mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.FieldTheory.Differential.Liouville

Goal state before:
pp: F : Type u_1
K : Type u_2
inst✝⁹ : Field F
inst✝⁸ : Field K
inst✝⁷ : Differential F
inst✝⁶ : Differential K
inst✝⁵ : Algebra F K
inst✝⁴ : DifferentialAlgebra F K
inst✝³ : CharZero F
inst✝² : FiniteDimensional F K
inst✝¹ : IsGalois F K
a : F
ι : Type
inst✝ : Fintype ι
c : ι → F
hc : ∀ (x : ι), (c x)′ = 0
u : ι → K
v : K
h : ↑a = ∑ x, ↑(c x) * logDeriv (u x) + v′
this : CharZero K
c₀ : ι → F := fun i ↦ c i / ↑(Fintype.card Gal(K/F))
u₁ : ι → K := fun i ↦ ∏ x, x (u i)
⊢ ∀ (i : ι), u₁ i ∈ fixedField ⊤

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Differential.Liouville

Goal state before:
pp: F : Type u_1
K : Type u_2
inst✝⁹ : Field F
inst✝⁸ : Field K
inst✝⁷ : Differential F
inst✝⁶ : Differential K
inst✝⁵ : Algebra F K
inst✝⁴ : DifferentialAlgebra F K
inst✝³ : CharZero F
inst✝² : FiniteDimensional F K
inst✝¹ : IsGalois F K
a : F
ι : Type
inst✝ : Fintype ι
c : ι → F
hc : ∀ (x : ι), (c x)′ = 0
u : ι → K
v : K
h : ↑a = ∑ x, ↑(c x) * logDeriv (u x) + v′
this : CharZero K
c₀ : ι → F := fun i ↦ c i / ↑(Fintype.card Gal(K/F))
u₁ : ι → K := fun i ↦ ∏ x, x (u i)
⊢ ∃ ι₀ x c₀, ∃ (_ : ∀ (x : ι₀), (c₀ x)′ = 0), ∃ u₀ v₀, a = ∑ x, c₀ x * logDeriv (u₀ x) + v₀′

Tactic:
have : ∀ i, u₁ i ∈ fixedField (⊤ : Subgroup (K ≃ₐ[F] K)) := by
      rintro i ⟨e, _⟩
      change e (u₁ i) = u₁ i
      simp only [u₁, map_prod]
      apply Fintype.prod_equiv (Equiv.mulLeft e)
      simp

Goal state after:
F : Type u_1
K : Type u_2
inst✝⁹ : Field F
inst✝⁸ : Field K
inst✝⁷ : Differential F
inst✝⁶ : Differential K
inst✝⁵ : Algebra F K
inst✝⁴ : DifferentialAlgebra F K
inst✝³ : CharZero F
inst✝² : FiniteDimensional F K
inst✝¹ : IsGalois F K
a : F
ι : Type
inst✝ : Fintype ι
c : ι → F
hc : ∀ (x : ι), (c x)′ = 0
u : ι → K
v : K
h : ↑a = ∑ x, ↑(c x) * logDeriv (u x) + v′
this✝ : CharZero K
c₀ : ι → F := fun i ↦ c i / ↑(Fintype.card Gal(K/F))
u₁ : ι → K := fun i ↦ ∏ x, x (u i)
this : ∀ (i : ι), u₁ i ∈ fixedField ⊤
⊢ ∃ ι₀ x c₀, ∃ (_ : ∀ (x : ι₀), (c₀ x)′ = 0), ∃ u₀ v₀, a = ∑ x, c₀ x * logDeriv (u₀ x) + v₀′
Module: Mathlib.FieldTheory.Differential.Liouville

