Documentation

Mathlib.RingTheory.Derivation.Basic

Derivations #

This file defines derivation. A derivation D from the R-algebra A to the A-module M is an R-linear map that satisfy the Leibniz rule D (a * b) = a * D b + D a * b.

Main results #

See RingTheory.Derivation.Lie for

and RingTheory.Derivation.ToSquareZero for

Future project #

structure Derivation (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] extends LinearMap :
Type (max u_2 u_3)

D : Derivation R A M is an R-linear map from A to M that satisfies the leibniz equality. We also require that D 1 = 0. See Derivation.mk' for a constructor that deduces this assumption from the Leibniz rule when M is cancellative.

TODO: update this when bimodules are defined.

  • toFun : AM
  • map_add' : ∀ (x y : A), (↑self).toFun (x + y) = (↑self).toFun x + (↑self).toFun y
  • map_smul' : ∀ (m : R) (x : A), (↑self).toFun (m x) = (RingHom.id R) m (↑self).toFun x
  • map_one_eq_zero' : self 1 = 0
  • leibniz' : ∀ (a b : A), self (a * b) = a self b + b self a
Instances For
    theorem Derivation.map_one_eq_zero' {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) :
    self 1 = 0
    theorem Derivation.leibniz' {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) :
    self (a * b) = a self b + b self a
    instance Derivation.instFunLike {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] :
    FunLike (Derivation R A M) A M
    Equations
    • Derivation.instFunLike = { coe := fun (D : Derivation R A M) => (↑D).toFun, coe_injective' := }
    instance Derivation.instAddMonoidHomClass {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] :
    Equations
    • =
    theorem Derivation.toFun_eq_coe {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) :
    (↑D).toFun = D
    def Derivation.Simps.apply {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) :
    AM

    See Note [custom simps projection]

    Equations
    Instances For
      instance Derivation.hasCoeToLinearMap {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] :
      Coe (Derivation R A M) (A →ₗ[R] M)
      Equations
      • Derivation.hasCoeToLinearMap = { coe := fun (D : Derivation R A M) => D }
      @[simp]
      theorem Derivation.mk_coe {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) :
      { toLinearMap := f, map_one_eq_zero' := h₁, leibniz' := h₂ } = f
      @[simp]
      theorem Derivation.coeFn_coe {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) :
      f = f
      theorem Derivation.coe_injective {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] :
      Function.Injective DFunLike.coe
      theorem Derivation.ext {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) :
      D1 = D2
      theorem Derivation.congr_fun {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) :
      D1 a = D2 a
      theorem Derivation.map_add {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) :
      D (a + b) = D a + D b
      theorem Derivation.map_zero {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) :
      D 0 = 0
      @[simp]
      theorem Derivation.map_smul {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) :
      D (r a) = r D a
      @[simp]
      theorem Derivation.leibniz {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) :
      D (a * b) = a D b + b D a
      @[simp]
      theorem Derivation.map_smul_of_tower {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) :
      D (r a) = r D a
      @[simp]
      theorem Derivation.map_one_eq_zero {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) :
      D 1 = 0
      @[simp]
      theorem Derivation.map_algebraMap {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) :
      D ((algebraMap R A) r) = 0
      @[simp]
      theorem Derivation.map_natCast {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 : ) :
      D n = 0
      @[simp]
      theorem Derivation.leibniz_pow {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 : ) :
      D (a ^ n) = n a ^ (n - 1) D a
      @[simp]
      theorem Derivation.map_aeval {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) :
      D ((Polynomial.aeval x) P) = (Polynomial.aeval x) (Polynomial.derivative P) D x
      theorem Derivation.eqOn_adjoin {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) :
      Set.EqOn D1 D2 (Algebra.adjoin R s)
      theorem Derivation.ext_of_adjoin_eq_top {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) :
      D1 = D2

      If adjoin of a set is the whole algebra, then any two derivations equal on this set are equal on the whole algebra.

