Documentation Verification Report

Derivation

📁 Source: Mathlib/Algebra/Polynomial/Derivation.lean

Statistics

MetricCount
DefinitionsDerivation, compAEval, Derivation, derivative', mkDerivation, mkDerivationEquiv, Derivation
7
TheoremscompAEval_apply, compAEval_eq, comp_aeval_eq, C_smul_derivation_apply, derivation_C, derivation_ext, derivation_ext_iff, derivative'_apply, mkDerivationEquiv_apply, mkDerivationEquiv_symm_apply, mkDerivation_X, mkDerivation_apply, mkDerivation_one_eq_derivative, mkDerivation_one_eq_derivative'
14
Total21

Derivation

Definitions

NameCategoryTheorems
compAEval 📖CompOp
2 mathmath: compAEval_eq, compAEval_apply

Theorems

NameKindAssumesProvesValidatesDepends On
compAEval_apply 📖mathematicalDFunLike.coe
Derivation
Polynomial
CommSemiring.toSemiring
Module.AEval
Polynomial.commSemiring
Module.AEval.instAddCommMonoid
Polynomial.algebraOfAlgebra
Algebra.id
Module.AEval.instModulePolynomial
Module.AEval.instModuleOrig
instFunLike
compAEval
LinearEquiv
RingHom.id
Semiring.toNonAssocSemiring
EquivLike.toFunLike
DFinsupp.instEquivLikeLinearEquiv
Module.AEval.of
AlgHom
Polynomial.semiring
AlgHom.funLike
Polynomial.aeval
compAEval_eq 📖mathematicalDFunLike.coe
Derivation
Polynomial
CommSemiring.toSemiring
Module.AEval
Polynomial.commSemiring
Module.AEval.instAddCommMonoid
Polynomial.algebraOfAlgebra
Algebra.id
Module.AEval.instModulePolynomial
Module.AEval.instModuleOrig
instFunLike
compAEval
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Polynomial.semiring
Module.toDistribMulAction
LinearMap
RingHom.id
Semiring.toNonAssocSemiring
NonUnitalNonAssocSemiring.toAddCommMonoid
NonAssocSemiring.toNonUnitalNonAssocSemiring
Polynomial.module
Semiring.toModule
LinearMap.instFunLike
Polynomial.derivative
LinearEquiv
RingHomInvPair.ids
EquivLike.toFunLike
DFinsupp.instEquivLikeLinearEquiv
Module.AEval.of
RingHomInvPair.ids
map_aeval
Module.AEval.of_aeval_smul
comp_aeval_eq 📖mathematicalDFunLike.coe
Derivation
instFunLike
AlgHom
Polynomial
CommSemiring.toSemiring
Polynomial.semiring
Polynomial.algebraOfAlgebra
Algebra.id
AlgHom.funLike
Polynomial.aeval
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Module.toDistribMulAction
LinearMap
RingHom.id
Semiring.toNonAssocSemiring
NonUnitalNonAssocSemiring.toAddCommMonoid
NonAssocSemiring.toNonUnitalNonAssocSemiring
Polynomial.module
Semiring.toModule
LinearMap.instFunLike
Polynomial.derivative
RingHomInvPair.ids
compAEval_eq
LinearEquiv.symm_apply_apply

ModuleCat

Definitions

NameCategoryTheorems
Derivation 📖CompOp

Polynomial

Definitions

NameCategoryTheorems
derivative' 📖CompOp
5 mathmath: derivative'_apply, KaehlerDifferential.polynomialEquiv_comp_D, mkDerivation_one_eq_derivative', Bivariate.pderiv_zero_equivMvPolynomial, Bivariate.Polynomial.Bivariate.pderiv_zero_equivMvPolynomial
mkDerivation 📖CompOp
5 mathmath: mkDerivation_one_eq_derivative', mkDerivationEquiv_apply, mkDerivation_X, mkDerivation_one_eq_derivative, mkDerivation_apply
mkDerivationEquiv 📖CompOp
2 mathmath: mkDerivationEquiv_symm_apply, mkDerivationEquiv_apply

