ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Algebra.FormallyEtale.of_equiv
theorem of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B 
Arguments: {R : Type u} [CommSemiring R] {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [Algebra.FormallyEtale R A] (e : A ≃ₐ[R] B)
Goal: Algebra.FormallyEtale R B
-- Algebra.FormallyEtale.comp
theorem comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B 
Arguments: (R : Type u) [CommSemiring R] (A : Type u) [CommSemiring A] [Algebra R A] (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B] [Algebra.FormallyEtale R A] [Algebra.FormallyEtale A B]
Goal: Algebra.FormallyEtale R B
-- Algebra.FormallyEtale.of_isLocalization
theorem of_isLocalization : FormallyEtale R Rₘ 
Arguments: {R Rₘ : Type u} [CommRing R] [CommRing Rₘ] (M : Submonoid R) [Algebra R Rₘ] [IsLocalization M Rₘ]
Goal: Algebra.FormallyEtale R Rₘ
-- Algebra.FormallyEtale.localization_base
theorem localization_base [FormallyEtale R Sₘ] : FormallyEtale Rₘ Sₘ 
Arguments: {R Rₘ Sₘ : Type u} [CommRing R] [CommRing Rₘ] [CommRing Sₘ] (M : Submonoid R) [Algebra R Sₘ] [Algebra R Rₘ] [Algebra Rₘ Sₘ] [IsScalarTower R Rₘ Sₘ] [IsLocalization M Rₘ] [Algebra.FormallyEtale R Sₘ]
Goal: Algebra.FormallyEtale Rₘ Sₘ
-- Algebra.FormallyEtale.localization_map
theorem localization_map [FormallyEtale R S] : FormallyEtale Rₘ Sₘ 
Arguments: {R S Rₘ Sₘ : Type u} [CommRing R] [CommRing S] [CommRing Rₘ] [CommRing Sₘ] (M : Submonoid R) [Algebra R S] [Algebra R Sₘ] [Algebra S Sₘ] [Algebra R Rₘ] [Algebra Rₘ Sₘ] [IsScalarTower R Rₘ Sₘ] [IsScalarTower R S Sₘ] [IsLocalization M Rₘ] [IsLocalization (Submonoid.map (algebraMap R S) M) Sₘ] [Algebra.FormallyEtale R S]
Goal: Algebra.FormallyEtale Rₘ Sₘ
-- Algebra.Etale.finitePresentation
class Etale : Prop where
  formallyEtale : FormallyEtale R A := by infer_instance
  finitePresentation : FinitePresentation R A := by infer_instance
Arguments: {R : Type u} [CommSemiring R] {A : Type u} [Semiring A] [Algebra R A] [self : Algebra.Etale R A]
Goal: Algebra.FinitePresentation R A
-- Algebra.Etale.formallyEtale
class Etale : Prop where
  formallyEtale : FormallyEtale R A := by infer_instance
  finitePresentation : FinitePresentation R A := by infer_instance
Arguments: {R : Type u} [CommSemiring R] {A : Type u} [Semiring A] [Algebra R A] [self : Algebra.Etale R A]
Goal: Algebra.FormallyEtale R A
-- Algebra.Etale.of_equiv
theorem of_equiv [Etale R A] (e : A ≃ₐ[R] B) : Etale R B 
Arguments: {R : Type u} [CommRing R] {A B : Type u} [CommRing A] [Algebra R A] [CommRing B] [Algebra R B] [Algebra.Etale R A] (e : A ≃ₐ[R] B)
Goal: Algebra.Etale R B
-- Algebra.Etale.comp
theorem comp [Algebra A B] [IsScalarTower R A B] [Etale R A] [Etale A B] : Etale R B 
Arguments: (R : Type u) [CommRing R] (A B : Type u) [CommRing A] [Algebra R A] [CommRing B] [Algebra R B] [Algebra A B] [IsScalarTower R A B] [Algebra.Etale R A] [Algebra.Etale A B]
Goal: Algebra.Etale R B
-- Algebra.Etale.of_isLocalization_Away
