Documentation Verification Report

Monic

πŸ“ Source: Mathlib/Algebra/Polynomial/Monic.lean

Statistics

MetricCount
DefinitionsMonic, Monic, Monic, MonicDegreeEq, map, mk
6
Theoremsmonic_map_iff, add_of_left, add_of_right, as_sum, comp, comp_X_add_C, comp_X_sub_C, degree_le_zero_iff_eq_one, degree_map, degree_mul_comm, eq_one_of_isUnit, eq_one_of_map_eq_one, irreducible_iff_lt_natDegree_lt, irreducible_iff_natDegree, irreducible_iff_natDegree', isRegular, isUnit_iff, map, mul, mul_left_eq_zero_iff, mul_left_ne_zero, mul_natDegree_lt_iff, mul_right_eq_zero_iff, mul_right_ne_zero, natDegree_eq_zero_iff_eq_one, natDegree_map, natDegree_mul, natDegree_mul', natDegree_mul_comm, natDegree_pow, nextCoeff_mul, nextCoeff_multiset_prod, nextCoeff_pow, nextCoeff_prod, not_dvd_of_degree_lt, not_dvd_of_natDegree_lt, not_irreducible_iff_exists_add_mul_eq_coeff, of_mul_monic_left, of_mul_monic_right, pow, sub_of_left, sub_of_right, coeff_of_ge, degree, map_coe, mk_coe, monic, natDegree, degree_smul_of_smul_regular, eq_of_monic_of_associated, irreducible_of_monic, isUnit_leadingCoeff_mul_left_eq_zero_iff, isUnit_leadingCoeff_mul_right_eq_zero_iff, leadingCoeff_map', leadingCoeff_of_injective, leadingCoeff_smul_of_smul_regular, monic_C_mul_of_mul_leadingCoeff_eq_one, monic_X_add_C, monic_X_pow_add, monic_X_pow_add_C, monic_X_pow_sub, monic_X_pow_sub_C, monic_X_sub_C, monic_finprod_of_monic, monic_mul_C_of_leadingCoeff_mul_eq_one, monic_multiset_prod_of_monic, monic_of_injective, monic_of_isUnit_leadingCoeff_inv_smul, monic_prod_of_monic, monic_zero_iff_subsingleton, monic_zero_iff_subsingleton', natDegree_pow_X_add_C, natDegree_smul_of_smul_regular, not_isUnit_X_add_C, not_isUnit_X_pow_sub_one, not_monic_zero, not_monic_zero_iff
77
Total83

AddMonoidAlgebra

Definitions

NameCategoryTheorems
Monic πŸ“–MathDef
2 mathmath: MvPolynomial.monic_esymm, monic_one

Function.Injective

Theorems

NameKindAssumesProvesValidatesDepends On
monic_map_iff πŸ“–mathematicalDFunLike.coe
RingHom
Semiring.toNonAssocSemiring
RingHom.instFunLike
Polynomial.Monic
Polynomial.map
β€”Polynomial.Monic.map
Polynomial.monic_of_injective

MonomialOrder

Definitions

NameCategoryTheorems
Monic πŸ“–MathDef
7 mathmath: monic_X_sub_C, Monic.of_subsingleton, monic_X_add_C, monic_monomial, monic_monomial_one, monic_one, monic_X

Polynomial

Definitions

NameCategoryTheorems
Monic πŸ“–MathDef
92 mathmath: monic_of_subsingleton, MonicDegreeEq.monic, monic_X_pow, LinearMap.exists_monic_and_aeval_eq_zero, monic_X_sub_C, FixedPoints.minpoly.monic, PowerBasis.minpolyGen_monic, WeierstrassCurve.Affine.monic_polynomial, monic_toSubring, FirstOrder.Field.realize_genericMonicPolyHasRoot, minpoly.monic, monic_X_pow_add_C, monic_multisetProd_X_sub_C, monic_restriction, monic_map_iff, IsDistinguishedAt.monic, monic_X_pow_add, exists_monic_irreducible_factor, Cubic.monic_of_c_eq_one, monic_finprod_X_sub_C, monic_of_isUnit_leadingCoeff_inv_smul, int_coeff_of_cyclotomic', FirstOrder.Field.lift_genericMonicPoly, monic_mapAlg_iff, monic_X_pow_sub_C, cyclotomic'.monic, not_monic_zero, ConjRootClass.monic_minpoly, Matrix.charpoly.univ_monic, exists_monic_aeval_eq_zero_forall_mem_pow_of_isIntegral, int_cyclotomic_rw, monic_X_add_C, LinearMap.charpoly_monic, monic_of_natDegree_le_of_coeff_eq_one, trinomial_monic, monic_C_mul_of_mul_leadingCoeff_eq_one, monic_integralNormalization, Irreducible.exists_dvd_monic_irreducible_of_isIntegral, monic_of_degree_le_of_coeff_eq_one, StandardEtalePair.monic_f, Function.Injective.monic_map_iff, prodXSubSMul.monic, monic_ascPochhammer, AlgebraicClosure.Monics.splits_finsetProd, Cubic.monic_of_a_eq_one', isMonicOfDegree_iff', Cubic.monic_of_b_eq_one', Cubic.monic_of_c_eq_one', AlgebraicClosure.toSplittingField_coeff, monic_expand_iff, monic_geom_sum_X, int_monic_iff, LinearRecurrence.charPoly_monic, monic_of_degree_le, monic_descPochhammer, cyclotomic.monic, normalize_eq_self_iff_monic, AlgebraicClosure.Monics.map_eq_prod, Monic.sub_of_right, Cubic.monic_of_a_eq_one, monic_freeMonic, monic_mul_C_of_leadingCoeff_mul_eq_one, monic_mul_leadingCoeff_inv, exists_monic_aeval_eq_zero_forall_mem_pow_of_mem_map, MulSemiringAction.monic_charpoly, LinearMap.polyCharpoly_monic, mem_normalizedFactors_iff, IsMonicOfDegree.monic, Monic.def, not_monic_zero_iff, monic_one, Cubic.monic_of_d_eq_one', Matrix.charpoly_monic, Lagrange.nodal_monic, Cubic.monic_of_d_eq_one, monic_zero_iff_subsingleton', monic_annIdealGenerator, Cubic.monic_of_b_eq_one, monic_zero_iff_subsingleton, monic_normalize, monic_prod_X_sub_C, minpolyDiv_monic, exists_monic_aeval_eq_zero_forall_mem_of_mem_map, monic_scaleRoots_iff, monic_X, LieAlgebra.engel_isBot_of_isMin.lieCharpoly_monic, LinearMap.exists_monic_and_coeff_mem_pow_and_aeval_eq_zero_of_range_le_smul, IsAdjoinRootMonic.monic, hermite_monic, RatFunc.monic_denom, monic_X_pow_sub, int_cyclotomic_spec
MonicDegreeEq πŸ“–CompOp
8 mathmath: UniversalCoprimeFactorizationRing.homEquiv_comp_snd, MvPolynomial.coe_mapEquivMonic_comp', UniversalCoprimeFactorizationRing.homEquiv_comp_fst, MvPolynomial.mapEquivMonic_symm_map_algebraMap, MvPolynomial.mapEquivMonic_symm_map, UniversalFactorizationRing.fromTensor_comp_universalFactorizationMap, UniversalFactorizationRing.fromTensor_comp_universalFactorizationMap', MvPolynomial.coe_mapEquivMonic_comp

Theorems

NameKindAssumesProvesValidatesDepends On
degree_smul_of_smul_regular πŸ“–mathematicalIsSMulRegular
SMulZeroClass.toSMul
MulZeroClass.toZero
NonUnitalNonAssocSemiring.toMulZeroClass
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
degree
Polynomial
instZero
smulZeroClass
β€”le_antisymm
degree_le_iff_coeff_zero
coeff_smul
degree_lt_iff_coeff_zero
le_rfl
smul_zero
eq_of_monic_of_associated πŸ“–β€”Monic
Associated
Polynomial
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
semiring
β€”β€”Monic.eq_one_of_isUnit
Monic.of_mul_monic_left
Units.isUnit
mul_one
irreducible_of_monic πŸ“–mathematicalMonic
CommSemiring.toSemiring
Irreducible
Polynomial
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
semiring
instOne
β€”Monic.eq_one_of_isUnit
Irreducible.isUnit_or_isUnit
IsUnit.of_mul_eq_one
instIsDedekindFiniteMonoid
Monic.eq_1
leadingCoeff_mul
leadingCoeff_C
mul_comm
mul_mul_mul_comm
C_mul
Monic.leadingCoeff
C_1
mul_one
isUnit_leadingCoeff_mul_left_eq_zero_iff πŸ“–mathematicalIsUnit
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
leadingCoeff
Polynomial
instMul
instZero
β€”Monic.mul_left_eq_zero_iff
monic_mul_C_of_leadingCoeff_mul_eq_one
IsUnit.mul_val_inv
mul_assoc
MulZeroClass.zero_mul
isUnit_leadingCoeff_mul_right_eq_zero_iff πŸ“–mathematicalIsUnit
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
leadingCoeff
Polynomial
instMul
instZero
β€”ext
coeff_smul
coeff_mul
Finset.mul_sum
Finset.sum_congr
mul_assoc
Monic.mul_right_eq_zero_iff
monic_of_isUnit_leadingCoeff_inv_smul
smul_eq_zero_iff_eq
MulZeroClass.mul_zero
leadingCoeff_map' πŸ“–mathematicalDFunLike.coe
RingHom
Semiring.toNonAssocSemiring
RingHom.instFunLike
leadingCoeff
map
β€”leadingCoeff_map_of_injective
leadingCoeff_of_injective πŸ“–mathematicalDFunLike.coe
RingHom
Semiring.toNonAssocSemiring
RingHom.instFunLike
leadingCoeff
map
β€”leadingCoeff_map_of_injective
leadingCoeff_smul_of_smul_regular πŸ“–mathematicalIsSMulRegular
SMulZeroClass.toSMul
MulZeroClass.toZero
NonUnitalNonAssocSemiring.toMulZeroClass
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
leadingCoeff
Polynomial
instZero
smulZeroClass
β€”leadingCoeff.eq_1
coeff_smul
natDegree_smul_of_smul_regular
monic_C_mul_of_mul_leadingCoeff_eq_one πŸ“–mathematicalDistrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
leadingCoeff
AddMonoidWithOne.toOne
AddCommMonoidWithOne.toAddMonoidWithOne
NonAssocSemiring.toAddCommMonoidWithOne
Monic
Polynomial
instMul
DFunLike.coe
RingHom
semiring
RingHom.instFunLike
C
β€”Mathlib.Tactic.Nontriviality.subsingleton_or_nontrivial_elim
Monic.leadingCoeff
leadingCoeff_mul'
leadingCoeff_C
NeZero.one
monic_X_add_C πŸ“–mathematicalβ€”Monic
Polynomial
instAdd
X
DFunLike.coe
RingHom
Semiring.toNonAssocSemiring
semiring
RingHom.instFunLike
C
β€”monic_X_pow_add_C
one_ne_zero
pow_one
monic_X_pow_add πŸ“–mathematicalWithBot
Preorder.toLT
WithBot.instPreorder
Nat.instPreorder
degree
AddMonoidWithOne.toNatCast
WithBot.addMonoidWithOne
Nat.instAddMonoidWithOne
Monic
Polynomial
instAdd
Monoid.toNatPow
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
semiring
X
β€”monic_of_degree_le
le_trans
degree_add_le
max_le
degree_X_pow_le
le_of_lt
coeff_add
coeff_X_pow
coeff_eq_zero_of_degree_lt
add_zero
monic_X_pow_add_C πŸ“–mathematicalβ€”Monic
Polynomial
instAdd
Monoid.toNatPow
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
semiring
X
DFunLike.coe
RingHom
Semiring.toNonAssocSemiring
RingHom.instFunLike
C
β€”monic_X_pow_add
lt_of_le_of_lt
degree_C_le
Nat.instCanonicallyOrderedAdd
IsStrictOrderedRing.noZeroDivisors
Nat.instIsStrictOrderedRing
CanonicallyOrderedAdd.toExistsAddOfLE
Nat.instNontrivial
WithBot.instIsOrderedRing
IsStrictOrderedRing.toIsOrderedRing
WithBot.nontrivial
monic_X_pow_sub πŸ“–mathematicalWithBot
Preorder.toLT
WithBot.instPreorder
Nat.instPreorder
degree
Ring.toSemiring
AddMonoidWithOne.toNatCast
WithBot.addMonoidWithOne
Nat.instAddMonoidWithOne
Monic
Polynomial
instSub
Monoid.toNatPow
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
semiring
X
β€”sub_eq_add_neg
monic_X_pow_add
degree_neg
monic_X_pow_sub_C πŸ“–mathematicalβ€”Monic
Ring.toSemiring
Polynomial
instSub
Monoid.toNatPow
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
semiring
X
DFunLike.coe
RingHom
Semiring.toNonAssocSemiring
RingHom.instFunLike
C
β€”map_neg
RingHomClass.toAddMonoidHomClass
RingHom.instRingHomClass
monic_X_pow_add_C
monic_X_sub_C πŸ“–mathematicalβ€”Monic
Ring.toSemiring
Polynomial
instSub
X
DFunLike.coe
RingHom
Semiring.toNonAssocSemiring
semiring
RingHom.instFunLike
C
β€”sub_eq_add_neg
C_neg
monic_X_add_C
monic_finprod_of_monic πŸ“–mathematicalMonic
CommSemiring.toSemiring
finprod
Polynomial
CommSemiring.toCommMonoid
commSemiring
β€”finprod_def
monic_prod_of_monic
Set.Finite.mem_toFinset
monic_one
monic_mul_C_of_leadingCoeff_mul_eq_one πŸ“–mathematicalDistrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
leadingCoeff
AddMonoidWithOne.toOne
AddCommMonoidWithOne.toAddMonoidWithOne
NonAssocSemiring.toAddCommMonoidWithOne
Monic
Polynomial
instMul
DFunLike.coe
RingHom
semiring
RingHom.instFunLike
C
β€”Mathlib.Tactic.Nontriviality.subsingleton_or_nontrivial_elim
Monic.leadingCoeff
leadingCoeff_mul'
leadingCoeff_C
NeZero.one
monic_multiset_prod_of_monic πŸ“–mathematicalMonic
CommSemiring.toSemiring
Multiset.prod
Polynomial
CommSemiring.toCommMonoid
commSemiring
Multiset.map
β€”Multiset.induction_on
instIsEmptyFalse
Multiset.map_cons
Multiset.prod_cons
Monic.mul
Multiset.mem_cons_self
Multiset.mem_cons_of_mem
monic_of_injective πŸ“–β€”DFunLike.coe
RingHom
Semiring.toNonAssocSemiring
RingHom.instFunLike
Monic
map
β€”β€”leadingCoeff_map_of_injective
Monic.leadingCoeff
RingHom.map_one
monic_of_isUnit_leadingCoeff_inv_smul πŸ“–mathematicalIsUnit
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
leadingCoeff
Monic
Units
Polynomial
Units.instSMul
SMulZeroClass.toSMul
instZero
smulZeroClass
DistribSMul.toSMulZeroClass
AddMonoid.toAddZeroClass
AddMonoidWithOne.toAddMonoid
AddCommMonoidWithOne.toAddMonoidWithOne
NonAssocSemiring.toAddCommMonoidWithOne
Semiring.toNonAssocSemiring
instDistribSMul
NonAssocSemiring.toNonUnitalNonAssocSemiring
Units.instInv
IsUnit.unit
β€”Monic.def
leadingCoeff_smul_of_smul_regular
isSMulRegular_of_group
Units.smul_def
IsUnit.val_inv_mul
monic_prod_of_monic πŸ“–mathematicalMonic
CommSemiring.toSemiring
Finset.prod
Polynomial
CommSemiring.toCommMonoid
commSemiring
β€”monic_multiset_prod_of_monic
monic_zero_iff_subsingleton πŸ“–mathematicalβ€”Monic
Polynomial
instZero
β€”subsingleton_iff_zero_eq_one
monic_zero_iff_subsingleton' πŸ“–mathematicalβ€”Monic
Polynomial
instZero
β€”monic_zero_iff_subsingleton
Unique.instSubsingleton
subsingleton_iff
natDegree_pow_X_add_C πŸ“–mathematicalβ€”natDegree
Polynomial
Monoid.toNatPow
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
semiring
instAdd
X
DFunLike.coe
RingHom
Semiring.toNonAssocSemiring
RingHom.instFunLike
C
β€”Monic.natDegree_pow
monic_X_add_C
natDegree_X_add_C
mul_one
natDegree_smul_of_smul_regular πŸ“–mathematicalIsSMulRegular
SMulZeroClass.toSMul
MulZeroClass.toZero
NonUnitalNonAssocSemiring.toMulZeroClass
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
natDegree
Polynomial
instZero
smulZeroClass
β€”smul_zero
WithBot.charZero
Nat.instCharZero
degree_eq_natDegree
Mathlib.Tactic.Contrapose.contraposeβ‚„
IsSMulRegular.polynomial
degree_smul_of_smul_regular
not_isUnit_X_add_C πŸ“–mathematicalβ€”IsUnit
Polynomial
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
semiring
instAdd
X
DFunLike.coe
RingHom
Semiring.toNonAssocSemiring
RingHom.instFunLike
C
β€”Monic.natDegree_mul'
monic_X_add_C
right_ne_zero_of_mul_eq_one
nontrivial
natDegree_X_add_C
natDegree_one
not_isUnit_X_pow_sub_one πŸ“–mathematicalβ€”IsUnit
Polynomial
Ring.toSemiring
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
semiring
instSub
Monoid.toNatPow
X
instOne
β€”eq_or_ne
pow_zero
sub_self
NeZero.one
nontrivial
natDegree_one
Monic.eq_one_of_isUnit
monic_X_pow_sub_C
natDegree_X_pow_sub_C
not_monic_zero πŸ“–mathematicalβ€”Monic
Polynomial
instZero
β€”not_monic_zero_iff
zero_ne_one
NeZero.one
not_monic_zero_iff πŸ“–mathematicalβ€”Monic
Polynomial
instZero
β€”Iff.not
monic_zero_iff_subsingleton
subsingleton_iff_zero_eq_one

Polynomial.Monic

Theorems

NameKindAssumesProvesValidatesDepends On
add_of_left πŸ“–mathematicalPolynomial.Monic
WithBot
Preorder.toLT
WithBot.instPreorder
Nat.instPreorder
Polynomial.degree
Polynomial
Polynomial.instAdd
β€”eq_1
add_comm
Polynomial.leadingCoeff_add_of_degree_lt
add_of_right πŸ“–mathematicalPolynomial.Monic
WithBot
Preorder.toLT
WithBot.instPreorder
Nat.instPreorder
Polynomial.degree
Polynomial
Polynomial.instAdd
β€”eq_1
Polynomial.leadingCoeff_add_of_degree_lt
as_sum πŸ“–mathematicalPolynomial.MonicPolynomial
Polynomial.instAdd
Monoid.toNatPow
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Polynomial.semiring
Polynomial.X
Polynomial.natDegree
Finset.sum
NonUnitalNonAssocSemiring.toAddCommMonoid
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
Finset.range
Polynomial.instMul
DFunLike.coe
RingHom
RingHom.instFunLike
Polynomial.C
Polynomial.coeff
β€”Polynomial.as_sum_range_C_mul_X_pow
Finset.sum_range_succ_comm
one_mul
comp πŸ“–mathematicalPolynomial.MonicPolynomial.compβ€”Mathlib.Tactic.Nontriviality.subsingleton_or_nontrivial_elim
Polynomial.natDegree_comp_eq_of_mul_ne_zero
leadingCoeff
one_pow
mul_one
NeZero.one
def
Polynomial.leadingCoeff.eq_1
Polynomial.coeff_comp_degree_mul_degree
comp_X_add_C πŸ“–mathematicalPolynomial.MonicPolynomial.comp
Polynomial
Polynomial.instAdd
Polynomial.X
DFunLike.coe
RingHom
Semiring.toNonAssocSemiring
Polynomial.semiring
RingHom.instFunLike
Polynomial.C
β€”Mathlib.Tactic.Nontriviality.subsingleton_or_nontrivial_elim
comp
Polynomial.monic_X_add_C
one_ne_zero
Polynomial.natDegree_X_add_C
comp_X_sub_C πŸ“–mathematicalPolynomial.Monic
Ring.toSemiring
Polynomial.comp
Polynomial
Polynomial.instSub
Polynomial.X
DFunLike.coe
RingHom
Semiring.toNonAssocSemiring
Polynomial.semiring
RingHom.instFunLike
Polynomial.C
β€”map_neg
RingHomClass.toAddMonoidHomClass
RingHom.instRingHomClass
comp_X_add_C
degree_le_zero_iff_eq_one πŸ“–mathematicalPolynomial.MonicWithBot
Preorder.toLE
WithBot.instPreorder
Nat.instPreorder
Polynomial.degree
WithBot.zero
MulZeroClass.toZero
Nat.instMulZeroClass
Polynomial
Polynomial.instOne
β€”natDegree_eq_zero
Polynomial.natDegree_eq_zero_iff_degree_le_zero
degree_map πŸ“–mathematicalPolynomial.MonicPolynomial.degree
Polynomial.map
β€”leadingCoeff
map_one
MonoidHomClass.toOneHomClass
MonoidWithZeroHomClass.toMonoidHomClass
RingHomClass.toMonoidWithZeroHomClass
RingHom.instRingHomClass
NeZero.one
degree_mul_comm πŸ“–mathematicalPolynomial.MonicPolynomial.degree
Polynomial
Polynomial.instMul
β€”MulZeroClass.mul_zero
MulZeroClass.zero_mul
Polynomial.degree_mul'
leadingCoeff
one_mul
Polynomial.leadingCoeff_ne_zero
degree_mul
add_comm
eq_one_of_isUnit πŸ“–mathematicalPolynomial.Monic
IsUnit
Polynomial
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Polynomial.semiring
Polynomial.instOneβ€”Mathlib.Tactic.Nontriviality.subsingleton_or_nontrivial_elim
Unique.instSubsingleton
IsUnit.exists_right_inv
natDegree_mul'
right_ne_zero_of_mul_eq_one
Polynomial.nontrivial
natDegree_eq_zero
add_eq_zero
Polynomial.natDegree_one
eq_one_of_map_eq_one πŸ“–β€”Polynomial.Monic
Polynomial.map
Polynomial
Polynomial.instOne
β€”β€”Mathlib.Tactic.Nontriviality.subsingleton_or_nontrivial_elim
Unique.instSubsingleton
Polynomial.degree_map_eq_of_leadingCoeff_ne_zero
leadingCoeff
RingHom.map_one
one_ne_zero
NeZero.one
Polynomial.degree_one
WithBot.coe_eq_coe
Polynomial.degree_eq_natDegree
ne_zero
Polynomial.leadingCoeff.eq_1
Polynomial.eq_C_of_degree_eq_zero
irreducible_iff_lt_natDegree_lt πŸ“–mathematicalPolynomial.Monic
CommSemiring.toSemiring
Irreducible
Polynomial
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Polynomial.semiring
semigroupDvd
SemigroupWithZero.toSemigroup
NonUnitalSemiring.toSemigroupWithZero
NonUnitalCommSemiring.toNonUnitalSemiring
CommSemiring.toNonUnitalCommSemiring
Polynomial.commSemiring
β€”irreducible_iff_natDegree'
of_mul_monic_left
mul_comm
dvd_mul_left
irreducible_iff_natDegree πŸ“–mathematicalPolynomial.Monic
CommSemiring.toSemiring
Irreducible
Polynomial
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Polynomial.semiring
Polynomial.natDegree
β€”Polynomial.irreducible_of_monic
natDegree_eq_zero
irreducible_iff_natDegree' πŸ“–mathematicalPolynomial.Monic
CommSemiring.toSemiring
Irreducible
Polynomial
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Polynomial.semiring
Finset
Finset.instMembership
Finset.Ioc
Nat.instPreorder
Nat.instLocallyFiniteOrder
Polynomial.natDegree
β€”irreducible_iff_natDegree
Nat.instAtLeastTwoHAddOfNat
zero_lt_two
Nat.instZeroLEOneClass
IsOrderedAddMonoid.toAddLeftMono
Nat.instIsOrderedAddMonoid
mul_two
natDegree_mul
add_le_add_iff_right
covariant_swap_add_of_covariant_add
IsRightCancelAdd.addRightReflectLE_of_addRightReflectLT
AddRightCancelSemigroup.toIsRightCancelAdd
contravariant_swap_add_of_contravariant_add
contravariant_lt_of_covariant_le
LT.lt.ne'
LT.lt.trans_le
Mathlib.Tactic.Contrapose.contrapose₁
Mathlib.Tactic.Push.not_forall_eq
le_total
mul_comm
add_le_add_right
add_le_add_left
Nat.instCanonicallyOrderedAdd
isRegular πŸ“–mathematicalPolynomial.Monic
Ring.toSemiring
IsRegular
Polynomial
Polynomial.instMul
β€”sub_eq_zero
mul_right_eq_zero_iff
mul_sub
sub_self
mul_left_eq_zero_iff
sub_mul
isUnit_iff πŸ“–mathematicalPolynomial.MonicIsUnit
Polynomial
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Polynomial.semiring
Polynomial.instOne
β€”eq_one_of_isUnit
isUnit_one
map πŸ“–mathematicalPolynomial.MonicPolynomial.mapβ€”subsingleton_or_nontrivial
Polynomial.leadingCoeff_map_eq_of_isUnit_leadingCoeff
RingHom.map_one
isUnit_one
mul πŸ“–mathematicalPolynomial.MonicPolynomial
Polynomial.instMul
β€”subsingleton_of_zero_eq_one
def
mul_one
Polynomial.leadingCoeff_mul'
one_mul
mul_left_eq_zero_iff πŸ“–mathematicalPolynomial.MonicPolynomial
Polynomial.instMul
Polynomial.instZero
β€”MulZeroClass.zero_mul
mul_left_ne_zero
mul_left_ne_zero πŸ“–β€”Polynomial.Monicβ€”β€”mul_one
Polynomial.degree_eq_bot
degree_mul
WithBot.add_eq_bot
LT.lt.ne'
lt_trans
not_le
degree_le_zero_iff_eq_one
mul_natDegree_lt_iff πŸ“–mathematicalPolynomial.MonicPolynomial.natDegree
Polynomial
Polynomial.instMul
Polynomial.instZero
β€”LT.lt.ne'
lt_of_le_of_ne
MulZeroClass.mul_zero
natDegree_eq_zero
natDegree_mul'
IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono
AddLeftCancelSemigroup.toIsLeftCancelAdd
IsOrderedAddMonoid.toAddLeftMono
Nat.instIsOrderedAddMonoid
contravariant_lt_of_covariant_le
Nat.instCanonicallyOrderedAdd
mul_right_eq_zero_iff πŸ“–mathematicalPolynomial.MonicPolynomial
Polynomial.instMul
Polynomial.instZero
β€”MulZeroClass.mul_zero
mul_right_ne_zero
mul_right_ne_zero πŸ“–β€”Polynomial.Monicβ€”β€”one_mul
Polynomial.degree_eq_bot
degree_mul_comm
degree_mul
WithBot.add_eq_bot
LT.lt.ne'
lt_trans
not_le
degree_le_zero_iff_eq_one
natDegree_eq_zero_iff_eq_one πŸ“–mathematicalPolynomial.MonicPolynomial.natDegree
Polynomial
Polynomial.instOne
β€”natDegree_eq_zero
natDegree_map πŸ“–mathematicalPolynomial.MonicPolynomial.natDegree
Polynomial.map
β€”le_antisymm
Polynomial.natDegree_map_le
Polynomial.le_natDegree_of_ne_zero
Polynomial.coeff_map
coeff_natDegree
map_one
MonoidHomClass.toOneHomClass
MonoidWithZeroHomClass.toMonoidHomClass
RingHomClass.toMonoidWithZeroHomClass
RingHom.instRingHomClass
one_ne_zero
NeZero.one
natDegree_mul πŸ“–mathematicalPolynomial.MonicPolynomial.natDegree
Polynomial
Polynomial.instMul
β€”Mathlib.Tactic.Nontriviality.subsingleton_or_nontrivial_elim
Polynomial.natDegree_of_subsingleton
add_zero
Polynomial.natDegree_mul'
leadingCoeff
mul_one
NeZero.one
natDegree_mul' πŸ“–mathematicalPolynomial.MonicPolynomial.natDegree
Polynomial
Polynomial.instMul
β€”Polynomial.natDegree_mul'
leadingCoeff
one_mul
natDegree_mul_comm πŸ“–mathematicalPolynomial.MonicPolynomial.natDegree
Polynomial
Polynomial.instMul
β€”MulZeroClass.mul_zero
MulZeroClass.zero_mul
natDegree_mul'
Polynomial.natDegree_mul'
leadingCoeff
mul_one
add_comm
natDegree_pow πŸ“–mathematicalPolynomial.MonicPolynomial.natDegree
Polynomial
Monoid.toNatPow
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Polynomial.semiring
β€”pow_zero
Polynomial.natDegree_one
MulZeroClass.zero_mul
pow_succ
natDegree_mul
pow
add_comm
nextCoeff_mul πŸ“–mathematicalPolynomial.MonicPolynomial.nextCoeff
Polynomial
Polynomial.instMul
Distrib.toAdd
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
β€”Mathlib.Tactic.Nontriviality.subsingleton_or_nontrivial_elim
Polynomial.reverse_mul
leadingCoeff
mul_one
NeZero.one
Polynomial.mul_coeff_one
Polynomial.coeff_zero_reverse
Polynomial.coeff_one_reverse
one_mul
add_comm
nextCoeff_multiset_prod πŸ“–mathematicalPolynomial.Monic
CommSemiring.toSemiring
Polynomial.nextCoeff
Multiset.prod
Polynomial
CommSemiring.toCommMonoid
Polynomial.commSemiring
Multiset.map
Multiset.sum
NonUnitalNonAssocSemiring.toAddCommMonoid
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
β€”Multiset.induction_on
Polynomial.C_1
Polynomial.nextCoeff_C_eq_zero
Multiset.map_cons
Multiset.prod_cons
Multiset.sum_cons
nextCoeff_mul
Multiset.mem_cons_self
Polynomial.monic_multiset_prod_of_monic
Multiset.mem_cons_of_mem
nextCoeff_pow πŸ“–mathematicalPolynomial.MonicPolynomial.nextCoeff
Polynomial
Monoid.toNatPow
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Polynomial.semiring
AddMonoid.toNatSMul
AddMonoidWithOne.toAddMonoid
AddCommMonoidWithOne.toAddMonoidWithOne
NonAssocSemiring.toAddCommMonoidWithOne
Semiring.toNonAssocSemiring
β€”pow_zero
zero_smul
map_one
MonoidHomClass.toOneHomClass
MonoidWithZeroHomClass.toMonoidHomClass
RingHomClass.toMonoidWithZeroHomClass
RingHom.instRingHomClass
Polynomial.nextCoeff_C_eq_zero
pow_succ
nextCoeff_mul
pow
succ_nsmul
nextCoeff_prod πŸ“–mathematicalPolynomial.Monic
CommSemiring.toSemiring
Polynomial.nextCoeff
Finset.prod
Polynomial
CommSemiring.toCommMonoid
Polynomial.commSemiring
Finset.sum
NonUnitalNonAssocSemiring.toAddCommMonoid
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
β€”nextCoeff_multiset_prod
not_dvd_of_degree_lt πŸ“–mathematicalPolynomial.Monic
WithBot
Preorder.toLT
WithBot.instPreorder
Nat.instPreorder
Polynomial.degree
Polynomial
semigroupDvd
SemigroupWithZero.toSemigroup
NonUnitalSemiring.toSemigroupWithZero
Semiring.toNonUnitalSemiring
Polynomial.semiring
β€”not_dvd_of_natDegree_lt
Polynomial.natDegree_lt_natDegree
not_dvd_of_natDegree_lt πŸ“–mathematicalPolynomial.Monic
Polynomial.natDegree
Polynomial
semigroupDvd
SemigroupWithZero.toSemigroup
NonUnitalSemiring.toSemigroupWithZero
Semiring.toNonUnitalSemiring
Polynomial.semiring
β€”LT.lt.not_ge
natDegree_mul'
right_ne_zero_of_mul
not_irreducible_iff_exists_add_mul_eq_coeff πŸ“–mathematicalPolynomial.Monic
CommSemiring.toSemiring
Polynomial.natDegree
Irreducible
Polynomial
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Polynomial.semiring
Polynomial.coeff
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
Distrib.toAdd
β€”subsingleton_or_nontrivial
Polynomial.natDegree_of_subsingleton
Nat.instCharZero
Nat.instAtLeastTwoHAddOfNat
irreducible_iff_natDegree'
Polynomial.natDegree_one
Mathlib.Tactic.Push.not_forall_eq
Polynomial.mul_coeff_zero
add_right_cancel
AddRightCancelSemigroup.toIsRightCancelAdd
Nat.instZeroLEOneClass
IsOrderedAddMonoid.toAddLeftMono
Nat.instIsOrderedAddMonoid
Nat.Ioc_succ_singleton
zero_add
natDegree_mul
nextCoeff_mul
Polynomial.monic_X_add_C
Polynomial.as_sum_range_C_mul_X_pow
Finset.sum_range_succ
Finset.sum_range_one
coeff_natDegree
Polynomial.C_mul
Polynomial.C_add
Polynomial.C_1
Mathlib.Tactic.Ring.of_eq
Mathlib.Tactic.Ring.mul_congr
Mathlib.Tactic.Ring.add_congr
Mathlib.Tactic.Ring.atom_pf
Mathlib.Tactic.Ring.add_pf_add_lt
Mathlib.Tactic.Ring.add_pf_zero_add
Mathlib.Tactic.Ring.add_mul
Mathlib.Tactic.Ring.mul_add
Mathlib.Tactic.Ring.mul_pp_pf_overlap
Mathlib.Meta.NormNum.IsNat.to_raw_eq
Mathlib.Meta.NormNum.isNat_add
Mathlib.Meta.NormNum.IsNat.of_raw
Mathlib.Tactic.Ring.one_mul
Mathlib.Tactic.Ring.mul_pf_left
Mathlib.Tactic.Ring.mul_pf_right
Mathlib.Tactic.Ring.mul_zero
Mathlib.Tactic.Ring.add_pf_add_zero
Mathlib.Tactic.Ring.add_pf_add_gt
Mathlib.Tactic.Ring.zero_mul
Mathlib.Tactic.Ring.pow_congr
Mathlib.Tactic.Ring.cast_zero
Mathlib.Meta.NormNum.isNat_ofNat
Mathlib.Tactic.Ring.pow_zero
Mathlib.Tactic.Ring.cast_pos
Mathlib.Tactic.Ring.pow_add
Mathlib.Tactic.Ring.pow_one_cast
Nat.cast_one
Mathlib.Tactic.Ring.single_pow
Mathlib.Tactic.Ring.mul_pow
Mathlib.Tactic.Ring.one_pow
IsStrictOrderedRing.toIsOrderedRing
Nat.instIsStrictOrderedRing
Nat.instNontrivial
Polynomial.natDegree_add_C
Polynomial.natDegree_X
of_mul_monic_left πŸ“–β€”Polynomial.Monic
Polynomial
Polynomial.instMul
β€”β€”Mathlib.Tactic.Contrapose.contrapose₁
def
Polynomial.leadingCoeff_monic_mul
of_mul_monic_right πŸ“–β€”Polynomial.Monic
Polynomial
Polynomial.instMul
β€”β€”Mathlib.Tactic.Contrapose.contrapose₁
def
Polynomial.leadingCoeff_mul_monic
pow πŸ“–mathematicalPolynomial.MonicPolynomial
Monoid.toNatPow
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Polynomial.semiring
β€”Polynomial.monic_one
pow_succ
mul
sub_of_left πŸ“–mathematicalPolynomial.Monic
Ring.toSemiring
WithBot
Preorder.toLT
WithBot.instPreorder
Nat.instPreorder
Polynomial.degree
Polynomial
Polynomial.instSub
β€”sub_eq_add_neg
add_of_left
Polynomial.degree_neg
sub_of_right πŸ“–mathematicalPolynomial.leadingCoeff
Ring.toSemiring
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
Ring.toAddCommGroup
AddMonoidWithOne.toOne
AddGroupWithOne.toAddMonoidWithOne
Ring.toAddGroupWithOne
WithBot
Preorder.toLT
WithBot.instPreorder
Nat.instPreorder
Polynomial.degree
Polynomial.Monic
Polynomial
Polynomial.instSub
β€”Polynomial.natDegree_neg
Polynomial.coeff_neg
neg_neg
sub_eq_add_neg
add_of_right
Polynomial.degree_neg