Goal state before:
pp: F : Type u_1
K : Type u_2
inst✝⁹ : Field F
inst✝⁸ : Field K
inst✝⁷ : Differential F
inst✝⁶ : Differential K
inst✝⁵ : Algebra F K
inst✝⁴ : DifferentialAlgebra F K
inst✝³ : CharZero F
inst✝² : FiniteDimensional F K
inst✝¹ : IsGalois F K
a : F
ι : Type
inst✝ : Fintype ι
c : ι → F
hc : ∀ (x : ι), (c x)′ = 0
u : ι → K
v : K
h : ↑a = ∑ x, ↑(c x) * logDeriv (u x) + v′
this : CharZero K
c₀ : ι → F := fun i ↦ c i / ↑(Fintype.card Gal(K/F))
u₁ : ι → K := fun i ↦ ∏ x, x (u i)
ffb : fixedField ⊤ = ⊥
u₀ : ι → F
hu₀ : ∀ (i : ι), (algebraMap F K) (u₀ i) = u₁ i
v₁ : K := (∑ x, x v) / ↑(Fintype.card Gal(K/F))
⊢ v₁ ∈ fixedField ⊤

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Galois.NormalBasis

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : Finite L
this✝ : Finite K
this : Fintype K
x : L
hx :
  Ideal.span {X ^ finrank K L - 1} =
    (toSpanSingleton K[X] (AEval' (frobeniusAlgHom K L).toLinearMap)
        ((AEval'.of (frobeniusAlgHom K L).toLinearMap) x)).ker
⊢ (↑(AEval'.of (frobeniusAlgHom K L).toLinearMap).symm ∘ₗ
        ↑K
          (liftQ (Ideal.span {X ^ finrank K L - 1})
            (toSpanSingleton K[X] (AEval' (frobeniusAlgHom K L).toLinearMap)
              ((AEval'.of (frobeniusAlgHom K L).toLinearMap) x))
            ⋯)).ker =
    ⊥

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Galois.NormalBasis

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : Finite L
⊢ ∃ x, LinearIndependent K fun σ ↦ σ x

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Galois.NormalBasis

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : FiniteDimensional K L
inst✝ : Infinite K
e : Module.Basis (Module.Free.ChooseBasisIndex K L) K L
M : Matrix Gal(L/K) Gal(L/K) (MvPolynomial (Module.Free.ChooseBasisIndex K L) L) :=
  Matrix.of fun i j ↦ ∑ k, i.symm (j (e k)) • MvPolynomial.X k
⊢ Submodule.span L (Set.range fun k j ↦ j (e k)) = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Galois.NormalBasis

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : FiniteDimensional K L
inst✝ : Infinite K
e : Module.Basis (Module.Free.ChooseBasisIndex K L) K L
M : Matrix Gal(L/K) Gal(L/K) (MvPolynomial (Module.Free.ChooseBasisIndex K L) L) :=
  Matrix.of fun i j ↦ ∑ k, i.symm (j (e k)) • MvPolynomial.X k
hq : Submodule.span L (Set.range fun k j ↦ j (e k)) = ⊤
⊢ ∃ c, (MvPolynomial.eval c) M.det = 1

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Galois.NormalBasis

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : FiniteDimensional K L
inst✝ : Infinite K
e : Module.Basis (Module.Free.ChooseBasisIndex K L) K L
M : Matrix Gal(L/K) Gal(L/K) (MvPolynomial (Module.Free.ChooseBasisIndex K L) L) :=
  Matrix.of fun i j ↦ ∑ k, i.symm (j (e k)) • MvPolynomial.X k
hq : Submodule.span L (Set.range fun k j ↦ j (e k)) = ⊤
c : Module.Free.ChooseBasisIndex K L → L
hc : (MvPolynomial.eval c) M.det = 1
h0 : M.det = 0
⊢ False

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Galois.NormalBasis

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : FiniteDimensional K L
inst✝ : Infinite K
e : Module.Basis (Module.Free.ChooseBasisIndex K L) K L
M : Matrix Gal(L/K) Gal(L/K) (MvPolynomial (Module.Free.ChooseBasisIndex K L) L) :=
  Matrix.of fun i j ↦ ∑ k, i.symm (j (e k)) • MvPolynomial.X k
hq : Submodule.span L (Set.range fun k j ↦ j (e k)) = ⊤
c : Module.Free.ChooseBasisIndex K L → L
hc : (MvPolynomial.eval c) M.det = 1
hM : M.det ≠ 0
⊢ ∃ b, (MvPolynomial.eval (⇑(algebraMap K L) ∘ b)) M.det ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Galois.NormalBasis