theorem of_isLocalization_Away (r : R) [IsLocalization.Away r A] : Etale R A 
Arguments: {R : Type u} [CommRing R] {A : Type u} [CommRing A] [Algebra R A] (r : R) [IsLocalization.Away r A]
Goal: Algebra.Etale R A
-- Submodule.eq_smul_of_le_smul_of_le_jacobson
theorem eq_smul_of_le_smul_of_le_jacobson {I J : Ideal R} {N : Submodule R M} (hN : N.FG)
    (hIN : N ≤ I • N) (hIjac : I ≤ jacobson J) : N = J • N 
Arguments: {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] {I J : Ideal R} {N : Submodule R M} (hN : N.FG) (hIN : N ≤ I • N) (hIjac : I ≤ J.jacobson)
Goal: N = J • N
-- Submodule.eq_bot_of_eq_ideal_smul_of_le_jacobson_annihilator
lemma eq_bot_of_eq_ideal_smul_of_le_jacobson_annihilator {I : Ideal R}
    {N : Submodule R M} (hN : FG N) (hIN : N = I • N)
    (hIjac : I ≤ N.annihilator.jacobson) : N = ⊥ 
Arguments: {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] {I : Ideal R} {N : Submodule R M} (hN : N.FG) (hIN : N = I • N) (hIjac : I ≤ N.annihilator.jacobson)
Goal: N = ⊥
-- Submodule.eq_bot_of_eq_pointwise_smul_of_mem_jacobson_annihilator
lemma eq_bot_of_eq_pointwise_smul_of_mem_jacobson_annihilator {r : R}
    {N : Submodule R M} (hN : FG N) (hrN : N = r • N)
    (hrJac : r ∈ N.annihilator.jacobson) : N = ⊥ 
Arguments: {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] {r : R} {N : Submodule R M} (hN : N.FG) (hrN : N = r • N) (hrJac : r ∈ N.annihilator.jacobson)
Goal: N = ⊥
-- Submodule.eq_bot_of_set_smul_eq_of_subset_jacobson_annihilator
lemma eq_bot_of_set_smul_eq_of_subset_jacobson_annihilator {s : Set R}
    {N : Submodule R M} (hN : FG N) (hsN : N = s • N)
    (hsJac : s ⊆ N.annihilator.jacobson) : N = ⊥ 
Arguments: {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] {s : Set R} {N : Submodule R M} (hN : N.FG) (hsN : N = s • N) (hsJac : s ⊆ ↑N.annihilator.jacobson)
Goal: N = ⊥
-- Submodule.top_ne_ideal_smul_of_le_jacobson_annihilator
lemma top_ne_ideal_smul_of_le_jacobson_annihilator [Nontrivial M]
    [Module.Finite R M] {I} (h : I ≤ (Module.annihilator R M).jacobson) :
    (⊤ : Submodule R M) ≠ I • ⊤ 
Arguments: {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] [Nontrivial M] [Module.Finite R M] {I : Ideal R} (h : I ≤ (Module.annihilator R M).jacobson)
Goal: ⊤ ≠ I • ⊤
-- Submodule.top_ne_set_smul_of_subset_jacobson_annihilator
lemma top_ne_set_smul_of_subset_jacobson_annihilator [Nontrivial M]
    [Module.Finite R M] {s : Set R}
    (h : s ⊆ (Module.annihilator R M).jacobson) :
    (⊤ : Submodule R M) ≠ s • ⊤ 
Arguments: {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] [Nontrivial M] [Module.Finite R M] {s : Set R} (h : s ⊆ ↑(Module.annihilator R M).jacobson)
Goal: ⊤ ≠ s • ⊤
-- Submodule.top_ne_pointwise_smul_of_mem_jacobson_annihilator
lemma top_ne_pointwise_smul_of_mem_jacobson_annihilator [Nontrivial M]
    [Module.Finite R M] {r} (h : r ∈ (Module.annihilator R M).jacobson) :
    (⊤ : Submodule R M) ≠ r • ⊤ 