Theorems

NameKindAssumesProvesValidatesDepends On
C_smul_derivation_apply 📖mathematicalPolynomial
CommSemiring.toSemiring
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
semiring
Module.toDistribMulAction
DFunLike.coe
RingHom
Semiring.toNonAssocSemiring
RingHom.instFunLike
C
Derivation
commSemiring
algebraOfAlgebra
Algebra.id
Derivation.instFunLike
Derivation.leibniz
derivation_C
smul_zero
add_zero
C_mul'
Derivation.map_smul
derivation_C 📖mathematicalDFunLike.coe
Derivation
Polynomial
CommSemiring.toSemiring
commSemiring
algebraOfAlgebra
Algebra.id
Derivation.instFunLike
RingHom
Semiring.toNonAssocSemiring
semiring
RingHom.instFunLike
C
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
Derivation.map_algebraMap
derivation_ext 📖DFunLike.coe
Derivation
Polynomial
CommSemiring.toSemiring
commSemiring
algebraOfAlgebra
Algebra.id
Derivation.instFunLike
X
Derivation.ext
Derivation.eqOn_adjoin
Set.eqOn_singleton
adjoin_X
derivation_ext_iff 📖mathematicalDFunLike.coe
Derivation
Polynomial
CommSemiring.toSemiring
commSemiring
algebraOfAlgebra
Algebra.id
Derivation.instFunLike
X
derivation_ext
derivative'_apply 📖mathematicalDFunLike.coe
Derivation
Polynomial
CommSemiring.toSemiring
commSemiring
NonUnitalNonAssocSemiring.toAddCommMonoid
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
semiring
algebraOfAlgebra
Algebra.id
Semiring.toModule
module
Derivation.instFunLike
derivative'
LinearMap
RingHom.id
LinearMap.instFunLike
derivative
mkDerivationEquiv_apply 📖mathematicalDFunLike.coe
LinearEquiv
CommSemiring.toSemiring
RingHom.id
Semiring.toNonAssocSemiring
RingHomInvPair.ids
Derivation
Polynomial
commSemiring
algebraOfAlgebra
Algebra.id
Derivation.instAddCommMonoid
Derivation.instModule
smulCommClass_self
CommSemiring.toCommMonoid
DistribMulAction.toMulAction
CommMonoid.toMonoid
AddCommMonoid.toAddMonoid
Module.toDistribMulAction
IsScalarTower.to_smulCommClass
EquivLike.toFunLike
DFinsupp.instEquivLikeLinearEquiv
mkDerivationEquiv
LinearMap
LinearMap.instFunLike
mkDerivation
RingHomInvPair.ids
smulCommClass_self
IsScalarTower.to_smulCommClass
mkDerivationEquiv_symm_apply 📖mathematicalDFunLike.coe
LinearEquiv
CommSemiring.toSemiring
RingHom.id
Semiring.toNonAssocSemiring
RingHomInvPair.ids
Derivation
Polynomial
commSemiring
algebraOfAlgebra
Algebra.id
Derivation.instAddCommMonoid
Derivation.instModule
smulCommClass_self
CommSemiring.toCommMonoid
DistribMulAction.toMulAction
CommMonoid.toMonoid
AddCommMonoid.toAddMonoid
Module.toDistribMulAction
IsScalarTower.to_smulCommClass
EquivLike.toFunLike
DFinsupp.instEquivLikeLinearEquiv
LinearEquiv.symm
mkDerivationEquiv
Derivation.instFunLike
X
RingHomInvPair.ids
smulCommClass_self
IsScalarTower.to_smulCommClass
