ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- HahnSeries.min_order_le_order_add
theorem min_order_le_order_add {Γ} [Zero Γ] [LinearOrder Γ] {x y : HahnSeries Γ R}
    (hxy : x + y ≠ 0) : min x.order y.order ≤ (x + y).order 
Arguments: {R : Type u_2} [AddMonoid R] {Γ : Type u_3} [Zero Γ] [LinearOrder Γ] {x y : HahnSeries Γ R} (hxy : x + y ≠ 0)
Goal: min x.order y.order ≤ (x + y).order
-- HahnSeries.orderTop_add_eq_left
theorem orderTop_add_eq_left {Γ} [LinearOrder Γ] {x y : HahnSeries Γ R}
    (hxy : x.orderTop < y.orderTop) : (x + y).orderTop = x.orderTop 
Arguments: {R : Type u_2} [AddMonoid R] {Γ : Type u_3} [LinearOrder Γ] {x y : HahnSeries Γ R} (hxy : x.orderTop < y.orderTop)
Goal: (x + y).orderTop = x.orderTop
-- HahnSeries.orderTop_add_eq_right
theorem orderTop_add_eq_right {Γ} [LinearOrder Γ] {x y : HahnSeries Γ R}
    (hxy : y.orderTop < x.orderTop) : (x + y).orderTop = y.orderTop 
Arguments: {R : Type u_2} [AddMonoid R] {Γ : Type u_3} [LinearOrder Γ] {x y : HahnSeries Γ R} (hxy : y.orderTop < x.orderTop)
Goal: (x + y).orderTop = y.orderTop
-- HahnSeries.leadingCoeff_add_eq_left
theorem leadingCoeff_add_eq_left {Γ} [LinearOrder Γ] {x y : HahnSeries Γ R}
    (hxy : x.orderTop < y.orderTop) : (x + y).leadingCoeff = x.leadingCoeff 
Arguments: {R : Type u_2} [AddMonoid R] {Γ : Type u_3} [LinearOrder Γ] {x y : HahnSeries Γ R} (hxy : x.orderTop < y.orderTop)
Goal: (x + y).leadingCoeff = x.leadingCoeff
-- HahnSeries.leadingCoeff_add_eq_right
theorem leadingCoeff_add_eq_right {Γ} [LinearOrder Γ] {x y : HahnSeries Γ R}
    (hxy : y.orderTop < x.orderTop) : (x + y).leadingCoeff = y.leadingCoeff 
Arguments: {R : Type u_2} [AddMonoid R] {Γ : Type u_3} [LinearOrder Γ] {x y : HahnSeries Γ R} (hxy : y.orderTop < x.orderTop)
Goal: (x + y).leadingCoeff = y.leadingCoeff
-- HahnSeries.single.addMonoidHom_apply_coeff
@[simps!]