Arguments: {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] [Nontrivial M] [Module.Finite R M] {r : R} (h : r ∈ (Module.annihilator R M).jacobson)
Goal: ⊤ ≠ r • ⊤
-- Submodule.eq_bot_of_le_smul_of_le_jacobson_bot
theorem eq_bot_of_le_smul_of_le_jacobson_bot (I : Ideal R) (N : Submodule R M) (hN : N.FG)
    (hIN : N ≤ I • N) (hIjac : I ≤ jacobson ⊥) : N = ⊥ 
Arguments: {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R) (N : Submodule R M) (hN : N.FG) (hIN : N ≤ I • N) (hIjac : I ≤ ⊥.jacobson)
Goal: N = ⊥
-- Submodule.sup_eq_sup_smul_of_le_smul_of_le_jacobson
theorem sup_eq_sup_smul_of_le_smul_of_le_jacobson {I J : Ideal R} {N N' : Submodule R M}
    (hN' : N'.FG) (hIJ : I ≤ jacobson J) (hNN : N' ≤ N ⊔ I • N') : N ⊔ N' = N ⊔ J • N' 
Arguments: {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] {I J : Ideal R} {N N' : Submodule R M} (hN' : N'.FG) (hIJ : I ≤ J.jacobson) (hNN : N' ≤ N ⊔ I • N')
Goal: N ⊔ N' = N ⊔ J • N'
-- Submodule.sup_smul_eq_sup_smul_of_le_smul_of_le_jacobson
theorem sup_smul_eq_sup_smul_of_le_smul_of_le_jacobson {I J : Ideal R} {N N' : Submodule R M}
    (hN' : N'.FG) (hIJ : I ≤ jacobson J) (hNN : N' ≤ N ⊔ I • N') : N ⊔ I • N' = N ⊔ J • N' 
Arguments: {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] {I J : Ideal R} {N N' : Submodule R M} (hN' : N'.FG) (hIJ : I ≤ J.jacobson) (hNN : N' ≤ N ⊔ I • N')
Goal: N ⊔ I • N' = N ⊔ J • N'
-- Submodule.le_of_le_smul_of_le_jacobson_bot
theorem le_of_le_smul_of_le_jacobson_bot {R M} [CommRing R] [AddCommGroup M] [Module R M]
    {I : Ideal R} {N N' : Submodule R M} (hN' : N'.FG)
    (hIJ : I ≤ jacobson ⊥) (hNN : N' ≤ N ⊔ I • N') : N' ≤ N 
Arguments: {R : Type u_3} {M : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] {I : Ideal R} {N N' : Submodule R M} (hN' : N'.FG) (hIJ : I ≤ ⊥.jacobson) (hNN : N' ≤ N ⊔ I • N')
Goal: N' ≤ N
-- Submodule.smul_le_of_le_smul_of_le_jacobson_bot
theorem smul_le_of_le_smul_of_le_jacobson_bot {I : Ideal R} {N N' : Submodule R M} (hN' : N'.FG)
    (hIJ : I ≤ jacobson ⊥) (hNN : N' ≤ N ⊔ I • N') : I • N' ≤ N 
Arguments: {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] {I : Ideal R} {N N' : Submodule R M} (hN' : N'.FG) (hIJ : I ≤ ⊥.jacobson) (hNN : N' ≤ N ⊔ I • N')
Goal: I • N' ≤ N
-- TwoSidedIdeal.mem_iff
lemma mem_iff (x : R) : x ∈ I ↔ I.ringCon x 0 
Arguments: {R : Type u_1} [NonUnitalNonAssocRing R] (I : TwoSidedIdeal R) (x : R)
Goal: x ∈ I ↔ I.ringCon x 0
-- TwoSidedIdeal.rel_iff
lemma rel_iff (x y : R) : I.ringCon x y ↔ x - y ∈ I 
Arguments: {R : Type u_1} [NonUnitalNonAssocRing R] (I : TwoSidedIdeal R) (x y : R)
Goal: I.ringCon x y ↔ x - y ∈ I
-- TwoSidedIdeal.coeOrderEmbedding_apply
@[simps]
def coeOrderEmbedding : TwoSidedIdeal R ↪o Set R 
Arguments: {R : Type u_1} [NonUnitalNonAssocRing R]
Goal: ∀ (a : TwoSidedIdeal R), TwoSidedIdeal.coeOrderEmbedding a = ↑a
-- TwoSidedIdeal.le_iff
lemma le_iff {I J : TwoSidedIdeal R} : I ≤ J ↔ (I : Set R) ⊆ (J : Set R) 
Arguments: {R : Type u_1} [NonUnitalNonAssocRing R] {I J : TwoSidedIdeal R}