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : FiniteDimensional K L
inst✝ : Infinite K
⊢ ∃ x, LinearIndependent K fun σ ↦ σ x

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.CardinalEmb

Goal state before:
pp: F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
rank_inf : Fact (ℵ₀ ≤ Module.rank F E)
inst✝ : Algebra.IsAlgebraic F E
i : (Module.rank F E).ord.ToType
ih : (y : (Module.rank F E).ord.ToType) → y < i → (Module.rank F E).ord.ToType
s : Set E := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
this✝ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
this : Module.rank F ↥(adjoin F s) = Module.rank F ↥⊤
⊢ (#(Module.rank F E).ord.ToType).ord = Ordinal.type fun x1 x2 ↦ x1 < x2

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Galois.NormalBasis

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : FiniteDimensional K L
⊢ ∃ x, LinearIndependent K fun σ ↦ σ x

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Galois.NormalBasis

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : FiniteDimensional K L
inst✝ : IsGalois K L
e : Gal(L/K)
⊢ (normalBasis K L) e = e ((normalBasis K L) 1)

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Differential.Liouville

Goal state before:
pp: F : Type u_1
K : Type u_2
inst✝⁹ : Field F
inst✝⁸ : Field K
inst✝⁷ : Differential F
inst✝⁶ : Differential K
inst✝⁵ : Algebra F K
inst✝⁴ : DifferentialAlgebra F K
inst✝³ : CharZero F
inst✝² : FiniteDimensional F K
inst✝¹ : IsGalois F K
a : F
ι : Type
inst✝ : Fintype ι
c : ι → F
hc : ∀ (x : ι), (c x)′ = 0
u : ι → K
v : K
h : ↑a = ∑ x, ↑(c x) * logDeriv (u x) + v′
⊢ ∃ ι₀ x c₀, ∃ (_ : ∀ (x : ι₀), (c₀ x)′ = 0), ∃ u₀ v₀, a = ∑ x, c₀ x * logDeriv (u₀ x) + v₀′

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.CardinalEmb

Goal state before:
pp: F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
rank_inf : Fact (ℵ₀ ≤ Module.rank F E)
inst✝ : Algebra.IsAlgebraic F E
i : (Module.rank F E).ord.ToType
ih : (y : (Module.rank F E).ord.ToType) → y < i → (Module.rank F E).ord.ToType
s : Set E := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.CardinalEmb

Goal state before:
pp: F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
rank_inf : Fact (ℵ₀ ≤ Module.rank F E)
inst✝ : Algebra.IsAlgebraic F E
i : (Module.rank F E).ord.ToType
⊢ IsLeast {k | b k ∉ adjoin F (⇑b ∘ φ '' Iio i)} (φ i)

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.CardinalEmb

Goal state before:
pp: F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
rank_inf : Fact (ℵ₀ ≤ Module.rank F E)
inst✝ : Algebra.IsAlgebraic F E
i j : (Module.rank F E).ord.ToType
h : i < j
⊢ φ i < φ j

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.CardinalEmb

Goal state before:
pp: F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
rank_inf : Fact (ℵ₀ ≤ Module.rank F E)
inst✝ : Algebra.IsAlgebraic F E
i : (Module.rank F E).ord.ToType
⊢ adjoin F (⇑b ∘ φ '' Iio i) = adjoin F (⇑b '' Iio (φ i))

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.CardinalEmb

Goal state before:
pp: F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
rank_inf : Fact (ℵ₀ ≤ Module.rank F E)
inst✝ : Algebra.IsAlgebraic F E
⊢ ⨆ i, adjoin F (⇑b ∘ φ '' Iio i) = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.CardinalEmb

Goal state before:
pp: F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
rank_inf : Fact (ℵ₀ ≤ Module.rank F E)
inst✝ : Algebra.IsAlgebraic F E
i : (Module.rank F E).ord.ToType
⊢ adjoin F (⇑b ∘ φ '' Iio (i⁺)) = IntermediateField.restrictScalars F (↥(adjoin F (⇑b ∘ φ '' Iio i)))⟮b (φ i)⟯