def single.addMonoidHom (a : Γ) : R →+ HahnSeries Γ R 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [AddMonoid R] (a : Γ) (r : R) (j : Γ)
Goal: ((HahnSeries.single.addMonoidHom a) r).coeff j = Pi.single a r j
-- HahnSeries.coeff.addMonoidHom_apply
@[simps]
def coeff.addMonoidHom (g : Γ) : HahnSeries Γ R →+ R 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [AddMonoid R] (g : Γ) (f : HahnSeries Γ R)
Goal: (HahnSeries.coeff.addMonoidHom g) f = f.coeff g
-- HahnSeries.embDomain_add
theorem embDomain_add (f : Γ ↪o Γ') (x y : HahnSeries Γ R) :
    embDomain f (x + y) = embDomain f x + embDomain f y 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [AddMonoid R] {Γ' : Type u_3} [PartialOrder Γ'] (f : Γ ↪o Γ') (x y : HahnSeries Γ R)
Goal: HahnSeries.embDomain f (x + y) = HahnSeries.embDomain f x + HahnSeries.embDomain f y
-- HahnSeries.neg_coeff'
@[simp]
theorem neg_coeff' {x : HahnSeries Γ R} : (-x).coeff = -x.coeff 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [AddGroup R] {x : HahnSeries Γ R}
Goal: (-x).coeff = -x.coeff
-- HahnSeries.neg_coeff
theorem neg_coeff {x : HahnSeries Γ R} {a : Γ} : (-x).coeff a = -x.coeff a 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [AddGroup R] {x : HahnSeries Γ R} {a : Γ}
Goal: (-x).coeff a = -x.coeff a
-- HahnSeries.support_neg
@[simp]
theorem support_neg {x : HahnSeries Γ R} : (-x).support = x.support 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [AddGroup R] {x : HahnSeries Γ R}
Goal: (-x).support = x.support
-- HahnSeries.sub_coeff'
@[simp]
theorem sub_coeff' {x y : HahnSeries Γ R} : (x - y).coeff = x.coeff - y.coeff 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [AddGroup R] {x y : HahnSeries Γ R}
Goal: (x - y).coeff = x.coeff - y.coeff
-- HahnSeries.sub_coeff
theorem sub_coeff {x y : HahnSeries Γ R} {a : Γ} : (x - y).coeff a = x.coeff a - y.coeff a 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [AddGroup R] {x y : HahnSeries Γ R} {a : Γ}
Goal: (x - y).coeff a = x.coeff a - y.coeff a
-- HahnSeries.orderTop_neg
theorem orderTop_neg {x : HahnSeries Γ R} : (-x).orderTop = x.orderTop 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [AddGroup R] {x : HahnSeries Γ R}
Goal: (-x).orderTop = x.orderTop
-- HahnSeries.order_neg
@[simp]
theorem order_neg [Zero Γ] {f : HahnSeries Γ R} : (-f).order = f.order 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [AddGroup R] [Zero Γ] {f : HahnSeries Γ R}
Goal: (-f).order = f.order
-- HahnSeries.min_orderTop_le_orderTop_sub
theorem min_orderTop_le_orderTop_sub {Γ} [LinearOrder Γ] {x y : HahnSeries Γ R} :
    min x.orderTop y.orderTop ≤ (x - y).orderTop 
Arguments: {R : Type u_2} [AddGroup R] {Γ : Type u_3} [LinearOrder Γ] {x y : HahnSeries Γ R}
Goal: min x.orderTop y.orderTop ≤ (x - y).orderTop
-- HahnSeries.orderTop_sub
theorem orderTop_sub {Γ} [LinearOrder Γ] {x y : HahnSeries Γ R}
    (hxy : x.orderTop < y.orderTop) : (x - y).orderTop = x.orderTop 
Arguments: {R : Type u_2} [AddGroup R] {Γ : Type u_3} [LinearOrder Γ] {x y : HahnSeries Γ R} (hxy : x.orderTop < y.orderTop)
Goal: (x - y).orderTop = x.orderTop