Goal: I ≤ J ↔ ↑I ⊆ ↑J
-- TwoSidedIdeal.ringCon_injective
lemma ringCon_injective : Function.Injective (TwoSidedIdeal.ringCon (R := R)) 
Arguments: {R : Type u_1} [NonUnitalNonAssocRing R]
Goal: Function.Injective TwoSidedIdeal.ringCon
-- TwoSidedIdeal.ringCon_le_iff
lemma ringCon_le_iff {I J : TwoSidedIdeal R} : I ≤ J ↔ I.ringCon ≤ J.ringCon 
Arguments: {R : Type u_1} [NonUnitalNonAssocRing R] {I J : TwoSidedIdeal R}
Goal: I ≤ J ↔ I.ringCon ≤ J.ringCon
-- TwoSidedIdeal.ext
@[ext]
lemma ext {I J : TwoSidedIdeal R} (h : ∀ x, x ∈ I ↔ x ∈ J) : I = J 
Arguments: {R : Type u_1} [NonUnitalNonAssocRing R] {I J : TwoSidedIdeal R} (h : ∀ (x : R), x ∈ I ↔ x ∈ J)
Goal: I = J
-- TwoSidedIdeal.lt_iff
lemma lt_iff (I J : TwoSidedIdeal R) : I < J ↔ (I : Set R) ⊂ (J : Set R) 
Arguments: {R : Type u_1} [NonUnitalNonAssocRing R] (I J : TwoSidedIdeal R)
Goal: I < J ↔ ↑I ⊂ ↑J
-- TwoSidedIdeal.zero_mem
lemma zero_mem : 0 ∈ I 
Arguments: {R : Type u_1} [NonUnitalNonAssocRing R] (I : TwoSidedIdeal R)
Goal: 0 ∈ I
-- TwoSidedIdeal.add_mem
lemma add_mem {x y} (hx : x ∈ I) (hy : y ∈ I) : x + y ∈ I 
Arguments: {R : Type u_1} [NonUnitalNonAssocRing R] (I : TwoSidedIdeal R) {x y : R} (hx : x ∈ I) (hy : y ∈ I)
Goal: x + y ∈ I
-- TwoSidedIdeal.neg_mem
lemma neg_mem {x} (hx : x ∈ I) : -x ∈ I 
Arguments: {R : Type u_1} [NonUnitalNonAssocRing R] (I : TwoSidedIdeal R) {x : R} (hx : x ∈ I)
Goal: -x ∈ I
-- TwoSidedIdeal.sub_mem
lemma sub_mem {x y} (hx : x ∈ I) (hy : y ∈ I) : x - y ∈ I 
Arguments: {R : Type u_1} [NonUnitalNonAssocRing R] (I : TwoSidedIdeal R) {x y : R} (hx : x ∈ I) (hy : y ∈ I)
Goal: x - y ∈ I
-- TwoSidedIdeal.mul_mem_left
lemma mul_mem_left (x y) (hy : y ∈ I) : x * y ∈ I 
Arguments: {R : Type u_1} [NonUnitalNonAssocRing R] (I : TwoSidedIdeal R) (x y : R) (hy : y ∈ I)
Goal: x * y ∈ I
-- TwoSidedIdeal.mul_mem_right
lemma mul_mem_right (x y) (hx : x ∈ I) : x * y ∈ I 
Arguments: {R : Type u_1} [NonUnitalNonAssocRing R] (I : TwoSidedIdeal R) (x y : R) (hx : x ∈ I)
Goal: x * y ∈ I
-- TwoSidedIdeal.nsmul_mem
lemma nsmul_mem {x} (n : ℕ) (hx : x ∈ I) : n • x ∈ I 
Arguments: {R : Type u_1} [NonUnitalNonAssocRing R] (I : TwoSidedIdeal R) {x : R} (n : ℕ) (hx : x ∈ I)
Goal: n • x ∈ I
-- TwoSidedIdeal.zsmul_mem
lemma zsmul_mem {x} (n : ℤ) (hx : x ∈ I) : n • x ∈ I 
Arguments: {R : Type u_1} [NonUnitalNonAssocRing R] (I : TwoSidedIdeal R) {x : R} (n : ℤ) (hx : x ∈ I)
Goal: n • x ∈ I
-- TwoSidedIdeal.mem_mk'
lemma mem_mk' (carrier : Set R)
    (zero_mem : 0 ∈ carrier)
    (add_mem : ∀ {x y}, x ∈ carrier → y ∈ carrier → x + y ∈ carrier)
    (neg_mem : ∀ {x}, x ∈ carrier → -x ∈ carrier)
    (mul_mem_left : ∀ {x y}, y ∈ carrier → x * y ∈ carrier)
    (mul_mem_right : ∀ {x y}, x ∈ carrier → x * y ∈ carrier)
    (x : R) :
    x ∈ mk' carrier zero_mem add_mem neg_mem mul_mem_left mul_mem_right ↔ x ∈ carrier 
