ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- AlgHom.algebraicIndependent_iff
theorem AlgHom.algebraicIndependent_iff (f : A →ₐ[R] A') (hf : Injective f) :
    AlgebraicIndependent R (f ∘ x) ↔ AlgebraicIndependent R x 
Arguments: {ι : Type u_1} {R : Type u_3} {A : Type u_5} {A' : Type u_6} {x : ι → A} [CommRing R] [CommRing A] [CommRing A'] [Algebra R A] [Algebra R A'] (f : A →ₐ[R] A') (hf : Function.Injective ⇑f)
Goal: AlgebraicIndependent R (⇑f ∘ x) ↔ AlgebraicIndependent R x
-- algebraicIndependent_of_subsingleton
@[nontriviality]
theorem algebraicIndependent_of_subsingleton [Subsingleton R] : AlgebraicIndependent R x 
Arguments: {ι : Type u_1} {R : Type u_3} {A : Type u_5} {x : ι → A} [CommRing R] [CommRing A] [Algebra R A] [Subsingleton R]
Goal: AlgebraicIndependent R x
-- algebraicIndependent_equiv
theorem algebraicIndependent_equiv (e : ι ≃ ι') {f : ι' → A} :
    AlgebraicIndependent R (f ∘ e) ↔ AlgebraicIndependent R f 
Arguments: {ι : Type u_1} {ι' : Type u_2} {R : Type u_3} {A : Type u_5} [CommRing R] [CommRing A] [Algebra R A] (e : ι ≃ ι') {f : ι' → A}
Goal: AlgebraicIndependent R (f ∘ ⇑e) ↔ AlgebraicIndependent R f
-- algebraicIndependent_equiv'
theorem algebraicIndependent_equiv' (e : ι ≃ ι') {f : ι' → A} {g : ι → A} (h : f ∘ e = g) :
    AlgebraicIndependent R g ↔ AlgebraicIndependent R f 
Arguments: {ι : Type u_1} {ι' : Type u_2} {R : Type u_3} {A : Type u_5} [CommRing R] [CommRing A] [Algebra R A] (e : ι ≃ ι') {f : ι' → A} {g : ι → A} (h : f ∘ ⇑e = g)
Goal: AlgebraicIndependent R g ↔ AlgebraicIndependent R f
-- algebraicIndependent_subtype_range
theorem algebraicIndependent_subtype_range {ι} {f : ι → A} (hf : Injective f) :
    AlgebraicIndependent R ((↑) : range f → A) ↔ AlgebraicIndependent R f 
Arguments: {R : Type u_3} {A : Type u_5} [CommRing R] [CommRing A] [Algebra R A] {ι : Type u_9} {f : ι → A} (hf : Function.Injective f)
Goal: AlgebraicIndependent R Subtype.val ↔ AlgebraicIndependent R f
-- algebraicIndependent_image
theorem algebraicIndependent_image {ι} {s : Set ι} {f : ι → A} (hf : Set.InjOn f s) :
    (AlgebraicIndependent R fun x : s => f x) ↔ AlgebraicIndependent R fun x : f '' s => (x : A) 
Arguments: {R : Type u_3} {A : Type u_5} [CommRing R] [CommRing A] [Algebra R A] {ι : Type u_9} {s : Set ι} {f : ι → A} (hf : Set.InjOn f s)
Goal: (AlgebraicIndependent R fun x => f ↑x) ↔ AlgebraicIndependent R fun x => ↑x
-- algebraicIndependent_adjoin
theorem algebraicIndependent_adjoin (hs : AlgebraicIndependent R x) :
    @AlgebraicIndependent ι R (adjoin R (range x))
      (fun i : ι => ⟨x i, subset_adjoin (mem_range_self i)⟩) _ _ _ 
Arguments: {ι : Type u_1} {R : Type u_3} {A : Type u_5} {x : ι → A} [CommRing R] [CommRing A] [Algebra R A] (hs : AlgebraicIndependent R x)
Goal: AlgebraicIndependent R fun i => ⟨x i, ⋯⟩
-- AlgebraicIndependent.restrictScalars
theorem AlgebraicIndependent.restrictScalars {K : Type*} [CommRing K] [Algebra R K] [Algebra K A]
    [IsScalarTower R K A] (hinj : Function.Injective (algebraMap R K))
    (ai : AlgebraicIndependent K x) : AlgebraicIndependent R x 
Arguments: {ι : Type u_1} {R : Type u_3} {A : Type u_5} {x : ι → A} [CommRing R] [CommRing A] [Algebra R A] {K : Type u_9} [CommRing K] [Algebra R K] [Algebra K A] [IsScalarTower R K A] (hinj : Function.Injective ⇑(algebraMap R K)) (ai : AlgebraicIndependent K x)
Goal: AlgebraicIndependent R x
-- algebraicIndependent_finset_map_embedding_subtype
theorem algebraicIndependent_finset_map_embedding_subtype (s : Set A)
    (li : AlgebraicIndependent R ((↑) : s → A)) (t : Finset s) :
    AlgebraicIndependent R ((↑) : Finset.map (Embedding.subtype s) t → A) 
Arguments: {R : Type u_3} {A : Type u_5} [CommRing R] [CommRing A] [Algebra R A] (s : Set A) (li : AlgebraicIndependent R Subtype.val) (t : Finset ↑s)
Goal: AlgebraicIndependent R Subtype.val
-- algebraicIndependent_bounded_of_finset_algebraicIndependent_bounded
theorem algebraicIndependent_bounded_of_finset_algebraicIndependent_bounded {n : ℕ}
    (H : ∀ s : Finset A, (AlgebraicIndependent R fun i : s => (i : A)) → s.card ≤ n) :
    ∀ s : Set A, AlgebraicIndependent R ((↑) : s → A) → Cardinal.mk s ≤ n 
Arguments: {R : Type u_3} {A : Type u_5} [CommRing R] [CommRing A] [Algebra R A] {n : ℕ} (H : ∀ (s : Finset A), (AlgebraicIndependent R fun i => ↑i) → s.card ≤ n) (s : Set A)
Goal: AlgebraicIndependent R Subtype.val → Cardinal.mk ↑s ≤ ↑n
-- AlgebraicIndependent.restrict_of_comp_subtype
theorem AlgebraicIndependent.restrict_of_comp_subtype {s : Set ι}
    (hs : AlgebraicIndependent R (x ∘ (↑) : s → A)) : AlgebraicIndependent R (s.restrict x) 
Arguments: {ι : Type u_1} {R : Type u_3} {A : Type u_5} {x : ι → A} [CommRing R] [CommRing A] [Algebra R A] {s : Set ι} (hs : AlgebraicIndependent R (x ∘ Subtype.val))
Goal: AlgebraicIndependent R (s.restrict x)
-- algebraicIndependent_empty_iff
theorem algebraicIndependent_empty_iff :
    AlgebraicIndependent R ((↑) : (∅ : Set A) → A) ↔ Injective (algebraMap R A) 
Arguments: (R : Type u_3) (A : Type u_5) [CommRing R] [CommRing A] [Algebra R A]
Goal: AlgebraicIndependent R Subtype.val ↔ Function.Injective ⇑(algebraMap R A)
-- AlgebraicIndependent.mono
theorem AlgebraicIndependent.mono {t s : Set A} (h : t ⊆ s)
    (hx : AlgebraicIndependent R ((↑) : s → A)) : AlgebraicIndependent R ((↑) : t → A) 
Arguments: {R : Type u_3} {A : Type u_5} [CommRing R] [CommRing A] [Algebra R A] {t s : Set A} (h : t ⊆ s) (hx : AlgebraicIndependent R Subtype.val)
Goal: AlgebraicIndependent R Subtype.val
-- AlgebraicIndependent.to_subtype_range
theorem AlgebraicIndependent.to_subtype_range {ι} {f : ι → A} (hf : AlgebraicIndependent R f) :
    AlgebraicIndependent R ((↑) : range f → A) 
Arguments: {R : Type u_3} {A : Type u_5} [CommRing R] [CommRing A] [Algebra R A] {ι : Type u_9} {f : ι → A} (hf : AlgebraicIndependent R f)
Goal: AlgebraicIndependent R Subtype.val
-- AlgebraicIndependent.to_subtype_range'
theorem AlgebraicIndependent.to_subtype_range' {ι} {f : ι → A} (hf : AlgebraicIndependent R f) {t}
    (ht : range f = t) : AlgebraicIndependent R ((↑) : t → A) 
Arguments: {R : Type u_3} {A : Type u_5} [CommRing R] [CommRing A] [Algebra R A] {ι : Type u_9} {f : ι → A} (hf : AlgebraicIndependent R f) {t : Set A} (ht : Set.range f = t)
Goal: AlgebraicIndependent R Subtype.val
-- algebraicIndependent_comp_subtype
theorem algebraicIndependent_comp_subtype {s : Set ι} :
    AlgebraicIndependent R (x ∘ (↑) : s → A) ↔
      ∀ p ∈ MvPolynomial.supported R s, aeval x p = 0 → p = 0 
Arguments: {ι : Type u_1} {R : Type u_3} {A : Type u_5} {x : ι → A} [CommRing R] [CommRing A] [Algebra R A] {s : Set ι}
Goal: AlgebraicIndependent R (x ∘ Subtype.val) ↔ ∀ p ∈ MvPolynomial.supported R s, (MvPolynomial.aeval x) p = 0 → p = 0
-- algebraicIndependent_subtype
theorem algebraicIndependent_subtype {s : Set A} :
    AlgebraicIndependent R ((↑) : s → A) ↔
      ∀ p : MvPolynomial A R, p ∈ MvPolynomial.supported R s → aeval id p = 0 → p = 0 
Arguments: {R : Type u_3} {A : Type u_5} [CommRing R] [CommRing A] [Algebra R A] {s : Set A}
Goal: AlgebraicIndependent R Subtype.val ↔ ∀ p ∈ MvPolynomial.supported R s, (MvPolynomial.aeval id) p = 0 → p = 0
-- algebraicIndependent_of_finite
theorem algebraicIndependent_of_finite (s : Set A)
    (H : ∀ t ⊆ s, t.Finite → AlgebraicIndependent R ((↑) : t → A)) :
    AlgebraicIndependent R ((↑) : s → A) 
Arguments: {R : Type u_3} {A : Type u_5} [CommRing R] [CommRing A] [Algebra R A] (s : Set A) (H : ∀ t ⊆ s, t.Finite → AlgebraicIndependent R Subtype.val)
Goal: AlgebraicIndependent R Subtype.val
-- AlgebraicIndependent.image_of_comp
theorem AlgebraicIndependent.image_of_comp {ι ι'} (s : Set ι) (f : ι → ι') (g : ι' → A)
    (hs : AlgebraicIndependent R fun x : s => g (f x)) :
    AlgebraicIndependent R fun x : f '' s => g x 
Arguments: {R : Type u_3} {A : Type u_5} [CommRing R] [CommRing A] [Algebra R A] {ι : Type u_9} {ι' : Type u_10} (s : Set ι) (f : ι → ι') (g : ι' → A) (hs : AlgebraicIndependent R fun x => g (f ↑x))
Goal: AlgebraicIndependent R fun x => g ↑x
-- AlgebraicIndependent.image
theorem AlgebraicIndependent.image {ι} {s : Set ι} {f : ι → A}
    (hs : AlgebraicIndependent R fun x : s => f x) :
    AlgebraicIndependent R fun x : f '' s => (x : A) 
Arguments: {R : Type u_3} {A : Type u_5} [CommRing R] [CommRing A] [Algebra R A] {ι : Type u_9} {s : Set ι} {f : ι → A} (hs : AlgebraicIndependent R fun x => f ↑x)
Goal: AlgebraicIndependent R fun x => ↑x
-- algebraicIndependent_iUnion_of_directed
theorem algebraicIndependent_iUnion_of_directed {η : Type*} [Nonempty η] {s : η → Set A}
    (hs : Directed (· ⊆ ·) s) (h : ∀ i, AlgebraicIndependent R ((↑) : s i → A)) :
    AlgebraicIndependent R ((↑) : (⋃ i, s i) → A) 
Arguments: {R : Type u_3} {A : Type u_5} [CommRing R] [CommRing A] [Algebra R A] {η : Type u_9} [Nonempty η] {s : η → Set A} (hs : Directed (fun x x_1 => x ⊆ x_1) s) (h : ∀ (i : η), AlgebraicIndependent R Subtype.val)
Goal: AlgebraicIndependent R Subtype.val
-- algebraicIndependent_sUnion_of_directed
theorem algebraicIndependent_sUnion_of_directed {s : Set (Set A)} (hsn : s.Nonempty)
    (hs : DirectedOn (· ⊆ ·) s) (h : ∀ a ∈ s, AlgebraicIndependent R ((↑) : a → A)) :
    AlgebraicIndependent R ((↑) : ⋃₀ s → A) 
Arguments: {R : Type u_3} {A : Type u_5} [CommRing R] [CommRing A] [Algebra R A] {s : Set (Set A)} (hsn : s.Nonempty) (hs : DirectedOn (fun x x_1 => x ⊆ x_1) s) (h : ∀ a ∈ s, AlgebraicIndependent R Subtype.val)
Goal: AlgebraicIndependent R Subtype.val
-- exists_maximal_algebraicIndependent
theorem exists_maximal_algebraicIndependent (s t : Set A) (hst : s ⊆ t)
    (hs : AlgebraicIndependent R ((↑) : s → A)) :
    ∃ u : Set A, AlgebraicIndependent R ((↑) : u → A) ∧ s ⊆ u ∧ u ⊆ t ∧
      ∀ x : Set A, AlgebraicIndependent R ((↑) : x → A) → u ⊆ x → x ⊆ t → x = u 
Arguments: {R : Type u_3} {A : Type u_5} [CommRing R] [CommRing A] [Algebra R A] (s t : Set A) (hst : s ⊆ t) (hs : AlgebraicIndependent R Subtype.val)
Goal: ∃ u,
  AlgebraicIndependent R Subtype.val ∧
    s ⊆ u ∧ u ⊆ t ∧ ∀ (x : Set A), AlgebraicIndependent R Subtype.val → u ⊆ x → x ⊆ t → x = u
-- AlgebraicIndependent.aevalEquiv_apply_coe
@[simps!]
def AlgebraicIndependent.aevalEquiv (hx : AlgebraicIndependent R x) :
    MvPolynomial ι R ≃ₐ[R] Algebra.adjoin R (range x) 
Arguments: {ι : Type u_1} {R : Type u_3} {A : Type u_5} {x : ι → A} [CommRing R] [CommRing A] [Algebra R A] (hx : AlgebraicIndependent R x) (a : MvPolynomial ι R)
Goal: ↑(hx.aevalEquiv a) = (MvPolynomial.aeval x) a
-- AlgebraicIndependent.aevalEquiv_symm_apply
@[simps!]
def AlgebraicIndependent.aevalEquiv (hx : AlgebraicIndependent R x) :
    MvPolynomial ι R ≃ₐ[R] Algebra.adjoin R (range x) 
Arguments: {ι : Type u_1} {R : Type u_3} {A : Type u_5} {x : ι → A} [CommRing R] [CommRing A] [Algebra R A] (hx : AlgebraicIndependent R x) (b : ↥(Algebra.adjoin R (Set.range x)))
Goal: hx.aevalEquiv.symm b = Function.surjInv ⋯ b
-- AlgebraicIndependent.algebraMap_aevalEquiv
theorem AlgebraicIndependent.algebraMap_aevalEquiv (hx : AlgebraicIndependent R x)
    (p : MvPolynomial ι R) :
    algebraMap (Algebra.adjoin R (range x)) A (hx.aevalEquiv p) = aeval x p 
Arguments: {ι : Type u_1} {R : Type u_3} {A : Type u_5} {x : ι → A} [CommRing R] [CommRing A] [Algebra R A] (hx : AlgebraicIndependent R x) (p : MvPolynomial ι R)
Goal: (algebraMap (↥(Algebra.adjoin R (Set.range x))) A) (hx.aevalEquiv p) = (MvPolynomial.aeval x) p
-- AlgebraicIndependent.aeval_repr
@[simp]
theorem AlgebraicIndependent.aeval_repr (p) : aeval x (hx.repr p) = p 
Arguments: {ι : Type u_1} {R : Type u_3} {A : Type u_5} {x : ι → A} [CommRing R] [CommRing A] [Algebra R A] (hx : AlgebraicIndependent R x) (p : ↥(Algebra.adjoin R (Set.range x)))
Goal: (MvPolynomial.aeval x) (hx.repr p) = ↑p
-- AlgebraicIndependent.aeval_comp_repr
theorem AlgebraicIndependent.aeval_comp_repr : (aeval x).comp hx.repr = Subalgebra.val _ 
Arguments: {ι : Type u_1} {R : Type u_3} {A : Type u_5} {x : ι → A} [CommRing R] [CommRing A] [Algebra R A] (hx : AlgebraicIndependent R x)
Goal: (MvPolynomial.aeval x).comp hx.repr = (Algebra.adjoin R (Set.range x)).val
-- AlgebraicIndependent.repr_ker
theorem AlgebraicIndependent.repr_ker :
    RingHom.ker (hx.repr : adjoin R (range x) →+* MvPolynomial ι R) = ⊥ 
Arguments: {ι : Type u_1} {R : Type u_3} {A : Type u_5} {x : ι → A} [CommRing R] [CommRing A] [Algebra R A] (hx : AlgebraicIndependent R x)
Goal: RingHom.ker ↑hx.repr = ⊥
-- AlgebraicIndependent.mvPolynomialOptionEquivPolynomialAdjoin_apply
@[simp]
theorem AlgebraicIndependent.mvPolynomialOptionEquivPolynomialAdjoin_apply
    (hx : AlgebraicIndependent R x) (y) :
    hx.mvPolynomialOptionEquivPolynomialAdjoin y =
      Polynomial.map (hx.aevalEquiv : MvPolynomial ι R →+* adjoin R (range x))
        (aeval (fun o : Option ι => o.elim Polynomial.X fun s : ι => Polynomial.C (X s)) y) 
Arguments: {ι : Type u_1} {R : Type u_3} {A : Type u_5} {x : ι → A} [CommRing R] [CommRing A] [Algebra R A] (hx : AlgebraicIndependent R x) (y : MvPolynomial (Option ι) R)
Goal: hx.mvPolynomialOptionEquivPolynomialAdjoin y =
  Polynomial.map (↑hx.aevalEquiv)
    ((MvPolynomial.aeval fun o => o.elim Polynomial.X fun s => Polynomial.C (MvPolynomial.X s)) y)
-- AlgebraicIndependent.mvPolynomialOptionEquivPolynomialAdjoin_C
theorem AlgebraicIndependent.mvPolynomialOptionEquivPolynomialAdjoin_C
    (hx : AlgebraicIndependent R x) (r) :
    hx.mvPolynomialOptionEquivPolynomialAdjoin (C r) = Polynomial.C (algebraMap _ _ r) 
Arguments: {ι : Type u_1} {R : Type u_3} {A : Type u_5} {x : ι → A} [CommRing R] [CommRing A] [Algebra R A] (hx : AlgebraicIndependent R x) (r : R)
Goal: hx.mvPolynomialOptionEquivPolynomialAdjoin (MvPolynomial.C r) =
  Polynomial.C ((algebraMap R ↥(Algebra.adjoin R (Set.range x))) r)
-- AlgebraicIndependent.mvPolynomialOptionEquivPolynomialAdjoin_X_none
theorem AlgebraicIndependent.mvPolynomialOptionEquivPolynomialAdjoin_X_none
    (hx : AlgebraicIndependent R x) :
    hx.mvPolynomialOptionEquivPolynomialAdjoin (X none) = Polynomial.X 
Arguments: {ι : Type u_1} {R : Type u_3} {A : Type u_5} {x : ι → A} [CommRing R] [CommRing A] [Algebra R A] (hx : AlgebraicIndependent R x)
Goal: hx.mvPolynomialOptionEquivPolynomialAdjoin (MvPolynomial.X none) = Polynomial.X
-- AlgebraicIndependent.mvPolynomialOptionEquivPolynomialAdjoin_X_some
theorem AlgebraicIndependent.mvPolynomialOptionEquivPolynomialAdjoin_X_some
    (hx : AlgebraicIndependent R x) (i) :
    hx.mvPolynomialOptionEquivPolynomialAdjoin (X (some i)) =
      Polynomial.C (hx.aevalEquiv (X i)) 
Arguments: {ι : Type u_1} {R : Type u_3} {A : Type u_5} {x : ι → A} [CommRing R] [CommRing A] [Algebra R A] (hx : AlgebraicIndependent R x) (i : ι)
Goal: hx.mvPolynomialOptionEquivPolynomialAdjoin (MvPolynomial.X (some i)) = Polynomial.C (hx.aevalEquiv (MvPolynomial.X i))
-- AlgebraicIndependent.aeval_comp_mvPolynomialOptionEquivPolynomialAdjoin
theorem AlgebraicIndependent.aeval_comp_mvPolynomialOptionEquivPolynomialAdjoin
    (hx : AlgebraicIndependent R x) (a : A) :
    RingHom.comp
        (↑(Polynomial.aeval a : Polynomial (adjoin R (Set.range x)) →ₐ[_] A) :
          Polynomial (adjoin R (Set.range x)) →+* A)
        hx.mvPolynomialOptionEquivPolynomialAdjoin.toRingHom =
      ↑(MvPolynomial.aeval fun o : Option ι => o.elim a x : MvPolynomial (Option ι) R →ₐ[R] A) 
Arguments: {ι : Type u_1} {R : Type u_3} {A : Type u_5} {x : ι → A} [CommRing R] [CommRing A] [Algebra R A] (hx : AlgebraicIndependent R x) (a : A)
Goal: (↑(Polynomial.aeval a)).comp hx.mvPolynomialOptionEquivPolynomialAdjoin.toRingHom =
  ↑(MvPolynomial.aeval fun o => o.elim a x)
-- AlgebraicIndependent.option_iff
theorem AlgebraicIndependent.option_iff (hx : AlgebraicIndependent R x) (a : A) :
    (AlgebraicIndependent R fun o : Option ι => o.elim a x) ↔
      ¬IsAlgebraic (adjoin R (Set.range x)) a 
Arguments: {ι : Type u_1} {R : Type u_3} {A : Type u_5} {x : ι → A} [CommRing R] [CommRing A] [Algebra R A] (hx : AlgebraicIndependent R x) (a : A)
Goal: (AlgebraicIndependent R fun o => o.elim a x) ↔ ¬IsAlgebraic (↥(Algebra.adjoin R (Set.range x))) a
-- exists_isTranscendenceBasis
theorem exists_isTranscendenceBasis (h : Injective (algebraMap R A)) :
    ∃ s : Set A, IsTranscendenceBasis R ((↑) : s → A) 
Arguments: (R : Type u_3) {A : Type u_5} [CommRing R] [CommRing A] [Algebra R A] (h : Function.Injective ⇑(algebraMap R A))
Goal: ∃ s, IsTranscendenceBasis R Subtype.val
-- AlgebraicIndependent.isTranscendenceBasis_iff
theorem AlgebraicIndependent.isTranscendenceBasis_iff {ι : Type w} {R : Type u} [CommRing R]
    [Nontrivial R] {A : Type v} [CommRing A] [Algebra R A] {x : ι → A}
    (i : AlgebraicIndependent R x) :
    IsTranscendenceBasis R x ↔
      ∀ (κ : Type v) (w : κ → A) (_ : AlgebraicIndependent R w) (j : ι → κ) (_ : w ∘ j = x),
        Surjective j 
Arguments: {ι : Type w} {R : Type u} [CommRing R] [Nontrivial R] {A : Type v} [CommRing A] [Algebra R A] {x : ι → A} (i : AlgebraicIndependent R x)
Goal: IsTranscendenceBasis R x ↔
  ∀ (κ : Type v) (w : κ → A), AlgebraicIndependent R w → ∀ (j : ι → κ), w ∘ j = x → Function.Surjective j
-- IsTranscendenceBasis.isAlgebraic
theorem IsTranscendenceBasis.isAlgebraic [Nontrivial R] (hx : IsTranscendenceBasis R x) :
    Algebra.IsAlgebraic (adjoin R (range x)) A 
Arguments: {ι : Type u_1} {R : Type u_3} {A : Type u_5} {x : ι → A} [CommRing R] [CommRing A] [Algebra R A] [Nontrivial R] (hx : IsTranscendenceBasis R x)
Goal: Algebra.IsAlgebraic (↥(Algebra.adjoin R (Set.range x))) A
-- algebraicIndependent_empty_type
theorem algebraicIndependent_empty_type [IsEmpty ι] [Nontrivial A] : AlgebraicIndependent K x 
Arguments: {ι : Type u_1} {K : Type u_4} {A : Type u_5} {x : ι → A} [CommRing A] [Field K] [Algebra K A] [IsEmpty ι] [Nontrivial A]
Goal: AlgebraicIndependent K x
-- algebraicIndependent_empty
theorem algebraicIndependent_empty [Nontrivial A] :
    AlgebraicIndependent K ((↑) : (∅ : Set A) → A) 
Arguments: {K : Type u_4} {A : Type u_5} [CommRing A] [Field K] [Algebra K A] [Nontrivial A]
Goal: AlgebraicIndependent K Subtype.val
-- Ideal.Quotient.ringHom_ext
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
    f = g 
Arguments: {R : Type u} [CommRing R] {I : Ideal R} {S : Type v} [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (Ideal.Quotient.mk I) = g.comp (Ideal.Quotient.mk I))
Goal: f = g
-- Ideal.Quotient.eq
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I 
Arguments: {R : Type u} [CommRing R] {I : Ideal R} {x y : R}
Goal: (Ideal.Quotient.mk I) x = (Ideal.Quotient.mk I) y ↔ x - y ∈ I
-- Ideal.Quotient.mk_eq_mk
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x 
Arguments: {R : Type u} [CommRing R] {I : Ideal R} (x : R)
Goal: Submodule.Quotient.mk x = (Ideal.Quotient.mk I) x
-- Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I 
Arguments: {R : Type u} [CommRing R] {a : R} {I : Ideal R}
Goal: (Ideal.Quotient.mk I) a = 0 ↔ a ∈ I
-- Ideal.Quotient.eq_zero_iff_dvd
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y 
Arguments: {R : Type u} [CommRing R] (x y : R)
Goal: (Ideal.Quotient.mk (Ideal.span {x})) y = 0 ↔ x ∣ y
-- Ideal.Quotient.mk_singleton_self
@[simp]
lemma mk_singleton_self (x : R) : mk (Ideal.span {x}) x = 0 
Arguments: {R : Type u} [CommRing R] (x : R)
Goal: (Ideal.Quotient.mk (Ideal.span {x})) x = 0
-- Ideal.Quotient.mk_eq_mk_iff_sub_mem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I 
Arguments: {R : Type u} [CommRing R] {I : Ideal R} (x y : R)
Goal: (Ideal.Quotient.mk I) x = (Ideal.Quotient.mk I) y ↔ x - y ∈ I
-- Ideal.Quotient.zero_eq_one_iff
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ 
Arguments: {R : Type u} [CommRing R] {I : Ideal R}
Goal: 0 = 1 ↔ I = ⊤
-- Ideal.Quotient.zero_ne_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ 
Arguments: {R : Type u} [CommRing R] {I : Ideal R}
Goal: 0 ≠ 1 ↔ I ≠ ⊤
-- Ideal.Quotient.nontrivial
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) 
Arguments: {R : Type u} [CommRing R] {I : Ideal R} (hI : I ≠ ⊤)
Goal: Nontrivial (R ⧸ I)
-- Ideal.Quotient.subsingleton_iff
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ 
Arguments: {R : Type u} [CommRing R] {I : Ideal R}
Goal: Subsingleton (R ⧸ I) ↔ I = ⊤
-- Ideal.Quotient.mk_surjective
theorem mk_surjective : Function.Surjective (mk I) 
Arguments: {R : Type u} [CommRing R] {I : Ideal R}
Goal: Function.Surjective ⇑(Ideal.Quotient.mk I)
-- Ideal.Quotient.quotient_ring_saturate
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
    mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s 
Arguments: {R : Type u} [CommRing R] (I : Ideal R) (s : Set R)
Goal: ⇑(Ideal.Quotient.mk I) ⁻¹' (⇑(Ideal.Quotient.mk I) '' s) = ⋃ x, (fun y => ↑x + y) '' s
-- Ideal.Quotient.isDomain_iff_prime
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime 
Arguments: {R : Type u} [CommRing R] (I : Ideal R)
Goal: IsDomain (R ⧸ I) ↔ I.IsPrime
-- Ideal.Quotient.exists_inv
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
    ∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 
Arguments: {R : Type u} [CommRing R] {I : Ideal R} [hI : I.IsMaximal] {a : R ⧸ I}
Goal: a ≠ 0 → ∃ b, a * b = 1
-- Ideal.Quotient.maximal_of_isField
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal 
Arguments: {R : Type u} [CommRing R] (I : Ideal R) (hqf : IsField (R ⧸ I))
Goal: I.IsMaximal
-- Ideal.Quotient.maximal_ideal_iff_isField_quotient
theorem maximal_ideal_iff_isField_quotient (I : Ideal R) : I.IsMaximal ↔ IsField (R ⧸ I) 
Arguments: {R : Type u} [CommRing R] (I : Ideal R)
Goal: I.IsMaximal ↔ IsField (R ⧸ I)
-- Ideal.Quotient.lift_mk
@[simp]
theorem lift_mk (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) :
    lift I f H (mk I a) = f a 
Arguments: {R : Type u} [CommRing R] {a : R} {S : Type v} [Semiring S] (I : Ideal R) (f : R →+* S) (H : ∀ a ∈ I, f a = 0)
Goal: (Ideal.Quotient.lift I f H) ((Ideal.Quotient.mk I) a) = f a
-- Ideal.Quotient.lift_surjective_of_surjective
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
    (hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) 
Arguments: {R : Type u} [CommRing R] {S : Type v} [Semiring S] (I : Ideal R) {f : R →+* S} (H : ∀ a ∈ I, f a = 0) (hf : Function.Surjective ⇑f)
Goal: Function.Surjective ⇑(Ideal.Quotient.lift I f H)
-- Ideal.Quotient.factor_mk
@[simp]
theorem factor_mk (S T : Ideal R) (H : S ≤ T) (x : R) : factor S T H (mk S x) = mk T x 
Arguments: {R : Type u} [CommRing R] (S T : Ideal R) (H : S ≤ T) (x : R)
Goal: (Ideal.Quotient.factor S T H) ((Ideal.Quotient.mk S) x) = (Ideal.Quotient.mk T) x
-- Ideal.Quotient.factor_comp_mk
@[simp]
theorem factor_comp_mk (S T : Ideal R) (H : S ≤ T) : (factor S T H).comp (mk S) = mk T 
Arguments: {R : Type u} [CommRing R] (S T : Ideal R) (H : S ≤ T)
Goal: (Ideal.Quotient.factor S T H).comp (Ideal.Quotient.mk S) = Ideal.Quotient.mk T
-- Ideal.quotEquivOfEq_mk
@[simp]
theorem quotEquivOfEq_mk {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) (x : R) :
    quotEquivOfEq h (Ideal.Quotient.mk I x) = Ideal.Quotient.mk J x 
Arguments: {R : Type u_1} [CommRing R] {I J : Ideal R} (h : I = J) (x : R)
Goal: (Ideal.quotEquivOfEq h) ((Ideal.Quotient.mk I) x) = (Ideal.Quotient.mk J) x
-- Ideal.quotEquivOfEq_symm
@[simp]
theorem quotEquivOfEq_symm {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) :
    (Ideal.quotEquivOfEq h).symm = Ideal.quotEquivOfEq h.symm 
Arguments: {R : Type u_1} [CommRing R] {I J : Ideal R} (h : I = J)
Goal: (Ideal.quotEquivOfEq h).symm = Ideal.quotEquivOfEq ⋯
-- Ideal.map_pi
theorem map_pi {ι : Type*} [Finite ι] {ι' : Type w} (x : ι → R) (hi : ∀ i, x i ∈ I)
    (f : (ι → R) →ₗ[R] ι' → R) (i : ι') : f x i ∈ I 
Arguments: {R : Type u} [CommRing R] (I : Ideal R) {ι : Type u_1} [Finite ι] {ι' : Type w} (x : ι → R) (hi : ∀ (i : ι), x i ∈ I) (f : (ι → R) →ₗ[R] ι' → R) (i : ι')
Goal: f x i ∈ I
-- Module.isTrivialRelation_iff_vanishesTrivially
theorem isTrivialRelation_iff_vanishesTrivially :
    IsTrivialRelation f x ↔ VanishesTrivially R f x 
Arguments: {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] {ι : Type u} [Fintype ι] {f : ι → R} {x : ι → M}
Goal: Module.IsTrivialRelation f x ↔ TensorProduct.VanishesTrivially R f x