ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Commute.isNilpotent_mul_right
theorem isNilpotent_mul_right (h : IsNilpotent y) : IsNilpotent (x * y) 
Arguments: {R : Type u_1} {x y : R} [Semiring R] (h_comm : Commute x y) (h : IsNilpotent y)
Goal: IsNilpotent (x * y)
-- RingHom.ker_isRadical_iff_reduced_of_surjective
theorem RingHom.ker_isRadical_iff_reduced_of_surjective {S F} [CommSemiring R] [CommRing S]
    [FunLike F R S] [RingHomClass F R S] {f : F} (hf : Function.Surjective f) :
    (RingHom.ker f).IsRadical ↔ IsReduced S 
Arguments: {R : Type u_1} {S : Type u_3} {F : Type u_4} [CommSemiring R] [CommRing S] [FunLike F R S] [RingHomClass F R S] {f : F} (hf : Function.Surjective ⇑f)
Goal: (RingHom.ker f).IsRadical ↔ IsReduced S
-- isRadical_iff_span_singleton
theorem isRadical_iff_span_singleton [CommSemiring R] :
    IsRadical y ↔ (Ideal.span ({y} : Set R)).IsRadical 
Arguments: {R : Type u_1} {y : R} [CommSemiring R]
Goal: IsRadical y ↔ (Ideal.span {y}).IsRadical
-- mem_nilradical
theorem mem_nilradical : x ∈ nilradical R ↔ IsNilpotent x 
Arguments: {R : Type u_1} [CommSemiring R] {x : R}
Goal: x ∈ nilradical R ↔ IsNilpotent x
-- nilradical_eq_sInf
theorem nilradical_eq_sInf (R : Type*) [CommSemiring R] :
    nilradical R = sInf { J : Ideal R | J.IsPrime } 
Arguments: (R : Type u_3) [CommSemiring R]
Goal: nilradical R = sInf {J | J.IsPrime}
-- nilpotent_iff_mem_prime
theorem nilpotent_iff_mem_prime : IsNilpotent x ↔ ∀ J : Ideal R, J.IsPrime → x ∈ J 
Arguments: {R : Type u_1} [CommSemiring R] {x : R}
Goal: IsNilpotent x ↔ ∀ (J : Ideal R), J.IsPrime → x ∈ J
-- nilradical_le_prime
theorem nilradical_le_prime (J : Ideal R) [H : J.IsPrime] : nilradical R ≤ J 
Arguments: {R : Type u_1} [CommSemiring R] (J : Ideal R) [H : J.IsPrime]
Goal: nilradical R ≤ J
-- nilradical_eq_zero
@[simp]
theorem nilradical_eq_zero (R : Type*) [CommSemiring R] [IsReduced R] : nilradical R = 0 
Arguments: (R : Type u_3) [CommSemiring R] [IsReduced R]
Goal: nilradical R = 0
-- LinearMap.isNilpotent_mulLeft_iff
@[simp]
theorem isNilpotent_mulLeft_iff (a : A) : IsNilpotent (mulLeft R a) ↔ IsNilpotent a 
Arguments: (R : Type u_1) {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] (a : A)
Goal: IsNilpotent (LinearMap.mulLeft R a) ↔ IsNilpotent a
-- LinearMap.isNilpotent_mulRight_iff
@[simp]
theorem isNilpotent_mulRight_iff (a : A) : IsNilpotent (mulRight R a) ↔ IsNilpotent a 
Arguments: (R : Type u_1) {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] (a : A)
Goal: IsNilpotent (LinearMap.mulRight R a) ↔ IsNilpotent a
-- LinearMap.isNilpotent_toMatrix_iff
@[simp]
lemma isNilpotent_toMatrix_iff (b : Basis ι R M) (f : M →ₗ[R] M) :
    IsNilpotent (toMatrix b b f) ↔ IsNilpotent f 
Arguments: {R : Type u_1} [CommSemiring R] {ι : Type u_3} {M : Type u_4} [Fintype ι] [DecidableEq ι] [AddCommMonoid M] [Module R M] (b : Basis ι R M) (f : M →ₗ[R] M)
Goal: IsNilpotent ((LinearMap.toMatrix b b) f) ↔ IsNilpotent f
-- Module.End.isNilpotent.restrict