mkDerivation_X 📖mathematicalDFunLike.coe
Derivation
Polynomial
CommSemiring.toSemiring
commSemiring
algebraOfAlgebra
Algebra.id
Derivation.instFunLike
LinearMap
RingHom.id
Semiring.toNonAssocSemiring
Derivation.instAddCommMonoid
Derivation.instModule
smulCommClass_self
CommSemiring.toCommMonoid
DistribMulAction.toMulAction
CommMonoid.toMonoid
AddCommMonoid.toAddMonoid
Module.toDistribMulAction
IsScalarTower.to_smulCommClass
LinearMap.instFunLike
mkDerivation
X
smulCommClass_self
IsScalarTower.to_smulCommClass
mkDerivation_apply
derivative_X
one_smul
mkDerivation_apply 📖mathematicalDFunLike.coe
Derivation
Polynomial
CommSemiring.toSemiring
commSemiring
algebraOfAlgebra
Algebra.id
Derivation.instFunLike
LinearMap
RingHom.id
Semiring.toNonAssocSemiring
Derivation.instAddCommMonoid
Derivation.instModule
smulCommClass_self
CommSemiring.toCommMonoid
DistribMulAction.toMulAction
CommMonoid.toMonoid
AddCommMonoid.toAddMonoid
Module.toDistribMulAction
IsScalarTower.to_smulCommClass
LinearMap.instFunLike
mkDerivation
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
semiring
NonUnitalNonAssocSemiring.toAddCommMonoid
NonAssocSemiring.toNonUnitalNonAssocSemiring
module
Semiring.toModule
derivative
smulCommClass_self
IsScalarTower.to_smulCommClass
mkDerivation_one_eq_derivative 📖mathematicalDFunLike.coe
Derivation
Polynomial
CommSemiring.toSemiring
commSemiring
NonUnitalNonAssocSemiring.toAddCommMonoid
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
semiring
algebraOfAlgebra
Algebra.id
Semiring.toModule
module
Derivation.instFunLike
LinearMap
RingHom.id
Derivation.instAddCommMonoid
Derivation.instModule
smulCommClass_self
CommSemiring.toCommMonoid
DistribMulAction.toMulAction
CommMonoid.toMonoid
AddCommMonoid.toAddMonoid
Module.toDistribMulAction
IsScalarTower.to_smulCommClass
IsScalarTower.right
LinearMap.instFunLike
mkDerivation
instOne
derivative
smulCommClass_self
IsScalarTower.to_smulCommClass
IsScalarTower.right
mkDerivation_one_eq_derivative'
mkDerivation_one_eq_derivative' 📖mathematicalDFunLike.coe
LinearMap
CommSemiring.toSemiring
RingHom.id
Semiring.toNonAssocSemiring
Polynomial
Derivation
commSemiring
NonUnitalNonAssocSemiring.toAddCommMonoid
NonAssocSemiring.toNonUnitalNonAssocSemiring
semiring
algebraOfAlgebra
Algebra.id
Semiring.toModule
module
Derivation.instAddCommMonoid
Derivation.instModule
smulCommClass_self
CommSemiring.toCommMonoid
DistribMulAction.toMulAction
CommMonoid.toMonoid
AddCommMonoid.toAddMonoid
Module.toDistribMulAction
IsScalarTower.to_smulCommClass
IsScalarTower.right
LinearMap.instFunLike
mkDerivation
instOne
derivative'
derivation_ext
smulCommClass_self
IsScalarTower.to_smulCommClass
IsScalarTower.right
mkDerivation_X
derivative_X

