mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.FieldTheory.Galois.NormalBasis

Goal state before:
pp: case h
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
⊢ LinearIndependent K
    ((fun σ ↦ σ x) ∘ ⇑((finCongr ⋯).trans (Equiv.ofBijective (fun n ↦ frobeniusAlgEquivOfAlgebraic K L ^ ↑n) ⋯)))

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
⊢ K

Tactic:
]

Goal state after:
case h
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
⊢ LinearIndependent K
    ((fun σ ↦ σ x) ∘ ⇑((finCongr ⋯).trans (Equiv.ofBijective (fun n ↦ frobeniusAlgEquivOfAlgebraic K L ^ ↑n) ⋯)))

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
⊢ K
Module: Mathlib.FieldTheory.Differential.Liouville

Goal state before:
pp: case refine_2
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
ffb : fixedField ⊤ = ⊥
⊢ (algebraMap F K) a =
    ∑ x, ↑(c x) / ↑(Fintype.card Gal(K/F)) * logDeriv (∏ x_1, x_1 (u x)) + (∑ x, x v)′ / ↑(Fintype.card Gal(K/F))

pp: case refine_2.h
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
ffb : fixedField ⊤ = ⊥
⊢ (↑(Fintype.card Gal(K/F)))′ = 0

Tactic:
]

Goal state after:
case refine_2
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
ffb : fixedField ⊤ = ⊥
⊢ (algebraMap F K) a =
    ∑ x, ↑(c x) / ↑(Fintype.card Gal(K/F)) * logDeriv (∏ x_1, x_1 (u x)) + (∑ x, x v)′ / ↑(Fintype.card Gal(K/F))

case refine_2.h
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
ffb : fixedField ⊤ = ⊥
⊢ (↑(Fintype.card Gal(K/F)))′ = 0
Module: Mathlib.FieldTheory.Differential.Liouville

Goal state before:
pp: case refine_2
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
ffb : fixedField ⊤ = ⊥
⊢ (algebraMap F K) a = (∑ i, ↑(c i) * logDeriv (∏ x, x (u i)) + ∑ x, (x v)′) / ↑(Fintype.card Gal(K/F))

Tactic:
]

Goal state after:
case refine_2
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
ffb : fixedField ⊤ = ⊥
⊢ (algebraMap F K) a = (∑ i, ↑(c i) * logDeriv (∏ x, x (u i)) + ∑ x, (x v)′) / ↑(Fintype.card Gal(K/F))
Module: Mathlib.FieldTheory.Galois.NormalBasis

Goal state before:
pp: case e'_4
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
e_1✝ : Fin (X ^ finrank K L - C 1).natDegree = Fin (AdjoinRoot.powerBasis ⋯).dim
i : Fin (X ^ finrank K L - C 1).natDegree
⊢ (AEval'.of (frobeniusAlgHom K L).toLinearMap) ((⇑(frobeniusAlgHom K L).toLinearMap)^[↑i] x) =
    (AEval'.of (frobeniusAlgHom K L).toLinearMap) ((⇑(frobeniusAlgEquivOfAlgebraic K L))^[↑(Fin.cast ⋯ i)] x)

Tactic:
]

Goal state after:
case e'_4
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
e_1✝ : Fin (X ^ finrank K L - C 1).natDegree = Fin (AdjoinRoot.powerBasis ⋯).dim
i : Fin (X ^ finrank K L - C 1).natDegree
⊢ (AEval'.of (frobeniusAlgHom K L).toLinearMap) ((⇑(frobeniusAlgHom K L).toLinearMap)^[↑i] x) =
    (AEval'.of (frobeniusAlgHom K L).toLinearMap) ((⇑(frobeniusAlgEquivOfAlgebraic K L))^[↑(Fin.cast ⋯ i)] x)
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:
]

Goal state after:
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)
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 : Module.Free.ChooseBasisIndex K L → K
hb : (M.map ⇑(MvPolynomial.eval (⇑(algebraMap K L) ∘ b))).det ≠ 0
a : Gal(L/K) → K
ha : ∑ i, a i • i (∑ k, b k • e k) = 0
⊢ (fun x ↦ (algebraMap K L) (a x)) = fun x ↦ 0