lemma isNilpotent.restrict {R M : Type*} [Semiring R] [AddCommMonoid M] [Module R M]
    {f : M →ₗ[R] M} {p : Submodule R M} (hf : MapsTo f p p) (hnil : IsNilpotent f) :
    IsNilpotent (f.restrict hf) 
Arguments: {R : Type u_3} {M : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] {f : M →ₗ[R] M} {p : Submodule R M} (hf : Set.MapsTo ⇑f ↑p ↑p) (hnil : IsNilpotent f)
Goal: IsNilpotent (f.restrict hf)
-- Module.End.IsNilpotent.mapQ
theorem IsNilpotent.mapQ (hnp : IsNilpotent f) : IsNilpotent (p.mapQ p f hp) 
Arguments: {R : Type u_1} {M : Type v} [Ring R] [AddCommGroup M] [Module R M] {f : Module.End R M} {p : Submodule R M} (hp : p ≤ Submodule.comap f p) (hnp : IsNilpotent f)
Goal: IsNilpotent (p.mapQ p f hp)
-- gcd_ne_zero_of_left
theorem gcd_ne_zero_of_left (hp : p ≠ 0) : GCDMonoid.gcd p q ≠ 0 
Arguments: {R : Type u_1} [EuclideanDomain R] [GCDMonoid R] {p q : R} (hp : p ≠ 0)
Goal: gcd p q ≠ 0
-- gcd_ne_zero_of_right
theorem gcd_ne_zero_of_right (hp : q ≠ 0) : GCDMonoid.gcd p q ≠ 0 
Arguments: {R : Type u_1} [EuclideanDomain R] [GCDMonoid R] {p q : R} (hp : q ≠ 0)
Goal: gcd p q ≠ 0
-- left_div_gcd_ne_zero
theorem left_div_gcd_ne_zero {p q : R} (hp : p ≠ 0) : p / GCDMonoid.gcd p q ≠ 0 
Arguments: {R : Type u_1} [EuclideanDomain R] [GCDMonoid R] {p q : R} (hp : p ≠ 0)
Goal: p / gcd p q ≠ 0
-- right_div_gcd_ne_zero
theorem right_div_gcd_ne_zero {p q : R} (hq : q ≠ 0) : q / GCDMonoid.gcd p q ≠ 0 
Arguments: {R : Type u_1} [EuclideanDomain R] [GCDMonoid R] {p q : R} (hq : q ≠ 0)
Goal: q / gcd p q ≠ 0
-- isCoprime_div_gcd_div_gcd
theorem isCoprime_div_gcd_div_gcd (hq : q ≠ 0) :
    IsCoprime (p / GCDMonoid.gcd p q) (q / GCDMonoid.gcd p q) 
Arguments: {R : Type u_1} [EuclideanDomain R] [GCDMonoid R] {p q : R} (hq : q ≠ 0)
Goal: IsCoprime (p / gcd p q) (q / gcd p q)
-- EuclideanDomain.span_gcd
theorem span_gcd [DecidableEq α] (x y : α) :
    span ({gcd x y} : Set α) = span ({x, y} : Set α) 
Arguments: {α : Type u_1} [EuclideanDomain α] [DecidableEq α] (x y : α)
Goal: Ideal.span {EuclideanDomain.gcd x y} = Ideal.span {x, y}
-- EuclideanDomain.gcd_isUnit_iff
theorem gcd_isUnit_iff [DecidableEq α] {x y : α} : IsUnit (gcd x y) ↔ IsCoprime x y 
Arguments: {α : Type u_1} [EuclideanDomain α] [DecidableEq α] {x y : α}
Goal: IsUnit (EuclideanDomain.gcd x y) ↔ IsCoprime x y
-- EuclideanDomain.isCoprime_of_dvd
theorem isCoprime_of_dvd {x y : α} (nonzero : ¬(x = 0 ∧ y = 0))
    (H : ∀ z ∈ nonunits α, z ≠ 0 → z ∣ x → ¬z ∣ y) : IsCoprime x y 
Arguments: {α : Type u_1} [EuclideanDomain α] {x y : α} (nonzero : ¬(x = 0 ∧ y = 0)) (H : ∀ z ∈ nonunits α, z ≠ 0 → z ∣ x → ¬z ∣ y)
Goal: IsCoprime x y
-- EuclideanDomain.dvd_or_coprime
theorem dvd_or_coprime (x y : α) (h : Irreducible x) :
    x ∣ y ∨ IsCoprime x y 
Arguments: {α : Type u_1} [EuclideanDomain α] (x y : α) (h : Irreducible x)
Goal: x ∣ y ∨ IsCoprime x y
-- IsNilpotent.neg
theorem IsNilpotent.neg [Ring R] (h : IsNilpotent x) : IsNilpotent (-x) 
Arguments: {R : Type u_1} {x : R} [Ring R] (h : IsNilpotent x)
Goal: IsNilpotent (-x)
-- isNilpotent_neg_iff
@[simp]