Polynomial.MonicDegreeEq

Definitions

NameCategoryTheorems
map πŸ“–CompOp
8 mathmath: Polynomial.UniversalCoprimeFactorizationRing.exists_liesOver_residueFieldMap_bijective, Polynomial.UniversalCoprimeFactorizationRing.homEquiv_comp_snd, MvPolynomial.coe_mapEquivMonic_comp', Polynomial.UniversalCoprimeFactorizationRing.homEquiv_comp_fst, map_coe, Polynomial.UniversalCoprimeFactorizationRing.factor₁_mul_factorβ‚‚, MvPolynomial.mapEquivMonic_symm_map_algebraMap, MvPolynomial.mapEquivMonic_symm_map
mk πŸ“–CompOpβ€”

Theorems

NameKindAssumesProvesValidatesDepends On
coeff_of_ge πŸ“–mathematicalβ€”Polynomial.coeff
Polynomial
AddMonoidWithOne.toOne
AddCommMonoidWithOne.toAddMonoidWithOne
NonAssocSemiring.toAddCommMonoidWithOne
Semiring.toNonAssocSemiring
MulZeroClass.toZero
NonUnitalNonAssocSemiring.toMulZeroClass
NonAssocSemiring.toNonUnitalNonAssocSemiring
β€”LE.le.lt_of_ne
degree πŸ“–mathematicalβ€”Polynomial.degree
Polynomial
Polynomial.coeff
AddMonoidWithOne.toOne
AddCommMonoidWithOne.toAddMonoidWithOne
NonAssocSemiring.toAddCommMonoidWithOne
Semiring.toNonAssocSemiring
WithBot
AddMonoidWithOne.toNatCast
WithBot.addMonoidWithOne
Nat.instAddMonoidWithOne
β€”Polynomial.degree_eq_of_le_of_coeff_ne_zero
Polynomial.degree_le_of_natDegree_le
Eq.le
natDegree
NeZero.one
map_coe πŸ“–mathematicalβ€”Polynomial
Polynomial.coeff
AddMonoidWithOne.toOne
AddCommMonoidWithOne.toAddMonoidWithOne
NonAssocSemiring.toAddCommMonoidWithOne
Semiring.toNonAssocSemiring
map
Polynomial.map
β€”β€”
mk_coe πŸ“–mathematicalPolynomial.Monic
Polynomial.natDegree
Polynomial
Polynomial.coeff
AddMonoidWithOne.toOne
AddCommMonoidWithOne.toAddMonoidWithOne
NonAssocSemiring.toAddCommMonoidWithOne
Semiring.toNonAssocSemiring
β€”β€”
monic πŸ“–mathematicalβ€”Polynomial.Monic
Polynomial
Polynomial.coeff
AddMonoidWithOne.toOne
AddCommMonoidWithOne.toAddMonoidWithOne
NonAssocSemiring.toAddCommMonoidWithOne
Semiring.toNonAssocSemiring
β€”Mathlib.Tactic.Nontriviality.subsingleton_or_nontrivial_elim
Polynomial.Monic.eq_1
Polynomial.leadingCoeff.eq_1
natDegree
natDegree πŸ“–mathematicalβ€”Polynomial.natDegree
Polynomial
Polynomial.coeff
AddMonoidWithOne.toOne
AddCommMonoidWithOne.toAddMonoidWithOne
NonAssocSemiring.toAddCommMonoidWithOne
Semiring.toNonAssocSemiring
β€”Polynomial.natDegree_eq_of_le_of_coeff_ne_zero
Polynomial.natDegree_le_iff_coeff_eq_zero
NeZero.one

---

← Back to Index