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.CardinalEmb

Goal state before:
pp: F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
rank_inf : Fact (ℵ₀ ≤ Module.rank F E)
inst✝ : Algebra.IsAlgebraic F E
i : (Module.rank F E).ord.ToType
f : ↥(adjoin F (⇑b ∘ φ '' Iio (i⁺))) →ₐ[F] Ē
⊢ ((succEquiv i) f).1 = f.comp (Subalgebra.inclusion ⋯)

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.CardinalEmb

Goal state before:
pp: F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
rank_inf : Fact (ℵ₀ ≤ Module.rank F E)
inst✝ : Algebra.IsAlgebraic F E
i j : WithTop (Module.rank F E).ord.ToType
h : i < j
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.CardinalEmb

Goal state before:
pp: F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
rank_inf : Fact (ℵ₀ ≤ Module.rank F E)
inst✝¹ : Algebra.IsAlgebraic F E
inst✝ : Algebra.IsSeparable F E
i : (Module.rank F E).ord.ToType
⊢ 2 ≤ #(Emb ↥(adjoin F (⇑b ∘ φ '' Iio i)) ↥(↥(adjoin F (⇑b ∘ φ '' Iio i)))⟮b (φ i)⟯)

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.CardinalEmb

Goal state before:
pp: F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
rank_inf : Fact (ℵ₀ ≤ Module.rank F E)
inst✝ : Algebra.IsAlgebraic F E
i : WithTop (Module.rank F E).ord.ToType
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.CardinalEmb

Goal state before:
pp: F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
rank_inf : Fact (ℵ₀ ≤ Module.rank F E)
inst✝ : Algebra.IsAlgebraic F E
i : WithTop (Module.rank F E).ord.ToType
f : ↥(filtration (i⁺)) →ₐ[F] Ē
⊢ ((equivSucc i) f).1 = embFunctor F E ⋯ f

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.CardinalEmb

Goal state before:
pp: F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
rank_inf : Fact (ℵ₀ ≤ Module.rank F E)
inst✝ : Algebra.IsAlgebraic F E
i : WithTop (Module.rank F E).ord.ToType
hi : IsSuccPrelimit i
⊢ ⨆ j, filtration ↑j = filtration i

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.CardinalEmb

Goal state before:
pp: F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
rank_inf : Fact (ℵ₀ ≤ Module.rank F E)
inst✝ : Algebra.IsAlgebraic F E
i : WithTop (Module.rank F E).ord.ToType
hi✝ : IsSuccPrelimit i
hi : ¬Nonempty ↑(Iio i)
⊢ filtration i = ⊥

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.IntermediateField.ExtendRight

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
F : IntermediateField K L
M : Type u_3
inst✝³ : Field M
inst✝² : Algebra K M
inst✝¹ : Algebra L M
inst✝ : IsScalarTower K L M
a : ↥(F.extendRight M)
⊢ (algebraMap (↥F) M) ((F.extendRightEquiv M).symm a) = ↑a

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.CardinalEmb

Goal state before:
pp: F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
rank_inf : Fact (ℵ₀ ≤ Module.rank F E)
inst✝ : Algebra.IsAlgebraic F E
i : WithTop (Module.rank F E).ord.ToType
hi : IsSuccPrelimit i
f : ↑(limit (embFunctor F E) i)
h : Nonempty ↑(Iio i)
⊢ (filtration i).toSubalgebra ≤ ⨆ x, ((fun j ↦ filtration ↑j) x).toSubalgebra

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.IntermediateField.ExtendRight

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝¹⁰ : Field K
inst✝⁹ : Field L
inst✝⁸ : Algebra K L
F : IntermediateField K L
M : Type u_3
inst✝⁷ : Field M
inst✝⁶ : Algebra K M
inst✝⁵ : Algebra L M
inst✝⁴ : IsScalarTower K L M
S : Type u_5
inst✝³ : CommRing S
inst✝² : Algebra S ↥F
inst✝¹ : Algebra S M
inst✝ : IsScalarTower S (↥F) M
s : S
⊢ (algebraMap (↥F) L) ((algebraMap S ↥F) s) ∈ ↑F.toSubsemiring

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.IntermediateField.ExtendRight