theorem isNilpotent_neg_iff [Ring R] : IsNilpotent (-x) ↔ IsNilpotent x 
Arguments: {R : Type u_1} {x : R} [Ring R]
Goal: IsNilpotent (-x) ↔ IsNilpotent x
-- IsNilpotent.smul
lemma IsNilpotent.smul [MonoidWithZero R] [MonoidWithZero S] [MulActionWithZero R S]
    [SMulCommClass R S S] [IsScalarTower R S S] {a : S} (ha : IsNilpotent a) (t : R) :
    IsNilpotent (t • a) 
Arguments: {R : Type u_1} {S : Type u_2} [MonoidWithZero R] [MonoidWithZero S] [MulActionWithZero R S] [SMulCommClass R S S] [IsScalarTower R S S] {a : S} (ha : IsNilpotent a) (t : R)
Goal: IsNilpotent (t • a)
-- IsNilpotent.isUnit_sub_one
theorem IsNilpotent.isUnit_sub_one [Ring R] {r : R} (hnil : IsNilpotent r) : IsUnit (r - 1) 
Arguments: {R : Type u_1} [Ring R] {r : R} (hnil : IsNilpotent r)
Goal: IsUnit (r - 1)
-- IsNilpotent.isUnit_one_sub
theorem IsNilpotent.isUnit_one_sub [Ring R] {r : R} (hnil : IsNilpotent r) : IsUnit (1 - r) 
Arguments: {R : Type u_1} [Ring R] {r : R} (hnil : IsNilpotent r)
Goal: IsUnit (1 - r)
-- IsNilpotent.isUnit_add_one
theorem IsNilpotent.isUnit_add_one [Ring R] {r : R} (hnil : IsNilpotent r) : IsUnit (r + 1) 
Arguments: {R : Type u_1} [Ring R] {r : R} (hnil : IsNilpotent r)
Goal: IsUnit (r + 1)
-- IsNilpotent.isUnit_one_add
theorem IsNilpotent.isUnit_one_add [Ring R] {r : R} (hnil : IsNilpotent r) : IsUnit (1 + r) 
Arguments: {R : Type u_1} [Ring R] {r : R} (hnil : IsNilpotent r)
Goal: IsUnit (1 + r)
-- IsNilpotent.isUnit_add_left_of_commute
theorem IsNilpotent.isUnit_add_left_of_commute [Ring R] {r u : R}
    (hnil : IsNilpotent r) (hu : IsUnit u) (h_comm : Commute r u) :
    IsUnit (u + r) 
Arguments: {R : Type u_1} [Ring R] {r u : R} (hnil : IsNilpotent r) (hu : IsUnit u) (h_comm : Commute r u)
Goal: IsUnit (u + r)
-- IsNilpotent.isUnit_add_right_of_commute
theorem IsNilpotent.isUnit_add_right_of_commute [Ring R] {r u : R}
    (hnil : IsNilpotent r) (hu : IsUnit u) (h_comm : Commute r u) :
    IsUnit (r + u) 
Arguments: {R : Type u_1} [Ring R] {r u : R} (hnil : IsNilpotent r) (hu : IsUnit u) (h_comm : Commute r u)
Goal: IsUnit (r + u)
-- Prime.isRadical
theorem Prime.isRadical [CommMonoidWithZero R] {y : R} (hy : Prime y) : IsRadical y 
Arguments: {R : Type u_1} [CommMonoidWithZero R] {y : R} (hy : Prime y)
Goal: IsRadical y
-- zero_isRadical_iff
theorem zero_isRadical_iff [MonoidWithZero R] : IsRadical (0 : R) ↔ IsReduced R 
Arguments: {R : Type u_1} [MonoidWithZero R]
Goal: IsRadical 0 ↔ IsReduced R
-- isReduced_iff_pow_one_lt
theorem isReduced_iff_pow_one_lt [MonoidWithZero R] (k : ℕ) (hk : 1 < k) :
    IsReduced R ↔ ∀ x : R, x ^ k = 0 → x = 0 
Arguments: {R : Type u_1} [MonoidWithZero R] (k : ℕ) (hk : 1 < k)
Goal: IsReduced R ↔ ∀ (x : R), x ^ k = 0 → x = 0
-- IsRadical.of_dvd
theorem IsRadical.of_dvd [CancelCommMonoidWithZero R] {x y : R} (hy : IsRadical y) (h0 : y ≠ 0)
    (hxy : x ∣ y) : IsRadical x 
Arguments: {R : Type u_1} [CancelCommMonoidWithZero R] {x y : R} (hy : IsRadical y) (h0 : y ≠ 0) (hxy : x ∣ y)
Goal: IsRadical x
-- Commute.add_pow_eq_zero_of_add_le_succ_of_pow_eq_zero
