pos.mpr mult_pos).le
      (normAtPlace_nonneg _ _))

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
f : Free.ChooseBasisIndex ℤ (𝓞 K) ≃ (K →+* ℂ) :=
  (canonicalEmbedding.latticeBasis K).indexEquiv (Pi.basisFun ℂ (K →+* ℂ))
e : index K ≃ Free.ChooseBasisIndex ℤ (𝓞 K) := (indexEquiv K).trans f.symm
M : Matrix (index K) (index K) ℝ := (mixedEmbedding.stdBasis K).toMatrix ⇑((latticeBasis K).reindex e.symm)
N : Matrix (K →+* ℂ) (K →+* ℂ) ℂ :=
  Algebra.embeddingsMatrixReindex ℚ ℂ (⇑(integralBasis K) ∘ ⇑f.symm) (RingHom.equivRatAlgHom K ℂ)
this : M.map ⇑ofRealHom = matrixToStdBasis K * ((reindex (indexEquiv K).symm (indexEquiv K).symm) N)ᵀ
⊢ 2⁻¹ ^ Fintype.card { w // w.IsComplex } * ↑(sqrt ‖(algebraMap ℚ ℂ) (Algebra.discr ℚ ⇑(integralBasis K))‖₊) =
    2⁻¹ ^ Fintype.card { w // w.IsComplex } * ↑(sqrt ‖discr K‖₊)

Tactic:
← coe_discr,

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
f : Free.ChooseBasisIndex ℤ (𝓞 K) ≃ (K →+* ℂ) :=
  (canonicalEmbedding.latticeBasis K).indexEquiv (Pi.basisFun ℂ (K →+* ℂ))
e : index K ≃ Free.ChooseBasisIndex ℤ (𝓞 K) := (indexEquiv K).trans f.symm
M : Matrix (index K) (index K) ℝ := (mixedEmbedding.stdBasis K).toMatrix ⇑((latticeBasis K).reindex e.symm)
N : Matrix (K →+* ℂ) (K →+* ℂ) ℂ :=
  Algebra.embeddingsMatrixReindex ℚ ℂ (⇑(integralBasis K) ∘ ⇑f.symm) (RingHom.equivRatAlgHom K ℂ)
this : M.map ⇑ofRealHom = matrixToStdBasis K * ((reindex (indexEquiv K).symm (indexEquiv K).symm) N)ᵀ
⊢ 2⁻¹ ^ Fintype.card { w // w.IsComplex } * ↑(sqrt ‖(algebraMap ℚ ℂ) ↑(discr K)‖₊) =
    2⁻¹ ^ Fintype.card { w // w.IsComplex } * ↑(sqrt ‖discr K‖₊)
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification

Goal state before:
pp: K : Type u_4
L : Type u_5
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
v : InfinitePlace K
inst✝¹ : NumberField K
inst✝ : NumberField L
this : Algebra K ℂ := v.embedding.toAlgebra
⊢ {φ | ComplexEmbedding.LiesOver φ v.embedding}.ncard = Fintype.card (L →ₐ[K] ℂ)

Tactic:
Set.ncard_eq_toFinset_card

Goal state after:
K : Type u_4
L : Type u_5
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
v : InfinitePlace K
inst✝¹ : NumberField K
inst✝ : NumberField L
this : Algebra K ℂ := v.embedding.toAlgebra
⊢ #⋯.toFinset = Fintype.card (L →ₐ[K] ℂ)
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic

Goal state before:
pp: K : Type u_1
inst✝ : Field K
s : Set { w // w.IsReal }
x : mixedSpace K
w : { w // w.IsReal }
hw : w ∈ s
⊢ ((negAt s) x).1 w = -x.1 w

Tactic:
simp_rw [negAt, prodCongr_apply, piCongrRight_apply, if_pos hw,
    ContinuousLinearEquiv.neg_apply]