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝¹⁰ : Field K
inst✝⁹ : Field L
inst✝⁸ : Algebra K L
F : IntermediateField K L
M : Type u_3
inst✝⁷ : Field M
inst✝⁶ : Algebra K M
inst✝⁵ : Algebra L M
inst✝⁴ : IsScalarTower K L M
S : Type u_5
inst✝³ : CommRing S
inst✝² : Algebra S ↥F
inst✝¹ : Algebra S M
inst✝ : IsScalarTower S (↥F) M
s : S
⊢ (algebraMap S M) s ∈ F.extendRight M

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.IntermediateField.ExtendRight

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝¹¹ : Field K
inst✝¹⁰ : Field L
inst✝⁹ : Algebra K L
F : IntermediateField K L
M : Type u_3
inst✝⁸ : Field M
inst✝⁷ : Algebra K M
inst✝⁶ : Algebra L M
inst✝⁵ : IsScalarTower K L M
R : Type u_4
S : Type u_5
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra S ↥F
inst✝¹ : Algebra S M
inst✝ : IsScalarTower S (↥F) M
s : S
x : ↥(F.extendRight M)
⊢ ↥(F.extendRight M)

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.IntermediateField.ExtendRight

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝¹¹ : Field K
inst✝¹⁰ : Field L
inst✝⁹ : Algebra K L
F : IntermediateField K L
M : Type u_3
inst✝⁸ : Field M
inst✝⁷ : Algebra K M
inst✝⁶ : Algebra L M
inst✝⁵ : IsScalarTower K L M
R : Type u_4
S : Type u_5
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra S ↥F
inst✝¹ : Algebra S M
inst✝ : IsScalarTower S (↥F) M
x✝¹ : S
x✝ : ↥(F.extendRight M)
⊢ ↑(((algebraMap S M).codRestrict (F.extendRight M).toSubalgebra ⋯) x✝¹ * x✝) =
    ↑(x✝ * ((algebraMap S M).codRestrict (F.extendRight M).toSubalgebra ⋯) x✝¹)

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.IntermediateField.ExtendRight

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝¹¹ : Field K
inst✝¹⁰ : Field L
inst✝⁹ : Algebra K L
F : IntermediateField K L
M : Type u_3
inst✝⁸ : Field M
inst✝⁷ : Algebra K M
inst✝⁶ : Algebra L M
inst✝⁵ : IsScalarTower K L M
R : Type u_4
S : Type u_5
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra S ↥F
inst✝¹ : Algebra S M
inst✝ : IsScalarTower S (↥F) M
s : S
x : ↥(F.extendRight M)
⊢ ↑(s • x) = ↑(((algebraMap S M).codRestrict (F.extendRight M).toSubalgebra ⋯) s * x)

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.IntermediateField.ExtendRight

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝¹³ : Field K
inst✝¹² : Field L
inst✝¹¹ : Algebra K L
F : IntermediateField K L
M : Type u_3
inst✝¹⁰ : Field M
inst✝⁹ : Algebra K M
inst✝⁸ : Algebra L M
inst✝⁷ : IsScalarTower K L M
R : Type u_4
S : Type u_5
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra S ↥F
inst✝³ : Algebra S M
inst✝² : IsScalarTower S (↥F) M
inst✝¹ : Algebra S K
inst✝ : IsScalarTower S K M
⊢ (F.extendRight M).algebra' = algebra F M

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.IntermediateField.ExtendRight

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝¹⁰ : Field K
inst✝⁹ : Field L
inst✝⁸ : Algebra K L
F : IntermediateField K L
M : Type u_3
inst✝⁷ : Field M
inst✝⁶ : Algebra K M
inst✝⁵ : Algebra L M
inst✝⁴ : IsScalarTower K L M
S : Type u_5
inst✝³ : CommRing S
inst✝² : Algebra S ↥F
inst✝¹ : Algebra S M
inst✝ : IsScalarTower S (↥F) M
a : ↥(F.extendRight M)
⊢ (algebraMap (↥F) M) ((F.extendRightEquiv' M S).symm a) = ↑a