      instance Derivation.instZero {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] :
      Equations
      • Derivation.instZero = { zero := { toLinearMap := 0, map_one_eq_zero' := , leibniz' := } }
      @[simp]
      theorem Derivation.coe_zero {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] :
      0 = 0
      @[simp]
      theorem Derivation.coe_zero_linearMap {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] :
      0 = 0
      theorem Derivation.zero_apply {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) :
      0 a = 0
      instance Derivation.instAdd {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] :
      Add (Derivation R A M)
      Equations
      • Derivation.instAdd = { add := fun (D1 D2 : Derivation R A M) => { toLinearMap := D1 + D2, map_one_eq_zero' := , leibniz' := } }
      @[simp]
      theorem Derivation.coe_add {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) :
      (D1 + D2) = D1 + D2
      @[simp]
      theorem Derivation.coe_add_linearMap {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) :
      (D1 + D2) = D1 + D2
      theorem Derivation.add_apply {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) :
      (D1 + D2) a = D1 a + D2 a
      instance Derivation.instInhabited {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] :
      Equations
      • Derivation.instInhabited = { default := 0 }
      instance Derivation.instSMul {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] :
      SMul S (Derivation R A M)
      Equations
      • Derivation.instSMul = { smul := fun (r : S) (D : Derivation R A M) => { toLinearMap := r D, map_one_eq_zero' := , leibniz' := } }
      @[simp]
      theorem Derivation.coe_smul {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) :
      (r D) = r D
      @[simp]
      theorem Derivation.coe_smul_linearMap {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) :
      (r D) = r D
      theorem Derivation.smul_apply {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) :
      (r D) a = r D a
      instance Derivation.instAddCommMonoid {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] :
      Equations
      def Derivation.coeFnAddMonoidHom {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] :
      Derivation R A M →+ AM

      coe_fn as an AddMonoidHom.

      Equations
      • Derivation.coeFnAddMonoidHom = { toFun := DFunLike.coe, map_zero' := , map_add' := }
      Instances For
        instance Derivation.instDistribMulAction {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] :
        Equations
        instance Derivation.instIsCentralScalar {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] [DistribMulAction Sᵐᵒᵖ M] [IsCentralScalar S M] :
        Equations
        • =
        instance Derivation.instIsScalarTower {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} {T : Type u_6} [Monoid S] [DistribMulAction S M] [SMulCommClass R S M] [SMulCommClass S A M] [Monoid T] [DistribMulAction T M] [SMulCommClass R T M] [SMulCommClass T A M] [SMul S T] [IsScalarTower S T M] :
        Equations
        • =
        instance Derivation.instSMulCommClass {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} {T : Type u_6} [Monoid S] [DistribMulAction S M] [SMulCommClass R S M] [SMulCommClass S A M] [Monoid T] [DistribMulAction T M] [SMulCommClass R T M] [SMulCommClass T A M] [SMulCommClass S T M] :
        Equations
        • =
        instance Derivation.instModule {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} [Semiring S] [Module S M] [SMulCommClass R S M] [SMulCommClass S A M] :
        Module S (Derivation R A M)
        Equations
        def LinearMap.compDer {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) :

        We can push forward derivations using linear maps, i.e., the composition of a derivation with a linear map is a derivation. Furthermore, this operation is linear on the spaces of derivations.

        Equations
        • f.compDer = { toFun := fun (D : Derivation R A M) => { toLinearMap := R f ∘ₗ D, map_one_eq_zero' := , leibniz' := }, map_add' := , map_smul' := }
        Instances For
          @[simp]
          theorem Derivation.coe_to_linearMap_comp {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) :
          (f.compDer D) = R f ∘ₗ D
          @[simp]
          theorem Derivation.coe_comp {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) :
          (f.compDer D) = (R f ∘ₗ D)
          def Derivation.llcomp {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] :

          The composition of a derivation with a linear map as a bilinear map

          Equations
          • Derivation.llcomp = { toFun := fun (f : M →ₗ[A] N) => f.compDer, map_add' := , map_smul' := }
          Instances For
            @[simp]
            theorem Derivation.llcomp_apply {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) :
            Derivation.llcomp f = f.compDer
            def LinearEquiv.compDer {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] (e : M ≃ₗ[A] N) :

            Pushing a derivation forward through a linear equivalence is an equivalence.

            Equations
            • e.compDer = { toLinearMap := (↑e).compDer, invFun := (↑e.symm).compDer, left_inv := , right_inv := }
            Instances For
              @[simp]
              theorem Derivation.linearEquiv_coe_to_linearMap_comp {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) :
              (e.compDer D) = R e ∘ₗ D
              @[simp]
              theorem Derivation.linearEquiv_coe_comp {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) :
              (e.compDer D) = (R e ∘ₗ D)
              def Derivation.compAlgebraMap {R : Type u_1} (A : Type u_2) {B : Type u_3} {M : Type u_4} [CommSemiring R] [CommSemiring A] [CommSemiring B] [AddCommMonoid M] [Algebra R A] [Algebra R B] [Module A M] [Module B M] [Module R M] [Algebra A B] [IsScalarTower R A B] [IsScalarTower A B M] (d : Derivation R B M) :

              For a tower R → A → B and an R-derivation B → M, we may compose with A → B to obtain an R-derivation A → M.

              Equations
              Instances For
                @[simp]
                theorem Derivation.compAlgebraMap_apply {R : Type u_1} (A : Type u_2) {B : Type u_3} {M : Type u_4} [CommSemiring R] [CommSemiring A] [CommSemiring B] [AddCommMonoid M] [Algebra R A] [Algebra R B] [Module A M] [Module B M] [Module R M] [Algebra A B] [IsScalarTower R A B] [IsScalarTower A B M] (d : Derivation R B M) (a✝ : A) :
                (Derivation.compAlgebraMap A d) a✝ = d ((algebraMap A B) a✝)
                def Derivation.restrictScalars (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} [CommSemiring S] [Algebra S A] [Module S M] [LinearMap.CompatibleSMul A M R S] (d : Derivation S A M) :

                If A is both an R-algebra and an S-algebra; M is both an R-module and an S-module, then an S-derivation A → M is also an R-derivation if it is also R-linear.

                Equations
                Instances For
                  theorem Derivation.coe_restrictScalars (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} [CommSemiring S] [Algebra S A] [Module S M] [LinearMap.CompatibleSMul A M R S] (d : Derivation S A M) :
                  @[simp]
                  theorem Derivation.restrictScalars_apply (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} [CommSemiring S] [Algebra S A] [Module S M] [LinearMap.CompatibleSMul A M R S] (d : Derivation S A M) (x : A) :
                  def Derivation.liftOfRightInverse {R : Type u_1} {A : Type u_2} {M : Type u_3} [CommSemiring R] [CommRing A] [CommRing M] [Algebra R A] [Algebra R M] {F : Type u_4} [FunLike F A M] [AlgHomClass F R A M] {f : F} {f_inv : MA} (hf : Function.RightInverse f_inv f) ⦃d : Derivation R A A (hd : ∀ (x : A), f x = 0f (d x) = 0) :

                  Lift a derivation via an algebra homomorphism f with a right inverse such that f(x) = 0 → f(d(x)) = 0. This gives the derivation f ∘ d ∘ f⁻¹. This is needed for an argument in Rosenlicht, M. Integration in finite terms.

                  Equations
                  • Derivation.liftOfRightInverse hf hd = { toFun := fun (x : M) => f (d (f_inv x)), map_add' := , map_smul' := , map_one_eq_zero' := , leibniz' := }
                  Instances For
                    @[simp]
                    theorem Derivation.liftOfRightInverse_apply {R : Type u_1} {A : Type u_2} {M : Type u_3} [CommSemiring R] [CommRing A] [CommRing M] [Algebra R A] [Algebra R M] {F : Type u_4} [FunLike F A M] [AlgHomClass F R A M] {f : F} {f_inv : MA} (hf : Function.RightInverse f_inv f) {d : Derivation R A A} (hd : ∀ (x : A), f x = 0f (d x) = 0) (x : A) :
                    (Derivation.liftOfRightInverse hf hd) (f x) = f (d x)
                    theorem Derivation.liftOfRightInverse_eq {R : Type u_1} {A : Type u_2} {M : Type u_3} [CommSemiring R] [CommRing A] [CommRing M] [Algebra R A] [Algebra R M] {F : Type u_4} [FunLike F A M] [AlgHomClass F R A M] {f : F} {f_inv₁ f_inv₂ : MA} (hf₁ : Function.RightInverse f_inv₁ f) (hf₂ : Function.RightInverse f_inv₂ f) :
                    @[reducible, inline]
                    noncomputable abbrev Derivation.liftOfSurjective {R : Type u_1} {A : Type u_2} {M : Type u_3} [CommSemiring R] [CommRing A] [CommRing M] [Algebra R A] [Algebra R M] {F : Type u_4} [FunLike F A M] [AlgHomClass F R A M] {f : F} (hf : Function.Surjective f) ⦃d : Derivation R A A (hd : ∀ (x : A), f x = 0f (d x) = 0) :

                    A noncomputable version of liftOfRightInverse for surjective homomorphisms.

                    Equations
                    Instances For
                      theorem Derivation.liftOfSurjective_apply {R : Type u_1} {A : Type u_2} {M : Type u_3} [CommSemiring R] [CommRing A] [CommRing M] [Algebra R A] [Algebra R M] {F : Type u_4} [FunLike F A M] [AlgHomClass F R A M] {f : F} (hf : Function.Surjective f) {d : Derivation R A A} (hd : ∀ (x : A), f x = 0f (d x) = 0) (x : A) :
                      (Derivation.liftOfSurjective hf hd) (f x) = f (d x)
                      def Derivation.mk' {R : Type u_1} [CommSemiring R] {A : Type u_2} [CommSemiring A] [Algebra R A] {M : Type u_3} [AddCancelCommMonoid M] [Module R M] [Module A M] (D : A →ₗ[R] M) (h : ∀ (a b : A), D (a * b) = a D b + b D a) :

                      Define Derivation R A M from a linear map when M is cancellative by verifying the Leibniz rule.

                      Equations
                      • Derivation.mk' D h = { toLinearMap := D, map_one_eq_zero' := , leibniz' := h }
                      Instances For
                        @[simp]
                        theorem Derivation.coe_mk' {R : Type u_1} [CommSemiring R] {A : Type u_2} [CommSemiring A] [Algebra R A] {M : Type u_3} [AddCancelCommMonoid M] [Module R M] [Module A M] (D : A →ₗ[R] M) (h : ∀ (a b : A), D (a * b) = a D b + b D a) :
                        (Derivation.mk' D h) = D
                        @[simp]
                        theorem Derivation.coe_mk'_linearMap {R : Type u_1} [CommSemiring R] {A : Type u_2} [CommSemiring A] [Algebra R A] {M : Type u_3} [AddCancelCommMonoid M] [Module R M] [Module A M] (D : A →ₗ[R] M) (h : ∀ (a b : A), D (a * b) = a D b + b D a) :
                        (Derivation.mk' D h) = D
                        theorem Derivation.map_neg {R : Type u_1} [CommRing R] {A : Type u_2} [CommRing A] [Algebra R A] {M : Type u_3} [AddCommGroup M] [Module A M] [Module R M] (D : Derivation R A M) (a : A) :
                        D (-a) = -D a
                        theorem Derivation.map_sub {R : Type u_1} [CommRing R] {A : Type u_2} [CommRing A] [Algebra R A] {M : Type u_3} [AddCommGroup M] [Module A M] [Module R M] (D : Derivation R A M) (a b : A) :
                        D (a - b) = D a - D b
                        @[simp]
                        theorem Derivation.map_intCast {R : Type u_1} [CommRing R] {A : Type u_2} [CommRing A] [Algebra R A] {M : Type u_3} [AddCommGroup M] [Module A M] [Module R M] (D : Derivation R A M) (n : ) :
                        D n = 0
                        @[deprecated Derivation.map_natCast]
                        theorem Derivation.map_coe_nat {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 : ) :
                        D n = 0

                        Alias of Derivation.map_natCast.

                        @[deprecated Derivation.map_intCast]
                        theorem Derivation.map_coe_int {R : Type u_1} [CommRing R] {A : Type u_2} [CommRing A] [Algebra R A] {M : Type u_3} [AddCommGroup M] [Module A M] [Module R M] (D : Derivation R A M) (n : ) :
                        D n = 0

                        Alias of Derivation.map_intCast.

                        theorem Derivation.leibniz_of_mul_eq_one {R : Type u_1} [CommRing R] {A : Type u_2} [CommRing A] [Algebra R A] {M : Type u_3} [AddCommGroup M] [Module A M] [Module R M] (D : Derivation R A M) {a b : A} (h : a * b = 1) :
                        D a = -a ^ 2 D b
                        theorem Derivation.leibniz_invOf {R : Type u_1} [CommRing R] {A : Type u_2} [CommRing A] [Algebra R A] {M : Type u_3} [AddCommGroup M] [Module A M] [Module R M] (D : Derivation R A M) (a : A) [Invertible a] :
                        D a = -a ^ 2 D a
                        theorem Derivation.leibniz_inv {R : Type u_1} [CommRing R] {M : Type u_3} [AddCommGroup M] [Module R M] {K : Type u_4} [Field K] [Module K M] [Algebra R K] (D : Derivation R K M) (a : K) :
                        D a⁻¹ = -a⁻¹ ^ 2 D a
                        theorem Derivation.leibniz_div {R : Type u_1} [CommRing R] {M : Type u_3} [AddCommGroup M] [Module R M] {K : Type u_4} [Field K] [Module K M] [Algebra R K] (D : Derivation R K M) (a b : K) :
                        D (a / b) = b⁻¹ ^ 2 (b D a - a D b)
                        theorem Derivation.leibniz_div_const {R : Type u_1} [CommRing R] {M : Type u_3} [AddCommGroup M] [Module R M] {K : Type u_4} [Field K] [Module K M] [Algebra R K] (D : Derivation R K M) (a b : K) (h : D b = 0) :
                        D (a / b) = b⁻¹ D a
                        theorem Derivation.leibniz_zpow {R : Type u_1} [CommRing R] {M : Type u_3} [AddCommGroup M] [Module R M] {K : Type u_4} [Field K] [Module K M] [Algebra R K] (D : Derivation R K M) (a : K) (n : ) :
                        D (a ^ n) = n a ^ (n - 1) D a
                        instance Derivation.instNeg {R : Type u_1} [CommRing R] {A : Type u_2} [CommRing A] [Algebra R A] {M : Type u_3} [AddCommGroup M] [Module A M] [Module R M] :
                        Neg (Derivation R A M)
                        Equations
                        @[simp]
                        theorem Derivation.coe_neg {R : Type u_1} [CommRing R] {A : Type u_2} [CommRing A] [Algebra R A] {M : Type u_3} [AddCommGroup M] [Module A M] [Module R M] (D : Derivation R A M) :
                        (-D) = -D
                        @[simp]
                        theorem Derivation.coe_neg_linearMap {R : Type u_1} [CommRing R] {A : Type u_2} [CommRing A] [Algebra R A] {M : Type u_3} [AddCommGroup M] [Module A M] [Module R M] (D : Derivation R A M) :
                        (-D) = -D
                        theorem Derivation.neg_apply {R : Type u_1} [CommRing R] {A : Type u_2} [CommRing A] [Algebra R A] {M : Type u_3} [AddCommGroup M] [Module A M] [Module R M] (D : Derivation R A M) (a : A) :
                        (-D) a = -D a
                        instance Derivation.instSub {R : Type u_1} [CommRing R] {A : Type u_2} [CommRing A] [Algebra R A] {M : Type u_3} [AddCommGroup M] [Module A M] [Module R M] :
                        Sub (Derivation R A M)
                        Equations
                        @[simp]
                        theorem Derivation.coe_sub {R : Type u_1} [CommRing R] {A : Type u_2} [CommRing A] [Algebra R A] {M : Type u_3} [AddCommGroup M] [Module A M] [Module R M] (D1 D2 : Derivation R A M) :
                        (D1 - D2) = D1 - D2
                        @[simp]
                        theorem Derivation.coe_sub_linearMap {R : Type u_1} [CommRing R] {A : Type u_2} [CommRing A] [Algebra R A] {M : Type u_3} [AddCommGroup M] [Module A M] [Module R M] (D1 D2 : Derivation R A M) :
                        (D1 - D2) = D1 - D2
                        theorem Derivation.sub_apply {R : Type u_1} [CommRing R] {A : Type u_2} [CommRing A] [Algebra R A] {M : Type u_3} [AddCommGroup M] [Module A M] [Module R M] {D1 D2 : Derivation R A M} (a : A) :
                        (D1 - D2) a = D1 a - D2 a
                        instance Derivation.instAddCommGroup {R : Type u_1} [CommRing R] {A : Type u_2} [CommRing A] [Algebra R A] {M : Type u_3} [AddCommGroup M] [Module A M] [Module R M] :
                        Equations