theorem add_pow_eq_zero_of_add_le_succ_of_pow_eq_zero (h_comm : Commute x y) {m n k : ℕ}
    (hx : x ^ m = 0) (hy : y ^ n = 0) (h : m + n ≤ k + 1) :
    (x + y) ^ k = 0 
Arguments: {R : Type u_1} {x y : R} [Semiring R] (h_comm : Commute x y) {m n k : ℕ} (hx : x ^ m = 0) (hy : y ^ n = 0) (h : m + n ≤ k + 1)
Goal: (x + y) ^ k = 0
-- Commute.add_pow_add_eq_zero_of_pow_eq_zero
theorem add_pow_add_eq_zero_of_pow_eq_zero (h_comm : Commute x y) {m n : ℕ}
    (hx : x ^ m = 0) (hy : y ^ n = 0) :
    (x + y) ^ (m + n - 1) = 0 
Arguments: {R : Type u_1} {x y : R} [Semiring R] (h_comm : Commute x y) {m n : ℕ} (hx : x ^ m = 0) (hy : y ^ n = 0)
Goal: (x + y) ^ (m + n - 1) = 0
-- Commute.isNilpotent_add
theorem isNilpotent_add (h_comm : Commute x y) (hx : IsNilpotent x) (hy : IsNilpotent y) :
    IsNilpotent (x + y) 
Arguments: {R : Type u_1} {x y : R} [Semiring R] (h_comm : Commute x y) (hx : IsNilpotent x) (hy : IsNilpotent y)
Goal: IsNilpotent (x + y)
-- Commute.isNilpotent_sum
protected lemma isNilpotent_sum {ι : Type*} {s : Finset ι} {f : ι → R}
    (hnp : ∀ i ∈ s, IsNilpotent (f i)) (h_comm : ∀ i j, i ∈ s → j ∈ s → Commute (f i) (f j)) :
    IsNilpotent (∑ i ∈ s, f i) 
Arguments: {R : Type u_1} [Semiring R] {ι : Type u_3} {s : Finset ι} {f : ι → R} (hnp : ∀ i ∈ s, IsNilpotent (f i)) (h_comm : ∀ (i j : ι), i ∈ s → j ∈ s → Commute (f i) (f j))
Goal: IsNilpotent (∑ i ∈ s, f i)
-- Commute.isNilpotent_mul_left_iff
protected lemma isNilpotent_mul_left_iff (h_comm : Commute x y) (hy : y ∈ nonZeroDivisorsLeft R) :
    IsNilpotent (x * y) ↔ IsNilpotent x 
Arguments: {R : Type u_1} {x y : R} [Semiring R] (h_comm : Commute x y) (hy : y ∈ nonZeroDivisorsLeft R)
Goal: IsNilpotent (x * y) ↔ IsNilpotent x
-- Commute.isNilpotent_mul_right_iff
protected lemma isNilpotent_mul_right_iff (h_comm : Commute x y) (hx : x ∈ nonZeroDivisorsRight R) :
    IsNilpotent (x * y) ↔ IsNilpotent y 
Arguments: {R : Type u_1} {x y : R} [Semiring R] (h_comm : Commute x y) (hx : x ∈ nonZeroDivisorsRight R)
Goal: IsNilpotent (x * y) ↔ IsNilpotent y
-- Commute.isNilpotent_sub
theorem isNilpotent_sub (h_comm : Commute x y) (hx : IsNilpotent x) (hy : IsNilpotent y) :
    IsNilpotent (x - y) 
Arguments: {R : Type u_1} {x y : R} [Ring R] (h_comm : Commute x y) (hx : IsNilpotent x) (hy : IsNilpotent y)
Goal: IsNilpotent (x - y)
-- isNilpotent_sum
lemma isNilpotent_sum {ι : Type*} {s : Finset ι} {f : ι → R}
    (hnp : ∀ i ∈ s, IsNilpotent (f i)) :
    IsNilpotent (∑ i ∈ s, f i) 
Arguments: {R : Type u_1} [CommSemiring R] {ι : Type u_3} {s : Finset ι} {f : ι → R} (hnp : ∀ i ∈ s, IsNilpotent (f i))
Goal: IsNilpotent (∑ i ∈ s, f i)
-- NoZeroSMulDivisors.isReduced
lemma NoZeroSMulDivisors.isReduced (R M : Type*)
    [MonoidWithZero R] [Zero M] [MulActionWithZero R M] [Nontrivial M] [NoZeroSMulDivisors R M] :
    IsReduced R 
Arguments: (R : Type u_3) (M : Type u_4) [MonoidWithZero R] [Zero M] [MulActionWithZero R M] [Nontrivial M] [NoZeroSMulDivisors R M]
Goal: IsReduced R
-- Derivation.leibniz'
structure Derivation (R : Type*) (A : Type*) (M : Type*)
    [CommSemiring R] [CommSemiring A] [AddCommMonoid M] [Algebra R A] [Module A M] [Module R M]
    extends A →ₗ[R] M where
  protected map_one_eq_zero' : toLinearMap 1 = 0
  protected leibniz' (a b : A) : toLinearMap (a * b) = a • toLinearMap b + b • toLinearMap a
Arguments: {R : Type u_1} {A : Type u_2} {M : Type u_3} [CommSemiring R] [CommSemiring A] [AddCommMonoid M] [Algebra R A] [Module A M] [Module R M] (self : Derivation R A M) (a b : A)
Goal: ↑self (a * b) = a • ↑self b + b • ↑self a
-- Derivation.map_one_eq_zero'
structure Derivation (R : Type*) (A : Type*) (M : Type*)
    [CommSemiring R] [CommSemiring A] [AddCommMonoid M] [Algebra R A] [Module A M] [Module R M]
    extends A →ₗ[R] M where
  protected map_one_eq_zero' : toLinearMap 1 = 0
  protected leibniz' (a b : A) : toLinearMap (a * b) = a • toLinearMap b + b • toLinearMap a
Arguments: {R : Type u_1} {A : Type u_2} {M : Type u_3} [CommSemiring R] [CommSemiring A] [AddCommMonoid M] [Algebra R A] [Module A M] [Module R M] (self : Derivation R A M)
Goal: ↑self 1 = 0
-- Derivation.toFun_eq_coe
theorem toFun_eq_coe : D.toFun = ⇑D 
Arguments: {R : Type u_1} {A : Type u_2} {M : Type u_4} [CommSemiring R] [CommSemiring A] [AddCommMonoid M] [Algebra R A] [Module A M] [Module R M] (D : Derivation R A M)
Goal: (↑D).toFun = ⇑D
-- Derivation.mk_coe
@[simp]
theorem mk_coe (f : A →ₗ[R] M) (h₁ h₂) : ((⟨f, h₁, h₂⟩ : Derivation R A M) : A → M) = f 
Arguments: {R : Type u_1} {A : Type u_2} {M : Type u_4} [CommSemiring R] [CommSemiring A] [AddCommMonoid M] [Algebra R A] [Module A M] [Module R M] (f : A →ₗ[R] M) (h₁ : f 1 = 0) (h₂ : ∀ (a b : A), f (a * b) = a • f b + b • f a)
Goal: ⇑{ toLinearMap := f, map_one_eq_zero' := h₁, leibniz' := h₂ } = ⇑f
-- Derivation.coeFn_coe
@[simp, norm_cast]
theorem coeFn_coe (f : Derivation R A M) : ⇑(f : A →ₗ[R] M) = f 
Arguments: {R : Type u_1} {A : Type u_2} {M : Type u_4} [CommSemiring R] [CommSemiring A] [AddCommMonoid M] [Algebra R A] [Module A M] [Module R M] (f : Derivation R A M)
Goal: ⇑↑f = ⇑f
-- Derivation.coe_injective
theorem coe_injective : @Function.Injective (Derivation R A M) (A → M) DFunLike.coe 
Arguments: {R : Type u_1} {A : Type u_2} {M : Type u_4} [CommSemiring R] [CommSemiring A] [AddCommMonoid M] [Algebra R A] [Module A M] [Module R M]
Goal: Function.Injective DFunLike.coe
-- Derivation.ext
@[ext]
theorem ext (H : ∀ a, D1 a = D2 a) : D1 = D2 
Arguments: {R : Type u_1} {A : Type u_2} {M : Type u_4} [CommSemiring R] [CommSemiring A] [AddCommMonoid M] [Algebra R A] [Module A M] [Module R M] {D1 D2 : Derivation R A M} (H : ∀ (a : A), D1 a = D2 a)
Goal: D1 = D2
-- Derivation.congr_fun
theorem congr_fun (h : D1 = D2) (a : A) : D1 a = D2 a 
Arguments: {R : Type u_1} {A : Type u_2} {M : Type u_4} [CommSemiring R] [CommSemiring A] [AddCommMonoid M] [Algebra R A] [Module A M] [Module R M] {D1 D2 : Derivation R A M} (h : D1 = D2) (a : A)
Goal: D1 a = D2 a
-- Derivation.map_add
protected theorem map_add : D (a + b) = D a + D b 
Arguments: {R : Type u_1} {A : Type u_2} {M : Type u_4} [CommSemiring R] [CommSemiring A] [AddCommMonoid M] [Algebra R A] [Module A M] [Module R M] (D : Derivation R A M) (a b : A)
Goal: D (a + b) = D a + D b
-- Derivation.map_zero
protected theorem map_zero : D 0 = 0 
Arguments: {R : Type u_1} {A : Type u_2} {M : Type u_4} [CommSemiring R] [CommSemiring A] [AddCommMonoid M] [Algebra R A] [Module A M] [Module R M] (D : Derivation R A M)
Goal: D 0 = 0
-- Derivation.map_smul
@[simp]
theorem map_smul : D (r • a) = r • D a 
Arguments: {R : Type u_1} {A : Type u_2} {M : Type u_4} [CommSemiring R] [CommSemiring A] [AddCommMonoid M] [Algebra R A] [Module A M] [Module R M] (D : Derivation R A M) (r : R) (a : A)
Goal: D (r • a) = r • D a
-- Derivation.leibniz
@[simp]
theorem leibniz : D (a * b) = a • D b + b • D a 
Arguments: {R : Type u_1} {A : Type u_2} {M : Type u_4} [CommSemiring R] [CommSemiring A] [AddCommMonoid M] [Algebra R A] [Module A M] [Module R M] (D : Derivation R A M) (a b : A)
Goal: D (a * b) = a • D b + b • D a
-- Derivation.map_smul_of_tower
@[simp]
theorem map_smul_of_tower {S : Type*} [SMul S A] [SMul S M] [LinearMap.CompatibleSMul A M S R]
    (D : Derivation R A M) (r : S) (a : A) : D (r • a) = r • D a 
Arguments: {R : Type u_1} {A : Type u_2} {M : Type u_4} [CommSemiring R] [CommSemiring A] [AddCommMonoid M] [Algebra R A] [Module A M] [Module R M] {S : Type u_5} [SMul S A] [SMul S M] [LinearMap.CompatibleSMul A M S R] (D : Derivation R A M) (r : S) (a : A)
Goal: D (r • a) = r • D a
-- Derivation.map_one_eq_zero
@[simp]
theorem map_one_eq_zero : D 1 = 0 
Arguments: {R : Type u_1} {A : Type u_2} {M : Type u_4} [CommSemiring R] [CommSemiring A] [AddCommMonoid M] [Algebra R A] [Module A M] [Module R M] (D : Derivation R A M)
Goal: D 1 = 0
-- Derivation.map_algebraMap
@[simp]
theorem map_algebraMap : D (algebraMap R A r) = 0 
Arguments: {R : Type u_1} {A : Type u_2} {M : Type u_4} [CommSemiring R] [CommSemiring A] [AddCommMonoid M] [Algebra R A] [Module A M] [Module R M] (D : Derivation R A M) (r : R)
Goal: D ((algebraMap R A) r) = 0
-- Derivation.map_natCast
@[simp]
theorem map_natCast (n : ℕ) : D (n : A) = 0 
Arguments: {R : Type u_1} {A : Type u_2} {M : Type u_4} [CommSemiring R] [CommSemiring A] [AddCommMonoid M] [Algebra R A] [Module A M] [Module R M] (D : Derivation R A M) (n : ℕ)
Goal: D ↑n = 0
-- Derivation.leibniz_pow
@[simp]
theorem leibniz_pow (n : ℕ) : D (a ^ n) = n • a ^ (n - 1) • D a 
Arguments: {R : Type u_1} {A : Type u_2} {M : Type u_4} [CommSemiring R] [CommSemiring A] [AddCommMonoid M] [Algebra R A] [Module A M] [Module R M] (D : Derivation R A M) (a : A) (n : ℕ)
Goal: D (a ^ n) = n • a ^ (n - 1) • D a
-- Derivation.map_aeval
@[simp]
theorem map_aeval (P : R[X]) (x : A) :
    D (aeval x P) = aeval x (derivative P) • D x 
Arguments: {R : Type u_1} {A : Type u_2} {M : Type u_4} [CommSemiring R] [CommSemiring A] [AddCommMonoid M] [Algebra R A] [Module A M] [Module R M] (D : Derivation R A M) (P : Polynomial R) (x : A)
Goal: D ((Polynomial.aeval x) P) = (Polynomial.aeval x) (Polynomial.derivative P) • D x
-- Derivation.eqOn_adjoin
theorem eqOn_adjoin {s : Set A} (h : Set.EqOn D1 D2 s) : Set.EqOn D1 D2 (adjoin R s) 
Arguments: {R : Type u_1} {A : Type u_2} {M : Type u_4} [CommSemiring R] [CommSemiring A] [AddCommMonoid M] [Algebra R A] [Module A M] [Module R M] {D1 D2 : Derivation R A M} {s : Set A} (h : Set.EqOn (⇑D1) (⇑D2) s)
Goal: Set.EqOn ⇑D1 ⇑D2 ↑(Algebra.adjoin R s)
-- Derivation.ext_of_adjoin_eq_top
theorem ext_of_adjoin_eq_top (s : Set A) (hs : adjoin R s = ⊤) (h : Set.EqOn D1 D2 s) : D1 = D2 
Arguments: {R : Type u_1} {A : Type u_2} {M : Type u_4} [CommSemiring R] [CommSemiring A] [AddCommMonoid M] [Algebra R A] [Module A M] [Module R M] {D1 D2 : Derivation R A M} (s : Set A) (hs : Algebra.adjoin R s = ⊤) (h : Set.EqOn (⇑D1) (⇑D2) s)
Goal: D1 = D2
-- Derivation.coe_zero
@[simp]
theorem coe_zero : ⇑(0 : Derivation R A M) = 0 
Arguments: {R : Type u_1} {A : Type u_2} {M : Type u_4} [CommSemiring R] [CommSemiring A] [AddCommMonoid M] [Algebra R A] [Module A M] [Module R M]
Goal: ⇑0 = 0
-- Derivation.coe_zero_linearMap
@[simp]
theorem coe_zero_linearMap : ↑(0 : Derivation R A M) = (0 : A →ₗ[R] M) 
Arguments: {R : Type u_1} {A : Type u_2} {M : Type u_4} [CommSemiring R] [CommSemiring A] [AddCommMonoid M] [Algebra R A] [Module A M] [Module R M]
Goal: ↑0 = 0
-- Derivation.zero_apply
theorem zero_apply (a : A) : (0 : Derivation R A M) a = 0 
Arguments: {R : Type u_1} {A : Type u_2} {M : Type u_4} [CommSemiring R] [CommSemiring A] [AddCommMonoid M] [Algebra R A] [Module A M] [Module R M] (a : A)
Goal: 0 a = 0
-- Derivation.coe_add
@[simp]
theorem coe_add (D1 D2 : Derivation R A M) : ⇑(D1 + D2) = D1 + D2 
Arguments: {R : Type u_1} {A : Type u_2} {M : Type u_4} [CommSemiring R] [CommSemiring A] [AddCommMonoid M] [Algebra R A] [Module A M] [Module R M] (D1 D2 : Derivation R A M)
Goal: ⇑(D1 + D2) = ⇑D1 + ⇑D2
-- Derivation.coe_add_linearMap
@[simp]
theorem coe_add_linearMap (D1 D2 : Derivation R A M) : ↑(D1 + D2) = (D1 + D2 : A →ₗ[R] M) 
Arguments: {R : Type u_1} {A : Type u_2} {M : Type u_4} [CommSemiring R] [CommSemiring A] [AddCommMonoid M] [Algebra R A] [Module A M] [Module R M] (D1 D2 : Derivation R A M)
Goal: ↑(D1 + D2) = ↑D1 + ↑D2
-- Derivation.add_apply
theorem add_apply : (D1 + D2) a = D1 a + D2 a 
Arguments: {R : Type u_1} {A : Type u_2} {M : Type u_4} [CommSemiring R] [CommSemiring A] [AddCommMonoid M] [Algebra R A] [Module A M] [Module R M] {D1 D2 : Derivation R A M} (a : A)
Goal: (D1 + D2) a = D1 a + D2 a
-- Derivation.coe_smul
@[simp]
theorem coe_smul (r : S) (D : Derivation R A M) : ⇑(r • D) = r • ⇑D 
Arguments: {R : Type u_1} {A : Type u_2} {M : Type u_4} [CommSemiring R] [CommSemiring A] [AddCommMonoid M] [Algebra R A] [Module A M] [Module R M] {S : Type u_5} [Monoid S] [DistribMulAction S M] [SMulCommClass R S M] [SMulCommClass S A M] (r : S) (D : Derivation R A M)
Goal: ⇑(r • D) = r • ⇑D
-- Derivation.coe_smul_linearMap
@[simp]
theorem coe_smul_linearMap (r : S) (D : Derivation R A M) : ↑(r • D) = r • (D : A →ₗ[R] M) 
Arguments: {R : Type u_1} {A : Type u_2} {M : Type u_4} [CommSemiring R] [CommSemiring A] [AddCommMonoid M] [Algebra R A] [Module A M] [Module R M] {S : Type u_5} [Monoid S] [DistribMulAction S M] [SMulCommClass R S M] [SMulCommClass S A M] (r : S) (D : Derivation R A M)
Goal: ↑(r • D) = r • ↑D
-- Derivation.smul_apply
theorem smul_apply (r : S) (D : Derivation R A M) : (r • D) a = r • D a 
Arguments: {R : Type u_1} {A : Type u_2} {M : Type u_4} [CommSemiring R] [CommSemiring A] [AddCommMonoid M] [Algebra R A] [Module A M] [Module R M] (a : A) {S : Type u_5} [Monoid S] [DistribMulAction S M] [SMulCommClass R S M] [SMulCommClass S A M] (r : S) (D : Derivation R A M)
Goal: (r • D) a = r • D a
-- Derivation.coe_to_linearMap_comp
@[simp]
theorem coe_to_linearMap_comp : (f.compDer D : A →ₗ[R] N) = (f : M →ₗ[R] N).comp (D : A →ₗ[R] M) 
Arguments: {R : Type u_1} {A : Type u_2} {M : Type u_4} [CommSemiring R] [CommSemiring A] [AddCommMonoid M] [Algebra R A] [Module A M] [Module R M] (D : Derivation R A M) {N : Type u_5} [AddCommMonoid N] [Module A N] [Module R N] [IsScalarTower R A M] [IsScalarTower R A N] (f : M →ₗ[A] N)
Goal: ↑(f.compDer D) = ↑R f ∘ₗ ↑D
-- Derivation.coe_comp
@[simp]
theorem coe_comp : (f.compDer D : A → N) = (f : M →ₗ[R] N).comp (D : A →ₗ[R] M) 
Arguments: {R : Type u_1} {A : Type u_2} {M : Type u_4} [CommSemiring R] [CommSemiring A] [AddCommMonoid M] [Algebra R A] [Module A M] [Module R M] (D : Derivation R A M) {N : Type u_5} [AddCommMonoid N] [Module A N] [Module R N] [IsScalarTower R A M] [IsScalarTower R A N] (f : M →ₗ[A] N)
Goal: ⇑(f.compDer D) = ⇑(↑R f ∘ₗ ↑D)
-- Derivation.llcomp_apply
@[simps]
def llcomp : (M →ₗ[A] N) →ₗ[A] Derivation R A M →ₗ[R] Derivation R A N 
Arguments: {R : Type u_1} {A : Type u_2} {M : Type u_4} [CommSemiring R] [CommSemiring A] [AddCommMonoid M] [Algebra R A] [Module A M] [Module R M] {N : Type u_5} [AddCommMonoid N] [Module A N] [Module R N] [IsScalarTower R A M] [IsScalarTower R A N] (f : M →ₗ[A] N)
Goal: Derivation.llcomp f = f.compDer
-- Derivation.linearEquiv_coe_to_linearMap_comp
@[simp]
theorem linearEquiv_coe_to_linearMap_comp :
    (e.compDer D : A →ₗ[R] N) = (e.toLinearMap : M →ₗ[R] N).comp (D : A →ₗ[R] M) 
Arguments: {R : Type u_1} {A : Type u_2} {M : Type u_4} [CommSemiring R] [CommSemiring A] [AddCommMonoid M] [Algebra R A] [Module A M] [Module R M] (D : Derivation R A M) {N : Type u_5} [AddCommMonoid N] [Module A N] [Module R N] [IsScalarTower R A M] [IsScalarTower R A N] (e : M ≃ₗ[A] N)
Goal: ↑(e.compDer D) = ↑R ↑e ∘ₗ ↑D
-- Derivation.linearEquiv_coe_comp
@[simp]
theorem linearEquiv_coe_comp :
    (e.compDer D : A → N) = (e.toLinearMap : M →ₗ[R] N).comp (D : A →ₗ[R] M) 
Arguments: {R : Type u_1} {A : Type u_2} {M : Type u_4} [CommSemiring R] [CommSemiring A] [AddCommMonoid M] [Algebra R A] [Module A M] [Module R M] (D : Derivation R A M) {N : Type u_5} [AddCommMonoid N] [Module A N] [Module R N] [IsScalarTower R A M] [IsScalarTower R A N] (e : M ≃ₗ[A] N)
Goal: ⇑(e.compDer D) = ⇑(↑R ↑e ∘ₗ ↑D)