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.CardinalEmb

Goal state before:
pp: F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
rank_inf : Fact (ℵ₀ ≤ Module.rank F E)
inst✝ : Algebra.IsAlgebraic F E
i : WithTop (Module.rank F E).ord.ToType
hi : IsSuccPrelimit i
f : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
h : Nonempty ↑(Iio i)
x : E
hx✝ : x ∈ (filtration i).toSubalgebra
hx : ∃ i_1, x ∈ ↑(filtration ↑i_1)
⊢ ↑⟨x, hx✝⟩ ∈ ((fun j ↦ filtration ↑j) ?m.322).toSubalgebra

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.IntermediateField.ExtendRight

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝¹⁵ : Field K
inst✝¹⁴ : Field L
inst✝¹³ : Algebra K L
F : IntermediateField K L
M : Type u_3
inst✝¹² : Field M
inst✝¹¹ : Algebra K M
inst✝¹⁰ : Algebra L M
inst✝⁹ : IsScalarTower K L M
R : Type u_4
S : Type u_5
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra S ↥F
inst✝⁵ : Algebra S M
inst✝⁴ : IsScalarTower S (↥F) M
inst✝³ : Algebra R ↥F
inst✝² : Algebra R M
inst✝¹ : IsScalarTower R (↥F) M
inst✝ : IsIntegralClosure S R ↥F
⊢ IsIntegralClosure S R ↥(F.extendRight M)

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.CardinalEmb

Goal state before:
pp: F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
rank_inf : Fact (ℵ₀ ≤ Module.rank F E)
inst✝ : Algebra.IsAlgebraic F E
i : WithTop (Module.rank F E).ord.ToType
hi : IsSuccPrelimit i
f : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.CardinalEmb

Goal state before:
pp: F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
rank_inf : Fact (ℵ₀ ≤ Module.rank F E)
inst✝ : Algebra.IsAlgebraic F E
i : WithTop (Module.rank F E).ord.ToType
hi : IsSuccPrelimit i
f : ↑(limit (embFunctor F E) i)
j : ↑(Iio i)
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.PerfectClosure

Goal state before:
pp: K : Type u
inst✝² : CommRing K
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : CharP K p
x1 x2 y : ℕ × K
H : R K p x1 x2
n : ℕ
x : K
⊢ R K p ((n, x).1 + y.1, (⇑(frobenius K p))^[y.1] (n, x).2 * (⇑(frobenius K p))^[(n, x).1] y.2)
    ((n + 1, (frobenius K p) x).1 + y.1,
      (⇑(frobenius K p))^[y.1] (n + 1, (frobenius K p) x).2 * (⇑(frobenius K p))^[(n + 1, (frobenius K p) x).1] y.2)

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.PerfectClosure

Goal state before:
pp: K : Type u
inst✝² : CommRing K
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : CharP K p
x y1 y2 : ℕ × K
H : R K p y1 y2
n : ℕ
y : K
⊢ R K p (x.1 + (n, y).1, (⇑(frobenius K p))^[(n, y).1] x.2 * (⇑(frobenius K p))^[x.1] (n, y).2)
    (x.1 + (n + 1, (frobenius K p) y).1,
      (⇑(frobenius K p))^[(n + 1, (frobenius K p) y).1] x.2 * (⇑(frobenius K p))^[x.1] (n + 1, (frobenius K p) y).2)

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Galois.Profinite

Goal state before:
pp: k : Type u_1
K : Type u_2
inst✝² : Field k
inst✝¹ : Field K
inst✝ : Algebra k K
L₁ L₂ L₃ : (FiniteGaloisIntermediateField k K)ᵒᵖ
f : L₁ ⟶ L₂
g : L₂ ⟶ L₃
⊢ finGaloisGroupMap (f ≫ g) = finGaloisGroupMap f ≫ finGaloisGroupMap g

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.CardinalEmb

Goal state before:
pp: F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Algebra.IsSeparable F E
rank_inf : ℵ₀ ≤ Module.rank F E
⊢ #(Emb F E) = 2 ^ Module.rank F E