Arguments: {R : Type u_1} [NonUnitalNonAssocRing R] (carrier : Set R) (zero_mem : 0 ∈ carrier) (add_mem : ∀ {x y : R}, x ∈ carrier → y ∈ carrier → x + y ∈ carrier) (neg_mem : ∀ {x : R}, x ∈ carrier → -x ∈ carrier) (mul_mem_left : ∀ {x y : R}, y ∈ carrier → x * y ∈ carrier) (mul_mem_right : ∀ {x y : R}, x ∈ carrier → x * y ∈ carrier) (x : R)
Goal: x ∈ TwoSidedIdeal.mk' carrier zero_mem ⋯ ⋯ ⋯ ⋯ ↔ x ∈ carrier
-- TwoSidedIdeal.coe_smul
@[simp]
lemma coe_smul {r : R} {x : I} : (r • x : R) = r * (x : R) 
Arguments: {R : Type u_1} [Ring R] (I : TwoSidedIdeal R) {r : R} {x : ↥I}
Goal: r • ↑x = r * ↑x
-- TwoSidedIdeal.coe_mop_smul
@[simp]
lemma coe_mop_smul {r : Rᵐᵒᵖ} {x : I} : (r • x : R) = (x : R) * r.unop 
Arguments: {R : Type u_1} [Ring R] (I : TwoSidedIdeal R) {r : Rᵐᵒᵖ} {x : ↥I}
Goal: r • ↑x = ↑x * MulOpposite.unop r
-- TwoSidedIdeal.subtype_apply
@[simps]
def subtype : I →ₗ[R] R 
Arguments: {R : Type u_1} [Ring R] (I : TwoSidedIdeal R) (x : ↥I)
Goal: I.subtype x = ↑x
-- TwoSidedIdeal.subtypeMop_apply
@[simps]
def subtypeMop : I →ₗ[Rᵐᵒᵖ] Rᵐᵒᵖ 
Arguments: {R : Type u_1} [Ring R] (I : TwoSidedIdeal R) (x : ↥I)
Goal: I.subtypeMop x = MulOpposite.op ↑x
-- Algebra.formallyUnramified_iff
@[mk_iff]
class FormallyUnramified : Prop where
  comp_injective :
    ∀ ⦃B : Type u⦄ [CommRing B],
      ∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
        Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
Arguments: (R : Type u) [CommSemiring R] (A : Type u) [Semiring A] [Algebra R A]
Goal: Algebra.FormallyUnramified R A ↔
  ∀ ⦃B : Type u⦄ [inst : CommRing B] [inst_1 : Algebra R B] (I : Ideal B),
    I ^ 2 = ⊥ → Function.Injective (Ideal.Quotient.mkₐ R I).comp
-- Algebra.FormallyUnramified.comp_injective
@[mk_iff]
class FormallyUnramified : Prop where
  comp_injective :
    ∀ ⦃B : Type u⦄ [CommRing B],
      ∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
        Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
Arguments: {R : Type u} [CommSemiring R] {A : Type u} [Semiring A] [Algebra R A] [self : Algebra.FormallyUnramified R A] ⦃B : Type u⦄ [CommRing B] [Algebra R B] (I : Ideal B)
Goal: I ^ 2 = ⊥ → Function.Injective (Ideal.Quotient.mkₐ R I).comp
-- Algebra.FormallyUnramified.lift_unique
theorem lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
    [FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
    (h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ 
Arguments: {R : Type u} [CommSemiring R] {A : Type u} [Semiring A] [Algebra R A] {B : Type u} [CommRing B] [_RB : Algebra R B] [Algebra.FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B) (h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂)
Goal: g₁ = g₂
-- Algebra.FormallyUnramified.ext
theorem ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
    (H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ 
Arguments: {R : Type u} [CommSemiring R] {A : Type u} [Semiring A] [Algebra R A] {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B) [Algebra.FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B} (H : ∀ (x : A), (Ideal.Quotient.mk I) (g₁ x) = (Ideal.Quotient.mk I) (g₂ x))
