Documentation Verification Report

Even

📁 Source: Mathlib/LinearAlgebra/CliffordAlgebra/Even.lean

Statistics

MetricCount
DefinitionsEvenHom, bilin, compr₂, even, aux, ι, instInhabitedEvenHomSubtypeMemSubalgebraEven, Even, Even, Even
10
Theoremscompr₂_bilin, contract, contract_mid, ext, ext_iff, algHom_ext, algHom_ext_iff, aux_algebraMap, aux_apply, aux_mul, aux_one, aux_ι, lift_symm_apply_bilin, lift_ι, even_toSubmodule
15
Total25

CliffordAlgebra

Definitions

NameCategoryTheorems
EvenHom 📖CompData
2 mathmath: even.lift_ι, even.lift_symm_apply_bilin
even 📖CompOp
20 mathmath: spinGroup.mem_even, even.lift.aux_mul, spinGroup.mem_iff, coe_toEven_reverse_involute, even.lift_ι, even.algHom_ext_iff, toEven_comp_ofEven, evenToNeg_ι, ofEven_comp_toEven, even_toSubmodule, even.lift_symm_apply_bilin, evenEquivEvenNeg_apply, toEven_ι, even.lift.aux_apply, even.lift.aux_ι, evenEquivEvenNeg_symm_apply, ofEven_ι, evenToNeg_comp_evenToNeg, even.lift.aux_one, even.lift.aux_algebraMap
instInhabitedEvenHomSubtypeMemSubalgebraEven 📖CompOp

Theorems

NameKindAssumesProvesValidatesDepends On
even_toSubmodule 📖mathematicalDFunLike.coe
OrderEmbedding
Subalgebra
CliffordAlgebra
CommRing.toCommSemiring
Ring.toSemiring
instRingCliffordAlgebra
instAlgebraCliffordAlgebra
Submodule
CommSemiring.toSemiring
NonUnitalNonAssocSemiring.toAddCommMonoid
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
Algebra.toModule
Preorder.toLE
PartialOrder.toPreorder
Subalgebra.instPartialOrder
Submodule.instPartialOrder
instFunLikeOrderEmbedding
Subalgebra.toSubmodule
even
evenOdd
ZMod
MulZeroClass.toZero
NonUnitalNonAssocSemiring.toMulZeroClass
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
CommRing.toNonUnitalCommRing
ZMod.commRing

CliffordAlgebra.EvenHom

Definitions

NameCategoryTheorems
bilin 📖CompOp
9 mathmath: compr₂_bilin, contract, CliffordAlgebra.even.lift_ι, CliffordAlgebra.evenToNeg_ι, ext_iff, CliffordAlgebra.even.lift_symm_apply_bilin, CliffordAlgebra.even.lift.aux_ι, CliffordAlgebra.ofEven_ι, contract_mid
compr₂ 📖CompOp
2 mathmath: compr₂_bilin, CliffordAlgebra.even.algHom_ext_iff

Theorems

NameKindAssumesProvesValidatesDepends On
compr₂_bilin 📖mathematicalbilin
compr₂
LinearMap.compr₂
CommRing.toCommSemiring
CommSemiring.toSemiring
AddCommGroup.toAddCommMonoid
NonUnitalNonAssocSemiring.toAddCommMonoid
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
Algebra.toModule
Ring.toSemiring
Semiring.toModule
AlgHom.toLinearMap
contract 📖mathematicalDFunLike.coe
LinearMap
CommSemiring.toSemiring
CommRing.toCommSemiring
RingHom.id
Semiring.toNonAssocSemiring
AddCommGroup.toAddCommMonoid
NonUnitalNonAssocSemiring.toAddCommMonoid
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
Algebra.toModule
Ring.toSemiring
LinearMap.instFunLike
LinearMap.addCommMonoid
LinearMap.module
smulCommClass_self
CommRing.toCommMonoid
DistribMulAction.toMulAction
CommMonoid.toMonoid
AddCommMonoid.toAddMonoid
Module.toDistribMulAction
CommRing.toRing
bilin
RingHom
RingHom.instFunLike
algebraMap
QuadraticForm
QuadraticMap.instFunLike
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toModule
contract_mid 📖mathematicalDistrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
DFunLike.coe
LinearMap
CommSemiring.toSemiring
CommRing.toCommSemiring
RingHom.id
Semiring.toNonAssocSemiring
AddCommGroup.toAddCommMonoid
NonUnitalNonAssocSemiring.toAddCommMonoid
Algebra.toModule
Ring.toSemiring
LinearMap.instFunLike
LinearMap.addCommMonoid
LinearMap.module
smulCommClass_self
CommRing.toCommMonoid
DistribMulAction.toMulAction
CommMonoid.toMonoid
AddCommMonoid.toAddMonoid
Module.toDistribMulAction
CommRing.toRing
bilin
Algebra.toSMul
QuadraticForm
QuadraticMap.instFunLike
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toModule
ext 📖bilinsmulCommClass_self
ext_iff 📖mathematicalbilinsmulCommClass_self
ext

CliffordAlgebra.even

Definitions

NameCategoryTheorems
ι 📖CompOp
6 mathmath: lift_ι, algHom_ext_iff, CliffordAlgebra.evenToNeg_ι, lift_symm_apply_bilin, lift.aux_ι, CliffordAlgebra.ofEven_ι

Theorems

NameKindAssumesProvesValidatesDepends On
algHom_ext 📖CliffordAlgebra.EvenHom.compr₂
CliffordAlgebra
Subalgebra
CommRing.toCommSemiring
Ring.toSemiring
instRingCliffordAlgebra
instAlgebraCliffordAlgebra
SetLike.instMembership
Subalgebra.instSetLike
CliffordAlgebra.even
Subalgebra.toRing
Subalgebra.algebra
ι
AlgHom.ext
CliffordAlgebra.even_induction
SetLike.algebraMap_mem_graded
SetLike.GradedMonoid.toGradedOne
CliffordAlgebra.evenOdd.gradedMonoid
AlgHom.commutes
map_add
SemilinearMapClass.toAddHomClass
NonUnitalAlgHomClass.instLinearMapClass
AlgHom.instNonUnitalAlgHomClassOfAlgHomClass
AlgHom.algHomClass
smulCommClass_self
LinearMap.congr_fun
CliffordAlgebra.EvenHom.ext_iff
map_mul
NonUnitalAlgSemiHomClass.toMulHomClass
algHom_ext_iff 📖mathematicalCliffordAlgebra.EvenHom.compr₂
CliffordAlgebra
Subalgebra
CommRing.toCommSemiring
Ring.toSemiring
instRingCliffordAlgebra
instAlgebraCliffordAlgebra
SetLike.instMembership
Subalgebra.instSetLike
CliffordAlgebra.even
Subalgebra.toRing
Subalgebra.algebra
ι
algHom_ext
lift_symm_apply_bilin 📖mathematicalCliffordAlgebra.EvenHom.bilin
DFunLike.coe
Equiv
AlgHom
CliffordAlgebra
Subalgebra
CommRing.toCommSemiring
Ring.toSemiring
instRingCliffordAlgebra
instAlgebraCliffordAlgebra
SetLike.instMembership
Subalgebra.instSetLike
CliffordAlgebra.even
Subalgebra.toSemiring
Subalgebra.algebra
CliffordAlgebra.EvenHom
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.symm
lift
LinearMap.compr₂
CommSemiring.toSemiring
AddCommGroup.toAddCommMonoid
NonUnitalNonAssocSemiring.toAddCommMonoid
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
Subalgebra.toRing
Algebra.toModule
Semiring.toModule
ι
AlgHom.toLinearMap
lift_ι 📖mathematicalDFunLike.coe
AlgHom
CliffordAlgebra
Subalgebra
CommRing.toCommSemiring
Ring.toSemiring
instRingCliffordAlgebra
instAlgebraCliffordAlgebra
SetLike.instMembership
Subalgebra.instSetLike
CliffordAlgebra.even
Subalgebra.toSemiring
Subalgebra.algebra
AlgHom.funLike
Equiv
CliffordAlgebra.EvenHom
EquivLike.toFunLike
Equiv.instEquivLike
lift
LinearMap
CommSemiring.toSemiring
RingHom.id
Semiring.toNonAssocSemiring
AddCommGroup.toAddCommMonoid
NonUnitalNonAssocSemiring.toAddCommMonoid
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
Subalgebra.toRing
Algebra.toModule
LinearMap.instFunLike
LinearMap.addCommMonoid
LinearMap.module
smulCommClass_self
CommRing.toCommMonoid
DistribMulAction.toMulAction
CommMonoid.toMonoid
AddCommMonoid.toAddMonoid
Module.toDistribMulAction
CommRing.toRing
CliffordAlgebra.EvenHom.bilin
ι
lift.aux_ι

CliffordAlgebra.even.lift

Definitions

NameCategoryTheorems
aux 📖CompOp
5 mathmath: aux_mul, aux_apply, aux_ι, aux_one, aux_algebraMap

Theorems

NameKindAssumesProvesValidatesDepends On
aux_algebraMap 📖mathematicalDFunLike.coe
LinearMap
CommSemiring.toSemiring
CommRing.toCommSemiring
RingHom.id
Semiring.toNonAssocSemiring
CliffordAlgebra
Subalgebra
Ring.toSemiring
instRingCliffordAlgebra
instAlgebraCliffordAlgebra
SetLike.instMembership
Subalgebra.instSetLike
CliffordAlgebra.even
NonUnitalNonAssocSemiring.toAddCommMonoid
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
Subalgebra.toRing
Subalgebra.instModuleSubtypeMem
Algebra.toModule
LinearMap.instFunLike
aux
RingHom
Subalgebra.toSemiring
RingHom.instFunLike
algebraMap
Subalgebra.algebra
smulCommClass_self
CliffordAlgebra.foldr_algebraMap
Algebra.algebraMap_eq_smul_one
aux_apply 📖mathematicalDFunLike.coe
LinearMap
CommSemiring.toSemiring
CommRing.toCommSemiring
RingHom.id
Semiring.toNonAssocSemiring
CliffordAlgebra
Subalgebra
Ring.toSemiring
instRingCliffordAlgebra
instAlgebraCliffordAlgebra
SetLike.instMembership
Subalgebra.instSetLike
CliffordAlgebra.even
NonUnitalNonAssocSemiring.toAddCommMonoid
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
Subalgebra.toRing
Subalgebra.instModuleSubtypeMem
Algebra.toModule
LinearMap.instFunLike
aux
AddCommGroup.toAddCommMonoid
Submodule
LinearMap.addCommMonoid
LinearMap.module
Submodule.setLike
NonAssocSemiring.toNonUnitalNonAssocSemiring
Prod.instAddCommMonoid
Submodule.addCommMonoid
Prod.instModule
Submodule.module
Prod.instAddCommGroup
Ring.toAddCommGroup
Submodule.addCommGroup
CommRing.toRing
LinearMap.addCommGroup
CliffordAlgebra.foldr
AddMonoidWithOne.toOne
AddGroupWithOne.toAddMonoidWithOne
Ring.toAddGroupWithOne
Submodule.zero
aux_mul 📖mathematicalDFunLike.coe
LinearMap
CommSemiring.toSemiring
CommRing.toCommSemiring
RingHom.id
Semiring.toNonAssocSemiring
CliffordAlgebra
Subalgebra
Ring.toSemiring
instRingCliffordAlgebra
instAlgebraCliffordAlgebra
SetLike.instMembership
Subalgebra.instSetLike
CliffordAlgebra.even
NonUnitalNonAssocSemiring.toAddCommMonoid
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
Subalgebra.toRing
Subalgebra.instModuleSubtypeMem
Algebra.toModule
LinearMap.instFunLike
aux
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
smulCommClass_self
CliffordAlgebra.foldr_mul
CliffordAlgebra.even_induction
CliffordAlgebra.foldr_algebraMap
aux_algebraMap
Algebra.smul_def
SetLike.algebraMap_mem_graded
SetLike.GradedMonoid.toGradedOne
CliffordAlgebra.evenOdd.gradedMonoid
Submodule.add_mem
map_add
SemilinearMapClass.toAddHomClass
LinearMap.semilinearMapClass
Prod.fst_add
AddMemClass.add_mem
AddSubmonoidClass.toAddMemClass
Submodule.addSubmonoidClass
Distrib.rightDistribClass
SetLike.mul_mem_graded
SetLike.GradedMonoid.toGradedMul
CliffordAlgebra.ι_mul_ι_mem_evenOdd_zero
zero_add
CliffordAlgebra.foldr_ι
aux_one 📖mathematicalDFunLike.coe
LinearMap
CommSemiring.toSemiring
CommRing.toCommSemiring
RingHom.id
Semiring.toNonAssocSemiring
CliffordAlgebra
Subalgebra
Ring.toSemiring
instRingCliffordAlgebra
instAlgebraCliffordAlgebra
SetLike.instMembership
Subalgebra.instSetLike
CliffordAlgebra.even
NonUnitalNonAssocSemiring.toAddCommMonoid
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
Subalgebra.toRing
Subalgebra.instModuleSubtypeMem
Algebra.toModule
LinearMap.instFunLike
aux
OneMemClass.one
AddMonoidWithOne.toOne
AddGroupWithOne.toAddMonoidWithOne
Ring.toAddGroupWithOne
AddSubmonoidWithOneClass.toOneMemClass
SubsemiringClass.addSubmonoidWithOneClass
Subalgebra.instSubsemiringClass
smulCommClass_self
CliffordAlgebra.foldr_one
aux_ι 📖mathematicalDFunLike.coe
LinearMap
CommSemiring.toSemiring
CommRing.toCommSemiring
RingHom.id
Semiring.toNonAssocSemiring
CliffordAlgebra
Subalgebra
Ring.toSemiring
instRingCliffordAlgebra
instAlgebraCliffordAlgebra
SetLike.instMembership
Subalgebra.instSetLike
CliffordAlgebra.even
NonUnitalNonAssocSemiring.toAddCommMonoid
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
Subalgebra.toRing
Subalgebra.instModuleSubtypeMem
Algebra.toModule
LinearMap.instFunLike
aux
AddCommGroup.toAddCommMonoid
Subalgebra.algebra
LinearMap.addCommMonoid
LinearMap.module
smulCommClass_self
CommRing.toCommMonoid
DistribMulAction.toMulAction
CommMonoid.toMonoid
AddCommMonoid.toAddMonoid
Module.toDistribMulAction
CommRing.toRing
CliffordAlgebra.EvenHom.bilin
CliffordAlgebra.even.ι
smulCommClass_self
CliffordAlgebra.foldr_mul
CliffordAlgebra.foldr_ι
mul_one

DirichletCharacter

Definitions

NameCategoryTheorems
Even 📖MathDef
3 mathmath: even_or_odd, not_even_and_odd, Odd.not_even

Function

Definitions

NameCategoryTheorems
Even 📖MathDef
11 mathmath: ValueDistribution.characteristic_even, ZMod.dft_even_iff, EisensteinSeries.e2Summand_even, Even.zero, Even.const, ValueDistribution.proximity_even, DirichletCharacter.Even.to_fun, Odd.smul_odd, Odd.mul_odd, locallyFinsuppWithin.logCounting_even, ValueDistribution.logCounting_even

(root)

Definitions

NameCategoryTheorems
Even 📖MathDef
192 mathmath: Int.even_pow, Function.Involutive.iterate_eq_id, neg_one_pow_eq_ite, Odd.add_odd, even_sub_one, Nat.even_mul_pred_self, zpow_eq_zpow_iff_of_ne_zero₀, Polynomial.Chebyshev.U_eval_zero, IsCyclotomicExtension.Rat.torsionOrder_eq, Nat.even_add_one, Even.all, WeierstrassCurve.coeff_preΨ', Fintype.card_fin_two, geom_sum_alternating_of_le_neg_one, even_ofMul_iff, Even.add_self, isSquare_toMul_iff, Fin.sum_neg_one_pow, even_subset_image_even, WeierstrassCurve.leadingCoeff_preΨ', Int.even_pow', WeierstrassCurve.ΨSq_odd, even_iff_exists_two_nsmul, Int.even_xor'_odd, Fin.even_of_odd, Polynomial.Chebyshev.T_eval_zero, Finset.even_sum_iff_even_card_odd, Even.zero, Nat.odd_sub', Int.negOnePow_eq_iff, even_two, even_op_iff, Odd.tsub_odd, hyperoperation_ge_four_zero, ZMod.eq_zero_iff_even, isSquare_ofAdd_iff, WeierstrassCurve.preΨ_odd, Int.not_even_iff, Set.odd_card_insert_iff, AddIrreducible.not_even, Int.odd_add, SimpleGraph.IsClique.even_iff_exists_isMatching, even_neg_two, Int.not_even_one, WeierstrassCurve.ΨSq_ofNat, SimpleGraph.ConnectedComponent.even_ncard_supp_sdiff_rep, WeierstrassCurve.coeff_preΨ, SimpleGraph.Subgraph.IsPerfectMatching.even_card, SimpleGraph.ConnectedComponent.even_card_of_isPerfectMatching, Int.even_iff, Nat.even_mul_succ_self, WeierstrassCurve.Ψ_ofNat, Nat.Prime.even_iff, NumberField.InfinitePlace.even_card_aut_of_not_isUnramified, AddSubmonoid.coe_even, preNormEDS'_odd, Nat.not_odd_iff_even, even_toAdd_iff, SimpleGraph.even_card_odd_degree_vertices, Nat.infinite_even_abundant, Int.isCompl_even_odd, Int.odd_sub, schnirelmannDensity_setOf_even, even_unop_iff, NumberField.InfinitePlace.even_finrank_of_not_isUnramifiedIn, WeierstrassCurve.leadingCoeff_preΨ, Set.even_card_insert_iff, Fin.even_iff_imp, Nat.Prime.even_sub_one, SimpleGraph.Walk.IsTrail.even_countP_edges_iff, Nat.not_even_iff_odd, Nat.even_add, SimpleGraph.Walk.IsEulerian.even_degree_iff, CoxeterSystem.getElem_alternatingWord, Nat.eq_sq_add_sq_iff, neg_one_pow_eq_one_iff_even, Int.fib_neg, Fin.even_add_one_iff_odd, Nat.even_pow', Nat.not_even_one, range_two_mul, Int.even_or_odd, Nat.not_even_iff, Int.not_even_two_mul_add_one, AlternatingGroup.card_of_cycleType, Int.even_add, Int.even_sign_iff, Fin.even_iff, Nat.even_pow, Nat.not_even_bit1, Even.two_nsmul, Int.even_mul_pred_self, geom_sum_eq_zero_iff_neg_one, Int.odd_sub', WeierstrassCurve.Φ_ofNat, Int.not_even_iff_odd, even_add_one, Int.even_sub_one, neg_one_zpow_eq_ite, Nat.even_largeSchroder, WeierstrassCurve.preΨ'_odd, ZMod.intCast_eq_zero_iff_even, even_two_mul, Nat.even_sub', geom_sum_neg_iff, even_sub_two, NumberField.InfinitePlace.even_finrank_of_not_isUnramified, WeierstrassCurve.natDegree_preΨ_le, geom_sum_alternating_of_lt_neg_one, ZMod.natCast_eq_zero_iff_even, neg_one_geom_sum, Int.even_add', CoxeterSystem.prod_alternatingWord_eq_mul_pow, Fin.even_succAbove_add_predAbove, Nat.even_xor_odd, preNormEDS_mul_complEDS₂, Odd.sub_odd, Odd.one_add, HomologicalComplex.alternatingConst_d, CoxeterSystem.listTake_alternatingWord, Nat.not_even_two_mul_add_one, Int.odd_add', Equiv.Perm.prod_list_swap_mem_alternatingGroup_iff_even_length, CoxeterSystem.alternatingWord_succ', WeierstrassCurve.natDegree_preΨ', odd_add_one, AddSubgroup.coe_even, AddSubgroup.mem_even, Nat.infinite_even_deficient, Set.exists_union_disjoint_cardinal_eq_iff, pow_eq_one_iff_of_ne_zero, Fin.odd_add_one_iff_even, Finset.exists_disjoint_union_of_even_card_iff, NumberField.InfinitePlace.even_nat_card_aut_of_not_isUnramified, Nat.even_add', SimpleGraph.even_ncard_image_val_supp_sdiff_image_val_rep_union, Nat.odd_add', even_add_two, Int.even_mul_succ_self, Nat.Odd.sub_odd, normEDS_ofNat, Int.even_mul, even_iff_two_dvd, SimpleGraph.two_colorable_iff_forall_loop_even, Polynomial.Chebyshev.eval_T_real_eq_one_iff, WeierstrassCurve.natDegree_preΨ'_le, Int.not_odd_iff_even, Odd.add_one, Polynomial.coeff_hermite, even_abs, pow_eq_one_iff_cases, Int.natAbs_even, even_iff_exists_add_self, preNormEDS_odd, Nat.frequently_even, Nat.even_iff, Nat.even_mul, List.IsChain.two_mul_count_bool_eq_ite, odd_sub_one, Nat.odd_sub, Nat.isCompl_even_odd, Even.of_isUnit_two, zpow_eq_one_iff_cases₀, pow_eq_pow_iff_of_ne_zero, NumberField.InfinitePlace.even_card_aut_of_not_isUnramifiedIn, zpow_eq_zpow_iff_cases₀, AlternatingGroup.map_subtype_of_cycleType, Nat.even_or_odd, AddSubsemigroup.mem_even, Nat.even_div, SimpleGraph.Subgraph.IsMatching.even_card, AddSubsemigroup.coe_even, even_iff_exists_two_mul, Int.even_sub, pow_eq_pow_iff_cases, Nat.even_xor, isUnit_two_iff_forall_even, zpow_eq_one_iff_of_ne_zero₀, WeierstrassCurve.Ψ_odd, even_neg, NumberField.Units.even_torsionOrder, Int.negOnePow_eq_one_iff, Int.even_add_one, AlternatingGroup.card_of_cycleType_mul_eq, SimpleGraph.Coloring.even_length_iff_congr, AddSubmonoid.mem_even, Int.even_coe_nat, Int.even_sub', Nat.totient_even, Nat.odd_add, WeierstrassCurve.natDegree_preΨ, Nat.even_sub

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