Tactic:
]

Goal state after:
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 : Module.Free.ChooseBasisIndex K L → K
hb : (M.map ⇑(MvPolynomial.eval (⇑(algebraMap K L) ∘ b))).det ≠ 0
a : Gal(L/K) → K
ha : ∑ i, a i • i (∑ k, b k • e k) = 0
⊢ (fun x ↦ (algebraMap K L) (a x)) = fun x ↦ 0
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 : adjoin F s = ⊤
⊢ False

Tactic:
]

Goal state after:
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 : adjoin F s = ⊤
⊢ False
Module: Mathlib.FieldTheory.Differential.Liouville

Goal state before:
pp: case refine_2
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
ffb : fixedField ⊤ = ⊥
⊢ (algebraMap F K) a * ↑(Fintype.card Gal(K/F)) = ∑ x, (∑ x_1, ↑(c x_1) * logDeriv (x (u x_1)) + (x v)′)

Tactic:
]

Goal state after:
case refine_2
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
ffb : fixedField ⊤ = ⊥
⊢ (algebraMap F K) a * ↑(Fintype.card Gal(K/F)) = ∑ x, (∑ x_1, ↑(c x_1) * logDeriv (x (u x_1)) + (x v)′)
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)
⊢ e ⋯.choose = e ⋯.choose

Tactic:
]

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✝¹ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
this✝ : Module.rank F ↥(adjoin F s) = Module.rank F ↥⊤
this : #↑(Iio i) < Module.rank F E
⊢ Module.rank F ↥(adjoin F s) < Module.rank F E

Tactic:
]

Goal state after:
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✝¹ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
this✝ : Module.rank F ↥(adjoin F s) = Module.rank F ↥⊤
this : #↑(Iio i) < Module.rank F E
⊢ Module.rank F ↥(adjoin F s) < Module.rank F E
Module: Mathlib.FieldTheory.Differential.Liouville

Goal state before:
pp: case e'_6
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
this : CharZero K
ffb : fixedField ⊤ = ⊥
e : Gal(K/F)
h : (algebraMap F K) a = ∑ x, (algebraMap F K) (c x) * e (logDeriv (u x)) + e v′
⊢ (e v)′ = (e v)′

Tactic:
]

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
          (range fun x ↦
            (⇑b ∘
                ⋯.fix fun i ih ↦
                  let s := range fun j ↦ b (ih ↑j ⋯);
                  ⋯.min {k | b k ∉ adjoin F s} ⋯)
              ↑x)}
    (let s :=
      range fun j ↦
        b
          ((fun y x ↦
              ⋯.fix
                (fun i ih ↦
                  let s := range fun j ↦ b (ih ↑j ⋯);
                  ⋯.min {k | b k ∉ adjoin F s} ⋯)
                y)
            ↑j ⋯);
    ⋯.min {k | b k ∉ adjoin F s} ⋯)

Tactic:
]

Goal state after:
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
          (range fun x ↦
            (⇑b ∘
                ⋯.fix fun i ih ↦
                  let s := range fun j ↦ b (ih ↑j ⋯);
                  ⋯.min {k | b k ∉ adjoin F s} ⋯)
              ↑x)}
    (let s :=
      range fun j ↦
        b
          ((fun y x ↦
              ⋯.fix
                (fun i ih ↦
                  let s := range fun j ↦ b (ih ↑j ⋯);
                  ⋯.min {k | b k ∉ adjoin F s} ⋯)
                y)
            ↑j ⋯);
    ⋯.min {k | b k ∉ adjoin F s} ⋯)
Module: Mathlib.FieldTheory.CardinalEmb

Goal state before:
pp: case refine_1
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
⊢ ⇑b '' φ '' Iio i ⊆ ⇑b '' Iio (φ i)

Tactic:
]