-- HahnSeries.leadingCoeff_sub
theorem leadingCoeff_sub {Γ} [LinearOrder Γ] {x y : HahnSeries Γ R}
    (hxy : x.orderTop < y.orderTop) : (x - y).leadingCoeff = x.leadingCoeff 
Arguments: {R : Type u_2} [AddGroup R] {Γ : Type u_3} [LinearOrder Γ] {x y : HahnSeries Γ R} (hxy : x.orderTop < y.orderTop)
Goal: (x - y).leadingCoeff = x.leadingCoeff
-- HahnSeries.smul_coeff
@[simp]
theorem smul_coeff {r : R} {x : HahnSeries Γ V} {a : Γ} : (r • x).coeff a = r • x.coeff a 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] {V : Type u_3} [Zero V] [SMulZeroClass R V] {r : R} {x : HahnSeries Γ V} {a : Γ}
Goal: (r • x).coeff a = r • x.coeff a
-- HahnSeries.orderTop_smul_not_lt
theorem orderTop_smul_not_lt (r : R) (x : HahnSeries Γ V) : ¬ (r • x).orderTop < x.orderTop 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] {V : Type u_3} [Zero V] [SMulZeroClass R V] (r : R) (x : HahnSeries Γ V)
Goal: ¬(r • x).orderTop < x.orderTop
-- HahnSeries.order_smul_not_lt
theorem order_smul_not_lt [Zero Γ] (r : R) (x : HahnSeries Γ V) (h : r • x ≠ 0) :
    ¬ (r • x).order < x.order 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] {V : Type u_3} [Zero V] [SMulZeroClass R V] [Zero Γ] (r : R) (x : HahnSeries Γ V) (h : r • x ≠ 0)
Goal: ¬(r • x).order < x.order
-- HahnSeries.le_order_smul
theorem le_order_smul {Γ} [Zero Γ] [LinearOrder Γ] (r : R) (x : HahnSeries Γ V) (h : r • x ≠ 0) :
    x.order ≤ (r • x).order 
Arguments: {R : Type u_2} {V : Type u_3} [Zero V] [SMulZeroClass R V] {Γ : Type u_4} [Zero Γ] [LinearOrder Γ] (r : R) (x : HahnSeries Γ V) (h : r • x ≠ 0)
Goal: x.order ≤ (r • x).order
-- HahnSeries.single.linearMap_apply
@[simps]
def single.linearMap (a : Γ) : V →ₗ[R] HahnSeries Γ V 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Semiring R] {V : Type u_3} [AddCommMonoid V] [Module R V] (a : Γ)
Goal: ∀ (a_1 : V), (HahnSeries.single.linearMap a) a_1 = (↑(HahnSeries.single.addMonoidHom a)).toFun a_1
-- HahnSeries.coeff.linearMap_apply
@[simps]
def coeff.linearMap (g : Γ) : HahnSeries Γ V →ₗ[R] V 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Semiring R] {V : Type u_3} [AddCommMonoid V] [Module R V] (g : Γ)
Goal: ∀ (a : HahnSeries Γ V), (HahnSeries.coeff.linearMap g) a = (↑(HahnSeries.coeff.addMonoidHom g)).toFun a
-- HahnSeries.embDomain_smul
theorem embDomain_smul (f : Γ ↪o Γ') (r : R) (x : HahnSeries Γ R) :
    embDomain f (r • x) = r • embDomain f x 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Semiring R] {Γ' : Type u_4} [PartialOrder Γ'] (f : Γ ↪o Γ') (r : R) (x : HahnSeries Γ R)
Goal: HahnSeries.embDomain f (r • x) = r • HahnSeries.embDomain f x
-- HahnSeries.embDomainLinearMap_apply
@[simps]
def embDomainLinearMap (f : Γ ↪o Γ') : HahnSeries Γ R →ₗ[R] HahnSeries Γ' R 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Semiring R] {Γ' : Type u_4} [PartialOrder Γ'] (f : Γ ↪o Γ')
Goal: ∀ (a : HahnSeries Γ R), (HahnSeries.embDomainLinearMap f) a = HahnSeries.embDomain f a
-- KaehlerDifferential.one_smul_sub_smul_one_mem_ideal
theorem KaehlerDifferential.one_smul_sub_smul_one_mem_ideal (a : S) :
    (1 : S) ⊗ₜ[R] a - a ⊗ₜ[R] (1 : S) ∈ KaehlerDifferential.ideal R S 
Arguments: (R : Type u) {S : Type v} [CommRing R] [CommRing S] [Algebra R S] (a : S)
Goal: 1 ⊗ₜ[R] a - a ⊗ₜ[R] 1 ∈ KaehlerDifferential.ideal R S
-- Derivation.tensorProductTo_tmul
theorem Derivation.tensorProductTo_tmul (D : Derivation R S M) (s t : S) :
    D.tensorProductTo (s ⊗ₜ t) = s • D t 
Arguments: {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] {M : Type u_1} [AddCommGroup M] [Module R M] [Module S M] [IsScalarTower R S M] (D : Derivation R S M) (s t : S)
Goal: D.tensorProductTo (s ⊗ₜ[R] t) = s • D t
-- Derivation.tensorProductTo_mul
theorem Derivation.tensorProductTo_mul (D : Derivation R S M) (x y : S ⊗[R] S) :
    D.tensorProductTo (x * y) =
      TensorProduct.lmul' (S := S) R x • D.tensorProductTo y +
        TensorProduct.lmul' (S := S) R y • D.tensorProductTo x 
Arguments: {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] {M : Type u_1} [AddCommGroup M] [Module R M] [Module S M] [IsScalarTower R S M] (D : Derivation R S M) (x y : TensorProduct R S S)
Goal: D.tensorProductTo (x * y) =
  (Algebra.TensorProduct.lmul' R) x • D.tensorProductTo y + (Algebra.TensorProduct.lmul' R) y • D.tensorProductTo x
-- KaehlerDifferential.submodule_span_range_eq_ideal
theorem KaehlerDifferential.submodule_span_range_eq_ideal :
    Submodule.span S (Set.range fun s : S => (1 : S) ⊗ₜ[R] s - s ⊗ₜ[R] (1 : S)) =
      (KaehlerDifferential.ideal R S).restrictScalars S 
Arguments: (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S]
Goal: Submodule.span S (Set.range fun s => 1 ⊗ₜ[R] s - s ⊗ₜ[R] 1) =
  Submodule.restrictScalars S (KaehlerDifferential.ideal R S)
-- KaehlerDifferential.span_range_eq_ideal
theorem KaehlerDifferential.span_range_eq_ideal :
    Ideal.span (Set.range fun s : S => (1 : S) ⊗ₜ[R] s - s ⊗ₜ[R] (1 : S)) =
      KaehlerDifferential.ideal R S 
Arguments: (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S]
Goal: Ideal.span (Set.range fun s => 1 ⊗ₜ[R] s - s ⊗ₜ[R] 1) = KaehlerDifferential.ideal R S
-- KaehlerDifferential.DLinearMap_apply
theorem KaehlerDifferential.DLinearMap_apply (s : S) :
    KaehlerDifferential.DLinearMap R S s =
      (KaehlerDifferential.ideal R S).toCotangent
        ⟨1 ⊗ₜ s - s ⊗ₜ 1, KaehlerDifferential.one_smul_sub_smul_one_mem_ideal R s⟩ 
Arguments: (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] (s : S)
Goal: (KaehlerDifferential.DLinearMap R S) s = (KaehlerDifferential.ideal R S).toCotangent ⟨1 ⊗ₜ[R] s - s ⊗ₜ[R] 1, ⋯⟩
-- KaehlerDifferential.D_apply
theorem KaehlerDifferential.D_apply (s : S) :
    KaehlerDifferential.D R S s =
      (KaehlerDifferential.ideal R S).toCotangent
        ⟨1 ⊗ₜ s - s ⊗ₜ 1, KaehlerDifferential.one_smul_sub_smul_one_mem_ideal R s⟩ 
Arguments: (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] (s : S)
Goal: (KaehlerDifferential.D R S) s = (KaehlerDifferential.ideal R S).toCotangent ⟨1 ⊗ₜ[R] s - s ⊗ₜ[R] 1, ⋯⟩
-- KaehlerDifferential.span_range_derivation
theorem KaehlerDifferential.span_range_derivation :
    Submodule.span S (Set.range <| KaehlerDifferential.D R S) = ⊤ 
Arguments: (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S]
Goal: Submodule.span S (Set.range ⇑(KaehlerDifferential.D R S)) = ⊤
-- KaehlerDifferential.subsingleton_of_surjective
lemma KaehlerDifferential.subsingleton_of_surjective (h : Function.Surjective (algebraMap R S)) :
    Subsingleton (Ω[S⁄R]) 
Arguments: (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] (h : Function.Surjective ⇑(algebraMap R S))
Goal: Subsingleton (Ω[S⁄R])
-- Derivation.liftKaehlerDifferential_apply
theorem Derivation.liftKaehlerDifferential_apply (D : Derivation R S M) (x) :
    D.liftKaehlerDifferential ((KaehlerDifferential.ideal R S).toCotangent x) =
      D.tensorProductTo x 
Arguments: {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] {M : Type u_1} [AddCommGroup M] [Module R M] [Module S M] [IsScalarTower R S M] (D : Derivation R S M) (x : ↥(KaehlerDifferential.ideal R S))
Goal: D.liftKaehlerDifferential ((KaehlerDifferential.ideal R S).toCotangent x) = D.tensorProductTo ↑x
-- Derivation.liftKaehlerDifferential_comp
theorem Derivation.liftKaehlerDifferential_comp (D : Derivation R S M) :
    D.liftKaehlerDifferential.compDer (KaehlerDifferential.D R S) = D 
Arguments: {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] {M : Type u_1} [AddCommGroup M] [Module R M] [Module S M] [IsScalarTower R S M] (D : Derivation R S M)
Goal: D.liftKaehlerDifferential.compDer (KaehlerDifferential.D R S) = D
-- Derivation.liftKaehlerDifferential_comp_D
@[simp]
theorem Derivation.liftKaehlerDifferential_comp_D (D' : Derivation R S M) (x : S) :
    D'.liftKaehlerDifferential (KaehlerDifferential.D R S x) = D' x 
Arguments: {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] {M : Type u_1} [AddCommGroup M] [Module R M] [Module S M] [IsScalarTower R S M] (D' : Derivation R S M) (x : S)
Goal: D'.liftKaehlerDifferential ((KaehlerDifferential.D R S) x) = D' x
-- Derivation.liftKaehlerDifferential_unique
@[ext]
theorem Derivation.liftKaehlerDifferential_unique (f f' : Ω[S⁄R] →ₗ[S] M)
    (hf : f.compDer (KaehlerDifferential.D R S) = f'.compDer (KaehlerDifferential.D R S)) :
    f = f' 
Arguments: {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] {M : Type u_1} [AddCommGroup M] [Module R M] [Module S M] [IsScalarTower R S M] (f f' : Ω[S⁄R] →ₗ[S] M) (hf : f.compDer (KaehlerDifferential.D R S) = f'.compDer (KaehlerDifferential.D R S))
Goal: f = f'
-- Derivation.liftKaehlerDifferential_D
theorem Derivation.liftKaehlerDifferential_D :
    (KaehlerDifferential.D R S).liftKaehlerDifferential = LinearMap.id 
Arguments: (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S]
Goal: (KaehlerDifferential.D R S).liftKaehlerDifferential = LinearMap.id
-- KaehlerDifferential.D_tensorProductTo
theorem KaehlerDifferential.D_tensorProductTo (x : KaehlerDifferential.ideal R S) :
    (KaehlerDifferential.D R S).tensorProductTo x =
      (KaehlerDifferential.ideal R S).toCotangent x 
Arguments: {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] (x : ↥(KaehlerDifferential.ideal R S))
Goal: (KaehlerDifferential.D R S).tensorProductTo ↑x = (KaehlerDifferential.ideal R S).toCotangent x
-- KaehlerDifferential.tensorProductTo_surjective
theorem KaehlerDifferential.tensorProductTo_surjective :
    Function.Surjective (KaehlerDifferential.D R S).tensorProductTo 
Arguments: (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S]
Goal: Function.Surjective ⇑(KaehlerDifferential.D R S).tensorProductTo
-- KaehlerDifferential.linearMapEquivDerivation_symm_apply
@[simps! symm_apply apply_apply]
def KaehlerDifferential.linearMapEquivDerivation : (Ω[S⁄R] →ₗ[S] M) ≃ₗ[S] Derivation R S M 
Arguments: (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] {M : Type u_1} [AddCommGroup M] [Module R M] [Module S M] [IsScalarTower R S M] (D : Derivation R S M)
Goal: (KaehlerDifferential.linearMapEquivDerivation R S).symm D = D.liftKaehlerDifferential
-- KaehlerDifferential.linearMapEquivDerivation_apply_apply
@[simps! symm_apply apply_apply]
def KaehlerDifferential.linearMapEquivDerivation : (Ω[S⁄R] →ₗ[S] M) ≃ₗ[S] Derivation R S M 
Arguments: (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] {M : Type u_1} [AddCommGroup M] [Module R M] [Module S M] [IsScalarTower R S M] (m : Ω[S⁄R] →ₗ[S] M)
Goal: ∀ (a : S), ((KaehlerDifferential.linearMapEquivDerivation R S) m) a = m ((KaehlerDifferential.D R S) a)
-- KaehlerDifferential.End_equiv_aux
theorem KaehlerDifferential.End_equiv_aux (f : S →ₐ[R] S ⊗ S ⧸ KaehlerDifferential.ideal R S ^ 2) :
    (Ideal.Quotient.mkₐ R (KaehlerDifferential.ideal R S).cotangentIdeal).comp f =
        IsScalarTower.toAlgHom R S _ ↔
      (TensorProduct.lmul' R : S ⊗[R] S →ₐ[R] S).kerSquareLift.comp f = AlgHom.id R S 
Arguments: (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] (f : S →ₐ[R] TensorProduct R S S ⧸ KaehlerDifferential.ideal R S ^ 2)
Goal: (Ideal.Quotient.mkₐ R (KaehlerDifferential.ideal R S).cotangentIdeal).comp f =
    IsScalarTower.toAlgHom R S
      ((TensorProduct R S S ⧸ KaehlerDifferential.ideal R S ^ 2) ⧸ (KaehlerDifferential.ideal R S).cotangentIdeal) ↔
  (Algebra.TensorProduct.lmul' R).kerSquareLift.comp f = AlgHom.id R S
-- KaehlerDifferential.ideal_fg
theorem KaehlerDifferential.ideal_fg [EssFiniteType R S] :
    (KaehlerDifferential.ideal R S).FG 
Arguments: (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] [Algebra.EssFiniteType R S]
Goal: (KaehlerDifferential.ideal R S).FG
-- KaehlerDifferential.kerTotal_mkQ_single_add
theorem KaehlerDifferential.kerTotal_mkQ_single_add (x y z) : (z𝖣x + y) = (z𝖣x) + z𝖣y 
Arguments: (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] (x y z : S)
Goal: (KaehlerDifferential.kerTotal R S).mkQ (Finsupp.single (x + y) z) =
  (KaehlerDifferential.kerTotal R S).mkQ (Finsupp.single x z) +
    (KaehlerDifferential.kerTotal R S).mkQ (Finsupp.single y z)
-- KaehlerDifferential.kerTotal_mkQ_single_mul
theorem KaehlerDifferential.kerTotal_mkQ_single_mul (x y z) :
    (z𝖣x * y) = ((z * x)𝖣y) + (z * y)𝖣x 
Arguments: (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] (x y z : S)
Goal: (KaehlerDifferential.kerTotal R S).mkQ (Finsupp.single (x * y) z) =
  (KaehlerDifferential.kerTotal R S).mkQ (Finsupp.single y (z * x)) +
    (KaehlerDifferential.kerTotal R S).mkQ (Finsupp.single x (z * y))
-- KaehlerDifferential.kerTotal_mkQ_single_algebraMap
theorem KaehlerDifferential.kerTotal_mkQ_single_algebraMap (x y) : (y𝖣algebraMap R S x) = 0 
Arguments: (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] (x : R) (y : S)
Goal: (KaehlerDifferential.kerTotal R S).mkQ (Finsupp.single ((algebraMap R S) x) y) = 0
-- KaehlerDifferential.kerTotal_mkQ_single_algebraMap_one
theorem KaehlerDifferential.kerTotal_mkQ_single_algebraMap_one (x) : (x𝖣1) = 0 
Arguments: (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] (x : S)
Goal: (KaehlerDifferential.kerTotal R S).mkQ (Finsupp.single 1 x) = 0
-- KaehlerDifferential.kerTotal_mkQ_single_smul
theorem KaehlerDifferential.kerTotal_mkQ_single_smul (r : R) (x y) : (y𝖣r • x) = r • y𝖣x 
Arguments: (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] (r : R) (x y : S)
Goal: (KaehlerDifferential.kerTotal R S).mkQ (Finsupp.single (r • x) y) =
  r • (KaehlerDifferential.kerTotal R S).mkQ (Finsupp.single x y)
-- KaehlerDifferential.derivationQuotKerTotal_apply
theorem KaehlerDifferential.derivationQuotKerTotal_apply (x) :
    KaehlerDifferential.derivationQuotKerTotal R S x = 1𝖣x 
Arguments: (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] (x : S)
Goal: (KaehlerDifferential.derivationQuotKerTotal R S) x = (KaehlerDifferential.kerTotal R S).mkQ (Finsupp.single x 1)
-- KaehlerDifferential.derivationQuotKerTotal_lift_comp_total
theorem KaehlerDifferential.derivationQuotKerTotal_lift_comp_total :
    (KaehlerDifferential.derivationQuotKerTotal R S).liftKaehlerDifferential.comp
        (Finsupp.total S (Ω[S⁄R]) S (KaehlerDifferential.D R S)) =
      Submodule.mkQ _ 
Arguments: (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S]
Goal: (KaehlerDifferential.derivationQuotKerTotal R S).liftKaehlerDifferential ∘ₗ
    Finsupp.total S (Ω[S⁄R]) S ⇑(KaehlerDifferential.D R S) =
  (KaehlerDifferential.kerTotal R S).mkQ
-- KaehlerDifferential.kerTotal_eq
theorem KaehlerDifferential.kerTotal_eq :
    LinearMap.ker (Finsupp.total S (Ω[S⁄R]) S (KaehlerDifferential.D R S)) =
      KaehlerDifferential.kerTotal R S 
Arguments: (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S]
Goal: LinearMap.ker (Finsupp.total S (Ω[S⁄R]) S ⇑(KaehlerDifferential.D R S)) = KaehlerDifferential.kerTotal R S
-- KaehlerDifferential.total_surjective
theorem KaehlerDifferential.total_surjective :
    Function.Surjective (Finsupp.total S (Ω[S⁄R]) S (KaehlerDifferential.D R S)) 
Arguments: (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S]
Goal: Function.Surjective ⇑(Finsupp.total S (Ω[S⁄R]) S ⇑(KaehlerDifferential.D R S))
-- KaehlerDifferential.quotKerTotalEquiv_symm_apply
@[simps!]
noncomputable def KaehlerDifferential.quotKerTotalEquiv :
    ((S →₀ S) ⧸ KaehlerDifferential.kerTotal R S) ≃ₗ[S] Ω[S⁄R] 
Arguments: (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] (a : Ω[S⁄R])
Goal: (KaehlerDifferential.quotKerTotalEquiv R S).symm a =
  (KaehlerDifferential.derivationQuotKerTotal R S).liftKaehlerDifferential a
-- KaehlerDifferential.quotKerTotalEquiv_apply
@[simps!]
noncomputable def KaehlerDifferential.quotKerTotalEquiv :
    ((S →₀ S) ⧸ KaehlerDifferential.kerTotal R S) ≃ₗ[S] Ω[S⁄R] 
Arguments: (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S]
Goal: ∀ (a : (S →₀ S) ⧸ (KaehlerDifferential.kerTotal R S).toAddSubgroup),
  (KaehlerDifferential.quotKerTotalEquiv R S) a =
    (QuotientAddGroup.lift (KaehlerDifferential.kerTotal R S).toAddSubgroup
        (Finsupp.total S (Ω[S⁄R]) S ⇑(KaehlerDifferential.D R S)).toAddMonoidHom ⋯)
      a
-- KaehlerDifferential.quotKerTotalEquiv_symm_comp_D
theorem KaehlerDifferential.quotKerTotalEquiv_symm_comp_D :
    (KaehlerDifferential.quotKerTotalEquiv R S).symm.toLinearMap.compDer
        (KaehlerDifferential.D R S) =
      KaehlerDifferential.derivationQuotKerTotal R S 
Arguments: (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S]
Goal: (↑(KaehlerDifferential.quotKerTotalEquiv R S).symm).compDer (KaehlerDifferential.D R S) =
  KaehlerDifferential.derivationQuotKerTotal R S
-- KaehlerDifferential.kerTotal_map
theorem KaehlerDifferential.kerTotal_map (h : Function.Surjective (algebraMap A B)) :
    (KaehlerDifferential.kerTotal R A).map finsupp_map ⊔
        Submodule.span A (Set.range fun x : S => .single (algebraMap S B x) (1 : B)) =
      (KaehlerDifferential.kerTotal S B).restrictScalars _ 
Arguments: (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] (A : Type u_2) (B : Type u_3) [CommRing A] [CommRing B] [Algebra R A] [Algebra R B] [Algebra A B] [Algebra S B] [IsScalarTower R A B] [IsScalarTower R S B] (h : Function.Surjective ⇑(algebraMap A B))
Goal: Submodule.map (Finsupp.mapRange.linearMap (Algebra.linearMap A B) ∘ₗ Finsupp.lmapDomain A A ⇑(algebraMap A B))
      (KaehlerDifferential.kerTotal R A) ⊔
    Submodule.span A (Set.range fun x => Finsupp.single ((algebraMap S B) x) 1) =
  Submodule.restrictScalars A (KaehlerDifferential.kerTotal S B)
-- KaehlerDifferential.kerTotal_map'
theorem KaehlerDifferential.kerTotal_map' (h : Function.Surjective (algebraMap A B)) :
    (KaehlerDifferential.kerTotal R A ⊔
      Submodule.span A (Set.range fun x ↦ .single (algebraMap R A x) 1)).map finsupp_map =
      (KaehlerDifferential.kerTotal R B).restrictScalars _ 
Arguments: (R : Type u) [CommRing R] (A : Type u_2) (B : Type u_3) [CommRing A] [CommRing B] [Algebra R A] [Algebra R B] [Algebra A B] [IsScalarTower R A B] (h : Function.Surjective ⇑(algebraMap A B))
Goal: Submodule.map (Finsupp.mapRange.linearMap (Algebra.linearMap A B) ∘ₗ Finsupp.lmapDomain A A ⇑(algebraMap A B))
    (KaehlerDifferential.kerTotal R A ⊔ Submodule.span A (Set.range fun x => Finsupp.single ((algebraMap R A) x) 1)) =
  Submodule.restrictScalars A (KaehlerDifferential.kerTotal R B)
-- KaehlerDifferential.map_compDer
theorem KaehlerDifferential.map_compDer :
    (KaehlerDifferential.map R S A B).compDer (KaehlerDifferential.D R A) =
      ((KaehlerDifferential.D S B).restrictScalars R).compAlgebraMap A 
Arguments: (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] (A : Type u_2) (B : Type u_3) [CommRing A] [CommRing B] [Algebra R A] [Algebra R B] [Algebra A B] [Algebra S B] [IsScalarTower R A B] [IsScalarTower R S B] [SMulCommClass S A B]
Goal: (KaehlerDifferential.map R S A B).compDer (KaehlerDifferential.D R A) =
  Derivation.compAlgebraMap A (Derivation.restrictScalars R (KaehlerDifferential.D S B))