Goal: g₁ = g₂
-- Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
    (f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
    (h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ 
Arguments: {R : Type u} [CommSemiring R] {A : Type u} [Semiring A] [Algebra R A] {B : Type u} [CommRing B] [Algebra R B] [Algebra.FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C) (hf : IsNilpotent (RingHom.ker f)) (g₁ g₂ : A →ₐ[R] B) (h : f.comp ↑g₁ = f.comp ↑g₂)
Goal: g₁ = g₂
-- Algebra.FormallyUnramified.ext'
theorem ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
    (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
    g₁ = g₂ 
Arguments: {R : Type u} [CommSemiring R] {A : Type u} [Semiring A] [Algebra R A] {B : Type u} [CommRing B] [Algebra R B] [Algebra.FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C) (hf : IsNilpotent (RingHom.ker f)) (g₁ g₂ : A →ₐ[R] B) (h : ∀ (x : A), f (g₁ x) = f (g₂ x))
Goal: g₁ = g₂
-- Algebra.FormallyUnramified.lift_unique'
theorem lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
    [Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
    (g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ 
Arguments: {R : Type u} [CommSemiring R] {A : Type u} [Semiring A] [Algebra R A] {B : Type u} [CommRing B] [Algebra R B] [Algebra.FormallyUnramified R A] {C : Type u} [CommRing C] [Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent (RingHom.ker ↑f)) (g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂)
Goal: g₁ = g₂
-- Algebra.FormallyUnramified.of_equiv
theorem of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
    FormallyUnramified R B 
Arguments: {R : Type u} [CommSemiring R] {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [Algebra.FormallyUnramified R A] (e : A ≃ₐ[R] B)
Goal: Algebra.FormallyUnramified R B
-- Algebra.FormallyUnramified.comp
theorem comp [FormallyUnramified R A] [FormallyUnramified A B] :
    FormallyUnramified R B 
Arguments: (R : Type u) [CommSemiring R] (A : Type u) [CommSemiring A] [Algebra R A] (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B] [Algebra.FormallyUnramified R A] [Algebra.FormallyUnramified A B]
Goal: Algebra.FormallyUnramified R B
-- Algebra.FormallyUnramified.of_comp
theorem of_comp [FormallyUnramified R B] : FormallyUnramified A B 
Arguments: (R : Type u) [CommSemiring R] (A : Type u) [CommSemiring A] [Algebra R A] (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B] [Algebra.FormallyUnramified R B]
Goal: Algebra.FormallyUnramified A B
-- Algebra.FormallyUnramified.of_isLocalization
theorem of_isLocalization : FormallyUnramified R Rₘ 
Arguments: {R Rₘ : Type u} [CommRing R] [CommRing Rₘ] (M : Submonoid R) [Algebra R Rₘ] [IsLocalization M Rₘ]
Goal: Algebra.FormallyUnramified R Rₘ
-- Algebra.FormallyUnramified.localization_base
-- @[nolint unusedArguments] -- Porting note: removed
theorem localization_base [FormallyUnramified R Sₘ] : FormallyUnramified Rₘ Sₘ 
Arguments: {R Rₘ Sₘ : Type u} [CommRing R] [CommRing Rₘ] [CommRing Sₘ] (M : Submonoid R) [Algebra R Sₘ] [Algebra R Rₘ] [Algebra Rₘ Sₘ] [IsScalarTower R Rₘ Sₘ] [Algebra.FormallyUnramified R Sₘ]
Goal: Algebra.FormallyUnramified Rₘ Sₘ
-- Algebra.FormallyUnramified.localization_map
theorem localization_map [FormallyUnramified R S] :
    FormallyUnramified Rₘ Sₘ 
Arguments: {R S Rₘ Sₘ : Type u} [CommRing R] [CommRing S] [CommRing Rₘ] [CommRing Sₘ] (M : Submonoid R) [Algebra R S] [Algebra R Sₘ] [Algebra S Sₘ] [Algebra R Rₘ] [Algebra Rₘ Sₘ] [IsScalarTower R Rₘ Sₘ] [IsScalarTower R S Sₘ] [IsLocalization (Submonoid.map (algebraMap R S) M) Sₘ] [Algebra.FormallyUnramified R S]
Goal: Algebra.FormallyUnramified Rₘ Sₘ
-- Algebra.Unramified.finiteType
class Unramified : Prop where
  formallyUnramified : FormallyUnramified R A := by infer_instance
  finiteType : FiniteType R A := by infer_instance
Arguments: {R : Type u} [CommSemiring R] {A : Type u} [Semiring A] [Algebra R A] [self : Algebra.Unramified R A]
Goal: Algebra.FiniteType R A
-- Algebra.Unramified.formallyUnramified
class Unramified : Prop where
  formallyUnramified : FormallyUnramified R A := by infer_instance
  finiteType : FiniteType R A := by infer_instance
Arguments: {R : Type u} [CommSemiring R] {A : Type u} [Semiring A] [Algebra R A] [self : Algebra.Unramified R A]
Goal: Algebra.FormallyUnramified R A
-- Algebra.Unramified.of_equiv
theorem of_equiv [Unramified R A] (e : A ≃ₐ[R] B) : Unramified R B 
Arguments: {R : Type u} [CommRing R] {A B : Type u} [CommRing A] [Algebra R A] [CommRing B] [Algebra R B] [Algebra.Unramified R A] (e : A ≃ₐ[R] B)
Goal: Algebra.Unramified R B
-- Algebra.Unramified.of_isLocalization_Away
theorem of_isLocalization_Away (r : R) [IsLocalization.Away r A] : Unramified R A 
Arguments: {R : Type u} [CommRing R] {A : Type u} [CommRing A] [Algebra R A] (r : R) [IsLocalization.Away r A]
Goal: Algebra.Unramified R A
-- Algebra.Unramified.comp
theorem comp [Algebra A B] [IsScalarTower R A B] [Unramified R A] [Unramified A B] :
    Unramified R B 
Arguments: (R : Type u) [CommRing R] (A B : Type u) [CommRing A] [Algebra R A] [CommRing B] [Algebra R B] [Algebra A B] [IsScalarTower R A B] [Algebra.Unramified R A] [Algebra.Unramified A B]
Goal: Algebra.Unramified R B
-- MvPolynomial.mem_ideal_span_monomial_image
theorem mem_ideal_span_monomial_image {x : MvPolynomial σ R} {s : Set (σ →₀ ℕ)} :
    x ∈ Ideal.span ((fun s => monomial s (1 : R)) '' s) ↔ ∀ xi ∈ x.support, ∃ si ∈ s, si ≤ xi 
Arguments: {σ : Type u_1} {R : Type u_2} [CommSemiring R] {x : MvPolynomial σ R} {s : Set (σ →₀ ℕ)}
Goal: x ∈ Ideal.span ((fun s => (MvPolynomial.monomial s) 1) '' s) ↔ ∀ xi ∈ x.support, ∃ si ∈ s, si ≤ xi
-- MvPolynomial.mem_ideal_span_monomial_image_iff_dvd
theorem mem_ideal_span_monomial_image_iff_dvd {x : MvPolynomial σ R} {s : Set (σ →₀ ℕ)} :
    x ∈ Ideal.span ((fun s => monomial s (1 : R)) '' s) ↔
      ∀ xi ∈ x.support, ∃ si ∈ s, monomial si 1 ∣ monomial xi (x.coeff xi) 
Arguments: {σ : Type u_1} {R : Type u_2} [CommSemiring R] {x : MvPolynomial σ R} {s : Set (σ →₀ ℕ)}
Goal: x ∈ Ideal.span ((fun s => (MvPolynomial.monomial s) 1) '' s) ↔
  ∀ xi ∈ x.support, ∃ si ∈ s, (MvPolynomial.monomial si) 1 ∣ (MvPolynomial.monomial xi) (MvPolynomial.coeff xi x)
-- MvPolynomial.mem_ideal_span_X_image
theorem mem_ideal_span_X_image {x : MvPolynomial σ R} {s : Set σ} :
    x ∈ Ideal.span (MvPolynomial.X '' s : Set (MvPolynomial σ R)) ↔
      ∀ m ∈ x.support, ∃ i ∈ s, (m : σ →₀ ℕ) i ≠ 0 
Arguments: {σ : Type u_1} {R : Type u_2} [CommSemiring R] {x : MvPolynomial σ R} {s : Set σ}
Goal: x ∈ Ideal.span (MvPolynomial.X '' s) ↔ ∀ m ∈ x.support, ∃ i ∈ s, m i ≠ 0
-- MvPolynomial.mapRange_eq_map
theorem mapRange_eq_map {R S : Type*} [CommSemiring R] [CommSemiring S] (p : MvPolynomial σ R)
    (f : R →+* S) : Finsupp.mapRange f f.map_zero p = map f p 
Arguments: (σ : Type u) {R : Type u_1} {S : Type u_2} [CommSemiring R] [CommSemiring S] (p : MvPolynomial σ R) (f : R →+* S)
Goal: Finsupp.mapRange ⇑f ⋯ p = (MvPolynomial.map f) p
-- MvPolynomial.restrictSupport_mono
theorem restrictSupport_mono {s t : Set (σ →₀ ℕ)} (h : s ⊆ t) :
    restrictSupport R s ≤ restrictSupport R t 
Arguments: {σ : Type u} (R : Type v) [CommSemiring R] {s t : Set (σ →₀ ℕ)} (h : s ⊆ t)
Goal: MvPolynomial.restrictSupport R s ≤ MvPolynomial.restrictSupport R t
-- MvPolynomial.mem_restrictTotalDegree
theorem mem_restrictTotalDegree (p : MvPolynomial σ R) :
    p ∈ restrictTotalDegree σ R m ↔ p.totalDegree ≤ m 
Arguments: (σ : Type u) {R : Type v} [CommSemiring R] (m : ℕ) (p : MvPolynomial σ R)
Goal: p ∈ MvPolynomial.restrictTotalDegree σ R m ↔ p.totalDegree ≤ m
-- MvPolynomial.mem_restrictDegree
theorem mem_restrictDegree (p : MvPolynomial σ R) (n : ℕ) :
    p ∈ restrictDegree σ R n ↔ ∀ s ∈ p.support, ∀ i, (s : σ →₀ ℕ) i ≤ n 
Arguments: (σ : Type u) {R : Type v} [CommSemiring R] (p : MvPolynomial σ R) (n : ℕ)
Goal: p ∈ MvPolynomial.restrictDegree σ R n ↔ ∀ s ∈ p.support, ∀ (i : σ), s i ≤ n
-- MvPolynomial.mem_restrictDegree_iff_sup
theorem mem_restrictDegree_iff_sup [DecidableEq σ] (p : MvPolynomial σ R) (n : ℕ) :
    p ∈ restrictDegree σ R n ↔ ∀ i, p.degrees.count i ≤ n 
Arguments: (σ : Type u) {R : Type v} [CommSemiring R] [DecidableEq σ] (p : MvPolynomial σ R) (n : ℕ)
Goal: p ∈ MvPolynomial.restrictDegree σ R n ↔ ∀ (i : σ), Multiset.count i p.degrees ≤ n
-- MvPolynomial.restrictTotalDegree_le_restrictDegree
theorem restrictTotalDegree_le_restrictDegree (m : ℕ) :
    restrictTotalDegree σ R m ≤ restrictDegree σ R m 
Arguments: (σ : Type u) (R : Type v) [CommSemiring R] (m : ℕ)
Goal: MvPolynomial.restrictTotalDegree σ R m ≤ MvPolynomial.restrictDegree σ R m
-- MvPolynomial.coe_basisMonomials
@[simp]
theorem coe_basisMonomials :
    (basisMonomials σ R : (σ →₀ ℕ) → MvPolynomial σ R) = fun s => monomial s 1 
Arguments: (σ : Type u) (R : Type v) [CommSemiring R]
Goal: ⇑(MvPolynomial.basisMonomials σ R) = fun s => (MvPolynomial.monomial s) 1