PresheafOfModules

Definitions

NameCategoryTheorems
Derivation 📖CompData

(root)

Definitions

NameCategoryTheorems
Derivation 📖CompData
222 mathmath: Derivation.map_natCast, KaehlerDifferential.kerCotangentToTensor_toCotangent, KaehlerDifferential.mvPolynomialBasis_apply, Derivation.apply_eval_eq, Algebra.PreSubmersivePresentation.jacobiMatrix_naive, LeftInvariantDerivation.lift_zero, derivationToSquareZeroOfLift_apply, Algebra.PreSubmersivePresentation.cotangentComplexAux_apply, Differential.implicitDeriv_C, Derivation.coe_add, Differential.algHom_deriv, KaehlerDifferential.polynomialEquiv_symm, Derivation.coe_to_linearMap_comp, KaehlerDifferential.tensorKaehlerEquiv_symm_D_tmul', Algebra.Generators.cotangentSpaceBasis_apply, MvPolynomial.IsWeightedHomogeneous.pderiv, KaehlerDifferential.mvPolynomialBasis_repr_D_X, PowerSeries.derivative_exp, LeftInvariantDerivation.coe_derivation, KaehlerDifferential.tensorKaehlerEquiv_symm_D_tmul, LeftInvariantDerivation.left_invariant', Algebra.PreSubmersivePresentation.cotangentComplexAux_zero_iff, KaehlerDifferential.polynomialEquiv_D, Derivation.mapCoeffs_X, KaehlerDifferential.kerToTensor_apply, PowerSeries.derivative_inv', MvPolynomial.pderiv_one, MvPolynomial.mkDerivation_X, MvPolynomial.pderiv_mul, Derivation.linearEquiv_coe_to_linearMap_comp, KaehlerDifferential.quotKerTotalEquiv_symm_comp_D, Polynomial.Bivariate.Polynomial.Bivariate.pderiv_one_equivMvPolynomial, LieRinehartAlgebra.Hom.anchor_derivation, Algebra.Generators.cotangentSpaceBasis_repr_one_tmul, MvPolynomial.pderiv_rename, Polynomial.derivative'_apply, LeftInvariantDerivation.lift_add, Derivation.add_apply, DifferentialAlgebra.deriv_algebraMap, Derivation.leibniz_div, LeftInvariantDerivation.lift_smul, Algebra.SubmersivePresentation.basisKaehlerOfIsCompl_apply, Derivation.bracket_eq_fun, KaehlerDifferential.kerTotal_eq, Algebra.Generators.H1Cotangent.δAux_C, MvPolynomial.X_mul_pderiv_monomial, Algebra.Extension.CotangentSpace.map_tmul, KaehlerDifferential.polynomialEquiv_comp_D, KaehlerDifferential.span_range_map_derivation_of_isLocalization, Derivation.mapCoeffs_C, Derivation.toFun_eq_coe, Derivation.coe_neg_linearMap, Derivation.comp_aeval_eq, Derivation.neg_apply, Derivation.instAddMonoidHomClass, Derivation.leibniz_inv, Derivation.map_one_eq_zero, Polynomial.mkDerivation_one_eq_derivative', KaehlerDifferential.linearMapEquivDerivation_apply_apply, LeftInvariantDerivation.evalAt_coe, MvPolynomial.pderiv_sumToIter, algebraMap.coe_deriv, Derivation.mapCoeffs_monomial, Algebra.PreSubmersivePresentation.jacobiMatrix_apply, KaehlerDifferential.linearMapEquivDerivation_symm_apply, KaehlerDifferential.span_range_derivation, Algebra.Generators.H1Cotangent.δAux_monomial, Differential.coeff_mapCoeffs, Algebra.Generators.cotangentSpaceBasis_repr_tmul, Derivation.leibniz_zpow, Derivation.instLieModule, Derivation.mk_coe, Algebra.Extension.Hom.sub_one_tmul, Algebra.SubmersivePresentation.linearIndependent_aeval_val_pderiv_relation, MvPolynomial.pderiv_X, Algebra.Generators.H1Cotangent.δAux_toAlgHom, KaehlerDifferential.D_apply, Derivation.coeFnAddMonoidHom_apply, Derivation.liftKaehlerDifferential_comp, KaehlerDifferential.ker_map, LeftInvariantDerivation.left_invariant'', Derivation.restrictScalars_apply, MvPolynomial.IsWeightedHomogeneous.sum_weight_X_mul_pderiv, Derivation.compAlgebraMapL_apply_apply, Polynomial.mkDerivationEquiv_symm_apply, Derivation.instIsScalarTower, Derivation.commutator_coe_linear_map, Derivation.coe_comp, Derivation.map_zero, KaehlerDifferential.derivationQuotKerTotal_apply, Differential.mapCoeffs_monomial, tensorKaehlerQuotKerSqEquiv_symm_tmul_D, Polynomial.mkDerivationEquiv_apply, Derivation.liftKaehlerDifferential_comp_D, Algebra.Generators.H1Cotangent.