Goal state after:
case refine_1
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
⊢ ⇑b '' φ '' Iio i ⊆ ⇑b '' Iio (φ i)
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 ∪ {(⇑b ∘ φ) i}) = adjoin F (⇑b ∘ φ '' Iio i ∪ {b (φ i)})

Tactic:
]

Goal state after:
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 ∪ {(⇑b ∘ φ) i}) = adjoin F (⇑b ∘ φ '' Iio i ∪ {b (φ i)})
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
⊢ 1 < finrank ↥(adjoin F (⇑b ∘ φ '' Iio i)) ↥(↥(adjoin F (⇑b ∘ φ '' Iio i)))⟮b (φ i)⟯

Tactic:
]

Goal state after:
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
⊢ 1 < finrank ↥(adjoin F (⇑b ∘ φ '' Iio i)) ↥(↥(adjoin F (⇑b ∘ φ '' Iio i)))⟮b (φ i)⟯
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
⊢ filtration ⊤ = filtration ⊤

Tactic:
]

Goal state after:
Module: Mathlib.FieldTheory.CardinalEmb

Goal state before:
pp: case top
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
hi : IsSuccPrelimit ⊤
⊢ ⨆ i, filtration ↑i = filtration ⊤

Tactic:
]

Goal state after:
case top
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
hi : IsSuccPrelimit ⊤
⊢ ⨆ i, filtration ↑i = filtration ⊤
Module: Mathlib.FieldTheory.CardinalEmb

Goal state before:
pp: case top
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
hi✝ : IsSuccPrelimit ⊤
hi : ¬Nonempty ↑(range WithTop.some)
this : Nonempty (Module.rank F E).ord.ToType
⊢ filtration ⊤ = ⊥

Tactic:
]

Goal state after:
case top
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
hi✝ : IsSuccPrelimit ⊤
hi : ¬Nonempty ↑(range WithTop.some)
this : Nonempty (Module.rank F E).ord.ToType
⊢ filtration ⊤ = ⊥
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)
⊢ ↑a = ↑a

Tactic:
]

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)
⊢ ⨆ i_1, (filtration ↑i_1).toSubalgebra ≤ ⨆ x, ((fun j ↦ filtration ↑j) x).toSubalgebra

Tactic:
]

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 L M) ((algebraMap (↥F) L) ((algebraMap S ↥F) s)) ∈ F.extendRight M

Tactic:
]

Goal state after:
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 L M) ((algebraMap (↥F) L) ((algebraMap S ↥F) s)) ∈ F.extendRight M
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)
⊢ (algebraMap S M) s * ↑x ∈ F.extendRight M

Tactic:
]

Goal state after:
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)
⊢ (algebraMap S M) s * ↑x ∈ F.extendRight M
Module: Mathlib.FieldTheory.IntermediateField.ExtendRight

Goal state before:
pp: case convert_2
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 = s • ↑x

Tactic:
]

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)
⊢ ↑a = ↑a

Tactic:
]

Goal state after:
Module: Mathlib.FieldTheory.CardinalEmb

Goal state before:
pp: case pos
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)
⊢ ((fun f ↦ Subalgebra.iSupLift (fun x ↦ ((fun j ↦ filtration ↑j) x).toSubalgebra) ⋯ ↑f ⋯ (filtration i).toSubalgebra ⋯)
        ((fun f ↦ ⟨fun j ↦ embFunctor F E ⋯ f, ⋯⟩) f))
      ⟨x, hx✝⟩ =
    f ⟨x, hx✝⟩

Tactic:
]

Goal state after:
case pos
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)
⊢ ((fun f ↦ Subalgebra.iSupLift (fun x ↦ ((fun j ↦ filtration ↑j) x).toSubalgebra) ⋯ ↑f ⋯ (filtration i).toSubalgebra ⋯)
        ((fun f ↦ ⟨fun j ↦ embFunctor F E ⋯ f, ⋯⟩) f))
      ⟨x, hx✝⟩ =
    f ⟨x, hx✝⟩
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
x : S
⊢ (algebraMap S M) x = (algebraMap S M) x

Tactic:
]

Goal state after: