Documentation Verification Report

Nat

πŸ“ Source: Mathlib/Algebra/Ring/Nat.lean

Statistics

MetricCount
DefinitionsinstAddCommMonoidWithOne, instAddMonoidWithOne, instCommSemiring, instDistrib, instNonAssocSemiring, instNonUnitalNonAssocSemiring, instNonUnitalSemiring, instSemiring
8
TheoremsinstCharZero, instIsDomain
2
Total10

Nat

Definitions

NameCategoryTheorems
instAddCommMonoidWithOne πŸ“–CompOpβ€”
instAddMonoidWithOne πŸ“–CompOp
42 mathmath: Tactic.NormNum.isInt_lcm, Mathlib.Meta.NormNum.IsNNRat.natFloor, Mathlib.Meta.NormNum.isNat_natAbs_pos, Mathlib.Tactic.Ring.coeff_one, Mathlib.Meta.NormNum.isNat_minFac_1, Tactic.NormNum.isNat_sqrt, ENat.lift_one, Mathlib.Meta.NormNum.isNat_descFactorial, Mathlib.Meta.NormNum.IsNNRat.natCeil, instCharZero, Mathlib.Meta.NormNum.isNat_natSub, Mathlib.Meta.NormNum.IsNat.natCeil, Mathlib.Tactic.Ring.natCast_nat, Mathlib.Tactic.Ring.atom_pf', ENat.lift_zero, Mathlib.Meta.NormNum.IsRat.natFloor, Mathlib.Tactic.Ring.pow_one_cast, Equiv.Perm.toList_formPerm_nontrivial, Mathlib.Meta.NormNum.isNat_minFac_3, Mathlib.Meta.NormNum.isNat_descFactorial_zero, Tactic.NormNum.isInt_gcd, Mathlib.Meta.NormNum.IsNat.raw_refl, Mathlib.Meta.NormNum.isNat_fib, Mathlib.Meta.NormNum.IsRat.natCeil, Mathlib.Meta.NormNum.isNat_natDiv, Mathlib.Tactic.Ring.const_pos, Mathlib.Meta.NormNum.isNat_ascFactorial, Mathlib.Meta.NormNum.isNat_natMod, Mathlib.Meta.NormNum.isNat_minFac_4, Tactic.NormNum.isNat_lcm, ArithmeticFunction.natCoe_nat, Mathlib.Meta.NormNum.IsInt.natCeil, Mathlib.Meta.NormNum.IsInt.natFloor, Tactic.NormNum.isNat_ratDen, Mathlib.Tactic.Ring.atom_pf, Tactic.NormNum.isNat_gcd, Mathlib.Meta.NormNum.isNat_minFac_2, Mathlib.Meta.NormNum.isNat_factorial, Mathlib.Meta.NormNum.isNat_natSucc, Mathlib.Meta.NormNum.isNat_natAbs_neg, Mathlib.Tactic.Ring.toProd_pf, Mathlib.Meta.NormNum.IsNat.natFloor
instCommSemiring πŸ“–CompOp
55 mathmath: Polynomial.degree_pow_le_of_le, DivisibleHull.archimedeanClassMk_mk_eq, QuadraticMap.canLift, ComplexShape.eulerCharSignsDownNat_Ο‡, uzpow_natCast, DivisibleHull.mk_add_mk_left, mem_subalgebraOfSubsemiring, uzpow_coe_nat, ComplexShape.Ξ΅_down_β„•, Polynomial.degree_comp, NonUnitalSubsemiring.unitization_apply, NNRat.isFractionRing, DivisibleHull.instIsStrictOrderedModuleNNRat, NumberField.InfinitePlace.ComplexEmbedding.conjugate_sign, subalgebraOfSubsemiring_toSubsemiring, nat_algebra_subsingleton, mem_maximalIdeal_iff, AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.mem_carrier_iff_of_mem_mul, DivisibleHull.coeAddMonoidHom_apply, DivisibleHull.instIsOrderedCancelAddMonoid, Subbimodule.coe_toSubbimoduleInt, Polynomial.evalRingHom_zero, DivisibleHull.qsmul_def, Subbimodule.coe_toSubbimoduleNat, QuadraticMap.canLift', DivisibleHull.qsmul_of_nonneg, DivisibleHull.coeOrderAddMonoidHom_apply, DivisibleHull.nsmul_mk, RingHom.toNatAlgHom_coe, Ideal.isPrime_nat_iff, ComplexShape.eulerCharSignsUpNat_Ο‡, DivisibleHull.mk_add_mk, DivisibleHull.archimedeanClassOrderIso_apply, ringKrullDim_nat, DivisibleHull.zsmul_mk, DivisibleHull.neg_mk, NonUnitalSubsemiring.unitization_range, DivisibleHull.qsmul_of_nonpos, DivisibleHull.archimedeanClassOrderIso_symm_apply, maximalIdeal_eq_span_two_three, MvPolynomial.counitNat_C, DivisibleHull.instIsStrictOrderedModuleRat, DivisibleHull.coe_add, mem_nonUnitalSubalgebraOfNonUnitalSubsemiring, RingHom.toNatAlgHom_apply, DivisibleHull.mk_zero, DivisibleHull.zero_qsmul, MvPolynomial.counitNat_X, fermatLastTheoremWith'_nat_int_tfae, MvPolynomial.counitNat_surjective, isCoprime_iff, PNat.isCoprime_iff, coe_maximalIdeal, DivisibleHull.nnqsmul_mk, Algebra.adjoin_nat
instDistrib πŸ“–CompOpβ€”
instNonAssocSemiring πŸ“–CompOp
20 mathmath: addMonoidHomLequivNat_symm_apply, AddMonoidHom.toNatLinearMap_injective, AddEquiv.toNatLinearEquiv_toAddEquiv, RingHom.ENatMap_apply, AddEquiv.toNatLinearEquiv_trans, castRingHom_nat, AddMonoidHom.coe_toNatLinearMap, AddCommGroup.DirectLimit.directedSystem, RingHom.injective_nat, coe_castRingHom, AddEquiv.coe_symm_toNatLinearEquiv, addMonoidHomLequivNat_apply, Ideal.map_comap_natCastRingHom_int, AddEquiv.coe_toNatLinearEquiv, LinearEquiv.toAddEquiv_toNatLinearEquiv, AddEquiv.toNatLinearEquiv_symm, CharP.quotient_iff_le_ker_natCast, MonoidWithZeroHom.ext_int_iff, MonoidWithZeroHomClass.ext_nnrat_iff, MonoidHom.ext_int_iff
instNonUnitalNonAssocSemiring πŸ“–CompOp
10 mathmath: floorDiv_eq_div, isLinearSet_iff_exists_matrix, instTrivialStar, ceilRoot_def, factorization_ceilRoot, isSemilinearSet_setOf_mulVec_eq, factorization_floorRoot, floorRoot_def, StarAddMonoid.toStarModuleNat, ceilDiv_eq_add_pred_div
instNonUnitalSemiring πŸ“–CompOp
2 mathmath: instStarOrderedRing, isQuasiregular_zero
instSemiring πŸ“–CompOp
259 mathmath: odd_fermatNumber, Submodule.toAddSubmonoid_toNatSubmodule, Polynomial.degree_pow_le_of_le, DivisibleHull.archimedeanClassMk_mk_eq, pow_eq_neg_pow_iff, iSupIndep.dfinsupp_lsum_injective, ZMod.isCyclic_units_iff_of_odd, Polynomial.ascPochhammer_smeval_neg_eq_descPochhammer, MvPowerSeries.weightedOrder_monomial, addMonoidHomLequivNat_symm_apply, MvPolynomial.weightedDegree_eq_zero_iff, ofDigits_lt_base_pow_length, PowerSeries.catalanSeries_coeff, PowerSeries.coeff_X_mul_largeSchroderSeries, neg_one_pow_eq_neg_one_iff_odd, AddCommMonoid.subsingletonNatModule, four_dvd_or_exists_odd_prime_and_dvd_of_two_lt, OreLocalization.nsmul_eq_nsmul, Finsupp.le_weight_of_ne_zero', ofDigits_one, Fintype.linearIndependent_iff', Odd.natAbs, Polynomial.Chebyshev.eval_T_real_eq_neg_one_iff, sum_le_ofDigits, DivisibleHull.mk_add_mk_left, odd_add_one, exists_mem_span_nat_finset_of_ge, PowerSeries.catalanSeries_constantCoeff, Finset.even_sum_iff_even_card_odd, odd_totient_iff, MvPolynomial.le_weightedTotalDegree, ZMod.natCast_eq_one_iff_odd, Set.odd_ncard_compl_iff, ofDigits_lt_base_pow_length', odd_sub', SimpleGraph.odd_card_odd_degree_vertices_ne, Submodule.biSup_eq_range_dfinsupp_lsum, AddMonoidHom.toNatLinearMap_injective, ZMod.isCyclic_units_iff, ofDigits_eq_sum_mapIdx, ascPochhammer_nat_eq_descFactorial, ofDigits_append_zero, infinite_odd_deficient, geom_sum_pos_iff, Set.odd_card_insert_iff, AddEquiv.toNatLinearEquiv_toAddEquiv, MvPowerSeries.weightedOrder_monomial_of_ne_zero, ZMod.natCast_ne_zero_iff_odd, infinite_odd_abundant, ofDigits_digits_append_digits, Equiv.Perm.sign_of_cycleType_eq_replicate, odd_totient_iff_eq_one, ofDigits_mod_pow_eq_ofDigits_take, pow_length_le_mul_ofDigits, subset_span_setGcd, Submodule.span_nat_eq, PowerSeries.coeff_X_mul_largeSchroderSeriesSeries_sq, AddSubmonoid.toNatSubmodule_toAddSubmonoid, AddSubmonoid.toNatSubmodule_closure, Polynomial.degree_comp, AlternatingGroup.card_of_cycleType_singleton, NonUnitalSubsemiring.unitization_apply, ComplexShape.down_nat_odd_add, Polynomial.smeval_neg_nat, bijOn_ofDigits, DFinsupp.sum_mapRange_index.linearMap, injOn_ofDigits, Int.odd_coe_nat, PowerSeries.largeSchroderSeries_eq_one_add_X_mul_largeSchroderSeries_add_X_mul_largeSchroderSeries_sq, MvPolynomial.weightedHomogeneousSubmodule_eq_finsupp_supported, finite_setOf_setGcd_dvd_and_mem_span, ofDigits_mod_eq_head!, ArithmeticFunction.zeta_mul_pow_eq_sigma, Submodule.mem_iSup_iff_exists_dfinsupp, not_odd_iff_even, Submodule.iSup_eq_range_dfinsupp_lsum, CategoryTheory.Functor.natLinear, DivisibleHull.instIsStrictOrderedModuleNNRat, ofDigits_div_pow_eq_ofDigits_drop, bijOn_ofDigits', ArithmeticFunction.zeta_mul_comm, SimpleGraph.even_card_odd_degree_vertices, Finsupp.finite_of_nat_weight_le, Even.sub_odd, Odd.mod_even, Finsupp.le_weight, Prime.odd_of_ne_two, Module.isTorsionFree_nat_iff_isAddTorsionFree, Set.even_card_insert_iff, instIsDomain, Ring.factorial_nsmul_multichoose_eq_ascPochhammer, ofDigits_monotone, not_even_iff_odd, Fin.odd_iff_of_even, frequently_odd, ofDigits_replicate_zero, PowerSeries.coeff_largeSchroderSeries, Fintype.linearIndependent_iff'β‚›, AddEquiv.toNatLinearEquiv_trans, AddMonoidHom.coe_toNatLinearMap, MvPowerSeries.le_weightedOrder_subst, isProperLinearSet_iff, ofDigits_cons, coprime_two_left, coprime_two_right, Polynomial.smeval_at_natCast, AddMonoid.FG.to_moduleFinite_nat, DivisibleHull.coeAddMonoidHom_apply, DivisibleHull.instIsOrderedCancelAddMonoid, MvPowerSeries.coeff_weightedHomogeneousComponent, Odd.of_dvd_nat, Ring.ascPochhammer_succ_succ, Fin.even_iff, sum_digits_ofDigits_eq_sum, instIsLocalRingNat, AddSubmonoid.toNatSubmodule_symm, floor_nat, Equiv.Perm.centralizer_le_alternating_iff, BinomialRing.factorial_nsmul_multichoose, Odd.sub_even, odd_pow_iff, Module.End.natCast_def, ceil_nat, MonoidHom.FixedPointFree.odd_orderOf_of_involutive, Odd.of_mul_right, Finset.odd_sum_iff_odd_card_odd, AddMonoid.End.natCast_def, DivisibleHull.qsmul_def, ofDigits_nil, ascPochhammer_nat_eval_succ, fermatLastTheoremWith_nat_int_rat_tfae, Ring.smeval_ascPochhammer_int_ofNat, odd_mul, PowerSeries.constantCoeff_largeSchroderSeries, fermatLastTheoremFor_iff_nat, ofDigits_reverse_cons, coe_ofDigits, even_sub', Fin.odd_iff_imp, DivisibleHull.qsmul_of_nonneg, Ring.smeval_ascPochhammer_nat_cast, DivisibleHull.coeOrderAddMonoidHom_apply, Submodule.fg_iff_addSubmonoid_fg, AddCommGroup.DirectLimit.directedSystem, DivisibleHull.nsmul_mk, ComplexShape.up_nat_odd_add, Polynomial.sumIDeriv_X, Polynomial.ascPochhammer_smeval_cast, Polynomial.Chebyshev.T_iterate_derivative_mem_span_T, MvPowerSeries.weightedOrder_eq_nat, instIsNoetherian, LieModule.chainTopCoeff_add_one, ofDigits_mod, Fin.even_succAbove_add_predAbove, setInvOn_digitsAppend_ofDigits, even_xor_odd, MvPowerSeries.map_iterateFrobenius_expand, Ideal.isPrime_nat_iff, MvPowerSeries.exists_coeff_ne_zero_and_weightedOrder, Submodule.span_nat_eq_addSubmonoidClosure, nat_smul_eq_nsmul, MvPowerSeries.weightedOrder_le, AddEquiv.toNatLinearEquiv_refl, Matrix.toLinearMapβ‚‚'_single, Fin.odd_iff, Prime.eq_two_or_odd', Int.natAbs_odd, DivisibleHull.mk_add_mk, MvPolynomial.weightedDecomposition.decompose'_eq, not_odd_iff, Finsupp.le_weight_of_ne_zero, Ring.smeval_ascPochhammer_neg_add, Module.Basis.toMatrix_eq_toMatrix_constr, even_add', ofDigits_append, DivisibleHull.archimedeanClassOrderIso_apply, odd_add', AddEquiv.coe_symm_toNatLinearEquiv, addMonoidHomLequivNat_apply, ArithmeticFunction.mul_zeta_apply, Function.Involutive.iterate_eq_self, AddMonoid.isTorsion_iff_isTorsion_nat, DivisibleHull.zsmul_mk, iSupIndep_iff_dfinsupp_lsum_injective, sub_one_mul_sum_div_pow_eq_sub_sum_digits, SimpleGraph.odd_ncard_oddComponents, Ring.smeval_ascPochhammer_succ_neg, MvPowerSeries.ne_zero_iff_exists_coeff_ne_zero_and_weight, DivisibleHull.neg_mk, NonUnitalSubsemiring.unitization_range, AddSubmonoid.coe_toNatSubmodule, DivisibleHull.qsmul_of_nonpos, PowerSeries.catalanSeries_sq_mul_X_add_one, Polynomial.ascPochhammer_smeval_eq_eval, Ideal.map_comap_natCastRingHom_int, Finsupp.weight_sub_single_add, digits_ofDigits, ascPochhammer_eval_cast, DivisibleHull.archimedeanClassOrderIso_symm_apply, ofDigits_singleton, instIsStrictOrderedRing, AddEquiv.coe_toNatLinearEquiv, ofDigits_reverse_zero_cons, MonoidHom.FixedPointFree.odd_card_of_involutive, Ring.smeval_ascPochhammer_neg_of_lt, DivisibleHull.coe_add, Module.Finite.iff_addMonoid_fg, mersenne_odd, Cubic.monic_of_d_eq_one', DistribSMul.toAddMonoidHom_eq_nsmulAddMonoidHom, ofDigits_append_replicate_zero, pow_eq_neg_one_iff, eq_two_pow_or_exists_odd_prime_and_dvd, odd_sub, ascPochhammer_nat_eq_ascFactorial, schnirelmannDensity_setOf_Odd, Coprime.odd_of_left, LinearEquiv.toAddEquiv_toNatLinearEquiv, SimpleGraph.Coloring.odd_length_iff_not_congr, self_mod_pow_eq_ofDigits_take, isCompl_even_odd, AddEquiv.toNatLinearEquiv_symm, MvPolynomial.coeff_weightedHomogeneousComponent, ofDigits_add_ofDigits_eq_ofDigits_zipWith_of_length_eq, Prime.odd_iff, ofDigits_modEq, ArithmeticFunction.natCoe_mul, CharP.quotient_iff_le_ker_natCast, exists_eq_two_pow_mul_odd, Submodule.mem_biSup_iff_exists_dfinsupp, Coprime.odd_of_right, ArithmeticFunction.zeta_mul_apply, Polynomial.descPochhammer_smeval_eq_ascPochhammer, ofDigits_modEq', mul_ofDigits, even_or_odd, Polynomial.Chebyshev.U_mem_span_T, Ring.smeval_ascPochhammer_self_neg, SimpleGraph.ConnectedComponent.odd_oddComponents_ncard_subset_supp, CoxeterSystem.IsReflection.odd_length, one_mem_span_iff, Finsupp.weight_eq_zero_iff_eq_zero, odd_iff, SimpleGraph.exists_ne_odd_degree_of_exists_odd_degree, SimpleGraph.Walk.IsEulerian.card_odd_degree, ofDigits_div_eq_ofDigits_tail, MvPolynomial.weightedHomogeneousComponent_apply, DivisibleHull.nnqsmul_mk, self_div_pow_eq_ofDigits_drop, lsum_comp_mapRange_toSpanSingleton, mapsTo_ofDigits, IsAddTorsionFree.to_isTorsionFree_nat, Odd.of_mul_left, Odd.mod_even_iff, not_odd_zero, Polynomial.Chebyshev.T_derivative_mem_span_T, odd_add, Equiv.Perm.OnCycleFactors.odd_of_centralizer_le_alternatingGroup, ofDigits_digits

Theorems

NameKindAssumesProvesValidatesDepends On
instCharZero πŸ“–mathematicalβ€”CharZero
instAddMonoidWithOne
β€”β€”
instIsDomain πŸ“–mathematicalβ€”IsDomain
instSemiring
β€”instIsCancelMulZero
instNontrivial

---

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