Goal state after:
Module: Mathlib.RingTheory.IntegralClosure.IntegralRestrict

Goal state before:
pp: A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_6
inst✝¹⁹ : CommRing A
inst✝¹⁸ : CommRing B
inst✝¹⁷ : Algebra A B
inst✝¹⁶ : Field K
inst✝¹⁵ : Field L
inst✝¹⁴ : Algebra A K
inst✝¹³ : IsFractionRing A K
inst✝¹² : Algebra K L
inst✝¹¹ : Algebra A L
inst✝¹⁰ : IsScalarTower A K L
inst✝⁹ : Algebra B L
inst✝⁸ : IsScalarTower A B L
inst✝⁷ : IsIntegralClosure B A L
inst✝⁶ : FiniteDimensional K L
inst✝⁵ : IsDomain A
inst✝⁴ : IsIntegrallyClosed A
inst✝³ : IsDomain B
inst✝² : IsIntegrallyClosed B
inst✝¹ : Module.Finite A B
inst✝ : IsTorsionFree A B
x : B
this✝³ : IsIntegralClosure B A (FractionRing B)
this✝² : Algebra.IsAlgebraic (FractionRing A) (FractionRing B)
this✝¹ : IsLocalization (algebraMapSubmonoid B A⁰) (FractionRing B)
this✝ : FiniteDimensional (FractionRing A) (FractionRing B)
this : IsFractionRing B L
⊢ (FractionRing.algEquiv A K).symm ((algebraMap A K) ((intTrace A B) x)) =
    (FractionRing.algEquiv A K).symm ((trace K L) ((algebraMap B L) x))

Tactic:
AlgEquiv.commutes,

Goal state after:
A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_6
inst✝¹⁹ : CommRing A
inst✝¹⁸ : CommRing B
inst✝¹⁷ : Algebra A B
inst✝¹⁶ : Field K
inst✝¹⁵ : Field L
inst✝¹⁴ : Algebra A K
inst✝¹³ : IsFractionRing A K
inst✝¹² : Algebra K L
inst✝¹¹ : Algebra A L
inst✝¹⁰ : IsScalarTower A K L
inst✝⁹ : Algebra B L
inst✝⁸ : IsScalarTower A B L
inst✝⁷ : IsIntegralClosure B A L
inst✝⁶ : FiniteDimensional K L
inst✝⁵ : IsDomain A
inst✝⁴ : IsIntegrallyClosed A
inst✝³ : IsDomain B
inst✝² : IsIntegrallyClosed B
inst✝¹ : Module.Finite A B
inst✝ : IsTorsionFree A B
x : B
this✝³ : IsIntegralClosure B A (FractionRing B)
this✝² : Algebra.IsAlgebraic (FractionRing A) (FractionRing B)
this✝¹ : IsLocalization (algebraMapSubmonoid B A⁰) (FractionRing B)
this✝ : FiniteDimensional (FractionRing A) (FractionRing B)
this : IsFractionRing B L
⊢ (algebraMap A (FractionRing A)) ((intTrace A B) x) =
    (FractionRing.algEquiv A K).symm ((trace K L) ((algebraMap B L) x))
Module: Mathlib.RingTheory.Invariant.Galois

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝¹⁹ : CommRing A
inst✝¹⁸ : CommRing B
inst✝¹⁷ : Algebra A B
G : Type u_3
inst✝¹⁶ : Finite G
inst✝¹⁵ : Group G
inst✝¹⁴ : MulSemiringAction G B
inst✝¹³ : Algebra.IsInvariant A B G
P : Ideal A
Q : Ideal B
inst✝¹² : Q.LiesOver P
inst✝¹¹ : P.IsPrime
inst✝¹⁰ : Q.IsPrime
K : Type u_4
L : Type u_5
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
inst✝⁶ : Algebra (A ⧸ P) K
inst✝⁵ : IsFractionRing (A ⧸ P) K
inst✝⁴ : Algebra (B ⧸ Q) L
inst✝³ : IsFractionRing (B ⧸ Q) L
inst✝² : Algebra (A ⧸ P) L
inst✝¹ : IsScalarTower (A ⧸ P) (B ⧸ Q) L
inst✝ : IsScalarTower (A ⧸ P) K L
this✝ : Algebra.IsIntegral A B
this : Algebra.IsAlgebraic K L
x y : B ⧸ Q
hy : y ∈ nonZeroDivisors (B ⧸ Q)
a : A
ha : (algebraMap A (A ⧸ P)) a ≠ 0
b : B
h : (algebraMap A (A ⧸ P)) a • x = y * (algebraMap B (B ⧸ Q)) b
val✝ : Fintype G
p : A[X]
hp : Polynomial.map (algebraMap A B) p = MulSemiringAction.charpoly G b
h_monic : p.Monic
h_eval : (aeval b) p = 0
q : A[X] := p.comp (C a * X)
d : L := (algebraMap (B ⧸ Q) L) x / (algebraMap (B ⧸ Q) L) y
comm₁ : (algebraMap K L).comp (algebraMap (A ⧸ P) K) = (algebraMap (B ⧸ Q) L).comp (algebraMap (A ⧸ P) (B ⧸ Q))
comm₂ : (algebraMap (A ⧸ P) (B ⧸ Q)).comp (algebraMap A (A ⧸ P)) = (algebraMap B (B ⧸ Q)).comp (algebraMap A B)
⊢ (aeval d) (Polynomial.map (algebraMap (A ⧸ P) K) (Polynomial.map (algebraMap A (A ⧸ P)) (p.comp (C a * X)))) = 0

Tactic:
map_comp,

Goal state after:
A : Type u_1
B : Type u_2
inst✝¹⁹ : CommRing A
inst✝¹⁸ : CommRing B
inst✝¹⁷ : Algebra A B
G : Type u_3
inst✝¹⁶ : Finite G
inst✝¹⁵ : Group G
inst✝¹⁴ : MulSemiringAction G B
inst✝¹³ : Algebra.IsInvariant A B G
P : Ideal A
Q : Ideal B
inst✝¹² : Q.LiesOver P
inst✝¹¹ : P.IsPrime
inst✝¹⁰ : Q.IsPrime
K : Type u_4
L : Type u_5
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
inst✝⁶ : Algebra (A ⧸ P) K
inst✝⁵ : IsFractionRing (A ⧸ P) K
inst✝⁴ : Algebra (B ⧸ Q) L
inst✝³ : IsFractionRing (B ⧸ Q) L
inst✝² : Algebra (A ⧸ P) L
inst✝¹ : IsScalarTower (A ⧸ P) (B ⧸ Q) L
inst✝ : IsScalarTower (A ⧸ P) K L
this✝ : Algebra.IsIntegral A B
this : Algebra.IsAlgebraic K L
x y : B ⧸ Q
hy : y ∈ nonZeroDivisors (B ⧸ Q)
a : A
ha : (algebraMap A (A ⧸ P)) a ≠ 0
b : B
h : (algebraMap A (A ⧸ P)) a • x = y * (algebraMap B (B ⧸ Q)) b
val✝ : Fintype G
p : A[X]
hp : Polynomial.map (algebraMap A B) p = MulSemiringAction.charpoly G b
h_monic : p.Monic
h_eval : (aeval b) p = 0
q : A[X] := p.comp (C a * X)
d : L := (algebraMap (B ⧸ Q) L) x / (algebraMap (B ⧸ Q) L) y
comm₁ : (algebraMap K L).comp (algebraMap (A ⧸ P) K) = (algebraMap (B ⧸ Q) L).comp (algebraMap (A ⧸ P) (B ⧸ Q))
comm₂ : (algebraMap (A ⧸ P) (B ⧸ Q)).comp (algebraMap A (A ⧸ P)) = (algebraMap B (B ⧸ Q)).comp (algebraMap A B)
⊢ (aeval d)
      ((Polynomial.map (algebraMap (A ⧸ P) K) (Polynomial.map (algebraMap A (A ⧸ P)) p)).comp
        (Polynomial.map (algebraMap (A ⧸ P) K) (Polynomial.map (algebraMap A (A ⧸ P)) (C a * X)))) =
    0
Module: Mathlib.RingTheory.IntegralClosure.IntegralRestrict

Goal state before:
pp: A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_6
inst✝¹⁹ : CommRing A
inst✝¹⁸ : CommRing B
inst✝¹⁷ : Algebra A B
inst✝¹⁶ : Field K
inst✝¹⁵ : Field L
inst✝¹⁴ : Algebra A K
inst✝¹³ : IsFractionRing A K
inst✝¹² : Algebra K L
inst✝¹¹ : Algebra A L
inst✝¹⁰ : IsScalarTower A K L
inst✝⁹ : Algebra B L
inst✝⁸ : IsScalarTower A B L
inst✝⁷ : IsIntegralClosure B A L
inst✝⁶ : FiniteDimensional K L
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : IsDomain A
inst✝³ : IsDomain B
inst✝² : IsIntegrallyClosed B
inst✝¹ : Algebra.IsIntegral A B
inst✝ : IsTorsionFree A B
x : B
this : IsFractionRing B L
⊢ (FractionRing.algEquiv A K).symm ((algebraMap A K) ((intNorm A B) x)) =
    (FractionRing.algEquiv A K).symm ((norm K) ((algebraMap B L) x))

Tactic:
AlgEquiv.commutes,

Goal state after:
A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_6
inst✝¹⁹ : CommRing A
inst✝¹⁸ : CommRing B
inst✝¹⁷ : Algebra A B
inst✝¹⁶ : Field K
inst✝¹⁵ : Field L
inst✝¹⁴ : Algebra A K
inst✝¹³ : IsFractionRing A K
inst✝¹² : Algebra K L
inst✝¹¹ : Algebra A L
inst✝¹⁰ : IsScalarTower A K L
inst✝⁹ : Algebra B L
inst✝⁸ : IsScalarTower A B L
inst✝⁷ : IsIntegralClosure B A L
inst✝⁶ : FiniteDimensional K L
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : IsDomain A
inst✝³ : IsDomain B
inst✝² : IsIntegrallyClosed B
inst✝¹ : Algebra.IsIntegral A B
inst✝ : IsTorsionFree A B
x : B
this : IsFractionRing B L
⊢ (algebraMap A (FractionRing A)) ((intNorm A B) x) = (FractionRing.algEquiv A K).symm ((norm K) ((algebraMap B L) x))
Module: Mathlib.NumberTheory.RamificationInertia.Galois

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝¹⁹ : CommRing A
inst✝¹⁸ : CommRing B
inst✝¹⁷ : Algebra A B
p : Ideal A
P : Ideal B
hPp : P.IsPrime
hp : P.LiesOver p
G : Type u_3
inst✝¹⁶ : Group G
inst✝¹⁵ : Finite G
inst✝¹⁴ : MulSemiringAction G B
inst✝¹³ : IsGaloisGroup G A B
C : Type u_4
inst✝¹² : CommRing C
inst✝¹¹ : Algebra A C
inst✝¹⁰ : Algebra B C
inst✝⁹ : Nonempty ↑(P.primesOver C)
inst✝⁸ : IsScalarTower A B C
GAC : Type u_5
inst✝⁷ : Group GAC
inst✝⁶ : Finite GAC
inst✝⁵ : MulSemiringAction GAC C
inst✝⁴ : IsGaloisGroup GAC A C
GBC : Type u_6
inst✝³ : Group GBC
inst✝² : Finite GBC
inst✝¹ : MulSemiringAction GBC C
inst✝ : IsGaloisGroup GBC B C
Q : Ideal C
left✝ : Q.IsPrime
right✝ : Q.LiesOver P
⊢ p.inertiaDegIn B * P.inertiaDegIn C = p.inertiaDegIn C

Tactic:
have : Q.LiesOver p := LiesOver.trans Q P p

Goal state after:
A : Type u_1
B : Type u_2
inst✝¹⁹ : CommRing A
inst✝¹⁸ : CommRing B
inst✝¹⁷ : Algebra A B
p : Ideal A
P : Ideal B
hPp : P.IsPrime
hp : P.LiesOver p
G : Type u_3
inst✝¹⁶ : Group G
inst✝¹⁵ : Finite G
inst✝¹⁴ : MulSemiringAction G B
inst✝¹³ : IsGaloisGroup G A B
C : Type u_4
inst✝¹² : CommRing C
inst✝¹¹ : Algebra A C
inst✝¹⁰ : Algebra B C
inst✝⁹ : Nonempty ↑(P.primesOver C)
inst✝⁸ : IsScalarTower A B C
GAC : Type u_5
inst✝⁷ : Group GAC
inst✝⁶ : Finite GAC
inst✝⁵ : MulSemiringAction GAC C
inst✝⁴ : IsGaloisGroup GAC A C
GBC : Type u_6
inst✝³ : Group GBC
inst✝² : Finite GBC
inst✝¹ : MulSemiringAction GBC C
inst✝ : IsGaloisGroup GBC B C
Q : Ideal C
left✝ : Q.IsPrime
right✝ : Q.LiesOver P
this : Q.LiesOver p
⊢ p.inertiaDegIn B * P.inertiaDegIn C = p.inertiaDegIn C