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Galois.Profinite

Goal state before:
pp: k✝ : Type u_1
K✝ : Type u_2
inst✝⁵ : Field k✝
inst✝⁴ : Field K✝
inst✝³ : Algebra k✝ K✝
k : Type u_3
K : Type u_4
inst✝² : Field k
inst✝¹ : Field K
inst✝ : Algebra k K
σ : Gal(K/k)
L₁ L₂ : (FiniteGaloisIntermediateField k K)ᵒᵖ
π : L₁ ⟶ L₂
⊢ (Hom.hom ((asProfiniteGaloisGroupFunctor k K).map π))
      ((fun L ↦ (AlgEquiv.restrictNormalHom ↥(unop L).toIntermediateField) σ) L₁) =
    (fun L ↦ (AlgEquiv.restrictNormalHom ↥(unop L).toIntermediateField) σ) L₂

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.PerfectClosure

Goal state before:
pp: K : Type u
inst✝² : CommRing K
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : CharP K p
e f g : PerfectClosure K p
x✝² : ℕ × K
m : ℕ
x : K
x✝¹ : ℕ × K
n : ℕ
y : K
x✝ : ℕ × K
s : ℕ
z : K
⊢ Quot.mk (R K p) (m, x) * Quot.mk (R K p) (n, y) * Quot.mk (R K p) (s, z) =
    Quot.mk (R K p) (m, x) * (Quot.mk (R K p) (n, y) * Quot.mk (R K p) (s, z))

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Galois.Profinite

Goal state before:
pp: k✝ : Type u_1
K✝ : Type u_2
inst✝⁵ : Field k✝
inst✝⁴ : Field K✝
inst✝³ : Algebra k✝ K✝
k : Type u_3
K : Type u_4
inst✝² : Field k
inst✝¹ : Field K
inst✝ : Algebra k K
⊢ ⟨fun L ↦ (AlgEquiv.restrictNormalHom ↥(unop L).toIntermediateField) 1, ⋯⟩ = 1

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.PerfectClosure

Goal state before:
pp: K : Type u
inst✝² : CommRing K
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : CharP K p
e : PerfectClosure K p
x✝ : ℕ × K
n : ℕ
x : K
⊢ ((0, 1).1 + (n, x).1, (⇑(frobenius K p))^[(n, x).1] (0, 1).2 * (⇑(frobenius K p))^[(0, 1).1] (n, x).2) = (n, x)

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Galois.Profinite

Goal state before:
pp: k✝ : Type u_1
K✝ : Type u_2
inst✝⁵ : Field k✝
inst✝⁴ : Field K✝
inst✝³ : Algebra k✝ K✝
k : Type u_3
K : Type u_4
inst✝² : Field k
inst✝¹ : Field K
inst✝ : Algebra k K
x y : Gal(K/k)
⊢ ⟨fun L ↦ (AlgEquiv.restrictNormalHom ↥(unop L).toIntermediateField) (x * y), ⋯⟩ =
    ⟨fun L ↦ (AlgEquiv.restrictNormalHom ↥(unop L).toIntermediateField) x, ⋯⟩ *
      ⟨fun L ↦ (AlgEquiv.restrictNormalHom ↥(unop L).toIntermediateField) y, ⋯⟩

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.PerfectClosure

Goal state before:
pp: K : Type u
inst✝² : CommRing K
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : CharP K p
e : PerfectClosure K p
x✝ : ℕ × K
n : ℕ
x : K
⊢ ((n, x).1 + (0, 1).1, (⇑(frobenius K p))^[(0, 1).1] (n, x).2 * (⇑(frobenius K p))^[(n, x).1] (0, 1).2) = (n, x)

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.PerfectClosure

Goal state before:
pp: K : Type u
inst✝² : CommRing K
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : CharP K p
e f : PerfectClosure K p
x✝¹ : ℕ × K
m : ℕ
x : K
x✝ : ℕ × K
n : ℕ
y : K
⊢ ((m, x).1 + (n, y).1, (⇑(frobenius K p))^[(n, y).1] (m, x).2 * (⇑(frobenius K p))^[(m, x).1] (n, y).2) =
    ((n, y).1 + (m, x).1, (⇑(frobenius K p))^[(m, x).1] (n, y).2 * (⇑(frobenius K p))^[(n, y).1] (m, x).2)

Tactic:
by

Goal state after: