ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- IsPurelyInseparable.tower_top
theorem IsPurelyInseparable.tower_top [Algebra E K] [IsScalarTower F E K]
    [h : IsPurelyInseparable F K] : IsPurelyInseparable E K 
Arguments: (F : Type u) (E : Type v) [Field F] [Field E] [Algebra F E] (K : Type w) [Field K] [Algebra F K] [Algebra E K] [IsScalarTower F E K] [h : IsPurelyInseparable F K]
Goal: IsPurelyInseparable E K
-- IsPurelyInseparable.trans
theorem IsPurelyInseparable.trans [Algebra E K] [IsScalarTower F E K]
    [h1 : IsPurelyInseparable F E] [h2 : IsPurelyInseparable E K] : IsPurelyInseparable F K 
Arguments: (F : Type u) (E : Type v) [Field F] [Field E] [Algebra F E] (K : Type w) [Field K] [Algebra F K] [Algebra E K] [IsScalarTower F E K] [h1 : IsPurelyInseparable F E] [h2 : IsPurelyInseparable E K]
Goal: IsPurelyInseparable F K
-- isPurelyInseparable_iff_natSepDegree_eq_one
theorem isPurelyInseparable_iff_natSepDegree_eq_one :
    IsPurelyInseparable F E ↔ ∀ x : E, (minpoly F x).natSepDegree = 1 
Arguments: (F : Type u) {E : Type v} [Field F] [Field E] [Algebra F E]
Goal: IsPurelyInseparable F E ↔ ∀ (x : E), (minpoly F x).natSepDegree = 1
-- IsPurelyInseparable.natSepDegree_eq_one
theorem IsPurelyInseparable.natSepDegree_eq_one [IsPurelyInseparable F E] (x : E) :
    (minpoly F x).natSepDegree = 1 
Arguments: (F : Type u) {E : Type v} [Field F] [Field E] [Algebra F E] [IsPurelyInseparable F E] (x : E)
Goal: (minpoly F x).natSepDegree = 1
-- isPurelyInseparable_iff_minpoly_eq_X_pow_sub_C
theorem isPurelyInseparable_iff_minpoly_eq_X_pow_sub_C (q : ℕ) [hF : ExpChar F q] :
    IsPurelyInseparable F E ↔ ∀ x : E, ∃ (n : ℕ) (y : F), minpoly F x = X ^ q ^ n - C y 
Arguments: (F : Type u) {E : Type v} [Field F] [Field E] [Algebra F E] (q : ℕ) [hF : ExpChar F q]
Goal: IsPurelyInseparable F E ↔ ∀ (x : E), ∃ n y, minpoly F x = Polynomial.X ^ q ^ n - Polynomial.C y
-- IsPurelyInseparable.minpoly_eq_X_pow_sub_C
theorem IsPurelyInseparable.minpoly_eq_X_pow_sub_C (q : ℕ) [ExpChar F q] [IsPurelyInseparable F E]
    (x : E) : ∃ (n : ℕ) (y : F), minpoly F x = X ^ q ^ n - C y 
Arguments: (F : Type u) {E : Type v} [Field F] [Field E] [Algebra F E] (q : ℕ) [ExpChar F q] [IsPurelyInseparable F E] (x : E)
Goal: ∃ n y, minpoly F x = Polynomial.X ^ q ^ n - Polynomial.C y
-- isPurelyInseparable_iff_minpoly_eq_X_sub_C_pow
theorem isPurelyInseparable_iff_minpoly_eq_X_sub_C_pow (q : ℕ) [hF : ExpChar F q] :
    IsPurelyInseparable F E ↔
    ∀ x : E, ∃ n : ℕ, (minpoly F x).map (algebraMap F E) = (X - C x) ^ q ^ n 
Arguments: (F : Type u) {E : Type v} [Field F] [Field E] [Algebra F E] (q : ℕ) [hF : ExpChar F q]
Goal: IsPurelyInseparable F E ↔
  ∀ (x : E), ∃ n, Polynomial.map (algebraMap F E) (minpoly F x) = (Polynomial.X - Polynomial.C x) ^ q ^ n
-- IsPurelyInseparable.minpoly_eq_X_sub_C_pow
theorem IsPurelyInseparable.minpoly_eq_X_sub_C_pow (q : ℕ) [ExpChar F q] [IsPurelyInseparable F E]
    (x : E) : ∃ n : ℕ, (minpoly F x).map (algebraMap F E) = (X - C x) ^ q ^ n 
Arguments: (F : Type u) {E : Type v} [Field F] [Field E] [Algebra F E] (q : ℕ) [ExpChar F q] [IsPurelyInseparable F E] (x : E)
Goal: ∃ n, Polynomial.map (algebraMap F E) (minpoly F x) = (Polynomial.X - Polynomial.C x) ^ q ^ n
-- isPurelyInseparable_of_finSepDegree_eq_one
theorem isPurelyInseparable_of_finSepDegree_eq_one [Algebra.IsAlgebraic F E]
    (hdeg : finSepDegree F E = 1) : IsPurelyInseparable F E 
Arguments: {F : Type u} {E : Type v} [Field F] [Field E] [Algebra F E] [Algebra.IsAlgebraic F E] (hdeg : Field.finSepDegree F E = 1)
Goal: IsPurelyInseparable F E
-- IsPurelyInseparable.injective_comp_algebraMap
theorem IsPurelyInseparable.injective_comp_algebraMap [IsPurelyInseparable F E]
    (L : Type w) [CommRing L] [IsReduced L] :
    Function.Injective fun f : E →+* L ↦ f.comp (algebraMap F E) 
Arguments: (F : Type u) (E : Type v) [Field F] [Field E] [Algebra F E] [IsPurelyInseparable F E] (L : Type w) [CommRing L] [IsReduced L]
Goal: Function.Injective fun f => f.comp (algebraMap F E)
-- IsPurelyInseparable.finSepDegree_eq_one
theorem IsPurelyInseparable.finSepDegree_eq_one [IsPurelyInseparable F E] :
    finSepDegree F E = 1 
Arguments: (F : Type u) (E : Type v) [Field F] [Field E] [Algebra F E] [IsPurelyInseparable F E]
Goal: Field.finSepDegree F E = 1
-- IsPurelyInseparable.sepDegree_eq_one
theorem IsPurelyInseparable.sepDegree_eq_one [IsPurelyInseparable F E] :
    sepDegree F E = 1 
Arguments: (F : Type u) (E : Type v) [Field F] [Field E] [Algebra F E] [IsPurelyInseparable F E]
Goal: Field.sepDegree F E = 1
-- IsPurelyInseparable.insepDegree_eq
theorem IsPurelyInseparable.insepDegree_eq [IsPurelyInseparable F E] :
    insepDegree F E = Module.rank F E 
Arguments: (F : Type u) (E : Type v) [Field F] [Field E] [Algebra F E] [IsPurelyInseparable F E]
Goal: Field.insepDegree F E = Module.rank F E
-- IsPurelyInseparable.finInsepDegree_eq
theorem IsPurelyInseparable.finInsepDegree_eq [IsPurelyInseparable F E] :
    finInsepDegree F E = finrank F E 
Arguments: (F : Type u) (E : Type v) [Field F] [Field E] [Algebra F E] [IsPurelyInseparable F E]
Goal: Field.finInsepDegree F E = FiniteDimensional.finrank F E
-- isPurelyInseparable_iff_finSepDegree_eq_one
theorem isPurelyInseparable_iff_finSepDegree_eq_one [Algebra.IsAlgebraic F E] :
    IsPurelyInseparable F E ↔ finSepDegree F E = 1 
Arguments: (F : Type u) (E : Type v) [Field F] [Field E] [Algebra F E] [Algebra.IsAlgebraic F E]
Goal: IsPurelyInseparable F E ↔ Field.finSepDegree F E = 1
-- isPurelyInseparable_iff_fd_isPurelyInseparable
theorem isPurelyInseparable_iff_fd_isPurelyInseparable [Algebra.IsAlgebraic F E] :
    IsPurelyInseparable F E ↔
    ∀ L : IntermediateField F E, FiniteDimensional F L → IsPurelyInseparable F L 
Arguments: {F : Type u} {E : Type v} [Field F] [Field E] [Algebra F E] [Algebra.IsAlgebraic F E]
Goal: IsPurelyInseparable F E ↔ ∀ (L : IntermediateField F E), FiniteDimensional F ↥L → IsPurelyInseparable F ↥L
-- separableClosure.isPurelyInseparable
theorem separableClosure.isPurelyInseparable [Algebra.IsAlgebraic F E] :
    IsPurelyInseparable (separableClosure F E) E 
Arguments: (F : Type u) (E : Type v) [Field F] [Field E] [Algebra F E] [Algebra.IsAlgebraic F E]
Goal: IsPurelyInseparable (↥(separableClosure F E)) E
-- separableClosure_le
theorem separableClosure_le (L : IntermediateField F E)
    [h : IsPurelyInseparable L E] : separableClosure F E ≤ L 
Arguments: (F : Type u) (E : Type v) [Field F] [Field E] [Algebra F E] (L : IntermediateField F E) [h : IsPurelyInseparable (↥L) E]
Goal: separableClosure F E ≤ L
-- separableClosure_le_iff
theorem separableClosure_le_iff [Algebra.IsAlgebraic F E] (L : IntermediateField F E) :
    separableClosure F E ≤ L ↔ IsPurelyInseparable L E 
Arguments: (F : Type u) (E : Type v) [Field F] [Field E] [Algebra F E] [Algebra.IsAlgebraic F E] (L : IntermediateField F E)
Goal: separableClosure F E ≤ L ↔ IsPurelyInseparable (↥L) E
-- eq_separableClosure
theorem eq_separableClosure (L : IntermediateField F E)
    [Algebra.IsSeparable F L] [IsPurelyInseparable L E] : L = separableClosure F E 
Arguments: (F : Type u) (E : Type v) [Field F] [Field E] [Algebra F E] (L : IntermediateField F E) [Algebra.IsSeparable F ↥L] [IsPurelyInseparable (↥L) E]
Goal: L = separableClosure F E
-- eq_separableClosure_iff
theorem eq_separableClosure_iff [Algebra.IsAlgebraic F E] (L : IntermediateField F E) :
    L = separableClosure F E ↔ Algebra.IsSeparable F L ∧ IsPurelyInseparable L E 
Arguments: (F : Type u) (E : Type v) [Field F] [Field E] [Algebra F E] [Algebra.IsAlgebraic F E] (L : IntermediateField F E)
Goal: L = separableClosure F E ↔ Algebra.IsSeparable F ↥L ∧ IsPurelyInseparable (↥L) E
-- IntermediateField.isPurelyInseparable_adjoin_simple_iff_natSepDegree_eq_one
theorem isPurelyInseparable_adjoin_simple_iff_natSepDegree_eq_one {x : E} :
    IsPurelyInseparable F F⟮x⟯ ↔ (minpoly F x).natSepDegree = 1 
Arguments: (F : Type u) (E : Type v) [Field F] [Field E] [Algebra F E] {x : E}
Goal: IsPurelyInseparable F ↥F⟮x⟯ ↔ (minpoly F x).natSepDegree = 1
-- IntermediateField.isPurelyInseparable_adjoin_simple_iff_pow_mem
theorem isPurelyInseparable_adjoin_simple_iff_pow_mem (q : ℕ) [hF : ExpChar F q] {x : E} :
    IsPurelyInseparable F F⟮x⟯ ↔ ∃ n : ℕ, x ^ q ^ n ∈ (algebraMap F E).range 
Arguments: (F : Type u) (E : Type v) [Field F] [Field E] [Algebra F E] (q : ℕ) [hF : ExpChar F q] {x : E}
Goal: IsPurelyInseparable F ↥F⟮x⟯ ↔ ∃ n, x ^ q ^ n ∈ (algebraMap F E).range
-- IntermediateField.isPurelyInseparable_adjoin_iff_pow_mem
theorem isPurelyInseparable_adjoin_iff_pow_mem (q : ℕ) [hF : ExpChar F q] {S : Set E} :
    IsPurelyInseparable F (adjoin F S) ↔ ∀ x ∈ S, ∃ n : ℕ, x ^ q ^ n ∈ (algebraMap F E).range 
Arguments: (F : Type u) (E : Type v) [Field F] [Field E] [Algebra F E] (q : ℕ) [hF : ExpChar F q] {S : Set E}
Goal: IsPurelyInseparable F ↥(IntermediateField.adjoin F S) ↔ ∀ x ∈ S, ∃ n, x ^ q ^ n ∈ (algebraMap F E).range
-- IntermediateField.adjoin_eq_adjoin_pow_expChar_pow_of_isSeparable
theorem adjoin_eq_adjoin_pow_expChar_pow_of_isSeparable (S : Set E)
    [Algebra.IsSeparable F (adjoin F S)] (q : ℕ) [ExpChar F q] (n : ℕ) :
    adjoin F S = adjoin F ((· ^ q ^ n) '' S) 
Arguments: (F : Type u) (E : Type v) [Field F] [Field E] [Algebra F E] (S : Set E) [Algebra.IsSeparable F ↥(IntermediateField.adjoin F S)] (q : ℕ) [ExpChar F q] (n : ℕ)
Goal: IntermediateField.adjoin F S = IntermediateField.adjoin F ((fun x => x ^ q ^ n) '' S)
-- IntermediateField.adjoin_eq_adjoin_pow_expChar_pow_of_isSeparable'
theorem adjoin_eq_adjoin_pow_expChar_pow_of_isSeparable' [Algebra.IsSeparable F E] (S : Set E)
    (q : ℕ) [ExpChar F q] (n : ℕ) : adjoin F S = adjoin F ((· ^ q ^ n) '' S) 
Arguments: (F : Type u) (E : Type v) [Field F] [Field E] [Algebra F E] [Algebra.IsSeparable F E] (S : Set E) (q : ℕ) [ExpChar F q] (n : ℕ)
Goal: IntermediateField.adjoin F S = IntermediateField.adjoin F ((fun x => x ^ q ^ n) '' S)
-- IntermediateField.adjoin_eq_adjoin_pow_expChar_of_isSeparable
theorem adjoin_eq_adjoin_pow_expChar_of_isSeparable (S : Set E) [Algebra.IsSeparable F (adjoin F S)]
    (q : ℕ) [ExpChar F q] : adjoin F S = adjoin F ((· ^ q) '' S) 
Arguments: (F : Type u) (E : Type v) [Field F] [Field E] [Algebra F E] (S : Set E) [Algebra.IsSeparable F ↥(IntermediateField.adjoin F S)] (q : ℕ) [ExpChar F q]
Goal: IntermediateField.adjoin F S = IntermediateField.adjoin F ((fun x => x ^ q) '' S)
-- IntermediateField.adjoin_eq_adjoin_pow_expChar_of_isSeparable'
theorem adjoin_eq_adjoin_pow_expChar_of_isSeparable' [Algebra.IsSeparable F E] (S : Set E)
    (q : ℕ) [ExpChar F q] : adjoin F S = adjoin F ((· ^ q) '' S) 
Arguments: (F : Type u) (E : Type v) [Field F] [Field E] [Algebra F E] [Algebra.IsSeparable F E] (S : Set E) (q : ℕ) [ExpChar F q]
Goal: IntermediateField.adjoin F S = IntermediateField.adjoin F ((fun x => x ^ q) '' S)
-- Field.span_map_pow_expChar_pow_eq_top_of_isSeparable
theorem Field.span_map_pow_expChar_pow_eq_top_of_isSeparable [Algebra.IsSeparable F E]
    (h : Submodule.span F (Set.range v) = ⊤) :
    Submodule.span F (Set.range (v · ^ q ^ n)) = ⊤ 
Arguments: {F : Type u} {E : Type v} [Field F] [Field E] [Algebra F E] (q n : ℕ) [hF : ExpChar F q] {ι : Type u_1} {v : ι → E} [Algebra.IsSeparable F E] (h : Submodule.span F (Set.range v) = ⊤)
Goal: Submodule.span F (Set.range fun x => v x ^ q ^ n) = ⊤
-- _private.Mathlib.FieldTheory.PurelyInseparable.0.LinearIndependent.map_pow_expChar_pow_of_fd_isSeparable
private theorem LinearIndependent.map_pow_expChar_pow_of_fd_isSeparable
    [FiniteDimensional F E] [Algebra.IsSeparable F E]
    (h : LinearIndependent F v) : LinearIndependent F (v · ^ q ^ n) 
Arguments: {F : Type u} {E : Type v} [Field F] [Field E] [Algebra F E] (q n : ℕ) [hF : ExpChar F q] {ι : Type u_1} {v : ι → E} [FiniteDimensional F E] [Algebra.IsSeparable F E] (h : LinearIndependent F v)
Goal: LinearIndependent F fun x => v x ^ q ^ n
-- LinearIndependent.map_pow_expChar_pow_of_isSeparable
theorem LinearIndependent.map_pow_expChar_pow_of_isSeparable [Algebra.IsSeparable F E]
    (h : LinearIndependent F v) : LinearIndependent F (v · ^ q ^ n) 
Arguments: {F : Type u} {E : Type v} [Field F] [Field E] [Algebra F E] (q n : ℕ) [hF : ExpChar F q] {ι : Type u_1} {v : ι → E} [Algebra.IsSeparable F E] (h : LinearIndependent F v)
Goal: LinearIndependent F fun x => v x ^ q ^ n
-- LinearIndependent.map_pow_expChar_pow_of_isIntegral'
theorem LinearIndependent.map_pow_expChar_pow_of_isIntegral'
    (hsep : ∀ i : ι, IsSeparable F (v i))
    (h : LinearIndependent F v) : LinearIndependent F (v · ^ q ^ n) 
Arguments: {F : Type u} {E : Type v} [Field F] [Field E] [Algebra F E] (q n : ℕ) [hF : ExpChar F q] {ι : Type u_1} {v : ι → E} (hsep : ∀ (i : ι), IsSeparable F (v i)) (h : LinearIndependent F v)
Goal: LinearIndependent F fun x => v x ^ q ^ n
-- isSepClosed_iff_isPurelyInseparable_algebraicClosure
theorem isSepClosed_iff_isPurelyInseparable_algebraicClosure [IsAlgClosure F E] :
    IsSepClosed F ↔ IsPurelyInseparable F E 
Arguments: (F : Type u) (E : Type v) [Field F] [Field E] [Algebra F E] [IsAlgClosure F E]
Goal: IsSepClosed F ↔ IsPurelyInseparable F E
-- Algebra.IsAlgebraic.isSepClosed
theorem Algebra.IsAlgebraic.isSepClosed [Algebra.IsAlgebraic F E]
    [IsSepClosed F] : IsSepClosed E 
Arguments: {F : Type u} {E : Type v} [Field F] [Field E] [Algebra F E] [Algebra.IsAlgebraic F E] [IsSepClosed F]
Goal: IsSepClosed E
-- perfectField_of_perfectClosure_eq_bot
theorem perfectField_of_perfectClosure_eq_bot [h : PerfectField E] (eq : perfectClosure F E = ⊥) :
    PerfectField F 
Arguments: (F : Type u) (E : Type v) [Field F] [Field E] [Algebra F E] [h : PerfectField E] (eq : perfectClosure F E = ⊥)
Goal: PerfectField F
-- perfectField_of_isSeparable_of_perfectField_top
theorem perfectField_of_isSeparable_of_perfectField_top [Algebra.IsSeparable F E] [PerfectField E] :
    PerfectField F 
Arguments: (F : Type u) (E : Type v) [Field F] [Field E] [Algebra F E] [Algebra.IsSeparable F E] [PerfectField E]
Goal: PerfectField F
-- perfectField_iff_isSeparable_algebraicClosure
theorem perfectField_iff_isSeparable_algebraicClosure [IsAlgClosure F E] :
    PerfectField F ↔ Algebra.IsSeparable F E 
Arguments: (F : Type u) (E : Type v) [Field F] [Field E] [Algebra F E] [IsAlgClosure F E]
Goal: PerfectField F ↔ Algebra.IsSeparable F E
-- Field.finSepDegree_eq
theorem finSepDegree_eq [Algebra.IsAlgebraic F E] :
    finSepDegree F E = Cardinal.toNat (sepDegree F E) 
Arguments: (F : Type u) (E : Type v) [Field F] [Field E] [Algebra F E] [Algebra.IsAlgebraic F E]
Goal: Field.finSepDegree F E = Cardinal.toNat (Field.sepDegree F E)
-- Field.finSepDegree_mul_finInsepDegree
theorem finSepDegree_mul_finInsepDegree : finSepDegree F E * finInsepDegree F E = finrank F E 
Arguments: (F : Type u) (E : Type v) [Field F] [Field E] [Algebra F E]
Goal: Field.finSepDegree F E * Field.finInsepDegree F E = FiniteDimensional.finrank F E
-- separableClosure.adjoin_eq_of_isAlgebraic_of_isSeparable
lemma adjoin_eq_of_isAlgebraic_of_isSeparable [Algebra.IsAlgebraic F E]
    [Algebra.IsSeparable E K] : adjoin E (separableClosure F K : Set K) = ⊤ 
Arguments: {F : Type u} {E : Type v} [Field F] [Field E] [Algebra F E] (K : Type w) [Field K] [Algebra F K] [Algebra E K] [IsScalarTower F E K] [Algebra.IsAlgebraic F E] [Algebra.IsSeparable E K]
Goal: IntermediateField.adjoin E ↑(separableClosure F K) = ⊤
-- separableClosure.adjoin_eq_of_isAlgebraic
theorem adjoin_eq_of_isAlgebraic [Algebra.IsAlgebraic F E] :
    adjoin E (separableClosure F K) = separableClosure E K 
Arguments: {F : Type u} {E : Type v} [Field F] [Field E] [Algebra F E] (K : Type w) [Field K] [Algebra F K] [Algebra E K] [IsScalarTower F E K] [Algebra.IsAlgebraic F E]
Goal: IntermediateField.adjoin E ↑(separableClosure F K) = separableClosure E K
-- LinearIndependent.map_of_isPurelyInseparable_of_isSeparable
theorem LinearIndependent.map_of_isPurelyInseparable_of_isSeparable [IsPurelyInseparable F E]
    {ι : Type*} {v : ι → K} (hsep : ∀ i : ι, IsSeparable F (v i))
    (h : LinearIndependent F v) : LinearIndependent E v 
Arguments: {F : Type u} (E : Type v) [Field F] [Field E] [Algebra F E] {K : Type w} [Field K] [Algebra F K] [Algebra E K] [IsScalarTower F E K] [IsPurelyInseparable F E] {ι : Type u_1} {v : ι → K} (hsep : ∀ (i : ι), IsSeparable F (v i)) (h : LinearIndependent F v)
Goal: LinearIndependent E v
-- Field.sepDegree_eq_of_isPurelyInseparable_of_isSeparable
lemma sepDegree_eq_of_isPurelyInseparable_of_isSeparable
    [IsPurelyInseparable F E] [Algebra.IsSeparable E K] : sepDegree F K = Module.rank E K 
Arguments: (F : Type u) (E : Type v) [Field F] [Field E] [Algebra F E] (K : Type w) [Field K] [Algebra F K] [Algebra E K] [IsScalarTower F E K] [IsPurelyInseparable F E] [Algebra.IsSeparable E K]
Goal: Field.sepDegree F K = Module.rank E K
-- Field.lift_rank_mul_lift_sepDegree_of_isSeparable
lemma lift_rank_mul_lift_sepDegree_of_isSeparable [Algebra.IsSeparable F E] :
    Cardinal.lift.{w} (Module.rank F E) * Cardinal.lift.{v} (sepDegree E K) =
    Cardinal.lift.{v} (sepDegree F K) 
Arguments: (F : Type u) (E : Type v) [Field F] [Field E] [Algebra F E] (K : Type w) [Field K] [Algebra F K] [Algebra E K] [IsScalarTower F E K] [Algebra.IsSeparable F E]
Goal: Cardinal.lift.{w, v} (Module.rank F E) * Cardinal.lift.{v, w} (Field.sepDegree E K) =
  Cardinal.lift.{v, w} (Field.sepDegree F K)
-- Field.rank_mul_sepDegree_of_isSeparable
lemma rank_mul_sepDegree_of_isSeparable (K : Type v) [Field K] [Algebra F K]
    [Algebra E K] [IsScalarTower F E K] [Algebra.IsSeparable F E] :
    Module.rank F E * sepDegree E K = sepDegree F K 
Arguments: (F : Type u) (E : Type v) [Field F] [Field E] [Algebra F E] (K : Type v) [Field K] [Algebra F K] [Algebra E K] [IsScalarTower F E K] [Algebra.IsSeparable F E]
Goal: Module.rank F E * Field.sepDegree E K = Field.sepDegree F K
-- Field.sepDegree_eq_of_isPurelyInseparable
lemma sepDegree_eq_of_isPurelyInseparable [IsPurelyInseparable F E] :
    sepDegree F K = sepDegree E K 
Arguments: (F : Type u) (E : Type v) [Field F] [Field E] [Algebra F E] (K : Type w) [Field K] [Algebra F K] [Algebra E K] [IsScalarTower F E K] [IsPurelyInseparable F E]
Goal: Field.sepDegree F K = Field.sepDegree E K
-- Field.lift_sepDegree_mul_lift_sepDegree_of_isAlgebraic
theorem lift_sepDegree_mul_lift_sepDegree_of_isAlgebraic [Algebra.IsAlgebraic F E] :
    Cardinal.lift.{w} (sepDegree F E) * Cardinal.lift.{v} (sepDegree E K) =
    Cardinal.lift.{v} (sepDegree F K) 
Arguments: (F : Type u) (E : Type v) [Field F] [Field E] [Algebra F E] (K : Type w) [Field K] [Algebra F K] [Algebra E K] [IsScalarTower F E K] [Algebra.IsAlgebraic F E]
Goal: Cardinal.lift.{w, v} (Field.sepDegree F E) * Cardinal.lift.{v, w} (Field.sepDegree E K) =
  Cardinal.lift.{v, w} (Field.sepDegree F K)
-- Field.sepDegree_mul_sepDegree_of_isAlgebraic
theorem sepDegree_mul_sepDegree_of_isAlgebraic (K : Type v) [Field K] [Algebra F K]
    [Algebra E K] [IsScalarTower F E K] [Algebra.IsAlgebraic F E] :
    sepDegree F E * sepDegree E K = sepDegree F K 
Arguments: (F : Type u) (E : Type v) [Field F] [Field E] [Algebra F E] (K : Type v) [Field K] [Algebra F K] [Algebra E K] [IsScalarTower F E K] [Algebra.IsAlgebraic F E]
Goal: Field.sepDegree F E * Field.sepDegree E K = Field.sepDegree F K
-- IntermediateField.sepDegree_adjoin_eq_of_isAlgebraic_of_isPurelyInseparable
theorem IntermediateField.sepDegree_adjoin_eq_of_isAlgebraic_of_isPurelyInseparable
    (S : Set K) [Algebra.IsAlgebraic F (adjoin F S)] [IsPurelyInseparable F E] :
    sepDegree E (adjoin E S) = sepDegree F (adjoin F S) 
Arguments: {F : Type u} (E : Type v) [Field F] [Field E] [Algebra F E] {K : Type w} [Field K] [Algebra F K] [Algebra E K] [IsScalarTower F E K] (S : Set K) [Algebra.IsAlgebraic F ↥(IntermediateField.adjoin F S)] [IsPurelyInseparable F E]
Goal: Field.sepDegree E ↥(IntermediateField.adjoin E S) = Field.sepDegree F ↥(IntermediateField.adjoin F S)
-- IntermediateField.sepDegree_adjoin_eq_of_isAlgebraic_of_isPurelyInseparable'
theorem IntermediateField.sepDegree_adjoin_eq_of_isAlgebraic_of_isPurelyInseparable'
    (S : IntermediateField F K) [Algebra.IsAlgebraic F S] [IsPurelyInseparable F E] :
    sepDegree E (adjoin E (S : Set K)) = sepDegree F S 
Arguments: {F : Type u} (E : Type v) [Field F] [Field E] [Algebra F E] {K : Type w} [Field K] [Algebra F K] [Algebra E K] [IsScalarTower F E K] (S : IntermediateField F K) [Algebra.IsAlgebraic F ↥S] [IsPurelyInseparable F E]
Goal: Field.sepDegree E ↥(IntermediateField.adjoin E ↑S) = Field.sepDegree F ↥S
-- minpoly.map_eq_of_isSeparable_of_isPurelyInseparable
theorem minpoly.map_eq_of_isSeparable_of_isPurelyInseparable (x : K)
    (hsep : IsSeparable F x) [IsPurelyInseparable F E] :
    (minpoly F x).map (algebraMap F E) = minpoly E x 
Arguments: {F : Type u} (E : Type v) [Field F] [Field E] [Algebra F E] {K : Type w} [Field K] [Algebra F K] [Algebra E K] [IsScalarTower F E K] (x : K) (hsep : IsSeparable F x) [IsPurelyInseparable F E]
Goal: Polynomial.map (algebraMap F E) (minpoly F x) = minpoly E x
-- Polynomial.Separable.map_irreducible_of_isPurelyInseparable
theorem Polynomial.Separable.map_irreducible_of_isPurelyInseparable {f : F[X]} (hsep : f.Separable)
    (hirr : Irreducible f) [IsPurelyInseparable F E] : Irreducible (f.map (algebraMap F E)) 
Arguments: {F : Type u} (E : Type v) [Field F] [Field E] [Algebra F E] {f : F[X]} (hsep : f.Separable) (hirr : Irreducible f) [IsPurelyInseparable F E]
Goal: Irreducible (Polynomial.map (algebraMap F E) f)
-- CategoryTheory.Sheaf.isSheaf_of_isLimit
theorem isSheaf_of_isLimit (F : K ⥤ Sheaf J D) (E : Cone (F ⋙ sheafToPresheaf J D))
    (hE : IsLimit E) : Presheaf.IsSheaf J E.pt 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {J : CategoryTheory.GrothendieckTopology C} {D : Type w} [CategoryTheory.Category.{w', w} D] {K : Type z} [CategoryTheory.Category.{z', z} K] [CategoryTheory.Limits.HasLimitsOfShape K D] (F : CategoryTheory.Functor K (CategoryTheory.Sheaf J D)) (E : CategoryTheory.Limits.Cone (F.comp (CategoryTheory.sheafToPresheaf J D))) (hE : CategoryTheory.Limits.IsLimit E)
Goal: CategoryTheory.Presheaf.IsSheaf J E.pt
-- CategoryTheory.Presheaf.equalizerSieve_apply
@[simps]
def equalizerSieve {F : Cᵒᵖ ⥤ D} {X : Cᵒᵖ} (x y : F.obj X) : Sieve X.unop 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {D : Type u'} [CategoryTheory.Category.{v', u'} D] [CategoryTheory.ConcreteCategory D] {F : CategoryTheory.Functor Cᵒᵖ D} {X : Cᵒᵖ} (x y : (CategoryTheory.forget D).obj (F.obj X))
Goal: ∀ (x_1 : C) (f : x_1 ⟶ Opposite.unop X),
  (CategoryTheory.Presheaf.equalizerSieve x y).arrows f = ((F.map f.op) x = (F.map f.op) y)
-- CategoryTheory.Presheaf.equalizerSieve_self_eq_top
@[simp]
lemma equalizerSieve_self_eq_top {F : Cᵒᵖ ⥤ D} {X : Cᵒᵖ} (x : F.obj X) :
    equalizerSieve x x = ⊤ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {D : Type u'} [CategoryTheory.Category.{v', u'} D] [CategoryTheory.ConcreteCategory D] {F : CategoryTheory.Functor Cᵒᵖ D} {X : Cᵒᵖ} (x : (CategoryTheory.forget D).obj (F.obj X))
Goal: CategoryTheory.Presheaf.equalizerSieve x x = ⊤
-- CategoryTheory.Presheaf.equalizerSieve_eq_top_iff
@[simp]
lemma equalizerSieve_eq_top_iff {F : Cᵒᵖ ⥤ D} {X : Cᵒᵖ} (x y : F.obj X) :
    equalizerSieve x y = ⊤ ↔ x = y 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {D : Type u'} [CategoryTheory.Category.{v', u'} D] [CategoryTheory.ConcreteCategory D] {F : CategoryTheory.Functor Cᵒᵖ D} {X : Cᵒᵖ} (x y : (CategoryTheory.forget D).obj (F.obj X))
Goal: CategoryTheory.Presheaf.equalizerSieve x y = ⊤ ↔ x = y
-- CategoryTheory.Presheaf.IsLocallyInjective.equalizerSieve_mem
class IsLocallyInjective : Prop where
  equalizerSieve_mem {X : Cᵒᵖ} (x y : F₁.obj X) (h : φ.app X x = φ.app X y) :
    equalizerSieve x y ∈ J X.unop
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {D : Type u'} [CategoryTheory.Category.{v', u'} D] [CategoryTheory.ConcreteCategory D] {J : CategoryTheory.GrothendieckTopology C} {F₁ F₂ : CategoryTheory.Functor Cᵒᵖ D} {φ : F₁ ⟶ F₂} [self : CategoryTheory.Presheaf.IsLocallyInjective J φ] {X : Cᵒᵖ} (x y : (CategoryTheory.forget D).obj (F₁.obj X)) (h : (φ.app X) x = (φ.app X) y)
Goal: CategoryTheory.Presheaf.equalizerSieve x y ∈ J.sieves (Opposite.unop X)