δAux_ofComp, Differential.mapCoeffs_C, Algebra.Presentation.differentials.comm₂₃', Derivation.coe_mk', PowerSeries.derivative_invOf, sectionOfRetractionKerToTensorAux_prop, Derivation.liftKaehlerDifferential_unique_iff, MvPolynomial.derivation_C, tensorKaehlerQuotKerSqEquiv_tmul_D, MvPolynomial.mkDerivation_monomial, derivationToSquareZeroEquivLift_symm_apply_apply_coe, Algebra.PreSubmersivePresentation.aevalDifferential_single, Derivation.leibniz, Derivation.llcomp_apply, Derivation.map_neg, Derivation.map_algebraMap, KaehlerDifferential.mvPolynomialBasis_repr_apply, Differential.mapCoeffs_X, Derivation.smul_apply, Polynomial.Bivariate.pderiv_zero_equivMvPolynomial, MvPolynomial.pderiv_C_mul, Derivation.leibniz_of_mul_eq_one, Algebra.Generators.toKaehler_tmul_D, Derivation.instIsCentralScalar, Derivation.coeFn_coe, sectionOfRetractionKerToTensor_algebraMap, Derivation.coe_sub_linearMap, Differential.algHom_deriv', Derivation.coe_smul, KaehlerDifferential.map_compDer, MvPolynomial.pderiv_monomial_single, PowerSeries.derivative_coe, MvPolynomial.pderiv_X_of_ne, Polynomial.Bivariate.Polynomial.Bivariate.pderiv_zero_equivMvPolynomial, MvPolynomial.pderiv_monomial, liftOfDerivationToSquareZero_apply, Polynomial.mkDerivation_X, Polynomial.derivation_ext_iff, Derivation.instSMulCommClass, KaehlerDifferential.derivationQuotKerTotal_lift_comp_linearCombination, Polynomial.mkDerivation_one_eq_derivative, Derivation.coe_restrictScalars, Algebra.Generators.repr_CotangentSpaceMap, MvPolynomial.pderiv_pow, Derivation.map_smul_of_tower, Derivation.map_aeval, Algebra.Generators.toKaehler_cotangentSpaceBasis, Algebra.SubmersivePresentation.basisKaehler_apply, KaehlerDifferential.polynomial_D_apply, Derivation.leibniz_pow, Differential.deriv_aeval_eq, MvPolynomial.pderiv_map, Polynomial.derivation_C, Derivation.map_add, KaehlerDifferential.derivationTensorProduct_algebraMap, MvPolynomial.pderiv_inr_universalFactorizationMap_X, Derivation.compAEval_eq, Derivation.tensorProductTo_tmul, derivationOfSectionOfKerSqZero_apply_coe, Differential.logDeriv_eq_zero, Derivation.map_smul, sectionOfRetractionKerToTensorAux_algebraMap, KaehlerDifferential.mvPolynomialBasis_repr_D, MvPolynomial.pderiv_eq_zero_of_notMem_vars, KaehlerDifferential.ker_map_of_surjective, Algebra.Generators.cotangentRestrict_mk, Derivation.coe_add_linearMap, KaehlerDifferential.tensorKaehlerEquivOfFormallyEtale_symm_D_algebraMap, Derivation.coe_smul_linearMap, KaehlerDifferential.mvPolynomialBasis_repr_comp_D, Derivation.mapCoeffs_apply, Derivation.ext_iff, derivationToSquareZeroEquivLift_apply_coe_apply, PowerSeries.coeff_derivative, liftOfDerivationToSquareZero_mk_apply', Differential.implicitDeriv_X, LeftInvariantDerivation.toDerivation_injective, Derivation.zero_apply, Derivation.sub_apply, PowerSeries.derivative_subst, Algebra.Extension.Hom.sub_tmul, Derivation.coe_sub, MvPolynomial.IsHomogeneous.sum_X_mul_pderiv, MvPolynomial.derivation_C_mul, MvPolynomial.aeval_sumElim_pderiv_inl, Differential.algEquiv_deriv', Derivation.commutator_apply, KaehlerDifferential.mvPolynomialBasis_repr_symm_single, Derivation.compAlgebraMap_apply, Algebra.IsStandardSmooth.iff_exists_basis_kaehlerDifferential, Derivation.coe_zero_linearMap, Derivation.congr_fun, Derivation.coe_zero, PowerSeries.derivative_C, KaehlerDifferential.map_D, LieRinehartAlgebra.instDerivation, LeftInvariantDerivation.commutator_coe_derivation, MvPolynomial.pderiv_X_self, KaehlerDifferential.tensorKaehlerEquiv_tmul_D, MvPolynomial.pderiv_C, Polynomial.Bivariate.pderiv_one_equivMvPolynomial, Derivation.leibniz_invOf, Derivation.linearEquiv_coe_comp, Derivation.coe_injective, Derivation.compAEval_apply, PowerSeries.trunc_derivative, Derivation.coe_neg, Derivation.map_sub, KaehlerDifferential.quotKerTotalEquiv_apply, PowerSeries.derivative_inv, PowerSeries.derivative_X, Differential.algEquiv_deriv, MvPolynomial.derivation_ext_iff, Polynomial.mkDerivation_apply, Derivation.evalAt_apply, MvPolynomial.IsHomogeneous.pderiv, Algebra.SubmersivePresentation.basisDeriv_apply, LieRinehartAlgebra.instLieRinehartRingDerivation, Algebra.Extension.cotangentComplex_mk, retractionOfSectionOfKerSqZero_tmul_D, KaehlerDifferential.linearCombination_surjective, derivationQuotKerSq_mk, Derivation.map_intCast, Polynomial.C_smul_derivation_apply, PowerSeries.derivative_pow, MvPolynomial.pderiv_inl_universalFactorizationMap_X, Derivation.apply_aeval_eq, PowerSeries.trunc_derivative'

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